Properties

Label 8470.2.a.cp.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.93185\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} +2.82843 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} +2.82843 q^{9} -1.00000 q^{10} -2.41421 q^{12} +5.59575 q^{13} -1.00000 q^{14} -2.41421 q^{15} +1.00000 q^{16} +0.449490 q^{17} -2.82843 q^{18} -2.27792 q^{19} +1.00000 q^{20} -2.41421 q^{21} +0.110988 q^{23} +2.41421 q^{24} +1.00000 q^{25} -5.59575 q^{26} +0.414214 q^{27} +1.00000 q^{28} -2.49938 q^{29} +2.41421 q^{30} +4.52004 q^{31} -1.00000 q^{32} -0.449490 q^{34} +1.00000 q^{35} +2.82843 q^{36} -6.42883 q^{37} +2.27792 q^{38} -13.5093 q^{39} -1.00000 q^{40} -7.67752 q^{41} +2.41421 q^{42} +1.34366 q^{43} +2.82843 q^{45} -0.110988 q^{46} +1.51399 q^{47} -2.41421 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.08516 q^{51} +5.59575 q^{52} -6.94887 q^{53} -0.414214 q^{54} -1.00000 q^{56} +5.49938 q^{57} +2.49938 q^{58} +3.97469 q^{59} -2.41421 q^{60} -4.29253 q^{61} -4.52004 q^{62} +2.82843 q^{63} +1.00000 q^{64} +5.59575 q^{65} -11.1708 q^{67} +0.449490 q^{68} -0.267949 q^{69} -1.00000 q^{70} -1.87780 q^{71} -2.82843 q^{72} -3.41421 q^{73} +6.42883 q^{74} -2.41421 q^{75} -2.27792 q^{76} +13.5093 q^{78} +11.2168 q^{79} +1.00000 q^{80} -9.48528 q^{81} +7.67752 q^{82} -16.6960 q^{83} -2.41421 q^{84} +0.449490 q^{85} -1.34366 q^{86} +6.03403 q^{87} +2.17209 q^{89} -2.82843 q^{90} +5.59575 q^{91} +0.110988 q^{92} -10.9123 q^{93} -1.51399 q^{94} -2.27792 q^{95} +2.41421 q^{96} -10.1270 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} - 4q^{8} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} - 4q^{8} - 4q^{10} - 4q^{12} - 4q^{14} - 4q^{15} + 4q^{16} - 8q^{17} + 12q^{19} + 4q^{20} - 4q^{21} - 8q^{23} + 4q^{24} + 4q^{25} - 4q^{27} + 4q^{28} + 8q^{29} + 4q^{30} - 4q^{32} + 8q^{34} + 4q^{35} - 16q^{37} - 12q^{38} - 8q^{39} - 4q^{40} - 8q^{41} + 4q^{42} + 8q^{43} + 8q^{46} - 16q^{47} - 4q^{48} + 4q^{49} - 4q^{50} + 8q^{51} + 4q^{54} - 4q^{56} + 4q^{57} - 8q^{58} - 8q^{59} - 4q^{60} + 8q^{61} + 4q^{64} - 8q^{68} - 8q^{69} - 4q^{70} - 8q^{71} - 8q^{73} + 16q^{74} - 4q^{75} + 12q^{76} + 8q^{78} + 24q^{79} + 4q^{80} - 4q^{81} + 8q^{82} - 8q^{83} - 4q^{84} - 8q^{85} - 8q^{86} - 16q^{87} - 8q^{92} + 16q^{93} + 16q^{94} + 12q^{95} + 4q^{96} - 8q^{97} - 4q^{98} + O(q^{100}) \)

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
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.41421 0.985599
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 2.82843 0.942809
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.41421 −0.696923
\(13\) 5.59575 1.55198 0.775991 0.630743i \(-0.217250\pi\)
0.775991 + 0.630743i \(0.217250\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.41421 −0.623347
\(16\) 1.00000 0.250000
\(17\) 0.449490 0.109017 0.0545086 0.998513i \(-0.482641\pi\)
0.0545086 + 0.998513i \(0.482641\pi\)
\(18\) −2.82843 −0.666667
\(19\) −2.27792 −0.522590 −0.261295 0.965259i \(-0.584150\pi\)
−0.261295 + 0.965259i \(0.584150\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.41421 −0.526825
\(22\) 0 0
\(23\) 0.110988 0.0231426 0.0115713 0.999933i \(-0.496317\pi\)
0.0115713 + 0.999933i \(0.496317\pi\)
\(24\) 2.41421 0.492799
\(25\) 1.00000 0.200000
\(26\) −5.59575 −1.09742
\(27\) 0.414214 0.0797154
\(28\) 1.00000 0.188982
\(29\) −2.49938 −0.464123 −0.232061 0.972701i \(-0.574547\pi\)
−0.232061 + 0.972701i \(0.574547\pi\)
\(30\) 2.41421 0.440773
\(31\) 4.52004 0.811824 0.405912 0.913912i \(-0.366954\pi\)
0.405912 + 0.913912i \(0.366954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.449490 −0.0770869
\(35\) 1.00000 0.169031
\(36\) 2.82843 0.471405
\(37\) −6.42883 −1.05689 −0.528446 0.848967i \(-0.677225\pi\)
−0.528446 + 0.848967i \(0.677225\pi\)
\(38\) 2.27792 0.369527
\(39\) −13.5093 −2.16323
\(40\) −1.00000 −0.158114
\(41\) −7.67752 −1.19903 −0.599513 0.800365i \(-0.704639\pi\)
−0.599513 + 0.800365i \(0.704639\pi\)
\(42\) 2.41421 0.372521
\(43\) 1.34366 0.204906 0.102453 0.994738i \(-0.467331\pi\)
0.102453 + 0.994738i \(0.467331\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) −0.110988 −0.0163643
\(47\) 1.51399 0.220838 0.110419 0.993885i \(-0.464781\pi\)
0.110419 + 0.993885i \(0.464781\pi\)
\(48\) −2.41421 −0.348462
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −1.08516 −0.151953
\(52\) 5.59575 0.775991
\(53\) −6.94887 −0.954500 −0.477250 0.878767i \(-0.658366\pi\)
−0.477250 + 0.878767i \(0.658366\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 5.49938 0.728410
\(58\) 2.49938 0.328184
\(59\) 3.97469 0.517461 0.258730 0.965950i \(-0.416696\pi\)
0.258730 + 0.965950i \(0.416696\pi\)
\(60\) −2.41421 −0.311674
\(61\) −4.29253 −0.549602 −0.274801 0.961501i \(-0.588612\pi\)
−0.274801 + 0.961501i \(0.588612\pi\)
\(62\) −4.52004 −0.574046
\(63\) 2.82843 0.356348
\(64\) 1.00000 0.125000
\(65\) 5.59575 0.694068
\(66\) 0 0
\(67\) −11.1708 −1.36474 −0.682368 0.731009i \(-0.739050\pi\)
−0.682368 + 0.731009i \(0.739050\pi\)
