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Properties

Label 2475.2.d.d
Level $2475$
Weight $2$
Character orbit 2475.d
Analytic conductor $19.763$
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] = [2475,2,Mod(2474,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2475.2474");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.d (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: \(19.7629745003\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} + 16 q^{16} + 48 q^{31} + 64 q^{34} - 16 q^{49} + 48 q^{64} - 64 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2474.1 − 2.52236i 0 −4.36231 0 0 −2.99045 5.95859i 0 0
2474.2 2.52236i 0 −4.36231 0 0 −2.99045 − 5.95859i 0 0
2474.3 − 1.95158i 0 −1.80865 0 0 −0.919382 − 0.373440i 0 0
2474.4 1.95158i 0 −1.80865 0 0 −0.919382 0.373440i 0 0
2474.5 − 1.95158i 0 −1.80865 0 0 0.919382 − 0.373440i 0 0
2474.6 1.95158i 0 −1.80865 0 0 0.919382 0.373440i 0 0
2474.7 − 2.52236i 0 −4.36231 0 0 −2.99045 5.95859i 0 0
2474.8 2.52236i 0 −4.36231 0 0 −2.99045 − 5.95859i 0 0
2474.9 − 0.308555i 0 1.90479 0 0 3.51321 − 1.20484i 0 0
2474.10 0.308555i 0 1.90479 0 0 3.51321 1.20484i 0 0
2474.11 − 1.31675i 0 0.266160 0 0 1.96706 − 2.98397i 0 0
2474.12 1.31675i 0 0.266160 0 0 1.96706 2.98397i 0 0
2474.13 − 0.308555i 0 1.90479 0 0 −3.51321 − 1.20484i 0 0
2474.14 0.308555i 0 1.90479 0 0 −3.51321 1.20484i 0 0
2474.15 − 1.31675i 0 0.266160 0 0 −1.96706 − 2.98397i 0 0
2474.16 1.31675i 0 0.266160 0 0 −1.96706 2.98397i 0 0
2474.17 − 1.31675i 0 0.266160 0 0 1.96706 − 2.98397i 0 0
2474.18 1.31675i 0 0.266160 0 0 1.96706 2.98397i 0 0
2474.19 − 0.308555i 0 1.90479 0 0 3.51321 − 1.20484i 0 0
2474.20 0.308555i 0 1.90479 0 0 3.51321 1.20484i 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2474.32
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
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.d.d 32
3.b odd 2 1 inner 2475.2.d.d 32
5.b even 2 1 inner 2475.2.d.d 32
5.c odd 4 1 2475.2.f.f ✓ 16
5.c odd 4 1 2475.2.f.g yes 16
11.b odd 2 1 inner 2475.2.d.d 32
15.d odd 2 1 inner 2475.2.d.d 32
15.e even 4 1 2475.2.f.f ✓ 16
15.e even 4 1 2475.2.f.g yes 16
33.d even 2 1 inner 2475.2.d.d 32
55.d odd 2 1 inner 2475.2.d.d 32
55.e even 4 1 2475.2.f.f ✓ 16
55.e even 4 1 2475.2.f.g yes 16
165.d even 2 1 inner 2475.2.d.d 32
165.l odd 4 1 2475.2.f.f ✓ 16
165.l odd 4 1 2475.2.f.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.d.d 32 1.a even 1 1 trivial
2475.2.d.d 32 3.b odd 2 1 inner
2475.2.d.d 32 5.b even 2 1 inner
2475.2.d.d 32 11.b odd 2 1 inner
2475.2.d.d 32 15.d odd 2 1 inner
2475.2.d.d 32 33.d even 2 1 inner
2475.2.d.d 32 55.d odd 2 1 inner
2475.2.d.d 32 165.d even 2 1 inner
2475.2.f.f ✓ 16 5.c odd 4 1
2475.2.f.f ✓ 16 15.e even 4 1
2475.2.f.f ✓ 16 55.e even 4 1
2475.2.f.f ✓ 16 165.l odd 4 1
2475.2.f.g yes 16 5.c odd 4 1
2475.2.f.g yes 16 15.e even 4 1
2475.2.f.g yes 16 55.e even 4 1
2475.2.f.g yes 16 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2475, [\chi])\):

\( T_{2}^{8} + 12T_{2}^{6} + 43T_{2}^{4} + 46T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{23}^{8} - 118T_{23}^{6} + 3525T_{23}^{4} - 27500T_{23}^{2} + 62500 \) Copy content Toggle raw display