Properties

Label 8034.2.a.bb.1.11
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 1.11
Root \(-1.50270\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.50270 q^{5} -1.00000 q^{6} +0.210711 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.50270 q^{5} -1.00000 q^{6} +0.210711 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.50270 q^{10} -5.97021 q^{11} +1.00000 q^{12} -1.00000 q^{13} -0.210711 q^{14} +1.50270 q^{15} +1.00000 q^{16} +5.33536 q^{17} -1.00000 q^{18} -3.08917 q^{19} +1.50270 q^{20} +0.210711 q^{21} +5.97021 q^{22} -1.61839 q^{23} -1.00000 q^{24} -2.74190 q^{25} +1.00000 q^{26} +1.00000 q^{27} +0.210711 q^{28} +5.82084 q^{29} -1.50270 q^{30} +1.12646 q^{31} -1.00000 q^{32} -5.97021 q^{33} -5.33536 q^{34} +0.316634 q^{35} +1.00000 q^{36} -4.46916 q^{37} +3.08917 q^{38} -1.00000 q^{39} -1.50270 q^{40} -1.90597 q^{41} -0.210711 q^{42} +3.41276 q^{43} -5.97021 q^{44} +1.50270 q^{45} +1.61839 q^{46} -2.58248 q^{47} +1.00000 q^{48} -6.95560 q^{49} +2.74190 q^{50} +5.33536 q^{51} -1.00000 q^{52} +3.85905 q^{53} -1.00000 q^{54} -8.97142 q^{55} -0.210711 q^{56} -3.08917 q^{57} -5.82084 q^{58} -9.92780 q^{59} +1.50270 q^{60} +8.82911 q^{61} -1.12646 q^{62} +0.210711 q^{63} +1.00000 q^{64} -1.50270 q^{65} +5.97021 q^{66} +3.01531 q^{67} +5.33536 q^{68} -1.61839 q^{69} -0.316634 q^{70} -4.95796 q^{71} -1.00000 q^{72} +9.33060 q^{73} +4.46916 q^{74} -2.74190 q^{75} -3.08917 q^{76} -1.25799 q^{77} +1.00000 q^{78} -0.938893 q^{79} +1.50270 q^{80} +1.00000 q^{81} +1.90597 q^{82} +1.74572 q^{83} +0.210711 q^{84} +8.01742 q^{85} -3.41276 q^{86} +5.82084 q^{87} +5.97021 q^{88} -4.95336 q^{89} -1.50270 q^{90} -0.210711 q^{91} -1.61839 q^{92} +1.12646 q^{93} +2.58248 q^{94} -4.64209 q^{95} -1.00000 q^{96} -10.7905 q^{97} +6.95560 q^{98} -5.97021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) 1.50270 0.672026 0.336013 0.941857i \(-0.390921\pi\)
0.336013 + 0.941857i \(0.390921\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.210711 0.0796411 0.0398206 0.999207i \(-0.487321\pi\)
0.0398206 + 0.999207i \(0.487321\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.50270 −0.475194
\(11\) −5.97021 −1.80009 −0.900044 0.435800i \(-0.856466\pi\)
−0.900044 + 0.435800i \(0.856466\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −0.210711 −0.0563148
\(15\) 1.50270 0.387994
\(16\) 1.00000 0.250000
\(17\) 5.33536 1.29401 0.647007 0.762484i \(-0.276020\pi\)
0.647007 + 0.762484i \(0.276020\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.08917 −0.708705 −0.354353 0.935112i \(-0.615299\pi\)
−0.354353 + 0.935112i \(0.615299\pi\)
\(20\) 1.50270 0.336013
\(21\) 0.210711 0.0459808
\(22\) 5.97021 1.27285
\(23\) −1.61839 −0.337457 −0.168729 0.985663i \(-0.553966\pi\)
−0.168729 + 0.985663i \(0.553966\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.74190 −0.548381
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0.210711 0.0398206
\(29\) 5.82084 1.08090 0.540451 0.841375i \(-0.318254\pi\)
0.540451 + 0.841375i \(0.318254\pi\)
\(30\) −1.50270 −0.274353
\(31\) 1.12646 0.202318 0.101159 0.994870i \(-0.467745\pi\)
0.101159 + 0.994870i \(0.467745\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.97021 −1.03928
\(34\) −5.33536 −0.915007
\(35\) 0.316634 0.0535209
\(36\) 1.00000 0.166667
\(37\) −4.46916 −0.734725 −0.367363 0.930078i \(-0.619739\pi\)
−0.367363 + 0.930078i \(0.619739\pi\)
\(38\) 3.08917 0.501130
\(39\) −1.00000 −0.160128
\(40\) −1.50270 −0.237597
\(41\) −1.90597 −0.297663 −0.148832 0.988863i \(-0.547551\pi\)
−0.148832 + 0.988863i \(0.547551\pi\)
\(42\) −0.210711 −0.0325134
\(43\) 3.41276 0.520441 0.260221 0.965549i \(-0.416205\pi\)
0.260221 + 0.965549i \(0.416205\pi\)
\(44\) −5.97021 −0.900044
\(45\) 1.50270 0.224009
\(46\) 1.61839 0.238618
\(47\) −2.58248 −0.376694 −0.188347 0.982103i \(-0.560313\pi\)
−0.188347 + 0.982103i \(0.560313\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.95560 −0.993657
\(50\) 2.74190 0.387764
\(51\) 5.33536 0.747100
\(52\) −1.00000 −0.138675
\(53\) 3.85905 0.530082 0.265041 0.964237i \(-0.414615\pi\)
0.265041 + 0.964237i \(0.414615\pi\)
\(54\) −1.00000 −0.136083
\(55\) −8.97142 −1.20971
\(56\) −0.210711 −0.0281574
\(57\) −3.08917 −0.409171
\(58\) −5.82084 −0.764313
\(59\) −9.92780 −1.29249 −0.646245 0.763130i \(-0.723661\pi\)
−0.646245 + 0.763130i \(0.723661\pi\)
\(60\) 1.50270 0.193997
\(61\) 8.82911 1.13045 0.565226 0.824936i \(-0.308789\pi\)
0.565226 + 0.824936i \(0.308789\pi\)
\(62\) −1.12646 −0.143061
\(63\) 0.210711 0.0265470
\(64\) 1.00000 0.125000
\(65\) −1.50270 −0.186386
\(66\) 5.97021 0.734882
\(67\) 3.01531 0.368379 0.184190 0.982891i \(-0.441034\pi\)
0.184190 + 0.982891i \(0.441034\pi\)
\(68\) 5.33536 0.647007
\(69\) −1.61839 −0.194831
\(70\) −0.316634 −0.0378450
\(71\) −4.95796 −0.588402 −0.294201 0.955744i \(-0.595053\pi\)
−0.294201 + 0.955744i \(0.595053\pi\)
\(72\) −1.00000 −0.117851
\(73\) 9.33060 1.09206 0.546032 0.837764i \(-0.316138\pi\)
0.546032 + 0.837764i \(0.316138\pi\)
\(74\) 4.46916 0.519529
\(75\) −2.74190 −0.316608
\(76\) −3.08917 −0.354353
\(77\) −1.25799 −0.143361
