# Properties

 Label 637.2.a.i.1.2 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.210756$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.21076 q^{2} -1.74483 q^{3} -0.534070 q^{4} +2.21076 q^{5} +2.11256 q^{6} +3.06814 q^{8} +0.0444180 q^{9} +O(q^{10})$$ $$q-1.21076 q^{2} -1.74483 q^{3} -0.534070 q^{4} +2.21076 q^{5} +2.11256 q^{6} +3.06814 q^{8} +0.0444180 q^{9} -2.67669 q^{10} -0.789244 q^{11} +0.931860 q^{12} -1.00000 q^{13} -3.85738 q^{15} -2.64663 q^{16} -1.74483 q^{17} -0.0537794 q^{18} +4.32331 q^{19} -1.18070 q^{20} +0.955582 q^{22} -1.11256 q^{23} -5.35337 q^{24} -0.112558 q^{25} +1.21076 q^{26} +5.15698 q^{27} -8.48965 q^{29} +4.67035 q^{30} +5.70041 q^{31} -2.93186 q^{32} +1.37709 q^{33} +2.11256 q^{34} -0.0237224 q^{36} +2.27890 q^{37} -5.23448 q^{38} +1.74483 q^{39} +6.78291 q^{40} +12.1363 q^{41} +8.06814 q^{43} +0.421512 q^{44} +0.0981974 q^{45} +1.34704 q^{46} +8.74483 q^{47} +4.61791 q^{48} +0.136281 q^{50} +3.04442 q^{51} +0.534070 q^{52} +7.95558 q^{53} -6.24384 q^{54} -1.74483 q^{55} -7.54343 q^{57} +10.2789 q^{58} +10.9556 q^{59} +2.06011 q^{60} -13.0681 q^{61} -6.90180 q^{62} +8.84302 q^{64} -2.21076 q^{65} -1.66732 q^{66} +6.55779 q^{67} +0.931860 q^{68} +1.94122 q^{69} +5.85738 q^{71} +0.136281 q^{72} +8.00936 q^{73} -2.75919 q^{74} +0.196395 q^{75} -2.30895 q^{76} -2.11256 q^{78} -6.91116 q^{79} -5.85105 q^{80} -9.13128 q^{81} -14.6941 q^{82} +3.14262 q^{83} -3.85738 q^{85} -9.76855 q^{86} +14.8130 q^{87} -2.42151 q^{88} +3.39145 q^{89} -0.118893 q^{90} +0.594184 q^{92} -9.94622 q^{93} -10.5878 q^{94} +9.55779 q^{95} +5.11559 q^{96} -0.0981974 q^{97} -0.0350567 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 4 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} - 6 q^{8} + 11 q^{9} + O(q^{10})$$ $$3 q - 2 q^{2} + 4 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} - 6 q^{8} + 11 q^{9} - 14 q^{10} - 4 q^{11} + 18 q^{12} - 3 q^{13} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 8 q^{18} + 7 q^{19} + 16 q^{20} - 8 q^{22} + q^{23} - 28 q^{24} + 4 q^{25} + 2 q^{26} + 22 q^{27} - 7 q^{29} - 24 q^{30} - 3 q^{31} - 24 q^{32} - 10 q^{33} + 2 q^{34} + 26 q^{36} - 10 q^{37} + 12 q^{38} - 4 q^{39} - 22 q^{40} + 6 q^{41} + 9 q^{43} - 2 q^{44} + 3 q^{45} - 28 q^{46} + 17 q^{47} + 16 q^{48} - 30 q^{50} + 20 q^{51} - 6 q^{52} + 13 q^{53} + 28 q^{54} + 4 q^{55} + 4 q^{57} + 14 q^{58} + 22 q^{59} + 42 q^{60} - 24 q^{61} - 18 q^{62} + 20 q^{64} - 5 q^{65} - 30 q^{66} - 14 q^{67} + 18 q^{68} + 2 q^{69} + 4 q^{71} - 30 q^{72} + 5 q^{73} + 8 q^{74} + 6 q^{75} - 8 q^{76} - 2 q^{78} + q^{79} + 40 q^{80} + 15 q^{81} + 20 q^{82} + 23 q^{83} + 2 q^{85} + 6 q^{86} + 20 q^{87} - 4 q^{88} - 11 q^{89} - 40 q^{90} + 30 q^{92} - 38 q^{93} - 16 q^{94} - 5 q^{95} - 52 q^{96} - 3 q^{97} - 30 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.21076 −0.856134 −0.428067 0.903747i $$-0.640805\pi$$
−0.428067 + 0.903747i $$0.640805\pi$$
$$3$$ −1.74483 −1.00738 −0.503688 0.863886i $$-0.668024\pi$$
−0.503688 + 0.863886i $$0.668024\pi$$
$$4$$ −0.534070 −0.267035
$$5$$ 2.21076 0.988680 0.494340 0.869269i $$-0.335410\pi$$
0.494340 + 0.869269i $$0.335410\pi$$
$$6$$ 2.11256 0.862448
$$7$$ 0 0
$$8$$ 3.06814 1.08475
$$9$$ 0.0444180 0.0148060
$$10$$ −2.67669 −0.846442
$$11$$ −0.789244 −0.237966 −0.118983 0.992896i $$-0.537963\pi$$
−0.118983 + 0.992896i $$0.537963\pi$$
$$12$$ 0.931860 0.269005
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −3.85738 −0.995972
$$16$$ −2.64663 −0.661657
$$17$$ −1.74483 −0.423182 −0.211591 0.977358i $$-0.567865\pi$$
−0.211591 + 0.977358i $$0.567865\pi$$
$$18$$ −0.0537794 −0.0126759
$$19$$ 4.32331 0.991836 0.495918 0.868369i $$-0.334832\pi$$
0.495918 + 0.868369i $$0.334832\pi$$
$$20$$ −1.18070 −0.264012
$$21$$ 0 0
$$22$$ 0.955582 0.203731
$$23$$ −1.11256 −0.231984 −0.115992 0.993250i $$-0.537005\pi$$
−0.115992 + 0.993250i $$0.537005\pi$$
$$24$$ −5.35337 −1.09275
$$25$$ −0.112558 −0.0225117
$$26$$ 1.21076 0.237449
$$27$$ 5.15698 0.992461
$$28$$ 0 0
$$29$$ −8.48965 −1.57649 −0.788244 0.615362i $$-0.789010\pi$$
−0.788244 + 0.615362i $$0.789010\pi$$
$$30$$ 4.67035 0.852686
$$31$$ 5.70041 1.02382 0.511912 0.859038i $$-0.328937\pi$$
0.511912 + 0.859038i $$0.328937\pi$$
$$32$$ −2.93186 −0.518284
$$33$$ 1.37709 0.239721
$$34$$ 2.11256 0.362301
$$35$$ 0 0
$$36$$ −0.0237224 −0.00395373
$$37$$ 2.27890 0.374648 0.187324 0.982298i $$-0.440019\pi$$
0.187324 + 0.982298i $$0.440019\pi$$
$$38$$ −5.23448 −0.849144
$$39$$ 1.74483 0.279396
$$40$$ 6.78291 1.07247
$$41$$ 12.1363 1.89537 0.947684 0.319209i $$-0.103417\pi$$
0.947684 + 0.319209i $$0.103417\pi$$
$$42$$ 0 0
$$43$$ 8.06814 1.23038 0.615190 0.788379i $$-0.289079\pi$$
0.615190 + 0.788379i $$0.289079\pi$$
$$44$$ 0.421512 0.0635453
$$45$$ 0.0981974 0.0146384
$$46$$ 1.34704 0.198610
$$47$$ 8.74483 1.27556 0.637782 0.770217i $$-0.279852\pi$$
0.637782 + 0.770217i $$0.279852\pi$$
$$48$$ 4.61791 0.666537
$$49$$ 0 0
$$50$$ 0.136281 0.0192730
$$51$$ 3.04442 0.426304
$$52$$ 0.534070 0.0740622
$$53$$ 7.95558 1.09278 0.546392 0.837530i $$-0.316001\pi$$
