# Properties

 Label 231.2.bc.a Level 231 Weight 2 Character orbit 231.bc Analytic conductor 1.845 Analytic rank 0 Dimension 224 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.bc (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$224$$ Relative dimension: $$28$$ over $$\Q(\zeta_{30})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$224q - 9q^{3} - 30q^{4} - 16q^{7} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$224q - 9q^{3} - 30q^{4} - 16q^{7} + 3q^{9} - 60q^{10} - 12q^{12} - 36q^{15} + 18q^{16} + 13q^{18} - 18q^{19} - 6q^{21} + 20q^{22} - 51q^{24} - 18q^{25} - 26q^{28} - 15q^{30} - 36q^{31} + 60q^{33} - 32q^{36} - 10q^{37} + 9q^{39} - 114q^{42} - 96q^{43} + 24q^{45} - 54q^{46} - 56q^{49} - 29q^{51} - 30q^{52} - 96q^{54} + 68q^{57} - 64q^{58} + 125q^{60} - 18q^{61} - 26q^{63} + 56q^{64} + 135q^{66} + 48q^{67} - 44q^{70} + 19q^{72} + 30q^{73} + 63q^{75} + 28q^{78} + 30q^{79} + 31q^{81} + 54q^{82} + 99q^{84} - 248q^{85} + 102q^{87} + 82q^{88} - 144q^{91} + 34q^{93} + 162q^{94} - 87q^{96} - 100q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.546758 + 2.57230i −1.67722 + 0.432351i −4.49067 1.99937i 0.108184 + 0.120151i −0.195099 4.55070i −0.360360 2.62110i 4.50682 6.20311i 2.62615 1.45030i −0.368214 + 0.212589i
5.2 −0.513957 + 2.41798i −0.550359 1.64229i −3.75538 1.67200i 1.71178 + 1.90112i 4.25388 0.486692i −1.49416 + 2.18346i 3.06697 4.22132i −2.39421 + 1.80769i −5.47665 + 3.16195i
5.3 −0.506364 + 2.38226i 0.353670 + 1.69556i −3.59165 1.59911i 1.14381 + 1.27033i −4.21834 0.0160368i 1.23243 + 2.34118i 2.76509 3.80582i −2.74983 + 1.19934i −3.60545 + 2.08161i
5.4 −0.459196 + 2.16035i 1.71801 0.220131i −2.62916 1.17058i −0.345466 0.383679i −0.313341 + 3.81257i 2.35485 + 1.20611i 1.13977 1.56876i 2.90308 0.756373i 0.987517 0.570143i
5.5 −0.447920 + 2.10730i 0.826050 1.52238i −2.41299 1.07433i −2.59584 2.88297i 2.83811 + 2.42264i −2.63172 + 0.272090i 0.812141 1.11782i −1.63528 2.51512i 7.23801 4.17887i
5.6 −0.396988 + 1.86768i −1.30026 1.14426i −1.50354 0.669417i −1.28280 1.42469i 2.65330 1.97420i 2.43431 1.03640i −0.397497 + 0.547108i 0.381326 + 2.97567i 3.17012 1.83027i
5.7 −0.380999 + 1.79246i 1.69357 0.363054i −1.24065 0.552375i 2.38945 + 2.65375i 0.00550992 + 3.17398i −1.61866 2.09283i −0.691439 + 0.951684i 2.73638 1.22972i −5.66712 + 3.27191i
5.8 −0.311281 + 1.46446i −0.472312 + 1.66641i −0.220656 0.0982422i −0.116479 0.129364i −2.29337 1.21040i −2.61507 0.401743i −1.54748 + 2.12992i −2.55384 1.57413i 0.225706 0.130311i
5.9 −0.234283 + 1.10222i −1.26866 + 1.17920i 0.667100 + 0.297012i 2.02306 + 2.24683i −1.00250 1.67460i 2.45218 0.993382i −1.80834 + 2.48897i 0.218989 2.99200i −2.95046 + 1.70345i
5.10 −0.228179 + 1.07350i −1.72896 0.103384i 0.726756 + 0.323573i −0.978563 1.08680i 0.505496 1.83245i 0.116723 + 2.64318i −1.80335 + 2.48210i 2.97862 + 0.357493i 1.38997 0.802500i
