Properties

Label 8034.2.a.bb.1.1
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.19123\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.19123 q^{5} -1.00000 q^{6} -2.48485 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.19123 q^{5} -1.00000 q^{6} -2.48485 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.19123 q^{10} -3.45337 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.48485 q^{14} -4.19123 q^{15} +1.00000 q^{16} +4.15125 q^{17} -1.00000 q^{18} -1.49203 q^{19} -4.19123 q^{20} -2.48485 q^{21} +3.45337 q^{22} -1.16579 q^{23} -1.00000 q^{24} +12.5664 q^{25} +1.00000 q^{26} +1.00000 q^{27} -2.48485 q^{28} +3.97697 q^{29} +4.19123 q^{30} +6.19959 q^{31} -1.00000 q^{32} -3.45337 q^{33} -4.15125 q^{34} +10.4146 q^{35} +1.00000 q^{36} +0.938276 q^{37} +1.49203 q^{38} -1.00000 q^{39} +4.19123 q^{40} -8.24288 q^{41} +2.48485 q^{42} -1.09636 q^{43} -3.45337 q^{44} -4.19123 q^{45} +1.16579 q^{46} +1.91281 q^{47} +1.00000 q^{48} -0.825500 q^{49} -12.5664 q^{50} +4.15125 q^{51} -1.00000 q^{52} -3.82574 q^{53} -1.00000 q^{54} +14.4739 q^{55} +2.48485 q^{56} -1.49203 q^{57} -3.97697 q^{58} +8.10947 q^{59} -4.19123 q^{60} +5.11955 q^{61} -6.19959 q^{62} -2.48485 q^{63} +1.00000 q^{64} +4.19123 q^{65} +3.45337 q^{66} +13.0998 q^{67} +4.15125 q^{68} -1.16579 q^{69} -10.4146 q^{70} +0.896009 q^{71} -1.00000 q^{72} -0.857920 q^{73} -0.938276 q^{74} +12.5664 q^{75} -1.49203 q^{76} +8.58112 q^{77} +1.00000 q^{78} +4.36696 q^{79} -4.19123 q^{80} +1.00000 q^{81} +8.24288 q^{82} -1.28939 q^{83} -2.48485 q^{84} -17.3989 q^{85} +1.09636 q^{86} +3.97697 q^{87} +3.45337 q^{88} +1.33376 q^{89} +4.19123 q^{90} +2.48485 q^{91} -1.16579 q^{92} +6.19959 q^{93} -1.91281 q^{94} +6.25345 q^{95} -1.00000 q^{96} -1.09589 q^{97} +0.825500 q^{98} -3.45337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) −4.19123 −1.87438 −0.937188 0.348824i \(-0.886581\pi\)
−0.937188 + 0.348824i \(0.886581\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.48485 −0.939187 −0.469593 0.882883i \(-0.655599\pi\)
−0.469593 + 0.882883i \(0.655599\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 4.19123 1.32538
\(11\) −3.45337 −1.04123 −0.520615 0.853791i \(-0.674297\pi\)
−0.520615 + 0.853791i \(0.674297\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.48485 0.664105
\(15\) −4.19123 −1.08217
\(16\) 1.00000 0.250000
\(17\) 4.15125 1.00683 0.503413 0.864046i \(-0.332077\pi\)
0.503413 + 0.864046i \(0.332077\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.49203 −0.342295 −0.171148 0.985245i \(-0.554748\pi\)
−0.171148 + 0.985245i \(0.554748\pi\)
\(20\) −4.19123 −0.937188
\(21\) −2.48485 −0.542240
\(22\) 3.45337 0.736261
\(23\) −1.16579 −0.243084 −0.121542 0.992586i \(-0.538784\pi\)
−0.121542 + 0.992586i \(0.538784\pi\)
\(24\) −1.00000 −0.204124
\(25\) 12.5664 2.51329
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −2.48485 −0.469593
\(29\) 3.97697 0.738505 0.369253 0.929329i \(-0.379614\pi\)
0.369253 + 0.929329i \(0.379614\pi\)
\(30\) 4.19123 0.765211
\(31\) 6.19959 1.11348 0.556740 0.830687i \(-0.312052\pi\)
0.556740 + 0.830687i \(0.312052\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.45337 −0.601155
\(34\) −4.15125 −0.711933
\(35\) 10.4146 1.76039
\(36\) 1.00000 0.166667
\(37\) 0.938276 0.154252 0.0771258 0.997021i \(-0.475426\pi\)
0.0771258 + 0.997021i \(0.475426\pi\)
\(38\) 1.49203 0.242039
\(39\) −1.00000 −0.160128
\(40\) 4.19123 0.662692
\(41\) −8.24288 −1.28732 −0.643661 0.765311i \(-0.722585\pi\)
−0.643661 + 0.765311i \(0.722585\pi\)
\(42\) 2.48485 0.383421
\(43\) −1.09636 −0.167193 −0.0835964 0.996500i \(-0.526641\pi\)
−0.0835964 + 0.996500i \(0.526641\pi\)
\(44\) −3.45337 −0.520615
\(45\) −4.19123 −0.624792
\(46\) 1.16579 0.171886
\(47\) 1.91281 0.279012 0.139506 0.990221i \(-0.455448\pi\)
0.139506 + 0.990221i \(0.455448\pi\)
\(48\) 1.00000 0.144338
\(49\) −0.825500 −0.117929
\(50\) −12.5664 −1.77716
\(51\) 4.15125 0.581291
\(52\) −1.00000 −0.138675
\(53\) −3.82574 −0.525505 −0.262753 0.964863i \(-0.584630\pi\)
−0.262753 + 0.964863i \(0.584630\pi\)
\(54\) −1.00000 −0.136083
\(55\) 14.4739 1.95166
\(56\) 2.48485 0.332053
\(57\) −1.49203 −0.197624
\(58\) −3.97697 −0.522202
\(59\) 8.10947 1.05576 0.527882 0.849318i \(-0.322986\pi\)
0.527882 + 0.849318i \(0.322986\pi\)
\(60\) −4.19123 −0.541086
\(61\) 5.11955 0.655491 0.327745 0.944766i \(-0.393711\pi\)
0.327745 + 0.944766i \(0.393711\pi\)
\(62\) −6.19959 −0.787349
\(63\) −2.48485 −0.313062
\(64\) 1.00000 0.125000
\(65\) 4.19123 0.519859
\(66\) 3.45337 0.425080
\(67\) 13.0998 1.60040 0.800198 0.599736i \(-0.204728\pi\)
0.800198 + 0.599736i \(0.204728\pi\)
\(68\) 4.15125 0.503413
\(69\) −1.16579 −0.140345
\(70\) −10.4146 −1.24478
\(71\) 0.896009 0.106337 0.0531683 0.998586i \(-0.483068\pi\)
0.0531683 + 0.998586i \(0.483068\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.857920 −0.100412 −0.0502059 0.998739i \(-0.515988\pi\)
−0.0502059 + 0.998739i \(0.515988\pi\)
\(74\) −0.938276 −0.109072
\(75\) 12.5664 1.45105
\(76\) −1.49203 −0.171148
\(77\) 8.58112 0.977909
