Properties

 Label 2151.2.a.j.1.6 Level $2151$ Weight $2$ Character 2151.1 Self dual yes Analytic conductor $17.176$ Analytic rank $1$ Dimension $20$ CM no Inner twists $1$

Learn more about

Newspace parameters

 Level: $$N$$ $$=$$ $$2151 = 3^{2} \cdot 239$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2151.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$17.1758214748$$ Analytic rank: $$1$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.6 Root $$1.61701$$ of defining polynomial Character $$\chi$$ $$=$$ 2151.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.61701 q^{2} +0.614711 q^{4} -1.46591 q^{5} +3.83563 q^{7} +2.24002 q^{8} +O(q^{10})$$ $$q-1.61701 q^{2} +0.614711 q^{4} -1.46591 q^{5} +3.83563 q^{7} +2.24002 q^{8} +2.37038 q^{10} -2.90374 q^{11} +2.33112 q^{13} -6.20223 q^{14} -4.85155 q^{16} -3.77139 q^{17} -1.21740 q^{19} -0.901109 q^{20} +4.69537 q^{22} +1.47665 q^{23} -2.85112 q^{25} -3.76944 q^{26} +2.35780 q^{28} -1.02391 q^{29} -1.72030 q^{31} +3.36495 q^{32} +6.09837 q^{34} -5.62267 q^{35} -1.08984 q^{37} +1.96855 q^{38} -3.28366 q^{40} +1.58037 q^{41} +0.433251 q^{43} -1.78496 q^{44} -2.38776 q^{46} -7.65919 q^{47} +7.71202 q^{49} +4.61028 q^{50} +1.43297 q^{52} -1.07471 q^{53} +4.25661 q^{55} +8.59188 q^{56} +1.65567 q^{58} +4.82708 q^{59} +10.7610 q^{61} +2.78174 q^{62} +4.26196 q^{64} -3.41720 q^{65} -6.11056 q^{67} -2.31832 q^{68} +9.09189 q^{70} -10.9308 q^{71} -12.1327 q^{73} +1.76228 q^{74} -0.748351 q^{76} -11.1377 q^{77} +14.7398 q^{79} +7.11192 q^{80} -2.55547 q^{82} +13.0905 q^{83} +5.52851 q^{85} -0.700570 q^{86} -6.50444 q^{88} +6.08868 q^{89} +8.94131 q^{91} +0.907716 q^{92} +12.3850 q^{94} +1.78460 q^{95} +7.99478 q^{97} -12.4704 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10})$$ $$20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.61701 −1.14340 −0.571698 0.820464i $$-0.693715\pi$$
−0.571698 + 0.820464i $$0.693715\pi$$
$$3$$ 0 0
$$4$$ 0.614711 0.307356
$$5$$ −1.46591 −0.655573 −0.327787 0.944752i $$-0.606303\pi$$
−0.327787 + 0.944752i $$0.606303\pi$$
$$6$$ 0 0
$$7$$ 3.83563 1.44973 0.724865 0.688891i $$-0.241902\pi$$
0.724865 + 0.688891i $$0.241902\pi$$
$$8$$ 2.24002 0.791967
$$9$$ 0 0
$$10$$ 2.37038 0.749580
$$11$$ −2.90374 −0.875511 −0.437755 0.899094i $$-0.644226\pi$$
−0.437755 + 0.899094i $$0.644226\pi$$
$$12$$ 0 0
$$13$$ 2.33112 0.646537 0.323268 0.946307i $$-0.395218\pi$$
0.323268 + 0.946307i $$0.395218\pi$$
$$14$$ −6.20223 −1.65762
$$15$$ 0 0
$$16$$ −4.85155 −1.21289
$$17$$ −3.77139 −0.914697 −0.457348 0.889288i $$-0.651201\pi$$
−0.457348 + 0.889288i $$0.651201\pi$$
$$18$$ 0 0
$$19$$ −1.21740 −0.279291 −0.139646 0.990202i $$-0.544596\pi$$
−0.139646 + 0.990202i $$0.544596\pi$$
$$20$$ −0.901109 −0.201494
$$21$$ 0 0
$$22$$ 4.69537 1.00106
$$23$$ 1.47665 0.307904 0.153952 0.988078i $$-0.450800\pi$$
0.153952 + 0.988078i $$0.450800\pi$$
$$24$$ 0 0
$$25$$ −2.85112 −0.570224
$$26$$ −3.76944 −0.739248
$$27$$ 0 0
$$28$$ 2.35780 0.445583
$$29$$ −1.02391 −0.190136 −0.0950678 0.995471i $$-0.530307\pi$$
−0.0950678 + 0.995471i $$0.530307\pi$$
$$30$$ 0 0
$$31$$ −1.72030 −0.308975 −0.154487 0.987995i $$-0.549373\pi$$
−0.154487 + 0.987995i $$0.549373\pi$$
$$32$$ 3.36495 0.594845
$$33$$ 0 0
$$34$$ 6.09837 1.04586
$$35$$ −5.62267 −0.950404
$$36$$ 0 0
$$37$$ −1.08984 −0.179169 −0.0895845 0.995979i $$-0.528554\pi$$
−0.0895845 + 0.995979i $$0.528554\pi$$
$$38$$ 1.96855 0.319341
$$39$$ 0 0
$$40$$ −3.28366 −0.519192
$$41$$ 1.58037 0.246812 0.123406 0.992356i $$-0.460618\pi$$
0.123406 + 0.992356i $$0.460618\pi$$
$$42$$ 0 0
$$43$$ 0.433251 0.0660702 0.0330351 0.999454i $$-0.489483\pi$$
0.0330351 + 0.999454i $$0.489483\pi$$
$$44$$ −1.78496 −0.269093
$$45$$ 0 0
$$46$$ −2.38776 −0.352056
$$47$$ −7.65919 −1.11721 −0.558604 0.829434i $$-0.688663\pi$$
−0.558604 + 0.829434i $$0.688663\pi$$
$$48$$ 0 0
$$49$$ 7.71202 1.10172
$$50$$ 4.61028 0.651992
$$51$$ 0 0
$$52$$ 1.43297 0.198717
$$53$$ −1.07471 −0.147623 −0.0738117 0.997272i $$-0.523516\pi$$
−0.0738117 + 0.997272i $$0.523516\pi$$
$$54$$ 0 0
$$55$$ 4.25661 0.573961
$$56$$ 8.59188 1.14814
$$57$$ 0 0
$$58$$ 1.65567 0.217400
$$59$$ 4.82708 0.628432 0.314216 0.949351i $$-0.398258\pi$$
0.314216 + 0.949351i $$0.398258\pi$$
$$60$$ 0 0
$$61$$ 10.7610 1.37780 0.688902 0.724854i $$-0.258093\pi$$
0.688902 + 0.724854i $$0.258093\pi$$
$$62$$ 2.78174 0.353281
$$63$$ 0 0
$$64$$ 4.26196 0.532745
$$65$$ −3.41720 −0.423852
$$66$$ 0 0
$$67$$ −6.11056 −0.746523 −0.373262 0.927726i $$-0.621761\pi$$
