Properties

Label 731.2.v.a
Level 731
Weight 2
Character orbit 731.v
Analytic conductor 5.837
Analytic rank 0
Dimension 512
CM No

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.v (of order \(24\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(64\) over \(\Q(\zeta_{24})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 512q - 16q^{2} - 4q^{3} - 12q^{5} - 4q^{6} - 4q^{7} + 16q^{8} - 28q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 512q - 16q^{2} - 4q^{3} - 12q^{5} - 4q^{6} - 4q^{7} + 16q^{8} - 28q^{9} - 4q^{10} - 16q^{11} + 32q^{12} + 12q^{14} - 4q^{15} - 496q^{16} - 4q^{17} - 32q^{18} - 4q^{19} + 12q^{20} - 16q^{22} + 36q^{24} + 4q^{25} + 36q^{26} - 112q^{27} + 28q^{28} - 4q^{29} - 12q^{31} + 16q^{32} + 32q^{33} + 12q^{34} - 96q^{35} - 52q^{36} + 28q^{37} - 40q^{39} + 40q^{40} - 32q^{41} + 32q^{42} + 44q^{43} + 112q^{44} + 16q^{45} - 20q^{46} - 104q^{48} + 16q^{49} - 16q^{50} - 16q^{51} + 88q^{52} - 8q^{53} - 104q^{54} - 44q^{56} - 40q^{57} - 32q^{58} - 40q^{59} - 32q^{60} - 8q^{61} + 16q^{62} + 8q^{63} - 56q^{65} - 100q^{66} - 72q^{67} + 12q^{68} - 80q^{69} - 48q^{70} + 44q^{71} + 52q^{73} - 68q^{74} + 96q^{75} - 8q^{76} + 36q^{77} + 40q^{78} + 60q^{79} + 76q^{80} - 136q^{82} - 24q^{83} + 96q^{84} - 56q^{85} - 40q^{86} - 56q^{87} - 24q^{88} + 96q^{90} - 44q^{91} - 44q^{92} - 4q^{93} + 88q^{94} - 20q^{95} + 40q^{96} - 152q^{97} - 88q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −1.99180 1.99180i 1.18947 + 0.912714i 5.93451i 0.559417 0.729046i −0.551246 4.18713i −0.274381 0.357581i 7.83675 7.83675i −0.194660 0.726482i −2.56636 + 0.337867i
36.2 −1.88749 1.88749i −2.07239 1.59020i 5.12526i 0.776315 1.01171i 0.910131 + 6.91313i −2.46818 3.21659i 5.89891 5.89891i 0.989611 + 3.69328i −3.37489 + 0.444313i
36.3 −1.86379 1.86379i 1.50648 + 1.15596i 4.94745i −2.08729 + 2.72021i −0.653291 4.96224i 3.04795 + 3.97216i 5.49344 5.49344i 0.156773 + 0.585083i 8.96020 1.17963i
36.4 −1.73754 1.73754i −0.135121 0.103682i 4.03807i −2.51475 + 3.27729i 0.0546263 + 0.414928i −2.29712 2.99367i 3.54122 3.54122i −0.768949 2.86976i 10.0639 1.32493i
36.5 −1.69640 1.69640i −0.879972 0.675226i 3.75554i 0.00391762 0.00510554i 0.347331 + 2.63824i 1.14324 + 1.48990i 2.97810 2.97810i −0.458037 1.70942i −0.0153069 + 0.00201519i
36.6 −1.65128 1.65128i 1.93435 + 1.48428i 3.45346i −0.809599 + 1.05509i −0.743194 5.64512i −1.90304 2.48008i 2.40007 2.40007i 0.762170 + 2.84446i 3.07913 0.405375i
36.7 −1.62967 1.62967i 0.979757 + 0.751794i 3.31164i 1.53210 1.99667i −0.371504 2.82185i 0.813343 + 1.05997i 2.13753 2.13753i −0.381728 1.42463i −5.75072 + 0.757096i
36.8 −1.62148 1.62148i 2.59315 + 1.98979i 3.25840i 2.22006 2.89323i −0.978330 7.43116i −0.233089 0.303768i 2.04047 2.04047i 1.98869 + 7.42189i −8.29110 + 1.09154i
36.9 −1.57826 1.57826i −2.41848 1.85577i 2.98178i 2.30449 3.00327i 0.888110 + 6.74586i 3.01947 + 3.93505i 1.54951 1.54951i 1.62873 + 6.07851i −8.37702 + 1.10286i
36.10 −1.55278 1.55278i −0.793419 0.608812i 2.82227i −1.19124 + 1.55246i 0.286655 + 2.17736i −0.933104 1.21604i 1.27681 1.27681i −0.517595 1.93169i 4.26037 0.560889i
36.11 −1.51428 1.51428i −0.591375 0.453778i 2.58612i 2.49753 3.25484i 0.208361 + 1.58266i −1.77330 2.31101i 0.887545 0.887545i −0.632647 2.36107i −8.71073 + 1.14679i
36.12 −1.48255 1.48255i −0.314627 0.241421i 2.39590i −0.727962 + 0.948699i 0.108530 + 0.824368i 1.97310 + 2.57139i 0.586944 0.586944i −0.735752 2.74586i 2.48573 0.327253i
36.13 −1.31498 1.31498i 2.01865 + 1.54896i 1.45837i −0.221616 + 0.288815i −0.617629 4.69135i 1.79020 + 2.33304i −0.712236 + 0.712236i 0.899197 + 3.35585i 0.671209 0.0883663i
36.14 −1.28524 1.28524i −2.29696 1.76252i 1.30370i 0.354450 0.461928i 0.686886 + 5.21742i −1.37111 1.78687i −0.894913 + 0.894913i 1.39309 + 5.19909i −1.04924 + 0.138136i
36.15 −1.17911 1.17911i −2.26634 1.73903i 0.780582i −2.39062 + 3.11551i 0.621763 + 4.72276i 1.24532 + 1.62293i −1.43782 + 1.43782i 1.33564 + 4.98469i 6.49231 0.854729i
36.16 −1.15817 1.15817i 0.857326 + 0.657849i 0.682728i 0.910213 1.18621i −0.231028 1.75483i −2.35231 3.06560i −1.52563 + 1.52563i −0.474215 1.76979i −2.42802 + 0.319655i
36.17 −1.08984 1.08984i 0.863300 + 0.662433i 0.375497i −0.296927 + 0.386963i −0.218912 1.66280i 0.200337 + 0.261085i −1.77045 + 1.77045i −0.469988 1.75402i 0.745330 0.0981246i
36.18 −1.07479 1.07479i −2.28358 1.75225i 0.310339i −0.670369 + 0.873642i 0.571063 + 4.33766i −1.27789 1.66538i −1.81603 + 1.81603i 1.36788 + 5.10501i 1.65949 0.218475i
36.19 −1.00051 1.00051i 1.10606 + 0.848712i 0.00204655i −1.93877 + 2.52666i −0.257483 1.95578i −0.432434 0.563559i −1.99898 + 1.99898i −0.273393 1.02032i 4.46772 0.588186i
36.20 −0.952104 0.952104i −1.16992 0.897715i 0.186997i 0.397683 0.518271i 0.259172 + 1.96861i 1.89147 + 2.46501i −2.08225 + 2.08225i −0.213625 0.797259i −0.872083 + 0.114812i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 638.64
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(731, [\chi])\).