Properties

Label 6762.2.a.cn.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.18398\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.183979 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.183979 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.183979 q^{10} +3.08861 q^{11} -1.00000 q^{12} +5.85838 q^{13} +0.183979 q^{15} +1.00000 q^{16} +6.36796 q^{17} +1.00000 q^{18} +3.41421 q^{19} -0.183979 q^{20} +3.08861 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.96615 q^{25} +5.85838 q^{26} -1.00000 q^{27} -4.68681 q^{29} +0.183979 q^{30} +3.41421 q^{31} +1.00000 q^{32} -3.08861 q^{33} +6.36796 q^{34} +1.00000 q^{36} -1.49042 q^{37} +3.41421 q^{38} -5.85838 q^{39} -0.183979 q^{40} +2.07621 q^{41} +4.94699 q^{43} +3.08861 q^{44} -0.183979 q^{45} -1.00000 q^{46} +5.01241 q^{47} -1.00000 q^{48} -4.96615 q^{50} -6.36796 q^{51} +5.85838 q^{52} -1.79972 q^{53} -1.00000 q^{54} -0.568241 q^{55} -3.41421 q^{57} -4.68681 q^{58} +7.91704 q^{59} +0.183979 q^{60} -2.30254 q^{61} +3.41421 q^{62} +1.00000 q^{64} -1.07782 q^{65} -3.08861 q^{66} -1.53953 q^{67} +6.36796 q^{68} +1.00000 q^{69} -3.08861 q^{71} +1.00000 q^{72} -5.13097 q^{73} -1.49042 q^{74} +4.96615 q^{75} +3.41421 q^{76} -5.85838 q^{78} -9.08861 q^{79} -0.183979 q^{80} +1.00000 q^{81} +2.07621 q^{82} +0.0175454 q^{83} -1.17157 q^{85} +4.94699 q^{86} +4.68681 q^{87} +3.08861 q^{88} -9.63380 q^{89} -0.183979 q^{90} -1.00000 q^{92} -3.41421 q^{93} +5.01241 q^{94} -0.628145 q^{95} -1.00000 q^{96} -4.60385 q^{97} +3.08861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 6q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 6q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + 6q^{10} - 4q^{12} + 10q^{13} - 6q^{15} + 4q^{16} + 12q^{17} + 4q^{18} + 8q^{19} + 6q^{20} - 4q^{23} - 4q^{24} + 6q^{25} + 10q^{26} - 4q^{27} + 6q^{29} - 6q^{30} + 8q^{31} + 4q^{32} + 12q^{34} + 4q^{36} - 6q^{37} + 8q^{38} - 10q^{39} + 6q^{40} + 14q^{41} - 6q^{43} + 6q^{45} - 4q^{46} + 2q^{47} - 4q^{48} + 6q^{50} - 12q^{51} + 10q^{52} - 4q^{53} - 4q^{54} + 8q^{55} - 8q^{57} + 6q^{58} + 8q^{59} - 6q^{60} + 12q^{61} + 8q^{62} + 4q^{64} + 6q^{65} - 4q^{67} + 12q^{68} + 4q^{69} + 4q^{72} + 12q^{73} - 6q^{74} - 6q^{75} + 8q^{76} - 10q^{78} - 24q^{79} + 6q^{80} + 4q^{81} + 14q^{82} + 16q^{83} - 16q^{85} - 6q^{86} - 6q^{87} + 12q^{89} + 6q^{90} - 4q^{92} - 8q^{93} + 2q^{94} + 12q^{95} - 4q^{96} + 30q^{97} + O(q^{100}) \)

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
\(5\) −0.183979 −0.0822781 −0.0411390 0.999153i \(-0.513099\pi\)
−0.0411390 + 0.999153i \(0.513099\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.183979 −0.0581794
\(11\) 3.08861 0.931252 0.465626 0.884982i \(-0.345829\pi\)
0.465626 + 0.884982i \(0.345829\pi\)
\(12\) −1.00000 −0.288675
\(13\) 5.85838 1.62482 0.812411 0.583085i \(-0.198155\pi\)
0.812411 + 0.583085i \(0.198155\pi\)
\(14\) 0 0
\(15\) 0.183979 0.0475033
\(16\) 1.00000 0.250000
\(17\) 6.36796 1.54446 0.772228 0.635345i \(-0.219142\pi\)
0.772228 + 0.635345i \(0.219142\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.41421 0.783274 0.391637 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(20\) −0.183979 −0.0411390
\(21\) 0 0
\(22\) 3.08861 0.658495
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.96615 −0.993230
\(26\) 5.85838 1.14892
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.68681 −0.870318 −0.435159 0.900354i \(-0.643308\pi\)
−0.435159 + 0.900354i \(0.643308\pi\)
\(30\) 0.183979 0.0335899
\(31\) 3.41421 0.613211 0.306605 0.951837i \(-0.400807\pi\)
0.306605 + 0.951837i \(0.400807\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.08861 −0.537659
\(34\) 6.36796 1.09210
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.49042 −0.245024 −0.122512 0.992467i \(-0.539095\pi\)
−0.122512 + 0.992467i \(0.539095\pi\)
\(38\) 3.41421 0.553859
\(39\) −5.85838 −0.938091
\(40\) −0.183979 −0.0290897
\(41\) 2.07621 0.324249 0.162125 0.986770i \(-0.448165\pi\)
0.162125 + 0.986770i \(0.448165\pi\)
\(42\) 0 0
\(43\) 4.94699 0.754409 0.377205 0.926130i \(-0.376885\pi\)
0.377205 + 0.926130i \(0.376885\pi\)
\(44\) 3.08861 0.465626
\(45\) −0.183979 −0.0274260
\(46\) −1.00000 −0.147442
\(47\) 5.01241 0.731135 0.365567 0.930785i \(-0.380875\pi\)
0.365567 + 0.930785i \(0.380875\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.96615 −0.702320
\(51\) −6.36796 −0.891693
\(52\) 5.85838 0.812411
\(53\) −1.79972 −0.247210 −0.123605 0.992331i \(-0.539446\pi\)
−0.123605 + 0.992331i \(0.539446\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.568241 −0.0766216
\(56\) 0 0
\(57\) −3.41421 −0.452224
\(58\) −4.68681 −0.615408
\(59\) 7.91704 1.03071 0.515355 0.856977i \(-0.327660\pi\)
0.515355 + 0.856977i \(0.327660\pi\)
\(60\) 0.183979 0.0237516
\(61\) −2.30254 −0.294811 −0.147405 0.989076i \(-0.547092\pi\)
−0.147405 + 0.989076i \(0.547092\pi\)
\(62\) 3.41421 0.433606
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.07782 −0.133687
