# Properties

 Label 7098.2.a.bl.1.2 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.6778153547$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.561553 q^{10} -2.56155 q^{11} +1.00000 q^{12} -1.00000 q^{14} +0.561553 q^{15} +1.00000 q^{16} +5.68466 q^{17} -1.00000 q^{18} -7.68466 q^{19} +0.561553 q^{20} +1.00000 q^{21} +2.56155 q^{22} -1.43845 q^{23} -1.00000 q^{24} -4.68466 q^{25} +1.00000 q^{27} +1.00000 q^{28} -5.68466 q^{29} -0.561553 q^{30} +10.2462 q^{31} -1.00000 q^{32} -2.56155 q^{33} -5.68466 q^{34} +0.561553 q^{35} +1.00000 q^{36} +3.43845 q^{37} +7.68466 q^{38} -0.561553 q^{40} -7.12311 q^{41} -1.00000 q^{42} -10.5616 q^{43} -2.56155 q^{44} +0.561553 q^{45} +1.43845 q^{46} +1.00000 q^{48} +1.00000 q^{49} +4.68466 q^{50} +5.68466 q^{51} -4.24621 q^{53} -1.00000 q^{54} -1.43845 q^{55} -1.00000 q^{56} -7.68466 q^{57} +5.68466 q^{58} +14.2462 q^{59} +0.561553 q^{60} -5.68466 q^{61} -10.2462 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.56155 q^{66} -1.12311 q^{67} +5.68466 q^{68} -1.43845 q^{69} -0.561553 q^{70} -8.00000 q^{71} -1.00000 q^{72} -0.561553 q^{73} -3.43845 q^{74} -4.68466 q^{75} -7.68466 q^{76} -2.56155 q^{77} -2.87689 q^{79} +0.561553 q^{80} +1.00000 q^{81} +7.12311 q^{82} -17.1231 q^{83} +1.00000 q^{84} +3.19224 q^{85} +10.5616 q^{86} -5.68466 q^{87} +2.56155 q^{88} -10.0000 q^{89} -0.561553 q^{90} -1.43845 q^{92} +10.2462 q^{93} -4.31534 q^{95} -1.00000 q^{96} +18.4924 q^{97} -1.00000 q^{98} -2.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - q^{11} + 2 q^{12} - 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 3 q^{19} - 3 q^{20} + 2 q^{21} + q^{22} - 7 q^{23} - 2 q^{24} + 3 q^{25} + 2 q^{27} + 2 q^{28} + q^{29} + 3 q^{30} + 4 q^{31} - 2 q^{32} - q^{33} + q^{34} - 3 q^{35} + 2 q^{36} + 11 q^{37} + 3 q^{38} + 3 q^{40} - 6 q^{41} - 2 q^{42} - 17 q^{43} - q^{44} - 3 q^{45} + 7 q^{46} + 2 q^{48} + 2 q^{49} - 3 q^{50} - q^{51} + 8 q^{53} - 2 q^{54} - 7 q^{55} - 2 q^{56} - 3 q^{57} - q^{58} + 12 q^{59} - 3 q^{60} + q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} + q^{66} + 6 q^{67} - q^{68} - 7 q^{69} + 3 q^{70} - 16 q^{71} - 2 q^{72} + 3 q^{73} - 11 q^{74} + 3 q^{75} - 3 q^{76} - q^{77} - 14 q^{79} - 3 q^{80} + 2 q^{81} + 6 q^{82} - 26 q^{83} + 2 q^{84} + 27 q^{85} + 17 q^{86} + q^{87} + q^{88} - 20 q^{89} + 3 q^{90} - 7 q^{92} + 4 q^{93} - 21 q^{95} - 2 q^{96} + 4 q^{97} - 2 q^{98} - q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0.561553 0.251134 0.125567 0.992085i $$-0.459925\pi$$
0.125567 + 0.992085i $$0.459925\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −0.561553 −0.177579
$$11$$ −2.56155 −0.772337 −0.386169 0.922428i $$-0.626202\pi$$
−0.386169 + 0.922428i $$0.626202\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −1.00000 −0.267261
$$15$$ 0.561553 0.144992
$$16$$ 1.00000 0.250000
$$17$$ 5.68466 1.37873 0.689366 0.724413i $$-0.257889\pi$$
0.689366 + 0.724413i $$0.257889\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −7.68466 −1.76298 −0.881491 0.472201i $$-0.843460\pi$$
−0.881491 + 0.472201i $$0.843460\pi$$
$$20$$ 0.561553 0.125567
$$21$$ 1.00000 0.218218
$$22$$ 2.56155 0.546125
$$23$$ −1.43845 −0.299937 −0.149968 0.988691i $$-0.547917\pi$$
−0.149968 + 0.988691i $$0.547917\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.68466 −0.936932
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 1.00000 0.188982
$$29$$ −5.68466 −1.05561 −0.527807 0.849364i $$-0.676986\pi$$
−0.527807 + 0.849364i $$0.676986\pi$$
$$30$$ −0.561553 −0.102525
$$31$$ 10.2462 1.84027 0.920137 0.391597i $$-0.128077\pi$$
0.920137 + 0.391597i $$0.128077\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.56155 −0.445909
$$34$$ −5.68466 −0.974911
$$35$$ 0.561553 0.0949197
$$36$$ 1.00000 0.166667
$$37$$ 3.43845 0.565277 0.282639 0.959226i $$-0.408790\pi$$
0.282639 + 0.959226i $$0.408790\pi$$
$$38$$ 7.68466 1.24662
$$39$$ 0 0
$$40$$ −0.561553 −0.0887893
$$41$$ −7.12311 −1.11244 −0.556221 0.831034i $$-0.687749\pi$$
−0.556221 + 0.831034i $$0.687749\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −10.5616 −1.61062 −0.805311 0.592853i $$-0.798002\pi$$
−0.805311 + 0.592853i $$0.798002\pi$$
$$44$$ −2.56155 −0.386169
$$45$$ 0.561553 0.0837114
$$46$$ 1.43845 0.212087
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 4.68466 0.662511
$$51$$ 5.68466 0.796011
$$52$$ 0 0
$$53$$ −4.24621 −0.583262 −0.291631 0.956531i $$-0.594198\pi$$
−0.291631 + 0.956531i $$0.594198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −1.43845 −0.193960
$$56$$ −1.00000 −0.133631
$$57$$ −7.68466 −1.01786
$$58$$ 5.68466 0.746432
$$59$$ 14.2462 1.85470 0.927349 0.374197i $$-0.122082\pi$$
0.927349 + 0.374197i $$0.122082\pi$$
$$60$$ 0.561553 0.0724962
$$61$$ −5.68466 −0.727846 −0.363923 0.931429i $$-0.618563\pi$$
−0.363923 + 0.931429i $$0.618563\pi$$
$$62$$ −10.2462 −1.30127
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.56155 0.315305
