Properties

Label 147.6.a.k.1.2
Level $147$
Weight $6$
Character 147.1
Self dual yes
Analytic conductor $23.576$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.5764215125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-7.38987\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.38987 q^{2} +9.00000 q^{3} +8.83040 q^{4} -38.7291 q^{5} +57.5088 q^{6} -148.051 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+6.38987 q^{2} +9.00000 q^{3} +8.83040 q^{4} -38.7291 q^{5} +57.5088 q^{6} -148.051 q^{8} +81.0000 q^{9} -247.474 q^{10} -576.390 q^{11} +79.4736 q^{12} -391.491 q^{13} -348.562 q^{15} -1228.60 q^{16} +1329.70 q^{17} +517.579 q^{18} -942.474 q^{19} -341.993 q^{20} -3683.05 q^{22} -1632.08 q^{23} -1332.46 q^{24} -1625.06 q^{25} -2501.58 q^{26} +729.000 q^{27} -1463.54 q^{29} -2227.26 q^{30} +3912.42 q^{31} -3112.95 q^{32} -5187.51 q^{33} +8496.61 q^{34} +715.262 q^{36} -16300.3 q^{37} -6022.28 q^{38} -3523.42 q^{39} +5733.86 q^{40} +13103.8 q^{41} +14733.5 q^{43} -5089.75 q^{44} -3137.05 q^{45} -10428.8 q^{46} +6814.52 q^{47} -11057.4 q^{48} -10383.9 q^{50} +11967.3 q^{51} -3457.02 q^{52} -2011.34 q^{53} +4658.21 q^{54} +22323.0 q^{55} -8482.26 q^{57} -9351.85 q^{58} -51453.1 q^{59} -3077.94 q^{60} -41097.8 q^{61} +24999.8 q^{62} +19423.8 q^{64} +15162.1 q^{65} -33147.5 q^{66} +50578.2 q^{67} +11741.8 q^{68} -14688.7 q^{69} +39970.6 q^{71} -11992.1 q^{72} +55686.6 q^{73} -104157. q^{74} -14625.5 q^{75} -8322.42 q^{76} -22514.2 q^{78} -63151.4 q^{79} +47582.4 q^{80} +6561.00 q^{81} +83731.7 q^{82} -45572.4 q^{83} -51498.1 q^{85} +94145.1 q^{86} -13171.9 q^{87} +85334.9 q^{88} -15686.7 q^{89} -20045.4 q^{90} -14411.9 q^{92} +35211.8 q^{93} +43543.9 q^{94} +36501.1 q^{95} -28016.5 q^{96} -3128.49 q^{97} -46687.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 18 q^{3} + 65 q^{4} + 33 q^{5} - 27 q^{6} - 375 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 18 q^{3} + 65 q^{4} + 33 q^{5} - 27 q^{6} - 375 q^{8} + 162 q^{9} - 921 q^{10} - 1137 q^{11} + 585 q^{12} - 925 q^{13} + 297 q^{15} - 895 q^{16} + 324 q^{17} - 243 q^{18} - 2311 q^{19} + 3687 q^{20} + 1581 q^{22} + 1596 q^{23} - 3375 q^{24} + 395 q^{25} + 2508 q^{26} + 1458 q^{27} - 2217 q^{29} - 8289 q^{30} - 4294 q^{31} + 1017 q^{32} - 10233 q^{33} + 17940 q^{34} + 5265 q^{36} - 19109 q^{37} + 6828 q^{38} - 8325 q^{39} - 10545 q^{40} + 12858 q^{41} - 2771 q^{43} - 36579 q^{44} + 2673 q^{45} - 40740 q^{46} + 23160 q^{47} - 8055 q^{48} - 29352 q^{50} + 2916 q^{51} - 33424 q^{52} - 31653 q^{53} - 2187 q^{54} - 17889 q^{55} - 20799 q^{57} - 2277 q^{58} - 41097 q^{59} + 33183 q^{60} - 42052 q^{61} + 102057 q^{62} - 30031 q^{64} - 23106 q^{65} + 14229 q^{66} + 30763 q^{67} - 44748 q^{68} + 14364 q^{69} + 102096 q^{71} - 30375 q^{72} + 28577 q^{73} - 77784 q^{74} + 3555 q^{75} - 85192 q^{76} + 22572 q^{78} - 18464 q^{79} + 71511 q^{80} + 13122 q^{81} + 86040 q^{82} - 61179 q^{83} - 123636 q^{85} + 258510 q^{86} - 19953 q^{87} + 212565 q^{88} - 29322 q^{89} - 74601 q^{90} + 166908 q^{92} - 38646 q^{93} - 109938 q^{94} - 61662 q^{95} + 9153 q^{96} + 9791 q^{97} - 92097 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.38987 1.12958 0.564790 0.825235i \(-0.308957\pi\)
0.564790 + 0.825235i \(0.308957\pi\)
\(3\) 9.00000 0.577350
\(4\) 8.83040 0.275950
\(5\) −38.7291 −0.692807 −0.346403 0.938086i \(-0.612597\pi\)
−0.346403 + 0.938086i \(0.612597\pi\)
\(6\) 57.5088 0.652163
\(7\) 0 0
\(8\) −148.051 −0.817872
\(9\) 81.0000 0.333333
\(10\) −247.474 −0.782580
\(11\) −576.390 −1.43627 −0.718133 0.695906i \(-0.755003\pi\)
−0.718133 + 0.695906i \(0.755003\pi\)
\(12\) 79.4736 0.159320
\(13\) −391.491 −0.642486 −0.321243 0.946997i \(-0.604101\pi\)
−0.321243 + 0.946997i \(0.604101\pi\)
\(14\) 0 0
\(15\) −348.562 −0.399992
\(16\) −1228.60 −1.19980
\(17\) 1329.70 1.11592 0.557958 0.829869i \(-0.311585\pi\)
0.557958 + 0.829869i \(0.311585\pi\)
\(18\) 517.579 0.376527
\(19\) −942.474 −0.598943 −0.299471 0.954105i \(-0.596810\pi\)
−0.299471 + 0.954105i \(0.596810\pi\)
\(20\) −341.993 −0.191180
\(21\) 0 0
\(22\) −3683.05 −1.62238
\(23\) −1632.08 −0.643312 −0.321656 0.946857i \(-0.604239\pi\)
−0.321656 + 0.946857i \(0.604239\pi\)
\(24\) −1332.46 −0.472199
\(25\) −1625.06 −0.520019
\(26\) −2501.58 −0.725739
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −1463.54 −0.323155 −0.161577 0.986860i \(-0.551658\pi\)
−0.161577 + 0.986860i \(0.551658\pi\)
\(30\) −2227.26 −0.451823
\(31\) 3912.42 0.731208 0.365604 0.930770i \(-0.380862\pi\)
0.365604 + 0.930770i \(0.380862\pi\)
\(32\) −3112.95 −0.537399
\(33\) −5187.51 −0.829228
\(34\) 8496.61 1.26052
\(35\) 0 0
\(36\) 715.262 0.0919833
\(37\) −16300.3 −1.95746 −0.978729 0.205160i \(-0.934229\pi\)
−0.978729 + 0.205160i \(0.934229\pi\)
\(38\) −6022.28 −0.676553
\(39\) −3523.42 −0.370939
\(40\) 5733.86 0.566627
\(41\) 13103.8 1.21741 0.608707 0.793395i \(-0.291688\pi\)
0.608707 + 0.793395i \(0.291688\pi\)
\(42\) 0 0
\(43\) 14733.5 1.21516 0.607582 0.794257i \(-0.292140\pi\)
0.607582 + 0.794257i \(0.292140\pi\)
\(44\) −5089.75 −0.396337
\(45\) −3137.05 −0.230936
\(46\) −10428.8 −0.726672
\(47\) 6814.52 0.449977 0.224989 0.974361i \(-0.427765\pi\)
