# Properties

 Label 588.2.b.b Level $588$ Weight $2$ Character orbit 588.b Analytic conductor $4.695$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 588.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.69520363885$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.562828176.1 Defining polynomial: $$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + \beta_{2} q^{4} + ( -\beta_{2} + \beta_{5} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{3} + \beta_{4} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + \beta_{2} q^{4} + ( -\beta_{2} + \beta_{5} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{3} + \beta_{4} ) q^{8} + q^{9} + ( -1 + \beta_{3} - \beta_{4} + \beta_{6} ) q^{10} + ( 1 - \beta_{1} + \beta_{3} - \beta_{7} ) q^{11} + \beta_{2} q^{12} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{13} + ( -\beta_{2} + \beta_{5} ) q^{15} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{16} + ( 1 - 2 \beta_{1} - \beta_{6} ) q^{17} -\beta_{1} q^{18} + ( 1 + \beta_{2} + \beta_{5} ) q^{19} + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{20} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{22} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} ) q^{23} + ( -\beta_{3} + \beta_{4} ) q^{24} + ( -\beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{25} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{26} + q^{27} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{29} + ( -1 + \beta_{3} - \beta_{4} + \beta_{6} ) q^{30} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{31} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{32} + ( 1 - \beta_{1} + \beta_{3} - \beta_{7} ) q^{33} + ( -4 + 2 \beta_{2} ) q^{34} + \beta_{2} q^{36} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} ) q^{37} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{38} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{39} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{40} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{41} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + ( 1 + \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{44} + ( -\beta_{2} + \beta_{5} ) q^{45} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{46} + ( -1 + 2 \beta_{1} - \beta_{6} ) q^{47} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{48} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{50} + ( 1 - 2 \beta_{1} - \beta_{6} ) q^{51} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{52} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{53} -\beta_{1} q^{54} + ( -1 + \beta_{1} + \beta_{3} + \beta_{7} ) q^{55} + ( 1 + \beta_{2} + \beta_{5} ) q^{57} + ( \beta_{1} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{58} + ( 4 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{7} ) q^{59} + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{60} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -7 + 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{62} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{64} + ( -1 + 2 \beta_{1} - \beta_{6} ) q^{65} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{66} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{67} + ( 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{68} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} ) q^{69} + ( 1 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{71} + ( -\beta_{3} + \beta_{4} ) q^{72} + ( 1 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{73} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{74} + ( -\beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{75} + ( 