Properties

Label 3630.2.a.bs.1.2
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.52625.1
Defining polynomial: \(x^{4} - x^{3} - 24 x^{2} + 19 x + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.55745\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.58059 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.58059 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -5.75608 q^{13} -1.58059 q^{14} -1.00000 q^{15} +1.00000 q^{16} +7.17549 q^{17} +1.00000 q^{18} +1.32139 q^{19} -1.00000 q^{20} -1.58059 q^{21} -0.358826 q^{23} +1.00000 q^{24} +1.00000 q^{25} -5.75608 q^{26} +1.00000 q^{27} -1.58059 q^{28} +9.67076 q^{29} -1.00000 q^{30} +7.19863 q^{31} +1.00000 q^{32} +7.17549 q^{34} +1.58059 q^{35} +1.00000 q^{36} +9.73294 q^{37} +1.32139 q^{38} -5.75608 q^{39} -1.00000 q^{40} -12.0064 q^{41} -1.58059 q^{42} -6.41156 q^{43} -1.00000 q^{45} -0.358826 q^{46} +6.52001 q^{47} +1.00000 q^{48} -4.50173 q^{49} +1.00000 q^{50} +7.17549 q^{51} -5.75608 q^{52} +2.34453 q^{53} +1.00000 q^{54} -1.58059 q^{56} +1.32139 q^{57} +9.67076 q^{58} +10.3262 q^{59} -1.00000 q^{60} +5.63332 q^{61} +7.19863 q^{62} -1.58059 q^{63} +1.00000 q^{64} +5.75608 q^{65} -5.07747 q^{67} +7.17549 q^{68} -0.358826 q^{69} +1.58059 q^{70} +9.11491 q^{71} +1.00000 q^{72} +6.94427 q^{73} +9.73294 q^{74} +1.00000 q^{75} +1.32139 q^{76} -5.75608 q^{78} +11.5265 q^{79} -1.00000 q^{80} +1.00000 q^{81} -12.0064 q^{82} +8.00000 q^{83} -1.58059 q^{84} -7.17549 q^{85} -6.41156 q^{86} +9.67076 q^{87} +7.16764 q^{89} -1.00000 q^{90} +9.09802 q^{91} -0.358826 q^{92} +7.19863 q^{93} +6.52001 q^{94} -1.32139 q^{95} +1.00000 q^{96} -1.92512 q^{97} -4.50173 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} - 4q^{10} + 4q^{12} + 2q^{13} + q^{14} - 4q^{15} + 4q^{16} + 11q^{17} + 4q^{18} + q^{19} - 4q^{20} + q^{21} + 4q^{24} + 4q^{25} + 2q^{26} + 4q^{27} + q^{28} + 9q^{29} - 4q^{30} + 17q^{31} + 4q^{32} + 11q^{34} - q^{35} + 4q^{36} + 8q^{37} + q^{38} + 2q^{39} - 4q^{40} - 11q^{41} + q^{42} + q^{43} - 4q^{45} + 10q^{47} + 4q^{48} + 31q^{49} + 4q^{50} + 11q^{51} + 2q^{52} + 11q^{53} + 4q^{54} + q^{56} + q^{57} + 9q^{58} + 10q^{59} - 4q^{60} - 10q^{61} + 17q^{62} + q^{63} + 4q^{64} - 2q^{65} + 9q^{67} + 11q^{68} - q^{70} + 10q^{71} + 4q^{72} - 8q^{73} + 8q^{74} + 4q^{75} + q^{76} + 2q^{78} - 7q^{79} - 4q^{80} + 4q^{81} - 11q^{82} + 32q^{83} + q^{84} - 11q^{85} + q^{86} + 9q^{87} - 23q^{89} - 4q^{90} + 48q^{91} + 17q^{93} + 10q^{94} - q^{95} + 4q^{96} - 2q^{97} + 31q^{98} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −1.58059 −0.597408 −0.298704 0.954346i \(-0.596554\pi\)
−0.298704 + 0.954346i \(0.596554\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) −5.75608 −1.59645 −0.798225 0.602360i \(-0.794227\pi\)
−0.798225 + 0.602360i \(0.794227\pi\)
\(14\) −1.58059 −0.422431
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 7.17549 1.74031 0.870156 0.492777i \(-0.164018\pi\)
0.870156 + 0.492777i \(0.164018\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.32139 0.303147 0.151573 0.988446i \(-0.451566\pi\)
0.151573 + 0.988446i \(0.451566\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.58059 −0.344914
\(22\) 0 0
\(23\) −0.358826 −0.0748205 −0.0374102 0.999300i \(-0.511911\pi\)
−0.0374102 + 0.999300i \(0.511911\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −5.75608 −1.12886
\(27\) 1.00000 0.192450
\(28\) −1.58059 −0.298704
\(29\) 9.67076 1.79582 0.897908 0.440184i \(-0.145087\pi\)
0.897908 + 0.440184i \(0.145087\pi\)
\(30\) −1.00000 −0.182574
\(31\) 7.19863 1.29291 0.646456 0.762951i \(-0.276250\pi\)
0.646456 + 0.762951i \(0.276250\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.17549 1.23059
\(35\) 1.58059 0.267169
\(36\) 1.00000 0.166667
\(37\) 9.73294 1.60009 0.800043 0.599943i \(-0.204810\pi\)
0.800043 + 0.599943i \(0.204810\pi\)
\(38\) 1.32139 0.214357
\(39\) −5.75608 −0.921711
\(40\) −1.00000 −0.158114
\(41\) −12.0064 −1.87509 −0.937546 0.347861i \(-0.886908\pi\)
−0.937546 + 0.347861i \(0.886908\pi\)
\(42\) −1.58059 −0.243891
\(43\) −6.41156 −0.977753 −0.488877 0.872353i \(-0.662593\pi\)
−0.488877 + 0.872353i \(0.662593\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −0.358826 −0.0529061
\(47\) 6.52001 0.951042 0.475521 0.879704i \(-0.342260\pi\)
0.475521 + 0.879704i \(0.342260\pi\)
\(48\) 1.00000 0.144338
\(49\) −4.50173 −0.643104
\(50\) 1.00000 0.141421
\(51\) 7.17549 1.00477
\(52\) −5.75608 −0.798225
\(53\) 2.34453 0.322045 0.161023 0.986951i \(-0.448521\pi\)
0.161023 + 0.986951i \(0.448521\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.58059 −0.211216
\(57\) 1.32139 0.175022
\(58\) 9.67076 1.26983
\(59\) 10.3262 1.34436 0.672181 0.740387i \(-0.265358\pi\)
0.672181 + 0.740387i \(0.265358\pi\)
\(60\) −1.00000 −0.129099
\(61\) 5.63332 0.721273 0.360637 0.932706i \(-0.382559\pi\)
0.360637 + 0.932706i \(0.382559\pi\)
\(62\) 7.19863 0.914227
\(63\) −1.58059 −0.199136
\(64\) 1.00000 0.125000
\(65\) 5.75608 0.713954
\(66\) 0 0
\(67\) −5.07747 −0.620311 −0.310156 0.950686i \(-0.600381\pi\)
