# Properties

 Label 1617.2.a.n.1.1 Level $1617$ Weight $2$ Character 1617.1 Self dual yes Analytic conductor $12.912$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9118100068$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1617.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -2.00000 q^{5} -2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -2.00000 q^{5} -2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +4.82843 q^{10} +1.00000 q^{11} +3.82843 q^{12} -0.828427 q^{13} -2.00000 q^{15} +3.00000 q^{16} -4.41421 q^{17} -2.41421 q^{18} +7.24264 q^{19} -7.65685 q^{20} -2.41421 q^{22} -7.00000 q^{23} -4.41421 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +3.24264 q^{29} +4.82843 q^{30} +5.65685 q^{31} +1.58579 q^{32} +1.00000 q^{33} +10.6569 q^{34} +3.82843 q^{36} -9.48528 q^{37} -17.4853 q^{38} -0.828427 q^{39} +8.82843 q^{40} -1.17157 q^{41} -2.75736 q^{43} +3.82843 q^{44} -2.00000 q^{45} +16.8995 q^{46} +9.82843 q^{47} +3.00000 q^{48} +2.41421 q^{50} -4.41421 q^{51} -3.17157 q^{52} -7.17157 q^{53} -2.41421 q^{54} -2.00000 q^{55} +7.24264 q^{57} -7.82843 q^{58} -8.65685 q^{59} -7.65685 q^{60} -4.00000 q^{61} -13.6569 q^{62} -9.82843 q^{64} +1.65685 q^{65} -2.41421 q^{66} -3.17157 q^{67} -16.8995 q^{68} -7.00000 q^{69} -4.17157 q^{71} -4.41421 q^{72} +0.343146 q^{73} +22.8995 q^{74} -1.00000 q^{75} +27.7279 q^{76} +2.00000 q^{78} +13.3137 q^{79} -6.00000 q^{80} +1.00000 q^{81} +2.82843 q^{82} -2.82843 q^{83} +8.82843 q^{85} +6.65685 q^{86} +3.24264 q^{87} -4.41421 q^{88} -14.1421 q^{89} +4.82843 q^{90} -26.7990 q^{92} +5.65685 q^{93} -23.7279 q^{94} -14.4853 q^{95} +1.58579 q^{96} -11.4853 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 6 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 4 q^{13} - 4 q^{15} + 6 q^{16} - 6 q^{17} - 2 q^{18} + 6 q^{19} - 4 q^{20} - 2 q^{22} - 14 q^{23} - 6 q^{24} - 2 q^{25} + 4 q^{26} + 2 q^{27} - 2 q^{29} + 4 q^{30} + 6 q^{32} + 2 q^{33} + 10 q^{34} + 2 q^{36} - 2 q^{37} - 18 q^{38} + 4 q^{39} + 12 q^{40} - 8 q^{41} - 14 q^{43} + 2 q^{44} - 4 q^{45} + 14 q^{46} + 14 q^{47} + 6 q^{48} + 2 q^{50} - 6 q^{51} - 12 q^{52} - 20 q^{53} - 2 q^{54} - 4 q^{55} + 6 q^{57} - 10 q^{58} - 6 q^{59} - 4 q^{60} - 8 q^{61} - 16 q^{62} - 14 q^{64} - 8 q^{65} - 2 q^{66} - 12 q^{67} - 14 q^{68} - 14 q^{69} - 14 q^{71} - 6 q^{72} + 12 q^{73} + 26 q^{74} - 2 q^{75} + 30 q^{76} + 4 q^{78} + 4 q^{79} - 12 q^{80} + 2 q^{81} + 12 q^{85} + 2 q^{86} - 2 q^{87} - 6 q^{88} + 4 q^{90} - 14 q^{92} - 22 q^{94} - 12 q^{95} + 6 q^{96} - 6 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 6 * q^8 + 2 * q^9 + 4 * q^10 + 2 * q^11 + 2 * q^12 + 4 * q^13 - 4 * q^15 + 6 * q^16 - 6 * q^17 - 2 * q^18 + 6 * q^19 - 4 * q^20 - 2 * q^22 - 14 * q^23 - 6 * q^24 - 2 * q^25 + 4 * q^26 + 2 * q^27 - 2 * q^29 + 4 * q^30 + 6 * q^32 + 2 * q^33 + 10 * q^34 + 2 * q^36 - 2 * q^37 - 18 * q^38 + 4 * q^39 + 12 * q^40 - 8 * q^41 - 14 * q^43 + 2 * q^44 - 4 * q^45 + 14 * q^46 + 14 * q^47 + 6 * q^48 + 2 * q^50 - 6 * q^51 - 12 * q^52 - 20 * q^53 - 2 * q^54 - 4 * q^55 + 6 * q^57 - 10 * q^58 - 6 * q^59 - 4 * q^60 - 8 * q^61 - 16 * q^62 - 14 * q^64 - 8 * q^65 - 2 * q^66 - 12 * q^67 - 14 * q^68 - 14 * q^69 - 14 * q^71 - 6 * q^72 + 12 * q^73 + 26 * q^74 - 2 * q^75 + 30 * q^76 + 4 * q^78 + 4 * q^79 - 12 * q^80 + 2 * q^81 + 12 * q^85 + 2 * q^86 - 2 * q^87 - 6 * q^88 + 4 * q^90 - 14 * q^92 - 22 * q^94 - 12 * q^95 + 6 * q^96 - 6 * q^97 + 2 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.41421 −1.70711 −0.853553 0.521005i $$-0.825557\pi$$
−0.853553 + 0.521005i $$0.825557\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 3.82843 1.91421
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ −2.41421 −0.985599
$$7$$ 0 0
$$8$$ −4.41421 −1.56066
$$9$$ 1.00000 0.333333
$$10$$ 4.82843 1.52688
$$11$$ 1.00000 0.301511
$$12$$ 3.82843 1.10517
$$13$$ −0.828427 −0.229764 −0.114882 0.993379i $$-0.536649\pi$$
−0.114882 + 0.993379i $$0.536649\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 3.00000 0.750000
$$17$$ −4.41421 −1.07060 −0.535302 0.844661i $$-0.679802\pi$$
−0.535302 + 0.844661i $$0.679802\pi$$
$$18$$ −2.41421 −0.569036
$$19$$ 7.24264 1.66158 0.830788 0.556589i $$-0.187890\pi$$
0.830788 + 0.556589i $$0.187890\pi$$
$$20$$ −7.65685 −1.71212
$$21$$ 0 0
$$22$$ −2.41421 −0.514712
$$23$$ −7.00000 −1.45960 −0.729800 0.683660i $$-0.760387\pi$$
−0.729800 + 0.683660i $$0.760387\pi$$
$$24$$ −4.41421 −0.901048
$$25$$ −1.00000 −0.200000
$$26$$ 2.00000 0.392232
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 3.24264 0.602143 0.301072 0.953602i $$-0.402656\pi$$
0.301072 + 0.953602i $$0.402656\pi$$
$$30$$ 4.82843 0.881546
$$31$$ 5.65685 1.01600 0.508001 0.861357i $$-0.330385\pi$$
0.508001 + 0.861357i $$0.330385\pi$$
$$32$$ 1.58579 0.280330
$$33$$ 1.00000 0.174078
$$34$$ 10.6569 1.82764
$$35$$ 0 0
$$36$$ 3.82843 0.638071
$$37$$ −9.48528 −1.55937 −0.779685 0.626172i $$-0.784621\pi$$
−0.779685 + 0.626172i $$0.784621\pi$$
$$38$$ −17.4853 −2.83649
$$39$$ −0.828427 −0.132655