0.546392 + 0.837530i $$0.316001\pi$$
$$54$$ −6.24384 −0.849679
$$55$$ −1.74483 −0.235272
$$56$$ 0 0
$$57$$ −7.54343 −0.999152
$$58$$ 10.2789 1.34969
$$59$$ 10.9556 1.42630 0.713148 0.701014i $$-0.247269\pi$$
0.713148 + 0.701014i $$0.247269\pi$$
$$60$$ 2.06011 0.265960
$$61$$ −13.0681 −1.67320 −0.836602 0.547811i $$-0.815461\pi$$
−0.836602 + 0.547811i $$0.815461\pi$$
$$62$$ −6.90180 −0.876530
$$63$$ 0 0
$$64$$ 8.84302 1.10538
$$65$$ −2.21076 −0.274211
$$66$$ −1.66732 −0.205233
$$67$$ 6.55779 0.801162 0.400581 0.916261i $$-0.368808\pi$$
0.400581 + 0.916261i $$0.368808\pi$$
$$68$$ 0.931860 0.113005
$$69$$ 1.94122 0.233696
$$70$$ 0 0
$$71$$ 5.85738 0.695144 0.347572 0.937653i $$-0.387006\pi$$
0.347572 + 0.937653i $$0.387006\pi$$
$$72$$ 0.136281 0.0160608
$$73$$ 8.00936 0.937425 0.468712 0.883351i $$-0.344718\pi$$
0.468712 + 0.883351i $$0.344718\pi$$
$$74$$ −2.75919 −0.320749
$$75$$ 0.196395 0.0226777
$$76$$ −2.30895 −0.264855
$$77$$ 0 0
$$78$$ −2.11256 −0.239200
$$79$$ −6.91116 −0.777567 −0.388783 0.921329i $$-0.627105\pi$$
−0.388783 + 0.921329i $$0.627105\pi$$
$$80$$ −5.85105 −0.654167
$$81$$ −9.13128 −1.01459
$$82$$ −14.6941 −1.62269
$$83$$ 3.14262 0.344947 0.172473 0.985014i $$-0.444824\pi$$
0.172473 + 0.985014i $$0.444824\pi$$
$$84$$ 0 0
$$85$$ −3.85738 −0.418392
$$86$$ −9.76855 −1.05337
$$87$$ 14.8130 1.58812
$$88$$ −2.42151 −0.258134
$$89$$ 3.39145 0.359493 0.179747 0.983713i $$-0.442472\pi$$
0.179747 + 0.983713i $$0.442472\pi$$
$$90$$ −0.118893 −0.0125324
$$91$$ 0 0
$$92$$ 0.594184 0.0619480
$$93$$ −9.94622 −1.03138
$$94$$ −10.5878 −1.09205
$$95$$ 9.55779 0.980609
$$96$$ 5.11559 0.522107
$$97$$ −0.0981974 −0.00997044 −0.00498522 0.999988i $$-0.501587\pi$$
−0.00498522 + 0.999988i $$0.501587\pi$$
$$98$$ 0 0
$$99$$ −0.0350567 −0.00352333
$$100$$ 0.0601141 0.00601141
$$101$$ 6.90180 0.686755 0.343378 0.939197i $$-0.388429\pi$$
0.343378 + 0.939197i $$0.388429\pi$$
$$102$$ −3.68605 −0.364973
$$103$$ −15.8223 −1.55902 −0.779510 0.626390i $$-0.784532\pi$$
−0.779510 + 0.626390i $$0.784532\pi$$
$$104$$ −3.06814 −0.300856
$$105$$ 0 0
$$106$$ −9.63227 −0.935569
$$107$$ −2.02372 −0.195641 −0.0978203 0.995204i $$-0.531187\pi$$
−0.0978203 + 0.995204i $$0.531187\pi$$
$$108$$ −2.75419 −0.265022
$$109$$ −17.3470 −1.66154 −0.830772 0.556612i $$-0.812101\pi$$
−0.830772 + 0.556612i $$0.812101\pi$$
$$110$$ 2.11256 0.201425
$$111$$ −3.97628 −0.377412
$$112$$ 0 0
$$113$$ 10.1807 0.957720 0.478860 0.877891i $$-0.341050\pi$$
0.478860 + 0.877891i $$0.341050\pi$$
$$114$$ 9.13325 0.855408
$$115$$ −2.45960 −0.229358
$$116$$ 4.53407 0.420978
$$117$$ −0.0444180 −0.00410645
$$118$$ −13.2645 −1.22110
$$119$$ 0 0
$$120$$ −11.8350 −1.08038
$$121$$ −10.3771 −0.943372
$$122$$ 15.8223 1.43249
$$123$$ −21.1757 −1.90935
$$124$$ −3.04442 −0.273397
$$125$$ −11.3026 −1.01094
$$126$$ 0 0
$$127$$ 16.4452 1.45928 0.729639 0.683832i $$-0.239688\pi$$
0.729639 + 0.683832i $$0.239688\pi$$
$$128$$ −4.84302 −0.428067
$$129$$ −14.0775 −1.23945
$$130$$ 2.67669 0.234761
$$131$$ 12.3327 1.07751 0.538755 0.842462i $$-0.318895\pi$$
0.538755 + 0.842462i $$0.318895\pi$$
$$132$$ −0.735465 −0.0640140
$$133$$ 0 0
$$134$$ −7.93989 −0.685902
$$135$$ 11.4008 0.981226
$$136$$ −5.35337 −0.459048
$$137$$ −3.34704 −0.285957 −0.142978 0.989726i $$-0.545668\pi$$
−0.142978 + 0.989726i $$0.545668\pi$$
$$138$$ −2.35034 −0.200075
$$139$$ −6.16634 −0.523022 −0.261511 0.965201i $$-0.584221\pi$$
−0.261511 + 0.965201i $$0.584221\pi$$
$$140$$ 0 0
$$141$$ −15.2582 −1.28497
$$142$$ −7.09186 −0.595136
$$143$$ 0.789244 0.0659999
$$144$$ −0.117558 −0.00979651
$$145$$ −18.7685 −1.55864
$$146$$ −9.69738 −0.802561
$$147$$ 0 0
$$148$$ −1.21709 −0.100044
$$149$$ 18.8367 1.54316 0.771581 0.636131i $$-0.219466\pi$$
0.771581 + 0.636131i $$0.219466\pi$$
$$150$$ −0.237786 −0.0194152
$$151$$ 20.1901 1.64304 0.821522 0.570177i $$-0.193125\pi$$
0.821522 + 0.570177i $$0.193125\pi$$
$$152$$ 13.2645 1.07590
$$153$$ −0.0775018 −0.00626565
$$154$$ 0 0
$$155$$ 12.6022 1.01223
$$156$$ −0.931860 −0.0746085
$$157$$ 13.8811 1.10783 0.553916 0.832572i $$-0.313133\pi$$
0.553916 + 0.832572i $$0.313133\pi$$
$$158$$ 8.36773 0.665701
$$159$$ −13.8811 −1.10084
$$160$$ −6.48163 −0.512418
$$161$$ 0 0
$$162$$ 11.0558 0.868622
$$163$$ 11.4897 0.899939 0.449970 0.893044i $$-0.351435\pi$$
0.449970 + 0.893044i $$0.351435\pi$$
$$164$$ −6.48163 −0.506130
$$165$$ 3.04442 0.237008
$$166$$ −3.80494 −0.295321
$$167$$ −12.9699 −1.00364 −0.501822 0.864971i $$-0.667337\pi$$
−0.501822 + 0.864971i $$0.667337\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 4.67035 0.358200
$$171$$ 0.192033 0.0146851
$$172$$ −4.30895 −0.328555
$$173$$ 2.48029 0.188573 0.0942865 0.995545i $$-0.469943\pi$$
0.0942865 + 0.995545i $$0.469943\pi$$
$$174$$ −17.9349 −1.35964
$$175$$ 0 0
$$176$$ 2.08884 0.157452
$$177$$ −19.1156 −1.43682
$$178$$ −4.10622 −0.307774
$$179$$ −8.15698 −0.609681 −0.304841 0.952403i $$-0.598603\pi$$
−0.304841 + 0.952403i $$0.598603\pi$$
$$180$$ −0.0524443 −0.00390897
