Properties

Label 8034.2.a.w.1.8
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-1.61063\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.40749 q^{5} -1.00000 q^{6} -0.998879 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.40749 q^{5} -1.00000 q^{6} -0.998879 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.40749 q^{10} +4.83053 q^{11} +1.00000 q^{12} +1.00000 q^{13} +0.998879 q^{14} +1.40749 q^{15} +1.00000 q^{16} -2.95365 q^{17} -1.00000 q^{18} +1.88860 q^{19} +1.40749 q^{20} -0.998879 q^{21} -4.83053 q^{22} -2.84489 q^{23} -1.00000 q^{24} -3.01896 q^{25} -1.00000 q^{26} +1.00000 q^{27} -0.998879 q^{28} -5.72981 q^{29} -1.40749 q^{30} -10.1479 q^{31} -1.00000 q^{32} +4.83053 q^{33} +2.95365 q^{34} -1.40592 q^{35} +1.00000 q^{36} -4.75750 q^{37} -1.88860 q^{38} +1.00000 q^{39} -1.40749 q^{40} -4.77235 q^{41} +0.998879 q^{42} +9.75344 q^{43} +4.83053 q^{44} +1.40749 q^{45} +2.84489 q^{46} -2.46105 q^{47} +1.00000 q^{48} -6.00224 q^{49} +3.01896 q^{50} -2.95365 q^{51} +1.00000 q^{52} -8.38795 q^{53} -1.00000 q^{54} +6.79894 q^{55} +0.998879 q^{56} +1.88860 q^{57} +5.72981 q^{58} -8.26441 q^{59} +1.40749 q^{60} -6.11324 q^{61} +10.1479 q^{62} -0.998879 q^{63} +1.00000 q^{64} +1.40749 q^{65} -4.83053 q^{66} -4.55266 q^{67} -2.95365 q^{68} -2.84489 q^{69} +1.40592 q^{70} -3.66759 q^{71} -1.00000 q^{72} -13.4965 q^{73} +4.75750 q^{74} -3.01896 q^{75} +1.88860 q^{76} -4.82512 q^{77} -1.00000 q^{78} -11.1098 q^{79} +1.40749 q^{80} +1.00000 q^{81} +4.77235 q^{82} +1.76274 q^{83} -0.998879 q^{84} -4.15724 q^{85} -9.75344 q^{86} -5.72981 q^{87} -4.83053 q^{88} -0.860992 q^{89} -1.40749 q^{90} -0.998879 q^{91} -2.84489 q^{92} -10.1479 q^{93} +2.46105 q^{94} +2.65819 q^{95} -1.00000 q^{96} -5.09415 q^{97} +6.00224 q^{98} +4.83053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.40749 0.629450 0.314725 0.949183i \(-0.398088\pi\)
0.314725 + 0.949183i \(0.398088\pi\)
\(6\) −1.00000 −0.408248
\(7\) −0.998879 −0.377541 −0.188770 0.982021i \(-0.560450\pi\)
−0.188770 + 0.982021i \(0.560450\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.40749 −0.445089
\(11\) 4.83053 1.45646 0.728230 0.685333i \(-0.240343\pi\)
0.728230 + 0.685333i \(0.240343\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0.998879 0.266962
\(15\) 1.40749 0.363413
\(16\) 1.00000 0.250000
\(17\) −2.95365 −0.716365 −0.358183 0.933652i \(-0.616603\pi\)
−0.358183 + 0.933652i \(0.616603\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.88860 0.433274 0.216637 0.976252i \(-0.430491\pi\)
0.216637 + 0.976252i \(0.430491\pi\)
\(20\) 1.40749 0.314725
\(21\) −0.998879 −0.217973
\(22\) −4.83053 −1.02987
\(23\) −2.84489 −0.593200 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.01896 −0.603792
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −0.998879 −0.188770
\(29\) −5.72981 −1.06400 −0.532000 0.846744i \(-0.678559\pi\)
−0.532000 + 0.846744i \(0.678559\pi\)
\(30\) −1.40749 −0.256972
\(31\) −10.1479 −1.82261 −0.911305 0.411731i \(-0.864924\pi\)
−0.911305 + 0.411731i \(0.864924\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.83053 0.840888
\(34\) 2.95365 0.506547
\(35\) −1.40592 −0.237643
\(36\) 1.00000 0.166667
\(37\) −4.75750 −0.782128 −0.391064 0.920363i \(-0.627893\pi\)
−0.391064 + 0.920363i \(0.627893\pi\)
\(38\) −1.88860 −0.306371
\(39\) 1.00000 0.160128
\(40\) −1.40749 −0.222544
\(41\) −4.77235 −0.745315 −0.372658 0.927969i \(-0.621553\pi\)
−0.372658 + 0.927969i \(0.621553\pi\)
\(42\) 0.998879 0.154130
\(43\) 9.75344 1.48739 0.743693 0.668521i \(-0.233072\pi\)
0.743693 + 0.668521i \(0.233072\pi\)
\(44\) 4.83053 0.728230
\(45\) 1.40749 0.209817
\(46\) 2.84489 0.419456
\(47\) −2.46105 −0.358981 −0.179491 0.983760i \(-0.557445\pi\)
−0.179491 + 0.983760i \(0.557445\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00224 −0.857463
\(50\) 3.01896 0.426946
\(51\) −2.95365 −0.413594
\(52\) 1.00000 0.138675
\(53\) −8.38795 −1.15217 −0.576087 0.817389i \(-0.695421\pi\)
−0.576087 + 0.817389i \(0.695421\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.79894 0.916769
\(56\) 0.998879 0.133481
\(57\) 1.88860 0.250151
\(58\) 5.72981 0.752362
\(59\) −8.26441 −1.07593 −0.537967 0.842966i \(-0.680808\pi\)
−0.537967 + 0.842966i \(0.680808\pi\)
\(60\) 1.40749 0.181707
\(61\) −6.11324 −0.782720 −0.391360 0.920238i \(-0.627995\pi\)
−0.391360 + 0.920238i \(0.627995\pi\)
\(62\) 10.1479 1.28878
\(63\) −0.998879 −0.125847
\(64\) 1.00000 0.125000
\(65\) 1.40749 0.174578
\(66\) −4.83053 −0.594597
\(67\) −4.55266 −0.556196 −0.278098 0.960553i \(-0.589704\pi\)
−0.278098 + 0.960553i \(0.589704\pi\)
\(68\) −2.95365 −0.358183
\(69\) −2.84489 −0.342484
\(70\) 1.40592 0.168039
\(71\) −3.66759 −0.435262 −0.217631 0.976031i \(-0.569833\pi\)
−0.217631 + 0.976031i \(0.569833\pi\)
\(72\) −1.00000 −0.117851
\(73\) −13.4965 −1.57964 −0.789821 0.613337i \(-0.789827\pi\)
−0.789821 + 0.613337i \(0.789827\pi\)
\(74\) 4.75750 0.553048
\(75\) −3.01896 −0.348600
\(76\) 1.88860 0.216637
\(77\) −4.82512 −0.549873
\(78\) −1.00000 −0.113228
