Properties

Label 7623.2.a.cz.1.4
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.60257\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.60257 q^{2} +0.568238 q^{4} -1.76461 q^{5} -1.00000 q^{7} +2.29450 q^{8} +O(q^{10})\) \(q-1.60257 q^{2} +0.568238 q^{4} -1.76461 q^{5} -1.00000 q^{7} +2.29450 q^{8} +2.82792 q^{10} +0.237832 q^{13} +1.60257 q^{14} -4.81358 q^{16} +6.14292 q^{17} +1.53812 q^{19} -1.00272 q^{20} -3.89707 q^{23} -1.88615 q^{25} -0.381142 q^{26} -0.568238 q^{28} -4.45543 q^{29} +8.52985 q^{31} +3.12511 q^{32} -9.84447 q^{34} +1.76461 q^{35} -3.64448 q^{37} -2.46495 q^{38} -4.04890 q^{40} -1.90659 q^{41} -11.7102 q^{43} +6.24534 q^{46} +8.55604 q^{47} +1.00000 q^{49} +3.02269 q^{50} +0.135145 q^{52} -3.14359 q^{53} -2.29450 q^{56} +7.14015 q^{58} +2.56187 q^{59} -6.78951 q^{61} -13.6697 q^{62} +4.61895 q^{64} -0.419680 q^{65} -1.62185 q^{67} +3.49064 q^{68} -2.82792 q^{70} +7.37151 q^{71} +4.60387 q^{73} +5.84054 q^{74} +0.874018 q^{76} -17.2504 q^{79} +8.49410 q^{80} +3.05546 q^{82} +9.16622 q^{83} -10.8399 q^{85} +18.7665 q^{86} +1.14303 q^{89} -0.237832 q^{91} -2.21446 q^{92} -13.7117 q^{94} -2.71418 q^{95} -4.68538 q^{97} -1.60257 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 20q^{4} - 12q^{7} + O(q^{10}) \) \( 12q + 20q^{4} - 12q^{7} - 20q^{10} - 20q^{13} + 28q^{16} - 12q^{19} + 32q^{25} - 20q^{28} - 16q^{31} - 24q^{34} + 4q^{37} - 48q^{40} - 16q^{43} - 24q^{46} + 12q^{49} - 96q^{52} + 20q^{58} - 44q^{61} + 76q^{64} + 20q^{70} - 52q^{73} + 8q^{79} - 68q^{82} - 72q^{85} + 20q^{91} - 20q^{94} - 32q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.60257 −1.13319 −0.566595 0.823997i \(-0.691740\pi\)
−0.566595 + 0.823997i \(0.691740\pi\)
\(3\) 0 0
\(4\) 0.568238 0.284119
\(5\) −1.76461 −0.789158 −0.394579 0.918862i \(-0.629110\pi\)
−0.394579 + 0.918862i \(0.629110\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.29450 0.811229
\(9\) 0 0
\(10\) 2.82792 0.894265
\(11\) 0 0
\(12\) 0 0
\(13\) 0.237832 0.0659626 0.0329813 0.999456i \(-0.489500\pi\)
0.0329813 + 0.999456i \(0.489500\pi\)
\(14\) 1.60257 0.428305
\(15\) 0 0
\(16\) −4.81358 −1.20340
\(17\) 6.14292 1.48988 0.744938 0.667133i \(-0.232479\pi\)
0.744938 + 0.667133i \(0.232479\pi\)
\(18\) 0 0
\(19\) 1.53812 0.352869 0.176434 0.984312i \(-0.443544\pi\)
0.176434 + 0.984312i \(0.443544\pi\)
\(20\) −1.00272 −0.224215
\(21\) 0 0
\(22\) 0 0
\(23\) −3.89707 −0.812596 −0.406298 0.913741i \(-0.633181\pi\)
−0.406298 + 0.913741i \(0.633181\pi\)
\(24\) 0 0
\(25\) −1.88615 −0.377230
\(26\) −0.381142 −0.0747482
\(27\) 0 0
\(28\) −0.568238 −0.107387
\(29\) −4.45543 −0.827352 −0.413676 0.910424i \(-0.635755\pi\)
−0.413676 + 0.910424i \(0.635755\pi\)
\(30\) 0 0
\(31\) 8.52985 1.53201 0.766003 0.642837i \(-0.222243\pi\)
0.766003 + 0.642837i \(0.222243\pi\)
\(32\) 3.12511 0.552446
\(33\) 0 0
\(34\) −9.84447 −1.68831
\(35\) 1.76461 0.298274
\(36\) 0 0
\(37\) −3.64448 −0.599149 −0.299574 0.954073i \(-0.596845\pi\)
−0.299574 + 0.954073i \(0.596845\pi\)
\(38\) −2.46495 −0.399867
\(39\) 0 0
\(40\) −4.04890 −0.640188
\(41\) −1.90659 −0.297760 −0.148880 0.988855i \(-0.547567\pi\)
−0.148880 + 0.988855i \(0.547567\pi\)
\(42\) 0 0
\(43\) −11.7102 −1.78579 −0.892896 0.450262i \(-0.851330\pi\)
−0.892896 + 0.450262i \(0.851330\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.24534 0.920826
\(47\) 8.55604 1.24803 0.624013 0.781414i \(-0.285501\pi\)
0.624013 + 0.781414i \(0.285501\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 3.02269 0.427473
\(51\) 0 0
\(52\) 0.135145 0.0187412
\(53\) −3.14359 −0.431805 −0.215903 0.976415i \(-0.569269\pi\)
−0.215903 + 0.976415i \(0.569269\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.29450 −0.306616
\(57\) 0 0
\(58\) 7.14015 0.937547
\(59\) 2.56187 0.333527 0.166764 0.985997i \(-0.446668\pi\)
0.166764 + 0.985997i \(0.446668\pi\)
\(60\) 0 0
\(61\) −6.78951 −0.869307 −0.434654 0.900598i \(-0.643129\pi\)
−0.434654 + 0.900598i \(0.643129\pi\)
\(62\) −13.6697 −1.73605
\(63\) 0 0
\(64\) 4.61895 0.577369
\(65\) −0.419680 −0.0520549
\(66\) 0 0
\(67\) −1.62185 −0.198141 −0.0990704 0.995080i \(-0.531587\pi\)
−0.0990704 + 0.995080i \(0.531587\pi\)
\(68\) 3.49064 0.423302
\(69\) 0 0
\(70\) −2.82792 −0.338001
\(71\) 7.37151 0.874837 0.437418 0.899258i \(-0.355893\pi\)
0.437418 + 0.899258i \(0.355893\pi\)
\(72\) 0 0
\(73\) 4.60387 0.538842 0.269421 0.963023i \(-0.413168\pi\)