\(78\) 1.00000 0.113228
\(79\) 4.36696 0.491322 0.245661 0.969356i \(-0.420995\pi\)
0.245661 + 0.969356i \(0.420995\pi\)
\(80\) −4.19123 −0.468594
\(81\) 1.00000 0.111111
\(82\) 8.24288 0.910274
\(83\) −1.28939 −0.141529 −0.0707644 0.997493i \(-0.522544\pi\)
−0.0707644 + 0.997493i \(0.522544\pi\)
\(84\) −2.48485 −0.271120
\(85\) −17.3989 −1.88717
\(86\) 1.09636 0.118223
\(87\) 3.97697 0.426376
\(88\) 3.45337 0.368130
\(89\) 1.33376 0.141378 0.0706891 0.997498i \(-0.477480\pi\)
0.0706891 + 0.997498i \(0.477480\pi\)
\(90\) 4.19123 0.441795
\(91\) 2.48485 0.260483
\(92\) −1.16579 −0.121542
\(93\) 6.19959 0.642868
\(94\) −1.91281 −0.197292
\(95\) 6.25345 0.641590
\(96\) −1.00000 −0.102062
\(97\) −1.09589 −0.111271 −0.0556355 0.998451i \(-0.517718\pi\)
−0.0556355 + 0.998451i \(0.517718\pi\)
\(98\) 0.825500 0.0833881
\(99\) −3.45337 −0.347077
\(100\) 12.5664 1.25664
\(101\) −12.2271 −1.21664 −0.608320 0.793692i \(-0.708156\pi\)
−0.608320 + 0.793692i \(0.708156\pi\)
\(102\) −4.15125 −0.411035
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 10.4146 1.01636
\(106\) 3.82574 0.371588
\(107\) −17.9304 −1.73340 −0.866700 0.498830i \(-0.833763\pi\)
−0.866700 + 0.498830i \(0.833763\pi\)
\(108\) 1.00000 0.0962250
\(109\) 18.1175 1.73534 0.867670 0.497140i \(-0.165617\pi\)
0.867670 + 0.497140i \(0.165617\pi\)
\(110\) −14.4739 −1.38003
\(111\) 0.938276 0.0890572
\(112\) −2.48485 −0.234797
\(113\) 16.6258 1.56402 0.782011 0.623264i \(-0.214194\pi\)
0.782011 + 0.623264i \(0.214194\pi\)
\(114\) 1.49203 0.139741
\(115\) 4.88610 0.455631
\(116\) 3.97697 0.369253
\(117\) −1.00000 −0.0924500
\(118\) −8.10947 −0.746538
\(119\) −10.3152 −0.945597
\(120\) 4.19123 0.382606
\(121\) 0.925764 0.0841604
\(122\) −5.11955 −0.463502
\(123\) −8.24288 −0.743236
\(124\) 6.19959 0.556740
\(125\) −31.7127 −2.83647
\(126\) 2.48485 0.221368
\(127\) −6.45305 −0.572615 −0.286308 0.958138i \(-0.592428\pi\)
−0.286308 + 0.958138i \(0.592428\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.09636 −0.0965288
\(130\) −4.19123 −0.367596
\(131\) −5.48297 −0.479049 −0.239525 0.970890i \(-0.576992\pi\)
−0.239525 + 0.970890i \(0.576992\pi\)
\(132\) −3.45337 −0.300577
\(133\) 3.70748 0.321479
\(134\) −13.0998 −1.13165
\(135\) −4.19123 −0.360724
\(136\) −4.15125 −0.355967
\(137\) 9.92674 0.848098 0.424049 0.905639i \(-0.360608\pi\)
0.424049 + 0.905639i \(0.360608\pi\)
\(138\) 1.16579 0.0992386
\(139\) 3.44571 0.292261 0.146131 0.989265i \(-0.453318\pi\)
0.146131 + 0.989265i \(0.453318\pi\)
\(140\) 10.4146 0.880195
\(141\) 1.91281 0.161088
\(142\) −0.896009 −0.0751914
\(143\) 3.45337 0.288785
\(144\) 1.00000 0.0833333
\(145\) −16.6684 −1.38424
\(146\) 0.857920 0.0710019
\(147\) −0.825500 −0.0680861
\(148\) 0.938276 0.0771258
\(149\) −4.22838 −0.346403 −0.173201 0.984886i \(-0.555411\pi\)
−0.173201 + 0.984886i \(0.555411\pi\)
\(150\) −12.5664 −1.02605
\(151\) −14.8402 −1.20768 −0.603840 0.797106i \(-0.706363\pi\)
−0.603840 + 0.797106i \(0.706363\pi\)
\(152\) 1.49203 0.121020
\(153\) 4.15125 0.335609
\(154\) −8.58112 −0.691486
\(155\) −25.9839 −2.08708
\(156\) −1.00000 −0.0800641
\(157\) 11.3687 0.907322 0.453661 0.891174i \(-0.350118\pi\)
0.453661 + 0.891174i \(0.350118\pi\)
\(158\) −4.36696 −0.347417
\(159\) −3.82574 −0.303401
\(160\) 4.19123 0.331346
\(161\) 2.89682 0.228301
\(162\) −1.00000 −0.0785674
\(163\) 17.4080 1.36350 0.681750 0.731585i \(-0.261219\pi\)
0.681750 + 0.731585i \(0.261219\pi\)
\(164\) −8.24288 −0.643661
\(165\) 14.4739 1.12679
\(166\) 1.28939 0.100076
\(167\) −19.8534 −1.53630 −0.768151 0.640268i \(-0.778823\pi\)
−0.768151 + 0.640268i \(0.778823\pi\)
\(168\) 2.48485 0.191711
\(169\) 1.00000 0.0769231
\(170\) 17.3989 1.33443
\(171\) −1.49203 −0.114098
\(172\) −1.09636 −0.0835964
\(173\) 8.38483 0.637487 0.318744 0.947841i \(-0.396739\pi\)
0.318744 + 0.947841i \(0.396739\pi\)
\(174\) −3.97697 −0.301493
\(175\) −31.2258 −2.36045
\(176\) −3.45337 −0.260308
\(177\) 8.10947 0.609545
\(178\) −1.33376 −0.0999695
\(179\) 24.2415 1.81189 0.905946 0.423394i \(-0.139161\pi\)
0.905946 + 0.423394i \(0.139161\pi\)
\(180\) −4.19123 −0.312396
\(181\) 1.00614 0.0747858 0.0373929 0.999301i \(-0.488095\pi\)
0.0373929 + 0.999301i \(0.488095\pi\)
\(182\) −2.48485 −0.184190
\(183\) 5.11955 0.378448
\(184\) 1.16579 0.0859431
\(185\) −3.93253 −0.289126
\(186\) −6.19959 −0.454576
\(187\) −14.3358 −1.04834
\(188\) 1.91281 0.139506
\(189\) −2.48485 −0.180747
\(190\) −6.25345 −0.453673
\(191\) −17.7258 −1.28259 −0.641295 0.767294i \(-0.721603\pi\)
−0.641295 + 0.767294i \(0.721603\pi\)
\(192\) 1.00000 0.0721688
\(193\) −8.92394 −0.642359 −0.321180 0.947018i \(-0.604079\pi\)
−0.321180 + 0.947018i \(0.604079\pi\)
\(194\) 1.09589 0.0786805
\(195\) 4.19123 0.300140
\(196\) −0.825500 −0.0589643
\(197\) −5.72916 −0.408186 −0.204093 0.978952i \(-0.565425\pi\)
−0.204093 + 0.978952i \(0.565425\pi\)
\(198\) 3.45337 0.245420
\(199\) 4.04033 0.286412 0.143206 0.989693i \(-0.454259\pi\)
0.143206 + 0.989693i \(0.454259\pi\)
\(200\) −12.5664 −0.888582
\(201\) 13.0998 0.923989
\(202\) 12.2271 0.860295
