Properties

Label 7098.2.a.ct.1.6
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.48406561.1
Defining polynomial: \(x^{6} - 3 x^{5} - 17 x^{4} + 39 x^{3} + 111 x^{2} - 131 x - 281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(3.41496\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.41496 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.41496 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.41496 q^{10} +3.24698 q^{11} +1.00000 q^{12} +1.00000 q^{14} +3.41496 q^{15} +1.00000 q^{16} +1.80194 q^{17} +1.00000 q^{18} +1.93665 q^{19} +3.41496 q^{20} +1.00000 q^{21} +3.24698 q^{22} +3.28422 q^{23} +1.00000 q^{24} +6.66194 q^{25} +1.00000 q^{27} +1.00000 q^{28} -3.64112 q^{29} +3.41496 q^{30} -3.10698 q^{31} +1.00000 q^{32} +3.24698 q^{33} +1.80194 q^{34} +3.41496 q^{35} +1.00000 q^{36} -4.80172 q^{37} +1.93665 q^{38} +3.41496 q^{40} -4.58189 q^{41} +1.00000 q^{42} +5.11205 q^{43} +3.24698 q^{44} +3.41496 q^{45} +3.28422 q^{46} -10.5966 q^{47} +1.00000 q^{48} +1.00000 q^{49} +6.66194 q^{50} +1.80194 q^{51} -2.46409 q^{53} +1.00000 q^{54} +11.0883 q^{55} +1.00000 q^{56} +1.93665 q^{57} -3.64112 q^{58} +13.2749 q^{59} +3.41496 q^{60} -1.76668 q^{61} -3.10698 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.24698 q^{66} -12.4092 q^{67} +1.80194 q^{68} +3.28422 q^{69} +3.41496 q^{70} -2.66429 q^{71} +1.00000 q^{72} +0.368396 q^{73} -4.80172 q^{74} +6.66194 q^{75} +1.93665 q^{76} +3.24698 q^{77} -15.0033 q^{79} +3.41496 q^{80} +1.00000 q^{81} -4.58189 q^{82} -1.43401 q^{83} +1.00000 q^{84} +6.15354 q^{85} +5.11205 q^{86} -3.64112 q^{87} +3.24698 q^{88} +5.10615 q^{89} +3.41496 q^{90} +3.28422 q^{92} -3.10698 q^{93} -10.5966 q^{94} +6.61356 q^{95} +1.00000 q^{96} -11.9553 q^{97} +1.00000 q^{98} +3.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 6q^{3} + 6q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 6q^{3} + 6q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} + 3q^{10} + 10q^{11} + 6q^{12} + 6q^{14} + 3q^{15} + 6q^{16} + 2q^{17} + 6q^{18} + 2q^{19} + 3q^{20} + 6q^{21} + 10q^{22} + 4q^{23} + 6q^{24} + 13q^{25} + 6q^{27} + 6q^{28} + 2q^{29} + 3q^{30} + 9q^{31} + 6q^{32} + 10q^{33} + 2q^{34} + 3q^{35} + 6q^{36} + 7q^{37} + 2q^{38} + 3q^{40} + 11q^{41} + 6q^{42} - 5q^{43} + 10q^{44} + 3q^{45} + 4q^{46} + 5q^{47} + 6q^{48} + 6q^{49} + 13q^{50} + 2q^{51} - 6q^{53} + 6q^{54} + 5q^{55} + 6q^{56} + 2q^{57} + 2q^{58} + 28q^{59} + 3q^{60} + 23q^{61} + 9q^{62} + 6q^{63} + 6q^{64} + 10q^{66} - 10q^{67} + 2q^{68} + 4q^{69} + 3q^{70} + 21q^{71} + 6q^{72} - 7q^{73} + 7q^{74} + 13q^{75} + 2q^{76} + 10q^{77} - 14q^{79} + 3q^{80} + 6q^{81} + 11q^{82} + 17q^{83} + 6q^{84} + q^{85} - 5q^{86} + 2q^{87} + 10q^{88} + 17q^{89} + 3q^{90} + 4q^{92} + 9q^{93} + 5q^{94} - 22q^{95} + 6q^{96} + 6q^{98} + 10q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 3.41496 1.52722 0.763608 0.645680i \(-0.223426\pi\)
0.763608 + 0.645680i \(0.223426\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.41496 1.07990
\(11\) 3.24698 0.979001 0.489501 0.872003i \(-0.337179\pi\)
0.489501 + 0.872003i \(0.337179\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 3.41496 0.881738
\(16\) 1.00000 0.250000
\(17\) 1.80194 0.437034 0.218517 0.975833i \(-0.429878\pi\)
0.218517 + 0.975833i \(0.429878\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.93665 0.444297 0.222149 0.975013i \(-0.428693\pi\)
0.222149 + 0.975013i \(0.428693\pi\)
\(20\) 3.41496 0.763608
\(21\) 1.00000 0.218218
\(22\) 3.24698 0.692258
\(23\) 3.28422 0.684808 0.342404 0.939553i \(-0.388759\pi\)
0.342404 + 0.939553i \(0.388759\pi\)
\(24\) 1.00000 0.204124
\(25\) 6.66194 1.33239
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −3.64112 −0.676139 −0.338069 0.941121i \(-0.609774\pi\)
−0.338069 + 0.941121i \(0.609774\pi\)
\(30\) 3.41496 0.623483
\(31\) −3.10698 −0.558030 −0.279015 0.960287i \(-0.590008\pi\)
−0.279015 + 0.960287i \(0.590008\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.24698 0.565227
\(34\) 1.80194 0.309030
\(35\) 3.41496 0.577233
\(36\) 1.00000 0.166667
\(37\) −4.80172 −0.789398 −0.394699 0.918810i \(-0.629151\pi\)
−0.394699 + 0.918810i \(0.629151\pi\)
\(38\) 1.93665 0.314165
\(39\) 0 0
\(40\) 3.41496 0.539952
\(41\) −4.58189 −0.715571 −0.357785 0.933804i \(-0.616468\pi\)
−0.357785 + 0.933804i \(0.616468\pi\)
\(42\) 1.00000 0.154303
\(43\) 5.11205 0.779581 0.389790 0.920904i \(-0.372547\pi\)
0.389790 + 0.920904i \(0.372547\pi\)
\(44\) 3.24698 0.489501
\(45\) 3.41496 0.509072
\(46\) 3.28422 0.484232
\(47\) −10.5966 −1.54567 −0.772836 0.634605i \(-0.781163\pi\)
−0.772836 + 0.634605i \(0.781163\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 6.66194 0.942140
\(51\) 1.80194 0.252322
\(52\) 0 0
\(53\) −2.46409 −0.338469 −0.169235 0.985576i \(-0.554130\pi\)
−0.169235 + 0.985576i \(0.554130\pi\)
\(54\) 1.00000 0.136083
\(55\) 11.0883 1.49515
\(56\) 1.00000 0.133631
\(57\) 1.93665 0.256515
\(58\) −3.64112 −0.478102
\(59\) 13.2749 1.72824 0.864120 0.503286i \(-0.167876\pi\)
0.864120 + 0.503286i \(0.167876\pi\)
