# Properties

 Label 666.2.c.b Level $666$ Weight $2$ Character orbit 666.c Analytic conductor $5.318$ Analytic rank $0$ Dimension $4$ 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.31803677462$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - q^{4} + ( - \beta_{2} + \beta_1) q^{5} - 2 q^{7} + \beta_{2} q^{8}+O(q^{10})$$ q - b2 * q^2 - q^4 + (-b2 + b1) * q^5 - 2 * q^7 + b2 * q^8 $$q - \beta_{2} q^{2} - q^{4} + ( - \beta_{2} + \beta_1) q^{5} - 2 q^{7} + \beta_{2} q^{8} + (\beta_{3} - 2) q^{10} + (\beta_{3} - 2) q^{11} + (2 \beta_{2} + \beta_1) q^{13} + 2 \beta_{2} q^{14} + q^{16} + (4 \beta_{2} + 2 \beta_1) q^{17} + (2 \beta_{2} - 2 \beta_1) q^{19} + (\beta_{2} - \beta_1) q^{20} + (\beta_{2} - \beta_1) q^{22} + ( - 2 \beta_{2} - \beta_1) q^{23} + (3 \beta_{3} - 4) q^{25} + (\beta_{3} + 1) q^{26} + 2 q^{28} + (2 \beta_{2} + \beta_1) q^{29} + (3 \beta_{2} + 3 \beta_1) q^{31} - \beta_{2} q^{32} + (2 \beta_{3} + 2) q^{34} + (2 \beta_{2} - 2 \beta_1) q^{35} + (\beta_{2} + 2 \beta_1 + 4) q^{37} + ( - 2 \beta_{3} + 4) q^{38} + ( - \beta_{3} + 2) q^{40} + ( - \beta_{3} - 7) q^{41} + 6 \beta_{2} q^{43} + ( - \beta_{3} + 2) q^{44} + ( - \beta_{3} - 1) q^{46} + (2 \beta_{3} + 2) q^{47} - 3 q^{49} + (\beta_{2} - 3 \beta_1) q^{50} + ( - 2 \beta_{2} - \beta_1) q^{52} + (2 \beta_{3} - 4) q^{53} + (6 \beta_{2} - 3 \beta_1) q^{55} - 2 \beta_{2} q^{56} + (\beta_{3} + 1) q^{58} + ( - 2 \beta_{2} + 2 \beta_1) q^{59} + ( - 11 \beta_{2} - \beta_1) q^{61} + 3 \beta_{3} q^{62} - q^{64} - 3 q^{65} + (3 \beta_{3} - 2) q^{67} + ( - 4 \beta_{2} - 2 \beta_1) q^{68} + ( - 2 \beta_{3} + 4) q^{70} + (4 \beta_{3} - 2) q^{71} + ( - 3 \beta_{3} + 4) q^{73} + (2 \beta_{3} - 4 \beta_{2} - 1) q^{74} + ( - 2 \beta_{2} + 2 \beta_1) q^{76} + ( - 2 \beta_{3} + 4) q^{77} + ( - 10 \beta_{2} + \beta_1) q^{79} + ( - \beta_{2} + \beta_1) q^{80} + (8 \beta_{2} + \beta_1) q^{82} + ( - 4 \beta_{3} - 4) q^{83} - 6 q^{85} + 6 q^{86} + ( - \beta_{2} + \beta_1) q^{88} + 6 \beta_{2} q^{89} + ( - 4 \beta_{2} - 2 \beta_1) q^{91} + (2 \beta_{2} + \beta_1) q^{92} + ( - 4 \beta_{2} - 2 \beta_1) q^{94} + ( - 6 \beta_{3} + 18) q^{95} + (10 \beta_{2} + 2 \beta_1) q^{97} + 3 \beta_{2} q^{98}+O(q^{100})$$ q - b2 * q^2 - q^4 + (-b2 + b1) * q^5 - 2 * q^7 + b2 * q^8 + (b3 - 2) * q^10 + (b3 - 2) * q^11 + (2*b2 + b1) * q^13 + 2*b2 * q^14 + q^16 + (4*b2 + 2*b1) * q^17 + (2*b2 - 2*b1) * q^19 + (b2 - b1) * q^20 + (b2 - b1) * q^22 + (-2*b2 - b1) * q^23 + (3*b3 - 4) * q^25 + (b3 + 1) * q^26 + 2 * q^28 + (2*b2 + b1) * q^29 + (3*b2 + 3*b1) * q^31 - b2 * q^32 + (2*b3 + 2) * q^34 + (2*b2 - 2*b1) * q^35 + (b2 + 2*b1 + 4) * q^37 + (-2*b3 + 4) * q^38 + (-b3 + 2) * q^40 + (-b3 - 7) * q^41 + 6*b2 * q^43 + (-b3 + 2) * q^44 + (-b3 - 1) * q^46 + (2*b3 + 2) * q^47 - 3 * q^49 + (b2 - 3*b1) * q^50 + (-2*b2 - b1) * q^52 + (2*b3 - 4) * q^53 + (6*b2 - 3*b1) * q^55 - 2*b2 * q^56 + (b3 + 1) * q^58 + (-2*b2 + 2*b1) * q^59 + (-11*b2 - b1) * q^61 + 3*b3 * q^62 - q^64 - 3 * q^65 + (3*b3 - 2) * q^67 + (-4*b2 - 2*b1) * q^68 + (-2*b3 + 4) * q^70 + (4*b3 - 2) * q^71 + (-3*b3 + 4) * q^73 + (2*b3 - 4*b2 - 1) * q^74 + (-2*b2 + 2*b1) * q^76 + (-2*b3 + 4) * q^77 + (-10*b2 + b1) * q^79 + (-b2 + b1) * q^80 + (8*b2 + b1) * q^82 + (-4*b3 - 4) * q^83 - 6 * q^85 + 6 * q^86 + (-b2 + b1) * q^88 + 6*b2 * q^89 + (-4*b2 - 2*b1) * q^91 + (2*b2 + b1) * q^92 + (-4*b2 - 2*b1) * q^94 + (-6*b3 + 18) * q^95 + (10*b2 + 2*b1) * q^97 + 3*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 8 q^{7}+O(q^{10})$$ 4 * q - 4 * q^4 - 8 * q^7 $$4 q - 4 q^{4} - 8 q^{7} - 6 q^{10} - 6 q^{11} + 4 q^{16} - 10 q^{25} + 6 q^{26} + 8 q^{28} + 12 q^{34} + 16 q^{37} + 12 q^{38} + 6 q^{40} - 30 q^{41} + 6 q^{44} - 6 q^{46} + 12 q^{47} - 12 q^{49} - 12 q^{53} + 6 q^{58} + 6 q^{62} - 4 q^{64} - 12 q^{65} - 2 q^{67} + 12 q^{70} + 10 q^{73} + 12 q^{77} - 24 q^{83} - 24 q^{85} + 24 q^{86} + 60 q^{95}+O(q^{100})$$ 4 * q - 4 * q^4 - 8 * q^7 - 6 * q^10 - 6 * q^11 + 4 * q^16 - 10 * q^25 + 6 * q^26 + 8 * q^28 + 12 * q^34 + 16 * q^37 + 12 * q^38 + 6 * q^40 - 30 * q^41 + 6 * q^44 - 6 * q^46 + 12 * q^47 - 12 * q^49 - 12 * q^53 + 6 * q^58 + 6 * q^62 - 4 * q^64 - 12 * q^65 - 2 * q^67 + 12 * q^70 + 10 * q^73 + 12 * q^77 - 24 * q^83 - 24 * q^85 + 24 * q^86 + 60 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 5$$ (v^3 + 6*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$ v^2 + 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ b3 - 6 $$\nu^{3}$$ $$=$$ $$5\beta_{2} - 6\beta_1$$ 5*b2 - 6*b1

