Properties

Label 177.6.a.d.1.7
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + 81977088 x^{5} - 3773728 x^{4} - 1245415104 x^{3} + 453320896 x^{2} + 6872784896 x - 6400833792\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.05732\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.05732 q^{2} -9.00000 q^{3} -30.8821 q^{4} +33.7931 q^{5} -9.51585 q^{6} -85.6226 q^{7} -66.4863 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+1.05732 q^{2} -9.00000 q^{3} -30.8821 q^{4} +33.7931 q^{5} -9.51585 q^{6} -85.6226 q^{7} -66.4863 q^{8} +81.0000 q^{9} +35.7300 q^{10} -33.3696 q^{11} +277.939 q^{12} -1043.41 q^{13} -90.5302 q^{14} -304.138 q^{15} +917.930 q^{16} +522.840 q^{17} +85.6427 q^{18} -1300.57 q^{19} -1043.60 q^{20} +770.603 q^{21} -35.2822 q^{22} +2439.93 q^{23} +598.377 q^{24} -1983.02 q^{25} -1103.22 q^{26} -729.000 q^{27} +2644.20 q^{28} +7069.65 q^{29} -321.570 q^{30} +2519.18 q^{31} +3098.10 q^{32} +300.326 q^{33} +552.807 q^{34} -2893.45 q^{35} -2501.45 q^{36} -6580.49 q^{37} -1375.12 q^{38} +9390.70 q^{39} -2246.78 q^{40} +19615.8 q^{41} +814.771 q^{42} +18389.9 q^{43} +1030.52 q^{44} +2737.24 q^{45} +2579.77 q^{46} +18005.2 q^{47} -8261.37 q^{48} -9475.78 q^{49} -2096.69 q^{50} -4705.56 q^{51} +32222.7 q^{52} +16541.3 q^{53} -770.784 q^{54} -1127.66 q^{55} +5692.73 q^{56} +11705.1 q^{57} +7474.86 q^{58} +3481.00 q^{59} +9392.42 q^{60} -29130.6 q^{61} +2663.57 q^{62} -6935.43 q^{63} -26098.1 q^{64} -35260.1 q^{65} +317.540 q^{66} -23137.8 q^{67} -16146.4 q^{68} -21959.3 q^{69} -3059.30 q^{70} -35612.7 q^{71} -5385.39 q^{72} +76489.3 q^{73} -6957.66 q^{74} +17847.2 q^{75} +40164.4 q^{76} +2857.19 q^{77} +9928.95 q^{78} +61578.6 q^{79} +31019.7 q^{80} +6561.00 q^{81} +20740.1 q^{82} -11802.0 q^{83} -23797.8 q^{84} +17668.4 q^{85} +19443.9 q^{86} -63626.8 q^{87} +2218.62 q^{88} +34947.8 q^{89} +2894.13 q^{90} +89339.6 q^{91} -75350.0 q^{92} -22672.6 q^{93} +19037.2 q^{94} -43950.4 q^{95} -27882.9 q^{96} -89284.3 q^{97} -10018.9 q^{98} -2702.94 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + O(q^{10}) \) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + 137q^{10} + 250q^{11} - 2214q^{12} + 1054q^{13} - 575q^{14} + 126q^{15} + 922q^{16} + 271q^{17} + 671q^{19} - 5491q^{20} - 3357q^{21} + 1094q^{22} + 3975q^{23} - 1107q^{24} + 15569q^{25} + 4622q^{26} - 9477q^{27} + 21214q^{28} - 10613q^{29} - 1233q^{30} + 25597q^{31} + 15966q^{32} - 2250q^{33} + 31796q^{34} + 6729q^{35} + 19926q^{36} + 17585q^{37} + 34903q^{38} - 9486q^{39} + 31382q^{40} + 12537q^{41} + 5175q^{42} + 26644q^{43} + 6654q^{44} - 1134q^{45} + 149005q^{46} + 52087q^{47} - 8298q^{48} + 95384q^{49} + 121821q^{50} - 2439q^{51} + 263630q^{52} + 20014q^{53} + 120932q^{55} + 126688q^{56} - 6039q^{57} + 86066q^{58} + 45253q^{59} + 49419q^{60} - 11667q^{61} + 164794q^{62} + 30213q^{63} + 151893q^{64} - 28674q^{65} - 9846q^{66} + 1106q^{67} - 4043q^{68} - 35775q^{69} + 56066q^{70} + 21230q^{71} + 9963q^{72} + 81131q^{73} + 102042q^{74} - 140121q^{75} + 73900q^{76} - 104655q^{77} - 41598q^{78} - 13470q^{79} - 191969q^{80} + 85293q^{81} + 79909q^{82} - 76149q^{83} - 190926q^{84} + 10035q^{85} - 321496q^{86} + 95517q^{87} - 276779q^{88} - 190205q^{89} + 11097q^{90} + 80601q^{91} + 45672q^{92} - 230373q^{93} + 36768q^{94} + 9875q^{95} - 143694q^{96} + 160850q^{97} - 116644q^{98} + 20250q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.05732 0.186909 0.0934545 0.995624i \(-0.470209\pi\)
0.0934545 + 0.995624i \(0.470209\pi\)
\(3\) −9.00000 −0.577350
\(4\) −30.8821 −0.965065
\(5\) 33.7931 0.604510 0.302255 0.953227i \(-0.402261\pi\)
0.302255 + 0.953227i \(0.402261\pi\)
\(6\) −9.51585 −0.107912
\(7\) −85.6226 −0.660455 −0.330227 0.943901i \(-0.607125\pi\)
−0.330227 + 0.943901i \(0.607125\pi\)
\(8\) −66.4863 −0.367288
\(9\) 81.0000 0.333333
\(10\) 35.7300 0.112988
\(11\) −33.3696 −0.0831513 −0.0415757 0.999135i \(-0.513238\pi\)
−0.0415757 + 0.999135i \(0.513238\pi\)
\(12\) 277.939 0.557181
\(13\) −1043.41 −1.71237 −0.856184 0.516671i \(-0.827171\pi\)
−0.856184 + 0.516671i \(0.827171\pi\)
\(14\) −90.5302 −0.123445
\(15\) −304.138 −0.349014
\(16\) 917.930 0.896416
\(17\) 522.840 0.438780 0.219390 0.975637i \(-0.429593\pi\)
0.219390 + 0.975637i \(0.429593\pi\)
\(18\) 85.6427 0.0623030
\(19\) −1300.57 −0.826514 −0.413257 0.910614i \(-0.635609\pi\)
−0.413257 + 0.910614i \(0.635609\pi\)
\(20\) −1043.60 −0.583391
\(21\) 770.603 0.381314
\(22\) −35.2822 −0.0155417
\(23\) 2439.93 0.961739 0.480869 0.876792i \(-0.340321\pi\)
0.480869 + 0.876792i \(0.340321\pi\)
\(24\) 598.377 0.212054
\(25\) −1983.02 −0.634568
\(26\) −1103.22 −0.320057
\(27\) −729.000 −0.192450
\(28\) 2644.20 0.637382
\(29\) 7069.65 1.56100 0.780500 0.625156i \(-0.214965\pi\)
0.780500 + 0.625156i \(0.214965\pi\)
\(30\) −321.570 −0.0652338
\(31\) 2519.18 0.470821 0.235410 0.971896i \(-0.424357\pi\)
0.235410 + 0.971896i \(0.424357\pi\)
\(32\) 3098.10 0.534836
\(33\) 300.326 0.0480074
\(34\) 552.807 0.0820118
\(35\) −2893.45 −0.399251
\(36\) −2501.45 −0.321688
\(37\) −6580.49 −0.790231 −0.395115 0.918632i \(-0.629295\pi\)
−0.395115 + 0.918632i \(0.629295\pi\)
\(38\) −1375.12 −0.154483
\(39\) 9390.70 0.988636
\(40\) −2246.78 −0.222029
\(41\) 19615.8 1.82241 0.911206 0.411951i \(-0.135153\pi\)
0.911206 + 0.411951i \(0.135153\pi\)
\(42\) 814.771 0.0712710
\(43\) 18389.9 1.51673 0.758364 0.651831i \(-0.225999\pi\)
0.758364 + 0.651831i \(0.225999\pi\)
