Properties

Label 731.2.f.c
Level 731
Weight 2
Character orbit 731.f
Analytic conductor 5.837
Analytic rank 0
Dimension 56
CM No

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56q - 4q^{3} - 60q^{4} - 2q^{5} + 8q^{6} + 4q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 4q^{3} - 60q^{4} - 2q^{5} + 8q^{6} + 4q^{7} + 4q^{10} - 6q^{11} - 14q^{12} + 16q^{13} + 6q^{14} + 44q^{16} + 8q^{17} - 16q^{18} + 12q^{20} + 28q^{21} + 14q^{22} + 6q^{23} + 62q^{24} - 4q^{27} + 34q^{28} - 12q^{29} - 96q^{30} + 14q^{31} - 44q^{33} + 6q^{34} - 32q^{35} + 8q^{37} + 72q^{38} - 32q^{39} - 84q^{40} + 24q^{41} + 4q^{44} + 48q^{45} - 4q^{46} - 8q^{47} + 38q^{48} - 64q^{50} + 28q^{51} - 48q^{52} + 34q^{54} + 56q^{55} - 26q^{56} - 102q^{57} + 76q^{58} - 40q^{61} + 34q^{62} + 4q^{63} - 204q^{64} + 18q^{65} - 20q^{67} + 30q^{68} - 16q^{69} - 26q^{71} + 144q^{72} + 8q^{73} - 80q^{74} + 142q^{75} - 32q^{78} - 22q^{79} - 32q^{80} - 32q^{81} + 100q^{82} - 20q^{84} - 2q^{85} + 12q^{86} + 34q^{88} + 68q^{89} - 10q^{90} - 28q^{91} + 68q^{92} - 62q^{95} + 62q^{96} - 2q^{97} - 32q^{98} + 66q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
259.1 2.81279i −1.57952 + 1.57952i −5.91177 −2.62660 + 2.62660i 4.44284 + 4.44284i 0.229862 + 0.229862i 11.0030i 1.98974i 7.38806 + 7.38806i
259.2 2.77460i 1.70485 1.70485i −5.69840 0.433295 0.433295i −4.73027 4.73027i −2.87350 2.87350i 10.2616i 2.81302i −1.20222 1.20222i
259.3 2.68403i 0.675148 0.675148i −5.20400 −0.573416 + 0.573416i −1.81211 1.81211i 1.88065 + 1.88065i 8.59962i 2.08835i 1.53907 + 1.53907i
259.4 2.23190i −0.335489 + 0.335489i −2.98138 −0.357194 + 0.357194i 0.748779 + 0.748779i 0.708232 + 0.708232i 2.19035i 2.77489i 0.797222 + 0.797222i
259.5 2.18921i 0.566370 0.566370i −2.79262 2.16421 2.16421i −1.23990 1.23990i 0.402819 + 0.402819i 1.73522i 2.35845i −4.73791 4.73791i
259.6 2.09057i 2.10973 2.10973i −2.37046 0.614601 0.614601i −4.41052 4.41052i 0.361953 + 0.361953i 0.774475i 5.90190i −1.28486 1.28486i
259.7 1.74414i −0.697298 + 0.697298i −1.04202 1.44943 1.44943i 1.21618 + 1.21618i 1.15181 + 1.15181i 1.67086i 2.02755i −2.52800 2.52800i
259.8 1.64399i −2.17617 + 2.17617i −0.702716 −2.09084 + 2.09084i 3.57761 + 3.57761i −2.29672 2.29672i 2.13273i 6.47143i 3.43732 + 3.43732i
259.9 1.52916i −0.854970 + 0.854970i −0.338329 −2.11844 + 2.11844i 1.30739 + 1.30739i 0.0998235 + 0.0998235i 2.54096i 1.53805i 3.23944 + 3.23944i
259.10 1.47026i −1.92339 + 1.92339i −0.161659 2.97219 2.97219i 2.82787 + 2.82787i −2.28989 2.28989i 2.70284i 4.39883i −4.36988 4.36988i
259.11 1.21542i 0.483456 0.483456i 0.522759 1.73874 1.73874i −0.587601 0.587601i 3.36984 + 3.36984i 3.06621i 2.53254i −2.11330 2.11330i
259.12 1.09376i 0.497569 0.497569i 0.803697 −2.18307 + 2.18307i −0.544219 0.544219i 0.291940 + 0.291940i 3.06656i 2.50485i 2.38774 + 2.38774i
259.13 0.607091i −1.55808 + 1.55808i 1.63144 1.22244 1.22244i 0.945900 + 0.945900i 1.59838 + 1.59838i 2.20462i 1.85526i −0.742133 0.742133i
259.14 0.600676i 0.381351 0.381351i 1.63919 0.0334344 0.0334344i −0.229068 0.229068i −3.21348 3.21348i 2.18597i 2.70914i −0.0200832 0.0200832i
259.15 0.395202i −2.06164 + 2.06164i 1.84382 −2.29638 + 2.29638i 0.814765 + 0.814765i 2.83331 + 2.83331i 1.51908i 5.50076i 0.907534 + 0.907534i
259.16 0.265604i 0.626669 0.626669i 1.92945 −0.906788 + 0.906788i 0.166446 + 0.166446i −0.565427 0.565427i 1.04368i 2.21457i −0.240847 0.240847i
259.17 0.401786i 1.92935 1.92935i 1.83857 1.46367 1.46367i 0.775185 + 0.775185i 0.274168 + 0.274168i 1.54228i 4.44476i 0.588082 + 0.588082i
259.18 0.478501i −0.772701 + 0.772701i 1.77104 2.77993 2.77993i −0.369738 0.369738i −2.40550 2.40550i 1.80444i 1.80587i 1.33020 + 1.33020i
259.19 0.831038i −0.711849 + 0.711849i 1.30938 0.619206 0.619206i −0.591574 0.591574i 2.80838 + 2.80838i 2.75022i 1.98654i 0.514584 + 0.514584i
259.20 1.30582i 2.39223 2.39223i 0.294832 −1.70398 + 1.70398i 3.12383 + 3.12383i 2.61268 + 2.61268i 2.99664i 8.44554i −2.22509 2.22509i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 302.28
Significant digits:
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Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\):

\(T_{2}^{56} + \cdots\)
\(T_{3}^{56} + \cdots\)