\(203\) −9.88220 −0.693594
\(204\) 4.15125 0.290646
\(205\) 34.5479 2.41293
\(206\) −1.00000 −0.0696733
\(207\) −1.16579 −0.0810279
\(208\) −1.00000 −0.0693375
\(209\) 5.15253 0.356408
\(210\) −10.4146 −0.718676
\(211\) −18.1802 −1.25158 −0.625789 0.779992i \(-0.715223\pi\)
−0.625789 + 0.779992i \(0.715223\pi\)
\(212\) −3.82574 −0.262753
\(213\) 0.896009 0.0613935
\(214\) 17.9304 1.22570
\(215\) 4.59509 0.313382
\(216\) −1.00000 −0.0680414
\(217\) −15.4051 −1.04577
\(218\) −18.1175 −1.22707
\(219\) −0.857920 −0.0579728
\(220\) 14.4739 0.975829
\(221\) −4.15125 −0.279243
\(222\) −0.938276 −0.0629730
\(223\) −15.7232 −1.05290 −0.526452 0.850205i \(-0.676478\pi\)
−0.526452 + 0.850205i \(0.676478\pi\)
\(224\) 2.48485 0.166026
\(225\) 12.5664 0.837763
\(226\) −16.6258 −1.10593
\(227\) 5.43218 0.360546 0.180273 0.983617i \(-0.442302\pi\)
0.180273 + 0.983617i \(0.442302\pi\)
\(228\) −1.49203 −0.0988121
\(229\) 14.2912 0.944386 0.472193 0.881495i \(-0.343462\pi\)
0.472193 + 0.881495i \(0.343462\pi\)
\(230\) −4.88610 −0.322180
\(231\) 8.58112 0.564596
\(232\) −3.97697 −0.261101
\(233\) −15.7787 −1.03370 −0.516848 0.856077i \(-0.672895\pi\)
−0.516848 + 0.856077i \(0.672895\pi\)
\(234\) 1.00000 0.0653720
\(235\) −8.01704 −0.522974
\(236\) 8.10947 0.527882
\(237\) 4.36696 0.283665
\(238\) 10.3152 0.668638
\(239\) −24.0758 −1.55733 −0.778667 0.627438i \(-0.784104\pi\)
−0.778667 + 0.627438i \(0.784104\pi\)
\(240\) −4.19123 −0.270543
\(241\) −24.3323 −1.56738 −0.783689 0.621153i \(-0.786665\pi\)
−0.783689 + 0.621153i \(0.786665\pi\)
\(242\) −0.925764 −0.0595104
\(243\) 1.00000 0.0641500
\(244\) 5.11955 0.327745
\(245\) 3.45987 0.221043
\(246\) 8.24288 0.525547
\(247\) 1.49203 0.0949356
\(248\) −6.19959 −0.393675
\(249\) −1.28939 −0.0817117
\(250\) 31.7127 2.00569
\(251\) 8.22652 0.519253 0.259627 0.965709i \(-0.416400\pi\)
0.259627 + 0.965709i \(0.416400\pi\)
\(252\) −2.48485 −0.156531
\(253\) 4.02590 0.253106
\(254\) 6.45305 0.404900
\(255\) −17.3989 −1.08956
\(256\) 1.00000 0.0625000
\(257\) 19.2253 1.19924 0.599620 0.800285i \(-0.295319\pi\)
0.599620 + 0.800285i \(0.295319\pi\)
\(258\) 1.09636 0.0682562
\(259\) −2.33148 −0.144871
\(260\) 4.19123 0.259929
\(261\) 3.97697 0.246168
\(262\) 5.48297 0.338739
\(263\) −26.5673 −1.63821 −0.819104 0.573644i \(-0.805529\pi\)
−0.819104 + 0.573644i \(0.805529\pi\)
\(264\) 3.45337 0.212540
\(265\) 16.0346 0.984995
\(266\) −3.70748 −0.227320
\(267\) 1.33376 0.0816248
\(268\) 13.0998 0.800198
\(269\) 3.93881 0.240154 0.120077 0.992765i \(-0.461686\pi\)
0.120077 + 0.992765i \(0.461686\pi\)
\(270\) 4.19123 0.255070
\(271\) −6.13049 −0.372401 −0.186200 0.982512i \(-0.559617\pi\)
−0.186200 + 0.982512i \(0.559617\pi\)
\(272\) 4.15125 0.251706
\(273\) 2.48485 0.150390
\(274\) −9.92674 −0.599696
\(275\) −43.3966 −2.61691
\(276\) −1.16579 −0.0701723
\(277\) −3.38074 −0.203129 −0.101564 0.994829i \(-0.532385\pi\)
−0.101564 + 0.994829i \(0.532385\pi\)
\(278\) −3.44571 −0.206660
\(279\) 6.19959 0.371160
\(280\) −10.4146 −0.622392
\(281\) −5.93074 −0.353798 −0.176899 0.984229i \(-0.556607\pi\)
−0.176899 + 0.984229i \(0.556607\pi\)
\(282\) −1.91281 −0.113906
\(283\) 23.4537 1.39418 0.697088 0.716985i \(-0.254479\pi\)
0.697088 + 0.716985i \(0.254479\pi\)
\(284\) 0.896009 0.0531683
\(285\) 6.25345 0.370422
\(286\) −3.45337 −0.204202
\(287\) 20.4824 1.20904
\(288\) −1.00000 −0.0589256
\(289\) 0.232872 0.0136984
\(290\) 16.6684 0.978803
\(291\) −1.09589 −0.0642424
\(292\) −0.857920 −0.0502059
\(293\) −29.0023 −1.69433 −0.847167 0.531327i \(-0.821693\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(294\) 0.825500 0.0481442
\(295\) −33.9887 −1.97890
\(296\) −0.938276 −0.0545362
\(297\) −3.45337 −0.200385
\(298\) 4.22838 0.244944
\(299\) 1.16579 0.0674193
\(300\) 12.5664 0.725524
\(301\) 2.72429 0.157025
\(302\) 14.8402 0.853959
\(303\) −12.2271 −0.702428
\(304\) −1.49203 −0.0855738
\(305\) −21.4572 −1.22864
\(306\) −4.15125 −0.237311
\(307\) 25.5987 1.46100 0.730499 0.682914i \(-0.239288\pi\)
0.730499 + 0.682914i \(0.239288\pi\)
\(308\) 8.58112 0.488955
\(309\) 1.00000 0.0568880
\(310\) 25.9839 1.47579
\(311\) −14.8637 −0.842842 −0.421421 0.906865i \(-0.638468\pi\)
−0.421421 + 0.906865i \(0.638468\pi\)
\(312\) 1.00000 0.0566139
\(313\) 11.2838 0.637798 0.318899 0.947789i \(-0.396687\pi\)
0.318899 + 0.947789i \(0.396687\pi\)
\(314\) −11.3687 −0.641574
\(315\) 10.4146 0.586796
\(316\) 4.36696 0.245661
\(317\) 3.43988 0.193203 0.0966013 0.995323i \(-0.469203\pi\)
0.0966013 + 0.995323i \(0.469203\pi\)
\(318\) 3.82574 0.214537
\(319\) −13.7340 −0.768954
\(320\) −4.19123 −0.234297
\(321\) −17.9304 −1.00078
\(322\) −2.89682 −0.161433
\(323\) −6.19379 −0.344632
\(324\) 1.00000 0.0555556
\(325\) −12.5664 −0.697061
\(326\) −17.4080 −0.964140
\(327\) 18.1175 1.00190
\(328\) 8.24288 0.455137
\(329\) −4.75306 −0.262045
\(330\) −14.4739 −0.796761
\(331\) −6.84967 −0.376492 −0.188246 0.982122i \(-0.560280\pi\)
−0.188246 + 0.982122i \(0.560280\pi\)
\(332\) −1.28939 −0.0707644
\(333\) 0.938276 0.0514172
\(334\) 19.8534 1.08633
