Properties

Label 7.6.a.b.1.1
Level 7
Weight 6
Character 7.1
Self dual yes
Analytic conductor 1.123
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.12268673869\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 7.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.725083 q^{2} +19.6495 q^{3} -31.4743 q^{4} -46.7492 q^{5} +14.2475 q^{6} +49.0000 q^{7} -46.0241 q^{8} +143.103 q^{9} +O(q^{10})\) \(q+0.725083 q^{2} +19.6495 q^{3} -31.4743 q^{4} -46.7492 q^{5} +14.2475 q^{6} +49.0000 q^{7} -46.0241 q^{8} +143.103 q^{9} -33.8970 q^{10} +666.090 q^{11} -618.453 q^{12} -650.640 q^{13} +35.5291 q^{14} -918.598 q^{15} +973.805 q^{16} +1186.89 q^{17} +103.762 q^{18} -1565.05 q^{19} +1471.40 q^{20} +962.826 q^{21} +482.970 q^{22} -1100.15 q^{23} -904.350 q^{24} -939.515 q^{25} -471.768 q^{26} -1962.93 q^{27} -1542.24 q^{28} +2396.72 q^{29} -666.060 q^{30} -2048.46 q^{31} +2178.86 q^{32} +13088.3 q^{33} +860.596 q^{34} -2290.71 q^{35} -4504.06 q^{36} +1077.54 q^{37} -1134.79 q^{38} -12784.7 q^{39} +2151.59 q^{40} +1098.21 q^{41} +698.128 q^{42} +16564.3 q^{43} -20964.7 q^{44} -6689.95 q^{45} -797.702 q^{46} -8298.39 q^{47} +19134.8 q^{48} +2401.00 q^{49} -681.226 q^{50} +23321.9 q^{51} +20478.4 q^{52} +5519.18 q^{53} -1423.28 q^{54} -31139.1 q^{55} -2255.18 q^{56} -30752.5 q^{57} +1737.82 q^{58} -14230.4 q^{59} +28912.2 q^{60} -14234.7 q^{61} -1485.30 q^{62} +7012.05 q^{63} -29581.9 q^{64} +30416.9 q^{65} +9490.12 q^{66} +19730.4 q^{67} -37356.6 q^{68} -21617.5 q^{69} -1660.95 q^{70} +64562.7 q^{71} -6586.18 q^{72} +28567.0 q^{73} +781.309 q^{74} -18461.0 q^{75} +49258.8 q^{76} +32638.4 q^{77} -9270.00 q^{78} -30633.4 q^{79} -45524.6 q^{80} -73344.6 q^{81} +796.293 q^{82} -675.946 q^{83} -30304.2 q^{84} -55486.3 q^{85} +12010.5 q^{86} +47094.4 q^{87} -30656.2 q^{88} +125971. q^{89} -4850.76 q^{90} -31881.3 q^{91} +34626.5 q^{92} -40251.1 q^{93} -6017.02 q^{94} +73164.9 q^{95} +42813.5 q^{96} -22906.8 q^{97} +1740.92 q^{98} +95319.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 9q^{2} - 6q^{3} + 5q^{4} - 18q^{5} - 198q^{6} + 98q^{7} - 9q^{8} + 558q^{9} + O(q^{10}) \) \( 2q + 9q^{2} - 6q^{3} + 5q^{4} - 18q^{5} - 198q^{6} + 98q^{7} - 9q^{8} + 558q^{9} + 204q^{10} + 396q^{11} - 1554q^{12} - 350q^{13} + 441q^{14} - 1656q^{15} + 113q^{16} + 1800q^{17} + 3537q^{18} - 3266q^{19} + 2520q^{20} - 294q^{21} - 1752q^{22} + 2088q^{23} - 1854q^{24} - 3238q^{25} + 2016q^{26} - 6372q^{27} + 245q^{28} + 6696q^{29} - 6768q^{30} - 20q^{31} - 6129q^{32} + 20016q^{33} + 5934q^{34} - 882q^{35} + 10629q^{36} + 6232q^{37} - 15210q^{38} - 20496q^{39} + 3216q^{40} - 6048q^{41} - 9702q^{42} - 3020q^{43} - 30816q^{44} + 5238q^{45} + 25584q^{46} + 11700q^{47} + 41214q^{48} + 4802q^{49} - 19701q^{50} + 7596q^{51} + 31444q^{52} + 9468q^{53} - 37908q^{54} - 38904q^{55} - 441q^{56} + 12876q^{57} + 37314q^{58} - 43938q^{59} + 2016q^{60} - 64754q^{61} + 15300q^{62} + 27342q^{63} - 70783q^{64} + 39060q^{65} + 66816q^{66} + 24784q^{67} - 14994q^{68} - 103392q^{69} + 9996q^{70} + 97416q^{71} + 8775q^{72} + 17452q^{73} + 43434q^{74} + 40494q^{75} - 12782q^{76} + 19404q^{77} - 73080q^{78} + 51256q^{79} - 70272q^{80} - 61074q^{81} - 58338q^{82} + 117558q^{83} - 76146q^{84} - 37860q^{85} - 150048q^{86} - 63180q^{87} - 40656q^{88} + 84276q^{89} + 93852q^{90} - 17150q^{91} + 150912q^{92} - 92280q^{93} + 159468q^{94} + 24264q^{95} + 255906q^{96} + 20776q^{97} + 21609q^{98} - 16740q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.725083 0.128178 0.0640889 0.997944i \(-0.479586\pi\)
0.0640889 + 0.997944i \(0.479586\pi\)
\(3\) 19.6495 1.26052 0.630258 0.776386i \(-0.282949\pi\)
0.630258 + 0.776386i \(0.282949\pi\)
\(4\) −31.4743 −0.983570
\(5\) −46.7492 −0.836275 −0.418137 0.908384i \(-0.637317\pi\)
−0.418137 + 0.908384i \(0.637317\pi\)
\(6\) 14.2475 0.161570
\(7\) 49.0000 0.377964
\(8\) −46.0241 −0.254250
\(9\) 143.103 0.588901
\(10\) −33.8970 −0.107192
\(11\) 666.090 1.65978 0.829891 0.557926i \(-0.188403\pi\)
0.829891 + 0.557926i \(0.188403\pi\)
\(12\) −618.453 −1.23981
\(13\) −650.640 −1.06778 −0.533890 0.845554i \(-0.679271\pi\)
−0.533890 + 0.845554i \(0.679271\pi\)
\(14\) 35.5291 0.0484466
\(15\) −918.598 −1.05414
\(16\) 973.805 0.950981
\(17\) 1186.89 0.996069 0.498035 0.867157i \(-0.334055\pi\)
0.498035 + 0.867157i \(0.334055\pi\)
\(18\) 103.762 0.0754840
\(19\) −1565.05 −0.994591 −0.497296 0.867581i \(-0.665674\pi\)
−0.497296 + 0.867581i \(0.665674\pi\)
\(20\) 1471.40 0.822535
\(21\) 962.826 0.476430
\(22\) 482.970 0.212747
\(23\) −1100.15 −0.433644 −0.216822 0.976211i \(-0.569569\pi\)
−0.216822 + 0.976211i \(0.569569\pi\)
\(24\) −904.350 −0.320486
\(25\) −939.515 −0.300645
\(26\) −471.768 −0.136866
\(27\) −1962.93 −0.518197
\(28\) −1542.24 −0.371755
\(29\) 2396.72 0.529203 0.264602 0.964358i \(-0.414760\pi\)
0.264602 + 0.964358i \(0.414760\pi\)
\(30\) −666.060 −0.135117
\(31\) −2048.46 −0.382844 −0.191422 0.981508i \(-0.561310\pi\)
−0.191422 + 0.981508i \(0.561310\pi\)
\(32\) 2178.86 0.376144
\(33\) 13088.3 2.09218
\(34\) 860.596 0.127674
\(35\) −2290.71 −0.316082
\(36\) −4504.06 −0.579226
\(37\) 1077.54 0.129399 0.0646995 0.997905i \(-0.479391\pi\)
0.0646995 + 0.997905i \(0.479391\pi\)
\(38\) −1134.79 −0.127484
\(39\) −12784.7 −1.34595
\(40\) 2151.59 0.212622
\(41\) 1098.21 0.102029 0.0510147 0.998698i \(-0.483754\pi\)
0.0510147 + 0.998698i \(0.483754\pi\)
\(42\) 698.128 0.0610678
\(43\) 16564.3 1.36616 0.683081 0.730343i \(-0.260640\pi\)
0.683081 + 0.730343i \(0.260640\pi\)
\(44\) −20964.7 −1.63251
