# Properties

 Label 666.2.x.c Level $666$ Weight $2$ Character orbit 666.x Analytic conductor $5.318$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$666 = 2 \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 666.x (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.31803677462$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{18} - \zeta_{18}^{4} ) q^{2} -\zeta_{18}^{5} q^{4} + ( -1 - \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{5} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{7} -\zeta_{18}^{3} q^{8} +O(q^{10})$$ $$q + ( \zeta_{18} - \zeta_{18}^{4} ) q^{2} -\zeta_{18}^{5} q^{4} + ( -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} -\zeta_{18}^{3} q^{8} + ( -2 - \zeta_{18}^{2} + 2 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{10} + ( -\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} + ( -1 + \zeta_{18} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{14} -\zeta_{18} q^{16} + 3 \zeta_{18}^{4} q^{17} + ( 3 \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} ) q^{19} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{20} + ( 1 + 3 \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{22} + ( 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} + ( \zeta_{18} + \zeta_{18}^{2} + 3 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{26} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{28} + ( 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} + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{32} + 3 \zeta_{18}^{2} q^{34} + ( -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} + ( -1 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{5} ) q^{38} + ( 1 + 2 \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{40} + ( -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} + ( 3 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{44} + ( 3 - 3 \zeta_{18}^{2} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{46} + ( 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} + ( 1 + 3 \zeta_{18} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{50} + ( 1 + 3 \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{52} + ( 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} + ( -\zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{56} + ( -5 - \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{5} ) q^{58} + ( -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} + ( 1 - 3 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 3 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{62} + ( -1 + \zeta_{18}^{3} ) q^{64} + ( 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} + 3 q^{68} + ( -3 + \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{70} + ( 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} + ( -4 + 4 \zeta_{18} + 2 \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{74} + ( 3 - \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{76} + ( 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} + ( 2 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{4} ) q^{80} + ( 2 \zeta_{18} + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{82} + ( 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} + ( 1 + 2 \zeta_{18} + 6 \zeta_{18}^{2} - 6 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{86} + ( 3 + \zeta_{18} - \zeta_{18}^{2} - 3 \zeta_{18}^{3} ) q^{88} + ( 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} + ( -3 + 2 \zeta_{18} - 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{92} + ( -4 + 4 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{94} + ( -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} + ( -2 + \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{5} + 6 q^{7} - 3 q^{8} + O(q^{10})$$ $$6 q - 3 q^{5} + 6 q^{7} - 3 q^{8} - 6 q^{10} + 9 q^{11} - 3 q^{14} + 9 q^{19} - 3 q^{20} + 3 q^{22} + 15 q^{23} + 21 q^{25} + 9 q^{26} - 3 q^{28} - 18 q^{31} - 3 q^{35} + 9 q^{37} - 6 q^{38} + 6 q^{40} - 6 q^{41} + 12 q^{43} + 3 q^{44} + 15 q^{46} + 3 q^{47} - 6 q^{50} + 9 q^{52} + 18 q^{53} - 18 q^{55} - 3 q^{56} - 18 q^{58} + 6 q^{59} - 12 q^{61} + 3 q^{62} - 3 q^{64} - 3 q^{67} + 18 q^{68} - 12 q^{70} + 6 q^{71} - 36 q^{73} - 18 q^{74} + 9 q^{76} + 15 q^{77} + 30 q^{79} + 12 q^{80} - 6 q^{82} - 6 q^{83} - 18 q^{85} - 12 q^{86} + 9 q^{88} + 33 q^{89} + 9 q^{91} - 12 q^{92} - 12 q^{94} - 51 q^{95} + 42 q^{97} - 9 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/666\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$\chi(n)$$ $$1$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.173648 − 0.984808i 0.939693 − 0.342020i −0.766044 − 0.642788i 0.939693 + 0.342020i −0.766044 + 0.642788i −0.173648 + 0.984808i
−0.939693 0.342020i 0 0.766044 + 0.642788i 0.0209445 0.118782i 0 0.233956 1.32683i −0.500000 0.866025i 0 −0.0603074 + 0.104455i
145.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −3.31908 1.20805i 0 0.826352 + 0.300767i −0.500000 + 0.866025i 0 −1.76604 3.05888i
181.1 0.173648 0.984808i 0 −0.939693 0.342020i 1.79813 1.50881i 0 1.93969 1.62760i −0.500000 + 0.866025i 0 −1.17365 2.03282i
271.1 0.766044 0.642788i 0 0.173648 0.984808i −3.31908 + 1.20805i 0 0.826352 0.300767i −0.500000 0.866025i 0 −1.76604 + 3.05888i
379.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 1.79813 + 1.50881i 0 1.93969 + 1.62760i −0.500000 0.866025i 0 −1.17365 + 2.03282i
451.1 −0.939693 + 0.342020i 0 0.766044 0.642788i 0.0209445 + 0.118782i 0 0.233956 + 1.32683i −0.500000 + 0.866025i 0 −0.0603074 0.104455i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.1 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 666.2.x.c 6
3.b odd 2 1 74.2.f.a 6
12.b even 2 1 592.2.bc.b 6
37.f even 9 1 inner 666.2.x.c 6
111.n odd 18 1 2738.2.a.p 3
111.p odd 18 1 74.2.f.a 6
111.p odd 18 1 2738.2.a.m 3
444.z even 18 1 592.2.bc.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.a 6 3.b odd 2 1
74.2.f.a 6 111.p odd 18 1
592.2.bc.b 6 12.b even 2 1
592.2.bc.b 6 444.z even 18 1
666.2.x.c 6 1.a even 1 1 trivial
666.2.x.c 6 37.f even 9 1 inner
2738.2.a.m 3 111.p odd 18 1
2738.2.a.p 3 111.n odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 3 T_{5}^{5} - 6 T_{5}^{4} - 8 T_{5}^{3} + 69 T_{5}^{2} - 3 T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(666, [\chi])$$.

## Hecke characteristic polynomials

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