Properties

Label 8470.2.a.cr.1.1
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 1.1
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 \(-0.782750\pi\)
−0.775991 + 0.630743i \(0.782750\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.41421 −0.623347
\(16\) 1.00000 0.250000
\(17\) −0.449490 −0.109017 −0.0545086 0.998513i \(-0.517359\pi\)
−0.0545086 + 0.998513i \(0.517359\pi\)
\(18\) 2.82843 0.666667
\(19\) 2.27792 0.522590 0.261295 0.965259i \(-0.415850\pi\)
0.261295 + 0.965259i \(0.415850\pi\)
\(20\) 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 \(-0.425453\pi\)
0.232061 + 0.972701i \(0.425453\pi\)
\(30\) −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 \(-0.295361\pi\)
0.599513 + 0.800365i \(0.295361\pi\)
\(42\) 2.41421 0.372521
\(43\) −1.34366 −0.204906 −0.102453 0.994738i \(-0.532669\pi\)
−0.102453 + 0.994738i \(0.532669\pi\)
\(44\) 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 \(-0.411388\pi\)
0.274801 + 0.961501i \(0.411388\pi\)
\(62\) 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 \(-0.435970\pi\)
0.199802 + 0.979836i \(0.435970\pi\)
\(74\) −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 \(-0.717353\pi\)
−0.630995 + 0.775787i \(0.717353\pi\)
\(80\) 1.00000 0.111803
\(81\) −9.48528 −1.05392
\(82\) 7.67752 0.847840
\(83\) 16.6960 1.83263 0.916315 0.400459i \(-0.131149\pi\)
0.916315 + 0.400459i \(0.131149\pi\)
\(84\) 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 \(-0.395779\pi\)
0.321601 + 0.946875i \(0.395779\pi\)
\(102\) 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 \(-0.593740\pi\)
−0.290253 + 0.956950i \(0.593740\pi\)
\(108\) 0.414214 0.0398577
\(109\) −1.01461 −0.0971822 −0.0485911 0.998819i \(-0.515473\pi\)
−0.0485911 + 0.998819i \(0.515473\pi\)
\(110\) 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 \(-0.379005\pi\)
0.371029 + 0.928621i \(0.379005\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.24389 0.285608
\(130\) −5.59575 −0.490780
\(131\) −10.5704 −0.923544 −0.461772 0.886999i \(-0.652786\pi\)
−0.461772 + 0.886999i \(0.652786\pi\)
\(132\) 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 \(-0.729205\pi\)
−0.659436 + 0.751761i \(0.729205\pi\)
\(140\) −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 \(-0.214948\pi\)
0.780532 + 0.625116i \(0.214948\pi\)
\(150\) −2.41421 −0.197120
\(151\) −23.3590 −1.90092 −0.950462 0.310840i \(-0.899390\pi\)
−0.950462 + 0.310840i \(0.899390\pi\)
\(152\) 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 \(-0.834747\pi\)
−0.868237 + 0.496149i \(0.834747\pi\)
\(168\) 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 \(-0.220514\pi\)
0.769482 + 0.638668i \(0.220514\pi\)
\(174\) −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 \(-0.269635\pi\)
0.662171 + 0.749352i \(0.269635\pi\)
\(194\) −10.1270 −0.727077
\(195\) 13.5093 0.967424
\(196\) 1.00000 0.0714286
\(197\) −4.24921 −0.302744 −0.151372 0.988477i \(-0.548369\pi\)
−0.151372 + 0.988477i \(0.548369\pi\)
\(198\) 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 \(-0.491643\pi\)
0.0262526 + 0.999655i \(0.491643\pi\)
\(212\) −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 \(-0.254713\pi\)
0.696560 + 0.717499i \(0.254713\pi\)
\(228\) −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 \(-0.627361\pi\)
−0.389525 + 0.921016i \(0.627361\pi\)
\(234\) −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 \(-0.762155\pi\)
−0.733587 + 0.679596i \(0.762155\pi\)
\(240\) −2.41421 −0.155837
\(241\) −8.72135 −0.561792 −0.280896 0.959738i \(-0.590632\pi\)
−0.280896 + 0.959738i \(0.590632\pi\)
