Properties

Label 731.2.bl.a
Level 731
Weight 2
Character orbit 731.bl
Analytic conductor 5.837
Analytic rank 0
Dimension 3072
CM No

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.bl (of order \(168\) and degree \(48\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(3072\)
Relative dimension: \(64\) over \(\Q(\zeta_{168})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{168}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3072q - 40q^{2} - 52q^{3} - 44q^{5} - 24q^{6} - 24q^{7} - 72q^{8} - 28q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 3072q - 40q^{2} - 52q^{3} - 44q^{5} - 24q^{6} - 24q^{7} - 72q^{8} - 28q^{9} - 52q^{10} - 40q^{11} - 88q^{12} - 68q^{14} - 52q^{15} + 384q^{16} - 52q^{17} - 80q^{18} - 52q^{19} - 68q^{20} - 40q^{22} - 56q^{23} - 120q^{24} - 60q^{25} - 92q^{26} + 56q^{27} - 196q^{28} - 52q^{29} - 44q^{31} - 72q^{32} - 32q^{33} - 68q^{34} - 16q^{35} + 24q^{36} - 56q^{37} - 16q^{39} - 96q^{40} - 24q^{41} - 144q^{42} - 44q^{43} + 40q^{45} - 260q^{46} + 48q^{48} - 44q^{49} + 464q^{50} - 40q^{51} - 200q^{52} - 48q^{53} + 48q^{54} - 208q^{56} - 16q^{57} - 24q^{58} - 16q^{59} - 24q^{60} - 76q^{61} - 72q^{62} - 64q^{63} + 44q^{66} - 264q^{67} - 68q^{68} - 32q^{69} - 8q^{70} - 100q^{71} - 80q^{73} + 180q^{74} + 240q^{75} - 440q^{76} - 92q^{77} - 264q^{78} + 136q^{79} - 104q^{80} + 80q^{82} + 108q^{83} - 208q^{84} - 336q^{85} - 72q^{86} - 56q^{87} - 284q^{88} - 152q^{90} - 320q^{91} + 296q^{92} - 248q^{93} + 80q^{94} - 36q^{95} + 800q^{96} + 96q^{97} - 52q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −2.72251 0.306753i −0.620611 0.373954i 5.36808 + 1.22523i 0.404374 0.264759i 1.57490 + 1.20847i −5.03855 0.663338i −9.06682 3.17262i −1.15649 2.18819i −1.18213 + 0.596764i
9.2 −2.63781 0.297210i −1.45043 0.873969i 4.91985 + 1.12292i −0.178314 + 0.116749i 3.56621 + 2.73645i 3.75466 + 0.494311i −7.63281 2.67084i −0.0618808 0.117084i 0.505058 0.254965i
9.3 −2.55439 0.287810i 1.48660 + 0.895766i 4.49220 + 1.02532i −2.99252 + 1.95932i −3.53955 2.71599i 0.541511 + 0.0712913i −6.32713 2.21396i 0.00578988 + 0.0109550i 8.20797 4.14357i
9.4 −2.54211 0.286426i 2.35858 + 1.42118i 4.43040 + 1.01121i 1.38702 0.908135i −5.58870 4.28836i 3.61781 + 0.476294i −6.14365 2.14976i 2.14134 + 4.05160i −3.78607 + 1.91129i
9.5 −2.52269 0.284239i 0.0154490 + 0.00930894i 4.33332 + 0.989053i 0.766036 0.501553i −0.0363272 0.0278748i −0.223013 0.0293603i −5.85813 2.04985i −1.40165 2.65206i −2.07503 + 1.04753i
9.6 −2.50024 0.281710i 0.980817 + 0.591000i 4.22200 + 0.963643i 2.79856 1.83232i −2.28579 1.75395i −0.0420865 0.00554079i −5.53481 1.93671i −0.789084 1.49302i −7.51326 + 3.79287i