\(68\) 0.449490 0.0545086
\(69\) −0.267949 −0.0322573
\(70\) −1.00000 −0.119523
\(71\) −1.87780 −0.222854 −0.111427 0.993773i \(-0.535542\pi\)
−0.111427 + 0.993773i \(0.535542\pi\)
\(72\) −2.82843 −0.333333
\(73\) −3.41421 −0.399603 −0.199802 0.979836i \(-0.564030\pi\)
−0.199802 + 0.979836i \(0.564030\pi\)
\(74\) 6.42883 0.747336
\(75\) −2.41421 −0.278769
\(76\) −2.27792 −0.261295
\(77\) 0 0
\(78\) 13.5093 1.52963
\(79\) 11.2168 1.26199 0.630995 0.775787i \(-0.282647\pi\)
0.630995 + 0.775787i \(0.282647\pi\)
\(80\) 1.00000 0.111803
\(81\) −9.48528 −1.05392
\(82\) 7.67752 0.847840
\(83\) −16.6960 −1.83263 −0.916315 0.400459i \(-0.868851\pi\)
−0.916315 + 0.400459i \(0.868851\pi\)
\(84\) −2.41421 −0.263412
\(85\) 0.449490 0.0487540
\(86\) −1.34366 −0.144891
\(87\) 6.03403 0.646916
\(88\) 0 0
\(89\) 2.17209 0.230241 0.115120 0.993352i \(-0.463275\pi\)
0.115120 + 0.993352i \(0.463275\pi\)
\(90\) −2.82843 −0.298142
\(91\) 5.59575 0.586594
\(92\) 0.110988 0.0115713
\(93\) −10.9123 −1.13156
\(94\) −1.51399 −0.156156
\(95\) −2.27792 −0.233709
\(96\) 2.41421 0.246400
\(97\) −10.1270 −1.02824 −0.514121 0.857718i \(-0.671882\pi\)
−0.514121 + 0.857718i \(0.671882\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.46410 −0.643202 −0.321601 0.946875i \(-0.604221\pi\)
−0.321601 + 0.946875i \(0.604221\pi\)
\(102\) 1.08516 0.107447
\(103\) −7.55051 −0.743974 −0.371987 0.928238i \(-0.621323\pi\)
−0.371987 + 0.928238i \(0.621323\pi\)
\(104\) −5.59575 −0.548709
\(105\) −2.41421 −0.235603
\(106\) 6.94887 0.674934
\(107\) 6.00481 0.580507 0.290253 0.956950i \(-0.406260\pi\)
0.290253 + 0.956950i \(0.406260\pi\)
\(108\) 0.414214 0.0398577
\(109\) 1.01461 0.0971822 0.0485911 0.998819i \(-0.484527\pi\)
0.0485911 + 0.998819i \(0.484527\pi\)
\(110\) 0 0
\(111\) 15.5206 1.47315
\(112\) 1.00000 0.0944911
\(113\) −5.83912 −0.549299 −0.274649 0.961544i \(-0.588562\pi\)
−0.274649 + 0.961544i \(0.588562\pi\)
\(114\) −5.49938 −0.515064
\(115\) 0.110988 0.0103497
\(116\) −2.49938 −0.232061
\(117\) 15.8272 1.46322
\(118\) −3.97469 −0.365900
\(119\) 0.449490 0.0412047
\(120\) 2.41421 0.220387
\(121\) 0 0
\(122\) 4.29253 0.388627
\(123\) 18.5352 1.67126
\(124\) 4.52004 0.405912
\(125\) 1.00000 0.0894427
\(126\) −2.82843 −0.251976
\(127\) −8.36257 −0.742058 −0.371029 0.928621i \(-0.620995\pi\)
−0.371029 + 0.928621i \(0.620995\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.24389 −0.285608
\(130\) −5.59575 −0.490780
\(131\) 10.5704 0.923544 0.461772 0.886999i \(-0.347214\pi\)
0.461772 + 0.886999i \(0.347214\pi\)
\(132\) 0 0
\(133\) −2.27792 −0.197520
\(134\) 11.1708 0.965014
\(135\) 0.414214 0.0356498
\(136\) −0.449490 −0.0385434
\(137\) −10.8895 −0.930355 −0.465178 0.885217i \(-0.654010\pi\)
−0.465178 + 0.885217i \(0.654010\pi\)
\(138\) 0.267949 0.0228093
\(139\) 15.5493 1.31887 0.659436 0.751761i \(-0.270795\pi\)
0.659436 + 0.751761i \(0.270795\pi\)
\(140\) 1.00000 0.0845154
\(141\) −3.65509 −0.307814
\(142\) 1.87780 0.157581
\(143\) 0 0
\(144\) 2.82843 0.235702
\(145\) −2.49938 −0.207562
\(146\) 3.41421 0.282562
\(147\) −2.41421 −0.199121
\(148\) −6.42883 −0.528446
\(149\) −19.0552 −1.56106 −0.780532 0.625116i \(-0.785052\pi\)
−0.780532 + 0.625116i \(0.785052\pi\)
\(150\) 2.41421 0.197120
\(151\) 23.3590 1.90092 0.950462 0.310840i \(-0.100610\pi\)
0.950462 + 0.310840i \(0.100610\pi\)
\(152\) 2.27792 0.184763
\(153\) 1.27135 0.102782
\(154\) 0 0
\(155\) 4.52004 0.363059
\(156\) −13.5093 −1.08161
\(157\) −16.6617 −1.32975 −0.664873 0.746957i \(-0.731514\pi\)
−0.664873 + 0.746957i \(0.731514\pi\)
\(158\) −11.2168 −0.892362
\(159\) 16.7761 1.33043
\(160\) −1.00000 −0.0790569
\(161\) 0.110988 0.00874709
\(162\) 9.48528 0.745234
\(163\) 2.01994 0.158214 0.0791068 0.996866i \(-0.474793\pi\)
0.0791068 + 0.996866i \(0.474793\pi\)
\(164\) −7.67752 −0.599513
\(165\) 0 0
\(166\) 16.6960 1.29586
\(167\) 22.4402 1.73647 0.868237 0.496149i \(-0.165253\pi\)
0.868237 + 0.496149i \(0.165253\pi\)
\(168\) 2.41421 0.186261
\(169\) 18.3125 1.40865
\(170\) −0.449490 −0.0344743
\(171\) −6.44292 −0.492703
\(172\) 1.34366 0.102453
\(173\) −20.2419 −1.53896 −0.769482 0.638668i \(-0.779486\pi\)
−0.769482 + 0.638668i \(0.779486\pi\)
\(174\) −6.03403 −0.457439
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −9.59575 −0.721261
\(178\) −2.17209 −0.162805
\(179\) −16.3132 −1.21931 −0.609653 0.792669i \(-0.708691\pi\)
−0.609653 + 0.792669i \(0.708691\pi\)
\(180\) 2.82843 0.210819
\(181\) 10.0659 0.748193 0.374097 0.927390i \(-0.377953\pi\)
0.374097 + 0.927390i \(0.377953\pi\)
\(182\) −5.59575 −0.414785
\(183\) 10.3631 0.766061
\(184\) −0.110988 −0.00818216
\(185\) −6.42883 −0.472657
\(186\) 10.9123 0.800132
\(187\) 0 0
\(188\) 1.51399 0.110419
\(189\) 0.414214 0.0301296
\(190\) 2.27792 0.165257
\(191\) −12.6211 −0.913228 −0.456614 0.889665i \(-0.650938\pi\)
−0.456614 + 0.889665i \(0.650938\pi\)
\(192\) −2.41421 −0.174231
\(193\) −18.3984 −1.32434 −0.662171 0.749352i \(-0.730365\pi\)
