# Properties

 Label 5070.2.a.bc.1.1 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.65685 q^{11} +1.00000 q^{12} +2.82843 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.82843 q^{17} -1.00000 q^{18} +2.82843 q^{19} -1.00000 q^{20} -2.82843 q^{21} -5.65685 q^{22} +8.48528 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -2.82843 q^{28} -3.17157 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +5.65685 q^{33} +4.82843 q^{34} +2.82843 q^{35} +1.00000 q^{36} +0.343146 q^{37} -2.82843 q^{38} +1.00000 q^{40} -3.65685 q^{41} +2.82843 q^{42} -1.65685 q^{43} +5.65685 q^{44} -1.00000 q^{45} -8.48528 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -4.82843 q^{51} -9.31371 q^{53} -1.00000 q^{54} -5.65685 q^{55} +2.82843 q^{56} +2.82843 q^{57} +3.17157 q^{58} +13.6569 q^{59} -1.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} -2.82843 q^{63} +1.00000 q^{64} -5.65685 q^{66} +5.65685 q^{67} -4.82843 q^{68} +8.48528 q^{69} -2.82843 q^{70} -5.65685 q^{71} -1.00000 q^{72} -2.48528 q^{73} -0.343146 q^{74} +1.00000 q^{75} +2.82843 q^{76} -16.0000 q^{77} +13.6569 q^{79} -1.00000 q^{80} +1.00000 q^{81} +3.65685 q^{82} -17.6569 q^{83} -2.82843 q^{84} +4.82843 q^{85} +1.65685 q^{86} -3.17157 q^{87} -5.65685 q^{88} +4.34315 q^{89} +1.00000 q^{90} +8.48528 q^{92} -4.00000 q^{93} -8.00000 q^{94} -2.82843 q^{95} -1.00000 q^{96} +8.82843 q^{97} -1.00000 q^{98} +5.65685 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} - 2q^{15} + 2q^{16} - 4q^{17} - 2q^{18} - 2q^{20} - 2q^{24} + 2q^{25} + 2q^{27} - 12q^{29} + 2q^{30} - 8q^{31} - 2q^{32} + 4q^{34} + 2q^{36} + 12q^{37} + 2q^{40} + 4q^{41} + 8q^{43} - 2q^{45} + 16q^{47} + 2q^{48} + 2q^{49} - 2q^{50} - 4q^{51} + 4q^{53} - 2q^{54} + 12q^{58} + 16q^{59} - 2q^{60} + 12q^{61} + 8q^{62} + 2q^{64} - 4q^{68} - 2q^{72} + 12q^{73} - 12q^{74} + 2q^{75} - 32q^{77} + 16q^{79} - 2q^{80} + 2q^{81} - 4q^{82} - 24q^{83} + 4q^{85} - 8q^{86} - 12q^{87} + 20q^{89} + 2q^{90} - 8q^{93} - 16q^{94} - 2q^{96} + 12q^{97} - 2q^{98} + O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −2.82843 −1.06904 −0.534522 0.845154i $$-0.679509\pi$$
−0.534522 + 0.845154i $$0.679509\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 5.65685 1.70561 0.852803 0.522233i $$-0.174901\pi$$
0.852803 + 0.522233i $$0.174901\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 2.82843 0.755929
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −4.82843 −1.17107 −0.585533 0.810649i $$-0.699115\pi$$
−0.585533 + 0.810649i $$0.699115\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 2.82843 0.648886 0.324443 0.945905i $$-0.394823\pi$$
0.324443 + 0.945905i $$0.394823\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −2.82843 −0.617213
$$22$$ −5.65685 −1.20605
$$23$$ 8.48528 1.76930 0.884652 0.466252i $$-0.154396\pi$$
0.884652 + 0.466252i $$0.154396\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ −2.82843 −0.534522
$$29$$ −3.17157 −0.588946 −0.294473 0.955660i $$-0.595144\pi$$
−0.294473 + 0.955660i $$0.595144\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 5.65685 0.984732
$$34$$ 4.82843 0.828068
$$35$$ 2.82843 0.478091
$$36$$ 1.00000 0.166667
$$37$$ 0.343146 0.0564128 0.0282064 0.999602i $$-0.491020\pi$$
0.0282064 + 0.999602i $$0.491020\pi$$
$$38$$ −2.82843 −0.458831
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −3.65685 −0.571105 −0.285552 0.958363i $$-0.592177\pi$$
−0.285552 + 0.958363i $$0.592177\pi$$
$$42$$ 2.82843 0.436436
$$43$$ −1.65685 −0.252668 −0.126334 0.991988i $$-0.540321\pi$$
−0.126334 + 0.991988i $$0.540321\pi$$
$$44$$ 5.65685 0.852803
$$45$$ −1.00000 −0.149071
$$46$$ −8.48528 −1.25109
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ −4.82843 −0.676115
$$52$$ 0 0
$$53$$ −9.31371 −1.27934 −0.639668 0.768651i $$-0.720928\pi$$
−0.639668 + 0.768651i $$0.720928\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −5.65685 −0.762770
$$56$$ 2.82843 0.377964
$$57$$ 2.82843 0.374634
$$58$$ 3.17157 0.416448
$$59$$ 13.6569 1.77797 0.888985 0.457935i $$-0.151411\pi$$
0.888985 + 0.457935i $$0.151411\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −2.82843 −0.356348
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −5.65685 −0.696311
$$67$$ 5.65685 0.691095 0.345547 0.938401i $$-0.387693\pi$$
0.345547 + 0.938401i $$0.387693\pi$$
$$68$$ −4.82843 −0.585533
$$69$$ 8.48528 1.02151
$$70$$ −2.82843 −0.338062
$$71$$ −5.65685 −0.671345 −0.335673 0.941979i $$-0.608964\pi$$
−0.335673 + 0.941979i $$0.608964\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −2.48528 −0.290880 −0.145440 0.989367i $$-0.546460\pi$$
−0.145440 + 0.989367i $$0.546460\pi$$
$$74$$ −0.343146 −0.0398899
$$75$$ 1.00000 0.115470
$$76$$ 2.82843 0.324443
$$77$$ −16.0000 −1.82337
$$78$$ 0 0
$$79$$ 13.6569 1.53652 0.768258 0.640140i $$-0.221124\pi$$
0.768258 + 0.640140i $$0.221124\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 3.65685 0.403832
$$83$$ −17.6569 −1.93809 −0.969046 0.246881i $$-0.920594\pi$$
