Properties

Label 8034.2.a.z.1.9
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + 6361 x^{6} - 14618 x^{5} - 7015 x^{4} + 15763 x^{3} + 1118 x^{2} - 6316 x + 1552\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-1.02849\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.02849 q^{5} +1.00000 q^{6} -4.96874 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.02849 q^{5} +1.00000 q^{6} -4.96874 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.02849 q^{10} +4.09511 q^{11} -1.00000 q^{12} +1.00000 q^{13} +4.96874 q^{14} -1.02849 q^{15} +1.00000 q^{16} +0.459045 q^{17} -1.00000 q^{18} +2.05940 q^{19} +1.02849 q^{20} +4.96874 q^{21} -4.09511 q^{22} +7.97226 q^{23} +1.00000 q^{24} -3.94221 q^{25} -1.00000 q^{26} -1.00000 q^{27} -4.96874 q^{28} +3.58639 q^{29} +1.02849 q^{30} -9.09347 q^{31} -1.00000 q^{32} -4.09511 q^{33} -0.459045 q^{34} -5.11029 q^{35} +1.00000 q^{36} -6.55603 q^{37} -2.05940 q^{38} -1.00000 q^{39} -1.02849 q^{40} +5.24275 q^{41} -4.96874 q^{42} +8.24715 q^{43} +4.09511 q^{44} +1.02849 q^{45} -7.97226 q^{46} +1.59375 q^{47} -1.00000 q^{48} +17.6884 q^{49} +3.94221 q^{50} -0.459045 q^{51} +1.00000 q^{52} +5.49115 q^{53} +1.00000 q^{54} +4.21177 q^{55} +4.96874 q^{56} -2.05940 q^{57} -3.58639 q^{58} -2.90936 q^{59} -1.02849 q^{60} -9.24844 q^{61} +9.09347 q^{62} -4.96874 q^{63} +1.00000 q^{64} +1.02849 q^{65} +4.09511 q^{66} -1.92754 q^{67} +0.459045 q^{68} -7.97226 q^{69} +5.11029 q^{70} +12.7767 q^{71} -1.00000 q^{72} -1.18160 q^{73} +6.55603 q^{74} +3.94221 q^{75} +2.05940 q^{76} -20.3476 q^{77} +1.00000 q^{78} +10.7487 q^{79} +1.02849 q^{80} +1.00000 q^{81} -5.24275 q^{82} -16.0640 q^{83} +4.96874 q^{84} +0.472123 q^{85} -8.24715 q^{86} -3.58639 q^{87} -4.09511 q^{88} +15.5497 q^{89} -1.02849 q^{90} -4.96874 q^{91} +7.97226 q^{92} +9.09347 q^{93} -1.59375 q^{94} +2.11807 q^{95} +1.00000 q^{96} -15.5491 q^{97} -17.6884 q^{98} +4.09511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + 3q^{10} + 5q^{11} - 14q^{12} + 14q^{13} - 4q^{14} + 3q^{15} + 14q^{16} - 7q^{17} - 14q^{18} + 31q^{19} - 3q^{20} - 4q^{21} - 5q^{22} + 14q^{23} + 14q^{24} + 11q^{25} - 14q^{26} - 14q^{27} + 4q^{28} + 9q^{29} - 3q^{30} + 23q^{31} - 14q^{32} - 5q^{33} + 7q^{34} - 2q^{35} + 14q^{36} + 12q^{37} - 31q^{38} - 14q^{39} + 3q^{40} - 5q^{41} + 4q^{42} - 2q^{43} + 5q^{44} - 3q^{45} - 14q^{46} - 11q^{47} - 14q^{48} + 52q^{49} - 11q^{50} + 7q^{51} + 14q^{52} + 6q^{53} + 14q^{54} + 30q^{55} - 4q^{56} - 31q^{57} - 9q^{58} - 4q^{59} + 3q^{60} + 12q^{61} - 23q^{62} + 4q^{63} + 14q^{64} - 3q^{65} + 5q^{66} + 24q^{67} - 7q^{68} - 14q^{69} + 2q^{70} + 20q^{71} - 14q^{72} + 2q^{73} - 12q^{74} - 11q^{75} + 31q^{76} - 28q^{77} + 14q^{78} + 59q^{79} - 3q^{80} + 14q^{81} + 5q^{82} + q^{83} - 4q^{84} - 29q^{85} + 2q^{86} - 9q^{87} - 5q^{88} + 6q^{89} + 3q^{90} + 4q^{91} + 14q^{92} - 23q^{93} + 11q^{94} - 58q^{95} + 14q^{96} - 6q^{97} - 52q^{98} + 5q^{99} + O(q^{100}) \)

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
\(5\) 1.02849 0.459954 0.229977 0.973196i \(-0.426135\pi\)
0.229977 + 0.973196i \(0.426135\pi\)
\(6\) 1.00000 0.408248
\(7\) −4.96874 −1.87801 −0.939004 0.343906i \(-0.888250\pi\)
−0.939004 + 0.343906i \(0.888250\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.02849 −0.325236
\(11\) 4.09511 1.23472 0.617361 0.786680i \(-0.288202\pi\)
0.617361 + 0.786680i \(0.288202\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 4.96874 1.32795
\(15\) −1.02849 −0.265554
\(16\) 1.00000 0.250000
\(17\) 0.459045 0.111335 0.0556674 0.998449i \(-0.482271\pi\)
0.0556674 + 0.998449i \(0.482271\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.05940 0.472459 0.236230 0.971697i \(-0.424088\pi\)
0.236230 + 0.971697i \(0.424088\pi\)
\(20\) 1.02849 0.229977
\(21\) 4.96874 1.08427
\(22\) −4.09511 −0.873081
\(23\) 7.97226 1.66233 0.831165 0.556025i \(-0.187674\pi\)
0.831165 + 0.556025i \(0.187674\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.94221 −0.788443
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −4.96874 −0.939004
\(29\) 3.58639 0.665975 0.332988 0.942931i \(-0.391943\pi\)
0.332988 + 0.942931i \(0.391943\pi\)
\(30\) 1.02849 0.187775
\(31\) −9.09347 −1.63324 −0.816618 0.577179i \(-0.804154\pi\)
−0.816618 + 0.577179i \(0.804154\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.09511 −0.712868
\(34\) −0.459045 −0.0787256
\(35\) −5.11029 −0.863797
\(36\) 1.00000 0.166667
\(37\) −6.55603 −1.07780 −0.538902 0.842368i \(-0.681161\pi\)
−0.538902 + 0.842368i \(0.681161\pi\)
\(38\) −2.05940 −0.334079
\(39\) −1.00000 −0.160128
\(40\) −1.02849 −0.162618
\(41\) 5.24275 0.818779 0.409390 0.912360i \(-0.365742\pi\)
0.409390 + 0.912360i \(0.365742\pi\)
\(42\) −4.96874 −0.766693
\(43\) 8.24715 1.25768 0.628840 0.777535i \(-0.283530\pi\)
0.628840 + 0.777535i \(0.283530\pi\)
\(44\) 4.09511 0.617361
\(45\) 1.02849 0.153318
\(46\) −7.97226 −1.17545
\(47\) 1.59375 0.232473 0.116236 0.993222i \(-0.462917\pi\)
0.116236 + 0.993222i \(0.462917\pi\)
\(48\) −1.00000 −0.144338
\(49\) 17.6884 2.52691
\(50\) 3.94221 0.557513
\(51\) −0.459045 −0.0642792
\(52\) 1.00000 0.138675
\(53\) 5.49115 0.754267 0.377133 0.926159i \(-0.376910\pi\)