\(60\) 3.41496 0.440869
\(61\) −1.76668 −0.226200 −0.113100 0.993584i \(-0.536078\pi\)
−0.113100 + 0.993584i \(0.536078\pi\)
\(62\) −3.10698 −0.394587
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.24698 0.399676
\(67\) −12.4092 −1.51603 −0.758013 0.652240i \(-0.773830\pi\)
−0.758013 + 0.652240i \(0.773830\pi\)
\(68\) 1.80194 0.218517
\(69\) 3.28422 0.395374
\(70\) 3.41496 0.408166
\(71\) −2.66429 −0.316193 −0.158097 0.987424i \(-0.550536\pi\)
−0.158097 + 0.987424i \(0.550536\pi\)
\(72\) 1.00000 0.117851
\(73\) 0.368396 0.0431175 0.0215587 0.999768i \(-0.493137\pi\)
0.0215587 + 0.999768i \(0.493137\pi\)
\(74\) −4.80172 −0.558189
\(75\) 6.66194 0.769254
\(76\) 1.93665 0.222149
\(77\) 3.24698 0.370028
\(78\) 0 0
\(79\) −15.0033 −1.68801 −0.844004 0.536336i \(-0.819808\pi\)
−0.844004 + 0.536336i \(0.819808\pi\)
\(80\) 3.41496 0.381804
\(81\) 1.00000 0.111111
\(82\) −4.58189 −0.505985
\(83\) −1.43401 −0.157403 −0.0787014 0.996898i \(-0.525077\pi\)
−0.0787014 + 0.996898i \(0.525077\pi\)
\(84\) 1.00000 0.109109
\(85\) 6.15354 0.667445
\(86\) 5.11205 0.551247
\(87\) −3.64112 −0.390369
\(88\) 3.24698 0.346129
\(89\) 5.10615 0.541251 0.270625 0.962685i \(-0.412770\pi\)
0.270625 + 0.962685i \(0.412770\pi\)
\(90\) 3.41496 0.359968
\(91\) 0 0
\(92\) 3.28422 0.342404
\(93\) −3.10698 −0.322179
\(94\) −10.5966 −1.09296
\(95\) 6.61356 0.678537
\(96\) 1.00000 0.102062
\(97\) −11.9553 −1.21387 −0.606936 0.794750i \(-0.707602\pi\)
−0.606936 + 0.794750i \(0.707602\pi\)
\(98\) 1.00000 0.101015
\(99\) 3.24698 0.326334
\(100\) 6.66194 0.666194
\(101\) −18.5354 −1.84434 −0.922170 0.386785i \(-0.873585\pi\)
−0.922170 + 0.386785i \(0.873585\pi\)
\(102\) 1.80194 0.178418
\(103\) 18.2577 1.79899 0.899493 0.436936i \(-0.143936\pi\)
0.899493 + 0.436936i \(0.143936\pi\)
\(104\) 0 0
\(105\) 3.41496 0.333266
\(106\) −2.46409 −0.239334
\(107\) −15.0210 −1.45213 −0.726066 0.687625i \(-0.758653\pi\)
−0.726066 + 0.687625i \(0.758653\pi\)
\(108\) 1.00000 0.0962250
\(109\) 11.7533 1.12577 0.562883 0.826536i \(-0.309692\pi\)
0.562883 + 0.826536i \(0.309692\pi\)
\(110\) 11.0883 1.05723
\(111\) −4.80172 −0.455759
\(112\) 1.00000 0.0944911
\(113\) 3.69957 0.348026 0.174013 0.984743i \(-0.444326\pi\)
0.174013 + 0.984743i \(0.444326\pi\)
\(114\) 1.93665 0.181384
\(115\) 11.2155 1.04585
\(116\) −3.64112 −0.338069
\(117\) 0 0
\(118\) 13.2749 1.22205
\(119\) 1.80194 0.165183
\(120\) 3.41496 0.311742
\(121\) −0.457123 −0.0415567
\(122\) −1.76668 −0.159948
\(123\) −4.58189 −0.413135
\(124\) −3.10698 −0.279015
\(125\) 5.67545 0.507627
\(126\) 1.00000 0.0890871
\(127\) 3.89724 0.345824 0.172912 0.984937i \(-0.444682\pi\)
0.172912 + 0.984937i \(0.444682\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.11205 0.450091
\(130\) 0 0
\(131\) −17.6740 −1.54418 −0.772091 0.635511i \(-0.780789\pi\)
−0.772091 + 0.635511i \(0.780789\pi\)
\(132\) 3.24698 0.282613
\(133\) 1.93665 0.167928
\(134\) −12.4092 −1.07199
\(135\) 3.41496 0.293913
\(136\) 1.80194 0.154515
\(137\) −14.9195 −1.27466 −0.637331 0.770590i \(-0.719962\pi\)
−0.637331 + 0.770590i \(0.719962\pi\)
\(138\) 3.28422 0.279572
\(139\) −3.27513 −0.277793 −0.138897 0.990307i \(-0.544356\pi\)
−0.138897 + 0.990307i \(0.544356\pi\)
\(140\) 3.41496 0.288617
\(141\) −10.5966 −0.892395
\(142\) −2.66429 −0.223582
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −12.4343 −1.03261
\(146\) 0.368396 0.0304887
\(147\) 1.00000 0.0824786
\(148\) −4.80172 −0.394699
\(149\) 8.92475 0.731144 0.365572 0.930783i \(-0.380873\pi\)
0.365572 + 0.930783i \(0.380873\pi\)
\(150\) 6.66194 0.543945
\(151\) −0.712364 −0.0579714 −0.0289857 0.999580i \(-0.509228\pi\)
−0.0289857 + 0.999580i \(0.509228\pi\)
\(152\) 1.93665 0.157083
\(153\) 1.80194 0.145678
\(154\) 3.24698 0.261649
\(155\) −10.6102 −0.852232
\(156\) 0 0
\(157\) −14.1436 −1.12879 −0.564393 0.825506i \(-0.690890\pi\)
−0.564393 + 0.825506i \(0.690890\pi\)
\(158\) −15.0033 −1.19360
\(159\) −2.46409 −0.195415
\(160\) 3.41496 0.269976
\(161\) 3.28422 0.258833
\(162\) 1.00000 0.0785674
\(163\) 15.6565 1.22632 0.613158 0.789960i \(-0.289899\pi\)
0.613158 + 0.789960i \(0.289899\pi\)
\(164\) −4.58189 −0.357785
\(165\) 11.0883 0.863223
\(166\) −1.43401 −0.111301
\(167\) 19.6904 1.52369 0.761845 0.647759i \(-0.224294\pi\)
0.761845 + 0.647759i \(0.224294\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 6.15354 0.471955
\(171\) 1.93665 0.148099
\(172\) 5.11205 0.389790
\(173\) 16.3915 1.24622 0.623112 0.782132i \(-0.285868\pi\)
0.623112 + 0.782132i \(0.285868\pi\)
\(174\) −3.64112 −0.276033
\(175\) 6.66194 0.503595
\(176\) 3.24698 0.244750
\(177\) 13.2749 0.997800
\(178\) 5.10615 0.382722
\(179\) −17.0906 −1.27741 −0.638704 0.769452i \(-0.720529\pi\)
−0.638704 + 0.769452i \(0.720529\pi\)
\(180\) 3.41496 0.254536
\(181\) 21.3647 1.58803 0.794015 0.607899i \(-0.207987\pi\)
0.794015 + 0.607899i \(0.207987\pi\)
\(182\) 0 0
\(183\) −1.76668 −0.130597
\(184\) 3.28422 0.242116
\(185\) −16.3977 −1.20558
\(186\) −3.10698 −0.227815