\(44\) 1030.52 0.0802464
\(45\) 2737.24 0.201503
\(46\) 2579.77 0.179758
\(47\) 18005.2 1.18892 0.594461 0.804124i \(-0.297365\pi\)
0.594461 + 0.804124i \(0.297365\pi\)
\(48\) −8261.37 −0.517546
\(49\) −9475.78 −0.563799
\(50\) −2096.69 −0.118606
\(51\) −4705.56 −0.253330
\(52\) 32222.7 1.65255
\(53\) 16541.3 0.808870 0.404435 0.914567i \(-0.367468\pi\)
0.404435 + 0.914567i \(0.367468\pi\)
\(54\) −770.784 −0.0359706
\(55\) −1127.66 −0.0502658
\(56\) 5692.73 0.242577
\(57\) 11705.1 0.477188
\(58\) 7474.86 0.291765
\(59\) 3481.00 0.130189
\(60\) 9392.42 0.336821
\(61\) −29130.6 −1.00236 −0.501182 0.865342i \(-0.667101\pi\)
−0.501182 + 0.865342i \(0.667101\pi\)
\(62\) 2663.57 0.0880006
\(63\) −6935.43 −0.220152
\(64\) −26098.1 −0.796450
\(65\) −35260.1 −1.03514
\(66\) 317.540 0.00897302
\(67\) −23137.8 −0.629702 −0.314851 0.949141i \(-0.601955\pi\)
−0.314851 + 0.949141i \(0.601955\pi\)
\(68\) −16146.4 −0.423451
\(69\) −21959.3 −0.555260
\(70\) −3059.30 −0.0746237
\(71\) −35612.7 −0.838414 −0.419207 0.907891i \(-0.637692\pi\)
−0.419207 + 0.907891i \(0.637692\pi\)
\(72\) −5385.39 −0.122429
\(73\) 76489.3 1.67994 0.839970 0.542633i \(-0.182573\pi\)
0.839970 + 0.542633i \(0.182573\pi\)
\(74\) −6957.66 −0.147701
\(75\) 17847.2 0.366368
\(76\) 40164.4 0.797640
\(77\) 2857.19 0.0549177
\(78\) 9928.95 0.184785
\(79\) 61578.6 1.11010 0.555050 0.831817i \(-0.312699\pi\)
0.555050 + 0.831817i \(0.312699\pi\)
\(80\) 31019.7 0.541892
\(81\) 6561.00 0.111111
\(82\) 20740.1 0.340625
\(83\) −11802.0 −0.188045 −0.0940224 0.995570i \(-0.529973\pi\)
−0.0940224 + 0.995570i \(0.529973\pi\)
\(84\) −23797.8 −0.367993
\(85\) 17668.4 0.265247
\(86\) 19443.9 0.283490
\(87\) −63626.8 −0.901243
\(88\) 2218.62 0.0305405
\(89\) 34947.8 0.467675 0.233838 0.972276i \(-0.424872\pi\)
0.233838 + 0.972276i \(0.424872\pi\)
\(90\) 2894.13 0.0376628
\(91\) 89339.6 1.13094
\(92\) −75350.0 −0.928140
\(93\) −22672.6 −0.271828
\(94\) 19037.2 0.222220
\(95\) −43950.4 −0.499636
\(96\) −27882.9 −0.308788
\(97\) −89284.3 −0.963487 −0.481744 0.876312i \(-0.659996\pi\)
−0.481744 + 0.876312i \(0.659996\pi\)
\(98\) −10018.9 −0.105379
\(99\) −2702.94 −0.0277171
\(100\) 61239.9 0.612399
\(101\) −25180.8 −0.245621 −0.122811 0.992430i \(-0.539191\pi\)
−0.122811 + 0.992430i \(0.539191\pi\)
\(102\) −4975.27 −0.0473496
\(103\) 133786. 1.24256 0.621282 0.783587i \(-0.286612\pi\)
0.621282 + 0.783587i \(0.286612\pi\)
\(104\) 69372.5 0.628933
\(105\) 26041.1 0.230508
\(106\) 17489.3 0.151185
\(107\) −71670.6 −0.605176 −0.302588 0.953122i \(-0.597851\pi\)
−0.302588 + 0.953122i \(0.597851\pi\)
\(108\) 22513.0 0.185727
\(109\) −101459. −0.817948 −0.408974 0.912546i \(-0.634113\pi\)
−0.408974 + 0.912546i \(0.634113\pi\)
\(110\) −1192.30 −0.00939512
\(111\) 59224.4 0.456240
\(112\) −78595.5 −0.592042
\(113\) 234915. 1.73067 0.865337 0.501190i \(-0.167104\pi\)
0.865337 + 0.501190i \(0.167104\pi\)
\(114\) 12376.0 0.0891907
\(115\) 82452.7 0.581380
\(116\) −218325. −1.50647
\(117\) −84516.3 −0.570789
\(118\) 3680.52 0.0243335
\(119\) −44766.9 −0.289794
\(120\) 20221.0 0.128189
\(121\) −159937. −0.993086
\(122\) −30800.3 −0.187351
\(123\) −176542. −1.05217
\(124\) −77797.6 −0.454372
\(125\) −172616. −0.988112
\(126\) −7332.94 −0.0411483
\(127\) 28866.5 0.158813 0.0794064 0.996842i \(-0.474698\pi\)
0.0794064 + 0.996842i \(0.474698\pi\)
\(128\) −126733. −0.683700
\(129\) −165509. −0.875684
\(130\) −37281.1 −0.193478
\(131\) 119417. 0.607977 0.303989 0.952676i \(-0.401682\pi\)
0.303989 + 0.952676i \(0.401682\pi\)
\(132\) −9274.70 −0.0463303
\(133\) 111358. 0.545875
\(134\) −24464.0 −0.117697
\(135\) −24635.2 −0.116338
\(136\) −34761.7 −0.161159
\(137\) −199935. −0.910097 −0.455049 0.890467i \(-0.650378\pi\)
−0.455049 + 0.890467i \(0.650378\pi\)
\(138\) −23218.0 −0.103783
\(139\) 13511.9 0.0593172 0.0296586 0.999560i \(-0.490558\pi\)
0.0296586 + 0.999560i \(0.490558\pi\)
\(140\) 89355.9 0.385304
\(141\) −162047. −0.686425
\(142\) −37653.9 −0.156707
\(143\) 34818.2 0.142386
\(144\) 74352.3 0.298805
\(145\) 238905. 0.943640
\(146\) 80873.4 0.313996
\(147\) 85282.0 0.325510
\(148\) 203219. 0.762624
\(149\) −448024. −1.65324 −0.826619 0.562762i \(-0.809739\pi\)
−0.826619 + 0.562762i \(0.809739\pi\)
\(150\) 18870.2 0.0684775
\(151\) −221792. −0.791597 −0.395799 0.918337i \(-0.629532\pi\)
−0.395799 + 0.918337i \(0.629532\pi\)
\(152\) 86470.2 0.303569
\(153\) 42350.0 0.146260
\(154\) 3020.95 0.0102646
\(155\) 85131.0 0.284616
\(156\) −290004. −0.954098
\(157\) 135887. 0.439974 0.219987 0.975503i \(-0.429398\pi\)
0.219987 + 0.975503i \(0.429398\pi\)
\(158\) 65108.0 0.207487
\(159\) −148871. −0.467001
\(160\) 104695. 0.323314
\(161\) −208913. −0.635185
\(162\) 6937.06 0.0207677
\(163\) 301769. 0.889623 0.444812 0.895624i \(-0.353271\pi\)
0.444812 + 0.895624i \(0.353271\pi\)
\(164\) −605777. −1.75875
\(165\) 10149.0 0.0290210
\(166\) −12478.5 −0.0351473
\(167\) 572098. 1.58737 0.793687 0.608326i \(-0.208159\pi\)
0.793687 + 0.608326i \(0.208159\pi\)
\(168\) −51234.5 −0.140052
\(169\) 717414. 1.93221
\(170\) 18681.1 0.0495770
\(171\) −105346. −0.275505
\(172\) −567918. −1.46374
\(173\) −279838. −0.710873 −0.355436 0.934700i \(-0.615668\pi\)
−0.355436 + 0.934700i \(0.615668\pi\)
\(174\) −67273.7 −0.168450
\(175\) 169792. 0.419103
\(176\) −30630.9 −0.0745381
\(177\) −31329.0 −0.0751646
\(178\) 36950.8 0.0874127
