Properties

Label 8034.2.a.ba.1.14
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: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(-3.78531\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.78531 q^{5} +1.00000 q^{6} -3.09768 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.78531 q^{5} +1.00000 q^{6} -3.09768 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.78531 q^{10} -1.99141 q^{11} -1.00000 q^{12} -1.00000 q^{13} +3.09768 q^{14} -3.78531 q^{15} +1.00000 q^{16} -7.51222 q^{17} -1.00000 q^{18} +6.13877 q^{19} +3.78531 q^{20} +3.09768 q^{21} +1.99141 q^{22} +6.69447 q^{23} +1.00000 q^{24} +9.32855 q^{25} +1.00000 q^{26} -1.00000 q^{27} -3.09768 q^{28} +4.16226 q^{29} +3.78531 q^{30} +3.05022 q^{31} -1.00000 q^{32} +1.99141 q^{33} +7.51222 q^{34} -11.7257 q^{35} +1.00000 q^{36} -0.695449 q^{37} -6.13877 q^{38} +1.00000 q^{39} -3.78531 q^{40} +6.91346 q^{41} -3.09768 q^{42} +5.14209 q^{43} -1.99141 q^{44} +3.78531 q^{45} -6.69447 q^{46} -8.19852 q^{47} -1.00000 q^{48} +2.59560 q^{49} -9.32855 q^{50} +7.51222 q^{51} -1.00000 q^{52} -5.97715 q^{53} +1.00000 q^{54} -7.53812 q^{55} +3.09768 q^{56} -6.13877 q^{57} -4.16226 q^{58} +5.70653 q^{59} -3.78531 q^{60} -9.33645 q^{61} -3.05022 q^{62} -3.09768 q^{63} +1.00000 q^{64} -3.78531 q^{65} -1.99141 q^{66} +0.628166 q^{67} -7.51222 q^{68} -6.69447 q^{69} +11.7257 q^{70} -6.41206 q^{71} -1.00000 q^{72} -5.18377 q^{73} +0.695449 q^{74} -9.32855 q^{75} +6.13877 q^{76} +6.16876 q^{77} -1.00000 q^{78} -5.82259 q^{79} +3.78531 q^{80} +1.00000 q^{81} -6.91346 q^{82} +15.0761 q^{83} +3.09768 q^{84} -28.4361 q^{85} -5.14209 q^{86} -4.16226 q^{87} +1.99141 q^{88} -17.5821 q^{89} -3.78531 q^{90} +3.09768 q^{91} +6.69447 q^{92} -3.05022 q^{93} +8.19852 q^{94} +23.2371 q^{95} +1.00000 q^{96} +2.07965 q^{97} -2.59560 q^{98} -1.99141 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

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
\(5\) 3.78531 1.69284 0.846420 0.532515i \(-0.178753\pi\)
0.846420 + 0.532515i \(0.178753\pi\)
\(6\) 1.00000 0.408248
\(7\) −3.09768 −1.17081 −0.585406 0.810740i \(-0.699065\pi\)
−0.585406 + 0.810740i \(0.699065\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.78531 −1.19702
\(11\) −1.99141 −0.600434 −0.300217 0.953871i \(-0.597059\pi\)
−0.300217 + 0.953871i \(0.597059\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 3.09768 0.827889
\(15\) −3.78531 −0.977362
\(16\) 1.00000 0.250000
\(17\) −7.51222 −1.82198 −0.910991 0.412427i \(-0.864681\pi\)
−0.910991 + 0.412427i \(0.864681\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.13877 1.40833 0.704165 0.710036i \(-0.251321\pi\)
0.704165 + 0.710036i \(0.251321\pi\)
\(20\) 3.78531 0.846420
\(21\) 3.09768 0.675969
\(22\) 1.99141 0.424571
\(23\) 6.69447 1.39589 0.697947 0.716149i \(-0.254097\pi\)
0.697947 + 0.716149i \(0.254097\pi\)
\(24\) 1.00000 0.204124
\(25\) 9.32855 1.86571
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −3.09768 −0.585406
\(29\) 4.16226 0.772913 0.386457 0.922308i \(-0.373699\pi\)
0.386457 + 0.922308i \(0.373699\pi\)
\(30\) 3.78531 0.691099
\(31\) 3.05022 0.547836 0.273918 0.961753i \(-0.411680\pi\)
0.273918 + 0.961753i \(0.411680\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.99141 0.346661
\(34\) 7.51222 1.28834
\(35\) −11.7257 −1.98200
\(36\) 1.00000 0.166667
\(37\) −0.695449 −0.114331 −0.0571656 0.998365i \(-0.518206\pi\)
−0.0571656 + 0.998365i \(0.518206\pi\)
\(38\) −6.13877 −0.995840
\(39\) 1.00000 0.160128
\(40\) −3.78531 −0.598510
\(41\) 6.91346 1.07970 0.539850 0.841761i \(-0.318481\pi\)
0.539850 + 0.841761i \(0.318481\pi\)
\(42\) −3.09768 −0.477982
\(43\) 5.14209 0.784162 0.392081 0.919931i \(-0.371755\pi\)
0.392081 + 0.919931i \(0.371755\pi\)
\(44\) −1.99141 −0.300217
\(45\) 3.78531 0.564280
\(46\) −6.69447 −0.987046
\(47\) −8.19852 −1.19588 −0.597938 0.801542i \(-0.704013\pi\)
−0.597938 + 0.801542i \(0.704013\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.59560 0.370800
\(50\) −9.32855 −1.31926
\(51\) 7.51222 1.05192
\(52\) −1.00000 −0.138675
\(53\) −5.97715 −0.821025 −0.410512 0.911855i \(-0.634650\pi\)
−0.410512 + 0.911855i \(0.634650\pi\)
\(54\) 1.00000 0.136083
\(55\) −7.53812 −1.01644
\(56\) 3.09768 0.413944
\(57\) −6.13877 −0.813100
\(58\) −4.16226 −0.546532
\(59\) 5.70653 0.742927 0.371463 0.928448i \(-0.378856\pi\)
0.371463 + 0.928448i \(0.378856\pi\)
\(60\) −3.78531 −0.488681
\(61\) −9.33645 −1.19541 −0.597705 0.801716i \(-0.703921\pi\)
−0.597705 + 0.801716i \(0.703921\pi\)
\(62\) −3.05022 −0.387378
\(63\) −3.09768 −0.390271
\(64\) 1.00000 0.125000
\(65\) −3.78531 −0.469510
\(66\) −1.99141 −0.245126
\(67\) 0.628166 0.0767427 0.0383713 0.999264i \(-0.487783\pi\)
0.0383713 + 0.999264i \(0.487783\pi\)
\(68\) −7.51222 −0.910991
\(69\) −6.69447 −0.805920
\(70\) 11.7257 1.40148
\(71\) −6.41206 −0.760972 −0.380486 0.924787i \(-0.624243\pi\)
−0.380486 + 0.924787i \(0.624243\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.18377 −0.606714 −0.303357 0.952877i \(-0.598108\pi\)
−0.303357 + 0.952877i \(0.598108\pi\)