−0.310156 + 0.950686i \(0.600381\pi\)
\(68\) 7.17549 0.870156
\(69\) −0.358826 −0.0431976
\(70\) 1.58059 0.188917
\(71\) 9.11491 1.08174 0.540870 0.841106i \(-0.318095\pi\)
0.540870 + 0.841106i \(0.318095\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.94427 0.812766 0.406383 0.913703i \(-0.366790\pi\)
0.406383 + 0.913703i \(0.366790\pi\)
\(74\) 9.73294 1.13143
\(75\) 1.00000 0.115470
\(76\) 1.32139 0.151573
\(77\) 0 0
\(78\) −5.75608 −0.651748
\(79\) 11.5265 1.29683 0.648414 0.761288i \(-0.275433\pi\)
0.648414 + 0.761288i \(0.275433\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −12.0064 −1.32589
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −1.58059 −0.172457
\(85\) −7.17549 −0.778291
\(86\) −6.41156 −0.691376
\(87\) 9.67076 1.03681
\(88\) 0 0
\(89\) 7.16764 0.759768 0.379884 0.925034i \(-0.375964\pi\)
0.379884 + 0.925034i \(0.375964\pi\)
\(90\) −1.00000 −0.105409
\(91\) 9.09802 0.953732
\(92\) −0.358826 −0.0374102
\(93\) 7.19863 0.746463
\(94\) 6.52001 0.672488
\(95\) −1.32139 −0.135571
\(96\) 1.00000 0.102062
\(97\) −1.92512 −0.195466 −0.0977331 0.995213i \(-0.531159\pi\)
−0.0977331 + 0.995213i \(0.531159\pi\)
\(98\) −4.50173 −0.454743
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −8.24092 −0.820002 −0.410001 0.912085i \(-0.634472\pi\)
−0.410001 + 0.912085i \(0.634472\pi\)
\(102\) 7.17549 0.710479
\(103\) −4.90843 −0.483642 −0.241821 0.970321i \(-0.577745\pi\)
−0.241821 + 0.970321i \(0.577745\pi\)
\(104\) −5.75608 −0.564430
\(105\) 1.58059 0.154250
\(106\) 2.34453 0.227720
\(107\) −1.64903 −0.159417 −0.0797086 0.996818i \(-0.525399\pi\)
−0.0797086 + 0.996818i \(0.525399\pi\)
\(108\) 1.00000 0.0962250
\(109\) −8.94427 −0.856706 −0.428353 0.903612i \(-0.640906\pi\)
−0.428353 + 0.903612i \(0.640906\pi\)
\(110\) 0 0
\(111\) 9.73294 0.923810
\(112\) −1.58059 −0.149352
\(113\) 0.0374409 0.00352214 0.00176107 0.999998i \(-0.499439\pi\)
0.00176107 + 0.999998i \(0.499439\pi\)
\(114\) 1.32139 0.123759
\(115\) 0.358826 0.0334607
\(116\) 9.67076 0.897908
\(117\) −5.75608 −0.532150
\(118\) 10.3262 0.950607
\(119\) −11.3415 −1.03968
\(120\) −1.00000 −0.0912871
\(121\) 0 0
\(122\) 5.63332 0.510017
\(123\) −12.0064 −1.08259
\(124\) 7.19863 0.646456
\(125\) −1.00000 −0.0894427
\(126\) −1.58059 −0.140810
\(127\) −16.5418 −1.46784 −0.733921 0.679234i \(-0.762312\pi\)
−0.733921 + 0.679234i \(0.762312\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.41156 −0.564506
\(130\) 5.75608 0.504842
\(131\) −12.5096 −1.09297 −0.546483 0.837470i \(-0.684034\pi\)
−0.546483 + 0.837470i \(0.684034\pi\)
\(132\) 0 0
\(133\) −2.08857 −0.181102
\(134\) −5.07747 −0.438626
\(135\) −1.00000 −0.0860663
\(136\) 7.17549 0.615293
\(137\) −12.3135 −1.05202 −0.526008 0.850480i \(-0.676312\pi\)
−0.526008 + 0.850480i \(0.676312\pi\)
\(138\) −0.358826 −0.0305453
\(139\) 12.9970 1.10239 0.551196 0.834376i \(-0.314172\pi\)
0.551196 + 0.834376i \(0.314172\pi\)
\(140\) 1.58059 0.133585
\(141\) 6.52001 0.549084
\(142\) 9.11491 0.764906
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −9.67076 −0.803113
\(146\) 6.94427 0.574712
\(147\) −4.50173 −0.371296
\(148\) 9.73294 0.800043
\(149\) −1.64762 −0.134979 −0.0674893 0.997720i \(-0.521499\pi\)
−0.0674893 + 0.997720i \(0.521499\pi\)
\(150\) 1.00000 0.0816497
\(151\) 6.74823 0.549163 0.274582 0.961564i \(-0.411461\pi\)
0.274582 + 0.961564i \(0.411461\pi\)
\(152\) 1.32139 0.107179
\(153\) 7.17549 0.580104
\(154\) 0 0
\(155\) −7.19863 −0.578208
\(156\) −5.75608 −0.460855
\(157\) 8.92898 0.712610 0.356305 0.934370i \(-0.384036\pi\)
0.356305 + 0.934370i \(0.384036\pi\)
\(158\) 11.5265 0.916996
\(159\) 2.34453 0.185933
\(160\) −1.00000 −0.0790569
\(161\) 0.567158 0.0446983
\(162\) 1.00000 0.0785674
\(163\) 9.72251 0.761525 0.380763 0.924673i \(-0.375661\pi\)
0.380763 + 0.924673i \(0.375661\pi\)
\(164\) −12.0064 −0.937546
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) −15.4643 −1.19666 −0.598331 0.801249i \(-0.704169\pi\)
−0.598331 + 0.801249i \(0.704169\pi\)
\(168\) −1.58059 −0.121945
\(169\) 20.1325 1.54865
\(170\) −7.17549 −0.550335
\(171\) 1.32139 0.101049
\(172\) −6.41156 −0.488877
\(173\) 18.4037 1.39921 0.699604 0.714531i \(-0.253360\pi\)
0.699604 + 0.714531i \(0.253360\pi\)
\(174\) 9.67076 0.733139
\(175\) −1.58059 −0.119482
\(176\) 0 0
\(177\) 10.3262 0.776168
\(178\) 7.16764 0.537237
\(179\) −18.5623 −1.38741 −0.693706 0.720258i \(-0.744023\pi\)
−0.693706 + 0.720258i \(0.744023\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 9.52786 0.708201 0.354100 0.935207i \(-0.384787\pi\)
0.354100 + 0.935207i \(0.384787\pi\)
\(182\) 9.09802 0.674390
\(183\) 5.63332 0.416427
\(184\) −0.358826 −0.0264530
\(185\) −9.73294 −0.715580
\(186\) 7.19863 0.527829
\(187\) 0 0
\(188\) 6.52001 0.475521
\(189\) −1.58059 −0.114971
\(190\) −1.32139 −0.0958634
\(191\) 20.0306 1.44936 0.724681 0.689085i \(-0.241987\pi\)
0.724681 + 0.689085i \(0.241987\pi\)
\(192\) 1.00000 0.0721688
\(193\) −7.82937 −0.563570 −0.281785 0.959478i \(-0.590926\pi\)