0.377133 + 0.926159i \(0.376910\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.21177 0.567915
\(56\) 4.96874 0.663976
\(57\) −2.05940 −0.272774
\(58\) −3.58639 −0.470916
\(59\) −2.90936 −0.378767 −0.189383 0.981903i \(-0.560649\pi\)
−0.189383 + 0.981903i \(0.560649\pi\)
\(60\) −1.02849 −0.132777
\(61\) −9.24844 −1.18414 −0.592071 0.805886i \(-0.701689\pi\)
−0.592071 + 0.805886i \(0.701689\pi\)
\(62\) 9.09347 1.15487
\(63\) −4.96874 −0.626003
\(64\) 1.00000 0.125000
\(65\) 1.02849 0.127568
\(66\) 4.09511 0.504074
\(67\) −1.92754 −0.235487 −0.117743 0.993044i \(-0.537566\pi\)
−0.117743 + 0.993044i \(0.537566\pi\)
\(68\) 0.459045 0.0556674
\(69\) −7.97226 −0.959747
\(70\) 5.11029 0.610797
\(71\) 12.7767 1.51632 0.758160 0.652069i \(-0.226099\pi\)
0.758160 + 0.652069i \(0.226099\pi\)
\(72\) −1.00000 −0.117851
\(73\) −1.18160 −0.138296 −0.0691481 0.997606i \(-0.522028\pi\)
−0.0691481 + 0.997606i \(0.522028\pi\)
\(74\) 6.55603 0.762123
\(75\) 3.94221 0.455208
\(76\) 2.05940 0.236230
\(77\) −20.3476 −2.31882
\(78\) 1.00000 0.113228
\(79\) 10.7487 1.20933 0.604663 0.796481i \(-0.293308\pi\)
0.604663 + 0.796481i \(0.293308\pi\)
\(80\) 1.02849 0.114988
\(81\) 1.00000 0.111111
\(82\) −5.24275 −0.578964
\(83\) −16.0640 −1.76325 −0.881627 0.471946i \(-0.843552\pi\)
−0.881627 + 0.471946i \(0.843552\pi\)
\(84\) 4.96874 0.542134
\(85\) 0.472123 0.0512089
\(86\) −8.24715 −0.889313
\(87\) −3.58639 −0.384501
\(88\) −4.09511 −0.436540
\(89\) 15.5497 1.64827 0.824133 0.566396i \(-0.191663\pi\)
0.824133 + 0.566396i \(0.191663\pi\)
\(90\) −1.02849 −0.108412
\(91\) −4.96874 −0.520866
\(92\) 7.97226 0.831165
\(93\) 9.09347 0.942949
\(94\) −1.59375 −0.164383
\(95\) 2.11807 0.217309
\(96\) 1.00000 0.102062
\(97\) −15.5491 −1.57877 −0.789386 0.613897i \(-0.789601\pi\)
−0.789386 + 0.613897i \(0.789601\pi\)
\(98\) −17.6884 −1.78680
\(99\) 4.09511 0.411574
\(100\) −3.94221 −0.394221
\(101\) −11.6401 −1.15824 −0.579118 0.815244i \(-0.696603\pi\)
−0.579118 + 0.815244i \(0.696603\pi\)
\(102\) 0.459045 0.0454523
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 5.11029 0.498713
\(106\) −5.49115 −0.533347
\(107\) 10.8455 1.04848 0.524239 0.851571i \(-0.324350\pi\)
0.524239 + 0.851571i \(0.324350\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.0922 −1.54136 −0.770679 0.637224i \(-0.780083\pi\)
−0.770679 + 0.637224i \(0.780083\pi\)
\(110\) −4.21177 −0.401577
\(111\) 6.55603 0.622270
\(112\) −4.96874 −0.469502
\(113\) −14.8459 −1.39658 −0.698291 0.715814i \(-0.746056\pi\)
−0.698291 + 0.715814i \(0.746056\pi\)
\(114\) 2.05940 0.192881
\(115\) 8.19937 0.764595
\(116\) 3.58639 0.332988
\(117\) 1.00000 0.0924500
\(118\) 2.90936 0.267828
\(119\) −2.28088 −0.209088
\(120\) 1.02849 0.0938877
\(121\) 5.76995 0.524541
\(122\) 9.24844 0.837314
\(123\) −5.24275 −0.472722
\(124\) −9.09347 −0.816618
\(125\) −9.19696 −0.822601
\(126\) 4.96874 0.442651
\(127\) 0.472953 0.0419678 0.0209839 0.999780i \(-0.493320\pi\)
0.0209839 + 0.999780i \(0.493320\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.24715 −0.726121
\(130\) −1.02849 −0.0902044
\(131\) 5.27085 0.460517 0.230258 0.973130i \(-0.426043\pi\)
0.230258 + 0.973130i \(0.426043\pi\)
\(132\) −4.09511 −0.356434
\(133\) −10.2326 −0.887282
\(134\) 1.92754 0.166514
\(135\) −1.02849 −0.0885181
\(136\) −0.459045 −0.0393628
\(137\) −3.17885 −0.271587 −0.135794 0.990737i \(-0.543358\pi\)
−0.135794 + 0.990737i \(0.543358\pi\)
\(138\) 7.97226 0.678644
\(139\) −0.950817 −0.0806473 −0.0403236 0.999187i \(-0.512839\pi\)
−0.0403236 + 0.999187i \(0.512839\pi\)
\(140\) −5.11029 −0.431898
\(141\) −1.59375 −0.134218
\(142\) −12.7767 −1.07220
\(143\) 4.09511 0.342451
\(144\) 1.00000 0.0833333
\(145\) 3.68855 0.306318
\(146\) 1.18160 0.0977901
\(147\) −17.6884 −1.45891
\(148\) −6.55603 −0.538902
\(149\) 6.42360 0.526242 0.263121 0.964763i \(-0.415248\pi\)
0.263121 + 0.964763i \(0.415248\pi\)
\(150\) −3.94221 −0.321880
\(151\) 12.5514 1.02142 0.510710 0.859753i \(-0.329383\pi\)
0.510710 + 0.859753i \(0.329383\pi\)
\(152\) −2.05940 −0.167040
\(153\) 0.459045 0.0371116
\(154\) 20.3476 1.63965
\(155\) −9.35253 −0.751213
\(156\) −1.00000 −0.0800641
\(157\) −16.7227 −1.33462 −0.667309 0.744781i \(-0.732554\pi\)
−0.667309 + 0.744781i \(0.732554\pi\)
\(158\) −10.7487 −0.855123
\(159\) −5.49115 −0.435476
\(160\) −1.02849 −0.0813091
\(161\) −39.6121 −3.12187
\(162\) −1.00000 −0.0785674
\(163\) 10.3522 0.810850 0.405425 0.914128i \(-0.367123\pi\)
0.405425 + 0.914128i \(0.367123\pi\)
\(164\) 5.24275 0.409390
\(165\) −4.21177 −0.327886
\(166\) 16.0640 1.24681
\(167\) −11.8605 −0.917797 −0.458898 0.888489i \(-0.651756\pi\)
−0.458898 + 0.888489i \(0.651756\pi\)
\(168\) −4.96874 −0.383347
\(169\) 1.00000 0.0769231
\(170\) −0.472123 −0.0362101
\(171\) 2.05940 0.157486
\(172\) 8.24715 0.628840
\(173\) 24.8743 1.89116 0.945581 0.325388i \(-0.105495\pi\)
0.945581 + 0.325388i \(0.105495\pi\)
\(174\) 3.58639 0.271883
\(175\) 19.5878 1.48070
\(176\) 4.09511 0.308681
\(177\) 2.90936 0.218681
\(178\) −15.5497 −1.16550
\(179\) 7.75139 0.579366 0.289683 0.957123i \(-0.406450\pi\)
0.289683 + 0.957123i \(0.406450\pi\)
\(180\) 1.02849 0.0766590
\(181\) 17.7205 1.31715 0.658577 0.752513i \(-0.271159\pi\)