\(187\) 5.85086 0.427857
\(188\) −10.5966 −0.772836
\(189\) 1.00000 0.0727393
\(190\) 6.61356 0.479798
\(191\) 16.2973 1.17923 0.589614 0.807685i \(-0.299280\pi\)
0.589614 + 0.807685i \(0.299280\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.7221 1.34765 0.673824 0.738892i \(-0.264651\pi\)
0.673824 + 0.738892i \(0.264651\pi\)
\(194\) −11.9553 −0.858338
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.35011 0.452426 0.226213 0.974078i \(-0.427365\pi\)
0.226213 + 0.974078i \(0.427365\pi\)
\(198\) 3.24698 0.230753
\(199\) 11.9820 0.849380 0.424690 0.905339i \(-0.360383\pi\)
0.424690 + 0.905339i \(0.360383\pi\)
\(200\) 6.66194 0.471070
\(201\) −12.4092 −0.875278
\(202\) −18.5354 −1.30414
\(203\) −3.64112 −0.255556
\(204\) 1.80194 0.126161
\(205\) −15.6470 −1.09283
\(206\) 18.2577 1.27207
\(207\) 3.28422 0.228269
\(208\) 0 0
\(209\) 6.28825 0.434967
\(210\) 3.41496 0.235654
\(211\) 9.42870 0.649099 0.324549 0.945869i \(-0.394787\pi\)
0.324549 + 0.945869i \(0.394787\pi\)
\(212\) −2.46409 −0.169235
\(213\) −2.66429 −0.182554
\(214\) −15.0210 −1.02681
\(215\) 17.4575 1.19059
\(216\) 1.00000 0.0680414
\(217\) −3.10698 −0.210916
\(218\) 11.7533 0.796037
\(219\) 0.368396 0.0248939
\(220\) 11.0883 0.747573
\(221\) 0 0
\(222\) −4.80172 −0.322270
\(223\) −15.9921 −1.07091 −0.535456 0.844563i \(-0.679860\pi\)
−0.535456 + 0.844563i \(0.679860\pi\)
\(224\) 1.00000 0.0668153
\(225\) 6.66194 0.444129
\(226\) 3.69957 0.246092
\(227\) 4.11815 0.273331 0.136665 0.990617i \(-0.456361\pi\)
0.136665 + 0.990617i \(0.456361\pi\)
\(228\) 1.93665 0.128258
\(229\) 0.403067 0.0266354 0.0133177 0.999911i \(-0.495761\pi\)
0.0133177 + 0.999911i \(0.495761\pi\)
\(230\) 11.2155 0.739527
\(231\) 3.24698 0.213636
\(232\) −3.64112 −0.239051
\(233\) −18.7561 −1.22875 −0.614376 0.789014i \(-0.710592\pi\)
−0.614376 + 0.789014i \(0.710592\pi\)
\(234\) 0 0
\(235\) −36.1869 −2.36058
\(236\) 13.2749 0.864120
\(237\) −15.0033 −0.974572
\(238\) 1.80194 0.116802
\(239\) −10.3497 −0.669466 −0.334733 0.942313i \(-0.608646\pi\)
−0.334733 + 0.942313i \(0.608646\pi\)
\(240\) 3.41496 0.220435
\(241\) −9.33396 −0.601253 −0.300627 0.953742i \(-0.597196\pi\)
−0.300627 + 0.953742i \(0.597196\pi\)
\(242\) −0.457123 −0.0293850
\(243\) 1.00000 0.0641500
\(244\) −1.76668 −0.113100
\(245\) 3.41496 0.218174
\(246\) −4.58189 −0.292131
\(247\) 0 0
\(248\) −3.10698 −0.197293
\(249\) −1.43401 −0.0908766
\(250\) 5.67545 0.358947
\(251\) −30.9745 −1.95509 −0.977547 0.210718i \(-0.932420\pi\)
−0.977547 + 0.210718i \(0.932420\pi\)
\(252\) 1.00000 0.0629941
\(253\) 10.6638 0.670428
\(254\) 3.89724 0.244535
\(255\) 6.15354 0.385350
\(256\) 1.00000 0.0625000
\(257\) −2.27305 −0.141789 −0.0708944 0.997484i \(-0.522585\pi\)
−0.0708944 + 0.997484i \(0.522585\pi\)
\(258\) 5.11205 0.318263
\(259\) −4.80172 −0.298364
\(260\) 0 0
\(261\) −3.64112 −0.225380
\(262\) −17.6740 −1.09190
\(263\) −18.5834 −1.14590 −0.572950 0.819590i \(-0.694201\pi\)
−0.572950 + 0.819590i \(0.694201\pi\)
\(264\) 3.24698 0.199838
\(265\) −8.41477 −0.516915
\(266\) 1.93665 0.118743
\(267\) 5.10615 0.312491
\(268\) −12.4092 −0.758013
\(269\) 26.7522 1.63111 0.815555 0.578679i \(-0.196432\pi\)
0.815555 + 0.578679i \(0.196432\pi\)
\(270\) 3.41496 0.207828
\(271\) 5.33182 0.323885 0.161943 0.986800i \(-0.448224\pi\)
0.161943 + 0.986800i \(0.448224\pi\)
\(272\) 1.80194 0.109259
\(273\) 0 0
\(274\) −14.9195 −0.901323
\(275\) 21.6312 1.30441
\(276\) 3.28422 0.197687
\(277\) 25.0045 1.50238 0.751188 0.660088i \(-0.229481\pi\)
0.751188 + 0.660088i \(0.229481\pi\)
\(278\) −3.27513 −0.196429
\(279\) −3.10698 −0.186010
\(280\) 3.41496 0.204083
\(281\) 13.8066 0.823633 0.411817 0.911267i \(-0.364894\pi\)
0.411817 + 0.911267i \(0.364894\pi\)
\(282\) −10.5966 −0.631018
\(283\) −22.0224 −1.30910 −0.654548 0.756021i \(-0.727141\pi\)
−0.654548 + 0.756021i \(0.727141\pi\)
\(284\) −2.66429 −0.158097
\(285\) 6.61356 0.391754
\(286\) 0 0
\(287\) −4.58189 −0.270460
\(288\) 1.00000 0.0589256
\(289\) −13.7530 −0.809001
\(290\) −12.4343 −0.730165
\(291\) −11.9553 −0.700830
\(292\) 0.368396 0.0215587
\(293\) −2.17514 −0.127073 −0.0635364 0.997980i \(-0.520238\pi\)
−0.0635364 + 0.997980i \(0.520238\pi\)
\(294\) 1.00000 0.0583212
\(295\) 45.3331 2.63939
\(296\) −4.80172 −0.279094
\(297\) 3.24698 0.188409
\(298\) 8.92475 0.516997
\(299\) 0 0
\(300\) 6.66194 0.384627
\(301\) 5.11205 0.294654
\(302\) −0.712364 −0.0409920
\(303\) −18.5354 −1.06483
\(304\) 1.93665 0.111074
\(305\) −6.03315 −0.345457
\(306\) 1.80194 0.103010
\(307\) 21.5338 1.22900 0.614500 0.788917i \(-0.289358\pi\)
0.614500 + 0.788917i \(0.289358\pi\)
\(308\) 3.24698 0.185014
\(309\) 18.2577 1.03864
\(310\) −10.6102 −0.602619
\(311\) −10.5043 −0.595644 −0.297822 0.954621i \(-0.596260\pi\)
−0.297822 + 0.954621i \(0.596260\pi\)
\(312\) 0 0
\(313\) −16.0754 −0.908635 −0.454317 0.890840i \(-0.650117\pi\)
−0.454317 + 0.890840i \(0.650117\pi\)
\(314\) −14.1436 −0.798172
\(315\) 3.41496 0.192411
\(316\) −15.0033 −0.844004
