Properties

Label 69.6.a.b.1.3
Level $69$
Weight $6$
Character 69.1
Self dual yes
Analytic conductor $11.066$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0664835671\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.5333.1
Defining polynomial: \(x^{3} - x^{2} - 11 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.20733\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.49429 q^{2} +9.00000 q^{3} -19.7900 q^{4} -59.5955 q^{5} +31.4486 q^{6} +96.8268 q^{7} -180.969 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.49429 q^{2} +9.00000 q^{3} -19.7900 q^{4} -59.5955 q^{5} +31.4486 q^{6} +96.8268 q^{7} -180.969 q^{8} +81.0000 q^{9} -208.244 q^{10} -731.078 q^{11} -178.110 q^{12} -631.879 q^{13} +338.341 q^{14} -536.360 q^{15} +0.920920 q^{16} +75.7854 q^{17} +283.037 q^{18} +2.35318 q^{19} +1179.39 q^{20} +871.441 q^{21} -2554.60 q^{22} +529.000 q^{23} -1628.72 q^{24} +426.625 q^{25} -2207.97 q^{26} +729.000 q^{27} -1916.20 q^{28} -2425.82 q^{29} -1874.19 q^{30} -349.683 q^{31} +5794.23 q^{32} -6579.70 q^{33} +264.816 q^{34} -5770.44 q^{35} -1602.99 q^{36} +8314.43 q^{37} +8.22270 q^{38} -5686.91 q^{39} +10784.9 q^{40} +4198.56 q^{41} +3045.07 q^{42} +9372.62 q^{43} +14468.0 q^{44} -4827.24 q^{45} +1848.48 q^{46} -2436.94 q^{47} +8.28828 q^{48} -7431.57 q^{49} +1490.75 q^{50} +682.069 q^{51} +12504.8 q^{52} -34979.0 q^{53} +2547.34 q^{54} +43569.0 q^{55} -17522.6 q^{56} +21.1787 q^{57} -8476.51 q^{58} +15086.9 q^{59} +10614.5 q^{60} +987.986 q^{61} -1221.89 q^{62} +7842.97 q^{63} +20217.2 q^{64} +37657.1 q^{65} -22991.4 q^{66} +44337.8 q^{67} -1499.79 q^{68} +4761.00 q^{69} -20163.6 q^{70} -33934.4 q^{71} -14658.5 q^{72} -42483.0 q^{73} +29053.0 q^{74} +3839.62 q^{75} -46.5694 q^{76} -70788.0 q^{77} -19871.7 q^{78} +51187.8 q^{79} -54.8827 q^{80} +6561.00 q^{81} +14671.0 q^{82} -88496.5 q^{83} -17245.8 q^{84} -4516.47 q^{85} +32750.6 q^{86} -21832.4 q^{87} +132303. q^{88} -34083.9 q^{89} -16867.8 q^{90} -61182.8 q^{91} -10468.9 q^{92} -3147.15 q^{93} -8515.36 q^{94} -140.239 q^{95} +52148.0 q^{96} -156803. q^{97} -25968.0 q^{98} -59217.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 8 q^{2} + 27 q^{3} + 22 q^{4} - 56 q^{5} - 72 q^{6} - 114 q^{7} - 510 q^{8} + 243 q^{9} + O(q^{10}) \) \( 3 q - 8 q^{2} + 27 q^{3} + 22 q^{4} - 56 q^{5} - 72 q^{6} - 114 q^{7} - 510 q^{8} + 243 q^{9} + 282 q^{10} - 376 q^{11} + 198 q^{12} - 858 q^{13} + 588 q^{14} - 504 q^{15} + 2738 q^{16} - 2548 q^{17} - 648 q^{18} - 2846 q^{19} - 4618 q^{20} - 1026 q^{21} - 5050 q^{22} + 1587 q^{23} - 4590 q^{24} + 753 q^{25} - 7788 q^{26} + 2187 q^{27} + 4736 q^{28} - 16370 q^{29} + 2538 q^{30} - 14756 q^{31} - 3878 q^{32} - 3384 q^{33} + 16520 q^{34} - 18520 q^{35} + 1782 q^{36} + 15874 q^{37} + 12438 q^{38} - 7722 q^{39} + 38270 q^{40} + 12606 q^{41} + 5292 q^{42} + 3154 q^{43} + 27114 q^{44} - 4536 q^{45} - 4232 q^{46} + 29928 q^{47} + 24642 q^{48} + 4471 q^{49} + 1452 q^{50} - 22932 q^{51} + 86856 q^{52} - 44084 q^{53} - 5832 q^{54} + 38360 q^{55} - 35704 q^{56} - 25614 q^{57} + 73316 q^{58} - 29300 q^{59} - 41562 q^{60} + 54010 q^{61} + 99908 q^{62} - 9234 q^{63} - 1582 q^{64} - 51216 q^{65} - 45450 q^{66} + 43390 q^{67} - 69840 q^{68} + 14283 q^{69} - 2476 q^{70} + 23424 q^{71} - 41310 q^{72} - 91402 q^{73} - 2294 q^{74} + 6777 q^{75} - 14274 q^{76} - 97208 q^{77} - 70092 q^{78} - 49398 q^{79} - 52626 q^{80} + 19683 q^{81} + 40152 q^{82} - 103936 q^{83} + 42624 q^{84} + 5888 q^{85} + 133634 q^{86} - 147330 q^{87} + 48898 q^{88} + 96112 q^{89} + 22842 q^{90} + 129228 q^{91} + 11638 q^{92} - 132804 q^{93} - 133688 q^{94} - 55928 q^{95} - 34902 q^{96} - 135318 q^{97} + 108440 q^{98} - 30456 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.49429 0.617709 0.308854 0.951109i \(-0.400054\pi\)
0.308854 + 0.951109i \(0.400054\pi\)
\(3\) 9.00000 0.577350
\(4\) −19.7900 −0.618436
\(5\) −59.5955 −1.06608 −0.533038 0.846091i \(-0.678950\pi\)
−0.533038 + 0.846091i \(0.678950\pi\)
\(6\) 31.4486 0.356634
\(7\) 96.8268 0.746879 0.373440 0.927654i \(-0.378178\pi\)
0.373440 + 0.927654i \(0.378178\pi\)
\(8\) −180.969 −0.999722
\(9\) 81.0000 0.333333
\(10\) −208.244 −0.658525
\(11\) −731.078 −1.82172 −0.910861 0.412713i \(-0.864581\pi\)
−0.910861 + 0.412713i \(0.864581\pi\)
\(12\) −178.110 −0.357054
\(13\) −631.879 −1.03699 −0.518496 0.855080i \(-0.673508\pi\)
−0.518496 + 0.855080i \(0.673508\pi\)
\(14\) 338.341 0.461354
\(15\) −536.360 −0.615500
\(16\) 0.920920 0.000899336 0
\(17\) 75.7854 0.0636009 0.0318005 0.999494i \(-0.489876\pi\)
0.0318005 + 0.999494i \(0.489876\pi\)
\(18\) 283.037 0.205903
\(19\) 2.35318 0.00149545 0.000747725 1.00000i \(-0.499762\pi\)
0.000747725 1.00000i \(0.499762\pi\)
\(20\) 1179.39 0.659300
\(21\) 871.441 0.431211
\(22\) −2554.60 −1.12529
\(23\) 529.000 0.208514
\(24\) −1628.72 −0.577190
\(25\) 426.625 0.136520
\(26\) −2207.97 −0.640559
\(27\) 729.000 0.192450
\(28\) −1916.20 −0.461897
\(29\) −2425.82 −0.535628 −0.267814 0.963471i \(-0.586301\pi\)
−0.267814 + 0.963471i \(0.586301\pi\)
\(30\) −1874.19 −0.380200
\(31\) −349.683 −0.0653537 −0.0326769 0.999466i \(-0.510403\pi\)
−0.0326769 + 0.999466i \(0.510403\pi\)
\(32\) 5794.23 1.00028
\(33\) −6579.70 −1.05177
\(34\) 264.816 0.0392868
\(35\) −5770.44 −0.796231
\(36\) −1602.99 −0.206145
\(37\) 8314.43 0.998454 0.499227 0.866471i \(-0.333617\pi\)
0.499227 + 0.866471i \(0.333617\pi\)
\(38\) 8.22270 0.000923753 0
\(39\) −5686.91 −0.598707
\(40\) 10784.9 1.06578
