Properties

Label 3675.2.a.p.1.1
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3675.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{9} -6.00000 q^{11} +2.00000 q^{12} -3.00000 q^{13} -4.00000 q^{16} -4.00000 q^{17} +2.00000 q^{18} -1.00000 q^{19} -12.0000 q^{22} +4.00000 q^{23} -6.00000 q^{26} +1.00000 q^{27} -8.00000 q^{29} -1.00000 q^{31} -8.00000 q^{32} -6.00000 q^{33} -8.00000 q^{34} +2.00000 q^{36} -7.00000 q^{37} -2.00000 q^{38} -3.00000 q^{39} +6.00000 q^{41} -1.00000 q^{43} -12.0000 q^{44} +8.00000 q^{46} +2.00000 q^{47} -4.00000 q^{48} -4.00000 q^{51} -6.00000 q^{52} -4.00000 q^{53} +2.00000 q^{54} -1.00000 q^{57} -16.0000 q^{58} +8.00000 q^{59} +14.0000 q^{61} -2.00000 q^{62} -8.00000 q^{64} -12.0000 q^{66} -7.00000 q^{67} -8.00000 q^{68} +4.00000 q^{69} +6.00000 q^{71} +1.00000 q^{73} -14.0000 q^{74} -2.00000 q^{76} -6.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} +12.0000 q^{82} +2.00000 q^{83} -2.00000 q^{86} -8.00000 q^{87} +12.0000 q^{89} +8.00000 q^{92} -1.00000 q^{93} +4.00000 q^{94} -8.00000 q^{96} -6.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 2.00000 0.577350
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 2.00000 0.471405
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −12.0000 −2.55841
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −8.00000 −1.41421
\(33\) −6.00000 −1.04447
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −2.00000 −0.324443
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −12.0000 −1.80907
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) −4.00000 −0.577350
\(49\) 0 0
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) −6.00000 −0.832050
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) −16.0000 −2.10090
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −12.0000 −1.47710
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) −8.00000 −0.970143
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −14.0000 −1.62747
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) −1.00000 −0.103695
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −8.00000 −0.792118
\(103\) −19.0000 −1.87213 −0.936063 0.351833i \(-0.885559\pi\)
−0.936063 + 0.351833i \(0.885559\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −8.00000 −0.777029
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 2.00000 0.192450
\(109\) −15.0000 −1.43674 −0.718370 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −16.0000 −1.48556
\(117\) −3.00000 −0.277350
\(118\) 16.0000 1.47292
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 28.0000 2.53500
\(123\) 6.00000 0.541002
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) −12.0000 −1.04447
\(133\) 0 0
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 8.00000 0.681005
\(139\) −21.0000 −1.78120 −0.890598 0.454791i \(-0.849714\pi\)
−0.890598 + 0.454791i \(0.849714\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 12.0000 1.00702
\(143\) 18.0000 1.50524
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −14.0000 −1.15079
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −2.00000 −0.159111
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 0 0
\(162\) 2.00000 0.157135
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −2.00000 −0.152499
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) −16.0000 −1.21296
\(175\) 0 0
\(176\) 24.0000 1.80907
\(177\) 8.00000 0.601317
\(178\) 24.0000 1.79888
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 0 0
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 24.0000 1.75505
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) −8.00000 −0.577350
\(193\) 9.00000 0.647834 0.323917 0.946085i \(-0.395000\pi\)
0.323917 + 0.946085i \(0.395000\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) −12.0000 −0.852803
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 20.0000 1.40720
\(203\) 0 0
\(204\) −8.00000 −0.560112
\(205\) 0 0
\(206\) −38.0000 −2.64759
\(207\) 4.00000 0.278019
\(208\) 12.0000 0.832050
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −8.00000 −0.549442
\(213\) 6.00000 0.411113
\(214\) −24.0000 −1.64061
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −30.0000 −2.03186
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −14.0000 −0.939618
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) −10.0000 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) −2.00000 −0.132453
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 16.0000 1.04151
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) 14.0000 0.905585 0.452792 0.891616i \(-0.350428\pi\)
