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Properties

Label 1344.2.s.d
Level $1344$
Weight $2$
Character orbit 1344.s
Analytic conductor $10.732$
Analytic rank $0$
Dimension $48$
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] = [1344,2,Mod(239,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.s (of order \(4\), 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: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{7} - 8 q^{19} + 12 q^{27} + 16 q^{37} + 24 q^{39} + 48 q^{43} + 20 q^{45} + 48 q^{49} + 32 q^{55} + 8 q^{61} + 16 q^{67} - 28 q^{69} + 12 q^{75} - 48 q^{85} - 56 q^{87} - 64 q^{93} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −1.71763 − 0.223071i 0 −2.27022 − 2.27022i 0 −1.00000 0 2.90048 + 0.766304i 0
239.2 0 −1.66117 + 0.490427i 0 2.27018 + 2.27018i 0 −1.00000 0 2.51896 − 1.62936i 0
239.3 0 −1.62672 + 0.594785i 0 −0.0563417 − 0.0563417i 0 −1.00000 0 2.29246 − 1.93510i 0
239.4 0 −1.45100 − 0.945839i 0 1.69539 + 1.69539i 0 −1.00000 0 1.21078 + 2.74482i 0
239.5 0 −1.34538 − 1.09085i 0 −1.07379 − 1.07379i 0 −1.00000 0 0.620087 + 2.93522i 0
239.6 0 −1.29441 − 1.15087i 0 0.186466 + 0.186466i 0 −1.00000 0 0.350987 + 2.97940i 0
239.7 0 −1.26449 + 1.18367i 0 1.97913 + 1.97913i 0 −1.00000 0 0.197870 − 2.99347i 0
239.8 0 −1.18367 + 1.26449i 0 −1.97913 − 1.97913i 0 −1.00000 0 −0.197870 − 2.99347i 0
239.9 0 −0.594785 + 1.62672i 0 0.0563417 + 0.0563417i 0 −1.00000 0 −2.29246 − 1.93510i 0
239.10 0 −0.490427 + 1.66117i 0 −2.27018 − 2.27018i 0 −1.00000 0 −2.51896 − 1.62936i 0
239.11 0 −0.399703 − 1.68530i 0 2.09830 + 2.09830i 0 −1.00000 0 −2.68048 + 1.34724i 0
239.12 0 −0.277087 − 1.70974i 0 0.459458 + 0.459458i 0 −1.00000 0 −2.84645 + 0.947494i 0
239.13 0 0.223071 + 1.71763i 0 2.27022 + 2.27022i 0 −1.00000 0 −2.90048 + 0.766304i 0
239.14 0 0.262630 − 1.71202i 0 −2.76208 − 2.76208i 0 −1.00000 0 −2.86205 − 0.899256i 0
239.15 0 0.505560 − 1.65663i 0 −2.00763 − 2.00763i 0 −1.00000 0 −2.48882 − 1.67505i 0
239.16 0 0.861124 − 1.50282i 0 1.19856 + 1.19856i 0 −1.00000 0 −1.51693 − 2.58823i 0
239.17 0 0.945839 + 1.45100i 0 −1.69539 − 1.69539i 0 −1.00000 0 −1.21078 + 2.74482i 0
239.18 0 1.09085 + 1.34538i 0 1.07379 + 1.07379i 0 −1.00000 0 −0.620087 + 2.93522i 0
239.19 0 1.15087 + 1.29441i 0 −0.186466 − 0.186466i 0 −1.00000 0 −0.350987 + 2.97940i 0
239.20 0 1.50282 − 0.861124i 0 −1.19856 − 1.19856i 0 −1.00000 0 1.51693 − 2.58823i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.s.d 48
3.b odd 2 1 inner 1344.2.s.d 48
4.b odd 2 1 336.2.s.d ✓ 48
12.b even 2 1 336.2.s.d ✓ 48
16.e even 4 1 336.2.s.d ✓ 48
16.f odd 4 1 inner 1344.2.s.d 48
48.i odd 4 1 336.2.s.d ✓ 48
48.k even 4 1 inner 1344.2.s.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.s.d ✓ 48 4.b odd 2 1
336.2.s.d ✓ 48 12.b even 2 1
336.2.s.d ✓ 48 16.e even 4 1
336.2.s.d ✓ 48 48.i odd 4 1
1344.2.s.d 48 1.a even 1 1 trivial
1344.2.s.d 48 3.b odd 2 1 inner
1344.2.s.d 48 16.f odd 4 1 inner
1344.2.s.d 48 48.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 696 T_{5}^{44} + 196224 T_{5}^{40} + 29529376 T_{5}^{36} + 2599544544 T_{5}^{32} + \cdots + 40960000 \) acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\). Copy content Toggle raw display