Properties

Label 8281.2.a.l
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

\(f(q)\) \(=\) \( q + 2q^{2} + 2q^{4} - 3q^{5} - 3q^{9} + O(q^{10}) \) \( q + 2q^{2} + 2q^{4} - 3q^{5} - 3q^{9} - 6q^{10} + 6q^{11} - 4q^{16} - 4q^{17} - 6q^{18} + 5q^{19} - 6q^{20} + 12q^{22} + 3q^{23} + 4q^{25} - 5q^{29} - 3q^{31} - 8q^{32} - 8q^{34} - 6q^{36} + 4q^{37} + 10q^{38} - 6q^{41} - q^{43} + 12q^{44} + 9q^{45} + 6q^{46} + 7q^{47} + 8q^{50} - 9q^{53} - 18q^{55} - 10q^{58} + 8q^{59} + 10q^{61} - 6q^{62} - 8q^{64} + 6q^{67} - 8q^{68} + 8q^{71} - 13q^{73} + 8q^{74} + 10q^{76} + 3q^{79} + 12q^{80} + 9q^{81} - 12q^{82} + 15q^{83} + 12q^{85} - 2q^{86} + 3q^{89} + 18q^{90} + 6q^{92} + 14q^{94} - 15q^{95} + 7q^{97} - 18q^{99} + O(q^{100}) \)

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
2.00000 0 2.00000 −3.00000 0 0 0 −3.00000 −6.00000
\(n\): 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.l 1
7.b odd 2 1 1183.2.a.b 1
13.b even 2 1 637.2.a.a 1
39.d odd 2 1 5733.2.a.l 1
91.b odd 2 1 91.2.a.a 1
91.i even 4 2 1183.2.c.b 2
91.r even 6 2 637.2.e.d 2
91.s odd 6 2 637.2.e.e 2
273.g even 2 1 819.2.a.f 1
364.h even 2 1 1456.2.a.g 1
455.h odd 2 1 2275.2.a.h 1
728.b even 2 1 5824.2.a.t 1
728.l odd 2 1 5824.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.a 1 91.b odd 2 1
637.2.a.a 1 13.b even 2 1
637.2.e.d 2 91.r even 6 2
637.2.e.e 2 91.s odd 6 2
819.2.a.f 1 273.g even 2 1
1183.2.a.b 1 7.b odd 2 1
1183.2.c.b 2 91.i even 4 2
1456.2.a.g 1 364.h even 2 1
2275.2.a.h 1 455.h odd 2 1
5733.2.a.l 1 39.d odd 2 1
5824.2.a.s 1 728.l odd 2 1
5824.2.a.t 1 728.b even 2 1
8281.2.a.l 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2} - 2 \)
\( T_{3} \)
\( T_{5} + 3 \)
\( T_{11} - 6 \)
\( T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( 3 + T \)
$7$ \( T \)
$11$ \( -6 + T \)
$13$ \( T \)
$17$ \( 4 + T \)
$19$ \( -5 + T \)
$23$ \( -3 + T \)
$29$ \( 5 + T \)
$31$ \( 3 + T \)
$37$ \( -4 + T \)
$41$ \( 6 + T \)
$43$ \( 1 + T \)
$47$ \( -7 + T \)
$53$ \( 9 + T \)
$59$ \( -8 + T \)
$61$ \( -10 + T \)
$67$ \( -6 + T \)
$71$ \( -8 + T \)
$73$ \( 13 + T \)
$79$ \( -3 + T \)
$83$ \( -15 + T \)
$89$ \( -3 + T \)
$97$ \( -7 + T \)
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