Properties

Label 8470.2.a.dh.1.4
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5\)
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 1.4
Root \(-0.211079\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.211079 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.211079 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.95545 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.211079 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.211079 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.95545 q^{9} -1.00000 q^{10} -0.211079 q^{12} -0.384053 q^{13} -1.00000 q^{14} +0.211079 q^{15} +1.00000 q^{16} -3.35151 q^{17} -2.95545 q^{18} -4.82002 q^{19} -1.00000 q^{20} +0.211079 q^{21} +0.683067 q^{23} -0.211079 q^{24} +1.00000 q^{25} -0.384053 q^{26} +1.25707 q^{27} -1.00000 q^{28} +1.60685 q^{29} +0.211079 q^{30} -1.25772 q^{31} +1.00000 q^{32} -3.35151 q^{34} +1.00000 q^{35} -2.95545 q^{36} +0.856889 q^{37} -4.82002 q^{38} +0.0810656 q^{39} -1.00000 q^{40} +9.62110 q^{41} +0.211079 q^{42} +9.55155 q^{43} +2.95545 q^{45} +0.683067 q^{46} -7.55052 q^{47} -0.211079 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.707434 q^{51} -0.384053 q^{52} -13.5386 q^{53} +1.25707 q^{54} -1.00000 q^{56} +1.01741 q^{57} +1.60685 q^{58} +7.26925 q^{59} +0.211079 q^{60} +3.89131 q^{61} -1.25772 q^{62} +2.95545 q^{63} +1.00000 q^{64} +0.384053 q^{65} +4.97372 q^{67} -3.35151 q^{68} -0.144181 q^{69} +1.00000 q^{70} +0.840435 q^{71} -2.95545 q^{72} -10.3485 q^{73} +0.856889 q^{74} -0.211079 q^{75} -4.82002 q^{76} +0.0810656 q^{78} +3.36771 q^{79} -1.00000 q^{80} +8.60099 q^{81} +9.62110 q^{82} -10.5195 q^{83} +0.211079 q^{84} +3.35151 q^{85} +9.55155 q^{86} -0.339174 q^{87} +15.0365 q^{89} +2.95545 q^{90} +0.384053 q^{91} +0.683067 q^{92} +0.265479 q^{93} -7.55052 q^{94} +4.82002 q^{95} -0.211079 q^{96} +11.2671 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} - 8q^{10} - q^{13} - 8q^{14} + 8q^{16} + 6q^{17} + 8q^{18} + 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} - 8q^{28} + 3q^{29} - 8q^{31} + 8q^{32} + 6q^{34} + 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} + 35q^{39} - 8q^{40} + 11q^{41} - 5q^{43} - 8q^{45} + 10q^{46} - 15q^{47} + 8q^{49} + 8q^{50} - 6q^{51} - q^{52} - 16q^{53} - 8q^{56} + 38q^{57} + 3q^{58} - 9q^{59} + 32q^{61} - 8q^{62} - 8q^{63} + 8q^{64} + q^{65} + 33q^{67} + 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} + 8q^{72} - 34q^{73} - 6q^{74} + 5q^{76} + 35q^{78} + 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} + 50q^{83} - 6q^{85} - 5q^{86} - 12q^{87} + q^{89} - 8q^{90} + q^{91} + 10q^{92} + 26q^{93} - 15q^{94} - 5q^{95} - 4q^{97} + 8q^{98} + O(q^{100}) \)

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
\(3\) −0.211079 −0.121867 −0.0609333 0.998142i \(-0.519408\pi\)
−0.0609333 + 0.998142i \(0.519408\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.211079 −0.0861728
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.95545 −0.985149
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −0.211079 −0.0609333
\(13\) −0.384053 −0.106517 −0.0532585 0.998581i \(-0.516961\pi\)
−0.0532585 + 0.998581i \(0.516961\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.211079 0.0545004
\(16\) 1.00000 0.250000
\(17\) −3.35151 −0.812860 −0.406430 0.913682i \(-0.633226\pi\)
−0.406430 + 0.913682i \(0.633226\pi\)
\(18\) −2.95545 −0.696605
\(19\) −4.82002 −1.10579 −0.552894 0.833251i \(-0.686477\pi\)
−0.552894 + 0.833251i \(0.686477\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.211079 0.0460613
\(22\) 0 0
\(23\) 0.683067 0.142429 0.0712147 0.997461i \(-0.477312\pi\)
0.0712147 + 0.997461i \(0.477312\pi\)
\(24\) −0.211079 −0.0430864
\(25\) 1.00000 0.200000
\(26\) −0.384053 −0.0753189
\(27\) 1.25707 0.241923
\(28\) −1.00000 −0.188982
\(29\) 1.60685 0.298385 0.149193 0.988808i \(-0.452333\pi\)
0.149193 + 0.988808i \(0.452333\pi\)
\(30\) 0.211079 0.0385376
\(31\) −1.25772 −0.225893 −0.112947 0.993601i \(-0.536029\pi\)
−0.112947 + 0.993601i \(0.536029\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.35151 −0.574779
\(35\) 1.00000 0.169031
\(36\) −2.95545 −0.492574
\(37\) 0.856889 0.140872 0.0704359 0.997516i \(-0.477561\pi\)
0.0704359 + 0.997516i \(0.477561\pi\)
\(38\) −4.82002 −0.781911
\(39\) 0.0810656 0.0129809
\(40\) −1.00000 −0.158114
\(41\) 9.62110 1.50256 0.751281 0.659982i \(-0.229436\pi\)
0.751281 + 0.659982i \(0.229436\pi\)
\(42\) 0.211079 0.0325702
\(43\) 9.55155 1.45660 0.728299 0.685259i \(-0.240311\pi\)
0.728299 + 0.685259i \(0.240311\pi\)
\(44\) 0 0
\(45\) 2.95545 0.440572
\(46\) 0.683067 0.100713
\(47\) −7.55052 −1.10136 −0.550678 0.834718i \(-0.685631\pi\)
−0.550678 + 0.834718i \(0.685631\pi\)
\(48\) −0.211079 −0.0304667
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0.707434 0.0990605
\(52\) −0.384053 −0.0532585
\(53\) −13.5386 −1.85967 −0.929837 0.367971i \(-0.880053\pi\)
−0.929837 + 0.367971i \(0.880053\pi\)
\(54\) 1.25707 0.171066
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 1.01741 0.134759
\(58\) 1.60685 0.210990
\(59\) 7.26925 0.946375 0.473188 0.880962i \(-0.343103\pi\)