−0.969046 + 0.246881i $$0.920594\pi$$
$$84$$ −2.82843 −0.308607
$$85$$ 4.82843 0.523716
$$86$$ 1.65685 0.178663
$$87$$ −3.17157 −0.340028
$$88$$ −5.65685 −0.603023
$$89$$ 4.34315 0.460373 0.230186 0.973147i $$-0.426066\pi$$
0.230186 + 0.973147i $$0.426066\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 8.48528 0.884652
$$93$$ −4.00000 −0.414781
$$94$$ −8.00000 −0.825137
$$95$$ −2.82843 −0.290191
$$96$$ −1.00000 −0.102062
$$97$$ 8.82843 0.896391 0.448195 0.893936i $$-0.352067\pi$$
0.448195 + 0.893936i $$0.352067\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 5.65685 0.568535
$$100$$ 1.00000 0.100000
$$101$$ −12.1421 −1.20819 −0.604094 0.796913i $$-0.706465\pi$$
−0.604094 + 0.796913i $$0.706465\pi$$
$$102$$ 4.82843 0.478086
$$103$$ 9.65685 0.951518 0.475759 0.879576i $$-0.342173\pi$$
0.475759 + 0.879576i $$0.342173\pi$$
$$104$$ 0 0
$$105$$ 2.82843 0.276026
$$106$$ 9.31371 0.904627
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −3.17157 −0.303782 −0.151891 0.988397i $$-0.548536\pi$$
−0.151891 + 0.988397i $$0.548536\pi$$
$$110$$ 5.65685 0.539360
$$111$$ 0.343146 0.0325700
$$112$$ −2.82843 −0.267261
$$113$$ −10.4853 −0.986372 −0.493186 0.869924i $$-0.664168\pi$$
−0.493186 + 0.869924i $$0.664168\pi$$
$$114$$ −2.82843 −0.264906
$$115$$ −8.48528 −0.791257
$$116$$ −3.17157 −0.294473
$$117$$ 0 0
$$118$$ −13.6569 −1.25722
$$119$$ 13.6569 1.25192
$$120$$ 1.00000 0.0912871
$$121$$ 21.0000 1.90909
$$122$$ −6.00000 −0.543214
$$123$$ −3.65685 −0.329727
$$124$$ −4.00000 −0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ 2.82843 0.251976
$$127$$ −1.65685 −0.147022 −0.0735110 0.997294i $$-0.523420\pi$$
−0.0735110 + 0.997294i $$0.523420\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −1.65685 −0.145878
$$130$$ 0 0
$$131$$ 22.1421 1.93457 0.967284 0.253697i $$-0.0816467\pi$$
0.967284 + 0.253697i $$0.0816467\pi$$
$$132$$ 5.65685 0.492366
$$133$$ −8.00000 −0.693688
$$134$$ −5.65685 −0.488678
$$135$$ −1.00000 −0.0860663
$$136$$ 4.82843 0.414034
$$137$$ −5.31371 −0.453981 −0.226990 0.973897i $$-0.572889\pi$$
−0.226990 + 0.973897i $$0.572889\pi$$
$$138$$ −8.48528 −0.722315
$$139$$ 17.6569 1.49763 0.748817 0.662776i $$-0.230622\pi$$
0.748817 + 0.662776i $$0.230622\pi$$
$$140$$ 2.82843 0.239046
$$141$$ 8.00000 0.673722
$$142$$ 5.65685 0.474713
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 3.17157 0.263385
$$146$$ 2.48528 0.205683
$$147$$ 1.00000 0.0824786
$$148$$ 0.343146 0.0282064
$$149$$ 7.65685 0.627274 0.313637 0.949543i $$-0.398453\pi$$
0.313637 + 0.949543i $$0.398453\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ −2.82843 −0.229416
$$153$$ −4.82843 −0.390355
$$154$$ 16.0000 1.28932
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −17.3137 −1.38178 −0.690892 0.722958i $$-0.742782\pi$$
−0.690892 + 0.722958i $$0.742782\pi$$
$$158$$ −13.6569 −1.08648
$$159$$ −9.31371 −0.738625
$$160$$ 1.00000 0.0790569
$$161$$ −24.0000 −1.89146
$$162$$ −1.00000 −0.0785674
$$163$$ 11.3137 0.886158 0.443079 0.896483i $$-0.353886\pi$$
0.443079 + 0.896483i $$0.353886\pi$$
$$164$$ −3.65685 −0.285552
$$165$$ −5.65685 −0.440386
$$166$$ 17.6569 1.37044
$$167$$ 24.9706 1.93228 0.966140 0.258018i $$-0.0830694\pi$$
0.966140 + 0.258018i $$0.0830694\pi$$
$$168$$ 2.82843 0.218218
$$169$$ 0 0
$$170$$ −4.82843 −0.370323
$$171$$ 2.82843 0.216295
$$172$$ −1.65685 −0.126334
$$173$$ 13.3137 1.01222 0.506111 0.862468i $$-0.331083\pi$$
0.506111 + 0.862468i $$0.331083\pi$$
$$174$$ 3.17157 0.240436
$$175$$ −2.82843 −0.213809
$$176$$ 5.65685 0.426401
$$177$$ 13.6569 1.02651
$$178$$ −4.34315 −0.325533
$$179$$ 24.4853 1.83012 0.915058 0.403322i $$-0.132145\pi$$
0.915058 + 0.403322i $$0.132145\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 3.65685 0.271812 0.135906 0.990722i $$-0.456606\pi$$
0.135906 + 0.990722i $$0.456606\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ −8.48528 −0.625543
$$185$$ −0.343146 −0.0252286
$$186$$ 4.00000 0.293294
$$187$$ −27.3137 −1.99738
$$188$$ 8.00000 0.583460
$$189$$ −2.82843 −0.205738
$$190$$ 2.82843 0.205196
$$191$$ −11.3137 −0.818631 −0.409316 0.912393i $$-0.634232\pi$$
−0.409316 + 0.912393i $$0.634232\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 14.4853 1.04267 0.521337 0.853351i $$-0.325434\pi$$
0.521337 + 0.853351i $$0.325434\pi$$
$$194$$ −8.82843 −0.633844
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −9.31371 −0.663574 −0.331787 0.943354i $$-0.607652\pi$$
−0.331787 + 0.943354i $$0.607652\pi$$
$$198$$ −5.65685 −0.402015
$$199$$ 21.6569 1.53521 0.767607 0.640921i $$-0.221447\pi$$
0.767607 + 0.640921i $$0.221447\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 5.65685 0.399004
$$202$$ 12.1421 0.854318
$$203$$ 8.97056 0.629610
$$204$$ −4.82843 −0.338058
$$205$$ 3.65685 0.255406
$$206$$ −9.65685 −0.672825
$$207$$ 8.48528 0.589768
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ −2.82843 −0.195180
