# Properties

 Label 7098.2.a.ct.1.2 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $6$ 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: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.48406561.1 Defining polynomial: $$x^{6} - 3 x^{5} - 17 x^{4} + 39 x^{3} + 111 x^{2} - 131 x - 281$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.10863$$ of defining polynomial Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.10863 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.10863 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.10863 q^{10} +1.55496 q^{11} +1.00000 q^{12} +1.00000 q^{14} -2.10863 q^{15} +1.00000 q^{16} -1.24698 q^{17} +1.00000 q^{18} +6.98503 q^{19} -2.10863 q^{20} +1.00000 q^{21} +1.55496 q^{22} -1.97111 q^{23} +1.00000 q^{24} -0.553673 q^{25} +1.00000 q^{27} +1.00000 q^{28} +6.02003 q^{29} -2.10863 q^{30} +2.75174 q^{31} +1.00000 q^{32} +1.55496 q^{33} -1.24698 q^{34} -2.10863 q^{35} +1.00000 q^{36} -2.76545 q^{37} +6.98503 q^{38} -2.10863 q^{40} -7.97320 q^{41} +1.00000 q^{42} -3.66462 q^{43} +1.55496 q^{44} -2.10863 q^{45} -1.97111 q^{46} +5.63759 q^{47} +1.00000 q^{48} +1.00000 q^{49} -0.553673 q^{50} -1.24698 q^{51} +8.81309 q^{53} +1.00000 q^{54} -3.27883 q^{55} +1.00000 q^{56} +6.98503 q^{57} +6.02003 q^{58} +5.40902 q^{59} -2.10863 q^{60} +3.42032 q^{61} +2.75174 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.55496 q^{66} -1.40756 q^{67} -1.24698 q^{68} -1.97111 q^{69} -2.10863 q^{70} +11.6269 q^{71} +1.00000 q^{72} -7.48979 q^{73} -2.76545 q^{74} -0.553673 q^{75} +6.98503 q^{76} +1.55496 q^{77} -16.4358 q^{79} -2.10863 q^{80} +1.00000 q^{81} -7.97320 q^{82} +16.7237 q^{83} +1.00000 q^{84} +2.62942 q^{85} -3.66462 q^{86} +6.02003 q^{87} +1.55496 q^{88} +10.9813 q^{89} -2.10863 q^{90} -1.97111 q^{92} +2.75174 q^{93} +5.63759 q^{94} -14.7289 q^{95} +1.00000 q^{96} -6.39487 q^{97} +1.00000 q^{98} +1.55496 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{2} + 6q^{3} + 6q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{2} + 6q^{3} + 6q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} + 3q^{10} + 10q^{11} + 6q^{12} + 6q^{14} + 3q^{15} + 6q^{16} + 2q^{17} + 6q^{18} + 2q^{19} + 3q^{20} + 6q^{21} + 10q^{22} + 4q^{23} + 6q^{24} + 13q^{25} + 6q^{27} + 6q^{28} + 2q^{29} + 3q^{30} + 9q^{31} + 6q^{32} + 10q^{33} + 2q^{34} + 3q^{35} + 6q^{36} + 7q^{37} + 2q^{38} + 3q^{40} + 11q^{41} + 6q^{42} - 5q^{43} + 10q^{44} + 3q^{45} + 4q^{46} + 5q^{47} + 6q^{48} + 6q^{49} + 13q^{50} + 2q^{51} - 6q^{53} + 6q^{54} + 5q^{55} + 6q^{56} + 2q^{57} + 2q^{58} + 28q^{59} + 3q^{60} + 23q^{61} + 9q^{62} + 6q^{63} + 6q^{64} + 10q^{66} - 10q^{67} + 2q^{68} + 4q^{69} + 3q^{70} + 21q^{71} + 6q^{72} - 7q^{73} + 7q^{74} + 13q^{75} + 2q^{76} + 10q^{77} - 14q^{79} + 3q^{80} + 6q^{81} + 11q^{82} + 17q^{83} + 6q^{84} + q^{85} - 5q^{86} + 2q^{87} + 10q^{88} + 17q^{89} + 3q^{90} + 4q^{92} + 9q^{93} + 5q^{94} - 22q^{95} + 6q^{96} + 6q^{98} + 10q^{99} + O(q^{100})$$

## 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
$$5$$ −2.10863 −0.943009 −0.471504 0.881864i $$-0.656289\pi$$
−0.471504 + 0.881864i $$0.656289\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −2.10863 −0.666808
$$11$$ 1.55496 0.468838 0.234419 0.972136i $$-0.424681\pi$$
0.234419 + 0.972136i $$0.424681\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 1.00000 0.267261
$$15$$ −2.10863 −0.544446
$$16$$ 1.00000 0.250000
$$17$$ −1.24698 −0.302437 −0.151218 0.988500i $$-0.548320\pi$$
−0.151218 + 0.988500i $$0.548320\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 6.98503 1.60248 0.801238 0.598346i $$-0.204175\pi$$
0.801238 + 0.598346i $$0.204175\pi$$
$$20$$ −2.10863 −0.471504
$$21$$ 1.00000 0.218218
$$22$$ 1.55496 0.331518
$$23$$ −1.97111 −0.411005 −0.205502 0.978657i $$-0.565883\pi$$
−0.205502 + 0.978657i $$0.565883\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −0.553673 −0.110735
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 1.00000 0.188982
$$29$$ 6.02003 1.11789 0.558945 0.829204i $$-0.311206\pi$$
0.558945 + 0.829204i $$0.311206\pi$$
$$30$$ −2.10863 −0.384982
$$31$$ 2.75174 0.494226 0.247113 0.968987i $$-0.420518\pi$$
0.247113 + 0.968987i $$0.420518\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.55496 0.270683
$$34$$ −1.24698 −0.213855
$$35$$ −2.10863 −0.356424
$$36$$ 1.00000 0.166667
$$37$$ −2.76545 −0.454638 −0.227319 0.973820i $$-0.572996\pi$$
−0.227319 + 0.973820i $$0.572996\pi$$
$$38$$ 6.98503 1.13312
$$39$$ 0 0
$$40$$ −2.10863 −0.333404
$$41$$ −7.97320 −1.24521 −0.622603 0.782538i $$-0.713925\pi$$
−0.622603 + 0.782538i $$0.713925\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −3.66462 −0.558849 −0.279425 0.960168i $$-0.590144\pi$$
−0.279425 + 0.960168i $$0.590144\pi$$
$$44$$ 1.55496 0.234419
$$45$$ −2.10863 −0.314336
$$46$$ −1.97111 −0.290624
$$47$$ 5.63759 0.822326 0.411163 0.911562i $$-0.365123\pi$$
0.411163 + 0.911562i $$0.365123\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ −0.553673 −0.0783012
$$51$$ −1.24698 −0.174612
$$52$$ 0 0
$$53$$ 8.81309 1.21057 0.605285 0.796009i $$-0.293059\pi$$
