Properties

Label 5070.2.b.t.1351.5
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 1351.5
Root \(1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.t.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.35690i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.35690i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.29590i q^{11} +1.00000 q^{12} -1.35690 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.04892 q^{17} +1.00000i q^{18} +3.24698i q^{19} -1.00000i q^{20} -1.35690i q^{21} -5.29590 q^{22} -1.33513 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.35690i q^{28} -5.13706 q^{29} +1.00000 q^{30} +2.80194i q^{31} +1.00000i q^{32} -5.29590i q^{33} +2.04892i q^{34} -1.35690 q^{35} -1.00000 q^{36} +2.24698i q^{37} -3.24698 q^{38} +1.00000 q^{40} +10.6310i q^{41} +1.35690 q^{42} -4.41789 q^{43} -5.29590i q^{44} +1.00000i q^{45} -1.33513i q^{46} -4.82371i q^{47} -1.00000 q^{48} +5.15883 q^{49} -1.00000i q^{50} -2.04892 q^{51} +14.2349 q^{53} -1.00000i q^{54} -5.29590 q^{55} +1.35690 q^{56} -3.24698i q^{57} -5.13706i q^{58} -13.8998i q^{59} +1.00000i q^{60} -3.62565 q^{61} -2.80194 q^{62} +1.35690i q^{63} -1.00000 q^{64} +5.29590 q^{66} +9.03684i q^{67} -2.04892 q^{68} +1.33513 q^{69} -1.35690i q^{70} +11.7506i q^{71} -1.00000i q^{72} +12.1032i q^{73} -2.24698 q^{74} +1.00000 q^{75} -3.24698i q^{76} -7.18598 q^{77} +0.259061 q^{79} +1.00000i q^{80} +1.00000 q^{81} -10.6310 q^{82} +3.02177i q^{83} +1.35690i q^{84} +2.04892i q^{85} -4.41789i q^{86} +5.13706 q^{87} +5.29590 q^{88} +5.47650i q^{89} -1.00000 q^{90} +1.33513 q^{92} -2.80194i q^{93} +4.82371 q^{94} -3.24698 q^{95} -1.00000i q^{96} -5.85086i q^{97} +5.15883i q^{98} +5.29590i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} + 6q^{12} + 6q^{16} - 6q^{17} - 4q^{22} - 6q^{23} - 6q^{25} - 6q^{27} - 20q^{29} + 6q^{30} - 6q^{36} - 10q^{38} + 6q^{40} - 38q^{43} - 6q^{48} + 14q^{49} + 6q^{51} + 38q^{53} - 4q^{55} + 2q^{61} - 8q^{62} - 6q^{64} + 4q^{66} + 6q^{68} + 6q^{69} - 4q^{74} + 6q^{75} - 14q^{77} + 30q^{79} + 6q^{81} - 34q^{82} + 20q^{87} + 4q^{88} - 6q^{90} + 6q^{92} + 14q^{94} - 10q^{95} + O(q^{100}) \)

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
\(7\) 1.35690i 0.512858i 0.966563 + 0.256429i \(0.0825461\pi\)
−0.966563 + 0.256429i \(0.917454\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 5.29590i 1.59677i 0.602145 + 0.798387i \(0.294313\pi\)
−0.602145 + 0.798387i \(0.705687\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.35690 −0.362646
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 2.04892 0.496935 0.248468 0.968640i \(-0.420073\pi\)
0.248468 + 0.968640i \(0.420073\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 3.24698i 0.744908i 0.928051 + 0.372454i \(0.121484\pi\)
−0.928051 + 0.372454i \(0.878516\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) − 1.35690i − 0.296099i
\(22\) −5.29590 −1.12909
\(23\) −1.33513 −0.278393 −0.139196 0.990265i \(-0.544452\pi\)
−0.139196 + 0.990265i \(0.544452\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 1.35690i − 0.256429i
\(29\) −5.13706 −0.953929 −0.476964 0.878923i \(-0.658263\pi\)
−0.476964 + 0.878923i \(0.658263\pi\)
\(30\) 1.00000 0.182574
\(31\) 2.80194i 0.503243i 0.967826 + 0.251621i \(0.0809638\pi\)
−0.967826 + 0.251621i \(0.919036\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 5.29590i − 0.921897i
\(34\) 2.04892i 0.351386i
\(35\) −1.35690 −0.229357
\(36\) −1.00000 −0.166667
\(37\) 2.24698i 0.369401i 0.982795 + 0.184701i \(0.0591315\pi\)
−0.982795 + 0.184701i \(0.940868\pi\)
\(38\) −3.24698 −0.526730
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 10.6310i 1.66029i 0.557550 + 0.830143i \(0.311742\pi\)
−0.557550 + 0.830143i \(0.688258\pi\)
\(42\) 1.35690 0.209374
\(43\) −4.41789 −0.673723 −0.336861 0.941554i \(-0.609365\pi\)
−0.336861 + 0.941554i \(0.609365\pi\)
\(44\) − 5.29590i − 0.798387i
\(45\) 1.00000i 0.149071i
\(46\) − 1.33513i − 0.196854i
\(47\) − 4.82371i − 0.703610i −0.936073 0.351805i \(-0.885568\pi\)
0.936073 0.351805i \(-0.114432\pi\)
\(48\) −1.00000 −0.144338
\(49\) 5.15883 0.736976
\(50\) − 1.00000i − 0.141421i
\(51\) −2.04892 −0.286906
\(52\) 0 0
\(53\) 14.2349 1.95531 0.977657 0.210207i \(-0.0674139\pi\)
0.977657 + 0.210207i \(0.0674139\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −5.29590 −0.714099
\(56\) 1.35690 0.181323
\(57\) − 3.24698i − 0.430073i
\(58\) − 5.13706i − 0.674529i
\(59\) − 13.8998i − 1.80960i −0.425841 0.904798i \(-0.640022\pi\)
0.425841 0.904798i \(-0.359978\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −3.62565 −0.464216 −0.232108 0.972690i \(-0.574562\pi\)
−0.232108 + 0.972690i \(0.574562\pi\)
\(62\) −2.80194 −0.355846
\(63\) 1.35690i 0.170953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 5.29590 0.651880
\(67\) 9.03684i 1.10403i 0.833836 + 0.552013i \(0.186140\pi\)
−0.833836 + 0.552013i \(0.813860\pi\)
\(68\) −2.04892 −0.248468
\(69\) 1.33513 0.160730
\(70\) − 1.35690i − 0.162180i
\(71\) 11.7506i 1.39454i 0.716807 + 0.697271i \(0.245603\pi\)
−0.716807 + 0.697271i \(0.754397\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 12.1032i 1.41657i 0.705925 + 0.708287i \(0.250532\pi\)
