# Properties

 Label 380.2.v.c Level $380$ Weight $2$ Character orbit 380.v Analytic conductor $3.034$ Analytic rank $0$ Dimension $216$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(7,380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 3, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("380.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.v (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$216$$ Relative dimension: $$54$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$216 q + 12 q^{6} - 36 q^{8}+O(q^{10})$$ 216 * q + 12 * q^6 - 36 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$216 q + 12 q^{6} - 36 q^{8} + 4 q^{10} - 20 q^{12} - 8 q^{13} + 8 q^{16} - 16 q^{17} + 12 q^{18} - 48 q^{20} + 40 q^{21} + 16 q^{22} - 16 q^{25} + 24 q^{26} + 8 q^{28} - 12 q^{30} - 20 q^{32} + 20 q^{33} - 56 q^{36} - 32 q^{37} - 18 q^{38} - 48 q^{41} + 54 q^{42} - 104 q^{45} - 24 q^{46} - 4 q^{48} + 8 q^{50} - 14 q^{52} - 16 q^{53} - 16 q^{56} + 12 q^{57} - 136 q^{58} + 50 q^{60} - 84 q^{61} + 42 q^{62} - 56 q^{65} + 28 q^{68} - 130 q^{70} + 80 q^{72} - 36 q^{73} + 96 q^{77} + 36 q^{78} + 6 q^{80} + 76 q^{81} - 16 q^{85} + 88 q^{86} + 104 q^{88} - 86 q^{90} + 28 q^{92} - 84 q^{93} - 128 q^{96} + 12 q^{97} + 34 q^{98}+O(q^{100})$$ 216 * q + 12 * q^6 - 36 * q^8 + 4 * q^10 - 20 * q^12 - 8 * q^13 + 8 * q^16 - 16 * q^17 + 12 * q^18 - 48 * q^20 + 40 * q^21 + 16 * q^22 - 16 * q^25 + 24 * q^26 + 8 * q^28 - 12 * q^30 - 20 * q^32 + 20 * q^33 - 56 * q^36 - 32 * q^37 - 18 * q^38 - 48 * q^41 + 54 * q^42 - 104 * q^45 - 24 * q^46 - 4 * q^48 + 8 * q^50 - 14 * q^52 - 16 * q^53 - 16 * q^56 + 12 * q^57 - 136 * q^58 + 50 * q^60 - 84 * q^61 + 42 * q^62 - 56 * q^65 + 28 * q^68 - 130 * q^70 + 80 * q^72 - 36 * q^73 + 96 * q^77 + 36 * q^78 + 6 * q^80 + 76 * q^81 - 16 * q^85 + 88 * q^86 + 104 * q^88 - 86 * q^90 + 28 * q^92 - 84 * q^93 - 128 * q^96 + 12 * q^97 + 34 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.41404 0.0219748i −0.838811 + 3.13048i 1.99903 + 0.0621466i 0.366696 + 2.20580i 1.25491 4.40821i −2.07593 + 2.07593i −2.82535 0.131806i −6.49825 3.75177i −0.470052 3.12715i
7.2 −1.41076 + 0.0987116i 0.456579 1.70398i 1.98051 0.278518i −1.07799 + 1.95906i −0.475923 + 2.44898i 0.164130 0.164130i −2.76654 + 0.588422i −0.0969920 0.0559983i 1.32741 2.87019i
7.3 −1.39566 0.228298i −0.00674551 + 0.0251746i 1.89576 + 0.637256i −2.17162 + 0.532967i 0.0151618 0.0335953i −3.02167 + 3.02167i −2.50036 1.32219i 2.59749 + 1.49966i 3.15253 0.248066i
7.4 −1.39454 0.235046i −0.550661 + 2.05509i 1.88951 + 0.655563i −1.04297 1.97793i 1.25096 2.73649i 0.179304 0.179304i −2.48091 1.35833i −1.32211 0.763319i 0.989568 + 3.00346i
7.5 −1.38538 0.284105i 0.297989 1.11211i 1.83857 + 0.787187i 0.561673 2.16438i −0.728785 + 1.45604i 1.56945 1.56945i −2.32348 1.61290i 1.45008 + 0.837205i −1.39304 + 2.83891i
7.6 −1.37248 + 0.341027i −0.280019 + 1.04505i 1.76740 0.936104i 1.80654 + 1.31773i 0.0279322 1.52980i 3.16094 3.16094i −2.10649 + 1.88752i 1.58437 + 0.914735i −2.92882 1.19249i
7.7 −1.33358 + 0.470716i 0.879814 3.28351i 1.55685 1.25547i −1.53557 1.62543i 0.372301 + 4.79295i −1.21752 + 1.21752i −1.48521 + 2.40710i −7.40928 4.27775i 2.81292 + 1.44482i