0.224989 + 0.974361i \(0.427765\pi\)
\(48\) −11057.4 −0.692706
\(49\) 0 0
\(50\) −10383.9 −0.587403
\(51\) 11967.3 0.644274
\(52\) −3457.02 −0.177294
\(53\) −2011.34 −0.0983550 −0.0491775 0.998790i \(-0.515660\pi\)
−0.0491775 + 0.998790i \(0.515660\pi\)
\(54\) 4658.21 0.217388
\(55\) 22323.0 0.995054
\(56\) 0 0
\(57\) −8482.26 −0.345800
\(58\) −9351.85 −0.365029
\(59\) −51453.1 −1.92434 −0.962170 0.272451i \(-0.912166\pi\)
−0.962170 + 0.272451i \(0.912166\pi\)
\(60\) −3077.94 −0.110378
\(61\) −41097.8 −1.41415 −0.707073 0.707141i \(-0.749985\pi\)
−0.707073 + 0.707141i \(0.749985\pi\)
\(62\) 24999.8 0.825958
\(63\) 0 0
\(64\) 19423.8 0.592766
\(65\) 15162.1 0.445119
\(66\) −33147.5 −0.936679
\(67\) 50578.2 1.37650 0.688250 0.725473i \(-0.258379\pi\)
0.688250 + 0.725473i \(0.258379\pi\)
\(68\) 11741.8 0.307937
\(69\) −14688.7 −0.371416
\(70\) 0 0
\(71\) 39970.6 0.941012 0.470506 0.882397i \(-0.344071\pi\)
0.470506 + 0.882397i \(0.344071\pi\)
\(72\) −11992.1 −0.272624
\(73\) 55686.6 1.22305 0.611524 0.791226i \(-0.290557\pi\)
0.611524 + 0.791226i \(0.290557\pi\)
\(74\) −104157. −2.21110
\(75\) −14625.5 −0.300233
\(76\) −8322.42 −0.165278
\(77\) 0 0
\(78\) −22514.2 −0.419006
\(79\) −63151.4 −1.13845 −0.569226 0.822181i \(-0.692757\pi\)
−0.569226 + 0.822181i \(0.692757\pi\)
\(80\) 47582.4 0.831231
\(81\) 6561.00 0.111111
\(82\) 83731.7 1.37517
\(83\) −45572.4 −0.726116 −0.363058 0.931766i \(-0.618267\pi\)
−0.363058 + 0.931766i \(0.618267\pi\)
\(84\) 0 0
\(85\) −51498.1 −0.773114
\(86\) 94145.1 1.37262
\(87\) −13171.9 −0.186574
\(88\) 85334.9 1.17468
\(89\) −15686.7 −0.209921 −0.104961 0.994476i \(-0.533472\pi\)
−0.104961 + 0.994476i \(0.533472\pi\)
\(90\) −20045.4 −0.260860
\(91\) 0 0
\(92\) −14411.9 −0.177522
\(93\) 35211.8 0.422163
\(94\) 43543.9 0.508285
\(95\) 36501.1 0.414951
\(96\) −28016.5 −0.310268
\(97\) −3128.49 −0.0337603 −0.0168801 0.999858i \(-0.505373\pi\)
−0.0168801 + 0.999858i \(0.505373\pi\)
\(98\) 0 0
\(99\) −46687.6 −0.478755
\(100\) −14349.9 −0.143499
\(101\) 169011. 1.64858 0.824292 0.566164i \(-0.191573\pi\)
0.824292 + 0.566164i \(0.191573\pi\)
\(102\) 76469.5 0.727759
\(103\) −112820. −1.04784 −0.523918 0.851769i \(-0.675530\pi\)
−0.523918 + 0.851769i \(0.675530\pi\)
\(104\) 57960.5 0.525471
\(105\) 0 0
\(106\) −12852.2 −0.111100
\(107\) −22309.5 −0.188378 −0.0941890 0.995554i \(-0.530026\pi\)
−0.0941890 + 0.995554i \(0.530026\pi\)
\(108\) 6437.36 0.0531066
\(109\) −83819.5 −0.675739 −0.337869 0.941193i \(-0.609706\pi\)
−0.337869 + 0.941193i \(0.609706\pi\)
\(110\) 142641. 1.12399
\(111\) −146703. −1.13014
\(112\) 0 0
\(113\) −40928.4 −0.301529 −0.150764 0.988570i \(-0.548174\pi\)
−0.150764 + 0.988570i \(0.548174\pi\)
\(114\) −54200.5 −0.390608
\(115\) 63208.9 0.445691
\(116\) −12923.7 −0.0891746
\(117\) −31710.8 −0.214162
\(118\) −328779. −2.17369
\(119\) 0 0
\(120\) 51604.8 0.327142
\(121\) 171174. 1.06286
\(122\) −262610. −1.59739
\(123\) 117934. 0.702875
\(124\) 34548.2 0.201777
\(125\) 183965. 1.05308
\(126\) 0 0
\(127\) 83270.1 0.458120 0.229060 0.973412i \(-0.426435\pi\)
0.229060 + 0.973412i \(0.426435\pi\)
\(128\) 223730. 1.20698
\(129\) 132601. 0.701575
\(130\) 96883.7 0.502797
\(131\) 166875. 0.849596 0.424798 0.905288i \(-0.360345\pi\)
0.424798 + 0.905288i \(0.360345\pi\)
\(132\) −45807.8 −0.228825
\(133\) 0 0
\(134\) 323188. 1.55487
\(135\) −28233.5 −0.133331
\(136\) −196863. −0.912676
\(137\) 38223.9 0.173994 0.0869969 0.996209i \(-0.472273\pi\)
0.0869969 + 0.996209i \(0.472273\pi\)
\(138\) −93858.9 −0.419544
\(139\) −106263. −0.466492 −0.233246 0.972418i \(-0.574935\pi\)
−0.233246 + 0.972418i \(0.574935\pi\)
\(140\) 0 0
\(141\) 61330.7 0.259795
\(142\) 255407. 1.06295
\(143\) 225652. 0.922780
\(144\) −99516.3 −0.399934
\(145\) 56681.7 0.223884
\(146\) 355830. 1.38153
\(147\) 0 0
\(148\) −143938. −0.540160
\(149\) 192556. 0.710543 0.355271 0.934763i \(-0.384388\pi\)
0.355271 + 0.934763i \(0.384388\pi\)
\(150\) −93455.2 −0.339137
\(151\) 141699. 0.505735 0.252868 0.967501i \(-0.418626\pi\)
0.252868 + 0.967501i \(0.418626\pi\)
\(152\) 139534. 0.489858
\(153\) 107706. 0.371972
\(154\) 0 0
\(155\) −151524. −0.506586
\(156\) −31113.2 −0.102361
\(157\) −565771. −1.83186 −0.915928 0.401342i \(-0.868544\pi\)
−0.915928 + 0.401342i \(0.868544\pi\)
\(158\) −403529. −1.28597
\(159\) −18102.1 −0.0567853
\(160\) 120562. 0.372314
\(161\) 0 0
\(162\) 41923.9 0.125509
\(163\) −430201. −1.26824 −0.634121 0.773233i \(-0.718638\pi\)
−0.634121 + 0.773233i \(0.718638\pi\)
\(164\) 115712. 0.335946
\(165\) 200907. 0.574495
\(166\) −291201. −0.820206
\(167\) 240265. 0.666653 0.333327 0.942811i \(-0.391829\pi\)
0.333327 + 0.942811i \(0.391829\pi\)
\(168\) 0 0
\(169\) −218028. −0.587212
\(170\) −329066. −0.873294
\(171\) −76340.4 −0.199648
\(172\) 130103. 0.335324
\(173\) 179300. 0.455476 0.227738 0.973722i \(-0.426867\pi\)
0.227738 + 0.973722i \(0.426867\pi\)
\(174\) −84166.7 −0.210750
\(175\) 0 0
\(176\) 708151. 1.72323
\(177\) −463078. −1.11102
\(178\) −100236. −0.237123
\(179\) −575559. −1.34263 −0.671317 0.741170i \(-0.734271\pi\)
−0.671317 + 0.741170i \(0.734271\pi\)
\(180\) −27701.4 −0.0637267
