Properties

 Label 1815.2.a.d.1.1 Level $1815$ Weight $2$ Character 1815.1 Self dual yes Analytic conductor $14.493$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1815.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -1.00000 q^{30} +5.00000 q^{32} -2.00000 q^{34} -1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -2.00000 q^{39} -3.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +7.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} +2.00000 q^{68} -8.00000 q^{71} -3.00000 q^{72} -10.0000 q^{73} -10.0000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -12.0000 q^{83} -2.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{94} -4.00000 q^{95} -5.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} +O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000 0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ −1.00000 −0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −2.00000 −0.320256
$$40$$ −3.00000 −0.474342
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 2.00000 0.280056
$$52$$ −2.00000 −0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 2.00000 0.262613
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ −1.00000 −0.115470
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ −4.00000 −0.431331
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ −4.00000 −0.410391
$$96$$ −5.00000 −0.510310
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 2.00000 0.184900
$$118$$ −4.00000 −0.368230
$$119$$ 0 0
$$120$$ 3.00000 0.273861
$$121$$ 0 0
$$122$$ 2.00000 0.181071
$$123$$ 10.0000 0.901670
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 4.00000 0.352180
$$130$$ 2.00000 0.175412
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ −1.00000 −0.0860663
$$136$$ 6.00000 0.514496
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ −8.00000 −0.671345
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 2.00000 0.166091
$$146$$ −10.0000 −0.827606
$$147$$ 7.00000 0.577350
$$148$$ 10.0000 0.821995
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 12.0000 0.973329
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 5.00000 0.395285
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −2.00000 −0.153393
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ −6.00000 −0.449719
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 2.00000 0.143592
$$195$$ −2.00000 −0.143223
$$196$$ 7.00000 0.500000
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ −12.0000 −0.846415
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ −2.00000 −0.140028
$$205$$ −10.0000 −0.698430
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 8.00000 0.548151
$$214$$ 12.0000 0.820303
$$215$$ −4.00000 −0.272798
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ −14.0000 −0.948200
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 10.0000 0.671156
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 2.00000 0.133038
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 8.00000 0.521862
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ −7.00000 −0.447214
$$246$$ 10.0000 0.637577
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 1.00000 0.0632456
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 2.00000 0.125245
$$256$$ −17.0000 −1.06250
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ −2.00000 −0.124035
$$261$$ 2.00000 0.123797
$$262$$ 12.0000 0.741362
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ −10.0000 −0.614295
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ −12.0000 −0.733017
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6.00000 −0.360505 −0.180253 0.983620i $$-0.557691\pi$$
−0.180253 + 0.983620i $$0.557691\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 4.00000 0.236940
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 5.00000 0.294628
$$289$$ −13.0000 −0.764706
$$290$$ 2.00000 0.117444
$$291$$ −2.00000 −0.117242
$$292$$ 10.0000 0.585206
