# Properties

 Label 666.2.bj.c Level $666$ Weight $2$ Character orbit 666.bj Analytic conductor $5.318$ Analytic rank $0$ Dimension $12$ 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(289,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 7]))

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

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

chi := DirichletCharacter("666.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$666 = 2 \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 666.bj (of order $$18$$, degree $$6$$, 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: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-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 $$\zeta_{36}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{9} + \zeta_{36}^{8} + \zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36}^{3} + \zeta_{36}^{2} - \zeta_{36}) q^{5} + ( - 2 \zeta_{36}^{11} + \zeta_{36}^{10} + \zeta_{36}^{5} - 1) q^{7} + \zeta_{36}^{3} q^{8}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 + (-z^9 + z^8 + z^7 + z^5 + z^3 + z^2 - z) * q^5 + (-2*z^11 + z^10 + z^5 - 1) * q^7 + z^3 * q^8 $$q + \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{9} + \zeta_{36}^{8} + \zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36}^{3} + \zeta_{36}^{2} - \zeta_{36}) q^{5} + ( - 2 \zeta_{36}^{11} + \zeta_{36}^{10} + \zeta_{36}^{5} - 1) q^{7} + \zeta_{36}^{3} q^{8} + ( - \zeta_{36}^{10} + \zeta_{36}^{9} + \zeta_{36}^{8} + \zeta_{36}^{6} + \zeta_{36}^{4} + \zeta_{36}^{3} - \zeta_{36}^{2}) q^{10} + (\zeta_{36}^{10} + 4 \zeta_{36}^{9} + \zeta_{36}^{8} - \zeta_{36}^{6} - 2 \zeta_{36}^{3} - \zeta_{36}^{2} + 1) q^{11} + (2 \zeta_{36}^{11} + \zeta_{36}^{10} + 2 \zeta_{36}^{8} + \zeta_{36}^{6} - \zeta_{36}^{4} - \zeta_{36}^{3} - \zeta_{36}^{2} + \zeta_{36}) q^{13} + (\zeta_{36}^{11} - \zeta_{36}^{6} - \zeta_{36} + 2) q^{14} + \zeta_{36}^{4} q^{16} + ( - 2 \zeta_{36}^{11} - \zeta_{36}^{10} - 2 \zeta_{36}^{7} - \zeta_{36}^{4} - 2 \zeta_{36}^{3}) q^{17} + ( - 2 \zeta_{36}^{11} + 2 \zeta_{36}^{10} - \zeta_{36}^{8} - 2 \zeta_{36}^{7} - \zeta_{36}^{6} + 2 \zeta_{36}^{5} - \zeta_{36}^{4} + \cdots - 1) q^{19}+ \cdots + (2 \zeta_{36}^{11} + 2 \zeta_{36}^{10} + 2 \zeta_{36}^{6} + 2 \zeta_{36}^{4} - \zeta_{36}^{3} + \zeta_{36} - 4) q^{98}+O(q^{100})$$ q + z * q^2 + z^2 * q^4 + (-z^9 + z^8 + z^7 + z^5 + z^3 + z^2 - z) * q^5 + (-2*z^11 + z^10 + z^5 - 1) * q^7 + z^3 * q^8 + (-z^10 + z^9 + z^8 + z^6 + z^4 + z^3 - z^2) * q^10 + (z^10 + 4*z^9 + z^8 - z^6 - 2*z^3 - z^2 + 1) * q^11 + (2*z^11 + z^10 + 2*z^8 + z^6 - z^4 - z^3 - z^2 + z) * q^13 + (z^11 - z^6 - z + 2) * q^14 + z^4 * q^16 + (-2*z^11 - z^10 - 2*z^7 - z^4 - 2*z^3) * q^17 + (-2*z^11 + 2*z^10 - z^8 - 2*z^7 - z^6 + 2*z^5 - z^4 + 2*z^3 + 2*z^2 + 2*z - 1) * q^19 + (-z^11 + z^10 + z^9 + z^7 + z^5 + z^4 - z^3) * q^20 + (z^11 + 4*z^10 + z^9 - z^7 - 2*z^4 - z^3 + z) * q^22 + (-3*z^11 + 2*z^10 + z^9 + z^8 - 2*z^7 + 2*z^5 - z^4 - z^3 - 2*z^2 + 3*z) * q^23 + (-2*z^11 + z^10 + 2*z^9 - z^8 + 4*z^7 - z^6 + 4*z^5 - 4*z^3 + 2*z^2 - 2*z - 1) * q^25 + (z^11 + 2*z^9 + z^7 + 2*z^6 - z^5 - z^4 - z^3 + z^2 - 2) * q^26 + (-z^7 + z^6 - z^2 + 2*z - 1) * q^28 + (2*z^10 - 2*z^8 - z^6 + z^5 - z^4 - z^3 + z^2 + z - 1) * q^29 + (-2*z^11 + 3*z^9 - z^8 - 4*z^7 - 2*z^5 - z^4 + z^2 + 2*z) * q^31 + z^5 * q^32 + (-z^11 - 2*z^8 - 2*z^6 - z^5 - 2*z^4 + 2) * q^34 + (-z^10 + 2*z^9 - 3*z^8 + 3*z^6 - 2*z^5 + 2*z^4 - 2*z^3 + 4*z - 3) * q^35 + (-5*z^11 + 2*z^10 - z^8 - z^7 + z^6 + 3*z^5 - z^4 - 2*z^3 + 2*z^2 + z + 2) * q^37 + (2*z^11 - z^9 - 2*z^8 - z^7 - z^5 + 2*z^4 + 2*z^3 + 2*z^2 - z + 2) * q^38 + (z^11 + z^10 + z^8 + z^5 - z^4 + 1) * q^40 + (-2*z^9 + 2*z^7 - 2*z^4 + 4*z^3 + 2*z^2 + 2*z - 2) * q^41 + (-2*z^11 - z^10 - z^8 + 2*z^7 + 4*z^5 - 4*z) * q^43 + (4*z^11 + z^10 - z^8 + z^6 - 2*z^5 - z^4 + z^2 - 1) * q^44 + (2*z^11 + z^10 + z^9 - 2*z^8 - z^6 - z^5 - z^4 - 2*z^3 + 3*z^2 + 3) * q^46 + (-z^11 - 4*z^10 - 2*z^9 + 3*z^8 - z^7 - z^6 - z^5 + 3*z^4 - 2*z^3 - 4*z^2 - z) * q^47 + (4*z^11 + 2*z^10 + 2*z^9 - 2*z^5 + 2*z^3 - z^2 + 1) * q^49 + (z^11 + 2*z^10 - z^9 + 4*z^8 - z^7 + 2*z^6 - 4*z^4 + 2*z^3 - 2*z^2 - z + 2) * q^50 + (2*z^10 + z^8 + 2*z^7 - z^5 - z^4 + z^3 - 2*z - 1) * q^52 + (-2*z^11 + 2*z^9 - z^8 - 3*z^7 + 4*z^6 + 4*z^5 + 4*z^4 + 2*z^3 + 4*z^2 - 3*z - 1) * q^53 + (6*z^11 - z^10 - z^9 + z^8 + 2*z^7 + 5*z^6 - 6*z^5 - 4*z^4 - z^3 - 2*z^2 - z - 4) * q^55 + (-z^8 + z^7 - z^3 + 2*z^2 - z) * q^56 + (2*z^11 - 2*z^9 - z^7 + z^6 - z^5 - z^4 + z^3 + z^2 - z) * q^58 + (-6*z^11 + 2*z^10 - 6*z^9 + z^8 + 2*z^7 + 2*z^6 + 2*z^5 - z^4 + 4*z^3 - 2*z^2 - 2*z - 1) * q^59 + (-2*z^11 + 2*z^10 + 2*z^9 + 4*z^8 + 2*z^7 + 2*z^6 + 2*z^4 - 2*z^2 - 2*z - 4) * q^61 + (3*z^10 - z^9 - 4*z^8 - 4*z^6 - z^5 + z^3 + 2*z^2 + 2) * q^62 + z^6 * q^64 + (3*z^10 + 4*z^8 + 4*z^7 - 