Properties

Label 8034.2.a.z.1.5
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 1.5
Root \(1.52247\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.52247 q^{5} +1.00000 q^{6} -3.40750 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.52247 q^{5} +1.00000 q^{6} -3.40750 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.52247 q^{10} +2.89222 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.40750 q^{14} +1.52247 q^{15} +1.00000 q^{16} +3.63214 q^{17} -1.00000 q^{18} +8.33542 q^{19} -1.52247 q^{20} +3.40750 q^{21} -2.89222 q^{22} -8.49258 q^{23} +1.00000 q^{24} -2.68210 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.40750 q^{28} -1.96589 q^{29} -1.52247 q^{30} +2.49196 q^{31} -1.00000 q^{32} -2.89222 q^{33} -3.63214 q^{34} +5.18780 q^{35} +1.00000 q^{36} +5.76014 q^{37} -8.33542 q^{38} -1.00000 q^{39} +1.52247 q^{40} +1.30953 q^{41} -3.40750 q^{42} -6.49471 q^{43} +2.89222 q^{44} -1.52247 q^{45} +8.49258 q^{46} +4.75831 q^{47} -1.00000 q^{48} +4.61105 q^{49} +2.68210 q^{50} -3.63214 q^{51} +1.00000 q^{52} +7.93365 q^{53} +1.00000 q^{54} -4.40330 q^{55} +3.40750 q^{56} -8.33542 q^{57} +1.96589 q^{58} -7.98118 q^{59} +1.52247 q^{60} +8.78947 q^{61} -2.49196 q^{62} -3.40750 q^{63} +1.00000 q^{64} -1.52247 q^{65} +2.89222 q^{66} +3.78863 q^{67} +3.63214 q^{68} +8.49258 q^{69} -5.18780 q^{70} +12.6153 q^{71} -1.00000 q^{72} -8.45160 q^{73} -5.76014 q^{74} +2.68210 q^{75} +8.33542 q^{76} -9.85523 q^{77} +1.00000 q^{78} +3.93397 q^{79} -1.52247 q^{80} +1.00000 q^{81} -1.30953 q^{82} +11.4679 q^{83} +3.40750 q^{84} -5.52980 q^{85} +6.49471 q^{86} +1.96589 q^{87} -2.89222 q^{88} -15.2946 q^{89} +1.52247 q^{90} -3.40750 q^{91} -8.49258 q^{92} -2.49196 q^{93} -4.75831 q^{94} -12.6904 q^{95} +1.00000 q^{96} -14.2627 q^{97} -4.61105 q^{98} +2.89222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + 3q^{10} + 5q^{11} - 14q^{12} + 14q^{13} - 4q^{14} + 3q^{15} + 14q^{16} - 7q^{17} - 14q^{18} + 31q^{19} - 3q^{20} - 4q^{21} - 5q^{22} + 14q^{23} + 14q^{24} + 11q^{25} - 14q^{26} - 14q^{27} + 4q^{28} + 9q^{29} - 3q^{30} + 23q^{31} - 14q^{32} - 5q^{33} + 7q^{34} - 2q^{35} + 14q^{36} + 12q^{37} - 31q^{38} - 14q^{39} + 3q^{40} - 5q^{41} + 4q^{42} - 2q^{43} + 5q^{44} - 3q^{45} - 14q^{46} - 11q^{47} - 14q^{48} + 52q^{49} - 11q^{50} + 7q^{51} + 14q^{52} + 6q^{53} + 14q^{54} + 30q^{55} - 4q^{56} - 31q^{57} - 9q^{58} - 4q^{59} + 3q^{60} + 12q^{61} - 23q^{62} + 4q^{63} + 14q^{64} - 3q^{65} + 5q^{66} + 24q^{67} - 7q^{68} - 14q^{69} + 2q^{70} + 20q^{71} - 14q^{72} + 2q^{73} - 12q^{74} - 11q^{75} + 31q^{76} - 28q^{77} + 14q^{78} + 59q^{79} - 3q^{80} + 14q^{81} + 5q^{82} + q^{83} - 4q^{84} - 29q^{85} + 2q^{86} - 9q^{87} - 5q^{88} + 6q^{89} + 3q^{90} + 4q^{91} + 14q^{92} - 23q^{93} + 11q^{94} - 58q^{95} + 14q^{96} - 6q^{97} - 52q^{98} + 5q^{99} + O(q^{100}) \)

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
\(5\) −1.52247 −0.680868 −0.340434 0.940268i \(-0.610574\pi\)
−0.340434 + 0.940268i \(0.610574\pi\)
\(6\) 1.00000 0.408248
\(7\) −3.40750 −1.28791 −0.643957 0.765062i \(-0.722708\pi\)
−0.643957 + 0.765062i \(0.722708\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.52247 0.481446
\(11\) 2.89222 0.872036 0.436018 0.899938i \(-0.356388\pi\)
0.436018 + 0.899938i \(0.356388\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 3.40750 0.910693
\(15\) 1.52247 0.393099
\(16\) 1.00000 0.250000
\(17\) 3.63214 0.880922 0.440461 0.897772i \(-0.354815\pi\)
0.440461 + 0.897772i \(0.354815\pi\)
\(18\) −1.00000 −0.235702
\(19\) 8.33542 1.91228 0.956139 0.292914i \(-0.0946251\pi\)
0.956139 + 0.292914i \(0.0946251\pi\)
\(20\) −1.52247 −0.340434
\(21\) 3.40750 0.743577
\(22\) −2.89222 −0.616623
\(23\) −8.49258 −1.77083 −0.885413 0.464806i \(-0.846124\pi\)
−0.885413 + 0.464806i \(0.846124\pi\)
\(24\) 1.00000 0.204124
\(25\) −2.68210 −0.536419
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −3.40750 −0.643957
\(29\) −1.96589 −0.365057 −0.182528 0.983201i \(-0.558428\pi\)
−0.182528 + 0.983201i \(0.558428\pi\)
\(30\) −1.52247 −0.277963
\(31\) 2.49196 0.447569 0.223784 0.974639i \(-0.428159\pi\)
0.223784 + 0.974639i \(0.428159\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.89222 −0.503470
\(34\) −3.63214 −0.622906
\(35\) 5.18780 0.876899
\(36\) 1.00000 0.166667
\(37\) 5.76014 0.946962 0.473481 0.880804i \(-0.342997\pi\)
0.473481 + 0.880804i \(0.342997\pi\)
\(38\) −8.33542 −1.35218
\(39\) −1.00000 −0.160128
\(40\) 1.52247 0.240723
\(41\) 1.30953 0.204514 0.102257 0.994758i \(-0.467394\pi\)
0.102257 + 0.994758i \(0.467394\pi\)
\(42\) −3.40750 −0.525789
\(43\) −6.49471 −0.990434 −0.495217 0.868769i \(-0.664912\pi\)
−0.495217 + 0.868769i \(0.664912\pi\)
\(44\) 2.89222 0.436018
\(45\) −1.52247 −0.226956
\(46\) 8.49258 1.25216
\(47\) 4.75831 0.694071 0.347035 0.937852i \(-0.387188\pi\)
0.347035 + 0.937852i \(0.387188\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.61105 0.658722
\(50\) 2.68210 0.379306
\(51\) −3.63214 −0.508601
\(52\) 1.00000 0.138675
\(53\) 7.93365 1.08977 0.544885 0.838511i \(-0.316573\pi\)
0.544885 + 0.838511i \(0.316573\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.40330 −0.593741
\(56\) 3.40750 0.455346
