Properties

Label 8470.2.a.dc.1.3
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.19898000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.935683\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.935683 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.935683 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.12450 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.935683 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.935683 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.12450 q^{9} +1.00000 q^{10} -0.935683 q^{12} +2.51397 q^{13} -1.00000 q^{14} -0.935683 q^{15} +1.00000 q^{16} -4.68602 q^{17} -2.12450 q^{18} -4.05554 q^{19} +1.00000 q^{20} +0.935683 q^{21} -3.46945 q^{23} -0.935683 q^{24} +1.00000 q^{25} +2.51397 q^{26} +4.79491 q^{27} -1.00000 q^{28} +7.46463 q^{29} -0.935683 q^{30} +1.54591 q^{31} +1.00000 q^{32} -4.68602 q^{34} -1.00000 q^{35} -2.12450 q^{36} -4.92622 q^{37} -4.05554 q^{38} -2.35228 q^{39} +1.00000 q^{40} -1.65079 q^{41} +0.935683 q^{42} +7.97092 q^{43} -2.12450 q^{45} -3.46945 q^{46} +1.77448 q^{47} -0.935683 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.38463 q^{51} +2.51397 q^{52} +7.14297 q^{53} +4.79491 q^{54} -1.00000 q^{56} +3.79470 q^{57} +7.46463 q^{58} -0.426837 q^{59} -0.935683 q^{60} -6.15201 q^{61} +1.54591 q^{62} +2.12450 q^{63} +1.00000 q^{64} +2.51397 q^{65} -5.99409 q^{67} -4.68602 q^{68} +3.24631 q^{69} -1.00000 q^{70} +6.53922 q^{71} -2.12450 q^{72} +0.855372 q^{73} -4.92622 q^{74} -0.935683 q^{75} -4.05554 q^{76} -2.35228 q^{78} +9.86178 q^{79} +1.00000 q^{80} +1.88697 q^{81} -1.65079 q^{82} +8.15674 q^{83} +0.935683 q^{84} -4.68602 q^{85} +7.97092 q^{86} -6.98453 q^{87} +1.00048 q^{89} -2.12450 q^{90} -2.51397 q^{91} -3.46945 q^{92} -1.44649 q^{93} +1.77448 q^{94} -4.05554 q^{95} -0.935683 q^{96} +11.3338 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} + 6q^{8} + 3q^{9} + 6q^{10} - q^{12} + 9q^{13} - 6q^{14} - q^{15} + 6q^{16} + 9q^{17} + 3q^{18} + 12q^{19} + 6q^{20} + q^{21} + 4q^{23} - q^{24} + 6q^{25} + 9q^{26} - 4q^{27} - 6q^{28} + 15q^{29} - q^{30} + 8q^{31} + 6q^{32} + 9q^{34} - 6q^{35} + 3q^{36} - 4q^{37} + 12q^{38} - 19q^{39} + 6q^{40} + 4q^{41} + q^{42} + 30q^{43} + 3q^{45} + 4q^{46} - 7q^{47} - q^{48} + 6q^{49} + 6q^{50} + 16q^{51} + 9q^{52} - 6q^{53} - 4q^{54} - 6q^{56} - 14q^{57} + 15q^{58} + 4q^{59} - q^{60} - 14q^{61} + 8q^{62} - 3q^{63} + 6q^{64} + 9q^{65} + 18q^{67} + 9q^{68} - 10q^{69} - 6q^{70} + 23q^{71} + 3q^{72} + 23q^{73} - 4q^{74} - q^{75} + 12q^{76} - 19q^{78} + 21q^{79} + 6q^{80} - 18q^{81} + 4q^{82} + 25q^{83} + q^{84} + 9q^{85} + 30q^{86} + 14q^{87} - 18q^{89} + 3q^{90} - 9q^{91} + 4q^{92} - 24q^{93} - 7q^{94} + 12q^{95} - q^{96} + 7q^{97} + 6q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.935683 −0.540217 −0.270109 0.962830i \(-0.587060\pi\)
−0.270109 + 0.962830i \(0.587060\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.935683 −0.381991
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.12450 −0.708165
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −0.935683 −0.270109
\(13\) 2.51397 0.697249 0.348625 0.937262i \(-0.386649\pi\)
0.348625 + 0.937262i \(0.386649\pi\)
\(14\) −1.00000 −0.267261
\(15\) −0.935683 −0.241592
\(16\) 1.00000 0.250000
\(17\) −4.68602 −1.13653 −0.568263 0.822847i \(-0.692384\pi\)
−0.568263 + 0.822847i \(0.692384\pi\)
\(18\) −2.12450 −0.500749
\(19\) −4.05554 −0.930405 −0.465203 0.885204i \(-0.654018\pi\)
−0.465203 + 0.885204i \(0.654018\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.935683 0.204183
\(22\) 0 0
\(23\) −3.46945 −0.723431 −0.361716 0.932289i \(-0.617809\pi\)
−0.361716 + 0.932289i \(0.617809\pi\)
\(24\) −0.935683 −0.190996
\(25\) 1.00000 0.200000
\(26\) 2.51397 0.493030
\(27\) 4.79491 0.922780
\(28\) −1.00000 −0.188982
\(29\) 7.46463 1.38615 0.693074 0.720867i \(-0.256256\pi\)
0.693074 + 0.720867i \(0.256256\pi\)
\(30\) −0.935683 −0.170832
\(31\) 1.54591 0.277654 0.138827 0.990317i \(-0.455667\pi\)
0.138827 + 0.990317i \(0.455667\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.68602 −0.803645
\(35\) −1.00000 −0.169031
\(36\) −2.12450 −0.354083
\(37\) −4.92622 −0.809866 −0.404933 0.914346i \(-0.632705\pi\)
−0.404933 + 0.914346i \(0.632705\pi\)
\(38\) −4.05554 −0.657896
\(39\) −2.35228 −0.376666
\(40\) 1.00000 0.158114
\(41\) −1.65079 −0.257810 −0.128905 0.991657i \(-0.541146\pi\)
−0.128905 + 0.991657i \(0.541146\pi\)
\(42\) 0.935683 0.144379
\(43\) 7.97092 1.21555 0.607777 0.794108i \(-0.292061\pi\)
0.607777 + 0.794108i \(0.292061\pi\)
\(44\) 0 0
\(45\) −2.12450 −0.316701
\(46\) −3.46945 −0.511543
\(47\) 1.77448 0.258834 0.129417 0.991590i \(-0.458689\pi\)
0.129417 + 0.991590i \(0.458689\pi\)
\(48\) −0.935683 −0.135054
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 4.38463 0.613971
\(52\) 2.51397 0.348625
\(53\) 7.14297 0.981162 0.490581 0.871395i \(-0.336784\pi\)
0.490581 + 0.871395i \(0.336784\pi\)
\(54\) 4.79491 0.652504
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 3.79470 0.502621
\(58\) 7.46463 0.980154