\(335\) −54.9043 −2.99974
\(336\) −2.48485 −0.135560
\(337\) 8.11014 0.441788 0.220894 0.975298i \(-0.429103\pi\)
0.220894 + 0.975298i \(0.429103\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 16.6258 0.902989
\(340\) −17.3989 −0.943586
\(341\) −21.4095 −1.15939
\(342\) 1.49203 0.0806798
\(343\) 19.4452 1.04994
\(344\) 1.09636 0.0591116
\(345\) 4.88610 0.263059
\(346\) −8.38483 −0.450772
\(347\) −15.6913 −0.842351 −0.421176 0.906979i \(-0.638382\pi\)
−0.421176 + 0.906979i \(0.638382\pi\)
\(348\) 3.97697 0.213188
\(349\) −16.5396 −0.885347 −0.442673 0.896683i \(-0.645970\pi\)
−0.442673 + 0.896683i \(0.645970\pi\)
\(350\) 31.2258 1.66909
\(351\) −1.00000 −0.0533761
\(352\) 3.45337 0.184065
\(353\) −28.2812 −1.50525 −0.752627 0.658447i \(-0.771214\pi\)
−0.752627 + 0.658447i \(0.771214\pi\)
\(354\) −8.10947 −0.431014
\(355\) −3.75538 −0.199315
\(356\) 1.33376 0.0706891
\(357\) −10.3152 −0.545941
\(358\) −24.2415 −1.28120
\(359\) −21.2405 −1.12103 −0.560516 0.828143i \(-0.689397\pi\)
−0.560516 + 0.828143i \(0.689397\pi\)
\(360\) 4.19123 0.220897
\(361\) −16.7738 −0.882834
\(362\) −1.00614 −0.0528815
\(363\) 0.925764 0.0485900
\(364\) 2.48485 0.130242
\(365\) 3.59574 0.188210
\(366\) −5.11955 −0.267603
\(367\) −4.96807 −0.259331 −0.129666 0.991558i \(-0.541390\pi\)
−0.129666 + 0.991558i \(0.541390\pi\)
\(368\) −1.16579 −0.0607710
\(369\) −8.24288 −0.429107
\(370\) 3.93253 0.204443
\(371\) 9.50640 0.493547
\(372\) 6.19959 0.321434
\(373\) −1.35351 −0.0700822 −0.0350411 0.999386i \(-0.511156\pi\)
−0.0350411 + 0.999386i \(0.511156\pi\)
\(374\) 14.3358 0.741287
\(375\) −31.7127 −1.63764
\(376\) −1.91281 −0.0986458
\(377\) −3.97697 −0.204824
\(378\) 2.48485 0.127807
\(379\) 1.33194 0.0684173 0.0342086 0.999415i \(-0.489109\pi\)
0.0342086 + 0.999415i \(0.489109\pi\)
\(380\) 6.25345 0.320795
\(381\) −6.45305 −0.330600
\(382\) 17.7258 0.906929
\(383\) −5.46332 −0.279162 −0.139581 0.990211i \(-0.544576\pi\)
−0.139581 + 0.990211i \(0.544576\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −35.9655 −1.83297
\(386\) 8.92394 0.454217
\(387\) −1.09636 −0.0557309
\(388\) −1.09589 −0.0556355
\(389\) 5.47837 0.277764 0.138882 0.990309i \(-0.455649\pi\)
0.138882 + 0.990309i \(0.455649\pi\)
\(390\) −4.19123 −0.212231
\(391\) −4.83948 −0.244743
\(392\) 0.825500 0.0416941
\(393\) −5.48297 −0.276579
\(394\) 5.72916 0.288631
\(395\) −18.3030 −0.920922
\(396\) −3.45337 −0.173538
\(397\) −32.9394 −1.65318 −0.826589 0.562806i \(-0.809722\pi\)
−0.826589 + 0.562806i \(0.809722\pi\)
\(398\) −4.04033 −0.202524
\(399\) 3.70748 0.185606
\(400\) 12.5664 0.628322
\(401\) 7.80301 0.389664 0.194832 0.980837i \(-0.437584\pi\)
0.194832 + 0.980837i \(0.437584\pi\)
\(402\) −13.0998 −0.653359
\(403\) −6.19959 −0.308824
\(404\) −12.2271 −0.608320
\(405\) −4.19123 −0.208264
\(406\) 9.88220 0.490445
\(407\) −3.24021 −0.160611
\(408\) −4.15125 −0.205517
\(409\) 25.9347 1.28239 0.641194 0.767378i \(-0.278439\pi\)
0.641194 + 0.767378i \(0.278439\pi\)
\(410\) −34.5479 −1.70620
\(411\) 9.92674 0.489650
\(412\) 1.00000 0.0492665
\(413\) −20.1509 −0.991559
\(414\) 1.16579 0.0572954
\(415\) 5.40413 0.265278
\(416\) 1.00000 0.0490290
\(417\) 3.44571 0.168737
\(418\) −5.15253 −0.252019
\(419\) −11.8178 −0.577339 −0.288670 0.957429i \(-0.593213\pi\)
−0.288670 + 0.957429i \(0.593213\pi\)
\(420\) 10.4146 0.508181
\(421\) −28.3038 −1.37944 −0.689720 0.724076i \(-0.742267\pi\)
−0.689720 + 0.724076i \(0.742267\pi\)
\(422\) 18.1802 0.884999
\(423\) 1.91281 0.0930041
\(424\) 3.82574 0.185794
\(425\) 52.1664 2.53044
\(426\) −0.896009 −0.0434118
\(427\) −12.7213 −0.615628
\(428\) −17.9304 −0.866700
\(429\) 3.45337 0.166730
\(430\) −4.59509 −0.221595
\(431\) −11.3047 −0.544528 −0.272264 0.962223i \(-0.587772\pi\)
−0.272264 + 0.962223i \(0.587772\pi\)
\(432\) 1.00000 0.0481125
\(433\) −13.6585 −0.656385 −0.328193 0.944611i \(-0.606439\pi\)
−0.328193 + 0.944611i \(0.606439\pi\)
\(434\) 15.4051 0.739468
\(435\) −16.6684 −0.799190
\(436\) 18.1175 0.867670
\(437\) 1.73939 0.0832064
\(438\) 0.857920 0.0409930
\(439\) −10.1305 −0.483503 −0.241751 0.970338i \(-0.577722\pi\)
−0.241751 + 0.970338i \(0.577722\pi\)
\(440\) −14.4739 −0.690015
\(441\) −0.825500 −0.0393095
\(442\) 4.15125 0.197455
\(443\) −24.1352 −1.14670 −0.573348 0.819312i \(-0.694356\pi\)
−0.573348 + 0.819312i \(0.694356\pi\)
\(444\) 0.938276 0.0445286
\(445\) −5.59010 −0.264996
\(446\) 15.7232 0.744515
\(447\) −4.22838 −0.199996
\(448\) −2.48485 −0.117398
\(449\) 34.6217 1.63390 0.816951 0.576708i \(-0.195663\pi\)
0.816951 + 0.576708i \(0.195663\pi\)
\(450\) −12.5664 −0.592388
\(451\) 28.4657 1.34040
\(452\) 16.6258 0.782011
\(453\) −14.8402 −0.697254
\(454\) −5.43218 −0.254945
\(455\) −10.4146 −0.488244
\(456\) 1.49203 0.0698707
\(457\) −31.4318 −1.47032 −0.735158 0.677896i \(-0.762892\pi\)
−0.735158 + 0.677896i \(0.762892\pi\)
\(458\) −14.2912 −0.667782
\(459\) 4.15125 0.193764
\(460\) 4.88610 0.227815
\(461\) 17.5107 0.815554 0.407777 0.913082i \(-0.366304\pi\)
0.407777 + 0.913082i \(0.366304\pi\)
\(462\) −8.58112 −0.399230