\(57\) −8.33542 −1.10405
\(58\) 1.96589 0.258134
\(59\) −7.98118 −1.03906 −0.519531 0.854452i \(-0.673893\pi\)
−0.519531 + 0.854452i \(0.673893\pi\)
\(60\) 1.52247 0.196550
\(61\) 8.78947 1.12538 0.562688 0.826669i \(-0.309767\pi\)
0.562688 + 0.826669i \(0.309767\pi\)
\(62\) −2.49196 −0.316479
\(63\) −3.40750 −0.429305
\(64\) 1.00000 0.125000
\(65\) −1.52247 −0.188839
\(66\) 2.89222 0.356007
\(67\) 3.78863 0.462855 0.231428 0.972852i \(-0.425660\pi\)
0.231428 + 0.972852i \(0.425660\pi\)
\(68\) 3.63214 0.440461
\(69\) 8.49258 1.02239
\(70\) −5.18780 −0.620061
\(71\) 12.6153 1.49716 0.748581 0.663044i \(-0.230736\pi\)
0.748581 + 0.663044i \(0.230736\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.45160 −0.989185 −0.494593 0.869125i \(-0.664683\pi\)
−0.494593 + 0.869125i \(0.664683\pi\)
\(74\) −5.76014 −0.669603
\(75\) 2.68210 0.309702
\(76\) 8.33542 0.956139
\(77\) −9.85523 −1.12311
\(78\) 1.00000 0.113228
\(79\) 3.93397 0.442607 0.221303 0.975205i \(-0.428969\pi\)
0.221303 + 0.975205i \(0.428969\pi\)
\(80\) −1.52247 −0.170217
\(81\) 1.00000 0.111111
\(82\) −1.30953 −0.144613
\(83\) 11.4679 1.25876 0.629382 0.777096i \(-0.283308\pi\)
0.629382 + 0.777096i \(0.283308\pi\)
\(84\) 3.40750 0.371789
\(85\) −5.52980 −0.599791
\(86\) 6.49471 0.700343
\(87\) 1.96589 0.210766
\(88\) −2.89222 −0.308311
\(89\) −15.2946 −1.62122 −0.810611 0.585585i \(-0.800865\pi\)
−0.810611 + 0.585585i \(0.800865\pi\)
\(90\) 1.52247 0.160482
\(91\) −3.40750 −0.357203
\(92\) −8.49258 −0.885413
\(93\) −2.49196 −0.258404
\(94\) −4.75831 −0.490782
\(95\) −12.6904 −1.30201
\(96\) 1.00000 0.102062
\(97\) −14.2627 −1.44816 −0.724080 0.689716i \(-0.757735\pi\)
−0.724080 + 0.689716i \(0.757735\pi\)
\(98\) −4.61105 −0.465787
\(99\) 2.89222 0.290679
\(100\) −2.68210 −0.268210
\(101\) −6.08371 −0.605352 −0.302676 0.953094i \(-0.597880\pi\)
−0.302676 + 0.953094i \(0.597880\pi\)
\(102\) 3.63214 0.359635
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −5.18780 −0.506278
\(106\) −7.93365 −0.770584
\(107\) −0.0118287 −0.00114352 −0.000571760 1.00000i \(-0.500182\pi\)
−0.000571760 1.00000i \(0.500182\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 9.52537 0.912365 0.456182 0.889886i \(-0.349216\pi\)
0.456182 + 0.889886i \(0.349216\pi\)
\(110\) 4.40330 0.419838
\(111\) −5.76014 −0.546729
\(112\) −3.40750 −0.321978
\(113\) −12.3005 −1.15713 −0.578565 0.815637i \(-0.696387\pi\)
−0.578565 + 0.815637i \(0.696387\pi\)
\(114\) 8.33542 0.780684
\(115\) 12.9297 1.20570
\(116\) −1.96589 −0.182528
\(117\) 1.00000 0.0924500
\(118\) 7.98118 0.734728
\(119\) −12.3765 −1.13455
\(120\) −1.52247 −0.138982
\(121\) −2.63508 −0.239553
\(122\) −8.78947 −0.795762
\(123\) −1.30953 −0.118076
\(124\) 2.49196 0.223784
\(125\) 11.6957 1.04610
\(126\) 3.40750 0.303564
\(127\) 9.53346 0.845958 0.422979 0.906139i \(-0.360984\pi\)
0.422979 + 0.906139i \(0.360984\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.49471 0.571828
\(130\) 1.52247 0.133529
\(131\) 4.16630 0.364011 0.182005 0.983298i \(-0.441741\pi\)
0.182005 + 0.983298i \(0.441741\pi\)
\(132\) −2.89222 −0.251735
\(133\) −28.4030 −2.46285
\(134\) −3.78863 −0.327288
\(135\) 1.52247 0.131033
\(136\) −3.63214 −0.311453
\(137\) −22.5972 −1.93061 −0.965304 0.261130i \(-0.915905\pi\)
−0.965304 + 0.261130i \(0.915905\pi\)
\(138\) −8.49258 −0.722937
\(139\) −1.82634 −0.154908 −0.0774541 0.996996i \(-0.524679\pi\)
−0.0774541 + 0.996996i \(0.524679\pi\)
\(140\) 5.18780 0.438449
\(141\) −4.75831 −0.400722
\(142\) −12.6153 −1.05865
\(143\) 2.89222 0.241859
\(144\) 1.00000 0.0833333
\(145\) 2.99300 0.248555
\(146\) 8.45160 0.699460
\(147\) −4.61105 −0.380313
\(148\) 5.76014 0.473481
\(149\) −7.97737 −0.653531 −0.326766 0.945105i \(-0.605959\pi\)
−0.326766 + 0.945105i \(0.605959\pi\)
\(150\) −2.68210 −0.218992
\(151\) −24.2779 −1.97570 −0.987852 0.155398i \(-0.950334\pi\)
−0.987852 + 0.155398i \(0.950334\pi\)
\(152\) −8.33542 −0.676092
\(153\) 3.63214 0.293641
\(154\) 9.85523 0.794157
\(155\) −3.79392 −0.304735
\(156\) −1.00000 −0.0800641
\(157\) 11.7313 0.936260 0.468130 0.883660i \(-0.344928\pi\)
0.468130 + 0.883660i \(0.344928\pi\)
\(158\) −3.93397 −0.312970
\(159\) −7.93365 −0.629179
\(160\) 1.52247 0.120362
\(161\) 28.9385 2.28067
\(162\) −1.00000 −0.0785674
\(163\) −11.2838 −0.883818 −0.441909 0.897060i \(-0.645699\pi\)
−0.441909 + 0.897060i \(0.645699\pi\)
\(164\) 1.30953 0.102257
\(165\) 4.40330 0.342797
\(166\) −11.4679 −0.890080
\(167\) −21.2258 −1.64250 −0.821249 0.570569i \(-0.806723\pi\)
−0.821249 + 0.570569i \(0.806723\pi\)
\(168\) −3.40750 −0.262894
\(169\) 1.00000 0.0769231
\(170\) 5.52980 0.424117
\(171\) 8.33542 0.637426
\(172\) −6.49471 −0.495217
\(173\) −10.3787 −0.789081 −0.394541 0.918878i \(-0.629096\pi\)
−0.394541 + 0.918878i \(0.629096\pi\)
\(174\) −1.96589 −0.149034
\(175\) 9.13924 0.690862
\(176\) 2.89222 0.218009
\(177\) 7.98118 0.599903
\(178\) 15.2946 1.14638
\(179\) −5.05976 −0.378184 −0.189092 0.981959i \(-0.560554\pi\)
−0.189092 + 0.981959i \(0.560554\pi\)
\(180\) −1.52247 −0.113478
\(181\) −13.1453 −0.977084 −0.488542 0.872540i \(-0.662471\pi\)
−0.488542 + 0.872540i \(0.662471\pi\)
\(182\) 3.40750 0.252581
\(183\) −8.78947 −0.649737