0.605285 + 0.796009i $$0.293059\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −3.27883 −0.442118
$$56$$ 1.00000 0.133631
$$57$$ 6.98503 0.925190
$$58$$ 6.02003 0.790468
$$59$$ 5.40902 0.704194 0.352097 0.935964i $$-0.385469\pi$$
0.352097 + 0.935964i $$0.385469\pi$$
$$60$$ −2.10863 −0.272223
$$61$$ 3.42032 0.437928 0.218964 0.975733i $$-0.429732\pi$$
0.218964 + 0.975733i $$0.429732\pi$$
$$62$$ 2.75174 0.349471
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.55496 0.191402
$$67$$ −1.40756 −0.171960 −0.0859802 0.996297i $$-0.527402\pi$$
−0.0859802 + 0.996297i $$0.527402\pi$$
$$68$$ −1.24698 −0.151218
$$69$$ −1.97111 −0.237294
$$70$$ −2.10863 −0.252030
$$71$$ 11.6269 1.37985 0.689927 0.723879i $$-0.257643\pi$$
0.689927 + 0.723879i $$0.257643\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −7.48979 −0.876613 −0.438307 0.898825i $$-0.644422\pi$$
−0.438307 + 0.898825i $$0.644422\pi$$
$$74$$ −2.76545 −0.321477
$$75$$ −0.553673 −0.0639327
$$76$$ 6.98503 0.801238
$$77$$ 1.55496 0.177204
$$78$$ 0 0
$$79$$ −16.4358 −1.84918 −0.924589 0.380967i $$-0.875591\pi$$
−0.924589 + 0.380967i $$0.875591\pi$$
$$80$$ −2.10863 −0.235752
$$81$$ 1.00000 0.111111
$$82$$ −7.97320 −0.880493
$$83$$ 16.7237 1.83566 0.917830 0.396974i $$-0.129940\pi$$
0.917830 + 0.396974i $$0.129940\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 2.62942 0.285201
$$86$$ −3.66462 −0.395166
$$87$$ 6.02003 0.645415
$$88$$ 1.55496 0.165759
$$89$$ 10.9813 1.16401 0.582005 0.813185i $$-0.302268\pi$$
0.582005 + 0.813185i $$0.302268\pi$$
$$90$$ −2.10863 −0.222269
$$91$$ 0 0
$$92$$ −1.97111 −0.205502
$$93$$ 2.75174 0.285342
$$94$$ 5.63759 0.581473
$$95$$ −14.7289 −1.51115
$$96$$ 1.00000 0.102062
$$97$$ −6.39487 −0.649301 −0.324651 0.945834i $$-0.605247\pi$$
−0.324651 + 0.945834i $$0.605247\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 1.55496 0.156279
$$100$$ −0.553673 −0.0553673
$$101$$ −1.28670 −0.128032 −0.0640158 0.997949i $$-0.520391\pi$$
−0.0640158 + 0.997949i $$0.520391\pi$$
$$102$$ −1.24698 −0.123469
$$103$$ 6.54173 0.644576 0.322288 0.946642i $$-0.395548\pi$$
0.322288 + 0.946642i $$0.395548\pi$$
$$104$$ 0 0
$$105$$ −2.10863 −0.205781
$$106$$ 8.81309 0.856003
$$107$$ 2.66334 0.257474 0.128737 0.991679i $$-0.458908\pi$$
0.128737 + 0.991679i $$0.458908\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −4.40973 −0.422376 −0.211188 0.977446i $$-0.567733\pi$$
−0.211188 + 0.977446i $$0.567733\pi$$
$$110$$ −3.27883 −0.312625
$$111$$ −2.76545 −0.262485
$$112$$ 1.00000 0.0944911
$$113$$ −5.81726 −0.547242 −0.273621 0.961838i $$-0.588221\pi$$
−0.273621 + 0.961838i $$0.588221\pi$$
$$114$$ 6.98503 0.654208
$$115$$ 4.15634 0.387581
$$116$$ 6.02003 0.558945
$$117$$ 0 0
$$118$$ 5.40902 0.497940
$$119$$ −1.24698 −0.114310
$$120$$ −2.10863 −0.192491
$$121$$ −8.58211 −0.780191
$$122$$ 3.42032 0.309662
$$123$$ −7.97320 −0.718920
$$124$$ 2.75174 0.247113
$$125$$ 11.7107 1.04743
$$126$$ 1.00000 0.0890871
$$127$$ −3.83276 −0.340103 −0.170051 0.985435i $$-0.554393\pi$$
−0.170051 + 0.985435i $$0.554393\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −3.66462 −0.322652
$$130$$ 0 0
$$131$$ 11.8617 1.03637 0.518183 0.855270i $$-0.326609\pi$$
0.518183 + 0.855270i $$0.326609\pi$$
$$132$$ 1.55496 0.135342
$$133$$ 6.98503 0.605679
$$134$$ −1.40756 −0.121594
$$135$$ −2.10863 −0.181482
$$136$$ −1.24698 −0.106928
$$137$$ 2.18760 0.186899 0.0934496 0.995624i $$-0.470211\pi$$
0.0934496 + 0.995624i $$0.470211\pi$$
$$138$$ −1.97111 −0.167792
$$139$$ 4.14041 0.351185 0.175592 0.984463i $$-0.443816\pi$$
0.175592 + 0.984463i $$0.443816\pi$$
$$140$$ −2.10863 −0.178212
$$141$$ 5.63759 0.474770
$$142$$ 11.6269 0.975704
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −12.6940 −1.05418
$$146$$ −7.48979 −0.619859
$$147$$ 1.00000 0.0824786
$$148$$ −2.76545 −0.227319
$$149$$ 16.1634 1.32416 0.662079 0.749434i $$-0.269674\pi$$
0.662079 + 0.749434i $$0.269674\pi$$
$$150$$ −0.553673 −0.0452072
$$151$$ 14.5820 1.18667 0.593333 0.804957i $$-0.297812\pi$$
0.593333 + 0.804957i $$0.297812\pi$$
$$152$$ 6.98503 0.566561
$$153$$ −1.24698 −0.100812
$$154$$ 1.55496 0.125302
$$155$$ −5.80240 −0.466060
$$156$$ 0 0
$$157$$ 5.49884 0.438855 0.219428 0.975629i $$-0.429581\pi$$
0.219428 + 0.975629i $$0.429581\pi$$
$$158$$ −16.4358 −1.30757
$$159$$ 8.81309 0.698923
$$160$$ −2.10863 −0.166702
$$161$$ −1.97111 −0.155345
$$162$$ 1.00000 0.0785674
$$163$$ 14.1960 1.11192 0.555958 0.831211i $$-0.312352\pi$$
0.555958 + 0.831211i $$0.312352\pi$$
$$164$$ −7.97320 −0.622603
$$165$$ −3.27883 −0.255257
$$166$$ 16.7237 1.29801
$$167$$ 24.5175 1.89722 0.948612 0.316442i $$-0.102488\pi$$
0.948612 + 0.316442i $$0.102488\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 0 0
$$170$$ 2.62942 0.201667
$$171$$ 6.98503 0.534159
$$172$$ −3.66462 −0.279425
$$173$$ −12.5776 −0.956259 −0.478129 0.878289i $$-0.658685\pi$$
−0.478129 + 0.878289i $$0.658685\pi$$
$$174$$ 6.02003 0.456377
$$175$$ −0.553673 −0.0418538
$$176$$ 1.55496 0.117209
$$177$$ 5.40902 0.406567
$$178$$ 10.9813 0.823080
$$179$$ 2.52767 0.188927 0.0944635 0.995528i $$-0.469886\pi$$
0.0944635 + 0.995528i $$0.469886\pi$$
$$180$$ −2.10863 −0.157168
