Properties

Label 731.2.a.d
Level 731
Weight 2
Character orbit 731.a
Self dual Yes
Analytic conductor 5.837
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(5.83706438776\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{3} + \beta_{4} ) q^{3} + ( \beta_{2} - \beta_{3} ) q^{4} + ( -1 - \beta_{1} - \beta_{4} ) q^{5} + ( \beta_{4} - \beta_{5} + \beta_{7} ) q^{6} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{7} + ( \beta_{5} + \beta_{6} ) q^{8} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{3} + \beta_{4} ) q^{3} + ( \beta_{2} - \beta_{3} ) q^{4} + ( -1 - \beta_{1} - \beta_{4} ) q^{5} + ( \beta_{4} - \beta_{5} + \beta_{7} ) q^{6} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{7} + ( \beta_{5} + \beta_{6} ) q^{8} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{9} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{10} + ( -\beta_{2} + \beta_{4} ) q^{11} + ( -1 - \beta_{2} + \beta_{5} ) q^{12} + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{14} + ( -1 - \beta_{4} + \beta_{5} + \beta_{6} ) q^{15} + ( \beta_{3} + \beta_{6} + \beta_{7} ) q^{16} + q^{17} + ( -\beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{18} + ( -\beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{6} ) q^{20} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{21} + ( -2 + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{22} + ( -2 + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{23} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{24} + ( -1 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} + ( -2 - \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{26} + ( -1 + \beta_{2} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{27} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{28} + ( -3 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{29} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{30} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{31} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{32} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{33} + \beta_{1} q^{34} + ( 2 + \beta_{2} + \beta_{5} + \beta_{6} ) q^{35} + ( -1 - \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{36} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{37} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{38} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{39} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{40} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( 2 \beta_{2} + 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{42} + q^{43} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{44} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{45} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{46} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{47} + ( -\beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{48} + ( -1 - \beta_{2} + 4 \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{49} + ( 3 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{50} + ( \beta_{3} + \beta_{4} ) q^{51} + ( 1 + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{52} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{53} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{54} + ( -2 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{55} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{56} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{57} + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{59} + ( 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{60} + ( -3 + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{61} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{62} + ( -1 + \beta_{1} + \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{63} + ( -3 + 2 \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{64} + ( 1 + 5 \beta_{1} - \beta_{2} + 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{65} + ( 1 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{66} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{67} + ( \beta_{2} - \beta_{3} ) q^{68} + ( -2 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{69} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{70} + ( -1 - \beta_{1} - 2 \beta_{2} - 5 \beta_{4} + \beta_{6} ) q^{71} + ( -1 - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{72} + ( -1 + 2 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{73} + ( 1 - 7 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{74} + ( 2 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{75} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{76} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{77} + ( -1 - 2 \beta_{2} - \beta_{3} - 