0.658577 + 0.752513i \(0.271159\pi\)
\(182\) 4.96874 0.368308
\(183\) 9.24844 0.683664
\(184\) −7.97226 −0.587723
\(185\) −6.74279 −0.495740
\(186\) −9.09347 −0.666766
\(187\) 1.87984 0.137468
\(188\) 1.59375 0.116236
\(189\) 4.96874 0.361423
\(190\) −2.11807 −0.153661
\(191\) 19.3541 1.40042 0.700209 0.713938i \(-0.253090\pi\)
0.700209 + 0.713938i \(0.253090\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −3.45493 −0.248691 −0.124346 0.992239i \(-0.539683\pi\)
−0.124346 + 0.992239i \(0.539683\pi\)
\(194\) 15.5491 1.11636
\(195\) −1.02849 −0.0736515
\(196\) 17.6884 1.26346
\(197\) −8.07696 −0.575459 −0.287730 0.957712i \(-0.592900\pi\)
−0.287730 + 0.957712i \(0.592900\pi\)
\(198\) −4.09511 −0.291027
\(199\) 10.6009 0.751477 0.375739 0.926726i \(-0.377389\pi\)
0.375739 + 0.926726i \(0.377389\pi\)
\(200\) 3.94221 0.278757
\(201\) 1.92754 0.135958
\(202\) 11.6401 0.818996
\(203\) −17.8198 −1.25071
\(204\) −0.459045 −0.0321396
\(205\) 5.39210 0.376601
\(206\) 1.00000 0.0696733
\(207\) 7.97226 0.554110
\(208\) 1.00000 0.0693375
\(209\) 8.43348 0.583356
\(210\) −5.11029 −0.352644
\(211\) −6.54374 −0.450490 −0.225245 0.974302i \(-0.572318\pi\)
−0.225245 + 0.974302i \(0.572318\pi\)
\(212\) 5.49115 0.377133
\(213\) −12.7767 −0.875448
\(214\) −10.8455 −0.741385
\(215\) 8.48210 0.578474
\(216\) 1.00000 0.0680414
\(217\) 45.1831 3.06723
\(218\) 16.0922 1.08990
\(219\) 1.18160 0.0798453
\(220\) 4.21177 0.283958
\(221\) 0.459045 0.0308787
\(222\) −6.55603 −0.440012
\(223\) −5.38319 −0.360485 −0.180243 0.983622i \(-0.557688\pi\)
−0.180243 + 0.983622i \(0.557688\pi\)
\(224\) 4.96874 0.331988
\(225\) −3.94221 −0.262814
\(226\) 14.8459 0.987532
\(227\) −5.14503 −0.341488 −0.170744 0.985315i \(-0.554617\pi\)
−0.170744 + 0.985315i \(0.554617\pi\)
\(228\) −2.05940 −0.136387
\(229\) 14.1375 0.934231 0.467116 0.884196i \(-0.345293\pi\)
0.467116 + 0.884196i \(0.345293\pi\)
\(230\) −8.19937 −0.540650
\(231\) 20.3476 1.33877
\(232\) −3.58639 −0.235458
\(233\) −8.36084 −0.547737 −0.273869 0.961767i \(-0.588303\pi\)
−0.273869 + 0.961767i \(0.588303\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 1.63916 0.106927
\(236\) −2.90936 −0.189383
\(237\) −10.7487 −0.698205
\(238\) 2.28088 0.147847
\(239\) 17.7749 1.14976 0.574882 0.818236i \(-0.305048\pi\)
0.574882 + 0.818236i \(0.305048\pi\)
\(240\) −1.02849 −0.0663886
\(241\) −1.63447 −0.105285 −0.0526426 0.998613i \(-0.516764\pi\)
−0.0526426 + 0.998613i \(0.516764\pi\)
\(242\) −5.76995 −0.370906
\(243\) −1.00000 −0.0641500
\(244\) −9.24844 −0.592071
\(245\) 18.1923 1.16226
\(246\) 5.24275 0.334265
\(247\) 2.05940 0.131037
\(248\) 9.09347 0.577436
\(249\) 16.0640 1.01802
\(250\) 9.19696 0.581667
\(251\) −18.7438 −1.18310 −0.591549 0.806269i \(-0.701483\pi\)
−0.591549 + 0.806269i \(0.701483\pi\)
\(252\) −4.96874 −0.313001
\(253\) 32.6473 2.05252
\(254\) −0.472953 −0.0296757
\(255\) −0.472123 −0.0295655
\(256\) 1.00000 0.0625000
\(257\) −21.6529 −1.35067 −0.675334 0.737512i \(-0.736000\pi\)
−0.675334 + 0.737512i \(0.736000\pi\)
\(258\) 8.24715 0.513445
\(259\) 32.5752 2.02412
\(260\) 1.02849 0.0637841
\(261\) 3.58639 0.221992
\(262\) −5.27085 −0.325634
\(263\) 16.9718 1.04653 0.523264 0.852170i \(-0.324714\pi\)
0.523264 + 0.852170i \(0.324714\pi\)
\(264\) 4.09511 0.252037
\(265\) 5.64758 0.346928
\(266\) 10.2326 0.627403
\(267\) −15.5497 −0.951627
\(268\) −1.92754 −0.117743
\(269\) 5.09786 0.310822 0.155411 0.987850i \(-0.450330\pi\)
0.155411 + 0.987850i \(0.450330\pi\)
\(270\) 1.02849 0.0625918
\(271\) −25.2640 −1.53468 −0.767338 0.641242i \(-0.778419\pi\)
−0.767338 + 0.641242i \(0.778419\pi\)
\(272\) 0.459045 0.0278337
\(273\) 4.96874 0.300722
\(274\) 3.17885 0.192041
\(275\) −16.1438 −0.973508
\(276\) −7.97226 −0.479874
\(277\) 30.4540 1.82981 0.914903 0.403675i \(-0.132267\pi\)
0.914903 + 0.403675i \(0.132267\pi\)
\(278\) 0.950817 0.0570262
\(279\) −9.09347 −0.544412
\(280\) 5.11029 0.305398
\(281\) −21.9475 −1.30928 −0.654638 0.755943i \(-0.727179\pi\)
−0.654638 + 0.755943i \(0.727179\pi\)
\(282\) 1.59375 0.0949066
\(283\) −14.0193 −0.833360 −0.416680 0.909053i \(-0.636807\pi\)
−0.416680 + 0.909053i \(0.636807\pi\)
\(284\) 12.7767 0.758160
\(285\) −2.11807 −0.125464
\(286\) −4.09511 −0.242149
\(287\) −26.0498 −1.53767
\(288\) −1.00000 −0.0589256
\(289\) −16.7893 −0.987605
\(290\) −3.68855 −0.216599
\(291\) 15.5491 0.911504
\(292\) −1.18160 −0.0691481
\(293\) 32.0087 1.86997 0.934985 0.354688i \(-0.115413\pi\)
0.934985 + 0.354688i \(0.115413\pi\)
\(294\) 17.6884 1.03161
\(295\) −2.99224 −0.174215
\(296\) 6.55603 0.381061
\(297\) −4.09511 −0.237623
\(298\) −6.42360 −0.372109
\(299\) 7.97226 0.461048
\(300\) 3.94221 0.227604
\(301\) −40.9780 −2.36193
\(302\) −12.5514 −0.722253
\(303\) 11.6401 0.668708
\(304\) 2.05940 0.118115
\(305\) −9.51191 −0.544650
\(306\) −0.459045 −0.0262419
\(307\) 8.88019 0.506819 0.253409 0.967359i \(-0.418448\pi\)
0.253409 + 0.967359i \(0.418448\pi\)
\(308\) −20.3476 −1.15941
\(309\) 1.00000 0.0568880
\(310\) 9.35253 0.531188
\(311\) 3.26270 0.185011 0.0925053 0.995712i \(-0.470512\pi\)
0.0925053 + 0.995712i \(0.470512\pi\)
\(312\) 1.00000 0.0566139
\(313\) 15.9412 0.901050 0.450525 0.892764i \(-0.351237\pi\)