0.473188 + 0.880962i \(0.343103\pi\)
\(60\) 0.211079 0.0272502
\(61\) 3.89131 0.498232 0.249116 0.968474i \(-0.419860\pi\)
0.249116 + 0.968474i \(0.419860\pi\)
\(62\) −1.25772 −0.159731
\(63\) 2.95545 0.372351
\(64\) 1.00000 0.125000
\(65\) 0.384053 0.0476359
\(66\) 0 0
\(67\) 4.97372 0.607637 0.303818 0.952730i \(-0.401738\pi\)
0.303818 + 0.952730i \(0.401738\pi\)
\(68\) −3.35151 −0.406430
\(69\) −0.144181 −0.0173574
\(70\) 1.00000 0.119523
\(71\) 0.840435 0.0997413 0.0498707 0.998756i \(-0.484119\pi\)
0.0498707 + 0.998756i \(0.484119\pi\)
\(72\) −2.95545 −0.348303
\(73\) −10.3485 −1.21120 −0.605602 0.795768i \(-0.707068\pi\)
−0.605602 + 0.795768i \(0.707068\pi\)
\(74\) 0.856889 0.0996114
\(75\) −0.211079 −0.0243733
\(76\) −4.82002 −0.552894
\(77\) 0 0
\(78\) 0.0810656 0.00917887
\(79\) 3.36771 0.378898 0.189449 0.981891i \(-0.439330\pi\)
0.189449 + 0.981891i \(0.439330\pi\)
\(80\) −1.00000 −0.111803
\(81\) 8.60099 0.955666
\(82\) 9.62110 1.06247
\(83\) −10.5195 −1.15467 −0.577334 0.816508i \(-0.695907\pi\)
−0.577334 + 0.816508i \(0.695907\pi\)
\(84\) 0.211079 0.0230306
\(85\) 3.35151 0.363522
\(86\) 9.55155 1.02997
\(87\) −0.339174 −0.0363632
\(88\) 0 0
\(89\) 15.0365 1.59386 0.796931 0.604070i \(-0.206455\pi\)
0.796931 + 0.604070i \(0.206455\pi\)
\(90\) 2.95545 0.311531
\(91\) 0.384053 0.0402597
\(92\) 0.683067 0.0712147
\(93\) 0.265479 0.0275289
\(94\) −7.55052 −0.778776
\(95\) 4.82002 0.494524
\(96\) −0.211079 −0.0215432
\(97\) 11.2671 1.14400 0.572001 0.820253i \(-0.306167\pi\)
0.572001 + 0.820253i \(0.306167\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0.707992 0.0704478 0.0352239 0.999379i \(-0.488786\pi\)
0.0352239 + 0.999379i \(0.488786\pi\)
\(102\) 0.707434 0.0700464
\(103\) 0.602041 0.0593209 0.0296604 0.999560i \(-0.490557\pi\)
0.0296604 + 0.999560i \(0.490557\pi\)
\(104\) −0.384053 −0.0376595
\(105\) −0.211079 −0.0205992
\(106\) −13.5386 −1.31499
\(107\) 10.2801 0.993819 0.496909 0.867802i \(-0.334468\pi\)
0.496909 + 0.867802i \(0.334468\pi\)
\(108\) 1.25707 0.120962
\(109\) −2.65530 −0.254331 −0.127166 0.991881i \(-0.540588\pi\)
−0.127166 + 0.991881i \(0.540588\pi\)
\(110\) 0 0
\(111\) −0.180872 −0.0171676
\(112\) −1.00000 −0.0944911
\(113\) 10.2264 0.962016 0.481008 0.876716i \(-0.340271\pi\)
0.481008 + 0.876716i \(0.340271\pi\)
\(114\) 1.01741 0.0952889
\(115\) −0.683067 −0.0636963
\(116\) 1.60685 0.149193
\(117\) 1.13505 0.104935
\(118\) 7.26925 0.669188
\(119\) 3.35151 0.307232
\(120\) 0.211079 0.0192688
\(121\) 0 0
\(122\) 3.89131 0.352303
\(123\) −2.03081 −0.183112
\(124\) −1.25772 −0.112947
\(125\) −1.00000 −0.0894427
\(126\) 2.95545 0.263292
\(127\) 21.0499 1.86788 0.933938 0.357434i \(-0.116348\pi\)
0.933938 + 0.357434i \(0.116348\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.01614 −0.177511
\(130\) 0.384053 0.0336836
\(131\) 12.9629 1.13257 0.566287 0.824208i \(-0.308379\pi\)
0.566287 + 0.824208i \(0.308379\pi\)
\(132\) 0 0
\(133\) 4.82002 0.417949
\(134\) 4.97372 0.429664
\(135\) −1.25707 −0.108191
\(136\) −3.35151 −0.287389
\(137\) 12.4469 1.06341 0.531704 0.846930i \(-0.321552\pi\)
0.531704 + 0.846930i \(0.321552\pi\)
\(138\) −0.144181 −0.0122735
\(139\) −12.0205 −1.01957 −0.509784 0.860302i \(-0.670275\pi\)
−0.509784 + 0.860302i \(0.670275\pi\)
\(140\) 1.00000 0.0845154
\(141\) 1.59376 0.134219
\(142\) 0.840435 0.0705278
\(143\) 0 0
\(144\) −2.95545 −0.246287
\(145\) −1.60685 −0.133442
\(146\) −10.3485 −0.856450
\(147\) −0.211079 −0.0174095
\(148\) 0.856889 0.0704359
\(149\) 12.1628 0.996419 0.498209 0.867057i \(-0.333991\pi\)
0.498209 + 0.867057i \(0.333991\pi\)
\(150\) −0.211079 −0.0172346
\(151\) −20.2628 −1.64896 −0.824481 0.565889i \(-0.808533\pi\)
−0.824481 + 0.565889i \(0.808533\pi\)
\(152\) −4.82002 −0.390955
\(153\) 9.90519 0.800787
\(154\) 0 0
\(155\) 1.25772 0.101023
\(156\) 0.0810656 0.00649044
\(157\) −10.0988 −0.805972 −0.402986 0.915206i \(-0.632028\pi\)
−0.402986 + 0.915206i \(0.632028\pi\)
\(158\) 3.36771 0.267921
\(159\) 2.85773 0.226632
\(160\) −1.00000 −0.0790569
\(161\) −0.683067 −0.0538332
\(162\) 8.60099 0.675758
\(163\) 20.9868 1.64381 0.821905 0.569625i \(-0.192911\pi\)
0.821905 + 0.569625i \(0.192911\pi\)
\(164\) 9.62110 0.751281
\(165\) 0 0
\(166\) −10.5195 −0.816474
\(167\) 21.8752 1.69276 0.846378 0.532582i \(-0.178778\pi\)
0.846378 + 0.532582i \(0.178778\pi\)
\(168\) 0.211079 0.0162851
\(169\) −12.8525 −0.988654
\(170\) 3.35151 0.257049
\(171\) 14.2453 1.08937
\(172\) 9.55155 0.728299
\(173\) 0.216594 0.0164673 0.00823366 0.999966i \(-0.497379\pi\)
0.00823366 + 0.999966i \(0.497379\pi\)
\(174\) −0.339174 −0.0257127
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −1.53439 −0.115332
\(178\) 15.0365 1.12703
\(179\) −17.0592 −1.27507 −0.637534 0.770422i \(-0.720045\pi\)
−0.637534 + 0.770422i \(0.720045\pi\)
\(180\) 2.95545 0.220286
\(181\) 3.45754 0.256997 0.128498 0.991710i \(-0.458984\pi\)
0.128498 + 0.991710i \(0.458984\pi\)
\(182\) 0.384053 0.0284679
\(183\) −0.821376 −0.0607178
\(184\) 0.683067 0.0503564
\(185\) −0.856889 −0.0629998
\(186\) 0.265479 0.0194658
\(187\) 0 0
