Properties

Label 6762.2.a.co.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
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: \(1\)
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 1.1
Root \(2.87996\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.87996 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.87996 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.87996 q^{10} +2.65867 q^{11} +1.00000 q^{12} -4.19292 q^{13} -3.87996 q^{15} +1.00000 q^{16} +1.75992 q^{17} +1.00000 q^{18} -0.585786 q^{19} -3.87996 q^{20} +2.65867 q^{22} -1.00000 q^{23} +1.00000 q^{24} +10.0541 q^{25} -4.19292 q^{26} +1.00000 q^{27} +2.63551 q^{29} -3.87996 q^{30} -0.585786 q^{31} +1.00000 q^{32} +2.65867 q^{33} +1.75992 q^{34} +1.00000 q^{36} -7.95284 q^{37} -0.585786 q^{38} -4.19292 q^{39} -3.87996 q^{40} -11.3671 q^{41} +2.85158 q^{43} +2.65867 q^{44} -3.87996 q^{45} -1.00000 q^{46} +4.70839 q^{47} +1.00000 q^{48} +10.0541 q^{50} +1.75992 q^{51} -4.19292 q^{52} -4.55560 q^{53} +1.00000 q^{54} -10.3155 q^{55} -0.585786 q^{57} +2.63551 q^{58} -1.83024 q^{59} -3.87996 q^{60} +1.80005 q^{61} -0.585786 q^{62} +1.00000 q^{64} +16.2684 q^{65} +2.65867 q^{66} +0.931495 q^{67} +1.75992 q^{68} -1.00000 q^{69} -2.65867 q^{71} +1.00000 q^{72} -1.02838 q^{73} -7.95284 q^{74} +10.0541 q^{75} -0.585786 q^{76} -4.19292 q^{78} -8.65867 q^{79} -3.87996 q^{80} +1.00000 q^{81} -11.3671 q^{82} -13.7297 q^{83} -6.82843 q^{85} +2.85158 q^{86} +2.63551 q^{87} +2.65867 q^{88} +0.216077 q^{89} -3.87996 q^{90} -1.00000 q^{92} -0.585786 q^{93} +4.70839 q^{94} +2.27283 q^{95} +1.00000 q^{96} -8.80527 q^{97} +2.65867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 6q^{5} + 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 6q^{5} + 4q^{6} + 4q^{8} + 4q^{9} - 6q^{10} + 4q^{12} - 10q^{13} - 6q^{15} + 4q^{16} - 12q^{17} + 4q^{18} - 8q^{19} - 6q^{20} - 4q^{23} + 4q^{24} + 6q^{25} - 10q^{26} + 4q^{27} + 6q^{29} - 6q^{30} - 8q^{31} + 4q^{32} - 12q^{34} + 4q^{36} - 6q^{37} - 8q^{38} - 10q^{39} - 6q^{40} - 14q^{41} - 6q^{43} - 6q^{45} - 4q^{46} - 2q^{47} + 4q^{48} + 6q^{50} - 12q^{51} - 10q^{52} - 4q^{53} + 4q^{54} - 8q^{55} - 8q^{57} + 6q^{58} - 8q^{59} - 6q^{60} - 12q^{61} - 8q^{62} + 4q^{64} + 6q^{65} - 4q^{67} - 12q^{68} - 4q^{69} + 4q^{72} - 12q^{73} - 6q^{74} + 6q^{75} - 8q^{76} - 10q^{78} - 24q^{79} - 6q^{80} + 4q^{81} - 14q^{82} - 16q^{83} - 16q^{85} - 6q^{86} + 6q^{87} - 12q^{89} - 6q^{90} - 4q^{92} - 8q^{93} - 2q^{94} + 12q^{95} + 4q^{96} - 30q^{97} + O(q^{100}) \)

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
\(5\) −3.87996 −1.73517 −0.867586 0.497288i \(-0.834329\pi\)
−0.867586 + 0.497288i \(0.834329\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.87996 −1.22695
\(11\) 2.65867 0.801618 0.400809 0.916162i \(-0.368729\pi\)
0.400809 + 0.916162i \(0.368729\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.19292 −1.16291 −0.581453 0.813580i \(-0.697516\pi\)
−0.581453 + 0.813580i \(0.697516\pi\)
\(14\) 0 0
\(15\) −3.87996 −1.00180
\(16\) 1.00000 0.250000
\(17\) 1.75992 0.426844 0.213422 0.976960i \(-0.431539\pi\)
0.213422 + 0.976960i \(0.431539\pi\)
\(18\) 1.00000 0.235702
\(19\) −0.585786 −0.134389 −0.0671943 0.997740i \(-0.521405\pi\)
−0.0671943 + 0.997740i \(0.521405\pi\)
\(20\) −3.87996 −0.867586
\(21\) 0 0
\(22\) 2.65867 0.566830
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 10.0541 2.01082
\(26\) −4.19292 −0.822299
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.63551 0.489402 0.244701 0.969599i \(-0.421310\pi\)
0.244701 + 0.969599i \(0.421310\pi\)
\(30\) −3.87996 −0.708381
\(31\) −0.585786 −0.105210 −0.0526052 0.998615i \(-0.516752\pi\)
−0.0526052 + 0.998615i \(0.516752\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.65867 0.462814
\(34\) 1.75992 0.301824
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.95284 −1.30744 −0.653719 0.756737i \(-0.726792\pi\)
−0.653719 + 0.756737i \(0.726792\pi\)
\(38\) −0.585786 −0.0950271
\(39\) −4.19292 −0.671404
\(40\) −3.87996 −0.613476
\(41\) −11.3671 −1.77524 −0.887618 0.460581i \(-0.847641\pi\)
−0.887618 + 0.460581i \(0.847641\pi\)
\(42\) 0 0
\(43\) 2.85158 0.434863 0.217431 0.976076i \(-0.430232\pi\)
0.217431 + 0.976076i \(0.430232\pi\)
\(44\) 2.65867 0.400809
\(45\) −3.87996 −0.578390
\(46\) −1.00000 −0.147442
\(47\) 4.70839 0.686789 0.343394 0.939191i \(-0.388423\pi\)
0.343394 + 0.939191i \(0.388423\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 10.0541 1.42186
\(51\) 1.75992 0.246438
\(52\) −4.19292 −0.581453
\(53\) −4.55560 −0.625760 −0.312880 0.949793i \(-0.601294\pi\)
−0.312880 + 0.949793i \(0.601294\pi\)
\(54\) 1.00000 0.136083
\(55\) −10.3155 −1.39094
\(56\) 0 0
\(57\) −0.585786 −0.0775893
\(58\) 2.63551 0.346059
\(59\) −1.83024 −0.238277 −0.119138 0.992878i \(-0.538013\pi\)
−0.119138 + 0.992878i \(0.538013\pi\)
\(60\) −3.87996 −0.500901
\(61\) 1.80005 0.230473 0.115236 0.993338i \(-0.463237\pi\)
