Properties

Label 49.6.a.g.1.3
Level $49$
Weight $6$
Character 49.1
Self dual yes
Analytic conductor $7.859$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.85880717084\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
Defining polynomial: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.40086\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.81507 q^{2} -6.54802 q^{3} -24.0754 q^{4} +45.9910 q^{5} -18.4331 q^{6} -157.856 q^{8} -200.123 q^{9} +O(q^{10})\) \(q+2.81507 q^{2} -6.54802 q^{3} -24.0754 q^{4} +45.9910 q^{5} -18.4331 q^{6} -157.856 q^{8} -200.123 q^{9} +129.468 q^{10} -551.781 q^{11} +157.646 q^{12} -1094.10 q^{13} -301.150 q^{15} +326.035 q^{16} +1180.71 q^{17} -563.362 q^{18} +1166.13 q^{19} -1107.25 q^{20} -1553.30 q^{22} +44.3851 q^{23} +1033.65 q^{24} -1009.82 q^{25} -3079.97 q^{26} +2901.58 q^{27} +3329.02 q^{29} -847.759 q^{30} -8784.01 q^{31} +5969.21 q^{32} +3613.07 q^{33} +3323.80 q^{34} +4818.05 q^{36} -2557.12 q^{37} +3282.73 q^{38} +7164.17 q^{39} -7259.97 q^{40} +12761.3 q^{41} -96.7714 q^{43} +13284.3 q^{44} -9203.89 q^{45} +124.947 q^{46} -7679.15 q^{47} -2134.88 q^{48} -2842.73 q^{50} -7731.33 q^{51} +26340.8 q^{52} -11953.3 q^{53} +8168.16 q^{54} -25377.0 q^{55} -7635.81 q^{57} +9371.43 q^{58} -9857.24 q^{59} +7250.30 q^{60} +38517.9 q^{61} -24727.6 q^{62} +6370.65 q^{64} -50318.7 q^{65} +10171.1 q^{66} -67548.9 q^{67} -28426.1 q^{68} -290.634 q^{69} -61374.6 q^{71} +31590.7 q^{72} -1850.40 q^{73} -7198.49 q^{74} +6612.34 q^{75} -28074.9 q^{76} +20167.7 q^{78} -8.52913 q^{79} +14994.7 q^{80} +29630.4 q^{81} +35923.9 q^{82} +95039.3 q^{83} +54302.3 q^{85} -272.419 q^{86} -21798.5 q^{87} +87102.1 q^{88} -53605.6 q^{89} -25909.6 q^{90} -1068.59 q^{92} +57517.9 q^{93} -21617.4 q^{94} +53631.4 q^{95} -39086.5 q^{96} -3110.79 q^{97} +110424. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9} - 1952 q^{11} - 4096 q^{15} - 1566 q^{16} - 5974 q^{18} + 3524 q^{22} - 7136 q^{23} + 2764 q^{25} - 3352 q^{29} + 25608 q^{30} + 27810 q^{32} + 27670 q^{36} - 9208 q^{37} + 2464 q^{39} + 20448 q^{43} + 1900 q^{44} + 56712 q^{46} - 43070 q^{50} - 67408 q^{51} - 102920 q^{53} - 15576 q^{57} + 96972 q^{58} - 87080 q^{60} - 40318 q^{64} - 63168 q^{65} - 22896 q^{67} - 153824 q^{71} + 77358 q^{72} + 17596 q^{74} + 133056 q^{78} - 90688 q^{79} - 17204 q^{81} + 272656 q^{85} - 161860 q^{86} + 154812 q^{88} - 212200 q^{92} + 247760 q^{93} + 108224 q^{95} - 42272 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.81507 0.497639 0.248820 0.968550i \(-0.419957\pi\)
0.248820 + 0.968550i \(0.419957\pi\)
\(3\) −6.54802 −0.420055 −0.210028 0.977695i \(-0.567355\pi\)
−0.210028 + 0.977695i \(0.567355\pi\)
\(4\) −24.0754 −0.752355
\(5\) 45.9910 0.822713 0.411356 0.911475i \(-0.365055\pi\)
0.411356 + 0.911475i \(0.365055\pi\)
\(6\) −18.4331 −0.209036
\(7\) 0 0
\(8\) −157.856 −0.872041
\(9\) −200.123 −0.823553
\(10\) 129.468 0.409414
\(11\) −551.781 −1.37494 −0.687472 0.726211i \(-0.741280\pi\)
−0.687472 + 0.726211i \(0.741280\pi\)
\(12\) 157.646 0.316031
\(13\) −1094.10 −1.79555 −0.897776 0.440453i \(-0.854818\pi\)
−0.897776 + 0.440453i \(0.854818\pi\)
\(14\) 0 0
\(15\) −301.150 −0.345585
\(16\) 326.035 0.318393
\(17\) 1180.71 0.990883 0.495442 0.868641i \(-0.335006\pi\)
0.495442 + 0.868641i \(0.335006\pi\)
\(18\) −563.362 −0.409833
\(19\) 1166.13 0.741074 0.370537 0.928818i \(-0.379174\pi\)
0.370537 + 0.928818i \(0.379174\pi\)
\(20\) −1107.25 −0.618972
\(21\) 0 0
\(22\) −1553.30 −0.684226
\(23\) 44.3851 0.0174951 0.00874757 0.999962i \(-0.497216\pi\)
0.00874757 + 0.999962i \(0.497216\pi\)
\(24\) 1033.65 0.366305
\(25\) −1009.82 −0.323143
\(26\) −3079.97 −0.893537
\(27\) 2901.58 0.765993
\(28\) 0 0
\(29\) 3329.02 0.735057 0.367529 0.930012i \(-0.380204\pi\)
0.367529 + 0.930012i \(0.380204\pi\)
\(30\) −847.759 −0.171977
\(31\) −8784.01 −1.64168 −0.820841 0.571157i \(-0.806495\pi\)
−0.820841 + 0.571157i \(0.806495\pi\)
\(32\) 5969.21 1.03049
\(33\) 3613.07 0.577552
\(34\) 3323.80 0.493102
\(35\) 0 0
\(36\) 4818.05 0.619605
\(37\) −2557.12 −0.307077 −0.153539 0.988143i \(-0.549067\pi\)
−0.153539 + 0.988143i \(0.549067\pi\)
\(38\) 3282.73 0.368787
\(39\) 7164.17 0.754231
\(40\) −7259.97 −0.717439
\(41\) 12761.3 1.18559 0.592794 0.805354i \(-0.298025\pi\)
0.592794 + 0.805354i \(0.298025\pi\)
\(42\) 0 0
\(43\) −96.7714 −0.00798135 −0.00399067 0.999992i \(-0.501270\pi\)
−0.00399067 + 0.999992i \(0.501270\pi\)
\(44\) 13284.3 1.03445
\(45\) −9203.89 −0.677548
\(46\) 124.947 0.00870627
\(47\) −7679.15 −0.507071 −0.253535 0.967326i \(-0.581593\pi\)
−0.253535 + 0.967326i \(0.581593\pi\)
\(48\) −2134.88 −0.133743
\(49\) 0 0
\(50\) −2842.73 −0.160809
\(51\) −7731.33 −0.416226
\(52\) 26340.8 1.35089
\(53\) −11953.3 −0.584520 −0.292260 0.956339i \(-0.594407\pi\)
−0.292260 + 0.956339i \(0.594407\pi\)
\(54\) 8168.16 0.381188
\(55\) −25377.0 −1.13118
\(56\) 0 0
\(57\) −7635.81 −0.311292
\(58\) 9371.43 0.365793
\(59\) −9857.24 −0.368659 −0.184330 0.982864i \(-0.559011\pi\)
−0.184330 + 0.982864i \(0.559011\pi\)
\(60\) 7250.30 0.260003
\(61\) 38517.9 1.32537 0.662686 0.748897i \(-0.269416\pi\)
