Properties

Label 4014.2.a.r.1.3
Level 4014
Weight 2
Character 4014.1
Self dual yes
Analytic conductor 32.052
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.356173.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1338)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.85688\)
Character \(\chi\) = 4014.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.43386 q^{5} -1.87103 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.43386 q^{5} -1.87103 q^{7} +1.00000 q^{8} -1.43386 q^{10} -5.50771 q^{11} +5.81260 q^{13} -1.87103 q^{14} +1.00000 q^{16} +3.16177 q^{17} -0.890321 q^{19} -1.43386 q^{20} -5.50771 q^{22} -1.68749 q^{23} -2.94404 q^{25} +5.81260 q^{26} -1.87103 q^{28} +6.78000 q^{29} +2.18675 q^{31} +1.00000 q^{32} +3.16177 q^{34} +2.68279 q^{35} +9.53205 q^{37} -0.890321 q^{38} -1.43386 q^{40} -5.50256 q^{41} -9.08158 q^{43} -5.50771 q^{44} -1.68749 q^{46} -4.43633 q^{47} -3.49926 q^{49} -2.94404 q^{50} +5.81260 q^{52} -9.50010 q^{53} +7.89729 q^{55} -1.87103 q^{56} +6.78000 q^{58} -14.8655 q^{59} -12.3227 q^{61} +2.18675 q^{62} +1.00000 q^{64} -8.33446 q^{65} -0.993472 q^{67} +3.16177 q^{68} +2.68279 q^{70} -4.42624 q^{71} +2.32096 q^{73} +9.53205 q^{74} -0.890321 q^{76} +10.3051 q^{77} -12.5579 q^{79} -1.43386 q^{80} -5.50256 q^{82} -15.1636 q^{83} -4.53354 q^{85} -9.08158 q^{86} -5.50771 q^{88} +3.41540 q^{89} -10.8755 q^{91} -1.68749 q^{92} -4.43633 q^{94} +1.27660 q^{95} -13.6143 q^{97} -3.49926 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} - 5q^{10} - 9q^{11} - q^{14} + 5q^{16} - 6q^{17} - 4q^{19} - 5q^{20} - 9q^{22} - 16q^{23} + 8q^{25} - q^{28} - 8q^{29} - q^{31} + 5q^{32} - 6q^{34} - 22q^{35} - 2q^{37} - 4q^{38} - 5q^{40} - 4q^{41} + 3q^{43} - 9q^{44} - 16q^{46} - 18q^{47} + 2q^{49} + 8q^{50} - 26q^{53} + q^{55} - q^{56} - 8q^{58} - 21q^{59} - 20q^{61} - q^{62} + 5q^{64} + 3q^{65} - 5q^{67} - 6q^{68} - 22q^{70} - 17q^{71} + 5q^{73} - 2q^{74} - 4q^{76} - 2q^{77} - 21q^{79} - 5q^{80} - 4q^{82} - 11q^{83} - 12q^{85} + 3q^{86} - 9q^{88} + 5q^{89} - 10q^{91} - 16q^{92} - 18q^{94} - 10q^{95} - 11q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.43386 −0.641242 −0.320621 0.947208i \(-0.603892\pi\)
−0.320621 + 0.947208i \(0.603892\pi\)
\(6\) 0 0
\(7\) −1.87103 −0.707182 −0.353591 0.935400i \(-0.615039\pi\)
−0.353591 + 0.935400i \(0.615039\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.43386 −0.453427
\(11\) −5.50771 −1.66064 −0.830319 0.557288i \(-0.811842\pi\)
−0.830319 + 0.557288i \(0.811842\pi\)
\(12\) 0 0
\(13\) 5.81260 1.61213 0.806063 0.591830i \(-0.201594\pi\)
0.806063 + 0.591830i \(0.201594\pi\)
\(14\) −1.87103 −0.500053
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.16177 0.766841 0.383421 0.923574i \(-0.374746\pi\)
0.383421 + 0.923574i \(0.374746\pi\)
\(18\) 0 0
\(19\) −0.890321 −0.204254 −0.102127 0.994771i \(-0.532565\pi\)
−0.102127 + 0.994771i \(0.532565\pi\)
\(20\) −1.43386 −0.320621
\(21\) 0 0
\(22\) −5.50771 −1.17425
\(23\) −1.68749 −0.351867 −0.175933 0.984402i \(-0.556294\pi\)
−0.175933 + 0.984402i \(0.556294\pi\)
\(24\) 0 0
\(25\) −2.94404 −0.588809
\(26\) 5.81260 1.13994
\(27\) 0 0
\(28\) −1.87103 −0.353591
\(29\) 6.78000 1.25901 0.629507 0.776995i \(-0.283257\pi\)
0.629507 + 0.776995i \(0.283257\pi\)
\(30\) 0 0
\(31\) 2.18675 0.392753 0.196376 0.980529i \(-0.437083\pi\)
0.196376 + 0.980529i \(0.437083\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.16177 0.542239
\(35\) 2.68279 0.453475
\(36\) 0 0
\(37\) 9.53205 1.56706 0.783530 0.621354i \(-0.213417\pi\)
0.783530 + 0.621354i \(0.213417\pi\)
\(38\) −0.890321 −0.144429
\(39\) 0 0
\(40\) −1.43386 −0.226713
\(41\) −5.50256 −0.859356 −0.429678 0.902982i \(-0.641373\pi\)
−0.429678 + 0.902982i \(0.641373\pi\)
\(42\) 0 0
\(43\) −9.08158 −1.38493 −0.692464 0.721453i \(-0.743475\pi\)
−0.692464 + 0.721453i \(0.743475\pi\)
\(44\) −5.50771 −0.830319
\(45\) 0 0
\(46\) −1.68749 −0.248808
\(47\) −4.43633 −0.647105 −0.323553 0.946210i \(-0.604877\pi\)
−0.323553 + 0.946210i \(0.604877\pi\)
\(48\) 0 0
\(49\) −3.49926 −0.499894
\(50\) −2.94404 −0.416351
\(51\) 0 0
\(52\) 5.81260 0.806063
\(53\) −9.50010 −1.30494 −0.652469 0.757815i \(-0.726267\pi\)
−0.652469 + 0.757815i \(0.726267\pi\)
\(54\) 0 0
\(55\) 7.89729 1.06487
\(56\) −1.87103 −0.250026
\(57\) 0 0
\(58\) 6.78000 0.890257
\(59\) −14.8655 −1.93533 −0.967663 0.252246i \(-0.918831\pi\)
−0.967663 + 0.252246i \(0.918831\pi\)
\(60\) 0 0
\(61\) −12.3227 −1.57776 −0.788880 0.614547i \(-0.789339\pi\)
−0.788880 + 0.614547i \(0.789339\pi\)
