Properties

 Label 2850.2.a.bm.1.3 Level $2850$ Weight $2$ Character 2850.1 Self dual yes Analytic conductor $22.757$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.35026 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.35026 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.962389 q^{11} -1.00000 q^{12} +1.61213 q^{13} +3.35026 q^{14} +1.00000 q^{16} +0.387873 q^{17} +1.00000 q^{18} +1.00000 q^{19} -3.35026 q^{21} -0.962389 q^{22} +0.962389 q^{23} -1.00000 q^{24} +1.61213 q^{26} -1.00000 q^{27} +3.35026 q^{28} +6.96239 q^{29} +3.35026 q^{31} +1.00000 q^{32} +0.962389 q^{33} +0.387873 q^{34} +1.00000 q^{36} +1.61213 q^{37} +1.00000 q^{38} -1.61213 q^{39} -9.27504 q^{41} -3.35026 q^{42} -6.18664 q^{43} -0.962389 q^{44} +0.962389 q^{46} -0.962389 q^{47} -1.00000 q^{48} +4.22425 q^{49} -0.387873 q^{51} +1.61213 q^{52} -6.00000 q^{53} -1.00000 q^{54} +3.35026 q^{56} -1.00000 q^{57} +6.96239 q^{58} +10.3127 q^{59} +11.9248 q^{61} +3.35026 q^{62} +3.35026 q^{63} +1.00000 q^{64} +0.962389 q^{66} +7.22425 q^{67} +0.387873 q^{68} -0.962389 q^{69} +7.22425 q^{71} +1.00000 q^{72} +3.22425 q^{73} +1.61213 q^{74} +1.00000 q^{76} -3.22425 q^{77} -1.61213 q^{78} -3.35026 q^{79} +1.00000 q^{81} -9.27504 q^{82} -15.0132 q^{83} -3.35026 q^{84} -6.18664 q^{86} -6.96239 q^{87} -0.962389 q^{88} +4.64974 q^{89} +5.40105 q^{91} +0.962389 q^{92} -3.35026 q^{93} -0.962389 q^{94} -1.00000 q^{96} +10.9624 q^{97} +4.22425 q^{98} -0.962389 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + 8q^{11} - 3q^{12} + 4q^{13} + 3q^{16} + 2q^{17} + 3q^{18} + 3q^{19} + 8q^{22} - 8q^{23} - 3q^{24} + 4q^{26} - 3q^{27} + 10q^{29} + 3q^{32} - 8q^{33} + 2q^{34} + 3q^{36} + 4q^{37} + 3q^{38} - 4q^{39} + 4q^{41} - 6q^{43} + 8q^{44} - 8q^{46} + 8q^{47} - 3q^{48} + 11q^{49} - 2q^{51} + 4q^{52} - 18q^{53} - 3q^{54} - 3q^{57} + 10q^{58} + 10q^{59} + 14q^{61} + 3q^{64} - 8q^{66} + 20q^{67} + 2q^{68} + 8q^{69} + 20q^{71} + 3q^{72} + 8q^{73} + 4q^{74} + 3q^{76} - 8q^{77} - 4q^{78} + 3q^{81} + 4q^{82} - 4q^{83} - 6q^{86} - 10q^{87} + 8q^{88} + 24q^{89} - 24q^{91} - 8q^{92} + 8q^{94} - 3q^{96} + 22q^{97} + 11q^{98} + 8q^{99} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 3.35026 1.26628 0.633140 0.774037i $$-0.281766\pi$$
0.633140 + 0.774037i $$0.281766\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −0.962389 −0.290171 −0.145086 0.989419i $$-0.546346\pi$$
−0.145086 + 0.989419i $$0.546346\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.61213 0.447124 0.223562 0.974690i $$-0.428232\pi$$
0.223562 + 0.974690i $$0.428232\pi$$
$$14$$ 3.35026 0.895395
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0.387873 0.0940731 0.0470365 0.998893i $$-0.485022\pi$$
0.0470365 + 0.998893i $$0.485022\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −3.35026 −0.731087
$$22$$ −0.962389 −0.205182
$$23$$ 0.962389 0.200672 0.100336 0.994954i $$-0.468008\pi$$
0.100336 + 0.994954i $$0.468008\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 1.61213 0.316164
$$27$$ −1.00000 −0.192450
$$28$$ 3.35026 0.633140
$$29$$ 6.96239 1.29288 0.646442 0.762964i $$-0.276256\pi$$
0.646442 + 0.762964i $$0.276256\pi$$
$$30$$ 0 0
$$31$$ 3.35026 0.601725 0.300862 0.953668i $$-0.402726\pi$$
0.300862 + 0.953668i $$0.402726\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0.962389 0.167530
$$34$$ 0.387873 0.0665197
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.61213 0.265032 0.132516 0.991181i $$-0.457694\pi$$
0.132516 + 0.991181i $$0.457694\pi$$
$$38$$ 1.00000 0.162221
$$39$$ −1.61213 −0.258147
$$40$$ 0 0
$$41$$ −9.27504 −1.44852 −0.724259 0.689528i $$-0.757818\pi$$
−0.724259 + 0.689528i $$0.757818\pi$$
$$42$$ −3.35026 −0.516957
$$43$$ −6.18664 −0.943454 −0.471727 0.881745i $$-0.656369\pi$$
−0.471727 + 0.881745i $$0.656369\pi$$
$$44$$ −0.962389 −0.145086
$$45$$ 0 0
$$46$$ 0.962389 0.141896
$$47$$ −0.962389 −0.140379 −0.0701894 0.997534i $$-0.522360\pi$$
−0.0701894 + 0.997534i $$0.522360\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 4.22425 0.603465
$$50$$ 0 0
$$51$$ −0.387873 −0.0543131
$$52$$ 1.61213 0.223562
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 3.35026 0.447698
$$57$$ −1.00000 −0.132453
$$58$$ 6.96239 0.914206
$$59$$ 10.3127 1.34259 0.671296 0.741189i $$-0.265738\pi$$
0.671296 + 0.741189i $$0.265738\pi$$
$$60$$ 0 0
$$61$$ 11.9248 1.52681 0.763406 0.645919i $$-0.223526\pi$$
0.763406 + 0.645919i $$0.223526\pi$$
$$62$$ 3.35026 0.425484
$$63$$ 3.35026 0.422093
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.962389 0.118462
$$67$$ 7.22425 0.882583 0.441292 0.897364i $$-0.354520\pi$$
0.441292 + 0.897364i $$0.354520\pi$$
$$68$$ 0.387873 0.0470365
$$69$$ −0.962389 −0.115858
$$70$$ 0 0
$$71$$ 7.22425 0.857361 0.428681 0.903456i $$-0.358979\pi$$
