# Properties

 Label 666.2.s.a Level $666$ Weight $2$ Character orbit 666.s Analytic conductor $5.318$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(307,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("666.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## 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: no Analytic conductor: $$5.31803677462$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} - 2) q^{5} + 2 \zeta_{12}^{2} q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 + (z^2 - 2) * q^5 + 2*z^2 * q^7 + z^3 * q^8 $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} - 2) q^{5} + 2 \zeta_{12}^{2} q^{7} + \zeta_{12}^{3} q^{8} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{10} + (\zeta_{12}^{3} - 2 \zeta_{12} + 3) q^{11} + (2 \zeta_{12}^{2} - 4) q^{13} + 2 \zeta_{12}^{3} q^{14} + (\zeta_{12}^{2} - 1) q^{16} + ( - \zeta_{12}^{2} + 6 \zeta_{12} - 1) q^{17} + (3 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{19} + ( - \zeta_{12}^{2} - 1) q^{20} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{22} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{23} + (2 \zeta_{12}^{2} - 2) q^{25} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{26} + (2 \zeta_{12}^{2} - 2) q^{28} + (10 \zeta_{12}^{2} - 5) q^{29} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} - \zeta_{12}) q^{34} + ( - 2 \zeta_{12}^{2} - 2) q^{35} + ( - 6 \zeta_{12}^{3} - \zeta_{12}^{2} + 6 \zeta_{12}) q^{37} + (\zeta_{12}^{3} - 2 \zeta_{12} - 3) q^{38} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{40} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 4 \zeta_{12}) q^{41} + (6 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{43} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12}) q^{44} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12} + 3) q^{46} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 3) q^{47} + ( - 3 \zeta_{12}^{2} + 3) q^{49} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{50} + ( - 2 \zeta_{12}^{2} - 2) q^{52} + ( - 4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 6) q^{53} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{55} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{56} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{58} + (2 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{59} + ( - \zeta_{12}^{2} + 2) q^{61} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} + 3) q^{62} - q^{64} + ( - 6 \zeta_{12}^{2} + 6) q^{65} + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 3 \zeta_{12}) q^{67} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{68} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{70} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{71} + 4 q^{73} + ( - \zeta_{12}^{3} + 6) q^{74} + ( - \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{76} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{77} + ( - 15 \zeta_{12}^{3} - \zeta_{12}^{2} + 15 \zeta_{12} + 2) q^{79} + ( - 2 \zeta_{12}^{2} + 1) q^{80} + (3 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{82} + ( - 10 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 5 \zeta_{12} + 3) q^{83} + (6 \zeta_{12}^{3} - 12 \zeta_{12} + 3) q^{85} + (8 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 4 \zeta_{12} - 6) q^{86} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{88} + ( - 3 \zeta_{12}^{2} - 12 \zeta_{12} - 3) q^{89} + ( - 4 \zeta_{12}^{2} - 4) q^{91} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{92} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{94} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{95} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{97} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{98}+O(q^{100})$$ q + z * q^2 + z^2 * q^4 + (z^2 - 2) * q^5 + 2*z^2 * q^7 + z^3 * q^8 + (z^3 - 2*z) * q^10 + (z^3 - 2*z + 3) * q^11 + (2*z^2 - 4) * q^13 + 2*z^3 * q^14 + (z^2 - 1) * q^16 + (-z^2 + 6*z - 1) * q^17 + (3*z^3 + z^2 - 3*z - 2) * q^19 + (-z^2 - 1) * q^20 + (-z^2 + 3*z - 1) * q^22 + (-3*z^3 + 2*z^2 - 1) * q^23 + (2*z^2 - 2) * q^25 + (2*z^3 - 4*z) * q^26 + (2*z^2 - 2) * q^28 + (10*z^2 - 5) * q^29 + (-3*z^3 - 2*z^2 + 1) * q^31 + (z^3 - z) * q^32 + (-z^3 + 6*z^2 - z) * q^34 + (-2*z^2 - 2) * q^35 + (-6*z^3 - z^2 + 6*z) * q^37 + (z^3 - 2*z - 3) * q^38 + (-z^3 - z) * q^40 + (-4*z^3 + 3*z^2 - 4*z) * q^41 + (6*z^3 + 8*z^2 - 4) * q^43 + (-z^3 + 3*z^2 - z) * q^44 + (2*z^3 - 3*z^2 - z + 3) * q^46 + (-z^3 + 2*z - 3) * q^47 + (-3*z^2 + 3) * q^49 + (2*z^3 - 2*z) * q^50 + (-2*z^2 - 2) * q^52 + (-4*z^3 - 6*z^2 + 2*z + 6) * q^53 + (-3*z^3 + 3*z^2 + 3*z - 6) * q^55 + (2*z^3 - 2*z) * q^56 + (10*z^3 - 5*z) * q^58 + (2*z^2 + 6*z + 2) * q^59 + (-z^2 + 2) * q^61 + (-2*z^3 - 3*z^2 + z + 3) * q^62 - q^64 + (-6*z^2 + 6) * q^65 + (-3*z^3 + 5*z^2 - 3*z) * q^67 + (6*z^3 - 2*z^2 + 1) * q^68 + (-2*z^3 - 2*z) * q^70 + (-2*z^3 - 2*z) * q^71 + 4 * q^73 + (-z^3 + 6) * q^74 + (-z^2 - 3*z - 1) * q^76 + (-2*z^3 + 6*z^2 - 2*z) * q^77 + (-15*z^3 - z^2 + 15*z + 2) * q^79 + (-2*z^2 + 1) * q^80 + (3*z^3 - 8*z^2 + 4) * q^82 + (-10*z^3 - 3*z^2 + 5*z + 3) * q^83 + (6*z^3 - 12*z + 3) * q^85 + (8*z^3 + 6*z^2 - 4*z - 6) * q^86 + (3*z^3 - 2*z^2 + 1) * q^88 + (-3*z^2 - 12*z - 3) * q^89 + (-4*z^2 - 4) * q^91 + (-3*z^3 + z^2 + 3*z - 2) * q^92 + (z^2 - 3*z + 1) * q^94 + (-6*z^3 - 3*z^2 + 3*z + 3) * q^95 + (6*z^3 - 2*z^2 + 1) * q^97 + (-3*z^3 + 3*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 6 q^{5} + 4 q^{7}+O(q^{10})$$ 4 * q + 2 * q^4 - 6 * q^5 + 4 * q^7 $$4 q + 2 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{11} - 12 q^{13} - 2 q^{16} - 6 q^{17} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 4 q^{25} - 4 q^{28} + 12 q^{34} - 12 q^{35} - 2 q^{37} - 12 q^{38} + 6 q^{41} + 6 q^{44} + 6 q^{46} - 12 q^{47} + 6 q^{49} - 12 q^{52} + 12 q^{53} - 18 q^{55} + 12 q^{59} + 6 q^{61} + 6 q^{62} - 4 q^{64} + 12 q^{65} + 10 q^{67} + 16 q^{73} + 24 q^{74} - 6 q^{76} + 12 q^{77} + 6 q^{79} + 6 q^{83} + 12 q^{85} - 12 q^{86} - 18 q^{89} - 24 q^{91} - 6 q^{92} + 6 q^{94} + 6 q^{95}+O(q^{100})$$ 4 * q + 2 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^11 - 12 * q^13 - 2 * q^16 - 6 * q^17 - 6 * q^19 - 6 * q^20 - 6 * q^22 - 4 * q^25 - 4 * q^28 + 12 * q^34 - 12 * q^35 - 2 * q^37 - 12 * q^38 + 6 * q^41 + 6 * q^44 + 6 * q^46 - 12 * q^47 + 6 * q^49 - 12 * q^52 + 12 * q^53 - 18 * q^55 + 12 * q^59 + 6 * q^61 + 6 * q^62 - 4 * q^64 + 12 * q^65 + 10 * q^67 + 16 * q^73 + 24 * q^74 - 6 * q^76 + 12 * q^77 + 6 * q^79 + 6 * q^83 + 12 * q^85 - 12 * q^86 - 18 * q^89 - 24 * q^91 - 6 * q^92 + 6 * q^94 + 6 * q^95

