Properties

Label 6762.2.a.cn.1.4
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 1.4
Root \(2.87996\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

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

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 \(-0.165671\pi\)
0.867586 + 0.497288i \(0.165671\pi\)
\(6\) −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 \(-0.302484\pi\)
0.581453 + 0.813580i \(0.302484\pi\)
\(14\) 0 0
\(15\) −3.87996 −1.00180
\(16\) 1.00000 0.250000
\(17\) −1.75992 −0.426844 −0.213422 0.976960i \(-0.568461\pi\)
−0.213422 + 0.976960i \(0.568461\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.585786 0.134389 0.0671943 0.997740i \(-0.478595\pi\)
0.0671943 + 0.997740i \(0.478595\pi\)
\(20\) 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 \(-0.483248\pi\)
0.0526052 + 0.998615i \(0.483248\pi\)
\(32\) 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 \(-0.152359\pi\)
0.887618 + 0.460581i \(0.152359\pi\)
\(42\) 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 \(-0.611577\pi\)
−0.343394 + 0.939191i \(0.611577\pi\)
\(48\) −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 \(-0.461987\pi\)
0.119138 + 0.992878i \(0.461987\pi\)
\(60\) −3.87996 −0.500901
\(61\) −1.80005 −0.230473 −0.115236 0.993338i \(-0.536763\pi\)
−0.115236 + 0.993338i \(0.536763\pi\)
\(62\) 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 \(-0.480832\pi\)
0.0601811 + 0.998187i \(0.480832\pi\)
\(74\) −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 \(-0.228356\pi\)
0.753517 + 0.657428i \(0.228356\pi\)
\(84\) 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 \(-0.503645\pi\)
−0.0114520 + 0.999934i \(0.503645\pi\)
\(90\) 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 \(-0.352485\pi\)
0.447020 + 0.894524i \(0.352485\pi\)
\(98\) 0 0
\(99\) 2.65867 0.267206
\(100\) 10.0541 1.00541
\(101\) −11.1439 −1.10886 −0.554432 0.832229i \(-0.687064\pi\)
−0.554432 + 0.832229i \(0.687064\pi\)
\(102\) 1.75992 0.174258
\(103\) 1.80708 0.178057 0.0890285 0.996029i \(-0.471624\pi\)
0.0890285 + 0.996029i \(0.471624\pi\)
\(104\) 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 \(-0.811843\pi\)
−0.830322 + 0.557284i \(0.811843\pi\)
\(132\) −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 \(-0.629645\pi\)
−0.396125 + 0.918197i \(0.629645\pi\)
\(140\) 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 \(-0.177260\pi\)
0.848908 + 0.528540i \(0.177260\pi\)
\(158\) −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 \(-0.179841\pi\)
0.844596 + 0.535404i \(0.179841\pi\)
\(168\) 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 \(-0.215694\pi\)
0.779066 + 0.626942i \(0.215694\pi\)
\(174\) −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 \(-0.202590\pi\)
0.804209 + 0.594347i \(0.202590\pi\)
\(182\) 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 \(-0.469054\pi\)
0.0970662 + 0.995278i \(0.469054\pi\)
\(200\) 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 \(-0.648617\pi\)
−0.450116 + 0.892970i \(0.648617\pi\)
\(224\) 0 0
\(225\) 10.0541 0.670273
\(226\) −4.61151 −0.306753
\(227\) −1.84721 −0.122604 −0.0613018 0.998119i \(-0.519525\pi\)
−0.0613018 + 0.998119i \(0.519525\pi\)
\(228\) −0.585786 −0.0387947
\(229\) −13.0471 −0.862174 −0.431087 0.902310i \(-0.641870\pi\)
−0.431087 + 0.902310i \(0.641870\pi\)
\(230\) −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 \(-0.306629\pi\)
0.570810 + 0.821082i \(0.306629\pi\)
\(242\) −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 \(-0.811738\pi\)
−0.830137 + 0.557560i \(0.811738\pi\)
\(252\) 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 \(-0.652325\pi\)