0.450525 + 0.892764i \(0.351237\pi\)
\(314\) 16.7227 0.943717
\(315\) −5.11029 −0.287932
\(316\) 10.7487 0.604663
\(317\) 31.6414 1.77716 0.888578 0.458725i \(-0.151694\pi\)
0.888578 + 0.458725i \(0.151694\pi\)
\(318\) 5.49115 0.307928
\(319\) 14.6867 0.822295
\(320\) 1.02849 0.0574942
\(321\) −10.8455 −0.605339
\(322\) 39.6121 2.20750
\(323\) 0.945359 0.0526012
\(324\) 1.00000 0.0555556
\(325\) −3.94221 −0.218675
\(326\) −10.3522 −0.573358
\(327\) 16.0922 0.889903
\(328\) −5.24275 −0.289482
\(329\) −7.91895 −0.436586
\(330\) 4.21177 0.231851
\(331\) 23.7022 1.30279 0.651396 0.758738i \(-0.274184\pi\)
0.651396 + 0.758738i \(0.274184\pi\)
\(332\) −16.0640 −0.881627
\(333\) −6.55603 −0.359268
\(334\) 11.8605 0.648980
\(335\) −1.98246 −0.108313
\(336\) 4.96874 0.271067
\(337\) −28.6757 −1.56207 −0.781033 0.624490i \(-0.785307\pi\)
−0.781033 + 0.624490i \(0.785307\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 14.8459 0.806316
\(340\) 0.472123 0.0256044
\(341\) −37.2388 −2.01659
\(342\) −2.05940 −0.111360
\(343\) −53.1079 −2.86756
\(344\) −8.24715 −0.444657
\(345\) −8.19937 −0.441439
\(346\) −24.8743 −1.33725
\(347\) 11.0875 0.595208 0.297604 0.954689i \(-0.403812\pi\)
0.297604 + 0.954689i \(0.403812\pi\)
\(348\) −3.58639 −0.192250
\(349\) −1.21549 −0.0650636 −0.0325318 0.999471i \(-0.510357\pi\)
−0.0325318 + 0.999471i \(0.510357\pi\)
\(350\) −19.5878 −1.04701
\(351\) −1.00000 −0.0533761
\(352\) −4.09511 −0.218270
\(353\) 35.0241 1.86415 0.932074 0.362269i \(-0.117998\pi\)
0.932074 + 0.362269i \(0.117998\pi\)
\(354\) −2.90936 −0.154631
\(355\) 13.1407 0.697437
\(356\) 15.5497 0.824133
\(357\) 2.28088 0.120717
\(358\) −7.75139 −0.409674
\(359\) −1.78365 −0.0941373 −0.0470687 0.998892i \(-0.514988\pi\)
−0.0470687 + 0.998892i \(0.514988\pi\)
\(360\) −1.02849 −0.0542061
\(361\) −14.7589 −0.776782
\(362\) −17.7205 −0.931369
\(363\) −5.76995 −0.302844
\(364\) −4.96874 −0.260433
\(365\) −1.21526 −0.0636098
\(366\) −9.24844 −0.483424
\(367\) 28.0568 1.46455 0.732276 0.681008i \(-0.238458\pi\)
0.732276 + 0.681008i \(0.238458\pi\)
\(368\) 7.97226 0.415583
\(369\) 5.24275 0.272926
\(370\) 6.74279 0.350541
\(371\) −27.2841 −1.41652
\(372\) 9.09347 0.471475
\(373\) 11.2886 0.584501 0.292251 0.956342i \(-0.405596\pi\)
0.292251 + 0.956342i \(0.405596\pi\)
\(374\) −1.87984 −0.0972043
\(375\) 9.19696 0.474929
\(376\) −1.59375 −0.0821915
\(377\) 3.58639 0.184708
\(378\) −4.96874 −0.255564
\(379\) 27.1091 1.39250 0.696250 0.717799i \(-0.254850\pi\)
0.696250 + 0.717799i \(0.254850\pi\)
\(380\) 2.11807 0.108655
\(381\) −0.472953 −0.0242301
\(382\) −19.3541 −0.990244
\(383\) −14.4162 −0.736632 −0.368316 0.929701i \(-0.620066\pi\)
−0.368316 + 0.929701i \(0.620066\pi\)
\(384\) 1.00000 0.0510310
\(385\) −20.9272 −1.06655
\(386\) 3.45493 0.175851
\(387\) 8.24715 0.419226
\(388\) −15.5491 −0.789386
\(389\) −15.2367 −0.772531 −0.386266 0.922388i \(-0.626235\pi\)
−0.386266 + 0.922388i \(0.626235\pi\)
\(390\) 1.02849 0.0520795
\(391\) 3.65963 0.185075
\(392\) −17.6884 −0.893399
\(393\) −5.27085 −0.265879
\(394\) 8.07696 0.406911
\(395\) 11.0549 0.556234
\(396\) 4.09511 0.205787
\(397\) −31.4044 −1.57614 −0.788071 0.615585i \(-0.788920\pi\)
−0.788071 + 0.615585i \(0.788920\pi\)
\(398\) −10.6009 −0.531375
\(399\) 10.2326 0.512272
\(400\) −3.94221 −0.197111
\(401\) −8.39107 −0.419030 −0.209515 0.977805i \(-0.567189\pi\)
−0.209515 + 0.977805i \(0.567189\pi\)
\(402\) −1.92754 −0.0961372
\(403\) −9.09347 −0.452978
\(404\) −11.6401 −0.579118
\(405\) 1.02849 0.0511060
\(406\) 17.8198 0.884383
\(407\) −26.8477 −1.33079
\(408\) 0.459045 0.0227261
\(409\) −14.6020 −0.722022 −0.361011 0.932562i \(-0.617568\pi\)
−0.361011 + 0.932562i \(0.617568\pi\)
\(410\) −5.39210 −0.266297
\(411\) 3.17885 0.156801
\(412\) −1.00000 −0.0492665
\(413\) 14.4559 0.711327
\(414\) −7.97226 −0.391815
\(415\) −16.5216 −0.811016
\(416\) −1.00000 −0.0490290
\(417\) 0.950817 0.0465617
\(418\) −8.43348 −0.412495
\(419\) −34.2605 −1.67373 −0.836867 0.547406i \(-0.815615\pi\)
−0.836867 + 0.547406i \(0.815615\pi\)
\(420\) 5.11029 0.249357
\(421\) 35.7527 1.74248 0.871239 0.490858i \(-0.163317\pi\)
0.871239 + 0.490858i \(0.163317\pi\)
\(422\) 6.54374 0.318544
\(423\) 1.59375 0.0774909
\(424\) −5.49115 −0.266674
\(425\) −1.80965 −0.0877811
\(426\) 12.7767 0.619035
\(427\) 45.9531 2.22383
\(428\) 10.8455 0.524239
\(429\) −4.09511 −0.197714
\(430\) −8.48210 −0.409043
\(431\) −7.35529 −0.354292 −0.177146 0.984185i \(-0.556686\pi\)
−0.177146 + 0.984185i \(0.556686\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −15.3502 −0.737684 −0.368842 0.929492i \(-0.620246\pi\)
−0.368842 + 0.929492i \(0.620246\pi\)
\(434\) −45.1831 −2.16886
\(435\) −3.68855 −0.176853
\(436\) −16.0922 −0.770679
\(437\) 16.4181 0.785383
\(438\) −1.18160 −0.0564592
\(439\) 28.9661 1.38248 0.691238 0.722627i \(-0.257066\pi\)
0.691238 + 0.722627i \(0.257066\pi\)
\(440\) −4.21177 −0.200788
\(441\) 17.6884 0.842304
\(442\) −0.459045 −0.0218346
\(443\) 19.1078 0.907838 0.453919 0.891043i \(-0.350026\pi\)
0.453919 + 0.891043i \(0.350026\pi\)
\(444\) 6.55603 0.311135
\(445\) 15.9927 0.758126
\(446\) 5.38319 0.254902
\(447\) −6.42360 −0.303826