2*z^6 - 3*z^4 - 2*z^2 - 8*z - 2) * q^65 + (-4*z^11 - 4*z^9 + 3*z^8 - 2*z^7 - 5*z^6 + 2*z^5 + 5*z^4 + 2*z^3 - 3*z^2 + 4*z) * q^67 + (-2*z^9 - 2*z^7 - 2*z^6 - 2*z^5 + 2*z + 1) * q^68 + (-z^11 + 2*z^10 - 3*z^9 + 3*z^7 - 2*z^6 + 2*z^5 - 2*z^4 + 4*z^2 - 3*z) * q^70 + (4*z^11 - 4*z^10 - 2*z^9 - 4*z^8 - 4*z^7 - 6*z^6 + 4*z^5 - 2*z^4 - 2*z^3 + 2*z + 2) * q^71 + (-2*z^10 - 2*z^9 + 5*z^8 - 6*z^7 + 6*z^5 - 3*z^4 + 4*z^3 - 3*z^2 + 6*z) * q^73 + (2*z^11 - z^9 - z^8 + z^7 - 2*z^6 - z^5 - 2*z^4 + 2*z^3 + z^2 + 2*z + 5) * q^74 + (-z^10 - 2*z^9 - z^8 + z^6 + 2*z^5 + 2*z^4 + 2*z^3 - z^2 + 2*z - 2) * q^76 + (-z^11 - z^10 - 3*z^9 + 5*z^8 - z^5 + z^4 + 3*z^3 - 3*z - 1) * q^77 + (z^10 - z^8 - z^7 - z^6 - 2*z^5 - z^4 - z^3 - z^2 + 1) * q^79 + (z^11 + z^9 + 2*z^6 - z^5 + z - 1) * q^80 + (-2*z^10 + 2*z^8 - 2*z^5 + 4*z^4 + 2*z^3 + 2*z^2 - 2*z) * q^82 + (-6*z^11 + z^10 - 3*z^8 - 2*z^6 - z^4 + 6*z^3 + 5*z^2 + 5) * q^83 + (-3*z^11 - 4*z^10 - 6*z^9 - 4*z^8 - 3*z^7 - 3*z^6 + 2*z^4 + 3*z^3 + 2*z^2 + 3*z + 3) * q^85 + (-z^11 - z^9 + 2*z^8 + 2*z^6 - 4*z^2 + 2) * q^86 + (z^11 - z^9 + z^7 + 2*z^6 - z^5 + z^3 - z - 4) * q^88 + (2*z^10 - z^8 + z^6 + 6*z^5 - z^4 - 6*z^3 - z^2 + 1) * q^89 + (-2*z^11 + 2*z^10 + 2*z^9 - 2*z^8 + 2*z^7 - 3*z^6 + 2*z^5 + 2*z^4 - 2*z^3 - z^2 + 1) * q^91 + (z^11 + z^10 - 2*z^9 - z^7 + z^6 - z^5 - 2*z^4 + 3*z^3 + 3*z - 2) * q^92 + (-4*z^11 - 2*z^10 + 3*z^9 - z^8 - z^7 - 2*z^6 + 3*z^5 - 2*z^4 - 4*z^3 - z^2 + 1) * q^94 + (-2*z^11 + 2*z^9 - z^8 + 4*z^7 + 2*z^6 + z^5 + 3*z^4 - z^3 + 2*z^2 + 4*z - 1) * q^95 + (4*z^11 - 4*z^10 - 2*z^9 + 6*z^7 - 2*z^6 - 6*z^5 + 2*z^3 + 4*z^2 - 4*z + 4) * q^97 + (2*z^11 + 2*z^10 + 2*z^6 + 2*z^4 - z^3 + z - 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{7}+O(q^{10})$$ 12 * q - 12 * q^7 $$12 q - 12 q^{7} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 18 q^{14} - 18 q^{19} - 18 q^{25} - 12 q^{26} - 6 q^{28} - 18 q^{29} + 12 q^{34} - 18 q^{35} + 30 q^{37} + 24 q^{38} + 12 q^{40} - 24 q^{41} - 6 q^{44} + 30 q^{46} - 6 q^{47} + 12 q^{49} + 36 q^{50} - 12 q^{52} + 12 q^{53} - 18 q^{55} + 6 q^{58} - 36 q^{61} + 6 q^{64} - 36 q^{65} - 30 q^{67} - 12 q^{70} - 12 q^{71} + 48 q^{74} - 18 q^{76} - 12 q^{77} + 6 q^{79} + 48 q^{83} + 18 q^{85} + 36 q^{86} - 36 q^{88} + 18 q^{89} - 6 q^{91} - 18 q^{92} + 36 q^{97} - 36 q^{98}+O(q^{100})$$ 12 * q - 12 * q^7 + 6 * q^10 + 6 * q^11 + 6 * q^13 + 18 * q^14 - 18 * q^19 - 18 * q^25 - 