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Properties

Label 1248.2.br.a
Level $1248$
Weight $2$
Character orbit 1248.br
Analytic conductor $9.965$
Analytic rank $0$
Dimension $56$
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(529,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1248.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.br (of order \(6\), 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: \(9.96533017226\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 312)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 28 q^{9} - 4 q^{17} - 64 q^{25} + 4 q^{41} - 36 q^{49} - 16 q^{55} + 32 q^{57} - 28 q^{65} + 64 q^{71} + 24 q^{73} - 80 q^{79} - 28 q^{81} - 24 q^{87} + 40 q^{89} + 40 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 0 −0.866025 − 0.500000i 0 − 4.17948i 0 1.23795 + 2.14419i 0 0.500000 + 0.866025i 0
529.2 0 −0.866025 − 0.500000i 0 − 2.89261i 0 −2.18031 − 3.77641i 0 0.500000 + 0.866025i 0
529.3 0 −0.866025 − 0.500000i 0 − 0.581590i 0 −1.02479 − 1.77499i 0 0.500000 + 0.866025i 0
529.4 0 −0.866025 − 0.500000i 0 − 0.0787347i 0 −0.680796 − 1.17917i 0 0.500000 + 0.866025i 0
529.5 0 −0.866025 − 0.500000i 0 0.369739i 0 1.16461 + 2.01717i 0 0.500000 + 0.866025i 0
529.6 0 −0.866025 − 0.500000i 0 − 1.62558i 0 0.510871 + 0.884855i 0 0.500000 + 0.866025i 0
529.7 0 −0.866025 − 0.500000i 0 − 1.76603i 0 2.26933 + 3.93060i 0 0.500000 + 0.866025i 0
529.8 0 −0.866025 − 0.500000i 0 − 1.47697i 0 −1.94969 − 3.37697i 0 0.500000 + 0.866025i 0
529.9 0 −0.866025 − 0.500000i 0 1.45280i 0 −1.04137 − 1.80370i 0 0.500000 + 0.866025i 0
529.10 0 −0.866025 − 0.500000i 0 − 2.42445i 0 0.755347 + 1.30830i 0 0.500000 + 0.866025i 0
529.11 0 −0.866025 − 0.500000i 0 3.04873i 0 1.36275 + 2.36035i 0 0.500000 + 0.866025i 0
529.12 0 −0.866025 − 0.500000i 0 3.18381i 0 0.0947723 + 0.164150i 0 0.500000 + 0.866025i 0
529.13 0 −0.866025 − 0.500000i 0 3.36712i 0 −2.12695 − 3.68399i 0 0.500000 + 0.866025i 0
529.14 0 −0.866025 − 0.500000i 0 3.60324i 0 1.60828 + 2.78562i 0 0.500000 + 0.866025i 0
529.15 0 0.866025 + 0.500000i 0 − 3.60324i 0 1.60828 + 2.78562i 0 0.500000 + 0.866025i 0
529.16 0 0.866025 + 0.500000i 0 − 3.36712i 0 −2.12695 − 3.68399i 0 0.500000 + 0.866025i 0
529.17 0 0.866025 + 0.500000i 0 − 3.18381i 0 0.0947723 + 0.164150i 0 0.500000 + 0.866025i 0
529.18 0 0.866025 + 0.500000i 0 − 3.04873i 0 1.36275 + 2.36035i 0 0.500000 + 0.866025i 0
529.19 0 0.866025 + 0.500000i 0 2.42445i 0 0.755347 + 1.30830i 0 0.500000 + 0.866025i 0
529.20 0 0.866025 + 0.500000i 0 − 1.45280i 0 −1.04137 − 1.80370i 0 0.500000 + 0.866025i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.c even 3 1 inner
104.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.2.br.a 56
4.b odd 2 1 312.2.bb.a ✓ 56
8.b even 2 1 inner 1248.2.br.a 56
8.d odd 2 1 312.2.bb.a ✓ 56
12.b even 2 1 936.2.be.c 56
13.c even 3 1 inner 1248.2.br.a 56
24.f even 2 1 936.2.be.c 56
52.j odd 6 1 312.2.bb.a ✓ 56
104.n odd 6 1 312.2.bb.a ✓ 56
104.r even 6 1 inner 1248.2.br.a 56
156.p even 6 1 936.2.be.c 56
312.bn even 6 1 936.2.be.c 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bb.a ✓ 56 4.b odd 2 1
312.2.bb.a ✓ 56 8.d odd 2 1
312.2.bb.a ✓ 56 52.j odd 6 1
312.2.bb.a ✓ 56 104.n odd 6 1
936.2.be.c 56 12.b even 2 1
936.2.be.c 56 24.f even 2 1
936.2.be.c 56 156.p even 6 1
936.2.be.c 56 312.bn even 6 1
1248.2.br.a 56 1.a even 1 1 trivial
1248.2.br.a 56 8.b even 2 1 inner
1248.2.br.a 56 13.c even 3 1 inner
1248.2.br.a 56 104.r even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1248, [\chi])\).