$$40$$ 8.82843 1.39590
$$41$$ −1.17157 −0.182969 −0.0914845 0.995807i $$-0.529161\pi$$
−0.0914845 + 0.995807i $$0.529161\pi$$
$$42$$ 0 0
$$43$$ −2.75736 −0.420493 −0.210247 0.977648i $$-0.567427\pi$$
−0.210247 + 0.977648i $$0.567427\pi$$
$$44$$ 3.82843 0.577157
$$45$$ −2.00000 −0.298142
$$46$$ 16.8995 2.49169
$$47$$ 9.82843 1.43362 0.716812 0.697267i $$-0.245601\pi$$
0.716812 + 0.697267i $$0.245601\pi$$
$$48$$ 3.00000 0.433013
$$49$$ 0 0
$$50$$ 2.41421 0.341421
$$51$$ −4.41421 −0.618114
$$52$$ −3.17157 −0.439818
$$53$$ −7.17157 −0.985091 −0.492546 0.870287i $$-0.663934\pi$$
−0.492546 + 0.870287i $$0.663934\pi$$
$$54$$ −2.41421 −0.328533
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 7.24264 0.959311
$$58$$ −7.82843 −1.02792
$$59$$ −8.65685 −1.12703 −0.563513 0.826107i $$-0.690551\pi$$
−0.563513 + 0.826107i $$0.690551\pi$$
$$60$$ −7.65685 −0.988496
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ −13.6569 −1.73442
$$63$$ 0 0
$$64$$ −9.82843 −1.22855
$$65$$ 1.65685 0.205507
$$66$$ −2.41421 −0.297169
$$67$$ −3.17157 −0.387469 −0.193735 0.981054i $$-0.562060\pi$$
−0.193735 + 0.981054i $$0.562060\pi$$
$$68$$ −16.8995 −2.04936
$$69$$ −7.00000 −0.842701
$$70$$ 0 0
$$71$$ −4.17157 −0.495075 −0.247537 0.968878i $$-0.579621\pi$$
−0.247537 + 0.968878i $$0.579621\pi$$
$$72$$ −4.41421 −0.520220
$$73$$ 0.343146 0.0401622 0.0200811 0.999798i $$-0.493608\pi$$
0.0200811 + 0.999798i $$0.493608\pi$$
$$74$$ 22.8995 2.66201
$$75$$ −1.00000 −0.115470
$$76$$ 27.7279 3.18061
$$77$$ 0 0
$$78$$ 2.00000 0.226455
$$79$$ 13.3137 1.49791 0.748955 0.662621i $$-0.230556\pi$$
0.748955 + 0.662621i $$0.230556\pi$$
$$80$$ −6.00000 −0.670820
$$81$$ 1.00000 0.111111
$$82$$ 2.82843 0.312348
$$83$$ −2.82843 −0.310460 −0.155230 0.987878i $$-0.549612\pi$$
−0.155230 + 0.987878i $$0.549612\pi$$
$$84$$ 0 0
$$85$$ 8.82843 0.957577
$$86$$ 6.65685 0.717827
$$87$$ 3.24264 0.347648
$$88$$ −4.41421 −0.470557
$$89$$ −14.1421 −1.49906 −0.749532 0.661968i $$-0.769721\pi$$
−0.749532 + 0.661968i $$0.769721\pi$$
$$90$$ 4.82843 0.508961
$$91$$ 0 0
$$92$$ −26.7990 −2.79399
$$93$$ 5.65685 0.586588
$$94$$ −23.7279 −2.44735
$$95$$ −14.4853 −1.48616
$$96$$ 1.58579 0.161849
$$97$$ −11.4853 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ −3.82843 −0.382843
$$101$$ 4.89949 0.487518 0.243759 0.969836i $$-0.421619\pi$$
0.243759 + 0.969836i $$0.421619\pi$$
$$102$$ 10.6569 1.05519
$$103$$ 12.4853 1.23021 0.615106 0.788445i $$-0.289113\pi$$
0.615106 + 0.788445i $$0.289113\pi$$
$$104$$ 3.65685 0.358584
$$105$$ 0 0
$$106$$ 17.3137 1.68166
$$107$$ 9.65685 0.933563 0.466782 0.884373i $$-0.345413\pi$$
0.466782 + 0.884373i $$0.345413\pi$$
$$108$$ 3.82843 0.368391
$$109$$ −6.82843 −0.654045 −0.327022 0.945017i $$-0.606045\pi$$
−0.327022 + 0.945017i $$0.606045\pi$$
$$110$$ 4.82843 0.460372
$$111$$ −9.48528 −0.900303
$$112$$ 0 0
$$113$$ −7.65685 −0.720296 −0.360148 0.932895i $$-0.617274\pi$$
−0.360148 + 0.932895i $$0.617274\pi$$
$$114$$ −17.4853 −1.63765
$$115$$ 14.0000 1.30551
$$116$$ 12.4142 1.15263
$$117$$ −0.828427 −0.0765881
$$118$$ 20.8995 1.92395
$$119$$ 0 0
$$120$$ 8.82843 0.805921
$$121$$ 1.00000 0.0909091
$$122$$ 9.65685 0.874291
$$123$$ −1.17157 −0.105637
$$124$$ 21.6569 1.94484
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −16.0711 −1.42608 −0.713038 0.701125i $$-0.752681\pi$$
−0.713038 + 0.701125i $$0.752681\pi$$
$$128$$ 20.5563 1.81694
$$129$$ −2.75736 −0.242772
$$130$$ −4.00000 −0.350823
$$131$$ −5.17157 −0.451842 −0.225921 0.974146i $$-0.572539\pi$$
−0.225921 + 0.974146i $$0.572539\pi$$
$$132$$ 3.82843 0.333222
$$133$$ 0 0
$$134$$ 7.65685 0.661451
$$135$$ −2.00000 −0.172133
$$136$$ 19.4853 1.67085
$$137$$ 0.485281 0.0414604 0.0207302 0.999785i $$-0.493401\pi$$
0.0207302 + 0.999785i $$0.493401\pi$$
$$138$$ 16.8995 1.43858
$$139$$ 8.41421 0.713684 0.356842 0.934165i $$-0.383853\pi$$
0.356842 + 0.934165i $$0.383853\pi$$
$$140$$ 0 0
$$141$$ 9.82843 0.827703
$$142$$ 10.0711 0.845145
$$143$$ −0.828427 −0.0692766
$$144$$ 3.00000 0.250000
$$145$$ −6.48528 −0.538573
$$146$$ −0.828427 −0.0685611
$$147$$ 0 0
$$148$$ −36.3137 −2.98497
$$149$$ −22.2132 −1.81978 −0.909888 0.414853i $$-0.863833\pi$$
−0.909888 + 0.414853i $$0.863833\pi$$
$$150$$ 2.41421 0.197120
$$151$$ −18.8995 −1.53802 −0.769010 0.639237i $$-0.779250\pi$$
−0.769010 + 0.639237i $$0.779250\pi$$
$$152$$ −31.9706 −2.59316
$$153$$ −4.41421 −0.356868
$$154$$ 0 0
$$155$$ −11.3137 −0.908739
$$156$$ −3.17157 −0.253929
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ −32.1421 −2.55709
$$159$$ −7.17157 −0.568743
$$160$$ −3.17157 −0.250735
$$161$$ 0 0
$$162$$ −2.41421 −0.189679
$$163$$ −20.9706 −1.64254 −0.821271 0.570539i $$-0.806734\pi$$
−0.821271 + 0.570539i $$0.806734\pi$$
$$164$$ −4.48528 −0.350242
$$165$$ −2.00000 −0.155700
$$166$$ 6.82843 0.529989
$$167$$ 5.17157 0.400188 0.200094 0.979777i $$-0.435875\pi$$
0.200094 + 0.979777i $$0.435875\pi$$
$$168$$ 0 0
$$169$$ −12.3137 −0.947208
$$170$$ −21.3137 −1.63469
$$171$$ 7.24264 0.553859
$$172$$ −10.5563 −0.804914
