Properties

Label 5070.2.b.z.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.z.1351.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.69202i q^{11} -1.00000 q^{12} +3.35690 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.939001 q^{17} -1.00000i q^{18} +4.85086i q^{19} -1.00000i q^{20} +3.35690i q^{21} +1.69202 q^{22} -4.04892 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.35690i q^{28} -8.12498 q^{29} +1.00000 q^{30} -4.08815i q^{31} -1.00000i q^{32} +1.69202i q^{33} +0.939001i q^{34} -3.35690 q^{35} -1.00000 q^{36} +11.6310i q^{37} +4.85086 q^{38} -1.00000 q^{40} -3.86294i q^{41} +3.35690 q^{42} -4.02177 q^{43} -1.69202i q^{44} +1.00000i q^{45} +4.04892i q^{46} -1.27413i q^{47} +1.00000 q^{48} -4.26875 q^{49} +1.00000i q^{50} -0.939001 q^{51} +5.74094 q^{53} -1.00000i q^{54} -1.69202 q^{55} -3.35690 q^{56} +4.85086i q^{57} +8.12498i q^{58} +0.417895i q^{59} -1.00000i q^{60} +0.198062 q^{61} -4.08815 q^{62} +3.35690i q^{63} -1.00000 q^{64} +1.69202 q^{66} -8.93900i q^{67} +0.939001 q^{68} -4.04892 q^{69} +3.35690i q^{70} +5.15883i q^{71} +1.00000i q^{72} -11.5308i q^{73} +11.6310 q^{74} -1.00000 q^{75} -4.85086i q^{76} -5.67994 q^{77} -4.94869 q^{79} +1.00000i q^{80} +1.00000 q^{81} -3.86294 q^{82} -13.1739i q^{83} -3.35690i q^{84} -0.939001i q^{85} +4.02177i q^{86} -8.12498 q^{87} -1.69202 q^{88} -9.47650i q^{89} +1.00000 q^{90} +4.04892 q^{92} -4.08815i q^{93} -1.27413 q^{94} -4.85086 q^{95} -1.00000i q^{96} -15.4547i q^{97} +4.26875i q^{98} +1.69202i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} - 6q^{4} + 6q^{9} + 6q^{10} - 6q^{12} + 12q^{14} + 6q^{16} + 14q^{17} - 6q^{23} - 6q^{25} + 6q^{27} + 6q^{30} - 12q^{35} - 6q^{36} + 2q^{38} - 6q^{40} + 12q^{42} - 18q^{43} + 6q^{48} - 10q^{49} + 14q^{51} + 6q^{53} - 12q^{56} + 10q^{61} - 32q^{62} - 6q^{64} - 14q^{68} - 6q^{69} + 40q^{74} - 6q^{75} + 14q^{77} + 34q^{79} + 6q^{81} - 34q^{82} + 6q^{90} + 6q^{92} + 14q^{94} - 2q^{95} + O(q^{100}) \)

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
\(7\) 3.35690i 1.26879i 0.773010 + 0.634394i \(0.218750\pi\)
−0.773010 + 0.634394i \(0.781250\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.69202i 0.510164i 0.966919 + 0.255082i \(0.0821024\pi\)
−0.966919 + 0.255082i \(0.917898\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 3.35690 0.897168
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −0.939001 −0.227741 −0.113871 0.993496i \(-0.536325\pi\)
−0.113871 + 0.993496i \(0.536325\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.85086i 1.11286i 0.830894 + 0.556431i \(0.187830\pi\)
−0.830894 + 0.556431i \(0.812170\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 3.35690i 0.732535i
\(22\) 1.69202 0.360740
\(23\) −4.04892 −0.844258 −0.422129 0.906536i \(-0.638717\pi\)
−0.422129 + 0.906536i \(0.638717\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 3.35690i − 0.634394i
\(29\) −8.12498 −1.50877 −0.754386 0.656432i \(-0.772065\pi\)
−0.754386 + 0.656432i \(0.772065\pi\)
\(30\) 1.00000 0.182574
\(31\) − 4.08815i − 0.734253i −0.930171 0.367126i \(-0.880342\pi\)
0.930171 0.367126i \(-0.119658\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 1.69202i 0.294543i
\(34\) 0.939001i 0.161037i
\(35\) −3.35690 −0.567419
\(36\) −1.00000 −0.166667
\(37\) 11.6310i 1.91213i 0.293157 + 0.956064i \(0.405294\pi\)
−0.293157 + 0.956064i \(0.594706\pi\)
\(38\) 4.85086 0.786913
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) − 3.86294i − 0.603289i −0.953420 0.301645i \(-0.902464\pi\)
0.953420 0.301645i \(-0.0975356\pi\)
\(42\) 3.35690 0.517980
\(43\) −4.02177 −0.613314 −0.306657 0.951820i \(-0.599210\pi\)
−0.306657 + 0.951820i \(0.599210\pi\)
\(44\) − 1.69202i − 0.255082i
\(45\) 1.00000i 0.149071i
\(46\) 4.04892i 0.596980i
\(47\) − 1.27413i − 0.185850i −0.995673 0.0929252i \(-0.970378\pi\)
0.995673 0.0929252i \(-0.0296218\pi\)
\(48\) 1.00000 0.144338
\(49\) −4.26875 −0.609821
\(50\) 1.00000i 0.141421i
\(51\) −0.939001 −0.131486
\(52\) 0 0
\(53\) 5.74094 0.788579 0.394289 0.918986i \(-0.370991\pi\)
0.394289 + 0.918986i \(0.370991\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −1.69202 −0.228152
\(56\) −3.35690 −0.448584
\(57\) 4.85086i 0.642511i
\(58\) 8.12498i 1.06686i
\(59\) 0.417895i 0.0544053i 0.999630 + 0.0272026i \(0.00865994\pi\)
−0.999630 + 0.0272026i \(0.991340\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 0.198062 0.0253593 0.0126796 0.999920i \(-0.495964\pi\)
0.0126796 + 0.999920i \(0.495964\pi\)
\(62\) −4.08815 −0.519195
\(63\) 3.35690i 0.422929i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.69202 0.208273
\(67\) − 8.93900i − 1.09207i −0.837761 0.546036i \(-0.816136\pi\)
0.837761 0.546036i \(-0.183864\pi\)
\(68\) 0.939001 0.113871
\(69\) −4.04892 −0.487432
\(70\) 3.35690i 0.401226i
\(71\) 5.15883i 0.612241i 0.951993 + 0.306120i \(0.0990310\pi\)
−0.951993 + 0.306120i \(0.900969\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 11.5308i − 1.34958i −0.738011 0.674789i \(-0.764235\pi\)
0.738011 0.674789i \(-0.235765\pi\)
\(74\) 11.6310 1.35208
\(75\) −1.00000 −0.115470
\(76\) − 4.85086i − 0.556431i
\(77\) −5.67994 −0.647289
\(78\) 0 0
\(79\) −4.94869 −0.556771 −0.278386 0.960469i \(-0.589799\pi\)