−0.373262 + 0.927726i $$0.621761\pi$$
$$68$$ −2.31832 −0.281137
$$69$$ 0 0
$$70$$ 9.09189 1.08669
$$71$$ −10.9308 −1.29725 −0.648626 0.761108i $$-0.724656\pi$$
−0.648626 + 0.761108i $$0.724656\pi$$
$$72$$ 0 0
$$73$$ −12.1327 −1.42003 −0.710015 0.704187i $$-0.751312\pi$$
−0.710015 + 0.704187i $$0.751312\pi$$
$$74$$ 1.76228 0.204861
$$75$$ 0 0
$$76$$ −0.748351 −0.0858418
$$77$$ −11.1377 −1.26925
$$78$$ 0 0
$$79$$ 14.7398 1.65835 0.829177 0.558986i $$-0.188810\pi$$
0.829177 + 0.558986i $$0.188810\pi$$
$$80$$ 7.11192 0.795137
$$81$$ 0 0
$$82$$ −2.55547 −0.282204
$$83$$ 13.0905 1.43686 0.718432 0.695597i $$-0.244860\pi$$
0.718432 + 0.695597i $$0.244860\pi$$
$$84$$ 0 0
$$85$$ 5.52851 0.599651
$$86$$ −0.700570 −0.0755444
$$87$$ 0 0
$$88$$ −6.50444 −0.693376
$$89$$ 6.08868 0.645398 0.322699 0.946502i $$-0.395410\pi$$
0.322699 + 0.946502i $$0.395410\pi$$
$$90$$ 0 0
$$91$$ 8.94131 0.937304
$$92$$ 0.907716 0.0946359
$$93$$ 0 0
$$94$$ 12.3850 1.27741
$$95$$ 1.78460 0.183096
$$96$$ 0 0
$$97$$ 7.99478 0.811747 0.405873 0.913929i $$-0.366967\pi$$
0.405873 + 0.913929i $$0.366967\pi$$
$$98$$ −12.4704 −1.25970
$$99$$ 0 0
$$100$$ −1.75261 −0.175261
$$101$$ −12.4894 −1.24274 −0.621371 0.783516i $$-0.713424\pi$$
−0.621371 + 0.783516i $$0.713424\pi$$
$$102$$ 0 0
$$103$$ −6.88824 −0.678719 −0.339359 0.940657i $$-0.610210\pi$$
−0.339359 + 0.940657i $$0.610210\pi$$
$$104$$ 5.22176 0.512036
$$105$$ 0 0
$$106$$ 1.73782 0.168792
$$107$$ −16.7073 −1.61515 −0.807577 0.589763i $$-0.799221\pi$$
−0.807577 + 0.589763i $$0.799221\pi$$
$$108$$ 0 0
$$109$$ −15.9013 −1.52307 −0.761534 0.648125i $$-0.775554\pi$$
−0.761534 + 0.648125i $$0.775554\pi$$
$$110$$ −6.88297 −0.656265
$$111$$ 0 0
$$112$$ −18.6087 −1.75836
$$113$$ −9.87403 −0.928870 −0.464435 0.885607i $$-0.653743\pi$$
−0.464435 + 0.885607i $$0.653743\pi$$
$$114$$ 0 0
$$115$$ −2.16464 −0.201853
$$116$$ −0.629410 −0.0584393
$$117$$ 0 0
$$118$$ −7.80542 −0.718548
$$119$$ −14.4656 −1.32606
$$120$$ 0 0
$$121$$ −2.56829 −0.233481
$$122$$ −17.4006 −1.57538
$$123$$ 0 0
$$124$$ −1.05749 −0.0949652
$$125$$ 11.5090 1.02940
$$126$$ 0 0
$$127$$ −10.3172 −0.915505 −0.457753 0.889080i $$-0.651345\pi$$
−0.457753 + 0.889080i $$0.651345\pi$$
$$128$$ −13.6215 −1.20398
$$129$$ 0 0
$$130$$ 5.52564 0.484631
$$131$$ 9.86768 0.862143 0.431072 0.902318i $$-0.358136\pi$$
0.431072 + 0.902318i $$0.358136\pi$$
$$132$$ 0 0
$$133$$ −4.66950 −0.404897
$$134$$ 9.88081 0.853572
$$135$$ 0 0
$$136$$ −8.44800 −0.724410
$$137$$ −3.24609 −0.277332 −0.138666 0.990339i $$-0.544282\pi$$
−0.138666 + 0.990339i $$0.544282\pi$$
$$138$$ 0 0
$$139$$ −14.2318 −1.20713 −0.603564 0.797315i $$-0.706253\pi$$
−0.603564 + 0.797315i $$0.706253\pi$$
$$140$$ −3.45632 −0.292112
$$141$$ 0 0
$$142$$ 17.6752 1.48327
$$143$$ −6.76897 −0.566050
$$144$$ 0 0
$$145$$ 1.50096 0.124648
$$146$$ 19.6187 1.62366
$$147$$ 0 0
$$148$$ −0.669938 −0.0550686
$$149$$ −3.90332 −0.319772 −0.159886 0.987135i $$-0.551113\pi$$
−0.159886 + 0.987135i $$0.551113\pi$$
$$150$$ 0 0
$$151$$ −0.699764 −0.0569460 −0.0284730 0.999595i $$-0.509064\pi$$
−0.0284730 + 0.999595i $$0.509064\pi$$
$$152$$ −2.72701 −0.221190
$$153$$ 0 0
$$154$$ 18.0097 1.45126
$$155$$ 2.52180 0.202556
$$156$$ 0 0
$$157$$ 1.91363 0.152725 0.0763623 0.997080i $$-0.475669\pi$$
0.0763623 + 0.997080i $$0.475669\pi$$
$$158$$ −23.8343 −1.89616
$$159$$ 0 0
$$160$$ −4.93270 −0.389964
$$161$$ 5.66389 0.446377
$$162$$ 0 0
$$163$$ −17.1905 −1.34647 −0.673233 0.739430i $$-0.735095\pi$$
−0.673233 + 0.739430i $$0.735095\pi$$
$$164$$ 0.971471 0.0758592
$$165$$ 0 0
$$166$$ −21.1673 −1.64290
$$167$$ −14.8924 −1.15241 −0.576205 0.817305i $$-0.695467\pi$$
−0.576205 + 0.817305i $$0.695467\pi$$
$$168$$ 0 0
$$169$$ −7.56588 −0.581990
$$170$$ −8.93963 −0.685639
$$171$$ 0 0
$$172$$ 0.266324 0.0203070
$$173$$ 11.7169 0.890823 0.445411 0.895326i $$-0.353057\pi$$
0.445411 + 0.895326i $$0.353057\pi$$
$$174$$ 0 0
$$175$$ −10.9358 −0.826671
$$176$$ 14.0876 1.06190
$$177$$ 0 0
$$178$$ −9.84543 −0.737946
$$179$$ 3.36761 0.251707 0.125854 0.992049i $$-0.459833\pi$$
0.125854 + 0.992049i $$0.459833\pi$$
$$180$$ 0 0
$$181$$ −6.66617 −0.495493 −0.247746 0.968825i $$-0.579690\pi$$
−0.247746 + 0.968825i $$0.579690\pi$$
$$182$$ −14.4582 −1.07171
$$183$$ 0 0
$$184$$ 3.30774 0.243850
$$185$$ 1.59761 0.117458
$$186$$ 0 0
$$187$$ 10.9511 0.800827
$$188$$ −4.70819 −0.343380
$$189$$ 0 0
$$190$$ −2.88571 −0.209351
$$191$$ 15.7911 1.14260 0.571300 0.820741i $$-0.306439\pi$$
0.571300 + 0.820741i $$0.306439\pi$$