\(78\) 1.00000 0.113228
\(79\) −0.938893 −0.105634 −0.0528168 0.998604i \(-0.516820\pi\)
−0.0528168 + 0.998604i \(0.516820\pi\)
\(80\) 1.50270 0.168007
\(81\) 1.00000 0.111111
\(82\) 1.90597 0.210480
\(83\) 1.74572 0.191618 0.0958091 0.995400i \(-0.469456\pi\)
0.0958091 + 0.995400i \(0.469456\pi\)
\(84\) 0.210711 0.0229904
\(85\) 8.01742 0.869612
\(86\) −3.41276 −0.368008
\(87\) 5.82084 0.624059
\(88\) 5.97021 0.636427
\(89\) −4.95336 −0.525055 −0.262527 0.964925i \(-0.584556\pi\)
−0.262527 + 0.964925i \(0.584556\pi\)
\(90\) −1.50270 −0.158398
\(91\) −0.210711 −0.0220885
\(92\) −1.61839 −0.168729
\(93\) 1.12646 0.116809
\(94\) 2.58248 0.266363
\(95\) −4.64209 −0.476268
\(96\) −1.00000 −0.102062
\(97\) −10.7905 −1.09561 −0.547804 0.836607i \(-0.684536\pi\)
−0.547804 + 0.836607i \(0.684536\pi\)
\(98\) 6.95560 0.702622
\(99\) −5.97021 −0.600029
\(100\) −2.74190 −0.274190
\(101\) 0.162439 0.0161633 0.00808167 0.999967i \(-0.497427\pi\)
0.00808167 + 0.999967i \(0.497427\pi\)
\(102\) −5.33536 −0.528279
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 0.316634 0.0309003
\(106\) −3.85905 −0.374824
\(107\) −1.59344 −0.154043 −0.0770216 0.997029i \(-0.524541\pi\)
−0.0770216 + 0.997029i \(0.524541\pi\)
\(108\) 1.00000 0.0962250
\(109\) 3.58061 0.342960 0.171480 0.985188i \(-0.445145\pi\)
0.171480 + 0.985188i \(0.445145\pi\)
\(110\) 8.97142 0.855391
\(111\) −4.46916 −0.424194
\(112\) 0.210711 0.0199103
\(113\) −18.2195 −1.71395 −0.856974 0.515360i \(-0.827658\pi\)
−0.856974 + 0.515360i \(0.827658\pi\)
\(114\) 3.08917 0.289328
\(115\) −2.43195 −0.226780
\(116\) 5.82084 0.540451
\(117\) −1.00000 −0.0924500
\(118\) 9.92780 0.913928
\(119\) 1.12422 0.103057
\(120\) −1.50270 −0.137177
\(121\) 24.6434 2.24031
\(122\) −8.82911 −0.799350
\(123\) −1.90597 −0.171856
\(124\) 1.12646 0.101159
\(125\) −11.6337 −1.04055
\(126\) −0.210711 −0.0187716
\(127\) −17.8739 −1.58605 −0.793026 0.609188i \(-0.791496\pi\)
−0.793026 + 0.609188i \(0.791496\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.41276 0.300477
\(130\) 1.50270 0.131795
\(131\) −8.17659 −0.714392 −0.357196 0.934030i \(-0.616267\pi\)
−0.357196 + 0.934030i \(0.616267\pi\)
\(132\) −5.97021 −0.519640
\(133\) −0.650922 −0.0564421
\(134\) −3.01531 −0.260483
\(135\) 1.50270 0.129331
\(136\) −5.33536 −0.457503
\(137\) −4.87565 −0.416555 −0.208278 0.978070i \(-0.566786\pi\)
−0.208278 + 0.978070i \(0.566786\pi\)
\(138\) 1.61839 0.137766
\(139\) 10.4014 0.882236 0.441118 0.897449i \(-0.354582\pi\)
0.441118 + 0.897449i \(0.354582\pi\)
\(140\) 0.316634 0.0267605
\(141\) −2.58248 −0.217484
\(142\) 4.95796 0.416063
\(143\) 5.97021 0.499254
\(144\) 1.00000 0.0833333
\(145\) 8.74695 0.726394
\(146\) −9.33060 −0.772206
\(147\) −6.95560 −0.573688
\(148\) −4.46916 −0.367363
\(149\) −7.01473 −0.574669 −0.287334 0.957830i \(-0.592769\pi\)
−0.287334 + 0.957830i \(0.592769\pi\)
\(150\) 2.74190 0.223876
\(151\) −9.45181 −0.769178 −0.384589 0.923088i \(-0.625657\pi\)
−0.384589 + 0.923088i \(0.625657\pi\)
\(152\) 3.08917 0.250565
\(153\) 5.33536 0.431338
\(154\) 1.25799 0.101371
\(155\) 1.69273 0.135963
\(156\) −1.00000 −0.0800641
\(157\) 7.64083 0.609805 0.304902 0.952384i \(-0.401376\pi\)
0.304902 + 0.952384i \(0.401376\pi\)
\(158\) 0.938893 0.0746943
\(159\) 3.85905 0.306043
\(160\) −1.50270 −0.118799
\(161\) −0.341012 −0.0268755
\(162\) −1.00000 −0.0785674
\(163\) −12.7215 −0.996428 −0.498214 0.867054i \(-0.666011\pi\)
−0.498214 + 0.867054i \(0.666011\pi\)
\(164\) −1.90597 −0.148832
\(165\) −8.97142 −0.698424
\(166\) −1.74572 −0.135495
\(167\) −13.8412 −1.07107 −0.535533 0.844514i \(-0.679889\pi\)
−0.535533 + 0.844514i \(0.679889\pi\)
\(168\) −0.210711 −0.0162567
\(169\) 1.00000 0.0769231
\(170\) −8.01742 −0.614908
\(171\) −3.08917 −0.236235
\(172\) 3.41276 0.260221
\(173\) −18.4111 −1.39977 −0.699886 0.714254i \(-0.746766\pi\)
−0.699886 + 0.714254i \(0.746766\pi\)
\(174\) −5.82084 −0.441276
\(175\) −0.577748 −0.0436737
\(176\) −5.97021 −0.450022
\(177\) −9.92780 −0.746219
\(178\) 4.95336 0.371270
\(179\) −22.9519 −1.71550 −0.857752 0.514063i \(-0.828140\pi\)
−0.857752 + 0.514063i \(0.828140\pi\)
\(180\) 1.50270 0.112004
\(181\) −12.2315 −0.909163 −0.454582 0.890705i \(-0.650211\pi\)
−0.454582 + 0.890705i \(0.650211\pi\)
\(182\) 0.210711 0.0156189
\(183\) 8.82911 0.652666
\(184\) 1.61839 0.119309
\(185\) −6.71579 −0.493754
\(186\) −1.12646 −0.0825961
\(187\) −31.8532 −2.32934
\(188\) −2.58248 −0.188347
\(189\) 0.210711 0.0153269
\(190\) 4.64209 0.336773
\(191\) 16.9153 1.22395 0.611974 0.790878i \(-0.290376\pi\)
0.611974 + 0.790878i \(0.290376\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.5001 −0.755816 −0.377908 0.925843i \(-0.623356\pi\)
−0.377908 + 0.925843i \(0.623356\pi\)
\(194\) 10.7905 0.774712
\(195\) −1.50270 −0.107610
\(196\) −6.95560 −0.496829
\(197\) −8.88348 −0.632922 −0.316461 0.948606i \(-0.602495\pi\)
−0.316461 + 0.948606i \(0.602495\pi\)
\(198\) 5.97021 0.424285
\(199\) 10.5193 0.745691 0.372845 0.927893i \(-0.378382\pi\)
0.372845 + 0.927893i \(0.378382\pi\)
\(200\) 2.74190 0.193882
\(201\) 3.01531 0.212684
