# Properties

 Label 403.2.bg.a Level 403 Weight 2 Character orbit 403.bg Analytic conductor 3.218 Analytic rank 0 Dimension 144 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.bg (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$36$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$144q - 8q^{2} - 12q^{4} - 8q^{5} - 16q^{7} - 28q^{8} + 64q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q - 8q^{2} - 12q^{4} - 8q^{5} - 16q^{7} - 28q^{8} + 64q^{9} - 12q^{10} + 68q^{16} - 52q^{18} - 4q^{19} - 44q^{20} - 64q^{28} - 4q^{31} - 28q^{32} + 36q^{33} + 28q^{35} - 12q^{36} - 28q^{39} - 48q^{40} - 8q^{41} + 12q^{45} - 24q^{47} + 48q^{49} + 40q^{50} + 48q^{56} + 8q^{59} + 42q^{62} - 84q^{63} - 80q^{66} + 4q^{67} - 24q^{69} - 96q^{70} - 80q^{71} + 104q^{72} - 116q^{76} + 24q^{78} - 112q^{80} + 16q^{81} + 108q^{82} - 92q^{87} + 64q^{93} + 4q^{94} - 132q^{95} + 120q^{97} - 160q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
123.1 −0.705571 2.63323i −1.33423 0.770317i −4.70400 + 2.71585i −0.754353 + 0.754353i −1.08703 + 4.05684i 0.397529 1.48360i 6.61515 + 6.61515i −0.313224 0.542519i 2.51863 + 1.45413i
123.2 −0.705571 2.63323i 1.33423 + 0.770317i −4.70400 + 2.71585i −0.754353 + 0.754353i 1.08703 4.05684i 0.397529 1.48360i 6.61515 + 6.61515i −0.313224 0.542519i 2.51863 + 1.45413i
123.3 −0.623506 2.32695i −1.53051 0.883643i −3.29391 + 1.90174i 2.78278 2.78278i −1.10191 + 4.11240i 0.171705 0.640810i 3.07213 + 3.07213i 0.0616504 + 0.106782i −8.21047 4.74032i
123.4 −0.623506 2.32695i 1.53051 + 0.883643i −3.29391 + 1.90174i 2.78278 2.78278i 1.10191 4.11240i 0.171705 0.640810i 3.07213 + 3.07213i 0.0616504 + 0.106782i −8.21047 4.74032i
123.5 −0.598354 2.23309i −1.62337 0.937250i −2.89661 + 1.67236i −1.13336 + 1.13336i −1.12162 + 4.18593i −0.972184 + 3.62824i 2.19825 + 2.19825i 0.256877 + 0.444923i 3.20903 + 1.85274i
123.6 −0.598354 2.23309i 1.62337 + 0.937250i −2.89661 + 1.67236i −1.13336 + 1.13336i 1.12162 4.18593i −0.972184 + 3.62824i 2.19825 + 2.19825i 0.256877 + 0.444923i 3.20903 + 1.85274i
123.7 −0.436335 1.62842i −0.707551 0.408505i −0.729326 + 0.421077i −2.22177 + 2.22177i −0.356490 + 1.33044i 0.484835 1.80943i −1.38026 1.38026i −1.16625 2.02000i 4.58742 + 2.64855i
123.8 −0.436335 1.62842i 0.707551 + 0.408505i −0.729326 + 0.421077i −2.22177 + 2.22177i 0.356490 1.33044i 0.484835 1.80943i −1.38026 1.38026i −1.16625 2.02000i 4.58742 + 2.64855i
123.9 −0.434186 1.62041i −1.90487 1.09978i −0.705145 + 0.407116i 0.649841 0.649841i −0.955016 + 3.56417i 0.921552 3.43928i −1.40658 1.40658i 0.919017 + 1.59178i −1.33516 0.770854i
123.10 −0.434186 1.62041i 1.90487 + 1.09978i −0.705145 + 0.407116i 0.649841 0.649841i 0.955016 3.56417i 0.921552 3.43928i −1.40658 1.40658i 0.919017 + 1.59178i −1.33516 0.770854i
123.11 −0.359643 1.34220i −2.72479 1.57316i 0.0598795 0.0345714i 1.15466 1.15466i −1.13155 + 4.22301i −1.21090 + 4.51915i −2.03306 2.03306i 3.44967 + 5.97500i −1.96506 1.13453i
123.12 −0.359643 1.34220i 2.72479 + 1.57316i 0.0598795 0.0345714i 1.15466 1.15466i 1.13155 4.22301i −1.21090 + 4.51915i −2.03306 2.03306i 3.44967 + 5.97500i −1.96506 1.13453i
123.13 −0.248585 0.927732i −0.327048 0.188821i 0.933159 0.538760i 1.52097 1.52097i −0.0938763 + 0.350351i −0.204081 + 0.761641i −2.09009 2.09009i −1.42869 2.47457i −1.78914 1.03296i
123.14 −0.248585 0.927732i 0.327048 + 0.188821i 0.933159 0.538760i 1.52097 1.52097i 0.0938763 0.350351i −0.204081 + 0.761641i −2.09009 2.09009i −1.42869 2.47457i −1.78914 1.03296i
123.15 −0.137428 0.512890i −2.33567 1.34850i 1.48788 0.859029i 0.158725 0.158725i −0.370644 + 1.38326i 0.491600 1.83468i −1.39599 1.39599i 2.13689 + 3.70121i −0.103222 0.0595950i
123.16 −0.137428 0.512890i 2.33567 + 1.34850i 1.48788 0.859029i 0.158725 0.158725i 0.370644 1.38326i 0.491600 1.83468i −1.39599 1.39599i 2.13689 + 3.70121i −0.103222 0.0595950i
123.17 −0.111049 0.414442i −0.896531 0.517612i 1.57262 0.907953i −1.95188 + 1.95188i −0.114961 + 0.429040i −0.976537 + 3.64449i −1.15772 1.15772i −0.964155 1.66997i 1.02569 + 0.592184i
123.18 −0.111049 0.414442i 0.896531 + 0.517612i 1.57262 0.907953i −1.95188 + 1.95188i 0.114961 0.429040i −0.976537 + 3.64449i −1.15772 1.15772i −0.964155 1.66997i 1.02569 + 0.592184i
123.19 0.0151761 + 0.0566379i −2.67732 1.54575i 1.72907 0.998281i −2.77132 + 2.77132i 0.0469169 0.175096i 0.222097 0.828876i 0.165705 + 0.165705i 3.27869 + 5.67887i −0.199020 0.114904i
123.20 0.0151761 + 0.0566379i 2.67732 + 1.54575i 1.72907 0.998281i −2.77132 + 2.77132i −0.0469169 + 0.175096i 0.222097 0.828876i 0.165705 + 0.165705i 3.27869 + 5.67887i −0.199020 0.114904i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 371.36 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.f odd 12 1 inner
31.b odd 2 1 inner
403.bg even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bg.a 144
13.f odd 12 1 inner 403.2.bg.a 144
31.b odd 2 1 inner 403.2.bg.a 144
403.bg even 12 1 inner 403.2.bg.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bg.a 144 1.a even 1 1 trivial
403.2.bg.a 144 13.f odd 12 1 inner
403.2.bg.a 144 31.b odd 2 1 inner
403.2.bg.a 144 403.bg even 12 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