$$192$$ 0 0
$$193$$ 8.45917 0.608904 0.304452 0.952528i $$-0.401527\pi$$
0.304452 + 0.952528i $$0.401527\pi$$
$$194$$ −12.9276 −0.928149
$$195$$ 0 0
$$196$$ 4.74066 0.338619
$$197$$ 4.68346 0.333683 0.166842 0.985984i $$-0.446643\pi$$
0.166842 + 0.985984i $$0.446643\pi$$
$$198$$ 0 0
$$199$$ −25.8363 −1.83149 −0.915743 0.401764i $$-0.868397\pi$$
−0.915743 + 0.401764i $$0.868397\pi$$
$$200$$ −6.38657 −0.451599
$$201$$ 0 0
$$202$$ 20.1955 1.42095
$$203$$ −3.92734 −0.275645
$$204$$ 0 0
$$205$$ −2.31667 −0.161804
$$206$$ 11.1383 0.776044
$$207$$ 0 0
$$208$$ −11.3096 −0.784177
$$209$$ 3.53502 0.244523
$$210$$ 0 0
$$211$$ 4.07485 0.280524 0.140262 0.990114i $$-0.455205\pi$$
0.140262 + 0.990114i $$0.455205\pi$$
$$212$$ −0.660639 −0.0453729
$$213$$ 0 0
$$214$$ 27.0158 1.84676
$$215$$ −0.635106 −0.0433138
$$216$$ 0 0
$$217$$ −6.59843 −0.447930
$$218$$ 25.7125 1.74147
$$219$$ 0 0
$$220$$ 2.61659 0.176410
$$221$$ −8.79157 −0.591385
$$222$$ 0 0
$$223$$ −2.44657 −0.163835 −0.0819173 0.996639i $$-0.526104\pi$$
−0.0819173 + 0.996639i $$0.526104\pi$$
$$224$$ 12.9067 0.862364
$$225$$ 0 0
$$226$$ 15.9664 1.06207
$$227$$ −2.74547 −0.182223 −0.0911115 0.995841i $$-0.529042\pi$$
−0.0911115 + 0.995841i $$0.529042\pi$$
$$228$$ 0 0
$$229$$ 13.7145 0.906277 0.453139 0.891440i $$-0.350304\pi$$
0.453139 + 0.891440i $$0.350304\pi$$
$$230$$ 3.50023 0.230798
$$231$$ 0 0
$$232$$ −2.29358 −0.150581
$$233$$ 5.36762 0.351645 0.175822 0.984422i $$-0.443742\pi$$
0.175822 + 0.984422i $$0.443742\pi$$
$$234$$ 0 0
$$235$$ 11.2277 0.732412
$$236$$ 2.96726 0.193152
$$237$$ 0 0
$$238$$ 23.3910 1.51622
$$239$$ −1.00000 −0.0646846
$$240$$ 0 0
$$241$$ −12.9180 −0.832120 −0.416060 0.909337i $$-0.636589\pi$$
−0.416060 + 0.909337i $$0.636589\pi$$
$$242$$ 4.15295 0.266962
$$243$$ 0 0
$$244$$ 6.61491 0.423476
$$245$$ −11.3051 −0.722256
$$246$$ 0 0
$$247$$ −2.83791 −0.180572
$$248$$ −3.85351 −0.244698
$$249$$ 0 0
$$250$$ −18.6101 −1.17701
$$251$$ −18.9396 −1.19546 −0.597729 0.801698i $$-0.703930\pi$$
−0.597729 + 0.801698i $$0.703930\pi$$
$$252$$ 0 0
$$253$$ −4.28782 −0.269573
$$254$$ 16.6830 1.04679
$$255$$ 0 0
$$256$$ 13.5022 0.843886
$$257$$ 15.3234 0.955847 0.477923 0.878402i $$-0.341390\pi$$
0.477923 + 0.878402i $$0.341390\pi$$
$$258$$ 0 0
$$259$$ −4.18023 −0.259747
$$260$$ −2.10059 −0.130273
$$261$$ 0 0
$$262$$ −15.9561 −0.985771
$$263$$ −27.1435 −1.67374 −0.836870 0.547402i $$-0.815617\pi$$
−0.836870 + 0.547402i $$0.815617\pi$$
$$264$$ 0 0
$$265$$ 1.57543 0.0967779
$$266$$ 7.55062 0.462958
$$267$$ 0 0
$$268$$ −3.75623 −0.229448
$$269$$ −12.7939 −0.780058 −0.390029 0.920802i $$-0.627535\pi$$
−0.390029 + 0.920802i $$0.627535\pi$$
$$270$$ 0 0
$$271$$ −28.4970 −1.73107 −0.865533 0.500851i $$-0.833020\pi$$
−0.865533 + 0.500851i $$0.833020\pi$$
$$272$$ 18.2971 1.10943
$$273$$ 0 0
$$274$$ 5.24895 0.317101
$$275$$ 8.27891 0.499237
$$276$$ 0 0
$$277$$ −2.41529 −0.145120 −0.0725602 0.997364i $$-0.523117\pi$$
−0.0725602 + 0.997364i $$0.523117\pi$$
$$278$$ 23.0130 1.38023
$$279$$ 0 0
$$280$$ −12.5949 −0.752689
$$281$$ 8.27813 0.493832 0.246916 0.969037i $$-0.420583\pi$$
0.246916 + 0.969037i $$0.420583\pi$$
$$282$$ 0 0
$$283$$ −0.142683 −0.00848163 −0.00424082 0.999991i $$-0.501350\pi$$
−0.00424082 + 0.999991i $$0.501350\pi$$
$$284$$ −6.71930 −0.398717
$$285$$ 0 0
$$286$$ 10.9455 0.647219
$$287$$ 6.06171 0.357811
$$288$$ 0 0
$$289$$ −2.77660 −0.163329
$$290$$ −2.42706 −0.142522
$$291$$ 0 0
$$292$$ −7.45813 −0.436454
$$293$$ −12.9861 −0.758657 −0.379328 0.925262i $$-0.623845\pi$$
−0.379328 + 0.925262i $$0.623845\pi$$
$$294$$ 0 0
$$295$$ −7.07605 −0.411983
$$296$$ −2.44127 −0.141896
$$297$$ 0 0
$$298$$ 6.31169 0.365627
$$299$$ 3.44226 0.199071
$$300$$ 0 0
$$301$$ 1.66179 0.0957839
$$302$$ 1.13152 0.0651119
$$303$$ 0 0
$$304$$ 5.90629 0.338749
$$305$$ −15.7746 −0.903252
$$306$$ 0 0
$$307$$ 7.69912 0.439412 0.219706 0.975566i $$-0.429490\pi$$
0.219706 + 0.975566i $$0.429490\pi$$
$$308$$ −6.84644 −0.390112
$$309$$ 0 0
$$310$$ −4.07776 −0.231601
$$311$$ 0.0557085 0.00315894 0.00157947 0.999999i $$-0.499497\pi$$
0.00157947 + 0.999999i $$0.499497\pi$$
$$312$$ 0 0
$$313$$ −2.73407 −0.154538 −0.0772692 0.997010i $$-0.524620\pi$$
−0.0772692 + 0.997010i $$0.524620\pi$$
$$314$$ −3.09436 −0.174625
$$315$$ 0 0
$$316$$ 9.06070 0.509704
$$317$$ −5.74313 −0.322566 −0.161283 0.986908i $$-0.551563\pi$$
−0.161283 + 0.986908i $$0.551563\pi$$
$$318$$ 0 0
$$319$$ 2.97317 0.166466
$$320$$ −6.24763 −0.349253
$$321$$ 0 0
