Properties

Label 8281.2.a.x
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} - \beta q^{3} + (2 \beta + 1) q^{4} + ( - 2 \beta - 1) q^{5} + ( - \beta - 2) q^{6} + (\beta + 3) q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} - \beta q^{3} + (2 \beta + 1) q^{4} + ( - 2 \beta - 1) q^{5} + ( - \beta - 2) q^{6} + (\beta + 3) q^{8} - q^{9} + ( - 3 \beta - 5) q^{10} + ( - \beta - 2) q^{11} + ( - \beta - 4) q^{12} + (\beta + 4) q^{15} + 3 q^{16} + (2 \beta - 3) q^{17} + ( - \beta - 1) q^{18} - 6 q^{19} + ( - 4 \beta - 9) q^{20} + ( - 3 \beta - 4) q^{22} + \beta q^{23} + ( - 3 \beta - 2) q^{24} + (4 \beta + 4) q^{25} + 4 \beta q^{27} + (2 \beta + 7) q^{29} + (5 \beta + 6) q^{30} + ( - \beta - 4) q^{31} + (\beta - 3) q^{32} + (2 \beta + 2) q^{33} + ( - \beta + 1) q^{34} + ( - 2 \beta - 1) q^{36} + ( - 6 \beta + 1) q^{37} + ( - 6 \beta - 6) q^{38} + ( - 7 \beta - 7) q^{40} + ( - 2 \beta - 3) q^{41} + ( - \beta + 2) q^{43} + ( - 5 \beta - 6) q^{44} + (2 \beta + 1) q^{45} + (\beta + 2) q^{46} + ( - 4 \beta - 2) q^{47} - 3 \beta q^{48} + (8 \beta + 12) q^{50} + (3 \beta - 4) q^{51} - 3 q^{53} + (4 \beta + 8) q^{54} + (5 \beta + 6) q^{55} + 6 \beta q^{57} + (9 \beta + 11) q^{58} + ( - 3 \beta + 6) q^{59} + (9 \beta + 8) q^{60} + (2 \beta + 7) q^{61} + ( - 5 \beta - 6) q^{62} + ( - 2 \beta - 7) q^{64} + (4 \beta + 6) q^{66} + 3 \beta q^{67} + ( - 4 \beta + 5) q^{68} - 2 q^{69} + ( - 4 \beta + 6) q^{71} + ( - \beta - 3) q^{72} + ( - 4 \beta + 5) q^{73} + ( - 5 \beta - 11) q^{74} + ( - 4 \beta - 8) q^{75} + ( - 12 \beta - 6) q^{76} + (3 \beta + 6) q^{79} + ( - 6 \beta - 3) q^{80} - 5 q^{81} + ( - 5 \beta - 7) q^{82} + (5 \beta + 6) q^{83} + (4 \beta - 5) q^{85} + \beta q^{86} + ( - 7 \beta - 4) q^{87} + ( - 5 \beta - 8) q^{88} + (8 \beta - 4) q^{89} + (3 \beta + 5) q^{90} + (\beta + 4) q^{92} + (4 \beta + 2) q^{93} + ( - 6 \beta - 10) q^{94} + (12 \beta + 6) q^{95} + (3 \beta - 2) q^{96} + ( - 2 \beta + 8) q^{97} + (\beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 6 q^{8} - 2 q^{9} - 10 q^{10} - 4 q^{11} - 8 q^{12} + 8 q^{15} + 6 q^{16} - 6 q^{17} - 2 q^{18} - 12 q^{19} - 18 q^{20} - 8 q^{22} - 4 q^{24} + 8 q^{25} + 14 q^{29} + 12 q^{30} - 8 q^{31} - 6 q^{32} + 4 q^{33} + 2 q^{34} - 2 q^{36} + 2 q^{37} - 12 q^{38} - 14 q^{40} - 6 q^{41} + 4 q^{43} - 12 q^{44} + 2 q^{45} + 4 q^{46} - 4 q^{47} + 24 q^{50} - 8 q^{51} - 6 q^{53} + 16 q^{54} + 12 q^{55} + 22 q^{58} + 12 q^{59} + 16 q^{60} + 14 q^{61} - 12 q^{62} - 14 q^{64} + 12 q^{66} + 10 q^{68} - 4 q^{69} + 12 q^{71} - 6 q^{72} + 10 q^{73} - 22 q^{74} - 16 q^{75} - 12 q^{76} + 12 q^{79} - 6 q^{80} - 10 q^{81} - 14 q^{82} + 12 q^{83} - 10 q^{85} - 8 q^{87} - 16 q^{88} - 8 q^{89} + 10 q^{90} + 8 q^{92} + 4 q^{93} - 20 q^{94} + 12 q^{95} - 4 q^{96} + 16 q^{97} + 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−0.414214 1.41421 −1.82843 1.82843 −0.585786 0 1.58579 −1.00000 −0.757359
1.2 2.41421 −1.41421 3.82843 −3.82843 −3.41421 0 4.41421 −1.00000 −9.24264
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} - 7 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 41 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 71 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 41 \) Copy content Toggle raw display
$67$ \( T^{2} - 18 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T - 7 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 14 \) Copy content Toggle raw display
$89$ \( T^{2} + 8T - 112 \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
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