Properties

Label 8624.2.a.cl
Level $8624$
Weight $2$
Character orbit 8624.a
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + \beta_{2} q^{9} + q^{11} + ( -4 + \beta_{1} ) q^{13} + ( 3 - 2 \beta_{1} ) q^{15} + ( -1 - \beta_{2} ) q^{17} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{19} + ( -4 + \beta_{2} ) q^{23} + ( 2 - 3 \beta_{1} ) q^{25} + ( -2 \beta_{1} + \beta_{2} ) q^{27} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{29} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{31} + \beta_{1} q^{33} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{39} + ( -2 + \beta_{1} + 4 \beta_{2} ) q^{41} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{43} + ( -3 + \beta_{2} ) q^{45} + ( 1 + \beta_{2} ) q^{47} + ( -2 \beta_{1} - \beta_{2} ) q^{51} + ( 6 - \beta_{1} + 2 \beta_{2} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} ) q^{55} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{57} + ( -3 + \beta_{1} - 6 \beta_{2} ) q^{59} + ( -8 + 2 \beta_{2} ) q^{61} + ( 7 - 6 \beta_{1} + 4 \beta_{2} ) q^{65} + ( 6 - 2 \beta_{1} ) q^{67} + ( -3 \beta_{1} + \beta_{2} ) q^{69} + ( -1 - 4 \beta_{1} + 3 \beta_{2} ) q^{71} + ( -5 - 5 \beta_{1} ) q^{73} + ( -9 + 2 \beta_{1} - 3 \beta_{2} ) q^{75} + ( -3 \beta_{1} + \beta_{2} ) q^{79} + ( -6 + \beta_{1} - 4 \beta_{2} ) q^{81} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{83} + ( 4 - \beta_{1} ) q^{85} + ( -9 - 3 \beta_{1} - 4 \beta_{2} ) q^{87} + ( \beta_{1} - \beta_{2} ) q^{89} + ( -9 + \beta_{1} - 4 \beta_{2} ) q^{93} + ( 6 - \beta_{1} - 4 \beta_{2} ) q^{95} + ( 4 - 3 \beta_{1} ) q^{97} + \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} - 2q^{5} + O(q^{10}) \) \( 3q + q^{3} - 2q^{5} + 3q^{11} - 11q^{13} + 7q^{15} - 3q^{17} + 11q^{19} - 12q^{23} + 3q^{25} - 2q^{27} - 9q^{29} + 3q^{31} + q^{33} - 4q^{37} + 5q^{39} - 5q^{41} - 2q^{43} - 9q^{45} + 3q^{47} - 2q^{51} + 17q^{53} - 2q^{55} + 20q^{57} - 8q^{59} - 24q^{61} + 15q^{65} + 16q^{67} - 3q^{69} - 7q^{71} - 20q^{73} - 25q^{75} - 3q^{79} - 17q^{81} + 11q^{83} + 11q^{85} - 30q^{87} + q^{89} - 26q^{93} + 17q^{95} + 9q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−1.91223
0.713538
2.19869
0 −1.91223 0 −3.56885 0 0 0 0.656620 0
1.2 0 0.713538 0 2.20440 0 0 0 −2.49086 0
1.3 0 2.19869 0 −0.635552 0 0 0 1.83424 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.cl 3
4.b odd 2 1 539.2.a.h 3
7.b odd 2 1 8624.2.a.ck 3
7.c even 3 2 1232.2.q.k 6
12.b even 2 1 4851.2.a.bo 3
28.d even 2 1 539.2.a.i 3
28.f even 6 2 539.2.e.l 6
28.g odd 6 2 77.2.e.b 6
44.c even 2 1 5929.2.a.v 3
84.h odd 2 1 4851.2.a.bn 3
84.n even 6 2 693.2.i.g 6
308.g odd 2 1 5929.2.a.w 3
308.n even 6 2 847.2.e.d 6
308.bb odd 30 8 847.2.n.e 24
308.bc even 30 8 847.2.n.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 28.g odd 6 2
539.2.a.h 3 4.b odd 2 1
539.2.a.i 3 28.d even 2 1
539.2.e.l 6 28.f even 6 2
693.2.i.g 6 84.n even 6 2
847.2.e.d 6 308.n even 6 2
847.2.n.d 24 308.bc even 30 8
847.2.n.e 24 308.bb odd 30 8
1232.2.q.k 6 7.c even 3 2
4851.2.a.bn 3 84.h odd 2 1
4851.2.a.bo 3 12.b even 2 1
5929.2.a.v 3 44.c even 2 1
5929.2.a.w 3 308.g odd 2 1
8624.2.a.ck 3 7.b odd 2 1
8624.2.a.cl 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

\( T_{3}^{3} - T_{3}^{2} - 4 T_{3} + 3 \)
\( T_{5}^{3} + 2 T_{5}^{2} - 7 T_{5} - 5 \)
\( T_{13}^{3} + 11 T_{13}^{2} + 36 T_{13} + 35 \)
\( T_{17}^{3} + 3 T_{17}^{2} - 2 T_{17} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 3 - 4 T - T^{2} + T^{3} \)
$5$ \( -5 - 7 T + 2 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( 35 + 36 T + 11 T^{2} + T^{3} \)
$17$ \( -7 - 2 T + 3 T^{2} + T^{3} \)
$19$ \( 57 + 20 T - 11 T^{2} + T^{3} \)
$23$ \( 47 + 43 T + 12 T^{2} + T^{3} \)
$29$ \( -53 - 20 T + 9 T^{2} + T^{3} \)
$31$ \( 107 - 44 T - 3 T^{2} + T^{3} \)
$37$ \( -152 - 36 T + 4 T^{2} + T^{3} \)
$41$ \( -109 - 80 T + 5 T^{2} + T^{3} \)
$43$ \( -41 - 25 T + 2 T^{2} + T^{3} \)
$47$ \( 7 - 2 T - 3 T^{2} + T^{3} \)
$53$ \( -21 + 74 T - 17 T^{2} + T^{3} \)
$59$ \( -1323 - 157 T + 8 T^{2} + T^{3} \)
$61$ \( 376 + 172 T + 24 T^{2} + T^{3} \)
$67$ \( -72 + 68 T - 16 T^{2} + T^{3} \)
$71$ \( -419 - 86 T + 7 T^{2} + T^{3} \)
$73$ \( -625 + 25 T + 20 T^{2} + T^{3} \)
$79$ \( -141 - 38 T + 3 T^{2} + T^{3} \)
$83$ \( 3 + 16 T - 11 T^{2} + T^{3} \)
$89$ \( 3 - 8 T - T^{2} + T^{3} \)
$97$ \( 47 - 12 T - 9 T^{2} + T^{3} \)
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