\(66\) −3.08861 −0.380182
\(67\) −1.53953 −0.188084 −0.0940419 0.995568i \(-0.529979\pi\)
−0.0940419 + 0.995568i \(0.529979\pi\)
\(68\) 6.36796 0.772228
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −3.08861 −0.366551 −0.183275 0.983062i \(-0.558670\pi\)
−0.183275 + 0.983062i \(0.558670\pi\)
\(72\) 1.00000 0.117851
\(73\) −5.13097 −0.600535 −0.300267 0.953855i \(-0.597076\pi\)
−0.300267 + 0.953855i \(0.597076\pi\)
\(74\) −1.49042 −0.173258
\(75\) 4.96615 0.573442
\(76\) 3.41421 0.391637
\(77\) 0 0
\(78\) −5.85838 −0.663331
\(79\) −9.08861 −1.02255 −0.511274 0.859418i \(-0.670826\pi\)
−0.511274 + 0.859418i \(0.670826\pi\)
\(80\) −0.183979 −0.0205695
\(81\) 1.00000 0.111111
\(82\) 2.07621 0.229279
\(83\) 0.0175454 0.00192586 0.000962929 1.00000i \(-0.499693\pi\)
0.000962929 1.00000i \(0.499693\pi\)
\(84\) 0 0
\(85\) −1.17157 −0.127075
\(86\) 4.94699 0.533448
\(87\) 4.68681 0.502478
\(88\) 3.08861 0.329247
\(89\) −9.63380 −1.02118 −0.510590 0.859824i \(-0.670573\pi\)
−0.510590 + 0.859824i \(0.670573\pi\)
\(90\) −0.183979 −0.0193931
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −3.41421 −0.354037
\(94\) 5.01241 0.516990
\(95\) −0.628145 −0.0644463
\(96\) −1.00000 −0.102062
\(97\) −4.60385 −0.467450 −0.233725 0.972303i \(-0.575092\pi\)
−0.233725 + 0.972303i \(0.575092\pi\)
\(98\) 0 0
\(99\) 3.08861 0.310417
\(100\) −4.96615 −0.496615
\(101\) 5.39667 0.536989 0.268494 0.963281i \(-0.413474\pi\)
0.268494 + 0.963281i \(0.413474\pi\)
\(102\) −6.36796 −0.630522
\(103\) 0.141621 0.0139543 0.00697717 0.999976i \(-0.497779\pi\)
0.00697717 + 0.999976i \(0.497779\pi\)
\(104\) 5.85838 0.574461
\(105\) 0 0
\(106\) −1.79972 −0.174804
\(107\) −16.5452 −1.59948 −0.799742 0.600344i \(-0.795030\pi\)
−0.799742 + 0.600344i \(0.795030\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 15.9431 1.52707 0.763536 0.645765i \(-0.223462\pi\)
0.763536 + 0.645765i \(0.223462\pi\)
\(110\) −0.568241 −0.0541797
\(111\) 1.49042 0.141464
\(112\) 0 0
\(113\) 1.42097 0.133673 0.0668366 0.997764i \(-0.478709\pi\)
0.0668366 + 0.997764i \(0.478709\pi\)
\(114\) −3.41421 −0.319770
\(115\) 0.183979 0.0171562
\(116\) −4.68681 −0.435159
\(117\) 5.85838 0.541607
\(118\) 7.91704 0.728823
\(119\) 0 0
\(120\) 0.183979 0.0167949
\(121\) −1.46047 −0.132770
\(122\) −2.30254 −0.208463
\(123\) −2.07621 −0.187205
\(124\) 3.41421 0.306605
\(125\) 1.83357 0.163999
\(126\) 0 0
\(127\) −12.0356 −1.06799 −0.533994 0.845488i \(-0.679309\pi\)
−0.533994 + 0.845488i \(0.679309\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.94699 −0.435558
\(130\) −1.07782 −0.0945311
\(131\) 2.47573 0.216306 0.108153 0.994134i \(-0.465506\pi\)
0.108153 + 0.994134i \(0.465506\pi\)
\(132\) −3.08861 −0.268829
\(133\) 0 0
\(134\) −1.53953 −0.132995
\(135\) 0.183979 0.0158344
\(136\) 6.36796 0.546048
\(137\) 17.6230 1.50563 0.752817 0.658229i \(-0.228694\pi\)
0.752817 + 0.658229i \(0.228694\pi\)
\(138\) 1.00000 0.0851257
\(139\) −17.9526 −1.52272 −0.761362 0.648327i \(-0.775469\pi\)
−0.761362 + 0.648327i \(0.775469\pi\)
\(140\) 0 0
\(141\) −5.01241 −0.422121
\(142\) −3.08861 −0.259191
\(143\) 18.0943 1.51312
\(144\) 1.00000 0.0833333
\(145\) 0.862276 0.0716081
\(146\) −5.13097 −0.424642
\(147\) 0 0
\(148\) −1.49042 −0.122512
\(149\) 1.33925 0.109716 0.0548578 0.998494i \(-0.482529\pi\)
0.0548578 + 0.998494i \(0.482529\pi\)
\(150\) 4.96615 0.405485
\(151\) 9.05476 0.736866 0.368433 0.929654i \(-0.379894\pi\)
0.368433 + 0.929654i \(0.379894\pi\)
\(152\) 3.41421 0.276929
\(153\) 6.36796 0.514819
\(154\) 0 0
\(155\) −0.628145 −0.0504538
\(156\) −5.85838 −0.469046
\(157\) −9.98421 −0.796827 −0.398413 0.917206i \(-0.630439\pi\)
−0.398413 + 0.917206i \(0.630439\pi\)
\(158\) −9.08861 −0.723051
\(159\) 1.79972 0.142727
\(160\) −0.183979 −0.0145448
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 2.07621 0.162125
\(165\) 0.568241 0.0442375
\(166\) 0.0175454 0.00136179
\(167\) −12.1845 −0.942864 −0.471432 0.881902i \(-0.656263\pi\)
−0.471432 + 0.881902i \(0.656263\pi\)
\(168\) 0 0
\(169\) 21.3206 1.64005
\(170\) −1.17157 −0.0898555
\(171\) 3.41421 0.261091
\(172\) 4.94699 0.377205
\(173\) −6.21555 −0.472559 −0.236280 0.971685i \(-0.575928\pi\)
−0.236280 + 0.971685i \(0.575928\pi\)
\(174\) 4.68681 0.355306
\(175\) 0 0
\(176\) 3.08861 0.232813
\(177\) −7.91704 −0.595081
\(178\) −9.63380 −0.722084
\(179\) 12.6422 0.944920 0.472460 0.881352i \(-0.343366\pi\)
0.472460 + 0.881352i \(0.343366\pi\)
\(180\) −0.183979 −0.0137130
\(181\) 18.4893 1.37430 0.687150 0.726515i \(-0.258861\pi\)
0.687150 + 0.726515i \(0.258861\pi\)
\(182\) 0 0
\(183\) 2.30254 0.170209
\(184\) −1.00000 −0.0737210
\(185\) 0.274207 0.0201601
\(186\) −3.41421 −0.250342
\(187\) 19.6682 1.43828
\(188\) 5.01241 0.365567
\(189\) 0 0
\(190\) −0.628145 −0.0455704
\(191\) −13.1230 −0.949545 −0.474773 0.880108i \(-0.657470\pi\)