0.115236 + 0.993338i \(0.463237\pi\)
\(62\) −0.585786 −0.0743950
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 16.2684 2.01784
\(66\) 2.65867 0.327259
\(67\) 0.931495 0.113800 0.0569001 0.998380i \(-0.481878\pi\)
0.0569001 + 0.998380i \(0.481878\pi\)
\(68\) 1.75992 0.213422
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −2.65867 −0.315526 −0.157763 0.987477i \(-0.550428\pi\)
−0.157763 + 0.987477i \(0.550428\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.02838 −0.120362 −0.0601811 0.998187i \(-0.519168\pi\)
−0.0601811 + 0.998187i \(0.519168\pi\)
\(74\) −7.95284 −0.924499
\(75\) 10.0541 1.16095
\(76\) −0.585786 −0.0671943
\(77\) 0 0
\(78\) −4.19292 −0.474755
\(79\) −8.65867 −0.974176 −0.487088 0.873353i \(-0.661941\pi\)
−0.487088 + 0.873353i \(0.661941\pi\)
\(80\) −3.87996 −0.433793
\(81\) 1.00000 0.111111
\(82\) −11.3671 −1.25528
\(83\) −13.7297 −1.50703 −0.753517 0.657428i \(-0.771644\pi\)
−0.753517 + 0.657428i \(0.771644\pi\)
\(84\) 0 0
\(85\) −6.82843 −0.740647
\(86\) 2.85158 0.307494
\(87\) 2.63551 0.282556
\(88\) 2.65867 0.283415
\(89\) 0.216077 0.0229041 0.0114520 0.999934i \(-0.496355\pi\)
0.0114520 + 0.999934i \(0.496355\pi\)
\(90\) −3.87996 −0.408984
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −0.585786 −0.0607432
\(94\) 4.70839 0.485633
\(95\) 2.27283 0.233187
\(96\) 1.00000 0.102062
\(97\) −8.80527 −0.894040 −0.447020 0.894524i \(-0.647515\pi\)
−0.447020 + 0.894524i \(0.647515\pi\)
\(98\) 0 0
\(99\) 2.65867 0.267206
\(100\) 10.0541 1.00541
\(101\) 11.1439 1.10886 0.554432 0.832229i \(-0.312936\pi\)
0.554432 + 0.832229i \(0.312936\pi\)
\(102\) 1.75992 0.174258
\(103\) −1.80708 −0.178057 −0.0890285 0.996029i \(-0.528376\pi\)
−0.0890285 + 0.996029i \(0.528376\pi\)
\(104\) −4.19292 −0.411150
\(105\) 0 0
\(106\) −4.55560 −0.442479
\(107\) −7.55741 −0.730602 −0.365301 0.930889i \(-0.619034\pi\)
−0.365301 + 0.930889i \(0.619034\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.81883 0.269995 0.134998 0.990846i \(-0.456897\pi\)
0.134998 + 0.990846i \(0.456897\pi\)
\(110\) −10.3155 −0.983546
\(111\) −7.95284 −0.754850
\(112\) 0 0
\(113\) −4.61151 −0.433814 −0.216907 0.976192i \(-0.569597\pi\)
−0.216907 + 0.976192i \(0.569597\pi\)
\(114\) −0.585786 −0.0548639
\(115\) 3.87996 0.361808
\(116\) 2.63551 0.244701
\(117\) −4.19292 −0.387635
\(118\) −1.83024 −0.168487
\(119\) 0 0
\(120\) −3.87996 −0.354190
\(121\) −3.93149 −0.357409
\(122\) 1.80005 0.162969
\(123\) −11.3671 −1.02493
\(124\) −0.585786 −0.0526052
\(125\) −19.6097 −1.75394
\(126\) 0 0
\(127\) −9.51025 −0.843898 −0.421949 0.906619i \(-0.638654\pi\)
−0.421949 + 0.906619i \(0.638654\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.85158 0.251068
\(130\) 16.2684 1.42683
\(131\) 19.0069 1.66064 0.830322 0.557284i \(-0.188157\pi\)
0.830322 + 0.557284i \(0.188157\pi\)
\(132\) 2.65867 0.231407
\(133\) 0 0
\(134\) 0.931495 0.0804689
\(135\) −3.87996 −0.333934
\(136\) 1.75992 0.150912
\(137\) −8.71095 −0.744227 −0.372113 0.928187i \(-0.621367\pi\)
−0.372113 + 0.928187i \(0.621367\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 9.34049 0.792250 0.396125 0.918197i \(-0.370355\pi\)
0.396125 + 0.918197i \(0.370355\pi\)
\(140\) 0 0
\(141\) 4.70839 0.396518
\(142\) −2.65867 −0.223110
\(143\) −11.1476 −0.932207
\(144\) 1.00000 0.0833333
\(145\) −10.2257 −0.849196
\(146\) −1.02838 −0.0851090
\(147\) 0 0
\(148\) −7.95284 −0.653719
\(149\) 1.62410 0.133052 0.0665259 0.997785i \(-0.478808\pi\)
0.0665259 + 0.997785i \(0.478808\pi\)
\(150\) 10.0541 0.820914
\(151\) −6.39543 −0.520453 −0.260226 0.965548i \(-0.583797\pi\)
−0.260226 + 0.965548i \(0.583797\pi\)
\(152\) −0.585786 −0.0475136
\(153\) 1.75992 0.142281
\(154\) 0 0
\(155\) 2.27283 0.182558
\(156\) −4.19292 −0.335702
\(157\) −21.2736 −1.69782 −0.848908 0.528540i \(-0.822740\pi\)
−0.848908 + 0.528540i \(0.822740\pi\)
\(158\) −8.65867 −0.688846
\(159\) −4.55560 −0.361282
\(160\) −3.87996 −0.306738
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −8.48528 −0.664619 −0.332309 0.943170i \(-0.607828\pi\)
−0.332309 + 0.943170i \(0.607828\pi\)
\(164\) −11.3671 −0.887618
\(165\) −10.3155 −0.803062
\(166\) −13.7297 −1.06563
\(167\) −21.8292 −1.68919 −0.844596 0.535404i \(-0.820159\pi\)
−0.844596 + 0.535404i \(0.820159\pi\)
\(168\) 0 0
\(169\) 4.58057 0.352351
\(170\) −6.82843 −0.523716
\(171\) −0.585786 −0.0447962
\(172\) 2.85158 0.217431
\(173\) −20.4940 −1.55813 −0.779066 0.626942i \(-0.784306\pi\)
−0.779066 + 0.626942i \(0.784306\pi\)
\(174\) 2.63551 0.199797
\(175\) 0 0
\(176\) 2.65867 0.200404
\(177\) −1.83024 −0.137569
\(178\) 0.216077 0.0161956
\(179\) −26.6166 −1.98942 −0.994710 0.102721i \(-0.967245\pi\)
−0.994710 + 0.102721i \(0.967245\pi\)
\(180\) −3.87996 −0.289195
\(181\) −21.6390 −1.60842 −0.804209 0.594347i \(-0.797410\pi\)
−0.804209 + 0.594347i \(0.797410\pi\)
\(182\) 0 0
\(183\) 1.80005 0.133064
\(184\) −1.00000 −0.0737210