−0.662171 + 0.749352i \(0.730365\pi\)
\(194\) 10.1270 0.727077
\(195\) −13.5093 −0.967424
\(196\) 1.00000 0.0714286
\(197\) 4.24921 0.302744 0.151372 0.988477i \(-0.451631\pi\)
0.151372 + 0.988477i \(0.451631\pi\)
\(198\) 0 0
\(199\) −16.8211 −1.19242 −0.596209 0.802829i \(-0.703327\pi\)
−0.596209 + 0.802829i \(0.703327\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 26.9688 1.90223
\(202\) 6.46410 0.454813
\(203\) −2.49938 −0.175422
\(204\) −1.08516 −0.0759767
\(205\) −7.67752 −0.536221
\(206\) 7.55051 0.526069
\(207\) 0.313922 0.0218191
\(208\) 5.59575 0.387996
\(209\) 0 0
\(210\) 2.41421 0.166597
\(211\) −0.762683 −0.0525052 −0.0262526 0.999655i \(-0.508357\pi\)
−0.0262526 + 0.999655i \(0.508357\pi\)
\(212\) −6.94887 −0.477250
\(213\) 4.53341 0.310624
\(214\) −6.00481 −0.410480
\(215\) 1.34366 0.0916369
\(216\) −0.414214 −0.0281837
\(217\) 4.52004 0.306840
\(218\) −1.01461 −0.0687182
\(219\) 8.24264 0.556986
\(220\) 0 0
\(221\) 2.51523 0.169193
\(222\) −15.5206 −1.04167
\(223\) 9.00532 0.603041 0.301521 0.953460i \(-0.402506\pi\)
0.301521 + 0.953460i \(0.402506\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.82843 0.188562
\(226\) 5.83912 0.388413
\(227\) −20.9895 −1.39312 −0.696560 0.717499i \(-0.745287\pi\)
−0.696560 + 0.717499i \(0.745287\pi\)
\(228\) 5.49938 0.364205
\(229\) 29.9109 1.97657 0.988283 0.152633i \(-0.0487751\pi\)
0.988283 + 0.152633i \(0.0487751\pi\)
\(230\) −0.110988 −0.00731834
\(231\) 0 0
\(232\) 2.49938 0.164092
\(233\) 11.8917 0.779050 0.389525 0.921016i \(-0.372639\pi\)
0.389525 + 0.921016i \(0.372639\pi\)
\(234\) −15.8272 −1.03466
\(235\) 1.51399 0.0987618
\(236\) 3.97469 0.258730
\(237\) −27.0798 −1.75902
\(238\) −0.449490 −0.0291361
\(239\) 22.6819 1.46717 0.733587 0.679596i \(-0.237845\pi\)
0.733587 + 0.679596i \(0.237845\pi\)
\(240\) −2.41421 −0.155837
\(241\) 8.72135 0.561792 0.280896 0.959738i \(-0.409368\pi\)
0.280896 + 0.959738i \(0.409368\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) −4.29253 −0.274801
\(245\) 1.00000 0.0638877
\(246\) −18.5352 −1.18176
\(247\) −12.7467 −0.811051
\(248\) −4.52004 −0.287023
\(249\) 40.3078 2.55440
\(250\) −1.00000 −0.0632456
\(251\) 0.728651 0.0459920 0.0229960 0.999736i \(-0.492679\pi\)
0.0229960 + 0.999736i \(0.492679\pi\)
\(252\) 2.82843 0.178174
\(253\) 0 0
\(254\) 8.36257 0.524714
\(255\) −1.08516 −0.0679556
\(256\) 1.00000 0.0625000
\(257\) 10.4203 0.649998 0.324999 0.945714i \(-0.394636\pi\)
0.324999 + 0.945714i \(0.394636\pi\)
\(258\) 3.24389 0.201955
\(259\) −6.42883 −0.399468
\(260\) 5.59575 0.347034
\(261\) −7.06931 −0.437579
\(262\) −10.5704 −0.653044
\(263\) −0.150909 −0.00930542 −0.00465271 0.999989i \(-0.501481\pi\)
−0.00465271 + 0.999989i \(0.501481\pi\)
\(264\) 0 0
\(265\) −6.94887 −0.426866
\(266\) 2.27792 0.139668
\(267\) −5.24389 −0.320921
\(268\) −11.1708 −0.682368
\(269\) 1.33726 0.0815340 0.0407670 0.999169i \(-0.487020\pi\)
0.0407670 + 0.999169i \(0.487020\pi\)
\(270\) −0.414214 −0.0252082
\(271\) 25.2500 1.53383 0.766913 0.641751i \(-0.221792\pi\)
0.766913 + 0.641751i \(0.221792\pi\)
\(272\) 0.449490 0.0272543
\(273\) −13.5093 −0.817623
\(274\) 10.8895 0.657860
\(275\) 0 0
\(276\) −0.267949 −0.0161286
\(277\) 4.14286 0.248921 0.124460 0.992225i \(-0.460280\pi\)
0.124460 + 0.992225i \(0.460280\pi\)
\(278\) −15.5493 −0.932583
\(279\) 12.7846 0.765395
\(280\) −1.00000 −0.0597614
\(281\) 9.33046 0.556608 0.278304 0.960493i \(-0.410228\pi\)
0.278304 + 0.960493i \(0.410228\pi\)
\(282\) 3.65509 0.217658
\(283\) 0.839639 0.0499114 0.0249557 0.999689i \(-0.492056\pi\)
0.0249557 + 0.999689i \(0.492056\pi\)
\(284\) −1.87780 −0.111427
\(285\) 5.49938 0.325755
\(286\) 0 0
\(287\) −7.67752 −0.453190
\(288\) −2.82843 −0.166667
\(289\) −16.7980 −0.988115
\(290\) 2.49938 0.146769
\(291\) 24.4488 1.43321
\(292\) −3.41421 −0.199802
\(293\) −14.6968 −0.858595 −0.429297 0.903163i \(-0.641239\pi\)
−0.429297 + 0.903163i \(0.641239\pi\)
\(294\) 2.41421 0.140800
\(295\) 3.97469 0.231415
\(296\) 6.42883 0.373668
\(297\) 0 0
\(298\) 19.0552 1.10384
\(299\) 0.621063 0.0359170
\(300\) −2.41421 −0.139385
\(301\) 1.34366 0.0774473
\(302\) −23.3590 −1.34416
\(303\) 15.6057 0.896525
\(304\) −2.27792 −0.130647
\(305\) −4.29253 −0.245789
\(306\) −1.27135 −0.0726782
\(307\) 21.2724 1.21408 0.607039 0.794672i \(-0.292357\pi\)
0.607039 + 0.794672i \(0.292357\pi\)
\(308\) 0 0
\(309\) 18.2285 1.03699
\(310\) −4.52004 −0.256721
\(311\) −15.4623 −0.876789 −0.438394 0.898783i \(-0.644453\pi\)
−0.438394 + 0.898783i \(0.644453\pi\)
\(312\) 13.5093 0.764816
\(313\) −10.0613 −0.568696 −0.284348 0.958721i \(-0.591777\pi\)
−0.284348 + 0.958721i \(0.591777\pi\)
\(314\) 16.6617 0.940272
\(315\) 2.82843 0.159364
\(316\) 11.2168 0.630995
\(317\) 12.3764 0.695131 0.347565 0.937656i \(-0.387008\pi\)
0.347565 + 0.937656i \(0.387008\pi\)
\(318\) −16.7761 −0.940754
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −14.4969 −0.809137
\(322\) −0.110988 −0.00618513
\(323\) −1.02390 −0.0569713