\(317\) 15.3743 0.863508 0.431754 0.901991i \(-0.357895\pi\)
0.431754 + 0.901991i \(0.357895\pi\)
\(318\) −2.46409 −0.138179
\(319\) −11.8226 −0.661941
\(320\) 3.41496 0.190902
\(321\) −15.0210 −0.838389
\(322\) 3.28422 0.183023
\(323\) 3.48972 0.194173
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 15.6565 0.867136
\(327\) 11.7533 0.649962
\(328\) −4.58189 −0.252992
\(329\) −10.5966 −0.584209
\(330\) 11.0883 0.610391
\(331\) −8.21249 −0.451399 −0.225700 0.974197i \(-0.572467\pi\)
−0.225700 + 0.974197i \(0.572467\pi\)
\(332\) −1.43401 −0.0787014
\(333\) −4.80172 −0.263133
\(334\) 19.6904 1.07741
\(335\) −42.3769 −2.31530
\(336\) 1.00000 0.0545545
\(337\) −18.4567 −1.00540 −0.502701 0.864460i \(-0.667660\pi\)
−0.502701 + 0.864460i \(0.667660\pi\)
\(338\) 0 0
\(339\) 3.69957 0.200933
\(340\) 6.15354 0.333723
\(341\) −10.0883 −0.546312
\(342\) 1.93665 0.104722
\(343\) 1.00000 0.0539949
\(344\) 5.11205 0.275623
\(345\) 11.2155 0.603821
\(346\) 16.3915 0.881214
\(347\) 31.3648 1.68375 0.841876 0.539671i \(-0.181451\pi\)
0.841876 + 0.539671i \(0.181451\pi\)
\(348\) −3.64112 −0.195184
\(349\) −6.96959 −0.373073 −0.186537 0.982448i \(-0.559726\pi\)
−0.186537 + 0.982448i \(0.559726\pi\)
\(350\) 6.66194 0.356096
\(351\) 0 0
\(352\) 3.24698 0.173065
\(353\) 13.0978 0.697127 0.348563 0.937285i \(-0.386670\pi\)
0.348563 + 0.937285i \(0.386670\pi\)
\(354\) 13.2749 0.705551
\(355\) −9.09845 −0.482895
\(356\) 5.10615 0.270625
\(357\) 1.80194 0.0953687
\(358\) −17.0906 −0.903264
\(359\) −8.01400 −0.422963 −0.211481 0.977382i \(-0.567829\pi\)
−0.211481 + 0.977382i \(0.567829\pi\)
\(360\) 3.41496 0.179984
\(361\) −15.2494 −0.802600
\(362\) 21.3647 1.12291
\(363\) −0.457123 −0.0239928
\(364\) 0 0
\(365\) 1.25806 0.0658497
\(366\) −1.76668 −0.0923460
\(367\) −25.7361 −1.34342 −0.671708 0.740816i \(-0.734439\pi\)
−0.671708 + 0.740816i \(0.734439\pi\)
\(368\) 3.28422 0.171202
\(369\) −4.58189 −0.238524
\(370\) −16.3977 −0.852474
\(371\) −2.46409 −0.127929
\(372\) −3.10698 −0.161089
\(373\) −0.973483 −0.0504050 −0.0252025 0.999682i \(-0.508023\pi\)
−0.0252025 + 0.999682i \(0.508023\pi\)
\(374\) 5.85086 0.302541
\(375\) 5.67545 0.293079
\(376\) −10.5966 −0.546478
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −5.27383 −0.270899 −0.135449 0.990784i \(-0.543248\pi\)
−0.135449 + 0.990784i \(0.543248\pi\)
\(380\) 6.61356 0.339269
\(381\) 3.89724 0.199662
\(382\) 16.2973 0.833840
\(383\) 11.1179 0.568100 0.284050 0.958810i \(-0.408322\pi\)
0.284050 + 0.958810i \(0.408322\pi\)
\(384\) 1.00000 0.0510310
\(385\) 11.0883 0.565112
\(386\) 18.7221 0.952931
\(387\) 5.11205 0.259860
\(388\) −11.9553 −0.606936
\(389\) −13.6798 −0.693595 −0.346797 0.937940i \(-0.612731\pi\)
−0.346797 + 0.937940i \(0.612731\pi\)
\(390\) 0 0
\(391\) 5.91797 0.299284
\(392\) 1.00000 0.0505076
\(393\) −17.6740 −0.891534
\(394\) 6.35011 0.319914
\(395\) −51.2358 −2.57795
\(396\) 3.24698 0.163167
\(397\) 16.3683 0.821500 0.410750 0.911748i \(-0.365267\pi\)
0.410750 + 0.911748i \(0.365267\pi\)
\(398\) 11.9820 0.600603
\(399\) 1.93665 0.0969536
\(400\) 6.66194 0.333097
\(401\) −35.9089 −1.79320 −0.896602 0.442837i \(-0.853972\pi\)
−0.896602 + 0.442837i \(0.853972\pi\)
\(402\) −12.4092 −0.618915
\(403\) 0 0
\(404\) −18.5354 −0.922170
\(405\) 3.41496 0.169691
\(406\) −3.64112 −0.180706
\(407\) −15.5911 −0.772822
\(408\) 1.80194 0.0892092
\(409\) 12.4416 0.615197 0.307599 0.951516i \(-0.400475\pi\)
0.307599 + 0.951516i \(0.400475\pi\)
\(410\) −15.6470 −0.772748
\(411\) −14.9195 −0.735927
\(412\) 18.2577 0.899493
\(413\) 13.2749 0.653213
\(414\) 3.28422 0.161411
\(415\) −4.89708 −0.240388
\(416\) 0 0
\(417\) −3.27513 −0.160384
\(418\) 6.28825 0.307568
\(419\) 9.73591 0.475630 0.237815 0.971310i \(-0.423569\pi\)
0.237815 + 0.971310i \(0.423569\pi\)
\(420\) 3.41496 0.166633
\(421\) 24.4352 1.19090 0.595449 0.803393i \(-0.296974\pi\)
0.595449 + 0.803393i \(0.296974\pi\)
\(422\) 9.42870 0.458982
\(423\) −10.5966 −0.515224
\(424\) −2.46409 −0.119667
\(425\) 12.0044 0.582299
\(426\) −2.66429 −0.129085
\(427\) −1.76668 −0.0854957
\(428\) −15.0210 −0.726066
\(429\) 0 0
\(430\) 17.4575 0.841873
\(431\) 38.2356 1.84174 0.920872 0.389864i \(-0.127478\pi\)
0.920872 + 0.389864i \(0.127478\pi\)
\(432\) 1.00000 0.0481125
\(433\) 31.9423 1.53505 0.767524 0.641021i \(-0.221489\pi\)
0.767524 + 0.641021i \(0.221489\pi\)
\(434\) −3.10698 −0.149140
\(435\) −12.4343 −0.596177
\(436\) 11.7533 0.562883
\(437\) 6.36038 0.304258
\(438\) 0.368396 0.0176026
\(439\) 20.6203 0.984152 0.492076 0.870552i \(-0.336238\pi\)
0.492076 + 0.870552i \(0.336238\pi\)
\(440\) 11.0883 0.528614
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −0.760370 −0.0361263 −0.0180631 0.999837i \(-0.505750\pi\)
−0.0180631 + 0.999837i \(0.505750\pi\)
\(444\) −4.80172 −0.227880
\(445\) 17.4373 0.826606
\(446\) −15.9921 −0.757250
\(447\) 8.92475 0.422126
\(448\) 1.00000 0.0472456
\(449\) 19.8729 0.937859 0.468929 0.883236i \(-0.344640\pi\)
0.468929 + 0.883236i \(0.344640\pi\)
\(450\) 6.66194 0.314047
\(451\) −14.8773 −0.700545