6 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{76} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{78} + ( -3 + 5 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{79} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{80} + q^{81} + ( 2 - 2 \beta_{1} + 4 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} ) q^{82} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -4 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{85} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{86} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{87} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{7} ) q^{88} + ( -2 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( -1 + \beta_{3} - \beta_{4} + \beta_{6} ) q^{90} + ( -4 + 2 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{92} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{93} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{95} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{96} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{97} + ( 1 - \beta_{1} + \beta_{3} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} + 8q^{3} + 2q^{4} - 2q^{6} + 4q^{8} + 8q^{9} + O(q^{10})$$ $$8q - 2q^{2} + 8q^{3} + 2q^{4} - 2q^{6} + 4q^{8} + 8q^{9} - 8q^{10} + 2q^{12} + 10q^{16} - 2q^{18} + 12q^{19} + 22q^{20} - 6q^{22} + 4q^{24} - 4q^{25} - 6q^{26} + 8q^{27} - 16q^{29} - 8q^{30} - 12q^{31} - 12q^{32} - 28q^{34} + 2q^{36} - 12q^{37} - 2q^{38} + 4q^{40} + 4q^{44} - 12q^{46} - 8q^{47} + 10q^{48} + 2q^{50} + 4q^{52} + 8q^{53} - 2q^{54} - 8q^{55} + 12q^{57} - 14q^{58} + 28q^{59} + 22q^{60} - 48q^{62} + 2q^{64} - 8q^{65} - 6q^{66} + 16q^{68} + 4q^{72} + 38q^{74} - 4q^{75} + 44q^{76} - 6q^{78} + 6q^{80} + 8q^{81} - 4q^{82} + 4q^{83} - 32q^{85} + 6q^{86} - 16q^{87} + 26q^{88} - 8q^{90} - 28q^{92} - 12q^{93} - 32q^{94} - 12q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 4 \nu^{4} + 2 \nu^{3} + 4 \nu - 16$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 4 \nu^{4} - 6 \nu^{3} + 4 \nu - 16$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + \nu^{5} - 3 \nu^{4} + 4 \nu^{3} + 2 \nu^{2} - 8 \nu + 12$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + \nu^{5} + 2 \nu^{4} - 6 \nu^{3} + 4 \nu^{2} + 4 \nu - 12$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{7} + \nu^{6} - 3 \nu^{4} + 6 \nu^{3} + 2 \nu^{2} - 8 \nu + 20$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 2 \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-4 \beta_{7} - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 7$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
391.1
 1.40376 + 0.171630i 1.40376 − 0.171630i 0.856419 + 1.12541i 0.856419 − 1.12541i 0.0777157 + 1.41208i 0.0777157 − 1.41208i −1.33790 + 0.458297i −1.33790 − 0.458297i
−1.40376 0.171630i 1.00000 1.94109 + 0.481855i 0.963711i −1.40376 0.171630i 0 −2.64212 1.00956i 1.00000 −0.165402 + 1.35282i
391.2 −1.40376 + 0.171630i 1.00000 1.94109 0.481855i 0.963711i −1.40376 + 0.171630i 0 −2.64212 + 1.00956i 1.00000 −0.165402 1.35282i
391.3 −0.856419 1.12541i 1.00000 −0.533092 + 1.92764i 3.85529i −0.856419 1.12541i 0 2.62594 1.05092i 1.00000 −4.33878 + 3.30174i
391.4 −0.856419 + 1.12541i 1.00000 −0.533092 1.92764i 3.85529i −0.856419 + 1.12541i 0 2.62594 + 1.05092i 1.00000 −4.33878 3.30174i
391.5 −0.0777157 1.41208i 1.00000 −1.98792 + 0.219481i 0.438962i −0.0777157 1.41208i 0 0.464416 + 2.79004i 1.00000 −0.619848 + 0.0341142i
391.6 −0.0777157 + 1.41208i 1.00000 −1.98792 0.219481i 0.438962i −0.0777157 + 1.41208i 0 0.464416 2.79004i 1.00000 −0.619848 0.0341142i