−0.278386 + 0.960469i \(0.589799\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) −3.86294 −0.426590
\(83\) − 13.1739i − 1.44602i −0.690836 0.723012i \(-0.742757\pi\)
0.690836 0.723012i \(-0.257243\pi\)
\(84\) − 3.35690i − 0.366267i
\(85\) − 0.939001i − 0.101849i
\(86\) 4.02177i 0.433679i
\(87\) −8.12498 −0.871089
\(88\) −1.69202 −0.180370
\(89\) − 9.47650i − 1.00451i −0.864720 0.502254i \(-0.832504\pi\)
0.864720 0.502254i \(-0.167496\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 4.04892 0.422129
\(93\) − 4.08815i − 0.423921i
\(94\) −1.27413 −0.131416
\(95\) −4.85086 −0.497687
\(96\) − 1.00000i − 0.102062i
\(97\) − 15.4547i − 1.56919i −0.620009 0.784595i \(-0.712871\pi\)
0.620009 0.784595i \(-0.287129\pi\)
\(98\) 4.26875i 0.431209i
\(99\) 1.69202i 0.170055i
\(100\) 1.00000 0.100000
\(101\) −5.38404 −0.535732 −0.267866 0.963456i \(-0.586318\pi\)
−0.267866 + 0.963456i \(0.586318\pi\)
\(102\) 0.939001i 0.0929750i
\(103\) −4.08575 −0.402581 −0.201291 0.979532i \(-0.564514\pi\)
−0.201291 + 0.979532i \(0.564514\pi\)
\(104\) 0 0
\(105\) −3.35690 −0.327599
\(106\) − 5.74094i − 0.557609i
\(107\) −16.1196 −1.55834 −0.779171 0.626812i \(-0.784359\pi\)
−0.779171 + 0.626812i \(0.784359\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.2325i 1.36323i 0.731713 + 0.681613i \(0.238721\pi\)
−0.731713 + 0.681613i \(0.761279\pi\)
\(110\) 1.69202i 0.161328i
\(111\) 11.6310i 1.10397i
\(112\) 3.35690i 0.317197i
\(113\) −13.3274 −1.25373 −0.626866 0.779127i \(-0.715663\pi\)
−0.626866 + 0.779127i \(0.715663\pi\)
\(114\) 4.85086 0.454324
\(115\) − 4.04892i − 0.377563i
\(116\) 8.12498 0.754386
\(117\) 0 0
\(118\) 0.417895 0.0384703
\(119\) − 3.15213i − 0.288955i
\(120\) −1.00000 −0.0912871
\(121\) 8.13706 0.739733
\(122\) − 0.198062i − 0.0179317i
\(123\) − 3.86294i − 0.348309i
\(124\) 4.08815i 0.367126i
\(125\) − 1.00000i − 0.0894427i
\(126\) 3.35690 0.299056
\(127\) 12.8509 1.14033 0.570164 0.821531i \(-0.306879\pi\)
0.570164 + 0.821531i \(0.306879\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.02177 −0.354097
\(130\) 0 0
\(131\) −0.570024 −0.0498032 −0.0249016 0.999690i \(-0.507927\pi\)
−0.0249016 + 0.999690i \(0.507927\pi\)
\(132\) − 1.69202i − 0.147272i
\(133\) −16.2838 −1.41199
\(134\) −8.93900 −0.772212
\(135\) 1.00000i 0.0860663i
\(136\) − 0.939001i − 0.0805187i
\(137\) 21.5405i 1.84033i 0.391533 + 0.920164i \(0.371945\pi\)
−0.391533 + 0.920164i \(0.628055\pi\)
\(138\) 4.04892i 0.344667i
\(139\) 7.09783 0.602030 0.301015 0.953619i \(-0.402674\pi\)
0.301015 + 0.953619i \(0.402674\pi\)
\(140\) 3.35690 0.283709
\(141\) − 1.27413i − 0.107301i
\(142\) 5.15883 0.432920
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 8.12498i − 0.674743i
\(146\) −11.5308 −0.954295
\(147\) −4.26875 −0.352081
\(148\) − 11.6310i − 0.956064i
\(149\) 1.14914i 0.0941416i 0.998892 + 0.0470708i \(0.0149886\pi\)
−0.998892 + 0.0470708i \(0.985011\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 0.0217703i 0.00177164i 1.00000 0.000885819i \(0.000281965\pi\)
−1.00000 0.000885819i \(0.999718\pi\)
\(152\) −4.85086 −0.393456
\(153\) −0.939001 −0.0759137
\(154\) 5.67994i 0.457703i
\(155\) 4.08815 0.328368
\(156\) 0 0
\(157\) −10.1588 −0.810763 −0.405382 0.914148i \(-0.632861\pi\)
−0.405382 + 0.914148i \(0.632861\pi\)
\(158\) 4.94869i 0.393697i
\(159\) 5.74094 0.455286
\(160\) 1.00000 0.0790569
\(161\) − 13.5918i − 1.07118i
\(162\) − 1.00000i − 0.0785674i
\(163\) 19.1511i 1.50003i 0.661422 + 0.750014i \(0.269953\pi\)
−0.661422 + 0.750014i \(0.730047\pi\)
\(164\) 3.86294i 0.301645i
\(165\) −1.69202 −0.131724
\(166\) −13.1739 −1.02249
\(167\) − 15.2174i − 1.17756i −0.808293 0.588780i \(-0.799608\pi\)
0.808293 0.588780i \(-0.200392\pi\)
\(168\) −3.35690 −0.258990
\(169\) 0 0
\(170\) −0.939001 −0.0720181
\(171\) 4.85086i 0.370954i
\(172\) 4.02177 0.306657
\(173\) −21.8213 −1.65904 −0.829522 0.558474i \(-0.811387\pi\)
−0.829522 + 0.558474i \(0.811387\pi\)
\(174\) 8.12498i 0.615953i
\(175\) − 3.35690i − 0.253757i
\(176\) 1.69202i 0.127541i
\(177\) 0.417895i 0.0314109i
\(178\) −9.47650 −0.710294
\(179\) 20.2513 1.51365 0.756826 0.653616i \(-0.226749\pi\)
0.756826 + 0.653616i \(0.226749\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) −14.4969 −1.07755 −0.538775 0.842450i \(-0.681113\pi\)
−0.538775 + 0.842450i \(0.681113\pi\)
\(182\) 0 0
\(183\) 0.198062 0.0146412
\(184\) − 4.04892i − 0.298490i
\(185\) −11.6310 −0.855130
\(186\) −4.08815 −0.299757
\(187\) − 1.58881i − 0.116185i
\(188\) 1.27413i 0.0929252i
\(189\) 3.35690i 0.244178i
\(190\) 4.85086i 0.351918i
\(191\) −10.1806 −0.736643 −0.368321 0.929699i \(-0.620067\pi\)
−0.368321 + 0.929699i \(0.620067\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 27.6407i 1.98962i 0.101741 + 0.994811i \(0.467559\pi\)
−0.101741 + 0.994811i \(0.532441\pi\)
\(194\) −15.4547 −1.10958
\(195\) 0 0
\(196\) 4.26875 0.304911
\(197\) 13.0261i 0.928070i 0.885817 + 0.464035i \(0.153599\pi\)
−0.885817 + 0.464035i \(0.846401\pi\)
\(198\) 1.69202 0.120247
\(199\) −6.67563 −0.473223 −0.236611 0.971604i \(-0.576037\pi\)
−0.236611 + 0.971604i \(0.576037\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 8.93900i − 0.630509i
