# Properties

 Label 588.2.i.f Level $588$ Weight $2$ Character orbit 588.i Analytic conductor $4.695$ Analytic rank $0$ 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: $$0$$ 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: no (minimal twist has level 84) 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.

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

## 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 0 0 0 0 −0.500000 0.866025i 0
373.1 0 0.500000 + 0.866025i 0 0 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.f 2
3.b odd 2 1 1764.2.k.d 2
4.b odd 2 1 2352.2.q.g 2
7.b odd 2 1 588.2.i.c 2
7.c even 3 1 588.2.a.c 1
7.c even 3 1 inner 588.2.i.f 2
7.d odd 6 1 84.2.a.b 1
7.d odd 6 1 588.2.i.c 2
21.c even 2 1 1764.2.k.e 2
21.g even 6 1 252.2.a.b 1
21.g even 6 1 1764.2.k.e 2
21.h odd 6 1 1764.2.a.g 1
21.h odd 6 1 1764.2.k.d 2
28.d even 2 1 2352.2.q.s 2
28.f even 6 1 336.2.a.b 1
28.f even 6 1 2352.2.q.s 2
28.g odd 6 1 2352.2.a.s 1
28.g odd 6 1 2352.2.q.g 2
35.i odd 6 1 2100.2.a.a 1
35.k even 12 2 2100.2.k.a 2
56.j odd 6 1 1344.2.a.f 1
56.k odd 6 1 9408.2.a.r 1
56.m even 6 1 1344.2.a.o 1
56.p even 6 1 9408.2.a.co 1
63.i even 6 1 2268.2.j.f 2
63.k odd 6 1 2268.2.j.i 2
63.s even 6 1 2268.2.j.f 2
63.t odd 6 1 2268.2.j.i 2
84.j odd 6 1 1008.2.a.g 1
84.n even 6 1 7056.2.a.x 1
105.p even 6 1 6300.2.a.p 1
105.w odd 12 2 6300.2.k.r 2
112.v even 12 2 5376.2.c.x 2
112.x odd 12 2 5376.2.c.i 2
140.s even 6 1 8400.2.a.ct 1
168.ba even 6 1 4032.2.a.u 1
168.be odd 6 1 4032.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.b 1 7.d odd 6 1
252.2.a.b 1 21.g even 6 1
336.2.a.b 1 28.f even 6 1
588.2.a.c 1 7.c even 3 1
588.2.i.c 2 7.b odd 2 1
588.2.i.c 2 7.d odd 6 1
588.2.i.f 2 1.a even 1 1 trivial
588.2.i.f 2 7.c even 3 1 inner
1008.2.a.g 1 84.j odd 6 1
1344.2.a.f 1 56.j odd 6 1
1344.2.a.o 1 56.m even 6 1
1764.2.a.g 1 21.h odd 6 1
1764.2.k.d 2 3.b odd 2 1
1764.2.k.d 2 21.h odd 6 1
1764.2.k.e 2 21.c even 2 1
1764.2.k.e 2 21.g even 6 1
2100.2.a.a 1 35.i odd 6 1
2100.2.k.a 2 35.k even 12 2
2268.2.j.f 2 63.i even 6 1
2268.2.j.f 2 63.s even 6 1
2268.2.j.i 2 63.k odd 6 1
2268.2.j.i 2 63.t odd 6 1
2352.2.a.s 1 28.g odd 6 1
2352.2.q.g 2 4.b odd 2 1
2352.2.q.g 2 28.g odd 6 1
2352.2.q.s 2 28.d even 2 1
2352.2.q.s 2 28.f even 6 1
4032.2.a.t 1 168.be odd 6 1
4032.2.a.u 1 168.ba even 6 1
5376.2.c.i 2 112.x odd 12 2
5376.2.c.x 2 112.v even 12 2
6300.2.a.p 1 105.p even 6 1
6300.2.k.r 2 105.w odd 12 2
7056.2.a.x 1 84.n even 6 1
8400.2.a.ct 1 140.s even 6 1
9408.2.a.r 1 56.k odd 6 1
9408.2.a.co 1 56.p even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + T^{2}$$
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 12 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4}$$
$53$ $$1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4}$$
$67$ $$1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$( 1 - 12 T + 83 T^{2} )^{2}$$
$89$ $$1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 10 T + 97 T^{2} )^{2}$$