Properties

Label 5070.2.b.c.1351.1
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.c.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +1.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000i q^{18} +1.00000i q^{20} +4.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} +1.00000 q^{30} -1.00000i q^{32} -6.00000i q^{34} -1.00000 q^{36} +6.00000i q^{37} +1.00000 q^{40} +2.00000i q^{41} +4.00000 q^{43} -1.00000i q^{45} -4.00000i q^{46} -1.00000 q^{48} +7.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} +1.00000i q^{54} +10.0000i q^{58} -1.00000i q^{60} +6.00000 q^{61} -1.00000 q^{64} +4.00000i q^{67} -6.00000 q^{68} -4.00000 q^{69} +16.0000i q^{71} +1.00000i q^{72} +2.00000i q^{73} +6.00000 q^{74} +1.00000 q^{75} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000i q^{83} -6.00000i q^{85} -4.00000i q^{86} +10.0000 q^{87} +6.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +1.00000i q^{96} +14.0000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} + 2q^{12} + 2q^{16} + 12q^{17} + 8q^{23} - 2q^{25} - 2q^{27} - 20q^{29} + 2q^{30} - 2q^{36} + 2q^{40} + 8q^{43} - 2q^{48} + 14q^{49} - 12q^{51} - 12q^{53} + 12q^{61} - 2q^{64} - 12q^{68} - 8q^{69} + 12q^{74} + 2q^{75} + 2q^{81} + 4q^{82} + 20q^{87} - 2q^{90} - 8q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 0 0
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) − 6.00000i − 1.02899i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 2.00000i 0.312348i 0.987730 + 0.156174i \(0.0499160\pi\)
−0.987730 + 0.156174i \(0.950084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) − 4.00000i − 0.589768i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 −0.144338
\(49\) 7.00000 1.00000
\(50\) 1.00000i 0.141421i
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000i 1.31306i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) −6.00000 −0.727607
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 16.0000i 1.89885i 0.313993 + 0.949425i \(0.398333\pi\)
−0.313993 + 0.949425i \(0.601667\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) − 6.00000i − 0.650791i
\(86\) − 4.00000i − 0.431331i
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000i 0.102062i
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.00000i 0.594089i
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000i 0.582772i
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 14.0000i − 1.34096i −0.741929 0.670478i \(-0.766089\pi\)
0.741929 0.670478i \(-0.233911\pi\)
\(110\) 0 0
\(111\) − 6.00000i − 0.569495i
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) − 4.00000i − 0.373002i
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 11.0000 1.00000
\(122\) − 6.00000i − 0.543214i
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 1.00000i 0.0860663i
\(136\) 6.00000i 0.514496i
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 4.00000i 0.340503i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 10.0000i 0.830455i
\(146\) 2.00000 0.165521
\(147\) −7.00000 −0.577350
\(148\) − 6.00000i − 0.493197i
\(149\) − 14.0000i − 1.14692i −0.819232 0.573462i \(-0.805600\pi\)
0.819232 0.573462i \(-0.194400\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 16.0000i 1.30206i 0.759051 + 0.651031i \(0.225663\pi\)
−0.759051 + 0.651031i \(0.774337\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) − 2.00000i − 0.156174i
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) − 24.0000i − 1.85718i −0.371113 0.928588i \(-0.621024\pi\)
0.371113 0.928588i \(-0.378976\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) − 10.0000i − 0.758098i
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 4.00000i 0.294884i
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 4.00000i − 0.282138i
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 2.00000 0.139686
\(206\) 12.0000i 0.836080i
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000 0.412082
\(213\) − 16.0000i − 1.09630i
\(214\) 4.00000i 0.273434i
\(215\) − 4.00000i − 0.272798i
\(216\) − 1.00000i − 0.0680414i
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) − 2.00000i − 0.135147i
\(220\) 0 0
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) − 10.0000i − 0.665190i
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) − 2.00000i − 0.132164i −0.997814 0.0660819i \(-0.978950\pi\)
0.997814 0.0660819i \(-0.0210498\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) − 10.0000i − 0.656532i
