Properties

Label 3630.2.a.bp.1.1
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +0.267949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +0.267949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +3.00000 q^{13} +0.267949 q^{14} +1.00000 q^{15} +1.00000 q^{16} +3.73205 q^{17} +1.00000 q^{18} +2.46410 q^{19} +1.00000 q^{20} +0.267949 q^{21} -1.26795 q^{23} +1.00000 q^{24} +1.00000 q^{25} +3.00000 q^{26} +1.00000 q^{27} +0.267949 q^{28} -3.00000 q^{29} +1.00000 q^{30} -2.19615 q^{31} +1.00000 q^{32} +3.73205 q^{34} +0.267949 q^{35} +1.00000 q^{36} +1.73205 q^{37} +2.46410 q^{38} +3.00000 q^{39} +1.00000 q^{40} +1.26795 q^{41} +0.267949 q^{42} +3.26795 q^{43} +1.00000 q^{45} -1.26795 q^{46} -4.92820 q^{47} +1.00000 q^{48} -6.92820 q^{49} +1.00000 q^{50} +3.73205 q^{51} +3.00000 q^{52} -8.73205 q^{53} +1.00000 q^{54} +0.267949 q^{56} +2.46410 q^{57} -3.00000 q^{58} +6.19615 q^{59} +1.00000 q^{60} +8.19615 q^{61} -2.19615 q^{62} +0.267949 q^{63} +1.00000 q^{64} +3.00000 q^{65} -4.53590 q^{67} +3.73205 q^{68} -1.26795 q^{69} +0.267949 q^{70} +3.92820 q^{71} +1.00000 q^{72} -4.19615 q^{73} +1.73205 q^{74} +1.00000 q^{75} +2.46410 q^{76} +3.00000 q^{78} +7.46410 q^{79} +1.00000 q^{80} +1.00000 q^{81} +1.26795 q^{82} +9.39230 q^{83} +0.267949 q^{84} +3.73205 q^{85} +3.26795 q^{86} -3.00000 q^{87} +4.39230 q^{89} +1.00000 q^{90} +0.803848 q^{91} -1.26795 q^{92} -2.19615 q^{93} -4.92820 q^{94} +2.46410 q^{95} +1.00000 q^{96} -15.2679 q^{97} -6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 6q^{13} + 4q^{14} + 2q^{15} + 2q^{16} + 4q^{17} + 2q^{18} - 2q^{19} + 2q^{20} + 4q^{21} - 6q^{23} + 2q^{24} + 2q^{25} + 6q^{26} + 2q^{27} + 4q^{28} - 6q^{29} + 2q^{30} + 6q^{31} + 2q^{32} + 4q^{34} + 4q^{35} + 2q^{36} - 2q^{38} + 6q^{39} + 2q^{40} + 6q^{41} + 4q^{42} + 10q^{43} + 2q^{45} - 6q^{46} + 4q^{47} + 2q^{48} + 2q^{50} + 4q^{51} + 6q^{52} - 14q^{53} + 2q^{54} + 4q^{56} - 2q^{57} - 6q^{58} + 2q^{59} + 2q^{60} + 6q^{61} + 6q^{62} + 4q^{63} + 2q^{64} + 6q^{65} - 16q^{67} + 4q^{68} - 6q^{69} + 4q^{70} - 6q^{71} + 2q^{72} + 2q^{73} + 2q^{75} - 2q^{76} + 6q^{78} + 8q^{79} + 2q^{80} + 2q^{81} + 6q^{82} - 2q^{83} + 4q^{84} + 4q^{85} + 10q^{86} - 6q^{87} - 12q^{89} + 2q^{90} + 12q^{91} - 6q^{92} + 6q^{93} + 4q^{94} - 2q^{95} + 2q^{96} - 34q^{97} + O(q^{100}) \)

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
\(7\) 0.267949 0.101275 0.0506376 0.998717i \(-0.483875\pi\)
0.0506376 + 0.998717i \(0.483875\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0.267949 0.0716124
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 3.73205 0.905155 0.452578 0.891725i \(-0.350505\pi\)
0.452578 + 0.891725i \(0.350505\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.46410 0.565304 0.282652 0.959223i \(-0.408786\pi\)
0.282652 + 0.959223i \(0.408786\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.267949 0.0584713
\(22\) 0 0
\(23\) −1.26795 −0.264386 −0.132193 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 3.00000 0.588348
\(27\) 1.00000 0.192450
\(28\) 0.267949 0.0506376
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 1.00000 0.182574
\(31\) −2.19615 −0.394441 −0.197220 0.980359i \(-0.563191\pi\)
−0.197220 + 0.980359i \(0.563191\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.73205 0.640041
\(35\) 0.267949 0.0452917
\(36\) 1.00000 0.166667
\(37\) 1.73205 0.284747 0.142374 0.989813i \(-0.454527\pi\)
0.142374 + 0.989813i \(0.454527\pi\)
\(38\) 2.46410 0.399730
\(39\) 3.00000 0.480384
\(40\) 1.00000 0.158114
\(41\) 1.26795 0.198020 0.0990102 0.995086i \(-0.468432\pi\)
0.0990102 + 0.995086i \(0.468432\pi\)
\(42\) 0.267949 0.0413455
\(43\) 3.26795 0.498358 0.249179 0.968458i \(-0.419839\pi\)
0.249179 + 0.968458i \(0.419839\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −1.26795 −0.186949
\(47\) −4.92820 −0.718852 −0.359426 0.933174i \(-0.617028\pi\)
−0.359426 + 0.933174i \(0.617028\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.92820 −0.989743
\(50\) 1.00000 0.141421
\(51\) 3.73205 0.522592
\(52\) 3.00000 0.416025
\(53\) −8.73205 −1.19944 −0.599720 0.800210i \(-0.704721\pi\)
−0.599720 + 0.800210i \(0.704721\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0.267949 0.0358062
\(57\) 2.46410 0.326378
\(58\) −3.00000 −0.393919
\(59\) 6.19615 0.806670 0.403335 0.915052i \(-0.367851\pi\)
0.403335 + 0.915052i \(0.367851\pi\)
\(60\) 1.00000 0.129099
\(61\) 8.19615 1.04941 0.524705 0.851284i \(-0.324176\pi\)
0.524705 + 0.851284i \(0.324176\pi\)
\(62\) −2.19615 −0.278912
\(63\) 0.267949 0.0337584
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −4.53590 −0.554148 −0.277074 0.960849i \(-0.589365\pi\)
−0.277074 + 0.960849i \(0.589365\pi\)
\(68\) 3.73205 0.452578
\(69\) −1.26795 −0.152643
\(70\) 0.267949 0.0320261
\(71\) 3.92820 0.466192 0.233096 0.972454i \(-0.425114\pi\)
0.233096 + 0.972454i \(0.425114\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.19615 −0.491122 −0.245561 0.969381i \(-0.578972\pi\)