\(188\) −7.55052 −0.550678
\(189\) −1.25707 −0.0914385
\(190\) 4.82002 0.349681
\(191\) 2.95839 0.214062 0.107031 0.994256i \(-0.465866\pi\)
0.107031 + 0.994256i \(0.465866\pi\)
\(192\) −0.211079 −0.0152333
\(193\) −9.47612 −0.682106 −0.341053 0.940044i \(-0.610784\pi\)
−0.341053 + 0.940044i \(0.610784\pi\)
\(194\) 11.2671 0.808931
\(195\) −0.0810656 −0.00580522
\(196\) 1.00000 0.0714286
\(197\) 5.06389 0.360787 0.180394 0.983594i \(-0.442263\pi\)
0.180394 + 0.983594i \(0.442263\pi\)
\(198\) 0 0
\(199\) −21.7104 −1.53901 −0.769506 0.638639i \(-0.779498\pi\)
−0.769506 + 0.638639i \(0.779498\pi\)
\(200\) 1.00000 0.0707107
\(201\) −1.04985 −0.0740507
\(202\) 0.707992 0.0498141
\(203\) −1.60685 −0.112779
\(204\) 0.707434 0.0495303
\(205\) −9.62110 −0.671967
\(206\) 0.602041 0.0419462
\(207\) −2.01877 −0.140314
\(208\) −0.384053 −0.0266293
\(209\) 0 0
\(210\) −0.211079 −0.0145659
\(211\) 15.9979 1.10134 0.550671 0.834723i \(-0.314372\pi\)
0.550671 + 0.834723i \(0.314372\pi\)
\(212\) −13.5386 −0.929837
\(213\) −0.177399 −0.0121551
\(214\) 10.2801 0.702736
\(215\) −9.55155 −0.651411
\(216\) 1.25707 0.0855329
\(217\) 1.25772 0.0853796
\(218\) −2.65530 −0.179839
\(219\) 2.18436 0.147605
\(220\) 0 0
\(221\) 1.28715 0.0865834
\(222\) −0.180872 −0.0121393
\(223\) 2.78387 0.186422 0.0932109 0.995646i \(-0.470287\pi\)
0.0932109 + 0.995646i \(0.470287\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.95545 −0.197030
\(226\) 10.2264 0.680248
\(227\) −15.7137 −1.04295 −0.521476 0.853266i \(-0.674618\pi\)
−0.521476 + 0.853266i \(0.674618\pi\)
\(228\) 1.01741 0.0673794
\(229\) 17.0716 1.12812 0.564060 0.825734i \(-0.309239\pi\)
0.564060 + 0.825734i \(0.309239\pi\)
\(230\) −0.683067 −0.0450401
\(231\) 0 0
\(232\) 1.60685 0.105495
\(233\) −7.21602 −0.472737 −0.236369 0.971663i \(-0.575957\pi\)
−0.236369 + 0.971663i \(0.575957\pi\)
\(234\) 1.13505 0.0742003
\(235\) 7.55052 0.492541
\(236\) 7.26925 0.473188
\(237\) −0.710855 −0.0461750
\(238\) 3.35151 0.217246
\(239\) −4.90570 −0.317324 −0.158662 0.987333i \(-0.550718\pi\)
−0.158662 + 0.987333i \(0.550718\pi\)
\(240\) 0.211079 0.0136251
\(241\) 23.5063 1.51417 0.757086 0.653315i \(-0.226622\pi\)
0.757086 + 0.653315i \(0.226622\pi\)
\(242\) 0 0
\(243\) −5.58671 −0.358387
\(244\) 3.89131 0.249116
\(245\) −1.00000 −0.0638877
\(246\) −2.03081 −0.129480
\(247\) 1.85114 0.117785
\(248\) −1.25772 −0.0798653
\(249\) 2.22046 0.140716
\(250\) −1.00000 −0.0632456
\(251\) −6.08639 −0.384170 −0.192085 0.981378i \(-0.561525\pi\)
−0.192085 + 0.981378i \(0.561525\pi\)
\(252\) 2.95545 0.186176
\(253\) 0 0
\(254\) 21.0499 1.32079
\(255\) −0.707434 −0.0443012
\(256\) 1.00000 0.0625000
\(257\) 1.76924 0.110362 0.0551811 0.998476i \(-0.482426\pi\)
0.0551811 + 0.998476i \(0.482426\pi\)
\(258\) −2.01614 −0.125519
\(259\) −0.856889 −0.0532445
\(260\) 0.384053 0.0238179
\(261\) −4.74897 −0.293954
\(262\) 12.9629 0.800851
\(263\) 20.9446 1.29150 0.645750 0.763549i \(-0.276545\pi\)
0.645750 + 0.763549i \(0.276545\pi\)
\(264\) 0 0
\(265\) 13.5386 0.831672
\(266\) 4.82002 0.295534
\(267\) −3.17389 −0.194239
\(268\) 4.97372 0.303818
\(269\) −28.7266 −1.75149 −0.875745 0.482774i \(-0.839629\pi\)
−0.875745 + 0.482774i \(0.839629\pi\)
\(270\) −1.25707 −0.0765029
\(271\) −4.59932 −0.279389 −0.139694 0.990195i \(-0.544612\pi\)
−0.139694 + 0.990195i \(0.544612\pi\)
\(272\) −3.35151 −0.203215
\(273\) −0.0810656 −0.00490631
\(274\) 12.4469 0.751943
\(275\) 0 0
\(276\) −0.144181 −0.00867869
\(277\) 23.9175 1.43707 0.718533 0.695493i \(-0.244814\pi\)
0.718533 + 0.695493i \(0.244814\pi\)
\(278\) −12.0205 −0.720944
\(279\) 3.71712 0.222538
\(280\) 1.00000 0.0597614
\(281\) 24.2413 1.44611 0.723056 0.690790i \(-0.242737\pi\)
0.723056 + 0.690790i \(0.242737\pi\)
\(282\) 1.59376 0.0949069
\(283\) 2.20835 0.131273 0.0656364 0.997844i \(-0.479092\pi\)
0.0656364 + 0.997844i \(0.479092\pi\)
\(284\) 0.840435 0.0498707
\(285\) −1.01741 −0.0602660
\(286\) 0 0
\(287\) −9.62110 −0.567915
\(288\) −2.95545 −0.174151
\(289\) −5.76741 −0.339259
\(290\) −1.60685 −0.0943578
\(291\) −2.37825 −0.139416
\(292\) −10.3485 −0.605602
\(293\) −21.8741 −1.27790 −0.638950 0.769248i \(-0.720631\pi\)
−0.638950 + 0.769248i \(0.720631\pi\)
\(294\) −0.211079 −0.0123104
\(295\) −7.26925 −0.423232
\(296\) 0.856889 0.0498057
\(297\) 0 0
\(298\) 12.1628 0.704575
\(299\) −0.262334 −0.0151711
\(300\) −0.211079 −0.0121867
\(301\) −9.55155 −0.550542
\(302\) −20.2628 −1.16599
\(303\) −0.149442 −0.00858524
\(304\) −4.82002 −0.276447
\(305\) −3.89131 −0.222816
\(306\) 9.90519 0.566242
\(307\) −0.917520 −0.0523656 −0.0261828 0.999657i \(-0.508335\pi\)
−0.0261828 + 0.999657i \(0.508335\pi\)
\(308\) 0 0
\(309\) −0.127078 −0.00722924
\(310\) 1.25772 0.0714337
\(311\) 2.96958 0.168390 0.0841949 0.996449i \(-0.473168\pi\)
0.0841949 + 0.996449i \(0.473168\pi\)
\(312\) 0.0810656 0.00458943
\(313\) −1.00800 −0.0569757 −0.0284878 0.999594i \(-0.509069\pi\)
−0.0284878 + 0.999594i \(0.509069\pi\)
\(314\) −10.0988 −0.569908
\(315\) −2.95545 −0.166520
\(316\) 3.36771 0.189449
\(317\) −3.47654 −0.195262 −0.0976309 0.995223i \(-0.531126\pi\)
