# Properties

 Label 588.2.i.a Level $588$ Weight $2$ Character orbit 588.i Analytic conductor $4.695$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 588.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.69520363885$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} -4 q^{13} + 2 q^{15} + ( -6 + 6 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + q^{27} -10 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} -6 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{39} -6 q^{41} + 4 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} -8 \zeta_{6} q^{47} -6 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{53} + 4 q^{55} + 8 q^{57} + ( 4 - 4 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} + 8 \zeta_{6} q^{65} + ( 8 - 8 \zeta_{6} ) q^{67} -6 q^{69} -10 q^{71} + ( -4 + 4 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} -4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + 12 q^{85} + ( 10 - 10 \zeta_{6} ) q^{87} + 14 \zeta_{6} q^{89} -4 \zeta_{6} q^{93} + ( -16 + 16 \zeta_{6} ) q^{95} + 4 q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 2q^{5} - q^{9} + O(q^{10})$$ $$2q - q^{3} - 2q^{5} - q^{9} - 2q^{11} - 8q^{13} + 4q^{15} - 6q^{17} - 8q^{19} + 6q^{23} + q^{25} + 2q^{27} - 20q^{29} - 4q^{31} - 2q^{33} - 6q^{37} + 4q^{39} - 12q^{41} + 8q^{43} - 2q^{45} - 8q^{47} - 6q^{51} - 2q^{53} + 8q^{55} + 16q^{57} + 4q^{59} + 8q^{61} + 8q^{65} + 8q^{67} - 12q^{69} - 20q^{71} - 4q^{73} + q^{75} - 4q^{79} - q^{81} + 24q^{83} + 24q^{85} + 10q^{87} + 14q^{89} - 4q^{93} - 16q^{95} + 8q^{97} + 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 −1.00000 1.73205i 0 0 0 −0.500000 0.866025i 0
373.1 0 −0.500000 0.866025i 0 −1.00000 + 1.73205i 0 0 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 + 2 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$36 + 6 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$64 + 8 T + T^{2}$$
$53$ $$4 + 2 T + T^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$64 - 8 T + T^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$16 + 4 T + T^{2}$$
$79$ $$16 + 4 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$196 - 14 T + T^{2}$$
$97$ $$( -4 + T )^{2}$$