$$67$$ −1.12311 −0.137209 −0.0686046 0.997644i $$-0.521855\pi$$
−0.0686046 + 0.997644i $$0.521855\pi$$
$$68$$ 5.68466 0.689366
$$69$$ −1.43845 −0.173169
$$70$$ −0.561553 −0.0671184
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −0.561553 −0.0657248 −0.0328624 0.999460i $$-0.510462\pi$$
−0.0328624 + 0.999460i $$0.510462\pi$$
$$74$$ −3.43845 −0.399711
$$75$$ −4.68466 −0.540938
$$76$$ −7.68466 −0.881491
$$77$$ −2.56155 −0.291916
$$78$$ 0 0
$$79$$ −2.87689 −0.323676 −0.161838 0.986817i $$-0.551742\pi$$
−0.161838 + 0.986817i $$0.551742\pi$$
$$80$$ 0.561553 0.0627835
$$81$$ 1.00000 0.111111
$$82$$ 7.12311 0.786615
$$83$$ −17.1231 −1.87951 −0.939753 0.341856i $$-0.888945\pi$$
−0.939753 + 0.341856i $$0.888945\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 3.19224 0.346247
$$86$$ 10.5616 1.13888
$$87$$ −5.68466 −0.609459
$$88$$ 2.56155 0.273062
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ −0.561553 −0.0591929
$$91$$ 0 0
$$92$$ −1.43845 −0.149968
$$93$$ 10.2462 1.06248
$$94$$ 0 0
$$95$$ −4.31534 −0.442745
$$96$$ −1.00000 −0.102062
$$97$$ 18.4924 1.87762 0.938811 0.344434i $$-0.111929\pi$$
0.938811 + 0.344434i $$0.111929\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ −2.56155 −0.257446
$$100$$ −4.68466 −0.468466
$$101$$ 3.12311 0.310761 0.155380 0.987855i $$-0.450340\pi$$
0.155380 + 0.987855i $$0.450340\pi$$
$$102$$ −5.68466 −0.562865
$$103$$ 11.6847 1.15132 0.575662 0.817688i $$-0.304744\pi$$
0.575662 + 0.817688i $$0.304744\pi$$
$$104$$ 0 0
$$105$$ 0.561553 0.0548019
$$106$$ 4.24621 0.412428
$$107$$ −17.1231 −1.65535 −0.827677 0.561205i $$-0.810338\pi$$
−0.827677 + 0.561205i $$0.810338\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −11.9309 −1.14277 −0.571385 0.820682i $$-0.693594\pi$$
−0.571385 + 0.820682i $$0.693594\pi$$
$$110$$ 1.43845 0.137151
$$111$$ 3.43845 0.326363
$$112$$ 1.00000 0.0944911
$$113$$ 12.2462 1.15203 0.576013 0.817440i $$-0.304608\pi$$
0.576013 + 0.817440i $$0.304608\pi$$
$$114$$ 7.68466 0.719734
$$115$$ −0.807764 −0.0753244
$$116$$ −5.68466 −0.527807
$$117$$ 0 0
$$118$$ −14.2462 −1.31147
$$119$$ 5.68466 0.521112
$$120$$ −0.561553 −0.0512625
$$121$$ −4.43845 −0.403495
$$122$$ 5.68466 0.514665
$$123$$ −7.12311 −0.642269
$$124$$ 10.2462 0.920137
$$125$$ −5.43845 −0.486430
$$126$$ −1.00000 −0.0890871
$$127$$ −13.1231 −1.16449 −0.582244 0.813014i $$-0.697825\pi$$
−0.582244 + 0.813014i $$0.697825\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −10.5616 −0.929893
$$130$$ 0 0
$$131$$ −5.43845 −0.475159 −0.237580 0.971368i $$-0.576354\pi$$
−0.237580 + 0.971368i $$0.576354\pi$$
$$132$$ −2.56155 −0.222955
$$133$$ −7.68466 −0.666344
$$134$$ 1.12311 0.0970215
$$135$$ 0.561553 0.0483308
$$136$$ −5.68466 −0.487455
$$137$$ −0.561553 −0.0479767 −0.0239883 0.999712i $$-0.507636\pi$$
−0.0239883 + 0.999712i $$0.507636\pi$$
$$138$$ 1.43845 0.122449
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0.561553 0.0474599
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −3.19224 −0.265101
$$146$$ 0.561553 0.0464744
$$147$$ 1.00000 0.0824786
$$148$$ 3.43845 0.282639
$$149$$ −23.6155 −1.93466 −0.967330 0.253522i $$-0.918411\pi$$
−0.967330 + 0.253522i $$0.918411\pi$$
$$150$$ 4.68466 0.382501
$$151$$ 8.80776 0.716766 0.358383 0.933575i $$-0.383328\pi$$
0.358383 + 0.933575i $$0.383328\pi$$
$$152$$ 7.68466 0.623308
$$153$$ 5.68466 0.459577
$$154$$ 2.56155 0.206416
$$155$$ 5.75379 0.462155
$$156$$ 0 0
$$157$$ −11.4384 −0.912887 −0.456444 0.889752i $$-0.650877\pi$$
−0.456444 + 0.889752i $$0.650877\pi$$
$$158$$ 2.87689 0.228873
$$159$$ −4.24621 −0.336746
$$160$$ −0.561553 −0.0443946
$$161$$ −1.43845 −0.113366
$$162$$ −1.00000 −0.0785674
$$163$$ 25.1231 1.96779 0.983897 0.178738i $$-0.0572014\pi$$
0.983897 + 0.178738i $$0.0572014\pi$$
$$164$$ −7.12311 −0.556221
$$165$$ −1.43845 −0.111983
$$166$$ 17.1231 1.32901
$$167$$ 5.93087 0.458944 0.229472 0.973315i $$-0.426300\pi$$
0.229472 + 0.973315i $$0.426300\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 0 0
$$170$$ −3.19224 −0.244833
$$171$$ −7.68466 −0.587661
$$172$$ −10.5616 −0.805311
$$173$$ 3.75379 0.285395 0.142698 0.989766i $$-0.454422\pi$$
0.142698 + 0.989766i $$0.454422\pi$$
$$174$$ 5.68466 0.430953
$$175$$ −4.68466 −0.354127
$$176$$ −2.56155 −0.193084
$$177$$ 14.2462 1.07081
$$178$$ 10.0000 0.749532
$$179$$ −16.4924 −1.23270 −0.616351 0.787472i $$-0.711390\pi$$
−0.616351 + 0.787472i $$0.711390\pi$$
$$180$$ 0.561553 0.0418557
$$181$$ 16.2462 1.20757 0.603786 0.797147i $$-0.293658\pi$$
0.603786 + 0.797147i $$0.293658\pi$$
$$182$$ 0 0
$$183$$ −5.68466 −0.420222
$$184$$ 1.43845 0.106044
$$185$$ 1.93087 0.141960
$$186$$ −10.2462 −0.751289
$$187$$ −14.5616 −1.06485
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 4.31534 0.313068
$$191$$ −13.9309 −1.00800 −0.504001 0.863703i $$-0.668139\pi$$
−0.504001 + 0.863703i $$0.668139\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 5.36932 0.386492 0.193246 0.981150i $$-0.438098\pi$$
0.193246 + 0.981150i $$0.438098\pi$$