\(79\) −11.1098 −1.24995 −0.624977 0.780643i \(-0.714892\pi\)
−0.624977 + 0.780643i \(0.714892\pi\)
\(80\) 1.40749 0.157363
\(81\) 1.00000 0.111111
\(82\) 4.77235 0.527018
\(83\) 1.76274 0.193486 0.0967429 0.995309i \(-0.469158\pi\)
0.0967429 + 0.995309i \(0.469158\pi\)
\(84\) −0.998879 −0.108987
\(85\) −4.15724 −0.450916
\(86\) −9.75344 −1.05174
\(87\) −5.72981 −0.614301
\(88\) −4.83053 −0.514936
\(89\) −0.860992 −0.0912650 −0.0456325 0.998958i \(-0.514530\pi\)
−0.0456325 + 0.998958i \(0.514530\pi\)
\(90\) −1.40749 −0.148363
\(91\) −0.998879 −0.104711
\(92\) −2.84489 −0.296600
\(93\) −10.1479 −1.05228
\(94\) 2.46105 0.253838
\(95\) 2.65819 0.272725
\(96\) −1.00000 −0.102062
\(97\) −5.09415 −0.517232 −0.258616 0.965980i \(-0.583266\pi\)
−0.258616 + 0.965980i \(0.583266\pi\)
\(98\) 6.00224 0.606318
\(99\) 4.83053 0.485487
\(100\) −3.01896 −0.301896
\(101\) −5.83551 −0.580655 −0.290327 0.956927i \(-0.593764\pi\)
−0.290327 + 0.956927i \(0.593764\pi\)
\(102\) 2.95365 0.292455
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −1.40592 −0.137203
\(106\) 8.38795 0.814710
\(107\) 13.0054 1.25728 0.628642 0.777695i \(-0.283611\pi\)
0.628642 + 0.777695i \(0.283611\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.24549 −0.119297 −0.0596484 0.998219i \(-0.518998\pi\)
−0.0596484 + 0.998219i \(0.518998\pi\)
\(110\) −6.79894 −0.648254
\(111\) −4.75750 −0.451562
\(112\) −0.998879 −0.0943852
\(113\) 9.96193 0.937139 0.468570 0.883427i \(-0.344770\pi\)
0.468570 + 0.883427i \(0.344770\pi\)
\(114\) −1.88860 −0.176883
\(115\) −4.00416 −0.373390
\(116\) −5.72981 −0.532000
\(117\) 1.00000 0.0924500
\(118\) 8.26441 0.760801
\(119\) 2.95034 0.270457
\(120\) −1.40749 −0.128486
\(121\) 12.3340 1.12128
\(122\) 6.11324 0.553466
\(123\) −4.77235 −0.430308
\(124\) −10.1479 −0.911305
\(125\) −11.2866 −1.00951
\(126\) 0.998879 0.0889872
\(127\) 3.66487 0.325204 0.162602 0.986692i \(-0.448011\pi\)
0.162602 + 0.986692i \(0.448011\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 9.75344 0.858743
\(130\) −1.40749 −0.123445
\(131\) 11.9908 1.04764 0.523822 0.851827i \(-0.324506\pi\)
0.523822 + 0.851827i \(0.324506\pi\)
\(132\) 4.83053 0.420444
\(133\) −1.88648 −0.163579
\(134\) 4.55266 0.393290
\(135\) 1.40749 0.121138
\(136\) 2.95365 0.253273
\(137\) 11.6193 0.992702 0.496351 0.868122i \(-0.334673\pi\)
0.496351 + 0.868122i \(0.334673\pi\)
\(138\) 2.84489 0.242173
\(139\) 10.1782 0.863301 0.431650 0.902041i \(-0.357931\pi\)
0.431650 + 0.902041i \(0.357931\pi\)
\(140\) −1.40592 −0.118822
\(141\) −2.46105 −0.207258
\(142\) 3.66759 0.307777
\(143\) 4.83053 0.403949
\(144\) 1.00000 0.0833333
\(145\) −8.06468 −0.669735
\(146\) 13.4965 1.11698
\(147\) −6.00224 −0.495056
\(148\) −4.75750 −0.391064
\(149\) 17.0885 1.39995 0.699973 0.714169i \(-0.253195\pi\)
0.699973 + 0.714169i \(0.253195\pi\)
\(150\) 3.01896 0.246497
\(151\) 13.2289 1.07655 0.538277 0.842768i \(-0.319076\pi\)
0.538277 + 0.842768i \(0.319076\pi\)
\(152\) −1.88860 −0.153186
\(153\) −2.95365 −0.238788
\(154\) 4.82512 0.388819
\(155\) −14.2831 −1.14724
\(156\) 1.00000 0.0800641
\(157\) 9.61127 0.767062 0.383531 0.923528i \(-0.374708\pi\)
0.383531 + 0.923528i \(0.374708\pi\)
\(158\) 11.1098 0.883852
\(159\) −8.38795 −0.665208
\(160\) −1.40749 −0.111272
\(161\) 2.84170 0.223957
\(162\) −1.00000 −0.0785674
\(163\) −7.98844 −0.625703 −0.312851 0.949802i \(-0.601284\pi\)
−0.312851 + 0.949802i \(0.601284\pi\)
\(164\) −4.77235 −0.372658
\(165\) 6.79894 0.529297
\(166\) −1.76274 −0.136815
\(167\) 3.41720 0.264431 0.132215 0.991221i \(-0.457791\pi\)
0.132215 + 0.991221i \(0.457791\pi\)
\(168\) 0.998879 0.0770652
\(169\) 1.00000 0.0769231
\(170\) 4.15724 0.318846
\(171\) 1.88860 0.144425
\(172\) 9.75344 0.743693
\(173\) −24.4518 −1.85904 −0.929519 0.368773i \(-0.879778\pi\)
−0.929519 + 0.368773i \(0.879778\pi\)
\(174\) 5.72981 0.434376
\(175\) 3.01558 0.227956
\(176\) 4.83053 0.364115
\(177\) −8.26441 −0.621191
\(178\) 0.860992 0.0645341
\(179\) −11.6219 −0.868664 −0.434332 0.900753i \(-0.643015\pi\)
−0.434332 + 0.900753i \(0.643015\pi\)
\(180\) 1.40749 0.104908
\(181\) 16.1688 1.20181 0.600907 0.799319i \(-0.294806\pi\)
0.600907 + 0.799319i \(0.294806\pi\)
\(182\) 0.998879 0.0740418
\(183\) −6.11324 −0.451903
\(184\) 2.84489 0.209728
\(185\) −6.69615 −0.492311
\(186\) 10.1479 0.744078
\(187\) −14.2677 −1.04336
\(188\) −2.46105 −0.179491
\(189\) −0.998879 −0.0726578
\(190\) −2.65819 −0.192845
\(191\) −15.3479 −1.11054 −0.555269 0.831671i \(-0.687385\pi\)
−0.555269 + 0.831671i \(0.687385\pi\)
\(192\) 1.00000 0.0721688
\(193\) 5.62711 0.405048 0.202524 0.979277i \(-0.435086\pi\)
0.202524 + 0.979277i \(0.435086\pi\)
\(194\) 5.09415 0.365739
\(195\) 1.40749 0.100793
\(196\) −6.00224 −0.428731
\(197\) −8.91051 −0.634848 −0.317424 0.948284i \(-0.602818\pi\)
−0.317424 + 0.948284i \(0.602818\pi\)
\(198\) −4.83053 −0.343291
\(199\) −21.5100 −1.52480 −0.762401 0.647105i \(-0.775980\pi\)
−0.762401 + 0.647105i \(0.775980\pi\)
\(200\) 3.01896 0.213473
\(201\) −4.55266 −0.321120
\(202\) 5.83551 0.410585
\(203\) 5.72339 0.401703