0.428681 + 0.903456i $$0.358979\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 3.22425 0.377370 0.188685 0.982038i $$-0.439577\pi$$
0.188685 + 0.982038i $$0.439577\pi$$
$$74$$ 1.61213 0.187406
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ −3.22425 −0.367438
$$78$$ −1.61213 −0.182537
$$79$$ −3.35026 −0.376934 −0.188467 0.982080i $$-0.560352\pi$$
−0.188467 + 0.982080i $$0.560352\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −9.27504 −1.02426
$$83$$ −15.0132 −1.64791 −0.823955 0.566655i $$-0.808237\pi$$
−0.823955 + 0.566655i $$0.808237\pi$$
$$84$$ −3.35026 −0.365544
$$85$$ 0 0
$$86$$ −6.18664 −0.667123
$$87$$ −6.96239 −0.746446
$$88$$ −0.962389 −0.102591
$$89$$ 4.64974 0.492871 0.246436 0.969159i $$-0.420741\pi$$
0.246436 + 0.969159i $$0.420741\pi$$
$$90$$ 0 0
$$91$$ 5.40105 0.566184
$$92$$ 0.962389 0.100336
$$93$$ −3.35026 −0.347406
$$94$$ −0.962389 −0.0992628
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 10.9624 1.11306 0.556531 0.830827i $$-0.312132\pi$$
0.556531 + 0.830827i $$0.312132\pi$$
$$98$$ 4.22425 0.426714
$$99$$ −0.962389 −0.0967237
$$100$$ 0 0
$$101$$ 2.72496 0.271144 0.135572 0.990768i $$-0.456713\pi$$
0.135572 + 0.990768i $$0.456713\pi$$
$$102$$ −0.387873 −0.0384052
$$103$$ −0.574515 −0.0566087 −0.0283043 0.999599i $$-0.509011\pi$$
−0.0283043 + 0.999599i $$0.509011\pi$$
$$104$$ 1.61213 0.158082
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −10.7005 −1.03446 −0.517229 0.855847i $$-0.673037\pi$$
−0.517229 + 0.855847i $$0.673037\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 10.1260 0.969896 0.484948 0.874543i $$-0.338839\pi$$
0.484948 + 0.874543i $$0.338839\pi$$
$$110$$ 0 0
$$111$$ −1.61213 −0.153016
$$112$$ 3.35026 0.316570
$$113$$ −20.5501 −1.93319 −0.966594 0.256311i $$-0.917493\pi$$
−0.966594 + 0.256311i $$0.917493\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 6.96239 0.646442
$$117$$ 1.61213 0.149041
$$118$$ 10.3127 0.949356
$$119$$ 1.29948 0.119123
$$120$$ 0 0
$$121$$ −10.0738 −0.915801
$$122$$ 11.9248 1.07962
$$123$$ 9.27504 0.836302
$$124$$ 3.35026 0.300862
$$125$$ 0 0
$$126$$ 3.35026 0.298465
$$127$$ −1.35026 −0.119816 −0.0599082 0.998204i $$-0.519081\pi$$
−0.0599082 + 0.998204i $$0.519081\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 6.18664 0.544703
$$130$$ 0 0
$$131$$ 12.4387 1.08677 0.543385 0.839483i $$-0.317142\pi$$
0.543385 + 0.839483i $$0.317142\pi$$
$$132$$ 0.962389 0.0837652
$$133$$ 3.35026 0.290505
$$134$$ 7.22425 0.624080
$$135$$ 0 0
$$136$$ 0.387873 0.0332598
$$137$$ 18.1622 1.55170 0.775851 0.630916i $$-0.217321\pi$$
0.775851 + 0.630916i $$0.217321\pi$$
$$138$$ −0.962389 −0.0819240
$$139$$ 8.77575 0.744349 0.372175 0.928163i $$-0.378612\pi$$
0.372175 + 0.928163i $$0.378612\pi$$
$$140$$ 0 0
$$141$$ 0.962389 0.0810477
$$142$$ 7.22425 0.606246
$$143$$ −1.55149 −0.129742
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 3.22425 0.266841
$$147$$ −4.22425 −0.348411
$$148$$ 1.61213 0.132516
$$149$$ 15.9756 1.30877 0.654385 0.756162i $$-0.272928\pi$$
0.654385 + 0.756162i $$0.272928\pi$$
$$150$$ 0 0
$$151$$ 18.4241 1.49933 0.749665 0.661818i $$-0.230215\pi$$
0.749665 + 0.661818i $$0.230215\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0.387873 0.0313577
$$154$$ −3.22425 −0.259818
$$155$$ 0 0
$$156$$ −1.61213 −0.129073
$$157$$ 13.7889 1.10048 0.550238 0.835008i $$-0.314537\pi$$
0.550238 + 0.835008i $$0.314537\pi$$
$$158$$ −3.35026 −0.266533
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 3.22425 0.254107
$$162$$ 1.00000 0.0785674
$$163$$ 3.73813 0.292793 0.146397 0.989226i $$-0.453232\pi$$
0.146397 + 0.989226i $$0.453232\pi$$
$$164$$ −9.27504 −0.724259
$$165$$ 0 0
$$166$$ −15.0132 −1.16525
$$167$$ −15.4763 −1.19759 −0.598795 0.800902i $$-0.704354\pi$$
−0.598795 + 0.800902i $$0.704354\pi$$
$$168$$ −3.35026 −0.258478
$$169$$ −10.4010 −0.800081
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ −6.18664 −0.471727
$$173$$ 1.47627 0.112239 0.0561194 0.998424i $$-0.482127\pi$$
0.0561194 + 0.998424i $$0.482127\pi$$
$$174$$ −6.96239 −0.527817
$$175$$ 0 0
$$176$$ −0.962389 −0.0725428
$$177$$ −10.3127 −0.775146
$$178$$ 4.64974 0.348513
$$179$$ −14.3127 −1.06978 −0.534889 0.844922i $$-0.679647\pi$$
−0.534889 + 0.844922i $$0.679647\pi$$
$$180$$ 0 0
$$181$$ −8.82653 −0.656071 −0.328035 0.944665i $$-0.606387\pi$$
−0.328035 + 0.944665i $$0.606387\pi$$
$$182$$ 5.40105 0.400352
$$183$$ −11.9248 −0.881505
$$184$$ 0.962389 0.0709482
$$185$$ 0 0
$$186$$ −3.35026 −0.245653
$$187$$ −0.373285 −0.0272973
$$188$$ −0.962389 −0.0701894
$$189$$ −3.35026 −0.243696
$$190$$ 0 0
$$191$$ 2.31265 0.167338 0.0836688 0.996494i $$-0.473336\pi$$
0.0836688 + 0.996494i $$0.473336\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −7.58769 −0.546174 −0.273087 0.961989i $$-0.588045\pi$$
−0.273087 + 0.961989i $$0.588045\pi$$
$$194$$ 10.9624 0.787054
$$195$$ 0 0
$$196$$ 4.22425 0.301732