\(41\) 4198.56 0.390068 0.195034 0.980796i \(-0.437518\pi\)
0.195034 + 0.980796i \(0.437518\pi\)
\(42\) 3045.07 0.266363
\(43\) 9372.62 0.773019 0.386509 0.922285i \(-0.373681\pi\)
0.386509 + 0.922285i \(0.373681\pi\)
\(44\) 14468.0 1.12662
\(45\) −4827.24 −0.355359
\(46\) 1848.48 0.128801
\(47\) −2436.94 −0.160916 −0.0804581 0.996758i \(-0.525638\pi\)
−0.0804581 + 0.996758i \(0.525638\pi\)
\(48\) 8.28828 0.000519232 0
\(49\) −7431.57 −0.442171
\(50\) 1490.75 0.0843296
\(51\) 682.069 0.0367200
\(52\) 12504.8 0.641313
\(53\) −34979.0 −1.71048 −0.855241 0.518231i \(-0.826591\pi\)
−0.855241 + 0.518231i \(0.826591\pi\)
\(54\) 2547.34 0.118878
\(55\) 43569.0 1.94210
\(56\) −17522.6 −0.746672
\(57\) 21.1787 0.000863399 0
\(58\) −8476.51 −0.330862
\(59\) 15086.9 0.564249 0.282125 0.959378i \(-0.408961\pi\)
0.282125 + 0.959378i \(0.408961\pi\)
\(60\) 10614.5 0.380647
\(61\) 987.986 0.0339959 0.0169979 0.999856i \(-0.494589\pi\)
0.0169979 + 0.999856i \(0.494589\pi\)
\(62\) −1221.89 −0.0403696
\(63\) 7842.97 0.248960
\(64\) 20217.2 0.616981
\(65\) 37657.1 1.10551
\(66\) −22991.4 −0.649689
\(67\) 44337.8 1.20667 0.603333 0.797490i \(-0.293839\pi\)
0.603333 + 0.797490i \(0.293839\pi\)
\(68\) −1499.79 −0.0393331
\(69\) 4761.00 0.120386
\(70\) −20163.6 −0.491839
\(71\) −33934.4 −0.798904 −0.399452 0.916754i \(-0.630800\pi\)
−0.399452 + 0.916754i \(0.630800\pi\)
\(72\) −14658.5 −0.333241
\(73\) −42483.0 −0.933058 −0.466529 0.884506i \(-0.654496\pi\)
−0.466529 + 0.884506i \(0.654496\pi\)
\(74\) 29053.0 0.616753
\(75\) 3839.62 0.0788198
\(76\) −46.5694 −0.000924841 0
\(77\) −70788.0 −1.36061
\(78\) −19871.7 −0.369827
\(79\) 51187.8 0.922782 0.461391 0.887197i \(-0.347351\pi\)
0.461391 + 0.887197i \(0.347351\pi\)
\(80\) −54.8827 −0.000958761 0
\(81\) 6561.00 0.111111
\(82\) 14671.0 0.240949
\(83\) −88496.5 −1.41004 −0.705019 0.709188i \(-0.749062\pi\)
−0.705019 + 0.709188i \(0.749062\pi\)
\(84\) −17245.8 −0.266676
\(85\) −4516.47 −0.0678035
\(86\) 32750.6 0.477500
\(87\) −21832.4 −0.309245
\(88\) 132303. 1.82122
\(89\) −34083.9 −0.456115 −0.228058 0.973648i \(-0.573237\pi\)
−0.228058 + 0.973648i \(0.573237\pi\)
\(90\) −16867.8 −0.219508
\(91\) −61182.8 −0.774508
\(92\) −10468.9 −0.128953
\(93\) −3147.15 −0.0377320
\(94\) −8515.36 −0.0993993
\(95\) −140.239 −0.00159427
\(96\) 52148.0 0.577510
\(97\) −156803. −1.69209 −0.846047 0.533108i \(-0.821024\pi\)
−0.846047 + 0.533108i \(0.821024\pi\)
\(98\) −25968.0 −0.273133
\(99\) −59217.3 −0.607241
\(100\) −8442.89 −0.0844289
\(101\) −148400. −1.44754 −0.723772 0.690039i \(-0.757593\pi\)
−0.723772 + 0.690039i \(0.757593\pi\)
\(102\) 2383.34 0.0226823
\(103\) 155116. 1.44067 0.720335 0.693626i \(-0.243988\pi\)
0.720335 + 0.693626i \(0.243988\pi\)
\(104\) 114350. 1.03670
\(105\) −51934.0 −0.459704
\(106\) −122227. −1.05658
\(107\) −177527. −1.49901 −0.749504 0.662000i \(-0.769708\pi\)
−0.749504 + 0.662000i \(0.769708\pi\)
\(108\) −14426.9 −0.119018
\(109\) −169314. −1.36498 −0.682492 0.730893i \(-0.739104\pi\)
−0.682492 + 0.730893i \(0.739104\pi\)
\(110\) 152243. 1.19965
\(111\) 74829.8 0.576458
\(112\) 89.1697 0.000671696 0
\(113\) 81710.2 0.601977 0.300989 0.953628i \(-0.402683\pi\)
0.300989 + 0.953628i \(0.402683\pi\)
\(114\) 74.0043 0.000533329 0
\(115\) −31526.0 −0.222292
\(116\) 48006.9 0.331252
\(117\) −51182.2 −0.345664
\(118\) 52718.1 0.348542
\(119\) 7338.06 0.0475022
\(120\) 97064.5 0.615329
\(121\) 373424. 2.31867
\(122\) 3452.31 0.0209996
\(123\) 37787.0 0.225206
\(124\) 6920.21 0.0404171
\(125\) 160811. 0.920536
\(126\) 27405.6 0.153785
\(127\) −317307. −1.74570 −0.872851 0.487986i \(-0.837732\pi\)
−0.872851 + 0.487986i \(0.837732\pi\)
\(128\) −114770. −0.619163
\(129\) 84353.6 0.446303
\(130\) 131585. 0.682885
\(131\) −154610. −0.787151 −0.393576 0.919292i \(-0.628762\pi\)
−0.393576 + 0.919292i \(0.628762\pi\)
\(132\) 130212. 0.650454
\(133\) 227.851 0.00111692
\(134\) 154929. 0.745368
\(135\) −43445.1 −0.205167
\(136\) −13714.8 −0.0635832
\(137\) 217676. 0.990852 0.495426 0.868650i \(-0.335012\pi\)
0.495426 + 0.868650i \(0.335012\pi\)
\(138\) 16636.3 0.0743634
\(139\) 236914. 1.04005 0.520023 0.854152i \(-0.325923\pi\)
0.520023 + 0.854152i \(0.325923\pi\)
\(140\) 114197. 0.492418
\(141\) −21932.4 −0.0929050
\(142\) −118577. −0.493490
\(143\) 461953. 1.88911
\(144\) 74.5945 0.000299779 0
\(145\) 144568. 0.571021
\(146\) −148448. −0.576358
\(147\) −66884.1 −0.255288
\(148\) −164542. −0.617480
\(149\) 434197. 1.60221 0.801107 0.598521i \(-0.204245\pi\)
0.801107 + 0.598521i \(0.204245\pi\)
\(150\) 13416.8 0.0486877
\(151\) −135760. −0.484541 −0.242270 0.970209i \(-0.577892\pi\)
−0.242270 + 0.970209i \(0.577892\pi\)
\(152\) −425.853 −0.00149503
\(153\) 6138.62 0.0212003
\(154\) −247354. −0.840459
\(155\) 20839.5 0.0696721
\(156\) 112544. 0.370262
\(157\) −355399. −1.15071 −0.575356 0.817903i \(-0.695136\pi\)
−0.575356 + 0.817903i \(0.695136\pi\)
\(158\) 178865. 0.570010
\(159\) −314811. −0.987547
\(160\) −345310. −1.06637
\(161\) 51221.4 0.155735
\(162\) 22926.0 0.0686343
\(163\) −514344. −1.51630 −0.758148 0.652082i \(-0.773896\pi\)
−0.758148 + 0.652082i \(0.773896\pi\)
\(164\) −83089.3 −0.241232
\(165\) 392121. 1.12127
\(166\) −309232. −0.870993
\(167\) 178285. 0.494680 0.247340 0.968929i \(-0.420444\pi\)
0.247340 + 0.968929i \(0.420444\pi\)
\(168\) −157704. −0.431091
\(169\) 27977.5 0.0753515
\(170\) −15781.8 −0.0418828
\(171\) 190.608 0.000498483 0
\(172\) −185484. −0.478063
\(173\) −535921. −1.36140 −0.680699 0.732563i \(-0.738324\pi\)