0.662686 + 0.748897i \(0.269416\pi\)
\(62\) −24727.6 −0.816965
\(63\) 0 0
\(64\) 6370.65 0.194417
\(65\) −50318.7 −1.47722
\(66\) 10171.1 0.287413
\(67\) −67548.9 −1.83836 −0.919182 0.393833i \(-0.871149\pi\)
−0.919182 + 0.393833i \(0.871149\pi\)
\(68\) −28426.1 −0.745496
\(69\) −290.634 −0.00734893
\(70\) 0 0
\(71\) −61374.6 −1.44492 −0.722458 0.691415i \(-0.756988\pi\)
−0.722458 + 0.691415i \(0.756988\pi\)
\(72\) 31590.7 0.718172
\(73\) −1850.40 −0.0406404 −0.0203202 0.999794i \(-0.506469\pi\)
−0.0203202 + 0.999794i \(0.506469\pi\)
\(74\) −7198.49 −0.152814
\(75\) 6612.34 0.135738
\(76\) −28074.9 −0.557551
\(77\) 0 0
\(78\) 20167.7 0.375335
\(79\) −8.52913 −0.000153758 0 −7.68788e−5 1.00000i \(-0.500024\pi\)
−7.68788e−5 1.00000i \(0.500024\pi\)
\(80\) 14994.7 0.261946
\(81\) 29630.4 0.501794
\(82\) 35923.9 0.589996
\(83\) 95039.3 1.51429 0.757143 0.653249i \(-0.226595\pi\)
0.757143 + 0.653249i \(0.226595\pi\)
\(84\) 0 0
\(85\) 54302.3 0.815212
\(86\) −272.419 −0.00397183
\(87\) −21798.5 −0.308765
\(88\) 87102.1 1.19901
\(89\) −53605.6 −0.717357 −0.358678 0.933461i \(-0.616773\pi\)
−0.358678 + 0.933461i \(0.616773\pi\)
\(90\) −25909.6 −0.337175
\(91\) 0 0
\(92\) −1068.59 −0.0131626
\(93\) 57517.9 0.689597
\(94\) −21617.4 −0.252338
\(95\) 53631.4 0.609691
\(96\) −39086.5 −0.432861
\(97\) −3110.79 −0.0335693 −0.0167846 0.999859i \(-0.505343\pi\)
−0.0167846 + 0.999859i \(0.505343\pi\)
\(98\) 0 0
\(99\) 110424. 1.13234
\(100\) 24311.9 0.243119
\(101\) −21835.9 −0.212994 −0.106497 0.994313i \(-0.533963\pi\)
−0.106497 + 0.994313i \(0.533963\pi\)
\(102\) −21764.3 −0.207130
\(103\) −65341.4 −0.606870 −0.303435 0.952852i \(-0.598134\pi\)
−0.303435 + 0.952852i \(0.598134\pi\)
\(104\) 172710. 1.56579
\(105\) 0 0
\(106\) −33649.5 −0.290880
\(107\) −108957. −0.920020 −0.460010 0.887914i \(-0.652154\pi\)
−0.460010 + 0.887914i \(0.652154\pi\)
\(108\) −69856.6 −0.576299
\(109\) 86728.7 0.699192 0.349596 0.936901i \(-0.386319\pi\)
0.349596 + 0.936901i \(0.386319\pi\)
\(110\) −71438.1 −0.562922
\(111\) 16744.1 0.128989
\(112\) 0 0
\(113\) −101496. −0.747746 −0.373873 0.927480i \(-0.621970\pi\)
−0.373873 + 0.927480i \(0.621970\pi\)
\(114\) −21495.4 −0.154911
\(115\) 2041.32 0.0143935
\(116\) −80147.3 −0.553024
\(117\) 218955. 1.47873
\(118\) −27748.9 −0.183459
\(119\) 0 0
\(120\) 47538.4 0.301364
\(121\) 143411. 0.890470
\(122\) 108431. 0.659557
\(123\) −83560.9 −0.498013
\(124\) 211478. 1.23513
\(125\) −190165. −1.08857
\(126\) 0 0
\(127\) −3094.61 −0.0170253 −0.00851267 0.999964i \(-0.502710\pi\)
−0.00851267 + 0.999964i \(0.502710\pi\)
\(128\) −173081. −0.933736
\(129\) 633.661 0.00335261
\(130\) −141651. −0.735124
\(131\) −253431. −1.29027 −0.645136 0.764067i \(-0.723199\pi\)
−0.645136 + 0.764067i \(0.723199\pi\)
\(132\) −86986.0 −0.434525
\(133\) 0 0
\(134\) −190155. −0.914842
\(135\) 133447. 0.630193
\(136\) −186383. −0.864091
\(137\) −97152.9 −0.442236 −0.221118 0.975247i \(-0.570971\pi\)
−0.221118 + 0.975247i \(0.570971\pi\)
\(138\) −818.156 −0.00365711
\(139\) −210308. −0.923249 −0.461624 0.887076i \(-0.652733\pi\)
−0.461624 + 0.887076i \(0.652733\pi\)
\(140\) 0 0
\(141\) 50283.2 0.212998
\(142\) −172774. −0.719047
\(143\) 603702. 2.46878
\(144\) −65247.2 −0.262214
\(145\) 153105. 0.604741
\(146\) −5209.01 −0.0202243
\(147\) 0 0
\(148\) 61563.7 0.231031
\(149\) 140406. 0.518109 0.259055 0.965863i \(-0.416589\pi\)
0.259055 + 0.965863i \(0.416589\pi\)
\(150\) 18614.2 0.0675486
\(151\) 119696. 0.427205 0.213603 0.976921i \(-0.431480\pi\)
0.213603 + 0.976921i \(0.431480\pi\)
\(152\) −184080. −0.646247
\(153\) −236289. −0.816045
\(154\) 0 0
\(155\) −403986. −1.35063
\(156\) −172480. −0.567450
\(157\) −97616.9 −0.316065 −0.158032 0.987434i \(-0.550515\pi\)
−0.158032 + 0.987434i \(0.550515\pi\)
\(158\) −24.0101 −7.65159e−5 0
\(159\) 78270.6 0.245531
\(160\) 274530. 0.847794
\(161\) 0 0
\(162\) 83411.8 0.249712
\(163\) 182678. 0.538539 0.269270 0.963065i \(-0.413218\pi\)
0.269270 + 0.963065i \(0.413218\pi\)
\(164\) −307232. −0.891984
\(165\) 166169. 0.475160
\(166\) 267542. 0.753568
\(167\) 451674. 1.25324 0.626619 0.779326i \(-0.284438\pi\)
0.626619 + 0.779326i \(0.284438\pi\)
\(168\) 0 0
\(169\) 825757. 2.22400
\(170\) 152865. 0.405682
\(171\) −233369. −0.610314
\(172\) 2329.81 0.00600481
\(173\) −371647. −0.944095 −0.472047 0.881573i \(-0.656485\pi\)
−0.472047 + 0.881573i \(0.656485\pi\)
\(174\) −61364.2 −0.153653
\(175\) 0 0
\(176\) −179900. −0.437773
\(177\) 64545.4 0.154857
\(178\) −150904. −0.356985
\(179\) 85003.4 0.198291 0.0991457 0.995073i \(-0.468389\pi\)
0.0991457 + 0.995073i \(0.468389\pi\)
\(180\) 221587. 0.509757
\(181\) 379442. 0.860892 0.430446 0.902616i \(-0.358356\pi\)
0.430446 + 0.902616i \(0.358356\pi\)
\(182\) 0 0
\(183\) −252216. −0.556730
\(184\) −7006.46 −0.0152565
\(185\) −117605. −0.252636
\(186\) 161917. 0.343171
\(187\) −651496. −1.36241
\(188\) 184878. 0.381497
\(189\) 0 0
\(190\) 150976. 0.303406
\(191\) −922196. −1.82911 −0.914555 0.404462i \(-0.867459\pi\)
−0.914555 + 0.404462i \(0.867459\pi\)
\(192\) −41715.1 −0.0816658