−0.474773 + 0.880108i \(0.657470\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.6038 1.05121 0.525604 0.850729i \(-0.323839\pi\)
0.525604 + 0.850729i \(0.323839\pi\)
\(194\) −4.60385 −0.330537
\(195\) 1.07782 0.0771843
\(196\) 0 0
\(197\) 17.5751 1.25218 0.626088 0.779752i \(-0.284655\pi\)
0.626088 + 0.779752i \(0.284655\pi\)
\(198\) 3.08861 0.219498
\(199\) −1.39791 −0.0990953 −0.0495476 0.998772i \(-0.515778\pi\)
−0.0495476 + 0.998772i \(0.515778\pi\)
\(200\) −4.96615 −0.351160
\(201\) 1.53953 0.108590
\(202\) 5.39667 0.379708
\(203\) 0 0
\(204\) −6.36796 −0.445846
\(205\) −0.381979 −0.0266786
\(206\) 0.141621 0.00986720
\(207\) −1.00000 −0.0695048
\(208\) 5.85838 0.406205
\(209\) 10.5452 0.729426
\(210\) 0 0
\(211\) −17.9419 −1.23517 −0.617584 0.786505i \(-0.711888\pi\)
−0.617584 + 0.786505i \(0.711888\pi\)
\(212\) −1.79972 −0.123605
\(213\) 3.08861 0.211628
\(214\) −16.5452 −1.13101
\(215\) −0.910144 −0.0620713
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 15.9431 1.07980
\(219\) 5.13097 0.346719
\(220\) −0.568241 −0.0383108
\(221\) 37.3059 2.50947
\(222\) 1.49042 0.100030
\(223\) 23.9013 1.60055 0.800273 0.599636i \(-0.204688\pi\)
0.800273 + 0.599636i \(0.204688\pi\)
\(224\) 0 0
\(225\) −4.96615 −0.331077
\(226\) 1.42097 0.0945213
\(227\) −8.81212 −0.584881 −0.292441 0.956284i \(-0.594467\pi\)
−0.292441 + 0.956284i \(0.594467\pi\)
\(228\) −3.41421 −0.226112
\(229\) −0.194772 −0.0128709 −0.00643544 0.999979i \(-0.502048\pi\)
−0.00643544 + 0.999979i \(0.502048\pi\)
\(230\) 0.183979 0.0121312
\(231\) 0 0
\(232\) −4.68681 −0.307704
\(233\) 9.79972 0.642001 0.321000 0.947079i \(-0.395981\pi\)
0.321000 + 0.947079i \(0.395981\pi\)
\(234\) 5.85838 0.382974
\(235\) −0.922179 −0.0601563
\(236\) 7.91704 0.515355
\(237\) 9.08861 0.590369
\(238\) 0 0
\(239\) 10.5835 0.684590 0.342295 0.939593i \(-0.388796\pi\)
0.342295 + 0.939593i \(0.388796\pi\)
\(240\) 0.183979 0.0118758
\(241\) 6.17847 0.397990 0.198995 0.980000i \(-0.436232\pi\)
0.198995 + 0.980000i \(0.436232\pi\)
\(242\) −1.46047 −0.0938825
\(243\) −1.00000 −0.0641500
\(244\) −2.30254 −0.147405
\(245\) 0 0
\(246\) −2.07621 −0.132374
\(247\) 20.0018 1.27268
\(248\) 3.41421 0.216803
\(249\) −0.0175454 −0.00111189
\(250\) 1.83357 0.115965
\(251\) 18.6845 1.17936 0.589678 0.807638i \(-0.299255\pi\)
0.589678 + 0.807638i \(0.299255\pi\)
\(252\) 0 0
\(253\) −3.08861 −0.194179
\(254\) −12.0356 −0.755181
\(255\) 1.17157 0.0733667
\(256\) 1.00000 0.0625000
\(257\) −1.76691 −0.110217 −0.0551084 0.998480i \(-0.517550\pi\)
−0.0551084 + 0.998480i \(0.517550\pi\)
\(258\) −4.94699 −0.307986
\(259\) 0 0
\(260\) −1.07782 −0.0668436
\(261\) −4.68681 −0.290106
\(262\) 2.47573 0.152951
\(263\) −7.58864 −0.467936 −0.233968 0.972244i \(-0.575171\pi\)
−0.233968 + 0.972244i \(0.575171\pi\)
\(264\) −3.08861 −0.190091
\(265\) 0.331111 0.0203400
\(266\) 0 0
\(267\) 9.63380 0.589579
\(268\) −1.53953 −0.0940419
\(269\) 24.6452 1.50264 0.751321 0.659937i \(-0.229417\pi\)
0.751321 + 0.659937i \(0.229417\pi\)
\(270\) 0.183979 0.0111966
\(271\) −18.8261 −1.14361 −0.571803 0.820391i \(-0.693756\pi\)
−0.571803 + 0.820391i \(0.693756\pi\)
\(272\) 6.36796 0.386114
\(273\) 0 0
\(274\) 17.6230 1.06464
\(275\) −15.3385 −0.924948
\(276\) 1.00000 0.0601929
\(277\) 11.8571 0.712426 0.356213 0.934405i \(-0.384068\pi\)
0.356213 + 0.934405i \(0.384068\pi\)
\(278\) −17.9526 −1.07673
\(279\) 3.41421 0.204404
\(280\) 0 0
\(281\) 8.48652 0.506264 0.253132 0.967432i \(-0.418539\pi\)
0.253132 + 0.967432i \(0.418539\pi\)
\(282\) −5.01241 −0.298484
\(283\) 5.34329 0.317626 0.158813 0.987309i \(-0.449233\pi\)
0.158813 + 0.987309i \(0.449233\pi\)
\(284\) −3.08861 −0.183275
\(285\) 0.628145 0.0372081
\(286\) 18.0943 1.06994
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 23.5509 1.38535
\(290\) 0.862276 0.0506346
\(291\) 4.60385 0.269882
\(292\) −5.13097 −0.300267
\(293\) 17.9148 1.04659 0.523296 0.852151i \(-0.324702\pi\)
0.523296 + 0.852151i \(0.324702\pi\)
\(294\) 0 0
\(295\) −1.45657 −0.0848049
\(296\) −1.49042 −0.0866289
\(297\) −3.08861 −0.179220
\(298\) 1.33925 0.0775806
\(299\) −5.85838 −0.338799
\(300\) 4.96615 0.286721
\(301\) 0 0
\(302\) 9.05476 0.521043
\(303\) −5.39667 −0.310030
\(304\) 3.41421 0.195819
\(305\) 0.423621 0.0242564
\(306\) 6.36796 0.364032
\(307\) −6.99339 −0.399134 −0.199567 0.979884i \(-0.563954\pi\)
−0.199567 + 0.979884i \(0.563954\pi\)
\(308\) 0 0
\(309\) −0.141621 −0.00805654
\(310\) −0.628145 −0.0356762
\(311\) 19.9013 1.12850 0.564248 0.825605i \(-0.309166\pi\)
0.564248 + 0.825605i \(0.309166\pi\)
\(312\) −5.85838 −0.331665
\(313\) 17.0440 0.963382 0.481691 0.876341i \(-0.340023\pi\)
0.481691 + 0.876341i \(0.340023\pi\)
\(314\) −9.98421 −0.563442
\(315\) 0 0
\(316\) −9.08861 −0.511274
\(317\) −6.47449 −0.363644 −0.181822 0.983332i \(-0.558199\pi\)
−0.181822 + 0.983332i \(0.558199\pi\)
\(318\) 1.79972 0.100923
\(319\) −14.4757 −0.810485
\(320\) −0.183979 −0.0102848
\(321\) 16.5452 0.923462