\(242\) 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 \(-0.498519\pi\)
0.00465271 + 0.999989i \(0.498519\pi\)
\(264\) 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 \(-0.778208\pi\)
−0.766913 + 0.641751i \(0.778208\pi\)
\(272\) −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 \(-0.539720\pi\)
−0.124460 + 0.992225i \(0.539720\pi\)
\(278\) −15.5493 −0.932583
\(279\) 12.7846 0.765395
\(280\) −1.00000 −0.0597614
\(281\) −9.33046 −0.556608 −0.278304 0.960493i \(-0.589772\pi\)
−0.278304 + 0.960493i \(0.589772\pi\)
\(282\) −3.65509 −0.217658
\(283\) −0.839639 −0.0499114 −0.0249557 0.999689i \(-0.507944\pi\)
−0.0249557 + 0.999689i \(0.507944\pi\)
\(284\) −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 \(-0.358761\pi\)
0.429297 + 0.903163i \(0.358761\pi\)
\(294\) −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 \(-0.707643\pi\)
−0.607039 + 0.794672i \(0.707643\pi\)
\(308\) 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 \(-0.539012\pi\)
−0.122253 + 0.992499i \(0.539012\pi\)
\(338\) 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 \(-0.731737\pi\)
−0.665396 + 0.746490i \(0.731737\pi\)
\(348\) −6.03403 −0.323458
\(349\) −5.59716 −0.299609 −0.149805 0.988716i \(-0.547864\pi\)
−0.149805 + 0.988716i \(0.547864\pi\)
\(350\) −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 \(-0.245102\pi\)
0.717903 + 0.696143i \(0.245102\pi\)
\(360\) 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 \(-0.475017\pi\)
0.0784043 + 0.996922i \(0.475017\pi\)
\(374\) 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 \(-0.380222\pi\)
0.367476 + 0.930033i \(0.380222\pi\)
\(410\) 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 \(-0.482222\pi\)
0.0558231 + 0.998441i \(0.482222\pi\)
\(432\) 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 \(-0.343312\pi\)
0.472609 + 0.881272i \(0.343312\pi\)
\(440\) 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 \(-0.723577\pi\)
−0.646041 + 0.763302i \(0.723577\pi\)
\(458\) 29.9109 1.39764
\(459\) −0.186185 −0.00869036
\(460\) 0.110988 0.00517485
\(461\) −31.6027 −1.47189 −0.735943 0.677044i \(-0.763261\pi\)
−0.735943 + 0.677044i \(0.763261\pi\)
\(462\) 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 \(-0.521255\pi\)
−0.0667245 + 0.997771i \(0.521255\pi\)
\(480\) −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 \(-0.745190\pi\)
−0.696340 + 0.717712i \(0.745190\pi\)
\(492\) −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 \(-0.338973\pi\)
0.484579 + 0.874747i \(0.338973\pi\)
\(504\) −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 \(-0.518006\pi\)
−0.0565379 + 0.998400i \(0.518006\pi\)
\(524\) −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 \(-0.133776\pi\)
0.912979 + 0.408007i \(0.133776\pi\)
\(542\) −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 \(-0.718649\pi\)
−0.634148 + 0.773212i \(0.718649\pi\)
\(548\) −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 \(-0.511082\pi\)
−0.0348079 + 0.999394i \(0.511082\pi\)
\(558\) 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 \(-0.457645\pi\)
0.132670 + 0.991160i \(0.457645\pi\)
\(564\) −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 \(-0.364055\pi\)
0.414218 + 0.910178i \(0.364055\pi\)
\(570\) −5.49938 −0.230344
\(571\) −15.9882 −0.669086 −0.334543 0.942380i \(-0.608582\pi\)
−0.334543 + 0.942380i \(0.608582\pi\)
\(572\) 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 \(-0.330285\pi\)
0.508270 + 0.861198i \(0.330285\pi\)
\(594\) 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 \(-0.722231\pi\)
