Properties

Label 8470.2.a.cp.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 \(1.93185\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.414214 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.414214 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.82843 q^{9} -1.00000 q^{10} +0.414214 q^{12} -2.13165 q^{13} -1.00000 q^{14} +0.414214 q^{15} +1.00000 q^{16} -4.44949 q^{17} +2.82843 q^{18} +8.27792 q^{19} +1.00000 q^{20} +0.414214 q^{21} -0.646887 q^{23} -0.414214 q^{24} +1.00000 q^{25} +2.13165 q^{26} -2.41421 q^{27} +1.00000 q^{28} -0.428825 q^{29} -0.414214 q^{30} -4.52004 q^{31} -1.00000 q^{32} +4.44949 q^{34} +1.00000 q^{35} -2.82843 q^{36} -8.49938 q^{37} -8.27792 q^{38} -0.882959 q^{39} -1.00000 q^{40} +10.6057 q^{41} -0.414214 q^{42} +2.65634 q^{43} -2.82843 q^{45} +0.646887 q^{46} +4.34242 q^{47} +0.414214 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.84304 q^{51} -2.13165 q^{52} +0.0206643 q^{53} +2.41421 q^{54} -1.00000 q^{56} +3.42883 q^{57} +0.428825 q^{58} -4.51059 q^{59} +0.414214 q^{60} +1.36433 q^{61} +4.52004 q^{62} -2.82843 q^{63} +1.00000 q^{64} -2.13165 q^{65} -2.68556 q^{67} -4.44949 q^{68} -0.267949 q^{69} -1.00000 q^{70} -9.05040 q^{71} +2.82843 q^{72} -0.585786 q^{73} +8.49938 q^{74} +0.414214 q^{75} +8.27792 q^{76} +0.882959 q^{78} +4.24728 q^{79} +1.00000 q^{80} +7.48528 q^{81} -10.6057 q^{82} -4.62446 q^{83} +0.414214 q^{84} -4.44949 q^{85} -2.65634 q^{86} -0.177625 q^{87} -2.17209 q^{89} +2.82843 q^{90} -2.13165 q^{91} -0.646887 q^{92} -1.87226 q^{93} -4.34242 q^{94} +8.27792 q^{95} -0.414214 q^{96} +13.0552 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} - 4q^{8} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} - 4q^{8} - 4q^{10} - 4q^{12} - 4q^{14} - 4q^{15} + 4q^{16} - 8q^{17} + 12q^{19} + 4q^{20} - 4q^{21} - 8q^{23} + 4q^{24} + 4q^{25} - 4q^{27} + 4q^{28} + 8q^{29} + 4q^{30} - 4q^{32} + 8q^{34} + 4q^{35} - 16q^{37} - 12q^{38} - 8q^{39} - 4q^{40} - 8q^{41} + 4q^{42} + 8q^{43} + 8q^{46} - 16q^{47} - 4q^{48} + 4q^{49} - 4q^{50} + 8q^{51} + 4q^{54} - 4q^{56} + 4q^{57} - 8q^{58} - 8q^{59} - 4q^{60} + 8q^{61} + 4q^{64} - 8q^{68} - 8q^{69} - 4q^{70} - 8q^{71} - 8q^{73} + 16q^{74} - 4q^{75} + 12q^{76} + 8q^{78} + 24q^{79} + 4q^{80} - 4q^{81} + 8q^{82} - 8q^{83} - 4q^{84} - 8q^{85} - 8q^{86} - 16q^{87} - 8q^{92} + 16q^{93} + 16q^{94} + 12q^{95} + 4q^{96} - 8q^{97} - 4q^{98} + O(q^{100}) \)

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
\(13\) −2.13165 −0.591214 −0.295607 0.955310i \(-0.595522\pi\)
−0.295607 + 0.955310i \(0.595522\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.414214 0.106949
\(16\) 1.00000 0.250000
\(17\) −4.44949 −1.07916 −0.539580 0.841934i \(-0.681417\pi\)
−0.539580 + 0.841934i \(0.681417\pi\)
\(18\) 2.82843 0.666667
\(19\) 8.27792 1.89908 0.949542 0.313639i \(-0.101548\pi\)
0.949542 + 0.313639i \(0.101548\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.414214 0.0903888
\(22\) 0 0
\(23\) −0.646887 −0.134885 −0.0674426 0.997723i \(-0.521484\pi\)
−0.0674426 + 0.997723i \(0.521484\pi\)
\(24\) −0.414214 −0.0845510
\(25\) 1.00000 0.200000
\(26\) 2.13165 0.418051
\(27\) −2.41421 −0.464616
\(28\) 1.00000 0.188982
\(29\) −0.428825 −0.0796309 −0.0398154 0.999207i \(-0.512677\pi\)
−0.0398154 + 0.999207i \(0.512677\pi\)
\(30\) −0.414214 −0.0756247
\(31\) −4.52004 −0.811824 −0.405912 0.913912i \(-0.633046\pi\)
−0.405912 + 0.913912i \(0.633046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.44949 0.763081
\(35\) 1.00000 0.169031
\(36\) −2.82843 −0.471405
\(37\) −8.49938 −1.39729 −0.698645 0.715469i \(-0.746213\pi\)
−0.698645 + 0.715469i \(0.746213\pi\)
\(38\) −8.27792 −1.34286
\(39\) −0.882959 −0.141387
\(40\) −1.00000 −0.158114
\(41\) 10.6057 1.65634 0.828168 0.560480i \(-0.189383\pi\)
0.828168 + 0.560480i \(0.189383\pi\)
\(42\) −0.414214 −0.0639145
\(43\) 2.65634 0.405088 0.202544 0.979273i \(-0.435079\pi\)
0.202544 + 0.979273i \(0.435079\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0.646887 0.0953782
\(47\) 4.34242 0.633407 0.316703 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(48\) 0.414214 0.0597866
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −1.84304 −0.258077
\(52\) −2.13165 −0.295607
\(53\) 0.0206643 0.00283846 0.00141923 0.999999i \(-0.499548\pi\)
0.00141923 + 0.999999i \(0.499548\pi\)
\(54\) 2.41421 0.328533
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 3.42883 0.454159
\(58\) 0.428825 0.0563075
\(59\) −4.51059 −0.587229 −0.293614 0.955924i \(-0.594858\pi\)
−0.293614 + 0.955924i \(0.594858\pi\)
\(60\) 0.414214 0.0534747
\(61\) 1.36433 0.174684 0.0873420 0.996178i \(-0.472163\pi\)
0.0873420 + 0.996178i \(0.472163\pi\)
\(62\) 4.52004 0.574046
\(63\) −2.82843 −0.356348
\(64\) 1.00000 0.125000
\(65\) −2.13165 −0.264399
\(66\) 0 0
\(67\) −2.68556 −0.328094 −0.164047 0.986453i \(-0.552455\pi\)
−0.164047 + 0.986453i \(0.552455\pi\)
\(68\) −4.44949 −0.539580
\(69\) −0.267949 −0.0322573
\(70\) −1.00000 −0.119523