$$194$$ −18.4924 −1.32768
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −19.1231 −1.36246 −0.681232 0.732067i $$-0.738556\pi$$
−0.681232 + 0.732067i $$0.738556\pi$$
$$198$$ 2.56155 0.182042
$$199$$ −21.9309 −1.55464 −0.777319 0.629107i $$-0.783421\pi$$
−0.777319 + 0.629107i $$0.783421\pi$$
$$200$$ 4.68466 0.331255
$$201$$ −1.12311 −0.0792178
$$202$$ −3.12311 −0.219741
$$203$$ −5.68466 −0.398985
$$204$$ 5.68466 0.398006
$$205$$ −4.00000 −0.279372
$$206$$ −11.6847 −0.814109
$$207$$ −1.43845 −0.0999790
$$208$$ 0 0
$$209$$ 19.6847 1.36162
$$210$$ −0.561553 −0.0387508
$$211$$ 4.80776 0.330980 0.165490 0.986211i $$-0.447079\pi$$
0.165490 + 0.986211i $$0.447079\pi$$
$$212$$ −4.24621 −0.291631
$$213$$ −8.00000 −0.548151
$$214$$ 17.1231 1.17051
$$215$$ −5.93087 −0.404482
$$216$$ −1.00000 −0.0680414
$$217$$ 10.2462 0.695558
$$218$$ 11.9309 0.808060
$$219$$ −0.561553 −0.0379462
$$220$$ −1.43845 −0.0969801
$$221$$ 0 0
$$222$$ −3.43845 −0.230773
$$223$$ −17.6155 −1.17962 −0.589812 0.807541i $$-0.700798\pi$$
−0.589812 + 0.807541i $$0.700798\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −4.68466 −0.312311
$$226$$ −12.2462 −0.814606
$$227$$ −1.12311 −0.0745431 −0.0372716 0.999305i $$-0.511867\pi$$
−0.0372716 + 0.999305i $$0.511867\pi$$
$$228$$ −7.68466 −0.508929
$$229$$ −11.7538 −0.776712 −0.388356 0.921509i $$-0.626957\pi$$
−0.388356 + 0.921509i $$0.626957\pi$$
$$230$$ 0.807764 0.0532624
$$231$$ −2.56155 −0.168538
$$232$$ 5.68466 0.373216
$$233$$ 2.63068 0.172342 0.0861709 0.996280i $$-0.472537\pi$$
0.0861709 + 0.996280i $$0.472537\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 14.2462 0.927349
$$237$$ −2.87689 −0.186874
$$238$$ −5.68466 −0.368482
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0.561553 0.0362481
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 4.43845 0.285314
$$243$$ 1.00000 0.0641500
$$244$$ −5.68466 −0.363923
$$245$$ 0.561553 0.0358763
$$246$$ 7.12311 0.454153
$$247$$ 0 0
$$248$$ −10.2462 −0.650635
$$249$$ −17.1231 −1.08513
$$250$$ 5.43845 0.343958
$$251$$ 12.8078 0.808419 0.404209 0.914666i $$-0.367547\pi$$
0.404209 + 0.914666i $$0.367547\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 3.68466 0.231652
$$254$$ 13.1231 0.823417
$$255$$ 3.19224 0.199906
$$256$$ 1.00000 0.0625000
$$257$$ 12.2462 0.763898 0.381949 0.924183i $$-0.375253\pi$$
0.381949 + 0.924183i $$0.375253\pi$$
$$258$$ 10.5616 0.657534
$$259$$ 3.43845 0.213655
$$260$$ 0 0
$$261$$ −5.68466 −0.351872
$$262$$ 5.43845 0.335988
$$263$$ −13.7538 −0.848095 −0.424047 0.905640i $$-0.639391\pi$$
−0.424047 + 0.905640i $$0.639391\pi$$
$$264$$ 2.56155 0.157653
$$265$$ −2.38447 −0.146477
$$266$$ 7.68466 0.471177
$$267$$ −10.0000 −0.611990
$$268$$ −1.12311 −0.0686046
$$269$$ 3.75379 0.228873 0.114436 0.993431i $$-0.463494\pi$$
0.114436 + 0.993431i $$0.463494\pi$$
$$270$$ −0.561553 −0.0341750
$$271$$ −13.1231 −0.797172 −0.398586 0.917131i $$-0.630499\pi$$
−0.398586 + 0.917131i $$0.630499\pi$$
$$272$$ 5.68466 0.344683
$$273$$ 0 0
$$274$$ 0.561553 0.0339246
$$275$$ 12.0000 0.723627
$$276$$ −1.43845 −0.0865843
$$277$$ 10.4924 0.630429 0.315214 0.949021i $$-0.397924\pi$$
0.315214 + 0.949021i $$0.397924\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ 10.2462 0.613425
$$280$$ −0.561553 −0.0335592
$$281$$ 0.246211 0.0146877 0.00734387 0.999973i $$-0.497662\pi$$
0.00734387 + 0.999973i $$0.497662\pi$$
$$282$$ 0 0
$$283$$ 1.75379 0.104252 0.0521260 0.998641i $$-0.483400\pi$$
0.0521260 + 0.998641i $$0.483400\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ −4.31534 −0.255619
$$286$$ 0 0
$$287$$ −7.12311 −0.420464
$$288$$ −1.00000 −0.0589256
$$289$$ 15.3153 0.900902
$$290$$ 3.19224 0.187455
$$291$$ 18.4924 1.08405
$$292$$ −0.561553 −0.0328624
$$293$$ 24.7386 1.44525 0.722623 0.691242i $$-0.242936\pi$$
0.722623 + 0.691242i $$0.242936\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 8.00000 0.465778
$$296$$ −3.43845 −0.199856
$$297$$ −2.56155 −0.148636
$$298$$ 23.6155 1.36801
$$299$$ 0 0
$$300$$ −4.68466 −0.270469
$$301$$ −10.5616 −0.608758
$$302$$ −8.80776 −0.506830
$$303$$ 3.12311 0.179418
$$304$$ −7.68466 −0.440745
$$305$$ −3.19224 −0.182787
$$306$$ −5.68466 −0.324970
$$307$$ −9.75379 −0.556678 −0.278339 0.960483i $$-0.589784\pi$$
−0.278339 + 0.960483i $$0.589784\pi$$
$$308$$ −2.56155 −0.145958
$$309$$ 11.6847 0.664717
$$310$$ −5.75379 −0.326793
$$311$$ −21.1231 −1.19778 −0.598891 0.800831i $$-0.704392\pi$$
−0.598891 + 0.800831i $$0.704392\pi$$
$$312$$ 0 0
$$313$$ 27.6155 1.56092 0.780461 0.625205i $$-0.214984\pi$$
0.780461 + 0.625205i $$0.214984\pi$$
$$314$$ 11.4384 0.645509
$$315$$ 0.561553 0.0316399
$$316$$ −2.87689 −0.161838
$$317$$ −11.1231 −0.624736 −0.312368 0.949961i $$-0.601122\pi$$
−0.312368 + 0.949961i $$0.601122\pi$$
$$318$$ 4.24621 0.238116
$$319$$ 14.5616 0.815290
$$320$$ 0.561553 0.0313918
$$321$$ −17.1231 −0.955719
$$322$$ 1.43845 0.0801615
$$323$$ −43.6847 −2.43068