\(193\) 505107. 0.976090 0.488045 0.872818i \(-0.337710\pi\)
0.488045 + 0.872818i \(0.337710\pi\)
\(194\) −8757.11 −0.0167054
\(195\) 329488. 0.620516
\(196\) 0 0
\(197\) 251505. 0.461723 0.230861 0.972987i \(-0.425846\pi\)
0.230861 + 0.972987i \(0.425846\pi\)
\(198\) 310853. 0.563497
\(199\) 208033. 0.372392 0.186196 0.982513i \(-0.440384\pi\)
0.186196 + 0.982513i \(0.440384\pi\)
\(200\) 159407. 0.281794
\(201\) 442311. 0.772215
\(202\) −61469.7 −0.105994
\(203\) 0 0
\(204\) 186135. 0.313150
\(205\) 586904. 0.975399
\(206\) −183941. −0.302002
\(207\) −8882.50 −0.0144082
\(208\) −356714. −0.571692
\(209\) −643446. −1.01893
\(210\) 0 0
\(211\) −640577. −0.990525 −0.495262 0.868744i \(-0.664928\pi\)
−0.495262 + 0.868744i \(0.664928\pi\)
\(212\) 287781. 0.439767
\(213\) 401882. 0.606945
\(214\) −306723. −0.457838
\(215\) −4450.62 −0.00656636
\(216\) −458032. −0.667977
\(217\) 0 0
\(218\) 244148. 0.347945
\(219\) 12116.4 0.0170712
\(220\) 610960. 0.851052
\(221\) −1.29182e6 −1.77918
\(222\) 47135.8 0.0641902
\(223\) −390135. −0.525354 −0.262677 0.964884i \(-0.584605\pi\)
−0.262677 + 0.964884i \(0.584605\pi\)
\(224\) 0 0
\(225\) 202089. 0.266126
\(226\) −285720. −0.372108
\(227\) −291353. −0.375279 −0.187639 0.982238i \(-0.560084\pi\)
−0.187639 + 0.982238i \(0.560084\pi\)
\(228\) 183835. 0.234202
\(229\) 1.23040e6 1.55045 0.775227 0.631682i \(-0.217635\pi\)
0.775227 + 0.631682i \(0.217635\pi\)
\(230\) 5746.45 0.00716276
\(231\) 0 0
\(232\) −525506. −0.641000
\(233\) 114279. 0.137903 0.0689517 0.997620i \(-0.478035\pi\)
0.0689517 + 0.997620i \(0.478035\pi\)
\(234\) 616373. 0.735875
\(235\) −353172. −0.417174
\(236\) 237317. 0.277363
\(237\) 55.8489 6.45867e−5 0
\(238\) 0 0
\(239\) −1.14782e6 −1.29981 −0.649906 0.760014i \(-0.725192\pi\)
−0.649906 + 0.760014i \(0.725192\pi\)
\(240\) −98185.4 −0.110032
\(241\) −812708. −0.901346 −0.450673 0.892689i \(-0.648816\pi\)
−0.450673 + 0.892689i \(0.648816\pi\)
\(242\) 403713. 0.443133
\(243\) −899104. −0.976775
\(244\) −927332. −0.997150
\(245\) 0 0
\(246\) −235230. −0.247831
\(247\) −1.27586e6 −1.33064
\(248\) 1.38661e6 1.43161
\(249\) −622318. −0.636084
\(250\) −535328. −0.541714
\(251\) 406772. 0.407537 0.203768 0.979019i \(-0.434681\pi\)
0.203768 + 0.979019i \(0.434681\pi\)
\(252\) 0 0
\(253\) −24490.8 −0.0240548
\(254\) −8711.54 −0.00847248
\(255\) −355572. −0.342434
\(256\) −691096. −0.659081
\(257\) 1.69712e6 1.60281 0.801403 0.598125i \(-0.204087\pi\)
0.801403 + 0.598125i \(0.204087\pi\)
\(258\) 1783.80 0.00166839
\(259\) 0 0
\(260\) 1.21144e6 1.11140
\(261\) −666215. −0.605359
\(262\) −713427. −0.642090
\(263\) 205694. 0.183372 0.0916859 0.995788i \(-0.470774\pi\)
0.0916859 + 0.995788i \(0.470774\pi\)
\(264\) −570346. −0.503649
\(265\) −549746. −0.480892
\(266\) 0 0
\(267\) 351010. 0.301330
\(268\) 1.62627e6 1.38310
\(269\) −1.73425e6 −1.46127 −0.730635 0.682769i \(-0.760776\pi\)
−0.730635 + 0.682769i \(0.760776\pi\)
\(270\) 375662. 0.313609
\(271\) −369042. −0.305248 −0.152624 0.988284i \(-0.548772\pi\)
−0.152624 + 0.988284i \(0.548772\pi\)
\(272\) 384954. 0.315491
\(273\) 0 0
\(274\) −273492. −0.220074
\(275\) 557201. 0.444304
\(276\) 6997.12 0.00552900
\(277\) 1.22767e6 0.961351 0.480676 0.876899i \(-0.340391\pi\)
0.480676 + 0.876899i \(0.340391\pi\)
\(278\) −592032. −0.459445
\(279\) 1.75789e6 1.35201
\(280\) 0 0
\(281\) 2.00671e6 1.51607 0.758035 0.652214i \(-0.226159\pi\)
0.758035 + 0.652214i \(0.226159\pi\)
\(282\) 141551. 0.105996
\(283\) 1.78581e6 1.32547 0.662734 0.748855i \(-0.269396\pi\)
0.662734 + 0.748855i \(0.269396\pi\)
\(284\) 1.47762e6 1.08709
\(285\) −351179. −0.256104
\(286\) 1.69947e6 1.22856
\(287\) 0 0
\(288\) −1.19458e6 −0.848660
\(289\) −25770.8 −0.0181503
\(290\) 431002. 0.300943
\(291\) 20369.5 0.0141009
\(292\) 44549.0 0.0305760
\(293\) −853248. −0.580639 −0.290319 0.956930i \(-0.593762\pi\)
−0.290319 + 0.956930i \(0.593762\pi\)
\(294\) 0 0
\(295\) −453345. −0.303301
\(296\) 403658. 0.267784
\(297\) −1.60104e6 −1.05320
\(298\) 395254. 0.257831
\(299\) −48561.6 −0.0314134
\(300\) −159194. −0.102123
\(301\) 0 0
\(302\) 336952. 0.212594
\(303\) 142982. 0.0894693
\(304\) 380198. 0.235953
\(305\) 1.77148e6 1.09040
\(306\) −665170. −0.406096
\(307\) 1.96068e6 1.18730 0.593652 0.804722i \(-0.297686\pi\)
0.593652 + 0.804722i \(0.297686\pi\)
\(308\) 0 0
\(309\) 427857. 0.254919
\(310\) −1.13725e6 −0.672128
\(311\) −863604. −0.506307 −0.253153 0.967426i \(-0.581468\pi\)
−0.253153 + 0.967426i \(0.581468\pi\)
\(312\) −1.13091e6 −0.657720
\(313\) −1.10047e6 −0.634918 −0.317459 0.948272i \(-0.602830\pi\)
−0.317459 + 0.948272i \(0.602830\pi\)
\(314\) −274799. −0.157286
\(315\) 0 0
\(316\) 205.342 0.000115680 0
\(317\) 1.49591e6 0.836097 0.418048 0.908425i \(-0.362714\pi\)
0.418048 + 0.908425i \(0.362714\pi\)
\(318\) 220337. 0.122186
\(319\) −1.83689e6 −1.01066
\(320\) 292993. 0.159949
\(321\) 713454. 0.386459
\(322\) 0 0
\(323\) 1.37686e6 0.734318
\(324\) −713363. −0.377527
\(325\) 1.10485e6 0.580221
\(326\) 514252. 0.267998
\(327\) −567901. −0.293699
\(328\) −2.01445e6 −1.03388
\(329\) 0 0