\(322\) 0 0
\(323\) 21.7416 1.20973
\(324\) 1.00000 0.0555556
\(325\) −29.0936 −1.61382
\(326\) 8.48528 0.469956
\(327\) −15.9431 −0.881655
\(328\) 2.07621 0.114639
\(329\) 0 0
\(330\) 0.568241 0.0312806
\(331\) −17.4909 −0.961389 −0.480694 0.876888i \(-0.659615\pi\)
−0.480694 + 0.876888i \(0.659615\pi\)
\(332\) 0.0175454 0.000962929 0
\(333\) −1.49042 −0.0816745
\(334\) −12.1845 −0.666706
\(335\) 0.283242 0.0154752
\(336\) 0 0
\(337\) −21.0999 −1.14939 −0.574693 0.818369i \(-0.694879\pi\)
−0.574693 + 0.818369i \(0.694879\pi\)
\(338\) 21.3206 1.15969
\(339\) −1.42097 −0.0771763
\(340\) −1.17157 −0.0635375
\(341\) 10.5452 0.571054
\(342\) 3.41421 0.184620
\(343\) 0 0
\(344\) 4.94699 0.266724
\(345\) −0.183979 −0.00990512
\(346\) −6.21555 −0.334150
\(347\) 12.1976 0.654800 0.327400 0.944886i \(-0.393828\pi\)
0.327400 + 0.944886i \(0.393828\pi\)
\(348\) 4.68681 0.251239
\(349\) 25.3353 1.35617 0.678084 0.734985i \(-0.262811\pi\)
0.678084 + 0.734985i \(0.262811\pi\)
\(350\) 0 0
\(351\) −5.85838 −0.312697
\(352\) 3.08861 0.164624
\(353\) −26.9816 −1.43608 −0.718042 0.696000i \(-0.754961\pi\)
−0.718042 + 0.696000i \(0.754961\pi\)
\(354\) −7.91704 −0.420786
\(355\) 0.568241 0.0301591
\(356\) −9.63380 −0.510590
\(357\) 0 0
\(358\) 12.6422 0.668159
\(359\) 3.43623 0.181357 0.0906786 0.995880i \(-0.471096\pi\)
0.0906786 + 0.995880i \(0.471096\pi\)
\(360\) −0.183979 −0.00969656
\(361\) −7.34315 −0.386481
\(362\) 18.4893 0.971777
\(363\) 1.46047 0.0766547
\(364\) 0 0
\(365\) 0.943993 0.0494108
\(366\) 2.30254 0.120356
\(367\) −21.8481 −1.14046 −0.570231 0.821485i \(-0.693146\pi\)
−0.570231 + 0.821485i \(0.693146\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.07621 0.108083
\(370\) 0.274207 0.0142553
\(371\) 0 0
\(372\) −3.41421 −0.177019
\(373\) −33.9436 −1.75753 −0.878767 0.477251i \(-0.841633\pi\)
−0.878767 + 0.477251i \(0.841633\pi\)
\(374\) 19.6682 1.01702
\(375\) −1.83357 −0.0946849
\(376\) 5.01241 0.258495
\(377\) −27.4571 −1.41411
\(378\) 0 0
\(379\) −28.7563 −1.47711 −0.738555 0.674193i \(-0.764491\pi\)
−0.738555 + 0.674193i \(0.764491\pi\)
\(380\) −0.628145 −0.0322231
\(381\) 12.0356 0.616603
\(382\) −13.1230 −0.671430
\(383\) 2.94648 0.150558 0.0752790 0.997163i \(-0.476015\pi\)
0.0752790 + 0.997163i \(0.476015\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.6038 0.743316
\(387\) 4.94699 0.251470
\(388\) −4.60385 −0.233725
\(389\) 17.7742 0.901187 0.450593 0.892729i \(-0.351212\pi\)
0.450593 + 0.892729i \(0.351212\pi\)
\(390\) 1.07782 0.0545776
\(391\) −6.36796 −0.322042
\(392\) 0 0
\(393\) −2.47573 −0.124884
\(394\) 17.5751 0.885423
\(395\) 1.67212 0.0841333
\(396\) 3.08861 0.155209
\(397\) 22.6026 1.13439 0.567196 0.823583i \(-0.308028\pi\)
0.567196 + 0.823583i \(0.308028\pi\)
\(398\) −1.39791 −0.0700709
\(399\) 0 0
\(400\) −4.96615 −0.248308
\(401\) −31.8047 −1.58825 −0.794126 0.607754i \(-0.792071\pi\)
−0.794126 + 0.607754i \(0.792071\pi\)
\(402\) 1.53953 0.0767849
\(403\) 20.0018 0.996358
\(404\) 5.39667 0.268494
\(405\) −0.183979 −0.00914201
\(406\) 0 0
\(407\) −4.60333 −0.228179
\(408\) −6.36796 −0.315261
\(409\) 11.3082 0.559154 0.279577 0.960123i \(-0.409806\pi\)
0.279577 + 0.960123i \(0.409806\pi\)
\(410\) −0.381979 −0.0188646
\(411\) −17.6230 −0.869279
\(412\) 0.141621 0.00697717
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −0.00322799 −0.000158456 0
\(416\) 5.85838 0.287231
\(417\) 17.9526 0.879145
\(418\) 10.5452 0.515782
\(419\) −30.3833 −1.48432 −0.742161 0.670222i \(-0.766199\pi\)
−0.742161 + 0.670222i \(0.766199\pi\)
\(420\) 0 0
\(421\) 18.4149 0.897486 0.448743 0.893661i \(-0.351872\pi\)
0.448743 + 0.893661i \(0.351872\pi\)
\(422\) −17.9419 −0.873396
\(423\) 5.01241 0.243712
\(424\) −1.79972 −0.0874020
\(425\) −31.6242 −1.53400
\(426\) 3.08861 0.149644
\(427\) 0 0
\(428\) −16.5452 −0.799742
\(429\) −18.0943 −0.873599
\(430\) −0.910144 −0.0438911
\(431\) −0.811922 −0.0391089 −0.0195544 0.999809i \(-0.506225\pi\)
−0.0195544 + 0.999809i \(0.506225\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 35.7020 1.71573 0.857865 0.513876i \(-0.171791\pi\)
0.857865 + 0.513876i \(0.171791\pi\)
\(434\) 0 0
\(435\) −0.862276 −0.0413429
\(436\) 15.9431 0.763536
\(437\) −3.41421 −0.163324
\(438\) 5.13097 0.245167
\(439\) 12.0193 0.573650 0.286825 0.957983i \(-0.407400\pi\)
0.286825 + 0.957983i \(0.407400\pi\)
\(440\) −0.568241 −0.0270898
\(441\) 0 0
\(442\) 37.3059 1.77446
\(443\) −36.0622 −1.71337 −0.856683 0.515844i \(-0.827479\pi\)
−0.856683 + 0.515844i \(0.827479\pi\)
\(444\) 1.49042 0.0707322
\(445\) 1.77242 0.0840208
\(446\) 23.9013 1.13176
\(447\) −1.33925 −0.0633443
\(448\) 0 0
\(449\) 0.0867970 0.00409620 0.00204810 0.999998i \(-0.499348\pi\)
0.00204810 + 0.999998i \(0.499348\pi\)
\(450\) −4.96615 −0.234107
\(451\) 6.41260 0.301958
\(452\) 1.42097 0.0668366
\(453\) −9.05476 −0.425430
\(454\) −8.81212 −0.413573
\(455\) 0 0
\(456\) −3.41421 −0.159885
