# Properties

 Label 666.2.a.g Level $666$ Weight $2$ Character orbit 666.a Self dual yes Analytic conductor $5.318$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$666 = 2 \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 666.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.31803677462$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 222) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 4 q^{5} + 3 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 4 * q^5 + 3 * q^7 + q^8 $$q + q^{2} + q^{4} + 4 q^{5} + 3 q^{7} + q^{8} + 4 q^{10} - 5 q^{11} + 3 q^{13} + 3 q^{14} + q^{16} - 3 q^{17} - 7 q^{19} + 4 q^{20} - 5 q^{22} - 9 q^{23} + 11 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{31} + q^{32} - 3 q^{34} + 12 q^{35} + q^{37} - 7 q^{38} + 4 q^{40} - 6 q^{41} + 4 q^{43} - 5 q^{44} - 9 q^{46} + 10 q^{47} + 2 q^{49} + 11 q^{50} + 3 q^{52} - 3 q^{53} - 20 q^{55} + 3 q^{56} + 4 q^{59} - 2 q^{61} - 2 q^{62} + q^{64} + 12 q^{65} + 6 q^{67} - 3 q^{68} + 12 q^{70} + 12 q^{71} + 13 q^{73} + q^{74} - 7 q^{76} - 15 q^{77} - 6 q^{79} + 4 q^{80} - 6 q^{82} - 5 q^{83} - 12 q^{85} + 4 q^{86} - 5 q^{88} - 11 q^{89} + 9 q^{91} - 9 q^{92} + 10 q^{94} - 28 q^{95} + 6 q^{97} + 2 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 4 * q^5 + 3 * q^7 + q^8 + 4 * q^10 - 5 * q^11 + 3 * q^13 + 3 * q^14 + q^16 - 3 * q^17 - 7 * q^19 + 4 * q^20 - 5 * q^22 - 9 * q^23 + 11 * q^25 + 3 * q^26 + 3 * q^28 - 2 * q^31 + q^32 - 3 * q^34 + 12 * q^35 + q^37 - 7 * q^38 + 4 * q^40 - 6 * q^41 + 4 * q^43 - 5 * q^44 - 9 * q^46 + 10 * q^47 + 2 * q^49 + 11 * q^50 + 3 * q^52 - 3 * q^53 - 20 * q^55 + 3 * q^56 + 4 * q^59 - 2 * q^61 - 2 * q^62 + q^64 + 12 * q^65 + 6 * q^67 - 3 * q^68 + 12 * q^70 + 12 * q^71 + 13 * q^73 + q^74 - 7 * q^76 - 15 * q^77 - 6 * q^79 + 4 * q^80 - 6 * q^82 - 5 * q^83 - 12 * q^85 + 4 * q^86 - 5 * q^88 - 11 * q^89 + 9 * q^91 - 9 * q^92 + 10 * q^94 - 28 * q^95 + 6 * q^97 + 2 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 1.00000 4.00000 0 3.00000 1.00000 0 4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$37$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.a.g 1
3.b odd 2 1 222.2.a.a 1
4.b odd 2 1 5328.2.a.v 1
12.b even 2 1 1776.2.a.f 1
15.d odd 2 1 5550.2.a.bh 1
24.f even 2 1 7104.2.a.m 1
24.h odd 2 1 7104.2.a.bb 1
111.d odd 2 1 8214.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.a 1 3.b odd 2 1
666.2.a.g 1 1.a even 1 1 trivial
1776.2.a.f 1 12.b even 2 1
5328.2.a.v 1 4.b odd 2 1
5550.2.a.bh 1 15.d odd 2 1
7104.2.a.m 1 24.f even 2 1
7104.2.a.bb 1 24.h odd 2 1
8214.2.a.i 1 111.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(666))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{7} - 3$$ T7 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 4$$
$7$ $$T - 3$$
$11$ $$T + 5$$
$13$ $$T - 3$$
$17$ $$T + 3$$
$19$ $$T + 7$$
$23$ $$T + 9$$
$29$ $$T$$
$31$ $$T + 2$$
$37$ $$T - 1$$
$41$ $$T + 6$$
$43$ $$T - 4$$
$47$ $$T - 10$$
$53$ $$T + 3$$
$59$ $$T - 4$$
$61$ $$T + 2$$
$67$ $$T - 6$$
$71$ $$T - 12$$
$73$ $$T - 13$$
$79$ $$T + 6$$
$83$ $$T + 5$$
$89$ $$T + 11$$
$97$ $$T - 6$$