Properties

Label 7350.2.a.c.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 1050)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +4.00000 q^{22} +2.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{29} -1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} -1.00000 q^{38} +1.00000 q^{39} +12.0000 q^{41} -8.00000 q^{43} -4.00000 q^{44} -2.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +2.00000 q^{51} -1.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{57} -4.00000 q^{58} +6.00000 q^{59} -13.0000 q^{61} +1.00000 q^{64} -4.00000 q^{66} -3.00000 q^{67} -2.00000 q^{68} -2.00000 q^{69} +16.0000 q^{71} -1.00000 q^{72} +11.0000 q^{73} +3.00000 q^{74} +1.00000 q^{76} -1.00000 q^{78} +13.0000 q^{79} +1.00000 q^{81} -12.0000 q^{82} +6.00000 q^{83} +8.00000 q^{86} -4.00000 q^{87} +4.00000 q^{88} -2.00000 q^{89} +2.00000 q^{92} -6.00000 q^{94} +1.00000 q^{96} -17.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −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 \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −1.00000 −0.162221
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −1.00000 −0.138675
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) −4.00000 −0.525226
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −2.00000 −0.242536
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −4.00000 −0.428845
\(88\) 4.00000 0.426401
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) −2.00000 −0.198030
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) −1.00000 −0.0924500
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 13.0000 1.17696
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 2.00000 0.174741 0.0873704 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 2.00000 0.170251
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) −16.0000 −1.34269
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −11.0000 −0.910366
\(147\) 0 0
\(148\) −3.00000 −0.246598
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 21.0000 1.67598 0.837991 0.545684i \(-0.183730\pi\)
0.837991 + 0.545684i \(0.183730\pi\)
\(158\) −13.0000 −1.03422
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −23.0000 −1.80150 −0.900750 0.434339i \(-0.856982\pi\)
−0.900750 + 0.434339i \(0.856982\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −6.00000 −0.450988
\(178\) 2.00000 0.149906
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 13.0000 0.960988
\(184\) −2.00000 −0.147442
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 17.0000 1.22053
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 4.00000 0.284268
\(199\) 15.0000 1.06332 0.531661 0.846957i \(-0.321568\pi\)
0.531661 + 0.846957i \(0.321568\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 7.00000 0.487713
\(207\) 2.00000 0.139010
\(208\) −1.00000 −0.0693375
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 3.00000 0.206529 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) −2.00000 −0.137361
\(213\) −16.0000 −1.09630
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 19.0000 1.28684
\(219\) −11.0000 −0.743311
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) −3.00000 −0.201347
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −13.0000 −0.844441
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 13.0000 0.837404 0.418702 0.908124i \(-0.362485\pi\)
0.418702 + 0.908124i \(0.362485\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) −1.00000 −0.0627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) −8.00000 −0.498058
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) −2.00000 −0.123560
\(263\) −32.0000 −1.97320 −0.986602 0.163144i \(-0.947836\pi\)
−0.986602 + 0.163144i \(0.947836\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) −3.00000 −0.183254
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 13.0000 0.779688
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 6.00000 0.357295
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) 11.0000 0.643726
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.00000 0.174371
\(297\) 4.00000 0.232104
\(298\) 16.0000 0.926855
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) −19.0000 −1.09333
\(303\) −12.0000 −0.689382
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 7.00000 0.398216
\(310\) 0 0
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 18.0000 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) −21.0000 −1.18510
\(315\) 0 0
\(316\) 13.0000 0.731307
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −2.00000 −0.112154
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 23.0000 1.27385
\(327\) 19.0000 1.05070
\(328\) −12.0000 −0.662589
\(329\) 0 0
\(330\) 0 0
\(331\) 21.0000 1.15426 0.577132 0.816651i \(-0.304172\pi\)
0.577132 + 0.816651i \(0.304172\pi\)
\(332\) 6.00000 0.329293
\(333\) −3.00000 −0.164399
\(334\) 10.0000 0.547176
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 12.0000 0.652714
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 22.0000 1.18102 0.590511 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) −4.00000 −0.214423
