Table of Clebsch-Gordan coefficients

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This is a table of Clebsch-Gordan coefficients. The overall sign of the coefficients for each set of constant $j 1$ , $j 2$ , $j$ is arbitrary to some degree and has been fixed according to the Condon-Shortley and Wigner sign convention as discussed by Baird and Biedenharn.

1 references
2 j1=1/2, j2=1/2
3 j1=1, j2=1/2
4 j1=1, j2=1
5 j1=3/2, j2=1/2
6 j1=3/2, j2=1
7 j1=3/2, j2=3/2
8 j1=2, j2=1/2
9 j1=2, j2=1
10 j1=2, j2=3/2
11 j1=2, j2=2
12 j1=5/2, j2=1/2
13 j1=5/2, j2=1
14 j1=5/2, j2=3/2
15 j1=5/2, j2=2

[edit] references

A kind of script to generate these coefficients can be found at : [1]

Tables with the same sign convention are in K Hagiwara et al, Phys Rev D 66 (2002) 010001 (PDF) and CGord (ASCII).

These are the answers to

$\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=?$ ,

explicitly $\delta_{m,m_1+m_2} \sqrt{\frac{(2j+1)(j+j_1-j_2)!(j-j_1+j_2)!(j_1+j_2-j)!(j+m)!(j-m)!} {(j_1+j_2+j+1)!(j_1-m_1)!(j_1+m_1)!(j_2-m_2)!(j_2+m_2)!}} \sum_{k=0}^{j+m} (-1)^{k+j_2+m_2}\frac{(j_2+j+m_1-k)!(j_1-m_1+k)!}{k!(j-j_1+j_2-k)!(j+m-k)!(k+j_1-j_2-m)!}$

For brevity, answers for m < 0 are omitted, use that (as far as I can see from the tables generated)

$\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=(-1)^{j-j_1-j_2}\langle j_1j_2;-m_1,-m_2|j_1j_2;j,-m\rangle$ .

[edit] j₁=1/2, j₂=1/2

m=1

m₁, m₂=

	1
1/2, 1/2	$1\!\,$

m=0

m₁, m₂=

	1	0
1/2, -1/2	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$
-1/2, 1/2	$\sqrt{\frac{1}{2}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$

[edit] j₁=1, j₂=1/2

m=3/2

m₁, m₂=

	3/2
1, 1/2	$1\!\,$

m=1/2

m₁, m₂=

	3/2	1/2
1, -1/2	$\sqrt{\frac{1}{3}}\!\,$	$\sqrt{\frac{2}{3}}\!\,$
0, 1/2	$\sqrt{\frac{2}{3}}\!\,$	$-\sqrt{\frac{1}{3}}\!\,$

[edit] j₁=1, j₂=1

m=2

m₁, m₂=

	2
1, 1	$1\!\,$

m=1

m₁, m₂=

	2	1
1, 0	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$
0, 1	$\sqrt{\frac{1}{2}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$

m=0

m₁, m₂=

	2	1	0
1, -1	$\sqrt{\frac{1}{6}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{3}}\!\,$
0, 0	$\sqrt{\frac{2}{3}}\!\,$	$0\!\,$	$-\sqrt{\frac{1}{3}}\!\,$
-1, 1	$\sqrt{\frac{1}{6}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{3}}\!\,$

[edit] j₁=3/2, j₂=1/2

m=2

m₁, m₂=

	2
3/2, 1/2	$1\!\,$

m=1

m₁, m₂=

	2	1
3/2, -1/2	$\frac{1}{2}\!\,$	$\sqrt{\frac{3}{4}}\!\,$
1/2, 1/2	$\sqrt{\frac{3}{4}}\!\,$	$-\frac{1}{2}\!\,$

m=0

m₁, m₂=

	2	1
1/2, -1/2	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$
-1/2, 1/2	$\sqrt{\frac{1}{2}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$

[edit] j₁=3/2, j₂=1

m=5/2

m₁, m₂=

	5/2
3/2, 1	$1\!\,$

m=3/2

m₁, m₂=

	5/2	3/2
3/2, 0	$\sqrt{\frac{2}{5}}\!\,$	$\sqrt{\frac{3}{5}}\!\,$
1/2, 1	$\sqrt{\frac{3}{5}}\!\,$	$-\sqrt{\frac{2}{5}}\!\,$

m=1/2

m₁, m₂=

	5/2	3/2	1/2
3/2, -1	$\sqrt{\frac{1}{10}}\!\,$	$\sqrt{\frac{2}{5}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$
1/2, 0	$\sqrt{\frac{3}{5}}\!\,$	$\sqrt{\frac{1}{15}}\!\,$	$-\sqrt{\frac{1}{3}}\!\,$
-1/2, 1	$\sqrt{\frac{3}{10}}\!\,$	$-\sqrt{\frac{8}{15}}\!\,$	$\sqrt{\frac{1}{6}}\!\,$