\(62\) 2.18675 0.277718
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.33446 −1.03376
\(66\) 0 0
\(67\) −0.993472 −0.121372 −0.0606860 0.998157i \(-0.519329\pi\)
−0.0606860 + 0.998157i \(0.519329\pi\)
\(68\) 3.16177 0.383421
\(69\) 0 0
\(70\) 2.68279 0.320655
\(71\) −4.42624 −0.525298 −0.262649 0.964891i \(-0.584596\pi\)
−0.262649 + 0.964891i \(0.584596\pi\)
\(72\) 0 0
\(73\) 2.32096 0.271648 0.135824 0.990733i \(-0.456632\pi\)
0.135824 + 0.990733i \(0.456632\pi\)
\(74\) 9.53205 1.10808
\(75\) 0 0
\(76\) −0.890321 −0.102127
\(77\) 10.3051 1.17437
\(78\) 0 0
\(79\) −12.5579 −1.41287 −0.706436 0.707777i \(-0.749698\pi\)
−0.706436 + 0.707777i \(0.749698\pi\)
\(80\) −1.43386 −0.160310
\(81\) 0 0
\(82\) −5.50256 −0.607657
\(83\) −15.1636 −1.66442 −0.832210 0.554460i \(-0.812925\pi\)
−0.832210 + 0.554460i \(0.812925\pi\)
\(84\) 0 0
\(85\) −4.53354 −0.491731
\(86\) −9.08158 −0.979292
\(87\) 0 0
\(88\) −5.50771 −0.587124
\(89\) 3.41540 0.362032 0.181016 0.983480i \(-0.442061\pi\)
0.181016 + 0.983480i \(0.442061\pi\)
\(90\) 0 0
\(91\) −10.8755 −1.14007
\(92\) −1.68749 −0.175933
\(93\) 0 0
\(94\) −4.43633 −0.457573
\(95\) 1.27660 0.130976
\(96\) 0 0
\(97\) −13.6143 −1.38232 −0.691160 0.722702i \(-0.742900\pi\)
−0.691160 + 0.722702i \(0.742900\pi\)
\(98\) −3.49926 −0.353479
\(99\) 0 0
\(100\) −2.94404 −0.294404
\(101\) 9.77079 0.972230 0.486115 0.873895i \(-0.338414\pi\)
0.486115 + 0.873895i \(0.338414\pi\)
\(102\) 0 0
\(103\) 7.03849 0.693523 0.346761 0.937953i \(-0.387281\pi\)
0.346761 + 0.937953i \(0.387281\pi\)
\(104\) 5.81260 0.569972
\(105\) 0 0
\(106\) −9.50010 −0.922731
\(107\) −12.3178 −1.19081 −0.595405 0.803425i \(-0.703009\pi\)
−0.595405 + 0.803425i \(0.703009\pi\)
\(108\) 0 0
\(109\) 15.5040 1.48502 0.742509 0.669836i \(-0.233636\pi\)
0.742509 + 0.669836i \(0.233636\pi\)
\(110\) 7.89729 0.752977
\(111\) 0 0
\(112\) −1.87103 −0.176795
\(113\) −8.44083 −0.794047 −0.397023 0.917808i \(-0.629957\pi\)
−0.397023 + 0.917808i \(0.629957\pi\)
\(114\) 0 0
\(115\) 2.41963 0.225632
\(116\) 6.78000 0.629507
\(117\) 0 0
\(118\) −14.8655 −1.36848
\(119\) −5.91575 −0.542296
\(120\) 0 0
\(121\) 19.3349 1.75772
\(122\) −12.3227 −1.11564
\(123\) 0 0
\(124\) 2.18675 0.196376
\(125\) 11.3907 1.01881
\(126\) 0 0
\(127\) 12.7041 1.12731 0.563654 0.826011i \(-0.309395\pi\)
0.563654 + 0.826011i \(0.309395\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −8.33446 −0.730980
\(131\) 12.8692 1.12439 0.562193 0.827006i \(-0.309958\pi\)
0.562193 + 0.827006i \(0.309958\pi\)
\(132\) 0 0
\(133\) 1.66581 0.144444
\(134\) −0.993472 −0.0858229
\(135\) 0 0
\(136\) 3.16177 0.271119
\(137\) 5.08158 0.434149 0.217074 0.976155i \(-0.430349\pi\)
0.217074 + 0.976155i \(0.430349\pi\)
\(138\) 0 0
\(139\) −6.73928 −0.571618 −0.285809 0.958287i \(-0.592262\pi\)
−0.285809 + 0.958287i \(0.592262\pi\)
\(140\) 2.68279 0.226737
\(141\) 0 0
\(142\) −4.42624 −0.371442
\(143\) −32.0141 −2.67716
\(144\) 0 0
\(145\) −9.72157 −0.807332
\(146\) 2.32096 0.192084
\(147\) 0 0
\(148\) 9.53205 0.783530
\(149\) −2.81176 −0.230349 −0.115174 0.993345i \(-0.536743\pi\)
−0.115174 + 0.993345i \(0.536743\pi\)
\(150\) 0 0
\(151\) −7.74113 −0.629965 −0.314982 0.949098i \(-0.601999\pi\)
−0.314982 + 0.949098i \(0.601999\pi\)
\(152\) −0.890321 −0.0722145
\(153\) 0 0
\(154\) 10.3051 0.830407
\(155\) −3.13550 −0.251850
\(156\) 0 0
\(157\) −19.2202 −1.53394 −0.766969 0.641684i \(-0.778236\pi\)
−0.766969 + 0.641684i \(0.778236\pi\)
\(158\) −12.5579 −0.999051
\(159\) 0 0
\(160\) −1.43386 −0.113357
\(161\) 3.15735 0.248834
\(162\) 0 0
\(163\) 4.23627 0.331810 0.165905 0.986142i \(-0.446945\pi\)
0.165905 + 0.986142i \(0.446945\pi\)
\(164\) −5.50256 −0.429678
\(165\) 0 0
\(166\) −15.1636 −1.17692
\(167\) −24.8132 −1.92011 −0.960053 0.279818i \(-0.909726\pi\)
−0.960053 + 0.279818i \(0.909726\pi\)
\(168\) 0 0
\(169\) 20.7863 1.59895
\(170\) −4.53354 −0.347706
\(171\) 0 0
\(172\) −9.08158 −0.692464
\(173\) −3.90530 −0.296915 −0.148457 0.988919i \(-0.547431\pi\)
−0.148457 + 0.988919i \(0.547431\pi\)
\(174\) 0 0
\(175\) 5.50838 0.416395
\(176\) −5.50771 −0.415160
\(177\) 0 0
\(178\) 3.41540 0.255995
\(179\) −6.73708 −0.503553 −0.251776 0.967785i \(-0.581015\pi\)
−0.251776 + 0.967785i \(0.581015\pi\)
\(180\) 0 0
\(181\) −3.43219 −0.255113 −0.127556 0.991831i \(-0.540713\pi\)
−0.127556 + 0.991831i \(0.540713\pi\)
\(182\) −10.8755 −0.806148
\(183\) 0 0
\(184\) −1.68749 −0.124404
\(185\) −13.6676 −1.00486
\(186\) 0 0
\(187\) −17.4141 −1.27345
