# Properties

 Label 588.3.g.b Level $588$ Weight $3$ Character orbit 588.g Analytic conductor $16.022$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 588.g (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 2 q^{5} + ( -4 + 2 \zeta_{6} ) q^{6} + 8 q^{8} -3 q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 2 q^{5} + ( -4 + 2 \zeta_{6} ) q^{6} + 8 q^{8} -3 q^{9} -4 \zeta_{6} q^{10} + ( -4 + 8 \zeta_{6} ) q^{11} + ( 4 + 4 \zeta_{6} ) q^{12} -2 q^{13} + ( 2 - 4 \zeta_{6} ) q^{15} -16 \zeta_{6} q^{16} -10 q^{17} + 6 \zeta_{6} q^{18} + ( 12 - 24 \zeta_{6} ) q^{19} + ( -8 + 8 \zeta_{6} ) q^{20} + ( 16 - 8 \zeta_{6} ) q^{22} + ( 16 - 32 \zeta_{6} ) q^{23} + ( 8 - 16 \zeta_{6} ) q^{24} -21 q^{25} + 4 \zeta_{6} q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} -26 q^{29} + ( -8 + 4 \zeta_{6} ) q^{30} + ( 4 - 8 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 12 q^{33} + 20 \zeta_{6} q^{34} + ( 12 - 12 \zeta_{6} ) q^{36} + 26 q^{37} + ( -48 + 24 \zeta_{6} ) q^{38} + ( -2 + 4 \zeta_{6} ) q^{39} + 16 q^{40} -58 q^{41} + ( 28 - 56 \zeta_{6} ) q^{43} + ( -16 - 16 \zeta_{6} ) q^{44} -6 q^{45} + ( -64 + 32 \zeta_{6} ) q^{46} + ( 40 - 80 \zeta_{6} ) q^{47} + ( -32 + 16 \zeta_{6} ) q^{48} + 42 \zeta_{6} q^{50} + ( -10 + 20 \zeta_{6} ) q^{51} + ( 8 - 8 \zeta_{6} ) q^{52} -74 q^{53} + ( 12 - 6 \zeta_{6} ) q^{54} + ( -8 + 16 \zeta_{6} ) q^{55} -36 q^{57} + 52 \zeta_{6} q^{58} + ( -52 + 104 \zeta_{6} ) q^{59} + ( 8 + 8 \zeta_{6} ) q^{60} -26 q^{61} + ( -16 + 8 \zeta_{6} ) q^{62} + 64 q^{64} -4 q^{65} -24 \zeta_{6} q^{66} + ( -4 + 8 \zeta_{6} ) q^{67} + ( 40 - 40 \zeta_{6} ) q^{68} -48 q^{69} -24 q^{72} + 46 q^{73} -52 \zeta_{6} q^{74} + ( -21 + 42 \zeta_{6} ) q^{75} + ( 48 + 48 \zeta_{6} ) q^{76} + ( 8 - 4 \zeta_{6} ) q^{78} + ( -68 + 136 \zeta_{6} ) q^{79} -32 \zeta_{6} q^{80} + 9 q^{81} + 116 \zeta_{6} q^{82} + ( 28 - 56 \zeta_{6} ) q^{83} -20 q^{85} + ( -112 + 56 \zeta_{6} ) q^{86} + ( -26 + 52 \zeta_{6} ) q^{87} + ( -32 + 64 \zeta_{6} ) q^{88} -82 q^{89} + 12 \zeta_{6} q^{90} + ( 64 + 64 \zeta_{6} ) q^{92} -12 q^{93} + ( -160 + 80 \zeta_{6} ) q^{94} + ( 24 - 48 \zeta_{6} ) q^{95} + ( 32 + 32 \zeta_{6} ) q^{96} -2 q^{97} + ( 12 - 24 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} + 4q^{5} - 6q^{6} + 16q^{8} - 6q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} + 4q^{5} - 6q^{6} + 16q^{8} - 6q^{9} - 4q^{10} + 12q^{12} - 4q^{13} - 16q^{16} - 20q^{17} + 6q^{18} - 8q^{20} + 24q^{22} - 42q^{25} + 4q^{26} - 52q^{29} - 12q^{30} - 32q^{32} + 24q^{33} + 20q^{34} + 12q^{36} + 52q^{37} - 72q^{38} + 32q^{40} - 116q^{41} - 48q^{44} - 12q^{45} - 96q^{46} - 48q^{48} + 42q^{50} + 8q^{52} - 148q^{53} + 18q^{54} - 72q^{57} + 52q^{58} + 24q^{60} - 52q^{61} - 24q^{62} + 128q^{64} - 8q^{65} - 24q^{66} + 40q^{68} - 96q^{69} - 48q^{72} + 92q^{73} - 52q^{74} + 144q^{76} + 12q^{78} - 32q^{80} + 18q^{81} + 116q^{82} - 40q^{85} - 168q^{86} - 164q^{89} + 12q^{90} + 192q^{92} - 24q^{93} - 240q^{94} + 96q^{96} - 4q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 1.73205i −2.00000 + 3.46410i 2.00000 −3.00000 + 1.73205i 0 8.00000 −3.00000 −2.00000 3.46410i
