Properties

Label 43.6.a.a.1.7
Level $43$
Weight $6$
Character 43.1
Self dual yes
Analytic conductor $6.897$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.89650425196\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(7.21373\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.21373 q^{2} -11.2683 q^{3} -4.81697 q^{4} -9.80186 q^{5} -58.7500 q^{6} -11.1041 q^{7} -191.954 q^{8} -116.025 q^{9} +O(q^{10})\) \(q+5.21373 q^{2} -11.2683 q^{3} -4.81697 q^{4} -9.80186 q^{5} -58.7500 q^{6} -11.1041 q^{7} -191.954 q^{8} -116.025 q^{9} -51.1043 q^{10} -557.274 q^{11} +54.2791 q^{12} +107.663 q^{13} -57.8938 q^{14} +110.450 q^{15} -846.654 q^{16} +329.005 q^{17} -604.924 q^{18} +2938.10 q^{19} +47.2153 q^{20} +125.124 q^{21} -2905.48 q^{22} -385.537 q^{23} +2163.00 q^{24} -3028.92 q^{25} +561.326 q^{26} +4045.61 q^{27} +53.4881 q^{28} -3309.15 q^{29} +575.859 q^{30} -5471.71 q^{31} +1728.30 q^{32} +6279.54 q^{33} +1715.34 q^{34} +108.841 q^{35} +558.890 q^{36} +4832.22 q^{37} +15318.5 q^{38} -1213.18 q^{39} +1881.51 q^{40} +1065.16 q^{41} +652.365 q^{42} -1849.00 q^{43} +2684.37 q^{44} +1137.26 q^{45} -2010.09 q^{46} -8991.95 q^{47} +9540.36 q^{48} -16683.7 q^{49} -15792.0 q^{50} -3707.33 q^{51} -518.609 q^{52} -10216.7 q^{53} +21092.7 q^{54} +5462.32 q^{55} +2131.47 q^{56} -33107.4 q^{57} -17253.0 q^{58} -27457.0 q^{59} -532.036 q^{60} -36692.9 q^{61} -28528.1 q^{62} +1288.35 q^{63} +36103.8 q^{64} -1055.30 q^{65} +32739.9 q^{66} -26272.2 q^{67} -1584.81 q^{68} +4344.35 q^{69} +567.467 q^{70} +55130.3 q^{71} +22271.5 q^{72} -9315.14 q^{73} +25193.9 q^{74} +34130.9 q^{75} -14152.7 q^{76} +6188.02 q^{77} -6325.20 q^{78} +56826.5 q^{79} +8298.78 q^{80} -17393.1 q^{81} +5553.44 q^{82} -5858.19 q^{83} -602.721 q^{84} -3224.86 q^{85} -9640.20 q^{86} +37288.5 q^{87} +106971. q^{88} -42815.7 q^{89} +5929.38 q^{90} -1195.50 q^{91} +1857.12 q^{92} +61657.0 q^{93} -46881.7 q^{94} -28798.8 q^{95} -19475.0 q^{96} -231.671 q^{97} -86984.4 q^{98} +64657.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} + O(q^{10}) \) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} - 617 q^{10} - 532 q^{11} - 4195 q^{12} - 2492 q^{13} - 4240 q^{14} - 1780 q^{15} + 1882 q^{16} - 2534 q^{17} - 3711 q^{18} - 1678 q^{19} - 2607 q^{20} - 2256 q^{21} + 11502 q^{22} - 2488 q^{23} + 19953 q^{24} + 4378 q^{25} + 4586 q^{26} - 8960 q^{27} + 18640 q^{28} - 4360 q^{29} + 25092 q^{30} + 5704 q^{31} - 18294 q^{32} - 12852 q^{33} + 30007 q^{34} + 5640 q^{35} + 67969 q^{36} - 3772 q^{37} - 6559 q^{38} + 11120 q^{39} + 14869 q^{40} - 10698 q^{41} + 78698 q^{42} - 14792 q^{43} - 356 q^{44} - 44912 q^{45} - 19389 q^{46} - 77864 q^{47} - 118727 q^{48} + 7188 q^{49} + 26877 q^{50} - 80246 q^{51} - 60736 q^{52} - 62352 q^{53} + 61026 q^{54} - 49552 q^{55} - 144528 q^{56} - 808 q^{57} + 52951 q^{58} - 26224 q^{59} + 101500 q^{60} - 82540 q^{61} - 9023 q^{62} - 61768 q^{63} + 153858 q^{64} - 5000 q^{65} - 48516 q^{66} + 27784 q^{67} + 40507 q^{68} - 93776 q^{69} + 185910 q^{70} - 9504 q^{71} - 186687 q^{72} + 14260 q^{73} - 15239 q^{74} + 167420 q^{75} + 1279 q^{76} - 218140 q^{77} + 264170 q^{78} + 160248 q^{79} - 1291 q^{80} + 161076 q^{81} - 47781 q^{82} - 77176 q^{83} + 16382 q^{84} + 141096 q^{85} + 22188 q^{86} + 268136 q^{87} + 129544 q^{88} - 265692 q^{89} + 48990 q^{90} + 401148 q^{91} + 190391 q^{92} - 123860 q^{93} + 248737 q^{94} + 135884 q^{95} + 950817 q^{96} + 144742 q^{97} - 292244 q^{98} + 239516 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.21373 0.921667 0.460833 0.887487i \(-0.347551\pi\)
