Properties

Label 731.2.a.f
Level 731
Weight 2
Character orbit 731.a
Self dual Yes
Analytic conductor 5.837
Analytic rank 0
Dimension 21
CM No

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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: \(0\)
Dimension: \(21\)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 21q + 2q^{2} - q^{3} + 32q^{4} - 3q^{5} + q^{6} + 5q^{7} + 6q^{8} + 34q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 21q + 2q^{2} - q^{3} + 32q^{4} - 3q^{5} + q^{6} + 5q^{7} + 6q^{8} + 34q^{9} + 12q^{10} + 2q^{11} - 5q^{12} + 26q^{13} - 3q^{14} + 5q^{15} + 62q^{16} - 21q^{17} - 10q^{18} + 16q^{19} - 27q^{20} + 36q^{21} - 14q^{22} - q^{23} + 15q^{24} + 40q^{25} - 3q^{26} - 16q^{27} + 25q^{28} + 15q^{29} + 38q^{30} + 18q^{31} + 14q^{32} + 14q^{33} - 2q^{34} - 5q^{35} + 73q^{36} + 12q^{37} + 19q^{38} - 11q^{39} + 41q^{40} - 45q^{42} + 21q^{43} - 34q^{44} - 18q^{45} + 28q^{46} - q^{47} - 36q^{48} + 40q^{49} + 24q^{50} + q^{51} + 23q^{52} + 39q^{53} - 83q^{54} - 2q^{55} - 10q^{56} + 3q^{57} + 19q^{58} + 4q^{59} - 35q^{60} + 50q^{61} + 5q^{62} - 37q^{63} + 120q^{64} - 8q^{65} - 37q^{66} + 16q^{67} - 32q^{68} + 33q^{69} - q^{70} + q^{71} - 54q^{72} + 15q^{73} + 52q^{74} + 11q^{75} - 15q^{76} + 13q^{77} - 100q^{78} + 56q^{79} - 100q^{80} + 97q^{81} - 11q^{82} + 61q^{84} + 3q^{85} + 2q^{86} - 8q^{87} - 56q^{88} + 5q^{89} - 69q^{90} - 4q^{91} - 27q^{92} + 17q^{93} - 47q^{94} - 9q^{95} + 81q^{96} - 28q^{97} + 18q^{98} - 19q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.79909 −2.92541 5.83488 −1.76305 8.18847 −2.49895 −10.7342 5.55801 4.93492
1.2 −2.71140 −0.220213 5.35167 −0.654741 0.597084 1.40135 −9.08771 −2.95151 1.77526
1.3 −2.48059 2.78565 4.15334 −4.06907 −6.91006 4.55433 −5.34155 4.75983 10.0937
1.4 −1.94262 3.44146 1.77376 0.802251 −6.68543 −1.20527 0.439504 8.84362 −1.55846
1.5 −1.79843 −0.497156 1.23435 −2.09348 0.894100 2.50164 1.37698 −2.75284 3.76498
1.6 −1.62264 −3.32613 0.632968 2.56619 5.39712 −2.29244 2.21820 8.06315 −4.16401
1.7 −1.58029 −0.808715 0.497302 3.43972 1.27800 2.59985 2.37469 −2.34598 −5.43574
1.8 −0.899791 0.468379 −1.19038 −0.603312 −0.421443 −2.56518 2.87067 −2.78062 0.542854
1.9 −0.387858 1.84237 −1.84957 2.41846 −0.714576 4.70313 1.49308 0.394314 −0.938020
1.10 −0.172452 −0.293380 −1.97026 −4.21864 0.0505941 −4.31664 0.684680 −2.91393 0.727513
1.11 0.169978 −1.63219 −1.97111 −3.37745 −0.277437 0.890456 −0.675002 −0.335949 −0.574094
1.12 0.549823 2.96872 −1.69769 −0.0524046 1.63227 −0.375563 −2.03308 5.81330 −0.0288132
1.13 0.715531 −2.68508 −1.48801 1.60306 −1.92126 −4.82676 −2.49578 4.20967 1.14704
1.14 1.10090 −1.62202 −0.788018 2.17965 −1.78568 4.08725 −3.06933 −0.369054 2.39957
1.15 1.35548 2.29949 −0.162684 4.25442 3.11690 −1.26534 −2.93147 2.28766 5.76676
1.16 2.02811 1.50881 2.11322 −1.59736 3.06003 3.53856 0.229620 −0.723487 −3.23961
1.17 2.03648 −3.27527 2.14723 −3.88495 −6.67000 −1.33230 0.299836 7.72736 −7.91160
1.18 2.29170 0.857983 3.25189 2.44469 1.96624 2.38002 2.86896 −2.26387 5.60249
1.19 2.68194 2.28817 5.19279 −3.20246 6.13674 1.43094 8.56287 2.23574 −8.58881
1.20 2.72856 −2.69681 5.44504 0.220730 −7.35840 2.22641 9.39999 4.27277 0.602275
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
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Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{21} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(731))\).