−0.680699 + 0.732563i \(0.738324\pi\)
\(174\) −76288.6 −0.191023
\(175\) 41308.7 0.101964
\(176\) −673.265 −0.00163834
\(177\) 135782. 0.325769
\(178\) −119099. −0.281746
\(179\) −58891.2 −0.137378 −0.0686891 0.997638i \(-0.521882\pi\)
−0.0686891 + 0.997638i \(0.521882\pi\)
\(180\) 95530.8 0.219767
\(181\) 829741. 1.88255 0.941274 0.337644i \(-0.109630\pi\)
0.941274 + 0.337644i \(0.109630\pi\)
\(182\) −213790. −0.478420
\(183\) 8891.88 0.0196275
\(184\) −95732.6 −0.208456
\(185\) −495502. −1.06443
\(186\) −10997.0 −0.0233074
\(187\) −55405.1 −0.115863
\(188\) 48226.9 0.0995164
\(189\) 70586.7 0.143737
\(190\) −490.036 −0.000984791 0
\(191\) 149335. 0.296195 0.148097 0.988973i \(-0.452685\pi\)
0.148097 + 0.988973i \(0.452685\pi\)
\(192\) 181955. 0.356214
\(193\) 913030. 1.76438 0.882189 0.470895i \(-0.156069\pi\)
0.882189 + 0.470895i \(0.156069\pi\)
\(194\) −547914. −1.04522
\(195\) 338914. 0.638268
\(196\) 147070. 0.273455
\(197\) −919205. −1.68751 −0.843756 0.536727i \(-0.819661\pi\)
−0.843756 + 0.536727i \(0.819661\pi\)
\(198\) −206922. −0.375098
\(199\) 493730. 0.883806 0.441903 0.897063i \(-0.354304\pi\)
0.441903 + 0.897063i \(0.354304\pi\)
\(200\) −77205.9 −0.136482
\(201\) 399040. 0.696669
\(202\) −518554. −0.894160
\(203\) −234884. −0.400050
\(204\) −13498.1 −0.0227090
\(205\) −250215. −0.415843
\(206\) 542021. 0.889914
\(207\) 42849.0 0.0695048
\(208\) −581.910 −0.000932604 0
\(209\) −1720.36 −0.00272430
\(210\) −181472. −0.283963
\(211\) −133387. −0.206257 −0.103129 0.994668i \(-0.532885\pi\)
−0.103129 + 0.994668i \(0.532885\pi\)
\(212\) 692234. 1.05782
\(213\) −305410. −0.461247
\(214\) −620329. −0.925950
\(215\) −558566. −0.824098
\(216\) −131926. −0.192397
\(217\) −33858.7 −0.0488114
\(218\) −591633. −0.843162
\(219\) −382347. −0.538701
\(220\) −862228. −1.20106
\(221\) −47887.2 −0.0659536
\(222\) 261477. 0.356083
\(223\) 362735. 0.488457 0.244229 0.969718i \(-0.421465\pi\)
0.244229 + 0.969718i \(0.421465\pi\)
\(224\) 561036. 0.747087
\(225\) 34556.6 0.0455067
\(226\) 285519. 0.371846
\(227\) −190962. −0.245970 −0.122985 0.992409i \(-0.539247\pi\)
−0.122985 + 0.992409i \(0.539247\pi\)
\(228\) −419.125 −0.000533957 0
\(229\) 745968. 0.940008 0.470004 0.882664i \(-0.344252\pi\)
0.470004 + 0.882664i \(0.344252\pi\)
\(230\) −110161. −0.137312
\(231\) −637092. −0.785547
\(232\) 438998. 0.535479
\(233\) 636804. 0.768450 0.384225 0.923239i \(-0.374469\pi\)
0.384225 + 0.923239i \(0.374469\pi\)
\(234\) −178845. −0.213520
\(235\) 145231. 0.171549
\(236\) −298570. −0.348952
\(237\) 460691. 0.532768
\(238\) 25641.3 0.0293425
\(239\) 1.44028e6 1.63099 0.815494 0.578765i \(-0.196465\pi\)
0.815494 + 0.578765i \(0.196465\pi\)
\(240\) −493.944 −0.000553541 0
\(241\) 1.13226e6 1.25575 0.627876 0.778313i \(-0.283924\pi\)
0.627876 + 0.778313i \(0.283924\pi\)
\(242\) 1.30485e6 1.43226
\(243\) 59049.0 0.0641500
\(244\) −19552.2 −0.0210243
\(245\) 442888. 0.471388
\(246\) 132039. 0.139112
\(247\) −1486.93 −0.00155077
\(248\) 63281.8 0.0653356
\(249\) −796469. −0.814086
\(250\) 561920. 0.568623
\(251\) −622529. −0.623700 −0.311850 0.950131i \(-0.600949\pi\)
−0.311850 + 0.950131i \(0.600949\pi\)
\(252\) −155212. −0.153966
\(253\) −386740. −0.379855
\(254\) −1.10876e6 −1.07834
\(255\) −40648.2 −0.0391463
\(256\) −1.04799e6 −0.999443
\(257\) −2.10304e6 −1.98616 −0.993082 0.117419i \(-0.962538\pi\)
−0.993082 + 0.117419i \(0.962538\pi\)
\(258\) 294756. 0.275685
\(259\) 805059. 0.745725
\(260\) −745233. −0.683689
\(261\) −196491. −0.178543
\(262\) −540250. −0.486230
\(263\) 838283. 0.747310 0.373655 0.927568i \(-0.378104\pi\)
0.373655 + 0.927568i \(0.378104\pi\)
\(264\) 1.19072e6 1.05148
\(265\) 2.08459e6 1.82350
\(266\) 796.178 0.000689932 0
\(267\) −306755. −0.263338
\(268\) −877443. −0.746246
\(269\) 897532. 0.756257 0.378128 0.925753i \(-0.376568\pi\)
0.378128 + 0.925753i \(0.376568\pi\)
\(270\) −151810. −0.126733
\(271\) −1.00088e6 −0.827864 −0.413932 0.910308i \(-0.635845\pi\)
−0.413932 + 0.910308i \(0.635845\pi\)
\(272\) 69.7923 5.71986e−5 0
\(273\) −550645. −0.447162
\(274\) 760622. 0.612058
\(275\) −311896. −0.248701
\(276\) −94220.0 −0.0744510
\(277\) 2.09974e6 1.64424 0.822120 0.569314i \(-0.192791\pi\)
0.822120 + 0.569314i \(0.192791\pi\)
\(278\) 827844. 0.642446
\(279\) −28324.3 −0.0217846
\(280\) 1.04427e6 0.796009
\(281\) 503227. 0.380188 0.190094 0.981766i \(-0.439121\pi\)
0.190094 + 0.981766i \(0.439121\pi\)
\(282\) −76638.3 −0.0573882
\(283\) −1.61771e6 −1.20070 −0.600349 0.799738i \(-0.704972\pi\)
−0.600349 + 0.799738i \(0.704972\pi\)
\(284\) 671561. 0.494071
\(285\) −1262.15 −0.000920449 0
\(286\) 1.61420e6 1.16692
\(287\) 406533. 0.291334
\(288\) 469332. 0.333426
\(289\) −1.41411e6 −0.995955
\(290\) 505162. 0.352725
\(291\) −1.41123e6 −0.976932
\(292\) 840738. 0.577037
\(293\) 1.04521e6 0.711270 0.355635 0.934625i \(-0.384265\pi\)
0.355635 + 0.934625i \(0.384265\pi\)
\(294\) −233712. −0.157693
\(295\) −899113. −0.601533
\(296\) −1.50465e6 −0.998176
\(297\) −532956. −0.350591
\(298\) 1.51721e6 0.989702
\(299\) −334264. −0.216228
\(300\) −75986.0 −0.0487450
\(301\) 907521. 0.577352
\(302\) −474385. −0.299305
\(303\) −1.33560e6 −0.835740
\(304\) 2.16710 1.34491e−6 0
\(305\) −58879.6 −0.0362422
\(306\) 21450.1 0.0130956
\(307\) −697539. −0.422399 −0.211199 0.977443i \(-0.567737\pi\)
−0.211199 + 0.977443i \(0.567737\pi\)
\(308\) 1.40089e6 0.841448
\(309\) 1.39605e6 0.831771
\(310\) 72819.4 0.0430371
\(311\) −2.68619e6 −1.57484 −0.787419 0.616419i \(-0.788583\pi\)