−0.460486 + 0.887667i \(0.652325\pi\)
\(258\) −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 \(-0.705727\pi\)
−0.602245 + 0.798311i \(0.705727\pi\)
\(270\) −3.87996 −0.236127
\(271\) 24.4966 1.48806 0.744031 0.668145i \(-0.232911\pi\)
0.744031 + 0.668145i \(0.232911\pi\)
\(272\) −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 \(-0.222594\pi\)
0.765292 + 0.643683i \(0.222594\pi\)
\(284\) −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 \(-0.772234\pi\)
−0.754734 + 0.656031i \(0.772234\pi\)
\(294\) 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 \(-0.685539\pi\)
−0.550437 + 0.834877i \(0.685539\pi\)
\(308\) 0 0
\(309\) −1.80708 −0.102801
\(310\) 2.27283 0.129088
\(311\) −17.4433 −0.989121 −0.494561 0.869143i \(-0.664671\pi\)
−0.494561 + 0.869143i \(0.664671\pi\)
\(312\) −4.19292 −0.237377
\(313\) −15.3225 −0.866076 −0.433038 0.901376i \(-0.642558\pi\)
−0.433038 + 0.901376i \(0.642558\pi\)
\(314\) 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 \(-0.0669229\pi\)
0.977980 + 0.208699i \(0.0669229\pi\)
\(350\) 0 0
\(351\) −4.19292 −0.223801
\(352\) 2.65867 0.141707
\(353\) 2.90096 0.154402 0.0772012 0.997016i \(-0.475402\pi\)
0.0772012 + 0.997016i \(0.475402\pi\)
\(354\) −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 \(-0.254771\pi\)
0.696429 + 0.717626i \(0.254771\pi\)
\(368\) −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 \(-0.211672\pi\)
0.786924 + 0.617050i \(0.211672\pi\)
\(384\) −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 \(-0.525674\pi\)
−0.0805687 + 0.996749i \(0.525674\pi\)
\(398\) 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 \(-0.466184\pi\)
0.106038 + 0.994362i \(0.466184\pi\)
\(410\) 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 \(-0.754362\pi\)
−0.716730 + 0.697350i \(0.754362\pi\)
\(420\) 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 \(-0.281514\pi\)
0.633752 + 0.773536i \(0.281514\pi\)
\(434\) 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 \(-0.437417\pi\)
0.195346 + 0.980734i \(0.437417\pi\)
\(440\) 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 \(-0.205282\pi\)
0.799152 + 0.601130i \(0.205282\pi\)
\(462\) 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 \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(468\) 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 \(-0.591926\pi\)
−0.284795 + 0.958588i \(0.591926\pi\)
\(480\) −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 \(-0.477316\pi\)
0.0712042 + 0.997462i \(0.477316\pi\)
\(504\) 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 \(-0.727674\pi\)
−0.655814 + 0.754923i \(0.727674\pi\)
\(510\) 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 \(-0.823984\pi\)
−0.850968 + 0.525217i \(0.823984\pi\)
\(522\) 2.63551 0.115353
\(523\) 41.6214 1.81997 0.909987 0.414636i \(-0.136091\pi\)
0.909987 + 0.414636i \(0.136091\pi\)
\(524\) −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 \(-0.507086\pi\)
−0.0222599 + 0.999752i \(0.507086\pi\)
\(564\) 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 \(-0.755538\pi\)
−0.719302 + 0.694697i \(0.755538\pi\)
\(578\) −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 \(-0.409562\pi\)
0.280313 + 0.959909i \(0.409562\pi\)
\(588\) 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 \(-0.629934\pi\)
−0.396958 + 0.917837i \(0.629934\pi\)
\(594\) −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 \(-0.704675\pi\)
−0.599604 + 0.800297i \(0.704675\pi\)
\(602\) 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 \(-0.823300\pi\)
−0.849837 + 0.527046i \(0.823300\pi\)
\(608\) 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 \(-0.542057\pi\)