\(179\) 421817. 0.983991 0.491995 0.870598i \(-0.336268\pi\)
0.491995 + 0.870598i \(0.336268\pi\)
\(180\) −84531.8 −0.194464
\(181\) −257718. −0.584720 −0.292360 0.956308i \(-0.594441\pi\)
−0.292360 + 0.956308i \(0.594441\pi\)
\(182\) 94460.2 0.211383
\(183\) 262176. 0.578715
\(184\) −162222. −0.353235
\(185\) −222375. −0.477702
\(186\) −23972.2 −0.0508072
\(187\) −17447.0 −0.0364851
\(188\) −556038. −1.14739
\(189\) 62418.8 0.127105
\(190\) −46469.5 −0.0933864
\(191\) −256126. −0.508008 −0.254004 0.967203i \(-0.581748\pi\)
−0.254004 + 0.967203i \(0.581748\pi\)
\(192\) 234883. 0.459831
\(193\) −788156. −1.52307 −0.761534 0.648125i \(-0.775553\pi\)
−0.761534 + 0.648125i \(0.775553\pi\)
\(194\) −94401.8 −0.180084
\(195\) 317341. 0.597640
\(196\) 292632. 0.544103
\(197\) 953753. 1.75094 0.875468 0.483275i \(-0.160553\pi\)
0.875468 + 0.483275i \(0.160553\pi\)
\(198\) −2857.86 −0.00518057
\(199\) 694899. 1.24391 0.621955 0.783053i \(-0.286339\pi\)
0.621955 + 0.783053i \(0.286339\pi\)
\(200\) 131844. 0.233069
\(201\) 208240. 0.363559
\(202\) −26624.1 −0.0459088
\(203\) −605321. −1.03097
\(204\) 145317. 0.244479
\(205\) 662879. 1.10167
\(206\) 141454. 0.232246
\(207\) 197634. 0.320580
\(208\) −957778. −1.53499
\(209\) 43399.5 0.0687257
\(210\) 27533.7 0.0430840
\(211\) 605261. 0.935915 0.467957 0.883751i \(-0.344990\pi\)
0.467957 + 0.883751i \(0.344990\pi\)
\(212\) −510828. −0.780612
\(213\) 320514. 0.484059
\(214\) −75778.5 −0.113113
\(215\) 621452. 0.916877
\(216\) 48468.5 0.0706847
\(217\) −215699. −0.310956
\(218\) −107275. −0.152882
\(219\) −688404. −0.969913
\(220\) 34824.6 0.0485097
\(221\) −545537. −0.751352
\(222\) 62619.0 0.0852753
\(223\) 1.35183e6 1.82037 0.910186 0.414199i \(-0.135938\pi\)
0.910186 + 0.414199i \(0.135938\pi\)
\(224\) −265268. −0.353235
\(225\) −160625. −0.211523
\(226\) 248380. 0.323479
\(227\) 289071. 0.372341 0.186170 0.982517i \(-0.440392\pi\)
0.186170 + 0.982517i \(0.440392\pi\)
\(228\) −361479. −0.460518
\(229\) 851974. 1.07359 0.536794 0.843713i \(-0.319635\pi\)
0.536794 + 0.843713i \(0.319635\pi\)
\(230\) 87178.7 0.108665
\(231\) −25714.7 −0.0317067
\(232\) −470035. −0.573337
\(233\) 1.51708e6 1.83071 0.915356 0.402646i \(-0.131910\pi\)
0.915356 + 0.402646i \(0.131910\pi\)
\(234\) −89360.5 −0.106686
\(235\) 608452. 0.718715
\(236\) −107501. −0.125641
\(237\) −554207. −0.640916
\(238\) −47332.8 −0.0541651
\(239\) −166996. −0.189109 −0.0945543 0.995520i \(-0.530143\pi\)
−0.0945543 + 0.995520i \(0.530143\pi\)
\(240\) −279177. −0.312862
\(241\) −228585. −0.253516 −0.126758 0.991934i \(-0.540457\pi\)
−0.126758 + 0.991934i \(0.540457\pi\)
\(242\) −169105. −0.185617
\(243\) −59049.0 −0.0641500
\(244\) 899614. 0.967346
\(245\) −320216. −0.340822
\(246\) −186661. −0.196660
\(247\) 1.35703e6 1.41530
\(248\) −167491. −0.172927
\(249\) 106218. 0.108568
\(250\) −182510. −0.184687
\(251\) 1.34491e6 1.34743 0.673717 0.738989i \(-0.264697\pi\)
0.673717 + 0.738989i \(0.264697\pi\)
\(252\) 214180. 0.212461
\(253\) −81419.3 −0.0799698
\(254\) 30521.1 0.0296835
\(255\) −159016. −0.153140
\(256\) 701141. 0.668660
\(257\) −193589. −0.182830 −0.0914151 0.995813i \(-0.529139\pi\)
−0.0914151 + 0.995813i \(0.529139\pi\)
\(258\) −174995. −0.163673
\(259\) 563438. 0.521912
\(260\) 1.08891e6 0.998981
\(261\) 572641. 0.520333
\(262\) 126261. 0.113636
\(263\) −1.29250e6 −1.15224 −0.576118 0.817367i \(-0.695433\pi\)
−0.576118 + 0.817367i \(0.695433\pi\)
\(264\) −19967.6 −0.0176326
\(265\) 558981. 0.488970
\(266\) 117741. 0.102029
\(267\) −314530. −0.270012
\(268\) 714544. 0.607704
\(269\) −2.00558e6 −1.68990 −0.844948 0.534848i \(-0.820369\pi\)
−0.844948 + 0.534848i \(0.820369\pi\)
\(270\) −26047.2 −0.0217446
\(271\) 1.28831e6 1.06561 0.532803 0.846240i \(-0.321139\pi\)
0.532803 + 0.846240i \(0.321139\pi\)
\(272\) 479930. 0.393329
\(273\) −804056. −0.652950
\(274\) −211395. −0.170105
\(275\) 66172.7 0.0527651
\(276\) 678150. 0.535862
\(277\) 1.13601e6 0.889572 0.444786 0.895637i \(-0.353280\pi\)
0.444786 + 0.895637i \(0.353280\pi\)
\(278\) 14286.4 0.0110869
\(279\) 204054. 0.156940
\(280\) 192375. 0.146640
\(281\) 1.45769e6 1.10128 0.550642 0.834742i \(-0.314383\pi\)
0.550642 + 0.834742i \(0.314383\pi\)
\(282\) −171335. −0.128299
\(283\) −2.50369e6 −1.85830 −0.929148 0.369709i \(-0.879457\pi\)
−0.929148 + 0.369709i \(0.879457\pi\)
\(284\) 1.09979e6 0.809124
\(285\) 395553. 0.288465
\(286\) 36813.9 0.0266132
\(287\) −1.67956e6 −1.20362
\(288\) 250946. 0.178279
\(289\) −1.14650e6 −0.807472
\(290\) 252599. 0.176375
\(291\) 803559. 0.556270
\(292\) −2.36215e6 −1.62125
\(293\) 1.43176e6 0.974322 0.487161 0.873312i \(-0.338032\pi\)
0.487161 + 0.873312i \(0.338032\pi\)
\(294\) 90170.1 0.0608407
\(295\) 117634. 0.0787005
\(296\) 437512. 0.290242
\(297\) 24326.4 0.0160025
\(298\) −473703. −0.309005
\(299\) −2.54585e6 −1.64685
\(300\) −551159. −0.353569
\(301\) −1.57459e6 −1.00173
\(302\) −234505. −0.147957
\(303\) 226627. 0.141810
\(304\) −1.19383e6 −0.740900
\(305\) −984415. −0.605938
\(306\) 44777.4 0.0273373
\(307\) −2.16520e6 −1.31115 −0.655574 0.755131i \(-0.727573\pi\)
−0.655574 + 0.755131i \(0.727573\pi\)
\(308\) −88235.9 −0.0529991
\(309\) −1.20408e6 −0.717394
\(310\) 90010.5 0.0531972
\(311\) 1.28475e6 0.753216 0.376608 0.926373i \(-0.377090\pi\)
0.376608 + 0.926373i \(0.377090\pi\)
\(312\) −624353. −0.363115
\(313\) −191903. −0.110719 −0.0553594 0.998466i \(-0.517630\pi\)
−0.0553594 + 0.998466i \(0.517630\pi\)