\(181\) −581006. −1.31821 −0.659105 0.752051i \(-0.729065\pi\)
−0.659105 + 0.752051i \(0.729065\pi\)
\(182\) 0 0
\(183\) −369880. −0.816458
\(184\) 241630. 0.526147
\(185\) 631297. 1.35614
\(186\) 224998. 0.476867
\(187\) −766426. −1.60275
\(188\) 60174.9 0.124171
\(189\) 0 0
\(190\) 233237. 0.468721
\(191\) −660560. −1.31017 −0.655087 0.755554i \(-0.727368\pi\)
−0.655087 + 0.755554i \(0.727368\pi\)
\(192\) 174814. 0.342234
\(193\) −557310. −1.07697 −0.538485 0.842635i \(-0.681003\pi\)
−0.538485 + 0.842635i \(0.681003\pi\)
\(194\) −19990.7 −0.0381349
\(195\) 136459. 0.256989
\(196\) 0 0
\(197\) −761400. −1.39781 −0.698904 0.715216i \(-0.746328\pi\)
−0.698904 + 0.715216i \(0.746328\pi\)
\(198\) −298327. −0.540792
\(199\) 135860. 0.243197 0.121598 0.992579i \(-0.461198\pi\)
0.121598 + 0.992579i \(0.461198\pi\)
\(200\) 240591. 0.425309
\(201\) 455204. 0.794723
\(202\) 1.07996e6 1.86221
\(203\) 0 0
\(204\) 105676. 0.177787
\(205\) −507499. −0.843433
\(206\) −720906. −1.18361
\(207\) −132198. −0.214437
\(208\) 480985. 0.770856
\(209\) 543232. 0.860240
\(210\) 0 0
\(211\) −991157. −1.53263 −0.766313 0.642467i \(-0.777911\pi\)
−0.766313 + 0.642467i \(0.777911\pi\)
\(212\) −17761.0 −0.0271411
\(213\) 359736. 0.543293
\(214\) −142555. −0.212788
\(215\) −570615. −0.841873
\(216\) −107929. −0.157400
\(217\) 0 0
\(218\) −535596. −0.763301
\(219\) 501180. 0.706128
\(220\) 197121. 0.274585
\(221\) −520566. −0.716960
\(222\) −937413. −1.27658
\(223\) −543344. −0.731666 −0.365833 0.930681i \(-0.619216\pi\)
−0.365833 + 0.930681i \(0.619216\pi\)
\(224\) 0 0
\(225\) −131630. −0.173340
\(226\) −261527. −0.340601
\(227\) −16.3341 −2.10393e−5 0 −1.05196e−5 1.00000i \(-0.500003\pi\)
−1.05196e−5 1.00000i \(0.500003\pi\)
\(228\) −74901.8 −0.0954234
\(229\) 77379.0 0.0975067 0.0487534 0.998811i \(-0.484475\pi\)
0.0487534 + 0.998811i \(0.484475\pi\)
\(230\) 403896. 0.503443
\(231\) 0 0
\(232\) 216679. 0.264299
\(233\) −103657. −0.125086 −0.0625432 0.998042i \(-0.519921\pi\)
−0.0625432 + 0.998042i \(0.519921\pi\)
\(234\) −202628. −0.241913
\(235\) −263920. −0.311747
\(236\) −454351. −0.531021
\(237\) −568362. −0.657286
\(238\) 0 0
\(239\) 689109. 0.780356 0.390178 0.920739i \(-0.372413\pi\)
0.390178 + 0.920739i \(0.372413\pi\)
\(240\) 428242. 0.479911
\(241\) −220296. −0.244323 −0.122161 0.992510i \(-0.538983\pi\)
−0.122161 + 0.992510i \(0.538983\pi\)
\(242\) 1.09378e6 1.20058
\(243\) 59049.0 0.0641500
\(244\) −362910. −0.390234
\(245\) 0 0
\(246\) 753585. 0.793953
\(247\) 368970. 0.384812
\(248\) −579236. −0.598035
\(249\) −410151. −0.419224
\(250\) 1.17551e6 1.18954
\(251\) −1.43641e6 −1.43912 −0.719558 0.694433i \(-0.755655\pi\)
−0.719558 + 0.694433i \(0.755655\pi\)
\(252\) 0 0
\(253\) 940714. 0.923966
\(254\) 532085. 0.517483
\(255\) −463482. −0.446358
\(256\) 808042. 0.770609
\(257\) 909197. 0.858668 0.429334 0.903146i \(-0.358748\pi\)
0.429334 + 0.903146i \(0.358748\pi\)
\(258\) 847306. 0.792485
\(259\) 0 0
\(260\) 133887. 0.122830
\(261\) −118547. −0.107718
\(262\) 1.06631e6 0.959687
\(263\) 749057. 0.667768 0.333884 0.942614i \(-0.391641\pi\)
0.333884 + 0.942614i \(0.391641\pi\)
\(264\) 768014. 0.678203
\(265\) 77897.4 0.0681410
\(266\) 0 0
\(267\) −141180. −0.121198
\(268\) 446626. 0.379845
\(269\) −669945. −0.564493 −0.282246 0.959342i \(-0.591080\pi\)
−0.282246 + 0.959342i \(0.591080\pi\)
\(270\) −180408. −0.150608
\(271\) −540659. −0.447199 −0.223599 0.974681i \(-0.571781\pi\)
−0.223599 + 0.974681i \(0.571781\pi\)
\(272\) −1.63367e6 −1.33888
\(273\) 0 0
\(274\) 244246. 0.196540
\(275\) 936668. 0.746885
\(276\) −129707. −0.102492
\(277\) −401910. −0.314723 −0.157362 0.987541i \(-0.550299\pi\)
−0.157362 + 0.987541i \(0.550299\pi\)
\(278\) −679005. −0.526940
\(279\) 316906. 0.243736
\(280\) 0 0
\(281\) −429139. −0.324214 −0.162107 0.986773i \(-0.551829\pi\)
−0.162107 + 0.986773i \(0.551829\pi\)
\(282\) 391895. 0.293459
\(283\) 340927. 0.253044 0.126522 0.991964i \(-0.459619\pi\)
0.126522 + 0.991964i \(0.459619\pi\)
\(284\) 352957. 0.259672
\(285\) 328510. 0.239572
\(286\) 1.44188e6 1.04235
\(287\) 0 0
\(288\) −252149. −0.179133
\(289\) 348246. 0.245268
\(290\) 362188. 0.252895
\(291\) −28156.5 −0.0194915
\(292\) 491735. 0.337500
\(293\) 388847. 0.264612 0.132306 0.991209i \(-0.457762\pi\)
0.132306 + 0.991209i \(0.457762\pi\)
\(294\) 0 0
\(295\) 1.99273e6 1.33319
\(296\) 2.41328e6 1.60095
\(297\) −420188. −0.276409
\(298\) 1.23040e6 0.802615
\(299\) 638945. 0.413319
\(300\) −129149. −0.0828493
\(301\) 0 0
\(302\) 905435. 0.571268
\(303\) 1.52110e6 0.951811
\(304\) 1.15792e6 0.718612
\(305\) 1.59168e6 0.979730
\(306\) 688225. 0.420172
\(307\) 2.35747e6 1.42758 0.713789 0.700361i \(-0.246978\pi\)
0.713789 + 0.700361i \(0.246978\pi\)
\(308\) 0 0
\(309\) −1.01538e6 −0.604969
\(310\) −968220. −0.572229
\(311\) 1.43663e6 0.842254 0.421127 0.907002i \(-0.361635\pi\)
0.421127 + 0.907002i \(0.361635\pi\)
\(312\) 521645. 0.303381
\(313\) 822800. 0.474715 0.237358 0.971422i \(-0.423719\pi\)
0.237358 + 0.971422i \(0.423719\pi\)
\(314\) −3.61520e6 −2.06923
\(315\) 0 0
\(316\) −557652. −0.314156
\(317\) −1.76693e6 −0.987580 −0.493790 0.869581i \(-0.664389\pi\)
−0.493790 + 0.869581i \(0.664389\pi\)