\(188\) −4.43633 −0.323553
\(189\) 0 0
\(190\) 1.27660 0.0926140
\(191\) 19.5410 1.41393 0.706967 0.707247i \(-0.250063\pi\)
0.706967 + 0.707247i \(0.250063\pi\)
\(192\) 0 0
\(193\) −5.55650 −0.399965 −0.199983 0.979799i \(-0.564089\pi\)
−0.199983 + 0.979799i \(0.564089\pi\)
\(194\) −13.6143 −0.977448
\(195\) 0 0
\(196\) −3.49926 −0.249947
\(197\) 21.9757 1.56571 0.782853 0.622207i \(-0.213764\pi\)
0.782853 + 0.622207i \(0.213764\pi\)
\(198\) 0 0
\(199\) 4.07733 0.289034 0.144517 0.989502i \(-0.453837\pi\)
0.144517 + 0.989502i \(0.453837\pi\)
\(200\) −2.94404 −0.208175
\(201\) 0 0
\(202\) 9.77079 0.687470
\(203\) −12.6856 −0.890351
\(204\) 0 0
\(205\) 7.88991 0.551055
\(206\) 7.03849 0.490395
\(207\) 0 0
\(208\) 5.81260 0.403031
\(209\) 4.90363 0.339191
\(210\) 0 0
\(211\) −10.0288 −0.690409 −0.345205 0.938527i \(-0.612190\pi\)
−0.345205 + 0.938527i \(0.612190\pi\)
\(212\) −9.50010 −0.652469
\(213\) 0 0
\(214\) −12.3178 −0.842030
\(215\) 13.0217 0.888074
\(216\) 0 0
\(217\) −4.09148 −0.277747
\(218\) 15.5040 1.05007
\(219\) 0 0
\(220\) 7.89729 0.532435
\(221\) 18.3781 1.23624
\(222\) 0 0
\(223\) −1.00000 −0.0669650
\(224\) −1.87103 −0.125013
\(225\) 0 0
\(226\) −8.44083 −0.561476
\(227\) 20.0364 1.32986 0.664930 0.746906i \(-0.268461\pi\)
0.664930 + 0.746906i \(0.268461\pi\)
\(228\) 0 0
\(229\) −18.4886 −1.22176 −0.610881 0.791723i \(-0.709185\pi\)
−0.610881 + 0.791723i \(0.709185\pi\)
\(230\) 2.41963 0.159546
\(231\) 0 0
\(232\) 6.78000 0.445129
\(233\) −6.03324 −0.395251 −0.197625 0.980278i \(-0.563323\pi\)
−0.197625 + 0.980278i \(0.563323\pi\)
\(234\) 0 0
\(235\) 6.36108 0.414951
\(236\) −14.8655 −0.967663
\(237\) 0 0
\(238\) −5.91575 −0.383461
\(239\) 24.6088 1.59181 0.795906 0.605420i \(-0.206995\pi\)
0.795906 + 0.605420i \(0.206995\pi\)
\(240\) 0 0
\(241\) 12.3387 0.794804 0.397402 0.917645i \(-0.369912\pi\)
0.397402 + 0.917645i \(0.369912\pi\)
\(242\) 19.3349 1.24289
\(243\) 0 0
\(244\) −12.3227 −0.788880
\(245\) 5.01745 0.320553
\(246\) 0 0
\(247\) −5.17508 −0.329282
\(248\) 2.18675 0.138859
\(249\) 0 0
\(250\) 11.3907 0.720408
\(251\) 13.9475 0.880361 0.440180 0.897909i \(-0.354914\pi\)
0.440180 + 0.897909i \(0.354914\pi\)
\(252\) 0 0
\(253\) 9.29424 0.584324
\(254\) 12.7041 0.797127
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.0207 −1.24885 −0.624427 0.781083i \(-0.714668\pi\)
−0.624427 + 0.781083i \(0.714668\pi\)
\(258\) 0 0
\(259\) −17.8347 −1.10820
\(260\) −8.33446 −0.516881
\(261\) 0 0
\(262\) 12.8692 0.795061
\(263\) 2.19289 0.135219 0.0676097 0.997712i \(-0.478463\pi\)
0.0676097 + 0.997712i \(0.478463\pi\)
\(264\) 0 0
\(265\) 13.6218 0.836781
\(266\) 1.66581 0.102138
\(267\) 0 0
\(268\) −0.993472 −0.0606860
\(269\) 26.2148 1.59835 0.799174 0.601100i \(-0.205271\pi\)
0.799174 + 0.601100i \(0.205271\pi\)
\(270\) 0 0
\(271\) 8.88235 0.539564 0.269782 0.962921i \(-0.413048\pi\)
0.269782 + 0.962921i \(0.413048\pi\)
\(272\) 3.16177 0.191710
\(273\) 0 0
\(274\) 5.08158 0.306989
\(275\) 16.2149 0.977798
\(276\) 0 0
\(277\) −25.3250 −1.52163 −0.760815 0.648968i \(-0.775201\pi\)
−0.760815 + 0.648968i \(0.775201\pi\)
\(278\) −6.73928 −0.404195
\(279\) 0 0
\(280\) 2.68279 0.160327
\(281\) −20.0533 −1.19628 −0.598139 0.801392i \(-0.704093\pi\)
−0.598139 + 0.801392i \(0.704093\pi\)
\(282\) 0 0
\(283\) 21.6602 1.28757 0.643783 0.765208i \(-0.277364\pi\)
0.643783 + 0.765208i \(0.277364\pi\)
\(284\) −4.42624 −0.262649
\(285\) 0 0
\(286\) −32.0141 −1.89304
\(287\) 10.2954 0.607721
\(288\) 0 0
\(289\) −7.00322 −0.411954
\(290\) −9.72157 −0.570870
\(291\) 0 0
\(292\) 2.32096 0.135824
\(293\) 18.1963 1.06304 0.531519 0.847046i \(-0.321621\pi\)
0.531519 + 0.847046i \(0.321621\pi\)
\(294\) 0 0
\(295\) 21.3151 1.24101
\(296\) 9.53205 0.554039
\(297\) 0 0
\(298\) −2.81176 −0.162881
\(299\) −9.80873 −0.567254
\(300\) 0 0
\(301\) 16.9919 0.979395
\(302\) −7.74113 −0.445452
\(303\) 0 0
\(304\) −0.890321 −0.0510634
\(305\) 17.6690 1.01173
\(306\) 0 0
\(307\) −9.63403 −0.549843 −0.274921 0.961467i \(-0.588652\pi\)
−0.274921 + 0.961467i \(0.588652\pi\)
\(308\) 10.3051 0.587186
\(309\) 0 0
\(310\) −3.13550 −0.178084
\(311\) 7.31701 0.414909 0.207455 0.978245i \(-0.433482\pi\)
0.207455 + 0.978245i \(0.433482\pi\)
\(312\) 0 0
\(313\) 20.3220 1.14867 0.574333 0.818622i \(-0.305261\pi\)
0.574333 + 0.818622i \(0.305261\pi\)
\(314\) −19.2202 −1.08466
\(315\) 0 0
\(316\) −12.5579 −0.706436
\(317\) 34.6867 1.94820 0.974099 0.226121i \(-0.0726045\pi\)
0.974099 + 0.226121i \(0.0726045\pi\)