0.460833 + 0.887487i \(0.347551\pi\)
\(3\) −11.2683 −0.722863 −0.361431 0.932399i \(-0.617712\pi\)
−0.361431 + 0.932399i \(0.617712\pi\)
\(4\) −4.81697 −0.150530
\(5\) −9.80186 −0.175341 −0.0876705 0.996150i \(-0.527942\pi\)
−0.0876705 + 0.996150i \(0.527942\pi\)
\(6\) −58.7500 −0.666239
\(7\) −11.1041 −0.0856521 −0.0428260 0.999083i \(-0.513636\pi\)
−0.0428260 + 0.999083i \(0.513636\pi\)
\(8\) −191.954 −1.06041
\(9\) −116.025 −0.477470
\(10\) −51.1043 −0.161606
\(11\) −557.274 −1.38863 −0.694316 0.719670i \(-0.744293\pi\)
−0.694316 + 0.719670i \(0.744293\pi\)
\(12\) 54.2791 0.108813
\(13\) 107.663 0.176688 0.0883441 0.996090i \(-0.471842\pi\)
0.0883441 + 0.996090i \(0.471842\pi\)
\(14\) −57.8938 −0.0789427
\(15\) 110.450 0.126747
\(16\) −846.654 −0.826810
\(17\) 329.005 0.276109 0.138054 0.990425i \(-0.455915\pi\)
0.138054 + 0.990425i \(0.455915\pi\)
\(18\) −604.924 −0.440068
\(19\) 2938.10 1.86716 0.933582 0.358365i \(-0.116666\pi\)
0.933582 + 0.358365i \(0.116666\pi\)
\(20\) 47.2153 0.0263941
\(21\) 125.124 0.0619147
\(22\) −2905.48 −1.27986
\(23\) −385.537 −0.151966 −0.0759830 0.997109i \(-0.524209\pi\)
−0.0759830 + 0.997109i \(0.524209\pi\)
\(24\) 2163.00 0.766528
\(25\) −3028.92 −0.969256
\(26\) 561.326 0.162848
\(27\) 4045.61 1.06801
\(28\) 53.4881 0.0128932
\(29\) −3309.15 −0.730670 −0.365335 0.930876i \(-0.619046\pi\)
−0.365335 + 0.930876i \(0.619046\pi\)
\(30\) 575.859 0.116819
\(31\) −5471.71 −1.02263 −0.511316 0.859393i \(-0.670842\pi\)
−0.511316 + 0.859393i \(0.670842\pi\)
\(32\) 1728.30 0.298362
\(33\) 6279.54 1.00379
\(34\) 1715.34 0.254480
\(35\) 108.841 0.0150183
\(36\) 558.890 0.0718737
\(37\) 4832.22 0.580286 0.290143 0.956983i \(-0.406297\pi\)
0.290143 + 0.956983i \(0.406297\pi\)
\(38\) 15318.5 1.72090
\(39\) −1213.18 −0.127721
\(40\) 1881.51 0.185933
\(41\) 1065.16 0.0989587 0.0494793 0.998775i \(-0.484244\pi\)
0.0494793 + 0.998775i \(0.484244\pi\)
\(42\) 652.365 0.0570647
\(43\) −1849.00 −0.152499
\(44\) 2684.37 0.209031
\(45\) 1137.26 0.0837200
\(46\) −2010.09 −0.140062
\(47\) −8991.95 −0.593758 −0.296879 0.954915i \(-0.595946\pi\)
−0.296879 + 0.954915i \(0.595946\pi\)
\(48\) 9540.36 0.597670
\(49\) −16683.7 −0.992664
\(50\) −15792.0 −0.893331
\(51\) −3707.33 −0.199589
\(52\) −518.609 −0.0265969
\(53\) −10216.7 −0.499598 −0.249799 0.968298i \(-0.580365\pi\)
−0.249799 + 0.968298i \(0.580365\pi\)
\(54\) 21092.7 0.984347
\(55\) 5462.32 0.243484
\(56\) 2131.47 0.0908260
\(57\) −33107.4 −1.34970
\(58\) −17253.0 −0.673434
\(59\) −27457.0 −1.02689 −0.513444 0.858123i \(-0.671631\pi\)
−0.513444 + 0.858123i \(0.671631\pi\)
\(60\) −532.036 −0.0190793
\(61\) −36692.9 −1.26258 −0.631288 0.775548i \(-0.717473\pi\)
−0.631288 + 0.775548i \(0.717473\pi\)
\(62\) −28528.1 −0.942525
\(63\) 1288.35 0.0408963
\(64\) 36103.8 1.10180
\(65\) −1055.30 −0.0309807
\(66\) 32739.9 0.925160
\(67\) −26272.2 −0.715004 −0.357502 0.933912i \(-0.616372\pi\)
−0.357502 + 0.933912i \(0.616372\pi\)
\(68\) −1584.81 −0.0415627
\(69\) 4344.35 0.109851
\(70\) 567.467 0.0138419
\(71\) 55130.3 1.29791 0.648955 0.760827i \(-0.275206\pi\)
0.648955 + 0.760827i \(0.275206\pi\)
\(72\) 22271.5 0.506311
\(73\) −9315.14 −0.204589 −0.102294 0.994754i \(-0.532618\pi\)
−0.102294 + 0.994754i \(0.532618\pi\)
\(74\) 25193.9 0.534830
\(75\) 34130.9 0.700639
\(76\) −14152.7 −0.281065
\(77\) 6188.02 0.118939
\(78\) −6325.20 −0.117717
\(79\) 56826.5 1.02443 0.512216 0.858856i \(-0.328825\pi\)
0.512216 + 0.858856i \(0.328825\pi\)