−0.245561 + 0.969381i \(0.578972\pi\)
\(74\) 1.73205 0.201347
\(75\) 1.00000 0.115470
\(76\) 2.46410 0.282652
\(77\) 0 0
\(78\) 3.00000 0.339683
\(79\) 7.46410 0.839777 0.419889 0.907576i \(-0.362069\pi\)
0.419889 + 0.907576i \(0.362069\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 1.26795 0.140022
\(83\) 9.39230 1.03094 0.515470 0.856908i \(-0.327617\pi\)
0.515470 + 0.856908i \(0.327617\pi\)
\(84\) 0.267949 0.0292357
\(85\) 3.73205 0.404798
\(86\) 3.26795 0.352392
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 4.39230 0.465583 0.232792 0.972527i \(-0.425214\pi\)
0.232792 + 0.972527i \(0.425214\pi\)
\(90\) 1.00000 0.105409
\(91\) 0.803848 0.0842661
\(92\) −1.26795 −0.132193
\(93\) −2.19615 −0.227730
\(94\) −4.92820 −0.508305
\(95\) 2.46410 0.252811
\(96\) 1.00000 0.102062
\(97\) −15.2679 −1.55023 −0.775113 0.631823i \(-0.782307\pi\)
−0.775113 + 0.631823i \(0.782307\pi\)
\(98\) −6.92820 −0.699854
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −12.4641 −1.24022 −0.620112 0.784513i \(-0.712913\pi\)
−0.620112 + 0.784513i \(0.712913\pi\)
\(102\) 3.73205 0.369528
\(103\) 16.3205 1.60811 0.804054 0.594557i \(-0.202672\pi\)
0.804054 + 0.594557i \(0.202672\pi\)
\(104\) 3.00000 0.294174
\(105\) 0.267949 0.0261492
\(106\) −8.73205 −0.848132
\(107\) 4.53590 0.438502 0.219251 0.975669i \(-0.429639\pi\)
0.219251 + 0.975669i \(0.429639\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.53590 0.626026 0.313013 0.949749i \(-0.398662\pi\)
0.313013 + 0.949749i \(0.398662\pi\)
\(110\) 0 0
\(111\) 1.73205 0.164399
\(112\) 0.267949 0.0253188
\(113\) −13.8564 −1.30350 −0.651751 0.758433i \(-0.725965\pi\)
−0.651751 + 0.758433i \(0.725965\pi\)
\(114\) 2.46410 0.230784
\(115\) −1.26795 −0.118237
\(116\) −3.00000 −0.278543
\(117\) 3.00000 0.277350
\(118\) 6.19615 0.570402
\(119\) 1.00000 0.0916698
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) 8.19615 0.742045
\(123\) 1.26795 0.114327
\(124\) −2.19615 −0.197220
\(125\) 1.00000 0.0894427
\(126\) 0.267949 0.0238708
\(127\) 20.3923 1.80952 0.904762 0.425917i \(-0.140048\pi\)
0.904762 + 0.425917i \(0.140048\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.26795 0.287727
\(130\) 3.00000 0.263117
\(131\) −5.66025 −0.494539 −0.247269 0.968947i \(-0.579533\pi\)
−0.247269 + 0.968947i \(0.579533\pi\)
\(132\) 0 0
\(133\) 0.660254 0.0572513
\(134\) −4.53590 −0.391842
\(135\) 1.00000 0.0860663
\(136\) 3.73205 0.320021
\(137\) 21.7846 1.86118 0.930592 0.366057i \(-0.119293\pi\)
0.930592 + 0.366057i \(0.119293\pi\)
\(138\) −1.26795 −0.107935
\(139\) −14.3205 −1.21465 −0.607325 0.794454i \(-0.707757\pi\)
−0.607325 + 0.794454i \(0.707757\pi\)
\(140\) 0.267949 0.0226458
\(141\) −4.92820 −0.415030
\(142\) 3.92820 0.329647
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −3.00000 −0.249136
\(146\) −4.19615 −0.347276
\(147\) −6.92820 −0.571429
\(148\) 1.73205 0.142374
\(149\) 20.3923 1.67060 0.835301 0.549792i \(-0.185293\pi\)
0.835301 + 0.549792i \(0.185293\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.7321 1.36163 0.680817 0.732453i \(-0.261625\pi\)
0.680817 + 0.732453i \(0.261625\pi\)
\(152\) 2.46410 0.199865
\(153\) 3.73205 0.301718
\(154\) 0 0
\(155\) −2.19615 −0.176399
\(156\) 3.00000 0.240192
\(157\) 2.80385 0.223771 0.111886 0.993721i \(-0.464311\pi\)
0.111886 + 0.993721i \(0.464311\pi\)
\(158\) 7.46410 0.593812
\(159\) −8.73205 −0.692497
\(160\) 1.00000 0.0790569
\(161\) −0.339746 −0.0267757
\(162\) 1.00000 0.0785674
\(163\) −20.0526 −1.57064 −0.785319 0.619092i \(-0.787501\pi\)
−0.785319 + 0.619092i \(0.787501\pi\)
\(164\) 1.26795 0.0990102
\(165\) 0 0
\(166\) 9.39230 0.728984
\(167\) −14.5359 −1.12482 −0.562411 0.826858i \(-0.690126\pi\)
−0.562411 + 0.826858i \(0.690126\pi\)
\(168\) 0.267949 0.0206727
\(169\) −4.00000 −0.307692
\(170\) 3.73205 0.286235
\(171\) 2.46410 0.188435
\(172\) 3.26795 0.249179
\(173\) −16.5885 −1.26120 −0.630599 0.776109i \(-0.717191\pi\)
−0.630599 + 0.776109i \(0.717191\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0.267949 0.0202551
\(176\) 0 0
\(177\) 6.19615 0.465731
\(178\) 4.39230 0.329217
\(179\) −17.1244 −1.27993 −0.639967 0.768402i \(-0.721052\pi\)
−0.639967 + 0.768402i \(0.721052\pi\)
\(180\) 1.00000 0.0745356
\(181\) −6.19615 −0.460556 −0.230278 0.973125i \(-0.573964\pi\)
−0.230278 + 0.973125i \(0.573964\pi\)
\(182\) 0.803848 0.0595851
\(183\) 8.19615 0.605877
\(184\) −1.26795 −0.0934745
\(185\) 1.73205 0.127343
\(186\) −2.19615 −0.161030
\(187\) 0 0
\(188\) −4.92820 −0.359426
\(189\) 0.267949 0.0194904
\(190\) 2.46410 0.178765
\(191\) 11.5359 0.834708 0.417354 0.908744i \(-0.362957\pi\)
0.417354 + 0.908744i \(0.362957\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.53590 0.182538 0.0912690 0.995826i \(-0.470908\pi\)
0.0912690 + 0.995826i \(0.470908\pi\)
\(194\) −15.2679 −1.09617
\(195\) 3.00000 0.214834
\(196\) −6.92820 −0.494872
