Properties

Label 4014.2.a.r.1.4
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 1.4
Root \(0.253142\)
Character \(\chi\) = 4014.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.686428 q^{5} +2.87549 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.686428 q^{5} +2.87549 q^{7} +1.00000 q^{8} -0.686428 q^{10} -5.51891 q^{11} +0.329853 q^{13} +2.87549 q^{14} +1.00000 q^{16} -4.44220 q^{17} -1.53786 q^{19} -0.686428 q^{20} -5.51891 q^{22} +3.17011 q^{23} -4.52882 q^{25} +0.329853 q^{26} +2.87549 q^{28} -7.03297 q^{29} -7.43855 q^{31} +1.00000 q^{32} -4.44220 q^{34} -1.97382 q^{35} -7.58613 q^{37} -1.53786 q^{38} -0.686428 q^{40} +2.51679 q^{41} +6.97369 q^{43} -5.51891 q^{44} +3.17011 q^{46} -6.00990 q^{47} +1.26844 q^{49} -4.52882 q^{50} +0.329853 q^{52} +0.840259 q^{53} +3.78834 q^{55} +2.87549 q^{56} -7.03297 q^{58} +3.14631 q^{59} -2.68741 q^{61} -7.43855 q^{62} +1.00000 q^{64} -0.226420 q^{65} -13.9815 q^{67} -4.44220 q^{68} -1.97382 q^{70} +6.67275 q^{71} +11.9575 q^{73} -7.58613 q^{74} -1.53786 q^{76} -15.8696 q^{77} +12.1543 q^{79} -0.686428 q^{80} +2.51679 q^{82} +5.22743 q^{83} +3.04925 q^{85} +6.97369 q^{86} -5.51891 q^{88} -8.29874 q^{89} +0.948489 q^{91} +3.17011 q^{92} -6.00990 q^{94} +1.05563 q^{95} +4.45112 q^{97} +1.26844 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} - 5q^{10} - 9q^{11} - q^{14} + 5q^{16} - 6q^{17} - 4q^{19} - 5q^{20} - 9q^{22} - 16q^{23} + 8q^{25} - q^{28} - 8q^{29} - q^{31} + 5q^{32} - 6q^{34} - 22q^{35} - 2q^{37} - 4q^{38} - 5q^{40} - 4q^{41} + 3q^{43} - 9q^{44} - 16q^{46} - 18q^{47} + 2q^{49} + 8q^{50} - 26q^{53} + q^{55} - q^{56} - 8q^{58} - 21q^{59} - 20q^{61} - q^{62} + 5q^{64} + 3q^{65} - 5q^{67} - 6q^{68} - 22q^{70} - 17q^{71} + 5q^{73} - 2q^{74} - 4q^{76} - 2q^{77} - 21q^{79} - 5q^{80} - 4q^{82} - 11q^{83} - 12q^{85} + 3q^{86} - 9q^{88} + 5q^{89} - 10q^{91} - 16q^{92} - 18q^{94} - 10q^{95} - 11q^{97} + 2q^{98} + O(q^{100}) \)

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
\(5\) −0.686428 −0.306980 −0.153490 0.988150i \(-0.549051\pi\)
−0.153490 + 0.988150i \(0.549051\pi\)
\(6\) 0 0
\(7\) 2.87549 1.08683 0.543416 0.839463i \(-0.317130\pi\)
0.543416 + 0.839463i \(0.317130\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −0.686428 −0.217068
\(11\) −5.51891 −1.66402 −0.832008 0.554764i \(-0.812808\pi\)
−0.832008 + 0.554764i \(0.812808\pi\)
\(12\) 0 0
\(13\) 0.329853 0.0914848 0.0457424 0.998953i \(-0.485435\pi\)
0.0457424 + 0.998953i \(0.485435\pi\)
\(14\) 2.87549 0.768507
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.44220 −1.07739 −0.538696 0.842500i \(-0.681083\pi\)
−0.538696 + 0.842500i \(0.681083\pi\)
\(18\) 0 0
\(19\) −1.53786 −0.352810 −0.176405 0.984318i \(-0.556447\pi\)
−0.176405 + 0.984318i \(0.556447\pi\)
\(20\) −0.686428 −0.153490
\(21\) 0 0
\(22\) −5.51891 −1.17664
\(23\) 3.17011 0.661014 0.330507 0.943804i \(-0.392780\pi\)
0.330507 + 0.943804i \(0.392780\pi\)
\(24\) 0 0
\(25\) −4.52882 −0.905763
\(26\) 0.329853 0.0646895
\(27\) 0 0
\(28\) 2.87549 0.543416
\(29\) −7.03297 −1.30599 −0.652995 0.757362i \(-0.726488\pi\)
−0.652995 + 0.757362i \(0.726488\pi\)
\(30\) 0 0
\(31\) −7.43855 −1.33600 −0.668002 0.744160i \(-0.732850\pi\)
−0.668002 + 0.744160i \(0.732850\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.44220 −0.761832
\(35\) −1.97382 −0.333636
\(36\) 0 0
\(37\) −7.58613 −1.24715 −0.623576 0.781762i \(-0.714321\pi\)
−0.623576 + 0.781762i \(0.714321\pi\)
\(38\) −1.53786 −0.249474
\(39\) 0 0
\(40\) −0.686428 −0.108534
\(41\) 2.51679 0.393056 0.196528 0.980498i \(-0.437033\pi\)
0.196528 + 0.980498i \(0.437033\pi\)
\(42\) 0 0
\(43\) 6.97369 1.06348 0.531739 0.846908i \(-0.321539\pi\)
0.531739 + 0.846908i \(0.321539\pi\)
\(44\) −5.51891 −0.832008
\(45\) 0 0
\(46\) 3.17011 0.467408
\(47\) −6.00990 −0.876634 −0.438317 0.898820i \(-0.644425\pi\)
−0.438317 + 0.898820i \(0.644425\pi\)
\(48\) 0 0
\(49\) 1.26844 0.181206
\(50\) −4.52882 −0.640471
\(51\) 0 0
\(52\) 0.329853 0.0457424
\(53\) 0.840259 0.115419 0.0577093 0.998333i \(-0.481620\pi\)
0.0577093 + 0.998333i \(0.481620\pi\)
\(54\) 0 0
\(55\) 3.78834 0.510819
\(56\) 2.87549 0.384253
\(57\) 0 0
\(58\) −7.03297 −0.923475
\(59\) 3.14631 0.409614 0.204807 0.978802i \(-0.434343\pi\)
0.204807 + 0.978802i \(0.434343\pi\)
\(60\) 0 0
\(61\) −2.68741 −0.344088 −0.172044 0.985089i \(-0.555037\pi\)
−0.172044 + 0.985089i \(0.555037\pi\)
\(62\) −7.43855 −0.944697
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.226420 −0.0280840