[edit] j₁=3/2, j₂=3/2

m=3

m₁, m₂=

	3
3/2, 3/2	$1\!\,$

m=2

m₁, m₂=

	3	2
3/2, 1/2	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$
1/2, 3/2	$\sqrt{\frac{1}{2}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$

m=1

m₁, m₂=

	3	2	1
3/2, -1/2	$\sqrt{\frac{1}{5}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{3}{10}}\!\,$
1/2, 1/2	$\sqrt{\frac{3}{5}}\!\,$	$0\!\,$	$-\sqrt{\frac{2}{5}}\!\,$
-1/2, 3/2	$\sqrt{\frac{1}{5}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{3}{10}}\!\,$

m=0

m₁, m₂=

	3	2	1	0
3/2, -3/2	$\sqrt{\frac{1}{20}}\!\,$	$\frac{1}{2}\!\,$	$\sqrt{\frac{9}{20}}\!\,$	$\frac{1}{2}\!\,$
1/2, -1/2	$\sqrt{\frac{9}{20}}\!\,$	$\frac{1}{2}\!\,$	$-\sqrt{\frac{1}{20}}\!\,$	$-\frac{1}{2}\!\,$
-1/2, 1/2	$\sqrt{\frac{9}{20}}\!\,$	$-\frac{1}{2}\!\,$	$-\sqrt{\frac{1}{20}}\!\,$	$\frac{1}{2}\!\,$
-3/2, 3/2	$\sqrt{\frac{1}{20}}\!\,$	$-\frac{1}{2}\!\,$	$\sqrt{\frac{9}{20}}\!\,$	$-\frac{1}{2}\!\,$

[edit] j₁=2, j₂=1/2

m=5/2

m₁, m₂=

	5/2
2, 1/2	$1\!\,$

m=3/2

m₁, m₂=

	5/2	3/2
2, -1/2	$\sqrt{\frac{1}{5}}\!\,$	$\sqrt{\frac{4}{5}}\!\,$
1, 1/2	$\sqrt{\frac{4}{5}}\!\,$	$-\sqrt{\frac{1}{5}}\!\,$

m=1/2

m₁, m₂=

	5/2	3/2
1, -1/2	$\sqrt{\frac{2}{5}}\!\,$	$\sqrt{\frac{3}{5}}\!\,$
0, 1/2	$\sqrt{\frac{3}{5}}\!\,$	$-\sqrt{\frac{2}{5}}\!\,$

[edit] j₁=2, j₂=1

m=3

m₁, m₂=

	3
2, 1	$1\!\,$

m=2

m₁, m₂=

	3	2
2, 0	$\sqrt{\frac{1}{3}}\!\,$	$\sqrt{\frac{2}{3}}\!\,$
1, 1	$\sqrt{\frac{2}{3}}\!\,$	$-\sqrt{\frac{1}{3}}\!\,$

m=1

m₁, m₂=

	3	2	1
2, -1	$\sqrt{\frac{1}{15}}\!\,$	$\sqrt{\frac{1}{3}}\!\,$	$\sqrt{\frac{3}{5}}\!\,$
1, 0	$\sqrt{\frac{8}{15}}\!\,$	$\sqrt{\frac{1}{6}}\!\,$	$-\sqrt{\frac{3}{10}}\!\,$
0, 1	$\sqrt{\frac{2}{5}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{10}}\!\,$

m=0

m₁, m₂=

	3	2	1
1, -1	$\sqrt{\frac{1}{5}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{3}{10}}\!\,$
0, 0	$\sqrt{\frac{3}{5}}\!\,$	$0\!\,$	$-\sqrt{\frac{2}{5}}\!\,$
-1, 1	$\sqrt{\frac{1}{5}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{3}{10}}\!\,$