\(71\) −9.05040 −1.07409 −0.537043 0.843555i \(-0.680459\pi\)
−0.537043 + 0.843555i \(0.680459\pi\)
\(72\) 2.82843 0.333333
\(73\) −0.585786 −0.0685611 −0.0342806 0.999412i \(-0.510914\pi\)
−0.0342806 + 0.999412i \(0.510914\pi\)
\(74\) 8.49938 0.988033
\(75\) 0.414214 0.0478293
\(76\) 8.27792 0.949542
\(77\) 0 0
\(78\) 0.882959 0.0999755
\(79\) 4.24728 0.477857 0.238928 0.971037i \(-0.423204\pi\)
0.238928 + 0.971037i \(0.423204\pi\)
\(80\) 1.00000 0.111803
\(81\) 7.48528 0.831698
\(82\) −10.6057 −1.17121
\(83\) −4.62446 −0.507601 −0.253800 0.967257i \(-0.581681\pi\)
−0.253800 + 0.967257i \(0.581681\pi\)
\(84\) 0.414214 0.0451944
\(85\) −4.44949 −0.482615
\(86\) −2.65634 −0.286440
\(87\) −0.177625 −0.0190434
\(88\) 0 0
\(89\) −2.17209 −0.230241 −0.115120 0.993352i \(-0.536725\pi\)
−0.115120 + 0.993352i \(0.536725\pi\)
\(90\) 2.82843 0.298142
\(91\) −2.13165 −0.223458
\(92\) −0.646887 −0.0674426
\(93\) −1.87226 −0.194145
\(94\) −4.34242 −0.447886
\(95\) 8.27792 0.849296
\(96\) −0.414214 −0.0422755
\(97\) 13.0552 1.32556 0.662778 0.748816i \(-0.269377\pi\)
0.662778 + 0.748816i \(0.269377\pi\)
\(98\) −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 \(0.604221\pi\)
\(102\) 1.84304 0.182488
\(103\) −12.4495 −1.22668 −0.613342 0.789817i \(-0.710175\pi\)
−0.613342 + 0.789817i \(0.710175\pi\)
\(104\) 2.13165 0.209026
\(105\) 0.414214 0.0404231
\(106\) −0.0206643 −0.00200710
\(107\) −10.0048 −0.967201 −0.483601 0.875289i \(-0.660671\pi\)
−0.483601 + 0.875289i \(0.660671\pi\)
\(108\) −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
\(111\) −3.52056 −0.334157
\(112\) 1.00000 0.0944911
\(113\) −3.76857 −0.354517 −0.177259 0.984164i \(-0.556723\pi\)
−0.177259 + 0.984164i \(0.556723\pi\)
\(114\) −3.42883 −0.321139
\(115\) −0.646887 −0.0603225
\(116\) −0.428825 −0.0398154
\(117\) 6.02922 0.557402
\(118\) 4.51059 0.415233
\(119\) −4.44949 −0.407884
\(120\) −0.414214 −0.0378124
\(121\) 0 0
\(122\) −1.36433 −0.123520
\(123\) 4.39303 0.396107
\(124\) −4.52004 −0.405912
\(125\) 1.00000 0.0894427
\(126\) 2.82843 0.251976
\(127\) −8.56564 −0.760077 −0.380039 0.924971i \(-0.624089\pi\)
−0.380039 + 0.924971i \(0.624089\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.10029 0.0968753
\(130\) 2.13165 0.186958
\(131\) −5.64224 −0.492965 −0.246483 0.969147i \(-0.579275\pi\)
−0.246483 + 0.969147i \(0.579275\pi\)
\(132\) 0 0
\(133\) 8.27792 0.717786
\(134\) 2.68556 0.231997
\(135\) −2.41421 −0.207782
\(136\) 4.44949 0.381541
\(137\) −1.64637 −0.140659 −0.0703295 0.997524i \(-0.522405\pi\)
−0.0703295 + 0.997524i \(0.522405\pi\)
\(138\) 0.267949 0.0228093
\(139\) 16.3071 1.38315 0.691577 0.722303i \(-0.256916\pi\)
0.691577 + 0.722303i \(0.256916\pi\)
\(140\) 1.00000 0.0845154
\(141\) 1.79869 0.151477
\(142\) 9.05040 0.759493
\(143\) 0 0
\(144\) −2.82843 −0.235702
\(145\) −0.428825 −0.0356120
\(146\) 0.585786 0.0484800
\(147\) 0.414214 0.0341638
\(148\) −8.49938 −0.698645
\(149\) 4.12701 0.338098 0.169049 0.985608i \(-0.445930\pi\)
0.169049 + 0.985608i \(0.445930\pi\)
\(150\) −0.414214 −0.0338204
\(151\) −11.8949 −0.967989 −0.483995 0.875071i \(-0.660815\pi\)
−0.483995 + 0.875071i \(0.660815\pi\)
\(152\) −8.27792 −0.671428
\(153\) 12.5851 1.01744
\(154\) 0 0
\(155\) −4.52004 −0.363059
\(156\) −0.882959 −0.0706933
\(157\) 10.6617 0.850893 0.425447 0.904984i \(-0.360117\pi\)
0.425447 + 0.904984i \(0.360117\pi\)
\(158\) −4.24728 −0.337896
\(159\) 0.00855944 0.000678807 0
\(160\) −1.00000 −0.0790569
\(161\) −0.646887 −0.0509818
\(162\) −7.48528 −0.588099
\(163\) −19.0917 −1.49538 −0.747690 0.664048i \(-0.768837\pi\)
−0.747690 + 0.664048i \(0.768837\pi\)
\(164\) 10.6057 0.828168
\(165\) 0 0
\(166\) 4.62446 0.358928
\(167\) −13.3684 −1.03448 −0.517239 0.855841i \(-0.673040\pi\)
−0.517239 + 0.855841i \(0.673040\pi\)
\(168\) −0.414214 −0.0319573
\(169\) −8.45606 −0.650466
\(170\) 4.44949 0.341260
\(171\) −23.4135 −1.79047
\(172\) 2.65634 0.202544
\(173\) 2.38551 0.181367 0.0906833 0.995880i \(-0.471095\pi\)
0.0906833 + 0.995880i \(0.471095\pi\)
\(174\) 0.177625 0.0134657
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −1.86835 −0.140434
\(178\) 2.17209 0.162805
\(179\) −3.68681 −0.275565 −0.137782 0.990463i \(-0.543997\pi\)
−0.137782 + 0.990463i \(0.543997\pi\)
\(180\) −2.82843 −0.210819
\(181\) −9.53001 −0.708360 −0.354180 0.935177i \(-0.615240\pi\)
−0.354180 + 0.935177i \(0.615240\pi\)
\(182\) 2.13165 0.158009
\(183\) 0.565122 0.0417750
\(184\) 0.646887 0.0476891
\(185\) −8.49938 −0.624887
\(186\) 1.87226 0.137281
\(187\) 0 0
\(188\) 4.34242 0.316703
\(189\) −2.41421 −0.175608
\(190\) −8.27792 −0.600543
\(191\) −13.3789 −0.968066 −0.484033 0.875050i \(-0.660829\pi\)
−0.484033 + 0.875050i \(0.660829\pi\)
\(192\) 0.414214 0.0298933
\(193\) −6.52985 −0.470029 −0.235014 0.971992i \(-0.575514\pi\)
−0.235014 + 0.971992i \(0.575514\pi\)
\(194\) −13.0552 −0.937310
\(195\) −0.882959 −0.0632300
\(196\) 1.00000 0.0714286
\(197\) −26.1056 −1.85995 −0.929974 0.367625i \(-0.880171\pi\)
