Properties

Label 8624.2.a.bs
Level $8624$
Weight $2$
Character orbit 8624.a
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1078)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} - 3 \beta q^{5} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 3 \beta q^{5} - q^{9} + q^{11} - 6 q^{15} + 4 \beta q^{17} - 6 q^{23} + 13 q^{25} - 4 \beta q^{27} + 2 q^{29} - \beta q^{31} + \beta q^{33} - 10 q^{37} + 8 \beta q^{41} + 8 q^{43} + 3 \beta q^{45} + 3 \beta q^{47} + 8 q^{51} + 8 q^{53} - 3 \beta q^{55} + \beta q^{59} + 2 \beta q^{61} - 2 q^{67} - 6 \beta q^{69} + 2 q^{71} - 6 \beta q^{73} + 13 \beta q^{75} - 16 q^{79} - 5 q^{81} - 12 \beta q^{83} - 24 q^{85} + 2 \beta q^{87} + 5 \beta q^{89} - 2 q^{93} + 7 \beta q^{97} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 2 q^{11} - 12 q^{15} - 12 q^{23} + 26 q^{25} + 4 q^{29} - 20 q^{37} + 16 q^{43} + 16 q^{51} + 16 q^{53} - 4 q^{67} + 4 q^{71} - 32 q^{79} - 10 q^{81} - 48 q^{85} - 4 q^{93} - 2 q^{99}+O(q^{100}) \) 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 −1.41421 0 4.24264 0 0 0 −1.00000 0
1.2 0 1.41421 0 −4.24264 0 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 8624.2.a.bs 2
4.b odd 2 1 1078.2.a.u 2
7.b odd 2 1 inner 8624.2.a.bs 2
12.b even 2 1 9702.2.a.cp 2
28.d even 2 1 1078.2.a.u 2
28.f even 6 2 1078.2.e.p 4
28.g odd 6 2 1078.2.e.p 4
84.h odd 2 1 9702.2.a.cp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.u 2 4.b odd 2 1
1078.2.a.u 2 28.d even 2 1
1078.2.e.p 4 28.f even 6 2
1078.2.e.p 4 28.g odd 6 2
8624.2.a.bs 2 1.a even 1 1 trivial
8624.2.a.bs 2 7.b odd 2 1 inner
9702.2.a.cp 2 12.b even 2 1
9702.2.a.cp 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 18 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 18 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 32 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 128 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 18 \) Copy content Toggle raw display
$53$ \( (T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 2 \) Copy content Toggle raw display
$61$ \( T^{2} - 8 \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 72 \) Copy content Toggle raw display
$79$ \( (T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 288 \) Copy content Toggle raw display
$89$ \( T^{2} - 50 \) Copy content Toggle raw display
$97$ \( T^{2} - 98 \) Copy content Toggle raw display
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