$$181$$ 3.60855 0.268221 0.134111 0.990966i $$-0.457182\pi$$
0.134111 + 0.990966i $$0.457182\pi$$
$$182$$ 0 0
$$183$$ 22.8016 1.68555
$$184$$ −3.41349 −0.251645
$$185$$ 5.03808 0.370407
$$186$$ 12.0424 0.882995
$$187$$ 1.37709 0.100703
$$188$$ −4.67035 −0.340620
$$189$$ 0 0
$$190$$ −11.5722 −0.839532
$$191$$ 0.337675 0.0244333 0.0122167 0.999925i $$-0.496111\pi$$
0.0122167 + 0.999925i $$0.496111\pi$$
$$192$$ −15.4295 −1.11353
$$193$$ 8.42151 0.606194 0.303097 0.952960i $$-0.401979\pi$$
0.303097 + 0.952960i $$0.401979\pi$$
$$194$$ 0.118893 0.00853603
$$195$$ 3.85738 0.276233
$$196$$ 0 0
$$197$$ −8.51035 −0.606337 −0.303169 0.952937i $$-0.598045\pi$$
−0.303169 + 0.952937i $$0.598045\pi$$
$$198$$ 0.0424451 0.00301644
$$199$$ −14.6166 −1.03614 −0.518071 0.855338i $$-0.673350\pi$$
−0.518071 + 0.855338i $$0.673350\pi$$
$$200$$ −0.345345 −0.0244196
$$201$$ −11.4422 −0.807071
$$202$$ −8.35640 −0.587954
$$203$$ 0 0
$$204$$ −1.62593 −0.113838
$$205$$ 26.8304 1.87391
$$206$$ 19.1570 1.33473
$$207$$ −0.0494177 −0.00343477
$$208$$ 2.64663 0.183511
$$209$$ −3.41215 −0.236023
$$210$$ 0 0
$$211$$ −23.8667 −1.64305 −0.821527 0.570169i $$-0.806878\pi$$
−0.821527 + 0.570169i $$0.806878\pi$$
$$212$$ −4.24884 −0.291811
$$213$$ −10.2201 −0.700271
$$214$$ 2.45023 0.167495
$$215$$ 17.8367 1.21645
$$216$$ 15.8223 1.07657
$$217$$ 0 0
$$218$$ 21.0030 1.42250
$$219$$ −13.9749 −0.944339
$$220$$ 0.931860 0.0628260
$$221$$ 1.74483 0.117370
$$222$$ 4.81430 0.323115
$$223$$ 1.27890 0.0856412 0.0428206 0.999083i $$-0.486366\pi$$
0.0428206 + 0.999083i $$0.486366\pi$$
$$224$$ 0 0
$$225$$ −0.00499963 −0.000333308 0
$$226$$ −12.3263 −0.819936
$$227$$ 9.57849 0.635747 0.317873 0.948133i $$-0.397031\pi$$
0.317873 + 0.948133i $$0.397031\pi$$
$$228$$ 4.02872 0.266809
$$229$$ −13.5134 −0.892989 −0.446494 0.894786i $$-0.647328\pi$$
−0.446494 + 0.894786i $$0.647328\pi$$
$$230$$ 2.97797 0.196361
$$231$$ 0 0
$$232$$ −26.0474 −1.71010
$$233$$ 28.9586 1.89714 0.948571 0.316565i $$-0.102530\pi$$
0.948571 + 0.316565i $$0.102530\pi$$
$$234$$ 0.0537794 0.00351567
$$235$$ 19.3327 1.26112
$$236$$ −5.85105 −0.380871
$$237$$ 12.0588 0.783302
$$238$$ 0 0
$$239$$ −24.1363 −1.56125 −0.780623 0.625002i $$-0.785098\pi$$
−0.780623 + 0.625002i $$0.785098\pi$$
$$240$$ 10.2091 0.658992
$$241$$ −9.92552 −0.639359 −0.319680 0.947526i $$-0.603575\pi$$
−0.319680 + 0.947526i $$0.603575\pi$$
$$242$$ 12.5641 0.807653
$$243$$ 0.461568 0.0296096
$$244$$ 6.97930 0.446804
$$245$$ 0 0
$$246$$ 25.6386 1.63466
$$247$$ −4.32331 −0.275086
$$248$$ 17.4897 1.11059
$$249$$ −5.48332 −0.347491
$$250$$ 13.6847 0.865497
$$251$$ 18.4990 1.16765 0.583824 0.811880i $$-0.301556\pi$$
0.583824 + 0.811880i $$0.301556\pi$$
$$252$$ 0 0
$$253$$ 0.878080 0.0552044
$$254$$ −19.9112 −1.24934
$$255$$ 6.73047 0.421478
$$256$$ −11.8223 −0.738895
$$257$$ 16.6767 1.04026 0.520132 0.854086i $$-0.325883\pi$$
0.520132 + 0.854086i $$0.325883\pi$$
$$258$$ 17.0444 1.06114
$$259$$ 0 0
$$260$$ 1.18070 0.0732238
$$261$$ −0.377094 −0.0233415
$$262$$ −14.9319 −0.922493
$$263$$ −13.4690 −0.830531 −0.415266 0.909700i $$-0.636311\pi$$
−0.415266 + 0.909700i $$0.636311\pi$$
$$264$$ 4.22512 0.260038
$$265$$ 17.5878 1.08041
$$266$$ 0 0
$$267$$ −5.91750 −0.362145
$$268$$ −3.50232 −0.213938
$$269$$ −30.5578 −1.86314 −0.931571 0.363560i $$-0.881561\pi$$
−0.931571 + 0.363560i $$0.881561\pi$$
$$270$$ −13.8036 −0.840061
$$271$$ 4.22512 0.256658 0.128329 0.991732i $$-0.459039\pi$$
0.128329 + 0.991732i $$0.459039\pi$$
$$272$$ 4.61791 0.280002
$$273$$ 0 0
$$274$$ 4.05244 0.244817
$$275$$ 0.0888361 0.00535702
$$276$$ −1.03675 −0.0624049
$$277$$ −3.00000 −0.180253 −0.0901263 0.995930i $$-0.528727\pi$$
−0.0901263 + 0.995930i $$0.528727\pi$$
$$278$$ 7.46593 0.447777
$$279$$ 0.253201 0.0151587
$$280$$ 0 0
$$281$$ −6.27890 −0.374568 −0.187284 0.982306i $$-0.559968\pi$$
−0.187284 + 0.982306i $$0.559968\pi$$
$$282$$ 18.4740 1.10011
$$283$$ 23.5371 1.39914 0.699568 0.714566i $$-0.253376\pi$$
0.699568 + 0.714566i $$0.253376\pi$$
$$284$$ −3.12825 −0.185628
$$285$$ −16.6767 −0.987842
$$286$$ −0.955582 −0.0565047
$$287$$ 0 0
$$288$$ −0.130227 −0.00767373
$$289$$ −13.9556 −0.820917
$$290$$ 22.7241 1.33441
$$291$$ 0.171337 0.0100440
$$292$$ −4.27756 −0.250325
$$293$$ 21.6166 1.26285 0.631427 0.775435i $$-0.282470\pi$$
0.631427 + 0.775435i $$0.282470\pi$$
$$294$$ 0 0
$$295$$ 24.2201 1.41015
$$296$$ 6.99197 0.406400
$$297$$ −4.07011 −0.236172
$$298$$ −22.8066 −1.32115
$$299$$ 1.11256 0.0643409
$$300$$ −0.104889 −0.00605575
$$301$$ 0 0
$$302$$ −24.4452 −1.40667
$$303$$ −12.0424 −0.691820
$$304$$ −11.4422 −0.656256
$$305$$ −28.8905 −1.65426
$$306$$ 0.0938357 0.00536423
$$307$$ −20.3945 −1.16397 −0.581987 0.813198i $$-0.697725\pi$$
−0.581987 + 0.813198i $$0.697725\pi$$
$$308$$ 0 0
$$309$$ 27.6072 1.57052
$$310$$ −15.2582 −0.866608
$$311$$ 12.9492 0.734284 0.367142 0.930165i $$-0.380336\pi$$
0.367142 + 0.930165i $$0.380336\pi$$
$$312$$ 5.35337 0.303075
$$313$$ −33.2345 −1.87852 −0.939262 0.343201i $$-0.888489\pi$$
−0.939262 + 0.343201i $$0.888489\pi$$
$$314$$ −16.8066 −0.948453