−0.787419 + 0.616419i \(0.788583\pi\)
\(312\) 1.02915e6 0.598541
\(313\) 1.62222e6 0.935944 0.467972 0.883743i \(-0.344985\pi\)
0.467972 + 0.883743i \(0.344985\pi\)
\(314\) −1.24186e6 −0.710805
\(315\) −467406. −0.265410
\(316\) −1.01301e6 −0.570682
\(317\) 648171. 0.362277 0.181139 0.983458i \(-0.442022\pi\)
0.181139 + 0.983458i \(0.442022\pi\)
\(318\) −1.10004e6 −0.610016
\(319\) 1.77346e6 0.975766
\(320\) −1.20486e6 −0.657749
\(321\) −1.59774e6 −0.865452
\(322\) 178982. 0.0961989
\(323\) 178.337 9.51120e−5 0
\(324\) −129842. −0.0687151
\(325\) −269575. −0.141570
\(326\) −1.79726e6 −0.936630
\(327\) −1.52383e6 −0.788074
\(328\) −759809. −0.389960
\(329\) −235961. −0.120185
\(330\) 1.37018e6 0.692618
\(331\) −1.66550e6 −0.835552 −0.417776 0.908550i \(-0.637190\pi\)
−0.417776 + 0.908550i \(0.637190\pi\)
\(332\) 1.75134e6 0.872019
\(333\) 673468. 0.332818
\(334\) 622980. 0.305568
\(335\) −2.64233e6 −1.28640
\(336\) 802.528 0.000387804 0
\(337\) −2.11943e6 −1.01659 −0.508294 0.861184i \(-0.669724\pi\)
−0.508294 + 0.861184i \(0.669724\pi\)
\(338\) 97761.3 0.0465452
\(339\) 735392. 0.347552
\(340\) 89380.8 0.0419321
\(341\) 255646. 0.119056
\(342\) 666.039 0.000307918 0
\(343\) −2.34694e6 −1.07713
\(344\) −1.69615e6 −0.772804
\(345\) −283734. −0.128341
\(346\) −1.87266e6 −0.840948
\(347\) −791717. −0.352977 −0.176488 0.984303i \(-0.556474\pi\)
−0.176488 + 0.984303i \(0.556474\pi\)
\(348\) 432062. 0.191248
\(349\) 743781. 0.326875 0.163437 0.986554i \(-0.447742\pi\)
0.163437 + 0.986554i \(0.447742\pi\)
\(350\) 144345. 0.0629840
\(351\) −460639. −0.199569
\(352\) −4.23603e6 −1.82223
\(353\) −646287. −0.276051 −0.138025 0.990429i \(-0.544076\pi\)
−0.138025 + 0.990429i \(0.544076\pi\)
\(354\) 474463. 0.201231
\(355\) 2.02234e6 0.851693
\(356\) 674519. 0.282078
\(357\) 66042.5 0.0274254
\(358\) −205783. −0.0848597
\(359\) 1.89862e6 0.777503 0.388752 0.921343i \(-0.372906\pi\)
0.388752 + 0.921343i \(0.372906\pi\)
\(360\) 873580. 0.355260
\(361\) −2.47609e6 −0.999998
\(362\) 2.89935e6 1.16287
\(363\) 3.36082e6 1.33869
\(364\) 1.21080e6 0.478983
\(365\) 2.53180e6 0.994711
\(366\) 31070.8 0.0121241
\(367\) 1.19082e6 0.461509 0.230754 0.973012i \(-0.425881\pi\)
0.230754 + 0.973012i \(0.425881\pi\)
\(368\) 487.167 0.000187525 0
\(369\) 340083. 0.130023
\(370\) −1.73143e6 −0.657507
\(371\) −3.38691e6 −1.27752
\(372\) 62281.9 0.0233348
\(373\) −254301. −0.0946402 −0.0473201 0.998880i \(-0.515068\pi\)
−0.0473201 + 0.998880i \(0.515068\pi\)
\(374\) −193601. −0.0715697
\(375\) 1.44730e6 0.531472
\(376\) 441010. 0.160871
\(377\) 1.53282e6 0.555442
\(378\) 246650. 0.0887876
\(379\) −5.16696e6 −1.84772 −0.923861 0.382728i \(-0.874985\pi\)
−0.923861 + 0.382728i \(0.874985\pi\)
\(380\) 2775.33 0.000985951 0
\(381\) −2.85576e6 −1.00788
\(382\) 521818. 0.182962
\(383\) 5.10331e6 1.77769 0.888843 0.458212i \(-0.151510\pi\)
0.888843 + 0.458212i \(0.151510\pi\)
\(384\) −1.03293e6 −0.357474
\(385\) 4.21865e6 1.45051
\(386\) 3.19039e6 1.08987
\(387\) 759183. 0.257673
\(388\) 3.10312e6 1.04645
\(389\) −1.00400e6 −0.336403 −0.168201 0.985753i \(-0.553796\pi\)
−0.168201 + 0.985753i \(0.553796\pi\)
\(390\) 1.18426e6 0.394264
\(391\) 40090.5 0.0132617
\(392\) 1.34488e6 0.442048
\(393\) −1.39149e6 −0.454462
\(394\) −3.21197e6 −1.04239
\(395\) −3.05057e6 −0.983757
\(396\) 1.17191e6 0.375540
\(397\) 3.62003e6 1.15275 0.576377 0.817184i \(-0.304466\pi\)
0.576377 + 0.817184i \(0.304466\pi\)
\(398\) 1.72523e6 0.545934
\(399\) 2050.66 0.000644855 0
\(400\) 392.887 0.000122777 0
\(401\) 1.18825e6 0.369016 0.184508 0.982831i \(-0.440931\pi\)
0.184508 + 0.982831i \(0.440931\pi\)
\(402\) 1.39436e6 0.430338
\(403\) 220957. 0.0677713
\(404\) 2.93684e6 0.895213
\(405\) −391006. −0.118453
\(406\) −820753. −0.247114
\(407\) −6.07850e6 −1.81891
\(408\) −123433. −0.0367098
\(409\) −64602.3 −0.0190959 −0.00954794 0.999954i \(-0.503039\pi\)
−0.00954794 + 0.999954i \(0.503039\pi\)
\(410\) −874324. −0.256870
\(411\) 1.95908e6 0.572068
\(412\) −3.06975e6 −0.890962
\(413\) 1.46082e6 0.421426
\(414\) 149727. 0.0429337
\(415\) 5.27399e6 1.50321
\(416\) −3.66125e6 −1.03728
\(417\) 2.13222e6 0.600471
\(418\) −6011.44 −0.00168282
\(419\) −2.92057e6 −0.812704 −0.406352 0.913717i \(-0.633199\pi\)
−0.406352 + 0.913717i \(0.633199\pi\)
\(420\) 1.02777e6 0.284298
\(421\) −2.99592e6 −0.823807 −0.411904 0.911227i \(-0.635136\pi\)
−0.411904 + 0.911227i \(0.635136\pi\)
\(422\) −466094. −0.127407
\(423\) −197392. −0.0536387
\(424\) 6.33012e6 1.71001
\(425\) 32331.9 0.00868280
\(426\) −1.06719e6 −0.284916
\(427\) 95663.6 0.0253908
\(428\) 3.51324e6 0.927040
\(429\) 4.15757e6 1.09068
\(430\) −1.95179e6 −0.509052
\(431\) 5.82982e6 1.51169 0.755844 0.654752i \(-0.227227\pi\)
0.755844 + 0.654752i \(0.227227\pi\)
\(432\) 671.351 0.000173077 0
\(433\) −2.98624e6 −0.765429 −0.382714 0.923867i \(-0.625011\pi\)
−0.382714 + 0.923867i \(0.625011\pi\)
\(434\) −118312. −0.0301512
\(435\) 1.30111e6 0.329679
\(436\) 3.35072e6 0.844155
\(437\) 1244.83 0.000311823 0
\(438\) −1.33603e6 −0.332760
\(439\) −5.68591e6 −1.40812 −0.704058 0.710142i \(-0.748631\pi\)
−0.704058 + 0.710142i \(0.748631\pi\)
\(440\) −7.88464e6 −1.94156
\(441\) −601957. −0.147390
\(442\) −167332. −0.0407401
\(443\) −68926.8 −0.0166870 −0.00834352 0.999965i \(-0.502656\pi\)
−0.00834352 + 0.999965i \(0.502656\pi\)
\(444\) −1.48088e6 −0.356502
\(445\) 2.03125e6 0.486254
\(446\) 1.26750e6 0.301724
\(447\) 3.90777e6 0.925039
