# Properties

 Label 864.2.i.f Level $864$ Weight $2$ Character orbit 864.i Analytic conductor $6.899$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.170772624.1 Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16$$ x^8 - 3*x^7 + 5*x^6 - 6*x^5 + 6*x^4 - 12*x^3 + 20*x^2 - 24*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{5} + ( - \beta_{7} + \beta_1) q^{7}+O(q^{10})$$ q + b6 * q^5 + (-b7 + b1) * q^7 $$q + \beta_{6} q^{5} + ( - \beta_{7} + \beta_1) q^{7} - \beta_{3} q^{11} + ( - \beta_{6} - 2 \beta_{4}) q^{13} + \beta_{5} q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{19} - \beta_{7} q^{23} + ( - \beta_{6} - \beta_{5} + 3 \beta_{4} - 4) q^{25} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - 3) q^{29} + ( - \beta_{7} - 2 \beta_{2}) q^{31} + ( - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{35} + 4 q^{37} + \beta_{4} q^{41} - \beta_{3} q^{43} + (\beta_{7} - \beta_1) q^{47} + (3 \beta_{6} - 5 \beta_{4}) q^{49} + 4 q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{55} + ( - 2 \beta_{7} + \beta_{2}) q^{59} + (3 \beta_{6} + 3 \beta_{5} + 8 \beta_{4} - 5) q^{61} + ( - \beta_{6} - \beta_{5} - 8 \beta_{4} + 7) q^{65} + ( - 2 \beta_{7} + \beta_{2}) q^{67} + 2 \beta_1 q^{71} + (\beta_{5} + 8) q^{73} + ( - 3 \beta_{6} - 6 \beta_{4}) q^{77} + ( - \beta_{7} + 2 \beta_{3} + \beta_1) q^{79} + ( - \beta_{7} - 2 \beta_{3} + \beta_1) q^{83} - 8 \beta_{4} q^{85} + ( - 2 \beta_{5} - 8) q^{89} + (2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{91} + 4 \beta_{7} q^{95} + (9 \beta_{4} - 9) q^{97}+O(q^{100})$$ q + b6 * q^5 + (-b7 + b1) * q^7 - b3 * q^11 + (-b6 - 2*b4) * q^13 + b5 * q^17 + (-b3 - b2 - b1) * q^19 - b7 * q^23 + (-b6 - b5 + 3*b4 - 4) * q^25 + (-b6 - b5 + 2*b4 - 3) * q^29 + (-b7 - 2*b2) * q^31 + (-2*b3 - 2*b2 - 3*b1) * q^35 + 4 * q^37 + b4 * q^41 - b3 * q^43 + (b7 - b1) * q^47 + (3*b6 - 5*b4) * q^49 + 4 * q^53 + (-2*b3 - 2*b2 + b1) * q^55 + (-2*b7 + b2) * q^59 + (3*b6 + 3*b5 + 8*b4 - 5) * q^61 + (-b6 - b5 - 8*b4 + 7) * q^65 + (-2*b7 + b2) * q^67 + 2*b1 * q^71 + (b5 + 8) * q^73 + (-3*b6 - 6*b4) * q^77 + (-b7 + 2*b3 + b1) * q^79 + (-b7 - 2*b3 + b1) * q^83 - 8*b4 * q^85 + (-2*b5 - 8) * q^89 + (2*b3 + 2*b2 + b1) * q^91 + 4*b7 * q^95 + (9*b4 - 9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{5}+O(q^{10})$$ 8 * q - 2 * q^5 $$8 q - 2 q^{5} - 6 q^{13} - 4 q^{17} - 14 q^{25} - 10 q^{29} + 32 q^{37} + 4 q^{41} - 26 q^{49} + 32 q^{53} - 26 q^{61} + 30 q^{65} + 60 q^{73} - 18 q^{77} - 32 q^{85} - 56 q^{89} - 36 q^{97}+O(q^{100})$$ 8 * q - 2 * q^5 - 6 * q^13 - 4 * q^17 - 14 * q^25 - 10 * q^29 + 32 * q^37 + 4 * q^41 - 26 * q^49 + 32 * q^53 - 26 * q^61 + 30 * q^65 + 60 * q^73 - 18 * q^77 - 32 * q^85 - 56 * q^89 - 36 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + 3\nu^{4} + 