$$322$$ −9.15855 −0.510386
$$323$$ 4.59130 0.255467
$$324$$ 0 0
$$325$$ −6.64630 −0.368671
$$326$$ 27.7972 1.53954
$$327$$ 0 0
$$328$$ 3.54006 0.195467
$$329$$ −29.3778 −1.61965
$$330$$ 0 0
$$331$$ −32.3302 −1.77703 −0.888514 0.458849i $$-0.848262\pi$$
−0.888514 + 0.458849i $$0.848262\pi$$
$$332$$ 8.04685 0.441628
$$333$$ 0 0
$$334$$ 24.0811 1.31766
$$335$$ 8.95750 0.489401
$$336$$ 0 0
$$337$$ −4.91558 −0.267769 −0.133884 0.990997i $$-0.542745\pi$$
−0.133884 + 0.990997i $$0.542745\pi$$
$$338$$ 12.2341 0.665446
$$339$$ 0 0
$$340$$ 3.39843 0.184306
$$341$$ 4.99530 0.270511
$$342$$ 0 0
$$343$$ 2.73104 0.147462
$$344$$ 0.970492 0.0523254
$$345$$ 0 0
$$346$$ −18.9464 −1.01856
$$347$$ 28.7741 1.54467 0.772337 0.635213i $$-0.219088\pi$$
0.772337 + 0.635213i $$0.219088\pi$$
$$348$$ 0 0
$$349$$ −12.6879 −0.679168 −0.339584 0.940576i $$-0.610286\pi$$
−0.339584 + 0.940576i $$0.610286\pi$$
$$350$$ 17.6833 0.945212
$$351$$ 0 0
$$352$$ −9.77094 −0.520793
$$353$$ 6.88724 0.366571 0.183285 0.983060i $$-0.441327\pi$$
0.183285 + 0.983060i $$0.441327\pi$$
$$354$$ 0 0
$$355$$ 16.0236 0.850443
$$356$$ 3.74278 0.198367
$$357$$ 0 0
$$358$$ −5.44545 −0.287801
$$359$$ 4.46094 0.235439 0.117720 0.993047i $$-0.462442\pi$$
0.117720 + 0.993047i $$0.462442\pi$$
$$360$$ 0 0
$$361$$ −17.5179 −0.921996
$$362$$ 10.7792 0.566545
$$363$$ 0 0
$$364$$ 5.49632 0.288085
$$365$$ 17.7855 0.930934
$$366$$ 0 0
$$367$$ 4.13856 0.216031 0.108016 0.994149i $$-0.465550\pi$$
0.108016 + 0.994149i $$0.465550\pi$$
$$368$$ −7.16406 −0.373453
$$369$$ 0 0
$$370$$ −2.58334 −0.134302
$$371$$ −4.12220 −0.214014
$$372$$ 0 0
$$373$$ −15.9172 −0.824161 −0.412080 0.911148i $$-0.635198\pi$$
−0.412080 + 0.911148i $$0.635198\pi$$
$$374$$ −17.7081 −0.915663
$$375$$ 0 0
$$376$$ −17.1568 −0.884792
$$377$$ −2.38686 −0.122930
$$378$$ 0 0
$$379$$ 35.7486 1.83628 0.918142 0.396252i $$-0.129689\pi$$
0.918142 + 0.396252i $$0.129689\pi$$
$$380$$ 1.09701 0.0562756
$$381$$ 0 0
$$382$$ −25.5342 −1.30645
$$383$$ 22.7410 1.16201 0.581005 0.813900i $$-0.302660\pi$$
0.581005 + 0.813900i $$0.302660\pi$$
$$384$$ 0 0
$$385$$ 16.3268 0.832089
$$386$$ −13.6785 −0.696219
$$387$$ 0 0
$$388$$ 4.91448 0.249495
$$389$$ 2.53628 0.128594 0.0642972 0.997931i $$-0.479519\pi$$
0.0642972 + 0.997931i $$0.479519\pi$$
$$390$$ 0 0
$$391$$ −5.56904 −0.281639
$$392$$ 17.2751 0.872524
$$393$$ 0 0
$$394$$ −7.57319 −0.381532
$$395$$ −21.6071 −1.08717
$$396$$ 0 0
$$397$$ −13.3410 −0.669564 −0.334782 0.942296i $$-0.608663\pi$$
−0.334782 + 0.942296i $$0.608663\pi$$
$$398$$ 41.7775 2.09412
$$399$$ 0 0
$$400$$ 13.8324 0.691618
$$401$$ 0.833758 0.0416359 0.0208179 0.999783i $$-0.493373\pi$$
0.0208179 + 0.999783i $$0.493373\pi$$
$$402$$ 0 0
$$403$$ −4.01023 −0.199764
$$404$$ −7.67738 −0.381964
$$405$$ 0 0
$$406$$ 6.35054 0.315172
$$407$$ 3.16462 0.156864
$$408$$ 0 0
$$409$$ 12.4602 0.616118 0.308059 0.951367i $$-0.400321\pi$$
0.308059 + 0.951367i $$0.400321\pi$$
$$410$$ 3.74608 0.185006
$$411$$ 0 0
$$412$$ −4.23428 −0.208608
$$413$$ 18.5149 0.911057
$$414$$ 0 0
$$415$$ −19.1894 −0.941969
$$416$$ 7.84411 0.384589
$$417$$ 0 0
$$418$$ −5.71615 −0.279586
$$419$$ 9.75869 0.476743 0.238372 0.971174i $$-0.423386\pi$$
0.238372 + 0.971174i $$0.423386\pi$$
$$420$$ 0 0
$$421$$ 16.1165 0.785470 0.392735 0.919652i $$-0.371529\pi$$
0.392735 + 0.919652i $$0.371529\pi$$
$$422$$ −6.58906 −0.320750
$$423$$ 0 0
$$424$$ −2.40738 −0.116913
$$425$$ 10.7527 0.521582
$$426$$ 0 0
$$427$$ 41.2752 1.99744
$$428$$ −10.2701 −0.496426
$$429$$ 0 0
$$430$$ 1.02697 0.0495249
$$431$$ 23.3638 1.12539 0.562697 0.826663i $$-0.309764\pi$$
0.562697 + 0.826663i $$0.309764\pi$$
$$432$$ 0 0
$$433$$ −16.8084 −0.807758 −0.403879 0.914812i $$-0.632338\pi$$
−0.403879 + 0.914812i $$0.632338\pi$$
$$434$$ 10.6697 0.512162
$$435$$ 0 0
$$436$$ −9.77471 −0.468124
$$437$$ −1.79768 −0.0859948
$$438$$ 0 0
$$439$$ 4.57370 0.218291 0.109145 0.994026i $$-0.465189\pi$$
0.109145 + 0.994026i $$0.465189\pi$$
$$440$$ 9.53490 0.454558
$$441$$ 0 0
$$442$$ 14.2160 0.676188
$$443$$ 8.43805 0.400904 0.200452 0.979704i $$-0.435759\pi$$
0.200452 + 0.979704i $$0.435759\pi$$
$$444$$ 0 0
$$445$$ −8.92543 −0.423106
$$446$$ 3.95612 0.187328
$$447$$ 0 0
$$448$$ 16.3473 0.772336
$$449$$ −11.3585 −0.536043 −0.268021 0.963413i $$-0.586370\pi$$
−0.268021 + 0.963413i $$0.586370\pi$$
$$450$$ 0 0
$$451$$ −4.58898 −0.216087
$$452$$ −6.06967 −0.285493
$$453$$ 0 0
$$454$$ 4.43944 0.208353
$$455$$ −13.1071 −0.614471
$$456$$ 0 0
$$457$$ 13.8039 0.645720 0.322860 0.946447i $$-0.395356\pi$$
