# Properties

 Label 588.2.o.a Level $588$ Weight $2$ Character orbit 588.o Analytic conductor $4.695$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.69520363885$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.432972864.2 Defining polynomial: $$x^{8} - x^{7} + x^{6} + 4 x^{5} - 6 x^{4} + 8 x^{3} + 4 x^{2} - 8 x + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} - \nu^{5} + \nu^{4} - 2 \nu^{2} + 4 \nu - 4$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + \nu^{5} - 2 \nu^{3} + 4 \nu^{2} - 4 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 5 \nu^{5} - 2 \nu^{3} + 8 \nu^{2} - 4 \nu + 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + \nu^{5} - 6 \nu^{4} - 2 \nu^{3} + 4 \nu^{2} - 4 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + \nu^{5} - 6 \nu^{4} - 2 \nu^{3} + 4 \nu^{2} - 20 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} + 3 \nu^{5} + 2 \nu^{4} - 6 \nu^{3} + 12 \nu^{2} - 4 \nu$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} - \nu^{4} + 6 \nu^{3} + 2 \nu^{2} - 4 \nu + 12$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} - \beta_{3} - 2 \beta_{1} - 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{6} + \beta_{5} - 3 \beta_{4} - 4 \beta_{2}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - 2 \beta_{6} - \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2 \beta_{1} - 4$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{7} + 3 \beta_{5} + 3 \beta_{3} + 10 \beta_{1} + 4$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-6 \beta_{6} - 3 \beta_{5} + \beta_{4} + 20 \beta_{2}$$$$)/2$$

## Character values

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

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −1.41156 − 0.0865986i 0.121053 + 1.40902i 0.630783 − 1.26575i 1.15972 + 0.809347i −1.41156 + 0.0865986i 0.121053 − 1.40902i 0.630783 + 1.26575i 1.15972 − 0.809347i
−1.41156 + 0.0865986i −0.500000 0.866025i 1.98500 0.244478i 1.46890 + 0.848071i 0.780776 + 1.17915i 0 −2.78078 + 0.516994i −0.500000 + 0.866025i −2.14688 1.06990i
19.2 0.121053 1.40902i −0.500000 0.866025i −1.97069 0.341134i 2.88831 + 1.66757i −1.28078 + 0.599676i 0 −0.719224 + 2.73546i −0.500000 + 0.866025i 2.69928 3.86783i
19.3 0.630783 + 1.26575i −0.500000 0.866025i −1.20422 + 1.59682i −1.46890 0.848071i 0.780776 1.17915i 0 −2.78078 0.516994i −0.500000 + 0.866025i 0.146883 2.39420i
19.4 1.15972 0.809347i −0.500000 0.866025i 0.689916 1.87724i −2.88831 1.66757i −1.28078 0.599676i 0 −0.719224 2.73546i −0.500000 + 0.866025i −4.69928 + 0.403728i
31.1 −1.41156 0.0865986i −0.500000 + 0.866025i 1.98500 + 0.244478i 1.46890 0.848071i 0.780776 1.17915i 0 −2.78078 0.516994i −0.500000 0.866025i −2.14688 + 1.06990i
31.2 0.121053 + 1.40902i −0.500000 + 0.866025i −1.97069 + 0.341134i 2.88831 1.66757i −1.28078 0.599676i 0 −0.719224 2.73546i −0.500000 0.866025i 2.69928 + 3.86783i
31.3 0.630783 1.26575i −0.500000 + 0.866025i −1.20422 1.59682i −1.46890 + 0.848071i 0.780776 + 1.17915i 0 −2.78078 + 0.516994i −0.500000 0.866025i 0.146883 + 2.39420i
31.4 1.15972 + 0.809347i −0.500000 + 0.866025i 0.689916 + 1.87724i −2.88831 + 1.66757i −1.28078 + 0.599676i 0 −0.719224 + 2.73546i −0.500000 0.866025i −4.69928 0.403728i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.4 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
28.d even 2 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.o.a 8
4.b odd 2 1 588.2.o.c 8
7.b odd 2 1 588.2.o.c 8
7.c even 3 1 84.2.b.b yes 4
7.c even 3 1 inner 588.2.o.a 8
7.d odd 6 1 84.2.b.a 4
7.d odd 6 1 588.2.o.c 8
21.g even 6 1 252.2.b.e 4
21.h odd 6 1 252.2.b.d 4
28.d even 2 1 inner 588.2.o.a 8
28.f even 6 1 84.2.b.b yes 4
28.f even 6 1 inner 588.2.o.a 8
28.g odd 6 1 84.2.b.a 4
28.g odd 6 1 588.2.o.c 8
56.j odd 6 1 1344.2.b.f 4
56.k odd 6 1 1344.2.b.f 4
56.m even 6 1 1344.2.b.e 4
56.p even 6 1 1344.2.b.e 4
84.j odd 6 1 252.2.b.d 4
84.n even 6 1 252.2.b.e 4
168.s odd 6 1 4032.2.b.j 4
168.v even 6 1 4032.2.b.n 4
168.ba even 6 1 4032.2.b.n 4
168.be odd 6 1 4032.2.b.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a 4 7.d odd 6 1
84.2.b.a 4 28.g odd 6 1
84.2.b.b yes 4 7.c even 3 1
84.2.b.b yes 4 28.f even 6 1
252.2.b.d 4 21.h odd 6 1
252.2.b.d 4 84.j odd 6 1
252.2.b.e 4 21.g even 6 1
252.2.b.e 4 84.n even 6 1
588.2.o.a 8 1.a even 1 1 trivial
588.2.o.a 8 7.c even 3 1 inner
588.2.o.a 8 28.d even 2 1 inner
588.2.o.a 8 28.f even 6 1 inner
588.2.o.c 8 4.b odd 2 1
588.2.o.c 8 7.b odd 2 1
588.2.o.c 8 7.d odd 6 1
588.2.o.c 8 28.g odd 6 1
1344.2.b.e 4 56.m even 6 1
1344.2.b.e 4 56.p even 6 1
1344.2.b.f 4 56.j odd 6 1
1344.2.b.f 4 56.k odd 6 1
4032.2.b.j 4 168.s odd 6 1
4032.2.b.j 4 168.be odd 6 1
4032.2.b.n 4 168.v even 6 1
4032.2.b.n 4 168.ba even 6 1

