# Properties

 Label 666.2.f.d Level $666$ Weight $2$ Character orbit 666.f Analytic conductor $5.318$ Analytic rank $0$ Dimension $2$ 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(343,666)]

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, 4]))

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

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

chi := DirichletCharacter("666.343");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$666 = 2 \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 666.f (of order $$3$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + 4 \zeta_{6} q^{7} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + 3*z * q^5 + 4*z * q^7 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + 4 \zeta_{6} q^{7} + q^{8} - 3 q^{10} + 6 q^{11} - 2 \zeta_{6} q^{13} - 4 q^{14} + (\zeta_{6} - 1) q^{16} + ( - 3 \zeta_{6} + 3) q^{17} - 2 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 3) q^{20} + (6 \zeta_{6} - 6) q^{22} - 6 q^{23} + (4 \zeta_{6} - 4) q^{25} + 2 q^{26} + ( - 4 \zeta_{6} + 4) q^{28} - 3 q^{29} + 2 q^{31} - \zeta_{6} q^{32} + 3 \zeta_{6} q^{34} + (12 \zeta_{6} - 12) q^{35} + (3 \zeta_{6} + 4) q^{37} + 2 q^{38} + 3 \zeta_{6} q^{40} + 3 \zeta_{6} q^{41} - 4 q^{43} - 6 \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{46} + 6 q^{47} + (9 \zeta_{6} - 9) q^{49} - 4 \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} + (6 \zeta_{6} - 6) q^{53} + 18 \zeta_{6} q^{55} + 4 \zeta_{6} q^{56} + ( - 3 \zeta_{6} + 3) q^{58} + \zeta_{6} q^{61} + (2 \zeta_{6} - 2) q^{62} + q^{64} + ( - 6 \zeta_{6} + 6) q^{65} - 2 \zeta_{6} q^{67} - 3 q^{68} - 12 \zeta_{6} q^{70} - 12 \zeta_{6} q^{71} - 10 q^{73} + (4 \zeta_{6} - 7) q^{74} + (2 \zeta_{6} - 2) q^{76} + 24 \zeta_{6} q^{77} - 14 \zeta_{6} q^{79} - 3 q^{80} - 3 q^{82} + ( - 6 \zeta_{6} + 6) q^{83} + 9 q^{85} + ( - 4 \zeta_{6} + 4) q^{86} + 6 q^{88} + ( - 3 \zeta_{6} + 3) q^{89} + ( - 8 \zeta_{6} + 8) q^{91} + 6 \zeta_{6} q^{92} + (6 \zeta_{6} - 6) q^{94} + ( - 6 \zeta_{6} + 6) q^{95} - 13 q^{97} - 9 \zeta_{6} q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + 3*z * q^5 + 4*z * q^7 + q^8 - 3 * q^10 + 6 * q^11 - 2*z * q^13 - 4 * q^14 + (z - 1) * q^16 + (-3*z + 3) * q^17 - 2*z * q^19 + (-3*z + 3) * q^20 + (6*z - 6) * q^22 - 6 * q^23 + (4*z - 4) * q^25 + 2 * q^26 + (-4*z + 4) * q^28 - 3 * q^29 + 2 * q^31 - z * q^32 + 3*z * q^34 + (12*z - 12) * q^35 + (3*z + 4) * q^37 + 2 * q^38 + 3*z * q^40 + 3*z * q^41 - 4 * q^43 - 6*z * q^44 + (-6*z + 6) * q^46 + 6 * q^47 + (9*z - 9) * q^49 - 4*z * q^50 + (2*z - 2) * q^52 + (6*z - 6) * q^53 + 18*z * q^55 + 4*z * q^56 + (-3*z + 3) * q^58 + z * q^61 + (2*z - 2) * q^62 + q^64 + (-6*z + 6) * q^65 - 2*z * q^67 - 3 * q^68 - 12*z * q^70 - 12*z * q^71 - 10 * q^73 + (4*z - 7) * q^74 + (2*z - 2) * q^76 + 24*z * q^77 - 14*z * q^79 - 3 * q^80 - 3 * q^82 + (-6*z + 6) * q^83 + 9 * q^85 + (-4*z + 4) * q^86 + 6 * q^88 + (-3*z + 3) * q^89 + (-8*z + 8) * q^91 + 6*z * q^92 + (6*z - 6) * q^94 + (-6*z + 6) * q^95 - 13 * q^97 - 9*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 3 q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 + 3 * q^5 + 4 * q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} + 3 q^{5} + 4 q^{7} + 2 q^{8} - 6 q^{10} + 12 q^{11} - 2 q^{13} - 8 q^{14} - q^{16} + 3 q^{17} - 2 q^{19} + 3 q^{20} - 6 q^{22} - 12 q^{23} - 4 q^{25} + 4 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} + 3 q^{34} - 12 q^{35} + 11 q^{37} + 4 q^{38} + 3 q^{40} + 3 q^{41} - 8 q^{43} - 6 q^{44} + 6 q^{46} + 12 q^{47} - 9 q^{49} - 4 q^{50} - 2 q^{52} - 6 q^{53} + 18 q^{55} + 4 q^{56} + 3 q^{58} + q^{61} - 2 q^{62} + 2 q^{64} + 6 q^{65} - 2 q^{67} - 6 q^{68} - 12 q^{70} - 12 q^{71} - 20 q^{73} - 10 q^{74} - 2 q^{76} + 24 q^{77} - 14 q^{79} - 6 q^{80} - 6 q^{82} + 6 q^{83} + 18 q^{85} + 4 q^{86} + 12 q^{88} + 3 q^{89} + 8 q^{91} + 6 q^{92} - 6 q^{94} + 6 q^{95} - 26 q^{97} - 9 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + 3 * q^5 + 4 * q^7 + 2 * q^8 - 6 * q^10 + 12 * q^11 - 2 * q^13 - 8 * q^14 - q^16 + 3 * q^17 - 2 * q^19 + 3 * q^20 - 6 * q^22 - 12 * q^23 - 4 * q^25 + 4 * q^26 + 4 * q^28 - 6 * q^29 + 4 * q^31 - q^32 + 3 * q^34 - 12 * q^35 + 11 * q^37 + 4 * q^38 + 3 * q^40 + 3 * q^41 - 8 * q^43 - 6 * q^44 + 6 * q^46 + 12 * q^47 - 9 * q^49 - 4 * q^50 - 2 * q^52 - 6 * q^53 + 18 * q^55 + 4 * q^56 + 3 * q^58 + q^61 - 2 * q^62 + 2 * q^64 + 6 * q^65 - 2 * q^67 - 6 * q^68 - 12 * q^70 - 12 * q^71 - 20 * q^73 - 10 * q^74 - 2 * q^76 + 24 * q^77 - 14 * q^79 - 6 * q^80 - 6 * q^82 + 6 * q^83 + 18 * q^85 + 4 * q^86 + 12 * q^88 + 3 * q^89 + 8 * q^91 + 6 * q^92 - 6 * q^94 + 6 * q^95 - 26 * q^97 - 9 * q^98

## Character values

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

 $$n$$ $$371$$ $$631$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
343.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.50000 + 2.59808i 0 2.00000 + 3.46410i 1.00000 0 −3.00000
433.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 2.00000 3.46410i 1.00000 0 −3.00000
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.f.d 2
3.b odd 2 1 74.2.c.b 2
12.b even 2 1 592.2.i.a 2
37.c even 3 1 inner 666.2.f.d 2
111.h odd 6 1 2738.2.a.c 1
111.i odd 6 1 74.2.c.b 2
111.i odd 6 1 2738.2.a.a 1
444.t even 6 1 592.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.b 2 3.b odd 2 1
74.2.c.b 2 111.i odd 6 1
592.2.i.a 2 12.b even 2 1
592.2.i.a 2 444.t even 6 1
666.2.f.d 2 1.a even 1 1 trivial
666.2.f.d 2 37.c even 3 1 inner
2738.2.a.a 1 111.i odd 6 1
2738.2.a.c 1 111.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T + 3)^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} - 11T + 37$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$(T + 4)^{2}$$
$47$ $$(T - 6)^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} + 12T + 144$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$T^{2} - 6T + 36$$
$89$ $$T^{2} - 3T + 9$$
$97$ $$(T + 13)^{2}$$