\(66\) 0 0
\(67\) −13.9815 −1.70811 −0.854057 0.520179i \(-0.825865\pi\)
−0.854057 + 0.520179i \(0.825865\pi\)
\(68\) −4.44220 −0.538696
\(69\) 0 0
\(70\) −1.97382 −0.235916
\(71\) 6.67275 0.791909 0.395955 0.918270i \(-0.370414\pi\)
0.395955 + 0.918270i \(0.370414\pi\)
\(72\) 0 0
\(73\) 11.9575 1.39952 0.699758 0.714380i \(-0.253291\pi\)
0.699758 + 0.714380i \(0.253291\pi\)
\(74\) −7.58613 −0.881870
\(75\) 0 0
\(76\) −1.53786 −0.176405
\(77\) −15.8696 −1.80851
\(78\) 0 0
\(79\) 12.1543 1.36747 0.683733 0.729732i \(-0.260355\pi\)
0.683733 + 0.729732i \(0.260355\pi\)
\(80\) −0.686428 −0.0767450
\(81\) 0 0
\(82\) 2.51679 0.277932
\(83\) 5.22743 0.573785 0.286892 0.957963i \(-0.407378\pi\)
0.286892 + 0.957963i \(0.407378\pi\)
\(84\) 0 0
\(85\) 3.04925 0.330738
\(86\) 6.97369 0.751992
\(87\) 0 0
\(88\) −5.51891 −0.588318
\(89\) −8.29874 −0.879665 −0.439833 0.898080i \(-0.644962\pi\)
−0.439833 + 0.898080i \(0.644962\pi\)
\(90\) 0 0
\(91\) 0.948489 0.0994286
\(92\) 3.17011 0.330507
\(93\) 0 0
\(94\) −6.00990 −0.619874
\(95\) 1.05563 0.108306
\(96\) 0 0
\(97\) 4.45112 0.451943 0.225971 0.974134i \(-0.427444\pi\)
0.225971 + 0.974134i \(0.427444\pi\)
\(98\) 1.26844 0.128132
\(99\) 0 0
\(100\) −4.52882 −0.452882
\(101\) 3.23632 0.322026 0.161013 0.986952i \(-0.448524\pi\)
0.161013 + 0.986952i \(0.448524\pi\)
\(102\) 0 0
\(103\) −12.7274 −1.25407 −0.627034 0.778992i \(-0.715731\pi\)
−0.627034 + 0.778992i \(0.715731\pi\)
\(104\) 0.329853 0.0323447
\(105\) 0 0
\(106\) 0.840259 0.0816132
\(107\) −4.52529 −0.437477 −0.218738 0.975784i \(-0.570194\pi\)
−0.218738 + 0.975784i \(0.570194\pi\)
\(108\) 0 0
\(109\) −10.5371 −1.00927 −0.504637 0.863332i \(-0.668374\pi\)
−0.504637 + 0.863332i \(0.668374\pi\)
\(110\) 3.78834 0.361204
\(111\) 0 0
\(112\) 2.87549 0.271708
\(113\) −2.93690 −0.276281 −0.138140 0.990413i \(-0.544112\pi\)
−0.138140 + 0.990413i \(0.544112\pi\)
\(114\) 0 0
\(115\) −2.17605 −0.202918
\(116\) −7.03297 −0.652995
\(117\) 0 0
\(118\) 3.14631 0.289641
\(119\) −12.7735 −1.17095
\(120\) 0 0
\(121\) 19.4584 1.76895
\(122\) −2.68741 −0.243307
\(123\) 0 0
\(124\) −7.43855 −0.668002
\(125\) 6.54085 0.585031
\(126\) 0 0
\(127\) 4.04306 0.358764 0.179382 0.983780i \(-0.442590\pi\)
0.179382 + 0.983780i \(0.442590\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −0.226420 −0.0198584
\(131\) −6.64749 −0.580794 −0.290397 0.956906i \(-0.593787\pi\)
−0.290397 + 0.956906i \(0.593787\pi\)
\(132\) 0 0
\(133\) −4.42211 −0.383446
\(134\) −13.9815 −1.20782
\(135\) 0 0
\(136\) −4.44220 −0.380916
\(137\) −10.9737 −0.937545 −0.468773 0.883319i \(-0.655304\pi\)
−0.468773 + 0.883319i \(0.655304\pi\)
\(138\) 0 0
\(139\) −15.2142 −1.29045 −0.645225 0.763992i \(-0.723237\pi\)
−0.645225 + 0.763992i \(0.723237\pi\)
\(140\) −1.97382 −0.166818
\(141\) 0 0
\(142\) 6.67275 0.559964
\(143\) −1.82043 −0.152232
\(144\) 0 0
\(145\) 4.82763 0.400913
\(146\) 11.9575 0.989607
\(147\) 0 0
\(148\) −7.58613 −0.623576
\(149\) −2.90167 −0.237714 −0.118857 0.992911i \(-0.537923\pi\)
−0.118857 + 0.992911i \(0.537923\pi\)
\(150\) 0 0
\(151\) 2.66399 0.216792 0.108396 0.994108i \(-0.465429\pi\)
0.108396 + 0.994108i \(0.465429\pi\)
\(152\) −1.53786 −0.124737
\(153\) 0 0
\(154\) −15.8696 −1.27881
\(155\) 5.10603 0.410126
\(156\) 0 0
\(157\) 15.2048 1.21347 0.606736 0.794903i \(-0.292478\pi\)
0.606736 + 0.794903i \(0.292478\pi\)
\(158\) 12.1543 0.966944
\(159\) 0 0
\(160\) −0.686428 −0.0542669
\(161\) 9.11562 0.718412
\(162\) 0 0
\(163\) 12.4990 0.978996 0.489498 0.872004i \(-0.337180\pi\)
0.489498 + 0.872004i \(0.337180\pi\)
\(164\) 2.51679 0.196528
\(165\) 0 0
\(166\) 5.22743 0.405727
\(167\) −6.88133 −0.532493 −0.266247 0.963905i \(-0.585784\pi\)
−0.266247 + 0.963905i \(0.585784\pi\)
\(168\) 0 0
\(169\) −12.8912 −0.991631
\(170\) 3.04925 0.233867
\(171\) 0 0
\(172\) 6.97369 0.531739
\(173\) 3.65630 0.277984 0.138992 0.990294i \(-0.455614\pi\)
0.138992 + 0.990294i \(0.455614\pi\)
\(174\) 0 0
\(175\) −13.0226 −0.984413
\(176\) −5.51891 −0.416004
\(177\) 0 0
\(178\) −8.29874 −0.622017
\(179\) −11.6410 −0.870090 −0.435045 0.900409i \(-0.643267\pi\)
−0.435045 + 0.900409i \(0.643267\pi\)
\(180\) 0 0
\(181\) −13.8301 −1.02798 −0.513990 0.857796i \(-0.671833\pi\)
−0.513990 + 0.857796i \(0.671833\pi\)
\(182\) 0.948489 0.0703067
\(183\) 0 0
\(184\) 3.17011 0.233704
\(185\) 5.20733 0.382851
\(186\) 0 0
\(187\) 24.5161 1.79280
\(188\) −6.00990 −0.438317
\(189\) 0 0
\(190\) 1.05563 0.0765836
\(191\) 15.4332 1.11671 0.558354 0.829603i \(-0.311433\pi\)