$$197$$ 18.8119 1.34030 0.670148 0.742228i $$-0.266231\pi$$
0.670148 + 0.742228i $$0.266231\pi$$
$$198$$ −0.962389 −0.0683940
$$199$$ −9.40105 −0.666423 −0.333211 0.942852i $$-0.608132\pi$$
−0.333211 + 0.942852i $$0.608132\pi$$
$$200$$ 0 0
$$201$$ −7.22425 −0.509560
$$202$$ 2.72496 0.191728
$$203$$ 23.3258 1.63715
$$204$$ −0.387873 −0.0271566
$$205$$ 0 0
$$206$$ −0.574515 −0.0400284
$$207$$ 0.962389 0.0668906
$$208$$ 1.61213 0.111781
$$209$$ −0.962389 −0.0665698
$$210$$ 0 0
$$211$$ 4.77575 0.328776 0.164388 0.986396i $$-0.447435\pi$$
0.164388 + 0.986396i $$0.447435\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −7.22425 −0.494998
$$214$$ −10.7005 −0.731473
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 11.2243 0.761952
$$218$$ 10.1260 0.685820
$$219$$ −3.22425 −0.217875
$$220$$ 0 0
$$221$$ 0.625301 0.0420623
$$222$$ −1.61213 −0.108199
$$223$$ 24.6761 1.65243 0.826216 0.563353i $$-0.190489\pi$$
0.826216 + 0.563353i $$0.190489\pi$$
$$224$$ 3.35026 0.223849
$$225$$ 0 0
$$226$$ −20.5501 −1.36697
$$227$$ −15.4763 −1.02720 −0.513598 0.858031i $$-0.671688\pi$$
−0.513598 + 0.858031i $$0.671688\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ 21.3258 1.40925 0.704625 0.709580i $$-0.251115\pi$$
0.704625 + 0.709580i $$0.251115\pi$$
$$230$$ 0 0
$$231$$ 3.22425 0.212140
$$232$$ 6.96239 0.457103
$$233$$ 9.01317 0.590473 0.295236 0.955424i $$-0.404602\pi$$
0.295236 + 0.955424i $$0.404602\pi$$
$$234$$ 1.61213 0.105388
$$235$$ 0 0
$$236$$ 10.3127 0.671296
$$237$$ 3.35026 0.217623
$$238$$ 1.29948 0.0842326
$$239$$ −0.135857 −0.00878787 −0.00439393 0.999990i $$-0.501399\pi$$
−0.00439393 + 0.999990i $$0.501399\pi$$
$$240$$ 0 0
$$241$$ −25.8496 −1.66512 −0.832558 0.553938i $$-0.813125\pi$$
−0.832558 + 0.553938i $$0.813125\pi$$
$$242$$ −10.0738 −0.647569
$$243$$ −1.00000 −0.0641500
$$244$$ 11.9248 0.763406
$$245$$ 0 0
$$246$$ 9.27504 0.591355
$$247$$ 1.61213 0.102577
$$248$$ 3.35026 0.212742
$$249$$ 15.0132 0.951421
$$250$$ 0 0
$$251$$ −10.1114 −0.638227 −0.319114 0.947716i $$-0.603385\pi$$
−0.319114 + 0.947716i $$0.603385\pi$$
$$252$$ 3.35026 0.211047
$$253$$ −0.926192 −0.0582292
$$254$$ −1.35026 −0.0847230
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −26.9986 −1.68413 −0.842063 0.539380i $$-0.818659\pi$$
−0.842063 + 0.539380i $$0.818659\pi$$
$$258$$ 6.18664 0.385164
$$259$$ 5.40105 0.335605
$$260$$ 0 0
$$261$$ 6.96239 0.430961
$$262$$ 12.4387 0.768463
$$263$$ −15.0376 −0.927259 −0.463629 0.886029i $$-0.653453\pi$$
−0.463629 + 0.886029i $$0.653453\pi$$
$$264$$ 0.962389 0.0592309
$$265$$ 0 0
$$266$$ 3.35026 0.205418
$$267$$ −4.64974 −0.284559
$$268$$ 7.22425 0.441292
$$269$$ −4.51388 −0.275216 −0.137608 0.990487i $$-0.543941\pi$$
−0.137608 + 0.990487i $$0.543941\pi$$
$$270$$ 0 0
$$271$$ 15.8496 0.962792 0.481396 0.876503i $$-0.340130\pi$$
0.481396 + 0.876503i $$0.340130\pi$$
$$272$$ 0.387873 0.0235183
$$273$$ −5.40105 −0.326886
$$274$$ 18.1622 1.09722
$$275$$ 0 0
$$276$$ −0.962389 −0.0579290
$$277$$ −10.3127 −0.619627 −0.309814 0.950797i $$-0.600267\pi$$
−0.309814 + 0.950797i $$0.600267\pi$$
$$278$$ 8.77575 0.526334
$$279$$ 3.35026 0.200575
$$280$$ 0 0
$$281$$ 24.3488 1.45253 0.726265 0.687415i $$-0.241254\pi$$
0.726265 + 0.687415i $$0.241254\pi$$
$$282$$ 0.962389 0.0573094
$$283$$ −26.2882 −1.56267 −0.781336 0.624111i $$-0.785461\pi$$
−0.781336 + 0.624111i $$0.785461\pi$$
$$284$$ 7.22425 0.428681
$$285$$ 0 0
$$286$$ −1.55149 −0.0917417
$$287$$ −31.0738 −1.83423
$$288$$ 1.00000 0.0589256
$$289$$ −16.8496 −0.991150
$$290$$ 0 0
$$291$$ −10.9624 −0.642627
$$292$$ 3.22425 0.188685
$$293$$ 13.0738 0.763780 0.381890 0.924208i $$-0.375273\pi$$
0.381890 + 0.924208i $$0.375273\pi$$
$$294$$ −4.22425 −0.246363
$$295$$ 0 0
$$296$$ 1.61213 0.0937030
$$297$$ 0.962389 0.0558435
$$298$$ 15.9756 0.925439
$$299$$ 1.55149 0.0897251
$$300$$ 0 0
$$301$$ −20.7269 −1.19468
$$302$$ 18.4241 1.06019
$$303$$ −2.72496 −0.156545
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0.387873 0.0221732
$$307$$ 7.07381 0.403724 0.201862 0.979414i $$-0.435301\pi$$
0.201862 + 0.979414i $$0.435301\pi$$
$$308$$ −3.22425 −0.183719
$$309$$ 0.574515 0.0326830
$$310$$ 0 0
$$311$$ 26.3127 1.49205 0.746027 0.665916i $$-0.231959\pi$$
0.746027 + 0.665916i $$0.231959\pi$$
$$312$$ −1.61213 −0.0912687
$$313$$ 18.7005 1.05702 0.528508 0.848928i $$-0.322752\pi$$
0.528508 + 0.848928i $$0.322752\pi$$
$$314$$ 13.7889 0.778154
$$315$$ 0 0
$$316$$ −3.35026 −0.188467
$$317$$ −26.4749 −1.48698 −0.743488 0.668749i $$-0.766830\pi$$
−0.743488 + 0.668749i $$0.766830\pi$$
$$318$$ 6.00000 0.336463
$$319$$ −6.70052 −0.375157
$$320$$ 0 0
$$321$$ 10.7005 0.597245
$$322$$ 3.22425 0.179681
$$323$$ 0.387873 0.0215818
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 3.73813 0.207036
$$327$$ −10.1260 −0.559970
$$328$$ −9.27504 −0.512128