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 7.00000 0.408248
$$295$$ −4.00000 −0.232889
$$296$$ 30.0000 1.74371
$$297$$ 0 0
$$298$$ −22.0000 −1.27443
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ 8.00000 0.460348
$$303$$ 6.00000 0.344691
$$304$$ 4.00000 0.229416
$$305$$ 2.00000 0.114520
$$306$$ −2.00000 −0.114332
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 6.00000 0.339683
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 10.0000 0.560772
$$319$$ 0 0
$$320$$ 7.00000 0.391312
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ −1.00000 −0.0555556
$$325$$ 2.00000 0.110940
$$326$$ −4.00000 −0.221540
$$327$$ 14.0000 0.774202
$$328$$ 30.0000 1.65647
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −10.0000 −0.547997
$$334$$ 0 0
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −2.00000 −0.108625
$$340$$ 2.00000 0.108465
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 2.00000 0.107211
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 4.00000 0.212598
$$355$$ −8.00000 −0.424596
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 20.0000 1.05703
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −3.00000 −0.157895
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ −2.00000 −0.104542
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ −10.0000 −0.519875
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ −24.0000 −1.23771
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 4.00000 0.205196
$$381$$ −8.00000 −0.409852
$$382$$ 16.0000 0.818631
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ −4.00000 −0.203331
$$388$$ −2.00000 −0.101535
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ 0 0
$$392$$ 21.0000 1.06066
$$393$$ −12.0000 −0.605320
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −6.00000 −0.297044
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ 6.00000 0.295958
$$412$$ 16.0000 0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 10.0000 0.490290
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 8.00000 0.388973
$$424$$ 30.0000 1.45693
$$425$$ −2.00000 −0.0970143
$$426$$ 8.00000 0.387601
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ −2.00000 −0.0958927
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ 10.0000 0.477818
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ −4.00000 −0.190261
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ −6.00000 −0.284427
$$446$$ 8.00000 0.378811
$$447$$ 22.0000 1.04056
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ −2.00000 −0.0940721
$$453$$ −8.00000 −0.375873
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ −12.0000 −0.561951
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ −14.0000 −0.645086
$$472$$ 12.0000 0.552345
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 16.0000 0.731823
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ −5.00000 −0.228218
$$481$$ −20.0000 −0.911922
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.00000 0.0908153
$$486$$ −1.00000 −0.0453609
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 4.00000 0.180886
$$490$$ −7.00000 −0.316228
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ −4.00000 −0.180151
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ 32.0000 1.42681 0.713405 0.700752i $$-0.247152\pi$$
0.713405 + 0.700752i $$0.247152\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ −8.00000 −0.354943
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ 0 0
$$512$$ −11.0000 −0.486136
$$513$$ 4.00000 0.176604
$$514$$ 18.0000 0.793946
$$515$$ −16.0000 −0.705044
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ −6.00000 −0.263117
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ −10.0000 −0.434372
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ 6.00000 0.259645
$$535$$ 12.0000 0.518805
$$536$$ −36.0000 −1.55496
$$537$$ −20.0000 −0.863064
$$538$$ 14.0000 0.603583
$$539$$ 0 0
$$540$$ 1.00000 0.0430331