\(204\) −2.95365 −0.206797
\(205\) −6.71705 −0.469139
\(206\) 1.00000 0.0696733
\(207\) −2.84489 −0.197733
\(208\) 1.00000 0.0693375
\(209\) 9.12294 0.631047
\(210\) 1.40592 0.0970174
\(211\) 25.6510 1.76589 0.882945 0.469476i \(-0.155557\pi\)
0.882945 + 0.469476i \(0.155557\pi\)
\(212\) −8.38795 −0.576087
\(213\) −3.66759 −0.251299
\(214\) −13.0054 −0.889034
\(215\) 13.7279 0.936235
\(216\) −1.00000 −0.0680414
\(217\) 10.1365 0.688110
\(218\) 1.24549 0.0843555
\(219\) −13.4965 −0.912007
\(220\) 6.79894 0.458385
\(221\) −2.95365 −0.198684
\(222\) 4.75750 0.319302
\(223\) 12.8137 0.858069 0.429034 0.903288i \(-0.358854\pi\)
0.429034 + 0.903288i \(0.358854\pi\)
\(224\) 0.998879 0.0667404
\(225\) −3.01896 −0.201264
\(226\) −9.96193 −0.662657
\(227\) −12.3807 −0.821736 −0.410868 0.911695i \(-0.634774\pi\)
−0.410868 + 0.911695i \(0.634774\pi\)
\(228\) 1.88860 0.125076
\(229\) 12.8890 0.851726 0.425863 0.904788i \(-0.359971\pi\)
0.425863 + 0.904788i \(0.359971\pi\)
\(230\) 4.00416 0.264026
\(231\) −4.82512 −0.317469
\(232\) 5.72981 0.376181
\(233\) −6.27793 −0.411281 −0.205641 0.978628i \(-0.565928\pi\)
−0.205641 + 0.978628i \(0.565928\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −3.46391 −0.225961
\(236\) −8.26441 −0.537967
\(237\) −11.1098 −0.721662
\(238\) −2.95034 −0.191242
\(239\) −20.3261 −1.31478 −0.657392 0.753549i \(-0.728341\pi\)
−0.657392 + 0.753549i \(0.728341\pi\)
\(240\) 1.40749 0.0908533
\(241\) −5.81756 −0.374742 −0.187371 0.982289i \(-0.559997\pi\)
−0.187371 + 0.982289i \(0.559997\pi\)
\(242\) −12.3340 −0.792862
\(243\) 1.00000 0.0641500
\(244\) −6.11324 −0.391360
\(245\) −8.44812 −0.539730
\(246\) 4.77235 0.304274
\(247\) 1.88860 0.120169
\(248\) 10.1479 0.644390
\(249\) 1.76274 0.111709
\(250\) 11.2866 0.713830
\(251\) 10.3172 0.651215 0.325607 0.945505i \(-0.394431\pi\)
0.325607 + 0.945505i \(0.394431\pi\)
\(252\) −0.998879 −0.0629235
\(253\) −13.7423 −0.863972
\(254\) −3.66487 −0.229954
\(255\) −4.15724 −0.260337
\(256\) 1.00000 0.0625000
\(257\) 3.22415 0.201117 0.100558 0.994931i \(-0.467937\pi\)
0.100558 + 0.994931i \(0.467937\pi\)
\(258\) −9.75344 −0.607223
\(259\) 4.75217 0.295285
\(260\) 1.40749 0.0872890
\(261\) −5.72981 −0.354667
\(262\) −11.9908 −0.740797
\(263\) −27.8650 −1.71823 −0.859115 0.511782i \(-0.828986\pi\)
−0.859115 + 0.511782i \(0.828986\pi\)
\(264\) −4.83053 −0.297299
\(265\) −11.8060 −0.725236
\(266\) 1.88648 0.115668
\(267\) −0.860992 −0.0526918
\(268\) −4.55266 −0.278098
\(269\) −5.47995 −0.334119 −0.167059 0.985947i \(-0.553427\pi\)
−0.167059 + 0.985947i \(0.553427\pi\)
\(270\) −1.40749 −0.0856573
\(271\) 25.0590 1.52223 0.761114 0.648618i \(-0.224653\pi\)
0.761114 + 0.648618i \(0.224653\pi\)
\(272\) −2.95365 −0.179091
\(273\) −0.998879 −0.0604549
\(274\) −11.6193 −0.701947
\(275\) −14.5832 −0.879399
\(276\) −2.84489 −0.171242
\(277\) 4.14155 0.248841 0.124421 0.992230i \(-0.460293\pi\)
0.124421 + 0.992230i \(0.460293\pi\)
\(278\) −10.1782 −0.610446
\(279\) −10.1479 −0.607537
\(280\) 1.40592 0.0840195
\(281\) −2.03599 −0.121457 −0.0607284 0.998154i \(-0.519342\pi\)
−0.0607284 + 0.998154i \(0.519342\pi\)
\(282\) 2.46105 0.146553
\(283\) 10.5521 0.627256 0.313628 0.949546i \(-0.398455\pi\)
0.313628 + 0.949546i \(0.398455\pi\)
\(284\) −3.66759 −0.217631
\(285\) 2.65819 0.157458
\(286\) −4.83053 −0.285635
\(287\) 4.76700 0.281387
\(288\) −1.00000 −0.0589256
\(289\) −8.27596 −0.486821
\(290\) 8.06468 0.473574
\(291\) −5.09415 −0.298624
\(292\) −13.4965 −0.789821
\(293\) 20.2309 1.18190 0.590950 0.806708i \(-0.298753\pi\)
0.590950 + 0.806708i \(0.298753\pi\)
\(294\) 6.00224 0.350058
\(295\) −11.6321 −0.677248
\(296\) 4.75750 0.276524
\(297\) 4.83053 0.280296
\(298\) −17.0885 −0.989912
\(299\) −2.84489 −0.164524
\(300\) −3.01896 −0.174300
\(301\) −9.74251 −0.561549
\(302\) −13.2289 −0.761238
\(303\) −5.83551 −0.335241
\(304\) 1.88860 0.108319
\(305\) −8.60434 −0.492683
\(306\) 2.95365 0.168849
\(307\) 9.28706 0.530041 0.265020 0.964243i \(-0.414621\pi\)
0.265020 + 0.964243i \(0.414621\pi\)
\(308\) −4.82512 −0.274937
\(309\) −1.00000 −0.0568880
\(310\) 14.2831 0.811223
\(311\) 18.0200 1.02182 0.510910 0.859634i \(-0.329309\pi\)
0.510910 + 0.859634i \(0.329309\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 26.1537 1.47829 0.739146 0.673545i \(-0.235229\pi\)
0.739146 + 0.673545i \(0.235229\pi\)
\(314\) −9.61127 −0.542395
\(315\) −1.40592 −0.0792144
\(316\) −11.1098 −0.624977
\(317\) 7.89534 0.443447 0.221723 0.975110i \(-0.428832\pi\)
0.221723 + 0.975110i \(0.428832\pi\)
\(318\) 8.38795 0.470373
\(319\) −27.6780 −1.54967
\(320\) 1.40749 0.0786813
\(321\) 13.0054 0.725893
\(322\) −2.84170 −0.158362
\(323\) −5.57826 −0.310383
\(324\) 1.00000 0.0555556
\(325\) −3.01896 −0.167462
\(326\) 7.98844 0.442439
\(327\) −1.24549 −0.0688760
\(328\) 4.77235 0.263509
\(329\) 2.45829 0.135530
\(330\) −6.79894 −0.374269
\(331\) 21.0364 1.15627 0.578134 0.815942i \(-0.303781\pi\)
0.578134 + 0.815942i \(0.303781\pi\)
\(332\) 1.76274 0.0967429
\(333\) −4.75750 −0.260709