−0.929974 + 0.367625i \(0.880171\pi\)
\(198\) 0 0
\(199\) −18.8917 −1.33920 −0.669598 0.742724i \(-0.733534\pi\)
−0.669598 + 0.742724i \(0.733534\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −1.11240 −0.0784624
\(202\) 6.46410 0.454813
\(203\) −0.428825 −0.0300976
\(204\) −1.84304 −0.129039
\(205\) 10.6057 0.740736
\(206\) 12.4495 0.867397
\(207\) 1.82967 0.127171
\(208\) −2.13165 −0.147804
\(209\) 0 0
\(210\) −0.414214 −0.0285835
\(211\) 16.7627 1.15399 0.576995 0.816747i \(-0.304225\pi\)
0.576995 + 0.816747i \(0.304225\pi\)
\(212\) 0.0206643 0.00141923
\(213\) −3.74880 −0.256864
\(214\) 10.0048 0.683915
\(215\) 2.65634 0.181161
\(216\) 2.41421 0.164266
\(217\) −4.52004 −0.306840
\(218\) −5.91359 −0.400519
\(219\) −0.242641 −0.0163961
\(220\) 0 0
\(221\) 9.48477 0.638014
\(222\) 3.52056 0.236284
\(223\) −17.0053 −1.13876 −0.569380 0.822074i \(-0.692817\pi\)
−0.569380 + 0.822074i \(0.692817\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.82843 −0.188562
\(226\) 3.76857 0.250682
\(227\) 14.0613 0.933279 0.466639 0.884448i \(-0.345465\pi\)
0.466639 + 0.884448i \(0.345465\pi\)
\(228\) 3.42883 0.227080
\(229\) −27.7673 −1.83491 −0.917457 0.397836i \(-0.869761\pi\)
−0.917457 + 0.397836i \(0.869761\pi\)
\(230\) 0.646887 0.0426544
\(231\) 0 0
\(232\) 0.428825 0.0281538
\(233\) 9.82113 0.643404 0.321702 0.946841i \(-0.395745\pi\)
0.321702 + 0.946841i \(0.395745\pi\)
\(234\) −6.02922 −0.394143
\(235\) 4.34242 0.283268
\(236\) −4.51059 −0.293614
\(237\) 1.75928 0.114278
\(238\) 4.44949 0.288418
\(239\) −4.28964 −0.277474 −0.138737 0.990329i \(-0.544304\pi\)
−0.138737 + 0.990329i \(0.544304\pi\)
\(240\) 0.414214 0.0267374
\(241\) 5.13505 0.330778 0.165389 0.986228i \(-0.447112\pi\)
0.165389 + 0.986228i \(0.447112\pi\)
\(242\) 0 0
\(243\) 10.3431 0.663513
\(244\) 1.36433 0.0873420
\(245\) 1.00000 0.0638877
\(246\) −4.39303 −0.280090
\(247\) −17.6456 −1.12277
\(248\) 4.52004 0.287023
\(249\) −1.91552 −0.121391
\(250\) −1.00000 −0.0632456
\(251\) −10.5851 −0.668123 −0.334061 0.942551i \(-0.608419\pi\)
−0.334061 + 0.942551i \(0.608419\pi\)
\(252\) −2.82843 −0.178174
\(253\) 0 0
\(254\) 8.56564 0.537456
\(255\) −1.84304 −0.115416
\(256\) 1.00000 0.0625000
\(257\) −4.27667 −0.266771 −0.133386 0.991064i \(-0.542585\pi\)
−0.133386 + 0.991064i \(0.542585\pi\)
\(258\) −1.10029 −0.0685012
\(259\) −8.49938 −0.528126
\(260\) −2.13165 −0.132199
\(261\) 1.21290 0.0750767
\(262\) 5.64224 0.348579
\(263\) −12.7773 −0.787882 −0.393941 0.919136i \(-0.628889\pi\)
−0.393941 + 0.919136i \(0.628889\pi\)
\(264\) 0 0
\(265\) 0.0206643 0.00126940
\(266\) −8.27792 −0.507552
\(267\) −0.899708 −0.0550613
\(268\) −2.68556 −0.164047
\(269\) −6.94495 −0.423441 −0.211721 0.977330i \(-0.567907\pi\)
−0.211721 + 0.977330i \(0.567907\pi\)
\(270\) 2.41421 0.146924
\(271\) 29.3911 1.78538 0.892690 0.450671i \(-0.148815\pi\)
0.892690 + 0.450671i \(0.148815\pi\)
\(272\) −4.44949 −0.269790
\(273\) −0.882959 −0.0534391
\(274\) 1.64637 0.0994609
\(275\) 0 0
\(276\) −0.267949 −0.0161286
\(277\) −9.99927 −0.600798 −0.300399 0.953814i \(-0.597120\pi\)
−0.300399 + 0.953814i \(0.597120\pi\)
\(278\) −16.3071 −0.978037
\(279\) 12.7846 0.765395
\(280\) −1.00000 −0.0597614
\(281\) −21.5792 −1.28731 −0.643653 0.765318i \(-0.722582\pi\)
−0.643653 + 0.765318i \(0.722582\pi\)
\(282\) −1.79869 −0.107110
\(283\) −11.2319 −0.667670 −0.333835 0.942632i \(-0.608343\pi\)
−0.333835 + 0.942632i \(0.608343\pi\)
\(284\) −9.05040 −0.537043
\(285\) 3.42883 0.203106
\(286\) 0 0
\(287\) 10.6057 0.626036
\(288\) 2.82843 0.166667
\(289\) 2.79796 0.164586
\(290\) 0.428825 0.0251815
\(291\) 5.40765 0.317002
\(292\) −0.585786 −0.0342806
\(293\) −16.7673 −0.979558 −0.489779 0.871847i \(-0.662922\pi\)
−0.489779 + 0.871847i \(0.662922\pi\)
\(294\) −0.414214 −0.0241574
\(295\) −4.51059 −0.262617
\(296\) 8.49938 0.494016
\(297\) 0 0
\(298\) −4.12701 −0.239071
\(299\) 1.37894 0.0797460
\(300\) 0.414214 0.0239146
\(301\) 2.65634 0.153109
\(302\) 11.8949 0.684472
\(303\) −2.67752 −0.153819
\(304\) 8.27792 0.474771
\(305\) 1.36433 0.0781211
\(306\) −12.5851 −0.719440
\(307\) 12.5840 0.718208 0.359104 0.933297i \(-0.383082\pi\)
0.359104 + 0.933297i \(0.383082\pi\)
\(308\) 0 0
\(309\) −5.15675 −0.293357
\(310\) 4.52004 0.256721
\(311\) −21.3223 −1.20907 −0.604537 0.796577i \(-0.706642\pi\)
−0.604537 + 0.796577i \(0.706642\pi\)
\(312\) 0.882959 0.0499877
\(313\) 24.9895 1.41249 0.706244 0.707968i \(-0.250388\pi\)
0.706244 + 0.707968i \(0.250388\pi\)
\(314\) −10.6617 −0.601672
\(315\) −2.82843 −0.159364
\(316\) 4.24728 0.238928
\(317\) 3.33636 0.187389 0.0936944 0.995601i \(-0.470132\pi\)
0.0936944 + 0.995601i \(0.470132\pi\)
\(318\) −0.00855944 −0.000479989 0
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −4.14413 −0.231303
\(322\) 0.646887 0.0360496
\(323\) −36.8325 −2.04942
\(324\) 7.48528 0.415849
\(325\) −2.13165 −0.118243
\(326\) 19.0917 1.05739
\(327\) 2.44949 0.135457