0.452792 + 0.891616i \(0.350428\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 50.0000 3.21412
\(243\) 1.00000 0.0641500
\(244\) 28.0000 1.79252
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 3.00000 0.190885
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) −10.0000 −0.627456
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) −4.00000 −0.247121
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) −14.0000 −0.855186
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 16.0000 0.970143
\(273\) 0 0
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 7.00000 0.420589 0.210295 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) −42.0000 −2.51899
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 4.00000 0.238197
\(283\) −7.00000 −0.416107 −0.208053 0.978117i \(-0.566713\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 36.0000 2.12872
\(287\) 0 0
\(288\) −8.00000 −0.471405
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 2.00000 0.117041
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −6.00000 −0.348155
\(298\) −8.00000 −0.463428
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 10.0000 0.574485
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −8.00000 −0.457330
\(307\) 3.00000 0.171219 0.0856095 0.996329i \(-0.472716\pi\)
0.0856095 + 0.996329i \(0.472716\pi\)
\(308\) 0 0
\(309\) −19.0000 −1.08087
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 11.0000 0.621757 0.310878 0.950450i \(-0.399377\pi\)
0.310878 + 0.950450i \(0.399377\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −20.0000 −1.12331 −0.561656 0.827371i \(-0.689836\pi\)
−0.561656 + 0.827371i \(0.689836\pi\)
\(318\) −8.00000 −0.448618
\(319\) 48.0000 2.68748
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) −15.0000 −0.829502
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9.00000 −0.494685 −0.247342 0.968928i \(-0.579557\pi\)
−0.247342 + 0.968928i \(0.579557\pi\)
\(332\) 4.00000 0.219529
\(333\) −7.00000 −0.383598
\(334\) −20.0000 −1.09435
\(335\) 0 0
\(336\) 0 0
\(337\) −25.0000 −1.36184 −0.680918 0.732359i \(-0.738419\pi\)
−0.680918 + 0.732359i \(0.738419\pi\)
\(338\) −8.00000 −0.435143
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 48.0000 2.58050
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −16.0000 −0.857690
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 48.0000 2.55841
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 16.0000 0.850390
\(355\) 0 0
\(356\) 24.0000 1.27200
\(357\) 0 0
\(358\) 36.0000 1.90266
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −26.0000 −1.36653
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) 28.0000 1.46358
\(367\) 19.0000 0.991792 0.495896 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) −16.0000 −0.834058
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) 48.0000 2.48202
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 0 0
\(381\) −5.00000 −0.256158
\(382\) 20.0000 1.02329
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) −1.00000 −0.0508329
\(388\) −12.0000 −0.609208
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) −2.00000 −0.100887
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) −37.0000 −1.85698 −0.928488 0.371361i \(-0.878891\pi\)
−0.928488 + 0.371361i \(0.878891\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) −14.0000 −0.698257
\(403\) 3.00000 0.149441
\(404\) 20.0000 0.995037
\(405\) 0 0
\(406\) 0 0
\(407\) 42.0000 2.08186
\(408\) 0 0
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) −38.0000 −1.87213
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 24.0000 1.17670
\(417\) −21.0000 −1.02837
\(418\) 12.0000 0.586939
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) −40.0000 −1.94717
\(423\) 2.00000 0.0972433
\(424\) 0 0
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 0 0
\(428\) −24.0000 −1.16008
\(429\) 18.0000 0.869048
\(430\) 0 0
\(431\) −2.00000 −0.0963366 −0.0481683 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) −4.00000 −0.192450
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −30.0000 −1.43674
\(437\) −4.00000 −0.191346
\(438\) 2.00000 0.0955637
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −14.0000 −0.664411
\(445\) 0 0
\(446\) −48.0000 −2.27287
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 12.0000 0.564433
\(453\) 8.00000 0.375873
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 15.0000 0.701670 0.350835 0.936437i \(-0.385898\pi\)
0.350835 + 0.936437i \(0.385898\pi\)
\(458\) −26.0000 −1.21490
