# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(1,666)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$5.31803677462$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{5} + ( - 2 \beta + 2) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + b * q^5 + (-2*b + 2) * q^7 + q^8 $$q + q^{2} + q^{4} + \beta q^{5} + ( - 2 \beta + 2) q^{7} + q^{8} + \beta q^{10} + \beta q^{11} + (\beta - 1) q^{13} + ( - 2 \beta + 2) q^{14} + q^{16} + 6 q^{17} + 2 q^{19} + \beta q^{20} + \beta q^{22} + ( - 3 \beta + 3) q^{23} + (\beta - 2) q^{25} + (\beta - 1) q^{26} + ( - 2 \beta + 2) q^{28} + (3 \beta - 3) q^{29} + ( - \beta + 2) q^{31} + q^{32} + 6 q^{34} - 6 q^{35} + q^{37} + 2 q^{38} + \beta q^{40} + ( - 3 \beta - 3) q^{41} + (2 \beta - 4) q^{43} + \beta q^{44} + ( - 3 \beta + 3) q^{46} - 2 \beta q^{47} + ( - 4 \beta + 9) q^{49} + (\beta - 2) q^{50} + (\beta - 1) q^{52} + 6 q^{53} + (\beta + 3) q^{55} + ( - 2 \beta + 2) q^{56} + (3 \beta - 3) q^{58} + ( - 2 \beta - 6) q^{59} + (5 \beta - 4) q^{61} + ( - \beta + 2) q^{62} + q^{64} + 3 q^{65} + ( - 5 \beta + 8) q^{67} + 6 q^{68} - 6 q^{70} - 6 q^{71} + ( - \beta - 10) q^{73} + q^{74} + 2 q^{76} - 6 q^{77} + (7 \beta - 7) q^{79} + \beta q^{80} + ( - 3 \beta - 3) q^{82} + (4 \beta - 12) q^{83} + 6 \beta q^{85} + (2 \beta - 4) q^{86} + \beta q^{88} + 4 \beta q^{89} + (2 \beta - 8) q^{91} + ( - 3 \beta + 3) q^{92} - 2 \beta q^{94} + 2 \beta q^{95} + ( - 8 \beta + 2) q^{97} + ( - 4 \beta + 9) q^{98} +O(q^{100})$$ q + q^2 + q^4 + b * q^5 + (-2*b + 2) * q^7 + q^8 + b * q^10 + b * q^11 + (b - 1) * q^13 + (-2*b + 2) * q^14 + q^16 + 6 * q^17 + 2 * q^19 + b * q^20 + b * q^22 + (-3*b + 3) * q^23 + (b - 2) * q^25 + (b - 1) * q^26 + (-2*b + 2) * q^28 + (3*b - 3) * q^29 + (-b + 2) * q^31 + q^32 + 6 * q^34 - 6 * q^35 + q^37 + 2 * q^38 + b * q^40 + (-3*b - 3) * q^41 + (2*b - 4) * q^43 + b * q^44 + (-3*b + 3) * q^46 - 2*b * q^47 + (-4*b + 9) * q^49 + (b - 2) * q^50 + (b - 1) * q^52 + 6 * q^53 + (b + 3) * q^55 + (-2*b + 2) * q^56 + (3*b - 3) * q^58 + (-2*b - 6) * q^59 + (5*b - 4) * q^61 + (-b + 2) * q^62 + q^64 + 3 * q^65 + (-5*b + 8) * q^67 + 6 * q^68 - 6 * q^70 - 6 * q^71 + (-b - 10) * q^73 + q^74 + 2 * q^76 - 6 * q^77 + (7*b - 7) * q^79 + b * q^80 + (-3*b - 3) * q^82 + (4*b - 12) * q^83 + 6*b * q^85 + (2*b - 4) * q^86 + b * q^88 + 4*b * q^89 + (2*b - 8) * q^91 + (-3*b + 3) * q^92 - 2*b * q^94 + 2*b * q^95 + (-8*b + 2) * q^97 + (-4*b + 9) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} + q^{11} - q^{13} + 2 q^{14} + 2 q^{16} + 12 q^{17} + 4 q^{19} + q^{20} + q^{22} + 3 q^{23} - 3 q^{25} - q^{26} + 2 q^{28} - 3 q^{29} + 3 q^{31} + 2 q^{32} + 12 q^{34} - 12 q^{35} + 2 q^{37} + 4 q^{38} + q^{40} - 9 q^{41} - 6 q^{43} + q^{44} + 3 q^{46} - 