# Properties

 Label 588.2.a.e Level $588$ Weight $2$ Character orbit 588.a Self dual yes Analytic conductor $4.695$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 588.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.69520363885$$ 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

 $$f(q)$$ $$=$$ $$q + q^{3} + 2q^{5} + q^{9} + O(q^{10})$$ $$q + q^{3} + 2q^{5} + q^{9} + 2q^{11} - 4q^{13} + 2q^{15} + 6q^{17} + 8q^{19} - 6q^{23} - q^{25} + q^{27} - 10q^{29} + 4q^{31} + 2q^{33} + 6q^{37} - 4q^{39} - 6q^{41} + 4q^{43} + 2q^{45} + 8q^{47} + 6q^{51} + 2q^{53} + 4q^{55} + 8q^{57} - 4q^{59} - 8q^{61} - 8q^{65} - 8q^{67} - 6q^{69} - 10q^{71} + 4q^{73} - q^{75} + 4q^{79} + q^{81} + 12q^{83} + 12q^{85} - 10q^{87} - 14q^{89} + 4q^{93} + 16q^{95} + 4q^{97} + 2q^{99} + O(q^{100})$$

## 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 1.00000 0 2.00000 0 0 0 1.00000 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 588.2.a.e yes 1
3.b odd 2 1 1764.2.a.b 1
4.b odd 2 1 2352.2.a.j 1
7.b odd 2 1 588.2.a.b 1
7.c even 3 2 588.2.i.a 2
7.d odd 6 2 588.2.i.g 2
8.b even 2 1 9408.2.a.l 1
8.d odd 2 1 9408.2.a.ca 1
12.b even 2 1 7056.2.a.n 1
21.c even 2 1 1764.2.a.i 1
21.g even 6 2 1764.2.k.c 2
21.h odd 6 2 1764.2.k.i 2
28.d even 2 1 2352.2.a.p 1
28.f even 6 2 2352.2.q.k 2
28.g odd 6 2 2352.2.q.p 2
56.e even 2 1 9408.2.a.bf 1
56.h odd 2 1 9408.2.a.cu 1
84.h odd 2 1 7056.2.a.bu 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.a.b 1 7.b odd 2 1
588.2.a.e yes 1 1.a even 1 1 trivial
588.2.i.a 2 7.c even 3 2
588.2.i.g 2 7.d odd 6 2
1764.2.a.b 1 3.b odd 2 1
1764.2.a.i 1 21.c even 2 1
1764.2.k.c 2 21.g even 6 2
1764.2.k.i 2 21.h odd 6 2
2352.2.a.j 1 4.b odd 2 1
2352.2.a.p 1 28.d even 2 1
2352.2.q.k 2 28.f even 6 2
2352.2.q.p 2 28.g odd 6 2
7056.2.a.n 1 12.b even 2 1
7056.2.a.bu 1 84.h odd 2 1
9408.2.a.l 1 8.b even 2 1
9408.2.a.bf 1 56.e even 2 1
9408.2.a.ca 1 8.d odd 2 1
9408.2.a.cu 1 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(588))$$:

 $$T_{5} - 2$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$-2 + T$$
$7$ $$T$$
$11$ $$-2 + T$$
$13$ $$4 + T$$
$17$ $$-6 + T$$
$19$ $$-8 + T$$
$23$ $$6 + T$$
$29$ $$10 + T$$
$31$ $$-4 + T$$
$37$ $$-6 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$-8 + T$$
$53$ $$-2 + T$$
$59$ $$4 + T$$
$61$ $$8 + T$$
$67$ $$8 + T$$
$71$ $$10 + T$$
$73$ $$-4 + T$$
$79$ $$-4 + T$$
$83$ $$-12 + T$$
$89$ $$14 + T$$
$97$ $$-4 + T$$