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Properties

Label 1344.2.s.c
Level $1344$
Weight $2$
Character orbit 1344.s
Analytic conductor $10.732$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} + 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} + 40 q^{7} + 24 q^{13} - 16 q^{19} - 4 q^{21} + 32 q^{27} + 24 q^{33} - 8 q^{37} + 64 q^{39} - 24 q^{43} - 28 q^{45} + 40 q^{49} + 32 q^{51} - 16 q^{55} - 48 q^{61} - 40 q^{67} + 4 q^{69} - 40 q^{75} + 56 q^{81} - 48 q^{85} - 32 q^{87} + 24 q^{91} + 56 q^{93} + 16 q^{97} - 24 q^{99}+O(q^{100}) \) 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 −1.73158 − 0.0404130i 0 0.0573500 + 0.0573500i 0 1.00000 0 2.99673 + 0.139957i 0
239.2 0 −1.72790 − 0.119805i 0 0.793669 + 0.793669i 0 1.00000 0 2.97129 + 0.414021i 0
239.3 0 −1.61364 − 0.629429i 0 −3.08691 − 3.08691i 0 1.00000 0 2.20764 + 2.03134i 0
239.4 0 −1.42461 + 0.985133i 0 −0.766553 − 0.766553i 0 1.00000 0 1.05903 − 2.80686i 0
239.5 0 −1.26429 + 1.18388i 0 −2.84277 − 2.84277i 0 1.00000 0 0.196841 − 2.99354i 0
239.6 0 −1.18388 + 1.26429i 0 2.84277 + 2.84277i 0 1.00000 0 −0.196841 − 2.99354i 0
239.7 0 −0.985133 + 1.42461i 0 0.766553 + 0.766553i 0 1.00000 0 −1.05903 − 2.80686i 0
239.8 0 −0.827739 − 1.52146i 0 −1.23935 − 1.23935i 0 1.00000 0 −1.62970 + 2.51875i 0
239.9 0 −0.567369 − 1.63649i 0 −0.132854 − 0.132854i 0 1.00000 0 −2.35619 + 1.85698i 0
239.10 0 −0.408017 − 1.68331i 0 2.26080 + 2.26080i 0 1.00000 0 −2.66704 + 1.37363i 0
239.11 0 0.0404130 + 1.73158i 0 −0.0573500 − 0.0573500i 0 1.00000 0 −2.99673 + 0.139957i 0
239.12 0 0.0806676 − 1.73017i 0 −1.66193 − 1.66193i 0 1.00000 0 −2.98699 − 0.279137i 0
239.13 0 0.119805 + 1.72790i 0 −0.793669 − 0.793669i 0 1.00000 0 −2.97129 + 0.414021i 0
239.14 0 0.629429 + 1.61364i 0 3.08691 + 3.08691i 0 1.00000 0 −2.20764 + 2.03134i 0
239.15 0 0.714681 − 1.57773i 0 1.31983 + 1.31983i 0 1.00000 0 −1.97846 − 2.25515i 0
239.16 0 1.52146 + 0.827739i 0 1.23935 + 1.23935i 0 1.00000 0 1.62970 + 2.51875i 0
239.17 0 1.57773 − 0.714681i 0 −1.31983 − 1.31983i 0 1.00000 0 1.97846 − 2.25515i 0
239.18 0 1.63649 + 0.567369i 0 0.132854 + 0.132854i 0 1.00000 0 2.35619 + 1.85698i 0
239.19 0 1.68331 + 0.408017i 0 −2.26080 − 2.26080i 0 1.00000 0 2.66704 + 1.37363i 0
239.20 0 1.73017 − 0.0806676i 0 1.66193 + 1.66193i 0 1.00000 0 2.98699 − 0.279137i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.20
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.c 40
3.b odd 2 1 inner 1344.2.s.c 40
4.b odd 2 1 336.2.s.c ✓ 40
12.b even 2 1 336.2.s.c ✓ 40
16.e even 4 1 336.2.s.c ✓ 40
16.f odd 4 1 inner 1344.2.s.c 40
48.i odd 4 1 336.2.s.c ✓ 40
48.k even 4 1 inner 1344.2.s.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.s.c ✓ 40 4.b odd 2 1
336.2.s.c ✓ 40 12.b even 2 1
336.2.s.c ✓ 40 16.e even 4 1
336.2.s.c ✓ 40 48.i odd 4 1
1344.2.s.c 40 1.a even 1 1 trivial
1344.2.s.c 40 3.b odd 2 1 inner
1344.2.s.c 40 16.f odd 4 1 inner
1344.2.s.c 40 48.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 784 T_{5}^{36} + 201200 T_{5}^{32} + 19415584 T_{5}^{28} + 699116768 T_{5}^{24} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\). Copy content Toggle raw display