\(318\) 0 0
\(319\) −37.3423 −2.09077
\(320\) −1.43386 −0.0801552
\(321\) 0 0
\(322\) 3.15735 0.175952
\(323\) −2.81499 −0.156630
\(324\) 0 0
\(325\) −17.1125 −0.949233
\(326\) 4.23627 0.234625
\(327\) 0 0
\(328\) −5.50256 −0.303828
\(329\) 8.30049 0.457621
\(330\) 0 0
\(331\) 20.8200 1.14437 0.572184 0.820125i \(-0.306096\pi\)
0.572184 + 0.820125i \(0.306096\pi\)
\(332\) −15.1636 −0.832210
\(333\) 0 0
\(334\) −24.8132 −1.35772
\(335\) 1.42450 0.0778288
\(336\) 0 0
\(337\) 17.8513 0.972421 0.486211 0.873842i \(-0.338379\pi\)
0.486211 + 0.873842i \(0.338379\pi\)
\(338\) 20.7863 1.13063
\(339\) 0 0
\(340\) −4.53354 −0.245865
\(341\) −12.0440 −0.652220
\(342\) 0 0
\(343\) 19.6444 1.06070
\(344\) −9.08158 −0.489646
\(345\) 0 0
\(346\) −3.90530 −0.209950
\(347\) −16.4617 −0.883709 −0.441855 0.897087i \(-0.645679\pi\)
−0.441855 + 0.897087i \(0.645679\pi\)
\(348\) 0 0
\(349\) −15.7220 −0.841581 −0.420791 0.907158i \(-0.638247\pi\)
−0.420791 + 0.907158i \(0.638247\pi\)
\(350\) 5.50838 0.294435
\(351\) 0 0
\(352\) −5.50771 −0.293562
\(353\) −23.2166 −1.23569 −0.617847 0.786298i \(-0.711995\pi\)
−0.617847 + 0.786298i \(0.711995\pi\)
\(354\) 0 0
\(355\) 6.34662 0.336843
\(356\) 3.41540 0.181016
\(357\) 0 0
\(358\) −6.73708 −0.356066
\(359\) 15.3354 0.809372 0.404686 0.914456i \(-0.367381\pi\)
0.404686 + 0.914456i \(0.367381\pi\)
\(360\) 0 0
\(361\) −18.2073 −0.958280
\(362\) −3.43219 −0.180392
\(363\) 0 0
\(364\) −10.8755 −0.570033
\(365\) −3.32793 −0.174192
\(366\) 0 0
\(367\) −1.82214 −0.0951152 −0.0475576 0.998868i \(-0.515144\pi\)
−0.0475576 + 0.998868i \(0.515144\pi\)
\(368\) −1.68749 −0.0879667
\(369\) 0 0
\(370\) −13.6676 −0.710547
\(371\) 17.7749 0.922828
\(372\) 0 0
\(373\) −24.9826 −1.29355 −0.646776 0.762680i \(-0.723883\pi\)
−0.646776 + 0.762680i \(0.723883\pi\)
\(374\) −17.4141 −0.900462
\(375\) 0 0
\(376\) −4.43633 −0.228786
\(377\) 39.4094 2.02969
\(378\) 0 0
\(379\) −9.99010 −0.513157 −0.256579 0.966523i \(-0.582595\pi\)
−0.256579 + 0.966523i \(0.582595\pi\)
\(380\) 1.27660 0.0654880
\(381\) 0 0
\(382\) 19.5410 0.999802
\(383\) 14.7719 0.754808 0.377404 0.926049i \(-0.376817\pi\)
0.377404 + 0.926049i \(0.376817\pi\)
\(384\) 0 0
\(385\) −14.7760 −0.753057
\(386\) −5.55650 −0.282818
\(387\) 0 0
\(388\) −13.6143 −0.691160
\(389\) −26.0548 −1.32103 −0.660516 0.750812i \(-0.729663\pi\)
−0.660516 + 0.750812i \(0.729663\pi\)
\(390\) 0 0
\(391\) −5.33547 −0.269826
\(392\) −3.49926 −0.176739
\(393\) 0 0
\(394\) 21.9757 1.10712
\(395\) 18.0062 0.905993
\(396\) 0 0
\(397\) −25.3046 −1.27000 −0.635000 0.772512i \(-0.719000\pi\)
−0.635000 + 0.772512i \(0.719000\pi\)
\(398\) 4.07733 0.204378
\(399\) 0 0
\(400\) −2.94404 −0.147202
\(401\) −16.9955 −0.848716 −0.424358 0.905495i \(-0.639500\pi\)
−0.424358 + 0.905495i \(0.639500\pi\)
\(402\) 0 0
\(403\) 12.7107 0.633167
\(404\) 9.77079 0.486115
\(405\) 0 0
\(406\) −12.6856 −0.629573
\(407\) −52.4998 −2.60232
\(408\) 0 0
\(409\) −30.6720 −1.51663 −0.758315 0.651888i \(-0.773977\pi\)
−0.758315 + 0.651888i \(0.773977\pi\)
\(410\) 7.88991 0.389655
\(411\) 0 0
\(412\) 7.03849 0.346761
\(413\) 27.8138 1.36863
\(414\) 0 0
\(415\) 21.7425 1.06730
\(416\) 5.81260 0.284986
\(417\) 0 0
\(418\) 4.90363 0.239844
\(419\) −17.4958 −0.854724 −0.427362 0.904081i \(-0.640557\pi\)
−0.427362 + 0.904081i \(0.640557\pi\)
\(420\) 0 0
\(421\) −12.6330 −0.615695 −0.307847 0.951436i \(-0.599609\pi\)
−0.307847 + 0.951436i \(0.599609\pi\)
\(422\) −10.0288 −0.488193
\(423\) 0 0
\(424\) −9.50010 −0.461365
\(425\) −9.30838 −0.451523
\(426\) 0 0
\(427\) 23.0561 1.11576
\(428\) −12.3178 −0.595405
\(429\) 0 0
\(430\) 13.0217 0.627963
\(431\) −14.6767 −0.706953 −0.353477 0.935443i \(-0.615001\pi\)
−0.353477 + 0.935443i \(0.615001\pi\)
\(432\) 0 0
\(433\) −22.5184 −1.08216 −0.541082 0.840970i \(-0.681985\pi\)
−0.541082 + 0.840970i \(0.681985\pi\)
\(434\) −4.09148 −0.196397
\(435\) 0 0
\(436\) 15.5040 0.742509
\(437\) 1.50241 0.0718701
\(438\) 0 0
\(439\) −8.52689 −0.406966 −0.203483 0.979078i \(-0.565226\pi\)
−0.203483 + 0.979078i \(0.565226\pi\)
\(440\) 7.89729 0.376489
\(441\) 0 0
\(442\) 18.3781 0.874157
\(443\) 10.6999 0.508368 0.254184 0.967156i \(-0.418193\pi\)
0.254184 + 0.967156i \(0.418193\pi\)
\(444\) 0 0
\(445\) −4.89721 −0.232150
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) −1.87103 −0.0883977
\(449\) 42.0615 1.98501 0.992503 0.122224i \(-0.0390025\pi\)
0.992503 + 0.122224i \(0.0390025\pi\)
\(450\) 0 0
\(451\) 30.3065 1.42708
\(452\) −8.44083 −0.397023