0.269421 + 0.963023i \(0.413168\pi\)
\(74\) 5.84054 0.678949
\(75\) 0 0
\(76\) 0.874018 0.100257
\(77\) 0 0
\(78\) 0 0
\(79\) −17.2504 −1.94082 −0.970409 0.241468i \(-0.922371\pi\)
−0.970409 + 0.241468i \(0.922371\pi\)
\(80\) 8.49410 0.949669
\(81\) 0 0
\(82\) 3.05546 0.337419
\(83\) 9.16622 1.00612 0.503062 0.864250i \(-0.332207\pi\)
0.503062 + 0.864250i \(0.332207\pi\)
\(84\) 0 0
\(85\) −10.8399 −1.17575
\(86\) 18.7665 2.02364
\(87\) 0 0
\(88\) 0 0
\(89\) 1.14303 0.121161 0.0605807 0.998163i \(-0.480705\pi\)
0.0605807 + 0.998163i \(0.480705\pi\)
\(90\) 0 0
\(91\) −0.237832 −0.0249315
\(92\) −2.21446 −0.230874
\(93\) 0 0
\(94\) −13.7117 −1.41425
\(95\) −2.71418 −0.278469
\(96\) 0 0
\(97\) −4.68538 −0.475728 −0.237864 0.971298i \(-0.576447\pi\)
−0.237864 + 0.971298i \(0.576447\pi\)
\(98\) −1.60257 −0.161884
\(99\) 0 0
\(100\) −1.07178 −0.107178
\(101\) 18.1433 1.80532 0.902661 0.430352i \(-0.141611\pi\)
0.902661 + 0.430352i \(0.141611\pi\)
\(102\) 0 0
\(103\) 4.75190 0.468219 0.234110 0.972210i \(-0.424783\pi\)
0.234110 + 0.972210i \(0.424783\pi\)
\(104\) 0.545705 0.0535108
\(105\) 0 0
\(106\) 5.03783 0.489317
\(107\) −2.16093 −0.208905 −0.104452 0.994530i \(-0.533309\pi\)
−0.104452 + 0.994530i \(0.533309\pi\)
\(108\) 0 0
\(109\) 4.21878 0.404086 0.202043 0.979377i \(-0.435242\pi\)
0.202043 + 0.979377i \(0.435242\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.81358 0.454841
\(113\) 7.53163 0.708516 0.354258 0.935148i \(-0.384733\pi\)
0.354258 + 0.935148i \(0.384733\pi\)
\(114\) 0 0
\(115\) 6.87682 0.641267
\(116\) −2.53174 −0.235066
\(117\) 0 0
\(118\) −4.10559 −0.377950
\(119\) −6.14292 −0.563120
\(120\) 0 0
\(121\) 0 0
\(122\) 10.8807 0.985090
\(123\) 0 0
\(124\) 4.84698 0.435272
\(125\) 12.1514 1.08685
\(126\) 0 0
\(127\) 3.86671 0.343115 0.171557 0.985174i \(-0.445120\pi\)
0.171557 + 0.985174i \(0.445120\pi\)
\(128\) −13.6524 −1.20671
\(129\) 0 0
\(130\) 0.672568 0.0589881
\(131\) −4.55591 −0.398052 −0.199026 0.979994i \(-0.563778\pi\)
−0.199026 + 0.979994i \(0.563778\pi\)
\(132\) 0 0
\(133\) −1.53812 −0.133372
\(134\) 2.59914 0.224531
\(135\) 0 0
\(136\) 14.0949 1.20863
\(137\) −3.34382 −0.285682 −0.142841 0.989746i \(-0.545624\pi\)
−0.142841 + 0.989746i \(0.545624\pi\)
\(138\) 0 0
\(139\) −10.7288 −0.910003 −0.455002 0.890491i \(-0.650361\pi\)
−0.455002 + 0.890491i \(0.650361\pi\)
\(140\) 1.00272 0.0847451
\(141\) 0 0
\(142\) −11.8134 −0.991356
\(143\) 0 0
\(144\) 0 0
\(145\) 7.86210 0.652912
\(146\) −7.37803 −0.610610
\(147\) 0 0
\(148\) −2.07093 −0.170229
\(149\) 20.4997 1.67940 0.839699 0.543052i \(-0.182731\pi\)
0.839699 + 0.543052i \(0.182731\pi\)
\(150\) 0 0
\(151\) −19.6500 −1.59909 −0.799547 0.600604i \(-0.794927\pi\)
−0.799547 + 0.600604i \(0.794927\pi\)
\(152\) 3.52922 0.286258
\(153\) 0 0
\(154\) 0 0
\(155\) −15.0519 −1.20899
\(156\) 0 0
\(157\) 13.0140 1.03863 0.519313 0.854584i \(-0.326188\pi\)
0.519313 + 0.854584i \(0.326188\pi\)
\(158\) 27.6450 2.19931
\(159\) 0 0
\(160\) −5.51460 −0.435967
\(161\) 3.89707 0.307133
\(162\) 0 0
\(163\) 8.09731 0.634230 0.317115 0.948387i \(-0.397286\pi\)
0.317115 + 0.948387i \(0.397286\pi\)
\(164\) −1.08340 −0.0845992
\(165\) 0 0
\(166\) −14.6895 −1.14013
\(167\) −4.97203 −0.384747 −0.192374 0.981322i \(-0.561619\pi\)
−0.192374 + 0.981322i \(0.561619\pi\)
\(168\) 0 0
\(169\) −12.9434 −0.995649
\(170\) 17.3717 1.33235
\(171\) 0 0
\(172\) −6.65419 −0.507377
\(173\) 14.0437 1.06772 0.533861 0.845572i \(-0.320741\pi\)
0.533861 + 0.845572i \(0.320741\pi\)
\(174\) 0 0
\(175\) 1.88615 0.142580
\(176\) 0 0
\(177\) 0 0
\(178\) −1.83180 −0.137299
\(179\) 19.2574 1.43937 0.719684 0.694301i \(-0.244286\pi\)
0.719684 + 0.694301i \(0.244286\pi\)
\(180\) 0 0
\(181\) −5.45622 −0.405558 −0.202779 0.979225i \(-0.564997\pi\)
−0.202779 + 0.979225i \(0.564997\pi\)
\(182\) 0.381142 0.0282522
\(183\) 0 0
\(184\) −8.94185 −0.659202
\(185\) 6.43108 0.472823
\(186\) 0 0
\(187\) 0 0
\(188\) 4.86187 0.354588
\(189\) 0 0
\(190\) 4.34967 0.315559
\(191\) 0.336721 0.0243643 0.0121822 0.999926i \(-0.496122\pi\)
0.0121822 + 0.999926i \(0.496122\pi\)
\(192\) 0 0
\(193\) −12.4807 −0.898378 −0.449189 0.893437i \(-0.648287\pi\)
−0.449189 + 0.893437i \(0.648287\pi\)
\(194\) 7.50866 0.539090
\(195\) 0 0
\(196\) 0.568238 0.0405884
\(197\) −20.9906 −1.49552 −0.747760 0.663969i \(-0.768871\pi\)
−0.747760 + 0.663969i \(0.768871\pi\)
\(198\) 0 0