7.8 −1.31685 + 0.515654i 0.129961 0.485021i 1.46820 1.35808i 2.06245 0.863894i 0.0789637 + 0.705716i −2.26251 + 2.26251i −1.23310 + 2.54548i 2.37972 + 1.37393i −2.27047 + 2.20113i
7.9 −1.29228 0.574473i 0.743999 2.77664i 1.33996 + 1.48476i 1.98459 + 1.03025i −2.55656 + 3.16078i −0.196102 + 0.196102i −0.878647 2.68849i −4.55813 2.63164i −1.97278 2.47146i
7.10 −1.28287 + 0.595184i −0.314724 + 1.17457i 1.29151 1.52709i −2.23301 + 0.116883i −0.295333 1.69414i 1.80880 1.80880i −0.747943 + 2.72774i 1.31752 + 0.760671i 2.79510 1.47900i
7.11 −1.19133 + 0.762063i −0.699452 + 2.61039i 0.838520 1.81573i 0.245430 2.22256i −1.15601 3.64286i −1.37846 + 1.37846i 0.384750 + 2.80214i −3.72684 2.15169i 1.40134 + 2.83483i
7.12 −1.13818 0.839368i −0.286879 + 1.07065i 0.590923 + 1.91071i 0.355287 + 2.20766i 1.22519 0.977795i 0.641806 0.641806i 0.931209 2.67074i 1.53409 + 0.885709i 1.44866 2.81094i
7.13 −1.04438 0.953560i −0.712111 + 2.65764i 0.181448 + 1.99175i 1.75700 1.38309i 3.27793 2.09653i 2.28576 2.28576i 1.70975 2.25316i −3.95785 2.28507i −3.15383 0.230937i
7.14 −0.953207 1.04470i −0.108090 + 0.403396i −0.182795 + 1.99163i −1.78057 1.35262i 0.524459 0.271598i −0.329987 + 0.329987i 2.25490 1.70747i 2.44703 + 1.41279i 0.284168 + 3.14948i
7.15 −0.910865 + 1.08182i 0.680944 2.54132i −0.340649 1.97078i 2.20047 0.397406i 2.12899 + 3.05146i 2.71258 2.71258i 2.44230 + 1.42659i −3.39654 1.96099i −1.57441 + 2.74249i
7.16 −0.853629 + 1.12753i 0.406597 1.51744i −0.542637 1.92498i −1.61426 + 1.54731i 1.36387 + 1.75378i 0.846718 0.846718i 2.63368 + 1.03138i 0.460773 + 0.266027i −0.366648 3.14095i
7.17 −0.824501 + 1.14900i −0.320669 + 1.19675i −0.640395 1.89470i 1.11922 + 1.93580i −1.11068 1.35517i −1.23765 + 1.23765i 2.70502 + 0.826371i 1.26868 + 0.732475i −3.14704 0.310087i
7.18 −0.691641 1.23354i 0.549358 2.05023i −1.04326 + 1.70634i −0.716472 + 2.11818i −2.90901 + 0.740368i −3.20712 + 3.20712i 2.82641 + 0.106737i −1.30358 0.752624i 3.10841 0.581217i
7.19 −0.672527 + 1.24407i 0.174995 0.653091i −1.09541 1.67334i −1.53174 1.62904i 0.694801 + 0.656927i −2.59798 + 2.59798i 2.81845 0.237403i 2.20217 + 1.27142i 3.05678 0.810016i
7.20 −0.575530 1.29181i 0.224197 0.836715i −1.33753 + 1.48695i 2.16715 + 0.550885i −1.20991 + 0.191935i 1.16634 1.16634i 2.69064 + 0.872049i 1.94825 + 1.12482i −0.535621 3.11659i
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.54 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.c even 3 1 inner
20.e even 4 1 inner
76.g odd 6 1 inner
95.m odd 12 1 inner
380.v even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.v.c 216
4.b odd 2 1 inner 380.2.v.c 216
5.c odd 4 1 inner 380.2.v.c 216
19.c even 3 1 inner 380.2.v.c 216
20.e even 4 1 inner 380.2.v.c 216
76.g odd 6 1 inner 380.2.v.c 216
95.m odd 12 1 inner 380.2.v.c 216
380.v even 12 1 inner 380.2.v.c 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.v.c 216 1.a even 1 1 trivial
380.2.v.c 216 4.b odd 2 1 inner
380.2.v.c 216 5.c odd 4 1 inner
380.2.v.c 216 19.c even 3 1 inner
380.2.v.c 216 20.e even 4 1 inner
380.2.v.c 216 76.g odd 6 1 inner
380.2.v.c 216 95.m odd 12 1 inner
380.2.v.c 216 380.v even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{216} - 694 T_{3}^{212} + 266097 T_{3}^{208} - 70048138 T_{3}^{204} + 13980235252 T_{3}^{200} + \cdots + 23\!\cdots\!36$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.