−0.0976309 + 0.995223i \(0.531126\pi\)
\(318\) 2.85773 0.160253
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −2.16992 −0.121113
\(322\) −0.683067 −0.0380658
\(323\) 16.1543 0.898851
\(324\) 8.60099 0.477833
\(325\) −0.384053 −0.0213034
\(326\) 20.9868 1.16235
\(327\) 0.560478 0.0309945
\(328\) 9.62110 0.531236
\(329\) 7.55052 0.416274
\(330\) 0 0
\(331\) 17.3885 0.955758 0.477879 0.878426i \(-0.341406\pi\)
0.477879 + 0.878426i \(0.341406\pi\)
\(332\) −10.5195 −0.577334
\(333\) −2.53249 −0.138780
\(334\) 21.8752 1.19696
\(335\) −4.97372 −0.271743
\(336\) 0.211079 0.0115153
\(337\) −16.4320 −0.895109 −0.447555 0.894257i \(-0.647705\pi\)
−0.447555 + 0.894257i \(0.647705\pi\)
\(338\) −12.8525 −0.699084
\(339\) −2.15858 −0.117238
\(340\) 3.35151 0.181761
\(341\) 0 0
\(342\) 14.2453 0.770298
\(343\) −1.00000 −0.0539949
\(344\) 9.55155 0.514985
\(345\) 0.144181 0.00776246
\(346\) 0.216594 0.0116442
\(347\) 32.0372 1.71985 0.859925 0.510421i \(-0.170511\pi\)
0.859925 + 0.510421i \(0.170511\pi\)
\(348\) −0.339174 −0.0181816
\(349\) 35.8054 1.91662 0.958309 0.285733i \(-0.0922370\pi\)
0.958309 + 0.285733i \(0.0922370\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −0.482781 −0.0257690
\(352\) 0 0
\(353\) −15.4312 −0.821322 −0.410661 0.911788i \(-0.634702\pi\)
−0.410661 + 0.911788i \(0.634702\pi\)
\(354\) −1.53439 −0.0815518
\(355\) −0.840435 −0.0446057
\(356\) 15.0365 0.796931
\(357\) −0.707434 −0.0374414
\(358\) −17.0592 −0.901609
\(359\) 16.8842 0.891115 0.445558 0.895253i \(-0.353005\pi\)
0.445558 + 0.895253i \(0.353005\pi\)
\(360\) 2.95545 0.155766
\(361\) 4.23260 0.222768
\(362\) 3.45754 0.181724
\(363\) 0 0
\(364\) 0.384053 0.0201298
\(365\) 10.3485 0.541667
\(366\) −0.821376 −0.0429340
\(367\) −8.94654 −0.467006 −0.233503 0.972356i \(-0.575019\pi\)
−0.233503 + 0.972356i \(0.575019\pi\)
\(368\) 0.683067 0.0356073
\(369\) −28.4346 −1.48025
\(370\) −0.856889 −0.0445476
\(371\) 13.5386 0.702891
\(372\) 0.265479 0.0137644
\(373\) 13.2408 0.685583 0.342792 0.939411i \(-0.388627\pi\)
0.342792 + 0.939411i \(0.388627\pi\)
\(374\) 0 0
\(375\) 0.211079 0.0109001
\(376\) −7.55052 −0.389388
\(377\) −0.617117 −0.0317831
\(378\) −1.25707 −0.0646568
\(379\) −15.8412 −0.813708 −0.406854 0.913493i \(-0.633374\pi\)
−0.406854 + 0.913493i \(0.633374\pi\)
\(380\) 4.82002 0.247262
\(381\) −4.44320 −0.227632
\(382\) 2.95839 0.151365
\(383\) −0.127136 −0.00649637 −0.00324818 0.999995i \(-0.501034\pi\)
−0.00324818 + 0.999995i \(0.501034\pi\)
\(384\) −0.211079 −0.0107716
\(385\) 0 0
\(386\) −9.47612 −0.482322
\(387\) −28.2291 −1.43497
\(388\) 11.2671 0.572001
\(389\) 10.0764 0.510891 0.255446 0.966823i \(-0.417778\pi\)
0.255446 + 0.966823i \(0.417778\pi\)
\(390\) −0.0810656 −0.00410491
\(391\) −2.28930 −0.115775
\(392\) 1.00000 0.0505076
\(393\) −2.73620 −0.138023
\(394\) 5.06389 0.255115
\(395\) −3.36771 −0.169448
\(396\) 0 0
\(397\) 8.42050 0.422613 0.211306 0.977420i \(-0.432228\pi\)
0.211306 + 0.977420i \(0.432228\pi\)
\(398\) −21.7104 −1.08825
\(399\) −1.01741 −0.0509340
\(400\) 1.00000 0.0500000
\(401\) −18.9810 −0.947866 −0.473933 0.880561i \(-0.657166\pi\)
−0.473933 + 0.880561i \(0.657166\pi\)
\(402\) −1.04985 −0.0523617
\(403\) 0.483031 0.0240615
\(404\) 0.707992 0.0352239
\(405\) −8.60099 −0.427387
\(406\) −1.60685 −0.0797469
\(407\) 0 0
\(408\) 0.707434 0.0350232
\(409\) 23.6931 1.17155 0.585775 0.810474i \(-0.300790\pi\)
0.585775 + 0.810474i \(0.300790\pi\)
\(410\) −9.62110 −0.475152
\(411\) −2.62728 −0.129594
\(412\) 0.602041 0.0296604
\(413\) −7.26925 −0.357696
\(414\) −2.01877 −0.0992170
\(415\) 10.5195 0.516384
\(416\) −0.384053 −0.0188297
\(417\) 2.53729 0.124251
\(418\) 0 0
\(419\) −2.81468 −0.137506 −0.0687531 0.997634i \(-0.521902\pi\)
−0.0687531 + 0.997634i \(0.521902\pi\)
\(420\) −0.211079 −0.0102996
\(421\) −2.78939 −0.135946 −0.0679732 0.997687i \(-0.521653\pi\)
−0.0679732 + 0.997687i \(0.521653\pi\)
\(422\) 15.9979 0.778766
\(423\) 22.3151 1.08500
\(424\) −13.5386 −0.657494
\(425\) −3.35151 −0.162572
\(426\) −0.177399 −0.00859499
\(427\) −3.89131 −0.188314
\(428\) 10.2801 0.496909
\(429\) 0 0
\(430\) −9.55155 −0.460617
\(431\) −6.27580 −0.302295 −0.151147 0.988511i \(-0.548297\pi\)
−0.151147 + 0.988511i \(0.548297\pi\)
\(432\) 1.25707 0.0604809
\(433\) 25.3703 1.21922 0.609609 0.792702i \(-0.291326\pi\)
0.609609 + 0.792702i \(0.291326\pi\)
\(434\) 1.25772 0.0603725
\(435\) 0.339174 0.0162621
\(436\) −2.65530 −0.127166
\(437\) −3.29240 −0.157497
\(438\) 2.18436 0.104373
\(439\) 14.4302 0.688715 0.344357 0.938839i \(-0.388097\pi\)
0.344357 + 0.938839i \(0.388097\pi\)
\(440\) 0 0
\(441\) −2.95545 −0.140736
\(442\) 1.28715 0.0612237
\(443\) 23.5142 1.11719 0.558597 0.829439i \(-0.311340\pi\)
0.558597 + 0.829439i \(0.311340\pi\)
\(444\) −0.180872 −0.00858379
\(445\) −15.0365 −0.712797
\(446\) 2.78387 0.131820
\(447\) −2.56732 −0.121430
\(448\) −1.00000 −0.0472456
\(449\) −6.33865 −0.299140 −0.149570 0.988751i \(-0.547789\pi\)
−0.149570 + 0.988751i \(0.547789\pi\)
\(450\) −2.95545 −0.139321
\(451\) 0 0
\(452\) 10.2264 0.481008
\(453\) 4.27706 0.200954
