# Properties

 Label 576.1.g.a Level $576$ Weight $1$ Character orbit 576.g Self dual yes Analytic conductor $0.287$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -4, 12 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 576.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.287461447277$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\zeta_{12})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.1728.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + O(q^{10})$$ $$q + 2q^{13} - q^{25} - 2q^{37} + q^{49} - 2q^{61} - 2q^{73} - 2q^{97} + O(q^{100})$$

## Expression as an eta quotient

$$f(z) = \dfrac{\eta(24z)^{6}}{\eta(12z)^{2}\eta(48z)^{2}}=q\prod_{n=1}^\infty(1 - q^{12n})^{-2}(1 - q^{24n})^{6}(1 - q^{48n})^{-2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0
0 0 0 0 0 0 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 CM by $$\Q(\sqrt{-1})$$
12.b even 2 1 RM by $$\Q(\sqrt{3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.1.g.a 1
3.b odd 2 1 CM 576.1.g.a 1
4.b odd 2 1 CM 576.1.g.a 1
8.b even 2 1 144.1.g.a 1
8.d odd 2 1 144.1.g.a 1
12.b even 2 1 RM 576.1.g.a 1
16.e even 4 2 2304.1.b.a 2
16.f odd 4 2 2304.1.b.a 2
24.f even 2 1 144.1.g.a 1
24.h odd 2 1 144.1.g.a 1
40.e odd 2 1 3600.1.e.b 1
40.f even 2 1 3600.1.e.b 1
40.i odd 4 2 3600.1.j.a 2
40.k even 4 2 3600.1.j.a 2
48.i odd 4 2 2304.1.b.a 2
48.k even 4 2 2304.1.b.a 2
72.j odd 6 2 1296.1.o.b 2
72.l even 6 2 1296.1.o.b 2
72.n even 6 2 1296.1.o.b 2
72.p odd 6 2 1296.1.o.b 2
120.i odd 2 1 3600.1.e.b 1
120.m even 2 1 3600.1.e.b 1
120.q odd 4 2 3600.1.j.a 2
120.w even 4 2 3600.1.j.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 8.b even 2 1
144.1.g.a 1 8.d odd 2 1
144.1.g.a 1 24.f even 2 1
144.1.g.a 1 24.h odd 2 1
576.1.g.a 1 1.a even 1 1 trivial
576.1.g.a 1 3.b odd 2 1 CM
576.1.g.a 1 4.b odd 2 1 CM
576.1.g.a 1 12.b even 2 1 RM
1296.1.o.b 2 72.j odd 6 2
1296.1.o.b 2 72.l even 6 2
1296.1.o.b 2 72.n even 6 2
1296.1.o.b 2 72.p odd 6 2
2304.1.b.a 2 16.e even 4 2
2304.1.b.a 2 16.f odd 4 2
2304.1.b.a 2 48.i odd 4 2
2304.1.b.a 2 48.k even 4 2
3600.1.e.b 1 40.e odd 2 1
3600.1.e.b 1 40.f even 2 1
3600.1.e.b 1 120.i odd 2 1
3600.1.e.b 1 120.m even 2 1
3600.1.j.a 2 40.i odd 4 2
3600.1.j.a 2 40.k even 4 2
3600.1.j.a 2 120.q odd 4 2
3600.1.j.a 2 120.w even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$-2 + T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$2 + T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$2 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$2 + T$$