# Properties

 Label 380.2.be.a Level $380$ Weight $2$ Character orbit 380.be Analytic conductor $3.034$ Analytic rank $0$ Dimension $120$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(51,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 0, 5]))

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

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

chi := DirichletCharacter("380.51");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.be (of order $$18$$, degree $$6$$, 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: $$120$$ Relative dimension: $$20$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$120 q - 3 q^{2} - 3 q^{4} + 3 q^{6} + 36 q^{8} + 6 q^{9}+O(q^{10})$$ 120 * q - 3 * q^2 - 3 * q^4 + 3 * q^6 + 36 * q^8 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$120 q - 3 q^{2} - 3 q^{4} + 3 q^{6} + 36 q^{8} + 6 q^{9} + 3 q^{10} - 12 q^{13} + 18 q^{14} - 15 q^{16} - 48 q^{17} - 12 q^{21} + 18 q^{24} - 69 q^{28} - 12 q^{30} - 33 q^{32} + 6 q^{33} - 18 q^{34} - 72 q^{36} + 18 q^{38} - 36 q^{41} - 27 q^{42} - 27 q^{44} - 60 q^{45} - 117 q^{46} + 45 q^{48} + 60 q^{49} - 15 q^{52} + 48 q^{53} + 69 q^{54} + 48 q^{57} + 6 q^{60} + 18 q^{62} - 24 q^{64} - 18 q^{65} + 168 q^{66} + 12 q^{68} - 72 q^{69} + 36 q^{70} - 42 q^{72} + 66 q^{74} + 24 q^{76} - 60 q^{77} - 54 q^{78} + 21 q^{80} - 66 q^{81} - 63 q^{82} + 252 q^{84} - 48 q^{85} - 45 q^{86} + 135 q^{88} - 3 q^{90} - 42 q^{92} + 96 q^{93} + 78 q^{96} - 54 q^{97} - 33 q^{98}+O(q^{100})$$ 120 * q - 3 * q^2 - 3 * q^4 + 3 * q^6 + 36 * q^8 + 6 * q^9 + 3 * q^10 - 12 * q^13 + 18 * q^14 - 15 * q^16 - 48 * q^17 - 12 * q^21 + 18 * q^24 - 69 * q^28 - 12 * q^30 - 33 * q^32 + 6 * q^33 - 18 * q^34 - 72 * q^36 + 18 * q^38 - 36 * q^41 - 27 * q^42 - 27 * q^44 - 60 * q^45 - 117 * q^46 + 45 * q^48 + 60 * q^49 - 15 * q^52 + 48 * q^53 + 69 * q^54 + 48 * q^57 + 6 * q^60 + 18 * q^62 - 24 * q^64 - 18 * q^65 + 168 * q^66 + 12 * q^68 - 72 * q^69 + 36 * q^70 - 42 * q^72 + 66 * q^74 + 24 * q^76 - 60 * q^77 - 54 * q^78 + 21 * q^80 - 66 * q^81 - 63 * q^82 + 252 * q^84 - 48 * q^85 - 45 * q^86 + 135 * q^88 - 3 * q^90 - 42 * q^92 + 96 * q^93 + 78 * q^96 - 54 * q^97 - 33 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1 −1.41417 0.0114942i 2.22536 + 0.809964i 1.99974 + 0.0325095i 0.173648 + 0.984808i −3.13772 1.17100i −1.02771 0.593350i −2.82759 0.0689592i 1.99805 + 1.67656i −0.234248 1.39468i
51.2 −1.36060 0.385715i 0.497127 + 0.180940i 1.70245 + 1.04960i 0.173648 + 0.984808i −0.606599 0.437935i −2.30321 1.32976i −1.91150 2.08475i −2.08374 1.74846i 0.143590 1.40691i
51.3 −1.28569 + 0.589063i −2.16384 0.787572i 1.30601 1.51471i 0.173648 + 0.984808i 3.24596 0.262061i −3.09267 1.78556i −0.786867 + 2.71677i 1.76379 + 1.48000i −0.803372 1.16387i
51.4 −1.23339 0.691917i −0.199361 0.0725616i 1.04250 + 1.70681i 0.173648 + 0.984808i 0.195684 + 0.227438i 3.46492 + 2.00047i −0.104843 2.82648i −2.26365 1.89943i 0.467229 1.33480i
51.5 −1.04516 0.952704i −2.85924 1.04068i 0.184712 + 1.99145i 0.173648 + 0.984808i 1.99690 + 3.81168i −0.0195893 0.0113099i 1.70421 2.25736i 4.79409 + 4.02272i 0.756740 1.19472i
51.6 −0.845155 + 1.13389i 0.191206 + 0.0695933i −0.571427 1.91663i 0.173648 + 0.984808i −0.240510 + 0.157990i −0.977956 0.564623i 2.65620 + 0.971911i −2.26642 1.90175i −1.26343 0.635416i