\(202\) −0.162439 −0.0114292
\(203\) 1.22651 0.0860843
\(204\) 5.33536 0.373550
\(205\) −2.86410 −0.200037
\(206\) −1.00000 −0.0696733
\(207\) −1.61839 −0.112486
\(208\) −1.00000 −0.0693375
\(209\) 18.4430 1.27573
\(210\) −0.316634 −0.0218498
\(211\) 9.22779 0.635267 0.317633 0.948214i \(-0.397112\pi\)
0.317633 + 0.948214i \(0.397112\pi\)
\(212\) 3.85905 0.265041
\(213\) −4.95796 −0.339714
\(214\) 1.59344 0.108925
\(215\) 5.12834 0.349750
\(216\) −1.00000 −0.0680414
\(217\) 0.237357 0.0161129
\(218\) −3.58061 −0.242509
\(219\) 9.33060 0.630504
\(220\) −8.97142 −0.604853
\(221\) −5.33536 −0.358895
\(222\) 4.46916 0.299950
\(223\) −3.58034 −0.239757 −0.119879 0.992789i \(-0.538251\pi\)
−0.119879 + 0.992789i \(0.538251\pi\)
\(224\) −0.210711 −0.0140787
\(225\) −2.74190 −0.182794
\(226\) 18.2195 1.21194
\(227\) −16.0110 −1.06269 −0.531345 0.847156i \(-0.678313\pi\)
−0.531345 + 0.847156i \(0.678313\pi\)
\(228\) −3.08917 −0.204586
\(229\) 1.00564 0.0664544 0.0332272 0.999448i \(-0.489422\pi\)
0.0332272 + 0.999448i \(0.489422\pi\)
\(230\) 2.43195 0.160358
\(231\) −1.25799 −0.0827695
\(232\) −5.82084 −0.382157
\(233\) −5.94744 −0.389630 −0.194815 0.980840i \(-0.562411\pi\)
−0.194815 + 0.980840i \(0.562411\pi\)
\(234\) 1.00000 0.0653720
\(235\) −3.88068 −0.253148
\(236\) −9.92780 −0.646245
\(237\) −0.938893 −0.0609876
\(238\) −1.12422 −0.0728722
\(239\) 12.4367 0.804466 0.402233 0.915537i \(-0.368234\pi\)
0.402233 + 0.915537i \(0.368234\pi\)
\(240\) 1.50270 0.0969986
\(241\) 8.96181 0.577281 0.288641 0.957438i \(-0.406797\pi\)
0.288641 + 0.957438i \(0.406797\pi\)
\(242\) −24.6434 −1.58414
\(243\) 1.00000 0.0641500
\(244\) 8.82911 0.565226
\(245\) −10.4522 −0.667764
\(246\) 1.90597 0.121521
\(247\) 3.08917 0.196559
\(248\) −1.12646 −0.0715303
\(249\) 1.74572 0.110631
\(250\) 11.6337 0.735782
\(251\) −24.8961 −1.57143 −0.785714 0.618590i \(-0.787704\pi\)
−0.785714 + 0.618590i \(0.787704\pi\)
\(252\) 0.210711 0.0132735
\(253\) 9.66212 0.607453
\(254\) 17.8739 1.12151
\(255\) 8.01742 0.502071
\(256\) 1.00000 0.0625000
\(257\) −1.94204 −0.121141 −0.0605706 0.998164i \(-0.519292\pi\)
−0.0605706 + 0.998164i \(0.519292\pi\)
\(258\) −3.41276 −0.212469
\(259\) −0.941699 −0.0585143
\(260\) −1.50270 −0.0931932
\(261\) 5.82084 0.360301
\(262\) 8.17659 0.505151
\(263\) −26.4527 −1.63114 −0.815571 0.578657i \(-0.803577\pi\)
−0.815571 + 0.578657i \(0.803577\pi\)
\(264\) 5.97021 0.367441
\(265\) 5.79898 0.356229
\(266\) 0.650922 0.0399106
\(267\) −4.95336 −0.303141
\(268\) 3.01531 0.184190
\(269\) 8.23376 0.502021 0.251011 0.967984i \(-0.419237\pi\)
0.251011 + 0.967984i \(0.419237\pi\)
\(270\) −1.50270 −0.0914512
\(271\) −13.7981 −0.838176 −0.419088 0.907946i \(-0.637650\pi\)
−0.419088 + 0.907946i \(0.637650\pi\)
\(272\) 5.33536 0.323504
\(273\) −0.210711 −0.0127528
\(274\) 4.87565 0.294549
\(275\) 16.3698 0.987134
\(276\) −1.61839 −0.0974155
\(277\) 6.74113 0.405035 0.202517 0.979279i \(-0.435088\pi\)
0.202517 + 0.979279i \(0.435088\pi\)
\(278\) −10.4014 −0.623835
\(279\) 1.12646 0.0674394
\(280\) −0.316634 −0.0189225
\(281\) −5.37631 −0.320724 −0.160362 0.987058i \(-0.551266\pi\)
−0.160362 + 0.987058i \(0.551266\pi\)
\(282\) 2.58248 0.153785
\(283\) −31.3050 −1.86089 −0.930446 0.366429i \(-0.880580\pi\)
−0.930446 + 0.366429i \(0.880580\pi\)
\(284\) −4.95796 −0.294201
\(285\) −4.64209 −0.274974
\(286\) −5.97021 −0.353026
\(287\) −0.401609 −0.0237062
\(288\) −1.00000 −0.0589256
\(289\) 11.4661 0.674474
\(290\) −8.74695 −0.513638
\(291\) −10.7905 −0.632550
\(292\) 9.33060 0.546032
\(293\) 0.0967142 0.00565010 0.00282505 0.999996i \(-0.499101\pi\)
0.00282505 + 0.999996i \(0.499101\pi\)
\(294\) 6.95560 0.405659
\(295\) −14.9185 −0.868587
\(296\) 4.46916 0.259765
\(297\) −5.97021 −0.346427
\(298\) 7.01473 0.406352
\(299\) 1.61839 0.0935938
\(300\) −2.74190 −0.158304
\(301\) 0.719105 0.0414485
\(302\) 9.45181 0.543891
\(303\) 0.162439 0.00933190
\(304\) −3.08917 −0.177176
\(305\) 13.2675 0.759693
\(306\) −5.33536 −0.305002
\(307\) 9.32491 0.532201 0.266100 0.963945i \(-0.414265\pi\)
0.266100 + 0.963945i \(0.414265\pi\)
\(308\) −1.25799 −0.0716805
\(309\) 1.00000 0.0568880
\(310\) −1.69273 −0.0961405
\(311\) 15.1452 0.858807 0.429404 0.903113i \(-0.358724\pi\)
0.429404 + 0.903113i \(0.358724\pi\)
\(312\) 1.00000 0.0566139
\(313\) 20.6758 1.16867 0.584334 0.811513i \(-0.301356\pi\)
0.584334 + 0.811513i \(0.301356\pi\)
\(314\) −7.64083 −0.431197
\(315\) 0.316634 0.0178403
\(316\) −0.938893 −0.0528168
\(317\) −0.556614 −0.0312625 −0.0156313 0.999878i \(-0.504976\pi\)
−0.0156313 + 0.999878i \(0.504976\pi\)
\(318\) −3.85905 −0.216405
\(319\) −34.7516 −1.94572
\(320\) 1.50270 0.0840033
\(321\) −1.59344 −0.0889369
\(322\) 0.341012 0.0190038
\(323\) −16.4819 −0.917075
\(324\) 1.00000 0.0555556
\(325\) 2.74190 0.152094
\(326\) 12.7215 0.704581
\(327\) 3.58061 0.198008
\(328\) 1.90597 0.105240
\(329\) −0.544156 −0.0300003
\(330\) 8.97142 0.493860
\(331\) −11.8691 −0.652387 −0.326193 0.945303i \(-0.605766\pi\)
−0.326193 + 0.945303i \(0.605766\pi\)
\(332\) 1.74572 0.0958091
\(333\) −4.46916 −0.244908