\(314\) 143675. 0.0822351
\(315\) −234370. −0.133084
\(316\) −1.90167e6 −1.07132
\(317\) 2.80920e6 1.57013 0.785064 0.619414i \(-0.212630\pi\)
0.785064 + 0.619414i \(0.212630\pi\)
\(318\) −157404. −0.0872867
\(319\) −235911. −0.129799
\(320\) −881935. −0.481462
\(321\) 645035. 0.349398
\(322\) −220887. −0.118722
\(323\) −679991. −0.362658
\(324\) −202617. −0.107229
\(325\) 2.06911e6 1.08661
\(326\) 319066. 0.166279
\(327\) 913134. 0.472243
\(328\) −1.30418e6 −0.669351
\(329\) −1.54165e6 −0.785230
\(330\) 10730.7 0.00542428
\(331\) −104756. −0.0525546 −0.0262773 0.999655i \(-0.508365\pi\)
−0.0262773 + 0.999655i \(0.508365\pi\)
\(332\) 364471. 0.181475
\(333\) −533020. −0.263410
\(334\) 604889. 0.296694
\(335\) −781899. −0.380661
\(336\) 707359. 0.341816
\(337\) −226616. −0.108697 −0.0543483 0.998522i \(-0.517308\pi\)
−0.0543483 + 0.998522i \(0.517308\pi\)
\(338\) 758534. 0.361147
\(339\) −2.11424e6 −0.999205
\(340\) −545637. −0.255980
\(341\) −84064.1 −0.0391493
\(342\) −111384. −0.0514943
\(343\) 2.25040e6 1.03282
\(344\) −1.22267e6 −0.557077
\(345\) −742075. −0.335660
\(346\) −295878. −0.132869
\(347\) −1.32731e6 −0.591766 −0.295883 0.955224i \(-0.595614\pi\)
−0.295883 + 0.955224i \(0.595614\pi\)
\(348\) 1.96493e6 0.869759
\(349\) −321857. −0.141449 −0.0707245 0.997496i \(-0.522531\pi\)
−0.0707245 + 0.997496i \(0.522531\pi\)
\(350\) 179524. 0.0783342
\(351\) 760647. 0.329545
\(352\) −103382. −0.0444723
\(353\) 982643. 0.419719 0.209860 0.977732i \(-0.432699\pi\)
0.209860 + 0.977732i \(0.432699\pi\)
\(354\) −33124.7 −0.0140489
\(355\) −1.20346e6 −0.506830
\(356\) −1.07926e6 −0.451337
\(357\) 402902. 0.167313
\(358\) 445994. 0.183917
\(359\) 937903. 0.384080 0.192040 0.981387i \(-0.438490\pi\)
0.192040 + 0.981387i \(0.438490\pi\)
\(360\) −181989. −0.0740098
\(361\) −784613. −0.316875
\(362\) −272489. −0.109289
\(363\) 1.43944e6 0.573358
\(364\) −2.75899e6 −1.09143
\(365\) 2.58481e6 1.01554
\(366\) 277203. 0.108167
\(367\) −4.14787e6 −1.60753 −0.803766 0.594946i \(-0.797174\pi\)
−0.803766 + 0.594946i \(0.797174\pi\)
\(368\) 2.23968e6 0.862118
\(369\) 1.58888e6 0.607471
\(370\) −235121. −0.0892868
\(371\) −1.41630e6 −0.534222
\(372\) 700178. 0.262332
\(373\) −2.43392e6 −0.905805 −0.452902 0.891560i \(-0.649611\pi\)
−0.452902 + 0.891560i \(0.649611\pi\)
\(374\) −18447.0 −0.00681939
\(375\) 1.55355e6 0.570487
\(376\) −1.19710e6 −0.436677
\(377\) −7.37655e6 −2.67301
\(378\) 65996.5 0.0237570
\(379\) −196709. −0.0703440 −0.0351720 0.999381i \(-0.511198\pi\)
−0.0351720 + 0.999381i \(0.511198\pi\)
\(380\) 1.35728e6 0.482181
\(381\) −259799. −0.0916906
\(382\) −270807. −0.0949513
\(383\) 3.93318e6 1.37008 0.685041 0.728504i \(-0.259784\pi\)
0.685041 + 0.728504i \(0.259784\pi\)
\(384\) 1.14060e6 0.394734
\(385\) 96553.3 0.0331983
\(386\) −833331. −0.284675
\(387\) 1.48958e6 0.505576
\(388\) 2.75729e6 0.929828
\(389\) 2.56292e6 0.858739 0.429369 0.903129i \(-0.358736\pi\)
0.429369 + 0.903129i \(0.358736\pi\)
\(390\) 335530. 0.111704
\(391\) 1.27569e6 0.421991
\(392\) 630009. 0.207077
\(393\) −1.07475e6 −0.351016
\(394\) 1.00842e6 0.327266
\(395\) 2.08093e6 0.671066
\(396\) 83472.3 0.0267488
\(397\) −4.68608e6 −1.49222 −0.746110 0.665822i \(-0.768081\pi\)
−0.746110 + 0.665822i \(0.768081\pi\)
\(398\) 734729. 0.232498
\(399\) −1.00222e6 −0.315161
\(400\) −1.82028e6 −0.568837
\(401\) 1.79185e6 0.556468 0.278234 0.960513i \(-0.410251\pi\)
0.278234 + 0.960513i \(0.410251\pi\)
\(402\) 220176. 0.0679524
\(403\) −2.62854e6 −0.806218
\(404\) 777635. 0.237040
\(405\) 221717. 0.0671678
\(406\) −640016. −0.192697
\(407\) 219588. 0.0657087
\(408\) 312855. 0.0930450
\(409\) 736446. 0.217687 0.108843 0.994059i \(-0.465285\pi\)
0.108843 + 0.994059i \(0.465285\pi\)
\(410\) 700874. 0.205911
\(411\) 1.79942e6 0.525445
\(412\) −4.13160e6 −1.19915
\(413\) −298052. −0.0859839
\(414\) 208962. 0.0599192
\(415\) −398827. −0.113675
\(416\) −3.23260e6 −0.915837
\(417\) −121607. −0.0342468
\(418\) 45887.0 0.0128455
\(419\) 5.71624e6 1.59065 0.795327 0.606181i \(-0.207299\pi\)
0.795327 + 0.606181i \(0.207299\pi\)
\(420\) −804203. −0.222455
\(421\) −547892. −0.150657 −0.0753286 0.997159i \(-0.524001\pi\)
−0.0753286 + 0.997159i \(0.524001\pi\)
\(422\) 639952. 0.174931
\(423\) 1.45842e6 0.396308
\(424\) −1.09977e6 −0.297088
\(425\) −1.03680e6 −0.278435
\(426\) 338885. 0.0904749
\(427\) 2.49424e6 0.662016
\(428\) 2.21334e6 0.584034
\(429\) −313364. −0.0822064
\(430\) 657071. 0.171373
\(431\) −1.20482e6 −0.312412 −0.156206 0.987724i \(-0.549926\pi\)
−0.156206 + 0.987724i \(0.549926\pi\)
\(432\) −669171. −0.172515
\(433\) −4.71811e6 −1.20934 −0.604669 0.796477i \(-0.706695\pi\)
−0.604669 + 0.796477i \(0.706695\pi\)
\(434\) −228062. −0.0581204
\(435\) −2.15015e6 −0.544811
\(436\) 3.13328e6 0.789373
\(437\) −3.17330e6 −0.794890
\(438\) −727861. −0.181286
\(439\) −3.47079e6 −0.859541 −0.429770 0.902938i \(-0.641406\pi\)
−0.429770 + 0.902938i \(0.641406\pi\)
\(440\) 74974.1 0.0184620
\(441\) −767538. −0.187933
\(442\) −576806. −0.140435
\(443\) −2.07936e6 −0.503409 −0.251705 0.967804i \(-0.580991\pi\)
−0.251705 + 0.967804i \(0.580991\pi\)
\(444\) −1.82897e6 −0.440301
\(445\) 1.18099e6 0.282714
\(446\) 1.42931e6 0.340244
\(447\) 4.03222e6 0.954498
\(448\) 2.23458e6 0.526019
\(449\) −3.92965e6 −0.919893 −0.459947 0.887947i \(-0.652131\pi\)
−0.459947 + 0.887947i \(0.652131\pi\)
\(450\) −169831. −0.0395355