$$211$$ 23.3137 1.60498 0.802491 0.596664i $$-0.203508\pi$$
0.802491 + 0.596664i $$0.203508\pi$$
$$212$$ −9.31371 −0.639668
$$213$$ −5.65685 −0.387601
$$214$$ 4.00000 0.273434
$$215$$ 1.65685 0.112997
$$216$$ −1.00000 −0.0680414
$$217$$ 11.3137 0.768025
$$218$$ 3.17157 0.214806
$$219$$ −2.48528 −0.167940
$$220$$ −5.65685 −0.381385
$$221$$ 0 0
$$222$$ −0.343146 −0.0230304
$$223$$ −5.17157 −0.346314 −0.173157 0.984894i $$-0.555397\pi$$
−0.173157 + 0.984894i $$0.555397\pi$$
$$224$$ 2.82843 0.188982
$$225$$ 1.00000 0.0666667
$$226$$ 10.4853 0.697471
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 2.82843 0.187317
$$229$$ 24.1421 1.59536 0.797679 0.603083i $$-0.206061\pi$$
0.797679 + 0.603083i $$0.206061\pi$$
$$230$$ 8.48528 0.559503
$$231$$ −16.0000 −1.05272
$$232$$ 3.17157 0.208224
$$233$$ 22.4853 1.47306 0.736530 0.676405i $$-0.236463\pi$$
0.736530 + 0.676405i $$0.236463\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 13.6569 0.888985
$$237$$ 13.6569 0.887108
$$238$$ −13.6569 −0.885242
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 17.3137 1.11527 0.557637 0.830085i $$-0.311708\pi$$
0.557637 + 0.830085i $$0.311708\pi$$
$$242$$ −21.0000 −1.34993
$$243$$ 1.00000 0.0641500
$$244$$ 6.00000 0.384111
$$245$$ −1.00000 −0.0638877
$$246$$ 3.65685 0.233153
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ −17.6569 −1.11896
$$250$$ 1.00000 0.0632456
$$251$$ 5.17157 0.326427 0.163213 0.986591i $$-0.447814\pi$$
0.163213 + 0.986591i $$0.447814\pi$$
$$252$$ −2.82843 −0.178174
$$253$$ 48.0000 3.01773
$$254$$ 1.65685 0.103960
$$255$$ 4.82843 0.302368
$$256$$ 1.00000 0.0625000
$$257$$ 0.828427 0.0516759 0.0258379 0.999666i $$-0.491775\pi$$
0.0258379 + 0.999666i $$0.491775\pi$$
$$258$$ 1.65685 0.103151
$$259$$ −0.970563 −0.0603078
$$260$$ 0 0
$$261$$ −3.17157 −0.196315
$$262$$ −22.1421 −1.36795
$$263$$ −0.485281 −0.0299237 −0.0149619 0.999888i $$-0.504763\pi$$
−0.0149619 + 0.999888i $$0.504763\pi$$
$$264$$ −5.65685 −0.348155
$$265$$ 9.31371 0.572137
$$266$$ 8.00000 0.490511
$$267$$ 4.34315 0.265796
$$268$$ 5.65685 0.345547
$$269$$ 2.48528 0.151530 0.0757651 0.997126i $$-0.475860\pi$$
0.0757651 + 0.997126i $$0.475860\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 15.3137 0.930242 0.465121 0.885247i $$-0.346011\pi$$
0.465121 + 0.885247i $$0.346011\pi$$
$$272$$ −4.82843 −0.292766
$$273$$ 0 0
$$274$$ 5.31371 0.321013
$$275$$ 5.65685 0.341121
$$276$$ 8.48528 0.510754
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ −17.6569 −1.05899
$$279$$ −4.00000 −0.239474
$$280$$ −2.82843 −0.169031
$$281$$ −19.6569 −1.17263 −0.586315 0.810083i $$-0.699422\pi$$
−0.586315 + 0.810083i $$0.699422\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −6.34315 −0.377061 −0.188530 0.982067i $$-0.560372\pi$$
−0.188530 + 0.982067i $$0.560372\pi$$
$$284$$ −5.65685 −0.335673
$$285$$ −2.82843 −0.167542
$$286$$ 0 0
$$287$$ 10.3431 0.610537
$$288$$ −1.00000 −0.0589256
$$289$$ 6.31371 0.371395
$$290$$ −3.17157 −0.186241
$$291$$ 8.82843 0.517532
$$292$$ −2.48528 −0.145440
$$293$$ −28.6274 −1.67243 −0.836216 0.548401i $$-0.815237\pi$$
−0.836216 + 0.548401i $$0.815237\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ −13.6569 −0.795133
$$296$$ −0.343146 −0.0199449
$$297$$ 5.65685 0.328244
$$298$$ −7.65685 −0.443550
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 4.68629 0.270113
$$302$$ 12.0000 0.690522
$$303$$ −12.1421 −0.697547
$$304$$ 2.82843 0.162221
$$305$$ −6.00000 −0.343559
$$306$$ 4.82843 0.276023
$$307$$ 10.3431 0.590315 0.295157 0.955449i $$-0.404628\pi$$
0.295157 + 0.955449i $$0.404628\pi$$
$$308$$ −16.0000 −0.911685
$$309$$ 9.65685 0.549359
$$310$$ −4.00000 −0.227185
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 2.97056 0.167906 0.0839531 0.996470i $$-0.473245\pi$$
0.0839531 + 0.996470i $$0.473245\pi$$
$$314$$ 17.3137 0.977069
$$315$$ 2.82843 0.159364
$$316$$ 13.6569 0.768258
$$317$$ −2.68629 −0.150877 −0.0754386 0.997150i $$-0.524036\pi$$
−0.0754386 + 0.997150i $$0.524036\pi$$
$$318$$ 9.31371 0.522287
$$319$$ −17.9411 −1.00451
$$320$$ −1.00000 −0.0559017
$$321$$ −4.00000 −0.223258
$$322$$ 24.0000 1.33747
$$323$$ −13.6569 −0.759888
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −11.3137 −0.626608
$$327$$ −3.17157 −0.175388
$$328$$ 3.65685 0.201916
$$329$$ −22.6274 −1.24749
$$330$$ 5.65685 0.311400
$$331$$ −8.48528 −0.466393 −0.233197 0.972430i $$-0.574919\pi$$
−0.233197 + 0.972430i $$0.574919\pi$$
$$332$$ −17.6569 −0.969046
$$333$$ 0.343146 0.0188043
$$334$$ −24.9706 −1.36633
$$335$$ −5.65685 −0.309067
$$336$$ −2.82843 −0.154303
$$337$$ −22.9706 −1.25129 −0.625643 0.780109i $$-0.715163\pi$$
−0.625643 + 0.780109i $$0.715163\pi$$
$$338$$ 0 0
$$339$$ −10.4853 −0.569482
$$340$$ 4.82843 0.261858
$$341$$ −22.6274 −1.22534
$$342$$ −2.82843 −0.152944
$$343$$ 16.9706 0.916324
$$344$$ 1.65685 0.0893316
$$345$$ −8.48528 −0.456832
$$346$$ −13.3137 −0.715749
$$347$$ 1.65685 0.0889446 0.0444723 0.999011i $$-0.485839\pi$$
