Properties

Label 8281.2.a.b
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 4q^{6} + q^{9} + O(q^{10}) \) \( q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 4q^{6} + q^{9} - 2q^{10} + 2q^{11} + 4q^{12} + 2q^{15} - 4q^{16} - 6q^{17} - 2q^{18} - 3q^{19} + 2q^{20} - 4q^{22} + 3q^{23} - 4q^{25} - 4q^{27} + 3q^{29} - 4q^{30} - 3q^{31} + 8q^{32} + 4q^{33} + 12q^{34} + 2q^{36} - 6q^{37} + 6q^{38} + 10q^{41} - q^{43} + 4q^{44} + q^{45} - 6q^{46} + 11q^{47} - 8q^{48} + 8q^{50} - 12q^{51} - 9q^{53} + 8q^{54} + 2q^{55} - 6q^{57} - 6q^{58} - 8q^{59} + 4q^{60} + 8q^{61} + 6q^{62} - 8q^{64} - 8q^{66} + 12q^{67} - 12q^{68} + 6q^{69} + 14q^{71} - 9q^{73} + 12q^{74} - 8q^{75} - 6q^{76} - 9q^{79} - 4q^{80} - 11q^{81} - 20q^{82} + 11q^{83} - 6q^{85} + 2q^{86} + 6q^{87} - 5q^{89} - 2q^{90} + 6q^{92} - 6q^{93} - 22q^{94} - 3q^{95} + 16q^{96} - 9q^{97} + 2q^{99} + O(q^{100}) \)

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
−2.00000 2.00000 2.00000 1.00000 −4.00000 0 0 1.00000 −2.00000
\(n\): 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.b 1
7.b odd 2 1 8281.2.a.a 1
13.b even 2 1 8281.2.a.m 1
13.d odd 4 2 637.2.c.c yes 2
91.b odd 2 1 8281.2.a.k 1
91.i even 4 2 637.2.c.a 2
91.z odd 12 4 637.2.r.a 4
91.bb even 12 4 637.2.r.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.a 2 91.i even 4 2
637.2.c.c yes 2 13.d odd 4 2
637.2.r.a 4 91.z odd 12 4
637.2.r.c 4 91.bb even 12 4
8281.2.a.a 1 7.b odd 2 1
8281.2.a.b 1 1.a even 1 1 trivial
8281.2.a.k 1 91.b odd 2 1
8281.2.a.m 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2} + 2 \)
\( T_{3} - 2 \)
\( T_{5} - 1 \)
\( T_{11} - 2 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -2 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( T \)
$17$ \( 6 + T \)
$19$ \( 3 + T \)
$23$ \( -3 + T \)
$29$ \( -3 + T \)
$31$ \( 3 + T \)
$37$ \( 6 + T \)
$41$ \( -10 + T \)
$43$ \( 1 + T \)
$47$ \( -11 + T \)
$53$ \( 9 + T \)
$59$ \( 8 + T \)
$61$ \( -8 + T \)
$67$ \( -12 + T \)
$71$ \( -14 + T \)
$73$ \( 9 + T \)
$79$ \( 9 + T \)
$83$ \( -11 + T \)
$89$ \( 5 + T \)
$97$ \( 9 + T \)
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