\(330\) 467777. 0.236458
\(331\) −2.74015e6 −1.37469 −0.687345 0.726331i \(-0.741224\pi\)
−0.687345 + 0.726331i \(0.741224\pi\)
\(332\) −2.28810e6 −1.13928
\(333\) 511741. 0.252894
\(334\) 1.27149e6 0.623661
\(335\) −3.10665e6 −1.51245
\(336\) 0 0
\(337\) −2.31353e6 −1.10968 −0.554842 0.831956i \(-0.687221\pi\)
−0.554842 + 0.831956i \(0.687221\pi\)
\(338\) 2.32457e6 1.10675
\(339\) 664600. 0.314095
\(340\) −1.30735e6 −0.613329
\(341\) 4.84685e6 2.25722
\(342\) −656951. −0.303716
\(343\) 0 0
\(344\) 15276.0 0.00696006
\(345\) −13366.6 −0.00604606
\(346\) −1.04621e6 −0.469819
\(347\) −3.05926e6 −1.36393 −0.681966 0.731384i \(-0.738875\pi\)
−0.681966 + 0.731384i \(0.738875\pi\)
\(348\) 524806. 0.232301
\(349\) 210232. 0.0923921 0.0461961 0.998932i \(-0.485290\pi\)
0.0461961 + 0.998932i \(0.485290\pi\)
\(350\) 0 0
\(351\) −3.17461e6 −1.37538
\(352\) −3.29370e6 −1.41686
\(353\) 3.76790e6 1.60939 0.804697 0.593686i \(-0.202328\pi\)
0.804697 + 0.593686i \(0.202328\pi\)
\(354\) 181700. 0.0770631
\(355\) −2.82268e6 −1.18875
\(356\) 1.29057e6 0.539707
\(357\) 0 0
\(358\) 239291. 0.0986776
\(359\) 1.00722e6 0.412465 0.206232 0.978503i \(-0.433880\pi\)
0.206232 + 0.978503i \(0.433880\pi\)
\(360\) 1.45289e6 0.590850
\(361\) −1.11625e6 −0.450809
\(362\) 1.06816e6 0.428414
\(363\) −939058. −0.374047
\(364\) 0 0
\(365\) −85101.8 −0.0334354
\(366\) −710005. −0.277050
\(367\) 1.52650e6 0.591603 0.295802 0.955249i \(-0.404413\pi\)
0.295802 + 0.955249i \(0.404413\pi\)
\(368\) 14471.1 0.00557034
\(369\) −2.55383e6 −0.976396
\(370\) −331066. −0.125722
\(371\) 0 0
\(372\) −1.38476e6 −0.518822
\(373\) 4.86297e6 1.80980 0.904898 0.425629i \(-0.139947\pi\)
0.904898 + 0.425629i \(0.139947\pi\)
\(374\) −1.83401e6 −0.677988
\(375\) 1.24520e6 0.457258
\(376\) 1.21220e6 0.442186
\(377\) −3.64227e6 −1.31983
\(378\) 0 0
\(379\) 630878. 0.225604 0.112802 0.993617i \(-0.464017\pi\)
0.112802 + 0.993617i \(0.464017\pi\)
\(380\) −1.29119e6 −0.458704
\(381\) 20263.5 0.00715159
\(382\) −2.59605e6 −0.910237
\(383\) −565644. −0.197036 −0.0985182 0.995135i \(-0.531410\pi\)
−0.0985182 + 0.995135i \(0.531410\pi\)
\(384\) 1.13334e6 0.392221
\(385\) 0 0
\(386\) 1.42191e6 0.485741
\(387\) 19366.2 0.00657306
\(388\) 74893.5 0.0252560
\(389\) 592212. 0.198428 0.0992140 0.995066i \(-0.468367\pi\)
0.0992140 + 0.995066i \(0.468367\pi\)
\(390\) 927532. 0.308793
\(391\) 52406.1 0.0173356
\(392\) 0 0
\(393\) 1.65947e6 0.541986
\(394\) 708005. 0.229771
\(395\) −392.264 −0.000126498 0
\(396\) −2.65851e6 −0.851922
\(397\) −1.34312e6 −0.427698 −0.213849 0.976867i \(-0.568600\pi\)
−0.213849 + 0.976867i \(0.568600\pi\)
\(398\) 585629. 0.185317
\(399\) 0 0
\(400\) −329238. −0.102887
\(401\) −3.68716e6 −1.14507 −0.572534 0.819881i \(-0.694040\pi\)
−0.572534 + 0.819881i \(0.694040\pi\)
\(402\) 1.24514e6 0.384284
\(403\) 9.61057e6 2.94772
\(404\) 525707. 0.160247
\(405\) 1.36273e6 0.412832
\(406\) 0 0
\(407\) 1.41097e6 0.422214
\(408\) 1.22044e6 0.362966
\(409\) −1.45630e6 −0.430470 −0.215235 0.976562i \(-0.569052\pi\)
−0.215235 + 0.976562i \(0.569052\pi\)
\(410\) 1.65218e6 0.485397
\(411\) 636159. 0.185764
\(412\) 1.57312e6 0.456582
\(413\) 0 0
\(414\) −25004.9 −0.00717008
\(415\) 4.37096e6 1.24582
\(416\) −6.53090e6 −1.85029
\(417\) 1.37710e6 0.387816
\(418\) −1.81135e6 −0.507062
\(419\) 2.92192e6 0.813080 0.406540 0.913633i \(-0.366735\pi\)
0.406540 + 0.913633i \(0.366735\pi\)
\(420\) 0 0
\(421\) 2.01999e6 0.555450 0.277725 0.960661i \(-0.410420\pi\)
0.277725 + 0.960661i \(0.410420\pi\)
\(422\) −1.80327e6 −0.492924
\(423\) 1.53678e6 0.417600
\(424\) 1.88691e6 0.509725
\(425\) −1.19231e6 −0.320197
\(426\) 1.13133e6 0.302040
\(427\) 0 0
\(428\) 2.62319e6 0.692182
\(429\) −3.95305e6 −1.03703
\(430\) −12528.8 −0.00326768
\(431\) −5.43800e6 −1.41009 −0.705043 0.709164i \(-0.749072\pi\)
−0.705043 + 0.709164i \(0.749072\pi\)
\(432\) 946016. 0.243887
\(433\) 3.77335e6 0.967179 0.483590 0.875295i \(-0.339333\pi\)
0.483590 + 0.875295i \(0.339333\pi\)
\(434\) 0 0
\(435\) −1.00253e6 −0.254025
\(436\) −2.08802e6 −0.526041
\(437\) 51758.6 0.0129652
\(438\) 34108.7 0.00849531
\(439\) −2.35150e6 −0.582350 −0.291175 0.956670i \(-0.594046\pi\)
−0.291175 + 0.956670i \(0.594046\pi\)
\(440\) 4.00591e6 0.986439
\(441\) 0 0
\(442\) −3.63656e6 −0.885391
\(443\) 4.80377e6 1.16298 0.581491 0.813553i \(-0.302470\pi\)
0.581491 + 0.813553i \(0.302470\pi\)
\(444\) −403120. −0.0970458
\(445\) −2.46538e6 −0.590179
\(446\) −1.09826e6 −0.261437
\(447\) −919383. −0.217634
\(448\) 0 0
\(449\) −2.76805e6 −0.647975 −0.323987 0.946061i \(-0.605024\pi\)
−0.323987 + 0.946061i \(0.605024\pi\)
\(450\) 568896. 0.132435
\(451\) −7.04142e6 −1.63012
\(452\) 2.44356e6 0.562571
\(453\) −783770. −0.179450
\(454\) −820179. −0.186754
\(455\) 0 0
\(456\) 1.20536e6 0.271459
\(457\) 241566. 0.0541061 0.0270530 0.999634i \(-0.491388\pi\)
0.0270530 + 0.999634i \(0.491388\pi\)
\(458\) 3.46368e6 0.771567
\(459\) 3.42594e6 0.759010
\(460\) −49145.4 −0.0108290
\(461\) −990579. −0.217088 −0.108544 0.994092i \(-0.534619\pi\)
−0.108544 + 0.994092i \(0.534619\pi\)
\(462\) 0 0