9.7 −2.31943 0.261336i −2.95396 1.77993i 3.36158 + 0.767259i −1.47851 + 0.968034i 6.38632 + 4.90040i −1.74587 0.229848i −3.19018 1.11629i 4.15589 + 7.86334i 3.68227 1.85889i
9.8 −2.29109 0.258143i −0.766885 0.462093i 3.23258 + 0.737816i −3.34001 + 2.18683i 1.63771 + 1.25666i −2.68123 0.352990i −2.86326 1.00190i −1.02722 1.94360i 8.21676 4.14801i
9.9 −2.25200 0.253740i 2.54761 + 1.53509i 3.05728 + 0.697804i 0.169346 0.110877i −5.34772 4.10345i −3.68545 0.485199i −2.42979 0.850219i 2.73203 + 5.16926i −0.409502 + 0.206726i
9.10 −2.13253 0.240278i −1.98754 1.19761i 2.54008 + 0.579756i −1.15205 + 0.754294i 3.95072 + 3.03150i 0.0673855 + 0.00887147i −1.22629 0.429097i 1.11425 + 2.10826i 2.63803 1.33174i
9.11 −2.06687 0.232880i −0.844141 0.508645i 2.26786 + 0.517625i 3.26327 2.13659i 1.62628 + 1.24789i 1.78259 + 0.234683i −0.640380 0.224079i −0.947951 1.79361i −7.24233 + 3.65610i
9.12 −1.98773 0.223963i 0.209805 + 0.126420i 1.95104 + 0.445312i −1.38577 + 0.907318i −0.388722 0.298277i 2.47032 + 0.325224i −0.00229218 0.000802070i −1.37377 2.59930i 2.95774 1.49314i
9.13 −1.75957 0.198256i 1.94400 + 1.17137i 1.10693 + 0.252650i −1.51027 + 0.988831i −3.18837 2.44652i 0.127518 + 0.0167880i 1.44504 + 0.505643i 1.00521 + 1.90194i 2.85347 1.44050i
9.14 −1.71083 0.192764i 1.34544 + 0.810704i 0.939930 + 0.214533i 1.48366 0.971406i −2.14554 1.64633i −3.99522 0.525981i 1.68338 + 0.589040i −0.248850 0.470847i −2.72554 + 1.37592i
9.15 −1.70400 0.191995i −1.59121 0.958800i 0.916901 + 0.209277i 0.260745 0.170719i 2.52735 + 1.93930i −1.40514 0.184990i 1.71489 + 0.600066i 0.210860 + 0.398966i −0.477086 + 0.240844i
9.16 −1.69079 0.190506i −1.80830 1.08960i 0.872615 + 0.199169i 2.36534 1.54868i 2.84987 + 2.18678i −2.81774 0.370962i 1.77455 + 0.620940i 0.680893 + 1.28831i −4.29432 + 2.16787i
9.17 −1.58124 0.178163i 0.373386 + 0.224987i 0.518729 + 0.118397i −0.176659 + 0.115666i −0.550330 0.422283i 4.75303 + 0.625749i 2.20476 + 0.771479i −1.31301 2.48433i 0.299949 0.151421i
9.18 −1.52146 0.171428i −2.06462 1.24405i 0.335610 + 0.0766007i 1.34334 0.879534i 2.92798 + 2.24672i 4.03118 + 0.530715i 2.39286 + 0.837297i 1.31318 + 2.48465i −2.19462 + 1.10789i
9.19 −1.36186 0.153444i −0.627515 0.378115i −0.118749 0.0271036i −2.79353 + 1.82903i 0.796566 + 0.611227i −4.13974 0.545007i 2.74469 + 0.960410i −1.15100 2.17780i 4.08504 2.06222i
9.20 −1.22085 0.137557i 2.07165 + 1.24829i −0.478291 0.109167i 1.32412 0.866951i −2.35747 1.80895i 3.11502 + 0.410100i 2.88818 + 1.01062i 1.33169 + 2.51968i −1.73581 + 0.876279i
See next 80 embeddings (of 3072 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 712.64
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 \(S_{2}^{\mathrm{new}}(731, [\chi])\).