Properties

Label 8470.2.a.cr.1.3
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 1.3
Root \(-0.517638\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.414214 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.414214 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.82843 q^{9} +1.00000 q^{10} +0.414214 q^{12} +0.696775 q^{13} -1.00000 q^{14} +0.414214 q^{15} +1.00000 q^{16} -0.449490 q^{17} -2.82843 q^{18} -3.37894 q^{19} +1.00000 q^{20} -0.414214 q^{21} -9.00997 q^{23} +0.414214 q^{24} +1.00000 q^{25} +0.696775 q^{26} -2.41421 q^{27} -1.00000 q^{28} -1.60040 q^{29} +0.414214 q^{30} +10.1769 q^{31} +1.00000 q^{32} -0.449490 q^{34} -1.00000 q^{35} -2.82843 q^{36} +3.32780 q^{37} -3.37894 q^{38} +0.288614 q^{39} +1.00000 q^{40} +6.12044 q^{41} -0.414214 q^{42} +7.14162 q^{43} -2.82843 q^{45} -9.00997 q^{46} -9.51399 q^{47} +0.414214 q^{48} +1.00000 q^{49} +1.00000 q^{50} -0.186185 q^{51} +0.696775 q^{52} -2.84909 q^{53} -2.41421 q^{54} -1.00000 q^{56} -1.39960 q^{57} -1.60040 q^{58} -7.97469 q^{59} +0.414214 q^{60} -8.29253 q^{61} +10.1769 q^{62} +2.82843 q^{63} +1.00000 q^{64} +0.696775 q^{65} +11.1708 q^{67} -0.449490 q^{68} -3.73205 q^{69} -1.00000 q^{70} -11.9202 q^{71} -2.82843 q^{72} +0.585786 q^{73} +3.32780 q^{74} +0.414214 q^{75} -3.37894 q^{76} +0.288614 q^{78} -10.5811 q^{79} +1.00000 q^{80} +7.48528 q^{81} +6.12044 q^{82} -17.5950 q^{83} -0.414214 q^{84} -0.449490 q^{85} +7.14162 q^{86} -0.662907 q^{87} -11.9700 q^{89} -2.82843 q^{90} -0.696775 q^{91} -9.00997 q^{92} +4.21541 q^{93} -9.51399 q^{94} -3.37894 q^{95} +0.414214 q^{96} -8.56993 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 4q^{7} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 4q^{7} + 4q^{8} + 4q^{10} - 4q^{12} - 4q^{14} - 4q^{15} + 4q^{16} + 8q^{17} - 12q^{19} + 4q^{20} + 4q^{21} - 8q^{23} - 4q^{24} + 4q^{25} - 4q^{27} - 4q^{28} - 8q^{29} - 4q^{30} + 4q^{32} + 8q^{34} - 4q^{35} - 16q^{37} - 12q^{38} + 8q^{39} + 4q^{40} + 8q^{41} + 4q^{42} - 8q^{43} - 8q^{46} - 16q^{47} - 4q^{48} + 4q^{49} + 4q^{50} - 8q^{51} - 4q^{54} - 4q^{56} - 4q^{57} - 8q^{58} - 8q^{59} - 4q^{60} - 8q^{61} + 4q^{64} + 8q^{68} - 8q^{69} - 4q^{70} - 8q^{71} + 8q^{73} - 16q^{74} - 4q^{75} - 12q^{76} + 8q^{78} - 24q^{79} + 4q^{80} - 4q^{81} + 8q^{82} + 8q^{83} + 4q^{84} + 8q^{85} - 8q^{86} + 16q^{87} - 8q^{92} + 16q^{93} - 16q^{94} - 12q^{95} - 4q^{96} - 8q^{97} + 4q^{98} + O(q^{100}) \)

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
\(3\) 0.414214 0.239146 0.119573 0.992825i \(-0.461847\pi\)
0.119573 + 0.992825i \(0.461847\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0.414214 0.169102
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.82843 −0.942809
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 0.414214 0.119573
\(13\) 0.696775 0.193251 0.0966253 0.995321i \(-0.469195\pi\)
0.0966253 + 0.995321i \(0.469195\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.414214 0.106949
\(16\) 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
\(19\) −3.37894 −0.775181 −0.387591 0.921832i \(-0.626693\pi\)
−0.387591 + 0.921832i \(0.626693\pi\)
\(20\) 1.00000 0.223607
\(21\) −0.414214 −0.0903888
\(22\) 0 0
\(23\) −9.00997 −1.87871 −0.939354 0.342949i \(-0.888574\pi\)
−0.939354 + 0.342949i \(0.888574\pi\)
\(24\) 0.414214 0.0845510
\(25\) 1.00000 0.200000
\(26\) 0.696775 0.136649
\(27\) −2.41421 −0.464616
\(28\) −1.00000 −0.188982
\(29\) −1.60040 −0.297187 −0.148593 0.988898i \(-0.547475\pi\)
−0.148593 + 0.988898i \(0.547475\pi\)
\(30\) 0.414214 0.0756247
\(31\) 10.1769 1.82782 0.913912 0.405912i \(-0.133046\pi\)
0.913912 + 0.405912i \(0.133046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.449490 −0.0770869
\(35\) −1.00000 −0.169031
\(36\) −2.82843 −0.471405
\(37\) 3.32780 0.547088 0.273544 0.961860i \(-0.411804\pi\)
0.273544 + 0.961860i \(0.411804\pi\)
\(38\) −3.37894 −0.548136
\(39\) 0.288614 0.0462151
\(40\) 1.00000 0.158114
\(41\) 6.12044 0.955852 0.477926 0.878400i \(-0.341389\pi\)
0.477926 + 0.878400i \(0.341389\pi\)
\(42\) −0.414214 −0.0639145
\(43\) 7.14162 1.08909 0.544543 0.838733i \(-0.316703\pi\)
0.544543 + 0.838733i \(0.316703\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) −9.00997 −1.32845
\(47\) −9.51399 −1.38776 −0.693879 0.720092i \(-0.744100\pi\)
−0.693879 + 0.720092i \(0.744100\pi\)
\(48\) 0.414214 0.0597866
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −0.186185 −0.0260711
\(52\) 0.696775 0.0966253
\(53\) −2.84909 −0.391353 −0.195676 0.980669i \(-0.562690\pi\)
−0.195676 + 0.980669i \(0.562690\pi\)
\(54\) −2.41421 −0.328533
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −1.39960 −0.185382
\(58\) −1.60040 −0.210143
\(59\) −7.97469 −1.03822 −0.519108 0.854709i \(-0.673736\pi\)
−0.519108 + 0.854709i \(0.673736\pi\)
\(60\) 0.414214 0.0534747
\(61\) −8.29253 −1.06175 −0.530875 0.847450i \(-0.678137\pi\)
−0.530875 + 0.847450i \(0.678137\pi\)
\(62\) 10.1769 1.29247
\(63\) 2.82843 0.356348
\(64\) 1.00000 0.125000
\(65\) 0.696775 0.0864243
\(66\) 0 0