\(328\) −10.6057 −0.585603
\(329\) 4.34242 0.239405
\(330\) 0 0
\(331\) 8.12044 0.446340 0.223170 0.974780i \(-0.428360\pi\)
0.223170 + 0.974780i \(0.428360\pi\)
\(332\) −4.62446 −0.253800
\(333\) 24.0399 1.31738
\(334\) 13.3684 0.731487
\(335\) −2.68556 −0.146728
\(336\) 0.414214 0.0225972
\(337\) 30.2961 1.65033 0.825167 0.564889i \(-0.191081\pi\)
0.825167 + 0.564889i \(0.191081\pi\)
\(338\) 8.45606 0.459949
\(339\) −1.56099 −0.0847815
\(340\) −4.44949 −0.241307
\(341\) 0 0
\(342\) 23.4135 1.26606
\(343\) 1.00000 0.0539949
\(344\) −2.65634 −0.143220
\(345\) −0.267949 −0.0144259
\(346\) −2.38551 −0.128246
\(347\) −21.5745 −1.15818 −0.579091 0.815263i \(-0.696592\pi\)
−0.579091 + 0.815263i \(0.696592\pi\)
\(348\) −0.177625 −0.00952172
\(349\) −29.4536 −1.57661 −0.788307 0.615282i \(-0.789042\pi\)
−0.788307 + 0.615282i \(0.789042\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 5.14626 0.274687
\(352\) 0 0
\(353\) 10.2578 0.545966 0.272983 0.962019i \(-0.411990\pi\)
0.272983 + 0.962019i \(0.411990\pi\)
\(354\) 1.86835 0.0993015
\(355\) −9.05040 −0.480346
\(356\) −2.17209 −0.115120
\(357\) −1.84304 −0.0975440
\(358\) 3.68681 0.194854
\(359\) 31.9893 1.68833 0.844164 0.536085i \(-0.180097\pi\)
0.844164 + 0.536085i \(0.180097\pi\)
\(360\) 2.82843 0.149071
\(361\) 49.5239 2.60652
\(362\) 9.53001 0.500886
\(363\) 0 0
\(364\) −2.13165 −0.111729
\(365\) −0.585786 −0.0306615
\(366\) −0.565122 −0.0295394
\(367\) 21.2734 1.11046 0.555232 0.831695i \(-0.312629\pi\)
0.555232 + 0.831695i \(0.312629\pi\)
\(368\) −0.646887 −0.0337213
\(369\) −29.9975 −1.56161
\(370\) 8.49938 0.441862
\(371\) 0.0206643 0.00107284
\(372\) −1.87226 −0.0970723
\(373\) 21.6695 1.12200 0.561002 0.827815i \(-0.310416\pi\)
0.561002 + 0.827815i \(0.310416\pi\)
\(374\) 0 0
\(375\) 0.414214 0.0213899
\(376\) −4.34242 −0.223943
\(377\) 0.914107 0.0470789
\(378\) 2.41421 0.124174
\(379\) −15.6617 −0.804486 −0.402243 0.915533i \(-0.631769\pi\)
−0.402243 + 0.915533i \(0.631769\pi\)
\(380\) 8.27792 0.424648
\(381\) −3.54800 −0.181770
\(382\) 13.3789 0.684526
\(383\) 17.5412 0.896315 0.448157 0.893955i \(-0.352080\pi\)
0.448157 + 0.893955i \(0.352080\pi\)
\(384\) −0.414214 −0.0211377
\(385\) 0 0
\(386\) 6.52985 0.332360
\(387\) −7.51326 −0.381921
\(388\) 13.0552 0.662778
\(389\) −19.6118 −0.994356 −0.497178 0.867648i \(-0.665630\pi\)
−0.497178 + 0.867648i \(0.665630\pi\)
\(390\) 0.882959 0.0447104
\(391\) 2.87832 0.145563
\(392\) −1.00000 −0.0505076
\(393\) −2.33709 −0.117891
\(394\) 26.1056 1.31518
\(395\) 4.24728 0.213704
\(396\) 0 0
\(397\) −1.78482 −0.0895777 −0.0447889 0.998996i \(-0.514262\pi\)
−0.0447889 + 0.998996i \(0.514262\pi\)
\(398\) 18.8917 0.946954
\(399\) 3.42883 0.171656
\(400\) 1.00000 0.0500000
\(401\) −27.8471 −1.39062 −0.695309 0.718711i \(-0.744733\pi\)
−0.695309 + 0.718711i \(0.744733\pi\)
\(402\) 1.11240 0.0554813
\(403\) 9.63516 0.479961
\(404\) −6.46410 −0.321601
\(405\) 7.48528 0.371947
\(406\) 0.428825 0.0212822
\(407\) 0 0
\(408\) 1.84304 0.0912440
\(409\) 17.0071 0.840946 0.420473 0.907305i \(-0.361864\pi\)
0.420473 + 0.907305i \(0.361864\pi\)
\(410\) −10.6057 −0.523779
\(411\) −0.681949 −0.0336381
\(412\) −12.4495 −0.613342
\(413\) −4.51059 −0.221952
\(414\) −1.82967 −0.0899235
\(415\) −4.62446 −0.227006
\(416\) 2.13165 0.104513
\(417\) 6.75464 0.330776
\(418\) 0 0
\(419\) 7.35112 0.359126 0.179563 0.983746i \(-0.442532\pi\)
0.179563 + 0.983746i \(0.442532\pi\)
\(420\) 0.414214 0.0202116
\(421\) 20.0267 0.976043 0.488022 0.872832i \(-0.337719\pi\)
0.488022 + 0.872832i \(0.337719\pi\)
\(422\) −16.7627 −0.815995
\(423\) −12.2822 −0.597181
\(424\) −0.0206643 −0.00100355
\(425\) −4.44949 −0.215832
\(426\) 3.74880 0.181630
\(427\) 1.36433 0.0660243
\(428\) −10.0048 −0.483601
\(429\) 0 0
\(430\) −2.65634 −0.128100
\(431\) −5.14626 −0.247887 −0.123943 0.992289i \(-0.539554\pi\)
−0.123943 + 0.992289i \(0.539554\pi\)
\(432\) −2.41421 −0.116154
\(433\) −41.3976 −1.98944 −0.994722 0.102611i \(-0.967280\pi\)
−0.994722 + 0.102611i \(0.967280\pi\)
\(434\) 4.52004 0.216969
\(435\) −0.177625 −0.00851648
\(436\) 5.91359 0.283210
\(437\) −5.35487 −0.256158
\(438\) 0.242641 0.0115938
\(439\) 21.6609 1.03382 0.516910 0.856040i \(-0.327082\pi\)
0.516910 + 0.856040i \(0.327082\pi\)
\(440\) 0 0
\(441\) −2.82843 −0.134687
\(442\) −9.48477 −0.451144
\(443\) 21.0771 1.00141 0.500703 0.865619i \(-0.333075\pi\)
0.500703 + 0.865619i \(0.333075\pi\)
\(444\) −3.52056 −0.167078
\(445\) −2.17209 −0.102967
\(446\) 17.0053 0.805225
\(447\) 1.70946 0.0808548
\(448\) 1.00000 0.0472456
\(449\) 40.9893 1.93440 0.967201 0.254011i \(-0.0817501\pi\)
0.967201 + 0.254011i \(0.0817501\pi\)
\(450\) 2.82843 0.133333
\(451\) 0 0
\(452\) −3.76857 −0.177259
\(453\) −4.92701 −0.231491
\(454\) −14.0613 −0.659928
\(455\) −2.13165 −0.0999334
\(456\) −3.42883 −0.160569
\(457\) 18.3784 0.859706 0.429853 0.902899i \(-0.358565\pi\)
0.429853 + 0.902899i \(0.358565\pi\)
\(458\) 27.7673 1.29748
\(459\) 10.7420 0.501394