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000i 1.03495i 0.855697 + 0.517477i \(0.173129\pi\)
−0.855697 + 0.517477i \(0.826871\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 14.0000i 0.901819i 0.892570 + 0.450910i \(0.148900\pi\)
−0.892570 + 0.450910i \(0.851100\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) − 7.00000i − 0.447214i
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 0 0
\(249\) − 4.00000i − 0.253490i
\(250\) 1.00000 0.0632456
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 12.0000i − 0.752947i
\(255\) 6.00000i 0.375735i
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) − 12.0000i − 0.741362i
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) − 4.00000i − 0.244339i
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 1.00000 0.0608581
\(271\) 24.0000i 1.45790i 0.684569 + 0.728948i \(0.259990\pi\)
−0.684569 + 0.728948i \(0.740010\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) − 16.0000i − 0.949425i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) 19.0000 1.11765
\(290\) 10.0000 0.587220
\(291\) − 14.0000i − 0.820695i
\(292\) − 2.00000i − 0.117041i
\(293\) − 26.0000i − 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) 7.00000i 0.408248i
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) − 6.00000i − 0.343559i
\(306\) − 6.00000i − 0.342997i
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 14.0000i 0.790066i
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 1.00000i 0.0559017i
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 14.0000i 0.774202i
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000i 0.439720i 0.975531 + 0.219860i \(0.0705600\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) 6.00000i 0.328798i
\(334\) −24.0000 −1.31322
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 6.00000i 0.325396i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000i 0.215666i
\(345\) 4.00000i 0.215353i
\(346\) − 14.0000i − 0.752645i
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −10.0000 −0.536056
\(349\) − 34.0000i − 1.81998i −0.414632 0.909989i \(-0.636090\pi\)
0.414632 0.909989i \(-0.363910\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 26.0000i − 1.38384i −0.721974 0.691920i \(-0.756765\pi\)
0.721974 0.691920i \(-0.243235\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) − 6.00000i − 0.317999i
\(357\) 0 0
\(358\) − 20.0000i − 1.05703i
\(359\) − 24.0000i − 1.26667i −0.773877 0.633336i \(-0.781685\pi\)
0.773877 0.633336i \(-0.218315\pi\)
\(360\) 1.00000 0.0527046
\(361\) 19.0000 1.00000
\(362\) − 10.0000i − 0.525588i
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 6.00000i 0.313625i
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 4.00000 0.208514
\(369\) 2.00000i 0.104116i
\(370\) − 6.00000i − 0.311925i
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) − 1.00000i − 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 24.0000i 1.23280i 0.787434 + 0.616399i \(0.211409\pi\)
−0.787434 + 0.616399i \(0.788591\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) − 24.0000i − 1.22795i
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) − 14.0000i − 0.710742i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 7.00000i 0.353553i
\(393\) −12.0000 −0.605320
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) − 18.0000i − 0.898877i −0.893311 0.449439i \(-0.851624\pi\)
0.893311 0.449439i \(-0.148376\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) − 1.00000i − 0.0496904i
\(406\) 0 0
\(407\) 0 0
\(408\) − 6.00000i − 0.297044i
\(409\) 2.00000i 0.0988936i 0.998777 + 0.0494468i \(0.0157458\pi\)
−0.998777 + 0.0494468i \(0.984254\pi\)
\(410\) − 2.00000i − 0.0987730i
\(411\) 6.00000i 0.295958i
\(412\) 12.0000 0.591198
\(413\) 0 0
\(414\) − 4.00000i − 0.196589i
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) − 22.0000i − 1.07221i −0.844150 0.536107i \(-0.819894\pi\)
0.844150 0.536107i \(-0.180106\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) − 6.00000i − 0.291386i
\(425\) −6.00000 −0.291043
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) − 24.0000i − 1.15604i −0.816023 0.578020i \(-0.803826\pi\)
0.816023 0.578020i \(-0.196174\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) − 10.0000i − 0.479463i
\(436\) 14.0000i 0.670478i
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 6.00000i 0.284747i
\(445\) 6.00000 0.284427
\(446\) 16.0000 0.757622
\(447\) 14.0000i 0.662177i
\(448\) 0 0