\(457\) −37.6005 −1.75888 −0.879438 0.476014i \(-0.842081\pi\)
−0.879438 + 0.476014i \(0.842081\pi\)
\(458\) −0.194772 −0.00910109
\(459\) −6.36796 −0.297231
\(460\) 0.183979 0.00857808
\(461\) −33.2804 −1.55002 −0.775011 0.631948i \(-0.782256\pi\)
−0.775011 + 0.631948i \(0.782256\pi\)
\(462\) 0 0
\(463\) −18.4036 −0.855286 −0.427643 0.903948i \(-0.640656\pi\)
−0.427643 + 0.903948i \(0.640656\pi\)
\(464\) −4.68681 −0.217580
\(465\) 0.628145 0.0291295
\(466\) 9.79972 0.453963
\(467\) 32.6327 1.51006 0.755031 0.655690i \(-0.227622\pi\)
0.755031 + 0.655690i \(0.227622\pi\)
\(468\) 5.85838 0.270804
\(469\) 0 0
\(470\) −0.922179 −0.0425370
\(471\) 9.98421 0.460048
\(472\) 7.91704 0.364411
\(473\) 15.2793 0.702545
\(474\) 9.08861 0.417454
\(475\) −16.9555 −0.777972
\(476\) 0 0
\(477\) −1.79972 −0.0824034
\(478\) 10.5835 0.484078
\(479\) −33.0578 −1.51045 −0.755224 0.655467i \(-0.772472\pi\)
−0.755224 + 0.655467i \(0.772472\pi\)
\(480\) 0.183979 0.00839747
\(481\) −8.73145 −0.398120
\(482\) 6.17847 0.281422
\(483\) 0 0
\(484\) −1.46047 −0.0663849
\(485\) 0.847013 0.0384609
\(486\) −1.00000 −0.0453609
\(487\) 24.1562 1.09462 0.547310 0.836930i \(-0.315652\pi\)
0.547310 + 0.836930i \(0.315652\pi\)
\(488\) −2.30254 −0.104231
\(489\) −8.48528 −0.383718
\(490\) 0 0
\(491\) 27.6320 1.24702 0.623508 0.781817i \(-0.285707\pi\)
0.623508 + 0.781817i \(0.285707\pi\)
\(492\) −2.07621 −0.0936026
\(493\) −29.8454 −1.34417
\(494\) 20.0018 0.899922
\(495\) −0.568241 −0.0255405
\(496\) 3.41421 0.153303
\(497\) 0 0
\(498\) −0.0175454 −0.000786228 0
\(499\) −33.9787 −1.52110 −0.760548 0.649282i \(-0.775070\pi\)
−0.760548 + 0.649282i \(0.775070\pi\)
\(500\) 1.83357 0.0819996
\(501\) 12.1845 0.544363
\(502\) 18.6845 0.833931
\(503\) 25.6451 1.14346 0.571729 0.820442i \(-0.306273\pi\)
0.571729 + 0.820442i \(0.306273\pi\)
\(504\) 0 0
\(505\) −0.992875 −0.0441824
\(506\) −3.08861 −0.137306
\(507\) −21.3206 −0.946881
\(508\) −12.0356 −0.533994
\(509\) 29.1631 1.29263 0.646315 0.763071i \(-0.276309\pi\)
0.646315 + 0.763071i \(0.276309\pi\)
\(510\) 1.17157 0.0518781
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −3.41421 −0.150741
\(514\) −1.76691 −0.0779350
\(515\) −0.0260553 −0.00114814
\(516\) −4.94699 −0.217779
\(517\) 15.4814 0.680871
\(518\) 0 0
\(519\) 6.21555 0.272832
\(520\) −1.07782 −0.0472656
\(521\) 43.0011 1.88391 0.941956 0.335736i \(-0.108985\pi\)
0.941956 + 0.335736i \(0.108985\pi\)
\(522\) −4.68681 −0.205136
\(523\) 19.3202 0.844815 0.422407 0.906406i \(-0.361185\pi\)
0.422407 + 0.906406i \(0.361185\pi\)
\(524\) 2.47573 0.108153
\(525\) 0 0
\(526\) −7.58864 −0.330881
\(527\) 21.7416 0.947078
\(528\) −3.08861 −0.134415
\(529\) 1.00000 0.0434783
\(530\) 0.331111 0.0143825
\(531\) 7.91704 0.343570
\(532\) 0 0
\(533\) 12.1632 0.526847
\(534\) 9.63380 0.416895
\(535\) 3.04397 0.131602
\(536\) −1.53953 −0.0664976
\(537\) −12.6422 −0.545550
\(538\) 24.6452 1.06253
\(539\) 0 0
\(540\) 0.183979 0.00791721
\(541\) −33.7203 −1.44975 −0.724874 0.688882i \(-0.758102\pi\)
−0.724874 + 0.688882i \(0.758102\pi\)
\(542\) −18.8261 −0.808652
\(543\) −18.4893 −0.793453
\(544\) 6.36796 0.273024
\(545\) −2.93320 −0.125644
\(546\) 0 0
\(547\) −27.6275 −1.18127 −0.590633 0.806940i \(-0.701122\pi\)
−0.590633 + 0.806940i \(0.701122\pi\)
\(548\) 17.6230 0.752817
\(549\) −2.30254 −0.0982702
\(550\) −15.3385 −0.654037
\(551\) −16.0018 −0.681698
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 11.8571 0.503761
\(555\) −0.274207 −0.0116394
\(556\) −17.9526 −0.761362
\(557\) −1.65685 −0.0702032 −0.0351016 0.999384i \(-0.511175\pi\)
−0.0351016 + 0.999384i \(0.511175\pi\)
\(558\) 3.41421 0.144535
\(559\) 28.9814 1.22578
\(560\) 0 0
\(561\) −19.6682 −0.830390
\(562\) 8.48652 0.357982
\(563\) −38.9217 −1.64035 −0.820176 0.572111i \(-0.806125\pi\)
−0.820176 + 0.572111i \(0.806125\pi\)
\(564\) −5.01241 −0.211060
\(565\) −0.261428 −0.0109984
\(566\) 5.34329 0.224595
\(567\) 0 0
\(568\) −3.08861 −0.129595
\(569\) 10.3093 0.432188 0.216094 0.976373i \(-0.430668\pi\)
0.216094 + 0.976373i \(0.430668\pi\)
\(570\) 0.628145 0.0263101
\(571\) 23.3137 0.975648 0.487824 0.872942i \(-0.337791\pi\)
0.487824 + 0.872942i \(0.337791\pi\)
\(572\) 18.0943 0.756559
\(573\) 13.1230 0.548220
\(574\) 0 0
\(575\) 4.96615 0.207103
\(576\) 1.00000 0.0416667
\(577\) −33.2716 −1.38512 −0.692558 0.721362i \(-0.743516\pi\)
−0.692558 + 0.721362i \(0.743516\pi\)
\(578\) 23.5509 0.979588
\(579\) −14.6038 −0.606915
\(580\) 0.862276 0.0358040
\(581\) 0 0
\(582\) 4.60385 0.190836
\(583\) −5.55863 −0.230215
\(584\) −5.13097 −0.212321
\(585\) −1.07782 −0.0445624
\(586\) 17.9148 0.740052
\(587\) −35.4328 −1.46247 −0.731234 0.682127i \(-0.761055\pi\)
−0.731234 + 0.682127i \(0.761055\pi\)
\(588\) 0 0
\(589\) 11.6569 0.480312
\(590\) −1.45657 −0.0599661
\(591\) −17.5751 −0.722945
\(592\) −1.49042 −0.0612559
\(593\) −8.28604 −0.340267 −0.170133 0.985421i \(-0.554420\pi\)