\(454\) −15.7137 −0.737478
\(455\) −0.384053 −0.0180047
\(456\) 1.01741 0.0476444
\(457\) −4.08335 −0.191011 −0.0955056 0.995429i \(-0.530447\pi\)
−0.0955056 + 0.995429i \(0.530447\pi\)
\(458\) 17.0716 0.797701
\(459\) −4.21308 −0.196650
\(460\) −0.683067 −0.0318482
\(461\) 27.7053 1.29037 0.645183 0.764028i \(-0.276781\pi\)
0.645183 + 0.764028i \(0.276781\pi\)
\(462\) 0 0
\(463\) −30.6895 −1.42626 −0.713131 0.701030i \(-0.752724\pi\)
−0.713131 + 0.701030i \(0.752724\pi\)
\(464\) 1.60685 0.0745964
\(465\) −0.265479 −0.0123113
\(466\) −7.21602 −0.334276
\(467\) −5.78418 −0.267660 −0.133830 0.991004i \(-0.542728\pi\)
−0.133830 + 0.991004i \(0.542728\pi\)
\(468\) 1.13505 0.0524675
\(469\) −4.97372 −0.229665
\(470\) 7.55052 0.348279
\(471\) 2.13165 0.0982212
\(472\) 7.26925 0.334594
\(473\) 0 0
\(474\) −0.710855 −0.0326506
\(475\) −4.82002 −0.221158
\(476\) 3.35151 0.153616
\(477\) 40.0127 1.83206
\(478\) −4.90570 −0.224382
\(479\) 6.65632 0.304135 0.152068 0.988370i \(-0.451407\pi\)
0.152068 + 0.988370i \(0.451407\pi\)
\(480\) 0.211079 0.00963441
\(481\) −0.329091 −0.0150052
\(482\) 23.5063 1.07068
\(483\) 0.144181 0.00656048
\(484\) 0 0
\(485\) −11.2671 −0.511613
\(486\) −5.58671 −0.253418
\(487\) −3.19787 −0.144910 −0.0724548 0.997372i \(-0.523083\pi\)
−0.0724548 + 0.997372i \(0.523083\pi\)
\(488\) 3.89131 0.176152
\(489\) −4.42987 −0.200326
\(490\) −1.00000 −0.0451754
\(491\) 2.50327 0.112971 0.0564854 0.998403i \(-0.482011\pi\)
0.0564854 + 0.998403i \(0.482011\pi\)
\(492\) −2.03081 −0.0915562
\(493\) −5.38538 −0.242545
\(494\) 1.85114 0.0832868
\(495\) 0 0
\(496\) −1.25772 −0.0564733
\(497\) −0.840435 −0.0376987
\(498\) 2.22046 0.0995010
\(499\) 21.6925 0.971088 0.485544 0.874212i \(-0.338622\pi\)
0.485544 + 0.874212i \(0.338622\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −4.61741 −0.206291
\(502\) −6.08639 −0.271649
\(503\) −1.50610 −0.0671536 −0.0335768 0.999436i \(-0.510690\pi\)
−0.0335768 + 0.999436i \(0.510690\pi\)
\(504\) 2.95545 0.131646
\(505\) −0.707992 −0.0315052
\(506\) 0 0
\(507\) 2.71290 0.120484
\(508\) 21.0499 0.933938
\(509\) 11.6817 0.517781 0.258891 0.965907i \(-0.416643\pi\)
0.258891 + 0.965907i \(0.416643\pi\)
\(510\) −0.707434 −0.0313257
\(511\) 10.3485 0.457792
\(512\) 1.00000 0.0441942
\(513\) −6.05911 −0.267516
\(514\) 1.76924 0.0780378
\(515\) −0.602041 −0.0265291
\(516\) −2.01614 −0.0887554
\(517\) 0 0
\(518\) −0.856889 −0.0376496
\(519\) −0.0457185 −0.00200682
\(520\) 0.384053 0.0168418
\(521\) −6.10991 −0.267680 −0.133840 0.991003i \(-0.542731\pi\)
−0.133840 + 0.991003i \(0.542731\pi\)
\(522\) −4.74897 −0.207857
\(523\) 5.86885 0.256627 0.128314 0.991734i \(-0.459044\pi\)
0.128314 + 0.991734i \(0.459044\pi\)
\(524\) 12.9629 0.566287
\(525\) 0.211079 0.00921226
\(526\) 20.9446 0.913228
\(527\) 4.21526 0.183619
\(528\) 0 0
\(529\) −22.5334 −0.979714
\(530\) 13.5386 0.588081
\(531\) −21.4839 −0.932320
\(532\) 4.82002 0.208974
\(533\) −3.69501 −0.160049
\(534\) −3.17389 −0.137347
\(535\) −10.2801 −0.444449
\(536\) 4.97372 0.214832
\(537\) 3.60085 0.155388
\(538\) −28.7266 −1.23849
\(539\) 0 0
\(540\) −1.25707 −0.0540957
\(541\) −2.51357 −0.108067 −0.0540333 0.998539i \(-0.517208\pi\)
−0.0540333 + 0.998539i \(0.517208\pi\)
\(542\) −4.59932 −0.197558
\(543\) −0.729814 −0.0313193
\(544\) −3.35151 −0.143695
\(545\) 2.65530 0.113740
\(546\) −0.0810656 −0.00346929
\(547\) −14.6785 −0.627608 −0.313804 0.949488i \(-0.601603\pi\)
−0.313804 + 0.949488i \(0.601603\pi\)
\(548\) 12.4469 0.531704
\(549\) −11.5006 −0.490832
\(550\) 0 0
\(551\) −7.74507 −0.329951
\(552\) −0.144181 −0.00613676
\(553\) −3.36771 −0.143210
\(554\) 23.9175 1.01616
\(555\) 0.180872 0.00767757
\(556\) −12.0205 −0.509784
\(557\) −21.3175 −0.903252 −0.451626 0.892207i \(-0.649156\pi\)
−0.451626 + 0.892207i \(0.649156\pi\)
\(558\) 3.71712 0.157358
\(559\) −3.66830 −0.155153
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 24.2413 1.02256
\(563\) 33.0588 1.39326 0.696630 0.717430i \(-0.254682\pi\)
0.696630 + 0.717430i \(0.254682\pi\)
\(564\) 1.59376 0.0671093
\(565\) −10.2264 −0.430227
\(566\) 2.20835 0.0928239
\(567\) −8.60099 −0.361208
\(568\) 0.840435 0.0352639
\(569\) 46.0666 1.93121 0.965606 0.260011i \(-0.0837263\pi\)
0.965606 + 0.260011i \(0.0837263\pi\)
\(570\) −1.01741 −0.0426145
\(571\) 24.0901 1.00814 0.504070 0.863663i \(-0.331835\pi\)
0.504070 + 0.863663i \(0.331835\pi\)
\(572\) 0 0
\(573\) −0.624456 −0.0260870
\(574\) −9.62110 −0.401577
\(575\) 0.683067 0.0284859
\(576\) −2.95545 −0.123144
\(577\) −9.62623 −0.400745 −0.200373 0.979720i \(-0.564215\pi\)
−0.200373 + 0.979720i \(0.564215\pi\)
\(578\) −5.76741 −0.239893
\(579\) 2.00021 0.0831260
\(580\) −1.60685 −0.0667210
\(581\) 10.5195 0.436424
\(582\) −2.37825 −0.0985818
\(583\) 0 0
\(584\) −10.3485 −0.428225
\(585\) −1.13505 −0.0469284
\(586\) −21.8741 −0.903612
\(587\) −48.2830 −1.99285 −0.996425 0.0844765i \(-0.973078\pi\)
−0.996425 + 0.0844765i \(0.973078\pi\)
\(588\) −0.211079 −0.00870476
\(589\) 6.06224 0.249790
\(590\) −7.26925 −0.299270
\(591\) −1.06888 −0.0439680
\(592\) 0.856889 0.0352179
