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Properties

Label 666.2.bs.b
Level $666$
Weight $2$
Character orbit 666.bs
Analytic conductor $5.318$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(17,666)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 12 q^{13} + 24 q^{19} + 12 q^{22} + 48 q^{31} + 72 q^{34} + 24 q^{37} + 72 q^{43} + 60 q^{46} + 12 q^{52} - 60 q^{55} + 12 q^{58} - 120 q^{61} + 36 q^{67} + 12 q^{70} - 24 q^{76} + 60 q^{79} + 96 q^{82}+ \cdots + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 17.8
Significant digits:
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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 \( T_{5}^{96} - 108 T_{5}^{92} + 5904 T_{5}^{90} + 21546 T_{5}^{88} - 906444 T_{5}^{86} + \cdots + 86\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display