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −25.1231 −1.39144
$$327$$ −11.9309 −0.659779
$$328$$ 7.12311 0.393308
$$329$$ 0 0
$$330$$ 1.43845 0.0791839
$$331$$ 11.3693 0.624914 0.312457 0.949932i $$-0.398848\pi$$
0.312457 + 0.949932i $$0.398848\pi$$
$$332$$ −17.1231 −0.939753
$$333$$ 3.43845 0.188426
$$334$$ −5.93087 −0.324523
$$335$$ −0.630683 −0.0344579
$$336$$ 1.00000 0.0545545
$$337$$ 8.56155 0.466377 0.233189 0.972431i $$-0.425084\pi$$
0.233189 + 0.972431i $$0.425084\pi$$
$$338$$ 0 0
$$339$$ 12.2462 0.665123
$$340$$ 3.19224 0.173123
$$341$$ −26.2462 −1.42131
$$342$$ 7.68466 0.415539
$$343$$ 1.00000 0.0539949
$$344$$ 10.5616 0.569441
$$345$$ −0.807764 −0.0434886
$$346$$ −3.75379 −0.201805
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ −5.68466 −0.304730
$$349$$ −24.2462 −1.29787 −0.648935 0.760844i $$-0.724785\pi$$
−0.648935 + 0.760844i $$0.724785\pi$$
$$350$$ 4.68466 0.250406
$$351$$ 0 0
$$352$$ 2.56155 0.136531
$$353$$ 2.49242 0.132658 0.0663291 0.997798i $$-0.478871\pi$$
0.0663291 + 0.997798i $$0.478871\pi$$
$$354$$ −14.2462 −0.757178
$$355$$ −4.49242 −0.238433
$$356$$ −10.0000 −0.529999
$$357$$ 5.68466 0.300864
$$358$$ 16.4924 0.871652
$$359$$ −22.7386 −1.20010 −0.600050 0.799963i $$-0.704852\pi$$
−0.600050 + 0.799963i $$0.704852\pi$$
$$360$$ −0.561553 −0.0295964
$$361$$ 40.0540 2.10810
$$362$$ −16.2462 −0.853882
$$363$$ −4.43845 −0.232958
$$364$$ 0 0
$$365$$ −0.315342 −0.0165057
$$366$$ 5.68466 0.297142
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ −1.43845 −0.0749842
$$369$$ −7.12311 −0.370814
$$370$$ −1.93087 −0.100381
$$371$$ −4.24621 −0.220452
$$372$$ 10.2462 0.531241
$$373$$ 23.6155 1.22277 0.611383 0.791335i $$-0.290614\pi$$
0.611383 + 0.791335i $$0.290614\pi$$
$$374$$ 14.5616 0.752960
$$375$$ −5.43845 −0.280840
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −1.00000 −0.0514344
$$379$$ −17.7538 −0.911951 −0.455975 0.889992i $$-0.650710\pi$$
−0.455975 + 0.889992i $$0.650710\pi$$
$$380$$ −4.31534 −0.221372
$$381$$ −13.1231 −0.672317
$$382$$ 13.9309 0.712765
$$383$$ 34.4233 1.75895 0.879474 0.475947i $$-0.157895\pi$$
0.879474 + 0.475947i $$0.157895\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −1.43845 −0.0733101
$$386$$ −5.36932 −0.273291
$$387$$ −10.5616 −0.536874
$$388$$ 18.4924 0.938811
$$389$$ 20.7386 1.05149 0.525745 0.850642i $$-0.323787\pi$$
0.525745 + 0.850642i $$0.323787\pi$$
$$390$$ 0 0
$$391$$ −8.17708 −0.413533
$$392$$ −1.00000 −0.0505076
$$393$$ −5.43845 −0.274333
$$394$$ 19.1231 0.963408
$$395$$ −1.61553 −0.0812860
$$396$$ −2.56155 −0.128723
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 21.9309 1.09930
$$399$$ −7.68466 −0.384714
$$400$$ −4.68466 −0.234233
$$401$$ 8.24621 0.411796 0.205898 0.978573i $$-0.433988\pi$$
0.205898 + 0.978573i $$0.433988\pi$$
$$402$$ 1.12311 0.0560154
$$403$$ 0 0
$$404$$ 3.12311 0.155380
$$405$$ 0.561553 0.0279038
$$406$$ 5.68466 0.282125
$$407$$ −8.80776 −0.436585
$$408$$ −5.68466 −0.281433
$$409$$ −23.9309 −1.18331 −0.591653 0.806193i $$-0.701524\pi$$
−0.591653 + 0.806193i $$0.701524\pi$$
$$410$$ 4.00000 0.197546
$$411$$ −0.561553 −0.0276994
$$412$$ 11.6847 0.575662
$$413$$ 14.2462 0.701010
$$414$$ 1.43845 0.0706958
$$415$$ −9.61553 −0.472008
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ −19.6847 −0.962808
$$419$$ −24.3153 −1.18788 −0.593941 0.804509i $$-0.702429\pi$$
−0.593941 + 0.804509i $$0.702429\pi$$
$$420$$ 0.561553 0.0274010
$$421$$ 20.2462 0.986740 0.493370 0.869820i $$-0.335765\pi$$
0.493370 + 0.869820i $$0.335765\pi$$
$$422$$ −4.80776 −0.234038
$$423$$ 0 0
$$424$$ 4.24621 0.206214
$$425$$ −26.6307 −1.29178
$$426$$ 8.00000 0.387601
$$427$$ −5.68466 −0.275100
$$428$$ −17.1231 −0.827677
$$429$$ 0 0
$$430$$ 5.93087 0.286012
$$431$$ 8.63068 0.415725 0.207863 0.978158i $$-0.433349\pi$$
0.207863 + 0.978158i $$0.433349\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −6.63068 −0.318650 −0.159325 0.987226i $$-0.550932\pi$$
−0.159325 + 0.987226i $$0.550932\pi$$
$$434$$ −10.2462 −0.491834
$$435$$ −3.19224 −0.153056
$$436$$ −11.9309 −0.571385
$$437$$ 11.0540 0.528783
$$438$$ 0.561553 0.0268320
$$439$$ 21.9309 1.04670 0.523352 0.852117i $$-0.324681\pi$$
0.523352 + 0.852117i $$0.324681\pi$$
$$440$$ 1.43845 0.0685753
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 19.3693 0.920264 0.460132 0.887851i $$-0.347802\pi$$
0.460132 + 0.887851i $$0.347802\pi$$
$$444$$ 3.43845 0.163181
$$445$$ −5.61553 −0.266202
$$446$$ 17.6155 0.834119
$$447$$ −23.6155 −1.11698
$$448$$ 1.00000 0.0472456
$$449$$ 1.68466 0.0795039 0.0397520 0.999210i $$-0.487343\pi$$
0.0397520 + 0.999210i $$0.487343\pi$$
$$450$$ 4.68466 0.220837
$$451$$ 18.2462 0.859181
$$452$$ 12.2462 0.576013
$$453$$ 8.80776 0.413825
$$454$$ 1.12311 0.0527100
$$455$$ 0 0
$$456$$ 7.68466 0.359867
$$457$$ −2.63068 −0.123058 −0.0615291 0.998105i $$-0.519598\pi$$
−0.0615291 + 0.998105i $$0.519598\pi$$
$$458$$ 11.7538 0.549218
$$459$$ 5.68466 0.265337
