# Properties

 Label 588.4.a.b Level $588$ Weight $4$ Character orbit 588.a Self dual yes Analytic conductor $34.693$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 588.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.6931230834$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} + 4q^{5} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} + 4q^{5} + 9q^{9} - 20q^{11} - 4q^{13} - 12q^{15} + 24q^{17} + 44q^{19} + 72q^{23} - 109q^{25} - 27q^{27} - 38q^{29} + 184q^{31} + 60q^{33} - 30q^{37} + 12q^{39} - 216q^{41} - 164q^{43} + 36q^{45} + 520q^{47} - 72q^{51} - 146q^{53} - 80q^{55} - 132q^{57} + 460q^{59} + 628q^{61} - 16q^{65} + 556q^{67} - 216q^{69} + 592q^{71} + 1024q^{73} + 327q^{75} - 104q^{79} + 81q^{81} - 324q^{83} + 96q^{85} + 114q^{87} + 896q^{89} - 552q^{93} + 176q^{95} - 920q^{97} - 180q^{99} + O(q^{100})$$

## 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 −3.00000 0 4.00000 0 0 0 9.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## 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 588.4.a.b 1
3.b odd 2 1 1764.4.a.d 1
4.b odd 2 1 2352.4.a.be 1
7.b odd 2 1 588.4.a.e yes 1
7.c even 3 2 588.4.i.g 2
7.d odd 6 2 588.4.i.b 2
21.c even 2 1 1764.4.a.i 1
21.g even 6 2 1764.4.k.g 2
21.h odd 6 2 1764.4.k.j 2
28.d even 2 1 2352.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.b 1 1.a even 1 1 trivial
588.4.a.e yes 1 7.b odd 2 1
588.4.i.b 2 7.d odd 6 2
588.4.i.g 2 7.c even 3 2
1764.4.a.d 1 3.b odd 2 1
1764.4.a.i 1 21.c even 2 1
1764.4.k.g 2 21.g even 6 2
1764.4.k.j 2 21.h odd 6 2
2352.4.a.g 1 28.d even 2 1
2352.4.a.be 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(588))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$20 + T$$
$13$ $$4 + T$$
$17$ $$-24 + T$$
$19$ $$-44 + T$$
$23$ $$-72 + T$$
$29$ $$38 + T$$
$31$ $$-184 + T$$
$37$ $$30 + T$$
$41$ $$216 + T$$
$43$ $$164 + T$$
$47$ $$-520 + T$$
$53$ $$146 + T$$
$59$ $$-460 + T$$
$61$ $$-628 + T$$
$67$ $$-556 + T$$
$71$ $$-592 + T$$
$73$ $$-1024 + T$$
$79$ $$104 + T$$
$83$ $$324 + T$$
$89$ $$-896 + T$$
$97$ $$920 + T$$