Properties

Label 8470.2.a.cr.1.4
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 1.4
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 \(-0.404478\pi\)
0.295607 + 0.955310i \(0.404478\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.414214 0.106949
\(16\) 1.00000 0.250000
\(17\) 4.44949 1.07916 0.539580 0.841934i \(-0.318583\pi\)
0.539580 + 0.841934i \(0.318583\pi\)
\(18\) −2.82843 −0.666667
\(19\) −8.27792 −1.89908 −0.949542 0.313639i \(-0.898452\pi\)
−0.949542 + 0.313639i \(0.898452\pi\)
\(20\) 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 \(-0.487323\pi\)
0.0398154 + 0.999207i \(0.487323\pi\)
\(30\) 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 \(-0.810617\pi\)
−0.828168 + 0.560480i \(0.810617\pi\)
\(42\) −0.414214 −0.0639145
\(43\) −2.65634 −0.405088 −0.202544 0.979273i \(-0.564921\pi\)
−0.202544 + 0.979273i \(0.564921\pi\)
\(44\) 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 \(-0.527837\pi\)
−0.0873420 + 0.996178i \(0.527837\pi\)
\(62\) −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 \(-0.489086\pi\)
0.0342806 + 0.999412i \(0.489086\pi\)
\(74\) −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 \(-0.576796\pi\)
−0.238928 + 0.971037i \(0.576796\pi\)
\(80\) 1.00000 0.111803
\(81\) 7.48528 0.831698
\(82\) −10.6057 −1.17121
\(83\) 4.62446 0.507601 0.253800 0.967257i \(-0.418319\pi\)
0.253800 + 0.967257i \(0.418319\pi\)
\(84\) −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 \(-0.395779\pi\)
0.321601 + 0.946875i \(0.395779\pi\)
\(102\) 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 \(-0.339329\pi\)
0.483601 + 0.875289i \(0.339329\pi\)
\(108\) −2.41421 −0.232308
\(109\) −5.91359 −0.566419 −0.283210 0.959058i \(-0.591399\pi\)
−0.283210 + 0.959058i \(0.591399\pi\)
\(110\) 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 \(-0.375911\pi\)
0.380039 + 0.924971i \(0.375911\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.10029 −0.0968753
\(130\) 2.13165 0.186958
\(131\) 5.64224 0.492965 0.246483 0.969147i \(-0.420725\pi\)
0.246483 + 0.969147i \(0.420725\pi\)
\(132\) 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 \(-0.743084\pi\)
−0.691577 + 0.722303i \(0.743084\pi\)
\(140\) −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 \(-0.554070\pi\)
−0.169049 + 0.985608i \(0.554070\pi\)
\(150\) 0.414214 0.0338204
\(151\) 11.8949 0.967989 0.483995 0.875071i \(-0.339185\pi\)
0.483995 + 0.875071i \(0.339185\pi\)
\(152\) −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 \(-0.326960\pi\)
0.517239 + 0.855841i \(0.326960\pi\)
\(168\) −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 \(-0.528905\pi\)
−0.0906833 + 0.995880i \(0.528905\pi\)
\(174\) 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 \(-0.424486\pi\)
0.235014 + 0.971992i \(0.424486\pi\)
\(194\) 13.0552 0.937310
\(195\) 0.882959 0.0632300
\(196\) 1.00000 0.0714286
\(197\) 26.1056 1.85995 0.929974 0.367625i \(-0.119829\pi\)
0.929974 + 0.367625i \(0.119829\pi\)
\(198\) 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 \(-0.695775\pi\)
−0.576995 + 0.816747i \(0.695775\pi\)
\(212\) 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 \(-0.654535\pi\)
−0.466639 + 0.884448i \(0.654535\pi\)
\(228\) −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 \(-0.604255\pi\)
−0.321702 + 0.946841i \(0.604255\pi\)
\(234\) −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 \(-0.455696\pi\)
0.138737 + 0.990329i \(0.455696\pi\)
\(240\) 0.414214 0.0267374
\(241\) −5.13505 −0.330778 −0.165389 0.986228i \(-0.552888\pi\)
−0.165389 + 0.986228i \(0.552888\pi\)
