Properties

Label 1344.2.w.b
Level $1344$
Weight $2$
Character orbit 1344.w
Analytic conductor $10.732$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(337,1344)]
 
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([0, 3, 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("1344.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.w (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: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 4 q^{29} + 24 q^{33} + 4 q^{37} - 20 q^{43} - 28 q^{49} + 8 q^{51} + 20 q^{53} - 40 q^{61} + 28 q^{63} + 16 q^{65} - 4 q^{67} - 16 q^{69} + 16 q^{75} - 4 q^{77} - 24 q^{79} - 28 q^{81} - 40 q^{83} + 48 q^{85} + 72 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −0.707107 0.707107i 0 −1.77267 + 1.77267i 0 1.00000i 0 1.00000i 0
337.2 0 −0.707107 0.707107i 0 −1.84737 + 1.84737i 0 1.00000i 0 1.00000i 0
337.3 0 −0.707107 0.707107i 0 −0.646579 + 0.646579i 0 1.00000i 0 1.00000i 0
337.4 0 −0.707107 0.707107i 0 −0.539395 + 0.539395i 0 1.00000i 0 1.00000i 0
337.5 0 −0.707107 0.707107i 0 1.25389 1.25389i 0 1.00000i 0 1.00000i 0
337.6 0 −0.707107 0.707107i 0 2.44528 2.44528i 0 1.00000i 0 1.00000i 0
337.7 0 −0.707107 0.707107i 0 2.52107 2.52107i 0 1.00000i 0 1.00000i 0
337.8 0 0.707107 + 0.707107i 0 −3.03597 + 3.03597i 0 1.00000i 0 1.00000i 0
337.9 0 0.707107 + 0.707107i 0 −0.805240 + 0.805240i 0 1.00000i 0 1.00000i 0
337.10 0 0.707107 + 0.707107i 0 −0.979857 + 0.979857i 0 1.00000i 0 1.00000i 0
337.11 0 0.707107 + 0.707107i 0 0.116928 0.116928i 0 1.00000i 0 1.00000i 0
337.12 0 0.707107 + 0.707107i 0 2.39875 2.39875i 0 1.00000i 0 1.00000i 0
337.13 0 0.707107 + 0.707107i 0 2.66347 2.66347i 0 1.00000i 0 1.00000i 0
337.14 0 0.707107 + 0.707107i 0 −1.77230 + 1.77230i 0 1.00000i 0 1.00000i 0
1009.1 0 −0.707107 + 0.707107i 0 −1.77267 1.77267i 0 1.00000i 0 1.00000i 0
1009.2 0 −0.707107 + 0.707107i 0 −1.84737 1.84737i 0 1.00000i 0 1.00000i 0
1009.3 0 −0.707107 + 0.707107i 0 −0.646579 0.646579i 0 1.00000i 0 1.00000i 0
1009.4 0 −0.707107 + 0.707107i 0 −0.539395 0.539395i 0 1.00000i 0 1.00000i 0
1009.5 0 −0.707107 + 0.707107i 0 1.25389 + 1.25389i 0 1.00000i 0 1.00000i 0
1009.6 0 −0.707107 + 0.707107i 0 2.44528 + 2.44528i 0 1.00000i 0 1.00000i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e 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.w.b 28
4.b odd 2 1 336.2.w.b 28
8.b even 2 1 2688.2.w.c 28
8.d odd 2 1 2688.2.w.d 28
16.e even 4 1 inner 1344.2.w.b 28
16.e even 4 1 2688.2.w.c 28
16.f odd 4 1 336.2.w.b 28
16.f odd 4 1 2688.2.w.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.w.b 28 4.b odd 2 1
336.2.w.b 28 16.f odd 4 1
1344.2.w.b 28 1.a even 1 1 trivial
1344.2.w.b 28 16.e even 4 1 inner
2688.2.w.c 28 8.b even 2 1
2688.2.w.c 28 16.e even 4 1
2688.2.w.d 28 8.d odd 2 1
2688.2.w.d 28 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} + 24 T_{5}^{25} + 560 T_{5}^{24} + 144 T_{5}^{23} + 288 T_{5}^{22} + 13392 T_{5}^{21} + 103880 T_{5}^{20} + 103872 T_{5}^{19} + 170496 T_{5}^{18} + 1865824 T_{5}^{17} + 9658432 T_{5}^{16} + \cdots + 12845056 \) acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\). Copy content Toggle raw display