\(451\) −654571. −0.151536
\(452\) −7.25468e6 −1.67021
\(453\) 1.99613e6 0.457029
\(454\) 305640. 0.0695938
\(455\) 3.01906e6 0.683666
\(456\) −778231. −0.175266
\(457\) −490813. −0.109932 −0.0549662 0.998488i \(-0.517505\pi\)
−0.0549662 + 0.998488i \(0.517505\pi\)
\(458\) 900806. 0.200663
\(459\) −381150. −0.0844432
\(460\) −2.54631e6 −0.561070
\(461\) −6.08206e6 −1.33290 −0.666452 0.745548i \(-0.732188\pi\)
−0.666452 + 0.745548i \(0.732188\pi\)
\(462\) −27188.6 −0.00592627
\(463\) 8.92213e6 1.93427 0.967133 0.254271i \(-0.0818354\pi\)
0.967133 + 0.254271i \(0.0818354\pi\)
\(464\) 6.48944e6 1.39930
\(465\) −766179. −0.164323
\(466\) 1.60404e6 0.342176
\(467\) 1.91638e6 0.406621 0.203310 0.979114i \(-0.434830\pi\)
0.203310 + 0.979114i \(0.434830\pi\)
\(468\) 2.61004e6 0.550849
\(469\) 1.98112e6 0.415890
\(470\) 643327. 0.134334
\(471\) −1.22298e6 −0.254019
\(472\) −231439. −0.0478169
\(473\) −613663. −0.126118
\(474\) −585972. −0.119793
\(475\) 2.57907e6 0.524479
\(476\) 1.38249e6 0.279670
\(477\) 1.33984e6 0.269623
\(478\) −176568. −0.0353461
\(479\) −2.38919e6 −0.475786 −0.237893 0.971291i \(-0.576457\pi\)
−0.237893 + 0.971291i \(0.576457\pi\)
\(480\) −942251. −0.186665
\(481\) 6.86616e6 1.35317
\(482\) −241686. −0.0473843
\(483\) 1.88021e6 0.366724
\(484\) 4.93920e6 0.958392
\(485\) −3.01720e6 −0.582437
\(486\) −62433.5 −0.0119902
\(487\) −876099. −0.167390 −0.0836952 0.996491i \(-0.526672\pi\)
−0.0836952 + 0.996491i \(0.526672\pi\)
\(488\) 1.93679e6 0.368156
\(489\) −2.71592e6 −0.513624
\(490\) −338570. −0.0637027
\(491\) 2.47004e6 0.462381 0.231190 0.972909i \(-0.425738\pi\)
0.231190 + 0.972909i \(0.425738\pi\)
\(492\) 5.45199e6 1.01541
\(493\) 3.69629e6 0.684935
\(494\) 1.43481e6 0.264532
\(495\) −91340.7 −0.0167553
\(496\) 2.31243e6 0.422051
\(497\) 3.04925e6 0.553735
\(498\) 112306. 0.0202923
\(499\) −1.91170e6 −0.343691 −0.171846 0.985124i \(-0.554973\pi\)
−0.171846 + 0.985124i \(0.554973\pi\)
\(500\) 5.33075e6 0.953593
\(501\) −5.14888e6 −0.916471
\(502\) 1.42199e6 0.251848
\(503\) 4.85098e6 0.854888 0.427444 0.904042i \(-0.359414\pi\)
0.427444 + 0.904042i \(0.359414\pi\)
\(504\) 461111. 0.0808591
\(505\) −850938. −0.148480
\(506\) −86086.0 −0.0149471
\(507\) −6.45673e6 −1.11556
\(508\) −891459. −0.153265
\(509\) 7.45465e6 1.27536 0.637680 0.770301i \(-0.279894\pi\)
0.637680 + 0.770301i \(0.279894\pi\)
\(510\) −168130. −0.0286233
\(511\) −6.54921e6 −1.10952
\(512\) 4.79679e6 0.808679
\(513\) 948117. 0.159063
\(514\) −204685. −0.0341726
\(515\) 4.52106e6 0.751142
\(516\) 5.11126e6 0.845092
\(517\) −600826. −0.0988605
\(518\) 595733. 0.0975500
\(519\) 2.51854e6 0.410423
\(520\) 2.34432e6 0.380196
\(521\) 3.66620e6 0.591727 0.295864 0.955230i \(-0.404393\pi\)
0.295864 + 0.955230i \(0.404393\pi\)
\(522\) 605463. 0.0972549
\(523\) 9.66704e6 1.54539 0.772697 0.634774i \(-0.218907\pi\)
0.772697 + 0.634774i \(0.218907\pi\)
\(524\) −3.68784e6 −0.586738
\(525\) −1.52812e6 −0.241969
\(526\) −1.36658e6 −0.215363
\(527\) 1.31713e6 0.206586
\(528\) 275678. 0.0430346
\(529\) −483104. −0.0750587
\(530\) 591020. 0.0913928
\(531\) 281961. 0.0433963
\(532\) −3.43897e6 −0.526805
\(533\) −2.04674e7 −3.12064
\(534\) −332558. −0.0504677
\(535\) −2.42197e6 −0.365835
\(536\) 1.53835e6 0.231282
\(537\) −3.79635e6 −0.568107
\(538\) −2.12054e6 −0.315857
\(539\) 316203. 0.0468807
\(540\) 760786. 0.112274
\(541\) 7.32547e6 1.07608 0.538038 0.842921i \(-0.319166\pi\)
0.538038 + 0.842921i \(0.319166\pi\)
\(542\) 1.36215e6 0.199171
\(543\) 2.31946e6 0.337588
\(544\) 1.61981e6 0.234675
\(545\) −3.42863e6 −0.494458
\(546\) −850142. −0.122042
\(547\) 6.36157e6 0.909067 0.454533 0.890730i \(-0.349806\pi\)
0.454533 + 0.890730i \(0.349806\pi\)
\(548\) 6.17441e6 0.878303
\(549\) −2.35958e6 −0.334121
\(550\) 69965.5 0.00986228
\(551\) −9.19458e6 −1.29019
\(552\) 1.45999e6 0.203941
\(553\) −5.27251e6 −0.733170
\(554\) 1.20112e6 0.166269
\(555\) 2.00138e6 0.275802
\(556\) −417277. −0.0572449
\(557\) 4.74906e6 0.648589 0.324295 0.945956i \(-0.394873\pi\)
0.324295 + 0.945956i \(0.394873\pi\)
\(558\) 215749. 0.0293335
\(559\) −1.91882e7 −2.59720
\(560\) −2.65599e6 −0.357895
\(561\) 157023. 0.0210647
\(562\) 1.54124e6 0.205840
\(563\) 3.15470e6 0.419457 0.209729 0.977760i \(-0.432742\pi\)
0.209729 + 0.977760i \(0.432742\pi\)
\(564\) 5.00435e6 0.662445
\(565\) 7.93853e6 1.04621
\(566\) −2.64720e6 −0.347332
\(567\) −561770. −0.0733839
\(568\) 2.36775e6 0.307940
\(569\) 8.64099e6 1.11888 0.559439 0.828872i \(-0.311017\pi\)
0.559439 + 0.828872i \(0.311017\pi\)
\(570\) 418225. 0.0539167
\(571\) 6.75178e6 0.866619 0.433310 0.901245i \(-0.357346\pi\)
0.433310 + 0.901245i \(0.357346\pi\)
\(572\) −1.07526e6 −0.137411
\(573\) 2.30514e6 0.293299
\(574\) −1.77582e6 −0.224967
\(575\) −4.83843e6 −0.610288
\(576\) −2.11394e6 −0.265483
\(577\) −80122.8 −0.0100188 −0.00500941 0.999987i \(-0.501595\pi\)
−0.00500941 + 0.999987i \(0.501595\pi\)
\(578\) −1.21221e6 −0.150924
\(579\) 7.09341e6 0.879343
\(580\) −7.37790e6 −0.910674
\(581\) 1.01052e6 0.124195
\(582\) 849616. 0.103972
\(583\) −551975. −0.0672586
\(584\) −5.08549e6 −0.617022
\(585\) −2.85607e6 −0.345048
\(586\) 1.51383e6 0.182110
\(587\) 8.20721e6 0.983106 0.491553 0.870848i \(-0.336429\pi\)
0.491553 + 0.870848i \(0.336429\pi\)
\(588\) −2.63369e6 −0.314138
\(589\) −3.27638e6 −0.389140
\(590\) 124376. 0.0147098
\(591\) −8.58378e6 −1.01090
\(592\) −6.04043e6 −0.708375
