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Properties

Label 666.2.q.b
Level $666$
Weight $2$
Character orbit 666.q
Analytic conductor $5.318$
Analytic rank $0$
Dimension $72$
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(295,666)] 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("666.295"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] 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: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{3} + 36 q^{4} + 4 q^{7} + 8 q^{9} + 8 q^{10} - 10 q^{11} + 2 q^{12} - 36 q^{16} - 12 q^{21} + 52 q^{25} - 4 q^{26} - 32 q^{27} + 8 q^{28} + 24 q^{30} - 38 q^{33} + 12 q^{34} - 2 q^{36} - 32 q^{37}+ \cdots + 74 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1 −0.866025 − 0.500000i −1.71655 − 0.231194i 0.500000 + 0.866025i −2.11882 + 1.22330i 1.37098 + 1.05850i −0.0163493 + 0.0283178i − 1.00000i 2.89310 + 0.793714i 2.44661
295.2 −0.866025 − 0.500000i −1.66312 + 0.483784i 0.500000 + 0.866025i 0.0135176 − 0.00780437i 1.68219 + 0.412588i 0.114323 − 0.198014i − 1.00000i 2.53191 − 1.60918i −0.0156087
295.3 −0.866025 − 0.500000i −1.61913 + 0.615160i 0.500000 + 0.866025i 3.22658 − 1.86286i 1.70979 + 0.276820i −2.33164 + 4.03852i − 1.00000i 2.24316 − 1.99205i −3.72573
295.4 −0.866025 − 0.500000i −1.52521 + 0.820815i 0.500000 + 0.866025i 1.79041 − 1.03369i 1.73128 + 0.0517575i 2.35442 − 4.07797i − 1.00000i 1.65252 − 2.50383i −2.06738
295.5 −0.866025 − 0.500000i −1.10736 − 1.33183i 0.500000 + 0.866025i 1.06178 − 0.613020i 0.293086 + 1.70707i −0.577625 + 1.00048i − 1.00000i −0.547521 + 2.94961i −1.22604
295.6 −0.866025 − 0.500000i −1.00570 − 1.41016i 0.500000 + 0.866025i −2.61658 + 1.51068i 0.165883 + 1.72409i 1.88563 − 3.26601i − 1.00000i −0.977123 + 2.83641i 3.02137
295.7 −0.866025 − 0.500000i −0.933613 + 1.45889i 0.500000 + 0.866025i −1.73582 + 1.00218i 1.53798 − 0.796631i −1.84656 + 3.19833i − 1.00000i −1.25673 − 2.72408i 2.00435
295.8 −0.866025 − 0.500000i −0.0766549 − 1.73035i 0.500000 + 0.866025i 0.893958 − 0.516127i −0.798792 + 1.53686i −1.34612 + 2.33155i − 1.00000i −2.98825 + 0.265280i −1.03225
295.9 −0.866025 − 0.500000i 0.134727 − 1.72680i 0.500000 + 0.866025i 3.47746 − 2.00771i −0.980078 + 1.42809i 1.62916 − 2.82178i − 1.00000i −2.96370 − 0.465293i −4.01543
295.10 −0.866025 − 0.500000i 0.489499 + 1.66144i 0.500000 + 0.866025i −1.91018 + 1.10284i 0.406802 − 1.68360i 2.33806 − 4.04964i − 1.00000i −2.52078 + 1.62655i 2.20569
295.11 −0.866025 − 0.500000i 0.630864 − 1.61308i 0.500000 + 0.866025i −1.26082 + 0.727936i −1.35288 + 1.08153i −0.555043 + 0.961363i − 1.00000i −2.20402 − 2.03526i 1.45587
295.12 −0.866025 − 0.500000i 0.706817 + 1.58127i 0.500000 + 0.866025i 2.09692 − 1.21066i 0.178512 − 1.72283i −0.633629 + 1.09748i − 1.00000i −2.00082 + 2.23534i −2.42132
295.13 −0.866025 − 0.500000i 0.843072 + 1.51302i 0.500000 + 0.866025i −3.56506 + 2.05829i 0.0263886 − 1.73185i −1.11612 + 1.93317i − 1.00000i −1.57846 + 2.55117i 4.11657
295.14 −0.866025 − 0.500000i 1.44553 + 0.954171i 0.500000 + 0.866025i −0.622305 + 0.359288i −0.774780 − 1.54910i 0.930527 − 1.61172i − 1.00000i 1.17911 + 2.75857i 0.718576
295.15 −0.866025 − 0.500000i 1.46455 − 0.924718i 0.500000 + 0.866025i −3.53299 + 2.03977i −1.73069 + 0.0685558i 0.189560 − 0.328327i − 1.00000i 1.28979 − 2.70858i 4.07955
295.16 −0.866025 − 0.500000i 1.55737 − 0.758019i 0.500000 + 0.866025i 0.313745 − 0.181141i −1.72773 − 0.122222i 1.42679 − 2.47128i − 1.00000i 1.85081 − 2.36103i −0.362282
295.17 −0.866025 − 0.500000i 1.64558 + 0.540424i 0.500000 + 0.866025i 3.09623 − 1.78761i −1.15490 − 1.29081i 0.750004 − 1.29904i − 1.00000i 2.41588 + 1.77862i −3.57521
295.18 −0.866025 − 0.500000i 1.72932 + 0.0971749i 0.500000 + 0.866025i −0.340064 + 0.196336i −1.44905 − 0.948817i −2.19539 + 3.80253i − 1.00000i 2.98111 + 0.336093i 0.392672
295.19 0.866025 + 0.500000i −1.71655 − 0.231194i 0.500000 + 0.866025i 2.11882 − 1.22330i −1.37098 − 1.05850i −0.0163493 + 0.0283178i 1.00000i 2.89310 + 0.793714i 2.44661
295.20 0.866025 + 0.500000i −1.66312 + 0.483784i 0.500000 + 0.866025i −0.0135176 + 0.00780437i −1.68219 − 0.412588i 0.114323 − 0.198014i 1.00000i 2.53191 − 1.60918i −0.0156087
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
37.b even 2 1 inner
333.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.q.b ✓ 72
3.b odd 2 1 1998.2.q.b 72
9.c even 3 1 inner 666.2.q.b ✓ 72
9.d odd 6 1 1998.2.q.b 72
37.b even 2 1 inner 666.2.q.b ✓ 72
111.d odd 2 1 1998.2.q.b 72
333.n odd 6 1 1998.2.q.b 72
333.q even 6 1 inner 666.2.q.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.q.b ✓ 72 1.a even 1 1 trivial
666.2.q.b ✓ 72 9.c even 3 1 inner
666.2.q.b ✓ 72 37.b even 2 1 inner
666.2.q.b ✓ 72 333.q even 6 1 inner
1998.2.q.b 72 3.b odd 2 1
1998.2.q.b 72 9.d odd 6 1
1998.2.q.b 72 111.d odd 2 1
1998.2.q.b 72 333.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 116 T_{5}^{70} + 7428 T_{5}^{68} - 328096 T_{5}^{66} + 11036840 T_{5}^{64} + \cdots + 34828517376 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display