Properties

Label 8281.2.a.e
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9} + 3 q^{11} - q^{16} - 7 q^{17} + 3 q^{18} - 7 q^{19} - 3 q^{22} - 6 q^{23} - 5 q^{25} - 5 q^{29} - 5 q^{32} + 7 q^{34} + 3 q^{36} - 8 q^{37} + 7 q^{38} + 2 q^{43} - 3 q^{44} + 6 q^{46} + 7 q^{47} + 5 q^{50} - 3 q^{53} + 5 q^{58} - 7 q^{59} + 7 q^{61} + 7 q^{64} + 3 q^{67} + 7 q^{68} + 5 q^{71} - 9 q^{72} + 14 q^{73} + 8 q^{74} + 7 q^{76} - 6 q^{79} + 9 q^{81} - 2 q^{86} + 9 q^{88} + 6 q^{92} - 7 q^{94} - 14 q^{97} - 9 q^{99}+O(q^{100}) \) 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
−1.00000 0 −1.00000 0 0 0 3.00000 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 8281.2.a.e 1
7.b odd 2 1 8281.2.a.f 1
7.d odd 6 2 1183.2.e.b 2
13.b even 2 1 637.2.a.d 1
39.d odd 2 1 5733.2.a.d 1
91.b odd 2 1 637.2.a.c 1
91.r even 6 2 637.2.e.a 2
91.s odd 6 2 91.2.e.a 2
273.g even 2 1 5733.2.a.c 1
273.ba even 6 2 819.2.j.b 2
364.x even 6 2 1456.2.r.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 91.s odd 6 2
637.2.a.c 1 91.b odd 2 1
637.2.a.d 1 13.b even 2 1
637.2.e.a 2 91.r even 6 2
819.2.j.b 2 273.ba even 6 2
1183.2.e.b 2 7.d odd 6 2
1456.2.r.g 2 364.x even 6 2
5733.2.a.c 1 273.g even 2 1
5733.2.a.d 1 39.d odd 2 1
8281.2.a.e 1 1.a even 1 1 trivial
8281.2.a.f 1 7.b odd 2 1

Hecke kernels

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

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{17} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 7 \) Copy content Toggle raw display
$19$ \( T + 7 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 5 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 8 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T - 7 \) Copy content Toggle raw display
$53$ \( T + 3 \) Copy content Toggle raw display
$59$ \( T + 7 \) Copy content Toggle raw display
$61$ \( T - 7 \) Copy content Toggle raw display
$67$ \( T - 3 \) Copy content Toggle raw display
$71$ \( T - 5 \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T + 6 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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