# Properties

 Label 666.2.f.j Level $666$ Weight $2$ Character orbit 666.f Analytic conductor $5.318$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.4406832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1$$ x^6 - x^5 + 6*x^4 + 7*x^3 + 24*x^2 + 5*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{2} + \beta_{3} q^{4} - \beta_{5} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} - q^{8}+O(q^{10})$$ q + (b3 + 1) * q^2 + b3 * q^4 - b5 * q^5 + (-b4 - b2) * q^7 - q^8 $$q + (\beta_{3} + 1) q^{2} + \beta_{3} q^{4} - \beta_{5} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} - q^{8} + \beta_1 q^{10} + ( - \beta_{2} - \beta_1 - 1) q^{11} + 2 \beta_{3} q^{13} - \beta_{2} q^{14} + ( - \beta_{3} - 1) q^{16} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{17} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_{2}) q^{19}+ \cdots + ( - \beta_{5} + \beta_{4} + \cdots + \beta_{2}) q^{98}+O(q^{100})$$ q + (b3 + 1) * q^2 + b3 * q^4 - b5 * q^5 + (-b4 - b2) * q^7 - q^8 + b1 * q^10 + (-b2 - b1 - 1) * q^11 + 2*b3 * q^13 - b2 * q^14 + (-b3 - 1) * q^16 + (-2*b4 - b3 - 1) * q^17 + (-b5 - b4 + 3*b3 - b2) * q^19 + (b5 + b1) * q^20 + (-b5 + b4 - b3 - b1 - 1) * q^22 + (b1 - 3) * q^23 + (b5 - 3*b4 - 7*b3 + b1 - 7) * q^25 - 2 * q^26 + b4 * q^28 + (b2 + 1) * q^29 + (-b2 + b1 + 1) * q^31 - b3 * q^32 + (-2*b4 - b3 - 2*b2) * q^34 + (-b5 - 2*b4 - b3 - b1 - 1) * q^35 + (b5 + 2*b4 - 2*b3 + b2 + b1 - 3) * q^37 + (-b2 + b1 - 3) * q^38 + b5 * q^40 + (b5 + b4 + 2*b3 + b2) * q^41 + (-2*b2 - b1 + 3) * q^43 + (-b5 + b4 - b3 + b2) * q^44 + (b5 - 3*b3 + b1 - 3) * q^46 + (-b2 + b1 + 5) * q^47 + (-b5 + b4 + 2*b3 - b1 + 2) * q^49 + (b5 - 3*b4 - 7*b3 - 3*b2) * q^50 + (-2*b3 - 2) * q^52 + (b5 - b4 + 7*b3 + b1 + 7) * q^53 + (-b5 + b4 + 11*b3 + b2) * q^55 + (b4 + b2) * q^56 + (-b4 + b3 + 1) * q^58 + (-2*b5 + 4*b4 + 2*b3 - 2*b1 + 2) * q^59 + (-b4 + 3*b3 - b2) * q^61 + (b5 + b4 + b3 + b1 + 1) * q^62 + q^64 + (2*b5 + 2*b1) * q^65 + (-b5 + b4 - 5*b3 + b2) * q^67 + (-2*b2 + 1) * q^68 + (-b5 - 2*b4 - b3 - 2*b2) * q^70 + (-b4 + 4*b3 - b2) * q^71 + (-b2 - b1 + 1) * q^73 + (b5 + b4 - 3*b3 + 2*b2 - 1) * q^74 + (b5 + b4 - 3*b3 + b1 - 3) * q^76 + (4*b4 - 4*b3 + 4*b2) * q^77 + (-b5 + 3*b4 - b3 + 3*b2) * q^79 - b1 * q^80 + (b2 - b1 - 2) * q^82 + (-2*b5 + b4 + 2*b3 - 2*b1 + 2) * q^83 + (4*b2 - 3*b1 - 2) * q^85 + (-b5 + 2*b4 + 3*b3 - b1 + 3) * q^86 + (b2 + b1 + 1) * q^88 + (-3*b5 - b4 + 4*b3 - 3*b1 + 4) * q^89 + 2*b4 * q^91 + (b5 - 3*b3) * q^92 + (b5 + b4 + 5*b3 + b1 + 5) * q^94 + (3*b5 - 5*b4 - 13*b3 + 3*b1 - 13) * q^95 + (-b2 + b1 + 4) * q^97 + (-b5 + b4 + 2*b3 + b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} - 3 q^{4} + q^{5} - 6 q^{8}+O(q^{10})$$ 6 * q + 