Properties

Label 8281.2.a.bo
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.105456.1
Defining polynomial: \(x^{4} - 13 x^{2} + 39\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{4} -\beta_{1} q^{5} + \beta_{3} q^{6} -3 q^{8} + ( 4 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{4} -\beta_{1} q^{5} + \beta_{3} q^{6} -3 q^{8} + ( 4 + \beta_{2} ) q^{9} -\beta_{3} q^{10} + ( 2 + 3 \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{3} ) q^{12} + ( -7 - \beta_{2} ) q^{15} + ( -2 - \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + ( 3 + 3 \beta_{2} ) q^{18} -\beta_{1} q^{19} + ( -\beta_{1} + \beta_{3} ) q^{20} + ( 9 - \beta_{2} ) q^{22} + ( 4 + 2 \beta_{2} ) q^{23} -3 \beta_{1} q^{24} + ( 2 + \beta_{2} ) q^{25} + ( \beta_{1} + \beta_{3} ) q^{27} + \beta_{2} q^{29} + ( -3 - 6 \beta_{2} ) q^{30} + ( \beta_{1} - \beta_{3} ) q^{31} + ( 3 - \beta_{2} ) q^{32} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{33} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{34} + ( 1 - 2 \beta_{2} ) q^{36} + ( -4 + 2 \beta_{2} ) q^{37} -\beta_{3} q^{38} + 3 \beta_{1} q^{40} -2 \beta_{3} q^{41} + ( -1 + 5 \beta_{2} ) q^{43} + ( -7 + 4 \beta_{2} ) q^{44} + ( -4 \beta_{1} - \beta_{3} ) q^{45} + ( 6 + 2 \beta_{2} ) q^{46} + ( 3 \beta_{1} + \beta_{3} ) q^{47} + ( -2 \beta_{1} - \beta_{3} ) q^{48} + ( 3 + \beta_{2} ) q^{50} + ( 4 - 5 \beta_{2} ) q^{51} + ( 7 + 2 \beta_{2} ) q^{53} + 3 \beta_{1} q^{54} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{55} + ( -7 - \beta_{2} ) q^{57} + ( 3 - \beta_{2} ) q^{58} + ( -\beta_{1} - \beta_{3} ) q^{59} + ( -4 + 5 \beta_{2} ) q^{60} + 2 \beta_{1} q^{61} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{62} + ( 1 + 6 \beta_{2} ) q^{64} + ( 9 \beta_{1} - \beta_{3} ) q^{66} - q^{67} + ( 4 \beta_{1} - 3 \beta_{3} ) q^{68} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{69} -4 q^{71} + ( -12 - 3 \beta_{2} ) q^{72} + 2 \beta_{1} q^{73} + ( 6 - 6 \beta_{2} ) q^{74} + ( 2 \beta_{1} + \beta_{3} ) q^{75} + ( -\beta_{1} + \beta_{3} ) q^{76} + ( -2 + 2 \beta_{2} ) q^{79} + ( 2 \beta_{1} + \beta_{3} ) q^{80} + ( -2 + 4 \beta_{2} ) q^{81} + ( -6 \beta_{1} + 2 \beta_{3} ) q^{82} + ( \beta_{1} + \beta_{3} ) q^{83} + ( -4 + 5 \beta_{2} ) q^{85} + ( 15 - 6 \beta_{2} ) q^{86} + \beta_{3} q^{87} + ( -6 - 9 \beta_{2} ) q^{88} -3 \beta_{1} q^{89} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{90} -2 q^{92} + ( 4 - 5 \beta_{2} ) q^{93} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{94} + ( 7 + \beta_{2} ) q^{95} + ( 3 \beta_{1} - \beta_{3} ) q^{96} + ( 4 \beta_{1} - \beta_{3} ) q^{97} + ( 17 + 11 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + 2 q^{11} - 26 q^{15} - 6 q^{16} + 6 q^{18} + 38 q^{22} + 