0.0444723 + 0.999011i $$0.485839\pi$$
$$348$$ −3.17157 −0.170014
$$349$$ 16.1421 0.864069 0.432034 0.901857i $$-0.357796\pi$$
0.432034 + 0.901857i $$0.357796\pi$$
$$350$$ 2.82843 0.151186
$$351$$ 0 0
$$352$$ −5.65685 −0.301511
$$353$$ 17.3137 0.921516 0.460758 0.887526i $$-0.347578\pi$$
0.460758 + 0.887526i $$0.347578\pi$$
$$354$$ −13.6569 −0.725854
$$355$$ 5.65685 0.300235
$$356$$ 4.34315 0.230186
$$357$$ 13.6569 0.722797
$$358$$ −24.4853 −1.29409
$$359$$ −28.2843 −1.49279 −0.746393 0.665505i $$-0.768216\pi$$
−0.746393 + 0.665505i $$0.768216\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −11.0000 −0.578947
$$362$$ −3.65685 −0.192200
$$363$$ 21.0000 1.10221
$$364$$ 0 0
$$365$$ 2.48528 0.130086
$$366$$ −6.00000 −0.313625
$$367$$ 14.3431 0.748706 0.374353 0.927286i $$-0.377865\pi$$
0.374353 + 0.927286i $$0.377865\pi$$
$$368$$ 8.48528 0.442326
$$369$$ −3.65685 −0.190368
$$370$$ 0.343146 0.0178393
$$371$$ 26.3431 1.36767
$$372$$ −4.00000 −0.207390
$$373$$ −25.3137 −1.31069 −0.655347 0.755328i $$-0.727478\pi$$
−0.655347 + 0.755328i $$0.727478\pi$$
$$374$$ 27.3137 1.41236
$$375$$ −1.00000 −0.0516398
$$376$$ −8.00000 −0.412568
$$377$$ 0 0
$$378$$ 2.82843 0.145479
$$379$$ 24.4853 1.25772 0.628862 0.777517i $$-0.283521\pi$$
0.628862 + 0.777517i $$0.283521\pi$$
$$380$$ −2.82843 −0.145095
$$381$$ −1.65685 −0.0848832
$$382$$ 11.3137 0.578860
$$383$$ 18.3431 0.937291 0.468645 0.883386i $$-0.344742\pi$$
0.468645 + 0.883386i $$0.344742\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 16.0000 0.815436
$$386$$ −14.4853 −0.737281
$$387$$ −1.65685 −0.0842226
$$388$$ 8.82843 0.448195
$$389$$ 10.4853 0.531625 0.265812 0.964025i $$-0.414360\pi$$
0.265812 + 0.964025i $$0.414360\pi$$
$$390$$ 0 0
$$391$$ −40.9706 −2.07197
$$392$$ −1.00000 −0.0505076
$$393$$ 22.1421 1.11692
$$394$$ 9.31371 0.469218
$$395$$ −13.6569 −0.687151
$$396$$ 5.65685 0.284268
$$397$$ 26.2843 1.31917 0.659585 0.751630i $$-0.270732\pi$$
0.659585 + 0.751630i $$0.270732\pi$$
$$398$$ −21.6569 −1.08556
$$399$$ −8.00000 −0.400501
$$400$$ 1.00000 0.0500000
$$401$$ −6.97056 −0.348093 −0.174047 0.984737i $$-0.555684\pi$$
−0.174047 + 0.984737i $$0.555684\pi$$
$$402$$ −5.65685 −0.282138
$$403$$ 0 0
$$404$$ −12.1421 −0.604094
$$405$$ −1.00000 −0.0496904
$$406$$ −8.97056 −0.445202
$$407$$ 1.94113 0.0962180
$$408$$ 4.82843 0.239043
$$409$$ −7.65685 −0.378607 −0.189304 0.981919i $$-0.560623\pi$$
−0.189304 + 0.981919i $$0.560623\pi$$
$$410$$ −3.65685 −0.180599
$$411$$ −5.31371 −0.262106
$$412$$ 9.65685 0.475759
$$413$$ −38.6274 −1.90073
$$414$$ −8.48528 −0.417029
$$415$$ 17.6569 0.866741
$$416$$ 0 0
$$417$$ 17.6569 0.864660
$$418$$ −16.0000 −0.782586
$$419$$ −5.17157 −0.252648 −0.126324 0.991989i $$-0.540318\pi$$
−0.126324 + 0.991989i $$0.540318\pi$$
$$420$$ 2.82843 0.138013
$$421$$ −4.14214 −0.201875 −0.100938 0.994893i $$-0.532184\pi$$
−0.100938 + 0.994893i $$0.532184\pi$$
$$422$$ −23.3137 −1.13489
$$423$$ 8.00000 0.388973
$$424$$ 9.31371 0.452314
$$425$$ −4.82843 −0.234213
$$426$$ 5.65685 0.274075
$$427$$ −16.9706 −0.821263
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ −1.65685 −0.0799006
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 10.9706 0.527212 0.263606 0.964630i $$-0.415088\pi$$
0.263606 + 0.964630i $$0.415088\pi$$
$$434$$ −11.3137 −0.543075
$$435$$ 3.17157 0.152065
$$436$$ −3.17157 −0.151891
$$437$$ 24.0000 1.14808
$$438$$ 2.48528 0.118751
$$439$$ −22.6274 −1.07995 −0.539974 0.841682i $$-0.681566\pi$$
−0.539974 + 0.841682i $$0.681566\pi$$
$$440$$ 5.65685 0.269680
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 41.6569 1.97918 0.989588 0.143926i $$-0.0459728\pi$$
0.989588 + 0.143926i $$0.0459728\pi$$
$$444$$ 0.343146 0.0162850
$$445$$ −4.34315 −0.205885
$$446$$ 5.17157 0.244881
$$447$$ 7.65685 0.362157
$$448$$ −2.82843 −0.133631
$$449$$ 30.2843 1.42920 0.714602 0.699532i $$-0.246608\pi$$
0.714602 + 0.699532i $$0.246608\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −20.6863 −0.974079
$$452$$ −10.4853 −0.493186
$$453$$ −12.0000 −0.563809
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ −2.82843 −0.132453
$$457$$ −15.1716 −0.709696 −0.354848 0.934924i $$-0.615467\pi$$
−0.354848 + 0.934924i $$0.615467\pi$$
$$458$$ −24.1421 −1.12809
$$459$$ −4.82843 −0.225372
$$460$$ −8.48528 −0.395628
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 16.0000 0.744387
$$463$$ −35.7990 −1.66372 −0.831860 0.554985i $$-0.812724\pi$$
−0.831860 + 0.554985i $$0.812724\pi$$
$$464$$ −3.17157 −0.147237
$$465$$ 4.00000 0.185496
$$466$$ −22.4853 −1.04161
$$467$$ −15.3137 −0.708634 −0.354317 0.935125i $$-0.615287\pi$$
−0.354317 + 0.935125i $$0.615287\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 8.00000 0.369012
$$471$$ −17.3137 −0.797774
$$472$$ −13.6569 −0.628608
$$473$$ −9.37258 −0.430952
$$474$$ −13.6569 −0.627280
$$475$$ 2.82843 0.129777
$$476$$ 13.6569 0.625961
$$477$$ −9.31371 −0.426445
$$478$$ 16.0000 0.731823