0.558354 + 0.829603i \(0.311433\pi\)
\(192\) 0 0
\(193\) −1.86292 −0.134096 −0.0670479 0.997750i \(-0.521358\pi\)
−0.0670479 + 0.997750i \(0.521358\pi\)
\(194\) 4.45112 0.319572
\(195\) 0 0
\(196\) 1.26844 0.0906028
\(197\) 18.0327 1.28477 0.642387 0.766381i \(-0.277944\pi\)
0.642387 + 0.766381i \(0.277944\pi\)
\(198\) 0 0
\(199\) 24.0118 1.70215 0.851077 0.525042i \(-0.175950\pi\)
0.851077 + 0.525042i \(0.175950\pi\)
\(200\) −4.52882 −0.320236
\(201\) 0 0
\(202\) 3.23632 0.227707
\(203\) −20.2232 −1.41939
\(204\) 0 0
\(205\) −1.72759 −0.120660
\(206\) −12.7274 −0.886760
\(207\) 0 0
\(208\) 0.329853 0.0228712
\(209\) 8.48734 0.587081
\(210\) 0 0
\(211\) 6.05163 0.416611 0.208306 0.978064i \(-0.433205\pi\)
0.208306 + 0.978064i \(0.433205\pi\)
\(212\) 0.840259 0.0577093
\(213\) 0 0
\(214\) −4.52529 −0.309343
\(215\) −4.78694 −0.326466
\(216\) 0 0
\(217\) −21.3895 −1.45201
\(218\) −10.5371 −0.713664
\(219\) 0 0
\(220\) 3.78834 0.255410
\(221\) −1.46527 −0.0985650
\(222\) 0 0
\(223\) −1.00000 −0.0669650
\(224\) 2.87549 0.192127
\(225\) 0 0
\(226\) −2.93690 −0.195360
\(227\) −24.3730 −1.61769 −0.808846 0.588021i \(-0.799907\pi\)
−0.808846 + 0.588021i \(0.799907\pi\)
\(228\) 0 0
\(229\) 7.57496 0.500567 0.250284 0.968173i \(-0.419476\pi\)
0.250284 + 0.968173i \(0.419476\pi\)
\(230\) −2.17605 −0.143485
\(231\) 0 0
\(232\) −7.03297 −0.461737
\(233\) −1.91430 −0.125410 −0.0627050 0.998032i \(-0.519973\pi\)
−0.0627050 + 0.998032i \(0.519973\pi\)
\(234\) 0 0
\(235\) 4.12537 0.269109
\(236\) 3.14631 0.204807
\(237\) 0 0
\(238\) −12.7735 −0.827984
\(239\) 18.8018 1.21619 0.608094 0.793865i \(-0.291934\pi\)
0.608094 + 0.793865i \(0.291934\pi\)
\(240\) 0 0
\(241\) −12.7202 −0.819378 −0.409689 0.912225i \(-0.634363\pi\)
−0.409689 + 0.912225i \(0.634363\pi\)
\(242\) 19.4584 1.25083
\(243\) 0 0
\(244\) −2.68741 −0.172044
\(245\) −0.870693 −0.0556265
\(246\) 0 0
\(247\) −0.507269 −0.0322767
\(248\) −7.43855 −0.472349
\(249\) 0 0
\(250\) 6.54085 0.413680
\(251\) −8.40005 −0.530206 −0.265103 0.964220i \(-0.585406\pi\)
−0.265103 + 0.964220i \(0.585406\pi\)
\(252\) 0 0
\(253\) −17.4956 −1.09994
\(254\) 4.04306 0.253684
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.1611 1.44475 0.722375 0.691502i \(-0.243051\pi\)
0.722375 + 0.691502i \(0.243051\pi\)
\(258\) 0 0
\(259\) −21.8138 −1.35545
\(260\) −0.226420 −0.0140420
\(261\) 0 0
\(262\) −6.64749 −0.410683
\(263\) −6.61626 −0.407976 −0.203988 0.978973i \(-0.565390\pi\)
−0.203988 + 0.978973i \(0.565390\pi\)
\(264\) 0 0
\(265\) −0.576778 −0.0354312
\(266\) −4.42211 −0.271137
\(267\) 0 0
\(268\) −13.9815 −0.854057
\(269\) 4.44005 0.270715 0.135357 0.990797i \(-0.456782\pi\)
0.135357 + 0.990797i \(0.456782\pi\)
\(270\) 0 0
\(271\) −6.05643 −0.367902 −0.183951 0.982935i \(-0.558889\pi\)
−0.183951 + 0.982935i \(0.558889\pi\)
\(272\) −4.44220 −0.269348
\(273\) 0 0
\(274\) −10.9737 −0.662945
\(275\) 24.9942 1.50720
\(276\) 0 0
\(277\) −25.7454 −1.54689 −0.773446 0.633862i \(-0.781469\pi\)
−0.773446 + 0.633862i \(0.781469\pi\)
\(278\) −15.2142 −0.912487
\(279\) 0 0
\(280\) −1.97382 −0.117958
\(281\) 14.7983 0.882790 0.441395 0.897313i \(-0.354484\pi\)
0.441395 + 0.897313i \(0.354484\pi\)
\(282\) 0 0
\(283\) −4.17776 −0.248342 −0.124171 0.992261i \(-0.539627\pi\)
−0.124171 + 0.992261i \(0.539627\pi\)
\(284\) 6.67275 0.395955
\(285\) 0 0
\(286\) −1.82043 −0.107644
\(287\) 7.23699 0.427186
\(288\) 0 0
\(289\) 2.73318 0.160775
\(290\) 4.82763 0.283488
\(291\) 0 0
\(292\) 11.9575 0.699758
\(293\) 12.6202 0.737281 0.368641 0.929572i \(-0.379823\pi\)
0.368641 + 0.929572i \(0.379823\pi\)
\(294\) 0 0
\(295\) −2.15971 −0.125743
\(296\) −7.58613 −0.440935
\(297\) 0 0
\(298\) −2.90167 −0.168089
\(299\) 1.04567 0.0604727
\(300\) 0 0
\(301\) 20.0528 1.15582
\(302\) 2.66399 0.153295
\(303\) 0 0
\(304\) −1.53786 −0.0882025
\(305\) 1.84472 0.105628
\(306\) 0 0
\(307\) −32.7514 −1.86922 −0.934612 0.355669i \(-0.884253\pi\)
−0.934612 + 0.355669i \(0.884253\pi\)
\(308\) −15.8696 −0.904253
\(309\) 0 0
\(310\) 5.10603 0.290003
\(311\) 5.09711 0.289031 0.144515 0.989503i \(-0.453838\pi\)
0.144515 + 0.989503i \(0.453838\pi\)
\(312\) 0 0
\(313\) 3.70513 0.209426 0.104713 0.994502i \(-0.466608\pi\)
0.104713 + 0.994502i \(0.466608\pi\)
\(314\) 15.2048 0.858055
\(315\) 0 0
\(316\) 12.1543 0.683733
\(317\) −11.1944 −0.628743 −0.314371 0.949300i \(-0.601794\pi\)
−0.314371 + 0.949300i \(0.601794\pi\)
\(318\) 0 0
\(319\) 38.8144 2.17319
\(320\) −0.686428 −0.0383725