0.322860 + 0.946447i $$0.395356\pi$$
$$458$$ −22.1764 −1.03623
$$459$$ 0 0
$$460$$ −1.33063 −0.0620408
$$461$$ −12.8220 −0.597179 −0.298589 0.954382i $$-0.596516\pi$$
−0.298589 + 0.954382i $$0.596516\pi$$
$$462$$ 0 0
$$463$$ −5.22053 −0.242618 −0.121309 0.992615i $$-0.538709\pi$$
−0.121309 + 0.992615i $$0.538709\pi$$
$$464$$ 4.96756 0.230613
$$465$$ 0 0
$$466$$ −8.67948 −0.402069
$$467$$ 8.47835 0.392331 0.196166 0.980571i $$-0.437151\pi$$
0.196166 + 0.980571i $$0.437151\pi$$
$$468$$ 0 0
$$469$$ −23.4378 −1.08226
$$470$$ −18.1552 −0.837437
$$471$$ 0 0
$$472$$ 10.8128 0.497698
$$473$$ −1.25805 −0.0578451
$$474$$ 0 0
$$475$$ 3.47096 0.159259
$$476$$ −8.89219 −0.407573
$$477$$ 0 0
$$478$$ 1.61701 0.0739602
$$479$$ −37.0880 −1.69459 −0.847297 0.531119i $$-0.821772\pi$$
−0.847297 + 0.531119i $$0.821772\pi$$
$$480$$ 0 0
$$481$$ −2.54056 −0.115839
$$482$$ 20.8885 0.951443
$$483$$ 0 0
$$484$$ −1.57876 −0.0717618
$$485$$ −11.7196 −0.532159
$$486$$ 0 0
$$487$$ 34.2956 1.55408 0.777041 0.629450i $$-0.216720\pi$$
0.777041 + 0.629450i $$0.216720\pi$$
$$488$$ 24.1049 1.09118
$$489$$ 0 0
$$490$$ 18.2804 0.825825
$$491$$ 1.07986 0.0487335 0.0243667 0.999703i $$-0.492243\pi$$
0.0243667 + 0.999703i $$0.492243\pi$$
$$492$$ 0 0
$$493$$ 3.86157 0.173916
$$494$$ 4.58893 0.206466
$$495$$ 0 0
$$496$$ 8.34613 0.374752
$$497$$ −41.9266 −1.88066
$$498$$ 0 0
$$499$$ 0.972126 0.0435183 0.0217592 0.999763i $$-0.493073\pi$$
0.0217592 + 0.999763i $$0.493073\pi$$
$$500$$ 7.07471 0.316391
$$501$$ 0 0
$$502$$ 30.6255 1.36688
$$503$$ −30.5203 −1.36083 −0.680417 0.732825i $$-0.738201\pi$$
−0.680417 + 0.732825i $$0.738201\pi$$
$$504$$ 0 0
$$505$$ 18.3083 0.814708
$$506$$ 6.93343 0.308229
$$507$$ 0 0
$$508$$ −6.34211 −0.281386
$$509$$ 20.2337 0.896845 0.448422 0.893822i $$-0.351986\pi$$
0.448422 + 0.893822i $$0.351986\pi$$
$$510$$ 0 0
$$511$$ −46.5367 −2.05866
$$512$$ 5.40993 0.239087
$$513$$ 0 0
$$514$$ −24.7780 −1.09291
$$515$$ 10.0975 0.444950
$$516$$ 0 0
$$517$$ 22.2403 0.978127
$$518$$ 6.75946 0.296993
$$519$$ 0 0
$$520$$ −7.65461 −0.335677
$$521$$ 17.2171 0.754296 0.377148 0.926153i $$-0.376905\pi$$
0.377148 + 0.926153i $$0.376905\pi$$
$$522$$ 0 0
$$523$$ −2.23753 −0.0978405 −0.0489202 0.998803i $$-0.515578\pi$$
−0.0489202 + 0.998803i $$0.515578\pi$$
$$524$$ 6.06577 0.264985
$$525$$ 0 0
$$526$$ 43.8912 1.91375
$$527$$ 6.48793 0.282618
$$528$$ 0 0
$$529$$ −20.8195 −0.905195
$$530$$ −2.54748 −0.110656
$$531$$ 0 0
$$532$$ −2.87039 −0.124447
$$533$$ 3.68403 0.159573
$$534$$ 0 0
$$535$$ 24.4913 1.05885
$$536$$ −13.6878 −0.591222
$$537$$ 0 0
$$538$$ 20.6878 0.891916
$$539$$ −22.3937 −0.964565
$$540$$ 0 0
$$541$$ −4.02257 −0.172944 −0.0864719 0.996254i $$-0.527559\pi$$
−0.0864719 + 0.996254i $$0.527559\pi$$
$$542$$ 46.0798 1.97930
$$543$$ 0 0
$$544$$ −12.6905 −0.544103
$$545$$ 23.3098 0.998483
$$546$$ 0 0
$$547$$ −16.4728 −0.704327 −0.352163 0.935939i $$-0.614554\pi$$
−0.352163 + 0.935939i $$0.614554\pi$$
$$548$$ −1.99541 −0.0852397
$$549$$ 0 0
$$550$$ −13.3871 −0.570826
$$551$$ 1.24651 0.0531033
$$552$$ 0 0
$$553$$ 56.5362 2.40417
$$554$$ 3.90553 0.165930
$$555$$ 0 0
$$556$$ −8.74846 −0.371018
$$557$$ −28.4924 −1.20726 −0.603631 0.797264i $$-0.706280\pi$$
−0.603631 + 0.797264i $$0.706280\pi$$
$$558$$ 0 0
$$559$$ 1.00996 0.0427168
$$560$$ 27.2787 1.15273
$$561$$ 0 0
$$562$$ −13.3858 −0.564646
$$563$$ 19.7584 0.832716 0.416358 0.909201i $$-0.363306\pi$$
0.416358 + 0.909201i $$0.363306\pi$$
$$564$$ 0 0
$$565$$ 14.4744 0.608942
$$566$$ 0.230720 0.00969787
$$567$$ 0 0
$$568$$ −24.4853 −1.02738
$$569$$ 9.62616 0.403550 0.201775 0.979432i $$-0.435329\pi$$
0.201775 + 0.979432i $$0.435329\pi$$
$$570$$ 0 0
$$571$$ 17.4099 0.728583 0.364292 0.931285i $$-0.381311\pi$$
0.364292 + 0.931285i $$0.381311\pi$$
$$572$$ −4.16096 −0.173979
$$573$$ 0 0
$$574$$ −9.80182 −0.409120
$$575$$ −4.21012 −0.175574
$$576$$ 0 0
$$577$$ 39.3271 1.63721 0.818605 0.574357i $$-0.194748\pi$$
0.818605 + 0.574357i $$0.194748\pi$$
$$578$$ 4.48978 0.186750
$$579$$ 0 0
$$580$$ 0.922656 0.0383112
$$581$$ 50.2101 2.08306
$$582$$ 0 0
$$583$$ 3.12069 0.129246
$$584$$ −27.1776 −1.12462
$$585$$ 0 0
$$586$$ 20.9986 0.867445
$$587$$ 26.7724 1.10502 0.552508 0.833507i $$-0.313671\pi$$
0.552508 + 0.833507i $$0.313671\pi$$
$$588$$ 0 0
$$589$$ 2.09430 0.0862941
$$590$$ 11.4420 0.471060
$$591$$ 0 0
$$592$$ 5.28743 0.217312
$$593$$ −44.4733 −1.82630 −0.913150 0.407623i $$-0.866358\pi$$
−0.913150 + 0.407623i $$0.866358\pi$$
$$594$$ 0 0