295.2 −1.00000 + 1.73205i 1.73205i −2.00000 3.46410i 2.00000 −3.00000 1.73205i 0 8.00000 −3.00000 −2.00000 + 3.46410i
 $$n$$: 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.g.b 2
4.b odd 2 1 inner 588.3.g.b 2
7.b odd 2 1 12.3.d.a 2
21.c even 2 1 36.3.d.c 2
28.d even 2 1 12.3.d.a 2
35.c odd 2 1 300.3.c.b 2
35.f even 4 2 300.3.f.a 4
56.e even 2 1 192.3.g.b 2
56.h odd 2 1 192.3.g.b 2
63.l odd 6 1 324.3.f.d 2
63.l odd 6 1 324.3.f.j 2
63.o even 6 1 324.3.f.a 2
63.o even 6 1 324.3.f.g 2
84.h odd 2 1 36.3.d.c 2
105.g even 2 1 900.3.c.e 2
105.k odd 4 2 900.3.f.c 4
112.j even 4 2 768.3.b.c 4
112.l odd 4 2 768.3.b.c 4
140.c even 2 1 300.3.c.b 2
140.j odd 4 2 300.3.f.a 4
168.e odd 2 1 576.3.g.e 2
168.i even 2 1 576.3.g.e 2
252.s odd 6 1 324.3.f.a 2
252.s odd 6 1 324.3.f.g 2
252.bi even 6 1 324.3.f.d 2
252.bi even 6 1 324.3.f.j 2
336.v odd 4 2 2304.3.b.l 4
336.y even 4 2 2304.3.b.l 4
420.o odd 2 1 900.3.c.e 2
420.w even 4 2 900.3.f.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 7.b odd 2 1
12.3.d.a 2 28.d even 2 1
36.3.d.c 2 21.c even 2 1
36.3.d.c 2 84.h odd 2 1
192.3.g.b 2 56.e even 2 1
192.3.g.b 2 56.h odd 2 1
300.3.c.b 2 35.c odd 2 1
300.3.c.b 2 140.c even 2 1
300.3.f.a 4 35.f even 4 2
300.3.f.a 4 140.j odd 4 2
324.3.f.a 2 63.o even 6 1
324.3.f.a 2 252.s odd 6 1
324.3.f.d 2 63.l odd 6 1
324.3.f.d 2 252.bi even 6 1
324.3.f.g 2 63.o even 6 1
324.3.f.g 2 252.s odd 6 1
324.3.f.j 2 63.l odd 6 1
324.3.f.j 2 252.bi even 6 1
576.3.g.e 2 168.e odd 2 1
576.3.g.e 2 168.i even 2 1
588.3.g.b 2 1.a even 1 1 trivial
588.3.g.b 2 4.b odd 2 1 inner
768.3.b.c 4 112.j even 4 2
768.3.b.c 4 112.l odd 4 2
900.3.c.e 2 105.g even 2 1
900.3.c.e 2 420.o odd 2 1
900.3.f.c 4 105.k odd 4 2
900.3.f.c 4 420.w even 4 2
2304.3.b.l 4 336.v odd 4 2
2304.3.b.l 4 336.y even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{3}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$48 + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$( 10 + T )^{2}$$
$19$ $$432 + T^{2}$$
$23$ $$768 + T^{2}$$
$29$ $$( 26 + T )^{2}$$
$31$ $$48 + T^{2}$$
$37$ $$( -26 + T )^{2}$$
$41$ $$( 58 + T )^{2}$$
$43$ $$2352 + T^{2}$$
$47$ $$4800 + T^{2}$$
$53$ $$( 74 + T )^{2}$$
$59$ $$8112 + T^{2}$$
$61$ $$( 26 + T )^{2}$$
$67$ $$48 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -46 + T )^{2}$$
$79$ $$13872 + T^{2}$$
$83$ $$2352 + T^{2}$$
$89$ $$( 82 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$