Properties

Label 8664.2.a.z
Level $8664$
Weight $2$
Character orbit 8664.a
Self dual yes
Analytic conductor $69.182$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{3} - \beta_1 q^{5} + (\beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_1 q^{5} + (\beta_1 + 1) q^{7} + q^{9} + ( - \beta_{2} - 2) q^{11} - q^{13} - \beta_1 q^{15} + (\beta_{2} + \beta_1) q^{17} + (\beta_1 + 1) q^{21} - \beta_1 q^{23} + (\beta_{2} + \beta_1 + 3) q^{25} + q^{27} + (\beta_{2} - \beta_1 - 4) q^{29} + ( - \beta_{2} - 5) q^{31} + ( - \beta_{2} - 2) q^{33} + ( - \beta_{2} - 2 \beta_1 - 8) q^{35} + (\beta_{2} + \beta_1 + 1) q^{37} - q^{39} + (2 \beta_1 - 4) q^{41} + ( - \beta_1 + 3) q^{43} - \beta_1 q^{45} - 6 q^{47} + (\beta_{2} + 3 \beta_1 + 2) q^{49} + (\beta_{2} + \beta_1) q^{51} + (\beta_{2} - 2 \beta_1) q^{53} + ( - \beta_{2} + 5 \beta_1) q^{55} + ( - 3 \beta_1 + 2) q^{59} + ( - 2 \beta_1 - 1) q^{61} + (\beta_1 + 1) q^{63} + \beta_1 q^{65} + ( - \beta_{2} + 1) q^{67} - \beta_1 q^{69} + (\beta_{2} + 3 \beta_1 - 2) q^{71} + (2 \beta_1 - 3) q^{73} + (\beta_{2} + \beta_1 + 3) q^{75} + ( - 5 \beta_1 - 2) q^{77} + (\beta_{2} - 2 \beta_1 - 9) q^{79} + q^{81} + ( - \beta_{2} + 3 \beta_1 - 4) q^{83} + ( - 4 \beta_1 - 8) q^{85} + (\beta_{2} - \beta_1 - 4) q^{87} + ( - \beta_{2} - 6) q^{89} + ( - \beta_1 - 1) q^{91} + ( - \beta_{2} - 5) q^{93} + (\beta_{2} + 3 \beta_1 - 2) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} - 3 q^{13} + 3 q^{21} + 9 q^{25} + 3 q^{27} - 12 q^{29} - 15 q^{31} - 6 q^{33} - 24 q^{35} + 3 q^{37} - 3 q^{39} - 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49} + 6 q^{59} - 3 q^{61} + 3 q^{63} + 3 q^{67} - 6 q^{71} - 9 q^{73} + 9 q^{75} - 6 q^{77} - 27 q^{79} + 3 q^{81} - 12 q^{83} - 24 q^{85} - 12 q^{87} - 18 q^{89} - 3 q^{91} - 15 q^{93} - 6 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 2\nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 8 ) / 4 \) 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
1.87939
−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.z 3
19.b odd 2 1 8664.2.a.x 3
19.d odd 6 2 456.2.q.f 6
57.f even 6 2 1368.2.s.j 6
76.f even 6 2 912.2.q.k 6
228.n odd 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.d odd 6 2
912.2.q.k 6 76.f even 6 2
1368.2.s.j 6 57.f even 6 2
2736.2.s.x 6 228.n odd 6 2
8664.2.a.x 3 19.b odd 2 1
8664.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8664))\):

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

Hecke characteristic polynomials

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