$$315$$ 0 0
$$316$$ 3.69105 0.207638
$$317$$ −4.87175 −0.273624 −0.136812 0.990597i $$-0.543686\pi$$
−0.136812 + 0.990597i $$0.543686\pi$$
$$318$$ 16.8066 0.942469
$$319$$ 6.70041 0.375151
$$320$$ 19.5498 1.09287
$$321$$ 3.53104 0.197084
$$322$$ 0 0
$$323$$ −7.54343 −0.419728
$$324$$ 4.87675 0.270930
$$325$$ 0.112558 0.00624362
$$326$$ −13.9112 −0.770468
$$327$$ 30.2676 1.67380
$$328$$ 37.2358 2.05600
$$329$$ 0 0
$$330$$ −3.68605 −0.202910
$$331$$ 6.87175 0.377705 0.188853 0.982005i $$-0.439523\pi$$
0.188853 + 0.982005i $$0.439523\pi$$
$$332$$ −1.67838 −0.0921129
$$333$$ 0.101224 0.00554705
$$334$$ 15.7034 0.859254
$$335$$ 14.4977 0.792093
$$336$$ 0 0
$$337$$ 11.0712 0.603085 0.301542 0.953453i $$-0.402499\pi$$
0.301542 + 0.953453i $$0.402499\pi$$
$$338$$ −1.21076 −0.0658564
$$339$$ −17.7635 −0.964784
$$340$$ 2.06011 0.111725
$$341$$ −4.49901 −0.243635
$$342$$ −0.232505 −0.0125724
$$343$$ 0 0
$$344$$ 24.7542 1.33466
$$345$$ 4.29157 0.231050
$$346$$ −3.00303 −0.161444
$$347$$ 5.62593 0.302016 0.151008 0.988533i $$-0.451748\pi$$
0.151008 + 0.988533i $$0.451748\pi$$
$$348$$ −7.91116 −0.424083
$$349$$ 18.4783 0.989122 0.494561 0.869143i $$-0.335329\pi$$
0.494561 + 0.869143i $$0.335329\pi$$
$$350$$ 0 0
$$351$$ −5.15698 −0.275259
$$352$$ 2.31395 0.123334
$$353$$ 4.67035 0.248578 0.124289 0.992246i $$-0.460335\pi$$
0.124289 + 0.992246i $$0.460335\pi$$
$$354$$ 23.1443 1.23011
$$355$$ 12.9492 0.687275
$$356$$ −1.81127 −0.0959974
$$357$$ 0 0
$$358$$ 9.87611 0.521968
$$359$$ −11.7211 −0.618616 −0.309308 0.950962i $$-0.600097\pi$$
−0.309308 + 0.950962i $$0.600097\pi$$
$$360$$ 0.301284 0.0158790
$$361$$ −0.308953 −0.0162607
$$362$$ −4.36907 −0.229633
$$363$$ 18.1062 0.950330
$$364$$ 0 0
$$365$$ 17.7067 0.926813
$$366$$ −27.6072 −1.44305
$$367$$ 19.9699 1.04242 0.521211 0.853428i $$-0.325480\pi$$
0.521211 + 0.853428i $$0.325480\pi$$
$$368$$ 2.94453 0.153494
$$369$$ 0.539070 0.0280629
$$370$$ −6.09989 −0.317118
$$371$$ 0 0
$$372$$ 5.31198 0.275413
$$373$$ −4.42651 −0.229196 −0.114598 0.993412i $$-0.536558\pi$$
−0.114598 + 0.993412i $$0.536558\pi$$
$$374$$ −1.66732 −0.0862153
$$375$$ 19.7211 1.01839
$$376$$ 26.8304 1.38367
$$377$$ 8.48965 0.437239
$$378$$ 0 0
$$379$$ −32.5702 −1.67302 −0.836509 0.547953i $$-0.815407\pi$$
−0.836509 + 0.547953i $$0.815407\pi$$
$$380$$ −5.10453 −0.261857
$$381$$ −28.6941 −1.47004
$$382$$ −0.408842 −0.0209182
$$383$$ 14.1964 0.725402 0.362701 0.931906i $$-0.381855\pi$$
0.362701 + 0.931906i $$0.381855\pi$$
$$384$$ 8.45023 0.431224
$$385$$ 0 0
$$386$$ −10.1964 −0.518983
$$387$$ 0.358371 0.0182170
$$388$$ 0.0524443 0.00266246
$$389$$ 17.3170 0.878006 0.439003 0.898486i $$-0.355332\pi$$
0.439003 + 0.898486i $$0.355332\pi$$
$$390$$ −4.67035 −0.236492
$$391$$ 1.94122 0.0981718
$$392$$ 0 0
$$393$$ −21.5184 −1.08546
$$394$$ 10.3040 0.519106
$$395$$ −15.2789 −0.768765
$$396$$ 0.0187227 0.000940852 0
$$397$$ 15.7241 0.789171 0.394586 0.918859i $$-0.370888\pi$$
0.394586 + 0.918859i $$0.370888\pi$$
$$398$$ 17.6971 0.887075
$$399$$ 0 0
$$400$$ 0.297900 0.0148950
$$401$$ 3.57849 0.178701 0.0893506 0.996000i $$-0.471521\pi$$
0.0893506 + 0.996000i $$0.471521\pi$$
$$402$$ 13.8537 0.690961
$$403$$ −5.70041 −0.283958
$$404$$ −3.68605 −0.183388
$$405$$ −20.1870 −1.00310
$$406$$ 0 0
$$407$$ −1.79861 −0.0891536
$$408$$ 9.34070 0.462434
$$409$$ −31.3501 −1.55016 −0.775080 0.631863i $$-0.782291\pi$$
−0.775080 + 0.631863i $$0.782291\pi$$
$$410$$ −32.4850 −1.60432
$$411$$ 5.84000 0.288066
$$412$$ 8.45023 0.416313
$$413$$ 0 0
$$414$$ 0.0598327 0.00294062
$$415$$ 6.94756 0.341042
$$416$$ 2.93186 0.143746
$$417$$ 10.7592 0.526880
$$418$$ 4.13128 0.202068
$$419$$ −28.7716 −1.40558 −0.702792 0.711396i $$-0.748063\pi$$
−0.702792 + 0.711396i $$0.748063\pi$$
$$420$$ 0 0
$$421$$ −0.190060 −0.00926297 −0.00463148 0.999989i $$-0.501474\pi$$
−0.00463148 + 0.999989i $$0.501474\pi$$
$$422$$ 28.8968 1.40667
$$423$$ 0.388428 0.0188860
$$424$$ 24.4088 1.18540
$$425$$ 0.196395 0.00952655
$$426$$ 12.3741 0.599526
$$427$$ 0 0
$$428$$ 1.08081 0.0522429
$$429$$ −1.37709 −0.0664867
$$430$$ −21.5959 −1.04145
$$431$$ 25.6734 1.23664 0.618322 0.785925i $$-0.287813\pi$$
0.618322 + 0.785925i $$0.287813\pi$$
$$432$$ −13.6486 −0.656669
$$433$$ 1.82233 0.0875755 0.0437877 0.999041i $$-0.486057\pi$$
0.0437877 + 0.999041i $$0.486057\pi$$
$$434$$ 0 0
$$435$$ 32.7479 1.57014
$$436$$ 9.26454 0.443691
$$437$$ −4.80994 −0.230091
$$438$$ 16.9202 0.808481
$$439$$ 0.735465 0.0351018 0.0175509 0.999846i $$-0.494413\pi$$
0.0175509 + 0.999846i $$0.494413\pi$$
$$440$$ −5.35337 −0.255212
$$441$$ 0 0
$$442$$ −2.11256 −0.100484
$$443$$ −3.71174 −0.176350 −0.0881751 0.996105i $$-0.528104\pi$$
−0.0881751 + 0.996105i $$0.528104\pi$$
$$444$$ 2.12361 0.100782
$$445$$ 7.49768 0.355424
$$446$$ −1.54843 −0.0733203
$$447$$ −32.8667 −1.55454
$$448$$ 0 0
$$449$$ −5.17570 −0.244256 −0.122128 0.992514i $$-0.538972\pi$$
−0.122128 + 0.992514i $$0.538972\pi$$
$$450$$ 0.00605333 0.000285357 0
$$451$$ −9.57849 −0.451033
$$452$$ −5.43721 −0.255745
