Properties

Label 8281.2.a.br
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $4$

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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: \(\Q(\sqrt{3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 11x^{2} + 12x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + q^{4} + \beta_{3} q^{5} + \beta_{2} q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + q^{4} + \beta_{3} q^{5} + \beta_{2} q^{8} - 3 q^{9} - \beta_1 q^{10} - 2 \beta_{2} q^{11} - 5 q^{16} + \beta_1 q^{17} + 3 \beta_{2} q^{18} - 2 \beta_{3} q^{19} + \beta_{3} q^{20} + 6 q^{22} - 4 q^{23} + 8 q^{25} + q^{29} + 3 \beta_{2} q^{32} - 3 \beta_{3} q^{34} - 3 q^{36} + \beta_{2} q^{37} + 2 \beta_1 q^{38} + \beta_1 q^{40} - \beta_{3} q^{41} - 6 q^{43} - 2 \beta_{2} q^{44} - 3 \beta_{3} q^{45} + 4 \beta_{2} q^{46} + 2 \beta_{3} q^{47} - 8 \beta_{2} q^{50} + 5 q^{53} - 2 \beta_1 q^{55} - \beta_{2} q^{58} + 2 \beta_{3} q^{59} + \beta_1 q^{61} + q^{64} + 8 \beta_{2} q^{67} + \beta_1 q^{68} + 6 \beta_{2} q^{71} - 3 \beta_{2} q^{72} - 3 \beta_{3} q^{73} - 3 q^{74} - 2 \beta_{3} q^{76} - 6 q^{79} - 5 \beta_{3} q^{80} + 9 q^{81} + \beta_1 q^{82} - 2 \beta_{3} q^{83} + 13 \beta_{2} q^{85} + 6 \beta_{2} q^{86} - 6 q^{88} + 2 \beta_{3} q^{89} + 3 \beta_1 q^{90} - 4 q^{92} - 2 \beta_1 q^{94} - 26 q^{95} - 2 \beta_{3} q^{97} + 6 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 12 q^{9} - 20 q^{16} + 24 q^{22} - 16 q^{23} + 32 q^{25} + 4 q^{29} - 12 q^{36} - 24 q^{43} + 20 q^{53} + 4 q^{64} - 12 q^{74} - 24 q^{79} + 36 q^{81} - 24 q^{88} - 16 q^{92} - 104 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 11x^{2} + 12x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} - 3\nu^{2} - 23\nu + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\nu^{3} + 6\nu^{2} + 48\nu - 25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{3} + 27\beta_{2} + 3\beta _1 + 19 ) / 2 \) 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
0.429275
4.03483
−3.03483
0.570725
−1.73205 0 1.00000 −3.60555 0 0 1.73205 −3.00000 6.24500
1.2 −1.73205 0 1.00000 3.60555 0 0 1.73205 −3.00000 −6.24500
1.3 1.73205 0 1.00000 −3.60555 0 0 −1.73205 −3.00000 −6.24500
1.4 1.73205 0 1.00000 3.60555 0 0 −1.73205 −3.00000 6.24500
\(n\): 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
13.b even 2 1 inner
91.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.br 4
7.b odd 2 1 inner 8281.2.a.br 4
13.b even 2 1 inner 8281.2.a.br 4
13.f odd 12 2 637.2.q.d 4
91.b odd 2 1 inner 8281.2.a.br 4
91.w even 12 2 637.2.u.f 4
91.x odd 12 2 637.2.k.e 4
91.ba even 12 2 637.2.k.e 4
91.bc even 12 2 637.2.q.d 4
91.bd odd 12 2 637.2.u.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.e 4 91.x odd 12 2
637.2.k.e 4 91.ba even 12 2
637.2.q.d 4 13.f odd 12 2
637.2.q.d 4 91.bc even 12 2
637.2.u.f 4 91.w even 12 2
637.2.u.f 4 91.bd odd 12 2
8281.2.a.br 4 1.a even 1 1 trivial
8281.2.a.br 4 7.b odd 2 1 inner
8281.2.a.br 4 13.b even 2 1 inner
8281.2.a.br 4 91.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} - 13 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{17}^{2} - 39 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 39)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{4} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$53$ \( (T - 5)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 39)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 117)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
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