\(460\) −0.646887 −0.0301612
\(461\) 15.0383 0.700403 0.350202 0.936674i \(-0.386113\pi\)
0.350202 + 0.936674i \(0.386113\pi\)
\(462\) 0 0
\(463\) −28.6376 −1.33090 −0.665451 0.746442i \(-0.731761\pi\)
−0.665451 + 0.746442i \(0.731761\pi\)
\(464\) −0.428825 −0.0199077
\(465\) −1.87226 −0.0868241
\(466\) −9.82113 −0.454955
\(467\) 9.16960 0.424318 0.212159 0.977235i \(-0.431950\pi\)
0.212159 + 0.977235i \(0.431950\pi\)
\(468\) 6.02922 0.278701
\(469\) −2.68556 −0.124008
\(470\) −4.34242 −0.200301
\(471\) 4.41621 0.203488
\(472\) 4.51059 0.207617
\(473\) 0 0
\(474\) −1.75928 −0.0808065
\(475\) 8.27792 0.379817
\(476\) −4.44949 −0.203942
\(477\) −0.0584475 −0.00267613
\(478\) 4.28964 0.196204
\(479\) −33.8489 −1.54659 −0.773297 0.634044i \(-0.781394\pi\)
−0.773297 + 0.634044i \(0.781394\pi\)
\(480\) −0.414214 −0.0189062
\(481\) 18.1177 0.826097
\(482\) −5.13505 −0.233895
\(483\) −0.267949 −0.0121921
\(484\) 0 0
\(485\) 13.0552 0.592807
\(486\) −10.3431 −0.469175
\(487\) −27.2216 −1.23353 −0.616765 0.787148i \(-0.711557\pi\)
−0.616765 + 0.787148i \(0.711557\pi\)
\(488\) −1.36433 −0.0617601
\(489\) −7.90805 −0.357615
\(490\) −1.00000 −0.0451754
\(491\) −33.7879 −1.52483 −0.762414 0.647089i \(-0.775986\pi\)
−0.762414 + 0.647089i \(0.775986\pi\)
\(492\) 4.39303 0.198053
\(493\) 1.90805 0.0859344
\(494\) 17.6456 0.793915
\(495\) 0 0
\(496\) −4.52004 −0.202956
\(497\) −9.05040 −0.405966
\(498\) 1.91552 0.0858363
\(499\) 14.6347 0.655140 0.327570 0.944827i \(-0.393770\pi\)
0.327570 + 0.944827i \(0.393770\pi\)
\(500\) 1.00000 0.0447214
\(501\) −5.53737 −0.247392
\(502\) 10.5851 0.472434
\(503\) −23.0486 −1.02769 −0.513844 0.857884i \(-0.671779\pi\)
−0.513844 + 0.857884i \(0.671779\pi\)
\(504\) 2.82843 0.125988
\(505\) −6.46410 −0.287649
\(506\) 0 0
\(507\) −3.50261 −0.155557
\(508\) −8.56564 −0.380039
\(509\) 2.55368 0.113190 0.0565949 0.998397i \(-0.481976\pi\)
0.0565949 + 0.998397i \(0.481976\pi\)
\(510\) 1.84304 0.0816111
\(511\) −0.585786 −0.0259137
\(512\) −1.00000 −0.0441942
\(513\) −19.9847 −0.882344
\(514\) 4.27667 0.188636
\(515\) −12.4495 −0.548590
\(516\) 1.10029 0.0484376
\(517\) 0 0
\(518\) 8.49938 0.373441
\(519\) 0.988109 0.0433732
\(520\) 2.13165 0.0934791
\(521\) 9.67198 0.423737 0.211869 0.977298i \(-0.432045\pi\)
0.211869 + 0.977298i \(0.432045\pi\)
\(522\) −1.21290 −0.0530873
\(523\) −26.0501 −1.13909 −0.569545 0.821960i \(-0.692880\pi\)
−0.569545 + 0.821960i \(0.692880\pi\)
\(524\) −5.64224 −0.246483
\(525\) 0.414214 0.0180778
\(526\) 12.7773 0.557117
\(527\) 20.1119 0.876087
\(528\) 0 0
\(529\) −22.5815 −0.981806
\(530\) −0.0206643 −0.000897600 0
\(531\) 12.7579 0.553645
\(532\) 8.27792 0.358893
\(533\) −22.6077 −0.979249
\(534\) 0.899708 0.0389342
\(535\) −10.0048 −0.432546
\(536\) 2.68556 0.115999
\(537\) −1.52713 −0.0659003
\(538\) 6.94495 0.299418
\(539\) 0 0
\(540\) −2.41421 −0.103891
\(541\) −40.9549 −1.76079 −0.880395 0.474241i \(-0.842723\pi\)
−0.880395 + 0.474241i \(0.842723\pi\)
\(542\) −29.3911 −1.26245
\(543\) −3.94746 −0.169402
\(544\) 4.44949 0.190770
\(545\) 5.91359 0.253310
\(546\) 0.882959 0.0377872
\(547\) 26.8345 1.14736 0.573680 0.819079i \(-0.305515\pi\)
0.573680 + 0.819079i \(0.305515\pi\)
\(548\) −1.64637 −0.0703295
\(549\) −3.85890 −0.164694
\(550\) 0 0
\(551\) −3.54978 −0.151226
\(552\) 0.267949 0.0114047
\(553\) 4.24728 0.180613
\(554\) 9.99927 0.424828
\(555\) −3.52056 −0.149439
\(556\) 16.3071 0.691577
\(557\) 19.9262 0.844301 0.422151 0.906526i \(-0.361275\pi\)
0.422151 + 0.906526i \(0.361275\pi\)
\(558\) −12.7846 −0.541216
\(559\) −5.66239 −0.239494
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 21.5792 0.910262
\(563\) 8.40103 0.354061 0.177031 0.984205i \(-0.443351\pi\)
0.177031 + 0.984205i \(0.443351\pi\)
\(564\) 1.79869 0.0757384
\(565\) −3.76857 −0.158545
\(566\) 11.2319 0.472114
\(567\) 7.48528 0.314352
\(568\) 9.05040 0.379746
\(569\) −29.5592 −1.23919 −0.619594 0.784923i \(-0.712703\pi\)
−0.619594 + 0.784923i \(0.712703\pi\)
\(570\) −3.42883 −0.143618
\(571\) −23.2036 −0.971041 −0.485520 0.874225i \(-0.661370\pi\)
−0.485520 + 0.874225i \(0.661370\pi\)
\(572\) 0 0
\(573\) −5.54174 −0.231509
\(574\) −10.6057 −0.442674
\(575\) −0.646887 −0.0269770
\(576\) −2.82843 −0.117851
\(577\) 9.45408 0.393579 0.196789 0.980446i \(-0.436948\pi\)
0.196789 + 0.980446i \(0.436948\pi\)
\(578\) −2.79796 −0.116380
\(579\) −2.70475 −0.112406
\(580\) −0.428825 −0.0178060
\(581\) −4.62446 −0.191855
\(582\) −5.40765 −0.224154
\(583\) 0 0
\(584\) 0.585786 0.0242400
\(585\) 6.02922 0.249278
\(586\) 16.7673 0.692652
\(587\) 20.1626 0.832199 0.416100 0.909319i \(-0.363397\pi\)
0.416100 + 0.909319i \(0.363397\pi\)
\(588\) 0.414214 0.0170819
\(589\) −37.4165 −1.54172
\(590\) 4.51059 0.185698
\(591\) −10.8133 −0.444800
\(592\) −8.49938 −0.349322
\(593\) −34.9585 −1.43557 −0.717786 0.696264i \(-0.754844\pi\)
−0.717786 + 0.696264i \(0.754844\pi\)
\(594\) 0 0
\(595\) −4.44949 −0.182411
\(596\) 4.12701 0.169049