## Character values

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

 $$n$$ $$371$$ $$631$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 − 2.79129i 1.79129i − 1.79129i 2.79129i
1.00000i 0 −1.00000 3.79129i 0 −2.00000 1.00000i 0 −3.79129
73.2 1.00000i 0 −1.00000 0.791288i 0 −2.00000 1.00000i 0 0.791288
73.3 1.00000i 0 −1.00000 0.791288i 0 −2.00000 1.00000i 0 0.791288
73.4 1.00000i 0 −1.00000 3.79129i 0 −2.00000 1.00000i 0 −3.79129
 $$n$$: e.g. 2-40 or 990-1000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.c.b 4
3.b odd 2 1 74.2.b.a 4
4.b odd 2 1 5328.2.h.m 4
12.b even 2 1 592.2.g.c 4
15.d odd 2 1 1850.2.d.e 4
15.e even 4 1 1850.2.c.g 4
15.e even 4 1 1850.2.c.h 4
24.f even 2 1 2368.2.g.h 4
24.h odd 2 1 2368.2.g.j 4
37.b even 2 1 inner 666.2.c.b 4
111.d odd 2 1 74.2.b.a 4
111.g even 4 1 2738.2.a.h 2
111.g even 4 1 2738.2.a.k 2
148.b odd 2 1 5328.2.h.m 4
444.g even 2 1 592.2.g.c 4
555.b odd 2 1 1850.2.d.e 4
555.n even 4 1 1850.2.c.g 4
555.n even 4 1 1850.2.c.h 4
888.c even 2 1 2368.2.g.h 4
888.i odd 2 1 2368.2.g.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 3.b odd 2 1
74.2.b.a 4 111.d odd 2 1
592.2.g.c 4 12.b even 2 1
592.2.g.c 4 444.g even 2 1
666.2.c.b 4 1.a even 1 1 trivial
666.2.c.b 4 37.b even 2 1 inner
1850.2.c.g 4 15.e even 4 1
1850.2.c.g 4 555.n even 4 1
1850.2.c.h 4 15.e even 4 1
1850.2.c.h 4 555.n even 4 1
1850.2.d.e 4 15.d odd 2 1
1850.2.d.e 4 555.b odd 2 1
2368.2.g.h 4 24.f even 2 1
2368.2.g.h 4 888.c even 2 1
2368.2.g.j 4 24.h odd 2 1
2368.2.g.j 4 888.i odd 2 1
2738.2.a.h 2 111.g even 4 1
2738.2.a.k 2 111.g even 4 1
5328.2.h.m 4 4.b odd 2 1
5328.2.h.m 4 148.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(666, [\chi])$$:

 $$T_{5}^{4} + 15T_{5}^{2} + 9$$ T5^4 + 15*T5^2 + 9 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 15T^{2} + 9$$
$7$ $$(T + 2)^{4}$$
$11$ $$(T^{2} + 3 T - 3)^{2}$$
$13$ $$T^{4} + 15T^{2} + 9$$
$17$ $$T^{4} + 60T^{2} + 144$$
$19$ $$T^{4} + 60T^{2} + 144$$
$23$ $$T^{4} + 15T^{2} + 9$$
$29$ $$T^{4} + 15T^{2} + 9$$
$31$ $$T^{4} + 99T^{2} + 2025$$
$37$ $$(T^{2} - 8 T + 37)^{2}$$
$41$ $$(T^{2} + 15 T + 51)^{2}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} - 6 T - 12)^{2}$$
$53$ $$(T^{2} + 6 T - 12)^{2}$$
$59$ $$T^{4} + 60T^{2} + 144$$
$61$ $$T^{4} + 231 T^{2} + 11025$$
$67$ $$(T^{2} + T - 47)^{2}$$
$71$ $$(T^{2} - 84)^{2}$$
$73$ $$(T^{2} - 5 T - 41)^{2}$$
$79$ $$T^{4} + 231 T^{2} + 11025$$
$83$ $$(T^{2} + 12 T - 48)^{2}$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$T^{4} + 204T^{2} + 3600$$