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) 0 0
\(463\) −3.00000 −0.139422 −0.0697109 0.997567i \(-0.522208\pi\)
−0.0697109 + 0.997567i \(0.522208\pi\)
\(464\) 32.0000 1.48556
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 22.0000 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) −2.00000 −0.0918630
\(475\) 0 0
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 28.0000 1.28069
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) 21.0000 0.957518
\(482\) 36.0000 1.63976
\(483\) 0 0
\(484\) 50.0000 2.27273
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 12.0000 0.541002
\(493\) 32.0000 1.44121
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 29.0000 1.29822 0.649109 0.760695i \(-0.275142\pi\)
0.649109 + 0.760695i \(0.275142\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) −24.0000 −1.07117
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −48.0000 −2.13386
\(507\) −4.00000 −0.177646
\(508\) −10.0000 −0.443678
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 32.0000 1.41421
\(513\) −1.00000 −0.0441511
\(514\) −36.0000 −1.58789
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) −16.0000 −0.700301
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 4.00000 0.174243
\(528\) 24.0000 1.04447
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −18.0000 −0.779667
\(534\) 24.0000 1.03858
\(535\) 0 0
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 20.0000 0.862261
\(539\) 0 0
\(540\) 0 0
\(541\) −3.00000 −0.128980 −0.0644900 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) 48.0000 2.06178
\(543\) −13.0000 −0.557883
\(544\) 32.0000 1.37199
\(545\) 0 0
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) −16.0000 −0.683486
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) −42.0000 −1.78120
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 3.00000 0.126886
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) −24.0000 −1.01238
\(563\) −26.0000 −1.09577 −0.547885 0.836554i \(-0.684567\pi\)
−0.547885 + 0.836554i \(0.684567\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −3.00000 −0.125546 −0.0627730 0.998028i \(-0.519994\pi\)
−0.0627730 + 0.998028i \(0.519994\pi\)
\(572\) 36.0000 1.50524
\(573\) 10.0000 0.417756
\(574\) 0 0
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) −29.0000 −1.20729 −0.603643 0.797255i \(-0.706285\pi\)
−0.603643 + 0.797255i \(0.706285\pi\)
\(578\) −2.00000 −0.0831890
\(579\) 9.00000 0.374027
\(580\) 0 0
\(581\) 0 0
\(582\) −12.0000 −0.497416
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) −32.0000 −1.32191
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 28.0000 1.15079
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) −8.00000 −0.327693
\(597\) −8.00000 −0.327418
\(598\) −24.0000 −0.981433
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) 33.0000 1.34610 0.673049 0.739598i \(-0.264984\pi\)
0.673049 + 0.739598i \(0.264984\pi\)
\(602\) 0 0
\(603\) −7.00000 −0.285062
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 20.0000 0.812444
\(607\) 35.0000 1.42061 0.710303 0.703896i \(-0.248558\pi\)
0.710303 + 0.703896i \(0.248558\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) −8.00000 −0.323381
\(613\) 30.0000 1.21169 0.605844 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 6.00000 0.242140
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −38.0000 −1.52858
\(619\) −3.00000 −0.120580 −0.0602901 0.998181i \(-0.519203\pi\)
−0.0602901 + 0.998181i \(0.519203\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 12.0000 0.480384
\(625\) 0 0
\(626\) 22.0000 0.879297
\(627\) 6.00000 0.239617
\(628\) 20.0000 0.798087
\(629\) 28.0000 1.11643
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) −40.0000 −1.58860
\(635\) 0 0
\(636\) −8.00000 −0.317221
\(637\) 0 0
\(638\) 96.0000 3.80068
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) −24.0000 −0.947204
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 30.0000 1.17942 0.589711 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −30.0000 −1.17309
\(655\) 0 0
\(656\) −24.0000 −0.937043
\(657\) 1.00000 0.0390137
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −23.0000 −0.894596 −0.447298 0.894385i \(-0.647614\pi\)
−0.447298 + 0.894385i \(0.647614\pi\)
\(662\) −18.0000 −0.699590
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 0 0
\(666\) −14.0000 −0.542489
\(667\) −32.0000 −1.23904
\(668\) −20.0000 −0.773823
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) −84.0000 −3.24278
\(672\) 0 0
