Properties

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

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{3}, \sqrt{7})\)
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{2} + 1) q^{3} - \beta_{2} q^{4} + \beta_{3} q^{5} + ( - 2 \beta_{3} - \beta_1) q^{6} + (\beta_{3} + \beta_1) q^{8} + (\beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{2} + 1) q^{3} - \beta_{2} q^{4} + \beta_{3} q^{5} + ( - 2 \beta_{3} - \beta_1) q^{6} + (\beta_{3} + \beta_1) q^{8} + (\beta_{2} + 3) q^{9} + ( - \beta_{2} + 2) q^{10} + (\beta_{3} + 2 \beta_1) q^{11} - 5 q^{12} + ( - 2 \beta_{3} - \beta_1) q^{15} + (\beta_{2} + 1) q^{16} - 3 q^{17} - \beta_1 q^{18} + 3 \beta_{3} q^{19} + (3 \beta_{3} + \beta_1) q^{20} - \beta_{2} q^{22} + (2 \beta_{2} - 2) q^{23} + ( - \beta_{3} + 2 \beta_1) q^{24} + ( - \beta_{2} - 3) q^{25} + 5 q^{27} + ( - \beta_{2} - 5) q^{29} + (2 \beta_{2} - 3) q^{30} + ( - 5 \beta_{3} - 5 \beta_1) q^{31} + ( - 4 \beta_{3} - 3 \beta_1) q^{32} + 5 \beta_1 q^{33} - 3 \beta_{3} q^{34} + ( - 2 \beta_{2} - 5) q^{36} + ( - 4 \beta_{3} - 4 \beta_1) q^{37} + ( - 3 \beta_{2} + 6) q^{38} + ( - \beta_{2} + 1) q^{40} + (2 \beta_{3} + 4 \beta_1) q^{41} + ( - 3 \beta_{2} - 4) q^{43} + (\beta_{3} - 3 \beta_1) q^{44} - \beta_1 q^{45} + ( - 8 \beta_{3} - 2 \beta_1) q^{46} + ( - 3 \beta_{3} - 5 \beta_1) q^{47} + (\beta_{2} + 6) q^{48} + \beta_1 q^{50} + ( - 3 \beta_{2} - 3) q^{51} + (4 \beta_{2} - 1) q^{53} + 5 \beta_{3} q^{54} - \beta_{2} q^{55} + ( - 6 \beta_{3} - 3 \beta_1) q^{57} + ( - 2 \beta_{3} + \beta_1) q^{58} + ( - 3 \beta_{3} + 5 \beta_1) q^{59} - 5 \beta_{3} q^{60} + ( - 6 \beta_{2} - 4) q^{61} + (5 \beta_{2} - 5) q^{62} + (2 \beta_{2} - 7) q^{64} - 5 q^{66} + (5 \beta_{3} - \beta_1) q^{67} + 3 \beta_{2} q^{68} + ( - 2 \beta_{2} + 8) q^{69} - 2 \beta_1 q^{71} + (\beta_{3} + 4 \beta_1) q^{72} + ( - 2 \beta_{3} - 2 \beta_1) q^{73} + (4 \beta_{2} - 4) q^{74} + ( - 3 \beta_{2} - 8) q^{75} + (9 \beta_{3} + 3 \beta_1) q^{76} - 6 q^{79} + ( - 2 \beta_{3} - \beta_1) q^{80} + (2 \beta_{2} - 4) q^{81} - 2 \beta_{2} q^{82} + ( - \beta_{3} + 3 \beta_1) q^{83} - 3 \beta_{3} q^{85} + (5 \beta_{3} + 3 \beta_1) q^{86} + ( - 5 \beta_{2} - 10) q^{87} + (\beta_{2} + 5) q^{88} + (3 \beta_{3} + 8 \beta_1) q^{89} + q^{90} + (4 \beta_{2} - 10) q^{92} + (5 \beta_{3} - 10 \beta_1) q^{93} + (3 \beta_{2} - 1) q^{94} + ( - 3 \beta_{2} + 6) q^{95} + (5 \beta_{3} - 5 \beta_1) q^{96} + (8 \beta_{3} + 5 \beta_1) q^{97} + (2 \beta_{3} + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{4} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{4} + 10 q^{9} + 10 q^{10} - 20 q^{12} + 2 q^{16} - 12 q^{17} + 2 q^{22} - 12 q^{23} - 10 q^{25} + 20 q^{27} - 18 q^{29} - 16 q^{30} - 16 q^{36} + 30 q^{38} + 6 q^{40} - 10 q^{43} + 22 q^{48} - 6 q^{51} - 12 q^{53} + 2 q^{55} - 4 q^{61} - 30 q^{62} - 32 q^{64} - 20 q^{66} - 6 q^{68} + 36 q^{69} - 24 q^{74} - 26 q^{75} - 24 q^{79} - 20 q^{81} + 4 q^{82} - 30 q^{87} + 18 q^{88} + 4 q^{90} - 48 q^{92} - 10 q^{94} + 30 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display

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
0.456850
2.18890
−2.18890
−0.456850
−2.18890 −1.79129 2.79129 −2.18890 3.92095 0 −1.73205 0.208712 4.79129
1.2 −0.456850 2.79129 −1.79129 −0.456850 −1.27520 0 1.73205 4.79129 0.208712
1.3 0.456850 2.79129 −1.79129 0.456850 1.27520 0 −1.73205 4.79129 0.208712
1.4 2.18890 −1.79129 2.79129 2.18890 −3.92095 0 1.73205 0.208712 4.79129
\(n\): 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.bs 4
7.b odd 2 1 8281.2.a.bq 4
13.b even 2 1 inner 8281.2.a.bs 4
13.f odd 12 2 637.2.q.e 4
91.b odd 2 1 8281.2.a.bq 4
91.w even 12 2 637.2.u.d 4
91.x odd 12 2 637.2.k.d 4
91.ba even 12 2 637.2.k.f 4
91.bc even 12 2 637.2.q.f yes 4
91.bd odd 12 2 637.2.u.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.d 4 91.x odd 12 2
637.2.k.f 4 91.ba even 12 2
637.2.q.e 4 13.f odd 12 2
637.2.q.f yes 4 91.bc even 12 2
637.2.u.d 4 91.w even 12 2
637.2.u.e 4 91.bd odd 12 2
8281.2.a.bq 4 7.b odd 2 1
8281.2.a.bq 4 91.b odd 2 1
8281.2.a.bs 4 1.a even 1 1 trivial
8281.2.a.bs 4 13.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 5T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} - 5 \) Copy content Toggle raw display
\( T_{5}^{4} - 5T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 17T_{11}^{2} + 25 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T - 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 17T^{2} + 25 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T + 3)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 45T^{2} + 81 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9 T + 15)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 68T^{2} + 400 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T - 41)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 110T^{2} + 1681 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 75)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 230 T^{2} + 11881 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 188)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 150T^{2} + 2601 \) Copy content Toggle raw display
$71$ \( T^{4} - 20T^{2} + 16 \) Copy content Toggle raw display
$73$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 62T^{2} + 625 \) Copy content Toggle raw display
$89$ \( T^{4} - 269T^{2} + 2209 \) Copy content Toggle raw display
$97$ \( T^{4} - 285 T^{2} + 12321 \) Copy content Toggle raw display
show more
show less