$$479$$ −11.3137 −0.516937 −0.258468 0.966020i $$-0.583218\pi$$
−0.258468 + 0.966020i $$0.583218\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −17.3137 −0.788618
$$483$$ −24.0000 −1.09204
$$484$$ 21.0000 0.954545
$$485$$ −8.82843 −0.400878
$$486$$ −1.00000 −0.0453609
$$487$$ 0.485281 0.0219902 0.0109951 0.999940i $$-0.496500\pi$$
0.0109951 + 0.999940i $$0.496500\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 11.3137 0.511624
$$490$$ 1.00000 0.0451754
$$491$$ −9.85786 −0.444879 −0.222440 0.974946i $$-0.571402\pi$$
−0.222440 + 0.974946i $$0.571402\pi$$
$$492$$ −3.65685 −0.164864
$$493$$ 15.3137 0.689695
$$494$$ 0 0
$$495$$ −5.65685 −0.254257
$$496$$ −4.00000 −0.179605
$$497$$ 16.0000 0.717698
$$498$$ 17.6569 0.791223
$$499$$ 16.4853 0.737983 0.368991 0.929433i $$-0.379703\pi$$
0.368991 + 0.929433i $$0.379703\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 24.9706 1.11560
$$502$$ −5.17157 −0.230819
$$503$$ −40.4853 −1.80515 −0.902575 0.430533i $$-0.858326\pi$$
−0.902575 + 0.430533i $$0.858326\pi$$
$$504$$ 2.82843 0.125988
$$505$$ 12.1421 0.540318
$$506$$ −48.0000 −2.13386
$$507$$ 0 0
$$508$$ −1.65685 −0.0735110
$$509$$ 14.6863 0.650958 0.325479 0.945549i $$-0.394474\pi$$
0.325479 + 0.945549i $$0.394474\pi$$
$$510$$ −4.82843 −0.213806
$$511$$ 7.02944 0.310964
$$512$$ −1.00000 −0.0441942
$$513$$ 2.82843 0.124878
$$514$$ −0.828427 −0.0365404
$$515$$ −9.65685 −0.425532
$$516$$ −1.65685 −0.0729389
$$517$$ 45.2548 1.99031
$$518$$ 0.970563 0.0426441
$$519$$ 13.3137 0.584407
$$520$$ 0 0
$$521$$ −6.97056 −0.305386 −0.152693 0.988274i $$-0.548795\pi$$
−0.152693 + 0.988274i $$0.548795\pi$$
$$522$$ 3.17157 0.138816
$$523$$ 34.6274 1.51415 0.757076 0.653327i $$-0.226627\pi$$
0.757076 + 0.653327i $$0.226627\pi$$
$$524$$ 22.1421 0.967284
$$525$$ −2.82843 −0.123443
$$526$$ 0.485281 0.0211593
$$527$$ 19.3137 0.841318
$$528$$ 5.65685 0.246183
$$529$$ 49.0000 2.13043
$$530$$ −9.31371 −0.404562
$$531$$ 13.6569 0.592657
$$532$$ −8.00000 −0.346844
$$533$$ 0 0
$$534$$ −4.34315 −0.187946
$$535$$ 4.00000 0.172935
$$536$$ −5.65685 −0.244339
$$537$$ 24.4853 1.05662
$$538$$ −2.48528 −0.107148
$$539$$ 5.65685 0.243658
$$540$$ −1.00000 −0.0430331
$$541$$ 2.48528 0.106851 0.0534253 0.998572i $$-0.482986\pi$$
0.0534253 + 0.998572i $$0.482986\pi$$
$$542$$ −15.3137 −0.657780
$$543$$ 3.65685 0.156931
$$544$$ 4.82843 0.207017
$$545$$ 3.17157 0.135855
$$546$$ 0 0
$$547$$ 23.3137 0.996822 0.498411 0.866941i $$-0.333917\pi$$
0.498411 + 0.866941i $$0.333917\pi$$
$$548$$ −5.31371 −0.226990
$$549$$ 6.00000 0.256074
$$550$$ −5.65685 −0.241209
$$551$$ −8.97056 −0.382159
$$552$$ −8.48528 −0.361158
$$553$$ −38.6274 −1.64260
$$554$$ −26.0000 −1.10463
$$555$$ −0.343146 −0.0145657
$$556$$ 17.6569 0.748817
$$557$$ −33.3137 −1.41155 −0.705774 0.708437i $$-0.749400\pi$$
−0.705774 + 0.708437i $$0.749400\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 0 0
$$560$$ 2.82843 0.119523
$$561$$ −27.3137 −1.15319
$$562$$ 19.6569 0.829174
$$563$$ −41.6569 −1.75563 −0.877814 0.479002i $$-0.840998\pi$$
−0.877814 + 0.479002i $$0.840998\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 10.4853 0.441119
$$566$$ 6.34315 0.266622
$$567$$ −2.82843 −0.118783
$$568$$ 5.65685 0.237356
$$569$$ 20.3431 0.852829 0.426415 0.904528i $$-0.359776\pi$$
0.426415 + 0.904528i $$0.359776\pi$$
$$570$$ 2.82843 0.118470
$$571$$ −12.9706 −0.542801 −0.271401 0.962466i $$-0.587487\pi$$
−0.271401 + 0.962466i $$0.587487\pi$$
$$572$$ 0 0
$$573$$ −11.3137 −0.472637
$$574$$ −10.3431 −0.431715
$$575$$ 8.48528 0.353861
$$576$$ 1.00000 0.0416667
$$577$$ −27.4558 −1.14300 −0.571501 0.820601i $$-0.693639\pi$$
−0.571501 + 0.820601i $$0.693639\pi$$
$$578$$ −6.31371 −0.262616
$$579$$ 14.4853 0.601988
$$580$$ 3.17157 0.131692
$$581$$ 49.9411 2.07191
$$582$$ −8.82843 −0.365950
$$583$$ −52.6863 −2.18204
$$584$$ 2.48528 0.102842
$$585$$ 0 0
$$586$$ 28.6274 1.18259
$$587$$ 42.6274 1.75942 0.879711 0.475509i $$-0.157736\pi$$
0.879711 + 0.475509i $$0.157736\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −11.3137 −0.466173
$$590$$ 13.6569 0.562244
$$591$$ −9.31371 −0.383115
$$592$$ 0.343146 0.0141032
$$593$$ 11.6569 0.478690 0.239345 0.970935i $$-0.423067\pi$$
0.239345 + 0.970935i $$0.423067\pi$$
$$594$$ −5.65685 −0.232104
$$595$$ −13.6569 −0.559876
$$596$$ 7.65685 0.313637
$$597$$ 21.6569 0.886356
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 6.68629 0.272740 0.136370 0.990658i $$-0.456456\pi$$
0.136370 + 0.990658i $$0.456456\pi$$
$$602$$ −4.68629 −0.190999
$$603$$ 5.65685 0.230365
$$604$$ −12.0000 −0.488273
$$605$$ −21.0000 −0.853771
$$606$$ 12.1421 0.493241
$$607$$ −4.97056 −0.201749 −0.100874 0.994899i $$-0.532164\pi$$
−0.100874 + 0.994899i $$0.532164\pi$$
$$608$$ −2.82843 −0.114708
$$609$$ 8.97056 0.363506
$$610$$ 6.00000 0.242933
$$611$$ 0 0
$$612$$ −4.82843 −0.195178
$$613$$ 34.2843 1.38473 0.692364 0.721548i $$-0.256569\pi$$
0.692364 + 0.721548i $$0.256569\pi$$
$$614$$ −10.3431 −0.417415