\(318\) −115670. −0.0641435
\(319\) 843572. 0.464136
\(320\) −752264. −0.410672
\(321\) −200785. −0.108760
\(322\) 0 0
\(323\) −1.25321e6 −0.668370
\(324\) 57936.2 0.0306611
\(325\) 636196. 0.334105
\(326\) −2.74893e6 −1.43258
\(327\) −754376. −0.390138
\(328\) −1.94003e6 −0.995689
\(329\) 0 0
\(330\) 1.28377e6 0.648937
\(331\) −3052.27 −0.00153128 −0.000765638 1.00000i \(-0.500244\pi\)
−0.000765638 1.00000i \(0.500244\pi\)
\(332\) −402422. −0.200372
\(333\) −1.32033e6 −0.652486
\(334\) 1.53526e6 0.753038
\(335\) −1.95885e6 −0.953649
\(336\) 0 0
\(337\) 2.02939e6 0.973398 0.486699 0.873570i \(-0.338201\pi\)
0.486699 + 0.873570i \(0.338201\pi\)
\(338\) −1.39317e6 −0.663302
\(339\) −368356. −0.174088
\(340\) −454748. −0.213341
\(341\) −2.25508e6 −1.05021
\(342\) −487805. −0.225518
\(343\) 0 0
\(344\) −2.18130e6 −0.993848
\(345\) 568880. 0.257320
\(346\) 1.14570e6 0.514496
\(347\) 3.78218e6 1.68624 0.843119 0.537727i \(-0.180717\pi\)
0.843119 + 0.537727i \(0.180717\pi\)
\(348\) −116313. −0.0514850
\(349\) −291147. −0.127953 −0.0639763 0.997951i \(-0.520378\pi\)
−0.0639763 + 0.997951i \(0.520378\pi\)
\(350\) 0 0
\(351\) −285397. −0.123646
\(352\) 1.79427e6 0.771848
\(353\) −385076. −0.164479 −0.0822394 0.996613i \(-0.526207\pi\)
−0.0822394 + 0.996613i \(0.526207\pi\)
\(354\) −2.95901e6 −1.25498
\(355\) −1.54803e6 −0.651939
\(356\) −138520. −0.0579277
\(357\) 0 0
\(358\) −3.67775e6 −1.51661
\(359\) −3.23014e6 −1.32277 −0.661385 0.750046i \(-0.730031\pi\)
−0.661385 + 0.750046i \(0.730031\pi\)
\(360\) 464443. 0.188876
\(361\) −1.58784e6 −0.641268
\(362\) −3.71255e6 −1.48902
\(363\) 1.54057e6 0.613641
\(364\) 0 0
\(365\) −2.15669e6 −0.847336
\(366\) −2.36349e6 −0.922254
\(367\) 479559. 0.185856 0.0929280 0.995673i \(-0.470377\pi\)
0.0929280 + 0.995673i \(0.470377\pi\)
\(368\) 2.00517e6 0.771847
\(369\) 1.06141e6 0.405805
\(370\) 4.03390e6 1.53187
\(371\) 0 0
\(372\) 310934. 0.116496
\(373\) 872666. 0.324770 0.162385 0.986727i \(-0.448081\pi\)
0.162385 + 0.986727i \(0.448081\pi\)
\(374\) −4.89736e6 −1.81043
\(375\) 1.65569e6 0.607996
\(376\) −1.00889e6 −0.368024
\(377\) 572965. 0.207622
\(378\) 0 0
\(379\) −2.43493e6 −0.870742 −0.435371 0.900251i \(-0.643383\pi\)
−0.435371 + 0.900251i \(0.643383\pi\)
\(380\) 322320. 0.114506
\(381\) 749431. 0.264496
\(382\) −4.22089e6 −1.47995
\(383\) 3.61169e6 1.25809 0.629047 0.777367i \(-0.283445\pi\)
0.629047 + 0.777367i \(0.283445\pi\)
\(384\) 2.01357e6 0.696848
\(385\) 0 0
\(386\) −3.56114e6 −1.21652
\(387\) 1.19341e6 0.405055
\(388\) −27625.9 −0.00931615
\(389\) −175232. −0.0587138 −0.0293569 0.999569i \(-0.509346\pi\)
−0.0293569 + 0.999569i \(0.509346\pi\)
\(390\) 871954. 0.290290
\(391\) −2.17018e6 −0.717882
\(392\) 0 0
\(393\) 1.50187e6 0.490515
\(394\) −4.86525e6 −1.57893
\(395\) 2.44579e6 0.788727
\(396\) −412270. −0.132112
\(397\) −1.88515e6 −0.600303 −0.300152 0.953892i \(-0.597037\pi\)
−0.300152 + 0.953892i \(0.597037\pi\)
\(398\) 868126. 0.274710
\(399\) 0 0
\(400\) 1.99654e6 0.623920
\(401\) 1.33983e6 0.416091 0.208046 0.978119i \(-0.433290\pi\)
0.208046 + 0.978119i \(0.433290\pi\)
\(402\) 2.90869e6 0.897703
\(403\) −1.53168e6 −0.469791
\(404\) 1.49243e6 0.454927
\(405\) −254101. −0.0769785
\(406\) 0 0
\(407\) 9.39535e6 2.81143
\(408\) −1.77177e6 −0.526934
\(409\) −6.58628e6 −1.94685 −0.973423 0.229013i \(-0.926450\pi\)
−0.973423 + 0.229013i \(0.926450\pi\)
\(410\) −3.24285e6 −0.952725
\(411\) 344015. 0.100455
\(412\) −996247. −0.289150
\(413\) 0 0
\(414\) −844730. −0.242224
\(415\) 1.76497e6 0.503058
\(416\) 1.21869e6 0.345271
\(417\) −956365. −0.269329
\(418\) 3.47118e6 0.971710
\(419\) 6.96869e6 1.93917 0.969585 0.244754i \(-0.0787071\pi\)
0.969585 + 0.244754i \(0.0787071\pi\)
\(420\) 0 0
\(421\) 3.84041e6 1.05602 0.528010 0.849238i \(-0.322938\pi\)
0.528010 + 0.849238i \(0.322938\pi\)
\(422\) −6.33336e6 −1.73122
\(423\) 551976. 0.149992
\(424\) 297781. 0.0804418
\(425\) −2.16084e6 −0.580297
\(426\) 2.29866e6 0.613693
\(427\) 0 0
\(428\) −197002. −0.0519829
\(429\) 2.03086e6 0.532767
\(430\) −3.64615e6 −0.950963
\(431\) 3.03636e6 0.787337 0.393668 0.919253i \(-0.371206\pi\)
0.393668 + 0.919253i \(0.371206\pi\)
\(432\) −895647. −0.230902
\(433\) 941529. 0.241332 0.120666 0.992693i \(-0.461497\pi\)
0.120666 + 0.992693i \(0.461497\pi\)
\(434\) 0 0
\(435\) 510135. 0.129259
\(436\) −740160. −0.186470
\(437\) 1.53819e6 0.385307
\(438\) 3.20247e6 0.797627
\(439\) −1.34109e6 −0.332122 −0.166061 0.986116i \(-0.553105\pi\)
−0.166061 + 0.986116i \(0.553105\pi\)
\(440\) −3.30494e6 −0.813827
\(441\) 0 0
\(442\) −3.32635e6 −0.809864
\(443\) −772341. −0.186982 −0.0934910 0.995620i \(-0.529803\pi\)
−0.0934910 + 0.995620i \(0.529803\pi\)
\(444\) −1.29545e6 −0.311862
\(445\) 607531. 0.145435
\(446\) −3.47190e6 −0.826475
\(447\) 1.73300e6 0.410232
\(448\) 0 0
\(449\) 2.25684e6 0.528304 0.264152 0.964481i \(-0.414908\pi\)
0.264152 + 0.964481i \(0.414908\pi\)
\(450\) −841097. −0.195801
\(451\) −7.55291e6 −1.74853
\(452\) −361414. −0.0832069
\(453\) 1.27529e6 0.291986
\(454\) −104.373 −2.37655e−5 0
\(455\) 0 0
\(456\) 1.25580e6 0.282820
\(457\) 4.28470e6 0.959688 0.479844 0.877354i \(-0.340693\pi\)