\(334\) 13.8412 0.757358
\(335\) 4.53110 0.247560
\(336\) 0.210711 0.0114952
\(337\) 12.4993 0.680881 0.340441 0.940266i \(-0.389424\pi\)
0.340441 + 0.940266i \(0.389424\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −18.2195 −0.989548
\(340\) 8.01742 0.434806
\(341\) −6.72521 −0.364191
\(342\) 3.08917 0.167043
\(343\) −2.94059 −0.158777
\(344\) −3.41276 −0.184004
\(345\) −2.43195 −0.130932
\(346\) 18.4111 0.989789
\(347\) −5.29113 −0.284042 −0.142021 0.989864i \(-0.545360\pi\)
−0.142021 + 0.989864i \(0.545360\pi\)
\(348\) 5.82084 0.312030
\(349\) 12.9672 0.694120 0.347060 0.937843i \(-0.387180\pi\)
0.347060 + 0.937843i \(0.387180\pi\)
\(350\) 0.577748 0.0308820
\(351\) −1.00000 −0.0533761
\(352\) 5.97021 0.318213
\(353\) 10.7941 0.574514 0.287257 0.957854i \(-0.407257\pi\)
0.287257 + 0.957854i \(0.407257\pi\)
\(354\) 9.92780 0.527657
\(355\) −7.45031 −0.395421
\(356\) −4.95336 −0.262527
\(357\) 1.12422 0.0594999
\(358\) 22.9519 1.21304
\(359\) 36.9472 1.95000 0.974999 0.222207i \(-0.0713262\pi\)
0.974999 + 0.222207i \(0.0713262\pi\)
\(360\) −1.50270 −0.0791990
\(361\) −9.45700 −0.497737
\(362\) 12.2315 0.642876
\(363\) 24.6434 1.29345
\(364\) −0.210711 −0.0110442
\(365\) 14.0211 0.733896
\(366\) −8.82911 −0.461505
\(367\) 10.4460 0.545279 0.272640 0.962116i \(-0.412103\pi\)
0.272640 + 0.962116i \(0.412103\pi\)
\(368\) −1.61839 −0.0843643
\(369\) −1.90597 −0.0992211
\(370\) 6.71579 0.349137
\(371\) 0.813143 0.0422163
\(372\) 1.12646 0.0584043
\(373\) 29.3786 1.52117 0.760583 0.649240i \(-0.224913\pi\)
0.760583 + 0.649240i \(0.224913\pi\)
\(374\) 31.8532 1.64709
\(375\) −11.6337 −0.600763
\(376\) 2.58248 0.133181
\(377\) −5.82084 −0.299788
\(378\) −0.210711 −0.0108378
\(379\) −23.7275 −1.21880 −0.609401 0.792862i \(-0.708590\pi\)
−0.609401 + 0.792862i \(0.708590\pi\)
\(380\) −4.64209 −0.238134
\(381\) −17.8739 −0.915708
\(382\) −16.9153 −0.865462
\(383\) −29.9546 −1.53061 −0.765306 0.643667i \(-0.777412\pi\)
−0.765306 + 0.643667i \(0.777412\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −1.89037 −0.0963423
\(386\) 10.5001 0.534442
\(387\) 3.41276 0.173480
\(388\) −10.7905 −0.547804
\(389\) 1.59115 0.0806743 0.0403371 0.999186i \(-0.487157\pi\)
0.0403371 + 0.999186i \(0.487157\pi\)
\(390\) 1.50270 0.0760920
\(391\) −8.63468 −0.436675
\(392\) 6.95560 0.351311
\(393\) −8.17659 −0.412454
\(394\) 8.88348 0.447543
\(395\) −1.41087 −0.0709886
\(396\) −5.97021 −0.300015
\(397\) 24.9523 1.25232 0.626160 0.779695i \(-0.284626\pi\)
0.626160 + 0.779695i \(0.284626\pi\)
\(398\) −10.5193 −0.527283
\(399\) −0.650922 −0.0325868
\(400\) −2.74190 −0.137095
\(401\) −17.1289 −0.855378 −0.427689 0.903926i \(-0.640672\pi\)
−0.427689 + 0.903926i \(0.640672\pi\)
\(402\) −3.01531 −0.150390
\(403\) −1.12646 −0.0561130
\(404\) 0.162439 0.00808167
\(405\) 1.50270 0.0746696
\(406\) −1.22651 −0.0608708
\(407\) 26.6818 1.32257
\(408\) −5.33536 −0.264140
\(409\) −14.3710 −0.710600 −0.355300 0.934752i \(-0.615621\pi\)
−0.355300 + 0.934752i \(0.615621\pi\)
\(410\) 2.86410 0.141448
\(411\) −4.87565 −0.240498
\(412\) 1.00000 0.0492665
\(413\) −2.09189 −0.102935
\(414\) 1.61839 0.0795395
\(415\) 2.62329 0.128772
\(416\) 1.00000 0.0490290
\(417\) 10.4014 0.509359
\(418\) −18.4430 −0.902078
\(419\) 2.17462 0.106237 0.0531187 0.998588i \(-0.483084\pi\)
0.0531187 + 0.998588i \(0.483084\pi\)
\(420\) 0.316634 0.0154502
\(421\) −12.9617 −0.631714 −0.315857 0.948807i \(-0.602292\pi\)
−0.315857 + 0.948807i \(0.602292\pi\)
\(422\) −9.22779 −0.449202
\(423\) −2.58248 −0.125565
\(424\) −3.85905 −0.187412
\(425\) −14.6290 −0.709613
\(426\) 4.95796 0.240214
\(427\) 1.86039 0.0900304
\(428\) −1.59344 −0.0770216
\(429\) 5.97021 0.288245
\(430\) −5.12834 −0.247311
\(431\) 35.9482 1.73157 0.865783 0.500420i \(-0.166821\pi\)
0.865783 + 0.500420i \(0.166821\pi\)
\(432\) 1.00000 0.0481125
\(433\) 40.0696 1.92562 0.962810 0.270179i \(-0.0870828\pi\)
0.962810 + 0.270179i \(0.0870828\pi\)
\(434\) −0.237357 −0.0113935
\(435\) 8.74695 0.419384
\(436\) 3.58061 0.171480
\(437\) 4.99948 0.239158
\(438\) −9.33060 −0.445833
\(439\) 11.9083 0.568352 0.284176 0.958772i \(-0.408280\pi\)
0.284176 + 0.958772i \(0.408280\pi\)
\(440\) 8.97142 0.427695
\(441\) −6.95560 −0.331219
\(442\) 5.33536 0.253777
\(443\) 18.5916 0.883316 0.441658 0.897184i \(-0.354390\pi\)
0.441658 + 0.897184i \(0.354390\pi\)
\(444\) −4.46916 −0.212097
\(445\) −7.44339 −0.352850
\(446\) 3.58034 0.169534
\(447\) −7.01473 −0.331785
\(448\) 0.210711 0.00995514
\(449\) 23.0858 1.08949 0.544744 0.838603i \(-0.316627\pi\)
0.544744 + 0.838603i \(0.316627\pi\)
\(450\) 2.74190 0.129255
\(451\) 11.3791 0.535820
\(452\) −18.2195 −0.856974
\(453\) −9.45181 −0.444085
\(454\) 16.0110 0.751435
\(455\) −0.316634 −0.0148440
\(456\) 3.08917 0.144664
\(457\) 9.15541 0.428272 0.214136 0.976804i \(-0.431306\pi\)
0.214136 + 0.976804i \(0.431306\pi\)
\(458\) −1.00564 −0.0469903
\(459\) 5.33536 0.249033
\(460\) −2.43195 −0.113390
\(461\) 24.0754 1.12130 0.560652 0.828052i \(-0.310551\pi\)
0.560652 + 0.828052i \(0.310551\pi\)
\(462\) 1.25799 0.0585269
