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Properties

Label 888.2.bd.a
Level $888$
Weight $2$
Character orbit 888.bd
Analytic conductor $7.091$
Analytic rank $0$
Dimension $296$
Inner twists $8$

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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] = [888,2,Mod(491,888)] 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("888.491"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bd (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: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(296\)
Relative dimension: \(148\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 296 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9} + 10 q^{12} + 6 q^{16} - 2 q^{18} - 4 q^{19} - 6 q^{22} + 16 q^{24} - 136 q^{25} - 8 q^{27} - 2 q^{28} - 4 q^{30} + 4 q^{33} - 18 q^{34} + 20 q^{36} + 6 q^{40}+ \cdots - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1 −1.41418 − 0.00966923i −1.43433 + 0.970921i 1.99981 + 0.0273481i −0.959953 − 1.66269i 2.03779 − 1.35919i −1.06973 + 0.617610i −2.82783 − 0.0580118i 1.11462 − 2.78525i 1.34147 + 2.36062i
491.2 −1.41394 − 0.0275953i −0.182924 − 1.72236i 1.99848 + 0.0780363i −1.59939 − 2.77022i 0.211116 + 2.44037i 4.22474 − 2.43915i −2.82358 − 0.165488i −2.93308 + 0.630125i 2.18500 + 3.96107i
491.3 −1.41393 + 0.0284145i 1.47267 − 0.911731i 1.99839 − 0.0803522i 0.0159807 + 0.0276793i −2.05634 + 1.33097i 2.54318 − 1.46831i −2.82329 + 0.170395i 1.33749 − 2.68535i −0.0233820 − 0.0386825i
491.4 −1.41258 + 0.0679039i −0.559026 − 1.63936i 1.99078 − 0.191840i 1.94347 + 3.36619i 0.900990 + 2.27777i 1.80784 − 1.04376i −2.79911 + 0.406171i −2.37498 + 1.83289i −2.97389 − 4.62306i
491.5 −1.41257 + 0.0682209i −1.53110 − 0.809772i 1.99069 − 0.192733i −0.385438 − 0.667599i 2.21803 + 1.03940i −0.687016 + 0.396649i −2.79884 + 0.408056i 1.68854 + 2.47969i 0.590002 + 0.916733i
491.6 −1.40319 − 0.176210i 0.537741 + 1.64646i 1.93790 + 0.494514i 0.330224 + 0.571965i −0.464431 − 2.40506i −2.83212 + 1.63513i −2.63211 − 1.03538i −2.42167 + 1.77074i −0.362582 − 0.860766i
491.7 −1.39965 − 0.202429i 0.845131 − 1.51187i 1.91805 + 0.566659i 1.13354 + 1.96336i −1.48893 + 1.94501i −2.50608 + 1.44689i −2.56989 − 1.18139i −1.57151 − 2.55546i −1.18913 − 2.97747i
491.8 −1.39656 + 0.222785i 1.70171 + 0.322773i 1.90073 − 0.622263i −1.38974 − 2.40710i −2.44844 − 0.0716546i −4.20477 + 2.42763i −2.51585 + 1.29248i 2.79164 + 1.09853i 2.47712 + 3.05204i
491.9 −1.37409 − 0.334497i −0.859704 + 1.50363i 1.77622 + 0.919255i 0.663694 + 1.14955i 1.68427 − 1.77855i −1.20310 + 0.694610i −2.13320 − 1.85728i −1.52182 − 2.58536i −0.527451 − 1.80159i
491.10 −1.37038 + 0.349366i 1.73031 − 0.0775977i 1.75589 − 0.957528i 0.186689 + 0.323355i −2.34408 + 0.710850i 1.87920 − 1.08496i −2.07171 + 1.92562i 2.98796 − 0.268537i −0.368805 − 0.377897i