\(74\) 0.695449 0.0808443
\(75\) −9.32855 −1.07717
\(76\) 6.13877 0.704165
\(77\) 6.16876 0.702995
\(78\) −1.00000 −0.113228
\(79\) −5.82259 −0.655093 −0.327547 0.944835i \(-0.606222\pi\)
−0.327547 + 0.944835i \(0.606222\pi\)
\(80\) 3.78531 0.423210
\(81\) 1.00000 0.111111
\(82\) −6.91346 −0.763463
\(83\) 15.0761 1.65482 0.827408 0.561601i \(-0.189814\pi\)
0.827408 + 0.561601i \(0.189814\pi\)
\(84\) 3.09768 0.337984
\(85\) −28.4361 −3.08432
\(86\) −5.14209 −0.554486
\(87\) −4.16226 −0.446242
\(88\) 1.99141 0.212285
\(89\) −17.5821 −1.86370 −0.931848 0.362848i \(-0.881805\pi\)
−0.931848 + 0.362848i \(0.881805\pi\)
\(90\) −3.78531 −0.399006
\(91\) 3.09768 0.324725
\(92\) 6.69447 0.697947
\(93\) −3.05022 −0.316293
\(94\) 8.19852 0.845612
\(95\) 23.2371 2.38408
\(96\) 1.00000 0.102062
\(97\) 2.07965 0.211157 0.105578 0.994411i \(-0.466331\pi\)
0.105578 + 0.994411i \(0.466331\pi\)
\(98\) −2.59560 −0.262195
\(99\) −1.99141 −0.200145
\(100\) 9.32855 0.932855
\(101\) 8.79029 0.874666 0.437333 0.899300i \(-0.355923\pi\)
0.437333 + 0.899300i \(0.355923\pi\)
\(102\) −7.51222 −0.743821
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 11.7257 1.14431
\(106\) 5.97715 0.580552
\(107\) 9.62233 0.930226 0.465113 0.885251i \(-0.346014\pi\)
0.465113 + 0.885251i \(0.346014\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −0.639705 −0.0612726 −0.0306363 0.999531i \(-0.509753\pi\)
−0.0306363 + 0.999531i \(0.509753\pi\)
\(110\) 7.53812 0.718731
\(111\) 0.695449 0.0660091
\(112\) −3.09768 −0.292703
\(113\) −3.39898 −0.319750 −0.159875 0.987137i \(-0.551109\pi\)
−0.159875 + 0.987137i \(0.551109\pi\)
\(114\) 6.13877 0.574948
\(115\) 25.3406 2.36303
\(116\) 4.16226 0.386457
\(117\) −1.00000 −0.0924500
\(118\) −5.70653 −0.525328
\(119\) 23.2704 2.13320
\(120\) 3.78531 0.345550
\(121\) −7.03427 −0.639479
\(122\) 9.33645 0.845282
\(123\) −6.91346 −0.623365
\(124\) 3.05022 0.273918
\(125\) 16.3849 1.46551
\(126\) 3.09768 0.275963
\(127\) −1.57275 −0.139559 −0.0697796 0.997562i \(-0.522230\pi\)
−0.0697796 + 0.997562i \(0.522230\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.14209 −0.452736
\(130\) 3.78531 0.331993
\(131\) 9.95245 0.869549 0.434775 0.900539i \(-0.356828\pi\)
0.434775 + 0.900539i \(0.356828\pi\)
\(132\) 1.99141 0.173330
\(133\) −19.0159 −1.64889
\(134\) −0.628166 −0.0542653
\(135\) −3.78531 −0.325787
\(136\) 7.51222 0.644168
\(137\) −20.7496 −1.77275 −0.886377 0.462964i \(-0.846786\pi\)
−0.886377 + 0.462964i \(0.846786\pi\)
\(138\) 6.69447 0.569872
\(139\) 18.3189 1.55379 0.776894 0.629631i \(-0.216794\pi\)
0.776894 + 0.629631i \(0.216794\pi\)
\(140\) −11.7257 −0.990999
\(141\) 8.19852 0.690440
\(142\) 6.41206 0.538088
\(143\) 1.99141 0.166530
\(144\) 1.00000 0.0833333
\(145\) 15.7554 1.30842
\(146\) 5.18377 0.429012
\(147\) −2.59560 −0.214082
\(148\) −0.695449 −0.0571656
\(149\) −0.0161712 −0.00132479 −0.000662397 1.00000i \(-0.500211\pi\)
−0.000662397 1.00000i \(0.500211\pi\)
\(150\) 9.32855 0.761673
\(151\) 8.80074 0.716194 0.358097 0.933684i \(-0.383426\pi\)
0.358097 + 0.933684i \(0.383426\pi\)
\(152\) −6.13877 −0.497920
\(153\) −7.51222 −0.607327
\(154\) −6.16876 −0.497093
\(155\) 11.5460 0.927399
\(156\) 1.00000 0.0800641
\(157\) −8.35461 −0.666770 −0.333385 0.942791i \(-0.608191\pi\)
−0.333385 + 0.942791i \(0.608191\pi\)
\(158\) 5.82259 0.463221
\(159\) 5.97715 0.474019
\(160\) −3.78531 −0.299255
\(161\) −20.7373 −1.63433
\(162\) −1.00000 −0.0785674
\(163\) −22.9785 −1.79982 −0.899909 0.436078i \(-0.856367\pi\)
−0.899909 + 0.436078i \(0.856367\pi\)
\(164\) 6.91346 0.539850
\(165\) 7.53812 0.586842
\(166\) −15.0761 −1.17013
\(167\) 22.5862 1.74778 0.873888 0.486127i \(-0.161591\pi\)
0.873888 + 0.486127i \(0.161591\pi\)
\(168\) −3.09768 −0.238991
\(169\) 1.00000 0.0769231
\(170\) 28.4361 2.18095
\(171\) 6.13877 0.469443
\(172\) 5.14209 0.392081
\(173\) 16.2875 1.23832 0.619158 0.785267i \(-0.287474\pi\)
0.619158 + 0.785267i \(0.287474\pi\)
\(174\) 4.16226 0.315540
\(175\) −28.8968 −2.18440
\(176\) −1.99141 −0.150109
\(177\) −5.70653 −0.428929
\(178\) 17.5821 1.31783
\(179\) 7.52632 0.562543 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(180\) 3.78531 0.282140
\(181\) −9.82936 −0.730610 −0.365305 0.930888i \(-0.619035\pi\)
−0.365305 + 0.930888i \(0.619035\pi\)
\(182\) −3.09768 −0.229615
\(183\) 9.33645 0.690170
\(184\) −6.69447 −0.493523
\(185\) −2.63249 −0.193544
\(186\) 3.05022 0.223653
\(187\) 14.9599 1.09398
\(188\) −8.19852 −0.597938
\(189\) 3.09768 0.225323
\(190\) −23.2371 −1.68580
\(191\) −0.940593 −0.0680589 −0.0340295 0.999421i \(-0.510834\pi\)
−0.0340295 + 0.999421i \(0.510834\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 23.5134 1.69253 0.846267 0.532759i \(-0.178845\pi\)
0.846267 + 0.532759i \(0.178845\pi\)
\(194\) −2.07965 −0.149310
\(195\) 3.78531 0.271071
\(196\) 2.59560 0.185400
\(197\) 24.1507 1.72066 0.860331 0.509735i \(-0.170257\pi\)
0.860331 + 0.509735i \(0.170257\pi\)
\(198\) 1.99141 0.141524
\(199\) 26.5695 1.88346 0.941732 0.336365i \(-0.109197\pi\)
0.941732 + 0.336365i \(0.109197\pi\)