\(45\) −6689.95 −0.492483
\(46\) −797.702 −0.0555835
\(47\) −8298.39 −0.547960 −0.273980 0.961735i \(-0.588340\pi\)
−0.273980 + 0.961735i \(0.588340\pi\)
\(48\) 19134.8 1.19873
\(49\) 2401.00 0.142857
\(50\) −681.226 −0.0385360
\(51\) 23321.9 1.25556
\(52\) 20478.4 1.05024
\(53\) 5519.18 0.269889 0.134944 0.990853i \(-0.456914\pi\)
0.134944 + 0.990853i \(0.456914\pi\)
\(54\) −1423.28 −0.0664213
\(55\) −31139.1 −1.38803
\(56\) −2255.18 −0.0960973
\(57\) −30752.5 −1.25370
\(58\) 1737.82 0.0678321
\(59\) −14230.4 −0.532216 −0.266108 0.963943i \(-0.585738\pi\)
−0.266108 + 0.963943i \(0.585738\pi\)
\(60\) 28912.2 1.03682
\(61\) −14234.7 −0.489807 −0.244904 0.969547i \(-0.578756\pi\)
−0.244904 + 0.969547i \(0.578756\pi\)
\(62\) −1485.30 −0.0490721
\(63\) 7012.05 0.222584
\(64\) −29581.9 −0.902768
\(65\) 30416.9 0.892958
\(66\) 9490.12 0.268171
\(67\) 19730.4 0.536970 0.268485 0.963284i \(-0.413477\pi\)
0.268485 + 0.963284i \(0.413477\pi\)
\(68\) −37356.6 −0.979704
\(69\) −21617.5 −0.546615
\(70\) −1660.95 −0.0405147
\(71\) 64562.7 1.51997 0.759986 0.649940i \(-0.225206\pi\)
0.759986 + 0.649940i \(0.225206\pi\)
\(72\) −6586.18 −0.149728
\(73\) 28567.0 0.627418 0.313709 0.949519i \(-0.398428\pi\)
0.313709 + 0.949519i \(0.398428\pi\)
\(74\) 781.309 0.0165861
\(75\) −18461.0 −0.378968
\(76\) 49258.8 0.978251
\(77\) 32638.4 0.627339
\(78\) −9270.00 −0.172521
\(79\) −30633.4 −0.552239 −0.276119 0.961123i \(-0.589049\pi\)
−0.276119 + 0.961123i \(0.589049\pi\)
\(80\) −45524.6 −0.795282
\(81\) −73344.6 −1.24210
\(82\) 796.293 0.0130779
\(83\) −675.946 −0.0107700 −0.00538501 0.999986i \(-0.501714\pi\)
−0.00538501 + 0.999986i \(0.501714\pi\)
\(84\) −30304.2 −0.468603
\(85\) −55486.3 −0.832987
\(86\) 12010.5 0.175111
\(87\) 47094.4 0.667069
\(88\) −30656.2 −0.421999
\(89\) 125971. 1.68576 0.842882 0.538098i \(-0.180857\pi\)
0.842882 + 0.538098i \(0.180857\pi\)
\(90\) −4850.76 −0.0631254
\(91\) −31881.3 −0.403583
\(92\) 34626.5 0.426520
\(93\) −40251.1 −0.482582
\(94\) −6017.02 −0.0702363
\(95\) 73164.9 0.831751
\(96\) 42813.5 0.474136
\(97\) −22906.8 −0.247192 −0.123596 0.992333i \(-0.539443\pi\)
−0.123596 + 0.992333i \(0.539443\pi\)
\(98\) 1740.92 0.0183111
\(99\) 95319.4 0.977447
\(100\) 29570.5 0.295705
\(101\) −181474. −1.77015 −0.885077 0.465444i \(-0.845895\pi\)
−0.885077 + 0.465444i \(0.845895\pi\)
\(102\) 16910.3 0.160935
\(103\) 64772.0 0.601581 0.300791 0.953690i \(-0.402749\pi\)
0.300791 + 0.953690i \(0.402749\pi\)
\(104\) 29945.1 0.271483
\(105\) −45011.3 −0.398427
\(106\) 4001.86 0.0345938
\(107\) −148170. −1.25112 −0.625562 0.780175i \(-0.715130\pi\)
−0.625562 + 0.780175i \(0.715130\pi\)
\(108\) 61781.7 0.509683
\(109\) −111294. −0.897237 −0.448618 0.893723i \(-0.648084\pi\)
−0.448618 + 0.893723i \(0.648084\pi\)
\(110\) −22578.5 −0.177915
\(111\) 21173.2 0.163110
\(112\) 47716.4 0.359437
\(113\) −43175.5 −0.318084 −0.159042 0.987272i \(-0.550840\pi\)
−0.159042 + 0.987272i \(0.550840\pi\)
\(114\) −22298.1 −0.160696
\(115\) 51431.2 0.362646
\(116\) −75435.0 −0.520509
\(117\) −93108.5 −0.628817
\(118\) −10318.2 −0.0682182
\(119\) 58157.8 0.376479
\(120\) 42277.6 0.268014
\(121\) 282625. 1.75488
\(122\) −10321.4 −0.0627824
\(123\) 21579.3 0.128610
\(124\) 64473.6 0.376554
\(125\) 190013. 1.08770
\(126\) 5084.31 0.0285303
\(127\) −131449. −0.723182 −0.361591 0.932337i \(-0.617766\pi\)
−0.361591 + 0.932337i \(0.617766\pi\)
\(128\) −91172.8 −0.491859
\(129\) 325480. 1.72207
\(130\) 22054.7 0.114457
\(131\) −349458. −1.77916 −0.889582 0.456775i \(-0.849005\pi\)
−0.889582 + 0.456775i \(0.849005\pi\)
\(132\) −411946. −2.05781
\(133\) −76687.5 −0.375920
\(134\) 14306.2 0.0688276
\(135\) 91765.2 0.433355
\(136\) −54625.7 −0.253250
\(137\) 386434. 1.75903 0.879516 0.475869i \(-0.157866\pi\)
0.879516 + 0.475869i \(0.157866\pi\)
\(138\) −15674.4 −0.0700639
\(139\) 17289.3 0.0758997 0.0379498 0.999280i \(-0.487917\pi\)
0.0379498 + 0.999280i \(0.487917\pi\)
\(140\) 72098.4 0.310889
\(141\) −163059. −0.690713
\(142\) 46813.3 0.194827
\(143\) −433384. −1.77228
\(144\) 139354. 0.560034
\(145\) −112045. −0.442559
\(146\) 20713.4 0.0804210
\(147\) 47178.5 0.180074
\(148\) −33914.9 −0.127273
\(149\) −112171. −0.413917 −0.206959 0.978350i \(-0.566357\pi\)
−0.206959 + 0.978350i \(0.566357\pi\)
\(150\) −13385.8 −0.0485752
\(151\) 30495.4 0.108841 0.0544205 0.998518i \(-0.482669\pi\)
0.0544205 + 0.998518i \(0.482669\pi\)
\(152\) 72030.1 0.252874
\(153\) 169848. 0.586586
\(154\) 23665.5 0.0804108
\(155\) 95763.6 0.320163
\(156\) 402390. 1.32384
\(157\) 523509. 1.69502 0.847510 0.530780i \(-0.178101\pi\)
0.847510 + 0.530780i \(0.178101\pi\)
\(158\) −22211.7 −0.0707847
\(159\) 108449. 0.340199
\(160\) −101860. −0.314560
\(161\) −53907.5 −0.163902
\(162\) −53180.9 −0.159209
\(163\) −439646. −1.29609 −0.648043 0.761604i \(-0.724412\pi\)
−0.648043 + 0.761604i \(0.724412\pi\)
\(164\) −34565.3 −0.100353
\(165\) −611869. −1.74964
\(166\) −490.117 −0.00138048
\(167\) 279353. 0.775107 0.387554 0.921847i \(-0.373320\pi\)
0.387554 + 0.921847i \(0.373320\pi\)
\(168\) −44313.2 −0.121132
\(169\) 52038.8 0.140156
\(170\) −40232.2 −0.106770
\(171\) −223964. −0.585716
\(172\) −521349. −1.34372
\(173\) 99699.4 0.253266 0.126633 0.991950i \(-0.459583\pi\)
0.126633 + 0.991950i \(0.459583\pi\)
\(174\) 34147.3 0.0855034
\(175\) −46036.2 −0.113633
\(176\) 648641. 1.57842
\(177\) −279621. −0.670867
\(178\) 91339.7 0.216077
\(179\) 329980. 0.769760 0.384880 0.922967i \(-0.374243\pi\)
0.384880 + 0.922967i \(0.374243\pi\)
\(180\) 210561. 0.484392
