# Properties

 Label 2760.2.p.a Level $2760$ Weight $2$ Character orbit 2760.p Analytic conductor $22.039$ Analytic rank $0$ Dimension $48$ 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(1241,2760)]

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, 1, 0, 1]))

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

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

chi := DirichletCharacter("2760.1241");

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.p (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: $$48$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 48 q^{5} - 2 q^{9}+O(q^{10})$$ 48 * q - 48 * q^5 - 2 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 48 q^{5} - 2 q^{9} + 4 q^{11} - 4 q^{21} - 8 q^{23} + 48 q^{25} - 4 q^{31} + 8 q^{33} + 12 q^{39} + 2 q^{45} - 68 q^{49} + 34 q^{51} - 4 q^{55} + 24 q^{57} - 20 q^{63} - 2 q^{69} + 8 q^{73} - 22 q^{81} + 16 q^{83} + 48 q^{87} - 8 q^{89} - 24 q^{93} + 20 q^{99}+O(q^{100})$$ 48 * q - 48 * q^5 - 2 * q^9 + 4 * q^11 - 4 * q^21 - 8 * q^23 + 48 * q^25 - 4 * q^31 + 8 * q^33 + 12 * q^39 + 2 * q^45 - 68 * q^49 + 34 * q^51 - 4 * q^55 + 24 * q^57 - 20 * q^63 - 2 * q^69 + 8 * q^73 - 22 * q^81 + 16 * q^83 + 48 * q^87 - 8 * q^89 - 24 * q^93 + 20 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1241.1 0 −1.73131 0.0505586i 0 −1.00000 0 2.89850i 0 2.99489 + 0.175066i 0
1241.2 0 −1.73131 + 0.0505586i 0 −1.00000 0 2.89850i 0 2.99489 0.175066i 0
1241.3 0 −1.72135 0.192263i 0 −1.00000 0 2.41479i 0 2.92607 + 0.661902i 0
1241.4 0 −1.72135 + 0.192263i 0 −1.00000 0 2.41479i 0 2.92607 0.661902i 0
1241.5 0 −1.70911 0.280952i 0 −1.00000 0 2.69510i 0 2.84213 + 0.960358i 0
1241.6 0 −1.70911 + 0.280952i 0 −1.00000 0 2.69510i 0 2.84213 0.960358i 0
1241.7 0 −1.37850 1.04868i 0 −1.00000 0 4.66363i 0 0.800541 + 2.89122i 0
1241.8 0 −1.37850 + 1.04868i 0 −1.00000 0 4.66363i 0 0.800541 2.89122i 0
1241.9 0 −1.33966 1.09786i 0 −1.00000 0 3.79178i 0 0.589396 + 2.94153i 0
1241.10 0 −1.33966 + 1.09786i 0 −1.00000 0 3.79178i 0 0.589396 2.94153i 0
1241.11 0 −1.27593 1.17132i 0 −1.00000 0 4.49556i 0 0.256020 + 2.98906i 0
1241.12 0 −1.27593 + 1.17132i 0 −1.00000 0 4.49556i 0 0.256020 2.98906i 0
1241.13 0 −1.27150 1.17613i 0 −1.00000 0 1.32290i 0 0.233436 + 2.99090i 0
1241.14 0 −1.27150 + 1.17613i 0 −1.00000 0 1.32290i 0 0.233436 2.99090i 0
1241.15 0 −0.891707 1.48488i 0 −1.00000 0 0.626160i 0 −1.40972 + 2.64815i 0
1241.16 0 −0.891707 + 1.48488i 0 −1.00000 0 0.626160i 0 −1.40972 2.64815i 0
1241.17 0 −0.779214 1.54688i 0 −1.00000 0 1.13322i 0 −1.78565 + 2.41070i 0
1241.18 0 −0.779214 + 1.54688i 0 −1.00000 0 1.13322i 0 −1.78565 2.41070i 0
1241.19 0 −0.491879 1.66074i 0 −1.00000 0 1.55251i 0 −2.51611 + 1.63377i 0
1241.20 0 −0.491879 + 1.66074i 0 −1.00000 0 1.55251i 0 −2.51611 1.63377i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1241.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c 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.p.a 48
3.b odd 2 1 2760.2.p.b yes 48
23.b odd 2 1 2760.2.p.b yes 48
69.c even 2 1 inner 2760.2.p.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.p.a 48 1.a even 1 1 trivial
2760.2.p.a 48 69.c even 2 1 inner
2760.2.p.b yes 48 3.b odd 2 1
2760.2.p.b yes 48 23.b odd 2 1