\(463\) −16.3129 −0.758127 −0.379063 0.925371i \(-0.623754\pi\)
−0.379063 + 0.925371i \(0.623754\pi\)
\(464\) 5.82084 0.270226
\(465\) 1.69273 0.0784984
\(466\) 5.94744 0.275510
\(467\) −11.2856 −0.522234 −0.261117 0.965307i \(-0.584091\pi\)
−0.261117 + 0.965307i \(0.584091\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0.635359 0.0293381
\(470\) 3.88068 0.179003
\(471\) 7.64083 0.352071
\(472\) 9.92780 0.456964
\(473\) −20.3749 −0.936840
\(474\) 0.938893 0.0431248
\(475\) 8.47022 0.388640
\(476\) 1.12422 0.0515284
\(477\) 3.85905 0.176694
\(478\) −12.4367 −0.568843
\(479\) −14.2312 −0.650238 −0.325119 0.945673i \(-0.605404\pi\)
−0.325119 + 0.945673i \(0.605404\pi\)
\(480\) −1.50270 −0.0685884
\(481\) 4.46916 0.203776
\(482\) −8.96181 −0.408199
\(483\) −0.341012 −0.0155166
\(484\) 24.6434 1.12016
\(485\) −16.2148 −0.736277
\(486\) −1.00000 −0.0453609
\(487\) 28.6776 1.29951 0.649753 0.760145i \(-0.274872\pi\)
0.649753 + 0.760145i \(0.274872\pi\)
\(488\) −8.82911 −0.399675
\(489\) −12.7215 −0.575288
\(490\) 10.4522 0.472180
\(491\) −11.8787 −0.536077 −0.268039 0.963408i \(-0.586375\pi\)
−0.268039 + 0.963408i \(0.586375\pi\)
\(492\) −1.90597 −0.0859280
\(493\) 31.0563 1.39870
\(494\) −3.08917 −0.138989
\(495\) −8.97142 −0.403235
\(496\) 1.12646 0.0505796
\(497\) −1.04469 −0.0468610
\(498\) −1.74572 −0.0782278
\(499\) −25.6731 −1.14928 −0.574642 0.818405i \(-0.694859\pi\)
−0.574642 + 0.818405i \(0.694859\pi\)
\(500\) −11.6337 −0.520276
\(501\) −13.8412 −0.618380
\(502\) 24.8961 1.11117
\(503\) −26.7132 −1.19108 −0.595541 0.803325i \(-0.703062\pi\)
−0.595541 + 0.803325i \(0.703062\pi\)
\(504\) −0.210711 −0.00938580
\(505\) 0.244097 0.0108622
\(506\) −9.66212 −0.429534
\(507\) 1.00000 0.0444116
\(508\) −17.8739 −0.793026
\(509\) 0.575983 0.0255300 0.0127650 0.999919i \(-0.495937\pi\)
0.0127650 + 0.999919i \(0.495937\pi\)
\(510\) −8.01742 −0.355017
\(511\) 1.96606 0.0869732
\(512\) −1.00000 −0.0441942
\(513\) −3.08917 −0.136390
\(514\) 1.94204 0.0856598
\(515\) 1.50270 0.0662167
\(516\) 3.41276 0.150238
\(517\) 15.4180 0.678081
\(518\) 0.941699 0.0413759
\(519\) −18.4111 −0.808159
\(520\) 1.50270 0.0658976
\(521\) −30.9555 −1.35618 −0.678092 0.734977i \(-0.737193\pi\)
−0.678092 + 0.734977i \(0.737193\pi\)
\(522\) −5.82084 −0.254771
\(523\) 3.58907 0.156939 0.0784695 0.996917i \(-0.474997\pi\)
0.0784695 + 0.996917i \(0.474997\pi\)
\(524\) −8.17659 −0.357196
\(525\) −0.577748 −0.0252150
\(526\) 26.4527 1.15339
\(527\) 6.01007 0.261803
\(528\) −5.97021 −0.259820
\(529\) −20.3808 −0.886123
\(530\) −5.79898 −0.251892
\(531\) −9.92780 −0.430830
\(532\) −0.650922 −0.0282210
\(533\) 1.90597 0.0825569
\(534\) 4.95336 0.214353
\(535\) −2.39445 −0.103521
\(536\) −3.01531 −0.130242
\(537\) −22.9519 −0.990447
\(538\) −8.23376 −0.354983
\(539\) 41.5264 1.78867
\(540\) 1.50270 0.0646657
\(541\) 6.41437 0.275775 0.137888 0.990448i \(-0.455969\pi\)
0.137888 + 0.990448i \(0.455969\pi\)
\(542\) 13.7981 0.592680
\(543\) −12.2315 −0.524906
\(544\) −5.33536 −0.228752
\(545\) 5.38057 0.230478
\(546\) 0.210711 0.00901758
\(547\) 1.09095 0.0466455 0.0233228 0.999728i \(-0.492575\pi\)
0.0233228 + 0.999728i \(0.492575\pi\)
\(548\) −4.87565 −0.208278
\(549\) 8.82911 0.376817
\(550\) −16.3698 −0.698009
\(551\) −17.9816 −0.766041
\(552\) 1.61839 0.0688832
\(553\) −0.197835 −0.00841278
\(554\) −6.74113 −0.286403
\(555\) −6.71579 −0.285069
\(556\) 10.4014 0.441118
\(557\) 4.09704 0.173597 0.0867985 0.996226i \(-0.472336\pi\)
0.0867985 + 0.996226i \(0.472336\pi\)
\(558\) −1.12646 −0.0476869
\(559\) −3.41276 −0.144344
\(560\) 0.316634 0.0133802
\(561\) −31.8532 −1.34484
\(562\) 5.37631 0.226786
\(563\) −26.0152 −1.09641 −0.548206 0.836343i \(-0.684689\pi\)
−0.548206 + 0.836343i \(0.684689\pi\)
\(564\) −2.58248 −0.108742
\(565\) −27.3784 −1.15182
\(566\) 31.3050 1.31585
\(567\) 0.210711 0.00884901
\(568\) 4.95796 0.208031
\(569\) −41.2205 −1.72805 −0.864026 0.503446i \(-0.832065\pi\)
−0.864026 + 0.503446i \(0.832065\pi\)
\(570\) 4.64209 0.194436
\(571\) −27.1897 −1.13785 −0.568927 0.822388i \(-0.692641\pi\)
−0.568927 + 0.822388i \(0.692641\pi\)
\(572\) 5.97021 0.249627
\(573\) 16.9153 0.706647
\(574\) 0.401609 0.0167628
\(575\) 4.43747 0.185055
\(576\) 1.00000 0.0416667
\(577\) 15.5388 0.646890 0.323445 0.946247i \(-0.395159\pi\)
0.323445 + 0.946247i \(0.395159\pi\)
\(578\) −11.4661 −0.476925
\(579\) −10.5001 −0.436370
\(580\) 8.74695 0.363197
\(581\) 0.367843 0.0152607
\(582\) 10.7905 0.447280
\(583\) −23.0394 −0.954193
\(584\) −9.33060 −0.386103
\(585\) −1.50270 −0.0621288
\(586\) −0.0967142 −0.00399523
\(587\) 39.9662 1.64958 0.824791 0.565438i \(-0.191293\pi\)
0.824791 + 0.565438i \(0.191293\pi\)
\(588\) −6.95560 −0.286844
\(589\) −3.47983 −0.143384
\(590\) 14.9185 0.614183
\(591\) −8.88348 −0.365418
\(592\) −4.46916 −0.183681
\(593\) 13.3409 0.547846 0.273923 0.961752i \(-0.411679\pi\)
0.273923 + 0.961752i \(0.411679\pi\)
\(594\) 5.97021 0.244961
\(595\) 1.68936 0.0692568
\(596\) −7.01473 −0.287334
\(597\) 10.5193 0.430525
\(598\) −1.61839 −0.0661808
\(599\) −6.78375 −0.277176 −0.138588 0.990350i \(-0.544256\pi\)