\(181\) −505810. −1.14760 −0.573800 0.818995i \(-0.694531\pi\)
−0.573800 + 0.818995i \(0.694531\pi\)
\(182\) −23116.6 −0.0517304
\(183\) −279706. −0.617410
\(184\) 50633.5 0.110254
\(185\) −50374.3 −0.108213
\(186\) −29185.4 −0.0618562
\(187\) 790578. 1.65326
\(188\) 261186. 0.538958
\(189\) −96183.4 −0.195860
\(190\) 53050.6 0.106612
\(191\) −63835.6 −0.126613 −0.0633067 0.997994i \(-0.520165\pi\)
−0.0633067 + 0.997994i \(0.520165\pi\)
\(192\) −581270. −1.13795
\(193\) 469355. 0.907001 0.453501 0.891256i \(-0.350175\pi\)
0.453501 + 0.891256i \(0.350175\pi\)
\(194\) −16609.3 −0.0316845
\(195\) 597676. 1.12559
\(196\) −75569.7 −0.140510
\(197\) 268021. 0.492043 0.246021 0.969264i \(-0.420877\pi\)
0.246021 + 0.969264i \(0.420877\pi\)
\(198\) 69114.5 0.125287
\(199\) −605167. −1.08328 −0.541642 0.840609i \(-0.682197\pi\)
−0.541642 + 0.840609i \(0.682197\pi\)
\(200\) 43240.3 0.0764388
\(201\) 387693. 0.676859
\(202\) −131584. −0.226894
\(203\) 117439. 0.200020
\(204\) −734039. −1.23493
\(205\) −51340.4 −0.0853246
\(206\) 46965.1 0.0771094
\(207\) −157435. −0.255374
\(208\) −633596. −1.01544
\(209\) −1.04246e6 −1.65080
\(210\) −32636.9 −0.0510694
\(211\) 335389. 0.518612 0.259306 0.965795i \(-0.416506\pi\)
0.259306 + 0.965795i \(0.416506\pi\)
\(212\) −173712. −0.265455
\(213\) 1.26862e6 1.91595
\(214\) −107435. −0.160366
\(215\) −774367. −1.14249
\(216\) 90341.9 0.131751
\(217\) −100374. −0.144702
\(218\) −80697.7 −0.115006
\(219\) 561327. 0.790871
\(220\) 980081. 1.36523
\(221\) −772240. −1.06358
\(222\) 15352.3 0.0209070
\(223\) 1.02526e6 1.38061 0.690305 0.723518i \(-0.257476\pi\)
0.690305 + 0.723518i \(0.257476\pi\)
\(224\) 106764. 0.142169
\(225\) −134447. −0.177050
\(226\) −31305.8 −0.0407713
\(227\) −504226. −0.649473 −0.324736 0.945805i \(-0.605276\pi\)
−0.324736 + 0.945805i \(0.605276\pi\)
\(228\) 967912. 1.23310
\(229\) −1.11939e6 −1.41057 −0.705283 0.708925i \(-0.749180\pi\)
−0.705283 + 0.708925i \(0.749180\pi\)
\(230\) 37291.9 0.0464831
\(231\) 641328. 0.790770
\(232\) −110307. −0.134550
\(233\) 770703. 0.930031 0.465015 0.885303i \(-0.346049\pi\)
0.465015 + 0.885303i \(0.346049\pi\)
\(234\) −67511.3 −0.0806004
\(235\) 387943. 0.458245
\(236\) 447892. 0.523472
\(237\) −601930. −0.696106
\(238\) 42169.2 0.0482562
\(239\) −171646. −0.194374 −0.0971871 0.995266i \(-0.530985\pi\)
−0.0971871 + 0.995266i \(0.530985\pi\)
\(240\) −894535. −1.00247
\(241\) −383779. −0.425637 −0.212818 0.977092i \(-0.568264\pi\)
−0.212818 + 0.977092i \(0.568264\pi\)
\(242\) 204926. 0.224936
\(243\) −964193. −1.04749
\(244\) 448028. 0.481760
\(245\) −112245. −0.119468
\(246\) 15646.8 0.0164849
\(247\) 1.01828e6 1.06201
\(248\) 94278.3 0.0973380
\(249\) −13282.0 −0.0135758
\(250\) 137775. 0.139418
\(251\) −1.57046e6 −1.57342 −0.786708 0.617325i \(-0.788216\pi\)
−0.786708 + 0.617325i \(0.788216\pi\)
\(252\) −220699. −0.218927
\(253\) −732801. −0.719755
\(254\) −95311.4 −0.0926959
\(255\) −1.09028e6 −1.04999
\(256\) 880513. 0.839723
\(257\) 790656. 0.746715 0.373357 0.927688i \(-0.378207\pi\)
0.373357 + 0.927688i \(0.378207\pi\)
\(258\) 236000. 0.220731
\(259\) 52799.7 0.0489082
\(260\) −957348. −0.878287
\(261\) 342978. 0.311648
\(262\) −253386. −0.228049
\(263\) 464416. 0.414017 0.207008 0.978339i \(-0.433627\pi\)
0.207008 + 0.978339i \(0.433627\pi\)
\(264\) −602379. −0.531936
\(265\) −258017. −0.225701
\(266\) −55604.8 −0.0481846
\(267\) 2.47527e6 2.12493
\(268\) −621001. −0.528147
\(269\) 1.99959e6 1.68484 0.842422 0.538818i \(-0.181129\pi\)
0.842422 + 0.538818i \(0.181129\pi\)
\(270\) 66537.4 0.0555464
\(271\) 1.61296e6 1.33414 0.667070 0.744995i \(-0.267548\pi\)
0.667070 + 0.744995i \(0.267548\pi\)
\(272\) 1.15580e6 0.947243
\(273\) −626452. −0.508723
\(274\) 280197. 0.225469
\(275\) −625801. −0.499005
\(276\) 680393. 0.537635
\(277\) −2.08119e6 −1.62972 −0.814860 0.579658i \(-0.803186\pi\)
−0.814860 + 0.579658i \(0.803186\pi\)
\(278\) 12536.2 0.00972865
\(279\) −293140. −0.225457
\(280\) 105428. 0.0803637
\(281\) −982035. −0.741927 −0.370964 0.928647i \(-0.620973\pi\)
−0.370964 + 0.928647i \(0.620973\pi\)
\(282\) −118231. −0.0885340
\(283\) −1.39622e6 −1.03630 −0.518152 0.855289i \(-0.673380\pi\)
−0.518152 + 0.855289i \(0.673380\pi\)
\(284\) −2.03206e6 −1.49500
\(285\) 1.43765e6 1.04844
\(286\) −314240. −0.227167
\(287\) 53812.3 0.0385635
\(288\) 311801. 0.221512
\(289\) −11140.3 −0.00784609
\(290\) −81241.7 −0.0567262
\(291\) −450107. −0.311590
\(292\) −899124. −0.617110
\(293\) 2.56205e6 1.74348 0.871742 0.489965i \(-0.162990\pi\)
0.871742 + 0.489965i \(0.162990\pi\)
\(294\) 34208.3 0.0230814
\(295\) 665260. 0.445078
\(296\) −49593.0 −0.0328996
\(297\) −1.30749e6 −0.860094
\(298\) −81333.0 −0.0530550
\(299\) 715803. 0.463037
\(300\) 581046. 0.372741
\(301\) 811651. 0.516361
\(302\) 22111.7 0.0139510
\(303\) −3.56588e6 −2.23131
\(304\) −1.52405e6 −0.945838
\(305\) 665463. 0.409613
\(306\) 123154. 0.0751873
\(307\) −884855. −0.535829 −0.267915 0.963443i \(-0.586334\pi\)
−0.267915 + 0.963443i \(0.586334\pi\)
\(308\) −1.02727e6 −0.617032
\(309\) 1.27274e6 0.758303
\(310\) 69436.5 0.0410378
\(311\) 1.80120e6 1.05599 0.527997 0.849246i \(-0.322943\pi\)
0.527997 + 0.849246i \(0.322943\pi\)
\(312\) 588406. 0.342208
\(313\) 950366. 0.548315 0.274158 0.961685i \(-0.411601\pi\)
0.274158 + 0.961685i \(0.411601\pi\)
\(314\) 379587. 0.217264
\(315\) −327807. −0.186141
\(316\) 964162. 0.543166
\(317\) 3.04277e6 1.70068 0.850338 0.526237i \(-0.176398\pi\)
0.850338 + 0.526237i \(0.176398\pi\)