−0.705925 + 0.708287i \(0.749468\pi\)
\(74\) −2.24698 −0.261206
\(75\) 1.00000 0.115470
\(76\) − 3.24698i − 0.372454i
\(77\) −7.18598 −0.818919
\(78\) 0 0
\(79\) 0.259061 0.0291467 0.0145733 0.999894i \(-0.495361\pi\)
0.0145733 + 0.999894i \(0.495361\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) −10.6310 −1.17400
\(83\) 3.02177i 0.331682i 0.986152 + 0.165841i \(0.0530339\pi\)
−0.986152 + 0.165841i \(0.946966\pi\)
\(84\) 1.35690i 0.148049i
\(85\) 2.04892i 0.222236i
\(86\) − 4.41789i − 0.476394i
\(87\) 5.13706 0.550751
\(88\) 5.29590 0.564545
\(89\) 5.47650i 0.580508i 0.956950 + 0.290254i \(0.0937398\pi\)
−0.956950 + 0.290254i \(0.906260\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 1.33513 0.139196
\(93\) − 2.80194i − 0.290547i
\(94\) 4.82371 0.497527
\(95\) −3.24698 −0.333133
\(96\) − 1.00000i − 0.102062i
\(97\) − 5.85086i − 0.594064i −0.954867 0.297032i \(-0.904003\pi\)
0.954867 0.297032i \(-0.0959969\pi\)
\(98\) 5.15883i 0.521121i
\(99\) 5.29590i 0.532258i
\(100\) 1.00000 0.100000
\(101\) 8.39612 0.835446 0.417723 0.908575i \(-0.362828\pi\)
0.417723 + 0.908575i \(0.362828\pi\)
\(102\) − 2.04892i − 0.202873i
\(103\) −13.6353 −1.34353 −0.671765 0.740765i \(-0.734463\pi\)
−0.671765 + 0.740765i \(0.734463\pi\)
\(104\) 0 0
\(105\) 1.35690 0.132419
\(106\) 14.2349i 1.38262i
\(107\) 3.92394 0.379341 0.189671 0.981848i \(-0.439258\pi\)
0.189671 + 0.981848i \(0.439258\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 0.554958i − 0.0531553i −0.999647 0.0265777i \(-0.991539\pi\)
0.999647 0.0265777i \(-0.00846093\pi\)
\(110\) − 5.29590i − 0.504944i
\(111\) − 2.24698i − 0.213274i
\(112\) 1.35690i 0.128215i
\(113\) 4.39373 0.413328 0.206664 0.978412i \(-0.433739\pi\)
0.206664 + 0.978412i \(0.433739\pi\)
\(114\) 3.24698 0.304108
\(115\) − 1.33513i − 0.124501i
\(116\) 5.13706 0.476964
\(117\) 0 0
\(118\) 13.8998 1.27958
\(119\) 2.78017i 0.254858i
\(120\) −1.00000 −0.0912871
\(121\) −17.0465 −1.54968
\(122\) − 3.62565i − 0.328251i
\(123\) − 10.6310i − 0.958567i
\(124\) − 2.80194i − 0.251621i
\(125\) − 1.00000i − 0.0894427i
\(126\) −1.35690 −0.120882
\(127\) 2.23490 0.198315 0.0991576 0.995072i \(-0.468385\pi\)
0.0991576 + 0.995072i \(0.468385\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.41789 0.388974
\(130\) 0 0
\(131\) −8.51573 −0.744023 −0.372011 0.928228i \(-0.621332\pi\)
−0.372011 + 0.928228i \(0.621332\pi\)
\(132\) 5.29590i 0.460949i
\(133\) −4.40581 −0.382032
\(134\) −9.03684 −0.780664
\(135\) − 1.00000i − 0.0860663i
\(136\) − 2.04892i − 0.175693i
\(137\) − 13.1685i − 1.12506i −0.826776 0.562531i \(-0.809828\pi\)
0.826776 0.562531i \(-0.190172\pi\)
\(138\) 1.33513i 0.113653i
\(139\) 7.17629 0.608685 0.304343 0.952563i \(-0.401563\pi\)
0.304343 + 0.952563i \(0.401563\pi\)
\(140\) 1.35690 0.114679
\(141\) 4.82371i 0.406229i
\(142\) −11.7506 −0.986091
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 5.13706i − 0.426610i
\(146\) −12.1032 −1.00167
\(147\) −5.15883 −0.425493
\(148\) − 2.24698i − 0.184701i
\(149\) − 9.66248i − 0.791581i −0.918341 0.395791i \(-0.870471\pi\)
0.918341 0.395791i \(-0.129529\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 7.35152i − 0.598258i −0.954213 0.299129i \(-0.903304\pi\)
0.954213 0.299129i \(-0.0966961\pi\)
\(152\) 3.24698 0.263365
\(153\) 2.04892 0.165645
\(154\) − 7.18598i − 0.579063i
\(155\) −2.80194 −0.225057
\(156\) 0 0
\(157\) 2.85325 0.227714 0.113857 0.993497i \(-0.463679\pi\)
0.113857 + 0.993497i \(0.463679\pi\)
\(158\) 0.259061i 0.0206098i
\(159\) −14.2349 −1.12890
\(160\) −1.00000 −0.0790569
\(161\) − 1.81163i − 0.142776i
\(162\) 1.00000i 0.0785674i
\(163\) − 15.1075i − 1.18331i −0.806190 0.591656i \(-0.798474\pi\)
0.806190 0.591656i \(-0.201526\pi\)
\(164\) − 10.6310i − 0.830143i
\(165\) 5.29590 0.412285
\(166\) −3.02177 −0.234535
\(167\) − 16.0881i − 1.24494i −0.782644 0.622469i \(-0.786130\pi\)
0.782644 0.622469i \(-0.213870\pi\)
\(168\) −1.35690 −0.104687
\(169\) 0 0
\(170\) −2.04892 −0.157145
\(171\) 3.24698i 0.248303i
\(172\) 4.41789 0.336861
\(173\) 15.4795 1.17688 0.588442 0.808540i \(-0.299742\pi\)
0.588442 + 0.808540i \(0.299742\pi\)
\(174\) 5.13706i 0.389440i
\(175\) − 1.35690i − 0.102572i
\(176\) 5.29590i 0.399193i
\(177\) 13.8998i 1.04477i
\(178\) −5.47650 −0.410481
\(179\) −19.0737 −1.42563 −0.712817 0.701351i \(-0.752581\pi\)
−0.712817 + 0.701351i \(0.752581\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) −2.16852 −0.161185 −0.0805925 0.996747i \(-0.525681\pi\)
−0.0805925 + 0.996747i \(0.525681\pi\)
\(182\) 0 0
\(183\) 3.62565 0.268015
\(184\) 1.33513i 0.0984268i
\(185\) −2.24698 −0.165201
\(186\) 2.80194 0.205448
\(187\) 10.8509i 0.793493i
\(188\) 4.82371i 0.351805i
\(189\) − 1.35690i − 0.0986997i
\(190\) − 3.24698i − 0.235561i
\(191\) −11.7409 −0.849545 −0.424772 0.905300i \(-0.639646\pi\)
−0.424772 + 0.905300i \(0.639646\pi\)
\(192\) 1.00000 0.0721688
\(193\) 24.6722i 1.77594i 0.459900 + 0.887971i \(0.347885\pi\)
−0.459900 + 0.887971i \(0.652115\pi\)
\(194\) 5.85086 0.420067
\(195\) 0 0
\(196\) −5.15883 −0.368488
\(197\) 26.2010i 1.86675i 0.358906 + 0.933374i \(0.383150\pi\)
−0.358906 + 0.933374i \(0.616850\pi\)
\(198\) −5.29590 −0.376363
