Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [547,2,Mod(4,547)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(547, base_ring=CyclotomicField(546))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("547.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 547 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 547.o (of order \(273\), degree \(144\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(4.36781699056\) |
Analytic rank: | \(0\) |
Dimension: | \(6480\) |
Relative dimension: | \(45\) over \(\Q(\zeta_{273})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{273}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −2.82015 | − | 0.0324547i | −0.0795262 | + | 0.348427i | 5.95271 | + | 0.137028i | 0.114891 | − | 0.215879i | 0.235584 | − | 0.980034i | −0.421000 | − | 1.03743i | −11.1458 | − | 0.384938i | 2.58783 | + | 1.24623i | −0.331017 | + | 0.605083i |
4.2 | −2.62113 | − | 0.0301644i | 0.701785 | − | 3.07472i | 4.86994 | + | 0.112103i | −0.110921 | + | 0.208419i | −1.93222 | + | 8.03808i | 0.862757 | + | 2.12600i | −7.52187 | − | 0.259781i | −6.25851 | − | 3.01394i | 0.297025 | − | 0.542947i |
4.3 | −2.54982 | − | 0.0293438i | 0.292466 | − | 1.28138i | 4.50126 | + | 0.103616i | −1.61382 | + | 3.03235i | −0.783337 | + | 3.25870i | 0.230485 | + | 0.567960i | −6.37743 | − | 0.220256i | 1.14652 | + | 0.552133i | 4.20394 | − | 7.68459i |
4.4 | −2.52384 | − | 0.0290448i | −0.715155 | + | 3.13330i | 4.36948 | + | 0.100583i | 1.84693 | − | 3.47035i | 1.89595 | − | 7.88719i | 1.46224 | + | 3.60324i | −5.97995 | − | 0.206528i | −6.60321 | − | 3.17994i | −4.76215 | + | 8.70498i |
4.5 | −2.46368 | − | 0.0283525i | −0.550970 | + | 2.41396i | 4.06944 | + | 0.0936762i | −1.20693 | + | 2.26781i | 1.42585 | − | 5.93159i | 0.105582 | + | 0.260175i | −5.09840 | − | 0.176082i | −2.82071 | − | 1.35838i | 3.03779 | − | 5.55293i |
4.6 | −2.28601 | − | 0.0263078i | 0.534963 | − | 2.34383i | 3.22568 | + | 0.0742532i | 1.59850 | − | 3.00356i | −1.28459 | + | 5.34393i | −1.61107 | − | 3.96999i | −2.80238 | − | 0.0967850i | −2.50442 | − | 1.20607i | −3.73320 | + | 6.82410i |
4.7 | −2.14370 | − | 0.0246701i | −0.0167644 | + | 0.0734497i | 2.59538 | + | 0.0597440i | 0.940362 | − | 1.76693i | 0.0377499 | − | 0.157041i | −0.448666 | − | 1.10560i | −1.27711 | − | 0.0441071i | 2.69779 | + | 1.29919i | −2.05945 | + | 3.76457i |
4.8 | −2.07856 | − | 0.0239204i | 0.521570 | − | 2.28515i | 2.32035 | + | 0.0534131i | −0.897784 | + | 1.68692i | −1.13877 | + | 4.73733i | −0.603206 | − | 1.48642i | −0.666796 | − | 0.0230290i | −2.24696 | − | 1.08208i | 1.90645 | − | 3.48489i |
4.9 | −1.95847 | − | 0.0225384i | 0.189262 | − | 0.829210i | 1.83561 | + | 0.0422546i | 0.0364523 | − | 0.0684934i | −0.389352 | + | 1.61971i | 1.35863 | + | 3.34794i | 0.320834 | + | 0.0110806i | 2.05114 | + | 0.987776i | −0.0729343 | + | 0.133320i |
4.10 | −1.91587 | − | 0.0220482i | −0.216762 | + | 0.949696i | 1.67061 | + | 0.0384563i | −0.340710 | + | 0.640190i | 0.436227 | − | 1.81472i | 1.64679 | + | 4.05800i | 0.629896 | + | 0.0217545i | 1.84797 | + | 0.889935i | 0.666871 | − | 1.21901i |