\(185\) 30.8567 2.26863
\(186\) −0.585786 −0.0429519
\(187\) 4.67904 0.342166
\(188\) 4.70839 0.343394
\(189\) 0 0
\(190\) 2.27283 0.164888
\(191\) 2.23645 0.161824 0.0809121 0.996721i \(-0.474217\pi\)
0.0809121 + 0.996721i \(0.474217\pi\)
\(192\) 1.00000 0.0721688
\(193\) 1.19473 0.0859986 0.0429993 0.999075i \(-0.486309\pi\)
0.0429993 + 0.999075i \(0.486309\pi\)
\(194\) −8.80527 −0.632181
\(195\) 16.2684 1.16500
\(196\) 0 0
\(197\) 12.5788 0.896199 0.448100 0.893984i \(-0.352101\pi\)
0.448100 + 0.893984i \(0.352101\pi\)
\(198\) 2.65867 0.188943
\(199\) −2.73858 −0.194132 −0.0970662 0.995278i \(-0.530946\pi\)
−0.0970662 + 0.995278i \(0.530946\pi\)
\(200\) 10.0541 0.710932
\(201\) 0.931495 0.0657026
\(202\) 11.1439 0.784085
\(203\) 0 0
\(204\) 1.75992 0.123219
\(205\) 44.1037 3.08034
\(206\) −1.80708 −0.125905
\(207\) −1.00000 −0.0695048
\(208\) −4.19292 −0.290727
\(209\) −1.55741 −0.107728
\(210\) 0 0
\(211\) 7.58654 0.522279 0.261139 0.965301i \(-0.415902\pi\)
0.261139 + 0.965301i \(0.415902\pi\)
\(212\) −4.55560 −0.312880
\(213\) −2.65867 −0.182169
\(214\) −7.55741 −0.516614
\(215\) −11.0640 −0.754561
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.81883 0.190916
\(219\) −1.02838 −0.0694912
\(220\) −10.3155 −0.695472
\(221\) −7.37921 −0.496379
\(222\) −7.95284 −0.533760
\(223\) 13.4433 0.900233 0.450116 0.892970i \(-0.351383\pi\)
0.450116 + 0.892970i \(0.351383\pi\)
\(224\) 0 0
\(225\) 10.0541 0.670273
\(226\) −4.61151 −0.306753
\(227\) 1.84721 0.122604 0.0613018 0.998119i \(-0.480475\pi\)
0.0613018 + 0.998119i \(0.480475\pi\)
\(228\) −0.585786 −0.0387947
\(229\) 13.0471 0.862174 0.431087 0.902310i \(-0.358130\pi\)
0.431087 + 0.902310i \(0.358130\pi\)
\(230\) 3.87996 0.255837
\(231\) 0 0
\(232\) 2.63551 0.173030
\(233\) 12.5556 0.822545 0.411272 0.911513i \(-0.365085\pi\)
0.411272 + 0.911513i \(0.365085\pi\)
\(234\) −4.19292 −0.274100
\(235\) −18.2684 −1.19170
\(236\) −1.83024 −0.119138
\(237\) −8.65867 −0.562441
\(238\) 0 0
\(239\) −24.2540 −1.56886 −0.784429 0.620219i \(-0.787044\pi\)
−0.784429 + 0.620219i \(0.787044\pi\)
\(240\) −3.87996 −0.250450
\(241\) −17.7227 −1.14162 −0.570810 0.821082i \(-0.693371\pi\)
−0.570810 + 0.821082i \(0.693371\pi\)
\(242\) −3.93149 −0.252726
\(243\) 1.00000 0.0641500
\(244\) 1.80005 0.115236
\(245\) 0 0
\(246\) −11.3671 −0.724737
\(247\) 2.45615 0.156281
\(248\) −0.585786 −0.0371975
\(249\) −13.7297 −0.870087
\(250\) −19.6097 −1.24023
\(251\) 26.3037 1.66027 0.830137 0.557560i \(-0.188262\pi\)
0.830137 + 0.557560i \(0.188262\pi\)
\(252\) 0 0
\(253\) −2.65867 −0.167149
\(254\) −9.51025 −0.596726
\(255\) −6.82843 −0.427613
\(256\) 1.00000 0.0625000
\(257\) 14.7643 0.920971 0.460486 0.887667i \(-0.347675\pi\)
0.460486 + 0.887667i \(0.347675\pi\)
\(258\) 2.85158 0.177532
\(259\) 0 0
\(260\) 16.2684 1.00892
\(261\) 2.63551 0.163134
\(262\) 19.0069 1.17425
\(263\) 3.81583 0.235294 0.117647 0.993055i \(-0.462465\pi\)
0.117647 + 0.993055i \(0.462465\pi\)
\(264\) 2.65867 0.163630
\(265\) 17.6755 1.08580
\(266\) 0 0
\(267\) 0.216077 0.0132237
\(268\) 0.931495 0.0569001
\(269\) 19.7551 1.20449 0.602245 0.798311i \(-0.294273\pi\)
0.602245 + 0.798311i \(0.294273\pi\)
\(270\) −3.87996 −0.236127
\(271\) −24.4966 −1.48806 −0.744031 0.668145i \(-0.767089\pi\)
−0.744031 + 0.668145i \(0.767089\pi\)
\(272\) 1.75992 0.106711
\(273\) 0 0
\(274\) −8.71095 −0.526248
\(275\) 26.7305 1.61191
\(276\) −1.00000 −0.0601929
\(277\) −2.21245 −0.132933 −0.0664667 0.997789i \(-0.521173\pi\)
−0.0664667 + 0.997789i \(0.521173\pi\)
\(278\) 9.34049 0.560205
\(279\) −0.585786 −0.0350701
\(280\) 0 0
\(281\) 3.92009 0.233853 0.116926 0.993141i \(-0.462696\pi\)
0.116926 + 0.993141i \(0.462696\pi\)
\(282\) 4.70839 0.280380
\(283\) −25.7484 −1.53058 −0.765292 0.643683i \(-0.777406\pi\)
−0.765292 + 0.643683i \(0.777406\pi\)
\(284\) −2.65867 −0.157763
\(285\) 2.27283 0.134631
\(286\) −11.1476 −0.659170
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.9027 −0.817804
\(290\) −10.2257 −0.600472
\(291\) −8.80527 −0.516174
\(292\) −1.02838 −0.0601811
\(293\) 25.8379 1.50947 0.754734 0.656031i \(-0.227766\pi\)
0.754734 + 0.656031i \(0.227766\pi\)
\(294\) 0 0
\(295\) 7.10126 0.413451
\(296\) −7.95284 −0.462249
\(297\) 2.65867 0.154271
\(298\) 1.62410 0.0940818
\(299\) 4.19292 0.242483
\(300\) 10.0541 0.580474
\(301\) 0 0
\(302\) −6.39543 −0.368016
\(303\) 11.1439 0.640203
\(304\) −0.585786 −0.0335972
\(305\) −6.98413 −0.399910
\(306\) 1.75992 0.100608
\(307\) 19.2889 1.10087 0.550437 0.834877i \(-0.314461\pi\)
0.550437 + 0.834877i \(0.314461\pi\)
\(308\) 0 0
\(309\) −1.80708 −0.102801
\(310\) 2.27283 0.129088
\(311\) 17.4433 0.989121 0.494561 0.869143i \(-0.335329\pi\)
0.494561 + 0.869143i \(0.335329\pi\)
\(312\) −4.19292 −0.237377
\(313\) 15.3225 0.866076 0.433038 0.901376i \(-0.357442\pi\)
0.433038 + 0.901376i \(0.357442\pi\)
\(314\) −21.2736 −1.20054
\(315\) 0 0
\(316\) −8.65867 −0.487088
