# Properties

 Label 380.3.b.a Level $380$ Weight $3$ Character orbit 380.b Analytic conductor $10.354$ Analytic rank $0$ Dimension $72$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 380.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3542500457$$ Analytic rank: $$0$$ Dimension: $$72$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72 q + 4 q^{2} + 12 q^{4} + 12 q^{6} + 16 q^{8} - 216 q^{9}+O(q^{10})$$ 72 * q + 4 * q^2 + 12 * q^4 + 12 * q^6 + 16 * q^8 - 216 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$72 q + 4 q^{2} + 12 q^{4} + 12 q^{6} + 16 q^{8} - 216 q^{9} - 80 q^{12} - 80 q^{14} + 4 q^{16} - 44 q^{18} - 40 q^{20} + 16 q^{21} + 160 q^{22} + 204 q^{24} + 360 q^{25} + 28 q^{26} + 20 q^{28} + 16 q^{29} + 40 q^{30} - 136 q^{32} - 96 q^{34} + 8 q^{36} - 192 q^{37} - 4 q^{42} - 40 q^{44} + 80 q^{45} - 232 q^{46} - 156 q^{48} - 504 q^{49} + 20 q^{50} + 228 q^{52} + 320 q^{53} + 92 q^{54} + 8 q^{56} + 380 q^{58} - 140 q^{60} - 168 q^{62} - 60 q^{64} - 40 q^{66} + 396 q^{68} - 48 q^{69} - 120 q^{70} - 284 q^{72} + 192 q^{74} - 640 q^{77} - 520 q^{78} + 120 q^{80} + 568 q^{81} - 240 q^{82} + 112 q^{84} + 688 q^{86} - 484 q^{88} + 240 q^{89} + 12 q^{92} + 512 q^{93} + 432 q^{94} + 300 q^{96} - 44 q^{98}+O(q^{100})$$ 72 * q + 4 * q^2 + 12 * q^4 + 12 * q^6 + 16 * q^8 - 216 * q^9 - 80 * q^12 - 80 * q^14 + 4 * q^16 - 44 * q^18 - 40 * q^20 + 16 * q^21 + 160 * q^22 + 204 * q^24 + 360 * q^25 + 28 * q^26 + 20 * q^28 + 16 * q^29 + 40 * q^30 - 136 * q^32 - 96 * q^34 + 8 * q^36 - 192 * q^37 - 4 * q^42 - 40 * q^44 + 80 * q^45 - 232 * q^46 - 156 * q^48 - 504 * q^49 + 20 * q^50 + 228 * q^52 + 320 * q^53 + 92 * q^54 + 8 * q^56 + 380 * q^58 - 140 * q^60 - 168 * q^62 - 60 * q^64 - 40 * q^66 + 396 * q^68 - 48 * q^69 - 120 * q^70 - 284 * q^72 + 192 * q^74 - 640 * q^77 - 520 * q^78 + 120 * q^80 + 568 * q^81 - 240 * q^82 + 112 * q^84 + 688 * q^86 - 484 * q^88 + 240 * q^89 + 12 * q^92 + 512 * q^93 + 432 * q^94 + 300 * q^96 - 44 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1 −2.00000 0.00104961i 3.00750i 4.00000 + 0.00419842i 2.23607 0.00315669 6.01499i 0.387511i −7.99999 0.0125953i −0.0450421 −4.47214 0.00234699i
191.2 −2.00000 + 0.00104961i 3.00750i 4.00000 0.00419842i 2.23607 0.00315669 + 6.01499i 0.387511i −7.99999 + 0.0125953i −0.0450421 −4.47214 + 0.00234699i
191.3 −1.99914 0.0586031i 4.03802i 3.99313 + 0.234312i −2.23607 0.236641 8.07258i 12.0049i −7.96910 0.702432i −7.30564 4.47022 + 0.131040i
191.4 −1.99914 + 0.0586031i 4.03802i 3.99313 0.234312i −2.23607 0.236641 + 8.07258i 12.0049i −7.96910 + 0.702432i −7.30564 4.47022 0.131040i
