Properties

Label 370.2.a.b.1.1
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -2.00000 q^{26} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -3.00000 q^{36} -1.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +3.00000 q^{45} -8.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +10.0000 q^{53} +4.00000 q^{55} +6.00000 q^{58} +4.00000 q^{59} +10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} -2.00000 q^{68} +3.00000 q^{72} +10.0000 q^{73} +1.00000 q^{74} -4.00000 q^{76} -4.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} +6.00000 q^{82} +2.00000 q^{85} -4.00000 q^{86} +4.00000 q^{88} +2.00000 q^{89} -3.00000 q^{90} +8.00000 q^{94} +4.00000 q^{95} +6.00000 q^{97} +7.00000 q^{98} +12.0000 q^{99} +O(q^{100})\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 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 \(0.577979\pi\)
\(18\) 3.00000 0.707107
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) −1.00000 −0.164399
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 9.00000 1.00000
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 7.00000 0.707107
\(99\) 12.0000 1.20605
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −6.00000 −0.554700
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) −3.00000 −0.250000
\(145\) 6.00000 0.498273
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 4.00000 0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 12.0000 0.917663
\(172\) 4.00000 0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 3.00000 0.223607
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −12.0000 −0.852803
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) −14.0000 −0.931266
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 6.00000 0.392232
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 7.00000 0.447214
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 18.0000 1.11417
\(262\) 12.0000 0.741362
\(263\) −32.0000 −1.97320 −0.986602 0.163144i \(-0.947836\pi\)
−0.986602 + 0.163144i \(0.947836\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 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 \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 4.00000 0.239904
\(279\) 12.0000 0.718421
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 3.00000 0.176777
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 24.0000 1.38104
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −10.0000 −0.572598
\(306\) −6.00000 −0.342997
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 9.00000 0.500000
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 3.00000 0.164399
\(334\) 16.0000 0.875481
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 16.0000 0.866449
\(342\) −12.0000 −0.648886
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) −1.00000 −0.0519875
\(371\) 0 0
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −12.0000 −0.609994
\(388\) 6.00000 0.304604
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 7.00000 0.353553
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 4.00000 0.201262
\(396\) 12.0000 0.603023
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 6.00000 0.298511
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −16.0000 −0.782586
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −28.0000 −1.36302
\(423\) 24.0000 1.16692
\(424\) −10.0000 −0.485643
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −4.00000 −0.190693
\(441\) 21.0000 1.00000
\(442\) 4.00000 0.190261
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 3.00000 0.141421
\(451\) 24.0000 1.13012
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 18.0000 0.841085
\(459\) 0 0
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −30.0000 −1.37361
\(478\) −12.0000 −0.548867
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) −7.00000 −0.316228
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) −12.0000 −0.539360
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) 0 0
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) −18.0000 −0.787839
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 10.0000 0.434372
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −6.00000 −0.256307
\(549\) −30.0000 −1.28037
\(550\) 4.00000 0.170561
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −12.0000 −0.508001
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −2.00000 −0.0843649
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) 46.0000 1.91501 0.957503 0.288425i \(-0.0931316\pi\)
0.957503 + 0.288425i \(0.0931316\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) −10.0000 −0.413803
\(585\) 6.00000 0.248069
\(586\) −18.0000 −0.743573
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 24.0000 0.977356
\(604\) −24.0000 −0.976546
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) −16.0000 −0.647291
\(612\) 6.00000 0.242536
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 32.0000 1.29141
\(615\) 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
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) −36.0000 −1.43314 −0.716569 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −14.0000 −0.554700
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) −9.00000 −0.353553
\(649\) −16.0000 −0.628055
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −6.00000 −0.234261
\(657\) −30.0000 −1.17041
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −3.00000 −0.116248
\(667\) 0 0
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 12.0000 0.458831
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 18.0000 0.681310
\(699\) 0 0
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) −2.00000 −0.0749532
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 0 0
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 10.0000 0.370117
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 32.0000 1.17874
\(738\) −18.0000 −0.662589
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) −6.00000 −0.216930
\(766\) 24.0000 0.867155
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −22.0000 −0.791285 −0.395643 0.918405i \(-0.629478\pi\)
−0.395643 + 0.918405i \(0.629478\pi\)
\(774\) 12.0000 0.431331
\(775\) −4.00000 −0.143684
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) 48.0000 1.71102 0.855508 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) 0 0
\(792\) −12.0000 −0.426401
\(793\) 20.0000 0.710221
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) 22.0000 0.776847
\(803\) −40.0000 −1.41157
\(804\) 0 0
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 9.00000 0.316228
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −4.00000 −0.140200
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 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
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 14.0000 0.485071
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −28.0000 −0.967244
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) 28.0000 0.963800
\(845\) 9.00000 0.309609
\(846\) −24.0000 −0.825137
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 8.00000 0.273434
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 20.0000 0.681203
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) −18.0000 −0.609557
\(873\) −18.0000 −0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −21.0000 −0.707107
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 56.0000 1.88030 0.940148 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.00000 0.0670402
\(891\) −36.0000 −1.20605
\(892\) −16.0000 −0.535720
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 24.0000 0.800445
\(900\) −3.00000 −0.100000
\(901\) −20.0000 −0.666297
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 20.0000 0.663723
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) −18.0000 −0.594737
\(917\) 0 0
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −10.0000 −0.329332
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 40.0000 1.31448
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) −22.0000 −0.720634
\(933\) 0 0
\(934\) −28.0000 −0.916188
\(935\) −8.00000 −0.261628
\(936\) 6.00000 0.196116
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 30.0000 0.971286
\(955\) 12.0000 0.388311
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −28.0000 −0.904639
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) 24.0000 0.773389
\(964\) 2.00000 0.0644157
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −50.0000 −1.59964 −0.799821 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 7.00000 0.223607
\(981\) −54.0000 −1.72409
\(982\) −20.0000 −0.638226
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 12.0000 0.381385
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 20.0000 0.633089
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 370.2.a.b.1.1 1
3.2 odd 2 3330.2.a.w.1.1 1
4.3 odd 2 2960.2.a.g.1.1 1
5.2 odd 4 1850.2.b.d.149.1 2
5.3 odd 4 1850.2.b.d.149.2 2
5.4 even 2 1850.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.b.1.1 1 1.1 even 1 trivial
1850.2.a.k.1.1 1 5.4 even 2
1850.2.b.d.149.1 2 5.2 odd 4
1850.2.b.d.149.2 2 5.3 odd 4
2960.2.a.g.1.1 1 4.3 odd 2
3330.2.a.w.1.1 1 3.2 odd 2