12 * q^26 - 6 * q^28 - 18 * q^29 + 12 * q^34 - 18 * q^35 + 30 * q^37 + 24 * q^38 + 12 * q^40 - 24 * q^41 - 6 * q^44 + 30 * q^46 - 6 * q^47 + 12 * q^49 + 36 * q^50 - 12 * q^52 + 12 * q^53 - 18 * q^55 + 6 * q^58 - 36 * q^61 + 6 * q^64 - 36 * q^65 - 30 * q^67 - 12 * q^70 - 12 * q^71 + 48 * q^74 - 18 * q^76 - 12 * q^77 + 6 * q^79 + 48 * q^83 + 18 * q^85 + 36 * q^86 - 36 * q^88 + 18 * q^89 - 6 * q^91 - 18 * q^92 + 36 * q^97 - 36 * q^98

## Character values

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

 $$n$$ $$371$$ $$631$$ $$\chi(n)$$ $$1$$ $$\zeta_{36}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.342020 − 0.939693i 0.342020 + 0.939693i −0.984808 + 0.173648i 0.984808 − 0.173648i −0.642788 − 0.766044i 0.642788 + 0.766044i −0.984808 − 0.173648i 0.984808 + 0.173648i −0.642788 + 0.766044i 0.642788 − 0.766044i −0.342020 + 0.939693i 0.342020 − 0.939693i
−0.342020 0.939693i 0 −0.766044 + 0.642788i −0.839712 + 0.148064i 0 0.240460 + 1.36372i 0.866025 + 0.500000i 0 0.426333 + 0.738430i
289.2 0.342020 + 0.939693i 0 −0.766044 + 0.642788i −2.57176 + 0.453471i 0 −0.361075 2.04776i −0.866025 0.500000i 0 −1.30572 2.26157i
361.1 −0.984808 + 0.173648i 0 0.939693 0.342020i 0.247315 0.294739i 0 −2.50048 2.09815i −0.866025 + 0.500000i 0 −0.192377 + 0.333207i
361.2 0.984808 0.173648i 0 0.939693 0.342020i 1.97937 2.35892i 0 0.153180 + 0.128533i 0.866025 0.500000i 0 1.53967 2.66679i
469.1 −0.642788 0.766044i 0 −0.173648 + 0.984808i 1.45842 + 4.00698i 0 −3.39364 + 1.23518i 0.866025 0.500000i 0 2.13207 3.69285i
469.2 0.642788 + 0.766044i 0 −0.173648 + 0.984808i −0.273629 0.751790i 0 −0.138449 + 0.0503913i −0.866025 + 0.500000i 0 0.400019 0.692853i
559.1 −0.984808 0.173648i 0 0.939693 + 0.342020i 0.247315 + 0.294739i 0 −2.50048 + 2.09815i −0.866025 0.500000i 0 −0.192377 0.333207i
559.2 0.984808 + 0.173648i 0 0.939693 + 0.342020i 1.97937 + 2.35892i 0 0.153180 0.128533i 0.866025 + 0.500000i 0 1.53967 + 2.66679i
595.1 −0.642788 + 0.766044i 0 −0.173648 0.984808i 1.45842 4.00698i 0 −3.39364 1.23518i 0.866025 + 0.500000i 0 2.13207 + 3.69285i
595.2 0.642788 0.766044i 0 −0.173648 0.984808i −0.273629 + 0.751790i 0 −0.138449 0.0503913i −0.866025 0.500000i 0 0.400019 + 0.692853i
613.1 −0.342020 + 0.939693i 0 −0.766044 0.642788i −0.839712 0.148064i 0 0.240460 1.36372i 0.866025 0.500000i 0 0.426333 0.738430i
