Properties

Label 7350.2.a.bn.1.1
Level 7350
Weight 2
Character 7350.1
Self dual yes
Analytic conductor 58.690
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.6900454856\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7350.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +7.00000 q^{19} -5.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} -6.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +5.00000 q^{37} +7.00000 q^{38} +1.00000 q^{39} -9.00000 q^{41} -10.0000 q^{43} -5.00000 q^{44} -3.00000 q^{46} +13.0000 q^{47} -1.00000 q^{48} -2.00000 q^{51} -1.00000 q^{52} +1.00000 q^{53} -1.00000 q^{54} -7.00000 q^{57} +4.00000 q^{59} -2.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} +5.00000 q^{66} -6.00000 q^{67} +2.00000 q^{68} +3.00000 q^{69} -2.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} +5.00000 q^{74} +7.00000 q^{76} +1.00000 q^{78} -14.0000 q^{79} +1.00000 q^{81} -9.00000 q^{82} +10.0000 q^{83} -10.0000 q^{86} -5.00000 q^{88} +10.0000 q^{89} -3.00000 q^{92} +6.00000 q^{93} +13.0000 q^{94} -1.00000 q^{96} -8.00000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.00000 0.870388
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 7.00000 1.13555
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 13.0000 1.89624 0.948122 0.317905i \(-0.102979\pi\)
0.948122 + 0.317905i \(0.102979\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −1.00000 −0.138675
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 2.00000 0.242536
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 5.00000 0.581238
\(75\) 0 0
\(76\) 7.00000 0.802955
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 0 0
\(88\) −5.00000 −0.533002
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 6.00000 0.622171
\(94\) 13.0000 1.34085
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) −2.00000 −0.198030
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 1.00000 0.0971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −7.00000 −0.655610
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −2.00000 −0.181071
\(123\) 9.00000 0.811503
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) −9.00000 −0.798621 −0.399310 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) −17.0000 −1.48530 −0.742648 0.669681i \(-0.766431\pi\)
−0.742648 + 0.669681i \(0.766431\pi\)
\(132\) 5.00000 0.435194
\(133\) 0 0
\(134\) −6.00000 −0.518321
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 3.00000 0.255377
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −13.0000 −1.09480
\(142\) −2.00000 −0.167836
\(143\) 5.00000 0.418121
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 5.00000 0.410997
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) 7.00000 0.567775
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) −14.0000 −1.11378
\(159\) −1.00000 −0.0793052
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 7.00000 0.535303
\(172\) −10.0000 −0.762493
\(173\) −7.00000 −0.532200 −0.266100 0.963945i \(-0.585735\pi\)
−0.266100 + 0.963945i \(0.585735\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) −4.00000 −0.300658
\(178\) 10.0000 0.749532
\(179\) −11.0000 −0.822179 −0.411089 0.911595i \(-0.634852\pi\)
−0.411089 + 0.911595i \(0.634852\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) −10.0000 −0.731272
\(188\) 13.0000 0.948122
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 0 0
\(197\) 27.0000 1.92367 0.961835 0.273629i \(-0.0882242\pi\)
0.961835 + 0.273629i \(0.0882242\pi\)
\(198\) −5.00000 −0.355335
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 6.00000 0.423207
\(202\) 8.00000 0.562878
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) −1.00000 −0.0693375
\(209\) −35.0000 −2.42100
\(210\) 0 0
\(211\) 19.0000 1.30801 0.654007 0.756489i \(-0.273087\pi\)
0.654007 + 0.756489i \(0.273087\pi\)
\(212\) 1.00000 0.0686803
\(213\) 2.00000 0.137038
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) −5.00000 −0.335578
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) −7.00000 −0.463586
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 14.0000 0.909398
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 14.0000 0.899954
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) −7.00000 −0.445399
\(248\) −6.00000 −0.381000
\(249\) −10.0000 −0.633724
\(250\) 0 0
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) −9.00000 −0.564710
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 10.0000 0.622573