\(463\) 6.20488e6 1.34518 0.672591 0.740014i \(-0.265181\pi\)
0.672591 + 0.740014i \(0.265181\pi\)
\(464\) 1.08538e6 0.234037
\(465\) 2.64531e6 0.567340
\(466\) 321702. 0.0686261
\(467\) 6.88497e6 1.46086 0.730432 0.682985i \(-0.239319\pi\)
0.730432 + 0.682985i \(0.239319\pi\)
\(468\) −5.27141e6 −1.11253
\(469\) 0 0
\(470\) −994206. −0.207602
\(471\) 639197. 0.132765
\(472\) 1.55603e6 0.321486
\(473\) 53396.6 0.0109739
\(474\) 157.219 3.21409e−5 0
\(475\) −1.17758e6 −0.239473
\(476\) 0 0
\(477\) 2.39214e6 0.481383
\(478\) −3.23121e6 −0.646838
\(479\) 5.41288e6 1.07793 0.538963 0.842329i \(-0.318816\pi\)
0.538963 + 0.842329i \(0.318816\pi\)
\(480\) −1.79763e6 −0.356120
\(481\) 2.79774e6 0.551373
\(482\) −2.28783e6 −0.448545
\(483\) 0 0
\(484\) −3.45268e6 −0.669950
\(485\) −143069. −0.0276179
\(486\) −2.53104e6 −0.486081
\(487\) −3.01043e6 −0.575182 −0.287591 0.957753i \(-0.592854\pi\)
−0.287591 + 0.957753i \(0.592854\pi\)
\(488\) −6.08029e6 −1.15578
\(489\) −1.19618e6 −0.226216
\(490\) 0 0
\(491\) −7.24498e6 −1.35623 −0.678115 0.734956i \(-0.737203\pi\)
−0.678115 + 0.734956i \(0.737203\pi\)
\(492\) 2.01176e6 0.374683
\(493\) 3.93062e6 0.728356
\(494\) −3.59163e6 −0.662177
\(495\) 5.07853e6 0.931591
\(496\) −2.86389e6 −0.522700
\(497\) 0 0
\(498\) −1.75187e6 −0.316540
\(499\) −5.12788e6 −0.921906 −0.460953 0.887425i \(-0.652492\pi\)
−0.460953 + 0.887425i \(0.652492\pi\)
\(500\) 4.57829e6 0.818989
\(501\) −2.95757e6 −0.526429
\(502\) 1.14509e6 0.202806
\(503\) −1.05978e7 −1.86766 −0.933830 0.357718i \(-0.883555\pi\)
−0.933830 + 0.357718i \(0.883555\pi\)
\(504\) 0 0
\(505\) −1.00426e6 −0.175233
\(506\) −68943.5 −0.0119706
\(507\) −5.40707e6 −0.934205
\(508\) 74503.8 0.0128091
\(509\) −8.78840e6 −1.50354 −0.751770 0.659425i \(-0.770800\pi\)
−0.751770 + 0.659425i \(0.770800\pi\)
\(510\) −1.00096e6 −0.170409
\(511\) 0 0
\(512\) 3.59310e6 0.605752
\(513\) 3.38361e6 0.567658
\(514\) 4.77753e6 0.797619
\(515\) −3.00512e6 −0.499280
\(516\) −15255.6 −0.00252235
\(517\) 4.23721e6 0.697194
\(518\) 0 0
\(519\) 2.43355e6 0.396572
\(520\) 7.94312e6 1.28820
\(521\) −162133. −0.0261684 −0.0130842 0.999914i \(-0.504165\pi\)
−0.0130842 + 0.999914i \(0.504165\pi\)
\(522\) −1.87544e6 −0.301250
\(523\) 7.14844e6 1.14277 0.571383 0.820684i \(-0.306407\pi\)
0.571383 + 0.820684i \(0.306407\pi\)
\(524\) 6.10144e6 0.970743
\(525\) 0 0
\(526\) 579044. 0.0912530
\(527\) −1.03714e7 −1.62671
\(528\) 1.17799e6 0.183889
\(529\) −6.43437e6 −0.999694
\(530\) −1.54758e6 −0.239311
\(531\) 1.97267e6 0.303611
\(532\) 0 0
\(533\) −1.39621e7 −2.12879
\(534\) 988120. 0.149953
\(535\) −5.01106e6 −0.756912
\(536\) 1.06630e7 1.60313
\(537\) −556604. −0.0832934
\(538\) −4.88203e6 −0.727185
\(539\) 0 0
\(540\) −3.21278e6 −0.474129
\(541\) −4.83604e6 −0.710390 −0.355195 0.934792i \(-0.615586\pi\)
−0.355195 + 0.934792i \(0.615586\pi\)
\(542\) −1.03888e6 −0.151903
\(543\) −2.48459e6 −0.361622
\(544\) 7.04793e6 1.02109
\(545\) 3.98874e6 0.575234
\(546\) 0 0
\(547\) 9.98777e6 1.42725 0.713626 0.700527i \(-0.247052\pi\)
0.713626 + 0.700527i \(0.247052\pi\)
\(548\) 2.33899e6 0.332719
\(549\) −7.70833e6 −1.09151
\(550\) 1.56856e6 0.221103
\(551\) 3.88205e6 0.544732
\(552\) 45878.4 0.00640856
\(553\) 0 0
\(554\) 3.45598e6 0.478406
\(555\) 770078. 0.106121
\(556\) 5.06324e6 0.694611
\(557\) 1.74619e6 0.238481 0.119241 0.992865i \(-0.461954\pi\)
0.119241 + 0.992865i \(0.461954\pi\)
\(558\) 4.94858e6 0.672814
\(559\) 105877. 0.0143309
\(560\) 0 0
\(561\) 4.26600e6 0.572287
\(562\) 5.64904e6 0.754456
\(563\) 755218. 0.100416 0.0502078 0.998739i \(-0.484012\pi\)
0.0502078 + 0.998739i \(0.484012\pi\)
\(564\) −1.21059e6 −0.160250
\(565\) −4.66792e6 −0.615181
\(566\) 5.02719e6 0.659605
\(567\) 0 0
\(568\) 9.68836e6 1.26003
\(569\) −4.39534e6 −0.569131 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(570\) −988594. −0.127447
\(571\) 1.16104e7 1.49024 0.745121 0.666930i \(-0.232392\pi\)
0.745121 + 0.666930i \(0.232392\pi\)
\(572\) −1.45344e7 −1.85740
\(573\) 6.03855e6 0.768327
\(574\) 0 0
\(575\) −44821.1 −0.00565344
\(576\) −1.27492e6 −0.160113
\(577\) −1.06643e7 −1.33350 −0.666748 0.745283i \(-0.732314\pi\)
−0.666748 + 0.745283i \(0.732314\pi\)
\(578\) −72546.6 −0.00903228
\(579\) −3.30745e6 −0.410012
\(580\) −3.68606e6 −0.454980
\(581\) 0 0
\(582\) 57341.7 0.00701719
\(583\) 6.59562e6 0.803682
\(584\) 292097. 0.0354401
\(585\) 1.00700e7 1.21657
\(586\) −2.40195e6 −0.288949
\(587\) 1.39482e7 1.67079 0.835396 0.549648i \(-0.185238\pi\)
0.835396 + 0.549648i \(0.185238\pi\)
\(588\) 0 0
\(589\) −1.02433e7 −1.21661
\(590\) −1.27620e6 −0.150934
\(591\) −1.64686e6 −0.193949
\(592\) −833711. −0.0977713
\(593\) −1.17933e7 −1.37720 −0.688600 0.725142i \(-0.741774\pi\)
−0.688600 + 0.725142i \(0.741774\pi\)
\(594\) −4.50703e6 −0.524113
\(595\) 0 0
\(596\) −3.38033e6 −0.389802
\(597\) −1.36220e6 −0.156425
\(598\) −136705. −0.0156326
\(599\) −4.38057e6 −0.498843 −0.249421 0.968395i \(-0.580240\pi\)
−0.249421 + 0.968395i \(0.580240\pi\)
\(600\) −1.04380e6 −0.118369
\(601\) −688570. −0.0777610 −0.0388805 0.999244i \(-0.512379\pi\)