$$595$$ 21.2053 0.869332
$$596$$ −2.39941 −0.0982838
$$597$$ 0 0
$$598$$ −5.56616 −0.227617
$$599$$ 19.6240 0.801815 0.400908 0.916118i $$-0.368695\pi$$
0.400908 + 0.916118i $$0.368695\pi$$
$$600$$ 0 0
$$601$$ 15.4787 0.631389 0.315695 0.948861i $$-0.397763\pi$$
0.315695 + 0.948861i $$0.397763\pi$$
$$602$$ −2.68712 −0.109519
$$603$$ 0 0
$$604$$ −0.430153 −0.0175027
$$605$$ 3.76488 0.153064
$$606$$ 0 0
$$607$$ 20.5122 0.832566 0.416283 0.909235i $$-0.363333\pi$$
0.416283 + 0.909235i $$0.363333\pi$$
$$608$$ −4.09650 −0.166135
$$609$$ 0 0
$$610$$ 25.5077 1.03277
$$611$$ −17.8545 −0.722316
$$612$$ 0 0
$$613$$ 32.1077 1.29682 0.648409 0.761292i $$-0.275435\pi$$
0.648409 + 0.761292i $$0.275435\pi$$
$$614$$ −12.4495 −0.502422
$$615$$ 0 0
$$616$$ −24.9486 −1.00521
$$617$$ 16.7973 0.676233 0.338117 0.941104i $$-0.390210\pi$$
0.338117 + 0.941104i $$0.390210\pi$$
$$618$$ 0 0
$$619$$ 21.4017 0.860206 0.430103 0.902780i $$-0.358477\pi$$
0.430103 + 0.902780i $$0.358477\pi$$
$$620$$ 1.55018 0.0622566
$$621$$ 0 0
$$622$$ −0.0900810 −0.00361192
$$623$$ 23.3539 0.935653
$$624$$ 0 0
$$625$$ −2.61552 −0.104621
$$626$$ 4.42100 0.176699
$$627$$ 0 0
$$628$$ 1.17633 0.0469407
$$629$$ 4.11022 0.163885
$$630$$ 0 0
$$631$$ −16.0747 −0.639924 −0.319962 0.947430i $$-0.603670\pi$$
−0.319962 + 0.947430i $$0.603670\pi$$
$$632$$ 33.0174 1.31336
$$633$$ 0 0
$$634$$ 9.28668 0.368821
$$635$$ 15.1241 0.600181
$$636$$ 0 0
$$637$$ 17.9777 0.712300
$$638$$ −4.80764 −0.190336
$$639$$ 0 0
$$640$$ 19.9679 0.789299
$$641$$ −35.1303 −1.38756 −0.693782 0.720185i $$-0.744057\pi$$
−0.693782 + 0.720185i $$0.744057\pi$$
$$642$$ 0 0
$$643$$ −1.14336 −0.0450898 −0.0225449 0.999746i $$-0.507177\pi$$
−0.0225449 + 0.999746i $$0.507177\pi$$
$$644$$ 3.48166 0.137197
$$645$$ 0 0
$$646$$ −7.42417 −0.292100
$$647$$ −11.6354 −0.457433 −0.228716 0.973493i $$-0.573453\pi$$
−0.228716 + 0.973493i $$0.573453\pi$$
$$648$$ 0 0
$$649$$ −14.0166 −0.550199
$$650$$ 10.7471 0.421537
$$651$$ 0 0
$$652$$ −10.5672 −0.413844
$$653$$ 2.96261 0.115936 0.0579680 0.998318i $$-0.481538\pi$$
0.0579680 + 0.998318i $$0.481538\pi$$
$$654$$ 0 0
$$655$$ −14.4651 −0.565198
$$656$$ −7.66725 −0.299356
$$657$$ 0 0
$$658$$ 47.5041 1.85190
$$659$$ 35.0365 1.36483 0.682415 0.730965i $$-0.260930\pi$$
0.682415 + 0.730965i $$0.260930\pi$$
$$660$$ 0 0
$$661$$ −30.3727 −1.18136 −0.590681 0.806905i $$-0.701141\pi$$
−0.590681 + 0.806905i $$0.701141\pi$$
$$662$$ 52.2782 2.03185
$$663$$ 0 0
$$664$$ 29.3229 1.13795
$$665$$ 6.84505 0.265440
$$666$$ 0 0
$$667$$ −1.51196 −0.0585435
$$668$$ −9.15453 −0.354199
$$669$$ 0 0
$$670$$ −14.4843 −0.559579
$$671$$ −31.2471 −1.20628
$$672$$ 0 0
$$673$$ 0.848634 0.0327124 0.0163562 0.999866i $$-0.494793\pi$$
0.0163562 + 0.999866i $$0.494793\pi$$
$$674$$ 7.94852 0.306166
$$675$$ 0 0
$$676$$ −4.65083 −0.178878
$$677$$ 1.77815 0.0683400 0.0341700 0.999416i $$-0.489121\pi$$
0.0341700 + 0.999416i $$0.489121\pi$$
$$678$$ 0 0
$$679$$ 30.6650 1.17681
$$680$$ 12.3840 0.474904
$$681$$ 0 0
$$682$$ −8.07744 −0.309301
$$683$$ 34.3363 1.31384 0.656922 0.753959i $$-0.271858\pi$$
0.656922 + 0.753959i $$0.271858\pi$$
$$684$$ 0 0
$$685$$ 4.75847 0.181812
$$686$$ −4.41611 −0.168608
$$687$$ 0 0
$$688$$ −2.10194 −0.0801357
$$689$$ −2.50529 −0.0954439
$$690$$ 0 0
$$691$$ 30.6872 1.16740 0.583698 0.811971i $$-0.301605\pi$$
0.583698 + 0.811971i $$0.301605\pi$$
$$692$$ 7.20254 0.273799
$$693$$ 0 0
$$694$$ −46.5279 −1.76618
$$695$$ 20.8625 0.791361
$$696$$ 0 0
$$697$$ −5.96020 −0.225759
$$698$$ 20.5164 0.776558
$$699$$ 0 0
$$700$$ −6.72237 −0.254082
$$701$$ −1.99537 −0.0753641 −0.0376820 0.999290i $$-0.511997\pi$$
−0.0376820 + 0.999290i $$0.511997\pi$$
$$702$$ 0 0
$$703$$ 1.32678 0.0500404
$$704$$ −12.3756 −0.466424
$$705$$ 0 0
$$706$$ −11.1367 −0.419136
$$707$$ −47.9047 −1.80164
$$708$$ 0 0
$$709$$ 33.9090 1.27348 0.636739 0.771080i $$-0.280283\pi$$
0.636739 + 0.771080i $$0.280283\pi$$
$$710$$ −25.9102 −0.972394
$$711$$ 0 0
$$712$$ 13.6388 0.511134
$$713$$ −2.54029 −0.0951345
$$714$$ 0 0
$$715$$ 9.92267 0.371087
$$716$$ 2.07011 0.0773636
$$717$$ 0 0
$$718$$ −7.21337 −0.269201
$$719$$ −24.1685 −0.901331 −0.450666 0.892693i $$-0.648813\pi$$
−0.450666 + 0.892693i $$0.648813\pi$$
$$720$$ 0 0
$$721$$ −26.4207 −0.983959
$$722$$ 28.3266 1.05421
$$723$$ 0 0
$$724$$ −4.09777 −0.152292
$$725$$ 2.91929 0.108420
$$726$$ 0 0
$$727$$ −46.2043 −1.71362 −0.856811 0.515631i $$-0.827558\pi$$
−0.856811 + 0.515631i $$0.827558\pi$$
$$728$$ 20.0287 0.742314
$$729$$ 0 0
