Properties

Label 8112.2.a.bx
Level $8112$
Weight $2$
Character orbit 8112.a
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $2$
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: \(2\)
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 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)
 

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

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + (\beta + 3) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta q^{5} + (\beta + 3) q^{7} + q^{9} + (\beta + 3) q^{11} + \beta q^{15} - 3 \beta q^{17} + ( - \beta + 3) q^{19} + (\beta + 3) q^{21} + ( - 3 \beta + 3) q^{23} - 2 q^{25} + q^{27} - 3 q^{29} + ( - 2 \beta + 6) q^{31} + (\beta + 3) q^{33} + (3 \beta + 3) q^{35} - 3 q^{37} + (2 \beta - 3) q^{41} + ( - 3 \beta - 1) q^{43} + \beta q^{45} + (\beta - 3) q^{47} + (6 \beta + 5) q^{49} - 3 \beta q^{51} + 3 q^{53} + (3 \beta + 3) q^{55} + ( - \beta + 3) q^{57} + 8 \beta q^{59} + ( - 3 \beta + 10) q^{61} + (\beta + 3) q^{63} + (\beta + 9) q^{67} + ( - 3 \beta + 3) q^{69} + (3 \beta + 3) q^{71} + 7 \beta q^{73} - 2 q^{75} + (6 \beta + 12) q^{77} + (6 \beta + 2) q^{79} + q^{81} + ( - 5 \beta - 3) q^{83} - 9 q^{85} - 3 q^{87} + (2 \beta - 6) q^{89} + ( - 2 \beta + 6) q^{93} + (3 \beta - 3) q^{95} - 6 q^{97} + (\beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{7} + 2 q^{9} + 6 q^{11} + 6 q^{19} + 6 q^{21} + 6 q^{23} - 4 q^{25} + 2 q^{27} - 6 q^{29} + 12 q^{31} + 6 q^{33} + 6 q^{35} - 6 q^{37} - 6 q^{41} - 2 q^{43} - 6 q^{47} + 10 q^{49} + 6 q^{53} + 6 q^{55} + 6 q^{57} + 20 q^{61} + 6 q^{63} + 18 q^{67} + 6 q^{69} + 6 q^{71} - 4 q^{75} + 24 q^{77} + 4 q^{79} + 2 q^{81} - 6 q^{83} - 18 q^{85} - 6 q^{87} - 12 q^{89} + 12 q^{93} - 6 q^{95} - 12 q^{97} + 6 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 1.00000 0 −1.73205 0 1.26795 0 1.00000 0
1.2 0 1.00000 0 1.73205 0 4.73205 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.bx 2
4.b odd 2 1 1014.2.a.j 2
12.b even 2 1 3042.2.a.s 2
13.b even 2 1 8112.2.a.bq 2
13.f odd 12 2 624.2.bv.d 4
39.k even 12 2 1872.2.by.k 4
52.b odd 2 1 1014.2.a.h 2
52.f even 4 2 1014.2.b.d 4
52.i odd 6 2 1014.2.e.j 4
52.j odd 6 2 1014.2.e.h 4
52.l even 12 2 78.2.i.b 4
52.l even 12 2 1014.2.i.f 4
156.h even 2 1 3042.2.a.v 2
156.l odd 4 2 3042.2.b.l 4
156.v odd 12 2 234.2.l.a 4
260.bc even 12 2 1950.2.bc.c 4
260.be odd 12 2 1950.2.y.a 4
260.bl odd 12 2 1950.2.y.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b 4 52.l even 12 2
234.2.l.a 4 156.v odd 12 2
624.2.bv.d 4 13.f odd 12 2
1014.2.a.h 2 52.b odd 2 1
1014.2.a.j 2 4.b odd 2 1
1014.2.b.d 4 52.f even 4 2
1014.2.e.h 4 52.j odd 6 2
1014.2.e.j 4 52.i odd 6 2
1014.2.i.f 4 52.l even 12 2
1872.2.by.k 4 39.k even 12 2
1950.2.y.a 4 260.be odd 12 2
1950.2.y.h 4 260.bl odd 12 2
1950.2.bc.c 4 260.bc even 12 2
3042.2.a.s 2 12.b even 2 1
3042.2.a.v 2 156.h even 2 1
3042.2.b.l 4 156.l odd 4 2
8112.2.a.bq 2 13.b even 2 1
8112.2.a.bx 2 1.a even 1 1 trivial

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
\( T_{7}^{2} - 6T_{7} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 6 \) 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
$7$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 27 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 18 \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$37$ \( (T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 3 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 192 \) Copy content Toggle raw display
$61$ \( T^{2} - 20T + 73 \) Copy content Toggle raw display
$67$ \( T^{2} - 18T + 78 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 18 \) Copy content Toggle raw display
$73$ \( T^{2} - 147 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 66 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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