\(67\) 11.1708 1.36474 0.682368 0.731009i \(-0.260950\pi\)
0.682368 + 0.731009i \(0.260950\pi\)
\(68\) −0.449490 −0.0545086
\(69\) −3.73205 −0.449286
\(70\) −1.00000 −0.119523
\(71\) −11.9202 −1.41466 −0.707331 0.706882i \(-0.750101\pi\)
−0.707331 + 0.706882i \(0.750101\pi\)
\(72\) −2.82843 −0.333333
\(73\) 0.585786 0.0685611 0.0342806 0.999412i \(-0.489086\pi\)
0.0342806 + 0.999412i \(0.489086\pi\)
\(74\) 3.32780 0.386849
\(75\) 0.414214 0.0478293
\(76\) −3.37894 −0.387591
\(77\) 0 0
\(78\) 0.288614 0.0326790
\(79\) −10.5811 −1.19047 −0.595236 0.803551i \(-0.702941\pi\)
−0.595236 + 0.803551i \(0.702941\pi\)
\(80\) 1.00000 0.111803
\(81\) 7.48528 0.831698
\(82\) 6.12044 0.675890
\(83\) −17.5950 −1.93130 −0.965652 0.259837i \(-0.916331\pi\)
−0.965652 + 0.259837i \(0.916331\pi\)
\(84\) −0.414214 −0.0451944
\(85\) −0.449490 −0.0487540
\(86\) 7.14162 0.770101
\(87\) −0.662907 −0.0710711
\(88\) 0 0
\(89\) −11.9700 −1.26882 −0.634411 0.772996i \(-0.718757\pi\)
−0.634411 + 0.772996i \(0.718757\pi\)
\(90\) −2.82843 −0.298142
\(91\) −0.696775 −0.0730418
\(92\) −9.00997 −0.939354
\(93\) 4.21541 0.437117
\(94\) −9.51399 −0.981293
\(95\) −3.37894 −0.346672
\(96\) 0.414214 0.0422755
\(97\) −8.56993 −0.870145 −0.435072 0.900396i \(-0.643277\pi\)
−0.435072 + 0.900396i \(0.643277\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −0.464102 −0.0461798 −0.0230899 0.999733i \(-0.507350\pi\)
−0.0230899 + 0.999733i \(0.507350\pi\)
\(102\) −0.186185 −0.0184350
\(103\) −7.55051 −0.743974 −0.371987 0.928238i \(-0.621323\pi\)
−0.371987 + 0.928238i \(0.621323\pi\)
\(104\) 0.696775 0.0683244
\(105\) −0.414214 −0.0404231
\(106\) −2.84909 −0.276728
\(107\) −14.4901 −1.40081 −0.700405 0.713746i \(-0.746997\pi\)
−0.700405 + 0.713746i \(0.746997\pi\)
\(108\) −2.41421 −0.232308
\(109\) 5.91359 0.566419 0.283210 0.959058i \(-0.408601\pi\)
0.283210 + 0.959058i \(0.408601\pi\)
\(110\) 0 0
\(111\) 1.37842 0.130834
\(112\) −1.00000 −0.0944911
\(113\) −19.0599 −1.79300 −0.896500 0.443043i \(-0.853899\pi\)
−0.896500 + 0.443043i \(0.853899\pi\)
\(114\) −1.39960 −0.131085
\(115\) −9.00997 −0.840184
\(116\) −1.60040 −0.148593
\(117\) −1.97078 −0.182198
\(118\) −7.97469 −0.734130
\(119\) 0.449490 0.0412047
\(120\) 0.414214 0.0378124
\(121\) 0 0
\(122\) −8.29253 −0.750770
\(123\) 2.53517 0.228589
\(124\) 10.1769 0.913912
\(125\) 1.00000 0.0894427
\(126\) 2.82843 0.251976
\(127\) 21.2334 1.88416 0.942078 0.335395i \(-0.108870\pi\)
0.942078 + 0.335395i \(0.108870\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.95816 0.260451
\(130\) 0.696775 0.0611112
\(131\) 7.67147 0.670259 0.335130 0.942172i \(-0.391220\pi\)
0.335130 + 0.942172i \(0.391220\pi\)
\(132\) 0 0
\(133\) 3.37894 0.292991
\(134\) 11.1708 0.965014
\(135\) −2.41421 −0.207782
\(136\) −0.449490 −0.0385434
\(137\) −0.211493 −0.0180691 −0.00903454 0.999959i \(-0.502876\pi\)
−0.00903454 + 0.999959i \(0.502876\pi\)
\(138\) −3.73205 −0.317693
\(139\) −7.34971 −0.623395 −0.311697 0.950181i \(-0.600897\pi\)
−0.311697 + 0.950181i \(0.600897\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −3.94082 −0.331877
\(142\) −11.9202 −1.00032
\(143\) 0 0
\(144\) −2.82843 −0.235702
\(145\) −1.60040 −0.132906
\(146\) 0.585786 0.0484800
\(147\) 0.414214 0.0341638
\(148\) 3.32780 0.273544
\(149\) 3.64173 0.298342 0.149171 0.988811i \(-0.452340\pi\)
0.149171 + 0.988811i \(0.452340\pi\)
\(150\) 0.414214 0.0338204
\(151\) 5.56099 0.452547 0.226274 0.974064i \(-0.427346\pi\)
0.226274 + 0.974064i \(0.427346\pi\)
\(152\) −3.37894 −0.274068
\(153\) 1.27135 0.102782
\(154\) 0 0
\(155\) 10.1769 0.817428
\(156\) 0.288614 0.0231076
\(157\) −13.8332 −1.10401 −0.552006 0.833840i \(-0.686138\pi\)
−0.552006 + 0.833840i \(0.686138\pi\)
\(158\) −10.5811 −0.841790
\(159\) −1.18013 −0.0935906
\(160\) 1.00000 0.0790569
\(161\) 9.00997 0.710085
\(162\) 7.48528 0.588099
\(163\) −16.2220 −1.27060 −0.635302 0.772264i \(-0.719124\pi\)
−0.635302 + 0.772264i \(0.719124\pi\)
\(164\) 6.12044 0.477926
\(165\) 0 0
\(166\) −17.5950 −1.36564
\(167\) −18.0547 −1.39711 −0.698557 0.715554i \(-0.746174\pi\)
−0.698557 + 0.715554i \(0.746174\pi\)
\(168\) −0.414214 −0.0319573
\(169\) −12.5145 −0.962654
\(170\) −0.449490 −0.0344743
\(171\) 9.55708 0.730848
\(172\) 7.14162 0.544543
\(173\) −16.2419 −1.23485 −0.617425 0.786630i \(-0.711824\pi\)
−0.617425 + 0.786630i \(0.711824\pi\)
\(174\) −0.662907 −0.0502548
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −3.30323 −0.248286
\(178\) −11.9700 −0.897193
\(179\) −13.4848 −1.00790 −0.503949 0.863733i \(-0.668120\pi\)
−0.503949 + 0.863733i \(0.668120\pi\)
\(180\) −2.82843 −0.210819
\(181\) 13.5300 1.00568 0.502839 0.864380i \(-0.332289\pi\)
0.502839 + 0.864380i \(0.332289\pi\)
\(182\) −0.696775 −0.0516484
\(183\) −3.43488 −0.253913
\(184\) −9.00997 −0.664224
\(185\) 3.32780 0.244665
\(186\) 4.21541 0.309089
\(187\) 0 0
\(188\) −9.51399 −0.693879
\(189\) 2.41421 0.175608
\(190\) −3.37894 −0.245134
\(191\) −18.2779 −1.32254 −0.661272 0.750146i \(-0.729983\pi\)