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 4.00000 0.213201
\(353\) 20.0000 1.06449 0.532246 0.846590i \(-0.321348\pi\)
0.532246 + 0.846590i \(0.321348\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 8.00000 0.422813
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 18.0000 0.946059
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) −13.0000 −0.679521
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 2.00000 0.104257
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −23.0000 −1.19089 −0.595447 0.803394i \(-0.703025\pi\)
−0.595447 + 0.803394i \(0.703025\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −1.00000 −0.0512316
\(382\) −20.0000 −1.02329
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −8.00000 −0.406663
\(388\) −17.0000 −0.863044
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −2.00000 −0.100887
\(394\) 0 0
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −15.0000 −0.751882
\(399\) 0 0
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) −3.00000 −0.149626
\(403\) 0 0
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) −2.00000 −0.0990148
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) −7.00000 −0.344865
\(413\) 0 0
\(414\) −2.00000 −0.0982946
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 13.0000 0.636613
\(418\) 4.00000 0.195646
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) −3.00000 −0.146038
\(423\) 6.00000 0.291730
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −19.0000 −0.909935
\(437\) 2.00000 0.0956730
\(438\) 11.0000 0.525600
\(439\) −23.0000 −1.09773 −0.548865 0.835911i \(-0.684940\pi\)
−0.548865 + 0.835911i \(0.684940\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 3.00000 0.142374
\(445\) 0 0
\(446\) −19.0000 −0.899676
\(447\) 16.0000 0.756774
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) −48.0000 −2.26023
\(452\) −18.0000 −0.846649
\(453\) −19.0000 −0.892698
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 7.00000 0.327089
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) 3.00000 0.139422 0.0697109 0.997567i \(-0.477792\pi\)
0.0697109 + 0.997567i \(0.477792\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) −30.0000 −1.38823 −0.694117 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −21.0000 −0.967629
\(472\) −6.00000 −0.276172
\(473\) 32.0000 1.47136
\(474\) 13.0000 0.597110
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 12.0000 0.548867
\(479\) −34.0000 −1.55350 −0.776750 0.629809i \(-0.783133\pi\)
−0.776750 + 0.629809i \(0.783133\pi\)
\(480\) 0 0
\(481\) 3.00000 0.136788
\(482\) −13.0000 −0.592134
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 13.0000 0.588482
\(489\) 23.0000 1.04010
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) −12.0000 −0.541002
\(493\) −8.00000 −0.360302
\(494\) 1.00000 0.0449921
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) −27.0000 −1.20869 −0.604343 0.796724i \(-0.706564\pi\)
−0.604343 + 0.796724i \(0.706564\pi\)
\(500\) 0 0
\(501\) 10.0000 0.446767
\(502\) −24.0000 −1.07117
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 12.0000 0.532939
\(508\) 1.00000 0.0443678
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −8.00000 −0.352865
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −4.00000 −0.175075
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 2.00000 0.0873704
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) −2.00000 −0.0865485
\(535\) 0 0
\(536\) 3.00000 0.129580
\(537\) 8.00000 0.345225
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 20.0000 0.859074
\(543\) 18.0000 0.772454
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −10.0000 −0.427179
\(549\) −13.0000 −0.554826
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 2.00000 0.0851257
\(553\) 0 0
\(554\) 1.00000 0.0424859
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 18.0000 0.759284
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) 13.0000 0.546431
\(567\) 0 0
\(568\) −16.0000 −0.671345
\(569\) −40.0000 −1.67689 −0.838444 0.544988i \(-0.816534\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 0 0
\(571\) −35.0000 −1.46470 −0.732352 0.680926i \(-0.761578\pi\)
−0.732352 + 0.680926i \(0.761578\pi\)
\(572\) 4.00000 0.167248
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 13.0000 0.540729
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) −17.0000 −0.704673
\(583\) 8.00000 0.331326
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −3.00000 −0.123299
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −16.0000 −0.655386
\(597\) −15.0000 −0.613909
\(598\) 2.00000 0.0817861
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −29.0000 −1.18293 −0.591467 0.806329i \(-0.701451\pi\)
−0.591467 + 0.806329i \(0.701451\pi\)
\(602\) 0 0
\(603\) −3.00000 −0.122169
\(604\) 19.0000 0.773099
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) −1.00000 −0.0405887 −0.0202944 0.999794i \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) −2.00000 −0.0808452