$$181$$ −12.7141 −0.945031 −0.472516 0.881322i $$-0.656654\pi$$
−0.472516 + 0.881322i $$0.656654\pi$$
$$182$$ 0 0
$$183$$ 3.42032 0.252838
$$184$$ −1.97111 −0.145312
$$185$$ 5.83132 0.428727
$$186$$ 2.75174 0.201767
$$187$$ −1.93900 −0.141794
$$188$$ 5.63759 0.411163
$$189$$ 1.00000 0.0727393
$$190$$ −14.7289 −1.06854
$$191$$ −2.77016 −0.200442 −0.100221 0.994965i $$-0.531955\pi$$
−0.100221 + 0.994965i $$0.531955\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 22.2930 1.60468 0.802341 0.596865i $$-0.203587\pi$$
0.802341 + 0.596865i $$0.203587\pi$$
$$194$$ −6.39487 −0.459125
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −8.75440 −0.623725 −0.311863 0.950127i $$-0.600953\pi$$
−0.311863 + 0.950127i $$0.600953\pi$$
$$198$$ 1.55496 0.110506
$$199$$ −14.5824 −1.03372 −0.516859 0.856070i $$-0.672899\pi$$
−0.516859 + 0.856070i $$0.672899\pi$$
$$200$$ −0.553673 −0.0391506
$$201$$ −1.40756 −0.0992814
$$202$$ −1.28670 −0.0905320
$$203$$ 6.02003 0.422523
$$204$$ −1.24698 −0.0873060
$$205$$ 16.8125 1.17424
$$206$$ 6.54173 0.455784
$$207$$ −1.97111 −0.137002
$$208$$ 0 0
$$209$$ 10.8614 0.751301
$$210$$ −2.10863 −0.145509
$$211$$ 13.1688 0.906575 0.453287 0.891364i $$-0.350251\pi$$
0.453287 + 0.891364i $$0.350251\pi$$
$$212$$ 8.81309 0.605285
$$213$$ 11.6269 0.796659
$$214$$ 2.66334 0.182062
$$215$$ 7.72733 0.527000
$$216$$ 1.00000 0.0680414
$$217$$ 2.75174 0.186800
$$218$$ −4.40973 −0.298665
$$219$$ −7.48979 −0.506113
$$220$$ −3.27883 −0.221059
$$221$$ 0 0
$$222$$ −2.76545 −0.185605
$$223$$ 4.36466 0.292280 0.146140 0.989264i $$-0.453315\pi$$
0.146140 + 0.989264i $$0.453315\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −0.553673 −0.0369116
$$226$$ −5.81726 −0.386958
$$227$$ 0.618667 0.0410624 0.0205312 0.999789i $$-0.493464\pi$$
0.0205312 + 0.999789i $$0.493464\pi$$
$$228$$ 6.98503 0.462595
$$229$$ −2.93024 −0.193636 −0.0968180 0.995302i $$-0.530866\pi$$
−0.0968180 + 0.995302i $$0.530866\pi$$
$$230$$ 4.15634 0.274061
$$231$$ 1.55496 0.102309
$$232$$ 6.02003 0.395234
$$233$$ −18.7941 −1.23124 −0.615620 0.788043i $$-0.711094\pi$$
−0.615620 + 0.788043i $$0.711094\pi$$
$$234$$ 0 0
$$235$$ −11.8876 −0.775461
$$236$$ 5.40902 0.352097
$$237$$ −16.4358 −1.06762
$$238$$ −1.24698 −0.0808297
$$239$$ −19.6751 −1.27267 −0.636337 0.771411i $$-0.719551\pi$$
−0.636337 + 0.771411i $$0.719551\pi$$
$$240$$ −2.10863 −0.136112
$$241$$ −5.57097 −0.358858 −0.179429 0.983771i $$-0.557425\pi$$
−0.179429 + 0.983771i $$0.557425\pi$$
$$242$$ −8.58211 −0.551679
$$243$$ 1.00000 0.0641500
$$244$$ 3.42032 0.218964
$$245$$ −2.10863 −0.134716
$$246$$ −7.97320 −0.508353
$$247$$ 0 0
$$248$$ 2.75174 0.174735
$$249$$ 16.7237 1.05982
$$250$$ 11.7107 0.740647
$$251$$ 10.6372 0.671413 0.335707 0.941967i $$-0.391025\pi$$
0.335707 + 0.941967i $$0.391025\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −3.06499 −0.192694
$$254$$ −3.83276 −0.240489
$$255$$ 2.62942 0.164661
$$256$$ 1.00000 0.0625000
$$257$$ −2.38334 −0.148669 −0.0743345 0.997233i $$-0.523683\pi$$
−0.0743345 + 0.997233i $$0.523683\pi$$
$$258$$ −3.66462 −0.228149
$$259$$ −2.76545 −0.171837
$$260$$ 0 0
$$261$$ 6.02003 0.372630
$$262$$ 11.8617 0.732821
$$263$$ 7.81509 0.481899 0.240949 0.970538i $$-0.422541\pi$$
0.240949 + 0.970538i $$0.422541\pi$$
$$264$$ 1.55496 0.0957011
$$265$$ −18.5836 −1.14158
$$266$$ 6.98503 0.428280
$$267$$ 10.9813 0.672042
$$268$$ −1.40756 −0.0859802
$$269$$ −7.21556 −0.439940 −0.219970 0.975507i $$-0.570596\pi$$
−0.219970 + 0.975507i $$0.570596\pi$$
$$270$$ −2.10863 −0.128327
$$271$$ −4.97552 −0.302241 −0.151121 0.988515i $$-0.548288\pi$$
−0.151121 + 0.988515i $$0.548288\pi$$
$$272$$ −1.24698 −0.0756092
$$273$$ 0 0
$$274$$ 2.18760 0.132158
$$275$$ −0.860939 −0.0519166
$$276$$ −1.97111 −0.118647
$$277$$ −13.1229 −0.788479 −0.394240 0.919008i $$-0.628992\pi$$
−0.394240 + 0.919008i $$0.628992\pi$$
$$278$$ 4.14041 0.248325
$$279$$ 2.75174 0.164742
$$280$$ −2.10863 −0.126015
$$281$$ −0.105985 −0.00632254 −0.00316127 0.999995i $$-0.501006\pi$$
−0.00316127 + 0.999995i $$0.501006\pi$$
$$282$$ 5.63759 0.335713
$$283$$ 29.5509 1.75662 0.878309 0.478094i $$-0.158672\pi$$
0.878309 + 0.478094i $$0.158672\pi$$
$$284$$ 11.6269 0.689927
$$285$$ −14.7289 −0.872462
$$286$$ 0 0
$$287$$ −7.97320 −0.470643
$$288$$ 1.00000 0.0589256
$$289$$ −15.4450 −0.908532
$$290$$ −12.6940 −0.745418
$$291$$ −6.39487 −0.374874
$$292$$ −7.48979 −0.438307
$$293$$ 12.1002 0.706903 0.353451 0.935453i $$-0.385008\pi$$
0.353451 + 0.935453i $$0.385008\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ −11.4056 −0.664061
$$296$$ −2.76545 −0.160739
$$297$$ 1.55496 0.0902278
$$298$$ 16.1634 0.936322
$$299$$ 0 0
$$300$$ −0.553673 −0.0319663
$$301$$ −3.66462 −0.211225
$$302$$ 14.5820 0.839099
$$303$$ −1.28670 −0.0739190
$$304$$ 6.98503 0.400619
$$305$$ −7.21220 −0.412969
$$306$$ −1.24698 −0.0712851
$$307$$ −11.8071 −0.673867 −0.336934 0.941528i $$-0.609390\pi$$
−0.336934 + 0.941528i $$0.609390\pi$$
$$308$$ 1.55496 0.0886020
$$309$$ 6.54173 0.372146
$$310$$ −5.80240 −0.329554
$$311$$ 2.36454 0.134080 0.0670402 0.997750i $$-0.478644\pi$$
0.0670402 + 0.997750i $$0.478644\pi$$
