# Properties

 Label 154.4.a.d.1.1 Level $154$ Weight $4$ Character 154.1 Self dual yes Analytic conductor $9.086$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 154.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.08629414088$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 154.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} +18.0000 q^{5} -4.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} -23.0000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} +18.0000 q^{5} -4.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} -23.0000 q^{9} +36.0000 q^{10} -11.0000 q^{11} -8.00000 q^{12} +56.0000 q^{13} +14.0000 q^{14} -36.0000 q^{15} +16.0000 q^{16} +36.0000 q^{17} -46.0000 q^{18} -28.0000 q^{19} +72.0000 q^{20} -14.0000 q^{21} -22.0000 q^{22} +180.000 q^{23} -16.0000 q^{24} +199.000 q^{25} +112.000 q^{26} +100.000 q^{27} +28.0000 q^{28} -54.0000 q^{29} -72.0000 q^{30} -334.000 q^{31} +32.0000 q^{32} +22.0000 q^{33} +72.0000 q^{34} +126.000 q^{35} -92.0000 q^{36} +386.000 q^{37} -56.0000 q^{38} -112.000 q^{39} +144.000 q^{40} -444.000 q^{41} -28.0000 q^{42} -316.000 q^{43} -44.0000 q^{44} -414.000 q^{45} +360.000 q^{46} -402.000 q^{47} -32.0000 q^{48} +49.0000 q^{49} +398.000 q^{50} -72.0000 q^{51} +224.000 q^{52} -486.000 q^{53} +200.000 q^{54} -198.000 q^{55} +56.0000 q^{56} +56.0000 q^{57} -108.000 q^{58} -282.000 q^{59} -144.000 q^{60} +380.000 q^{61} -668.000 q^{62} -161.000 q^{63} +64.0000 q^{64} +1008.00 q^{65} +44.0000 q^{66} +176.000 q^{67} +144.000 q^{68} -360.000 q^{69} +252.000 q^{70} -324.000 q^{71} -184.000 q^{72} +800.000 q^{73} +772.000 q^{74} -398.000 q^{75} -112.000 q^{76} -77.0000 q^{77} -224.000 q^{78} -1144.00 q^{79} +288.000 q^{80} +421.000 q^{81} -888.000 q^{82} +468.000 q^{83} -56.0000 q^{84} +648.000 q^{85} -632.000 q^{86} +108.000 q^{87} -88.0000 q^{88} -870.000 q^{89} -828.000 q^{90} +392.000 q^{91} +720.000 q^{92} +668.000 q^{93} -804.000 q^{94} -504.000 q^{95} -64.0000 q^{96} -1330.00 q^{97} +98.0000 q^{98} +253.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ −2.00000 −0.384900 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 18.0000 1.60997 0.804984 0.593296i $$-0.202174\pi$$
0.804984 + 0.593296i $$0.202174\pi$$
$$6$$ −4.00000 −0.272166
$$7$$ 7.00000 0.377964
$$8$$ 8.00000 0.353553
$$9$$ −23.0000 −0.851852
$$10$$ 36.0000 1.13842
$$11$$ −11.0000 −0.301511
$$12$$ −8.00000 −0.192450
$$13$$ 56.0000 1.19474 0.597369 0.801966i $$-0.296213\pi$$
0.597369 + 0.801966i $$0.296213\pi$$
$$14$$ 14.0000 0.267261
$$15$$ −36.0000 −0.619677
$$16$$ 16.0000 0.250000
$$17$$ 36.0000 0.513605 0.256802 0.966464i $$-0.417331\pi$$
0.256802 + 0.966464i $$0.417331\pi$$
$$18$$ −46.0000 −0.602350
$$19$$ −28.0000 −0.338086 −0.169043 0.985609i $$-0.554068\pi$$
−0.169043 + 0.985609i $$0.554068\pi$$
$$20$$ 72.0000 0.804984
$$21$$ −14.0000 −0.145479
$$22$$ −22.0000 −0.213201
$$23$$ 180.000 1.63185 0.815926 0.578156i $$-0.196228\pi$$
0.815926 + 0.578156i $$0.196228\pi$$
$$24$$ −16.0000 −0.136083
$$25$$ 199.000 1.59200
$$26$$ 112.000 0.844808
$$27$$ 100.000 0.712778
$$28$$ 28.0000 0.188982
$$29$$ −54.0000 −0.345778 −0.172889 0.984941i $$-0.555310\pi$$
−0.172889 + 0.984941i $$0.555310\pi$$
$$30$$ −72.0000 −0.438178
$$31$$ −334.000 −1.93510 −0.967551 0.252675i $$-0.918690\pi$$
−0.967551 + 0.252675i $$0.918690\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 22.0000 0.116052
$$34$$ 72.0000 0.363173
$$35$$ 126.000 0.608511
$$36$$ −92.0000 −0.425926
$$37$$ 386.000 1.71508 0.857541 0.514416i $$-0.171991\pi$$
0.857541 + 0.514416i $$0.171991\pi$$
$$38$$ −56.0000 −0.239063
$$39$$ −112.000 −0.459855
$$40$$ 144.000 0.569210
$$41$$ −444.000 −1.69125 −0.845624 0.533779i $$-0.820771\pi$$
−0.845624 + 0.533779i $$0.820771\pi$$
$$42$$ −28.0000 −0.102869
$$43$$ −316.000 −1.12069 −0.560344 0.828260i $$-0.689331\pi$$
−0.560344 + 0.828260i $$0.689331\pi$$
$$44$$ −44.0000 −0.150756
$$45$$ −414.000 −1.37146
$$46$$ 360.000 1.15389
$$47$$ −402.000 −1.24761 −0.623806 0.781580i $$-0.714414\pi$$
−0.623806 + 0.781580i $$0.714414\pi$$
$$48$$ −32.0000 −0.0962250
$$49$$ 49.0000 0.142857
$$50$$ 398.000 1.12571
$$51$$ −72.0000 −0.197687
$$52$$ 224.000 0.597369
$$53$$ −486.000 −1.25957 −0.629785 0.776769i $$-0.716857\pi$$
−0.629785 + 0.776769i $$0.716857\pi$$
$$54$$ 200.000 0.504010
$$55$$ −198.000 −0.485424
$$56$$ 56.0000 0.133631
$$57$$ 56.0000 0.130129
$$58$$ −108.000 −0.244502
$$59$$ −282.000 −0.622259 −0.311129 0.950368i $$-0.600707\pi$$
−0.311129 + 0.950368i $$0.600707\pi$$
$$60$$ −144.000 −0.309839
$$61$$ 380.000 0.797607 0.398803 0.917036i $$-0.369426\pi$$
0.398803 + 0.917036i $$0.369426\pi$$
$$62$$ −668.000 −1.36832
$$63$$ −161.000 −0.321970
$$64$$ 64.0000 0.125000
$$65$$ 1008.00 1.92349
$$66$$ 44.0000 0.0820610
$$67$$ 176.000 0.320923 0.160461 0.987042i $$-0.448702\pi$$
0.160461 + 0.987042i $$0.448702\pi$$
$$68$$ 144.000 0.256802
$$69$$ −360.000 −0.628100
$$70$$ 252.000 0.430282
$$71$$ −324.000 −0.541574 −0.270787 0.962639i $$-0.587284\pi$$
−0.270787 + 0.962639i $$0.587284\pi$$
$$72$$ −184.000 −0.301175
$$73$$ 800.000 1.28264 0.641321 0.767272i $$-0.278387\pi$$
