# Properties

 Label 731.2.k.a Level 731 Weight 2 Character orbit 731.k Analytic conductor 5.837 Analytic rank 0 Dimension 180 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.k (of order $$7$$ and degree $$6$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$180$$ Relative dimension: $$30$$ over $$\Q(\zeta_{7})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$180q - 4q^{2} - 6q^{3} - 34q^{4} - 6q^{5} - 14q^{6} + 66q^{7} + 2q^{8} - 26q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$180q - 4q^{2} - 6q^{3} - 34q^{4} - 6q^{5} - 14q^{6} + 66q^{7} + 2q^{8} - 26q^{9} - 8q^{10} - 12q^{11} - 24q^{12} - 5q^{13} - 8q^{14} - q^{15} - 58q^{16} - 30q^{17} + 36q^{18} - 24q^{19} - 36q^{20} - 28q^{21} + 5q^{22} + 27q^{23} - 46q^{24} - 38q^{25} + 6q^{26} - 36q^{27} - 54q^{28} + 28q^{29} - 21q^{30} + 36q^{31} + 11q^{32} + 5q^{33} - 4q^{34} + 6q^{35} + 192q^{36} + 164q^{37} + 32q^{38} - 34q^{39} + 2q^{40} + 42q^{41} - 2q^{42} - 22q^{43} + 14q^{44} + 58q^{45} - 53q^{46} + 21q^{47} - 20q^{48} + 166q^{49} + 18q^{50} - 6q^{51} + 54q^{52} + 8q^{53} - 154q^{54} - 5q^{55} - 35q^{56} + 8q^{57} + 9q^{58} - 13q^{59} - 88q^{60} - 34q^{61} - 36q^{62} - 31q^{63} - 18q^{64} + 9q^{65} + 222q^{66} - 48q^{67} - 27q^{68} + 2q^{69} - 64q^{70} - 42q^{71} + 77q^{72} - 44q^{73} - 35q^{74} - 21q^{75} + 44q^{76} - 32q^{77} + 124q^{78} - 96q^{79} + 490q^{80} - 12q^{81} - 130q^{82} - 8q^{83} + 170q^{84} + 22q^{85} - 12q^{86} - 90q^{87} - 122q^{88} - 81q^{89} + 49q^{90} - 27q^{91} - 116q^{92} + 18q^{93} - 124q^{94} + 117q^{95} + 52q^{96} - 79q^{97} - 54q^{98} - 39q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −0.598997 + 2.62438i −0.0443239 0.194196i −4.72662 2.27622i 2.45202 3.07473i 0.536193 −2.50591 5.44818 6.83180i 2.66716 1.28444i 6.60051 + 8.27677i
35.2 −0.578512 + 2.53463i −0.698098 3.05857i −4.28771 2.06485i 2.03783 2.55536i 8.15618 3.05620 4.47222 5.60798i −6.16458 + 2.96870i 5.29796 + 6.64344i
35.3 −0.570082 + 2.49769i −0.272791 1.19518i −4.11153 1.98001i −0.726540 + 0.911052i 3.14070 −3.39707 4.09469 5.13458i 1.34887 0.649583i −1.86134 2.33405i
35.4 −0.534521 + 2.34189i 0.544712 + 2.38654i −3.39680 1.63581i −2.43138 + 3.04885i −5.88018 1.85392 2.65117 3.32446i −2.69596 + 1.29830i −5.84046 7.32370i
35.5 −0.506396 + 2.21867i 0.238867 + 1.04654i −2.86411 1.37928i 0.0785430 0.0984899i −2.44289 3.77944 1.67276 2.09757i 1.66471 0.801683i 0.178742 + 0.224136i
35.6 −0.455706 + 1.99658i −0.326366 1.42990i −1.97672 0.951940i 1.20644 1.51282i 3.00364 3.96037 0.247708 0.310617i 0.764797 0.368307i 2.47069 + 3.09815i
35.7 −0.396271 + 1.73618i −0.0213391 0.0934929i −1.05534 0.508225i −1.30201 + 1.63267i 0.170776 −1.70084 −0.920084 + 1.15375i 2.69462 1.29766i −2.31865 2.90749i
35.8 −0.376167 + 1.64810i 0.312756 + 1.37027i −0.772785 0.372154i 1.51766 1.90309i −2.37599 1.98101 −1.20395 + 1.50971i 0.923074 0.444529i 2.56558 + 3.21713i
35.9 −0.370259 + 1.62221i −0.724410 3.17385i −0.692544 0.333512i −2.53002 + 3.17255i 5.41688 −2.21279 −1.27744 + 1.60186i −6.84564 + 3.29669i −4.20978 5.27890i
35.10 −0.277295 + 1.21491i −0.676534 2.96409i 0.402822 + 0.193989i −0.276528 + 0.346755i 3.78870 2.21699 −1.90131 + 2.38417i −5.62522 + 2.70896i −0.344596 0.432110i
35.11 −0.265151 + 1.16170i 0.558598 + 2.44738i 0.522689 + 0.251714i 0.802903 1.00681i −2.99124 −4.26133 −1.91688 + 2.40369i −2.97472 + 1.43255i 0.956722 + 1.19969i
35.12 −0.257740 + 1.12923i −0.136286 0.597106i 0.593200 + 0.285670i 1.65651 2.07719i 0.709399 −2.51623 −1.91982 + 2.40738i 2.36494 1.13890i 1.91869 + 2.40596i
35.13 −0.160022 + 0.701103i 0.119356 + 0.522931i 1.33600 + 0.643383i −0.180888 + 0.226826i −0.385728 −0.492965 −1.56161 + 1.95820i 2.44370 1.17682i −0.130083 0.163118i
35.14 −0.104116 + 0.456161i −0.128875 0.564637i 1.60469 + 0.772780i −2.20664 + 2.76704i 0.270983 0.0588619 −1.10304 + 1.38317i 2.40070 1.15612i −1.03247 1.29468i
35.15 −0.100864 + 0.441914i 0.452538 + 1.98270i 1.61682 + 0.778621i −1.88332 + 2.36161i −0.921827 4.55915 −1.07239 + 1.34474i −1.02340 + 0.492843i −0.853668 1.07047i
35.16 0.00525619 0.0230289i −0.382698 1.67671i 1.80144 + 0.867525i 2.01270 2.52384i −0.0406243 0.0151017 0.0589018 0.0738605i 0.0380061 0.0183028i −0.0475421 0.0596159i
35.17 0.0448375 0.196446i 0.586118 + 2.56795i 1.76536 + 0.850151i 2.40408 3.01462i 0.530744 0.970344 0.497427 0.623754i −3.54794 + 1.70860i −0.484418 0.607441i
35.18 0.129219 0.566146i −0.516629 2.26350i 1.49811 + 0.721454i −2.00450 + 2.51357i −1.34823 2.36842 1.32616 1.66295i −2.15362 + 1.03713i 1.16403 + 1.45964i
35.19 0.142029 0.622270i 0.0629732 + 0.275904i 1.43489 + 0.691006i 0.259053 0.324842i 0.180631 2.83831 1.42970 1.79279i 2.63075 1.26690i −0.165347 0.207338i
35.20 0.173572 0.760469i 0.719456 + 3.15214i 1.25375 + 0.603775i −1.24535 + 1.56163i 2.52199 −1.13105 1.64944 2.06834i −6.71549 + 3.23401i 0.971409 + 1.21811i
See next 80 embeddings (of 180 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 613.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{180} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$.