$$312$$ 0 0
$$313$$ −6.67399 −0.377236 −0.188618 0.982051i $$-0.560401\pi$$
−0.188618 + 0.982051i $$0.560401\pi$$
$$314$$ 5.49884 0.310318
$$315$$ −2.10863 −0.118808
$$316$$ −16.4358 −0.924589
$$317$$ 0.860447 0.0483275 0.0241638 0.999708i $$-0.492308\pi$$
0.0241638 + 0.999708i $$0.492308\pi$$
$$318$$ 8.81309 0.494213
$$319$$ 9.36089 0.524109
$$320$$ −2.10863 −0.117876
$$321$$ 2.66334 0.148653
$$322$$ −1.97111 −0.109846
$$323$$ −8.71019 −0.484648
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 14.1960 0.786243
$$327$$ −4.40973 −0.243859
$$328$$ −7.97320 −0.440247
$$329$$ 5.63759 0.310810
$$330$$ −3.27883 −0.180494
$$331$$ 15.4980 0.851848 0.425924 0.904759i $$-0.359949\pi$$
0.425924 + 0.904759i $$0.359949\pi$$
$$332$$ 16.7237 0.917830
$$333$$ −2.76545 −0.151546
$$334$$ 24.5175 1.34154
$$335$$ 2.96802 0.162160
$$336$$ 1.00000 0.0545545
$$337$$ −15.6615 −0.853135 −0.426568 0.904456i $$-0.640277\pi$$
−0.426568 + 0.904456i $$0.640277\pi$$
$$338$$ 0 0
$$339$$ −5.81726 −0.315950
$$340$$ 2.62942 0.142600
$$341$$ 4.27883 0.231712
$$342$$ 6.98503 0.377707
$$343$$ 1.00000 0.0539949
$$344$$ −3.66462 −0.197583
$$345$$ 4.15634 0.223770
$$346$$ −12.5776 −0.676177
$$347$$ −13.0233 −0.699129 −0.349564 0.936912i $$-0.613670\pi$$
−0.349564 + 0.936912i $$0.613670\pi$$
$$348$$ 6.02003 0.322707
$$349$$ 26.4749 1.41717 0.708584 0.705626i $$-0.249334\pi$$
0.708584 + 0.705626i $$0.249334\pi$$
$$350$$ −0.553673 −0.0295951
$$351$$ 0 0
$$352$$ 1.55496 0.0828795
$$353$$ 21.5831 1.14875 0.574375 0.818592i $$-0.305245\pi$$
0.574375 + 0.818592i $$0.305245\pi$$
$$354$$ 5.40902 0.287486
$$355$$ −24.5167 −1.30121
$$356$$ 10.9813 0.582005
$$357$$ −1.24698 −0.0659972
$$358$$ 2.52767 0.133592
$$359$$ 4.67664 0.246824 0.123412 0.992356i $$-0.460616\pi$$
0.123412 + 0.992356i $$0.460616\pi$$
$$360$$ −2.10863 −0.111135
$$361$$ 29.7907 1.56793
$$362$$ −12.7141 −0.668238
$$363$$ −8.58211 −0.450444
$$364$$ 0 0
$$365$$ 15.7932 0.826654
$$366$$ 3.42032 0.178783
$$367$$ 31.3614 1.63705 0.818525 0.574471i $$-0.194792\pi$$
0.818525 + 0.574471i $$0.194792\pi$$
$$368$$ −1.97111 −0.102751
$$369$$ −7.97320 −0.415068
$$370$$ 5.83132 0.303156
$$371$$ 8.81309 0.457553
$$372$$ 2.75174 0.142671
$$373$$ 28.2046 1.46038 0.730190 0.683244i $$-0.239432\pi$$
0.730190 + 0.683244i $$0.239432\pi$$
$$374$$ −1.93900 −0.100263
$$375$$ 11.7107 0.604735
$$376$$ 5.63759 0.290736
$$377$$ 0 0
$$378$$ 1.00000 0.0514344
$$379$$ −32.5590 −1.67245 −0.836223 0.548390i $$-0.815241\pi$$
−0.836223 + 0.548390i $$0.815241\pi$$
$$380$$ −14.7289 −0.755574
$$381$$ −3.83276 −0.196358
$$382$$ −2.77016 −0.141734
$$383$$ 8.63110 0.441029 0.220514 0.975384i $$-0.429226\pi$$
0.220514 + 0.975384i $$0.429226\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ −3.27883 −0.167105
$$386$$ 22.2930 1.13468
$$387$$ −3.66462 −0.186283
$$388$$ −6.39487 −0.324651
$$389$$ −26.3682 −1.33692 −0.668460 0.743748i $$-0.733046\pi$$
−0.668460 + 0.743748i $$0.733046\pi$$
$$390$$ 0 0
$$391$$ 2.45793 0.124303
$$392$$ 1.00000 0.0505076
$$393$$ 11.8617 0.598346
$$394$$ −8.75440 −0.441040
$$395$$ 34.6571 1.74379
$$396$$ 1.55496 0.0781396
$$397$$ −12.0701 −0.605779 −0.302889 0.953026i $$-0.597951\pi$$
−0.302889 + 0.953026i $$0.597951\pi$$
$$398$$ −14.5824 −0.730949
$$399$$ 6.98503 0.349689
$$400$$ −0.553673 −0.0276837
$$401$$ −18.5164 −0.924664 −0.462332 0.886707i $$-0.652987\pi$$
−0.462332 + 0.886707i $$0.652987\pi$$
$$402$$ −1.40756 −0.0702025
$$403$$ 0 0
$$404$$ −1.28670 −0.0640158
$$405$$ −2.10863 −0.104779
$$406$$ 6.02003 0.298769
$$407$$ −4.30016 −0.213151
$$408$$ −1.24698 −0.0617347
$$409$$ −35.8394 −1.77214 −0.886072 0.463547i $$-0.846576\pi$$
−0.886072 + 0.463547i $$0.846576\pi$$
$$410$$ 16.8125 0.830313
$$411$$ 2.18760 0.107906
$$412$$ 6.54173 0.322288
$$413$$ 5.40902 0.266160
$$414$$ −1.97111 −0.0968748
$$415$$ −35.2640 −1.73104
$$416$$ 0 0
$$417$$ 4.14041 0.202757
$$418$$ 10.8614 0.531250
$$419$$ 27.0079 1.31942 0.659710 0.751520i $$-0.270679\pi$$
0.659710 + 0.751520i $$0.270679\pi$$
$$420$$ −2.10863 −0.102891
$$421$$ −26.2859 −1.28110 −0.640549 0.767918i $$-0.721293\pi$$
−0.640549 + 0.767918i $$0.721293\pi$$
$$422$$ 13.1688 0.641045
$$423$$ 5.63759 0.274109
$$424$$ 8.81309 0.428001
$$425$$ 0.690419 0.0334903
$$426$$ 11.6269 0.563323
$$427$$ 3.42032 0.165521
$$428$$ 2.66334 0.128737
$$429$$ 0 0
$$430$$ 7.72733 0.372645
$$431$$ −6.58585 −0.317229 −0.158615 0.987341i $$-0.550703\pi$$
−0.158615 + 0.987341i $$0.550703\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 25.4472 1.22291 0.611457 0.791278i $$-0.290584\pi$$
0.611457 + 0.791278i $$0.290584\pi$$
$$434$$ 2.75174 0.132088
$$435$$ −12.6940 −0.608632
$$436$$ −4.40973 −0.211188
$$437$$ −13.7683 −0.658625
$$438$$ −7.48979 −0.357876
$$439$$ 23.8445 1.13803 0.569017 0.822326i $$-0.307324\pi$$
0.569017 + 0.822326i $$0.307324\pi$$
$$440$$ −3.27883 −0.156312
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −0.966959 −0.0459416 −0.0229708 0.999736i $$-0.507312\pi$$
−0.0229708 + 0.999736i $$0.507312\pi$$
$$444$$ −2.76545 −0.131243
$$445$$ −23.1554 −1.09767
$$446$$ 4.36466 0.206673
$$447$$ 16.1634 0.764503