0.641321 + 0.767272i $$0.278387\pi$$
$$74$$ 772.000 1.21275
$$75$$ −398.000 −0.612761
$$76$$ −112.000 −0.169043
$$77$$ −77.0000 −0.113961
$$78$$ −224.000 −0.325167
$$79$$ −1144.00 −1.62924 −0.814621 0.579994i $$-0.803055\pi$$
−0.814621 + 0.579994i $$0.803055\pi$$
$$80$$ 288.000 0.402492
$$81$$ 421.000 0.577503
$$82$$ −888.000 −1.19589
$$83$$ 468.000 0.618912 0.309456 0.950914i $$-0.399853\pi$$
0.309456 + 0.950914i $$0.399853\pi$$
$$84$$ −56.0000 −0.0727393
$$85$$ 648.000 0.826888
$$86$$ −632.000 −0.792445
$$87$$ 108.000 0.133090
$$88$$ −88.0000 −0.106600
$$89$$ −870.000 −1.03618 −0.518089 0.855327i $$-0.673356\pi$$
−0.518089 + 0.855327i $$0.673356\pi$$
$$90$$ −828.000 −0.969765
$$91$$ 392.000 0.451569
$$92$$ 720.000 0.815926
$$93$$ 668.000 0.744821
$$94$$ −804.000 −0.882194
$$95$$ −504.000 −0.544309
$$96$$ −64.0000 −0.0680414
$$97$$ −1330.00 −1.39218 −0.696088 0.717957i $$-0.745078\pi$$
−0.696088 + 0.717957i $$0.745078\pi$$
$$98$$ 98.0000 0.101015
$$99$$ 253.000 0.256843
$$100$$ 796.000 0.796000
$$101$$ −120.000 −0.118222 −0.0591111 0.998251i $$-0.518827\pi$$
−0.0591111 + 0.998251i $$0.518827\pi$$
$$102$$ −144.000 −0.139786
$$103$$ −1210.00 −1.15752 −0.578761 0.815497i $$-0.696464\pi$$
−0.578761 + 0.815497i $$0.696464\pi$$
$$104$$ 448.000 0.422404
$$105$$ −252.000 −0.234216
$$106$$ −972.000 −0.890651
$$107$$ 1236.00 1.11672 0.558358 0.829600i $$-0.311432\pi$$
0.558358 + 0.829600i $$0.311432\pi$$
$$108$$ 400.000 0.356389
$$109$$ −694.000 −0.609845 −0.304923 0.952377i $$-0.598631\pi$$
−0.304923 + 0.952377i $$0.598631\pi$$
$$110$$ −396.000 −0.343247
$$111$$ −772.000 −0.660135
$$112$$ 112.000 0.0944911
$$113$$ 978.000 0.814181 0.407091 0.913388i $$-0.366543\pi$$
0.407091 + 0.913388i $$0.366543\pi$$
$$114$$ 112.000 0.0920154
$$115$$ 3240.00 2.62723
$$116$$ −216.000 −0.172889
$$117$$ −1288.00 −1.01774
$$118$$ −564.000 −0.440003
$$119$$ 252.000 0.194124
$$120$$ −288.000 −0.219089
$$121$$ 121.000 0.0909091
$$122$$ 760.000 0.563993
$$123$$ 888.000 0.650961
$$124$$ −1336.00 −0.967551
$$125$$ 1332.00 0.953102
$$126$$ −322.000 −0.227667
$$127$$ −1216.00 −0.849626 −0.424813 0.905281i $$-0.639660\pi$$
−0.424813 + 0.905281i $$0.639660\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 632.000 0.431353
$$130$$ 2016.00 1.36011
$$131$$ 1680.00 1.12048 0.560238 0.828332i $$-0.310710\pi$$
0.560238 + 0.828332i $$0.310710\pi$$
$$132$$ 88.0000 0.0580259
$$133$$ −196.000 −0.127785
$$134$$ 352.000 0.226927
$$135$$ 1800.00 1.14755
$$136$$ 288.000 0.181587
$$137$$ 1062.00 0.662283 0.331142 0.943581i $$-0.392566\pi$$
0.331142 + 0.943581i $$0.392566\pi$$
$$138$$ −720.000 −0.444134
$$139$$ −508.000 −0.309986 −0.154993 0.987916i $$-0.549535\pi$$
−0.154993 + 0.987916i $$0.549535\pi$$
$$140$$ 504.000 0.304256
$$141$$ 804.000 0.480206
$$142$$ −648.000 −0.382950
$$143$$ −616.000 −0.360227
$$144$$ −368.000 −0.212963
$$145$$ −972.000 −0.556691
$$146$$ 1600.00 0.906965
$$147$$ −98.0000 −0.0549857
$$148$$ 1544.00 0.857541
$$149$$ 2598.00 1.42843 0.714216 0.699925i $$-0.246783\pi$$
0.714216 + 0.699925i $$0.246783\pi$$
$$150$$ −796.000 −0.433288
$$151$$ 2648.00 1.42709 0.713547 0.700607i $$-0.247088\pi$$
0.713547 + 0.700607i $$0.247088\pi$$
$$152$$ −224.000 −0.119532
$$153$$ −828.000 −0.437515
$$154$$ −154.000 −0.0805823
$$155$$ −6012.00 −3.11545
$$156$$ −448.000 −0.229928
$$157$$ −790.000 −0.401585 −0.200793 0.979634i $$-0.564352\pi$$
−0.200793 + 0.979634i $$0.564352\pi$$
$$158$$ −2288.00 −1.15205
$$159$$ 972.000 0.484809
$$160$$ 576.000 0.284605
$$161$$ 1260.00 0.616782
$$162$$ 842.000 0.408357
$$163$$ −160.000 −0.0768845 −0.0384422 0.999261i $$-0.512240\pi$$
−0.0384422 + 0.999261i $$0.512240\pi$$
$$164$$ −1776.00 −0.845624
$$165$$ 396.000 0.186840
$$166$$ 936.000 0.437637
$$167$$ 264.000 0.122329 0.0611645 0.998128i $$-0.480519\pi$$
0.0611645 + 0.998128i $$0.480519\pi$$
$$168$$ −112.000 −0.0514344
$$169$$ 939.000 0.427401
$$170$$ 1296.00 0.584698
$$171$$ 644.000 0.287999
$$172$$ −1264.00 −0.560344
$$173$$ 1632.00 0.717218 0.358609 0.933488i $$-0.383251\pi$$
0.358609 + 0.933488i $$0.383251\pi$$
$$174$$ 216.000 0.0941087
$$175$$ 1393.00 0.601719
$$176$$ −176.000 −0.0753778
$$177$$ 564.000 0.239508
$$178$$ −1740.00 −0.732688
$$179$$ −708.000 −0.295634 −0.147817 0.989015i $$-0.547225\pi$$
−0.147817 + 0.989015i $$0.547225\pi$$
$$180$$ −1656.00 −0.685728
$$181$$ 902.000 0.370415 0.185208 0.982699i $$-0.440704\pi$$
0.185208 + 0.982699i $$0.440704\pi$$
$$182$$ 784.000 0.319307
$$183$$ −760.000 −0.306999
$$184$$ 1440.00 0.576947
$$185$$ 6948.00 2.76123
$$186$$ 1336.00 0.526668
$$187$$ −396.000 −0.154858
$$188$$ −1608.00 −0.623806
$$189$$ 700.000 0.269405
$$190$$ −1008.00 −0.384884
$$191$$ 1824.00 0.690995 0.345497 0.938420i $$-0.387710\pi$$
0.345497 + 0.938420i $$0.387710\pi$$
$$192$$ −128.000 −0.0481125
$$193$$ 2090.00 0.779490 0.389745 0.920923i $$-0.372563\pi$$
0.389745 + 0.920923i $$0.372563\pi$$
$$194$$ −2660.00 −0.984417
$$195$$ −2016.00 −0.740353
$$196$$ 196.000 0.0714286
$$197$$ −1602.00 −0.579380 −0.289690 0.957121i $$-0.593552\pi$$
−0.289690 + 0.957121i $$0.593552\pi$$