\(452\) 3.69957 0.174013
\(453\) −0.712364 −0.0334698
\(454\) 4.11815 0.193274
\(455\) 0 0
\(456\) 1.93665 0.0906918
\(457\) 39.0448 1.82644 0.913220 0.407466i \(-0.133587\pi\)
0.913220 + 0.407466i \(0.133587\pi\)
\(458\) 0.403067 0.0188341
\(459\) 1.80194 0.0841073
\(460\) 11.2155 0.522925
\(461\) −25.0785 −1.16802 −0.584011 0.811746i \(-0.698517\pi\)
−0.584011 + 0.811746i \(0.698517\pi\)
\(462\) 3.24698 0.151063
\(463\) 11.3140 0.525807 0.262904 0.964822i \(-0.415320\pi\)
0.262904 + 0.964822i \(0.415320\pi\)
\(464\) −3.64112 −0.169035
\(465\) −10.6102 −0.492036
\(466\) −18.7561 −0.868858
\(467\) 12.6861 0.587043 0.293521 0.955952i \(-0.405173\pi\)
0.293521 + 0.955952i \(0.405173\pi\)
\(468\) 0 0
\(469\) −12.4092 −0.573004
\(470\) −36.1869 −1.66918
\(471\) −14.1436 −0.651705
\(472\) 13.2749 0.611025
\(473\) 16.5987 0.763211
\(474\) −15.0033 −0.689127
\(475\) 12.9018 0.591976
\(476\) 1.80194 0.0825917
\(477\) −2.46409 −0.112823
\(478\) −10.3497 −0.473384
\(479\) 5.31845 0.243006 0.121503 0.992591i \(-0.461229\pi\)
0.121503 + 0.992591i \(0.461229\pi\)
\(480\) 3.41496 0.155871
\(481\) 0 0
\(482\) −9.33396 −0.425150
\(483\) 3.28422 0.149437
\(484\) −0.457123 −0.0207783
\(485\) −40.8267 −1.85385
\(486\) 1.00000 0.0453609
\(487\) −31.2192 −1.41468 −0.707338 0.706875i \(-0.750104\pi\)
−0.707338 + 0.706875i \(0.750104\pi\)
\(488\) −1.76668 −0.0799739
\(489\) 15.6565 0.708014
\(490\) 3.41496 0.154272
\(491\) 24.9801 1.12734 0.563668 0.826001i \(-0.309390\pi\)
0.563668 + 0.826001i \(0.309390\pi\)
\(492\) −4.58189 −0.206567
\(493\) −6.56107 −0.295496
\(494\) 0 0
\(495\) 11.0883 0.498382
\(496\) −3.10698 −0.139507
\(497\) −2.66429 −0.119510
\(498\) −1.43401 −0.0642595
\(499\) 8.22457 0.368182 0.184091 0.982909i \(-0.441066\pi\)
0.184091 + 0.982909i \(0.441066\pi\)
\(500\) 5.67545 0.253814
\(501\) 19.6904 0.879703
\(502\) −30.9745 −1.38246
\(503\) −28.4248 −1.26740 −0.633700 0.773579i \(-0.718465\pi\)
−0.633700 + 0.773579i \(0.718465\pi\)
\(504\) 1.00000 0.0445435
\(505\) −63.2975 −2.81670
\(506\) 10.6638 0.474064
\(507\) 0 0
\(508\) 3.89724 0.172912
\(509\) 8.58953 0.380724 0.190362 0.981714i \(-0.439034\pi\)
0.190362 + 0.981714i \(0.439034\pi\)
\(510\) 6.15354 0.272483
\(511\) 0.368396 0.0162969
\(512\) 1.00000 0.0441942
\(513\) 1.93665 0.0855050
\(514\) −2.27305 −0.100260
\(515\) 62.3493 2.74744
\(516\) 5.11205 0.225046
\(517\) −34.4069 −1.51322
\(518\) −4.80172 −0.210975
\(519\) 16.3915 0.719508
\(520\) 0 0
\(521\) 24.0161 1.05216 0.526082 0.850434i \(-0.323660\pi\)
0.526082 + 0.850434i \(0.323660\pi\)
\(522\) −3.64112 −0.159367
\(523\) −28.7110 −1.25544 −0.627721 0.778438i \(-0.716012\pi\)
−0.627721 + 0.778438i \(0.716012\pi\)
\(524\) −17.6740 −0.772091
\(525\) 6.66194 0.290751
\(526\) −18.5834 −0.810274
\(527\) −5.59858 −0.243878
\(528\) 3.24698 0.141307
\(529\) −12.2139 −0.531038
\(530\) −8.41477 −0.365514
\(531\) 13.2749 0.576080
\(532\) 1.93665 0.0839642
\(533\) 0 0
\(534\) 5.10615 0.220965
\(535\) −51.2960 −2.21772
\(536\) −12.4092 −0.535996
\(537\) −17.0906 −0.737512
\(538\) 26.7522 1.15337
\(539\) 3.24698 0.139857
\(540\) 3.41496 0.146956
\(541\) −13.3317 −0.573173 −0.286587 0.958054i \(-0.592521\pi\)
−0.286587 + 0.958054i \(0.592521\pi\)
\(542\) 5.33182 0.229021
\(543\) 21.3647 0.916849
\(544\) 1.80194 0.0772574
\(545\) 40.1372 1.71929
\(546\) 0 0
\(547\) 9.82136 0.419931 0.209965 0.977709i \(-0.432665\pi\)
0.209965 + 0.977709i \(0.432665\pi\)
\(548\) −14.9195 −0.637331
\(549\) −1.76668 −0.0754002
\(550\) 21.6312 0.922356
\(551\) −7.05156 −0.300406
\(552\) 3.28422 0.139786
\(553\) −15.0033 −0.638007
\(554\) 25.0045 1.06234
\(555\) −16.3977 −0.696042
\(556\) −3.27513 −0.138897
\(557\) 25.6623 1.08735 0.543674 0.839297i \(-0.317033\pi\)
0.543674 + 0.839297i \(0.317033\pi\)
\(558\) −3.10698 −0.131529
\(559\) 0 0
\(560\) 3.41496 0.144308
\(561\) 5.85086 0.247023
\(562\) 13.8066 0.582396
\(563\) −39.5263 −1.66584 −0.832918 0.553396i \(-0.813331\pi\)
−0.832918 + 0.553396i \(0.813331\pi\)
\(564\) −10.5966 −0.446197
\(565\) 12.6339 0.531511
\(566\) −22.0224 −0.925670
\(567\) 1.00000 0.0419961
\(568\) −2.66429 −0.111791
\(569\) 4.74188 0.198790 0.0993949 0.995048i \(-0.468309\pi\)
0.0993949 + 0.995048i \(0.468309\pi\)
\(570\) 6.61356 0.277012
\(571\) −7.02759 −0.294095 −0.147048 0.989129i \(-0.546977\pi\)
−0.147048 + 0.989129i \(0.546977\pi\)
\(572\) 0 0
\(573\) 16.2973 0.680828
\(574\) −4.58189 −0.191244
\(575\) 21.8793 0.912429
\(576\) 1.00000 0.0416667
\(577\) 10.9656 0.456503 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(578\) −13.7530 −0.572050
\(579\) 18.7221 0.778065
\(580\) −12.4343 −0.516305
\(581\) −1.43401 −0.0594927
\(582\) −11.9553 −0.495562
\(583\) −8.00086 −0.331362
\(584\) 0.368396 0.0152443
\(585\) 0 0
\(586\) −2.17514 −0.0898541
\(587\) −28.8685 −1.19153 −0.595766 0.803158i \(-0.703151\pi\)
−0.595766 + 0.803158i \(0.703151\pi\)
\(588\) 1.00000 0.0412393
\(589\) −6.01712 −0.247931
\(590\) 45.3331 1.86633
\(591\) 6.35011 0.261208
\(592\) −4.80172 −0.197350