\(80\) 8298.78 0.144974
\(81\) −17393.1 −0.294553
\(82\) 5553.44 0.0912069
\(83\) −5858.19 −0.0933401 −0.0466700 0.998910i \(-0.514861\pi\)
−0.0466700 + 0.998910i \(0.514861\pi\)
\(84\) −602.721 −0.00932004
\(85\) −3224.86 −0.0484132
\(86\) −9640.20 −0.140553
\(87\) 37288.5 0.528174
\(88\) 106971. 1.47251
\(89\) −42815.7 −0.572965 −0.286483 0.958085i \(-0.592486\pi\)
−0.286483 + 0.958085i \(0.592486\pi\)
\(90\) 5929.38 0.0771619
\(91\) −1195.50 −0.0151337
\(92\) 1857.12 0.0228755
\(93\) 61657.0 0.739222
\(94\) −46881.7 −0.547247
\(95\) −28798.8 −0.327390
\(96\) −19475.0 −0.215675
\(97\) −231.671 −0.00250002 −0.00125001 0.999999i \(-0.500398\pi\)
−0.00125001 + 0.999999i \(0.500398\pi\)
\(98\) −86984.4 −0.914905
\(99\) 64657.8 0.663029
\(100\) 14590.2 0.145902
\(101\) −65329.5 −0.637245 −0.318622 0.947882i \(-0.603220\pi\)
−0.318622 + 0.947882i \(0.603220\pi\)
\(102\) −19329.0 −0.183954
\(103\) 210856. 1.95837 0.979183 0.202980i \(-0.0650628\pi\)
0.979183 + 0.202980i \(0.0650628\pi\)
\(104\) −20666.3 −0.187361
\(105\) −1226.45 −0.0108562
\(106\) −53267.2 −0.460463
\(107\) −158342. −1.33702 −0.668508 0.743705i \(-0.733067\pi\)
−0.668508 + 0.743705i \(0.733067\pi\)
\(108\) −19487.6 −0.160768
\(109\) −85822.7 −0.691888 −0.345944 0.938255i \(-0.612441\pi\)
−0.345944 + 0.938255i \(0.612441\pi\)
\(110\) 28479.1 0.224411
\(111\) −54450.9 −0.419467
\(112\) 9401.32 0.0708180
\(113\) 101903. 0.750745 0.375372 0.926874i \(-0.377515\pi\)
0.375372 + 0.926874i \(0.377515\pi\)
\(114\) −172613. −1.24398
\(115\) 3778.98 0.0266459
\(116\) 15940.1 0.109988
\(117\) −12491.6 −0.0843633
\(118\) −143153. −0.946448
\(119\) −3653.30 −0.0236493
\(120\) −21201.4 −0.134404
\(121\) 149503. 0.928299
\(122\) −191307. −1.16367
\(123\) −12002.5 −0.0715335
\(124\) 26357.1 0.153937
\(125\) 60319.9 0.345291
\(126\) 6717.13 0.0376927
\(127\) 109901. 0.604635 0.302318 0.953207i \(-0.402240\pi\)
0.302318 + 0.953207i \(0.402240\pi\)
\(128\) 132930. 0.717131
\(129\) 20835.1 0.110236
\(130\) −5502.04 −0.0285539
\(131\) −79310.4 −0.403787 −0.201893 0.979408i \(-0.564709\pi\)
−0.201893 + 0.979408i \(0.564709\pi\)
\(132\) −30248.4 −0.151101
\(133\) −32624.9 −0.159926
\(134\) −136976. −0.658996
\(135\) −39654.5 −0.187266
\(136\) −63153.8 −0.292787
\(137\) −360169. −1.63948 −0.819738 0.572738i \(-0.805881\pi\)
−0.819738 + 0.572738i \(0.805881\pi\)
\(138\) 22650.3 0.101246
\(139\) 330826. 1.45232 0.726160 0.687525i \(-0.241303\pi\)
0.726160 + 0.687525i \(0.241303\pi\)
\(140\) −524.283 −0.00226071
\(141\) 101324. 0.429205
\(142\) 287435. 1.19624
\(143\) −59997.8 −0.245355
\(144\) 98233.1 0.394777
\(145\) 32435.8 0.128116
\(146\) −48566.7 −0.188563
\(147\) 187997. 0.717560
\(148\) −23276.6 −0.0873506
\(149\) 24179.7 0.0892249 0.0446125 0.999004i \(-0.485795\pi\)
0.0446125 + 0.999004i \(0.485795\pi\)
\(150\) 177949. 0.645755
\(151\) 278667. 0.994589 0.497295 0.867582i \(-0.334327\pi\)
0.497295 + 0.867582i \(0.334327\pi\)
\(152\) −563979. −1.97995
\(153\) −38172.8 −0.131833
\(154\) 32262.7 0.109622
\(155\) 53633.0 0.179309
\(156\) 5843.85 0.0192259
\(157\) −102228. −0.330995 −0.165497 0.986210i \(-0.552923\pi\)
−0.165497 + 0.986210i \(0.552923\pi\)
\(158\) 296279. 0.944186
\(159\) 115125. 0.361141
\(160\) −16940.5 −0.0523151
\(161\) 4281.04 0.0130162
\(162\) −90682.9 −0.271480
\(163\) 444216. 1.30956 0.654780 0.755819i \(-0.272761\pi\)
0.654780 + 0.755819i \(0.272761\pi\)
\(164\) −5130.83 −0.0148963
\(165\) −61551.2 −0.176006
\(166\) −30543.0 −0.0860285
