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Properties

Label 666.2.h.c
Level $666$
Weight $2$
Character orbit 666.h
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(121,666)] 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("666.121"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.h (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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 38 q^{2} - 2 q^{3} + 38 q^{4} - 4 q^{5} - 2 q^{6} + q^{7} + 38 q^{8} - 2 q^{9} - 4 q^{10} - 2 q^{11} - 2 q^{12} + 10 q^{13} + q^{14} - 4 q^{15} + 38 q^{16} - 2 q^{18} - 8 q^{19} - 4 q^{20} + 4 q^{21}+ \cdots - 5 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 1.00000 −1.72390 + 0.167877i 1.00000 −3.70230 −1.72390 + 0.167877i −0.330849 + 0.573047i 1.00000 2.94363 − 0.578805i −3.70230
121.2 1.00000 −1.68864 + 0.385344i 1.00000 3.96916 −1.68864 + 0.385344i 1.05023 − 1.81906i 1.00000 2.70302 − 1.30142i 3.96916
121.3 1.00000 −1.61518 − 0.625457i 1.00000 2.52032 −1.61518 − 0.625457i −1.45985 + 2.52854i 1.00000 2.21761 + 2.02045i 2.52032
121.4 1.00000 −1.34882 + 1.08659i 1.00000 −0.495406 −1.34882 + 1.08659i −2.03036 + 3.51669i 1.00000 0.638625 − 2.93124i −0.495406
121.5 1.00000 −1.12306 − 1.31861i 1.00000 0.457071 −1.12306 − 1.31861i 2.24881 − 3.89506i 1.00000 −0.477485 + 2.96176i 0.457071
121.6 1.00000 −1.09407 − 1.34276i 1.00000 −2.22000 −1.09407 − 1.34276i −0.959017 + 1.66107i 1.00000 −0.606007 + 2.93816i −2.22000
121.7 1.00000 −1.07244 + 1.36010i 1.00000 1.22505 −1.07244 + 1.36010i 1.29500 − 2.24301i 1.00000 −0.699738 − 2.91725i 1.22505
121.8 1.00000 −0.543399 + 1.64460i 1.00000 −3.51311 −0.543399 + 1.64460i 2.20815 − 3.82462i 1.00000 −2.40943 − 1.78735i −3.51311
121.9 1.00000 −0.367479 − 1.69262i 1.00000 −2.43204 −0.367479 − 1.69262i −0.0737984 + 0.127823i 1.00000 −2.72992 + 1.24400i −2.43204
121.10 1.00000 −0.266458 − 1.71143i 1.00000 2.46736 −0.266458 − 1.71143i −0.202832 + 0.351316i 1.00000 −2.85800 + 0.912048i 2.46736
121.11 1.00000 −0.124130 + 1.72760i 1.00000 1.34325 −0.124130 + 1.72760i −0.00815087 + 0.0141177i 1.00000 −2.96918 − 0.428893i 1.34325
121.12 1.00000 0.600169 + 1.62475i 1.00000 −2.29481 0.600169 + 1.62475i −1.88616 + 3.26692i 1.00000 −2.27960 + 1.95024i −2.29481
121.13 1.00000 0.725216 − 1.57292i 1.00000 −1.02077 0.725216 − 1.57292i 0.251154 − 0.435011i 1.00000 −1.94812 − 2.28141i −1.02077
121.14 1.00000 1.13949 + 1.30444i 1.00000 2.72110 1.13949 + 1.30444i −0.0957449 + 0.165835i 1.00000 −0.403115 + 2.97279i 2.72110
121.15 1.00000 1.30864 − 1.13466i 1.00000 0.219776 1.30864 − 1.13466i 1.50224 − 2.60196i 1.00000 0.425075 − 2.96973i 0.219776
121.16 1.00000 1.34292 + 1.09388i 1.00000 −2.55391 1.34292 + 1.09388i 1.70038 − 2.94514i 1.00000 0.606853 + 2.93798i −2.55391
121.17 1.00000 1.38847 − 1.03545i 1.00000 2.81288 1.38847 − 1.03545i −1.72500 + 2.98779i 1.00000 0.855671 − 2.87538i 2.81288
121.18 1.00000 1.73083 + 0.0651450i 1.00000 0.204308 1.73083 + 0.0651450i 1.30647 − 2.26288i 1.00000 2.99151 + 0.225509i 0.204308
121.19 1.00000 1.73185 − 0.0264065i 1.00000 −1.70793 1.73185 − 0.0264065i −2.29068 + 3.96757i 1.00000 2.99861 − 0.0914641i −1.70793
655.1 1.00000 −1.72390 − 0.167877i 1.00000 −3.70230 −1.72390 − 0.167877i −0.330849 − 0.573047i 1.00000 2.94363 + 0.578805i −3.70230
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.19
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{19} + 2 T_{5}^{18} - 48 T_{5}^{17} - 94 T_{5}^{16} + 928 T_{5}^{15} + 1755 T_{5}^{14} + \cdots + 2268 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display