\(593\) 17.5637 0.721255 0.360628 0.932710i \(-0.382563\pi\)
0.360628 + 0.932710i \(0.382563\pi\)
\(594\) 3.24698 0.133225
\(595\) 6.15354 0.252271
\(596\) 8.92475 0.365572
\(597\) 11.9820 0.490390
\(598\) 0 0
\(599\) 9.82575 0.401469 0.200735 0.979646i \(-0.435667\pi\)
0.200735 + 0.979646i \(0.435667\pi\)
\(600\) 6.66194 0.271972
\(601\) 22.3197 0.910438 0.455219 0.890380i \(-0.349561\pi\)
0.455219 + 0.890380i \(0.349561\pi\)
\(602\) 5.11205 0.208352
\(603\) −12.4092 −0.505342
\(604\) −0.712364 −0.0289857
\(605\) −1.56106 −0.0634660
\(606\) −18.5354 −0.752948
\(607\) 29.0228 1.17800 0.588999 0.808134i \(-0.299522\pi\)
0.588999 + 0.808134i \(0.299522\pi\)
\(608\) 1.93665 0.0785414
\(609\) −3.64112 −0.147546
\(610\) −6.03315 −0.244275
\(611\) 0 0
\(612\) 1.80194 0.0728390
\(613\) −36.4576 −1.47251 −0.736255 0.676704i \(-0.763408\pi\)
−0.736255 + 0.676704i \(0.763408\pi\)
\(614\) 21.5338 0.869035
\(615\) −15.6470 −0.630946
\(616\) 3.24698 0.130825
\(617\) −33.4147 −1.34522 −0.672612 0.739995i \(-0.734828\pi\)
−0.672612 + 0.739995i \(0.734828\pi\)
\(618\) 18.2577 0.734433
\(619\) −12.8283 −0.515614 −0.257807 0.966196i \(-0.583000\pi\)
−0.257807 + 0.966196i \(0.583000\pi\)
\(620\) −10.6102 −0.426116
\(621\) 3.28422 0.131791
\(622\) −10.5043 −0.421184
\(623\) 5.10615 0.204574
\(624\) 0 0
\(625\) −13.9283 −0.557131
\(626\) −16.0754 −0.642502
\(627\) 6.28825 0.251129
\(628\) −14.1436 −0.564393
\(629\) −8.65240 −0.344994
\(630\) 3.41496 0.136055
\(631\) 28.1757 1.12166 0.560828 0.827933i \(-0.310483\pi\)
0.560828 + 0.827933i \(0.310483\pi\)
\(632\) −15.0033 −0.596801
\(633\) 9.42870 0.374757
\(634\) 15.3743 0.610592
\(635\) 13.3089 0.528148
\(636\) −2.46409 −0.0977076
\(637\) 0 0
\(638\) −11.8226 −0.468063
\(639\) −2.66429 −0.105398
\(640\) 3.41496 0.134988
\(641\) 48.0176 1.89658 0.948291 0.317404i \(-0.102811\pi\)
0.948291 + 0.317404i \(0.102811\pi\)
\(642\) −15.0210 −0.592830
\(643\) 8.71604 0.343727 0.171864 0.985121i \(-0.445021\pi\)
0.171864 + 0.985121i \(0.445021\pi\)
\(644\) 3.28422 0.129417
\(645\) 17.4575 0.687386
\(646\) 3.48972 0.137301
\(647\) 27.8665 1.09555 0.547773 0.836627i \(-0.315476\pi\)
0.547773 + 0.836627i \(0.315476\pi\)
\(648\) 1.00000 0.0392837
\(649\) 43.1032 1.69195
\(650\) 0 0
\(651\) −3.10698 −0.121772
\(652\) 15.6565 0.613158
\(653\) −12.1077 −0.473810 −0.236905 0.971533i \(-0.576133\pi\)
−0.236905 + 0.971533i \(0.576133\pi\)
\(654\) 11.7533 0.459592
\(655\) −60.3559 −2.35830
\(656\) −4.58189 −0.178893
\(657\) 0.368396 0.0143725
\(658\) −10.5966 −0.413098
\(659\) −23.4234 −0.912448 −0.456224 0.889865i \(-0.650798\pi\)
−0.456224 + 0.889865i \(0.650798\pi\)
\(660\) 11.0883 0.431611
\(661\) 5.85897 0.227888 0.113944 0.993487i \(-0.463652\pi\)
0.113944 + 0.993487i \(0.463652\pi\)
\(662\) −8.21249 −0.319187
\(663\) 0 0
\(664\) −1.43401 −0.0556503
\(665\) 6.61356 0.256463
\(666\) −4.80172 −0.186063
\(667\) −11.9582 −0.463025
\(668\) 19.6904 0.761845
\(669\) −15.9921 −0.618292
\(670\) −42.3769 −1.63716
\(671\) −5.73638 −0.221451
\(672\) 1.00000 0.0385758
\(673\) −12.6299 −0.486845 −0.243423 0.969920i \(-0.578270\pi\)
−0.243423 + 0.969920i \(0.578270\pi\)
\(674\) −18.4567 −0.710927
\(675\) 6.66194 0.256418
\(676\) 0 0
\(677\) 14.8183 0.569515 0.284757 0.958600i \(-0.408087\pi\)
0.284757 + 0.958600i \(0.408087\pi\)
\(678\) 3.69957 0.142081
\(679\) −11.9553 −0.458801
\(680\) 6.15354 0.235978
\(681\) 4.11815 0.157808
\(682\) −10.0883 −0.386301
\(683\) −26.9967 −1.03300 −0.516499 0.856288i \(-0.672765\pi\)
−0.516499 + 0.856288i \(0.672765\pi\)
\(684\) 1.93665 0.0740495
\(685\) −50.9496 −1.94668
\(686\) 1.00000 0.0381802
\(687\) 0.403067 0.0153780
\(688\) 5.11205 0.194895
\(689\) 0 0
\(690\) 11.2155 0.426966
\(691\) −14.4624 −0.550174 −0.275087 0.961419i \(-0.588707\pi\)
−0.275087 + 0.961419i \(0.588707\pi\)
\(692\) 16.3915 0.623112
\(693\) 3.24698 0.123343
\(694\) 31.3648 1.19059
\(695\) −11.1844 −0.424250
\(696\) −3.64112 −0.138016
\(697\) −8.25628 −0.312729
\(698\) −6.96959 −0.263803
\(699\) −18.7561 −0.709420
\(700\) 6.66194 0.251798
\(701\) 31.2297 1.17953 0.589764 0.807575i \(-0.299221\pi\)
0.589764 + 0.807575i \(0.299221\pi\)
\(702\) 0 0
\(703\) −9.29923 −0.350727
\(704\) 3.24698 0.122375
\(705\) −36.1869 −1.36288
\(706\) 13.0978 0.492943
\(707\) −18.5354 −0.697095
\(708\) 13.2749 0.498900
\(709\) 23.8268 0.894834 0.447417 0.894325i \(-0.352344\pi\)
0.447417 + 0.894325i \(0.352344\pi\)
\(710\) −9.09845 −0.341459
\(711\) −15.0033 −0.562670
\(712\) 5.10615 0.191361
\(713\) −10.2040 −0.382143
\(714\) 1.80194 0.0674358
\(715\) 0 0
\(716\) −17.0906 −0.638704
\(717\) −10.3497 −0.386516
\(718\) −8.01400 −0.299080
\(719\) 37.7147 1.40652 0.703261 0.710931i \(-0.251726\pi\)
0.703261 + 0.710931i \(0.251726\pi\)
\(720\) 3.41496 0.127268
\(721\) 18.2577 0.679952
\(722\) −15.2494 −0.567524
\(723\) −9.33396 −0.347134
\(724\) 21.3647 0.794015
\(725\) −24.2569 −0.900879
\(726\) −0.457123 −0.0169654
\(727\) 40.5041 1.50222 0.751108 0.660180i \(-0.229520\pi\)