\(593\) −22.5234 −0.924924 −0.462462 0.886639i \(-0.653034\pi\)
−0.462462 + 0.886639i \(0.653034\pi\)
\(594\) 0 0
\(595\) −3.35151 −0.137398
\(596\) 12.1628 0.498209
\(597\) 4.58263 0.187554
\(598\) −0.262334 −0.0107276
\(599\) −26.8704 −1.09789 −0.548947 0.835857i \(-0.684971\pi\)
−0.548947 + 0.835857i \(0.684971\pi\)
\(600\) −0.211079 −0.00861728
\(601\) 42.6062 1.73794 0.868972 0.494862i \(-0.164781\pi\)
0.868972 + 0.494862i \(0.164781\pi\)
\(602\) −9.55155 −0.389292
\(603\) −14.6996 −0.598613
\(604\) −20.2628 −0.824481
\(605\) 0 0
\(606\) −0.149442 −0.00607068
\(607\) −16.8446 −0.683701 −0.341851 0.939754i \(-0.611054\pi\)
−0.341851 + 0.939754i \(0.611054\pi\)
\(608\) −4.82002 −0.195478
\(609\) 0.339174 0.0137440
\(610\) −3.89131 −0.157555
\(611\) 2.89980 0.117313
\(612\) 9.90519 0.400394
\(613\) −29.9604 −1.21009 −0.605044 0.796192i \(-0.706845\pi\)
−0.605044 + 0.796192i \(0.706845\pi\)
\(614\) −0.917520 −0.0370281
\(615\) 2.03081 0.0818903
\(616\) 0 0
\(617\) −9.21777 −0.371093 −0.185547 0.982635i \(-0.559406\pi\)
−0.185547 + 0.982635i \(0.559406\pi\)
\(618\) −0.127078 −0.00511184
\(619\) −34.1356 −1.37202 −0.686012 0.727590i \(-0.740640\pi\)
−0.686012 + 0.727590i \(0.740640\pi\)
\(620\) 1.25772 0.0505113
\(621\) 0.858664 0.0344570
\(622\) 2.96958 0.119070
\(623\) −15.0365 −0.602423
\(624\) 0.0810656 0.00324522
\(625\) 1.00000 0.0400000
\(626\) −1.00800 −0.0402879
\(627\) 0 0
\(628\) −10.0988 −0.402986
\(629\) −2.87187 −0.114509
\(630\) −2.95545 −0.117748
\(631\) −19.6717 −0.783119 −0.391559 0.920153i \(-0.628064\pi\)
−0.391559 + 0.920153i \(0.628064\pi\)
\(632\) 3.36771 0.133960
\(633\) −3.37683 −0.134217
\(634\) −3.47654 −0.138071
\(635\) −21.0499 −0.835340
\(636\) 2.85773 0.113316
\(637\) −0.384053 −0.0152167
\(638\) 0 0
\(639\) −2.48386 −0.0982600
\(640\) −1.00000 −0.0395285
\(641\) −6.78428 −0.267963 −0.133982 0.990984i \(-0.542776\pi\)
−0.133982 + 0.990984i \(0.542776\pi\)
\(642\) −2.16992 −0.0856401
\(643\) 7.64781 0.301600 0.150800 0.988564i \(-0.451815\pi\)
0.150800 + 0.988564i \(0.451815\pi\)
\(644\) −0.683067 −0.0269166
\(645\) 2.01614 0.0793853
\(646\) 16.1543 0.635583
\(647\) 12.4504 0.489475 0.244737 0.969589i \(-0.421298\pi\)
0.244737 + 0.969589i \(0.421298\pi\)
\(648\) 8.60099 0.337879
\(649\) 0 0
\(650\) −0.384053 −0.0150638
\(651\) −0.265479 −0.0104049
\(652\) 20.9868 0.821905
\(653\) 32.5664 1.27442 0.637212 0.770689i \(-0.280088\pi\)
0.637212 + 0.770689i \(0.280088\pi\)
\(654\) 0.560478 0.0219164
\(655\) −12.9629 −0.506503
\(656\) 9.62110 0.375641
\(657\) 30.5845 1.19322
\(658\) 7.55052 0.294350
\(659\) 13.3486 0.519990 0.259995 0.965610i \(-0.416279\pi\)
0.259995 + 0.965610i \(0.416279\pi\)
\(660\) 0 0
\(661\) 37.5491 1.46049 0.730246 0.683184i \(-0.239405\pi\)
0.730246 + 0.683184i \(0.239405\pi\)
\(662\) 17.3885 0.675823
\(663\) −0.271692 −0.0105516
\(664\) −10.5195 −0.408237
\(665\) −4.82002 −0.186912
\(666\) −2.53249 −0.0981320
\(667\) 1.09759 0.0424988
\(668\) 21.8752 0.846378
\(669\) −0.587617 −0.0227186
\(670\) −4.97372 −0.192152
\(671\) 0 0
\(672\) 0.211079 0.00814256
\(673\) −6.12768 −0.236205 −0.118102 0.993001i \(-0.537681\pi\)
−0.118102 + 0.993001i \(0.537681\pi\)
\(674\) −16.4320 −0.632938
\(675\) 1.25707 0.0483847
\(676\) −12.8525 −0.494327
\(677\) 15.7476 0.605229 0.302615 0.953113i \(-0.402141\pi\)
0.302615 + 0.953113i \(0.402141\pi\)
\(678\) −2.15858 −0.0828996
\(679\) −11.2671 −0.432392
\(680\) 3.35151 0.128524
\(681\) 3.31683 0.127101
\(682\) 0 0
\(683\) −15.0157 −0.574559 −0.287280 0.957847i \(-0.592751\pi\)
−0.287280 + 0.957847i \(0.592751\pi\)
\(684\) 14.2453 0.544683
\(685\) −12.4469 −0.475570
\(686\) −1.00000 −0.0381802
\(687\) −3.60345 −0.137480
\(688\) 9.55155 0.364150
\(689\) 5.19955 0.198087
\(690\) 0.144181 0.00548889
\(691\) 30.5545 1.16235 0.581174 0.813779i \(-0.302594\pi\)
0.581174 + 0.813779i \(0.302594\pi\)
\(692\) 0.216594 0.00823366
\(693\) 0 0
\(694\) 32.0372 1.21612
\(695\) 12.0205 0.455965
\(696\) −0.339174 −0.0128563
\(697\) −32.2452 −1.22137
\(698\) 35.8054 1.35525
\(699\) 1.52315 0.0576109
\(700\) −1.00000 −0.0377964
\(701\) −3.92911 −0.148400 −0.0742002 0.997243i \(-0.523640\pi\)
−0.0742002 + 0.997243i \(0.523640\pi\)
\(702\) −0.482781 −0.0182214
\(703\) −4.13022 −0.155774
\(704\) 0 0
\(705\) −1.59376 −0.0600244
\(706\) −15.4312 −0.580763
\(707\) −0.707992 −0.0266268
\(708\) −1.53439 −0.0576658
\(709\) 24.1255 0.906054 0.453027 0.891497i \(-0.350344\pi\)
0.453027 + 0.891497i \(0.350344\pi\)
\(710\) −0.840435 −0.0315410
\(711\) −9.95310 −0.373270
\(712\) 15.0365 0.563515
\(713\) −0.859107 −0.0321738
\(714\) −0.707434 −0.0264750
\(715\) 0 0
\(716\) −17.0592 −0.637534
\(717\) 1.03549 0.0386712
\(718\) 16.8842 0.630114
\(719\) −43.8622 −1.63578 −0.817892 0.575371i \(-0.804858\pi\)
−0.817892 + 0.575371i \(0.804858\pi\)
\(720\) 2.95545 0.110143
\(721\) −0.602041 −0.0224212
\(722\) 4.23260 0.157521
\(723\) −4.96169 −0.184527
\(724\) 3.45754 0.128498
\(725\) 1.60685 0.0596771
\(726\) 0 0
\(727\) −20.7183 −0.768400 −0.384200 0.923250i \(-0.625523\pi\)
−0.384200 + 0.923250i \(0.625523\pi\)