\(321\) 0 0
\(322\) 9.11562 0.507994
\(323\) 6.83150 0.380115
\(324\) 0 0
\(325\) −1.49384 −0.0828635
\(326\) 12.4990 0.692255
\(327\) 0 0
\(328\) 2.51679 0.138966
\(329\) −17.2814 −0.952755
\(330\) 0 0
\(331\) −30.7807 −1.69186 −0.845930 0.533293i \(-0.820954\pi\)
−0.845930 + 0.533293i \(0.820954\pi\)
\(332\) 5.22743 0.286892
\(333\) 0 0
\(334\) −6.88133 −0.376530
\(335\) 9.59731 0.524357
\(336\) 0 0
\(337\) −28.0781 −1.52951 −0.764755 0.644322i \(-0.777140\pi\)
−0.764755 + 0.644322i \(0.777140\pi\)
\(338\) −12.8912 −0.701189
\(339\) 0 0
\(340\) 3.04925 0.165369
\(341\) 41.0527 2.22313
\(342\) 0 0
\(343\) −16.4810 −0.889893
\(344\) 6.97369 0.375996
\(345\) 0 0
\(346\) 3.65630 0.196564
\(347\) −4.81474 −0.258469 −0.129234 0.991614i \(-0.541252\pi\)
−0.129234 + 0.991614i \(0.541252\pi\)
\(348\) 0 0
\(349\) 20.7579 1.11114 0.555571 0.831469i \(-0.312500\pi\)
0.555571 + 0.831469i \(0.312500\pi\)
\(350\) −13.0226 −0.696085
\(351\) 0 0
\(352\) −5.51891 −0.294159
\(353\) 3.21876 0.171317 0.0856587 0.996325i \(-0.472701\pi\)
0.0856587 + 0.996325i \(0.472701\pi\)
\(354\) 0 0
\(355\) −4.58036 −0.243100
\(356\) −8.29874 −0.439833
\(357\) 0 0
\(358\) −11.6410 −0.615246
\(359\) −25.5107 −1.34641 −0.673203 0.739458i \(-0.735082\pi\)
−0.673203 + 0.739458i \(0.735082\pi\)
\(360\) 0 0
\(361\) −16.6350 −0.875525
\(362\) −13.8301 −0.726892
\(363\) 0 0
\(364\) 0.948489 0.0497143
\(365\) −8.20794 −0.429623
\(366\) 0 0
\(367\) −24.4579 −1.27669 −0.638346 0.769750i \(-0.720381\pi\)
−0.638346 + 0.769750i \(0.720381\pi\)
\(368\) 3.17011 0.165254
\(369\) 0 0
\(370\) 5.20733 0.270716
\(371\) 2.41616 0.125441
\(372\) 0 0
\(373\) −9.79830 −0.507337 −0.253668 0.967291i \(-0.581637\pi\)
−0.253668 + 0.967291i \(0.581637\pi\)
\(374\) 24.5161 1.26770
\(375\) 0 0
\(376\) −6.00990 −0.309937
\(377\) −2.31985 −0.119478
\(378\) 0 0
\(379\) 23.3632 1.20009 0.600043 0.799968i \(-0.295150\pi\)
0.600043 + 0.799968i \(0.295150\pi\)
\(380\) 1.05563 0.0541528
\(381\) 0 0
\(382\) 15.4332 0.789632
\(383\) −26.1424 −1.33582 −0.667908 0.744244i \(-0.732810\pi\)
−0.667908 + 0.744244i \(0.732810\pi\)
\(384\) 0 0
\(385\) 10.8933 0.555175
\(386\) −1.86292 −0.0948201
\(387\) 0 0
\(388\) 4.45112 0.225971
\(389\) −3.73549 −0.189397 −0.0946985 0.995506i \(-0.530189\pi\)
−0.0946985 + 0.995506i \(0.530189\pi\)
\(390\) 0 0
\(391\) −14.0823 −0.712172
\(392\) 1.26844 0.0640659
\(393\) 0 0
\(394\) 18.0327 0.908472
\(395\) −8.34305 −0.419785
\(396\) 0 0
\(397\) 17.6199 0.884316 0.442158 0.896937i \(-0.354213\pi\)
0.442158 + 0.896937i \(0.354213\pi\)
\(398\) 24.0118 1.20360
\(399\) 0 0
\(400\) −4.52882 −0.226441
\(401\) 22.5232 1.12475 0.562376 0.826881i \(-0.309887\pi\)
0.562376 + 0.826881i \(0.309887\pi\)
\(402\) 0 0
\(403\) −2.45363 −0.122224
\(404\) 3.23632 0.161013
\(405\) 0 0
\(406\) −20.2232 −1.00366
\(407\) 41.8672 2.07528
\(408\) 0 0
\(409\) −12.4542 −0.615821 −0.307910 0.951415i \(-0.599630\pi\)
−0.307910 + 0.951415i \(0.599630\pi\)
\(410\) −1.72759 −0.0853197
\(411\) 0 0
\(412\) −12.7274 −0.627034
\(413\) 9.04717 0.445182
\(414\) 0 0
\(415\) −3.58825 −0.176140
\(416\) 0.329853 0.0161724
\(417\) 0 0
\(418\) 8.48734 0.415129
\(419\) 30.5170 1.49086 0.745428 0.666586i \(-0.232245\pi\)
0.745428 + 0.666586i \(0.232245\pi\)
\(420\) 0 0
\(421\) 5.17369 0.252150 0.126075 0.992021i \(-0.459762\pi\)
0.126075 + 0.992021i \(0.459762\pi\)
\(422\) 6.05163 0.294589
\(423\) 0 0
\(424\) 0.840259 0.0408066
\(425\) 20.1179 0.975863
\(426\) 0 0
\(427\) −7.72762 −0.373966
\(428\) −4.52529 −0.218738
\(429\) 0 0
\(430\) −4.78694 −0.230847
\(431\) 11.8685 0.571686 0.285843 0.958277i \(-0.407726\pi\)
0.285843 + 0.958277i \(0.407726\pi\)
\(432\) 0 0
\(433\) 2.77072 0.133152 0.0665762 0.997781i \(-0.478792\pi\)
0.0665762 + 0.997781i \(0.478792\pi\)
\(434\) −21.3895 −1.02673
\(435\) 0 0
\(436\) −10.5371 −0.504637
\(437\) −4.87520 −0.233212
\(438\) 0 0
\(439\) −13.5404 −0.646248 −0.323124 0.946357i \(-0.604733\pi\)
−0.323124 + 0.946357i \(0.604733\pi\)
\(440\) 3.78834 0.180602
\(441\) 0 0
\(442\) −1.46527 −0.0696960
\(443\) −38.7298 −1.84011 −0.920054 0.391791i \(-0.871856\pi\)
−0.920054 + 0.391791i \(0.871856\pi\)
\(444\) 0 0
\(445\) 5.69649 0.270040
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) 2.87549 0.135854
\(449\) 35.2746 1.66471 0.832356 0.554241i \(-0.186992\pi\)
0.832356 + 0.554241i \(0.186992\pi\)
\(450\) 0 0
\(451\) −13.8899 −0.654051
\(452\) −2.93690 −0.138140
\(453\) 0 0
\(454\) −24.3730 −1.14388
\(455\) −0.651069 −0.0305226