−0.170133 + 0.985421i \(0.554420\pi\)
\(594\) −3.08861 −0.126727
\(595\) 0 0
\(596\) 1.33925 0.0548578
\(597\) 1.39791 0.0572127
\(598\) −5.85838 −0.239567
\(599\) 17.3392 0.708463 0.354231 0.935158i \(-0.384743\pi\)
0.354231 + 0.935158i \(0.384743\pi\)
\(600\) 4.96615 0.202742
\(601\) −18.1214 −0.739186 −0.369593 0.929194i \(-0.620503\pi\)
−0.369593 + 0.929194i \(0.620503\pi\)
\(602\) 0 0
\(603\) −1.53953 −0.0626946
\(604\) 9.05476 0.368433
\(605\) 0.268696 0.0109240
\(606\) −5.39667 −0.219225
\(607\) −23.3663 −0.948411 −0.474205 0.880414i \(-0.657265\pi\)
−0.474205 + 0.880414i \(0.657265\pi\)
\(608\) 3.41421 0.138465
\(609\) 0 0
\(610\) 0.423621 0.0171519
\(611\) 29.3646 1.18796
\(612\) 6.36796 0.257409
\(613\) −37.9353 −1.53219 −0.766096 0.642726i \(-0.777803\pi\)
−0.766096 + 0.642726i \(0.777803\pi\)
\(614\) −6.99339 −0.282230
\(615\) 0.381979 0.0154029
\(616\) 0 0
\(617\) 32.6274 1.31353 0.656765 0.754095i \(-0.271924\pi\)
0.656765 + 0.754095i \(0.271924\pi\)
\(618\) −0.141621 −0.00569683
\(619\) −31.1150 −1.25062 −0.625308 0.780378i \(-0.715027\pi\)
−0.625308 + 0.780378i \(0.715027\pi\)
\(620\) −0.628145 −0.0252269
\(621\) 1.00000 0.0401286
\(622\) 19.9013 0.797968
\(623\) 0 0
\(624\) −5.85838 −0.234523
\(625\) 24.4934 0.979737
\(626\) 17.0440 0.681214
\(627\) −10.5452 −0.421134
\(628\) −9.98421 −0.398413
\(629\) −9.49093 −0.378428
\(630\) 0 0
\(631\) 19.9556 0.794422 0.397211 0.917727i \(-0.369978\pi\)
0.397211 + 0.917727i \(0.369978\pi\)
\(632\) −9.08861 −0.361526
\(633\) 17.9419 0.713125
\(634\) −6.47449 −0.257135
\(635\) 2.21430 0.0878719
\(636\) 1.79972 0.0713634
\(637\) 0 0
\(638\) −14.4757 −0.573100
\(639\) −3.08861 −0.122184
\(640\) −0.183979 −0.00727242
\(641\) −1.72123 −0.0679844 −0.0339922 0.999422i \(-0.510822\pi\)
−0.0339922 + 0.999422i \(0.510822\pi\)
\(642\) 16.5452 0.652986
\(643\) 6.02885 0.237755 0.118877 0.992909i \(-0.462070\pi\)
0.118877 + 0.992909i \(0.462070\pi\)
\(644\) 0 0
\(645\) 0.910144 0.0358369
\(646\) 21.7416 0.855411
\(647\) 19.9498 0.784309 0.392155 0.919899i \(-0.371730\pi\)
0.392155 + 0.919899i \(0.371730\pi\)
\(648\) 1.00000 0.0392837
\(649\) 24.4527 0.959851
\(650\) −29.0936 −1.14114
\(651\) 0 0
\(652\) 8.48528 0.332309
\(653\) 41.8484 1.63765 0.818827 0.574040i \(-0.194625\pi\)
0.818827 + 0.574040i \(0.194625\pi\)
\(654\) −15.9431 −0.623424
\(655\) −0.455483 −0.0177972
\(656\) 2.07621 0.0810623
\(657\) −5.13097 −0.200178
\(658\) 0 0
\(659\) 17.6721 0.688408 0.344204 0.938895i \(-0.388149\pi\)
0.344204 + 0.938895i \(0.388149\pi\)
\(660\) 0.568241 0.0221188
\(661\) −45.0019 −1.75037 −0.875186 0.483787i \(-0.839261\pi\)
−0.875186 + 0.483787i \(0.839261\pi\)
\(662\) −17.4909 −0.679804
\(663\) −37.3059 −1.44884
\(664\) 0.0175454 0.000680894 0
\(665\) 0 0
\(666\) −1.49042 −0.0577526
\(667\) 4.68681 0.181474
\(668\) −12.1845 −0.471432
\(669\) −23.9013 −0.924076
\(670\) 0.283242 0.0109426
\(671\) −7.11167 −0.274543
\(672\) 0 0
\(673\) 18.0948 0.697503 0.348751 0.937215i \(-0.386606\pi\)
0.348751 + 0.937215i \(0.386606\pi\)
\(674\) −21.0999 −0.812739
\(675\) 4.96615 0.191147
\(676\) 21.3206 0.820023
\(677\) 34.4000 1.32210 0.661050 0.750341i \(-0.270111\pi\)
0.661050 + 0.750341i \(0.270111\pi\)
\(678\) −1.42097 −0.0545719
\(679\) 0 0
\(680\) −1.17157 −0.0449278
\(681\) 8.81212 0.337681
\(682\) 10.5452 0.403796
\(683\) −39.2300 −1.50110 −0.750548 0.660816i \(-0.770210\pi\)
−0.750548 + 0.660816i \(0.770210\pi\)
\(684\) 3.41421 0.130546
\(685\) −3.24227 −0.123881
\(686\) 0 0
\(687\) 0.194772 0.00743101
\(688\) 4.94699 0.188602
\(689\) −10.5434 −0.401673
\(690\) −0.183979 −0.00700397
\(691\) −23.2054 −0.882776 −0.441388 0.897316i \(-0.645514\pi\)
−0.441388 + 0.897316i \(0.645514\pi\)
\(692\) −6.21555 −0.236280
\(693\) 0 0
\(694\) 12.1976 0.463013
\(695\) 3.30292 0.125287
\(696\) 4.68681 0.177653
\(697\) 13.2212 0.500789
\(698\) 25.3353 0.958955
\(699\) −9.79972 −0.370659
\(700\) 0 0
\(701\) −6.82271 −0.257690 −0.128845 0.991665i \(-0.541127\pi\)
−0.128845 + 0.991665i \(0.541127\pi\)
\(702\) −5.85838 −0.221110
\(703\) −5.08861 −0.191921
\(704\) 3.08861 0.116406
\(705\) 0.922179 0.0347313
\(706\) −26.9816 −1.01546
\(707\) 0 0
\(708\) −7.91704 −0.297541
\(709\) −36.6309 −1.37570 −0.687852 0.725851i \(-0.741446\pi\)
−0.687852 + 0.725851i \(0.741446\pi\)
\(710\) 0.568241 0.0213257
\(711\) −9.08861 −0.340850
\(712\) −9.63380 −0.361042
\(713\) −3.41421 −0.127863
\(714\) 0 0
\(715\) −3.32897 −0.124496
\(716\) 12.6422 0.472460
\(717\) −10.5835 −0.395248
\(718\) 3.43623 0.128239
\(719\) −43.5746 −1.62506 −0.812529 0.582920i \(-0.801910\pi\)
−0.812529 + 0.582920i \(0.801910\pi\)
\(720\) −0.183979 −0.00685651
\(721\) 0 0
\(722\) −7.34315 −0.273284
\(723\) −6.17847 −0.229780
\(724\) 18.4893 0.687150
\(725\) 23.2754 0.864426
\(726\) 1.46047 0.0542031
\(727\) 15.5859 0.578050 0.289025 0.957322i \(-0.406669\pi\)
0.289025 + 0.957322i \(0.406669\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0.943993 0.0349387
