# Properties

 Label 1089.3.b.a.485.2 Level $1089$ Weight $3$ Character 1089.485 Analytic conductor $29.673$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1089.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.6731007888$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 485.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 1089.485 Dual form 1089.3.b.a.485.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.41421i q^{2} +2.00000 q^{4} -5.65685i q^{5} -1.00000 q^{7} +8.48528i q^{8} +O(q^{10})$$ $$q+1.41421i q^{2} +2.00000 q^{4} -5.65685i q^{5} -1.00000 q^{7} +8.48528i q^{8} +8.00000 q^{10} -16.0000 q^{13} -1.41421i q^{14} -4.00000 q^{16} -11.3137i q^{17} -33.0000 q^{19} -11.3137i q^{20} -24.0416i q^{23} -7.00000 q^{25} -22.6274i q^{26} -2.00000 q^{28} +12.7279i q^{29} +31.0000 q^{31} +28.2843i q^{32} +16.0000 q^{34} +5.65685i q^{35} -57.0000 q^{37} -46.6690i q^{38} +48.0000 q^{40} -15.5563i q^{41} +34.0000 q^{46} +60.8112i q^{47} -48.0000 q^{49} -9.89949i q^{50} -32.0000 q^{52} -89.0955i q^{53} -8.48528i q^{56} -18.0000 q^{58} +69.2965i q^{59} -105.000 q^{61} +43.8406i q^{62} -56.0000 q^{64} +90.5097i q^{65} -103.000 q^{67} -22.6274i q^{68} -8.00000 q^{70} -118.794i q^{71} +47.0000 q^{73} -80.6102i q^{74} -66.0000 q^{76} -23.0000 q^{79} +22.6274i q^{80} +22.0000 q^{82} -106.066i q^{83} -64.0000 q^{85} +50.9117i q^{89} +16.0000 q^{91} -48.0833i q^{92} -86.0000 q^{94} +186.676i q^{95} -25.0000 q^{97} -67.8823i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 2 q^{7}+O(q^{10})$$ 2 * q + 4 * q^4 - 2 * q^7 $$2 q + 4 q^{4} - 2 q^{7} + 16 q^{10} - 32 q^{13} - 8 q^{16} - 66 q^{19} - 14 q^{25} - 4 q^{28} + 62 q^{31} + 32 q^{34} - 114 q^{37} + 96 q^{40} + 68 q^{46} - 96 q^{49} - 64 q^{52} - 36 q^{58} - 210 q^{61} - 112 q^{64} - 206 q^{67} - 16 q^{70} + 94 q^{73} - 132 q^{76} - 46 q^{79} + 44 q^{82} - 128 q^{85} + 32 q^{91} - 172 q^{94} - 50 q^{97}+O(q^{100})$$ 2 * q + 4 * q^4 - 2 * q^7 + 16 * q^10 - 32 * q^13 - 8 * q^16 - 66 * q^19 - 14 * q^25 - 4 * q^28 + 62 * q^31 + 32 * q^34 - 114 * q^37 + 96 * q^40 + 68 * q^46 - 96 * q^49 - 64 * q^52 - 36 * q^58 - 210 * q^61 - 112 * q^64 - 206 * q^67 - 16 * q^70 + 94 * q^73 - 132 * q^76 - 46 * q^79 + 44 * q^82 - 128 * q^85 + 32 * q^91 - 172 * q^94 - 50 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.41421i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 0 0
$$4$$ 2.00000 0.500000
$$5$$ − 5.65685i − 1.13137i −0.824621 0.565685i $$-0.808612\pi$$
0.824621 0.565685i $$-0.191388\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.142857 −0.0714286 0.997446i $$-0.522756\pi$$
−0.0714286 + 0.997446i $$0.522756\pi$$
$$8$$ 8.48528i 1.06066i
$$9$$ 0 0
$$10$$ 8.00000 0.800000
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −16.0000 −1.23077 −0.615385 0.788227i $$-0.710999\pi$$
−0.615385 + 0.788227i $$0.710999\pi$$
$$14$$ − 1.41421i − 0.101015i
$$15$$ 0 0
$$16$$ −4.00000 −0.250000
$$17$$ − 11.3137i − 0.665512i −0.943013 0.332756i $$-0.892021\pi$$
0.943013 0.332756i $$-0.107979\pi$$
$$18$$ 0 0
$$19$$ −33.0000 −1.73684 −0.868421 0.495827i $$-0.834865\pi$$
−0.868421 + 0.495827i $$0.834865\pi$$
$$20$$ − 11.3137i − 0.565685i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 24.0416i − 1.04529i −0.852551 0.522644i $$-0.824946\pi$$
0.852551 0.522644i $$-0.175054\pi$$
$$24$$ 0 0
$$25$$ −7.00000 −0.280000
$$26$$ − 22.6274i − 0.870285i
$$27$$ 0 0
$$28$$ −2.00000 −0.0714286
$$29$$ 12.7279i 0.438894i 0.975624 + 0.219447i $$0.0704253\pi$$
−0.975624 + 0.219447i $$0.929575\pi$$
$$30$$ 0 0
$$31$$ 31.0000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$32$$ 28.2843i 0.883883i
$$33$$ 0 0
$$34$$ 16.0000 0.470588
$$35$$ 5.65685i 0.161624i
$$36$$ 0 0
$$37$$ −57.0000 −1.54054 −0.770270 0.637718i $$-0.779879\pi$$
−0.770270 + 0.637718i $$0.779879\pi$$
$$38$$ − 46.6690i − 1.22813i
$$39$$ 0 0
$$40$$ 48.0000 1.20000
$$41$$ − 15.5563i − 0.379423i −0.981840 0.189712i $$-0.939245\pi$$
0.981840 0.189712i $$-0.0607553\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 34.0000 0.739130
$$47$$ 60.8112i 1.29385i 0.762552 + 0.646927i $$0.223946\pi$$
−0.762552 + 0.646927i $$0.776054\pi$$
$$48$$ 0 0
$$49$$ −48.0000 −0.979592
$$50$$ − 9.89949i − 0.197990i
$$51$$ 0 0
$$52$$ −32.0000 −0.615385
$$53$$ − 89.0955i − 1.68105i −0.541776 0.840523i $$-0.682248\pi$$
0.541776 0.840523i $$-0.317752\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ − 8.48528i − 0.151523i
$$57$$ 0 0
$$58$$ −18.0000 −0.310345
$$59$$ 69.2965i 1.17452i 0.809400 + 0.587258i $$0.199793\pi$$
−0.809400 + 0.587258i $$0.800207\pi$$
$$60$$ 0 0
$$61$$ −105.000 −1.72131 −0.860656 0.509187i $$-0.829946\pi$$
−0.860656 + 0.509187i $$0.829946\pi$$
$$62$$ 43.8406i 0.707107i