391.7 1.33790 0.458297i 1.00000 1.57993 1.22631i 2.45262i 1.33790 0.458297i 0 1.55176 2.36475i 1.00000 1.12403 + 3.28134i
391.8 1.33790 + 0.458297i 1.00000 1.57993 + 1.22631i 2.45262i 1.33790 + 0.458297i 0 1.55176 + 2.36475i 1.00000 1.12403 3.28134i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.b.b 8
3.b odd 2 1 1764.2.b.i 8
4.b odd 2 1 588.2.b.a 8
7.b odd 2 1 588.2.b.a 8
7.c even 3 1 84.2.o.a 8
7.c even 3 1 588.2.o.b 8
7.d odd 6 1 84.2.o.b yes 8
7.d odd 6 1 588.2.o.d 8
12.b even 2 1 1764.2.b.j 8
21.c even 2 1 1764.2.b.j 8
21.g even 6 1 252.2.bf.f 8
21.h odd 6 1 252.2.bf.g 8
28.d even 2 1 inner 588.2.b.b 8
28.f even 6 1 84.2.o.a 8
28.f even 6 1 588.2.o.b 8
28.g odd 6 1 84.2.o.b yes 8
28.g odd 6 1 588.2.o.d 8
56.j odd 6 1 1344.2.bl.i 8
56.k odd 6 1 1344.2.bl.i 8
56.m even 6 1 1344.2.bl.j 8
56.p even 6 1 1344.2.bl.j 8
84.h odd 2 1 1764.2.b.i 8
84.j odd 6 1 252.2.bf.g 8
84.n even 6 1 252.2.bf.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a 8 7.c even 3 1
84.2.o.a 8 28.f even 6 1
84.2.o.b yes 8 7.d odd 6 1
84.2.o.b yes 8 28.g odd 6 1
252.2.bf.f 8 21.g even 6 1
252.2.bf.f 8 84.n even 6 1
252.2.bf.g 8 21.h odd 6 1
252.2.bf.g 8 84.j odd 6 1
588.2.b.a 8 4.b odd 2 1
588.2.b.a 8 7.b odd 2 1
588.2.b.b 8 1.a even 1 1 trivial
588.2.b.b 8 28.d even 2 1 inner
588.2.o.b 8 7.c even 3 1
588.2.o.b 8 28.f even 6 1
588.2.o.d 8 7.d odd 6 1
588.2.o.d 8 28.g odd 6 1
1344.2.bl.i 8 56.j odd 6 1
1344.2.bl.i 8 56.k odd 6 1
1344.2.bl.j 8 56.m even 6 1
1344.2.bl.j 8 56.p even 6 1
1764.2.b.i 8 3.b odd 2 1
1764.2.b.i 8 84.h odd 2 1
1764.2.b.j 8 12.b even 2 1
1764.2.b.j 8 21.c even 2 1

## Hecke kernels

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

 $$T_{5}^{8} + 22 T_{5}^{6} + 113 T_{5}^{4} + 104 T_{5}^{2} + 16$$ $$T_{19}^{4} - 6 T_{19}^{3} - 7 T_{19}^{2} + 60 T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 16 T + 4 T^{2} - 4 T^{3} - 6 T^{4} - 2 T^{5} + T^{6} + 2 T^{7} + T^{8}$$
$3$ $$( -1 + T )^{8}$$
$5$ $$16 + 104 T^{2} + 113 T^{4} + 22 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$400 + 568 T^{2} + 257 T^{4} + 38 T^{6} + T^{8}$$
$13$ $$256 + 1936 T^{2} + 473 T^{4} + 38 T^{6} + T^{8}$$
$17$ $$1024 + 2560 T^{2} + 848 T^{4} + 56 T^{6} + T^{8}$$
$19$ $$( 4 + 60 T - 7 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$23$ $$16384 + 13312 T^{2} + 1856 T^{4} + 80 T^{6} + T^{8}$$
$29$ $$( -512 - 352 T - 45 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$31$ $$( 2043 - 270 T - 84 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$37$ $$( -596 - 360 T - 43 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$41$ $$350464 + 311552 T^{2} + 14048 T^{4} + 208 T^{6} + T^{8}$$
$43$ $$1073296 + 140152 T^{2} + 6593 T^{4} + 134 T^{6} + T^{8}$$
$47$ $$( 64 - 16 T - 28 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$53$ $$( -8 - 116 T - 61 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$59$ $$( 1192 + 460 T - 27 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$61$ $$1048576 + 557056 T^{2} + 22784 T^{4} + 272 T^{6} + T^{8}$$
$67$ $$4129024 + 680416 T^{2} + 21809 T^{4} + 254 T^{6} + T^{8}$$
$71$ $$200704 + 173312 T^{2} + 19856 T^{4} + 280 T^{6} + T^{8}$$
$73$ $$952576 + 338912 T^{2} + 14681 T^{4} + 214 T^{6} + T^{8}$$
$79$ $$241081 + 198820 T^{2} + 39950 T^{4} + 404 T^{6} + T^{8}$$
$83$ $$( 196 - 304 T - 103 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$89$ $$4096 + 15872 T^{2} + 8336 T^{4} + 184 T^{6} + T^{8}$$
$97$ $$246016 + 86176 T^{2} + 8249 T^{4} + 182 T^{6} + T^{8}$$