\(324\) −9.48528 −0.526960
\(325\) 5.59575 0.310397
\(326\) −2.01994 −0.111874
\(327\) −2.44949 −0.135457
\(328\) 7.67752 0.423920
\(329\) 1.51399 0.0834690
\(330\) 0 0
\(331\) 6.80776 0.374188 0.187094 0.982342i \(-0.440093\pi\)
0.187094 + 0.982342i \(0.440093\pi\)
\(332\) −16.6960 −0.916315
\(333\) −18.1835 −0.996448
\(334\) −22.4402 −1.22787
\(335\) −11.1708 −0.610328
\(336\) −2.41421 −0.131706
\(337\) 4.48852 0.244505 0.122253 0.992499i \(-0.460988\pi\)
0.122253 + 0.992499i \(0.460988\pi\)
\(338\) −18.3125 −0.996067
\(339\) 14.0969 0.765638
\(340\) 0.449490 0.0243770
\(341\) 0 0
\(342\) 6.44292 0.348393
\(343\) 1.00000 0.0539949
\(344\) −1.34366 −0.0724454
\(345\) −0.267949 −0.0144259
\(346\) 20.2419 1.08821
\(347\) 24.7899 1.33079 0.665396 0.746490i \(-0.268263\pi\)
0.665396 + 0.746490i \(0.268263\pi\)
\(348\) 6.03403 0.323458
\(349\) 5.59716 0.299609 0.149805 0.988716i \(-0.452136\pi\)
0.149805 + 0.988716i \(0.452136\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.31784 0.123717
\(352\) 0 0
\(353\) −3.32956 −0.177215 −0.0886074 0.996067i \(-0.528242\pi\)
−0.0886074 + 0.996067i \(0.528242\pi\)
\(354\) 9.59575 0.510009
\(355\) −1.87780 −0.0996633
\(356\) 2.17209 0.115120
\(357\) −1.08516 −0.0574330
\(358\) 16.3132 0.862179
\(359\) −27.2046 −1.43581 −0.717903 0.696143i \(-0.754898\pi\)
−0.717903 + 0.696143i \(0.754898\pi\)
\(360\) −2.82843 −0.149071
\(361\) −13.8111 −0.726900
\(362\) −10.0659 −0.529052
\(363\) 0 0
\(364\) 5.59575 0.293297
\(365\) −3.41421 −0.178708
\(366\) −10.3631 −0.541687
\(367\) −36.2016 −1.88971 −0.944855 0.327489i \(-0.893798\pi\)
−0.944855 + 0.327489i \(0.893798\pi\)
\(368\) 0.110988 0.00578566
\(369\) −21.7153 −1.13045
\(370\) 6.42883 0.334219
\(371\) −6.94887 −0.360767
\(372\) −10.9123 −0.565779
\(373\) −3.02848 −0.156809 −0.0784043 0.996922i \(-0.524983\pi\)
−0.0784043 + 0.996922i \(0.524983\pi\)
\(374\) 0 0
\(375\) −2.41421 −0.124669
\(376\) −1.51399 −0.0780781
\(377\) −13.9859 −0.720311
\(378\) −0.414214 −0.0213048
\(379\) 11.6617 0.599019 0.299510 0.954093i \(-0.403177\pi\)
0.299510 + 0.954093i \(0.403177\pi\)
\(380\) −2.27792 −0.116855
\(381\) 20.1890 1.03431
\(382\) 12.6211 0.645750
\(383\) −8.46943 −0.432767 −0.216384 0.976308i \(-0.569426\pi\)
−0.216384 + 0.976308i \(0.569426\pi\)
\(384\) 2.41421 0.123200
\(385\) 0 0
\(386\) 18.3984 0.936452
\(387\) 3.80045 0.193188
\(388\) −10.1270 −0.514121
\(389\) 10.5400 0.534398 0.267199 0.963641i \(-0.413902\pi\)
0.267199 + 0.963641i \(0.413902\pi\)
\(390\) 13.5093 0.684072
\(391\) 0.0498881 0.00252295
\(392\) −1.00000 −0.0505076
\(393\) −25.5193 −1.28728
\(394\) −4.24921 −0.214072
\(395\) 11.2168 0.564379
\(396\) 0 0
\(397\) −25.9280 −1.30129 −0.650644 0.759383i \(-0.725501\pi\)
−0.650644 + 0.759383i \(0.725501\pi\)
\(398\) 16.8211 0.843167
\(399\) 5.49938 0.275313
\(400\) 1.00000 0.0500000
\(401\) 3.06251 0.152934 0.0764672 0.997072i \(-0.475636\pi\)
0.0764672 + 0.997072i \(0.475636\pi\)
\(402\) −26.9688 −1.34508
\(403\) 25.2930 1.25994
\(404\) −6.46410 −0.321601
\(405\) −9.48528 −0.471327
\(406\) 2.49938 0.124042
\(407\) 0 0
\(408\) 1.08516 0.0537236
\(409\) −14.8635 −0.734952 −0.367476 0.930033i \(-0.619778\pi\)
−0.367476 + 0.930033i \(0.619778\pi\)
\(410\) 7.67752 0.379166
\(411\) 26.2896 1.29677
\(412\) −7.55051 −0.371987
\(413\) 3.97469 0.195582
\(414\) −0.313922 −0.0154284
\(415\) −16.6960 −0.819577
\(416\) −5.59575 −0.274354
\(417\) −37.5392 −1.83830
\(418\) 0 0
\(419\) −30.5280 −1.49139 −0.745696 0.666286i \(-0.767883\pi\)
−0.745696 + 0.666286i \(0.767883\pi\)
\(420\) −2.41421 −0.117802
\(421\) 21.5425 1.04992 0.524958 0.851128i \(-0.324081\pi\)
0.524958 + 0.851128i \(0.324081\pi\)
\(422\) 0.762683 0.0371268
\(423\) 4.28221 0.208208
\(424\) 6.94887 0.337467
\(425\) 0.449490 0.0218035
\(426\) −4.53341 −0.219644
\(427\) −4.29253 −0.207730
\(428\) 6.00481 0.290253
\(429\) 0 0
\(430\) −1.34366 −0.0647971
\(431\) −2.31784 −0.111646 −0.0558231 0.998441i \(-0.517778\pi\)
−0.0558231 + 0.998441i \(0.517778\pi\)
\(432\) 0.414214 0.0199289
\(433\) −15.3870 −0.739451 −0.369726 0.929141i \(-0.620548\pi\)
−0.369726 + 0.929141i \(0.620548\pi\)
\(434\) −4.52004 −0.216969
\(435\) 6.03403 0.289310
\(436\) 1.01461 0.0485911
\(437\) −0.252822 −0.0120941
\(438\) −8.24264 −0.393849
\(439\) −19.8045 −0.945218 −0.472609 0.881272i \(-0.656688\pi\)
−0.472609 + 0.881272i \(0.656688\pi\)
\(440\) 0 0
\(441\) 2.82843 0.134687
\(442\) −2.51523 −0.119637
\(443\) 15.4203 0.732640 0.366320 0.930489i \(-0.380618\pi\)
0.366320 + 0.930489i \(0.380618\pi\)
\(444\) 15.5206 0.736573
\(445\) 2.17209 0.102967
\(446\) −9.00532 −0.426414
\(447\) 46.0034 2.17588
\(448\) 1.00000 0.0472456
\(449\) −18.2046 −0.859130 −0.429565 0.903036i \(-0.641333\pi\)
−0.429565 + 0.903036i \(0.641333\pi\)
\(450\) −2.82843 −0.133333
\(451\) 0 0
\(452\) −5.83912 −0.274649
\(453\) −56.3935 −2.64960
\(454\) 20.9895 0.985085
\(455\) 5.59575 0.262333
\(456\) −5.49938 −0.257532
\(457\) 27.6216 1.29208 0.646041 0.763302i \(-0.276423\pi\)