\(199\) −19.6722 −1.39452 −0.697262 0.716817i \(-0.745598\pi\)
−0.697262 + 0.716817i \(0.745598\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) − 9.03684i − 0.637409i
\(202\) 8.39612i 0.590749i
\(203\) − 6.97046i − 0.489230i
\(204\) 2.04892 0.143453
\(205\) −10.6310 −0.742503
\(206\) − 13.6353i − 0.950019i
\(207\) −1.33513 −0.0927976
\(208\) 0 0
\(209\) −17.1957 −1.18945
\(210\) 1.35690i 0.0936347i
\(211\) −16.2935 −1.12169 −0.560846 0.827920i \(-0.689524\pi\)
−0.560846 + 0.827920i \(0.689524\pi\)
\(212\) −14.2349 −0.977657
\(213\) − 11.7506i − 0.805140i
\(214\) 3.92394i 0.268235i
\(215\) − 4.41789i − 0.301298i
\(216\) 1.00000i 0.0680414i
\(217\) −3.80194 −0.258092
\(218\) 0.554958 0.0375865
\(219\) − 12.1032i − 0.817859i
\(220\) 5.29590 0.357049
\(221\) 0 0
\(222\) 2.24698 0.150807
\(223\) − 2.28083i − 0.152736i −0.997080 0.0763679i \(-0.975668\pi\)
0.997080 0.0763679i \(-0.0243323\pi\)
\(224\) −1.35690 −0.0906614
\(225\) −1.00000 −0.0666667
\(226\) 4.39373i 0.292267i
\(227\) − 3.37196i − 0.223805i −0.993719 0.111902i \(-0.964306\pi\)
0.993719 0.111902i \(-0.0356944\pi\)
\(228\) 3.24698i 0.215036i
\(229\) − 18.3763i − 1.21434i −0.794573 0.607169i \(-0.792305\pi\)
0.794573 0.607169i \(-0.207695\pi\)
\(230\) 1.33513 0.0880356
\(231\) 7.18598 0.472803
\(232\) 5.13706i 0.337265i
\(233\) −10.0858 −0.660740 −0.330370 0.943852i \(-0.607173\pi\)
−0.330370 + 0.943852i \(0.607173\pi\)
\(234\) 0 0
\(235\) 4.82371 0.314664
\(236\) 13.8998i 0.904798i
\(237\) −0.259061 −0.0168278
\(238\) −2.78017 −0.180211
\(239\) − 20.0834i − 1.29908i −0.760325 0.649542i \(-0.774961\pi\)
0.760325 0.649542i \(-0.225039\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 2.99462i − 0.192901i −0.995338 0.0964503i \(-0.969251\pi\)
0.995338 0.0964503i \(-0.0307489\pi\)
\(242\) − 17.0465i − 1.09579i
\(243\) −1.00000 −0.0641500
\(244\) 3.62565 0.232108
\(245\) 5.15883i 0.329586i
\(246\) 10.6310 0.677809
\(247\) 0 0
\(248\) 2.80194 0.177923
\(249\) − 3.02177i − 0.191497i
\(250\) 1.00000 0.0632456
\(251\) 7.20237 0.454610 0.227305 0.973824i \(-0.427009\pi\)
0.227305 + 0.973824i \(0.427009\pi\)
\(252\) − 1.35690i − 0.0854764i
\(253\) − 7.07069i − 0.444530i
\(254\) 2.23490i 0.140230i
\(255\) − 2.04892i − 0.128308i
\(256\) 1.00000 0.0625000
\(257\) 18.8291 1.17453 0.587263 0.809396i \(-0.300205\pi\)
0.587263 + 0.809396i \(0.300205\pi\)
\(258\) 4.41789i 0.275046i
\(259\) −3.04892 −0.189451
\(260\) 0 0
\(261\) −5.13706 −0.317976
\(262\) − 8.51573i − 0.526104i
\(263\) −17.9245 −1.10527 −0.552637 0.833422i \(-0.686378\pi\)
−0.552637 + 0.833422i \(0.686378\pi\)
\(264\) −5.29590 −0.325940
\(265\) 14.2349i 0.874443i
\(266\) − 4.40581i − 0.270138i
\(267\) − 5.47650i − 0.335156i
\(268\) − 9.03684i − 0.552013i
\(269\) −22.4045 −1.36603 −0.683013 0.730406i \(-0.739331\pi\)
−0.683013 + 0.730406i \(0.739331\pi\)
\(270\) 1.00000 0.0608581
\(271\) 23.2325i 1.41127i 0.708573 + 0.705637i \(0.249339\pi\)
−0.708573 + 0.705637i \(0.750661\pi\)
\(272\) 2.04892 0.124234
\(273\) 0 0
\(274\) 13.1685 0.795540
\(275\) − 5.29590i − 0.319355i
\(276\) −1.33513 −0.0803651
\(277\) 9.78448 0.587892 0.293946 0.955822i \(-0.405031\pi\)
0.293946 + 0.955822i \(0.405031\pi\)
\(278\) 7.17629i 0.430405i
\(279\) 2.80194i 0.167748i
\(280\) 1.35690i 0.0810900i
\(281\) − 21.6286i − 1.29026i −0.764075 0.645128i \(-0.776804\pi\)
0.764075 0.645128i \(-0.223196\pi\)
\(282\) −4.82371 −0.287248
\(283\) −3.38942 −0.201480 −0.100740 0.994913i \(-0.532121\pi\)
−0.100740 + 0.994913i \(0.532121\pi\)
\(284\) − 11.7506i − 0.697271i
\(285\) 3.24698 0.192334
\(286\) 0 0
\(287\) −14.4252 −0.851492
\(288\) 1.00000i 0.0589256i
\(289\) −12.8019 −0.753055
\(290\) 5.13706 0.301659
\(291\) 5.85086i 0.342983i
\(292\) − 12.1032i − 0.708287i
\(293\) − 34.1280i − 1.99378i −0.0788243 0.996889i \(-0.525117\pi\)
0.0788243 0.996889i \(-0.474883\pi\)
\(294\) − 5.15883i − 0.300869i
\(295\) 13.8998 0.809276
\(296\) 2.24698 0.130603
\(297\) − 5.29590i − 0.307299i
\(298\) 9.66248 0.559733
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) − 5.99462i − 0.345524i
\(302\) 7.35152 0.423032
\(303\) −8.39612 −0.482345
\(304\) 3.24698i 0.186227i
\(305\) − 3.62565i − 0.207604i
\(306\) 2.04892i 0.117129i
\(307\) 15.6601i 0.893768i 0.894592 + 0.446884i \(0.147466\pi\)
−0.894592 + 0.446884i \(0.852534\pi\)
\(308\) 7.18598 0.409459
\(309\) 13.6353 0.775687
\(310\) − 2.80194i − 0.159139i
\(311\) −25.7972 −1.46282 −0.731411 0.681937i \(-0.761138\pi\)
−0.731411 + 0.681937i \(0.761138\pi\)
\(312\) 0 0
\(313\) −15.6243 −0.883139 −0.441569 0.897227i \(-0.645578\pi\)
−0.441569 + 0.897227i \(0.645578\pi\)
\(314\) 2.85325i 0.161018i
\(315\) −1.35690 −0.0764524
\(316\) −0.259061 −0.0145733
\(317\) − 9.25965i − 0.520074i −0.965599 0.260037i \(-0.916265\pi\)
0.965599 0.260037i \(-0.0837347\pi\)
\(318\) − 14.2349i − 0.798253i
\(319\) − 27.2054i − 1.52321i
\(320\) − 1.00000i − 0.0559017i
\(321\) −3.92394 −0.219013
\(322\) 1.81163 0.100958
\(323\) 6.65279i 0.370171i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 15.1075 0.836728
\(327\) 0.554958i 0.0306893i
\(328\) 10.6310 0.587000
\(329\) 6.54527 0.360852
\(330\) 5.29590i 0.291530i
