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Properties

Label 1248.2.m.c
Level $1248$
Weight $2$
Character orbit 1248.m
Analytic conductor $9.965$
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] = [1248,2,Mod(337,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1248.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.m (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: \(9.96533017226\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 312)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{9} + 32 q^{17} + 40 q^{25} + 8 q^{39} - 72 q^{49} - 16 q^{55} + 64 q^{79} + 24 q^{81} + 128 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1 0 − 1.00000i 0 −1.21406 0 4.92861i 0 −1.00000 0
337.2 0 − 1.00000i 0 1.21406 0 − 4.92861i 0 −1.00000 0
337.3 0 1.00000i 0 −1.21406 0 − 4.92861i 0 −1.00000 0
337.4 0 1.00000i 0 1.21406 0 4.92861i 0 −1.00000 0
337.5 0 − 1.00000i 0 −3.44141 0 2.84073i 0 −1.00000 0
337.6 0 − 1.00000i 0 3.44141 0 − 2.84073i 0 −1.00000 0
337.7 0 1.00000i 0 −3.44141 0 − 2.84073i 0 −1.00000 0
337.8 0 1.00000i 0 3.44141 0 2.84073i 0 −1.00000 0
337.9 0 − 1.00000i 0 −0.172184 0 − 1.91402i 0 −1.00000 0
337.10 0 − 1.00000i 0 0.172184 0 1.91402i 0 −1.00000 0
337.11 0 1.00000i 0 −0.172184 0 1.91402i 0 −1.00000 0
337.12 0 1.00000i 0 0.172184 0 − 1.91402i 0 −1.00000 0
337.13 0 − 1.00000i 0 −0.859867 0 − 2.42446i 0 −1.00000 0
337.14 0 − 1.00000i 0 0.859867 0 2.42446i 0 −1.00000 0
337.15 0 1.00000i 0 −0.859867 0 2.42446i 0 −1.00000 0
337.16 0 1.00000i 0 0.859867 0 − 2.42446i 0 −1.00000 0
337.17 0 − 1.00000i 0 −3.48777 0 0.644203i 0 −1.00000 0
337.18 0 − 1.00000i 0 3.48777 0 − 0.644203i 0 −1.00000 0
337.19 0 1.00000i 0 −3.48777 0 − 0.644203i 0 −1.00000 0
337.20 0 1.00000i 0 3.48777 0 0.644203i 0 −1.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.b even 2 1 inner
104.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.2.m.c 24
3.b odd 2 1 3744.2.m.i 24
4.b odd 2 1 312.2.m.c ✓ 24
8.b even 2 1 inner 1248.2.m.c 24
8.d odd 2 1 312.2.m.c ✓ 24
12.b even 2 1 936.2.m.i 24
13.b even 2 1 inner 1248.2.m.c 24
24.f even 2 1 936.2.m.i 24
24.h odd 2 1 3744.2.m.i 24
39.d odd 2 1 3744.2.m.i 24
52.b odd 2 1 312.2.m.c ✓ 24
104.e even 2 1 inner 1248.2.m.c 24
104.h odd 2 1 312.2.m.c ✓ 24
156.h even 2 1 936.2.m.i 24
312.b odd 2 1 3744.2.m.i 24
312.h even 2 1 936.2.m.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.m.c ✓ 24 4.b odd 2 1
312.2.m.c ✓ 24 8.d odd 2 1
312.2.m.c ✓ 24 52.b odd 2 1
312.2.m.c ✓ 24 104.h odd 2 1
936.2.m.i 24 12.b even 2 1
936.2.m.i 24 24.f even 2 1
936.2.m.i 24 156.h even 2 1
936.2.m.i 24 312.h even 2 1
1248.2.m.c 24 1.a even 1 1 trivial
1248.2.m.c 24 8.b even 2 1 inner
1248.2.m.c 24 13.b even 2 1 inner
1248.2.m.c 24 104.e even 2 1 inner
3744.2.m.i 24 3.b odd 2 1
3744.2.m.i 24 24.h odd 2 1
3744.2.m.i 24 39.d odd 2 1
3744.2.m.i 24 312.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 40T_{5}^{10} + 560T_{5}^{8} - 3088T_{5}^{6} + 4992T_{5}^{4} - 2304T_{5}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\). Copy content Toggle raw display