\(448\) −4.96874 −0.234751
\(449\) 13.3828 0.631575 0.315787 0.948830i \(-0.397731\pi\)
0.315787 + 0.948830i \(0.397731\pi\)
\(450\) 3.94221 0.185838
\(451\) 21.4696 1.01097
\(452\) −14.8459 −0.698291
\(453\) −12.5514 −0.589717
\(454\) 5.14503 0.241468
\(455\) −5.11029 −0.239574
\(456\) 2.05940 0.0964403
\(457\) 28.3863 1.32786 0.663928 0.747796i \(-0.268888\pi\)
0.663928 + 0.747796i \(0.268888\pi\)
\(458\) −14.1375 −0.660601
\(459\) −0.459045 −0.0214264
\(460\) 8.19937 0.382298
\(461\) 21.3334 0.993594 0.496797 0.867867i \(-0.334509\pi\)
0.496797 + 0.867867i \(0.334509\pi\)
\(462\) −20.3476 −0.946654
\(463\) 17.6985 0.822519 0.411259 0.911518i \(-0.365089\pi\)
0.411259 + 0.911518i \(0.365089\pi\)
\(464\) 3.58639 0.166494
\(465\) 9.35253 0.433713
\(466\) 8.36084 0.387309
\(467\) 2.08932 0.0966821 0.0483411 0.998831i \(-0.484607\pi\)
0.0483411 + 0.998831i \(0.484607\pi\)
\(468\) 1.00000 0.0462250
\(469\) 9.57747 0.442246
\(470\) −1.63916 −0.0756086
\(471\) 16.7227 0.770542
\(472\) 2.90936 0.133914
\(473\) 33.7730 1.55289
\(474\) 10.7487 0.493705
\(475\) −8.11860 −0.372507
\(476\) −2.28088 −0.104544
\(477\) 5.49115 0.251422
\(478\) −17.7749 −0.813006
\(479\) 42.6843 1.95030 0.975148 0.221552i \(-0.0711124\pi\)
0.975148 + 0.221552i \(0.0711124\pi\)
\(480\) 1.02849 0.0469438
\(481\) −6.55603 −0.298929
\(482\) 1.63447 0.0744479
\(483\) 39.6121 1.80241
\(484\) 5.76995 0.262270
\(485\) −15.9921 −0.726162
\(486\) 1.00000 0.0453609
\(487\) −10.8026 −0.489512 −0.244756 0.969585i \(-0.578708\pi\)
−0.244756 + 0.969585i \(0.578708\pi\)
\(488\) 9.24844 0.418657
\(489\) −10.3522 −0.468145
\(490\) −18.1923 −0.821844
\(491\) −3.24078 −0.146254 −0.0731271 0.997323i \(-0.523298\pi\)
−0.0731271 + 0.997323i \(0.523298\pi\)
\(492\) −5.24275 −0.236361
\(493\) 1.64631 0.0741462
\(494\) −2.05940 −0.0926569
\(495\) 4.21177 0.189305
\(496\) −9.09347 −0.408309
\(497\) −63.4843 −2.84766
\(498\) −16.0640 −0.719846
\(499\) 42.1087 1.88504 0.942521 0.334146i \(-0.108448\pi\)
0.942521 + 0.334146i \(0.108448\pi\)
\(500\) −9.19696 −0.411300
\(501\) 11.8605 0.529890
\(502\) 18.7438 0.836577
\(503\) 33.9257 1.51267 0.756336 0.654183i \(-0.226988\pi\)
0.756336 + 0.654183i \(0.226988\pi\)
\(504\) 4.96874 0.221325
\(505\) −11.9717 −0.532735
\(506\) −32.6473 −1.45135
\(507\) −1.00000 −0.0444116
\(508\) 0.472953 0.0209839
\(509\) 15.7968 0.700181 0.350090 0.936716i \(-0.386151\pi\)
0.350090 + 0.936716i \(0.386151\pi\)
\(510\) 0.472123 0.0209059
\(511\) 5.87108 0.259721
\(512\) −1.00000 −0.0441942
\(513\) −2.05940 −0.0909248
\(514\) 21.6529 0.955066
\(515\) −1.02849 −0.0453206
\(516\) −8.24715 −0.363061
\(517\) 6.52660 0.287039
\(518\) −32.5752 −1.43127
\(519\) −24.8743 −1.09186
\(520\) −1.02849 −0.0451022
\(521\) 4.30005 0.188388 0.0941942 0.995554i \(-0.469973\pi\)
0.0941942 + 0.995554i \(0.469973\pi\)
\(522\) −3.58639 −0.156972
\(523\) −12.3676 −0.540796 −0.270398 0.962749i \(-0.587155\pi\)
−0.270398 + 0.962749i \(0.587155\pi\)
\(524\) 5.27085 0.230258
\(525\) −19.5878 −0.854883
\(526\) −16.9718 −0.740008
\(527\) −4.17432 −0.181836
\(528\) −4.09511 −0.178217
\(529\) 40.5569 1.76334
\(530\) −5.64758 −0.245315
\(531\) −2.90936 −0.126256
\(532\) −10.2326 −0.443641
\(533\) 5.24275 0.227089
\(534\) 15.5497 0.672902
\(535\) 11.1545 0.482251
\(536\) 1.92754 0.0832572
\(537\) −7.75139 −0.334497
\(538\) −5.09786 −0.219784
\(539\) 72.4360 3.12004
\(540\) −1.02849 −0.0442591
\(541\) 33.6022 1.44467 0.722336 0.691542i \(-0.243068\pi\)
0.722336 + 0.691542i \(0.243068\pi\)
\(542\) 25.2640 1.08518
\(543\) −17.7205 −0.760459
\(544\) −0.459045 −0.0196814
\(545\) −16.5507 −0.708953
\(546\) −4.96874 −0.212643
\(547\) 24.7409 1.05785 0.528923 0.848670i \(-0.322596\pi\)
0.528923 + 0.848670i \(0.322596\pi\)
\(548\) −3.17885 −0.135794
\(549\) −9.24844 −0.394714
\(550\) 16.1438 0.688374
\(551\) 7.38581 0.314646
\(552\) 7.97226 0.339322
\(553\) −53.4077 −2.27112
\(554\) −30.4540 −1.29387
\(555\) 6.74279 0.286216
\(556\) −0.950817 −0.0403236
\(557\) 4.47790 0.189735 0.0948674 0.995490i \(-0.469757\pi\)
0.0948674 + 0.995490i \(0.469757\pi\)
\(558\) 9.09347 0.384957
\(559\) 8.24715 0.348817
\(560\) −5.11029 −0.215949
\(561\) −1.87984 −0.0793670
\(562\) 21.9475 0.925798
\(563\) −3.44667 −0.145260 −0.0726298 0.997359i \(-0.523139\pi\)
−0.0726298 + 0.997359i \(0.523139\pi\)
\(564\) −1.59375 −0.0671091
\(565\) −15.2688 −0.642363
\(566\) 14.0193 0.589275
\(567\) −4.96874 −0.208668
\(568\) −12.7767 −0.536100
\(569\) −15.1304 −0.634301 −0.317151 0.948375i \(-0.602726\pi\)
−0.317151 + 0.948375i \(0.602726\pi\)
\(570\) 2.11807 0.0887162
\(571\) 14.3366 0.599970 0.299985 0.953944i \(-0.403018\pi\)
0.299985 + 0.953944i \(0.403018\pi\)
\(572\) 4.09511 0.171225
\(573\) −19.3541 −0.808531
\(574\) 26.0498 1.08730
\(575\) −31.4283 −1.31065
\(576\) 1.00000 0.0416667
\(577\) −29.2586 −1.21805 −0.609025 0.793151i \(-0.708439\pi\)
−0.609025 + 0.793151i \(0.708439\pi\)
\(578\) 16.7893 0.698342
\(579\) 3.45493 0.143582
\(580\) 3.68855 0.153159
\(581\) 79.8179 3.31141
\(582\) −15.5491 −0.644531
\(583\) 22.4869 0.931311
\(584\) 1.18160 0.0488951
\(585\) 1.02849 0.0425227
\(586\) −32.0087 −1.32227
\(587\) −34.5674 −1.42675 −0.713374 0.700783i \(-0.752834\pi\)