\(242\) 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 \(-0.371111\pi\)
0.393941 + 0.919136i \(0.371111\pi\)
\(264\) 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 \(-0.851185\pi\)
−0.892690 + 0.450671i \(0.851185\pi\)
\(272\) 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 \(-0.402880\pi\)
0.300399 + 0.953814i \(0.402880\pi\)
\(278\) −16.3071 −0.978037
\(279\) 12.7846 0.765395
\(280\) −1.00000 −0.0597614
\(281\) 21.5792 1.28731 0.643653 0.765318i \(-0.277418\pi\)
0.643653 + 0.765318i \(0.277418\pi\)
\(282\) 1.79869 0.107110
\(283\) 11.2319 0.667670 0.333835 0.942632i \(-0.391657\pi\)
0.333835 + 0.942632i \(0.391657\pi\)
\(284\) −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 \(-0.337078\pi\)
0.489779 + 0.871847i \(0.337078\pi\)
\(294\) 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 \(-0.616918\pi\)
−0.359104 + 0.933297i \(0.616918\pi\)
\(308\) 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 \(-0.808919\pi\)
−0.825167 + 0.564889i \(0.808919\pi\)
\(338\) −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 \(-0.303408\pi\)
0.579091 + 0.815263i \(0.303408\pi\)
\(348\) 0.177625 0.00952172
\(349\) 29.4536 1.57661 0.788307 0.615282i \(-0.210958\pi\)
0.788307 + 0.615282i \(0.210958\pi\)
\(350\) −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 \(-0.819903\pi\)
−0.844164 + 0.536085i \(0.819903\pi\)
\(360\) −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 \(-0.689584\pi\)
−0.561002 + 0.827815i \(0.689584\pi\)
\(374\) 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 \(-0.638136\pi\)
−0.420473 + 0.907305i \(0.638136\pi\)
\(410\) −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 \(-0.460446\pi\)
0.123943 + 0.992289i \(0.460446\pi\)
\(432\) −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 \(-0.672918\pi\)
−0.516910 + 0.856040i \(0.672918\pi\)
\(440\) 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 \(-0.641435\pi\)
−0.429853 + 0.902899i \(0.641435\pi\)
\(458\) −27.7673 −1.29748
\(459\) −10.7420 −0.501394
\(460\) −0.646887 −0.0301612
\(461\) −15.0383 −0.700403 −0.350202 0.936674i \(-0.613887\pi\)
−0.350202 + 0.936674i \(0.613887\pi\)
\(462\) 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 \(-0.218606\pi\)
0.773297 + 0.634044i \(0.218606\pi\)
\(480\) 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 \(-0.224014\pi\)
0.762414 + 0.647089i \(0.224014\pi\)
\(492\) −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 \(-0.328221\pi\)
0.513844 + 0.857884i \(0.328221\pi\)
\(504\) 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 \(-0.307120\pi\)
0.569545 + 0.821960i \(0.307120\pi\)
\(524\) 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 \(-0.157277\pi\)
0.880395 + 0.474241i \(0.157277\pi\)
\(542\) −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 \(-0.694485\pi\)
−0.573680 + 0.819079i \(0.694485\pi\)
\(548\) −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 \(-0.638725\pi\)
−0.422151 + 0.906526i \(0.638725\pi\)
\(558\) 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 \(-0.556649\pi\)
−0.177031 + 0.984205i \(0.556649\pi\)
\(564\) 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 \(-0.287297\pi\)
0.619594 + 0.784923i \(0.287297\pi\)
\(570\) −3.42883 −0.143618
\(571\) 23.2036 0.971041 0.485520 0.874225i \(-0.338630\pi\)
0.485520 + 0.874225i \(0.338630\pi\)
\(572\) 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 \(-0.245156\pi\)
0.717786 + 0.696264i \(0.245156\pi\)
\(594\) 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 \(-0.434965\pi\)
