# Properties

 Label 588.1.c.b Level $588$ Weight $1$ Character orbit 588.c Self dual yes Analytic conductor $0.293$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 588.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.293450227428$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.588.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.588.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{9}+O(q^{10})$$ q + q^3 + q^9 $$q + q^{3} + q^{9} - q^{13} - q^{19} + q^{25} + q^{27} - q^{31} - q^{37} - q^{39} - q^{43} - q^{57} + 2 q^{61} - q^{67} - q^{73} + q^{75} - q^{79} + q^{81} - q^{93} + 2 q^{97}+O(q^{100})$$ q + q^3 + q^9 - q^13 - q^19 + q^25 + q^27 - q^31 - q^37 - q^39 - q^43 - q^57 + 2 * q^61 - q^67 - q^73 + q^75 - q^79 + q^81 - q^93 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 0
0 1.00000 0 0 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.1.c.b 1
3.b odd 2 1 CM 588.1.c.b 1
4.b odd 2 1 2352.1.d.a 1
7.b odd 2 1 588.1.c.a 1
7.c even 3 2 84.1.p.a 2
7.d odd 6 2 588.1.p.a 2
12.b even 2 1 2352.1.d.a 1
21.c even 2 1 588.1.c.a 1
21.g even 6 2 588.1.p.a 2
21.h odd 6 2 84.1.p.a 2
28.d even 2 1 2352.1.d.b 1
28.f even 6 2 2352.1.bn.a 2
28.g odd 6 2 336.1.bn.a 2
35.j even 6 2 2100.1.bn.c 2
35.l odd 12 4 2100.1.bh.a 4
56.k odd 6 2 1344.1.bn.a 2
56.p even 6 2 1344.1.bn.b 2
63.g even 3 2 2268.1.m.a 2
63.h even 3 2 2268.1.bh.b 2
63.j odd 6 2 2268.1.bh.b 2
63.n odd 6 2 2268.1.m.a 2
84.h odd 2 1 2352.1.d.b 1
84.j odd 6 2 2352.1.bn.a 2
84.n even 6 2 336.1.bn.a 2
105.o odd 6 2 2100.1.bn.c 2
105.x even 12 4 2100.1.bh.a 4
168.s odd 6 2 1344.1.bn.b 2
168.v even 6 2 1344.1.bn.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 7.c even 3 2
84.1.p.a 2 21.h odd 6 2
336.1.bn.a 2 28.g odd 6 2
336.1.bn.a 2 84.n even 6 2
588.1.c.a 1 7.b odd 2 1
588.1.c.a 1 21.c even 2 1
588.1.c.b 1 1.a even 1 1 trivial
588.1.c.b 1 3.b odd 2 1 CM
588.1.p.a 2 7.d odd 6 2
588.1.p.a 2 21.g even 6 2
1344.1.bn.a 2 56.k odd 6 2
1344.1.bn.a 2 168.v even 6 2
1344.1.bn.b 2 56.p even 6 2
1344.1.bn.b 2 168.s odd 6 2
2100.1.bh.a 4 35.l odd 12 4
2100.1.bh.a 4 105.x even 12 4
2100.1.bn.c 2 35.j even 6 2
2100.1.bn.c 2 105.o odd 6 2
2268.1.m.a 2 63.g even 3 2
2268.1.m.a 2 63.n odd 6 2
2268.1.bh.b 2 63.h even 3 2
2268.1.bh.b 2 63.j odd 6 2
2352.1.d.a 1 4.b odd 2 1
2352.1.d.a 1 12.b even 2 1
2352.1.d.b 1 28.d even 2 1
2352.1.d.b 1 84.h odd 2 1
2352.1.bn.a 2 28.f even 6 2
2352.1.bn.a 2 84.j odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

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