# Properties

 Label 2760.2.k.f Level $2760$ Weight $2$ Character orbit 2760.k Analytic conductor $22.039$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(2209,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2760.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2760.k (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: $$22.0387109579$$ Analytic rank: $$0$$ Dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 2 q^{5} - 22 q^{9}+O(q^{10})$$ 22 * q - 2 * q^5 - 22 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 2 q^{5} - 22 q^{9} - 16 q^{19} + 6 q^{21} + 2 q^{25} - 22 q^{29} + 6 q^{31} + 14 q^{35} - 12 q^{39} + 6 q^{41} + 2 q^{45} - 68 q^{49} + 10 q^{51} + 12 q^{55} - 18 q^{59} + 16 q^{61} + 44 q^{65} - 22 q^{69} - 30 q^{71} + 4 q^{75} + 36 q^{79} + 22 q^{81} + 34 q^{85} - 16 q^{89} + 28 q^{91} + 20 q^{95}+O(q^{100})$$ 22 * q - 2 * q^5 - 22 * q^9 - 16 * q^19 + 6 * q^21 + 2 * q^25 - 22 * q^29 + 6 * q^31 + 14 * q^35 - 12 * q^39 + 6 * q^41 + 2 * q^45 - 68 * q^49 + 10 * q^51 + 12 * q^55 - 18 * q^59 + 16 * q^61 + 44 * q^65 - 22 * q^69 - 30 * q^71 + 4 * q^75 + 36 * q^79 + 22 * q^81 + 34 * q^85 - 16 * q^89 + 28 * q^91 + 20 * q^95

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2209.1 0 1.00000i 0 −2.22799 + 0.189879i 0 4.23369i 0 −1.00000 0
2209.2 0 1.00000i 0 −2.06524 + 0.857190i 0 3.22845i 0 −1.00000 0
2209.3 0 1.00000i 0 −1.61965 1.54167i 0 4.12396i 0 −1.00000 0
2209.4 0 1.00000i 0 −0.887264 + 2.05250i 0 1.58269i 0 −1.00000 0
2209.5 0 1.00000i 0 −0.885149 2.05341i 0 1.25349i 0 −1.00000 0
2209.6 0 1.00000i 0 −0.488181 2.18213i 0 4.05083i 0 −1.00000 0
2209.7 0 1.00000i 0 −0.299974 + 2.21586i 0 3.84917i 0 −1.00000 0
2209.8 0 1.00000i 0 1.46815 + 1.68658i 0 4.80325i 0 −1.00000 0
2209.9 0 1.00000i 0 1.78427 1.34773i 0 0.198322i 0 −1.00000 0
2209.10 0 1.00000i 0 2.08908 + 0.797325i 0 2.60191i 0 −1.00000 0
2209.11 0 1.00000i 0 2.13195 0.674383i 0 0.681214i 0 −1.00000 0
2209.12 0 1.00000i 0 −2.22799 0.189879i 0 4.23369i 0 −1.00000 0
2209.13 0 1.00000i 0 −2.06524 0.857190i 0 3.22845i 0 −1.00000 0
2209.14 0 1.00000i 0 −1.61965 + 1.54167i 0 4.12396i 0 −1.00000 0
2209.15 0 1.00000i 0 −0.887264 2.05250i 0 1.58269i 0 −1.00000 0
2209.16 0 1.00000i 0 −0.885149 + 2.05341i 0 1.25349i 0 −1.00000 0
2209.17 0 1.00000i 0 −0.488181 + 2.18213i 0 4.05083i 0 −1.00000 0
2209.18 0 1.00000i 0 −0.299974 2.21586i 0 3.84917i 0 −1.00000 0
2209.19 0 1.00000i 0 1.46815 1.68658i 0 4.80325i 0 −1.00000 0
2209.20 0 1.00000i 0 1.78427 + 1.34773i 0 0.198322i 0 −1.00000 0
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2209.22 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.2.k.f 22
5.b even 2 1 inner 2760.2.k.f 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.k.f 22 1.a even 1 1 trivial
2760.2.k.f 22 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{22} + 111 T_{7}^{20} + 5263 T_{7}^{18} + 139013 T_{7}^{16} + 2235156 T_{7}^{14} + 22436528 T_{7}^{12} + 138653632 T_{7}^{10} + 501435200 T_{7}^{8} + 965632000 T_{7}^{6} + 857090048 T_{7}^{4} + \cdots + 8667136$$ acting on $$S_{2}^{\mathrm{new}}(2760, [\chi])$$.