$$173$$ −14.8284 −1.12738 −0.563692 0.825985i $$-0.690620\pi$$
−0.563692 + 0.825985i $$0.690620\pi$$
$$174$$ −7.82843 −0.593472
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ −8.65685 −0.650689
$$178$$ 34.1421 2.55906
$$179$$ 17.8284 1.33256 0.666280 0.745702i $$-0.267886\pi$$
0.666280 + 0.745702i $$0.267886\pi$$
$$180$$ −7.65685 −0.570708
$$181$$ −11.6569 −0.866447 −0.433224 0.901286i $$-0.642624\pi$$
−0.433224 + 0.901286i $$0.642624\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 30.8995 2.27794
$$185$$ 18.9706 1.39474
$$186$$ −13.6569 −1.00137
$$187$$ −4.41421 −0.322799
$$188$$ 37.6274 2.74426
$$189$$ 0 0
$$190$$ 34.9706 2.53703
$$191$$ −21.6569 −1.56703 −0.783517 0.621370i $$-0.786577\pi$$
−0.783517 + 0.621370i $$0.786577\pi$$
$$192$$ −9.82843 −0.709306
$$193$$ −12.1421 −0.874010 −0.437005 0.899459i $$-0.643961\pi$$
−0.437005 + 0.899459i $$0.643961\pi$$
$$194$$ 27.7279 1.99075
$$195$$ 1.65685 0.118650
$$196$$ 0 0
$$197$$ −16.4142 −1.16946 −0.584732 0.811226i $$-0.698800\pi$$
−0.584732 + 0.811226i $$0.698800\pi$$
$$198$$ −2.41421 −0.171571
$$199$$ −19.7990 −1.40351 −0.701757 0.712417i $$-0.747601\pi$$
−0.701757 + 0.712417i $$0.747601\pi$$
$$200$$ 4.41421 0.312132
$$201$$ −3.17157 −0.223706
$$202$$ −11.8284 −0.832245
$$203$$ 0 0
$$204$$ −16.8995 −1.18320
$$205$$ 2.34315 0.163652
$$206$$ −30.1421 −2.10010
$$207$$ −7.00000 −0.486534
$$208$$ −2.48528 −0.172323
$$209$$ 7.24264 0.500984
$$210$$ 0 0
$$211$$ 14.9706 1.03062 0.515308 0.857005i $$-0.327678\pi$$
0.515308 + 0.857005i $$0.327678\pi$$
$$212$$ −27.4558 −1.88568
$$213$$ −4.17157 −0.285831
$$214$$ −23.3137 −1.59369
$$215$$ 5.51472 0.376101
$$216$$ −4.41421 −0.300349
$$217$$ 0 0
$$218$$ 16.4853 1.11652
$$219$$ 0.343146 0.0231876
$$220$$ −7.65685 −0.516225
$$221$$ 3.65685 0.245987
$$222$$ 22.8995 1.53691
$$223$$ −22.9706 −1.53822 −0.769111 0.639115i $$-0.779301\pi$$
−0.769111 + 0.639115i $$0.779301\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 18.4853 1.22962
$$227$$ 14.9706 0.993631 0.496816 0.867856i $$-0.334503\pi$$
0.496816 + 0.867856i $$0.334503\pi$$
$$228$$ 27.7279 1.83633
$$229$$ −15.6569 −1.03463 −0.517317 0.855794i $$-0.673069\pi$$
−0.517317 + 0.855794i $$0.673069\pi$$
$$230$$ −33.7990 −2.22864
$$231$$ 0 0
$$232$$ −14.3137 −0.939741
$$233$$ 10.4142 0.682258 0.341129 0.940017i $$-0.389191\pi$$
0.341129 + 0.940017i $$0.389191\pi$$
$$234$$ 2.00000 0.130744
$$235$$ −19.6569 −1.28227
$$236$$ −33.1421 −2.15737
$$237$$ 13.3137 0.864818
$$238$$ 0 0
$$239$$ −6.48528 −0.419498 −0.209749 0.977755i $$-0.567265\pi$$
−0.209749 + 0.977755i $$0.567265\pi$$
$$240$$ −6.00000 −0.387298
$$241$$ 7.31371 0.471117 0.235559 0.971860i $$-0.424308\pi$$
0.235559 + 0.971860i $$0.424308\pi$$
$$242$$ −2.41421 −0.155192
$$243$$ 1.00000 0.0641500
$$244$$ −15.3137 −0.980360
$$245$$ 0 0
$$246$$ 2.82843 0.180334
$$247$$ −6.00000 −0.381771
$$248$$ −24.9706 −1.58563
$$249$$ −2.82843 −0.179244
$$250$$ −28.9706 −1.83226
$$251$$ −2.51472 −0.158728 −0.0793638 0.996846i $$-0.525289\pi$$
−0.0793638 + 0.996846i $$0.525289\pi$$
$$252$$ 0 0
$$253$$ −7.00000 −0.440086
$$254$$ 38.7990 2.43447
$$255$$ 8.82843 0.552858
$$256$$ −29.9706 −1.87316
$$257$$ 21.4558 1.33838 0.669189 0.743092i $$-0.266641\pi$$
0.669189 + 0.743092i $$0.266641\pi$$
$$258$$ 6.65685 0.414438
$$259$$ 0 0
$$260$$ 6.34315 0.393385
$$261$$ 3.24264 0.200714
$$262$$ 12.4853 0.771343
$$263$$ 18.0000 1.10993 0.554964 0.831875i $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ −4.41421 −0.271676
$$265$$ 14.3431 0.881092
$$266$$ 0 0
$$267$$ −14.1421 −0.865485
$$268$$ −12.1421 −0.741699
$$269$$ −27.7990 −1.69493 −0.847467 0.530848i $$-0.821874\pi$$
−0.847467 + 0.530848i $$0.821874\pi$$
$$270$$ 4.82843 0.293849
$$271$$ −22.9706 −1.39536 −0.697681 0.716408i $$-0.745785\pi$$
−0.697681 + 0.716408i $$0.745785\pi$$
$$272$$ −13.2426 −0.802953
$$273$$ 0 0
$$274$$ −1.17157 −0.0707773
$$275$$ −1.00000 −0.0603023
$$276$$ −26.7990 −1.61311
$$277$$ −25.6569 −1.54157 −0.770785 0.637095i $$-0.780136\pi$$
−0.770785 + 0.637095i $$0.780136\pi$$
$$278$$ −20.3137 −1.21834
$$279$$ 5.65685 0.338667
$$280$$ 0 0
$$281$$ 14.4142 0.859880 0.429940 0.902857i $$-0.358535\pi$$
0.429940 + 0.902857i $$0.358535\pi$$
$$282$$ −23.7279 −1.41298
$$283$$ 20.3431 1.20927 0.604637 0.796501i $$-0.293318\pi$$
0.604637 + 0.796501i $$0.293318\pi$$
$$284$$ −15.9706 −0.947679
$$285$$ −14.4853 −0.858034
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ 1.58579 0.0934434
$$289$$ 2.48528 0.146193
$$290$$ 15.6569 0.919402
$$291$$ −11.4853 −0.673279
$$292$$ 1.31371 0.0768790
$$293$$ 5.72792 0.334629 0.167314 0.985904i $$-0.446491\pi$$
0.167314 + 0.985904i $$0.446491\pi$$
$$294$$ 0 0
$$295$$ 17.3137 1.00804
$$296$$ 41.8701 2.43365
$$297$$ 1.00000 0.0580259
$$298$$ 53.6274 3.10655
$$299$$ 5.79899 0.335364
$$300$$ −3.82843 −0.221034
$$301$$ 0 0
$$302$$ 45.6274 2.62556
$$303$$ 4.89949 0.281469
$$304$$ 21.7279 1.24618
$$305$$ 8.00000 0.458079
$$306$$ 10.6569 0.609212
$$307$$ −17.3137 −0.988146 −0.494073 0.869421i $$-0.664492\pi$$
