# Properties

 Label 380.3.p.a Level $380$ Weight $3$ Character orbit 380.p Analytic conductor $10.354$ Analytic rank $0$ Dimension $232$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 380.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3542500457$$ Analytic rank: $$0$$ Dimension: $$232$$ Relative dimension: $$116$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10})$$ 232 * q - 2 * q^5 + 8 * q^6 - 328 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44} + 128 q^{45} + 68 q^{46} + 1320 q^{49} - 156 q^{50} - 44 q^{54} - 400 q^{56} + 146 q^{60} - 68 q^{61} - 324 q^{64} - 204 q^{65} + 58 q^{66} + 440 q^{69} + 62 q^{70} - 212 q^{74} + 246 q^{76} + 28 q^{80} - 1116 q^{81} + 96 q^{84} - 46 q^{85} - 28 q^{86} - 60 q^{89} + 482 q^{90} - 756 q^{94} - 628 q^{96}+O(q^{100})$$ 232 * q - 2 * q^5 + 8 * q^6 - 328 * q^9 + 20 * q^14 + 12 * q^16 + 92 * q^20 - 40 * q^21 - 134 * q^24 - 2 * q^25 + 28 * q^26 - 4 * q^29 + 268 * q^30 - 70 * q^34 + 12 * q^36 - 42 * q^40 - 12 * q^41 + 98 * q^44 + 128 * q^45 + 68 * q^46 + 1320 * q^49 - 156 * q^50 - 44 * q^54 - 400 * q^56 + 146 * q^60 - 68 * q^61 - 324 * q^64 - 204 * q^65 + 58 * q^66 + 440 * q^69 + 62 * q^70 - 212 * q^74 + 246 * q^76 + 28 * q^80 - 1116 * q^81 + 96 * q^84 - 46 * q^85 - 28 * q^86 - 60 * q^89 + 482 * q^90 - 756 * q^94 - 628 * q^96