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 10.0000 0.429141
$$544$$ −10.0000 −0.428746
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −6.00000 −0.254916
$$555$$ 10.0000 0.424476
$$556$$ −4.00000 −0.169638
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 2.00000 0.0841406
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 24.0000 1.00702
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 4.00000 0.167542
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ −16.0000 −0.668410
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 2.00000 0.0831172
$$580$$ −2.00000 −0.0830455
$$581$$ 0 0
$$582$$ −2.00000 −0.0829027
$$583$$ 0 0
$$584$$ 30.0000 1.24141
$$585$$ 2.00000 0.0826898
$$586$$ −6.00000 −0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −7.00000 −0.288675
$$589$$ 0 0
$$590$$ −4.00000 −0.164677
$$591$$ 6.00000 0.246807
$$592$$ 10.0000 0.410997
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 3.00000 0.122474
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ −20.0000 −0.811107
$$609$$ 0 0
$$610$$ 2.00000 0.0809776
$$611$$ 16.0000 0.647291
$$612$$ 2.00000 0.0808452
$$613$$ −22.0000 −0.888572 −0.444286 0.895885i $$-0.646543\pi$$
−0.444286 + 0.895885i $$0.646543\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 10.0000 0.403239
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 16.0000 0.643614
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ −2.00000 −0.0794301
$$635$$ 8.00000 0.317470
$$636$$ −10.0000 −0.396526
$$637$$ −14.0000 −0.554700
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ −3.00000 −0.118585
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 0 0
$$645$$ 4.00000 0.157500
$$646$$ 8.00000 0.314756
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ 0 0
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 46.0000 1.80012 0.900060 0.435767i $$-0.143523\pi$$
0.900060 + 0.435767i $$0.143523\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 12.0000 0.468879
$$656$$ 10.0000 0.390434
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 12.0000 0.466393
$$663$$ 4.00000 0.155347
$$664$$ 36.0000 1.39707
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −8.00000 −0.309298
$$670$$ 12.0000 0.463600
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 30.0000 1.15642 0.578208 0.815890i $$-0.303752\pi$$
0.578208 + 0.815890i $$0.303752\pi$$
$$674$$ 14.0000 0.539260
$$675$$ −1.00000 −0.0384900
$$676$$ 9.00000 0.346154
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ 0 0
$$680$$ 6.00000 0.230089
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ −6.00000 −0.228914
$$688$$ 4.00000 0.152499
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ 4.00000 0.151729
$$696$$ 6.00000 0.227429
$$697$$ 20.0000 0.757554
$$698$$ 2.00000 0.0757011
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 40.0000 1.50863
$$704$$ 0 0
$$705$$ −8.00000 −0.301297
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ −4.00000 −0.150329
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 0 0
$$712$$ 18.0000 0.674579
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ −16.0000 −0.597531
$$718$$ 24.0000 0.895672
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ −14.0000 −0.520666
$$724$$ 10.0000 0.371647
$$725$$ 2.00000 0.0742781
$$726$$ 0 0
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −10.0000 −0.370117
$$731$$ 8.00000 0.295891
$$732$$ 2.00000 0.0739221
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 7.00000 0.258199
$$736$$ 0 0
$$737$$ 0 0
$$738$$ −10.0000 −0.368105
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 10.0000 0.367607
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ −22.0000 −0.806018
$$746$$ 26.0000 0.951928
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −12.0000 −0.437304
$$754$$ 4.00000 0.145671
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 12.0000 0.435286
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ −2.00000 −0.0723102