\(318\) 78634.6 0.0436060
\(319\) 1.59643e6 0.878362
\(320\) 1.38293e6 0.754962
\(321\) −2.91146e6 −1.57706
\(322\) −39087.4 −0.0210086
\(323\) −1.85755e6 −0.990682
\(324\) 2.30847e6 1.22169
\(325\) 611286. 0.321023
\(326\) −318780. −0.166129
\(327\) −2.18688e6 −1.13098
\(328\) −50544.1 −0.0259409
\(329\) −406621. −0.207110
\(330\) −443655. −0.224265
\(331\) −2.19616e6 −1.10178 −0.550889 0.834579i \(-0.685711\pi\)
−0.550889 + 0.834579i \(0.685711\pi\)
\(332\) 21274.9 0.0105931
\(333\) 154200. 0.0762032
\(334\) 202554. 0.0993515
\(335\) −922382. −0.449054
\(336\) 937604. 0.453076
\(337\) −2.41491e6 −1.15832 −0.579158 0.815216i \(-0.696618\pi\)
−0.579158 + 0.815216i \(0.696618\pi\)
\(338\) 37732.5 0.0179648
\(339\) −848377. −0.400950
\(340\) 1.74639e6 0.819302
\(341\) −1.36446e6 −0.635438
\(342\) −162392. −0.0750757
\(343\) 117649. 0.0539949
\(344\) −762357. −0.347346
\(345\) 1.01060e6 0.457121
\(346\) 72290.3 0.0324631
\(347\) −1.08833e6 −0.485219 −0.242609 0.970124i \(-0.578003\pi\)
−0.242609 + 0.970124i \(0.578003\pi\)
\(348\) −1.48226e6 −0.656110
\(349\) 2.79267e6 1.22731 0.613657 0.789573i \(-0.289698\pi\)
0.613657 + 0.789573i \(0.289698\pi\)
\(350\) −33380.1 −0.0145652
\(351\) 1.27716e6 0.553321
\(352\) 1.45132e6 0.624317
\(353\) 2.53134e6 1.08122 0.540610 0.841273i \(-0.318193\pi\)
0.540610 + 0.841273i \(0.318193\pi\)
\(354\) −202748. −0.0859902
\(355\) −3.01825e6 −1.27111
\(356\) −3.96485e6 −1.65807
\(357\) 1.14277e6 0.474558
\(358\) 239263. 0.0986661
\(359\) 1.09028e6 0.446480 0.223240 0.974763i \(-0.428337\pi\)
0.223240 + 0.974763i \(0.428337\pi\)
\(360\) 307899. 0.125214
\(361\) −26712.8 −0.0107883
\(362\) −366754. −0.147097
\(363\) 5.55343e6 2.21205
\(364\) 1.00344e6 0.396953
\(365\) −1.33548e6 −0.524694
\(366\) −202810. −0.0791382
\(367\) 188070. 0.0728879 0.0364439 0.999336i \(-0.488397\pi\)
0.0364439 + 0.999336i \(0.488397\pi\)
\(368\) −1.07133e6 −0.412387
\(369\) 157157. 0.0600853
\(370\) −36525.6 −0.0138705
\(371\) 270440. 0.102008
\(372\) 1.26687e6 0.474653
\(373\) −1.79371e6 −0.667545 −0.333772 0.942654i \(-0.608322\pi\)
−0.333772 + 0.942654i \(0.608322\pi\)
\(374\) 573234. 0.211911
\(375\) 3.73366e6 1.37106
\(376\) 381926. 0.139319
\(377\) −1.55940e6 −0.565073
\(378\) −69740.9 −0.0251049
\(379\) 3.58806e6 1.28310 0.641551 0.767080i \(-0.278291\pi\)
0.641551 + 0.767080i \(0.278291\pi\)
\(380\) −2.30281e6 −0.818086
\(381\) −2.58291e6 −0.911583
\(382\) −46286.1 −0.0162290
\(383\) −3.42457e6 −1.19291 −0.596457 0.802645i \(-0.703425\pi\)
−0.596457 + 0.802645i \(0.703425\pi\)
\(384\) −1.79150e6 −0.619996
\(385\) −1.52582e6 −0.524627
\(386\) 340321. 0.116257
\(387\) 2.37040e6 0.804534
\(388\) 720974. 0.243131
\(389\) 8625.27 0.00289001 0.00144500 0.999999i \(-0.499540\pi\)
0.00144500 + 0.999999i \(0.499540\pi\)
\(390\) 433365. 0.144275
\(391\) −1.30576e6 −0.431940
\(392\) −110504. −0.0363214
\(393\) −6.86667e6 −2.24267
\(394\) 194337. 0.0630690
\(395\) 1.43208e6 0.461823
\(396\) −3.00011e6 −0.961388
\(397\) 1.25709e6 0.400306 0.200153 0.979765i \(-0.435856\pi\)
0.200153 + 0.979765i \(0.435856\pi\)
\(398\) −438796. −0.138853
\(399\) −1.50687e6 −0.473853
\(400\) −914904. −0.285908
\(401\) 1.42670e6 0.443070 0.221535 0.975152i \(-0.428893\pi\)
0.221535 + 0.975152i \(0.428893\pi\)
\(402\) 281110. 0.0867583
\(403\) 1.33281e6 0.408794
\(404\) 5.71176e6 1.74107
\(405\) 3.42880e6 1.03873
\(406\) 85153.2 0.0256381
\(407\) 717741. 0.214774
\(408\) −1.07337e6 −0.319226
\(409\) −3.06529e6 −0.906073 −0.453036 0.891492i \(-0.649659\pi\)
−0.453036 + 0.891492i \(0.649659\pi\)
\(410\) −37226.0 −0.0109367
\(411\) 7.59324e6 2.21729
\(412\) −2.03865e6 −0.591698
\(413\) −697291. −0.201159
\(414\) −114154. −0.0327332
\(415\) 31599.9 0.00900670
\(416\) −1.41765e6 −0.401640
\(417\) 339726. 0.0956728
\(418\) −755873. −0.211596
\(419\) −248240. −0.0690776 −0.0345388 0.999403i \(-0.510996\pi\)
−0.0345388 + 0.999403i \(0.510996\pi\)
\(420\) 1.41670e6 0.391881
\(421\) 5.96280e6 1.63963 0.819814 0.572630i \(-0.194077\pi\)
0.819814 + 0.572630i \(0.194077\pi\)
\(422\) 243185. 0.0664746
\(423\) −1.18752e6 −0.322695
\(424\) −254015. −0.0686192
\(425\) −1.11510e6 −0.299463
\(426\) 919857. 0.245582
\(427\) −697503. −0.185130
\(428\) 4.66353e6 1.23057
\(429\) −8.51579e6 −2.23399
\(430\) −561481. −0.146441
\(431\) −4.93538e6 −1.27976 −0.639879 0.768476i \(-0.721015\pi\)
−0.639879 + 0.768476i \(0.721015\pi\)
\(432\) −1.91151e6 −0.492795
\(433\) 4.15513e6 1.06504 0.532519 0.846418i \(-0.321245\pi\)
0.532519 + 0.846418i \(0.321245\pi\)
\(434\) −72779.7 −0.0185475
\(435\) −2.20162e6 −0.557853
\(436\) 3.50291e6 0.882496
\(437\) 1.72180e6 0.431299
\(438\) 407008. 0.101372
\(439\) −227955. −0.0564531 −0.0282265 0.999602i \(-0.508986\pi\)
−0.0282265 + 0.999602i \(0.508986\pi\)
\(440\) 1.43315e6 0.352907
\(441\) 343590. 0.0841287
\(442\) −559938. −0.136328
\(443\) −1.98462e6 −0.480472 −0.240236 0.970715i \(-0.577225\pi\)
−0.240236 + 0.970715i \(0.577225\pi\)
\(444\) −666411. −0.160430
\(445\) −5.88906e6 −1.40976
\(446\) 743397. 0.176963
\(447\) −2.20410e6 −0.521749
\(448\) −1.44951e6 −0.341214
\(449\) 2.61077e6 0.611157 0.305579 0.952167i \(-0.401150\pi\)
0.305579 + 0.952167i \(0.401150\pi\)
\(450\) −97485.5 −0.0226939
\(451\) 731506. 0.169347
\(452\) 1.35892e6 0.312858
\(453\) 599220. 0.137196
\(454\) −365606. −0.0832480
\(455\) 1.49043e6 0.337506
\(456\) 1.41536e6 0.318752
\(457\) −4.09917e6 −0.918132 −0.459066 0.888402i \(-0.651816\pi\)
−0.459066 + 0.888402i \(0.651816\pi\)
\(458\) −811652. −0.180803
\(459\) −2.32979e6 −0.516160
