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Properties

Label 1248.2.h.c
Level $1248$
Weight $2$
Character orbit 1248.h
Analytic conductor $9.965$
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] = [1248,2,Mod(623,1248)]
 
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([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1248.623");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.h (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: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} - 28 q^{9} + 56 q^{25} + 16 q^{27} - 40 q^{43} + 8 q^{49} + 52 q^{51} + 8 q^{75} - 76 q^{81} + 56 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
623.1 0 −1.21097 − 1.23837i 0 − 2.18412i 0 0.621089 0 −0.0671052 + 2.99925i 0
623.2 0 −1.21097 − 1.23837i 0 2.18412i 0 −0.621089 0 −0.0671052 + 2.99925i 0
623.3 0 −1.21097 − 1.23837i 0 − 2.18412i 0 0.621089 0 −0.0671052 + 2.99925i 0
623.4 0 −1.21097 − 1.23837i 0 2.18412i 0 −0.621089 0 −0.0671052 + 2.99925i 0
623.5 0 −1.21097 + 1.23837i 0 2.18412i 0 0.621089 0 −0.0671052 − 2.99925i 0
623.6 0 −1.21097 + 1.23837i 0 − 2.18412i 0 −0.621089 0 −0.0671052 − 2.99925i 0
623.7 0 −1.21097 + 1.23837i 0 2.18412i 0 0.621089 0 −0.0671052 − 2.99925i 0
623.8 0 −1.21097 + 1.23837i 0 − 2.18412i 0 −0.621089 0 −0.0671052 − 2.99925i 0
623.9 0 −0.423309 − 1.67953i 0 0.349197i 0 −2.86091 0 −2.64162 + 1.42192i 0
623.10 0 −0.423309 − 1.67953i 0 − 0.349197i 0 2.86091 0 −2.64162 + 1.42192i 0
623.11 0 −0.423309 − 1.67953i 0 0.349197i 0 −2.86091 0 −2.64162 + 1.42192i 0
623.12 0 −0.423309 − 1.67953i 0 − 0.349197i 0 2.86091 0 −2.64162 + 1.42192i 0
623.13 0 −0.423309 + 1.67953i 0 − 0.349197i 0 −2.86091 0 −2.64162 − 1.42192i 0
623.14 0 −0.423309 + 1.67953i 0 0.349197i 0 2.86091 0 −2.64162 − 1.42192i 0
623.15 0 −0.423309 + 1.67953i 0 − 0.349197i 0 −2.86091 0 −2.64162 − 1.42192i 0
623.16 0 −0.423309 + 1.67953i 0 0.349197i 0 2.86091 0 −2.64162 − 1.42192i 0
623.17 0 0.662918 − 1.60017i 0 2.38561i 0 2.63829 0 −2.12108 − 2.12156i 0
623.18 0 0.662918 − 1.60017i 0 − 2.38561i 0 −2.63829 0 −2.12108 − 2.12156i 0
623.19 0 0.662918 − 1.60017i 0 2.38561i 0 2.63829 0 −2.12108 − 2.12156i 0
623.20 0 0.662918 − 1.60017i 0 − 2.38561i 0 −2.63829 0 −2.12108 − 2.12156i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 623.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
13.b even 2 1 inner
24.f even 2 1 inner
39.d odd 2 1 inner
104.h odd 2 1 inner
312.h 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.h.c 32
3.b odd 2 1 inner 1248.2.h.c 32
4.b odd 2 1 312.2.h.c ✓ 32
8.b even 2 1 312.2.h.c ✓ 32
8.d odd 2 1 inner 1248.2.h.c 32
12.b even 2 1 312.2.h.c ✓ 32
13.b even 2 1 inner 1248.2.h.c 32
24.f even 2 1 inner 1248.2.h.c 32
24.h odd 2 1 312.2.h.c ✓ 32
39.d odd 2 1 inner 1248.2.h.c 32
52.b odd 2 1 312.2.h.c ✓ 32
104.e even 2 1 312.2.h.c ✓ 32
104.h odd 2 1 inner 1248.2.h.c 32
156.h even 2 1 312.2.h.c ✓ 32
312.b odd 2 1 312.2.h.c ✓ 32
312.h even 2 1 inner 1248.2.h.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.h.c ✓ 32 4.b odd 2 1
312.2.h.c ✓ 32 8.b even 2 1
312.2.h.c ✓ 32 12.b even 2 1
312.2.h.c ✓ 32 24.h odd 2 1
312.2.h.c ✓ 32 52.b odd 2 1
312.2.h.c ✓ 32 104.e even 2 1
312.2.h.c ✓ 32 156.h even 2 1
312.2.h.c ✓ 32 312.b odd 2 1
1248.2.h.c 32 1.a even 1 1 trivial
1248.2.h.c 32 3.b odd 2 1 inner
1248.2.h.c 32 8.d odd 2 1 inner
1248.2.h.c 32 13.b even 2 1 inner
1248.2.h.c 32 24.f even 2 1 inner
1248.2.h.c 32 39.d odd 2 1 inner
1248.2.h.c 32 104.h odd 2 1 inner
1248.2.h.c 32 312.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 13T_{5}^{6} + 54T_{5}^{4} + 72T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\). Copy content Toggle raw display