\(200\) −9.32855 −0.659628
\(201\) −0.628166 −0.0443074
\(202\) −8.79029 −0.618483
\(203\) −12.8934 −0.904936
\(204\) 7.51222 0.525961
\(205\) 26.1696 1.82776
\(206\) −1.00000 −0.0696733
\(207\) 6.69447 0.465298
\(208\) −1.00000 −0.0693375
\(209\) −12.2248 −0.845609
\(210\) −11.7257 −0.809147
\(211\) 20.4472 1.40764 0.703821 0.710377i \(-0.251476\pi\)
0.703821 + 0.710377i \(0.251476\pi\)
\(212\) −5.97715 −0.410512
\(213\) 6.41206 0.439347
\(214\) −9.62233 −0.657769
\(215\) 19.4644 1.32746
\(216\) 1.00000 0.0680414
\(217\) −9.44860 −0.641413
\(218\) 0.639705 0.0433263
\(219\) 5.18377 0.350287
\(220\) −7.53812 −0.508220
\(221\) 7.51222 0.505327
\(222\) −0.695449 −0.0466755
\(223\) 19.4070 1.29959 0.649793 0.760111i \(-0.274855\pi\)
0.649793 + 0.760111i \(0.274855\pi\)
\(224\) 3.09768 0.206972
\(225\) 9.32855 0.621903
\(226\) 3.39898 0.226097
\(227\) −15.5048 −1.02909 −0.514544 0.857464i \(-0.672039\pi\)
−0.514544 + 0.857464i \(0.672039\pi\)
\(228\) −6.13877 −0.406550
\(229\) 23.4503 1.54964 0.774818 0.632184i \(-0.217841\pi\)
0.774818 + 0.632184i \(0.217841\pi\)
\(230\) −25.3406 −1.67091
\(231\) −6.16876 −0.405875
\(232\) −4.16226 −0.273266
\(233\) −1.15040 −0.0753649 −0.0376825 0.999290i \(-0.511998\pi\)
−0.0376825 + 0.999290i \(0.511998\pi\)
\(234\) 1.00000 0.0653720
\(235\) −31.0339 −2.02443
\(236\) 5.70653 0.371463
\(237\) 5.82259 0.378218
\(238\) −23.2704 −1.50840
\(239\) 22.2279 1.43780 0.718901 0.695112i \(-0.244645\pi\)
0.718901 + 0.695112i \(0.244645\pi\)
\(240\) −3.78531 −0.244341
\(241\) −0.626194 −0.0403367 −0.0201684 0.999797i \(-0.506420\pi\)
−0.0201684 + 0.999797i \(0.506420\pi\)
\(242\) 7.03427 0.452180
\(243\) −1.00000 −0.0641500
\(244\) −9.33645 −0.597705
\(245\) 9.82515 0.627706
\(246\) 6.91346 0.440786
\(247\) −6.13877 −0.390601
\(248\) −3.05022 −0.193689
\(249\) −15.0761 −0.955409
\(250\) −16.3849 −1.03627
\(251\) −10.8422 −0.684355 −0.342178 0.939635i \(-0.611164\pi\)
−0.342178 + 0.939635i \(0.611164\pi\)
\(252\) −3.09768 −0.195135
\(253\) −13.3315 −0.838143
\(254\) 1.57275 0.0986833
\(255\) 28.4361 1.78074
\(256\) 1.00000 0.0625000
\(257\) −27.6196 −1.72286 −0.861432 0.507874i \(-0.830432\pi\)
−0.861432 + 0.507874i \(0.830432\pi\)
\(258\) 5.14209 0.320133
\(259\) 2.15428 0.133860
\(260\) −3.78531 −0.234755
\(261\) 4.16226 0.257638
\(262\) −9.95245 −0.614864
\(263\) −15.2570 −0.940786 −0.470393 0.882457i \(-0.655888\pi\)
−0.470393 + 0.882457i \(0.655888\pi\)
\(264\) −1.99141 −0.122563
\(265\) −22.6254 −1.38986
\(266\) 19.0159 1.16594
\(267\) 17.5821 1.07601
\(268\) 0.628166 0.0383713
\(269\) −1.32901 −0.0810311 −0.0405155 0.999179i \(-0.512900\pi\)
−0.0405155 + 0.999179i \(0.512900\pi\)
\(270\) 3.78531 0.230366
\(271\) −23.7281 −1.44138 −0.720688 0.693259i \(-0.756174\pi\)
−0.720688 + 0.693259i \(0.756174\pi\)
\(272\) −7.51222 −0.455495
\(273\) −3.09768 −0.187480
\(274\) 20.7496 1.25353
\(275\) −18.5770 −1.12024
\(276\) −6.69447 −0.402960
\(277\) 20.9041 1.25600 0.628002 0.778212i \(-0.283873\pi\)
0.628002 + 0.778212i \(0.283873\pi\)
\(278\) −18.3189 −1.09869
\(279\) 3.05022 0.182612
\(280\) 11.7257 0.700742
\(281\) −7.33988 −0.437861 −0.218930 0.975740i \(-0.570257\pi\)
−0.218930 + 0.975740i \(0.570257\pi\)
\(282\) −8.19852 −0.488215
\(283\) −1.97292 −0.117278 −0.0586389 0.998279i \(-0.518676\pi\)
−0.0586389 + 0.998279i \(0.518676\pi\)
\(284\) −6.41206 −0.380486
\(285\) −23.2371 −1.37645
\(286\) −1.99141 −0.117755
\(287\) −21.4157 −1.26413
\(288\) −1.00000 −0.0589256
\(289\) 39.4335 2.31962
\(290\) −15.7554 −0.925192
\(291\) −2.07965 −0.121911
\(292\) −5.18377 −0.303357
\(293\) 9.65800 0.564226 0.282113 0.959381i \(-0.408965\pi\)
0.282113 + 0.959381i \(0.408965\pi\)
\(294\) 2.59560 0.151379
\(295\) 21.6010 1.25766
\(296\) 0.695449 0.0404222
\(297\) 1.99141 0.115554
\(298\) 0.0161712 0.000936771 0
\(299\) −6.69447 −0.387151
\(300\) −9.32855 −0.538584
\(301\) −15.9285 −0.918106
\(302\) −8.80074 −0.506426
\(303\) −8.79029 −0.504989
\(304\) 6.13877 0.352083
\(305\) −35.3413 −2.02364
\(306\) 7.51222 0.429445
\(307\) −14.6185 −0.834324 −0.417162 0.908832i \(-0.636975\pi\)
−0.417162 + 0.908832i \(0.636975\pi\)
\(308\) 6.16876 0.351498
\(309\) −1.00000 −0.0568880
\(310\) −11.5460 −0.655770
\(311\) 18.8745 1.07027 0.535136 0.844766i \(-0.320260\pi\)
0.535136 + 0.844766i \(0.320260\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 22.3554 1.26360 0.631801 0.775130i \(-0.282316\pi\)
0.631801 + 0.775130i \(0.282316\pi\)
\(314\) 8.35461 0.471478
\(315\) −11.7257 −0.660666
\(316\) −5.82259 −0.327547
\(317\) −18.8230 −1.05720 −0.528602 0.848870i \(-0.677284\pi\)
−0.528602 + 0.848870i \(0.677284\pi\)
\(318\) −5.97715 −0.335182
\(319\) −8.28879 −0.464083
\(320\) 3.78531 0.211605
\(321\) −9.62233 −0.537066
\(322\) 20.7373 1.15565
\(323\) −46.1158 −2.56595
\(324\) 1.00000 0.0555556
\(325\) −9.32855 −0.517455
\(326\) 22.9785 1.27266
\(327\) 0.639705 0.0353758
\(328\) −6.91346 −0.381732
\(329\) 25.3964 1.40015
\(330\) −7.53812 −0.414960
\(331\) 30.7156 1.68828 0.844141 0.536121i \(-0.180111\pi\)