\(731\) 31.5022 1.16515
\(732\) 2.30254 0.0851045
\(733\) 1.66485 0.0614926 0.0307463 0.999527i \(-0.490212\pi\)
0.0307463 + 0.999527i \(0.490212\pi\)
\(734\) −21.8481 −0.806428
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −4.75502 −0.175153
\(738\) 2.07621 0.0764262
\(739\) 27.3856 1.00740 0.503699 0.863879i \(-0.331972\pi\)
0.503699 + 0.863879i \(0.331972\pi\)
\(740\) 0.274207 0.0100800
\(741\) −20.0018 −0.734783
\(742\) 0 0
\(743\) −28.1775 −1.03373 −0.516866 0.856066i \(-0.672902\pi\)
−0.516866 + 0.856066i \(0.672902\pi\)
\(744\) −3.41421 −0.125171
\(745\) −0.246394 −0.00902718
\(746\) −33.9436 −1.24276
\(747\) 0.0175454 0.000641953 0
\(748\) 19.6682 0.719139
\(749\) 0 0
\(750\) −1.83357 −0.0669524
\(751\) 22.1439 0.808042 0.404021 0.914750i \(-0.367612\pi\)
0.404021 + 0.914750i \(0.367612\pi\)
\(752\) 5.01241 0.182784
\(753\) −18.6845 −0.680902
\(754\) −27.4571 −0.999928
\(755\) −1.66589 −0.0606279
\(756\) 0 0
\(757\) −11.1748 −0.406155 −0.203078 0.979163i \(-0.565094\pi\)
−0.203078 + 0.979163i \(0.565094\pi\)
\(758\) −28.7563 −1.04447
\(759\) 3.08861 0.112110
\(760\) −0.628145 −0.0227852
\(761\) −8.27141 −0.299838 −0.149919 0.988698i \(-0.547901\pi\)
−0.149919 + 0.988698i \(0.547901\pi\)
\(762\) 12.0356 0.436004
\(763\) 0 0
\(764\) −13.1230 −0.474773
\(765\) −1.17157 −0.0423583
\(766\) 2.94648 0.106461
\(767\) 46.3810 1.67472
\(768\) −1.00000 −0.0360844
\(769\) −5.44007 −0.196174 −0.0980869 0.995178i \(-0.531272\pi\)
−0.0980869 + 0.995178i \(0.531272\pi\)
\(770\) 0 0
\(771\) 1.76691 0.0636337
\(772\) 14.6038 0.525604
\(773\) 42.6671 1.53463 0.767314 0.641272i \(-0.221593\pi\)
0.767314 + 0.641272i \(0.221593\pi\)
\(774\) 4.94699 0.177816
\(775\) −16.9555 −0.609060
\(776\) −4.60385 −0.165268
\(777\) 0 0
\(778\) 17.7742 0.637235
\(779\) 7.08861 0.253976
\(780\) 1.07782 0.0385922
\(781\) −9.53953 −0.341351
\(782\) −6.36796 −0.227718
\(783\) 4.68681 0.167493
\(784\) 0 0
\(785\) 1.83689 0.0655614
\(786\) −2.47573 −0.0883064
\(787\) −7.86897 −0.280498 −0.140249 0.990116i \(-0.544790\pi\)
−0.140249 + 0.990116i \(0.544790\pi\)
\(788\) 17.5751 0.626088
\(789\) 7.58864 0.270163
\(790\) 1.67212 0.0594913
\(791\) 0 0
\(792\) 3.08861 0.109749
\(793\) −13.4892 −0.479015
\(794\) 22.6026 0.802136
\(795\) −0.331111 −0.0117433
\(796\) −1.39791 −0.0495476
\(797\) 13.0748 0.463133 0.231567 0.972819i \(-0.425615\pi\)
0.231567 + 0.972819i \(0.425615\pi\)
\(798\) 0 0
\(799\) 31.9188 1.12921
\(800\) −4.96615 −0.175580
\(801\) −9.63380 −0.340394
\(802\) −31.8047 −1.12306
\(803\) −15.8476 −0.559249
\(804\) 1.53953 0.0542951
\(805\) 0 0
\(806\) 20.0018 0.704532
\(807\) −24.6452 −0.867551
\(808\) 5.39667 0.189854
\(809\) −28.0188 −0.985088 −0.492544 0.870288i \(-0.663933\pi\)
−0.492544 + 0.870288i \(0.663933\pi\)
\(810\) −0.183979 −0.00646438
\(811\) −17.1104 −0.600828 −0.300414 0.953809i \(-0.597125\pi\)
−0.300414 + 0.953809i \(0.597125\pi\)
\(812\) 0 0
\(813\) 18.8261 0.660262
\(814\) −4.60333 −0.161347
\(815\) −1.56112 −0.0546835
\(816\) −6.36796 −0.222923
\(817\) 16.8901 0.590909
\(818\) 11.3082 0.395382
\(819\) 0 0
\(820\) −0.381979 −0.0133393
\(821\) 18.0857 0.631197 0.315598 0.948893i \(-0.397795\pi\)
0.315598 + 0.948893i \(0.397795\pi\)
\(822\) −17.6230 −0.614673
\(823\) −24.7219 −0.861751 −0.430876 0.902411i \(-0.641795\pi\)
−0.430876 + 0.902411i \(0.641795\pi\)
\(824\) 0.141621 0.00493360
\(825\) 15.3385 0.534019
\(826\) 0 0
\(827\) 9.39843 0.326815 0.163408 0.986559i \(-0.447751\pi\)
0.163408 + 0.986559i \(0.447751\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −2.43566 −0.0845938 −0.0422969 0.999105i \(-0.513468\pi\)
−0.0422969 + 0.999105i \(0.513468\pi\)
\(830\) −0.00322799 −0.000112045 0
\(831\) −11.8571 −0.411320
\(832\) 5.85838 0.203103
\(833\) 0 0
\(834\) 17.9526 0.621649
\(835\) 2.24170 0.0775771
\(836\) 10.5452 0.364713
\(837\) −3.41421 −0.118012
\(838\) −30.3833 −1.04957
\(839\) −6.62007 −0.228550 −0.114275 0.993449i \(-0.536455\pi\)
−0.114275 + 0.993449i \(0.536455\pi\)
\(840\) 0 0
\(841\) −7.03385 −0.242546
\(842\) 18.4149 0.634618
\(843\) −8.48652 −0.292291
\(844\) −17.9419 −0.617584
\(845\) −3.92255 −0.134940
\(846\) 5.01241 0.172330
\(847\) 0 0
\(848\) −1.79972 −0.0618026
\(849\) −5.34329 −0.183381
\(850\) −31.6242 −1.08470
\(851\) 1.49042 0.0510909
\(852\) 3.08861 0.105814
\(853\) 7.63256 0.261334 0.130667 0.991426i \(-0.458288\pi\)
0.130667 + 0.991426i \(0.458288\pi\)
\(854\) 0 0
\(855\) −0.628145 −0.0214821
\(856\) −16.5452 −0.565503
\(857\) 14.0772 0.480869 0.240435 0.970665i \(-0.422710\pi\)
0.240435 + 0.970665i \(0.422710\pi\)
\(858\) −18.0943 −0.617728
\(859\) −2.16468 −0.0738578 −0.0369289 0.999318i \(-0.511758\pi\)
−0.0369289 + 0.999318i \(0.511758\pi\)
\(860\) −0.910144 −0.0310357
\(861\) 0 0
\(862\) −0.811922 −0.0276542
\(863\) −4.31054 −0.146732 −0.0733662 0.997305i \(-0.523374\pi\)