2 q^{47} + 14 q^{49} - 3 q^{50} - q^{52} + 12 q^{53} + 7 q^{55} + 2 q^{56} - 3 q^{58} - 14 q^{59} - 3 q^{61} + 3 q^{62} + 2 q^{64} + 6 q^{65} + 11 q^{67} + 12 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 2 q^{74} + 4 q^{76} - 12 q^{77} - 7 q^{79} + q^{80} - 9 q^{82} - 20 q^{83} + 6 q^{85} - 6 q^{86} + q^{88} + 4 q^{89} - 14 q^{91} + 3 q^{92} - 2 q^{94} + 2 q^{95} - 4 q^{97} + 14 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 + 2 * q^8 + q^10 + q^11 - q^13 + 2 * q^14 + 2 * q^16 + 12 * q^17 + 4 * q^19 + q^20 + q^22 + 3 * q^23 - 3 * q^25 - q^26 + 2 * q^28 - 3 * q^29 + 3 * q^31 + 2 * q^32 + 12 * q^34 - 12 * q^35 + 2 * q^37 + 4 * q^38 + q^40 - 9 * q^41 - 6 * q^43 + q^44 + 3 * q^46 - 2 * q^47 + 14 * q^49 - 3 * q^50 - q^52 + 12 * q^53 + 7 * q^55 + 2 * q^56 - 3 * q^58 - 14 * q^59 - 3 * q^61 + 3 * q^62 + 2 * q^64 + 6 * q^65 + 11 * q^67 + 12 * q^68 - 12 * q^70 - 12 * q^71 - 21 * q^73 + 2 * q^74 + 4 * q^76 - 12 * q^77 - 7 * q^79 + q^80 - 9 * q^82 - 20 * q^83 + 6 * q^85 - 6 * q^86 + q^88 + 4 * q^89 - 14 * q^91 + 3 * q^92 - 2 * q^94 + 2 * q^95 - 4 * q^97 + 14 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.30278 2.30278
1.00000 0 1.00000 −1.30278 0 4.60555 1.00000 0 −1.30278
1.2 1.00000 0 1.00000 2.30278 0 −2.60555 1.00000 0 2.30278
 $$n$$: 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.j 2
3.b odd 2 1 74.2.a.a 2
4.b odd 2 1 5328.2.a.bf 2
12.b even 2 1 592.2.a.f 2
15.d odd 2 1 1850.2.a.u 2
15.e even 4 2 1850.2.b.i 4
21.c even 2 1 3626.2.a.a 2
24.f even 2 1 2368.2.a.ba 2
24.h odd 2 1 2368.2.a.s 2
33.d even 2 1 8954.2.a.p 2
111.d odd 2 1 2738.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 3.b odd 2 1
592.2.a.f 2 12.b even 2 1
666.2.a.j 2 1.a even 1 1 trivial
1850.2.a.u 2 15.d odd 2 1
1850.2.b.i 4 15.e even 4 2
2368.2.a.s 2 24.h odd 2 1
2368.2.a.ba 2 24.f even 2 1
2738.2.a.l 2 111.d odd 2 1
3626.2.a.a 2 21.c even 2 1
5328.2.a.bf 2 4.b odd 2 1
8954.2.a.p 2 33.d even 2 1

## Hecke kernels

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

 $$T_{5}^{2} - T_{5} - 3$$ T5^2 - T5 - 3 $$T_{7}^{2} - 2T_{7} - 12$$ T7^2 - 2*T7 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T - 3$$
$7$ $$T^{2} - 2T - 12$$
$11$ $$T^{2} - T - 3$$
$13$ $$T^{2} + T - 3$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} - 3T - 27$$
$29$ $$T^{2} + 3T - 27$$
$31$ $$T^{2} - 3T - 1$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 9T - 9$$
$43$ $$T^{2} + 6T - 4$$
$47$ $$T^{2} + 2T - 12$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 14T + 36$$
$61$ $$T^{2} + 3T - 79$$
$67$ $$T^{2} - 11T - 51$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 21T + 107$$
$79$ $$T^{2} + 7T - 147$$
$83$ $$T^{2} + 20T + 48$$
$89$ $$T^{2} - 4T - 48$$
$97$ $$T^{2} + 4T - 204$$