# Properties

 Label 380.2.d.b Level $380$ Weight $2$ Character orbit 380.d Analytic conductor $3.034$ Analytic rank $0$ Dimension $40$ Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(379,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("380.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.d (of order $$2$$, degree $$1$$, 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: $$40$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 4 q^{5} + 8 q^{6} - 8 q^{9}+O(q^{10})$$ 40 * q - 4 * q^5 + 8 * q^6 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 4 q^{5} + 8 q^{6} - 8 q^{9} - 8 q^{16} - 20 q^{20} - 40 q^{24} - 84 q^{25} - 24 q^{26} + 24 q^{30} + 24 q^{36} - 40 q^{44} - 12 q^{45} + 128 q^{49} - 120 q^{54} + 24 q^{61} + 72 q^{64} + 112 q^{66} + 32 q^{74} + 56 q^{76} + 96 q^{80} - 72 q^{81} + 44 q^{85} - 40 q^{96}+O(q^{100})$$ 40 * q - 4 * q^5 + 8 * q^6 - 8 * q^9 - 8 * q^16 - 20 * q^20 - 40 * q^24 - 84 * q^25 - 24 * q^26 + 24 * q^30 + 24 * q^36 - 40 * q^44 - 12 * q^45 + 128 * q^49 - 120 * q^54 + 24 * q^61 + 72 * q^64 + 112 * q^66 + 32 * q^74 + 56 * q^76 + 96 * q^80 - 72 * q^81 + 44 * q^85 - 40 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1 −1.39672 0.221721i 2.04365i 1.90168 + 0.619368i 0.665544 2.13473i −0.453121 + 2.85442i 2.07881 −2.51879 1.28673i −1.17650 −1.40290 + 2.83406i
379.2 −1.39672 0.221721i 2.04365i 1.90168 + 0.619368i 0.665544 + 2.13473i −0.453121 + 2.85442i −2.07881 −2.51879 1.28673i −1.17650 −0.456267 3.12919i
379.3 −1.39672 + 0.221721i 2.04365i 1.90168 0.619368i 0.665544 2.13473i −0.453121 2.85442i −2.07881 −2.51879 + 1.28673i −1.17650 −0.456267 + 3.12919i
379.4 −1.39672 + 0.221721i 2.04365i 1.90168 0.619368i 0.665544 + 2.13473i −0.453121 2.85442i 2.07881 −2.51879 + 1.28673i −1.17650 −1.40290 2.83406i
379.5 −1.27059 0.620975i 0.701989i 1.22878 + 1.57800i −1.10449 1.94425i 0.435918 0.891938i 1.18642 −0.581371 2.76803i 2.50721 0.196019 + 3.15620i
379.6 −1.27059 0.620975i 0.701989i 1.22878 + 1.57800i −1.10449 + 1.94425i 0.435918 0.891938i −1.18642 −0.581371 2.76803i 2.50721 2.61068 1.78448i
379.7 −1.27059 + 0.620975i 0.701989i 1.22878 1.57800i −1.10449 1.94425i 0.435918 + 0.891938i −1.18642 −0.581371 + 2.76803i 2.50721 2.61068 + 1.78448i
379.8 −1.27059 + 0.620975i 0.701989i 1.22878 1.57800i −1.10449 + 1.94425i 0.435918 + 0.891938i 1.18642 −0.581371 + 2.76803i 2.50721 0.196019 3.15620i
379.9 −0.948256 1.04920i 1.76151i −0.201622 + 1.98981i −1.85854 1.24332i −1.84816 + 1.67036i −4.29252 2.27889 1.67531i −0.102903 0.457886 + 3.12895i
379.10 −0.948256 1.04920i 1.76151i −0.201622 + 1.98981i −1.85854 + 1.24332i −1.84816 + 1.67036i 4.29252 2.27889 1.67531i −0.102903 3.06685 + 0.770986i
379.11 −0.948256 + 1.04920i 1.76151i −0.201622 1.98981i −1.85854 1.24332i −1.84816 1.67036i 4.29252 2.27889 + 1.67531i −0.102903 3.06685 0.770986i
379.12 −0.948256 + 1.04920i 1.76151i −0.201622 1.98981i −1.85854 + 1.24332i −1.84816 1.67036i −4.29252 2.27889 + 1.67531i −0.102903 0.457886 3.12895i
379.13 −0.584780 1.28765i 2.81267i −1.31606 + 1.50598i 0.390055 2.20178i 3.62172 1.64479i −4.13073 2.70878 + 0.813956i −4.91111 −3.06322 + 0.785308i
379.14 −0.584780 1.28765i 2.81267i −1.31606 + 1.50598i 0.390055 + 2.20178i 3.62172 1.64479i 4.13073 2.70878 + 0.813956i −4.91111 2.60702 1.78981i
379.15 −0.584780 + 1.28765i 2.81267i −1.31606 1.50598i 0.390055 2.20178i 3.62172 + 1.64479i 4.13073 2.70878 0.813956i −4.91111 2.60702 + 1.78981i
379.16 −0.584780 + 1.28765i 2.81267i −1.31606 1.50598i 0.390055 + 2.20178i 3.62172 + 1.64479i −4.13073 2.70878 0.813956i −4.91111 −3.06322 0.785308i
379.17 −0.440015 1.34402i 0.562756i −1.61277 + 1.18278i 1.40743 1.73757i −0.756355 + 0.247621i 3.12767 2.29932 + 1.64716i 2.68331 −2.95462 1.12705i
379.18 −0.440015 1.34402i 0.562756i −1.61277 + 1.18278i 1.40743 + 1.73757i −0.756355 + 0.247621i −3.12767 2.29932 + 1.64716i 2.68331 1.71604 2.65617i
379.19 −0.440015 + 1.34402i 0.562756i −1.61277 1.18278i 1.40743 1.73757i −0.756355 0.247621i −3.12767 2.29932 1.64716i 2.68331 1.71604 + 2.65617i
379.20 −0.440015 + 1.34402i 0.562756i −1.61277 1.18278i 1.40743 + 1.73757i −0.756355 0.247621i 3.12767 2.29932 1.64716i 2.68331 −2.95462 + 1.12705i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.40 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.b odd 2 1 inner
20.d odd 2 1 inner
76.d even 2 1 inner
95.d odd 2 1 inner
380.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.d.b 40
4.b odd 2 1 inner 380.2.d.b 40
5.b even 2 1 inner 380.2.d.b 40
19.b odd 2 1 inner 380.2.d.b 40
20.d odd 2 1 inner 380.2.d.b 40
76.d even 2 1 inner 380.2.d.b 40
95.d odd 2 1 inner 380.2.d.b 40
380.d even 2 1 inner 380.2.d.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.d.b 40 1.a even 1 1 trivial
380.2.d.b 40 4.b odd 2 1 inner
380.2.d.b 40 5.b even 2 1 inner
380.2.d.b 40 19.b odd 2 1 inner
380.2.d.b 40 20.d odd 2 1 inner
380.2.d.b 40 76.d even 2 1 inner
380.2.d.b 40 95.d odd 2 1 inner
380.2.d.b 40 380.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} + 16T_{3}^{8} + 83T_{3}^{6} + 162T_{3}^{4} + 94T_{3}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.