0.844141 + 0.536121i \(0.180111\pi\)
\(332\) 15.0761 0.827408
\(333\) −0.695449 −0.0381104
\(334\) −22.5862 −1.23586
\(335\) 2.37780 0.129913
\(336\) 3.09768 0.168992
\(337\) −26.9596 −1.46858 −0.734290 0.678836i \(-0.762485\pi\)
−0.734290 + 0.678836i \(0.762485\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 3.39898 0.184607
\(340\) −28.4361 −1.54216
\(341\) −6.07425 −0.328939
\(342\) −6.13877 −0.331947
\(343\) 13.6434 0.736674
\(344\) −5.14209 −0.277243
\(345\) −25.3406 −1.36429
\(346\) −16.2875 −0.875621
\(347\) −33.3690 −1.79134 −0.895670 0.444720i \(-0.853303\pi\)
−0.895670 + 0.444720i \(0.853303\pi\)
\(348\) −4.16226 −0.223121
\(349\) 16.5858 0.887817 0.443909 0.896072i \(-0.353591\pi\)
0.443909 + 0.896072i \(0.353591\pi\)
\(350\) 28.8968 1.54460
\(351\) 1.00000 0.0533761
\(352\) 1.99141 0.106143
\(353\) 16.7491 0.891465 0.445733 0.895166i \(-0.352943\pi\)
0.445733 + 0.895166i \(0.352943\pi\)
\(354\) 5.70653 0.303299
\(355\) −24.2716 −1.28820
\(356\) −17.5821 −0.931848
\(357\) −23.2704 −1.23160
\(358\) −7.52632 −0.397778
\(359\) 25.7189 1.35739 0.678697 0.734419i \(-0.262545\pi\)
0.678697 + 0.734419i \(0.262545\pi\)
\(360\) −3.78531 −0.199503
\(361\) 18.6845 0.983394
\(362\) 9.82936 0.516619
\(363\) 7.03427 0.369203
\(364\) 3.09768 0.162362
\(365\) −19.6222 −1.02707
\(366\) −9.33645 −0.488024
\(367\) 14.3211 0.747557 0.373779 0.927518i \(-0.378062\pi\)
0.373779 + 0.927518i \(0.378062\pi\)
\(368\) 6.69447 0.348974
\(369\) 6.91346 0.359900
\(370\) 2.63249 0.136857
\(371\) 18.5153 0.961266
\(372\) −3.05022 −0.158147
\(373\) 24.5108 1.26912 0.634561 0.772872i \(-0.281181\pi\)
0.634561 + 0.772872i \(0.281181\pi\)
\(374\) −14.9599 −0.773560
\(375\) −16.3849 −0.846113
\(376\) 8.19852 0.422806
\(377\) −4.16226 −0.214368
\(378\) −3.09768 −0.159327
\(379\) 7.15639 0.367599 0.183800 0.982964i \(-0.441160\pi\)
0.183800 + 0.982964i \(0.441160\pi\)
\(380\) 23.2371 1.19204
\(381\) 1.57275 0.0805746
\(382\) 0.940593 0.0481249
\(383\) −0.745504 −0.0380935 −0.0190467 0.999819i \(-0.506063\pi\)
−0.0190467 + 0.999819i \(0.506063\pi\)
\(384\) 1.00000 0.0510310
\(385\) 23.3506 1.19006
\(386\) −23.5134 −1.19680
\(387\) 5.14209 0.261387
\(388\) 2.07965 0.105578
\(389\) 19.9258 1.01028 0.505140 0.863038i \(-0.331441\pi\)
0.505140 + 0.863038i \(0.331441\pi\)
\(390\) −3.78531 −0.191676
\(391\) −50.2904 −2.54329
\(392\) −2.59560 −0.131098
\(393\) −9.95245 −0.502034
\(394\) −24.1507 −1.21669
\(395\) −22.0403 −1.10897
\(396\) −1.99141 −0.100072
\(397\) 6.58175 0.330329 0.165164 0.986266i \(-0.447185\pi\)
0.165164 + 0.986266i \(0.447185\pi\)
\(398\) −26.5695 −1.33181
\(399\) 19.0159 0.951987
\(400\) 9.32855 0.466428
\(401\) 25.0372 1.25030 0.625149 0.780506i \(-0.285038\pi\)
0.625149 + 0.780506i \(0.285038\pi\)
\(402\) 0.628166 0.0313301
\(403\) −3.05022 −0.151942
\(404\) 8.79029 0.437333
\(405\) 3.78531 0.188093
\(406\) 12.8934 0.639886
\(407\) 1.38493 0.0686483
\(408\) −7.51222 −0.371910
\(409\) 18.2340 0.901614 0.450807 0.892622i \(-0.351136\pi\)
0.450807 + 0.892622i \(0.351136\pi\)
\(410\) −26.1696 −1.29242
\(411\) 20.7496 1.02350
\(412\) 1.00000 0.0492665
\(413\) −17.6770 −0.869827
\(414\) −6.69447 −0.329015
\(415\) 57.0677 2.80134
\(416\) 1.00000 0.0490290
\(417\) −18.3189 −0.897080
\(418\) 12.2248 0.597936
\(419\) 31.4533 1.53659 0.768297 0.640094i \(-0.221105\pi\)
0.768297 + 0.640094i \(0.221105\pi\)
\(420\) 11.7257 0.572154
\(421\) −17.7470 −0.864933 −0.432467 0.901650i \(-0.642357\pi\)
−0.432467 + 0.901650i \(0.642357\pi\)
\(422\) −20.4472 −0.995354
\(423\) −8.19852 −0.398625
\(424\) 5.97715 0.290276
\(425\) −70.0782 −3.39929
\(426\) −6.41206 −0.310665
\(427\) 28.9213 1.39960
\(428\) 9.62233 0.465113
\(429\) −1.99141 −0.0961464
\(430\) −19.4644 −0.938657
\(431\) −17.6107 −0.848279 −0.424140 0.905597i \(-0.639423\pi\)
−0.424140 + 0.905597i \(0.639423\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 21.0987 1.01394 0.506970 0.861964i \(-0.330766\pi\)
0.506970 + 0.861964i \(0.330766\pi\)
\(434\) 9.44860 0.453547
\(435\) −15.7554 −0.755416
\(436\) −0.639705 −0.0306363
\(437\) 41.0958 1.96588
\(438\) −5.18377 −0.247690
\(439\) 3.92954 0.187547 0.0937734 0.995594i \(-0.470107\pi\)
0.0937734 + 0.995594i \(0.470107\pi\)
\(440\) 7.53812 0.359366
\(441\) 2.59560 0.123600
\(442\) −7.51222 −0.357320
\(443\) 24.9289 1.18441 0.592205 0.805788i \(-0.298258\pi\)
0.592205 + 0.805788i \(0.298258\pi\)
\(444\) 0.695449 0.0330046
\(445\) −66.5536 −3.15494
\(446\) −19.4070 −0.918946
\(447\) 0.0161712 0.000764870 0
\(448\) −3.09768 −0.146351
\(449\) 15.4427 0.728784 0.364392 0.931246i \(-0.381277\pi\)
0.364392 + 0.931246i \(0.381277\pi\)
\(450\) −9.32855 −0.439752
\(451\) −13.7676 −0.648289
\(452\) −3.39898 −0.159875
\(453\) −8.80074 −0.413495
\(454\) 15.5048 0.727675
\(455\) 11.7257 0.549707
\(456\) 6.13877 0.287474
\(457\) 21.6250 1.01158 0.505788 0.862658i \(-0.331202\pi\)
0.505788 + 0.862658i \(0.331202\pi\)
\(458\) −23.4503 −1.09576
\(459\) 7.51222 0.350640
\(460\) 25.3406 1.18151
\(461\) −22.2887 −1.03809 −0.519043 0.854748i \(-0.673712\pi\)
