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Properties

Label 666.2.g.c
Level $666$
Weight $2$
Character orbit 666.g
Analytic conductor $5.318$
Analytic rank $0$
Dimension $38$
Inner twists $2$

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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(211,666)] 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("666.211"); 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([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [38] 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: \(38\)
Relative dimension: \(19\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 19 q^{2} + q^{3} - 19 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 38 q^{8} + q^{9} - 4 q^{10} - 2 q^{11} + q^{12} - 5 q^{13} + q^{14} + 11 q^{15} - 19 q^{16} + q^{18} - 8 q^{19} + 2 q^{20} - 8 q^{21}+ \cdots - 29 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1 −0.500000 + 0.866025i −1.72479 + 0.158402i −0.500000 − 0.866025i 0.510385 + 0.884013i 0.725216 − 1.57292i −0.502308 1.00000 2.94982 − 0.546423i −1.02077
211.2 −0.500000 + 0.866025i −1.63697 − 0.565983i −0.500000 − 0.866025i −0.109888 − 0.190332i 1.30864 − 1.13466i −3.00448 1.00000 2.35933 + 1.85299i 0.219776
211.3 −0.500000 + 0.866025i −1.59096 − 0.684719i −0.500000 − 0.866025i −1.40644 − 2.43602i 1.38847 − 1.03545i 3.45000 1.00000 2.06232 + 2.17872i 2.81288
211.4 −0.500000 + 0.866025i −1.34892 + 1.08648i −0.500000 − 0.866025i −1.23368 − 2.13679i −0.266458 − 1.71143i 0.405665 1.00000 0.639143 − 2.93113i 2.46736
211.5 −0.500000 + 0.866025i −1.28211 + 1.16456i −0.500000 − 0.866025i 1.21602 + 2.10621i −0.367479 − 1.69262i 0.147597 1.00000 0.287621 − 2.98618i −2.43204
211.6 −0.500000 + 0.866025i −0.888793 − 1.48662i −0.500000 − 0.866025i 0.853967 + 1.47911i 1.73185 − 0.0264065i 4.58135 1.00000 −1.42009 + 2.64260i −1.70793
211.7 −0.500000 + 0.866025i −0.808995 − 1.53151i −0.500000 − 0.866025i −0.102154 − 0.176936i 1.73083 + 0.0651450i −2.61295 1.00000 −1.69105 + 2.47797i 0.204308
211.8 −0.500000 + 0.866025i −0.615827 + 1.61888i −0.500000 − 0.866025i 1.11000 + 1.92257i −1.09407 − 1.34276i 1.91803 1.00000 −2.24151 − 1.99389i −2.22000
211.9 −0.500000 + 0.866025i −0.580424 + 1.63190i −0.500000 − 0.866025i −0.228535 − 0.395835i −1.12306 − 1.31861i −4.49762 1.00000 −2.32622 − 1.89439i 0.457071
211.10 −0.500000 + 0.866025i 0.265927 + 1.71151i −0.500000 − 0.866025i −1.26016 − 2.18266i −1.61518 − 0.625457i 2.91970 1.00000 −2.85857 + 0.910277i 2.52032
211.11 −0.500000 + 0.866025i 0.275869 − 1.70994i −0.500000 − 0.866025i 1.27695 + 2.21175i 1.34292 + 1.09388i −3.40076 1.00000 −2.84779 − 0.943440i −2.55391
211.12 −0.500000 + 0.866025i 0.559930 − 1.63905i −0.500000 − 0.866025i −1.36055 − 2.35654i 1.13949 + 1.30444i 0.191490 1.00000 −2.37296 − 1.83550i 2.72110
211.13 −0.500000 + 0.866025i 1.00733 + 1.40900i −0.500000 − 0.866025i 1.85115 + 3.20629i −1.72390 + 0.167877i 0.661698 1.00000 −0.970558 + 2.83866i −3.70230
211.14 −0.500000 + 0.866025i 1.10699 − 1.33213i −0.500000 − 0.866025i 1.14740 + 1.98736i 0.600169 + 1.62475i 3.77231 1.00000 −0.549162 − 2.94931i −2.29481
211.15 −0.500000 + 0.866025i 1.17804 + 1.26973i −0.500000 − 0.866025i −1.98458 − 3.43739i −1.68864 + 0.385344i −2.10047 1.00000 −0.224452 + 2.99159i 3.96916
211.16 −0.500000 + 0.866025i 1.55821 − 0.756299i −0.500000 − 0.866025i −0.671627 − 1.16329i −0.124130 + 1.72760i 0.0163017 1.00000 1.85602 − 2.35694i 1.34325
211.17 −0.500000 + 0.866025i 1.61543 + 0.624814i −0.500000 − 0.866025i 0.247703 + 0.429034i −1.34882 + 1.08659i 4.06073 1.00000 2.21921 + 2.01868i −0.495406
211.18 −0.500000 + 0.866025i 1.69597 − 0.351704i −0.500000 − 0.866025i 1.75655 + 3.04244i −0.543399 + 1.64460i −4.41629 1.00000 2.75261 − 1.19296i −3.51311
211.19 −0.500000 + 0.866025i 1.71410 + 0.248712i −0.500000 − 0.866025i −0.612527 − 1.06093i −1.07244 + 1.36010i −2.59000 1.00000 2.87628 + 0.852635i 1.22505
565.1 −0.500000 − 0.866025i −1.72479 − 0.158402i −0.500000 + 0.866025i 0.510385 − 0.884013i 0.725216 + 1.57292i −0.502308 1.00000 2.94982 + 0.546423i −1.02077
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.19
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
333.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.g.c ✓ 38
3.b odd 2 1 1998.2.g.c 38
9.c even 3 1 666.2.h.c yes 38
9.d odd 6 1 1998.2.h.c 38
37.c even 3 1 666.2.h.c yes 38
111.i odd 6 1 1998.2.h.c 38
333.g even 3 1 inner 666.2.g.c ✓ 38
333.u odd 6 1 1998.2.g.c 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.g.c ✓ 38 1.a even 1 1 trivial
666.2.g.c ✓ 38 333.g even 3 1 inner
666.2.h.c yes 38 9.c even 3 1
666.2.h.c yes 38 37.c even 3 1
1998.2.g.c 38 3.b odd 2 1
1998.2.g.c 38 333.u odd 6 1
1998.2.h.c 38 9.d odd 6 1
1998.2.h.c 38 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{38} - 2 T_{5}^{37} + 52 T_{5}^{36} - 92 T_{5}^{35} + 1564 T_{5}^{34} - 2555 T_{5}^{33} + \cdots + 5143824 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display