[edit] j₁=2, j₂=3/2

m=7/2

m₁, m₂=

	7/2
2, 3/2	$1\!\,$

m=5/2

m₁, m₂=

	7/2	5/2
2, 1/2	$\sqrt{\frac{3}{7}}\!\,$	$\sqrt{\frac{4}{7}}\!\,$
1, 3/2	$\sqrt{\frac{4}{7}}\!\,$	$-\sqrt{\frac{3}{7}}\!\,$

m=3/2

m₁, m₂=

	7/2	5/2	3/2
2, -1/2	$\sqrt{\frac{1}{7}}\!\,$	$\sqrt{\frac{16}{35}}\!\,$	$\sqrt{\frac{2}{5}}\!\,$
1, 1/2	$\sqrt{\frac{4}{7}}\!\,$	$\sqrt{\frac{1}{35}}\!\,$	$-\sqrt{\frac{2}{5}}\!\,$
0, 3/2	$\sqrt{\frac{2}{7}}\!\,$	$-\sqrt{\frac{18}{35}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$

m=1/2

m₁, m₂=

	7/2	5/2	3/2	1/2
2, -3/2	$\sqrt{\frac{1}{35}}\!\,$	$\sqrt{\frac{6}{35}}\!\,$	$\sqrt{\frac{2}{5}}\!\,$	$\sqrt{\frac{2}{5}}\!\,$
1, -1/2	$\sqrt{\frac{12}{35}}\!\,$	$\sqrt{\frac{5}{14}}\!\,$	$0\!\,$	$-\sqrt{\frac{3}{10}}\!\,$
0, 1/2	$\sqrt{\frac{18}{35}}\!\,$	$-\sqrt{\frac{3}{35}}\!\,$	$-\sqrt{\frac{1}{5}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$
-1, 3/2	$\sqrt{\frac{4}{35}}\!\,$	$-\sqrt{\frac{27}{70}}\!\,$	$\sqrt{\frac{2}{5}}\!\,$	$-\sqrt{\frac{1}{10}}\!\,$

[edit] j₁=2, j₂=2

m=4

m₁, m₂=

	4
2, 2	$1\!\,$

m=3

m₁, m₂=

	4	3
2, 1	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$
1, 2	$\sqrt{\frac{1}{2}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$

m=2

m₁, m₂=

	4	3	2
2, 0	$\sqrt{\frac{3}{14}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{2}{7}}\!\,$
1, 1	$\sqrt{\frac{4}{7}}\!\,$	$0\!\,$	$-\sqrt{\frac{3}{7}}\!\,$
0, 2	$\sqrt{\frac{3}{14}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{2}{7}}\!\,$

m=1

m₁, m₂=

	4	3	2	1
2, -1	$\sqrt{\frac{1}{14}}\!\,$	$\sqrt{\frac{3}{10}}\!\,$	$\sqrt{\frac{3}{7}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$
1, 0	$\sqrt{\frac{3}{7}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$	$-\sqrt{\frac{1}{14}}\!\,$	$-\sqrt{\frac{3}{10}}\!\,$
0, 1	$\sqrt{\frac{3}{7}}\!\,$	$-\sqrt{\frac{1}{5}}\!\,$	$-\sqrt{\frac{1}{14}}\!\,$	$\sqrt{\frac{3}{10}}\!\,$
-1, 2	$\sqrt{\frac{1}{14}}\!\,$	$-\sqrt{\frac{3}{10}}\!\,$	$\sqrt{\frac{3}{7}}\!\,$	$-\sqrt{\frac{1}{5}}\!\,$

m=0

m₁, m₂=

	4	3	2	1	0
2, -2	$\sqrt{\frac{1}{70}}\!\,$	$\sqrt{\frac{1}{10}}\!\,$	$\sqrt{\frac{2}{7}}\!\,$	$\sqrt{\frac{2}{5}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$
1, -1	$\sqrt{\frac{8}{35}}\!\,$	$\sqrt{\frac{2}{5}}\!\,$	$\sqrt{\frac{1}{14}}\!\,$	$-\sqrt{\frac{1}{10}}\!\,$	$-\sqrt{\frac{1}{5}}\!\,$
0, 0	$\sqrt{\frac{18}{35}}\!\,$	$0\!\,$	$-\sqrt{\frac{2}{7}}\!\,$	$0\!\,$	$\sqrt{\frac{1}{5}}\!\,$
-1, 1	$\sqrt{\frac{8}{35}}\!\,$	$-\sqrt{\frac{2}{5}}\!\,$	$\sqrt{\frac{1}{14}}\!\,$	$\sqrt{\frac{1}{10}}\!\,$	$-\sqrt{\frac{1}{5}}\!\,$
-2, 2	$\sqrt{\frac{1}{70}}\!\,$	$-\sqrt{\frac{1}{10}}\!\,$	$\sqrt{\frac{2}{7}}\!\,$	$-\sqrt{\frac{2}{5}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$

