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Properties

Label 1512.2.c.f
Level $1512$
Weight $2$
Character orbit 1512.c
Analytic conductor $12.073$
Analytic rank $0$
Dimension $24$
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] = [1512,2,Mod(757,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1512.757");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.c (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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{7} + 20 q^{10} - 4 q^{16} + 4 q^{22} - 24 q^{25} - 16 q^{31} + 4 q^{34} + 12 q^{40} - 52 q^{46} + 24 q^{49} + 12 q^{52} - 8 q^{55} - 28 q^{58} + 24 q^{64} + 20 q^{70} - 24 q^{76} + 32 q^{79} + 44 q^{82} - 60 q^{88} + 12 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1 −1.41290 − 0.0608900i 0 1.99258 + 0.172063i − 3.11390i 0 1.00000 −2.80485 − 0.364437i 0 −0.189605 + 4.39964i
757.2 −1.41290 + 0.0608900i 0 1.99258 − 0.172063i 3.11390i 0 1.00000 −2.80485 + 0.364437i 0 −0.189605 − 4.39964i
757.3 −1.29659 − 0.564669i 0 1.36230 + 1.46429i 1.53368i 0 1.00000 −0.939505 − 2.66783i 0 0.866019 − 1.98855i
757.4 −1.29659 + 0.564669i 0 1.36230 − 1.46429i − 1.53368i 0 1.00000 −0.939505 + 2.66783i 0 0.866019 + 1.98855i
757.5 −1.09864 − 0.890504i 0 0.414007 + 1.95668i − 1.58470i 0 1.00000 1.28759 − 2.51836i 0 −1.41118 + 1.74101i
757.6 −1.09864 + 0.890504i 0 0.414007 − 1.95668i 1.58470i 0 1.00000 1.28759 + 2.51836i 0 −1.41118 − 1.74101i
757.7 −0.796644 − 1.16849i 0 −0.730716 + 1.86173i 2.52623i 0 1.00000 2.75753 − 0.629309i 0 2.95186 − 2.01251i
757.8 −0.796644 + 1.16849i 0 −0.730716 − 1.86173i − 2.52623i 0 1.00000 2.75753 + 0.629309i 0 2.95186 + 2.01251i
757.9 −0.671239 − 1.24476i 0 −1.09888 + 1.67107i 3.66698i 0 1.00000 2.81770 + 0.246156i 0 4.56453 − 2.46142i
757.10 −0.671239 + 1.24476i 0 −1.09888 − 1.67107i − 3.66698i 0 1.00000 2.81770 − 0.246156i 0 4.56453 + 2.46142i
757.11 −0.174217 − 1.40344i 0 −1.93930 + 0.489007i − 1.26947i 0 1.00000 1.02415 + 2.63650i 0 −1.78162 + 0.221163i
757.12 −0.174217 + 1.40344i 0 −1.93930 − 0.489007i 1.26947i 0 1.00000 1.02415 − 2.63650i 0 −1.78162 − 0.221163i
757.13 0.174217 − 1.40344i 0 −1.93930 − 0.489007i − 1.26947i 0 1.00000 −1.02415 + 2.63650i 0 −1.78162 − 0.221163i
757.14 0.174217 + 1.40344i 0 −1.93930 + 0.489007i 1.26947i 0 1.00000 −1.02415 − 2.63650i 0 −1.78162 + 0.221163i
757.15 0.671239 − 1.24476i 0 −1.09888 − 1.67107i 3.66698i 0 1.00000 −2.81770 + 0.246156i 0 4.56453 + 2.46142i
757.16 0.671239 + 1.24476i 0 −1.09888 + 1.67107i − 3.66698i 0 1.00000 −2.81770 − 0.246156i 0 4.56453 − 2.46142i
757.17 0.796644 − 1.16849i 0 −0.730716 − 1.86173i 2.52623i 0 1.00000 −2.75753 − 0.629309i 0 2.95186 + 2.01251i
757.18 0.796644 + 1.16849i 0 −0.730716 + 1.86173i − 2.52623i 0 1.00000 −2.75753 + 0.629309i 0 2.95186 − 2.01251i
757.19 1.09864 − 0.890504i 0 0.414007 − 1.95668i − 1.58470i 0 1.00000 −1.28759 − 2.51836i 0 −1.41118 − 1.74101i
757.20 1.09864 + 0.890504i 0 0.414007 + 1.95668i 1.58470i 0 1.00000 −1.28759 + 2.51836i 0 −1.41118 + 1.74101i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.c.f ✓ 24
3.b odd 2 1 inner 1512.2.c.f ✓ 24
4.b odd 2 1 6048.2.c.g 24
8.b even 2 1 inner 1512.2.c.f ✓ 24
8.d odd 2 1 6048.2.c.g 24
12.b even 2 1 6048.2.c.g 24
24.f even 2 1 6048.2.c.g 24
24.h odd 2 1 inner 1512.2.c.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.f ✓ 24 1.a even 1 1 trivial
1512.2.c.f ✓ 24 3.b odd 2 1 inner
1512.2.c.f ✓ 24 8.b even 2 1 inner
1512.2.c.f ✓ 24 24.h odd 2 1 inner
6048.2.c.g 24 4.b odd 2 1
6048.2.c.g 24 8.d odd 2 1
6048.2.c.g 24 12.b even 2 1
6048.2.c.g 24 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\):

\( T_{5}^{12} + 36T_{5}^{10} + 483T_{5}^{8} + 3048T_{5}^{6} + 9491T_{5}^{4} + 14084T_{5}^{2} + 7921 \) Copy content Toggle raw display
\( T_{17}^{12} - 124T_{17}^{10} + 5788T_{17}^{8} - 128800T_{17}^{6} + 1407472T_{17}^{4} - 6700992T_{17}^{2} + 8156736 \) Copy content Toggle raw display