\(460\) −1.61876e6 −0.356687
\(461\) −2.62378e6 −0.575009 −0.287505 0.957779i \(-0.592826\pi\)
−0.287505 + 0.957779i \(0.592826\pi\)
\(462\) 465016. 0.101359
\(463\) −4.28563e6 −0.929100 −0.464550 0.885547i \(-0.653784\pi\)
−0.464550 + 0.885547i \(0.653784\pi\)
\(464\) 2.33394e6 0.503262
\(465\) 1.88171e6 0.403571
\(466\) 558824. 0.119209
\(467\) 990118. 0.210085 0.105042 0.994468i \(-0.466502\pi\)
0.105042 + 0.994468i \(0.466502\pi\)
\(468\) 2.93052e6 0.618486
\(469\) 966792. 0.202955
\(470\) 281291. 0.0587369
\(471\) 1.02867e7 2.13660
\(472\) 654942. 0.135316
\(473\) 1.10333e7 2.26753
\(474\) −436449. −0.0892253
\(475\) 1.47039e6 0.299019
\(476\) −1.83047e6 −0.370293
\(477\) 789812. 0.158938
\(478\) −124457. −0.0249144
\(479\) 2.57976e6 0.513737 0.256868 0.966446i \(-0.417309\pi\)
0.256868 + 0.966446i \(0.417309\pi\)
\(480\) −2.00150e6 −0.396508
\(481\) −701093. −0.138170
\(482\) −278272. −0.0545571
\(483\) −1.05926e6 −0.206601
\(484\) −8.89540e6 −1.72604
\(485\) 1.07087e6 0.206720
\(486\) −699120. −0.134264
\(487\) 3.21474e6 0.614219 0.307109 0.951674i \(-0.400638\pi\)
0.307109 + 0.951674i \(0.400638\pi\)
\(488\) 655141. 0.124533
\(489\) −8.63882e6 −1.63374
\(490\) −81386.7 −0.0153131
\(491\) −7.86108e6 −1.47156 −0.735781 0.677220i \(-0.763185\pi\)
−0.735781 + 0.677220i \(0.763185\pi\)
\(492\) −679192. −0.126497
\(493\) 2.84465e6 0.527123
\(494\) 738341. 0.136125
\(495\) −4.45610e6 −0.817415
\(496\) −1.99480e6 −0.364078
\(497\) 3.16357e6 0.574495
\(498\) −9630.55 −0.00174011
\(499\) −1.35382e6 −0.243395 −0.121697 0.992567i \(-0.538834\pi\)
−0.121697 + 0.992567i \(0.538834\pi\)
\(500\) −5.98051e6 −1.06983
\(501\) 5.48914e6 0.977035
\(502\) −1.13872e6 −0.201677
\(503\) 3.85775e6 0.679851 0.339926 0.940452i \(-0.389598\pi\)
0.339926 + 0.940452i \(0.389598\pi\)
\(504\) −322723. −0.0565918
\(505\) 8.48376e6 1.48034
\(506\) −531341. −0.0922565
\(507\) 1.02254e6 0.176669
\(508\) 4.13726e6 0.711301
\(509\) −1.06060e7 −1.81451 −0.907253 0.420585i \(-0.861825\pi\)
−0.907253 + 0.420585i \(0.861825\pi\)
\(510\) −790542. −0.134586
\(511\) 1.39978e6 0.237142
\(512\) 3.55598e6 0.599493
\(513\) 3.07208e6 0.515394
\(514\) 573291. 0.0957122
\(515\) −3.02804e6 −0.503087
\(516\) −1.02443e7 −1.69378
\(517\) −5.52747e6 −0.909495
\(518\) 38284.1 0.00626895
\(519\) 1.95904e6 0.319246
\(520\) −1.39991e6 −0.227034
\(521\) −9.17989e6 −1.48164 −0.740821 0.671703i \(-0.765563\pi\)
−0.740821 + 0.671703i \(0.765563\pi\)
\(522\) 248687. 0.0399464
\(523\) 9.05585e6 1.44769 0.723844 0.689964i \(-0.242373\pi\)
0.723844 + 0.689964i \(0.242373\pi\)
\(524\) 1.09989e7 1.74993
\(525\) −904589. −0.143236
\(526\) 336740. 0.0530677
\(527\) −2.43130e6 −0.381339
\(528\) 1.27455e7 1.98963
\(529\) −5.22601e6 −0.811953
\(530\) −187084. −0.0289299
\(531\) −2.03642e6 −0.313422
\(532\) 2.41368e6 0.369744
\(533\) −714539. −0.108945
\(534\) 1.79478e6 0.272369
\(535\) 6.92681e6 1.04628
\(536\) −908075. −0.136524
\(537\) 6.48394e6 0.970295
\(538\) 1.44987e6 0.215960
\(539\) 1.59928e6 0.237112
\(540\) −2.88824e6 −0.426235
\(541\) 1.21783e7 1.78894 0.894468 0.447132i \(-0.147555\pi\)
0.894468 + 0.447132i \(0.147555\pi\)
\(542\) 1.16953e6 0.171007
\(543\) −9.93891e6 −1.44657
\(544\) 2.58608e6 0.374666
\(545\) 5.20292e6 0.750336
\(546\) −454230. −0.0652070
\(547\) −9.00451e6 −1.28674 −0.643372 0.765554i \(-0.722465\pi\)
−0.643372 + 0.765554i \(0.722465\pi\)
\(548\) −1.21627e7 −1.73013
\(549\) −2.03703e6 −0.288448
\(550\) −453758. −0.0639613
\(551\) −3.75099e6 −0.526341
\(552\) 994924. 0.138977
\(553\) −1.50103e6 −0.208727
\(554\) −1.50904e6 −0.208894
\(555\) −989830. −0.136404
\(556\) −544167. −0.0746527
\(557\) 1.54461e6 0.210950 0.105475 0.994422i \(-0.466364\pi\)
0.105475 + 0.994422i \(0.466364\pi\)
\(558\) −212551. −0.0288986
\(559\) −1.07774e7 −1.45876
\(560\) −2.23070e6 −0.300588
\(561\) 1.55345e7 2.08396
\(562\) −712057. −0.0950986
\(563\) −1.18748e7 −1.57890 −0.789449 0.613816i \(-0.789634\pi\)
−0.789449 + 0.613816i \(0.789634\pi\)
\(564\) 5.13217e6 0.679365
\(565\) 2.01842e6 0.266005
\(566\) −1.01237e6 −0.132831
\(567\) −3.59388e6 −0.469468
\(568\) −2.97144e6 −0.386452
\(569\) −1.54308e6 −0.199806 −0.0999031 0.994997i \(-0.531853\pi\)
−0.0999031 + 0.994997i \(0.531853\pi\)
\(570\) 1.04242e6 0.134386
\(571\) −7.01812e6 −0.900804 −0.450402 0.892826i \(-0.648719\pi\)
−0.450402 + 0.892826i \(0.648719\pi\)
\(572\) 1.36404e7 1.74317
\(573\) −1.25434e6 −0.159598
\(574\) 39018.4 0.00494298
\(575\) 1.03361e6 0.130373
\(576\) −4.23326e6 −0.531641
\(577\) −5.37543e6 −0.672162 −0.336081 0.941833i \(-0.609102\pi\)
−0.336081 + 0.941833i \(0.609102\pi\)
\(578\) −8077.66 −0.00100569
\(579\) 9.22259e6 1.14329
\(580\) 3.52652e6 0.435288
\(581\) −33121.4 −0.00407069
\(582\) −326365. −0.0399389
\(583\) 3.67627e6 0.447957
\(584\) −1.31477e6 −0.159521
\(585\) 4.35274e6 0.525864
\(586\) 1.85770e6 0.223476
\(587\) −2.06682e6 −0.247575 −0.123788 0.992309i \(-0.539504\pi\)
−0.123788 + 0.992309i \(0.539504\pi\)
\(588\) −1.48491e6 −0.177115
\(589\) 3.20594e6 0.380774
\(590\) 482369. 0.0570492
\(591\) 5.26648e6 0.620228
\(592\) 1.04932e6 0.123056
\(593\) −5.46947e6 −0.638717 −0.319358 0.947634i \(-0.603467\pi\)
−0.319358 + 0.947634i \(0.603467\pi\)
\(594\) −948035. −0.110245
\(595\) −2.71883e6 −0.314840
\(596\) 3.53049e6 0.407117
\(597\) −1.18912e7 −1.36550
\(598\) 519016. 0.0593510
\(599\) 1.09943e7 1.25199 0.625996 0.779826i \(-0.284693\pi\)
0.625996 + 0.779826i \(0.284693\pi\)
\(600\) 849651. 0.0963524
\(601\) 1.58788e7 1.79322 0.896608 0.442826i \(-0.146024\pi\)