\(334\) −3.41720 −0.186981
\(335\) −6.40784 −0.350098
\(336\) −0.998879 −0.0544933
\(337\) 26.1538 1.42469 0.712344 0.701831i \(-0.247634\pi\)
0.712344 + 0.701831i \(0.247634\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 9.96193 0.541058
\(340\) −4.15724 −0.225458
\(341\) −49.0196 −2.65456
\(342\) −1.88860 −0.102124
\(343\) 12.9877 0.701268
\(344\) −9.75344 −0.525870
\(345\) −4.00416 −0.215577
\(346\) 24.4518 1.31454
\(347\) −20.9493 −1.12462 −0.562308 0.826928i \(-0.690087\pi\)
−0.562308 + 0.826928i \(0.690087\pi\)
\(348\) −5.72981 −0.307150
\(349\) 33.7856 1.80850 0.904251 0.427002i \(-0.140430\pi\)
0.904251 + 0.427002i \(0.140430\pi\)
\(350\) −3.01558 −0.161189
\(351\) 1.00000 0.0533761
\(352\) −4.83053 −0.257468
\(353\) −8.70939 −0.463554 −0.231777 0.972769i \(-0.574454\pi\)
−0.231777 + 0.972769i \(0.574454\pi\)
\(354\) 8.26441 0.439249
\(355\) −5.16210 −0.273976
\(356\) −0.860992 −0.0456325
\(357\) 2.95034 0.156148
\(358\) 11.6219 0.614238
\(359\) −26.5369 −1.40056 −0.700281 0.713867i \(-0.746942\pi\)
−0.700281 + 0.713867i \(0.746942\pi\)
\(360\) −1.40749 −0.0741814
\(361\) −15.4332 −0.812273
\(362\) −16.1688 −0.849811
\(363\) 12.3340 0.647369
\(364\) −0.998879 −0.0523555
\(365\) −18.9962 −0.994306
\(366\) 6.11324 0.319544
\(367\) 9.83164 0.513207 0.256604 0.966517i \(-0.417396\pi\)
0.256604 + 0.966517i \(0.417396\pi\)
\(368\) −2.84489 −0.148300
\(369\) −4.77235 −0.248438
\(370\) 6.69615 0.348116
\(371\) 8.37855 0.434992
\(372\) −10.1479 −0.526142
\(373\) −36.6010 −1.89513 −0.947565 0.319563i \(-0.896464\pi\)
−0.947565 + 0.319563i \(0.896464\pi\)
\(374\) 14.2677 0.737765
\(375\) −11.2866 −0.582839
\(376\) 2.46105 0.126919
\(377\) −5.72981 −0.295100
\(378\) 0.998879 0.0513768
\(379\) −23.6372 −1.21416 −0.607080 0.794640i \(-0.707659\pi\)
−0.607080 + 0.794640i \(0.707659\pi\)
\(380\) 2.65819 0.136362
\(381\) 3.66487 0.187757
\(382\) 15.3479 0.785269
\(383\) 29.2380 1.49399 0.746996 0.664829i \(-0.231496\pi\)
0.746996 + 0.664829i \(0.231496\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −6.79132 −0.346118
\(386\) −5.62711 −0.286412
\(387\) 9.75344 0.495795
\(388\) −5.09415 −0.258616
\(389\) 9.48255 0.480784 0.240392 0.970676i \(-0.422724\pi\)
0.240392 + 0.970676i \(0.422724\pi\)
\(390\) −1.40749 −0.0712712
\(391\) 8.40279 0.424948
\(392\) 6.00224 0.303159
\(393\) 11.9908 0.604858
\(394\) 8.91051 0.448905
\(395\) −15.6370 −0.786784
\(396\) 4.83053 0.242743
\(397\) 6.03291 0.302783 0.151392 0.988474i \(-0.451625\pi\)
0.151392 + 0.988474i \(0.451625\pi\)
\(398\) 21.5100 1.07820
\(399\) −1.88648 −0.0944422
\(400\) −3.01896 −0.150948
\(401\) −10.9056 −0.544601 −0.272300 0.962212i \(-0.587784\pi\)
−0.272300 + 0.962212i \(0.587784\pi\)
\(402\) 4.55266 0.227066
\(403\) −10.1479 −0.505501
\(404\) −5.83551 −0.290327
\(405\) 1.40749 0.0699389
\(406\) −5.72339 −0.284047
\(407\) −22.9812 −1.13914
\(408\) 2.95365 0.146227
\(409\) −20.6537 −1.02126 −0.510631 0.859800i \(-0.670588\pi\)
−0.510631 + 0.859800i \(0.670588\pi\)
\(410\) 6.71705 0.331731
\(411\) 11.6193 0.573137
\(412\) −1.00000 −0.0492665
\(413\) 8.25515 0.406209
\(414\) 2.84489 0.139819
\(415\) 2.48104 0.121790
\(416\) −1.00000 −0.0490290
\(417\) 10.1782 0.498427
\(418\) −9.12294 −0.446217
\(419\) 27.0666 1.32229 0.661145 0.750258i \(-0.270071\pi\)
0.661145 + 0.750258i \(0.270071\pi\)
\(420\) −1.40592 −0.0686017
\(421\) −25.4862 −1.24212 −0.621062 0.783762i \(-0.713298\pi\)
−0.621062 + 0.783762i \(0.713298\pi\)
\(422\) −25.6510 −1.24867
\(423\) −2.46105 −0.119660
\(424\) 8.38795 0.407355
\(425\) 8.91695 0.432536
\(426\) 3.66759 0.177695
\(427\) 6.10638 0.295509
\(428\) 13.0054 0.628642
\(429\) 4.83053 0.233220
\(430\) −13.7279 −0.662018
\(431\) 30.4313 1.46582 0.732911 0.680324i \(-0.238161\pi\)
0.732911 + 0.680324i \(0.238161\pi\)
\(432\) 1.00000 0.0481125
\(433\) 37.2374 1.78952 0.894758 0.446551i \(-0.147348\pi\)
0.894758 + 0.446551i \(0.147348\pi\)
\(434\) −10.1365 −0.486567
\(435\) −8.06468 −0.386672
\(436\) −1.24549 −0.0596484
\(437\) −5.37285 −0.257018
\(438\) 13.4965 0.644886
\(439\) −28.7436 −1.37186 −0.685929 0.727668i \(-0.740604\pi\)
−0.685929 + 0.727668i \(0.740604\pi\)
\(440\) −6.79894 −0.324127
\(441\) −6.00224 −0.285821
\(442\) 2.95365 0.140491
\(443\) −9.15518 −0.434976 −0.217488 0.976063i \(-0.569786\pi\)
−0.217488 + 0.976063i \(0.569786\pi\)
\(444\) −4.75750 −0.225781
\(445\) −1.21184 −0.0574468
\(446\) −12.8137 −0.606746
\(447\) 17.0885 0.808260
\(448\) −0.998879 −0.0471926
\(449\) 35.7192 1.68569 0.842847 0.538154i \(-0.180878\pi\)
0.842847 + 0.538154i \(0.180878\pi\)
\(450\) 3.01896 0.142315
\(451\) −23.0530 −1.08552
\(452\) 9.96193 0.468570
\(453\) 13.2289 0.621548
\(454\) 12.3807 0.581055
\(455\) −1.40592 −0.0659104
\(456\) −1.88860 −0.0884417
\(457\) 22.8348 1.06817 0.534083 0.845432i \(-0.320657\pi\)
0.534083 + 0.845432i \(0.320657\pi\)
\(458\) −12.8890 −0.602261
\(459\) −2.95365 −0.137865
\(460\) −4.00416 −0.186695
\(461\) −24.7047 −1.15061 −0.575305 0.817939i \(-0.695117\pi\)
−0.575305 + 0.817939i \(0.695117\pi\)
\(462\) 4.82512 0.224485
