# Properties

 Label 390.2.n.a Level $390$ Weight $2$ Character orbit 390.n Analytic conductor $3.114$ Analytic rank $0$ Dimension $56$ Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(239,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

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

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

chi := DirichletCharacter("390.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.n (of order $$4$$, degree $$2$$, 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: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56 q+O(q^{10})$$ 56 * q $$\operatorname{Tr}(f)(q) =$$ $$56 q + 8 q^{15} - 56 q^{16} + 32 q^{19} + 8 q^{21} + 8 q^{31} - 24 q^{34} - 24 q^{39} - 12 q^{45} + 8 q^{46} - 24 q^{54} - 16 q^{55} - 8 q^{60} + 48 q^{61} - 16 q^{66} + 32 q^{70} - 32 q^{76} - 64 q^{79} + 64 q^{81} + 8 q^{84} - 72 q^{85} - 48 q^{91} + 64 q^{94} + 56 q^{99}+O(q^{100})$$ 56 * q + 8 * q^15 - 56 * q^16 + 32 * q^19 + 8 * q^21 + 8 * q^31 - 24 * q^34 - 24 * q^39 - 12 * q^45 + 8 * q^46 - 24 * q^54 - 16 * q^55 - 8 * q^60 + 48 * q^61 - 16 * q^66 + 32 * q^70 - 32 * q^76 - 64 * q^79 + 64 * q^81 + 8 * q^84 - 72 * q^85 - 48 * q^91 + 64 * q^94 + 56 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1 −0.707107 + 0.707107i −1.71366 0.251734i 1.00000i −1.77539 1.35941i 1.38974 1.03374i 0.242355 0.242355i 0.707107 + 0.707107i 2.87326 + 0.862772i 2.21664 0.294144i
239.2 −0.707107 + 0.707107i −1.71366 + 0.251734i 1.00000i 1.35941 + 1.77539i 1.03374 1.38974i −0.242355 + 0.242355i 0.707107 + 0.707107i 2.87326 0.862772i −2.21664 0.294144i
239.3 −0.707107 + 0.707107i −1.08285 1.35182i 1.00000i 2.23041 + 0.158919i 1.72158 + 0.190189i 2.29387 2.29387i 0.707107 + 0.707107i −0.654851 + 2.92766i −1.68951 + 1.46477i
239.4 −0.707107 + 0.707107i −1.08285 + 1.35182i 1.00000i −0.158919 2.23041i −0.190189 1.72158i −2.29387 + 2.29387i 0.707107 + 0.707107i −0.654851 2.92766i 1.68951 + 1.46477i
239.5 −0.707107 + 0.707107i −1.01321 1.40478i 1.00000i −1.62409 + 1.53698i 1.70978 + 0.276879i −0.656637 + 0.656637i 0.707107 + 0.707107i −0.946804 + 2.84668i 0.0615952 2.23522i
239.6 −0.707107 + 0.707107i −1.01321 + 1.40478i 1.00000i −1.53698 + 1.62409i −0.276879 1.70978i 0.656637 0.656637i 0.707107 + 0.707107i −0.946804 2.84668i −0.0615952 2.23522i
239.7 −0.707107 + 0.707107i 0.164052 1.72426i 1.00000i −0.365747 2.20595i 1.10324 + 1.33524i 2.08554 2.08554i 0.707107 + 0.707107i −2.94617 0.565738i 1.81847 + 1.30122i
239.8 −0.707107 + 0.707107i 0.164052 + 1.72426i 1.00000i 2.20595 + 0.365747i −1.33524 1.10324i −2.08554 + 2.08554i 0.707107 + 0.707107i −2.94617 + 0.565738i −1.81847 + 1.30122i
239.9 −0.707107 + 0.707107i 0.452930 1.67178i 1.00000i 0.934355 + 2.03150i 0.861859 + 1.50240i −3.48345 + 3.48345i 0.707107 + 0.707107i −2.58971 1.51440i −2.09717 0.775797i