−0.138588 + 0.990350i \(0.544256\pi\)
\(600\) 2.74190 0.111938
\(601\) −19.6675 −0.802254 −0.401127 0.916022i \(-0.631381\pi\)
−0.401127 + 0.916022i \(0.631381\pi\)
\(602\) −0.719105 −0.0293085
\(603\) 3.01531 0.122793
\(604\) −9.45181 −0.384589
\(605\) 37.0316 1.50555
\(606\) −0.162439 −0.00659865
\(607\) −47.8269 −1.94123 −0.970617 0.240631i \(-0.922646\pi\)
−0.970617 + 0.240631i \(0.922646\pi\)
\(608\) 3.08917 0.125283
\(609\) 1.22651 0.0497008
\(610\) −13.2675 −0.537184
\(611\) 2.58248 0.104476
\(612\) 5.33536 0.215669
\(613\) 4.52465 0.182749 0.0913744 0.995817i \(-0.470874\pi\)
0.0913744 + 0.995817i \(0.470874\pi\)
\(614\) −9.32491 −0.376323
\(615\) −2.86410 −0.115492
\(616\) 1.25799 0.0506857
\(617\) 31.8462 1.28208 0.641040 0.767508i \(-0.278503\pi\)
0.641040 + 0.767508i \(0.278503\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 25.0095 1.00522 0.502608 0.864515i \(-0.332374\pi\)
0.502608 + 0.864515i \(0.332374\pi\)
\(620\) 1.69273 0.0679816
\(621\) −1.61839 −0.0649437
\(622\) −15.1452 −0.607269
\(623\) −1.04372 −0.0418160
\(624\) −1.00000 −0.0400320
\(625\) −3.77243 −0.150897
\(626\) −20.6758 −0.826373
\(627\) 18.4430 0.736544
\(628\) 7.64083 0.304902
\(629\) −23.8446 −0.950745
\(630\) −0.316634 −0.0126150
\(631\) 21.0089 0.836350 0.418175 0.908367i \(-0.362670\pi\)
0.418175 + 0.908367i \(0.362670\pi\)
\(632\) 0.938893 0.0373471
\(633\) 9.22779 0.366772
\(634\) 0.556614 0.0221060
\(635\) −26.8590 −1.06587
\(636\) 3.85905 0.153021
\(637\) 6.95560 0.275591
\(638\) 34.7516 1.37583
\(639\) −4.95796 −0.196134
\(640\) −1.50270 −0.0593993
\(641\) 1.92961 0.0762149 0.0381075 0.999274i \(-0.487867\pi\)
0.0381075 + 0.999274i \(0.487867\pi\)
\(642\) 1.59344 0.0628879
\(643\) 19.5628 0.771481 0.385740 0.922607i \(-0.373946\pi\)
0.385740 + 0.922607i \(0.373946\pi\)
\(644\) −0.341012 −0.0134377
\(645\) 5.12834 0.201928
\(646\) 16.4819 0.648470
\(647\) −18.5003 −0.727320 −0.363660 0.931532i \(-0.618473\pi\)
−0.363660 + 0.931532i \(0.618473\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 59.2711 2.32659
\(650\) −2.74190 −0.107546
\(651\) 0.237357 0.00930276
\(652\) −12.7215 −0.498214
\(653\) −0.532652 −0.0208443 −0.0104221 0.999946i \(-0.503318\pi\)
−0.0104221 + 0.999946i \(0.503318\pi\)
\(654\) −3.58061 −0.140013
\(655\) −12.2869 −0.480090
\(656\) −1.90597 −0.0744158
\(657\) 9.33060 0.364021
\(658\) 0.544156 0.0212134
\(659\) 42.0313 1.63731 0.818654 0.574287i \(-0.194720\pi\)
0.818654 + 0.574287i \(0.194720\pi\)
\(660\) −8.97142 −0.349212
\(661\) 22.7122 0.883401 0.441700 0.897163i \(-0.354375\pi\)
0.441700 + 0.897163i \(0.354375\pi\)
\(662\) 11.8691 0.461307
\(663\) −5.33536 −0.207208
\(664\) −1.74572 −0.0677473
\(665\) −0.978138 −0.0379305
\(666\) 4.46916 0.173176
\(667\) −9.42037 −0.364758
\(668\) −13.8412 −0.535533
\(669\) −3.58034 −0.138424
\(670\) −4.53110 −0.175052
\(671\) −52.7117 −2.03491
\(672\) −0.210711 −0.00812834
\(673\) −1.45877 −0.0562313 −0.0281156 0.999605i \(-0.508951\pi\)
−0.0281156 + 0.999605i \(0.508951\pi\)
\(674\) −12.4993 −0.481456
\(675\) −2.74190 −0.105536
\(676\) 1.00000 0.0384615
\(677\) −24.7516 −0.951281 −0.475641 0.879640i \(-0.657784\pi\)
−0.475641 + 0.879640i \(0.657784\pi\)
\(678\) 18.2195 0.699716
\(679\) −2.27367 −0.0872554
\(680\) −8.01742 −0.307454
\(681\) −16.0110 −0.613544
\(682\) 6.72521 0.257522
\(683\) −37.0495 −1.41766 −0.708830 0.705380i \(-0.750777\pi\)
−0.708830 + 0.705380i \(0.750777\pi\)
\(684\) −3.08917 −0.118118
\(685\) −7.32662 −0.279936
\(686\) 2.94059 0.112272
\(687\) 1.00564 0.0383675
\(688\) 3.41276 0.130110
\(689\) −3.85905 −0.147018
\(690\) 2.43195 0.0925826
\(691\) 31.9134 1.21404 0.607021 0.794685i \(-0.292364\pi\)
0.607021 + 0.794685i \(0.292364\pi\)
\(692\) −18.4111 −0.699886
\(693\) −1.25799 −0.0477870
\(694\) 5.29113 0.200848
\(695\) 15.6302 0.592886
\(696\) −5.82084 −0.220638
\(697\) −10.1691 −0.385181
\(698\) −12.9672 −0.490817
\(699\) −5.94744 −0.224953
\(700\) −0.577748 −0.0218368
\(701\) −17.1796 −0.648863 −0.324431 0.945909i \(-0.605173\pi\)
−0.324431 + 0.945909i \(0.605173\pi\)
\(702\) 1.00000 0.0377426
\(703\) 13.8060 0.520703
\(704\) −5.97021 −0.225011
\(705\) −3.88068 −0.146155
\(706\) −10.7941 −0.406243
\(707\) 0.0342277 0.00128727
\(708\) −9.92780 −0.373110
\(709\) −44.2304 −1.66111 −0.830555 0.556937i \(-0.811976\pi\)
−0.830555 + 0.556937i \(0.811976\pi\)
\(710\) 7.45031 0.279605
\(711\) −0.938893 −0.0352112
\(712\) 4.95336 0.185635
\(713\) −1.82305 −0.0682738
\(714\) −1.12422 −0.0420728
\(715\) 8.97142 0.335512
\(716\) −22.9519 −0.857752
\(717\) 12.4367 0.464459
\(718\) −36.9472 −1.37886
\(719\) 38.6635 1.44191 0.720954 0.692983i \(-0.243704\pi\)
0.720954 + 0.692983i \(0.243704\pi\)
\(720\) 1.50270 0.0560022
\(721\) 0.210711 0.00784727
\(722\) 9.45700 0.351953
\(723\) 8.96181 0.333293
\(724\) −12.2315 −0.454582
\(725\) −15.9602 −0.592746
\(726\) −24.6434 −0.914604
\(727\) −23.5817 −0.874598 −0.437299 0.899316i \(-0.644065\pi\)
−0.437299 + 0.899316i \(0.644065\pi\)
\(728\) 0.210711 0.00780945
\(729\) 1.00000 0.0370370
\(730\) −14.0211 −0.518943
\(731\) 18.2083 0.673459
