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Properties

Label 820.2.k.c
Level $820$
Weight $2$
Character orbit 820.k
Analytic conductor $6.548$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [820,2,Mod(83,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("820.83"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [108] 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: \(6.54773296574\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 12 q^{2} - 4 q^{6} - 24 q^{8} + 4 q^{10} - 16 q^{13} + 52 q^{16} - 8 q^{17} + 18 q^{18} + 38 q^{20} + 72 q^{21} + 10 q^{22} - 12 q^{25} + 24 q^{26} - 58 q^{28} - 70 q^{30} - 38 q^{32} + 8 q^{33}+ \cdots + 122 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 −1.41421 + 0.00282635i 1.90181 − 1.90181i 1.99998 − 0.00799412i −1.72661 + 1.42085i −2.68419 + 2.69494i −0.433364 − 0.433364i −2.82838 + 0.0169580i − 4.23379i 2.43777 − 2.01427i
83.2 −1.40937 + 0.116919i −0.715888 + 0.715888i 1.97266 − 0.329564i −1.84456 + 1.26397i 0.925252 − 1.09265i 0.636079 + 0.636079i −2.74168 + 0.695119i 1.97501i 2.45189 − 1.99706i
83.3 −1.40383 + 0.171082i −0.252309 + 0.252309i 1.94146 − 0.480339i 2.11789 + 0.717329i 0.311033 − 0.397364i −2.38627 − 2.38627i −2.64330 + 1.00646i 2.87268i −3.09587 − 0.644674i
83.4 −1.39987 − 0.200915i 0.926056 − 0.926056i 1.91927 + 0.562509i 1.95896 − 1.07818i −1.48242 + 1.11030i 0.815570 + 0.815570i −2.57371 − 1.17305i 1.28484i −2.95891 + 1.11572i
83.5 −1.36592 − 0.366410i −1.28172 + 1.28172i 1.73149 + 1.00097i −0.124111 − 2.23262i 2.22036 − 1.28109i 1.75500 + 1.75500i −1.99831 − 2.00169i − 0.285599i −0.648529 + 3.09506i
83.6 −1.35506 − 0.404723i −2.33700 + 2.33700i 1.67240 + 1.09685i −1.95358 − 1.08790i 4.11263 − 2.22095i −3.18120 − 3.18120i −1.82229 − 2.16316i − 7.92318i 2.20693 + 2.26483i
83.7 −1.35278 + 0.412299i 0.638078 − 0.638078i 1.66002 − 1.11550i −0.848351 − 2.06889i −0.600099 + 1.12626i −1.94968 − 1.94968i −1.78572 + 2.19345i 2.18571i 2.00063 + 2.44897i
83.8 −1.35150 + 0.416481i 2.20659 − 2.20659i 1.65309 − 1.12575i 2.01642 + 0.966467i −2.06319 + 3.90120i 2.48515 + 2.48515i −1.76529 + 2.20992i − 6.73806i −3.12770 − 0.466377i
83.9 −1.35102 − 0.418033i −1.41800 + 1.41800i 1.65050 + 1.12954i 0.647982 + 2.14012i 2.50851 − 1.32297i −1.07473 − 1.07473i −1.75767 − 2.21599i − 1.02144i 0.0192065 − 3.16222i
83.10 −1.20951 − 0.732860i 0.993723 − 0.993723i 0.925834 + 1.77280i −1.63476 − 1.52564i −1.93018 + 0.473660i 2.18364 + 2.18364i 0.179410 − 2.82273i 1.02503i 0.859183 + 3.04332i
83.11 −1.12977 − 0.850658i 1.73461 − 1.73461i 0.552763 + 1.92210i 2.10007 + 0.767920i −3.43527 + 0.484152i −1.19221 − 1.19221i 1.01055 − 2.64174i − 3.01775i −1.71936 − 2.65401i
83.12 −0.958818 − 1.03955i −1.58218 + 1.58218i −0.161335 + 1.99348i −2.15223 + 0.606551i 3.16177 + 0.127734i −0.113172 − 0.113172i 2.22702 − 1.74367i − 2.00656i 2.69414 + 1.65578i
83.13 −0.896192 − 1.09400i 1.47545 − 1.47545i −0.393681 + 1.96087i 1.51870 − 1.64120i −2.93643 − 0.291859i 2.76703 + 2.76703i 2.49801 − 1.32663i − 1.35390i −3.15653 − 0.190630i
83.14 −0.866971 − 1.11730i −0.278914 + 0.278914i −0.496724 + 1.93733i −0.644595 + 2.14114i 0.553441 + 0.0698207i 1.69878 + 1.69878i 2.59523 − 1.12462i 2.84441i 2.95115 − 1.13610i
83.15 −0.788955 − 1.17369i −2.13185 + 2.13185i −0.755100 + 1.85198i 0.905998 − 2.04430i 4.18407 + 0.820199i 0.682120 + 0.682120i 2.76939 − 0.574874i − 6.08957i −3.11417 + 0.549501i
83.16 −0.607244 − 1.27721i 1.25708 − 1.25708i −1.26251 + 1.55115i −0.966894 + 2.01621i −2.36890 − 0.842196i −2.94147 − 2.94147i 2.74779 + 0.670560i − 0.160500i 3.16226 + 0.0105901i
83.17 −0.530735 − 1.31085i −0.00659099 + 0.00659099i −1.43664 + 1.39143i −1.13558 − 1.92625i 0.0121378 + 0.00514171i −1.72131 − 1.72131i 2.58642 + 1.14474i 2.99991i −1.92233 + 2.51091i
83.18 −0.416481 + 1.35150i −2.20659 + 2.20659i −1.65309 − 1.12575i 2.01642 + 0.966467i −2.06319 − 3.90120i −2.48515 − 2.48515i 2.20992 − 1.76529i − 6.73806i −2.14598 + 2.32267i
83.19 −0.412299 + 1.35278i −0.638078 + 0.638078i −1.66002 − 1.11550i −0.848351 − 2.06889i −0.600099 − 1.12626i 1.94968 + 1.94968i 2.19345 − 1.78572i 2.18571i 3.14852 − 0.294630i
83.20 −0.238964 − 1.39388i −0.444020 + 0.444020i −1.88579 + 0.666174i 1.32043 − 1.80457i 0.725014 + 0.512805i −0.174110 − 0.174110i 1.37920 + 2.46937i 2.60569i −2.83089 − 1.40929i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.54
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.k.c ✓ 108
4.b odd 2 1 inner 820.2.k.c ✓ 108
5.c odd 4 1 inner 820.2.k.c ✓ 108
20.e even 4 1 inner 820.2.k.c ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.k.c ✓ 108 1.a even 1 1 trivial
820.2.k.c ✓ 108 4.b odd 2 1 inner
820.2.k.c ✓ 108 5.c odd 4 1 inner
820.2.k.c ✓ 108 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{108} + 688 T_{3}^{104} + 212112 T_{3}^{100} + 38811208 T_{3}^{96} + 4706920982 T_{3}^{92} + \cdots + 25600000000 \) acting on \(S_{2}^{\mathrm{new}}(820, [\chi])\). Copy content Toggle raw display