Properties

 Label 400.2.be.a Level $400$ Weight $2$ Character orbit 400.be Analytic conductor $3.194$ Analytic rank $0$ Dimension $464$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.be (of order $$20$$, degree $$8$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.19401608085$$ Analytic rank: $$0$$ Dimension: $$464$$ Relative dimension: $$58$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8} - 14 q^{10} - 6 q^{11} + 10 q^{12} - 6 q^{13} + 6 q^{14} - 16 q^{15} - 6 q^{16} - 12 q^{17} - 24 q^{18} - 6 q^{19} - 22 q^{20} + 12 q^{21} + 10 q^{22} - 16 q^{24} + 4 q^{26} - 18 q^{27} + 18 q^{28} - 6 q^{29} - 6 q^{30} + 12 q^{31} - 36 q^{32} - 12 q^{33} - 30 q^{34} + 44 q^{35} - 82 q^{36} - 6 q^{37} - 76 q^{38} - 52 q^{40} - 10 q^{42} - 48 q^{43} + 36 q^{44} - 12 q^{45} - 14 q^{46} - 12 q^{47} - 116 q^{48} - 400 q^{49} + 10 q^{50} - 4 q^{51} + 32 q^{52} - 6 q^{53} - 30 q^{54} + 36 q^{56} + 26 q^{58} - 6 q^{59} + 48 q^{60} - 6 q^{61} - 34 q^{62} + 72 q^{63} - 24 q^{64} + 16 q^{65} + 92 q^{66} + 30 q^{67} + 28 q^{68} - 18 q^{69} + 60 q^{70} + 22 q^{72} + 28 q^{74} - 26 q^{75} - 76 q^{76} + 36 q^{77} + 14 q^{78} - 52 q^{79} - 34 q^{80} + 72 q^{81} + 56 q^{82} - 46 q^{83} + 112 q^{84} + 2 q^{85} - 46 q^{86} - 136 q^{88} - 42 q^{90} + 36 q^{91} - 4 q^{93} + 50 q^{94} - 40 q^{95} - 66 q^{96} - 12 q^{97} - 8 q^{98} - 40 q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1 −1.41077 0.0986613i 2.12918 0.337229i 1.98053 + 0.278376i 0.968273 + 2.01555i −3.03705 + 0.265684i 1.96518i −2.76661 0.588127i 1.56651 0.508991i −1.16715 2.93901i
21.2 −1.39702 0.219860i −1.62017 + 0.256610i 1.90332 + 0.614297i −1.31621 + 1.80765i 2.31983 0.00227828i 2.84840i −2.52392 1.27665i −0.294054 + 0.0955441i 2.23620 2.23593i
21.3 −1.38373 0.292059i 2.72156 0.431053i 1.82940 + 0.808261i −1.03145 1.98396i −3.89179 0.198398i 2.87694i −2.29533 1.65271i 4.36791 1.41922i 0.847808 + 3.04651i
21.4 −1.37734 + 0.320840i 0.993702 0.157387i 1.79412 0.883811i −2.20186 0.389615i −1.31817 + 0.535595i 0.222360i −2.18755 + 1.79293i −1.89050 + 0.614260i 3.15771 0.169814i
21.5 −1.37167 + 0.344287i −2.96765 + 0.470029i 1.76293 0.944493i −1.94195 1.10853i 3.90880 1.66645i 4.09088i −2.09298 + 1.90248i 5.73284 1.86271i 3.04536 + 0.851938i
21.6 −1.35531 + 0.403902i 0.307556 0.0487120i 1.67373 1.09482i 1.69021 + 1.46397i −0.397158 + 0.190242i 1.72856i −1.82622 + 2.15985i −2.76095 + 0.897088i −2.88205 1.30145i
21.7 −1.35172 0.415745i 0.141890 0.0224731i 1.65431 + 1.12395i 0.679981 2.13017i −0.201139 0.0286125i 3.99719i −1.76890 2.20704i −2.83354 + 0.920674i −1.80475 + 2.59670i
21.8 −1.34997 + 0.421416i −1.51383 + 0.239767i 1.64482 1.13779i 0.647118 2.14038i 1.94258 0.961631i 3.05783i −1.74096 + 2.22914i −0.618970 + 0.201115i 0.0284042 + 3.16215i