\(167\) −467231. −1.29640 −0.648202 0.761469i \(-0.724479\pi\)
−0.648202 + 0.761469i \(0.724479\pi\)
\(168\) −24018.1 −0.0656547
\(169\) −359702. −0.968781
\(170\) −16813.6 −0.0446208
\(171\) −340893. −0.891514
\(172\) 8906.58 0.0229557
\(173\) 296435. 0.753033 0.376516 0.926410i \(-0.377122\pi\)
0.376516 + 0.926410i \(0.377122\pi\)
\(174\) 194412. 0.486801
\(175\) 33633.5 0.0830188
\(176\) 471818. 1.14814
\(177\) 309394. 0.742299
\(178\) −223230. −0.528083
\(179\) 211576. 0.493553 0.246777 0.969072i \(-0.420629\pi\)
0.246777 + 0.969072i \(0.420629\pi\)
\(180\) −5478.16 −0.0126024
\(181\) −187769. −0.426018 −0.213009 0.977050i \(-0.568326\pi\)
−0.213009 + 0.977050i \(0.568326\pi\)
\(182\) −6233.01 −0.0139482
\(183\) 413467. 0.912669
\(184\) 74005.3 0.161146
\(185\) −47364.7 −0.101748
\(186\) 321463. 0.681316
\(187\) −183346. −0.383413
\(188\) 43314.0 0.0893786
\(189\) −44922.8 −0.0914771
\(190\) −150149. −0.301745
\(191\) −655959. −1.30105 −0.650524 0.759486i \(-0.725450\pi\)
−0.650524 + 0.759486i \(0.725450\pi\)
\(192\) −406829. −0.796451
\(193\) 450394. 0.870361 0.435180 0.900343i \(-0.356685\pi\)
0.435180 + 0.900343i \(0.356685\pi\)
\(194\) −1207.87 −0.00230418
\(195\) 11891.4 0.0223948
\(196\) 80364.9 0.149426
\(197\) −973133. −1.78652 −0.893258 0.449545i \(-0.851586\pi\)
−0.893258 + 0.449545i \(0.851586\pi\)
\(198\) 337109. 0.611092
\(199\) 259840. 0.465129 0.232564 0.972581i \(-0.425288\pi\)
0.232564 + 0.972581i \(0.425288\pi\)
\(200\) 581414. 1.02780
\(201\) 296043. 0.516850
\(202\) −340611. −0.587327
\(203\) 36745.1 0.0625834
\(204\) 17858.1 0.0300441
\(205\) −10440.5 −0.0173515
\(206\) 1.09935e6 1.80496
\(207\) 44732.0 0.0725591
\(208\) −91153.2 −0.146088
\(209\) −1.63733e6 −2.59280
\(210\) −6394.39 −0.0100058
\(211\) −621161. −0.960502 −0.480251 0.877131i \(-0.659454\pi\)
−0.480251 + 0.877131i \(0.659454\pi\)
\(212\) 49213.5 0.0752047
\(213\) −621226. −0.938211
\(214\) −825553. −1.23228
\(215\) 18123.6 0.0267392
\(216\) −776570. −1.13252
\(217\) 60758.4 0.0875905
\(218\) −447457. −0.637690
\(219\) 104966. 0.147890
\(220\) −26311.8 −0.0366517
\(221\) 35421.6 0.0487852
\(222\) −283893. −0.386609
\(223\) −779614. −1.04983 −0.524913 0.851156i \(-0.675902\pi\)
−0.524913 + 0.851156i \(0.675902\pi\)
\(224\) −19191.2 −0.0255553
\(225\) 351431. 0.462790
\(226\) 531297. 0.691937
\(227\) 795424. 1.02455 0.512276 0.858821i \(-0.328803\pi\)
0.512276 + 0.858821i \(0.328803\pi\)
\(228\) 159477. 0.203171
\(229\) −702980. −0.885838 −0.442919 0.896562i \(-0.646057\pi\)
−0.442919 + 0.896562i \(0.646057\pi\)
\(230\) 19702.6 0.0245586
\(231\) −69728.6 −0.0859767
\(232\) 635204. 0.774807
\(233\) −219327. −0.264669 −0.132334 0.991205i \(-0.542247\pi\)
−0.132334 + 0.991205i \(0.542247\pi\)
\(234\) −65127.9 −0.0777548
\(235\) 88137.9 0.104110
\(236\) 132260. 0.154578
\(237\) −640339. −0.740524
\(238\) −19047.3 −0.0217968
\(239\) 1.75148e6 1.98340 0.991701 0.128565i \(-0.0410372\pi\)
0.991701 + 0.128565i \(0.0410372\pi\)
\(240\) −93513.3 −0.104796
\(241\) 985600. 1.09310 0.546548 0.837428i \(-0.315942\pi\)
0.546548 + 0.837428i \(0.315942\pi\)
\(242\) 779471. 0.855582
\(243\) −787092. −0.855086
\(244\) 176749. 0.190056
\(245\) 163531. 0.174055
\(246\) −62578.0 −0.0659301
\(247\) 316324. 0.329906
\(248\) 1.05032e6 1.08440
\(249\) 66011.9 0.0674721
\(250\) 314492. 0.318243
\(251\) 126142. 0.126379 0.0631894 0.998002i \(-0.479873\pi\)
0.0631894 + 0.998002i \(0.479873\pi\)
\(252\) −6205.96 −0.00615613
\(253\) 214850. 0.211025
\(254\) 572996. 0.557272