\(449\) 14.0000i 0.660701i 0.943858 + 0.330350i \(0.107167\pi\)
−0.943858 + 0.330350i \(0.892833\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 0 0
\(452\) −10.0000 −0.470360
\(453\) − 16.0000i − 0.751746i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000i 0.280668i 0.990104 + 0.140334i \(0.0448177\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) −2.00000 −0.0934539
\(459\) −6.00000 −0.280056
\(460\) 4.00000i 0.186501i
\(461\) − 14.0000i − 0.652045i −0.945362 0.326023i \(-0.894291\pi\)
0.945362 0.326023i \(-0.105709\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) 18.0000i 0.833834i
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 16.0000 0.731823
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 14.0000 0.635707
\(486\) 1.00000i 0.0453609i
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) 6.00000i 0.271607i
\(489\) − 4.00000i − 0.180886i
\(490\) −7.00000 −0.316228
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) −60.0000 −2.70226
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) − 40.0000i − 1.79065i −0.445418 0.895323i \(-0.646945\pi\)
0.445418 0.895323i \(-0.353055\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) 24.0000i 1.07224i
\(502\) − 4.00000i − 0.178529i
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 6.00000i 0.266996i
\(506\) 0 0
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) − 38.0000i − 1.68432i −0.539227 0.842160i \(-0.681284\pi\)
0.539227 0.842160i \(-0.318716\pi\)
\(510\) 6.00000 0.265684
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) − 6.00000i − 0.264649i
\(515\) 12.0000i 0.528783i
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 10.0000i 0.437688i
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 28.0000i 1.22086i
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −6.00000 −0.259645
\(535\) 4.00000i 0.172935i
\(536\) −4.00000 −0.172774
\(537\) −20.0000 −0.863064
\(538\) − 14.0000i − 0.603583i
\(539\) 0 0
\(540\) − 1.00000i − 0.0430331i
\(541\) 22.0000i 0.945854i 0.881102 + 0.472927i \(0.156803\pi\)
−0.881102 + 0.472927i \(0.843197\pi\)
\(542\) 24.0000 1.03089
\(543\) −10.0000 −0.429141
\(544\) − 6.00000i − 0.257248i
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) − 4.00000i − 0.170251i
\(553\) 0 0
\(554\) 2.00000i 0.0849719i
\(555\) −6.00000 −0.254686
\(556\) −4.00000 −0.169638
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) − 10.0000i − 0.420703i
\(566\) − 12.0000i − 0.504398i
\(567\) 0 0
\(568\) −16.0000 −0.671345
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) −1.00000 −0.0416667
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 14.0000i 0.581820i
\(580\) − 10.0000i − 0.415227i
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 7.00000 0.288675
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000i 0.904959i
\(592\) 6.00000i 0.246598i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.0000i 0.573462i
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) − 16.0000i − 0.651031i
\(605\) − 11.0000i − 0.447214i
\(606\) − 6.00000i − 0.243733i
\(607\) −36.0000 −1.46119 −0.730597 0.682808i \(-0.760758\pi\)
−0.730597 + 0.682808i \(0.760758\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) − 22.0000i − 0.888572i −0.895885 0.444286i \(-0.853457\pi\)
0.895885 0.444286i \(-0.146543\pi\)
\(614\) 4.00000 0.161427
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) − 12.0000i − 0.482711i
\(619\) − 40.0000i − 1.60774i −0.594808 0.803868i \(-0.702772\pi\)
0.594808 0.803868i \(-0.297228\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) − 16.0000i − 0.641542i
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 26.0000i − 1.03917i
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 36.0000i 1.43541i
\(630\) 0 0
\(631\) 48.0000i 1.91085i 0.295234 + 0.955425i \(0.404602\pi\)
−0.295234 + 0.955425i \(0.595398\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 2.00000 0.0794301
\(635\) − 12.0000i − 0.476205i
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 16.0000i 0.632950i
\(640\) 1.00000 0.0395285
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) − 4.00000i − 0.157867i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 4.00000i 0.157500i
\(646\) 0 0
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) − 4.00000i − 0.156652i
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 14.0000 0.547443
\(655\) − 12.0000i − 0.468879i
\(656\) 2.00000i 0.0780869i