$$460$$ −0.807764 −0.0376622
$$461$$ −9.05398 −0.421686 −0.210843 0.977520i $$-0.567621\pi$$
−0.210843 + 0.977520i $$0.567621\pi$$
$$462$$ 2.56155 0.119174
$$463$$ −3.68466 −0.171241 −0.0856203 0.996328i $$-0.527287\pi$$
−0.0856203 + 0.996328i $$0.527287\pi$$
$$464$$ −5.68466 −0.263904
$$465$$ 5.75379 0.266826
$$466$$ −2.63068 −0.121864
$$467$$ −15.6847 −0.725799 −0.362900 0.931828i $$-0.618213\pi$$
−0.362900 + 0.931828i $$0.618213\pi$$
$$468$$ 0 0
$$469$$ −1.12311 −0.0518602
$$470$$ 0 0
$$471$$ −11.4384 −0.527056
$$472$$ −14.2462 −0.655735
$$473$$ 27.0540 1.24394
$$474$$ 2.87689 0.132140
$$475$$ 36.0000 1.65179
$$476$$ 5.68466 0.260556
$$477$$ −4.24621 −0.194421
$$478$$ 16.0000 0.731823
$$479$$ 21.3002 0.973230 0.486615 0.873616i $$-0.338231\pi$$
0.486615 + 0.873616i $$0.338231\pi$$
$$480$$ −0.561553 −0.0256313
$$481$$ 0 0
$$482$$ −14.0000 −0.637683
$$483$$ −1.43845 −0.0654516
$$484$$ −4.43845 −0.201748
$$485$$ 10.3845 0.471535
$$486$$ −1.00000 −0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 5.68466 0.257332
$$489$$ 25.1231 1.13611
$$490$$ −0.561553 −0.0253684
$$491$$ −17.1231 −0.772755 −0.386377 0.922341i $$-0.626274\pi$$
−0.386377 + 0.922341i $$0.626274\pi$$
$$492$$ −7.12311 −0.321134
$$493$$ −32.3153 −1.45541
$$494$$ 0 0
$$495$$ −1.43845 −0.0646534
$$496$$ 10.2462 0.460068
$$497$$ −8.00000 −0.358849
$$498$$ 17.1231 0.767305
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ −5.43845 −0.243215
$$501$$ 5.93087 0.264972
$$502$$ −12.8078 −0.571638
$$503$$ 5.12311 0.228428 0.114214 0.993456i $$-0.463565\pi$$
0.114214 + 0.993456i $$0.463565\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 1.75379 0.0780426
$$506$$ −3.68466 −0.163803
$$507$$ 0 0
$$508$$ −13.1231 −0.582244
$$509$$ −25.0540 −1.11050 −0.555249 0.831684i $$-0.687377\pi$$
−0.555249 + 0.831684i $$0.687377\pi$$
$$510$$ −3.19224 −0.141355
$$511$$ −0.561553 −0.0248416
$$512$$ −1.00000 −0.0441942
$$513$$ −7.68466 −0.339286
$$514$$ −12.2462 −0.540157
$$515$$ 6.56155 0.289137
$$516$$ −10.5616 −0.464946
$$517$$ 0 0
$$518$$ −3.43845 −0.151077
$$519$$ 3.75379 0.164773
$$520$$ 0 0
$$521$$ −9.68466 −0.424293 −0.212146 0.977238i $$-0.568045\pi$$
−0.212146 + 0.977238i $$0.568045\pi$$
$$522$$ 5.68466 0.248811
$$523$$ 38.2462 1.67239 0.836195 0.548432i $$-0.184775\pi$$
0.836195 + 0.548432i $$0.184775\pi$$
$$524$$ −5.43845 −0.237580
$$525$$ −4.68466 −0.204455
$$526$$ 13.7538 0.599694
$$527$$ 58.2462 2.53724
$$528$$ −2.56155 −0.111477
$$529$$ −20.9309 −0.910038
$$530$$ 2.38447 0.103575
$$531$$ 14.2462 0.618233
$$532$$ −7.68466 −0.333172
$$533$$ 0 0
$$534$$ 10.0000 0.432742
$$535$$ −9.61553 −0.415716
$$536$$ 1.12311 0.0485108
$$537$$ −16.4924 −0.711701
$$538$$ −3.75379 −0.161837
$$539$$ −2.56155 −0.110334
$$540$$ 0.561553 0.0241654
$$541$$ 4.06913 0.174946 0.0874728 0.996167i $$-0.472121\pi$$
0.0874728 + 0.996167i $$0.472121\pi$$
$$542$$ 13.1231 0.563686
$$543$$ 16.2462 0.697192
$$544$$ −5.68466 −0.243728
$$545$$ −6.69981 −0.286988
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −0.561553 −0.0239883
$$549$$ −5.68466 −0.242615
$$550$$ −12.0000 −0.511682
$$551$$ 43.6847 1.86103
$$552$$ 1.43845 0.0612244
$$553$$ −2.87689 −0.122338
$$554$$ −10.4924 −0.445780
$$555$$ 1.93087 0.0819609
$$556$$ 12.0000 0.508913
$$557$$ −21.3693 −0.905447 −0.452724 0.891651i $$-0.649548\pi$$
−0.452724 + 0.891651i $$0.649548\pi$$
$$558$$ −10.2462 −0.433757
$$559$$ 0 0
$$560$$ 0.561553 0.0237299
$$561$$ −14.5616 −0.614789
$$562$$ −0.246211 −0.0103858
$$563$$ 31.0540 1.30877 0.654385 0.756162i $$-0.272928\pi$$
0.654385 + 0.756162i $$0.272928\pi$$
$$564$$ 0 0
$$565$$ 6.87689 0.289313
$$566$$ −1.75379 −0.0737172
$$567$$ 1.00000 0.0419961
$$568$$ 8.00000 0.335673
$$569$$ −41.2311 −1.72850 −0.864248 0.503066i $$-0.832205\pi$$
−0.864248 + 0.503066i $$0.832205\pi$$
$$570$$ 4.31534 0.180750
$$571$$ 1.75379 0.0733938 0.0366969 0.999326i $$-0.488316\pi$$
0.0366969 + 0.999326i $$0.488316\pi$$
$$572$$ 0 0
$$573$$ −13.9309 −0.581970
$$574$$ 7.12311 0.297313
$$575$$ 6.73863 0.281020
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$578$$ −15.3153 −0.637034
$$579$$ 5.36932 0.223141
$$580$$ −3.19224 −0.132550
$$581$$ −17.1231 −0.710386
$$582$$ −18.4924 −0.766536
$$583$$ 10.8769 0.450475
$$584$$ 0.561553 0.0232372
$$585$$ 0 0
$$586$$ −24.7386 −1.02194
$$587$$ 11.3693 0.469262 0.234631 0.972085i $$-0.424612\pi$$
0.234631 + 0.972085i $$0.424612\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −78.7386 −3.24437
$$590$$ −8.00000 −0.329355
$$591$$ −19.1231 −0.786619
$$592$$ 3.43845 0.141319
$$593$$ 3.75379 0.154150 0.0770748 0.997025i $$-0.475442\pi$$
0.0770748 + 0.997025i $$0.475442\pi$$
$$594$$ 2.56155 0.105102
$$595$$ 3.19224 0.130869
$$596$$ −23.6155 −0.967330
$$597$$ −21.9309 −0.897571
$$598$$ 0 0
$$599$$ 17.4384 0.712516 0.356258 0.934388i $$-0.384052\pi$$
0.356258 + 0.934388i $$0.384052\pi$$
$$600$$ 4.68466 0.191250
$$601$$ −19.1231 −0.780048 −0.390024 0.920805i $$-0.627533\pi$$