$$329$$ −3.22425 −0.177759
$$330$$ 0 0
$$331$$ −30.7005 −1.68745 −0.843727 0.536773i $$-0.819643\pi$$
−0.843727 + 0.536773i $$0.819643\pi$$
$$332$$ −15.0132 −0.823955
$$333$$ 1.61213 0.0883440
$$334$$ −15.4763 −0.846824
$$335$$ 0 0
$$336$$ −3.35026 −0.182772
$$337$$ −6.81194 −0.371070 −0.185535 0.982638i $$-0.559402\pi$$
−0.185535 + 0.982638i $$0.559402\pi$$
$$338$$ −10.4010 −0.565742
$$339$$ 20.5501 1.11613
$$340$$ 0 0
$$341$$ −3.22425 −0.174603
$$342$$ 1.00000 0.0540738
$$343$$ −9.29948 −0.502125
$$344$$ −6.18664 −0.333561
$$345$$ 0 0
$$346$$ 1.47627 0.0793648
$$347$$ −18.3879 −0.987113 −0.493556 0.869714i $$-0.664303\pi$$
−0.493556 + 0.869714i $$0.664303\pi$$
$$348$$ −6.96239 −0.373223
$$349$$ 31.1490 1.66737 0.833685 0.552241i $$-0.186227\pi$$
0.833685 + 0.552241i $$0.186227\pi$$
$$350$$ 0 0
$$351$$ −1.61213 −0.0860490
$$352$$ −0.962389 −0.0512955
$$353$$ −7.61213 −0.405153 −0.202576 0.979266i $$-0.564931\pi$$
−0.202576 + 0.979266i $$0.564931\pi$$
$$354$$ −10.3127 −0.548111
$$355$$ 0 0
$$356$$ 4.64974 0.246436
$$357$$ −1.29948 −0.0687756
$$358$$ −14.3127 −0.756447
$$359$$ −35.3112 −1.86366 −0.931828 0.362901i $$-0.881786\pi$$
−0.931828 + 0.362901i $$0.881786\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −8.82653 −0.463912
$$363$$ 10.0738 0.528738
$$364$$ 5.40105 0.283092
$$365$$ 0 0
$$366$$ −11.9248 −0.623318
$$367$$ −31.9756 −1.66911 −0.834555 0.550924i $$-0.814275\pi$$
−0.834555 + 0.550924i $$0.814275\pi$$
$$368$$ 0.962389 0.0501680
$$369$$ −9.27504 −0.482839
$$370$$ 0 0
$$371$$ −20.1016 −1.04362
$$372$$ −3.35026 −0.173703
$$373$$ −26.4894 −1.37157 −0.685786 0.727804i $$-0.740541\pi$$
−0.685786 + 0.727804i $$0.740541\pi$$
$$374$$ −0.373285 −0.0193021
$$375$$ 0 0
$$376$$ −0.962389 −0.0496314
$$377$$ 11.2243 0.578078
$$378$$ −3.35026 −0.172319
$$379$$ −19.3258 −0.992701 −0.496350 0.868122i $$-0.665327\pi$$
−0.496350 + 0.868122i $$0.665327\pi$$
$$380$$ 0 0
$$381$$ 1.35026 0.0691760
$$382$$ 2.31265 0.118325
$$383$$ −3.37470 −0.172439 −0.0862195 0.996276i $$-0.527479\pi$$
−0.0862195 + 0.996276i $$0.527479\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −7.58769 −0.386203
$$387$$ −6.18664 −0.314485
$$388$$ 10.9624 0.556531
$$389$$ −11.3503 −0.575481 −0.287741 0.957708i $$-0.592904\pi$$
−0.287741 + 0.957708i $$0.592904\pi$$
$$390$$ 0 0
$$391$$ 0.373285 0.0188778
$$392$$ 4.22425 0.213357
$$393$$ −12.4387 −0.627447
$$394$$ 18.8119 0.947732
$$395$$ 0 0
$$396$$ −0.962389 −0.0483618
$$397$$ 18.8364 0.945371 0.472685 0.881231i $$-0.343285\pi$$
0.472685 + 0.881231i $$0.343285\pi$$
$$398$$ −9.40105 −0.471232
$$399$$ −3.35026 −0.167723
$$400$$ 0 0
$$401$$ −4.12601 −0.206043 −0.103022 0.994679i $$-0.532851\pi$$
−0.103022 + 0.994679i $$0.532851\pi$$
$$402$$ −7.22425 −0.360313
$$403$$ 5.40105 0.269045
$$404$$ 2.72496 0.135572
$$405$$ 0 0
$$406$$ 23.3258 1.15764
$$407$$ −1.55149 −0.0769046
$$408$$ −0.387873 −0.0192026
$$409$$ 2.52373 0.124790 0.0623952 0.998052i $$-0.480126\pi$$
0.0623952 + 0.998052i $$0.480126\pi$$
$$410$$ 0 0
$$411$$ −18.1622 −0.895875
$$412$$ −0.574515 −0.0283043
$$413$$ 34.5501 1.70010
$$414$$ 0.962389 0.0472988
$$415$$ 0 0
$$416$$ 1.61213 0.0790410
$$417$$ −8.77575 −0.429750
$$418$$ −0.962389 −0.0470720
$$419$$ 7.51247 0.367008 0.183504 0.983019i $$-0.441256\pi$$
0.183504 + 0.983019i $$0.441256\pi$$
$$420$$ 0 0
$$421$$ 3.67750 0.179230 0.0896152 0.995976i $$-0.471436\pi$$
0.0896152 + 0.995976i $$0.471436\pi$$
$$422$$ 4.77575 0.232480
$$423$$ −0.962389 −0.0467929
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −7.22425 −0.350016
$$427$$ 39.9511 1.93337
$$428$$ −10.7005 −0.517229
$$429$$ 1.55149 0.0749068
$$430$$ 0 0
$$431$$ −10.3272 −0.497446 −0.248723 0.968575i $$-0.580011\pi$$
−0.248723 + 0.968575i $$0.580011\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −31.5877 −1.51801 −0.759004 0.651086i $$-0.774314\pi$$
−0.759004 + 0.651086i $$0.774314\pi$$
$$434$$ 11.2243 0.538781
$$435$$ 0 0
$$436$$ 10.1260 0.484948
$$437$$ 0.962389 0.0460373
$$438$$ −3.22425 −0.154061
$$439$$ −38.1524 −1.82091 −0.910456 0.413605i $$-0.864269\pi$$
−0.910456 + 0.413605i $$0.864269\pi$$
$$440$$ 0 0
$$441$$ 4.22425 0.201155
$$442$$ 0.625301 0.0297425
$$443$$ −16.3127 −0.775037 −0.387519 0.921862i $$-0.626668\pi$$
−0.387519 + 0.921862i $$0.626668\pi$$
$$444$$ −1.61213 −0.0765082
$$445$$ 0 0
$$446$$ 24.6761 1.16845
$$447$$ −15.9756 −0.755618
$$448$$ 3.35026 0.158285
$$449$$ −37.5271 −1.77101 −0.885506 0.464629i $$-0.846188\pi$$
−0.885506 + 0.464629i $$0.846188\pi$$
$$450$$ 0 0
$$451$$ 8.92619 0.420318
$$452$$ −20.5501 −0.966594
$$453$$ −18.4241 −0.865638
$$454$$ −15.4763 −0.726337
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ 21.3258 0.996490
$$459$$ −0.387873 −0.0181044
$$460$$ 0 0
$$461$$ 12.3780 0.576502 0.288251 0.957555i $$-0.406926\pi$$
0.288251 + 0.957555i $$0.406926\pi$$