\(184\) 8.49258 0.626081
\(185\) −8.76962 −0.644756
\(186\) 2.49196 0.182719
\(187\) 10.5049 0.768196
\(188\) 4.75831 0.347035
\(189\) 3.40750 0.247859
\(190\) 12.6904 0.920658
\(191\) 10.6105 0.767750 0.383875 0.923385i \(-0.374589\pi\)
0.383875 + 0.923385i \(0.374589\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −13.6729 −0.984200 −0.492100 0.870539i \(-0.663771\pi\)
−0.492100 + 0.870539i \(0.663771\pi\)
\(194\) 14.2627 1.02400
\(195\) 1.52247 0.109026
\(196\) 4.61105 0.329361
\(197\) 13.0622 0.930646 0.465323 0.885141i \(-0.345938\pi\)
0.465323 + 0.885141i \(0.345938\pi\)
\(198\) −2.89222 −0.205541
\(199\) 21.9234 1.55411 0.777054 0.629434i \(-0.216713\pi\)
0.777054 + 0.629434i \(0.216713\pi\)
\(200\) 2.68210 0.189653
\(201\) −3.78863 −0.267230
\(202\) 6.08371 0.428048
\(203\) 6.69877 0.470162
\(204\) −3.63214 −0.254300
\(205\) −1.99371 −0.139247
\(206\) 1.00000 0.0696733
\(207\) −8.49258 −0.590275
\(208\) 1.00000 0.0693375
\(209\) 24.1078 1.66757
\(210\) 5.18780 0.357992
\(211\) 15.1150 1.04056 0.520280 0.853995i \(-0.325827\pi\)
0.520280 + 0.853995i \(0.325827\pi\)
\(212\) 7.93365 0.544885
\(213\) −12.6153 −0.864386
\(214\) 0.0118287 0.000808591 0
\(215\) 9.88798 0.674355
\(216\) 1.00000 0.0680414
\(217\) −8.49135 −0.576430
\(218\) −9.52537 −0.645139
\(219\) 8.45160 0.571107
\(220\) −4.40330 −0.296871
\(221\) 3.63214 0.244324
\(222\) 5.76014 0.386596
\(223\) 18.7880 1.25813 0.629067 0.777351i \(-0.283437\pi\)
0.629067 + 0.777351i \(0.283437\pi\)
\(224\) 3.40750 0.227673
\(225\) −2.68210 −0.178806
\(226\) 12.3005 0.818214
\(227\) 6.60501 0.438390 0.219195 0.975681i \(-0.429657\pi\)
0.219195 + 0.975681i \(0.429657\pi\)
\(228\) −8.33542 −0.552027
\(229\) 4.77427 0.315493 0.157746 0.987480i \(-0.449577\pi\)
0.157746 + 0.987480i \(0.449577\pi\)
\(230\) −12.9297 −0.852557
\(231\) 9.85523 0.648426
\(232\) 1.96589 0.129067
\(233\) −7.86773 −0.515432 −0.257716 0.966221i \(-0.582970\pi\)
−0.257716 + 0.966221i \(0.582970\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −7.24437 −0.472570
\(236\) −7.98118 −0.519531
\(237\) −3.93397 −0.255539
\(238\) 12.3765 0.802249
\(239\) −12.9810 −0.839673 −0.419837 0.907600i \(-0.637913\pi\)
−0.419837 + 0.907600i \(0.637913\pi\)
\(240\) 1.52247 0.0982748
\(241\) 11.8220 0.761519 0.380760 0.924674i \(-0.375663\pi\)
0.380760 + 0.924674i \(0.375663\pi\)
\(242\) 2.63508 0.169390
\(243\) −1.00000 −0.0641500
\(244\) 8.78947 0.562688
\(245\) −7.02017 −0.448502
\(246\) 1.30953 0.0834923
\(247\) 8.33542 0.530370
\(248\) −2.49196 −0.158240
\(249\) −11.4679 −0.726747
\(250\) −11.6957 −0.739703
\(251\) 10.2546 0.647267 0.323634 0.946183i \(-0.395096\pi\)
0.323634 + 0.946183i \(0.395096\pi\)
\(252\) −3.40750 −0.214652
\(253\) −24.5624 −1.54422
\(254\) −9.53346 −0.598183
\(255\) 5.52980 0.346290
\(256\) 1.00000 0.0625000
\(257\) 7.88163 0.491643 0.245821 0.969315i \(-0.420942\pi\)
0.245821 + 0.969315i \(0.420942\pi\)
\(258\) −6.49471 −0.404343
\(259\) −19.6277 −1.21961
\(260\) −1.52247 −0.0944193
\(261\) −1.96589 −0.121686
\(262\) −4.16630 −0.257395
\(263\) −19.5054 −1.20276 −0.601379 0.798964i \(-0.705382\pi\)
−0.601379 + 0.798964i \(0.705382\pi\)
\(264\) 2.89222 0.178004
\(265\) −12.0787 −0.741989
\(266\) 28.4030 1.74150
\(267\) 15.2946 0.936013
\(268\) 3.78863 0.231428
\(269\) 29.6814 1.80971 0.904853 0.425724i \(-0.139980\pi\)
0.904853 + 0.425724i \(0.139980\pi\)
\(270\) −1.52247 −0.0926543
\(271\) 1.48750 0.0903590 0.0451795 0.998979i \(-0.485614\pi\)
0.0451795 + 0.998979i \(0.485614\pi\)
\(272\) 3.63214 0.220231
\(273\) 3.40750 0.206231
\(274\) 22.5972 1.36515
\(275\) −7.75720 −0.467777
\(276\) 8.49258 0.511193
\(277\) 32.2188 1.93584 0.967921 0.251255i \(-0.0808432\pi\)
0.967921 + 0.251255i \(0.0808432\pi\)
\(278\) 1.82634 0.109537
\(279\) 2.49196 0.149190
\(280\) −5.18780 −0.310031
\(281\) 15.9702 0.952700 0.476350 0.879256i \(-0.341960\pi\)
0.476350 + 0.879256i \(0.341960\pi\)
\(282\) 4.75831 0.283353
\(283\) 15.5347 0.923440 0.461720 0.887026i \(-0.347232\pi\)
0.461720 + 0.887026i \(0.347232\pi\)
\(284\) 12.6153 0.748581
\(285\) 12.6904 0.751714
\(286\) −2.89222 −0.171020
\(287\) −4.46221 −0.263396
\(288\) −1.00000 −0.0589256
\(289\) −3.80759 −0.223976
\(290\) −2.99300 −0.175755
\(291\) 14.2627 0.836096
\(292\) −8.45160 −0.494593
\(293\) 0.645648 0.0377191 0.0188596 0.999822i \(-0.493996\pi\)
0.0188596 + 0.999822i \(0.493996\pi\)
\(294\) 4.61105 0.268922
\(295\) 12.1511 0.707463
\(296\) −5.76014 −0.334802
\(297\) −2.89222 −0.167823
\(298\) 7.97737 0.462117
\(299\) −8.49258 −0.491139
\(300\) 2.68210 0.154851
\(301\) 22.1307 1.27559
\(302\) 24.2779 1.39703
\(303\) 6.08371 0.349500
\(304\) 8.33542 0.478069
\(305\) −13.3817 −0.766232
\(306\) −3.63214 −0.207635
\(307\) 17.3345 0.989331 0.494666 0.869083i \(-0.335291\pi\)
0.494666 + 0.869083i \(0.335291\pi\)
\(308\) −9.85523 −0.561554
\(309\) 1.00000 0.0568880
\(310\) 3.79392 0.215480
\(311\) 1.26256 0.0715933 0.0357966 0.999359i \(-0.488603\pi\)
0.0357966 + 0.999359i \(0.488603\pi\)
\(312\) 1.00000 0.0566139
\(313\) 22.2729 1.25894 0.629469 0.777026i \(-0.283273\pi\)
0.629469 + 0.777026i \(0.283273\pi\)
\(314\) −11.7313 −0.662036
\(315\) 5.18780 0.292300