−0.390024 + 0.920805i $$0.627533\pi$$
$$602$$ 10.5616 0.430457
$$603$$ −1.12311 −0.0457364
$$604$$ 8.80776 0.358383
$$605$$ −2.49242 −0.101331
$$606$$ −3.12311 −0.126867
$$607$$ −22.5616 −0.915745 −0.457873 0.889018i $$-0.651388\pi$$
−0.457873 + 0.889018i $$0.651388\pi$$
$$608$$ 7.68466 0.311654
$$609$$ −5.68466 −0.230354
$$610$$ 3.19224 0.129250
$$611$$ 0 0
$$612$$ 5.68466 0.229789
$$613$$ 25.5464 1.03181 0.515905 0.856646i $$-0.327456\pi$$
0.515905 + 0.856646i $$0.327456\pi$$
$$614$$ 9.75379 0.393631
$$615$$ −4.00000 −0.161296
$$616$$ 2.56155 0.103208
$$617$$ −28.4233 −1.14428 −0.572139 0.820156i $$-0.693886\pi$$
−0.572139 + 0.820156i $$0.693886\pi$$
$$618$$ −11.6847 −0.470026
$$619$$ −31.6847 −1.27351 −0.636757 0.771065i $$-0.719725\pi$$
−0.636757 + 0.771065i $$0.719725\pi$$
$$620$$ 5.75379 0.231078
$$621$$ −1.43845 −0.0577229
$$622$$ 21.1231 0.846959
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ 20.3693 0.814773
$$626$$ −27.6155 −1.10374
$$627$$ 19.6847 0.786130
$$628$$ −11.4384 −0.456444
$$629$$ 19.5464 0.779366
$$630$$ −0.561553 −0.0223728
$$631$$ −5.93087 −0.236104 −0.118052 0.993007i $$-0.537665\pi$$
−0.118052 + 0.993007i $$0.537665\pi$$
$$632$$ 2.87689 0.114437
$$633$$ 4.80776 0.191091
$$634$$ 11.1231 0.441755
$$635$$ −7.36932 −0.292442
$$636$$ −4.24621 −0.168373
$$637$$ 0 0
$$638$$ −14.5616 −0.576497
$$639$$ −8.00000 −0.316475
$$640$$ −0.561553 −0.0221973
$$641$$ −3.75379 −0.148266 −0.0741329 0.997248i $$-0.523619\pi$$
−0.0741329 + 0.997248i $$0.523619\pi$$
$$642$$ 17.1231 0.675795
$$643$$ −36.8078 −1.45156 −0.725778 0.687929i $$-0.758520\pi$$
−0.725778 + 0.687929i $$0.758520\pi$$
$$644$$ −1.43845 −0.0566828
$$645$$ −5.93087 −0.233528
$$646$$ 43.6847 1.71875
$$647$$ 2.24621 0.0883077 0.0441538 0.999025i $$-0.485941\pi$$
0.0441538 + 0.999025i $$0.485941\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −36.4924 −1.43245
$$650$$ 0 0
$$651$$ 10.2462 0.401581
$$652$$ 25.1231 0.983897
$$653$$ 11.9309 0.466891 0.233446 0.972370i $$-0.425000\pi$$
0.233446 + 0.972370i $$0.425000\pi$$
$$654$$ 11.9309 0.466534
$$655$$ −3.05398 −0.119329
$$656$$ −7.12311 −0.278111
$$657$$ −0.561553 −0.0219083
$$658$$ 0 0
$$659$$ −3.36932 −0.131250 −0.0656250 0.997844i $$-0.520904\pi$$
−0.0656250 + 0.997844i $$0.520904\pi$$
$$660$$ −1.43845 −0.0559915
$$661$$ −16.2462 −0.631904 −0.315952 0.948775i $$-0.602324\pi$$
−0.315952 + 0.948775i $$0.602324\pi$$
$$662$$ −11.3693 −0.441881
$$663$$ 0 0
$$664$$ 17.1231 0.664505
$$665$$ −4.31534 −0.167342
$$666$$ −3.43845 −0.133237
$$667$$ 8.17708 0.316618
$$668$$ 5.93087 0.229472
$$669$$ −17.6155 −0.681056
$$670$$ 0.630683 0.0243654
$$671$$ 14.5616 0.562143
$$672$$ −1.00000 −0.0385758
$$673$$ −13.1922 −0.508523 −0.254262 0.967135i $$-0.581832\pi$$
−0.254262 + 0.967135i $$0.581832\pi$$
$$674$$ −8.56155 −0.329779
$$675$$ −4.68466 −0.180313
$$676$$ 0 0
$$677$$ −30.4924 −1.17192 −0.585959 0.810340i $$-0.699282\pi$$
−0.585959 + 0.810340i $$0.699282\pi$$
$$678$$ −12.2462 −0.470313
$$679$$ 18.4924 0.709674
$$680$$ −3.19224 −0.122417
$$681$$ −1.12311 −0.0430375
$$682$$ 26.2462 1.00502
$$683$$ −47.6847 −1.82460 −0.912301 0.409519i $$-0.865696\pi$$
−0.912301 + 0.409519i $$0.865696\pi$$
$$684$$ −7.68466 −0.293830
$$685$$ −0.315342 −0.0120486
$$686$$ −1.00000 −0.0381802
$$687$$ −11.7538 −0.448435
$$688$$ −10.5616 −0.402655
$$689$$ 0 0
$$690$$ 0.807764 0.0307511
$$691$$ 16.4924 0.627401 0.313701 0.949522i $$-0.398431\pi$$
0.313701 + 0.949522i $$0.398431\pi$$
$$692$$ 3.75379 0.142698
$$693$$ −2.56155 −0.0973053
$$694$$ 20.0000 0.759190
$$695$$ 6.73863 0.255611
$$696$$ 5.68466 0.215476
$$697$$ −40.4924 −1.53376
$$698$$ 24.2462 0.917733
$$699$$ 2.63068 0.0995016
$$700$$ −4.68466 −0.177063
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ −26.4233 −0.996573
$$704$$ −2.56155 −0.0965422
$$705$$ 0 0
$$706$$ −2.49242 −0.0938036
$$707$$ 3.12311 0.117456
$$708$$ 14.2462 0.535405
$$709$$ −0.246211 −0.00924666 −0.00462333 0.999989i $$-0.501472\pi$$
−0.00462333 + 0.999989i $$0.501472\pi$$
$$710$$ 4.49242 0.168598
$$711$$ −2.87689 −0.107892
$$712$$ 10.0000 0.374766
$$713$$ −14.7386 −0.551966
$$714$$ −5.68466 −0.212743
$$715$$ 0 0
$$716$$ −16.4924 −0.616351
$$717$$ −16.0000 −0.597531
$$718$$ 22.7386 0.848598
$$719$$ 34.8769 1.30069 0.650344 0.759640i $$-0.274625\pi$$
0.650344 + 0.759640i $$0.274625\pi$$
$$720$$ 0.561553 0.0209278
$$721$$ 11.6847 0.435159
$$722$$ −40.0540 −1.49065
$$723$$ 14.0000 0.520666
$$724$$ 16.2462 0.603786
$$725$$ 26.6307 0.989039
$$726$$ 4.43845 0.164726
$$727$$ 21.9309 0.813371 0.406685 0.913568i $$-0.366684\pi$$
0.406685 + 0.913568i $$0.366684\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0.315342 0.0116713
$$731$$ −60.0388 −2.22062
$$732$$ −5.68466 −0.210111
$$733$$ 0.384472 0.0142008 0.00710040 0.999975i $$-0.497740\pi$$
0.00710040 + 0.999975i $$0.497740\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0.561553 0.0207132
$$736$$ 1.43845 0.0530219
$$737$$ 2.87689 0.105972