$$448$$ 1.00000 0.0472456
$$449$$ −32.7563 −1.54586 −0.772932 0.634489i $$-0.781210\pi$$
−0.772932 + 0.634489i $$0.781210\pi$$
$$450$$ −0.553673 −0.0261004
$$451$$ −12.3980 −0.583799
$$452$$ −5.81726 −0.273621
$$453$$ 14.5820 0.685122
$$454$$ 0.618667 0.0290355
$$455$$ 0 0
$$456$$ 6.98503 0.327104
$$457$$ 2.38368 0.111504 0.0557520 0.998445i $$-0.482244\pi$$
0.0557520 + 0.998445i $$0.482244\pi$$
$$458$$ −2.93024 −0.136921
$$459$$ −1.24698 −0.0582040
$$460$$ 4.15634 0.193791
$$461$$ 36.9292 1.71996 0.859982 0.510325i $$-0.170475\pi$$
0.859982 + 0.510325i $$0.170475\pi$$
$$462$$ 1.55496 0.0723432
$$463$$ −29.8161 −1.38567 −0.692836 0.721095i $$-0.743639\pi$$
−0.692836 + 0.721095i $$0.743639\pi$$
$$464$$ 6.02003 0.279473
$$465$$ −5.80240 −0.269080
$$466$$ −18.7941 −0.870618
$$467$$ 9.34853 0.432598 0.216299 0.976327i $$-0.430601\pi$$
0.216299 + 0.976327i $$0.430601\pi$$
$$468$$ 0 0
$$469$$ −1.40756 −0.0649949
$$470$$ −11.8876 −0.548334
$$471$$ 5.49884 0.253373
$$472$$ 5.40902 0.248970
$$473$$ −5.69833 −0.262010
$$474$$ −16.4358 −0.754923
$$475$$ −3.86743 −0.177450
$$476$$ −1.24698 −0.0571552
$$477$$ 8.81309 0.403523
$$478$$ −19.6751 −0.899917
$$479$$ 17.5839 0.803429 0.401715 0.915765i $$-0.368414\pi$$
0.401715 + 0.915765i $$0.368414\pi$$
$$480$$ −2.10863 −0.0962454
$$481$$ 0 0
$$482$$ −5.57097 −0.253751
$$483$$ −1.97111 −0.0896886
$$484$$ −8.58211 −0.390096
$$485$$ 13.4844 0.612297
$$486$$ 1.00000 0.0453609
$$487$$ 35.9216 1.62776 0.813882 0.581030i $$-0.197350\pi$$
0.813882 + 0.581030i $$0.197350\pi$$
$$488$$ 3.42032 0.154831
$$489$$ 14.1960 0.641965
$$490$$ −2.10863 −0.0952583
$$491$$ −28.4463 −1.28376 −0.641882 0.766804i $$-0.721846\pi$$
−0.641882 + 0.766804i $$0.721846\pi$$
$$492$$ −7.97320 −0.359460
$$493$$ −7.50685 −0.338092
$$494$$ 0 0
$$495$$ −3.27883 −0.147373
$$496$$ 2.75174 0.123557
$$497$$ 11.6269 0.521536
$$498$$ 16.7237 0.749405
$$499$$ −41.0934 −1.83960 −0.919798 0.392393i $$-0.871647\pi$$
−0.919798 + 0.392393i $$0.871647\pi$$
$$500$$ 11.7107 0.523716
$$501$$ 24.5175 1.09536
$$502$$ 10.6372 0.474761
$$503$$ −39.3325 −1.75375 −0.876874 0.480720i $$-0.840375\pi$$
−0.876874 + 0.480720i $$0.840375\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 2.71318 0.120735
$$506$$ −3.06499 −0.136256
$$507$$ 0 0
$$508$$ −3.83276 −0.170051
$$509$$ −39.3970 −1.74624 −0.873121 0.487504i $$-0.837907\pi$$
−0.873121 + 0.487504i $$0.837907\pi$$
$$510$$ 2.62942 0.116433
$$511$$ −7.48979 −0.331329
$$512$$ 1.00000 0.0441942
$$513$$ 6.98503 0.308397
$$514$$ −2.38334 −0.105125
$$515$$ −13.7941 −0.607841
$$516$$ −3.66462 −0.161326
$$517$$ 8.76621 0.385537
$$518$$ −2.76545 −0.121507
$$519$$ −12.5776 −0.552096
$$520$$ 0 0
$$521$$ −14.1675 −0.620689 −0.310344 0.950624i $$-0.600444\pi$$
−0.310344 + 0.950624i $$0.600444\pi$$
$$522$$ 6.02003 0.263489
$$523$$ −35.5333 −1.55376 −0.776882 0.629647i $$-0.783200\pi$$
−0.776882 + 0.629647i $$0.783200\pi$$
$$524$$ 11.8617 0.518183
$$525$$ −0.553673 −0.0241643
$$526$$ 7.81509 0.340754
$$527$$ −3.43136 −0.149472
$$528$$ 1.55496 0.0676709
$$529$$ −19.1147 −0.831075
$$530$$ −18.5836 −0.807218
$$531$$ 5.40902 0.234731
$$532$$ 6.98503 0.302840
$$533$$ 0 0
$$534$$ 10.9813 0.475205
$$535$$ −5.61599 −0.242801
$$536$$ −1.40756 −0.0607972
$$537$$ 2.52767 0.109077
$$538$$ −7.21556 −0.311085
$$539$$ 1.55496 0.0669768
$$540$$ −2.10863 −0.0907411
$$541$$ −26.1033 −1.12227 −0.561134 0.827725i $$-0.689635\pi$$
−0.561134 + 0.827725i $$0.689635\pi$$
$$542$$ −4.97552 −0.213717
$$543$$ −12.7141 −0.545614
$$544$$ −1.24698 −0.0534638
$$545$$ 9.29850 0.398304
$$546$$ 0 0
$$547$$ 30.0222 1.28366 0.641828 0.766848i $$-0.278176\pi$$
0.641828 + 0.766848i $$0.278176\pi$$
$$548$$ 2.18760 0.0934496
$$549$$ 3.42032 0.145976
$$550$$ −0.860939 −0.0367106
$$551$$ 42.0501 1.79139
$$552$$ −1.97111 −0.0838960
$$553$$ −16.4358 −0.698923
$$554$$ −13.1229 −0.557539
$$555$$ 5.83132 0.247526
$$556$$ 4.14041 0.175592
$$557$$ 2.05456 0.0870546 0.0435273 0.999052i $$-0.486140\pi$$
0.0435273 + 0.999052i $$0.486140\pi$$
$$558$$ 2.75174 0.116490
$$559$$ 0 0
$$560$$ −2.10863 −0.0891059
$$561$$ −1.93900 −0.0818647
$$562$$ −0.105985 −0.00447071
$$563$$ −2.42156 −0.102057 −0.0510283 0.998697i $$-0.516250\pi$$
−0.0510283 + 0.998697i $$0.516250\pi$$
$$564$$ 5.63759 0.237385
$$565$$ 12.2665 0.516054
$$566$$ 29.5509 1.24212
$$567$$ 1.00000 0.0419961
$$568$$ 11.6269 0.487852
$$569$$ 6.71089 0.281335 0.140668 0.990057i $$-0.455075\pi$$
0.140668 + 0.990057i $$0.455075\pi$$
$$570$$ −14.7289 −0.616924
$$571$$ 2.23277 0.0934384 0.0467192 0.998908i $$-0.485123\pi$$
0.0467192 + 0.998908i $$0.485123\pi$$
$$572$$ 0 0
$$573$$ −2.77016 −0.115725
$$574$$ −7.97320 −0.332795
$$575$$ 1.09135 0.0455125
$$576$$ 1.00000 0.0416667
$$577$$ −38.5277 −1.60393 −0.801964 0.597372i $$-0.796212\pi$$
−0.801964 + 0.597372i $$0.796212\pi$$
$$578$$ −15.4450 −0.642429
$$579$$ 22.2930 0.926464
$$580$$ −12.6940 −0.527090
$$581$$ 16.7237 0.693814
$$582$$ −6.39487 −0.265076
$$583$$ 13.7040 0.567561
$$584$$ −7.48979 −0.309930
$$585$$ 0 0
$$586$$ 12.1002 0.499856
$$587$$ 24.0684 0.993409 0.496704 0.867920i $$-0.334543\pi$$