−0.661272 + 0.750146i \(0.729983\pi\)
\(192\) 0.414214 0.0298933
\(193\) 14.2986 1.02923 0.514617 0.857420i \(-0.327934\pi\)
0.514617 + 0.857420i \(0.327934\pi\)
\(194\) −8.56993 −0.615285
\(195\) 0.288614 0.0206680
\(196\) 1.00000 0.0714286
\(197\) 7.35023 0.523682 0.261841 0.965111i \(-0.415670\pi\)
0.261841 + 0.965111i \(0.415670\pi\)
\(198\) 0 0
\(199\) 13.7201 0.972593 0.486296 0.873794i \(-0.338348\pi\)
0.486296 + 0.873794i \(0.338348\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.62712 0.326372
\(202\) −0.464102 −0.0326541
\(203\) 1.60040 0.112326
\(204\) −0.186185 −0.0130355
\(205\) 6.12044 0.427470
\(206\) −7.55051 −0.526069
\(207\) 25.4840 1.77126
\(208\) 0.696775 0.0483126
\(209\) 0 0
\(210\) −0.414214 −0.0285835
\(211\) −2.06574 −0.142212 −0.0711059 0.997469i \(-0.522653\pi\)
−0.0711059 + 0.997469i \(0.522653\pi\)
\(212\) −2.84909 −0.195676
\(213\) −4.93749 −0.338311
\(214\) −14.4901 −0.990522
\(215\) 7.14162 0.487054
\(216\) −2.41421 −0.164266
\(217\) −10.1769 −0.690853
\(218\) 5.91359 0.400519
\(219\) 0.242641 0.0163961
\(220\) 0 0
\(221\) −0.313193 −0.0210676
\(222\) 1.37842 0.0925136
\(223\) −2.30839 −0.154581 −0.0772904 0.997009i \(-0.524627\pi\)
−0.0772904 + 0.997009i \(0.524627\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.82843 −0.188562
\(226\) −19.0599 −1.26784
\(227\) 8.40441 0.557820 0.278910 0.960317i \(-0.410027\pi\)
0.278910 + 0.960317i \(0.410027\pi\)
\(228\) −1.39960 −0.0926909
\(229\) 15.4830 1.02315 0.511573 0.859240i \(-0.329063\pi\)
0.511573 + 0.859240i \(0.329063\pi\)
\(230\) −9.00997 −0.594100
\(231\) 0 0
\(232\) −1.60040 −0.105071
\(233\) 12.9927 0.851180 0.425590 0.904916i \(-0.360067\pi\)
0.425590 + 0.904916i \(0.360067\pi\)
\(234\) −1.97078 −0.128834
\(235\) −9.51399 −0.620624
\(236\) −7.97469 −0.519108
\(237\) −4.38285 −0.284697
\(238\) 0.449490 0.0291361
\(239\) 24.4799 1.58347 0.791737 0.610862i \(-0.209177\pi\)
0.791737 + 0.610862i \(0.209177\pi\)
\(240\) 0.414214 0.0267374
\(241\) 13.6203 0.877363 0.438681 0.898643i \(-0.355446\pi\)
0.438681 + 0.898643i \(0.355446\pi\)
\(242\) 0 0
\(243\) 10.3431 0.663513
\(244\) −8.29253 −0.530875
\(245\) 1.00000 0.0638877
\(246\) 2.53517 0.161636
\(247\) −2.35436 −0.149804
\(248\) 10.1769 0.646234
\(249\) −7.28810 −0.461864
\(250\) 1.00000 0.0632456
\(251\) 3.27135 0.206486 0.103243 0.994656i \(-0.467078\pi\)
0.103243 + 0.994656i \(0.467078\pi\)
\(252\) 2.82843 0.178174
\(253\) 0 0
\(254\) 21.2334 1.33230
\(255\) −0.186185 −0.0116593
\(256\) 1.00000 0.0625000
\(257\) 24.2767 1.51434 0.757169 0.653219i \(-0.226582\pi\)
0.757169 + 0.653219i \(0.226582\pi\)
\(258\) 2.95816 0.184167
\(259\) −3.32780 −0.206780
\(260\) 0.696775 0.0432121
\(261\) 4.52661 0.280190
\(262\) 7.67147 0.473945
\(263\) −3.94887 −0.243498 −0.121749 0.992561i \(-0.538850\pi\)
−0.121749 + 0.992561i \(0.538850\pi\)
\(264\) 0 0
\(265\) −2.84909 −0.175018
\(266\) 3.37894 0.207176
\(267\) −4.95816 −0.303434
\(268\) 11.1708 0.682368
\(269\) 2.25866 0.137713 0.0688565 0.997627i \(-0.478065\pi\)
0.0688565 + 0.997627i \(0.478065\pi\)
\(270\) −2.41421 −0.146924
\(271\) 15.0479 0.914096 0.457048 0.889442i \(-0.348907\pi\)
0.457048 + 0.889442i \(0.348907\pi\)
\(272\) −0.449490 −0.0272543
\(273\) −0.288614 −0.0174677
\(274\) −0.211493 −0.0127768
\(275\) 0 0
\(276\) −3.73205 −0.224643
\(277\) −3.85714 −0.231753 −0.115876 0.993264i \(-0.536968\pi\)
−0.115876 + 0.993264i \(0.536968\pi\)
\(278\) −7.34971 −0.440807
\(279\) −28.7846 −1.72329
\(280\) −1.00000 −0.0597614
\(281\) −22.2655 −1.32825 −0.664123 0.747623i \(-0.731195\pi\)
−0.664123 + 0.747623i \(0.731195\pi\)
\(282\) −3.94082 −0.234673
\(283\) 5.73862 0.341125 0.170563 0.985347i \(-0.445441\pi\)
0.170563 + 0.985347i \(0.445441\pi\)
\(284\) −11.9202 −0.707331
\(285\) −1.39960 −0.0829052
\(286\) 0 0
\(287\) −6.12044 −0.361278
\(288\) −2.82843 −0.166667
\(289\) −16.7980 −0.988115
\(290\) −1.60040 −0.0939786
\(291\) −3.54978 −0.208092
\(292\) 0.585786 0.0342806
\(293\) 8.40425 0.490981 0.245491 0.969399i \(-0.421051\pi\)
0.245491 + 0.969399i \(0.421051\pi\)
\(294\) 0.414214 0.0241574
\(295\) −7.97469 −0.464304
\(296\) 3.32780 0.193425
\(297\) 0 0
\(298\) 3.64173 0.210960
\(299\) −6.27792 −0.363061
\(300\) 0.414214 0.0239146
\(301\) −7.14162 −0.411636
\(302\) 5.56099 0.319999
\(303\) −0.192237 −0.0110437
\(304\) −3.37894 −0.193795
\(305\) −8.29253 −0.474829
\(306\) 1.27135 0.0726782
\(307\) 20.8683 1.19102 0.595508 0.803349i \(-0.296951\pi\)
0.595508 + 0.803349i \(0.296951\pi\)
\(308\) 0 0
\(309\) −3.12752 −0.177919
\(310\) 10.1769 0.578009
\(311\) −20.1336 −1.14167 −0.570835 0.821065i \(-0.693380\pi\)
−0.570835 + 0.821065i \(0.693380\pi\)
\(312\) 0.288614 0.0163395
\(313\) −11.3326 −0.640557 −0.320279 0.947323i \(-0.603777\pi\)
−0.320279 + 0.947323i \(0.603777\pi\)
\(314\) −13.8332 −0.780655
\(315\) 2.82843 0.159364
\(316\) −10.5811 −0.595236
\(317\) −9.67951 −0.543655 −0.271828 0.962346i \(-0.587628\pi\)
−0.271828 + 0.962346i \(0.587628\pi\)
\(318\) −1.18013 −0.0661785