\(202\) 5.38404i 0.378820i
\(203\) − 27.2747i − 1.91431i
\(204\) 0.939001 0.0657432
\(205\) 3.86294 0.269799
\(206\) 4.08575i 0.284668i
\(207\) −4.04892 −0.281419
\(208\) 0 0
\(209\) −8.20775 −0.567742
\(210\) 3.35690i 0.231648i
\(211\) −11.3056 −0.778309 −0.389154 0.921173i \(-0.627233\pi\)
−0.389154 + 0.921173i \(0.627233\pi\)
\(212\) −5.74094 −0.394289
\(213\) 5.15883i 0.353477i
\(214\) 16.1196i 1.10191i
\(215\) − 4.02177i − 0.274282i
\(216\) 1.00000i 0.0680414i
\(217\) 13.7235 0.931611
\(218\) 14.2325 0.963947
\(219\) − 11.5308i − 0.779179i
\(220\) 1.69202 0.114076
\(221\) 0 0
\(222\) 11.6310 0.780623
\(223\) − 18.0911i − 1.21147i −0.795666 0.605736i \(-0.792879\pi\)
0.795666 0.605736i \(-0.207121\pi\)
\(224\) 3.35690 0.224292
\(225\) −1.00000 −0.0666667
\(226\) 13.3274i 0.886523i
\(227\) 9.37196i 0.622039i 0.950404 + 0.311019i \(0.100670\pi\)
−0.950404 + 0.311019i \(0.899330\pi\)
\(228\) − 4.85086i − 0.321256i
\(229\) 21.2664i 1.40532i 0.711526 + 0.702660i \(0.248005\pi\)
−0.711526 + 0.702660i \(0.751995\pi\)
\(230\) −4.04892 −0.266978
\(231\) −5.67994 −0.373713
\(232\) − 8.12498i − 0.533431i
\(233\) −2.03146 −0.133085 −0.0665427 0.997784i \(-0.521197\pi\)
−0.0665427 + 0.997784i \(0.521197\pi\)
\(234\) 0 0
\(235\) 1.27413 0.0831149
\(236\) − 0.417895i − 0.0272026i
\(237\) −4.94869 −0.321452
\(238\) −3.15213 −0.204322
\(239\) 11.9903i 0.775589i 0.921746 + 0.387794i \(0.126763\pi\)
−0.921746 + 0.387794i \(0.873237\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 21.7627i 1.40186i 0.713230 + 0.700930i \(0.247231\pi\)
−0.713230 + 0.700930i \(0.752769\pi\)
\(242\) − 8.13706i − 0.523070i
\(243\) 1.00000 0.0641500
\(244\) −0.198062 −0.0126796
\(245\) − 4.26875i − 0.272720i
\(246\) −3.86294 −0.246292
\(247\) 0 0
\(248\) 4.08815 0.259598
\(249\) − 13.1739i − 0.834862i
\(250\) −1.00000 −0.0632456
\(251\) −13.7506 −0.867932 −0.433966 0.900929i \(-0.642886\pi\)
−0.433966 + 0.900929i \(0.642886\pi\)
\(252\) − 3.35690i − 0.211465i
\(253\) − 6.85086i − 0.430710i
\(254\) − 12.8509i − 0.806334i
\(255\) − 0.939001i − 0.0588025i
\(256\) 1.00000 0.0625000
\(257\) 24.1782 1.50820 0.754098 0.656762i \(-0.228075\pi\)
0.754098 + 0.656762i \(0.228075\pi\)
\(258\) 4.02177i 0.250384i
\(259\) −39.0441 −2.42608
\(260\) 0 0
\(261\) −8.12498 −0.502924
\(262\) 0.570024i 0.0352162i
\(263\) 0.960771 0.0592437 0.0296218 0.999561i \(-0.490570\pi\)
0.0296218 + 0.999561i \(0.490570\pi\)
\(264\) −1.69202 −0.104137
\(265\) 5.74094i 0.352663i
\(266\) 16.2838i 0.998425i
\(267\) − 9.47650i − 0.579952i
\(268\) 8.93900i 0.546036i
\(269\) 10.0761 0.614348 0.307174 0.951653i \(-0.400617\pi\)
0.307174 + 0.951653i \(0.400617\pi\)
\(270\) 1.00000 0.0608581
\(271\) − 14.9584i − 0.908657i −0.890834 0.454328i \(-0.849879\pi\)
0.890834 0.454328i \(-0.150121\pi\)
\(272\) −0.939001 −0.0569353
\(273\) 0 0
\(274\) 21.5405 1.30131
\(275\) − 1.69202i − 0.102033i
\(276\) 4.04892 0.243716
\(277\) 13.0465 0.783890 0.391945 0.919989i \(-0.371802\pi\)
0.391945 + 0.919989i \(0.371802\pi\)
\(278\) − 7.09783i − 0.425700i
\(279\) − 4.08815i − 0.244751i
\(280\) − 3.35690i − 0.200613i
\(281\) 16.2881i 0.971668i 0.874051 + 0.485834i \(0.161484\pi\)
−0.874051 + 0.485834i \(0.838516\pi\)
\(282\) −1.27413 −0.0758731
\(283\) 12.3773 0.735756 0.367878 0.929874i \(-0.380084\pi\)
0.367878 + 0.929874i \(0.380084\pi\)
\(284\) − 5.15883i − 0.306120i
\(285\) −4.85086 −0.287340
\(286\) 0 0
\(287\) 12.9675 0.765446
\(288\) − 1.00000i − 0.0589256i
\(289\) −16.1183 −0.948134
\(290\) −8.12498 −0.477115
\(291\) − 15.4547i − 0.905972i
\(292\) 11.5308i 0.674789i
\(293\) − 1.73663i − 0.101455i −0.998713 0.0507274i \(-0.983846\pi\)
0.998713 0.0507274i \(-0.0161540\pi\)
\(294\) 4.26875i 0.248959i
\(295\) −0.417895 −0.0243308
\(296\) −11.6310 −0.676039
\(297\) 1.69202i 0.0981810i
\(298\) 1.14914 0.0665682
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 13.5007i − 0.778165i
\(302\) 0.0217703 0.00125274
\(303\) −5.38404 −0.309305
\(304\) 4.85086i 0.278216i
\(305\) 0.198062i 0.0113410i
\(306\) 0.939001i 0.0536791i
\(307\) 25.4403i 1.45195i 0.687720 + 0.725976i \(0.258612\pi\)
−0.687720 + 0.725976i \(0.741388\pi\)
\(308\) 5.67994 0.323645
\(309\) −4.08575 −0.232430
\(310\) − 4.08815i − 0.232191i
\(311\) 10.0325 0.568892 0.284446 0.958692i \(-0.408190\pi\)
0.284446 + 0.958692i \(0.408190\pi\)
\(312\) 0 0
\(313\) −22.3274 −1.26202 −0.631008 0.775776i \(-0.717359\pi\)
−0.631008 + 0.775776i \(0.717359\pi\)
\(314\) 10.1588i 0.573296i
\(315\) −3.35690 −0.189140
\(316\) 4.94869 0.278386
\(317\) 5.33811i 0.299818i 0.988700 + 0.149909i \(0.0478981\pi\)
−0.988700 + 0.149909i \(0.952102\pi\)
\(318\) − 5.74094i − 0.321936i
\(319\) − 13.7476i − 0.769720i
\(320\) − 1.00000i − 0.0559017i
\(321\) −16.1196 −0.899709
\(322\) −13.5918 −0.757441
\(323\) − 4.55496i − 0.253445i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 19.1511 1.06068
\(327\) 14.2325i 0.787059i
\(328\) 3.86294 0.213295
\(329\) 4.27711 0.235805
\(330\) 1.69202i 0.0931427i
\(331\) 4.49827i 0.247247i 0.992329 + 0.123624i \(0.0394516\pi\)
−0.992329 + 0.123624i \(0.960548\pi\)
\(332\) 13.1739i 0.723012i
\(333\) 11.6310i 0.637376i
