Properties

Label 8281.2.a.bv
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{23})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 24x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{2} q^{3} - q^{4} + (\beta_{2} - \beta_1) q^{5} + \beta_{2} q^{6} - 3 q^{8} - q^{9} + (\beta_{2} - \beta_1) q^{10} + (\beta_{3} + 1) q^{11} - \beta_{2} q^{12} + ( - \beta_{3} + 1) q^{15}+ \cdots + ( - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{16} - 4 q^{18} + 4 q^{22} - 12 q^{23} + 28 q^{25} - 16 q^{29} + 4 q^{30} + 20 q^{32} + 4 q^{36} - 8 q^{37} + 12 q^{43} - 4 q^{44}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 24x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 13\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11\beta_{2} + 13\beta_1 \) 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
2.68406
−4.09827
4.09827
−2.68406
1.00000 −1.41421 −1.00000 −4.09827 −1.41421 0 −3.00000 −1.00000 −4.09827
1.2 1.00000 −1.41421 −1.00000 2.68406 −1.41421 0 −3.00000 −1.00000 2.68406
1.3 1.00000 1.41421 −1.00000 −2.68406 1.41421 0 −3.00000 −1.00000 −2.68406
1.4 1.00000 1.41421 −1.00000 4.09827 1.41421 0 −3.00000 −1.00000 4.09827
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)
\(13\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.bv 4
7.b odd 2 1 inner 8281.2.a.bv 4
13.b even 2 1 8281.2.a.bn 4
13.c even 3 2 637.2.f.g 8
91.b odd 2 1 8281.2.a.bn 4
91.g even 3 2 637.2.g.h 8
91.h even 3 2 637.2.h.k 8
91.m odd 6 2 637.2.g.h 8
91.n odd 6 2 637.2.f.g 8
91.v odd 6 2 637.2.h.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.g 8 13.c even 3 2
637.2.f.g 8 91.n odd 6 2
637.2.g.h 8 91.g even 3 2
637.2.g.h 8 91.m odd 6 2
637.2.h.k 8 91.h even 3 2
637.2.h.k 8 91.v odd 6 2
8281.2.a.bn 4 13.b even 2 1
8281.2.a.bn 4 91.b odd 2 1
8281.2.a.bv 4 1.a even 1 1 trivial
8281.2.a.bv 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{4} - 24T_{5}^{2} + 121 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 22 \) Copy content Toggle raw display
\( T_{17}^{4} - 32T_{17}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 24T^{2} + 121 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 22)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 32T^{2} + 49 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 14)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T - 7)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T - 19)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 104T^{2} + 841 \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T - 14)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 83)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 156T^{2} + 196 \) Copy content Toggle raw display
$61$ \( T^{4} - 72T^{2} + 169 \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T - 22)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 192T^{2} + 5329 \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 26)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 372 T^{2} + 33124 \) Copy content Toggle raw display
$97$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
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