$$730$$ −28.7592 −1.06443
$$731$$ −1.63396 −0.0604342
$$732$$ 0 0
$$733$$ −3.13031 −0.115621 −0.0578104 0.998328i $$-0.518412\pi$$
−0.0578104 + 0.998328i $$0.518412\pi$$
$$734$$ −6.69208 −0.247009
$$735$$ 0 0
$$736$$ 4.96887 0.183155
$$737$$ 17.7435 0.653589
$$738$$ 0 0
$$739$$ −46.7714 −1.72052 −0.860258 0.509859i $$-0.829697\pi$$
−0.860258 + 0.509859i $$0.829697\pi$$
$$740$$ 0.982067 0.0361015
$$741$$ 0 0
$$742$$ 6.66563 0.244703
$$743$$ −41.7626 −1.53212 −0.766061 0.642768i $$-0.777786\pi$$
−0.766061 + 0.642768i $$0.777786\pi$$
$$744$$ 0 0
$$745$$ 5.72190 0.209634
$$746$$ 25.7382 0.942342
$$747$$ 0 0
$$748$$ 6.73179 0.246139
$$749$$ −64.0828 −2.34154
$$750$$ 0 0
$$751$$ 4.01683 0.146576 0.0732881 0.997311i $$-0.476651\pi$$
0.0732881 + 0.997311i $$0.476651\pi$$
$$752$$ 37.1590 1.35505
$$753$$ 0 0
$$754$$ 3.85957 0.140557
$$755$$ 1.02579 0.0373323
$$756$$ 0 0
$$757$$ 34.0626 1.23803 0.619014 0.785380i $$-0.287532\pi$$
0.619014 + 0.785380i $$0.287532\pi$$
$$758$$ −57.8058 −2.09960
$$759$$ 0 0
$$760$$ 3.99754 0.145006
$$761$$ −17.7142 −0.642139 −0.321070 0.947056i $$-0.604042\pi$$
−0.321070 + 0.947056i $$0.604042\pi$$
$$762$$ 0 0
$$763$$ −60.9914 −2.20804
$$764$$ 9.70694 0.351185
$$765$$ 0 0
$$766$$ −36.7723 −1.32864
$$767$$ 11.2525 0.406305
$$768$$ 0 0
$$769$$ 19.4368 0.700907 0.350454 0.936580i $$-0.386027\pi$$
0.350454 + 0.936580i $$0.386027\pi$$
$$770$$ −26.4005 −0.951407
$$771$$ 0 0
$$772$$ 5.19994 0.187150
$$773$$ −15.5259 −0.558428 −0.279214 0.960229i $$-0.590074\pi$$
−0.279214 + 0.960229i $$0.590074\pi$$
$$774$$ 0 0
$$775$$ 4.90478 0.176185
$$776$$ 17.9085 0.642877
$$777$$ 0 0
$$778$$ −4.10118 −0.147034
$$779$$ −1.92395 −0.0689326
$$780$$ 0 0
$$781$$ 31.7403 1.13576
$$782$$ 9.00518 0.322024
$$783$$ 0 0
$$784$$ −37.4153 −1.33626
$$785$$ −2.80521 −0.100122
$$786$$ 0 0
$$787$$ −25.2433 −0.899828 −0.449914 0.893072i $$-0.648545\pi$$
−0.449914 + 0.893072i $$0.648545\pi$$
$$788$$ 2.87898 0.102559
$$789$$ 0 0
$$790$$ 34.9389 1.24307
$$791$$ −37.8731 −1.34661
$$792$$ 0 0
$$793$$ 25.0852 0.890801
$$794$$ 21.5724 0.765577
$$795$$ 0 0
$$796$$ −15.8819 −0.562918
$$797$$ −39.2256 −1.38944 −0.694721 0.719279i $$-0.744472\pi$$
−0.694721 + 0.719279i $$0.744472\pi$$
$$798$$ 0 0
$$799$$ 28.8858 1.02191
$$800$$ −9.59388 −0.339195
$$801$$ 0 0
$$802$$ −1.34819 −0.0476063
$$803$$ 35.2303 1.24325
$$804$$ 0 0
$$805$$ −8.30273 −0.292633
$$806$$ 6.48456 0.228409
$$807$$ 0 0
$$808$$ −27.9765 −0.984211
$$809$$ −21.5713 −0.758405 −0.379203 0.925314i $$-0.623802\pi$$
−0.379203 + 0.925314i $$0.623802\pi$$
$$810$$ 0 0
$$811$$ 2.52560 0.0886858 0.0443429 0.999016i $$-0.485881\pi$$
0.0443429 + 0.999016i $$0.485881\pi$$
$$812$$ −2.41418 −0.0847211
$$813$$ 0 0
$$814$$ −5.11721 −0.179358
$$815$$ 25.1997 0.882707
$$816$$ 0 0
$$817$$ −0.527441 −0.0184528
$$818$$ −20.1482 −0.704467
$$819$$ 0 0
$$820$$ −1.42409 −0.0497312
$$821$$ −21.8608 −0.762949 −0.381474 0.924379i $$-0.624584\pi$$
−0.381474 + 0.924379i $$0.624584\pi$$
$$822$$ 0 0
$$823$$ 15.6288 0.544785 0.272393 0.962186i $$-0.412185\pi$$
0.272393 + 0.962186i $$0.412185\pi$$
$$824$$ −15.4298 −0.537523
$$825$$ 0 0
$$826$$ −29.9387 −1.04170
$$827$$ −1.15424 −0.0401368 −0.0200684 0.999799i $$-0.506388\pi$$
−0.0200684 + 0.999799i $$0.506388\pi$$
$$828$$ 0 0
$$829$$ 22.5972 0.784832 0.392416 0.919788i $$-0.371639\pi$$
0.392416 + 0.919788i $$0.371639\pi$$
$$830$$ 31.0293 1.07704
$$831$$ 0 0
$$832$$ 9.93514 0.344439
$$833$$ −29.0851 −1.00774
$$834$$ 0 0
$$835$$ 21.8309 0.755489
$$836$$ 2.17302 0.0751554
$$837$$ 0 0
$$838$$ −15.7799 −0.545106
$$839$$ 26.1778 0.903757 0.451878 0.892080i $$-0.350754\pi$$
0.451878 + 0.892080i $$0.350754\pi$$
$$840$$ 0 0
$$841$$ −27.9516 −0.963848
$$842$$ −26.0605 −0.898103
$$843$$ 0 0
$$844$$ 2.50486 0.0862207
$$845$$ 11.0909 0.381537
$$846$$ 0 0
$$847$$ −9.85102 −0.338485
$$848$$ 5.21403 0.179051
$$849$$ 0 0
$$850$$ −17.3872 −0.596375
$$851$$ −1.60932 −0.0551668
$$852$$ 0 0
$$853$$ 20.2252 0.692497 0.346249 0.938143i $$-0.387455\pi$$
0.346249 + 0.938143i $$0.387455\pi$$
$$854$$ −66.7422 −2.28387
$$855$$ 0 0
$$856$$ −37.4247 −1.27915
$$857$$ −24.4336 −0.834636 −0.417318 0.908761i $$-0.637030\pi$$
−0.417318 + 0.908761i $$0.637030\pi$$
$$858$$ 0 0
$$859$$ 36.4278 1.24290 0.621451 0.783453i $$-0.286543\pi$$
0.621451 + 0.783453i $$0.286543\pi$$
$$860$$ −0.390406 −0.0133128
$$861$$ 0 0
$$862$$ −37.7794 −1.28677
$$863$$ 18.5506 0.631471 0.315735 0.948847i $$-0.397749\pi$$
0.315735 + 0.948847i $$0.397749\pi$$
$$864$$ 0 0
