↑

Properties

Label 2112.2.q.a
Level $2112$
Weight $2$
Character orbit 2112.q
Analytic conductor $16.864$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2112,2,Mod(175,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2112.175");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.q (of order \(4\), degree \(2\), 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: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 528)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{11} - 32 q^{23} + 32 q^{37} - 96 q^{49} - 32 q^{53} - 32 q^{55} - 32 q^{59} + 16 q^{67} - 64 q^{71} + 32 q^{75} + 16 q^{77} - 96 q^{81} + 16 q^{91} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1 0 −0.707107 + 0.707107i 0 −2.23973 + 2.23973i 0 − 2.85710i 0 − 1.00000i 0
175.2 0 −0.707107 + 0.707107i 0 −0.682950 + 0.682950i 0 − 4.75800i 0 − 1.00000i 0
175.3 0 −0.707107 + 0.707107i 0 2.65957 − 2.65957i 0 − 1.21761i 0 − 1.00000i 0
175.4 0 −0.707107 + 0.707107i 0 2.11423 − 2.11423i 0 − 4.78222i 0 − 1.00000i 0
175.5 0 −0.707107 + 0.707107i 0 2.39017 − 2.39017i 0 − 3.58593i 0 − 1.00000i 0
175.6 0 −0.707107 + 0.707107i 0 0.355264 − 0.355264i 0 − 2.95587i 0 − 1.00000i 0
175.7 0 −0.707107 + 0.707107i 0 0.671872 − 0.671872i 0 − 2.65911i 0 − 1.00000i 0
175.8 0 −0.707107 + 0.707107i 0 −1.28258 + 1.28258i 0 0.0390261i 0 − 1.00000i 0
175.9 0 −0.707107 + 0.707107i 0 1.18924 − 1.18924i 0 − 0.0841373i 0 − 1.00000i 0
175.10 0 −0.707107 + 0.707107i 0 −0.490377 + 0.490377i 0 1.20432i 0 − 1.00000i 0
175.11 0 −0.707107 + 0.707107i 0 1.18924 − 1.18924i 0 0.0841373i 0 − 1.00000i 0
175.12 0 −0.707107 + 0.707107i 0 −0.490377 + 0.490377i 0 − 1.20432i 0 − 1.00000i 0
175.13 0 −0.707107 + 0.707107i 0 −1.28258 + 1.28258i 0 − 0.0390261i 0 − 1.00000i 0
175.14 0 −0.707107 + 0.707107i 0 −1.55533 + 1.55533i 0 3.21056i 0 − 1.00000i 0
175.15 0 −0.707107 + 0.707107i 0 −2.23973 + 2.23973i 0 2.85710i 0 − 1.00000i 0
175.16 0 −0.707107 + 0.707107i 0 0.355264 − 0.355264i 0 2.95587i 0 − 1.00000i 0
175.17 0 −0.707107 + 0.707107i 0 0.671872 − 0.671872i 0 2.65911i 0 − 1.00000i 0
175.18 0 −0.707107 + 0.707107i 0 −3.12937 + 3.12937i 0 0.642297i 0 − 1.00000i 0
175.19 0 −0.707107 + 0.707107i 0 −0.682950 + 0.682950i 0 4.75800i 0 − 1.00000i 0
175.20 0 −0.707107 + 0.707107i 0 −3.12937 + 3.12937i 0 − 0.642297i 0 − 1.00000i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
16.f odd 4 1 inner
176.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2112.2.q.a 96
4.b odd 2 1 528.2.q.a ✓ 96
11.b odd 2 1 inner 2112.2.q.a 96
16.e even 4 1 528.2.q.a ✓ 96
16.f odd 4 1 inner 2112.2.q.a 96
44.c even 2 1 528.2.q.a ✓ 96
176.i even 4 1 inner 2112.2.q.a 96
176.l odd 4 1 528.2.q.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.q.a ✓ 96 4.b odd 2 1
528.2.q.a ✓ 96 16.e even 4 1
528.2.q.a ✓ 96 44.c even 2 1
528.2.q.a ✓ 96 176.l odd 4 1
2112.2.q.a 96 1.a even 1 1 trivial
2112.2.q.a 96 11.b odd 2 1 inner
2112.2.q.a 96 16.f odd 4 1 inner
2112.2.q.a 96 176.i even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(2112, [\chi])\).