# Properties

 Label 588.3.d.c Level $588$ Weight $3$ Character orbit 588.d Analytic conductor $16.022$ Analytic rank $0$ Dimension $8$ 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.0218395444$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( \beta_{4} + \beta_{5} ) q^{5} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( \beta_{4} + \beta_{5} ) q^{5} -3 q^{9} + ( \beta_{1} - \beta_{6} ) q^{11} + ( -\beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{13} + ( -3 \beta_{1} + \beta_{3} ) q^{15} + ( 4 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{17} + ( 8 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{19} + ( -2 - 9 \beta_{1} + 2 \beta_{3} + \beta_{6} ) q^{23} + ( 9 - \beta_{1} - 2 \beta_{6} ) q^{25} -3 \beta_{2} q^{27} + ( 10 - 5 \beta_{1} + 6 \beta_{3} - \beta_{6} ) q^{29} + ( 4 \beta_{2} - 4 \beta_{4} - 14 \beta_{5} - 3 \beta_{7} ) q^{31} + ( \beta_{5} - 3 \beta_{7} ) q^{33} + ( 16 + 9 \beta_{1} + 4 \beta_{3} + 6 \beta_{6} ) q^{37} + ( 12 \beta_{1} - \beta_{3} - 2 \beta_{6} ) q^{39} + ( -14 \beta_{2} + 7 \beta_{4} - 13 \beta_{5} - 5 \beta_{7} ) q^{41} + ( -14 + 8 \beta_{1} + 8 \beta_{3} - 2 \beta_{6} ) q^{43} + ( -3 \beta_{4} - 3 \beta_{5} ) q^{45} + ( 22 \beta_{2} - 8 \beta_{4} + 2 \beta_{5} + 7 \beta_{7} ) q^{47} + ( -12 + 9 \beta_{1} + 3 \beta_{3} ) q^{51} + ( -18 - 26 \beta_{1} - 4 \beta_{3} - 6 \beta_{6} ) q^{53} + ( -6 \beta_{4} - 10 \beta_{5} + \beta_{7} ) q^{55} + ( -24 - 6 \beta_{1} + 2 \beta_{3} - \beta_{6} ) q^{57} + ( 14 \beta_{2} + 12 \beta_{4} - 22 \beta_{5} + 5 \beta_{7} ) q^{59} + ( 12 \beta_{2} - 5 \beta_{4} + 14 \beta_{5} - 8 \beta_{7} ) q^{61} + ( 30 - 19 \beta_{1} + 4 \beta_{3} + 5 \beta_{6} ) q^{65} + ( -8 + 12 \beta_{1} + 4 \beta_{3} - 6 \beta_{6} ) q^{67} + ( -2 \beta_{2} - 6 \beta_{4} - 9 \beta_{5} + 3 \beta_{7} ) q^{69} + ( 28 + 43 \beta_{1} + 8 \beta_{3} + \beta_{6} ) q^{71} + ( 28 \beta_{2} - 15 \beta_{4} - 6 \beta_{5} + 8 \beta_{7} ) q^{73} + ( 9 \beta_{2} - \beta_{5} - 6 \beta_{7} ) q^{75} + ( -54 + 28 \beta_{1} - 4 \beta_{6} ) q^{79} + 9 q^{81} + ( 16 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} - 14 \beta_{7} ) q^{83} + ( -12 - 15 \beta_{1} + 4 \beta_{3} ) q^{85} + ( 10 \beta_{2} - 18 \beta_{4} - 5 \beta_{5} - 3 \beta_{7} ) q^{87} + ( -8 \beta_{2} + 11 \beta_{4} + 23 \beta_{5} + 16 \beta_{7} ) q^{89} + ( -12 + 42 \beta_{1} - 4 \beta_{3} + 3 \beta_{6} ) q^{93} + ( -34 - 36 \beta_{1} + 10 \beta_{3} - 4 \beta_{6} ) q^{95} + ( -28 \beta_{2} + 3 \beta_{4} - 10 \beta_{5} + 4 \beta_{7} ) q^{97} + ( -3 \beta_{1} + 3 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 24 q^{9} + O(q^{10})$$ $$8 q - 24 q^{9} - 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} - 112 q^{43} - 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} - 64 q^{67} + 224 q^{71} - 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} - 272 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 20$$$$)/14$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 9$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{7} + 35 \nu^{5} - 126 \nu^{3} + 134 \nu$$$$)/14$$ $$\beta_{4}$$ $$=$$ $$($$$$10 \nu^{7} - 35 \nu^{5} + 126 \nu^{3} - 10 \nu$$$$)/14$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} - 12 \nu^{2} + 4$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} - 28 \nu^{5} + 91 \nu^{3} - 96 \nu$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{7} - 28 \nu^{5} + 91 \nu^{3} - 8 \nu$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + 3 \beta_{4} + 3 \beta_{3}$$$$)/14$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{2} - \beta_{1} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{7} + 8 \beta_{4}$$$$)/7$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{2} + 2 \beta_{1} - 3$$ $$\nu^{5}$$ $$=$$ $$($$$$-9 \beta_{7} - 9 \beta_{6} + 13 \beta_{4} - 13 \beta_{3}$$$$)/7$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{1} - 20$$ $$\nu^{7}$$ $$=$$ $$($$$$31 \beta_{7} - 31 \beta_{6} - 44 \beta_{4} - 44 \beta_{3}$$$$)/7$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −0.662827 − 0.382683i −1.60021 − 0.923880i 0.662827 + 0.382683i 1.60021 + 0.923880i 1.60021 − 0.923880i 0.662827 − 0.382683i −1.60021 + 0.923880i −0.662827 + 0.382683i