\(453\) 0 0
\(454\) 20.0364 0.940353
\(455\) 15.5940 0.731058
\(456\) 0 0
\(457\) −8.63916 −0.404123 −0.202061 0.979373i \(-0.564764\pi\)
−0.202061 + 0.979373i \(0.564764\pi\)
\(458\) −18.4886 −0.863916
\(459\) 0 0
\(460\) 2.41963 0.112816
\(461\) 12.8499 0.598480 0.299240 0.954178i \(-0.403267\pi\)
0.299240 + 0.954178i \(0.403267\pi\)
\(462\) 0 0
\(463\) −14.9786 −0.696115 −0.348058 0.937473i \(-0.613159\pi\)
−0.348058 + 0.937473i \(0.613159\pi\)
\(464\) 6.78000 0.314753
\(465\) 0 0
\(466\) −6.03324 −0.279484
\(467\) −6.56107 −0.303610 −0.151805 0.988410i \(-0.548509\pi\)
−0.151805 + 0.988410i \(0.548509\pi\)
\(468\) 0 0
\(469\) 1.85881 0.0858320
\(470\) 6.36108 0.293415
\(471\) 0 0
\(472\) −14.8655 −0.684241
\(473\) 50.0187 2.29986
\(474\) 0 0
\(475\) 2.62114 0.120266
\(476\) −5.91575 −0.271148
\(477\) 0 0
\(478\) 24.6088 1.12558
\(479\) 13.4894 0.616347 0.308173 0.951330i \(-0.400282\pi\)
0.308173 + 0.951330i \(0.400282\pi\)
\(480\) 0 0
\(481\) 55.4060 2.52630
\(482\) 12.3387 0.562011
\(483\) 0 0
\(484\) 19.3349 0.878859
\(485\) 19.5210 0.886402
\(486\) 0 0
\(487\) −32.9194 −1.49172 −0.745861 0.666102i \(-0.767962\pi\)
−0.745861 + 0.666102i \(0.767962\pi\)
\(488\) −12.3227 −0.557822
\(489\) 0 0
\(490\) 5.01745 0.226665
\(491\) 28.1030 1.26827 0.634135 0.773222i \(-0.281356\pi\)
0.634135 + 0.773222i \(0.281356\pi\)
\(492\) 0 0
\(493\) 21.4368 0.965464
\(494\) −5.17508 −0.232838
\(495\) 0 0
\(496\) 2.18675 0.0981882
\(497\) 8.28162 0.371481
\(498\) 0 0
\(499\) 35.1778 1.57478 0.787388 0.616457i \(-0.211433\pi\)
0.787388 + 0.616457i \(0.211433\pi\)
\(500\) 11.3907 0.509405
\(501\) 0 0
\(502\) 13.9475 0.622509
\(503\) −20.6999 −0.922964 −0.461482 0.887150i \(-0.652682\pi\)
−0.461482 + 0.887150i \(0.652682\pi\)
\(504\) 0 0
\(505\) −14.0100 −0.623435
\(506\) 9.29424 0.413179
\(507\) 0 0
\(508\) 12.7041 0.563654
\(509\) −19.1836 −0.850299 −0.425150 0.905123i \(-0.639778\pi\)
−0.425150 + 0.905123i \(0.639778\pi\)
\(510\) 0 0
\(511\) −4.34258 −0.192104
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −20.0207 −0.883074
\(515\) −10.0922 −0.444716
\(516\) 0 0
\(517\) 24.4340 1.07461
\(518\) −17.8347 −0.783613
\(519\) 0 0
\(520\) −8.33446 −0.365490
\(521\) 11.0551 0.484331 0.242165 0.970235i \(-0.422142\pi\)
0.242165 + 0.970235i \(0.422142\pi\)
\(522\) 0 0
\(523\) −32.4419 −1.41859 −0.709293 0.704913i \(-0.750986\pi\)
−0.709293 + 0.704913i \(0.750986\pi\)
\(524\) 12.8692 0.562193
\(525\) 0 0
\(526\) 2.19289 0.0956146
\(527\) 6.91401 0.301179
\(528\) 0 0
\(529\) −20.1524 −0.876190
\(530\) 13.6218 0.591694
\(531\) 0 0
\(532\) 1.66581 0.0722222
\(533\) −31.9842 −1.38539
\(534\) 0 0
\(535\) 17.6621 0.763598
\(536\) −0.993472 −0.0429115
\(537\) 0 0
\(538\) 26.2148 1.13020
\(539\) 19.2729 0.830143
\(540\) 0 0
\(541\) 19.6356 0.844200 0.422100 0.906549i \(-0.361293\pi\)
0.422100 + 0.906549i \(0.361293\pi\)
\(542\) 8.88235 0.381530
\(543\) 0 0
\(544\) 3.16177 0.135560
\(545\) −22.2306 −0.952255
\(546\) 0 0
\(547\) −18.3055 −0.782688 −0.391344 0.920245i \(-0.627990\pi\)
−0.391344 + 0.920245i \(0.627990\pi\)
\(548\) 5.08158 0.217074
\(549\) 0 0
\(550\) 16.2149 0.691408
\(551\) −6.03637 −0.257158
\(552\) 0 0
\(553\) 23.4961 0.999157
\(554\) −25.3250 −1.07596
\(555\) 0 0
\(556\) −6.73928 −0.285809
\(557\) −29.8604 −1.26523 −0.632614 0.774467i \(-0.718018\pi\)
−0.632614 + 0.774467i \(0.718018\pi\)
\(558\) 0 0
\(559\) −52.7876 −2.23268
\(560\) 2.68279 0.113369
\(561\) 0 0
\(562\) −20.0533 −0.845896
\(563\) −2.85418 −0.120290 −0.0601448 0.998190i \(-0.519156\pi\)
−0.0601448 + 0.998190i \(0.519156\pi\)
\(564\) 0 0
\(565\) 12.1030 0.509176
\(566\) 21.6602 0.910447
\(567\) 0 0
\(568\) −4.42624 −0.185721
\(569\) −21.5560 −0.903673 −0.451837 0.892101i \(-0.649231\pi\)
−0.451837 + 0.892101i \(0.649231\pi\)
\(570\) 0 0
\(571\) −31.3753 −1.31302 −0.656508 0.754319i \(-0.727967\pi\)
−0.656508 + 0.754319i \(0.727967\pi\)
\(572\) −32.0141 −1.33858
\(573\) 0 0
\(574\) 10.2954 0.429724
\(575\) 4.96806 0.207182
\(576\) 0 0
\(577\) −13.1804 −0.548707 −0.274354 0.961629i \(-0.588464\pi\)
−0.274354 + 0.961629i \(0.588464\pi\)
\(578\) −7.00322 −0.291296
\(579\) 0 0
\(580\) −9.72157 −0.403666
\(581\) 28.3715 1.17705
\(582\) 0 0
\(583\) 52.3238 2.16703
\(584\) 2.32096 0.0960420
\(585\) 0 0
\(586\) 18.1963 0.751681
\(587\) 25.8281 1.06604 0.533020 0.846103i \(-0.321057\pi\)
0.533020 + 0.846103i \(0.321057\pi\)
\(588\) 0 0
\(589\) −1.94691 −0.0802211
\(590\) 21.3151 0.877528
\(591\) 0 0
\(592\) 9.53205 0.391765
