Properties

Label 731.2.a.c
Level 731
Weight 2
Character orbit 731.a
Self dual Yes
Analytic conductor 5.837
Analytic rank 1
Dimension 6
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: \(6\)
Coefficient field: 6.6.2460365.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 3 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{5} + 6 \nu^{3} - 7 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 6 \nu^{3} + \nu^{2} + 7 \nu - 2 \)
\(\beta_{5}\)\(=\)\( -\nu^{5} + \nu^{4} + 6 \nu^{3} - 5 \nu^{2} - 7 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 5 \beta_{4} + 4 \beta_{3} + 6\)
\(\nu^{5}\)\(=\)\(-\beta_{3} + 6 \beta_{2} + 11 \beta_{1}\)

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
−2.08764
2.05953
1.48737
−1.30704
0.668929
0.178849
−2.35826 1.68213 3.56138 0.252087 −3.96689 −3.41152 −3.68213 −0.170443 −0.594485
1.2 −2.24167 0.297851 3.02506 1.49772 −0.667683 2.00334 −2.29785 −2.91128 −3.35738
1.3 −0.212277 −2.83954 −1.95494 −1.65901 0.602769 2.53919 0.839542 5.06300 0.352169
1.4 0.291653 −0.858197 −1.91494 3.99528 −0.250296 −2.74049 −1.14180 −2.26350 1.16523
1.5 1.55253 0.467968 0.410361 −1.37639 0.726536 −3.35157 −2.46797 −2.78101 −2.13690
1.6 1.96801 −1.75021 1.87307 0.290323 −3.44443 −2.03895 −0.249790 0.0632334 0.571360
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 \( T_{2}^{6} + T_{2}^{5} - 8 T_{2}^{4} - 4 T_{2}^{3} + 17 T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(731))\).