0.896608 + 0.442826i \(0.146024\pi\)
\(602\) 588514. 0.0661859
\(603\) 2.82348e6 0.316222
\(604\) −959820. −0.107053
\(605\) −1.32125e7 −1.46756
\(606\) −2.58555e6 −0.286004
\(607\) 5.33262e6 0.587447 0.293724 0.955890i \(-0.405105\pi\)
0.293724 + 0.955890i \(0.405105\pi\)
\(608\) −3.41003e6 −0.374110
\(609\) 2.30762e6 0.252128
\(610\) 482516. 0.0525033
\(611\) 5.39926e6 0.585102
\(612\) −5.34584e6 −0.576949
\(613\) −8.91838e6 −0.958594 −0.479297 0.877653i \(-0.659108\pi\)
−0.479297 + 0.877653i \(0.659108\pi\)
\(614\) −641593. −0.0686814
\(615\) −1.00881e6 −0.107553
\(616\) −1.50215e6 −0.159501
\(617\) 9.63586e6 1.01901 0.509504 0.860468i \(-0.329829\pi\)
0.509504 + 0.860468i \(0.329829\pi\)
\(618\) 922841. 0.0971976
\(619\) 1.21747e7 1.27712 0.638560 0.769572i \(-0.279530\pi\)
0.638560 + 0.769572i \(0.279530\pi\)
\(620\) −3.01409e6 −0.314903
\(621\) 2.15952e6 0.224713
\(622\) 1.30602e6 0.135355
\(623\) 6.17260e6 0.637159
\(624\) −1.24498e7 −1.27998
\(625\) −5.94695e6 −0.608968
\(626\) 689094. 0.0702818
\(627\) −2.04839e7 −2.08087
\(628\) −1.64770e7 −1.66717
\(629\) 1.27893e6 0.128890
\(630\) −237687. −0.0238591
\(631\) −1.30854e7 −1.30832 −0.654161 0.756356i \(-0.726978\pi\)
−0.654161 + 0.756356i \(0.726978\pi\)
\(632\) 1.40987e6 0.140407
\(633\) 6.59023e6 0.653719
\(634\) 2.20626e6 0.217989
\(635\) 6.14513e6 0.604779
\(636\) −3.41336e6 −0.334610
\(637\) −1.56219e6 −0.152540
\(638\) 1.15754e6 0.112586
\(639\) 9.23911e6 0.895113
\(640\) 4.26226e6 0.411329
\(641\) −441107. −0.0424032 −0.0212016 0.999775i \(-0.506749\pi\)
−0.0212016 + 0.999775i \(0.506749\pi\)
\(642\) −2.11105e6 −0.202144
\(643\) −4.18888e6 −0.399550 −0.199775 0.979842i \(-0.564021\pi\)
−0.199775 + 0.979842i \(0.564021\pi\)
\(644\) 1.69670e6 0.161209
\(645\) −1.52159e7 −1.44012
\(646\) −1.34688e6 −0.126983
\(647\) 1.87822e7 1.76395 0.881973 0.471300i \(-0.156215\pi\)
0.881973 + 0.471300i \(0.156215\pi\)
\(648\) 3.37562e6 0.315803
\(649\) −9.47874e6 −0.883362
\(650\) 443233. 0.0411480
\(651\) −1.97231e6 −0.182399
\(652\) 1.38375e7 1.27479
\(653\) −1.51733e6 −0.139251 −0.0696254 0.997573i \(-0.522180\pi\)
−0.0696254 + 0.997573i \(0.522180\pi\)
\(654\) −1.58567e6 −0.144967
\(655\) 1.63368e7 1.48787
\(656\) 1.06944e6 0.0970281
\(657\) 4.08802e6 0.369487
\(658\) −294834. −0.0265468
\(659\) 1.84809e7 1.65772 0.828859 0.559458i \(-0.188991\pi\)
0.828859 + 0.559458i \(0.188991\pi\)
\(660\) 1.92581e7 1.72089
\(661\) −1.03952e7 −0.925403 −0.462702 0.886514i \(-0.653120\pi\)
−0.462702 + 0.886514i \(0.653120\pi\)
\(662\) −1.59240e6 −0.141223
\(663\) −1.51741e7 −1.34066
\(664\) 31109.8 0.00273828
\(665\) 3.58508e6 0.314372
\(666\) 111808. 0.00976756
\(667\) −2.63676e6 −0.229486
\(668\) −8.79242e6 −0.762372
\(669\) 2.01458e7 1.74028
\(670\) −668803. −0.0575587
\(671\) −9.48162e6 −0.812973
\(672\) 2.09786e6 0.179207
\(673\) −1.10398e7 −0.939556 −0.469778 0.882785i \(-0.655666\pi\)
−0.469778 + 0.882785i \(0.655666\pi\)
\(674\) −1.75101e6 −0.148470
\(675\) 1.84420e6 0.155793
\(676\) −1.63788e6 −0.137853
\(677\) −8.23485e6 −0.690532 −0.345266 0.938505i \(-0.612211\pi\)
−0.345266 + 0.938505i \(0.612211\pi\)
\(678\) −615144. −0.0513928
\(679\) −1.12243e6 −0.0934298
\(680\) 2.55371e6 0.211787
\(681\) −9.90780e6 −0.818671
\(682\) −989343. −0.0814490
\(683\) 1.98437e7 1.62768 0.813842 0.581086i \(-0.197372\pi\)
0.813842 + 0.581086i \(0.197372\pi\)
\(684\) 7.04909e6 0.576093
\(685\) −1.80655e7 −1.47103
\(686\) 85305.3 0.00692095
\(687\) −2.19955e7 −1.77804
\(688\) 1.61304e7 1.29919
\(689\) −3.59100e6 −0.288182
\(690\) 732767. 0.0585927
\(691\) −2.37019e6 −0.188837 −0.0944185 0.995533i \(-0.530099\pi\)
−0.0944185 + 0.995533i \(0.530099\pi\)
\(692\) −3.13796e6 −0.249105
\(693\) 4.67065e6 0.369440
\(694\) −789131. −0.0621943
\(695\) −808259. −0.0634730
\(696\) −2.16748e6 −0.169602
\(697\) 1.30346e6 0.101628
\(698\) 2.02491e6 0.157314
\(699\) 1.51439e7 1.17232
\(700\) 1.44896e6 0.111766
\(701\) −1.56833e7 −1.20543 −0.602714 0.797957i \(-0.705914\pi\)
−0.602714 + 0.797957i \(0.705914\pi\)
\(702\) 926045. 0.0709234
\(703\) −1.68641e6 −0.128699
\(704\) −1.97042e7 −1.49840
\(705\) 7.62288e6 0.577626
\(706\) 1.83543e6 0.138588
\(707\) −8.89223e6 −0.669056
\(708\) 8.80085e6 0.659844
\(709\) 3.68544e6 0.275343 0.137671 0.990478i \(-0.456038\pi\)
0.137671 + 0.990478i \(0.456038\pi\)
\(710\) −2.18848e6 −0.162928
\(711\) −4.38373e6 −0.325214
\(712\) −5.79772e6 −0.428605
\(713\) 2.25361e6 0.166018
\(714\) 828604. 0.0608277
\(715\) 2.02604e7 1.48212
\(716\) −1.03859e7 −0.757113
\(717\) −3.37276e6 −0.245012
\(718\) 790544. 0.0572289
\(719\) −1.56427e7 −1.12847 −0.564233 0.825615i \(-0.690828\pi\)
−0.564233 + 0.825615i \(0.690828\pi\)
\(720\) −6.51470e6 −0.468342
\(721\) 3.17383e6 0.227376
\(722\) −19369.0 −0.00138282
\(723\) −7.54107e6 −0.536522
\(724\) 1.59200e7 1.12875
\(725\) −2.25175e6 −0.159102
\(726\) 4.02670e6 0.283536
\(727\) 1.85908e7 1.30456 0.652279 0.757979i \(-0.273813\pi\)
0.652279 + 0.757979i \(0.273813\pi\)
\(728\) 1.46731e6 0.102611
\(729\) −1.12318e6 −0.0782767
\(730\) −968335. −0.0672541
\(731\) 1.96601e7 1.36079
\(732\) 8.80353e6 0.607266
\(733\) 2.49466e7 1.71495 0.857476 0.514524i \(-0.172031\pi\)
0.857476 + 0.514524i \(0.172031\pi\)
\(734\) 136367. 0.00934260
\(735\) −2.20555e6 −0.150591
\(736\) −2.39708e6 −0.163113
\(737\) 1.31422e7 0.891253
\(738\) 113952. 0.00770159
\(739\) −2.42944e7 −1.63642 −0.818211 0.574918i \(-0.805034\pi\)
−0.818211 + 0.574918i \(0.805034\pi\)
\(740\) 1.58549e6 0.106435