\(334\) −15.2174 −0.832661
\(335\) 8.93900 0.488390
\(336\) 3.35690i 0.183134i
\(337\) 11.5676 0.630129 0.315064 0.949070i \(-0.397974\pi\)
0.315064 + 0.949070i \(0.397974\pi\)
\(338\) 0 0
\(339\) −13.3274 −0.723843
\(340\) 0.939001i 0.0509245i
\(341\) 6.91723 0.374589
\(342\) 4.85086 0.262304
\(343\) 9.16852i 0.495054i
\(344\) − 4.02177i − 0.216839i
\(345\) − 4.04892i − 0.217986i
\(346\) 21.8213i 1.17312i
\(347\) 27.1226 1.45602 0.728008 0.685568i \(-0.240446\pi\)
0.728008 + 0.685568i \(0.240446\pi\)
\(348\) 8.12498 0.435545
\(349\) 12.8562i 0.688178i 0.938937 + 0.344089i \(0.111812\pi\)
−0.938937 + 0.344089i \(0.888188\pi\)
\(350\) −3.35690 −0.179434
\(351\) 0 0
\(352\) 1.69202 0.0901850
\(353\) − 24.6262i − 1.31072i −0.755316 0.655361i \(-0.772516\pi\)
0.755316 0.655361i \(-0.227484\pi\)
\(354\) 0.417895 0.0222109
\(355\) −5.15883 −0.273802
\(356\) 9.47650i 0.502254i
\(357\) − 3.15213i − 0.166828i
\(358\) − 20.2513i − 1.07031i
\(359\) − 23.8629i − 1.25944i −0.776823 0.629719i \(-0.783170\pi\)
0.776823 0.629719i \(-0.216830\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −4.53079 −0.238463
\(362\) 14.4969i 0.761942i
\(363\) 8.13706 0.427085
\(364\) 0 0
\(365\) 11.5308 0.603549
\(366\) − 0.198062i − 0.0103529i
\(367\) 33.1933 1.73267 0.866337 0.499459i \(-0.166468\pi\)
0.866337 + 0.499459i \(0.166468\pi\)
\(368\) −4.04892 −0.211064
\(369\) − 3.86294i − 0.201096i
\(370\) 11.6310i 0.604668i
\(371\) 19.2717i 1.00054i
\(372\) 4.08815i 0.211960i
\(373\) −10.8465 −0.561613 −0.280806 0.959764i \(-0.590602\pi\)
−0.280806 + 0.959764i \(0.590602\pi\)
\(374\) −1.58881 −0.0821554
\(375\) − 1.00000i − 0.0516398i
\(376\) 1.27413 0.0657081
\(377\) 0 0
\(378\) 3.35690 0.172660
\(379\) 6.09485i 0.313071i 0.987672 + 0.156536i \(0.0500326\pi\)
−0.987672 + 0.156536i \(0.949967\pi\)
\(380\) 4.85086 0.248844
\(381\) 12.8509 0.658369
\(382\) 10.1806i 0.520885i
\(383\) 7.95407i 0.406434i 0.979134 + 0.203217i \(0.0651397\pi\)
−0.979134 + 0.203217i \(0.934860\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 5.67994i − 0.289477i
\(386\) 27.6407 1.40688
\(387\) −4.02177 −0.204438
\(388\) 15.4547i 0.784595i
\(389\) −1.50365 −0.0762380 −0.0381190 0.999273i \(-0.512137\pi\)
−0.0381190 + 0.999273i \(0.512137\pi\)
\(390\) 0 0
\(391\) 3.80194 0.192272
\(392\) − 4.26875i − 0.215604i
\(393\) −0.570024 −0.0287539
\(394\) 13.0261 0.656245
\(395\) − 4.94869i − 0.248996i
\(396\) − 1.69202i − 0.0850273i
\(397\) 34.9627i 1.75473i 0.479826 + 0.877364i \(0.340700\pi\)
−0.479826 + 0.877364i \(0.659300\pi\)
\(398\) 6.67563i 0.334619i
\(399\) −16.2838 −0.815210
\(400\) −1.00000 −0.0500000
\(401\) − 31.1148i − 1.55380i −0.629624 0.776900i \(-0.716791\pi\)
0.629624 0.776900i \(-0.283209\pi\)
\(402\) −8.93900 −0.445837
\(403\) 0 0
\(404\) 5.38404 0.267866
\(405\) 1.00000i 0.0496904i
\(406\) −27.2747 −1.35362
\(407\) −19.6799 −0.975498
\(408\) − 0.939001i − 0.0464875i
\(409\) 32.7318i 1.61849i 0.587474 + 0.809243i \(0.300122\pi\)
−0.587474 + 0.809243i \(0.699878\pi\)
\(410\) − 3.86294i − 0.190777i
\(411\) 21.5405i 1.06251i
\(412\) 4.08575 0.201291
\(413\) −1.40283 −0.0690287
\(414\) 4.04892i 0.198993i
\(415\) 13.1739 0.646681
\(416\) 0 0
\(417\) 7.09783 0.347582
\(418\) 8.20775i 0.401454i
\(419\) 13.0694 0.638480 0.319240 0.947674i \(-0.396572\pi\)
0.319240 + 0.947674i \(0.396572\pi\)
\(420\) 3.35690 0.163800
\(421\) − 16.4789i − 0.803132i −0.915830 0.401566i \(-0.868466\pi\)
0.915830 0.401566i \(-0.131534\pi\)
\(422\) 11.3056i 0.550347i
\(423\) − 1.27413i − 0.0619502i
\(424\) 5.74094i 0.278805i
\(425\) 0.939001 0.0455482
\(426\) 5.15883 0.249946
\(427\) 0.664874i 0.0321755i
\(428\) 16.1196 0.779171
\(429\) 0 0
\(430\) −4.02177 −0.193947
\(431\) 41.0640i 1.97798i 0.147974 + 0.988991i \(0.452725\pi\)
−0.147974 + 0.988991i \(0.547275\pi\)
\(432\) 1.00000 0.0481125
\(433\) 6.04593 0.290549 0.145275 0.989391i \(-0.453593\pi\)
0.145275 + 0.989391i \(0.453593\pi\)
\(434\) − 13.7235i − 0.658748i
\(435\) − 8.12498i − 0.389563i
\(436\) − 14.2325i − 0.681613i
\(437\) − 19.6407i − 0.939543i
\(438\) −11.5308 −0.550963
\(439\) 26.2543 1.25305 0.626524 0.779402i \(-0.284477\pi\)
0.626524 + 0.779402i \(0.284477\pi\)
\(440\) − 1.69202i − 0.0806640i
\(441\) −4.26875 −0.203274
\(442\) 0 0
\(443\) −39.3846 −1.87122 −0.935610 0.353035i \(-0.885150\pi\)
−0.935610 + 0.353035i \(0.885150\pi\)
\(444\) − 11.6310i − 0.551984i
\(445\) 9.47650 0.449229
\(446\) −18.0911 −0.856640
\(447\) 1.14914i 0.0543527i
\(448\) − 3.35690i − 0.158598i
\(449\) − 9.39075i − 0.443177i −0.975140 0.221588i \(-0.928876\pi\)
0.975140 0.221588i \(-0.0711241\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 6.53617 0.307776
\(452\) 13.3274 0.626866
\(453\) 0.0217703i 0.00102286i
\(454\) 9.37196 0.439848
\(455\) 0 0
\(456\) −4.85086 −0.227162
\(457\) 2.66487i 0.124658i 0.998056 + 0.0623288i \(0.0198527\pi\)
−0.998056 + 0.0623288i \(0.980147\pi\)
\(458\) 21.2664 0.993712
\(459\) −0.939001 −0.0438288
\(460\) 4.04892i 0.188782i
\(461\) 24.5284i 1.14240i 0.820810 + 0.571201i \(0.193522\pi\)
−0.820810 + 0.571201i \(0.806478\pi\)
\(462\) 5.67994i 0.264255i
\(463\) − 4.50498i − 0.209364i −0.994506 0.104682i \(-0.966618\pi\)
