Properties

 Label 576.4.a.n Level $576$ Weight $4$ Character orbit 576.a Self dual yes Analytic conductor $33.985$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 96) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{5} - 12 q^{7} + O(q^{10})$$ $$q + 2 q^{5} - 12 q^{7} - 60 q^{11} + 42 q^{13} - 10 q^{17} + 132 q^{19} - 48 q^{23} - 121 q^{25} + 226 q^{29} + 252 q^{31} - 24 q^{35} + 362 q^{37} + 94 q^{41} - 228 q^{43} - 408 q^{47} - 199 q^{49} + 346 q^{53} - 120 q^{55} + 300 q^{59} + 466 q^{61} + 84 q^{65} + 204 q^{67} + 1056 q^{71} + 330 q^{73} + 720 q^{77} - 612 q^{79} - 564 q^{83} - 20 q^{85} + 1510 q^{89} - 504 q^{91} + 264 q^{95} + 594 q^{97} + O(q^{100})$$

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 2.00000 0 −12.0000 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

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.4.a.n 1
3.b odd 2 1 192.4.a.j 1
4.b odd 2 1 576.4.a.o 1
8.b even 2 1 288.4.a.e 1
8.d odd 2 1 288.4.a.f 1
12.b even 2 1 192.4.a.d 1
24.f even 2 1 96.4.a.e yes 1
24.h odd 2 1 96.4.a.b 1
48.i odd 4 2 768.4.d.m 2
48.k even 4 2 768.4.d.d 2
120.i odd 2 1 2400.4.a.t 1
120.m even 2 1 2400.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.4.a.b 1 24.h odd 2 1
96.4.a.e yes 1 24.f even 2 1
192.4.a.d 1 12.b even 2 1
192.4.a.j 1 3.b odd 2 1
288.4.a.e 1 8.b even 2 1
288.4.a.f 1 8.d odd 2 1
576.4.a.n 1 1.a even 1 1 trivial
576.4.a.o 1 4.b odd 2 1
768.4.d.d 2 48.k even 4 2
768.4.d.m 2 48.i odd 4 2
2400.4.a.c 1 120.m even 2 1
2400.4.a.t 1 120.i odd 2 1

Hecke kernels

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

 $$T_{5} - 2$$ $$T_{7} + 12$$ $$T_{11} + 60$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-2 + T$$
$7$ $$12 + T$$
$11$ $$60 + T$$
$13$ $$-42 + T$$
$17$ $$10 + T$$
$19$ $$-132 + T$$
$23$ $$48 + T$$
$29$ $$-226 + T$$
$31$ $$-252 + T$$
$37$ $$-362 + T$$
$41$ $$-94 + T$$
$43$ $$228 + T$$
$47$ $$408 + T$$
$53$ $$-346 + T$$
$59$ $$-300 + T$$
$61$ $$-466 + T$$
$67$ $$-204 + T$$
$71$ $$-1056 + T$$
$73$ $$-330 + T$$
$79$ $$612 + T$$
$83$ $$564 + T$$
$89$ $$-1510 + T$$
$97$ $$-594 + T$$