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 30.0000i 1.16686i 0.812162 + 0.583432i \(0.198291\pi\)
−0.812162 + 0.583432i \(0.801709\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −40.0000 −1.54881
\(668\) 24.0000i 0.928588i
\(669\) − 16.0000i − 0.618596i
\(670\) − 4.00000i − 0.154533i
\(671\) 0 0
\(672\) 0 0
\(673\) −42.0000 −1.61898 −0.809491 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) 18.0000i 0.693334i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 10.0000i 0.384048i
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) − 20.0000i − 0.766402i
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 4.00000 0.152277
\(691\) 32.0000i 1.21734i 0.793424 + 0.608669i \(0.208296\pi\)
−0.793424 + 0.608669i \(0.791704\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) − 12.0000i − 0.455514i
\(695\) − 4.00000i − 0.151729i
\(696\) 10.0000i 0.379049i
\(697\) 12.0000i 0.454532i
\(698\) −34.0000 −1.28692
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) 0 0
\(708\) 0 0
\(709\) − 2.00000i − 0.0751116i −0.999295 0.0375558i \(-0.988043\pi\)
0.999295 0.0375558i \(-0.0119572\pi\)
\(710\) − 16.0000i − 0.600469i
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) − 16.0000i − 0.597531i
\(718\) −24.0000 −0.895672
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 0 0
\(722\) − 19.0000i − 0.707107i
\(723\) − 14.0000i − 0.520666i
\(724\) −10.0000 −0.371647
\(725\) 10.0000 0.371391
\(726\) 11.0000i 0.408248i
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 2.00000i − 0.0740233i
\(731\) 24.0000 0.887672
\(732\) 6.00000 0.221766
\(733\) 50.0000i 1.84679i 0.383849 + 0.923396i \(0.374598\pi\)
−0.383849 + 0.923396i \(0.625402\pi\)
\(734\) − 20.0000i − 0.738213i
\(735\) 7.00000i 0.258199i
\(736\) − 4.00000i − 0.147442i
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) 40.0000i 1.47142i 0.677295 + 0.735712i \(0.263152\pi\)
−0.677295 + 0.735712i \(0.736848\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) − 48.0000i − 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) − 10.0000i − 0.366126i
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 24.0000 0.871719
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000i 0.797499i 0.917060 + 0.398750i \(0.130556\pi\)
−0.917060 + 0.398750i \(0.869444\pi\)
\(762\) 12.0000i 0.434714i
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) − 6.00000i − 0.216930i
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 34.0000i 1.22607i 0.790055 + 0.613036i \(0.210052\pi\)
−0.790055 + 0.613036i \(0.789948\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 14.0000i 0.503871i
\(773\) 26.0000i 0.935155i 0.883952 + 0.467578i \(0.154873\pi\)
−0.883952 + 0.467578i \(0.845127\pi\)
\(774\) − 4.00000i − 0.143777i
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) − 18.0000i − 0.645331i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) − 24.0000i − 0.858238i
\(783\) 10.0000 0.357371
\(784\) 7.00000 0.250000
\(785\) 14.0000i 0.499681i
\(786\) 12.0000i 0.428026i
\(787\) 36.0000i 1.28326i 0.767014 + 0.641631i \(0.221742\pi\)
−0.767014 + 0.641631i \(0.778258\pi\)
\(788\) 22.0000i 0.783718i
\(789\) 28.0000 0.996826
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) − 6.00000i − 0.212798i
\(796\) −16.0000 −0.567105
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 0.0353553i
\(801\) 6.00000i 0.212000i
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 4.00000i 0.141069i
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) − 6.00000i − 0.211079i
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 32.0000i 1.12367i 0.827249 + 0.561836i \(0.189905\pi\)
−0.827249 + 0.561836i \(0.810095\pi\)
\(812\) 0 0
\(813\) − 24.0000i − 0.841717i
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) −6.00000 −0.210042
\(817\) 0 0
\(818\) 2.00000 0.0699284
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) − 22.0000i − 0.767805i −0.923374 0.383903i \(-0.874580\pi\)
0.923374 0.383903i \(-0.125420\pi\)
\(822\) 6.00000 0.209274
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) − 12.0000i − 0.418040i
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) −4.00000 −0.139010
\(829\) −54.0000 −1.87550 −0.937749 0.347314i \(-0.887094\pi\)
−0.937749 + 0.347314i \(0.887094\pi\)
\(830\) − 4.00000i − 0.138842i
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 42.0000 1.45521
\(834\) 4.00000i 0.138509i
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) − 12.0000i − 0.414533i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −22.0000 −0.758170
