# Properties

 Label 576.3.h.b Level $576$ Weight $3$ Character orbit 576.h Analytic conductor $15.695$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 576.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.6948632272$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{12}\cdot 3^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{5} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{7} +O(q^{10})$$ $$q + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{5} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{7} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{11} + ( 6 - 12 \zeta_{24}^{4} ) q^{13} + ( -15 \zeta_{24} + 15 \zeta_{24}^{3} + 15 \zeta_{24}^{5} ) q^{17} + 20 \zeta_{24}^{6} q^{19} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{23} + 29 q^{25} + ( 15 \zeta_{24} + 15 \zeta_{24}^{3} + 15 \zeta_{24}^{5} - 30 \zeta_{24}^{7} ) q^{29} + ( -60 \zeta_{24}^{2} + 30 \zeta_{24}^{6} ) q^{31} + ( -54 \zeta_{24} - 54 \zeta_{24}^{3} + 54 \zeta_{24}^{5} ) q^{35} + ( -24 + 48 \zeta_{24}^{4} ) q^{37} + ( -51 \zeta_{24} + 51 \zeta_{24}^{3} + 51 \zeta_{24}^{5} ) q^{41} + 40 \zeta_{24}^{6} q^{43} + ( 30 \zeta_{24} - 30 \zeta_{24}^{3} + 30 \zeta_{24}^{5} + 60 \zeta_{24}^{7} ) q^{47} + 59 q^{49} + ( -15 \zeta_{24} - 15 \zeta_{24}^{3} - 15 \zeta_{24}^{5} + 30 \zeta_{24}^{7} ) q^{53} + ( 72 \zeta_{24}^{2} - 36 \zeta_{24}^{6} ) q^{55} + ( -24 \zeta_{24} - 24 \zeta_{24}^{3} + 24 \zeta_{24}^{5} ) q^{59} + ( -54 \zeta_{24} + 54 \zeta_{24}^{3} + 54 \zeta_{24}^{5} ) q^{65} + 100 \zeta_{24}^{6} q^{67} + ( -30 \zeta_{24} + 30 \zeta_{24}^{3} - 30 \zeta_{24}^{5} - 60 \zeta_{24}^{7} ) q^{71} + 20 q^{73} + ( -36 \zeta_{24} - 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} + 72 \zeta_{24}^{7} ) q^{77} + ( 60 \zeta_{24}^{2} - 30 \zeta_{24}^{6} ) q^{79} + ( -90 \zeta_{24} - 90 \zeta_{24}^{3} + 90 \zeta_{24}^{5} ) q^{83} + ( 90 - 180 \zeta_{24}^{4} ) q^{85} + ( -9 \zeta_{24} + 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} ) q^{89} -108 \zeta_{24}^{6} q^{91} + ( -60 \zeta_{24} + 60 \zeta_{24}^{3} - 60 \zeta_{24}^{5} - 120 \zeta_{24}^{7} ) q^{95} + 40 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 232q^{25} + 472q^{49} + 160q^{73} + 320q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 − 0.258819i −0.965926 + 0.258819i
0 0 0 −7.34847 0 −10.3923 0 0 0
161.2 0 0 0 −7.34847 0 −10.3923 0 0 0
161.3 0 0 0 −7.34847 0 10.3923 0 0 0
161.4 0 0 0 −7.34847 0 10.3923 0 0 0
161.5 0 0 0 7.34847 0 −10.3923 0 0 0
161.6 0 0 0 7.34847 0 −10.3923 0 0 0
161.7 0 0 0 7.34847 0 10.3923 0 0 0
161.8 0 0 0 7.34847 0 10.3923 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.8 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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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 576.3.h.b 8
3.b odd 2 1 inner 576.3.h.b 8
4.b odd 2 1 inner 576.3.h.b 8
8.b even 2 1 inner 576.3.h.b 8
8.d odd 2 1 inner 576.3.h.b 8
12.b even 2 1 inner 576.3.h.b 8
16.e even 4 1 2304.3.e.g 4
16.e even 4 1 2304.3.e.j 4
16.f odd 4 1 2304.3.e.g 4
16.f odd 4 1 2304.3.e.j 4
24.f even 2 1 inner 576.3.h.b 8
24.h odd 2 1 inner 576.3.h.b 8
48.i odd 4 1 2304.3.e.g 4
48.i odd 4 1 2304.3.e.j 4
48.k even 4 1 2304.3.e.g 4
48.k even 4 1 2304.3.e.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.h.b 8 1.a even 1 1 trivial
576.3.h.b 8 3.b odd 2 1 inner
576.3.h.b 8 4.b odd 2 1 inner
576.3.h.b 8 8.b even 2 1 inner
576.3.h.b 8 8.d odd 2 1 inner
576.3.h.b 8 12.b even 2 1 inner
576.3.h.b 8 24.f even 2 1 inner
576.3.h.b 8 24.h odd 2 1 inner
2304.3.e.g 4 16.e even 4 1
2304.3.e.g 4 16.f odd 4 1
2304.3.e.g 4 48.i odd 4 1
2304.3.e.g 4 48.k even 4 1
2304.3.e.j 4 16.e even 4 1
2304.3.e.j 4 16.f odd 4 1
2304.3.e.j 4 48.i odd 4 1
2304.3.e.j 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 54$$ acting on $$S_{3}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( -54 + T^{2} )^{4}$$
$7$ $$( -108 + T^{2} )^{4}$$
$11$ $$( -72 + T^{2} )^{4}$$
$13$ $$( 108 + T^{2} )^{4}$$
$17$ $$( 450 + T^{2} )^{4}$$
$19$ $$( 400 + T^{2} )^{4}$$
$23$ $$( 216 + T^{2} )^{4}$$
$29$ $$( -1350 + T^{2} )^{4}$$
$31$ $$( -2700 + T^{2} )^{4}$$
$37$ $$( 1728 + T^{2} )^{4}$$
$41$ $$( 5202 + T^{2} )^{4}$$
$43$ $$( 1600 + T^{2} )^{4}$$
$47$ $$( 5400 + T^{2} )^{4}$$
$53$ $$( -1350 + T^{2} )^{4}$$
$59$ $$( -1152 + T^{2} )^{4}$$
$61$ $$T^{8}$$
$67$ $$( 10000 + T^{2} )^{4}$$
$71$ $$( 5400 + T^{2} )^{4}$$
$73$ $$( -20 + T )^{8}$$
$79$ $$( -2700 + T^{2} )^{4}$$
$83$ $$( -16200 + T^{2} )^{4}$$
$89$ $$( 162 + T^{2} )^{4}$$
$97$ $$( -40 + T )^{8}$$