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −17.0000 −1.05026
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 5.00000 0.307729
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −6.00000 −0.366508
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −8.00000 −0.479808
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −11.0000 −0.656205 −0.328102 0.944642i \(-0.606409\pi\)
−0.328102 + 0.944642i \(0.606409\pi\)
\(282\) −13.0000 −0.774139
\(283\) −26.0000 −1.54554 −0.772770 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) −4.00000 −0.234082
\(293\) −1.00000 −0.0584206 −0.0292103 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.00000 0.290619
\(297\) 5.00000 0.290129
\(298\) −6.00000 −0.347571
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 0 0
\(302\) −22.0000 −1.26596
\(303\) −8.00000 −0.459588
\(304\) 7.00000 0.401478
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −26.0000 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) 1.00000 0.0566139
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) −14.0000 −0.787562
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 14.0000 0.778981
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 18.0000 0.995402
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) −15.0000 −0.824475 −0.412237 0.911077i \(-0.635253\pi\)
−0.412237 + 0.911077i \(0.635253\pi\)
\(332\) 10.0000 0.548821
\(333\) 5.00000 0.273998
\(334\) 19.0000 1.03963
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −12.0000 −0.652714
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 7.00000 0.378517
\(343\) 0 0
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) −7.00000 −0.376322
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 0 0
\(349\) −24.0000 −1.28469 −0.642345 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −5.00000 −0.266501
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −11.0000 −0.581368
\(359\) −28.0000 −1.47778 −0.738892 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −2.00000 −0.105118
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 37.0000 1.93138 0.965692 0.259690i \(-0.0836203\pi\)
0.965692 + 0.259690i \(0.0836203\pi\)
\(368\) −3.00000 −0.156386
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 0 0
\(372\) 6.00000 0.311086
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −10.0000 −0.517088
\(375\) 0 0
\(376\) 13.0000 0.670424
\(377\) 0 0
\(378\) 0 0
\(379\) −1.00000 −0.0513665 −0.0256833 0.999670i \(-0.508176\pi\)
−0.0256833 + 0.999670i \(0.508176\pi\)
\(380\) 0 0
\(381\) 9.00000 0.461084
\(382\) −16.0000 −0.818631
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) −10.0000 −0.508329
\(388\) −8.00000 −0.406138
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 17.0000 0.857537
\(394\) 27.0000 1.36024
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 6.00000 0.299253
\(403\) 6.00000 0.298881
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) −25.0000 −1.23920
\(408\) −2.00000 −0.0990148
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) 0 0
\(413\) 0 0
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 8.00000 0.391762
\(418\) −35.0000 −1.71191
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 19.0000 0.924906
\(423\) 13.0000 0.632082
\(424\) 1.00000 0.0485643
\(425\) 0 0
\(426\) 2.00000 0.0969003
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −5.00000 −0.241402
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) −21.0000 −1.00457
\(438\) 4.00000 0.191127
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) −5.00000 −0.237289
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 45.0000 2.11897
\(452\) −6.00000 −0.282216
\(453\) 22.0000 1.03365
\(454\) −14.0000 −0.657053
\(455\) 0 0
\(456\) −7.00000 −0.327805
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −4.00000 −0.186908
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 15.0000 0.697109 0.348555 0.937288i \(-0.386673\pi\)
0.348555 + 0.937288i \(0.386673\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.00000 −0.0925490 −0.0462745 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) 13.0000 0.599008
\(472\) 4.00000 0.184115
\(473\) 50.0000 2.29900
\(474\) 14.0000 0.643041
\(475\) 0 0
\(476\) 0 0
\(477\) 1.00000 0.0457869
\(478\) 20.0000 0.914779
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) −1.00000 −0.0455488
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 9.00000 0.405751
\(493\) 0 0
\(494\) −7.00000 −0.314945