613.2 0.342020 0.939693i 0 −0.766044 0.642788i −2.57176 0.453471i 0 −0.361075 + 2.04776i −0.866025 + 0.500000i 0 −1.30572 + 2.26157i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.2 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.h even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.bj.c 12
3.b odd 2 1 74.2.h.a 12
12.b even 2 1 592.2.bq.b 12
37.h even 18 1 inner 666.2.bj.c 12
111.n odd 18 1 74.2.h.a 12
111.q even 36 1 2738.2.a.r 6
111.q even 36 1 2738.2.a.s 6
444.ba even 18 1 592.2.bq.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.h.a 12 3.b odd 2 1
74.2.h.a 12 111.n odd 18 1
592.2.bq.b 12 12.b even 2 1
592.2.bq.b 12 444.ba even 18 1
666.2.bj.c 12 1.a even 1 1 trivial
666.2.bj.c 12 37.h even 18 1 inner
2738.2.a.r 6 111.q even 36 1
2738.2.a.s 6 111.q even 36 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} + 9 T_{5}^{10} + 72 T_{5}^{9} + 36 T_{5}^{8} + 162 T_{5}^{7} + 1476 T_{5}^{6} + 2268 T_{5}^{5} + 1701 T_{5}^{4} + 810 T_{5}^{3} + 81 T_{5}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(666, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{6} + 1$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 9 T^{10} + 72 T^{9} + 36 T^{8} + \cdots + 81$$
$7$ $$T^{12} + 12 T^{11} + 66 T^{10} + 218 T^{9} + \cdots + 1$$
$11$ $$T^{12} - 6 T^{11} + 63 T^{10} + \cdots + 408321$$
$13$ $$T^{12} - 6 T^{11} + 30 T^{10} + \cdots + 288369$$
$17$ $$T^{12} + 36 T^{10} - 234 T^{9} + \cdots + 81$$
$19$ $$T^{12} + 18 T^{11} + 135 T^{10} + \cdots + 10439361$$
$23$ $$T^{12} - 63 T^{10} + 3303 T^{8} + \cdots + 431649$$
$29$ $$T^{12} + 18 T^{11} + 126 T^{10} + \cdots + 110889$$
$31$ $$T^{12} + 210 T^{10} + \cdots + 317445489$$
$37$ $$T^{12} - 30 T^{11} + \cdots + 2565726409$$
$41$ $$T^{12} + 24 T^{11} + 216 T^{10} + \cdots + 331776$$
$43$ $$T^{12} + 156 T^{10} + 8910 T^{8} + \cdots + 2277081$$
$47$ $$T^{12} + 6 T^{11} + \cdots + 2027430729$$
$53$ $$T^{12} - 12 T^{11} + \cdots + 45041148441$$
$59$ $$T^{12} - 1404 T^{9} + \cdots + 17477104401$$
$61$ $$T^{12} + 36 T^{11} + 756 T^{10} + \cdots + 331776$$
$67$ $$T^{12} + 30 T^{11} + \cdots + 59697637561$$
$71$ $$T^{12} + 12 T^{11} + \cdots + 4131885551616$$
$73$ $$(T^{6} - 366 T^{4} + 322 T^{3} + \cdots + 94609)^{2}$$
$79$ $$T^{12} - 6 T^{11} + 12 T^{10} + \cdots + 47961$$
$83$ $$T^{12} - 48 T^{11} + 1008 T^{10} + \cdots + 110889$$
$89$ $$T^{12} - 18 T^{11} + \cdots + 687331089$$
$97$ $$T^{12} - 36 T^{11} + \cdots + 27455164416$$