\(331\) 21.8582i 1.20143i 0.799462 + 0.600716i \(0.205118\pi\)
−0.799462 + 0.600716i \(0.794882\pi\)
\(332\) − 3.02177i − 0.165841i
\(333\) 2.24698i 0.123134i
\(334\) 16.0881 0.880304
\(335\) −9.03684 −0.493735
\(336\) − 1.35690i − 0.0740247i
\(337\) −14.5013 −0.789934 −0.394967 0.918695i \(-0.629244\pi\)
−0.394967 + 0.918695i \(0.629244\pi\)
\(338\) 0 0
\(339\) −4.39373 −0.238635
\(340\) − 2.04892i − 0.111118i
\(341\) −14.8388 −0.803565
\(342\) −3.24698 −0.175577
\(343\) 16.4983i 0.890823i
\(344\) 4.41789i 0.238197i
\(345\) 1.33513i 0.0718807i
\(346\) 15.4795i 0.832182i
\(347\) 7.84117 0.420936 0.210468 0.977601i \(-0.432501\pi\)
0.210468 + 0.977601i \(0.432501\pi\)
\(348\) −5.13706 −0.275375
\(349\) − 20.4252i − 1.09334i −0.837350 0.546668i \(-0.815896\pi\)
0.837350 0.546668i \(-0.184104\pi\)
\(350\) 1.35690 0.0725291
\(351\) 0 0
\(352\) −5.29590 −0.282272
\(353\) 13.2905i 0.707383i 0.935362 + 0.353692i \(0.115074\pi\)
−0.935362 + 0.353692i \(0.884926\pi\)
\(354\) −13.8998 −0.738765
\(355\) −11.7506 −0.623659
\(356\) − 5.47650i − 0.290254i
\(357\) − 2.78017i − 0.147142i
\(358\) − 19.0737i − 1.00807i
\(359\) 12.8267i 0.676967i 0.940972 + 0.338483i \(0.109914\pi\)
−0.940972 + 0.338483i \(0.890086\pi\)
\(360\) 1.00000 0.0527046
\(361\) 8.45712 0.445112
\(362\) − 2.16852i − 0.113975i
\(363\) 17.0465 0.894710
\(364\) 0 0
\(365\) −12.1032 −0.633511
\(366\) 3.62565i 0.189516i
\(367\) −17.9554 −0.937264 −0.468632 0.883393i \(-0.655253\pi\)
−0.468632 + 0.883393i \(0.655253\pi\)
\(368\) −1.33513 −0.0695982
\(369\) 10.6310i 0.553429i
\(370\) − 2.24698i − 0.116815i
\(371\) 19.3153i 1.00280i
\(372\) 2.80194i 0.145274i
\(373\) 27.8732 1.44322 0.721610 0.692300i \(-0.243402\pi\)
0.721610 + 0.692300i \(0.243402\pi\)
\(374\) −10.8509 −0.561084
\(375\) 1.00000i 0.0516398i
\(376\) −4.82371 −0.248764
\(377\) 0 0
\(378\) 1.35690 0.0697912
\(379\) − 13.8702i − 0.712466i −0.934397 0.356233i \(-0.884061\pi\)
0.934397 0.356233i \(-0.115939\pi\)
\(380\) 3.24698 0.166567
\(381\) −2.23490 −0.114497
\(382\) − 11.7409i − 0.600719i
\(383\) 26.8984i 1.37445i 0.726446 + 0.687223i \(0.241171\pi\)
−0.726446 + 0.687223i \(0.758829\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 7.18598i − 0.366231i
\(386\) −24.6722 −1.25578
\(387\) −4.41789 −0.224574
\(388\) 5.85086i 0.297032i
\(389\) 0.187309 0.00949693 0.00474846 0.999989i \(-0.498489\pi\)
0.00474846 + 0.999989i \(0.498489\pi\)
\(390\) 0 0
\(391\) −2.73556 −0.138343
\(392\) − 5.15883i − 0.260560i
\(393\) 8.51573 0.429562
\(394\) −26.2010 −1.31999
\(395\) 0.259061i 0.0130348i
\(396\) − 5.29590i − 0.266129i
\(397\) − 8.58211i − 0.430724i −0.976534 0.215362i \(-0.930907\pi\)
0.976534 0.215362i \(-0.0690931\pi\)
\(398\) − 19.6722i − 0.986077i
\(399\) 4.40581 0.220567
\(400\) −1.00000 −0.0500000
\(401\) − 23.9748i − 1.19724i −0.801032 0.598621i \(-0.795715\pi\)
0.801032 0.598621i \(-0.204285\pi\)
\(402\) 9.03684 0.450716
\(403\) 0 0
\(404\) −8.39612 −0.417723
\(405\) 1.00000i 0.0496904i
\(406\) 6.97046 0.345938
\(407\) −11.8998 −0.589850
\(408\) 2.04892i 0.101437i
\(409\) − 8.37926i − 0.414328i −0.978306 0.207164i \(-0.933577\pi\)
0.978306 0.207164i \(-0.0664233\pi\)
\(410\) − 10.6310i − 0.525029i
\(411\) 13.1685i 0.649555i
\(412\) 13.6353 0.671765
\(413\) 18.8605 0.928067
\(414\) − 1.33513i − 0.0656178i
\(415\) −3.02177 −0.148333
\(416\) 0 0
\(417\) −7.17629 −0.351425
\(418\) − 17.1957i − 0.841068i
\(419\) 37.0887 1.81190 0.905952 0.423381i \(-0.139157\pi\)
0.905952 + 0.423381i \(0.139157\pi\)
\(420\) −1.35690 −0.0662097
\(421\) 4.90515i 0.239062i 0.992830 + 0.119531i \(0.0381391\pi\)
−0.992830 + 0.119531i \(0.961861\pi\)
\(422\) − 16.2935i − 0.793155i
\(423\) − 4.82371i − 0.234537i
\(424\) − 14.2349i − 0.691308i
\(425\) −2.04892 −0.0993871
\(426\) 11.7506 0.569320
\(427\) − 4.91962i − 0.238077i
\(428\) −3.92394 −0.189671
\(429\) 0 0
\(430\) 4.41789 0.213050
\(431\) − 6.06531i − 0.292156i −0.989273 0.146078i \(-0.953335\pi\)
0.989273 0.146078i \(-0.0466650\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 14.8877 0.715457 0.357728 0.933826i \(-0.383551\pi\)
0.357728 + 0.933826i \(0.383551\pi\)
\(434\) − 3.80194i − 0.182499i
\(435\) 5.13706i 0.246303i
\(436\) 0.554958i 0.0265777i
\(437\) − 4.33513i − 0.207377i
\(438\) 12.1032 0.578314
\(439\) 21.1336 1.00865 0.504326 0.863513i \(-0.331741\pi\)
0.504326 + 0.863513i \(0.331741\pi\)
\(440\) 5.29590i 0.252472i
\(441\) 5.15883 0.245659
\(442\) 0 0
\(443\) 28.0683 1.33356 0.666782 0.745252i \(-0.267671\pi\)
0.666782 + 0.745252i \(0.267671\pi\)
\(444\) 2.24698i 0.106637i
\(445\) −5.47650 −0.259611
\(446\) 2.28083 0.108000
\(447\) 9.66248i 0.457020i
\(448\) − 1.35690i − 0.0641073i
\(449\) − 19.2446i − 0.908208i −0.890949 0.454104i \(-0.849959\pi\)
0.890949 0.454104i \(-0.150041\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) −56.3008 −2.65110
\(452\) −4.39373 −0.206664
\(453\) 7.35152i 0.345404i
\(454\) 3.37196 0.158254
\(455\) 0 0
\(456\) −3.24698 −0.152054
\(457\) − 37.8485i − 1.77048i −0.465138 0.885238i \(-0.653995\pi\)
0.465138 0.885238i \(-0.346005\pi\)
\(458\) 18.3763 0.858667
\(459\) −2.04892 −0.0956353
\(460\) 1.33513i 0.0622506i
\(461\) − 11.2228i − 0.522699i −0.965244 0.261349i \(-0.915833\pi\)