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −6.00199 −0.334999
\(322\) 9.00997 0.502106
\(323\) 1.51880 0.0845082
\(324\) 7.48528 0.415849
\(325\) 0.696775 0.0386501
\(326\) −16.2220 −0.898452
\(327\) 2.44949 0.135457
\(328\) 6.12044 0.337945
\(329\) 9.51399 0.524523
\(330\) 0 0
\(331\) −8.60572 −0.473013 −0.236507 0.971630i \(-0.576002\pi\)
−0.236507 + 0.971630i \(0.576002\pi\)
\(332\) −17.5950 −0.965652
\(333\) −9.41245 −0.515799
\(334\) −18.0547 −0.987909
\(335\) 11.1708 0.610328
\(336\) −0.414214 −0.0225972
\(337\) 24.7814 1.34993 0.674964 0.737851i \(-0.264159\pi\)
0.674964 + 0.737851i \(0.264159\pi\)
\(338\) −12.5145 −0.680699
\(339\) −7.89485 −0.428789
\(340\) −0.449490 −0.0243770
\(341\) 0 0
\(342\) 9.55708 0.516788
\(343\) −1.00000 −0.0539949
\(344\) 7.14162 0.385050
\(345\) −3.73205 −0.200927
\(346\) −16.2419 −0.873171
\(347\) −28.6040 −1.53554 −0.767771 0.640725i \(-0.778634\pi\)
−0.767771 + 0.640725i \(0.778634\pi\)
\(348\) −0.662907 −0.0355355
\(349\) −13.7967 −0.738521 −0.369261 0.929326i \(-0.620389\pi\)
−0.369261 + 0.929326i \(0.620389\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −1.68216 −0.0897872
\(352\) 0 0
\(353\) 18.0265 0.959454 0.479727 0.877418i \(-0.340736\pi\)
0.479727 + 0.877418i \(0.340736\pi\)
\(354\) −3.30323 −0.175564
\(355\) −11.9202 −0.632656
\(356\) −11.9700 −0.634411
\(357\) 0.186185 0.00985394
\(358\) −13.4848 −0.712692
\(359\) 8.39127 0.442875 0.221437 0.975175i \(-0.428925\pi\)
0.221437 + 0.975175i \(0.428925\pi\)
\(360\) −2.82843 −0.149071
\(361\) −7.58278 −0.399094
\(362\) 13.5300 0.711122
\(363\) 0 0
\(364\) −0.696775 −0.0365209
\(365\) 0.585786 0.0306615
\(366\) −3.43488 −0.179544
\(367\) −20.7882 −1.08513 −0.542566 0.840013i \(-0.682547\pi\)
−0.542566 + 0.840013i \(0.682547\pi\)
\(368\) −9.00997 −0.469677
\(369\) −17.3112 −0.901186
\(370\) 3.32780 0.173004
\(371\) 2.84909 0.147917
\(372\) 4.21541 0.218559
\(373\) 17.8705 0.925300 0.462650 0.886541i \(-0.346899\pi\)
0.462650 + 0.886541i \(0.346899\pi\)
\(374\) 0 0
\(375\) 0.414214 0.0213899
\(376\) −9.51399 −0.490647
\(377\) −1.11512 −0.0574314
\(378\) 2.41421 0.124174
\(379\) 8.83324 0.453733 0.226866 0.973926i \(-0.427152\pi\)
0.226866 + 0.973926i \(0.427152\pi\)
\(380\) −3.37894 −0.173336
\(381\) 8.79514 0.450589
\(382\) −18.2779 −0.935180
\(383\) 9.77249 0.499351 0.249675 0.968330i \(-0.419676\pi\)
0.249675 + 0.968330i \(0.419676\pi\)
\(384\) 0.414214 0.0211377
\(385\) 0 0
\(386\) 14.2986 0.727779
\(387\) −20.1996 −1.02680
\(388\) −8.56993 −0.435072
\(389\) −2.04508 −0.103690 −0.0518448 0.998655i \(-0.516510\pi\)
−0.0518448 + 0.998655i \(0.516510\pi\)
\(390\) 0.288614 0.0146145
\(391\) 4.04989 0.204812
\(392\) 1.00000 0.0505076
\(393\) 3.17763 0.160290
\(394\) 7.35023 0.370299
\(395\) −10.5811 −0.532395
\(396\) 0 0
\(397\) 35.7259 1.79303 0.896517 0.443009i \(-0.146089\pi\)
0.896517 + 0.443009i \(0.146089\pi\)
\(398\) 13.7201 0.687727
\(399\) 1.39960 0.0700677
\(400\) 1.00000 0.0500000
\(401\) 12.5334 0.625889 0.312944 0.949771i \(-0.398685\pi\)
0.312944 + 0.949771i \(0.398685\pi\)
\(402\) 4.62712 0.230780
\(403\) 7.09100 0.353228
\(404\) −0.464102 −0.0230899
\(405\) 7.48528 0.371947
\(406\) 1.60040 0.0794264
\(407\) 0 0
\(408\) −0.186185 −0.00921752
\(409\) −35.7625 −1.76834 −0.884170 0.467165i \(-0.845275\pi\)
−0.884170 + 0.467165i \(0.845275\pi\)
\(410\) 6.12044 0.302267
\(411\) −0.0876034 −0.00432116
\(412\) −7.55051 −0.371987
\(413\) 7.97469 0.392409
\(414\) 25.4840 1.25247
\(415\) −17.5950 −0.863706
\(416\) 0.696775 0.0341622
\(417\) −3.04435 −0.149083
\(418\) 0 0
\(419\) 9.13416 0.446233 0.223116 0.974792i \(-0.428377\pi\)
0.223116 + 0.974792i \(0.428377\pi\)
\(420\) −0.414214 −0.0202116
\(421\) −31.3404 −1.52744 −0.763720 0.645548i \(-0.776629\pi\)
−0.763720 + 0.645548i \(0.776629\pi\)
\(422\) −2.06574 −0.100559
\(423\) 26.9096 1.30839
\(424\) −2.84909 −0.138364
\(425\) −0.449490 −0.0218035
\(426\) −4.93749 −0.239222
\(427\) 8.29253 0.401304
\(428\) −14.4901 −0.700405
\(429\) 0 0
\(430\) 7.14162 0.344400
\(431\) 1.68216 0.0810269 0.0405135 0.999179i \(-0.487101\pi\)
0.0405135 + 0.999179i \(0.487101\pi\)
\(432\) −2.41421 −0.116154
\(433\) −5.91608 −0.284309 −0.142154 0.989845i \(-0.545403\pi\)
−0.142154 + 0.989845i \(0.545403\pi\)
\(434\) −10.1769 −0.488507
\(435\) −0.662907 −0.0317839
\(436\) 5.91359 0.283210
\(437\) 30.4441 1.45634
\(438\) 0.242641 0.0115938
\(439\) 16.6904 0.796588 0.398294 0.917258i \(-0.369602\pi\)
0.398294 + 0.917258i \(0.369602\pi\)
\(440\) 0 0
\(441\) −2.82843 −0.134687
\(442\) −0.313193 −0.0148971
\(443\) −27.4203 −1.30278 −0.651389 0.758744i \(-0.725813\pi\)
−0.651389 + 0.758744i \(0.725813\pi\)
\(444\) 1.37842 0.0654170
\(445\) −11.9700 −0.567435
\(446\) −2.30839 −0.109305
\(447\) 1.50845 0.0713474
\(448\) −1.00000 −0.0472456
\(449\) 0.608726 0.0287276 0.0143638 0.999897i \(-0.495428\pi\)
0.0143638 + 0.999897i \(0.495428\pi\)
\(450\) −2.82843 −0.133333
\(451\) 0 0
\(452\) −19.0599 −0.896500
\(453\) 2.30344 0.108225
\(454\) 8.40441 0.394438
