# 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 $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$547$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 547.o (of order $$273$$, degree $$144$$, minimal)

## Newform invariants

 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.

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

## 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.

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 545.45 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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