−0.713374 + 0.700783i \(0.752834\pi\)
\(588\) −17.6884 −0.729457
\(589\) −18.7271 −0.771637
\(590\) 2.99224 0.123189
\(591\) 8.07696 0.332242
\(592\) −6.55603 −0.269451
\(593\) 44.3419 1.82090 0.910452 0.413615i \(-0.135734\pi\)
0.910452 + 0.413615i \(0.135734\pi\)
\(594\) 4.09511 0.168025
\(595\) −2.34585 −0.0961707
\(596\) 6.42360 0.263121
\(597\) −10.6009 −0.433865
\(598\) −7.97226 −0.326010
\(599\) −5.66048 −0.231281 −0.115641 0.993291i \(-0.536892\pi\)
−0.115641 + 0.993291i \(0.536892\pi\)
\(600\) −3.94221 −0.160940
\(601\) 27.2086 1.10986 0.554931 0.831896i \(-0.312745\pi\)
0.554931 + 0.831896i \(0.312745\pi\)
\(602\) 40.9780 1.67014
\(603\) −1.92754 −0.0784957
\(604\) 12.5514 0.510710
\(605\) 5.93432 0.241264
\(606\) −11.6401 −0.472848
\(607\) 27.1662 1.10264 0.551321 0.834293i \(-0.314124\pi\)
0.551321 + 0.834293i \(0.314124\pi\)
\(608\) −2.05940 −0.0835198
\(609\) 17.8198 0.722096
\(610\) 9.51191 0.385126
\(611\) 1.59375 0.0644763
\(612\) 0.459045 0.0185558
\(613\) 2.06093 0.0832401 0.0416200 0.999134i \(-0.486748\pi\)
0.0416200 + 0.999134i \(0.486748\pi\)
\(614\) −8.88019 −0.358375
\(615\) −5.39210 −0.217430
\(616\) 20.3476 0.819826
\(617\) 3.32466 0.133846 0.0669229 0.997758i \(-0.478682\pi\)
0.0669229 + 0.997758i \(0.478682\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −10.2630 −0.412506 −0.206253 0.978499i \(-0.566127\pi\)
−0.206253 + 0.978499i \(0.566127\pi\)
\(620\) −9.35253 −0.375607
\(621\) −7.97226 −0.319916
\(622\) −3.26270 −0.130822
\(623\) −77.2625 −3.09546
\(624\) −1.00000 −0.0400320
\(625\) 10.2521 0.410084
\(626\) −15.9412 −0.637138
\(627\) −8.43348 −0.336801
\(628\) −16.7227 −0.667309
\(629\) −3.00951 −0.119997
\(630\) 5.11029 0.203599
\(631\) −32.1948 −1.28165 −0.640827 0.767685i \(-0.721409\pi\)
−0.640827 + 0.767685i \(0.721409\pi\)
\(632\) −10.7487 −0.427561
\(633\) 6.54374 0.260090
\(634\) −31.6414 −1.25664
\(635\) 0.486426 0.0193032
\(636\) −5.49115 −0.217738
\(637\) 17.6884 0.700840
\(638\) −14.6867 −0.581450
\(639\) 12.7767 0.505440
\(640\) −1.02849 −0.0406546
\(641\) 13.5087 0.533560 0.266780 0.963757i \(-0.414040\pi\)
0.266780 + 0.963757i \(0.414040\pi\)
\(642\) 10.8455 0.428039
\(643\) −19.4901 −0.768613 −0.384307 0.923205i \(-0.625559\pi\)
−0.384307 + 0.923205i \(0.625559\pi\)
\(644\) −39.6121 −1.56093
\(645\) −8.48210 −0.333982
\(646\) −0.945359 −0.0371946
\(647\) 44.1340 1.73509 0.867544 0.497361i \(-0.165697\pi\)
0.867544 + 0.497361i \(0.165697\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −11.9142 −0.467672
\(650\) 3.94221 0.154626
\(651\) −45.1831 −1.77087
\(652\) 10.3522 0.405425
\(653\) −41.1137 −1.60890 −0.804452 0.594017i \(-0.797541\pi\)
−0.804452 + 0.594017i \(0.797541\pi\)
\(654\) −16.0922 −0.629257
\(655\) 5.42101 0.211816
\(656\) 5.24275 0.204695
\(657\) −1.18160 −0.0460987
\(658\) 7.91895 0.308713
\(659\) 46.5601 1.81372 0.906861 0.421429i \(-0.138471\pi\)
0.906861 + 0.421429i \(0.138471\pi\)
\(660\) −4.21177 −0.163943
\(661\) −37.4969 −1.45846 −0.729230 0.684269i \(-0.760122\pi\)
−0.729230 + 0.684269i \(0.760122\pi\)
\(662\) −23.7022 −0.921214
\(663\) −0.459045 −0.0178278
\(664\) 16.0640 0.623405
\(665\) −10.5241 −0.408109
\(666\) 6.55603 0.254041
\(667\) 28.5916 1.10707
\(668\) −11.8605 −0.458898
\(669\) 5.38319 0.208126
\(670\) 1.98246 0.0765889
\(671\) −37.8734 −1.46209
\(672\) −4.96874 −0.191673
\(673\) 11.2045 0.431903 0.215951 0.976404i \(-0.430715\pi\)
0.215951 + 0.976404i \(0.430715\pi\)
\(674\) 28.6757 1.10455
\(675\) 3.94221 0.151736
\(676\) 1.00000 0.0384615
\(677\) −49.1615 −1.88943 −0.944716 0.327890i \(-0.893662\pi\)
−0.944716 + 0.327890i \(0.893662\pi\)
\(678\) −14.8459 −0.570152
\(679\) 77.2595 2.96495
\(680\) −0.472123 −0.0181051
\(681\) 5.14503 0.197158
\(682\) 37.2388 1.42595
\(683\) −20.0565 −0.767440 −0.383720 0.923449i \(-0.625357\pi\)
−0.383720 + 0.923449i \(0.625357\pi\)
\(684\) 2.05940 0.0787432
\(685\) −3.26940 −0.124918
\(686\) 53.1079 2.02767
\(687\) −14.1375 −0.539379
\(688\) 8.24715 0.314420
\(689\) 5.49115 0.209196
\(690\) 8.19937 0.312145
\(691\) 15.0158 0.571227 0.285614 0.958345i \(-0.407803\pi\)
0.285614 + 0.958345i \(0.407803\pi\)
\(692\) 24.8743 0.945581
\(693\) −20.3476 −0.772940
\(694\) −11.0875 −0.420876
\(695\) −0.977904 −0.0370940
\(696\) 3.58639 0.135942
\(697\) 2.40666 0.0911587
\(698\) 1.21549 0.0460069
\(699\) 8.36084 0.316236
\(700\) 19.5878 0.740351
\(701\) 38.1436 1.44066 0.720331 0.693630i \(-0.243990\pi\)
0.720331 + 0.693630i \(0.243990\pi\)
\(702\) 1.00000 0.0377426
\(703\) −13.5015 −0.509218
\(704\) 4.09511 0.154340
\(705\) −1.63916 −0.0617342
\(706\) −35.0241 −1.31815
\(707\) 57.8368 2.17518
\(708\) 2.90936 0.109341
\(709\) 5.87299 0.220565 0.110282 0.993900i \(-0.464824\pi\)
0.110282 + 0.993900i \(0.464824\pi\)
\(710\) −13.1407 −0.493162
\(711\) 10.7487 0.403109
\(712\) −15.5497 −0.582750
\(713\) −72.4955 −2.71498
\(714\) −2.28088 −0.0853597
\(715\) 4.21177 0.157511
\(716\) 7.75139 0.289683
\(717\) −17.7749 −0.663816
\(718\) 1.78365 0.0665651
\(719\) −5.49463 −0.204915 −0.102458 0.994737i \(-0.532671\pi\)
−0.102458 + 0.994737i \(0.532671\pi\)
\(720\) 1.02849 0.0383295
\(721\) 4.96874 0.185046
\(722\) 14.7589 0.549268