\(843\) − 6.00000i − 0.206651i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −12.0000 −0.411839
\(850\) 6.00000i 0.205798i
\(851\) 24.0000i 0.822709i
\(852\) 16.0000i 0.548151i
\(853\) 38.0000i 1.30110i 0.759465 + 0.650548i \(0.225461\pi\)
−0.759465 + 0.650548i \(0.774539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) − 4.00000i − 0.136717i
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 4.00000i 0.136399i
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 14.0000i − 0.476014i
\(866\) 2.00000i 0.0679628i
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) −10.0000 −0.339032
\(871\) 0 0
\(872\) 14.0000 0.474100
\(873\) 14.0000i 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) 2.00000i 0.0675737i
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) − 24.0000i − 0.809961i
\(879\) 26.0000i 0.876958i
\(880\) 0 0
\(881\) −34.0000 −1.14549 −0.572745 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 4.00000i 0.134383i
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) − 6.00000i − 0.201120i
\(891\) 0 0
\(892\) − 16.0000i − 0.535720i
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) − 20.0000i − 0.668526i
\(896\) 0 0
\(897\) 0 0
\(898\) 14.0000 0.467186
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 10.0000i 0.332595i
\(905\) − 10.0000i − 0.332411i
\(906\) −16.0000 −0.531564
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 6.00000i 0.198354i
\(916\) 2.00000i 0.0660819i
\(917\) 0 0
\(918\) 6.00000i 0.198030i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 4.00000 0.131876
\(921\) − 4.00000i − 0.131804i
\(922\) −14.0000 −0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) − 6.00000i − 0.197279i
\(926\) 24.0000 0.788689
\(927\) −12.0000 −0.394132
\(928\) 10.0000i 0.328266i
\(929\) − 46.0000i − 1.50921i −0.656179 0.754606i \(-0.727828\pi\)
0.656179 0.754606i \(-0.272172\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −16.0000 −0.523816
\(934\) − 28.0000i − 0.916188i
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 10.0000i 0.325991i 0.986627 + 0.162995i \(0.0521156\pi\)
−0.986627 + 0.162995i \(0.947884\pi\)
\(942\) − 14.0000i − 0.456145i
\(943\) 8.00000i 0.260516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 2.00000i − 0.0648544i
\(952\) 0 0
\(953\) −10.0000 −0.323932 −0.161966 0.986796i \(-0.551783\pi\)
−0.161966 + 0.986796i \(0.551783\pi\)
\(954\) 6.00000i 0.194257i
\(955\) − 24.0000i − 0.776622i
\(956\) − 16.0000i − 0.517477i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) − 1.00000i − 0.0322749i
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) − 14.0000i − 0.450910i
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 11.0000i 0.353553i
\(969\) 0 0
\(970\) − 14.0000i − 0.449513i
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) − 2.00000i − 0.0639857i −0.999488 0.0319928i \(-0.989815\pi\)
0.999488 0.0319928i \(-0.0101854\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 7.00000i 0.223607i
\(981\) − 14.0000i − 0.446986i
\(982\) 20.0000i 0.638226i
\(983\) − 16.0000i − 0.510321i −0.966899 0.255160i \(-0.917872\pi\)
0.966899 0.255160i \(-0.0821283\pi\)
\(984\) 2.00000 0.0637577
\(985\) −22.0000 −0.700978
\(986\) 60.0000i 1.91079i
\(987\) 0 0
\(988\) 0 0
\(989\) 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
\(993\) − 8.00000i − 0.253872i
\(994\) 0 0
\(995\) − 16.0000i − 0.507234i
\(996\) 4.00000i 0.126745i
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −40.0000 −1.26618
\(999\) − 6.00000i − 0.189832i
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 5070.2.b.c.1351.1 2
13.5 odd 4 390.2.a.a.1.1 1
13.8 odd 4 5070.2.a.s.1.1 1
13.12 even 2 inner 5070.2.b.c.1351.2 2
39.5 even 4 1170.2.a.m.1.1 1
52.31 even 4 3120.2.a.q.1.1 1
65.18 even 4 1950.2.e.l.1249.2 2
65.44 odd 4 1950.2.a.y.1.1 1
65.57 even 4 1950.2.e.l.1249.1 2
156.83 odd 4 9360.2.a.bn.1.1 1
195.44 even 4 5850.2.a.m.1.1 1
195.83 odd 4 5850.2.e.p.5149.1 2
195.122 odd 4 5850.2.e.p.5149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.a.1.1 1 13.5 odd 4
1170.2.a.m.1.1 1 39.5 even 4
1950.2.a.y.1.1 1 65.44 odd 4
1950.2.e.l.1249.1 2 65.57 even 4
1950.2.e.l.1249.2 2 65.18 even 4
3120.2.a.q.1.1 1 52.31 even 4
5070.2.a.s.1.1 1 13.8 odd 4
5070.2.b.c.1351.1 2 1.1 even 1 trivial
5070.2.b.c.1351.2 2 13.12 even 2 inner
5850.2.a.m.1.1 1 195.44 even 4
5850.2.e.p.5149.1 2 195.83 odd 4
5850.2.e.p.5149.2 2 195.122 odd 4
9360.2.a.bn.1.1 1 156.83 odd 4