# Properties

 Label 1840.2.m.g Level $1840$ Weight $2$ Character orbit 1840.m Analytic conductor $14.692$ Analytic rank $0$ Dimension $40$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1839,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("1840.1839");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.m (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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 80 q^{9}+O(q^{10})$$ 40 * q + 80 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100})$$ 40 * q + 80 * q^9 - 24 * q^25 + 24 * q^41 - 16 * q^49 + 80 * q^69 + 40 * q^81 - 8 * q^85

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1839.1 0 −3.28652 0 1.54120 + 1.62009i 0 2.73174i 0 7.80121 0
1839.2 0 −3.28652 0 −1.54120 + 1.62009i 0 2.73174i 0 7.80121 0
1839.3 0 −3.28652 0 −1.54120 1.62009i 0 2.73174i 0 7.80121 0
1839.4 0 −3.28652 0 1.54120 1.62009i 0 2.73174i 0 7.80121 0
1839.5 0 −2.49894 0 −2.19676 + 0.417445i 0 3.04922i 0 3.24468 0
1839.6 0 −2.49894 0 2.19676 + 0.417445i 0 3.04922i 0 3.24468 0
1839.7 0 −2.49894 0 2.19676 0.417445i 0 3.04922i 0 3.24468 0
1839.8 0 −2.49894 0 −2.19676 0.417445i 0 3.04922i 0 3.24468 0
1839.9 0 −2.20329 0 1.71768 1.43163i 0 0.642617i 0 1.85447 0
1839.10 0 −2.20329 0 −1.71768 + 1.43163i 0 0.642617i 0 1.85447 0
1839.11 0 −2.20329 0 −1.71768 1.43163i 0 0.642617i 0 1.85447 0
1839.12 0 −2.20329 0 1.71768 + 1.43163i 0 0.642617i 0 1.85447 0
1839.13 0 −1.49850 0 0.689327 + 2.12716i 0 4.18735i 0 −0.754490 0
1839.14 0 −1.49850 0 −0.689327 2.12716i 0 4.18735i 0 −0.754490 0
1839.15 0 −1.49850 0 −0.689327 + 2.12716i 0 4.18735i 0 −0.754490 0
1839.16 0 −1.49850 0 0.689327 2.12716i 0 4.18735i 0 −0.754490 0
1839.17 0 −0.924187 0 0.611037 2.15096i 0 1.51428i 0 −2.14588 0
1839.18 0 −0.924187 0 −0.611037 2.15096i 0 1.51428i 0 −2.14588 0
1839.19 0 −0.924187 0 −0.611037 + 2.15096i 0 1.51428i 0 −2.14588 0
1839.20 0 −0.924187 0 0.611037 + 2.15096i 0 1.51428i 0 −2.14588 0
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1839.40 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
20.d odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
115.c odd 2 1 inner
460.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.m.g 40
4.b odd 2 1 inner 1840.2.m.g 40
5.b even 2 1 inner 1840.2.m.g 40
20.d odd 2 1 inner 1840.2.m.g 40
23.b odd 2 1 inner 1840.2.m.g 40
92.b even 2 1 inner 1840.2.m.g 40
115.c odd 2 1 inner 1840.2.m.g 40
460.g even 2 1 inner 1840.2.m.g 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.g 40 1.a even 1 1 trivial
1840.2.m.g 40 4.b odd 2 1 inner
1840.2.m.g 40 5.b even 2 1 inner
1840.2.m.g 40 20.d odd 2 1 inner
1840.2.m.g 40 23.b odd 2 1 inner
1840.2.m.g 40 92.b even 2 1 inner
1840.2.m.g 40 115.c odd 2 1 inner
1840.2.m.g 40 460.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

 $$T_{3}^{10} - 25T_{3}^{8} + 220T_{3}^{6} - 835T_{3}^{4} + 1303T_{3}^{2} - 628$$ T3^10 - 25*T3^8 + 220*T3^6 - 835*T3^4 + 1303*T3^2 - 628 $$T_{7}^{10} + 37T_{7}^{8} + 457T_{7}^{6} + 2232T_{7}^{4} + 3636T_{7}^{2} + 1152$$ T7^10 + 37*T7^8 + 457*T7^6 + 2232*T7^4 + 3636*T7^2 + 1152