$$198$$ 506.000 0.181615
$$199$$ −3274.00 −1.16627 −0.583135 0.812375i $$-0.698174\pi$$
−0.583135 + 0.812375i $$0.698174\pi$$
$$200$$ 1592.00 0.562857
$$201$$ −352.000 −0.123523
$$202$$ −240.000 −0.0835957
$$203$$ −378.000 −0.130692
$$204$$ −288.000 −0.0988433
$$205$$ −7992.00 −2.72286
$$206$$ −2420.00 −0.818492
$$207$$ −4140.00 −1.39010
$$208$$ 896.000 0.298685
$$209$$ 308.000 0.101937
$$210$$ −504.000 −0.165616
$$211$$ −4948.00 −1.61438 −0.807190 0.590291i $$-0.799013\pi$$
−0.807190 + 0.590291i $$0.799013\pi$$
$$212$$ −1944.00 −0.629785
$$213$$ 648.000 0.208452
$$214$$ 2472.00 0.789638
$$215$$ −5688.00 −1.80427
$$216$$ 800.000 0.252005
$$217$$ −2338.00 −0.731400
$$218$$ −1388.00 −0.431226
$$219$$ −1600.00 −0.493689
$$220$$ −792.000 −0.242712
$$221$$ 2016.00 0.613624
$$222$$ −1544.00 −0.466786
$$223$$ 2342.00 0.703282 0.351641 0.936135i $$-0.385624\pi$$
0.351641 + 0.936135i $$0.385624\pi$$
$$224$$ 224.000 0.0668153
$$225$$ −4577.00 −1.35615
$$226$$ 1956.00 0.575713
$$227$$ 2064.00 0.603491 0.301746 0.953388i $$-0.402431\pi$$
0.301746 + 0.953388i $$0.402431\pi$$
$$228$$ 224.000 0.0650647
$$229$$ −1666.00 −0.480753 −0.240376 0.970680i $$-0.577271\pi$$
−0.240376 + 0.970680i $$0.577271\pi$$
$$230$$ 6480.00 1.85773
$$231$$ 154.000 0.0438634
$$232$$ −432.000 −0.122251
$$233$$ 4158.00 1.16910 0.584549 0.811359i $$-0.301272\pi$$
0.584549 + 0.811359i $$0.301272\pi$$
$$234$$ −2576.00 −0.719651
$$235$$ −7236.00 −2.00862
$$236$$ −1128.00 −0.311129
$$237$$ 2288.00 0.627095
$$238$$ 504.000 0.137267
$$239$$ 72.0000 0.0194866 0.00974329 0.999953i $$-0.496899\pi$$
0.00974329 + 0.999953i $$0.496899\pi$$
$$240$$ −576.000 −0.154919
$$241$$ 6860.00 1.83357 0.916787 0.399376i $$-0.130773\pi$$
0.916787 + 0.399376i $$0.130773\pi$$
$$242$$ 242.000 0.0642824
$$243$$ −3542.00 −0.935059
$$244$$ 1520.00 0.398803
$$245$$ 882.000 0.229996
$$246$$ 1776.00 0.460299
$$247$$ −1568.00 −0.403925
$$248$$ −2672.00 −0.684162
$$249$$ −936.000 −0.238219
$$250$$ 2664.00 0.673945
$$251$$ −150.000 −0.0377208 −0.0188604 0.999822i $$-0.506004\pi$$
−0.0188604 + 0.999822i $$0.506004\pi$$
$$252$$ −644.000 −0.160985
$$253$$ −1980.00 −0.492022
$$254$$ −2432.00 −0.600777
$$255$$ −1296.00 −0.318269
$$256$$ 256.000 0.0625000
$$257$$ −2430.00 −0.589802 −0.294901 0.955528i $$-0.595287\pi$$
−0.294901 + 0.955528i $$0.595287\pi$$
$$258$$ 1264.00 0.305012
$$259$$ 2702.00 0.648240
$$260$$ 4032.00 0.961746
$$261$$ 1242.00 0.294551
$$262$$ 3360.00 0.792296
$$263$$ 3048.00 0.714630 0.357315 0.933984i $$-0.383692\pi$$
0.357315 + 0.933984i $$0.383692\pi$$
$$264$$ 176.000 0.0410305
$$265$$ −8748.00 −2.02787
$$266$$ −392.000 −0.0903574
$$267$$ 1740.00 0.398825
$$268$$ 704.000 0.160461
$$269$$ −3834.00 −0.869008 −0.434504 0.900670i $$-0.643076\pi$$
−0.434504 + 0.900670i $$0.643076\pi$$
$$270$$ 3600.00 0.811441
$$271$$ −3508.00 −0.786331 −0.393166 0.919468i $$-0.628620\pi$$
−0.393166 + 0.919468i $$0.628620\pi$$
$$272$$ 576.000 0.128401
$$273$$ −784.000 −0.173809
$$274$$ 2124.00 0.468305
$$275$$ −2189.00 −0.480006
$$276$$ −1440.00 −0.314050
$$277$$ 8294.00 1.79905 0.899527 0.436864i $$-0.143911\pi$$
0.899527 + 0.436864i $$0.143911\pi$$
$$278$$ −1016.00 −0.219193
$$279$$ 7682.00 1.64842
$$280$$ 1008.00 0.215141
$$281$$ 8022.00 1.70303 0.851517 0.524327i $$-0.175683\pi$$
0.851517 + 0.524327i $$0.175683\pi$$
$$282$$ 1608.00 0.339557
$$283$$ 392.000 0.0823392 0.0411696 0.999152i $$-0.486892\pi$$
0.0411696 + 0.999152i $$0.486892\pi$$
$$284$$ −1296.00 −0.270787
$$285$$ 1008.00 0.209504
$$286$$ −1232.00 −0.254719
$$287$$ −3108.00 −0.639231
$$288$$ −736.000 −0.150588
$$289$$ −3617.00 −0.736210
$$290$$ −1944.00 −0.393640
$$291$$ 2660.00 0.535849
$$292$$ 3200.00 0.641321
$$293$$ −2748.00 −0.547918 −0.273959 0.961741i $$-0.588333\pi$$
−0.273959 + 0.961741i $$0.588333\pi$$
$$294$$ −196.000 −0.0388808
$$295$$ −5076.00 −1.00182
$$296$$ 3088.00 0.606373
$$297$$ −1100.00 −0.214911
$$298$$ 5196.00 1.01005
$$299$$ 10080.0 1.94964
$$300$$ −1592.00 −0.306381
$$301$$ −2212.00 −0.423580
$$302$$ 5296.00 1.00911
$$303$$ 240.000 0.0455038
$$304$$ −448.000 −0.0845216
$$305$$ 6840.00 1.28412
$$306$$ −1656.00 −0.309370
$$307$$ −3064.00 −0.569615 −0.284807 0.958585i $$-0.591930\pi$$
−0.284807 + 0.958585i $$0.591930\pi$$
$$308$$ −308.000 −0.0569803
$$309$$ 2420.00 0.445531
$$310$$ −12024.0 −2.20296
$$311$$ 4062.00 0.740627 0.370313 0.928907i $$-0.379250\pi$$
0.370313 + 0.928907i $$0.379250\pi$$
$$312$$ −896.000 −0.162583
$$313$$ −4870.00 −0.879453 −0.439726 0.898132i $$-0.644925\pi$$
−0.439726 + 0.898132i $$0.644925\pi$$
$$314$$ −1580.00 −0.283964
$$315$$ −2898.00 −0.518361
$$316$$ −4576.00 −0.814621
$$317$$ 4806.00 0.851520 0.425760 0.904836i $$-0.360007\pi$$
0.425760 + 0.904836i $$0.360007\pi$$
$$318$$ 1944.00 0.342812
$$319$$ 594.000 0.104256
$$320$$ 1152.00 0.201246
$$321$$ −2472.00 −0.429824
$$322$$ 2520.00 0.436131
$$323$$ −1008.00 −0.173643
$$324$$ 1684.00 0.288752
$$325$$ 11144.0 1.90202
$$326$$ −320.000 −0.0543655
$$327$$ 1388.00 0.234730
$$328$$ −3552.00 −0.597946
$$329$$ −2814.00 −0.471553
$$330$$ 792.000 0.132116