−0.0388805 + 0.999244i \(0.512379\pi\)
\(602\) 0 0
\(603\) 1.35181e7 1.51399
\(604\) −2.88172e6 −0.321410
\(605\) 6.59563e6 0.732601
\(606\) 402504. 0.0445235
\(607\) 9.37319e6 1.03256 0.516281 0.856420i \(-0.327316\pi\)
0.516281 + 0.856420i \(0.327316\pi\)
\(608\) 6.96085e6 0.763666
\(609\) 0 0
\(610\) 4.98684e6 0.542626
\(611\) 8.40174e6 0.910472
\(612\) 5.68874e6 0.613956
\(613\) −2.16685e6 −0.232904 −0.116452 0.993196i \(-0.537152\pi\)
−0.116452 + 0.993196i \(0.537152\pi\)
\(614\) 5.51947e6 0.590849
\(615\) −3.84306e6 −0.409722
\(616\) 0 0
\(617\) −5.07951e6 −0.537166 −0.268583 0.963256i \(-0.586555\pi\)
−0.268583 + 0.963256i \(0.586555\pi\)
\(618\) 1.20445e6 0.126858
\(619\) 2.19034e6 0.229766 0.114883 0.993379i \(-0.463351\pi\)
0.114883 + 0.993379i \(0.463351\pi\)
\(620\) 9.72611e6 1.01616
\(621\) 128787. 0.0134012
\(622\) −2.43111e6 −0.251958
\(623\) 0 0
\(624\) 2.33577e6 0.240142
\(625\) −5.59018e6 −0.572435
\(626\) −3.09790e6 −0.315960
\(627\) 4.21329e6 0.428009
\(628\) 2.35016e6 0.237793
\(629\) −3.01923e6 −0.304278
\(630\) 0 0
\(631\) −7.18693e6 −0.718572 −0.359286 0.933228i \(-0.616980\pi\)
−0.359286 + 0.933228i \(0.616980\pi\)
\(632\) 1346.38 0.000134083 0
\(633\) 4.19451e6 0.416075
\(634\) 4.21109e6 0.416075
\(635\) −142324. −0.0140070
\(636\) −1.88439e6 −0.184726
\(637\) 0 0
\(638\) −5.17097e6 −0.502945
\(639\) 1.22825e7 1.18997
\(640\) −7.96017e6 −0.768197
\(641\) −1.76500e7 −1.69668 −0.848340 0.529452i \(-0.822398\pi\)
−0.848340 + 0.529452i \(0.822398\pi\)
\(642\) 2.00843e6 0.192317
\(643\) −898309. −0.0856837 −0.0428419 0.999082i \(-0.513641\pi\)
−0.0428419 + 0.999082i \(0.513641\pi\)
\(644\) 0 0
\(645\) 29142.7 0.00275823
\(646\) 3.87597e6 0.365425
\(647\) 1.38642e6 0.130207 0.0651035 0.997879i \(-0.479262\pi\)
0.0651035 + 0.997879i \(0.479262\pi\)
\(648\) −4.67735e6 −0.437585
\(649\) 5.43904e6 0.506886
\(650\) 3.11022e6 0.288741
\(651\) 0 0
\(652\) −4.39804e6 −0.405173
\(653\) −1.75425e7 −1.60994 −0.804968 0.593318i \(-0.797818\pi\)
−0.804968 + 0.593318i \(0.797818\pi\)
\(654\) −1.59868e6 −0.146156
\(655\) −1.16556e7 −1.06152
\(656\) 4.16062e6 0.377484
\(657\) 370308. 0.0334696
\(658\) 0 0
\(659\) 9.87522e6 0.885795 0.442898 0.896572i \(-0.353950\pi\)
0.442898 + 0.896572i \(0.353950\pi\)
\(660\) −4.00058e6 −0.357489
\(661\) 8.06792e6 0.718221 0.359110 0.933295i \(-0.383080\pi\)
0.359110 + 0.933295i \(0.383080\pi\)
\(662\) −7.71373e6 −0.684100
\(663\) 8.45884e6 0.747355
\(664\) −1.50025e7 −1.32052
\(665\) 0 0
\(666\) 1.44059e6 0.125850
\(667\) 147759. 0.0128599
\(668\) −1.08742e7 −0.942880
\(669\) 2.55461e6 0.220678
\(670\) −8.74544e6 −0.752652
\(671\) −2.12534e7 −1.82231
\(672\) 0 0
\(673\) −1.12772e7 −0.959762 −0.479881 0.877334i \(-0.659320\pi\)
−0.479881 + 0.877334i \(0.659320\pi\)
\(674\) −6.51274e6 −0.552223
\(675\) −2.93008e6 −0.247526
\(676\) −1.98804e7 −1.67324
\(677\) −5.20372e6 −0.436357 −0.218179 0.975909i \(-0.570012\pi\)
−0.218179 + 0.975909i \(0.570012\pi\)
\(678\) 1.87090e6 0.156306
\(679\) 0 0
\(680\) −8.57196e6 −0.710899
\(681\) 1.90778e6 0.157638
\(682\) 1.36442e7 1.12328
\(683\) 6.05915e6 0.497004 0.248502 0.968631i \(-0.420062\pi\)
0.248502 + 0.968631i \(0.420062\pi\)
\(684\) 5.61845e6 0.459173
\(685\) −4.46816e6 −0.363833
\(686\) 0 0
\(687\) −8.05671e6 −0.651277
\(688\) −31550.9 −0.00254121
\(689\) 1.30781e7 1.04954
\(690\) −37627.9 −0.00300876
\(691\) −7.36498e6 −0.586781 −0.293391 0.955993i \(-0.594784\pi\)
−0.293391 + 0.955993i \(0.594784\pi\)
\(692\) 8.94754e6 0.710295
\(693\) 0 0
\(694\) −8.61204e6 −0.678746
\(695\) −9.67228e6 −0.759569
\(696\) 3.44102e6 0.269255
\(697\) 1.50674e7 1.17478
\(698\) 591818. 0.0459779
\(699\) −748298. −0.0579271
\(700\) 0 0
\(701\) 7.80919e6 0.600221 0.300110 0.953904i \(-0.402977\pi\)
0.300110 + 0.953904i \(0.402977\pi\)
\(702\) −8.93676e6 −0.684443
\(703\) −2.98193e6 −0.227567
\(704\) −3.51520e6 −0.267312
\(705\) 2.31258e6 0.175236
\(706\) 1.06069e7 0.800898
\(707\) 0 0
\(708\) −1.55395e6 −0.116508
\(709\) 1.75650e7 1.31230 0.656150 0.754631i \(-0.272184\pi\)
0.656150 + 0.754631i \(0.272184\pi\)
\(710\) −7.94606e6 −0.591569
\(711\) 1706.88 0.000126628 0
\(712\) 8.46198e6 0.625564
\(713\) −389879. −0.0287214
\(714\) 0 0
\(715\) 2.77649e7 2.03110
\(716\) −2.04649e6 −0.149186
\(717\) 7.51597e6 0.545993
\(718\) 2.83539e6 0.205259
\(719\) 8.09220e6 0.583773 0.291887 0.956453i \(-0.405717\pi\)
0.291887 + 0.956453i \(0.405717\pi\)
\(720\) −3.00079e6 −0.215727
\(721\) 0 0
\(722\) −3.14232e6 −0.224341
\(723\) 5.32162e6 0.378615
\(724\) −9.13520e6 −0.647697
\(725\) −3.36172e6 −0.237529
\(726\) −2.64352e6 −0.186140
\(727\) −1.51986e7 −1.06652 −0.533258 0.845952i \(-0.679033\pi\)
−0.533258 + 0.845952i \(0.679033\pi\)
\(728\) 0 0
\(729\) −1.31285e6 −0.0914944
\(730\) −239568. −0.0166388
\(731\) −114259. −0.00790858
\(732\) 6.07218e6 0.418858
\(733\) −5.83402e6 −0.401059 −0.200530 0.979688i \(-0.564266\pi\)
−0.200530 + 0.979688i \(0.564266\pi\)
\(734\) 4.29720e6 0.294405
\(735\) 0 0
\(736\) 264944. 0.0180285
\(737\) 3.72722e7 2.52765
\(738\) −7.18921e6 −0.485893
\(739\) 6.47719e6 0.436290 0.218145 0.975916i \(-0.429999\pi\)