−0.131743 + 0.991284i \(0.542057\pi\)
\(620\) 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 \(-0.457777\pi\)
0.132260 + 0.991215i \(0.457777\pi\)
\(644\) 0 0
\(645\) −11.0640 −0.435646
\(646\) −1.03094 −0.0405617
\(647\) 3.62154 0.142377 0.0711887 0.997463i \(-0.477321\pi\)
0.0711887 + 0.997463i \(0.477321\pi\)
\(648\) 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 \(-0.751654\pi\)
−0.710771 + 0.703424i \(0.751654\pi\)
\(662\) 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 \(-0.668818\pi\)
−0.505842 + 0.862626i \(0.668818\pi\)
\(678\) 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 \(-0.814350\pi\)
−0.834685 + 0.550728i \(0.814350\pi\)
\(692\) 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 \(-0.399618\pi\)
0.310158 + 0.950685i \(0.399618\pi\)
\(720\) 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 \(-0.329621\pi\)
0.510067 + 0.860135i \(0.329621\pi\)
\(728\) 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 \(-0.502638\pi\)
−0.00828795 + 0.999966i \(0.502638\pi\)
\(734\) 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 \(-0.313031\pi\)
0.554183 + 0.832395i \(0.313031\pi\)
\(762\) 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 \(-0.548581\pi\)
−0.152029 + 0.988376i \(0.548581\pi\)
\(770\) 0 0
\(771\) 14.7643 0.531723
\(772\) 1.19473 0.0429993
\(773\) 15.0072 0.539770 0.269885 0.962893i \(-0.413014\pi\)
0.269885 + 0.962893i \(0.413014\pi\)
\(774\) 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 \(-0.721574\pi\)
−0.641227 + 0.767351i \(0.721574\pi\)
\(788\) 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 \(-0.278867\pi\)
0.640162 + 0.768240i \(0.278867\pi\)
\(798\) 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 \(-0.513307\pi\)
−0.0417929 + 0.999126i \(0.513307\pi\)
\(812\) 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 \(-0.638962\pi\)
−0.422825 + 0.906211i \(0.638962\pi\)
\(830\) 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 \(-0.282878\pi\)
0.630431 + 0.776246i \(0.282878\pi\)
\(840\) 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 \(-0.578104\pi\)
−0.242916 + 0.970047i \(0.578104\pi\)
\(854\) 0 0
\(855\) 2.27283 0.0777291
\(856\) −7.55741 −0.258307
\(857\) −36.5314 −1.24789 −0.623944 0.781469i \(-0.714471\pi\)
−0.623944 + 0.781469i \(0.714471\pi\)
\(858\) −11.1476 −0.380572
\(859\) −1.93415 −0.0659924 −0.0329962 0.999455i \(-0.510505\pi\)
−0.0329962 + 0.999455i \(0.510505\pi\)
\(860\) 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 \(-0.525308\pi\)
−0.0794249 + 0.996841i \(0.525308\pi\)
\(882\) 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 \(-0.614467\pi\)
−0.351909 + 0.936034i \(0.614467\pi\)
\(888\) 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 \(-0.854646\pi\)
−0.897538 + 0.440937i \(0.854646\pi\)
\(930\) −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 \(-0.718851\pi\)
−0.634637 + 0.772810i \(0.718851\pi\)
\(938\) 0 0
\(939\) 15.3225 0.500029
\(940\) −18.2684 −0.595848
\(941\) 19.1134 0.623079 0.311540 0.950233i \(-0.399155\pi\)
0.311540 + 0.950233i \(0.399155\pi\)
\(942\) −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 \(-0.533603\pi\)
−0.105371 + 0.994433i \(0.533603\pi\)
\(972\) −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 \(-0.344687\pi\)
0.468800 + 0.883304i \(0.344687\pi\)
\(984\) −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 \(-0.736419\pi\)
−0.676302 + 0.736624i \(0.736419\pi\)
\(998\) −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.cn.1.4 4
7.6 odd 2 6762.2.a.co.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.cn.1.4 4 1.1 even 1 trivial
6762.2.a.co.1.1 yes 4 7.6 odd 2