# Properties

 Label 403.2.v.a Level 403 Weight 2 Character orbit 403.v Analytic conductor 3.218 Analytic rank 0 Dimension 70 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.v (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$70$$ Relative dimension: $$35$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

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

## 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 −2.41811 + 1.39609i 1.64523 2.89816 5.01976i 1.92977 1.11415i −3.97834 + 2.29689i 3.37818 1.95039i 10.6000i −0.293224 −3.11092 + 5.38828i
36.2 −2.31373 + 1.33583i −1.83062 2.56889 4.44945i −2.88237 + 1.66414i 4.23556 2.44540i −0.362798 + 0.209462i 8.38310i 0.351166 4.44601 7.70072i
36.3 −2.12624 + 1.22758i −0.797547 2.01392 3.48821i 3.14114 1.81354i 1.69577 0.979055i −4.27757 + 2.46966i 4.97868i −2.36392 −4.45254 + 7.71202i
36.4 −2.01769 + 1.16491i 2.06545 1.71404 2.96880i −0.697719 + 0.402828i −4.16742 + 2.40606i −3.06444 + 1.76926i 3.32715i 1.26608 0.938518 1.62556i
36.5 −1.90799 + 1.10158i −1.41212 1.42696 2.47156i 1.04549 0.603615i 2.69431 1.55556i 0.354004 0.204384i 1.88131i −1.00592 −1.32986 + 2.30339i
36.6 −1.88902 + 1.09062i −0.144762 1.37892 2.38836i −1.56313 + 0.902471i 0.273457 0.157881i 2.20942 1.27561i 1.65304i −2.97904 1.96851 3.40956i
36.7 −1.86211 + 1.07509i 3.06359 1.31164 2.27183i −2.43528 + 1.40601i −5.70475 + 3.29364i 1.87718 1.08379i 1.34017i 6.38559 3.02317 5.23629i
36.8 −1.77854 + 1.02684i −3.13535 1.10881 1.92051i −0.188573 + 0.108873i 5.57636 3.21951i −0.439305 + 0.253633i 0.446916i 6.83044 0.223590 0.387269i
36.9 −1.42222 + 0.821121i 1.94597 0.348479 0.603584i 3.04464 1.75782i −2.76760 + 1.59787i 0.662294 0.382376i 2.13991i 0.786781 −2.88677 + 5.00004i
36.10 −1.06311 + 0.613789i 0.887230 −0.246526 + 0.426995i −0.237568 + 0.137160i −0.943226 + 0.544572i −1.34961 + 0.779199i 3.06042i −2.21282 0.168374 0.291633i
36.11 −1.01226 + 0.584427i −1.01318 −0.316890 + 0.548870i −2.14676 + 1.23943i 1.02560 0.592128i −3.71079 + 2.14243i 3.07850i −1.97347 1.44872 2.50925i
36.12 −1.00051 + 0.577645i 2.38906 −0.332651 + 0.576169i 0.697454 0.402675i −2.39028 + 1.38003i −1.54805 + 0.893766i 3.07920i 2.70761 −0.465207 + 0.805762i
36.13 −0.918845 + 0.530496i −1.70695 −0.437149 + 0.757164i −0.822265 + 0.474735i 1.56843 0.905531i 3.56148 2.05622i 3.04960i −0.0863111 0.503689 0.872416i
36.14 −0.793251 + 0.457984i −1.36967 −0.580502 + 1.00546i 3.20188 1.84861i 1.08649 0.627285i 1.32037 0.762317i 2.89538i −1.12401 −1.69327 + 2.93282i
36.15 −0.700842 + 0.404632i 1.07124 −0.672547 + 1.16488i −3.37908 + 1.95091i −0.750767 + 0.433456i 2.06539 1.19246i 2.70706i −1.85245 1.57880 2.73456i
36.16 −0.605874 + 0.349801i −2.90142 −0.755278 + 1.30818i 1.79550 1.03663i 1.75789 1.01492i −0.132266 + 0.0763637i 2.45599i 5.41822 −0.725232 + 1.25614i
36.17 −0.135075 + 0.0779857i 3.20789 −0.987836 + 1.71098i 1.02567 0.592170i −0.433307 + 0.250170i 3.25567 1.87966i 0.620091i 7.29058 −0.0923615 + 0.159975i
36.18 −0.0898111 + 0.0518525i −0.467545 −0.994623 + 1.72274i 1.88817 1.09014i 0.0419908 0.0242434i −2.42419 + 1.39960i 0.413704i −2.78140 −0.113053 + 0.195813i
36.19 0.163021 0.0941205i −3.04781 −0.982283 + 1.70136i −2.94789 + 1.70197i −0.496859 + 0.286862i −1.16818 + 0.674451i 0.746294i 6.28917 −0.320380 + 0.554914i
36.20 0.388678 0.224403i 0.0944788 −0.899286 + 1.55761i −0.0856400 + 0.0494443i 0.0367218 0.0212014i −2.25811 + 1.30372i 1.70483i −2.99107 −0.0221909 + 0.0384358i
See all 70 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 56.35 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.v even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.v.a yes 70
13.e even 6 1 403.2.s.a 70
31.c even 3 1 403.2.s.a 70
403.v even 6 1 inner 403.2.v.a yes 70

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.s.a 70 13.e even 6 1
403.2.s.a 70 31.c even 3 1
403.2.v.a yes 70 1.a even 1 1 trivial
403.2.v.a yes 70 403.v even 6 1 inner

## 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