$$462$$ 3.22425 0.150006
$$463$$ −32.4504 −1.50810 −0.754049 0.656818i $$-0.771902\pi$$
−0.754049 + 0.656818i $$0.771902\pi$$
$$464$$ 6.96239 0.323221
$$465$$ 0 0
$$466$$ 9.01317 0.417527
$$467$$ −7.53690 −0.348766 −0.174383 0.984678i $$-0.555793\pi$$
−0.174383 + 0.984678i $$0.555793\pi$$
$$468$$ 1.61213 0.0745206
$$469$$ 24.2031 1.11760
$$470$$ 0 0
$$471$$ −13.7889 −0.635360
$$472$$ 10.3127 0.474678
$$473$$ 5.95395 0.273763
$$474$$ 3.35026 0.153883
$$475$$ 0 0
$$476$$ 1.29948 0.0595614
$$477$$ −6.00000 −0.274721
$$478$$ −0.135857 −0.00621396
$$479$$ 15.2097 0.694947 0.347474 0.937690i $$-0.387040\pi$$
0.347474 + 0.937690i $$0.387040\pi$$
$$480$$ 0 0
$$481$$ 2.59895 0.118502
$$482$$ −25.8496 −1.17741
$$483$$ −3.22425 −0.146709
$$484$$ −10.0738 −0.457900
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 1.19982 0.0543689 0.0271844 0.999630i $$-0.491346\pi$$
0.0271844 + 0.999630i $$0.491346\pi$$
$$488$$ 11.9248 0.539809
$$489$$ −3.73813 −0.169044
$$490$$ 0 0
$$491$$ −5.11283 −0.230739 −0.115369 0.993323i $$-0.536805\pi$$
−0.115369 + 0.993323i $$0.536805\pi$$
$$492$$ 9.27504 0.418151
$$493$$ 2.70052 0.121625
$$494$$ 1.61213 0.0725330
$$495$$ 0 0
$$496$$ 3.35026 0.150431
$$497$$ 24.2031 1.08566
$$498$$ 15.0132 0.672756
$$499$$ 14.2981 0.640069 0.320035 0.947406i $$-0.396305\pi$$
0.320035 + 0.947406i $$0.396305\pi$$
$$500$$ 0 0
$$501$$ 15.4763 0.691429
$$502$$ −10.1114 −0.451295
$$503$$ −11.5125 −0.513316 −0.256658 0.966502i $$-0.582621\pi$$
−0.256658 + 0.966502i $$0.582621\pi$$
$$504$$ 3.35026 0.149233
$$505$$ 0 0
$$506$$ −0.926192 −0.0411742
$$507$$ 10.4010 0.461927
$$508$$ −1.35026 −0.0599082
$$509$$ −31.9610 −1.41665 −0.708323 0.705889i $$-0.750548\pi$$
−0.708323 + 0.705889i $$0.750548\pi$$
$$510$$ 0 0
$$511$$ 10.8021 0.477857
$$512$$ 1.00000 0.0441942
$$513$$ −1.00000 −0.0441511
$$514$$ −26.9986 −1.19086
$$515$$ 0 0
$$516$$ 6.18664 0.272352
$$517$$ 0.926192 0.0407339
$$518$$ 5.40105 0.237308
$$519$$ −1.47627 −0.0648010
$$520$$ 0 0
$$521$$ −33.2750 −1.45781 −0.728903 0.684617i $$-0.759969\pi$$
−0.728903 + 0.684617i $$0.759969\pi$$
$$522$$ 6.96239 0.304735
$$523$$ 9.29948 0.406638 0.203319 0.979113i $$-0.434827\pi$$
0.203319 + 0.979113i $$0.434827\pi$$
$$524$$ 12.4387 0.543385
$$525$$ 0 0
$$526$$ −15.0376 −0.655671
$$527$$ 1.29948 0.0566061
$$528$$ 0.962389 0.0418826
$$529$$ −22.0738 −0.959731
$$530$$ 0 0
$$531$$ 10.3127 0.447531
$$532$$ 3.35026 0.145252
$$533$$ −14.9525 −0.647666
$$534$$ −4.64974 −0.201214
$$535$$ 0 0
$$536$$ 7.22425 0.312040
$$537$$ 14.3127 0.617636
$$538$$ −4.51388 −0.194607
$$539$$ −4.06537 −0.175108
$$540$$ 0 0
$$541$$ 28.5501 1.22746 0.613732 0.789515i $$-0.289668\pi$$
0.613732 + 0.789515i $$0.289668\pi$$
$$542$$ 15.8496 0.680797
$$543$$ 8.82653 0.378783
$$544$$ 0.387873 0.0166299
$$545$$ 0 0
$$546$$ −5.40105 −0.231143
$$547$$ 28.4749 1.21750 0.608748 0.793363i $$-0.291672\pi$$
0.608748 + 0.793363i $$0.291672\pi$$
$$548$$ 18.1622 0.775851
$$549$$ 11.9248 0.508937
$$550$$ 0 0
$$551$$ 6.96239 0.296608
$$552$$ −0.962389 −0.0409620
$$553$$ −11.2243 −0.477304
$$554$$ −10.3127 −0.438143
$$555$$ 0 0
$$556$$ 8.77575 0.372175
$$557$$ 4.88717 0.207076 0.103538 0.994626i $$-0.466984\pi$$
0.103538 + 0.994626i $$0.466984\pi$$
$$558$$ 3.35026 0.141828
$$559$$ −9.97365 −0.421841
$$560$$ 0 0
$$561$$ 0.373285 0.0157601
$$562$$ 24.3488 1.02709
$$563$$ −30.8021 −1.29815 −0.649077 0.760723i $$-0.724845\pi$$
−0.649077 + 0.760723i $$0.724845\pi$$
$$564$$ 0.962389 0.0405239
$$565$$ 0 0
$$566$$ −26.2882 −1.10498
$$567$$ 3.35026 0.140698
$$568$$ 7.22425 0.303123
$$569$$ −24.1260 −1.01141 −0.505707 0.862705i $$-0.668768\pi$$
−0.505707 + 0.862705i $$0.668768\pi$$
$$570$$ 0 0
$$571$$ −5.67276 −0.237398 −0.118699 0.992930i $$-0.537872\pi$$
−0.118699 + 0.992930i $$0.537872\pi$$
$$572$$ −1.55149 −0.0648712
$$573$$ −2.31265 −0.0966124
$$574$$ −31.0738 −1.29700
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 32.0000 1.33218 0.666089 0.745873i $$-0.267967\pi$$
0.666089 + 0.745873i $$0.267967\pi$$
$$578$$ −16.8496 −0.700849
$$579$$ 7.58769 0.315334
$$580$$ 0 0
$$581$$ −50.2981 −2.08672
$$582$$ −10.9624 −0.454406
$$583$$ 5.77433 0.239148
$$584$$ 3.22425 0.133421
$$585$$ 0 0
$$586$$ 13.0738 0.540074
$$587$$ 13.6121 0.561833 0.280916 0.959732i $$-0.409362\pi$$
0.280916 + 0.959732i $$0.409362\pi$$
$$588$$ −4.22425 −0.174205
$$589$$ 3.35026 0.138045
$$590$$ 0 0
$$591$$ −18.8119 −0.773820
$$592$$ 1.61213 0.0662580
$$593$$ −23.4617 −0.963456 −0.481728 0.876321i $$-0.659991\pi$$
−0.481728 + 0.876321i $$0.659991\pi$$
$$594$$ 0.962389 0.0394873
$$595$$ 0 0
$$596$$ 15.9756 0.654385
$$597$$ 9.40105 0.384759
$$598$$ 1.55149 0.0634452
$$599$$ 10.0263 0.409665 0.204833 0.978797i $$-0.434335\pi$$
0.204833 + 0.978797i $$0.434335\pi$$
$$600$$ 0 0
$$601$$ 11.7743 0.480285 0.240143 0.970738i $$-0.422806\pi$$