\(741\) 2.00088e7 1.33868
\(742\) 196091. 0.0130752
\(743\) 4.16541e6 0.276812 0.138406 0.990376i \(-0.455802\pi\)
0.138406 + 0.990376i \(0.455802\pi\)
\(744\) 1.85252e6 0.122696
\(745\) 5.24388e6 0.346148
\(746\) −1.30059e6 −0.0855644
\(747\) −96729.9 −0.00634248
\(748\) −2.48828e7 −1.62610
\(749\) −7.26032e6 −0.472880
\(750\) 2.70721e6 0.175739
\(751\) −2.36434e7 −1.52972 −0.764858 0.644199i \(-0.777191\pi\)
−0.764858 + 0.644199i \(0.777191\pi\)
\(752\) −8.08101e6 −0.521100
\(753\) −3.08588e7 −1.98332
\(754\) −1.13070e6 −0.0724298
\(755\) −1.42564e6 −0.0910209
\(756\) 3.02730e6 0.192642
\(757\) 1.50108e7 0.952062 0.476031 0.879429i \(-0.342075\pi\)
0.476031 + 0.879429i \(0.342075\pi\)
\(758\) 2.60164e6 0.164465
\(759\) −1.43992e7 −0.907262
\(760\) −3.36735e6 −0.211472
\(761\) 2.92191e6 0.182897 0.0914483 0.995810i \(-0.470850\pi\)
0.0914483 + 0.995810i \(0.470850\pi\)
\(762\) −1.87282e6 −0.116845
\(763\) −5.45343e6 −0.339124
\(764\) 2.00918e6 0.124533
\(765\) −7.94025e6 −0.490547
\(766\) −2.48310e6 −0.152905
\(767\) 9.25887e6 0.568290
\(768\) 1.73016e7 1.05848
\(769\) 1.42847e7 0.871073 0.435536 0.900171i \(-0.356559\pi\)
0.435536 + 0.900171i \(0.356559\pi\)
\(770\) −1.10634e6 −0.0672455
\(771\) 1.55360e7 0.941246
\(772\) −1.47726e7 −0.892100
\(773\) −1.09012e7 −0.656186 −0.328093 0.944645i \(-0.606406\pi\)
−0.328093 + 0.944645i \(0.606406\pi\)
\(774\) 1.71874e6 0.103123
\(775\) 1.92455e6 0.115100
\(776\) 1.05426e6 0.0628485
\(777\) 1.03749e6 0.0616496
\(778\) 6254.04 0.000370434 0
\(779\) −1.71875e6 −0.101478
\(780\) −1.88114e7 −1.10710
\(781\) 4.30045e7 2.52282
\(782\) −946787. −0.0553650
\(783\) −4.70459e6 −0.274231
\(784\) 2.33811e6 0.135854
\(785\) −2.44736e7 −1.41750
\(786\) −4.97890e6 −0.287460
\(787\) −2.56449e7 −1.47593 −0.737963 0.674841i \(-0.764212\pi\)
−0.737963 + 0.674841i \(0.764212\pi\)
\(788\) −8.43576e6 −0.483959
\(789\) 9.12555e6 0.521875
\(790\) 1.03838e6 0.0591955
\(791\) −2.11560e6 −0.120224
\(792\) −4.38699e6 −0.248516
\(793\) 9.26169e6 0.523007
\(794\) 911498. 0.0513103
\(795\) −5.06991e6 −0.284500
\(796\) 1.90472e7 1.06549
\(797\) 7.73086e6 0.431104 0.215552 0.976492i \(-0.430845\pi\)
0.215552 + 0.976492i \(0.430845\pi\)
\(798\) −1.09261e6 −0.0607375
\(799\) −9.84931e6 −0.545807
\(800\) −2.04707e6 −0.113086
\(801\) 1.80269e7 0.992748
\(802\) 1.03448e6 0.0567917
\(803\) 1.90282e7 1.04138
\(804\) −1.22024e7 −0.665738
\(805\) 2.52013e6 0.137067
\(806\) 966395. 0.0523983
\(807\) 3.92909e7 2.12377
\(808\) 8.35218e6 0.450061
\(809\) −1.87811e7 −1.00890 −0.504452 0.863440i \(-0.668305\pi\)
−0.504452 + 0.863440i \(0.668305\pi\)
\(810\) 2.48616e6 0.133143
\(811\) −9.00729e6 −0.480886 −0.240443 0.970663i \(-0.577293\pi\)
−0.240443 + 0.970663i \(0.577293\pi\)
\(812\) −3.69632e6 −0.196734
\(813\) 3.16939e7 1.68170
\(814\) 520422. 0.0275293
\(815\) 2.05531e7 1.08388
\(816\) 2.27110e7 1.19402
\(817\) −2.59240e7 −1.35877
\(818\) −2.22259e6 −0.116138
\(819\) −4.56231e6 −0.237671
\(820\) 1.61590e6 0.0839228
\(821\) −9.27965e6 −0.480478 −0.240239 0.970714i \(-0.577226\pi\)
−0.240239 + 0.970714i \(0.577226\pi\)
\(822\) 5.50572e6 0.284207
\(823\) −1.08308e7 −0.557393 −0.278697 0.960379i \(-0.589902\pi\)
−0.278697 + 0.960379i \(0.589902\pi\)
\(824\) −2.98107e6 −0.152952
\(825\) −1.22967e7 −0.629004
\(826\) −505593. −0.0257841
\(827\) −2.05230e7 −1.04346 −0.521731 0.853110i \(-0.674714\pi\)
−0.521731 + 0.853110i \(0.674714\pi\)
\(828\) 4.95515e6 0.251178
\(829\) −1.42216e7 −0.718724 −0.359362 0.933198i \(-0.617006\pi\)
−0.359362 + 0.933198i \(0.617006\pi\)
\(830\) 22912.6 0.00115446
\(831\) −4.08944e7 −2.05429
\(832\) 1.92472e7 0.963958
\(833\) 2.84973e6 0.142296
\(834\) 246329. 0.0122631
\(835\) −1.30595e7 −0.648202
\(836\) 3.28108e7 1.62368
\(837\) 4.02097e6 0.198389
\(838\) −179995. −0.00885421
\(839\) −9.29934e6 −0.456087 −0.228043 0.973651i \(-0.573233\pi\)
−0.228043 + 0.973651i \(0.573233\pi\)
\(840\) 2.07160e6 0.101300
\(841\) −1.47669e7 −0.719944
\(842\) 4.32353e6 0.210164
\(843\) −1.92965e7 −0.935211
\(844\) −1.05561e7 −0.510092
\(845\) −2.43277e6 −0.117209
\(846\) −861053. −0.0413623
\(847\) 1.38486e7 0.663281
\(848\) 5.37461e6 0.256659
\(849\) −2.74350e7 −1.30628
\(850\) −808543. −0.0383845
\(851\) −1.18546e6 −0.0561131
\(852\) −3.99290e7 −1.88447
\(853\) −3.07436e6 −0.144671 −0.0723357 0.997380i \(-0.523045\pi\)
−0.0723357 + 0.997380i \(0.523045\pi\)
\(854\) −505747. −0.0237295
\(855\) 1.04701e7 0.489819
\(856\) 6.81938e6 0.318098
\(857\) 3.45835e7 1.60848 0.804242 0.594302i \(-0.202572\pi\)
0.804242 + 0.594302i \(0.202572\pi\)
\(858\) −6.17465e6 −0.286348
\(859\) 1.63022e7 0.753814 0.376907 0.926251i \(-0.376988\pi\)
0.376907 + 0.926251i \(0.376988\pi\)
\(860\) 2.43726e7 1.12372
\(861\) 1.05738e6 0.0486099
\(862\) −3.57856e6 −0.164036
\(863\) 2.56962e7 1.17447 0.587235 0.809416i \(-0.300216\pi\)
0.587235 + 0.809416i \(0.300216\pi\)
\(864\) −4.27694e6 −0.194917
\(865\) −4.66087e6 −0.211800
\(866\) 3.01281e6 0.136514
\(867\) −218902. −0.00989012
\(868\) 3.15921e6 0.142324
\(869\) −2.04046e7 −0.916596
\(870\) −1.59636e6 −0.0715043
\(871\) −1.28374e7 −0.573366
\(872\) 5.12222e6 0.228122
\(873\) −3.27803e6 −0.145572
\(874\) 1.24844e6 0.0552829
\(875\) 9.31062e6 0.411111
\(876\) −1.76673e7 −0.777877
\(877\) 3.30060e7 1.44908 0.724542 0.689230i \(-0.242051\pi\)
0.724542 + 0.689230i \(0.242051\pi\)
\(878\) −165286. −0.00723603
\(879\) 5.03429e7 2.19769
\(880\) −3.03235e7 −1.31999
\(881\) 2.26705e6 0.0984060 0.0492030 0.998789i \(-0.484332\pi\)
0.0492030 + 0.998789i \(0.484332\pi\)