\(316\) 3.93397 0.221303
\(317\) −1.98445 −0.111458 −0.0557288 0.998446i \(-0.517748\pi\)
−0.0557288 + 0.998446i \(0.517748\pi\)
\(318\) 7.93365 0.444897
\(319\) −5.68578 −0.318343
\(320\) −1.52247 −0.0851084
\(321\) 0.0118287 0.000660212 0
\(322\) −28.9385 −1.61268
\(323\) 30.2754 1.68457
\(324\) 1.00000 0.0555556
\(325\) −2.68210 −0.148776
\(326\) 11.2838 0.624954
\(327\) −9.52537 −0.526754
\(328\) −1.30953 −0.0723065
\(329\) −16.2139 −0.893904
\(330\) −4.40330 −0.242394
\(331\) 7.80176 0.428823 0.214412 0.976743i \(-0.431217\pi\)
0.214412 + 0.976743i \(0.431217\pi\)
\(332\) 11.4679 0.629382
\(333\) 5.76014 0.315654
\(334\) 21.2258 1.16142
\(335\) −5.76806 −0.315143
\(336\) 3.40750 0.185894
\(337\) 12.8629 0.700687 0.350343 0.936621i \(-0.386065\pi\)
0.350343 + 0.936621i \(0.386065\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 12.3005 0.668069
\(340\) −5.52980 −0.299896
\(341\) 7.20728 0.390296
\(342\) −8.33542 −0.450728
\(343\) 8.14033 0.439537
\(344\) 6.49471 0.350171
\(345\) −12.9297 −0.696110
\(346\) 10.3787 0.557965
\(347\) −14.1100 −0.757464 −0.378732 0.925506i \(-0.623640\pi\)
−0.378732 + 0.925506i \(0.623640\pi\)
\(348\) 1.96589 0.105383
\(349\) 20.8994 1.11872 0.559361 0.828924i \(-0.311047\pi\)
0.559361 + 0.828924i \(0.311047\pi\)
\(350\) −9.13924 −0.488513
\(351\) −1.00000 −0.0533761
\(352\) −2.89222 −0.154156
\(353\) −4.18829 −0.222920 −0.111460 0.993769i \(-0.535553\pi\)
−0.111460 + 0.993769i \(0.535553\pi\)
\(354\) −7.98118 −0.424195
\(355\) −19.2064 −1.01937
\(356\) −15.2946 −0.810611
\(357\) 12.3765 0.655034
\(358\) 5.05976 0.267416
\(359\) −27.7433 −1.46423 −0.732117 0.681179i \(-0.761468\pi\)
−0.732117 + 0.681179i \(0.761468\pi\)
\(360\) 1.52247 0.0802410
\(361\) 50.4793 2.65680
\(362\) 13.1453 0.690903
\(363\) 2.63508 0.138306
\(364\) −3.40750 −0.178602
\(365\) 12.8673 0.673504
\(366\) 8.78947 0.459433
\(367\) 5.85048 0.305392 0.152696 0.988273i \(-0.451204\pi\)
0.152696 + 0.988273i \(0.451204\pi\)
\(368\) −8.49258 −0.442706
\(369\) 1.30953 0.0681712
\(370\) 8.76962 0.455911
\(371\) −27.0339 −1.40353
\(372\) −2.49196 −0.129202
\(373\) 15.8596 0.821177 0.410588 0.911821i \(-0.365323\pi\)
0.410588 + 0.911821i \(0.365323\pi\)
\(374\) −10.5049 −0.543197
\(375\) −11.6957 −0.603965
\(376\) −4.75831 −0.245391
\(377\) −1.96589 −0.101249
\(378\) −3.40750 −0.175263
\(379\) 13.7262 0.705070 0.352535 0.935799i \(-0.385320\pi\)
0.352535 + 0.935799i \(0.385320\pi\)
\(380\) −12.6904 −0.651004
\(381\) −9.53346 −0.488414
\(382\) −10.6105 −0.542881
\(383\) 11.4824 0.586721 0.293360 0.956002i \(-0.405226\pi\)
0.293360 + 0.956002i \(0.405226\pi\)
\(384\) 1.00000 0.0510310
\(385\) 15.0042 0.764687
\(386\) 13.6729 0.695935
\(387\) −6.49471 −0.330145
\(388\) −14.2627 −0.724080
\(389\) −22.4333 −1.13741 −0.568707 0.822540i \(-0.692556\pi\)
−0.568707 + 0.822540i \(0.692556\pi\)
\(390\) −1.52247 −0.0770931
\(391\) −30.8462 −1.55996
\(392\) −4.61105 −0.232893
\(393\) −4.16630 −0.210162
\(394\) −13.0622 −0.658066
\(395\) −5.98934 −0.301356
\(396\) 2.89222 0.145339
\(397\) 25.1274 1.26111 0.630554 0.776145i \(-0.282828\pi\)
0.630554 + 0.776145i \(0.282828\pi\)
\(398\) −21.9234 −1.09892
\(399\) 28.4030 1.42193
\(400\) −2.68210 −0.134105
\(401\) 15.4267 0.770374 0.385187 0.922838i \(-0.374137\pi\)
0.385187 + 0.922838i \(0.374137\pi\)
\(402\) 3.78863 0.188960
\(403\) 2.49196 0.124133
\(404\) −6.08371 −0.302676
\(405\) −1.52247 −0.0756519
\(406\) −6.69877 −0.332455
\(407\) 16.6596 0.825785
\(408\) 3.63214 0.179818
\(409\) −33.1984 −1.64156 −0.820778 0.571247i \(-0.806460\pi\)
−0.820778 + 0.571247i \(0.806460\pi\)
\(410\) 1.99371 0.0984623
\(411\) 22.5972 1.11464
\(412\) −1.00000 −0.0492665
\(413\) 27.1959 1.33822
\(414\) 8.49258 0.417388
\(415\) −17.4595 −0.857051
\(416\) −1.00000 −0.0490290
\(417\) 1.82634 0.0894362
\(418\) −24.1078 −1.17915
\(419\) 7.81068 0.381577 0.190788 0.981631i \(-0.438896\pi\)
0.190788 + 0.981631i \(0.438896\pi\)
\(420\) −5.18780 −0.253139
\(421\) −9.14589 −0.445743 −0.222872 0.974848i \(-0.571543\pi\)
−0.222872 + 0.974848i \(0.571543\pi\)
\(422\) −15.1150 −0.735788
\(423\) 4.75831 0.231357
\(424\) −7.93365 −0.385292
\(425\) −9.74174 −0.472544
\(426\) 12.6153 0.611213
\(427\) −29.9501 −1.44939
\(428\) −0.0118287 −0.000571760 0
\(429\) −2.89222 −0.139638
\(430\) −9.88798 −0.476841
\(431\) 14.0382 0.676194 0.338097 0.941111i \(-0.390217\pi\)
0.338097 + 0.941111i \(0.390217\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 13.8082 0.663580 0.331790 0.943353i \(-0.392347\pi\)
0.331790 + 0.943353i \(0.392347\pi\)
\(434\) 8.49135 0.407598
\(435\) −2.99300 −0.143504
\(436\) 9.52537 0.456182
\(437\) −70.7893 −3.38631
\(438\) −8.45160 −0.403833
\(439\) −4.89323 −0.233541 −0.116771 0.993159i \(-0.537254\pi\)
−0.116771 + 0.993159i \(0.537254\pi\)
\(440\) 4.40330 0.209919
\(441\) 4.61105 0.219574
\(442\) −3.63214 −0.172763
\(443\) −6.91668 −0.328621 −0.164311 0.986409i \(-0.552540\pi\)
−0.164311 + 0.986409i \(0.552540\pi\)
\(444\) −5.76014 −0.273364
\(445\) 23.2855 1.10384
\(446\) −18.7880 −0.889636
\(447\) 7.97737 0.377317
\(448\) −3.40750 −0.160989
\(449\) −12.9209 −0.609773 −0.304887 0.952389i \(-0.598619\pi\)