\(255\) 36338.7 0.0349961
\(256\) −462259. −0.440845
\(257\) 1.55933e6 1.47267 0.736335 0.676617i \(-0.236555\pi\)
0.736335 + 0.676617i \(0.236555\pi\)
\(258\) 108629. 0.101600
\(259\) −53657.4 −0.0497027
\(260\) 5083.33 0.00466353
\(261\) 383944. 0.348873
\(262\) −413503. −0.372157
\(263\) −107289. −0.0956460 −0.0478230 0.998856i \(-0.515228\pi\)
−0.0478230 + 0.998856i \(0.515228\pi\)
\(264\) −1.20538e6 −1.06442
\(265\) 100143. 0.0876001
\(266\) −170098. −0.147399
\(267\) 482461. 0.414175
\(268\) 126552. 0.107630
\(269\) 829468. 0.698906 0.349453 0.936954i \(-0.386367\pi\)
0.349453 + 0.936954i \(0.386367\pi\)
\(270\) −206748. −0.172596
\(271\) 2.22835e6 1.84315 0.921575 0.388200i \(-0.126903\pi\)
0.921575 + 0.388200i \(0.126903\pi\)
\(272\) −278553. −0.228289
\(273\) 13471.3 0.0109396
\(274\) −1.87783e6 −1.51105
\(275\) 1.68794e6 1.34594
\(276\) −20926.6 −0.0165358
\(277\) −1.57467e6 −1.23308 −0.616538 0.787325i \(-0.711465\pi\)
−0.616538 + 0.787325i \(0.711465\pi\)
\(278\) 1.72484e6 1.33856
\(279\) 634856. 0.488275
\(280\) −20892.4 −0.0159255
\(281\) −2.26025e6 −1.70762 −0.853809 0.520586i \(-0.825713\pi\)
−0.853809 + 0.520586i \(0.825713\pi\)
\(282\) 528277. 0.395584
\(283\) −1.81290e6 −1.34558 −0.672789 0.739835i \(-0.734904\pi\)
−0.672789 + 0.739835i \(0.734904\pi\)
\(284\) −265561. −0.195375
\(285\) 324514. 0.236658
\(286\) −312812. −0.226136
\(287\) −11827.6 −0.00847602
\(288\) −200526. −0.142459
\(289\) −1.31161e6 −0.923764
\(290\) 169112. 0.118081
\(291\) 2610.55 0.00180717
\(292\) 44870.7 0.0307968
\(293\) −1.92999e6 −1.31336 −0.656682 0.754168i \(-0.728041\pi\)
−0.656682 + 0.754168i \(0.728041\pi\)
\(294\) 980167. 0.661351
\(295\) 269130. 0.180056
\(296\) −927563. −0.615338
\(297\) −2.25451e6 −1.48307
\(298\) 126067. 0.0822356
\(299\) −41508.0 −0.0268506
\(300\) −164407. −0.105467
\(301\) 20531.5 0.0130618
\(302\) 1.45290e6 0.916680
\(303\) 736154. 0.460640
\(304\) −2.48755e6 −1.54379
\(305\) 359659. 0.221381
\(306\) −199023. −0.121507
\(307\) −2.35372e6 −1.42531 −0.712654 0.701515i \(-0.752507\pi\)
−0.712654 + 0.701515i \(0.752507\pi\)
\(308\) −29807.5 −0.0179040
\(309\) −2.37600e6 −1.41563
\(310\) 279628. 0.165263
\(311\) 1.25933e6 0.738312 0.369156 0.929367i \(-0.379647\pi\)
0.369156 + 0.929367i \(0.379647\pi\)
\(312\) 232875. 0.135436
\(313\) −1.21791e6 −0.702676 −0.351338 0.936249i \(-0.614273\pi\)
−0.351338 + 0.936249i \(0.614273\pi\)
\(314\) −532990. −0.305067
\(315\) −12628.3 −0.00717079
\(316\) −273732. −0.154208
\(317\) 689890. 0.385595 0.192798 0.981239i \(-0.438244\pi\)
0.192798 + 0.981239i \(0.438244\pi\)
\(318\) 600231. 0.332852
\(319\) 1.84410e6 1.01463
\(320\) −353884. −0.193191
\(321\) 1.78425e6 0.966480
\(322\) 22320.2 0.0119966
\(323\) 966649. 0.515540
\(324\) 83782.0 0.0443392
\(325\) −326103. −0.171256
\(326\) 2.31603e6 1.20698
\(327\) 967077. 0.500140
\(328\) −204461. −0.104936
\(329\) 99847.5 0.0508566
\(330\) −320911. −0.162218
\(331\) 1.93331e6 0.969912 0.484956 0.874538i \(-0.338836\pi\)
0.484956 + 0.874538i \(0.338836\pi\)
\(332\) 28218.7 0.0140505
\(333\) −560658. −0.277069
\(334\) −2.43602e6 −1.19485
\(335\) 257516. 0.125370
\(336\) −105937. −0.0511917
\(337\) 347418. 0.166639 0.0833196 0.996523i \(-0.473448\pi\)
0.0833196 + 0.996523i \(0.473448\pi\)
\(338\) −1.87539e6 −0.892893
\(339\) −1.14828e6 −0.542686
\(340\) 15534.1 0.00728765
\(341\) 3.04924e6 1.42006
\(342\) −1.77733e6 −0.821678
\(343\) 371884. 0.170676
\(344\) 354923. 0.161710
\(345\) −42582.7 −0.0192613
\(346\) 1.54553e6 0.694045