\(463\) 9.59246 0.445799 0.222900 0.974841i \(-0.428448\pi\)
0.222900 + 0.974841i \(0.428448\pi\)
\(464\) −5.72981 −0.266000
\(465\) −14.2831 −0.662361
\(466\) 6.27793 0.290820
\(467\) −20.3961 −0.943817 −0.471908 0.881648i \(-0.656435\pi\)
−0.471908 + 0.881648i \(0.656435\pi\)
\(468\) 1.00000 0.0462250
\(469\) 4.54756 0.209987
\(470\) 3.46391 0.159778
\(471\) 9.61127 0.442864
\(472\) 8.26441 0.380400
\(473\) 47.1143 2.16632
\(474\) 11.1098 0.510292
\(475\) −5.70161 −0.261608
\(476\) 2.95034 0.135229
\(477\) −8.38795 −0.384058
\(478\) 20.3261 0.929693
\(479\) −6.02670 −0.275367 −0.137684 0.990476i \(-0.543966\pi\)
−0.137684 + 0.990476i \(0.543966\pi\)
\(480\) −1.40749 −0.0642430
\(481\) −4.75750 −0.216923
\(482\) 5.81756 0.264983
\(483\) 2.84170 0.129302
\(484\) 12.3340 0.560638
\(485\) −7.16998 −0.325572
\(486\) −1.00000 −0.0453609
\(487\) 23.6141 1.07006 0.535029 0.844834i \(-0.320301\pi\)
0.535029 + 0.844834i \(0.320301\pi\)
\(488\) 6.11324 0.276733
\(489\) −7.98844 −0.361250
\(490\) 8.44812 0.381647
\(491\) −18.4702 −0.833550 −0.416775 0.909010i \(-0.636840\pi\)
−0.416775 + 0.909010i \(0.636840\pi\)
\(492\) −4.77235 −0.215154
\(493\) 16.9239 0.762212
\(494\) −1.88860 −0.0849721
\(495\) 6.79894 0.305590
\(496\) −10.1479 −0.455653
\(497\) 3.66347 0.164329
\(498\) −1.76274 −0.0789902
\(499\) 24.8331 1.11168 0.555841 0.831288i \(-0.312396\pi\)
0.555841 + 0.831288i \(0.312396\pi\)
\(500\) −11.2866 −0.504754
\(501\) 3.41720 0.152669
\(502\) −10.3172 −0.460478
\(503\) −21.4279 −0.955423 −0.477712 0.878517i \(-0.658534\pi\)
−0.477712 + 0.878517i \(0.658534\pi\)
\(504\) 0.998879 0.0444936
\(505\) −8.21344 −0.365493
\(506\) 13.7423 0.610920
\(507\) 1.00000 0.0444116
\(508\) 3.66487 0.162602
\(509\) −44.6685 −1.97989 −0.989947 0.141437i \(-0.954828\pi\)
−0.989947 + 0.141437i \(0.954828\pi\)
\(510\) 4.15724 0.184086
\(511\) 13.4813 0.596379
\(512\) −1.00000 −0.0441942
\(513\) 1.88860 0.0833837
\(514\) −3.22415 −0.142211
\(515\) −1.40749 −0.0620216
\(516\) 9.75344 0.429371
\(517\) −11.8882 −0.522841
\(518\) −4.75217 −0.208798
\(519\) −24.4518 −1.07332
\(520\) −1.40749 −0.0617227
\(521\) 23.6255 1.03505 0.517525 0.855668i \(-0.326853\pi\)
0.517525 + 0.855668i \(0.326853\pi\)
\(522\) 5.72981 0.250787
\(523\) −14.6631 −0.641175 −0.320587 0.947219i \(-0.603880\pi\)
−0.320587 + 0.947219i \(0.603880\pi\)
\(524\) 11.9908 0.523822
\(525\) 3.01558 0.131611
\(526\) 27.8650 1.21497
\(527\) 29.9732 1.30565
\(528\) 4.83053 0.210222
\(529\) −14.9066 −0.648114
\(530\) 11.8060 0.512819
\(531\) −8.26441 −0.358645
\(532\) −1.88648 −0.0817894
\(533\) −4.77235 −0.206713
\(534\) 0.860992 0.0372588
\(535\) 18.3051 0.791398
\(536\) 4.55266 0.196645
\(537\) −11.6219 −0.501523
\(538\) 5.47995 0.236258
\(539\) −28.9940 −1.24886
\(540\) 1.40749 0.0605689
\(541\) −14.5293 −0.624663 −0.312331 0.949973i \(-0.601110\pi\)
−0.312331 + 0.949973i \(0.601110\pi\)
\(542\) −25.0590 −1.07638
\(543\) 16.1688 0.693868
\(544\) 2.95365 0.126637
\(545\) −1.75303 −0.0750914
\(546\) 0.998879 0.0427481
\(547\) 7.36417 0.314869 0.157435 0.987529i \(-0.449678\pi\)
0.157435 + 0.987529i \(0.449678\pi\)
\(548\) 11.6193 0.496351
\(549\) −6.11324 −0.260907
\(550\) 14.5832 0.621829
\(551\) −10.8213 −0.461004
\(552\) 2.84489 0.121086
\(553\) 11.0974 0.471909
\(554\) −4.14155 −0.175957
\(555\) −6.69615 −0.284236
\(556\) 10.1782 0.431650
\(557\) 31.0943 1.31751 0.658753 0.752359i \(-0.271084\pi\)
0.658753 + 0.752359i \(0.271084\pi\)
\(558\) 10.1479 0.429593
\(559\) 9.75344 0.412527
\(560\) −1.40592 −0.0594108
\(561\) −14.2677 −0.602383
\(562\) 2.03599 0.0858830
\(563\) −14.1457 −0.596171 −0.298086 0.954539i \(-0.596348\pi\)
−0.298086 + 0.954539i \(0.596348\pi\)
\(564\) −2.46105 −0.103629
\(565\) 14.0213 0.589883
\(566\) −10.5521 −0.443537
\(567\) −0.998879 −0.0419490
\(568\) 3.66759 0.153888
\(569\) −29.1790 −1.22325 −0.611623 0.791149i \(-0.709483\pi\)
−0.611623 + 0.791149i \(0.709483\pi\)
\(570\) −2.65819 −0.111339
\(571\) −28.3905 −1.18810 −0.594052 0.804426i \(-0.702473\pi\)
−0.594052 + 0.804426i \(0.702473\pi\)
\(572\) 4.83053 0.201975
\(573\) −15.3479 −0.641169
\(574\) −4.76700 −0.198971
\(575\) 8.58860 0.358169
\(576\) 1.00000 0.0416667
\(577\) 24.1055 1.00352 0.501762 0.865006i \(-0.332685\pi\)
0.501762 + 0.865006i \(0.332685\pi\)
\(578\) 8.27596 0.344234
\(579\) 5.62711 0.233855
\(580\) −8.06468 −0.334868
\(581\) −1.76076 −0.0730488
\(582\) 5.09415 0.211159
\(583\) −40.5182 −1.67809
\(584\) 13.4965 0.558488
\(585\) 1.40749 0.0581927
\(586\) −20.2309 −0.835729
\(587\) −20.9165 −0.863318 −0.431659 0.902037i \(-0.642072\pi\)
−0.431659 + 0.902037i \(0.642072\pi\)
\(588\) −6.00224 −0.247528
\(589\) −19.1653 −0.789690
\(590\) 11.6321 0.478886
\(591\) −8.91051 −0.366529
\(592\) −4.75750 −0.195532
\(593\) 31.4689 1.29227 0.646137 0.763222i \(-0.276384\pi\)
0.646137 + 0.763222i \(0.276384\pi\)
\(594\) −4.83053 −0.198199
\(595\) 4.15258 0.170239
\(596\) 17.0885 0.699973
\(597\) −21.5100 −0.880345
\(598\) 2.84489 0.116336
\(599\) −23.6003 −0.964284 −0.482142 0.876093i \(-0.660141\pi\)