$$865$$ −17.1759 −0.583999
$$866$$ 27.1792 0.923588
$$867$$ 0 0
$$868$$ −4.05613 −0.137674
$$869$$ −42.8005 −1.45191
$$870$$ 0 0
$$871$$ −14.2444 −0.482655
$$872$$ −35.6193 −1.20622
$$873$$ 0 0
$$874$$ 2.90687 0.0983262
$$875$$ 44.1442 1.49235
$$876$$ 0 0
$$877$$ 39.5945 1.33701 0.668505 0.743708i $$-0.266934\pi$$
0.668505 + 0.743708i $$0.266934\pi$$
$$878$$ −7.39570 −0.249593
$$879$$ 0 0
$$880$$ −20.6512 −0.696151
$$881$$ −32.1516 −1.08321 −0.541607 0.840632i $$-0.682184\pi$$
−0.541607 + 0.840632i $$0.682184\pi$$
$$882$$ 0 0
$$883$$ 20.8275 0.700900 0.350450 0.936581i $$-0.386029\pi$$
0.350450 + 0.936581i $$0.386029\pi$$
$$884$$ −5.40428 −0.181766
$$885$$ 0 0
$$886$$ −13.6444 −0.458392
$$887$$ −34.9748 −1.17434 −0.587170 0.809464i $$-0.699758\pi$$
−0.587170 + 0.809464i $$0.699758\pi$$
$$888$$ 0 0
$$889$$ −39.5730 −1.32724
$$890$$ 14.4325 0.483778
$$891$$ 0 0
$$892$$ −1.50394 −0.0503555
$$893$$ 9.32432 0.312027
$$894$$ 0 0
$$895$$ −4.93660 −0.165012
$$896$$ −52.2470 −1.74545
$$897$$ 0 0
$$898$$ 18.3668 0.612909
$$899$$ 1.76144 0.0587472
$$900$$ 0 0
$$901$$ 4.05317 0.135031
$$902$$ 7.42042 0.247073
$$903$$ 0 0
$$904$$ −22.1180 −0.735635
$$905$$ 9.77198 0.324832
$$906$$ 0 0
$$907$$ −28.3094 −0.939999 −0.470000 0.882667i $$-0.655746\pi$$
−0.470000 + 0.882667i $$0.655746\pi$$
$$908$$ −1.68767 −0.0560073
$$909$$ 0 0
$$910$$ 21.1943 0.702584
$$911$$ 47.1094 1.56081 0.780403 0.625277i $$-0.215014\pi$$
0.780403 + 0.625277i $$0.215014\pi$$
$$912$$ 0 0
$$913$$ −38.0113 −1.25799
$$914$$ −22.3210 −0.738314
$$915$$ 0 0
$$916$$ 8.43044 0.278549
$$917$$ 37.8487 1.24987
$$918$$ 0 0
$$919$$ −17.0107 −0.561131 −0.280565 0.959835i $$-0.590522\pi$$
−0.280565 + 0.959835i $$0.590522\pi$$
$$920$$ −4.84883 −0.159861
$$921$$ 0 0
$$922$$ 20.7332 0.682812
$$923$$ −25.4811 −0.838720
$$924$$ 0 0
$$925$$ 3.10727 0.102166
$$926$$ 8.44163 0.277409
$$927$$ 0 0
$$928$$ −3.44541 −0.113101
$$929$$ 0.496932 0.0163038 0.00815190 0.999967i $$-0.497405\pi$$
0.00815190 + 0.999967i $$0.497405\pi$$
$$930$$ 0 0
$$931$$ −9.38864 −0.307700
$$932$$ 3.29954 0.108080
$$933$$ 0 0
$$934$$ −13.7095 −0.448590
$$935$$ −16.0533 −0.525001
$$936$$ 0 0
$$937$$ 8.22697 0.268763 0.134382 0.990930i $$-0.457095\pi$$
0.134382 + 0.990930i $$0.457095\pi$$
$$938$$ 37.8991 1.23745
$$939$$ 0 0
$$940$$ 6.90177 0.225111
$$941$$ −50.1600 −1.63517 −0.817584 0.575809i $$-0.804687\pi$$
−0.817584 + 0.575809i $$0.804687\pi$$
$$942$$ 0 0
$$943$$ 2.33366 0.0759944
$$944$$ −23.4188 −0.762218
$$945$$ 0 0
$$946$$ 2.03427 0.0661399
$$947$$ −11.6253 −0.377773 −0.188886 0.981999i $$-0.560488\pi$$
−0.188886 + 0.981999i $$0.560488\pi$$
$$948$$ 0 0
$$949$$ −28.2829 −0.918101
$$950$$ −5.61257 −0.182096
$$951$$ 0 0
$$952$$ −32.4034 −1.05020
$$953$$ −27.5219 −0.891522 −0.445761 0.895152i $$-0.647067\pi$$
−0.445761 + 0.895152i $$0.647067\pi$$
$$954$$ 0 0
$$955$$ −23.1482 −0.749058
$$956$$ −0.614711 −0.0198812
$$957$$ 0 0
$$958$$ 59.9716 1.93759
$$959$$ −12.4508 −0.402057
$$960$$ 0 0
$$961$$ −28.0406 −0.904534
$$962$$ 4.10809 0.132450
$$963$$ 0 0
$$964$$ −7.94082 −0.255757
$$965$$ −12.4003 −0.399181
$$966$$ 0 0
$$967$$ −41.9315 −1.34843 −0.674213 0.738537i $$-0.735517\pi$$
−0.674213 + 0.738537i $$0.735517\pi$$
$$968$$ −5.75304 −0.184910
$$969$$ 0 0
$$970$$ 18.9507 0.608469
$$971$$ 44.8246 1.43849 0.719244 0.694757i $$-0.244488\pi$$
0.719244 + 0.694757i $$0.244488\pi$$
$$972$$ 0 0
$$973$$ −54.5880 −1.75001
$$974$$ −55.4562 −1.77693
$$975$$ 0 0
$$976$$ −52.2075 −1.67112
$$977$$ 36.6672 1.17309 0.586544 0.809917i $$-0.300488\pi$$
0.586544 + 0.809917i $$0.300488\pi$$
$$978$$ 0 0
$$979$$ −17.6799 −0.565053
$$980$$ −6.94937 −0.221989
$$981$$ 0 0
$$982$$ −1.74614 −0.0557217
$$983$$ 43.4827 1.38688 0.693441 0.720513i $$-0.256094\pi$$
0.693441 + 0.720513i $$0.256094\pi$$
$$984$$ 0 0
$$985$$ −6.86552 −0.218754
$$986$$ −6.24419 −0.198856
$$987$$ 0 0
$$988$$ −1.74450 −0.0554998
$$989$$ 0.639762 0.0203433
$$990$$ 0 0
$$991$$ −4.85590 −0.154253 −0.0771263 0.997021i $$-0.524574\pi$$
−0.0771263 + 0.997021i $$0.524574\pi$$
$$992$$ −5.78872 −0.183792
$$993$$ 0 0
$$994$$ 67.7956 2.15034
$$995$$ 37.8736 1.20067
$$996$$ 0 0
$$997$$ −10.2069 −0.323257 −0.161628 0.986852i $$-0.551675\pi$$
−0.161628 + 0.986852i $$0.551675\pi$$
$$998$$ −1.57193 −0.0497587
$$999$$ 0 0
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 2151.2.a.j.1.6 20
3.2 odd 2 2151.2.a.k.1.15 yes 20

By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.6 20 1.1 even 1 trivial
2151.2.a.k.1.15 yes 20 3.2 odd 2