0.994506 0.104682i \(-0.0333825\pi\)
\(464\) −8.12498 −0.377193
\(465\) 4.08815 0.189583
\(466\) 2.03146i 0.0941055i
\(467\) −21.3317 −0.987112 −0.493556 0.869714i \(-0.664303\pi\)
−0.493556 + 0.869714i \(0.664303\pi\)
\(468\) 0 0
\(469\) 30.0073 1.38561
\(470\) − 1.27413i − 0.0587711i
\(471\) −10.1588 −0.468094
\(472\) −0.417895 −0.0192352
\(473\) − 6.80492i − 0.312891i
\(474\) 4.94869i 0.227301i
\(475\) − 4.85086i − 0.222572i
\(476\) 3.15213i 0.144478i
\(477\) 5.74094 0.262860
\(478\) 11.9903 0.548424
\(479\) − 32.9517i − 1.50560i −0.658249 0.752800i \(-0.728703\pi\)
0.658249 0.752800i \(-0.271297\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 21.7627 0.991264
\(483\) − 13.5918i − 0.618448i
\(484\) −8.13706 −0.369867
\(485\) 15.4547 0.701763
\(486\) − 1.00000i − 0.0453609i
\(487\) 9.33034i 0.422798i 0.977400 + 0.211399i \(0.0678020\pi\)
−0.977400 + 0.211399i \(0.932198\pi\)
\(488\) 0.198062i 0.00896586i
\(489\) 19.1511i 0.866041i
\(490\) −4.26875 −0.192842
\(491\) 20.4349 0.922213 0.461107 0.887345i \(-0.347453\pi\)
0.461107 + 0.887345i \(0.347453\pi\)
\(492\) 3.86294i 0.174155i
\(493\) 7.62937 0.343609
\(494\) 0 0
\(495\) −1.69202 −0.0760507
\(496\) − 4.08815i − 0.183563i
\(497\) −17.3177 −0.776804
\(498\) −13.1739 −0.590337
\(499\) − 3.54958i − 0.158901i −0.996839 0.0794505i \(-0.974683\pi\)
0.996839 0.0794505i \(-0.0253166\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 15.2174i − 0.679865i
\(502\) 13.7506i 0.613721i
\(503\) −19.8146 −0.883490 −0.441745 0.897141i \(-0.645640\pi\)
−0.441745 + 0.897141i \(0.645640\pi\)
\(504\) −3.35690 −0.149528
\(505\) − 5.38404i − 0.239587i
\(506\) −6.85086 −0.304558
\(507\) 0 0
\(508\) −12.8509 −0.570164
\(509\) − 32.4325i − 1.43754i −0.695246 0.718772i \(-0.744704\pi\)
0.695246 0.718772i \(-0.255296\pi\)
\(510\) −0.939001 −0.0415797
\(511\) 38.7077 1.71233
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.85086i 0.214170i
\(514\) − 24.1782i − 1.06646i
\(515\) − 4.08575i − 0.180040i
\(516\) 4.02177 0.177049
\(517\) 2.15585 0.0948142
\(518\) 39.0441i 1.71550i
\(519\) −21.8213 −0.957849
\(520\) 0 0
\(521\) 7.87694 0.345095 0.172547 0.985001i \(-0.444800\pi\)
0.172547 + 0.985001i \(0.444800\pi\)
\(522\) 8.12498i 0.355621i
\(523\) −23.4276 −1.02442 −0.512208 0.858861i \(-0.671172\pi\)
−0.512208 + 0.858861i \(0.671172\pi\)
\(524\) 0.570024 0.0249016
\(525\) − 3.35690i − 0.146507i
\(526\) − 0.960771i − 0.0418916i
\(527\) 3.83877i 0.167220i
\(528\) 1.69202i 0.0736358i
\(529\) −6.60627 −0.287229
\(530\) 5.74094 0.249370
\(531\) 0.417895i 0.0181351i
\(532\) 16.2838 0.705993
\(533\) 0 0
\(534\) −9.47650 −0.410088
\(535\) − 16.1196i − 0.696911i
\(536\) 8.93900 0.386106
\(537\) 20.2513 0.873908
\(538\) − 10.0761i − 0.434410i
\(539\) − 7.22282i − 0.311109i
\(540\) − 1.00000i − 0.0430331i
\(541\) − 1.21552i − 0.0522593i −0.999659 0.0261297i \(-0.991682\pi\)
0.999659 0.0261297i \(-0.00831828\pi\)
\(542\) −14.9584 −0.642517
\(543\) −14.4969 −0.622123
\(544\) 0.939001i 0.0402593i
\(545\) −14.2325 −0.609654
\(546\) 0 0
\(547\) −15.1371 −0.647214 −0.323607 0.946192i \(-0.604896\pi\)
−0.323607 + 0.946192i \(0.604896\pi\)
\(548\) − 21.5405i − 0.920164i
\(549\) 0.198062 0.00845309
\(550\) −1.69202 −0.0721480
\(551\) − 39.4131i − 1.67905i
\(552\) − 4.04892i − 0.172333i
\(553\) − 16.6122i − 0.706424i
\(554\) − 13.0465i − 0.554294i
\(555\) −11.6310 −0.493709
\(556\) −7.09783 −0.301015
\(557\) 0.147817i 0.00626321i 0.999995 + 0.00313160i \(0.000996822\pi\)
−0.999995 + 0.00313160i \(0.999003\pi\)
\(558\) −4.08815 −0.173065
\(559\) 0 0
\(560\) −3.35690 −0.141855
\(561\) − 1.58881i − 0.0670796i
\(562\) 16.2881 0.687073
\(563\) −40.6945 −1.71507 −0.857535 0.514426i \(-0.828005\pi\)
−0.857535 + 0.514426i \(0.828005\pi\)
\(564\) 1.27413i 0.0536504i
\(565\) − 13.3274i − 0.560686i
\(566\) − 12.3773i − 0.520258i
\(567\) 3.35690i 0.140976i
\(568\) −5.15883 −0.216460
\(569\) 26.2433 1.10017 0.550087 0.835107i \(-0.314594\pi\)
0.550087 + 0.835107i \(0.314594\pi\)
\(570\) 4.85086i 0.203180i
\(571\) −2.31575 −0.0969110 −0.0484555 0.998825i \(-0.515430\pi\)
−0.0484555 + 0.998825i \(0.515430\pi\)
\(572\) 0 0
\(573\) −10.1806 −0.425301
\(574\) − 12.9675i − 0.541252i
\(575\) 4.04892 0.168852
\(576\) −1.00000 −0.0416667
\(577\) 14.5241i 0.604646i 0.953206 + 0.302323i \(0.0977621\pi\)
−0.953206 + 0.302323i \(0.902238\pi\)
\(578\) 16.1183i 0.670432i
\(579\) 27.6407i 1.14871i
\(580\) 8.12498i 0.337372i
\(581\) 44.2234 1.83470
\(582\) −15.4547 −0.640619
\(583\) 9.71379i 0.402304i
\(584\) 11.5308 0.477148
\(585\) 0 0
\(586\) −1.73663 −0.0717394
\(587\) 32.9028i 1.35804i 0.734119 + 0.679021i \(0.237596\pi\)
−0.734119 + 0.679021i \(0.762404\pi\)
\(588\) 4.26875 0.176040
\(589\) 19.8310 0.817122
\(590\) 0.417895i 0.0172045i
\(591\) 13.0261i 0.535821i
\(592\) 11.6310i 0.478032i
\(593\) − 36.0146i − 1.47894i −0.673188 0.739471i \(-0.735076\pi\)
0.673188 0.739471i \(-0.264924\pi\)
\(594\) 1.69202 0.0694245
\(595\) 3.15213 0.129225
\(596\) − 1.14914i − 0.0470708i
\(597\) −6.67563 −0.273215
\(598\) 0 0
\(599\) −29.4644 −1.20388 −0.601942 0.798540i \(-0.705606\pi\)
−0.601942 + 0.798540i \(0.705606\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −28.4004 −1.15848 −0.579239 0.815158i \(-0.696650\pi\)