0.479844 + 0.877354i \(0.340693\pi\)
\(458\) 494442. 0.110142
\(459\) 969352. 0.214758
\(460\) 558160. 0.122988
\(461\) 3.10462e6 0.680387 0.340193 0.940355i \(-0.389507\pi\)
0.340193 + 0.940355i \(0.389507\pi\)
\(462\) 0 0
\(463\) −3.53386e6 −0.766121 −0.383060 0.923723i \(-0.625130\pi\)
−0.383060 + 0.923723i \(0.625130\pi\)
\(464\) 1.79811e6 0.387722
\(465\) −1.36372e6 −0.292477
\(466\) −662356. −0.141295
\(467\) −2.72459e6 −0.578109 −0.289054 0.957313i \(-0.593341\pi\)
−0.289054 + 0.957313i \(0.593341\pi\)
\(468\) −280019. −0.0590980
\(469\) 0 0
\(470\) −1.68641e6 −0.352143
\(471\) −5.09194e6 −1.05762
\(472\) 7.61767e6 1.57386
\(473\) −8.49224e6 −1.74530
\(474\) −3.63176e6 −0.742457
\(475\) 1.53158e6 0.311461
\(476\) 0 0
\(477\) −162919. −0.0327850
\(478\) 4.40331e6 0.881475
\(479\) −978685. −0.194896 −0.0974482 0.995241i \(-0.531068\pi\)
−0.0974482 + 0.995241i \(0.531068\pi\)
\(480\) 1.08505e6 0.214955
\(481\) 6.38144e6 1.25764
\(482\) −1.40766e6 −0.275982
\(483\) 0 0
\(484\) 1.51154e6 0.293296
\(485\) 121164. 0.0233893
\(486\) 377315. 0.0724626
\(487\) 3.92744e6 0.750390 0.375195 0.926946i \(-0.377576\pi\)
0.375195 + 0.926946i \(0.377576\pi\)
\(488\) 6.08456e6 1.15659
\(489\) −3.87181e6 −0.732220
\(490\) 0 0
\(491\) 2.63241e6 0.492777 0.246388 0.969171i \(-0.420756\pi\)
0.246388 + 0.969171i \(0.420756\pi\)
\(492\) 1.04141e6 0.193958
\(493\) −1.94607e6 −0.360614
\(494\) 2.35767e6 0.434676
\(495\) 1.80817e6 0.331685
\(496\) −4.80678e6 −0.877305
\(497\) 0 0
\(498\) −2.62081e6 −0.473546
\(499\) 2.12544e6 0.382118 0.191059 0.981579i \(-0.438808\pi\)
0.191059 + 0.981579i \(0.438808\pi\)
\(500\) 1.62449e6 0.290597
\(501\) 2.16239e6 0.384892
\(502\) −9.17850e6 −1.62560
\(503\) −2.60929e6 −0.459835 −0.229917 0.973210i \(-0.573846\pi\)
−0.229917 + 0.973210i \(0.573846\pi\)
\(504\) 0 0
\(505\) −6.54564e6 −1.14215
\(506\) 6.01104e6 1.04369
\(507\) −1.96225e6 −0.339027
\(508\) 735308. 0.126418
\(509\) 1.00182e7 1.71394 0.856970 0.515366i \(-0.172344\pi\)
0.856970 + 0.515366i \(0.172344\pi\)
\(510\) −2.96159e6 −0.504196
\(511\) 0 0
\(512\) −1.99607e6 −0.336512
\(513\) −687063. −0.115267
\(514\) 5.80965e6 0.969933
\(515\) 4.36942e6 0.725948
\(516\) 1.17092e6 0.193600
\(517\) −3.92782e6 −0.646287
\(518\) 0 0
\(519\) 1.61370e6 0.262969
\(520\) −2.24476e6 −0.364050
\(521\) 4.17941e6 0.674560 0.337280 0.941404i \(-0.390493\pi\)
0.337280 + 0.941404i \(0.390493\pi\)
\(522\) −757500. −0.121676
\(523\) −3.60525e6 −0.576343 −0.288172 0.957579i \(-0.593047\pi\)
−0.288172 + 0.957579i \(0.593047\pi\)
\(524\) 1.47357e6 0.234446
\(525\) 0 0
\(526\) 4.78637e6 0.754297
\(527\) 5.20234e6 0.815967
\(528\) 6.37336e6 0.994909
\(529\) −3.77266e6 −0.586150
\(530\) 497754. 0.0769707
\(531\) −4.16770e6 −0.641446
\(532\) 0 0
\(533\) −5.13003e6 −0.782172
\(534\) −902122. −0.136903
\(535\) 864025. 0.130509
\(536\) −7.48814e6 −1.12580
\(537\) −5.18003e6 −0.775170
\(538\) −4.28086e6 −0.637640
\(539\) 0 0
\(540\) −249313. −0.0367926
\(541\) −6.68166e6 −0.981501 −0.490751 0.871300i \(-0.663277\pi\)
−0.490751 + 0.871300i \(0.663277\pi\)
\(542\) −3.45474e6 −0.505146
\(543\) −5.22906e6 −0.761069
\(544\) −4.13929e6 −0.599692
\(545\) 3.24625e6 0.468156
\(546\) 0 0
\(547\) 8.69076e6 1.24191 0.620954 0.783847i \(-0.286745\pi\)
0.620954 + 0.783847i \(0.286745\pi\)
\(548\) 337532. 0.0480136
\(549\) −3.32892e6 −0.471382
\(550\) 5.98518e6 0.843666
\(551\) 1.37935e6 0.193551
\(552\) 2.17467e6 0.303771
\(553\) 0 0
\(554\) −2.56815e6 −0.355505
\(555\) 5.68167e6 0.782967
\(556\) −938342. −0.128728
\(557\) 6.24742e6 0.853223 0.426612 0.904435i \(-0.359707\pi\)
0.426612 + 0.904435i \(0.359707\pi\)
\(558\) 2.02499e6 0.275319
\(559\) −5.76803e6 −0.780725
\(560\) 0 0
\(561\) −6.89783e6 −0.925349
\(562\) −2.74214e6 −0.366226
\(563\) 1.19045e7 1.58285 0.791423 0.611269i \(-0.209341\pi\)
0.791423 + 0.611269i \(0.209341\pi\)
\(564\) 541574. 0.0716903
\(565\) 1.58512e6 0.208901
\(566\) 2.17848e6 0.285833
\(567\) 0 0
\(568\) −5.91768e6 −0.769627
\(569\) 21414.4 0.00277284 0.00138642 0.999999i \(-0.499559\pi\)
0.00138642 + 0.999999i \(0.499559\pi\)
\(570\) 2.09914e6 0.270616
\(571\) 7.11647e6 0.913428 0.456714 0.889614i \(-0.349026\pi\)
0.456714 + 0.889614i \(0.349026\pi\)
\(572\) 1.99259e6 0.254641
\(573\) −5.94504e6 −0.756429
\(574\) 0 0
\(575\) 2.65223e6 0.334534
\(576\) 1.57333e6 0.197589
\(577\) 1.06652e7 1.33361 0.666805 0.745232i \(-0.267661\pi\)
0.666805 + 0.745232i \(0.267661\pi\)
\(578\) 2.22524e6 0.277050
\(579\) −5.01579e6 −0.621789
\(580\) 500522. 0.0617808
\(581\) 0 0
\(582\) −179916. −0.0220172
\(583\) 1.15932e6 0.141264
\(584\) −8.24444e6 −1.00030
\(585\) 1.22813e6 0.148373
\(586\) 2.48468e6 0.298901
\(587\) 1.30101e7 1.55843 0.779213 0.626759i \(-0.215619\pi\)
0.779213 + 0.626759i \(0.215619\pi\)
\(588\) 0 0
\(589\) −3.68735e6 −0.437952
\(590\) 1.27333e7 1.50595
\(591\) −6.85260e6 −0.807025
\(592\) 2.00265e7 2.34856
\(593\) −4.26086e6 −0.497578 −0.248789 0.968558i \(-0.580033\pi\)
−0.248789 + 0.968558i \(0.580033\pi\)
\(594\) −2.68495e6 −0.312226
\(595\) 0 0
\(596\) 1.70034e6 0.196074
\(597\) 1.22274e6 0.140410
\(598\) 4.08277e6 0.466877
\(599\) −1.37958e7 −1.57101 −0.785507 0.618853i \(-0.787598\pi\)