$$331$$ 6620.00 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 1872.00 0.309456
$$333$$ −8878.00 −1.46100
$$334$$ 528.000 0.0864996
$$335$$ 3168.00 0.516676
$$336$$ −224.000 −0.0363696
$$337$$ 1094.00 0.176837 0.0884184 0.996083i $$-0.471819\pi$$
0.0884184 + 0.996083i $$0.471819\pi$$
$$338$$ 1878.00 0.302218
$$339$$ −1956.00 −0.313379
$$340$$ 2592.00 0.413444
$$341$$ 3674.00 0.583455
$$342$$ 1288.00 0.203646
$$343$$ 343.000 0.0539949
$$344$$ −2528.00 −0.396223
$$345$$ −6480.00 −1.01122
$$346$$ 3264.00 0.507150
$$347$$ 3468.00 0.536519 0.268259 0.963347i $$-0.413552\pi$$
0.268259 + 0.963347i $$0.413552\pi$$
$$348$$ 432.000 0.0665449
$$349$$ −8188.00 −1.25586 −0.627928 0.778272i $$-0.716097\pi$$
−0.627928 + 0.778272i $$0.716097\pi$$
$$350$$ 2786.00 0.425480
$$351$$ 5600.00 0.851584
$$352$$ −352.000 −0.0533002
$$353$$ −5070.00 −0.764444 −0.382222 0.924070i $$-0.624841\pi$$
−0.382222 + 0.924070i $$0.624841\pi$$
$$354$$ 1128.00 0.169357
$$355$$ −5832.00 −0.871917
$$356$$ −3480.00 −0.518089
$$357$$ −504.000 −0.0747185
$$358$$ −1416.00 −0.209044
$$359$$ 1656.00 0.243455 0.121727 0.992564i $$-0.461157\pi$$
0.121727 + 0.992564i $$0.461157\pi$$
$$360$$ −3312.00 −0.484883
$$361$$ −6075.00 −0.885698
$$362$$ 1804.00 0.261923
$$363$$ −242.000 −0.0349909
$$364$$ 1568.00 0.225784
$$365$$ 14400.0 2.06501
$$366$$ −1520.00 −0.217081
$$367$$ 10166.0 1.44594 0.722971 0.690878i $$-0.242776\pi$$
0.722971 + 0.690878i $$0.242776\pi$$
$$368$$ 2880.00 0.407963
$$369$$ 10212.0 1.44069
$$370$$ 13896.0 1.95248
$$371$$ −3402.00 −0.476073
$$372$$ 2672.00 0.372411
$$373$$ −2722.00 −0.377855 −0.188927 0.981991i $$-0.560501\pi$$
−0.188927 + 0.981991i $$0.560501\pi$$
$$374$$ −792.000 −0.109501
$$375$$ −2664.00 −0.366849
$$376$$ −3216.00 −0.441097
$$377$$ −3024.00 −0.413114
$$378$$ 1400.00 0.190498
$$379$$ −5872.00 −0.795843 −0.397921 0.917420i $$-0.630268\pi$$
−0.397921 + 0.917420i $$0.630268\pi$$
$$380$$ −2016.00 −0.272154
$$381$$ 2432.00 0.327021
$$382$$ 3648.00 0.488607
$$383$$ 12330.0 1.64500 0.822498 0.568768i $$-0.192580\pi$$
0.822498 + 0.568768i $$0.192580\pi$$
$$384$$ −256.000 −0.0340207
$$385$$ −1386.00 −0.183473
$$386$$ 4180.00 0.551182
$$387$$ 7268.00 0.954659
$$388$$ −5320.00 −0.696088
$$389$$ −14586.0 −1.90113 −0.950565 0.310526i $$-0.899495\pi$$
−0.950565 + 0.310526i $$0.899495\pi$$
$$390$$ −4032.00 −0.523508
$$391$$ 6480.00 0.838127
$$392$$ 392.000 0.0505076
$$393$$ −3360.00 −0.431271
$$394$$ −3204.00 −0.409683
$$395$$ −20592.0 −2.62303
$$396$$ 1012.00 0.128421
$$397$$ 1874.00 0.236910 0.118455 0.992959i $$-0.462206\pi$$
0.118455 + 0.992959i $$0.462206\pi$$
$$398$$ −6548.00 −0.824677
$$399$$ 392.000 0.0491843
$$400$$ 3184.00 0.398000
$$401$$ 13338.0 1.66102 0.830509 0.557006i $$-0.188050\pi$$
0.830509 + 0.557006i $$0.188050\pi$$
$$402$$ −704.000 −0.0873441
$$403$$ −18704.0 −2.31194
$$404$$ −480.000 −0.0591111
$$405$$ 7578.00 0.929763
$$406$$ −756.000 −0.0924129
$$407$$ −4246.00 −0.517116
$$408$$ −576.000 −0.0698928
$$409$$ −8200.00 −0.991354 −0.495677 0.868507i $$-0.665080\pi$$
−0.495677 + 0.868507i $$0.665080\pi$$
$$410$$ −15984.0 −1.92535
$$411$$ −2124.00 −0.254913
$$412$$ −4840.00 −0.578761
$$413$$ −1974.00 −0.235192
$$414$$ −8280.00 −0.982946
$$415$$ 8424.00 0.996429
$$416$$ 1792.00 0.211202
$$417$$ 1016.00 0.119314
$$418$$ 616.000 0.0720803
$$419$$ −7362.00 −0.858370 −0.429185 0.903216i $$-0.641199\pi$$
−0.429185 + 0.903216i $$0.641199\pi$$
$$420$$ −1008.00 −0.117108
$$421$$ −11710.0 −1.35561 −0.677803 0.735243i $$-0.737068\pi$$
−0.677803 + 0.735243i $$0.737068\pi$$
$$422$$ −9896.00 −1.14154
$$423$$ 9246.00 1.06278
$$424$$ −3888.00 −0.445325
$$425$$ 7164.00 0.817659
$$426$$ 1296.00 0.147398
$$427$$ 2660.00 0.301467
$$428$$ 4944.00 0.558358
$$429$$ 1232.00 0.138652
$$430$$ −11376.0 −1.27581
$$431$$ −936.000 −0.104607 −0.0523034 0.998631i $$-0.516656\pi$$
−0.0523034 + 0.998631i $$0.516656\pi$$
$$432$$ 1600.00 0.178195
$$433$$ 9038.00 1.00309 0.501546 0.865131i $$-0.332765\pi$$
0.501546 + 0.865131i $$0.332765\pi$$
$$434$$ −4676.00 −0.517178
$$435$$ 1944.00 0.214270
$$436$$ −2776.00 −0.304923
$$437$$ −5040.00 −0.551707
$$438$$ −3200.00 −0.349091
$$439$$ 1964.00 0.213523 0.106762 0.994285i $$-0.465952\pi$$
0.106762 + 0.994285i $$0.465952\pi$$
$$440$$ −1584.00 −0.171623
$$441$$ −1127.00 −0.121693
$$442$$ 4032.00 0.433897
$$443$$ 10068.0 1.07979 0.539893 0.841734i $$-0.318465\pi$$
0.539893 + 0.841734i $$0.318465\pi$$
$$444$$ −3088.00 −0.330068
$$445$$ −15660.0 −1.66821
$$446$$ 4684.00 0.497296
$$447$$ −5196.00 −0.549804
$$448$$ 448.000 0.0472456
$$449$$ 3270.00 0.343699 0.171849 0.985123i $$-0.445026\pi$$
0.171849 + 0.985123i $$0.445026\pi$$
$$450$$ −9154.00 −0.958942
$$451$$ 4884.00 0.509930
$$452$$ 3912.00 0.407091
$$453$$ −5296.00 −0.549289
$$454$$ 4128.00 0.426733
$$455$$ 7056.00 0.727012
$$456$$ 448.000 0.0460077
$$457$$ −15526.0 −1.58922 −0.794612 0.607117i $$-0.792326\pi$$
−0.794612 + 0.607117i $$0.792326\pi$$
$$458$$ −3332.00 −0.339944
$$459$$ 3600.00 0.366086
$$460$$ 12960.0 1.31362
$$461$$ 10548.0 1.06566 0.532830 0.846222i $$-0.321128\pi$$
0.532830 + 0.846222i $$0.321128\pi$$
