# Properties

 Label 403.2.bp.a Level 403 Weight 2 Character orbit 403.bp Analytic conductor 3.218 Analytic rank 0 Dimension 280 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.bp (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$280$$ Relative dimension: $$35$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$280q - 9q^{2} - 9q^{3} - 35q^{4} - 21q^{6} - 3q^{7} - 45q^{8} - 63q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$280q - 9q^{2} - 9q^{3} - 35q^{4} - 21q^{6} - 3q^{7} - 45q^{8} - 63q^{9} - 24q^{10} - 9q^{11} - 8q^{12} - 6q^{13} + 4q^{14} + 23q^{16} - 21q^{17} - 45q^{18} - 27q^{19} - 75q^{20} + 76q^{21} - 22q^{22} - 10q^{23} - 15q^{24} + 96q^{25} - 15q^{26} - 24q^{27} + 5q^{28} + 13q^{29} + 36q^{30} - 2q^{31} - 141q^{32} - 3q^{33} + 9q^{34} + 32q^{35} + 97q^{36} - 49q^{38} + 15q^{39} - 75q^{40} - 33q^{41} - 16q^{42} + 21q^{43} - 18q^{44} - 27q^{45} + 51q^{46} - 68q^{48} - 24q^{49} + 90q^{50} + 47q^{51} + 73q^{52} + 12q^{53} - 33q^{54} - 65q^{55} + 25q^{56} + 105q^{57} - 3q^{58} + 12q^{59} - 90q^{60} - 57q^{61} + 12q^{62} + 201q^{63} + 13q^{64} + 11q^{65} + 22q^{66} - 45q^{67} + 142q^{68} - 139q^{69} - 15q^{71} - 15q^{72} - 9q^{73} + 4q^{74} - 75q^{75} - 80q^{76} - 24q^{77} - 104q^{78} + 54q^{79} - 21q^{80} - 107q^{81} + 43q^{82} - 54q^{83} - 15q^{84} - 117q^{85} - 84q^{86} - 21q^{87} + 49q^{88} - 9q^{89} + 11q^{90} - 10q^{91} + 266q^{92} - 22q^{93} + 33q^{94} + 75q^{95} - 204q^{96} - 10q^{97} + 168q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1 −2.76032 0.290121i −0.0839252 + 0.0609752i 5.57887 + 1.18583i −1.92798 + 1.11312i 0.249350 0.143962i −0.0225283 0.0202846i −9.77606 3.17644i −0.923726 + 2.84293i 5.64476 2.51321i
10.2 −2.69911 0.283688i −2.38863 + 1.73544i 5.24842 + 1.11559i 1.43529 0.828667i 6.93950 4.00652i −1.43143 1.28887i −8.68728 2.82267i 1.76675 5.43750i −4.10910 + 1.82949i
10.3 −2.60859 0.274174i 2.47351 1.79711i 4.77329 + 1.01459i 0.288360 0.166485i −6.94509 + 4.00975i 2.54091 + 2.28784i −7.18423 2.33430i 1.96159 6.03714i −0.797859 + 0.355230i
10.4 −2.19585 0.230793i −0.847098 + 0.615453i 2.81219 + 0.597750i 2.76986 1.59918i 2.00214 1.15594i 3.22981 + 2.90814i −1.83743 0.597018i −0.588258 + 1.81047i −6.45127 + 2.87229i
10.5 −2.18417 0.229566i 0.117173 0.0851309i 2.76161 + 0.586998i 0.0491735 0.0283903i −0.275468 + 0.159042i −2.29210 2.06382i −1.71965 0.558748i −0.920569 + 2.83322i −0.113921 + 0.0507208i
10.6 −2.06615 0.217161i −1.25606 + 0.912580i 2.26552 + 0.481551i −2.40278 + 1.38724i 2.79338 1.61276i 0.0422354 + 0.0380290i −0.624627 0.202953i −0.182170 + 0.560662i 5.26575 2.34446i
10.7 −2.02543 0.212881i 1.58944 1.15479i 2.10075 + 0.446527i 1.71112 0.987913i −3.46513 + 2.00059i −1.42691 1.28479i −0.286037 0.0929391i 0.265713 0.817780i −3.67605 + 1.63668i
10.8 −1.94065 0.203970i 1.97372 1.43400i 1.76822 + 0.375846i −2.63433 + 1.52093i −4.12280 + 2.38030i −1.07378 0.966833i 0.356842 + 0.115945i 0.912197 2.80745i 5.42253 2.41427i