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) −10.0000 −0.448111
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) −19.0000 −0.848857
\(502\) −3.00000 −0.133897
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 15.0000 0.666831
\(507\) 12.0000 0.532939
\(508\) −9.00000 −0.399310
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −7.00000 −0.309058
\(514\) 10.0000 0.441081
\(515\) 0 0
\(516\) 10.0000 0.440225
\(517\) −65.0000 −2.85870
\(518\) 0 0
\(519\) 7.00000 0.307266
\(520\) 0 0
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) −17.0000 −0.742648
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −12.0000 −0.522728
\(528\) 5.00000 0.217597
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 9.00000 0.389833
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) −6.00000 −0.259161
\(537\) 11.0000 0.474685
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) 0 0
\(541\) 4.00000 0.171973 0.0859867 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(542\) 8.00000 0.343629
\(543\) 2.00000 0.0858282
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 14.0000 0.598597 0.299298 0.954160i \(-0.403247\pi\)
0.299298 + 0.954160i \(0.403247\pi\)
\(548\) 4.00000 0.170872
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 3.00000 0.127688
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 39.0000 1.65248 0.826242 0.563316i \(-0.190475\pi\)
0.826242 + 0.563316i \(0.190475\pi\)
\(558\) −6.00000 −0.254000
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) 10.0000 0.422200
\(562\) −11.0000 −0.464007
\(563\) −30.0000 −1.26435 −0.632175 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) −13.0000 −0.547399
\(565\) 0 0
\(566\) −26.0000 −1.09286
\(567\) 0 0
\(568\) −2.00000 −0.0839181
\(569\) −3.00000 −0.125767 −0.0628833 0.998021i \(-0.520030\pi\)
−0.0628833 + 0.998021i \(0.520030\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 5.00000 0.209061
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −24.0000 −0.999133 −0.499567 0.866276i \(-0.666507\pi\)
−0.499567 + 0.866276i \(0.666507\pi\)
\(578\) −13.0000 −0.540729
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 0 0
\(582\) 8.00000 0.331611
\(583\) −5.00000 −0.207079
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) −1.00000 −0.0413096
\(587\) −2.00000 −0.0825488 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) 0 0
\(589\) −42.0000 −1.73058
\(590\) 0 0
\(591\) −27.0000 −1.11063
\(592\) 5.00000 0.205499
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 14.0000 0.572982
\(598\) 3.00000 0.122679
\(599\) −28.0000 −1.14405 −0.572024 0.820237i \(-0.693842\pi\)
−0.572024 + 0.820237i \(0.693842\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) −22.0000 −0.895167
\(605\) 0 0
\(606\) −8.00000 −0.324978
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 7.00000 0.283887
\(609\) 0 0
\(610\) 0 0
\(611\) −13.0000 −0.525924
\(612\) 2.00000 0.0808452
\(613\) −19.0000 −0.767403 −0.383701 0.923457i \(-0.625351\pi\)
−0.383701 + 0.923457i \(0.625351\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 15.0000 0.602901 0.301450 0.953482i \(-0.402529\pi\)
0.301450 + 0.953482i \(0.402529\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) −26.0000 −1.04251
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 35.0000 1.39777
\(628\) −13.0000 −0.518756
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) 18.0000 0.716569 0.358284 0.933613i \(-0.383362\pi\)
0.358284 + 0.933613i \(0.383362\pi\)
\(632\) −14.0000 −0.556890
\(633\) −19.0000 −0.755182
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −1.00000 −0.0396526
\(637\) 0 0
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 12.0000 0.473602
\(643\) −38.0000 −1.49857 −0.749287 0.662246i \(-0.769604\pi\)
−0.749287 + 0.662246i \(0.769604\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 14.0000 0.550823
\(647\) 1.00000 0.0393141 0.0196570 0.999807i \(-0.493743\pi\)
0.0196570 + 0.999807i \(0.493743\pi\)
\(648\) 1.00000 0.0392837
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −5.00000 −0.195665 −0.0978326 0.995203i \(-0.531191\pi\)
−0.0978326 + 0.995203i \(0.531191\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −15.0000 −0.582992
\(663\) 2.00000 0.0776736
\(664\) 10.0000 0.388075
\(665\) 0 0
\(666\) 5.00000 0.193746
\(667\) 0 0
\(668\) 19.0000 0.735132
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −36.0000 −1.38770 −0.693849 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −33.0000 −1.26829 −0.634147 0.773213i \(-0.718648\pi\)