\(317\) 27.4123 1.53963 0.769814 0.638268i \(-0.220349\pi\)
0.769814 + 0.638268i \(0.220349\pi\)
\(318\) −4.55560 −0.255465
\(319\) 7.00694 0.392313
\(320\) −3.87996 −0.216896
\(321\) −7.55741 −0.421814
\(322\) 0 0
\(323\) −1.03094 −0.0573629
\(324\) 1.00000 0.0555556
\(325\) −42.1560 −2.33839
\(326\) −8.48528 −0.469956
\(327\) 2.81883 0.155882
\(328\) −11.3671 −0.627641
\(329\) 0 0
\(330\) −10.3155 −0.567851
\(331\) 5.99638 0.329591 0.164795 0.986328i \(-0.447304\pi\)
0.164795 + 0.986328i \(0.447304\pi\)
\(332\) −13.7297 −0.753517
\(333\) −7.95284 −0.435813
\(334\) −21.8292 −1.19444
\(335\) −3.61416 −0.197463
\(336\) 0 0
\(337\) −7.63648 −0.415985 −0.207993 0.978130i \(-0.566693\pi\)
−0.207993 + 0.978130i \(0.566693\pi\)
\(338\) 4.58057 0.249150
\(339\) −4.61151 −0.250463
\(340\) −6.82843 −0.370323
\(341\) −1.55741 −0.0843385
\(342\) −0.585786 −0.0316757
\(343\) 0 0
\(344\) 2.85158 0.153747
\(345\) 3.87996 0.208890
\(346\) −20.4940 −1.10177
\(347\) 32.7660 1.75897 0.879486 0.475925i \(-0.157887\pi\)
0.879486 + 0.475925i \(0.157887\pi\)
\(348\) 2.63551 0.141278
\(349\) −36.5403 −1.95596 −0.977980 0.208699i \(-0.933077\pi\)
−0.977980 + 0.208699i \(0.933077\pi\)
\(350\) 0 0
\(351\) −4.19292 −0.223801
\(352\) 2.65867 0.141707
\(353\) −2.90096 −0.154402 −0.0772012 0.997016i \(-0.524598\pi\)
−0.0772012 + 0.997016i \(0.524598\pi\)
\(354\) −1.83024 −0.0972761
\(355\) 10.3155 0.547491
\(356\) 0.216077 0.0114520
\(357\) 0 0
\(358\) −26.6166 −1.40673
\(359\) −26.5499 −1.40125 −0.700626 0.713529i \(-0.747096\pi\)
−0.700626 + 0.713529i \(0.747096\pi\)
\(360\) −3.87996 −0.204492
\(361\) −18.6569 −0.981940
\(362\) −21.6390 −1.13732
\(363\) −3.93149 −0.206350
\(364\) 0 0
\(365\) 3.99006 0.208849
\(366\) 1.80005 0.0940902
\(367\) −26.6833 −1.39286 −0.696429 0.717626i \(-0.745229\pi\)
−0.696429 + 0.717626i \(0.745229\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −11.3671 −0.591745
\(370\) 30.8567 1.60416
\(371\) 0 0
\(372\) −0.585786 −0.0303716
\(373\) 9.13038 0.472753 0.236377 0.971662i \(-0.424040\pi\)
0.236377 + 0.971662i \(0.424040\pi\)
\(374\) 4.67904 0.241948
\(375\) −19.6097 −1.01264
\(376\) 4.70839 0.242817
\(377\) −11.0505 −0.569128
\(378\) 0 0
\(379\) −33.9288 −1.74281 −0.871404 0.490567i \(-0.836790\pi\)
−0.871404 + 0.490567i \(0.836790\pi\)
\(380\) 2.27283 0.116594
\(381\) −9.51025 −0.487225
\(382\) 2.23645 0.114427
\(383\) −30.8008 −1.57385 −0.786924 0.617050i \(-0.788328\pi\)
−0.786924 + 0.617050i \(0.788328\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 1.19473 0.0608102
\(387\) 2.85158 0.144954
\(388\) −8.80527 −0.447020
\(389\) −2.38221 −0.120783 −0.0603915 0.998175i \(-0.519235\pi\)
−0.0603915 + 0.998175i \(0.519235\pi\)
\(390\) 16.2684 0.823780
\(391\) −1.75992 −0.0890031
\(392\) 0 0
\(393\) 19.0069 0.958773
\(394\) 12.5788 0.633709
\(395\) 33.5953 1.69036
\(396\) 2.65867 0.133603
\(397\) 3.21064 0.161137 0.0805687 0.996749i \(-0.474326\pi\)
0.0805687 + 0.996749i \(0.474326\pi\)
\(398\) −2.73858 −0.137272
\(399\) 0 0
\(400\) 10.0541 0.502705
\(401\) 36.2591 1.81069 0.905346 0.424675i \(-0.139612\pi\)
0.905346 + 0.424675i \(0.139612\pi\)
\(402\) 0.931495 0.0464587
\(403\) 2.45615 0.122350
\(404\) 11.1439 0.554432
\(405\) −3.87996 −0.192797
\(406\) 0 0
\(407\) −21.1439 −1.04807
\(408\) 1.75992 0.0871291
\(409\) −4.28896 −0.212075 −0.106038 0.994362i \(-0.533816\pi\)
−0.106038 + 0.994362i \(0.533816\pi\)
\(410\) 44.1037 2.17813
\(411\) −8.71095 −0.429679
\(412\) −1.80708 −0.0890285
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 53.2708 2.61496
\(416\) −4.19292 −0.205575
\(417\) 9.34049 0.457406
\(418\) −1.55741 −0.0761754
\(419\) 29.3422 1.43346 0.716730 0.697350i \(-0.245638\pi\)
0.716730 + 0.697350i \(0.245638\pi\)
\(420\) 0 0
\(421\) −5.27186 −0.256935 −0.128467 0.991714i \(-0.541006\pi\)
−0.128467 + 0.991714i \(0.541006\pi\)
\(422\) 7.58654 0.369307
\(423\) 4.70839 0.228930
\(424\) −4.55560 −0.221239
\(425\) 17.6944 0.858306
\(426\) −2.65867 −0.128813
\(427\) 0 0
\(428\) −7.55741 −0.365301
\(429\) −11.1476 −0.538210
\(430\) −11.0640 −0.533555
\(431\) −6.70463 −0.322951 −0.161475 0.986877i \(-0.551625\pi\)
−0.161475 + 0.986877i \(0.551625\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.3751 −1.26750 −0.633752 0.773536i \(-0.718486\pi\)
−0.633752 + 0.773536i \(0.718486\pi\)
\(434\) 0 0
\(435\) −10.2257 −0.490283
\(436\) 2.81883 0.134998
\(437\) 0.585786 0.0280220
\(438\) −1.02838 −0.0491377
\(439\) −8.18589 −0.390691 −0.195346 0.980734i \(-0.562583\pi\)
−0.195346 + 0.980734i \(0.562583\pi\)
\(440\) −10.3155 −0.491773
\(441\) 0 0
\(442\) −7.37921 −0.350993
\(443\) 3.45037 0.163932 0.0819660 0.996635i \(-0.473880\pi\)
0.0819660 + 0.996635i \(0.473880\pi\)
\(444\) −7.95284 −0.377425
\(445\) −0.838369 −0.0397425
\(446\) 13.4433 0.636561
\(447\) 1.62410 0.0768175
\(448\) 0 0
\(449\) 39.1515 1.84767 0.923837 0.382787i \(-0.125036\pi\)
0.923837 + 0.382787i \(0.125036\pi\)