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
159.1 −1.99940 + 0.0491004i 2.13447 + 3.69701i 3.99518 0.196342i 3.83161 3.21229i −4.44918 7.28700i −2.09290 −7.97831 + 0.588731i −4.61194 + 7.98812i −7.50318 + 6.61077i
159.2 −1.99929 0.0533207i −2.05947 3.56711i 3.99431 + 0.213207i 4.96777 + 0.566763i 3.92728 + 7.24149i −2.21466 −7.97442 0.639242i −3.98284 + 6.89847i −9.90180 1.39801i
159.3 −1.99798 0.0898151i 0.192588 + 0.333573i 3.98387 + 0.358898i 0.758288 + 4.94217i −0.354828 0.683770i 5.98368 −7.92746 1.07488i 4.42582 7.66574i −1.07117 9.94246i
159.4 −1.99491 + 0.142649i 1.65352 + 2.86398i 3.95930 0.569141i −4.35243 2.46097i −3.70716 5.47750i −11.5548 −7.81725 + 1.70017i −0.968248 + 1.67706i 9.03374 + 4.28854i
159.5 −1.99219 0.176550i 1.30777 + 2.26512i 3.93766 + 0.703443i −3.13801 + 3.89267i −2.20541 4.74343i −0.966370 −7.72038 2.09659i 1.07950 1.86975i 6.93877 7.20093i
159.6 −1.97327 + 0.325875i −0.501671 0.868919i 3.78761 1.28608i −2.04342 4.56338i 1.27309 + 1.55113i 6.90502 −7.05489 + 3.77207i 3.99665 6.92241i 5.51932 + 8.33889i
159.7 −1.97102 0.339262i −1.97599 3.42251i 3.76980 + 1.33738i −0.693065 4.95173i 2.73357 + 7.41619i −2.56913 −6.97662 3.91494i −3.30904 + 5.73143i −0.313891 + 9.99507i
159.8 −1.97064 0.341417i −2.91038 5.04092i 3.76687 + 1.34562i −4.21638 + 2.68741i 4.01426 + 10.9275i 4.09062 −6.96373 3.93782i −12.4406 + 21.5477i 9.22650 3.85638i
159.9 −1.95148 0.437846i 2.80787 + 4.86337i 3.61658 + 1.70890i 4.08358 + 2.88520i −3.35010 10.7202i 7.39194 −6.30947 4.91840i −11.2682 + 19.5171i −6.70576 7.41841i
159.10 −1.94086 + 0.482776i 0.480142 + 0.831631i 3.53385 1.87400i 4.92739 + 0.849034i −1.33338 1.38227i −5.58137 −5.95398 + 5.34323i 4.03893 6.99563i −9.97325 + 0.730972i
159.11 −1.92793 + 0.532049i −1.00965 1.74877i 3.43385 2.05151i −4.76464 + 1.51598i 2.87698 + 2.83433i 3.07442 −5.52873 + 5.78214i 2.46120 4.26292i 8.37933 5.45773i
159.12 −1.87977 0.682971i 1.12368 + 1.94628i 3.06710 + 2.56766i 2.49657 4.33210i −0.783020 4.42601i −2.72652 −4.01181 6.92137i 1.97467 3.42023i −7.65169 + 6.43829i
159.13 −1.85215 + 0.754679i 2.72599 + 4.72155i 2.86092 2.79556i −4.08217 2.88720i −8.61220 6.68778i 7.14895 −3.18910 + 7.33687i −10.3621 + 17.9476i 9.73969 + 2.26680i
159.14 −1.83289 0.800316i −0.529478 0.917082i 2.71899 + 2.93379i 4.84073 1.25194i 0.236520 + 2.10466i 13.3882 −2.63566 7.55336i 3.93931 6.82308i −9.87448 1.57944i
159.15 −1.82919 0.808734i −0.866507 1.50083i 2.69190 + 2.95866i −4.28911 2.56974i 0.371233 + 3.44609i −10.8203 −2.53123 7.58900i 2.99833 5.19326i 5.76738 + 8.16929i
159.16 −1.82898 0.809227i −0.0580338 0.100518i 2.69030 + 2.96011i 2.00334 + 4.58111i 0.0248009 + 0.230807i −7.81754 −2.52509 7.59104i 4.49326 7.78256i 0.0430974 9.99991i
159.17 −1.76297 + 0.944431i −1.63664 2.83475i 2.21610 3.33000i −0.368392 + 4.98641i 5.56256 + 3.45187i −8.62575 −0.761952 + 7.96363i −0.857187 + 1.48469i −4.05986 9.13879i
159.18 −1.74433 + 0.978432i 2.25815 + 3.91123i 2.08534 3.41341i 2.67107 + 4.22675i −7.76582 4.61301i −11.4308 −0.297731 + 7.99446i −5.69848 + 9.87006i −8.79480 4.75936i
159.19 −1.73403 + 0.996556i 1.05335 + 1.82446i 2.01375 3.45613i 2.78233 4.15435i −3.64473 2.11396i 8.55585 −0.0476908 + 7.99986i 2.28089 3.95061i −0.684605 + 9.97654i
159.20 −1.73006 + 1.00344i −1.05335 1.82446i 1.98622 3.47202i 2.78233 4.15435i 3.65311 + 2.09945i −8.55585 0.0476908 + 7.99986i 2.28089 3.95061i −0.644950 + 9.97918i
See next 80 embeddings (of 232 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.116 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.b even 2 1 inner
19.c even 3 1 inner
20.d odd 2 1 inner
76.g odd 6 1 inner
95.i even 6 1 inner
380.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.p.a 232
4.b odd 2 1 inner 380.3.p.a 232
5.b even 2 1 inner 380.3.p.a 232
19.c even 3 1 inner 380.3.p.a 232
20.d odd 2 1 inner 380.3.p.a 232
76.g odd 6 1 inner 380.3.p.a 232
95.i even 6 1 inner 380.3.p.a 232
380.p odd 6 1 inner 380.3.p.a 232

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.p.a 232 1.a even 1 1 trivial
380.3.p.a 232 4.b odd 2 1 inner
380.3.p.a 232 5.b even 2 1 inner
380.3.p.a 232 19.c even 3 1 inner
380.3.p.a 232 20.d odd 2 1 inner
380.3.p.a 232 76.g odd 6 1 inner
380.3.p.a 232 95.i even 6 1 inner
380.3.p.a 232 380.p odd 6 1 inner

## Hecke kernels

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