−0.634147 + 0.773213i \(0.718648\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 30.0000 1.14876
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 7.00000 0.267652
\(685\) 0 0
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) −10.0000 −0.381246
\(689\) −1.00000 −0.0380970
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −7.00000 −0.266100
\(693\) 0 0
\(694\) 16.0000 0.607352
\(695\) 0 0
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) −24.0000 −0.908413
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 1.00000 0.0377426
\(703\) 35.0000 1.32005
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) 10.0000 0.374766
\(713\) 18.0000 0.674105
\(714\) 0 0
\(715\) 0 0
\(716\) −11.0000 −0.411089
\(717\) −20.0000 −0.746914
\(718\) −28.0000 −1.04495
\(719\) −2.00000 −0.0745874 −0.0372937 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 30.0000 1.11648
\(723\) 1.00000 0.0371904
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) −53.0000 −1.96566 −0.982831 0.184510i \(-0.940930\pi\)
−0.982831 + 0.184510i \(0.940930\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 2.00000 0.0739221
\(733\) 21.0000 0.775653 0.387826 0.921732i \(-0.373226\pi\)
0.387826 + 0.921732i \(0.373226\pi\)
\(734\) 37.0000 1.36569
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 30.0000 1.10506
\(738\) −9.00000 −0.331295
\(739\) 47.0000 1.72892 0.864461 0.502699i \(-0.167660\pi\)
0.864461 + 0.502699i \(0.167660\pi\)
\(740\) 0 0
\(741\) 7.00000 0.257151
\(742\) 0 0
\(743\) 31.0000 1.13728 0.568640 0.822587i \(-0.307470\pi\)
0.568640 + 0.822587i \(0.307470\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 10.0000 0.365881
\(748\) −10.0000 −0.365636
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 13.0000 0.474061
\(753\) 3.00000 0.109326
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −1.00000 −0.0363216
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 9.00000 0.326036
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) −4.00000 −0.144432
\(768\) −1.00000 −0.0360844
\(769\) 51.0000 1.83911 0.919554 0.392965i \(-0.128551\pi\)
0.919554 + 0.392965i \(0.128551\pi\)
\(770\) 0 0
\(771\) −10.0000 −0.360141
\(772\) 18.0000 0.647834
\(773\) 37.0000 1.33080 0.665399 0.746488i \(-0.268262\pi\)
0.665399 + 0.746488i \(0.268262\pi\)
\(774\) −10.0000 −0.359443
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −63.0000 −2.25721
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) −6.00000 −0.214560
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 17.0000 0.606370
\(787\) −38.0000 −1.35455 −0.677277 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) 27.0000 0.961835
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) −5.00000 −0.177667
\(793\) 2.00000 0.0710221
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 26.0000 0.919814
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) −27.0000 −0.953403
\(803\) 20.0000 0.705785
\(804\) 6.00000 0.211604
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) −14.0000 −0.492823
\(808\) 8.00000 0.281439
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 0 0
\(811\) 11.0000 0.386262 0.193131 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) −25.0000 −0.876250
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −70.0000 −2.44899
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −4.00000 −0.139516
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) −3.00000 −0.104257
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) −35.0000 −1.21050
\(837\) 6.00000 0.207390
\(838\) 3.00000 0.103633
\(839\) 2.00000 0.0690477 0.0345238 0.999404i \(-0.489009\pi\)
0.0345238 + 0.999404i \(0.489009\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −20.0000 −0.689246
\(843\) 11.0000 0.378860
\(844\) 19.0000 0.654007
\(845\) 0 0
\(846\) 13.0000 0.446949
\(847\) 0 0
\(848\) 1.00000 0.0343401
\(849\) 26.0000 0.892318
\(850\) 0 0
\(851\) −15.0000 −0.514193
\(852\) 2.00000 0.0685189
\(853\) 49.0000 1.67773 0.838864 0.544341i \(-0.183220\pi\)
0.838864 + 0.544341i \(0.183220\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 56.0000 1.91292 0.956462 0.291858i \(-0.0942733\pi\)
0.956462 + 0.291858i \(0.0942733\pi\)
\(858\) −5.00000 −0.170697
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) −15.0000 −0.510606 −0.255303 0.966861i \(-0.582175\pi\)
−0.255303 + 0.966861i \(0.582175\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −4.00000 −0.135926