\(197\) −6.53590 −0.465663 −0.232832 0.972517i \(-0.574799\pi\)
−0.232832 + 0.972517i \(0.574799\pi\)
\(198\) 0 0
\(199\) 4.39230 0.311362 0.155681 0.987807i \(-0.450243\pi\)
0.155681 + 0.987807i \(0.450243\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.53590 −0.319938
\(202\) −12.4641 −0.876971
\(203\) −0.803848 −0.0564190
\(204\) 3.73205 0.261296
\(205\) 1.26795 0.0885574
\(206\) 16.3205 1.13710
\(207\) −1.26795 −0.0881286
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) 0.267949 0.0184903
\(211\) −23.7846 −1.63740 −0.818700 0.574221i \(-0.805305\pi\)
−0.818700 + 0.574221i \(0.805305\pi\)
\(212\) −8.73205 −0.599720
\(213\) 3.92820 0.269156
\(214\) 4.53590 0.310068
\(215\) 3.26795 0.222872
\(216\) 1.00000 0.0680414
\(217\) −0.588457 −0.0399471
\(218\) 6.53590 0.442667
\(219\) −4.19615 −0.283550
\(220\) 0 0
\(221\) 11.1962 0.753135
\(222\) 1.73205 0.116248
\(223\) −0.464102 −0.0310785 −0.0155393 0.999879i \(-0.504947\pi\)
−0.0155393 + 0.999879i \(0.504947\pi\)
\(224\) 0.267949 0.0179031
\(225\) 1.00000 0.0666667
\(226\) −13.8564 −0.921714
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) 2.46410 0.163189
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −1.26795 −0.0836061
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) −25.8564 −1.69391 −0.846955 0.531665i \(-0.821567\pi\)
−0.846955 + 0.531665i \(0.821567\pi\)
\(234\) 3.00000 0.196116
\(235\) −4.92820 −0.321481
\(236\) 6.19615 0.403335
\(237\) 7.46410 0.484846
\(238\) 1.00000 0.0648204
\(239\) −9.73205 −0.629514 −0.314757 0.949172i \(-0.601923\pi\)
−0.314757 + 0.949172i \(0.601923\pi\)
\(240\) 1.00000 0.0645497
\(241\) −5.33975 −0.343963 −0.171982 0.985100i \(-0.555017\pi\)
−0.171982 + 0.985100i \(0.555017\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 8.19615 0.524705
\(245\) −6.92820 −0.442627
\(246\) 1.26795 0.0808415
\(247\) 7.39230 0.470361
\(248\) −2.19615 −0.139456
\(249\) 9.39230 0.595213
\(250\) 1.00000 0.0632456
\(251\) 25.4641 1.60728 0.803640 0.595116i \(-0.202894\pi\)
0.803640 + 0.595116i \(0.202894\pi\)
\(252\) 0.267949 0.0168792
\(253\) 0 0
\(254\) 20.3923 1.27953
\(255\) 3.73205 0.233710
\(256\) 1.00000 0.0625000
\(257\) −28.3205 −1.76658 −0.883292 0.468823i \(-0.844678\pi\)
−0.883292 + 0.468823i \(0.844678\pi\)
\(258\) 3.26795 0.203454
\(259\) 0.464102 0.0288379
\(260\) 3.00000 0.186052
\(261\) −3.00000 −0.185695
\(262\) −5.66025 −0.349692
\(263\) −0.875644 −0.0539945 −0.0269973 0.999636i \(-0.508595\pi\)
−0.0269973 + 0.999636i \(0.508595\pi\)
\(264\) 0 0
\(265\) −8.73205 −0.536406
\(266\) 0.660254 0.0404828
\(267\) 4.39230 0.268805
\(268\) −4.53590 −0.277074
\(269\) −19.5885 −1.19433 −0.597165 0.802119i \(-0.703706\pi\)
−0.597165 + 0.802119i \(0.703706\pi\)
\(270\) 1.00000 0.0608581
\(271\) 3.85641 0.234260 0.117130 0.993117i \(-0.462631\pi\)
0.117130 + 0.993117i \(0.462631\pi\)
\(272\) 3.73205 0.226289
\(273\) 0.803848 0.0486511
\(274\) 21.7846 1.31606
\(275\) 0 0
\(276\) −1.26795 −0.0763216
\(277\) 5.07180 0.304735 0.152367 0.988324i \(-0.451310\pi\)
0.152367 + 0.988324i \(0.451310\pi\)
\(278\) −14.3205 −0.858887
\(279\) −2.19615 −0.131480
\(280\) 0.267949 0.0160130
\(281\) −12.0526 −0.718995 −0.359498 0.933146i \(-0.617052\pi\)
−0.359498 + 0.933146i \(0.617052\pi\)
\(282\) −4.92820 −0.293470
\(283\) 18.2487 1.08477 0.542387 0.840129i \(-0.317521\pi\)
0.542387 + 0.840129i \(0.317521\pi\)
\(284\) 3.92820 0.233096
\(285\) 2.46410 0.145961
\(286\) 0 0
\(287\) 0.339746 0.0200546
\(288\) 1.00000 0.0589256
\(289\) −3.07180 −0.180694
\(290\) −3.00000 −0.176166
\(291\) −15.2679 −0.895023
\(292\) −4.19615 −0.245561
\(293\) −5.07180 −0.296298 −0.148149 0.988965i \(-0.547331\pi\)
−0.148149 + 0.988965i \(0.547331\pi\)
\(294\) −6.92820 −0.404061
\(295\) 6.19615 0.360754
\(296\) 1.73205 0.100673
\(297\) 0 0
\(298\) 20.3923 1.18129
\(299\) −3.80385 −0.219982
\(300\) 1.00000 0.0577350
\(301\) 0.875644 0.0504713
\(302\) 16.7321 0.962821
\(303\) −12.4641 −0.716044
\(304\) 2.46410 0.141326
\(305\) 8.19615 0.469310
\(306\) 3.73205 0.213347
\(307\) 3.26795 0.186512 0.0932559 0.995642i \(-0.470273\pi\)
0.0932559 + 0.995642i \(0.470273\pi\)
\(308\) 0 0
\(309\) 16.3205 0.928441
\(310\) −2.19615 −0.124733
\(311\) −13.8564 −0.785725 −0.392862 0.919597i \(-0.628515\pi\)
−0.392862 + 0.919597i \(0.628515\pi\)
\(312\) 3.00000 0.169842
\(313\) 5.12436 0.289646 0.144823 0.989458i \(-0.453739\pi\)
0.144823 + 0.989458i \(0.453739\pi\)
\(314\) 2.80385 0.158230
\(315\) 0.267949 0.0150972
\(316\) 7.46410 0.419889
\(317\) 9.12436 0.512475 0.256237 0.966614i \(-0.417517\pi\)
0.256237 + 0.966614i \(0.417517\pi\)
\(318\) −8.73205 −0.489669
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 4.53590 0.253169
\(322\) −0.339746 −0.0189333
\(323\) 9.19615 0.511688
\(324\) 1.00000 0.0555556
\(325\) 3.00000 0.166410
\(326\) −20.0526 −1.11061
\(327\) 6.53590 0.361436
\(328\) 1.26795 0.0700108
\(329\) −1.32051 −0.0728020
\(330\) 0 0