0.240143 + 0.970738i $$0.422806\pi$$
$$602$$ −20.7269 −0.844764
$$603$$ 7.22425 0.294194
$$604$$ 18.4241 0.749665
$$605$$ 0 0
$$606$$ −2.72496 −0.110694
$$607$$ −38.4993 −1.56264 −0.781319 0.624132i $$-0.785453\pi$$
−0.781319 + 0.624132i $$0.785453\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ −23.3258 −0.945210
$$610$$ 0 0
$$611$$ −1.55149 −0.0627667
$$612$$ 0.387873 0.0156788
$$613$$ 7.61213 0.307451 0.153725 0.988114i $$-0.450873\pi$$
0.153725 + 0.988114i $$0.450873\pi$$
$$614$$ 7.07381 0.285476
$$615$$ 0 0
$$616$$ −3.22425 −0.129909
$$617$$ 29.5369 1.18911 0.594555 0.804055i $$-0.297328\pi$$
0.594555 + 0.804055i $$0.297328\pi$$
$$618$$ 0.574515 0.0231104
$$619$$ −22.5501 −0.906364 −0.453182 0.891418i $$-0.649711\pi$$
−0.453182 + 0.891418i $$0.649711\pi$$
$$620$$ 0 0
$$621$$ −0.962389 −0.0386193
$$622$$ 26.3127 1.05504
$$623$$ 15.5778 0.624113
$$624$$ −1.61213 −0.0645367
$$625$$ 0 0
$$626$$ 18.7005 0.747423
$$627$$ 0.962389 0.0384341
$$628$$ 13.7889 0.550238
$$629$$ 0.625301 0.0249324
$$630$$ 0 0
$$631$$ −37.9248 −1.50976 −0.754881 0.655862i $$-0.772305\pi$$
−0.754881 + 0.655862i $$0.772305\pi$$
$$632$$ −3.35026 −0.133266
$$633$$ −4.77575 −0.189819
$$634$$ −26.4749 −1.05145
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 6.81003 0.269823
$$638$$ −6.70052 −0.265276
$$639$$ 7.22425 0.285787
$$640$$ 0 0
$$641$$ 6.67609 0.263690 0.131845 0.991270i $$-0.457910\pi$$
0.131845 + 0.991270i $$0.457910\pi$$
$$642$$ 10.7005 0.422316
$$643$$ 25.2605 0.996175 0.498087 0.867127i $$-0.334036\pi$$
0.498087 + 0.867127i $$0.334036\pi$$
$$644$$ 3.22425 0.127053
$$645$$ 0 0
$$646$$ 0.387873 0.0152607
$$647$$ 16.9135 0.664939 0.332469 0.943114i $$-0.392118\pi$$
0.332469 + 0.943114i $$0.392118\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −9.92478 −0.389582
$$650$$ 0 0
$$651$$ −11.2243 −0.439913
$$652$$ 3.73813 0.146397
$$653$$ 24.1114 0.943553 0.471776 0.881718i $$-0.343613\pi$$
0.471776 + 0.881718i $$0.343613\pi$$
$$654$$ −10.1260 −0.395958
$$655$$ 0 0
$$656$$ −9.27504 −0.362129
$$657$$ 3.22425 0.125790
$$658$$ −3.22425 −0.125694
$$659$$ −20.3879 −0.794199 −0.397099 0.917776i $$-0.629983\pi$$
−0.397099 + 0.917776i $$0.629983\pi$$
$$660$$ 0 0
$$661$$ 17.6023 0.684649 0.342325 0.939582i $$-0.388786\pi$$
0.342325 + 0.939582i $$0.388786\pi$$
$$662$$ −30.7005 −1.19321
$$663$$ −0.625301 −0.0242847
$$664$$ −15.0132 −0.582624
$$665$$ 0 0
$$666$$ 1.61213 0.0624686
$$667$$ 6.70052 0.259445
$$668$$ −15.4763 −0.598795
$$669$$ −24.6761 −0.954033
$$670$$ 0 0
$$671$$ −11.4763 −0.443036
$$672$$ −3.35026 −0.129239
$$673$$ 44.3634 1.71008 0.855042 0.518558i $$-0.173531\pi$$
0.855042 + 0.518558i $$0.173531\pi$$
$$674$$ −6.81194 −0.262386
$$675$$ 0 0
$$676$$ −10.4010 −0.400040
$$677$$ 8.70052 0.334388 0.167194 0.985924i $$-0.446529\pi$$
0.167194 + 0.985924i $$0.446529\pi$$
$$678$$ 20.5501 0.789221
$$679$$ 36.7269 1.40945
$$680$$ 0 0
$$681$$ 15.4763 0.593052
$$682$$ −3.22425 −0.123463
$$683$$ −37.8759 −1.44928 −0.724641 0.689127i $$-0.757994\pi$$
−0.724641 + 0.689127i $$0.757994\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ −9.29948 −0.355056
$$687$$ −21.3258 −0.813631
$$688$$ −6.18664 −0.235864
$$689$$ −9.67276 −0.368503
$$690$$ 0 0
$$691$$ −0.775746 −0.0295108 −0.0147554 0.999891i $$-0.504697\pi$$
−0.0147554 + 0.999891i $$0.504697\pi$$
$$692$$ 1.47627 0.0561194
$$693$$ −3.22425 −0.122479
$$694$$ −18.3879 −0.697994
$$695$$ 0 0
$$696$$ −6.96239 −0.263909
$$697$$ −3.59754 −0.136266
$$698$$ 31.1490 1.17901
$$699$$ −9.01317 −0.340910
$$700$$ 0 0
$$701$$ 42.3752 1.60049 0.800245 0.599674i $$-0.204703\pi$$
0.800245 + 0.599674i $$0.204703\pi$$
$$702$$ −1.61213 −0.0608458
$$703$$ 1.61213 0.0608025
$$704$$ −0.962389 −0.0362714
$$705$$ 0 0
$$706$$ −7.61213 −0.286486
$$707$$ 9.12933 0.343344
$$708$$ −10.3127 −0.387573
$$709$$ 36.2784 1.36246 0.681231 0.732068i $$-0.261445\pi$$
0.681231 + 0.732068i $$0.261445\pi$$
$$710$$ 0 0
$$711$$ −3.35026 −0.125645
$$712$$ 4.64974 0.174256
$$713$$ 3.22425 0.120749
$$714$$ −1.29948 −0.0486317
$$715$$ 0 0
$$716$$ −14.3127 −0.534889
$$717$$ 0.135857 0.00507368
$$718$$ −35.3112 −1.31780
$$719$$ −42.5355 −1.58631 −0.793153 0.609022i $$-0.791562\pi$$
−0.793153 + 0.609022i $$0.791562\pi$$
$$720$$ 0 0
$$721$$ −1.92478 −0.0716824
$$722$$ 1.00000 0.0372161
$$723$$ 25.8496 0.961355
$$724$$ −8.82653 −0.328035
$$725$$ 0 0
$$726$$ 10.0738 0.373874
$$727$$ −0.378024 −0.0140201 −0.00701007 0.999975i $$-0.502231\pi$$
−0.00701007 + 0.999975i $$0.502231\pi$$
$$728$$ 5.40105 0.200176
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −2.39963 −0.0887536
$$732$$ −11.9248 −0.440752
$$733$$ 26.0118 0.960766 0.480383 0.877059i $$-0.340498\pi$$
0.480383 + 0.877059i $$0.340498\pi$$
$$734$$ −31.9756 −1.18024
$$735$$ 0 0
$$736$$ 0.962389 0.0354741
$$737$$ −6.95254 −0.256100