$$766$$ −24.0000 −0.867155
$$767$$ −8.00000 −0.288863
$$768$$ 17.0000 0.613435
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 2.00000 0.0719816
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ 40.0000 1.43315
$$780$$ 2.00000 0.0716115
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 7.00000 0.250000
$$785$$ 14.0000 0.499681
$$786$$ −12.0000 −0.428026
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ −2.00000 −0.0709773
$$795$$ 10.0000 0.354663
$$796$$ 8.00000 0.283552
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 5.00000 0.176777
$$801$$ −6.00000 −0.212000
$$802$$ 18.0000 0.635602
$$803$$ 0 0
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ 18.0000 0.633238
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ 0 0
$$815$$ −4.00000 −0.140114
$$816$$ −2.00000 −0.0700140
$$817$$ 16.0000 0.559769
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ 10.0000 0.349215
$$821$$ −54.0000 −1.88461 −0.942306 0.334751i $$-0.891348\pi$$
−0.942306 + 0.334751i $$0.891348\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 48.0000 1.67216
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ 6.00000 0.208138
$$832$$ 14.0000 0.485363
$$833$$ 14.0000 0.485071
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 4.00000 0.138178
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −26.0000 −0.896019
$$843$$ −6.00000 −0.206651
$$844$$ 20.0000 0.688428
$$845$$ −9.00000 −0.309609
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ 10.0000 0.343401
$$849$$ −12.0000 −0.411839
$$850$$ −2.00000 −0.0685994
$$851$$ 0 0
$$852$$ −8.00000 −0.274075
$$853$$ −6.00000 −0.205436 −0.102718 0.994711i $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ −36.0000 −1.23045
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −56.0000 −1.90626 −0.953131 0.302558i $$-0.902160\pi$$
−0.953131 + 0.302558i $$0.902160\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 18.0000 0.612018
$$866$$ −14.0000 −0.475739
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −2.00000 −0.0678064
$$871$$ 24.0000 0.813209
$$872$$ 42.0000 1.42230
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ −30.0000 −1.01303 −0.506514 0.862232i $$-0.669066\pi$$
−0.506514 + 0.862232i $$0.669066\pi$$
$$878$$ −40.0000 −1.34993
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 4.00000 0.134459
$$886$$ −12.0000 −0.403148
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ −30.0000 −1.00673
$$889$$ 0 0
$$890$$ −6.00000 −0.201120
$$891$$ 0 0
$$892$$ −8.00000 −0.267860
$$893$$ −32.0000 −1.07084
$$894$$ 22.0000 0.735790
$$895$$ 20.0000 0.668526
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 2.00000 0.0667409
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ 20.0000 0.666297
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ −10.0000 −0.332411
$$906$$ −8.00000 −0.265782
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 0 0
$$914$$ −10.0000 −0.330771
$$915$$ −2.00000 −0.0661180
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ 2.00000 0.0660098
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ 18.0000 0.592798
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ 24.0000 0.788689
$$927$$ −16.0000 −0.525509
$$928$$ 10.0000 0.328266
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ −6.00000 −0.196537
$$933$$ 24.0000 0.785725
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ 54.0000 1.76410 0.882052 0.471153i $$-0.156162\pi$$
0.882052 + 0.471153i $$0.156162\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ −8.00000 −0.260931
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ −14.0000 −0.456145
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ −20.0000 −0.649227
$$950$$ −4.00000 −0.129777
$$951$$ 2.00000 0.0648544
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 16.0000 0.517748
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −7.00000 −0.225924
$$961$$ −31.0000 −1.00000
$$962$$ −20.0000 −0.644826
$$963$$ 12.0000 0.386695
