Properties

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

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + 2 \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta q^{5} + 2 \beta q^{7} + q^{9} - 2 \beta q^{11} + \beta q^{15} - 3 q^{17} + 2 \beta q^{19} + 2 \beta q^{21} + 6 q^{23} - 2 q^{25} + q^{27} + 9 q^{29} - 2 \beta q^{33} + 6 q^{35} - 3 \beta q^{37} + 5 \beta q^{41} + 2 q^{43} + \beta q^{45} - 2 \beta q^{47} + 5 q^{49} - 3 q^{51} - 9 q^{53} - 6 q^{55} + 2 \beta q^{57} + 8 \beta q^{59} - 11 q^{61} + 2 \beta q^{63} + 6 \beta q^{67} + 6 q^{69} - 6 \beta q^{71} + 3 \beta q^{73} - 2 q^{75} - 12 q^{77} + 8 q^{79} + q^{81} - 2 \beta q^{83} - 3 \beta q^{85} + 9 q^{87} - 4 \beta q^{89} + 6 q^{95} + 4 \beta q^{97} - 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{9} - 6 q^{17} + 12 q^{23} - 4 q^{25} + 2 q^{27} + 18 q^{29} + 12 q^{35} + 4 q^{43} + 10 q^{49} - 6 q^{51} - 18 q^{53} - 12 q^{55} - 22 q^{61} + 12 q^{69} - 4 q^{75} - 24 q^{77} + 16 q^{79} + 2 q^{81} + 18 q^{87} + 12 q^{95}+O(q^{100}) \) 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
−1.73205
1.73205
0 1.00000 0 −1.73205 0 −3.46410 0 1.00000 0
1.2 0 1.00000 0 1.73205 0 3.46410 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8112.2.a.bt 2
4.b odd 2 1 2028.2.a.h 2
12.b even 2 1 6084.2.a.u 2
13.b even 2 1 inner 8112.2.a.bt 2
13.f odd 12 2 624.2.bv.a 2
39.k even 12 2 1872.2.by.b 2
52.b odd 2 1 2028.2.a.h 2
52.f even 4 2 2028.2.b.b 2
52.i odd 6 2 2028.2.i.h 4
52.j odd 6 2 2028.2.i.h 4
52.l even 12 2 156.2.q.a 2
52.l even 12 2 2028.2.q.a 2
156.h even 2 1 6084.2.a.u 2
156.l odd 4 2 6084.2.b.c 2
156.v odd 12 2 468.2.t.c 2
260.bc even 12 2 3900.2.cd.a 2
260.be odd 12 2 3900.2.bw.e 4
260.bl odd 12 2 3900.2.bw.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.q.a 2 52.l even 12 2
468.2.t.c 2 156.v odd 12 2
624.2.bv.a 2 13.f odd 12 2
1872.2.by.b 2 39.k even 12 2
2028.2.a.h 2 4.b odd 2 1
2028.2.a.h 2 52.b odd 2 1
2028.2.b.b 2 52.f even 4 2
2028.2.i.h 4 52.i odd 6 2
2028.2.i.h 4 52.j odd 6 2
2028.2.q.a 2 52.l even 12 2
3900.2.bw.e 4 260.be odd 12 2
3900.2.bw.e 4 260.bl odd 12 2
3900.2.cd.a 2 260.bc even 12 2
6084.2.a.u 2 12.b even 2 1
6084.2.a.u 2 156.h even 2 1
6084.2.b.c 2 156.l odd 4 2
8112.2.a.bt 2 1.a even 1 1 trivial
8112.2.a.bt 2 13.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 12 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 27 \) Copy content Toggle raw display
$41$ \( T^{2} - 75 \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 12 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 192 \) Copy content Toggle raw display
$61$ \( (T + 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 108 \) Copy content Toggle raw display
$71$ \( T^{2} - 108 \) Copy content Toggle raw display
$73$ \( T^{2} - 27 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 12 \) Copy content Toggle raw display
$89$ \( T^{2} - 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 48 \) Copy content Toggle raw display
show more
show less