\(59\) −0.426837 −0.0555694 −0.0277847 0.999614i \(-0.508845\pi\)
−0.0277847 + 0.999614i \(0.508845\pi\)
\(60\) −0.935683 −0.120796
\(61\) −6.15201 −0.787684 −0.393842 0.919178i \(-0.628854\pi\)
−0.393842 + 0.919178i \(0.628854\pi\)
\(62\) 1.54591 0.196331
\(63\) 2.12450 0.267661
\(64\) 1.00000 0.125000
\(65\) 2.51397 0.311819
\(66\) 0 0
\(67\) −5.99409 −0.732295 −0.366147 0.930557i \(-0.619323\pi\)
−0.366147 + 0.930557i \(0.619323\pi\)
\(68\) −4.68602 −0.568263
\(69\) 3.24631 0.390810
\(70\) −1.00000 −0.119523
\(71\) 6.53922 0.776063 0.388031 0.921646i \(-0.373155\pi\)
0.388031 + 0.921646i \(0.373155\pi\)
\(72\) −2.12450 −0.250374
\(73\) 0.855372 0.100114 0.0500568 0.998746i \(-0.484060\pi\)
0.0500568 + 0.998746i \(0.484060\pi\)
\(74\) −4.92622 −0.572662
\(75\) −0.935683 −0.108043
\(76\) −4.05554 −0.465203
\(77\) 0 0
\(78\) −2.35228 −0.266343
\(79\) 9.86178 1.10954 0.554768 0.832005i \(-0.312807\pi\)
0.554768 + 0.832005i \(0.312807\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.88697 0.209664
\(82\) −1.65079 −0.182300
\(83\) 8.15674 0.895319 0.447659 0.894204i \(-0.352258\pi\)
0.447659 + 0.894204i \(0.352258\pi\)
\(84\) 0.935683 0.102091
\(85\) −4.68602 −0.508270
\(86\) 7.97092 0.859526
\(87\) −6.98453 −0.748820
\(88\) 0 0
\(89\) 1.00048 0.106051 0.0530255 0.998593i \(-0.483114\pi\)
0.0530255 + 0.998593i \(0.483114\pi\)
\(90\) −2.12450 −0.223942
\(91\) −2.51397 −0.263535
\(92\) −3.46945 −0.361716
\(93\) −1.44649 −0.149994
\(94\) 1.77448 0.183023
\(95\) −4.05554 −0.416090
\(96\) −0.935683 −0.0954978
\(97\) 11.3338 1.15078 0.575388 0.817880i \(-0.304851\pi\)
0.575388 + 0.817880i \(0.304851\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −3.16660 −0.315088 −0.157544 0.987512i \(-0.550358\pi\)
−0.157544 + 0.987512i \(0.550358\pi\)
\(102\) 4.38463 0.434143
\(103\) 15.0475 1.48267 0.741335 0.671135i \(-0.234193\pi\)
0.741335 + 0.671135i \(0.234193\pi\)
\(104\) 2.51397 0.246515
\(105\) 0.935683 0.0913134
\(106\) 7.14297 0.693787
\(107\) 14.4377 1.39574 0.697872 0.716222i \(-0.254130\pi\)
0.697872 + 0.716222i \(0.254130\pi\)
\(108\) 4.79491 0.461390
\(109\) 12.1869 1.16729 0.583646 0.812008i \(-0.301626\pi\)
0.583646 + 0.812008i \(0.301626\pi\)
\(110\) 0 0
\(111\) 4.60938 0.437503
\(112\) −1.00000 −0.0944911
\(113\) −11.7294 −1.10341 −0.551706 0.834039i \(-0.686023\pi\)
−0.551706 + 0.834039i \(0.686023\pi\)
\(114\) 3.79470 0.355407
\(115\) −3.46945 −0.323528
\(116\) 7.46463 0.693074
\(117\) −5.34092 −0.493768
\(118\) −0.426837 −0.0392935
\(119\) 4.68602 0.429567
\(120\) −0.935683 −0.0854158
\(121\) 0 0
\(122\) −6.15201 −0.556977
\(123\) 1.54462 0.139274
\(124\) 1.54591 0.138827
\(125\) 1.00000 0.0894427
\(126\) 2.12450 0.189265
\(127\) −3.68316 −0.326828 −0.163414 0.986558i \(-0.552251\pi\)
−0.163414 + 0.986558i \(0.552251\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.45826 −0.656663
\(130\) 2.51397 0.220490
\(131\) −8.57781 −0.749447 −0.374724 0.927137i \(-0.622262\pi\)
−0.374724 + 0.927137i \(0.622262\pi\)
\(132\) 0 0
\(133\) 4.05554 0.351660
\(134\) −5.99409 −0.517811
\(135\) 4.79491 0.412680
\(136\) −4.68602 −0.401823
\(137\) 1.63086 0.139334 0.0696671 0.997570i \(-0.477806\pi\)
0.0696671 + 0.997570i \(0.477806\pi\)
\(138\) 3.24631 0.276344
\(139\) 23.1799 1.96609 0.983046 0.183360i \(-0.0586974\pi\)
0.983046 + 0.183360i \(0.0586974\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −1.66035 −0.139827
\(142\) 6.53922 0.548759
\(143\) 0 0
\(144\) −2.12450 −0.177041
\(145\) 7.46463 0.619904
\(146\) 0.855372 0.0707911
\(147\) −0.935683 −0.0771739
\(148\) −4.92622 −0.404933
\(149\) 1.43386 0.117466 0.0587332 0.998274i \(-0.481294\pi\)
0.0587332 + 0.998274i \(0.481294\pi\)
\(150\) −0.935683 −0.0763982
\(151\) −17.7952 −1.44815 −0.724075 0.689722i \(-0.757733\pi\)
−0.724075 + 0.689722i \(0.757733\pi\)
\(152\) −4.05554 −0.328948
\(153\) 9.95543 0.804849
\(154\) 0 0
\(155\) 1.54591 0.124171
\(156\) −2.35228 −0.188333
\(157\) −7.29144 −0.581920 −0.290960 0.956735i \(-0.593975\pi\)
−0.290960 + 0.956735i \(0.593975\pi\)
\(158\) 9.86178 0.784561
\(159\) −6.68356 −0.530041
\(160\) 1.00000 0.0790569
\(161\) 3.46945 0.273431
\(162\) 1.88697 0.148255
\(163\) 9.41304 0.737286 0.368643 0.929571i \(-0.379822\pi\)
0.368643 + 0.929571i \(0.379822\pi\)
\(164\) −1.65079 −0.128905
\(165\) 0 0
\(166\) 8.15674 0.633086
\(167\) 0.289918 0.0224345 0.0112173 0.999937i \(-0.496429\pi\)
0.0112173 + 0.999937i \(0.496429\pi\)
\(168\) 0.935683 0.0721895
\(169\) −6.67997 −0.513844
\(170\) −4.68602 −0.359401
\(171\) 8.61598 0.658881
\(172\) 7.97092 0.607777
\(173\) 10.4188 0.792126 0.396063 0.918223i \(-0.370376\pi\)
0.396063 + 0.918223i \(0.370376\pi\)
\(174\) −6.98453 −0.529496
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0.399384 0.0300196
\(178\) 1.00048 0.0749893
\(179\) −14.9115 −1.11454 −0.557270 0.830331i \(-0.688151\pi\)
−0.557270 + 0.830331i \(0.688151\pi\)
\(180\) −2.12450 −0.158351
\(181\) 4.37330 0.325065 0.162533 0.986703i \(-0.448034\pi\)
0.162533 + 0.986703i \(0.448034\pi\)
\(182\) −2.51397 −0.186348