−0.281785 + 0.959478i \(0.590926\pi\)
\(194\) −1.92512 −0.138215
\(195\) 5.75608 0.412201
\(196\) −4.50173 −0.321552
\(197\) −20.6798 −1.47337 −0.736687 0.676234i \(-0.763611\pi\)
−0.736687 + 0.676234i \(0.763611\pi\)
\(198\) 0 0
\(199\) −3.12921 −0.221824 −0.110912 0.993830i \(-0.535377\pi\)
−0.110912 + 0.993830i \(0.535377\pi\)
\(200\) 1.00000 0.0707107
\(201\) −5.07747 −0.358137
\(202\) −8.24092 −0.579829
\(203\) −15.2855 −1.07283
\(204\) 7.17549 0.502385
\(205\) 12.0064 0.838567
\(206\) −4.90843 −0.341986
\(207\) −0.358826 −0.0249402
\(208\) −5.75608 −0.399112
\(209\) 0 0
\(210\) 1.58059 0.109071
\(211\) −14.3047 −0.984776 −0.492388 0.870376i \(-0.663876\pi\)
−0.492388 + 0.870376i \(0.663876\pi\)
\(212\) 2.34453 0.161023
\(213\) 9.11491 0.624543
\(214\) −1.64903 −0.112725
\(215\) 6.41156 0.437264
\(216\) 1.00000 0.0680414
\(217\) −11.3781 −0.772396
\(218\) −8.94427 −0.605783
\(219\) 6.94427 0.469250
\(220\) 0 0
\(221\) −41.3027 −2.77832
\(222\) 9.73294 0.653232
\(223\) 5.58059 0.373704 0.186852 0.982388i \(-0.440171\pi\)
0.186852 + 0.982388i \(0.440171\pi\)
\(224\) −1.58059 −0.105608
\(225\) 1.00000 0.0666667
\(226\) 0.0374409 0.00249053
\(227\) −4.68905 −0.311223 −0.155612 0.987818i \(-0.549735\pi\)
−0.155612 + 0.987818i \(0.549735\pi\)
\(228\) 1.32139 0.0875109
\(229\) 8.39725 0.554906 0.277453 0.960739i \(-0.410510\pi\)
0.277453 + 0.960739i \(0.410510\pi\)
\(230\) 0.358826 0.0236603
\(231\) 0 0
\(232\) 9.67076 0.634917
\(233\) −7.27351 −0.476503 −0.238252 0.971203i \(-0.576574\pi\)
−0.238252 + 0.971203i \(0.576574\pi\)
\(234\) −5.75608 −0.376287
\(235\) −6.52001 −0.425319
\(236\) 10.3262 0.672181
\(237\) 11.5265 0.748724
\(238\) −11.3415 −0.735162
\(239\) −21.7883 −1.40937 −0.704683 0.709523i \(-0.748911\pi\)
−0.704683 + 0.709523i \(0.748911\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −4.54057 −0.292484 −0.146242 0.989249i \(-0.546718\pi\)
−0.146242 + 0.989249i \(0.546718\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 5.63332 0.360637
\(245\) 4.50173 0.287605
\(246\) −12.0064 −0.765503
\(247\) −7.60600 −0.483958
\(248\) 7.19863 0.457113
\(249\) 8.00000 0.506979
\(250\) −1.00000 −0.0632456
\(251\) 4.20906 0.265674 0.132837 0.991138i \(-0.457591\pi\)
0.132837 + 0.991138i \(0.457591\pi\)
\(252\) −1.58059 −0.0995680
\(253\) 0 0
\(254\) −16.5418 −1.03792
\(255\) −7.17549 −0.449346
\(256\) 1.00000 0.0625000
\(257\) −14.7945 −0.922856 −0.461428 0.887178i \(-0.652663\pi\)
−0.461428 + 0.887178i \(0.652663\pi\)
\(258\) −6.41156 −0.399166
\(259\) −15.3838 −0.955904
\(260\) 5.75608 0.356977
\(261\) 9.67076 0.598605
\(262\) −12.5096 −0.772844
\(263\) 7.90358 0.487355 0.243678 0.969856i \(-0.421646\pi\)
0.243678 + 0.969856i \(0.421646\pi\)
\(264\) 0 0
\(265\) −2.34453 −0.144023
\(266\) −2.08857 −0.128059
\(267\) 7.16764 0.438652
\(268\) −5.07747 −0.310156
\(269\) 17.9469 1.09424 0.547120 0.837054i \(-0.315724\pi\)
0.547120 + 0.837054i \(0.315724\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 5.86454 0.356245 0.178123 0.984008i \(-0.442998\pi\)
0.178123 + 0.984008i \(0.442998\pi\)
\(272\) 7.17549 0.435078
\(273\) 9.09802 0.550637
\(274\) −12.3135 −0.743888
\(275\) 0 0
\(276\) −0.358826 −0.0215988
\(277\) −8.43371 −0.506732 −0.253366 0.967370i \(-0.581538\pi\)
−0.253366 + 0.967370i \(0.581538\pi\)
\(278\) 12.9970 0.779508
\(279\) 7.19863 0.430971
\(280\) 1.58059 0.0944585
\(281\) −5.85024 −0.348996 −0.174498 0.984658i \(-0.555830\pi\)
−0.174498 + 0.984658i \(0.555830\pi\)
\(282\) 6.52001 0.388261
\(283\) −11.0312 −0.655736 −0.327868 0.944724i \(-0.606330\pi\)
−0.327868 + 0.944724i \(0.606330\pi\)
\(284\) 9.11491 0.540870
\(285\) −1.32139 −0.0782721
\(286\) 0 0
\(287\) 18.9773 1.12020
\(288\) 1.00000 0.0589256
\(289\) 34.4876 2.02868
\(290\) −9.67076 −0.567887
\(291\) −1.92512 −0.112852
\(292\) 6.94427 0.406383
\(293\) −3.22962 −0.188676 −0.0943382 0.995540i \(-0.530073\pi\)
−0.0943382 + 0.995540i \(0.530073\pi\)
\(294\) −4.50173 −0.262546
\(295\) −10.3262 −0.601217
\(296\) 9.73294 0.565716
\(297\) 0 0
\(298\) −1.64762 −0.0954443
\(299\) 2.06543 0.119447
\(300\) 1.00000 0.0577350
\(301\) 10.1341 0.584118
\(302\) 6.74823 0.388317
\(303\) −8.24092 −0.473429
\(304\) 1.32139 0.0757867
\(305\) −5.63332 −0.322563
\(306\) 7.17549 0.410195
\(307\) 22.9531 1.31000 0.655002 0.755627i \(-0.272668\pi\)
0.655002 + 0.755627i \(0.272668\pi\)
\(308\) 0 0
\(309\) −4.90843 −0.279231
\(310\) −7.19863 −0.408855
\(311\) 4.20121 0.238229 0.119114 0.992881i \(-0.461994\pi\)
0.119114 + 0.992881i \(0.461994\pi\)
\(312\) −5.75608 −0.325874
\(313\) −18.3816 −1.03899 −0.519493 0.854474i \(-0.673879\pi\)
−0.519493 + 0.854474i \(0.673879\pi\)
\(314\) 8.92898 0.503892
\(315\) 1.58059 0.0890563
\(316\) 11.5265 0.648414
\(317\) −25.3311 −1.42274 −0.711368 0.702820i \(-0.751924\pi\)
−0.711368 + 0.702820i \(0.751924\pi\)
\(318\) 2.34453 0.131474
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −1.64903 −0.0920396
\(322\) 0.567158 0.0316065
\(323\) 9.48158 0.527569
\(324\) 1.00000 0.0555556
\(325\) −5.75608 −0.319290