0.965244 0.261349i \(-0.0841675\pi\)
\(462\) 7.18598i 0.334322i
\(463\) − 25.0726i − 1.16522i −0.812751 0.582611i \(-0.802031\pi\)
0.812751 0.582611i \(-0.197969\pi\)
\(464\) −5.13706 −0.238482
\(465\) 2.80194 0.129937
\(466\) − 10.0858i − 0.467213i
\(467\) −36.9657 −1.71057 −0.855284 0.518160i \(-0.826617\pi\)
−0.855284 + 0.518160i \(0.826617\pi\)
\(468\) 0 0
\(469\) −12.2620 −0.566209
\(470\) 4.82371i 0.222501i
\(471\) −2.85325 −0.131471
\(472\) −13.8998 −0.639789
\(473\) − 23.3967i − 1.07578i
\(474\) − 0.259061i − 0.0118991i
\(475\) − 3.24698i − 0.148982i
\(476\) − 2.78017i − 0.127429i
\(477\) 14.2349 0.651771
\(478\) 20.0834 0.918592
\(479\) − 31.4590i − 1.43740i −0.695320 0.718700i \(-0.744737\pi\)
0.695320 0.718700i \(-0.255263\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 2.99462 0.136401
\(483\) 1.81163i 0.0824319i
\(484\) 17.0465 0.774842
\(485\) 5.85086 0.265674
\(486\) − 1.00000i − 0.0453609i
\(487\) 18.8713i 0.855140i 0.903982 + 0.427570i \(0.140630\pi\)
−0.903982 + 0.427570i \(0.859370\pi\)
\(488\) 3.62565i 0.164125i
\(489\) 15.1075i 0.683186i
\(490\) −5.15883 −0.233052
\(491\) 16.3961 0.739947 0.369973 0.929042i \(-0.379367\pi\)
0.369973 + 0.929042i \(0.379367\pi\)
\(492\) 10.6310i 0.479284i
\(493\) −10.5254 −0.474041
\(494\) 0 0
\(495\) −5.29590 −0.238033
\(496\) 2.80194i 0.125811i
\(497\) −15.9444 −0.715203
\(498\) 3.02177 0.135409
\(499\) 40.2452i 1.80162i 0.434212 + 0.900811i \(0.357027\pi\)
−0.434212 + 0.900811i \(0.642973\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 16.0881i 0.718765i
\(502\) 7.20237i 0.321458i
\(503\) 18.4004 0.820435 0.410217 0.911988i \(-0.365453\pi\)
0.410217 + 0.911988i \(0.365453\pi\)
\(504\) 1.35690 0.0604409
\(505\) 8.39612i 0.373623i
\(506\) 7.07069 0.314330
\(507\) 0 0
\(508\) −2.23490 −0.0991576
\(509\) − 35.1377i − 1.55745i −0.627366 0.778725i \(-0.715867\pi\)
0.627366 0.778725i \(-0.284133\pi\)
\(510\) 2.04892 0.0907276
\(511\) −16.4228 −0.726502
\(512\) 1.00000i 0.0441942i
\(513\) − 3.24698i − 0.143358i
\(514\) 18.8291i 0.830515i
\(515\) − 13.6353i − 0.600845i
\(516\) −4.41789 −0.194487
\(517\) 25.5459 1.12351
\(518\) − 3.04892i − 0.133962i
\(519\) −15.4795 −0.679474
\(520\) 0 0
\(521\) −35.2610 −1.54481 −0.772406 0.635129i \(-0.780947\pi\)
−0.772406 + 0.635129i \(0.780947\pi\)
\(522\) − 5.13706i − 0.224843i
\(523\) 14.3043 0.625482 0.312741 0.949839i \(-0.398753\pi\)
0.312741 + 0.949839i \(0.398753\pi\)
\(524\) 8.51573 0.372011
\(525\) 1.35690i 0.0592198i
\(526\) − 17.9245i − 0.781546i
\(527\) 5.74094i 0.250079i
\(528\) − 5.29590i − 0.230474i
\(529\) −21.2174 −0.922497
\(530\) −14.2349 −0.618324
\(531\) − 13.8998i − 0.603199i
\(532\) 4.40581 0.191016
\(533\) 0 0
\(534\) 5.47650 0.236991
\(535\) 3.92394i 0.169647i
\(536\) 9.03684 0.390332
\(537\) 19.0737 0.823090
\(538\) − 22.4045i − 0.965926i
\(539\) 27.3207i 1.17678i
\(540\) 1.00000i 0.0430331i
\(541\) 44.2422i 1.90212i 0.309006 + 0.951060i \(0.400004\pi\)
−0.309006 + 0.951060i \(0.599996\pi\)
\(542\) −23.2325 −0.997922
\(543\) 2.16852 0.0930602
\(544\) 2.04892i 0.0878466i
\(545\) 0.554958 0.0237718
\(546\) 0 0
\(547\) 6.34481 0.271285 0.135642 0.990758i \(-0.456690\pi\)
0.135642 + 0.990758i \(0.456690\pi\)
\(548\) 13.1685i 0.562531i
\(549\) −3.62565 −0.154739
\(550\) 5.29590 0.225818
\(551\) − 16.6799i − 0.710589i
\(552\) − 1.33513i − 0.0568267i
\(553\) 0.351519i 0.0149481i
\(554\) 9.78448i 0.415703i
\(555\) 2.24698 0.0953790
\(556\) −7.17629 −0.304343
\(557\) − 31.3884i − 1.32997i −0.746858 0.664984i \(-0.768438\pi\)
0.746858 0.664984i \(-0.231562\pi\)
\(558\) −2.80194 −0.118615
\(559\) 0 0
\(560\) −1.35690 −0.0573393
\(561\) − 10.8509i − 0.458123i
\(562\) 21.6286 0.912349
\(563\) 44.4989 1.87540 0.937702 0.347441i \(-0.112949\pi\)
0.937702 + 0.347441i \(0.112949\pi\)
\(564\) − 4.82371i − 0.203115i
\(565\) 4.39373i 0.184846i
\(566\) − 3.38942i − 0.142468i
\(567\) 1.35690i 0.0569843i
\(568\) 11.7506 0.493045
\(569\) −30.5394 −1.28028 −0.640140 0.768259i \(-0.721123\pi\)
−0.640140 + 0.768259i \(0.721123\pi\)
\(570\) 3.24698i 0.136001i
\(571\) 32.7614 1.37102 0.685511 0.728063i \(-0.259579\pi\)
0.685511 + 0.728063i \(0.259579\pi\)
\(572\) 0 0
\(573\) 11.7409 0.490485
\(574\) − 14.4252i − 0.602096i
\(575\) 1.33513 0.0556786
\(576\) −1.00000 −0.0416667
\(577\) 36.5870i 1.52314i 0.648084 + 0.761569i \(0.275570\pi\)
−0.648084 + 0.761569i \(0.724430\pi\)
\(578\) − 12.8019i − 0.532490i
\(579\) − 24.6722i − 1.02534i
\(580\) 5.13706i 0.213305i
\(581\) −4.10023 −0.170106
\(582\) −5.85086 −0.242526
\(583\) 75.3866i 3.12219i
\(584\) 12.1032 0.500834
\(585\) 0 0
\(586\) 34.1280 1.40981
\(587\) 22.4873i 0.928148i 0.885796 + 0.464074i \(0.153613\pi\)
−0.885796 + 0.464074i \(0.846387\pi\)
\(588\) 5.15883 0.212747
\(589\) −9.09783 −0.374870
\(590\) 13.8998i 0.572245i
\(591\) − 26.2010i − 1.07777i
\(592\) 2.24698i 0.0923503i
\(593\) − 39.5749i − 1.62515i −0.582858 0.812574i \(-0.698066\pi\)
0.582858 0.812574i \(-0.301934\pi\)
\(594\) 5.29590 0.217293
\(595\) −2.78017 −0.113976
\(596\) 9.66248i 0.395791i
\(597\) 19.6722 0.805128
\(598\) 0 0
\(599\) −31.3424 −1.28062 −0.640308 0.768118i \(-0.721193\pi\)
