# Properties

 Label 592.2.bc.b Level $592$ Weight $2$ Character orbit 592.bc Analytic conductor $4.727$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$592 = 2^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 592.bc (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.72714379966$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{18} + \zeta_{18}^{3} ) q^{3} + ( 1 + \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{5} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{18} + \zeta_{18}^{3} ) q^{3} + ( 1 + \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{5} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{9} + ( -\zeta_{18} + \zeta_{18}^{2} + 3 \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( -1 - \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{13} + ( 3 + \zeta_{18} + 3 \zeta_{18}^{2} ) q^{15} -3 \zeta_{18}^{4} q^{17} + ( -3 \zeta_{18} + \zeta_{18}^{2} - 3 \zeta_{18}^{3} ) q^{19} + ( 1 - 2 \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{21} + ( 5 - 2 \zeta_{18} + 3 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{23} + ( 4 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{25} + ( -1 + 3 \zeta_{18} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{27} + ( -4 \zeta_{18} + 5 \zeta_{18}^{2} + 5 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{29} + ( 3 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{31} + ( -2 - \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{33} + ( -2 + \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{35} + ( -1 - 2 \zeta_{18} - 4 \zeta_{18}^{2} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{37} + ( -5 - 3 \zeta_{18} - 4 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{39} + ( 4 + 2 \zeta_{18} - 6 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{41} + ( -2 - \zeta_{18} - \zeta_{18}^{2} - 5 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{43} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{45} + ( 1 - 4 \zeta_{18}^{2} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{47} + ( -1 - 4 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{49} + ( 3 \zeta_{18} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{51} + ( -5 + 3 \zeta_{18} - \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{5} ) q^{53} + ( 2 + 2 \zeta_{18} + 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{55} + ( 3 - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{57} + ( -3 + \zeta_{18} + 5 \zeta_{18}^{2} + 8 \zeta_{18}^{3} - 8 \zeta_{18}^{5} ) q^{59} + ( 4 \zeta_{18} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{61} + ( \zeta_{18} - 2 \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{63} + ( -4 + 4 \zeta_{18}^{2} + 8 \zeta_{18}^{3} + 9 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{65} + ( 1 + \zeta_{18} + 7 \zeta_{18}^{2} - \zeta_{18}^{3} - 7 \zeta_{18}^{5} ) q^{67} + ( 6 + 6 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 8 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{69} + ( 4 - 6 \zeta_{18} - 8 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{71} + ( -6 - \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{73} + ( 5 + 5 \zeta_{18} + 5 \zeta_{18}^{2} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{75} + ( -2 - \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{77} + ( -6 - 6 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{79} + ( 5 + 5 \zeta_{18} + 3 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{81} + ( 1 - \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{83} + ( -3 \zeta_{18} - 6 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{85} + ( 4 - 5 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{87} + ( -8 - \zeta_{18} - 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{5} ) q^{89} + ( -3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{91} + ( -1 + 3 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{93} + ( -9 - \zeta_{18} - 8 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{95} + ( 14 + 2 \zeta_{18}^{2} - 14 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{97} + ( -4 + 4 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{3} + 3q^{5} - 6q^{7} - 3q^{9} + O(q^{10})$$ $$6q + 3q^{3} + 3q^{5} - 6q^{7} - 3q^{9} + 9q^{11} + 18q^{15} - 9q^{19} + 3q^{21} + 15q^{23} + 21q^{25} - 3q^{27} + 18q^{31} - 3q^{33} - 3q^{35} + 9q^{37} - 18q^{39} + 6q^{41} - 12q^{43} + 3q^{47} - 18q^{53} + 18q^{55} + 12q^{57} + 6q^{59} - 12q^{61} - 6q^{63} + 3q^{67} + 42q^{69} + 6q^{71} - 36q^{73} + 30q^{75} - 15q^{77} - 30q^{79} + 12q^{81} - 6q^{83} - 18q^{85} + 27q^{87} - 33q^{89} - 9q^{91} + 3q^{93} - 51q^{95} + 42q^{97} - 27q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$$.