\(723\) 1.63447 0.0607864
\(724\) 17.7205 0.658577
\(725\) −14.1383 −0.525083
\(726\) 5.76995 0.214143
\(727\) 16.2167 0.601446 0.300723 0.953712i \(-0.402772\pi\)
0.300723 + 0.953712i \(0.402772\pi\)
\(728\) 4.96874 0.184154
\(729\) 1.00000 0.0370370
\(730\) 1.21526 0.0449789
\(731\) 3.78582 0.140024
\(732\) 9.24844 0.341832
\(733\) −35.8501 −1.32415 −0.662076 0.749436i \(-0.730325\pi\)
−0.662076 + 0.749436i \(0.730325\pi\)
\(734\) −28.0568 −1.03559
\(735\) −18.1923 −0.671033
\(736\) −7.97226 −0.293861
\(737\) −7.89351 −0.290761
\(738\) −5.24275 −0.192988
\(739\) 11.4133 0.419846 0.209923 0.977718i \(-0.432679\pi\)
0.209923 + 0.977718i \(0.432679\pi\)
\(740\) −6.74279 −0.247870
\(741\) −2.05940 −0.0756540
\(742\) 27.2841 1.00163
\(743\) −39.2609 −1.44034 −0.720172 0.693795i \(-0.755937\pi\)
−0.720172 + 0.693795i \(0.755937\pi\)
\(744\) −9.09347 −0.333383
\(745\) 6.60660 0.242047
\(746\) −11.2886 −0.413305
\(747\) −16.0640 −0.587752
\(748\) 1.87984 0.0687338
\(749\) −53.8887 −1.96905
\(750\) −9.19696 −0.335825
\(751\) −27.9898 −1.02136 −0.510681 0.859770i \(-0.670607\pi\)
−0.510681 + 0.859770i \(0.670607\pi\)
\(752\) 1.59375 0.0581182
\(753\) 18.7438 0.683062
\(754\) −3.58639 −0.130608
\(755\) 12.9090 0.469806
\(756\) 4.96874 0.180711
\(757\) 34.3539 1.24862 0.624308 0.781179i \(-0.285381\pi\)
0.624308 + 0.781179i \(0.285381\pi\)
\(758\) −27.1091 −0.984647
\(759\) −32.6473 −1.18502
\(760\) −2.11807 −0.0768305
\(761\) −34.5760 −1.25338 −0.626689 0.779269i \(-0.715591\pi\)
−0.626689 + 0.779269i \(0.715591\pi\)
\(762\) 0.472953 0.0171333
\(763\) 79.9582 2.89468
\(764\) 19.3541 0.700209
\(765\) 0.472123 0.0170696
\(766\) 14.4162 0.520878
\(767\) −2.90936 −0.105051
\(768\) −1.00000 −0.0360844
\(769\) −48.7406 −1.75763 −0.878816 0.477161i \(-0.841666\pi\)
−0.878816 + 0.477161i \(0.841666\pi\)
\(770\) 20.9272 0.754164
\(771\) 21.6529 0.779808
\(772\) −3.45493 −0.124346
\(773\) −24.0093 −0.863556 −0.431778 0.901980i \(-0.642114\pi\)
−0.431778 + 0.901980i \(0.642114\pi\)
\(774\) −8.24715 −0.296438
\(775\) 35.8484 1.28771
\(776\) 15.5491 0.558180
\(777\) −32.5752 −1.16863
\(778\) 15.2367 0.546262
\(779\) 10.7969 0.386840
\(780\) −1.02849 −0.0368258
\(781\) 52.3222 1.87223
\(782\) −3.65963 −0.130868
\(783\) −3.58639 −0.128167
\(784\) 17.6884 0.631728
\(785\) −17.1991 −0.613863
\(786\) 5.27085 0.188005
\(787\) 42.5645 1.51726 0.758630 0.651522i \(-0.225869\pi\)
0.758630 + 0.651522i \(0.225869\pi\)
\(788\) −8.07696 −0.287730
\(789\) −16.9718 −0.604214
\(790\) −11.0549 −0.393317
\(791\) 73.7652 2.62279
\(792\) −4.09511 −0.145513
\(793\) −9.24844 −0.328422
\(794\) 31.4044 1.11450
\(795\) −5.64758 −0.200299
\(796\) 10.6009 0.375739
\(797\) −31.2353 −1.10641 −0.553205 0.833045i \(-0.686595\pi\)
−0.553205 + 0.833045i \(0.686595\pi\)
\(798\) −10.2326 −0.362231
\(799\) 0.731605 0.0258823
\(800\) 3.94221 0.139378
\(801\) 15.5497 0.549422
\(802\) 8.39107 0.296299
\(803\) −4.83880 −0.170757
\(804\) 1.92754 0.0679792
\(805\) −40.7406 −1.43592
\(806\) 9.09347 0.320304
\(807\) −5.09786 −0.179453
\(808\) 11.6401 0.409498
\(809\) −21.3621 −0.751050 −0.375525 0.926812i \(-0.622538\pi\)
−0.375525 + 0.926812i \(0.622538\pi\)
\(810\) −1.02849 −0.0361374
\(811\) 37.5767 1.31949 0.659747 0.751487i \(-0.270663\pi\)
0.659747 + 0.751487i \(0.270663\pi\)
\(812\) −17.8198 −0.625353
\(813\) 25.2640 0.886046
\(814\) 26.8477 0.941010
\(815\) 10.6472 0.372954
\(816\) −0.459045 −0.0160698
\(817\) 16.9842 0.594202
\(818\) 14.6020 0.510546
\(819\) −4.96874 −0.173622
\(820\) 5.39210 0.188300
\(821\) 16.8858 0.589320 0.294660 0.955602i \(-0.404794\pi\)
0.294660 + 0.955602i \(0.404794\pi\)
\(822\) −3.17885 −0.110875
\(823\) 21.1535 0.737363 0.368682 0.929556i \(-0.379809\pi\)
0.368682 + 0.929556i \(0.379809\pi\)
\(824\) 1.00000 0.0348367
\(825\) 16.1438 0.562055
\(826\) −14.4559 −0.502984
\(827\) 28.1599 0.979215 0.489607 0.871943i \(-0.337140\pi\)
0.489607 + 0.871943i \(0.337140\pi\)
\(828\) 7.97226 0.277055
\(829\) 37.7971 1.31275 0.656374 0.754436i \(-0.272090\pi\)
0.656374 + 0.754436i \(0.272090\pi\)
\(830\) 16.5216 0.573475
\(831\) −30.4540 −1.05644
\(832\) 1.00000 0.0346688
\(833\) 8.11977 0.281334
\(834\) −0.950817 −0.0329241
\(835\) −12.1984 −0.422144
\(836\) 8.43348 0.291678
\(837\) 9.09347 0.314316
\(838\) 34.2605 1.18351
\(839\) −26.5497 −0.916597 −0.458298 0.888798i \(-0.651541\pi\)
−0.458298 + 0.888798i \(0.651541\pi\)
\(840\) −5.11029 −0.176322
\(841\) −16.1378 −0.556477
\(842\) −35.7527 −1.23212
\(843\) 21.9475 0.755911
\(844\) −6.54374 −0.225245
\(845\) 1.02849 0.0353811
\(846\) −1.59375 −0.0547944
\(847\) −28.6694 −0.985091
\(848\) 5.49115 0.188567
\(849\) 14.0193 0.481141
\(850\) 1.80965 0.0620706
\(851\) −52.2663 −1.79167
\(852\) −12.7767 −0.437724
\(853\) 15.7644 0.539764 0.269882 0.962893i \(-0.413015\pi\)
0.269882 + 0.962893i \(0.413015\pi\)
\(854\) −45.9531 −1.57248
\(855\) 2.11807 0.0724364
\(856\) −10.8455 −0.370693
\(857\) 15.3382 0.523944 0.261972 0.965075i \(-0.415627\pi\)
0.261972 + 0.965075i \(0.415627\pi\)
\(858\) 4.09511 0.139805
\(859\) 38.6477 1.31864 0.659322 0.751861i \(-0.270843\pi\)
0.659322 + 0.751861i \(0.270843\pi\)
\(860\) 8.48210 0.289237
\(861\) 26.0498 0.887776
