# Properties

 Label 576.4.a.p Level $576$ Weight $4$ Character orbit 576.a Self dual yes Analytic conductor $33.985$ Analytic rank $1$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 576.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.9851001633$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4 q^{5} + O(q^{10})$$ $$q + 4 q^{5} - 18 q^{13} - 104 q^{17} - 109 q^{25} + 284 q^{29} - 214 q^{37} - 472 q^{41} - 343 q^{49} + 572 q^{53} - 830 q^{61} - 72 q^{65} - 1098 q^{73} - 416 q^{85} + 176 q^{89} - 594 q^{97} + O(q^{100})$$

## 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 4.00000 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.4.a.p 1
3.b odd 2 1 576.4.a.i 1
4.b odd 2 1 CM 576.4.a.p 1
8.b even 2 1 288.4.a.d 1
8.d odd 2 1 288.4.a.d 1
12.b even 2 1 576.4.a.i 1
24.f even 2 1 288.4.a.g yes 1
24.h odd 2 1 288.4.a.g yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.4.a.d 1 8.b even 2 1
288.4.a.d 1 8.d odd 2 1
288.4.a.g yes 1 24.f even 2 1
288.4.a.g yes 1 24.h odd 2 1
576.4.a.i 1 3.b odd 2 1
576.4.a.i 1 12.b even 2 1
576.4.a.p 1 1.a even 1 1 trivial
576.4.a.p 1 4.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(576))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$18 + T$$
$17$ $$104 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-284 + T$$
$31$ $$T$$
$37$ $$214 + T$$
$41$ $$472 + T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$-572 + T$$
$59$ $$T$$
$61$ $$830 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$1098 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$-176 + T$$
$97$ $$594 + T$$