$$63$$ 0 0
$$64$$ −56.0000 −0.875000
$$65$$ 90.5097i 1.39246i
$$66$$ 0 0
$$67$$ −103.000 −1.53731 −0.768657 0.639662i $$-0.779075\pi$$
−0.768657 + 0.639662i $$0.779075\pi$$
$$68$$ − 22.6274i − 0.332756i
$$69$$ 0 0
$$70$$ −8.00000 −0.114286
$$71$$ − 118.794i − 1.67315i −0.547849 0.836577i $$-0.684553\pi$$
0.547849 0.836577i $$-0.315447\pi$$
$$72$$ 0 0
$$73$$ 47.0000 0.643836 0.321918 0.946768i $$-0.395673\pi$$
0.321918 + 0.946768i $$0.395673\pi$$
$$74$$ − 80.6102i − 1.08933i
$$75$$ 0 0
$$76$$ −66.0000 −0.868421
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −23.0000 −0.291139 −0.145570 0.989348i $$-0.546501\pi$$
−0.145570 + 0.989348i $$0.546501\pi$$
$$80$$ 22.6274i 0.282843i
$$81$$ 0 0
$$82$$ 22.0000 0.268293
$$83$$ − 106.066i − 1.27790i −0.769247 0.638952i $$-0.779368\pi$$
0.769247 0.638952i $$-0.220632\pi$$
$$84$$ 0 0
$$85$$ −64.0000 −0.752941
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 50.9117i 0.572041i 0.958223 + 0.286021i $$0.0923326\pi$$
−0.958223 + 0.286021i $$0.907667\pi$$
$$90$$ 0 0
$$91$$ 16.0000 0.175824
$$92$$ − 48.0833i − 0.522644i
$$93$$ 0 0
$$94$$ −86.0000 −0.914894
$$95$$ 186.676i 1.96501i
$$96$$ 0 0
$$97$$ −25.0000 −0.257732 −0.128866 0.991662i $$-0.541134\pi$$
−0.128866 + 0.991662i $$0.541134\pi$$
$$98$$ − 67.8823i − 0.692676i
$$99$$ 0 0
$$100$$ −14.0000 −0.140000
$$101$$ 151.321i 1.49823i 0.662442 + 0.749113i $$0.269520\pi$$
−0.662442 + 0.749113i $$0.730480\pi$$
$$102$$ 0 0
$$103$$ −25.0000 −0.242718 −0.121359 0.992609i $$-0.538725\pi$$
−0.121359 + 0.992609i $$0.538725\pi$$
$$104$$ − 135.765i − 1.30543i
$$105$$ 0 0
$$106$$ 126.000 1.18868
$$107$$ − 131.522i − 1.22918i −0.788848 0.614588i $$-0.789322\pi$$
0.788848 0.614588i $$-0.210678\pi$$
$$108$$ 0 0
$$109$$ 119.000 1.09174 0.545872 0.837869i $$-0.316199\pi$$
0.545872 + 0.837869i $$0.316199\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.00000 0.0357143
$$113$$ 12.7279i 0.112636i 0.998413 + 0.0563182i $$0.0179361\pi$$
−0.998413 + 0.0563182i $$0.982064\pi$$
$$114$$ 0 0
$$115$$ −136.000 −1.18261
$$116$$ 25.4558i 0.219447i
$$117$$ 0 0
$$118$$ −98.0000 −0.830508
$$119$$ 11.3137i 0.0950732i
$$120$$ 0 0
$$121$$ 0 0
$$122$$ − 148.492i − 1.21715i
$$123$$ 0 0
$$124$$ 62.0000 0.500000
$$125$$ − 101.823i − 0.814587i
$$126$$ 0 0
$$127$$ 79.0000 0.622047 0.311024 0.950402i $$-0.399328\pi$$
0.311024 + 0.950402i $$0.399328\pi$$
$$128$$ 33.9411i 0.265165i
$$129$$ 0 0
$$130$$ −128.000 −0.984615
$$131$$ 114.551i 0.874437i 0.899355 + 0.437219i $$0.144036\pi$$
−0.899355 + 0.437219i $$0.855964\pi$$
$$132$$ 0 0
$$133$$ 33.0000 0.248120
$$134$$ − 145.664i − 1.08704i
$$135$$ 0 0
$$136$$ 96.0000 0.705882
$$137$$ 90.5097i 0.660655i 0.943866 + 0.330327i $$0.107159\pi$$
−0.943866 + 0.330327i $$0.892841\pi$$
$$138$$ 0 0
$$139$$ −58.0000 −0.417266 −0.208633 0.977994i $$-0.566901\pi$$
−0.208633 + 0.977994i $$0.566901\pi$$
$$140$$ 11.3137i 0.0808122i
$$141$$ 0 0
$$142$$ 168.000 1.18310
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 72.0000 0.496552
$$146$$ 66.4680i 0.455261i
$$147$$ 0 0
$$148$$ −114.000 −0.770270
$$149$$ − 260.215i − 1.74641i −0.487352 0.873206i $$-0.662037\pi$$
0.487352 0.873206i $$-0.337963\pi$$
$$150$$ 0 0
$$151$$ 64.0000 0.423841 0.211921 0.977287i $$-0.432028\pi$$
0.211921 + 0.977287i $$0.432028\pi$$
$$152$$ − 280.014i − 1.84220i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 175.362i − 1.13137i
$$156$$ 0 0
$$157$$ 257.000 1.63694 0.818471 0.574547i $$-0.194822\pi$$
0.818471 + 0.574547i $$0.194822\pi$$
$$158$$ − 32.5269i − 0.205867i
$$159$$ 0 0
$$160$$ 160.000 1.00000
$$161$$ 24.0416i 0.149327i
$$162$$ 0 0
$$163$$ 79.0000 0.484663 0.242331 0.970194i $$-0.422088\pi$$
0.242331 + 0.970194i $$0.422088\pi$$
$$164$$ − 31.1127i − 0.189712i
$$165$$ 0 0
$$166$$ 150.000 0.903614
$$167$$ 29.6985i 0.177835i 0.996039 + 0.0889176i $$0.0283408\pi$$
−0.996039 + 0.0889176i $$0.971659\pi$$
$$168$$ 0 0
$$169$$ 87.0000 0.514793
$$170$$ − 90.5097i − 0.532410i
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 67.8823i − 0.392383i −0.980566 0.196191i $$-0.937143\pi$$
0.980566 0.196191i $$-0.0628574\pi$$
$$174$$ 0 0
$$175$$ 7.00000 0.0400000
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −72.0000 −0.404494
$$179$$ − 199.404i − 1.11399i −0.830516 0.556995i $$-0.811954\pi$$
0.830516 0.556995i $$-0.188046\pi$$
$$180$$ 0 0
$$181$$ −287.000 −1.58564 −0.792818 0.609459i $$-0.791387\pi$$
−0.792818 + 0.609459i $$0.791387\pi$$
$$182$$ 22.6274i 0.124326i
$$183$$ 0 0
$$184$$ 204.000 1.10870
$$185$$ 322.441i 1.74292i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 121.622i 0.646927i
$$189$$ 0 0
$$190$$ −264.000 −1.38947
$$191$$ − 164.049i − 0.858894i −0.903092 0.429447i $$-0.858708\pi$$