−0.519043 + 0.854748i \(0.673712\pi\)
\(462\) 6.16876 0.286997
\(463\) 10.6074 0.492969 0.246485 0.969147i \(-0.420725\pi\)
0.246485 + 0.969147i \(0.420725\pi\)
\(464\) 4.16226 0.193228
\(465\) −11.5460 −0.535434
\(466\) 1.15040 0.0532911
\(467\) −1.80493 −0.0835224 −0.0417612 0.999128i \(-0.513297\pi\)
−0.0417612 + 0.999128i \(0.513297\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −1.94586 −0.0898512
\(470\) 31.0339 1.43149
\(471\) 8.35461 0.384960
\(472\) −5.70653 −0.262664
\(473\) −10.2400 −0.470837
\(474\) −5.82259 −0.267441
\(475\) 57.2658 2.62754
\(476\) 23.2704 1.06660
\(477\) −5.97715 −0.273675
\(478\) −22.2279 −1.01668
\(479\) −37.0283 −1.69187 −0.845934 0.533288i \(-0.820956\pi\)
−0.845934 + 0.533288i \(0.820956\pi\)
\(480\) 3.78531 0.172775
\(481\) 0.695449 0.0317097
\(482\) 0.626194 0.0285224
\(483\) 20.7373 0.943581
\(484\) −7.03427 −0.319739
\(485\) 7.87212 0.357455
\(486\) 1.00000 0.0453609
\(487\) −29.9059 −1.35516 −0.677582 0.735447i \(-0.736972\pi\)
−0.677582 + 0.735447i \(0.736972\pi\)
\(488\) 9.33645 0.422641
\(489\) 22.9785 1.03913
\(490\) −9.82515 −0.443855
\(491\) 13.9979 0.631715 0.315857 0.948807i \(-0.397708\pi\)
0.315857 + 0.948807i \(0.397708\pi\)
\(492\) −6.91346 −0.311683
\(493\) −31.2679 −1.40823
\(494\) 6.13877 0.276196
\(495\) −7.53812 −0.338813
\(496\) 3.05022 0.136959
\(497\) 19.8625 0.890955
\(498\) 15.0761 0.675576
\(499\) −1.26434 −0.0565998 −0.0282999 0.999599i \(-0.509009\pi\)
−0.0282999 + 0.999599i \(0.509009\pi\)
\(500\) 16.3849 0.732755
\(501\) −22.5862 −1.00908
\(502\) 10.8422 0.483912
\(503\) −31.4677 −1.40307 −0.701537 0.712633i \(-0.747502\pi\)
−0.701537 + 0.712633i \(0.747502\pi\)
\(504\) 3.09768 0.137981
\(505\) 33.2739 1.48067
\(506\) 13.3315 0.592656
\(507\) −1.00000 −0.0444116
\(508\) −1.57275 −0.0697796
\(509\) 22.3915 0.992484 0.496242 0.868184i \(-0.334713\pi\)
0.496242 + 0.868184i \(0.334713\pi\)
\(510\) −28.4361 −1.25917
\(511\) 16.0576 0.710348
\(512\) −1.00000 −0.0441942
\(513\) −6.13877 −0.271033
\(514\) 27.6196 1.21825
\(515\) 3.78531 0.166801
\(516\) −5.14209 −0.226368
\(517\) 16.3266 0.718045
\(518\) −2.15428 −0.0946535
\(519\) −16.2875 −0.714942
\(520\) 3.78531 0.165997
\(521\) 8.92353 0.390947 0.195474 0.980709i \(-0.437376\pi\)
0.195474 + 0.980709i \(0.437376\pi\)
\(522\) −4.16226 −0.182177
\(523\) 20.5910 0.900381 0.450191 0.892933i \(-0.351356\pi\)
0.450191 + 0.892933i \(0.351356\pi\)
\(524\) 9.95245 0.434775
\(525\) 28.8968 1.26116
\(526\) 15.2570 0.665236
\(527\) −22.9139 −0.998147
\(528\) 1.99141 0.0866652
\(529\) 21.8160 0.948521
\(530\) 22.6254 0.982783
\(531\) 5.70653 0.247642
\(532\) −19.0159 −0.824445
\(533\) −6.91346 −0.299455
\(534\) −17.5821 −0.760851
\(535\) 36.4235 1.57472
\(536\) −0.628166 −0.0271326
\(537\) −7.52632 −0.324784
\(538\) 1.32901 0.0572976
\(539\) −5.16892 −0.222641
\(540\) −3.78531 −0.162894
\(541\) −13.3913 −0.575737 −0.287869 0.957670i \(-0.592947\pi\)
−0.287869 + 0.957670i \(0.592947\pi\)
\(542\) 23.7281 1.01921
\(543\) 9.82936 0.421818
\(544\) 7.51222 0.322084
\(545\) −2.42148 −0.103725
\(546\) 3.09768 0.132568
\(547\) 4.44388 0.190007 0.0950033 0.995477i \(-0.469714\pi\)
0.0950033 + 0.995477i \(0.469714\pi\)
\(548\) −20.7496 −0.886377
\(549\) −9.33645 −0.398470
\(550\) 18.5770 0.792127
\(551\) 25.5512 1.08852
\(552\) 6.69447 0.284936
\(553\) 18.0365 0.766991
\(554\) −20.9041 −0.888129
\(555\) 2.63249 0.111743
\(556\) 18.3189 0.776894
\(557\) −22.9310 −0.971618 −0.485809 0.874065i \(-0.661475\pi\)
−0.485809 + 0.874065i \(0.661475\pi\)
\(558\) −3.05022 −0.129126
\(559\) −5.14209 −0.217487
\(560\) −11.7257 −0.495500
\(561\) −14.9599 −0.631609
\(562\) 7.33988 0.309614
\(563\) 19.5396 0.823496 0.411748 0.911298i \(-0.364918\pi\)
0.411748 + 0.911298i \(0.364918\pi\)
\(564\) 8.19852 0.345220
\(565\) −12.8662 −0.541285
\(566\) 1.97292 0.0829279
\(567\) −3.09768 −0.130090
\(568\) 6.41206 0.269044
\(569\) 21.4904 0.900925 0.450462 0.892795i \(-0.351259\pi\)
0.450462 + 0.892795i \(0.351259\pi\)
\(570\) 23.2371 0.973296
\(571\) 36.5476 1.52947 0.764735 0.644345i \(-0.222870\pi\)
0.764735 + 0.644345i \(0.222870\pi\)
\(572\) 1.99141 0.0832652
\(573\) 0.940593 0.0392938
\(574\) 21.4157 0.893872
\(575\) 62.4498 2.60433
\(576\) 1.00000 0.0416667
\(577\) 19.8309 0.825573 0.412787 0.910828i \(-0.364556\pi\)
0.412787 + 0.910828i \(0.364556\pi\)
\(578\) −39.4335 −1.64022
\(579\) −23.5134 −0.977185
\(580\) 15.7554 0.654209
\(581\) −46.7009 −1.93748
\(582\) 2.07965 0.0862044
\(583\) 11.9030 0.492971
\(584\) 5.18377 0.214506
\(585\) −3.78531 −0.156503
\(586\) −9.65800 −0.398968
\(587\) 7.15353 0.295258 0.147629 0.989043i \(-0.452836\pi\)
0.147629 + 0.989043i \(0.452836\pi\)
\(588\) −2.59560 −0.107041
\(589\) 18.7246 0.771534
\(590\) −21.6010 −0.889298
\(591\) −24.1507 −0.993425
\(592\) −0.695449 −0.0285828
\(593\) −17.2185 −0.707078 −0.353539 0.935420i \(-0.615022\pi\)
−0.353539 + 0.935420i \(0.615022\pi\)
\(594\) −1.99141 −0.0817087
\(595\) 88.0858 3.61116
\(596\) −0.0161712 −0.000662397 0
\(597\) −26.5695 −1.08742
\(598\) 6.69447 0.273757