\(597\) −7.82519 −0.320264
\(598\) −1.37894 −0.0563889
\(599\) 24.4400 0.998591 0.499295 0.866432i \(-0.333592\pi\)
0.499295 + 0.866432i \(0.333592\pi\)
\(600\) −0.414214 −0.0169102
\(601\) −9.94812 −0.405792 −0.202896 0.979200i \(-0.565035\pi\)
−0.202896 + 0.979200i \(0.565035\pi\)
\(602\) −2.65634 −0.108264
\(603\) 7.59592 0.309330
\(604\) −11.8949 −0.483995
\(605\) 0 0
\(606\) 2.67752 0.108767
\(607\) −32.6542 −1.32539 −0.662696 0.748889i \(-0.730588\pi\)
−0.662696 + 0.748889i \(0.730588\pi\)
\(608\) −8.27792 −0.335714
\(609\) −0.177625 −0.00719774
\(610\) −1.36433 −0.0552399
\(611\) −9.25652 −0.374479
\(612\) 12.5851 0.508721
\(613\) −39.0744 −1.57820 −0.789100 0.614264i \(-0.789453\pi\)
−0.789100 + 0.614264i \(0.789453\pi\)
\(614\) −12.5840 −0.507850
\(615\) 4.39303 0.177144
\(616\) 0 0
\(617\) −5.10384 −0.205473 −0.102736 0.994709i \(-0.532760\pi\)
−0.102736 + 0.994709i \(0.532760\pi\)
\(618\) 5.15675 0.207435
\(619\) 8.03596 0.322992 0.161496 0.986873i \(-0.448368\pi\)
0.161496 + 0.986873i \(0.448368\pi\)
\(620\) −4.52004 −0.181529
\(621\) 1.56172 0.0626698
\(622\) 21.3223 0.854945
\(623\) −2.17209 −0.0870229
\(624\) −0.882959 −0.0353467
\(625\) 1.00000 0.0400000
\(626\) −24.9895 −0.998780
\(627\) 0 0
\(628\) 10.6617 0.425447
\(629\) 37.8179 1.50790
\(630\) 2.82843 0.112687
\(631\) 2.37486 0.0945416 0.0472708 0.998882i \(-0.484948\pi\)
0.0472708 + 0.998882i \(0.484948\pi\)
\(632\) −4.24728 −0.168948
\(633\) 6.94333 0.275973
\(634\) −3.33636 −0.132504
\(635\) −8.56564 −0.339917
\(636\) 0.00855944 0.000339404 0
\(637\) −2.13165 −0.0844591
\(638\) 0 0
\(639\) 25.5984 1.01266
\(640\) −1.00000 −0.0395285
\(641\) −41.6595 −1.64545 −0.822725 0.568440i \(-0.807547\pi\)
−0.822725 + 0.568440i \(0.807547\pi\)
\(642\) 4.14413 0.163556
\(643\) 36.9681 1.45788 0.728939 0.684578i \(-0.240014\pi\)
0.728939 + 0.684578i \(0.240014\pi\)
\(644\) −0.646887 −0.0254909
\(645\) 1.10029 0.0433239
\(646\) 36.8325 1.44916
\(647\) −14.5229 −0.570952 −0.285476 0.958386i \(-0.592152\pi\)
−0.285476 + 0.958386i \(0.592152\pi\)
\(648\) −7.48528 −0.294050
\(649\) 0 0
\(650\) 2.13165 0.0836103
\(651\) −1.87226 −0.0733798
\(652\) −19.0917 −0.747690
\(653\) −49.0424 −1.91918 −0.959588 0.281408i \(-0.909199\pi\)
−0.959588 + 0.281408i \(0.909199\pi\)
\(654\) −2.44949 −0.0957826
\(655\) −5.64224 −0.220461
\(656\) 10.6057 0.414084
\(657\) 1.65685 0.0646400
\(658\) −4.34242 −0.169285
\(659\) 14.0139 0.545903 0.272951 0.962028i \(-0.412000\pi\)
0.272951 + 0.962028i \(0.412000\pi\)
\(660\) 0 0
\(661\) 44.9342 1.74774 0.873869 0.486161i \(-0.161603\pi\)
0.873869 + 0.486161i \(0.161603\pi\)
\(662\) −8.12044 −0.315610
\(663\) 3.92872 0.152579
\(664\) 4.62446 0.179464
\(665\) 8.27792 0.321004
\(666\) −24.0399 −0.931526
\(667\) 0.277401 0.0107410
\(668\) −13.3684 −0.517239
\(669\) −7.04384 −0.272330
\(670\) 2.68556 0.103752
\(671\) 0 0
\(672\) −0.414214 −0.0159786
\(673\) −43.1679 −1.66400 −0.832000 0.554775i \(-0.812804\pi\)
−0.832000 + 0.554775i \(0.812804\pi\)
\(674\) −30.2961 −1.16696
\(675\) −2.41421 −0.0929231
\(676\) −8.45606 −0.325233
\(677\) −7.26720 −0.279301 −0.139651 0.990201i \(-0.544598\pi\)
−0.139651 + 0.990201i \(0.544598\pi\)
\(678\) 1.56099 0.0599496
\(679\) 13.0552 0.501013
\(680\) 4.44949 0.170630
\(681\) 5.82437 0.223190
\(682\) 0 0
\(683\) 8.44196 0.323023 0.161511 0.986871i \(-0.448363\pi\)
0.161511 + 0.986871i \(0.448363\pi\)
\(684\) −23.4135 −0.895237
\(685\) −1.64637 −0.0629046
\(686\) −1.00000 −0.0381802
\(687\) −11.5016 −0.438813
\(688\) 2.65634 0.101272
\(689\) −0.0440491 −0.00167814
\(690\) 0.267949 0.0102007
\(691\) 42.7272 1.62542 0.812710 0.582669i \(-0.197992\pi\)
0.812710 + 0.582669i \(0.197992\pi\)
\(692\) 2.38551 0.0906833
\(693\) 0 0
\(694\) 21.5745 0.818958
\(695\) 16.3071 0.618565
\(696\) 0.177625 0.00673287
\(697\) −47.1901 −1.78745
\(698\) 29.4536 1.11483
\(699\) 4.06805 0.153868
\(700\) 1.00000 0.0377964
\(701\) 32.9814 1.24569 0.622846 0.782345i \(-0.285976\pi\)
0.622846 + 0.782345i \(0.285976\pi\)
\(702\) −5.14626 −0.194233
\(703\) −70.3571 −2.65357
\(704\) 0 0
\(705\) 1.79869 0.0677425
\(706\) −10.2578 −0.386056
\(707\) −6.46410 −0.243108
\(708\) −1.86835 −0.0702168
\(709\) 19.1550 0.719381 0.359690 0.933072i \(-0.382882\pi\)
0.359690 + 0.933072i \(0.382882\pi\)
\(710\) 9.05040 0.339656
\(711\) −12.0131 −0.450528
\(712\) 2.17209 0.0814025
\(713\) 2.92395 0.109503
\(714\) 1.84304 0.0689740
\(715\) 0 0
\(716\) −3.68681 −0.137782
\(717\) −1.77683 −0.0663569
\(718\) −31.9893 −1.19383
\(719\) 39.4644 1.47177 0.735887 0.677104i \(-0.236765\pi\)
0.735887 + 0.677104i \(0.236765\pi\)
\(720\) −2.82843 −0.105409
\(721\) −12.4495 −0.463643
\(722\) −49.5239 −1.84309
\(723\) 2.12701 0.0791043
\(724\) −9.53001 −0.354180
\(725\) −0.428825 −0.0159262
\(726\) 0 0
\(727\) −25.9129 −0.961055 −0.480527 0.876980i \(-0.659555\pi\)
−0.480527 + 0.876980i \(0.659555\pi\)
\(728\) 2.13165 0.0790043
\(729\) −18.1716 −0.673021
\(730\) 0.585786 0.0216809
