# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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