−0.0733662 + 0.997305i \(0.523374\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 1.14353 0.0388813
\(866\) 35.7020 1.21320
\(867\) −23.5509 −0.799830
\(868\) 0 0
\(869\) −28.0712 −0.952251
\(870\) −0.862276 −0.0292339
\(871\) −9.01916 −0.305603
\(872\) 15.9431 0.539901
\(873\) −4.60385 −0.155817
\(874\) −3.41421 −0.115487
\(875\) 0 0
\(876\) 5.13097 0.173359
\(877\) −13.8015 −0.466043 −0.233021 0.972472i \(-0.574861\pi\)
−0.233021 + 0.972472i \(0.574861\pi\)
\(878\) 12.0193 0.405632
\(879\) −17.9148 −0.604250
\(880\) −0.568241 −0.0191554
\(881\) −23.6955 −0.798320 −0.399160 0.916881i \(-0.630698\pi\)
−0.399160 + 0.916881i \(0.630698\pi\)
\(882\) 0 0
\(883\) 54.3484 1.82897 0.914485 0.404620i \(-0.132596\pi\)
0.914485 + 0.404620i \(0.132596\pi\)
\(884\) 37.3059 1.25473
\(885\) 1.45657 0.0489621
\(886\) −36.0622 −1.21153
\(887\) 14.5547 0.488697 0.244349 0.969687i \(-0.421426\pi\)
0.244349 + 0.969687i \(0.421426\pi\)
\(888\) 1.49042 0.0500152
\(889\) 0 0
\(890\) 1.77242 0.0594117
\(891\) 3.08861 0.103472
\(892\) 23.9013 0.800273
\(893\) 17.1134 0.572679
\(894\) −1.33925 −0.0447912
\(895\) −2.32590 −0.0777462
\(896\) 0 0
\(897\) 5.85838 0.195606
\(898\) 0.0867970 0.00289645
\(899\) −16.0018 −0.533688
\(900\) −4.96615 −0.165538
\(901\) −11.4605 −0.381806
\(902\) 6.41260 0.213516
\(903\) 0 0
\(904\) 1.42097 0.0472606
\(905\) −3.40165 −0.113075
\(906\) −9.05476 −0.300824
\(907\) −6.69884 −0.222431 −0.111216 0.993796i \(-0.535474\pi\)
−0.111216 + 0.993796i \(0.535474\pi\)
\(908\) −8.81212 −0.292441
\(909\) 5.39667 0.178996
\(910\) 0 0
\(911\) −14.3660 −0.475966 −0.237983 0.971269i \(-0.576486\pi\)
−0.237983 + 0.971269i \(0.576486\pi\)
\(912\) −3.41421 −0.113056
\(913\) 0.0541910 0.00179346
\(914\) −37.6005 −1.24371
\(915\) −0.423621 −0.0140045
\(916\) −0.194772 −0.00643544
\(917\) 0 0
\(918\) −6.36796 −0.210174
\(919\) 26.4314 0.871891 0.435945 0.899973i \(-0.356414\pi\)
0.435945 + 0.899973i \(0.356414\pi\)
\(920\) 0.183979 0.00606562
\(921\) 6.99339 0.230440
\(922\) −33.2804 −1.09603
\(923\) −18.0943 −0.595580
\(924\) 0 0
\(925\) 7.40165 0.243365
\(926\) −18.4036 −0.604779
\(927\) 0.141621 0.00465144
\(928\) −4.68681 −0.153852
\(929\) −42.6352 −1.39882 −0.699408 0.714723i \(-0.746553\pi\)
−0.699408 + 0.714723i \(0.746553\pi\)
\(930\) 0.628145 0.0205977
\(931\) 0 0
\(932\) 9.79972 0.321000
\(933\) −19.9013 −0.651538
\(934\) 32.6327 1.06777
\(935\) −3.61854 −0.118339
\(936\) 5.85838 0.191487
\(937\) 15.7651 0.515025 0.257512 0.966275i \(-0.417097\pi\)
0.257512 + 0.966275i \(0.417097\pi\)
\(938\) 0 0
\(939\) −17.0440 −0.556209
\(940\) −0.922179 −0.0300782
\(941\) −24.8387 −0.809718 −0.404859 0.914379i \(-0.632679\pi\)
−0.404859 + 0.914379i \(0.632679\pi\)
\(942\) 9.98421 0.325303
\(943\) −2.07621 −0.0676106
\(944\) 7.91704 0.257678
\(945\) 0 0
\(946\) 15.2793 0.496774
\(947\) −27.4245 −0.891176 −0.445588 0.895238i \(-0.647005\pi\)
−0.445588 + 0.895238i \(0.647005\pi\)
\(948\) 9.08861 0.295184
\(949\) −30.0592 −0.975762
\(950\) −16.9555 −0.550109
\(951\) 6.47449 0.209950
\(952\) 0 0
\(953\) −13.2045 −0.427735 −0.213867 0.976863i \(-0.568606\pi\)
−0.213867 + 0.976863i \(0.568606\pi\)
\(954\) −1.79972 −0.0582680
\(955\) 2.41436 0.0781267
\(956\) 10.5835 0.342295
\(957\) 14.4757 0.467934
\(958\) −33.0578 −1.06805
\(959\) 0 0
\(960\) 0.183979 0.00593791
\(961\) −19.3431 −0.623972
\(962\) −8.73145 −0.281513
\(963\) −16.5452 −0.533161
\(964\) 6.17847 0.198995
\(965\) −2.68681 −0.0864913
\(966\) 0 0
\(967\) −45.8950 −1.47588 −0.737942 0.674864i \(-0.764202\pi\)
−0.737942 + 0.674864i \(0.764202\pi\)
\(968\) −1.46047 −0.0469412
\(969\) −21.7416 −0.698440
\(970\) 0.847013 0.0271959
\(971\) 22.5411 0.723380 0.361690 0.932298i \(-0.382200\pi\)
0.361690 + 0.932298i \(0.382200\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 24.1562 0.774013
\(975\) 29.0936 0.931741
\(976\) −2.30254 −0.0737027
\(977\) 25.9073 0.828847 0.414424 0.910084i \(-0.363983\pi\)
0.414424 + 0.910084i \(0.363983\pi\)
\(978\) −8.48528 −0.271329
\(979\) −29.7551 −0.950976
\(980\) 0 0
\(981\) 15.9431 0.509024
\(982\) 27.6320 0.881774
\(983\) 34.7320 1.10778 0.553889 0.832591i \(-0.313143\pi\)
0.553889 + 0.832591i \(0.313143\pi\)
\(984\) −2.07621 −0.0661871
\(985\) −3.23346 −0.103027
\(986\) −29.8454 −0.950471
\(987\) 0 0
\(988\) 20.0018 0.636341
\(989\) −4.94699 −0.157305
\(990\) −0.568241 −0.0180599
\(991\) −32.9187 −1.04570 −0.522849 0.852425i \(-0.675131\pi\)
−0.522849 + 0.852425i \(0.675131\pi\)
\(992\) 3.41421 0.108401
\(993\) 17.4909 0.555058
\(994\) 0 0
\(995\) 0.257187 0.00815337
\(996\) −0.0175454 −0.000555947 0
\(997\) −12.6320 −0.400059 −0.200029 0.979790i \(-0.564104\pi\)
−0.200029 + 0.979790i \(0.564104\pi\)
\(998\) −33.9787 −1.07558
\(999\) 1.49042 0.0471548
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 6762.2.a.cn.1.2 4
7.6 odd 2 6762.2.a.co.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.cn.1.2 4 1.1 even 1 trivial
6762.2.a.co.1.3 yes 4 7.6 odd 2