## Character values

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

 $$n$$ $$371$$ $$631$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i −1.50000 0.866025i 0 1.00000 1.73205i 1.00000i 0 1.73205
307.2 0.866025 0.500000i 0 0.500000 0.866025i −1.50000 0.866025i 0 1.00000 1.73205i 1.00000i 0 −1.73205
397.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.50000 + 0.866025i 0 1.00000 + 1.73205i 1.00000i 0 1.73205
397.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.50000 + 0.866025i 0 1.00000 + 1.73205i 1.00000i 0 −1.73205
 $$n$$: 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.s.a 4
3.b odd 2 1 74.2.e.b 4
12.b even 2 1 592.2.w.e 4
37.e even 6 1 inner 666.2.s.a 4
111.h odd 6 1 74.2.e.b 4
111.m even 12 1 2738.2.a.e 2
111.m even 12 1 2738.2.a.i 2
444.p even 6 1 592.2.w.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.e.b 4 3.b odd 2 1
74.2.e.b 4 111.h odd 6 1
592.2.w.e 4 12.b even 2 1
592.2.w.e 4 444.p even 6 1
666.2.s.a 4 1.a even 1 1 trivial
666.2.s.a 4 37.e even 6 1 inner
2738.2.a.e 2 111.m even 12 1
2738.2.a.i 2 111.m even 12 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 3 T + 3)^{2}$$
$7$ $$(T^{2} - 2 T + 4)^{2}$$
$11$ $$(T^{2} - 6 T + 6)^{2}$$
$13$ $$(T^{2} + 6 T + 12)^{2}$$
$17$ $$T^{4} + 6 T^{3} - 21 T^{2} + \cdots + 1089$$
$19$ $$T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36$$
$23$ $$T^{4} + 24T^{2} + 36$$
$29$ $$(T^{2} + 75)^{2}$$
$31$ $$T^{4} + 24T^{2} + 36$$
$37$ $$T^{4} + 2 T^{3} - 33 T^{2} + \cdots + 1369$$
$41$ $$T^{4} - 6 T^{3} + 75 T^{2} + \cdots + 1521$$
$43$ $$T^{4} + 168T^{2} + 144$$
$47$ $$(T^{2} + 6 T + 6)^{2}$$
$53$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$59$ $$T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576$$
$61$ $$(T^{2} - 3 T + 3)^{2}$$
$67$ $$T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4$$
$71$ $$T^{4} + 12T^{2} + 144$$
$73$ $$(T - 4)^{4}$$
$79$ $$T^{4} - 6 T^{3} - 210 T^{2} + \cdots + 49284$$
$83$ $$T^{4} - 6 T^{3} + 102 T^{2} + \cdots + 4356$$
$89$ $$T^{4} + 18 T^{3} - 9 T^{2} + \cdots + 13689$$
$97$ $$T^{4} + 78T^{2} + 1089$$