$$738$$ −9.27504 −0.341419
$$739$$ −44.8773 −1.65084 −0.825419 0.564520i $$-0.809061\pi$$
−0.825419 + 0.564520i $$0.809061\pi$$
$$740$$ 0 0
$$741$$ −1.61213 −0.0592230
$$742$$ −20.1016 −0.737952
$$743$$ −4.67418 −0.171479 −0.0857394 0.996318i $$-0.527325\pi$$
−0.0857394 + 0.996318i $$0.527325\pi$$
$$744$$ −3.35026 −0.122827
$$745$$ 0 0
$$746$$ −26.4894 −0.969847
$$747$$ −15.0132 −0.549303
$$748$$ −0.373285 −0.0136486
$$749$$ −35.8496 −1.30991
$$750$$ 0 0
$$751$$ −6.57452 −0.239907 −0.119954 0.992779i $$-0.538275\pi$$
−0.119954 + 0.992779i $$0.538275\pi$$
$$752$$ −0.962389 −0.0350947
$$753$$ 10.1114 0.368481
$$754$$ 11.2243 0.408763
$$755$$ 0 0
$$756$$ −3.35026 −0.121848
$$757$$ −15.5633 −0.565656 −0.282828 0.959171i $$-0.591273\pi$$
−0.282828 + 0.959171i $$0.591273\pi$$
$$758$$ −19.3258 −0.701946
$$759$$ 0.926192 0.0336186
$$760$$ 0 0
$$761$$ −23.8759 −0.865501 −0.432750 0.901514i $$-0.642457\pi$$
−0.432750 + 0.901514i $$0.642457\pi$$
$$762$$ 1.35026 0.0489148
$$763$$ 33.9248 1.22816
$$764$$ 2.31265 0.0836688
$$765$$ 0 0
$$766$$ −3.37470 −0.121933
$$767$$ 16.6253 0.600305
$$768$$ −1.00000 −0.0360844
$$769$$ 30.4749 1.09895 0.549476 0.835510i $$-0.314827\pi$$
0.549476 + 0.835510i $$0.314827\pi$$
$$770$$ 0 0
$$771$$ 26.9986 0.972330
$$772$$ −7.58769 −0.273087
$$773$$ 16.3272 0.587250 0.293625 0.955921i $$-0.405138\pi$$
0.293625 + 0.955921i $$0.405138\pi$$
$$774$$ −6.18664 −0.222374
$$775$$ 0 0
$$776$$ 10.9624 0.393527
$$777$$ −5.40105 −0.193761
$$778$$ −11.3503 −0.406927
$$779$$ −9.27504 −0.332313
$$780$$ 0 0
$$781$$ −6.95254 −0.248781
$$782$$ 0.373285 0.0133486
$$783$$ −6.96239 −0.248815
$$784$$ 4.22425 0.150866
$$785$$ 0 0
$$786$$ −12.4387 −0.443672
$$787$$ 30.9525 1.10334 0.551669 0.834063i $$-0.313991\pi$$
0.551669 + 0.834063i $$0.313991\pi$$
$$788$$ 18.8119 0.670148
$$789$$ 15.0376 0.535353
$$790$$ 0 0
$$791$$ −68.8481 −2.44796
$$792$$ −0.962389 −0.0341970
$$793$$ 19.2243 0.682673
$$794$$ 18.8364 0.668478
$$795$$ 0 0
$$796$$ −9.40105 −0.333211
$$797$$ 14.8773 0.526982 0.263491 0.964662i $$-0.415126\pi$$
0.263491 + 0.964662i $$0.415126\pi$$
$$798$$ −3.35026 −0.118598
$$799$$ −0.373285 −0.0132059
$$800$$ 0 0
$$801$$ 4.64974 0.164290
$$802$$ −4.12601 −0.145694
$$803$$ −3.10299 −0.109502
$$804$$ −7.22425 −0.254780
$$805$$ 0 0
$$806$$ 5.40105 0.190244
$$807$$ 4.51388 0.158896
$$808$$ 2.72496 0.0958638
$$809$$ 25.4471 0.894672 0.447336 0.894366i $$-0.352373\pi$$
0.447336 + 0.894366i $$0.352373\pi$$
$$810$$ 0 0
$$811$$ −53.3522 −1.87345 −0.936724 0.350069i $$-0.886158\pi$$
−0.936724 + 0.350069i $$0.886158\pi$$
$$812$$ 23.3258 0.818576
$$813$$ −15.8496 −0.555868
$$814$$ −1.55149 −0.0543798
$$815$$ 0 0
$$816$$ −0.387873 −0.0135783
$$817$$ −6.18664 −0.216443
$$818$$ 2.52373 0.0882402
$$819$$ 5.40105 0.188728
$$820$$ 0 0
$$821$$ −27.9756 −0.976354 −0.488177 0.872745i $$-0.662338\pi$$
−0.488177 + 0.872745i $$0.662338\pi$$
$$822$$ −18.1622 −0.633480
$$823$$ −7.87399 −0.274470 −0.137235 0.990539i $$-0.543822\pi$$
−0.137235 + 0.990539i $$0.543822\pi$$
$$824$$ −0.574515 −0.0200142
$$825$$ 0 0
$$826$$ 34.5501 1.20215
$$827$$ 40.1016 1.39447 0.697234 0.716843i $$-0.254414\pi$$
0.697234 + 0.716843i $$0.254414\pi$$
$$828$$ 0.962389 0.0334453
$$829$$ 47.2262 1.64023 0.820116 0.572197i $$-0.193909\pi$$
0.820116 + 0.572197i $$0.193909\pi$$
$$830$$ 0 0
$$831$$ 10.3127 0.357742
$$832$$ 1.61213 0.0558904
$$833$$ 1.63847 0.0567698
$$834$$ −8.77575 −0.303879
$$835$$ 0 0
$$836$$ −0.962389 −0.0332849
$$837$$ −3.35026 −0.115802
$$838$$ 7.51247 0.259514
$$839$$ −3.89843 −0.134589 −0.0672944 0.997733i $$-0.521437\pi$$
−0.0672944 + 0.997733i $$0.521437\pi$$
$$840$$ 0 0
$$841$$ 19.4749 0.671547
$$842$$ 3.67750 0.126735
$$843$$ −24.3488 −0.838619
$$844$$ 4.77575 0.164388
$$845$$ 0 0
$$846$$ −0.962389 −0.0330876
$$847$$ −33.7499 −1.15966
$$848$$ −6.00000 −0.206041
$$849$$ 26.2882 0.902209
$$850$$ 0 0
$$851$$ 1.55149 0.0531845
$$852$$ −7.22425 −0.247499
$$853$$ 35.1900 1.20488 0.602441 0.798164i $$-0.294195\pi$$
0.602441 + 0.798164i $$0.294195\pi$$
$$854$$ 39.9511 1.36710
$$855$$ 0 0
$$856$$ −10.7005 −0.365736
$$857$$ −12.1768 −0.415951 −0.207976 0.978134i $$-0.566687\pi$$
−0.207976 + 0.978134i $$0.566687\pi$$
$$858$$ 1.55149 0.0529671
$$859$$ 57.3522 1.95683 0.978415 0.206648i $$-0.0662554\pi$$
0.978415 + 0.206648i $$0.0662554\pi$$
$$860$$ 0 0
$$861$$ 31.0738 1.05899
$$862$$ −10.3272 −0.351747
$$863$$ 14.0752 0.479126 0.239563 0.970881i $$-0.422996\pi$$
0.239563 + 0.970881i $$0.422996\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −31.5877 −1.07339
$$867$$ 16.8496 0.572241
$$868$$ 11.2243 0.380976
$$869$$ 3.22425 0.109375
$$870$$ 0 0
$$871$$ 11.6464 0.394624
$$872$$ 10.1260 0.342910
$$873$$ 10.9624 0.371021
$$874$$ 0.962389 0.0325533
$$875$$ 0 0
$$876$$ −3.22425 −0.108937