−0.494073 + 0.869421i $$0.664492\pi$$
$$308$$ 0 0
$$309$$ 12.4853 0.710263
$$310$$ 27.3137 1.55131
$$311$$ 29.6274 1.68002 0.840008 0.542573i $$-0.182550\pi$$
0.840008 + 0.542573i $$0.182550\pi$$
$$312$$ 3.65685 0.207029
$$313$$ 4.79899 0.271255 0.135627 0.990760i $$-0.456695\pi$$
0.135627 + 0.990760i $$0.456695\pi$$
$$314$$ −16.8995 −0.953694
$$315$$ 0 0
$$316$$ 50.9706 2.86732
$$317$$ −25.3137 −1.42176 −0.710880 0.703314i $$-0.751703\pi$$
−0.710880 + 0.703314i $$0.751703\pi$$
$$318$$ 17.3137 0.970905
$$319$$ 3.24264 0.181553
$$320$$ 19.6569 1.09885
$$321$$ 9.65685 0.538993
$$322$$ 0 0
$$323$$ −31.9706 −1.77889
$$324$$ 3.82843 0.212690
$$325$$ 0.828427 0.0459529
$$326$$ 50.6274 2.80399
$$327$$ −6.82843 −0.377613
$$328$$ 5.17157 0.285552
$$329$$ 0 0
$$330$$ 4.82843 0.265796
$$331$$ 22.4853 1.23590 0.617951 0.786216i $$-0.287963\pi$$
0.617951 + 0.786216i $$0.287963\pi$$
$$332$$ −10.8284 −0.594287
$$333$$ −9.48528 −0.519790
$$334$$ −12.4853 −0.683164
$$335$$ 6.34315 0.346563
$$336$$ 0 0
$$337$$ 3.85786 0.210151 0.105076 0.994464i $$-0.466492\pi$$
0.105076 + 0.994464i $$0.466492\pi$$
$$338$$ 29.7279 1.61699
$$339$$ −7.65685 −0.415863
$$340$$ 33.7990 1.83301
$$341$$ 5.65685 0.306336
$$342$$ −17.4853 −0.945496
$$343$$ 0 0
$$344$$ 12.1716 0.656247
$$345$$ 14.0000 0.753735
$$346$$ 35.7990 1.92457
$$347$$ 14.9706 0.803662 0.401831 0.915714i $$-0.368374\pi$$
0.401831 + 0.915714i $$0.368374\pi$$
$$348$$ 12.4142 0.665472
$$349$$ 10.9706 0.587241 0.293620 0.955922i $$-0.405140\pi$$
0.293620 + 0.955922i $$0.405140\pi$$
$$350$$ 0 0
$$351$$ −0.828427 −0.0442182
$$352$$ 1.58579 0.0845227
$$353$$ 13.3137 0.708617 0.354309 0.935129i $$-0.384716\pi$$
0.354309 + 0.935129i $$0.384716\pi$$
$$354$$ 20.8995 1.11080
$$355$$ 8.34315 0.442808
$$356$$ −54.1421 −2.86953
$$357$$ 0 0
$$358$$ −43.0416 −2.27482
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 8.82843 0.465299
$$361$$ 33.4558 1.76083
$$362$$ 28.1421 1.47912
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ −0.686292 −0.0359221
$$366$$ 9.65685 0.504772
$$367$$ −4.14214 −0.216218 −0.108109 0.994139i $$-0.534480\pi$$
−0.108109 + 0.994139i $$0.534480\pi$$
$$368$$ −21.0000 −1.09470
$$369$$ −1.17157 −0.0609896
$$370$$ −45.7990 −2.38098
$$371$$ 0 0
$$372$$ 21.6569 1.12286
$$373$$ −1.65685 −0.0857887 −0.0428943 0.999080i $$-0.513658\pi$$
−0.0428943 + 0.999080i $$0.513658\pi$$
$$374$$ 10.6569 0.551053
$$375$$ 12.0000 0.619677
$$376$$ −43.3848 −2.23740
$$377$$ −2.68629 −0.138351
$$378$$ 0 0
$$379$$ 18.8284 0.967151 0.483576 0.875303i $$-0.339338\pi$$
0.483576 + 0.875303i $$0.339338\pi$$
$$380$$ −55.4558 −2.84482
$$381$$ −16.0711 −0.823346
$$382$$ 52.2843 2.67510
$$383$$ 2.31371 0.118225 0.0591125 0.998251i $$-0.481173\pi$$
0.0591125 + 0.998251i $$0.481173\pi$$
$$384$$ 20.5563 1.04901
$$385$$ 0 0
$$386$$ 29.3137 1.49203
$$387$$ −2.75736 −0.140164
$$388$$ −43.9706 −2.23227
$$389$$ −11.7990 −0.598233 −0.299116 0.954217i $$-0.596692\pi$$
−0.299116 + 0.954217i $$0.596692\pi$$
$$390$$ −4.00000 −0.202548
$$391$$ 30.8995 1.56265
$$392$$ 0 0
$$393$$ −5.17157 −0.260871
$$394$$ 39.6274 1.99640
$$395$$ −26.6274 −1.33977
$$396$$ 3.82843 0.192386
$$397$$ 17.4853 0.877561 0.438781 0.898594i $$-0.355411\pi$$
0.438781 + 0.898594i $$0.355411\pi$$
$$398$$ 47.7990 2.39595
$$399$$ 0 0
$$400$$ −3.00000 −0.150000
$$401$$ 7.51472 0.375267 0.187634 0.982239i $$-0.439918\pi$$
0.187634 + 0.982239i $$0.439918\pi$$
$$402$$ 7.65685 0.381889
$$403$$ −4.68629 −0.233441
$$404$$ 18.7574 0.933214
$$405$$ −2.00000 −0.0993808
$$406$$ 0 0
$$407$$ −9.48528 −0.470168
$$408$$ 19.4853 0.964665
$$409$$ −7.79899 −0.385635 −0.192818 0.981235i $$-0.561763\pi$$
−0.192818 + 0.981235i $$0.561763\pi$$
$$410$$ −5.65685 −0.279372
$$411$$ 0.485281 0.0239372
$$412$$ 47.7990 2.35489
$$413$$ 0 0
$$414$$ 16.8995 0.830565
$$415$$ 5.65685 0.277684
$$416$$ −1.31371 −0.0644099
$$417$$ 8.41421 0.412046
$$418$$ −17.4853 −0.855233
$$419$$ 14.7990 0.722978 0.361489 0.932376i $$-0.382269\pi$$
0.361489 + 0.932376i $$0.382269\pi$$
$$420$$ 0 0
$$421$$ −7.00000 −0.341159 −0.170580 0.985344i $$-0.554564\pi$$
−0.170580 + 0.985344i $$0.554564\pi$$
$$422$$ −36.1421 −1.75937
$$423$$ 9.82843 0.477874
$$424$$ 31.6569 1.53739
$$425$$ 4.41421 0.214121
$$426$$ 10.0711 0.487945
$$427$$ 0 0
$$428$$ 36.9706 1.78704
$$429$$ −0.828427 −0.0399968
$$430$$ −13.3137 −0.642044
$$431$$ 6.97056 0.335760 0.167880 0.985807i $$-0.446308\pi$$
0.167880 + 0.985807i $$0.446308\pi$$
$$432$$ 3.00000 0.144338
$$433$$ 0.857864 0.0412263 0.0206132 0.999788i $$-0.493438\pi$$
0.0206132 + 0.999788i $$0.493438\pi$$
$$434$$ 0 0
$$435$$ −6.48528 −0.310945
$$436$$ −26.1421 −1.25198
$$437$$ −50.6985 −2.42524
$$438$$ −0.828427 −0.0395838
$$439$$ 30.8995 1.47475 0.737376 0.675482i $$-0.236065\pi$$
0.737376 + 0.675482i $$0.236065\pi$$
$$440$$ 8.82843 0.420879
$$441$$ 0 0
$$442$$ −8.82843 −0.419925
$$443$$ 33.6274 1.59769 0.798843 0.601539i $$-0.205446\pi$$
0.798843 + 0.601539i $$0.205446\pi$$
$$444$$ −36.3137 −1.72337
$$445$$ 28.2843 1.34080