\(347\) −4.28700e6 −1.91130 −0.955652 0.294498i \(-0.904847\pi\)
−0.955652 + 0.294498i \(0.904847\pi\)
\(348\) −179618. −0.0795062
\(349\) 2.58234e6 1.13488 0.567439 0.823415i \(-0.307934\pi\)
0.567439 + 0.823415i \(0.307934\pi\)
\(350\) 175356. 0.0765156
\(351\) 435562. 0.188704
\(352\) −963136. −0.414315
\(353\) −2.31876e6 −0.990419 −0.495210 0.868774i \(-0.664909\pi\)
−0.495210 + 0.868774i \(0.664909\pi\)
\(354\) 1.61310e6 0.684152
\(355\) −540380. −0.227577
\(356\) 206242. 0.0862486
\(357\) 41166.5 0.0170952
\(358\) 1.10310e6 0.454892
\(359\) −1.41572e6 −0.579752 −0.289876 0.957064i \(-0.593614\pi\)
−0.289876 + 0.957064i \(0.593614\pi\)
\(360\) −218302. −0.0887771
\(361\) 6.15632e6 2.48630
\(362\) −978979. −0.392647
\(363\) −1.68465e6 −0.671033
\(364\) 5758.68 0.00227808
\(365\) 91305.7 0.0358728
\(366\) 2.15571e6 0.841177
\(367\) 2.64620e6 1.02555 0.512776 0.858523i \(-0.328617\pi\)
0.512776 + 0.858523i \(0.328617\pi\)
\(368\) 326416. 0.125647
\(369\) −123585. −0.0472498
\(370\) −246947. −0.0937777
\(371\) 113447. 0.0427916
\(372\) −297000. −0.111275
\(373\) 3.49265e6 1.29982 0.649910 0.760011i \(-0.274806\pi\)
0.649910 + 0.760011i \(0.274806\pi\)
\(374\) −955917. −0.353379
\(375\) −679703. −0.249598
\(376\) 1.72604e6 0.629624
\(377\) −356273. −0.129101
\(378\) −234216. −0.0843114
\(379\) 1.42204e6 0.508526 0.254263 0.967135i \(-0.418167\pi\)
0.254263 + 0.967135i \(0.418167\pi\)
\(380\) 138723. 0.0492822
\(381\) −1.23840e6 −0.437068
\(382\) −3.41999e6 −1.19913
\(383\) −2.81442e6 −0.980373 −0.490187 0.871618i \(-0.663071\pi\)
−0.490187 + 0.871618i \(0.663071\pi\)
\(384\) −1.49790e6 −0.518387
\(385\) −60654.1 −0.0208549
\(386\) 2.34823e6 0.802182
\(387\) 214530. 0.0728134
\(388\) 1115.95 0.000376329 0
\(389\) −165325. −0.0553943 −0.0276971 0.999616i \(-0.508817\pi\)
−0.0276971 + 0.999616i \(0.508817\pi\)
\(390\) 61998.7 0.0206405
\(391\) −126844. −0.0419591
\(392\) 3.20250e6 1.05263
\(393\) 893695. 0.291882
\(394\) −5.07366e6 −1.64657
\(395\) −557006. −0.179625
\(396\) −311455. −0.0998061
\(397\) −3.79445e6 −1.20830 −0.604148 0.796872i \(-0.706486\pi\)
−0.604148 + 0.796872i \(0.706486\pi\)
\(398\) 1.35474e6 0.428694
\(399\) 367628. 0.115605
\(400\) 2.56445e6 0.801390
\(401\) 4.59396e6 1.42668 0.713339 0.700819i \(-0.247182\pi\)
0.713339 + 0.700819i \(0.247182\pi\)
\(402\) 1.54349e6 0.476363
\(403\) −589100. −0.180687
\(404\) 314690. 0.0959246
\(405\) 170485. 0.0516473
\(406\) 191579. 0.0576811
\(407\) −2.69287e6 −0.805804
\(408\) 711637. 0.211645
\(409\) 3.83956e6 1.13494 0.567471 0.823393i \(-0.307922\pi\)
0.567471 + 0.823393i \(0.307922\pi\)
\(410\) −54434.1 −0.0159923
\(411\) 4.05850e6 1.18512
\(412\) −1.01569e6 −0.294793
\(413\) 304885. 0.0879551
\(414\) 233221. 0.0668753
\(415\) 57421.2 0.0163663
\(416\) 186074. 0.0527171
\(417\) −3.72785e6 −1.04983
\(418\) −8.53658e6 −2.38970
\(419\) 2.87922e6 0.801199 0.400600 0.916253i \(-0.368802\pi\)
0.400600 + 0.916253i \(0.368802\pi\)
\(420\) 5907.78 0.00163419
\(421\) 1.41982e6 0.390417 0.195208 0.980762i \(-0.437462\pi\)
0.195208 + 0.980762i \(0.437462\pi\)
\(422\) −3.23857e6 −0.885262
\(423\) 1.04329e6 0.283501
\(424\) 1.96114e6 0.529777
\(425\) −996531. −0.267620
\(426\) −3.23891e6 −0.864718
\(427\) 407441. 0.108142
\(428\) 762729. 0.201262
\(429\) 676074. 0.177358
\(430\) 94491.8 0.0246447
\(431\) −4.77968e6 −1.23938 −0.619691 0.784846i \(-0.712742\pi\)
−0.619691 + 0.784846i \(0.712742\pi\)
\(432\) −3.42523e6 −0.883040
\(433\) 213516. 0.0547282 0.0273641 0.999626i \(-0.491289\pi\)