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −44.0000 −1.77137 −0.885687 0.464283i \(-0.846312\pi\)
−0.885687 + 0.464283i \(0.846312\pi\)
\(618\) −7.00000 −0.281581
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) −2.00000 −0.0802572
\(622\) 14.0000 0.561349
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −18.0000 −0.719425
\(627\) 4.00000 0.159745
\(628\) 21.0000 0.837991
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) −13.0000 −0.517112
\(633\) −3.00000 −0.119239
\(634\) 22.0000 0.873732
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 0 0
\(638\) 16.0000 0.633446
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) −6.00000 −0.236801
\(643\) −11.0000 −0.433798 −0.216899 0.976194i \(-0.569594\pi\)
−0.216899 + 0.976194i \(0.569594\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) −23.0000 −0.900750
\(653\) 16.0000 0.626128 0.313064 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) −19.0000 −0.742959
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −25.0000 −0.972387 −0.486194 0.873851i \(-0.661615\pi\)
−0.486194 + 0.873851i \(0.661615\pi\)
\(662\) −21.0000 −0.816188
\(663\) −2.00000 −0.0776736
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) 8.00000 0.309761
\(668\) −10.0000 −0.386912
\(669\) −19.0000 −0.734582
\(670\) 0 0
\(671\) 52.0000 2.00744
\(672\) 0 0
\(673\) 9.00000 0.346925 0.173462 0.984841i \(-0.444505\pi\)
0.173462 + 0.984841i \(0.444505\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) −18.0000 −0.691286
\(679\) 0 0
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) 7.00000 0.267067
\(688\) −8.00000 −0.304997
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −39.0000 −1.48363 −0.741815 0.670605i \(-0.766035\pi\)
−0.741815 + 0.670605i \(0.766035\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −22.0000 −0.835109
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) −24.0000 −0.909065
\(698\) −2.00000 −0.0757011
\(699\) −4.00000 −0.151294
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −3.00000 −0.113147
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −20.0000 −0.752710
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) −11.0000 −0.413114 −0.206557 0.978435i \(-0.566226\pi\)
−0.206557 + 0.978435i \(0.566226\pi\)
\(710\) 0 0
\(711\) 13.0000 0.487538
\(712\) 2.00000 0.0749532
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −8.00000 −0.298974
\(717\) 12.0000 0.448148
\(718\) 6.00000 0.223918
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.0000 0.669891
\(723\) −13.0000 −0.483475
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 1.00000 0.0370879 0.0185440 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 13.0000 0.480494
\(733\) 35.0000 1.29275 0.646377 0.763018i \(-0.276283\pi\)
0.646377 + 0.763018i \(0.276283\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 12.0000 0.442026
\(738\) −12.0000 −0.441726
\(739\) −1.00000 −0.0367856 −0.0183928 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 23.0000 0.842090
\(747\) 6.00000 0.219529
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) 19.0000 0.693320 0.346660 0.937991i \(-0.387316\pi\)
0.346660 + 0.937991i \(0.387316\pi\)
\(752\) 6.00000 0.218797
\(753\) −24.0000 −0.874609
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) −1.00000 −0.0363456 −0.0181728 0.999835i \(-0.505785\pi\)
−0.0181728 + 0.999835i \(0.505785\pi\)
\(758\) 5.00000 0.181608
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 1.00000 0.0362262
\(763\) 0 0
\(764\) 20.0000 0.723575
\(765\) 0 0
\(766\) −14.0000 −0.505841
\(767\) −6.00000 −0.216647
\(768\) −1.00000 −0.0360844
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) 10.0000 0.359908
\(773\) 32.0000 1.15096 0.575480 0.817816i \(-0.304815\pi\)
0.575480 + 0.817816i \(0.304815\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 17.0000 0.610264
\(777\) 0 0
\(778\) 20.0000 0.717035
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) 4.00000 0.143040
\(783\) −4.00000 −0.142948
\(784\) 0 0
\(785\) 0 0
\(786\) 2.00000 0.0713376
\(787\) −23.0000 −0.819861 −0.409931 0.912117i \(-0.634447\pi\)
−0.409931 + 0.912117i \(0.634447\pi\)
\(788\) 0 0
\(789\) 32.0000 1.13923
\(790\) 0 0
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) 13.0000 0.461644
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 15.0000 0.531661
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) −4.00000 −0.141245
\(803\) −44.0000 −1.55273
\(804\) 3.00000 0.105802
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) 14.0000 0.492214 0.246107 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) −5.00000 −0.175574 −0.0877869 0.996139i \(-0.527979\pi\)
−0.0877869 + 0.996139i \(0.527979\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) −8.00000 −0.279885
\(818\) 11.0000 0.384606
\(819\) 0 0
\(820\) 0 0
\(821\) −44.0000 −1.53561 −0.767805 0.640683i \(-0.778651\pi\)
−0.767805 + 0.640683i \(0.778651\pi\)
\(822\) −10.0000 −0.348790
\(823\) −9.00000 −0.313720 −0.156860 0.987621i \(-0.550137\pi\)
−0.156860 + 0.987621i \(0.550137\pi\)
\(824\) 7.00000 0.243857
\(825\) 0 0
\(826\) 0 0