51.7 −0.767584 + 1.18778i 2.18403 + 0.794921i −0.821629 1.82344i 0.173648 + 0.984808i −2.62061 + 1.98397i 3.27406 + 1.89028i 2.79651 + 0.423730i 1.83994 + 1.54390i −1.30302 0.549668i
51.8 −0.756740 1.19472i 2.85924 + 1.04068i −0.854689 + 1.80818i 0.173648 + 0.984808i −0.920386 4.20350i 0.0195893 + 0.0113099i 2.80703 0.347210i 4.79409 + 4.02272i 1.04516 0.952704i
51.9 −0.467229 1.33480i 0.199361 + 0.0725616i −1.56339 + 1.24732i 0.173648 + 0.984808i 0.00370794 0.300011i −3.46492 2.00047i 2.39538 + 1.50404i −2.26365 1.89943i 1.23339 0.691917i
51.10 −0.298614 + 1.38233i −1.20014 0.436814i −1.82166 0.825564i 0.173648 + 0.984808i 0.962198 1.52855i −0.435003 0.251149i 1.68517 2.27161i −1.04861 0.879888i −1.41318 0.0540382i
51.11 −0.143590 1.40691i −0.497127 0.180940i −1.95876 + 0.404034i 0.173648 + 0.984808i −0.183182 + 0.725392i 2.30321 + 1.32976i 0.849696 + 2.69778i −2.08374 1.74846i 1.36060 0.385715i
51.12 0.234248 1.39468i −2.22536 0.809964i −1.89026 0.653401i 0.173648 + 0.984808i −1.65093 + 2.91393i 1.02771 + 0.593350i −1.35407 + 2.48324i 1.99805 + 1.67656i 1.41417 0.0114942i
51.13 0.262366 + 1.38966i 2.24675 + 0.817752i −1.86233 + 0.729199i 0.173648 + 0.984808i −0.546929 + 3.33678i −0.386990 0.223429i −1.50195 2.39669i 2.08105 + 1.74621i −1.32299 + 0.499692i
51.14 0.579400 + 1.29008i −2.37127 0.863072i −1.32859 + 1.49494i 0.173648 + 0.984808i −0.260486 3.55918i 3.42776 + 1.97902i −2.69837 0.847816i 2.57989 + 2.16479i −1.16986 + 0.794617i
51.15 0.803372 1.16387i 2.16384 + 0.787572i −0.709186 1.87004i 0.173648 + 0.984808i 2.65500 1.88571i 3.09267 + 1.78556i −2.74623 0.676939i 1.76379 + 1.48000i 1.28569 + 0.589063i
51.16 1.16986 + 0.794617i 2.37127 + 0.863072i 0.737168 + 1.85919i 0.173648 + 0.984808i 2.08825 + 2.89393i −3.42776 1.97902i −0.614956 + 2.76077i 2.57989 + 2.16479i −0.579400 + 1.29008i
51.17 1.26343 0.635416i −0.191206 0.0695933i 1.19249 1.60560i 0.173648 + 0.984808i −0.285795 + 0.0335694i 0.977956 + 0.564623i 0.486400 2.78629i −2.26642 1.90175i 0.845155 + 1.13389i
51.18 1.30302 0.549668i −2.18403 0.794921i 1.39573 1.43246i 0.173648 + 0.984808i −3.28278 + 0.164690i −3.27406 1.89028i 1.03129 2.63371i 1.83994 + 1.54390i 0.767584 + 1.18778i
51.19 1.32299 + 0.499692i −2.24675 0.817752i 1.50062 + 1.32218i 0.173648 + 0.984808i −2.56381 2.20456i 0.386990 + 0.223429i 1.32462 + 2.49908i 2.08105 + 1.74621i −0.262366 + 1.38966i
51.20 1.41318 0.0540382i 1.20014 + 0.436814i 1.99416 0.152732i 0.173648 + 0.984808i 1.71962 + 0.552444i 0.435003 + 0.251149i 2.80985 0.323598i −1.04861 0.879888i 0.298614 + 1.38233i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 51.20 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
19.f odd 18 1 inner
76.k even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.be.a 120
4.b odd 2 1 inner 380.2.be.a 120
19.f odd 18 1 inner 380.2.be.a 120
76.k even 18 1 inner 380.2.be.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.be.a 120 1.a even 1 1 trivial
380.2.be.a 120 4.b odd 2 1 inner
380.2.be.a 120 19.f odd 18 1 inner
380.2.be.a 120 76.k even 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{120} - 3 T_{3}^{118} + 48 T_{3}^{116} + 1856 T_{3}^{114} - 7803 T_{3}^{112} + 75999 T_{3}^{110} + \cdots + 7676563456$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.