# Properties

 Label 576.2.a.f Level $576$ Weight $2$ Character orbit 576.a Self dual yes Analytic conductor $4.599$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{7} + O(q^{10})$$ $$q + 4q^{7} - 2q^{13} + 8q^{19} - 5q^{25} + 4q^{31} + 10q^{37} + 8q^{43} + 9q^{49} - 14q^{61} - 16q^{67} - 10q^{73} + 4q^{79} - 8q^{91} + 14q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 4.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.a.f 1
3.b odd 2 1 CM 576.2.a.f 1
4.b odd 2 1 576.2.a.e 1
8.b even 2 1 144.2.a.a 1
8.d odd 2 1 36.2.a.a 1
12.b even 2 1 576.2.a.e 1
16.e even 4 2 2304.2.d.a 2
16.f odd 4 2 2304.2.d.q 2
24.f even 2 1 36.2.a.a 1
24.h odd 2 1 144.2.a.a 1
40.e odd 2 1 900.2.a.g 1
40.f even 2 1 3600.2.a.e 1
40.i odd 4 2 3600.2.f.m 2
40.k even 4 2 900.2.d.b 2
48.i odd 4 2 2304.2.d.a 2
48.k even 4 2 2304.2.d.q 2
56.e even 2 1 1764.2.a.e 1
56.h odd 2 1 7056.2.a.bb 1
56.k odd 6 2 1764.2.k.h 2
56.m even 6 2 1764.2.k.g 2
72.j odd 6 2 1296.2.i.h 2
72.l even 6 2 324.2.e.c 2
72.n even 6 2 1296.2.i.h 2
72.p odd 6 2 324.2.e.c 2
88.g even 2 1 4356.2.a.g 1
104.h odd 2 1 6084.2.a.i 1
104.m even 4 2 6084.2.b.f 2
120.i odd 2 1 3600.2.a.e 1
120.m even 2 1 900.2.a.g 1
120.q odd 4 2 900.2.d.b 2
120.w even 4 2 3600.2.f.m 2
168.e odd 2 1 1764.2.a.e 1
168.i even 2 1 7056.2.a.bb 1
168.v even 6 2 1764.2.k.h 2
168.be odd 6 2 1764.2.k.g 2
264.p odd 2 1 4356.2.a.g 1
312.h even 2 1 6084.2.a.i 1
312.w odd 4 2 6084.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 8.d odd 2 1
36.2.a.a 1 24.f even 2 1
144.2.a.a 1 8.b even 2 1
144.2.a.a 1 24.h odd 2 1
324.2.e.c 2 72.l even 6 2
324.2.e.c 2 72.p odd 6 2
576.2.a.e 1 4.b odd 2 1
576.2.a.e 1 12.b even 2 1
576.2.a.f 1 1.a even 1 1 trivial
576.2.a.f 1 3.b odd 2 1 CM
900.2.a.g 1 40.e odd 2 1
900.2.a.g 1 120.m even 2 1
900.2.d.b 2 40.k even 4 2
900.2.d.b 2 120.q odd 4 2
1296.2.i.h 2 72.j odd 6 2
1296.2.i.h 2 72.n even 6 2
1764.2.a.e 1 56.e even 2 1
1764.2.a.e 1 168.e odd 2 1
1764.2.k.g 2 56.m even 6 2
1764.2.k.g 2 168.be odd 6 2
1764.2.k.h 2 56.k odd 6 2
1764.2.k.h 2 168.v even 6 2
2304.2.d.a 2 16.e even 4 2
2304.2.d.a 2 48.i odd 4 2
2304.2.d.q 2 16.f odd 4 2
2304.2.d.q 2 48.k even 4 2
3600.2.a.e 1 40.f even 2 1
3600.2.a.e 1 120.i odd 2 1
3600.2.f.m 2 40.i odd 4 2
3600.2.f.m 2 120.w even 4 2
4356.2.a.g 1 88.g even 2 1
4356.2.a.g 1 264.p odd 2 1
6084.2.a.i 1 104.h odd 2 1
6084.2.a.i 1 312.h even 2 1
6084.2.b.f 2 104.m even 4 2
6084.2.b.f 2 312.w odd 4 2
7056.2.a.bb 1 56.h odd 2 1
7056.2.a.bb 1 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(576))$$:

 $$T_{5}$$ $$T_{7} - 4$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-4 + T$$
$11$ $$T$$
$13$ $$2 + T$$
$17$ $$T$$
$19$ $$-8 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$-4 + T$$
$37$ $$-10 + T$$
$41$ $$T$$
$43$ $$-8 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$14 + T$$
$67$ $$16 + T$$
$71$ $$T$$
$73$ $$10 + T$$
$79$ $$-4 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$-14 + T$$