\(450\) 10.0541 0.473955
\(451\) −30.2212 −1.42306
\(452\) −4.61151 −0.216907
\(453\) −6.39543 −0.300484
\(454\) 1.84721 0.0866939
\(455\) 0 0
\(456\) −0.585786 −0.0274320
\(457\) 16.7872 0.785274 0.392637 0.919694i \(-0.371563\pi\)
0.392637 + 0.919694i \(0.371563\pi\)
\(458\) 13.0471 0.609649
\(459\) 1.75992 0.0821461
\(460\) 3.87996 0.180904
\(461\) −34.3170 −1.59830 −0.799152 0.601130i \(-0.794718\pi\)
−0.799152 + 0.601130i \(0.794718\pi\)
\(462\) 0 0
\(463\) −7.75033 −0.360188 −0.180094 0.983649i \(-0.557640\pi\)
−0.180094 + 0.983649i \(0.557640\pi\)
\(464\) 2.63551 0.122350
\(465\) 2.27283 0.105400
\(466\) 12.5556 0.581627
\(467\) −19.9023 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(468\) −4.19292 −0.193818
\(469\) 0 0
\(470\) −18.2684 −0.842657
\(471\) −21.2736 −0.980235
\(472\) −1.83024 −0.0842436
\(473\) 7.58141 0.348594
\(474\) −8.65867 −0.397706
\(475\) −5.88955 −0.270231
\(476\) 0 0
\(477\) −4.55560 −0.208587
\(478\) −24.2540 −1.10935
\(479\) 12.4661 0.569591 0.284795 0.958588i \(-0.408074\pi\)
0.284795 + 0.958588i \(0.408074\pi\)
\(480\) −3.87996 −0.177095
\(481\) 33.3456 1.52043
\(482\) −17.7227 −0.807247
\(483\) 0 0
\(484\) −3.93149 −0.178704
\(485\) 34.1641 1.55131
\(486\) 1.00000 0.0453609
\(487\) −40.4859 −1.83459 −0.917296 0.398205i \(-0.869633\pi\)
−0.917296 + 0.398205i \(0.869633\pi\)
\(488\) 1.80005 0.0814845
\(489\) −8.48528 −0.383718
\(490\) 0 0
\(491\) 35.7599 1.61382 0.806911 0.590673i \(-0.201138\pi\)
0.806911 + 0.590673i \(0.201138\pi\)
\(492\) −11.3671 −0.512466
\(493\) 4.63829 0.208898
\(494\) 2.45615 0.110508
\(495\) −10.3155 −0.463648
\(496\) −0.585786 −0.0263026
\(497\) 0 0
\(498\) −13.7297 −0.615244
\(499\) −18.3291 −0.820523 −0.410261 0.911968i \(-0.634562\pi\)
−0.410261 + 0.911968i \(0.634562\pi\)
\(500\) −19.6097 −0.876972
\(501\) −21.8292 −0.975256
\(502\) 26.3037 1.17399
\(503\) −3.19389 −0.142408 −0.0712042 0.997462i \(-0.522684\pi\)
−0.0712042 + 0.997462i \(0.522684\pi\)
\(504\) 0 0
\(505\) −43.2381 −1.92407
\(506\) −2.65867 −0.118192
\(507\) 4.58057 0.203430
\(508\) −9.51025 −0.421949
\(509\) 29.5917 1.31163 0.655814 0.754923i \(-0.272326\pi\)
0.655814 + 0.754923i \(0.272326\pi\)
\(510\) −6.82843 −0.302368
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −0.585786 −0.0258631
\(514\) 14.7643 0.651225
\(515\) 7.01140 0.308959
\(516\) 2.85158 0.125534
\(517\) 12.5180 0.550542
\(518\) 0 0
\(519\) −20.4940 −0.899588
\(520\) 16.2684 0.713415
\(521\) 38.8474 1.70194 0.850968 0.525217i \(-0.176016\pi\)
0.850968 + 0.525217i \(0.176016\pi\)
\(522\) 2.63551 0.115353
\(523\) −41.6214 −1.81997 −0.909987 0.414636i \(-0.863909\pi\)
−0.909987 + 0.414636i \(0.863909\pi\)
\(524\) 19.0069 0.830322
\(525\) 0 0
\(526\) 3.81583 0.166378
\(527\) −1.03094 −0.0449084
\(528\) 2.65867 0.115704
\(529\) 1.00000 0.0434783
\(530\) 17.6755 0.767777
\(531\) −1.83024 −0.0794256
\(532\) 0 0
\(533\) 47.6611 2.06443
\(534\) 0.216077 0.00935055
\(535\) 29.3225 1.26772
\(536\) 0.931495 0.0402344
\(537\) −26.6166 −1.14859
\(538\) 19.7551 0.851703
\(539\) 0 0
\(540\) −3.87996 −0.166967
\(541\) 4.70185 0.202148 0.101074 0.994879i \(-0.467772\pi\)
0.101074 + 0.994879i \(0.467772\pi\)
\(542\) −24.4966 −1.05222
\(543\) −21.6390 −0.928620
\(544\) 1.75992 0.0754560
\(545\) −10.9370 −0.468488
\(546\) 0 0
\(547\) 39.5764 1.69217 0.846083 0.533052i \(-0.178955\pi\)
0.846083 + 0.533052i \(0.178955\pi\)
\(548\) −8.71095 −0.372113
\(549\) 1.80005 0.0768243
\(550\) 26.7305 1.13979
\(551\) −1.54385 −0.0657700
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −2.21245 −0.0939981
\(555\) 30.8567 1.30979
\(556\) 9.34049 0.396125
\(557\) 9.65685 0.409174 0.204587 0.978848i \(-0.434415\pi\)
0.204587 + 0.978848i \(0.434415\pi\)
\(558\) −0.585786 −0.0247983
\(559\) −11.9565 −0.505705
\(560\) 0 0
\(561\) 4.67904 0.197549
\(562\) 3.92009 0.165359
\(563\) 1.05635 0.0445199 0.0222599 0.999752i \(-0.492914\pi\)
0.0222599 + 0.999752i \(0.492914\pi\)
\(564\) 4.70839 0.198259
\(565\) 17.8925 0.752742
\(566\) −25.7484 −1.08229
\(567\) 0 0
\(568\) −2.65867 −0.111555
\(569\) 6.60276 0.276802 0.138401 0.990376i \(-0.455804\pi\)
0.138401 + 0.990376i \(0.455804\pi\)
\(570\) 2.27283 0.0951983
\(571\) 0.686292 0.0287204 0.0143602 0.999897i \(-0.495429\pi\)
0.0143602 + 0.999897i \(0.495429\pi\)
\(572\) −11.1476 −0.466103
\(573\) 2.23645 0.0934292
\(574\) 0 0
\(575\) −10.0541 −0.419285
\(576\) 1.00000 0.0416667
\(577\) 34.5565 1.43860 0.719302 0.694697i \(-0.244462\pi\)
0.719302 + 0.694697i \(0.244462\pi\)
\(578\) −13.9027 −0.578275
\(579\) 1.19473 0.0496513
\(580\) −10.2257 −0.424598
\(581\) 0 0
\(582\) −8.80527 −0.364990
\(583\) −12.1118 −0.501620
\(584\) −1.02838 −0.0425545
\(585\) 16.2684 0.672614
\(586\) 25.8379 1.06735
\(587\) −13.5829 −0.560627 −0.280313 0.959909i \(-0.590438\pi\)
−0.280313 + 0.959909i \(0.590438\pi\)
\(588\) 0 0
\(589\) 0.343146 0.0141391
\(590\) 7.10126 0.292354