\(593\) 3.34469e6 0.390588 0.195294 0.980745i \(-0.437434\pi\)
0.195294 + 0.980745i \(0.437434\pi\)
\(594\) 25720.7 0.00299101
\(595\) −1.51281e6 −0.175183
\(596\) 1.38359e7 1.59548
\(597\) −6.25409e6 −0.718172
\(598\) −2.69177e6 −0.307811
\(599\) −1.54909e7 −1.76405 −0.882024 0.471204i \(-0.843819\pi\)
−0.882024 + 0.471204i \(0.843819\pi\)
\(600\) −1.18660e6 −0.134563
\(601\) −6.02417e6 −0.680317 −0.340158 0.940368i \(-0.610481\pi\)
−0.340158 + 0.940368i \(0.610481\pi\)
\(602\) −1.66484e6 −0.187232
\(603\) −1.87416e6 −0.209901
\(604\) 6.84941e6 0.763943
\(605\) −5.40479e6 −0.600330
\(606\) 239617. 0.0265055
\(607\) −1.80813e7 −1.99186 −0.995930 0.0901349i \(-0.971270\pi\)
−0.995930 + 0.0901349i \(0.971270\pi\)
\(608\) −4.02930e6 −0.442050
\(609\) 5.44789e6 0.595231
\(610\) −1.04084e6 −0.113255
\(611\) −1.87868e7 −2.03587
\(612\) −1.30786e6 −0.141150
\(613\) 1.63982e7 1.76256 0.881281 0.472593i \(-0.156682\pi\)
0.881281 + 0.472593i \(0.156682\pi\)
\(614\) −2.28930e6 −0.245065
\(615\) −5.96592e6 −0.636047
\(616\) −189964. −0.0201706
\(617\) −1.39449e7 −1.47470 −0.737350 0.675511i \(-0.763923\pi\)
−0.737350 + 0.675511i \(0.763923\pi\)
\(618\) −1.27309e6 −0.134087
\(619\) 3.47959e6 0.365007 0.182504 0.983205i \(-0.441580\pi\)
0.182504 + 0.983205i \(0.441580\pi\)
\(620\) −2.62902e6 −0.274673
\(621\) −1.77871e6 −0.185087
\(622\) 1.35839e6 0.140783
\(623\) −2.99232e6 −0.308878
\(624\) 8.62000e6 0.886229
\(625\) 363714. 0.0372443
\(626\) −202902. −0.0206943
\(627\) −390596. −0.0396788
\(628\) −4.19646e6 −0.424604
\(629\) −3.44054e6 −0.346737
\(630\) −247803. −0.0248746
\(631\) 1.90874e6 0.190842 0.0954210 0.995437i \(-0.469580\pi\)
0.0954210 + 0.995437i \(0.469580\pi\)
\(632\) −4.09413e6 −0.407726
\(633\) −5.44735e6 −0.540351
\(634\) 2.97022e6 0.293471
\(635\) 975491. 0.0960039
\(636\) 4.59745e6 0.450687
\(637\) 9.88714e6 0.965432
\(638\) −249433. −0.0242606
\(639\) −2.88463e6 −0.279471
\(640\) −4.28271e6 −0.413303
\(641\) 1.01908e7 0.979632 0.489816 0.871826i \(-0.337064\pi\)
0.489816 + 0.871826i \(0.337064\pi\)
\(642\) 682006. 0.0653057
\(643\) 1.55416e7 1.48241 0.741205 0.671279i \(-0.234255\pi\)
0.741205 + 0.671279i \(0.234255\pi\)
\(644\) 6.45166e6 0.612995
\(645\) −5.59306e6 −0.529359
\(646\) −718966. −0.0677839
\(647\) −3.01771e6 −0.283411 −0.141705 0.989909i \(-0.545259\pi\)
−0.141705 + 0.989909i \(0.545259\pi\)
\(648\) −436216. −0.0408098
\(649\) −116160. −0.0108254
\(650\) 2.18771e6 0.203098
\(651\) 1.94129e6 0.179530
\(652\) −9.31927e6 −0.858544
\(653\) −5.97669e6 −0.548502 −0.274251 0.961658i \(-0.588430\pi\)
−0.274251 + 0.961658i \(0.588430\pi\)
\(654\) 965472. 0.0882664
\(655\) 4.03547e6 0.367528
\(656\) 1.80059e7 1.63364
\(657\) 6.19563e6 0.559980
\(658\) −1.63001e6 −0.146766
\(659\) −3.91411e6 −0.351091 −0.175546 0.984471i \(-0.556169\pi\)
−0.175546 + 0.984471i \(0.556169\pi\)
\(660\) −313421. −0.0280071
\(661\) 7.35500e6 0.654755 0.327378 0.944894i \(-0.393835\pi\)
0.327378 + 0.944894i \(0.393835\pi\)
\(662\) −110761. −0.00982293
\(663\) 4.90983e6 0.433794
\(664\) 784673. 0.0690667
\(665\) 3.76314e6 0.329987
\(666\) −563571. −0.0492337
\(667\) 1.72494e7 1.50127
\(668\) −1.76676e7 −1.53192
\(669\) −1.21665e7 −1.05099
\(670\) −826715. −0.0711490
\(671\) 972077. 0.0833478
\(672\) 2.38741e6 0.203940
\(673\) −1.68761e6 −0.143627 −0.0718134 0.997418i \(-0.522879\pi\)
−0.0718134 + 0.997418i \(0.522879\pi\)
\(674\) −239605. −0.0203164
\(675\) 1.44562e6 0.122123
\(676\) −2.21553e7 −1.86470
\(677\) −8.90194e6 −0.746471 −0.373236 0.927737i \(-0.621752\pi\)
−0.373236 + 0.927737i \(0.621752\pi\)
\(678\) −2.23542e6 −0.186760
\(679\) 7.64475e6 0.636340
\(680\) −1.17471e6 −0.0974220
\(681\) −2.60164e6 −0.214971
\(682\) −88882.3 −0.00731736
\(683\) 6.04714e6 0.496019 0.248010 0.968758i \(-0.420224\pi\)
0.248010 + 0.968758i \(0.420224\pi\)
\(684\) 3.25331e6 0.265880
\(685\) −6.75643e6 −0.550163
\(686\) 2.37938e6 0.193043
\(687\) −7.66776e6 −0.619836
\(688\) 1.68806e7 1.35962
\(689\) −1.72593e7 −1.38508
\(690\) −784608. −0.0627379
\(691\) −3.65486e6 −0.291189 −0.145595 0.989344i \(-0.546509\pi\)
−0.145595 + 0.989344i \(0.546509\pi\)
\(692\) 8.64199e6 0.686039
\(693\) 231432. 0.0183059
\(694\) −1.40339e6 −0.110606
\(695\) 456610. 0.0358578
\(696\) 4.23031e6 0.331016
\(697\) 1.02559e7 0.799637
\(698\) −340305. −0.0264381
\(699\) −1.36538e7 −1.05696
\(700\) −5.24352e6 −0.404462
\(701\) −1.63664e7 −1.25793 −0.628966 0.777432i \(-0.716522\pi\)
−0.628966 + 0.777432i \(0.716522\pi\)
\(702\) 804245. 0.0615950
\(703\) 8.55840e6 0.653137
\(704\) 870882. 0.0662258
\(705\) −5.47607e6 −0.414951
\(706\) 1.03896e6 0.0784493
\(707\) 2.15604e6 0.162222
\(708\) 967505. 0.0725387
\(709\) −2.45108e7 −1.83123 −0.915613 0.402061i \(-0.868294\pi\)
−0.915613 + 0.402061i \(0.868294\pi\)
\(710\) −1.27244e6 −0.0947310
\(711\) 4.98786e6 0.370033
\(712\) −2.32355e6 −0.171772
\(713\) 6.14662e6 0.452806
\(714\) 425995. 0.0312722
\(715\) 1.17662e6 0.0860735
\(716\) −1.30266e7 −0.949615
\(717\) 1.50296e6 0.109182
\(718\) 991661. 0.0717880
\(719\) 1.16640e7 0.841446 0.420723 0.907189i \(-0.361776\pi\)
0.420723 + 0.907189i \(0.361776\pi\)
\(720\) 2.51260e6 0.180631
\(721\) −1.14551e7 −0.820657
\(722\) −829584. −0.0592267
\(723\) 2.05726e6 0.146367
\(724\) 7.95886e6 0.564293
\(725\) −1.40193e7 −0.990560
\(726\) 1.52194e6 0.107166
\(727\) 1.76785e7 1.24054 0.620268 0.784390i \(-0.287024\pi\)
