# Properties

 Label 380.2.bh.a Level $380$ Weight $2$ Character orbit 380.bh 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(13,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([0, 27, 10]))

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

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

chi := DirichletCharacter("380.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.bh (of order $$36$$, degree $$12$$, 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: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$120 q + 6 q^{7}+O(q^{10})$$ 120 * q + 6 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61} + 36 q^{63} + 90 q^{65} - 24 q^{67} + 18 q^{73} - 36 q^{77} - 30 q^{83} - 24 q^{85} - 72 q^{87} - 144 q^{91} - 132 q^{93} - 12 q^{95} - 60 q^{97}+O(q^{100})$$ 120 * q + 6 * q^7 + 18 * q^15 - 18 * q^17 + 48 * q^21 - 36 * q^23 - 24 * q^25 - 60 * q^33 - 18 * q^35 - 12 * q^41 - 36 * q^43 + 18 * q^45 - 24 * q^47 + 96 * q^51 - 18 * q^53 + 72 * q^55 - 6 * q^57 - 24 * q^61 + 36 * q^63 + 90 * q^65 - 24 * q^67 + 18 * q^73 - 36 * q^77 - 30 * q^83 - 24 * q^85 - 72 * q^87 - 144 * q^91 - 132 * q^93 - 12 * q^95 - 60 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 0 −2.74765 + 1.28125i 0 −1.16017 1.91154i 0 −0.370262 1.38184i 0 3.97961 4.74272i 0
13.2 0 −2.38123 + 1.11039i 0 1.56340 + 1.59868i 0 0.558906 + 2.08587i 0 2.50894 2.99004i 0
13.3 0 −1.41596 + 0.660272i 0 −0.949889 + 2.02428i 0 −1.12143 4.18524i 0 −0.359387 + 0.428300i 0
13.4 0 −0.927188 + 0.432355i 0 0.140904 2.23162i 0 −0.231730 0.864827i 0 −1.25562 + 1.49639i 0
13.5 0 −0.362674 + 0.169118i 0 1.57660 1.58566i 0 1.00773 + 3.76091i 0 −1.82543 + 2.17546i 0
13.6 0 0.224420 0.104649i 0 −2.15088 + 0.611320i 0 0.357605 + 1.33460i 0 −1.88895 + 2.25116i 0
13.7 0 0.959290 0.447324i 0 1.57989 + 1.58239i 0 0.204631 + 0.763694i 0 −1.20823 + 1.43991i 0
13.8 0 1.80092 0.839782i 0 −1.74340 1.40020i 0 −0.755503 2.81958i 0 0.609708 0.726622i 0
13.9 0 1.84872 0.862074i 0 2.22179 + 0.252247i 0 −1.05366 3.93231i 0 0.746246 0.889341i 0
13.10 0 3.00135 1.39955i 0 −0.904597 + 2.04492i 0 1.03769 + 3.87270i 0 5.12098 6.10295i 0
33.1 0 −1.61138 2.30128i 0 1.99179 1.01625i 0 4.91485 + 1.31693i 0 −1.67331 + 4.59739i 0
33.2 0 −1.41957 2.02735i 0 0.187939 2.22816i 0 −3.90736 1.04697i 0 −1.06893 + 2.93686i 0
33.3 0 −1.02333 1.46147i 0 0.710865 + 2.12006i 0 −2.65380 0.711084i 0 −0.0626300 + 0.172074i 0
33.4 0 −0.886233 1.26567i 0 −2.22818 0.187668i 0 0.477272 + 0.127885i 0 0.209544 0.575718i 0
33.5 0 0.0411373 + 0.0587502i 0 −2.20598 + 0.365568i 0 0.553309 + 0.148259i 0 1.02430 2.81424i 0
33.6 0 0.230238 + 0.328814i 0 −0.244120 2.22270i 0 1.71755 + 0.460216i 0 0.970951 2.66767i 0
33.7 0 0.377219 + 0.538725i 0 2.02714 + 0.943776i 0 1.97160 + 0.528288i 0 0.878130 2.41264i 0
33.8 0 0.935220 + 1.33563i 0 −0.351064 + 2.20834i 0 −4.10912 1.10104i 0 0.116783 0.320859i 0
33.9 0 1.60791 + 2.29633i 0 1.74139 1.40270i 0 −0.546238 0.146364i 0 −1.66171 + 4.56550i 0
33.10 0 1.74879 + 2.49753i 0 −0.863736 + 2.06251i 0 2.94797 + 0.789905i 0 −2.15333 + 5.91624i 0
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.f odd 18 1 inner
95.r 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.bh.a 120
5.c odd 4 1 inner 380.2.bh.a 120
19.f odd 18 1 inner 380.2.bh.a 120
95.r even 36 1 inner 380.2.bh.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.bh.a 120 1.a even 1 1 trivial
380.2.bh.a 120 5.c odd 4 1 inner
380.2.bh.a 120 19.f odd 18 1 inner
380.2.bh.a 120 95.r even 36 1 inner

## Hecke kernels

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