−0.482142 + 0.876093i \(0.660141\pi\)
\(600\) 3.01896 0.123249
\(601\) 31.7300 1.29429 0.647147 0.762365i \(-0.275962\pi\)
0.647147 + 0.762365i \(0.275962\pi\)
\(602\) 9.74251 0.397075
\(603\) −4.55266 −0.185399
\(604\) 13.2289 0.538277
\(605\) 17.3601 0.705787
\(606\) 5.83551 0.237051
\(607\) −14.8494 −0.602718 −0.301359 0.953511i \(-0.597440\pi\)
−0.301359 + 0.953511i \(0.597440\pi\)
\(608\) −1.88860 −0.0765928
\(609\) 5.72339 0.231924
\(610\) 8.60434 0.348380
\(611\) −2.46105 −0.0995634
\(612\) −2.95365 −0.119394
\(613\) 23.6567 0.955484 0.477742 0.878500i \(-0.341455\pi\)
0.477742 + 0.878500i \(0.341455\pi\)
\(614\) −9.28706 −0.374795
\(615\) −6.71705 −0.270858
\(616\) 4.82512 0.194409
\(617\) −9.47246 −0.381347 −0.190673 0.981654i \(-0.561067\pi\)
−0.190673 + 0.981654i \(0.561067\pi\)
\(618\) 1.00000 0.0402259
\(619\) 26.5081 1.06545 0.532726 0.846288i \(-0.321168\pi\)
0.532726 + 0.846288i \(0.321168\pi\)
\(620\) −14.2831 −0.573621
\(621\) −2.84489 −0.114161
\(622\) −18.0200 −0.722536
\(623\) 0.860027 0.0344562
\(624\) 1.00000 0.0400320
\(625\) −0.791061 −0.0316424
\(626\) −26.1537 −1.04531
\(627\) 9.12294 0.364335
\(628\) 9.61127 0.383531
\(629\) 14.0520 0.560289
\(630\) 1.40592 0.0560130
\(631\) 38.7613 1.54306 0.771531 0.636192i \(-0.219491\pi\)
0.771531 + 0.636192i \(0.219491\pi\)
\(632\) 11.1098 0.441926
\(633\) 25.6510 1.01954
\(634\) −7.89534 −0.313564
\(635\) 5.15828 0.204700
\(636\) −8.38795 −0.332604
\(637\) −6.00224 −0.237817
\(638\) 27.6780 1.09578
\(639\) −3.66759 −0.145087
\(640\) −1.40749 −0.0556361
\(641\) 21.2063 0.837600 0.418800 0.908079i \(-0.362451\pi\)
0.418800 + 0.908079i \(0.362451\pi\)
\(642\) −13.0054 −0.513284
\(643\) −15.7977 −0.622999 −0.311500 0.950246i \(-0.600831\pi\)
−0.311500 + 0.950246i \(0.600831\pi\)
\(644\) 2.84170 0.111979
\(645\) 13.7279 0.540536
\(646\) 5.57826 0.219474
\(647\) −31.0349 −1.22011 −0.610054 0.792360i \(-0.708852\pi\)
−0.610054 + 0.792360i \(0.708852\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −39.9215 −1.56706
\(650\) 3.01896 0.118413
\(651\) 10.1365 0.397280
\(652\) −7.98844 −0.312851
\(653\) −35.0335 −1.37097 −0.685483 0.728089i \(-0.740409\pi\)
−0.685483 + 0.728089i \(0.740409\pi\)
\(654\) 1.24549 0.0487027
\(655\) 16.8770 0.659440
\(656\) −4.77235 −0.186329
\(657\) −13.4965 −0.526547
\(658\) −2.45829 −0.0958342
\(659\) −49.0836 −1.91202 −0.956012 0.293327i \(-0.905237\pi\)
−0.956012 + 0.293327i \(0.905237\pi\)
\(660\) 6.79894 0.264648
\(661\) −24.0650 −0.936020 −0.468010 0.883723i \(-0.655029\pi\)
−0.468010 + 0.883723i \(0.655029\pi\)
\(662\) −21.0364 −0.817605
\(663\) −2.95365 −0.114710
\(664\) −1.76274 −0.0684076
\(665\) −2.65521 −0.102965
\(666\) 4.75750 0.184349
\(667\) 16.3007 0.631164
\(668\) 3.41720 0.132215
\(669\) 12.8137 0.495406
\(670\) 6.40784 0.247556
\(671\) −29.5302 −1.14000
\(672\) 0.998879 0.0385326
\(673\) 9.91676 0.382263 0.191131 0.981564i \(-0.438784\pi\)
0.191131 + 0.981564i \(0.438784\pi\)
\(674\) −26.1538 −1.00741
\(675\) −3.01896 −0.116200
\(676\) 1.00000 0.0384615
\(677\) 17.8633 0.686543 0.343272 0.939236i \(-0.388465\pi\)
0.343272 + 0.939236i \(0.388465\pi\)
\(678\) −9.96193 −0.382585
\(679\) 5.08844 0.195276
\(680\) 4.15724 0.159423
\(681\) −12.3807 −0.474429
\(682\) 49.0196 1.87706
\(683\) 3.17969 0.121668 0.0608338 0.998148i \(-0.480624\pi\)
0.0608338 + 0.998148i \(0.480624\pi\)
\(684\) 1.88860 0.0722124
\(685\) 16.3541 0.624857
\(686\) −12.9877 −0.495871
\(687\) 12.8890 0.491744
\(688\) 9.75344 0.371846
\(689\) −8.38795 −0.319555
\(690\) 4.00416 0.152436
\(691\) 22.1457 0.842462 0.421231 0.906953i \(-0.361598\pi\)
0.421231 + 0.906953i \(0.361598\pi\)
\(692\) −24.4518 −0.929519
\(693\) −4.82512 −0.183291
\(694\) 20.9493 0.795224
\(695\) 14.3257 0.543405
\(696\) 5.72981 0.217188
\(697\) 14.0958 0.533918
\(698\) −33.7856 −1.27880
\(699\) −6.27793 −0.237453
\(700\) 3.01558 0.113978
\(701\) −21.3476 −0.806289 −0.403144 0.915136i \(-0.632083\pi\)
−0.403144 + 0.915136i \(0.632083\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −8.98501 −0.338876
\(704\) 4.83053 0.182057
\(705\) −3.46391 −0.130458
\(706\) 8.70939 0.327782
\(707\) 5.82897 0.219221
\(708\) −8.26441 −0.310596
\(709\) 16.7407 0.628709 0.314355 0.949306i \(-0.398212\pi\)
0.314355 + 0.949306i \(0.398212\pi\)
\(710\) 5.16210 0.193730
\(711\) −11.1098 −0.416652
\(712\) 0.860992 0.0322670
\(713\) 28.8695 1.08117
\(714\) −2.95034 −0.110414
\(715\) 6.79894 0.254266
\(716\) −11.6219 −0.434332
\(717\) −20.3261 −0.759091
\(718\) 26.5369 0.990347
\(719\) 13.5812 0.506494 0.253247 0.967402i \(-0.418502\pi\)
0.253247 + 0.967402i \(0.418502\pi\)
\(720\) 1.40749 0.0524542
\(721\) 0.998879 0.0372002
\(722\) 15.4332 0.574364
\(723\) −5.81756 −0.216357
\(724\) 16.1688 0.600907
\(725\) 17.2981 0.642435
\(726\) −12.3340 −0.457759
\(727\) 20.6522 0.765947 0.382974 0.923759i \(-0.374900\pi\)
0.382974 + 0.923759i \(0.374900\pi\)
\(728\) 0.998879 0.0370209
\(729\) 1.00000 0.0370370
\(730\) 18.9962 0.703081
\(731\) −28.8082 −1.06551
\(732\) −6.11324 −0.225952