\(331\) 14.5167 0.797908 0.398954 0.916971i \(-0.369373\pi\)
0.398954 + 0.916971i \(0.369373\pi\)
\(332\) 9.39230 0.515470
\(333\) 1.73205 0.0949158
\(334\) −14.5359 −0.795369
\(335\) −4.53590 −0.247823
\(336\) 0.267949 0.0146178
\(337\) 26.7321 1.45619 0.728094 0.685478i \(-0.240407\pi\)
0.728094 + 0.685478i \(0.240407\pi\)
\(338\) −4.00000 −0.217571
\(339\) −13.8564 −0.752577
\(340\) 3.73205 0.202399
\(341\) 0 0
\(342\) 2.46410 0.133243
\(343\) −3.73205 −0.201512
\(344\) 3.26795 0.176196
\(345\) −1.26795 −0.0682641
\(346\) −16.5885 −0.891801
\(347\) 4.60770 0.247354 0.123677 0.992323i \(-0.460531\pi\)
0.123677 + 0.992323i \(0.460531\pi\)
\(348\) −3.00000 −0.160817
\(349\) −19.8564 −1.06289 −0.531445 0.847093i \(-0.678351\pi\)
−0.531445 + 0.847093i \(0.678351\pi\)
\(350\) 0.267949 0.0143225
\(351\) 3.00000 0.160128
\(352\) 0 0
\(353\) −6.92820 −0.368751 −0.184376 0.982856i \(-0.559026\pi\)
−0.184376 + 0.982856i \(0.559026\pi\)
\(354\) 6.19615 0.329322
\(355\) 3.92820 0.208487
\(356\) 4.39230 0.232792
\(357\) 1.00000 0.0529256
\(358\) −17.1244 −0.905050
\(359\) −20.5359 −1.08384 −0.541922 0.840429i \(-0.682303\pi\)
−0.541922 + 0.840429i \(0.682303\pi\)
\(360\) 1.00000 0.0527046
\(361\) −12.9282 −0.680432
\(362\) −6.19615 −0.325663
\(363\) 0 0
\(364\) 0.803848 0.0421331
\(365\) −4.19615 −0.219637
\(366\) 8.19615 0.428420
\(367\) −12.8564 −0.671099 −0.335549 0.942023i \(-0.608922\pi\)
−0.335549 + 0.942023i \(0.608922\pi\)
\(368\) −1.26795 −0.0660964
\(369\) 1.26795 0.0660068
\(370\) 1.73205 0.0900450
\(371\) −2.33975 −0.121474
\(372\) −2.19615 −0.113865
\(373\) 19.9282 1.03184 0.515922 0.856636i \(-0.327450\pi\)
0.515922 + 0.856636i \(0.327450\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −4.92820 −0.254153
\(377\) −9.00000 −0.463524
\(378\) 0.267949 0.0137818
\(379\) −37.4449 −1.92341 −0.961707 0.274081i \(-0.911626\pi\)
−0.961707 + 0.274081i \(0.911626\pi\)
\(380\) 2.46410 0.126406
\(381\) 20.3923 1.04473
\(382\) 11.5359 0.590228
\(383\) −3.07180 −0.156961 −0.0784807 0.996916i \(-0.525007\pi\)
−0.0784807 + 0.996916i \(0.525007\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.53590 0.129074
\(387\) 3.26795 0.166119
\(388\) −15.2679 −0.775113
\(389\) −10.1436 −0.514301 −0.257150 0.966371i \(-0.582784\pi\)
−0.257150 + 0.966371i \(0.582784\pi\)
\(390\) 3.00000 0.151911
\(391\) −4.73205 −0.239310
\(392\) −6.92820 −0.349927
\(393\) −5.66025 −0.285522
\(394\) −6.53590 −0.329274
\(395\) 7.46410 0.375560
\(396\) 0 0
\(397\) −25.9808 −1.30394 −0.651969 0.758246i \(-0.726057\pi\)
−0.651969 + 0.758246i \(0.726057\pi\)
\(398\) 4.39230 0.220166
\(399\) 0.660254 0.0330540
\(400\) 1.00000 0.0500000
\(401\) 0.339746 0.0169661 0.00848305 0.999964i \(-0.497300\pi\)
0.00848305 + 0.999964i \(0.497300\pi\)
\(402\) −4.53590 −0.226230
\(403\) −6.58846 −0.328194
\(404\) −12.4641 −0.620112
\(405\) 1.00000 0.0496904
\(406\) −0.803848 −0.0398943
\(407\) 0 0
\(408\) 3.73205 0.184764
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 1.26795 0.0626195
\(411\) 21.7846 1.07456
\(412\) 16.3205 0.804054
\(413\) 1.66025 0.0816958
\(414\) −1.26795 −0.0623163
\(415\) 9.39230 0.461050
\(416\) 3.00000 0.147087
\(417\) −14.3205 −0.701278
\(418\) 0 0
\(419\) −4.53590 −0.221593 −0.110797 0.993843i \(-0.535340\pi\)
−0.110797 + 0.993843i \(0.535340\pi\)
\(420\) 0.267949 0.0130746
\(421\) 34.0526 1.65962 0.829810 0.558046i \(-0.188449\pi\)
0.829810 + 0.558046i \(0.188449\pi\)
\(422\) −23.7846 −1.15782
\(423\) −4.92820 −0.239617
\(424\) −8.73205 −0.424066
\(425\) 3.73205 0.181031
\(426\) 3.92820 0.190322
\(427\) 2.19615 0.106279
\(428\) 4.53590 0.219251
\(429\) 0 0
\(430\) 3.26795 0.157595
\(431\) −18.1244 −0.873019 −0.436510 0.899700i \(-0.643786\pi\)
−0.436510 + 0.899700i \(0.643786\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.92820 0.140720 0.0703602 0.997522i \(-0.477585\pi\)
0.0703602 + 0.997522i \(0.477585\pi\)
\(434\) −0.588457 −0.0282469
\(435\) −3.00000 −0.143839
\(436\) 6.53590 0.313013
\(437\) −3.12436 −0.149458
\(438\) −4.19615 −0.200500
\(439\) −8.92820 −0.426120 −0.213060 0.977039i \(-0.568343\pi\)
−0.213060 + 0.977039i \(0.568343\pi\)
\(440\) 0 0
\(441\) −6.92820 −0.329914
\(442\) 11.1962 0.532547
\(443\) 5.19615 0.246877 0.123438 0.992352i \(-0.460608\pi\)
0.123438 + 0.992352i \(0.460608\pi\)
\(444\) 1.73205 0.0821995
\(445\) 4.39230 0.208215
\(446\) −0.464102 −0.0219758
\(447\) 20.3923 0.964523
\(448\) 0.267949 0.0126594
\(449\) 3.46410 0.163481 0.0817405 0.996654i \(-0.473952\pi\)
0.0817405 + 0.996654i \(0.473952\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −13.8564 −0.651751
\(453\) 16.7321 0.786140
\(454\) 10.3923 0.487735
\(455\) 0.803848 0.0376850
\(456\) 2.46410 0.115392
\(457\) 28.4449 1.33059 0.665297 0.746579i \(-0.268305\pi\)
0.665297 + 0.746579i \(0.268305\pi\)
\(458\) −12.0000 −0.560723
\(459\) 3.73205 0.174197
\(460\) −1.26795 −0.0591184
\(461\) 31.2487 1.45540 0.727699 0.685897i \(-0.240590\pi\)