\(731\) −11.8194 −0.437155
\(732\) 0.565122 0.0208875
\(733\) 47.2935 1.74683 0.873413 0.486981i \(-0.161902\pi\)
0.873413 + 0.486981i \(0.161902\pi\)
\(734\) −21.2734 −0.785217
\(735\) 0.414214 0.0152785
\(736\) 0.646887 0.0238446
\(737\) 0 0
\(738\) 29.9975 1.10422
\(739\) 34.1019 1.25446 0.627229 0.778835i \(-0.284189\pi\)
0.627229 + 0.778835i \(0.284189\pi\)
\(740\) −8.49938 −0.312443
\(741\) −7.30906 −0.268505
\(742\) −0.0206643 −0.000758611 0
\(743\) 37.2951 1.36823 0.684113 0.729376i \(-0.260190\pi\)
0.684113 + 0.729376i \(0.260190\pi\)
\(744\) 1.87226 0.0686405
\(745\) 4.12701 0.151202
\(746\) −21.6695 −0.793376
\(747\) 13.0800 0.478571
\(748\) 0 0
\(749\) −10.0048 −0.365568
\(750\) −0.414214 −0.0151249
\(751\) 27.1160 0.989475 0.494738 0.869042i \(-0.335264\pi\)
0.494738 + 0.869042i \(0.335264\pi\)
\(752\) 4.34242 0.158352
\(753\) −4.38447 −0.159779
\(754\) −0.914107 −0.0332898
\(755\) −11.8949 −0.432898
\(756\) −2.41421 −0.0878041
\(757\) 1.81278 0.0658868 0.0329434 0.999457i \(-0.489512\pi\)
0.0329434 + 0.999457i \(0.489512\pi\)
\(758\) 15.6617 0.568857
\(759\) 0 0
\(760\) −8.27792 −0.300272
\(761\) −31.5856 −1.14498 −0.572488 0.819913i \(-0.694022\pi\)
−0.572488 + 0.819913i \(0.694022\pi\)
\(762\) 3.54800 0.128531
\(763\) 5.91359 0.214086
\(764\) −13.3789 −0.484033
\(765\) 12.5851 0.455014
\(766\) −17.5412 −0.633790
\(767\) 9.61501 0.347178
\(768\) 0.414214 0.0149466
\(769\) 33.4286 1.20547 0.602733 0.797943i \(-0.294078\pi\)
0.602733 + 0.797943i \(0.294078\pi\)
\(770\) 0 0
\(771\) −1.77146 −0.0637974
\(772\) −6.52985 −0.235014
\(773\) −47.6294 −1.71311 −0.856555 0.516055i \(-0.827400\pi\)
−0.856555 + 0.516055i \(0.827400\pi\)
\(774\) 7.51326 0.270059
\(775\) −4.52004 −0.162365
\(776\) −13.0552 −0.468655
\(777\) −3.52056 −0.126299
\(778\) 19.6118 0.703116
\(779\) 87.7933 3.14552
\(780\) −0.882959 −0.0316150
\(781\) 0 0
\(782\) −2.87832 −0.102928
\(783\) 1.03528 0.0369978
\(784\) 1.00000 0.0357143
\(785\) 10.6617 0.380531
\(786\) 2.33709 0.0833614
\(787\) −35.3497 −1.26008 −0.630041 0.776562i \(-0.716962\pi\)
−0.630041 + 0.776562i \(0.716962\pi\)
\(788\) −26.1056 −0.929974
\(789\) −5.29253 −0.188419
\(790\) −4.24728 −0.151112
\(791\) −3.76857 −0.133995
\(792\) 0 0
\(793\) −2.90827 −0.103276
\(794\) 1.78482 0.0633410
\(795\) 0.00855944 0.000303572 0
\(796\) −18.8917 −0.669598
\(797\) −11.6576 −0.412933 −0.206467 0.978454i \(-0.566197\pi\)
−0.206467 + 0.978454i \(0.566197\pi\)
\(798\) −3.42883 −0.121379
\(799\) −19.3215 −0.683547
\(800\) −1.00000 −0.0353553
\(801\) 6.14359 0.217073
\(802\) 27.8471 0.983316
\(803\) 0 0
\(804\) −1.11240 −0.0392312
\(805\) −0.646887 −0.0227998
\(806\) −9.63516 −0.339384
\(807\) −2.87669 −0.101264
\(808\) 6.46410 0.227406
\(809\) 14.5052 0.509976 0.254988 0.966944i \(-0.417928\pi\)
0.254988 + 0.966944i \(0.417928\pi\)
\(810\) −7.48528 −0.263006
\(811\) −30.8587 −1.08359 −0.541797 0.840509i \(-0.682256\pi\)
−0.541797 + 0.840509i \(0.682256\pi\)
\(812\) −0.428825 −0.0150488
\(813\) 12.1742 0.426967
\(814\) 0 0
\(815\) −19.0917 −0.668754
\(816\) −1.84304 −0.0645193
\(817\) 21.9890 0.769296
\(818\) −17.0071 −0.594639
\(819\) 6.02922 0.210678
\(820\) 10.6057 0.370368
\(821\) 8.11042 0.283056 0.141528 0.989934i \(-0.454799\pi\)
0.141528 + 0.989934i \(0.454799\pi\)
\(822\) 0.681949 0.0237857
\(823\) −46.4122 −1.61783 −0.808914 0.587926i \(-0.799944\pi\)
−0.808914 + 0.587926i \(0.799944\pi\)
\(824\) 12.4495 0.433699
\(825\) 0 0
\(826\) 4.51059 0.156943
\(827\) 22.7819 0.792204 0.396102 0.918207i \(-0.370363\pi\)
0.396102 + 0.918207i \(0.370363\pi\)
\(828\) 1.82967 0.0635855
\(829\) −6.20021 −0.215342 −0.107671 0.994187i \(-0.534339\pi\)
−0.107671 + 0.994187i \(0.534339\pi\)
\(830\) 4.62446 0.160517
\(831\) −4.14183 −0.143679
\(832\) −2.13165 −0.0739018
\(833\) −4.44949 −0.154166
\(834\) −6.75464 −0.233894
\(835\) −13.3684 −0.462633
\(836\) 0 0
\(837\) 10.9123 0.377186
\(838\) −7.35112 −0.253940
\(839\) −4.54727 −0.156989 −0.0784947 0.996915i \(-0.525011\pi\)
−0.0784947 + 0.996915i \(0.525011\pi\)
\(840\) −0.414214 −0.0142917
\(841\) −28.8161 −0.993659
\(842\) −20.0267 −0.690167
\(843\) −8.93838 −0.307854
\(844\) 16.7627 0.576995
\(845\) −8.45606 −0.290897
\(846\) 12.2822 0.422271
\(847\) 0 0
\(848\) 0.0206643 0.000709615 0
\(849\) −4.65242 −0.159671
\(850\) 4.44949 0.152616
\(851\) 5.49813 0.188474
\(852\) −3.74880 −0.128432
\(853\) 26.8475 0.919241 0.459620 0.888115i \(-0.347985\pi\)
0.459620 + 0.888115i \(0.347985\pi\)
\(854\) −1.36433 −0.0466863
\(855\) −23.4135 −0.800724
\(856\) 10.0048 0.341957
\(857\) −40.8363 −1.39494 −0.697470 0.716614i \(-0.745691\pi\)
−0.697470 + 0.716614i \(0.745691\pi\)
\(858\) 0 0
\(859\) 1.34824 0.0460013 0.0230006 0.999735i \(-0.492678\pi\)
0.0230006 + 0.999735i \(0.492678\pi\)
\(860\) 2.65634 0.0905804
\(861\) 4.39303 0.149714
\(862\) 5.14626 0.175282
\(863\) −40.8376 −1.39013 −0.695064 0.718948i \(-0.744624\pi\)
−0.695064 + 0.718948i \(0.744624\pi\)
\(864\) 2.41421 0.0821332