\(862\) 7.35529 0.250522
\(863\) −2.61626 −0.0890585 −0.0445293 0.999008i \(-0.514179\pi\)
−0.0445293 + 0.999008i \(0.514179\pi\)
\(864\) 1.00000 0.0340207
\(865\) 25.5830 0.869847
\(866\) 15.3502 0.521621
\(867\) 16.7893 0.570194
\(868\) 45.1831 1.53361
\(869\) 44.0173 1.49318
\(870\) 3.68855 0.125054
\(871\) −1.92754 −0.0653123
\(872\) 16.0922 0.544952
\(873\) −15.5491 −0.526257
\(874\) −16.4181 −0.555350
\(875\) 45.6973 1.54485
\(876\) 1.18160 0.0399226
\(877\) 53.7139 1.81379 0.906895 0.421356i \(-0.138446\pi\)
0.906895 + 0.421356i \(0.138446\pi\)
\(878\) −28.9661 −0.977558
\(879\) −32.0087 −1.07963
\(880\) 4.21177 0.141979
\(881\) 20.3295 0.684917 0.342459 0.939533i \(-0.388740\pi\)
0.342459 + 0.939533i \(0.388740\pi\)
\(882\) −17.6884 −0.595599
\(883\) 2.47317 0.0832289 0.0416145 0.999134i \(-0.486750\pi\)
0.0416145 + 0.999134i \(0.486750\pi\)
\(884\) 0.459045 0.0154394
\(885\) 2.99224 0.100583
\(886\) −19.1078 −0.641938
\(887\) 47.5265 1.59578 0.797892 0.602800i \(-0.205948\pi\)
0.797892 + 0.602800i \(0.205948\pi\)
\(888\) −6.55603 −0.220006
\(889\) −2.34998 −0.0788158
\(890\) −15.9927 −0.536076
\(891\) 4.09511 0.137191
\(892\) −5.38319 −0.180243
\(893\) 3.28218 0.109834
\(894\) 6.42360 0.214837
\(895\) 7.97221 0.266482
\(896\) 4.96874 0.165994
\(897\) −7.97226 −0.266186
\(898\) −13.3828 −0.446591
\(899\) −32.6127 −1.08769
\(900\) −3.94221 −0.131407
\(901\) 2.52068 0.0839762
\(902\) −21.4696 −0.714861
\(903\) 40.9780 1.36366
\(904\) 14.8459 0.493766
\(905\) 18.2253 0.605830
\(906\) 12.5514 0.416993
\(907\) −42.7178 −1.41842 −0.709211 0.704996i \(-0.750949\pi\)
−0.709211 + 0.704996i \(0.750949\pi\)
\(908\) −5.14503 −0.170744
\(909\) −11.6401 −0.386078
\(910\) 5.11029 0.169404
\(911\) −14.2876 −0.473368 −0.236684 0.971587i \(-0.576061\pi\)
−0.236684 + 0.971587i \(0.576061\pi\)
\(912\) −2.05940 −0.0681936
\(913\) −65.7839 −2.17713
\(914\) −28.3863 −0.938936
\(915\) 9.51191 0.314454
\(916\) 14.1375 0.467116
\(917\) −26.1895 −0.864854
\(918\) 0.459045 0.0151508
\(919\) −17.1869 −0.566944 −0.283472 0.958980i \(-0.591486\pi\)
−0.283472 + 0.958980i \(0.591486\pi\)
\(920\) −8.19937 −0.270325
\(921\) −8.88019 −0.292612
\(922\) −21.3334 −0.702577
\(923\) 12.7767 0.420551
\(924\) 20.3476 0.669385
\(925\) 25.8453 0.849787
\(926\) −17.6985 −0.581609
\(927\) −1.00000 −0.0328443
\(928\) −3.58639 −0.117729
\(929\) 18.2463 0.598642 0.299321 0.954152i \(-0.403240\pi\)
0.299321 + 0.954152i \(0.403240\pi\)
\(930\) −9.35253 −0.306681
\(931\) 36.4275 1.19386
\(932\) −8.36084 −0.273869
\(933\) −3.26270 −0.106816
\(934\) −2.08932 −0.0683646
\(935\) 1.93339 0.0632288
\(936\) −1.00000 −0.0326860
\(937\) 21.6826 0.708339 0.354169 0.935181i \(-0.384764\pi\)
0.354169 + 0.935181i \(0.384764\pi\)
\(938\) −9.57747 −0.312715
\(939\) −15.9412 −0.520221
\(940\) 1.63916 0.0534634
\(941\) 54.5765 1.77914 0.889571 0.456796i \(-0.151003\pi\)
0.889571 + 0.456796i \(0.151003\pi\)
\(942\) −16.7227 −0.544856
\(943\) 41.7965 1.36108
\(944\) −2.90936 −0.0946917
\(945\) 5.11029 0.166238
\(946\) −33.7730 −1.09806
\(947\) −32.0403 −1.04117 −0.520585 0.853810i \(-0.674286\pi\)
−0.520585 + 0.853810i \(0.674286\pi\)
\(948\) −10.7487 −0.349102
\(949\) −1.18160 −0.0383564
\(950\) 8.11860 0.263402
\(951\) −31.6414 −1.02604
\(952\) 2.28088 0.0739237
\(953\) 52.5012 1.70068 0.850341 0.526232i \(-0.176396\pi\)
0.850341 + 0.526232i \(0.176396\pi\)
\(954\) −5.49115 −0.177782
\(955\) 19.9055 0.644127
\(956\) 17.7749 0.574882
\(957\) −14.6867 −0.474752
\(958\) −42.6843 −1.37907
\(959\) 15.7949 0.510043
\(960\) −1.02849 −0.0331943
\(961\) 51.6913 1.66746
\(962\) 6.55603 0.211375
\(963\) 10.8455 0.349492
\(964\) −1.63447 −0.0526426
\(965\) −3.55335 −0.114386
\(966\) −39.6121 −1.27450
\(967\) −2.86221 −0.0920424 −0.0460212 0.998940i \(-0.514654\pi\)
−0.0460212 + 0.998940i \(0.514654\pi\)
\(968\) −5.76995 −0.185453
\(969\) −0.945359 −0.0303693
\(970\) 15.9921 0.513474
\(971\) −0.791041 −0.0253857 −0.0126928 0.999919i \(-0.504040\pi\)
−0.0126928 + 0.999919i \(0.504040\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 4.72437 0.151456
\(974\) 10.8026 0.346137
\(975\) 3.94221 0.126252
\(976\) −9.24844 −0.296035
\(977\) −1.03227 −0.0330252 −0.0165126 0.999864i \(-0.505256\pi\)
−0.0165126 + 0.999864i \(0.505256\pi\)
\(978\) 10.3522 0.331028
\(979\) 63.6778 2.03515
\(980\) 18.1923 0.581132
\(981\) −16.0922 −0.513786
\(982\) 3.24078 0.103417
\(983\) 37.2289 1.18742 0.593709 0.804679i \(-0.297663\pi\)
0.593709 + 0.804679i \(0.297663\pi\)
\(984\) 5.24275 0.167133
\(985\) −8.30705 −0.264685
\(986\) −1.64631 −0.0524293
\(987\) 7.91895 0.252063
\(988\) 2.05940 0.0655183
\(989\) 65.7484 2.09068
\(990\) −4.21177 −0.133859
\(991\) 27.2031 0.864134 0.432067 0.901842i \(-0.357784\pi\)
0.432067 + 0.901842i \(0.357784\pi\)
\(992\) 9.09347 0.288718
\(993\) −23.7022 −0.752168
\(994\) 63.4843 2.01360
\(995\) 10.9029 0.345645
\(996\) 16.0640 0.509008
\(997\) 18.8700 0.597620 0.298810 0.954313i \(-0.403410\pi\)
0.298810 + 0.954313i \(0.403410\pi\)
\(998\) −42.1087 −1.33293
\(999\) 6.55603 0.207423
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.z.1.9 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.z.1.9 14 1.1 even 1 trivial