\(183\) 5.75633 0.425520
\(184\) −3.46945 −0.255772
\(185\) −4.92622 −0.362183
\(186\) −1.44649 −0.106062
\(187\) 0 0
\(188\) 1.77448 0.129417
\(189\) −4.79491 −0.348778
\(190\) −4.05554 −0.294220
\(191\) −10.6197 −0.768416 −0.384208 0.923246i \(-0.625526\pi\)
−0.384208 + 0.923246i \(0.625526\pi\)
\(192\) −0.935683 −0.0675271
\(193\) −7.98145 −0.574517 −0.287259 0.957853i \(-0.592744\pi\)
−0.287259 + 0.957853i \(0.592744\pi\)
\(194\) 11.3338 0.813722
\(195\) −2.35228 −0.168450
\(196\) 1.00000 0.0714286
\(197\) 7.28621 0.519121 0.259560 0.965727i \(-0.416422\pi\)
0.259560 + 0.965727i \(0.416422\pi\)
\(198\) 0 0
\(199\) 14.1853 1.00557 0.502783 0.864413i \(-0.332309\pi\)
0.502783 + 0.864413i \(0.332309\pi\)
\(200\) 1.00000 0.0707107
\(201\) 5.60857 0.395598
\(202\) −3.16660 −0.222801
\(203\) −7.46463 −0.523914
\(204\) 4.38463 0.306985
\(205\) −1.65079 −0.115296
\(206\) 15.0475 1.04841
\(207\) 7.37084 0.512309
\(208\) 2.51397 0.174312
\(209\) 0 0
\(210\) 0.935683 0.0645683
\(211\) 23.2770 1.60246 0.801229 0.598358i \(-0.204180\pi\)
0.801229 + 0.598358i \(0.204180\pi\)
\(212\) 7.14297 0.490581
\(213\) −6.11864 −0.419242
\(214\) 14.4377 0.986940
\(215\) 7.97092 0.543612
\(216\) 4.79491 0.326252
\(217\) −1.54591 −0.104943
\(218\) 12.1869 0.825400
\(219\) −0.800357 −0.0540831
\(220\) 0 0
\(221\) −11.7805 −0.792442
\(222\) 4.60938 0.309362
\(223\) −8.09748 −0.542248 −0.271124 0.962544i \(-0.587395\pi\)
−0.271124 + 0.962544i \(0.587395\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.12450 −0.141633
\(226\) −11.7294 −0.780230
\(227\) 18.8354 1.25015 0.625074 0.780565i \(-0.285069\pi\)
0.625074 + 0.780565i \(0.285069\pi\)
\(228\) 3.79470 0.251310
\(229\) 10.8721 0.718451 0.359225 0.933251i \(-0.383041\pi\)
0.359225 + 0.933251i \(0.383041\pi\)
\(230\) −3.46945 −0.228769
\(231\) 0 0
\(232\) 7.46463 0.490077
\(233\) 8.97639 0.588063 0.294031 0.955796i \(-0.405003\pi\)
0.294031 + 0.955796i \(0.405003\pi\)
\(234\) −5.34092 −0.349147
\(235\) 1.77448 0.115754
\(236\) −0.426837 −0.0277847
\(237\) −9.22750 −0.599391
\(238\) 4.68602 0.303749
\(239\) 9.78125 0.632696 0.316348 0.948643i \(-0.397543\pi\)
0.316348 + 0.948643i \(0.397543\pi\)
\(240\) −0.935683 −0.0603981
\(241\) −23.3855 −1.50639 −0.753195 0.657797i \(-0.771488\pi\)
−0.753195 + 0.657797i \(0.771488\pi\)
\(242\) 0 0
\(243\) −16.1503 −1.03604
\(244\) −6.15201 −0.393842
\(245\) 1.00000 0.0638877
\(246\) 1.54462 0.0984813
\(247\) −10.1955 −0.648724
\(248\) 1.54591 0.0981656
\(249\) −7.63213 −0.483666
\(250\) 1.00000 0.0632456
\(251\) 26.0939 1.64703 0.823517 0.567291i \(-0.192009\pi\)
0.823517 + 0.567291i \(0.192009\pi\)
\(252\) 2.12450 0.133831
\(253\) 0 0
\(254\) −3.68316 −0.231102
\(255\) 4.38463 0.274576
\(256\) 1.00000 0.0625000
\(257\) 5.51935 0.344287 0.172144 0.985072i \(-0.444931\pi\)
0.172144 + 0.985072i \(0.444931\pi\)
\(258\) −7.45826 −0.464331
\(259\) 4.92622 0.306101
\(260\) 2.51397 0.155910
\(261\) −15.8586 −0.981622
\(262\) −8.57781 −0.529939
\(263\) 27.1823 1.67613 0.838067 0.545568i \(-0.183686\pi\)
0.838067 + 0.545568i \(0.183686\pi\)
\(264\) 0 0
\(265\) 7.14297 0.438789
\(266\) 4.05554 0.248661
\(267\) −0.936135 −0.0572905
\(268\) −5.99409 −0.366147
\(269\) −8.14480 −0.496597 −0.248299 0.968684i \(-0.579871\pi\)
−0.248299 + 0.968684i \(0.579871\pi\)
\(270\) 4.79491 0.291809
\(271\) 6.30590 0.383056 0.191528 0.981487i \(-0.438656\pi\)
0.191528 + 0.981487i \(0.438656\pi\)
\(272\) −4.68602 −0.284132
\(273\) 2.35228 0.142366
\(274\) 1.63086 0.0985241
\(275\) 0 0
\(276\) 3.24631 0.195405
\(277\) 26.1396 1.57057 0.785287 0.619132i \(-0.212515\pi\)
0.785287 + 0.619132i \(0.212515\pi\)
\(278\) 23.1799 1.39024
\(279\) −3.28429 −0.196625
\(280\) −1.00000 −0.0597614
\(281\) −14.7356 −0.879055 −0.439527 0.898229i \(-0.644854\pi\)
−0.439527 + 0.898229i \(0.644854\pi\)
\(282\) −1.66035 −0.0988723
\(283\) −21.5060 −1.27840 −0.639199 0.769041i \(-0.720734\pi\)
−0.639199 + 0.769041i \(0.720734\pi\)
\(284\) 6.53922 0.388031
\(285\) 3.79470 0.224779
\(286\) 0 0
\(287\) 1.65079 0.0974432
\(288\) −2.12450 −0.125187
\(289\) 4.95876 0.291692
\(290\) 7.46463 0.438338
\(291\) −10.6049 −0.621669
\(292\) 0.855372 0.0500568
\(293\) 6.11604 0.357303 0.178651 0.983912i \(-0.442827\pi\)
0.178651 + 0.983912i \(0.442827\pi\)
\(294\) −0.935683 −0.0545702
\(295\) −0.426837 −0.0248514
\(296\) −4.92622 −0.286331
\(297\) 0 0
\(298\) 1.43386 0.0830613
\(299\) −8.72209 −0.504412
\(300\) −0.935683 −0.0540217
\(301\) −7.97092 −0.459436
\(302\) −17.7952 −1.02400
\(303\) 2.96293 0.170216
\(304\) −4.05554 −0.232601
\(305\) −6.15201 −0.352263
\(306\) 9.95543 0.569114
\(307\) −24.8721 −1.41952 −0.709762 0.704441i \(-0.751198\pi\)
−0.709762 + 0.704441i \(0.751198\pi\)
\(308\) 0 0
\(309\) −14.0797 −0.800963
\(310\) 1.54591 0.0878020
\(311\) 3.37946 0.191632 0.0958158 0.995399i \(-0.469454\pi\)
0.0958158 + 0.995399i \(0.469454\pi\)
\(312\) −2.35228 −0.133172
\(313\) −20.7401 −1.17230 −0.586150 0.810203i \(-0.699357\pi\)
−0.586150 + 0.810203i \(0.699357\pi\)
\(314\) −7.29144 −0.411480
\(315\) 2.12450 0.119702
