Properties

 Label 576.4.a.e 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

 $$f(q)$$ $$=$$ $$q - 14 q^{5} - 36 q^{7} + O(q^{10})$$ $$q - 14 q^{5} - 36 q^{7} - 36 q^{11} - 54 q^{13} + 22 q^{17} - 36 q^{19} + 144 q^{23} + 71 q^{25} + 50 q^{29} - 108 q^{31} + 504 q^{35} - 214 q^{37} + 446 q^{41} - 252 q^{43} - 72 q^{47} + 953 q^{49} - 22 q^{53} + 504 q^{55} - 684 q^{59} + 466 q^{61} + 756 q^{65} + 180 q^{67} - 576 q^{71} - 54 q^{73} + 1296 q^{77} - 972 q^{79} - 684 q^{83} - 308 q^{85} - 346 q^{89} + 1944 q^{91} + 504 q^{95} - 1134 q^{97} + O(q^{100})$$

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 −14.0000 0 −36.0000 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.e 1
3.b odd 2 1 192.4.a.e 1
4.b odd 2 1 576.4.a.f 1
8.b even 2 1 288.4.a.j 1
8.d odd 2 1 288.4.a.k 1
12.b even 2 1 192.4.a.k 1
24.f even 2 1 96.4.a.a 1
24.h odd 2 1 96.4.a.d yes 1
48.i odd 4 2 768.4.d.p 2
48.k even 4 2 768.4.d.a 2
120.i odd 2 1 2400.4.a.k 1
120.m even 2 1 2400.4.a.l 1

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

Hecke kernels

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

 $$T_{5} + 14$$ $$T_{7} + 36$$ $$T_{11} + 36$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$14 + T$$
$7$ $$36 + T$$
$11$ $$36 + T$$
$13$ $$54 + T$$
$17$ $$-22 + T$$
$19$ $$36 + T$$
$23$ $$-144 + T$$
$29$ $$-50 + T$$
$31$ $$108 + T$$
$37$ $$214 + T$$
$41$ $$-446 + T$$
$43$ $$252 + T$$
$47$ $$72 + T$$
$53$ $$22 + T$$
$59$ $$684 + T$$
$61$ $$-466 + T$$
$67$ $$-180 + T$$
$71$ $$576 + T$$
$73$ $$54 + T$$
$79$ $$972 + T$$
$83$ $$684 + T$$
$89$ $$346 + T$$
$97$ $$1134 + T$$