\(463\) 10.4776 0.486933 0.243467 0.969909i \(-0.421715\pi\)
0.243467 + 0.969909i \(0.421715\pi\)
\(464\) 3.97697 0.184626
\(465\) −25.9839 −1.20498
\(466\) 15.7787 0.730934
\(467\) 0.734108 0.0339705 0.0169852 0.999856i \(-0.494593\pi\)
0.0169852 + 0.999856i \(0.494593\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −32.5511 −1.50307
\(470\) 8.01704 0.369799
\(471\) 11.3687 0.523843
\(472\) −8.10947 −0.373269
\(473\) 3.78613 0.174086
\(474\) −4.36696 −0.200581
\(475\) −18.7495 −0.860287
\(476\) −10.3152 −0.472799
\(477\) −3.82574 −0.175168
\(478\) 24.0758 1.10120
\(479\) 40.2600 1.83952 0.919762 0.392475i \(-0.128381\pi\)
0.919762 + 0.392475i \(0.128381\pi\)
\(480\) 4.19123 0.191303
\(481\) −0.938276 −0.0427817
\(482\) 24.3323 1.10830
\(483\) 2.89682 0.131810
\(484\) 0.925764 0.0420802
\(485\) 4.59314 0.208564
\(486\) −1.00000 −0.0453609
\(487\) 10.8019 0.489482 0.244741 0.969589i \(-0.421297\pi\)
0.244741 + 0.969589i \(0.421297\pi\)
\(488\) −5.11955 −0.231751
\(489\) 17.4080 0.787217
\(490\) −3.45987 −0.156301
\(491\) 21.2973 0.961134 0.480567 0.876958i \(-0.340431\pi\)
0.480567 + 0.876958i \(0.340431\pi\)
\(492\) −8.24288 −0.371618
\(493\) 16.5094 0.743546
\(494\) −1.49203 −0.0671296
\(495\) 14.4739 0.650553
\(496\) 6.19959 0.278370
\(497\) −2.22645 −0.0998700
\(498\) 1.28939 0.0577789
\(499\) −26.0360 −1.16553 −0.582765 0.812640i \(-0.698029\pi\)
−0.582765 + 0.812640i \(0.698029\pi\)
\(500\) −31.7127 −1.41824
\(501\) −19.8534 −0.886985
\(502\) −8.22652 −0.367168
\(503\) −16.2381 −0.724020 −0.362010 0.932174i \(-0.617909\pi\)
−0.362010 + 0.932174i \(0.617909\pi\)
\(504\) 2.48485 0.110684
\(505\) 51.2466 2.28044
\(506\) −4.02590 −0.178973
\(507\) 1.00000 0.0444116
\(508\) −6.45305 −0.286308
\(509\) 2.48262 0.110040 0.0550202 0.998485i \(-0.482478\pi\)
0.0550202 + 0.998485i \(0.482478\pi\)
\(510\) 17.3989 0.770434
\(511\) 2.13180 0.0943055
\(512\) −1.00000 −0.0441942
\(513\) −1.49203 −0.0658747
\(514\) −19.2253 −0.847990
\(515\) −4.19123 −0.184688
\(516\) −1.09636 −0.0482644
\(517\) −6.60565 −0.290516
\(518\) 2.33148 0.102439
\(519\) 8.38483 0.368053
\(520\) −4.19123 −0.183798
\(521\) −1.51427 −0.0663415 −0.0331707 0.999450i \(-0.510561\pi\)
−0.0331707 + 0.999450i \(0.510561\pi\)
\(522\) −3.97697 −0.174067
\(523\) 11.1517 0.487628 0.243814 0.969822i \(-0.421601\pi\)
0.243814 + 0.969822i \(0.421601\pi\)
\(524\) −5.48297 −0.239525
\(525\) −31.2258 −1.36280
\(526\) 26.5673 1.15839
\(527\) 25.7361 1.12108
\(528\) −3.45337 −0.150289
\(529\) −21.6409 −0.940910
\(530\) −16.0346 −0.696497
\(531\) 8.10947 0.351921
\(532\) 3.70748 0.160740
\(533\) 8.24288 0.357039
\(534\) −1.33376 −0.0577174
\(535\) 75.1506 3.24904
\(536\) −13.0998 −0.565825
\(537\) 24.2415 1.04610
\(538\) −3.93881 −0.169814
\(539\) 2.85076 0.122791
\(540\) −4.19123 −0.180362
\(541\) 23.9397 1.02925 0.514623 0.857416i \(-0.327932\pi\)
0.514623 + 0.857416i \(0.327932\pi\)
\(542\) 6.13049 0.263327
\(543\) 1.00614 0.0431776
\(544\) −4.15125 −0.177983
\(545\) −75.9346 −3.25268
\(546\) −2.48485 −0.106342
\(547\) 26.2096 1.12064 0.560321 0.828275i \(-0.310678\pi\)
0.560321 + 0.828275i \(0.310678\pi\)
\(548\) 9.92674 0.424049
\(549\) 5.11955 0.218497
\(550\) 43.3966 1.85044
\(551\) −5.93376 −0.252787
\(552\) 1.16579 0.0496193
\(553\) −10.8513 −0.461443
\(554\) 3.38074 0.143634
\(555\) −3.93253 −0.166927
\(556\) 3.44571 0.146131
\(557\) −8.02154 −0.339884 −0.169942 0.985454i \(-0.554358\pi\)
−0.169942 + 0.985454i \(0.554358\pi\)
\(558\) −6.19959 −0.262450
\(559\) 1.09636 0.0463709
\(560\) 10.4146 0.440097
\(561\) −14.3358 −0.605258
\(562\) 5.93074 0.250173
\(563\) 13.6134 0.573736 0.286868 0.957970i \(-0.407386\pi\)
0.286868 + 0.957970i \(0.407386\pi\)
\(564\) 1.91281 0.0805439
\(565\) −69.6826 −2.93157
\(566\) −23.4537 −0.985831
\(567\) −2.48485 −0.104354
\(568\) −0.896009 −0.0375957
\(569\) 41.8088 1.75272 0.876358 0.481659i \(-0.159966\pi\)
0.876358 + 0.481659i \(0.159966\pi\)
\(570\) −6.25345 −0.261928
\(571\) 3.05692 0.127928 0.0639640 0.997952i \(-0.479626\pi\)
0.0639640 + 0.997952i \(0.479626\pi\)
\(572\) 3.45337 0.144393
\(573\) −17.7258 −0.740504
\(574\) −20.4824 −0.854917
\(575\) −14.6498 −0.610940
\(576\) 1.00000 0.0416667
\(577\) 5.43693 0.226342 0.113171 0.993576i \(-0.463899\pi\)
0.113171 + 0.993576i \(0.463899\pi\)
\(578\) −0.232872 −0.00968622
\(579\) −8.92394 −0.370866
\(580\) −16.6684 −0.692118
\(581\) 3.20394 0.132922
\(582\) 1.09589 0.0454262
\(583\) 13.2117 0.547172
\(584\) 0.857920 0.0355010
\(585\) 4.19123 0.173286
\(586\) 29.0023 1.19807
\(587\) 20.2161 0.834406 0.417203 0.908813i \(-0.363010\pi\)
0.417203 + 0.908813i \(0.363010\pi\)
\(588\) −0.825500 −0.0340431
\(589\) −9.24998 −0.381139
\(590\) 33.9887 1.39929
\(591\) −5.72916 −0.235666
\(592\) 0.938276 0.0385629
\(593\) 36.5479 1.50084 0.750422 0.660959i \(-0.229850\pi\)
0.750422 + 0.660959i \(0.229850\pi\)
\(594\) 3.45337 0.141693
\(595\) 43.2336 1.77241
\(596\) −4.22838 −0.173201
\(597\) 4.04033 0.165360
\(598\) −1.16579 −0.0476727
\(599\) 5.09064 0.207998 0.103999 0.994577i \(-0.466836\pi\)