\(728\) 0.384053 0.0142339
\(729\) −24.6237 −0.911991
\(730\) 10.3485 0.383016
\(731\) −32.0121 −1.18401
\(732\) −0.821376 −0.0303589
\(733\) −33.0173 −1.21952 −0.609761 0.792585i \(-0.708735\pi\)
−0.609761 + 0.792585i \(0.708735\pi\)
\(734\) −8.94654 −0.330223
\(735\) 0.211079 0.00778578
\(736\) 0.683067 0.0251782
\(737\) 0 0
\(738\) −28.4346 −1.04669
\(739\) 5.21925 0.191993 0.0959967 0.995382i \(-0.469396\pi\)
0.0959967 + 0.995382i \(0.469396\pi\)
\(740\) −0.856889 −0.0314999
\(741\) −0.390738 −0.0143541
\(742\) 13.5386 0.497019
\(743\) 32.5419 1.19385 0.596923 0.802299i \(-0.296390\pi\)
0.596923 + 0.802299i \(0.296390\pi\)
\(744\) 0.265479 0.00973292
\(745\) −12.1628 −0.445612
\(746\) 13.2408 0.484781
\(747\) 31.0899 1.13752
\(748\) 0 0
\(749\) −10.2801 −0.375628
\(750\) 0.211079 0.00770753
\(751\) 47.1955 1.72219 0.861094 0.508446i \(-0.169780\pi\)
0.861094 + 0.508446i \(0.169780\pi\)
\(752\) −7.55052 −0.275339
\(753\) 1.28471 0.0468175
\(754\) −0.617117 −0.0224741
\(755\) 20.2628 0.737438
\(756\) −1.25707 −0.0457192
\(757\) −36.8265 −1.33848 −0.669241 0.743045i \(-0.733381\pi\)
−0.669241 + 0.743045i \(0.733381\pi\)
\(758\) −15.8412 −0.575378
\(759\) 0 0
\(760\) 4.82002 0.174841
\(761\) 6.08704 0.220655 0.110328 0.993895i \(-0.464810\pi\)
0.110328 + 0.993895i \(0.464810\pi\)
\(762\) −4.44320 −0.160960
\(763\) 2.65530 0.0961282
\(764\) 2.95839 0.107031
\(765\) −9.90519 −0.358123
\(766\) −0.127136 −0.00459363
\(767\) −2.79177 −0.100805
\(768\) −0.211079 −0.00761667
\(769\) −42.1342 −1.51940 −0.759700 0.650274i \(-0.774654\pi\)
−0.759700 + 0.650274i \(0.774654\pi\)
\(770\) 0 0
\(771\) −0.373450 −0.0134495
\(772\) −9.47612 −0.341053
\(773\) −16.3152 −0.586818 −0.293409 0.955987i \(-0.594790\pi\)
−0.293409 + 0.955987i \(0.594790\pi\)
\(774\) −28.2291 −1.01467
\(775\) −1.25772 −0.0451786
\(776\) 11.2671 0.404466
\(777\) 0.180872 0.00648873
\(778\) 10.0764 0.361255
\(779\) −46.3739 −1.66152
\(780\) −0.0810656 −0.00290261
\(781\) 0 0
\(782\) −2.28930 −0.0818653
\(783\) 2.01993 0.0721864
\(784\) 1.00000 0.0357143
\(785\) 10.0988 0.360442
\(786\) −2.73620 −0.0975970
\(787\) −0.381541 −0.0136005 −0.00680023 0.999977i \(-0.502165\pi\)
−0.00680023 + 0.999977i \(0.502165\pi\)
\(788\) 5.06389 0.180394
\(789\) −4.42097 −0.157391
\(790\) −3.36771 −0.119818
\(791\) −10.2264 −0.363608
\(792\) 0 0
\(793\) −1.49447 −0.0530702
\(794\) 8.42050 0.298832
\(795\) −2.85773 −0.101353
\(796\) −21.7104 −0.769506
\(797\) −20.6764 −0.732397 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(798\) −1.01741 −0.0360158
\(799\) 25.3056 0.895248
\(800\) 1.00000 0.0353553
\(801\) −44.4394 −1.57019
\(802\) −18.9810 −0.670242
\(803\) 0 0
\(804\) −1.04985 −0.0370253
\(805\) 0.683067 0.0240749
\(806\) 0.483031 0.0170140
\(807\) 6.06358 0.213448
\(808\) 0.707992 0.0249071
\(809\) 0.587236 0.0206461 0.0103231 0.999947i \(-0.496714\pi\)
0.0103231 + 0.999947i \(0.496714\pi\)
\(810\) −8.60099 −0.302208
\(811\) −13.5752 −0.476691 −0.238345 0.971180i \(-0.576605\pi\)
−0.238345 + 0.971180i \(0.576605\pi\)
\(812\) −1.60685 −0.0563895
\(813\) 0.970821 0.0340482
\(814\) 0 0
\(815\) −20.9868 −0.735134
\(816\) 0.707434 0.0247651
\(817\) −46.0387 −1.61069
\(818\) 23.6931 0.828411
\(819\) −1.13505 −0.0396617
\(820\) −9.62110 −0.335983
\(821\) −0.584317 −0.0203928 −0.0101964 0.999948i \(-0.503246\pi\)
−0.0101964 + 0.999948i \(0.503246\pi\)
\(822\) −2.62728 −0.0916368
\(823\) 26.6347 0.928428 0.464214 0.885723i \(-0.346337\pi\)
0.464214 + 0.885723i \(0.346337\pi\)
\(824\) 0.602041 0.0209731
\(825\) 0 0
\(826\) −7.26925 −0.252929
\(827\) −13.1807 −0.458339 −0.229170 0.973386i \(-0.573601\pi\)
−0.229170 + 0.973386i \(0.573601\pi\)
\(828\) −2.01877 −0.0701570
\(829\) −20.0631 −0.696820 −0.348410 0.937342i \(-0.613278\pi\)
−0.348410 + 0.937342i \(0.613278\pi\)
\(830\) 10.5195 0.365138
\(831\) −5.04850 −0.175130
\(832\) −0.384053 −0.0133146
\(833\) −3.35151 −0.116123
\(834\) 2.53729 0.0878590
\(835\) −21.8752 −0.757024
\(836\) 0 0
\(837\) −1.58104 −0.0546489
\(838\) −2.81468 −0.0972316
\(839\) 9.70087 0.334911 0.167456 0.985880i \(-0.446445\pi\)
0.167456 + 0.985880i \(0.446445\pi\)
\(840\) −0.211079 −0.00728293
\(841\) −26.4180 −0.910966
\(842\) −2.78939 −0.0961287
\(843\) −5.11683 −0.176233
\(844\) 15.9979 0.550671
\(845\) 12.8525 0.442140
\(846\) 22.3151 0.767210
\(847\) 0 0
\(848\) −13.5386 −0.464919
\(849\) −0.466137 −0.0159978
\(850\) −3.35151 −0.114956
\(851\) 0.585313 0.0200643
\(852\) −0.177399 −0.00607757
\(853\) −5.16515 −0.176851 −0.0884256 0.996083i \(-0.528184\pi\)
−0.0884256 + 0.996083i \(0.528184\pi\)
\(854\) −3.89131 −0.133158
\(855\) −14.2453 −0.487179
\(856\) 10.2801 0.351368
\(857\) 5.43985 0.185822 0.0929109 0.995674i \(-0.470383\pi\)
0.0929109 + 0.995674i \(0.470383\pi\)
\(858\) 0 0
\(859\) 24.5615 0.838028 0.419014 0.907980i \(-0.362376\pi\)
0.419014 + 0.907980i \(0.362376\pi\)
\(860\) −9.55155 −0.325705
\(861\) 2.03081 0.0692100
\(862\) −6.27580 −0.213755
\(863\) −18.2185 −0.620166 −0.310083 0.950710i \(-0.600357\pi\)
−0.310083 + 0.950710i \(0.600357\pi\)
\(864\) 1.25707 0.0427664