\(456\) 0 0
\(457\) −15.7179 −0.735251 −0.367626 0.929974i \(-0.619829\pi\)
−0.367626 + 0.929974i \(0.619829\pi\)
\(458\) 7.57496 0.353955
\(459\) 0 0
\(460\) −2.17605 −0.101459
\(461\) 15.4964 0.721739 0.360869 0.932616i \(-0.382480\pi\)
0.360869 + 0.932616i \(0.382480\pi\)
\(462\) 0 0
\(463\) 34.0979 1.58466 0.792331 0.610091i \(-0.208867\pi\)
0.792331 + 0.610091i \(0.208867\pi\)
\(464\) −7.03297 −0.326498
\(465\) 0 0
\(466\) −1.91430 −0.0886783
\(467\) −5.76270 −0.266666 −0.133333 0.991071i \(-0.542568\pi\)
−0.133333 + 0.991071i \(0.542568\pi\)
\(468\) 0 0
\(469\) −40.2037 −1.85644
\(470\) 4.12537 0.190289
\(471\) 0 0
\(472\) 3.14631 0.144821
\(473\) −38.4872 −1.76964
\(474\) 0 0
\(475\) 6.96470 0.319562
\(476\) −12.7735 −0.585473
\(477\) 0 0
\(478\) 18.8018 0.859975
\(479\) 0.892156 0.0407637 0.0203818 0.999792i \(-0.493512\pi\)
0.0203818 + 0.999792i \(0.493512\pi\)
\(480\) 0 0
\(481\) −2.50231 −0.114095
\(482\) −12.7202 −0.579387
\(483\) 0 0
\(484\) 19.4584 0.884473
\(485\) −3.05538 −0.138737
\(486\) 0 0
\(487\) 4.83587 0.219134 0.109567 0.993979i \(-0.465054\pi\)
0.109567 + 0.993979i \(0.465054\pi\)
\(488\) −2.68741 −0.121653
\(489\) 0 0
\(490\) −0.870693 −0.0393339
\(491\) 14.5280 0.655640 0.327820 0.944740i \(-0.393686\pi\)
0.327820 + 0.944740i \(0.393686\pi\)
\(492\) 0 0
\(493\) 31.2419 1.40706
\(494\) −0.507269 −0.0228231
\(495\) 0 0
\(496\) −7.43855 −0.334001
\(497\) 19.1874 0.860673
\(498\) 0 0
\(499\) 42.7043 1.91171 0.955854 0.293842i \(-0.0949339\pi\)
0.955854 + 0.293842i \(0.0949339\pi\)
\(500\) 6.54085 0.292516
\(501\) 0 0
\(502\) −8.40005 −0.374912
\(503\) 28.7298 1.28100 0.640499 0.767959i \(-0.278727\pi\)
0.640499 + 0.767959i \(0.278727\pi\)
\(504\) 0 0
\(505\) −2.22150 −0.0988556
\(506\) −17.4956 −0.777773
\(507\) 0 0
\(508\) 4.04306 0.179382
\(509\) 31.9400 1.41572 0.707858 0.706355i \(-0.249662\pi\)
0.707858 + 0.706355i \(0.249662\pi\)
\(510\) 0 0
\(511\) 34.3836 1.52104
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 23.1611 1.02159
\(515\) 8.73644 0.384974
\(516\) 0 0
\(517\) 33.1681 1.45873
\(518\) −21.8138 −0.958445
\(519\) 0 0
\(520\) −0.226420 −0.00992919
\(521\) −34.5500 −1.51366 −0.756832 0.653609i \(-0.773254\pi\)
−0.756832 + 0.653609i \(0.773254\pi\)
\(522\) 0 0
\(523\) 20.3878 0.891498 0.445749 0.895158i \(-0.352937\pi\)
0.445749 + 0.895158i \(0.352937\pi\)
\(524\) −6.64749 −0.290397
\(525\) 0 0
\(526\) −6.61626 −0.288483
\(527\) 33.0436 1.43940
\(528\) 0 0
\(529\) −12.9504 −0.563060
\(530\) −0.576778 −0.0250536
\(531\) 0 0
\(532\) −4.42211 −0.191723
\(533\) 0.830169 0.0359586
\(534\) 0 0
\(535\) 3.10629 0.134297
\(536\) −13.9815 −0.603910
\(537\) 0 0
\(538\) 4.44005 0.191424
\(539\) −7.00041 −0.301529
\(540\) 0 0
\(541\) −26.5035 −1.13948 −0.569738 0.821827i \(-0.692955\pi\)
−0.569738 + 0.821827i \(0.692955\pi\)
\(542\) −6.05643 −0.260146
\(543\) 0 0
\(544\) −4.44220 −0.190458
\(545\) 7.23298 0.309827
\(546\) 0 0
\(547\) −0.362415 −0.0154958 −0.00774788 0.999970i \(-0.502466\pi\)
−0.00774788 + 0.999970i \(0.502466\pi\)
\(548\) −10.9737 −0.468773
\(549\) 0 0
\(550\) 24.9942 1.06575
\(551\) 10.8158 0.460766
\(552\) 0 0
\(553\) 34.9496 1.48621
\(554\) −25.7454 −1.09382
\(555\) 0 0
\(556\) −15.2142 −0.645225
\(557\) 17.2544 0.731092 0.365546 0.930793i \(-0.380882\pi\)
0.365546 + 0.930793i \(0.380882\pi\)
\(558\) 0 0
\(559\) 2.30029 0.0972920
\(560\) −1.97382 −0.0834090
\(561\) 0 0
\(562\) 14.7983 0.624227
\(563\) −38.1428 −1.60753 −0.803764 0.594948i \(-0.797173\pi\)
−0.803764 + 0.594948i \(0.797173\pi\)
\(564\) 0 0
\(565\) 2.01597 0.0848126
\(566\) −4.17776 −0.175604
\(567\) 0 0
\(568\) 6.67275 0.279982
\(569\) 9.79988 0.410832 0.205416 0.978675i \(-0.434145\pi\)
0.205416 + 0.978675i \(0.434145\pi\)
\(570\) 0 0
\(571\) 28.2196 1.18095 0.590476 0.807055i \(-0.298940\pi\)
0.590476 + 0.807055i \(0.298940\pi\)
\(572\) −1.82043 −0.0761160
\(573\) 0 0
\(574\) 7.23699 0.302066
\(575\) −14.3569 −0.598722
\(576\) 0 0
\(577\) 28.2068 1.17427 0.587133 0.809491i \(-0.300257\pi\)
0.587133 + 0.809491i \(0.300257\pi\)
\(578\) 2.73318 0.113685
\(579\) 0 0
\(580\) 4.82763 0.200456
\(581\) 15.0314 0.623608
\(582\) 0 0
\(583\) −4.63732 −0.192058
\(584\) 11.9575 0.494803
\(585\) 0 0
\(586\) 12.6202 0.521337
\(587\) 12.2413 0.505251 0.252626 0.967564i \(-0.418706\pi\)
0.252626 + 0.967564i \(0.418706\pi\)
\(588\) 0 0
\(589\) 11.4395 0.471355
\(590\) −2.15971 −0.0889140
\(591\) 0 0
\(592\) −7.58613 −0.311788
\(593\) −28.6547 −1.17671 −0.588353 0.808604i \(-0.700223\pi\)