0.218145 + 0.975916i \(0.429999\pi\)
\(740\) 2.83138e6 0.190072
\(741\) 8.35433e6 0.558941
\(742\) 0 0
\(743\) −1.50899e7 −1.00280 −0.501401 0.865215i \(-0.667182\pi\)
−0.501401 + 0.865215i \(0.667182\pi\)
\(744\) −9.07955e6 −0.601357
\(745\) 6.45744e6 0.426255
\(746\) 1.36896e7 0.900626
\(747\) −1.90196e7 −1.24710
\(748\) 1.56850e7 1.02502
\(749\) 0 0
\(750\) 3.50534e6 0.227550
\(751\) −2.13997e6 −0.138455 −0.0692273 0.997601i \(-0.522053\pi\)
−0.0692273 + 0.997601i \(0.522053\pi\)
\(752\) −2.50367e6 −0.161448
\(753\) −2.66355e6 −0.171188
\(754\) −1.02533e7 −0.656801
\(755\) 5.50494e6 0.351467
\(756\) 0 0
\(757\) 2.10943e7 1.33791 0.668954 0.743304i \(-0.266742\pi\)
0.668954 + 0.743304i \(0.266742\pi\)
\(758\) 1.77597e6 0.112270
\(759\) 160366. 0.0101044
\(760\) −8.46605e6 −0.531675
\(761\) −9.79958e6 −0.613403 −0.306701 0.951806i \(-0.599225\pi\)
−0.306701 + 0.951806i \(0.599225\pi\)
\(762\) 57043.3 0.00355891
\(763\) 0 0
\(764\) 2.22022e7 1.37614
\(765\) −1.08672e7 −0.671371
\(766\) −1.59233e6 −0.0980530
\(767\) 1.07848e7 0.661947
\(768\) 4.52531e6 0.276850
\(769\) 3.23493e7 1.97265 0.986323 0.164825i \(-0.0527060\pi\)
0.986323 + 0.164825i \(0.0527060\pi\)
\(770\) 0 0
\(771\) −1.11128e7 −0.673267
\(772\) −1.21606e7 −0.734366
\(773\) −9.19713e6 −0.553609 −0.276805 0.960926i \(-0.589276\pi\)
−0.276805 + 0.960926i \(0.589276\pi\)
\(774\) 54517.4 0.00327102
\(775\) 8.87030e6 0.530499
\(776\) 491058. 0.0292738
\(777\) 0 0
\(778\) 1.66712e6 0.0987456
\(779\) 1.48812e7 0.878609
\(780\) −7.93254e6 −0.466848
\(781\) 3.38653e7 1.98668
\(782\) 147527. 0.00862690
\(783\) 9.65941e6 0.563049
\(784\) 0 0
\(785\) −4.48950e6 −0.260030
\(786\) 4.67153e6 0.269713
\(787\) −1.46950e7 −0.845731 −0.422866 0.906192i \(-0.638976\pi\)
−0.422866 + 0.906192i \(0.638976\pi\)
\(788\) −6.05508e6 −0.347379
\(789\) −1.34689e6 −0.0770263
\(790\) −1104.25 −6.29506e−5 0
\(791\) 0 0
\(792\) −1.74312e7 −0.987446
\(793\) −4.21423e7 −2.37977
\(794\) −3.78097e6 −0.212839
\(795\) 3.59975e6 0.202001
\(796\) −5.00847e6 −0.280171
\(797\) −2.33344e7 −1.30122 −0.650610 0.759412i \(-0.725487\pi\)
−0.650610 + 0.759412i \(0.725487\pi\)
\(798\) 0 0
\(799\) −9.06688e6 −0.502448
\(800\) −6.02785e6 −0.332995
\(801\) 1.07277e7 0.590782
\(802\) −1.03796e7 −0.569831
\(803\) 1.02101e6 0.0558783
\(804\) −1.06488e7 −0.580980
\(805\) 0 0
\(806\) 2.70545e7 1.46690
\(807\) 1.13559e7 0.613814
\(808\) 3.44693e6 0.185740
\(809\) 1.69301e6 0.0909468 0.0454734 0.998966i \(-0.485520\pi\)
0.0454734 + 0.998966i \(0.485520\pi\)
\(810\) 3.83620e6 0.205442
\(811\) 2.12400e7 1.13397 0.566987 0.823727i \(-0.308109\pi\)
0.566987 + 0.823727i \(0.308109\pi\)
\(812\) 0 0
\(813\) 2.41649e6 0.128221
\(814\) 3.97199e6 0.210110
\(815\) 8.40155e6 0.443063
\(816\) −2.52068e6 −0.132524
\(817\) −112848. −0.00591477
\(818\) −4.09960e6 −0.214219
\(819\) 0 0
\(820\) −1.41299e7 −0.733847
\(821\) 8.73550e6 0.452304 0.226152 0.974092i \(-0.427385\pi\)
0.226152 + 0.974092i \(0.427385\pi\)
\(822\) 1.79083e6 0.0924433
\(823\) −3.27964e7 −1.68782 −0.843910 0.536485i \(-0.819752\pi\)
−0.843910 + 0.536485i \(0.819752\pi\)
\(824\) 1.03146e7 0.529215
\(825\) −3.64856e6 −0.186632
\(826\) 0 0
\(827\) 1.31248e7 0.667311 0.333656 0.942695i \(-0.391718\pi\)
0.333656 + 0.942695i \(0.391718\pi\)
\(828\) 213849. 0.0108401
\(829\) −2.05402e7 −1.03805 −0.519026 0.854759i \(-0.673705\pi\)
−0.519026 + 0.854759i \(0.673705\pi\)
\(830\) 1.23046e7 0.619970
\(831\) −8.03880e6 −0.403821
\(832\) −6.97012e6 −0.349085
\(833\) 0 0
\(834\) 3.87664e6 0.192992
\(835\) 2.07729e7 1.03106
\(836\) 1.54912e7 0.766601
\(837\) −2.54875e7 −1.25752
\(838\) 8.22542e6 0.404621
\(839\) −2.83736e7 −1.39159 −0.695793 0.718243i \(-0.744947\pi\)
−0.695793 + 0.718243i \(0.744947\pi\)
\(840\) 0 0
\(841\) −9.42879e6 −0.459691
\(842\) 5.68643e6 0.276414
\(843\) −1.31400e7 −0.636834
\(844\) 1.54221e7 0.745226
\(845\) 3.79774e7 1.82972
\(846\) 4.32614e6 0.207814
\(847\) 0 0
\(848\) −3.89720e6 −0.186107
\(849\) −1.16935e7 −0.556770
\(850\) −3.35645e6 −0.159343
\(851\) −113498. −0.00537236
\(852\) −9.67545e6 −0.456638
\(853\) −1.02093e7 −0.480424 −0.240212 0.970720i \(-0.577217\pi\)
−0.240212 + 0.970720i \(0.577217\pi\)
\(854\) 0 0
\(855\) −1.07329e7 −0.502113
\(856\) 1.71996e7 0.802295
\(857\) 8.33206e6 0.387525 0.193763 0.981048i \(-0.437931\pi\)
0.193763 + 0.981048i \(0.437931\pi\)
\(858\) −1.11281e7 −0.516064
\(859\) −3.12766e7 −1.44623 −0.723113 0.690729i \(-0.757290\pi\)
−0.723113 + 0.690729i \(0.757290\pi\)
\(860\) 107150. 0.00494023
\(861\) 0 0
\(862\) −1.53084e7 −0.701714
\(863\) −3.73573e7 −1.70745 −0.853726 0.520722i \(-0.825663\pi\)
−0.853726 + 0.520722i \(0.825663\pi\)
\(864\) 1.73201e7 0.789345
\(865\) −1.70924e7 −0.776719
\(866\) 1.06222e7 0.481306
\(867\) 168747. 0.00762411
\(868\) 0 0
\(869\) 4706.21 0.000211408 0
\(870\) −2.82221e6 −0.126413
\(871\) 7.39051e7 3.30088
\(872\) −1.36907e7 −0.609724
\(873\) 622543. 0.0276461
\(874\) 145704. 0.00645199
\(875\) 0 0
\(876\) −291708. −0.0128436
\(877\) 38996.7 0.00171210 0.000856049 1.00000i \(-0.499728\pi\)