\(732\) 8.82911 0.326333
\(733\) −16.0518 −0.592886 −0.296443 0.955051i \(-0.595801\pi\)
−0.296443 + 0.955051i \(0.595801\pi\)
\(734\) −10.4460 −0.385571
\(735\) −10.4522 −0.385533
\(736\) 1.61839 0.0596546
\(737\) −18.0021 −0.663115
\(738\) 1.90597 0.0701599
\(739\) −3.84802 −0.141552 −0.0707759 0.997492i \(-0.522548\pi\)
−0.0707759 + 0.997492i \(0.522548\pi\)
\(740\) −6.71579 −0.246877
\(741\) 3.08917 0.113484
\(742\) −0.813143 −0.0298514
\(743\) −41.7331 −1.53104 −0.765519 0.643413i \(-0.777518\pi\)
−0.765519 + 0.643413i \(0.777518\pi\)
\(744\) −1.12646 −0.0412980
\(745\) −10.5410 −0.386193
\(746\) −29.3786 −1.07563
\(747\) 1.74572 0.0638727
\(748\) −31.8532 −1.16467
\(749\) −0.335754 −0.0122682
\(750\) 11.6337 0.424804
\(751\) 8.47303 0.309185 0.154593 0.987978i \(-0.450594\pi\)
0.154593 + 0.987978i \(0.450594\pi\)
\(752\) −2.58248 −0.0941734
\(753\) −24.8961 −0.907264
\(754\) 5.82084 0.211982
\(755\) −14.2032 −0.516907
\(756\) 0.210711 0.00766347
\(757\) 20.7148 0.752894 0.376447 0.926438i \(-0.377146\pi\)
0.376447 + 0.926438i \(0.377146\pi\)
\(758\) 23.7275 0.861823
\(759\) 9.66212 0.350713
\(760\) 4.64209 0.168386
\(761\) −24.1299 −0.874708 −0.437354 0.899290i \(-0.644084\pi\)
−0.437354 + 0.899290i \(0.644084\pi\)
\(762\) 17.8739 0.647503
\(763\) 0.754472 0.0273137
\(764\) 16.9153 0.611974
\(765\) 8.01742 0.289871
\(766\) 29.9546 1.08231
\(767\) 9.92780 0.358472
\(768\) 1.00000 0.0360844
\(769\) −13.0541 −0.470742 −0.235371 0.971906i \(-0.575631\pi\)
−0.235371 + 0.971906i \(0.575631\pi\)
\(770\) 1.89037 0.0681243
\(771\) −1.94204 −0.0699410
\(772\) −10.5001 −0.377908
\(773\) 34.6144 1.24499 0.622497 0.782622i \(-0.286118\pi\)
0.622497 + 0.782622i \(0.286118\pi\)
\(774\) −3.41276 −0.122669
\(775\) −3.08865 −0.110948
\(776\) 10.7905 0.387356
\(777\) −0.941699 −0.0337833
\(778\) −1.59115 −0.0570453
\(779\) 5.88789 0.210956
\(780\) −1.50270 −0.0538051
\(781\) 29.6001 1.05917
\(782\) 8.63468 0.308776
\(783\) 5.82084 0.208020
\(784\) −6.95560 −0.248414
\(785\) 11.4818 0.409805
\(786\) 8.17659 0.291649
\(787\) 17.9302 0.639141 0.319571 0.947563i \(-0.396461\pi\)
0.319571 + 0.947563i \(0.396461\pi\)
\(788\) −8.88348 −0.316461
\(789\) −26.4527 −0.941740
\(790\) 1.41087 0.0501965
\(791\) −3.83904 −0.136501
\(792\) 5.97021 0.212142
\(793\) −8.82911 −0.313531
\(794\) −24.9523 −0.885524
\(795\) 5.79898 0.205669
\(796\) 10.5193 0.372845
\(797\) 26.2528 0.929922 0.464961 0.885331i \(-0.346068\pi\)
0.464961 + 0.885331i \(0.346068\pi\)
\(798\) 0.650922 0.0230424
\(799\) −13.7785 −0.487447
\(800\) 2.74190 0.0969410
\(801\) −4.95336 −0.175018
\(802\) 17.1289 0.604843
\(803\) −55.7057 −1.96581
\(804\) 3.01531 0.106342
\(805\) −0.512437 −0.0180610
\(806\) 1.12646 0.0396779
\(807\) 8.23376 0.289842
\(808\) −0.162439 −0.00571460
\(809\) −23.2678 −0.818054 −0.409027 0.912522i \(-0.634132\pi\)
−0.409027 + 0.912522i \(0.634132\pi\)
\(810\) −1.50270 −0.0527994
\(811\) 20.2770 0.712023 0.356012 0.934482i \(-0.384136\pi\)
0.356012 + 0.934482i \(0.384136\pi\)
\(812\) 1.22651 0.0430421
\(813\) −13.7981 −0.483921
\(814\) −26.6818 −0.935198
\(815\) −19.1166 −0.669626
\(816\) 5.33536 0.186775
\(817\) −10.5426 −0.368839
\(818\) 14.3710 0.502470
\(819\) −0.210711 −0.00736282
\(820\) −2.86410 −0.100019
\(821\) 22.7486 0.793933 0.396967 0.917833i \(-0.370063\pi\)
0.396967 + 0.917833i \(0.370063\pi\)
\(822\) 4.87565 0.170058
\(823\) 40.4001 1.40826 0.704130 0.710071i \(-0.251337\pi\)
0.704130 + 0.710071i \(0.251337\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 16.3698 0.569922
\(826\) 2.09189 0.0727863
\(827\) −40.6620 −1.41396 −0.706978 0.707235i \(-0.749942\pi\)
−0.706978 + 0.707235i \(0.749942\pi\)
\(828\) −1.61839 −0.0562429
\(829\) 8.74279 0.303650 0.151825 0.988407i \(-0.451485\pi\)
0.151825 + 0.988407i \(0.451485\pi\)
\(830\) −2.62329 −0.0910558
\(831\) 6.74113 0.233847
\(832\) −1.00000 −0.0346688
\(833\) −37.1106 −1.28581
\(834\) −10.4014 −0.360171
\(835\) −20.7992 −0.719784
\(836\) 18.4430 0.637866
\(837\) 1.12646 0.0389362
\(838\) −2.17462 −0.0751211
\(839\) 22.4549 0.775228 0.387614 0.921822i \(-0.373299\pi\)
0.387614 + 0.921822i \(0.373299\pi\)
\(840\) −0.316634 −0.0109249
\(841\) 4.88214 0.168350
\(842\) 12.9617 0.446689
\(843\) −5.37631 −0.185170
\(844\) 9.22779 0.317633
\(845\) 1.50270 0.0516943
\(846\) 2.58248 0.0887875
\(847\) 5.19264 0.178421
\(848\) 3.85905 0.132520
\(849\) −31.3050 −1.07439
\(850\) 14.6290 0.501772
\(851\) 7.23283 0.247938
\(852\) −4.95796 −0.169857
\(853\) −44.6610 −1.52917 −0.764583 0.644526i \(-0.777055\pi\)
−0.764583 + 0.644526i \(0.777055\pi\)
\(854\) −1.86039 −0.0636611
\(855\) −4.64209 −0.158756
\(856\) 1.59344 0.0544625
\(857\) 2.68975 0.0918802 0.0459401 0.998944i \(-0.485372\pi\)
0.0459401 + 0.998944i \(0.485372\pi\)
\(858\) −5.97021 −0.203820
\(859\) 29.0991 0.992849 0.496424 0.868080i \(-0.334646\pi\)
0.496424 + 0.868080i \(0.334646\pi\)
\(860\) 5.12834 0.174875
\(861\) −0.401609 −0.0136868
\(862\) −35.9482 −1.22440
\(863\) −5.09589 −0.173466 −0.0867331 0.996232i \(-0.527643\pi\)
−0.0867331 + 0.996232i \(0.527643\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −27.6663 −0.940684
