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Properties

Label 666.2.t.a
Level $666$
Weight $2$
Character orbit 666.t
Analytic conductor $5.318$
Analytic rank $0$
Dimension $76$
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(85,666)] 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("666.85"); 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, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 2 q^{3} + 38 q^{4} + 4 q^{7} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 6 q^{13} + 6 q^{15} - 38 q^{16} - 12 q^{21} - 12 q^{23} + 50 q^{25} - 24 q^{26} + 4 q^{27} + 2 q^{28} - 18 q^{29} - 12 q^{30} - 6 q^{31}+ \cdots - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 −0.866025 − 0.500000i −1.72745 − 0.126222i 0.500000 + 0.866025i 0.767303 − 0.443003i 1.43290 + 0.973034i −2.39076 − 1.00000i 2.96814 + 0.436082i −0.886006
85.2 −0.866025 − 0.500000i −1.64295 + 0.548374i 0.500000 + 0.866025i −1.10747 + 0.639396i 1.69702 + 0.346570i 4.50026 − 1.00000i 2.39857 − 1.80190i 1.27879
85.3 −0.866025 − 0.500000i −1.52427 − 0.822567i 0.500000 + 0.866025i −2.98479 + 1.72327i 0.908769 + 1.47450i −0.480194 − 1.00000i 1.64677 + 2.50762i 3.44654
85.4 −0.866025 − 0.500000i −1.47659 + 0.905358i 0.500000 + 0.866025i 3.27608 − 1.89144i 1.73145 − 0.0457663i 3.79962 − 1.00000i 1.36065 − 2.67369i −3.78289
85.5 −0.866025 − 0.500000i −1.36763 − 1.06282i 0.500000 + 0.866025i 1.21790 − 0.703156i 0.652996 + 1.60424i 1.68325 − 1.00000i 0.740837 + 2.90709i −1.40631
85.6 −0.866025 − 0.500000i −1.23926 + 1.21006i 0.500000 + 0.866025i −3.39345 + 1.95921i 1.67826 − 0.428312i −2.65550 − 1.00000i 0.0715207 − 2.99915i 3.91841
85.7 −0.866025 − 0.500000i −0.900303 + 1.47968i 0.500000 + 0.866025i −0.716641 + 0.413753i 1.51953 − 0.831290i −0.481710 − 1.00000i −1.37891 − 2.66432i 0.827506
85.8 −0.866025 − 0.500000i −0.453459 − 1.67164i 0.500000 + 0.866025i 3.82230 − 2.20681i −0.443112 + 1.67441i −3.96031 − 1.00000i −2.58875 + 1.51604i −4.41362
85.9 −0.866025 − 0.500000i −0.272455 − 1.71049i 0.500000 + 0.866025i −1.37876 + 0.796025i −0.619290 + 1.61755i −3.27769 − 1.00000i −2.85154 + 0.932064i 1.59205
85.10 −0.866025 − 0.500000i −0.0651010 + 1.73083i 0.500000 + 0.866025i −1.00174 + 0.578356i 0.921793 − 1.46639i 3.69476 − 1.00000i −2.99152 − 0.225357i 1.15671
85.11 −0.866025 − 0.500000i 0.241567 − 1.71512i 0.500000 + 0.866025i −2.77599 + 1.60272i −1.06676 + 1.36456i 3.40838 − 1.00000i −2.88329 − 0.828634i 3.20544
85.12 −0.866025 − 0.500000i 0.342832 − 1.69778i 0.500000 + 0.866025i 1.79709 − 1.03755i −1.14579 + 1.29891i 2.92436 − 1.00000i −2.76493 − 1.16411i −2.07510
85.13 −0.866025 − 0.500000i 0.615689 + 1.61893i 0.500000 + 0.866025i 0.323001 − 0.186485i 0.276261 − 1.70988i −0.0420982 − 1.00000i −2.24185 + 1.99351i −0.372969
85.14 −0.866025 − 0.500000i 1.31903 + 1.12257i 0.500000 + 0.866025i 3.57279 − 2.06275i −0.581033 − 1.63169i −2.48600 − 1.00000i 0.479695 + 2.96140i −4.12550
85.15 −0.866025 − 0.500000i 1.40836 − 1.00823i 0.500000 + 0.866025i −0.619054 + 0.357411i −1.72379 + 0.168971i −5.16921 − 1.00000i 0.966953 − 2.83989i 0.714822
85.16 −0.866025 − 0.500000i 1.44939 − 0.948302i 0.500000 + 0.866025i 1.90796 − 1.10156i −1.72936 + 0.0965604i 1.75157 − 1.00000i 1.20145 − 2.74891i −2.20312
85.17 −0.866025 − 0.500000i 1.47890 + 0.901585i 0.500000 + 0.866025i −1.64226 + 0.948160i −0.829972 − 1.52025i −2.22626 − 1.00000i 1.37429 + 2.66671i 1.89632
85.18 −0.866025 − 0.500000i 1.60359 + 0.654601i 0.500000 + 0.866025i −2.32443 + 1.34201i −1.06145 − 1.36870i 2.44933 − 1.00000i 2.14299 + 2.09942i 2.68401
85.19 −0.866025 − 0.500000i 1.71011 − 0.274834i 0.500000 + 0.866025i 1.26015 − 0.727547i −1.61841 − 0.617040i −0.0417734 − 1.00000i 2.84893 − 0.939992i −1.45509
85.20 0.866025 + 0.500000i −1.72874 + 0.107007i 0.500000 + 0.866025i −2.85491 + 1.64828i −1.55064 − 0.771700i 2.83098 1.00000i 2.97710 − 0.369975i −3.29656
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.38
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
333.t 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.t.a yes 76
3.b odd 2 1 1998.2.t.a 76
9.c even 3 1 666.2.k.a ✓ 76
9.d odd 6 1 1998.2.k.a 76
37.e even 6 1 666.2.k.a ✓ 76
111.h odd 6 1 1998.2.k.a 76
333.o odd 6 1 1998.2.t.a 76
333.t even 6 1 inner 666.2.t.a yes 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.k.a ✓ 76 9.c even 3 1
666.2.k.a ✓ 76 37.e even 6 1
666.2.t.a yes 76 1.a even 1 1 trivial
666.2.t.a yes 76 333.t even 6 1 inner
1998.2.k.a 76 9.d odd 6 1
1998.2.k.a 76 111.h odd 6 1
1998.2.t.a 76 3.b odd 2 1
1998.2.t.a 76 333.o odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(666, [\chi])\).