\(448\) 1.95757e6 0.460810
\(449\) 4.63362e6 1.08469 0.542343 0.840157i \(-0.317537\pi\)
0.542343 + 0.840157i \(0.317537\pi\)
\(450\) 120751. 0.0281099
\(451\) −3.06948e6 −0.710596
\(452\) −1.61704e6 −0.372284
\(453\) −1.22184e6 −0.279750
\(454\) −667275. −0.151937
\(455\) 3.64622e6 0.825685
\(456\) −3832.68 −0.000863159 0
\(457\) −4.33421e6 −0.970777 −0.485388 0.874299i \(-0.661322\pi\)
−0.485388 + 0.874299i \(0.661322\pi\)
\(458\) 2.60663e6 0.580651
\(459\) 55247.6 0.0122400
\(460\) 623899. 0.137474
\(461\) 2.08958e6 0.457938 0.228969 0.973434i \(-0.426465\pi\)
0.228969 + 0.973434i \(0.426465\pi\)
\(462\) −2.22618e6 −0.485239
\(463\) −1.24932e6 −0.270846 −0.135423 0.990788i \(-0.543239\pi\)
−0.135423 + 0.990788i \(0.543239\pi\)
\(464\) −2233.99 −0.000481710 0
\(465\) 187556. 0.0402252
\(466\) 2.22517e6 0.474678
\(467\) −157302. −0.0333765 −0.0166883 0.999861i \(-0.505312\pi\)
−0.0166883 + 0.999861i \(0.505312\pi\)
\(468\) 1.01289e6 0.213771
\(469\) 4.29309e6 0.901234
\(470\) 507477. 0.105967
\(471\) −3.19859e6 −0.664364
\(472\) −2.73027e6 −0.564092
\(473\) −6.85212e6 −1.40823
\(474\) 1.60979e6 0.329096
\(475\) 1003.93 0.000204159 0
\(476\) −145220. −0.0293771
\(477\) −2.83330e6 −0.570160
\(478\) 5.03274e6 1.00748
\(479\) 2.74855e6 0.547350 0.273675 0.961822i \(-0.411761\pi\)
0.273675 + 0.961822i \(0.411761\pi\)
\(480\) −3.10779e6 −0.615671
\(481\) −5.25371e6 −1.03539
\(482\) 3.95645e6 0.775689
\(483\) 460992. 0.0899137
\(484\) −7.39005e6 −1.43395
\(485\) 9.34475e6 1.80390
\(486\) 206334. 0.0396260
\(487\) −1.01838e7 −1.94576 −0.972878 0.231319i \(-0.925696\pi\)
−0.972878 + 0.231319i \(0.925696\pi\)
\(488\) −178795. −0.0339864
\(489\) −4.62909e6 −0.875434
\(490\) 1.54758e6 0.291181
\(491\) −8.81893e6 −1.65087 −0.825434 0.564499i \(-0.809069\pi\)
−0.825434 + 0.564499i \(0.809069\pi\)
\(492\) −747804. −0.139276
\(493\) −183842. −0.0340664
\(494\) −5195.75 −0.000957924 0
\(495\) 3.52909e6 0.647365
\(496\) −322.030 −5.87750e−5 0
\(497\) −3.28576e6 −0.596685
\(498\) −2.78309e6 −0.502868
\(499\) −7.27705e6 −1.30829 −0.654145 0.756369i \(-0.726971\pi\)
−0.654145 + 0.756369i \(0.726971\pi\)
\(500\) −3.18244e6 −0.569293
\(501\) 1.60457e6 0.285604
\(502\) −2.17530e6 −0.385265
\(503\) −2.82184e6 −0.497293 −0.248646 0.968594i \(-0.579986\pi\)
−0.248646 + 0.968594i \(0.579986\pi\)
\(504\) −1.41933e6 −0.248891
\(505\) 8.84400e6 1.54319
\(506\) −1.35138e6 −0.234640
\(507\) 251797. 0.0435042
\(508\) 6.27949e6 1.07961
\(509\) −4.78063e6 −0.817881 −0.408941 0.912561i \(-0.634102\pi\)
−0.408941 + 0.912561i \(0.634102\pi\)
\(510\) −142037. −0.0241810
\(511\) −4.11350e6 −0.696882
\(512\) 10669.1 0.00179867
\(513\) 1715.47 0.000287800 0
\(514\) −7.34864e6 −1.22687
\(515\) −9.24424e6 −1.53587
\(516\) −1.66935e6 −0.276010
\(517\) 1.78159e6 0.293145
\(518\) 2.81311e6 0.460640
\(519\) −4.82329e6 −0.786004
\(520\) −6.81477e6 −1.10521
\(521\) 7.11976e6 1.14914 0.574568 0.818457i \(-0.305170\pi\)
0.574568 + 0.818457i \(0.305170\pi\)
\(522\) −686597. −0.110287
\(523\) 4.97980e6 0.796082 0.398041 0.917368i \(-0.369690\pi\)
0.398041 + 0.917368i \(0.369690\pi\)
\(524\) 3.05972e6 0.486803
\(525\) 371779. 0.0588689
\(526\) 2.92920e6 0.461620
\(527\) −26500.9 −0.00415656
\(528\) −6059.38 −0.000945896 0
\(529\) 279841. 0.0434783
\(530\) 7.28417e6 1.12639
\(531\) 1.22204e6 0.188083
\(532\) −4509.17 −0.000690744 0
\(533\) −2.65298e6 −0.404498
\(534\) −1.07189e6 −0.162666
\(535\) 1.05798e7 1.59806
\(536\) −8.02376e6 −1.20633
\(537\) −530021. −0.0793153
\(538\) 3.13624e6 0.467146
\(539\) 5.43306e6 0.805513
\(540\) 859777. 0.126882
\(541\) 2.34995e6 0.345196 0.172598 0.984992i \(-0.444784\pi\)
0.172598 + 0.984992i \(0.444784\pi\)
\(542\) −3.49737e6 −0.511379
\(543\) 7.46767e6 1.08689
\(544\) 439118. 0.0636186
\(545\) 1.00904e7 1.45518
\(546\) −1.92411e6 −0.276216
\(547\) 1.27753e7 1.82559 0.912795 0.408419i \(-0.133920\pi\)
0.912795 + 0.408419i \(0.133920\pi\)
\(548\) −4.30779e6 −0.612778
\(549\) 80026.9 0.0113320
\(550\) −1.08986e6 −0.153625
\(551\) −5708.40 −0.000801005 0
\(552\) −861593. −0.120352
\(553\) 4.95636e6 0.689207
\(554\) 7.33708e6 1.01566
\(555\) −4.45952e6 −0.614548
\(556\) −4.68851e6 −0.643203
\(557\) −6.13111e6 −0.837339 −0.418670 0.908139i \(-0.637503\pi\)
−0.418670 + 0.908139i \(0.637503\pi\)
\(558\) −98973.4 −0.0134565
\(559\) −5.92236e6 −0.801614
\(560\) −5314.12 −0.000716079 0
\(561\) −498646. −0.0668937
\(562\) 1.75842e6 0.234845
\(563\) 1.30613e7 1.73667 0.868334 0.495979i \(-0.165191\pi\)
0.868334 + 0.495979i \(0.165191\pi\)
\(564\) 434042. 0.0574558
\(565\) −4.86956e6 −0.641754
\(566\) −5.65274e6 −0.741682
\(567\) 635281. 0.0829866
\(568\) 6.14108e6 0.798682
\(569\) −5.03952e6 −0.652542 −0.326271 0.945276i \(-0.605792\pi\)
−0.326271 + 0.945276i \(0.605792\pi\)
\(570\) −4410.33 −0.000568569 0
\(571\) −7.67025e6 −0.984508 −0.492254 0.870452i \(-0.663827\pi\)
−0.492254 + 0.870452i \(0.663827\pi\)
\(572\) −9.14202e6 −1.16829
\(573\) 1.34401e6 0.171008
\(574\) 1.42054e6 0.179960
\(575\) 225685. 0.0284664
\(576\) 1.63759e6 0.205660
\(577\) 1.25985e7 1.57536 0.787681 0.616083i \(-0.211281\pi\)
0.787681 + 0.616083i \(0.211281\pi\)
\(578\) −4.94132e6 −0.615210
\(579\) 8.21727e6 1.01866
\(580\) −2.86099e6 −0.353140
\(581\) −8.56883e6 −1.05313
\(582\) −4.93123e6 −0.603459
\(583\) 2.55724e7 3.11602
\(584\) 7.68811e6 0.932798
\(585\) 3.05023e6 0.368504
\(586\) 3.65226e6 0.439357
\(587\) 5.11295e6 0.612457 0.306229 0.951958i \(-0.400933\pi\)
0.306229 + 0.951958i \(0.400933\pi\)