0.903092 0.429447i $$-0.141292\pi$$
$$192$$ 0 0
$$193$$ −145.000 −0.751295 −0.375648 0.926763i $$-0.622580\pi$$
−0.375648 + 0.926763i $$0.622580\pi$$
$$194$$ − 35.3553i − 0.182244i
$$195$$ 0 0
$$196$$ −96.0000 −0.489796
$$197$$ − 45.2548i − 0.229720i −0.993382 0.114860i $$-0.963358\pi$$
0.993382 0.114860i $$-0.0366419\pi$$
$$198$$ 0 0
$$199$$ −57.0000 −0.286432 −0.143216 0.989691i $$-0.545744\pi$$
−0.143216 + 0.989691i $$0.545744\pi$$
$$200$$ − 59.3970i − 0.296985i
$$201$$ 0 0
$$202$$ −214.000 −1.05941
$$203$$ − 12.7279i − 0.0626991i
$$204$$ 0 0
$$205$$ −88.0000 −0.429268
$$206$$ − 35.3553i − 0.171628i
$$207$$ 0 0
$$208$$ 64.0000 0.307692
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 105.000 0.497630 0.248815 0.968551i $$-0.419959\pi$$
0.248815 + 0.968551i $$0.419959\pi$$
$$212$$ − 178.191i − 0.840523i
$$213$$ 0 0
$$214$$ 186.000 0.869159
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −31.0000 −0.142857
$$218$$ 168.291i 0.771979i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 181.019i 0.819092i
$$222$$ 0 0
$$223$$ 199.000 0.892377 0.446188 0.894939i $$-0.352781\pi$$
0.446188 + 0.894939i $$0.352781\pi$$
$$224$$ − 28.2843i − 0.126269i
$$225$$ 0 0
$$226$$ −18.0000 −0.0796460
$$227$$ − 186.676i − 0.822362i −0.911554 0.411181i $$-0.865116\pi$$
0.911554 0.411181i $$-0.134884\pi$$
$$228$$ 0 0
$$229$$ −72.0000 −0.314410 −0.157205 0.987566i $$-0.550248\pi$$
−0.157205 + 0.987566i $$0.550248\pi$$
$$230$$ − 192.333i − 0.836231i
$$231$$ 0 0
$$232$$ −108.000 −0.465517
$$233$$ − 131.522i − 0.564472i −0.959345 0.282236i $$-0.908924\pi$$
0.959345 0.282236i $$-0.0910760\pi$$
$$234$$ 0 0
$$235$$ 344.000 1.46383
$$236$$ 138.593i 0.587258i
$$237$$ 0 0
$$238$$ −16.0000 −0.0672269
$$239$$ − 4.24264i − 0.0177516i −0.999961 0.00887582i $$-0.997175\pi$$
0.999961 0.00887582i $$-0.00282530\pi$$
$$240$$ 0 0
$$241$$ −160.000 −0.663900 −0.331950 0.943297i $$-0.607707\pi$$
−0.331950 + 0.943297i $$0.607707\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −210.000 −0.860656
$$245$$ 271.529i 1.10828i
$$246$$ 0 0
$$247$$ 528.000 2.13765
$$248$$ 263.044i 1.06066i
$$249$$ 0 0
$$250$$ 144.000 0.576000
$$251$$ 108.894i 0.433842i 0.976189 + 0.216921i $$0.0696015\pi$$
−0.976189 + 0.216921i $$0.930399\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 111.723i 0.439854i
$$255$$ 0 0
$$256$$ −272.000 −1.06250
$$257$$ 261.630i 1.01801i 0.860762 + 0.509007i $$0.169987\pi$$
−0.860762 + 0.509007i $$0.830013\pi$$
$$258$$ 0 0
$$259$$ 57.0000 0.220077
$$260$$ 181.019i 0.696228i
$$261$$ 0 0
$$262$$ −162.000 −0.618321
$$263$$ 439.820i 1.67232i 0.548485 + 0.836160i $$0.315205\pi$$
−0.548485 + 0.836160i $$0.684795\pi$$
$$264$$ 0 0
$$265$$ −504.000 −1.90189
$$266$$ 46.6690i 0.175448i
$$267$$ 0 0
$$268$$ −206.000 −0.768657
$$269$$ 244.659i 0.909513i 0.890616 + 0.454756i $$0.150274\pi$$
−0.890616 + 0.454756i $$0.849726\pi$$
$$270$$ 0 0
$$271$$ −344.000 −1.26937 −0.634686 0.772770i $$-0.718871\pi$$
−0.634686 + 0.772770i $$0.718871\pi$$
$$272$$ 45.2548i 0.166378i
$$273$$ 0 0
$$274$$ −128.000 −0.467153
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 95.0000 0.342960 0.171480 0.985188i $$-0.445145\pi$$
0.171480 + 0.985188i $$0.445145\pi$$
$$278$$ − 82.0244i − 0.295052i
$$279$$ 0 0
$$280$$ −48.0000 −0.171429
$$281$$ 390.323i 1.38905i 0.719469 + 0.694525i $$0.244385\pi$$
−0.719469 + 0.694525i $$0.755615\pi$$
$$282$$ 0 0
$$283$$ 335.000 1.18375 0.591873 0.806031i $$-0.298389\pi$$
0.591873 + 0.806031i $$0.298389\pi$$
$$284$$ − 237.588i − 0.836577i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 15.5563i 0.0542033i
$$288$$ 0 0
$$289$$ 161.000 0.557093
$$290$$ 101.823i 0.351115i
$$291$$ 0 0
$$292$$ 94.0000 0.321918
$$293$$ − 560.029i − 1.91136i −0.294407 0.955680i $$-0.595122\pi$$
0.294407 0.955680i $$-0.404878\pi$$
$$294$$ 0 0
$$295$$ 392.000 1.32881
$$296$$ − 483.661i − 1.63399i
$$297$$ 0 0
$$298$$ 368.000 1.23490
$$299$$ 384.666i 1.28651i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 90.5097i 0.299701i
$$303$$ 0 0
$$304$$ 132.000 0.434211
$$305$$ 593.970i 1.94744i
$$306$$ 0 0
$$307$$ −311.000 −1.01303 −0.506515 0.862231i $$-0.669066\pi$$
−0.506515 + 0.862231i $$0.669066\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 248.000 0.800000
$$311$$ 16.9706i 0.0545677i 0.999628 + 0.0272839i $$0.00868580\pi$$
−0.999628 + 0.0272839i $$0.991314\pi$$
$$312$$ 0 0
$$313$$ 200.000 0.638978 0.319489 0.947590i $$-0.396489\pi$$
0.319489 + 0.947590i $$0.396489\pi$$
$$314$$ 363.453i 1.15749i
$$315$$ 0 0
$$316$$ −46.0000 −0.145570
$$317$$ 282.843i 0.892248i 0.894971 + 0.446124i $$0.147196\pi$$
−0.894971 + 0.446124i $$0.852804\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 316.784i 0.989949i