3 * q^2 - 3 * q^4 + q^5 - 6 * q^8 $$6 q + 3 q^{2} - 3 q^{4} + q^{5} - 6 q^{8} + 2 q^{10} - 8 q^{11} - 6 q^{13} - 3 q^{16} - 3 q^{17} - 8 q^{19} + q^{20} - 4 q^{22} - 16 q^{23} - 20 q^{25} - 12 q^{26} + 6 q^{29} + 8 q^{31} + 3 q^{32} + 3 q^{34} - 4 q^{35} - 11 q^{37} - 16 q^{38} - q^{40} - 7 q^{41} + 16 q^{43} + 4 q^{44} - 8 q^{46} + 32 q^{47} + 5 q^{49} + 20 q^{50} - 6 q^{52} + 22 q^{53} - 32 q^{55} + 3 q^{58} + 4 q^{59} - 9 q^{61} + 4 q^{62} + 6 q^{64} + 2 q^{65} + 16 q^{67} + 6 q^{68} + 4 q^{70} - 12 q^{71} + 4 q^{73} + 2 q^{74} - 8 q^{76} + 12 q^{77} + 4 q^{79} - 2 q^{80} - 14 q^{82} + 4 q^{83} - 18 q^{85} + 8 q^{86} + 8 q^{88} + 9 q^{89} + 8 q^{92} + 16 q^{94} - 36 q^{95} + 26 q^{97} - 5 q^{98}+O(q^{100})$$ 6 * q + 3 * q^2 - 3 * q^4 + q^5 - 6 * q^8 + 2 * q^10 - 8 * q^11 - 6 * q^13 - 3 * q^16 - 3 * q^17 - 8 * q^19 + q^20 - 4 * q^22 - 16 * q^23 - 20 * q^25 - 12 * q^26 + 6 * q^29 + 8 * q^31 + 3 * q^32 + 3 * q^34 - 4 * q^35 - 11 * q^37 - 16 * q^38 - q^40 - 7 * q^41 + 16 * q^43 + 4 * q^44 - 8 * q^46 + 32 * q^47 + 5 * q^49 + 20 * q^50 - 6 * q^52 + 22 * q^53 - 32 * q^55 + 3 * q^58 + 4 * q^59 - 9 * q^61 + 4 * q^62 + 6 * q^64 + 2 * q^65 + 16 * q^67 + 6 * q^68 + 4 * q^70 - 12 * q^71 + 4 * q^73 + 2 * q^74 - 8 * q^76 + 12 * q^77 + 4 * q^79 - 2 * q^80 - 14 * q^82 + 4 * q^83 - 18 * q^85 + 8 * q^86 + 8 * q^88 + 9 * q^89 + 8 * q^92 + 16 * q^94 - 36 * q^95 + 26 * q^97 - 5 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -5\nu^{5} + 30\nu^{4} - 31\nu^{3} + 120\nu^{2} + 25\nu + 595 ) / 149$$ (-5*v^5 + 30*v^4 - 31*v^3 + 120*v^2 + 25*v + 595) / 149 $$\beta_{2}$$ $$=$$ $$( 7\nu^{5} - 42\nu^{4} + 103\nu^{3} - 168\nu^{2} - 35\nu - 386 ) / 149$$ (7*v^5 - 42*v^4 + 103*v^3 - 168*v^2 - 35*v - 386) / 149 $$\beta_{3}$$ $$=$$ $$( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 6 ) / 149$$ (30*v^5 - 31*v^4 + 186*v^3 + 174*v^2 + 744*v + 6) / 149 $$\beta_{4}$$ $$=$$ $$( 88\nu^{5} - 81\nu^{4} + 486\nu^{3} + 719\nu^{2} + 1944\nu + 405 ) / 149$$ (88*v^5 - 81*v^4 + 486*v^3 + 719*v^2 + 1944*v + 405) / 149 $$\beta_{5}$$ $$=$$ $$( 125\nu^{5} - 154\nu^{4} + 775\nu^{3} + 725\nu^{2} + 2951\nu + 25 ) / 149$$ (125*v^5 - 154*v^4 + 775*v^3 + 725*v^2 + 2951*v + 25) / 149
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} ) / 2$$ (b5 - b4 - b3 - b2) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 4\beta_{3} + \beta _1 - 4$$ b5 - 4*b3 + b1 - 4 $$\nu^{3}$$ $$=$$ $$( 5\beta_{2} + 7\beta _1 - 15 ) / 2$$ (5*b2 + 7*b1 - 15) / 2 $$\nu^{4}$$ $$=$$ $$-9\beta_{5} + 3\beta_{4} + 28\beta_{3} + 3\beta_{2}$$ -9*b5 + 3*b4 + 28*b3 + 3*b2 $$\nu^{5}$$ $$=$$ $$( -55\beta_{5} + 31\beta_{4} + 139\beta_{3} - 55\beta _1 + 139 ) / 2$$ (-55*b5 + 31*b4 + 139*b3 - 55*b1 + 139) / 2