## Hecke kernels

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

 $$T_{5}^{8} - 14 T_{5}^{6} + 164 T_{5}^{4} - 448 T_{5}^{2} + 1024$$ $$T_{11}^{8} - 10 T_{11}^{6} + 92 T_{11}^{4} - 80 T_{11}^{2} + 64$$ $$T_{19}^{4} - 6 T_{19}^{3} + 44 T_{19}^{2} + 48 T_{19} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 8 T + 4 T^{2} + 8 T^{3} - 6 T^{4} + 4 T^{5} + T^{6} - T^{7} + T^{8}$$
$3$ $$( 1 + T + T^{2} )^{4}$$
$5$ $$1024 - 448 T^{2} + 164 T^{4} - 14 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$64 - 80 T^{2} + 92 T^{4} - 10 T^{6} + T^{8}$$
$13$ $$( 128 + 40 T^{2} + T^{4} )^{2}$$
$17$ $$262144 - 23552 T^{2} + 1604 T^{4} - 46 T^{6} + T^{8}$$
$19$ $$( 64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$23$ $$64 - 80 T^{2} + 92 T^{4} - 10 T^{6} + T^{8}$$
$29$ $$( 2 + T )^{8}$$
$31$ $$T^{8}$$
$37$ $$( 64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$41$ $$( 128 + 62 T^{2} + T^{4} )^{2}$$
$43$ $$( 5408 + 148 T^{2} + T^{4} )^{2}$$
$47$ $$( 4096 - 256 T + 80 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$53$ $$( 2704 - 416 T + 116 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$59$ $$( 16 - 4 T + T^{2} )^{4}$$
$61$ $$4194304 - 229376 T^{2} + 10496 T^{4} - 112 T^{6} + T^{8}$$
$67$ $$262144 - 63488 T^{2} + 14864 T^{4} - 124 T^{6} + T^{8}$$
$71$ $$( 2312 + 170 T^{2} + T^{4} )^{2}$$
$73$ $$262144 - 28672 T^{2} + 2624 T^{4} - 56 T^{6} + T^{8}$$
$79$ $$16384 - 3584 T^{2} + 656 T^{4} - 28 T^{6} + T^{8}$$
$83$ $$( -64 - 4 T + T^{2} )^{4}$$
$89$ $$16384 - 7936 T^{2} + 3716 T^{4} - 62 T^{6} + T^{8}$$
$97$ $$( 8192 + 184 T^{2} + T^{4} )^{2}$$