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Properties

Label 1152.2.r.g
Level $1152$
Weight $2$
Character orbit 1152.r
Analytic conductor $9.199$
Analytic rank $0$
Dimension $24$
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] = [1152,2,Mod(193,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.r (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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{9} - 40 q^{17} + 12 q^{25} - 28 q^{33} - 28 q^{41} - 12 q^{49} + 4 q^{57} + 16 q^{65} + 24 q^{73} + 44 q^{81} + 96 q^{89} - 36 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 0 −1.68221 − 0.412500i 0 −0.379899 − 0.219335i 0 2.14376 + 3.71310i 0 2.65969 + 1.38783i 0
193.2 0 −1.68221 − 0.412500i 0 0.379899 + 0.219335i 0 −2.14376 − 3.71310i 0 2.65969 + 1.38783i 0
193.3 0 −1.37296 + 1.05593i 0 −3.35903 − 1.93934i 0 1.14978 + 1.99147i 0 0.770012 − 2.89950i 0
193.4 0 −1.37296 + 1.05593i 0 3.35903 + 1.93934i 0 −1.14978 − 1.99147i 0 0.770012 − 2.89950i 0
193.5 0 −0.187483 − 1.72187i 0 −1.43964 − 0.831176i 0 0.286876 + 0.496884i 0 −2.92970 + 0.645646i 0
193.6 0 −0.187483 − 1.72187i 0 1.43964 + 0.831176i 0 −0.286876 − 0.496884i 0 −2.92970 + 0.645646i 0
193.7 0 0.187483 + 1.72187i 0 −1.43964 − 0.831176i 0 −0.286876 − 0.496884i 0 −2.92970 + 0.645646i 0
193.8 0 0.187483 + 1.72187i 0 1.43964 + 0.831176i 0 0.286876 + 0.496884i 0 −2.92970 + 0.645646i 0
193.9 0 1.37296 − 1.05593i 0 −3.35903 − 1.93934i 0 −1.14978 − 1.99147i 0 0.770012 − 2.89950i 0
193.10 0 1.37296 − 1.05593i 0 3.35903 + 1.93934i 0 1.14978 + 1.99147i 0 0.770012 − 2.89950i 0
193.11 0 1.68221 + 0.412500i 0 −0.379899 − 0.219335i 0 −2.14376 − 3.71310i 0 2.65969 + 1.38783i 0
193.12 0 1.68221 + 0.412500i 0 0.379899 + 0.219335i 0 2.14376 + 3.71310i 0 2.65969 + 1.38783i 0
961.1 0 −1.68221 + 0.412500i 0 −0.379899 + 0.219335i 0 2.14376 − 3.71310i 0 2.65969 − 1.38783i 0
961.2 0 −1.68221 + 0.412500i 0 0.379899 − 0.219335i 0 −2.14376 + 3.71310i 0 2.65969 − 1.38783i 0
961.3 0 −1.37296 − 1.05593i 0 −3.35903 + 1.93934i 0 1.14978 − 1.99147i 0 0.770012 + 2.89950i 0
961.4 0 −1.37296 − 1.05593i 0 3.35903 − 1.93934i 0 −1.14978 + 1.99147i 0 0.770012 + 2.89950i 0
961.5 0 −0.187483 + 1.72187i 0 −1.43964 + 0.831176i 0 0.286876 − 0.496884i 0 −2.92970 − 0.645646i 0
961.6 0 −0.187483 + 1.72187i 0 1.43964 − 0.831176i 0 −0.286876 + 0.496884i 0 −2.92970 − 0.645646i 0
961.7 0 0.187483 − 1.72187i 0 −1.43964 + 0.831176i 0 −0.286876 + 0.496884i 0 −2.92970 − 0.645646i 0
961.8 0 0.187483 − 1.72187i 0 1.43964 − 0.831176i 0 0.286876 − 0.496884i 0 −2.92970 − 0.645646i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.r.g ✓ 24
3.b odd 2 1 3456.2.r.h 24
4.b odd 2 1 inner 1152.2.r.g ✓ 24
8.b even 2 1 inner 1152.2.r.g ✓ 24
8.d odd 2 1 inner 1152.2.r.g ✓ 24
9.c even 3 1 inner 1152.2.r.g ✓ 24
9.d odd 6 1 3456.2.r.h 24
12.b even 2 1 3456.2.r.h 24
24.f even 2 1 3456.2.r.h 24
24.h odd 2 1 3456.2.r.h 24
36.f odd 6 1 inner 1152.2.r.g ✓ 24
36.h even 6 1 3456.2.r.h 24
72.j odd 6 1 3456.2.r.h 24
72.l even 6 1 3456.2.r.h 24
72.n even 6 1 inner 1152.2.r.g ✓ 24
72.p odd 6 1 inner 1152.2.r.g ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.r.g ✓ 24 1.a even 1 1 trivial
1152.2.r.g ✓ 24 4.b odd 2 1 inner
1152.2.r.g ✓ 24 8.b even 2 1 inner
1152.2.r.g ✓ 24 8.d odd 2 1 inner
1152.2.r.g ✓ 24 9.c even 3 1 inner
1152.2.r.g ✓ 24 36.f odd 6 1 inner
1152.2.r.g ✓ 24 72.n even 6 1 inner
1152.2.r.g ✓ 24 72.p odd 6 1 inner
3456.2.r.h 24 3.b odd 2 1
3456.2.r.h 24 9.d odd 6 1
3456.2.r.h 24 12.b even 2 1
3456.2.r.h 24 24.f even 2 1
3456.2.r.h 24 24.h odd 2 1
3456.2.r.h 24 36.h even 6 1
3456.2.r.h 24 72.j odd 6 1
3456.2.r.h 24 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{12} - 18T_{5}^{10} + 279T_{5}^{8} - 794T_{5}^{6} + 1881T_{5}^{4} - 360T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{12} + 24T_{7}^{10} + 471T_{7}^{8} + 2456T_{7}^{6} + 10257T_{7}^{4} + 3360T_{7}^{2} + 1024 \) Copy content Toggle raw display
\( T_{11}^{12} - 33T_{11}^{10} + 816T_{11}^{8} - 8959T_{11}^{6} + 73704T_{11}^{4} - 6825T_{11}^{2} + 625 \) Copy content Toggle raw display