−0.588353 + 0.808604i \(0.700223\pi\)
\(594\) 0 0
\(595\) 8.76810 0.359457
\(596\) −2.90167 −0.118857
\(597\) 0 0
\(598\) 1.04567 0.0427607
\(599\) 6.02231 0.246065 0.123032 0.992403i \(-0.460738\pi\)
0.123032 + 0.992403i \(0.460738\pi\)
\(600\) 0 0
\(601\) 33.1083 1.35052 0.675258 0.737582i \(-0.264032\pi\)
0.675258 + 0.737582i \(0.264032\pi\)
\(602\) 20.0528 0.817290
\(603\) 0 0
\(604\) 2.66399 0.108396
\(605\) −13.3568 −0.543031
\(606\) 0 0
\(607\) 35.7642 1.45163 0.725813 0.687892i \(-0.241464\pi\)
0.725813 + 0.687892i \(0.241464\pi\)
\(608\) −1.53786 −0.0623686
\(609\) 0 0
\(610\) 1.84472 0.0746903
\(611\) −1.98238 −0.0801987
\(612\) 0 0
\(613\) −23.7521 −0.959339 −0.479669 0.877449i \(-0.659243\pi\)
−0.479669 + 0.877449i \(0.659243\pi\)
\(614\) −32.7514 −1.32174
\(615\) 0 0
\(616\) −15.8696 −0.639404
\(617\) 37.5379 1.51122 0.755609 0.655023i \(-0.227341\pi\)
0.755609 + 0.655023i \(0.227341\pi\)
\(618\) 0 0
\(619\) 46.6282 1.87415 0.937073 0.349132i \(-0.113524\pi\)
0.937073 + 0.349132i \(0.113524\pi\)
\(620\) 5.10603 0.205063
\(621\) 0 0
\(622\) 5.09711 0.204376
\(623\) −23.8630 −0.956049
\(624\) 0 0
\(625\) 18.1543 0.726170
\(626\) 3.70513 0.148087
\(627\) 0 0
\(628\) 15.2048 0.606736
\(629\) 33.6991 1.34367
\(630\) 0 0
\(631\) 28.3815 1.12985 0.564926 0.825142i \(-0.308905\pi\)
0.564926 + 0.825142i \(0.308905\pi\)
\(632\) 12.1543 0.483472
\(633\) 0 0
\(634\) −11.1944 −0.444588
\(635\) −2.77527 −0.110133
\(636\) 0 0
\(637\) 0.418399 0.0165776
\(638\) 38.8144 1.53668
\(639\) 0 0
\(640\) −0.686428 −0.0271335
\(641\) −11.2437 −0.444098 −0.222049 0.975036i \(-0.571274\pi\)
−0.222049 + 0.975036i \(0.571274\pi\)
\(642\) 0 0
\(643\) −0.473139 −0.0186588 −0.00932940 0.999956i \(-0.502970\pi\)
−0.00932940 + 0.999956i \(0.502970\pi\)
\(644\) 9.11562 0.359206
\(645\) 0 0
\(646\) 6.83150 0.268782
\(647\) 44.1533 1.73585 0.867923 0.496699i \(-0.165455\pi\)
0.867923 + 0.496699i \(0.165455\pi\)
\(648\) 0 0
\(649\) −17.3642 −0.681604
\(650\) −1.49384 −0.0585934
\(651\) 0 0
\(652\) 12.4990 0.489498
\(653\) −17.7560 −0.694847 −0.347423 0.937708i \(-0.612943\pi\)
−0.347423 + 0.937708i \(0.612943\pi\)
\(654\) 0 0
\(655\) 4.56302 0.178292
\(656\) 2.51679 0.0982640
\(657\) 0 0
\(658\) −17.2814 −0.673700
\(659\) −28.1314 −1.09584 −0.547921 0.836530i \(-0.684581\pi\)
−0.547921 + 0.836530i \(0.684581\pi\)
\(660\) 0 0
\(661\) 13.7180 0.533567 0.266784 0.963756i \(-0.414039\pi\)
0.266784 + 0.963756i \(0.414039\pi\)
\(662\) −30.7807 −1.19633
\(663\) 0 0
\(664\) 5.22743 0.202864
\(665\) 3.03546 0.117710
\(666\) 0 0
\(667\) −22.2953 −0.863278
\(668\) −6.88133 −0.266247
\(669\) 0 0
\(670\) 9.59731 0.370776
\(671\) 14.8316 0.572567
\(672\) 0 0
\(673\) 12.1670 0.469002 0.234501 0.972116i \(-0.424654\pi\)
0.234501 + 0.972116i \(0.424654\pi\)
\(674\) −28.0781 −1.08153
\(675\) 0 0
\(676\) −12.8912 −0.495815
\(677\) −42.3323 −1.62696 −0.813481 0.581592i \(-0.802430\pi\)
−0.813481 + 0.581592i \(0.802430\pi\)
\(678\) 0 0
\(679\) 12.7992 0.491186
\(680\) 3.04925 0.116934
\(681\) 0 0
\(682\) 41.0527 1.57199
\(683\) −32.6141 −1.24794 −0.623972 0.781446i \(-0.714482\pi\)
−0.623972 + 0.781446i \(0.714482\pi\)
\(684\) 0 0
\(685\) 7.53265 0.287808
\(686\) −16.4810 −0.629249
\(687\) 0 0
\(688\) 6.97369 0.265869
\(689\) 0.277162 0.0105590
\(690\) 0 0
\(691\) 6.87327 0.261472 0.130736 0.991417i \(-0.458266\pi\)
0.130736 + 0.991417i \(0.458266\pi\)
\(692\) 3.65630 0.138992
\(693\) 0 0
\(694\) −4.81474 −0.182765
\(695\) 10.4434 0.396143
\(696\) 0 0
\(697\) −11.1801 −0.423475
\(698\) 20.7579 0.785697
\(699\) 0 0
\(700\) −13.0226 −0.492207
\(701\) 30.3022 1.14450 0.572249 0.820080i \(-0.306071\pi\)
0.572249 + 0.820080i \(0.306071\pi\)
\(702\) 0 0
\(703\) 11.6664 0.440008
\(704\) −5.51891 −0.208002
\(705\) 0 0
\(706\) 3.21876 0.121140
\(707\) 9.30601 0.349989
\(708\) 0 0
\(709\) −45.6605 −1.71482 −0.857408 0.514638i \(-0.827926\pi\)
−0.857408 + 0.514638i \(0.827926\pi\)
\(710\) −4.58036 −0.171898
\(711\) 0 0
\(712\) −8.29874 −0.311009
\(713\) −23.5810 −0.883117
\(714\) 0 0
\(715\) 1.24959 0.0467322
\(716\) −11.6410 −0.435045
\(717\) 0 0
\(718\) −25.5107 −0.952053
\(719\) 4.43820 0.165517 0.0827584 0.996570i \(-0.473627\pi\)
0.0827584 + 0.996570i \(0.473627\pi\)
\(720\) 0 0
\(721\) −36.5975 −1.36296
\(722\) −16.6350 −0.619090
\(723\) 0 0
\(724\) −13.8301 −0.513990
\(725\) 31.8510 1.18292
\(726\) 0 0
\(727\) 3.18844 0.118253 0.0591263 0.998251i \(-0.481169\pi\)
0.0591263 + 0.998251i \(0.481169\pi\)