[edit] j₁=5/2, j₂=1/2

m=3

m₁, m₂=

	3
5/2, 1/2	$1\!\,$

m=2

m₁, m₂=

	3	2
5/2, -1/2	$\sqrt{\frac{1}{6}}\!\,$	$\sqrt{\frac{5}{6}}\!\,$
3/2, 1/2	$\sqrt{\frac{5}{6}}\!\,$	$-\sqrt{\frac{1}{6}}\!\,$

m=1

m₁, m₂=

	3	2
3/2, -1/2	$\sqrt{\frac{1}{3}}\!\,$	$\sqrt{\frac{2}{3}}\!\,$
1/2, 1/2	$\sqrt{\frac{2}{3}}\!\,$	$-\sqrt{\frac{1}{3}}\!\,$

m=0

m₁, m₂=

	3	2
1/2, -1/2	$\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$
-1/2, 1/2	$\sqrt{\frac{1}{2}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$

[edit] j₁=5/2, j₂=1

m=7/2

m₁, m₂=

	7/2
5/2, 1	$1\!\,$

m=5/2

m₁, m₂=

	7/2	5/2
5/2, 0	$\sqrt{\frac{2}{7}}\!\,$	$\sqrt{\frac{5}{7}}\!\,$
3/2, 1	$\sqrt{\frac{5}{7}}\!\,$	$-\sqrt{\frac{2}{7}}\!\,$

m=3/2

m₁, m₂=

	7/2	5/2	3/2
5/2, -1	$\sqrt{\frac{1}{21}}\!\,$	$\sqrt{\frac{2}{7}}\!\,$	$\sqrt{\frac{2}{3}}\!\,$
3/2, 0	$\sqrt{\frac{10}{21}}\!\,$	$\sqrt{\frac{9}{35}}\!\,$	$-\sqrt{\frac{4}{15}}\!\,$
1/2, 1	$\sqrt{\frac{10}{21}}\!\,$	$-\sqrt{\frac{16}{35}}\!\,$	$\sqrt{\frac{1}{15}}\!\,$

m=1/2

m₁, m₂=

	7/2	5/2	3/2
3/2, -1	$\sqrt{\frac{1}{7}}\!\,$	$\sqrt{\frac{16}{35}}\!\,$	$\sqrt{\frac{2}{5}}\!\,$
1/2, 0	$\sqrt{\frac{4}{7}}\!\,$	$\sqrt{\frac{1}{35}}\!\,$	$-\sqrt{\frac{2}{5}}\!\,$
-1/2, 1	$\sqrt{\frac{2}{7}}\!\,$	$-\sqrt{\frac{18}{35}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$

[edit] j₁=5/2, j₂=3/2

m=4

m₁, m₂=

	4
5/2, 3/2	$1\!\,$

m=3

m₁, m₂=

	4	3
5/2, 1/2	$\sqrt{\frac{3}{8}}\!\,$	$\sqrt{\frac{5}{8}}\!\,$
3/2, 3/2	$\sqrt{\frac{5}{8}}\!\,$	$-\sqrt{\frac{3}{8}}\!\,$

m=2

m₁, m₂=

	4	3	2
5/2, -1/2	$\sqrt{\frac{3}{28}}\!\,$	$\sqrt{\frac{5}{12}}\!\,$	$\sqrt{\frac{10}{21}}\!\,$
3/2, 1/2	$\sqrt{\frac{15}{28}}\!\,$	$\sqrt{\frac{1}{12}}\!\,$	$-\sqrt{\frac{8}{21}}\!\,$
1/2, 3/2	$\sqrt{\frac{5}{14}}\!\,$	$-\sqrt{\frac{1}{2}}\!\,$	$\sqrt{\frac{1}{7}}\!\,$

m=1

m₁, m₂=

	4	3	2	1
5/2, -3/2	$\sqrt{\frac{1}{56}}\!\,$	$\sqrt{\frac{1}{8}}\!\,$	$\sqrt{\frac{5}{14}}\!\,$	$\sqrt{\frac{1}{2}}\!\,$
3/2, -1/2	$\sqrt{\frac{15}{56}}\!\,$	$\sqrt{\frac{49}{120}}\!\,$	$\sqrt{\frac{1}{42}}\!\,$	$-\sqrt{\frac{3}{10}}\!\,$
1/2, 1/2	$\sqrt{\frac{15}{28}}\!\,$	$-\sqrt{\frac{1}{60}}\!\,$	$-\sqrt{\frac{25}{84}}\!\,$	$\sqrt{\frac{3}{20}}\!\,$
-1/2, 3/2	$\sqrt{\frac{5}{28}}\!\,$	$-\sqrt{\frac{9}{20}}\!\,$	$\sqrt{\frac{9}{28}}\!\,$	$-\sqrt{\frac{1}{20}}\!\,$