\(455\) −0.696775 −0.0326653
\(456\) −1.39960 −0.0655424
\(457\) −13.4794 −0.630542 −0.315271 0.949002i \(-0.602095\pi\)
−0.315271 + 0.949002i \(0.602095\pi\)
\(458\) 15.4830 0.723473
\(459\) 1.08516 0.0506511
\(460\) −9.00997 −0.420092
\(461\) −19.5891 −0.912356 −0.456178 0.889888i \(-0.650782\pi\)
−0.456178 + 0.889888i \(0.650782\pi\)
\(462\) 0 0
\(463\) −24.3330 −1.13085 −0.565424 0.824800i \(-0.691288\pi\)
−0.565424 + 0.824800i \(0.691288\pi\)
\(464\) −1.60040 −0.0742966
\(465\) 4.21541 0.195485
\(466\) 12.9927 0.601875
\(467\) −28.3412 −1.31147 −0.655737 0.754990i \(-0.727642\pi\)
−0.655737 + 0.754990i \(0.727642\pi\)
\(468\) −1.97078 −0.0910992
\(469\) −11.1708 −0.515822
\(470\) −9.51399 −0.438848
\(471\) −5.72991 −0.264020
\(472\) −7.97469 −0.367065
\(473\) 0 0
\(474\) −4.38285 −0.201311
\(475\) −3.37894 −0.155036
\(476\) 0.449490 0.0206023
\(477\) 8.05845 0.368971
\(478\) 24.4799 1.11968
\(479\) 26.9207 1.23004 0.615019 0.788513i \(-0.289149\pi\)
0.615019 + 0.788513i \(0.289149\pi\)
\(480\) 0.414214 0.0189062
\(481\) 2.31873 0.105725
\(482\) 13.6203 0.620389
\(483\) 3.73205 0.169814
\(484\) 0 0
\(485\) −8.56993 −0.389140
\(486\) 10.3431 0.469175
\(487\) 10.5353 0.477402 0.238701 0.971093i \(-0.423278\pi\)
0.238701 + 0.971093i \(0.423278\pi\)
\(488\) −8.29253 −0.375385
\(489\) −6.71936 −0.303860
\(490\) 1.00000 0.0451754
\(491\) −12.3321 −0.556540 −0.278270 0.960503i \(-0.589761\pi\)
−0.278270 + 0.960503i \(0.589761\pi\)
\(492\) 2.53517 0.114294
\(493\) 0.719363 0.0323985
\(494\) −2.35436 −0.105928
\(495\) 0 0
\(496\) 10.1769 0.456956
\(497\) 11.9202 0.534692
\(498\) −7.28810 −0.326587
\(499\) 37.4485 1.67643 0.838214 0.545342i \(-0.183600\pi\)
0.838214 + 0.545342i \(0.183600\pi\)
\(500\) 1.00000 0.0447214
\(501\) −7.47850 −0.334115
\(502\) 3.27135 0.146007
\(503\) −7.53393 −0.335921 −0.167961 0.985794i \(-0.553718\pi\)
−0.167961 + 0.985794i \(0.553718\pi\)
\(504\) 2.82843 0.125988
\(505\) −0.464102 −0.0206523
\(506\) 0 0
\(507\) −5.18368 −0.230215
\(508\) 21.2334 0.942078
\(509\) 33.7306 1.49508 0.747541 0.664216i \(-0.231234\pi\)
0.747541 + 0.664216i \(0.231234\pi\)
\(510\) −0.186185 −0.00824440
\(511\) −0.585786 −0.0259137
\(512\) 1.00000 0.0441942
\(513\) 8.15748 0.360161
\(514\) 24.2767 1.07080
\(515\) −7.55051 −0.332715
\(516\) 2.95816 0.130226
\(517\) 0 0
\(518\) −3.32780 −0.146215
\(519\) −6.72762 −0.295310
\(520\) 0.696775 0.0305556
\(521\) 17.4407 0.764092 0.382046 0.924143i \(-0.375220\pi\)
0.382046 + 0.924143i \(0.375220\pi\)
\(522\) 4.52661 0.198124
\(523\) −11.7069 −0.511907 −0.255954 0.966689i \(-0.582389\pi\)
−0.255954 + 0.966689i \(0.582389\pi\)
\(524\) 7.67147 0.335130
\(525\) −0.414214 −0.0180778
\(526\) −3.94887 −0.172179
\(527\) −4.57441 −0.199264
\(528\) 0 0
\(529\) 58.1795 2.52954
\(530\) −2.84909 −0.123757
\(531\) 22.5558 0.978840
\(532\) 3.37894 0.146496
\(533\) 4.26457 0.184719
\(534\) −4.95816 −0.214560
\(535\) −14.4901 −0.626461
\(536\) 11.1708 0.482507
\(537\) −5.58557 −0.241035
\(538\) 2.25866 0.0973778
\(539\) 0 0
\(540\) −2.41421 −0.103891
\(541\) −24.2686 −1.04339 −0.521695 0.853132i \(-0.674700\pi\)
−0.521695 + 0.853132i \(0.674700\pi\)
\(542\) 15.0479 0.646363
\(543\) 5.60431 0.240504
\(544\) −0.449490 −0.0192717
\(545\) 5.91359 0.253310
\(546\) −0.288614 −0.0123515
\(547\) 21.6629 0.926240 0.463120 0.886296i \(-0.346730\pi\)
0.463120 + 0.886296i \(0.346730\pi\)
\(548\) −0.211493 −0.00903454
\(549\) 23.4548 1.00103
\(550\) 0 0
\(551\) 5.40765 0.230373
\(552\) −3.73205 −0.158847
\(553\) 10.5811 0.449956
\(554\) −3.85714 −0.163874
\(555\) 1.37842 0.0585108
\(556\) −7.34971 −0.311697
\(557\) 31.4409 1.33220 0.666098 0.745864i \(-0.267963\pi\)
0.666098 + 0.745864i \(0.267963\pi\)
\(558\) −28.7846 −1.21855
\(559\) 4.97610 0.210467
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −22.2655 −0.939212
\(563\) 44.4010 1.87128 0.935640 0.352955i \(-0.114823\pi\)
0.935640 + 0.352955i \(0.114823\pi\)
\(564\) −3.94082 −0.165939
\(565\) −19.0599 −0.801854
\(566\) 5.73862 0.241212
\(567\) −7.48528 −0.314352
\(568\) −11.9202 −0.500159
\(569\) 2.44077 0.102322 0.0511611 0.998690i \(-0.483708\pi\)
0.0511611 + 0.998690i \(0.483708\pi\)
\(570\) −1.39960 −0.0586229
\(571\) 4.79639 0.200723 0.100361 0.994951i \(-0.468000\pi\)
0.100361 + 0.994951i \(0.468000\pi\)
\(572\) 0 0
\(573\) −7.57096 −0.316281
\(574\) −6.12044 −0.255462
\(575\) −9.00997 −0.375742
\(576\) −2.82843 −0.117851
\(577\) −23.9982 −0.999060 −0.499530 0.866297i \(-0.666494\pi\)
−0.499530 + 0.866297i \(0.666494\pi\)
\(578\) −16.7980 −0.698703
\(579\) 5.92267 0.246138
\(580\) −1.60040 −0.0664529
\(581\) 17.5950 0.729965
\(582\) −3.54978 −0.147143
\(583\) 0 0
\(584\) 0.585786 0.0242400
\(585\) −1.97078 −0.0814816
\(586\) 8.40425 0.347176
\(587\) 13.2344 0.546241 0.273121 0.961980i \(-0.411944\pi\)
0.273121 + 0.961980i \(0.411944\pi\)
\(588\) 0.414214 0.0170819
\(589\) −34.3871 −1.41690
\(590\) −7.97469 −0.328313
\(591\) 3.04456 0.125237
\(592\) 3.32780 0.136772
\(593\) 36.6395 1.50460 0.752302 0.658818i \(-0.228943\pi\)