\(326\) 9.72251 0.538480
\(327\) −8.94427 −0.494619
\(328\) −12.0064 −0.662945
\(329\) −10.3055 −0.568160
\(330\) 0 0
\(331\) 26.1159 1.43546 0.717730 0.696322i \(-0.245181\pi\)
0.717730 + 0.696322i \(0.245181\pi\)
\(332\) 8.00000 0.439057
\(333\) 9.73294 0.533362
\(334\) −15.4643 −0.846168
\(335\) 5.07747 0.277412
\(336\) −1.58059 −0.0862284
\(337\) 18.2012 0.991483 0.495742 0.868470i \(-0.334896\pi\)
0.495742 + 0.868470i \(0.334896\pi\)
\(338\) 20.1325 1.09506
\(339\) 0.0374409 0.00203351
\(340\) −7.17549 −0.389145
\(341\) 0 0
\(342\) 1.32139 0.0714523
\(343\) 18.1795 0.981603
\(344\) −6.41156 −0.345688
\(345\) 0.358826 0.0193186
\(346\) 18.4037 0.989389
\(347\) 18.3816 0.986773 0.493387 0.869810i \(-0.335759\pi\)
0.493387 + 0.869810i \(0.335759\pi\)
\(348\) 9.67076 0.518407
\(349\) 29.2204 1.56413 0.782065 0.623197i \(-0.214166\pi\)
0.782065 + 0.623197i \(0.214166\pi\)
\(350\) −1.58059 −0.0844863
\(351\) −5.75608 −0.307237
\(352\) 0 0
\(353\) −4.41156 −0.234803 −0.117402 0.993085i \(-0.537457\pi\)
−0.117402 + 0.993085i \(0.537457\pi\)
\(354\) 10.3262 0.548833
\(355\) −9.11491 −0.483769
\(356\) 7.16764 0.379884
\(357\) −11.3415 −0.600257
\(358\) −18.5623 −0.981048
\(359\) −22.1803 −1.17063 −0.585317 0.810805i \(-0.699030\pi\)
−0.585317 + 0.810805i \(0.699030\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −17.2539 −0.908102
\(362\) 9.52786 0.500773
\(363\) 0 0
\(364\) 9.09802 0.476866
\(365\) −6.94427 −0.363480
\(366\) 5.63332 0.294458
\(367\) −7.49129 −0.391042 −0.195521 0.980700i \(-0.562640\pi\)
−0.195521 + 0.980700i \(0.562640\pi\)
\(368\) −0.358826 −0.0187051
\(369\) −12.0064 −0.625031
\(370\) −9.73294 −0.505991
\(371\) −3.70574 −0.192392
\(372\) 7.19863 0.373231
\(373\) 7.50173 0.388425 0.194212 0.980960i \(-0.437785\pi\)
0.194212 + 0.980960i \(0.437785\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 6.52001 0.336244
\(377\) −55.6657 −2.86693
\(378\) −1.58059 −0.0812969
\(379\) −22.6589 −1.16391 −0.581955 0.813221i \(-0.697712\pi\)
−0.581955 + 0.813221i \(0.697712\pi\)
\(380\) −1.32139 −0.0677856
\(381\) −16.5418 −0.847460
\(382\) 20.0306 1.02485
\(383\) −6.48742 −0.331492 −0.165746 0.986168i \(-0.553003\pi\)
−0.165746 + 0.986168i \(0.553003\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −7.82937 −0.398504
\(387\) −6.41156 −0.325918
\(388\) −1.92512 −0.0977331
\(389\) 21.5824 1.09427 0.547137 0.837043i \(-0.315718\pi\)
0.547137 + 0.837043i \(0.315718\pi\)
\(390\) 5.75608 0.291470
\(391\) −2.57475 −0.130211
\(392\) −4.50173 −0.227371
\(393\) −12.5096 −0.631025
\(394\) −20.6798 −1.04183
\(395\) −11.5265 −0.579959
\(396\) 0 0
\(397\) 16.5235 0.829289 0.414644 0.909984i \(-0.363906\pi\)
0.414644 + 0.909984i \(0.363906\pi\)
\(398\) −3.12921 −0.156853
\(399\) −2.08857 −0.104559
\(400\) 1.00000 0.0500000
\(401\) 14.5672 0.727449 0.363725 0.931507i \(-0.381505\pi\)
0.363725 + 0.931507i \(0.381505\pi\)
\(402\) −5.07747 −0.253241
\(403\) −41.4359 −2.06407
\(404\) −8.24092 −0.410001
\(405\) −1.00000 −0.0496904
\(406\) −15.2855 −0.758609
\(407\) 0 0
\(408\) 7.17549 0.355240
\(409\) −22.6287 −1.11892 −0.559458 0.828859i \(-0.688991\pi\)
−0.559458 + 0.828859i \(0.688991\pi\)
\(410\) 12.0064 0.592956
\(411\) −12.3135 −0.607382
\(412\) −4.90843 −0.241821
\(413\) −16.3216 −0.803132
\(414\) −0.358826 −0.0176354
\(415\) −8.00000 −0.392705
\(416\) −5.75608 −0.282215
\(417\) 12.9970 0.636466
\(418\) 0 0
\(419\) 5.41057 0.264324 0.132162 0.991228i \(-0.457808\pi\)
0.132162 + 0.991228i \(0.457808\pi\)
\(420\) 1.58059 0.0771250
\(421\) 3.48158 0.169682 0.0848410 0.996395i \(-0.472962\pi\)
0.0848410 + 0.996395i \(0.472962\pi\)
\(422\) −14.3047 −0.696342
\(423\) 6.52001 0.317014
\(424\) 2.34453 0.113860
\(425\) 7.17549 0.348062
\(426\) 9.11491 0.441619
\(427\) −8.90399 −0.430894
\(428\) −1.64903 −0.0797086
\(429\) 0 0
\(430\) 6.41156 0.309193
\(431\) −12.2169 −0.588468 −0.294234 0.955733i \(-0.595065\pi\)
−0.294234 + 0.955733i \(0.595065\pi\)
\(432\) 1.00000 0.0481125
\(433\) 9.48356 0.455751 0.227875 0.973690i \(-0.426822\pi\)
0.227875 + 0.973690i \(0.426822\pi\)
\(434\) −11.3781 −0.546166
\(435\) −9.67076 −0.463678
\(436\) −8.94427 −0.428353
\(437\) −0.474148 −0.0226816
\(438\) 6.94427 0.331810
\(439\) 7.54960 0.360323 0.180161 0.983637i \(-0.442338\pi\)
0.180161 + 0.983637i \(0.442338\pi\)
\(440\) 0 0
\(441\) −4.50173 −0.214368
\(442\) −41.3027 −1.96457
\(443\) −23.5713 −1.11991 −0.559954 0.828524i \(-0.689181\pi\)
−0.559954 + 0.828524i \(0.689181\pi\)
\(444\) 9.73294 0.461905
\(445\) −7.16764 −0.339779
\(446\) 5.58059 0.264249
\(447\) −1.64762 −0.0779299
\(448\) −1.58059 −0.0746760
\(449\) −1.98956 −0.0938933 −0.0469467 0.998897i \(-0.514949\pi\)
−0.0469467 + 0.998897i \(0.514949\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 0.0374409 0.00176107
\(453\) 6.74823 0.317059
\(454\) −4.68905 −0.220068
\(455\) −9.09802 −0.426522
\(456\) 1.32139 0.0618795
\(457\) −29.9634 −1.40163 −0.700815 0.713343i \(-0.747180\pi\)
−0.700815 + 0.713343i \(0.747180\pi\)
\(458\) 8.39725 0.392378