−0.640308 + 0.768118i \(0.721193\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −32.9487 −1.34401 −0.672003 0.740549i \(-0.734566\pi\)
−0.672003 + 0.740549i \(0.734566\pi\)
\(602\) 5.99462 0.244323
\(603\) 9.03684i 0.368008i
\(604\) 7.35152i 0.299129i
\(605\) − 17.0465i − 0.693040i
\(606\) − 8.39612i − 0.341069i
\(607\) −40.1739 −1.63061 −0.815304 0.579033i \(-0.803430\pi\)
−0.815304 + 0.579033i \(0.803430\pi\)
\(608\) −3.24698 −0.131682
\(609\) 6.97046i 0.282457i
\(610\) 3.62565 0.146798
\(611\) 0 0
\(612\) −2.04892 −0.0828226
\(613\) 11.3588i 0.458778i 0.973335 + 0.229389i \(0.0736728\pi\)
−0.973335 + 0.229389i \(0.926327\pi\)
\(614\) −15.6601 −0.631990
\(615\) 10.6310 0.428684
\(616\) 7.18598i 0.289531i
\(617\) 20.9172i 0.842096i 0.907038 + 0.421048i \(0.138338\pi\)
−0.907038 + 0.421048i \(0.861662\pi\)
\(618\) 13.6353i 0.548494i
\(619\) 14.0562i 0.564967i 0.959272 + 0.282483i \(0.0911582\pi\)
−0.959272 + 0.282483i \(0.908842\pi\)
\(620\) 2.80194 0.112529
\(621\) 1.33513 0.0535767
\(622\) − 25.7972i − 1.03437i
\(623\) −7.43104 −0.297718
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 15.6243i − 0.624473i
\(627\) 17.1957 0.686729
\(628\) −2.85325 −0.113857
\(629\) 4.60388i 0.183569i
\(630\) − 1.35690i − 0.0540600i
\(631\) 40.6601i 1.61865i 0.587359 + 0.809326i \(0.300168\pi\)
−0.587359 + 0.809326i \(0.699832\pi\)
\(632\) − 0.259061i − 0.0103049i
\(633\) 16.2935 0.647609
\(634\) 9.25965 0.367748
\(635\) 2.23490i 0.0886892i
\(636\) 14.2349 0.564450
\(637\) 0 0
\(638\) 27.2054 1.07707
\(639\) 11.7506i 0.464848i
\(640\) 1.00000 0.0395285
\(641\) 1.92633 0.0760854 0.0380427 0.999276i \(-0.487888\pi\)
0.0380427 + 0.999276i \(0.487888\pi\)
\(642\) − 3.92394i − 0.154865i
\(643\) 3.37973i 0.133284i 0.997777 + 0.0666418i \(0.0212285\pi\)
−0.997777 + 0.0666418i \(0.978772\pi\)
\(644\) 1.81163i 0.0713881i
\(645\) 4.41789i 0.173954i
\(646\) −6.65279 −0.261751
\(647\) 10.8181 0.425302 0.212651 0.977128i \(-0.431790\pi\)
0.212651 + 0.977128i \(0.431790\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 73.6118 2.88951
\(650\) 0 0
\(651\) 3.80194 0.149010
\(652\) 15.1075i 0.591656i
\(653\) 33.7646 1.32131 0.660656 0.750689i \(-0.270278\pi\)
0.660656 + 0.750689i \(0.270278\pi\)
\(654\) −0.554958 −0.0217006
\(655\) − 8.51573i − 0.332737i
\(656\) 10.6310i 0.415072i
\(657\) 12.1032i 0.472191i
\(658\) 6.54527i 0.255161i
\(659\) −16.8592 −0.656742 −0.328371 0.944549i \(-0.606500\pi\)
−0.328371 + 0.944549i \(0.606500\pi\)
\(660\) −5.29590 −0.206143
\(661\) 4.71140i 0.183252i 0.995793 + 0.0916261i \(0.0292065\pi\)
−0.995793 + 0.0916261i \(0.970794\pi\)
\(662\) −21.8582 −0.849541
\(663\) 0 0
\(664\) 3.02177 0.117267
\(665\) − 4.40581i − 0.170850i
\(666\) −2.24698 −0.0870687
\(667\) 6.85862 0.265567
\(668\) 16.0881i 0.622469i
\(669\) 2.28083i 0.0881820i
\(670\) − 9.03684i − 0.349123i
\(671\) − 19.2010i − 0.741248i
\(672\) 1.35690 0.0523434
\(673\) −3.12797 −0.120574 −0.0602871 0.998181i \(-0.519202\pi\)
−0.0602871 + 0.998181i \(0.519202\pi\)
\(674\) − 14.5013i − 0.558567i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −1.50737 −0.0579329 −0.0289664 0.999580i \(-0.509222\pi\)
−0.0289664 + 0.999580i \(0.509222\pi\)
\(678\) − 4.39373i − 0.168740i
\(679\) 7.93900 0.304671
\(680\) 2.04892 0.0785724
\(681\) 3.37196i 0.129214i
\(682\) − 14.8388i − 0.568206i
\(683\) 31.0726i 1.18896i 0.804110 + 0.594480i \(0.202642\pi\)
−0.804110 + 0.594480i \(0.797358\pi\)
\(684\) − 3.24698i − 0.124151i
\(685\) 13.1685 0.503143
\(686\) −16.4983 −0.629907
\(687\) 18.3763i 0.701099i
\(688\) −4.41789 −0.168431
\(689\) 0 0
\(690\) −1.33513 −0.0508274
\(691\) 26.7469i 1.01750i 0.860914 + 0.508750i \(0.169892\pi\)
−0.860914 + 0.508750i \(0.830108\pi\)
\(692\) −15.4795 −0.588442
\(693\) −7.18598 −0.272973
\(694\) 7.84117i 0.297647i
\(695\) 7.17629i 0.272212i
\(696\) − 5.13706i − 0.194720i
\(697\) 21.7821i 0.825055i
\(698\) 20.4252 0.773105
\(699\) 10.0858 0.381478
\(700\) 1.35690i 0.0512858i
\(701\) −11.2054 −0.423221 −0.211610 0.977354i \(-0.567871\pi\)
−0.211610 + 0.977354i \(0.567871\pi\)
\(702\) 0 0
\(703\) −7.29590 −0.275170
\(704\) − 5.29590i − 0.199597i
\(705\) −4.82371 −0.181671
\(706\) −13.2905 −0.500195
\(707\) 11.3927i 0.428465i
\(708\) − 13.8998i − 0.522385i
\(709\) 26.9933i 1.01375i 0.862018 + 0.506877i \(0.169200\pi\)
−0.862018 + 0.506877i \(0.830800\pi\)
\(710\) − 11.7506i − 0.440993i
\(711\) 0.259061 0.00971555
\(712\) 5.47650 0.205241
\(713\) − 3.74094i − 0.140099i
\(714\) 2.78017 0.104045
\(715\) 0 0
\(716\) 19.0737 0.712817
\(717\) 20.0834i 0.750027i
\(718\) −12.8267 −0.478688
\(719\) 15.4601 0.576565 0.288282 0.957545i \(-0.406916\pi\)
0.288282 + 0.957545i \(0.406916\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 18.5017i − 0.689040i
\(722\) 8.45712i 0.314742i
\(723\) 2.99462i 0.111371i
\(724\) 2.16852 0.0805925
\(725\) 5.13706 0.190786
\(726\) 17.0465i 0.632656i
\(727\) 10.9148 0.404809 0.202404 0.979302i \(-0.435124\pi\)
0.202404 + 0.979302i \(0.435124\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 12.1032i − 0.447960i
\(731\) −9.05190 −0.334797
\(732\) −3.62565 −0.134008
\(733\) 13.5539i 0.500624i 0.968165 + 0.250312i \(0.0805333\pi\)
