# Properties

 Label 3040.2.f.a Level $3040$ Weight $2$ Character orbit 3040.f Analytic conductor $24.275$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(1521,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3040.1521");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3040.f (of order $$2$$, degree $$1$$, not 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: $$24.2745222145$$ Analytic rank: $$0$$ Dimension: $$28$$ Twist minimal: no (minimal twist has level 760) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 4 q^{7} - 12 q^{9}+O(q^{10})$$ 28 * q - 4 * q^7 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 4 q^{7} - 12 q^{9} - 24 q^{17} - 4 q^{23} - 28 q^{25} - 40 q^{33} + 24 q^{39} + 32 q^{41} + 20 q^{47} - 36 q^{49} - 8 q^{55} - 20 q^{63} - 12 q^{65} + 8 q^{71} + 88 q^{73} - 40 q^{79} - 28 q^{81} - 48 q^{87} + 48 q^{89} - 28 q^{95} - 116 q^{97}+O(q^{100})$$ 28 * q - 4 * q^7 - 12 * q^9 - 24 * q^17 - 4 * q^23 - 28 * q^25 - 40 * q^33 + 24 * q^39 + 32 * q^41 + 20 * q^47 - 36 * q^49 - 8 * q^55 - 20 * q^63 - 12 * q^65 + 8 * q^71 + 88 * q^73 - 40 * q^79 - 28 * q^81 - 48 * q^87 + 48 * q^89 - 28 * q^95 - 116 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1521.1 0 3.19922i 0 1.00000i 0 0.255739 0 −7.23503 0
1521.2 0 2.72205i 0 1.00000i 0 1.81393 0 −4.40958 0
1521.3 0 2.50151i 0 1.00000i 0 −0.698520 0 −3.25757 0
1521.4 0 2.48816i 0 1.00000i 0 −3.58757 0 −3.19093 0
1521.5 0 2.02371i 0 1.00000i 0 −1.28404 0 −1.09538 0
1521.6 0 1.92849i 0 1.00000i 0 −0.210397 0 −0.719076 0
1521.7 0 1.74129i 0 1.00000i 0 4.26759 0 −0.0320903 0
1521.8 0 1.71180i 0 1.00000i 0 2.02692 0 0.0697451 0
1521.9 0 1.53286i 0 1.00000i 0 0.194868 0 0.650334 0
1521.10 0 0.823852i 0 1.00000i 0 0.523009 0 2.32127 0
1521.11 0 0.782983i 0 1.00000i 0 3.28528 0 2.38694 0
1521.12 0 0.496172i 0 1.00000i 0 −1.35432 0 2.75381 0
1521.13 0 0.485865i 0 1.00000i 0 −3.31744 0 2.76394 0
1521.14 0 0.0798397i 0 1.00000i 0 −3.91506 0 2.99363 0
1521.15 0 0.0798397i 0 1.00000i 0 −3.91506 0 2.99363 0
1521.16 0 0.485865i 0 1.00000i 0 −3.31744 0 2.76394 0
1521.17 0 0.496172i 0 1.00000i 0 −1.35432 0 2.75381 0
1521.18 0 0.782983i 0 1.00000i 0 3.28528 0 2.38694 0
1521.19 0 0.823852i 0 1.00000i 0 0.523009 0 2.32127 0
1521.20 0 1.53286i 0 1.00000i 0 0.194868 0 0.650334 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1521.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.2.f.a 28
4.b odd 2 1 760.2.f.a 28
8.b even 2 1 inner 3040.2.f.a 28
8.d odd 2 1 760.2.f.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.f.a 28 4.b odd 2 1
760.2.f.a 28 8.d odd 2 1
3040.2.f.a 28 1.a even 1 1 trivial
3040.2.f.a 28 8.b even 2 1 inner