# Properties

 Label 1900.2.l.d Level $1900$ Weight $2$ Character orbit 1900.l Analytic conductor $15.172$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(493,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1900.493");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.l (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: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q+O(q^{10})$$ 24 * q $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{11} + 24 q^{61} + 24 q^{81}+O(q^{100})$$ 24 * q - 24 * q^11 + 24 * q^61 + 24 * q^81

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
493.1 0 −2.05446 + 2.05446i 0 0 0 2.28491 2.28491i 0 5.44162i 0
493.2 0 −2.05446 + 2.05446i 0 0 0 −2.28491 + 2.28491i 0 5.44162i 0
493.3 0 −1.05614 + 1.05614i 0 0 0 −1.45445 + 1.45445i 0 0.769150i 0
493.4 0 −1.05614 + 1.05614i 0 0 0 1.45445 1.45445i 0 0.769150i 0
493.5 0 −0.814717 + 0.814717i 0 0 0 −1.28987 + 1.28987i 0 1.67247i 0
493.6 0 −0.814717 + 0.814717i 0 0 0 1.28987 1.28987i 0 1.67247i 0
493.7 0 0.814717 0.814717i 0 0 0 −1.28987 + 1.28987i 0 1.67247i 0
493.8 0 0.814717 0.814717i 0 0 0 1.28987 1.28987i 0 1.67247i 0
493.9 0 1.05614 1.05614i 0 0 0 −1.45445 + 1.45445i 0 0.769150i 0
493.10 0 1.05614 1.05614i 0 0 0 1.45445 1.45445i 0 0.769150i 0
493.11 0 2.05446 2.05446i 0 0 0 2.28491 2.28491i 0 5.44162i 0
493.12 0 2.05446 2.05446i 0 0 0 −2.28491 + 2.28491i 0 5.44162i 0
1557.1 0 −2.05446 2.05446i 0 0 0 2.28491 + 2.28491i 0 5.44162i 0
1557.2 0 −2.05446 2.05446i 0 0 0 −2.28491 2.28491i 0 5.44162i 0
1557.3 0 −1.05614 1.05614i 0 0 0 −1.45445 1.45445i 0 0.769150i 0
1557.4 0 −1.05614 1.05614i 0 0 0 1.45445 + 1.45445i 0 0.769150i 0
1557.5 0 −0.814717 0.814717i 0 0 0 −1.28987 1.28987i 0 1.67247i 0
1557.6 0 −0.814717 0.814717i 0 0 0 1.28987 + 1.28987i 0 1.67247i 0
1557.7 0 0.814717 + 0.814717i 0 0 0 −1.28987 1.28987i 0 1.67247i 0
1557.8 0 0.814717 + 0.814717i 0 0 0 1.28987 + 1.28987i 0 1.67247i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.12 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.b even 2 1 inner
5.c odd 4 2 inner
19.b odd 2 1 inner
95.d odd 2 1 inner
95.g even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.l.d 24
5.b even 2 1 inner 1900.2.l.d 24
5.c odd 4 2 inner 1900.2.l.d 24
19.b odd 2 1 inner 1900.2.l.d 24
95.d odd 2 1 inner 1900.2.l.d 24
95.g even 4 2 inner 1900.2.l.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.l.d 24 1.a even 1 1 trivial
1900.2.l.d 24 5.b even 2 1 inner
1900.2.l.d 24 5.c odd 4 2 inner
1900.2.l.d 24 19.b odd 2 1 inner
1900.2.l.d 24 95.d odd 2 1 inner
1900.2.l.d 24 95.g even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 78T_{3}^{8} + 489T_{3}^{4} + 625$$ acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$.