\(593\) 7.80648 0.320574 0.160287 0.987070i \(-0.448758\pi\)
0.160287 + 0.987070i \(0.448758\pi\)
\(594\) 0 0
\(595\) 8.48236 0.347743
\(596\) −2.81176 −0.115174
\(597\) 0 0
\(598\) −9.80873 −0.401109
\(599\) −12.2236 −0.499442 −0.249721 0.968318i \(-0.580339\pi\)
−0.249721 + 0.968318i \(0.580339\pi\)
\(600\) 0 0
\(601\) −18.5172 −0.755333 −0.377666 0.925942i \(-0.623273\pi\)
−0.377666 + 0.925942i \(0.623273\pi\)
\(602\) 16.9919 0.692537
\(603\) 0 0
\(604\) −7.74113 −0.314982
\(605\) −27.7236 −1.12712
\(606\) 0 0
\(607\) 38.9871 1.58244 0.791219 0.611533i \(-0.209447\pi\)
0.791219 + 0.611533i \(0.209447\pi\)
\(608\) −0.890321 −0.0361073
\(609\) 0 0
\(610\) 17.6690 0.715398
\(611\) −25.7866 −1.04321
\(612\) 0 0
\(613\) −27.2786 −1.10177 −0.550886 0.834580i \(-0.685710\pi\)
−0.550886 + 0.834580i \(0.685710\pi\)
\(614\) −9.63403 −0.388798
\(615\) 0 0
\(616\) 10.3051 0.415203
\(617\) 29.6233 1.19259 0.596295 0.802765i \(-0.296639\pi\)
0.596295 + 0.802765i \(0.296639\pi\)
\(618\) 0 0
\(619\) 22.0162 0.884908 0.442454 0.896791i \(-0.354108\pi\)
0.442454 + 0.896791i \(0.354108\pi\)
\(620\) −3.13550 −0.125925
\(621\) 0 0
\(622\) 7.31701 0.293385
\(623\) −6.39031 −0.256022
\(624\) 0 0
\(625\) −1.61239 −0.0644956
\(626\) 20.3220 0.812230
\(627\) 0 0
\(628\) −19.2202 −0.766969
\(629\) 30.1381 1.20169
\(630\) 0 0
\(631\) 28.6537 1.14069 0.570344 0.821406i \(-0.306810\pi\)
0.570344 + 0.821406i \(0.306810\pi\)
\(632\) −12.5579 −0.499525
\(633\) 0 0
\(634\) 34.6867 1.37758
\(635\) −18.2159 −0.722877
\(636\) 0 0
\(637\) −20.3398 −0.805892
\(638\) −37.3423 −1.47839
\(639\) 0 0
\(640\) −1.43386 −0.0566783
\(641\) −0.654574 −0.0258541 −0.0129270 0.999916i \(-0.504115\pi\)
−0.0129270 + 0.999916i \(0.504115\pi\)
\(642\) 0 0
\(643\) 37.4692 1.47764 0.738821 0.673902i \(-0.235383\pi\)
0.738821 + 0.673902i \(0.235383\pi\)
\(644\) 3.15735 0.124417
\(645\) 0 0
\(646\) −2.81499 −0.110754
\(647\) 24.4525 0.961327 0.480663 0.876905i \(-0.340396\pi\)
0.480663 + 0.876905i \(0.340396\pi\)
\(648\) 0 0
\(649\) 81.8750 3.21388
\(650\) −17.1125 −0.671209
\(651\) 0 0
\(652\) 4.23627 0.165905
\(653\) 7.10115 0.277889 0.138945 0.990300i \(-0.455629\pi\)
0.138945 + 0.990300i \(0.455629\pi\)
\(654\) 0 0
\(655\) −18.4526 −0.721004
\(656\) −5.50256 −0.214839
\(657\) 0 0
\(658\) 8.30049 0.323587
\(659\) −27.0572 −1.05400 −0.527000 0.849865i \(-0.676683\pi\)
−0.527000 + 0.849865i \(0.676683\pi\)
\(660\) 0 0
\(661\) −1.87803 −0.0730471 −0.0365235 0.999333i \(-0.511628\pi\)
−0.0365235 + 0.999333i \(0.511628\pi\)
\(662\) 20.8200 0.809190
\(663\) 0 0
\(664\) −15.1636 −0.588462
\(665\) −2.38854 −0.0926238
\(666\) 0 0
\(667\) −11.4412 −0.443005
\(668\) −24.8132 −0.960053
\(669\) 0 0
\(670\) 1.42450 0.0550333
\(671\) 67.8699 2.62009
\(672\) 0 0
\(673\) 27.8003 1.07162 0.535811 0.844338i \(-0.320006\pi\)
0.535811 + 0.844338i \(0.320006\pi\)
\(674\) 17.8513 0.687606
\(675\) 0 0
\(676\) 20.7863 0.799474
\(677\) 5.33651 0.205099 0.102549 0.994728i \(-0.467300\pi\)
0.102549 + 0.994728i \(0.467300\pi\)
\(678\) 0 0
\(679\) 25.4727 0.977552
\(680\) −4.53354 −0.173853
\(681\) 0 0
\(682\) −12.0440 −0.461189
\(683\) −29.6960 −1.13629 −0.568144 0.822929i \(-0.692338\pi\)
−0.568144 + 0.822929i \(0.692338\pi\)
\(684\) 0 0
\(685\) −7.28628 −0.278394
\(686\) 19.6444 0.750026
\(687\) 0 0
\(688\) −9.08158 −0.346232
\(689\) −55.2203 −2.10372
\(690\) 0 0
\(691\) 0.856810 0.0325946 0.0162973 0.999867i \(-0.494812\pi\)
0.0162973 + 0.999867i \(0.494812\pi\)
\(692\) −3.90530 −0.148457
\(693\) 0 0
\(694\) −16.4617 −0.624877
\(695\) 9.66318 0.366545
\(696\) 0 0
\(697\) −17.3978 −0.658990
\(698\) −15.7220 −0.595088
\(699\) 0 0
\(700\) 5.50838 0.208197
\(701\) −5.75969 −0.217541 −0.108770 0.994067i \(-0.534691\pi\)
−0.108770 + 0.994067i \(0.534691\pi\)
\(702\) 0 0
\(703\) −8.48658 −0.320078
\(704\) −5.50771 −0.207580
\(705\) 0 0
\(706\) −23.2166 −0.873768
\(707\) −18.2814 −0.687543
\(708\) 0 0
\(709\) 18.1581 0.681942 0.340971 0.940074i \(-0.389244\pi\)
0.340971 + 0.940074i \(0.389244\pi\)
\(710\) 6.34662 0.238184
\(711\) 0 0
\(712\) 3.41540 0.127998
\(713\) −3.69014 −0.138197
\(714\) 0 0
\(715\) 45.9038 1.71671
\(716\) −6.73708 −0.251776
\(717\) 0 0
\(718\) 15.3354 0.572312
\(719\) −35.6878 −1.33093 −0.665466 0.746428i \(-0.731767\pi\)
−0.665466 + 0.746428i \(0.731767\pi\)
\(720\) 0 0
\(721\) −13.1692 −0.490446
\(722\) −18.2073 −0.677607
\(723\) 0 0
\(724\) −3.43219 −0.127556
\(725\) −19.9606 −0.741318
\(726\) 0 0
\(727\) 37.3071 1.38365 0.691823 0.722068i \(-0.256808\pi\)