$$321$$ 0 0
$$322$$ −34.0000 −0.105590
$$323$$ 373.352i 1.15589i
$$324$$ 0 0
$$325$$ 112.000 0.344615
$$326$$ 111.723i 0.342708i
$$327$$ 0 0
$$328$$ 132.000 0.402439
$$329$$ − 60.8112i − 0.184836i
$$330$$ 0 0
$$331$$ 577.000 1.74320 0.871601 0.490216i $$-0.163082\pi$$
0.871601 + 0.490216i $$0.163082\pi$$
$$332$$ − 212.132i − 0.638952i
$$333$$ 0 0
$$334$$ −42.0000 −0.125749
$$335$$ 582.656i 1.73927i
$$336$$ 0 0
$$337$$ 289.000 0.857567 0.428783 0.903407i $$-0.358942\pi$$
0.428783 + 0.903407i $$0.358942\pi$$
$$338$$ 123.037i 0.364014i
$$339$$ 0 0
$$340$$ −128.000 −0.376471
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 97.0000 0.282799
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 96.0000 0.277457
$$347$$ − 541.644i − 1.56093i −0.625198 0.780467i $$-0.714982\pi$$
0.625198 0.780467i $$-0.285018\pi$$
$$348$$ 0 0
$$349$$ −513.000 −1.46991 −0.734957 0.678114i $$-0.762798\pi$$
−0.734957 + 0.678114i $$0.762798\pi$$
$$350$$ 9.89949i 0.0282843i
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 254.558i − 0.721129i −0.932734 0.360564i $$-0.882584\pi$$
0.932734 0.360564i $$-0.117416\pi$$
$$354$$ 0 0
$$355$$ −672.000 −1.89296
$$356$$ 101.823i 0.286021i
$$357$$ 0 0
$$358$$ 282.000 0.787709
$$359$$ − 452.548i − 1.26058i −0.776360 0.630290i $$-0.782936\pi$$
0.776360 0.630290i $$-0.217064\pi$$
$$360$$ 0 0
$$361$$ 728.000 2.01662
$$362$$ − 405.879i − 1.12121i
$$363$$ 0 0
$$364$$ 32.0000 0.0879121
$$365$$ − 265.872i − 0.728417i
$$366$$ 0 0
$$367$$ 160.000 0.435967 0.217984 0.975952i $$-0.430052\pi$$
0.217984 + 0.975952i $$0.430052\pi$$
$$368$$ 96.1665i 0.261322i
$$369$$ 0 0
$$370$$ −456.000 −1.23243
$$371$$ 89.0955i 0.240149i
$$372$$ 0 0
$$373$$ −127.000 −0.340483 −0.170241 0.985402i $$-0.554455\pi$$
−0.170241 + 0.985402i $$0.554455\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −516.000 −1.37234
$$377$$ − 203.647i − 0.540177i
$$378$$ 0 0
$$379$$ 454.000 1.19789 0.598945 0.800790i $$-0.295587\pi$$
0.598945 + 0.800790i $$0.295587\pi$$
$$380$$ 373.352i 0.982506i
$$381$$ 0 0
$$382$$ 232.000 0.607330
$$383$$ − 74.9533i − 0.195701i −0.995201 0.0978503i $$-0.968803\pi$$
0.995201 0.0978503i $$-0.0311966\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ − 205.061i − 0.531246i
$$387$$ 0 0
$$388$$ −50.0000 −0.128866
$$389$$ 4.24264i 0.0109065i 0.999985 + 0.00545327i $$0.00173584\pi$$
−0.999985 + 0.00545327i $$0.998264\pi$$
$$390$$ 0 0
$$391$$ −272.000 −0.695652
$$392$$ − 407.294i − 1.03901i
$$393$$ 0 0
$$394$$ 64.0000 0.162437
$$395$$ 130.108i 0.329386i
$$396$$ 0 0
$$397$$ 481.000 1.21159 0.605793 0.795622i $$-0.292856\pi$$
0.605793 + 0.795622i $$0.292856\pi$$
$$398$$ − 80.6102i − 0.202538i
$$399$$ 0 0
$$400$$ 28.0000 0.0700000
$$401$$ 429.921i 1.07212i 0.844179 + 0.536061i $$0.180088\pi$$
−0.844179 + 0.536061i $$0.819912\pi$$
$$402$$ 0 0
$$403$$ −496.000 −1.23077
$$404$$ 302.642i 0.749113i
$$405$$ 0 0
$$406$$ 18.0000 0.0443350
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −177.000 −0.432763 −0.216381 0.976309i $$-0.569425\pi$$
−0.216381 + 0.976309i $$0.569425\pi$$
$$410$$ − 124.451i − 0.303539i
$$411$$ 0 0
$$412$$ −50.0000 −0.121359
$$413$$ − 69.2965i − 0.167788i
$$414$$ 0 0
$$415$$ −600.000 −1.44578
$$416$$ − 452.548i − 1.08786i
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 306.884i 0.732421i 0.930532 + 0.366210i $$0.119345\pi$$
−0.930532 + 0.366210i $$0.880655\pi$$
$$420$$ 0 0
$$421$$ 160.000 0.380048 0.190024 0.981779i $$-0.439143\pi$$
0.190024 + 0.981779i $$0.439143\pi$$
$$422$$ 148.492i 0.351878i
$$423$$ 0 0
$$424$$ 756.000 1.78302
$$425$$ 79.1960i 0.186343i
$$426$$ 0 0
$$427$$ 105.000 0.245902
$$428$$ − 263.044i − 0.614588i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 691.550i 1.60453i 0.596971 + 0.802263i $$0.296371\pi$$
−0.596971 + 0.802263i $$0.703629\pi$$
$$432$$ 0 0
$$433$$ −143.000 −0.330254 −0.165127 0.986272i $$-0.552803\pi$$
−0.165127 + 0.986272i $$0.552803\pi$$
$$434$$ − 43.8406i − 0.101015i
$$435$$ 0 0
$$436$$ 238.000 0.545872
$$437$$ 793.374i 1.81550i
$$438$$ 0 0
$$439$$ 335.000 0.763098 0.381549 0.924349i $$-0.375391\pi$$
0.381549 + 0.924349i $$0.375391\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −256.000 −0.579186
$$443$$ 130.108i 0.293697i 0.989159 + 0.146848i $$0.0469129\pi$$
−0.989159 + 0.146848i $$0.953087\pi$$
$$444$$ 0 0
$$445$$ 288.000 0.647191
$$446$$ 281.428i 0.631006i
$$447$$ 0 0
$$448$$ 56.0000 0.125000
$$449$$ 854.185i 1.90242i 0.308550 + 0.951208i $$0.400156\pi$$
−0.308550 + 0.951208i $$0.599844\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 25.4558i 0.0563182i
$$453$$ 0 0
$$454$$ 264.000 0.581498
$$455$$ − 90.5097i − 0.198922i
$$456$$ 0 0