0.620268 + 0.784390i \(0.287024\pi\)
\(728\) −5.93985e6 −0.415382
\(729\) 531441. 0.0370370
\(730\) 2.73297e6 0.189813
\(731\) 9.61497e6 0.665510
\(732\) −8.09653e6 −0.558497
\(733\) 1.31251e6 0.0902279 0.0451140 0.998982i \(-0.485635\pi\)
0.0451140 + 0.998982i \(0.485635\pi\)
\(734\) −4.38561e6 −0.300462
\(735\) 2.88195e6 0.196774
\(736\) 7.55914e6 0.514373
\(737\) 772099. 0.0523606
\(738\) 1.67995e6 0.113542
\(739\) 2.62184e6 0.176602 0.0883010 0.996094i \(-0.471856\pi\)
0.0883010 + 0.996094i \(0.471856\pi\)
\(740\) 6.86741e6 0.461014
\(741\) −1.22133e7 −0.817122
\(742\) −1.49748e6 −0.0998509
\(743\) 7.41645e6 0.492861 0.246430 0.969161i \(-0.420742\pi\)
0.246430 + 0.969161i \(0.420742\pi\)
\(744\) 1.50742e6 0.0998394
\(745\) −1.51401e7 −0.999399
\(746\) −2.57343e6 −0.169303
\(747\) −955964. −0.0626816
\(748\) 538798. 0.0352105
\(749\) 6.13662e6 0.399691
\(750\) 1.64259e6 0.106629
\(751\) 1.89341e7 1.22503 0.612514 0.790460i \(-0.290158\pi\)
0.612514 + 0.790460i \(0.290158\pi\)
\(752\) 1.65275e7 1.06577
\(753\) −1.21042e7 −0.777942
\(754\) −7.79935e6 −0.499609
\(755\) −7.49506e6 −0.478528
\(756\) −1.92762e6 −0.122664
\(757\) −3.95313e6 −0.250727 −0.125364 0.992111i \(-0.540010\pi\)
−0.125364 + 0.992111i \(0.540010\pi\)
\(758\) −207984. −0.0131479
\(759\) 732774. 0.0461706
\(760\) 2.92210e6 0.183510
\(761\) 2.27391e7 1.42335 0.711675 0.702509i \(-0.247937\pi\)
0.711675 + 0.702509i \(0.247937\pi\)
\(762\) −274690. −0.0171378
\(763\) 8.68721e6 0.540218
\(764\) 7.90971e6 0.490261
\(765\) 1.43114e6 0.0884155
\(766\) 4.15862e6 0.256081
\(767\) −3.63212e6 −0.222931
\(768\) −6.31027e6 −0.386051
\(769\) −6.06544e6 −0.369867 −0.184934 0.982751i \(-0.559207\pi\)
−0.184934 + 0.982751i \(0.559207\pi\)
\(770\) 102087. 0.00620505
\(771\) 1.74230e6 0.105557
\(772\) 2.43399e7 1.46986
\(773\) 2.81598e7 1.69504 0.847521 0.530762i \(-0.178094\pi\)
0.847521 + 0.530762i \(0.178094\pi\)
\(774\) 1.57496e6 0.0944967
\(775\) −4.99560e6 −0.298768
\(776\) 5.93618e6 0.353878
\(777\) −5.07095e6 −0.301326
\(778\) 2.70982e6 0.160506
\(779\) −2.55118e7 −1.50625
\(780\) −9.80016e6 −0.576762
\(781\) 1.18838e6 0.0697153
\(782\) 1.34881e6 0.0788740
\(783\) −5.15377e6 −0.300414
\(784\) −8.69810e6 −0.505399
\(785\) 4.59203e6 0.265969
\(786\) −1.13635e6 −0.0656080
\(787\) 1.35985e7 0.782627 0.391313 0.920257i \(-0.372021\pi\)
0.391313 + 0.920257i \(0.372021\pi\)
\(788\) −2.94539e7 −1.68977
\(789\) 1.16325e7 0.665243
\(790\) 2.20020e6 0.125428
\(791\) −2.01141e7 −1.14303
\(792\) 179708. 0.0101802
\(793\) 3.03952e7 1.71642
\(794\) −4.95467e6 −0.278909
\(795\) −5.03083e6 −0.282307
\(796\) −2.14599e7 −1.20045
\(797\) 3.47277e6 0.193656 0.0968279 0.995301i \(-0.469130\pi\)
0.0968279 + 0.995301i \(0.469130\pi\)
\(798\) −1.05967e6 −0.0589064
\(799\) 9.41384e6 0.521675
\(800\) −6.14361e6 −0.339390
\(801\) 2.83077e6 0.155892
\(802\) 1.89455e6 0.104009
\(803\) −2.55242e6 −0.139689
\(804\) −6.43089e6 −0.350858
\(805\) −7.05981e6 −0.383976
\(806\) −2.77920e6 −0.150689
\(807\) 1.80503e7 0.975662
\(808\) 1.67418e6 0.0902138
\(809\) −4.11626e6 −0.221122 −0.110561 0.993869i \(-0.535265\pi\)
−0.110561 + 0.993869i \(0.535265\pi\)
\(810\) 234425. 0.0125543
\(811\) −1.46103e7 −0.780022 −0.390011 0.920810i \(-0.627529\pi\)
−0.390011 + 0.920810i \(0.627529\pi\)
\(812\) 1.86936e7 0.994953
\(813\) −1.15948e7 −0.615227
\(814\) 232174. 0.0122815
\(815\) 1.01977e7 0.537786
\(816\) −4.31937e6 −0.227089
\(817\) −2.39174e7 −1.25360
\(818\) 778656. 0.0406876
\(819\) 7.23650e6 0.376981
\(820\) −2.04711e7 −1.06318
\(821\) −2.36586e7 −1.22498 −0.612492 0.790477i \(-0.709833\pi\)
−0.612492 + 0.790477i \(0.709833\pi\)
\(822\) 1.90255e6 0.0982104
\(823\) 2.73951e6 0.140985 0.0704925 0.997512i \(-0.477543\pi\)
0.0704925 + 0.997512i \(0.477543\pi\)
\(824\) −8.89495e6 −0.456379
\(825\) −595554. −0.0304640
\(826\) −315135. −0.0160712
\(827\) 2.71855e7 1.38221 0.691103 0.722756i \(-0.257125\pi\)
0.691103 + 0.722756i \(0.257125\pi\)
\(828\) −6.10335e6 −0.309380
\(829\) −2.81195e6 −0.142109 −0.0710545 0.997472i \(-0.522636\pi\)
−0.0710545 + 0.997472i \(0.522636\pi\)
\(830\) −421687. −0.0212469
\(831\) −1.02241e7 −0.513595
\(832\) 2.72310e7 1.36382
\(833\) −4.95432e6 −0.247384
\(834\) −128578. −0.00640103
\(835\) 1.93330e7 0.959583
\(836\) −1.34027e6 −0.0663248
\(837\) −1.83648e6 −0.0906094
\(838\) 6.04388e6 0.297308
\(839\) −2.10069e6 −0.103028 −0.0515142 0.998672i \(-0.516405\pi\)
−0.0515142 + 0.998672i \(0.516405\pi\)
\(840\) −1.73137e6 −0.0846628
\(841\) 2.94688e7 1.43672
\(842\) −579296. −0.0281592
\(843\) −1.31192e7 −0.635827
\(844\) −1.86917e7 −0.903219
\(845\) 2.42437e7 1.16804
\(846\) 1.54201e6 0.0740734
\(847\) 1.36943e7 0.655888
\(848\) 1.51837e7 0.725084
\(849\) 2.25332e7 1.07289
\(850\) −1.09623e6 −0.0520421
\(851\) −1.60559e7 −0.759995
\(852\) −9.89814e6 −0.467148
\(853\) 1.91254e7 0.899991 0.449996 0.893031i \(-0.351426\pi\)
0.449996 + 0.893031i \(0.351426\pi\)
\(854\) 2.63720e6 0.123737
\(855\) −3.55998e6 −0.166545
\(856\) 4.76511e6 0.222274
\(857\) −1.08164e7 −0.503072 −0.251536 0.967848i \(-0.580936\pi\)
−0.251536 + 0.967848i \(0.580936\pi\)
\(858\) −331325. −0.0153651
\(859\) −4.44481e6 −0.205528 −0.102764 0.994706i \(-0.532769\pi\)
−0.102764 + 0.994706i \(0.532769\pi\)
\(860\) −1.91917e7 −0.884846
\(861\) 1.51160e7 0.694911
\(862\) −1.27387e6 −0.0583926
\(863\) 3.43899e7 1.57183 0.785913 0.618337i \(-0.212193\pi\)