0.000856049 1.00000i \(0.499728\pi\)
\(878\) −6.61965e6 −0.289800
\(879\) 5.58708e6 0.243900
\(880\) −8.27378e6 −0.360162
\(881\) −3.15554e7 −1.36973 −0.684864 0.728671i \(-0.740138\pi\)
−0.684864 + 0.728671i \(0.740138\pi\)
\(882\) 0 0
\(883\) −3.42253e7 −1.47722 −0.738611 0.674132i \(-0.764518\pi\)
−0.738611 + 0.674132i \(0.764518\pi\)
\(884\) 3.11010e7 1.33858
\(885\) 2.96851e6 0.127403
\(886\) 1.35230e7 0.578745
\(887\) 2.69886e7 1.15178 0.575892 0.817526i \(-0.304655\pi\)
0.575892 + 0.817526i \(0.304655\pi\)
\(888\) −2.64316e6 −0.112484
\(889\) 0 0
\(890\) −6.94022e6 −0.293696
\(891\) −1.63495e7 −0.689938
\(892\) 9.39263e6 0.395253
\(893\) −8.95486e6 −0.375777
\(894\) −2.58813e6 −0.108303
\(895\) 3.90940e6 0.163137
\(896\) 0 0
\(897\) 317982. 0.0131954
\(898\) −7.79226e6 −0.322458
\(899\) −2.92421e7 −1.20673
\(900\) −4.86538e6 −0.200221
\(901\) −1.41135e7 −0.579191
\(902\) −1.98221e7 −0.811211
\(903\) 0 0
\(904\) 1.60218e7 0.652065
\(905\) 1.74509e7 0.708267
\(906\) −2.20637e6 −0.0893013
\(907\) 1.92103e7 0.775381 0.387690 0.921790i \(-0.373273\pi\)
0.387690 + 0.921790i \(0.373273\pi\)
\(908\) 7.01442e6 0.282343
\(909\) 4.36988e6 0.175412
\(910\) 0 0
\(911\) 2.86013e7 1.14180 0.570899 0.821020i \(-0.306595\pi\)
0.570899 + 0.821020i \(0.306595\pi\)
\(912\) −2.48954e6 −0.0991133
\(913\) −5.24408e7 −2.08206
\(914\) 680027. 0.0269253
\(915\) −1.15997e7 −0.458029
\(916\) −2.96224e7 −1.16649
\(917\) 0 0
\(918\) 9.64426e6 0.377713
\(919\) −4.21754e7 −1.64729 −0.823645 0.567106i \(-0.808063\pi\)
−0.823645 + 0.567106i \(0.808063\pi\)
\(920\) −322235. −0.0125517
\(921\) −1.28386e7 −0.498733
\(922\) −2.78855e6 −0.108032
\(923\) 6.71498e7 2.59442
\(924\) 0 0
\(925\) 2.58224e6 0.0992299
\(926\) 1.74672e7 0.669416
\(927\) 1.30764e7 0.499790
\(928\) 1.98716e7 0.757466
\(929\) 3.01886e7 1.14763 0.573817 0.818983i \(-0.305462\pi\)
0.573817 + 0.818983i \(0.305462\pi\)
\(930\) 7.44673e6 0.282331
\(931\) 0 0
\(932\) −2.75130e6 −0.103752
\(933\) 5.65489e6 0.212677
\(934\) 1.93817e7 0.726984
\(935\) −2.99630e7 −1.12087
\(936\) −3.45634e7 −1.28952
\(937\) 3.64068e6 0.135467 0.0677335 0.997703i \(-0.478423\pi\)
0.0677335 + 0.997703i \(0.478423\pi\)
\(938\) 0 0
\(939\) 7.20589e6 0.266701
\(940\) 8.50275e6 0.313863
\(941\) −1.88601e7 −0.694336 −0.347168 0.937803i \(-0.612857\pi\)
−0.347168 + 0.937803i \(0.612857\pi\)
\(942\) 1.79939e6 0.0660689
\(943\) 566410. 0.0207420
\(944\) −3.21380e6 −0.117379
\(945\) 0 0
\(946\) 150315. 0.00546104
\(947\) −1.82172e7 −0.660094 −0.330047 0.943965i \(-0.607065\pi\)
−0.330047 + 0.943965i \(0.607065\pi\)
\(948\) −1344.58 −4.85922e−5 0
\(949\) 2.02452e6 0.0729720
\(950\) −3.31498e6 −0.119171
\(951\) −9.79522e6 −0.351207
\(952\) 0 0
\(953\) 2.76898e7 0.987616 0.493808 0.869571i \(-0.335604\pi\)
0.493808 + 0.869571i \(0.335604\pi\)
\(954\) 6.73406e6 0.239555
\(955\) −4.24127e7 −1.50483
\(956\) 2.76343e7 0.977921
\(957\) 1.20280e7 0.424534
\(958\) 1.52376e7 0.536419
\(959\) 0 0
\(960\) −1.91852e6 −0.0671875
\(961\) 4.85298e7 1.69512
\(962\) 7.87585e6 0.274385
\(963\) 2.18049e7 0.757685
\(964\) 1.95662e7 0.678133
\(965\) 2.32304e7 0.803042
\(966\) 0 0
\(967\) −2.44768e7 −0.841761 −0.420881 0.907116i \(-0.638279\pi\)
−0.420881 + 0.907116i \(0.638279\pi\)
\(968\) −2.26383e7 −0.776526
\(969\) −9.01571e6 −0.308454
\(970\) −402749. −0.0137437
\(971\) −9.50151e6 −0.323403 −0.161702 0.986840i \(-0.551698\pi\)
−0.161702 + 0.986840i \(0.551698\pi\)
\(972\) 2.16463e7 0.734881
\(973\) 0 0
\(974\) −8.47457e6 −0.286233
\(975\) −7.23455e6 −0.243725
\(976\) 1.25582e7 0.421990
\(977\) 4.69012e7 1.57198 0.785991 0.618238i \(-0.212153\pi\)
0.785991 + 0.618238i \(0.212153\pi\)
\(978\) −3.36733e6 −0.112574
\(979\) 2.95785e7 0.986325
\(980\) 0 0
\(981\) −1.73564e7 −0.575822
\(982\) −2.03951e7 −0.674914
\(983\) −2.35382e7 −0.776945 −0.388473 0.921460i \(-0.626997\pi\)
−0.388473 + 0.921460i \(0.626997\pi\)
\(984\) 1.31906e7 0.434288
\(985\) 1.15670e7 0.379865
\(986\) 1.10650e7 0.362458
\(987\) 0 0
\(988\) 3.07167e7 1.00111
\(989\) −4295.21 −0.000139635 0
\(990\) 1.42964e7 0.463596
\(991\) −2.64104e7 −0.854261 −0.427130 0.904190i \(-0.640475\pi\)
−0.427130 + 0.904190i \(0.640475\pi\)
\(992\) −5.24336e7 −1.69173
\(993\) 1.79426e7 0.577446
\(994\) 0 0
\(995\) 9.56766e6 0.306371
\(996\) 1.49825e7 0.478561
\(997\) 1.95164e7 0.621815 0.310907 0.950440i \(-0.399367\pi\)
0.310907 + 0.950440i \(0.399367\pi\)
\(998\) −1.44354e7 −0.458777
\(999\) −7.41970e6 −0.235219
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 49.6.a.g.1.3 4
3.2 odd 2 441.6.a.z.1.1 4
4.3 odd 2 784.6.a.bf.1.3 4
7.2 even 3 49.6.c.h.18.2 8
7.3 odd 6 49.6.c.h.30.1 8
7.4 even 3 49.6.c.h.30.2 8
7.5 odd 6 49.6.c.h.18.1 8
7.6 odd 2 inner 49.6.a.g.1.4 yes 4
21.20 even 2 441.6.a.z.1.2 4
28.27 even 2 784.6.a.bf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.3 4 1.1 even 1 trivial
49.6.a.g.1.4 yes 4 7.6 odd 2 inner
49.6.c.h.18.1 8 7.5 odd 6
49.6.c.h.18.2 8 7.2 even 3
49.6.c.h.30.1 8 7.3 odd 6
49.6.c.h.30.2 8 7.4 even 3
441.6.a.z.1.1 4 3.2 odd 2
441.6.a.z.1.2 4 21.20 even 2
784.6.a.bf.1.2 4 28.27 even 2
784.6.a.bf.1.3 4 4.3 odd 2