\(591\) 12.5788 0.517421
\(592\) −7.95284 −0.326860
\(593\) 19.3331 0.793916 0.396958 0.917837i \(-0.370066\pi\)
0.396958 + 0.917837i \(0.370066\pi\)
\(594\) 2.65867 0.109086
\(595\) 0 0
\(596\) 1.62410 0.0665259
\(597\) −2.73858 −0.112082
\(598\) 4.19292 0.171461
\(599\) 17.6241 0.720101 0.360051 0.932933i \(-0.382759\pi\)
0.360051 + 0.932933i \(0.382759\pi\)
\(600\) 10.0541 0.410457
\(601\) 29.3990 1.19921 0.599604 0.800297i \(-0.295325\pi\)
0.599604 + 0.800297i \(0.295325\pi\)
\(602\) 0 0
\(603\) 0.931495 0.0379334
\(604\) −6.39543 −0.260226
\(605\) 15.2540 0.620165
\(606\) 11.1439 0.452692
\(607\) 41.8755 1.69967 0.849837 0.527046i \(-0.176700\pi\)
0.849837 + 0.527046i \(0.176700\pi\)
\(608\) −0.585786 −0.0237568
\(609\) 0 0
\(610\) −6.98413 −0.282779
\(611\) −19.7419 −0.798671
\(612\) 1.75992 0.0711406
\(613\) −2.75333 −0.111206 −0.0556031 0.998453i \(-0.517708\pi\)
−0.0556031 + 0.998453i \(0.517708\pi\)
\(614\) 19.2889 0.778435
\(615\) 44.1037 1.77843
\(616\) 0 0
\(617\) −12.6274 −0.508361 −0.254180 0.967157i \(-0.581806\pi\)
−0.254180 + 0.967157i \(0.581806\pi\)
\(618\) −1.80708 −0.0726915
\(619\) 6.55547 0.263486 0.131743 0.991284i \(-0.457943\pi\)
0.131743 + 0.991284i \(0.457943\pi\)
\(620\) 2.27283 0.0912790
\(621\) −1.00000 −0.0401286
\(622\) 17.4433 0.699414
\(623\) 0 0
\(624\) −4.19292 −0.167851
\(625\) 25.8144 1.03257
\(626\) 15.3225 0.612409
\(627\) −1.55741 −0.0621970
\(628\) −21.2736 −0.848908
\(629\) −13.9964 −0.558072
\(630\) 0 0
\(631\) 6.20202 0.246898 0.123449 0.992351i \(-0.460604\pi\)
0.123449 + 0.992351i \(0.460604\pi\)
\(632\) −8.65867 −0.344423
\(633\) 7.58654 0.301538
\(634\) 27.4123 1.08868
\(635\) 36.8994 1.46431
\(636\) −4.55560 −0.180641
\(637\) 0 0
\(638\) 7.00694 0.277407
\(639\) −2.65867 −0.105175
\(640\) −3.87996 −0.153369
\(641\) 42.4796 1.67784 0.838922 0.544252i \(-0.183186\pi\)
0.838922 + 0.544252i \(0.183186\pi\)
\(642\) −7.55741 −0.298267
\(643\) −6.70754 −0.264520 −0.132260 0.991215i \(-0.542223\pi\)
−0.132260 + 0.991215i \(0.542223\pi\)
\(644\) 0 0
\(645\) −11.0640 −0.435646
\(646\) −1.03094 −0.0405617
\(647\) −3.62154 −0.142377 −0.0711887 0.997463i \(-0.522679\pi\)
−0.0711887 + 0.997463i \(0.522679\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.86599 −0.191007
\(650\) −42.1560 −1.65349
\(651\) 0 0
\(652\) −8.48528 −0.332309
\(653\) 11.4998 0.450023 0.225011 0.974356i \(-0.427758\pi\)
0.225011 + 0.974356i \(0.427758\pi\)
\(654\) 2.81883 0.110225
\(655\) −73.7462 −2.88150
\(656\) −11.3671 −0.443809
\(657\) −1.02838 −0.0401208
\(658\) 0 0
\(659\) −17.5953 −0.685415 −0.342708 0.939442i \(-0.611344\pi\)
−0.342708 + 0.939442i \(0.611344\pi\)
\(660\) −10.3155 −0.401531
\(661\) 36.5477 1.42154 0.710771 0.703424i \(-0.248346\pi\)
0.710771 + 0.703424i \(0.248346\pi\)
\(662\) 5.99638 0.233056
\(663\) −7.37921 −0.286585
\(664\) −13.7297 −0.532817
\(665\) 0 0
\(666\) −7.95284 −0.308166
\(667\) −2.63551 −0.102047
\(668\) −21.8292 −0.844596
\(669\) 13.4433 0.519750
\(670\) −3.61416 −0.139627
\(671\) 4.78574 0.184751
\(672\) 0 0
\(673\) −18.8016 −0.724750 −0.362375 0.932032i \(-0.618034\pi\)
−0.362375 + 0.932032i \(0.618034\pi\)
\(674\) −7.63648 −0.294146
\(675\) 10.0541 0.386982
\(676\) 4.58057 0.176176
\(677\) 26.3232 1.01168 0.505842 0.862626i \(-0.331182\pi\)
0.505842 + 0.862626i \(0.331182\pi\)
\(678\) −4.61151 −0.177104
\(679\) 0 0
\(680\) −6.82843 −0.261858
\(681\) 1.84721 0.0707853
\(682\) −1.55741 −0.0596363
\(683\) 31.8381 1.21825 0.609125 0.793074i \(-0.291521\pi\)
0.609125 + 0.793074i \(0.291521\pi\)
\(684\) −0.585786 −0.0223981
\(685\) 33.7981 1.29136
\(686\) 0 0
\(687\) 13.0471 0.497777
\(688\) 2.85158 0.108716
\(689\) 19.1013 0.727700
\(690\) 3.87996 0.147708
\(691\) 43.8825 1.66937 0.834685 0.550728i \(-0.185650\pi\)
0.834685 + 0.550728i \(0.185650\pi\)
\(692\) −20.4940 −0.779066
\(693\) 0 0
\(694\) 32.7660 1.24378
\(695\) −36.2407 −1.37469
\(696\) 2.63551 0.0998987
\(697\) −20.0051 −0.757748
\(698\) −36.5403 −1.38307
\(699\) 12.5556 0.474896
\(700\) 0 0
\(701\) −29.6317 −1.11917 −0.559586 0.828772i \(-0.689040\pi\)
−0.559586 + 0.828772i \(0.689040\pi\)
\(702\) −4.19292 −0.158252
\(703\) 4.65867 0.175705
\(704\) 2.65867 0.100202
\(705\) −18.2684 −0.688026
\(706\) −2.90096 −0.109179
\(707\) 0 0
\(708\) −1.83024 −0.0687846
\(709\) 43.7151 1.64176 0.820878 0.571104i \(-0.193485\pi\)
0.820878 + 0.571104i \(0.193485\pi\)
\(710\) 10.3155 0.387135
\(711\) −8.65867 −0.324725
\(712\) 0.216077 0.00809781
\(713\) 0.585786 0.0219379
\(714\) 0 0
\(715\) 43.2521 1.61754
\(716\) −26.6166 −0.994710
\(717\) −24.2540 −0.905780
\(718\) −26.5499 −0.990835
\(719\) −16.6333 −0.620316 −0.310158 0.950685i \(-0.600382\pi\)
−0.310158 + 0.950685i \(0.600382\pi\)
\(720\) −3.87996 −0.144598
\(721\) 0 0
\(722\) −18.6569 −0.694336
\(723\) −17.7227 −0.659114
\(724\) −21.6390 −0.804209
\(725\) 26.4977 0.984098
\(726\) −3.93149 −0.145911