\(599\) −38.9159 −1.59006 −0.795030 0.606570i \(-0.792545\pi\)
−0.795030 + 0.606570i \(0.792545\pi\)
\(600\) 9.32855 0.380837
\(601\) −39.4760 −1.61026 −0.805129 0.593100i \(-0.797904\pi\)
−0.805129 + 0.593100i \(0.797904\pi\)
\(602\) 15.9285 0.649199
\(603\) 0.628166 0.0255809
\(604\) 8.80074 0.358097
\(605\) −26.6269 −1.08254
\(606\) 8.79029 0.357081
\(607\) −21.5363 −0.874129 −0.437065 0.899430i \(-0.643982\pi\)
−0.437065 + 0.899430i \(0.643982\pi\)
\(608\) −6.13877 −0.248960
\(609\) 12.8934 0.522465
\(610\) 35.3413 1.43093
\(611\) 8.19852 0.331676
\(612\) −7.51222 −0.303664
\(613\) 39.3064 1.58757 0.793786 0.608197i \(-0.208107\pi\)
0.793786 + 0.608197i \(0.208107\pi\)
\(614\) 14.6185 0.589956
\(615\) −26.1696 −1.05526
\(616\) −6.16876 −0.248546
\(617\) −23.0807 −0.929192 −0.464596 0.885523i \(-0.653800\pi\)
−0.464596 + 0.885523i \(0.653800\pi\)
\(618\) 1.00000 0.0402259
\(619\) 11.6374 0.467748 0.233874 0.972267i \(-0.424860\pi\)
0.233874 + 0.972267i \(0.424860\pi\)
\(620\) 11.5460 0.463700
\(621\) −6.69447 −0.268640
\(622\) −18.8745 −0.756797
\(623\) 54.4636 2.18204
\(624\) 1.00000 0.0400320
\(625\) 15.3791 0.615165
\(626\) −22.3554 −0.893502
\(627\) 12.2248 0.488213
\(628\) −8.35461 −0.333385
\(629\) 5.22437 0.208309
\(630\) 11.7257 0.467161
\(631\) 13.7616 0.547841 0.273920 0.961752i \(-0.411680\pi\)
0.273920 + 0.961752i \(0.411680\pi\)
\(632\) 5.82259 0.231610
\(633\) −20.4472 −0.812703
\(634\) 18.8230 0.747556
\(635\) −5.95335 −0.236252
\(636\) 5.97715 0.237009
\(637\) −2.59560 −0.102842
\(638\) 8.28879 0.328156
\(639\) −6.41206 −0.253657
\(640\) −3.78531 −0.149627
\(641\) −10.5027 −0.414833 −0.207417 0.978253i \(-0.566506\pi\)
−0.207417 + 0.978253i \(0.566506\pi\)
\(642\) 9.62233 0.379763
\(643\) −36.6778 −1.44643 −0.723216 0.690622i \(-0.757337\pi\)
−0.723216 + 0.690622i \(0.757337\pi\)
\(644\) −20.7373 −0.817165
\(645\) −19.4644 −0.766410
\(646\) 46.1158 1.81440
\(647\) −31.8916 −1.25379 −0.626895 0.779104i \(-0.715674\pi\)
−0.626895 + 0.779104i \(0.715674\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −11.3641 −0.446078
\(650\) 9.32855 0.365896
\(651\) 9.44860 0.370320
\(652\) −22.9785 −0.899909
\(653\) 21.2444 0.831359 0.415680 0.909511i \(-0.363544\pi\)
0.415680 + 0.909511i \(0.363544\pi\)
\(654\) −0.639705 −0.0250144
\(655\) 37.6731 1.47201
\(656\) 6.91346 0.269925
\(657\) −5.18377 −0.202238
\(658\) −25.3964 −0.990053
\(659\) −32.4731 −1.26497 −0.632486 0.774572i \(-0.717965\pi\)
−0.632486 + 0.774572i \(0.717965\pi\)
\(660\) 7.53812 0.293421
\(661\) 7.26850 0.282712 0.141356 0.989959i \(-0.454854\pi\)
0.141356 + 0.989959i \(0.454854\pi\)
\(662\) −30.7156 −1.19380
\(663\) −7.51222 −0.291751
\(664\) −15.0761 −0.585066
\(665\) −71.9811 −2.79131
\(666\) 0.695449 0.0269481
\(667\) 27.8642 1.07891
\(668\) 22.5862 0.873888
\(669\) −19.4070 −0.750317
\(670\) −2.37780 −0.0918625
\(671\) 18.5927 0.717765
\(672\) −3.09768 −0.119495
\(673\) −8.81926 −0.339957 −0.169979 0.985448i \(-0.554370\pi\)
−0.169979 + 0.985448i \(0.554370\pi\)
\(674\) 26.9596 1.03844
\(675\) −9.32855 −0.359056
\(676\) 1.00000 0.0384615
\(677\) 23.8125 0.915187 0.457594 0.889161i \(-0.348711\pi\)
0.457594 + 0.889161i \(0.348711\pi\)
\(678\) −3.39898 −0.130537
\(679\) −6.44209 −0.247225
\(680\) 28.4361 1.09047
\(681\) 15.5048 0.594144
\(682\) 6.07425 0.232595
\(683\) 22.4760 0.860020 0.430010 0.902824i \(-0.358510\pi\)
0.430010 + 0.902824i \(0.358510\pi\)
\(684\) 6.13877 0.234722
\(685\) −78.5435 −3.00099
\(686\) −13.6434 −0.520907
\(687\) −23.4503 −0.894683
\(688\) 5.14209 0.196040
\(689\) 5.97715 0.227711
\(690\) 25.3406 0.964702
\(691\) 29.1132 1.10752 0.553759 0.832677i \(-0.313193\pi\)
0.553759 + 0.832677i \(0.313193\pi\)
\(692\) 16.2875 0.619158
\(693\) 6.16876 0.234332
\(694\) 33.3690 1.26667
\(695\) 69.3426 2.63032
\(696\) 4.16226 0.157770
\(697\) −51.9354 −1.96719
\(698\) −16.5858 −0.627782
\(699\) 1.15040 0.0435120
\(700\) −28.8968 −1.09220
\(701\) 16.7388 0.632217 0.316109 0.948723i \(-0.397624\pi\)
0.316109 + 0.948723i \(0.397624\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −4.26920 −0.161016
\(704\) −1.99141 −0.0750543
\(705\) 31.0339 1.16880
\(706\) −16.7491 −0.630361
\(707\) −27.2295 −1.02407
\(708\) −5.70653 −0.214464
\(709\) 25.3836 0.953303 0.476651 0.879092i \(-0.341850\pi\)
0.476651 + 0.879092i \(0.341850\pi\)
\(710\) 24.2716 0.910898
\(711\) −5.82259 −0.218364
\(712\) 17.5821 0.658916
\(713\) 20.4196 0.764721
\(714\) 23.2704 0.870874
\(715\) 7.53812 0.281910
\(716\) 7.52632 0.281272
\(717\) −22.2279 −0.830116
\(718\) −25.7189 −0.959822
\(719\) −32.2426 −1.20245 −0.601224 0.799081i \(-0.705320\pi\)
−0.601224 + 0.799081i \(0.705320\pi\)
\(720\) 3.78531 0.141070
\(721\) −3.09768 −0.115364
\(722\) −18.6845 −0.695364
\(723\) 0.626194 0.0232884
\(724\) −9.82936 −0.365305
\(725\) 38.8279 1.44203
\(726\) −7.03427 −0.261066
\(727\) −38.8037 −1.43915 −0.719576 0.694414i \(-0.755664\pi\)
−0.719576 + 0.694414i \(0.755664\pi\)
\(728\) −3.09768 −0.114808
\(729\) 1.00000 0.0370370
\(730\) 19.6222 0.726249
\(731\) −38.6285 −1.42873