$$446$$ 55.4558 2.62591
$$447$$ −22.2132 −1.05065
$$448$$ 0 0
$$449$$ −4.00000 −0.188772 −0.0943858 0.995536i $$-0.530089\pi$$
−0.0943858 + 0.995536i $$0.530089\pi$$
$$450$$ 2.41421 0.113807
$$451$$ −1.17157 −0.0551672
$$452$$ −29.3137 −1.37880
$$453$$ −18.8995 −0.887976
$$454$$ −36.1421 −1.69623
$$455$$ 0 0
$$456$$ −31.9706 −1.49716
$$457$$ 18.8284 0.880757 0.440378 0.897812i $$-0.354844\pi$$
0.440378 + 0.897812i $$0.354844\pi$$
$$458$$ 37.7990 1.76623
$$459$$ −4.41421 −0.206038
$$460$$ 53.5980 2.49902
$$461$$ 6.75736 0.314722 0.157361 0.987541i $$-0.449701\pi$$
0.157361 + 0.987541i $$0.449701\pi$$
$$462$$ 0 0
$$463$$ 18.6274 0.865689 0.432845 0.901468i $$-0.357510\pi$$
0.432845 + 0.901468i $$0.357510\pi$$
$$464$$ 9.72792 0.451607
$$465$$ −11.3137 −0.524661
$$466$$ −25.1421 −1.16469
$$467$$ −8.31371 −0.384713 −0.192356 0.981325i $$-0.561613\pi$$
−0.192356 + 0.981325i $$0.561613\pi$$
$$468$$ −3.17157 −0.146606
$$469$$ 0 0
$$470$$ 47.4558 2.18897
$$471$$ 7.00000 0.322543
$$472$$ 38.2132 1.75891
$$473$$ −2.75736 −0.126784
$$474$$ −32.1421 −1.47634
$$475$$ −7.24264 −0.332315
$$476$$ 0 0
$$477$$ −7.17157 −0.328364
$$478$$ 15.6569 0.716128
$$479$$ 14.0000 0.639676 0.319838 0.947472i $$-0.396371\pi$$
0.319838 + 0.947472i $$0.396371\pi$$
$$480$$ −3.17157 −0.144762
$$481$$ 7.85786 0.358288
$$482$$ −17.6569 −0.804248
$$483$$ 0 0
$$484$$ 3.82843 0.174019
$$485$$ 22.9706 1.04304
$$486$$ −2.41421 −0.109511
$$487$$ −32.6274 −1.47849 −0.739245 0.673437i $$-0.764817\pi$$
−0.739245 + 0.673437i $$0.764817\pi$$
$$488$$ 17.6569 0.799288
$$489$$ −20.9706 −0.948322
$$490$$ 0 0
$$491$$ −1.51472 −0.0683583 −0.0341791 0.999416i $$-0.510882\pi$$
−0.0341791 + 0.999416i $$0.510882\pi$$
$$492$$ −4.48528 −0.202212
$$493$$ −14.3137 −0.644657
$$494$$ 14.4853 0.651724
$$495$$ −2.00000 −0.0898933
$$496$$ 16.9706 0.762001
$$497$$ 0 0
$$498$$ 6.82843 0.305989
$$499$$ −31.7990 −1.42352 −0.711759 0.702424i $$-0.752101\pi$$
−0.711759 + 0.702424i $$0.752101\pi$$
$$500$$ 45.9411 2.05455
$$501$$ 5.17157 0.231049
$$502$$ 6.07107 0.270965
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ −9.79899 −0.436049
$$506$$ 16.8995 0.751274
$$507$$ −12.3137 −0.546871
$$508$$ −61.5269 −2.72982
$$509$$ −12.0000 −0.531891 −0.265945 0.963988i $$-0.585684\pi$$
−0.265945 + 0.963988i $$0.585684\pi$$
$$510$$ −21.3137 −0.943787
$$511$$ 0 0
$$512$$ 31.2426 1.38074
$$513$$ 7.24264 0.319770
$$514$$ −51.7990 −2.28476
$$515$$ −24.9706 −1.10033
$$516$$ −10.5563 −0.464717
$$517$$ 9.82843 0.432254
$$518$$ 0 0
$$519$$ −14.8284 −0.650896
$$520$$ −7.31371 −0.320727
$$521$$ 31.1716 1.36565 0.682826 0.730581i $$-0.260751\pi$$
0.682826 + 0.730581i $$0.260751\pi$$
$$522$$ −7.82843 −0.342641
$$523$$ −22.2843 −0.974423 −0.487212 0.873284i $$-0.661986\pi$$
−0.487212 + 0.873284i $$0.661986\pi$$
$$524$$ −19.7990 −0.864923
$$525$$ 0 0
$$526$$ −43.4558 −1.89476
$$527$$ −24.9706 −1.08773
$$528$$ 3.00000 0.130558
$$529$$ 26.0000 1.13043
$$530$$ −34.6274 −1.50412
$$531$$ −8.65685 −0.375675
$$532$$ 0 0
$$533$$ 0.970563 0.0420397
$$534$$ 34.1421 1.47747
$$535$$ −19.3137 −0.835004
$$536$$ 14.0000 0.604708
$$537$$ 17.8284 0.769353
$$538$$ 67.1127 2.89343
$$539$$ 0 0
$$540$$ −7.65685 −0.329499
$$541$$ 19.6569 0.845114 0.422557 0.906336i $$-0.361133\pi$$
0.422557 + 0.906336i $$0.361133\pi$$
$$542$$ 55.4558 2.38203
$$543$$ −11.6569 −0.500243
$$544$$ −7.00000 −0.300123
$$545$$ 13.6569 0.584995
$$546$$ 0 0
$$547$$ −24.2132 −1.03528 −0.517641 0.855598i $$-0.673190\pi$$
−0.517641 + 0.855598i $$0.673190\pi$$
$$548$$ 1.85786 0.0793640
$$549$$ −4.00000 −0.170716
$$550$$ 2.41421 0.102942
$$551$$ 23.4853 1.00051
$$552$$ 30.8995 1.31517
$$553$$ 0 0
$$554$$ 61.9411 2.63163
$$555$$ 18.9706 0.805256
$$556$$ 32.2132 1.36614
$$557$$ −8.55635 −0.362544 −0.181272 0.983433i $$-0.558021\pi$$
−0.181272 + 0.983433i $$0.558021\pi$$
$$558$$ −13.6569 −0.578141
$$559$$ 2.28427 0.0966144
$$560$$ 0 0
$$561$$ −4.41421 −0.186368
$$562$$ −34.7990 −1.46791
$$563$$ 24.8284 1.04639 0.523197 0.852212i $$-0.324739\pi$$
0.523197 + 0.852212i $$0.324739\pi$$
$$564$$ 37.6274 1.58440
$$565$$ 15.3137 0.644253
$$566$$ −49.1127 −2.06436
$$567$$ 0 0
$$568$$ 18.4142 0.772643
$$569$$ 1.24264 0.0520942 0.0260471 0.999661i $$-0.491708\pi$$
0.0260471 + 0.999661i $$0.491708\pi$$
$$570$$ 34.9706 1.46476
$$571$$ −3.92893 −0.164421 −0.0822103 0.996615i $$-0.526198\pi$$
−0.0822103 + 0.996615i $$0.526198\pi$$
$$572$$ −3.17157 −0.132610
$$573$$ −21.6569 −0.904728
$$574$$ 0 0
$$575$$ 7.00000 0.291920
$$576$$ −9.82843 −0.409518
$$577$$ 7.65685 0.318759 0.159380 0.987217i $$-0.449051\pi$$
0.159380 + 0.987217i $$0.449051\pi$$
$$578$$ −6.00000 −0.249567
$$579$$ −12.1421 −0.504610
$$580$$ −24.8284 −1.03094
$$581$$ 0 0
$$582$$ 27.7279 1.14936
$$583$$ −7.17157 −0.297016
$$584$$ −1.51472 −0.0626795
$$585$$ 1.65685 0.0685025
$$586$$ −13.8284 −0.571247
$$587$$ 8.68629 0.358522 0.179261 0.983802i $$-0.442629\pi$$
0.179261 + 0.983802i $$0.442629\pi$$
$$588$$ 0 0
$$589$$ 40.9706 1.68816
$$590$$ −41.7990 −1.72084
$$591$$ −16.4142 −0.675191