−0.968165 + 0.250312i \(0.919467\pi\)
\(734\) − 17.9554i − 0.662746i
\(735\) − 5.15883i − 0.190286i
\(736\) − 1.33513i − 0.0492134i
\(737\) −47.8582 −1.76288
\(738\) −10.6310 −0.391333
\(739\) 15.4679i 0.568995i 0.958677 + 0.284498i \(0.0918268\pi\)
−0.958677 + 0.284498i \(0.908173\pi\)
\(740\) 2.24698 0.0826006
\(741\) 0 0
\(742\) −19.3153 −0.709086
\(743\) − 40.2040i − 1.47494i −0.675378 0.737471i \(-0.736020\pi\)
0.675378 0.737471i \(-0.263980\pi\)
\(744\) −2.80194 −0.102724
\(745\) 9.66248 0.354006
\(746\) 27.8732i 1.02051i
\(747\) 3.02177i 0.110561i
\(748\) − 10.8509i − 0.396747i
\(749\) 5.32437i 0.194548i
\(750\) −1.00000 −0.0365148
\(751\) 19.8465 0.724211 0.362105 0.932137i \(-0.382058\pi\)
0.362105 + 0.932137i \(0.382058\pi\)
\(752\) − 4.82371i − 0.175903i
\(753\) −7.20237 −0.262469
\(754\) 0 0
\(755\) 7.35152 0.267549
\(756\) 1.35690i 0.0493498i
\(757\) 39.8883 1.44976 0.724882 0.688873i \(-0.241894\pi\)
0.724882 + 0.688873i \(0.241894\pi\)
\(758\) 13.8702 0.503790
\(759\) 7.07069i 0.256650i
\(760\) 3.24698i 0.117780i
\(761\) 31.8756i 1.15549i 0.816217 + 0.577745i \(0.196067\pi\)
−0.816217 + 0.577745i \(0.803933\pi\)
\(762\) − 2.23490i − 0.0809618i
\(763\) 0.753020 0.0272612
\(764\) 11.7409 0.424772
\(765\) 2.04892i 0.0740788i
\(766\) −26.8984 −0.971880
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 43.0672i 1.55304i 0.630090 + 0.776522i \(0.283018\pi\)
−0.630090 + 0.776522i \(0.716982\pi\)
\(770\) 7.18598 0.258965
\(771\) −18.8291 −0.678113
\(772\) − 24.6722i − 0.887971i
\(773\) − 33.1282i − 1.19154i −0.803155 0.595770i \(-0.796847\pi\)
0.803155 0.595770i \(-0.203153\pi\)
\(774\) − 4.41789i − 0.158798i
\(775\) − 2.80194i − 0.100649i
\(776\) −5.85086 −0.210033
\(777\) 3.04892 0.109379
\(778\) 0.187309i 0.00671534i
\(779\) −34.5187 −1.23676
\(780\) 0 0
\(781\) −62.2301 −2.22677
\(782\) − 2.73556i − 0.0978235i
\(783\) 5.13706 0.183584
\(784\) 5.15883 0.184244
\(785\) 2.85325i 0.101837i
\(786\) 8.51573i 0.303746i
\(787\) 35.9138i 1.28019i 0.768297 + 0.640094i \(0.221105\pi\)
−0.768297 + 0.640094i \(0.778895\pi\)
\(788\) − 26.2010i − 0.933374i
\(789\) 17.9245 0.638130
\(790\) −0.259061 −0.00921698
\(791\) 5.96184i 0.211978i
\(792\) 5.29590 0.188182
\(793\) 0 0
\(794\) 8.58211 0.304568
\(795\) − 14.2349i − 0.504860i
\(796\) 19.6722 0.697262
\(797\) 29.7918 1.05528 0.527639 0.849468i \(-0.323077\pi\)
0.527639 + 0.849468i \(0.323077\pi\)
\(798\) 4.40581i 0.155964i
\(799\) − 9.88338i − 0.349649i
\(800\) − 1.00000i − 0.0353553i
\(801\) 5.47650i 0.193503i
\(802\) 23.9748 0.846579
\(803\) −64.0974 −2.26195
\(804\) 9.03684i 0.318705i
\(805\) 1.81163 0.0638514
\(806\) 0 0
\(807\) 22.4045 0.788675
\(808\) − 8.39612i − 0.295375i
\(809\) 1.52648 0.0536683 0.0268341 0.999640i \(-0.491457\pi\)
0.0268341 + 0.999640i \(0.491457\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 51.6039i − 1.81206i −0.423217 0.906029i \(-0.639099\pi\)
0.423217 0.906029i \(-0.360901\pi\)
\(812\) 6.97046i 0.244615i
\(813\) − 23.2325i − 0.814800i
\(814\) − 11.8998i − 0.417087i
\(815\) 15.1075 0.529193
\(816\) −2.04892 −0.0717265
\(817\) − 14.3448i − 0.501862i
\(818\) 8.37926 0.292974
\(819\) 0 0
\(820\) 10.6310 0.371251
\(821\) 17.4795i 0.610038i 0.952346 + 0.305019i \(0.0986629\pi\)
−0.952346 + 0.305019i \(0.901337\pi\)
\(822\) −13.1685 −0.459305
\(823\) 25.5566 0.890848 0.445424 0.895320i \(-0.353053\pi\)
0.445424 + 0.895320i \(0.353053\pi\)
\(824\) 13.6353i 0.475009i
\(825\) 5.29590i 0.184379i
\(826\) 18.8605i 0.656242i
\(827\) 9.16182i 0.318588i 0.987231 + 0.159294i \(0.0509217\pi\)
−0.987231 + 0.159294i \(0.949078\pi\)
\(828\) 1.33513 0.0463988
\(829\) −48.6493 −1.68966 −0.844831 0.535034i \(-0.820299\pi\)
−0.844831 + 0.535034i \(0.820299\pi\)
\(830\) − 3.02177i − 0.104887i
\(831\) −9.78448 −0.339420
\(832\) 0 0
\(833\) 10.5700 0.366230
\(834\) − 7.17629i − 0.248495i
\(835\) 16.0881 0.556753
\(836\) 17.1957 0.594725
\(837\) − 2.80194i − 0.0968491i
\(838\) 37.0887i 1.28121i
\(839\) 8.71246i 0.300788i 0.988626 + 0.150394i \(0.0480542\pi\)
−0.988626 + 0.150394i \(0.951946\pi\)
\(840\) − 1.35690i − 0.0468174i
\(841\) −2.61058 −0.0900200
\(842\) −4.90515 −0.169043
\(843\) 21.6286i 0.744930i
\(844\) 16.2935 0.560846
\(845\) 0 0
\(846\) 4.82371 0.165842
\(847\) − 23.1304i − 0.794769i
\(848\) 14.2349 0.488828
\(849\) 3.38942 0.116325
\(850\) − 2.04892i − 0.0702773i
\(851\) − 3.00000i − 0.102839i
\(852\) 11.7506i 0.402570i
\(853\) 5.41060i 0.185255i 0.995701 + 0.0926277i \(0.0295266\pi\)
−0.995701 + 0.0926277i \(0.970473\pi\)
\(854\) 4.91962 0.168346
\(855\) −3.24698 −0.111044
\(856\) − 3.92394i − 0.134117i
\(857\) −0.165275 −0.00564570 −0.00282285 0.999996i \(-0.500899\pi\)
−0.00282285 + 0.999996i \(0.500899\pi\)
\(858\) 0 0
\(859\) 47.0853 1.60653 0.803264 0.595622i \(-0.203095\pi\)
0.803264 + 0.595622i \(0.203095\pi\)
\(860\) 4.41789i 0.150649i
\(861\) 14.4252 0.491609
\(862\) 6.06531 0.206585
\(863\) − 31.2553i − 1.06394i −0.846762 0.531972i \(-0.821451\pi\)
0.846762 0.531972i \(-0.178549\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 15.4795i 0.526318i
\(866\) 14.8877i 0.505904i
\(867\) 12.8019 0.434777
\(868\) 3.80194 0.129046
\(869\) 1.37196i 0.0465406i