$$462$$ 308.000 0.0310161
$$463$$ −3796.00 −0.381026 −0.190513 0.981685i $$-0.561015\pi$$
−0.190513 + 0.981685i $$0.561015\pi$$
$$464$$ −864.000 −0.0864444
$$465$$ 12024.0 1.19914
$$466$$ 8316.00 0.826677
$$467$$ 7122.00 0.705711 0.352855 0.935678i $$-0.385211\pi$$
0.352855 + 0.935678i $$0.385211\pi$$
$$468$$ −5152.00 −0.508870
$$469$$ 1232.00 0.121297
$$470$$ −14472.0 −1.42031
$$471$$ 1580.00 0.154570
$$472$$ −2256.00 −0.220002
$$473$$ 3476.00 0.337900
$$474$$ 4576.00 0.443423
$$475$$ −5572.00 −0.538233
$$476$$ 1008.00 0.0970622
$$477$$ 11178.0 1.07297
$$478$$ 144.000 0.0137791
$$479$$ 2292.00 0.218631 0.109315 0.994007i $$-0.465134\pi$$
0.109315 + 0.994007i $$0.465134\pi$$
$$480$$ −1152.00 −0.109545
$$481$$ 21616.0 2.04907
$$482$$ 13720.0 1.29653
$$483$$ −2520.00 −0.237400
$$484$$ 484.000 0.0454545
$$485$$ −23940.0 −2.24136
$$486$$ −7084.00 −0.661187
$$487$$ 5132.00 0.477522 0.238761 0.971078i $$-0.423259\pi$$
0.238761 + 0.971078i $$0.423259\pi$$
$$488$$ 3040.00 0.281997
$$489$$ 320.000 0.0295928
$$490$$ 1764.00 0.162631
$$491$$ 4188.00 0.384932 0.192466 0.981304i $$-0.438351\pi$$
0.192466 + 0.981304i $$0.438351\pi$$
$$492$$ 3552.00 0.325481
$$493$$ −1944.00 −0.177593
$$494$$ −3136.00 −0.285618
$$495$$ 4554.00 0.413509
$$496$$ −5344.00 −0.483776
$$497$$ −2268.00 −0.204696
$$498$$ −1872.00 −0.168446
$$499$$ 3848.00 0.345211 0.172605 0.984991i $$-0.444781\pi$$
0.172605 + 0.984991i $$0.444781\pi$$
$$500$$ 5328.00 0.476551
$$501$$ −528.000 −0.0470844
$$502$$ −300.000 −0.0266726
$$503$$ −1068.00 −0.0946715 −0.0473358 0.998879i $$-0.515073\pi$$
−0.0473358 + 0.998879i $$0.515073\pi$$
$$504$$ −1288.00 −0.113833
$$505$$ −2160.00 −0.190334
$$506$$ −3960.00 −0.347912
$$507$$ −1878.00 −0.164507
$$508$$ −4864.00 −0.424813
$$509$$ −6162.00 −0.536593 −0.268297 0.963336i $$-0.586461\pi$$
−0.268297 + 0.963336i $$0.586461\pi$$
$$510$$ −2592.00 −0.225050
$$511$$ 5600.00 0.484793
$$512$$ 512.000 0.0441942
$$513$$ −2800.00 −0.240981
$$514$$ −4860.00 −0.417053
$$515$$ −21780.0 −1.86358
$$516$$ 2528.00 0.215676
$$517$$ 4422.00 0.376169
$$518$$ 5404.00 0.458375
$$519$$ −3264.00 −0.276057
$$520$$ 8064.00 0.680057
$$521$$ −20946.0 −1.76135 −0.880673 0.473725i $$-0.842909\pi$$
−0.880673 + 0.473725i $$0.842909\pi$$
$$522$$ 2484.00 0.208279
$$523$$ −4696.00 −0.392623 −0.196311 0.980542i $$-0.562896\pi$$
−0.196311 + 0.980542i $$0.562896\pi$$
$$524$$ 6720.00 0.560238
$$525$$ −2786.00 −0.231602
$$526$$ 6096.00 0.505320
$$527$$ −12024.0 −0.993878
$$528$$ 352.000 0.0290129
$$529$$ 20233.0 1.66294
$$530$$ −17496.0 −1.43392
$$531$$ 6486.00 0.530072
$$532$$ −784.000 −0.0638923
$$533$$ −24864.0 −2.02060
$$534$$ 3480.00 0.282012
$$535$$ 22248.0 1.79788
$$536$$ 1408.00 0.113463
$$537$$ 1416.00 0.113789
$$538$$ −7668.00 −0.614481
$$539$$ −539.000 −0.0430730
$$540$$ 7200.00 0.573775
$$541$$ 19358.0 1.53838 0.769192 0.639018i $$-0.220659\pi$$
0.769192 + 0.639018i $$0.220659\pi$$
$$542$$ −7016.00 −0.556020
$$543$$ −1804.00 −0.142573
$$544$$ 1152.00 0.0907934
$$545$$ −12492.0 −0.981832
$$546$$ −1568.00 −0.122901
$$547$$ 18020.0 1.40855 0.704277 0.709925i $$-0.251271\pi$$
0.704277 + 0.709925i $$0.251271\pi$$
$$548$$ 4248.00 0.331142
$$549$$ −8740.00 −0.679443
$$550$$ −4378.00 −0.339416
$$551$$ 1512.00 0.116903
$$552$$ −2880.00 −0.222067
$$553$$ −8008.00 −0.615795
$$554$$ 16588.0 1.27212
$$555$$ −13896.0 −1.06280
$$556$$ −2032.00 −0.154993
$$557$$ 14622.0 1.11231 0.556153 0.831080i $$-0.312277\pi$$
0.556153 + 0.831080i $$0.312277\pi$$
$$558$$ 15364.0 1.16561
$$559$$ −17696.0 −1.33893
$$560$$ 2016.00 0.152128
$$561$$ 792.000 0.0596048
$$562$$ 16044.0 1.20423
$$563$$ −2244.00 −0.167981 −0.0839905 0.996467i $$-0.526767\pi$$
−0.0839905 + 0.996467i $$0.526767\pi$$
$$564$$ 3216.00 0.240103
$$565$$ 17604.0 1.31081
$$566$$ 784.000 0.0582226
$$567$$ 2947.00 0.218276
$$568$$ −2592.00 −0.191475
$$569$$ −3258.00 −0.240039 −0.120020 0.992772i $$-0.538296\pi$$
−0.120020 + 0.992772i $$0.538296\pi$$
$$570$$ 2016.00 0.148142
$$571$$ −6604.00 −0.484008 −0.242004 0.970275i $$-0.577805\pi$$
−0.242004 + 0.970275i $$0.577805\pi$$
$$572$$ −2464.00 −0.180114
$$573$$ −3648.00 −0.265964
$$574$$ −6216.00 −0.452005
$$575$$ 35820.0 2.59791
$$576$$ −1472.00 −0.106481
$$577$$ −16594.0 −1.19726 −0.598628 0.801027i $$-0.704287\pi$$
−0.598628 + 0.801027i $$0.704287\pi$$
$$578$$ −7234.00 −0.520579
$$579$$ −4180.00 −0.300026
$$580$$ −3888.00 −0.278346
$$581$$ 3276.00 0.233927
$$582$$ 5320.00 0.378902
$$583$$ 5346.00 0.379775
$$584$$ 6400.00 0.453483
$$585$$ −23184.0 −1.63853
$$586$$ −5496.00 −0.387436
$$587$$ −19062.0 −1.34033 −0.670164 0.742213i $$-0.733776\pi$$
−0.670164 + 0.742213i $$0.733776\pi$$
$$588$$ −392.000 −0.0274929
$$589$$ 9352.00 0.654232
$$590$$ −10152.0 −0.708392
$$591$$ 3204.00 0.223003
$$592$$ 6176.00 0.428770
$$593$$ −4776.00 −0.330737 −0.165368 0.986232i $$-0.552881\pi$$
−0.165368 + 0.986232i $$0.552881\pi$$
$$594$$ −2200.00 −0.151965
$$595$$ 4536.00 0.312534
$$596$$ 10392.0 0.714216
$$597$$ 6548.00 0.448897
$$598$$ 20160.0 1.37860
$$599$$ 7956.00 0.542693 0.271347 0.962482i $$-0.412531\pi$$