0.646041 + 0.763302i \(0.276423\pi\)
\(458\) −29.9109 −1.39764
\(459\) 0.186185 0.00869036
\(460\) 0.110988 0.00517485
\(461\) 31.6027 1.47189 0.735943 0.677044i \(-0.236739\pi\)
0.735943 + 0.677044i \(0.236739\pi\)
\(462\) 0 0
\(463\) 3.02990 0.140812 0.0704058 0.997518i \(-0.477571\pi\)
0.0704058 + 0.997518i \(0.477571\pi\)
\(464\) −2.49938 −0.116031
\(465\) −10.9123 −0.506048
\(466\) −11.8917 −0.550872
\(467\) −3.45679 −0.159961 −0.0799805 0.996796i \(-0.525486\pi\)
−0.0799805 + 0.996796i \(0.525486\pi\)
\(468\) 15.8272 0.731612
\(469\) −11.1708 −0.515822
\(470\) −1.51399 −0.0698351
\(471\) 40.2248 1.85346
\(472\) −3.97469 −0.182950
\(473\) 0 0
\(474\) 27.0798 1.24382
\(475\) −2.27792 −0.104518
\(476\) 0.449490 0.0206023
\(477\) −19.6544 −0.899912
\(478\) −22.6819 −1.03745
\(479\) 2.92067 0.133449 0.0667245 0.997771i \(-0.478745\pi\)
0.0667245 + 0.997771i \(0.478745\pi\)
\(480\) 2.41421 0.110193
\(481\) −35.9741 −1.64028
\(482\) −8.72135 −0.397247
\(483\) −0.267949 −0.0121921
\(484\) 0 0
\(485\) −10.1270 −0.459844
\(486\) −21.6569 −0.982375
\(487\) −4.24248 −0.192245 −0.0961225 0.995370i \(-0.530644\pi\)
−0.0961225 + 0.995370i \(0.530644\pi\)
\(488\) 4.29253 0.194314
\(489\) −4.87656 −0.220525
\(490\) −1.00000 −0.0451754
\(491\) 30.8597 1.39268 0.696340 0.717712i \(-0.254810\pi\)
0.696340 + 0.717712i \(0.254810\pi\)
\(492\) 18.5352 0.835630
\(493\) −1.12344 −0.0505974
\(494\) 12.7467 0.573499
\(495\) 0 0
\(496\) 4.52004 0.202956
\(497\) −1.87780 −0.0842308
\(498\) −40.3078 −1.80624
\(499\) −38.3475 −1.71667 −0.858336 0.513089i \(-0.828501\pi\)
−0.858336 + 0.513089i \(0.828501\pi\)
\(500\) 1.00000 0.0447214
\(501\) −54.1754 −2.42038
\(502\) −0.728651 −0.0325213
\(503\) −21.7360 −0.969159 −0.484579 0.874747i \(-0.661027\pi\)
−0.484579 + 0.874747i \(0.661027\pi\)
\(504\) −2.82843 −0.125988
\(505\) −6.46410 −0.287649
\(506\) 0 0
\(507\) −44.2102 −1.96344
\(508\) −8.36257 −0.371029
\(509\) −25.7306 −1.14049 −0.570244 0.821475i \(-0.693151\pi\)
−0.570244 + 0.821475i \(0.693151\pi\)
\(510\) 1.08516 0.0480519
\(511\) −3.41421 −0.151036
\(512\) −1.00000 −0.0441942
\(513\) −0.943544 −0.0416585
\(514\) −10.4203 −0.459618
\(515\) −7.55051 −0.332715
\(516\) −3.24389 −0.142804
\(517\) 0 0
\(518\) 6.42883 0.282466
\(519\) 48.8683 2.14508
\(520\) −5.59575 −0.245390
\(521\) −6.74378 −0.295450 −0.147725 0.989028i \(-0.547195\pi\)
−0.147725 + 0.989028i \(0.547195\pi\)
\(522\) 7.06931 0.309415
\(523\) 2.58595 0.113076 0.0565379 0.998400i \(-0.481994\pi\)
0.0565379 + 0.998400i \(0.481994\pi\)
\(524\) 10.5704 0.461772
\(525\) −2.41421 −0.105365
\(526\) 0.150909 0.00657993
\(527\) 2.03171 0.0885028
\(528\) 0 0
\(529\) −22.9877 −0.999464
\(530\) 6.94887 0.301840
\(531\) 11.2421 0.487867
\(532\) −2.27792 −0.0987602
\(533\) −42.9615 −1.86087
\(534\) 5.24389 0.226925
\(535\) 6.00481 0.259610
\(536\) 11.1708 0.482507
\(537\) 39.3835 1.69952
\(538\) −1.33726 −0.0576532
\(539\) 0 0
\(540\) 0.414214 0.0178249
\(541\) −42.4707 −1.82596 −0.912979 0.408007i \(-0.866224\pi\)
−0.912979 + 0.408007i \(0.866224\pi\)
\(542\) −25.2500 −1.08458
\(543\) −24.3013 −1.04287
\(544\) −0.449490 −0.0192717
\(545\) 1.01461 0.0434612
\(546\) 13.5093 0.578147
\(547\) 29.6629 1.26830 0.634148 0.773212i \(-0.281351\pi\)
0.634148 + 0.773212i \(0.281351\pi\)
\(548\) −10.8895 −0.465178
\(549\) −12.1411 −0.518170
\(550\) 0 0
\(551\) 5.69337 0.242546
\(552\) 0.267949 0.0114047
\(553\) 11.2168 0.476988
\(554\) −4.14286 −0.176013
\(555\) 15.5206 0.658811
\(556\) 15.5493 0.659436
\(557\) 1.64299 0.0696157 0.0348079 0.999394i \(-0.488918\pi\)
0.0348079 + 0.999394i \(0.488918\pi\)
\(558\) −12.7846 −0.541216
\(559\) 7.51880 0.318011
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −9.33046 −0.393582
\(563\) −6.29591 −0.265341 −0.132670 0.991160i \(-0.542355\pi\)
−0.132670 + 0.991160i \(0.542355\pi\)
\(564\) −3.65509 −0.153907
\(565\) −5.83912 −0.245654
\(566\) −0.839639 −0.0352927
\(567\) −9.48528 −0.398344
\(568\) 1.87780 0.0787907
\(569\) −19.7613 −0.828436 −0.414218 0.910178i \(-0.635945\pi\)
−0.414218 + 0.910178i \(0.635945\pi\)
\(570\) −5.49938 −0.230344
\(571\) 15.9882 0.669086 0.334543 0.942380i \(-0.391418\pi\)
0.334543 + 0.942380i \(0.391418\pi\)
\(572\) 0 0
\(573\) 30.4699 1.27290
\(574\) 7.67752 0.320453
\(575\) 0.110988 0.00462853
\(576\) 2.82843 0.117851
\(577\) −35.5977 −1.48195 −0.740975 0.671532i \(-0.765636\pi\)
−0.740975 + 0.671532i \(0.765636\pi\)
\(578\) 16.7980 0.698703
\(579\) 44.4176 1.84593
\(580\) −2.49938 −0.103781
\(581\) −16.6960 −0.692669
\(582\) −24.4488 −1.01343
\(583\) 0 0
\(584\) 3.41421 0.141281
\(585\) 15.8272 0.654373
\(586\) 14.6968 0.607118
\(587\) −39.2344 −1.61938 −0.809688 0.586860i \(-0.800364\pi\)
−0.809688 + 0.586860i \(0.800364\pi\)
\(588\) −2.41421 −0.0995605
\(589\) −10.2963 −0.424251
\(590\) −3.97469 −0.163635
\(591\) −10.2585 −0.421978
\(592\) −6.42883 −0.264223
\(593\) −24.7544 −1.01654 −0.508270 0.861198i \(-0.669715\pi\)
−0.508270 + 0.861198i \(0.669715\pi\)
