# Properties

 Label 864.2.c.b Level $864$ Weight $2$ Character orbit 864.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + ( 1 - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + ( 1 - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{7} + ( 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{11} + ( 1 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{13} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} + ( -1 + 2 \zeta_{24}^{4} ) q^{19} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{23} + ( -3 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{25} + ( 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{29} + ( -2 + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{31} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{35} + ( -3 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{37} + ( 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{41} + ( 2 - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{43} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{47} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{49} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{53} + 4 \zeta_{24}^{6} q^{55} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{59} + ( 1 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{61} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 10 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{65} + ( -1 + 2 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{67} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{71} + ( 3 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{73} + ( -2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{77} + ( -3 + 6 \zeta_{24}^{4} - 10 \zeta_{24}^{6} ) q^{79} + ( 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{83} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{85} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{89} + ( 5 - 10 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{91} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{95} + 7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 8 q^{13} - 24 q^{25} - 24 q^{37} + 8 q^{61} + 24 q^{73} + 56 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
863.1
 −0.965926 − 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 − 0.965926i 0.258819 + 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i
0 0 0 3.86370i 0 3.73205i 0 0 0
863.2 0 0 0 3.86370i 0 3.73205i 0 0 0
863.3 0 0 0 1.03528i 0 0.267949i 0 0 0
863.4 0 0 0 1.03528i 0 0.267949i 0 0 0
863.5 0 0 0 1.03528i 0 0.267949i 0 0 0
863.6 0 0 0 1.03528i 0 0.267949i 0 0 0
863.7 0 0 0 3.86370i 0 3.73205i 0 0 0
863.8 0 0 0 3.86370i 0 3.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 863.8 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
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.c.b 8
3.b odd 2 1 inner 864.2.c.b 8
4.b odd 2 1 inner 864.2.c.b 8
8.b even 2 1 1728.2.c.f 8
8.d odd 2 1 1728.2.c.f 8
9.c even 3 1 2592.2.s.c 8
9.c even 3 1 2592.2.s.g 8
9.d odd 6 1 2592.2.s.c 8
9.d odd 6 1 2592.2.s.g 8
12.b even 2 1 inner 864.2.c.b 8
24.f even 2 1 1728.2.c.f 8
24.h odd 2 1 1728.2.c.f 8
36.f odd 6 1 2592.2.s.c 8
36.f odd 6 1 2592.2.s.g 8
36.h even 6 1 2592.2.s.c 8
36.h even 6 1 2592.2.s.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.b 8 1.a even 1 1 trivial
864.2.c.b 8 3.b odd 2 1 inner
864.2.c.b 8 4.b odd 2 1 inner
864.2.c.b 8 12.b even 2 1 inner
1728.2.c.f 8 8.b even 2 1
1728.2.c.f 8 8.d odd 2 1
1728.2.c.f 8 24.f even 2 1
1728.2.c.f 8 24.h odd 2 1
2592.2.s.c 8 9.c even 3 1
2592.2.s.c 8 9.d odd 6 1
2592.2.s.c 8 36.f odd 6 1
2592.2.s.c 8 36.h even 6 1
2592.2.s.g 8 9.c even 3 1
2592.2.s.g 8 9.d odd 6 1
2592.2.s.g 8 36.f odd 6 1
2592.2.s.g 8 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 16 T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 16 + 16 T^{2} + T^{4} )^{2}$$
$7$ $$( 1 + 14 T^{2} + T^{4} )^{2}$$
$11$ $$( 16 - 16 T^{2} + T^{4} )^{2}$$
$13$ $$( -11 - 2 T + T^{2} )^{4}$$
$17$ $$( 144 + 48 T^{2} + T^{4} )^{2}$$
$19$ $$( 3 + T^{2} )^{4}$$
$23$ $$( 2704 - 112 T^{2} + T^{4} )^{2}$$
$29$ $$( 256 + 64 T^{2} + T^{4} )^{2}$$
$31$ $$( 16 + 56 T^{2} + T^{4} )^{2}$$
$37$ $$( -3 + 6 T + T^{2} )^{4}$$
$41$ $$( 256 + 64 T^{2} + T^{4} )^{2}$$
$43$ $$( 16 + 56 T^{2} + T^{4} )^{2}$$
$47$ $$( 1936 - 112 T^{2} + T^{4} )^{2}$$
$53$ $$( 2304 + 192 T^{2} + T^{4} )^{2}$$
$59$ $$( 8464 - 208 T^{2} + T^{4} )^{2}$$
$61$ $$( -107 - 2 T + T^{2} )^{4}$$
$67$ $$( 3721 + 134 T^{2} + T^{4} )^{2}$$
$71$ $$( -128 + T^{2} )^{4}$$
$73$ $$( -39 - 6 T + T^{2} )^{4}$$
$79$ $$( 5329 + 254 T^{2} + T^{4} )^{2}$$
$83$ $$( 256 - 64 T^{2} + T^{4} )^{2}$$
$89$ $$( 144 + 48 T^{2} + T^{4} )^{2}$$
$97$ $$( -7 + T )^{8}$$