0.496704 + 0.867920i $$0.334543\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 19.2210 0.791986
$$590$$ −11.4056 −0.469562
$$591$$ −8.75440 −0.360108
$$592$$ −2.76545 −0.113659
$$593$$ 18.5679 0.762492 0.381246 0.924474i $$-0.375495\pi$$
0.381246 + 0.924474i $$0.375495\pi$$
$$594$$ 1.55496 0.0638007
$$595$$ 2.62942 0.107796
$$596$$ 16.1634 0.662079
$$597$$ −14.5824 −0.596818
$$598$$ 0 0
$$599$$ 38.4810 1.57229 0.786145 0.618042i $$-0.212074\pi$$
0.786145 + 0.618042i $$0.212074\pi$$
$$600$$ −0.553673 −0.0226036
$$601$$ −46.5198 −1.89758 −0.948791 0.315905i $$-0.897692\pi$$
−0.948791 + 0.315905i $$0.897692\pi$$
$$602$$ −3.66462 −0.149359
$$603$$ −1.40756 −0.0573201
$$604$$ 14.5820 0.593333
$$605$$ 18.0965 0.735727
$$606$$ −1.28670 −0.0522687
$$607$$ −3.32130 −0.134807 −0.0674037 0.997726i $$-0.521472\pi$$
−0.0674037 + 0.997726i $$0.521472\pi$$
$$608$$ 6.98503 0.283280
$$609$$ 6.02003 0.243944
$$610$$ −7.21220 −0.292013
$$611$$ 0 0
$$612$$ −1.24698 −0.0504062
$$613$$ 6.34784 0.256387 0.128193 0.991749i $$-0.459082\pi$$
0.128193 + 0.991749i $$0.459082\pi$$
$$614$$ −11.8071 −0.476496
$$615$$ 16.8125 0.677947
$$616$$ 1.55496 0.0626510
$$617$$ 31.3868 1.26358 0.631792 0.775138i $$-0.282320\pi$$
0.631792 + 0.775138i $$0.282320\pi$$
$$618$$ 6.54173 0.263147
$$619$$ −8.74463 −0.351476 −0.175738 0.984437i $$-0.556231\pi$$
−0.175738 + 0.984437i $$0.556231\pi$$
$$620$$ −5.80240 −0.233030
$$621$$ −1.97111 −0.0790979
$$622$$ 2.36454 0.0948092
$$623$$ 10.9813 0.439955
$$624$$ 0 0
$$625$$ −21.9251 −0.877003
$$626$$ −6.67399 −0.266746
$$627$$ 10.8614 0.433764
$$628$$ 5.49884 0.219428
$$629$$ 3.44846 0.137499
$$630$$ −2.10863 −0.0840099
$$631$$ −37.0302 −1.47415 −0.737074 0.675812i $$-0.763793\pi$$
−0.737074 + 0.675812i $$0.763793\pi$$
$$632$$ −16.4358 −0.653783
$$633$$ 13.1688 0.523411
$$634$$ 0.860447 0.0341727
$$635$$ 8.08188 0.320720
$$636$$ 8.81309 0.349462
$$637$$ 0 0
$$638$$ 9.36089 0.370601
$$639$$ 11.6269 0.459951
$$640$$ −2.10863 −0.0833510
$$641$$ −4.96318 −0.196034 −0.0980169 0.995185i $$-0.531250\pi$$
−0.0980169 + 0.995185i $$0.531250\pi$$
$$642$$ 2.66334 0.105113
$$643$$ 0.0131905 0.000520184 0 0.000260092 1.00000i $$-0.499917\pi$$
0.000260092 1.00000i $$0.499917\pi$$
$$644$$ −1.97111 −0.0776726
$$645$$ 7.72733 0.304263
$$646$$ −8.71019 −0.342698
$$647$$ 16.2940 0.640582 0.320291 0.947319i $$-0.396219\pi$$
0.320291 + 0.947319i $$0.396219\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 8.41079 0.330153
$$650$$ 0 0
$$651$$ 2.75174 0.107849
$$652$$ 14.1960 0.555958
$$653$$ −29.3247 −1.14756 −0.573782 0.819008i $$-0.694524\pi$$
−0.573782 + 0.819008i $$0.694524\pi$$
$$654$$ −4.40973 −0.172434
$$655$$ −25.0121 −0.977302
$$656$$ −7.97320 −0.311301
$$657$$ −7.48979 −0.292204
$$658$$ 5.63759 0.219776
$$659$$ −33.7303 −1.31395 −0.656974 0.753914i $$-0.728164\pi$$
−0.656974 + 0.753914i $$0.728164\pi$$
$$660$$ −3.27883 −0.127628
$$661$$ 11.4642 0.445907 0.222953 0.974829i $$-0.428430\pi$$
0.222953 + 0.974829i $$0.428430\pi$$
$$662$$ 15.4980 0.602348
$$663$$ 0 0
$$664$$ 16.7237 0.649004
$$665$$ −14.7289 −0.571161
$$666$$ −2.76545 −0.107159
$$667$$ −11.8661 −0.459458
$$668$$ 24.5175 0.948612
$$669$$ 4.36466 0.168748
$$670$$ 2.96802 0.114665
$$671$$ 5.31846 0.205317
$$672$$ 1.00000 0.0385758
$$673$$ −16.8424 −0.649227 −0.324614 0.945847i $$-0.605234\pi$$
−0.324614 + 0.945847i $$0.605234\pi$$
$$674$$ −15.6615 −0.603258
$$675$$ −0.553673 −0.0213109
$$676$$ 0 0
$$677$$ −14.5299 −0.558429 −0.279215 0.960229i $$-0.590074\pi$$
−0.279215 + 0.960229i $$0.590074\pi$$
$$678$$ −5.81726 −0.223410
$$679$$ −6.39487 −0.245413
$$680$$ 2.62942 0.100834
$$681$$ 0.618667 0.0237074
$$682$$ 4.27883 0.163845
$$683$$ −0.888021 −0.0339792 −0.0169896 0.999856i $$-0.505408\pi$$
−0.0169896 + 0.999856i $$0.505408\pi$$
$$684$$ 6.98503 0.267079
$$685$$ −4.61284 −0.176248
$$686$$ 1.00000 0.0381802
$$687$$ −2.93024 −0.111796
$$688$$ −3.66462 −0.139712
$$689$$ 0 0
$$690$$ 4.15634 0.158229
$$691$$ −32.1864 −1.22443 −0.612214 0.790692i $$-0.709721\pi$$
−0.612214 + 0.790692i $$0.709721\pi$$
$$692$$ −12.5776 −0.478129
$$693$$ 1.55496 0.0590680
$$694$$ −13.0233 −0.494359
$$695$$ −8.73060 −0.331170
$$696$$ 6.02003 0.228189
$$697$$ 9.94242 0.376596
$$698$$ 26.4749 1.00209
$$699$$ −18.7941 −0.710857
$$700$$ −0.553673 −0.0209269
$$701$$ −29.3407 −1.10818 −0.554091 0.832456i $$-0.686934\pi$$
−0.554091 + 0.832456i $$0.686934\pi$$
$$702$$ 0 0
$$703$$ −19.3168 −0.728546
$$704$$ 1.55496 0.0586047
$$705$$ −11.8876 −0.447713
$$706$$ 21.5831 0.812289
$$707$$ −1.28670 −0.0483914
$$708$$ 5.40902 0.203283
$$709$$ −34.9450 −1.31239 −0.656194 0.754592i $$-0.727835\pi$$
−0.656194 + 0.754592i $$0.727835\pi$$
$$710$$ −24.5167 −0.920097
$$711$$ −16.4358 −0.616392
$$712$$ 10.9813 0.411540
$$713$$ −5.42397 −0.203129
$$714$$ −1.24698 −0.0466670
$$715$$ 0 0
$$716$$ 2.52767 0.0944635
$$717$$ −19.6751 −0.734779
$$718$$ 4.67664 0.174531
$$719$$ −39.7736 −1.48331 −0.741653 0.670784i $$-0.765958\pi$$
−0.741653 + 0.670784i $$0.765958\pi$$
$$720$$ −2.10863 −0.0785841
$$721$$ 6.54173 0.243627
$$722$$ 29.7907 1.10869
$$723$$ −5.57097 −0.207187
$$724$$ −12.7141 −0.472516