0.103999 + 0.994577i \(0.466836\pi\)
\(600\) −12.5664 −0.513023
\(601\) 4.56186 0.186082 0.0930410 0.995662i \(-0.470341\pi\)
0.0930410 + 0.995662i \(0.470341\pi\)
\(602\) −2.72429 −0.111034
\(603\) 13.0998 0.533465
\(604\) −14.8402 −0.603840
\(605\) −3.88010 −0.157748
\(606\) 12.2271 0.496691
\(607\) −20.6660 −0.838806 −0.419403 0.907800i \(-0.637761\pi\)
−0.419403 + 0.907800i \(0.637761\pi\)
\(608\) 1.49203 0.0605098
\(609\) −9.88220 −0.400447
\(610\) 21.4572 0.868777
\(611\) −1.91281 −0.0773841
\(612\) 4.15125 0.167804
\(613\) 27.3650 1.10526 0.552630 0.833426i \(-0.313624\pi\)
0.552630 + 0.833426i \(0.313624\pi\)
\(614\) −25.5987 −1.03308
\(615\) 34.5479 1.39310
\(616\) −8.58112 −0.345743
\(617\) −42.5586 −1.71334 −0.856672 0.515861i \(-0.827472\pi\)
−0.856672 + 0.515861i \(0.827472\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −15.1000 −0.606920 −0.303460 0.952844i \(-0.598142\pi\)
−0.303460 + 0.952844i \(0.598142\pi\)
\(620\) −25.9839 −1.04354
\(621\) −1.16579 −0.0467815
\(622\) 14.8637 0.595979
\(623\) −3.31420 −0.132781
\(624\) −1.00000 −0.0400320
\(625\) 70.0832 2.80333
\(626\) −11.2838 −0.450992
\(627\) 5.15253 0.205772
\(628\) 11.3687 0.453661
\(629\) 3.89502 0.155305
\(630\) −10.4146 −0.414928
\(631\) −11.6996 −0.465755 −0.232878 0.972506i \(-0.574814\pi\)
−0.232878 + 0.972506i \(0.574814\pi\)
\(632\) −4.36696 −0.173708
\(633\) −18.1802 −0.722599
\(634\) −3.43988 −0.136615
\(635\) 27.0462 1.07330
\(636\) −3.82574 −0.151700
\(637\) 0.825500 0.0327075
\(638\) 13.7340 0.543733
\(639\) 0.896009 0.0354456
\(640\) 4.19123 0.165673
\(641\) 6.02988 0.238166 0.119083 0.992884i \(-0.462005\pi\)
0.119083 + 0.992884i \(0.462005\pi\)
\(642\) 17.9304 0.707657
\(643\) 40.2778 1.58840 0.794201 0.607655i \(-0.207890\pi\)
0.794201 + 0.607655i \(0.207890\pi\)
\(644\) 2.89682 0.114151
\(645\) 4.59509 0.180931
\(646\) 6.19379 0.243691
\(647\) 34.4114 1.35285 0.676426 0.736511i \(-0.263528\pi\)
0.676426 + 0.736511i \(0.263528\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −28.0050 −1.09929
\(650\) 12.5664 0.492896
\(651\) −15.4051 −0.603773
\(652\) 17.4080 0.681750
\(653\) −18.8545 −0.737835 −0.368917 0.929462i \(-0.620272\pi\)
−0.368917 + 0.929462i \(0.620272\pi\)
\(654\) −18.1175 −0.708450
\(655\) 22.9804 0.897918
\(656\) −8.24288 −0.321831
\(657\) −0.857920 −0.0334706
\(658\) 4.75306 0.185294
\(659\) −33.2202 −1.29408 −0.647038 0.762458i \(-0.723992\pi\)
−0.647038 + 0.762458i \(0.723992\pi\)
\(660\) 14.4739 0.563395
\(661\) 16.0955 0.626041 0.313021 0.949746i \(-0.398659\pi\)
0.313021 + 0.949746i \(0.398659\pi\)
\(662\) 6.84967 0.266220
\(663\) −4.15125 −0.161221
\(664\) 1.28939 0.0500380
\(665\) −15.5389 −0.602573
\(666\) −0.938276 −0.0363575
\(667\) −4.63631 −0.179519
\(668\) −19.8534 −0.768151
\(669\) −15.7232 −0.607894
\(670\) 54.9043 2.12114
\(671\) −17.6797 −0.682517
\(672\) 2.48485 0.0958553
\(673\) 16.4323 0.633419 0.316709 0.948523i \(-0.397422\pi\)
0.316709 + 0.948523i \(0.397422\pi\)
\(674\) −8.11014 −0.312391
\(675\) 12.5664 0.483683
\(676\) 1.00000 0.0384615
\(677\) −14.8423 −0.570436 −0.285218 0.958463i \(-0.592066\pi\)
−0.285218 + 0.958463i \(0.592066\pi\)
\(678\) −16.6258 −0.638509
\(679\) 2.72313 0.104504
\(680\) 17.3989 0.667216
\(681\) 5.43218 0.208162
\(682\) 21.4095 0.819812
\(683\) 30.6457 1.17262 0.586312 0.810086i \(-0.300579\pi\)
0.586312 + 0.810086i \(0.300579\pi\)
\(684\) −1.49203 −0.0570492
\(685\) −41.6053 −1.58966
\(686\) −19.4452 −0.742422
\(687\) 14.2912 0.545242
\(688\) −1.09636 −0.0417982
\(689\) 3.82574 0.145749
\(690\) −4.88610 −0.186010
\(691\) 9.94855 0.378461 0.189230 0.981933i \(-0.439401\pi\)
0.189230 + 0.981933i \(0.439401\pi\)
\(692\) 8.38483 0.318744
\(693\) 8.58112 0.325970
\(694\) 15.6913 0.595632
\(695\) −14.4418 −0.547808
\(696\) −3.97697 −0.150747
\(697\) −34.2183 −1.29611
\(698\) 16.5396 0.626035
\(699\) −15.7787 −0.596805
\(700\) −31.2258 −1.18022
\(701\) 0.606614 0.0229115 0.0114557 0.999934i \(-0.496353\pi\)
0.0114557 + 0.999934i \(0.496353\pi\)
\(702\) 1.00000 0.0377426
\(703\) −1.39994 −0.0527996
\(704\) −3.45337 −0.130154
\(705\) −8.01704 −0.301939
\(706\) 28.2812 1.06438
\(707\) 30.3825 1.14265
\(708\) 8.10947 0.304773
\(709\) −12.3399 −0.463436 −0.231718 0.972783i \(-0.574435\pi\)
−0.231718 + 0.972783i \(0.574435\pi\)
\(710\) 3.75538 0.140937
\(711\) 4.36696 0.163774
\(712\) −1.33376 −0.0499848
\(713\) −7.22742 −0.270669
\(714\) 10.3152 0.386038
\(715\) −14.4739 −0.541292
\(716\) 24.2415 0.905946
\(717\) −24.0758 −0.899127
\(718\) 21.2405 0.792690
\(719\) −25.0439 −0.933980 −0.466990 0.884263i \(-0.654662\pi\)
−0.466990 + 0.884263i \(0.654662\pi\)
\(720\) −4.19123 −0.156198
\(721\) −2.48485 −0.0925408
\(722\) 16.7738 0.624258
\(723\) −24.3323 −0.904926
\(724\) 1.00614 0.0373929
\(725\) 49.9764 1.85608
\(726\) −0.925764 −0.0343583
\(727\) −36.0438 −1.33679 −0.668396 0.743806i \(-0.733019\pi\)
−0.668396 + 0.743806i \(0.733019\pi\)
\(728\) −2.48485 −0.0920948
\(729\) 1.00000 0.0370370
\(730\) −3.59574 −0.133084
\(731\) −4.55125 −0.168334
\(732\) 5.11955 0.189224