239.10 −0.707107 + 0.707107i 0.452930 + 1.67178i 1.00000i −2.03150 0.934355i −1.50240 0.861859i 3.48345 3.48345i 0.707107 + 0.707107i −2.58971 + 1.51440i 2.09717 0.775797i
239.11 −0.707107 + 0.707107i 1.46346 0.926431i 1.00000i 0.989929 2.00500i −0.379740 + 1.68991i −1.27923 + 1.27923i 0.707107 + 0.707107i 1.28345 2.71160i 0.717766 + 2.11774i
239.12 −0.707107 + 0.707107i 1.46346 + 0.926431i 1.00000i 2.00500 0.989929i −1.68991 + 0.379740i 1.27923 1.27923i 0.707107 + 0.707107i 1.28345 + 2.71160i −0.717766 + 2.11774i
239.13 −0.707107 + 0.707107i 1.72928 0.0979139i 1.00000i 0.00363329 + 2.23607i −1.15355 + 1.29202i 1.76860 1.76860i 0.707107 + 0.707107i 2.98083 0.338641i −1.58371 1.57857i
239.14 −0.707107 + 0.707107i 1.72928 + 0.0979139i 1.00000i −2.23607 0.00363329i −1.29202 + 1.15355i −1.76860 + 1.76860i 0.707107 + 0.707107i 2.98083 + 0.338641i 1.58371 1.57857i
239.15 0.707107 0.707107i −1.72928 0.0979139i 1.00000i −0.00363329 2.23607i −1.29202 + 1.15355i 1.76860 1.76860i −0.707107 0.707107i 2.98083 + 0.338641i −1.58371 1.57857i
239.16 0.707107 0.707107i −1.72928 + 0.0979139i 1.00000i 2.23607 + 0.00363329i −1.15355 + 1.29202i −1.76860 + 1.76860i −0.707107 0.707107i 2.98083 0.338641i 1.58371 1.57857i
239.17 0.707107 0.707107i −1.46346 0.926431i 1.00000i −0.989929 + 2.00500i −1.68991 + 0.379740i −1.27923 + 1.27923i −0.707107 0.707107i 1.28345 + 2.71160i 0.717766 + 2.11774i
239.18 0.707107 0.707107i −1.46346 + 0.926431i 1.00000i −2.00500 + 0.989929i −0.379740 + 1.68991i 1.27923 1.27923i −0.707107 0.707107i 1.28345 2.71160i −0.717766 + 2.11774i
239.19 0.707107 0.707107i −0.452930 1.67178i 1.00000i −0.934355 2.03150i −1.50240 0.861859i −3.48345 + 3.48345i −0.707107 0.707107i −2.58971 + 1.51440i −2.09717 0.775797i
239.20 0.707107 0.707107i −0.452930 + 1.67178i 1.00000i 2.03150 + 0.934355i 0.861859 + 1.50240i 3.48345 3.48345i −0.707107 0.707107i −2.58971 1.51440i 2.09717 0.775797i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
13.d odd 4 1 inner
15.d odd 2 1 inner
39.f even 4 1 inner
65.g odd 4 1 inner
195.n even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.n.a 56
3.b odd 2 1 inner 390.2.n.a 56
5.b even 2 1 inner 390.2.n.a 56
13.d odd 4 1 inner 390.2.n.a 56
15.d odd 2 1 inner 390.2.n.a 56
39.f even 4 1 inner 390.2.n.a 56
65.g odd 4 1 inner 390.2.n.a 56
195.n even 4 1 inner 390.2.n.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.n.a 56 1.a even 1 1 trivial
390.2.n.a 56 3.b odd 2 1 inner
390.2.n.a 56 5.b even 2 1 inner
390.2.n.a 56 13.d odd 4 1 inner
390.2.n.a 56 15.d odd 2 1 inner
390.2.n.a 56 39.f even 4 1 inner
390.2.n.a 56 65.g odd 4 1 inner
390.2.n.a 56 195.n even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(390, [\chi])$$.