491.11 −1.36637 + 0.364745i 0.369564 + 1.69217i 1.73392 − 0.996753i −1.97369 − 3.41853i −1.12217 − 2.17732i 0.305942 − 0.176636i −2.00561 + 1.99437i −2.72685 + 1.25073i 3.94368 + 3.95108i
491.12 −1.35113 + 0.417678i 1.03591 + 1.38813i 1.65109 − 1.12867i 1.53012 + 2.65024i −1.97943 − 1.44286i −1.21937 + 0.704001i −1.75941 + 2.21460i −0.853787 + 2.87594i −3.17433 − 2.94172i
491.13 −1.34708 − 0.430552i 1.04983 − 1.37763i 1.62925 + 1.15998i −1.48905 − 2.57912i −2.00735 + 1.40377i −1.43848 + 0.830505i −1.69530 − 2.26406i −0.795715 − 2.89255i 0.895430 + 4.11539i
491.14 −1.34210 − 0.445837i 1.43562 + 0.969021i 1.60246 + 1.19671i −0.892905 − 1.54656i −1.49472 − 1.94057i 2.10868 − 1.21745i −1.61712 − 2.32054i 1.12200 + 2.78229i 0.508854 + 2.47372i
491.15 −1.33618 − 0.463285i −1.04268 − 1.38304i 1.57073 + 1.23806i −0.239306 − 0.414490i 0.752466 + 2.33105i 0.465002 − 0.268469i −1.52520 − 2.38197i −0.825621 + 2.88416i 0.127728 + 0.664698i
491.16 −1.33155 − 0.476417i 0.00963414 + 1.73202i 1.54605 + 1.26875i 1.35469 + 2.34638i 0.812337 − 2.31087i 3.21811 − 1.85797i −1.45420 − 2.42597i −2.99981 + 0.0333731i −0.685976 − 3.76972i
491.17 −1.31790 + 0.512979i −1.03107 − 1.39172i 1.47371 − 1.35211i 0.338103 + 0.585612i 2.07277 + 1.30523i −4.18334 + 2.41525i −1.24859 + 2.53792i −0.873773 + 2.86993i −0.745992 − 0.598336i
491.18 −1.31464 + 0.521261i −0.153731 + 1.72521i 1.45657 − 1.37055i 0.719868 + 1.24685i −0.697186 − 2.34818i 2.94347 − 1.69941i −1.20046 + 2.56103i −2.95273 − 0.530439i −1.59630 − 1.26392i
491.19 −1.30891 + 0.535487i −0.0101877 − 1.73202i 1.42651 − 1.40181i −1.50279 − 2.60290i 0.940809 + 2.26161i −1.55855 + 0.899828i −1.11652 + 2.59873i −2.99979 + 0.0352908i 3.36084 + 2.60225i
491.20 −1.30831 − 0.536950i −1.71402 + 0.249255i 1.42337 + 1.40500i 1.11601 + 1.93298i 2.37632 + 0.594242i 3.43766 − 1.98473i −1.10780 − 2.60246i 2.87574 − 0.854457i −0.422173 − 3.12819i
See next 80 embeddings (of 296 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 491.148
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
37.c even 3 1 inner
111.i odd 6 1 inner
296.p odd 6 1 inner
888.bd even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 888.2.bd.a ✓ 296
3.b odd 2 1 inner 888.2.bd.a ✓ 296
8.d odd 2 1 inner 888.2.bd.a ✓ 296
24.f even 2 1 inner 888.2.bd.a ✓ 296
37.c even 3 1 inner 888.2.bd.a ✓ 296
111.i odd 6 1 inner 888.2.bd.a ✓ 296
296.p odd 6 1 inner 888.2.bd.a ✓ 296
888.bd even 6 1 inner 888.2.bd.a ✓ 296
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.bd.a ✓ 296 1.a even 1 1 trivial
888.2.bd.a ✓ 296 3.b odd 2 1 inner
888.2.bd.a ✓ 296 8.d odd 2 1 inner
888.2.bd.a ✓ 296 24.f even 2 1 inner
888.2.bd.a ✓ 296 37.c even 3 1 inner
888.2.bd.a ✓ 296 111.i odd 6 1 inner
888.2.bd.a ✓ 296 296.p odd 6 1 inner
888.2.bd.a ✓ 296 888.bd even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(888, [\chi])\).