$$615$$ 3.65685 0.147459
$$616$$ 16.0000 0.644658
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ −9.65685 −0.388456
$$619$$ 29.1716 1.17250 0.586252 0.810129i $$-0.300603\pi$$
0.586252 + 0.810129i $$0.300603\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 8.48528 0.340503
$$622$$ 24.0000 0.962312
$$623$$ −12.2843 −0.492159
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −2.97056 −0.118728
$$627$$ 16.0000 0.638978
$$628$$ −17.3137 −0.690892
$$629$$ −1.65685 −0.0660631
$$630$$ −2.82843 −0.112687
$$631$$ 22.3431 0.889467 0.444733 0.895663i $$-0.353298\pi$$
0.444733 + 0.895663i $$0.353298\pi$$
$$632$$ −13.6569 −0.543240
$$633$$ 23.3137 0.926637
$$634$$ 2.68629 0.106686
$$635$$ 1.65685 0.0657503
$$636$$ −9.31371 −0.369313
$$637$$ 0 0
$$638$$ 17.9411 0.710296
$$639$$ −5.65685 −0.223782
$$640$$ 1.00000 0.0395285
$$641$$ 40.6274 1.60469 0.802343 0.596863i $$-0.203586\pi$$
0.802343 + 0.596863i $$0.203586\pi$$
$$642$$ 4.00000 0.157867
$$643$$ 39.5980 1.56159 0.780796 0.624786i $$-0.214814\pi$$
0.780796 + 0.624786i $$0.214814\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 1.65685 0.0652386
$$646$$ 13.6569 0.537322
$$647$$ −8.48528 −0.333591 −0.166795 0.985992i $$-0.553342\pi$$
−0.166795 + 0.985992i $$0.553342\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 77.2548 3.03252
$$650$$ 0 0
$$651$$ 11.3137 0.443419
$$652$$ 11.3137 0.443079
$$653$$ 14.2843 0.558987 0.279493 0.960148i $$-0.409834\pi$$
0.279493 + 0.960148i $$0.409834\pi$$
$$654$$ 3.17157 0.124018
$$655$$ −22.1421 −0.865165
$$656$$ −3.65685 −0.142776
$$657$$ −2.48528 −0.0969601
$$658$$ 22.6274 0.882109
$$659$$ −24.4853 −0.953811 −0.476906 0.878955i $$-0.658242\pi$$
−0.476906 + 0.878955i $$0.658242\pi$$
$$660$$ −5.65685 −0.220193
$$661$$ −20.1421 −0.783438 −0.391719 0.920085i $$-0.628120\pi$$
−0.391719 + 0.920085i $$0.628120\pi$$
$$662$$ 8.48528 0.329790
$$663$$ 0 0
$$664$$ 17.6569 0.685219
$$665$$ 8.00000 0.310227
$$666$$ −0.343146 −0.0132966
$$667$$ −26.9117 −1.04202
$$668$$ 24.9706 0.966140
$$669$$ −5.17157 −0.199945
$$670$$ 5.65685 0.218543
$$671$$ 33.9411 1.31028
$$672$$ 2.82843 0.109109
$$673$$ −12.6274 −0.486751 −0.243376 0.969932i $$-0.578255\pi$$
−0.243376 + 0.969932i $$0.578255\pi$$
$$674$$ 22.9706 0.884793
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 23.6569 0.909207 0.454603 0.890694i $$-0.349781\pi$$
0.454603 + 0.890694i $$0.349781\pi$$
$$678$$ 10.4853 0.402685
$$679$$ −24.9706 −0.958282
$$680$$ −4.82843 −0.185162
$$681$$ −4.00000 −0.153280
$$682$$ 22.6274 0.866449
$$683$$ −22.3431 −0.854937 −0.427468 0.904030i $$-0.640594\pi$$
−0.427468 + 0.904030i $$0.640594\pi$$
$$684$$ 2.82843 0.108148
$$685$$ 5.31371 0.203026
$$686$$ −16.9706 −0.647939
$$687$$ 24.1421 0.921080
$$688$$ −1.65685 −0.0631670
$$689$$ 0 0
$$690$$ 8.48528 0.323029
$$691$$ −11.7990 −0.448855 −0.224427 0.974491i $$-0.572051\pi$$
−0.224427 + 0.974491i $$0.572051\pi$$
$$692$$ 13.3137 0.506111
$$693$$ −16.0000 −0.607790
$$694$$ −1.65685 −0.0628933
$$695$$ −17.6569 −0.669763
$$696$$ 3.17157 0.120218
$$697$$ 17.6569 0.668801
$$698$$ −16.1421 −0.610989
$$699$$ 22.4853 0.850471
$$700$$ −2.82843 −0.106904
$$701$$ −28.1421 −1.06291 −0.531457 0.847085i $$-0.678355\pi$$
−0.531457 + 0.847085i $$0.678355\pi$$
$$702$$ 0 0
$$703$$ 0.970563 0.0366055
$$704$$ 5.65685 0.213201
$$705$$ −8.00000 −0.301297
$$706$$ −17.3137 −0.651610
$$707$$ 34.3431 1.29161
$$708$$ 13.6569 0.513256
$$709$$ 12.8284 0.481782 0.240891 0.970552i $$-0.422560\pi$$
0.240891 + 0.970552i $$0.422560\pi$$
$$710$$ −5.65685 −0.212298
$$711$$ 13.6569 0.512172
$$712$$ −4.34315 −0.162766
$$713$$ −33.9411 −1.27111
$$714$$ −13.6569 −0.511095
$$715$$ 0 0
$$716$$ 24.4853 0.915058
$$717$$ −16.0000 −0.597531
$$718$$ 28.2843 1.05556
$$719$$ −18.3431 −0.684084 −0.342042 0.939685i $$-0.611118\pi$$
−0.342042 + 0.939685i $$0.611118\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ −27.3137 −1.01722
$$722$$ 11.0000 0.409378
$$723$$ 17.3137 0.643904
$$724$$ 3.65685 0.135906
$$725$$ −3.17157 −0.117789
$$726$$ −21.0000 −0.779383
$$727$$ 21.9411 0.813751 0.406876 0.913484i $$-0.366618\pi$$
0.406876 + 0.913484i $$0.366618\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −2.48528 −0.0919844
$$731$$ 8.00000 0.295891
$$732$$ 6.00000 0.221766
$$733$$ 11.6569 0.430556 0.215278 0.976553i $$-0.430934\pi$$
0.215278 + 0.976553i $$0.430934\pi$$
$$734$$ −14.3431 −0.529415
$$735$$ −1.00000 −0.0368856
$$736$$ −8.48528 −0.312772
$$737$$ 32.0000 1.17874
$$738$$ 3.65685 0.134611
$$739$$ −14.1421 −0.520227 −0.260113 0.965578i $$-0.583760\pi$$
−0.260113 + 0.965578i $$0.583760\pi$$
$$740$$ −0.343146 −0.0126143
$$741$$ 0 0
$$742$$ −26.3431 −0.967087
$$743$$ −20.2843 −0.744158 −0.372079 0.928201i $$-0.621355\pi$$
−0.372079 + 0.928201i $$0.621355\pi$$
$$744$$ 4.00000 0.146647
$$745$$ −7.65685 −0.280525
$$746$$ 25.3137 0.926801
$$747$$ −17.6569 −0.646031
$$748$$ −27.3137 −0.998688
$$749$$ 11.3137 0.413394