\(199\) −18.6504 −1.32209 −0.661045 0.750346i \(-0.729887\pi\)
−0.661045 + 0.750346i \(0.729887\pi\)
\(200\) −4.32778 −0.306020
\(201\) 0 0
\(202\) −29.0759 −2.04577
\(203\) 4.45543 0.312710
\(204\) 0 0
\(205\) 3.36440 0.234980
\(206\) −7.61527 −0.530581
\(207\) 0 0
\(208\) −1.14482 −0.0793791
\(209\) 0 0
\(210\) 0 0
\(211\) 12.1517 0.836558 0.418279 0.908319i \(-0.362634\pi\)
0.418279 + 0.908319i \(0.362634\pi\)
\(212\) −1.78631 −0.122684
\(213\) 0 0
\(214\) 3.46304 0.236729
\(215\) 20.6640 1.40927
\(216\) 0 0
\(217\) −8.52985 −0.579044
\(218\) −6.76090 −0.457906
\(219\) 0 0
\(220\) 0 0
\(221\) 1.46098 0.0982762
\(222\) 0 0
\(223\) 22.0300 1.47524 0.737618 0.675219i \(-0.235951\pi\)
0.737618 + 0.675219i \(0.235951\pi\)
\(224\) −3.12511 −0.208805
\(225\) 0 0
\(226\) −12.0700 −0.802883
\(227\) −0.991072 −0.0657798 −0.0328899 0.999459i \(-0.510471\pi\)
−0.0328899 + 0.999459i \(0.510471\pi\)
\(228\) 0 0
\(229\) 26.2425 1.73415 0.867076 0.498176i \(-0.165997\pi\)
0.867076 + 0.498176i \(0.165997\pi\)
\(230\) −11.0206 −0.726677
\(231\) 0 0
\(232\) −10.2230 −0.671172
\(233\) −10.3858 −0.680394 −0.340197 0.940354i \(-0.610494\pi\)
−0.340197 + 0.940354i \(0.610494\pi\)
\(234\) 0 0
\(235\) −15.0981 −0.984890
\(236\) 1.45575 0.0947614
\(237\) 0 0
\(238\) 9.84447 0.638122
\(239\) −7.75162 −0.501411 −0.250705 0.968063i \(-0.580663\pi\)
−0.250705 + 0.968063i \(0.580663\pi\)
\(240\) 0 0
\(241\) 5.25863 0.338738 0.169369 0.985553i \(-0.445827\pi\)
0.169369 + 0.985553i \(0.445827\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −3.85805 −0.246987
\(245\) −1.76461 −0.112737
\(246\) 0 0
\(247\) 0.365814 0.0232762
\(248\) 19.5718 1.24281
\(249\) 0 0
\(250\) −19.4735 −1.23161
\(251\) −22.0821 −1.39381 −0.696905 0.717164i \(-0.745440\pi\)
−0.696905 + 0.717164i \(0.745440\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −6.19668 −0.388814
\(255\) 0 0
\(256\) 12.6411 0.790068
\(257\) 19.1651 1.19549 0.597745 0.801687i \(-0.296064\pi\)
0.597745 + 0.801687i \(0.296064\pi\)
\(258\) 0 0
\(259\) 3.64448 0.226457
\(260\) −0.238478 −0.0147898
\(261\) 0 0
\(262\) 7.30118 0.451068
\(263\) −20.1639 −1.24336 −0.621679 0.783272i \(-0.713549\pi\)
−0.621679 + 0.783272i \(0.713549\pi\)
\(264\) 0 0
\(265\) 5.54721 0.340763
\(266\) 2.46495 0.151136
\(267\) 0 0
\(268\) −0.921598 −0.0562955
\(269\) −26.6866 −1.62711 −0.813556 0.581487i \(-0.802471\pi\)
−0.813556 + 0.581487i \(0.802471\pi\)
\(270\) 0 0
\(271\) 12.5714 0.763659 0.381830 0.924233i \(-0.375294\pi\)
0.381830 + 0.924233i \(0.375294\pi\)
\(272\) −29.5694 −1.79291
\(273\) 0 0
\(274\) 5.35871 0.323732
\(275\) 0 0
\(276\) 0 0
\(277\) −3.65785 −0.219779 −0.109890 0.993944i \(-0.535050\pi\)
−0.109890 + 0.993944i \(0.535050\pi\)
\(278\) 17.1936 1.03121
\(279\) 0 0
\(280\) 4.04890 0.241968
\(281\) −4.74032 −0.282784 −0.141392 0.989954i \(-0.545158\pi\)
−0.141392 + 0.989954i \(0.545158\pi\)
\(282\) 0 0
\(283\) 4.64873 0.276338 0.138169 0.990409i \(-0.455878\pi\)
0.138169 + 0.990409i \(0.455878\pi\)
\(284\) 4.18877 0.248558
\(285\) 0 0
\(286\) 0 0
\(287\) 1.90659 0.112543
\(288\) 0 0
\(289\) 20.7355 1.21973
\(290\) −12.5996 −0.739873
\(291\) 0 0
\(292\) 2.61609 0.153095
\(293\) −8.26249 −0.482700 −0.241350 0.970438i \(-0.577590\pi\)
−0.241350 + 0.970438i \(0.577590\pi\)
\(294\) 0 0
\(295\) −4.52071 −0.263206
\(296\) −8.36226 −0.486047
\(297\) 0 0
\(298\) −32.8522 −1.90308
\(299\) −0.926848 −0.0536010
\(300\) 0 0
\(301\) 11.7102 0.674966
\(302\) 31.4905 1.81208
\(303\) 0 0
\(304\) −7.40387 −0.424641
\(305\) 11.9808 0.686020
\(306\) 0 0
\(307\) −17.8463 −1.01854 −0.509272 0.860606i \(-0.670085\pi\)
−0.509272 + 0.860606i \(0.670085\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 24.1217 1.37002
\(311\) 12.6005 0.714511 0.357256 0.934007i \(-0.383712\pi\)
0.357256 + 0.934007i \(0.383712\pi\)
\(312\) 0 0
\(313\) −27.7579 −1.56897 −0.784484 0.620149i \(-0.787072\pi\)
−0.784484 + 0.620149i \(0.787072\pi\)
\(314\) −20.8558 −1.17696
\(315\) 0 0
\(316\) −9.80231 −0.551423
\(317\) 3.05179 0.171406 0.0857028 0.996321i \(-0.472686\pi\)
0.0857028 + 0.996321i \(0.472686\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.15065 −0.455635
\(321\) 0 0
\(322\) −6.24534 −0.348039
\(323\) 9.44855 0.525731
\(324\) 0 0
\(325\) −0.448586 −0.0248831
\(326\) −12.9765 −0.718703
\(327\) 0 0
\(328\) −4.37469 −0.241552
\(329\) −8.55604 −0.471710
\(330\) 0 0