−0.304887 + 0.952389i \(0.598619\pi\)
\(450\) 2.68210 0.126435
\(451\) 3.78743 0.178343
\(452\) −12.3005 −0.578565
\(453\) 24.2779 1.14067
\(454\) −6.60501 −0.309988
\(455\) 5.18780 0.243208
\(456\) 8.33542 0.390342
\(457\) −25.8588 −1.20962 −0.604811 0.796369i \(-0.706752\pi\)
−0.604811 + 0.796369i \(0.706752\pi\)
\(458\) −4.77427 −0.223087
\(459\) −3.63214 −0.169534
\(460\) 12.9297 0.602849
\(461\) −0.0963236 −0.00448624 −0.00224312 0.999997i \(-0.500714\pi\)
−0.00224312 + 0.999997i \(0.500714\pi\)
\(462\) −9.85523 −0.458507
\(463\) 35.9141 1.66907 0.834534 0.550956i \(-0.185737\pi\)
0.834534 + 0.550956i \(0.185737\pi\)
\(464\) −1.96589 −0.0912642
\(465\) 3.79392 0.175939
\(466\) 7.86773 0.364465
\(467\) −24.8147 −1.14829 −0.574143 0.818755i \(-0.694665\pi\)
−0.574143 + 0.818755i \(0.694665\pi\)
\(468\) 1.00000 0.0462250
\(469\) −12.9098 −0.596117
\(470\) 7.24437 0.334158
\(471\) −11.7313 −0.540550
\(472\) 7.98118 0.367364
\(473\) −18.7841 −0.863694
\(474\) 3.93397 0.180693
\(475\) −22.3564 −1.02578
\(476\) −12.3765 −0.567276
\(477\) 7.93365 0.363257
\(478\) 12.9810 0.593739
\(479\) 7.77361 0.355186 0.177593 0.984104i \(-0.443169\pi\)
0.177593 + 0.984104i \(0.443169\pi\)
\(480\) −1.52247 −0.0694908
\(481\) 5.76014 0.262640
\(482\) −11.8220 −0.538475
\(483\) −28.9385 −1.31675
\(484\) −2.63508 −0.119777
\(485\) 21.7145 0.986005
\(486\) 1.00000 0.0453609
\(487\) −2.38601 −0.108120 −0.0540601 0.998538i \(-0.517216\pi\)
−0.0540601 + 0.998538i \(0.517216\pi\)
\(488\) −8.78947 −0.397881
\(489\) 11.2838 0.510273
\(490\) 7.02017 0.317139
\(491\) 39.7482 1.79381 0.896906 0.442222i \(-0.145810\pi\)
0.896906 + 0.442222i \(0.145810\pi\)
\(492\) −1.30953 −0.0590380
\(493\) −7.14038 −0.321587
\(494\) −8.33542 −0.375028
\(495\) −4.40330 −0.197914
\(496\) 2.49196 0.111892
\(497\) −42.9866 −1.92821
\(498\) 11.4679 0.513888
\(499\) 31.1133 1.39282 0.696412 0.717643i \(-0.254779\pi\)
0.696412 + 0.717643i \(0.254779\pi\)
\(500\) 11.6957 0.523049
\(501\) 21.2258 0.948297
\(502\) −10.2546 −0.457687
\(503\) −26.4300 −1.17845 −0.589227 0.807967i \(-0.700568\pi\)
−0.589227 + 0.807967i \(0.700568\pi\)
\(504\) 3.40750 0.151782
\(505\) 9.26224 0.412164
\(506\) 24.5624 1.09193
\(507\) −1.00000 −0.0444116
\(508\) 9.53346 0.422979
\(509\) 0.544742 0.0241453 0.0120726 0.999927i \(-0.496157\pi\)
0.0120726 + 0.999927i \(0.496157\pi\)
\(510\) −5.52980 −0.244864
\(511\) 28.7988 1.27399
\(512\) −1.00000 −0.0441942
\(513\) −8.33542 −0.368018
\(514\) −7.88163 −0.347644
\(515\) 1.52247 0.0670879
\(516\) 6.49471 0.285914
\(517\) 13.7621 0.605255
\(518\) 19.6277 0.862391
\(519\) 10.3787 0.455576
\(520\) 1.52247 0.0667646
\(521\) 43.8413 1.92072 0.960360 0.278762i \(-0.0899241\pi\)
0.960360 + 0.278762i \(0.0899241\pi\)
\(522\) 1.96589 0.0860447
\(523\) 34.7287 1.51858 0.759291 0.650752i \(-0.225546\pi\)
0.759291 + 0.650752i \(0.225546\pi\)
\(524\) 4.16630 0.182005
\(525\) −9.13924 −0.398869
\(526\) 19.5054 0.850478
\(527\) 9.05113 0.394274
\(528\) −2.89222 −0.125868
\(529\) 49.1239 2.13582
\(530\) 12.0787 0.524666
\(531\) −7.98118 −0.346354
\(532\) −28.4030 −1.23142
\(533\) 1.30953 0.0567219
\(534\) −15.2946 −0.661861
\(535\) 0.0180087 0.000778586 0
\(536\) −3.78863 −0.163644
\(537\) 5.05976 0.218345
\(538\) −29.6814 −1.27966
\(539\) 13.3362 0.574429
\(540\) 1.52247 0.0655165
\(541\) 5.48239 0.235706 0.117853 0.993031i \(-0.462399\pi\)
0.117853 + 0.993031i \(0.462399\pi\)
\(542\) −1.48750 −0.0638935
\(543\) 13.1453 0.564120
\(544\) −3.63214 −0.155727
\(545\) −14.5021 −0.621200
\(546\) −3.40750 −0.145828
\(547\) 15.2348 0.651392 0.325696 0.945475i \(-0.394401\pi\)
0.325696 + 0.945475i \(0.394401\pi\)
\(548\) −22.5972 −0.965304
\(549\) 8.78947 0.375126
\(550\) 7.75720 0.330768
\(551\) −16.3865 −0.698090
\(552\) −8.49258 −0.361468
\(553\) −13.4050 −0.570039
\(554\) −32.2188 −1.36885
\(555\) 8.76962 0.372250
\(556\) −1.82634 −0.0774541
\(557\) −43.2263 −1.83156 −0.915779 0.401682i \(-0.868426\pi\)
−0.915779 + 0.401682i \(0.868426\pi\)
\(558\) −2.49196 −0.105493
\(559\) −6.49471 −0.274697
\(560\) 5.18780 0.219225
\(561\) −10.5049 −0.443518
\(562\) −15.9702 −0.673661
\(563\) −16.1975 −0.682642 −0.341321 0.939947i \(-0.610874\pi\)
−0.341321 + 0.939947i \(0.610874\pi\)
\(564\) −4.75831 −0.200361
\(565\) 18.7270 0.787852
\(566\) −15.5347 −0.652971
\(567\) −3.40750 −0.143102
\(568\) −12.6153 −0.529326
\(569\) 21.6497 0.907601 0.453801 0.891103i \(-0.350068\pi\)
0.453801 + 0.891103i \(0.350068\pi\)
\(570\) −12.6904 −0.531542
\(571\) 23.3965 0.979112 0.489556 0.871972i \(-0.337159\pi\)
0.489556 + 0.871972i \(0.337159\pi\)
\(572\) 2.89222 0.120930
\(573\) −10.6105 −0.443261
\(574\) 4.46221 0.186249
\(575\) 22.7779 0.949905
\(576\) 1.00000 0.0416667
\(577\) −12.7028 −0.528823 −0.264412 0.964410i \(-0.585178\pi\)
−0.264412 + 0.964410i \(0.585178\pi\)
\(578\) 3.80759 0.158375
\(579\) 13.6729 0.568228
\(580\) 2.99300 0.124278
\(581\) −39.0768 −1.62118
\(582\) −14.2627 −0.591209
\(583\) 22.9458 0.950319
\(584\) 8.45160 0.349730
\(585\) −1.52247 −0.0629462
\(586\) −0.645648 −0.0266715
\(587\) 5.24679 0.216558 0.108279 0.994121i \(-0.465466\pi\)
0.108279 + 0.994121i \(0.465466\pi\)
\(588\) −4.61105 −0.190157