0.727699 + 0.685897i \(0.240590\pi\)
\(462\) 0 0
\(463\) −4.53590 −0.210801 −0.105401 0.994430i \(-0.533612\pi\)
−0.105401 + 0.994430i \(0.533612\pi\)
\(464\) −3.00000 −0.139272
\(465\) −2.19615 −0.101844
\(466\) −25.8564 −1.19777
\(467\) −5.87564 −0.271892 −0.135946 0.990716i \(-0.543407\pi\)
−0.135946 + 0.990716i \(0.543407\pi\)
\(468\) 3.00000 0.138675
\(469\) −1.21539 −0.0561215
\(470\) −4.92820 −0.227321
\(471\) 2.80385 0.129194
\(472\) 6.19615 0.285201
\(473\) 0 0
\(474\) 7.46410 0.342838
\(475\) 2.46410 0.113061
\(476\) 1.00000 0.0458349
\(477\) −8.73205 −0.399813
\(478\) −9.73205 −0.445134
\(479\) −11.7321 −0.536051 −0.268026 0.963412i \(-0.586371\pi\)
−0.268026 + 0.963412i \(0.586371\pi\)
\(480\) 1.00000 0.0456435
\(481\) 5.19615 0.236924
\(482\) −5.33975 −0.243219
\(483\) −0.339746 −0.0154590
\(484\) 0 0
\(485\) −15.2679 −0.693282
\(486\) 1.00000 0.0453609
\(487\) −0.464102 −0.0210305 −0.0105152 0.999945i \(-0.503347\pi\)
−0.0105152 + 0.999945i \(0.503347\pi\)
\(488\) 8.19615 0.371022
\(489\) −20.0526 −0.906808
\(490\) −6.92820 −0.312984
\(491\) 4.73205 0.213554 0.106777 0.994283i \(-0.465947\pi\)
0.106777 + 0.994283i \(0.465947\pi\)
\(492\) 1.26795 0.0571636
\(493\) −11.1962 −0.504249
\(494\) 7.39230 0.332596
\(495\) 0 0
\(496\) −2.19615 −0.0986102
\(497\) 1.05256 0.0472137
\(498\) 9.39230 0.420879
\(499\) 37.4449 1.67626 0.838131 0.545469i \(-0.183648\pi\)
0.838131 + 0.545469i \(0.183648\pi\)
\(500\) 1.00000 0.0447214
\(501\) −14.5359 −0.649416
\(502\) 25.4641 1.13652
\(503\) −20.7321 −0.924396 −0.462198 0.886777i \(-0.652939\pi\)
−0.462198 + 0.886777i \(0.652939\pi\)
\(504\) 0.267949 0.0119354
\(505\) −12.4641 −0.554645
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) 20.3923 0.904762
\(509\) −35.8564 −1.58931 −0.794654 0.607063i \(-0.792348\pi\)
−0.794654 + 0.607063i \(0.792348\pi\)
\(510\) 3.73205 0.165258
\(511\) −1.12436 −0.0497386
\(512\) 1.00000 0.0441942
\(513\) 2.46410 0.108793
\(514\) −28.3205 −1.24916
\(515\) 16.3205 0.719168
\(516\) 3.26795 0.143863
\(517\) 0 0
\(518\) 0.464102 0.0203915
\(519\) −16.5885 −0.728152
\(520\) 3.00000 0.131559
\(521\) 8.58846 0.376267 0.188134 0.982143i \(-0.439756\pi\)
0.188134 + 0.982143i \(0.439756\pi\)
\(522\) −3.00000 −0.131306
\(523\) −22.7321 −0.994003 −0.497002 0.867750i \(-0.665566\pi\)
−0.497002 + 0.867750i \(0.665566\pi\)
\(524\) −5.66025 −0.247269
\(525\) 0.267949 0.0116943
\(526\) −0.875644 −0.0381799
\(527\) −8.19615 −0.357030
\(528\) 0 0
\(529\) −21.3923 −0.930100
\(530\) −8.73205 −0.379296
\(531\) 6.19615 0.268890
\(532\) 0.660254 0.0286256
\(533\) 3.80385 0.164763
\(534\) 4.39230 0.190074
\(535\) 4.53590 0.196104
\(536\) −4.53590 −0.195921
\(537\) −17.1244 −0.738970
\(538\) −19.5885 −0.844518
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −22.5885 −0.971154 −0.485577 0.874194i \(-0.661390\pi\)
−0.485577 + 0.874194i \(0.661390\pi\)
\(542\) 3.85641 0.165647
\(543\) −6.19615 −0.265902
\(544\) 3.73205 0.160010
\(545\) 6.53590 0.279967
\(546\) 0.803848 0.0344015
\(547\) 28.9282 1.23688 0.618440 0.785832i \(-0.287765\pi\)
0.618440 + 0.785832i \(0.287765\pi\)
\(548\) 21.7846 0.930592
\(549\) 8.19615 0.349803
\(550\) 0 0
\(551\) −7.39230 −0.314923
\(552\) −1.26795 −0.0539675
\(553\) 2.00000 0.0850487
\(554\) 5.07180 0.215480
\(555\) 1.73205 0.0735215
\(556\) −14.3205 −0.607325
\(557\) 10.3923 0.440336 0.220168 0.975462i \(-0.429339\pi\)
0.220168 + 0.975462i \(0.429339\pi\)
\(558\) −2.19615 −0.0929705
\(559\) 9.80385 0.414659
\(560\) 0.267949 0.0113229
\(561\) 0 0
\(562\) −12.0526 −0.508407
\(563\) −35.1051 −1.47950 −0.739752 0.672879i \(-0.765057\pi\)
−0.739752 + 0.672879i \(0.765057\pi\)
\(564\) −4.92820 −0.207515
\(565\) −13.8564 −0.582943
\(566\) 18.2487 0.767051
\(567\) 0.267949 0.0112528
\(568\) 3.92820 0.164824
\(569\) −36.1962 −1.51742 −0.758711 0.651428i \(-0.774170\pi\)
−0.758711 + 0.651428i \(0.774170\pi\)
\(570\) 2.46410 0.103210
\(571\) −10.6795 −0.446923 −0.223461 0.974713i \(-0.571736\pi\)
−0.223461 + 0.974713i \(0.571736\pi\)
\(572\) 0 0
\(573\) 11.5359 0.481919
\(574\) 0.339746 0.0141807
\(575\) −1.26795 −0.0528771
\(576\) 1.00000 0.0416667
\(577\) 17.3205 0.721062 0.360531 0.932747i \(-0.382595\pi\)
0.360531 + 0.932747i \(0.382595\pi\)
\(578\) −3.07180 −0.127770
\(579\) 2.53590 0.105388
\(580\) −3.00000 −0.124568
\(581\) 2.51666 0.104409
\(582\) −15.2679 −0.632877
\(583\) 0 0
\(584\) −4.19615 −0.173638
\(585\) 3.00000 0.124035
\(586\) −5.07180 −0.209514
\(587\) −42.9090 −1.77104 −0.885521 0.464599i \(-0.846199\pi\)
−0.885521 + 0.464599i \(0.846199\pi\)
\(588\) −6.92820 −0.285714
\(589\) −5.41154 −0.222979
\(590\) 6.19615 0.255092
\(591\) −6.53590 −0.268851
\(592\) 1.73205 0.0711868
\(593\) 8.92820 0.366637 0.183319 0.983054i \(-0.441316\pi\)
0.183319 + 0.983054i \(0.441316\pi\)
\(594\) 0 0
\(595\) 1.00000 0.0409960
\(596\) 20.3923 0.835301
\(597\) 4.39230 0.179765
\(598\) −3.80385 −0.155551