\(331\) 22.7323 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(332\) 5.20859 0.285859
\(333\) 0 0
\(334\) 7.96804 0.435992
\(335\) 2.86194 0.156364
\(336\) 0 0
\(337\) −11.2682 −0.613815 −0.306908 0.951739i \(-0.599294\pi\)
−0.306908 + 0.951739i \(0.599294\pi\)
\(338\) 20.7428 1.12826
\(339\) 0 0
\(340\) −6.15962 −0.334052
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −26.8691 −1.44869
\(345\) 0 0
\(346\) −22.5060 −1.20993
\(347\) −27.8666 −1.49596 −0.747980 0.663722i \(-0.768976\pi\)
−0.747980 + 0.663722i \(0.768976\pi\)
\(348\) 0 0
\(349\) 7.54901 0.404089 0.202045 0.979376i \(-0.435241\pi\)
0.202045 + 0.979376i \(0.435241\pi\)
\(350\) −3.02269 −0.161570
\(351\) 0 0
\(352\) 0 0
\(353\) −11.2041 −0.596334 −0.298167 0.954514i \(-0.596375\pi\)
−0.298167 + 0.954514i \(0.596375\pi\)
\(354\) 0 0
\(355\) −13.0078 −0.690384
\(356\) 0.649515 0.0344242
\(357\) 0 0
\(358\) −30.8614 −1.63108
\(359\) 14.2242 0.750725 0.375362 0.926878i \(-0.377518\pi\)
0.375362 + 0.926878i \(0.377518\pi\)
\(360\) 0 0
\(361\) −16.6342 −0.875483
\(362\) 8.74398 0.459574
\(363\) 0 0
\(364\) −0.135145 −0.00708352
\(365\) −8.12403 −0.425231
\(366\) 0 0
\(367\) −16.3499 −0.853455 −0.426728 0.904380i \(-0.640334\pi\)
−0.426728 + 0.904380i \(0.640334\pi\)
\(368\) 18.7589 0.977875
\(369\) 0 0
\(370\) −10.3063 −0.535798
\(371\) 3.14359 0.163207
\(372\) 0 0
\(373\) −22.6303 −1.17175 −0.585877 0.810400i \(-0.699250\pi\)
−0.585877 + 0.810400i \(0.699250\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 19.6319 1.01244
\(377\) −1.05964 −0.0545743
\(378\) 0 0
\(379\) 1.82350 0.0936671 0.0468335 0.998903i \(-0.485087\pi\)
0.0468335 + 0.998903i \(0.485087\pi\)
\(380\) −1.54230 −0.0791184
\(381\) 0 0
\(382\) −0.539620 −0.0276094
\(383\) 11.1434 0.569404 0.284702 0.958616i \(-0.408105\pi\)
0.284702 + 0.958616i \(0.408105\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 20.0012 1.01803
\(387\) 0 0
\(388\) −2.66241 −0.135163
\(389\) 28.6576 1.45300 0.726499 0.687168i \(-0.241146\pi\)
0.726499 + 0.687168i \(0.241146\pi\)
\(390\) 0 0
\(391\) −23.9394 −1.21067
\(392\) 2.29450 0.115890
\(393\) 0 0
\(394\) 33.6390 1.69471
\(395\) 30.4402 1.53161
\(396\) 0 0
\(397\) −9.37451 −0.470493 −0.235247 0.971936i \(-0.575590\pi\)
−0.235247 + 0.971936i \(0.575590\pi\)
\(398\) 29.8886 1.49818
\(399\) 0 0
\(400\) 9.07914 0.453957
\(401\) −17.1824 −0.858048 −0.429024 0.903293i \(-0.641142\pi\)
−0.429024 + 0.903293i \(0.641142\pi\)
\(402\) 0 0
\(403\) 2.02867 0.101055
\(404\) 10.3097 0.512926
\(405\) 0 0
\(406\) −7.14015 −0.354360
\(407\) 0 0
\(408\) 0 0
\(409\) 0.548440 0.0271186 0.0135593 0.999908i \(-0.495684\pi\)
0.0135593 + 0.999908i \(0.495684\pi\)
\(410\) −5.39169 −0.266276
\(411\) 0 0
\(412\) 2.70021 0.133030
\(413\) −2.56187 −0.126061
\(414\) 0 0
\(415\) −16.1748 −0.793990
\(416\) 0.743249 0.0364408
\(417\) 0 0
\(418\) 0 0
\(419\) −5.31528 −0.259668 −0.129834 0.991536i \(-0.541444\pi\)
−0.129834 + 0.991536i \(0.541444\pi\)
\(420\) 0 0
\(421\) 9.52820 0.464376 0.232188 0.972671i \(-0.425412\pi\)
0.232188 + 0.972671i \(0.425412\pi\)
\(422\) −19.4740 −0.947978
\(423\) 0 0
\(424\) −7.21298 −0.350293
\(425\) −11.5865 −0.562026
\(426\) 0 0
\(427\) 6.78951 0.328567
\(428\) −1.22792 −0.0593537
\(429\) 0 0
\(430\) −33.1155 −1.59697
\(431\) −35.6144 −1.71548 −0.857742 0.514081i \(-0.828133\pi\)
−0.857742 + 0.514081i \(0.828133\pi\)
\(432\) 0 0
\(433\) 1.15006 0.0552682 0.0276341 0.999618i \(-0.491203\pi\)
0.0276341 + 0.999618i \(0.491203\pi\)
\(434\) 13.6697 0.656167
\(435\) 0 0
\(436\) 2.39727 0.114808
\(437\) −5.99417 −0.286740
\(438\) 0 0
\(439\) −33.3954 −1.59388 −0.796938 0.604061i \(-0.793548\pi\)
−0.796938 + 0.604061i \(0.793548\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.34133 −0.111366
\(443\) 20.8554 0.990871 0.495435 0.868645i \(-0.335009\pi\)
0.495435 + 0.868645i \(0.335009\pi\)
\(444\) 0 0
\(445\) −2.01701 −0.0956155
\(446\) −35.3046 −1.67172
\(447\) 0 0
\(448\) −4.61895 −0.218225
\(449\) −32.8374 −1.54969 −0.774846 0.632150i \(-0.782173\pi\)
−0.774846 + 0.632150i \(0.782173\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.27975 0.201303
\(453\) 0 0
\(454\) 1.58826 0.0745410
\(455\) 0.419680 0.0196749
\(456\) 0 0
\(457\) −22.9232 −1.07230 −0.536151 0.844122i \(-0.680122\pi\)
−0.536151 + 0.844122i \(0.680122\pi\)
\(458\) −42.0555 −1.96512
\(459\) 0 0
\(460\) 3.90767 0.182196