\(594\) 0 0
\(595\) 0.449490 0.0184273
\(596\) −19.0552 −0.780532
\(597\) 40.6098 1.66205
\(598\) −0.621063 −0.0253971
\(599\) −35.5118 −1.45097 −0.725486 0.688237i \(-0.758385\pi\)
−0.725486 + 0.688237i \(0.758385\pi\)
\(600\) 2.41421 0.0985599
\(601\) 31.5173 1.28562 0.642810 0.766026i \(-0.277769\pi\)
0.642810 + 0.766026i \(0.277769\pi\)
\(602\) −1.34366 −0.0547635
\(603\) −31.5959 −1.28669
\(604\) 23.3590 0.950462
\(605\) 0 0
\(606\) −15.6057 −0.633939
\(607\) 11.0849 0.449924 0.224962 0.974368i \(-0.427774\pi\)
0.224962 + 0.974368i \(0.427774\pi\)
\(608\) 2.27792 0.0923817
\(609\) 6.03403 0.244511
\(610\) 4.29253 0.173799
\(611\) 8.47191 0.342737
\(612\) 1.27135 0.0513912
\(613\) 19.3616 0.782008 0.391004 0.920389i \(-0.372128\pi\)
0.391004 + 0.920389i \(0.372128\pi\)
\(614\) −21.2724 −0.858483
\(615\) 18.5352 0.747410
\(616\) 0 0
\(617\) 39.7449 1.60007 0.800034 0.599955i \(-0.204815\pi\)
0.800034 + 0.599955i \(0.204815\pi\)
\(618\) −18.2285 −0.733260
\(619\) −35.5001 −1.42687 −0.713434 0.700723i \(-0.752861\pi\)
−0.713434 + 0.700723i \(0.752861\pi\)
\(620\) 4.52004 0.181529
\(621\) 0.0459728 0.00184483
\(622\) 15.4623 0.619983
\(623\) 2.17209 0.0870229
\(624\) −13.5093 −0.540807
\(625\) 1.00000 0.0400000
\(626\) 10.0613 0.402129
\(627\) 0 0
\(628\) −16.6617 −0.664873
\(629\) −2.88969 −0.115220
\(630\) −2.82843 −0.112687
\(631\) 31.7687 1.26469 0.632347 0.774686i \(-0.282092\pi\)
0.632347 + 0.774686i \(0.282092\pi\)
\(632\) −11.2168 −0.446181
\(633\) 1.84128 0.0731843
\(634\) −12.3764 −0.491532
\(635\) −8.36257 −0.331858
\(636\) 16.7761 0.665214
\(637\) 5.59575 0.221712
\(638\) 0 0
\(639\) −5.31122 −0.210109
\(640\) −1.00000 −0.0395285
\(641\) 48.4441 1.91343 0.956713 0.291034i \(-0.0939992\pi\)
0.956713 + 0.291034i \(0.0939992\pi\)
\(642\) 14.4969 0.572147
\(643\) −5.25526 −0.207247 −0.103624 0.994617i \(-0.533044\pi\)
−0.103624 + 0.994617i \(0.533044\pi\)
\(644\) 0.110988 0.00437355
\(645\) −3.24389 −0.127728
\(646\) 1.02390 0.0402848
\(647\) 49.1639 1.93283 0.966416 0.256984i \(-0.0827287\pi\)
0.966416 + 0.256984i \(0.0827287\pi\)
\(648\) 9.48528 0.372617
\(649\) 0 0
\(650\) −5.59575 −0.219484
\(651\) −10.9123 −0.427689
\(652\) 2.01994 0.0791068
\(653\) −35.4550 −1.38746 −0.693732 0.720234i \(-0.744035\pi\)
−0.693732 + 0.720234i \(0.744035\pi\)
\(654\) 2.44949 0.0957826
\(655\) 10.5704 0.413021
\(656\) −7.67752 −0.299757
\(657\) −9.65685 −0.376750
\(658\) −1.51399 −0.0590215
\(659\) −15.5831 −0.607031 −0.303515 0.952827i \(-0.598160\pi\)
−0.303515 + 0.952827i \(0.598160\pi\)
\(660\) 0 0
\(661\) −22.5419 −0.876778 −0.438389 0.898785i \(-0.644451\pi\)
−0.438389 + 0.898785i \(0.644451\pi\)
\(662\) −6.80776 −0.264591
\(663\) −6.07231 −0.235829
\(664\) 16.6960 0.647932
\(665\) −2.27792 −0.0883338
\(666\) 18.1835 0.704595
\(667\) −0.277401 −0.0107410
\(668\) 22.4402 0.868237
\(669\) −21.7408 −0.840547
\(670\) 11.1708 0.431567
\(671\) 0 0
\(672\) 2.41421 0.0931303
\(673\) 42.2397 1.62822 0.814111 0.580710i \(-0.197225\pi\)
0.814111 + 0.580710i \(0.197225\pi\)
\(674\) −4.48852 −0.172891
\(675\) 0.414214 0.0159431
\(676\) 18.3125 0.704326
\(677\) 27.2287 1.04648 0.523242 0.852184i \(-0.324722\pi\)
0.523242 + 0.852184i \(0.324722\pi\)
\(678\) −14.0969 −0.541388
\(679\) −10.1270 −0.388639
\(680\) −0.449490 −0.0172371
\(681\) 50.6731 1.94180
\(682\) 0 0
\(683\) −33.2266 −1.27138 −0.635690 0.771945i \(-0.719284\pi\)
−0.635690 + 0.771945i \(0.719284\pi\)
\(684\) −6.44292 −0.246351
\(685\) −10.8895 −0.416067
\(686\) −1.00000 −0.0381802
\(687\) −72.2112 −2.75503
\(688\) 1.34366 0.0512266
\(689\) −38.8842 −1.48137
\(690\) 0.267949 0.0102007
\(691\) 3.12921 0.119041 0.0595204 0.998227i \(-0.481043\pi\)
0.0595204 + 0.998227i \(0.481043\pi\)
\(692\) −20.2419 −0.769482
\(693\) 0 0
\(694\) −24.7899 −0.941012
\(695\) 15.5493 0.589817
\(696\) −6.03403 −0.228719
\(697\) −3.45097 −0.130715
\(698\) −5.59716 −0.211856
\(699\) −28.7091 −1.08588
\(700\) 1.00000 0.0377964
\(701\) −8.48402 −0.320437 −0.160218 0.987082i \(-0.551220\pi\)
−0.160218 + 0.987082i \(0.551220\pi\)
\(702\) −2.31784 −0.0874811
\(703\) 14.6443 0.552321
\(704\) 0 0
\(705\) −3.65509 −0.137659
\(706\) 3.32956 0.125310
\(707\) −6.46410 −0.243108
\(708\) −9.59575 −0.360631
\(709\) 1.62962 0.0612017 0.0306009 0.999532i \(-0.490258\pi\)
0.0306009 + 0.999532i \(0.490258\pi\)
\(710\) 1.87780 0.0704726
\(711\) 31.7259 1.18982
\(712\) −2.17209 −0.0814025
\(713\) 0.501671 0.0187877
\(714\) 1.08516 0.0406113
\(715\) 0 0
\(716\) −16.3132 −0.609653
\(717\) −54.7591 −2.04501
\(718\) 27.2046 1.01527
\(719\) −23.4644 −0.875076 −0.437538 0.899200i \(-0.644149\pi\)
−0.437538 + 0.899200i \(0.644149\pi\)
\(720\) 2.82843 0.105409
\(721\) −7.55051 −0.281196
\(722\) 13.8111 0.513996
\(723\) −21.0552 −0.783052
\(724\) 10.0659 0.374097
\(725\) −2.49938 −0.0928246
\(726\) 0 0
\(727\) −6.87175 −0.254859 −0.127429 0.991848i \(-0.540673\pi\)
−0.127429 + 0.991848i \(0.540673\pi\)
\(728\) −5.59575 −0.207392
\(729\) −23.8284 −0.882534