$$453$$ −35.2281 −1.65516
$$454$$ −11.5972 −0.544284
$$455$$ 0 0
$$456$$ −23.1443 −1.08383
$$457$$ −34.1299 −1.59653 −0.798266 0.602305i $$-0.794249\pi$$
−0.798266 + 0.602305i $$0.794249\pi$$
$$458$$ 16.3614 0.764518
$$459$$ −8.99803 −0.419992
$$460$$ 1.31360 0.0612467
$$461$$ −11.4008 −0.530989 −0.265494 0.964112i $$-0.585535\pi$$
−0.265494 + 0.964112i $$0.585535\pi$$
$$462$$ 0 0
$$463$$ −30.0124 −1.39479 −0.697397 0.716685i $$-0.745659\pi$$
−0.697397 + 0.716685i $$0.745659\pi$$
$$464$$ 22.4690 1.04310
$$465$$ −21.9887 −1.01970
$$466$$ −35.0618 −1.62421
$$467$$ 34.1663 1.58103 0.790515 0.612443i $$-0.209813\pi$$
0.790515 + 0.612443i $$0.209813\pi$$
$$468$$ 0.0237224 0.00109657
$$469$$ 0 0
$$470$$ −23.4072 −1.07969
$$471$$ −24.2201 −1.11600
$$472$$ 33.6133 1.54718
$$473$$ −6.36773 −0.292789
$$474$$ −14.6002 −0.670611
$$475$$ −0.486625 −0.0223279
$$476$$ 0 0
$$477$$ 0.353371 0.0161798
$$478$$ 29.2231 1.33664
$$479$$ −12.8761 −0.588324 −0.294162 0.955756i $$-0.595041\pi$$
−0.294162 + 0.955756i $$0.595041\pi$$
$$480$$ 11.3093 0.516197
$$481$$ −2.27890 −0.103909
$$482$$ 12.0174 0.547377
$$483$$ 0 0
$$484$$ 5.54210 0.251913
$$485$$ −0.217091 −0.00985758
$$486$$ −0.558846 −0.0253498
$$487$$ −21.9399 −0.994191 −0.497096 0.867696i $$-0.665600\pi$$
−0.497096 + 0.867696i $$0.665600\pi$$
$$488$$ −40.0949 −1.81501
$$489$$ −20.0474 −0.906577
$$490$$ 0 0
$$491$$ 4.11256 0.185597 0.0927986 0.995685i $$-0.470419\pi$$
0.0927986 + 0.995685i $$0.470419\pi$$
$$492$$ 11.3093 0.509863
$$493$$ 14.8130 0.667142
$$494$$ 5.23448 0.235510
$$495$$ −0.0775018 −0.00348344
$$496$$ −15.0869 −0.677420
$$497$$ 0 0
$$498$$ 6.63896 0.297499
$$499$$ 19.0331 0.852038 0.426019 0.904714i $$-0.359916\pi$$
0.426019 + 0.904714i $$0.359916\pi$$
$$500$$ 6.03639 0.269956
$$501$$ 22.6303 1.01105
$$502$$ −22.3978 −0.999662
$$503$$ −35.8698 −1.59935 −0.799677 0.600430i $$-0.794996\pi$$
−0.799677 + 0.600430i $$0.794996\pi$$
$$504$$ 0 0
$$505$$ 15.2582 0.678981
$$506$$ −1.06314 −0.0472624
$$507$$ −1.74483 −0.0774904
$$508$$ −8.78291 −0.389679
$$509$$ 17.5465 0.777733 0.388867 0.921294i $$-0.372867\pi$$
0.388867 + 0.921294i $$0.372867\pi$$
$$510$$ −8.14895 −0.360842
$$511$$ 0 0
$$512$$ 24.0000 1.06066
$$513$$ 22.2952 0.984358
$$514$$ −20.1914 −0.890604
$$515$$ −34.9793 −1.54137
$$516$$ 7.51837 0.330978
$$517$$ −6.90180 −0.303541
$$518$$ 0 0
$$519$$ −4.32768 −0.189964
$$520$$ −6.78291 −0.297450
$$521$$ −8.84302 −0.387420 −0.193710 0.981059i $$-0.562052\pi$$
−0.193710 + 0.981059i $$0.562052\pi$$
$$522$$ 0.456568 0.0199835
$$523$$ 21.9112 0.958108 0.479054 0.877785i $$-0.340980\pi$$
0.479054 + 0.877785i $$0.340980\pi$$
$$524$$ −6.58651 −0.287733
$$525$$ 0 0
$$526$$ 16.3076 0.711046
$$527$$ −9.94622 −0.433264
$$528$$ −3.64466 −0.158613
$$529$$ −21.7622 −0.946183
$$530$$ −21.2946 −0.924978
$$531$$ 0.486625 0.0211177
$$532$$ 0 0
$$533$$ −12.1363 −0.525681
$$534$$ 7.16465 0.310045
$$535$$ −4.47396 −0.193426
$$536$$ 20.1202 0.869061
$$537$$ 14.2325 0.614178
$$538$$ 36.9980 1.59510
$$539$$ 0 0
$$540$$ −6.08884 −0.262022
$$541$$ 22.8654 0.983061 0.491530 0.870860i $$-0.336438\pi$$
0.491530 + 0.870860i $$0.336438\pi$$
$$542$$ −5.11559 −0.219733
$$543$$ −6.29628 −0.270199
$$544$$ 5.11559 0.219329
$$545$$ −38.3501 −1.64274
$$546$$ 0 0
$$547$$ −35.2676 −1.50793 −0.753966 0.656913i $$-0.771862\pi$$
−0.753966 + 0.656913i $$0.771862\pi$$
$$548$$ 1.78755 0.0763605
$$549$$ −0.580461 −0.0247735
$$550$$ −0.107559 −0.00458632
$$551$$ −36.7034 −1.56362
$$552$$ 5.95594 0.253502
$$553$$ 0 0
$$554$$ 3.63227 0.154320
$$555$$ −8.79058 −0.373139
$$556$$ 3.29326 0.139665
$$557$$ 39.8698 1.68934 0.844668 0.535290i $$-0.179798\pi$$
0.844668 + 0.535290i $$0.179798\pi$$
$$558$$ −0.306565 −0.0129779
$$559$$ −8.06814 −0.341246
$$560$$ 0 0
$$561$$ −2.40279 −0.101446
$$562$$ 7.60221 0.320680
$$563$$ 25.5972 1.07879 0.539397 0.842052i $$-0.318652\pi$$
0.539397 + 0.842052i $$0.318652\pi$$
$$564$$ 8.14895 0.343133
$$565$$ 22.5070 0.946878
$$566$$ −28.4977 −1.19785
$$567$$ 0 0
$$568$$ 17.9713 0.754058
$$569$$ −25.0949 −1.05203 −0.526016 0.850475i $$-0.676315\pi$$
−0.526016 + 0.850475i $$0.676315\pi$$
$$570$$ 20.1914 0.845724
$$571$$ −31.2488 −1.30772 −0.653862 0.756614i $$-0.726852\pi$$
−0.653862 + 0.756614i $$0.726852\pi$$
$$572$$ −0.421512 −0.0176243
$$573$$ −0.589185 −0.0246135
$$574$$ 0 0
$$575$$ 0.125228 0.00522236
$$576$$ 0.392790 0.0163662
$$577$$ −12.8717 −0.535858 −0.267929 0.963439i $$-0.586339\pi$$
−0.267929 + 0.963439i $$0.586339\pi$$
$$578$$ 16.8968 0.702814
$$579$$ −14.6941 −0.610665
$$580$$ 10.0237 0.416212
$$581$$ 0 0
$$582$$ −0.207448 −0.00859899
$$583$$ −6.27890 −0.260045
$$584$$ 24.5738 1.01687
$$585$$ −0.0981974 −0.00405996
$$586$$ −26.1724 −1.08117
$$587$$ 9.96692 0.411379 0.205689 0.978617i $$-0.434056\pi$$
0.205689 + 0.978617i $$0.434056\pi$$
$$588$$ 0 0
$$589$$ 24.6447 1.01547
$$590$$ −29.3246 −1.20728
$$591$$ 14.8491 0.610809
$$592$$ −6.03139 −0.247889
$$593$$ −20.5909 −0.845566 −0.422783 0.906231i $$-0.638947\pi$$
−0.422783 + 0.906231i $$0.638947\pi$$
$$594$$ 4.92791 0.202195