\(733\) 34.1416 1.26105 0.630525 0.776169i \(-0.282840\pi\)
0.630525 + 0.776169i \(0.282840\pi\)
\(734\) 4.96807 0.183375
\(735\) 3.45987 0.127619
\(736\) 1.16579 0.0429716
\(737\) −45.2385 −1.66638
\(738\) 8.24288 0.303425
\(739\) −34.8852 −1.28327 −0.641636 0.767010i \(-0.721744\pi\)
−0.641636 + 0.767010i \(0.721744\pi\)
\(740\) −3.93253 −0.144563
\(741\) 1.49203 0.0548111
\(742\) −9.50640 −0.348991
\(743\) −28.9668 −1.06269 −0.531345 0.847156i \(-0.678313\pi\)
−0.531345 + 0.847156i \(0.678313\pi\)
\(744\) −6.19959 −0.227288
\(745\) 17.7221 0.649289
\(746\) 1.35351 0.0495556
\(747\) −1.28939 −0.0471763
\(748\) −14.3358 −0.524169
\(749\) 44.5545 1.62799
\(750\) 31.7127 1.15798
\(751\) −19.0257 −0.694258 −0.347129 0.937817i \(-0.612843\pi\)
−0.347129 + 0.937817i \(0.612843\pi\)
\(752\) 1.91281 0.0697531
\(753\) 8.22652 0.299791
\(754\) 3.97697 0.144833
\(755\) 62.1988 2.26365
\(756\) −2.48485 −0.0903733
\(757\) 53.9813 1.96198 0.980991 0.194052i \(-0.0621631\pi\)
0.980991 + 0.194052i \(0.0621631\pi\)
\(758\) −1.33194 −0.0483783
\(759\) 4.02590 0.146131
\(760\) −6.25345 −0.226836
\(761\) −2.12579 −0.0770600 −0.0385300 0.999257i \(-0.512268\pi\)
−0.0385300 + 0.999257i \(0.512268\pi\)
\(762\) 6.45305 0.233769
\(763\) −45.0193 −1.62981
\(764\) −17.7258 −0.641295
\(765\) −17.3989 −0.629057
\(766\) 5.46332 0.197398
\(767\) −8.10947 −0.292816
\(768\) 1.00000 0.0360844
\(769\) −25.5752 −0.922264 −0.461132 0.887332i \(-0.652557\pi\)
−0.461132 + 0.887332i \(0.652557\pi\)
\(770\) 35.9655 1.29611
\(771\) 19.2253 0.692381
\(772\) −8.92394 −0.321180
\(773\) 10.8883 0.391626 0.195813 0.980641i \(-0.437265\pi\)
0.195813 + 0.980641i \(0.437265\pi\)
\(774\) 1.09636 0.0394077
\(775\) 77.9068 2.79850
\(776\) 1.09589 0.0393403
\(777\) −2.33148 −0.0836414
\(778\) −5.47837 −0.196409
\(779\) 12.2986 0.440644
\(780\) 4.19123 0.150070
\(781\) −3.09425 −0.110721
\(782\) 4.83948 0.173060
\(783\) 3.97697 0.142125
\(784\) −0.825500 −0.0294822
\(785\) −47.6489 −1.70066
\(786\) 5.48297 0.195571
\(787\) 8.06590 0.287518 0.143759 0.989613i \(-0.454081\pi\)
0.143759 + 0.989613i \(0.454081\pi\)
\(788\) −5.72916 −0.204093
\(789\) −26.5673 −0.945820
\(790\) 18.3030 0.651190
\(791\) −41.3126 −1.46891
\(792\) 3.45337 0.122710
\(793\) −5.11955 −0.181800
\(794\) 32.9394 1.16897
\(795\) 16.0346 0.568687
\(796\) 4.04033 0.143206
\(797\) −9.83867 −0.348503 −0.174252 0.984701i \(-0.555751\pi\)
−0.174252 + 0.984701i \(0.555751\pi\)
\(798\) −3.70748 −0.131243
\(799\) 7.94056 0.280917
\(800\) −12.5664 −0.444291
\(801\) 1.33376 0.0471261
\(802\) −7.80301 −0.275534
\(803\) 2.96271 0.104552
\(804\) 13.0998 0.461994
\(805\) −12.1412 −0.427922
\(806\) 6.19959 0.218371
\(807\) 3.93881 0.138653
\(808\) 12.2271 0.430147
\(809\) −23.3970 −0.822595 −0.411298 0.911501i \(-0.634924\pi\)
−0.411298 + 0.911501i \(0.634924\pi\)
\(810\) 4.19123 0.147265
\(811\) 24.8923 0.874087 0.437044 0.899440i \(-0.356026\pi\)
0.437044 + 0.899440i \(0.356026\pi\)
\(812\) −9.88220 −0.346797
\(813\) −6.13049 −0.215006
\(814\) 3.24021 0.113569
\(815\) −72.9610 −2.55571
\(816\) 4.15125 0.145323
\(817\) 1.63580 0.0572293
\(818\) −25.9347 −0.906786
\(819\) 2.48485 0.0868278
\(820\) 34.5479 1.20646
\(821\) 7.39290 0.258014 0.129007 0.991644i \(-0.458821\pi\)
0.129007 + 0.991644i \(0.458821\pi\)
\(822\) −9.92674 −0.346235
\(823\) −37.2278 −1.29768 −0.648839 0.760925i \(-0.724745\pi\)
−0.648839 + 0.760925i \(0.724745\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −43.3966 −1.51087
\(826\) 20.1509 0.701138
\(827\) 37.7529 1.31280 0.656399 0.754414i \(-0.272079\pi\)
0.656399 + 0.754414i \(0.272079\pi\)
\(828\) −1.16579 −0.0405140
\(829\) −17.6399 −0.612660 −0.306330 0.951925i \(-0.599101\pi\)
−0.306330 + 0.951925i \(0.599101\pi\)
\(830\) −5.40413 −0.187580
\(831\) −3.38074 −0.117277
\(832\) −1.00000 −0.0346688
\(833\) −3.42686 −0.118734
\(834\) −3.44571 −0.119315
\(835\) 83.2103 2.87961
\(836\) 5.15253 0.178204
\(837\) 6.19959 0.214289
\(838\) 11.8178 0.408241
\(839\) −36.9424 −1.27539 −0.637696 0.770288i \(-0.720112\pi\)
−0.637696 + 0.770288i \(0.720112\pi\)
\(840\) −10.4146 −0.359338
\(841\) −13.1837 −0.454610
\(842\) 28.3038 0.975412
\(843\) −5.93074 −0.204265
\(844\) −18.1802 −0.625789
\(845\) −4.19123 −0.144183
\(846\) −1.91281 −0.0657638
\(847\) −2.30039 −0.0790423
\(848\) −3.82574 −0.131376
\(849\) 23.4537 0.804928
\(850\) −52.1664 −1.78929
\(851\) −1.09383 −0.0374961
\(852\) 0.896009 0.0306967
\(853\) −37.4443 −1.28207 −0.641034 0.767512i \(-0.721494\pi\)
−0.641034 + 0.767512i \(0.721494\pi\)
\(854\) 12.7213 0.435315
\(855\) 6.25345 0.213863
\(856\) 17.9304 0.612849
\(857\) −9.08161 −0.310222 −0.155111 0.987897i \(-0.549574\pi\)
−0.155111 + 0.987897i \(0.549574\pi\)
\(858\) −3.45337 −0.117896
\(859\) −17.8016 −0.607384 −0.303692 0.952770i \(-0.598219\pi\)
−0.303692 + 0.952770i \(0.598219\pi\)
\(860\) 4.59509 0.156691
\(861\) 20.4824 0.698037
\(862\) 11.3047 0.385040
\(863\) 57.5943 1.96053 0.980266 0.197681i \(-0.0633410\pi\)
0.980266 + 0.197681i \(0.0633410\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −35.1428 −1.19489