\(588\) 1.32363e6 0.157879
\(589\) −822.869 −9.77333e−5 0
\(590\) −3.14176e6 −0.371572
\(591\) −8.27285e6 −0.974286
\(592\) 7656.92 0.000897945 0
\(593\) 5.45932e6 0.637532 0.318766 0.947833i \(-0.396732\pi\)
0.318766 + 0.947833i \(0.396732\pi\)
\(594\) −1.86230e6 −0.216563
\(595\) −437315. −0.0506410
\(596\) −8.59273e6 −0.990867
\(597\) 4.44357e6 0.510265
\(598\) −1.16801e6 −0.133566
\(599\) −22496.6 −0.00256183 −0.00128091 0.999999i \(-0.500408\pi\)
−0.00128091 + 0.999999i \(0.500408\pi\)
\(600\) −694853. −0.0787979
\(601\) 1.00912e6 0.113962 0.0569808 0.998375i \(-0.481853\pi\)
0.0569808 + 0.998375i \(0.481853\pi\)
\(602\) 3.17114e6 0.356635
\(603\) 3.59136e6 0.402222
\(604\) 2.68669e6 0.299657
\(605\) −2.22544e7 −2.47188
\(606\) −4.66698e6 −0.516244
\(607\) −68503.9 −0.00754646 −0.00377323 0.999993i \(-0.501201\pi\)
−0.00377323 + 0.999993i \(0.501201\pi\)
\(608\) 13634.9 0.00149587
\(609\) −2.11396e6 −0.230969
\(610\) −205742. −0.0223871
\(611\) 1.53985e6 0.166869
\(612\) −121483. −0.0131110
\(613\) 7.67884e6 0.825362 0.412681 0.910876i \(-0.364593\pi\)
0.412681 + 0.910876i \(0.364593\pi\)
\(614\) −2.43740e6 −0.260919
\(615\) −2.25194e6 −0.240087
\(616\) 1.28104e7 1.36023
\(617\) −1.85528e6 −0.196199 −0.0980995 0.995177i \(-0.531276\pi\)
−0.0980995 + 0.995177i \(0.531276\pi\)
\(618\) 4.87819e6 0.513792
\(619\) −2.88941e6 −0.303098 −0.151549 0.988450i \(-0.548426\pi\)
−0.151549 + 0.988450i \(0.548426\pi\)
\(620\) −412414. −0.0430878
\(621\) 385641. 0.0401286
\(622\) −9.38632e6 −0.972791
\(623\) −3.30024e6 −0.340663
\(624\) −5237.19 −0.000538439 0
\(625\) −1.09168e7 −1.11788
\(626\) 5.66851e6 0.578140
\(627\) −15483.3 −0.00157287
\(628\) 7.03332e6 0.711642
\(629\) 630112. 0.0635026
\(630\) −1.63325e6 −0.163946
\(631\) −1.22946e7 −1.22925 −0.614625 0.788819i \(-0.710693\pi\)
−0.614625 + 0.788819i \(0.710693\pi\)
\(632\) −9.26341e6 −0.922525
\(633\) −1.20049e6 −0.119083
\(634\) 2.26489e6 0.223782
\(635\) 1.89101e7 1.86105
\(636\) 6.23010e6 0.610735
\(637\) 4.69585e6 0.458528
\(638\) 6.19699e6 0.602739
\(639\) −2.74869e6 −0.266301
\(640\) 6.83980e6 0.660076
\(641\) −7.87250e6 −0.756776 −0.378388 0.925647i \(-0.623522\pi\)
−0.378388 + 0.925647i \(0.623522\pi\)
\(642\) −5.58296e6 −0.534597
\(643\) 3.14575e6 0.300052 0.150026 0.988682i \(-0.452064\pi\)
0.150026 + 0.988682i \(0.452064\pi\)
\(644\) −1.01367e6 −0.0963122
\(645\) −5.02710e6 −0.475793
\(646\) 623.161 5.87515e−5 0
\(647\) 4.14059e6 0.388867 0.194434 0.980916i \(-0.437713\pi\)
0.194434 + 0.980916i \(0.437713\pi\)
\(648\) −1.18734e6 −0.111080
\(649\) −1.10297e7 −1.02791
\(650\) −941973. −0.0874490
\(651\) −304728. −0.0281813
\(652\) 1.01788e7 0.937733
\(653\) 1.09894e7 1.00854 0.504268 0.863547i \(-0.331762\pi\)
0.504268 + 0.863547i \(0.331762\pi\)
\(654\) −5.32470e6 −0.486800
\(655\) 9.21404e6 0.839164
\(656\) 3866.54 0.000350803 0
\(657\) −3.44113e6 −0.311019
\(658\) −824515. −0.0742393
\(659\) 1.78286e7 1.59920 0.799602 0.600530i \(-0.205044\pi\)
0.799602 + 0.600530i \(0.205044\pi\)
\(660\) −7.76005e6 −0.693434
\(661\) 4.39680e6 0.391411 0.195705 0.980663i \(-0.437300\pi\)
0.195705 + 0.980663i \(0.437300\pi\)
\(662\) −5.81972e6 −0.516128
\(663\) −430985. −0.0380783
\(664\) 1.60151e7 1.40965
\(665\) −13578.9 −0.00119072
\(666\) 2.35329e6 0.205584
\(667\) −1.28326e6 −0.111686
\(668\) −3.52826e6 −0.305928
\(669\) 3.26461e6 0.282011
\(670\) −9.23307e6 −0.794619
\(671\) −722295. −0.0619311
\(672\) 5.04933e6 0.431331
\(673\) −1.98954e7 −1.69322 −0.846612 0.532210i \(-0.821362\pi\)
−0.846612 + 0.532210i \(0.821362\pi\)
\(674\) −7.40591e6 −0.627955
\(675\) 311010. 0.0262733
\(676\) −553673. −0.0466001
\(677\) −796059. −0.0667535 −0.0333767 0.999443i \(-0.510626\pi\)
−0.0333767 + 0.999443i \(0.510626\pi\)
\(678\) 2.56967e6 0.214686
\(679\) −1.51827e7 −1.26379
\(680\) 817341. 0.0677846
\(681\) −1.71865e6 −0.142011
\(682\) 893300. 0.0735421
\(683\) −9.97449e6 −0.818162 −0.409081 0.912498i \(-0.634151\pi\)
−0.409081 + 0.912498i \(0.634151\pi\)
\(684\) −3772.12 −0.000308280 0
\(685\) −1.29725e7 −1.05632
\(686\) −8.20089e6 −0.665351
\(687\) 6.71371e6 0.542714
\(688\) 8631.44 0.000695204 0
\(689\) 2.21025e7 1.77375
\(690\) −991449. −0.0792771
\(691\) −6.67183e6 −0.531557 −0.265779 0.964034i \(-0.585629\pi\)
−0.265779 + 0.964034i \(0.585629\pi\)
\(692\) 1.06058e7 0.841938
\(693\) −5.73383e6 −0.453536
\(694\) −2.76649e6 −0.218037
\(695\) −1.41190e7 −1.10877
\(696\) 3.95098e6 0.309159
\(697\) 318190. 0.0248087
\(698\) 2.59898e6 0.201913
\(699\) 5.73123e6 0.443665
\(700\) −817498. −0.0630582
\(701\) 5.64132e6 0.433597 0.216798 0.976216i \(-0.430439\pi\)
0.216798 + 0.976216i \(0.430439\pi\)
\(702\) −1.60961e6 −0.123276
\(703\) 19565.4 0.00149314
\(704\) −1.47804e7 −1.12397
\(705\) 1.30708e6 0.0990439
\(706\) −2.25831e6 −0.170519
\(707\) −1.43691e7 −1.08114
\(708\) −2.68713e6 −0.201468
\(709\) −1.86950e6 −0.139672 −0.0698361 0.997558i \(-0.522248\pi\)
−0.0698361 + 0.997558i \(0.522248\pi\)
\(710\) 7.06663e6 0.526098
\(711\) 4.14622e6 0.307594
\(712\) 6.16813e6 0.455988
\(713\) −184982. −0.0136272
\(714\) 230772. 0.0169409
\(715\) −2.75303e7 −2.01394
\(716\) 1.16545e6 0.0849596
\(717\) 1.29625e7 0.941652
\(718\) 6.63433e6 0.480270
\(719\) 1.87348e7 1.35153 0.675766 0.737116i \(-0.263813\pi\)
0.675766 + 0.737116i \(0.263813\pi\)
\(720\) −4445.50 −0.000319587 0
\(721\) 1.50194e7 1.07601
\(722\) −8.65218e6 −0.617707
\(723\) 1.01904e7 0.725009
\(724\) −1.64205e7 −1.16424
\(725\) −1.03492e6 −0.0731240