12 q^{23} + 6 q^{25} - 2 q^{29} + 14 q^{32} + 8 q^{36} - 20 q^{37} - 14 q^{43} - 36 q^{44} + 20 q^{46} + 10 q^{50} + 26 q^{51} + 24 q^{53} - 26 q^{57} + 14 q^{58} - 26 q^{60} - 8 q^{64} - 4 q^{67} - 16 q^{71} - 42 q^{72} + 36 q^{74} - 12 q^{79} - 16 q^{81} - 26 q^{85} + 72 q^{86} - 6 q^{88} - 8 q^{92} + 26 q^{93} + 26 q^{95} + 46 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 13 x^{2} + 39\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 7 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 7 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 7 \beta_{1}\)

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
−2.16731
2.16731
−2.88145
2.88145
−2.30278 −2.16731 3.30278 2.16731 4.99082 0 −3.00000 1.69722 −4.99082
1.2 −2.30278 2.16731 3.30278 −2.16731 −4.99082 0 −3.00000 1.69722 4.99082
1.3 1.30278 −2.88145 −0.302776 2.88145 −3.75389 0 −3.00000 5.30278 3.75389
1.4 1.30278 2.88145 −0.302776 −2.88145 3.75389 0 −3.00000 5.30278 −3.75389
\(n\): 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.bo 4
7.b odd 2 1 inner 8281.2.a.bo 4
13.b even 2 1 8281.2.a.bu 4
13.e even 6 2 637.2.f.h 8
91.b odd 2 1 8281.2.a.bu 4
91.k even 6 2 637.2.h.j 8
91.l odd 6 2 637.2.h.j 8
91.p odd 6 2 637.2.g.i 8
91.t odd 6 2 637.2.f.h 8
91.u even 6 2 637.2.g.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.h 8 13.e even 6 2
637.2.f.h 8 91.t odd 6 2
637.2.g.i 8 91.p odd 6 2
637.2.g.i 8 91.u even 6 2
637.2.h.j 8 91.k even 6 2
637.2.h.j 8 91.l odd 6 2
8281.2.a.bo 4 1.a even 1 1 trivial
8281.2.a.bo 4 7.b odd 2 1 inner
8281.2.a.bu 4 13.b even 2 1
8281.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{2} + T_{2} - 3 \)
\( T_{3}^{4} - 13 T_{3}^{2} + 39 \)
\( T_{5}^{4} - 13 T_{5}^{2} + 39 \)
\( T_{11}^{2} - T_{11} - 29 \)
\( T_{17}^{4} - 52 T_{17}^{2} + 39 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -3 + T + T^{2} )^{2} \)
$3$ \( 39 - 13 T^{2} + T^{4} \)
$5$ \( 39 - 13 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -29 - T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 39 - 52 T^{2} + T^{4} \)
$19$ \( 39 - 13 T^{2} + T^{4} \)
$23$ \( ( -4 - 6 T + T^{2} )^{2} \)
$29$ \( ( -3 + T + T^{2} )^{2} \)
$31$ \( 39 - 52 T^{2} + T^{4} \)
$37$ \( ( 12 + 10 T + T^{2} )^{2} \)
$41$ \( 5616 - 156 T^{2} + T^{4} \)
$43$ \( ( -69 + 7 T + T^{2} )^{2} \)
$47$ \( 351 - 156 T^{2} + T^{4} \)
$53$ \( ( 23 - 12 T + T^{2} )^{2} \)
$59$ \( 351 - 52 T^{2} + T^{4} \)
$61$ \( 624 - 52 T^{2} + T^{4} \)
$67$ \( ( 1 + T )^{4} \)
$71$ \( ( 4 + T )^{4} \)
$73$ \( 624 - 52 T^{2} + T^{4} \)
$79$ \( ( -4 + 6 T + T^{2} )^{2} \)
$83$ \( 351 - 52 T^{2} + T^{4} \)
$89$ \( 3159 - 117 T^{2} + T^{4} \)
$97$ \( 11271 - 247 T^{2} + T^{4} \)
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