\(827\) −40.0000 −1.39094 −0.695468 0.718557i \(-0.744803\pi\)
−0.695468 + 0.718557i \(0.744803\pi\)
\(828\) 2.00000 0.0695048
\(829\) −3.00000 −0.104194 −0.0520972 0.998642i \(-0.516591\pi\)
−0.0520972 + 0.998642i \(0.516591\pi\)
\(830\) 0 0
\(831\) 1.00000 0.0346896
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −13.0000 −0.450153
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 16.0000 0.552711
\(839\) 50.0000 1.72619 0.863096 0.505040i \(-0.168522\pi\)
0.863096 + 0.505040i \(0.168522\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −1.00000 −0.0344623
\(843\) 18.0000 0.619953
\(844\) 3.00000 0.103264
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) 13.0000 0.446159
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) −16.0000 −0.548151
\(853\) 38.0000 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 4.00000 0.136558
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −6.00000 −0.204361
\(863\) 34.0000 1.15737 0.578687 0.815550i \(-0.303565\pi\)
0.578687 + 0.815550i \(0.303565\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −52.0000 −1.76398
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 19.0000 0.643421
\(873\) −17.0000 −0.575363
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) −11.0000 −0.371656
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 23.0000 0.776212
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 46.0000 1.54453 0.772264 0.635301i \(-0.219124\pi\)
0.772264 + 0.635301i \(0.219124\pi\)
\(888\) −3.00000 −0.100673
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 19.0000 0.636167
\(893\) 6.00000 0.200782
\(894\) −16.0000 −0.535120
\(895\) 0 0
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) −2.00000 −0.0667409
\(899\) 0 0
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 48.0000 1.59823
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 19.0000 0.631233
\(907\) 47.0000 1.56061 0.780305 0.625400i \(-0.215064\pi\)
0.780305 + 0.625400i \(0.215064\pi\)
\(908\) 8.00000 0.265489
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −46.0000 −1.52405 −0.762024 0.647549i \(-0.775794\pi\)
−0.762024 + 0.647549i \(0.775794\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −24.0000 −0.794284
\(914\) 37.0000 1.22385
\(915\) 0 0
\(916\) −7.00000 −0.231287
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) −26.0000 −0.856264
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) −3.00000 −0.0985861
\(927\) −7.00000 −0.229910
\(928\) −4.00000 −0.131306
\(929\) −20.0000 −0.656179 −0.328089 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 4.00000 0.131024
\(933\) 14.0000 0.458339
\(934\) 30.0000 0.981630
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −46.0000 −1.50275 −0.751377 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 21.0000 0.684217
\(943\) 24.0000 0.781548
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) 38.0000 1.23483 0.617417 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) −13.0000 −0.422220
\(949\) −11.0000 −0.357075
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 4.00000 0.129573 0.0647864 0.997899i \(-0.479363\pi\)
0.0647864 + 0.997899i \(0.479363\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 16.0000 0.517207
\(958\) 34.0000 1.09849
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −3.00000 −0.0967239
\(963\) −6.00000 −0.193347
\(964\) 13.0000 0.418702
\(965\) 0 0
\(966\) 0 0
\(967\) 11.0000 0.353736 0.176868 0.984235i \(-0.443403\pi\)
0.176868 + 0.984235i \(0.443403\pi\)
\(968\) −5.00000 −0.160706
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) 16.0000 0.513464 0.256732 0.966483i \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −13.0000 −0.416120
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) −23.0000 −0.735459
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) −19.0000 −0.606623
\(982\) −18.0000 −0.574403
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) −1.00000 −0.0318142
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) −21.0000 −0.666415
\(994\) 0 0
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) −31.0000 −0.981780 −0.490890 0.871222i \(-0.663328\pi\)
−0.490890 + 0.871222i \(0.663328\pi\)
\(998\) 27.0000 0.854670
\(999\) 3.00000 0.0949158
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.c.1.1 1
5.4 even 2 7350.2.a.cl.1.1 1
7.2 even 3 1050.2.i.t.151.1 yes 2
7.4 even 3 1050.2.i.t.751.1 yes 2
7.6 odd 2 7350.2.a.y.1.1 1
35.2 odd 12 1050.2.o.k.949.1 4
35.4 even 6 1050.2.i.a.751.1 yes 2
35.9 even 6 1050.2.i.a.151.1 2
35.18 odd 12 1050.2.o.k.499.1 4
35.23 odd 12 1050.2.o.k.949.2 4
35.32 odd 12 1050.2.o.k.499.2 4
35.34 odd 2 7350.2.a.bp.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.i.a.151.1 2 35.9 even 6
1050.2.i.a.751.1 yes 2 35.4 even 6
1050.2.i.t.151.1 yes 2 7.2 even 3
1050.2.i.t.751.1 yes 2 7.4 even 3
1050.2.o.k.499.1 4 35.18 odd 12
1050.2.o.k.499.2 4 35.32 odd 12
1050.2.o.k.949.1 4 35.2 odd 12
1050.2.o.k.949.2 4 35.23 odd 12
7350.2.a.c.1.1 1 1.1 even 1 trivial
7350.2.a.y.1.1 1 7.6 odd 2
7350.2.a.bp.1.1 1 35.34 odd 2
7350.2.a.cl.1.1 1 5.4 even 2