$$457$$ −264.000 −0.577681 −0.288840 0.957377i $$-0.593270\pi$$
−0.288840 + 0.957377i $$0.593270\pi$$
$$458$$ − 101.823i − 0.222322i
$$459$$ 0 0
$$460$$ −272.000 −0.591304
$$461$$ 123.037i 0.266891i 0.991056 + 0.133445i $$0.0426041\pi$$
−0.991056 + 0.133445i $$0.957396\pi$$
$$462$$ 0 0
$$463$$ −608.000 −1.31317 −0.656587 0.754250i $$-0.728001\pi$$
−0.656587 + 0.754250i $$0.728001\pi$$
$$464$$ − 50.9117i − 0.109723i
$$465$$ 0 0
$$466$$ 186.000 0.399142
$$467$$ − 424.264i − 0.908488i −0.890877 0.454244i $$-0.849909\pi$$
0.890877 0.454244i $$-0.150091\pi$$
$$468$$ 0 0
$$469$$ 103.000 0.219616
$$470$$ 486.489i 1.03508i
$$471$$ 0 0
$$472$$ −588.000 −1.24576
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 231.000 0.486316
$$476$$ 22.6274i 0.0475366i
$$477$$ 0 0
$$478$$ 6.00000 0.0125523
$$479$$ 830.143i 1.73308i 0.499111 + 0.866538i $$0.333660\pi$$
−0.499111 + 0.866538i $$0.666340\pi$$
$$480$$ 0 0
$$481$$ 912.000 1.89605
$$482$$ − 226.274i − 0.469448i
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 141.421i 0.291590i
$$486$$ 0 0
$$487$$ 56.0000 0.114990 0.0574949 0.998346i $$-0.481689\pi$$
0.0574949 + 0.998346i $$0.481689\pi$$
$$488$$ − 890.955i − 1.82573i
$$489$$ 0 0
$$490$$ −384.000 −0.783673
$$491$$ − 278.600i − 0.567414i −0.958911 0.283707i $$-0.908436\pi$$
0.958911 0.283707i $$-0.0915642\pi$$
$$492$$ 0 0
$$493$$ 144.000 0.292089
$$494$$ 746.705i 1.51155i
$$495$$ 0 0
$$496$$ −124.000 −0.250000
$$497$$ 118.794i 0.239022i
$$498$$ 0 0
$$499$$ −439.000 −0.879760 −0.439880 0.898057i $$-0.644979\pi$$
−0.439880 + 0.898057i $$0.644979\pi$$
$$500$$ − 203.647i − 0.407294i
$$501$$ 0 0
$$502$$ −154.000 −0.306773
$$503$$ − 605.283i − 1.20335i −0.798742 0.601673i $$-0.794501\pi$$
0.798742 0.601673i $$-0.205499\pi$$
$$504$$ 0 0
$$505$$ 856.000 1.69505
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 158.000 0.311024
$$509$$ 352.139i 0.691825i 0.938267 + 0.345913i $$0.112431\pi$$
−0.938267 + 0.345913i $$0.887569\pi$$
$$510$$ 0 0
$$511$$ −47.0000 −0.0919765
$$512$$ − 248.902i − 0.486136i
$$513$$ 0 0
$$514$$ −370.000 −0.719844
$$515$$ 141.421i 0.274605i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 80.6102i 0.155618i
$$519$$ 0 0
$$520$$ −768.000 −1.47692
$$521$$ 581.242i 1.11563i 0.829966 + 0.557814i $$0.188360\pi$$
−0.829966 + 0.557814i $$0.811640\pi$$
$$522$$ 0 0
$$523$$ 385.000 0.736138 0.368069 0.929799i $$-0.380019\pi$$
0.368069 + 0.929799i $$0.380019\pi$$
$$524$$ 229.103i 0.437219i
$$525$$ 0 0
$$526$$ −622.000 −1.18251
$$527$$ − 350.725i − 0.665512i
$$528$$ 0 0
$$529$$ −49.0000 −0.0926276
$$530$$ − 712.764i − 1.34484i
$$531$$ 0 0
$$532$$ 66.0000 0.124060
$$533$$ 248.902i 0.466982i
$$534$$ 0 0
$$535$$ −744.000 −1.39065
$$536$$ − 873.984i − 1.63057i
$$537$$ 0 0
$$538$$ −346.000 −0.643123
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −88.0000 −0.162662 −0.0813309 0.996687i $$-0.525917\pi$$
−0.0813309 + 0.996687i $$0.525917\pi$$
$$542$$ − 486.489i − 0.897582i
$$543$$ 0 0
$$544$$ 320.000 0.588235
$$545$$ − 673.166i − 1.23517i
$$546$$ 0 0
$$547$$ 424.000 0.775137 0.387569 0.921841i $$-0.373315\pi$$
0.387569 + 0.921841i $$0.373315\pi$$
$$548$$ 181.019i 0.330327i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 420.021i − 0.762289i
$$552$$ 0 0
$$553$$ 23.0000 0.0415913
$$554$$ 134.350i 0.242510i
$$555$$ 0 0
$$556$$ −116.000 −0.208633
$$557$$ 209.304i 0.375769i 0.982191 + 0.187885i $$0.0601631\pi$$
−0.982191 + 0.187885i $$0.939837\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ − 22.6274i − 0.0404061i
$$561$$ 0 0
$$562$$ −552.000 −0.982206
$$563$$ − 469.519i − 0.833959i −0.908916 0.416979i $$-0.863089\pi$$
0.908916 0.416979i $$-0.136911\pi$$
$$564$$ 0 0
$$565$$ 72.0000 0.127434
$$566$$ 473.762i 0.837035i
$$567$$ 0 0
$$568$$ 1008.00 1.77465
$$569$$ − 831.558i − 1.46144i −0.682679 0.730718i $$-0.739185\pi$$
0.682679 0.730718i $$-0.260815\pi$$
$$570$$ 0 0
$$571$$ −503.000 −0.880911 −0.440455 0.897775i $$-0.645183\pi$$
−0.440455 + 0.897775i $$0.645183\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −22.0000 −0.0383275
$$575$$ 168.291i 0.292681i
$$576$$ 0 0
$$577$$ −615.000 −1.06586 −0.532929 0.846160i $$-0.678909\pi$$
−0.532929 + 0.846160i $$0.678909\pi$$
$$578$$ 227.688i 0.393925i
$$579$$ 0 0
$$580$$ 144.000 0.248276
$$581$$ 106.066i 0.182558i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 398.808i 0.682891i
$$585$$ 0 0
$$586$$ 792.000 1.35154
$$587$$ − 5.65685i − 0.00963689i −0.999988 0.00481844i $$-0.998466\pi$$
0.999988 0.00481844i $$-0.00153376\pi$$
$$588$$ 0 0
$$589$$ −1023.00 −1.73684
$$590$$ 554.372i 0.939613i
$$591$$ 0 0
$$592$$ 228.000 0.385135
$$593$$ 287.085i 0.484124i 0.970261 + 0.242062i $$0.0778237\pi$$
−0.970261 + 0.242062i $$0.922176\pi$$