−0.785507 + 0.618853i \(0.787598\pi\)
\(600\) 2.16532e6 0.245552
\(601\) −4.99695e6 −0.564311 −0.282155 0.959369i \(-0.591049\pi\)
−0.282155 + 0.959369i \(0.591049\pi\)
\(602\) 0 0
\(603\) 4.09683e6 0.458833
\(604\) 1.25126e6 0.139558
\(605\) −6.62942e6 −0.736355
\(606\) 9.71962e6 1.07515
\(607\) 3.04946e6 0.335932 0.167966 0.985793i \(-0.446280\pi\)
0.167966 + 0.985793i \(0.446280\pi\)
\(608\) 2.93387e6 0.321871
\(609\) 0 0
\(610\) 1.01706e7 1.10668
\(611\) −2.66782e6 −0.289104
\(612\) 951085. 0.102646
\(613\) −7.03625e6 −0.756293 −0.378147 0.925746i \(-0.623438\pi\)
−0.378147 + 0.925746i \(0.623438\pi\)
\(614\) 1.50639e7 1.61256
\(615\) −4.56749e6 −0.486956
\(616\) 0 0
\(617\) 1.00066e7 1.05822 0.529108 0.848554i \(-0.322527\pi\)
0.529108 + 0.848554i \(0.322527\pi\)
\(618\) −6.48815e6 −0.683360
\(619\) 6.55067e6 0.687161 0.343581 0.939123i \(-0.388360\pi\)
0.343581 + 0.939123i \(0.388360\pi\)
\(620\) −1.33802e6 −0.139792
\(621\) −1.18979e6 −0.123805
\(622\) 9.17986e6 0.951393
\(623\) 0 0
\(624\) 4.32886e6 0.445054
\(625\) −2.04650e6 −0.209561
\(626\) 5.25758e6 0.536229
\(627\) 4.88909e6 0.496660
\(628\) −4.99598e6 −0.505501
\(629\) −2.16746e7 −2.18436
\(630\) 0 0
\(631\) 2.22672e6 0.222635 0.111317 0.993785i \(-0.464493\pi\)
0.111317 + 0.993785i \(0.464493\pi\)
\(632\) 9.34960e6 0.931109
\(633\) −8.92042e6 −0.884863
\(634\) −1.12905e7 −1.11555
\(635\) −3.22497e6 −0.317389
\(636\) −159849. −0.0156699
\(637\) 0 0
\(638\) 5.39031e6 0.524279
\(639\) 3.23762e6 0.313671
\(640\) −8.66484e6 −0.836201
\(641\) −1.59340e7 −1.53172 −0.765859 0.643008i \(-0.777686\pi\)
−0.765859 + 0.643008i \(0.777686\pi\)
\(642\) −1.28299e6 −0.122853
\(643\) 1.49933e7 1.43011 0.715056 0.699067i \(-0.246401\pi\)
0.715056 + 0.699067i \(0.246401\pi\)
\(644\) 0 0
\(645\) −5.13553e6 −0.486056
\(646\) −8.00783e6 −0.754977
\(647\) −1.29805e7 −1.21908 −0.609540 0.792755i \(-0.708646\pi\)
−0.609540 + 0.792755i \(0.708646\pi\)
\(648\) −971360. −0.0908747
\(649\) 2.96571e7 2.76386
\(650\) 4.06521e6 0.377398
\(651\) 0 0
\(652\) −3.79885e6 −0.349972
\(653\) −1.65492e7 −1.51878 −0.759389 0.650637i \(-0.774502\pi\)
−0.759389 + 0.650637i \(0.774502\pi\)
\(654\) −4.82036e6 −0.440692
\(655\) −6.46291e6 −0.588606
\(656\) −1.60993e7 −1.46066
\(657\) 4.51062e6 0.407683
\(658\) 0 0
\(659\) 5.86879e6 0.526423 0.263212 0.964738i \(-0.415218\pi\)
0.263212 + 0.964738i \(0.415218\pi\)
\(660\) 1.77409e6 0.158532
\(661\) −7.27687e6 −0.647800 −0.323900 0.946091i \(-0.604994\pi\)
−0.323900 + 0.946091i \(0.604994\pi\)
\(662\) −19503.6 −0.00172970
\(663\) −4.68509e6 −0.413937
\(664\) 6.74702e6 0.593870
\(665\) 0 0
\(666\) −8.43672e6 −0.737035
\(667\) 2.38862e6 0.207889
\(668\) 2.12164e6 0.183963
\(669\) −4.89010e6 −0.422428
\(670\) −1.25168e7 −1.07722
\(671\) 2.36884e7 2.03109
\(672\) 0 0
\(673\) −1.82417e7 −1.55248 −0.776241 0.630437i \(-0.782876\pi\)
−0.776241 + 0.630437i \(0.782876\pi\)
\(674\) 1.29675e7 1.09953
\(675\) −1.18467e6 −0.100078
\(676\) −1.92527e6 −0.162041
\(677\) 7.76406e6 0.651054 0.325527 0.945533i \(-0.394458\pi\)
0.325527 + 0.945533i \(0.394458\pi\)
\(678\) −2.35374e6 −0.196646
\(679\) 0 0
\(680\) 7.62432e6 0.632308
\(681\) −147.007 −1.21470e−5 0
\(682\) −1.44096e7 −1.18629
\(683\) 8.56324e6 0.702403 0.351201 0.936300i \(-0.385773\pi\)
0.351201 + 0.936300i \(0.385773\pi\)
\(684\) −674116. −0.0550927
\(685\) −1.48038e6 −0.120544
\(686\) 0 0
\(687\) 696411. 0.0562955
\(688\) −1.81015e7 −1.45796
\(689\) 787423. 0.0631917
\(690\) 3.63507e6 0.290663
\(691\) −1.64509e7 −1.31068 −0.655338 0.755336i \(-0.727474\pi\)
−0.655338 + 0.755336i \(0.727474\pi\)
\(692\) 1.58329e6 0.125689
\(693\) 0 0
\(694\) 2.41677e7 1.90474
\(695\) 4.11546e6 0.323189
\(696\) 1.95011e6 0.152593
\(697\) 1.74242e7 1.35853
\(698\) −1.86039e6 −0.144533
\(699\) −932916. −0.0722187
\(700\) 0 0
\(701\) −1.66928e7 −1.28302 −0.641512 0.767113i \(-0.721693\pi\)
−0.641512 + 0.767113i \(0.721693\pi\)
\(702\) −1.82365e6 −0.139669
\(703\) 1.53626e7 1.17240
\(704\) −1.11957e7 −0.851370
\(705\) −2.37528e6 −0.179987
\(706\) −2.46059e6 −0.185792
\(707\) 0 0
\(708\) −4.08916e6 −0.306585
\(709\) 5.61779e6 0.419711 0.209855 0.977732i \(-0.432701\pi\)
0.209855 + 0.977732i \(0.432701\pi\)
\(710\) −9.89167e6 −0.736417
\(711\) −5.11526e6 −0.379484
\(712\) 2.32242e6 0.171689
\(713\) −6.38537e6 −0.470395
\(714\) 0 0
\(715\) −8.73927e6 −0.639308
\(716\) −5.08242e6 −0.370500
\(717\) 6.20198e6 0.450539
\(718\) −2.06401e7 −1.49417
\(719\) −1.03718e7 −0.748227 −0.374113 0.927383i \(-0.622053\pi\)
−0.374113 + 0.927383i \(0.622053\pi\)
\(720\) 3.85418e6 0.277077
\(721\) 0 0
\(722\) −1.01461e7 −0.724363
\(723\) −1.98266e6 −0.141060
\(724\) −5.13052e6 −0.363760
\(725\) 2.37835e6 0.168047
\(726\) 9.84403e6 0.693156
\(727\) −1.15369e7 −0.809565 −0.404783 0.914413i \(-0.632653\pi\)
−0.404783 + 0.914413i \(0.632653\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) −1.37810e7 −0.957134
\(731\) 1.95911e7 1.35602
\(732\) −3.26619e6 −0.225301
\(733\) −1.49470e7 −1.02753 −0.513764 0.857932i \(-0.671749\pi\)
−0.513764 + 0.857932i \(0.671749\pi\)
\(734\) 3.06432e6 0.209939
\(735\) 0 0
\(736\) 5.08058e6 0.345715
\(737\) −2.91528e7 −1.97702