\(866\) 13.6585 0.464134
\(867\) 0.232872 0.00790876
\(868\) −15.4051 −0.522883
\(869\) −15.0807 −0.511579
\(870\) 16.6684 0.565112
\(871\) −13.0998 −0.443870
\(872\) −18.1175 −0.613535
\(873\) −1.09589 −0.0370904
\(874\) −1.73939 −0.0588358
\(875\) 78.8015 2.66398
\(876\) −0.857920 −0.0289864
\(877\) −29.6978 −1.00282 −0.501412 0.865208i \(-0.667186\pi\)
−0.501412 + 0.865208i \(0.667186\pi\)
\(878\) 10.1305 0.341888
\(879\) −29.0023 −0.978224
\(880\) 14.4739 0.487914
\(881\) 48.3041 1.62741 0.813703 0.581281i \(-0.197448\pi\)
0.813703 + 0.581281i \(0.197448\pi\)
\(882\) 0.825500 0.0277960
\(883\) −22.8351 −0.768463 −0.384231 0.923237i \(-0.625534\pi\)
−0.384231 + 0.923237i \(0.625534\pi\)
\(884\) −4.15125 −0.139622
\(885\) −33.9887 −1.14252
\(886\) 24.1352 0.810837
\(887\) −32.2578 −1.08311 −0.541556 0.840664i \(-0.682165\pi\)
−0.541556 + 0.840664i \(0.682165\pi\)
\(888\) −0.938276 −0.0314865
\(889\) 16.0349 0.537793
\(890\) 5.59010 0.187381
\(891\) −3.45337 −0.115692
\(892\) −15.7232 −0.526452
\(893\) −2.85397 −0.0955046
\(894\) 4.22838 0.141418
\(895\) −101.602 −3.39617
\(896\) 2.48485 0.0830131
\(897\) 1.16579 0.0389246
\(898\) −34.6217 −1.15534
\(899\) 24.6556 0.822311
\(900\) 12.5664 0.418881
\(901\) −15.8816 −0.529092
\(902\) −28.4657 −0.947805
\(903\) 2.72429 0.0906586
\(904\) −16.6258 −0.552965
\(905\) −4.21697 −0.140177
\(906\) 14.8402 0.493033
\(907\) −1.51747 −0.0503868 −0.0251934 0.999683i \(-0.508020\pi\)
−0.0251934 + 0.999683i \(0.508020\pi\)
\(908\) 5.43218 0.180273
\(909\) −12.2271 −0.405547
\(910\) 10.4146 0.345241
\(911\) 37.8108 1.25273 0.626364 0.779531i \(-0.284543\pi\)
0.626364 + 0.779531i \(0.284543\pi\)
\(912\) −1.49203 −0.0494061
\(913\) 4.45274 0.147364
\(914\) 31.4318 1.03967
\(915\) −21.4572 −0.709354
\(916\) 14.2912 0.472193
\(917\) 13.6244 0.449916
\(918\) −4.15125 −0.137012
\(919\) 58.1081 1.91681 0.958405 0.285412i \(-0.0921304\pi\)
0.958405 + 0.285412i \(0.0921304\pi\)
\(920\) −4.88610 −0.161090
\(921\) 25.5987 0.843507
\(922\) −17.5107 −0.576684
\(923\) −0.896009 −0.0294925
\(924\) 8.58112 0.282298
\(925\) 11.7908 0.387679
\(926\) −10.4776 −0.344314
\(927\) 1.00000 0.0328443
\(928\) −3.97697 −0.130551
\(929\) 8.37459 0.274761 0.137381 0.990518i \(-0.456132\pi\)
0.137381 + 0.990518i \(0.456132\pi\)
\(930\) 25.9839 0.852047
\(931\) 1.23167 0.0403664
\(932\) −15.7787 −0.516848
\(933\) −14.8637 −0.486615
\(934\) −0.734108 −0.0240207
\(935\) 60.0847 1.96498
\(936\) 1.00000 0.0326860
\(937\) 12.3552 0.403625 0.201813 0.979424i \(-0.435317\pi\)
0.201813 + 0.979424i \(0.435317\pi\)
\(938\) 32.5511 1.06283
\(939\) 11.2838 0.368233
\(940\) −8.01704 −0.261487
\(941\) 20.5177 0.668859 0.334429 0.942421i \(-0.391456\pi\)
0.334429 + 0.942421i \(0.391456\pi\)
\(942\) −11.3687 −0.370413
\(943\) 9.60946 0.312927
\(944\) 8.10947 0.263941
\(945\) 10.4146 0.338787
\(946\) −3.78613 −0.123098
\(947\) −7.31607 −0.237740 −0.118870 0.992910i \(-0.537927\pi\)
−0.118870 + 0.992910i \(0.537927\pi\)
\(948\) 4.36696 0.141832
\(949\) 0.857920 0.0278492
\(950\) 18.7495 0.608314
\(951\) 3.43988 0.111546
\(952\) 10.3152 0.334319
\(953\) 25.3981 0.822726 0.411363 0.911471i \(-0.365053\pi\)
0.411363 + 0.911471i \(0.365053\pi\)
\(954\) 3.82574 0.123863
\(955\) 74.2928 2.40406
\(956\) −24.0758 −0.778667
\(957\) −13.7340 −0.443956
\(958\) −40.2600 −1.30074
\(959\) −24.6665 −0.796523
\(960\) −4.19123 −0.135271
\(961\) 7.43497 0.239838
\(962\) 0.938276 0.0302512
\(963\) −17.9304 −0.577800
\(964\) −24.3323 −0.783689
\(965\) 37.4023 1.20402
\(966\) −2.89682 −0.0932035
\(967\) −46.8399 −1.50627 −0.753134 0.657867i \(-0.771459\pi\)
−0.753134 + 0.657867i \(0.771459\pi\)
\(968\) −0.925764 −0.0297552
\(969\) −6.19379 −0.198973
\(970\) −4.59314 −0.147477
\(971\) −40.4508 −1.29813 −0.649063 0.760734i \(-0.724839\pi\)
−0.649063 + 0.760734i \(0.724839\pi\)
\(972\) 1.00000 0.0320750
\(973\) −8.56209 −0.274488
\(974\) −10.8019 −0.346116
\(975\) −12.5664 −0.402448
\(976\) 5.11955 0.163873
\(977\) 7.12306 0.227887 0.113943 0.993487i \(-0.463652\pi\)
0.113943 + 0.993487i \(0.463652\pi\)
\(978\) −17.4080 −0.556647
\(979\) −4.60597 −0.147207
\(980\) 3.45987 0.110521
\(981\) 18.1175 0.578447
\(982\) −21.2973 −0.679624
\(983\) −35.6158 −1.13597 −0.567984 0.823040i \(-0.692276\pi\)
−0.567984 + 0.823040i \(0.692276\pi\)
\(984\) 8.24288 0.262774
\(985\) 24.0123 0.765094
\(986\) −16.5094 −0.525767
\(987\) −4.75306 −0.151292
\(988\) 1.49203 0.0474678
\(989\) 1.27812 0.0406419
\(990\) −14.4739 −0.460010
\(991\) 5.41675 0.172069 0.0860344 0.996292i \(-0.472581\pi\)
0.0860344 + 0.996292i \(0.472581\pi\)
\(992\) −6.19959 −0.196837
\(993\) −6.84967 −0.217368
\(994\) 2.22645 0.0706187
\(995\) −16.9340 −0.536843
\(996\) −1.28939 −0.0408558
\(997\) −49.4466 −1.56599 −0.782995 0.622028i \(-0.786309\pi\)
−0.782995 + 0.622028i \(0.786309\pi\)
\(998\) 26.0360 0.824155
\(999\) 0.938276 0.0296857
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.bb.1.1 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.1 14 1.1 even 1 trivial