\(599\) −13.6077 −0.555995 −0.277998 0.960582i \(-0.589671\pi\)
−0.277998 + 0.960582i \(0.589671\pi\)
\(600\) 1.00000 0.0408248
\(601\) 12.1436 0.495348 0.247674 0.968843i \(-0.420334\pi\)
0.247674 + 0.968843i \(0.420334\pi\)
\(602\) 0.875644 0.0356886
\(603\) −4.53590 −0.184716
\(604\) 16.7321 0.680817
\(605\) 0 0
\(606\) −12.4641 −0.506320
\(607\) 39.5885 1.60685 0.803423 0.595409i \(-0.203010\pi\)
0.803423 + 0.595409i \(0.203010\pi\)
\(608\) 2.46410 0.0999325
\(609\) −0.803848 −0.0325735
\(610\) 8.19615 0.331853
\(611\) −14.7846 −0.598121
\(612\) 3.73205 0.150859
\(613\) −35.6410 −1.43953 −0.719764 0.694219i \(-0.755750\pi\)
−0.719764 + 0.694219i \(0.755750\pi\)
\(614\) 3.26795 0.131884
\(615\) 1.26795 0.0511286
\(616\) 0 0
\(617\) −13.7846 −0.554947 −0.277474 0.960733i \(-0.589497\pi\)
−0.277474 + 0.960733i \(0.589497\pi\)
\(618\) 16.3205 0.656507
\(619\) 31.9808 1.28542 0.642708 0.766112i \(-0.277811\pi\)
0.642708 + 0.766112i \(0.277811\pi\)
\(620\) −2.19615 −0.0881996
\(621\) −1.26795 −0.0508810
\(622\) −13.8564 −0.555591
\(623\) 1.17691 0.0471521
\(624\) 3.00000 0.120096
\(625\) 1.00000 0.0400000
\(626\) 5.12436 0.204810
\(627\) 0 0
\(628\) 2.80385 0.111886
\(629\) 6.46410 0.257741
\(630\) 0.267949 0.0106754
\(631\) 28.1962 1.12247 0.561236 0.827656i \(-0.310326\pi\)
0.561236 + 0.827656i \(0.310326\pi\)
\(632\) 7.46410 0.296906
\(633\) −23.7846 −0.945353
\(634\) 9.12436 0.362374
\(635\) 20.3923 0.809244
\(636\) −8.73205 −0.346248
\(637\) −20.7846 −0.823516
\(638\) 0 0
\(639\) 3.92820 0.155397
\(640\) 1.00000 0.0395285
\(641\) −4.92820 −0.194652 −0.0973262 0.995253i \(-0.531029\pi\)
−0.0973262 + 0.995253i \(0.531029\pi\)
\(642\) 4.53590 0.179018
\(643\) 10.1436 0.400024 0.200012 0.979793i \(-0.435902\pi\)
0.200012 + 0.979793i \(0.435902\pi\)
\(644\) −0.339746 −0.0133879
\(645\) 3.26795 0.128675
\(646\) 9.19615 0.361818
\(647\) 43.3205 1.70310 0.851552 0.524269i \(-0.175662\pi\)
0.851552 + 0.524269i \(0.175662\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 3.00000 0.117670
\(651\) −0.588457 −0.0230635
\(652\) −20.0526 −0.785319
\(653\) −15.8038 −0.618452 −0.309226 0.950989i \(-0.600070\pi\)
−0.309226 + 0.950989i \(0.600070\pi\)
\(654\) 6.53590 0.255574
\(655\) −5.66025 −0.221164
\(656\) 1.26795 0.0495051
\(657\) −4.19615 −0.163707
\(658\) −1.32051 −0.0514788
\(659\) −2.39230 −0.0931910 −0.0465955 0.998914i \(-0.514837\pi\)
−0.0465955 + 0.998914i \(0.514837\pi\)
\(660\) 0 0
\(661\) −7.80385 −0.303534 −0.151767 0.988416i \(-0.548496\pi\)
−0.151767 + 0.988416i \(0.548496\pi\)
\(662\) 14.5167 0.564206
\(663\) 11.1962 0.434823
\(664\) 9.39230 0.364492
\(665\) 0.660254 0.0256036
\(666\) 1.73205 0.0671156
\(667\) 3.80385 0.147286
\(668\) −14.5359 −0.562411
\(669\) −0.464102 −0.0179432
\(670\) −4.53590 −0.175237
\(671\) 0 0
\(672\) 0.267949 0.0103364
\(673\) 9.85641 0.379937 0.189968 0.981790i \(-0.439161\pi\)
0.189968 + 0.981790i \(0.439161\pi\)
\(674\) 26.7321 1.02968
\(675\) 1.00000 0.0384900
\(676\) −4.00000 −0.153846
\(677\) 4.05256 0.155752 0.0778762 0.996963i \(-0.475186\pi\)
0.0778762 + 0.996963i \(0.475186\pi\)
\(678\) −13.8564 −0.532152
\(679\) −4.09103 −0.157000
\(680\) 3.73205 0.143118
\(681\) 10.3923 0.398234
\(682\) 0 0
\(683\) −30.1244 −1.15268 −0.576338 0.817211i \(-0.695519\pi\)
−0.576338 + 0.817211i \(0.695519\pi\)
\(684\) 2.46410 0.0942173
\(685\) 21.7846 0.832347
\(686\) −3.73205 −0.142490
\(687\) −12.0000 −0.457829
\(688\) 3.26795 0.124589
\(689\) −26.1962 −0.997994
\(690\) −1.26795 −0.0482700
\(691\) 12.9474 0.492544 0.246272 0.969201i \(-0.420794\pi\)
0.246272 + 0.969201i \(0.420794\pi\)
\(692\) −16.5885 −0.630599
\(693\) 0 0
\(694\) 4.60770 0.174906
\(695\) −14.3205 −0.543208
\(696\) −3.00000 −0.113715
\(697\) 4.73205 0.179239
\(698\) −19.8564 −0.751576
\(699\) −25.8564 −0.977979
\(700\) 0.267949 0.0101275
\(701\) −41.5359 −1.56879 −0.784395 0.620262i \(-0.787026\pi\)
−0.784395 + 0.620262i \(0.787026\pi\)
\(702\) 3.00000 0.113228
\(703\) 4.26795 0.160969
\(704\) 0 0
\(705\) −4.92820 −0.185607
\(706\) −6.92820 −0.260746
\(707\) −3.33975 −0.125604
\(708\) 6.19615 0.232866
\(709\) −21.6603 −0.813468 −0.406734 0.913547i \(-0.633332\pi\)
−0.406734 + 0.913547i \(0.633332\pi\)
\(710\) 3.92820 0.147423
\(711\) 7.46410 0.279926
\(712\) 4.39230 0.164609
\(713\) 2.78461 0.104284
\(714\) 1.00000 0.0374241
\(715\) 0 0
\(716\) −17.1244 −0.639967
\(717\) −9.73205 −0.363450
\(718\) −20.5359 −0.766393
\(719\) 31.4641 1.17341 0.586706 0.809800i \(-0.300424\pi\)
0.586706 + 0.809800i \(0.300424\pi\)
\(720\) 1.00000 0.0372678
\(721\) 4.37307 0.162862
\(722\) −12.9282 −0.481138
\(723\) −5.33975 −0.198587
\(724\) −6.19615 −0.230278
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) −49.4974 −1.83576 −0.917879 0.396861i \(-0.870100\pi\)
−0.917879 + 0.396861i \(0.870100\pi\)
\(728\) 0.803848 0.0297926
\(729\) 1.00000 0.0370370
\(730\) −4.19615 −0.155307
\(731\) 12.1962 0.451091