\(459\) 7.17549 0.334923
\(460\) 0.358826 0.0167304
\(461\) −12.0143 −0.559562 −0.279781 0.960064i \(-0.590262\pi\)
−0.279781 + 0.960064i \(0.590262\pi\)
\(462\) 0 0
\(463\) 4.85895 0.225815 0.112907 0.993606i \(-0.463984\pi\)
0.112907 + 0.993606i \(0.463984\pi\)
\(464\) 9.67076 0.448954
\(465\) −7.19863 −0.333828
\(466\) −7.27351 −0.336939
\(467\) −15.3878 −0.712063 −0.356031 0.934474i \(-0.615870\pi\)
−0.356031 + 0.934474i \(0.615870\pi\)
\(468\) −5.75608 −0.266075
\(469\) 8.02541 0.370579
\(470\) −6.52001 −0.300746
\(471\) 8.92898 0.411426
\(472\) 10.3262 0.475304
\(473\) 0 0
\(474\) 11.5265 0.529428
\(475\) 1.32139 0.0606293
\(476\) −11.3415 −0.519838
\(477\) 2.34453 0.107348
\(478\) −21.7883 −0.996572
\(479\) 3.62362 0.165567 0.0827836 0.996568i \(-0.473619\pi\)
0.0827836 + 0.996568i \(0.473619\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −56.0236 −2.55446
\(482\) −4.54057 −0.206817
\(483\) 0.567158 0.0258066
\(484\) 0 0
\(485\) 1.92512 0.0874151
\(486\) 1.00000 0.0453609
\(487\) 3.26664 0.148026 0.0740129 0.997257i \(-0.476419\pi\)
0.0740129 + 0.997257i \(0.476419\pi\)
\(488\) 5.63332 0.255009
\(489\) 9.72251 0.439667
\(490\) 4.50173 0.203367
\(491\) −34.2745 −1.54679 −0.773393 0.633927i \(-0.781442\pi\)
−0.773393 + 0.633927i \(0.781442\pi\)
\(492\) −12.0064 −0.541293
\(493\) 69.3924 3.12528
\(494\) −7.60600 −0.342210
\(495\) 0 0
\(496\) 7.19863 0.323228
\(497\) −14.4070 −0.646240
\(498\) 8.00000 0.358489
\(499\) 38.2077 1.71041 0.855205 0.518290i \(-0.173431\pi\)
0.855205 + 0.518290i \(0.173431\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −15.4643 −0.690893
\(502\) 4.20906 0.187860
\(503\) 20.9173 0.932655 0.466327 0.884612i \(-0.345577\pi\)
0.466327 + 0.884612i \(0.345577\pi\)
\(504\) −1.58059 −0.0704052
\(505\) 8.24092 0.366716
\(506\) 0 0
\(507\) 20.1325 0.894114
\(508\) −16.5418 −0.733921
\(509\) −34.6279 −1.53486 −0.767428 0.641135i \(-0.778464\pi\)
−0.767428 + 0.641135i \(0.778464\pi\)
\(510\) −7.17549 −0.317736
\(511\) −10.9761 −0.485553
\(512\) 1.00000 0.0441942
\(513\) 1.32139 0.0583406
\(514\) −14.7945 −0.652558
\(515\) 4.90843 0.216291
\(516\) −6.41156 −0.282253
\(517\) 0 0
\(518\) −15.3838 −0.675926
\(519\) 18.4037 0.807833
\(520\) 5.75608 0.252421
\(521\) −14.2077 −0.622449 −0.311224 0.950336i \(-0.600739\pi\)
−0.311224 + 0.950336i \(0.600739\pi\)
\(522\) 9.67076 0.423278
\(523\) 1.54076 0.0673729 0.0336864 0.999432i \(-0.489275\pi\)
0.0336864 + 0.999432i \(0.489275\pi\)
\(524\) −12.5096 −0.546483
\(525\) −1.58059 −0.0689827
\(526\) 7.90358 0.344612
\(527\) 51.6537 2.25007
\(528\) 0 0
\(529\) −22.8712 −0.994402
\(530\) −2.34453 −0.101840
\(531\) 10.3262 0.448121
\(532\) −2.08857 −0.0905511
\(533\) 69.1101 2.99349
\(534\) 7.16764 0.310174
\(535\) 1.64903 0.0712936
\(536\) −5.07747 −0.219313
\(537\) −18.5623 −0.801023
\(538\) 17.9469 0.773744
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 21.1644 0.909928 0.454964 0.890510i \(-0.349652\pi\)
0.454964 + 0.890510i \(0.349652\pi\)
\(542\) 5.86454 0.251903
\(543\) 9.52786 0.408880
\(544\) 7.17549 0.307646
\(545\) 8.94427 0.383131
\(546\) 9.09802 0.389359
\(547\) 25.8720 1.10621 0.553103 0.833113i \(-0.313444\pi\)
0.553103 + 0.833113i \(0.313444\pi\)
\(548\) −12.3135 −0.526008
\(549\) 5.63332 0.240424
\(550\) 0 0
\(551\) 12.7788 0.544395
\(552\) −0.358826 −0.0152727
\(553\) −18.2186 −0.774736
\(554\) −8.43371 −0.358314
\(555\) −9.73294 −0.413140
\(556\) 12.9970 0.551196
\(557\) 9.76811 0.413888 0.206944 0.978353i \(-0.433648\pi\)
0.206944 + 0.978353i \(0.433648\pi\)
\(558\) 7.19863 0.304742
\(559\) 36.9054 1.56093
\(560\) 1.58059 0.0667923
\(561\) 0 0
\(562\) −5.85024 −0.246777
\(563\) 21.9348 0.924443 0.462221 0.886765i \(-0.347053\pi\)
0.462221 + 0.886765i \(0.347053\pi\)
\(564\) 6.52001 0.274542
\(565\) −0.0374409 −0.00157515
\(566\) −11.0312 −0.463675
\(567\) −1.58059 −0.0663787
\(568\) 9.11491 0.382453
\(569\) 41.7166 1.74885 0.874426 0.485159i \(-0.161238\pi\)
0.874426 + 0.485159i \(0.161238\pi\)
\(570\) −1.32139 −0.0553467
\(571\) −23.0928 −0.966402 −0.483201 0.875510i \(-0.660526\pi\)
−0.483201 + 0.875510i \(0.660526\pi\)
\(572\) 0 0
\(573\) 20.0306 0.836789
\(574\) 18.9773 0.792098
\(575\) −0.358826 −0.0149641
\(576\) 1.00000 0.0416667
\(577\) −30.2552 −1.25954 −0.629771 0.776781i \(-0.716851\pi\)
−0.629771 + 0.776781i \(0.716851\pi\)
\(578\) 34.4876 1.43450
\(579\) −7.82937 −0.325377
\(580\) −9.67076 −0.401557
\(581\) −12.6447 −0.524592
\(582\) −1.92512 −0.0797987
\(583\) 0 0
\(584\) 6.94427 0.287356
\(585\) 5.75608 0.237985
\(586\) −3.22962 −0.133414
\(587\) 5.40670 0.223159 0.111579 0.993756i \(-0.464409\pi\)
0.111579 + 0.993756i \(0.464409\pi\)
\(588\) −4.50173 −0.185648
\(589\) 9.51216 0.391942
\(590\) −10.3262 −0.425124
\(591\) −20.6798 −0.850653
\(592\) 9.73294 0.400021
\(593\) −33.4413 −1.37327 −0.686636 0.727002i \(-0.740913\pi\)
−0.686636 + 0.727002i \(0.740913\pi\)
\(594\) 0 0
\(595\) 11.3415 0.464957
\(596\) −1.64762 −0.0674893
\(597\) −3.12921 −0.128070
\(598\) 2.06543 0.0844618