m=0

m₁, m₂=

	4	3	2	1
3/2, -3/2	$\sqrt{\frac{1}{14}}\!\,$	$\sqrt{\frac{3}{10}}\!\,$	$\sqrt{\frac{3}{7}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$
1/2, -1/2	$\sqrt{\frac{3}{7}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$	$-\sqrt{\frac{1}{14}}\!\,$	$-\sqrt{\frac{3}{10}}\!\,$
-1/2, 1/2	$\sqrt{\frac{3}{7}}\!\,$	$-\sqrt{\frac{1}{5}}\!\,$	$-\sqrt{\frac{1}{14}}\!\,$	$\sqrt{\frac{3}{10}}\!\,$
-3/2, 3/2	$\sqrt{\frac{1}{14}}\!\,$	$-\sqrt{\frac{3}{10}}\!\,$	$\sqrt{\frac{3}{7}}\!\,$	$-\sqrt{\frac{1}{5}}\!\,$

[edit] j₁=5/2, j₂=2

m=9/2

m₁, m₂=

	9/2
5/2, 2	$1\!\,$

m=7/2

m₁, m₂=

	9/2	7/2
5/2, 1	$\frac{2}{3}\!\,$	$\sqrt{\frac{5}{9}}\!\,$
3/2, 2	$\sqrt{\frac{5}{9}}\!\,$	$-\frac{2}{3}\!\,$

m=5/2

m₁, m₂=

	9/2	7/2	5/2
5/2, 0	$\sqrt{\frac{1}{6}}\!\,$	$\sqrt{\frac{10}{21}}\!\,$	$\sqrt{\frac{5}{14}}\!\,$
3/2, 1	$\sqrt{\frac{5}{9}}\!\,$	$\sqrt{\frac{1}{63}}\!\,$	$-\sqrt{\frac{3}{7}}\!\,$
1/2, 2	$\sqrt{\frac{5}{18}}\!\,$	$-\sqrt{\frac{32}{63}}\!\,$	$\sqrt{\frac{3}{14}}\!\,$

m=3/2

m₁, m₂=

	9/2	7/2	5/2	3/2
5/2, -1	$\sqrt{\frac{1}{21}}\!\,$	$\sqrt{\frac{5}{21}}\!\,$	$\sqrt{\frac{3}{7}}\!\,$	$\sqrt{\frac{2}{7}}\!\,$
3/2, 0	$\sqrt{\frac{5}{14}}\!\,$	$\sqrt{\frac{2}{7}}\!\,$	$-\sqrt{\frac{1}{70}}\!\,$	$-\sqrt{\frac{12}{35}}\!\,$
1/2, 1	$\sqrt{\frac{10}{21}}\!\,$	$-\sqrt{\frac{2}{21}}\!\,$	$-\sqrt{\frac{6}{35}}\!\,$	$\sqrt{\frac{9}{35}}\!\,$
-1/2, 2	$\sqrt{\frac{5}{42}}\!\,$	$-\sqrt{\frac{8}{21}}\!\,$	$\sqrt{\frac{27}{70}}\!\,$	$-\sqrt{\frac{4}{35}}\!\,$

m=1/2

m₁, m₂=

	9/2	7/2	5/2	3/2	1/2
5/2, -2	$\sqrt{\frac{1}{126}}\!\,$	$\sqrt{\frac{4}{63}}\!\,$	$\sqrt{\frac{3}{14}}\!\,$	$\sqrt{\frac{8}{21}}\!\,$	$\sqrt{\frac{1}{3}}\!\,$
3/2, -1	$\sqrt{\frac{10}{63}}\!\,$	$\sqrt{\frac{121}{315}}\!\,$	$\sqrt{\frac{6}{35}}\!\,$	$-\sqrt{\frac{2}{105}}\!\,$	$-\sqrt{\frac{4}{15}}\!\,$
1/2, 0	$\sqrt{\frac{10}{21}}\!\,$	$\sqrt{\frac{4}{105}}\!\,$	$-\sqrt{\frac{8}{35}}\!\,$	$-\sqrt{\frac{2}{35}}\!\,$	$\sqrt{\frac{1}{5}}\!\,$
-1/2, 1	$\sqrt{\frac{20}{63}}\!\,$	$-\sqrt{\frac{14}{45}}\!\,$	$0\!\,$	$\sqrt{\frac{5}{21}}\!\,$	$-\sqrt{\frac{2}{15}}\!\,$
-3/2, 2	$\sqrt{\frac{5}{126}}\!\,$	$-\sqrt{\frac{64}{315}}\!\,$	$\sqrt{\frac{27}{70}}\!\,$	$-\sqrt{\frac{32}{105}}\!\,$	$\sqrt{\frac{1}{15}}\!\,$