5.11 −0.170302 + 0.801208i 1.42625 + 0.982761i 1.21416 + 0.540578i −0.800599 0.889155i −1.03029 + 0.975354i −2.04538 + 1.67822i −1.60281 + 2.20608i 1.06836 + 2.80332i 0.848742 0.490022i
5.12 −0.148555 + 0.698897i 0.250717 1.71381i 1.36070 + 0.605824i 1.08864 + 1.20906i 1.16053 + 0.429821i 1.71199 + 2.01719i −1.46551 + 2.01709i −2.87428 0.859364i −1.00673 + 0.581238i
5.13 −0.0876680 + 0.412446i 1.04792 1.37908i 1.66467 + 0.741157i −0.908673 1.00918i 0.476926 + 0.553112i 0.442873 2.60842i −0.947316 + 1.30387i −0.803721 2.89033i 0.495895 0.286305i
5.14 −0.0516949 + 0.243205i −1.39546 1.02601i 1.77061 + 0.788328i 2.20131 + 2.44480i 0.321668 0.286345i −1.85363 1.88787i −0.575550 + 0.792176i 0.894626 + 2.86350i −0.708384 + 0.408986i
5.15 0.0516949 0.243205i −1.57829 + 0.713452i 1.77061 + 0.788328i −2.20131 2.44480i 0.0919260 + 0.420730i −1.85363 1.88787i 0.575550 0.792176i 1.98197 2.25206i −0.708384 + 0.408986i
5.16 0.0876680 0.412446i 0.738295 + 1.56682i 1.66467 + 0.741157i 0.908673 + 1.00918i 0.710952 0.167147i 0.442873 2.60842i 0.947316 1.30387i −1.90984 + 2.31355i 0.495895 0.286305i
5.17 0.148555 0.698897i −0.111082 + 1.72849i 1.36070 + 0.605824i −1.08864 1.20906i 1.19153 + 0.334410i 1.71199 + 2.01719i 1.46551 2.01709i −2.97532 0.384008i −1.00673 + 0.581238i
5.18 0.170302 0.801208i 1.59941 0.664752i 1.21416 + 0.540578i 0.800599 + 0.889155i −0.260223 1.39467i −2.04538 + 1.67822i 1.60281 2.20608i 2.11621 2.12642i 0.848742 0.490022i
5.19 0.228179 1.07350i −1.71268 0.258347i 0.726756 + 0.323573i 0.978563 + 1.08680i −0.668132 + 1.77961i 0.116723 + 2.64318i 1.80335 2.48210i 2.86651 + 0.884929i 1.38997 0.802500i
5.20 0.234283 1.10222i −0.995767 1.41720i 0.667100 + 0.297012i −2.02306 2.24683i −1.79535 + 0.765524i 2.45218 0.993382i 1.80834 2.48897i −1.01690 + 2.82240i −2.95046 + 1.70345i
See next 80 embeddings (of 224 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 185.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
11.c even 5 1 inner
21.g even 6 1 inner
33.h odd 10 1 inner
77.p odd 30 1 inner
231.bc even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.bc.a 224
3.b odd 2 1 inner 231.2.bc.a 224
7.d odd 6 1 inner 231.2.bc.a 224
11.c even 5 1 inner 231.2.bc.a 224
21.g even 6 1 inner 231.2.bc.a 224
33.h odd 10 1 inner 231.2.bc.a 224
77.p odd 30 1 inner 231.2.bc.a 224
231.bc even 30 1 inner 231.2.bc.a 224

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.bc.a 224 1.a even 1 1 trivial
231.2.bc.a 224 3.b odd 2 1 inner
231.2.bc.a 224 7.d odd 6 1 inner
231.2.bc.a 224 11.c even 5 1 inner
231.2.bc.a 224 21.g even 6 1 inner
231.2.bc.a 224 33.h odd 10 1 inner
231.2.bc.a 224 77.p odd 30 1 inner
231.2.bc.a 224 231.bc even 30 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database