\(673\) 37.0000 1.42625 0.713123 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) −50.0000 −1.92593
\(675\) 0 0
\(676\) −8.00000 −0.307692
\(677\) 16.0000 0.614930 0.307465 0.951559i \(-0.400519\pi\)
0.307465 + 0.951559i \(0.400519\pi\)
\(678\) 12.0000 0.460857
\(679\) 0 0
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) 12.0000 0.459504
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 0 0
\(687\) −13.0000 −0.495981
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −27.0000 −1.02713 −0.513564 0.858051i \(-0.671675\pi\)
−0.513564 + 0.858051i \(0.671675\pi\)
\(692\) 48.0000 1.82469
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) 0 0
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) −4.00000 −0.151402
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −44.0000 −1.66186 −0.830929 0.556379i \(-0.812190\pi\)
−0.830929 + 0.556379i \(0.812190\pi\)
\(702\) −6.00000 −0.226455
\(703\) 7.00000 0.264010
\(704\) 48.0000 1.80907
\(705\) 0 0
\(706\) −36.0000 −1.35488
\(707\) 0 0
\(708\) 16.0000 0.601317
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 36.0000 1.34538
\(717\) 14.0000 0.522840
\(718\) −48.0000 −1.79134
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −36.0000 −1.33978
\(723\) 18.0000 0.669427
\(724\) −26.0000 −0.966282
\(725\) 0 0
\(726\) 50.0000 1.85567
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 28.0000 1.03491
\(733\) −43.0000 −1.58824 −0.794121 0.607760i \(-0.792068\pi\)
−0.794121 + 0.607760i \(0.792068\pi\)
\(734\) 38.0000 1.40261
\(735\) 0 0
\(736\) −32.0000 −1.17954
\(737\) 42.0000 1.54709
\(738\) 12.0000 0.441726
\(739\) 41.0000 1.50821 0.754105 0.656754i \(-0.228071\pi\)
0.754105 + 0.656754i \(0.228071\pi\)
\(740\) 0 0
\(741\) 3.00000 0.110208
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 2.00000 0.0731762
\(748\) 48.0000 1.75505
\(749\) 0 0
\(750\) 0 0
\(751\) 29.0000 1.05823 0.529113 0.848552i \(-0.322525\pi\)
0.529113 + 0.848552i \(0.322525\pi\)
\(752\) −8.00000 −0.291730
\(753\) −12.0000 −0.437304
\(754\) 48.0000 1.74806
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 22.0000 0.799076
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) −10.0000 −0.362262
\(763\) 0 0
\(764\) 20.0000 0.723575
\(765\) 0 0
\(766\) 56.0000 2.02336
\(767\) −24.0000 −0.866590
\(768\) 16.0000 0.577350
\(769\) 49.0000 1.76699 0.883493 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 18.0000 0.647834
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 12.0000 0.430221
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) −32.0000 −1.14432
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −24.0000 −0.854965
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −42.0000 −1.49146
\(794\) −74.0000 −2.62616
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −36.0000 −1.27519 −0.637593 0.770374i \(-0.720070\pi\)
−0.637593 + 0.770374i \(0.720070\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) −24.0000 −0.847469
\(803\) −6.00000 −0.211735
\(804\) −14.0000 −0.493742
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 48.0000 1.68551 0.842754 0.538299i \(-0.180933\pi\)
0.842754 + 0.538299i \(0.180933\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 84.0000 2.94420
\(815\) 0 0
\(816\) 16.0000 0.560112
\(817\) 1.00000 0.0349856
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) −16.0000 −0.558064
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 8.00000 0.278019
\(829\) −57.0000 −1.97969 −0.989846 0.142145i \(-0.954600\pi\)
−0.989846 + 0.142145i \(0.954600\pi\)
\(830\) 0 0
\(831\) 7.00000 0.242827
\(832\) 24.0000 0.832050
\(833\) 0 0
\(834\) −42.0000 −1.45434
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) −1.00000 −0.0345651
\(838\) −12.0000 −0.414533
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 2.00000 0.0689246
\(843\) −12.0000 −0.413302
\(844\) −40.0000 −1.37686
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) 16.0000 0.549442
\(849\) −7.00000 −0.240239
\(850\) 0 0
\(851\) −28.0000 −0.959828
\(852\) 12.0000 0.411113
\(853\) −9.00000 −0.308154 −0.154077 0.988059i \(-0.549240\pi\)
−0.154077 + 0.988059i \(0.549240\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 36.0000 1.22902
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −4.00000 −0.136241
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −8.00000 −0.272166
\(865\) 0 0
\(866\) −10.0000 −0.339814
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 21.0000 0.711558
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −32.0000 −1.07995
\(879\) −16.0000 −0.539667
