Properties

 Label 666.2.bs.b Level $666$ Weight $2$ Character orbit 666.bs Analytic conductor $5.318$ Analytic rank $0$ Dimension $96$ 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: $$96$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96 q+O(q^{10})$$ 96 * q $$\operatorname{Tr}(f)(q) =$$ $$96 q - 12 q^{13} + 24 q^{19} + 12 q^{22} + 48 q^{31} + 72 q^{34} + 24 q^{37} + 72 q^{43} + 60 q^{46} + 12 q^{52} - 60 q^{55} + 12 q^{58} - 120 q^{61} + 36 q^{67} + 12 q^{70} - 24 q^{76} + 60 q^{79} + 96 q^{82} - 108 q^{85} - 24 q^{88} + 216 q^{91} - 60 q^{94} + 24 q^{97}+O(q^{100})$$ 96 * q - 12 * q^13 + 24 * q^19 + 12 * q^22 + 48 * q^31 + 72 * q^34 + 24 * q^37 + 72 * q^43 + 60 * q^46 + 12 * q^52 - 60 * q^55 + 12 * q^58 - 120 * q^61 + 36 * q^67 + 12 * q^70 - 24 * q^76 + 60 * q^79 + 96 * q^82 - 108 * q^85 - 24 * q^88 + 216 * q^91 - 60 * q^94 + 24 * 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 −3.25585 + 0.284850i 0 −0.761384 0.638877i 0.965926 + 0.258819i 0 1.63414 2.83042i
17.2 −0.573576 + 0.819152i 0 −0.342020 0.939693i −1.60885 + 0.140756i 0 1.53998 + 1.29220i 0.965926 + 0.258819i 0 0.807495 1.39862i
17.3 −0.573576 + 0.819152i 0 −0.342020 0.939693i 1.31281 0.114856i 0 −1.79970 1.51012i 0.965926 + 0.258819i 0 −0.658910 + 1.14127i
17.4 −0.573576 + 0.819152i 0 −0.342020 0.939693i 3.55188 0.310750i 0 3.11397 + 2.61293i 0.965926 + 0.258819i 0 −1.78273 + 3.08777i
17.5 0.573576 0.819152i 0 −0.342020 0.939693i −3.55188 + 0.310750i 0 3.11397 + 2.61293i −0.965926 0.258819i 0 −1.78273 + 3.08777i
17.6 0.573576 0.819152i 0 −0.342020 0.939693i −1.31281 + 0.114856i 0 −1.79970 1.51012i −0.965926 0.258819i 0 −0.658910 + 1.14127i
17.7 0.573576 0.819152i 0 −0.342020 0.939693i 1.60885 0.140756i 0 1.53998 + 1.29220i −0.965926 0.258819i 0 0.807495 1.39862i
17.8 0.573576 0.819152i 0 −0.342020 0.939693i 3.25585 0.284850i 0 −0.761384 0.638877i −0.965926 0.258819i 0 1.63414 2.83042i
35.1 −0.996195 0.0871557i 0 0.984808 + 0.173648i −1.84075 + 3.94750i 0 −3.67900 + 1.33905i −0.965926 0.258819i 0 2.17779 3.77205i
35.2 −0.996195 0.0871557i 0 0.984808 + 0.173648i −0.252650 + 0.541810i 0 1.12060 0.407863i −0.965926 0.258819i 0 0.298910 0.517728i
35.3 −0.996195 0.0871557i 0 0.984808 + 0.173648i 0.898297 1.92640i 0 −4.07456 + 1.48302i −0.965926 0.258819i 0 −1.06278 + 1.84078i
35.4 −0.996195 0.0871557i 0 0.984808 + 0.173648i 1.19510 2.56290i 0 4.06568 1.47979i −0.965926 0.258819i 0 −1.41393 + 2.44899i
35.5 0.996195 + 0.0871557i 0 0.984808 + 0.173648i −1.19510 + 2.56290i 0 4.06568 1.47979i 0.965926 + 0.258819i 0 −1.41393 + 2.44899i
35.6 0.996195 + 0.0871557i 0 0.984808 + 0.173648i −0.898297 + 1.92640i 0 −4.07456 + 1.48302i 0.965926 + 0.258819i 0 −1.06278 + 1.84078i
35.7 0.996195 + 0.0871557i 0 0.984808 + 0.173648i 0.252650 0.541810i 0 1.12060 0.407863i 0.965926 + 0.258819i 0 0.298910 0.517728i
35.8 0.996195 + 0.0871557i 0 0.984808 + 0.173648i 1.84075 3.94750i 0 −3.67900 + 1.33905i 0.965926 + 0.258819i 0 2.17779 3.77205i
89.1 −0.906308 + 0.422618i 0 0.642788 0.766044i −3.46881 + 2.42889i 0 −0.619757 + 3.51482i −0.258819 + 0.965926i 0 2.11732 3.66730i
89.2 −0.906308 + 0.422618i 0 0.642788 0.766044i −1.52173 + 1.06553i 0 0.800320 4.53884i −0.258819 + 0.965926i 0 0.928845 1.60881i
89.3 −0.906308 + 0.422618i 0 0.642788 0.766044i 2.38993 1.67345i 0 0.280864 1.59286i −0.258819 + 0.965926i 0 −1.45878 + 2.52669i
89.4 −0.906308 + 0.422618i 0 0.642788 0.766044i 2.60061 1.82097i 0 −0.588545 + 3.33781i −0.258819 + 0.965926i 0 −1.58738 + 2.74942i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 647.8 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.b 96
3.b odd 2 1 inner 666.2.bs.b 96
37.i odd 36 1 inner 666.2.bs.b 96
111.q even 36 1 inner 666.2.bs.b 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.bs.b 96 1.a even 1 1 trivial
666.2.bs.b 96 3.b odd 2 1 inner
666.2.bs.b 96 37.i odd 36 1 inner
666.2.bs.b 96 111.q even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{96} - 108 T_{5}^{92} + 5904 T_{5}^{90} + 21546 T_{5}^{88} - 906444 T_{5}^{86} - 11888208 T_{5}^{84} + 518678964 T_{5}^{82} - 2868475761 T_{5}^{80} - 183722290344 T_{5}^{78} + \cdots + 86\!\cdots\!81$$ acting on $$S_{2}^{\mathrm{new}}(666, [\chi])$$.