$$964$$ −14.0000 −0.450910
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 2.00000 0.0642161
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 32.0000 1.02535
$$975$$ −2.00000 −0.0640513
$$976$$ −2.00000 −0.0640184
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 0 0
$$980$$ 7.00000 0.223607
$$981$$ −14.0000 −0.446986
$$982$$ −28.0000 −0.893516
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ −30.0000 −0.956365
$$985$$ −6.00000 −0.191176
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ −12.0000 −0.380808
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ −12.0000 −0.380235
$$997$$ −54.0000 −1.71020 −0.855099 0.518465i $$-0.826503\pi$$
−0.855099 + 0.518465i $$0.826503\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 10.0000 0.316386
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1815.2.a.d.1.1 1
3.2 odd 2 5445.2.a.c.1.1 1
5.4 even 2 9075.2.a.g.1.1 1
11.10 odd 2 15.2.a.a.1.1 1
33.32 even 2 45.2.a.a.1.1 1
44.43 even 2 240.2.a.d.1.1 1
55.32 even 4 75.2.b.b.49.1 2
55.43 even 4 75.2.b.b.49.2 2
55.54 odd 2 75.2.a.b.1.1 1
77.10 even 6 735.2.i.d.226.1 2
77.32 odd 6 735.2.i.e.226.1 2
77.54 even 6 735.2.i.d.361.1 2
77.65 odd 6 735.2.i.e.361.1 2
77.76 even 2 735.2.a.c.1.1 1
88.21 odd 2 960.2.a.l.1.1 1
88.43 even 2 960.2.a.a.1.1 1
99.32 even 6 405.2.e.c.136.1 2
99.43 odd 6 405.2.e.f.271.1 2
99.65 even 6 405.2.e.c.271.1 2
99.76 odd 6 405.2.e.f.136.1 2
132.131 odd 2 720.2.a.c.1.1 1
143.142 odd 2 2535.2.a.j.1.1 1
165.32 odd 4 225.2.b.b.199.2 2
165.98 odd 4 225.2.b.b.199.1 2
165.164 even 2 225.2.a.b.1.1 1
176.21 odd 4 3840.2.k.m.1921.2 2
176.43 even 4 3840.2.k.r.1921.1 2
176.109 odd 4 3840.2.k.m.1921.1 2
176.131 even 4 3840.2.k.r.1921.2 2
187.186 odd 2 4335.2.a.c.1.1 1
209.208 even 2 5415.2.a.j.1.1 1
220.43 odd 4 1200.2.f.h.49.2 2
220.87 odd 4 1200.2.f.h.49.1 2
220.219 even 2 1200.2.a.e.1.1 1
231.230 odd 2 2205.2.a.i.1.1 1
253.252 even 2 7935.2.a.d.1.1 1
264.131 odd 2 2880.2.a.bc.1.1 1
264.197 even 2 2880.2.a.y.1.1 1
385.384 even 2 3675.2.a.j.1.1 1
429.428 even 2 7605.2.a.g.1.1 1
440.43 odd 4 4800.2.f.c.3649.1 2
440.109 odd 2 4800.2.a.t.1.1 1
440.197 even 4 4800.2.f.bf.3649.1 2
440.219 even 2 4800.2.a.bz.1.1 1
440.307 odd 4 4800.2.f.c.3649.2 2
440.373 even 4 4800.2.f.bf.3649.2 2
660.263 even 4 3600.2.f.e.2449.1 2
660.527 even 4 3600.2.f.e.2449.2 2
660.659 odd 2 3600.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 11.10 odd 2
45.2.a.a.1.1 1 33.32 even 2
75.2.a.b.1.1 1 55.54 odd 2
75.2.b.b.49.1 2 55.32 even 4
75.2.b.b.49.2 2 55.43 even 4
225.2.a.b.1.1 1 165.164 even 2
225.2.b.b.199.1 2 165.98 odd 4
225.2.b.b.199.2 2 165.32 odd 4
240.2.a.d.1.1 1 44.43 even 2
405.2.e.c.136.1 2 99.32 even 6
405.2.e.c.271.1 2 99.65 even 6
405.2.e.f.136.1 2 99.76 odd 6
405.2.e.f.271.1 2 99.43 odd 6
720.2.a.c.1.1 1 132.131 odd 2
735.2.a.c.1.1 1 77.76 even 2
735.2.i.d.226.1 2 77.10 even 6
735.2.i.d.361.1 2 77.54 even 6
735.2.i.e.226.1 2 77.32 odd 6
735.2.i.e.361.1 2 77.65 odd 6
960.2.a.a.1.1 1 88.43 even 2
960.2.a.l.1.1 1 88.21 odd 2
1200.2.a.e.1.1 1 220.219 even 2
1200.2.f.h.49.1 2 220.87 odd 4
1200.2.f.h.49.2 2 220.43 odd 4
1815.2.a.d.1.1 1 1.1 even 1 trivial
2205.2.a.i.1.1 1 231.230 odd 2
2535.2.a.j.1.1 1 143.142 odd 2
2880.2.a.y.1.1 1 264.197 even 2
2880.2.a.bc.1.1 1 264.131 odd 2
3600.2.a.u.1.1 1 660.659 odd 2
3600.2.f.e.2449.1 2 660.263 even 4
3600.2.f.e.2449.2 2 660.527 even 4
3675.2.a.j.1.1 1 385.384 even 2
3840.2.k.m.1921.1 2 176.109 odd 4
3840.2.k.m.1921.2 2 176.21 odd 4
3840.2.k.r.1921.1 2 176.43 even 4
3840.2.k.r.1921.2 2 176.131 even 4
4335.2.a.c.1.1 1 187.186 odd 2
4800.2.a.t.1.1 1 440.109 odd 2
4800.2.a.bz.1.1 1 440.219 even 2
4800.2.f.c.3649.1 2 440.43 odd 4
4800.2.f.c.3649.2 2 440.307 odd 4
4800.2.f.bf.3649.1 2 440.197 even 4
4800.2.f.bf.3649.2 2 440.373 even 4
5415.2.a.j.1.1 1 209.208 even 2
5445.2.a.c.1.1 1 3.2 odd 2
7605.2.a.g.1.1 1 429.428 even 2
7935.2.a.d.1.1 1 253.252 even 2
9075.2.a.g.1.1 1 5.4 even 2