$$592$$ −28.4558 −1.16953
$$593$$ −22.2721 −0.914605 −0.457302 0.889311i $$-0.651184\pi$$
−0.457302 + 0.889311i $$0.651184\pi$$
$$594$$ −2.41421 −0.0990564
$$595$$ 0 0
$$596$$ −85.0416 −3.48344
$$597$$ −19.7990 −0.810319
$$598$$ −14.0000 −0.572503
$$599$$ −1.37258 −0.0560822 −0.0280411 0.999607i $$-0.508927\pi$$
−0.0280411 + 0.999607i $$0.508927\pi$$
$$600$$ 4.41421 0.180210
$$601$$ −14.4853 −0.590867 −0.295433 0.955363i $$-0.595464\pi$$
−0.295433 + 0.955363i $$0.595464\pi$$
$$602$$ 0 0
$$603$$ −3.17157 −0.129156
$$604$$ −72.3553 −2.94410
$$605$$ −2.00000 −0.0813116
$$606$$ −11.8284 −0.480497
$$607$$ −18.6863 −0.758453 −0.379227 0.925304i $$-0.623810\pi$$
−0.379227 + 0.925304i $$0.623810\pi$$
$$608$$ 11.4853 0.465790
$$609$$ 0 0
$$610$$ −19.3137 −0.781989
$$611$$ −8.14214 −0.329396
$$612$$ −16.8995 −0.683122
$$613$$ 25.1716 1.01667 0.508335 0.861159i $$-0.330261\pi$$
0.508335 + 0.861159i $$0.330261\pi$$
$$614$$ 41.7990 1.68687
$$615$$ 2.34315 0.0944848
$$616$$ 0 0
$$617$$ 13.1716 0.530268 0.265134 0.964212i $$-0.414584\pi$$
0.265134 + 0.964212i $$0.414584\pi$$
$$618$$ −30.1421 −1.21249
$$619$$ 34.6274 1.39179 0.695897 0.718142i $$-0.255007\pi$$
0.695897 + 0.718142i $$0.255007\pi$$
$$620$$ −43.3137 −1.73952
$$621$$ −7.00000 −0.280900
$$622$$ −71.5269 −2.86797
$$623$$ 0 0
$$624$$ −2.48528 −0.0994909
$$625$$ −19.0000 −0.760000
$$626$$ −11.5858 −0.463061
$$627$$ 7.24264 0.289243
$$628$$ 26.7990 1.06940
$$629$$ 41.8701 1.66947
$$630$$ 0 0
$$631$$ 26.2843 1.04636 0.523180 0.852222i $$-0.324745\pi$$
0.523180 + 0.852222i $$0.324745\pi$$
$$632$$ −58.7696 −2.33773
$$633$$ 14.9706 0.595026
$$634$$ 61.1127 2.42710
$$635$$ 32.1421 1.27552
$$636$$ −27.4558 −1.08870
$$637$$ 0 0
$$638$$ −7.82843 −0.309930
$$639$$ −4.17157 −0.165025
$$640$$ −41.1127 −1.62512
$$641$$ −12.4853 −0.493139 −0.246569 0.969125i $$-0.579303\pi$$
−0.246569 + 0.969125i $$0.579303\pi$$
$$642$$ −23.3137 −0.920119
$$643$$ 2.00000 0.0788723 0.0394362 0.999222i $$-0.487444\pi$$
0.0394362 + 0.999222i $$0.487444\pi$$
$$644$$ 0 0
$$645$$ 5.51472 0.217142
$$646$$ 77.1838 3.03675
$$647$$ −19.3137 −0.759300 −0.379650 0.925130i $$-0.623956\pi$$
−0.379650 + 0.925130i $$0.623956\pi$$
$$648$$ −4.41421 −0.173407
$$649$$ −8.65685 −0.339811
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ −80.2843 −3.14417
$$653$$ 31.1127 1.21753 0.608767 0.793349i $$-0.291664\pi$$
0.608767 + 0.793349i $$0.291664\pi$$
$$654$$ 16.4853 0.644626
$$655$$ 10.3431 0.404140
$$656$$ −3.51472 −0.137227
$$657$$ 0.343146 0.0133874
$$658$$ 0 0
$$659$$ 34.4853 1.34336 0.671678 0.740843i $$-0.265574\pi$$
0.671678 + 0.740843i $$0.265574\pi$$
$$660$$ −7.65685 −0.298043
$$661$$ 11.9706 0.465601 0.232800 0.972525i $$-0.425211\pi$$
0.232800 + 0.972525i $$0.425211\pi$$
$$662$$ −54.2843 −2.10982
$$663$$ 3.65685 0.142020
$$664$$ 12.4853 0.484523
$$665$$ 0 0
$$666$$ 22.8995 0.887337
$$667$$ −22.6985 −0.878889
$$668$$ 19.7990 0.766046
$$669$$ −22.9706 −0.888093
$$670$$ −15.3137 −0.591620
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ 27.1716 1.04739 0.523694 0.851907i $$-0.324554\pi$$
0.523694 + 0.851907i $$0.324554\pi$$
$$674$$ −9.31371 −0.358751
$$675$$ −1.00000 −0.0384900
$$676$$ −47.1421 −1.81316
$$677$$ −32.0122 −1.23033 −0.615164 0.788399i $$-0.710910\pi$$
−0.615164 + 0.788399i $$0.710910\pi$$
$$678$$ 18.4853 0.709923
$$679$$ 0 0
$$680$$ −38.9706 −1.49445
$$681$$ 14.9706 0.573673
$$682$$ −13.6569 −0.522948
$$683$$ 3.20101 0.122483 0.0612416 0.998123i $$-0.480494\pi$$
0.0612416 + 0.998123i $$0.480494\pi$$
$$684$$ 27.7279 1.06020
$$685$$ −0.970563 −0.0370833
$$686$$ 0 0
$$687$$ −15.6569 −0.597346
$$688$$ −8.27208 −0.315370
$$689$$ 5.94113 0.226339
$$690$$ −33.7990 −1.28671
$$691$$ −11.4558 −0.435801 −0.217900 0.975971i $$-0.569921\pi$$
−0.217900 + 0.975971i $$0.569921\pi$$
$$692$$ −56.7696 −2.15805
$$693$$ 0 0
$$694$$ −36.1421 −1.37194
$$695$$ −16.8284 −0.638339
$$696$$ −14.3137 −0.542560
$$697$$ 5.17157 0.195887
$$698$$ −26.4853 −1.00248
$$699$$ 10.4142 0.393902
$$700$$ 0 0
$$701$$ −2.89949 −0.109512 −0.0547562 0.998500i $$-0.517438\pi$$
−0.0547562 + 0.998500i $$0.517438\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −68.6985 −2.59101
$$704$$ −9.82843 −0.370423
$$705$$ −19.6569 −0.740320
$$706$$ −32.1421 −1.20969
$$707$$ 0 0
$$708$$ −33.1421 −1.24556
$$709$$ 25.7696 0.967796 0.483898 0.875124i $$-0.339221\pi$$
0.483898 + 0.875124i $$0.339221\pi$$
$$710$$ −20.1421 −0.755921
$$711$$ 13.3137 0.499303
$$712$$ 62.4264 2.33953
$$713$$ −39.5980 −1.48296
$$714$$ 0 0
$$715$$ 1.65685 0.0619628
$$716$$ 68.2548 2.55080
$$717$$ −6.48528 −0.242197
$$718$$ 19.3137 0.720781
$$719$$ 11.2010 0.417727 0.208864 0.977945i $$-0.433024\pi$$
0.208864 + 0.977945i $$0.433024\pi$$
$$720$$ −6.00000 −0.223607
$$721$$ 0 0
$$722$$ −80.7696 −3.00593
$$723$$ 7.31371 0.272000
$$724$$ −44.6274 −1.65856
$$725$$ −3.24264 −0.120429
$$726$$ −2.41421 −0.0895999
$$727$$ 46.0833 1.70913 0.854567 0.519342i $$-0.173823\pi$$
0.854567 + 0.519342i $$0.173823\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 1.65685 0.0613229