\(316\) 9.86178 0.554768
\(317\) 7.46200 0.419108 0.209554 0.977797i \(-0.432799\pi\)
0.209554 + 0.977797i \(0.432799\pi\)
\(318\) −6.68356 −0.374795
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −13.5091 −0.754005
\(322\) 3.46945 0.193345
\(323\) 19.0043 1.05743
\(324\) 1.88697 0.104832
\(325\) 2.51397 0.139450
\(326\) 9.41304 0.521340
\(327\) −11.4031 −0.630591
\(328\) −1.65079 −0.0911498
\(329\) −1.77448 −0.0978301
\(330\) 0 0
\(331\) −8.26702 −0.454396 −0.227198 0.973849i \(-0.572956\pi\)
−0.227198 + 0.973849i \(0.572956\pi\)
\(332\) 8.15674 0.447659
\(333\) 10.4657 0.573519
\(334\) 0.289918 0.0158636
\(335\) −5.99409 −0.327492
\(336\) 0.935683 0.0510457
\(337\) −0.309107 −0.0168381 −0.00841906 0.999965i \(-0.502680\pi\)
−0.00841906 + 0.999965i \(0.502680\pi\)
\(338\) −6.67997 −0.363342
\(339\) 10.9750 0.596082
\(340\) −4.68602 −0.254135
\(341\) 0 0
\(342\) 8.61598 0.465899
\(343\) −1.00000 −0.0539949
\(344\) 7.97092 0.429763
\(345\) 3.24631 0.174775
\(346\) 10.4188 0.560118
\(347\) 6.12313 0.328707 0.164353 0.986402i \(-0.447446\pi\)
0.164353 + 0.986402i \(0.447446\pi\)
\(348\) −6.98453 −0.374410
\(349\) −19.7838 −1.05900 −0.529501 0.848309i \(-0.677621\pi\)
−0.529501 + 0.848309i \(0.677621\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 12.0542 0.643408
\(352\) 0 0
\(353\) −30.8256 −1.64068 −0.820340 0.571876i \(-0.806216\pi\)
−0.820340 + 0.571876i \(0.806216\pi\)
\(354\) 0.399384 0.0212270
\(355\) 6.53922 0.347066
\(356\) 1.00048 0.0530255
\(357\) −4.38463 −0.232059
\(358\) −14.9115 −0.788099
\(359\) −25.2431 −1.33228 −0.666140 0.745827i \(-0.732055\pi\)
−0.666140 + 0.745827i \(0.732055\pi\)
\(360\) −2.12450 −0.111971
\(361\) −2.55258 −0.134347
\(362\) 4.37330 0.229856
\(363\) 0 0
\(364\) −2.51397 −0.131768
\(365\) 0.855372 0.0447722
\(366\) 5.75633 0.300888
\(367\) −8.87025 −0.463023 −0.231512 0.972832i \(-0.574367\pi\)
−0.231512 + 0.972832i \(0.574367\pi\)
\(368\) −3.46945 −0.180858
\(369\) 3.50710 0.182572
\(370\) −4.92622 −0.256102
\(371\) −7.14297 −0.370845
\(372\) −1.44649 −0.0749968
\(373\) 23.0860 1.19535 0.597675 0.801739i \(-0.296091\pi\)
0.597675 + 0.801739i \(0.296091\pi\)
\(374\) 0 0
\(375\) −0.935683 −0.0483185
\(376\) 1.77448 0.0915116
\(377\) 18.7658 0.966490
\(378\) −4.79491 −0.246623
\(379\) 31.6315 1.62480 0.812399 0.583101i \(-0.198161\pi\)
0.812399 + 0.583101i \(0.198161\pi\)
\(380\) −4.05554 −0.208045
\(381\) 3.44627 0.176558
\(382\) −10.6197 −0.543352
\(383\) 1.84985 0.0945228 0.0472614 0.998883i \(-0.484951\pi\)
0.0472614 + 0.998883i \(0.484951\pi\)
\(384\) −0.935683 −0.0477489
\(385\) 0 0
\(386\) −7.98145 −0.406245
\(387\) −16.9342 −0.860813
\(388\) 11.3338 0.575388
\(389\) 15.6624 0.794116 0.397058 0.917793i \(-0.370031\pi\)
0.397058 + 0.917793i \(0.370031\pi\)
\(390\) −2.35228 −0.119112
\(391\) 16.2579 0.822198
\(392\) 1.00000 0.0505076
\(393\) 8.02612 0.404864
\(394\) 7.28621 0.367074
\(395\) 9.86178 0.496200
\(396\) 0 0
\(397\) −14.8207 −0.743827 −0.371914 0.928267i \(-0.621298\pi\)
−0.371914 + 0.928267i \(0.621298\pi\)
\(398\) 14.1853 0.711043
\(399\) −3.79470 −0.189973
\(400\) 1.00000 0.0500000
\(401\) 14.4322 0.720710 0.360355 0.932815i \(-0.382656\pi\)
0.360355 + 0.932815i \(0.382656\pi\)
\(402\) 5.60857 0.279730
\(403\) 3.88638 0.193594
\(404\) −3.16660 −0.157544
\(405\) 1.88697 0.0937645
\(406\) −7.46463 −0.370463
\(407\) 0 0
\(408\) 4.38463 0.217071
\(409\) 38.0339 1.88066 0.940328 0.340270i \(-0.110518\pi\)
0.940328 + 0.340270i \(0.110518\pi\)
\(410\) −1.65079 −0.0815268
\(411\) −1.52597 −0.0752707
\(412\) 15.0475 0.741335
\(413\) 0.426837 0.0210033
\(414\) 7.37084 0.362257
\(415\) 8.15674 0.400399
\(416\) 2.51397 0.123257
\(417\) −21.6890 −1.06212
\(418\) 0 0
\(419\) 24.2138 1.18292 0.591460 0.806334i \(-0.298552\pi\)
0.591460 + 0.806334i \(0.298552\pi\)
\(420\) 0.935683 0.0456567
\(421\) 38.6678 1.88455 0.942276 0.334837i \(-0.108681\pi\)
0.942276 + 0.334837i \(0.108681\pi\)
\(422\) 23.2770 1.13311
\(423\) −3.76987 −0.183297
\(424\) 7.14297 0.346893
\(425\) −4.68602 −0.227305
\(426\) −6.11864 −0.296449
\(427\) 6.15201 0.297717
\(428\) 14.4377 0.697872
\(429\) 0 0
\(430\) 7.97092 0.384392
\(431\) 23.6029 1.13691 0.568455 0.822714i \(-0.307541\pi\)
0.568455 + 0.822714i \(0.307541\pi\)
\(432\) 4.79491 0.230695
\(433\) −30.6532 −1.47310 −0.736550 0.676384i \(-0.763546\pi\)
−0.736550 + 0.676384i \(0.763546\pi\)
\(434\) −1.54591 −0.0742062
\(435\) −6.98453 −0.334883
\(436\) 12.1869 0.583646
\(437\) 14.0705 0.673084
\(438\) −0.800357 −0.0382426
\(439\) −23.4152 −1.11755 −0.558774 0.829320i \(-0.688728\pi\)
−0.558774 + 0.829320i \(0.688728\pi\)
\(440\) 0 0
\(441\) −2.12450 −0.101166
\(442\) −11.7805 −0.560341
\(443\) 0.833237 0.0395883 0.0197941 0.999804i \(-0.493699\pi\)
0.0197941 + 0.999804i \(0.493699\pi\)
\(444\) 4.60938 0.218752
\(445\) 1.00048 0.0474274
\(446\) −8.09748 −0.383427
\(447\) −1.34164 −0.0634573
\(448\) −1.00000 −0.0472456
\(449\) −5.98030 −0.282228 −0.141114 0.989993i \(-0.545068\pi\)
−0.141114 + 0.989993i \(0.545068\pi\)
\(450\) −2.12450 −0.100150