$$738$$ 7.12311 0.262205
$$739$$ −42.1080 −1.54897 −0.774483 0.632595i $$-0.781990\pi$$
−0.774483 + 0.632595i $$0.781990\pi$$
$$740$$ 1.93087 0.0709802
$$741$$ 0 0
$$742$$ 4.24621 0.155883
$$743$$ 46.1080 1.69154 0.845768 0.533550i $$-0.179143\pi$$
0.845768 + 0.533550i $$0.179143\pi$$
$$744$$ −10.2462 −0.375644
$$745$$ −13.2614 −0.485859
$$746$$ −23.6155 −0.864626
$$747$$ −17.1231 −0.626502
$$748$$ −14.5616 −0.532423
$$749$$ −17.1231 −0.625665
$$750$$ 5.43845 0.198584
$$751$$ 10.2462 0.373890 0.186945 0.982370i $$-0.440141\pi$$
0.186945 + 0.982370i $$0.440141\pi$$
$$752$$ 0 0
$$753$$ 12.8078 0.466741
$$754$$ 0 0
$$755$$ 4.94602 0.180004
$$756$$ 1.00000 0.0363696
$$757$$ 1.50758 0.0547938 0.0273969 0.999625i $$-0.491278\pi$$
0.0273969 + 0.999625i $$0.491278\pi$$
$$758$$ 17.7538 0.644847
$$759$$ 3.68466 0.133745
$$760$$ 4.31534 0.156534
$$761$$ 3.12311 0.113212 0.0566062 0.998397i $$-0.481972\pi$$
0.0566062 + 0.998397i $$0.481972\pi$$
$$762$$ 13.1231 0.475400
$$763$$ −11.9309 −0.431926
$$764$$ −13.9309 −0.504001
$$765$$ 3.19224 0.115416
$$766$$ −34.4233 −1.24376
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 20.5616 0.741469 0.370734 0.928739i $$-0.379106\pi$$
0.370734 + 0.928739i $$0.379106\pi$$
$$770$$ 1.43845 0.0518380
$$771$$ 12.2462 0.441037
$$772$$ 5.36932 0.193246
$$773$$ −24.0691 −0.865706 −0.432853 0.901464i $$-0.642493\pi$$
−0.432853 + 0.901464i $$0.642493\pi$$
$$774$$ 10.5616 0.379627
$$775$$ −48.0000 −1.72421
$$776$$ −18.4924 −0.663839
$$777$$ 3.43845 0.123354
$$778$$ −20.7386 −0.743516
$$779$$ 54.7386 1.96122
$$780$$ 0 0
$$781$$ 20.4924 0.733277
$$782$$ 8.17708 0.292412
$$783$$ −5.68466 −0.203153
$$784$$ 1.00000 0.0357143
$$785$$ −6.42329 −0.229257
$$786$$ 5.43845 0.193983
$$787$$ 33.3002 1.18702 0.593512 0.804825i $$-0.297741\pi$$
0.593512 + 0.804825i $$0.297741\pi$$
$$788$$ −19.1231 −0.681232
$$789$$ −13.7538 −0.489648
$$790$$ 1.61553 0.0574779
$$791$$ 12.2462 0.435425
$$792$$ 2.56155 0.0910208
$$793$$ 0 0
$$794$$ −18.0000 −0.638796
$$795$$ −2.38447 −0.0845685
$$796$$ −21.9309 −0.777319
$$797$$ 6.63068 0.234871 0.117435 0.993081i $$-0.462533\pi$$
0.117435 + 0.993081i $$0.462533\pi$$
$$798$$ 7.68466 0.272034
$$799$$ 0 0
$$800$$ 4.68466 0.165628
$$801$$ −10.0000 −0.353333
$$802$$ −8.24621 −0.291184
$$803$$ 1.43845 0.0507617
$$804$$ −1.12311 −0.0396089
$$805$$ −0.807764 −0.0284699
$$806$$ 0 0
$$807$$ 3.75379 0.132140
$$808$$ −3.12311 −0.109870
$$809$$ 30.4924 1.07206 0.536028 0.844200i $$-0.319924\pi$$
0.536028 + 0.844200i $$0.319924\pi$$
$$810$$ −0.561553 −0.0197310
$$811$$ −37.4384 −1.31464 −0.657321 0.753611i $$-0.728310\pi$$
−0.657321 + 0.753611i $$0.728310\pi$$
$$812$$ −5.68466 −0.199492
$$813$$ −13.1231 −0.460247
$$814$$ 8.80776 0.308712
$$815$$ 14.1080 0.494180
$$816$$ 5.68466 0.199003
$$817$$ 81.1619 2.83950
$$818$$ 23.9309 0.836723
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ −55.6155 −1.94100 −0.970498 0.241111i $$-0.922488\pi$$
−0.970498 + 0.241111i $$0.922488\pi$$
$$822$$ 0.561553 0.0195864
$$823$$ −28.4924 −0.993183 −0.496592 0.867984i $$-0.665415\pi$$
−0.496592 + 0.867984i $$0.665415\pi$$
$$824$$ −11.6847 −0.407054
$$825$$ 12.0000 0.417786
$$826$$ −14.2462 −0.495689
$$827$$ 1.93087 0.0671429 0.0335715 0.999436i $$-0.489312\pi$$
0.0335715 + 0.999436i $$0.489312\pi$$
$$828$$ −1.43845 −0.0499895
$$829$$ 26.3153 0.913970 0.456985 0.889475i $$-0.348929\pi$$
0.456985 + 0.889475i $$0.348929\pi$$
$$830$$ 9.61553 0.333760
$$831$$ 10.4924 0.363978
$$832$$ 0 0
$$833$$ 5.68466 0.196962
$$834$$ −12.0000 −0.415526
$$835$$ 3.33050 0.115257
$$836$$ 19.6847 0.680808
$$837$$ 10.2462 0.354161
$$838$$ 24.3153 0.839960
$$839$$ −38.7386 −1.33741 −0.668703 0.743530i $$-0.733150\pi$$
−0.668703 + 0.743530i $$0.733150\pi$$
$$840$$ −0.561553 −0.0193754
$$841$$ 3.31534 0.114322
$$842$$ −20.2462 −0.697731
$$843$$ 0.246211 0.00847997
$$844$$ 4.80776 0.165490
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −4.43845 −0.152507
$$848$$ −4.24621 −0.145815
$$849$$ 1.75379 0.0601899
$$850$$ 26.6307 0.913425
$$851$$ −4.94602 −0.169548
$$852$$ −8.00000 −0.274075
$$853$$ 2.63068 0.0900729 0.0450364 0.998985i $$-0.485660\pi$$
0.0450364 + 0.998985i $$0.485660\pi$$
$$854$$ 5.68466 0.194525
$$855$$ −4.31534 −0.147582
$$856$$ 17.1231 0.585256
$$857$$ −1.50758 −0.0514979 −0.0257489 0.999668i $$-0.508197\pi$$
−0.0257489 + 0.999668i $$0.508197\pi$$
$$858$$ 0 0
$$859$$ 22.2462 0.759031 0.379515 0.925185i $$-0.376091\pi$$
0.379515 + 0.925185i $$0.376091\pi$$
$$860$$ −5.93087 −0.202241
$$861$$ −7.12311 −0.242755
$$862$$ −8.63068 −0.293962
$$863$$ −7.36932 −0.250854 −0.125427 0.992103i $$-0.540030\pi$$
−0.125427 + 0.992103i $$0.540030\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 2.10795 0.0716725
$$866$$ 6.63068 0.225320
$$867$$ 15.3153 0.520136
$$868$$ 10.2462 0.347779
$$869$$ 7.36932 0.249987
$$870$$ 3.19224 0.108227
$$871$$ 0 0
$$872$$ 11.9309 0.404030
$$873$$ 18.4924 0.625874
$$874$$ −11.0540 −0.373906