0 1.73205i 0 5.82798i 0 0 0 −3.00000 0
97.2 0 1.73205i 0 0.480662i 0 0 0 −3.00000 0
97.3 0 1.73205i 0 0.929003i 0 0 0 −3.00000 0
97.4 0 1.73205i 0 5.37964i 0 0 0 −3.00000 0
97.5 0 1.73205i 0 5.37964i 0 0 0 −3.00000 0
97.6 0 1.73205i 0 0.929003i 0 0 0 −3.00000 0
97.7 0 1.73205i 0 0.480662i 0 0 0 −3.00000 0
97.8 0 1.73205i 0 5.82798i 0 0 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.8 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.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.d.c 8
3.b odd 2 1 1764.3.d.h 8
4.b odd 2 1 2352.3.f.j 8
7.b odd 2 1 inner 588.3.d.c 8
7.c even 3 1 588.3.m.e 8
7.c even 3 1 588.3.m.f 8
7.d odd 6 1 588.3.m.e 8
7.d odd 6 1 588.3.m.f 8
21.c even 2 1 1764.3.d.h 8
21.g even 6 1 1764.3.z.l 8
21.g even 6 1 1764.3.z.m 8
21.h odd 6 1 1764.3.z.l 8
21.h odd 6 1 1764.3.z.m 8
28.d even 2 1 2352.3.f.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.d.c 8 1.a even 1 1 trivial
588.3.d.c 8 7.b odd 2 1 inner
588.3.m.e 8 7.c even 3 1
588.3.m.e 8 7.d odd 6 1
588.3.m.f 8 7.c even 3 1
588.3.m.f 8 7.d odd 6 1
1764.3.d.h 8 3.b odd 2 1
1764.3.d.h 8 21.c even 2 1
1764.3.z.l 8 21.g even 6 1
1764.3.z.l 8 21.h odd 6 1
1764.3.z.m 8 21.g even 6 1
1764.3.z.m 8 21.h odd 6 1
2352.3.f.j 8 4.b odd 2 1
2352.3.f.j 8 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 64 T_{5}^{6} + 1052 T_{5}^{4} + 1088 T_{5}^{2} + 196$$ acting on $$S_{3}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 3 + T^{2} )^{4}$$
$5$ $$196 + 1088 T^{2} + 1052 T^{4} + 64 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 3292 - 48 T - 124 T^{2} + T^{4} )^{2}$$
$13$ $$2979076 + 1723280 T^{2} + 123284 T^{4} + 712 T^{6} + T^{8}$$
$17$ $$168428484 + 9208512 T^{2} + 142236 T^{4} + 768 T^{6} + T^{8}$$
$19$ $$285745216 + 13750144 T^{2} + 215120 T^{4} + 1136 T^{6} + T^{8}$$
$23$ $$( -37988 - 9856 T - 612 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$29$ $$( 468892 + 65440 T - 1924 T^{2} - 40 T^{3} + T^{4} )^{2}$$
$31$ $$2052452416 + 8799202688 T^{2} + 13249616 T^{4} + 6448 T^{6} + T^{8}$$
$37$ $$( -546236 + 229184 T - 3492 T^{2} - 64 T^{3} + T^{4} )^{2}$$
$41$ $$26697826996036 + 51769047200 T^{2} + 34716572 T^{4} + 9808 T^{6} + T^{8}$$
$43$ $$( 192784 - 203168 T - 3784 T^{2} + 56 T^{3} + T^{4} )^{2}$$
$47$ $$8881401308224 + 35468743424 T^{2} + 35126384 T^{4} + 11488 T^{6} + T^{8}$$
$53$ $$( -8464112 - 487968 T - 5464 T^{2} + 72 T^{3} + T^{4} )^{2}$$
$59$ $$372481662446656 + 485083751680 T^{2} + 172522352 T^{4} + 22688 T^{6} + T^{8}$$
$61$ $$454276 + 310020368 T^{2} + 12768596 T^{4} + 13192 T^{6} + T^{8}$$
$67$ $$( -553856 - 264448 T - 6048 T^{2} + 32 T^{3} + T^{4} )^{2}$$
$71$ $$( -7722596 + 634384 T - 6460 T^{2} - 112 T^{3} + T^{4} )^{2}$$
$73$ $$3712242251524 + 29259770512 T^{2} + 56366228 T^{4} + 22472 T^{6} + T^{8}$$
$79$ $$( -7050224 + 62304 T + 12440 T^{2} + 216 T^{3} + T^{4} )^{2}$$
$83$ $$61585579131904 + 132200402944 T^{2} + 89759168 T^{4} + 19712 T^{6} + T^{8}$$
$89$ $$1748734585879876 + 1918328929216 T^{2} + 495721436 T^{4} + 41600 T^{6} + T^{8}$$
$97$ $$5315948141956 + 28759544464 T^{2} + 36017108 T^{4} + 13640 T^{6} + T^{8}$$