0.751108 + 0.660180i \(0.229520\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.25806 0.0465627
\(731\) 9.21160 0.340703
\(732\) −1.76668 −0.0652985
\(733\) 46.1126 1.70321 0.851604 0.524185i \(-0.175630\pi\)
0.851604 + 0.524185i \(0.175630\pi\)
\(734\) −25.7361 −0.949938
\(735\) 3.41496 0.125963
\(736\) 3.28422 0.121058
\(737\) −40.2924 −1.48419
\(738\) −4.58189 −0.168662
\(739\) −12.9995 −0.478195 −0.239097 0.970996i \(-0.576852\pi\)
−0.239097 + 0.970996i \(0.576852\pi\)
\(740\) −16.3977 −0.602790
\(741\) 0 0
\(742\) −2.46409 −0.0904597
\(743\) 31.1171 1.14158 0.570788 0.821098i \(-0.306638\pi\)
0.570788 + 0.821098i \(0.306638\pi\)
\(744\) −3.10698 −0.113907
\(745\) 30.4777 1.11662
\(746\) −0.973483 −0.0356417
\(747\) −1.43401 −0.0524676
\(748\) 5.85086 0.213928
\(749\) −15.0210 −0.548854
\(750\) 5.67545 0.207238
\(751\) −49.5868 −1.80945 −0.904725 0.425997i \(-0.859924\pi\)
−0.904725 + 0.425997i \(0.859924\pi\)
\(752\) −10.5966 −0.386418
\(753\) −30.9745 −1.12877
\(754\) 0 0
\(755\) −2.43269 −0.0885348
\(756\) 1.00000 0.0363696
\(757\) −24.1697 −0.878462 −0.439231 0.898374i \(-0.644749\pi\)
−0.439231 + 0.898374i \(0.644749\pi\)
\(758\) −5.27383 −0.191554
\(759\) 10.6638 0.387072
\(760\) 6.61356 0.239899
\(761\) 52.4508 1.90134 0.950670 0.310204i \(-0.100397\pi\)
0.950670 + 0.310204i \(0.100397\pi\)
\(762\) 3.89724 0.141182
\(763\) 11.7533 0.425500
\(764\) 16.2973 0.589614
\(765\) 6.15354 0.222482
\(766\) 11.1179 0.401707
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 21.4549 0.773685 0.386842 0.922146i \(-0.373566\pi\)
0.386842 + 0.922146i \(0.373566\pi\)
\(770\) 11.0883 0.399595
\(771\) −2.27305 −0.0818618
\(772\) 18.7221 0.673824
\(773\) −19.1744 −0.689654 −0.344827 0.938666i \(-0.612062\pi\)
−0.344827 + 0.938666i \(0.612062\pi\)
\(774\) 5.11205 0.183749
\(775\) −20.6985 −0.743512
\(776\) −11.9553 −0.429169
\(777\) −4.80172 −0.172261
\(778\) −13.6798 −0.490446
\(779\) −8.87349 −0.317926
\(780\) 0 0
\(781\) −8.65090 −0.309554
\(782\) 5.91797 0.211626
\(783\) −3.64112 −0.130123
\(784\) 1.00000 0.0357143
\(785\) −48.2999 −1.72390
\(786\) −17.6740 −0.630410
\(787\) 0.0274054 0.000976897 0 0.000488449 1.00000i \(-0.499845\pi\)
0.000488449 1.00000i \(0.499845\pi\)
\(788\) 6.35011 0.226213
\(789\) −18.5834 −0.661586
\(790\) −51.2358 −1.82289
\(791\) 3.69957 0.131542
\(792\) 3.24698 0.115376
\(793\) 0 0
\(794\) 16.3683 0.580888
\(795\) −8.41477 −0.298441
\(796\) 11.9820 0.424690
\(797\) 6.95916 0.246506 0.123253 0.992375i \(-0.460667\pi\)
0.123253 + 0.992375i \(0.460667\pi\)
\(798\) 1.93665 0.0685565
\(799\) −19.0944 −0.675512
\(800\) 6.66194 0.235535
\(801\) 5.10615 0.180417
\(802\) −35.9089 −1.26799
\(803\) 1.19617 0.0422120
\(804\) −12.4092 −0.437639
\(805\) 11.2155 0.395294
\(806\) 0 0
\(807\) 26.7522 0.941722
\(808\) −18.5354 −0.652072
\(809\) −35.2083 −1.23786 −0.618930 0.785446i \(-0.712433\pi\)
−0.618930 + 0.785446i \(0.712433\pi\)
\(810\) 3.41496 0.119989
\(811\) 25.6464 0.900567 0.450284 0.892886i \(-0.351323\pi\)
0.450284 + 0.892886i \(0.351323\pi\)
\(812\) −3.64112 −0.127778
\(813\) 5.33182 0.186995
\(814\) −15.5911 −0.546467
\(815\) 53.4665 1.87285
\(816\) 1.80194 0.0630804
\(817\) 9.90024 0.346366
\(818\) 12.4416 0.435010
\(819\) 0 0
\(820\) −15.6470 −0.546415
\(821\) 31.8661 1.11214 0.556068 0.831137i \(-0.312309\pi\)
0.556068 + 0.831137i \(0.312309\pi\)
\(822\) −14.9195 −0.520379
\(823\) −48.6456 −1.69568 −0.847839 0.530254i \(-0.822097\pi\)
−0.847839 + 0.530254i \(0.822097\pi\)
\(824\) 18.2577 0.636037
\(825\) 21.6312 0.753101
\(826\) 13.2749 0.461891
\(827\) −3.18286 −0.110679 −0.0553394 0.998468i \(-0.517624\pi\)
−0.0553394 + 0.998468i \(0.517624\pi\)
\(828\) 3.28422 0.114135
\(829\) −0.573876 −0.0199315 −0.00996576 0.999950i \(-0.503172\pi\)
−0.00996576 + 0.999950i \(0.503172\pi\)
\(830\) −4.89708 −0.169980
\(831\) 25.0045 0.867398
\(832\) 0 0
\(833\) 1.80194 0.0624334
\(834\) −3.27513 −0.113409
\(835\) 67.2419 2.32700
\(836\) 6.28825 0.217484
\(837\) −3.10698 −0.107393
\(838\) 9.73591 0.336321
\(839\) −30.7253 −1.06076 −0.530378 0.847762i \(-0.677950\pi\)
−0.530378 + 0.847762i \(0.677950\pi\)
\(840\) 3.41496 0.117827
\(841\) −15.7423 −0.542836
\(842\) 24.4352 0.842092
\(843\) 13.8066 0.475525
\(844\) 9.42870 0.324549
\(845\) 0 0
\(846\) −10.5966 −0.364319
\(847\) −0.457123 −0.0157069
\(848\) −2.46409 −0.0846173
\(849\) −22.0224 −0.755806
\(850\) 12.0044 0.411747
\(851\) −15.7699 −0.540586
\(852\) −2.66429 −0.0912772
\(853\) −23.5695 −0.807005 −0.403502 0.914979i \(-0.632207\pi\)
−0.403502 + 0.914979i \(0.632207\pi\)
\(854\) −1.76668 −0.0604546
\(855\) 6.61356 0.226179
\(856\) −15.0210 −0.513406
\(857\) −18.7309 −0.639834 −0.319917 0.947446i \(-0.603655\pi\)
−0.319917 + 0.947446i \(0.603655\pi\)
\(858\) 0 0
\(859\) −35.0464 −1.19577 −0.597884 0.801582i \(-0.703992\pi\)
−0.597884 + 0.801582i \(0.703992\pi\)
\(860\) 17.4575 0.595294
\(861\) −4.58189 −0.156150
\(862\) 38.2356 1.30231
\(863\) 35.2615 1.20032 0.600158 0.799881i \(-0.295104\pi\)
0.600158 + 0.799881i \(0.295104\pi\)
