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Properties

Label 1008.2.cx.k
Level $1008$
Weight $2$
Character orbit 1008.cx
Analytic conductor $8.049$
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] = [1008,2,Mod(223,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cx (of order \(6\), degree \(2\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{9} - 14 q^{21} + 16 q^{25} + 8 q^{37} + 2 q^{49} - 96 q^{53} + 20 q^{57} - 24 q^{65} - 18 q^{77} - 40 q^{81} - 12 q^{85} - 56 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1 0 −1.69540 − 0.354449i 0 −2.63271 + 1.52000i 0 −0.770727 + 2.53100i 0 2.74873 + 1.20186i 0
223.2 0 −1.69540 − 0.354449i 0 2.63271 − 1.52000i 0 −1.80655 + 1.93297i 0 2.74873 + 1.20186i 0
223.3 0 −1.37099 + 1.05848i 0 −0.973047 + 0.561789i 0 2.64036 + 0.168790i 0 0.759236 − 2.90234i 0
223.4 0 −1.37099 + 1.05848i 0 0.973047 − 0.561789i 0 −1.46636 − 2.20223i 0 0.759236 − 2.90234i 0
223.5 0 −1.26474 − 1.18340i 0 −0.830715 + 0.479614i 0 −0.711914 − 2.54817i 0 0.199131 + 2.99338i 0
223.6 0 −1.26474 − 1.18340i 0 0.830715 − 0.479614i 0 2.56274 − 0.657550i 0 0.199131 + 2.99338i 0
223.7 0 −0.382689 + 1.68925i 0 −3.07115 + 1.77313i 0 −2.63565 + 0.230944i 0 −2.70710 − 1.29291i 0
223.8 0 −0.382689 + 1.68925i 0 3.07115 − 1.77313i 0 1.11782 + 2.39801i 0 −2.70710 − 1.29291i 0
223.9 0 0.382689 − 1.68925i 0 −3.07115 + 1.77313i 0 2.63565 − 0.230944i 0 −2.70710 − 1.29291i 0
223.10 0 0.382689 − 1.68925i 0 3.07115 − 1.77313i 0 −1.11782 − 2.39801i 0 −2.70710 − 1.29291i 0
223.11 0 1.26474 + 1.18340i 0 −0.830715 + 0.479614i 0 0.711914 + 2.54817i 0 0.199131 + 2.99338i 0
223.12 0 1.26474 + 1.18340i 0 0.830715 − 0.479614i 0 −2.56274 + 0.657550i 0 0.199131 + 2.99338i 0
223.13 0 1.37099 − 1.05848i 0 −0.973047 + 0.561789i 0 −2.64036 − 0.168790i 0 0.759236 − 2.90234i 0
223.14 0 1.37099 − 1.05848i 0 0.973047 − 0.561789i 0 1.46636 + 2.20223i 0 0.759236 − 2.90234i 0
223.15 0 1.69540 + 0.354449i 0 −2.63271 + 1.52000i 0 0.770727 − 2.53100i 0 2.74873 + 1.20186i 0
223.16 0 1.69540 + 0.354449i 0 2.63271 − 1.52000i 0 1.80655 − 1.93297i 0 2.74873 + 1.20186i 0
895.1 0 −1.69540 + 0.354449i 0 −2.63271 − 1.52000i 0 −0.770727 − 2.53100i 0 2.74873 − 1.20186i 0
895.2 0 −1.69540 + 0.354449i 0 2.63271 + 1.52000i 0 −1.80655 − 1.93297i 0 2.74873 − 1.20186i 0
895.3 0 −1.37099 − 1.05848i 0 −0.973047 − 0.561789i 0 2.64036 − 0.168790i 0 0.759236 + 2.90234i 0
895.4 0 −1.37099 − 1.05848i 0 0.973047 + 0.561789i 0 −1.46636 + 2.20223i 0 0.759236 + 2.90234i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
28.d even 2 1 inner
36.f odd 6 1 inner
63.l odd 6 1 inner
252.bi even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.cx.k ✓ 32
3.b odd 2 1 3024.2.cx.k 32
4.b odd 2 1 inner 1008.2.cx.k ✓ 32
7.b odd 2 1 inner 1008.2.cx.k ✓ 32
9.c even 3 1 inner 1008.2.cx.k ✓ 32
9.d odd 6 1 3024.2.cx.k 32
12.b even 2 1 3024.2.cx.k 32
21.c even 2 1 3024.2.cx.k 32
28.d even 2 1 inner 1008.2.cx.k ✓ 32
36.f odd 6 1 inner 1008.2.cx.k ✓ 32
36.h even 6 1 3024.2.cx.k 32
63.l odd 6 1 inner 1008.2.cx.k ✓ 32
63.o even 6 1 3024.2.cx.k 32
84.h odd 2 1 3024.2.cx.k 32
252.s odd 6 1 3024.2.cx.k 32
252.bi even 6 1 inner 1008.2.cx.k ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.cx.k ✓ 32 1.a even 1 1 trivial
1008.2.cx.k ✓ 32 4.b odd 2 1 inner
1008.2.cx.k ✓ 32 7.b odd 2 1 inner
1008.2.cx.k ✓ 32 9.c even 3 1 inner
1008.2.cx.k ✓ 32 28.d even 2 1 inner
1008.2.cx.k ✓ 32 36.f odd 6 1 inner
1008.2.cx.k ✓ 32 63.l odd 6 1 inner
1008.2.cx.k ✓ 32 252.bi even 6 1 inner
3024.2.cx.k 32 3.b odd 2 1
3024.2.cx.k 32 9.d odd 6 1
3024.2.cx.k 32 12.b even 2 1
3024.2.cx.k 32 21.c even 2 1
3024.2.cx.k 32 36.h even 6 1
3024.2.cx.k 32 63.o even 6 1
3024.2.cx.k 32 84.h odd 2 1
3024.2.cx.k 32 252.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{16} - 24 T_{5}^{14} + 411 T_{5}^{12} - 3402 T_{5}^{10} + 20394 T_{5}^{8} - 39555 T_{5}^{6} + \cdots + 18225 \) Copy content Toggle raw display
\( T_{11}^{16} - 45 T_{11}^{14} + 1716 T_{11}^{12} - 12681 T_{11}^{10} + 67806 T_{11}^{8} - 176958 T_{11}^{6} + \cdots + 18225 \) Copy content Toggle raw display