6\nu^{2} + 4\nu ) / 4$$ (-v^7 + 3*v^4 + 6*v^2 + 4*v) / 4 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 3\nu^{6} + 3\nu^{5} - 6\nu^{4} + 6\nu^{3} + 4\nu - 24 ) / 8$$ (-v^7 + 3*v^6 + 3*v^5 - 6*v^4 + 6*v^3 + 4*v - 24) / 8 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 3\nu^{6} - 9\nu^{5} + 6\nu^{4} - 6\nu^{3} + 24\nu^{2} - 44\nu + 48 ) / 8$$ (-v^7 + 3*v^6 - 9*v^5 + 6*v^4 - 6*v^3 + 24*v^2 - 44*v + 48) / 8 $$\beta_{4}$$ $$=$$ $$( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 24 ) / 8$$ (3*v^7 - 5*v^6 + 7*v^5 - 6*v^4 + 10*v^3 - 24*v^2 + 28*v - 24) / 8 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 4\nu^{6} - 6\nu^{5} + 5\nu^{4} - 6\nu^{3} + 14\nu^{2} - 20\nu + 20 ) / 4$$ (-v^7 + 4*v^6 - 6*v^5 + 5*v^4 - 6*v^3 + 14*v^2 - 20*v + 20) / 4 $$\beta_{6}$$ $$=$$ $$( -2\nu^{7} + 3\nu^{6} - 5\nu^{5} + 4\nu^{4} - 5\nu^{3} + 13\nu^{2} - 18\nu + 20 ) / 2$$ (-2*v^7 + 3*v^6 - 5*v^5 + 4*v^4 - 5*v^3 + 13*v^2 - 18*v + 20) / 2 $$\beta_{7}$$ $$=$$ $$( 2\nu^{7} - 3\nu^{6} + 6\nu^{5} - 3\nu^{4} + 6\nu^{3} - 15\nu^{2} + 16\nu - 24 ) / 2$$ (2*v^7 - 3*v^6 + 6*v^5 - 3*v^4 + 6*v^3 - 15*v^2 + 16*v - 24) / 2
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 + 2 ) / 6$$ (-b7 - b6 + b5 + b4 - b3 + b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} - 3\beta_{6} - 3\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta_1 ) / 6$$ (-b7 - 3*b6 - 3*b4 + 2*b3 + b2 + 2*b1) / 6 $$\nu^{3}$$ $$=$$ $$( 2\beta_{6} + \beta_{5} + 10\beta_{4} + 3\beta_{3} + 3\beta_{2} + 3\beta _1 - 4 ) / 6$$ (2*b6 + b5 + 10*b4 + 3*b3 + 3*b2 + 3*b1 - 4) / 6 $$\nu^{4}$$ $$=$$ $$( 3\beta_{7} + 3\beta_{6} + 3\beta_{5} + 3\beta_{4} - \beta_{3} - 2\beta_{2} + 3\beta_1 ) / 6$$ (3*b7 + 3*b6 + 3*b5 + 3*b4 - b3 - 2*b2 + 3*b1) / 6 $$\nu^{5}$$ $$=$$ $$( 5\beta_{7} - \beta_{6} - 2\beta_{5} - 17\beta_{4} + \beta_{2} + 32 ) / 6$$ (5*b7 - b6 - 2*b5 - 17*b4 + b2 + 32) / 6 $$\nu^{6}$$ $$=$$ $$( 2\beta_{7} + 9\beta_{5} - 5\beta_{3} + 5\beta_{2} - \beta _1 + 24 ) / 6$$ (2*b7 + 9*b5 - 5*b3 + 5*b2 - b1 + 24) / 6 $$\nu^{7}$$ $$=$$ $$( -\beta_{7} - 13\beta_{6} + 13\beta_{5} - 5\beta_{4} + 5\beta_{3} + \beta _1 + 8 ) / 6$$ (-b7 - 13*b6 + 13*b5 - 5*b4 + 5*b3 + b1 + 8) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.774115 + 1.18353i 1.41203 + 0.0786378i −1.02187 + 0.977642i 0.335728 − 1.37379i 0.774115 − 1.18353i 1.41203 − 0.0786378i −1.02187 − 0.977642i 0.335728 + 1.37379i
0 0 0 −1.68614 2.92048i 0 −2.35143 + 4.07279i 0 0 0
289.2 0 0 0 −1.68614 2.92048i 0 2.35143 4.07279i 0 0 0
289.3 0 0 0 1.18614 + 2.05446i 0 −1.10489 + 1.91373i 0 0 0
289.4 0 0 0 1.18614 + 2.05446i 0 1.10489 1.91373i 0 0 0
577.1 0 0 0 −1.68614 + 2.92048i 0 −2.35143 4.07279i 0 0 0
577.2 0 0 0 −1.68614 + 2.92048i 0 2.35143 + 4.07279i 0 0 0
577.3 0 0 0 1.18614 2.05446i 0 −1.10489 1.91373i 0 0 0