21.9 −1.32191 0.502558i −2.84258 + 0.450220i 1.49487 + 1.32867i 2.19785 + 0.411629i 3.98388 + 0.833413i 0.416200i −1.30834 2.50764i 5.02439 1.63252i −2.69849 1.64868i
21.10 −1.21160 0.729394i 0.742359 0.117578i 0.935969 + 1.76747i 2.23507 0.0667519i −0.985206 0.399014i 3.12186i 0.155159 2.82417i −2.31590 + 0.752481i −2.75671 1.54937i
21.11 −1.12780 0.853266i −1.06205 + 0.168212i 0.543875 + 1.92463i −2.22886 0.179410i 1.34131 + 0.716502i 0.652480i 1.02884 2.63467i −1.75351 + 0.569750i 2.36063 + 2.10415i
21.12 −1.10053 + 0.888164i 0.211293 0.0334655i 0.422329 1.95490i −1.64970 + 1.50947i −0.202811 + 0.224492i 3.25198i 1.27149 + 2.52652i −2.80964 + 0.912909i 0.474885 3.12642i
21.13 −1.09297 + 0.897446i 3.32580 0.526756i 0.389180 1.96177i −0.862133 + 2.06318i −3.16228 + 3.56046i 3.15347i 1.33522 + 2.49343i 7.93033 2.57672i −0.909308 3.02872i
21.14 −1.03320 + 0.965656i −3.03259 + 0.480315i 0.135016 1.99544i 0.812902 + 2.08307i 2.66946 3.42470i 2.37186i 1.78741 + 2.19207i 6.11271 1.98614i −2.85142 1.36725i
21.15 −0.981124 + 1.01853i −1.08432 + 0.171740i −0.0747909 1.99860i 2.17279 + 0.528170i 0.888935 1.27291i 3.35884i 2.10901 + 1.88470i −1.70690 + 0.554607i −2.66974 + 1.69485i
21.16 −0.919880 1.07416i 3.26199 0.516648i −0.307643 + 1.97620i 2.23583 0.0328668i −3.55560 3.02864i 1.19699i 2.40575 1.48741i 7.52046 2.44355i −2.09200 2.37140i
21.17 −0.755128 1.19573i 2.38103 0.377117i −0.859564 + 1.80587i −2.20176 0.390215i −2.24891 2.56230i 2.82512i 2.80842 0.335850i 2.67389 0.868801i 1.19601 + 2.92738i
21.18 −0.741429 + 1.20428i −1.38564 + 0.219464i −0.900567 1.78577i 0.522765 2.17410i 0.763059 1.83141i 1.14575i 2.81827 + 0.239491i −0.981331 + 0.318854i 2.23063 + 2.24149i
21.19 −0.740418 1.20490i −1.22547 + 0.194095i −0.903562 + 1.78426i 0.515282 + 2.17589i 1.14122 + 1.33285i 3.96942i 2.81886 0.232396i −1.38907 + 0.451337i 2.24020 2.23193i
21.20 −0.737939 1.20642i 0.372869 0.0590566i −0.910891 + 1.78053i −0.553147 2.16657i −0.346402 0.406256i 3.39917i 2.82024 0.215005i −2.71763 + 0.883010i −2.20560 + 2.26612i
See next 80 embeddings (of 464 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 381.58 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
25.d even 5 1 inner
400.be even 20 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.be.a 464
16.e even 4 1 inner 400.2.be.a 464
25.d even 5 1 inner 400.2.be.a 464
400.be even 20 1 inner 400.2.be.a 464

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.be.a 464 1.a even 1 1 trivial
400.2.be.a 464 16.e even 4 1 inner
400.2.be.a 464 25.d even 5 1 inner
400.2.be.a 464 400.be even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(400, [\chi])$$.