\(461\) −19.5054 −0.908455 −0.454228 0.890886i \(-0.650085\pi\)
−0.454228 + 0.890886i \(0.650085\pi\)
\(462\) 0 0
\(463\) −6.22668 −0.289378 −0.144689 0.989477i \(-0.546218\pi\)
−0.144689 + 0.989477i \(0.546218\pi\)
\(464\) 21.4466 0.995632
\(465\) 0 0
\(466\) 16.6439 0.771016
\(467\) 5.20609 0.240909 0.120455 0.992719i \(-0.461565\pi\)
0.120455 + 0.992719i \(0.461565\pi\)
\(468\) 0 0
\(469\) 1.62185 0.0748902
\(470\) 24.1958 1.11607
\(471\) 0 0
\(472\) 5.87822 0.270567
\(473\) 0 0
\(474\) 0 0
\(475\) −2.90113 −0.133113
\(476\) −3.49064 −0.159993
\(477\) 0 0
\(478\) 12.4225 0.568193
\(479\) 25.9938 1.18769 0.593844 0.804580i \(-0.297609\pi\)
0.593844 + 0.804580i \(0.297609\pi\)
\(480\) 0 0
\(481\) −0.866772 −0.0395214
\(482\) −8.42733 −0.383855
\(483\) 0 0
\(484\) 0 0
\(485\) 8.26787 0.375424
\(486\) 0 0
\(487\) 37.1070 1.68148 0.840739 0.541441i \(-0.182121\pi\)
0.840739 + 0.541441i \(0.182121\pi\)
\(488\) −15.5785 −0.705207
\(489\) 0 0
\(490\) 2.82792 0.127752
\(491\) 27.7241 1.25117 0.625585 0.780156i \(-0.284860\pi\)
0.625585 + 0.780156i \(0.284860\pi\)
\(492\) 0 0
\(493\) −27.3693 −1.23265
\(494\) −0.586243 −0.0263763
\(495\) 0 0
\(496\) −41.0591 −1.84361
\(497\) −7.37151 −0.330657
\(498\) 0 0
\(499\) −18.9850 −0.849885 −0.424942 0.905220i \(-0.639706\pi\)
−0.424942 + 0.905220i \(0.639706\pi\)
\(500\) 6.90487 0.308795
\(501\) 0 0
\(502\) 35.3881 1.57945
\(503\) −23.2430 −1.03635 −0.518177 0.855273i \(-0.673389\pi\)
−0.518177 + 0.855273i \(0.673389\pi\)
\(504\) 0 0
\(505\) −32.0158 −1.42468
\(506\) 0 0
\(507\) 0 0
\(508\) 2.19721 0.0974853
\(509\) −28.8227 −1.27754 −0.638771 0.769397i \(-0.720557\pi\)
−0.638771 + 0.769397i \(0.720557\pi\)
\(510\) 0 0
\(511\) −4.60387 −0.203663
\(512\) 7.04659 0.311418
\(513\) 0 0
\(514\) −30.7135 −1.35472
\(515\) −8.38526 −0.369499
\(516\) 0 0
\(517\) 0 0
\(518\) −5.84054 −0.256619
\(519\) 0 0
\(520\) −0.962957 −0.0422285
\(521\) −41.2984 −1.80931 −0.904657 0.426140i \(-0.859873\pi\)
−0.904657 + 0.426140i \(0.859873\pi\)
\(522\) 0 0
\(523\) −21.0083 −0.918629 −0.459315 0.888274i \(-0.651905\pi\)
−0.459315 + 0.888274i \(0.651905\pi\)
\(524\) −2.58884 −0.113094
\(525\) 0 0
\(526\) 32.3141 1.40896
\(527\) 52.3982 2.28250
\(528\) 0 0
\(529\) −7.81281 −0.339687
\(530\) −8.88981 −0.386149
\(531\) 0 0
\(532\) −0.874018 −0.0378935
\(533\) −0.453449 −0.0196410
\(534\) 0 0
\(535\) 3.81319 0.164859
\(536\) −3.72134 −0.160738
\(537\) 0 0
\(538\) 42.7672 1.84383
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00282 −0.301075 −0.150537 0.988604i \(-0.548100\pi\)
−0.150537 + 0.988604i \(0.548100\pi\)
\(542\) −20.1466 −0.865371
\(543\) 0 0
\(544\) 19.1973 0.823076
\(545\) −7.44450 −0.318887
\(546\) 0 0
\(547\) 15.6735 0.670149 0.335075 0.942192i \(-0.391238\pi\)
0.335075 + 0.942192i \(0.391238\pi\)
\(548\) −1.90008 −0.0811676
\(549\) 0 0
\(550\) 0 0
\(551\) −6.85299 −0.291947
\(552\) 0 0
\(553\) 17.2504 0.733560
\(554\) 5.86197 0.249051
\(555\) 0 0
\(556\) −6.09650 −0.258549
\(557\) 43.2989 1.83463 0.917317 0.398157i \(-0.130350\pi\)
0.917317 + 0.398157i \(0.130350\pi\)
\(558\) 0 0
\(559\) −2.78506 −0.117796
\(560\) −8.49410 −0.358941
\(561\) 0 0
\(562\) 7.59671 0.320448
\(563\) 10.3329 0.435482 0.217741 0.976007i \(-0.430131\pi\)
0.217741 + 0.976007i \(0.430131\pi\)
\(564\) 0 0
\(565\) −13.2904 −0.559131
\(566\) −7.44992 −0.313144
\(567\) 0 0
\(568\) 16.9139 0.709693
\(569\) 21.1095 0.884957 0.442479 0.896779i \(-0.354099\pi\)
0.442479 + 0.896779i \(0.354099\pi\)
\(570\) 0 0
\(571\) 34.8603 1.45886 0.729430 0.684056i \(-0.239786\pi\)
0.729430 + 0.684056i \(0.239786\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −3.05546 −0.127532
\(575\) 7.35047 0.306536
\(576\) 0 0
\(577\) 13.5157 0.562668 0.281334 0.959610i \(-0.409223\pi\)
0.281334 + 0.959610i \(0.409223\pi\)
\(578\) −33.2301 −1.38219
\(579\) 0 0
\(580\) 4.46754 0.185504
\(581\) −9.16622 −0.380279
\(582\) 0 0
\(583\) 0 0
\(584\) 10.5636 0.437124
\(585\) 0 0
\(586\) 13.2412 0.546990
\(587\) 2.25845 0.0932161 0.0466081 0.998913i \(-0.485159\pi\)
0.0466081 + 0.998913i \(0.485159\pi\)
\(588\) 0 0
\(589\) 13.1199 0.540597
\(590\) 7.24476 0.298262
\(591\) 0 0
\(592\) 17.5430 0.721013
\(593\) 30.2674 1.24293 0.621467 0.783440i \(-0.286537\pi\)
0.621467 + 0.783440i \(0.286537\pi\)
\(594\) 0 0
\(595\) 10.8399 0.444391
\(596\) 11.6487 0.477149
\(597\) 0 0
\(598\) 1.48534 0.0607401
\(599\) −21.3459 −0.872169 −0.436084 0.899906i \(-0.643635\pi\)