\(589\) 20.7715 0.855876
\(590\) −12.1511 −0.500252
\(591\) −13.0622 −0.537309
\(592\) 5.76014 0.236740
\(593\) −10.7972 −0.443389 −0.221695 0.975116i \(-0.571159\pi\)
−0.221695 + 0.975116i \(0.571159\pi\)
\(594\) 2.89222 0.118669
\(595\) 18.8428 0.772480
\(596\) −7.97737 −0.326766
\(597\) −21.9234 −0.897265
\(598\) 8.49258 0.347287
\(599\) 42.3898 1.73200 0.866001 0.500042i \(-0.166682\pi\)
0.866001 + 0.500042i \(0.166682\pi\)
\(600\) −2.68210 −0.109496
\(601\) −34.5825 −1.41065 −0.705324 0.708885i \(-0.749199\pi\)
−0.705324 + 0.708885i \(0.749199\pi\)
\(602\) −22.1307 −0.901981
\(603\) 3.78863 0.154285
\(604\) −24.2779 −0.987852
\(605\) 4.01183 0.163104
\(606\) −6.08371 −0.247134
\(607\) 33.3540 1.35380 0.676899 0.736076i \(-0.263324\pi\)
0.676899 + 0.736076i \(0.263324\pi\)
\(608\) −8.33542 −0.338046
\(609\) −6.69877 −0.271448
\(610\) 13.3817 0.541808
\(611\) 4.75831 0.192501
\(612\) 3.63214 0.146820
\(613\) −9.48303 −0.383016 −0.191508 0.981491i \(-0.561338\pi\)
−0.191508 + 0.981491i \(0.561338\pi\)
\(614\) −17.3345 −0.699563
\(615\) 1.99371 0.0803941
\(616\) 9.85523 0.397078
\(617\) −40.8304 −1.64377 −0.821885 0.569653i \(-0.807077\pi\)
−0.821885 + 0.569653i \(0.807077\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 39.1180 1.57228 0.786142 0.618046i \(-0.212075\pi\)
0.786142 + 0.618046i \(0.212075\pi\)
\(620\) −3.79392 −0.152368
\(621\) 8.49258 0.340796
\(622\) −1.26256 −0.0506241
\(623\) 52.1163 2.08799
\(624\) −1.00000 −0.0400320
\(625\) −4.39587 −0.175835
\(626\) −22.2729 −0.890203
\(627\) −24.1078 −0.962775
\(628\) 11.7313 0.468130
\(629\) 20.9216 0.834200
\(630\) −5.18780 −0.206687
\(631\) 6.22753 0.247914 0.123957 0.992288i \(-0.460442\pi\)
0.123957 + 0.992288i \(0.460442\pi\)
\(632\) −3.93397 −0.156485
\(633\) −15.1150 −0.600768
\(634\) 1.98445 0.0788125
\(635\) −14.5144 −0.575985
\(636\) −7.93365 −0.314590
\(637\) 4.61105 0.182697
\(638\) 5.68578 0.225102
\(639\) 12.6153 0.499054
\(640\) 1.52247 0.0601808
\(641\) 0.0610278 0.00241045 0.00120523 0.999999i \(-0.499616\pi\)
0.00120523 + 0.999999i \(0.499616\pi\)
\(642\) −0.0118287 −0.000466840 0
\(643\) −24.5399 −0.967757 −0.483879 0.875135i \(-0.660772\pi\)
−0.483879 + 0.875135i \(0.660772\pi\)
\(644\) 28.9385 1.14034
\(645\) −9.88798 −0.389339
\(646\) −30.2754 −1.19117
\(647\) 27.2088 1.06969 0.534845 0.844950i \(-0.320370\pi\)
0.534845 + 0.844950i \(0.320370\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −23.0833 −0.906099
\(650\) 2.68210 0.105200
\(651\) 8.49135 0.332802
\(652\) −11.2838 −0.441909
\(653\) −14.8510 −0.581166 −0.290583 0.956850i \(-0.593849\pi\)
−0.290583 + 0.956850i \(0.593849\pi\)
\(654\) 9.52537 0.372471
\(655\) −6.34304 −0.247843
\(656\) 1.30953 0.0511284
\(657\) −8.45160 −0.329728
\(658\) 16.2139 0.632085
\(659\) −4.94968 −0.192812 −0.0964060 0.995342i \(-0.530735\pi\)
−0.0964060 + 0.995342i \(0.530735\pi\)
\(660\) 4.40330 0.171398
\(661\) 34.5141 1.34244 0.671222 0.741256i \(-0.265770\pi\)
0.671222 + 0.741256i \(0.265770\pi\)
\(662\) −7.80176 −0.303224
\(663\) −3.63214 −0.141060
\(664\) −11.4679 −0.445040
\(665\) 43.2425 1.67687
\(666\) −5.76014 −0.223201
\(667\) 16.6955 0.646452
\(668\) −21.2258 −0.821249
\(669\) −18.7880 −0.726384
\(670\) 5.76806 0.222840
\(671\) 25.4211 0.981369
\(672\) −3.40750 −0.131447
\(673\) −20.6255 −0.795054 −0.397527 0.917591i \(-0.630131\pi\)
−0.397527 + 0.917591i \(0.630131\pi\)
\(674\) −12.8629 −0.495460
\(675\) 2.68210 0.103234
\(676\) 1.00000 0.0384615
\(677\) 2.12877 0.0818155 0.0409077 0.999163i \(-0.486975\pi\)
0.0409077 + 0.999163i \(0.486975\pi\)
\(678\) −12.3005 −0.472396
\(679\) 48.6002 1.86511
\(680\) 5.52980 0.212058
\(681\) −6.60501 −0.253104
\(682\) −7.20728 −0.275981
\(683\) −13.9481 −0.533708 −0.266854 0.963737i \(-0.585984\pi\)
−0.266854 + 0.963737i \(0.585984\pi\)
\(684\) 8.33542 0.318713
\(685\) 34.4034 1.31449
\(686\) −8.14033 −0.310799
\(687\) −4.77427 −0.182150
\(688\) −6.49471 −0.247609
\(689\) 7.93365 0.302248
\(690\) 12.9297 0.492224
\(691\) −5.07598 −0.193099 −0.0965497 0.995328i \(-0.530781\pi\)
−0.0965497 + 0.995328i \(0.530781\pi\)
\(692\) −10.3787 −0.394541
\(693\) −9.85523 −0.374369
\(694\) 14.1100 0.535608
\(695\) 2.78054 0.105472
\(696\) −1.96589 −0.0745169
\(697\) 4.75638 0.180161
\(698\) −20.8994 −0.791055
\(699\) 7.86773 0.297585
\(700\) 9.13924 0.345431
\(701\) 19.6370 0.741678 0.370839 0.928697i \(-0.379070\pi\)
0.370839 + 0.928697i \(0.379070\pi\)
\(702\) 1.00000 0.0377426
\(703\) 48.0132 1.81085
\(704\) 2.89222 0.109005
\(705\) 7.24437 0.272839
\(706\) 4.18829 0.157629
\(707\) 20.7302 0.779641
\(708\) 7.98118 0.299951
\(709\) 12.6875 0.476487 0.238244 0.971205i \(-0.423428\pi\)
0.238244 + 0.971205i \(0.423428\pi\)
\(710\) 19.2064 0.720802
\(711\) 3.93397 0.147536
\(712\) 15.2946 0.573188
\(713\) −21.1632 −0.792567
\(714\) −12.3765 −0.463179
\(715\) −4.40330 −0.164674
\(716\) −5.05976 −0.189092
\(717\) 12.9810 0.484786
\(718\) 27.7433 1.03537
\(719\) 25.7246 0.959365 0.479683 0.877442i \(-0.340752\pi\)
0.479683 + 0.877442i \(0.340752\pi\)
\(720\) −1.52247 −0.0567390
\(721\) 3.40750 0.126902
\(722\) −50.4793 −1.87864
\(723\) −11.8220 −0.439663
\(724\) −13.1453 −0.488542