0.202896 + 0.979200i \(0.434965\pi\)
\(602\) 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 \(-0.269412\pi\)
0.662696 + 0.748889i \(0.269412\pi\)
\(608\) −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 \(-0.210547\pi\)
0.789100 + 0.614264i \(0.210547\pi\)
\(614\) −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 \(-0.588000\pi\)
−0.272951 + 0.962028i \(0.588000\pi\)
\(660\) 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 \(-0.187196\pi\)
0.832000 + 0.554775i \(0.187196\pi\)
\(674\) −30.2961 −1.16696
\(675\) −2.41421 −0.0929231
\(676\) −8.45606 −0.325233
\(677\) 7.26720 0.279301 0.139651 0.990201i \(-0.455402\pi\)
0.139651 + 0.990201i \(0.455402\pi\)
\(678\) −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 \(-0.714024\pi\)
−0.622846 + 0.782345i \(0.714024\pi\)
\(702\) −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 \(-0.838098\pi\)
−0.873413 + 0.486981i \(0.838098\pi\)
\(734\) 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 \(-0.715811\pi\)
−0.627229 + 0.778835i \(0.715811\pi\)
\(740\) −8.49938 −0.312443
\(741\) −7.30906 −0.268505
\(742\) −0.0206643 −0.000758611 0
\(743\) −37.2951 −1.36823 −0.684113 0.729376i \(-0.739810\pi\)
−0.684113 + 0.729376i \(0.739810\pi\)
\(744\) −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 \(-0.305978\pi\)
0.572488 + 0.819913i \(0.305978\pi\)
\(762\) 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 \(-0.705922\pi\)
−0.602733 + 0.797943i \(0.705922\pi\)
\(770\) 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 \(-0.283038\pi\)
0.630041 + 0.776562i \(0.283038\pi\)
\(788\) 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 \(-0.582072\pi\)
−0.254988 + 0.966944i \(0.582072\pi\)
\(810\) 7.48528 0.263006
\(811\) 30.8587 1.08359 0.541797 0.840509i \(-0.317744\pi\)
0.541797 + 0.840509i \(0.317744\pi\)
\(812\) −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 \(-0.545201\pi\)
−0.141528 + 0.989934i \(0.545201\pi\)
\(822\) −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 \(-0.629637\pi\)
−0.396102 + 0.918207i \(0.629637\pi\)
\(828\) 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 \(-0.652015\pi\)
−0.459620 + 0.888115i \(0.652015\pi\)
\(854\) 1.36433 0.0466863
\(855\) 23.4135 0.800724
\(856\) 10.0048 0.341957
\(857\) 40.8363 1.39494 0.697470 0.716614i \(-0.254309\pi\)
0.697470 + 0.716614i \(0.254309\pi\)
\(858\) 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 \(-0.475658\pi\)
0.0763967 + 0.997078i \(0.475658\pi\)
\(878\) −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 \(-0.489426\pi\)
0.0332125 + 0.999448i \(0.489426\pi\)
\(888\) −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 \(-0.794711\pi\)
−0.799138 + 0.601147i \(0.794711\pi\)
\(920\) −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 \(-0.475212\pi\)
0.0777950 + 0.996969i \(0.475212\pi\)
\(938\) 2.68556 0.0876867
\(939\) 10.3510 0.337791
\(940\) 4.34242 0.141634
\(941\) 25.4100 0.828341 0.414171 0.910199i \(-0.364072\pi\)
0.414171 + 0.910199i \(0.364072\pi\)
\(942\) 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 \(-0.348923\pi\)
0.457002 + 0.889466i \(0.348923\pi\)
\(954\) −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 \(-0.646473\pi\)
−0.444090 + 0.895982i \(0.646473\pi\)
\(968\) 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 \(-0.337868\pi\)
0.487613 + 0.873060i \(0.337868\pi\)
\(998\) 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.cr.1.4 yes 4
11.10 odd 2 8470.2.a.cp.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cp.1.3 4 11.10 odd 2
8470.2.a.cr.1.4 yes 4 1.1 even 1 trivial