\(730\) 3.41421 0.126366
\(731\) 0.603962 0.0223383
\(732\) 10.3631 0.383030
\(733\) −33.9730 −1.25482 −0.627411 0.778688i \(-0.715885\pi\)
−0.627411 + 0.778688i \(0.715885\pi\)
\(734\) 36.2016 1.33623
\(735\) −2.41421 −0.0890496
\(736\) −0.110988 −0.00409108
\(737\) 0 0
\(738\) 21.7153 0.799351
\(739\) −29.0301 −1.06789 −0.533944 0.845520i \(-0.679291\pi\)
−0.533944 + 0.845520i \(0.679291\pi\)
\(740\) −6.42883 −0.236328
\(741\) 30.7732 1.13048
\(742\) 6.94887 0.255101
\(743\) −47.1515 −1.72982 −0.864911 0.501925i \(-0.832625\pi\)
−0.864911 + 0.501925i \(0.832625\pi\)
\(744\) 10.9123 0.400066
\(745\) −19.0552 −0.698129
\(746\) 3.02848 0.110880
\(747\) −47.2235 −1.72782
\(748\) 0 0
\(749\) 6.00481 0.219411
\(750\) 2.41421 0.0881546
\(751\) −21.1160 −0.770532 −0.385266 0.922805i \(-0.625890\pi\)
−0.385266 + 0.922805i \(0.625890\pi\)
\(752\) 1.51399 0.0552095
\(753\) −1.75912 −0.0641059
\(754\) 13.9859 0.509337
\(755\) 23.3590 0.850119
\(756\) 0.414214 0.0150648
\(757\) 11.2590 0.409216 0.204608 0.978844i \(-0.434408\pi\)
0.204608 + 0.978844i \(0.434408\pi\)
\(758\) −11.6617 −0.423571
\(759\) 0 0
\(760\) 2.27792 0.0826287
\(761\) −10.2708 −0.372317 −0.186159 0.982520i \(-0.559604\pi\)
−0.186159 + 0.982520i \(0.559604\pi\)
\(762\) −20.1890 −0.731371
\(763\) 1.01461 0.0367314
\(764\) −12.6211 −0.456614
\(765\) 1.27135 0.0459657
\(766\) 8.46943 0.306013
\(767\) 22.2414 0.803090
\(768\) −2.41421 −0.0871154
\(769\) 11.3560 0.409508 0.204754 0.978813i \(-0.434361\pi\)
0.204754 + 0.978813i \(0.434361\pi\)
\(770\) 0 0
\(771\) −25.1567 −0.905998
\(772\) −18.3984 −0.662171
\(773\) −41.0116 −1.47508 −0.737542 0.675301i \(-0.764014\pi\)
−0.737542 + 0.675301i \(0.764014\pi\)
\(774\) −3.80045 −0.136604
\(775\) 4.52004 0.162365
\(776\) 10.1270 0.363538
\(777\) 15.5206 0.556797
\(778\) −10.5400 −0.377876
\(779\) 17.4887 0.626599
\(780\) −13.5093 −0.483712
\(781\) 0 0
\(782\) −0.0498881 −0.00178399
\(783\) −1.03528 −0.0369978
\(784\) 1.00000 0.0357143
\(785\) −16.6617 −0.594680
\(786\) 25.5193 0.910244
\(787\) −45.1477 −1.60934 −0.804671 0.593721i \(-0.797658\pi\)
−0.804671 + 0.593721i \(0.797658\pi\)
\(788\) 4.24921 0.151372
\(789\) 0.364326 0.0129703
\(790\) −11.2168 −0.399076
\(791\) −5.83912 −0.207615
\(792\) 0 0
\(793\) −24.0199 −0.852973
\(794\) 25.9280 0.920150
\(795\) 16.7761 0.594985
\(796\) −16.8211 −0.596209
\(797\) −34.8398 −1.23409 −0.617045 0.786928i \(-0.711670\pi\)
−0.617045 + 0.786928i \(0.711670\pi\)
\(798\) −5.49938 −0.194676
\(799\) 0.680523 0.0240752
\(800\) −1.00000 −0.0353553
\(801\) 6.14359 0.217073
\(802\) −3.06251 −0.108141
\(803\) 0 0
\(804\) 26.9688 0.951116
\(805\) 0.110988 0.00391182
\(806\) −25.2930 −0.890909
\(807\) −3.22842 −0.113646
\(808\) 6.46410 0.227406
\(809\) −23.5770 −0.828924 −0.414462 0.910067i \(-0.636030\pi\)
−0.414462 + 0.910067i \(0.636030\pi\)
\(810\) 9.48528 0.333279
\(811\) −14.9977 −0.526641 −0.263321 0.964708i \(-0.584818\pi\)
−0.263321 + 0.964708i \(0.584818\pi\)
\(812\) −2.49938 −0.0877110
\(813\) −60.9588 −2.13792
\(814\) 0 0
\(815\) 2.01994 0.0707552
\(816\) −1.08516 −0.0379883
\(817\) −3.06075 −0.107082
\(818\) 14.8635 0.519690
\(819\) 15.8272 0.553047
\(820\) −7.67752 −0.268111
\(821\) −38.2540 −1.33507 −0.667537 0.744576i \(-0.732651\pi\)
−0.667537 + 0.744576i \(0.732651\pi\)
\(822\) −26.2896 −0.916957
\(823\) −25.3006 −0.881923 −0.440961 0.897526i \(-0.645362\pi\)
−0.440961 + 0.897526i \(0.645362\pi\)
\(824\) 7.55051 0.263034
\(825\) 0 0
\(826\) −3.97469 −0.138297
\(827\) −29.9973 −1.04311 −0.521554 0.853218i \(-0.674647\pi\)
−0.521554 + 0.853218i \(0.674647\pi\)
\(828\) 0.313922 0.0109095
\(829\) 44.3053 1.53879 0.769394 0.638775i \(-0.220558\pi\)
0.769394 + 0.638775i \(0.220558\pi\)
\(830\) 16.6960 0.579528
\(831\) −10.0018 −0.346957
\(832\) 5.59575 0.193998
\(833\) 0.449490 0.0155739
\(834\) 37.5392 1.29988
\(835\) 22.4402 0.776575
\(836\) 0 0
\(837\) 1.87226 0.0647149
\(838\) 30.5280 1.05457
\(839\) 33.3319 1.15074 0.575372 0.817892i \(-0.304857\pi\)
0.575372 + 0.817892i \(0.304857\pi\)
\(840\) 2.41421 0.0832983
\(841\) −22.7531 −0.784590
\(842\) −21.5425 −0.742403
\(843\) −22.5257 −0.775827
\(844\) −0.762683 −0.0262526
\(845\) 18.3125 0.629968
\(846\) −4.28221 −0.147225
\(847\) 0 0
\(848\) −6.94887 −0.238625
\(849\) −2.02707 −0.0695688
\(850\) −0.449490 −0.0154174
\(851\) −0.713524 −0.0244593
\(852\) 4.53341 0.155312
\(853\) −11.3834 −0.389760 −0.194880 0.980827i \(-0.562432\pi\)
−0.194880 + 0.980827i \(0.562432\pi\)
\(854\) 4.29253 0.146887
\(855\) −6.44292 −0.220343
\(856\) −6.00481 −0.205240
\(857\) −37.8048 −1.29139 −0.645693 0.763597i \(-0.723431\pi\)
−0.645693 + 0.763597i \(0.723431\pi\)
\(858\) 0 0
\(859\) −57.8457 −1.97367 −0.986834 0.161735i \(-0.948291\pi\)
−0.986834 + 0.161735i \(0.948291\pi\)
\(860\) 1.34366 0.0458185
\(861\) 18.5352 0.631677
\(862\) 2.31784 0.0789458
\(863\) 45.1248 1.53607 0.768033 0.640410i \(-0.221236\pi\)
0.768033 + 0.640410i \(0.221236\pi\)
\(864\) −0.414214 −0.0140918