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 70.0000 2.37459
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) −18.0000 −0.609557
\(873\) −8.00000 −0.270759
\(874\) −21.0000 −0.710336
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) 27.0000 0.911725 0.455863 0.890050i \(-0.349331\pi\)
0.455863 + 0.890050i \(0.349331\pi\)
\(878\) 0 0
\(879\) 1.00000 0.0337292
\(880\) 0 0
\(881\) −3.00000 −0.101073 −0.0505363 0.998722i \(-0.516093\pi\)
−0.0505363 + 0.998722i \(0.516093\pi\)
\(882\) 0 0
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −5.00000 −0.167789
\(889\) 0 0
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) −16.0000 −0.535720
\(893\) 91.0000 3.04520
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 9.00000 0.300334
\(899\) 0 0
\(900\) 0 0
\(901\) 2.00000 0.0666297
\(902\) 45.0000 1.49834
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) −14.0000 −0.464606
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) −7.00000 −0.231793
\(913\) −50.0000 −1.65476
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) −12.0000 −0.395199
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) 0 0
\(926\) 15.0000 0.492931
\(927\) 0 0
\(928\) 0 0
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 26.0000 0.851202
\(934\) −2.00000 −0.0654420
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 13.0000 0.423563
\(943\) 27.0000 0.879241
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 50.0000 1.62564
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 14.0000 0.454699
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 1.00000 0.0323762
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −5.00000 −0.161206
\(963\) −12.0000 −0.386695
\(964\) −1.00000 −0.0322078
\(965\) 0 0
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 14.0000 0.449977
\(969\) −14.0000 −0.449745
\(970\) 0 0
\(971\) 39.0000 1.25157 0.625785 0.779996i \(-0.284779\pi\)
0.625785 + 0.779996i \(0.284779\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −12.0000 −0.383718
\(979\) −50.0000 −1.59801
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 24.0000 0.765871
\(983\) −33.0000 −1.05254 −0.526268 0.850319i \(-0.676409\pi\)
−0.526268 + 0.850319i \(0.676409\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −7.00000 −0.222700
\(989\) 30.0000 0.953945
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) −6.00000 −0.190500
\(993\) 15.0000 0.476011
\(994\) 0 0
\(995\) 0 0
\(996\) −10.0000 −0.316862
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −28.0000 −0.886325
\(999\) −5.00000 −0.158193
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 7350.2.a.bn.1.1 1
5.2 odd 4 1470.2.g.f.589.2 2
5.3 odd 4 1470.2.g.f.589.1 2
5.4 even 2 7350.2.a.t.1.1 1
7.2 even 3 1050.2.i.f.151.1 2
7.4 even 3 1050.2.i.f.751.1 2
7.6 odd 2 7350.2.a.ch.1.1 1
35.2 odd 12 210.2.n.a.109.2 yes 4
35.3 even 12 1470.2.n.i.79.2 4
35.4 even 6 1050.2.i.o.751.1 2
35.9 even 6 1050.2.i.o.151.1 2
35.12 even 12 1470.2.n.i.949.2 4
35.13 even 4 1470.2.g.a.589.1 2
35.17 even 12 1470.2.n.i.79.1 4
35.18 odd 12 210.2.n.a.79.2 yes 4
35.23 odd 12 210.2.n.a.109.1 yes 4
35.27 even 4 1470.2.g.a.589.2 2
35.32 odd 12 210.2.n.a.79.1 4
35.33 even 12 1470.2.n.i.949.1 4
35.34 odd 2 7350.2.a.b.1.1 1
105.2 even 12 630.2.u.c.109.1 4
105.23 even 12 630.2.u.c.109.2 4
105.32 even 12 630.2.u.c.289.2 4
105.53 even 12 630.2.u.c.289.1 4
140.23 even 12 1680.2.di.a.529.1 4
140.67 even 12 1680.2.di.a.289.1 4
140.107 even 12 1680.2.di.a.529.2 4
140.123 even 12 1680.2.di.a.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.n.a.79.1 4 35.32 odd 12
210.2.n.a.79.2 yes 4 35.18 odd 12
210.2.n.a.109.1 yes 4 35.23 odd 12
210.2.n.a.109.2 yes 4 35.2 odd 12
630.2.u.c.109.1 4 105.2 even 12
630.2.u.c.109.2 4 105.23 even 12
630.2.u.c.289.1 4 105.53 even 12
630.2.u.c.289.2 4 105.32 even 12
1050.2.i.f.151.1 2 7.2 even 3
1050.2.i.f.751.1 2 7.4 even 3
1050.2.i.o.151.1 2 35.9 even 6
1050.2.i.o.751.1 2 35.4 even 6
1470.2.g.a.589.1 2 35.13 even 4
1470.2.g.a.589.2 2 35.27 even 4
1470.2.g.f.589.1 2 5.3 odd 4
1470.2.g.f.589.2 2 5.2 odd 4
1470.2.n.i.79.1 4 35.17 even 12
1470.2.n.i.79.2 4 35.3 even 12
1470.2.n.i.949.1 4 35.33 even 12
1470.2.n.i.949.2 4 35.12 even 12
1680.2.di.a.289.1 4 140.67 even 12
1680.2.di.a.289.2 4 140.123 even 12
1680.2.di.a.529.1 4 140.23 even 12
1680.2.di.a.529.2 4 140.107 even 12
7350.2.a.b.1.1 1 35.34 odd 2
7350.2.a.t.1.1 1 5.4 even 2
7350.2.a.bn.1.1 1 1.1 even 1 trivial
7350.2.a.ch.1.1 1 7.6 odd 2