\(738\) 6.78227e6 0.458389
\(739\) 9.01364e6 0.607140 0.303570 0.952809i \(-0.401821\pi\)
0.303570 + 0.952809i \(0.401821\pi\)
\(740\) 5.57460e6 0.374227
\(741\) 3.32073e6 0.222171
\(742\) 0 0
\(743\) −2.10239e7 −1.39714 −0.698571 0.715541i \(-0.746180\pi\)
−0.698571 + 0.715541i \(0.746180\pi\)
\(744\) −5.21312e6 −0.345275
\(745\) −7.45750e6 −0.492269
\(746\) 5.57622e6 0.366854
\(747\) −3.69136e6 −0.242039
\(748\) −6.76785e6 −0.442279
\(749\) 0 0
\(750\) 1.05796e7 0.686779
\(751\) 4.04219e6 0.261527 0.130764 0.991414i \(-0.458257\pi\)
0.130764 + 0.991414i \(0.458257\pi\)
\(752\) −8.37230e6 −0.539884
\(753\) −1.29277e7 −0.830874
\(754\) 3.66117e6 0.234526
\(755\) −5.48786e6 −0.350377
\(756\) 0 0
\(757\) −1.82059e7 −1.15471 −0.577353 0.816495i \(-0.695914\pi\)
−0.577353 + 0.816495i \(0.695914\pi\)
\(758\) −1.55589e7 −0.983572
\(759\) 8.46642e6 0.533452
\(760\) −5.40402e6 −0.339377
\(761\) 1.89200e7 1.18429 0.592146 0.805831i \(-0.298281\pi\)
0.592146 + 0.805831i \(0.298281\pi\)
\(762\) 4.78876e6 0.298769
\(763\) 0 0
\(764\) −5.83301e6 −0.361542
\(765\) −4.17134e6 −0.257705
\(766\) 2.30782e7 1.42112
\(767\) 2.01434e7 1.23636
\(768\) 7.27238e6 0.444911
\(769\) 1.12831e7 0.688037 0.344019 0.938963i \(-0.388212\pi\)
0.344019 + 0.938963i \(0.388212\pi\)
\(770\) 0 0
\(771\) 8.18277e6 0.495752
\(772\) −4.92127e6 −0.297190
\(773\) 3.68565e6 0.221853 0.110926 0.993829i \(-0.464618\pi\)
0.110926 + 0.993829i \(0.464618\pi\)
\(774\) 7.62575e6 0.457541
\(775\) −6.35791e6 −0.380242
\(776\) 463176. 0.0276116
\(777\) 0 0
\(778\) −1.11971e6 −0.0663219
\(779\) −1.23500e7 −0.729161
\(780\) 1.20499e6 0.0709162
\(781\) −2.30387e7 −1.35154
\(782\) −1.38671e7 −0.810905
\(783\) −1.06692e6 −0.0621912
\(784\) 0 0
\(785\) 2.19118e7 1.26912
\(786\) 9.59677e6 0.554075
\(787\) 1.40748e7 0.810039 0.405019 0.914308i \(-0.367265\pi\)
0.405019 + 0.914308i \(0.367265\pi\)
\(788\) −6.72347e6 −0.385725
\(789\) 6.74151e6 0.385536
\(790\) 1.56283e7 0.890930
\(791\) 0 0
\(792\) 6.91213e6 0.391560
\(793\) 1.60894e7 0.908569
\(794\) −1.20459e7 −0.678090
\(795\) 701077. 0.0393412
\(796\) 1.19970e6 0.0671102
\(797\) 1.75191e7 0.976937 0.488469 0.872582i \(-0.337556\pi\)
0.488469 + 0.872582i \(0.337556\pi\)
\(798\) 0 0
\(799\) 9.06127e6 0.502137
\(800\) 5.05873e6 0.279458
\(801\) −1.27062e6 −0.0699737
\(802\) 8.56134e6 0.470008
\(803\) −3.20972e7 −1.75662
\(804\) 4.01963e6 0.219304
\(805\) 0 0
\(806\) −9.78721e6 −0.530666
\(807\) −6.02950e6 −0.325910
\(808\) −2.50222e7 −1.34833
\(809\) 814521. 0.0437553 0.0218777 0.999761i \(-0.493036\pi\)
0.0218777 + 0.999761i \(0.493036\pi\)
\(810\) −1.62367e6 −0.0869534
\(811\) −1.26533e7 −0.675540 −0.337770 0.941229i \(-0.609673\pi\)
−0.337770 + 0.941229i \(0.609673\pi\)
\(812\) 0 0
\(813\) −4.86593e6 −0.258190
\(814\) 6.00350e7 3.17573
\(815\) 1.66613e7 0.878647
\(816\) −1.47030e7 −0.773001
\(817\) −1.38859e7 −0.727813
\(818\) −4.20854e7 −2.19912
\(819\) 0 0
\(820\) −4.48142e6 −0.232745
\(821\) 4.27393e6 0.221294 0.110647 0.993860i \(-0.464708\pi\)
0.110647 + 0.993860i \(0.464708\pi\)
\(822\) 2.19821e6 0.113472
\(823\) 1.70463e7 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(824\) 1.67031e7 0.856996
\(825\) 8.43001e6 0.431214
\(826\) 0 0
\(827\) 2.60828e6 0.132614 0.0663071 0.997799i \(-0.478878\pi\)
0.0663071 + 0.997799i \(0.478878\pi\)
\(828\) −1.16736e6 −0.0591740
\(829\) −2.13865e7 −1.08082 −0.540409 0.841403i \(-0.681730\pi\)
−0.540409 + 0.841403i \(0.681730\pi\)
\(830\) 1.12780e7 0.568244
\(831\) −3.61719e6 −0.181706
\(832\) −7.60423e6 −0.380844
\(833\) 0 0
\(834\) −6.11104e6 −0.304229
\(835\) −9.30525e6 −0.461862
\(836\) 4.79696e6 0.237383
\(837\) 2.85215e6 0.140721
\(838\) 4.45290e7 2.19045
\(839\) −771393. −0.0378330 −0.0189165 0.999821i \(-0.506022\pi\)
−0.0189165 + 0.999821i \(0.506022\pi\)
\(840\) 0 0
\(841\) −1.83692e7 −0.895571
\(842\) 2.45397e7 1.19286
\(843\) −3.86225e6 −0.187185
\(844\) −8.75231e6 −0.422928
\(845\) 8.44401e6 0.406824
\(846\) 3.52705e6 0.169428
\(847\) 0 0
\(848\) 2.47113e6 0.118006
\(849\) 3.06834e6 0.146095
\(850\) −1.38075e7 −0.655492
\(851\) 2.66034e7 1.25926
\(852\) 3.17661e6 0.149922
\(853\) −2.94032e7 −1.38364 −0.691818 0.722072i \(-0.743190\pi\)
−0.691818 + 0.722072i \(0.743190\pi\)
\(854\) 0 0
\(855\) 2.95659e6 0.138317
\(856\) 3.30293e6 0.154069
\(857\) 2.65010e7 1.23257 0.616283 0.787525i \(-0.288638\pi\)
0.616283 + 0.787525i \(0.288638\pi\)
\(858\) 1.29770e7 0.601803
\(859\) 3.67519e7 1.69940 0.849702 0.527264i \(-0.176782\pi\)
0.849702 + 0.527264i \(0.176782\pi\)
\(860\) −5.03876e6 −0.232315
\(861\) 0 0
\(862\) 1.94020e7 0.889360
\(863\) 1.18189e7 0.540196 0.270098 0.962833i \(-0.412944\pi\)
0.270098 + 0.962833i \(0.412944\pi\)
\(864\) −2.26934e6 −0.103423
\(865\) −6.94413e6 −0.315557
\(866\) 6.01625e6 0.272603
\(867\) 3.13421e6 0.141606
\(868\) 0 0
\(869\) 3.63998e7 1.63512
\(870\) 3.25970e6 0.146009
\(871\) −1.98009e7 −0.884382
\(872\) 1.24095e7 0.552668
\(873\) −253408. −0.0112534
\(874\) 9.82884e6 0.435235
\(875\) 0 0
\(876\) 4.42562e6 0.194856
\(877\) 7.93509e6 0.348380 0.174190 0.984712i \(-0.444269\pi\)
0.174190 + 0.984712i \(0.444269\pi\)
\(878\) −8.56939e6 −0.375158