\(882\) 249131. 0.0107834
\(883\) 1.97779e7 0.853649 0.426825 0.904334i \(-0.359632\pi\)
0.426825 + 0.904334i \(0.359632\pi\)
\(884\) 2.43057e7 1.04611
\(885\) 1.30720e7 0.561029
\(886\) −1.43901e6 −0.0615858
\(887\) −36468.3 −0.00155635 −0.000778173 1.00000i \(-0.500248\pi\)
−0.000778173 1.00000i \(0.500248\pi\)
\(888\) −974478. −0.0414705
\(889\) −6.44100e6 −0.273337
\(890\) −4.27005e6 −0.180700
\(891\) −4.88541e7 −2.06161
\(892\) −3.22692e7 −1.35793
\(893\) 1.29874e7 0.544997
\(894\) −1.59815e6 −0.0668767
\(895\) −1.54263e7 −0.643730
\(896\) −4.46747e6 −0.185905
\(897\) 1.40652e7 0.583665
\(898\) 1.89303e6 0.0783367
\(899\) −4.90958e6 −0.202602
\(900\) 4.23163e6 0.174141
\(901\) 6.55068e6 0.268828
\(902\) 530403. 0.0217065
\(903\) 1.59485e7 0.650881
\(904\) 1.98711e6 0.0808727
\(905\) 2.36462e7 0.959709
\(906\) 434484. 0.0175854
\(907\) 1.62298e7 0.655079 0.327540 0.944837i \(-0.393781\pi\)
0.327540 + 0.944837i \(0.393781\pi\)
\(908\) 1.58702e7 0.638802
\(909\) −2.59695e7 −1.04245
\(910\) 1.08068e6 0.0432608
\(911\) −2.61699e7 −1.04474 −0.522368 0.852720i \(-0.674951\pi\)
−0.522368 + 0.852720i \(0.674951\pi\)
\(912\) −2.99469e7 −1.19224
\(913\) −450241. −0.0178759
\(914\) −2.97224e6 −0.117684
\(915\) 1.30760e7 0.516324
\(916\) 3.52320e7 1.38739
\(917\) −1.71234e7 −0.672461
\(918\) −1.68929e6 −0.0661602
\(919\) 4.05973e6 0.158565 0.0792826 0.996852i \(-0.474737\pi\)
0.0792826 + 0.996852i \(0.474737\pi\)
\(920\) −2.36708e6 −0.0922025
\(921\) −1.73870e7 −0.675421
\(922\) −1.90246e6 −0.0737034
\(923\) −4.20070e7 −1.62300
\(924\) −2.01853e7 −0.777778
\(925\) −1.01237e6 −0.0389031
\(926\) −3.10744e6 −0.119090
\(927\) 9.26907e6 0.354272
\(928\) 5.22212e6 0.199057
\(929\) 3.69518e6 0.140474 0.0702370 0.997530i \(-0.477624\pi\)
0.0702370 + 0.997530i \(0.477624\pi\)
\(930\) 1.36439e6 0.0517288
\(931\) −3.75769e6 −0.142084
\(932\) −2.42573e7 −0.914751
\(933\) 3.53927e7 1.33110
\(934\) 717918. 0.0269282
\(935\) −3.69589e7 −1.38258
\(936\) 4.28523e6 0.159877
\(937\) −1.91384e7 −0.712126 −0.356063 0.934462i \(-0.615881\pi\)
−0.356063 + 0.934462i \(0.615881\pi\)
\(938\) 701004. 0.0260144
\(939\) 1.86742e7 0.691160
\(940\) −1.22102e7 −0.450717
\(941\) 1.20954e7 0.445294 0.222647 0.974899i \(-0.428530\pi\)
0.222647 + 0.974899i \(0.428530\pi\)
\(942\) 7.45870e6 0.273865
\(943\) −1.20820e6 −0.0442445
\(944\) −1.38577e7 −0.506127
\(945\) 4.49649e6 0.163793
\(946\) 8.00007e6 0.290647
\(947\) −1.95969e7 −0.710087 −0.355044 0.934850i \(-0.615534\pi\)
−0.355044 + 0.934850i \(0.615534\pi\)
\(948\) 1.89453e7 0.684669
\(949\) −1.85868e7 −0.669945
\(950\) 1.06615e6 0.0383275
\(951\) 5.97890e7 2.14373
\(952\) −2.67666e6 −0.0957196
\(953\) 4.23265e7 1.50966 0.754832 0.655918i \(-0.227718\pi\)
0.754832 + 0.655918i \(0.227718\pi\)
\(954\) 572679. 0.0203723
\(955\) 2.98426e6 0.105884
\(956\) 5.40243e6 0.191181
\(957\) 3.13691e7 1.10719
\(958\) 1.87054e6 0.0658496
\(959\) 1.89353e7 0.664852
\(960\) 2.71739e7 0.951642
\(961\) −2.44330e7 −0.853430
\(962\) −508351. −0.0177103
\(963\) −2.12035e7 −0.736788
\(964\) 1.20792e7 0.418644
\(965\) −2.19419e7 −0.758502
\(966\) −768048. −0.0264817
\(967\) 1.39211e6 0.0478749 0.0239375 0.999713i \(-0.492380\pi\)
0.0239375 + 0.999713i \(0.492380\pi\)
\(968\) −1.30075e7 −0.446176
\(969\) −3.64999e7 −1.24877
\(970\) 776471. 0.0264970
\(971\) 4.41877e7 1.50402 0.752009 0.659153i \(-0.229085\pi\)
0.752009 + 0.659153i \(0.229085\pi\)
\(972\) 3.03473e7 1.03028
\(973\) 847175. 0.0286874
\(974\) 2.33095e6 0.0787292
\(975\) 1.20115e7 0.404654
\(976\) −1.38619e7 −0.465798
\(977\) −5.60457e7 −1.87848 −0.939239 0.343264i \(-0.888467\pi\)
−0.939239 + 0.343264i \(0.888467\pi\)
\(978\) −6.26386e6 −0.209409
\(979\) 8.39082e7 2.79800
\(980\) 3.53282e6 0.117505
\(981\) −1.59266e7 −0.528384
\(982\) −5.69993e6 −0.188621
\(983\) −5.00560e7 −1.65224 −0.826118 0.563497i \(-0.809456\pi\)
−0.826118 + 0.563497i \(0.809456\pi\)
\(984\) −993166. −0.0326990
\(985\) −1.25298e7 −0.411483
\(986\) 2.06261e6 0.0675654
\(987\) −7.98990e6 −0.261065
\(988\) −3.20497e7 −1.04456
\(989\) −1.82233e7 −0.592428
\(990\) −3.23104e6 −0.104774
\(991\) 1.59116e7 0.514670 0.257335 0.966322i \(-0.417156\pi\)
0.257335 + 0.966322i \(0.417156\pi\)
\(992\) −4.46330e6 −0.144005
\(993\) −4.31534e7 −1.38881
\(994\) 2.29385e6 0.0736375
\(995\) 2.82911e7 0.905924
\(996\) 418041. 0.0133528
\(997\) 4.25995e7 1.35727 0.678635 0.734476i \(-0.262572\pi\)
0.678635 + 0.734476i \(0.262572\pi\)
\(998\) −981634. −0.0311978
\(999\) −2.11514e6 −0.0670542
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 7.6.a.b.1.1 2
3.2 odd 2 63.6.a.f.1.2 2
4.3 odd 2 112.6.a.h.1.1 2
5.2 odd 4 175.6.b.c.99.3 4
5.3 odd 4 175.6.b.c.99.2 4
5.4 even 2 175.6.a.c.1.2 2
7.2 even 3 49.6.c.e.18.2 4
7.3 odd 6 49.6.c.d.30.2 4
7.4 even 3 49.6.c.e.30.2 4
7.5 odd 6 49.6.c.d.18.2 4
7.6 odd 2 49.6.a.f.1.1 2
8.3 odd 2 448.6.a.u.1.2 2
8.5 even 2 448.6.a.w.1.1 2
11.10 odd 2 847.6.a.c.1.2 2
12.11 even 2 1008.6.a.bq.1.2 2
21.20 even 2 441.6.a.l.1.2 2
28.27 even 2 784.6.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.b.1.1 2 1.1 even 1 trivial
49.6.a.f.1.1 2 7.6 odd 2
49.6.c.d.18.2 4 7.5 odd 6
49.6.c.d.30.2 4 7.3 odd 6
49.6.c.e.18.2 4 7.2 even 3
49.6.c.e.30.2 4 7.4 even 3
63.6.a.f.1.2 2 3.2 odd 2
112.6.a.h.1.1 2 4.3 odd 2
175.6.a.c.1.2 2 5.4 even 2
175.6.b.c.99.2 4 5.3 odd 4
175.6.b.c.99.3 4 5.2 odd 4
441.6.a.l.1.2 2 21.20 even 2
448.6.a.u.1.2 2 8.3 odd 2
448.6.a.w.1.1 2 8.5 even 2
784.6.a.v.1.2 2 28.27 even 2
847.6.a.c.1.2 2 11.10 odd 2
1008.6.a.bq.1.2 2 12.11 even 2