\(451\) 0 0
\(452\) −11.7294 −0.551706
\(453\) 16.6506 0.782315
\(454\) 18.8354 0.883989
\(455\) −2.51397 −0.117857
\(456\) 3.79470 0.177703
\(457\) 41.9762 1.96356 0.981782 0.190010i \(-0.0608519\pi\)
0.981782 + 0.190010i \(0.0608519\pi\)
\(458\) 10.8721 0.508021
\(459\) −22.4690 −1.04876
\(460\) −3.46945 −0.161764
\(461\) 32.1474 1.49725 0.748626 0.662992i \(-0.230714\pi\)
0.748626 + 0.662992i \(0.230714\pi\)
\(462\) 0 0
\(463\) −8.14625 −0.378588 −0.189294 0.981920i \(-0.560620\pi\)
−0.189294 + 0.981920i \(0.560620\pi\)
\(464\) 7.46463 0.346537
\(465\) −1.44649 −0.0670792
\(466\) 8.97639 0.415823
\(467\) 7.67994 0.355385 0.177693 0.984086i \(-0.443137\pi\)
0.177693 + 0.984086i \(0.443137\pi\)
\(468\) −5.34092 −0.246884
\(469\) 5.99409 0.276781
\(470\) 1.77448 0.0818505
\(471\) 6.82248 0.314363
\(472\) −0.426837 −0.0196468
\(473\) 0 0
\(474\) −9.22750 −0.423833
\(475\) −4.05554 −0.186081
\(476\) 4.68602 0.214783
\(477\) −15.1752 −0.694825
\(478\) 9.78125 0.447384
\(479\) 22.7225 1.03822 0.519109 0.854708i \(-0.326264\pi\)
0.519109 + 0.854708i \(0.326264\pi\)
\(480\) −0.935683 −0.0427079
\(481\) −12.3844 −0.564678
\(482\) −23.3855 −1.06518
\(483\) −3.24631 −0.147712
\(484\) 0 0
\(485\) 11.3338 0.514643
\(486\) −16.1503 −0.732594
\(487\) 33.4458 1.51557 0.757786 0.652503i \(-0.226281\pi\)
0.757786 + 0.652503i \(0.226281\pi\)
\(488\) −6.15201 −0.278488
\(489\) −8.80763 −0.398295
\(490\) 1.00000 0.0451754
\(491\) −6.10991 −0.275736 −0.137868 0.990451i \(-0.544025\pi\)
−0.137868 + 0.990451i \(0.544025\pi\)
\(492\) 1.54462 0.0696368
\(493\) −34.9794 −1.57539
\(494\) −10.1955 −0.458717
\(495\) 0 0
\(496\) 1.54591 0.0694136
\(497\) −6.53922 −0.293324
\(498\) −7.63213 −0.342004
\(499\) 21.1193 0.945430 0.472715 0.881215i \(-0.343274\pi\)
0.472715 + 0.881215i \(0.343274\pi\)
\(500\) 1.00000 0.0447214
\(501\) −0.271271 −0.0121195
\(502\) 26.0939 1.16463
\(503\) 25.5325 1.13844 0.569219 0.822186i \(-0.307246\pi\)
0.569219 + 0.822186i \(0.307246\pi\)
\(504\) 2.12450 0.0946326
\(505\) −3.16660 −0.140912
\(506\) 0 0
\(507\) 6.25033 0.277587
\(508\) −3.68316 −0.163414
\(509\) −22.0002 −0.975144 −0.487572 0.873083i \(-0.662117\pi\)
−0.487572 + 0.873083i \(0.662117\pi\)
\(510\) 4.38463 0.194155
\(511\) −0.855372 −0.0378394
\(512\) 1.00000 0.0441942
\(513\) −19.4459 −0.858559
\(514\) 5.51935 0.243448
\(515\) 15.0475 0.663070
\(516\) −7.45826 −0.328332
\(517\) 0 0
\(518\) 4.92622 0.216446
\(519\) −9.74870 −0.427920
\(520\) 2.51397 0.110245
\(521\) −27.5451 −1.20677 −0.603387 0.797449i \(-0.706182\pi\)
−0.603387 + 0.797449i \(0.706182\pi\)
\(522\) −15.8586 −0.694111
\(523\) −2.58863 −0.113193 −0.0565963 0.998397i \(-0.518025\pi\)
−0.0565963 + 0.998397i \(0.518025\pi\)
\(524\) −8.57781 −0.374724
\(525\) 0.935683 0.0408366
\(526\) 27.1823 1.18521
\(527\) −7.24418 −0.315561
\(528\) 0 0
\(529\) −10.9629 −0.476647
\(530\) 7.14297 0.310271
\(531\) 0.906814 0.0393524
\(532\) 4.05554 0.175830
\(533\) −4.15004 −0.179758
\(534\) −0.936135 −0.0405105
\(535\) 14.4377 0.624196
\(536\) −5.99409 −0.258905
\(537\) 13.9525 0.602094
\(538\) −8.14480 −0.351147
\(539\) 0 0
\(540\) 4.79491 0.206340
\(541\) −16.6493 −0.715809 −0.357905 0.933758i \(-0.616509\pi\)
−0.357905 + 0.933758i \(0.616509\pi\)
\(542\) 6.30590 0.270861
\(543\) −4.09203 −0.175606
\(544\) −4.68602 −0.200911
\(545\) 12.1869 0.522029
\(546\) 2.35228 0.100668
\(547\) 32.2258 1.37788 0.688938 0.724821i \(-0.258077\pi\)
0.688938 + 0.724821i \(0.258077\pi\)
\(548\) 1.63086 0.0696671
\(549\) 13.0699 0.557811
\(550\) 0 0
\(551\) −30.2731 −1.28968
\(552\) 3.24631 0.138172
\(553\) −9.86178 −0.419366
\(554\) 26.1396 1.11056
\(555\) 4.60938 0.195657
\(556\) 23.1799 0.983046
\(557\) 43.4129 1.83946 0.919732 0.392548i \(-0.128406\pi\)
0.919732 + 0.392548i \(0.128406\pi\)
\(558\) −3.28429 −0.139035
\(559\) 20.0386 0.847544
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −14.7356 −0.621586
\(563\) 12.0403 0.507437 0.253718 0.967278i \(-0.418346\pi\)
0.253718 + 0.967278i \(0.418346\pi\)
\(564\) −1.66035 −0.0699133
\(565\) −11.7294 −0.493461
\(566\) −21.5060 −0.903964
\(567\) −1.88697 −0.0792455
\(568\) 6.53922 0.274380
\(569\) 26.3405 1.10425 0.552126 0.833761i \(-0.313817\pi\)
0.552126 + 0.833761i \(0.313817\pi\)
\(570\) 3.79470 0.158943
\(571\) −22.2032 −0.929176 −0.464588 0.885527i \(-0.653797\pi\)
−0.464588 + 0.885527i \(0.653797\pi\)
\(572\) 0 0
\(573\) 9.93670 0.415112
\(574\) 1.65079 0.0689027
\(575\) −3.46945 −0.144686
\(576\) −2.12450 −0.0885207
\(577\) 33.9425 1.41304 0.706522 0.707691i \(-0.250263\pi\)
0.706522 + 0.707691i \(0.250263\pi\)
\(578\) 4.95876 0.206257
\(579\) 7.46811 0.310364
\(580\) 7.46463 0.309952
\(581\) −8.15674 −0.338399
\(582\) −10.6049 −0.439587
\(583\) 0 0
\(584\) 0.855372 0.0353955
\(585\) −5.34092 −0.220820
\(586\) 6.11604 0.252651
\(587\) −10.6550 −0.439779 −0.219889 0.975525i \(-0.570570\pi\)
−0.219889 + 0.975525i \(0.570570\pi\)
\(588\) −0.935683 −0.0385869
\(589\) −6.26952 −0.258331
\(590\) −0.426837 −0.0175726
\(591\) −6.81758 −0.280438