$$594$$ 0 0
$$595$$ 64.0000 0.107563
$$596$$ − 520.431i − 0.873206i
$$597$$ 0 0
$$598$$ −544.000 −0.909699
$$599$$ − 956.008i − 1.59601i −0.602653 0.798004i $$-0.705890\pi$$
0.602653 0.798004i $$-0.294110\pi$$
$$600$$ 0 0
$$601$$ 601.000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 128.000 0.211921
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 1024.00 1.68699 0.843493 0.537141i $$-0.180496\pi$$
0.843493 + 0.537141i $$0.180496\pi$$
$$608$$ − 933.381i − 1.53517i
$$609$$ 0 0
$$610$$ −840.000 −1.37705
$$611$$ − 972.979i − 1.59244i
$$612$$ 0 0
$$613$$ 721.000 1.17618 0.588091 0.808795i $$-0.299880\pi$$
0.588091 + 0.808795i $$0.299880\pi$$
$$614$$ − 439.820i − 0.716320i
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 50.9117i 0.0825149i 0.999149 + 0.0412574i $$0.0131364\pi$$
−0.999149 + 0.0412574i $$0.986864\pi$$
$$618$$ 0 0
$$619$$ 152.000 0.245557 0.122779 0.992434i $$-0.460819\pi$$
0.122779 + 0.992434i $$0.460819\pi$$
$$620$$ − 350.725i − 0.565685i
$$621$$ 0 0
$$622$$ −24.0000 −0.0385852
$$623$$ − 50.9117i − 0.0817202i
$$624$$ 0 0
$$625$$ −751.000 −1.20160
$$626$$ 282.843i 0.451825i
$$627$$ 0 0
$$628$$ 514.000 0.818471
$$629$$ 644.881i 1.02525i
$$630$$ 0 0
$$631$$ 544.000 0.862124 0.431062 0.902322i $$-0.358139\pi$$
0.431062 + 0.902322i $$0.358139\pi$$
$$632$$ − 195.161i − 0.308800i
$$633$$ 0 0
$$634$$ −400.000 −0.630915
$$635$$ − 446.891i − 0.703766i
$$636$$ 0 0
$$637$$ 768.000 1.20565
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 192.000 0.300000
$$641$$ 814.587i 1.27081i 0.772180 + 0.635403i $$0.219166\pi$$
−0.772180 + 0.635403i $$0.780834\pi$$
$$642$$ 0 0
$$643$$ −1231.00 −1.91446 −0.957232 0.289322i $$-0.906570\pi$$
−0.957232 + 0.289322i $$0.906570\pi$$
$$644$$ 48.0833i 0.0746634i
$$645$$ 0 0
$$646$$ −528.000 −0.817337
$$647$$ 158.392i 0.244810i 0.992480 + 0.122405i $$0.0390606\pi$$
−0.992480 + 0.122405i $$0.960939\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 158.392i 0.243680i
$$651$$ 0 0
$$652$$ 158.000 0.242331
$$653$$ − 264.458i − 0.404989i −0.979283 0.202495i $$-0.935095\pi$$
0.979283 0.202495i $$-0.0649049\pi$$
$$654$$ 0 0
$$655$$ 648.000 0.989313
$$656$$ 62.2254i 0.0948558i
$$657$$ 0 0
$$658$$ 86.0000 0.130699
$$659$$ − 490.732i − 0.744662i −0.928100 0.372331i $$-0.878559\pi$$
0.928100 0.372331i $$-0.121441\pi$$
$$660$$ 0 0
$$661$$ −521.000 −0.788200 −0.394100 0.919068i $$-0.628944\pi$$
−0.394100 + 0.919068i $$0.628944\pi$$
$$662$$ 816.001i 1.23263i
$$663$$ 0 0
$$664$$ 900.000 1.35542
$$665$$ − 186.676i − 0.280716i
$$666$$ 0 0
$$667$$ 306.000 0.458771
$$668$$ 59.3970i 0.0889176i
$$669$$ 0 0
$$670$$ −824.000 −1.22985
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −239.000 −0.355126 −0.177563 0.984109i $$-0.556821\pi$$
−0.177563 + 0.984109i $$0.556821\pi$$
$$674$$ 408.708i 0.606391i
$$675$$ 0 0
$$676$$ 174.000 0.257396
$$677$$ − 305.470i − 0.451211i −0.974219 0.225606i $$-0.927564\pi$$
0.974219 0.225606i $$-0.0724361\pi$$
$$678$$ 0 0
$$679$$ 25.0000 0.0368189
$$680$$ − 543.058i − 0.798615i
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 1101.67i − 1.61299i −0.591241 0.806495i $$-0.701362\pi$$
0.591241 0.806495i $$-0.298638\pi$$
$$684$$ 0 0
$$685$$ 512.000 0.747445
$$686$$ 137.179i 0.199969i
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 1425.53i 2.06898i
$$690$$ 0 0
$$691$$ −919.000 −1.32996 −0.664978 0.746863i $$-0.731559\pi$$
−0.664978 + 0.746863i $$0.731559\pi$$
$$692$$ − 135.765i − 0.196191i
$$693$$ 0 0
$$694$$ 766.000 1.10375
$$695$$ 328.098i 0.472083i
$$696$$ 0 0
$$697$$ −176.000 −0.252511
$$698$$ − 725.492i − 1.03939i
$$699$$ 0 0
$$700$$ 14.0000 0.0200000
$$701$$ − 646.296i − 0.921962i −0.887410 0.460981i $$-0.847498\pi$$
0.887410 0.460981i $$-0.152502\pi$$
$$702$$ 0 0
$$703$$ 1881.00 2.67568
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 360.000 0.509915
$$707$$ − 151.321i − 0.214032i
$$708$$ 0 0
$$709$$ 518.000 0.730606 0.365303 0.930889i $$-0.380965\pi$$
0.365303 + 0.930889i $$0.380965\pi$$
$$710$$ − 950.352i − 1.33852i
$$711$$ 0 0
$$712$$ −432.000 −0.606742
$$713$$ − 745.291i − 1.04529i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ − 398.808i − 0.556995i
$$717$$ 0 0
$$718$$ 640.000 0.891365
$$719$$ − 182.434i − 0.253732i −0.991920 0.126866i $$-0.959508\pi$$
0.991920 0.126866i $$-0.0404919\pi$$
$$720$$ 0 0
$$721$$ 25.0000 0.0346741
$$722$$ 1029.55i 1.42597i
$$723$$ 0 0
$$724$$ −574.000 −0.792818
$$725$$ − 89.0955i − 0.122890i
$$726$$ 0 0
$$727$$ −114.000 −0.156809 −0.0784044 0.996922i $$-0.524983\pi$$
−0.0784044 + 0.996922i $$0.524983\pi$$
$$728$$ 135.765i 0.186490i
$$729$$ 0 0
$$730$$ 376.000 0.515068
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 234.000 0.319236 0.159618 0.987179i $$-0.448974\pi$$
