# Properties

 Label 576.2.d.c Level $576$ Weight $2$ Character orbit 576.d Analytic conductor $4.599$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( 4 - 8 \zeta_{12}^{2} ) q^{13} -8 \zeta_{12}^{3} q^{19} + 5 q^{25} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{31} + ( 4 - 8 \zeta_{12}^{2} ) q^{37} + 8 \zeta_{12}^{3} q^{43} + 5 q^{49} + ( -4 + 8 \zeta_{12}^{2} ) q^{61} -16 \zeta_{12}^{3} q^{67} -10 q^{73} + ( 20 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{79} + 24 \zeta_{12}^{3} q^{91} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 20q^{25} + 20q^{49} - 40q^{73} - 56q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 0 0 0 −3.46410 0 0 0
289.2 0 0 0 0 0 −3.46410 0 0 0
289.3 0 0 0 0 0 3.46410 0 0 0
289.4 0 0 0 0 0 3.46410 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-3})$$
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 576.2.d.c 4
3.b odd 2 1 CM 576.2.d.c 4
4.b odd 2 1 inner 576.2.d.c 4
8.b even 2 1 inner 576.2.d.c 4
8.d odd 2 1 inner 576.2.d.c 4
12.b even 2 1 inner 576.2.d.c 4
16.e even 4 1 2304.2.a.v 2
16.e even 4 1 2304.2.a.w 2
16.f odd 4 1 2304.2.a.v 2
16.f odd 4 1 2304.2.a.w 2
24.f even 2 1 inner 576.2.d.c 4
24.h odd 2 1 inner 576.2.d.c 4
48.i odd 4 1 2304.2.a.v 2
48.i odd 4 1 2304.2.a.w 2
48.k even 4 1 2304.2.a.v 2
48.k even 4 1 2304.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.d.c 4 1.a even 1 1 trivial
576.2.d.c 4 3.b odd 2 1 CM
576.2.d.c 4 4.b odd 2 1 inner
576.2.d.c 4 8.b even 2 1 inner
576.2.d.c 4 8.d odd 2 1 inner
576.2.d.c 4 12.b even 2 1 inner
576.2.d.c 4 24.f even 2 1 inner
576.2.d.c 4 24.h odd 2 1 inner
2304.2.a.v 2 16.e even 4 1
2304.2.a.v 2 16.f odd 4 1
2304.2.a.v 2 48.i odd 4 1
2304.2.a.v 2 48.k even 4 1
2304.2.a.w 2 16.e even 4 1
2304.2.a.w 2 16.f odd 4 1
2304.2.a.w 2 48.i odd 4 1
2304.2.a.w 2 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$:

 $$T_{5}$$ $$T_{7}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -12 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 48 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 64 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( -108 + T^{2} )^{2}$$
$37$ $$( 48 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 48 + T^{2} )^{2}$$
$67$ $$( 256 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 10 + T )^{4}$$
$79$ $$( -300 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$( 14 + T )^{4}$$