\(599\) 6.37638 0.260532 0.130266 0.991479i \(-0.458417\pi\)
0.130266 + 0.991479i \(0.458417\pi\)
\(600\) 1.00000 0.0408248
\(601\) −5.76294 −0.235075 −0.117538 0.993068i \(-0.537500\pi\)
−0.117538 + 0.993068i \(0.537500\pi\)
\(602\) 10.1341 0.413033
\(603\) −5.07747 −0.206770
\(604\) 6.74823 0.274582
\(605\) 0 0
\(606\) −8.24092 −0.334765
\(607\) 45.5427 1.84852 0.924261 0.381760i \(-0.124682\pi\)
0.924261 + 0.381760i \(0.124682\pi\)
\(608\) 1.32139 0.0535893
\(609\) −15.2855 −0.619401
\(610\) −5.63332 −0.228087
\(611\) −37.5297 −1.51829
\(612\) 7.17549 0.290052
\(613\) −31.1552 −1.25835 −0.629173 0.777265i \(-0.716606\pi\)
−0.629173 + 0.777265i \(0.716606\pi\)
\(614\) 22.9531 0.926312
\(615\) 12.0064 0.484147
\(616\) 0 0
\(617\) −30.2761 −1.21887 −0.609435 0.792836i \(-0.708604\pi\)
−0.609435 + 0.792836i \(0.708604\pi\)
\(618\) −4.90843 −0.197446
\(619\) −37.1225 −1.49208 −0.746040 0.665901i \(-0.768047\pi\)
−0.746040 + 0.665901i \(0.768047\pi\)
\(620\) −7.19863 −0.289104
\(621\) −0.358826 −0.0143992
\(622\) 4.20121 0.168453
\(623\) −11.3291 −0.453891
\(624\) −5.75608 −0.230428
\(625\) 1.00000 0.0400000
\(626\) −18.3816 −0.734675
\(627\) 0 0
\(628\) 8.92898 0.356305
\(629\) 69.8386 2.78465
\(630\) 1.58059 0.0629723
\(631\) −34.9837 −1.39268 −0.696339 0.717713i \(-0.745189\pi\)
−0.696339 + 0.717713i \(0.745189\pi\)
\(632\) 11.5265 0.458498
\(633\) −14.3047 −0.568561
\(634\) −25.3311 −1.00603
\(635\) 16.5418 0.656439
\(636\) 2.34453 0.0929665
\(637\) 25.9123 1.02668
\(638\) 0 0
\(639\) 9.11491 0.360580
\(640\) −1.00000 −0.0395285
\(641\) 7.87357 0.310987 0.155494 0.987837i \(-0.450303\pi\)
0.155494 + 0.987837i \(0.450303\pi\)
\(642\) −1.64903 −0.0650818
\(643\) 45.7923 1.80587 0.902936 0.429774i \(-0.141407\pi\)
0.902936 + 0.429774i \(0.141407\pi\)
\(644\) 0.567158 0.0223492
\(645\) 6.41156 0.252455
\(646\) 9.48158 0.373048
\(647\) −31.0543 −1.22087 −0.610436 0.792066i \(-0.709006\pi\)
−0.610436 + 0.792066i \(0.709006\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −5.75608 −0.225772
\(651\) −11.3781 −0.445943
\(652\) 9.72251 0.380763
\(653\) −22.9084 −0.896476 −0.448238 0.893914i \(-0.647948\pi\)
−0.448238 + 0.893914i \(0.647948\pi\)
\(654\) −8.94427 −0.349749
\(655\) 12.5096 0.488790
\(656\) −12.0064 −0.468773
\(657\) 6.94427 0.270922
\(658\) −10.3055 −0.401750
\(659\) −17.9907 −0.700820 −0.350410 0.936596i \(-0.613958\pi\)
−0.350410 + 0.936596i \(0.613958\pi\)
\(660\) 0 0
\(661\) −36.6576 −1.42582 −0.712909 0.701257i \(-0.752623\pi\)
−0.712909 + 0.701257i \(0.752623\pi\)
\(662\) 26.1159 1.01502
\(663\) −41.3027 −1.60406
\(664\) 8.00000 0.310460
\(665\) 2.08857 0.0809914
\(666\) 9.73294 0.377144
\(667\) −3.47012 −0.134364
\(668\) −15.4643 −0.598331
\(669\) 5.58059 0.215758
\(670\) 5.07747 0.196160
\(671\) 0 0
\(672\) −1.58059 −0.0609727
\(673\) −1.53106 −0.0590180 −0.0295090 0.999565i \(-0.509394\pi\)
−0.0295090 + 0.999565i \(0.509394\pi\)
\(674\) 18.2012 0.701084
\(675\) 1.00000 0.0384900
\(676\) 20.1325 0.774326
\(677\) 1.73361 0.0666281 0.0333141 0.999445i \(-0.489394\pi\)
0.0333141 + 0.999445i \(0.489394\pi\)
\(678\) 0.0374409 0.00143791
\(679\) 3.04283 0.116773
\(680\) −7.17549 −0.275167
\(681\) −4.68905 −0.179685
\(682\) 0 0
\(683\) −15.3161 −0.586055 −0.293027 0.956104i \(-0.594663\pi\)
−0.293027 + 0.956104i \(0.594663\pi\)
\(684\) 1.32139 0.0505244
\(685\) 12.3135 0.470476
\(686\) 18.1795 0.694098
\(687\) 8.39725 0.320375
\(688\) −6.41156 −0.244438
\(689\) −13.4953 −0.514129
\(690\) 0.358826 0.0136603
\(691\) 25.1487 0.956703 0.478352 0.878168i \(-0.341234\pi\)
0.478352 + 0.878168i \(0.341234\pi\)
\(692\) 18.4037 0.699604
\(693\) 0 0
\(694\) 18.3816 0.697754
\(695\) −12.9970 −0.493004
\(696\) 9.67076 0.366569
\(697\) −86.1521 −3.26324
\(698\) 29.2204 1.10601
\(699\) −7.27351 −0.275109
\(700\) −1.58059 −0.0597408
\(701\) 22.0129 0.831416 0.415708 0.909498i \(-0.363534\pi\)
0.415708 + 0.909498i \(0.363534\pi\)
\(702\) −5.75608 −0.217249
\(703\) 12.8610 0.485061
\(704\) 0 0
\(705\) −6.52001 −0.245558
\(706\) −4.41156 −0.166031
\(707\) 13.0255 0.489876
\(708\) 10.3262 0.388084
\(709\) 34.0129 1.27738 0.638691 0.769464i \(-0.279476\pi\)
0.638691 + 0.769464i \(0.279476\pi\)
\(710\) −9.11491 −0.342076
\(711\) 11.5265 0.432276
\(712\) 7.16764 0.268619
\(713\) −2.58306 −0.0967362
\(714\) −11.3415 −0.424446
\(715\) 0 0
\(716\) −18.5623 −0.693706
\(717\) −21.7883 −0.813697
\(718\) −22.1803 −0.827763
\(719\) 37.4450 1.39646 0.698232 0.715872i \(-0.253971\pi\)
0.698232 + 0.715872i \(0.253971\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 7.75823 0.288932
\(722\) −17.2539 −0.642125
\(723\) −4.54057 −0.168866
\(724\) 9.52786 0.354100
\(725\) 9.67076 0.359163
\(726\) 0 0
\(727\) −36.1119 −1.33932 −0.669658 0.742669i \(-0.733559\pi\)
−0.669658 + 0.742669i \(0.733559\pi\)
\(728\) 9.09802 0.337195
\(729\) 1.00000 0.0370370
\(730\) −6.94427 −0.257019
\(731\) −46.0060 −1.70159
\(732\) 5.63332 0.208214
\(733\) 17.8774 0.660318 0.330159 0.943925i \(-0.392898\pi\)
0.330159 + 0.943925i \(0.392898\pi\)