0.271347 + 0.962482i $$0.412531\pi$$
$$600$$ −3184.00 −0.216644
$$601$$ 14348.0 0.973822 0.486911 0.873452i $$-0.338124\pi$$
0.486911 + 0.873452i $$0.338124\pi$$
$$602$$ −4424.00 −0.299516
$$603$$ −4048.00 −0.273379
$$604$$ 10592.0 0.713547
$$605$$ 2178.00 0.146361
$$606$$ 480.000 0.0321760
$$607$$ 24488.0 1.63746 0.818729 0.574180i $$-0.194679\pi$$
0.818729 + 0.574180i $$0.194679\pi$$
$$608$$ −896.000 −0.0597658
$$609$$ 756.000 0.0503032
$$610$$ 13680.0 0.908011
$$611$$ −22512.0 −1.49057
$$612$$ −3312.00 −0.218758
$$613$$ −19654.0 −1.29497 −0.647486 0.762078i $$-0.724179\pi$$
−0.647486 + 0.762078i $$0.724179\pi$$
$$614$$ −6128.00 −0.402778
$$615$$ 15984.0 1.04803
$$616$$ −616.000 −0.0402911
$$617$$ 2694.00 0.175780 0.0878901 0.996130i $$-0.471988\pi$$
0.0878901 + 0.996130i $$0.471988\pi$$
$$618$$ 4840.00 0.315038
$$619$$ 10178.0 0.660886 0.330443 0.943826i $$-0.392802\pi$$
0.330443 + 0.943826i $$0.392802\pi$$
$$620$$ −24048.0 −1.55773
$$621$$ 18000.0 1.16315
$$622$$ 8124.00 0.523702
$$623$$ −6090.00 −0.391638
$$624$$ −1792.00 −0.114964
$$625$$ −899.000 −0.0575360
$$626$$ −9740.00 −0.621867
$$627$$ −616.000 −0.0392355
$$628$$ −3160.00 −0.200793
$$629$$ 13896.0 0.880874
$$630$$ −5796.00 −0.366537
$$631$$ −7648.00 −0.482507 −0.241254 0.970462i $$-0.577559\pi$$
−0.241254 + 0.970462i $$0.577559\pi$$
$$632$$ −9152.00 −0.576024
$$633$$ 9896.00 0.621375
$$634$$ 9612.00 0.602116
$$635$$ −21888.0 −1.36787
$$636$$ 3888.00 0.242404
$$637$$ 2744.00 0.170677
$$638$$ 1188.00 0.0737200
$$639$$ 7452.00 0.461340
$$640$$ 2304.00 0.142302
$$641$$ 270.000 0.0166371 0.00831853 0.999965i $$-0.497352\pi$$
0.00831853 + 0.999965i $$0.497352\pi$$
$$642$$ −4944.00 −0.303932
$$643$$ 16250.0 0.996637 0.498318 0.866994i $$-0.333951\pi$$
0.498318 + 0.866994i $$0.333951\pi$$
$$644$$ 5040.00 0.308391
$$645$$ 11376.0 0.694464
$$646$$ −2016.00 −0.122784
$$647$$ 10242.0 0.622341 0.311170 0.950354i $$-0.399279\pi$$
0.311170 + 0.950354i $$0.399279\pi$$
$$648$$ 3368.00 0.204178
$$649$$ 3102.00 0.187618
$$650$$ 22288.0 1.34493
$$651$$ 4676.00 0.281516
$$652$$ −640.000 −0.0384422
$$653$$ −17322.0 −1.03807 −0.519037 0.854752i $$-0.673709\pi$$
−0.519037 + 0.854752i $$0.673709\pi$$
$$654$$ 2776.00 0.165979
$$655$$ 30240.0 1.80393
$$656$$ −7104.00 −0.422812
$$657$$ −18400.0 −1.09262
$$658$$ −5628.00 −0.333438
$$659$$ 11676.0 0.690186 0.345093 0.938569i $$-0.387847\pi$$
0.345093 + 0.938569i $$0.387847\pi$$
$$660$$ 1584.00 0.0934199
$$661$$ −20710.0 −1.21865 −0.609323 0.792922i $$-0.708559\pi$$
−0.609323 + 0.792922i $$0.708559\pi$$
$$662$$ 13240.0 0.777322
$$663$$ −4032.00 −0.236184
$$664$$ 3744.00 0.218818
$$665$$ −3528.00 −0.205729
$$666$$ −17756.0 −1.03308
$$667$$ −9720.00 −0.564258
$$668$$ 1056.00 0.0611645
$$669$$ −4684.00 −0.270693
$$670$$ 6336.00 0.365345
$$671$$ −4180.00 −0.240487
$$672$$ −448.000 −0.0257172
$$673$$ −10354.0 −0.593042 −0.296521 0.955026i $$-0.595827\pi$$
−0.296521 + 0.955026i $$0.595827\pi$$
$$674$$ 2188.00 0.125042
$$675$$ 19900.0 1.13474
$$676$$ 3756.00 0.213701
$$677$$ −10920.0 −0.619926 −0.309963 0.950749i $$-0.600317\pi$$
−0.309963 + 0.950749i $$0.600317\pi$$
$$678$$ −3912.00 −0.221592
$$679$$ −9310.00 −0.526193
$$680$$ 5184.00 0.292349
$$681$$ −4128.00 −0.232284
$$682$$ 7348.00 0.412565
$$683$$ 27804.0 1.55767 0.778836 0.627227i $$-0.215810\pi$$
0.778836 + 0.627227i $$0.215810\pi$$
$$684$$ 2576.00 0.144000
$$685$$ 19116.0 1.06626
$$686$$ 686.000 0.0381802
$$687$$ 3332.00 0.185042
$$688$$ −5056.00 −0.280172
$$689$$ −27216.0 −1.50486
$$690$$ −12960.0 −0.715042
$$691$$ −25834.0 −1.42225 −0.711123 0.703068i $$-0.751813\pi$$
−0.711123 + 0.703068i $$0.751813\pi$$
$$692$$ 6528.00 0.358609
$$693$$ 1771.00 0.0970775
$$694$$ 6936.00 0.379376
$$695$$ −9144.00 −0.499067
$$696$$ 864.000 0.0470544
$$697$$ −15984.0 −0.868633
$$698$$ −16376.0 −0.888024
$$699$$ −8316.00 −0.449986
$$700$$ 5572.00 0.300860
$$701$$ −10590.0 −0.570583 −0.285292 0.958441i $$-0.592090\pi$$
−0.285292 + 0.958441i $$0.592090\pi$$
$$702$$ 11200.0 0.602161
$$703$$ −10808.0 −0.579846
$$704$$ −704.000 −0.0376889
$$705$$ 14472.0 0.773116
$$706$$ −10140.0 −0.540544
$$707$$ −840.000 −0.0446838
$$708$$ 2256.00 0.119754
$$709$$ −6802.00 −0.360302 −0.180151 0.983639i $$-0.557659\pi$$
−0.180151 + 0.983639i $$0.557659\pi$$
$$710$$ −11664.0 −0.616538
$$711$$ 26312.0 1.38787
$$712$$ −6960.00 −0.366344
$$713$$ −60120.0 −3.15780
$$714$$ −1008.00 −0.0528340
$$715$$ −11088.0 −0.579955
$$716$$ −2832.00 −0.147817
$$717$$ −144.000 −0.00750039
$$718$$ 3312.00 0.172149
$$719$$ 23010.0 1.19350 0.596751 0.802426i $$-0.296458\pi$$
0.596751 + 0.802426i $$0.296458\pi$$
$$720$$ −6624.00 −0.342864
$$721$$ −8470.00 −0.437502
$$722$$ −12150.0 −0.626283
$$723$$ −13720.0 −0.705743
$$724$$ 3608.00 0.185208
$$725$$ −10746.0 −0.550478
$$726$$ −484.000 −0.0247423
$$727$$ 4682.00 0.238853 0.119426 0.992843i $$-0.461894\pi$$
0.119426 + 0.992843i $$0.461894\pi$$
$$728$$ 3136.00 0.159654
$$729$$ −4283.00 −0.217599
$$730$$ 28800.0 1.46019
$$731$$ −11376.0 −0.575590
$$732$$ −3040.00 −0.153499
$$733$$ −17860.0 −0.899965 −0.449982 0.893037i $$-0.648570\pi$$
