# 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 $1$

# Learn more

## 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: $$x^{3} - 3x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta_1 q^{5} + (\beta_1 + 1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 - b1 * q^5 + (b1 + 1) * q^7 + q^9 $$q + q^{3} - \beta_1 q^{5} + (\beta_1 + 1) q^{7} + q^{9} + ( - \beta_{2} - 2) q^{11} - q^{13} - \beta_1 q^{15} + (\beta_{2} + \beta_1) q^{17} + (\beta_1 + 1) q^{21} - \beta_1 q^{23} + (\beta_{2} + \beta_1 + 3) q^{25} + q^{27} + (\beta_{2} - \beta_1 - 4) q^{29} + ( - \beta_{2} - 5) q^{31} + ( - \beta_{2} - 2) q^{33} + ( - \beta_{2} - 2 \beta_1 - 8) q^{35} + (\beta_{2} + \beta_1 + 1) q^{37} - q^{39} + (2 \beta_1 - 4) q^{41} + ( - \beta_1 + 3) q^{43} - \beta_1 q^{45} - 6 q^{47} + (\beta_{2} + 3 \beta_1 + 2) q^{49} + (\beta_{2} + \beta_1) q^{51} + (\beta_{2} - 2 \beta_1) q^{53} + ( - \beta_{2} + 5 \beta_1) q^{55} + ( - 3 \beta_1 + 2) q^{59} + ( - 2 \beta_1 - 1) q^{61} + (\beta_1 + 1) q^{63} + \beta_1 q^{65} + ( - \beta_{2} + 1) q^{67} - \beta_1 q^{69} + (\beta_{2} + 3 \beta_1 - 2) q^{71} + (2 \beta_1 - 3) q^{73} + (\beta_{2} + \beta_1 + 3) q^{75} + ( - 5 \beta_1 - 2) q^{77} + (\beta_{2} - 2 \beta_1 - 9) q^{79} + q^{81} + ( - \beta_{2} + 3 \beta_1 - 4) q^{83} + ( - 4 \beta_1 - 8) q^{85} + (\beta_{2} - \beta_1 - 4) q^{87} + ( - \beta_{2} - 6) q^{89} + ( - \beta_1 - 1) q^{91} + ( - \beta_{2} - 5) q^{93} + (\beta_{2} + 3 \beta_1 - 2) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100})$$ q + q^3 - b1 * q^5 + (b1 + 1) * q^7 + q^9 + (-b2 - 2) * q^11 - q^13 - b1 * q^15 + (b2 + b1) * q^17 + (b1 + 1) * q^21 - b1 * q^23 + (b2 + b1 + 3) * q^25 + q^27 + (b2 - b1 - 4) * q^29 + (-b2 - 5) * q^31 + (-b2 - 2) * q^33 + (-b2 - 2*b1 - 8) * q^35 + (b2 + b1 + 1) * q^37 - q^39 + (2*b1 - 4) * q^41 + (-b1 + 3) * q^43 - b1 * q^45 - 6 * q^47 + (b2 + 3*b1 + 2) * q^49 + (b2 + b1) * q^51 + (b2 - 2*b1) * q^53 + (-b2 + 5*b1) * q^55 + (-3*b1 + 2) * q^59 + (-2*b1 - 1) * q^61 + (b1 + 1) * q^63 + b1 * q^65 + (-b2 + 1) * q^67 - b1 * q^69 + (b2 + 3*b1 - 2) * q^71 + (2*b1 - 3) * q^73 + (b2 + b1 + 3) * q^75 + (-5*b1 - 2) * q^77 + (b2 - 2*b1 - 9) * q^79 + q^81 + (-b2 + 3*b1 - 4) * q^83 + (-4*b1 - 8) * q^85 + (b2 - b1 - 4) * q^87 + (-b2 - 6) * q^89 + (-b1 - 1) * q^91 + (-b2 - 5) * q^93 + (b2 + 3*b1 - 2) * q^97 + (-b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} - 3 q^{13} + 3 q^{21} + 9 q^{25} + 3 q^{27} - 12 q^{29} - 15 q^{31} - 6 q^{33} - 24 q^{35} + 3 q^{37} - 3 q^{39} - 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49} + 6 q^{59} - 3 q^{61} + 3 q^{63} + 3 q^{67} - 6 q^{71} - 9 q^{73} + 9 q^{75} - 6 q^{77} - 27 q^{79} + 3 q^{81} - 12 q^{83} - 24 q^{85} - 12 q^{87} - 18 q^{89} - 3 q^{91} - 15 q^{93} - 6 q^{97} - 6 q^{99}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$4\nu^{2} - 2\nu - 8$$ 4*v^2 - 2*v - 8
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 8 ) / 4$$ (b2 + b1 + 8) / 4

## 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.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8664))$$:

 $$T_{5}^{3} - 12T_{5} + 8$$ T5^3 - 12*T5 + 8 $$T_{7}^{3} - 3T_{7}^{2} - 9T_{7} + 3$$ T7^3 - 3*T7^2 - 9*T7 + 3 $$T_{13} + 1$$ T13 + 1 $$T_{29}^{3} + 12T_{29}^{2} - 192$$ T29^3 + 12*T29^2 - 192

## Hecke characteristic polynomials

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