4.11 | −1.82406 | − | 0.0209916i | −0.585104 | + | 2.56351i | 1.32728 | + | 0.0305533i | −1.34163 | + | 2.52090i | 1.12108 | − | 4.66371i | −0.483254 | − | 1.19083i | 1.22578 | + | 0.0423345i | −3.52632 | − | 1.69819i | 2.50012 | − | 4.57011i |
4.12 | −1.48739 | − | 0.0171172i | −0.358018 | + | 1.56858i | 0.212570 | + | 0.00489324i | 1.55631 | − | 2.92429i | 0.559362 | − | 2.32696i | −0.286333 | − | 0.705581i | 2.65712 | + | 0.0917681i | 0.370646 | + | 0.178494i | −2.36490 | + | 4.32292i |
4.13 | −1.37136 | − | 0.0157819i | 0.455404 | − | 1.99525i | −0.119088 | − | 0.00274133i | −0.496687 | + | 0.933269i | −0.656012 | + | 2.72903i | 0.132038 | + | 0.325368i | 2.90454 | + | 0.100313i | −1.07074 | − | 0.515640i | 0.695866 | − | 1.27201i |
4.14 | −1.30259 | − | 0.0149904i | −0.660348 | + | 2.89317i | −0.302966 | − | 0.00697409i | 0.484844 | − | 0.911015i | 0.903530 | − | 3.75871i | −1.50505 | − | 3.70874i | 2.99833 | + | 0.103552i | −5.23149 | − | 2.51935i | −0.645207 | + | 1.17941i |
4.15 | −1.17840 | − | 0.0135612i | 0.739679 | − | 3.24075i | −0.611031 | − | 0.0140656i | 1.18492 | − | 2.22646i | −0.915585 | + | 3.80886i | 1.56661 | + | 3.86044i | 3.07540 | + | 0.106214i | −7.25240 | − | 3.49257i | −1.42651 | + | 2.60758i |
4.16 | −0.956173 | − | 0.0110038i | −0.143432 | + | 0.628415i | −1.08532 | − | 0.0249835i | −1.82781 | + | 3.43443i | 0.144060 | − | 0.599296i | 0.431500 | + | 1.06330i | 2.94882 | + | 0.101842i | 2.32857 | + | 1.12138i | 1.78550 | − | 3.26380i |
4.17 | −0.956113 | − | 0.0110031i | 0.269915 | − | 1.18258i | −1.08544 | − | 0.0249862i | 2.02066 | − | 3.79679i | −0.271082 | + | 1.12771i | 0.434254 | + | 1.07009i | 2.94874 | + | 0.101840i | 1.37727 | + | 0.663260i | −1.97375 | + | 3.60793i |
4.18 | −0.866936 | − | 0.00997685i | −0.129306 | + | 0.566525i | −1.24799 | − | 0.0287280i | −0.554911 | + | 1.04267i | 0.117752 | − | 0.489851i | −1.27973 | − | 3.15351i | 2.81460 | + | 0.0972069i | 2.39868 | + | 1.15514i | 0.491475 | − | 0.898393i |
4.19 | −0.764594 | − | 0.00879909i | 0.320949 | − | 1.40617i | −1.41494 | − | 0.0325711i | 0.427309 | − | 0.802909i | −0.257769 | + | 1.07233i | −1.22649 | − | 3.02231i | 2.60995 | + | 0.0901391i | 0.828601 | + | 0.399033i | −0.333783 | + | 0.610140i |
4.20 | −0.418462 | − | 0.00481573i | −0.456265 | + | 1.99903i | −1.82438 | − | 0.0419962i | 0.863080 | − | 1.62172i | 0.200556 | − | 0.834318i | 1.13544 | + | 2.79794i | 1.59971 | + | 0.0552488i | −1.08502 | − | 0.522518i | −0.368976 | + | 0.674470i |
See next 80 embeddings (of 6480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
547.o | even | 273 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 547.2.o.a | ✓ | 6480 |
547.o | even | 273 | 1 | inner | 547.2.o.a | ✓ | 6480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
547.2.o.a | ✓ | 6480 | 1.a | even | 1 | 1 | trivial |
547.2.o.a | ✓ | 6480 | 547.o | even | 273 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(547, [\chi])\).