$$750$$ 1.00000 0.0365148
$$751$$ −11.3137 −0.412843 −0.206422 0.978463i $$-0.566182\pi$$
−0.206422 + 0.978463i $$0.566182\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 5.17157 0.188463
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ −2.82843 −0.102869
$$757$$ −47.9411 −1.74245 −0.871225 0.490884i $$-0.836674\pi$$
−0.871225 + 0.490884i $$0.836674\pi$$
$$758$$ −24.4853 −0.889345
$$759$$ 48.0000 1.74229
$$760$$ 2.82843 0.102598
$$761$$ −16.3431 −0.592439 −0.296219 0.955120i $$-0.595726\pi$$
−0.296219 + 0.955120i $$0.595726\pi$$
$$762$$ 1.65685 0.0600215
$$763$$ 8.97056 0.324756
$$764$$ −11.3137 −0.409316
$$765$$ 4.82843 0.174572
$$766$$ −18.3431 −0.662765
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ −16.0000 −0.576600
$$771$$ 0.828427 0.0298351
$$772$$ 14.4853 0.521337
$$773$$ 30.6863 1.10371 0.551855 0.833940i $$-0.313920\pi$$
0.551855 + 0.833940i $$0.313920\pi$$
$$774$$ 1.65685 0.0595544
$$775$$ −4.00000 −0.143684
$$776$$ −8.82843 −0.316922
$$777$$ −0.970563 −0.0348187
$$778$$ −10.4853 −0.375916
$$779$$ −10.3431 −0.370582
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 40.9706 1.46510
$$783$$ −3.17157 −0.113343
$$784$$ 1.00000 0.0357143
$$785$$ 17.3137 0.617953
$$786$$ −22.1421 −0.789784
$$787$$ 24.0000 0.855508 0.427754 0.903895i $$-0.359305\pi$$
0.427754 + 0.903895i $$0.359305\pi$$
$$788$$ −9.31371 −0.331787
$$789$$ −0.485281 −0.0172765
$$790$$ 13.6569 0.485889
$$791$$ 29.6569 1.05448
$$792$$ −5.65685 −0.201008
$$793$$ 0 0
$$794$$ −26.2843 −0.932794
$$795$$ 9.31371 0.330323
$$796$$ 21.6569 0.767607
$$797$$ −28.6274 −1.01404 −0.507018 0.861936i $$-0.669252\pi$$
−0.507018 + 0.861936i $$0.669252\pi$$
$$798$$ 8.00000 0.283197
$$799$$ −38.6274 −1.36654
$$800$$ −1.00000 −0.0353553
$$801$$ 4.34315 0.153458
$$802$$ 6.97056 0.246139
$$803$$ −14.0589 −0.496127
$$804$$ 5.65685 0.199502
$$805$$ 24.0000 0.845889
$$806$$ 0 0
$$807$$ 2.48528 0.0874860
$$808$$ 12.1421 0.427159
$$809$$ −9.31371 −0.327453 −0.163726 0.986506i $$-0.552351\pi$$
−0.163726 + 0.986506i $$0.552351\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 30.1421 1.05843 0.529217 0.848487i $$-0.322486\pi$$
0.529217 + 0.848487i $$0.322486\pi$$
$$812$$ 8.97056 0.314805
$$813$$ 15.3137 0.537075
$$814$$ −1.94113 −0.0680364
$$815$$ −11.3137 −0.396302
$$816$$ −4.82843 −0.169029
$$817$$ −4.68629 −0.163953
$$818$$ 7.65685 0.267716
$$819$$ 0 0
$$820$$ 3.65685 0.127703
$$821$$ 22.2843 0.777726 0.388863 0.921296i $$-0.372868\pi$$
0.388863 + 0.921296i $$0.372868\pi$$
$$822$$ 5.31371 0.185337
$$823$$ −19.0294 −0.663324 −0.331662 0.943398i $$-0.607609\pi$$
−0.331662 + 0.943398i $$0.607609\pi$$
$$824$$ −9.65685 −0.336412
$$825$$ 5.65685 0.196946
$$826$$ 38.6274 1.34402
$$827$$ 1.65685 0.0576145 0.0288072 0.999585i $$-0.490829\pi$$
0.0288072 + 0.999585i $$0.490829\pi$$
$$828$$ 8.48528 0.294884
$$829$$ −30.6863 −1.06578 −0.532889 0.846185i $$-0.678894\pi$$
−0.532889 + 0.846185i $$0.678894\pi$$
$$830$$ −17.6569 −0.612878
$$831$$ 26.0000 0.901930
$$832$$ 0 0
$$833$$ −4.82843 −0.167295
$$834$$ −17.6569 −0.611407
$$835$$ −24.9706 −0.864142
$$836$$ 16.0000 0.553372
$$837$$ −4.00000 −0.138260
$$838$$ 5.17157 0.178649
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ −2.82843 −0.0975900
$$841$$ −18.9411 −0.653142
$$842$$ 4.14214 0.142747
$$843$$ −19.6569 −0.677018
$$844$$ 23.3137 0.802491
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ −59.3970 −2.04090
$$848$$ −9.31371 −0.319834
$$849$$ −6.34315 −0.217696
$$850$$ 4.82843 0.165614
$$851$$ 2.91169 0.0998114
$$852$$ −5.65685 −0.193801
$$853$$ 18.2843 0.626042 0.313021 0.949746i $$-0.398659\pi$$
0.313021 + 0.949746i $$0.398659\pi$$
$$854$$ 16.9706 0.580721
$$855$$ −2.82843 −0.0967302
$$856$$ 4.00000 0.136717
$$857$$ −15.1716 −0.518251 −0.259126 0.965844i $$-0.583434\pi$$
−0.259126 + 0.965844i $$0.583434\pi$$
$$858$$ 0 0
$$859$$ −29.9411 −1.02158 −0.510789 0.859706i $$-0.670647\pi$$
−0.510789 + 0.859706i $$0.670647\pi$$
$$860$$ 1.65685 0.0564983
$$861$$ 10.3431 0.352493
$$862$$ −16.0000 −0.544962
$$863$$ −28.2843 −0.962808 −0.481404 0.876499i $$-0.659873\pi$$
−0.481404 + 0.876499i $$0.659873\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −13.3137 −0.452680
$$866$$ −10.9706 −0.372795
$$867$$ 6.31371 0.214425
$$868$$ 11.3137 0.384012
$$869$$ 77.2548 2.62069
$$870$$ −3.17157 −0.107526
$$871$$ 0 0
$$872$$ 3.17157 0.107403
$$873$$ 8.82843 0.298797
$$874$$ −24.0000 −0.811812
$$875$$ 2.82843 0.0956183
$$876$$ −2.48528 −0.0839699
$$877$$ −39.2548 −1.32554 −0.662771 0.748822i $$-0.730620\pi$$
−0.662771 + 0.748822i $$0.730620\pi$$
$$878$$ 22.6274 0.763638
$$879$$ −28.6274 −0.965579
$$880$$ −5.65685 −0.190693
$$881$$ 46.2843 1.55936 0.779678 0.626180i $$-0.215383\pi$$
0.779678 + 0.626180i $$0.215383\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 8.68629 0.292317 0.146158 0.989261i $$-0.453309\pi$$
0.146158 + 0.989261i $$0.453309\pi$$
$$884$$ 0 0
$$885$$ −13.6569 −0.459070
$$886$$ −41.6569 −1.39949
