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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{7} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{7} - 60 q^{9} + 24 q^{17} - 4 q^{23} - 44 q^{25} + 40 q^{33} + 24 q^{39} - 32 q^{41} + 20 q^{47} + 108 q^{49} - 8 q^{55} - 20 q^{63} + 12 q^{65} + 8 q^{71} - 88 q^{73} - 40 q^{79} + 116 q^{81} - 48 q^{87} - 64 q^{89} + 44 q^{95} + 116 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:

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