Properties

Label 8281.2.a.s
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.73205 −1.00000 1.00000 1.73205 1.73205 0 1.73205 −2.00000 −3.00000
1.2 1.73205 −1.00000 1.00000 −1.73205 −1.73205 0 −1.73205 −2.00000 −3.00000
\(n\): 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
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.s 2
7.b odd 2 1 8281.2.a.w 2
7.c even 3 2 1183.2.e.e 4
13.b even 2 1 inner 8281.2.a.s 2
13.f odd 12 2 637.2.q.c 2
91.b odd 2 1 8281.2.a.w 2
91.r even 6 2 1183.2.e.e 4
91.w even 12 2 637.2.u.a 2
91.x odd 12 2 91.2.k.a 2
91.ba even 12 2 637.2.k.b 2
91.bc even 12 2 637.2.q.b 2
91.bd odd 12 2 91.2.u.a yes 2
273.bv even 12 2 819.2.bm.a 2
273.bw even 12 2 819.2.do.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 91.x odd 12 2
91.2.u.a yes 2 91.bd odd 12 2
637.2.k.b 2 91.ba even 12 2
637.2.q.b 2 91.bc even 12 2
637.2.q.c 2 13.f odd 12 2
637.2.u.a 2 91.w even 12 2
819.2.bm.a 2 273.bv even 12 2
819.2.do.c 2 273.bw even 12 2
1183.2.e.e 4 7.c even 3 2
1183.2.e.e 4 91.r even 6 2
8281.2.a.s 2 1.a even 1 1 trivial
8281.2.a.s 2 13.b even 2 1 inner
8281.2.a.w 2 7.b odd 2 1
8281.2.a.w 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{2} - 3 \)
\( T_{3} + 1 \)
\( T_{5}^{2} - 3 \)
\( T_{11}^{2} - 27 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -3 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -27 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( -3 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( -3 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( -27 + T^{2} \)
$43$ \( ( 11 + T )^{2} \)
$47$ \( -75 + T^{2} \)
$53$ \( ( 9 + T )^{2} \)
$59$ \( -12 + T^{2} \)
$61$ \( ( -7 + T )^{2} \)
$67$ \( -75 + T^{2} \)
$71$ \( -3 + T^{2} \)
$73$ \( -75 + T^{2} \)
$79$ \( ( 5 + T )^{2} \)
$83$ \( -12 + T^{2} \)
$89$ \( -48 + T^{2} \)
$97$ \( -27 + T^{2} \)
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