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Properties

Label 666.2.k.a
Level $666$
Weight $2$
Character orbit 666.k
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(175,666)] 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("666.175"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.k (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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 4 q^{3} - 76 q^{4} - 2 q^{7} - 4 q^{9} - 4 q^{11} - 4 q^{12} + 12 q^{15} + 76 q^{16} + 6 q^{21} + 12 q^{23} - 100 q^{25} - 24 q^{26} + 4 q^{27} + 2 q^{28} + 18 q^{29} - 12 q^{30} + 6 q^{31} - 32 q^{33}+ \cdots - 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1 − 1.00000i −1.70988 − 0.276261i −1.00000 − 0.372969i −0.276261 + 1.70988i 0.0210491 − 0.0364581i 1.00000i 2.84736 + 0.944746i −0.372969
175.2 − 1.00000i −1.63169 + 0.581033i −1.00000 − 4.12550i 0.581033 + 1.63169i 1.24300 − 2.15294i 1.00000i 2.32480 − 1.89613i −4.12550
175.3 − 1.00000i −1.52025 + 0.829972i −1.00000 1.89632i 0.829972 + 1.52025i 1.11313 − 1.92800i 1.00000i 1.62229 − 2.52352i 1.89632
175.4 − 1.00000i −1.46639 − 0.921793i −1.00000 1.15671i −0.921793 + 1.46639i −1.84738 + 3.19976i 1.00000i 1.30060 + 2.70341i 1.15671
175.5 − 1.00000i −1.36870 + 1.06145i −1.00000 2.68401i 1.06145 + 1.36870i −1.22466 + 2.12118i 1.00000i 0.746656 − 2.90560i 2.68401
175.6 − 1.00000i −0.831290 − 1.51953i −1.00000 0.827506i −1.51953 + 0.831290i 0.240855 − 0.417173i 1.00000i −1.61792 + 2.52633i 0.827506
175.7 − 1.00000i −0.617040 + 1.61841i −1.00000 − 1.45509i 1.61841 + 0.617040i 0.0208867 − 0.0361769i 1.00000i −2.23852 − 1.99725i −1.45509
175.8 − 1.00000i −0.428312 − 1.67826i −1.00000 3.91841i −1.67826 + 0.428312i 1.32775 − 2.29973i 1.00000i −2.63310 + 1.43763i 3.91841
175.9 − 1.00000i −0.0457663 − 1.73145i −1.00000 − 3.78289i −1.73145 + 0.0457663i −1.89981 + 3.29056i 1.00000i −2.99581 + 0.158484i −3.78289
175.10 − 1.00000i 0.0965604 + 1.72936i −1.00000 − 2.20312i 1.72936 − 0.0965604i −0.875783 + 1.51690i 1.00000i −2.98135 + 0.333975i −2.20312
175.11 − 1.00000i 0.168971 + 1.72379i −1.00000 0.714822i 1.72379 − 0.168971i 2.58461 − 4.47667i 1.00000i −2.94290 + 0.582542i 0.714822
175.12 − 1.00000i 0.346570 − 1.69702i −1.00000 1.27879i −1.69702 − 0.346570i −2.25013 + 3.89734i 1.00000i −2.75978 − 1.17627i 1.27879
175.13 − 1.00000i 0.973034 − 1.43290i −1.00000 − 0.886006i −1.43290 − 0.973034i 1.19538 − 2.07046i 1.00000i −1.10641 − 2.78852i −0.886006
175.14 − 1.00000i 1.29891 + 1.14579i −1.00000 − 2.07510i 1.14579 − 1.29891i −1.46218 + 2.53257i 1.00000i 0.374317 + 2.97656i −2.07510
175.15 − 1.00000i 1.36456 + 1.06676i −1.00000 3.20544i 1.06676 − 1.36456i −1.70419 + 2.95174i 1.00000i 0.724027 + 2.91132i 3.20544
175.16 − 1.00000i 1.47450 − 0.908769i −1.00000 3.44654i −0.908769 − 1.47450i 0.240097 − 0.415860i 1.00000i 1.34828 − 2.67995i 3.44654
175.17 − 1.00000i 1.60424 − 0.652996i −1.00000 − 1.40631i −0.652996 − 1.60424i −0.841625 + 1.45774i 1.00000i 2.14719 − 2.09513i −1.40631
175.18 − 1.00000i 1.61755 + 0.619290i −1.00000 1.59205i 0.619290 − 1.61755i 1.63885 − 2.83857i 1.00000i 2.23296 + 2.00347i 1.59205
175.19 − 1.00000i 1.67441 + 0.443112i −1.00000 − 4.41362i 0.443112 − 1.67441i 1.98016 − 3.42973i 1.00000i 2.60730 + 1.48390i −4.41362
175.20 1.00000i −1.72572 + 0.147954i −1.00000 − 1.23967i −0.147954 − 1.72572i −1.37235 + 2.37697i − 1.00000i 2.95622 − 0.510656i 1.23967
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.38
Significant digits:
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Inner twists

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

Hecke kernels

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