\(726\) 1.17437e7 0.826918
\(727\) 7.88165e6 0.553071 0.276536 0.961004i \(-0.410814\pi\)
0.276536 + 0.961004i \(0.410814\pi\)
\(728\) 1.10722e7 0.774292
\(729\) 531441. 0.0370370
\(730\) 8.84683e6 0.614442
\(731\) 710308. 0.0491647
\(732\) −175970. −0.0121384
\(733\) 196627. 0.0135171 0.00675854 0.999977i \(-0.497849\pi\)
0.00675854 + 0.999977i \(0.497849\pi\)
\(734\) 4.16106e6 0.285078
\(735\) 3.98599e6 0.272156
\(736\) 3.06515e6 0.208572
\(737\) −3.24144e7 −2.19821
\(738\) 1.18835e6 0.0803162
\(739\) −1.89161e7 −1.27415 −0.637074 0.770803i \(-0.719855\pi\)
−0.637074 + 0.770803i \(0.719855\pi\)
\(740\) 9.80597e6 0.658281
\(741\) −13382.3 −0.000895337 0
\(742\) −1.18348e7 −0.789137
\(743\) −5.29482e6 −0.351867 −0.175934 0.984402i \(-0.556294\pi\)
−0.175934 + 0.984402i \(0.556294\pi\)
\(744\) 569536. 0.0377215
\(745\) −2.58762e7 −1.70808
\(746\) −888600. −0.0584601
\(747\) −7.16822e6 −0.470013
\(748\) 1.09646e6 0.0716540
\(749\) −1.71893e7 −1.11958
\(750\) 5.05728e6 0.328295
\(751\) −1.23104e7 −0.796478 −0.398239 0.917282i \(-0.630379\pi\)
−0.398239 + 0.917282i \(0.630379\pi\)
\(752\) −2244.23 −0.000144718 0
\(753\) −5.60276e6 −0.360093
\(754\) 5.35612e6 0.343101
\(755\) 8.09070e6 0.516558
\(756\) −1.39691e6 −0.0888922
\(757\) −7.33051e6 −0.464938 −0.232469 0.972604i \(-0.574680\pi\)
−0.232469 + 0.972604i \(0.574680\pi\)
\(758\) −1.80548e7 −1.14135
\(759\) −3.48066e6 −0.219310
\(760\) 25379.0 0.00159382
\(761\) 3.11913e6 0.195242 0.0976208 0.995224i \(-0.468877\pi\)
0.0976208 + 0.995224i \(0.468877\pi\)
\(762\) −9.97886e6 −0.622577
\(763\) −1.63942e7 −1.01948
\(764\) −2.95533e6 −0.183178
\(765\) −365834. −0.0226012
\(766\) 1.78324e7 1.09809
\(767\) −9.53311e6 −0.585122
\(768\) −9.43193e6 −0.577029
\(769\) −7.07903e6 −0.431676 −0.215838 0.976429i \(-0.569248\pi\)
−0.215838 + 0.976429i \(0.569248\pi\)
\(770\) 1.47412e7 0.895993
\(771\) −1.89274e7 −1.14671
\(772\) −1.80688e7 −1.09116
\(773\) 1.91702e7 1.15393 0.576964 0.816770i \(-0.304237\pi\)
0.576964 + 0.816770i \(0.304237\pi\)
\(774\) 2.65280e6 0.159167
\(775\) −149184. −0.00892209
\(776\) 2.83765e7 1.69162
\(777\) 7.24553e6 0.430544
\(778\) −3.50826e6 −0.207799
\(779\) 9879.99 0.000583328 0
\(780\) −6.70710e6 −0.394728
\(781\) 2.48087e7 1.45538
\(782\) 140088. 0.00819187
\(783\) −1.76842e6 −0.103082
\(784\) −6843.88 −0.000397660 0
\(785\) 2.11802e7 1.22675
\(786\) −4.86225e6 −0.280725
\(787\) 8.38766e6 0.482730 0.241365 0.970434i \(-0.422405\pi\)
0.241365 + 0.970434i \(0.422405\pi\)
\(788\) 1.81910e7 1.04362
\(789\) 7.54454e6 0.431460
\(790\) −1.06596e7 −0.607675
\(791\) 7.91173e6 0.449604
\(792\) 1.07165e7 0.607072
\(793\) −624287. −0.0352534
\(794\) 1.26494e7 0.712066
\(795\) 1.87613e7 1.05280
\(796\) −9.77089e6 −0.546577
\(797\) −1.36638e7 −0.761950 −0.380975 0.924585i \(-0.624412\pi\)
−0.380975 + 0.924585i \(0.624412\pi\)
\(798\) 7165.60 0.000398332 0
\(799\) −184684. −0.0102344
\(800\) 2.47196e6 0.136558
\(801\) −2.76080e6 −0.152038
\(802\) 4.15207e6 0.227944
\(803\) 3.10584e7 1.69977
\(804\) −7.89698e6 −0.430845
\(805\) −3.05256e6 −0.166026
\(806\) 772088. 0.0418629
\(807\) 8.07779e6 0.436625
\(808\) 2.68559e7 1.44714
\(809\) 5.16198e6 0.277297 0.138648 0.990342i \(-0.455724\pi\)
0.138648 + 0.990342i \(0.455724\pi\)
\(810\) −1.36629e6 −0.0731694
\(811\) 1.20275e7 0.642131 0.321065 0.947057i \(-0.395959\pi\)
0.321065 + 0.947057i \(0.395959\pi\)
\(812\) 4.64835e6 0.247405
\(813\) −9.00793e6 −0.477968
\(814\) −2.12400e7 −1.12355
\(815\) 3.06526e7 1.61649
\(816\) 628.131 3.30236e−5 0
\(817\) 22055.5 0.00115601
\(818\) −225739. −0.0117957
\(819\) −4.95580e6 −0.258169
\(820\) 4.95175e6 0.257172
\(821\) −4.25894e6 −0.220518 −0.110259 0.993903i \(-0.535168\pi\)
−0.110259 + 0.993903i \(0.535168\pi\)
\(822\) 6.84560e6 0.353372
\(823\) 1.63473e7 0.841290 0.420645 0.907225i \(-0.361804\pi\)
0.420645 + 0.907225i \(0.361804\pi\)
\(824\) −2.80712e7 −1.44027
\(825\) −2.80707e6 −0.143588
\(826\) 5.10452e6 0.260318
\(827\) −1.62566e7 −0.826542 −0.413271 0.910608i \(-0.635614\pi\)
−0.413271 + 0.910608i \(0.635614\pi\)
\(828\) −847980. −0.0429843
\(829\) 3.87989e7 1.96080 0.980400 0.197018i \(-0.0631256\pi\)
0.980400 + 0.197018i \(0.0631256\pi\)
\(830\) 1.84289e7 0.928545
\(831\) 1.88976e7 0.949302
\(832\) −1.27748e7 −0.639804
\(833\) −563205. −0.0281225
\(834\) 7.45060e6 0.370916
\(835\) −1.06250e7 −0.527367
\(836\) 34045.9 0.00168480
\(837\) −254919. −0.0125773
\(838\) −1.02053e7 −0.502014
\(839\) −2.96779e7 −1.45555 −0.727776 0.685815i \(-0.759446\pi\)
−0.727776 + 0.685815i \(0.759446\pi\)
\(840\) 9.39844e6 0.459576
\(841\) −1.46265e7 −0.713102
\(842\) −1.04686e7 −0.508873
\(843\) 4.52904e6 0.219502
\(844\) 2.63973e6 0.127557
\(845\) −1.66733e6 −0.0803305
\(846\) −689744. −0.0331331
\(847\) 3.61575e7 1.73177
\(848\) −32212.9 −0.00153830
\(849\) −1.45594e7 −0.693224
\(850\) 112977. 0.00536344
\(851\) 4.39833e6 0.208192
\(852\) 6.04405e6 0.285252
\(853\) −3.20704e7 −1.50915 −0.754573 0.656216i \(-0.772156\pi\)
−0.754573 + 0.656216i \(0.772156\pi\)
\(854\) 334276. 0.0156841
\(855\) −11359.4 −0.000531422 0
\(856\) 3.21268e7 1.49859
\(857\) −1.10540e7 −0.514123 −0.257061 0.966395i \(-0.582754\pi\)
−0.257061 + 0.966395i \(0.582754\pi\)
\(858\) 1.45278e7 0.673721
\(859\) 1.49699e6 0.0692206 0.0346103 0.999401i \(-0.488981\pi\)
0.0346103 + 0.999401i \(0.488981\pi\)
\(860\) 1.10540e7 0.509652
\(861\) 3.65880e6 0.168202
\(862\) 2.03711e7 0.933783
\(863\) 7.26672e6 0.332132 0.166066 0.986115i \(-0.446893\pi\)