−0.579239 + 0.815158i \(0.696650\pi\)
\(602\) −13.5007 −0.550246
\(603\) − 8.93900i − 0.364024i
\(604\) − 0.0217703i 0 0.000885819i
\(605\) 8.13706i 0.330819i
\(606\) 5.38404i 0.218712i
\(607\) 15.6450 0.635012 0.317506 0.948256i \(-0.397155\pi\)
0.317506 + 0.948256i \(0.397155\pi\)
\(608\) 4.85086 0.196728
\(609\) − 27.2747i − 1.10523i
\(610\) 0.198062 0.00801931
\(611\) 0 0
\(612\) 0.939001 0.0379569
\(613\) 17.6329i 0.712188i 0.934450 + 0.356094i \(0.115892\pi\)
−0.934450 + 0.356094i \(0.884108\pi\)
\(614\) 25.4403 1.02669
\(615\) 3.86294 0.155769
\(616\) − 5.67994i − 0.228851i
\(617\) − 10.8582i − 0.437133i −0.975822 0.218566i \(-0.929862\pi\)
0.975822 0.218566i \(-0.0701380\pi\)
\(618\) 4.08575i 0.164353i
\(619\) − 34.2083i − 1.37495i −0.726208 0.687475i \(-0.758719\pi\)
0.726208 0.687475i \(-0.241281\pi\)
\(620\) −4.08815 −0.164184
\(621\) −4.04892 −0.162477
\(622\) − 10.0325i − 0.402268i
\(623\) 31.8116 1.27451
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 22.3274i 0.892381i
\(627\) −8.20775 −0.327786
\(628\) 10.1588 0.405382
\(629\) − 10.9215i − 0.435470i
\(630\) 3.35690i 0.133742i
\(631\) 31.9571i 1.27219i 0.771611 + 0.636095i \(0.219451\pi\)
−0.771611 + 0.636095i \(0.780549\pi\)
\(632\) − 4.94869i − 0.196848i
\(633\) −11.3056 −0.449357
\(634\) 5.33811 0.212003
\(635\) 12.8509i 0.509971i
\(636\) −5.74094 −0.227643
\(637\) 0 0
\(638\) −13.7476 −0.544274
\(639\) 5.15883i 0.204080i
\(640\) −1.00000 −0.0395285
\(641\) 45.8297 1.81016 0.905082 0.425238i \(-0.139810\pi\)
0.905082 + 0.425238i \(0.139810\pi\)
\(642\) 16.1196i 0.636190i
\(643\) − 20.6418i − 0.814032i −0.913421 0.407016i \(-0.866569\pi\)
0.913421 0.407016i \(-0.133431\pi\)
\(644\) 13.5918i 0.535592i
\(645\) − 4.02177i − 0.158357i
\(646\) −4.55496 −0.179212
\(647\) −1.40389 −0.0551928 −0.0275964 0.999619i \(-0.508785\pi\)
−0.0275964 + 0.999619i \(0.508785\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −0.707087 −0.0277556
\(650\) 0 0
\(651\) 13.7235 0.537866
\(652\) − 19.1511i − 0.750014i
\(653\) −4.54229 −0.177753 −0.0888767 0.996043i \(-0.528328\pi\)
−0.0888767 + 0.996043i \(0.528328\pi\)
\(654\) 14.2325 0.556535
\(655\) − 0.570024i − 0.0222727i
\(656\) − 3.86294i − 0.150822i
\(657\) − 11.5308i − 0.449859i
\(658\) − 4.27711i − 0.166739i
\(659\) 17.4161 0.678435 0.339217 0.940708i \(-0.389838\pi\)
0.339217 + 0.940708i \(0.389838\pi\)
\(660\) 1.69202 0.0658618
\(661\) 43.3682i 1.68683i 0.537263 + 0.843415i \(0.319458\pi\)
−0.537263 + 0.843415i \(0.680542\pi\)
\(662\) 4.49827 0.174830
\(663\) 0 0
\(664\) 13.1739 0.511246
\(665\) − 16.2838i − 0.631459i
\(666\) 11.6310 0.450693
\(667\) 32.8974 1.27379
\(668\) 15.2174i 0.588780i
\(669\) − 18.0911i − 0.699443i
\(670\) − 8.93900i − 0.345344i
\(671\) 0.335126i 0.0129374i
\(672\) 3.35690 0.129495
\(673\) 3.49875 0.134867 0.0674334 0.997724i \(-0.478519\pi\)
0.0674334 + 0.997724i \(0.478519\pi\)
\(674\) − 11.5676i − 0.445568i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −13.3284 −0.512253 −0.256126 0.966643i \(-0.582446\pi\)
−0.256126 + 0.966643i \(0.582446\pi\)
\(678\) 13.3274i 0.511834i
\(679\) 51.8799 1.99097
\(680\) 0.939001 0.0360090
\(681\) 9.37196i 0.359134i
\(682\) − 6.91723i − 0.264874i
\(683\) − 24.1739i − 0.924989i −0.886622 0.462494i \(-0.846955\pi\)
0.886622 0.462494i \(-0.153045\pi\)
\(684\) − 4.85086i − 0.185477i
\(685\) −21.5405 −0.823020
\(686\) 9.16852 0.350056
\(687\) 21.2664i 0.811362i
\(688\) −4.02177 −0.153329
\(689\) 0 0
\(690\) −4.04892 −0.154140
\(691\) 37.8810i 1.44106i 0.693423 + 0.720530i \(0.256102\pi\)
−0.693423 + 0.720530i \(0.743898\pi\)
\(692\) 21.8213 0.829522
\(693\) −5.67994 −0.215763
\(694\) − 27.1226i − 1.02956i
\(695\) 7.09783i 0.269236i
\(696\) − 8.12498i − 0.307977i
\(697\) 3.62730i 0.137394i
\(698\) 12.8562 0.486616
\(699\) −2.03146 −0.0768368
\(700\) 3.35690i 0.126879i
\(701\) 21.5967 0.815696 0.407848 0.913050i \(-0.366279\pi\)
0.407848 + 0.913050i \(0.366279\pi\)
\(702\) 0 0
\(703\) −56.4204 −2.12794
\(704\) − 1.69202i − 0.0637705i
\(705\) 1.27413 0.0479864
\(706\) −24.6262 −0.926821
\(707\) − 18.0737i − 0.679730i
\(708\) − 0.417895i − 0.0157054i
\(709\) 41.3062i 1.55129i 0.631172 + 0.775643i \(0.282574\pi\)
−0.631172 + 0.775643i \(0.717426\pi\)
\(710\) 5.15883i 0.193608i
\(711\) −4.94869 −0.185590
\(712\) 9.47650 0.355147
\(713\) 16.5526i 0.619898i
\(714\) −3.15213 −0.117965
\(715\) 0 0
\(716\) −20.2513 −0.756826
\(717\) 11.9903i 0.447786i
\(718\) −23.8629 −0.890557
\(719\) 35.5338 1.32519 0.662593 0.748980i \(-0.269456\pi\)
0.662593 + 0.748980i \(0.269456\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 13.7154i − 0.510790i
\(722\) 4.53079i 0.168619i
\(723\) 21.7627i 0.809364i
\(724\) 14.4969 0.538775
\(725\) 8.12498 0.301754
\(726\) − 8.13706i − 0.301995i
\(727\) −19.6635 −0.729281 −0.364640 0.931148i \(-0.618808\pi\)
−0.364640 + 0.931148i \(0.618808\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 11.5308i − 0.426774i
\(731\) 3.77645 0.139677
\(732\) −0.198062 −0.00732059
\(733\) 46.8702i 1.73119i 0.500743 + 0.865596i \(0.333060\pi\)
−0.500743 + 0.865596i \(0.666940\pi\)