0.785913 + 0.618337i \(0.212193\pi\)
\(864\) −2.25852e6 −0.102929
\(865\) −9.45661e6 −0.429730
\(866\) −4.98853e6 −0.226036
\(867\) 1.03185e7 0.466194
\(868\) 6.66123e6 0.300092
\(869\) −2.05485e6 −0.0923062
\(870\) −2.27339e6 −0.101830
\(871\) 2.41423e7 1.07828
\(872\) 6.74566e6 0.300423
\(873\) −7.23203e6 −0.321162
\(874\) −3.35518e6 −0.148572
\(875\) 1.47798e7 0.652604
\(876\) 2.12593e7 0.936030
\(877\) 2.34437e7 1.02926 0.514632 0.857411i \(-0.327929\pi\)
0.514632 + 0.857411i \(0.327929\pi\)
\(878\) −3.66972e6 −0.160656
\(879\) −1.28859e7 −0.562525
\(880\) −1.03511e6 −0.0450590
\(881\) 1.52635e7 0.662542 0.331271 0.943536i \(-0.392523\pi\)
0.331271 + 0.943536i \(0.392523\pi\)
\(882\) −811531. −0.0351264
\(883\) 1.88369e7 0.813034 0.406517 0.913643i \(-0.366743\pi\)
0.406517 + 0.913643i \(0.366743\pi\)
\(884\) 1.68473e7 0.725104
\(885\) −1.05870e6 −0.0454377
\(886\) −2.19855e6 −0.0940917
\(887\) 3.45462e7 1.47432 0.737160 0.675718i \(-0.236166\pi\)
0.737160 + 0.675718i \(0.236166\pi\)
\(888\) −3.93761e6 −0.167572
\(889\) −2.47163e6 −0.104889
\(890\) 1.24868e6 0.0528418
\(891\) −218938. −0.00923903
\(892\) −4.17474e7 −1.75678
\(893\) −2.34171e7 −0.982661
\(894\) 4.26333e6 0.178404
\(895\) 1.42545e7 0.594832
\(896\) 1.08512e7 0.451553
\(897\) 2.29126e7 0.950810
\(898\) −4.15488e6 −0.171936
\(899\) 1.78097e7 0.734951
\(900\) 4.96043e6 0.204133
\(901\) 8.64843e6 0.354916
\(902\) −692089. −0.0283234
\(903\) 1.41713e7 0.578349
\(904\) −1.56186e7 −0.635656
\(905\) −8.70909e6 −0.353469
\(906\) 2.11054e6 0.0854228
\(907\) −1.49504e7 −0.603440 −0.301720 0.953397i \(-0.597561\pi\)
−0.301720 + 0.953397i \(0.597561\pi\)
\(908\) −8.92713e6 −0.359333
\(909\) −2.03964e6 −0.0818738
\(910\) 3.19211e6 0.127783
\(911\) −4.82835e7 −1.92754 −0.963768 0.266741i \(-0.914053\pi\)
−0.963768 + 0.266741i \(0.914053\pi\)
\(912\) 1.07445e7 0.427759
\(913\) 393829. 0.0156362
\(914\) −518945. −0.0205473
\(915\) 8.85973e6 0.349839
\(916\) −2.63107e7 −1.03608
\(917\) −1.02248e7 −0.401541
\(918\) −402997. −0.0157832
\(919\) 2.12186e7 0.828758 0.414379 0.910105i \(-0.363999\pi\)
0.414379 + 0.910105i \(0.363999\pi\)
\(920\) −5.48198e6 −0.213534
\(921\) 1.94868e7 0.756991
\(922\) −6.43067e6 −0.249132
\(923\) 3.71587e7 1.43567
\(924\) 794123. 0.0305991
\(925\) 1.30493e7 0.501455
\(926\) 9.43352e6 0.361532
\(927\) 1.08367e7 0.414188
\(928\) 2.19025e7 0.834879
\(929\) 1.47484e6 0.0560667 0.0280333 0.999607i \(-0.491076\pi\)
0.0280333 + 0.999607i \(0.491076\pi\)
\(930\) −810094. −0.0307134
\(931\) 1.23239e7 0.465988
\(932\) −4.68507e7 −1.76676
\(933\) −1.15628e7 −0.434869
\(934\) 2.02622e6 0.0760011
\(935\) −589587. −0.0220556
\(936\) 5.61918e6 0.209644
\(937\) −2.87501e7 −1.06977 −0.534885 0.844925i \(-0.679645\pi\)
−0.534885 + 0.844925i \(0.679645\pi\)
\(938\) 2.09467e6 0.0777335
\(939\) 1.72713e6 0.0639235
\(940\) −1.87903e7 −0.693607
\(941\) 4.42692e7 1.62978 0.814888 0.579618i \(-0.196798\pi\)
0.814888 + 0.579618i \(0.196798\pi\)
\(942\) −1.29308e6 −0.0474785
\(943\) 4.78611e7 1.75268
\(944\) 3.19531e6 0.116703
\(945\) 2.10933e6 0.0768360
\(946\) −648836. −0.0235726
\(947\) −8.37611e6 −0.303506 −0.151753 0.988418i \(-0.548492\pi\)
−0.151753 + 0.988418i \(0.548492\pi\)
\(948\) 1.71151e7 0.618526
\(949\) −7.98098e7 −2.87668
\(950\) 2.72689e6 0.0980299
\(951\) −2.52828e7 −0.906514
\(952\) 2.97638e6 0.106438
\(953\) 1.04083e7 0.371232 0.185616 0.982622i \(-0.440572\pi\)
0.185616 + 0.982622i \(0.440572\pi\)
\(954\) 1.41664e6 0.0503950
\(955\) −8.65531e6 −0.307096
\(956\) 5.15719e6 0.182502
\(957\) 2.12320e6 0.0749396
\(958\) −2.52613e6 −0.0889287
\(959\) 1.71190e7 0.601078
\(960\) 7.93742e6 0.277972
\(961\) −2.22829e7 −0.778328
\(962\) 7.25970e6 0.252919
\(963\) −5.80532e6 −0.201725
\(964\) 7.05917e6 0.244659
\(965\) −2.66343e7 −0.920709
\(966\) 1.98798e6 0.0685440
\(967\) −3.41560e7 −1.17463 −0.587315 0.809358i \(-0.699815\pi\)
−0.587315 + 0.809358i \(0.699815\pi\)
\(968\) 1.06336e7 0.364749
\(969\) 6.11992e6 0.209380
\(970\) −3.19013e6 −0.108863
\(971\) −2.80041e7 −0.953176 −0.476588 0.879127i \(-0.658127\pi\)
−0.476588 + 0.879127i \(0.658127\pi\)
\(972\) 1.82356e6 0.0619090
\(973\) −1.15693e6 −0.0391763
\(974\) −926314. −0.0312868
\(975\) −1.86220e7 −0.627357
\(976\) −2.67399e7 −0.898534
\(977\) −4.46833e6 −0.149764 −0.0748822 0.997192i \(-0.523858\pi\)
−0.0748822 + 0.997192i \(0.523858\pi\)
\(978\) −2.87159e6 −0.0960010
\(979\) −1.16619e6 −0.0388878
\(980\) 9.88894e6 0.328916
\(981\) −8.21821e6 −0.272649
\(982\) 2.61161e6 0.0864231
\(983\) −2.39250e7 −0.789711 −0.394856 0.918743i \(-0.629205\pi\)
−0.394856 + 0.918743i \(0.629205\pi\)
\(984\) 1.17376e7 0.386450
\(985\) 3.22303e7 1.05846
\(986\) 3.90815e6 0.128020
\(987\) 1.38749e7 0.453353
\(988\) −4.19079e7 −1.36585
\(989\) 4.48700e7 1.45870
\(990\) −96576.0 −0.00313171
\(991\) −7.93514e6 −0.256667 −0.128334 0.991731i \(-0.540963\pi\)
−0.128334 + 0.991731i \(0.540963\pi\)
\(992\) 7.80469e6 0.251812
\(993\) 942808. 0.0303424
\(994\) 3.22402e6 0.103498
\(995\) 2.34828e7 0.751956
\(996\) −3.28024e6 −0.104775
\(997\) −7.07884e6 −0.225540 −0.112770 0.993621i \(-0.535972\pi\)
−0.112770 + 0.993621i \(0.535972\pi\)
\(998\) −2.02127e6 −0.0642390
\(999\) 4.79718e6 0.152080
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 177.6.a.d.1.7 13
3.2 odd 2 531.6.a.e.1.7 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.d.1.7 13 1.1 even 1 trivial
531.6.a.e.1.7 13 3.2 odd 2