$$595$$ 0 0
$$596$$ −10.0601 −0.412078
$$597$$ 25.5034 1.04378
$$598$$ −1.34704 −0.0550844
$$599$$ −12.5371 −0.512252 −0.256126 0.966643i $$-0.582446\pi$$
−0.256126 + 0.966643i $$0.582446\pi$$
$$600$$ 0.602567 0.0245997
$$601$$ 1.82233 0.0743343 0.0371672 0.999309i $$-0.488167\pi$$
0.0371672 + 0.999309i $$0.488167\pi$$
$$602$$ 0 0
$$603$$ 0.291284 0.0118620
$$604$$ −10.7829 −0.438750
$$605$$ −22.9412 −0.932693
$$606$$ 14.5805 0.592291
$$607$$ −30.9780 −1.25736 −0.628678 0.777665i $$-0.716404\pi$$
−0.628678 + 0.777665i $$0.716404\pi$$
$$608$$ −12.6754 −0.514053
$$609$$ 0 0
$$610$$ 34.9793 1.41627
$$611$$ −8.74483 −0.353778
$$612$$ 0.0413914 0.00167315
$$613$$ 12.4502 0.502860 0.251430 0.967875i $$-0.419099\pi$$
0.251430 + 0.967875i $$0.419099\pi$$
$$614$$ 24.6927 0.996518
$$615$$ −46.8143 −1.88773
$$616$$ 0 0
$$617$$ 31.7809 1.27945 0.639726 0.768603i $$-0.279048\pi$$
0.639726 + 0.768603i $$0.279048\pi$$
$$618$$ −33.4256 −1.34457
$$619$$ −4.22512 −0.169822 −0.0849109 0.996389i $$-0.527061\pi$$
−0.0849109 + 0.996389i $$0.527061\pi$$
$$620$$ −6.73047 −0.270302
$$621$$ −5.73744 −0.230235
$$622$$ −15.6784 −0.628646
$$623$$ 0 0
$$624$$ −4.61791 −0.184864
$$625$$ −24.4245 −0.976982
$$626$$ 40.2388 1.60827
$$627$$ 5.95361 0.237764
$$628$$ −7.41349 −0.295830
$$629$$ −3.97628 −0.158545
$$630$$ 0 0
$$631$$ 7.31198 0.291085 0.145543 0.989352i $$-0.453507\pi$$
0.145543 + 0.989352i $$0.453507\pi$$
$$632$$ −21.2044 −0.843467
$$633$$ 41.6433 1.65517
$$634$$ 5.89849 0.234259
$$635$$ 36.3564 1.44276
$$636$$ 7.41349 0.293964
$$637$$ 0 0
$$638$$ −8.11256 −0.321179
$$639$$ 0.260174 0.0102923
$$640$$ −10.7067 −0.423221
$$641$$ −23.0474 −0.910319 −0.455160 0.890410i $$-0.650418\pi$$
−0.455160 + 0.890410i $$0.650418\pi$$
$$642$$ −4.27523 −0.168730
$$643$$ 48.9379 1.92992 0.964961 0.262392i $$-0.0845113\pi$$
0.964961 + 0.262392i $$0.0845113\pi$$
$$644$$ 0 0
$$645$$ −31.1219 −1.22542
$$646$$ 9.13325 0.359343
$$647$$ 33.4309 1.31430 0.657152 0.753758i $$-0.271761\pi$$
0.657152 + 0.753758i $$0.271761\pi$$
$$648$$ −28.0161 −1.10057
$$649$$ −8.64663 −0.339410
$$650$$ −0.136281 −0.00534537
$$651$$ 0 0
$$652$$ −6.13628 −0.240315
$$653$$ −10.4629 −0.409445 −0.204723 0.978820i $$-0.565629\pi$$
−0.204723 + 0.978820i $$0.565629\pi$$
$$654$$ −36.6466 −1.43300
$$655$$ 27.2645 1.06531
$$656$$ −32.1202 −1.25408
$$657$$ 0.355760 0.0138795
$$658$$ 0 0
$$659$$ −8.73849 −0.340403 −0.170202 0.985409i $$-0.554442\pi$$
−0.170202 + 0.985409i $$0.554442\pi$$
$$660$$ −1.62593 −0.0632894
$$661$$ 4.61354 0.179446 0.0897230 0.995967i $$-0.471402\pi$$
0.0897230 + 0.995967i $$0.471402\pi$$
$$662$$ −8.32001 −0.323366
$$663$$ −3.04442 −0.118235
$$664$$ 9.64199 0.374182
$$665$$ 0 0
$$666$$ −0.122558 −0.00474901
$$667$$ 9.44523 0.365721
$$668$$ 6.92686 0.268008
$$669$$ −2.23145 −0.0862729
$$670$$ −17.5531 −0.678137
$$671$$ 10.3140 0.398166
$$672$$ 0 0
$$673$$ 9.83802 0.379228 0.189614 0.981859i $$-0.439276\pi$$
0.189614 + 0.981859i $$0.439276\pi$$
$$674$$ −13.4045 −0.516321
$$675$$ −0.580461 −0.0223420
$$676$$ −0.534070 −0.0205412
$$677$$ −4.69541 −0.180459 −0.0902296 0.995921i $$-0.528760\pi$$
−0.0902296 + 0.995921i $$0.528760\pi$$
$$678$$ 21.5073 0.825984
$$679$$ 0 0
$$680$$ −11.8350 −0.453851
$$681$$ −16.7128 −0.640436
$$682$$ 5.44721 0.208584
$$683$$ 11.0207 0.421695 0.210848 0.977519i $$-0.432378\pi$$
0.210848 + 0.977519i $$0.432378\pi$$
$$684$$ −0.102559 −0.00392145
$$685$$ −7.39948 −0.282720
$$686$$ 0 0
$$687$$ 23.5785 0.899575
$$688$$ −21.3534 −0.814090
$$689$$ −7.95558 −0.303084
$$690$$ −5.19604 −0.197810
$$691$$ 38.7635 1.47463 0.737317 0.675546i $$-0.236092\pi$$
0.737317 + 0.675546i $$0.236092\pi$$
$$692$$ −1.32465 −0.0503556
$$693$$ 0 0
$$694$$ −6.81163 −0.258566
$$695$$ −13.6323 −0.517101
$$696$$ 45.4483 1.72271
$$697$$ −21.1757 −0.802087
$$698$$ −22.3727 −0.846820
$$699$$ −50.5277 −1.91113
$$700$$ 0 0
$$701$$ −32.0681 −1.21120 −0.605598 0.795770i $$-0.707066\pi$$
−0.605598 + 0.795770i $$0.707066\pi$$
$$702$$ 6.24384 0.235659
$$703$$ 9.85238 0.371590
$$704$$ −6.97930 −0.263042
$$705$$ −33.7322 −1.27043
$$706$$ −5.65465 −0.212816
$$707$$ 0 0
$$708$$ 10.2091 0.383680
$$709$$ 38.3451 1.44008 0.720040 0.693933i $$-0.244124\pi$$
0.720040 + 0.693933i $$0.244124\pi$$
$$710$$ −15.6784 −0.588399
$$711$$ −0.306980 −0.0115127
$$712$$ 10.4055 0.389961
$$713$$ −6.34204 −0.237511
$$714$$ 0 0
$$715$$ 1.74483 0.0652528
$$716$$ 4.35640 0.162806
$$717$$ 42.1136 1.57276
$$718$$ 14.1914 0.529618
$$719$$ 8.72413 0.325355 0.162678 0.986679i $$-0.447987\pi$$
0.162678 + 0.986679i $$0.447987\pi$$
$$720$$ −0.259892 −0.00968561
$$721$$ 0 0
$$722$$ 0.374067 0.0139213
$$723$$ 17.3183 0.644075
$$724$$ −1.92722 −0.0716244
$$725$$ 0.955582 0.0354894
$$726$$ −21.9222 −0.813610
$$727$$ −26.6754 −0.989334 −0.494667 0.869083i $$-0.664710\pi$$
−0.494667 + 0.869083i $$0.664710\pi$$
$$728$$ 0 0
$$729$$ 26.5885 0.984759
$$730$$ −21.4385 −0.793476
$$731$$ −14.0775 −0.520675
$$732$$ −12.1777 −0.450100
$$733$$ −26.7211 −0.986966 −0.493483 0.869755i $$-0.664277\pi$$