\(734\) − 33.1933i − 1.22519i
\(735\) − 4.26875i − 0.157455i
\(736\) 4.04892i 0.149245i
\(737\) 15.1250 0.557136
\(738\) −3.86294 −0.142197
\(739\) − 8.57779i − 0.315539i −0.987476 0.157770i \(-0.949570\pi\)
0.987476 0.157770i \(-0.0504303\pi\)
\(740\) 11.6310 0.427565
\(741\) 0 0
\(742\) 19.2717 0.707488
\(743\) 44.9506i 1.64908i 0.565805 + 0.824539i \(0.308565\pi\)
−0.565805 + 0.824539i \(0.691435\pi\)
\(744\) 4.08815 0.149879
\(745\) −1.14914 −0.0421014
\(746\) 10.8465i 0.397120i
\(747\) − 13.1739i − 0.482008i
\(748\) 1.58881i 0.0580926i
\(749\) − 54.1118i − 1.97720i
\(750\) −1.00000 −0.0365148
\(751\) −44.0374 −1.60695 −0.803474 0.595339i \(-0.797018\pi\)
−0.803474 + 0.595339i \(0.797018\pi\)
\(752\) − 1.27413i − 0.0464626i
\(753\) −13.7506 −0.501101
\(754\) 0 0
\(755\) −0.0217703 −0.000792301 0
\(756\) − 3.35690i − 0.122089i
\(757\) −13.2798 −0.482661 −0.241331 0.970443i \(-0.577584\pi\)
−0.241331 + 0.970443i \(0.577584\pi\)
\(758\) 6.09485 0.221375
\(759\) − 6.85086i − 0.248670i
\(760\) − 4.85086i − 0.175959i
\(761\) 9.11231i 0.330321i 0.986267 + 0.165160i \(0.0528142\pi\)
−0.986267 + 0.165160i \(0.947186\pi\)
\(762\) − 12.8509i − 0.465537i
\(763\) −47.7770 −1.72964
\(764\) 10.1806 0.368321
\(765\) − 0.939001i − 0.0339497i
\(766\) 7.95407 0.287392
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 31.7784i 1.14596i 0.819570 + 0.572979i \(0.194212\pi\)
−0.819570 + 0.572979i \(0.805788\pi\)
\(770\) −5.67994 −0.204691
\(771\) 24.1782 0.870757
\(772\) − 27.6407i − 0.994811i
\(773\) − 7.11960i − 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408677\pi\)
\(774\) 4.02177i 0.144560i
\(775\) 4.08815i 0.146851i
\(776\) 15.4547 0.554792
\(777\) −39.0441 −1.40070
\(778\) 1.50365i 0.0539084i
\(779\) 18.7385 0.671378
\(780\) 0 0
\(781\) −8.72886 −0.312343
\(782\) − 3.80194i − 0.135957i
\(783\) −8.12498 −0.290363
\(784\) −4.26875 −0.152455
\(785\) − 10.1588i − 0.362584i
\(786\) 0.570024i 0.0203321i
\(787\) − 44.2121i − 1.57599i −0.615682 0.787995i \(-0.711119\pi\)
0.615682 0.787995i \(-0.288881\pi\)
\(788\) − 13.0261i − 0.464035i
\(789\) 0.960771 0.0342044
\(790\) −4.94869 −0.176066
\(791\) − 44.7385i − 1.59072i
\(792\) −1.69202 −0.0601234
\(793\) 0 0
\(794\) 34.9627 1.24078
\(795\) 5.74094i 0.203610i
\(796\) 6.67563 0.236611
\(797\) −19.2241 −0.680954 −0.340477 0.940253i \(-0.610589\pi\)
−0.340477 + 0.940253i \(0.610589\pi\)
\(798\) 16.2838i 0.576441i
\(799\) 1.19641i 0.0423258i
\(800\) 1.00000i 0.0353553i
\(801\) − 9.47650i − 0.334836i
\(802\) −31.1148 −1.09870
\(803\) 19.5104 0.688505
\(804\) 8.93900i 0.315254i
\(805\) 13.5918 0.479048
\(806\) 0 0
\(807\) 10.0761 0.354694
\(808\) − 5.38404i − 0.189410i
\(809\) 0.0988996 0.00347713 0.00173856 0.999998i \(-0.499447\pi\)
0.00173856 + 0.999998i \(0.499447\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 6.70304i − 0.235376i −0.993051 0.117688i \(-0.962452\pi\)
0.993051 0.117688i \(-0.0375482\pi\)
\(812\) 27.2747i 0.957155i
\(813\) − 14.9584i − 0.524613i
\(814\) 19.6799i 0.689782i
\(815\) −19.1511 −0.670833
\(816\) −0.939001 −0.0328716
\(817\) − 19.5090i − 0.682534i
\(818\) 32.7318 1.14444
\(819\) 0 0
\(820\) −3.86294 −0.134900
\(821\) 23.7450i 0.828706i 0.910116 + 0.414353i \(0.135992\pi\)
−0.910116 + 0.414353i \(0.864008\pi\)
\(822\) 21.5405 0.751311
\(823\) −18.0049 −0.627611 −0.313806 0.949487i \(-0.601604\pi\)
−0.313806 + 0.949487i \(0.601604\pi\)
\(824\) − 4.08575i − 0.142334i
\(825\) − 1.69202i − 0.0589086i
\(826\) 1.40283i 0.0488107i
\(827\) 40.4566i 1.40682i 0.710787 + 0.703408i \(0.248339\pi\)
−0.710787 + 0.703408i \(0.751661\pi\)
\(828\) 4.04892 0.140710
\(829\) −8.56704 −0.297546 −0.148773 0.988871i \(-0.547532\pi\)
−0.148773 + 0.988871i \(0.547532\pi\)
\(830\) − 13.1739i − 0.457273i
\(831\) 13.0465 0.452579
\(832\) 0 0
\(833\) 4.00836 0.138881
\(834\) − 7.09783i − 0.245778i
\(835\) 15.2174 0.526621
\(836\) 8.20775 0.283871
\(837\) − 4.08815i − 0.141307i
\(838\) − 13.0694i − 0.451474i
\(839\) − 6.75600i − 0.233243i −0.993176 0.116622i \(-0.962794\pi\)
0.993176 0.116622i \(-0.0372065\pi\)
\(840\) − 3.35690i − 0.115824i
\(841\) 37.0153 1.27639
\(842\) −16.4789 −0.567900
\(843\) 16.2881i 0.560993i
\(844\) 11.3056 0.389154
\(845\) 0 0
\(846\) −1.27413 −0.0438054
\(847\) 27.3153i 0.938564i
\(848\) 5.74094 0.197145
\(849\) 12.3773 0.424789
\(850\) − 0.939001i − 0.0322075i
\(851\) − 47.0930i − 1.61433i
\(852\) − 5.15883i − 0.176739i
\(853\) − 37.1269i − 1.27120i −0.772018 0.635600i \(-0.780753\pi\)
0.772018 0.635600i \(-0.219247\pi\)
\(854\) 0.664874 0.0227515
\(855\) −4.85086 −0.165896
\(856\) − 16.1196i − 0.550957i
\(857\) 24.3937 0.833274 0.416637 0.909073i \(-0.363209\pi\)
0.416637 + 0.909073i \(0.363209\pi\)
\(858\) 0 0
\(859\) 43.9366 1.49910 0.749549 0.661949i \(-0.230270\pi\)
0.749549 + 0.661949i \(0.230270\pi\)
\(860\) 4.02177i 0.137141i
\(861\) 12.9675 0.441930
\(862\) 41.0640 1.39864
\(863\) − 16.3846i − 0.557739i −0.960329 0.278870i \(-0.910040\pi\)
0.960329 0.278870i \(-0.0899598\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 21.8213i − 0.741947i
\(866\) − 6.04593i − 0.205449i
\(867\) −16.1183 −0.547405
\(868\) −13.7235 −0.465805
\(869\) − 8.37329i − 0.284044i