\(725\) 5.27271 0.195824
\(726\) −2.63508 −0.0977972
\(727\) 37.9623 1.40794 0.703972 0.710228i \(-0.251408\pi\)
0.703972 + 0.710228i \(0.251408\pi\)
\(728\) 3.40750 0.126290
\(729\) 1.00000 0.0370370
\(730\) −12.8673 −0.476239
\(731\) −23.5897 −0.872496
\(732\) −8.78947 −0.324868
\(733\) 15.9783 0.590173 0.295087 0.955470i \(-0.404651\pi\)
0.295087 + 0.955470i \(0.404651\pi\)
\(734\) −5.85048 −0.215945
\(735\) 7.02017 0.258943
\(736\) 8.49258 0.313041
\(737\) 10.9575 0.403626
\(738\) −1.30953 −0.0482043
\(739\) −21.1633 −0.778504 −0.389252 0.921131i \(-0.627266\pi\)
−0.389252 + 0.921131i \(0.627266\pi\)
\(740\) −8.76962 −0.322378
\(741\) −8.33542 −0.306209
\(742\) 27.0339 0.992446
\(743\) −11.0708 −0.406148 −0.203074 0.979163i \(-0.565093\pi\)
−0.203074 + 0.979163i \(0.565093\pi\)
\(744\) 2.49196 0.0913596
\(745\) 12.1453 0.444968
\(746\) −15.8596 −0.580660
\(747\) 11.4679 0.419588
\(748\) 10.5049 0.384098
\(749\) 0.0403062 0.00147276
\(750\) 11.6957 0.427068
\(751\) −34.9978 −1.27709 −0.638545 0.769585i \(-0.720463\pi\)
−0.638545 + 0.769585i \(0.720463\pi\)
\(752\) 4.75831 0.173518
\(753\) −10.2546 −0.373700
\(754\) 1.96589 0.0715935
\(755\) 36.9622 1.34519
\(756\) 3.40750 0.123930
\(757\) 17.5111 0.636451 0.318225 0.948015i \(-0.396913\pi\)
0.318225 + 0.948015i \(0.396913\pi\)
\(758\) −13.7262 −0.498560
\(759\) 24.5624 0.891558
\(760\) 12.6904 0.460329
\(761\) 2.75012 0.0996916 0.0498458 0.998757i \(-0.484127\pi\)
0.0498458 + 0.998757i \(0.484127\pi\)
\(762\) 9.53346 0.345361
\(763\) −32.4577 −1.17505
\(764\) 10.6105 0.383875
\(765\) −5.52980 −0.199930
\(766\) −11.4824 −0.414874
\(767\) −7.98118 −0.288184
\(768\) −1.00000 −0.0360844
\(769\) −27.0234 −0.974487 −0.487244 0.873266i \(-0.661998\pi\)
−0.487244 + 0.873266i \(0.661998\pi\)
\(770\) −15.0042 −0.540716
\(771\) −7.88163 −0.283850
\(772\) −13.6729 −0.492100
\(773\) 17.7262 0.637568 0.318784 0.947827i \(-0.396726\pi\)
0.318784 + 0.947827i \(0.396726\pi\)
\(774\) 6.49471 0.233448
\(775\) −6.68367 −0.240085
\(776\) 14.2627 0.512002
\(777\) 19.6277 0.704139
\(778\) 22.4333 0.804273
\(779\) 10.9155 0.391087
\(780\) 1.52247 0.0545130
\(781\) 36.4862 1.30558
\(782\) 30.8462 1.10306
\(783\) 1.96589 0.0702552
\(784\) 4.61105 0.164681
\(785\) −17.8605 −0.637469
\(786\) 4.16630 0.148607
\(787\) 22.2536 0.793254 0.396627 0.917980i \(-0.370181\pi\)
0.396627 + 0.917980i \(0.370181\pi\)
\(788\) 13.0622 0.465323
\(789\) 19.5054 0.694412
\(790\) 5.98934 0.213091
\(791\) 41.9138 1.49028
\(792\) −2.89222 −0.102770
\(793\) 8.78947 0.312123
\(794\) −25.1274 −0.891739
\(795\) 12.0787 0.428388
\(796\) 21.9234 0.777054
\(797\) 51.2856 1.81663 0.908315 0.418287i \(-0.137369\pi\)
0.908315 + 0.418287i \(0.137369\pi\)
\(798\) −28.4030 −1.00545
\(799\) 17.2828 0.611423
\(800\) 2.68210 0.0948264
\(801\) −15.2946 −0.540407
\(802\) −15.4267 −0.544737
\(803\) −24.4439 −0.862605
\(804\) −3.78863 −0.133615
\(805\) −44.0578 −1.55283
\(806\) −2.49196 −0.0877755
\(807\) −29.6814 −1.04483
\(808\) 6.08371 0.214024
\(809\) 37.3055 1.31159 0.655795 0.754939i \(-0.272334\pi\)
0.655795 + 0.754939i \(0.272334\pi\)
\(810\) 1.52247 0.0534940
\(811\) 19.1085 0.670991 0.335495 0.942042i \(-0.391096\pi\)
0.335495 + 0.942042i \(0.391096\pi\)
\(812\) 6.69877 0.235081
\(813\) −1.48750 −0.0521688
\(814\) −16.6596 −0.583918
\(815\) 17.1793 0.601763
\(816\) −3.63214 −0.127150
\(817\) −54.1362 −1.89399
\(818\) 33.1984 1.16076
\(819\) −3.40750 −0.119068
\(820\) −1.99371 −0.0696234
\(821\) −36.5634 −1.27607 −0.638036 0.770007i \(-0.720253\pi\)
−0.638036 + 0.770007i \(0.720253\pi\)
\(822\) −22.5972 −0.788167
\(823\) 21.8196 0.760585 0.380293 0.924866i \(-0.375823\pi\)
0.380293 + 0.924866i \(0.375823\pi\)
\(824\) 1.00000 0.0348367
\(825\) 7.75720 0.270071
\(826\) −27.1959 −0.946266
\(827\) −15.4165 −0.536085 −0.268043 0.963407i \(-0.586377\pi\)
−0.268043 + 0.963407i \(0.586377\pi\)
\(828\) −8.49258 −0.295138
\(829\) −39.4438 −1.36994 −0.684970 0.728571i \(-0.740185\pi\)
−0.684970 + 0.728571i \(0.740185\pi\)
\(830\) 17.4595 0.606027
\(831\) −32.2188 −1.11766
\(832\) 1.00000 0.0346688
\(833\) 16.7480 0.580283
\(834\) −1.82634 −0.0632410
\(835\) 32.3155 1.11832
\(836\) 24.1078 0.833787
\(837\) −2.49196 −0.0861347
\(838\) −7.81068 −0.269816
\(839\) −37.6597 −1.30016 −0.650079 0.759867i \(-0.725264\pi\)
−0.650079 + 0.759867i \(0.725264\pi\)
\(840\) 5.18780 0.178996
\(841\) −25.1353 −0.866733
\(842\) 9.14589 0.315188
\(843\) −15.9702 −0.550042
\(844\) 15.1150 0.520280
\(845\) −1.52247 −0.0523744
\(846\) −4.75831 −0.163594
\(847\) 8.97905 0.308524
\(848\) 7.93365 0.272443
\(849\) −15.5347 −0.533149
\(850\) 9.74174 0.334139
\(851\) −48.9185 −1.67690
\(852\) −12.6153 −0.432193
\(853\) 19.8404 0.679322 0.339661 0.940548i \(-0.389688\pi\)
0.339661 + 0.940548i \(0.389688\pi\)
\(854\) 29.9501 1.02487
\(855\) −12.6904 −0.434003
\(856\) 0.0118287 0.000404295 0
\(857\) 21.3369 0.728855 0.364427 0.931232i \(-0.381265\pi\)
0.364427 + 0.931232i \(0.381265\pi\)
\(858\) 2.89222 0.0987386
\(859\) −49.6574 −1.69429 −0.847144 0.531363i \(-0.821680\pi\)
−0.847144 + 0.531363i \(0.821680\pi\)
\(860\) 9.88798 0.337177
\(861\) 4.46221 0.152072
\(862\) −14.0382 −0.478141