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −7.00000 −0.235569 −0.117784 0.993039i \(-0.537579\pi\)
−0.117784 + 0.993039i \(0.537579\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 72.0000 2.41889
\(887\) 10.0000 0.335767 0.167884 0.985807i \(-0.446307\pi\)
0.167884 + 0.985807i \(0.446307\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) −48.0000 −1.60716
\(893\) −2.00000 −0.0669274
\(894\) −8.00000 −0.267560
\(895\) 0 0
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 60.0000 2.00223
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) −72.0000 −2.39734
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) −31.0000 −1.02934 −0.514669 0.857389i \(-0.672085\pi\)
−0.514669 + 0.857389i \(0.672085\pi\)
\(908\) −20.0000 −0.663723
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 4.00000 0.132453
\(913\) −12.0000 −0.397142
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) −8.00000 −0.264039
\(919\) −9.00000 −0.296883 −0.148441 0.988921i \(-0.547426\pi\)
−0.148441 + 0.988921i \(0.547426\pi\)
\(920\) 0 0
\(921\) 3.00000 0.0988534
\(922\) 16.0000 0.526932
\(923\) −18.0000 −0.592477
\(924\) 0 0
\(925\) 0 0
\(926\) −6.00000 −0.197172
\(927\) −19.0000 −0.624042
\(928\) 64.0000 2.10090
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −12.0000 −0.393073
\(933\) −6.00000 −0.196431
\(934\) 44.0000 1.43972
\(935\) 0 0
\(936\) 0 0
\(937\) −29.0000 −0.947389 −0.473694 0.880689i \(-0.657080\pi\)
−0.473694 + 0.880689i \(0.657080\pi\)
\(938\) 0 0
\(939\) 11.0000 0.358971
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 20.0000 0.651635
\(943\) 24.0000 0.781548
\(944\) −32.0000 −1.04151
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) −26.0000 −0.844886 −0.422443 0.906389i \(-0.638827\pi\)
−0.422443 + 0.906389i \(0.638827\pi\)
\(948\) −2.00000 −0.0649570
\(949\) −3.00000 −0.0973841
\(950\) 0 0
\(951\) −20.0000 −0.648544
\(952\) 0 0
\(953\) −4.00000 −0.129573 −0.0647864 0.997899i \(-0.520637\pi\)
−0.0647864 + 0.997899i \(0.520637\pi\)
\(954\) −8.00000 −0.259010
\(955\) 0 0
\(956\) 28.0000 0.905585
\(957\) 48.0000 1.55162
\(958\) 8.00000 0.258468
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 42.0000 1.35413
\(963\) −12.0000 −0.386695
\(964\) 36.0000 1.15948
\(965\) 0 0
\(966\) 0 0
\(967\) −55.0000 −1.76868 −0.884340 0.466843i \(-0.845391\pi\)
−0.884340 + 0.466843i \(0.845391\pi\)
\(968\) 0 0
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) 52.0000 1.66876 0.834380 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) 2.00000 0.0641500
\(973\) 0 0
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) −56.0000 −1.79252
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) 24.0000 0.767435
\(979\) −72.0000 −2.30113
\(980\) 0 0
\(981\) −15.0000 −0.478913
\(982\) −24.0000 −0.765871
\(983\) 32.0000 1.02064 0.510321 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 64.0000 2.03818
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −15.0000 −0.476491 −0.238245 0.971205i \(-0.576572\pi\)
−0.238245 + 0.971205i \(0.576572\pi\)
\(992\) 8.00000 0.254000
\(993\) −9.00000 −0.285606
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −25.0000 −0.791758 −0.395879 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(998\) 58.0000 1.83596
\(999\) −7.00000 −0.221470
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 3675.2.a.p.1.1 1
5.4 even 2 735.2.a.a.1.1 1
7.3 odd 6 525.2.i.a.226.1 2
7.5 odd 6 525.2.i.a.151.1 2
7.6 odd 2 3675.2.a.o.1.1 1
15.14 odd 2 2205.2.a.m.1.1 1
35.3 even 12 525.2.r.d.499.2 4
35.4 even 6 735.2.i.f.226.1 2
35.9 even 6 735.2.i.f.361.1 2
35.12 even 12 525.2.r.d.424.2 4
35.17 even 12 525.2.r.d.499.1 4
35.19 odd 6 105.2.i.b.46.1 yes 2
35.24 odd 6 105.2.i.b.16.1 2
35.33 even 12 525.2.r.d.424.1 4
35.34 odd 2 735.2.a.b.1.1 1
105.59 even 6 315.2.j.a.226.1 2
105.89 even 6 315.2.j.a.46.1 2
105.104 even 2 2205.2.a.k.1.1 1
140.19 even 6 1680.2.bg.l.1201.1 2
140.59 even 6 1680.2.bg.l.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.i.b.16.1 2 35.24 odd 6
105.2.i.b.46.1 yes 2 35.19 odd 6
315.2.j.a.46.1 2 105.89 even 6
315.2.j.a.226.1 2 105.59 even 6
525.2.i.a.151.1 2 7.5 odd 6
525.2.i.a.226.1 2 7.3 odd 6
525.2.r.d.424.1 4 35.33 even 12
525.2.r.d.424.2 4 35.12 even 12
525.2.r.d.499.1 4 35.17 even 12
525.2.r.d.499.2 4 35.3 even 12
735.2.a.a.1.1 1 5.4 even 2
735.2.a.b.1.1 1 35.34 odd 2
735.2.i.f.226.1 2 35.4 even 6
735.2.i.f.361.1 2 35.9 even 6
1680.2.bg.l.961.1 2 140.59 even 6
1680.2.bg.l.1201.1 2 140.19 even 6
2205.2.a.k.1.1 1 105.104 even 2
2205.2.a.m.1.1 1 15.14 odd 2
3675.2.a.o.1.1 1 7.6 odd 2
3675.2.a.p.1.1 1 1.1 even 1 trivial