Properties

Label 8664.2.a.x
Level $8664$
Weight $2$
Character orbit 8664.a
Self dual yes
Analytic conductor $69.182$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8664.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.1823883112\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 456)
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.

\(f(q)\) \(=\) \( q - q^{3} -\beta_{1} q^{5} + ( 1 + \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -\beta_{1} q^{5} + ( 1 + \beta_{1} ) q^{7} + q^{9} + ( -2 - \beta_{2} ) q^{11} + q^{13} + \beta_{1} q^{15} + ( \beta_{1} + \beta_{2} ) q^{17} + ( -1 - \beta_{1} ) q^{21} -\beta_{1} q^{23} + ( 3 + \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( 4 + \beta_{1} - \beta_{2} ) q^{29} + ( 5 + \beta_{2} ) q^{31} + ( 2 + \beta_{2} ) q^{33} + ( -8 - 2 \beta_{1} - \beta_{2} ) q^{35} + ( -1 - \beta_{1} - \beta_{2} ) q^{37} - q^{39} + ( 4 - 2 \beta_{1} ) q^{41} + ( 3 - \beta_{1} ) q^{43} -\beta_{1} q^{45} -6 q^{47} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{49} + ( -\beta_{1} - \beta_{2} ) q^{51} + ( 2 \beta_{1} - \beta_{2} ) q^{53} + ( 5 \beta_{1} - \beta_{2} ) q^{55} + ( -2 + 3 \beta_{1} ) q^{59} + ( -1 - 2 \beta_{1} ) q^{61} + ( 1 + \beta_{1} ) q^{63} -\beta_{1} q^{65} + ( -1 + \beta_{2} ) q^{67} + \beta_{1} q^{69} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{71} + ( -3 + 2 \beta_{1} ) q^{73} + ( -3 - \beta_{1} - \beta_{2} ) q^{75} + ( -2 - 5 \beta_{1} ) q^{77} + ( 9 + 2 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{83} + ( -8 - 4 \beta_{1} ) q^{85} + ( -4 - \beta_{1} + \beta_{2} ) q^{87} + ( 6 + \beta_{2} ) q^{89} + ( 1 + \beta_{1} ) q^{91} + ( -5 - \beta_{2} ) q^{93} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 3 q^{21} + 9 q^{25} - 3 q^{27} + 12 q^{29} + 15 q^{31} + 6 q^{33} - 24 q^{35} - 3 q^{37} - 3 q^{39} + 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49} - 6 q^{59} - 3 q^{61} + 3 q^{63} - 3 q^{67} + 6 q^{71} - 9 q^{73} - 9 q^{75} - 6 q^{77} + 27 q^{79} + 3 q^{81} - 12 q^{83} - 24 q^{85} - 12 q^{87} + 18 q^{89} + 3 q^{91} - 15 q^{93} + 6 q^{97} - 6 q^{99} + O(q^{100}) \)

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
1.87939
−0.347296
−1.53209
0 −1.00000 0 −3.75877 0 4.75877 0 1.00000 0
1.2 0 −1.00000 0 0.694593 0 0.305407 0 1.00000 0
1.3 0 −1.00000 0 3.06418 0 −2.06418 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)

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 8664.2.a.x 3
19.b odd 2 1 8664.2.a.z 3
19.c even 3 2 456.2.q.f 6
57.h odd 6 2 1368.2.s.j 6
76.g odd 6 2 912.2.q.k 6
228.m even 6 2 2736.2.s.x 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.f 6 19.c even 3 2
912.2.q.k 6 76.g odd 6 2
1368.2.s.j 6 57.h odd 6 2
2736.2.s.x 6 228.m even 6 2
8664.2.a.x 3 1.a even 1 1 trivial
8664.2.a.z 3 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8664))\):

\( T_{5}^{3} - 12 T_{5} + 8 \)
\( T_{7}^{3} - 3 T_{7}^{2} - 9 T_{7} + 3 \)
\( T_{13} - 1 \)
\( T_{29}^{3} - 12 T_{29}^{2} + 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 8 - 12 T + T^{3} \)
$7$ \( 3 - 9 T - 3 T^{2} + T^{3} \)
$11$ \( -136 - 24 T + 6 T^{2} + T^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( 64 - 48 T + T^{3} \)
$19$ \( T^{3} \)
$23$ \( 8 - 12 T + T^{3} \)
$29$ \( 192 - 12 T^{2} + T^{3} \)
$31$ \( 127 + 39 T - 15 T^{2} + T^{3} \)
$37$ \( -111 - 45 T + 3 T^{2} + T^{3} \)
$41$ \( 192 - 12 T^{2} + T^{3} \)
$43$ \( 17 + 15 T - 9 T^{2} + T^{3} \)
$47$ \( ( 6 + T )^{3} \)
$53$ \( 296 - 84 T + T^{3} \)
$59$ \( -424 - 96 T + 6 T^{2} + T^{3} \)
$61$ \( 17 - 45 T + 3 T^{2} + T^{3} \)
$67$ \( 37 - 33 T + 3 T^{2} + T^{3} \)
$71$ \( 856 - 132 T - 6 T^{2} + T^{3} \)
$73$ \( -181 - 21 T + 9 T^{2} + T^{3} \)
$79$ \( 323 + 159 T - 27 T^{2} + T^{3} \)
$83$ \( 64 - 96 T + 12 T^{2} + T^{3} \)
$89$ \( 72 + 72 T - 18 T^{2} + T^{3} \)
$97$ \( 856 - 132 T - 6 T^{2} + T^{3} \)
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