\(592\) −4.92622 −0.202466
\(593\) −44.1567 −1.81330 −0.906649 0.421886i \(-0.861368\pi\)
−0.906649 + 0.421886i \(0.861368\pi\)
\(594\) 0 0
\(595\) 4.68602 0.192108
\(596\) 1.43386 0.0587332
\(597\) −13.2729 −0.543224
\(598\) −8.72209 −0.356673
\(599\) 25.7621 1.05261 0.526306 0.850295i \(-0.323577\pi\)
0.526306 + 0.850295i \(0.323577\pi\)
\(600\) −0.935683 −0.0381991
\(601\) −48.8171 −1.99129 −0.995645 0.0932309i \(-0.970281\pi\)
−0.995645 + 0.0932309i \(0.970281\pi\)
\(602\) −7.97092 −0.324870
\(603\) 12.7344 0.518586
\(604\) −17.7952 −0.724075
\(605\) 0 0
\(606\) 2.96293 0.120361
\(607\) 23.4113 0.950237 0.475118 0.879922i \(-0.342405\pi\)
0.475118 + 0.879922i \(0.342405\pi\)
\(608\) −4.05554 −0.164474
\(609\) 6.98453 0.283028
\(610\) −6.15201 −0.249088
\(611\) 4.46098 0.180472
\(612\) 9.95543 0.402424
\(613\) 34.3936 1.38914 0.694572 0.719424i \(-0.255594\pi\)
0.694572 + 0.719424i \(0.255594\pi\)
\(614\) −24.8721 −1.00376
\(615\) 1.54462 0.0622851
\(616\) 0 0
\(617\) −33.9149 −1.36536 −0.682681 0.730717i \(-0.739186\pi\)
−0.682681 + 0.730717i \(0.739186\pi\)
\(618\) −14.0797 −0.566367
\(619\) −46.5283 −1.87013 −0.935066 0.354474i \(-0.884660\pi\)
−0.935066 + 0.354474i \(0.884660\pi\)
\(620\) 1.54591 0.0620854
\(621\) −16.6357 −0.667568
\(622\) 3.37946 0.135504
\(623\) −1.00048 −0.0400835
\(624\) −2.35228 −0.0941665
\(625\) 1.00000 0.0400000
\(626\) −20.7401 −0.828941
\(627\) 0 0
\(628\) −7.29144 −0.290960
\(629\) 23.0844 0.920434
\(630\) 2.12450 0.0846420
\(631\) −49.1136 −1.95518 −0.977591 0.210514i \(-0.932486\pi\)
−0.977591 + 0.210514i \(0.932486\pi\)
\(632\) 9.86178 0.392281
\(633\) −21.7799 −0.865675
\(634\) 7.46200 0.296354
\(635\) −3.68316 −0.146162
\(636\) −6.68356 −0.265020
\(637\) 2.51397 0.0996070
\(638\) 0 0
\(639\) −13.8926 −0.549581
\(640\) 1.00000 0.0395285
\(641\) −10.5819 −0.417959 −0.208979 0.977920i \(-0.567014\pi\)
−0.208979 + 0.977920i \(0.567014\pi\)
\(642\) −13.5091 −0.533162
\(643\) −27.2101 −1.07306 −0.536531 0.843881i \(-0.680266\pi\)
−0.536531 + 0.843881i \(0.680266\pi\)
\(644\) 3.46945 0.136716
\(645\) −7.45826 −0.293669
\(646\) 19.0043 0.747716
\(647\) −32.6756 −1.28461 −0.642304 0.766450i \(-0.722021\pi\)
−0.642304 + 0.766450i \(0.722021\pi\)
\(648\) 1.88697 0.0741273
\(649\) 0 0
\(650\) 2.51397 0.0986059
\(651\) 1.44649 0.0566923
\(652\) 9.41304 0.368643
\(653\) −35.8727 −1.40381 −0.701904 0.712272i \(-0.747666\pi\)
−0.701904 + 0.712272i \(0.747666\pi\)
\(654\) −11.4031 −0.445895
\(655\) −8.57781 −0.335163
\(656\) −1.65079 −0.0644526
\(657\) −1.81723 −0.0708971
\(658\) −1.77448 −0.0691763
\(659\) 2.91875 0.113698 0.0568492 0.998383i \(-0.481895\pi\)
0.0568492 + 0.998383i \(0.481895\pi\)
\(660\) 0 0
\(661\) −45.9224 −1.78617 −0.893087 0.449883i \(-0.851466\pi\)
−0.893087 + 0.449883i \(0.851466\pi\)
\(662\) −8.26702 −0.321307
\(663\) 11.0228 0.428091
\(664\) 8.15674 0.316543
\(665\) 4.05554 0.157267
\(666\) 10.4657 0.405539
\(667\) −25.8982 −1.00278
\(668\) 0.289918 0.0112173
\(669\) 7.57668 0.292931
\(670\) −5.99409 −0.231572
\(671\) 0 0
\(672\) 0.935683 0.0360948
\(673\) 17.1172 0.659818 0.329909 0.944013i \(-0.392982\pi\)
0.329909 + 0.944013i \(0.392982\pi\)
\(674\) −0.309107 −0.0119063
\(675\) 4.79491 0.184556
\(676\) −6.67997 −0.256922
\(677\) 0.309569 0.0118977 0.00594886 0.999982i \(-0.498106\pi\)
0.00594886 + 0.999982i \(0.498106\pi\)
\(678\) 10.9750 0.421494
\(679\) −11.3338 −0.434953
\(680\) −4.68602 −0.179701
\(681\) −17.6240 −0.675352
\(682\) 0 0
\(683\) 44.6187 1.70729 0.853644 0.520856i \(-0.174387\pi\)
0.853644 + 0.520856i \(0.174387\pi\)
\(684\) 8.61598 0.329440
\(685\) 1.63086 0.0623121
\(686\) −1.00000 −0.0381802
\(687\) −10.1729 −0.388119
\(688\) 7.97092 0.303888
\(689\) 17.9572 0.684115
\(690\) 3.24631 0.123585
\(691\) 1.84640 0.0702406 0.0351203 0.999383i \(-0.488819\pi\)
0.0351203 + 0.999383i \(0.488819\pi\)
\(692\) 10.4188 0.396063
\(693\) 0 0
\(694\) 6.12313 0.232431
\(695\) 23.1799 0.879263
\(696\) −6.98453 −0.264748
\(697\) 7.73564 0.293008
\(698\) −19.7838 −0.748828
\(699\) −8.39906 −0.317682
\(700\) −1.00000 −0.0377964
\(701\) 13.9594 0.527240 0.263620 0.964627i \(-0.415083\pi\)
0.263620 + 0.964627i \(0.415083\pi\)
\(702\) 12.0542 0.454958
\(703\) 19.9785 0.753503
\(704\) 0 0
\(705\) −1.66035 −0.0625323
\(706\) −30.8256 −1.16014
\(707\) 3.16660 0.119092
\(708\) 0.399384 0.0150098
\(709\) −12.2455 −0.459891 −0.229945 0.973204i \(-0.573855\pi\)
−0.229945 + 0.973204i \(0.573855\pi\)
\(710\) 6.53922 0.245413
\(711\) −20.9513 −0.785736
\(712\) 1.00048 0.0374947
\(713\) −5.36348 −0.200864
\(714\) −4.38463 −0.164091
\(715\) 0 0
\(716\) −14.9115 −0.557270
\(717\) −9.15215 −0.341793
\(718\) −25.2431 −0.942064
\(719\) −11.5631 −0.431230 −0.215615 0.976478i \(-0.569176\pi\)
−0.215615 + 0.976478i \(0.569176\pi\)
\(720\) −2.12450 −0.0791753
\(721\) −15.0475 −0.560396
\(722\) −2.55258 −0.0949973
\(723\) 21.8814 0.813778
\(724\) 4.37330 0.162533
\(725\) 7.46463 0.277229
\(726\) 0 0
\(727\) −11.2457 −0.417080 −0.208540 0.978014i \(-0.566871\pi\)
−0.208540 + 0.978014i \(0.566871\pi\)
