Properties

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

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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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{23})\)
Defining polynomial: \(x^{4} - 24 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{2} q^{3} - q^{4} -\beta_{1} q^{5} -\beta_{2} q^{6} + 3 q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{2} q^{3} - q^{4} -\beta_{1} q^{5} -\beta_{2} q^{6} + 3 q^{8} - q^{9} + \beta_{1} q^{10} + ( -1 + \beta_{3} ) q^{11} -\beta_{2} q^{12} + ( -1 - \beta_{3} ) q^{15} - q^{16} + ( -\beta_{1} + 2 \beta_{2} ) q^{17} + q^{18} + 2 \beta_{2} q^{19} + \beta_{1} q^{20} + ( 1 - \beta_{3} ) q^{22} + ( -3 - \beta_{3} ) q^{23} + 3 \beta_{2} q^{24} + ( 7 + \beta_{3} ) q^{25} -4 \beta_{2} q^{27} + ( -4 + \beta_{3} ) q^{29} + ( 1 + \beta_{3} ) q^{30} + \beta_{2} q^{31} -5 q^{32} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( \beta_{1} - 2 \beta_{2} ) q^{34} + q^{36} + ( 2 - \beta_{3} ) q^{37} -2 \beta_{2} q^{38} -3 \beta_{1} q^{40} + ( \beta_{1} - 5 \beta_{2} ) q^{41} + ( 3 + \beta_{3} ) q^{43} + ( 1 - \beta_{3} ) q^{44} + \beta_{1} q^{45} + ( 3 + \beta_{3} ) q^{46} -2 \beta_{2} q^{47} -\beta_{2} q^{48} + ( -7 - \beta_{3} ) q^{50} + ( 3 - \beta_{3} ) q^{51} + ( -3 - 2 \beta_{3} ) q^{53} + 4 \beta_{2} q^{54} -11 \beta_{2} q^{55} + 4 q^{57} + ( 4 - \beta_{3} ) q^{58} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{59} + ( 1 + \beta_{3} ) q^{60} + ( \beta_{1} + 3 \beta_{2} ) q^{61} -\beta_{2} q^{62} + 7 q^{64} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{66} + ( -1 + \beta_{3} ) q^{67} + ( \beta_{1} - 2 \beta_{2} ) q^{68} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{69} -6 q^{71} -3 q^{72} + ( \beta_{1} + 6 \beta_{2} ) q^{73} + ( -2 + \beta_{3} ) q^{74} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{75} -2 \beta_{2} q^{76} + ( -7 + \beta_{3} ) q^{79} + \beta_{1} q^{80} -5 q^{81} + ( -\beta_{1} + 5 \beta_{2} ) q^{82} + 7 \beta_{2} q^{83} + ( 10 - \beta_{3} ) q^{85} + ( -3 - \beta_{3} ) q^{86} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{87} + ( -3 + 3 \beta_{3} ) q^{88} + ( 4 \beta_{1} - \beta_{2} ) q^{89} -\beta_{1} q^{90} + ( 3 + \beta_{3} ) q^{92} + 2 q^{93} + 2 \beta_{2} q^{94} + ( -2 - 2 \beta_{3} ) q^{95} -5 \beta_{2} q^{96} + 3 \beta_{2} q^{97} + ( 1 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{4} + 12q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{4} + 12q^{8} - 4q^{9} - 4q^{11} - 4q^{15} - 4q^{16} + 4q^{18} + 4q^{22} - 12q^{23} + 28q^{25} - 16q^{29} + 4q^{30} - 20q^{32} + 4q^{36} + 8q^{37} + 12q^{43} + 4q^{44} + 12q^{46} - 28q^{50} + 12q^{51} - 12q^{53} + 16q^{57} + 16q^{58} + 4q^{60} + 28q^{64} - 4q^{67} - 24q^{71} - 12q^{72} - 8q^{74} - 28q^{79} - 20q^{81} + 40q^{85} - 12q^{86} - 12q^{88} + 12q^{92} + 8q^{93} - 8q^{95} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 24 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 13 \nu \)\()/11\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 12\)
\(\nu^{3}\)\(=\)\(11 \beta_{2} + 13 \beta_{1}\)

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
2.68406
−4.09827
4.09827
−2.68406
−1.00000 −1.41421 −1.00000 −2.68406 1.41421 0 3.00000 −1.00000 2.68406
1.2 −1.00000 −1.41421 −1.00000 4.09827 1.41421 0 3.00000 −1.00000 −4.09827
1.3 −1.00000 1.41421 −1.00000 −4.09827 −1.41421 0 3.00000 −1.00000 4.09827
1.4 −1.00000 1.41421 −1.00000 2.68406 −1.41421 0 3.00000 −1.00000 −2.68406
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.bn 4
7.b odd 2 1 inner 8281.2.a.bn 4
13.b even 2 1 8281.2.a.bv 4
13.e even 6 2 637.2.f.g 8
91.b odd 2 1 8281.2.a.bv 4
91.k even 6 2 637.2.h.k 8
91.l odd 6 2 637.2.h.k 8
91.p odd 6 2 637.2.g.h 8
91.t odd 6 2 637.2.f.g 8
91.u even 6 2 637.2.g.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.g 8 13.e even 6 2
637.2.f.g 8 91.t odd 6 2
637.2.g.h 8 91.p odd 6 2
637.2.g.h 8 91.u even 6 2
637.2.h.k 8 91.k even 6 2
637.2.h.k 8 91.l odd 6 2
8281.2.a.bn 4 1.a even 1 1 trivial
8281.2.a.bn 4 7.b odd 2 1 inner
8281.2.a.bv 4 13.b even 2 1
8281.2.a.bv 4 91.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2} + 1 \)
\( T_{3}^{2} - 2 \)
\( T_{5}^{4} - 24 T_{5}^{2} + 121 \)
\( T_{11}^{2} + 2 T_{11} - 22 \)
\( T_{17}^{4} - 32 T_{17}^{2} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( -2 + T^{2} )^{2} \)
$5$ \( 121 - 24 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -22 + 2 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 49 - 32 T^{2} + T^{4} \)
$19$ \( ( -8 + T^{2} )^{2} \)
$23$ \( ( -14 + 6 T + T^{2} )^{2} \)
$29$ \( ( -7 + 8 T + T^{2} )^{2} \)
$31$ \( ( -2 + T^{2} )^{2} \)
$37$ \( ( -19 - 4 T + T^{2} )^{2} \)
$41$ \( 841 - 104 T^{2} + T^{4} \)
$43$ \( ( -14 - 6 T + T^{2} )^{2} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( ( -83 + 6 T + T^{2} )^{2} \)
$59$ \( 196 - 156 T^{2} + T^{4} \)
$61$ \( 169 - 72 T^{2} + T^{4} \)
$67$ \( ( -22 + 2 T + T^{2} )^{2} \)
$71$ \( ( 6 + T )^{4} \)
$73$ \( 5329 - 192 T^{2} + T^{4} \)
$79$ \( ( 26 + 14 T + T^{2} )^{2} \)
$83$ \( ( -98 + T^{2} )^{2} \)
$89$ \( 33124 - 372 T^{2} + T^{4} \)
$97$ \( ( -18 + T^{2} )^{2} \)
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