−0.436084 + 0.899906i \(0.643635\pi\)
\(600\) 0 0
\(601\) −43.9786 −1.79392 −0.896962 0.442108i \(-0.854231\pi\)
−0.896962 + 0.442108i \(0.854231\pi\)
\(602\) −18.7665 −0.764865
\(603\) 0 0
\(604\) −11.1659 −0.454333
\(605\) 0 0
\(606\) 0 0
\(607\) −18.7123 −0.759507 −0.379753 0.925088i \(-0.623991\pi\)
−0.379753 + 0.925088i \(0.623991\pi\)
\(608\) 4.80679 0.194941
\(609\) 0 0
\(610\) −19.2001 −0.777391
\(611\) 2.03490 0.0823231
\(612\) 0 0
\(613\) 26.9777 1.08962 0.544810 0.838560i \(-0.316602\pi\)
0.544810 + 0.838560i \(0.316602\pi\)
\(614\) 28.6000 1.15420
\(615\) 0 0
\(616\) 0 0
\(617\) −29.1127 −1.17203 −0.586016 0.810300i \(-0.699304\pi\)
−0.586016 + 0.810300i \(0.699304\pi\)
\(618\) 0 0
\(619\) 9.41996 0.378620 0.189310 0.981917i \(-0.439375\pi\)
0.189310 + 0.981917i \(0.439375\pi\)
\(620\) −8.55303 −0.343498
\(621\) 0 0
\(622\) −20.1933 −0.809677
\(623\) −1.14303 −0.0457947
\(624\) 0 0
\(625\) −12.0117 −0.480467
\(626\) 44.4840 1.77794
\(627\) 0 0
\(628\) 7.39502 0.295093
\(629\) −22.3877 −0.892657
\(630\) 0 0
\(631\) −1.99811 −0.0795436 −0.0397718 0.999209i \(-0.512663\pi\)
−0.0397718 + 0.999209i \(0.512663\pi\)
\(632\) −39.5810 −1.57445
\(633\) 0 0
\(634\) −4.89071 −0.194235
\(635\) −6.82323 −0.270772
\(636\) 0 0
\(637\) 0.237832 0.00942323
\(638\) 0 0
\(639\) 0 0
\(640\) 24.0912 0.952288
\(641\) −45.5981 −1.80102 −0.900508 0.434840i \(-0.856805\pi\)
−0.900508 + 0.434840i \(0.856805\pi\)
\(642\) 0 0
\(643\) −18.0073 −0.710140 −0.355070 0.934840i \(-0.615543\pi\)
−0.355070 + 0.934840i \(0.615543\pi\)
\(644\) 2.21446 0.0872621
\(645\) 0 0
\(646\) −15.1420 −0.595753
\(647\) 47.3237 1.86049 0.930243 0.366944i \(-0.119596\pi\)
0.930243 + 0.366944i \(0.119596\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.718892 0.0281973
\(651\) 0 0
\(652\) 4.60119 0.180197
\(653\) −5.87593 −0.229943 −0.114972 0.993369i \(-0.536678\pi\)
−0.114972 + 0.993369i \(0.536678\pi\)
\(654\) 0 0
\(655\) 8.03941 0.314126
\(656\) 9.17755 0.358323
\(657\) 0 0
\(658\) 13.7117 0.534537
\(659\) −40.3862 −1.57322 −0.786612 0.617448i \(-0.788167\pi\)
−0.786612 + 0.617448i \(0.788167\pi\)
\(660\) 0 0
\(661\) 22.8932 0.890442 0.445221 0.895421i \(-0.353125\pi\)
0.445221 + 0.895421i \(0.353125\pi\)
\(662\) −36.4301 −1.41590
\(663\) 0 0
\(664\) 21.0319 0.816197
\(665\) 2.71418 0.105251
\(666\) 0 0
\(667\) 17.3631 0.672303
\(668\) −2.82530 −0.109314
\(669\) 0 0
\(670\) −4.58646 −0.177190
\(671\) 0 0
\(672\) 0 0
\(673\) −16.1286 −0.621712 −0.310856 0.950457i \(-0.600616\pi\)
−0.310856 + 0.950457i \(0.600616\pi\)
\(674\) 18.0580 0.695569
\(675\) 0 0
\(676\) −7.35495 −0.282883
\(677\) 27.0862 1.04101 0.520504 0.853859i \(-0.325744\pi\)
0.520504 + 0.853859i \(0.325744\pi\)
\(678\) 0 0
\(679\) 4.68538 0.179808
\(680\) −24.8721 −0.953801
\(681\) 0 0
\(682\) 0 0
\(683\) −20.7408 −0.793623 −0.396811 0.917900i \(-0.629883\pi\)
−0.396811 + 0.917900i \(0.629883\pi\)
\(684\) 0 0
\(685\) 5.90054 0.225448
\(686\) 1.60257 0.0611865
\(687\) 0 0
\(688\) 56.3681 2.14901
\(689\) −0.747645 −0.0284830
\(690\) 0 0
\(691\) −16.8328 −0.640348 −0.320174 0.947359i \(-0.603741\pi\)
−0.320174 + 0.947359i \(0.603741\pi\)
\(692\) 7.98015 0.303360
\(693\) 0 0
\(694\) 44.6583 1.69521
\(695\) 18.9321 0.718136
\(696\) 0 0
\(697\) −11.7121 −0.443626
\(698\) −12.0978 −0.457910
\(699\) 0 0
\(700\) 1.07178 0.0405095
\(701\) −39.9602 −1.50928 −0.754638 0.656141i \(-0.772188\pi\)
−0.754638 + 0.656141i \(0.772188\pi\)
\(702\) 0 0
\(703\) −5.60565 −0.211421
\(704\) 0 0
\(705\) 0 0
\(706\) 17.9554 0.675760
\(707\) −18.1433 −0.682348
\(708\) 0 0
\(709\) −2.60837 −0.0979595 −0.0489797 0.998800i \(-0.515597\pi\)
−0.0489797 + 0.998800i \(0.515597\pi\)
\(710\) 20.8460 0.782336
\(711\) 0 0
\(712\) 2.62270 0.0982897
\(713\) −33.2415 −1.24490
\(714\) 0 0
\(715\) 0 0
\(716\) 10.9428 0.408952
\(717\) 0 0
\(718\) −22.7953 −0.850714
\(719\) −38.7626 −1.44560 −0.722800 0.691057i \(-0.757145\pi\)
−0.722800 + 0.691057i \(0.757145\pi\)
\(720\) 0 0
\(721\) −4.75190 −0.176970
\(722\) 26.6575 0.992089
\(723\) 0 0
\(724\) −3.10043 −0.115227
\(725\) 8.40361 0.312102
\(726\) 0 0
\(727\) −51.3510 −1.90450 −0.952251 0.305317i \(-0.901238\pi\)
−0.952251 + 0.305317i \(0.901238\pi\)
\(728\) −0.545705 −0.0202252
\(729\) 0 0
\(730\) 13.0193 0.481868
\(731\) −71.9350 −2.66061
\(732\) 0 0
\(733\) −48.8445 −1.80411 −0.902057 0.431617i \(-0.857943\pi\)
