# Properties

 Label 864.2.d.c Level $864$ Weight $2$ Character orbit 864.d 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.629407744.1 Defining polynomial: $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} -\beta_{7} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} -\beta_{7} q^{7} -\beta_{2} q^{11} + ( \beta_{3} + \beta_{6} ) q^{13} -\beta_{5} q^{17} + ( -\beta_{3} - 2 \beta_{6} ) q^{19} + ( \beta_{4} - \beta_{5} ) q^{23} + ( -1 + 2 \beta_{7} ) q^{25} -2 \beta_{2} q^{29} -2 q^{31} + ( 2 \beta_{1} - \beta_{2} ) q^{35} + ( -\beta_{3} + \beta_{6} ) q^{37} + 2 \beta_{4} q^{41} + 2 \beta_{6} q^{43} + ( -\beta_{4} - \beta_{5} ) q^{47} -2 \beta_{2} q^{53} + ( -2 + 2 \beta_{7} ) q^{55} + ( 4 \beta_{1} + \beta_{2} ) q^{59} + ( \beta_{3} - 3 \beta_{6} ) q^{61} + ( -2 \beta_{4} - \beta_{5} ) q^{65} + \beta_{3} q^{67} + ( 1 - 4 \beta_{7} ) q^{73} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 6 - \beta_{7} ) q^{79} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -4 \beta_{3} - 2 \beta_{6} ) q^{85} + \beta_{5} q^{89} + ( 3 \beta_{3} + 2 \beta_{6} ) q^{91} + ( 3 \beta_{4} + \beta_{5} ) q^{95} + ( -3 - 2 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q - 8 q^{25} - 16 q^{31} - 16 q^{55} + 8 q^{73} + 48 q^{79} - 24 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{3} - 4 \nu$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{5} - 2 \nu^{3} + 16 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 6 \nu^{3} + 4 \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{4} + 6 \nu^{2} - 8$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{4} - 2 \nu^{2} - 4$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{7} + \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{4} - 3 \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$\beta_{7} + \beta_{6} + 2 \beta_{3} + 5$$ $$\nu^{7}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{2} + 4 \beta_{1}$$

## Character values

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

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 1.38255 − 0.297594i −1.38255 − 0.297594i 0.767178 − 1.18804i −0.767178 − 1.18804i −0.767178 + 1.18804i 0.767178 + 1.18804i −1.38255 + 0.297594i 1.38255 + 0.297594i
0 0 0 3.36028i 0 2.64575 0 0 0
433.2 0 0 0 3.36028i 0 2.64575 0 0 0
433.3 0 0 0 0.841723i 0 −2.64575 0 0 0
433.4 0 0 0 0.841723i 0 −2.64575 0 0 0
433.5 0 0 0 0.841723i 0 −2.64575 0 0 0
433.6 0 0 0 0.841723i 0 −2.64575 0 0 0
433.7 0 0 0 3.36028i 0 2.64575 0 0 0
433.8 0 0 0 3.36028i 0 2.64575 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 433.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.d.c 8
3.b odd 2 1 inner 864.2.d.c 8
4.b odd 2 1 216.2.d.c 8
8.b even 2 1 inner 864.2.d.c 8
8.d odd 2 1 216.2.d.c 8
9.c even 3 2 2592.2.r.q 16
9.d odd 6 2 2592.2.r.q 16
12.b even 2 1 216.2.d.c 8
16.e even 4 1 6912.2.a.cd 4
16.e even 4 1 6912.2.a.ci 4
16.f odd 4 1 6912.2.a.cc 4
16.f odd 4 1 6912.2.a.cj 4
24.f even 2 1 216.2.d.c 8
24.h odd 2 1 inner 864.2.d.c 8
36.f odd 6 2 648.2.n.q 16
36.h even 6 2 648.2.n.q 16
48.i odd 4 1 6912.2.a.cd 4
48.i odd 4 1 6912.2.a.ci 4
48.k even 4 1 6912.2.a.cc 4
48.k even 4 1 6912.2.a.cj 4
72.j odd 6 2 2592.2.r.q 16
72.l even 6 2 648.2.n.q 16
72.n even 6 2 2592.2.r.q 16
72.p odd 6 2 648.2.n.q 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.c 8 4.b odd 2 1
216.2.d.c 8 8.d odd 2 1
216.2.d.c 8 12.b even 2 1
216.2.d.c 8 24.f even 2 1
648.2.n.q 16 36.f odd 6 2
648.2.n.q 16 36.h even 6 2
648.2.n.q 16 72.l even 6 2
648.2.n.q 16 72.p odd 6 2
864.2.d.c 8 1.a even 1 1 trivial
864.2.d.c 8 3.b odd 2 1 inner
864.2.d.c 8 8.b even 2 1 inner
864.2.d.c 8 24.h odd 2 1 inner
2592.2.r.q 16 9.c even 3 2
2592.2.r.q 16 9.d odd 6 2
2592.2.r.q 16 72.j odd 6 2
2592.2.r.q 16 72.n even 6 2
6912.2.a.cc 4 16.f odd 4 1
6912.2.a.cc 4 48.k even 4 1
6912.2.a.cd 4 16.e even 4 1
6912.2.a.cd 4 48.i odd 4 1
6912.2.a.ci 4 16.e even 4 1
6912.2.a.ci 4 48.i odd 4 1
6912.2.a.cj 4 16.f odd 4 1
6912.2.a.cj 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12 T_{5}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 8 + 12 T^{2} + T^{4} )^{2}$$
$7$ $$( -7 + T^{2} )^{4}$$
$11$ $$( 72 + 20 T^{2} + T^{4} )^{2}$$
$13$ $$( 9 + 22 T^{2} + T^{4} )^{2}$$
$17$ $$( 648 - 52 T^{2} + T^{4} )^{2}$$
$19$ $$( 729 + 58 T^{2} + T^{4} )^{2}$$
$23$ $$( 72 - 76 T^{2} + T^{4} )^{2}$$
$29$ $$( 1152 + 80 T^{2} + T^{4} )^{2}$$
$31$ $$( 2 + T )^{8}$$
$37$ $$( 81 + 46 T^{2} + T^{4} )^{2}$$
$41$ $$( 4608 - 160 T^{2} + T^{4} )^{2}$$
$43$ $$( 576 + 64 T^{2} + T^{4} )^{2}$$
$47$ $$( 648 - 108 T^{2} + T^{4} )^{2}$$
$53$ $$( 1152 + 80 T^{2} + T^{4} )^{2}$$
$59$ $$( 6728 + 180 T^{2} + T^{4} )^{2}$$
$61$ $$( 729 + 198 T^{2} + T^{4} )^{2}$$
$67$ $$( 9 + T^{2} )^{4}$$
$71$ $$T^{8}$$
$73$ $$( -111 - 2 T + T^{2} )^{4}$$
$79$ $$( 29 - 12 T + T^{2} )^{4}$$
$83$ $$( 512 + 96 T^{2} + T^{4} )^{2}$$
$89$ $$( 648 - 52 T^{2} + T^{4} )^{2}$$
$97$ $$( -19 + 6 T + T^{2} )^{4}$$