# Properties

 Label 3040.2.f.b Level $3040$ Weight $2$ Character orbit 3040.f Analytic conductor $24.275$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

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: $$44$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 4 q^{7} - 60 q^{9}+O(q^{10})$$ 44 * q - 4 * q^7 - 60 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 4 q^{7} - 60 q^{9} + 24 q^{17} - 4 q^{23} - 44 q^{25} + 40 q^{33} + 24 q^{39} - 32 q^{41} + 20 q^{47} + 108 q^{49} - 8 q^{55} - 20 q^{63} + 12 q^{65} + 8 q^{71} - 88 q^{73} - 40 q^{79} + 116 q^{81} - 48 q^{87} - 64 q^{89} + 44 q^{95} + 116 q^{97}+O(q^{100})$$ 44 * q - 4 * q^7 - 60 * q^9 + 24 * q^17 - 4 * q^23 - 44 * q^25 + 40 * q^33 + 24 * q^39 - 32 * q^41 + 20 * q^47 + 108 * q^49 - 8 * q^55 - 20 * q^63 + 12 * q^65 + 8 * q^71 - 88 * q^73 - 40 * q^79 + 116 * q^81 - 48 * q^87 - 64 * q^89 + 44 * 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 3.43603i 0 1.00000i 0 −3.17203 0 −8.80634 0
1521.2 0 3.28479i 0 1.00000i 0 −3.22488 0 −7.78982 0
1521.3 0 3.07860i 0 1.00000i 0 1.19412 0 −6.47775 0
1521.4 0 3.06281i 0 1.00000i 0 2.81229 0 −6.38080 0
1521.5 0 2.94384i 0 1.00000i 0 3.06731 0 −5.66617 0
1521.6 0 2.81164i 0 1.00000i 0 3.31936 0 −4.90534 0
1521.7 0 2.59701i 0 1.00000i 0 4.55480 0 −3.74448 0
1521.8 0 2.28177i 0 1.00000i 0 −2.16697 0 −2.20646 0
1521.9 0 2.26423i 0 1.00000i 0 1.28214 0 −2.12673 0
1521.10 0 1.93855i 0 1.00000i 0 −4.42246 0 −0.757964 0
1521.11 0 1.80571i 0 1.00000i 0 −4.63974 0 −0.260578 0
1521.12 0 1.80124i 0 1.00000i 0 −4.97691 0 −0.244448 0
1521.13 0 1.70163i 0 1.00000i 0 −1.91794 0 0.104446 0
1521.14 0 1.40980i 0 1.00000i 0 1.07761 0 1.01245 0
1521.15 0 1.22456i 0 1.00000i 0 4.59099 0 1.50045 0
1521.16 0 1.16289i 0 1.00000i 0 −0.494296 0 1.64768 0
1521.17 0 1.00586i 0 1.00000i 0 4.51738 0 1.98825 0
1521.18 0 0.915439i 0 1.00000i 0 2.66013 0 2.16197 0
1521.19 0 0.777674i 0 1.00000i 0 −2.42638 0 2.39522 0
1521.20 0 0.530865i 0 1.00000i 0 −1.59876 0 2.71818 0
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1521.44 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.b 44
4.b odd 2 1 760.2.f.b 44
8.b even 2 1 inner 3040.2.f.b 44
8.d odd 2 1 760.2.f.b 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.f.b 44 4.b odd 2 1
760.2.f.b 44 8.d odd 2 1
3040.2.f.b 44 1.a even 1 1 trivial
3040.2.f.b 44 8.b even 2 1 inner