\(879\) 3.49963e6 0.152774
\(880\) −2.74260e7 −1.19387
\(881\) −4.20152e7 −1.82375 −0.911877 0.410464i \(-0.865367\pi\)
−0.911877 + 0.410464i \(0.865367\pi\)
\(882\) 0 0
\(883\) 2.12461e7 0.917016 0.458508 0.888690i \(-0.348384\pi\)
0.458508 + 0.888690i \(0.348384\pi\)
\(884\) −4.59681e6 −0.197845
\(885\) 1.79346e7 0.769720
\(886\) −4.93516e6 −0.211211
\(887\) −1.88490e7 −0.804415 −0.402208 0.915548i \(-0.631757\pi\)
−0.402208 + 0.915548i \(0.631757\pi\)
\(888\) 2.17195e7 0.924309
\(889\) 0 0
\(890\) 3.88204e6 0.164280
\(891\) −3.78169e6 −0.159585
\(892\) −4.79795e6 −0.201903
\(893\) −6.42251e6 −0.269511
\(894\) 1.10736e7 0.463390
\(895\) 2.22909e7 0.930186
\(896\) 0 0
\(897\) 5.75050e6 0.238630
\(898\) 1.44209e7 0.596762
\(899\) −5.72600e6 −0.236294
\(900\) −1.16234e6 −0.0478331
\(901\) −2.67448e6 −0.109756
\(902\) −4.82621e7 −1.97510
\(903\) 0 0
\(904\) 6.05948e6 0.246612
\(905\) 2.25018e7 0.913264
\(906\) 8.14892e6 0.329822
\(907\) −6.19446e6 −0.250026 −0.125013 0.992155i \(-0.539897\pi\)
−0.125013 + 0.992155i \(0.539897\pi\)
\(908\) −144.237 −5.80578e−6 0
\(909\) 1.36899e7 0.549528
\(910\) 0 0
\(911\) −2.50171e7 −0.998712 −0.499356 0.866397i \(-0.666430\pi\)
−0.499356 + 0.866397i \(0.666430\pi\)
\(912\) 1.04213e7 0.414891
\(913\) 2.62674e7 1.04290
\(914\) 2.73787e7 1.08404
\(915\) 1.43251e7 0.565647
\(916\) 683288. 0.0269070
\(917\) 0 0
\(918\) 6.19403e6 0.242586
\(919\) 1.54992e7 0.605370 0.302685 0.953091i \(-0.402117\pi\)
0.302685 + 0.953091i \(0.402117\pi\)
\(920\) −9.35812e6 −0.364518
\(921\) 2.12172e7 0.824212
\(922\) 1.98381e7 0.768551
\(923\) −1.56481e7 −0.604587
\(924\) 0 0
\(925\) 2.64890e7 1.01791
\(926\) −2.25809e7 −0.865394
\(927\) −9.13843e6 −0.349279
\(928\) 4.55594e6 0.173663
\(929\) −3.75285e7 −1.42667 −0.713333 0.700826i \(-0.752815\pi\)
−0.713333 + 0.700826i \(0.752815\pi\)
\(930\) −8.71398e6 −0.330377
\(931\) 0 0
\(932\) −915335. −0.0345176
\(933\) 1.29297e7 0.486276
\(934\) −1.74098e7 −0.653020
\(935\) 2.96830e7 1.11040
\(936\) 4.69480e6 0.175157
\(937\) 1.08298e7 0.402969 0.201485 0.979492i \(-0.435423\pi\)
0.201485 + 0.979492i \(0.435423\pi\)
\(938\) 0 0
\(939\) 7.40520e6 0.274077
\(940\) −2.33052e6 −0.0860267
\(941\) −3.24591e7 −1.19498 −0.597492 0.801875i \(-0.703836\pi\)
−0.597492 + 0.801875i \(0.703836\pi\)
\(942\) −3.25368e7 −1.19467
\(943\) −2.13865e7 −0.783177
\(944\) 6.32151e7 2.30883
\(945\) 0 0
\(946\) −5.42643e7 −1.97145
\(947\) 7.53877e6 0.273165 0.136583 0.990629i \(-0.456388\pi\)
0.136583 + 0.990629i \(0.456388\pi\)
\(948\) −5.01886e6 −0.181378
\(949\) −2.18008e7 −0.785792
\(950\) 9.78656e6 0.351821
\(951\) −1.59024e7 −0.570180
\(952\) 0 0
\(953\) −3.01356e7 −1.07485 −0.537424 0.843312i \(-0.680602\pi\)
−0.537424 + 0.843312i \(0.680602\pi\)
\(954\) −1.04103e6 −0.0370333
\(955\) 2.55829e7 0.907697
\(956\) 6.08510e6 0.215339
\(957\) 7.59215e6 0.267969
\(958\) −6.25366e6 −0.220151
\(959\) 0 0
\(960\) −6.77038e6 −0.237102
\(961\) −1.33221e7 −0.465335
\(962\) 4.07765e7 1.42060
\(963\) −1.80707e6 −0.0627926
\(964\) −1.94530e6 −0.0674208
\(965\) 2.15841e7 0.746132
\(966\) 0 0
\(967\) 2.88021e6 0.0990509 0.0495255 0.998773i \(-0.484229\pi\)
0.0495255 + 0.998773i \(0.484229\pi\)
\(968\) −2.53425e7 −0.869282
\(969\) −1.12789e7 −0.385883
\(970\) 774220. 0.0264201
\(971\) 2.74490e7 0.934283 0.467142 0.884183i \(-0.345284\pi\)
0.467142 + 0.884183i \(0.345284\pi\)
\(972\) 521426. 0.0177022
\(973\) 0 0
\(974\) 2.50958e7 0.847626
\(975\) 5.72577e6 0.192896
\(976\) 5.04927e7 1.69669
\(977\) −5.88524e7 −1.97255 −0.986274 0.165118i \(-0.947200\pi\)
−0.986274 + 0.165118i \(0.947200\pi\)
\(978\) −2.47403e7 −0.827101
\(979\) 9.04164e6 0.301502
\(980\) 0 0
\(981\) −6.78938e6 −0.225246
\(982\) 1.68208e7 0.556631
\(983\) −1.86077e7 −0.614198 −0.307099 0.951678i \(-0.599358\pi\)
−0.307099 + 0.951678i \(0.599358\pi\)
\(984\) −1.74603e7 −0.574862
\(985\) 2.94883e7 0.968410
\(986\) −1.24352e7 −0.407342
\(987\) 0 0
\(988\) 3.25815e6 0.106189
\(989\) −2.40462e7 −0.781729
\(990\) 1.15539e7 0.374664
\(991\) 2.47154e7 0.799435 0.399718 0.916638i \(-0.369108\pi\)
0.399718 + 0.916638i \(0.369108\pi\)
\(992\) −1.21792e7 −0.392951
\(993\) −27470.5 −0.000884083 0
\(994\) 0 0
\(995\) −5.26172e6 −0.168488
\(996\) −3.62180e6 −0.115685
\(997\) 1.79581e7 0.572167 0.286084 0.958205i \(-0.407646\pi\)
0.286084 + 0.958205i \(0.407646\pi\)
\(998\) 1.35813e7 0.431633
\(999\) −1.18829e7 −0.376713
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 147.6.a.k.1.2 2
3.2 odd 2 441.6.a.s.1.1 2
7.2 even 3 147.6.e.l.67.1 4
7.3 odd 6 21.6.e.b.16.1 yes 4
7.4 even 3 147.6.e.l.79.1 4
7.5 odd 6 21.6.e.b.4.1 4
7.6 odd 2 147.6.a.i.1.2 2
21.5 even 6 63.6.e.c.46.2 4
21.17 even 6 63.6.e.c.37.2 4
21.20 even 2 441.6.a.t.1.1 2
28.3 even 6 336.6.q.e.289.1 4
28.19 even 6 336.6.q.e.193.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.1 4 7.5 odd 6
21.6.e.b.16.1 yes 4 7.3 odd 6
63.6.e.c.37.2 4 21.17 even 6
63.6.e.c.46.2 4 21.5 even 6
147.6.a.i.1.2 2 7.6 odd 2
147.6.a.k.1.2 2 1.1 even 1 trivial
147.6.e.l.67.1 4 7.2 even 3
147.6.e.l.79.1 4 7.4 even 3
336.6.q.e.193.1 4 28.19 even 6
336.6.q.e.289.1 4 28.3 even 6
441.6.a.s.1.1 2 3.2 odd 2
441.6.a.t.1.1 2 21.20 even 2