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Properties

Label 2940.2.f.a
Level $2940$
Weight $2$
Character orbit 2940.f
Analytic conductor $23.476$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2940,2,Mod(1469,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2940.1469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.f (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: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 420)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{15} - 12 q^{25} - 48 q^{39} + 20 q^{51} + 56 q^{79} + 40 q^{81} - 4 q^{85} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1469.1 0 −1.69264 − 0.367385i 0 −2.03939 − 0.916997i 0 0 0 2.73006 + 1.24370i 0
1469.2 0 −1.69264 − 0.367385i 0 2.03939 − 0.916997i 0 0 0 2.73006 + 1.24370i 0
1469.3 0 −1.69264 + 0.367385i 0 −2.03939 + 0.916997i 0 0 0 2.73006 − 1.24370i 0
1469.4 0 −1.69264 + 0.367385i 0 2.03939 + 0.916997i 0 0 0 2.73006 − 1.24370i 0
1469.5 0 −1.51070 − 0.847215i 0 −1.16828 + 1.90660i 0 0 0 1.56445 + 2.55978i 0
1469.6 0 −1.51070 − 0.847215i 0 1.16828 + 1.90660i 0 0 0 1.56445 + 2.55978i 0
1469.7 0 −1.51070 + 0.847215i 0 −1.16828 − 1.90660i 0 0 0 1.56445 − 2.55978i 0
1469.8 0 −1.51070 + 0.847215i 0 1.16828 − 1.90660i 0 0 0 1.56445 − 2.55978i 0
1469.9 0 −0.917271 − 1.46922i 0 −0.536879 − 2.17066i 0 0 0 −1.31723 + 2.69535i 0
1469.10 0 −0.917271 − 1.46922i 0 0.536879 − 2.17066i 0 0 0 −1.31723 + 2.69535i 0
1469.11 0 −0.917271 + 1.46922i 0 −0.536879 + 2.17066i 0 0 0 −1.31723 − 2.69535i 0
1469.12 0 −0.917271 + 1.46922i 0 0.536879 + 2.17066i 0 0 0 −1.31723 − 2.69535i 0
1469.13 0 −0.106586 − 1.72877i 0 −1.85412 + 1.24989i 0 0 0 −2.97728 + 0.368524i 0
1469.14 0 −0.106586 − 1.72877i 0 1.85412 + 1.24989i 0 0 0 −2.97728 + 0.368524i 0
1469.15 0 −0.106586 + 1.72877i 0 −1.85412 − 1.24989i 0 0 0 −2.97728 − 0.368524i 0
1469.16 0 −0.106586 + 1.72877i 0 1.85412 − 1.24989i 0 0 0 −2.97728 − 0.368524i 0
1469.17 0 0.106586 − 1.72877i 0 −1.85412 + 1.24989i 0 0 0 −2.97728 − 0.368524i 0
1469.18 0 0.106586 − 1.72877i 0 1.85412 + 1.24989i 0 0 0 −2.97728 − 0.368524i 0
1469.19 0 0.106586 + 1.72877i 0 −1.85412 − 1.24989i 0 0 0 −2.97728 + 0.368524i 0
1469.20 0 0.106586 + 1.72877i 0 1.85412 − 1.24989i 0 0 0 −2.97728 + 0.368524i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1469.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.2.f.a 32
3.b odd 2 1 inner 2940.2.f.a 32
5.b even 2 1 inner 2940.2.f.a 32
7.b odd 2 1 inner 2940.2.f.a 32
7.c even 3 1 420.2.bn.a ✓ 32
7.d odd 6 1 420.2.bn.a ✓ 32
15.d odd 2 1 inner 2940.2.f.a 32
21.c even 2 1 inner 2940.2.f.a 32
21.g even 6 1 420.2.bn.a ✓ 32
21.h odd 6 1 420.2.bn.a ✓ 32
35.c odd 2 1 inner 2940.2.f.a 32
35.i odd 6 1 420.2.bn.a ✓ 32
35.j even 6 1 420.2.bn.a ✓ 32
35.k even 12 2 2100.2.bi.n 32
35.l odd 12 2 2100.2.bi.n 32
105.g even 2 1 inner 2940.2.f.a 32
105.o odd 6 1 420.2.bn.a ✓ 32
105.p even 6 1 420.2.bn.a ✓ 32
105.w odd 12 2 2100.2.bi.n 32
105.x even 12 2 2100.2.bi.n 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bn.a ✓ 32 7.c even 3 1
420.2.bn.a ✓ 32 7.d odd 6 1
420.2.bn.a ✓ 32 21.g even 6 1
420.2.bn.a ✓ 32 21.h odd 6 1
420.2.bn.a ✓ 32 35.i odd 6 1
420.2.bn.a ✓ 32 35.j even 6 1
420.2.bn.a ✓ 32 105.o odd 6 1
420.2.bn.a ✓ 32 105.p even 6 1
2100.2.bi.n 32 35.k even 12 2
2100.2.bi.n 32 35.l odd 12 2
2100.2.bi.n 32 105.w odd 12 2
2100.2.bi.n 32 105.x even 12 2
2940.2.f.a 32 1.a even 1 1 trivial
2940.2.f.a 32 3.b odd 2 1 inner
2940.2.f.a 32 5.b even 2 1 inner
2940.2.f.a 32 7.b odd 2 1 inner
2940.2.f.a 32 15.d odd 2 1 inner
2940.2.f.a 32 21.c even 2 1 inner
2940.2.f.a 32 35.c odd 2 1 inner
2940.2.f.a 32 105.g even 2 1 inner