$$887$$ −23.5147 −0.789547 −0.394773 0.918778i $$-0.629177\pi$$
−0.394773 + 0.918778i $$0.629177\pi$$
$$888$$ −0.343146 −0.0115152
$$889$$ 4.68629 0.157173
$$890$$ 4.34315 0.145583
$$891$$ 5.65685 0.189512
$$892$$ −5.17157 −0.173157
$$893$$ 22.6274 0.757198
$$894$$ −7.65685 −0.256084
$$895$$ −24.4853 −0.818453
$$896$$ 2.82843 0.0944911
$$897$$ 0 0
$$898$$ −30.2843 −1.01060
$$899$$ 12.6863 0.423112
$$900$$ 1.00000 0.0333333
$$901$$ 44.9706 1.49819
$$902$$ 20.6863 0.688778
$$903$$ 4.68629 0.155950
$$904$$ 10.4853 0.348735
$$905$$ −3.65685 −0.121558
$$906$$ 12.0000 0.398673
$$907$$ −48.2843 −1.60325 −0.801626 0.597825i $$-0.796032\pi$$
−0.801626 + 0.597825i $$0.796032\pi$$
$$908$$ −4.00000 −0.132745
$$909$$ −12.1421 −0.402729
$$910$$ 0 0
$$911$$ −8.97056 −0.297208 −0.148604 0.988897i $$-0.547478\pi$$
−0.148604 + 0.988897i $$0.547478\pi$$
$$912$$ 2.82843 0.0936586
$$913$$ −99.8823 −3.30562
$$914$$ 15.1716 0.501831
$$915$$ −6.00000 −0.198354
$$916$$ 24.1421 0.797679
$$917$$ −62.6274 −2.06814
$$918$$ 4.82843 0.159362
$$919$$ −25.9411 −0.855719 −0.427859 0.903845i $$-0.640732\pi$$
−0.427859 + 0.903845i $$0.640732\pi$$
$$920$$ 8.48528 0.279751
$$921$$ 10.3431 0.340818
$$922$$ 14.0000 0.461065
$$923$$ 0 0
$$924$$ −16.0000 −0.526361
$$925$$ 0.343146 0.0112826
$$926$$ 35.7990 1.17643
$$927$$ 9.65685 0.317173
$$928$$ 3.17157 0.104112
$$929$$ −45.5980 −1.49602 −0.748011 0.663687i $$-0.768991\pi$$
−0.748011 + 0.663687i $$0.768991\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ 2.82843 0.0926980
$$932$$ 22.4853 0.736530
$$933$$ −24.0000 −0.785725
$$934$$ 15.3137 0.501080
$$935$$ 27.3137 0.893254
$$936$$ 0 0
$$937$$ −28.6274 −0.935217 −0.467608 0.883936i $$-0.654884\pi$$
−0.467608 + 0.883936i $$0.654884\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 2.97056 0.0969407
$$940$$ −8.00000 −0.260931
$$941$$ −21.0294 −0.685540 −0.342770 0.939419i $$-0.611365\pi$$
−0.342770 + 0.939419i $$0.611365\pi$$
$$942$$ 17.3137 0.564111
$$943$$ −31.0294 −1.01046
$$944$$ 13.6569 0.444493
$$945$$ 2.82843 0.0920087
$$946$$ 9.37258 0.304729
$$947$$ −41.6569 −1.35367 −0.676833 0.736137i $$-0.736648\pi$$
−0.676833 + 0.736137i $$0.736648\pi$$
$$948$$ 13.6569 0.443554
$$949$$ 0 0
$$950$$ −2.82843 −0.0917663
$$951$$ −2.68629 −0.0871090
$$952$$ −13.6569 −0.442621
$$953$$ −56.1421 −1.81862 −0.909311 0.416117i $$-0.863391\pi$$
−0.909311 + 0.416117i $$0.863391\pi$$
$$954$$ 9.31371 0.301542
$$955$$ 11.3137 0.366103
$$956$$ −16.0000 −0.517477
$$957$$ −17.9411 −0.579954
$$958$$ 11.3137 0.365529
$$959$$ 15.0294 0.485326
$$960$$ −1.00000 −0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ 17.3137 0.557637
$$965$$ −14.4853 −0.466298
$$966$$ 24.0000 0.772187
$$967$$ 24.4853 0.787394 0.393697 0.919240i $$-0.371196\pi$$
0.393697 + 0.919240i $$0.371196\pi$$
$$968$$ −21.0000 −0.674966
$$969$$ −13.6569 −0.438721
$$970$$ 8.82843 0.283464
$$971$$ 32.4853 1.04250 0.521251 0.853403i $$-0.325465\pi$$
0.521251 + 0.853403i $$0.325465\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −49.9411 −1.60104
$$974$$ −0.485281 −0.0155494
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 19.6569 0.628878 0.314439 0.949278i $$-0.398184\pi$$
0.314439 + 0.949278i $$0.398184\pi$$
$$978$$ −11.3137 −0.361773
$$979$$ 24.5685 0.785214
$$980$$ −1.00000 −0.0319438
$$981$$ −3.17157 −0.101261
$$982$$ 9.85786 0.314577
$$983$$ 13.6569 0.435586 0.217793 0.975995i $$-0.430114\pi$$
0.217793 + 0.975995i $$0.430114\pi$$
$$984$$ 3.65685 0.116576
$$985$$ 9.31371 0.296759
$$986$$ −15.3137 −0.487688
$$987$$ −22.6274 −0.720239
$$988$$ 0 0
$$989$$ −14.0589 −0.447046
$$990$$ 5.65685 0.179787
$$991$$ 58.9117 1.87139 0.935696 0.352808i $$-0.114773\pi$$
0.935696 + 0.352808i $$0.114773\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −8.48528 −0.269272
$$994$$ −16.0000 −0.507489
$$995$$ −21.6569 −0.686568
$$996$$ −17.6569 −0.559479
$$997$$ 38.6863 1.22521 0.612604 0.790390i $$-0.290122\pi$$
0.612604 + 0.790390i $$0.290122\pi$$
$$998$$ −16.4853 −0.521832
$$999$$ 0.343146 0.0108567
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 5070.2.a.bc.1.1 2
13.5 odd 4 5070.2.b.q.1351.3 4
13.8 odd 4 5070.2.b.q.1351.2 4
13.12 even 2 390.2.a.h.1.2 2
39.38 odd 2 1170.2.a.o.1.2 2
52.51 odd 2 3120.2.a.bc.1.1 2
65.12 odd 4 1950.2.e.o.1249.4 4
65.38 odd 4 1950.2.e.o.1249.1 4
65.64 even 2 1950.2.a.bd.1.1 2
156.155 even 2 9360.2.a.ch.1.1 2
195.38 even 4 5850.2.e.bk.5149.3 4
195.77 even 4 5850.2.e.bk.5149.2 4
195.194 odd 2 5850.2.a.cl.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.2 2 13.12 even 2
1170.2.a.o.1.2 2 39.38 odd 2
1950.2.a.bd.1.1 2 65.64 even 2
1950.2.e.o.1249.1 4 65.38 odd 4
1950.2.e.o.1249.4 4 65.12 odd 4
3120.2.a.bc.1.1 2 52.51 odd 2
5070.2.a.bc.1.1 2 1.1 even 1 trivial
5070.2.b.q.1351.2 4 13.8 odd 4
5070.2.b.q.1351.3 4 13.5 odd 4
5850.2.a.cl.1.1 2 195.194 odd 2
5850.2.e.bk.5149.2 4 195.77 even 4
5850.2.e.bk.5149.3 4 195.38 even 4
9360.2.a.ch.1.1 2 156.155 even 2