0.0273641 + 0.999626i \(0.491289\pi\)
\(434\) 316778. 0.0807293
\(435\) −365497. −0.0926106
\(436\) 413405. 0.104150
\(437\) −1.13275e6 −0.283745
\(438\) 547264. 0.136305
\(439\) 555761. 0.137634 0.0688172 0.997629i \(-0.478077\pi\)
0.0688172 + 0.997629i \(0.478077\pi\)
\(440\) −1.04851e6 −0.258192
\(441\) 1.93573e6 0.473967
\(442\) 184679. 0.0449637
\(443\) −1.23542e6 −0.299092 −0.149546 0.988755i \(-0.547781\pi\)
−0.149546 + 0.988755i \(0.547781\pi\)
\(444\) 262289. 0.0631425
\(445\) 419674. 0.100464
\(446\) −4.06470e6 −0.967591
\(447\) −272465. −0.0644974
\(448\) −400900. −0.0943715
\(449\) 1.61043e6 0.376986 0.188493 0.982074i \(-0.439640\pi\)
0.188493 + 0.982074i \(0.439640\pi\)
\(450\) 1.83227e6 0.426538
\(451\) −593584. −0.137417
\(452\) −490866. −0.113010
\(453\) −3.14011e6 −0.718951
\(454\) 4.14713e6 0.944295
\(455\) 11718.1 0.00265356
\(456\) 6.35510e6 1.43123
\(457\) 4.44189e6 0.994894 0.497447 0.867494i \(-0.334271\pi\)
0.497447 + 0.867494i \(0.334271\pi\)
\(458\) −3.66515e6 −0.816447
\(459\) 1.33102e6 0.294886
\(460\) −18203.2 −0.00401101
\(461\) −8.65627e6 −1.89705 −0.948524 0.316705i \(-0.897423\pi\)
−0.948524 + 0.316705i \(0.897423\pi\)
\(462\) −363546. −0.0792419
\(463\) −8.50880e6 −1.84466 −0.922329 0.386404i \(-0.873717\pi\)
−0.922329 + 0.386404i \(0.873717\pi\)
\(464\) 2.80170e6 0.604126
\(465\) −604353. −0.129616
\(466\) −1.14351e6 −0.243936
\(467\) −692449. −0.146925 −0.0734625 0.997298i \(-0.523405\pi\)
−0.0734625 + 0.997298i \(0.523405\pi\)
\(468\) 60171.7 0.0126992
\(469\) 291728. 0.0612416
\(470\) 459527. 0.0959548
\(471\) 1.15194e6 0.239264
\(472\) 5.27048e6 1.08892
\(473\) 1.03040e6 0.211764
\(474\) −3.33856e6 −0.682517
\(475\) −8.89927e6 −1.80976
\(476\) 17597.8 0.00355993
\(477\) 1.18539e6 0.238543
\(478\) 9.13176e6 1.82804
\(479\) −7.63881e6 −1.52120 −0.760601 0.649220i \(-0.775095\pi\)
−0.760601 + 0.649220i \(0.775095\pi\)
\(480\) 190891. 0.0378166
\(481\) 520250. 0.102530
\(482\) 5.13865e6 1.00747
\(483\) −48240.1 −0.00940893
\(484\) −720154. −0.139737
\(485\) 2270.81 0.000438356 0
\(486\) −4.10369e6 −0.788104
\(487\) 4.82019e6 0.920962 0.460481 0.887670i \(-0.347677\pi\)
0.460481 + 0.887670i \(0.347677\pi\)
\(488\) 7.04335e6 1.33884
\(489\) −5.00557e6 −0.946633
\(490\) 852609. 0.160420
\(491\) 66664.0 0.0124792 0.00623961 0.999981i \(-0.498014\pi\)
0.00623961 + 0.999981i \(0.498014\pi\)
\(492\) 57815.8 0.0107680
\(493\) −1.08873e6 −0.201744
\(494\) 1.64923e6 0.304063
\(495\) −633767. −0.116256
\(496\) 4.63265e6 0.845522
\(497\) −612172. −0.111169
\(498\) 344169. 0.0621868
\(499\) −2.93211e6 −0.527144 −0.263572 0.964640i \(-0.584901\pi\)
−0.263572 + 0.964640i \(0.584901\pi\)
\(500\) −290559. −0.0519768
\(501\) 5.26490e6 0.937122
\(502\) 657669. 0.116479
\(503\) −3.90093e6 −0.687462 −0.343731 0.939068i \(-0.611691\pi\)
−0.343731 + 0.939068i \(0.611691\pi\)
\(504\) −247305. −0.0433666
\(505\) 640351. 0.111735
\(506\) 1.12017e6 0.194495
\(507\) 4.05323e6 0.700296
\(508\) −529391. −0.0910159
\(509\) 2.09591e6 0.358573 0.179287 0.983797i \(-0.442621\pi\)
0.179287 + 0.983797i \(0.442621\pi\)
\(510\) 189460. 0.0322547
\(511\) 103436. 0.0175235
\(512\) −6.66386e6 −1.12344
\(513\) 1.18864e7 1.99414
\(514\) 8.12994e6 1.35731
\(515\) −2.06678e6 −0.343382
\(516\) −100362. −0.0165938
\(517\) 5.01098e6 0.824511
\(518\) −279755. −0.0458093
\(519\) −3.34032e6 −0.544339
\(520\) 202568. 0.0328521
\(521\) 3.34502e6 0.539889 0.269944 0.962876i \(-0.412995\pi\)
0.269944 + 0.962876i \(0.412995\pi\)