\(865\) −20.2419 −0.688246
\(866\) 15.3870 0.522871
\(867\) 40.5539 1.37728
\(868\) 4.52004 0.153420
\(869\) 0 0
\(870\) −6.03403 −0.204573
\(871\) −62.5093 −2.11805
\(872\) −1.01461 −0.0343591
\(873\) −28.6435 −0.969436
\(874\) 0.252822 0.00855183
\(875\) 1.00000 0.0338062
\(876\) 8.24264 0.278493
\(877\) 20.5249 0.693075 0.346537 0.938036i \(-0.387357\pi\)
0.346537 + 0.938036i \(0.387357\pi\)
\(878\) 19.8045 0.668370
\(879\) 35.4812 1.19675
\(880\) 0 0
\(881\) −8.43131 −0.284058 −0.142029 0.989862i \(-0.545363\pi\)
−0.142029 + 0.989862i \(0.545363\pi\)
\(882\) −2.82843 −0.0952381
\(883\) 51.5287 1.73408 0.867040 0.498239i \(-0.166020\pi\)
0.867040 + 0.498239i \(0.166020\pi\)
\(884\) 2.51523 0.0845965
\(885\) −9.59575 −0.322558
\(886\) −15.4203 −0.518055
\(887\) −28.9499 −0.972042 −0.486021 0.873947i \(-0.661552\pi\)
−0.486021 + 0.873947i \(0.661552\pi\)
\(888\) −15.5206 −0.520836
\(889\) −8.36257 −0.280471
\(890\) −2.17209 −0.0728086
\(891\) 0 0
\(892\) 9.00532 0.301521
\(893\) −3.44874 −0.115408
\(894\) −46.0034 −1.53858
\(895\) −16.3132 −0.545290
\(896\) −1.00000 −0.0334077
\(897\) −1.49938 −0.0500628
\(898\) 18.2046 0.607497
\(899\) −11.2973 −0.376786
\(900\) 2.82843 0.0942809
\(901\) −3.12344 −0.104057
\(902\) 0 0
\(903\) −3.24389 −0.107950
\(904\) 5.83912 0.194206
\(905\) 10.0659 0.334602
\(906\) 56.3935 1.87355
\(907\) −5.81255 −0.193003 −0.0965013 0.995333i \(-0.530765\pi\)
−0.0965013 + 0.995333i \(0.530765\pi\)
\(908\) −20.9895 −0.696560
\(909\) −18.2832 −0.606417
\(910\) −5.59575 −0.185497
\(911\) 21.8126 0.722683 0.361341 0.932434i \(-0.382319\pi\)
0.361341 + 0.932434i \(0.382319\pi\)
\(912\) 5.49938 0.182103
\(913\) 0 0
\(914\) −27.6216 −0.913641
\(915\) 10.3631 0.342593
\(916\) 29.9109 0.988283
\(917\) 10.5704 0.349067
\(918\) −0.186185 −0.00614501
\(919\) 48.0456 1.58488 0.792440 0.609950i \(-0.208811\pi\)
0.792440 + 0.609950i \(0.208811\pi\)
\(920\) −0.110988 −0.00365917
\(921\) −51.3561 −1.69224
\(922\) −31.6027 −1.04078
\(923\) −10.5077 −0.345865
\(924\) 0 0
\(925\) −6.42883 −0.211378
\(926\) −3.02990 −0.0995688
\(927\) −21.3561 −0.701425
\(928\) 2.49938 0.0820461
\(929\) 28.8259 0.945748 0.472874 0.881130i \(-0.343217\pi\)
0.472874 + 0.881130i \(0.343217\pi\)
\(930\) 10.9123 0.357830
\(931\) −2.27792 −0.0746557
\(932\) 11.8917 0.389525
\(933\) 37.3294 1.22211
\(934\) 3.45679 0.113110
\(935\) 0 0
\(936\) −15.8272 −0.517328
\(937\) 12.7627 0.416939 0.208469 0.978029i \(-0.433152\pi\)
0.208469 + 0.978029i \(0.433152\pi\)
\(938\) 11.1708 0.364741
\(939\) 24.2900 0.792675
\(940\) 1.51399 0.0493809
\(941\) −20.1593 −0.657173 −0.328586 0.944474i \(-0.606572\pi\)
−0.328586 + 0.944474i \(0.606572\pi\)
\(942\) −40.2248 −1.31060
\(943\) −0.852114 −0.0277486
\(944\) 3.97469 0.129365
\(945\) 0.414214 0.0134744
\(946\) 0 0
\(947\) 1.46868 0.0477256 0.0238628 0.999715i \(-0.492404\pi\)
0.0238628 + 0.999715i \(0.492404\pi\)
\(948\) −27.0798 −0.879511
\(949\) −19.1051 −0.620178
\(950\) 2.27792 0.0739054
\(951\) −29.8794 −0.968906
\(952\) −0.449490 −0.0145680
\(953\) −38.5687 −1.24936 −0.624681 0.780880i \(-0.714771\pi\)
−0.624681 + 0.780880i \(0.714771\pi\)
\(954\) 19.6544 0.636334
\(955\) −12.6211 −0.408408
\(956\) 22.6819 0.733587
\(957\) 0 0
\(958\) −2.92067 −0.0943627
\(959\) −10.8895 −0.351641
\(960\) −2.41421 −0.0779184
\(961\) −10.5692 −0.340943
\(962\) 35.9741 1.15985
\(963\) 16.9842 0.547307
\(964\) 8.72135 0.280896
\(965\) −18.3984 −0.592264
\(966\) 0.267949 0.00862112
\(967\) −44.4040 −1.42794 −0.713968 0.700178i \(-0.753104\pi\)
−0.713968 + 0.700178i \(0.753104\pi\)
\(968\) 0 0
\(969\) 2.47191 0.0794093
\(970\) 10.1270 0.325159
\(971\) −17.4270 −0.559260 −0.279630 0.960108i \(-0.590212\pi\)
−0.279630 + 0.960108i \(0.590212\pi\)
\(972\) 21.6569 0.694644
\(973\) 15.5493 0.498487
\(974\) 4.24248 0.135938
\(975\) −13.5093 −0.432645
\(976\) −4.29253 −0.137400
\(977\) −1.26471 −0.0404618 −0.0202309 0.999795i \(-0.506440\pi\)
−0.0202309 + 0.999795i \(0.506440\pi\)
\(978\) 4.87656 0.155935
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 2.86976 0.0916242
\(982\) −30.8597 −0.984774
\(983\) −30.6312 −0.976983 −0.488492 0.872569i \(-0.662453\pi\)
−0.488492 + 0.872569i \(0.662453\pi\)
\(984\) −18.5352 −0.590880
\(985\) 4.24921 0.135391
\(986\) 1.12344 0.0357778
\(987\) −3.65509 −0.116343
\(988\) −12.7467 −0.405525
\(989\) 0.149131 0.00474207
\(990\) 0 0
\(991\) 26.3840 0.838116 0.419058 0.907959i \(-0.362360\pi\)
0.419058 + 0.907959i \(0.362360\pi\)
\(992\) −4.52004 −0.143511
\(993\) −16.4354 −0.521561
\(994\) 1.87780 0.0595602
\(995\) −16.8211 −0.533266
\(996\) 40.3078 1.27720
\(997\) −50.2403 −1.59113 −0.795563 0.605871i \(-0.792825\pi\)
−0.795563 + 0.605871i \(0.792825\pi\)
\(998\) 38.3475 1.21387
\(999\) −2.66291 −0.0842506
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.cp.1.2 4
11.10 odd 2 8470.2.a.cr.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cp.1.2 4 1.1 even 1 trivial
8470.2.a.cr.1.1 yes 4 11.10 odd 2