## Character values

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

 $$n$$ $$371$$ $$631$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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
 1.43310 + 2.48220i −0.827721 − 1.43366i −0.105378 − 0.182520i 1.43310 − 2.48220i −0.827721 + 1.43366i −0.105378 + 0.182520i
0.500000 0.866025i 0 −0.500000 0.866025i −2.10755 3.65038i 0 0.258652 + 0.447998i −1.00000 0 −4.21509
343.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.629755 + 1.09077i 0 −1.52569 2.64257i −1.00000 0 1.25951
343.3 0.500000 0.866025i 0 −0.500000 0.866025i 1.97779 + 3.42563i 0 1.26704 + 2.19457i −1.00000 0 3.95558
433.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.10755 + 3.65038i 0 0.258652 0.447998i −1.00000 0 −4.21509
433.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.629755 1.09077i 0 −1.52569 + 2.64257i −1.00000 0 1.25951
433.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.97779 3.42563i 0 1.26704 2.19457i −1.00000 0 3.95558
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 343.3 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.j 6
3.b odd 2 1 74.2.c.c 6
12.b even 2 1 592.2.i.e 6
37.c even 3 1 inner 666.2.f.j 6
111.h odd 6 1 2738.2.a.n 3
111.i odd 6 1 74.2.c.c 6
111.i odd 6 1 2738.2.a.o 3
444.t even 6 1 592.2.i.e 6

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

## Hecke kernels

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

 $$T_{5}^{6} - T_{5}^{5} + 18T_{5}^{4} - 25T_{5}^{3} + 310T_{5}^{2} - 357T_{5} + 441$$ T5^6 - T5^5 + 18*T5^4 - 25*T5^3 + 310*T5^2 - 357*T5 + 441 $$T_{7}^{6} + 8T_{7}^{4} - 8T_{7}^{3} + 64T_{7}^{2} - 32T_{7} + 16$$ T7^6 + 8*T7^4 - 8*T7^3 + 64*T7^2 - 32*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{5} + \cdots + 441$$
$7$ $$T^{6} + 8 T^{4} + \cdots + 16$$
$11$ $$(T^{3} + 4 T^{2} - 16 T - 48)^{2}$$
$13$ $$(T^{2} + 2 T + 4)^{3}$$
$17$ $$T^{6} + 3 T^{5} + \cdots + 3969$$
$19$ $$T^{6} + 8 T^{5} + \cdots + 12544$$
$23$ $$(T^{3} + 8 T^{2} + 4 T - 12)^{2}$$
$29$ $$(T^{3} - 3 T^{2} - 5 T + 3)^{2}$$
$31$ $$(T^{3} - 4 T^{2} - 24 T - 16)^{2}$$
$37$ $$T^{6} + 11 T^{5} + \cdots + 50653$$
$41$ $$T^{6} + 7 T^{5} + \cdots + 9$$
$43$ $$(T^{3} - 8 T^{2} + \cdots + 148)^{2}$$
$47$ $$(T^{3} - 16 T^{2} + \cdots - 48)^{2}$$
$53$ $$T^{6} - 22 T^{5} + \cdots + 46656$$
$59$ $$T^{6} - 4 T^{5} + \cdots + 451584$$
$61$ $$T^{6} + 9 T^{5} + \cdots + 49$$
$67$ $$T^{6} - 16 T^{5} + \cdots + 256$$
$71$ $$T^{6} + 12 T^{5} + \cdots + 1296$$
$73$ $$(T^{3} - 2 T^{2} - 20 T - 8)^{2}$$
$79$ $$T^{6} - 4 T^{5} + \cdots + 20736$$
$83$ $$T^{6} - 4 T^{5} + \cdots + 17424$$
$89$ $$T^{6} - 9 T^{5} + \cdots + 301401$$
$97$ $$(T^{3} - 13 T^{2} + 27 T - 7)^{2}$$