Properties

 Label 666.2.bs.a Level $666$ Weight $2$ Character orbit 666.bs Analytic conductor $5.318$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(17,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([18, 7]))

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

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

chi := DirichletCharacter("666.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$666 = 2 \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 666.bs (of order $$36$$, degree $$12$$, 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: $$72$$ Relative dimension: $$6$$ over $$\Q(\zeta_{36})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q+O(q^{10})$$ 72 * q $$\operatorname{Tr}(f)(q) =$$ $$72 q + 12 q^{13} - 24 q^{19} - 12 q^{22} + 72 q^{34} + 72 q^{37} + 24 q^{40} + 24 q^{43} + 36 q^{46} - 48 q^{49} - 12 q^{52} + 60 q^{55} + 120 q^{61} + 60 q^{67} - 60 q^{70} + 24 q^{76} - 12 q^{79} - 48 q^{82} + 108 q^{85} - 24 q^{88} - 168 q^{91} - 84 q^{94} - 264 q^{97}+O(q^{100})$$ 72 * q + 12 * q^13 - 24 * q^19 - 12 * q^22 + 72 * q^34 + 72 * q^37 + 24 * q^40 + 24 * q^43 + 36 * q^46 - 48 * q^49 - 12 * q^52 + 60 * q^55 + 120 * q^61 + 60 * q^67 - 60 * q^70 + 24 * q^76 - 12 * q^79 - 48 * q^82 + 108 * q^85 - 24 * q^88 - 168 * q^91 - 84 * q^94 - 264 * q^97

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −0.573576 + 0.819152i 0 −0.342020 0.939693i −0.980582 + 0.0857898i 0 1.16963 + 0.981439i 0.965926 + 0.258819i 0 0.492164 0.852453i
17.2 −0.573576 + 0.819152i 0 −0.342020 0.939693i −0.718088 + 0.0628246i 0 −2.07081 1.73762i 0.965926 + 0.258819i 0 0.360415 0.624258i
17.3 −0.573576 + 0.819152i 0 −0.342020 0.939693i 3.66079 0.320278i 0 −1.19170 0.999952i 0.965926 + 0.258819i 0 −1.83739 + 3.18245i
17.4 0.573576 0.819152i 0 −0.342020 0.939693i −3.66079 + 0.320278i 0 −1.19170 0.999952i −0.965926 0.258819i 0 −1.83739 + 3.18245i
17.5 0.573576 0.819152i 0 −0.342020 0.939693i 0.718088 0.0628246i 0 −2.07081 1.73762i −0.965926 0.258819i 0 0.360415 0.624258i
17.6 0.573576 0.819152i 0 −0.342020 0.939693i 0.980582 0.0857898i 0 1.16963 + 0.981439i −0.965926 0.258819i 0 0.492164 0.852453i
35.1 −0.996195 0.0871557i 0 0.984808 + 0.173648i −0.753592 + 1.61608i 0 4.63847 1.68826i −0.965926 0.258819i 0 0.891575 1.54425i
35.2 −0.996195 0.0871557i 0 0.984808 + 0.173648i −0.567891 + 1.21785i 0 −0.300852 + 0.109501i −0.965926 0.258819i 0 0.671872 1.16372i
35.3 −0.996195 0.0871557i 0 0.984808 + 0.173648i 0.778175 1.66880i 0 −1.77033 + 0.644346i −0.965926 0.258819i 0 −0.920660 + 1.59463i
35.4 0.996195 + 0.0871557i 0 0.984808 + 0.173648i −0.778175 + 1.66880i 0 −1.77033 + 0.644346i 0.965926 + 0.258819i 0 −0.920660 + 1.59463i
35.5 0.996195 + 0.0871557i 0 0.984808 + 0.173648i 0.567891 1.21785i 0 −0.300852 + 0.109501i 0.965926 + 0.258819i 0 0.671872 1.16372i
35.6 0.996195 + 0.0871557i 0 0.984808 + 0.173648i 0.753592 1.61608i 0 4.63847 1.68826i 0.965926 + 0.258819i 0 0.891575 1.54425i
89.1 −0.906308 + 0.422618i 0 0.642788 0.766044i −2.27330 + 1.59178i 0 0.235550 1.33587i −0.258819 + 0.965926i 0 1.38759 2.40338i
89.2 −0.906308 + 0.422618i 0 0.642788 0.766044i 0.733836 0.513837i 0 −0.650195 + 3.68744i −0.258819 + 0.965926i 0 −0.447924 + 0.775827i
89.3 −0.906308 + 0.422618i 0 0.642788 0.766044i 2.09980 1.47030i 0 0.541765 3.07250i −0.258819 + 0.965926i 0 −1.28169 + 2.21995i
89.4 0.906308 0.422618i 0 0.642788 0.766044i −2.09980 + 1.47030i 0 0.541765 3.07250i 0.258819 0.965926i 0 −1.28169 + 2.21995i
89.5 0.906308 0.422618i 0 0.642788 0.766044i −0.733836 + 0.513837i 0 −0.650195 + 3.68744i 0.258819 0.965926i 0 −0.447924 + 0.775827i
89.6 0.906308 0.422618i 0 0.642788 0.766044i 2.27330 1.59178i 0 0.235550 1.33587i 0.258819 0.965926i 0 1.38759 2.40338i
143.1 −0.422618 + 0.906308i 0 −0.642788 0.766044i −2.39204 + 3.41619i 0 −0.629322 3.56906i 0.965926 0.258819i 0 −2.08520 3.61167i
143.2 −0.422618 + 0.906308i 0 −0.642788 0.766044i 0.411306 0.587405i 0 0.509688 + 2.89058i 0.965926 0.258819i 0 0.358545 + 0.621018i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 647.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.i odd 36 1 inner
111.q even 36 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.bs.a 72
3.b odd 2 1 inner 666.2.bs.a 72
37.i odd 36 1 inner 666.2.bs.a 72
111.q even 36 1 inner 666.2.bs.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.bs.a 72 1.a even 1 1 trivial
666.2.bs.a 72 3.b odd 2 1 inner
666.2.bs.a 72 37.i odd 36 1 inner
666.2.bs.a 72 111.q even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{72} + 327 T_{5}^{68} - 6192 T_{5}^{66} + 28017 T_{5}^{64} - 932364 T_{5}^{62} + 6643081 T_{5}^{60} + 6446736 T_{5}^{58} - 698278833 T_{5}^{56} + 10632994704 T_{5}^{54} + \cdots + 570386655107361$$ acting on $$S_{2}^{\mathrm{new}}(666, [\chi])$$.