\(870\) −5.13706 −0.174163
\(871\) 0 0
\(872\) −0.554958 −0.0187933
\(873\) − 5.85086i − 0.198021i
\(874\) 4.33513 0.146638
\(875\) 1.35690 0.0458715
\(876\) 12.1032i 0.408930i
\(877\) 16.9898i 0.573706i 0.957975 + 0.286853i \(0.0926091\pi\)
−0.957975 + 0.286853i \(0.907391\pi\)
\(878\) 21.1336i 0.713225i
\(879\) 34.1280i 1.15111i
\(880\) −5.29590 −0.178525
\(881\) 3.29159 0.110896 0.0554482 0.998462i \(-0.482341\pi\)
0.0554482 + 0.998462i \(0.482341\pi\)
\(882\) 5.15883i 0.173707i
\(883\) −17.3157 −0.582721 −0.291361 0.956613i \(-0.594108\pi\)
−0.291361 + 0.956613i \(0.594108\pi\)
\(884\) 0 0
\(885\) −13.8998 −0.467236
\(886\) 28.0683i 0.942973i
\(887\) 15.6775 0.526401 0.263200 0.964741i \(-0.415222\pi\)
0.263200 + 0.964741i \(0.415222\pi\)
\(888\) −2.24698 −0.0754037
\(889\) 3.03252i 0.101708i
\(890\) − 5.47650i − 0.183573i
\(891\) 5.29590i 0.177419i
\(892\) 2.28083i 0.0763679i
\(893\) 15.6625 0.524125
\(894\) −9.66248 −0.323162
\(895\) − 19.0737i − 0.637563i
\(896\) 1.35690 0.0453307
\(897\) 0 0
\(898\) 19.2446 0.642200
\(899\) − 14.3937i − 0.480058i
\(900\) 1.00000 0.0333333
\(901\) 29.1661 0.971665
\(902\) − 56.3008i − 1.87461i
\(903\) 5.99462i 0.199489i
\(904\) − 4.39373i − 0.146133i
\(905\) − 2.16852i − 0.0720841i
\(906\) −7.35152 −0.244238
\(907\) 25.8877 0.859587 0.429793 0.902927i \(-0.358586\pi\)
0.429793 + 0.902927i \(0.358586\pi\)
\(908\) 3.37196i 0.111902i
\(909\) 8.39612 0.278482
\(910\) 0 0
\(911\) −37.5220 −1.24316 −0.621579 0.783351i \(-0.713509\pi\)
−0.621579 + 0.783351i \(0.713509\pi\)
\(912\) − 3.24698i − 0.107518i
\(913\) −16.0030 −0.529621
\(914\) 37.8485 1.25192
\(915\) 3.62565i 0.119860i
\(916\) 18.3763i 0.607169i
\(917\) − 11.5550i − 0.381578i
\(918\) − 2.04892i − 0.0676243i
\(919\) −5.56358 −0.183526 −0.0917628 0.995781i \(-0.529250\pi\)
−0.0917628 + 0.995781i \(0.529250\pi\)
\(920\) −1.33513 −0.0440178
\(921\) − 15.6601i − 0.516017i
\(922\) 11.2228 0.369604
\(923\) 0 0
\(924\) −7.18598 −0.236401
\(925\) − 2.24698i − 0.0738802i
\(926\) 25.0726 0.823937
\(927\) −13.6353 −0.447843
\(928\) − 5.13706i − 0.168632i
\(929\) 40.2602i 1.32090i 0.750872 + 0.660448i \(0.229634\pi\)
−0.750872 + 0.660448i \(0.770366\pi\)
\(930\) 2.80194i 0.0918792i
\(931\) 16.7506i 0.548980i
\(932\) 10.0858 0.330370
\(933\) 25.7972 0.844561
\(934\) − 36.9657i − 1.20955i
\(935\) −10.8509 −0.354861
\(936\) 0 0
\(937\) −30.5356 −0.997554 −0.498777 0.866730i \(-0.666217\pi\)
−0.498777 + 0.866730i \(0.666217\pi\)
\(938\) − 12.2620i − 0.400370i
\(939\) 15.6243 0.509880
\(940\) −4.82371 −0.157332
\(941\) − 41.6461i − 1.35762i −0.734312 0.678812i \(-0.762495\pi\)
0.734312 0.678812i \(-0.237505\pi\)
\(942\) − 2.85325i − 0.0929638i
\(943\) − 14.1938i − 0.462212i
\(944\) − 13.8998i − 0.452399i
\(945\) 1.35690 0.0441398
\(946\) 23.3967 0.760693
\(947\) 21.7724i 0.707508i 0.935339 + 0.353754i \(0.115095\pi\)
−0.935339 + 0.353754i \(0.884905\pi\)
\(948\) 0.259061 0.00841392
\(949\) 0 0
\(950\) 3.24698 0.105346
\(951\) 9.25965i 0.300265i
\(952\) 2.78017 0.0901057
\(953\) 41.9560 1.35909 0.679544 0.733635i \(-0.262178\pi\)
0.679544 + 0.733635i \(0.262178\pi\)
\(954\) 14.2349i 0.460872i
\(955\) − 11.7409i − 0.379928i
\(956\) 20.0834i 0.649542i
\(957\) 27.2054i 0.879424i
\(958\) 31.4590 1.01640
\(959\) 17.8683 0.576998
\(960\) 1.00000i 0.0322749i
\(961\) 23.1491 0.746747
\(962\) 0 0
\(963\) 3.92394 0.126447
\(964\) 2.99462i 0.0964503i
\(965\) −24.6722 −0.794225
\(966\) −1.81163 −0.0582881
\(967\) − 16.7041i − 0.537168i −0.963256 0.268584i \(-0.913444\pi\)
0.963256 0.268584i \(-0.0865556\pi\)
\(968\) 17.0465i 0.547896i
\(969\) − 6.65279i − 0.213718i
\(970\) 5.85086i 0.187860i
\(971\) −20.7275 −0.665178 −0.332589 0.943072i \(-0.607922\pi\)
−0.332589 + 0.943072i \(0.607922\pi\)
\(972\) 1.00000 0.0320750
\(973\) 9.73748i 0.312169i
\(974\) −18.8713 −0.604675
\(975\) 0 0
\(976\) −3.62565 −0.116054
\(977\) 33.4547i 1.07031i 0.844753 + 0.535156i \(0.179747\pi\)
−0.844753 + 0.535156i \(0.820253\pi\)
\(978\) −15.1075 −0.483085
\(979\) −29.0030 −0.926939
\(980\) − 5.15883i − 0.164793i
\(981\) − 0.554958i − 0.0177184i
\(982\) 16.3961i 0.523221i
\(983\) 31.8605i 1.01619i 0.861300 + 0.508097i \(0.169651\pi\)
−0.861300 + 0.508097i \(0.830349\pi\)
\(984\) −10.6310 −0.338905
\(985\) −26.2010 −0.834835
\(986\) − 10.5254i − 0.335198i
\(987\) −6.54527 −0.208338
\(988\) 0 0
\(989\) 5.89844 0.187560
\(990\) − 5.29590i − 0.168315i
\(991\) −21.0925 −0.670024 −0.335012 0.942214i \(-0.608740\pi\)
−0.335012 + 0.942214i \(0.608740\pi\)
\(992\) −2.80194 −0.0889616
\(993\) − 21.8582i − 0.693647i
\(994\) − 15.9444i − 0.505725i
\(995\) − 19.6722i − 0.623650i
\(996\) 3.02177i 0.0957485i
\(997\) 17.7832 0.563198 0.281599 0.959532i \(-0.409135\pi\)
0.281599 + 0.959532i \(0.409135\pi\)
\(998\) −40.2452 −1.27394
\(999\) − 2.24698i − 0.0710913i
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.t.1351.5 6
13.5 odd 4 5070.2.a.bu.1.2 yes 3
13.8 odd 4 5070.2.a.bj.1.2 3
13.12 even 2 inner 5070.2.b.t.1351.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bj.1.2 3 13.8 odd 4
5070.2.a.bu.1.2 yes 3 13.5 odd 4
5070.2.b.t.1351.2 6 13.12 even 2 inner
5070.2.b.t.1351.5 6 1.1 even 1 trivial