# Properties

 Label 403.2.y.a Level 403 Weight 2 Character orbit 403.y Analytic conductor 3.218 Analytic rank 0 Dimension 136 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.y (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$136$$ Relative dimension: $$34$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$136q - 10q^{3} + 22q^{4} - 32q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$136q - 10q^{3} + 22q^{4} - 32q^{9} + 6q^{10} + 48q^{12} - 16q^{13} - 4q^{14} - 46q^{16} + 12q^{17} - 44q^{22} + 16q^{23} - 92q^{25} - 48q^{26} - 52q^{27} - 10q^{29} + 28q^{35} - 62q^{38} - 31q^{39} + 12q^{40} + 22q^{42} - 66q^{43} + 64q^{48} + 58q^{49} + 28q^{51} + 27q^{52} + 44q^{55} - 208q^{56} + 4q^{61} + 72q^{62} + 10q^{64} + 7q^{65} + 38q^{66} + 44q^{68} + 46q^{69} + 44q^{74} + 34q^{75} + 48q^{77} + 5q^{78} + 48q^{79} - 12q^{81} + 8q^{82} - 60q^{87} - 160q^{88} + 142q^{90} - 80q^{91} + 28q^{92} - 108q^{94} + 60q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1 −1.59762 + 2.19894i −0.583936 + 0.424254i −1.66490 5.12402i 2.47143i 1.96183i −0.903099 + 0.293435i 8.75727 + 2.84541i −0.766062 + 2.35770i −5.43453 3.94842i
64.2 −1.47859 + 2.03510i −2.57815 + 1.87313i −1.33738 4.11602i 1.88262i 8.01636i −2.52898 + 0.821714i 5.56913 + 1.80952i 2.21116 6.80524i 3.83131 + 2.78361i
64.3 −1.44841 + 1.99357i 1.15476 0.838982i −1.25838 3.87289i 1.37198i 3.51728i −4.14268 + 1.34604i 4.85636 + 1.57793i −0.297472 + 0.915526i 2.73514 + 1.98720i
64.4 −1.37219 + 1.88866i 1.37165 0.996564i −1.06609 3.28109i 0.910017i 3.95806i 1.80401 0.586159i 3.21924 + 1.04600i −0.0387606 + 0.119293i −1.71871 1.24872i
64.5 −1.33163 + 1.83284i −1.79188 + 1.30188i −0.968010 2.97923i 0.557599i 5.01786i 4.29621 1.39592i 2.44022 + 0.792875i 0.588906 1.81247i −1.02199 0.742519i
64.6 −1.20132 + 1.65347i −0.773570 + 0.562032i −0.672771 2.07058i 3.18878i 1.95426i 3.58968 1.16636i 0.344303 + 0.111871i −0.644520 + 1.98363i 5.27256 + 3.83074i
64.7 −1.07240 + 1.47603i −0.526166 + 0.382282i −0.410588 1.26366i 2.39403i 1.18659i −1.79656 + 0.583739i −1.16484 0.378479i −0.796340 + 2.45088i 3.53365 + 2.56735i
64.8 −0.987540 + 1.35923i −1.39528 + 1.01373i −0.254242 0.782477i 3.09070i 2.89760i −3.01663 + 0.980163i −1.88110 0.611207i −0.00790000 + 0.0243137i −4.20097 3.05219i
64.9 −0.918537 + 1.26426i 1.35395 0.983703i −0.136603 0.420420i 3.21606i 2.61531i 1.56427 0.508261i −2.31545 0.752336i −0.0615391 + 0.189398i −4.06592 2.95406i
64.10 −0.801121 + 1.10265i 1.77414 1.28899i 0.0439947 + 0.135402i 3.51161i 2.98889i 0.360511 0.117137i −2.77703 0.902311i 0.559035 1.72053i 3.87208 + 2.81323i
64.11 −0.675748 + 0.930087i −0.384489 + 0.279347i 0.209607 + 0.645103i 1.11176i 0.546376i 0.340635 0.110679i −2.92841 0.951498i −0.857254 + 2.63836i −1.03404 0.751273i
64.12 −0.618027 + 0.850641i 2.69208 1.95591i 0.276401 + 0.850674i 1.05598i 3.49880i 0.656905 0.213441i −2.89442 0.940454i 2.49466 7.67776i −0.898257 0.652622i
64.13 −0.493698 + 0.679518i −2.07909 + 1.51055i 0.400028 + 1.23116i 0.0344567i 2.15854i −1.65083 + 0.536388i −2.63173 0.855101i 1.11382 3.42798i −0.0234140 0.0170112i
64.14 −0.336818 + 0.463590i −2.27344 + 1.65175i 0.516564 + 1.58982i 3.22494i 1.61029i 1.88146 0.611323i −2.00098 0.650158i 1.51320 4.65716i 1.49505 + 1.08622i
64.15 −0.169061 + 0.232692i −0.823965 + 0.598646i 0.592470 + 1.82343i 4.12859i 0.292938i 4.35350 1.41454i −1.07156 0.348169i −0.606509 + 1.86664i −0.960691 0.697983i
64.16 −0.160682 + 0.221160i 0.324832 0.236004i 0.594941 + 1.83104i 0.569297i 0.109762i −3.20061 + 1.03994i −1.02053 0.331590i −0.877233 + 2.69985i −0.125906 0.0914759i
64.17 −0.0184596 + 0.0254075i 1.61150 1.17082i 0.617729 + 1.90118i 2.11185i 0.0625570i 3.19880 1.03935i −0.119444 0.0388096i 0.299052 0.920387i 0.0536567 + 0.0389839i
64.18 0.0184596 0.0254075i 1.61150 1.17082i 0.617729 + 1.90118i 2.11185i 0.0625570i −3.19880 + 1.03935i 0.119444 + 0.0388096i 0.299052 0.920387i 0.0536567 + 0.0389839i
64.19 0.160682 0.221160i 0.324832 0.236004i 0.594941 + 1.83104i 0.569297i 0.109762i 3.20061 1.03994i 1.02053 + 0.331590i −0.877233 + 2.69985i −0.125906 0.0914759i
64.20 0.169061 0.232692i −0.823965 + 0.598646i 0.592470 + 1.82343i 4.12859i 0.292938i −4.35350 + 1.41454i 1.07156 + 0.348169i −0.606509 + 1.86664i −0.960691 0.697983i
See next 80 embeddings (of 136 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 376.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
31.d even 5 1 inner
403.y even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.y.a 136
13.b even 2 1 inner 403.2.y.a 136
31.d even 5 1 inner 403.2.y.a 136
403.y even 10 1 inner 403.2.y.a 136

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.y.a 136 1.a even 1 1 trivial
403.2.y.a 136 13.b even 2 1 inner
403.2.y.a 136 31.d even 5 1 inner
403.2.y.a 136 403.y even 10 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