\(732\) 8.19615 0.302939
\(733\) −9.85641 −0.364055 −0.182027 0.983293i \(-0.558266\pi\)
−0.182027 + 0.983293i \(0.558266\pi\)
\(734\) −12.8564 −0.474539
\(735\) −6.92820 −0.255551
\(736\) −1.26795 −0.0467372
\(737\) 0 0
\(738\) 1.26795 0.0466739
\(739\) 9.53590 0.350784 0.175392 0.984499i \(-0.443881\pi\)
0.175392 + 0.984499i \(0.443881\pi\)
\(740\) 1.73205 0.0636715
\(741\) 7.39230 0.271563
\(742\) −2.33975 −0.0858948
\(743\) 4.05256 0.148674 0.0743370 0.997233i \(-0.476316\pi\)
0.0743370 + 0.997233i \(0.476316\pi\)
\(744\) −2.19615 −0.0805149
\(745\) 20.3923 0.747116
\(746\) 19.9282 0.729623
\(747\) 9.39230 0.343646
\(748\) 0 0
\(749\) 1.21539 0.0444094
\(750\) 1.00000 0.0365148
\(751\) 38.1962 1.39380 0.696899 0.717170i \(-0.254563\pi\)
0.696899 + 0.717170i \(0.254563\pi\)
\(752\) −4.92820 −0.179713
\(753\) 25.4641 0.927963
\(754\) −9.00000 −0.327761
\(755\) 16.7321 0.608942
\(756\) 0.267949 0.00974522
\(757\) 3.46410 0.125905 0.0629525 0.998017i \(-0.479948\pi\)
0.0629525 + 0.998017i \(0.479948\pi\)
\(758\) −37.4449 −1.36006
\(759\) 0 0
\(760\) 2.46410 0.0893824
\(761\) 47.3205 1.71537 0.857684 0.514178i \(-0.171903\pi\)
0.857684 + 0.514178i \(0.171903\pi\)
\(762\) 20.3923 0.738735
\(763\) 1.75129 0.0634009
\(764\) 11.5359 0.417354
\(765\) 3.73205 0.134933
\(766\) −3.07180 −0.110989
\(767\) 18.5885 0.671190
\(768\) 1.00000 0.0360844
\(769\) −34.5167 −1.24470 −0.622351 0.782738i \(-0.713822\pi\)
−0.622351 + 0.782738i \(0.713822\pi\)
\(770\) 0 0
\(771\) −28.3205 −1.01994
\(772\) 2.53590 0.0912690
\(773\) 20.3397 0.731570 0.365785 0.930699i \(-0.380801\pi\)
0.365785 + 0.930699i \(0.380801\pi\)
\(774\) 3.26795 0.117464
\(775\) −2.19615 −0.0788881
\(776\) −15.2679 −0.548087
\(777\) 0.464102 0.0166496
\(778\) −10.1436 −0.363665
\(779\) 3.12436 0.111942
\(780\) 3.00000 0.107417
\(781\) 0 0
\(782\) −4.73205 −0.169218
\(783\) −3.00000 −0.107211
\(784\) −6.92820 −0.247436
\(785\) 2.80385 0.100074
\(786\) −5.66025 −0.201895
\(787\) −20.9808 −0.747883 −0.373942 0.927452i \(-0.621994\pi\)
−0.373942 + 0.927452i \(0.621994\pi\)
\(788\) −6.53590 −0.232832
\(789\) −0.875644 −0.0311738
\(790\) 7.46410 0.265561
\(791\) −3.71281 −0.132012
\(792\) 0 0
\(793\) 24.5885 0.873162
\(794\) −25.9808 −0.922023
\(795\) −8.73205 −0.309694
\(796\) 4.39230 0.155681
\(797\) 47.3731 1.67804 0.839020 0.544100i \(-0.183129\pi\)
0.839020 + 0.544100i \(0.183129\pi\)
\(798\) 0.660254 0.0233727
\(799\) −18.3923 −0.650673
\(800\) 1.00000 0.0353553
\(801\) 4.39230 0.155194
\(802\) 0.339746 0.0119968
\(803\) 0 0
\(804\) −4.53590 −0.159969
\(805\) −0.339746 −0.0119745
\(806\) −6.58846 −0.232069
\(807\) −19.5885 −0.689546
\(808\) −12.4641 −0.438486
\(809\) −22.7321 −0.799216 −0.399608 0.916686i \(-0.630854\pi\)
−0.399608 + 0.916686i \(0.630854\pi\)
\(810\) 1.00000 0.0351364
\(811\) −13.0000 −0.456492 −0.228246 0.973604i \(-0.573299\pi\)
−0.228246 + 0.973604i \(0.573299\pi\)
\(812\) −0.803848 −0.0282095
\(813\) 3.85641 0.135250
\(814\) 0 0
\(815\) −20.0526 −0.702411
\(816\) 3.73205 0.130648
\(817\) 8.05256 0.281723
\(818\) 14.0000 0.489499
\(819\) 0.803848 0.0280887
\(820\) 1.26795 0.0442787
\(821\) −47.3205 −1.65150 −0.825749 0.564038i \(-0.809247\pi\)
−0.825749 + 0.564038i \(0.809247\pi\)
\(822\) 21.7846 0.759826
\(823\) 1.39230 0.0485327 0.0242663 0.999706i \(-0.492275\pi\)
0.0242663 + 0.999706i \(0.492275\pi\)
\(824\) 16.3205 0.568552
\(825\) 0 0
\(826\) 1.66025 0.0577676
\(827\) −28.5692 −0.993449 −0.496725 0.867908i \(-0.665464\pi\)
−0.496725 + 0.867908i \(0.665464\pi\)
\(828\) −1.26795 −0.0440643
\(829\) 37.1244 1.28938 0.644691 0.764443i \(-0.276986\pi\)
0.644691 + 0.764443i \(0.276986\pi\)
\(830\) 9.39230 0.326012
\(831\) 5.07180 0.175939
\(832\) 3.00000 0.104006
\(833\) −25.8564 −0.895871
\(834\) −14.3205 −0.495879
\(835\) −14.5359 −0.503036
\(836\) 0 0
\(837\) −2.19615 −0.0759101
\(838\) −4.53590 −0.156690
\(839\) −11.7846 −0.406850 −0.203425 0.979091i \(-0.565207\pi\)
−0.203425 + 0.979091i \(0.565207\pi\)
\(840\) 0.267949 0.00924513
\(841\) −20.0000 −0.689655
\(842\) 34.0526 1.17353
\(843\) −12.0526 −0.415112
\(844\) −23.7846 −0.818700
\(845\) −4.00000 −0.137604
\(846\) −4.92820 −0.169435
\(847\) 0 0
\(848\) −8.73205 −0.299860
\(849\) 18.2487 0.626294
\(850\) 3.73205 0.128008
\(851\) −2.19615 −0.0752831
\(852\) 3.92820 0.134578
\(853\) 40.3205 1.38055 0.690274 0.723548i \(-0.257490\pi\)
0.690274 + 0.723548i \(0.257490\pi\)
\(854\) 2.19615 0.0751508
\(855\) 2.46410 0.0842705
\(856\) 4.53590 0.155034
\(857\) −27.3397 −0.933908 −0.466954 0.884282i \(-0.654649\pi\)
−0.466954 + 0.884282i \(0.654649\pi\)
\(858\) 0 0
\(859\) −22.3923 −0.764016 −0.382008 0.924159i \(-0.624767\pi\)
−0.382008 + 0.924159i \(0.624767\pi\)
\(860\) 3.26795 0.111436
\(861\) 0.339746 0.0115785
\(862\) −18.1244 −0.617318
\(863\) −36.2487 −1.23392 −0.616960 0.786994i \(-0.711636\pi\)
−0.616960 + 0.786994i \(0.711636\pi\)
\(864\) 1.00000 0.0340207
\(865\) −16.5885 −0.564024