−0.493483 + 0.869755i $$0.664277\pi$$
$$734$$ −24.1787 −0.892453
$$735$$ 0 0
$$736$$ 3.26187 0.120234
$$737$$ −5.17570 −0.190649
$$738$$ −0.652682 −0.0240256
$$739$$ 36.0538 1.32626 0.663130 0.748504i $$-0.269228\pi$$
0.663130 + 0.748504i $$0.269228\pi$$
$$740$$ −2.69069 −0.0989117
$$741$$ 7.54343 0.277115
$$742$$ 0 0
$$743$$ −2.96058 −0.108613 −0.0543066 0.998524i $$-0.517295\pi$$
−0.0543066 + 0.998524i $$0.517295\pi$$
$$744$$ −30.5164 −1.11879
$$745$$ 41.6433 1.52569
$$746$$ 5.35942 0.196222
$$747$$ 0.139589 0.00510729
$$748$$ −0.735465 −0.0268913
$$749$$ 0 0
$$750$$ −23.8774 −0.871881
$$751$$ 43.1550 1.57475 0.787374 0.616475i $$-0.211440\pi$$
0.787374 + 0.616475i $$0.211440\pi$$
$$752$$ −23.1443 −0.843986
$$753$$ −32.2776 −1.17626
$$754$$ −10.2789 −0.374335
$$755$$ 44.6353 1.62444
$$756$$ 0 0
$$757$$ 18.7335 0.680880 0.340440 0.940266i $$-0.389424\pi$$
0.340440 + 0.940266i $$0.389424\pi$$
$$758$$ 39.4345 1.43233
$$759$$ −1.53210 −0.0556116
$$760$$ 29.3246 1.06372
$$761$$ 4.22948 0.153318 0.0766592 0.997057i $$-0.475575\pi$$
0.0766592 + 0.997057i $$0.475575\pi$$
$$762$$ 34.7415 1.25855
$$763$$ 0 0
$$764$$ −0.180342 −0.00652456
$$765$$ −0.171337 −0.00619472
$$766$$ −17.1884 −0.621041
$$767$$ −10.9556 −0.395583
$$768$$ 20.6279 0.744345
$$769$$ 21.1299 0.761965 0.380983 0.924582i $$-0.375586\pi$$
0.380983 + 0.924582i $$0.375586\pi$$
$$770$$ 0 0
$$771$$ −29.0979 −1.04794
$$772$$ −4.49768 −0.161875
$$773$$ −33.0742 −1.18960 −0.594798 0.803875i $$-0.702768\pi$$
−0.594798 + 0.803875i $$0.702768\pi$$
$$774$$ −0.433900 −0.0155962
$$775$$ −0.641629 −0.0230480
$$776$$ −0.301284 −0.0108154
$$777$$ 0 0
$$778$$ −20.9666 −0.751690
$$779$$ 52.4690 1.87990
$$780$$ −2.06011 −0.0737639
$$781$$ −4.62291 −0.165421
$$782$$ −2.35034 −0.0840482
$$783$$ −43.7809 −1.56460
$$784$$ 0 0
$$785$$ 30.6877 1.09529
$$786$$ 26.0535 0.929297
$$787$$ −6.23948 −0.222413 −0.111207 0.993797i $$-0.535472\pi$$
−0.111207 + 0.993797i $$0.535472\pi$$
$$788$$ 4.54512 0.161913
$$789$$ 23.5010 0.836657
$$790$$ 18.4990 0.658165
$$791$$ 0 0
$$792$$ −0.107559 −0.00382194
$$793$$ 13.0681 0.464063
$$794$$ −19.0381 −0.675636
$$795$$ −30.6877 −1.08838
$$796$$ 7.80628 0.276686
$$797$$ 32.1837 1.14001 0.570003 0.821643i $$-0.306942\pi$$
0.570003 + 0.821643i $$0.306942\pi$$
$$798$$ 0 0
$$799$$ −15.2582 −0.539796
$$800$$ 0.330006 0.0116675
$$801$$ 0.150642 0.00532267
$$802$$ −4.33268 −0.152992
$$803$$ −6.32134 −0.223075
$$804$$ 6.11094 0.215516
$$805$$ 0 0
$$806$$ 6.90180 0.243106
$$807$$ 53.3180 1.87688
$$808$$ 21.1757 0.744958
$$809$$ 47.8016 1.68062 0.840308 0.542109i $$-0.182374\pi$$
0.840308 + 0.542109i $$0.182374\pi$$
$$810$$ 24.4416 0.858789
$$811$$ 37.4957 1.31665 0.658326 0.752733i $$-0.271265\pi$$
0.658326 + 0.752733i $$0.271265\pi$$
$$812$$ 0 0
$$813$$ −7.37209 −0.258551
$$814$$ 2.17767 0.0763274
$$815$$ 25.4008 0.889752
$$816$$ −8.05744 −0.282067
$$817$$ 34.8811 1.22034
$$818$$ 37.9573 1.32714
$$819$$ 0 0
$$820$$ −14.3293 −0.500401
$$821$$ 39.6447 1.38361 0.691804 0.722085i $$-0.256816\pi$$
0.691804 + 0.722085i $$0.256816\pi$$
$$822$$ −7.07081 −0.246623
$$823$$ −17.4659 −0.608824 −0.304412 0.952540i $$-0.598460\pi$$
−0.304412 + 0.952540i $$0.598460\pi$$
$$824$$ −48.5451 −1.69115
$$825$$ −0.155004 −0.00539653
$$826$$ 0 0
$$827$$ −1.53104 −0.0532396 −0.0266198 0.999646i $$-0.508474\pi$$
−0.0266198 + 0.999646i $$0.508474\pi$$
$$828$$ 0.0263925 0.000917203 0
$$829$$ −37.2218 −1.29277 −0.646383 0.763013i $$-0.723719\pi$$
−0.646383 + 0.763013i $$0.723719\pi$$
$$830$$ −8.41179 −0.291978
$$831$$ 5.23448 0.181582
$$832$$ −8.84302 −0.306577
$$833$$ 0 0
$$834$$ −13.0267 −0.451079
$$835$$ −28.6734 −0.992283
$$836$$ 1.82233 0.0630265
$$837$$ 29.3969 1.01610
$$838$$ 34.8354 1.20337
$$839$$ −7.37209 −0.254513 −0.127256 0.991870i $$-0.540617\pi$$
−0.127256 + 0.991870i $$0.540617\pi$$
$$840$$ 0 0
$$841$$ 43.0742 1.48532
$$842$$ 0.230116 0.00793034
$$843$$ 10.9556 0.377330
$$844$$ 12.7465 0.438753
$$845$$ 2.21076 0.0760523
$$846$$ −0.470292 −0.0161690
$$847$$ 0 0
$$848$$ −21.0555 −0.723048
$$849$$ −41.0681 −1.40945
$$850$$ −0.237786 −0.00815600
$$851$$ −2.53541 −0.0869126
$$852$$ 5.45826 0.186997
$$853$$ 4.57215 0.156548 0.0782738 0.996932i $$-0.475059\pi$$
0.0782738 + 0.996932i $$0.475059\pi$$
$$854$$ 0 0
$$855$$ 0.424538 0.0145189
$$856$$ −6.20906 −0.212221
$$857$$ −52.0348 −1.77747 −0.888737 0.458417i $$-0.848416\pi$$
−0.888737 + 0.458417i $$0.848416\pi$$
$$858$$ 1.66732 0.0569215
$$859$$ 42.8905 1.46340 0.731702 0.681625i $$-0.238726\pi$$
0.731702 + 0.681625i $$0.238726\pi$$
$$860$$ −9.52604 −0.324835
$$861$$ 0 0
$$862$$ −31.0842 −1.05873
$$863$$ −7.82233 −0.266275 −0.133138 0.991098i $$-0.542505\pi$$
−0.133138 + 0.991098i $$0.542505\pi$$
$$864$$ −15.1195 −0.514377
$$865$$ 5.48332 0.186438
$$866$$ −2.20639 −0.0749763
$$867$$ 24.3501 0.826972
$$868$$ 0 0
$$869$$ 5.45460 0.185034
$$870$$ −39.6497 −1.34425
$$871$$ −6.55779 −0.222202
$$872$$ −53.2231 −1.80236
$$873$$ −0.00436174 −0.000147622 0
$$874$$ 5.82366 0.196988
$$875$$ 0 0
$$876$$ 7.46360 0.252172