$$877$$ −18.8627 −0.636949 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$878$$ −38.1524 −1.28758
$$879$$ −13.0738 −0.440969
$$880$$ 0 0
$$881$$ −19.1490 −0.645147 −0.322574 0.946544i $$-0.604548\pi$$
−0.322574 + 0.946544i $$0.604548\pi$$
$$882$$ 4.22425 0.142238
$$883$$ −42.9135 −1.44415 −0.722077 0.691812i $$-0.756813\pi$$
−0.722077 + 0.691812i $$0.756813\pi$$
$$884$$ 0.625301 0.0210311
$$885$$ 0 0
$$886$$ −16.3127 −0.548034
$$887$$ −22.7005 −0.762209 −0.381104 0.924532i $$-0.624456\pi$$
−0.381104 + 0.924532i $$0.624456\pi$$
$$888$$ −1.61213 −0.0540994
$$889$$ −4.52373 −0.151721
$$890$$ 0 0
$$891$$ −0.962389 −0.0322412
$$892$$ 24.6761 0.826216
$$893$$ −0.962389 −0.0322051
$$894$$ −15.9756 −0.534303
$$895$$ 0 0
$$896$$ 3.35026 0.111924
$$897$$ −1.55149 −0.0518028
$$898$$ −37.5271 −1.25229
$$899$$ 23.3258 0.777960
$$900$$ 0 0
$$901$$ −2.32724 −0.0775316
$$902$$ 8.92619 0.297210
$$903$$ 20.7269 0.689747
$$904$$ −20.5501 −0.683485
$$905$$ 0 0
$$906$$ −18.4241 −0.612099
$$907$$ −14.5501 −0.483127 −0.241564 0.970385i $$-0.577660\pi$$
−0.241564 + 0.970385i $$0.577660\pi$$
$$908$$ −15.4763 −0.513598
$$909$$ 2.72496 0.0903813
$$910$$ 0 0
$$911$$ −0.998585 −0.0330846 −0.0165423 0.999863i $$-0.505266\pi$$
−0.0165423 + 0.999863i $$0.505266\pi$$
$$912$$ −1.00000 −0.0331133
$$913$$ 14.4485 0.478176
$$914$$ 8.00000 0.264616
$$915$$ 0 0
$$916$$ 21.3258 0.704625
$$917$$ 41.6728 1.37616
$$918$$ −0.387873 −0.0128017
$$919$$ −33.7743 −1.11411 −0.557056 0.830475i $$-0.688069\pi$$
−0.557056 + 0.830475i $$0.688069\pi$$
$$920$$ 0 0
$$921$$ −7.07381 −0.233090
$$922$$ 12.3780 0.407649
$$923$$ 11.6464 0.383346
$$924$$ 3.22425 0.106070
$$925$$ 0 0
$$926$$ −32.4504 −1.06639
$$927$$ −0.574515 −0.0188696
$$928$$ 6.96239 0.228552
$$929$$ 14.8773 0.488109 0.244054 0.969762i $$-0.421522\pi$$
0.244054 + 0.969762i $$0.421522\pi$$
$$930$$ 0 0
$$931$$ 4.22425 0.138444
$$932$$ 9.01317 0.295236
$$933$$ −26.3127 −0.861437
$$934$$ −7.53690 −0.246615
$$935$$ 0 0
$$936$$ 1.61213 0.0526940
$$937$$ −22.2981 −0.728446 −0.364223 0.931312i $$-0.618665\pi$$
−0.364223 + 0.931312i $$0.618665\pi$$
$$938$$ 24.2031 0.790261
$$939$$ −18.7005 −0.610269
$$940$$ 0 0
$$941$$ −36.3634 −1.18541 −0.592707 0.805418i $$-0.701941\pi$$
−0.592707 + 0.805418i $$0.701941\pi$$
$$942$$ −13.7889 −0.449267
$$943$$ −8.92619 −0.290677
$$944$$ 10.3127 0.335648
$$945$$ 0 0
$$946$$ 5.95395 0.193580
$$947$$ −50.3390 −1.63580 −0.817899 0.575362i $$-0.804861\pi$$
−0.817899 + 0.575362i $$0.804861\pi$$
$$948$$ 3.35026 0.108811
$$949$$ 5.19791 0.168731
$$950$$ 0 0
$$951$$ 26.4749 0.858506
$$952$$ 1.29948 0.0421163
$$953$$ 16.3272 0.528891 0.264446 0.964401i $$-0.414811\pi$$
0.264446 + 0.964401i $$0.414811\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −0.135857 −0.00439393
$$957$$ 6.70052 0.216597
$$958$$ 15.2097 0.491402
$$959$$ 60.8481 1.96489
$$960$$ 0 0
$$961$$ −19.7757 −0.637927
$$962$$ 2.59895 0.0837936
$$963$$ −10.7005 −0.344820
$$964$$ −25.8496 −0.832558
$$965$$ 0 0
$$966$$ −3.22425 −0.103739
$$967$$ −0.276454 −0.00889015 −0.00444507 0.999990i $$-0.501415\pi$$
−0.00444507 + 0.999990i $$0.501415\pi$$
$$968$$ −10.0738 −0.323784
$$969$$ −0.387873 −0.0124603
$$970$$ 0 0
$$971$$ 37.9102 1.21660 0.608298 0.793709i $$-0.291853\pi$$
0.608298 + 0.793709i $$0.291853\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 29.4010 0.942554
$$974$$ 1.19982 0.0384446
$$975$$ 0 0
$$976$$ 11.9248 0.381703
$$977$$ −28.3996 −0.908585 −0.454292 0.890853i $$-0.650108\pi$$
−0.454292 + 0.890853i $$0.650108\pi$$
$$978$$ −3.73813 −0.119532
$$979$$ −4.47486 −0.143017
$$980$$ 0 0
$$981$$ 10.1260 0.323299
$$982$$ −5.11283 −0.163157
$$983$$ 0.926192 0.0295409 0.0147705 0.999891i $$-0.495298\pi$$
0.0147705 + 0.999891i $$0.495298\pi$$
$$984$$ 9.27504 0.295677
$$985$$ 0 0
$$986$$ 2.70052 0.0860022
$$987$$ 3.22425 0.102629
$$988$$ 1.61213 0.0512886
$$989$$ −5.95395 −0.189325
$$990$$ 0 0
$$991$$ 30.5256 0.969679 0.484839 0.874603i $$-0.338878\pi$$
0.484839 + 0.874603i $$0.338878\pi$$
$$992$$ 3.35026 0.106371
$$993$$ 30.7005 0.974252
$$994$$ 24.2031 0.767677
$$995$$ 0 0
$$996$$ 15.0132 0.475711
$$997$$ −21.0132 −0.665494 −0.332747 0.943016i $$-0.607975\pi$$
−0.332747 + 0.943016i $$0.607975\pi$$
$$998$$ 14.2981 0.452597
$$999$$ −1.61213 −0.0510054
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 2850.2.a.bm.1.3 3
3.2 odd 2 8550.2.a.ce.1.3 3
5.2 odd 4 570.2.d.c.229.4 yes 6
5.3 odd 4 570.2.d.c.229.1 6
5.4 even 2 2850.2.a.bl.1.1 3
15.2 even 4 1710.2.d.f.1369.3 6
15.8 even 4 1710.2.d.f.1369.6 6
15.14 odd 2 8550.2.a.cq.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.c.229.1 6 5.3 odd 4
570.2.d.c.229.4 yes 6 5.2 odd 4
1710.2.d.f.1369.3 6 15.2 even 4
1710.2.d.f.1369.6 6 15.8 even 4
2850.2.a.bl.1.1 3 5.4 even 2
2850.2.a.bm.1.3 3 1.1 even 1 trivial
8550.2.a.ce.1.3 3 3.2 odd 2
8550.2.a.cq.1.1 3 15.14 odd 2