\(728\) 0.948489 0.0351533
\(729\) 0 0
\(730\) −8.20794 −0.303790
\(731\) −30.9785 −1.14578
\(732\) 0 0
\(733\) 7.01336 0.259045 0.129522 0.991577i \(-0.458656\pi\)
0.129522 + 0.991577i \(0.458656\pi\)
\(734\) −24.4579 −0.902758
\(735\) 0 0
\(736\) 3.17011 0.116852
\(737\) 77.1628 2.84233
\(738\) 0 0
\(739\) −24.9587 −0.918122 −0.459061 0.888405i \(-0.651814\pi\)
−0.459061 + 0.888405i \(0.651814\pi\)
\(740\) 5.20733 0.191425
\(741\) 0 0
\(742\) 2.41616 0.0886999
\(743\) −28.8093 −1.05691 −0.528456 0.848961i \(-0.677229\pi\)
−0.528456 + 0.848961i \(0.677229\pi\)
\(744\) 0 0
\(745\) 1.99179 0.0729735
\(746\) −9.79830 −0.358741
\(747\) 0 0
\(748\) 24.5161 0.896399
\(749\) −13.0124 −0.475464
\(750\) 0 0
\(751\) −51.0813 −1.86398 −0.931992 0.362479i \(-0.881931\pi\)
−0.931992 + 0.362479i \(0.881931\pi\)
\(752\) −6.00990 −0.219159
\(753\) 0 0
\(754\) −2.31985 −0.0844838
\(755\) −1.82864 −0.0665509
\(756\) 0 0
\(757\) 15.0243 0.546068 0.273034 0.962004i \(-0.411973\pi\)
0.273034 + 0.962004i \(0.411973\pi\)
\(758\) 23.3632 0.848588
\(759\) 0 0
\(760\) 1.05563 0.0382918
\(761\) −6.31216 −0.228816 −0.114408 0.993434i \(-0.536497\pi\)
−0.114408 + 0.993434i \(0.536497\pi\)
\(762\) 0 0
\(763\) −30.2994 −1.09691
\(764\) 15.4332 0.558354
\(765\) 0 0
\(766\) −26.1424 −0.944564
\(767\) 1.03782 0.0374735
\(768\) 0 0
\(769\) 24.2449 0.874293 0.437147 0.899390i \(-0.355989\pi\)
0.437147 + 0.899390i \(0.355989\pi\)
\(770\) 10.8933 0.392568
\(771\) 0 0
\(772\) −1.86292 −0.0670479
\(773\) −14.1903 −0.510390 −0.255195 0.966890i \(-0.582140\pi\)
−0.255195 + 0.966890i \(0.582140\pi\)
\(774\) 0 0
\(775\) 33.6878 1.21010
\(776\) 4.45112 0.159786
\(777\) 0 0
\(778\) −3.73549 −0.133924
\(779\) −3.87047 −0.138674
\(780\) 0 0
\(781\) −36.8263 −1.31775
\(782\) −14.0823 −0.503582
\(783\) 0 0
\(784\) 1.26844 0.0453014
\(785\) −10.4370 −0.372512
\(786\) 0 0
\(787\) −17.2608 −0.615280 −0.307640 0.951503i \(-0.599539\pi\)
−0.307640 + 0.951503i \(0.599539\pi\)
\(788\) 18.0327 0.642387
\(789\) 0 0
\(790\) −8.34305 −0.296833
\(791\) −8.44503 −0.300271
\(792\) 0 0
\(793\) −0.886451 −0.0314788
\(794\) 17.6199 0.625306
\(795\) 0 0
\(796\) 24.0118 0.851077
\(797\) 13.2887 0.470710 0.235355 0.971909i \(-0.424375\pi\)
0.235355 + 0.971909i \(0.424375\pi\)
\(798\) 0 0
\(799\) 26.6972 0.944479
\(800\) −4.52882 −0.160118
\(801\) 0 0
\(802\) 22.5232 0.795320
\(803\) −65.9922 −2.32882
\(804\) 0 0
\(805\) −6.25722 −0.220538
\(806\) −2.45363 −0.0864254
\(807\) 0 0
\(808\) 3.23632 0.113853
\(809\) 9.08056 0.319255 0.159628 0.987177i \(-0.448971\pi\)
0.159628 + 0.987177i \(0.448971\pi\)
\(810\) 0 0
\(811\) −49.9105 −1.75260 −0.876298 0.481770i \(-0.839994\pi\)
−0.876298 + 0.481770i \(0.839994\pi\)
\(812\) −20.2232 −0.709697
\(813\) 0 0
\(814\) 41.8672 1.46745
\(815\) −8.57965 −0.300532
\(816\) 0 0
\(817\) −10.7246 −0.375206
\(818\) −12.4542 −0.435451
\(819\) 0 0
\(820\) −1.72759 −0.0603301
\(821\) −9.08016 −0.316900 −0.158450 0.987367i \(-0.550650\pi\)
−0.158450 + 0.987367i \(0.550650\pi\)
\(822\) 0 0
\(823\) 19.7767 0.689371 0.344686 0.938718i \(-0.387986\pi\)
0.344686 + 0.938718i \(0.387986\pi\)
\(824\) −12.7274 −0.443380
\(825\) 0 0
\(826\) 9.04717 0.314791
\(827\) −37.8945 −1.31772 −0.658860 0.752266i \(-0.728961\pi\)
−0.658860 + 0.752266i \(0.728961\pi\)
\(828\) 0 0
\(829\) −6.13822 −0.213189 −0.106595 0.994303i \(-0.533995\pi\)
−0.106595 + 0.994303i \(0.533995\pi\)
\(830\) −3.58825 −0.124550
\(831\) 0 0
\(832\) 0.329853 0.0114356
\(833\) −5.63467 −0.195230
\(834\) 0 0
\(835\) 4.72354 0.163465
\(836\) 8.48734 0.293541
\(837\) 0 0
\(838\) 30.5170 1.05419
\(839\) −37.2877 −1.28731 −0.643657 0.765314i \(-0.722584\pi\)
−0.643657 + 0.765314i \(0.722584\pi\)
\(840\) 0 0
\(841\) 20.4627 0.705610
\(842\) 5.17369 0.178297
\(843\) 0 0
\(844\) 6.05163 0.208306
\(845\) 8.84888 0.304411
\(846\) 0 0
\(847\) 55.9525 1.92255
\(848\) 0.840259 0.0288546
\(849\) 0 0
\(850\) 20.1179 0.690039
\(851\) −24.0489 −0.824385
\(852\) 0 0
\(853\) −5.19329 −0.177815 −0.0889075 0.996040i \(-0.528338\pi\)
−0.0889075 + 0.996040i \(0.528338\pi\)
\(854\) −7.72762 −0.264434
\(855\) 0 0
\(856\) −4.52529 −0.154671
\(857\) −33.5933 −1.14753 −0.573763 0.819022i \(-0.694517\pi\)
−0.573763 + 0.819022i \(0.694517\pi\)
\(858\) 0 0
\(859\) 10.4008 0.354872 0.177436 0.984132i \(-0.443220\pi\)
0.177436 + 0.984132i \(0.443220\pi\)
\(860\) −4.78694 −0.163233
\(861\) 0 0
\(862\) 11.8685 0.404243
\(863\) 37.4393 1.27445 0.637224 0.770679i \(-0.280083\pi\)