$$725$$ −3.33313 −0.123789
$$726$$ −8.58211 −0.318512
$$727$$ 28.3096 1.04995 0.524973 0.851119i $$-0.324075\pi$$
0.524973 + 0.851119i $$0.324075\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 15.7932 0.584533
$$731$$ 4.56971 0.169017
$$732$$ 3.42032 0.126419
$$733$$ 15.4937 0.572272 0.286136 0.958189i $$-0.407629\pi$$
0.286136 + 0.958189i $$0.407629\pi$$
$$734$$ 31.3614 1.15757
$$735$$ −2.10863 −0.0777780
$$736$$ −1.97111 −0.0726561
$$737$$ −2.18869 −0.0806215
$$738$$ −7.97320 −0.293498
$$739$$ 46.7102 1.71826 0.859132 0.511755i $$-0.171004\pi$$
0.859132 + 0.511755i $$0.171004\pi$$
$$740$$ 5.83132 0.214364
$$741$$ 0 0
$$742$$ 8.81309 0.323539
$$743$$ 37.7024 1.38317 0.691584 0.722296i $$-0.256913\pi$$
0.691584 + 0.722296i $$0.256913\pi$$
$$744$$ 2.75174 0.100884
$$745$$ −34.0827 −1.24869
$$746$$ 28.2046 1.03264
$$747$$ 16.7237 0.611887
$$748$$ −1.93900 −0.0708969
$$749$$ 2.66334 0.0973161
$$750$$ 11.7107 0.427612
$$751$$ −30.7925 −1.12364 −0.561818 0.827261i $$-0.689898\pi$$
−0.561818 + 0.827261i $$0.689898\pi$$
$$752$$ 5.63759 0.205582
$$753$$ 10.6372 0.387641
$$754$$ 0 0
$$755$$ −30.7480 −1.11904
$$756$$ 1.00000 0.0363696
$$757$$ 8.21818 0.298695 0.149347 0.988785i $$-0.452283\pi$$
0.149347 + 0.988785i $$0.452283\pi$$
$$758$$ −32.5590 −1.18260
$$759$$ −3.06499 −0.111252
$$760$$ −14.7289 −0.534272
$$761$$ −42.8880 −1.55469 −0.777345 0.629074i $$-0.783434\pi$$
−0.777345 + 0.629074i $$0.783434\pi$$
$$762$$ −3.83276 −0.138846
$$763$$ −4.40973 −0.159643
$$764$$ −2.77016 −0.100221
$$765$$ 2.62942 0.0950669
$$766$$ 8.63110 0.311854
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 49.1724 1.77320 0.886602 0.462534i $$-0.153060\pi$$
0.886602 + 0.462534i $$0.153060\pi$$
$$770$$ −3.27883 −0.118161
$$771$$ −2.38334 −0.0858340
$$772$$ 22.2930 0.802341
$$773$$ −10.3709 −0.373016 −0.186508 0.982453i $$-0.559717\pi$$
−0.186508 + 0.982453i $$0.559717\pi$$
$$774$$ −3.66462 −0.131722
$$775$$ −1.52356 −0.0547280
$$776$$ −6.39487 −0.229563
$$777$$ −2.76545 −0.0992101
$$778$$ −26.3682 −0.945345
$$779$$ −55.6931 −1.99541
$$780$$ 0 0
$$781$$ 18.0793 0.646927
$$782$$ 2.45793 0.0878955
$$783$$ 6.02003 0.215138
$$784$$ 1.00000 0.0357143
$$785$$ −11.5950 −0.413844
$$786$$ 11.8617 0.423095
$$787$$ −23.5512 −0.839509 −0.419754 0.907638i $$-0.637884\pi$$
−0.419754 + 0.907638i $$0.637884\pi$$
$$788$$ −8.75440 −0.311863
$$789$$ 7.81509 0.278224
$$790$$ 34.6571 1.23305
$$791$$ −5.81726 −0.206838
$$792$$ 1.55496 0.0552530
$$793$$ 0 0
$$794$$ −12.0701 −0.428350
$$795$$ −18.5836 −0.659091
$$796$$ −14.5824 −0.516859
$$797$$ −26.7104 −0.946132 −0.473066 0.881027i $$-0.656853\pi$$
−0.473066 + 0.881027i $$0.656853\pi$$
$$798$$ 6.98503 0.247267
$$799$$ −7.02995 −0.248702
$$800$$ −0.553673 −0.0195753
$$801$$ 10.9813 0.388004
$$802$$ −18.5164 −0.653836
$$803$$ −11.6463 −0.410989
$$804$$ −1.40756 −0.0496407
$$805$$ 4.15634 0.146492
$$806$$ 0 0
$$807$$ −7.21556 −0.254000
$$808$$ −1.28670 −0.0452660
$$809$$ −3.80663 −0.133834 −0.0669169 0.997759i $$-0.521316\pi$$
−0.0669169 + 0.997759i $$0.521316\pi$$
$$810$$ −2.10863 −0.0740898
$$811$$ −0.849234 −0.0298206 −0.0149103 0.999889i $$-0.504746\pi$$
−0.0149103 + 0.999889i $$0.504746\pi$$
$$812$$ 6.02003 0.211262
$$813$$ −4.97552 −0.174499
$$814$$ −4.30016 −0.150721
$$815$$ −29.9341 −1.04855
$$816$$ −1.24698 −0.0436530
$$817$$ −25.5975 −0.895543
$$818$$ −35.8394 −1.25310
$$819$$ 0 0
$$820$$ 16.8125 0.587120
$$821$$ 20.0311 0.699090 0.349545 0.936919i $$-0.386336\pi$$
0.349545 + 0.936919i $$0.386336\pi$$
$$822$$ 2.18760 0.0763013
$$823$$ 36.6174 1.27640 0.638201 0.769870i $$-0.279679\pi$$
0.638201 + 0.769870i $$0.279679\pi$$
$$824$$ 6.54173 0.227892
$$825$$ −0.860939 −0.0299740
$$826$$ 5.40902 0.188204
$$827$$ −14.3761 −0.499907 −0.249953 0.968258i $$-0.580415\pi$$
−0.249953 + 0.968258i $$0.580415\pi$$
$$828$$ −1.97111 −0.0685008
$$829$$ −6.10322 −0.211974 −0.105987 0.994368i $$-0.533800\pi$$
−0.105987 + 0.994368i $$0.533800\pi$$
$$830$$ −35.2640 −1.22403
$$831$$ −13.1229 −0.455229
$$832$$ 0 0
$$833$$ −1.24698 −0.0432053
$$834$$ 4.14041 0.143371
$$835$$ −51.6984 −1.78910
$$836$$ 10.8614 0.375650
$$837$$ 2.75174 0.0951139
$$838$$ 27.0079 0.932971
$$839$$ −53.2223 −1.83744 −0.918718 0.394914i $$-0.870774\pi$$
−0.918718 + 0.394914i $$0.870774\pi$$
$$840$$ −2.10863 −0.0727547
$$841$$ 7.24072 0.249680
$$842$$ −26.2859 −0.905873
$$843$$ −0.105985 −0.00365032
$$844$$ 13.1688 0.453287
$$845$$ 0 0
$$846$$ 5.63759 0.193824
$$847$$ −8.58211 −0.294885
$$848$$ 8.81309 0.302643
$$849$$ 29.5509 1.01418
$$850$$ 0.690419 0.0236812
$$851$$ 5.45101 0.186858
$$852$$ 11.6269 0.398329
$$853$$ 13.3932 0.458573 0.229287 0.973359i $$-0.426361\pi$$
0.229287 + 0.973359i $$0.426361\pi$$
$$854$$ 3.42032 0.117041
$$855$$ −14.7289 −0.503716
$$856$$ 2.66334 0.0910309
$$857$$ −3.82155 −0.130541 −0.0652707 0.997868i $$-0.520791\pi$$
−0.0652707 + 0.997868i $$0.520791\pi$$
$$858$$ 0 0
$$859$$ −44.8782 −1.53122 −0.765612 0.643302i $$-0.777564\pi$$
−0.765612 + 0.643302i $$0.777564\pi$$
$$860$$ 7.72733 0.263500
$$861$$ −7.97320 −0.271726
$$862$$ −6.58585 −0.224315
$$863$$ 8.69475 0.295973 0.147986 0.988989i $$-0.452721\pi$$