$$731$$ 12.1716 0.450182
$$732$$ −15.3137 −0.566011
$$733$$ 9.65685 0.356684 0.178342 0.983969i $$-0.442927\pi$$
0.178342 + 0.983969i $$0.442927\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ −11.1005 −0.409170
$$737$$ −3.17157 −0.116826
$$738$$ 2.82843 0.104116
$$739$$ −39.6569 −1.45880 −0.729400 0.684087i $$-0.760201\pi$$
−0.729400 + 0.684087i $$0.760201\pi$$
$$740$$ 72.6274 2.66984
$$741$$ −6.00000 −0.220416
$$742$$ 0 0
$$743$$ 33.7990 1.23996 0.619982 0.784616i $$-0.287140\pi$$
0.619982 + 0.784616i $$0.287140\pi$$
$$744$$ −24.9706 −0.915465
$$745$$ 44.4264 1.62766
$$746$$ 4.00000 0.146450
$$747$$ −2.82843 −0.103487
$$748$$ −16.8995 −0.617907
$$749$$ 0 0
$$750$$ −28.9706 −1.05786
$$751$$ −29.3137 −1.06967 −0.534836 0.844956i $$-0.679627\pi$$
−0.534836 + 0.844956i $$0.679627\pi$$
$$752$$ 29.4853 1.07522
$$753$$ −2.51472 −0.0916414
$$754$$ 6.48528 0.236180
$$755$$ 37.7990 1.37565
$$756$$ 0 0
$$757$$ −7.68629 −0.279363 −0.139682 0.990196i $$-0.544608\pi$$
−0.139682 + 0.990196i $$0.544608\pi$$
$$758$$ −45.4558 −1.65103
$$759$$ −7.00000 −0.254084
$$760$$ 63.9411 2.31939
$$761$$ −0.201010 −0.00728661 −0.00364331 0.999993i $$-0.501160\pi$$
−0.00364331 + 0.999993i $$0.501160\pi$$
$$762$$ 38.7990 1.40554
$$763$$ 0 0
$$764$$ −82.9117 −2.99964
$$765$$ 8.82843 0.319192
$$766$$ −5.58579 −0.201823
$$767$$ 7.17157 0.258950
$$768$$ −29.9706 −1.08147
$$769$$ 33.7990 1.21882 0.609411 0.792854i $$-0.291406\pi$$
0.609411 + 0.792854i $$0.291406\pi$$
$$770$$ 0 0
$$771$$ 21.4558 0.772713
$$772$$ −46.4853 −1.67304
$$773$$ −51.6569 −1.85797 −0.928984 0.370120i $$-0.879317\pi$$
−0.928984 + 0.370120i $$0.879317\pi$$
$$774$$ 6.65685 0.239276
$$775$$ −5.65685 −0.203200
$$776$$ 50.6985 1.81997
$$777$$ 0 0
$$778$$ 28.4853 1.02125
$$779$$ −8.48528 −0.304017
$$780$$ 6.34315 0.227121
$$781$$ −4.17157 −0.149271
$$782$$ −74.5980 −2.66762
$$783$$ 3.24264 0.115883
$$784$$ 0 0
$$785$$ −14.0000 −0.499681
$$786$$ 12.4853 0.445335
$$787$$ 29.5269 1.05252 0.526260 0.850323i $$-0.323594\pi$$
0.526260 + 0.850323i $$0.323594\pi$$
$$788$$ −62.8406 −2.23860
$$789$$ 18.0000 0.640817
$$790$$ 64.2843 2.28713
$$791$$ 0 0
$$792$$ −4.41421 −0.156852
$$793$$ 3.31371 0.117673
$$794$$ −42.2132 −1.49809
$$795$$ 14.3431 0.508699
$$796$$ −75.7990 −2.68662
$$797$$ 11.1716 0.395717 0.197859 0.980231i $$-0.436601\pi$$
0.197859 + 0.980231i $$0.436601\pi$$
$$798$$ 0 0
$$799$$ −43.3848 −1.53484
$$800$$ −1.58579 −0.0560660
$$801$$ −14.1421 −0.499688
$$802$$ −18.1421 −0.640621
$$803$$ 0.343146 0.0121094
$$804$$ −12.1421 −0.428220
$$805$$ 0 0
$$806$$ 11.3137 0.398508
$$807$$ −27.7990 −0.978571
$$808$$ −21.6274 −0.760850
$$809$$ 10.8284 0.380707 0.190354 0.981716i $$-0.439037\pi$$
0.190354 + 0.981716i $$0.439037\pi$$
$$810$$ 4.82843 0.169654
$$811$$ 14.9706 0.525688 0.262844 0.964838i $$-0.415340\pi$$
0.262844 + 0.964838i $$0.415340\pi$$
$$812$$ 0 0
$$813$$ −22.9706 −0.805613
$$814$$ 22.8995 0.802627
$$815$$ 41.9411 1.46913
$$816$$ −13.2426 −0.463585
$$817$$ −19.9706 −0.698682
$$818$$ 18.8284 0.658321
$$819$$ 0 0
$$820$$ 8.97056 0.313266
$$821$$ 38.8284 1.35512 0.677561 0.735467i $$-0.263037\pi$$
0.677561 + 0.735467i $$0.263037\pi$$
$$822$$ −1.17157 −0.0408633
$$823$$ −33.1716 −1.15629 −0.578144 0.815935i $$-0.696223\pi$$
−0.578144 + 0.815935i $$0.696223\pi$$
$$824$$ −55.1127 −1.91994
$$825$$ −1.00000 −0.0348155
$$826$$ 0 0
$$827$$ −37.3137 −1.29752 −0.648762 0.760991i $$-0.724713\pi$$
−0.648762 + 0.760991i $$0.724713\pi$$
$$828$$ −26.7990 −0.931329
$$829$$ −6.17157 −0.214348 −0.107174 0.994240i $$-0.534180\pi$$
−0.107174 + 0.994240i $$0.534180\pi$$
$$830$$ −13.6569 −0.474036
$$831$$ −25.6569 −0.890026
$$832$$ 8.14214 0.282278
$$833$$ 0 0
$$834$$ −20.3137 −0.703406
$$835$$ −10.3431 −0.357939
$$836$$ 27.7279 0.958990
$$837$$ 5.65685 0.195529
$$838$$ −35.7279 −1.23420
$$839$$ 8.97056 0.309698 0.154849 0.987938i $$-0.450511\pi$$
0.154849 + 0.987938i $$0.450511\pi$$
$$840$$ 0 0
$$841$$ −18.4853 −0.637423
$$842$$ 16.8995 0.582395
$$843$$ 14.4142 0.496452
$$844$$ 57.3137 1.97282
$$845$$ 24.6274 0.847209
$$846$$ −23.7279 −0.815783
$$847$$ 0 0
$$848$$ −21.5147 −0.738818
$$849$$ 20.3431 0.698175
$$850$$ −10.6569 −0.365527
$$851$$ 66.3970 2.27606
$$852$$ −15.9706 −0.547142
$$853$$ −12.4853 −0.427488 −0.213744 0.976890i $$-0.568566\pi$$
−0.213744 + 0.976890i $$0.568566\pi$$
$$854$$ 0 0
$$855$$ −14.4853 −0.495386
$$856$$ −42.6274 −1.45698
$$857$$ −44.3553 −1.51515 −0.757575 0.652748i $$-0.773616\pi$$
−0.757575 + 0.652748i $$0.773616\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ −24.4853 −0.835427 −0.417714 0.908579i $$-0.637168\pi$$
−0.417714 + 0.908579i $$0.637168\pi$$
$$860$$ 21.1127 0.719937
$$861$$ 0 0
$$862$$ −16.8284 −0.573179
$$863$$ 2.62742 0.0894383 0.0447192 0.999000i $$-0.485761\pi$$
0.0447192 + 0.999000i $$0.485761\pi$$
$$864$$ 1.58579 0.0539496
$$865$$ 29.6569 1.00836
$$866$$ −2.07107 −0.0703777
$$867$$ 2.48528 0.0844046
$$868$$ 0 0
$$869$$ 13.3137 0.451637
$$870$$ 15.6569 0.530817