\(727\) −27.5058 −1.02013 −0.510067 0.860135i \(-0.670379\pi\)
−0.510067 + 0.860135i \(0.670379\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 3.99006 0.147679
\(731\) 5.01857 0.185618
\(732\) 1.80005 0.0665318
\(733\) 0.448776 0.0165759 0.00828795 0.999966i \(-0.497362\pi\)
0.00828795 + 0.999966i \(0.497362\pi\)
\(734\) −26.6833 −0.984899
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 2.47653 0.0912243
\(738\) −11.3671 −0.418427
\(739\) 42.0614 1.54725 0.773626 0.633642i \(-0.218441\pi\)
0.773626 + 0.633642i \(0.218441\pi\)
\(740\) 30.8567 1.13432
\(741\) 2.45615 0.0902291
\(742\) 0 0
\(743\) −45.5005 −1.66925 −0.834625 0.550818i \(-0.814316\pi\)
−0.834625 + 0.550818i \(0.814316\pi\)
\(744\) −0.585786 −0.0214760
\(745\) −6.30146 −0.230868
\(746\) 9.13038 0.334287
\(747\) −13.7297 −0.502345
\(748\) 4.67904 0.171083
\(749\) 0 0
\(750\) −19.6097 −0.716045
\(751\) −23.6860 −0.864314 −0.432157 0.901798i \(-0.642247\pi\)
−0.432157 + 0.901798i \(0.642247\pi\)
\(752\) 4.70839 0.171697
\(753\) 26.3037 0.958559
\(754\) −11.0505 −0.402434
\(755\) 24.8140 0.903075
\(756\) 0 0
\(757\) 36.4424 1.32452 0.662261 0.749273i \(-0.269597\pi\)
0.662261 + 0.749273i \(0.269597\pi\)
\(758\) −33.9288 −1.23235
\(759\) −2.65867 −0.0965035
\(760\) 2.27283 0.0824442
\(761\) −30.5757 −1.10837 −0.554183 0.832395i \(-0.686969\pi\)
−0.554183 + 0.832395i \(0.686969\pi\)
\(762\) −9.51025 −0.344520
\(763\) 0 0
\(764\) 2.23645 0.0809121
\(765\) −6.82843 −0.246882
\(766\) −30.8008 −1.11288
\(767\) 7.67404 0.277094
\(768\) 1.00000 0.0360844
\(769\) 8.43180 0.304059 0.152029 0.988376i \(-0.451419\pi\)
0.152029 + 0.988376i \(0.451419\pi\)
\(770\) 0 0
\(771\) 14.7643 0.531723
\(772\) 1.19473 0.0429993
\(773\) −15.0072 −0.539770 −0.269885 0.962893i \(-0.586986\pi\)
−0.269885 + 0.962893i \(0.586986\pi\)
\(774\) 2.85158 0.102498
\(775\) −5.88955 −0.211559
\(776\) −8.80527 −0.316091
\(777\) 0 0
\(778\) −2.38221 −0.0854065
\(779\) 6.65867 0.238571
\(780\) 16.2684 0.582501
\(781\) −7.06851 −0.252931
\(782\) −1.75992 −0.0629347
\(783\) 2.63551 0.0941854
\(784\) 0 0
\(785\) 82.5407 2.94600
\(786\) 19.0069 0.677955
\(787\) 35.9774 1.28245 0.641227 0.767351i \(-0.278426\pi\)
0.641227 + 0.767351i \(0.278426\pi\)
\(788\) 12.5788 0.448100
\(789\) 3.81583 0.135847
\(790\) 33.5953 1.19527
\(791\) 0 0
\(792\) 2.65867 0.0944716
\(793\) −7.54747 −0.268018
\(794\) 3.21064 0.113941
\(795\) 17.6755 0.626887
\(796\) −2.73858 −0.0970662
\(797\) −36.1450 −1.28032 −0.640162 0.768240i \(-0.721133\pi\)
−0.640162 + 0.768240i \(0.721133\pi\)
\(798\) 0 0
\(799\) 8.28639 0.293152
\(800\) 10.0541 0.355466
\(801\) 0.216077 0.00763469
\(802\) 36.2591 1.28035
\(803\) −2.73411 −0.0964846
\(804\) 0.931495 0.0328513
\(805\) 0 0
\(806\) 2.45615 0.0865144
\(807\) 19.7551 0.695413
\(808\) 11.1439 0.392043
\(809\) 31.0261 1.09082 0.545410 0.838169i \(-0.316374\pi\)
0.545410 + 0.838169i \(0.316374\pi\)
\(810\) −3.87996 −0.136328
\(811\) 2.38036 0.0835859 0.0417929 0.999126i \(-0.486693\pi\)
0.0417929 + 0.999126i \(0.486693\pi\)
\(812\) 0 0
\(813\) −24.4966 −0.859133
\(814\) −21.1439 −0.741095
\(815\) 32.9226 1.15323
\(816\) 1.75992 0.0616096
\(817\) −1.67042 −0.0584406
\(818\) −4.28896 −0.149960
\(819\) 0 0
\(820\) 44.1037 1.54017
\(821\) −53.2725 −1.85922 −0.929612 0.368540i \(-0.879858\pi\)
−0.929612 + 0.368540i \(0.879858\pi\)
\(822\) −8.71095 −0.303829
\(823\) −44.8240 −1.56247 −0.781233 0.624240i \(-0.785409\pi\)
−0.781233 + 0.624240i \(0.785409\pi\)
\(824\) −1.80708 −0.0629527
\(825\) 26.7305 0.930636
\(826\) 0 0
\(827\) −24.6878 −0.858479 −0.429239 0.903191i \(-0.641218\pi\)
−0.429239 + 0.903191i \(0.641218\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 24.3483 0.845650 0.422825 0.906211i \(-0.361038\pi\)
0.422825 + 0.906211i \(0.361038\pi\)
\(830\) 53.2708 1.84906
\(831\) −2.21245 −0.0767491
\(832\) −4.19292 −0.145363
\(833\) 0 0
\(834\) 9.34049 0.323435
\(835\) 84.6964 2.93104
\(836\) −1.55741 −0.0538642
\(837\) −0.585786 −0.0202477
\(838\) 29.3422 1.01361
\(839\) −36.5215 −1.26086 −0.630431 0.776246i \(-0.717122\pi\)
−0.630431 + 0.776246i \(0.717122\pi\)
\(840\) 0 0
\(841\) −22.0541 −0.760486
\(842\) −5.27186 −0.181680
\(843\) 3.92009 0.135015
\(844\) 7.58654 0.261139
\(845\) −17.7724 −0.611390
\(846\) 4.70839 0.161878
\(847\) 0 0
\(848\) −4.55560 −0.156440
\(849\) −25.7484 −0.883684
\(850\) 17.6944 0.606914
\(851\) 7.95284 0.272620
\(852\) −2.65867 −0.0910844
\(853\) 14.1893 0.485832 0.242916 0.970047i \(-0.421896\pi\)
0.242916 + 0.970047i \(0.421896\pi\)
\(854\) 0 0
\(855\) 2.27283 0.0777291
\(856\) −7.55741 −0.258307
\(857\) 36.5314 1.24789 0.623944 0.781469i \(-0.285529\pi\)
0.623944 + 0.781469i \(0.285529\pi\)
\(858\) −11.1476 −0.380572
\(859\) 1.93415 0.0659924 0.0329962 0.999455i \(-0.489495\pi\)
0.0329962 + 0.999455i \(0.489495\pi\)
\(860\) −11.0640 −0.377281
\(861\) 0 0
\(862\) −6.70463 −0.228361
\(863\) −13.0081 −0.442802 −0.221401 0.975183i \(-0.571063\pi\)