\(728\) −2.51397 −0.0931738
\(729\) 9.45068 0.350025
\(730\) 0.855372 0.0316587
\(731\) −37.3519 −1.38151
\(732\) 5.75633 0.212760
\(733\) 13.3226 0.492081 0.246041 0.969260i \(-0.420870\pi\)
0.246041 + 0.969260i \(0.420870\pi\)
\(734\) −8.87025 −0.327407
\(735\) −0.935683 −0.0345132
\(736\) −3.46945 −0.127886
\(737\) 0 0
\(738\) 3.50710 0.129098
\(739\) 42.5012 1.56343 0.781716 0.623635i \(-0.214345\pi\)
0.781716 + 0.623635i \(0.214345\pi\)
\(740\) −4.92622 −0.181092
\(741\) 9.53976 0.350452
\(742\) −7.14297 −0.262227
\(743\) −22.0875 −0.810311 −0.405156 0.914248i \(-0.632783\pi\)
−0.405156 + 0.914248i \(0.632783\pi\)
\(744\) −1.44649 −0.0530308
\(745\) 1.43386 0.0525326
\(746\) 23.0860 0.845240
\(747\) −17.3290 −0.634034
\(748\) 0 0
\(749\) −14.4377 −0.527542
\(750\) −0.935683 −0.0341663
\(751\) 20.2698 0.739657 0.369829 0.929100i \(-0.379416\pi\)
0.369829 + 0.929100i \(0.379416\pi\)
\(752\) 1.77448 0.0647085
\(753\) −24.4157 −0.889756
\(754\) 18.7658 0.683412
\(755\) −17.7952 −0.647632
\(756\) −4.79491 −0.174389
\(757\) −5.51054 −0.200284 −0.100142 0.994973i \(-0.531930\pi\)
−0.100142 + 0.994973i \(0.531930\pi\)
\(758\) 31.6315 1.14891
\(759\) 0 0
\(760\) −4.05554 −0.147110
\(761\) −33.9696 −1.23140 −0.615699 0.787981i \(-0.711126\pi\)
−0.615699 + 0.787981i \(0.711126\pi\)
\(762\) 3.44627 0.124845
\(763\) −12.1869 −0.441195
\(764\) −10.6197 −0.384208
\(765\) 9.95543 0.359939
\(766\) 1.84985 0.0668377
\(767\) −1.07305 −0.0387458
\(768\) −0.935683 −0.0337636
\(769\) −0.669422 −0.0241400 −0.0120700 0.999927i \(-0.503842\pi\)
−0.0120700 + 0.999927i \(0.503842\pi\)
\(770\) 0 0
\(771\) −5.16436 −0.185990
\(772\) −7.98145 −0.287259
\(773\) −32.7311 −1.17725 −0.588627 0.808405i \(-0.700331\pi\)
−0.588627 + 0.808405i \(0.700331\pi\)
\(774\) −16.9342 −0.608687
\(775\) 1.54591 0.0555309
\(776\) 11.3338 0.406861
\(777\) −4.60938 −0.165361
\(778\) 15.6624 0.561525
\(779\) 6.69486 0.239868
\(780\) −2.35228 −0.0842251
\(781\) 0 0
\(782\) 16.2579 0.581382
\(783\) 35.7922 1.27911
\(784\) 1.00000 0.0357143
\(785\) −7.29144 −0.260243
\(786\) 8.02612 0.286282
\(787\) 16.9929 0.605732 0.302866 0.953033i \(-0.402056\pi\)
0.302866 + 0.953033i \(0.402056\pi\)
\(788\) 7.28621 0.259560
\(789\) −25.4340 −0.905476
\(790\) 9.86178 0.350866
\(791\) 11.7294 0.417050
\(792\) 0 0
\(793\) −15.4659 −0.549212
\(794\) −14.8207 −0.525965
\(795\) −6.68356 −0.237041
\(796\) 14.1853 0.502783
\(797\) 46.4576 1.64561 0.822806 0.568322i \(-0.192407\pi\)
0.822806 + 0.568322i \(0.192407\pi\)
\(798\) −3.79470 −0.134331
\(799\) −8.31523 −0.294172
\(800\) 1.00000 0.0353553
\(801\) −2.12552 −0.0751016
\(802\) 14.4322 0.509619
\(803\) 0 0
\(804\) 5.60857 0.197799
\(805\) 3.46945 0.122282
\(806\) 3.88638 0.136892
\(807\) 7.62096 0.268270
\(808\) −3.16660 −0.111401
\(809\) −2.84705 −0.100097 −0.0500484 0.998747i \(-0.515938\pi\)
−0.0500484 + 0.998747i \(0.515938\pi\)
\(810\) 1.88697 0.0663015
\(811\) −1.46624 −0.0514866 −0.0257433 0.999669i \(-0.508195\pi\)
−0.0257433 + 0.999669i \(0.508195\pi\)
\(812\) −7.46463 −0.261957
\(813\) −5.90032 −0.206933
\(814\) 0 0
\(815\) 9.41304 0.329724
\(816\) 4.38463 0.153493
\(817\) −32.3264 −1.13096
\(818\) 38.0339 1.32982
\(819\) 5.34092 0.186627
\(820\) −1.65079 −0.0576482
\(821\) 19.6491 0.685757 0.342879 0.939380i \(-0.388598\pi\)
0.342879 + 0.939380i \(0.388598\pi\)
\(822\) −1.52597 −0.0532244
\(823\) −17.0752 −0.595203 −0.297601 0.954690i \(-0.596187\pi\)
−0.297601 + 0.954690i \(0.596187\pi\)
\(824\) 15.0475 0.524203
\(825\) 0 0
\(826\) 0.426837 0.0148516
\(827\) −15.9823 −0.555759 −0.277880 0.960616i \(-0.589632\pi\)
−0.277880 + 0.960616i \(0.589632\pi\)
\(828\) 7.37084 0.256154
\(829\) 17.6458 0.612863 0.306432 0.951893i \(-0.400865\pi\)
0.306432 + 0.951893i \(0.400865\pi\)
\(830\) 8.15674 0.283125
\(831\) −24.4584 −0.848451
\(832\) 2.51397 0.0871561
\(833\) −4.68602 −0.162361
\(834\) −21.6890 −0.751030
\(835\) 0.289918 0.0100330
\(836\) 0 0
\(837\) 7.41251 0.256214
\(838\) 24.2138 0.836451
\(839\) 7.96223 0.274887 0.137443 0.990510i \(-0.456111\pi\)
0.137443 + 0.990510i \(0.456111\pi\)
\(840\) 0.935683 0.0322841
\(841\) 26.7207 0.921404
\(842\) 38.6678 1.33258
\(843\) 13.7879 0.474880
\(844\) 23.2770 0.801229
\(845\) −6.67997 −0.229798
\(846\) −3.76987 −0.129611
\(847\) 0 0
\(848\) 7.14297 0.245291
\(849\) 20.1228 0.690613
\(850\) −4.68602 −0.160729
\(851\) 17.0913 0.585882
\(852\) −6.11864 −0.209621
\(853\) 39.3256 1.34648 0.673242 0.739422i \(-0.264901\pi\)
0.673242 + 0.739422i \(0.264901\pi\)
\(854\) 6.15201 0.210517
\(855\) 8.61598 0.294660
\(856\) 14.4377 0.493470
\(857\) 42.5288 1.45276 0.726379 0.687295i \(-0.241202\pi\)
0.726379 + 0.687295i \(0.241202\pi\)
\(858\) 0 0
\(859\) 36.8680 1.25792 0.628960 0.777438i \(-0.283481\pi\)
0.628960 + 0.777438i \(0.283481\pi\)
\(860\) 7.97092 0.271806
\(861\) −1.54462 −0.0526405
\(862\) 23.6029 0.803917
\(863\) 25.0975 0.854327 0.427164 0.904174i \(-0.359513\pi\)
0.427164 + 0.904174i \(0.359513\pi\)
\(864\) 4.79491 0.163126
\(865\) 10.4188 0.354250
\(866\) −30.6532 −1.04164
