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Properties

Label 3040.2.f.a
Level $3040$
Weight $2$
Character orbit 3040.f
Analytic conductor $24.275$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3040,2,Mod(1521,3040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

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

Copy content comment:embeddings in the coefficient field
 
Copy content 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:

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