0.752302 + 0.658818i \(0.228943\pi\)
\(594\) 0 0
\(595\) 0.449490 0.0184273
\(596\) 3.64173 0.149171
\(597\) 5.68306 0.232592
\(598\) −6.27792 −0.256723
\(599\) 2.81485 0.115011 0.0575057 0.998345i \(-0.481685\pi\)
0.0575057 + 0.998345i \(0.481685\pi\)
\(600\) 0.414214 0.0169102
\(601\) 27.0224 1.10227 0.551134 0.834417i \(-0.314195\pi\)
0.551134 + 0.834417i \(0.314195\pi\)
\(602\) −7.14162 −0.291071
\(603\) −31.5959 −1.28669
\(604\) 5.56099 0.226274
\(605\) 0 0
\(606\) −0.192237 −0.00780910
\(607\) −18.7130 −0.759538 −0.379769 0.925081i \(-0.623997\pi\)
−0.379769 + 0.925081i \(0.623997\pi\)
\(608\) −3.37894 −0.137034
\(609\) 0.662907 0.0268623
\(610\) −8.29253 −0.335755
\(611\) −6.62911 −0.268185
\(612\) 1.27135 0.0513912
\(613\) −13.1333 −0.530449 −0.265224 0.964187i \(-0.585446\pi\)
−0.265224 + 0.964187i \(0.585446\pi\)
\(614\) 20.8683 0.842176
\(615\) 2.53517 0.102228
\(616\) 0 0
\(617\) −0.553017 −0.0222636 −0.0111318 0.999938i \(-0.503543\pi\)
−0.0111318 + 0.999938i \(0.503543\pi\)
\(618\) −3.12752 −0.125807
\(619\) −17.8938 −0.719213 −0.359607 0.933104i \(-0.617089\pi\)
−0.359607 + 0.933104i \(0.617089\pi\)
\(620\) 10.1769 0.408714
\(621\) 21.7520 0.872877
\(622\) −20.1336 −0.807283
\(623\) 11.9700 0.479570
\(624\) 0.288614 0.0115538
\(625\) 1.00000 0.0400000
\(626\) −11.3326 −0.452942
\(627\) 0 0
\(628\) −13.8332 −0.552006
\(629\) −1.49581 −0.0596420
\(630\) 2.82843 0.112687
\(631\) 45.6251 1.81631 0.908154 0.418636i \(-0.137492\pi\)
0.908154 + 0.418636i \(0.137492\pi\)
\(632\) −10.5811 −0.420895
\(633\) −0.855659 −0.0340094
\(634\) −9.67951 −0.384422
\(635\) 21.2334 0.842620
\(636\) −1.18013 −0.0467953
\(637\) 0.696775 0.0276072
\(638\) 0 0
\(639\) 33.7153 1.33376
\(640\) 1.00000 0.0395285
\(641\) −23.2522 −0.918408 −0.459204 0.888331i \(-0.651865\pi\)
−0.459204 + 0.888331i \(0.651865\pi\)
\(642\) −6.00199 −0.236880
\(643\) −10.3407 −0.407796 −0.203898 0.978992i \(-0.565361\pi\)
−0.203898 + 0.978992i \(0.565361\pi\)
\(644\) 9.00997 0.355042
\(645\) 2.95816 0.116477
\(646\) 1.51880 0.0597563
\(647\) 14.5229 0.570952 0.285476 0.958386i \(-0.407848\pi\)
0.285476 + 0.958386i \(0.407848\pi\)
\(648\) 7.48528 0.294050
\(649\) 0 0
\(650\) 0.696775 0.0273297
\(651\) −4.21541 −0.165215
\(652\) −16.2220 −0.635302
\(653\) −15.2419 −0.596461 −0.298231 0.954494i \(-0.596396\pi\)
−0.298231 + 0.954494i \(0.596396\pi\)
\(654\) 2.44949 0.0957826
\(655\) 7.67147 0.299749
\(656\) 6.12044 0.238963
\(657\) −1.65685 −0.0646400
\(658\) 9.51399 0.370894
\(659\) −45.7851 −1.78354 −0.891768 0.452494i \(-0.850535\pi\)
−0.891768 + 0.452494i \(0.850535\pi\)
\(660\) 0 0
\(661\) −4.64993 −0.180861 −0.0904307 0.995903i \(-0.528824\pi\)
−0.0904307 + 0.995903i \(0.528824\pi\)
\(662\) −8.60572 −0.334471
\(663\) −0.129729 −0.00503825
\(664\) −17.5950 −0.682819
\(665\) 3.37894 0.131030
\(666\) −9.41245 −0.364725
\(667\) 14.4195 0.558327
\(668\) −18.0547 −0.698557
\(669\) −0.956164 −0.0369675
\(670\) 11.1708 0.431567
\(671\) 0 0
\(672\) −0.414214 −0.0159786
\(673\) 21.5428 0.830413 0.415206 0.909727i \(-0.363709\pi\)
0.415206 + 0.909727i \(0.363709\pi\)
\(674\) 24.7814 0.954543
\(675\) −2.41421 −0.0929231
\(676\) −12.5145 −0.481327
\(677\) 44.5318 1.71150 0.855748 0.517393i \(-0.173098\pi\)
0.855748 + 0.517393i \(0.173098\pi\)
\(678\) −7.89485 −0.303200
\(679\) 8.56993 0.328884
\(680\) −0.449490 −0.0172371
\(681\) 3.48122 0.133401
\(682\) 0 0
\(683\) 24.3276 0.930869 0.465435 0.885082i \(-0.345898\pi\)
0.465435 + 0.885082i \(0.345898\pi\)
\(684\) 9.55708 0.365424
\(685\) −0.211493 −0.00808074
\(686\) −1.00000 −0.0381802
\(687\) 6.41327 0.244682
\(688\) 7.14162 0.272272
\(689\) −1.98517 −0.0756291
\(690\) −3.73205 −0.142077
\(691\) 28.8708 1.09830 0.549148 0.835725i \(-0.314952\pi\)
0.549148 + 0.835725i \(0.314952\pi\)
\(692\) −16.2419 −0.617425
\(693\) 0 0
\(694\) −28.6040 −1.08579
\(695\) −7.34971 −0.278791
\(696\) −0.662907 −0.0251274
\(697\) −2.75108 −0.104204
\(698\) −13.7967 −0.522213
\(699\) 5.38175 0.203557
\(700\) −1.00000 −0.0377964
\(701\) 40.0109 1.51119 0.755595 0.655039i \(-0.227348\pi\)
0.755595 + 0.655039i \(0.227348\pi\)
\(702\) −1.68216 −0.0634891
\(703\) −11.2444 −0.424092
\(704\) 0 0
\(705\) −3.94082 −0.148420
\(706\) 18.0265 0.678436
\(707\) 0.464102 0.0174543
\(708\) −3.30323 −0.124143
\(709\) −16.3266 −0.613157 −0.306578 0.951845i \(-0.599184\pi\)
−0.306578 + 0.951845i \(0.599184\pi\)
\(710\) −11.9202 −0.447356
\(711\) 29.9280 1.12239
\(712\) −11.9700 −0.448596
\(713\) −91.6935 −3.43395
\(714\) 0.186185 0.00696779
\(715\) 0 0
\(716\) −13.4848 −0.503949
\(717\) 10.1399 0.378682
\(718\) 8.39127 0.313160
\(719\) −29.1213 −1.08604 −0.543020 0.839720i \(-0.682719\pi\)
−0.543020 + 0.839720i \(0.682719\pi\)
\(720\) −2.82843 −0.105409
\(721\) 7.55051 0.281196
\(722\) −7.58278 −0.282202
\(723\) 5.64173 0.209818
\(724\) 13.5300 0.502839
\(725\) −1.60040 −0.0594373
\(726\) 0 0
\(727\) −0.229273 −0.00850327 −0.00425164 0.999991i \(-0.501353\pi\)
−0.00425164 + 0.999991i \(0.501353\pi\)