\(734\) −7.49129 −0.276509
\(735\) 4.50173 0.166049
\(736\) −0.358826 −0.0132265
\(737\) 0 0
\(738\) −12.0064 −0.441964
\(739\) 29.8358 1.09753 0.548764 0.835977i \(-0.315099\pi\)
0.548764 + 0.835977i \(0.315099\pi\)
\(740\) −9.73294 −0.357790
\(741\) −7.60600 −0.279413
\(742\) −3.70574 −0.136042
\(743\) 22.1285 0.811815 0.405908 0.913914i \(-0.366955\pi\)
0.405908 + 0.913914i \(0.366955\pi\)
\(744\) 7.19863 0.263914
\(745\) 1.64762 0.0603642
\(746\) 7.50173 0.274658
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 2.60644 0.0952372
\(750\) −1.00000 −0.0365148
\(751\) −13.9334 −0.508438 −0.254219 0.967147i \(-0.581818\pi\)
−0.254219 + 0.967147i \(0.581818\pi\)
\(752\) 6.52001 0.237760
\(753\) 4.20906 0.153387
\(754\) −55.6657 −2.02722
\(755\) −6.74823 −0.245593
\(756\) −1.58059 −0.0574856
\(757\) −47.8361 −1.73863 −0.869317 0.494255i \(-0.835441\pi\)
−0.869317 + 0.494255i \(0.835441\pi\)
\(758\) −22.6589 −0.823009
\(759\) 0 0
\(760\) −1.32139 −0.0479317
\(761\) −30.7019 −1.11294 −0.556472 0.830866i \(-0.687846\pi\)
−0.556472 + 0.830866i \(0.687846\pi\)
\(762\) −16.5418 −0.599244
\(763\) 14.1373 0.511803
\(764\) 20.0306 0.724681
\(765\) −7.17549 −0.259430
\(766\) −6.48742 −0.234400
\(767\) −59.4387 −2.14621
\(768\) 1.00000 0.0360844
\(769\) −7.23993 −0.261079 −0.130539 0.991443i \(-0.541671\pi\)
−0.130539 + 0.991443i \(0.541671\pi\)
\(770\) 0 0
\(771\) −14.7945 −0.532811
\(772\) −7.82937 −0.281785
\(773\) 52.3755 1.88382 0.941908 0.335872i \(-0.109031\pi\)
0.941908 + 0.335872i \(0.109031\pi\)
\(774\) −6.41156 −0.230459
\(775\) 7.19863 0.258582
\(776\) −1.92512 −0.0691077
\(777\) −15.3838 −0.551891
\(778\) 21.5824 0.773768
\(779\) −15.8651 −0.568428
\(780\) 5.75608 0.206101
\(781\) 0 0
\(782\) −2.57475 −0.0920730
\(783\) 9.67076 0.345605
\(784\) −4.50173 −0.160776
\(785\) −8.92898 −0.318689
\(786\) −12.5096 −0.446202
\(787\) 40.2155 1.43353 0.716764 0.697316i \(-0.245622\pi\)
0.716764 + 0.697316i \(0.245622\pi\)
\(788\) −20.6798 −0.736687
\(789\) 7.90358 0.281375
\(790\) −11.5265 −0.410093
\(791\) −0.0591788 −0.00210415
\(792\) 0 0
\(793\) −32.4259 −1.15148
\(794\) 16.5235 0.586396
\(795\) −2.34453 −0.0831517
\(796\) −3.12921 −0.110912
\(797\) 8.24479 0.292045 0.146023 0.989281i \(-0.453353\pi\)
0.146023 + 0.989281i \(0.453353\pi\)
\(798\) −2.08857 −0.0739347
\(799\) 46.7843 1.65511
\(800\) 1.00000 0.0353553
\(801\) 7.16764 0.253256
\(802\) 14.5672 0.514384
\(803\) 0 0
\(804\) −5.07747 −0.179068
\(805\) −0.567158 −0.0199897
\(806\) −41.4359 −1.45952
\(807\) 17.9469 0.631759
\(808\) −8.24092 −0.289915
\(809\) 0.436292 0.0153392 0.00766961 0.999971i \(-0.497559\pi\)
0.00766961 + 0.999971i \(0.497559\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −8.03703 −0.282218 −0.141109 0.989994i \(-0.545067\pi\)
−0.141109 + 0.989994i \(0.545067\pi\)
\(812\) −15.2855 −0.536417
\(813\) 5.86454 0.205678
\(814\) 0 0
\(815\) −9.72251 −0.340564
\(816\) 7.17549 0.251192
\(817\) −8.47214 −0.296403
\(818\) −22.6287 −0.791193
\(819\) 9.09802 0.317911
\(820\) 12.0064 0.419283
\(821\) −10.8726 −0.379456 −0.189728 0.981837i \(-0.560761\pi\)
−0.189728 + 0.981837i \(0.560761\pi\)
\(822\) −12.3135 −0.429484
\(823\) 6.87466 0.239635 0.119818 0.992796i \(-0.461769\pi\)
0.119818 + 0.992796i \(0.461769\pi\)
\(824\) −4.90843 −0.170993
\(825\) 0 0
\(826\) −16.3216 −0.567900
\(827\) −7.40070 −0.257348 −0.128674 0.991687i \(-0.541072\pi\)
−0.128674 + 0.991687i \(0.541072\pi\)
\(828\) −0.358826 −0.0124701
\(829\) −2.50871 −0.0871311 −0.0435656 0.999051i \(-0.513872\pi\)
−0.0435656 + 0.999051i \(0.513872\pi\)
\(830\) −8.00000 −0.277684
\(831\) −8.43371 −0.292562
\(832\) −5.75608 −0.199556
\(833\) −32.3021 −1.11920
\(834\) 12.9970 0.450049
\(835\) 15.4643 0.535164
\(836\) 0 0
\(837\) 7.19863 0.248821
\(838\) 5.41057 0.186905
\(839\) −10.7450 −0.370960 −0.185480 0.982648i \(-0.559384\pi\)
−0.185480 + 0.982648i \(0.559384\pi\)
\(840\) 1.58059 0.0545356
\(841\) 64.5237 2.22495
\(842\) 3.48158 0.119983
\(843\) −5.85024 −0.201493
\(844\) −14.3047 −0.492388
\(845\) −20.1325 −0.692578
\(846\) 6.52001 0.224163
\(847\) 0 0
\(848\) 2.34453 0.0805113
\(849\) −11.0312 −0.378589
\(850\) 7.17549 0.246117
\(851\) −3.49244 −0.119719
\(852\) 9.11491 0.312272
\(853\) −7.70175 −0.263703 −0.131852 0.991269i \(-0.542092\pi\)
−0.131852 + 0.991269i \(0.542092\pi\)
\(854\) −8.90399 −0.304688
\(855\) −1.32139 −0.0451904
\(856\) −1.64903 −0.0563625
\(857\) 7.18839 0.245551 0.122775 0.992434i \(-0.460821\pi\)
0.122775 + 0.992434i \(0.460821\pi\)
\(858\) 0 0
\(859\) 40.4661 1.38068 0.690342 0.723483i \(-0.257460\pi\)
0.690342 + 0.723483i \(0.257460\pi\)
\(860\) 6.41156 0.218632
\(861\) 18.9773 0.646745
\(862\) −12.2169 −0.416110
\(863\) −49.6424 −1.68985 −0.844923 0.534888i \(-0.820354\pi\)
−0.844923 + 0.534888i \(0.820354\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.4037 −0.625745
\(866\) 9.48356 0.322264
\(867\) 34.4876 1.17126
\(868\) −11.3781 −0.386198
\(869\) 0 0
\(870\) −9.67076 −0.327870
\(871\) 29.2263 0.990296