577.4 0 0 0 1.18614 2.05446i 0 1.10489 + 1.91373i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.i.f 8
3.b odd 2 1 288.2.i.f 8
4.b odd 2 1 inner 864.2.i.f 8
8.b even 2 1 1728.2.i.n 8
8.d odd 2 1 1728.2.i.n 8
9.c even 3 1 inner 864.2.i.f 8
9.c even 3 1 2592.2.a.x 4
9.d odd 6 1 288.2.i.f 8
9.d odd 6 1 2592.2.a.u 4
12.b even 2 1 288.2.i.f 8
24.f even 2 1 576.2.i.n 8
24.h odd 2 1 576.2.i.n 8
36.f odd 6 1 inner 864.2.i.f 8
36.f odd 6 1 2592.2.a.x 4
36.h even 6 1 288.2.i.f 8
36.h even 6 1 2592.2.a.u 4
72.j odd 6 1 576.2.i.n 8
72.j odd 6 1 5184.2.a.cf 4
72.l even 6 1 576.2.i.n 8
72.l even 6 1 5184.2.a.cf 4
72.n even 6 1 1728.2.i.n 8
72.n even 6 1 5184.2.a.cc 4
72.p odd 6 1 1728.2.i.n 8
72.p odd 6 1 5184.2.a.cc 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 3.b odd 2 1
288.2.i.f 8 9.d odd 6 1
288.2.i.f 8 12.b even 2 1
288.2.i.f 8 36.h even 6 1
576.2.i.n 8 24.f even 2 1
576.2.i.n 8 24.h odd 2 1
576.2.i.n 8 72.j odd 6 1
576.2.i.n 8 72.l even 6 1
864.2.i.f 8 1.a even 1 1 trivial
864.2.i.f 8 4.b odd 2 1 inner
864.2.i.f 8 9.c even 3 1 inner
864.2.i.f 8 36.f odd 6 1 inner
1728.2.i.n 8 8.b even 2 1
1728.2.i.n 8 8.d odd 2 1
1728.2.i.n 8 72.n even 6 1
1728.2.i.n 8 72.p odd 6 1
2592.2.a.u 4 9.d odd 6 1
2592.2.a.u 4 36.h even 6 1
2592.2.a.x 4 9.c even 3 1
2592.2.a.x 4 36.f odd 6 1
5184.2.a.cc 4 72.n even 6 1
5184.2.a.cc 4 72.p odd 6 1
5184.2.a.cf 4 72.j odd 6 1
5184.2.a.cf 4 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{4} + T_{5}^{3} + 9T_{5}^{2} - 8T_{5} + 64$$ T5^4 + T5^3 + 9*T5^2 - 8*T5 + 64 $$T_{7}^{8} + 27T_{7}^{6} + 621T_{7}^{4} + 2916T_{7}^{2} + 11664$$ T7^8 + 27*T7^6 + 621*T7^4 + 2916*T7^2 + 11664

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + T^{3} + 9 T^{2} - 8 T + 64)^{2}$$
$7$ $$T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664$$
$11$ $$T^{8} + 36 T^{6} + 1269 T^{4} + \cdots + 729$$
$13$ $$(T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36)^{2}$$
$17$ $$(T^{2} + T - 8)^{4}$$
$19$ $$(T^{4} - 45 T^{2} + 432)^{2}$$
$23$ $$T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664$$
$29$ $$(T^{4} + 5 T^{3} + 27 T^{2} - 10 T + 4)^{2}$$
$31$ $$T^{8} + 135 T^{6} + \cdots + 15116544$$
$37$ $$(T - 4)^{8}$$
$41$ $$(T^{2} - T + 1)^{4}$$
$43$ $$T^{8} + 36 T^{6} + 1269 T^{4} + \cdots + 729$$
$47$ $$T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664$$
$53$ $$(T - 4)^{8}$$
$59$ $$T^{8} + 180 T^{6} + \cdots + 60886809$$
$61$ $$(T^{4} + 13 T^{3} + 201 T^{2} - 416 T + 1024)^{2}$$
$67$ $$T^{8} + 180 T^{6} + \cdots + 60886809$$
$71$ $$(T^{4} - 108 T^{2} + 1728)^{2}$$
$73$ $$(T^{2} - 15 T + 48)^{4}$$
$79$ $$T^{8} + 135 T^{6} + \cdots + 15116544$$
$83$ $$T^{8} + 207 T^{6} + 41121 T^{4} + \cdots + 2985984$$
$89$ $$(T^{2} + 14 T + 16)^{4}$$
$97$ $$(T^{2} + 9 T + 81)^{4}$$