191.5 −1.98112 0.274141i 4.09700i 3.84969 + 1.08622i −2.23607 1.12316 8.11666i 4.68200i −7.32894 3.20729i −7.78542 4.42992 + 0.612999i
191.6 −1.98112 + 0.274141i 4.09700i 3.84969 1.08622i −2.23607 1.12316 + 8.11666i 4.68200i −7.32894 + 3.20729i −7.78542 4.42992 0.612999i
191.7 −1.93566 0.503224i 0.768825i 3.49353 + 1.94814i 2.23607 0.386891 1.48818i 10.1723i −5.78193 5.52895i 8.40891 −4.32826 1.12524i
191.8 −1.93566 + 0.503224i 0.768825i 3.49353 1.94814i 2.23607 0.386891 + 1.48818i 10.1723i −5.78193 + 5.52895i 8.40891 −4.32826 + 1.12524i
191.9 −1.80036 0.871032i 0.512340i 2.48261 + 3.13635i −2.23607 −0.446265 + 0.922398i 3.99381i −1.73774 7.80899i 8.73751 4.02573 + 1.94769i
191.10 −1.80036 + 0.871032i 0.512340i 2.48261 3.13635i −2.23607 −0.446265 0.922398i 3.99381i −1.73774 + 7.80899i 8.73751 4.02573 1.94769i
191.11 −1.70466 1.04601i 1.85712i 1.81172 + 3.56618i −2.23607 −1.94256 + 3.16575i 8.17539i 0.641893 7.97421i 5.55112 3.81173 + 2.33895i
191.12 −1.70466 + 1.04601i 1.85712i 1.81172 3.56618i −2.23607 −1.94256 3.16575i 8.17539i 0.641893 + 7.97421i 5.55112 3.81173 2.33895i
191.13 −1.68205 1.08200i 2.49923i 1.65856 + 3.63994i 2.23607 2.70416 4.20382i 6.10491i 1.14864 7.91711i 2.75385 −3.76117 2.41942i
191.14 −1.68205 + 1.08200i 2.49923i 1.65856 3.63994i 2.23607 2.70416 + 4.20382i 6.10491i 1.14864 + 7.91711i 2.75385 −3.76117 + 2.41942i
191.15 −1.65600 1.12146i 5.88343i 1.48464 + 3.71428i 2.23607 6.59806 9.74294i 1.76155i 1.70687 7.81579i −25.6148 −3.70292 2.50767i
191.16 −1.65600 + 1.12146i 5.88343i 1.48464 3.71428i 2.23607 6.59806 + 9.74294i 1.76155i 1.70687 + 7.81579i −25.6148 −3.70292 + 2.50767i
191.17 −1.32631 1.49696i 0.452282i −0.481798 + 3.97088i 2.23607 0.677050 0.599867i 5.37605i 6.58327 4.54539i 8.79544 −2.96572 3.34731i
191.18 −1.32631 + 1.49696i 0.452282i −0.481798 3.97088i 2.23607 0.677050 + 0.599867i 5.37605i 6.58327 + 4.54539i 8.79544 −2.96572 + 3.34731i
191.19 −1.30085 1.51914i 4.10739i −0.615584 + 3.95235i −2.23607 6.23970 5.34309i 8.55175i 6.80496 4.20625i −7.87063 2.90879 + 3.39690i
191.20 −1.30085 + 1.51914i 4.10739i −0.615584 3.95235i −2.23607 6.23970 + 5.34309i 8.55175i 6.80496 + 4.20625i −7.87063 2.90879 3.39690i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.72 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.b.a 72
4.b odd 2 1 inner 380.3.b.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.b.a 72 1.a even 1 1 trivial
380.3.b.a 72 4.b odd 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(380, [\chi])$$.