# Properties

 Label 380.2.bj.a Level $380$ Weight $2$ Character orbit 380.bj Analytic conductor $3.034$ Analytic rank $0$ Dimension $672$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.bj (of order $$36$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$672$$ Relative dimension: $$56$$ over $$\Q(\zeta_{36})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$672 q - 12 q^{2} - 24 q^{5} - 36 q^{6} - 6 q^{8}+O(q^{10})$$ 672 * q - 12 * q^2 - 24 * q^5 - 36 * q^6 - 6 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$672 q - 12 q^{2} - 24 q^{5} - 36 q^{6} - 6 q^{8} - 12 q^{10} - 6 q^{12} - 24 q^{13} - 36 q^{16} - 24 q^{17} - 24 q^{18} + 36 q^{20} - 48 q^{21} - 24 q^{22} - 24 q^{25} - 60 q^{26} - 24 q^{28} - 6 q^{30} + 18 q^{32} - 60 q^{33} + 24 q^{36} - 48 q^{37} - 114 q^{38} - 42 q^{40} - 24 q^{41} - 48 q^{42} - 12 q^{45} - 12 q^{46} - 96 q^{48} - 6 q^{50} - 12 q^{52} - 24 q^{53} - 48 q^{56} - 24 q^{57} + 120 q^{58} - 12 q^{60} - 48 q^{61} + 36 q^{62} - 12 q^{65} - 96 q^{66} - 6 q^{68} - 12 q^{70} + 120 q^{72} - 24 q^{73} - 96 q^{76} - 360 q^{77} - 126 q^{78} + 48 q^{80} - 48 q^{81} + 228 q^{82} - 24 q^{85} - 132 q^{86} - 102 q^{88} + 78 q^{90} + 108 q^{92} - 60 q^{93} - 144 q^{96} - 24 q^{97} + 18 q^{98}+O(q^{100})$$ 672 * q - 12 * q^2 - 24 * q^5 - 36 * q^6 - 6 * q^8 - 12 * q^10 - 6 * q^12 - 24 * q^13 - 36 * q^16 - 24 * q^17 - 24 * q^18 + 36 * q^20 - 48 * q^21 - 24 * q^22 - 24 * q^25 - 60 * q^26 - 24 * q^28 - 6 * q^30 + 18 * q^32 - 60 * q^33 + 24 * q^36 - 48 * q^37 - 114 * q^38 - 42 * q^40 - 24 * q^41 - 48 * q^42 - 12 * q^45 - 12 * q^46 - 96 * q^48 - 6 * q^50 - 12 * q^52 - 24 * q^53 - 48 * q^56 - 24 * q^57 + 120 * q^58 - 12 * q^60 - 48 * q^61 + 36 * q^62 - 12 * q^65 - 96 * q^66 - 6 * q^68 - 12 * q^70 + 120 * q^72 - 24 * q^73 - 96 * q^76 - 360 * q^77 - 126 * q^78 + 48 * q^80 - 48 * q^81 + 228 * q^82 - 24 * q^85 - 132 * q^86 - 102 * q^88 + 78 * q^90 + 108 * q^92 - 60 * q^93 - 144 * q^96 - 24 * q^97 + 18 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −1.41421 0.00173027i −2.30458 1.61369i 1.99999 + 0.00489393i −1.01610 + 1.99187i 3.25638 + 2.28608i 0.362300 0.0970781i −2.82841 0.0103816i 1.68106 + 4.61866i 1.44042 2.81517i
23.2 −1.41416 + 0.0118355i −0.0933215 0.0653444i 1.99972 0.0334747i 1.39136 1.75046i 0.132745 + 0.0913032i 4.78721 1.28273i −2.82754 + 0.0710065i −1.02162 2.80688i −1.94689 + 2.49191i
23.3 −1.40256 0.181148i −1.67151 1.17040i 1.93437 + 0.508144i −0.660038 2.13643i 2.13238 + 1.94436i −2.38759 + 0.639752i −2.62103 1.06311i 0.398042 + 1.09361i 0.538734 + 3.11605i
23.4 −1.39453 + 0.235114i 1.39631 + 0.977708i 1.88944 0.655749i −1.14825 1.91873i −2.17707 1.03515i −2.67634 + 0.717123i −2.48071 + 1.35870i −0.0322876 0.0887093i 2.05239 + 2.40576i
23.5 −1.37359 + 0.336528i −0.322158 0.225578i 1.77350 0.924503i 0.973777 + 2.01290i 0.518426 + 0.201436i −2.45904 + 0.658899i −2.12494 + 1.86672i −0.973160 2.67373i −2.01497 2.43719i
23.6 −1.37318 + 0.338181i 2.50164 + 1.75167i 1.77127 0.928769i 2.16402 0.563029i −4.02760 1.55936i −0.675461 + 0.180989i −2.11818 + 1.87438i 2.16382 + 5.94503i −2.78120 + 1.50497i
23.7 −1.34500 0.436992i −0.109637 0.0767686i 1.61808 + 1.17551i −1.98421 + 1.03097i 0.113915 + 0.151165i 2.66363 0.713718i −1.66263 2.28816i −1.01993 2.80224i 3.11930 0.519577i