\(865\) 2.38551 0.0811096
\(866\) 41.3976 1.40675
\(867\) 1.15895 0.0393601
\(868\) −4.52004 −0.153420
\(869\) 0 0
\(870\) 0.177625 0.00602206
\(871\) 5.72469 0.193974
\(872\) −5.91359 −0.200259
\(873\) −36.9257 −1.24975
\(874\) 5.35487 0.181131
\(875\) 1.00000 0.0338062
\(876\) −0.242641 −0.00819807
\(877\) −4.52485 −0.152793 −0.0763967 0.997078i \(-0.524342\pi\)
−0.0763967 + 0.997078i \(0.524342\pi\)
\(878\) −21.6609 −0.731021
\(879\) −6.94525 −0.234258
\(880\) 0 0
\(881\) −18.7841 −0.632852 −0.316426 0.948617i \(-0.602483\pi\)
−0.316426 + 0.948617i \(0.602483\pi\)
\(882\) 2.82843 0.0952381
\(883\) −27.8159 −0.936080 −0.468040 0.883707i \(-0.655040\pi\)
−0.468040 + 0.883707i \(0.655040\pi\)
\(884\) 9.48477 0.319007
\(885\) −1.86835 −0.0628038
\(886\) −21.0771 −0.708100
\(887\) −1.97830 −0.0664250 −0.0332125 0.999448i \(-0.510574\pi\)
−0.0332125 + 0.999448i \(0.510574\pi\)
\(888\) 3.52056 0.118142
\(889\) −8.56564 −0.287282
\(890\) 2.17209 0.0728086
\(891\) 0 0
\(892\) −17.0053 −0.569380
\(893\) 35.9462 1.20289
\(894\) −1.70946 −0.0571730
\(895\) −3.68681 −0.123236
\(896\) −1.00000 −0.0334077
\(897\) 0.571175 0.0190710
\(898\) −40.9893 −1.36783
\(899\) 1.93831 0.0646462
\(900\) −2.82843 −0.0942809
\(901\) −0.0919456 −0.00306315
\(902\) 0 0
\(903\) 1.10029 0.0366154
\(904\) 3.76857 0.125341
\(905\) −9.53001 −0.316788
\(906\) 4.92701 0.163689
\(907\) 29.2382 0.970838 0.485419 0.874282i \(-0.338667\pi\)
0.485419 + 0.874282i \(0.338667\pi\)
\(908\) 14.0613 0.466639
\(909\) 18.2832 0.606417
\(910\) 2.13165 0.0706636
\(911\) 7.11563 0.235751 0.117876 0.993028i \(-0.462392\pi\)
0.117876 + 0.993028i \(0.462392\pi\)
\(912\) 3.42883 0.113540
\(913\) 0 0
\(914\) −18.3784 −0.607904
\(915\) 0.565122 0.0186824
\(916\) −27.7673 −0.917457
\(917\) −5.64224 −0.186323
\(918\) −10.7420 −0.354539
\(919\) 48.4518 1.59828 0.799138 0.601147i \(-0.205289\pi\)
0.799138 + 0.601147i \(0.205289\pi\)
\(920\) 0.646887 0.0213272
\(921\) 5.21247 0.171757
\(922\) −15.0383 −0.495260
\(923\) 19.2923 0.635014
\(924\) 0 0
\(925\) −8.49938 −0.279458
\(926\) 28.6376 0.941090
\(927\) 35.2125 1.15653
\(928\) 0.428825 0.0140769
\(929\) 14.8869 0.488423 0.244211 0.969722i \(-0.421471\pi\)
0.244211 + 0.969722i \(0.421471\pi\)
\(930\) 1.87226 0.0613939
\(931\) 8.27792 0.271298
\(932\) 9.82113 0.321702
\(933\) −8.83197 −0.289146
\(934\) −9.16960 −0.300038
\(935\) 0 0
\(936\) −6.02922 −0.197071
\(937\) −4.76268 −0.155590 −0.0777950 0.996969i \(-0.524788\pi\)
−0.0777950 + 0.996969i \(0.524788\pi\)
\(938\) 2.68556 0.0876867
\(939\) 10.3510 0.337791
\(940\) 4.34242 0.141634
\(941\) −25.4100 −0.828341 −0.414171 0.910199i \(-0.635928\pi\)
−0.414171 + 0.910199i \(0.635928\pi\)
\(942\) −4.41621 −0.143888
\(943\) −6.86070 −0.223415
\(944\) −4.51059 −0.146807
\(945\) −2.41421 −0.0785344
\(946\) 0 0
\(947\) −59.0379 −1.91847 −0.959237 0.282603i \(-0.908802\pi\)
−0.959237 + 0.282603i \(0.908802\pi\)
\(948\) 1.75928 0.0571389
\(949\) 1.24869 0.0405343
\(950\) −8.27792 −0.268571
\(951\) 1.38197 0.0448134
\(952\) 4.44949 0.144209
\(953\) −28.2159 −0.914003 −0.457002 0.889466i \(-0.651077\pi\)
−0.457002 + 0.889466i \(0.651077\pi\)
\(954\) 0.0584475 0.00189231
\(955\) −13.3789 −0.432932
\(956\) −4.28964 −0.138737
\(957\) 0 0
\(958\) 33.8489 1.09361
\(959\) −1.64637 −0.0531641
\(960\) 0.414214 0.0133687
\(961\) −10.5692 −0.340943
\(962\) −18.1177 −0.584139
\(963\) 28.2979 0.911886
\(964\) 5.13505 0.165389
\(965\) −6.52985 −0.210203
\(966\) 0.267949 0.00862112
\(967\) 27.6194 0.888179 0.444090 0.895982i \(-0.353527\pi\)
0.444090 + 0.895982i \(0.353527\pi\)
\(968\) 0 0
\(969\) −15.2565 −0.490110
\(970\) −13.0552 −0.419178
\(971\) 30.2501 0.970772 0.485386 0.874300i \(-0.338679\pi\)
0.485386 + 0.874300i \(0.338679\pi\)
\(972\) 10.3431 0.331757
\(973\) 16.3071 0.522783
\(974\) 27.2216 0.872237
\(975\) −0.882959 −0.0282773
\(976\) 1.36433 0.0436710
\(977\) 41.5134 1.32813 0.664066 0.747674i \(-0.268829\pi\)
0.664066 + 0.747674i \(0.268829\pi\)
\(978\) 7.90805 0.252872
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −16.7262 −0.534025
\(982\) 33.7879 1.07822
\(983\) 10.6312 0.339082 0.169541 0.985523i \(-0.445771\pi\)
0.169541 + 0.985523i \(0.445771\pi\)
\(984\) −4.39303 −0.140045
\(985\) −26.1056 −0.831794
\(986\) −1.90805 −0.0607648
\(987\) 1.79869 0.0572529
\(988\) −17.6456 −0.561383
\(989\) −1.71835 −0.0546403
\(990\) 0 0
\(991\) −44.8815 −1.42571 −0.712853 0.701313i \(-0.752597\pi\)
−0.712853 + 0.701313i \(0.752597\pi\)
\(992\) 4.52004 0.143511
\(993\) 3.36360 0.106741
\(994\) 9.05040 0.287061
\(995\) −18.8917 −0.598907
\(996\) −1.91552 −0.0606954
\(997\) −30.7930 −0.975225 −0.487613 0.873060i \(-0.662132\pi\)
−0.487613 + 0.873060i \(0.662132\pi\)
\(998\) −14.6347 −0.463254
\(999\) 20.5193 0.649202
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.1.3 4
11.10 odd 2 8470.2.a.cr.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cp.1.3 4 1.1 even 1 trivial
8470.2.a.cr.1.4 yes 4 11.10 odd 2