\(732\) 9.33645 0.345085
\(733\) 33.8085 1.24874 0.624372 0.781127i \(-0.285355\pi\)
0.624372 + 0.781127i \(0.285355\pi\)
\(734\) −14.3211 −0.528603
\(735\) −9.82515 −0.362406
\(736\) −6.69447 −0.246762
\(737\) −1.25094 −0.0460789
\(738\) −6.91346 −0.254488
\(739\) 12.3286 0.453515 0.226758 0.973951i \(-0.427187\pi\)
0.226758 + 0.973951i \(0.427187\pi\)
\(740\) −2.63249 −0.0967722
\(741\) 6.13877 0.225513
\(742\) −18.5153 −0.679717
\(743\) 5.10838 0.187408 0.0937042 0.995600i \(-0.470129\pi\)
0.0937042 + 0.995600i \(0.470129\pi\)
\(744\) 3.05022 0.111827
\(745\) −0.0612128 −0.00224267
\(746\) −24.5108 −0.897405
\(747\) 15.0761 0.551606
\(748\) 14.9599 0.546990
\(749\) −29.8069 −1.08912
\(750\) 16.3849 0.598292
\(751\) −35.7973 −1.30626 −0.653132 0.757244i \(-0.726545\pi\)
−0.653132 + 0.757244i \(0.726545\pi\)
\(752\) −8.19852 −0.298969
\(753\) 10.8422 0.395113
\(754\) 4.16226 0.151581
\(755\) 33.3135 1.21240
\(756\) 3.09768 0.112661
\(757\) 46.0475 1.67363 0.836813 0.547488i \(-0.184416\pi\)
0.836813 + 0.547488i \(0.184416\pi\)
\(758\) −7.15639 −0.259932
\(759\) 13.3315 0.483902
\(760\) −23.2371 −0.842899
\(761\) 30.1154 1.09168 0.545841 0.837889i \(-0.316210\pi\)
0.545841 + 0.837889i \(0.316210\pi\)
\(762\) −1.57275 −0.0569748
\(763\) 1.98160 0.0717387
\(764\) −0.940593 −0.0340295
\(765\) −28.4361 −1.02811
\(766\) 0.745504 0.0269361
\(767\) −5.70653 −0.206051
\(768\) −1.00000 −0.0360844
\(769\) −3.24995 −0.117196 −0.0585981 0.998282i \(-0.518663\pi\)
−0.0585981 + 0.998282i \(0.518663\pi\)
\(770\) −23.3506 −0.841499
\(771\) 27.6196 0.994695
\(772\) 23.5134 0.846267
\(773\) −19.3295 −0.695233 −0.347617 0.937637i \(-0.613009\pi\)
−0.347617 + 0.937637i \(0.613009\pi\)
\(774\) −5.14209 −0.184829
\(775\) 28.4541 1.02210
\(776\) −2.07965 −0.0746552
\(777\) −2.15428 −0.0772842
\(778\) −19.9258 −0.714375
\(779\) 42.4401 1.52057
\(780\) 3.78531 0.135536
\(781\) 12.7691 0.456913
\(782\) 50.2904 1.79838
\(783\) −4.16226 −0.148747
\(784\) 2.59560 0.0927001
\(785\) −31.6248 −1.12874
\(786\) 9.95245 0.354992
\(787\) −6.83414 −0.243611 −0.121805 0.992554i \(-0.538868\pi\)
−0.121805 + 0.992554i \(0.538868\pi\)
\(788\) 24.1507 0.860331
\(789\) 15.2570 0.543163
\(790\) 22.0403 0.784159
\(791\) 10.5290 0.374367
\(792\) 1.99141 0.0707618
\(793\) 9.33645 0.331547
\(794\) −6.58175 −0.233578
\(795\) 22.6254 0.802439
\(796\) 26.5695 0.941732
\(797\) 37.5899 1.33150 0.665752 0.746173i \(-0.268111\pi\)
0.665752 + 0.746173i \(0.268111\pi\)
\(798\) −19.0159 −0.673156
\(799\) 61.5891 2.17886
\(800\) −9.32855 −0.329814
\(801\) −17.5821 −0.621232
\(802\) −25.0372 −0.884094
\(803\) 10.3230 0.364292
\(804\) −0.628166 −0.0221537
\(805\) −78.4971 −2.76666
\(806\) 3.05022 0.107439
\(807\) 1.32901 0.0467833
\(808\) −8.79029 −0.309241
\(809\) 10.1718 0.357620 0.178810 0.983884i \(-0.442775\pi\)
0.178810 + 0.983884i \(0.442775\pi\)
\(810\) −3.78531 −0.133002
\(811\) 30.6570 1.07651 0.538257 0.842781i \(-0.319083\pi\)
0.538257 + 0.842781i \(0.319083\pi\)
\(812\) −12.8934 −0.452468
\(813\) 23.7281 0.832179
\(814\) −1.38493 −0.0485417
\(815\) −86.9808 −3.04680
\(816\) 7.51222 0.262980
\(817\) 31.5661 1.10436
\(818\) −18.2340 −0.637537
\(819\) 3.09768 0.108242
\(820\) 26.1696 0.913880
\(821\) −48.2456 −1.68378 −0.841892 0.539646i \(-0.818558\pi\)
−0.841892 + 0.539646i \(0.818558\pi\)
\(822\) −20.7496 −0.723724
\(823\) 46.0802 1.60626 0.803128 0.595807i \(-0.203168\pi\)
0.803128 + 0.595807i \(0.203168\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 18.5770 0.646769
\(826\) 17.6770 0.615061
\(827\) 27.8281 0.967679 0.483840 0.875157i \(-0.339242\pi\)
0.483840 + 0.875157i \(0.339242\pi\)
\(828\) 6.69447 0.232649
\(829\) −10.5642 −0.366912 −0.183456 0.983028i \(-0.558728\pi\)
−0.183456 + 0.983028i \(0.558728\pi\)
\(830\) −57.0677 −1.98085
\(831\) −20.9041 −0.725154
\(832\) −1.00000 −0.0346688
\(833\) −19.4987 −0.675591
\(834\) 18.3189 0.634331
\(835\) 85.4959 2.95871
\(836\) −12.2248 −0.422805
\(837\) −3.05022 −0.105431
\(838\) −31.4533 −1.08654
\(839\) −40.4266 −1.39568 −0.697841 0.716253i \(-0.745856\pi\)
−0.697841 + 0.716253i \(0.745856\pi\)
\(840\) −11.7257 −0.404574
\(841\) −11.6756 −0.402605
\(842\) 17.7470 0.611600
\(843\) 7.33988 0.252799
\(844\) 20.4472 0.703821
\(845\) 3.78531 0.130219
\(846\) 8.19852 0.281871
\(847\) 21.7899 0.748710
\(848\) −5.97715 −0.205256
\(849\) 1.97292 0.0677104
\(850\) 70.0782 2.40366
\(851\) −4.65567 −0.159594
\(852\) 6.41206 0.219674
\(853\) −8.79975 −0.301298 −0.150649 0.988587i \(-0.548136\pi\)
−0.150649 + 0.988587i \(0.548136\pi\)
\(854\) −28.9213 −0.989667
\(855\) 23.2371 0.794693
\(856\) −9.62233 −0.328884
\(857\) −11.7712 −0.402096 −0.201048 0.979581i \(-0.564435\pi\)
−0.201048 + 0.979581i \(0.564435\pi\)
\(858\) 1.99141 0.0679858
\(859\) 47.8822 1.63372 0.816859 0.576837i \(-0.195713\pi\)
0.816859 + 0.576837i \(0.195713\pi\)
\(860\) 19.4644 0.663731
\(861\) 21.4157 0.729843
\(862\) 17.6107 0.599824
\(863\) 43.5993 1.48414 0.742069 0.670324i \(-0.233845\pi\)
0.742069 + 0.670324i \(0.233845\pi\)
\(864\) 1.00000 0.0340207