0.637224 + 0.770679i \(0.280083\pi\)
\(864\) 0 0
\(865\) −2.50979 −0.0853354
\(866\) 2.77072 0.0941530
\(867\) 0 0
\(868\) −21.3895 −0.726006
\(869\) −67.0785 −2.27548
\(870\) 0 0
\(871\) −4.61185 −0.156266
\(872\) −10.5371 −0.356832
\(873\) 0 0
\(874\) −4.87520 −0.164906
\(875\) 18.8081 0.635831
\(876\) 0 0
\(877\) 49.1976 1.66129 0.830643 0.556805i \(-0.187973\pi\)
0.830643 + 0.556805i \(0.187973\pi\)
\(878\) −13.5404 −0.456966
\(879\) 0 0
\(880\) 3.78834 0.127705
\(881\) 42.7892 1.44161 0.720803 0.693140i \(-0.243773\pi\)
0.720803 + 0.693140i \(0.243773\pi\)
\(882\) 0 0
\(883\) −12.3874 −0.416870 −0.208435 0.978036i \(-0.566837\pi\)
−0.208435 + 0.978036i \(0.566837\pi\)
\(884\) −1.46527 −0.0492825
\(885\) 0 0
\(886\) −38.7298 −1.30115
\(887\) 1.39064 0.0466930 0.0233465 0.999727i \(-0.492568\pi\)
0.0233465 + 0.999727i \(0.492568\pi\)
\(888\) 0 0
\(889\) 11.6258 0.389916
\(890\) 5.69649 0.190947
\(891\) 0 0
\(892\) −1.00000 −0.0334825
\(893\) 9.24241 0.309285
\(894\) 0 0
\(895\) 7.99071 0.267100
\(896\) 2.87549 0.0960634
\(897\) 0 0
\(898\) 35.2746 1.17713
\(899\) 52.3151 1.74481
\(900\) 0 0
\(901\) −3.73260 −0.124351
\(902\) −13.8899 −0.462484
\(903\) 0 0
\(904\) −2.93690 −0.0976799
\(905\) 9.49335 0.315570
\(906\) 0 0
\(907\) −16.8327 −0.558921 −0.279460 0.960157i \(-0.590156\pi\)
−0.279460 + 0.960157i \(0.590156\pi\)
\(908\) −24.3730 −0.808846
\(909\) 0 0
\(910\) −0.651069 −0.0215827
\(911\) −25.6813 −0.850860 −0.425430 0.904991i \(-0.639877\pi\)
−0.425430 + 0.904991i \(0.639877\pi\)
\(912\) 0 0
\(913\) −28.8497 −0.954787
\(914\) −15.7179 −0.519901
\(915\) 0 0
\(916\) 7.57496 0.250284
\(917\) −19.1148 −0.631226
\(918\) 0 0
\(919\) −34.0182 −1.12216 −0.561079 0.827762i \(-0.689614\pi\)
−0.561079 + 0.827762i \(0.689614\pi\)
\(920\) −2.17605 −0.0717424
\(921\) 0 0
\(922\) 15.4964 0.510346
\(923\) 2.20103 0.0724476
\(924\) 0 0
\(925\) 34.3562 1.12963
\(926\) 34.0979 1.12053
\(927\) 0 0
\(928\) −7.03297 −0.230869
\(929\) 16.0961 0.528096 0.264048 0.964510i \(-0.414942\pi\)
0.264048 + 0.964510i \(0.414942\pi\)
\(930\) 0 0
\(931\) −1.95069 −0.0639312
\(932\) −1.91430 −0.0627050
\(933\) 0 0
\(934\) −5.76270 −0.188561
\(935\) −16.8286 −0.550353
\(936\) 0 0
\(937\) −11.1780 −0.365169 −0.182584 0.983190i \(-0.558446\pi\)
−0.182584 + 0.983190i \(0.558446\pi\)
\(938\) −40.2037 −1.31270
\(939\) 0 0
\(940\) 4.12537 0.134555
\(941\) 55.8606 1.82101 0.910503 0.413504i \(-0.135695\pi\)
0.910503 + 0.413504i \(0.135695\pi\)
\(942\) 0 0
\(943\) 7.97849 0.259815
\(944\) 3.14631 0.102404
\(945\) 0 0
\(946\) −38.4872 −1.25133
\(947\) −33.1983 −1.07880 −0.539400 0.842050i \(-0.681349\pi\)
−0.539400 + 0.842050i \(0.681349\pi\)
\(948\) 0 0
\(949\) 3.94421 0.128034
\(950\) 6.96470 0.225965
\(951\) 0 0
\(952\) −12.7735 −0.413992
\(953\) 52.1647 1.68978 0.844891 0.534939i \(-0.179665\pi\)
0.844891 + 0.534939i \(0.179665\pi\)
\(954\) 0 0
\(955\) −10.5938 −0.342807
\(956\) 18.8018 0.608094
\(957\) 0 0
\(958\) 0.892156 0.0288243
\(959\) −31.5547 −1.01896
\(960\) 0 0
\(961\) 24.3321 0.784905
\(962\) −2.50231 −0.0806777
\(963\) 0 0
\(964\) −12.7202 −0.409689
\(965\) 1.27876 0.0411648
\(966\) 0 0
\(967\) −6.32108 −0.203272 −0.101636 0.994822i \(-0.532408\pi\)
−0.101636 + 0.994822i \(0.532408\pi\)
\(968\) 19.4584 0.625417
\(969\) 0 0
\(970\) −3.05538 −0.0981022
\(971\) −24.4643 −0.785098 −0.392549 0.919731i \(-0.628407\pi\)
−0.392549 + 0.919731i \(0.628407\pi\)
\(972\) 0 0
\(973\) −43.7482 −1.40250
\(974\) 4.83587 0.154951
\(975\) 0 0
\(976\) −2.68741 −0.0860220
\(977\) 23.1983 0.742181 0.371090 0.928597i \(-0.378984\pi\)
0.371090 + 0.928597i \(0.378984\pi\)
\(978\) 0 0
\(979\) 45.8001 1.46378
\(980\) −0.870693 −0.0278133
\(981\) 0 0
\(982\) 14.5280 0.463608
\(983\) 44.9757 1.43450 0.717252 0.696814i \(-0.245400\pi\)
0.717252 + 0.696814i \(0.245400\pi\)
\(984\) 0 0
\(985\) −12.3781 −0.394400
\(986\) 31.2419 0.994945
\(987\) 0 0
\(988\) −0.507269 −0.0161384
\(989\) 22.1074 0.702974
\(990\) 0 0
\(991\) −54.4523 −1.72973 −0.864867 0.502001i \(-0.832597\pi\)
−0.864867 + 0.502001i \(0.832597\pi\)
\(992\) −7.43855 −0.236174
\(993\) 0 0
\(994\) 19.1874 0.608588
\(995\) −16.4824 −0.522527
\(996\) 0 0
\(997\) 20.4748 0.648444 0.324222 0.945981i \(-0.394898\pi\)
0.324222 + 0.945981i \(0.394898\pi\)
\(998\) 42.7043 1.35178
\(999\) 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.1.4 5
3.2 odd 2 1338.2.a.h.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1338.2.a.h.1.2 5 3.2 odd 2
4014.2.a.r.1.4 5 1.1 even 1 trivial