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Properties

Label 187.3.s.a
Level $187$
Weight $3$
Character orbit 187.s
Analytic conductor $5.095$
Analytic rank $0$
Dimension $1088$
Inner twists $4$

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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] = [187,3,Mod(3,187)] 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("187.3"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(187, base_ring=CyclotomicField(80)) chi = DirichletCharacter(H, H._module([64, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 187.s (of order \(80\), degree \(32\), 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: \(5.09538094354\)
Analytic rank: \(0\)
Dimension: \(1088\)
Relative dimension: \(34\) over \(\Q(\zeta_{80})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{80}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1088 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 24 q^{8} - 24 q^{9} - 64 q^{10} - 72 q^{11} - 256 q^{12} + 48 q^{13} - 88 q^{14} - 24 q^{15} - 24 q^{17} + 80 q^{18} - 24 q^{19}+ \cdots + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.96295 + 3.20325i −0.276503 − 2.33616i −4.59166 − 9.01164i −0.300077 + 7.63747i 8.02608 + 3.70008i −5.93333 − 7.52639i 22.8986 + 1.80216i 3.37012 − 0.809094i −23.8757 − 15.9532i
3.2 −1.90176 + 3.10340i 0.280815 + 2.37260i −4.19840 − 8.23982i 0.168607 − 4.29133i −7.89715 − 3.64064i −4.85649 − 6.16043i 19.0417 + 1.49861i 3.20097 − 0.768486i 12.9970 + 8.68435i
3.3 −1.85912 + 3.03382i −0.0796775 − 0.673192i −3.93173 − 7.71645i 0.0162409 − 0.413358i 2.19047 + 1.00982i 5.67410 + 7.19757i 16.5312 + 1.30103i 8.30449 − 1.99373i 1.22386 + 0.817755i
3.4 −1.84103 + 3.00428i −0.678183 − 5.72994i −3.82038 − 7.49792i 0.265647 − 6.76116i 18.4629 + 8.51153i 0.562533 + 0.713570i 15.5087 + 1.22056i −23.6210 + 5.67089i 19.8234 + 13.2456i
3.5 −1.78514 + 2.91308i 0.592058 + 5.00227i −3.48336 − 6.83647i −0.124830 + 3.17714i −15.6289 − 7.20503i 3.94234 + 5.00084i 12.5094 + 0.984513i −15.9209 + 3.82226i −9.03242 − 6.03527i
3.6 −1.42017 + 2.31750i 0.204124 + 1.72464i −1.53798 − 3.01846i 0.375968 − 9.56903i −4.28674 − 1.97622i 4.37900 + 5.55474i −1.65913 − 0.130577i 5.81862 − 1.39693i 21.6423 + 14.4609i
3.7 −1.41704 + 2.31239i −0.354174 − 2.99240i −1.52320 − 2.98946i −0.0337247 + 0.858350i 7.42149 + 3.42135i −0.0696112 − 0.0883014i −1.74347 − 0.137214i −0.0777044 + 0.0186552i −1.93705 − 1.29430i
3.8 −1.24072 + 2.02467i 0.105228 + 0.889063i −0.743947 − 1.46008i −0.344120 + 8.75844i −1.93062 − 0.890027i 4.39533 + 5.57544i −5.58988 − 0.439933i 7.97197 − 1.91390i −17.3060 − 11.5635i
3.9 −1.16906 + 1.90774i −0.281713 − 2.38018i −0.456793 − 0.896507i 0.0198603 − 0.505478i 4.87010 + 2.24515i −0.00883067 − 0.0112016i −6.67788 − 0.525561i 3.16543 − 0.759952i 0.941102 + 0.628824i
3.10 −1.05916 + 1.72839i 0.199385 + 1.68459i −0.0495482 − 0.0972439i 0.128367 − 3.26716i −3.12281 − 1.43963i −7.23020 − 9.17146i −7.86285 − 0.618820i 5.95323 − 1.42925i 5.51097 + 3.68231i
3.11 −1.05760 + 1.72584i 0.427270 + 3.60999i −0.0440575 − 0.0864677i −0.202821 + 5.16214i −6.68215 − 3.08051i −4.39210 − 5.57135i −7.87568 − 0.619829i −4.09810 + 0.983868i −8.69453 − 5.80950i
3.12 −0.881751 + 1.43889i 0.676582 + 5.71641i 0.523052 + 1.02655i 0.194023 − 4.93823i −8.82185 − 4.06693i 0.375553 + 0.476387i −8.66774 − 0.682166i −23.4683 + 5.63424i 6.93447 + 4.63346i
3.13 −0.647755 + 1.05704i −0.616836 − 5.21162i 1.11822 + 2.19462i −0.278676 + 7.09277i 5.90845 + 2.72383i −7.03699 − 8.92637i −7.98775 − 0.628649i −18.0292 + 4.32842i −7.31683 − 4.88895i
3.14 −0.592575 + 0.966994i −0.576717 − 4.87266i 1.23203 + 2.41799i −0.0804210 + 2.04685i 5.05358 + 2.32973i 8.01783 + 10.1706i −7.59074 − 0.597404i −14.6588 + 3.51928i −1.93164 − 1.29068i
3.15 −0.400743 + 0.653953i −0.440503 − 3.72179i 1.54890 + 3.03989i 0.348897 − 8.88002i 2.61041 + 1.20341i −1.81000 − 2.29597i −5.66709 − 0.446010i −4.90638 + 1.17792i 5.66729 + 3.78677i
3.16 −0.276611 + 0.451388i 0.0904024 + 0.763806i 1.68872 + 3.31431i 0.133991 − 3.41029i −0.369779 − 0.170471i 2.36738 + 3.00301i −4.07423 − 0.320649i 8.17610 − 1.96291i 1.50230 + 1.00380i
3.17 −0.215376 + 0.351462i 0.435183 + 3.67684i 1.73882 + 3.41263i −0.0424735 + 1.08102i −1.38600 − 0.638953i 5.03097 + 6.38176i −3.21765 − 0.253234i −4.57845 + 1.09919i −0.370790 − 0.247754i
3.18 −0.0865527 + 0.141241i 0.0768910 + 0.649649i 1.80350 + 3.53958i −0.217601 + 5.53833i −0.0984123 − 0.0453687i 0.171889 + 0.218040i −1.31660 − 0.103618i 8.33520 − 2.00110i −0.763406 − 0.510091i
3.19 0.0700640 − 0.114334i −0.130069 − 1.09894i 1.80780 + 3.54800i 0.0660040 − 1.67992i −0.134760 − 0.0621251i −5.88909 − 7.47028i 1.06704 + 0.0839781i 7.56057 − 1.81513i −0.187447 − 0.125248i
3.20 0.511749 − 0.835099i 0.519863 + 4.39230i 1.38046 + 2.70930i −0.281909 + 7.17507i 3.93404 + 1.81362i −6.91546 − 8.77222i 6.87462 + 0.541044i −10.2707 + 2.46577i 5.84763 + 3.90726i
See next 80 embeddings (of 1088 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.e odd 16 1 inner
187.s odd 80 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.3.s.a ✓ 1088
11.c even 5 1 inner 187.3.s.a ✓ 1088
17.e odd 16 1 inner 187.3.s.a ✓ 1088
187.s odd 80 1 inner 187.3.s.a ✓ 1088
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.3.s.a ✓ 1088 1.a even 1 1 trivial
187.3.s.a ✓ 1088 11.c even 5 1 inner
187.3.s.a ✓ 1088 17.e odd 16 1 inner
187.3.s.a ✓ 1088 187.s odd 80 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(187, [\chi])\).