−0.221401 + 0.975183i \(0.571063\pi\)
\(864\) 1.00000 0.0340207
\(865\) 79.5160 2.70363
\(866\) −26.3751 −0.896261
\(867\) −13.9027 −0.472160
\(868\) 0 0
\(869\) −23.0205 −0.780917
\(870\) −10.2257 −0.346683
\(871\) −3.90568 −0.132339
\(872\) 2.81883 0.0954578
\(873\) −8.80527 −0.298013
\(874\) 0.585786 0.0198145
\(875\) 0 0
\(876\) −1.02838 −0.0347456
\(877\) 0.988246 0.0333707 0.0166854 0.999861i \(-0.494689\pi\)
0.0166854 + 0.999861i \(0.494689\pi\)
\(878\) −8.18589 −0.276260
\(879\) 25.8379 0.871492
\(880\) −10.3155 −0.347736
\(881\) 4.71492 0.158850 0.0794249 0.996841i \(-0.474692\pi\)
0.0794249 + 0.996841i \(0.474692\pi\)
\(882\) 0 0
\(883\) 13.0253 0.438337 0.219168 0.975687i \(-0.429666\pi\)
0.219168 + 0.975687i \(0.429666\pi\)
\(884\) −7.37921 −0.248190
\(885\) 7.10126 0.238706
\(886\) 3.45037 0.115917
\(887\) 20.9615 0.703818 0.351909 0.936034i \(-0.385533\pi\)
0.351909 + 0.936034i \(0.385533\pi\)
\(888\) −7.95284 −0.266880
\(889\) 0 0
\(890\) −0.838369 −0.0281022
\(891\) 2.65867 0.0890687
\(892\) 13.4433 0.450116
\(893\) −2.75811 −0.0922966
\(894\) 1.62410 0.0543182
\(895\) 103.271 3.45199
\(896\) 0 0
\(897\) 4.19292 0.139997
\(898\) 39.1515 1.30650
\(899\) −1.54385 −0.0514901
\(900\) 10.0541 0.335137
\(901\) −8.01750 −0.267102
\(902\) −30.2212 −1.00626
\(903\) 0 0
\(904\) −4.61151 −0.153376
\(905\) 83.9586 2.79088
\(906\) −6.39543 −0.212474
\(907\) −28.6969 −0.952865 −0.476432 0.879211i \(-0.658070\pi\)
−0.476432 + 0.879211i \(0.658070\pi\)
\(908\) 1.84721 0.0613018
\(909\) 11.1439 0.369621
\(910\) 0 0
\(911\) 48.5199 1.60754 0.803768 0.594943i \(-0.202825\pi\)
0.803768 + 0.594943i \(0.202825\pi\)
\(912\) −0.585786 −0.0193973
\(913\) −36.5028 −1.20807
\(914\) 16.7872 0.555272
\(915\) −6.98413 −0.230888
\(916\) 13.0471 0.431087
\(917\) 0 0
\(918\) 1.75992 0.0580861
\(919\) −8.80492 −0.290448 −0.145224 0.989399i \(-0.546390\pi\)
−0.145224 + 0.989399i \(0.546390\pi\)
\(920\) 3.87996 0.127919
\(921\) 19.2889 0.635590
\(922\) −34.3170 −1.13017
\(923\) 11.1476 0.366927
\(924\) 0 0
\(925\) −79.9586 −2.62902
\(926\) −7.75033 −0.254692
\(927\) −1.80708 −0.0593523
\(928\) 2.63551 0.0865148
\(929\) 54.7130 1.79508 0.897538 0.440937i \(-0.145354\pi\)
0.897538 + 0.440937i \(0.145354\pi\)
\(930\) 2.27283 0.0745290
\(931\) 0 0
\(932\) 12.5556 0.411272
\(933\) 17.4433 0.571069
\(934\) −19.9023 −0.651222
\(935\) −18.1545 −0.593716
\(936\) −4.19292 −0.137050
\(937\) 38.8531 1.26927 0.634637 0.772810i \(-0.281149\pi\)
0.634637 + 0.772810i \(0.281149\pi\)
\(938\) 0 0
\(939\) 15.3225 0.500029
\(940\) −18.2684 −0.595848
\(941\) −19.1134 −0.623079 −0.311540 0.950233i \(-0.600845\pi\)
−0.311540 + 0.950233i \(0.600845\pi\)
\(942\) −21.2736 −0.693131
\(943\) 11.3671 0.370162
\(944\) −1.83024 −0.0595692
\(945\) 0 0
\(946\) 7.58141 0.246493
\(947\) 13.6992 0.445164 0.222582 0.974914i \(-0.428551\pi\)
0.222582 + 0.974914i \(0.428551\pi\)
\(948\) −8.65867 −0.281220
\(949\) 4.31190 0.139970
\(950\) −5.88955 −0.191082
\(951\) 27.4123 0.888905
\(952\) 0 0
\(953\) −39.6603 −1.28472 −0.642361 0.766402i \(-0.722045\pi\)
−0.642361 + 0.766402i \(0.722045\pi\)
\(954\) −4.55560 −0.147493
\(955\) −8.67736 −0.280793
\(956\) −24.2540 −0.784429
\(957\) 7.00694 0.226502
\(958\) 12.4661 0.402761
\(959\) 0 0
\(960\) −3.87996 −0.125225
\(961\) −30.6569 −0.988931
\(962\) 33.3456 1.07511
\(963\) −7.55741 −0.243534
\(964\) −17.7227 −0.570810
\(965\) −4.63551 −0.149222
\(966\) 0 0
\(967\) 18.1953 0.585120 0.292560 0.956247i \(-0.405493\pi\)
0.292560 + 0.956247i \(0.405493\pi\)
\(968\) −3.93149 −0.126363
\(969\) −1.03094 −0.0331185
\(970\) 34.1641 1.09694
\(971\) 6.56691 0.210742 0.105371 0.994433i \(-0.466397\pi\)
0.105371 + 0.994433i \(0.466397\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −40.4859 −1.29725
\(975\) −42.1560 −1.35007
\(976\) 1.80005 0.0576182
\(977\) −56.9952 −1.82344 −0.911719 0.410814i \(-0.865245\pi\)
−0.911719 + 0.410814i \(0.865245\pi\)
\(978\) −8.48528 −0.271329
\(979\) 0.574476 0.0183603
\(980\) 0 0
\(981\) 2.81883 0.0899985
\(982\) 35.7599 1.14114
\(983\) −29.3964 −0.937599 −0.468800 0.883304i \(-0.655313\pi\)
−0.468800 + 0.883304i \(0.655313\pi\)
\(984\) −11.3671 −0.362368
\(985\) −48.8051 −1.55506
\(986\) 4.63829 0.147713
\(987\) 0 0
\(988\) 2.45615 0.0781407
\(989\) −2.85158 −0.0906751
\(990\) −10.3155 −0.327849
\(991\) −31.2354 −0.992225 −0.496112 0.868258i \(-0.665240\pi\)
−0.496112 + 0.868258i \(0.665240\pi\)
\(992\) −0.585786 −0.0185987
\(993\) 5.99638 0.190289
\(994\) 0 0
\(995\) 10.6256 0.336853
\(996\) −13.7297 −0.435043
\(997\) 42.7089 1.35260 0.676302 0.736624i \(-0.263581\pi\)
0.676302 + 0.736624i \(0.263581\pi\)
\(998\) −18.3291 −0.580197
\(999\) −7.95284 −0.251617
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.co.1.1 yes 4
7.6 odd 2 6762.2.a.cn.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.cn.1.4 4 7.6 odd 2
6762.2.a.co.1.1 yes 4 1.1 even 1 trivial