0.691823 + 0.722068i \(0.256808\pi\)
\(728\) −10.8755 −0.403074
\(729\) 0 0
\(730\) −3.32793 −0.123172
\(731\) −28.7138 −1.06202
\(732\) 0 0
\(733\) 36.6158 1.35244 0.676218 0.736701i \(-0.263618\pi\)
0.676218 + 0.736701i \(0.263618\pi\)
\(734\) −1.82214 −0.0672566
\(735\) 0 0
\(736\) −1.68749 −0.0622019
\(737\) 5.47176 0.201555
\(738\) 0 0
\(739\) 13.4619 0.495204 0.247602 0.968862i \(-0.420357\pi\)
0.247602 + 0.968862i \(0.420357\pi\)
\(740\) −13.6676 −0.502432
\(741\) 0 0
\(742\) 17.7749 0.652538
\(743\) 8.49550 0.311670 0.155835 0.987783i \(-0.450193\pi\)
0.155835 + 0.987783i \(0.450193\pi\)
\(744\) 0 0
\(745\) 4.03168 0.147709
\(746\) −24.9826 −0.914679
\(747\) 0 0
\(748\) −17.4141 −0.636723
\(749\) 23.0470 0.842120
\(750\) 0 0
\(751\) −26.1796 −0.955309 −0.477654 0.878548i \(-0.658513\pi\)
−0.477654 + 0.878548i \(0.658513\pi\)
\(752\) −4.43633 −0.161776
\(753\) 0 0
\(754\) 39.4094 1.43521
\(755\) 11.0997 0.403960
\(756\) 0 0
\(757\) −6.61208 −0.240320 −0.120160 0.992755i \(-0.538341\pi\)
−0.120160 + 0.992755i \(0.538341\pi\)
\(758\) −9.99010 −0.362857
\(759\) 0 0
\(760\) 1.27660 0.0463070
\(761\) 32.5736 1.18079 0.590396 0.807114i \(-0.298972\pi\)
0.590396 + 0.807114i \(0.298972\pi\)
\(762\) 0 0
\(763\) −29.0085 −1.05018
\(764\) 19.5410 0.706967
\(765\) 0 0
\(766\) 14.7719 0.533730
\(767\) −86.4074 −3.11999
\(768\) 0 0
\(769\) −11.9619 −0.431356 −0.215678 0.976464i \(-0.569196\pi\)
−0.215678 + 0.976464i \(0.569196\pi\)
\(770\) −14.7760 −0.532492
\(771\) 0 0
\(772\) −5.55650 −0.199983
\(773\) −34.8936 −1.25504 −0.627518 0.778602i \(-0.715929\pi\)
−0.627518 + 0.778602i \(0.715929\pi\)
\(774\) 0 0
\(775\) −6.43790 −0.231256
\(776\) −13.6143 −0.488724
\(777\) 0 0
\(778\) −26.0548 −0.934111
\(779\) 4.89905 0.175527
\(780\) 0 0
\(781\) 24.3785 0.872330
\(782\) −5.33547 −0.190796
\(783\) 0 0
\(784\) −3.49926 −0.124974
\(785\) 27.5591 0.983625
\(786\) 0 0
\(787\) −8.06959 −0.287650 −0.143825 0.989603i \(-0.545940\pi\)
−0.143825 + 0.989603i \(0.545940\pi\)
\(788\) 21.9757 0.782853
\(789\) 0 0
\(790\) 18.0062 0.640633
\(791\) 15.7930 0.561535
\(792\) 0 0
\(793\) −71.6269 −2.54355
\(794\) −25.3046 −0.898026
\(795\) 0 0
\(796\) 4.07733 0.144517
\(797\) −8.25052 −0.292248 −0.146124 0.989266i \(-0.546680\pi\)
−0.146124 + 0.989266i \(0.546680\pi\)
\(798\) 0 0
\(799\) −14.0266 −0.496227
\(800\) −2.94404 −0.104088
\(801\) 0 0
\(802\) −16.9955 −0.600133
\(803\) −12.7832 −0.451108
\(804\) 0 0
\(805\) −4.52720 −0.159563
\(806\) 12.7107 0.447716
\(807\) 0 0
\(808\) 9.77079 0.343735
\(809\) −12.2391 −0.430305 −0.215152 0.976581i \(-0.569025\pi\)
−0.215152 + 0.976581i \(0.569025\pi\)
\(810\) 0 0
\(811\) 52.3012 1.83654 0.918272 0.395950i \(-0.129585\pi\)
0.918272 + 0.395950i \(0.129585\pi\)
\(812\) −12.6856 −0.445176
\(813\) 0 0
\(814\) −52.4998 −1.84012
\(815\) −6.07421 −0.212770
\(816\) 0 0
\(817\) 8.08552 0.282876
\(818\) −30.6720 −1.07242
\(819\) 0 0
\(820\) 7.88991 0.275528
\(821\) 7.67481 0.267853 0.133926 0.990991i \(-0.457241\pi\)
0.133926 + 0.990991i \(0.457241\pi\)
\(822\) 0 0
\(823\) 47.9518 1.67149 0.835747 0.549115i \(-0.185035\pi\)
0.835747 + 0.549115i \(0.185035\pi\)
\(824\) 7.03849 0.245197
\(825\) 0 0
\(826\) 27.8138 0.967766
\(827\) 28.3630 0.986278 0.493139 0.869951i \(-0.335849\pi\)
0.493139 + 0.869951i \(0.335849\pi\)
\(828\) 0 0
\(829\) 14.6558 0.509016 0.254508 0.967071i \(-0.418087\pi\)
0.254508 + 0.967071i \(0.418087\pi\)
\(830\) 21.7425 0.754692
\(831\) 0 0
\(832\) 5.81260 0.201516
\(833\) −11.0638 −0.383340
\(834\) 0 0
\(835\) 35.5787 1.23125
\(836\) 4.90363 0.169596
\(837\) 0 0
\(838\) −17.4958 −0.604381
\(839\) 29.2760 1.01072 0.505361 0.862908i \(-0.331359\pi\)
0.505361 + 0.862908i \(0.331359\pi\)
\(840\) 0 0
\(841\) 16.9683 0.585115
\(842\) −12.6330 −0.435362
\(843\) 0 0
\(844\) −10.0288 −0.345205
\(845\) −29.8047 −1.02531
\(846\) 0 0
\(847\) −36.1761 −1.24303
\(848\) −9.50010 −0.326235
\(849\) 0 0
\(850\) −9.30838 −0.319275
\(851\) −16.0853 −0.551397
\(852\) 0 0
\(853\) −53.1026 −1.81820 −0.909099 0.416581i \(-0.863228\pi\)
−0.909099 + 0.416581i \(0.863228\pi\)
\(854\) 23.0561 0.788964
\(855\) 0 0
\(856\) −12.3178 −0.421015
\(857\) −23.0183 −0.786292 −0.393146 0.919476i \(-0.628613\pi\)
−0.393146 + 0.919476i \(0.628613\pi\)
\(858\) 0 0
\(859\) 49.5835 1.69177 0.845884 0.533368i \(-0.179074\pi\)
0.845884 + 0.533368i \(0.179074\pi\)
\(860\) 13.0217 0.444037
\(861\) 0 0
\(862\) −14.6767 −0.499891
\(863\) −4.09527 −0.139405 −0.0697023 0.997568i \(-0.522205\pi\)