23.8 −1.31289 0.525665i 1.78355 + 1.24886i 1.44735 + 1.38028i −1.31769 + 1.80657i −1.68512 2.57716i −4.05211 + 1.08576i −1.17465 2.57297i 0.595352 + 1.63572i 2.67963 1.67916i
23.9 −1.28719 0.585784i 1.15005 + 0.805273i 1.31371 + 1.50803i 2.01826 + 0.962606i −1.00862 1.71022i 0.399624 0.107079i −0.807621 2.71067i −0.351912 0.966869i −2.03401 2.42132i
23.10 −1.21069 0.730902i −2.10842 1.47633i 0.931564 + 1.76980i 2.23455 + 0.0823462i 1.47360 + 3.32843i −1.24717 + 0.334178i 0.165710 2.82357i 1.23982 + 3.40636i −2.64517 1.73293i
23.11 −1.14173 + 0.834541i −2.50164 1.75167i 0.607081 1.90564i 2.16402 0.563029i 4.31804 0.0877964i 0.675461 0.180989i 0.897212 + 2.68235i 2.16382 + 5.94503i −2.00085 + 2.44879i
23.12 −1.14072 + 0.835915i 0.322158 + 0.225578i 0.602493 1.90709i 0.973777 + 2.01290i −0.556057 + 0.0119755i 2.45904 0.658899i 0.906891 + 2.67909i −0.973160 2.67373i −2.79342 1.48216i
23.13 −1.12405 0.858201i 2.71916 + 1.90398i 0.526984 + 1.92932i −1.53955 1.62166i −1.42248 4.47376i 3.91237 1.04832i 1.06339 2.62092i 2.74265 + 7.53538i 0.338817 + 3.14407i
23.14 −1.07650 + 0.917145i −1.39631 0.977708i 0.317689 1.97461i −1.14825 1.91873i 2.39982 0.228122i 2.67634 0.717123i 1.46901 + 2.41702i −0.0322876 0.0887093i 2.99584 + 1.01239i
23.15 −0.986683 1.01314i −0.566781 0.396864i −0.0529140 + 1.99930i −1.91076 1.16147i 0.157153 + 0.965809i 0.335634 0.0899329i 2.07778 1.91907i −0.862321 2.36921i 0.708583 + 3.08187i
23.16 −0.918074 + 1.07570i 0.0933215 + 0.0653444i −0.314282 1.97515i 1.39136 1.75046i −0.155967 + 0.0403954i −4.78721 + 1.28273i 2.41321 + 1.47526i −1.02162 2.80688i 0.605609 + 3.10375i
23.17 −0.907713 + 1.08446i 2.30458 + 1.61369i −0.352115 1.96876i −1.01610 + 1.99187i −3.84188 + 1.03447i −0.362300 + 0.0970781i 2.45466 + 1.40521i 1.68106 + 4.61866i −1.23778 2.90997i
23.18 −0.895026 1.09496i 0.336022 + 0.235285i −0.397856 + 1.96003i 1.19920 1.88731i −0.0431218 0.578516i −1.38682 + 0.371598i 2.50224 1.31864i −0.968509 2.66096i −3.13983 + 0.376122i
23.19 −0.762783 + 1.19087i 1.67151 + 1.17040i −0.836324 1.81674i −0.660038 2.13643i −2.66880 + 1.09778i 2.38759 0.639752i 2.80143 + 0.389833i 0.398042 + 1.09361i 3.04767 + 0.843619i
23.20 −0.631329 1.26547i −2.52409 1.76739i −1.20285 + 1.59786i −2.23416 + 0.0923380i −0.643051 + 4.30997i 1.52327 0.408158i 2.78144 + 0.513399i 2.22131 + 6.10300i 1.52734 + 2.76898i
See next 80 embeddings (of 672 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 367.56 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
19.e even 9 1 inner
20.e even 4 1 inner
76.l odd 18 1 inner
95.q odd 36 1 inner
380.bj even 36 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.bj.a 672
4.b odd 2 1 inner 380.2.bj.a 672
5.c odd 4 1 inner 380.2.bj.a 672
19.e even 9 1 inner 380.2.bj.a 672
20.e even 4 1 inner 380.2.bj.a 672
76.l odd 18 1 inner 380.2.bj.a 672
95.q odd 36 1 inner 380.2.bj.a 672
380.bj even 36 1 inner 380.2.bj.a 672

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.bj.a 672 1.a even 1 1 trivial
380.2.bj.a 672 4.b odd 2 1 inner
380.2.bj.a 672 5.c odd 4 1 inner
380.2.bj.a 672 19.e even 9 1 inner
380.2.bj.a 672 20.e even 4 1 inner
380.2.bj.a 672 76.l odd 18 1 inner
380.2.bj.a 672 95.q odd 36 1 inner
380.2.bj.a 672 380.bj even 36 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(380, [\chi])$$.