# Properties

 Label 588.3.p.b Level $588$ Weight $3$ Character orbit 588.p Analytic conductor $16.022$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 588.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.0218395444$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -3 \zeta_{6} q^{3} + ( -9 + 9 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q -3 \zeta_{6} q^{3} + ( -9 + 9 \zeta_{6} ) q^{9} + 22 q^{13} + ( 26 - 26 \zeta_{6} ) q^{19} -25 \zeta_{6} q^{25} + 27 q^{27} -46 \zeta_{6} q^{31} + ( -26 + 26 \zeta_{6} ) q^{37} -66 \zeta_{6} q^{39} -22 q^{43} -78 q^{57} + ( 74 - 74 \zeta_{6} ) q^{61} -122 \zeta_{6} q^{67} -46 \zeta_{6} q^{73} + ( -75 + 75 \zeta_{6} ) q^{75} + ( 142 - 142 \zeta_{6} ) q^{79} -81 \zeta_{6} q^{81} + ( -138 + 138 \zeta_{6} ) q^{93} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 9q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 9q^{9} + 44q^{13} + 26q^{19} - 25q^{25} + 54q^{27} - 46q^{31} - 26q^{37} - 66q^{39} - 44q^{43} - 156q^{57} + 74q^{61} - 122q^{67} - 46q^{73} - 75q^{75} + 142q^{79} - 81q^{81} - 138q^{93} - 4q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 2.59808i 0 0 0 0 0 −4.50000 7.79423i 0
569.1 0 −1.50000 2.59808i 0 0 0 0 0 −4.50000 + 7.79423i 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 $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.p.b 2
3.b odd 2 1 CM 588.3.p.b 2
7.b odd 2 1 588.3.p.c 2
7.c even 3 1 588.3.c.c 1
7.c even 3 1 inner 588.3.p.b 2
7.d odd 6 1 12.3.c.a 1
7.d odd 6 1 588.3.p.c 2
21.c even 2 1 588.3.p.c 2
21.g even 6 1 12.3.c.a 1
21.g even 6 1 588.3.p.c 2
21.h odd 6 1 588.3.c.c 1
21.h odd 6 1 inner 588.3.p.b 2
28.f even 6 1 48.3.e.a 1
35.i odd 6 1 300.3.g.b 1
35.k even 12 2 300.3.b.a 2
56.j odd 6 1 192.3.e.b 1
56.m even 6 1 192.3.e.a 1
63.i even 6 1 324.3.g.b 2
63.k odd 6 1 324.3.g.b 2
63.s even 6 1 324.3.g.b 2
63.t odd 6 1 324.3.g.b 2
77.i even 6 1 1452.3.e.b 1
84.j odd 6 1 48.3.e.a 1
105.p even 6 1 300.3.g.b 1
105.w odd 12 2 300.3.b.a 2
112.v even 12 2 768.3.h.b 2
112.x odd 12 2 768.3.h.a 2
140.s even 6 1 1200.3.l.b 1
140.x odd 12 2 1200.3.c.c 2
168.ba even 6 1 192.3.e.b 1
168.be odd 6 1 192.3.e.a 1
231.k odd 6 1 1452.3.e.b 1
252.n even 6 1 1296.3.q.b 2
252.r odd 6 1 1296.3.q.b 2
252.bj even 6 1 1296.3.q.b 2
252.bn odd 6 1 1296.3.q.b 2
336.bo even 12 2 768.3.h.a 2
336.br odd 12 2 768.3.h.b 2
420.be odd 6 1 1200.3.l.b 1
420.br even 12 2 1200.3.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 7.d odd 6 1
12.3.c.a 1 21.g even 6 1
48.3.e.a 1 28.f even 6 1
48.3.e.a 1 84.j odd 6 1
192.3.e.a 1 56.m even 6 1
192.3.e.a 1 168.be odd 6 1
192.3.e.b 1 56.j odd 6 1
192.3.e.b 1 168.ba even 6 1
300.3.b.a 2 35.k even 12 2
300.3.b.a 2 105.w odd 12 2
300.3.g.b 1 35.i odd 6 1
300.3.g.b 1 105.p even 6 1
324.3.g.b 2 63.i even 6 1
324.3.g.b 2 63.k odd 6 1
324.3.g.b 2 63.s even 6 1
324.3.g.b 2 63.t odd 6 1
588.3.c.c 1 7.c even 3 1
588.3.c.c 1 21.h odd 6 1
588.3.p.b 2 1.a even 1 1 trivial
588.3.p.b 2 3.b odd 2 1 CM
588.3.p.b 2 7.c even 3 1 inner
588.3.p.b 2 21.h odd 6 1 inner
588.3.p.c 2 7.b odd 2 1
588.3.p.c 2 7.d odd 6 1
588.3.p.c 2 21.c even 2 1
588.3.p.c 2 21.g even 6 1
768.3.h.a 2 112.x odd 12 2
768.3.h.a 2 336.bo even 12 2
768.3.h.b 2 112.v even 12 2
768.3.h.b 2 336.br odd 12 2
1200.3.c.c 2 140.x odd 12 2
1200.3.c.c 2 420.br even 12 2
1200.3.l.b 1 140.s even 6 1
1200.3.l.b 1 420.be odd 6 1
1296.3.q.b 2 252.n even 6 1
1296.3.q.b 2 252.r odd 6 1
1296.3.q.b 2 252.bj even 6 1
1296.3.q.b 2 252.bn odd 6 1
1452.3.e.b 1 77.i even 6 1
1452.3.e.b 1 231.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(588, [\chi])$$:

 $$T_{5}$$ $$T_{13} - 22$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -22 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$676 - 26 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$2116 + 46 T + T^{2}$$
$37$ $$676 + 26 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 22 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$5476 - 74 T + T^{2}$$
$67$ $$14884 + 122 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$2116 + 46 T + T^{2}$$
$79$ $$20164 - 142 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 2 + T )^{2}$$
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