0.166066 + 0.986115i \(0.446893\pi\)
\(864\) 4.22399e6 0.192503
\(865\) 3.19385e7 1.45136
\(866\) −1.04348e7 −0.472812
\(867\) −1.27270e7 −0.575015
\(868\) 670062. 0.0301867
\(869\) −3.74223e7 −1.68105
\(870\) 4.54646e6 0.203646
\(871\) −2.80161e7 −1.25130
\(872\) 3.06406e7 1.36460
\(873\) −1.27010e7 −0.564032
\(874\) 4349.81 0.000192616 0
\(875\) 1.55708e7 0.687529
\(876\) 7.56664e6 0.333152
\(877\) −3.36708e6 −0.147827 −0.0739137 0.997265i \(-0.523549\pi\)
−0.0739137 + 0.997265i \(0.523549\pi\)
\(878\) −1.98682e7 −0.869806
\(879\) 9.40688e6 0.410652
\(880\) 40123.6 0.00174660
\(881\) −1.40164e7 −0.608409 −0.304204 0.952607i \(-0.598391\pi\)
−0.304204 + 0.952607i \(0.598391\pi\)
\(882\) −2.10341e6 −0.0910443
\(883\) −4.00921e7 −1.73044 −0.865220 0.501392i \(-0.832821\pi\)
−0.865220 + 0.501392i \(0.832821\pi\)
\(884\) 947685. 0.0407881
\(885\) −8.09202e6 −0.347295
\(886\) −240850. −0.0103077
\(887\) 3.57779e7 1.52688 0.763442 0.645877i \(-0.223508\pi\)
0.763442 + 0.645877i \(0.223508\pi\)
\(888\) −1.35419e7 −0.576297
\(889\) −3.07238e7 −1.30383
\(890\) 7.09777e6 0.300363
\(891\) −4.79660e6 −0.202414
\(892\) −7.17850e6 −0.302080
\(893\) −5734.56 −0.000240642 0
\(894\) 1.36549e7 0.571405
\(895\) 3.50965e6 0.146456
\(896\) −1.11129e7 −0.462440
\(897\) −3.00837e6 −0.124839
\(898\) 1.61912e7 0.670020
\(899\) 848268. 0.0350053
\(900\) −683874. −0.0281430
\(901\) −2.65090e6 −0.108788
\(902\) −1.07256e7 −0.438941
\(903\) 8.16769e6 0.333334
\(904\) −1.47870e7 −0.601810
\(905\) −4.94488e7 −2.00694
\(906\) −4.26947e6 −0.172804
\(907\) 2.64880e6 0.106913 0.0534565 0.998570i \(-0.482976\pi\)
0.0534565 + 0.998570i \(0.482976\pi\)
\(908\) 3.77912e6 0.152116
\(909\) −1.20204e7 −0.482515
\(910\) 1.27409e7 0.510033
\(911\) −3.49769e7 −1.39632 −0.698161 0.715941i \(-0.745998\pi\)
−0.698161 + 0.715941i \(0.745998\pi\)
\(912\) 19.5039 7.76486e−7 0
\(913\) 6.46979e7 2.56870
\(914\) −1.51450e7 −0.599657
\(915\) −529916. −0.0209245
\(916\) −1.47627e7 −0.581335
\(917\) −1.49704e7 −0.587907
\(918\) 193051. 0.00756075
\(919\) −2.60166e7 −1.01616 −0.508079 0.861310i \(-0.669644\pi\)
−0.508079 + 0.861310i \(0.669644\pi\)
\(920\) 5.70523e6 0.222231
\(921\) −6.27785e6 −0.243872
\(922\) 7.30160e6 0.282872
\(923\) 2.14424e7 0.828457
\(924\) 1.26080e7 0.485810
\(925\) 3.54714e6 0.136309
\(926\) −4.36550e6 −0.167304
\(927\) 1.25644e7 0.480223
\(928\) −1.40557e7 −0.535777
\(929\) 4.76233e7 1.81042 0.905212 0.424960i \(-0.139712\pi\)
0.905212 + 0.424960i \(0.139712\pi\)
\(930\) 655374. 0.0248475
\(931\) −17487.9 −0.000661245 0
\(932\) −1.26023e7 −0.475237
\(933\) −2.41757e7 −0.909233
\(934\) −549657. −0.0206170
\(935\) 3.30189e6 0.123519
\(936\) 9.26238e6 0.345568
\(937\) −5.81079e6 −0.216215 −0.108108 0.994139i \(-0.534479\pi\)
−0.108108 + 0.994139i \(0.534479\pi\)
\(938\) 1.50013e7 0.556700
\(939\) 1.46000e7 0.540367
\(940\) −2.87411e6 −0.106092
\(941\) −6.47870e6 −0.238514 −0.119257 0.992863i \(-0.538051\pi\)
−0.119257 + 0.992863i \(0.538051\pi\)
\(942\) −1.11768e7 −0.410383
\(943\) 2.22104e6 0.0813349
\(944\) 13893.9 0.000507450 0
\(945\) −4.20665e6 −0.153235
\(946\) −2.39433e7 −0.869873
\(947\) −4.16257e6 −0.150830 −0.0754149 0.997152i \(-0.524028\pi\)
−0.0754149 + 0.997152i \(0.524028\pi\)
\(948\) −9.11705e6 −0.329483
\(949\) 2.68441e7 0.967573
\(950\) 3508.01 0.000126111 0
\(951\) 5.83354e6 0.209161
\(952\) −1.32796e6 −0.0474890
\(953\) −2.93375e7 −1.04638 −0.523191 0.852216i \(-0.675259\pi\)
−0.523191 + 0.852216i \(0.675259\pi\)
\(954\) −9.90037e6 −0.352193
\(955\) −8.89968e6 −0.315766
\(956\) −2.85030e7 −1.00866
\(957\) 1.59612e7 0.563359
\(958\) 9.60422e6 0.338103
\(959\) 2.10768e7 0.740047
\(960\) −1.08437e7 −0.379751
\(961\) −2.85069e7 −0.995729
\(962\) −1.83580e7 −0.639568
\(963\) −1.43796e7 −0.499669
\(964\) −2.24074e7 −0.776603
\(965\) −5.44125e7 −1.88096
\(966\) 1.61084e6 0.0555405
\(967\) 4.34493e6 0.149423 0.0747113 0.997205i \(-0.476196\pi\)
0.0747113 + 0.997205i \(0.476196\pi\)
\(968\) −6.75783e7 −2.31803
\(969\) 1605.03 5.49129e−5 0
\(970\) 3.26532e7 1.11429
\(971\) −1.84656e7 −0.628514 −0.314257 0.949338i \(-0.601755\pi\)
−0.314257 + 0.949338i \(0.601755\pi\)
\(972\) −1.16858e6 −0.0396727
\(973\) 2.29396e7 0.776790
\(974\) −3.55852e7 −1.20191
\(975\) −2.42618e6 −0.0817355
\(976\) 909.857 3.05737e−5 0
\(977\) 4.05887e7 1.36041 0.680203 0.733023i \(-0.261891\pi\)
0.680203 + 0.733023i \(0.261891\pi\)
\(978\) −1.61754e7 −0.540763
\(979\) 2.49180e7 0.830915
\(980\) −8.76474e6 −0.291524
\(981\) −1.37145e7 −0.454995
\(982\) −3.08159e7 −1.01975
\(983\) 5.29637e7 1.74822 0.874108 0.485732i \(-0.161447\pi\)
0.874108 + 0.485732i \(0.161447\pi\)
\(984\) −6.83828e6 −0.225143
\(985\) 5.47805e7 1.79902
\(986\) −642396. −0.0210431
\(987\) −2.12365e6 −0.0693888
\(988\) 29426.2 0.000959052 0
\(989\) 4.95812e6 0.161186
\(990\) 1.23316e7 0.399883
\(991\) 1.26953e7 0.410636 0.205318 0.978695i \(-0.434177\pi\)
0.205318 + 0.978695i \(0.434177\pi\)
\(992\) −2.02614e6 −0.0653719
\(993\) −1.49895e7 −0.482406
\(994\) −1.14814e7 −0.368577
\(995\) −2.94241e7 −0.942205
\(996\) 1.57621e7 0.503460
\(997\) 4.65074e7 1.48178 0.740890 0.671626i \(-0.234404\pi\)
0.740890 + 0.671626i \(0.234404\pi\)
\(998\) −2.54281e7 −0.808142
\(999\) 6.06122e6 0.192153
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 69.6.a.b.1.3 3
3.2 odd 2 207.6.a.c.1.1 3
4.3 odd 2 1104.6.a.i.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.b.1.3 3 1.1 even 1 trivial
207.6.a.c.1.1 3 3.2 odd 2
1104.6.a.i.1.1 3 4.3 odd 2