Properties

Label 8112.2.a.c
Level $8112$
Weight $2$
Character orbit 8112.a
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8112,2,Mod(1,8112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("8112.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.a (trivial)

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: yes
Analytic conductor: \(64.7746461197\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} - 3 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 3 q^{5} + 2 q^{7} + q^{9} + 6 q^{11} + 3 q^{15} - 3 q^{17} + 2 q^{19} - 2 q^{21} + 6 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 4 q^{31} - 6 q^{33} - 6 q^{35} + 7 q^{37} + 3 q^{41} + 10 q^{43} - 3 q^{45} + 6 q^{47} - 3 q^{49} + 3 q^{51} + 3 q^{53} - 18 q^{55} - 2 q^{57} - 7 q^{61} + 2 q^{63} - 10 q^{67} - 6 q^{69} + 6 q^{71} + 13 q^{73} - 4 q^{75} + 12 q^{77} + 4 q^{79} + q^{81} - 6 q^{83} + 9 q^{85} - 3 q^{87} - 18 q^{89} + 4 q^{93} - 6 q^{95} - 14 q^{97} + 6 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −1.00000 0 −3.00000 0 2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8112.2.a.c 1
4.b odd 2 1 1014.2.a.f 1
12.b even 2 1 3042.2.a.h 1
13.b even 2 1 8112.2.a.m 1
13.e even 6 2 624.2.q.g 2
39.h odd 6 2 1872.2.t.c 2
52.b odd 2 1 1014.2.a.c 1
52.f even 4 2 1014.2.b.c 2
52.i odd 6 2 78.2.e.a 2
52.j odd 6 2 1014.2.e.a 2
52.l even 12 4 1014.2.i.b 4
156.h even 2 1 3042.2.a.i 1
156.l odd 4 2 3042.2.b.h 2
156.r even 6 2 234.2.h.a 2
260.w odd 6 2 1950.2.i.m 2
260.bg even 12 4 1950.2.z.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 52.i odd 6 2
234.2.h.a 2 156.r even 6 2
624.2.q.g 2 13.e even 6 2
1014.2.a.c 1 52.b odd 2 1
1014.2.a.f 1 4.b odd 2 1
1014.2.b.c 2 52.f even 4 2
1014.2.e.a 2 52.j odd 6 2
1014.2.i.b 4 52.l even 12 4
1872.2.t.c 2 39.h odd 6 2
1950.2.i.m 2 260.w odd 6 2
1950.2.z.g 4 260.bg even 12 4
3042.2.a.h 1 12.b even 2 1
3042.2.a.i 1 156.h even 2 1
3042.2.b.h 2 156.l odd 4 2
8112.2.a.c 1 1.a even 1 1 trivial
8112.2.a.m 1 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):

\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T - 3 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 7 \) Copy content Toggle raw display
$41$ \( T - 3 \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T - 3 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 7 \) Copy content Toggle raw display
$67$ \( T + 10 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T - 13 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 18 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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