Properties

Label 8281.2.a.u
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $2$
CM discriminant -91
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{4} -\beta q^{5} -3 q^{9} +O(q^{10})\) \( q -2 q^{4} -\beta q^{5} -3 q^{9} + 4 q^{16} -\beta q^{19} + 2 \beta q^{20} - q^{23} + 8 q^{25} -5 q^{29} + 3 \beta q^{31} + 6 q^{36} -2 \beta q^{41} -9 q^{43} + 3 \beta q^{45} + \beta q^{47} + 11 q^{53} + 4 \beta q^{59} -8 q^{64} -3 \beta q^{73} + 2 \beta q^{76} + 15 q^{79} -4 \beta q^{80} + 9 q^{81} + 5 \beta q^{83} + \beta q^{89} + 2 q^{92} + 13 q^{95} + 5 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 6 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{4} - 6 q^{9} + 8 q^{16} - 2 q^{23} + 16 q^{25} - 10 q^{29} + 12 q^{36} - 18 q^{43} + 22 q^{53} - 16 q^{64} + 30 q^{79} + 18 q^{81} + 4 q^{92} + 26 q^{95} + O(q^{100}) \)

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
2.30278
−1.30278
0 0 −2.00000 −3.60555 0 0 0 −3.00000 0
1.2 0 0 −2.00000 3.60555 0 0 0 −3.00000 0
\(n\): 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
91.b odd 2 1 CM by \(\Q(\sqrt{-91}) \)
7.b odd 2 1 inner
13.b even 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.u 2
7.b odd 2 1 inner 8281.2.a.u 2
13.b even 2 1 inner 8281.2.a.u 2
13.d odd 4 2 637.2.c.b 2
91.b odd 2 1 CM 8281.2.a.u 2
91.i even 4 2 637.2.c.b 2
91.z odd 12 4 637.2.r.b 4
91.bb even 12 4 637.2.r.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.b 2 13.d odd 4 2
637.2.c.b 2 91.i even 4 2
637.2.r.b 4 91.z odd 12 4
637.2.r.b 4 91.bb even 12 4
8281.2.a.u 2 1.a even 1 1 trivial
8281.2.a.u 2 7.b odd 2 1 inner
8281.2.a.u 2 13.b even 2 1 inner
8281.2.a.u 2 91.b odd 2 1 CM

Hecke kernels

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

\( T_{2} \)
\( T_{3} \)
\( T_{5}^{2} - 13 \)
\( T_{11} \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -13 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( -13 + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( -117 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( -52 + T^{2} \)
$43$ \( ( 9 + T )^{2} \)
$47$ \( -13 + T^{2} \)
$53$ \( ( -11 + T )^{2} \)
$59$ \( -208 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( -117 + T^{2} \)
$79$ \( ( -15 + T )^{2} \)
$83$ \( -325 + T^{2} \)
$89$ \( -13 + T^{2} \)
$97$ \( -325 + T^{2} \)
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