$$875$$ −5.43845 −0.183853
$$876$$ −0.561553 −0.0189731
$$877$$ 23.7538 0.802108 0.401054 0.916054i $$-0.368644\pi$$
0.401054 + 0.916054i $$0.368644\pi$$
$$878$$ −21.9309 −0.740131
$$879$$ 24.7386 0.834413
$$880$$ −1.43845 −0.0484900
$$881$$ 26.1771 0.881928 0.440964 0.897525i $$-0.354637\pi$$
0.440964 + 0.897525i $$0.354637\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −2.56155 −0.0862031 −0.0431016 0.999071i $$-0.513724\pi$$
−0.0431016 + 0.999071i $$0.513724\pi$$
$$884$$ 0 0
$$885$$ 8.00000 0.268917
$$886$$ −19.3693 −0.650725
$$887$$ −19.5076 −0.655000 −0.327500 0.944851i $$-0.606206\pi$$
−0.327500 + 0.944851i $$0.606206\pi$$
$$888$$ −3.43845 −0.115387
$$889$$ −13.1231 −0.440135
$$890$$ 5.61553 0.188233
$$891$$ −2.56155 −0.0858152
$$892$$ −17.6155 −0.589812
$$893$$ 0 0
$$894$$ 23.6155 0.789821
$$895$$ −9.26137 −0.309573
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −1.68466 −0.0562178
$$899$$ −58.2462 −1.94262
$$900$$ −4.68466 −0.156155
$$901$$ −24.1383 −0.804162
$$902$$ −18.2462 −0.607532
$$903$$ −10.5616 −0.351466
$$904$$ −12.2462 −0.407303
$$905$$ 9.12311 0.303262
$$906$$ −8.80776 −0.292618
$$907$$ −40.4924 −1.34453 −0.672264 0.740311i $$-0.734678\pi$$
−0.672264 + 0.740311i $$0.734678\pi$$
$$908$$ −1.12311 −0.0372716
$$909$$ 3.12311 0.103587
$$910$$ 0 0
$$911$$ 50.4233 1.67060 0.835299 0.549796i $$-0.185294\pi$$
0.835299 + 0.549796i $$0.185294\pi$$
$$912$$ −7.68466 −0.254464
$$913$$ 43.8617 1.45161
$$914$$ 2.63068 0.0870153
$$915$$ −3.19224 −0.105532
$$916$$ −11.7538 −0.388356
$$917$$ −5.43845 −0.179593
$$918$$ −5.68466 −0.187622
$$919$$ 10.8769 0.358796 0.179398 0.983777i $$-0.442585\pi$$
0.179398 + 0.983777i $$0.442585\pi$$
$$920$$ 0.807764 0.0266312
$$921$$ −9.75379 −0.321398
$$922$$ 9.05398 0.298177
$$923$$ 0 0
$$924$$ −2.56155 −0.0842689
$$925$$ −16.1080 −0.529626
$$926$$ 3.68466 0.121085
$$927$$ 11.6847 0.383775
$$928$$ 5.68466 0.186608
$$929$$ 15.6155 0.512329 0.256164 0.966633i $$-0.417541\pi$$
0.256164 + 0.966633i $$0.417541\pi$$
$$930$$ −5.75379 −0.188674
$$931$$ −7.68466 −0.251855
$$932$$ 2.63068 0.0861709
$$933$$ −21.1231 −0.691539
$$934$$ 15.6847 0.513218
$$935$$ −8.17708 −0.267419
$$936$$ 0 0
$$937$$ 18.9848 0.620208 0.310104 0.950703i $$-0.399636\pi$$
0.310104 + 0.950703i $$0.399636\pi$$
$$938$$ 1.12311 0.0366707
$$939$$ 27.6155 0.901199
$$940$$ 0 0
$$941$$ 34.0000 1.10837 0.554184 0.832394i $$-0.313030\pi$$
0.554184 + 0.832394i $$0.313030\pi$$
$$942$$ 11.4384 0.372685
$$943$$ 10.2462 0.333663
$$944$$ 14.2462 0.463675
$$945$$ 0.561553 0.0182673
$$946$$ −27.0540 −0.879601
$$947$$ −45.7926 −1.48806 −0.744030 0.668146i $$-0.767088\pi$$
−0.744030 + 0.668146i $$0.767088\pi$$
$$948$$ −2.87689 −0.0934372
$$949$$ 0 0
$$950$$ −36.0000 −1.16799
$$951$$ −11.1231 −0.360691
$$952$$ −5.68466 −0.184241
$$953$$ −13.3693 −0.433075 −0.216537 0.976274i $$-0.569476\pi$$
−0.216537 + 0.976274i $$0.569476\pi$$
$$954$$ 4.24621 0.137476
$$955$$ −7.82292 −0.253144
$$956$$ −16.0000 −0.517477
$$957$$ 14.5616 0.470708
$$958$$ −21.3002 −0.688178
$$959$$ −0.561553 −0.0181335
$$960$$ 0.561553 0.0181240
$$961$$ 73.9848 2.38661
$$962$$ 0 0
$$963$$ −17.1231 −0.551784
$$964$$ 14.0000 0.450910
$$965$$ 3.01515 0.0970613
$$966$$ 1.43845 0.0462813
$$967$$ 17.4384 0.560783 0.280391 0.959886i $$-0.409536\pi$$
0.280391 + 0.959886i $$0.409536\pi$$
$$968$$ 4.43845 0.142657
$$969$$ −43.6847 −1.40335
$$970$$ −10.3845 −0.333425
$$971$$ −14.2462 −0.457183 −0.228591 0.973522i $$-0.573412\pi$$
−0.228591 + 0.973522i $$0.573412\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 12.0000 0.384702
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −5.68466 −0.181961
$$977$$ 10.3153 0.330017 0.165009 0.986292i $$-0.447235\pi$$
0.165009 + 0.986292i $$0.447235\pi$$
$$978$$ −25.1231 −0.803348
$$979$$ 25.6155 0.818676
$$980$$ 0.561553 0.0179381
$$981$$ −11.9309 −0.380923
$$982$$ 17.1231 0.546420
$$983$$ −32.1771 −1.02629 −0.513145 0.858302i $$-0.671520\pi$$
−0.513145 + 0.858302i $$0.671520\pi$$
$$984$$ 7.12311 0.227076
$$985$$ −10.7386 −0.342161
$$986$$ 32.3153 1.02913
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 15.1922 0.483085
$$990$$ 1.43845 0.0457169
$$991$$ 33.6155 1.06783 0.533916 0.845537i $$-0.320720\pi$$
0.533916 + 0.845537i $$0.320720\pi$$
$$992$$ −10.2462 −0.325318
$$993$$ 11.3693 0.360794
$$994$$ 8.00000 0.253745
$$995$$ −12.3153 −0.390423
$$996$$ −17.1231 −0.542566
$$997$$ −8.73863 −0.276755 −0.138378 0.990380i $$-0.544189\pi$$
−0.138378 + 0.990380i $$0.544189\pi$$
$$998$$ 36.0000 1.13956
$$999$$ 3.43845 0.108788
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 7098.2.a.bl.1.2 2
13.12 even 2 546.2.a.j.1.1 2
39.38 odd 2 1638.2.a.u.1.2 2
52.51 odd 2 4368.2.a.be.1.1 2
91.90 odd 2 3822.2.a.bo.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.j.1.1 2 13.12 even 2
1638.2.a.u.1.2 2 39.38 odd 2
3822.2.a.bo.1.2 2 91.90 odd 2
4368.2.a.be.1.1 2 52.51 odd 2
7098.2.a.bl.1.2 2 1.1 even 1 trivial