10.9 −1.83649 0.193022i −2.43858 + 1.77173i 1.37913 + 0.293142i −0.615405 + 0.355305i 4.82040 2.78306i 1.21282 + 1.09203i 1.03628 + 0.336709i 1.88059 5.78785i 1.19876 0.533725i
10.10 −1.48870 0.156468i −1.43929 + 1.04571i 0.235442 + 0.0500448i 3.43058 1.98065i 2.30629 1.33154i −1.81022 1.62993i 2.50460 + 0.813793i 0.0510066 0.156982i −5.41700 + 2.41181i
10.11 −1.34554 0.141422i 1.22820 0.892341i −0.165811 0.0352442i 0.772095 0.445769i −1.77880 + 1.02699i 2.87973 + 2.59292i 2.79159 + 0.907044i −0.214843 + 0.661218i −1.10193 + 0.490610i
10.12 −1.17237 0.123221i −1.03818 + 0.754283i −0.597037 0.126904i −2.09944 + 1.21211i 1.31007 0.756370i 2.47362 + 2.22725i 2.92656 + 0.950898i −0.418173 + 1.28701i 2.61067 1.16234i
10.13 −1.06147 0.111565i 1.27207 0.924209i −0.842029 0.178979i −2.77463 + 1.60194i −1.45336 + 0.839101i 0.448540 + 0.403867i 2.90397 + 0.943557i −0.163064 + 0.501860i 3.12390 1.39085i
10.14 −0.794088 0.0834620i −0.0739898 + 0.0537568i −1.33269 0.283271i 0.698098 0.403047i 0.0632411 0.0365123i −2.33566 2.10304i 2.55339 + 0.829648i −0.924466 + 2.84521i −0.587991 + 0.261790i
10.15 −0.501537 0.0527137i −0.192422 + 0.139802i −1.70753 0.362948i 0.660141 0.381132i 0.103876 0.0599729i −1.06122 0.955523i 1.79650 + 0.583717i −0.909570 + 2.79937i −0.351176 + 0.156354i
10.16 −0.349700 0.0367550i 1.55724 1.13140i −1.83536 0.390117i 3.40354 1.96504i −0.586153 + 0.338415i 1.19593 + 1.07682i 1.29632 + 0.421200i 0.217879 0.670563i −1.26245 + 0.562077i
10.17 −0.228252 0.0239903i −2.32883 + 1.69199i −1.90477 0.404872i −2.58167 + 1.49053i 0.572152 0.330332i −2.78454 2.50721i 0.861608 + 0.279953i 1.63355 5.02756i 0.625030 0.278281i
10.18 0.0702218 + 0.00738061i 2.77994 2.01975i −1.95142 0.414787i −0.986518 + 0.569566i 0.210119 0.121312i 2.56944 + 2.31354i −0.268276 0.0871682i 2.72165 8.37638i −0.0734788 + 0.0327149i
10.19 0.0782606 + 0.00822552i −1.50523 + 1.09361i −1.95024 0.414536i −0.722573 + 0.417178i −0.126795 + 0.0732053i −0.267898 0.241217i −0.298897 0.0971177i 0.142670 0.439093i −0.0599805 + 0.0267050i
10.20 0.342148 + 0.0359612i −2.36245 + 1.71642i −1.84052 0.391215i 2.15569 1.24459i −0.870032 + 0.502313i 3.18311 + 2.86608i −1.27005 0.412665i 1.70801 5.25672i 0.782322 0.348312i
See next 80 embeddings (of 280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.35 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bp even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bp.a 280
13.e even 6 1 403.2.bt.a yes 280
31.g even 15 1 403.2.bt.a yes 280
403.bp even 30 1 inner 403.2.bp.a 280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bp.a 280 1.a even 1 1 trivial
403.2.bp.a 280 403.bp even 30 1 inner
403.2.bt.a yes 280 13.e even 6 1
403.2.bt.a yes 280 31.g even 15 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database