−0.902057 + 0.431617i \(0.857943\pi\)
\(734\) 26.2018 0.967127
\(735\) 0 0
\(736\) −12.1788 −0.448916
\(737\) 0 0
\(738\) 0 0
\(739\) −15.0365 −0.553127 −0.276563 0.960996i \(-0.589196\pi\)
−0.276563 + 0.960996i \(0.589196\pi\)
\(740\) 3.65438 0.134338
\(741\) 0 0
\(742\) −5.03783 −0.184945
\(743\) −45.1304 −1.65567 −0.827836 0.560970i \(-0.810428\pi\)
−0.827836 + 0.560970i \(0.810428\pi\)
\(744\) 0 0
\(745\) −36.1739 −1.32531
\(746\) 36.2668 1.32782
\(747\) 0 0
\(748\) 0 0
\(749\) 2.16093 0.0789585
\(750\) 0 0
\(751\) 6.78179 0.247471 0.123736 0.992315i \(-0.460513\pi\)
0.123736 + 0.992315i \(0.460513\pi\)
\(752\) −41.1852 −1.50187
\(753\) 0 0
\(754\) 1.69815 0.0618431
\(755\) 34.6746 1.26194
\(756\) 0 0
\(757\) 49.9448 1.81528 0.907638 0.419754i \(-0.137884\pi\)
0.907638 + 0.419754i \(0.137884\pi\)
\(758\) −2.92230 −0.106143
\(759\) 0 0
\(760\) −6.22770 −0.225902
\(761\) −28.2853 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(762\) 0 0
\(763\) −4.21878 −0.152730
\(764\) 0.191338 0.00692236
\(765\) 0 0
\(766\) −17.8582 −0.645242
\(767\) 0.609294 0.0220003
\(768\) 0 0
\(769\) 51.9422 1.87308 0.936542 0.350555i \(-0.114007\pi\)
0.936542 + 0.350555i \(0.114007\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.09198 −0.255246
\(773\) 4.11535 0.148019 0.0740095 0.997258i \(-0.476420\pi\)
0.0740095 + 0.997258i \(0.476420\pi\)
\(774\) 0 0
\(775\) −16.0886 −0.577919
\(776\) −10.7506 −0.385924
\(777\) 0 0
\(778\) −45.9258 −1.64652
\(779\) −2.93257 −0.105070
\(780\) 0 0
\(781\) 0 0
\(782\) 38.3646 1.37192
\(783\) 0 0
\(784\) −4.81358 −0.171914
\(785\) −22.9646 −0.819640
\(786\) 0 0
\(787\) −0.444368 −0.0158400 −0.00792001 0.999969i \(-0.502521\pi\)
−0.00792001 + 0.999969i \(0.502521\pi\)
\(788\) −11.9277 −0.424906
\(789\) 0 0
\(790\) −48.7826 −1.73561
\(791\) −7.53163 −0.267794
\(792\) 0 0
\(793\) −1.61476 −0.0573418
\(794\) 15.0233 0.533158
\(795\) 0 0
\(796\) −10.5978 −0.375631
\(797\) 16.0300 0.567810 0.283905 0.958852i \(-0.408370\pi\)
0.283905 + 0.958852i \(0.408370\pi\)
\(798\) 0 0
\(799\) 52.5591 1.85941
\(800\) −5.89442 −0.208399
\(801\) 0 0
\(802\) 27.5360 0.972331
\(803\) 0 0
\(804\) 0 0
\(805\) −6.87682 −0.242376
\(806\) −3.25109 −0.114515
\(807\) 0 0
\(808\) 41.6298 1.46453
\(809\) −24.1194 −0.847993 −0.423997 0.905664i \(-0.639373\pi\)
−0.423997 + 0.905664i \(0.639373\pi\)
\(810\) 0 0
\(811\) 29.6041 1.03954 0.519769 0.854307i \(-0.326018\pi\)
0.519769 + 0.854307i \(0.326018\pi\)
\(812\) 2.53174 0.0888468
\(813\) 0 0
\(814\) 0 0
\(815\) −14.2886 −0.500507
\(816\) 0 0
\(817\) −18.0117 −0.630151
\(818\) −0.878914 −0.0307305
\(819\) 0 0
\(820\) 1.91178 0.0667621
\(821\) 44.3006 1.54610 0.773051 0.634344i \(-0.218730\pi\)
0.773051 + 0.634344i \(0.218730\pi\)
\(822\) 0 0
\(823\) 24.1312 0.841162 0.420581 0.907255i \(-0.361826\pi\)
0.420581 + 0.907255i \(0.361826\pi\)
\(824\) 10.9033 0.379833
\(825\) 0 0
\(826\) 4.10559 0.142852
\(827\) −30.3326 −1.05477 −0.527383 0.849627i \(-0.676827\pi\)
−0.527383 + 0.849627i \(0.676827\pi\)
\(828\) 0 0
\(829\) −46.1560 −1.60306 −0.801532 0.597951i \(-0.795982\pi\)
−0.801532 + 0.597951i \(0.795982\pi\)
\(830\) 25.9213 0.899741
\(831\) 0 0
\(832\) 1.09853 0.0380848
\(833\) 6.14292 0.212840
\(834\) 0 0
\(835\) 8.77370 0.303626
\(836\) 0 0
\(837\) 0 0
\(838\) 8.51812 0.294253
\(839\) −39.3482 −1.35845 −0.679226 0.733930i \(-0.737684\pi\)
−0.679226 + 0.733930i \(0.737684\pi\)
\(840\) 0 0
\(841\) −9.14915 −0.315488
\(842\) −15.2696 −0.526226
\(843\) 0 0
\(844\) 6.90505 0.237682
\(845\) 22.8401 0.785724
\(846\) 0 0
\(847\) 0 0
\(848\) 15.1319 0.519633
\(849\) 0 0
\(850\) 18.5682 0.636882
\(851\) 14.2028 0.486866
\(852\) 0 0
\(853\) −12.9953 −0.444950 −0.222475 0.974938i \(-0.571414\pi\)
−0.222475 + 0.974938i \(0.571414\pi\)
\(854\) −10.8807 −0.372329
\(855\) 0 0
\(856\) −4.95825 −0.169470
\(857\) 29.0953 0.993877 0.496939 0.867786i \(-0.334457\pi\)
0.496939 + 0.867786i \(0.334457\pi\)
\(858\) 0 0
\(859\) 23.4404 0.799775 0.399888 0.916564i \(-0.369049\pi\)
0.399888 + 0.916564i \(0.369049\pi\)
\(860\) 11.7421 0.400401
\(861\) 0 0
\(862\) 57.0746 1.94397
\(863\) −30.9751 −1.05441 −0.527203 0.849740i \(-0.676759\pi\)
−0.527203 + 0.849740i \(0.676759\pi\)
\(864\) 0 0
\(865\) −24.7816 −0.842601
\(866\) −1.84305 −0.0626294
\(867\) 0 0
\(868\) −4.84698 −0.164517
\(869\) 0 0
\(870\) 0 0
\(871\) −0.385728 −0.0130699
\(872\) 9.68000 0.327806
