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Properties

Label 1568.2.t.h
Level $1568$
Weight $2$
Character orbit 1568.t
Analytic conductor $12.521$
Analytic rank $0$
Dimension $24$
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] = [1568,2,Mod(177,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.t (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: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 392)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{9} + 32 q^{15} + 32 q^{23} + 12 q^{25} + 32 q^{39} - 8 q^{65} - 96 q^{71} - 80 q^{79} + 4 q^{81} - 64 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1 0 −2.48442 + 1.43438i 0 −2.76990 − 1.59920i 0 0 0 2.61491 − 4.52915i 0
177.2 0 −2.48442 + 1.43438i 0 −2.76990 − 1.59920i 0 0 0 2.61491 − 4.52915i 0
177.3 0 −1.61829 + 0.934317i 0 2.02949 + 1.17173i 0 0 0 0.245898 − 0.425908i 0
177.4 0 −1.61829 + 0.934317i 0 2.02949 + 1.17173i 0 0 0 0.245898 − 0.425908i 0
177.5 0 −0.456937 + 0.263813i 0 1.30721 + 0.754717i 0 0 0 −1.36081 + 2.35698i 0
177.6 0 −0.456937 + 0.263813i 0 1.30721 + 0.754717i 0 0 0 −1.36081 + 2.35698i 0
177.7 0 0.456937 − 0.263813i 0 −1.30721 − 0.754717i 0 0 0 −1.36081 + 2.35698i 0
177.8 0 0.456937 − 0.263813i 0 −1.30721 − 0.754717i 0 0 0 −1.36081 + 2.35698i 0
177.9 0 1.61829 − 0.934317i 0 −2.02949 − 1.17173i 0 0 0 0.245898 − 0.425908i 0
177.10 0 1.61829 − 0.934317i 0 −2.02949 − 1.17173i 0 0 0 0.245898 − 0.425908i 0
177.11 0 2.48442 − 1.43438i 0 2.76990 + 1.59920i 0 0 0 2.61491 − 4.52915i 0
177.12 0 2.48442 − 1.43438i 0 2.76990 + 1.59920i 0 0 0 2.61491 − 4.52915i 0
753.1 0 −2.48442 − 1.43438i 0 −2.76990 + 1.59920i 0 0 0 2.61491 + 4.52915i 0
753.2 0 −2.48442 − 1.43438i 0 −2.76990 + 1.59920i 0 0 0 2.61491 + 4.52915i 0
753.3 0 −1.61829 − 0.934317i 0 2.02949 − 1.17173i 0 0 0 0.245898 + 0.425908i 0
753.4 0 −1.61829 − 0.934317i 0 2.02949 − 1.17173i 0 0 0 0.245898 + 0.425908i 0
753.5 0 −0.456937 − 0.263813i 0 1.30721 − 0.754717i 0 0 0 −1.36081 − 2.35698i 0
753.6 0 −0.456937 − 0.263813i 0 1.30721 − 0.754717i 0 0 0 −1.36081 − 2.35698i 0
753.7 0 0.456937 + 0.263813i 0 −1.30721 + 0.754717i 0 0 0 −1.36081 − 2.35698i 0
753.8 0 0.456937 + 0.263813i 0 −1.30721 + 0.754717i 0 0 0 −1.36081 − 2.35698i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.t.h 24
4.b odd 2 1 392.2.p.h 24
7.b odd 2 1 inner 1568.2.t.h 24
7.c even 3 1 1568.2.b.g 12
7.c even 3 1 inner 1568.2.t.h 24
7.d odd 6 1 1568.2.b.g 12
7.d odd 6 1 inner 1568.2.t.h 24
8.b even 2 1 inner 1568.2.t.h 24
8.d odd 2 1 392.2.p.h 24
28.d even 2 1 392.2.p.h 24
28.f even 6 1 392.2.b.g ✓ 12
28.f even 6 1 392.2.p.h 24
28.g odd 6 1 392.2.b.g ✓ 12
28.g odd 6 1 392.2.p.h 24
56.e even 2 1 392.2.p.h 24
56.h odd 2 1 inner 1568.2.t.h 24
56.j odd 6 1 1568.2.b.g 12
56.j odd 6 1 inner 1568.2.t.h 24
56.k odd 6 1 392.2.b.g ✓ 12
56.k odd 6 1 392.2.p.h 24
56.m even 6 1 392.2.b.g ✓ 12
56.m even 6 1 392.2.p.h 24
56.p even 6 1 1568.2.b.g 12
56.p even 6 1 inner 1568.2.t.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.b.g ✓ 12 28.f even 6 1
392.2.b.g ✓ 12 28.g odd 6 1
392.2.b.g ✓ 12 56.k odd 6 1
392.2.b.g ✓ 12 56.m even 6 1
392.2.p.h 24 4.b odd 2 1
392.2.p.h 24 8.d odd 2 1
392.2.p.h 24 28.d even 2 1
392.2.p.h 24 28.f even 6 1
392.2.p.h 24 28.g odd 6 1
392.2.p.h 24 56.e even 2 1
392.2.p.h 24 56.k odd 6 1
392.2.p.h 24 56.m even 6 1
1568.2.b.g 12 7.c even 3 1
1568.2.b.g 12 7.d odd 6 1
1568.2.b.g 12 56.j odd 6 1
1568.2.b.g 12 56.p even 6 1
1568.2.t.h 24 1.a even 1 1 trivial
1568.2.t.h 24 7.b odd 2 1 inner
1568.2.t.h 24 7.c even 3 1 inner
1568.2.t.h 24 7.d odd 6 1 inner
1568.2.t.h 24 8.b even 2 1 inner
1568.2.t.h 24 56.h odd 2 1 inner
1568.2.t.h 24 56.j odd 6 1 inner
1568.2.t.h 24 56.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{12} - 12T_{3}^{10} + 112T_{3}^{8} - 368T_{3}^{6} + 928T_{3}^{4} - 256T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{12} + 86T_{17}^{10} + 5336T_{17}^{8} + 155256T_{17}^{6} + 3301728T_{17}^{4} + 22561120T_{17}^{2} + 119946304 \) Copy content Toggle raw display