# Properties

 Label 588.4.i.g Level $588$ Weight $4$ Character orbit 588.i Analytic conductor $34.693$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 588.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.6931230834$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} -4 \zeta_{6} q^{5} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} -4 \zeta_{6} q^{5} -9 \zeta_{6} q^{9} + ( 20 - 20 \zeta_{6} ) q^{11} -4 q^{13} -12 q^{15} + ( -24 + 24 \zeta_{6} ) q^{17} -44 \zeta_{6} q^{19} -72 \zeta_{6} q^{23} + ( 109 - 109 \zeta_{6} ) q^{25} -27 q^{27} -38 q^{29} + ( -184 + 184 \zeta_{6} ) q^{31} -60 \zeta_{6} q^{33} + 30 \zeta_{6} q^{37} + ( -12 + 12 \zeta_{6} ) q^{39} -216 q^{41} -164 q^{43} + ( -36 + 36 \zeta_{6} ) q^{45} -520 \zeta_{6} q^{47} + 72 \zeta_{6} q^{51} + ( 146 - 146 \zeta_{6} ) q^{53} -80 q^{55} -132 q^{57} + ( -460 + 460 \zeta_{6} ) q^{59} -628 \zeta_{6} q^{61} + 16 \zeta_{6} q^{65} + ( -556 + 556 \zeta_{6} ) q^{67} -216 q^{69} + 592 q^{71} + ( -1024 + 1024 \zeta_{6} ) q^{73} -327 \zeta_{6} q^{75} + 104 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} -324 q^{83} + 96 q^{85} + ( -114 + 114 \zeta_{6} ) q^{87} -896 \zeta_{6} q^{89} + 552 \zeta_{6} q^{93} + ( -176 + 176 \zeta_{6} ) q^{95} -920 q^{97} -180 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 4q^{5} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 4q^{5} - 9q^{9} + 20q^{11} - 8q^{13} - 24q^{15} - 24q^{17} - 44q^{19} - 72q^{23} + 109q^{25} - 54q^{27} - 76q^{29} - 184q^{31} - 60q^{33} + 30q^{37} - 12q^{39} - 432q^{41} - 328q^{43} - 36q^{45} - 520q^{47} + 72q^{51} + 146q^{53} - 160q^{55} - 264q^{57} - 460q^{59} - 628q^{61} + 16q^{65} - 556q^{67} - 432q^{69} + 1184q^{71} - 1024q^{73} - 327q^{75} + 104q^{79} - 81q^{81} - 648q^{83} + 192q^{85} - 114q^{87} - 896q^{89} + 552q^{93} - 176q^{95} - 1840q^{97} - 360q^{99} + O(q^{100})$$

## 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 1.50000 2.59808i 0 −2.00000 3.46410i 0 0 0 −4.50000 7.79423i 0
373.1 0 1.50000 + 2.59808i 0 −2.00000 + 3.46410i 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
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 588.4.i.g 2
3.b odd 2 1 1764.4.k.j 2
7.b odd 2 1 588.4.i.b 2
7.c even 3 1 588.4.a.b 1
7.c even 3 1 inner 588.4.i.g 2
7.d odd 6 1 588.4.a.e yes 1
7.d odd 6 1 588.4.i.b 2
21.c even 2 1 1764.4.k.g 2
21.g even 6 1 1764.4.a.i 1
21.g even 6 1 1764.4.k.g 2
21.h odd 6 1 1764.4.a.d 1
21.h odd 6 1 1764.4.k.j 2
28.f even 6 1 2352.4.a.g 1
28.g odd 6 1 2352.4.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.b 1 7.c even 3 1
588.4.a.e yes 1 7.d odd 6 1
588.4.i.b 2 7.b odd 2 1
588.4.i.b 2 7.d odd 6 1
588.4.i.g 2 1.a even 1 1 trivial
588.4.i.g 2 7.c even 3 1 inner
1764.4.a.d 1 21.h odd 6 1
1764.4.a.i 1 21.g even 6 1
1764.4.k.g 2 21.c even 2 1
1764.4.k.g 2 21.g even 6 1
1764.4.k.j 2 3.b odd 2 1
1764.4.k.j 2 21.h odd 6 1
2352.4.a.g 1 28.f even 6 1
2352.4.a.be 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4 T_{5} + 16$$ acting on $$S_{4}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$400 - 20 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$576 + 24 T + T^{2}$$
$19$ $$1936 + 44 T + T^{2}$$
$23$ $$5184 + 72 T + T^{2}$$
$29$ $$( 38 + T )^{2}$$
$31$ $$33856 + 184 T + T^{2}$$
$37$ $$900 - 30 T + T^{2}$$
$41$ $$( 216 + T )^{2}$$
$43$ $$( 164 + T )^{2}$$
$47$ $$270400 + 520 T + T^{2}$$
$53$ $$21316 - 146 T + T^{2}$$
$59$ $$211600 + 460 T + T^{2}$$
$61$ $$394384 + 628 T + T^{2}$$
$67$ $$309136 + 556 T + T^{2}$$
$71$ $$( -592 + T )^{2}$$
$73$ $$1048576 + 1024 T + T^{2}$$
$79$ $$10816 - 104 T + T^{2}$$
$83$ $$( 324 + T )^{2}$$
$89$ $$802816 + 896 T + T^{2}$$
$97$ $$( 920 + T )^{2}$$