−0.0697023 + 0.997568i \(0.522205\pi\)
\(864\) 0 0
\(865\) 5.59966 0.190394
\(866\) −22.5184 −0.765206
\(867\) 0 0
\(868\) −4.09148 −0.138874
\(869\) 69.1652 2.34627
\(870\) 0 0
\(871\) −5.77466 −0.195667
\(872\) 15.5040 0.525033
\(873\) 0 0
\(874\) 1.50241 0.0508198
\(875\) −21.3122 −0.720484
\(876\) 0 0
\(877\) −6.30454 −0.212889 −0.106445 0.994319i \(-0.533947\pi\)
−0.106445 + 0.994319i \(0.533947\pi\)
\(878\) −8.52689 −0.287769
\(879\) 0 0
\(880\) 7.89729 0.266218
\(881\) 20.8997 0.704128 0.352064 0.935976i \(-0.385480\pi\)
0.352064 + 0.935976i \(0.385480\pi\)
\(882\) 0 0
\(883\) 0.269804 0.00907964 0.00453982 0.999990i \(-0.498555\pi\)
0.00453982 + 0.999990i \(0.498555\pi\)
\(884\) 18.3781 0.618122
\(885\) 0 0
\(886\) 10.6999 0.359471
\(887\) 40.5158 1.36039 0.680193 0.733033i \(-0.261896\pi\)
0.680193 + 0.733033i \(0.261896\pi\)
\(888\) 0 0
\(889\) −23.7697 −0.797212
\(890\) −4.89721 −0.164155
\(891\) 0 0
\(892\) −1.00000 −0.0334825
\(893\) 3.94976 0.132174
\(894\) 0 0
\(895\) 9.66003 0.322899
\(896\) −1.87103 −0.0625066
\(897\) 0 0
\(898\) 42.0615 1.40361
\(899\) 14.8262 0.494481
\(900\) 0 0
\(901\) −30.0371 −1.00068
\(902\) 30.3065 1.00910
\(903\) 0 0
\(904\) −8.44083 −0.280738
\(905\) 4.92128 0.163589
\(906\) 0 0
\(907\) −18.9554 −0.629405 −0.314703 0.949190i \(-0.601905\pi\)
−0.314703 + 0.949190i \(0.601905\pi\)
\(908\) 20.0364 0.664930
\(909\) 0 0
\(910\) 15.5940 0.516936
\(911\) −3.74007 −0.123914 −0.0619571 0.998079i \(-0.519734\pi\)
−0.0619571 + 0.998079i \(0.519734\pi\)
\(912\) 0 0
\(913\) 83.5167 2.76400
\(914\) −8.63916 −0.285758
\(915\) 0 0
\(916\) −18.4886 −0.610881
\(917\) −24.0786 −0.795145
\(918\) 0 0
\(919\) 13.1779 0.434700 0.217350 0.976094i \(-0.430259\pi\)
0.217350 + 0.976094i \(0.430259\pi\)
\(920\) 2.41963 0.0797729
\(921\) 0 0
\(922\) 12.8499 0.423189
\(923\) −25.7280 −0.846847
\(924\) 0 0
\(925\) −28.0628 −0.922699
\(926\) −14.9786 −0.492228
\(927\) 0 0
\(928\) 6.78000 0.222564
\(929\) 46.8151 1.53595 0.767976 0.640478i \(-0.221264\pi\)
0.767976 + 0.640478i \(0.221264\pi\)
\(930\) 0 0
\(931\) 3.11546 0.102105
\(932\) −6.03324 −0.197625
\(933\) 0 0
\(934\) −6.56107 −0.214685
\(935\) 24.9694 0.816587
\(936\) 0 0
\(937\) 25.6778 0.838859 0.419429 0.907788i \(-0.362230\pi\)
0.419429 + 0.907788i \(0.362230\pi\)
\(938\) 1.85881 0.0606924
\(939\) 0 0
\(940\) 6.36108 0.207476
\(941\) −15.3764 −0.501255 −0.250628 0.968084i \(-0.580637\pi\)
−0.250628 + 0.968084i \(0.580637\pi\)
\(942\) 0 0
\(943\) 9.28555 0.302379
\(944\) −14.8655 −0.483832
\(945\) 0 0
\(946\) 50.0187 1.62625
\(947\) −27.2853 −0.886653 −0.443326 0.896360i \(-0.646202\pi\)
−0.443326 + 0.896360i \(0.646202\pi\)
\(948\) 0 0
\(949\) 13.4908 0.437930
\(950\) 2.62114 0.0850411
\(951\) 0 0
\(952\) −5.91575 −0.191731
\(953\) 41.1242 1.33214 0.666072 0.745887i \(-0.267974\pi\)
0.666072 + 0.745887i \(0.267974\pi\)
\(954\) 0 0
\(955\) −28.0190 −0.906674
\(956\) 24.6088 0.795906
\(957\) 0 0
\(958\) 13.4894 0.435823
\(959\) −9.50777 −0.307022
\(960\) 0 0
\(961\) −26.2181 −0.845745
\(962\) 55.4060 1.78636
\(963\) 0 0
\(964\) 12.3387 0.397402
\(965\) 7.96724 0.256475
\(966\) 0 0
\(967\) 43.0261 1.38363 0.691813 0.722077i \(-0.256812\pi\)
0.691813 + 0.722077i \(0.256812\pi\)
\(968\) 19.3349 0.621447
\(969\) 0 0
\(970\) 19.5210 0.626781
\(971\) −45.7690 −1.46880 −0.734398 0.678719i \(-0.762535\pi\)
−0.734398 + 0.678719i \(0.762535\pi\)
\(972\) 0 0
\(973\) 12.6094 0.404238
\(974\) −32.9194 −1.05481
\(975\) 0 0
\(976\) −12.3227 −0.394440
\(977\) −37.1672 −1.18908 −0.594542 0.804064i \(-0.702667\pi\)
−0.594542 + 0.804064i \(0.702667\pi\)
\(978\) 0 0
\(979\) −18.8111 −0.601204
\(980\) 5.01745 0.160277
\(981\) 0 0
\(982\) 28.1030 0.896802
\(983\) 35.8574 1.14367 0.571837 0.820367i \(-0.306231\pi\)
0.571837 + 0.820367i \(0.306231\pi\)
\(984\) 0 0
\(985\) −31.5101 −1.00400
\(986\) 21.4368 0.682686
\(987\) 0 0
\(988\) −5.17508 −0.164641
\(989\) 15.3251 0.487310
\(990\) 0 0
\(991\) −6.53157 −0.207482 −0.103741 0.994604i \(-0.533081\pi\)
−0.103741 + 0.994604i \(0.533081\pi\)
\(992\) 2.18675 0.0694295
\(993\) 0 0
\(994\) 8.28162 0.262677
\(995\) −5.84632 −0.185341
\(996\) 0 0
\(997\) −49.4426 −1.56586 −0.782932 0.622107i \(-0.786277\pi\)
−0.782932 + 0.622107i \(0.786277\pi\)
\(998\) 35.1778 1.11354
\(999\) 0 0
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 4014.2.a.r.1.3 5
3.2 odd 2 1338.2.a.h.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1338.2.a.h.1.3 5 3.2 odd 2
4014.2.a.r.1.3 5 1.1 even 1 trivial