\(733\) −24.6696 −0.911194 −0.455597 0.890186i \(-0.650574\pi\)
−0.455597 + 0.890186i \(0.650574\pi\)
\(734\) −9.83164 −0.362892
\(735\) −8.44812 −0.311613
\(736\) 2.84489 0.104864
\(737\) −21.9918 −0.810077
\(738\) 4.77235 0.175673
\(739\) 30.0209 1.10434 0.552169 0.833732i \(-0.313800\pi\)
0.552169 + 0.833732i \(0.313800\pi\)
\(740\) −6.69615 −0.246155
\(741\) 1.88860 0.0693794
\(742\) −8.37855 −0.307586
\(743\) −27.9466 −1.02526 −0.512631 0.858609i \(-0.671329\pi\)
−0.512631 + 0.858609i \(0.671329\pi\)
\(744\) 10.1479 0.372039
\(745\) 24.0520 0.881197
\(746\) 36.6010 1.34006
\(747\) 1.76274 0.0644953
\(748\) −14.2677 −0.521679
\(749\) −12.9909 −0.474676
\(750\) 11.2866 0.412130
\(751\) −1.09991 −0.0401365 −0.0200682 0.999799i \(-0.506388\pi\)
−0.0200682 + 0.999799i \(0.506388\pi\)
\(752\) −2.46105 −0.0897453
\(753\) 10.3172 0.375979
\(754\) 5.72981 0.208668
\(755\) 18.6196 0.677637
\(756\) −0.998879 −0.0363289
\(757\) 47.2731 1.71817 0.859084 0.511834i \(-0.171034\pi\)
0.859084 + 0.511834i \(0.171034\pi\)
\(758\) 23.6372 0.858541
\(759\) −13.7423 −0.498814
\(760\) −2.65819 −0.0964227
\(761\) −48.2098 −1.74760 −0.873802 0.486282i \(-0.838353\pi\)
−0.873802 + 0.486282i \(0.838353\pi\)
\(762\) −3.66487 −0.132764
\(763\) 1.24410 0.0450394
\(764\) −15.3479 −0.555269
\(765\) −4.15724 −0.150305
\(766\) −29.2380 −1.05641
\(767\) −8.26441 −0.298411
\(768\) 1.00000 0.0360844
\(769\) −44.9515 −1.62099 −0.810497 0.585743i \(-0.800803\pi\)
−0.810497 + 0.585743i \(0.800803\pi\)
\(770\) 6.79132 0.244742
\(771\) 3.22415 0.116115
\(772\) 5.62711 0.202524
\(773\) 20.0462 0.721011 0.360506 0.932757i \(-0.382604\pi\)
0.360506 + 0.932757i \(0.382604\pi\)
\(774\) −9.75344 −0.350580
\(775\) 30.6360 1.10048
\(776\) 5.09415 0.182869
\(777\) 4.75217 0.170483
\(778\) −9.48255 −0.339966
\(779\) −9.01305 −0.322926
\(780\) 1.40749 0.0503964
\(781\) −17.7164 −0.633942
\(782\) −8.40279 −0.300483
\(783\) −5.72981 −0.204767
\(784\) −6.00224 −0.214366
\(785\) 13.5278 0.482828
\(786\) −11.9908 −0.427699
\(787\) −14.6857 −0.523489 −0.261744 0.965137i \(-0.584298\pi\)
−0.261744 + 0.965137i \(0.584298\pi\)
\(788\) −8.91051 −0.317424
\(789\) −27.8650 −0.992021
\(790\) 15.6370 0.556341
\(791\) −9.95076 −0.353808
\(792\) −4.83053 −0.171645
\(793\) −6.11324 −0.217087
\(794\) −6.03291 −0.214100
\(795\) −11.8060 −0.418715
\(796\) −21.5100 −0.762401
\(797\) 37.0049 1.31078 0.655391 0.755290i \(-0.272504\pi\)
0.655391 + 0.755290i \(0.272504\pi\)
\(798\) 1.88648 0.0667807
\(799\) 7.26908 0.257161
\(800\) 3.01896 0.106736
\(801\) −0.860992 −0.0304217
\(802\) 10.9056 0.385091
\(803\) −65.1951 −2.30069
\(804\) −4.55266 −0.160560
\(805\) 3.99967 0.140970
\(806\) 10.1479 0.357443
\(807\) −5.47995 −0.192904
\(808\) 5.83551 0.205292
\(809\) −34.4491 −1.21117 −0.605583 0.795782i \(-0.707060\pi\)
−0.605583 + 0.795782i \(0.707060\pi\)
\(810\) −1.40749 −0.0494543
\(811\) −25.1527 −0.883230 −0.441615 0.897205i \(-0.645594\pi\)
−0.441615 + 0.897205i \(0.645594\pi\)
\(812\) 5.72339 0.200852
\(813\) 25.0590 0.878859
\(814\) 22.9812 0.805492
\(815\) −11.2437 −0.393849
\(816\) −2.95365 −0.103398
\(817\) 18.4203 0.644446
\(818\) 20.6537 0.722141
\(819\) −0.998879 −0.0349037
\(820\) −6.71705 −0.234569
\(821\) 51.1856 1.78639 0.893194 0.449671i \(-0.148459\pi\)
0.893194 + 0.449671i \(0.148459\pi\)
\(822\) −11.6193 −0.405269
\(823\) 9.93393 0.346275 0.173138 0.984898i \(-0.444609\pi\)
0.173138 + 0.984898i \(0.444609\pi\)
\(824\) 1.00000 0.0348367
\(825\) −14.5832 −0.507721
\(826\) −8.25515 −0.287233
\(827\) −4.04881 −0.140791 −0.0703954 0.997519i \(-0.522426\pi\)
−0.0703954 + 0.997519i \(0.522426\pi\)
\(828\) −2.84489 −0.0988666
\(829\) −44.1314 −1.53275 −0.766374 0.642395i \(-0.777941\pi\)
−0.766374 + 0.642395i \(0.777941\pi\)
\(830\) −2.48104 −0.0861183
\(831\) 4.14155 0.143669
\(832\) 1.00000 0.0346688
\(833\) 17.7285 0.614257
\(834\) −10.1782 −0.352441
\(835\) 4.80969 0.166446
\(836\) 9.12294 0.315523
\(837\) −10.1479 −0.350762
\(838\) −27.0666 −0.935001
\(839\) −25.9048 −0.894332 −0.447166 0.894451i \(-0.647567\pi\)
−0.447166 + 0.894451i \(0.647567\pi\)
\(840\) 1.40592 0.0485087
\(841\) 3.83078 0.132096
\(842\) 25.4862 0.878314
\(843\) −2.03599 −0.0701231
\(844\) 25.6510 0.882945
\(845\) 1.40749 0.0484193
\(846\) 2.46105 0.0846126
\(847\) −12.3202 −0.423327
\(848\) −8.38795 −0.288043
\(849\) 10.5521 0.362147
\(850\) −8.91695 −0.305849
\(851\) 13.5345 0.463958
\(852\) −3.66759 −0.125649
\(853\) 9.38918 0.321479 0.160740 0.986997i \(-0.448612\pi\)
0.160740 + 0.986997i \(0.448612\pi\)
\(854\) −6.10638 −0.208956
\(855\) 2.65819 0.0909082
\(856\) −13.0054 −0.444517
\(857\) −22.2294 −0.759342 −0.379671 0.925122i \(-0.623963\pi\)
−0.379671 + 0.925122i \(0.623963\pi\)
\(858\) −4.83053 −0.164912
\(859\) 37.6415 1.28431 0.642156 0.766574i \(-0.278040\pi\)
0.642156 + 0.766574i \(0.278040\pi\)
\(860\) 13.7279 0.468118
\(861\) 4.76700 0.162459
\(862\) −30.4313 −1.03649
\(863\) 33.2589 1.13215 0.566073 0.824355i \(-0.308462\pi\)
0.566073 + 0.824355i \(0.308462\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −34.4158 −1.17017