\(865\) 61.6532 2.09627
\(866\) −21.0987 −0.716963
\(867\) −39.4335 −1.33923
\(868\) −9.44860 −0.320706
\(869\) 11.5952 0.393340
\(870\) 15.7554 0.534160
\(871\) −0.628166 −0.0212846
\(872\) 0.639705 0.0216631
\(873\) 2.07965 0.0703856
\(874\) −41.0958 −1.39009
\(875\) −50.7551 −1.71584
\(876\) 5.18377 0.175143
\(877\) 37.1295 1.25377 0.626886 0.779111i \(-0.284329\pi\)
0.626886 + 0.779111i \(0.284329\pi\)
\(878\) −3.92954 −0.132616
\(879\) −9.65800 −0.325756
\(880\) −7.53812 −0.254110
\(881\) −9.10636 −0.306801 −0.153400 0.988164i \(-0.549022\pi\)
−0.153400 + 0.988164i \(0.549022\pi\)
\(882\) −2.59560 −0.0873985
\(883\) 4.05430 0.136438 0.0682190 0.997670i \(-0.478268\pi\)
0.0682190 + 0.997670i \(0.478268\pi\)
\(884\) 7.51222 0.252663
\(885\) −21.6010 −0.726108
\(886\) −24.9289 −0.837504
\(887\) −2.52846 −0.0848974 −0.0424487 0.999099i \(-0.513516\pi\)
−0.0424487 + 0.999099i \(0.513516\pi\)
\(888\) −0.695449 −0.0233377
\(889\) 4.87188 0.163398
\(890\) 66.5536 2.23088
\(891\) −1.99141 −0.0667149
\(892\) 19.4070 0.649793
\(893\) −50.3288 −1.68419
\(894\) −0.0161712 −0.000540845 0
\(895\) 28.4894 0.952296
\(896\) 3.09768 0.103486
\(897\) 6.69447 0.223522
\(898\) −15.4427 −0.515328
\(899\) 12.6958 0.423430
\(900\) 9.32855 0.310952
\(901\) 44.9017 1.49589
\(902\) 13.7676 0.458409
\(903\) 15.9285 0.530069
\(904\) 3.39898 0.113049
\(905\) −37.2071 −1.23681
\(906\) 8.80074 0.292385
\(907\) −33.9238 −1.12642 −0.563210 0.826314i \(-0.690434\pi\)
−0.563210 + 0.826314i \(0.690434\pi\)
\(908\) −15.5048 −0.514544
\(909\) 8.79029 0.291555
\(910\) −11.7257 −0.388702
\(911\) 11.7450 0.389129 0.194565 0.980890i \(-0.437671\pi\)
0.194565 + 0.980890i \(0.437671\pi\)
\(912\) −6.13877 −0.203275
\(913\) −30.0228 −0.993608
\(914\) −21.6250 −0.715293
\(915\) 35.3413 1.16835
\(916\) 23.4503 0.774818
\(917\) −30.8295 −1.01808
\(918\) −7.51222 −0.247940
\(919\) 20.3117 0.670022 0.335011 0.942214i \(-0.391260\pi\)
0.335011 + 0.942214i \(0.391260\pi\)
\(920\) −25.3406 −0.835456
\(921\) 14.6185 0.481697
\(922\) 22.2887 0.734038
\(923\) 6.41206 0.211056
\(924\) −6.16876 −0.202937
\(925\) −6.48753 −0.213309
\(926\) −10.6074 −0.348582
\(927\) 1.00000 0.0328443
\(928\) −4.16226 −0.136633
\(929\) −36.1337 −1.18551 −0.592755 0.805383i \(-0.701960\pi\)
−0.592755 + 0.805383i \(0.701960\pi\)
\(930\) 11.5460 0.378609
\(931\) 15.9338 0.522209
\(932\) −1.15040 −0.0376825
\(933\) −18.8745 −0.617922
\(934\) 1.80493 0.0590592
\(935\) 56.6280 1.85193
\(936\) 1.00000 0.0326860
\(937\) −53.8613 −1.75957 −0.879785 0.475371i \(-0.842314\pi\)
−0.879785 + 0.475371i \(0.842314\pi\)
\(938\) 1.94586 0.0635344
\(939\) −22.3554 −0.729541
\(940\) −31.0339 −1.01221
\(941\) −28.1101 −0.916365 −0.458182 0.888858i \(-0.651499\pi\)
−0.458182 + 0.888858i \(0.651499\pi\)
\(942\) −8.35461 −0.272208
\(943\) 46.2820 1.50715
\(944\) 5.70653 0.185732
\(945\) 11.7257 0.381436
\(946\) 10.2400 0.332932
\(947\) −18.4966 −0.601058 −0.300529 0.953773i \(-0.597163\pi\)
−0.300529 + 0.953773i \(0.597163\pi\)
\(948\) 5.82259 0.189109
\(949\) 5.18377 0.168272
\(950\) −57.2658 −1.85795
\(951\) 18.8230 0.610377
\(952\) −23.2704 −0.754199
\(953\) −51.4008 −1.66504 −0.832518 0.553998i \(-0.813102\pi\)
−0.832518 + 0.553998i \(0.813102\pi\)
\(954\) 5.97715 0.193517
\(955\) −3.56043 −0.115213
\(956\) 22.2279 0.718901
\(957\) 8.28879 0.267939
\(958\) 37.0283 1.19633
\(959\) 64.2754 2.07556
\(960\) −3.78531 −0.122170
\(961\) −21.6962 −0.699876
\(962\) −0.695449 −0.0224222
\(963\) 9.62233 0.310075
\(964\) −0.626194 −0.0201684
\(965\) 89.0056 2.86519
\(966\) −20.7373 −0.667212
\(967\) 31.7026 1.01949 0.509744 0.860326i \(-0.329740\pi\)
0.509744 + 0.860326i \(0.329740\pi\)
\(968\) 7.03427 0.226090
\(969\) 46.1158 1.48145
\(970\) −7.87212 −0.252759
\(971\) 4.46345 0.143239 0.0716195 0.997432i \(-0.477183\pi\)
0.0716195 + 0.997432i \(0.477183\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −56.7460 −1.81919
\(974\) 29.9059 0.958246
\(975\) 9.32855 0.298753
\(976\) −9.33645 −0.298852
\(977\) 8.96137 0.286700 0.143350 0.989672i \(-0.454213\pi\)
0.143350 + 0.989672i \(0.454213\pi\)
\(978\) −22.9785 −0.734772
\(979\) 35.0132 1.11903
\(980\) 9.82515 0.313853
\(981\) −0.639705 −0.0204242
\(982\) −13.9979 −0.446690
\(983\) −27.9612 −0.891823 −0.445912 0.895077i \(-0.647120\pi\)
−0.445912 + 0.895077i \(0.647120\pi\)
\(984\) 6.91346 0.220393
\(985\) 91.4177 2.91281
\(986\) 31.2679 0.995771
\(987\) −25.3964 −0.808375
\(988\) −6.13877 −0.195300
\(989\) 34.4236 1.09461
\(990\) 7.53812 0.239577
\(991\) −3.92413 −0.124654 −0.0623271 0.998056i \(-0.519852\pi\)
−0.0623271 + 0.998056i \(0.519852\pi\)
\(992\) −3.05022 −0.0968446
\(993\) −30.7156 −0.974730
\(994\) −19.8625 −0.630000
\(995\) 100.574 3.18840
\(996\) −15.0761 −0.477704
\(997\) 21.5902 0.683768 0.341884 0.939742i \(-0.388935\pi\)
0.341884 + 0.939742i \(0.388935\pi\)
\(998\) 1.26434 0.0400221
\(999\) 0.695449 0.0220030
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.ba.1.14 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.14 14 1.1 even 1 trivial