\(728\) −0.696775 −0.0258242
\(729\) −18.1716 −0.673021
\(730\) 0.585786 0.0216809
\(731\) −3.21009 −0.118729
\(732\) −3.43488 −0.126957
\(733\) −25.0740 −0.926130 −0.463065 0.886324i \(-0.653250\pi\)
−0.463065 + 0.886324i \(0.653250\pi\)
\(734\) −20.7882 −0.767305
\(735\) 0.414214 0.0152785
\(736\) −9.00997 −0.332112
\(737\) 0 0
\(738\) −17.3112 −0.637235
\(739\) 7.95973 0.292803 0.146402 0.989225i \(-0.453231\pi\)
0.146402 + 0.989225i \(0.453231\pi\)
\(740\) 3.32780 0.122333
\(741\) −0.975207 −0.0358251
\(742\) 2.84909 0.104593
\(743\) −11.9597 −0.438759 −0.219379 0.975640i \(-0.570403\pi\)
−0.219379 + 0.975640i \(0.570403\pi\)
\(744\) 4.21541 0.154544
\(745\) 3.64173 0.133423
\(746\) 17.8705 0.654286
\(747\) 49.7662 1.82085
\(748\) 0 0
\(749\) 14.4901 0.529456
\(750\) 0.414214 0.0151249
\(751\) −26.7728 −0.976954 −0.488477 0.872577i \(-0.662447\pi\)
−0.488477 + 0.872577i \(0.662447\pi\)
\(752\) −9.51399 −0.346940
\(753\) 1.35504 0.0493803
\(754\) −1.11512 −0.0406102
\(755\) 5.56099 0.202385
\(756\) 2.41421 0.0878041
\(757\) −15.7539 −0.572586 −0.286293 0.958142i \(-0.592423\pi\)
−0.286293 + 0.958142i \(0.592423\pi\)
\(758\) 8.83324 0.320838
\(759\) 0 0
\(760\) −3.37894 −0.122567
\(761\) 27.5271 0.997858 0.498929 0.866643i \(-0.333727\pi\)
0.498929 + 0.866643i \(0.333727\pi\)
\(762\) 8.79514 0.318614
\(763\) −5.91359 −0.214086
\(764\) −18.2779 −0.661272
\(765\) 1.27135 0.0459657
\(766\) 9.77249 0.353094
\(767\) −5.55656 −0.200636
\(768\) 0.414214 0.0149466
\(769\) −27.3409 −0.985940 −0.492970 0.870046i \(-0.664089\pi\)
−0.492970 + 0.870046i \(0.664089\pi\)
\(770\) 0 0
\(771\) 10.0557 0.362148
\(772\) 14.2986 0.514617
\(773\) −37.4833 −1.34818 −0.674090 0.738649i \(-0.735464\pi\)
−0.674090 + 0.738649i \(0.735464\pi\)
\(774\) −20.1996 −0.726058
\(775\) 10.1769 0.365565
\(776\) −8.56993 −0.307643
\(777\) −1.37842 −0.0494506
\(778\) −2.04508 −0.0733197
\(779\) −20.6806 −0.740959
\(780\) 0.288614 0.0103340
\(781\) 0 0
\(782\) 4.04989 0.144824
\(783\) 3.86370 0.138077
\(784\) 1.00000 0.0357143
\(785\) −13.8332 −0.493729
\(786\) 3.17763 0.113342
\(787\) −3.34973 −0.119405 −0.0597025 0.998216i \(-0.519015\pi\)
−0.0597025 + 0.998216i \(0.519015\pi\)
\(788\) 7.35023 0.261841
\(789\) −1.63567 −0.0582316
\(790\) −10.5811 −0.376460
\(791\) 19.0599 0.677690
\(792\) 0 0
\(793\) −5.77802 −0.205184
\(794\) 35.7259 1.26787
\(795\) −1.18013 −0.0418550
\(796\) 13.7201 0.486296
\(797\) 22.1429 0.784341 0.392171 0.919893i \(-0.371724\pi\)
0.392171 + 0.919893i \(0.371724\pi\)
\(798\) 1.39960 0.0495454
\(799\) 4.27644 0.151290
\(800\) 1.00000 0.0353553
\(801\) 33.8564 1.19626
\(802\) 12.5334 0.442570
\(803\) 0 0
\(804\) 4.62712 0.163186
\(805\) 9.00997 0.317560
\(806\) 7.09100 0.249770
\(807\) 0.935568 0.0329336
\(808\) −0.464102 −0.0163270
\(809\) 2.22095 0.0780843 0.0390421 0.999238i \(-0.487569\pi\)
0.0390421 + 0.999238i \(0.487569\pi\)
\(810\) 7.48528 0.263006
\(811\) 46.3962 1.62919 0.814595 0.580031i \(-0.196959\pi\)
0.814595 + 0.580031i \(0.196959\pi\)
\(812\) 1.60040 0.0561630
\(813\) 6.23305 0.218603
\(814\) 0 0
\(815\) −16.2220 −0.568231
\(816\) −0.186185 −0.00651777
\(817\) −24.1311 −0.844240
\(818\) −35.7625 −1.25041
\(819\) 1.97078 0.0688645
\(820\) 6.12044 0.213735
\(821\) 35.1399 1.22639 0.613195 0.789932i \(-0.289884\pi\)
0.613195 + 0.789932i \(0.289884\pi\)
\(822\) −0.0876034 −0.00305552
\(823\) −8.90147 −0.310286 −0.155143 0.987892i \(-0.549584\pi\)
−0.155143 + 0.987892i \(0.549584\pi\)
\(824\) −7.55051 −0.263034
\(825\) 0 0
\(826\) 7.97469 0.277475
\(827\) 22.4976 0.782319 0.391159 0.920323i \(-0.372074\pi\)
0.391159 + 0.920323i \(0.372074\pi\)
\(828\) 25.4840 0.885632
\(829\) −5.11350 −0.177599 −0.0887995 0.996050i \(-0.528303\pi\)
−0.0887995 + 0.996050i \(0.528303\pi\)
\(830\) −17.5950 −0.610732
\(831\) −1.59768 −0.0554228
\(832\) 0.696775 0.0241563
\(833\) −0.449490 −0.0155739
\(834\) −3.04435 −0.105417
\(835\) −18.0547 −0.624809
\(836\) 0 0
\(837\) −24.5692 −0.849236
\(838\) 9.13416 0.315534
\(839\) 4.06199 0.140236 0.0701178 0.997539i \(-0.477662\pi\)
0.0701178 + 0.997539i \(0.477662\pi\)
\(840\) −0.414214 −0.0142917
\(841\) −26.4387 −0.911680
\(842\) −31.3404 −1.08006
\(843\) −9.22266 −0.317645
\(844\) −2.06574 −0.0711059
\(845\) −12.5145 −0.430512
\(846\) 26.9096 0.925172
\(847\) 0 0
\(848\) −2.84909 −0.0978382
\(849\) 2.37701 0.0815789
\(850\) −0.449490 −0.0154174
\(851\) −29.9834 −1.02782
\(852\) −4.93749 −0.169156
\(853\) 40.3033 1.37996 0.689980 0.723828i \(-0.257619\pi\)
0.689980 + 0.723828i \(0.257619\pi\)
\(854\) 8.29253 0.283764
\(855\) 9.55708 0.326845
\(856\) −14.4901 −0.495261
\(857\) 25.7912 0.881009 0.440505 0.897750i \(-0.354800\pi\)
0.440505 + 0.897750i \(0.354800\pi\)
\(858\) 0 0
\(859\) 30.2497 1.03211 0.516054 0.856556i \(-0.327400\pi\)
0.516054 + 0.856556i \(0.327400\pi\)
\(860\) 7.14162 0.243527
\(861\) −2.53517 −0.0863983
\(862\) 1.68216 0.0572947
\(863\) 16.2691 0.553806 0.276903 0.960898i \(-0.410692\pi\)
0.276903 + 0.960898i \(0.410692\pi\)