\(863\) 56.4664 1.92214 0.961069 0.276310i \(-0.0891115\pi\)
0.961069 + 0.276310i \(0.0891115\pi\)
\(864\) 1.00000 0.0340207
\(865\) 15.8013 0.537260
\(866\) −13.8082 −0.469222
\(867\) 3.80759 0.129313
\(868\) −8.49135 −0.288215
\(869\) 11.3779 0.385969
\(870\) 2.99300 0.101472
\(871\) 3.78863 0.128373
\(872\) −9.52537 −0.322570
\(873\) −14.2627 −0.482720
\(874\) 70.7893 2.39448
\(875\) −39.8532 −1.34728
\(876\) 8.45160 0.285553
\(877\) −42.8506 −1.44696 −0.723482 0.690344i \(-0.757459\pi\)
−0.723482 + 0.690344i \(0.757459\pi\)
\(878\) 4.89323 0.165138
\(879\) −0.645648 −0.0217772
\(880\) −4.40330 −0.148435
\(881\) −28.9109 −0.974033 −0.487017 0.873393i \(-0.661915\pi\)
−0.487017 + 0.873393i \(0.661915\pi\)
\(882\) −4.61105 −0.155262
\(883\) −23.6224 −0.794957 −0.397479 0.917611i \(-0.630115\pi\)
−0.397479 + 0.917611i \(0.630115\pi\)
\(884\) 3.63214 0.122162
\(885\) −12.1511 −0.408454
\(886\) 6.91668 0.232370
\(887\) −17.9173 −0.601605 −0.300803 0.953686i \(-0.597255\pi\)
−0.300803 + 0.953686i \(0.597255\pi\)
\(888\) 5.76014 0.193298
\(889\) −32.4853 −1.08952
\(890\) −23.2855 −0.780531
\(891\) 2.89222 0.0968929
\(892\) 18.7880 0.629067
\(893\) 39.6625 1.32726
\(894\) −7.97737 −0.266803
\(895\) 7.70331 0.257493
\(896\) 3.40750 0.113837
\(897\) 8.49258 0.283559
\(898\) 12.9209 0.431175
\(899\) −4.89892 −0.163388
\(900\) −2.68210 −0.0894032
\(901\) 28.8161 0.960003
\(902\) −3.78743 −0.126108
\(903\) −22.1307 −0.736465
\(904\) 12.3005 0.409107
\(905\) 20.0133 0.665265
\(906\) −24.2779 −0.806578
\(907\) 11.3808 0.377894 0.188947 0.981987i \(-0.439493\pi\)
0.188947 + 0.981987i \(0.439493\pi\)
\(908\) 6.60501 0.219195
\(909\) −6.08371 −0.201784
\(910\) −5.18780 −0.171974
\(911\) 14.2585 0.472406 0.236203 0.971704i \(-0.424097\pi\)
0.236203 + 0.971704i \(0.424097\pi\)
\(912\) −8.33542 −0.276013
\(913\) 33.1676 1.09769
\(914\) 25.8588 0.855333
\(915\) 13.3817 0.442385
\(916\) 4.77427 0.157746
\(917\) −14.1967 −0.468815
\(918\) 3.63214 0.119878
\(919\) 26.4965 0.874040 0.437020 0.899452i \(-0.356034\pi\)
0.437020 + 0.899452i \(0.356034\pi\)
\(920\) −12.9297 −0.426279
\(921\) −17.3345 −0.571191
\(922\) 0.0963236 0.00317225
\(923\) 12.6153 0.415238
\(924\) 9.85523 0.324213
\(925\) −15.4493 −0.507969
\(926\) −35.9141 −1.18021
\(927\) −1.00000 −0.0328443
\(928\) 1.96589 0.0645335
\(929\) 12.1480 0.398562 0.199281 0.979942i \(-0.436139\pi\)
0.199281 + 0.979942i \(0.436139\pi\)
\(930\) −3.79392 −0.124408
\(931\) 38.4351 1.25966
\(932\) −7.86773 −0.257716
\(933\) −1.26256 −0.0413344
\(934\) 24.8147 0.811961
\(935\) −15.9934 −0.523040
\(936\) −1.00000 −0.0326860
\(937\) 6.14465 0.200737 0.100368 0.994950i \(-0.467998\pi\)
0.100368 + 0.994950i \(0.467998\pi\)
\(938\) 12.9098 0.421519
\(939\) −22.2729 −0.726848
\(940\) −7.24437 −0.236285
\(941\) −11.1205 −0.362519 −0.181259 0.983435i \(-0.558017\pi\)
−0.181259 + 0.983435i \(0.558017\pi\)
\(942\) 11.7313 0.382227
\(943\) −11.1213 −0.362158
\(944\) −7.98118 −0.259765
\(945\) −5.18780 −0.168759
\(946\) 18.7841 0.610724
\(947\) −33.8107 −1.09870 −0.549349 0.835593i \(-0.685124\pi\)
−0.549349 + 0.835593i \(0.685124\pi\)
\(948\) −3.93397 −0.127770
\(949\) −8.45160 −0.274351
\(950\) 22.3564 0.725338
\(951\) 1.98445 0.0643501
\(952\) 12.3765 0.401125
\(953\) 15.5173 0.502656 0.251328 0.967902i \(-0.419133\pi\)
0.251328 + 0.967902i \(0.419133\pi\)
\(954\) −7.93365 −0.256861
\(955\) −16.1541 −0.522736
\(956\) −12.9810 −0.419837
\(957\) 5.68578 0.183795
\(958\) −7.77361 −0.251154
\(959\) 76.9999 2.48646
\(960\) 1.52247 0.0491374
\(961\) −24.7901 −0.799682
\(962\) −5.76014 −0.185714
\(963\) −0.0118287 −0.000381173 0
\(964\) 11.8220 0.380760
\(965\) 20.8166 0.670110
\(966\) 28.9385 0.931080
\(967\) 30.3374 0.975586 0.487793 0.872959i \(-0.337802\pi\)
0.487793 + 0.872959i \(0.337802\pi\)
\(968\) 2.63508 0.0846948
\(969\) −30.2754 −0.972586
\(970\) −21.7145 −0.697211
\(971\) 5.26170 0.168856 0.0844280 0.996430i \(-0.473094\pi\)
0.0844280 + 0.996430i \(0.473094\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 6.22325 0.199508
\(974\) 2.38601 0.0764526
\(975\) 2.68210 0.0858958
\(976\) 8.78947 0.281344
\(977\) −44.4704 −1.42274 −0.711368 0.702820i \(-0.751924\pi\)
−0.711368 + 0.702820i \(0.751924\pi\)
\(978\) −11.2838 −0.360817
\(979\) −44.2352 −1.41376
\(980\) −7.02017 −0.224251
\(981\) 9.52537 0.304122
\(982\) −39.7482 −1.26842
\(983\) −54.8819 −1.75046 −0.875230 0.483706i \(-0.839290\pi\)
−0.875230 + 0.483706i \(0.839290\pi\)
\(984\) 1.30953 0.0417462
\(985\) −19.8868 −0.633647
\(986\) 7.14038 0.227396
\(987\) 16.2139 0.516095
\(988\) 8.33542 0.265185
\(989\) 55.1569 1.75389
\(990\) 4.40330 0.139946
\(991\) −32.0269 −1.01737 −0.508683 0.860954i \(-0.669868\pi\)
−0.508683 + 0.860954i \(0.669868\pi\)
\(992\) −2.49196 −0.0791198
\(993\) −7.80176 −0.247581
\(994\) 42.9866 1.36345
\(995\) −33.3776 −1.05814
\(996\) −11.4679 −0.363374
\(997\) 51.3065 1.62489 0.812446 0.583036i \(-0.198135\pi\)
0.812446 + 0.583036i \(0.198135\pi\)
\(998\) −31.1133 −0.984875
\(999\) −5.76014 −0.182243
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.1.5 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.z.1.5 14 1.1 even 1 trivial