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Properties

Label 547.2.o.a
Level $547$
Weight $2$
Character orbit 547.o
Analytic conductor $4.368$
Analytic rank $0$
Dimension $6480$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9} - 152 q^{10} - 161 q^{11} - 99 q^{12} - 158 q^{13} - 174 q^{14} - 71 q^{15} - 235 q^{16} - 146 q^{17} + 33 q^{18} - 144 q^{19} - 138 q^{20} - 98 q^{21} - 226 q^{22} - 115 q^{23} - 123 q^{24} - 208 q^{25} - 65 q^{26} + 131 q^{27} - 48 q^{28} - 207 q^{29} - 56 q^{30} - 215 q^{31} - 151 q^{32} - 63 q^{33} - 70 q^{34} - 192 q^{35} - 17 q^{36} - 124 q^{37} - 61 q^{38} - 268 q^{39} - 51 q^{40} + 533 q^{41} + 292 q^{42} - 166 q^{43} - 20 q^{44} + 51 q^{45} - 233 q^{46} - 96 q^{47} - 211 q^{48} - 213 q^{49} + 12 q^{50} - 213 q^{51} - 279 q^{52} - 178 q^{53} + 173 q^{54} - 233 q^{55} + 170 q^{56} - 22 q^{57} + 36 q^{58} - 133 q^{59} + 225 q^{60} - 117 q^{61} - 176 q^{62} + 202 q^{63} + 154 q^{64} - 320 q^{65} - 124 q^{66} - 78 q^{67} + 441 q^{68} + 203 q^{69} - 522 q^{70} - 195 q^{71} + 564 q^{72} - 254 q^{73} + 22 q^{74} - 326 q^{75} - 186 q^{76} - 182 q^{77} - 100 q^{78} - 268 q^{79} - 299 q^{80} - 776 q^{81} - 242 q^{82} + 78 q^{83} + 162 q^{84} - 144 q^{85} - 132 q^{86} - 84 q^{87} + 333 q^{88} - 167 q^{89} - 134 q^{90} - 268 q^{91} - 261 q^{92} - 21 q^{93} + 297 q^{94} - 65 q^{95} - 567 q^{96} + 427 q^{97} - 237 q^{98} + 237 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.45
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(547, [\chi])\).