0.147986 + 0.988989i $$0.452721\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 26.5216 0.901760
$$866$$ 25.4472 0.864731
$$867$$ −15.4450 −0.524541
$$868$$ 2.75174 0.0934000
$$869$$ −25.5571 −0.866964
$$870$$ −12.6940 −0.430368
$$871$$ 0 0
$$872$$ −4.40973 −0.149332
$$873$$ −6.39487 −0.216434
$$874$$ −13.7683 −0.465718
$$875$$ 11.7107 0.395892
$$876$$ −7.48979 −0.253056
$$877$$ 21.9730 0.741976 0.370988 0.928638i $$-0.379019\pi$$
0.370988 + 0.928638i $$0.379019\pi$$
$$878$$ 23.8445 0.804712
$$879$$ 12.1002 0.408131
$$880$$ −3.27883 −0.110529
$$881$$ −29.7224 −1.00137 −0.500687 0.865628i $$-0.666919\pi$$
−0.500687 + 0.865628i $$0.666919\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 35.0900 1.18087 0.590436 0.807084i $$-0.298956\pi$$
0.590436 + 0.807084i $$0.298956\pi$$
$$884$$ 0 0
$$885$$ −11.4056 −0.383396
$$886$$ −0.966959 −0.0324856
$$887$$ 25.4402 0.854200 0.427100 0.904204i $$-0.359535\pi$$
0.427100 + 0.904204i $$0.359535\pi$$
$$888$$ −2.76545 −0.0928025
$$889$$ −3.83276 −0.128547
$$890$$ −23.1554 −0.776172
$$891$$ 1.55496 0.0520931
$$892$$ 4.36466 0.146140
$$893$$ 39.3787 1.31776
$$894$$ 16.1634 0.540586
$$895$$ −5.32992 −0.178160
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −32.7563 −1.09309
$$899$$ 16.5655 0.552491
$$900$$ −0.553673 −0.0184558
$$901$$ −10.9897 −0.366121
$$902$$ −12.3980 −0.412808
$$903$$ −3.66462 −0.121951
$$904$$ −5.81726 −0.193479
$$905$$ 26.8093 0.891172
$$906$$ 14.5820 0.484454
$$907$$ 52.5340 1.74436 0.872181 0.489183i $$-0.162705\pi$$
0.872181 + 0.489183i $$0.162705\pi$$
$$908$$ 0.618667 0.0205312
$$909$$ −1.28670 −0.0426772
$$910$$ 0 0
$$911$$ −15.7819 −0.522879 −0.261439 0.965220i $$-0.584197\pi$$
−0.261439 + 0.965220i $$0.584197\pi$$
$$912$$ 6.98503 0.231298
$$913$$ 26.0046 0.860626
$$914$$ 2.38368 0.0788452
$$915$$ −7.21220 −0.238428
$$916$$ −2.93024 −0.0968180
$$917$$ 11.8617 0.391709
$$918$$ −1.24698 −0.0411565
$$919$$ −49.8498 −1.64439 −0.822197 0.569203i $$-0.807252\pi$$
−0.822197 + 0.569203i $$0.807252\pi$$
$$920$$ 4.15634 0.137031
$$921$$ −11.8071 −0.389057
$$922$$ 36.9292 1.21620
$$923$$ 0 0
$$924$$ 1.55496 0.0511544
$$925$$ 1.53116 0.0503442
$$926$$ −29.8161 −0.979818
$$927$$ 6.54173 0.214859
$$928$$ 6.02003 0.197617
$$929$$ −19.1901 −0.629606 −0.314803 0.949157i $$-0.601938\pi$$
−0.314803 + 0.949157i $$0.601938\pi$$
$$930$$ −5.80240 −0.190268
$$931$$ 6.98503 0.228925
$$932$$ −18.7941 −0.615620
$$933$$ 2.36454 0.0774114
$$934$$ 9.34853 0.305893
$$935$$ 4.08864 0.133713
$$936$$ 0 0
$$937$$ −23.4063 −0.764650 −0.382325 0.924028i $$-0.624876\pi$$
−0.382325 + 0.924028i $$0.624876\pi$$
$$938$$ −1.40756 −0.0459584
$$939$$ −6.67399 −0.217797
$$940$$ −11.8876 −0.387730
$$941$$ 3.86671 0.126051 0.0630256 0.998012i $$-0.479925\pi$$
0.0630256 + 0.998012i $$0.479925\pi$$
$$942$$ 5.49884 0.179162
$$943$$ 15.7161 0.511785
$$944$$ 5.40902 0.176048
$$945$$ −2.10863 −0.0685938
$$946$$ −5.69833 −0.185269
$$947$$ −48.4114 −1.57316 −0.786579 0.617489i $$-0.788150\pi$$
−0.786579 + 0.617489i $$0.788150\pi$$
$$948$$ −16.4358 −0.533811
$$949$$ 0 0
$$950$$ −3.86743 −0.125476
$$951$$ 0.860447 0.0279019
$$952$$ −1.24698 −0.0404148
$$953$$ 22.6556 0.733885 0.366943 0.930244i $$-0.380405\pi$$
0.366943 + 0.930244i $$0.380405\pi$$
$$954$$ 8.81309 0.285334
$$955$$ 5.84125 0.189018
$$956$$ −19.6751 −0.636337
$$957$$ 9.36089 0.302595
$$958$$ 17.5839 0.568110
$$959$$ 2.18760 0.0706413
$$960$$ −2.10863 −0.0680558
$$961$$ −23.4280 −0.755740
$$962$$ 0 0
$$963$$ 2.66334 0.0858248
$$964$$ −5.57097 −0.179429
$$965$$ −47.0077 −1.51323
$$966$$ −1.97111 −0.0634194
$$967$$ −19.7448 −0.634950 −0.317475 0.948267i $$-0.602835\pi$$
−0.317475 + 0.948267i $$0.602835\pi$$
$$968$$ −8.58211 −0.275839
$$969$$ −8.71019 −0.279812
$$970$$ 13.4844 0.432959
$$971$$ −12.7120 −0.407947 −0.203974 0.978976i $$-0.565386\pi$$
−0.203974 + 0.978976i $$0.565386\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 4.14041 0.132735
$$974$$ 35.9216 1.15100
$$975$$ 0 0
$$976$$ 3.42032 0.109482
$$977$$ −41.1062 −1.31510 −0.657552 0.753409i $$-0.728408\pi$$
−0.657552 + 0.753409i $$0.728408\pi$$
$$978$$ 14.1960 0.453938
$$979$$ 17.0754 0.545732
$$980$$ −2.10863 −0.0673578
$$981$$ −4.40973 −0.140792
$$982$$ −28.4463 −0.907758
$$983$$ 12.6710 0.404144 0.202072 0.979371i $$-0.435233\pi$$
0.202072 + 0.979371i $$0.435233\pi$$
$$984$$ −7.97320 −0.254176
$$985$$ 18.4598 0.588178
$$986$$ −7.50685 −0.239067
$$987$$ 5.63759 0.179446
$$988$$ 0 0
$$989$$ 7.22337 0.229690
$$990$$ −3.27883 −0.104208
$$991$$ 42.4770 1.34932 0.674662 0.738126i $$-0.264289\pi$$
0.674662 + 0.738126i $$0.264289\pi$$
$$992$$ 2.75174 0.0873677
$$993$$ 15.4980 0.491815
$$994$$ 11.6269 0.368781
$$995$$ 30.7489 0.974806
$$996$$ 16.7237 0.529909
$$997$$ 8.78235 0.278140 0.139070 0.990283i $$-0.455589\pi$$
0.139070 + 0.990283i $$0.455589\pi$$
$$998$$ −41.0934 −1.30079
$$999$$ −2.76545 −0.0874951
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.ct.1.2 yes 6
13.12 even 2 7098.2.a.cr.1.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cr.1.5 6 13.12 even 2
7098.2.a.ct.1.2 yes 6 1.1 even 1 trivial