 $$n$$ $$113$$ $$149$$ $$223$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 −0.766044 − 0.642788i 0.939693 + 0.342020i −0.173648 + 0.984808i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.173648 − 0.984808i
0 −0.266044 1.50881i 0 −1.79813 + 1.50881i 0 −1.93969 + 1.62760i 0 0.613341 0.223238i 0
49.1 0 1.43969 + 1.20805i 0 3.31908 1.20805i 0 −0.826352 + 0.300767i 0 0.0923963 + 0.524005i 0
81.1 0 0.326352 + 0.118782i 0 −0.0209445 0.118782i 0 −0.233956 1.32683i 0 −2.20574 1.85083i 0
145.1 0 1.43969 1.20805i 0 3.31908 + 1.20805i 0 −0.826352 0.300767i 0 0.0923963 0.524005i 0
305.1 0 −0.266044 + 1.50881i 0 −1.79813 1.50881i 0 −1.93969 1.62760i 0 0.613341 + 0.223238i 0
497.1 0 0.326352 0.118782i 0 −0.0209445 + 0.118782i 0 −0.233956 + 1.32683i 0 −2.20574 + 1.85083i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 497.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.bc.b 6
4.b odd 2 1 74.2.f.a 6
12.b even 2 1 666.2.x.c 6
37.f even 9 1 inner 592.2.bc.b 6
148.o odd 18 1 2738.2.a.p 3
148.p odd 18 1 74.2.f.a 6
148.p odd 18 1 2738.2.a.m 3
444.z even 18 1 666.2.x.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.a 6 4.b odd 2 1
74.2.f.a 6 148.p odd 18 1
592.2.bc.b 6 1.a even 1 1 trivial
592.2.bc.b 6 37.f even 9 1 inner
666.2.x.c 6 12.b even 2 1
666.2.x.c 6 444.z even 18 1
2738.2.a.m 3 148.p odd 18 1
2738.2.a.p 3 148.o odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 3 T_{3}^{5} + 6 T_{3}^{4} - 8 T_{3}^{3} + 12 T_{3}^{2} - 6 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(592, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$5$ $$1 + 3 T + 69 T^{2} + 8 T^{3} - 6 T^{4} - 3 T^{5} + T^{6}$$
$7$ $$9 + 27 T + 36 T^{2} + 30 T^{3} + 18 T^{4} + 6 T^{5} + T^{6}$$
$11$ $$289 - 408 T + 423 T^{2} - 182 T^{3} + 57 T^{4} - 9 T^{5} + T^{6}$$
$13$ $$81 + 81 T - 90 T^{3} + 36 T^{4} + T^{6}$$
$17$ $$729 - 27 T^{3} + T^{6}$$
$19$ $$2809 + 3339 T + 1530 T^{2} + 352 T^{3} + 63 T^{4} + 9 T^{5} + T^{6}$$
$23$ $$9 - 162 T + 2871 T^{2} - 804 T^{3} + 171 T^{4} - 15 T^{5} + T^{6}$$
$29$ $$29241 - 10773 T + 3969 T^{2} - 342 T^{3} + 63 T^{4} + T^{6}$$
$31$ $$( -17 + 24 T - 9 T^{2} + T^{3} )^{2}$$
$37$ $$50653 - 12321 T + 1998 T^{2} - 305 T^{3} + 54 T^{4} - 9 T^{5} + T^{6}$$
$41$ $$207936 - 16416 T - 288 T^{2} - 192 T^{3} + 36 T^{4} - 6 T^{5} + T^{6}$$
$43$ $$( -467 - 81 T + 6 T^{2} + T^{3} )^{2}$$
$47$ $$289 + 765 T + 1974 T^{2} + 169 T^{3} + 54 T^{4} - 3 T^{5} + T^{6}$$
$53$ $$81 + 567 T + 1134 T^{2} + 72 T^{3} + 144 T^{4} + 18 T^{5} + T^{6}$$
$59$ $$687241 - 62175 T - 4134 T^{2} - 260 T^{3} + 120 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$4096 - 3072 T + 1536 T^{2} - 512 T^{3} + 48 T^{4} + 12 T^{5} + T^{6}$$
$67$ $$103041 + 5778 T + 2439 T^{2} - 267 T^{3} - 36 T^{4} - 3 T^{5} + T^{6}$$
$71$ $$207936 - 32832 T + 30816 T^{2} - 624 T^{3} - 144 T^{4} - 6 T^{5} + T^{6}$$
$73$ $$( 53 + 87 T + 18 T^{2} + T^{3} )^{2}$$
$79$ $$45369 + 26838 T + 9756 T^{2} + 2265 T^{3} + 360 T^{4} + 30 T^{5} + T^{6}$$
$83$ $$32041 + 2685 T + 1218 T^{2} + 28 T^{3} - 12 T^{4} + 6 T^{5} + T^{6}$$
$89$ $$687241 + 203934 T + 33171 T^{2} + 4661 T^{3} + 516 T^{4} + 33 T^{5} + T^{6}$$
$97$ $$6594624 - 1479168 T + 223920 T^{2} - 19056 T^{3} + 1188 T^{4} - 42 T^{5} + T^{6}$$