$$877$$ −34.7191 −1.17238 −0.586191 0.810173i $$-0.699373\pi$$
−0.586191 + 0.810173i $$0.699373\pi$$
$$878$$ −0.890468 −0.0300518
$$879$$ −37.7172 −1.27217
$$880$$ 4.61791 0.155670
$$881$$ 11.4422 0.385498 0.192749 0.981248i $$-0.438260\pi$$
0.192749 + 0.981248i $$0.438260\pi$$
$$882$$ 0 0
$$883$$ −31.0217 −1.04396 −0.521982 0.852956i $$-0.674807\pi$$
−0.521982 + 0.852956i $$0.674807\pi$$
$$884$$ −0.931860 −0.0313418
$$885$$ −42.2599 −1.42055
$$886$$ 4.49401 0.150979
$$887$$ 12.8304 0.430801 0.215401 0.976526i $$-0.430894\pi$$
0.215401 + 0.976526i $$0.430894\pi$$
$$888$$ −12.1998 −0.409398
$$889$$ 0 0
$$890$$ −9.07786 −0.304291
$$891$$ 7.20681 0.241437
$$892$$ −0.683020 −0.0228692
$$893$$ 37.8066 1.26515
$$894$$ 39.7936 1.33090
$$895$$ −18.0331 −0.602780
$$896$$ 0 0
$$897$$ −1.94122 −0.0648155
$$898$$ 6.26651 0.209116
$$899$$ −48.3945 −1.61405
$$900$$ 0.00267015 8.90050e−5 0
$$901$$ −13.8811 −0.462447
$$902$$ 11.5972 0.386145
$$903$$ 0 0
$$904$$ 31.2358 1.03889
$$905$$ 7.97761 0.265185
$$906$$ 42.6527 1.41704
$$907$$ 10.1620 0.337423 0.168711 0.985665i $$-0.446039\pi$$
0.168711 + 0.985665i $$0.446039\pi$$
$$908$$ −5.11559 −0.169767
$$909$$ 0.306565 0.0101681
$$910$$ 0 0
$$911$$ 34.4008 1.13975 0.569875 0.821731i $$-0.306992\pi$$
0.569875 + 0.821731i $$0.306992\pi$$
$$912$$ 19.9647 0.661096
$$913$$ −2.48029 −0.0820856
$$914$$ 41.3230 1.36684
$$915$$ 50.4088 1.66646
$$916$$ 7.21709 0.238459
$$917$$ 0 0
$$918$$ 10.8944 0.359569
$$919$$ 8.15500 0.269009 0.134504 0.990913i $$-0.457056\pi$$
0.134504 + 0.990913i $$0.457056\pi$$
$$920$$ −7.54638 −0.248797
$$921$$ 35.5848 1.17256
$$922$$ 13.8036 0.454598
$$923$$ −5.85738 −0.192798
$$924$$ 0 0
$$925$$ −0.256509 −0.00843396
$$926$$ 36.3377 1.19413
$$927$$ −0.702797 −0.0230829
$$928$$ 24.8905 0.817070
$$929$$ 4.80994 0.157809 0.0789045 0.996882i $$-0.474858\pi$$
0.0789045 + 0.996882i $$0.474858\pi$$
$$930$$ 26.6229 0.872999
$$931$$ 0 0
$$932$$ −15.4659 −0.506603
$$933$$ −22.5942 −0.739700
$$934$$ −41.3671 −1.35357
$$935$$ 3.04442 0.0995631
$$936$$ −0.136281 −0.00445448
$$937$$ −47.1931 −1.54173 −0.770865 0.636998i $$-0.780176\pi$$
−0.770865 + 0.636998i $$0.780176\pi$$
$$938$$ 0 0
$$939$$ 57.9884 1.89238
$$940$$ −10.3250 −0.336765
$$941$$ 27.5702 0.898762 0.449381 0.893340i $$-0.351645\pi$$
0.449381 + 0.893340i $$0.351645\pi$$
$$942$$ 29.3246 0.955449
$$943$$ −13.5023 −0.439696
$$944$$ −28.9954 −0.943718
$$945$$ 0 0
$$946$$ 7.70977 0.250666
$$947$$ −27.8698 −0.905646 −0.452823 0.891600i $$-0.649583\pi$$
−0.452823 + 0.891600i $$0.649583\pi$$
$$948$$ −6.44023 −0.209169
$$949$$ −8.00936 −0.259995
$$950$$ 0.589185 0.0191157
$$951$$ 8.50035 0.275643
$$952$$ 0 0
$$953$$ −27.1076 −0.878100 −0.439050 0.898463i $$-0.644685\pi$$
−0.439050 + 0.898463i $$0.644685\pi$$
$$954$$ −0.427846 −0.0138520
$$955$$ 0.746518 0.0241567
$$956$$ 12.8905 0.416908
$$957$$ −11.6910 −0.377918
$$958$$ 15.5898 0.503684
$$959$$ 0 0
$$960$$ −34.1109 −1.10093
$$961$$ 1.49465 0.0482146
$$962$$ 2.75919 0.0889598
$$963$$ −0.0898898 −0.00289666
$$964$$ 5.30093 0.170731
$$965$$ 18.6179 0.599332
$$966$$ 0 0
$$967$$ −4.45657 −0.143314 −0.0716568 0.997429i $$-0.522829\pi$$
−0.0716568 + 0.997429i $$0.522829\pi$$
$$968$$ −31.8384 −1.02332
$$969$$ 13.1620 0.422824
$$970$$ 0.262844 0.00843940
$$971$$ −18.9192 −0.607146 −0.303573 0.952808i $$-0.598180\pi$$
−0.303573 + 0.952808i $$0.598180\pi$$
$$972$$ −0.246510 −0.00790680
$$973$$ 0 0
$$974$$ 26.5638 0.851161
$$975$$ −0.196395 −0.00628967
$$976$$ 34.5865 1.10709
$$977$$ 17.4359 0.557823 0.278911 0.960317i $$-0.410026\pi$$
0.278911 + 0.960317i $$0.410026\pi$$
$$978$$ 24.2726 0.776151
$$979$$ −2.67669 −0.0855472
$$980$$ 0 0
$$981$$ −0.770521 −0.0246009
$$982$$ −4.97930 −0.158896
$$983$$ −15.3815 −0.490592 −0.245296 0.969448i $$-0.578885\pi$$
−0.245296 + 0.969448i $$0.578885\pi$$
$$984$$ −64.9700 −2.07117
$$985$$ −18.8143 −0.599473
$$986$$ −17.9349 −0.571163
$$987$$ 0 0
$$988$$ 2.30895 0.0734576
$$989$$ −8.97628 −0.285429
$$990$$ 0.0938357 0.00298229
$$991$$ 2.42651 0.0770807 0.0385403 0.999257i $$-0.487729\pi$$
0.0385403 + 0.999257i $$0.487729\pi$$
$$992$$ −16.7128 −0.530632
$$993$$ −11.9900 −0.380491
$$994$$ 0 0
$$995$$ −32.3137 −1.02441
$$996$$ 2.92848 0.0927923
$$997$$ −15.0869 −0.477806 −0.238903 0.971043i $$-0.576788\pi$$
−0.238903 + 0.971043i $$0.576788\pi$$
$$998$$ −23.0444 −0.729458
$$999$$ 11.7522 0.371824
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 637.2.a.i.1.2 yes 3
3.2 odd 2 5733.2.a.bd.1.2 3
7.2 even 3 637.2.e.k.508.2 6
7.3 odd 6 637.2.e.l.79.2 6
7.4 even 3 637.2.e.k.79.2 6
7.5 odd 6 637.2.e.l.508.2 6
7.6 odd 2 637.2.a.h.1.2 3
13.12 even 2 8281.2.a.bk.1.2 3
21.20 even 2 5733.2.a.be.1.2 3
91.90 odd 2 8281.2.a.bh.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.2 3 7.6 odd 2
637.2.a.i.1.2 yes 3 1.1 even 1 trivial
637.2.e.k.79.2 6 7.4 even 3
637.2.e.k.508.2 6 7.2 even 3
637.2.e.l.79.2 6 7.3 odd 6
637.2.e.l.508.2 6 7.5 odd 6
5733.2.a.bd.1.2 3 3.2 odd 2
5733.2.a.be.1.2 3 21.20 even 2
8281.2.a.bh.1.2 3 91.90 odd 2
8281.2.a.bk.1.2 3 13.12 even 2