\(866\) 2.92820 0.0995044
\(867\) −3.07180 −0.104324
\(868\) −0.588457 −0.0199735
\(869\) 0 0
\(870\) −3.00000 −0.101710
\(871\) −13.6077 −0.461079
\(872\) 6.53590 0.221333
\(873\) −15.2679 −0.516742
\(874\) −3.12436 −0.105683
\(875\) 0.267949 0.00905834
\(876\) −4.19615 −0.141775
\(877\) −35.6410 −1.20351 −0.601756 0.798680i \(-0.705532\pi\)
−0.601756 + 0.798680i \(0.705532\pi\)
\(878\) −8.92820 −0.301312
\(879\) −5.07180 −0.171067
\(880\) 0 0
\(881\) −8.44486 −0.284515 −0.142257 0.989830i \(-0.545436\pi\)
−0.142257 + 0.989830i \(0.545436\pi\)
\(882\) −6.92820 −0.233285
\(883\) 47.1244 1.58586 0.792930 0.609312i \(-0.208554\pi\)
0.792930 + 0.609312i \(0.208554\pi\)
\(884\) 11.1962 0.376567
\(885\) 6.19615 0.208281
\(886\) 5.19615 0.174568
\(887\) 3.21539 0.107962 0.0539811 0.998542i \(-0.482809\pi\)
0.0539811 + 0.998542i \(0.482809\pi\)
\(888\) 1.73205 0.0581238
\(889\) 5.46410 0.183260
\(890\) 4.39230 0.147230
\(891\) 0 0
\(892\) −0.464102 −0.0155393
\(893\) −12.1436 −0.406370
\(894\) 20.3923 0.682021
\(895\) −17.1244 −0.572404
\(896\) 0.267949 0.00895155
\(897\) −3.80385 −0.127007
\(898\) 3.46410 0.115599
\(899\) 6.58846 0.219737
\(900\) 1.00000 0.0333333
\(901\) −32.5885 −1.08568
\(902\) 0 0
\(903\) 0.875644 0.0291396
\(904\) −13.8564 −0.460857
\(905\) −6.19615 −0.205967
\(906\) 16.7321 0.555885
\(907\) 54.7846 1.81909 0.909547 0.415602i \(-0.136429\pi\)
0.909547 + 0.415602i \(0.136429\pi\)
\(908\) 10.3923 0.344881
\(909\) −12.4641 −0.413408
\(910\) 0.803848 0.0266473
\(911\) 52.5692 1.74170 0.870848 0.491552i \(-0.163570\pi\)
0.870848 + 0.491552i \(0.163570\pi\)
\(912\) 2.46410 0.0815946
\(913\) 0 0
\(914\) 28.4449 0.940872
\(915\) 8.19615 0.270956
\(916\) −12.0000 −0.396491
\(917\) −1.51666 −0.0500845
\(918\) 3.73205 0.123176
\(919\) −31.6603 −1.04438 −0.522188 0.852831i \(-0.674884\pi\)
−0.522188 + 0.852831i \(0.674884\pi\)
\(920\) −1.26795 −0.0418030
\(921\) 3.26795 0.107683
\(922\) 31.2487 1.02912
\(923\) 11.7846 0.387895
\(924\) 0 0
\(925\) 1.73205 0.0569495
\(926\) −4.53590 −0.149059
\(927\) 16.3205 0.536036
\(928\) −3.00000 −0.0984798
\(929\) −15.1244 −0.496214 −0.248107 0.968733i \(-0.579808\pi\)
−0.248107 + 0.968733i \(0.579808\pi\)
\(930\) −2.19615 −0.0720147
\(931\) −17.0718 −0.559506
\(932\) −25.8564 −0.846955
\(933\) −13.8564 −0.453638
\(934\) −5.87564 −0.192257
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) 49.7128 1.62405 0.812023 0.583625i \(-0.198366\pi\)
0.812023 + 0.583625i \(0.198366\pi\)
\(938\) −1.21539 −0.0396839
\(939\) 5.12436 0.167227
\(940\) −4.92820 −0.160740
\(941\) −6.07180 −0.197935 −0.0989675 0.995091i \(-0.531554\pi\)
−0.0989675 + 0.995091i \(0.531554\pi\)
\(942\) 2.80385 0.0913543
\(943\) −1.60770 −0.0523538
\(944\) 6.19615 0.201668
\(945\) 0.267949 0.00871639
\(946\) 0 0
\(947\) −18.8038 −0.611043 −0.305521 0.952185i \(-0.598831\pi\)
−0.305521 + 0.952185i \(0.598831\pi\)
\(948\) 7.46410 0.242423
\(949\) −12.5885 −0.408639
\(950\) 2.46410 0.0799460
\(951\) 9.12436 0.295878
\(952\) 1.00000 0.0324102
\(953\) −31.8564 −1.03193 −0.515965 0.856610i \(-0.672567\pi\)
−0.515965 + 0.856610i \(0.672567\pi\)
\(954\) −8.73205 −0.282711
\(955\) 11.5359 0.373293
\(956\) −9.73205 −0.314757
\(957\) 0 0
\(958\) −11.7321 −0.379045
\(959\) 5.83717 0.188492
\(960\) 1.00000 0.0322749
\(961\) −26.1769 −0.844417
\(962\) 5.19615 0.167531
\(963\) 4.53590 0.146167
\(964\) −5.33975 −0.171982
\(965\) 2.53590 0.0816335
\(966\) −0.339746 −0.0109311
\(967\) 28.5359 0.917653 0.458826 0.888526i \(-0.348270\pi\)
0.458826 + 0.888526i \(0.348270\pi\)
\(968\) 0 0
\(969\) 9.19615 0.295423
\(970\) −15.2679 −0.490224
\(971\) −2.53590 −0.0813809 −0.0406904 0.999172i \(-0.512956\pi\)
−0.0406904 + 0.999172i \(0.512956\pi\)
\(972\) 1.00000 0.0320750
\(973\) −3.83717 −0.123014
\(974\) −0.464102 −0.0148708
\(975\) 3.00000 0.0960769
\(976\) 8.19615 0.262352
\(977\) 10.0000 0.319928 0.159964 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) −20.0526 −0.641210
\(979\) 0 0
\(980\) −6.92820 −0.221313
\(981\) 6.53590 0.208675
\(982\) 4.73205 0.151006
\(983\) 1.41154 0.0450212 0.0225106 0.999747i \(-0.492834\pi\)
0.0225106 + 0.999747i \(0.492834\pi\)
\(984\) 1.26795 0.0404207
\(985\) −6.53590 −0.208251
\(986\) −11.1962 −0.356558
\(987\) −1.32051 −0.0420322
\(988\) 7.39230 0.235181
\(989\) −4.14359 −0.131759
\(990\) 0 0
\(991\) 50.7846 1.61323 0.806613 0.591080i \(-0.201298\pi\)
0.806613 + 0.591080i \(0.201298\pi\)
\(992\) −2.19615 −0.0697279
\(993\) 14.5167 0.460672
\(994\) 1.05256 0.0333851
\(995\) 4.39230 0.139245
\(996\) 9.39230 0.297607
\(997\) 38.6077 1.22272 0.611359 0.791353i \(-0.290623\pi\)
0.611359 + 0.791353i \(0.290623\pi\)
\(998\) 37.4449 1.18530
\(999\) 1.73205 0.0547997
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.bp.1.1 yes 2
11.10 odd 2 3630.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3630.2.a.bh.1.2 2 11.10 odd 2
3630.2.a.bp.1.1 yes 2 1.1 even 1 trivial