\(866\) −40.0696 −1.36162
\(867\) 11.4661 0.389408
\(868\) 0.237357 0.00805643
\(869\) 5.60539 0.190150
\(870\) −8.74695 −0.296549
\(871\) −3.01531 −0.102170
\(872\) −3.58061 −0.121255
\(873\) −10.7905 −0.365203
\(874\) −4.99948 −0.169110
\(875\) −2.45135 −0.0828708
\(876\) 9.33060 0.315252
\(877\) −4.48966 −0.151605 −0.0758025 0.997123i \(-0.524152\pi\)
−0.0758025 + 0.997123i \(0.524152\pi\)
\(878\) −11.9083 −0.401886
\(879\) 0.0967142 0.00326209
\(880\) −8.97142 −0.302426
\(881\) −22.8961 −0.771391 −0.385695 0.922626i \(-0.626038\pi\)
−0.385695 + 0.922626i \(0.626038\pi\)
\(882\) 6.95560 0.234207
\(883\) 5.50760 0.185345 0.0926727 0.995697i \(-0.470459\pi\)
0.0926727 + 0.995697i \(0.470459\pi\)
\(884\) −5.33536 −0.179448
\(885\) −14.9185 −0.501479
\(886\) −18.5916 −0.624599
\(887\) 10.8054 0.362809 0.181404 0.983409i \(-0.441936\pi\)
0.181404 + 0.983409i \(0.441936\pi\)
\(888\) 4.46916 0.149975
\(889\) −3.76622 −0.126315
\(890\) 7.44339 0.249503
\(891\) −5.97021 −0.200010
\(892\) −3.58034 −0.119879
\(893\) 7.97773 0.266965
\(894\) 7.01473 0.234608
\(895\) −34.4897 −1.15286
\(896\) −0.210711 −0.00703935
\(897\) 1.61839 0.0540364
\(898\) −23.0858 −0.770384
\(899\) 6.55694 0.218686
\(900\) −2.74190 −0.0913968
\(901\) 20.5894 0.685933
\(902\) −11.3791 −0.378882
\(903\) 0.719105 0.0239303
\(904\) 18.2195 0.605972
\(905\) −18.3803 −0.610982
\(906\) 9.45181 0.314015
\(907\) 4.33306 0.143877 0.0719385 0.997409i \(-0.477081\pi\)
0.0719385 + 0.997409i \(0.477081\pi\)
\(908\) −16.0110 −0.531345
\(909\) 0.162439 0.00538778
\(910\) 0.316634 0.0104963
\(911\) 14.5252 0.481242 0.240621 0.970619i \(-0.422649\pi\)
0.240621 + 0.970619i \(0.422649\pi\)
\(912\) −3.08917 −0.102293
\(913\) −10.4223 −0.344929
\(914\) −9.15541 −0.302834
\(915\) 13.2675 0.438609
\(916\) 1.00564 0.0332272
\(917\) −1.72289 −0.0568949
\(918\) −5.33536 −0.176093
\(919\) 46.0785 1.51999 0.759994 0.649930i \(-0.225202\pi\)
0.759994 + 0.649930i \(0.225202\pi\)
\(920\) 2.43195 0.0801789
\(921\) 9.32491 0.307266
\(922\) −24.0754 −0.792881
\(923\) 4.95796 0.163193
\(924\) −1.25799 −0.0413847
\(925\) 12.2540 0.402909
\(926\) 16.3129 0.536077
\(927\) 1.00000 0.0328443
\(928\) −5.82084 −0.191078
\(929\) −13.9868 −0.458892 −0.229446 0.973321i \(-0.573691\pi\)
−0.229446 + 0.973321i \(0.573691\pi\)
\(930\) −1.69273 −0.0555067
\(931\) 21.4871 0.704210
\(932\) −5.94744 −0.194815
\(933\) 15.1452 0.495833
\(934\) 11.2856 0.369275
\(935\) −47.8657 −1.56538
\(936\) 1.00000 0.0326860
\(937\) −35.6776 −1.16554 −0.582769 0.812638i \(-0.698031\pi\)
−0.582769 + 0.812638i \(0.698031\pi\)
\(938\) −0.635359 −0.0207452
\(939\) 20.6758 0.674731
\(940\) −3.88068 −0.126574
\(941\) −11.8984 −0.387877 −0.193938 0.981014i \(-0.562126\pi\)
−0.193938 + 0.981014i \(0.562126\pi\)
\(942\) −7.64083 −0.248952
\(943\) 3.08461 0.100449
\(944\) −9.92780 −0.323122
\(945\) 0.316634 0.0103001
\(946\) 20.3749 0.662446
\(947\) 44.4092 1.44310 0.721552 0.692360i \(-0.243429\pi\)
0.721552 + 0.692360i \(0.243429\pi\)
\(948\) −0.938893 −0.0304938
\(949\) −9.33060 −0.302884
\(950\) −8.47022 −0.274810
\(951\) −0.556614 −0.0180494
\(952\) −1.12422 −0.0364361
\(953\) 30.5537 0.989731 0.494866 0.868969i \(-0.335217\pi\)
0.494866 + 0.868969i \(0.335217\pi\)
\(954\) −3.85905 −0.124941
\(955\) 25.4186 0.822525
\(956\) 12.4367 0.402233
\(957\) −34.7516 −1.12336
\(958\) 14.2312 0.459788
\(959\) −1.02735 −0.0331749
\(960\) 1.50270 0.0484993
\(961\) −29.7311 −0.959067
\(962\) −4.46916 −0.144091
\(963\) −1.59344 −0.0513478
\(964\) 8.96181 0.288641
\(965\) −15.7785 −0.507928
\(966\) 0.341012 0.0109719
\(967\) 30.4137 0.978038 0.489019 0.872273i \(-0.337355\pi\)
0.489019 + 0.872273i \(0.337355\pi\)
\(968\) −24.6434 −0.792070
\(969\) −16.4819 −0.529474
\(970\) 16.2148 0.520626
\(971\) 22.7762 0.730922 0.365461 0.930827i \(-0.380911\pi\)
0.365461 + 0.930827i \(0.380911\pi\)
\(972\) 1.00000 0.0320750
\(973\) 2.19169 0.0702623
\(974\) −28.6776 −0.918889
\(975\) 2.74190 0.0878112
\(976\) 8.82911 0.282613
\(977\) −38.1820 −1.22155 −0.610776 0.791804i \(-0.709142\pi\)
−0.610776 + 0.791804i \(0.709142\pi\)
\(978\) 12.7215 0.406790
\(979\) 29.5726 0.945144
\(980\) −10.4522 −0.333882
\(981\) 3.58061 0.114320
\(982\) 11.8787 0.379064
\(983\) −32.1683 −1.02601 −0.513005 0.858386i \(-0.671468\pi\)
−0.513005 + 0.858386i \(0.671468\pi\)
\(984\) 1.90597 0.0607603
\(985\) −13.3492 −0.425340
\(986\) −31.0563 −0.989033
\(987\) −0.544156 −0.0173207
\(988\) 3.08917 0.0982797
\(989\) −5.52317 −0.175627
\(990\) 8.97142 0.285130
\(991\) −16.4259 −0.521785 −0.260892 0.965368i \(-0.584017\pi\)
−0.260892 + 0.965368i \(0.584017\pi\)
\(992\) −1.12646 −0.0357652
\(993\) −11.8691 −0.376656
\(994\) 1.04469 0.0331357
\(995\) 15.8073 0.501124
\(996\) 1.74572 0.0553154
\(997\) 10.6046 0.335851 0.167925 0.985800i \(-0.446293\pi\)
0.167925 + 0.985800i \(0.446293\pi\)
\(998\) 25.6731 0.812667
\(999\) −4.46916 −0.141398
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.1.11 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.11 14 1.1 even 1 trivial