\(864\) 1.00000 0.0340207
\(865\) 55.9764 1.90325
\(866\) 31.9423 1.08544
\(867\) −13.7530 −0.467077
\(868\) −3.10698 −0.105458
\(869\) −48.7156 −1.65256
\(870\) −12.4343 −0.421561
\(871\) 0 0
\(872\) 11.7533 0.398019
\(873\) −11.9553 −0.404624
\(874\) 6.36038 0.215143
\(875\) 5.67545 0.191865
\(876\) 0.368396 0.0124469
\(877\) 31.7171 1.07101 0.535506 0.844532i \(-0.320121\pi\)
0.535506 + 0.844532i \(0.320121\pi\)
\(878\) 20.6203 0.695900
\(879\) −2.17514 −0.0733656
\(880\) 11.0883 0.373786
\(881\) −16.3131 −0.549602 −0.274801 0.961501i \(-0.588612\pi\)
−0.274801 + 0.961501i \(0.588612\pi\)
\(882\) 1.00000 0.0336718
\(883\) −20.4148 −0.687014 −0.343507 0.939150i \(-0.611615\pi\)
−0.343507 + 0.939150i \(0.611615\pi\)
\(884\) 0 0
\(885\) 45.3331 1.52386
\(886\) −0.760370 −0.0255451
\(887\) −8.78093 −0.294835 −0.147417 0.989074i \(-0.547096\pi\)
−0.147417 + 0.989074i \(0.547096\pi\)
\(888\) −4.80172 −0.161135
\(889\) 3.89724 0.130709
\(890\) 17.4373 0.584499
\(891\) 3.24698 0.108778
\(892\) −15.9921 −0.535456
\(893\) −20.5219 −0.686738
\(894\) 8.92475 0.298488
\(895\) −58.3635 −1.95088
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 19.8729 0.663166
\(899\) 11.3129 0.377306
\(900\) 6.66194 0.222065
\(901\) −4.44014 −0.147923
\(902\) −14.8773 −0.495360
\(903\) 5.11205 0.170119
\(904\) 3.69957 0.123046
\(905\) 72.9597 2.42526
\(906\) −0.712364 −0.0236667
\(907\) 12.4859 0.414588 0.207294 0.978279i \(-0.433534\pi\)
0.207294 + 0.978279i \(0.433534\pi\)
\(908\) 4.11815 0.136665
\(909\) −18.5354 −0.614780
\(910\) 0 0
\(911\) −15.1753 −0.502780 −0.251390 0.967886i \(-0.580888\pi\)
−0.251390 + 0.967886i \(0.580888\pi\)
\(912\) 1.93665 0.0641288
\(913\) −4.65620 −0.154098
\(914\) 39.0448 1.29149
\(915\) −6.03315 −0.199450
\(916\) 0.403067 0.0133177
\(917\) −17.6740 −0.583646
\(918\) 1.80194 0.0594728
\(919\) −0.144447 −0.00476487 −0.00238243 0.999997i \(-0.500758\pi\)
−0.00238243 + 0.999997i \(0.500758\pi\)
\(920\) 11.2155 0.369764
\(921\) 21.5338 0.709564
\(922\) −25.0785 −0.825916
\(923\) 0 0
\(924\) 3.24698 0.106818
\(925\) −31.9888 −1.05178
\(926\) 11.3140 0.371802
\(927\) 18.2577 0.599662
\(928\) −3.64112 −0.119526
\(929\) 40.2284 1.31985 0.659926 0.751331i \(-0.270588\pi\)
0.659926 + 0.751331i \(0.270588\pi\)
\(930\) −10.6102 −0.347922
\(931\) 1.93665 0.0634710
\(932\) −18.7561 −0.614376
\(933\) −10.5043 −0.343895
\(934\) 12.6861 0.415102
\(935\) 19.9804 0.653430
\(936\) 0 0
\(937\) 29.4246 0.961261 0.480630 0.876923i \(-0.340408\pi\)
0.480630 + 0.876923i \(0.340408\pi\)
\(938\) −12.4092 −0.405175
\(939\) −16.0754 −0.524600
\(940\) −36.1869 −1.18029
\(941\) −23.6938 −0.772395 −0.386197 0.922416i \(-0.626212\pi\)
−0.386197 + 0.922416i \(0.626212\pi\)
\(942\) −14.1436 −0.460825
\(943\) −15.0479 −0.490028
\(944\) 13.2749 0.432060
\(945\) 3.41496 0.111089
\(946\) 16.5987 0.539671
\(947\) 16.5847 0.538932 0.269466 0.963010i \(-0.413153\pi\)
0.269466 + 0.963010i \(0.413153\pi\)
\(948\) −15.0033 −0.487286
\(949\) 0 0
\(950\) 12.9018 0.418590
\(951\) 15.3743 0.498546
\(952\) 1.80194 0.0584011
\(953\) 17.5290 0.567820 0.283910 0.958851i \(-0.408368\pi\)
0.283910 + 0.958851i \(0.408368\pi\)
\(954\) −2.46409 −0.0797779
\(955\) 55.6545 1.80094
\(956\) −10.3497 −0.334733
\(957\) −11.8226 −0.382172
\(958\) 5.31845 0.171831
\(959\) −14.9195 −0.481777
\(960\) 3.41496 0.110217
\(961\) −21.3467 −0.688603
\(962\) 0 0
\(963\) −15.0210 −0.484044
\(964\) −9.33396 −0.300627
\(965\) 63.9353 2.05815
\(966\) 3.28422 0.105668
\(967\) −44.7198 −1.43809 −0.719045 0.694963i \(-0.755421\pi\)
−0.719045 + 0.694963i \(0.755421\pi\)
\(968\) −0.457123 −0.0146925
\(969\) 3.48972 0.112106
\(970\) −40.8267 −1.31087
\(971\) −31.6941 −1.01711 −0.508555 0.861029i \(-0.669820\pi\)
−0.508555 + 0.861029i \(0.669820\pi\)
\(972\) 1.00000 0.0320750
\(973\) −3.27513 −0.104996
\(974\) −31.2192 −1.00033
\(975\) 0 0
\(976\) −1.76668 −0.0565501
\(977\) 43.1638 1.38093 0.690467 0.723364i \(-0.257405\pi\)
0.690467 + 0.723364i \(0.257405\pi\)
\(978\) 15.6565 0.500641
\(979\) 16.5796 0.529885
\(980\) 3.41496 0.109087
\(981\) 11.7533 0.375255
\(982\) 24.9801 0.797147
\(983\) −58.3576 −1.86132 −0.930659 0.365887i \(-0.880766\pi\)
−0.930659 + 0.365887i \(0.880766\pi\)
\(984\) −4.58189 −0.146065
\(985\) 21.6853 0.690953
\(986\) −6.56107 −0.208947
\(987\) −10.5966 −0.337293
\(988\) 0 0
\(989\) 16.7891 0.533863
\(990\) 11.0883 0.352409
\(991\) −38.6339 −1.22725 −0.613623 0.789599i \(-0.710288\pi\)
−0.613623 + 0.789599i \(0.710288\pi\)
\(992\) −3.10698 −0.0986467
\(993\) −8.21249 −0.260615
\(994\) −2.66429 −0.0845062
\(995\) 40.9180 1.29719
\(996\) −1.43401 −0.0454383
\(997\) 13.4648 0.426433 0.213217 0.977005i \(-0.431606\pi\)
0.213217 + 0.977005i \(0.431606\pi\)
\(998\) 8.22457 0.260344
\(999\) −4.80172 −0.151920
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 7098.2.a.ct.1.6 yes 6
13.12 even 2 7098.2.a.cr.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cr.1.1 6 13.12 even 2
7098.2.a.ct.1.6 yes 6 1.1 even 1 trivial