Properties

Label 8281.2.a.g
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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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

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

\(f(q)\) \(=\) \( q - q^{2} + 3q^{3} - q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 6q^{9} + O(q^{10}) \) \( q - q^{2} + 3q^{3} - q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 6q^{9} - 3q^{10} + 3q^{11} - 3q^{12} + 9q^{15} - q^{16} + 2q^{17} - 6q^{18} - q^{19} - 3q^{20} - 3q^{22} + 9q^{24} + 4q^{25} + 9q^{27} + 7q^{29} - 9q^{30} + 3q^{31} - 5q^{32} + 9q^{33} - 2q^{34} - 6q^{36} - 2q^{37} + q^{38} + 9q^{40} + 3q^{41} - 7q^{43} - 3q^{44} + 18q^{45} + q^{47} - 3q^{48} - 4q^{50} + 6q^{51} + 3q^{53} - 9q^{54} + 9q^{55} - 3q^{57} - 7q^{58} - 4q^{59} - 9q^{60} + 13q^{61} - 3q^{62} + 7q^{64} - 9q^{66} + 3q^{67} - 2q^{68} - 13q^{71} + 18q^{72} - 13q^{73} + 2q^{74} + 12q^{75} + q^{76} - 3q^{79} - 3q^{80} + 9q^{81} - 3q^{82} + 6q^{85} + 7q^{86} + 21q^{87} + 9q^{88} + 6q^{89} - 18q^{90} + 9q^{93} - q^{94} - 3q^{95} - 15q^{96} - 5q^{97} + 18q^{99} + O(q^{100}) \)

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.

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 3.00000 −1.00000 3.00000 −3.00000 0 3.00000 6.00000 −3.00000
\(n\): 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.g 1
7.b odd 2 1 8281.2.a.c 1
7.d odd 6 2 1183.2.e.c 2
13.b even 2 1 8281.2.a.j 1
13.e even 6 2 637.2.f.a 2
91.b odd 2 1 8281.2.a.i 1
91.k even 6 2 637.2.h.a 2
91.l odd 6 2 91.2.h.a yes 2
91.p odd 6 2 91.2.g.a 2
91.s odd 6 2 1183.2.e.a 2
91.t odd 6 2 637.2.f.b 2
91.u even 6 2 637.2.g.a 2
273.y even 6 2 819.2.n.c 2
273.br even 6 2 819.2.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 91.p odd 6 2
91.2.h.a yes 2 91.l odd 6 2
637.2.f.a 2 13.e even 6 2
637.2.f.b 2 91.t odd 6 2
637.2.g.a 2 91.u even 6 2
637.2.h.a 2 91.k even 6 2
819.2.n.c 2 273.y even 6 2
819.2.s.a 2 273.br even 6 2
1183.2.e.a 2 91.s odd 6 2
1183.2.e.c 2 7.d odd 6 2
8281.2.a.c 1 7.b odd 2 1
8281.2.a.g 1 1.a even 1 1 trivial
8281.2.a.i 1 91.b odd 2 1
8281.2.a.j 1 13.b even 2 1

Hecke kernels

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

\( T_{2} + 1 \)
\( T_{3} - 3 \)
\( T_{5} - 3 \)
\( T_{11} - 3 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -3 + T \)
$5$ \( -3 + T \)
$7$ \( T \)
$11$ \( -3 + T \)
$13$ \( T \)
$17$ \( -2 + T \)
$19$ \( 1 + T \)
$23$ \( T \)
$29$ \( -7 + T \)
$31$ \( -3 + T \)
$37$ \( 2 + T \)
$41$ \( -3 + T \)
$43$ \( 7 + T \)
$47$ \( -1 + T \)
$53$ \( -3 + T \)
$59$ \( 4 + T \)
$61$ \( -13 + T \)
$67$ \( -3 + T \)
$71$ \( 13 + T \)
$73$ \( 13 + T \)
$79$ \( 3 + T \)
$83$ \( T \)
$89$ \( -6 + T \)
$97$ \( 5 + T \)
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