−0.449982 + 0.893037i $$0.648570\pi$$
$$734$$ 20332.0 1.02244
$$735$$ −1764.00 −0.0885253
$$736$$ 5760.00 0.288473
$$737$$ −1936.00 −0.0967618
$$738$$ 20424.0 1.01872
$$739$$ 6860.00 0.341474 0.170737 0.985317i $$-0.445385\pi$$
0.170737 + 0.985317i $$0.445385\pi$$
$$740$$ 27792.0 1.38061
$$741$$ 3136.00 0.155471
$$742$$ −6804.00 −0.336634
$$743$$ −22752.0 −1.12341 −0.561703 0.827339i $$-0.689853\pi$$
−0.561703 + 0.827339i $$0.689853\pi$$
$$744$$ 5344.00 0.263334
$$745$$ 46764.0 2.29973
$$746$$ −5444.00 −0.267184
$$747$$ −10764.0 −0.527221
$$748$$ −1584.00 −0.0774288
$$749$$ 8652.00 0.422079
$$750$$ −5328.00 −0.259401
$$751$$ 7364.00 0.357811 0.178906 0.983866i $$-0.442744\pi$$
0.178906 + 0.983866i $$0.442744\pi$$
$$752$$ −6432.00 −0.311903
$$753$$ 300.000 0.0145187
$$754$$ −6048.00 −0.292116
$$755$$ 47664.0 2.29758
$$756$$ 2800.00 0.134702
$$757$$ −34378.0 −1.65058 −0.825290 0.564709i $$-0.808989\pi$$
−0.825290 + 0.564709i $$0.808989\pi$$
$$758$$ −11744.0 −0.562746
$$759$$ 3960.00 0.189379
$$760$$ −4032.00 −0.192442
$$761$$ 27456.0 1.30786 0.653929 0.756556i $$-0.273120\pi$$
0.653929 + 0.756556i $$0.273120\pi$$
$$762$$ 4864.00 0.231239
$$763$$ −4858.00 −0.230500
$$764$$ 7296.00 0.345497
$$765$$ −14904.0 −0.704386
$$766$$ 24660.0 1.16319
$$767$$ −15792.0 −0.743437
$$768$$ −512.000 −0.0240563
$$769$$ 7952.00 0.372895 0.186448 0.982465i $$-0.440303\pi$$
0.186448 + 0.982465i $$0.440303\pi$$
$$770$$ −2772.00 −0.129735
$$771$$ 4860.00 0.227015
$$772$$ 8360.00 0.389745
$$773$$ −4986.00 −0.231997 −0.115999 0.993249i $$-0.537007\pi$$
−0.115999 + 0.993249i $$0.537007\pi$$
$$774$$ 14536.0 0.675046
$$775$$ −66466.0 −3.08068
$$776$$ −10640.0 −0.492208
$$777$$ −5404.00 −0.249508
$$778$$ −29172.0 −1.34430
$$779$$ 12432.0 0.571788
$$780$$ −8064.00 −0.370176
$$781$$ 3564.00 0.163291
$$782$$ 12960.0 0.592645
$$783$$ −5400.00 −0.246463
$$784$$ 784.000 0.0357143
$$785$$ −14220.0 −0.646540
$$786$$ −6720.00 −0.304955
$$787$$ −42748.0 −1.93622 −0.968108 0.250534i $$-0.919394\pi$$
−0.968108 + 0.250534i $$0.919394\pi$$
$$788$$ −6408.00 −0.289690
$$789$$ −6096.00 −0.275061
$$790$$ −41184.0 −1.85476
$$791$$ 6846.00 0.307732
$$792$$ 2024.00 0.0908077
$$793$$ 21280.0 0.952932
$$794$$ 3748.00 0.167521
$$795$$ 17496.0 0.780527
$$796$$ −13096.0 −0.583135
$$797$$ 35610.0 1.58265 0.791324 0.611397i $$-0.209392\pi$$
0.791324 + 0.611397i $$0.209392\pi$$
$$798$$ 784.000 0.0347786
$$799$$ −14472.0 −0.640779
$$800$$ 6368.00 0.281428
$$801$$ 20010.0 0.882670
$$802$$ 26676.0 1.17452
$$803$$ −8800.00 −0.386731
$$804$$ −1408.00 −0.0617616
$$805$$ 22680.0 0.993000
$$806$$ −37408.0 −1.63479
$$807$$ 7668.00 0.334481
$$808$$ −960.000 −0.0417979
$$809$$ −17046.0 −0.740798 −0.370399 0.928873i $$-0.620779\pi$$
−0.370399 + 0.928873i $$0.620779\pi$$
$$810$$ 15156.0 0.657441
$$811$$ −2176.00 −0.0942166 −0.0471083 0.998890i $$-0.515001\pi$$
−0.0471083 + 0.998890i $$0.515001\pi$$
$$812$$ −1512.00 −0.0653458
$$813$$ 7016.00 0.302659
$$814$$ −8492.00 −0.365657
$$815$$ −2880.00 −0.123782
$$816$$ −1152.00 −0.0494217
$$817$$ 8848.00 0.378889
$$818$$ −16400.0 −0.700993
$$819$$ −9016.00 −0.384670
$$820$$ −31968.0 −1.36143
$$821$$ 2094.00 0.0890147 0.0445074 0.999009i $$-0.485828\pi$$
0.0445074 + 0.999009i $$0.485828\pi$$
$$822$$ −4248.00 −0.180251
$$823$$ 7328.00 0.310374 0.155187 0.987885i $$-0.450402\pi$$
0.155187 + 0.987885i $$0.450402\pi$$
$$824$$ −9680.00 −0.409246
$$825$$ 4378.00 0.184754
$$826$$ −3948.00 −0.166306
$$827$$ −12492.0 −0.525259 −0.262630 0.964897i $$-0.584590\pi$$
−0.262630 + 0.964897i $$0.584590\pi$$
$$828$$ −16560.0 −0.695048
$$829$$ −37486.0 −1.57050 −0.785249 0.619180i $$-0.787465\pi$$
−0.785249 + 0.619180i $$0.787465\pi$$
$$830$$ 16848.0 0.704581
$$831$$ −16588.0 −0.692456
$$832$$ 3584.00 0.149342
$$833$$ 1764.00 0.0733721
$$834$$ 2032.00 0.0843674
$$835$$ 4752.00 0.196946
$$836$$ 1232.00 0.0509684
$$837$$ −33400.0 −1.37930
$$838$$ −14724.0 −0.606960
$$839$$ −17574.0 −0.723149 −0.361574 0.932343i $$-0.617761\pi$$
−0.361574 + 0.932343i $$0.617761\pi$$
$$840$$ −2016.00 −0.0828079
$$841$$ −21473.0 −0.880438
$$842$$ −23420.0 −0.958559
$$843$$ −16044.0 −0.655498
$$844$$ −19792.0 −0.807190
$$845$$ 16902.0 0.688102
$$846$$ 18492.0 0.751499
$$847$$ 847.000 0.0343604
$$848$$ −7776.00 −0.314893
$$849$$ −784.000 −0.0316924
$$850$$ 14328.0 0.578172
$$851$$ 69480.0 2.79876
$$852$$ 2592.00 0.104226
$$853$$ 9440.00 0.378921 0.189460 0.981888i $$-0.439326\pi$$
0.189460 + 0.981888i $$0.439326\pi$$
$$854$$ 5320.00 0.213169
$$855$$ 11592.0 0.463670
$$856$$ 9888.00 0.394819
$$857$$ −28440.0 −1.13360 −0.566798 0.823857i $$-0.691818\pi$$
−0.566798 + 0.823857i $$0.691818\pi$$
$$858$$ 2464.00 0.0980415
$$859$$ −24334.0 −0.966549 −0.483274 0.875469i $$-0.660553\pi$$
−0.483274 + 0.875469i $$0.660553\pi$$
$$860$$ −22752.0 −0.902136
$$861$$ 6216.00 0.246040
$$862$$ −1872.00 −0.0739682
$$863$$ 39264.0 1.54874 0.774370 0.632733i $$-0.218067\pi$$
0.774370 + 0.632733i $$0.218067\pi$$
$$864$$ 3200.00 0.126003
$$865$$ 29376.0 1.15470
$$866$$ 18076.0 0.709293
$$867$$ 7234.00 0.283367
$$868$$ −9352.00 −0.365700
$$869$$ 12584.0 0.491235