\(865\) −0.216594 −0.00736441
\(866\) 25.3703 0.862118
\(867\) 1.21738 0.0413444
\(868\) 1.25772 0.0426898
\(869\) 0 0
\(870\) 0.339174 0.0114991
\(871\) −1.91017 −0.0647237
\(872\) −2.65530 −0.0899197
\(873\) −33.2993 −1.12701
\(874\) −3.29240 −0.111367
\(875\) 1.00000 0.0338062
\(876\) 2.18436 0.0738027
\(877\) −10.5936 −0.357722 −0.178861 0.983874i \(-0.557241\pi\)
−0.178861 + 0.983874i \(0.557241\pi\)
\(878\) 14.4302 0.486995
\(879\) 4.61718 0.155733
\(880\) 0 0
\(881\) 56.7215 1.91100 0.955498 0.294996i \(-0.0953185\pi\)
0.955498 + 0.294996i \(0.0953185\pi\)
\(882\) −2.95545 −0.0995150
\(883\) 41.7759 1.40587 0.702935 0.711254i \(-0.251873\pi\)
0.702935 + 0.711254i \(0.251873\pi\)
\(884\) 1.28715 0.0432917
\(885\) 1.53439 0.0515779
\(886\) 23.5142 0.789975
\(887\) 0.460010 0.0154456 0.00772282 0.999970i \(-0.497542\pi\)
0.00772282 + 0.999970i \(0.497542\pi\)
\(888\) −0.180872 −0.00606965
\(889\) −21.0499 −0.705991
\(890\) −15.0365 −0.504023
\(891\) 0 0
\(892\) 2.78387 0.0932109
\(893\) 36.3936 1.21787
\(894\) −2.56732 −0.0858642
\(895\) 17.0592 0.570228
\(896\) −1.00000 −0.0334077
\(897\) 0.0553732 0.00184886
\(898\) −6.33865 −0.211524
\(899\) −2.02097 −0.0674032
\(900\) −2.95545 −0.0985149
\(901\) 45.3748 1.51165
\(902\) 0 0
\(903\) 2.01614 0.0670928
\(904\) 10.2264 0.340124
\(905\) −3.45754 −0.114932
\(906\) 4.27706 0.142096
\(907\) −36.2499 −1.20366 −0.601829 0.798625i \(-0.705561\pi\)
−0.601829 + 0.798625i \(0.705561\pi\)
\(908\) −15.7137 −0.521476
\(909\) −2.09243 −0.0694016
\(910\) −0.384053 −0.0127312
\(911\) 54.8407 1.81695 0.908477 0.417935i \(-0.137246\pi\)
0.908477 + 0.417935i \(0.137246\pi\)
\(912\) 1.01741 0.0336897
\(913\) 0 0
\(914\) −4.08335 −0.135065
\(915\) 0.821376 0.0271538
\(916\) 17.0716 0.564060
\(917\) −12.9629 −0.428073
\(918\) −4.21308 −0.139052
\(919\) 17.7239 0.584658 0.292329 0.956318i \(-0.405570\pi\)
0.292329 + 0.956318i \(0.405570\pi\)
\(920\) −0.683067 −0.0225201
\(921\) 0.193669 0.00638162
\(922\) 27.7053 0.912426
\(923\) −0.322771 −0.0106241
\(924\) 0 0
\(925\) 0.856889 0.0281743
\(926\) −30.6895 −1.00852
\(927\) −1.77930 −0.0584399
\(928\) 1.60685 0.0527476
\(929\) −8.59746 −0.282074 −0.141037 0.990004i \(-0.545044\pi\)
−0.141037 + 0.990004i \(0.545044\pi\)
\(930\) −0.265479 −0.00870539
\(931\) −4.82002 −0.157970
\(932\) −7.21602 −0.236369
\(933\) −0.626818 −0.0205211
\(934\) −5.78418 −0.189264
\(935\) 0 0
\(936\) 1.13505 0.0371002
\(937\) −54.1754 −1.76983 −0.884916 0.465751i \(-0.845784\pi\)
−0.884916 + 0.465751i \(0.845784\pi\)
\(938\) −4.97372 −0.162398
\(939\) 0.212768 0.00694343
\(940\) 7.55052 0.246271
\(941\) 42.0976 1.37234 0.686171 0.727440i \(-0.259290\pi\)
0.686171 + 0.727440i \(0.259290\pi\)
\(942\) 2.13165 0.0694528
\(943\) 6.57185 0.214009
\(944\) 7.26925 0.236594
\(945\) 1.25707 0.0408925
\(946\) 0 0
\(947\) 32.3240 1.05039 0.525194 0.850982i \(-0.323993\pi\)
0.525194 + 0.850982i \(0.323993\pi\)
\(948\) −0.710855 −0.0230875
\(949\) 3.97438 0.129014
\(950\) −4.82002 −0.156382
\(951\) 0.733825 0.0237959
\(952\) 3.35151 0.108623
\(953\) 40.3551 1.30723 0.653615 0.756827i \(-0.273252\pi\)
0.653615 + 0.756827i \(0.273252\pi\)
\(954\) 40.0127 1.29546
\(955\) −2.95839 −0.0957314
\(956\) −4.90570 −0.158662
\(957\) 0 0
\(958\) 6.65632 0.215056
\(959\) −12.4469 −0.401930
\(960\) 0.211079 0.00681256
\(961\) −29.4181 −0.948972
\(962\) −0.329091 −0.0106103
\(963\) −30.3824 −0.979059
\(964\) 23.5063 0.757086
\(965\) 9.47612 0.305047
\(966\) 0.144181 0.00463896
\(967\) 1.52424 0.0490161 0.0245081 0.999700i \(-0.492198\pi\)
0.0245081 + 0.999700i \(0.492198\pi\)
\(968\) 0 0
\(969\) −3.40984 −0.109540
\(970\) −11.2671 −0.361765
\(971\) 30.9119 0.992011 0.496006 0.868319i \(-0.334800\pi\)
0.496006 + 0.868319i \(0.334800\pi\)
\(972\) −5.58671 −0.179194
\(973\) 12.0205 0.385361
\(974\) −3.19787 −0.102466
\(975\) 0.0810656 0.00259618
\(976\) 3.89131 0.124558
\(977\) 57.1348 1.82790 0.913952 0.405823i \(-0.133015\pi\)
0.913952 + 0.405823i \(0.133015\pi\)
\(978\) −4.42987 −0.141652
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 7.84759 0.250554
\(982\) 2.50327 0.0798825
\(983\) −17.0792 −0.544741 −0.272370 0.962193i \(-0.587808\pi\)
−0.272370 + 0.962193i \(0.587808\pi\)
\(984\) −2.03081 −0.0647400
\(985\) −5.06389 −0.161349
\(986\) −5.38538 −0.171506
\(987\) −1.59376 −0.0507299
\(988\) 1.85114 0.0588926
\(989\) 6.52435 0.207462
\(990\) 0 0
\(991\) 56.3798 1.79096 0.895482 0.445097i \(-0.146831\pi\)
0.895482 + 0.445097i \(0.146831\pi\)
\(992\) −1.25772 −0.0399327
\(993\) −3.67035 −0.116475
\(994\) −0.840435 −0.0266570
\(995\) 21.7104 0.688267
\(996\) 2.22046 0.0703578
\(997\) 38.8679 1.23096 0.615479 0.788153i \(-0.288962\pi\)
0.615479 + 0.788153i \(0.288962\pi\)
\(998\) 21.6925 0.686663
\(999\) 1.07717 0.0340802
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.dh.1.4 8
11.5 even 5 770.2.n.k.421.3 16
11.9 even 5 770.2.n.k.631.3 yes 16
11.10 odd 2 8470.2.a.dg.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.421.3 16 11.5 even 5
770.2.n.k.631.3 yes 16 11.9 even 5
8470.2.a.dg.1.4 8 11.10 odd 2
8470.2.a.dh.1.4 8 1.1 even 1 trivial