Properties

Label 8624.2.a.bx
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: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2156)
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.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} -\beta q^{5} + 5 q^{9} +O(q^{10})\) \( q + 2 \beta q^{3} -\beta q^{5} + 5 q^{9} + q^{11} + \beta q^{13} -4 q^{15} -5 \beta q^{17} + 2 \beta q^{19} -4 q^{23} -3 q^{25} + 4 \beta q^{27} -4 \beta q^{31} + 2 \beta q^{33} -8 q^{37} + 4 q^{39} -7 \beta q^{41} -4 q^{43} -5 \beta q^{45} -20 q^{51} -6 q^{53} -\beta q^{55} + 8 q^{57} + 6 \beta q^{59} + \beta q^{61} -2 q^{65} + 8 q^{67} -8 \beta q^{69} -8 q^{71} + \beta q^{73} -6 \beta q^{75} -16 q^{79} + q^{81} -2 \beta q^{83} + 10 q^{85} + 11 \beta q^{89} -16 q^{93} -4 q^{95} -7 \beta q^{97} + 5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{9} + O(q^{10}) \) \( 2 q + 10 q^{9} + 2 q^{11} - 8 q^{15} - 8 q^{23} - 6 q^{25} - 16 q^{37} + 8 q^{39} - 8 q^{43} - 40 q^{51} - 12 q^{53} + 16 q^{57} - 4 q^{65} + 16 q^{67} - 16 q^{71} - 32 q^{79} + 2 q^{81} + 20 q^{85} - 32 q^{93} - 8 q^{95} + 10 q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −2.82843 0 1.41421 0 0 0 5.00000 0
1.2 0 2.82843 0 −1.41421 0 0 0 5.00000 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.bx 2
4.b odd 2 1 2156.2.a.f 2
7.b odd 2 1 inner 8624.2.a.bx 2
28.d even 2 1 2156.2.a.f 2
28.f even 6 2 2156.2.i.f 4
28.g odd 6 2 2156.2.i.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2156.2.a.f 2 4.b odd 2 1
2156.2.a.f 2 28.d even 2 1
2156.2.i.f 4 28.f even 6 2
2156.2.i.f 4 28.g odd 6 2
8624.2.a.bx 2 1.a even 1 1 trivial
8624.2.a.bx 2 7.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 8 \)
\( T_{5}^{2} - 2 \)
\( T_{13}^{2} - 2 \)
\( T_{17}^{2} - 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( -2 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -2 + T^{2} \)
$17$ \( -50 + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( -98 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -72 + T^{2} \)
$61$ \( -2 + T^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( -2 + T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( -8 + T^{2} \)
$89$ \( -242 + T^{2} \)
$97$ \( -98 + T^{2} \)
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