$$871$$ 2.62742 0.0890266
$$872$$ 30.1421 1.02074
$$873$$ −11.4853 −0.388718
$$874$$ 122.397 4.14014
$$875$$ 0 0
$$876$$ 1.31371 0.0443861
$$877$$ 2.48528 0.0839220 0.0419610 0.999119i $$-0.486639\pi$$
0.0419610 + 0.999119i $$0.486639\pi$$
$$878$$ −74.5980 −2.51756
$$879$$ 5.72792 0.193198
$$880$$ −6.00000 −0.202260
$$881$$ 16.6863 0.562175 0.281088 0.959682i $$-0.409305\pi$$
0.281088 + 0.959682i $$0.409305\pi$$
$$882$$ 0 0
$$883$$ 39.6569 1.33456 0.667280 0.744807i $$-0.267459\pi$$
0.667280 + 0.744807i $$0.267459\pi$$
$$884$$ 14.0000 0.470871
$$885$$ 17.3137 0.581994
$$886$$ −81.1838 −2.72742
$$887$$ −13.3137 −0.447031 −0.223515 0.974700i $$-0.571753\pi$$
−0.223515 + 0.974700i $$0.571753\pi$$
$$888$$ 41.8701 1.40507
$$889$$ 0 0
$$890$$ −68.2843 −2.28889
$$891$$ 1.00000 0.0335013
$$892$$ −87.9411 −2.94449
$$893$$ 71.1838 2.38207
$$894$$ 53.6274 1.79357
$$895$$ −35.6569 −1.19188
$$896$$ 0 0
$$897$$ 5.79899 0.193623
$$898$$ 9.65685 0.322253
$$899$$ 18.3431 0.611778
$$900$$ −3.82843 −0.127614
$$901$$ 31.6569 1.05464
$$902$$ 2.82843 0.0941763
$$903$$ 0 0
$$904$$ 33.7990 1.12414
$$905$$ 23.3137 0.774974
$$906$$ 45.6274 1.51587
$$907$$ −17.5147 −0.581567 −0.290783 0.956789i $$-0.593916\pi$$
−0.290783 + 0.956789i $$0.593916\pi$$
$$908$$ 57.3137 1.90202
$$909$$ 4.89949 0.162506
$$910$$ 0 0
$$911$$ −4.51472 −0.149579 −0.0747897 0.997199i $$-0.523829\pi$$
−0.0747897 + 0.997199i $$0.523829\pi$$
$$912$$ 21.7279 0.719483
$$913$$ −2.82843 −0.0936073
$$914$$ −45.4558 −1.50355
$$915$$ 8.00000 0.264472
$$916$$ −59.9411 −1.98051
$$917$$ 0 0
$$918$$ 10.6569 0.351729
$$919$$ 56.3553 1.85899 0.929496 0.368833i $$-0.120243\pi$$
0.929496 + 0.368833i $$0.120243\pi$$
$$920$$ −61.7990 −2.03745
$$921$$ −17.3137 −0.570506
$$922$$ −16.3137 −0.537263
$$923$$ 3.45584 0.113750
$$924$$ 0 0
$$925$$ 9.48528 0.311874
$$926$$ −44.9706 −1.47782
$$927$$ 12.4853 0.410070
$$928$$ 5.14214 0.168799
$$929$$ −22.8284 −0.748976 −0.374488 0.927232i $$-0.622182\pi$$
−0.374488 + 0.927232i $$0.622182\pi$$
$$930$$ 27.3137 0.895652
$$931$$ 0 0
$$932$$ 39.8701 1.30599
$$933$$ 29.6274 0.969958
$$934$$ 20.0711 0.656745
$$935$$ 8.82843 0.288720
$$936$$ 3.65685 0.119528
$$937$$ 5.85786 0.191368 0.0956840 0.995412i $$-0.469496\pi$$
0.0956840 + 0.995412i $$0.469496\pi$$
$$938$$ 0 0
$$939$$ 4.79899 0.156609
$$940$$ −75.2548 −2.45454
$$941$$ −12.0711 −0.393506 −0.196753 0.980453i $$-0.563040\pi$$
−0.196753 + 0.980453i $$0.563040\pi$$
$$942$$ −16.8995 −0.550615
$$943$$ 8.20101 0.267062
$$944$$ −25.9706 −0.845270
$$945$$ 0 0
$$946$$ 6.65685 0.216433
$$947$$ 37.4853 1.21811 0.609054 0.793129i $$-0.291549\pi$$
0.609054 + 0.793129i $$0.291549\pi$$
$$948$$ 50.9706 1.65545
$$949$$ −0.284271 −0.00922784
$$950$$ 17.4853 0.567297
$$951$$ −25.3137 −0.820853
$$952$$ 0 0
$$953$$ −14.1421 −0.458109 −0.229054 0.973414i $$-0.573563\pi$$
−0.229054 + 0.973414i $$0.573563\pi$$
$$954$$ 17.3137 0.560552
$$955$$ 43.3137 1.40160
$$956$$ −24.8284 −0.803009
$$957$$ 3.24264 0.104820
$$958$$ −33.7990 −1.09200
$$959$$ 0 0
$$960$$ 19.6569 0.634422
$$961$$ 1.00000 0.0322581
$$962$$ −18.9706 −0.611635
$$963$$ 9.65685 0.311188
$$964$$ 28.0000 0.901819
$$965$$ 24.2843 0.781738
$$966$$ 0 0
$$967$$ −21.0416 −0.676653 −0.338327 0.941029i $$-0.609861\pi$$
−0.338327 + 0.941029i $$0.609861\pi$$
$$968$$ −4.41421 −0.141878
$$969$$ −31.9706 −1.02704
$$970$$ −55.4558 −1.78058
$$971$$ 4.97056 0.159513 0.0797565 0.996814i $$-0.474586\pi$$
0.0797565 + 0.996814i $$0.474586\pi$$
$$972$$ 3.82843 0.122797
$$973$$ 0 0
$$974$$ 78.7696 2.52394
$$975$$ 0.828427 0.0265309
$$976$$ −12.0000 −0.384111
$$977$$ −12.8284 −0.410418 −0.205209 0.978718i $$-0.565787\pi$$
−0.205209 + 0.978718i $$0.565787\pi$$
$$978$$ 50.6274 1.61889
$$979$$ −14.1421 −0.451985
$$980$$ 0 0
$$981$$ −6.82843 −0.218015
$$982$$ 3.65685 0.116695
$$983$$ −2.02944 −0.0647290 −0.0323645 0.999476i $$-0.510304\pi$$
−0.0323645 + 0.999476i $$0.510304\pi$$
$$984$$ 5.17157 0.164864
$$985$$ 32.8284 1.04600
$$986$$ 34.5563 1.10050
$$987$$ 0 0
$$988$$ −22.9706 −0.730791
$$989$$ 19.3015 0.613752
$$990$$ 4.82843 0.153457
$$991$$ −24.6863 −0.784186 −0.392093 0.919926i $$-0.628249\pi$$
−0.392093 + 0.919926i $$0.628249\pi$$
$$992$$ 8.97056 0.284816
$$993$$ 22.4853 0.713549
$$994$$ 0 0
$$995$$ 39.5980 1.25534
$$996$$ −10.8284 −0.343112
$$997$$ 9.85786 0.312202 0.156101 0.987741i $$-0.450108\pi$$
0.156101 + 0.987741i $$0.450108\pi$$
$$998$$ 76.7696 2.43010
$$999$$ −9.48528 −0.300101
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 1617.2.a.n.1.1 2
3.2 odd 2 4851.2.a.be.1.2 2
7.2 even 3 231.2.i.d.67.2 4
7.4 even 3 231.2.i.d.100.2 yes 4
7.6 odd 2 1617.2.a.m.1.1 2
21.2 odd 6 693.2.i.f.298.1 4
21.11 odd 6 693.2.i.f.100.1 4
21.20 even 2 4851.2.a.bd.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.i.d.67.2 4 7.2 even 3
231.2.i.d.100.2 yes 4 7.4 even 3
693.2.i.f.100.1 4 21.11 odd 6
693.2.i.f.298.1 4 21.2 odd 6
1617.2.a.m.1.1 2 7.6 odd 2
1617.2.a.n.1.1 2 1.1 even 1 trivial
4851.2.a.bd.1.2 2 21.20 even 2
4851.2.a.be.1.2 2 3.2 odd 2