\(870\) −8.12498 −0.275463
\(871\) 0 0
\(872\) −14.2325 −0.481973
\(873\) − 15.4547i − 0.523063i
\(874\) −19.6407 −0.664357
\(875\) 3.35690 0.113484
\(876\) 11.5308i 0.389589i
\(877\) − 28.1347i − 0.950040i −0.879975 0.475020i \(-0.842441\pi\)
0.879975 0.475020i \(-0.157559\pi\)
\(878\) − 26.2543i − 0.886039i
\(879\) − 1.73663i − 0.0585750i
\(880\) −1.69202 −0.0570380
\(881\) −39.1454 −1.31884 −0.659421 0.751773i \(-0.729199\pi\)
−0.659421 + 0.751773i \(0.729199\pi\)
\(882\) 4.26875i 0.143736i
\(883\) −46.2833 −1.55756 −0.778779 0.627298i \(-0.784161\pi\)
−0.778779 + 0.627298i \(0.784161\pi\)
\(884\) 0 0
\(885\) −0.417895 −0.0140474
\(886\) 39.3846i 1.32315i
\(887\) −3.08097 −0.103449 −0.0517244 0.998661i \(-0.516472\pi\)
−0.0517244 + 0.998661i \(0.516472\pi\)
\(888\) −11.6310 −0.390312
\(889\) 43.1390i 1.44684i
\(890\) − 9.47650i − 0.317653i
\(891\) 1.69202i 0.0566849i
\(892\) 18.0911i 0.605736i
\(893\) 6.18060 0.206826
\(894\) 1.14914 0.0384332
\(895\) 20.2513i 0.676926i
\(896\) −3.35690 −0.112146
\(897\) 0 0
\(898\) −9.39075 −0.313373
\(899\) 33.2161i 1.10782i
\(900\) 1.00000 0.0333333
\(901\) −5.39075 −0.179592
\(902\) − 6.53617i − 0.217631i
\(903\) − 13.5007i − 0.449274i
\(904\) − 13.3274i − 0.443261i
\(905\) − 14.4969i − 0.481895i
\(906\) 0.0217703 0.000723269 0
\(907\) 2.61224 0.0867379 0.0433689 0.999059i \(-0.486191\pi\)
0.0433689 + 0.999059i \(0.486191\pi\)
\(908\) − 9.37196i − 0.311019i
\(909\) −5.38404 −0.178577
\(910\) 0 0
\(911\) 24.9302 0.825973 0.412987 0.910737i \(-0.364486\pi\)
0.412987 + 0.910737i \(0.364486\pi\)
\(912\) 4.85086i 0.160628i
\(913\) 22.2905 0.737709
\(914\) 2.66487 0.0881462
\(915\) 0.198062i 0.00654774i
\(916\) − 21.2664i − 0.702660i
\(917\) − 1.91351i − 0.0631897i
\(918\) 0.939001i 0.0309917i
\(919\) −54.4462 −1.79602 −0.898008 0.439980i \(-0.854986\pi\)
−0.898008 + 0.439980i \(0.854986\pi\)
\(920\) 4.04892 0.133489
\(921\) 25.4403i 0.838285i
\(922\) 24.5284 0.807800
\(923\) 0 0
\(924\) 5.67994 0.186856
\(925\) − 11.6310i − 0.382426i
\(926\) −4.50498 −0.148043
\(927\) −4.08575 −0.134194
\(928\) 8.12498i 0.266716i
\(929\) 40.8267i 1.33948i 0.742596 + 0.669740i \(0.233595\pi\)
−0.742596 + 0.669740i \(0.766405\pi\)
\(930\) − 4.08815i − 0.134056i
\(931\) − 20.7071i − 0.678647i
\(932\) 2.03146 0.0665427
\(933\) 10.0325 0.328450
\(934\) 21.3317i 0.697993i
\(935\) 1.58881 0.0519596
\(936\) 0 0
\(937\) −46.9571 −1.53402 −0.767010 0.641635i \(-0.778256\pi\)
−0.767010 + 0.641635i \(0.778256\pi\)
\(938\) − 30.0073i − 0.979773i
\(939\) −22.3274 −0.728626
\(940\) −1.27413 −0.0415574
\(941\) − 35.8525i − 1.16876i −0.811481 0.584379i \(-0.801338\pi\)
0.811481 0.584379i \(-0.198662\pi\)
\(942\) 10.1588i 0.330993i
\(943\) 15.6407i 0.509332i
\(944\) 0.417895i 0.0136013i
\(945\) −3.35690 −0.109200
\(946\) −6.80492 −0.221247
\(947\) − 8.73019i − 0.283693i −0.989889 0.141846i \(-0.954696\pi\)
0.989889 0.141846i \(-0.0453039\pi\)
\(948\) 4.94869 0.160726
\(949\) 0 0
\(950\) −4.85086 −0.157383
\(951\) 5.33811i 0.173100i
\(952\) 3.15213 0.102161
\(953\) −10.2107 −0.330758 −0.165379 0.986230i \(-0.552885\pi\)
−0.165379 + 0.986230i \(0.552885\pi\)
\(954\) − 5.74094i − 0.185870i
\(955\) − 10.1806i − 0.329437i
\(956\) − 11.9903i − 0.387794i
\(957\) − 13.7476i − 0.444398i
\(958\) −32.9517 −1.06462
\(959\) −72.3092 −2.33498
\(960\) − 1.00000i − 0.0322749i
\(961\) 14.2871 0.460873
\(962\) 0 0
\(963\) −16.1196 −0.519447
\(964\) − 21.7627i − 0.700930i
\(965\) −27.6407 −0.889786
\(966\) −13.5918 −0.437309
\(967\) 34.6907i 1.11558i 0.829983 + 0.557789i \(0.188350\pi\)
−0.829983 + 0.557789i \(0.811650\pi\)
\(968\) 8.13706i 0.261535i
\(969\) − 4.55496i − 0.146326i
\(970\) − 15.4547i − 0.496221i
\(971\) 34.8039 1.11691 0.558454 0.829535i \(-0.311395\pi\)
0.558454 + 0.829535i \(0.311395\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 23.8267i 0.763849i
\(974\) 9.33034 0.298963
\(975\) 0 0
\(976\) 0.198062 0.00633982
\(977\) − 23.3086i − 0.745707i −0.927890 0.372854i \(-0.878379\pi\)
0.927890 0.372854i \(-0.121621\pi\)
\(978\) 19.1511 0.612383
\(979\) 16.0344 0.512463
\(980\) 4.26875i 0.136360i
\(981\) 14.2325i 0.454409i
\(982\) − 20.4349i − 0.652103i
\(983\) 41.3639i 1.31930i 0.751571 + 0.659652i \(0.229296\pi\)
−0.751571 + 0.659652i \(0.770704\pi\)
\(984\) 3.86294 0.123146
\(985\) −13.0261 −0.415045
\(986\) − 7.62937i − 0.242969i
\(987\) 4.27711 0.136142
\(988\) 0 0
\(989\) 16.2838 0.517795
\(990\) 1.69202i 0.0537760i
\(991\) 49.4209 1.56991 0.784953 0.619555i \(-0.212687\pi\)
0.784953 + 0.619555i \(0.212687\pi\)
\(992\) −4.08815 −0.129799
\(993\) 4.49827i 0.142748i
\(994\) 17.3177i 0.549283i
\(995\) − 6.67563i − 0.211632i
\(996\) 13.1739i 0.417431i
\(997\) 44.1879 1.39944 0.699722 0.714415i \(-0.253307\pi\)
0.699722 + 0.714415i \(0.253307\pi\)
\(998\) −3.54958 −0.112360
\(999\) 11.6310i 0.367989i
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.z.1351.2 6
13.5 odd 4 5070.2.a.bp.1.2 3
13.8 odd 4 5070.2.a.bw.1.2 yes 3
13.12 even 2 inner 5070.2.b.z.1351.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bp.1.2 3 13.5 odd 4
5070.2.a.bw.1.2 yes 3 13.8 odd 4
5070.2.b.z.1351.2 6 1.1 even 1 trivial
5070.2.b.z.1351.5 6 13.12 even 2 inner