Properties

Label 8112.2.a.be
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 24)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + 2 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 2 q^{5} + q^{9} + 4 q^{11} + 2 q^{15} + 2 q^{17} - 4 q^{19} + 8 q^{23} - q^{25} + q^{27} + 6 q^{29} + 8 q^{31} + 4 q^{33} - 6 q^{37} + 6 q^{41} - 4 q^{43} + 2 q^{45} - 7 q^{49} + 2 q^{51} - 2 q^{53} + 8 q^{55} - 4 q^{57} + 4 q^{59} - 2 q^{61} - 4 q^{67} + 8 q^{69} + 8 q^{71} - 10 q^{73} - q^{75} + 8 q^{79} + q^{81} - 4 q^{83} + 4 q^{85} + 6 q^{87} + 6 q^{89} + 8 q^{93} - 8 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 1.00000 0 2.00000 0 0 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.be 1
4.b odd 2 1 4056.2.a.i 1
13.b even 2 1 48.2.a.a 1
39.d odd 2 1 144.2.a.b 1
52.b odd 2 1 24.2.a.a 1
52.f even 4 2 4056.2.c.e 2
65.d even 2 1 1200.2.a.d 1
65.h odd 4 2 1200.2.f.b 2
91.b odd 2 1 2352.2.a.i 1
91.r even 6 2 2352.2.q.l 2
91.s odd 6 2 2352.2.q.r 2
104.e even 2 1 192.2.a.b 1
104.h odd 2 1 192.2.a.d 1
117.n odd 6 2 1296.2.i.e 2
117.t even 6 2 1296.2.i.m 2
143.d odd 2 1 5808.2.a.s 1
156.h even 2 1 72.2.a.a 1
195.e odd 2 1 3600.2.a.v 1
195.s even 4 2 3600.2.f.r 2
208.o odd 4 2 768.2.d.e 2
208.p even 4 2 768.2.d.d 2
260.g odd 2 1 600.2.a.h 1
260.p even 4 2 600.2.f.e 2
273.g even 2 1 7056.2.a.q 1
312.b odd 2 1 576.2.a.b 1
312.h even 2 1 576.2.a.d 1
364.h even 2 1 1176.2.a.i 1
364.x even 6 2 1176.2.q.a 2
364.bl odd 6 2 1176.2.q.i 2
468.x even 6 2 648.2.i.b 2
468.bg odd 6 2 648.2.i.g 2
520.b odd 2 1 4800.2.a.q 1
520.p even 2 1 4800.2.a.cc 1
520.bc even 4 2 4800.2.f.d 2
520.bg odd 4 2 4800.2.f.bg 2
572.b even 2 1 2904.2.a.c 1
624.v even 4 2 2304.2.d.i 2
624.bi odd 4 2 2304.2.d.k 2
728.b even 2 1 9408.2.a.h 1
728.l odd 2 1 9408.2.a.cc 1
780.d even 2 1 1800.2.a.m 1
780.w odd 4 2 1800.2.f.c 2
884.h odd 2 1 6936.2.a.p 1
988.g even 2 1 8664.2.a.j 1
1092.d odd 2 1 3528.2.a.d 1
1092.by even 6 2 3528.2.s.j 2
1092.ct odd 6 2 3528.2.s.y 2
1716.m odd 2 1 8712.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 52.b odd 2 1
48.2.a.a 1 13.b even 2 1
72.2.a.a 1 156.h even 2 1
144.2.a.b 1 39.d odd 2 1
192.2.a.b 1 104.e even 2 1
192.2.a.d 1 104.h odd 2 1
576.2.a.b 1 312.b odd 2 1
576.2.a.d 1 312.h even 2 1
600.2.a.h 1 260.g odd 2 1
600.2.f.e 2 260.p even 4 2
648.2.i.b 2 468.x even 6 2
648.2.i.g 2 468.bg odd 6 2
768.2.d.d 2 208.p even 4 2
768.2.d.e 2 208.o odd 4 2
1176.2.a.i 1 364.h even 2 1
1176.2.q.a 2 364.x even 6 2
1176.2.q.i 2 364.bl odd 6 2
1200.2.a.d 1 65.d even 2 1
1200.2.f.b 2 65.h odd 4 2
1296.2.i.e 2 117.n odd 6 2
1296.2.i.m 2 117.t even 6 2
1800.2.a.m 1 780.d even 2 1
1800.2.f.c 2 780.w odd 4 2
2304.2.d.i 2 624.v even 4 2
2304.2.d.k 2 624.bi odd 4 2
2352.2.a.i 1 91.b odd 2 1
2352.2.q.l 2 91.r even 6 2
2352.2.q.r 2 91.s odd 6 2
2904.2.a.c 1 572.b even 2 1
3528.2.a.d 1 1092.d odd 2 1
3528.2.s.j 2 1092.by even 6 2
3528.2.s.y 2 1092.ct odd 6 2
3600.2.a.v 1 195.e odd 2 1
3600.2.f.r 2 195.s even 4 2
4056.2.a.i 1 4.b odd 2 1
4056.2.c.e 2 52.f even 4 2
4800.2.a.q 1 520.b odd 2 1
4800.2.a.cc 1 520.p even 2 1
4800.2.f.d 2 520.bc even 4 2
4800.2.f.bg 2 520.bg odd 4 2
5808.2.a.s 1 143.d odd 2 1
6936.2.a.p 1 884.h odd 2 1
7056.2.a.q 1 273.g even 2 1
8112.2.a.be 1 1.a even 1 1 trivial
8664.2.a.j 1 988.g even 2 1
8712.2.a.u 1 1716.m odd 2 1
9408.2.a.h 1 728.b even 2 1
9408.2.a.cc 1 728.l odd 2 1

Hecke kernels

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

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 4 \) 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
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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