\(864\) −2.41421 −0.0821332
\(865\) −16.2419 −0.552242
\(866\) −5.91608 −0.201037
\(867\) −6.95794 −0.236304
\(868\) −10.1769 −0.345426
\(869\) 0 0
\(870\) −0.662907 −0.0224746
\(871\) 7.78356 0.263736
\(872\) 5.91359 0.200259
\(873\) 24.2394 0.820380
\(874\) 30.4441 1.02979
\(875\) −1.00000 −0.0338062
\(876\) 0.242641 0.00819807
\(877\) −34.6670 −1.17062 −0.585310 0.810809i \(-0.699027\pi\)
−0.585310 + 0.810809i \(0.699027\pi\)
\(878\) 16.6904 0.563273
\(879\) 3.48115 0.117416
\(880\) 0 0
\(881\) −15.0738 −0.507849 −0.253924 0.967224i \(-0.581721\pi\)
−0.253924 + 0.967224i \(0.581721\pi\)
\(882\) −2.82843 −0.0952381
\(883\) −1.63994 −0.0551884 −0.0275942 0.999619i \(-0.508785\pi\)
−0.0275942 + 0.999619i \(0.508785\pi\)
\(884\) −0.313193 −0.0105338
\(885\) −3.30323 −0.111037
\(886\) −27.4203 −0.921202
\(887\) −14.7479 −0.495185 −0.247592 0.968864i \(-0.579639\pi\)
−0.247592 + 0.968864i \(0.579639\pi\)
\(888\) 1.37842 0.0462568
\(889\) −21.2334 −0.712144
\(890\) −11.9700 −0.401237
\(891\) 0 0
\(892\) −2.30839 −0.0772904
\(893\) 32.1472 1.07576
\(894\) 1.50845 0.0504502
\(895\) −13.4848 −0.450746
\(896\) −1.00000 −0.0334077
\(897\) −2.60040 −0.0868248
\(898\) 0.608726 0.0203135
\(899\) −16.2871 −0.543205
\(900\) −2.82843 −0.0942809
\(901\) 1.28064 0.0426642
\(902\) 0 0
\(903\) −2.95816 −0.0984413
\(904\) −19.0599 −0.633921
\(905\) 13.5300 0.449753
\(906\) 2.30344 0.0765266
\(907\) −55.5813 −1.84555 −0.922774 0.385342i \(-0.874083\pi\)
−0.922774 + 0.385342i \(0.874083\pi\)
\(908\) 8.40441 0.278910
\(909\) 1.31268 0.0435388
\(910\) −0.696775 −0.0230979
\(911\) 14.8844 0.493141 0.246571 0.969125i \(-0.420696\pi\)
0.246571 + 0.969125i \(0.420696\pi\)
\(912\) −1.39960 −0.0463454
\(913\) 0 0
\(914\) −13.4794 −0.445860
\(915\) −3.43488 −0.113554
\(916\) 15.4830 0.511573
\(917\) −7.67147 −0.253334
\(918\) 1.08516 0.0358158
\(919\) −39.1462 −1.29131 −0.645657 0.763628i \(-0.723416\pi\)
−0.645657 + 0.763628i \(0.723416\pi\)
\(920\) −9.00997 −0.297050
\(921\) 8.64393 0.284827
\(922\) −19.5891 −0.645133
\(923\) −8.30566 −0.273384
\(924\) 0 0
\(925\) 3.32780 0.109418
\(926\) −24.3330 −0.799631
\(927\) 21.3561 0.701425
\(928\) −1.60040 −0.0525356
\(929\) 6.76998 0.222116 0.111058 0.993814i \(-0.464576\pi\)
0.111058 + 0.993814i \(0.464576\pi\)
\(930\) 4.21541 0.138229
\(931\) −3.37894 −0.110740
\(932\) 12.9927 0.425590
\(933\) −8.33960 −0.273026
\(934\) −28.3412 −0.927351
\(935\) 0 0
\(936\) −1.97078 −0.0644168
\(937\) −9.93426 −0.324538 −0.162269 0.986747i \(-0.551881\pi\)
−0.162269 + 0.986747i \(0.551881\pi\)
\(938\) −11.1708 −0.364741
\(939\) −4.69412 −0.153187
\(940\) −9.51399 −0.310312
\(941\) −55.3511 −1.80439 −0.902197 0.431324i \(-0.858047\pi\)
−0.902197 + 0.431324i \(0.858047\pi\)
\(942\) −5.72991 −0.186691
\(943\) −55.1450 −1.79577
\(944\) −7.97469 −0.259554
\(945\) 2.41421 0.0785344
\(946\) 0 0
\(947\) 11.9252 0.387517 0.193758 0.981049i \(-0.437932\pi\)
0.193758 + 0.981049i \(0.437932\pi\)
\(948\) −4.38285 −0.142348
\(949\) 0.408161 0.0132495
\(950\) −3.37894 −0.109627
\(951\) −4.00938 −0.130013
\(952\) 0.449490 0.0145680
\(953\) 31.9262 1.03419 0.517096 0.855928i \(-0.327013\pi\)
0.517096 + 0.855928i \(0.327013\pi\)
\(954\) 8.05845 0.260902
\(955\) −18.2779 −0.591460
\(956\) 24.4799 0.791737
\(957\) 0 0
\(958\) 26.9207 0.869768
\(959\) 0.211493 0.00682947
\(960\) 0.414214 0.0133687
\(961\) 72.5692 2.34094
\(962\) 2.31873 0.0747589
\(963\) 40.9842 1.32070
\(964\) 13.6203 0.438681
\(965\) 14.2986 0.460288
\(966\) 3.73205 0.120077
\(967\) −38.6060 −1.24149 −0.620743 0.784014i \(-0.713169\pi\)
−0.620743 + 0.784014i \(0.713169\pi\)
\(968\) 0 0
\(969\) 0.629107 0.0202098
\(970\) −8.56993 −0.275164
\(971\) 22.2352 0.713561 0.356780 0.934188i \(-0.383874\pi\)
0.356780 + 0.934188i \(0.383874\pi\)
\(972\) 10.3431 0.331757
\(973\) 7.34971 0.235621
\(974\) 10.5353 0.337574
\(975\) 0.288614 0.00924303
\(976\) −8.29253 −0.265437
\(977\) −17.0281 −0.544779 −0.272389 0.962187i \(-0.587814\pi\)
−0.272389 + 0.962187i \(0.587814\pi\)
\(978\) −6.71936 −0.214862
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −16.7262 −0.534025
\(982\) −12.3321 −0.393533
\(983\) −33.4596 −1.06720 −0.533598 0.845738i \(-0.679160\pi\)
−0.533598 + 0.845738i \(0.679160\pi\)
\(984\) 2.53517 0.0808182
\(985\) 7.35023 0.234198
\(986\) 0.719363 0.0229092
\(987\) 3.94082 0.125438
\(988\) −2.35436 −0.0749021
\(989\) −64.3458 −2.04608
\(990\) 0 0
\(991\) 18.3129 0.581729 0.290864 0.956764i \(-0.406057\pi\)
0.290864 + 0.956764i \(0.406057\pi\)
\(992\) 10.1769 0.323117
\(993\) −3.56461 −0.113119
\(994\) 11.9202 0.378084
\(995\) 13.7201 0.434957
\(996\) −7.28810 −0.230932
\(997\) −48.5331 −1.53706 −0.768530 0.639814i \(-0.779012\pi\)
−0.768530 + 0.639814i \(0.779012\pi\)
\(998\) 37.4485 1.18541
\(999\) −8.03403 −0.254186
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.1.3 yes 4
11.10 odd 2 8470.2.a.cp.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cp.1.4 4 11.10 odd 2
8470.2.a.cr.1.3 yes 4 1.1 even 1 trivial