0.159618 + 0.987179i $$0.448974\pi$$
$$734$$ 226.274i 0.308275i
$$735$$ 0 0
$$736$$ 680.000 0.923913
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −1241.00 −1.67930 −0.839648 0.543131i $$-0.817239\pi$$
−0.839648 + 0.543131i $$0.817239\pi$$
$$740$$ 644.881i 0.871461i
$$741$$ 0 0
$$742$$ −126.000 −0.169811
$$743$$ 141.421i 0.190338i 0.995461 + 0.0951691i $$0.0303392\pi$$
−0.995461 + 0.0951691i $$0.969661\pi$$
$$744$$ 0 0
$$745$$ −1472.00 −1.97584
$$746$$ − 179.605i − 0.240758i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 131.522i 0.175597i
$$750$$ 0 0
$$751$$ −113.000 −0.150466 −0.0752330 0.997166i $$-0.523970\pi$$
−0.0752330 + 0.997166i $$0.523970\pi$$
$$752$$ − 243.245i − 0.323464i
$$753$$ 0 0
$$754$$ 288.000 0.381963
$$755$$ − 362.039i − 0.479521i
$$756$$ 0 0
$$757$$ −55.0000 −0.0726552 −0.0363276 0.999340i $$-0.511566\pi$$
−0.0363276 + 0.999340i $$0.511566\pi$$
$$758$$ 642.053i 0.847036i
$$759$$ 0 0
$$760$$ −1584.00 −2.08421
$$761$$ − 882.469i − 1.15962i −0.814752 0.579809i $$-0.803127\pi$$
0.814752 0.579809i $$-0.196873\pi$$
$$762$$ 0 0
$$763$$ −119.000 −0.155963
$$764$$ − 328.098i − 0.429447i
$$765$$ 0 0
$$766$$ 106.000 0.138381
$$767$$ − 1108.74i − 1.44556i
$$768$$ 0 0
$$769$$ −41.0000 −0.0533160 −0.0266580 0.999645i $$-0.508487\pi$$
−0.0266580 + 0.999645i $$0.508487\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −290.000 −0.375648
$$773$$ 339.411i 0.439083i 0.975603 + 0.219542i $$0.0704562\pi$$
−0.975603 + 0.219542i $$0.929544\pi$$
$$774$$ 0 0
$$775$$ −217.000 −0.280000
$$776$$ − 212.132i − 0.273366i
$$777$$ 0 0
$$778$$ −6.00000 −0.00771208
$$779$$ 513.360i 0.658998i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 384.666i − 0.491900i
$$783$$ 0 0
$$784$$ 192.000 0.244898
$$785$$ − 1453.81i − 1.85199i
$$786$$ 0 0
$$787$$ −1024.00 −1.30114 −0.650572 0.759445i $$-0.725471\pi$$
−0.650572 + 0.759445i $$0.725471\pi$$
$$788$$ − 90.5097i − 0.114860i
$$789$$ 0 0
$$790$$ −184.000 −0.232911
$$791$$ − 12.7279i − 0.0160909i
$$792$$ 0 0
$$793$$ 1680.00 2.11854
$$794$$ 680.237i 0.856721i
$$795$$ 0 0
$$796$$ −114.000 −0.143216
$$797$$ − 7.07107i − 0.00887211i −0.999990 0.00443605i $$-0.998588\pi$$
0.999990 0.00443605i $$-0.00141204\pi$$
$$798$$ 0 0
$$799$$ 688.000 0.861076
$$800$$ − 197.990i − 0.247487i
$$801$$ 0 0
$$802$$ −608.000 −0.758105
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 136.000 0.168944
$$806$$ − 701.450i − 0.870285i
$$807$$ 0 0
$$808$$ −1284.00 −1.58911
$$809$$ − 933.381i − 1.15375i −0.816834 0.576873i $$-0.804273\pi$$
0.816834 0.576873i $$-0.195727\pi$$
$$810$$ 0 0
$$811$$ −1513.00 −1.86560 −0.932799 0.360397i $$-0.882641\pi$$
−0.932799 + 0.360397i $$0.882641\pi$$
$$812$$ − 25.4558i − 0.0313496i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 446.891i − 0.548333i
$$816$$ 0 0
$$817$$ 0 0
$$818$$ − 250.316i − 0.306010i
$$819$$ 0 0
$$820$$ −176.000 −0.214634
$$821$$ 926.310i 1.12827i 0.825682 + 0.564135i $$0.190790\pi$$
−0.825682 + 0.564135i $$0.809210\pi$$
$$822$$ 0 0
$$823$$ 615.000 0.747266 0.373633 0.927577i $$-0.378112\pi$$
0.373633 + 0.927577i $$0.378112\pi$$
$$824$$ − 212.132i − 0.257442i
$$825$$ 0 0
$$826$$ 98.0000 0.118644
$$827$$ 1084.70i 1.31161i 0.754930 + 0.655805i $$0.227671\pi$$
−0.754930 + 0.655805i $$0.772329\pi$$
$$828$$ 0 0
$$829$$ 97.0000 0.117008 0.0585042 0.998287i $$-0.481367\pi$$
0.0585042 + 0.998287i $$0.481367\pi$$
$$830$$ − 848.528i − 1.02232i
$$831$$ 0 0
$$832$$ 896.000 1.07692
$$833$$ 543.058i 0.651930i
$$834$$ 0 0
$$835$$ 168.000 0.201198
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −434.000 −0.517900
$$839$$ − 1363.30i − 1.62491i −0.583022 0.812456i $$-0.698130\pi$$
0.583022 0.812456i $$-0.301870\pi$$
$$840$$ 0 0
$$841$$ 679.000 0.807372
$$842$$ 226.274i 0.268734i
$$843$$ 0 0
$$844$$ 210.000 0.248815
$$845$$ − 492.146i − 0.582422i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 356.382i 0.420262i
$$849$$ 0 0
$$850$$ −112.000 −0.131765
$$851$$ 1370.37i 1.61031i
$$852$$ 0 0
$$853$$ −41.0000 −0.0480657 −0.0240328 0.999711i $$-0.507651\pi$$
−0.0240328 + 0.999711i $$0.507651\pi$$
$$854$$ 148.492i 0.173879i
$$855$$ 0 0
$$856$$ 1116.00 1.30374
$$857$$ 401.637i 0.468654i 0.972158 + 0.234327i $$0.0752887\pi$$
−0.972158 + 0.234327i $$0.924711\pi$$
$$858$$ 0 0
$$859$$ −1425.00 −1.65891 −0.829453 0.558577i $$-0.811348\pi$$
−0.829453 + 0.558577i $$0.811348\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −978.000 −1.13457
$$863$$ − 1107.33i − 1.28312i −0.767074 0.641558i $$-0.778288\pi$$
0.767074 0.641558i $$-0.221712\pi$$
$$864$$ 0 0
$$865$$ −384.000 −0.443931
$$866$$ − 202.233i − 0.233525i
$$867$$ 0 0
$$868$$ −62.0000 −0.0714286