\(872\) −8.94427 −0.302891
\(873\) −1.92512 −0.0651554
\(874\) −0.474148 −0.0160383
\(875\) 1.58059 0.0534338
\(876\) 6.94427 0.234625
\(877\) −9.08046 −0.306626 −0.153313 0.988178i \(-0.548994\pi\)
−0.153313 + 0.988178i \(0.548994\pi\)
\(878\) 7.54960 0.254787
\(879\) −3.22962 −0.108932
\(880\) 0 0
\(881\) −9.85152 −0.331906 −0.165953 0.986134i \(-0.553070\pi\)
−0.165953 + 0.986134i \(0.553070\pi\)
\(882\) −4.50173 −0.151581
\(883\) −18.5456 −0.624110 −0.312055 0.950064i \(-0.601017\pi\)
−0.312055 + 0.950064i \(0.601017\pi\)
\(884\) −41.3027 −1.38916
\(885\) −10.3262 −0.347113
\(886\) −23.5713 −0.791895
\(887\) −22.6172 −0.759413 −0.379706 0.925107i \(-0.623975\pi\)
−0.379706 + 0.925107i \(0.623975\pi\)
\(888\) 9.73294 0.326616
\(889\) 26.1458 0.876901
\(890\) −7.16764 −0.240260
\(891\) 0 0
\(892\) 5.58059 0.186852
\(893\) 8.61545 0.288305
\(894\) −1.64762 −0.0551048
\(895\) 18.5623 0.620469
\(896\) −1.58059 −0.0528039
\(897\) 2.06543 0.0689628
\(898\) −1.98956 −0.0663926
\(899\) 69.6162 2.32183
\(900\) 1.00000 0.0333333
\(901\) 16.8231 0.560459
\(902\) 0 0
\(903\) 10.1341 0.337240
\(904\) 0.0374409 0.00124526
\(905\) −9.52786 −0.316717
\(906\) 6.74823 0.224195
\(907\) −34.4273 −1.14314 −0.571569 0.820554i \(-0.693665\pi\)
−0.571569 + 0.820554i \(0.693665\pi\)
\(908\) −4.68905 −0.155612
\(909\) −8.24092 −0.273334
\(910\) −9.09802 −0.301596
\(911\) 2.38155 0.0789043 0.0394522 0.999221i \(-0.487439\pi\)
0.0394522 + 0.999221i \(0.487439\pi\)
\(912\) 1.32139 0.0437554
\(913\) 0 0
\(914\) −29.9634 −0.991102
\(915\) −5.63332 −0.186232
\(916\) 8.39725 0.277453
\(917\) 19.7726 0.652947
\(918\) 7.17549 0.236826
\(919\) −7.11351 −0.234653 −0.117326 0.993093i \(-0.537432\pi\)
−0.117326 + 0.993093i \(0.537432\pi\)
\(920\) 0.358826 0.0118302
\(921\) 22.9531 0.756331
\(922\) −12.0143 −0.395670
\(923\) −52.4661 −1.72694
\(924\) 0 0
\(925\) 9.73294 0.320017
\(926\) 4.85895 0.159675
\(927\) −4.90843 −0.161214
\(928\) 9.67076 0.317458
\(929\) 6.54057 0.214589 0.107295 0.994227i \(-0.465781\pi\)
0.107295 + 0.994227i \(0.465781\pi\)
\(930\) −7.19863 −0.236052
\(931\) −5.94851 −0.194955
\(932\) −7.27351 −0.238252
\(933\) 4.20121 0.137541
\(934\) −15.3878 −0.503504
\(935\) 0 0
\(936\) −5.75608 −0.188143
\(937\) −36.8231 −1.20296 −0.601479 0.798888i \(-0.705422\pi\)
−0.601479 + 0.798888i \(0.705422\pi\)
\(938\) 8.02541 0.262039
\(939\) −18.3816 −0.599859
\(940\) −6.52001 −0.212659
\(941\) 9.59908 0.312921 0.156460 0.987684i \(-0.449992\pi\)
0.156460 + 0.987684i \(0.449992\pi\)
\(942\) 8.92898 0.290922
\(943\) 4.30823 0.140295
\(944\) 10.3262 0.336090
\(945\) 1.58059 0.0514167
\(946\) 0 0
\(947\) −35.5427 −1.15498 −0.577492 0.816396i \(-0.695968\pi\)
−0.577492 + 0.816396i \(0.695968\pi\)
\(948\) 11.5265 0.374362
\(949\) −39.9718 −1.29754
\(950\) 1.32139 0.0428714
\(951\) −25.3311 −0.821417
\(952\) −11.3415 −0.367581
\(953\) −30.5476 −0.989534 −0.494767 0.869026i \(-0.664747\pi\)
−0.494767 + 0.869026i \(0.664747\pi\)
\(954\) 2.34453 0.0759068
\(955\) −20.0306 −0.648174
\(956\) −21.7883 −0.704683
\(957\) 0 0
\(958\) 3.62362 0.117074
\(959\) 19.4627 0.628483
\(960\) −1.00000 −0.0322749
\(961\) 20.8202 0.671620
\(962\) −56.0236 −1.80627
\(963\) −1.64903 −0.0531391
\(964\) −4.54057 −0.146242
\(965\) 7.82937 0.252036
\(966\) 0.567158 0.0182480
\(967\) 22.9241 0.737190 0.368595 0.929590i \(-0.379839\pi\)
0.368595 + 0.929590i \(0.379839\pi\)
\(968\) 0 0
\(969\) 9.48158 0.304592
\(970\) 1.92512 0.0618118
\(971\) −52.1763 −1.67442 −0.837208 0.546885i \(-0.815814\pi\)
−0.837208 + 0.546885i \(0.815814\pi\)
\(972\) 1.00000 0.0320750
\(973\) −20.5430 −0.658577
\(974\) 3.26664 0.104670
\(975\) −5.75608 −0.184342
\(976\) 5.63332 0.180318
\(977\) 7.49646 0.239833 0.119916 0.992784i \(-0.461737\pi\)
0.119916 + 0.992784i \(0.461737\pi\)
\(978\) 9.72251 0.310891
\(979\) 0 0
\(980\) 4.50173 0.143802
\(981\) −8.94427 −0.285569
\(982\) −34.2745 −1.09374
\(983\) −57.8539 −1.84525 −0.922626 0.385695i \(-0.873962\pi\)
−0.922626 + 0.385695i \(0.873962\pi\)
\(984\) −12.0064 −0.382752
\(985\) 20.6798 0.658913
\(986\) 69.3924 2.20991
\(987\) −10.3055 −0.328027
\(988\) −7.60600 −0.241979
\(989\) 2.30063 0.0731559
\(990\) 0 0
\(991\) 15.1292 0.480595 0.240298 0.970699i \(-0.422755\pi\)
0.240298 + 0.970699i \(0.422755\pi\)
\(992\) 7.19863 0.228557
\(993\) 26.1159 0.828763
\(994\) −14.4070 −0.456961
\(995\) 3.12921 0.0992026
\(996\) 8.00000 0.253490
\(997\) 20.9268 0.662758 0.331379 0.943498i \(-0.392486\pi\)
0.331379 + 0.943498i \(0.392486\pi\)
\(998\) 38.2077 1.20944
\(999\) 9.73294 0.307937
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 3630.2.a.bs.1.2 4
11.7 odd 10 330.2.m.f.181.1 yes 8
11.8 odd 10 330.2.m.f.31.1 8
11.10 odd 2 3630.2.a.bq.1.3 4
33.8 even 10 990.2.n.i.361.1 8
33.29 even 10 990.2.n.i.181.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.m.f.31.1 8 11.8 odd 10
330.2.m.f.181.1 yes 8 11.7 odd 10
990.2.n.i.181.1 8 33.29 even 10
990.2.n.i.361.1 8 33.8 even 10
3630.2.a.bq.1.3 4 11.10 odd 2
3630.2.a.bs.1.2 4 1.1 even 1 trivial