Properties

Label 8624.2.a.bi
Level $8624$
Weight $2$
Character orbit 8624.a
Self dual yes
Analytic conductor $68.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 616)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( 2 - \beta ) q^{5} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( 2 - \beta ) q^{5} + ( 1 + \beta ) q^{9} - q^{11} + ( 2 - 2 \beta ) q^{13} + ( 4 - \beta ) q^{15} + 2 q^{17} -2 \beta q^{19} + ( 4 - \beta ) q^{23} + ( 3 - 3 \beta ) q^{25} + ( -4 + \beta ) q^{27} -2 q^{29} + ( -8 - \beta ) q^{31} + \beta q^{33} + ( 2 + \beta ) q^{37} + 8 q^{39} + 10 q^{41} -4 q^{43} -2 q^{45} + ( 4 - 4 \beta ) q^{47} -2 \beta q^{51} + ( 6 - 4 \beta ) q^{53} + ( -2 + \beta ) q^{55} + ( 8 + 2 \beta ) q^{57} + ( -8 + \beta ) q^{59} + ( -6 + 4 \beta ) q^{61} + ( 12 - 4 \beta ) q^{65} + \beta q^{67} + ( 4 - 3 \beta ) q^{69} + ( 4 - 3 \beta ) q^{71} + ( -6 + 4 \beta ) q^{73} + 12 q^{75} -2 \beta q^{79} -7 q^{81} -8 q^{83} + ( 4 - 2 \beta ) q^{85} + 2 \beta q^{87} + ( 10 + \beta ) q^{89} + ( 4 + 9 \beta ) q^{93} + ( 8 - 2 \beta ) q^{95} + ( -6 - \beta ) q^{97} + ( -1 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 2 q - q^{3} + 3 q^{5} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 7 q^{15} + 4 q^{17} - 2 q^{19} + 7 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 17 q^{31} + q^{33} + 5 q^{37} + 16 q^{39} + 20 q^{41} - 8 q^{43} - 4 q^{45} + 4 q^{47} - 2 q^{51} + 8 q^{53} - 3 q^{55} + 18 q^{57} - 15 q^{59} - 8 q^{61} + 20 q^{65} + q^{67} + 5 q^{69} + 5 q^{71} - 8 q^{73} + 24 q^{75} - 2 q^{79} - 14 q^{81} - 16 q^{83} + 6 q^{85} + 2 q^{87} + 21 q^{89} + 17 q^{93} + 14 q^{95} - 13 q^{97} - 3 q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
0 −2.56155 0 −0.561553 0 0 0 3.56155 0
1.2 0 1.56155 0 3.56155 0 0 0 −0.561553 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

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 8624.2.a.bi 2
4.b odd 2 1 4312.2.a.t 2
7.b odd 2 1 1232.2.a.o 2
28.d even 2 1 616.2.a.f 2
56.e even 2 1 4928.2.a.bs 2
56.h odd 2 1 4928.2.a.bo 2
84.h odd 2 1 5544.2.a.bf 2
308.g odd 2 1 6776.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.a.f 2 28.d even 2 1
1232.2.a.o 2 7.b odd 2 1
4312.2.a.t 2 4.b odd 2 1
4928.2.a.bo 2 56.h odd 2 1
4928.2.a.bs 2 56.e even 2 1
5544.2.a.bf 2 84.h odd 2 1
6776.2.a.l 2 308.g odd 2 1
8624.2.a.bi 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{2} + T_{3} - 4 \)
\( T_{5}^{2} - 3 T_{5} - 2 \)
\( T_{13}^{2} - 2 T_{13} - 16 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 + T + T^{2} \)
$5$ \( -2 - 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -16 - 2 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( -16 + 2 T + T^{2} \)
$23$ \( 8 - 7 T + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 68 + 17 T + T^{2} \)
$37$ \( 2 - 5 T + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( -64 - 4 T + T^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( 52 + 15 T + T^{2} \)
$61$ \( -52 + 8 T + T^{2} \)
$67$ \( -4 - T + T^{2} \)
$71$ \( -32 - 5 T + T^{2} \)
$73$ \( -52 + 8 T + T^{2} \)
$79$ \( -16 + 2 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 106 - 21 T + T^{2} \)
$97$ \( 38 + 13 T + T^{2} \)
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