$$870$$ 3888.00 0.151512
$$871$$ 9856.00 0.383419
$$872$$ −5552.00 −0.215613
$$873$$ 30590.0 1.18593
$$874$$ −10080.0 −0.390116
$$875$$ 9324.00 0.360239
$$876$$ −6400.00 −0.246845
$$877$$ 32114.0 1.23650 0.618251 0.785981i $$-0.287841\pi$$
0.618251 + 0.785981i $$0.287841\pi$$
$$878$$ 3928.00 0.150984
$$879$$ 5496.00 0.210894
$$880$$ −3168.00 −0.121356
$$881$$ 41454.0 1.58527 0.792634 0.609698i $$-0.208709\pi$$
0.792634 + 0.609698i $$0.208709\pi$$
$$882$$ −2254.00 −0.0860500
$$883$$ 2876.00 0.109609 0.0548047 0.998497i $$-0.482546\pi$$
0.0548047 + 0.998497i $$0.482546\pi$$
$$884$$ 8064.00 0.306812
$$885$$ 10152.0 0.385600
$$886$$ 20136.0 0.763524
$$887$$ 13932.0 0.527385 0.263693 0.964607i $$-0.415060\pi$$
0.263693 + 0.964607i $$0.415060\pi$$
$$888$$ −6176.00 −0.233393
$$889$$ −8512.00 −0.321129
$$890$$ −31320.0 −1.17961
$$891$$ −4631.00 −0.174124
$$892$$ 9368.00 0.351641
$$893$$ 11256.0 0.421800
$$894$$ −10392.0 −0.388770
$$895$$ −12744.0 −0.475961
$$896$$ 896.000 0.0334077
$$897$$ −20160.0 −0.750416
$$898$$ 6540.00 0.243032
$$899$$ 18036.0 0.669115
$$900$$ −18308.0 −0.678074
$$901$$ −17496.0 −0.646921
$$902$$ 9768.00 0.360575
$$903$$ 4424.00 0.163036
$$904$$ 7824.00 0.287857
$$905$$ 16236.0 0.596357
$$906$$ −10592.0 −0.388406
$$907$$ −19768.0 −0.723689 −0.361844 0.932239i $$-0.617853\pi$$
−0.361844 + 0.932239i $$0.617853\pi$$
$$908$$ 8256.00 0.301746
$$909$$ 2760.00 0.100708
$$910$$ 14112.0 0.514075
$$911$$ 43836.0 1.59424 0.797119 0.603822i $$-0.206356\pi$$
0.797119 + 0.603822i $$0.206356\pi$$
$$912$$ 896.000 0.0325324
$$913$$ −5148.00 −0.186609
$$914$$ −31052.0 −1.12375
$$915$$ −13680.0 −0.494259
$$916$$ −6664.00 −0.240376
$$917$$ 11760.0 0.423500
$$918$$ 7200.00 0.258862
$$919$$ 31544.0 1.13225 0.566127 0.824318i $$-0.308441\pi$$
0.566127 + 0.824318i $$0.308441\pi$$
$$920$$ 25920.0 0.928866
$$921$$ 6128.00 0.219245
$$922$$ 21096.0 0.753536
$$923$$ −18144.0 −0.647039
$$924$$ 616.000 0.0219317
$$925$$ 76814.0 2.73041
$$926$$ −7592.00 −0.269426
$$927$$ 27830.0 0.986038
$$928$$ −1728.00 −0.0611254
$$929$$ 11118.0 0.392648 0.196324 0.980539i $$-0.437100\pi$$
0.196324 + 0.980539i $$0.437100\pi$$
$$930$$ 24048.0 0.847919
$$931$$ −1372.00 −0.0482980
$$932$$ 16632.0 0.584549
$$933$$ −8124.00 −0.285067
$$934$$ 14244.0 0.499013
$$935$$ −7128.00 −0.249316
$$936$$ −10304.0 −0.359826
$$937$$ 10568.0 0.368454 0.184227 0.982884i $$-0.441022\pi$$
0.184227 + 0.982884i $$0.441022\pi$$
$$938$$ 2464.00 0.0857702
$$939$$ 9740.00 0.338501
$$940$$ −28944.0 −1.00431
$$941$$ 14964.0 0.518398 0.259199 0.965824i $$-0.416541\pi$$
0.259199 + 0.965824i $$0.416541\pi$$
$$942$$ 3160.00 0.109298
$$943$$ −79920.0 −2.75987
$$944$$ −4512.00 −0.155565
$$945$$ 12600.0 0.433733
$$946$$ 6952.00 0.238931
$$947$$ 3324.00 0.114061 0.0570304 0.998372i $$-0.481837\pi$$
0.0570304 + 0.998372i $$0.481837\pi$$
$$948$$ 9152.00 0.313548
$$949$$ 44800.0 1.53242
$$950$$ −11144.0 −0.380589
$$951$$ −9612.00 −0.327750
$$952$$ 2016.00 0.0686333
$$953$$ 3906.00 0.132768 0.0663839 0.997794i $$-0.478854\pi$$
0.0663839 + 0.997794i $$0.478854\pi$$
$$954$$ 22356.0 0.758703
$$955$$ 32832.0 1.11248
$$956$$ 288.000 0.00974329
$$957$$ −1188.00 −0.0401281
$$958$$ 4584.00 0.154595
$$959$$ 7434.00 0.250319
$$960$$ −2304.00 −0.0774597
$$961$$ 81765.0 2.74462
$$962$$ 43232.0 1.44891
$$963$$ −28428.0 −0.951277
$$964$$ 27440.0 0.916787
$$965$$ 37620.0 1.25495
$$966$$ −5040.00 −0.167867
$$967$$ −36448.0 −1.21209 −0.606044 0.795431i $$-0.707244\pi$$
−0.606044 + 0.795431i $$0.707244\pi$$
$$968$$ 968.000 0.0321412
$$969$$ 2016.00 0.0668351
$$970$$ −47880.0 −1.58488
$$971$$ −20526.0 −0.678384 −0.339192 0.940717i $$-0.610154\pi$$
−0.339192 + 0.940717i $$0.610154\pi$$
$$972$$ −14168.0 −0.467530
$$973$$ −3556.00 −0.117164
$$974$$ 10264.0 0.337659
$$975$$ −22288.0 −0.732089
$$976$$ 6080.00 0.199402
$$977$$ 37434.0 1.22581 0.612907 0.790155i $$-0.290000\pi$$
0.612907 + 0.790155i $$0.290000\pi$$
$$978$$ 640.000 0.0209253
$$979$$ 9570.00 0.312419
$$980$$ 3528.00 0.114998
$$981$$ 15962.0 0.519498
$$982$$ 8376.00 0.272188
$$983$$ −52194.0 −1.69352 −0.846760 0.531975i $$-0.821450\pi$$
−0.846760 + 0.531975i $$0.821450\pi$$
$$984$$ 7104.00 0.230150
$$985$$ −28836.0 −0.932783
$$986$$ −3888.00 −0.125577
$$987$$ 5628.00 0.181501
$$988$$ −6272.00 −0.201962
$$989$$ −56880.0 −1.82880
$$990$$ 9108.00 0.292395
$$991$$ −15220.0 −0.487870 −0.243935 0.969792i $$-0.578438\pi$$
−0.243935 + 0.969792i $$0.578438\pi$$
$$992$$ −10688.0 −0.342081
$$993$$ −13240.0 −0.423121
$$994$$ −4536.00 −0.144742
$$995$$ −58932.0 −1.87766
$$996$$ −3744.00 −0.119110
$$997$$ 37664.0 1.19642 0.598210 0.801339i $$-0.295879\pi$$
0.598210 + 0.801339i $$0.295879\pi$$
$$998$$ 7696.00 0.244101
$$999$$ 38600.0 1.22247
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 154.4.a.d.1.1 1
3.2 odd 2 1386.4.a.a.1.1 1
4.3 odd 2 1232.4.a.f.1.1 1
7.6 odd 2 1078.4.a.g.1.1 1
11.10 odd 2 1694.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
154.4.a.d.1.1 1 1.1 even 1 trivial
1078.4.a.g.1.1 1 7.6 odd 2
1232.4.a.f.1.1 1 4.3 odd 2
1386.4.a.a.1.1 1 3.2 odd 2
1694.4.a.c.1.1 1 11.10 odd 2