\(867\) −4.63983 −0.157577
\(868\) −1.54591 −0.0524717
\(869\) 0 0
\(870\) −6.98453 −0.236798
\(871\) −15.0690 −0.510592
\(872\) 12.1869 0.412700
\(873\) −24.0787 −0.814940
\(874\) 14.0705 0.475942
\(875\) −1.00000 −0.0338062
\(876\) −0.800357 −0.0270416
\(877\) 54.6433 1.84517 0.922586 0.385792i \(-0.126072\pi\)
0.922586 + 0.385792i \(0.126072\pi\)
\(878\) −23.4152 −0.790226
\(879\) −5.72268 −0.193021
\(880\) 0 0
\(881\) −49.1584 −1.65619 −0.828094 0.560590i \(-0.810575\pi\)
−0.828094 + 0.560590i \(0.810575\pi\)
\(882\) −2.12450 −0.0715355
\(883\) −26.9252 −0.906105 −0.453053 0.891484i \(-0.649665\pi\)
−0.453053 + 0.891484i \(0.649665\pi\)
\(884\) −11.7805 −0.396221
\(885\) 0.399384 0.0134252
\(886\) 0.833237 0.0279931
\(887\) −31.8126 −1.06816 −0.534082 0.845433i \(-0.679342\pi\)
−0.534082 + 0.845433i \(0.679342\pi\)
\(888\) 4.60938 0.154681
\(889\) 3.68316 0.123529
\(890\) 1.00048 0.0335363
\(891\) 0 0
\(892\) −8.09748 −0.271124
\(893\) −7.19646 −0.240820
\(894\) −1.34164 −0.0448711
\(895\) −14.9115 −0.498438
\(896\) −1.00000 −0.0334077
\(897\) 8.16112 0.272492
\(898\) −5.98030 −0.199565
\(899\) 11.5397 0.384870
\(900\) −2.12450 −0.0708165
\(901\) −33.4721 −1.11512
\(902\) 0 0
\(903\) 7.45826 0.248195
\(904\) −11.7294 −0.390115
\(905\) 4.37330 0.145374
\(906\) 16.6506 0.553180
\(907\) 27.8459 0.924609 0.462304 0.886721i \(-0.347023\pi\)
0.462304 + 0.886721i \(0.347023\pi\)
\(908\) 18.8354 0.625074
\(909\) 6.72743 0.223135
\(910\) −2.51397 −0.0833372
\(911\) −54.9974 −1.82214 −0.911072 0.412247i \(-0.864744\pi\)
−0.911072 + 0.412247i \(0.864744\pi\)
\(912\) 3.79470 0.125655
\(913\) 0 0
\(914\) 41.9762 1.38845
\(915\) 5.75633 0.190298
\(916\) 10.8721 0.359225
\(917\) 8.57781 0.283264
\(918\) −22.4690 −0.741588
\(919\) −16.9015 −0.557529 −0.278764 0.960360i \(-0.589925\pi\)
−0.278764 + 0.960360i \(0.589925\pi\)
\(920\) −3.46945 −0.114384
\(921\) 23.2724 0.766851
\(922\) 32.1474 1.05872
\(923\) 16.4394 0.541109
\(924\) 0 0
\(925\) −4.92622 −0.161973
\(926\) −8.14625 −0.267702
\(927\) −31.9683 −1.04998
\(928\) 7.46463 0.245039
\(929\) −16.9182 −0.555069 −0.277535 0.960716i \(-0.589517\pi\)
−0.277535 + 0.960716i \(0.589517\pi\)
\(930\) −1.44649 −0.0474321
\(931\) −4.05554 −0.132915
\(932\) 8.97639 0.294031
\(933\) −3.16210 −0.103523
\(934\) 7.67994 0.251295
\(935\) 0 0
\(936\) −5.34092 −0.174573
\(937\) 5.70923 0.186512 0.0932562 0.995642i \(-0.470272\pi\)
0.0932562 + 0.995642i \(0.470272\pi\)
\(938\) 5.99409 0.195714
\(939\) 19.4062 0.633297
\(940\) 1.77448 0.0578770
\(941\) 34.2699 1.11717 0.558584 0.829448i \(-0.311345\pi\)
0.558584 + 0.829448i \(0.311345\pi\)
\(942\) 6.82248 0.222288
\(943\) 5.72735 0.186508
\(944\) −0.426837 −0.0138924
\(945\) −4.79491 −0.155978
\(946\) 0 0
\(947\) −29.4856 −0.958154 −0.479077 0.877773i \(-0.659029\pi\)
−0.479077 + 0.877773i \(0.659029\pi\)
\(948\) −9.22750 −0.299695
\(949\) 2.15038 0.0698042
\(950\) −4.05554 −0.131579
\(951\) −6.98207 −0.226409
\(952\) 4.68602 0.151875
\(953\) 43.8435 1.42023 0.710116 0.704085i \(-0.248643\pi\)
0.710116 + 0.704085i \(0.248643\pi\)
\(954\) −15.1752 −0.491316
\(955\) −10.6197 −0.343646
\(956\) 9.78125 0.316348
\(957\) 0 0
\(958\) 22.7225 0.734131
\(959\) −1.63086 −0.0526633
\(960\) −0.935683 −0.0301991
\(961\) −28.6102 −0.922908
\(962\) −12.3844 −0.399288
\(963\) −30.6728 −0.988418
\(964\) −23.3855 −0.753195
\(965\) −7.98145 −0.256932
\(966\) −3.24631 −0.104448
\(967\) 43.1164 1.38653 0.693265 0.720683i \(-0.256172\pi\)
0.693265 + 0.720683i \(0.256172\pi\)
\(968\) 0 0
\(969\) −17.7820 −0.571242
\(970\) 11.3338 0.363908
\(971\) −55.6910 −1.78721 −0.893604 0.448856i \(-0.851831\pi\)
−0.893604 + 0.448856i \(0.851831\pi\)
\(972\) −16.1503 −0.518022
\(973\) −23.1799 −0.743113
\(974\) 33.4458 1.07167
\(975\) −2.35228 −0.0753332
\(976\) −6.15201 −0.196921
\(977\) −9.63540 −0.308264 −0.154132 0.988050i \(-0.549258\pi\)
−0.154132 + 0.988050i \(0.549258\pi\)
\(978\) −8.80763 −0.281637
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −25.8910 −0.826636
\(982\) −6.10991 −0.194975
\(983\) 54.5716 1.74056 0.870281 0.492555i \(-0.163937\pi\)
0.870281 + 0.492555i \(0.163937\pi\)
\(984\) 1.54462 0.0492407
\(985\) 7.28621 0.232158
\(986\) −34.9794 −1.11397
\(987\) 1.66035 0.0528495
\(988\) −10.1955 −0.324362
\(989\) −27.6547 −0.879370
\(990\) 0 0
\(991\) −50.0492 −1.58986 −0.794932 0.606699i \(-0.792494\pi\)
−0.794932 + 0.606699i \(0.792494\pi\)
\(992\) 1.54591 0.0490828
\(993\) 7.73531 0.245473
\(994\) −6.53922 −0.207411
\(995\) 14.1853 0.449703
\(996\) −7.63213 −0.241833
\(997\) −22.8919 −0.724994 −0.362497 0.931985i \(-0.618076\pi\)
−0.362497 + 0.931985i \(0.618076\pi\)
\(998\) 21.1193 0.668520
\(999\) −23.6208 −0.747328
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 8470.2.a.dc.1.3 6
11.7 odd 10 770.2.n.j.71.2 12
11.8 odd 10 770.2.n.j.141.2 yes 12
11.10 odd 2 8470.2.a.cw.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.j.71.2 12 11.7 odd 10
770.2.n.j.141.2 yes 12 11.8 odd 10
8470.2.a.cw.1.3 6 11.10 odd 2
8470.2.a.dc.1.3 6 1.1 even 1 trivial