\(873\) 0 0
\(874\) 9.60609 0.324931
\(875\) −12.1514 −0.410791
\(876\) 0 0
\(877\) −45.4618 −1.53514 −0.767568 0.640967i \(-0.778533\pi\)
−0.767568 + 0.640967i \(0.778533\pi\)
\(878\) 53.5186 1.80616
\(879\) 0 0
\(880\) 0 0
\(881\) 48.3625 1.62937 0.814687 0.579901i \(-0.196909\pi\)
0.814687 + 0.579901i \(0.196909\pi\)
\(882\) 0 0
\(883\) −13.9195 −0.468428 −0.234214 0.972185i \(-0.575252\pi\)
−0.234214 + 0.972185i \(0.575252\pi\)
\(884\) 0.830184 0.0279221
\(885\) 0 0
\(886\) −33.4223 −1.12284
\(887\) −13.9089 −0.467015 −0.233508 0.972355i \(-0.575020\pi\)
−0.233508 + 0.972355i \(0.575020\pi\)
\(888\) 0 0
\(889\) −3.86671 −0.129685
\(890\) 3.23241 0.108350
\(891\) 0 0
\(892\) 12.5183 0.419142
\(893\) 13.1602 0.440390
\(894\) 0 0
\(895\) −33.9819 −1.13589
\(896\) 13.6524 0.456095
\(897\) 0 0
\(898\) 52.6243 1.75609
\(899\) −38.0041 −1.26751
\(900\) 0 0
\(901\) −19.3108 −0.643337
\(902\) 0 0
\(903\) 0 0
\(904\) 17.2813 0.574769
\(905\) 9.62810 0.320049
\(906\) 0 0
\(907\) 46.8698 1.55629 0.778144 0.628086i \(-0.216162\pi\)
0.778144 + 0.628086i \(0.216162\pi\)
\(908\) −0.563165 −0.0186893
\(909\) 0 0
\(910\) −0.672568 −0.0222954
\(911\) 17.2331 0.570957 0.285478 0.958385i \(-0.407848\pi\)
0.285478 + 0.958385i \(0.407848\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 36.7361 1.21512
\(915\) 0 0
\(916\) 14.9120 0.492705
\(917\) 4.55591 0.150450
\(918\) 0 0
\(919\) 20.0621 0.661786 0.330893 0.943668i \(-0.392650\pi\)
0.330893 + 0.943668i \(0.392650\pi\)
\(920\) 15.7789 0.520214
\(921\) 0 0
\(922\) 31.2587 1.02945
\(923\) 1.75318 0.0577066
\(924\) 0 0
\(925\) 6.87403 0.226017
\(926\) 9.97870 0.327921
\(927\) 0 0
\(928\) −13.9237 −0.457068
\(929\) −49.7071 −1.63084 −0.815419 0.578871i \(-0.803493\pi\)
−0.815419 + 0.578871i \(0.803493\pi\)
\(930\) 0 0
\(931\) 1.53812 0.0504099
\(932\) −5.90158 −0.193313
\(933\) 0 0
\(934\) −8.34314 −0.272996
\(935\) 0 0
\(936\) 0 0
\(937\) −40.9723 −1.33851 −0.669254 0.743034i \(-0.733386\pi\)
−0.669254 + 0.743034i \(0.733386\pi\)
\(938\) −2.59914 −0.0848648
\(939\) 0 0
\(940\) −8.57930 −0.279826
\(941\) 57.7002 1.88097 0.940486 0.339832i \(-0.110370\pi\)
0.940486 + 0.339832i \(0.110370\pi\)
\(942\) 0 0
\(943\) 7.43014 0.241959
\(944\) −12.3318 −0.401365
\(945\) 0 0
\(946\) 0 0
\(947\) 27.9921 0.909621 0.454811 0.890588i \(-0.349707\pi\)
0.454811 + 0.890588i \(0.349707\pi\)
\(948\) 0 0
\(949\) 1.09495 0.0355434
\(950\) 4.64926 0.150842
\(951\) 0 0
\(952\) −14.0949 −0.456820
\(953\) −14.9272 −0.483538 −0.241769 0.970334i \(-0.577728\pi\)
−0.241769 + 0.970334i \(0.577728\pi\)
\(954\) 0 0
\(955\) −0.594182 −0.0192273
\(956\) −4.40476 −0.142460
\(957\) 0 0
\(958\) −41.6570 −1.34588
\(959\) 3.34382 0.107978
\(960\) 0 0
\(961\) 41.7583 1.34704
\(962\) 1.38907 0.0447853
\(963\) 0 0
\(964\) 2.98815 0.0962419
\(965\) 22.0235 0.708962
\(966\) 0 0
\(967\) −40.7409 −1.31014 −0.655070 0.755568i \(-0.727361\pi\)
−0.655070 + 0.755568i \(0.727361\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −13.2499 −0.425427
\(971\) −32.4367 −1.04094 −0.520471 0.853879i \(-0.674244\pi\)
−0.520471 + 0.853879i \(0.674244\pi\)
\(972\) 0 0
\(973\) 10.7288 0.343949
\(974\) −59.4666 −1.90543
\(975\) 0 0
\(976\) 32.6818 1.04612
\(977\) −43.6622 −1.39688 −0.698438 0.715670i \(-0.746121\pi\)
−0.698438 + 0.715670i \(0.746121\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.00272 −0.0320307
\(981\) 0 0
\(982\) −44.4298 −1.41781
\(983\) −46.6583 −1.48817 −0.744085 0.668085i \(-0.767114\pi\)
−0.744085 + 0.668085i \(0.767114\pi\)
\(984\) 0 0
\(985\) 37.0403 1.18020
\(986\) 43.8613 1.39683
\(987\) 0 0
\(988\) 0.207869 0.00661320
\(989\) 45.6356 1.45113
\(990\) 0 0
\(991\) 23.0760 0.733033 0.366516 0.930412i \(-0.380550\pi\)
0.366516 + 0.930412i \(0.380550\pi\)
\(992\) 26.6567 0.846351
\(993\) 0 0
\(994\) 11.8134 0.374697
\(995\) 32.9106 1.04334
\(996\) 0 0
\(997\) −29.4136 −0.931538 −0.465769 0.884906i \(-0.654222\pi\)
−0.465769 + 0.884906i \(0.654222\pi\)
\(998\) 30.4248 0.963081
\(999\) 0 0
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 7623.2.a.cz.1.4 12
3.2 odd 2 inner 7623.2.a.cz.1.9 yes 12
11.10 odd 2 7623.2.a.da.1.9 yes 12
33.32 even 2 7623.2.a.da.1.4 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7623.2.a.cz.1.4 12 1.1 even 1 trivial
7623.2.a.cz.1.9 yes 12 3.2 odd 2 inner
7623.2.a.da.1.4 yes 12 33.32 even 2
7623.2.a.da.1.9 yes 12 11.10 odd 2