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 1648.00 1.89208
$$872$$ 1009.75i 1.15797i
$$873$$ 0 0
$$874$$ −1122.00 −1.28375
$$875$$ 101.823i 0.116370i
$$876$$ 0 0
$$877$$ 1593.00 1.81642 0.908210 0.418515i $$-0.137449\pi$$
0.908210 + 0.418515i $$0.137449\pi$$
$$878$$ 473.762i 0.539592i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 646.296i 0.733593i 0.930301 + 0.366797i $$0.119546\pi$$
−0.930301 + 0.366797i $$0.880454\pi$$
$$882$$ 0 0
$$883$$ −1145.00 −1.29672 −0.648358 0.761336i $$-0.724544\pi$$
−0.648358 + 0.761336i $$0.724544\pi$$
$$884$$ 362.039i 0.409546i
$$885$$ 0 0
$$886$$ −184.000 −0.207675
$$887$$ 439.820i 0.495852i 0.968779 + 0.247926i $$0.0797489\pi$$
−0.968779 + 0.247926i $$0.920251\pi$$
$$888$$ 0 0
$$889$$ −79.0000 −0.0888639
$$890$$ 407.294i 0.457633i
$$891$$ 0 0
$$892$$ 398.000 0.446188
$$893$$ − 2006.77i − 2.24722i
$$894$$ 0 0
$$895$$ −1128.00 −1.26034
$$896$$ − 33.9411i − 0.0378807i
$$897$$ 0 0
$$898$$ −1208.00 −1.34521
$$899$$ 394.566i 0.438894i
$$900$$ 0 0
$$901$$ −1008.00 −1.11876
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −108.000 −0.119469
$$905$$ 1623.52i 1.79394i
$$906$$ 0 0
$$907$$ −87.0000 −0.0959206 −0.0479603 0.998849i $$-0.515272\pi$$
−0.0479603 + 0.998849i $$0.515272\pi$$
$$908$$ − 373.352i − 0.411181i
$$909$$ 0 0
$$910$$ 128.000 0.140659
$$911$$ − 1493.41i − 1.63931i −0.572859 0.819654i $$-0.694166\pi$$
0.572859 0.819654i $$-0.305834\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ − 373.352i − 0.408482i
$$915$$ 0 0
$$916$$ −144.000 −0.157205
$$917$$ − 114.551i − 0.124920i
$$918$$ 0 0
$$919$$ −865.000 −0.941240 −0.470620 0.882336i $$-0.655970\pi$$
−0.470620 + 0.882336i $$0.655970\pi$$
$$920$$ − 1154.00i − 1.25435i
$$921$$ 0 0
$$922$$ −174.000 −0.188720
$$923$$ 1900.70i 2.05927i
$$924$$ 0 0
$$925$$ 399.000 0.431351
$$926$$ − 859.842i − 0.928555i
$$927$$ 0 0
$$928$$ −360.000 −0.387931
$$929$$ − 1192.18i − 1.28330i −0.766999 0.641648i $$-0.778251\pi$$
0.766999 0.641648i $$-0.221749\pi$$
$$930$$ 0 0
$$931$$ 1584.00 1.70140
$$932$$ − 263.044i − 0.282236i
$$933$$ 0 0
$$934$$ 600.000 0.642398
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −473.000 −0.504803 −0.252401 0.967623i $$-0.581220\pi$$
−0.252401 + 0.967623i $$0.581220\pi$$
$$938$$ 145.664i 0.155292i
$$939$$ 0 0
$$940$$ 688.000 0.731915
$$941$$ 114.551i 0.121734i 0.998146 + 0.0608668i $$0.0193865\pi$$
−0.998146 + 0.0608668i $$0.980614\pi$$
$$942$$ 0 0
$$943$$ −374.000 −0.396607
$$944$$ − 277.186i − 0.293629i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 818.830i − 0.864656i −0.901716 0.432328i $$-0.857692\pi$$
0.901716 0.432328i $$-0.142308\pi$$
$$948$$ 0 0
$$949$$ −752.000 −0.792413
$$950$$ 326.683i 0.343877i
$$951$$ 0 0
$$952$$ −96.0000 −0.100840
$$953$$ − 1042.28i − 1.09368i −0.837238 0.546839i $$-0.815831\pi$$
0.837238 0.546839i $$-0.184169\pi$$
$$954$$ 0 0
$$955$$ −928.000 −0.971728
$$956$$ − 8.48528i − 0.00887582i
$$957$$ 0 0
$$958$$ −1174.00 −1.22547
$$959$$ − 90.5097i − 0.0943792i
$$960$$ 0 0
$$961$$ 0 0
$$962$$ 1289.76i 1.34071i
$$963$$ 0 0
$$964$$ −320.000 −0.331950
$$965$$ 820.244i 0.849994i
$$966$$ 0 0
$$967$$ −937.000 −0.968976 −0.484488 0.874798i $$-0.660994\pi$$
−0.484488 + 0.874798i $$0.660994\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −200.000 −0.206186
$$971$$ − 694.379i − 0.715117i −0.933891 0.357559i $$-0.883609\pi$$
0.933891 0.357559i $$-0.116391\pi$$
$$972$$ 0 0
$$973$$ 58.0000 0.0596095
$$974$$ 79.1960i 0.0813100i
$$975$$ 0 0
$$976$$ 420.000 0.430328
$$977$$ − 708.521i − 0.725201i −0.931945 0.362600i $$-0.881889\pi$$
0.931945 0.362600i $$-0.118111\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 543.058i 0.554141i
$$981$$ 0 0
$$982$$ 394.000 0.401222
$$983$$ 547.301i 0.556766i 0.960470 + 0.278383i $$0.0897984\pi$$
−0.960470 + 0.278383i $$0.910202\pi$$
$$984$$ 0 0
$$985$$ −256.000 −0.259898
$$986$$ 203.647i 0.206538i
$$987$$ 0 0
$$988$$ 1056.00 1.06883
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 286.000 0.288597 0.144299 0.989534i $$-0.453907\pi$$
0.144299 + 0.989534i $$0.453907\pi$$
$$992$$ 876.812i 0.883883i
$$993$$ 0 0
$$994$$ −168.000 −0.169014
$$995$$ 322.441i 0.324061i
$$996$$ 0 0
$$997$$ 265.000 0.265797 0.132899 0.991130i $$-0.457572\pi$$
0.132899 + 0.991130i $$0.457572\pi$$
$$998$$ − 620.840i − 0.622084i
$$999$$ 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 1089.3.b.a.485.2 yes 2
3.2 odd 2 inner 1089.3.b.a.485.1 2
11.10 odd 2 1089.3.b.b.485.1 yes 2
33.32 even 2 1089.3.b.b.485.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.b.a.485.1 2 3.2 odd 2 inner
1089.3.b.a.485.2 yes 2 1.1 even 1 trivial
1089.3.b.b.485.1 yes 2 11.10 odd 2
1089.3.b.b.485.2 yes 2 33.32 even 2