\(866\) −37.2374 −1.26538
\(867\) −8.27596 −0.281066
\(868\) 10.1365 0.344055
\(869\) −53.6664 −1.82051
\(870\) 8.06468 0.273418
\(871\) −4.55266 −0.154261
\(872\) 1.24549 0.0421778
\(873\) −5.09415 −0.172411
\(874\) 5.37285 0.181739
\(875\) 11.2740 0.381130
\(876\) −13.4965 −0.456003
\(877\) −35.4445 −1.19688 −0.598438 0.801169i \(-0.704212\pi\)
−0.598438 + 0.801169i \(0.704212\pi\)
\(878\) 28.7436 0.970050
\(879\) 20.2309 0.682370
\(880\) 6.79894 0.229192
\(881\) −9.58825 −0.323036 −0.161518 0.986870i \(-0.551639\pi\)
−0.161518 + 0.986870i \(0.551639\pi\)
\(882\) 6.00224 0.202106
\(883\) −31.9681 −1.07581 −0.537907 0.843004i \(-0.680785\pi\)
−0.537907 + 0.843004i \(0.680785\pi\)
\(884\) −2.95365 −0.0993420
\(885\) −11.6321 −0.391009
\(886\) 9.15518 0.307574
\(887\) 25.2887 0.849113 0.424556 0.905402i \(-0.360430\pi\)
0.424556 + 0.905402i \(0.360430\pi\)
\(888\) 4.75750 0.159651
\(889\) −3.66076 −0.122778
\(890\) 1.21184 0.0406210
\(891\) 4.83053 0.161829
\(892\) 12.8137 0.429034
\(893\) −4.64794 −0.155537
\(894\) −17.0885 −0.571526
\(895\) −16.3578 −0.546781
\(896\) 0.998879 0.0333702
\(897\) −2.84489 −0.0949880
\(898\) −35.7192 −1.19197
\(899\) 58.1454 1.93926
\(900\) −3.01896 −0.100632
\(901\) 24.7751 0.825377
\(902\) 23.0530 0.767580
\(903\) −9.74251 −0.324210
\(904\) −9.96193 −0.331329
\(905\) 22.7574 0.756482
\(906\) −13.2289 −0.439501
\(907\) 24.8400 0.824798 0.412399 0.911003i \(-0.364691\pi\)
0.412399 + 0.911003i \(0.364691\pi\)
\(908\) −12.3807 −0.410868
\(909\) −5.83551 −0.193552
\(910\) 1.40592 0.0466057
\(911\) −52.5136 −1.73985 −0.869926 0.493182i \(-0.835834\pi\)
−0.869926 + 0.493182i \(0.835834\pi\)
\(912\) 1.88860 0.0625378
\(913\) 8.51497 0.281804
\(914\) −22.8348 −0.755307
\(915\) −8.60434 −0.284451
\(916\) 12.8890 0.425863
\(917\) −11.9774 −0.395529
\(918\) 2.95365 0.0974850
\(919\) −17.4881 −0.576878 −0.288439 0.957498i \(-0.593136\pi\)
−0.288439 + 0.957498i \(0.593136\pi\)
\(920\) 4.00416 0.132013
\(921\) 9.28706 0.306019
\(922\) 24.7047 0.813605
\(923\) −3.66759 −0.120720
\(924\) −4.82512 −0.158735
\(925\) 14.3627 0.472243
\(926\) −9.59246 −0.315228
\(927\) −1.00000 −0.0328443
\(928\) 5.72981 0.188090
\(929\) −7.34782 −0.241074 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(930\) 14.2831 0.468360
\(931\) −11.3358 −0.371517
\(932\) −6.27793 −0.205641
\(933\) 18.0200 0.589948
\(934\) 20.3961 0.667379
\(935\) −20.0817 −0.656741
\(936\) −1.00000 −0.0326860
\(937\) 36.0157 1.17658 0.588291 0.808649i \(-0.299801\pi\)
0.588291 + 0.808649i \(0.299801\pi\)
\(938\) −4.54756 −0.148483
\(939\) 26.1537 0.853493
\(940\) −3.46391 −0.112980
\(941\) 2.80335 0.0913865 0.0456933 0.998956i \(-0.485450\pi\)
0.0456933 + 0.998956i \(0.485450\pi\)
\(942\) −9.61127 −0.313152
\(943\) 13.5768 0.442121
\(944\) −8.26441 −0.268984
\(945\) −1.40592 −0.0457345
\(946\) −47.1143 −1.53182
\(947\) −17.8646 −0.580522 −0.290261 0.956947i \(-0.593742\pi\)
−0.290261 + 0.956947i \(0.593742\pi\)
\(948\) −11.1098 −0.360831
\(949\) −13.4965 −0.438114
\(950\) 5.70161 0.184985
\(951\) 7.89534 0.256024
\(952\) −2.95034 −0.0956210
\(953\) 37.0971 1.20169 0.600846 0.799365i \(-0.294831\pi\)
0.600846 + 0.799365i \(0.294831\pi\)
\(954\) 8.38795 0.271570
\(955\) −21.6021 −0.699028
\(956\) −20.3261 −0.657392
\(957\) −27.6780 −0.894704
\(958\) 6.02670 0.194714
\(959\) −11.6063 −0.374786
\(960\) 1.40749 0.0454267
\(961\) 71.9792 2.32191
\(962\) 4.75750 0.153388
\(963\) 13.0054 0.419095
\(964\) −5.81756 −0.187371
\(965\) 7.92013 0.254958
\(966\) −2.84170 −0.0914301
\(967\) −12.1837 −0.391800 −0.195900 0.980624i \(-0.562763\pi\)
−0.195900 + 0.980624i \(0.562763\pi\)
\(968\) −12.3340 −0.396431
\(969\) −5.57826 −0.179199
\(970\) 7.16998 0.230214
\(971\) −57.5801 −1.84783 −0.923917 0.382592i \(-0.875031\pi\)
−0.923917 + 0.382592i \(0.875031\pi\)
\(972\) 1.00000 0.0320750
\(973\) −10.1668 −0.325931
\(974\) −23.6141 −0.756645
\(975\) −3.01896 −0.0966842
\(976\) −6.11324 −0.195680
\(977\) −32.1063 −1.02717 −0.513585 0.858039i \(-0.671683\pi\)
−0.513585 + 0.858039i \(0.671683\pi\)
\(978\) 7.98844 0.255442
\(979\) −4.15905 −0.132924
\(980\) −8.44812 −0.269865
\(981\) −1.24549 −0.0397656
\(982\) 18.4702 0.589409
\(983\) 42.9319 1.36931 0.684657 0.728865i \(-0.259952\pi\)
0.684657 + 0.728865i \(0.259952\pi\)
\(984\) 4.77235 0.152137
\(985\) −12.5415 −0.399605
\(986\) −16.9239 −0.538966
\(987\) 2.45829 0.0782483
\(988\) 1.88860 0.0600843
\(989\) −27.7474 −0.882317
\(990\) −6.79894 −0.216085
\(991\) −26.2680 −0.834429 −0.417215 0.908808i \(-0.636994\pi\)
−0.417215 + 0.908808i \(0.636994\pi\)
\(992\) 10.1479 0.322195
\(993\) 21.0364 0.667571
\(994\) −3.66347 −0.116198
\(995\) −30.2752 −0.959787
\(996\) 1.76274 0.0558545
\(997\) −60.1384 −1.90460 −0.952301 0.305160i \(-0.901290\pi\)
−0.952301 + 0.305160i \(0.901290\pi\)
\(998\) −24.8331 −0.786078
\(999\) −4.75750 −0.150521
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8034.2.a.w.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.8 12 1.1 even 1 trivial