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Properties

Label 9360.2.k.c
Level $9360$
Weight $2$
Character orbit 9360.k
Analytic conductor $74.740$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9360,2,Mod(4031,9360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("9360.4031");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9360.k (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: \(74.7399762919\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
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
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{13} - 32 q^{25} + 56 q^{37} - 72 q^{49} + 24 q^{61} - 16 q^{73} - 8 q^{85} - 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4031.1 0 0 0 − 1.00000i 0 − 0.284645i 0 0 0
4031.2 0 0 0 − 1.00000i 0 0.284645i 0 0 0
4031.3 0 0 0 1.00000i 0 0.284645i 0 0 0
4031.4 0 0 0 1.00000i 0 − 0.284645i 0 0 0
4031.5 0 0 0 − 1.00000i 0 5.18018i 0 0 0
4031.6 0 0 0 − 1.00000i 0 − 5.18018i 0 0 0
4031.7 0 0 0 1.00000i 0 − 5.18018i 0 0 0
4031.8 0 0 0 1.00000i 0 5.18018i 0 0 0
4031.9 0 0 0 − 1.00000i 0 2.68441i 0 0 0
4031.10 0 0 0 − 1.00000i 0 − 2.68441i 0 0 0
4031.11 0 0 0 1.00000i 0 − 2.68441i 0 0 0
4031.12 0 0 0 1.00000i 0 2.68441i 0 0 0
4031.13 0 0 0 − 1.00000i 0 − 0.770560i 0 0 0
4031.14 0 0 0 − 1.00000i 0 0.770560i 0 0 0
4031.15 0 0 0 1.00000i 0 0.770560i 0 0 0
4031.16 0 0 0 1.00000i 0 − 0.770560i 0 0 0
4031.17 0 0 0 − 1.00000i 0 0.634820i 0 0 0
4031.18 0 0 0 − 1.00000i 0 − 0.634820i 0 0 0
4031.19 0 0 0 1.00000i 0 − 0.634820i 0 0 0
4031.20 0 0 0 1.00000i 0 0.634820i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4031.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9360.2.k.c ✓ 32
3.b odd 2 1 inner 9360.2.k.c ✓ 32
4.b odd 2 1 inner 9360.2.k.c ✓ 32
12.b even 2 1 inner 9360.2.k.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9360.2.k.c ✓ 32 1.a even 1 1 trivial
9360.2.k.c ✓ 32 3.b odd 2 1 inner
9360.2.k.c ✓ 32 4.b odd 2 1 inner
9360.2.k.c ✓ 32 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 74 T_{7}^{14} + 2033 T_{7}^{12} + 25912 T_{7}^{10} + 157864 T_{7}^{8} + 416880 T_{7}^{6} + \cdots + 5184 \) acting on \(S_{2}^{\mathrm{new}}(9360, [\chi])\). Copy content Toggle raw display