6 \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{78} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{79} + ( 3 + 2 \beta_{4} - 2 \beta_{6} ) q^{80} + ( -2 + \beta_{2} - \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{81} + ( 4 + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{82} + ( -1 - \beta_{1} - \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{83} + ( 3 + \beta_{1} + 5 \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{84} + ( -1 - \beta_{1} - \beta_{4} ) q^{85} + \beta_{1} q^{86} + ( 4 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - \beta_{5} - 3 \beta_{7} ) q^{87} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - \beta_{7} ) q^{88} + ( 2 + 6 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{89} + ( 1 + 4 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{90} + ( -1 + 3 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{91} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{92} + ( 1 + 5 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{93} + ( -1 + 4 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{94} + ( 5 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{95} + ( 5 + \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{96} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{97} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{98} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{2} - 3q^{3} + 3q^{4} - 7q^{5} - q^{6} - 5q^{7} - 3q^{8} + 5q^{9} + O(q^{10}) \) \( 8q + q^{2} - 3q^{3} + 3q^{4} - 7q^{5} - q^{6} - 5q^{7} - 3q^{8} + 5q^{9} - 12q^{10} - 4q^{11} - 13q^{12} - 12q^{13} + q^{14} - 9q^{15} - 3q^{16} + 8q^{17} + 5q^{18} - 5q^{20} - 20q^{21} - 14q^{22} - 9q^{23} - q^{24} - 7q^{25} - 17q^{26} - 12q^{27} + q^{28} - 27q^{29} + 10q^{30} - 12q^{31} + 5q^{32} + 10q^{33} + q^{34} + 15q^{35} - 4q^{36} - 24q^{37} - q^{38} + 3q^{39} - 9q^{40} - 8q^{41} - 9q^{42} + 8q^{43} - 16q^{44} + 10q^{45} - 14q^{46} + 15q^{47} + 10q^{48} - 7q^{49} + 21q^{50} - 3q^{51} + q^{52} - 23q^{53} - 19q^{54} - 14q^{55} - 20q^{56} - 13q^{57} - 7q^{58} + 16q^{59} - 3q^{60} - 34q^{61} + 15q^{62} + 9q^{63} - 25q^{64} + 10q^{65} + 15q^{66} + 3q^{68} - 19q^{69} + 11q^{70} - 3q^{71} - 19q^{72} - 3q^{73} - 4q^{74} + 27q^{75} + 13q^{76} - 3q^{77} + 4q^{78} - 24q^{79} + 20q^{80} - 8q^{81} + 33q^{82} - 8q^{83} + 17q^{84} - 7q^{85} + q^{86} + 48q^{87} + 16q^{88} + 23q^{89} + 11q^{90} - 16q^{91} + 49q^{92} + 17q^{93} - 11q^{94} + 3q^{95} + 37q^{96} - 10q^{97} + 29q^{98} - 15q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 9 x^{6} + 9 x^{5} + 21 x^{4} - 21 x^{3} - 8 x^{2} + 7 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{7} - \nu^{6} - 9 \nu^{5} + 9 \nu^{4} + 21 \nu^{3} - 20 \nu^{2} - 8 \nu + 4 \)
\(\beta_{3}\)\(=\)\( \nu^{7} - \nu^{6} - 9 \nu^{5} + 9 \nu^{4} + 21 \nu^{3} - 21 \nu^{2} - 8 \nu + 6 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{7} - \nu^{6} - 18 \nu^{5} + 9 \nu^{4} + 43 \nu^{3} - 21 \nu^{2} - 22 \nu + 5 \)
\(\beta_{5}\)\(=\)\( -4 \nu^{7} + 3 \nu^{6} + 37 \nu^{5} - 27 \nu^{4} - 92 \nu^{3} + 62 \nu^{2} + 48 \nu - 16 \)
\(\beta_{6}\)\(=\)\( 4 \nu^{7} - 3 \nu^{6} - 37 \nu^{5} + 27 \nu^{4} + 93 \nu^{3} - 62 \nu^{2} - 52 \nu + 16 \)
\(\beta_{7}\)\(=\)\( -5 \nu^{7} + 4 \nu^{6} + 46 \nu^{5} - 35 \nu^{4} - 114 \nu^{3} + 77 \nu^{2} + 60 \nu - 18 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} - 5 \beta_{3} + 6 \beta_{2} + 8\)
\(\nu^{5}\)\(=\)\(7 \beta_{6} + 8 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 18 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(9 \beta_{7} + 8 \beta_{6} - \beta_{5} + \beta_{4} - 26 \beta_{3} + 33 \beta_{2} + 2 \beta_{1} + 37\)
\(\nu^{7}\)\(=\)\(41 \beta_{6} + 50 \beta_{5} + 10 \beta_{4} + 8 \beta_{3} + 9 \beta_{2} + 88 \beta_{1} + 10\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.31200
−1.78724
−0.700276
0.235409
0.298156
1.26872
1.66158
2.33565
−2.31200 −1.95532 3.34533 1.83479 4.52069 0.437329 −3.11039 0.823276 −4.24203
1.2 −1.78724 1.17799 1.19423 −1.95027 −2.10535 −1.57661 1.44011 −1.61234 3.48560
1.3 −0.700276 −2.17405 −1.50961 −0.553685 1.52243 −0.939298 2.45770 1.72649 0.387732
1.4 0.235409 2.48067 −1.94458 −0.468153 0.583972 −5.07261 −0.928590 3.15374 −0.110207
1.5 0.298156 0.0958345 −1.91110 0.959960 0.0285736 1.32998 −1.16612 −2.99082 0.286218
1.6 1.26872 1.44891 −0.390345 −3.92944 1.83826 0.416048 −3.03268 −0.900661 −4.98536
1.7 1.66158 −3.15537 0.760849 0.0956277 −5.24290 3.17624 −2.05895 6.95637 0.158893
1.8 2.33565 −0.918667 3.45524 −2.98883 −2.14568 −2.77109 3.39893 −2.15605 −6.98085
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(43\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(731))\).