−0.642810 + 0.766026i \(0.722231\pi\)
\(602\) 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 \(-0.572226\pi\)
−0.224962 + 0.974368i \(0.572226\pi\)
\(608\) 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 \(-0.627872\pi\)
−0.391004 + 0.920389i \(0.627872\pi\)
\(614\) −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 \(-0.401840\pi\)
0.303515 + 0.952827i \(0.401840\pi\)
\(660\) 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 \(-0.802775\pi\)
−0.814111 + 0.580710i \(0.802775\pi\)
\(674\) −4.48852 −0.172891
\(675\) 0.414214 0.0159431
\(676\) 18.3125 0.704326
\(677\) −27.2287 −1.04648 −0.523242 0.852184i \(-0.675278\pi\)
−0.523242 + 0.852184i \(0.675278\pi\)
\(678\) 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 \(-0.448780\pi\)
0.160218 + 0.987082i \(0.448780\pi\)
\(702\) −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 \(-0.284115\pi\)
0.627411 + 0.778688i \(0.284115\pi\)
\(734\) −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 \(-0.320709\pi\)
0.533944 + 0.845520i \(0.320709\pi\)
\(740\) −6.42883 −0.236328
\(741\) 30.7732 1.13048
\(742\) 6.94887 0.255101
\(743\) 47.1515 1.72982 0.864911 0.501925i \(-0.167375\pi\)
0.864911 + 0.501925i \(0.167375\pi\)
\(744\) −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 \(-0.440396\pi\)
0.186159 + 0.982520i \(0.440396\pi\)
\(762\) −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 \(-0.565639\pi\)
−0.204754 + 0.978813i \(0.565639\pi\)
\(770\) 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 \(-0.202342\pi\)
0.804671 + 0.593721i \(0.202342\pi\)
\(788\) −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 \(-0.363970\pi\)
0.414462 + 0.910067i \(0.363970\pi\)
\(810\) −9.48528 −0.333279
\(811\) 14.9977 0.526641 0.263321 0.964708i \(-0.415182\pi\)
0.263321 + 0.964708i \(0.415182\pi\)
\(812\) −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 \(-0.267349\pi\)
0.667537 + 0.744576i \(0.267349\pi\)
\(822\) 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 \(-0.325353\pi\)
0.521554 + 0.853218i \(0.325353\pi\)
\(828\) 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 \(-0.437568\pi\)
0.194880 + 0.980827i \(0.437568\pi\)
\(854\) −4.29253 −0.146887
\(855\) 6.44292 0.220343
\(856\) −6.00481 −0.205240
\(857\) 37.8048 1.29139 0.645693 0.763597i \(-0.276569\pi\)
0.645693 + 0.763597i \(0.276569\pi\)
\(858\) 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 \(-0.612643\pi\)
−0.346537 + 0.938036i \(0.612643\pi\)
\(878\) 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 \(-0.338448\pi\)
0.486021 + 0.873947i \(0.338448\pi\)
\(888\) 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 \(-0.791189\pi\)
−0.792440 + 0.609950i \(0.791189\pi\)
\(920\) 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 \(-0.566848\pi\)
−0.208469 + 0.978029i \(0.566848\pi\)
\(938\) 11.1708 0.364741
\(939\) 24.2900 0.792675
\(940\) 1.51399 0.0493809
\(941\) 20.1593 0.657173 0.328586 0.944474i \(-0.393428\pi\)
0.328586 + 0.944474i \(0.393428\pi\)
\(942\) 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 \(-0.285229\pi\)
0.624681 + 0.780880i \(0.285229\pi\)
\(954\) −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 \(-0.246896\pi\)
0.713968 + 0.700178i \(0.246896\pi\)
\(968\) 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 \(-0.207175\pi\)
0.795563 + 0.605871i \(0.207175\pi\)
\(998\) −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.cr.1.1 yes 4
11.10 odd 2 8470.2.a.cp.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cp.1.2 4 11.10 odd 2
8470.2.a.cr.1.1 yes 4 1.1 even 1 trivial