\(522\) 2.00178e6 0.321544
\(523\) −1.04450e7 −1.66976 −0.834878 0.550434i \(-0.814462\pi\)
−0.834878 + 0.550434i \(0.814462\pi\)
\(524\) 382036. 0.0607821
\(525\) −378992. −0.0600112
\(526\) −559378. −0.0881538
\(527\) −1.80022e6 −0.282357
\(528\) −5.31660e6 −0.829944
\(529\) −6.28770e6 −0.976906
\(530\) 522117. 0.0807381
\(531\) 3.18570e6 0.490308
\(532\) 157153. 0.0240738
\(533\) 114678. 0.0174848
\(534\) 2.51542e6 0.381731
\(535\) 1.55205e6 0.234434
\(536\) 5.04304e6 0.758195
\(537\) −2.38411e6 −0.356771
\(538\) 4.32463e6 0.644159
\(539\) 9.29739e6 1.37844
\(540\) 191014. 0.0281891
\(541\) 5.88573e6 0.864584 0.432292 0.901734i \(-0.357705\pi\)
0.432292 + 0.901734i \(0.357705\pi\)
\(542\) 1.16180e7 1.69877
\(543\) 2.11584e6 0.307952
\(544\) 568618. 0.0823803
\(545\) 841222. 0.121316
\(546\) 70235.6 0.0100827
\(547\) −2.94922e6 −0.421444 −0.210722 0.977546i \(-0.567581\pi\)
−0.210722 + 0.977546i \(0.567581\pi\)
\(548\) 1.73492e6 0.246791
\(549\) 4.25730e6 0.602842
\(550\) 8.80048e6 1.24051
\(551\) −9.72260e6 −1.36428
\(552\) −833915. −0.116486
\(553\) −631007. −0.0877448
\(554\) −8.20990e6 −1.13648
\(555\) 533720. 0.0735498
\(556\) −1.59358e6 −0.218618
\(557\) −1.44836e7 −1.97806 −0.989029 0.147723i \(-0.952805\pi\)
−0.989029 + 0.147723i \(0.952805\pi\)
\(558\) 3.30997e6 0.450027
\(559\) −199069. −0.0269447
\(560\) −92150.4 −0.0124173
\(561\) 2.06600e6 0.277155
\(562\) −1.17843e7 −1.57385
\(563\) 6.77740e6 0.901139 0.450570 0.892741i \(-0.351221\pi\)
0.450570 + 0.892741i \(0.351221\pi\)
\(564\) −488076. −0.0646084
\(565\) −998842. −0.131636
\(566\) −9.45200e6 −1.24017
\(567\) 193134. 0.0252291
\(568\) −1.05825e7 −1.37631
\(569\) 1.03288e7 1.33742 0.668710 0.743523i \(-0.266847\pi\)
0.668710 + 0.743523i \(0.266847\pi\)
\(570\) 1.69193e6 0.218120
\(571\) −9.52868e6 −1.22304 −0.611522 0.791227i \(-0.709443\pi\)
−0.611522 + 0.791227i \(0.709443\pi\)
\(572\) 289007. 0.0369334
\(573\) 7.39155e6 0.940478
\(574\) −61666.0 −0.00781206
\(575\) 1.16776e6 0.147294
\(576\) −4.18895e6 −0.526076
\(577\) 5.58573e6 0.698458 0.349229 0.937037i \(-0.386444\pi\)
0.349229 + 0.937037i \(0.386444\pi\)
\(578\) −6.83840e6 −0.851403
\(579\) −5.07518e6 −0.629151
\(580\) −156242. −0.0192854
\(581\) 65049.9 0.00799478
\(582\) 13610.7 0.00166561
\(583\) 5.69350e6 0.693758
\(584\) 1.78808e6 0.216947
\(585\) 122441. 0.0147923
\(586\) −1.00624e7 −1.21048
\(587\) −1.34089e7 −1.60619 −0.803096 0.595849i \(-0.796816\pi\)
−0.803096 + 0.595849i \(0.796816\pi\)
\(588\) −905577. −0.108014
\(589\) −1.60764e7 −1.90942
\(590\) 1.40317e6 0.165951
\(591\) 1.09656e7 1.29141
\(592\) −4.09121e6 −0.479786
\(593\) −1.46278e7 −1.70821 −0.854106 0.520099i \(-0.825895\pi\)
−0.854106 + 0.520099i \(0.825895\pi\)
\(594\) −1.17544e7 −1.36690
\(595\) 35809.1 0.00414669
\(596\) −116473. −0.0134311
\(597\) −2.92796e6 −0.336224
\(598\) −216412. −0.0247473
\(599\) 1.02632e7 1.16873 0.584367 0.811489i \(-0.301343\pi\)
0.584367 + 0.811489i \(0.301343\pi\)
\(600\) −6.55155e6 −0.742961
\(601\) 9.07707e6 1.02508 0.512542 0.858662i \(-0.328704\pi\)
0.512542 + 0.858662i \(0.328704\pi\)
\(602\) 107046. 0.0120386
\(603\) 3.04823e6 0.341393
\(604\) −1.34233e6 −0.149716
\(605\) −1.46541e6 −0.162769
\(606\) 3.83811e6 0.424557
\(607\) −1.20707e7 −1.32972 −0.664862 0.746966i \(-0.731510\pi\)
−0.664862 + 0.746966i \(0.731510\pi\)
\(608\) 5.07791e6 0.557091
\(609\) −414055. −0.0452392
\(610\) 1.87517e6 0.204040
\(611\) −968100. −0.104910
\(612\) 183877. 0.0198449
\(613\) 1.40958e6 0.151509 0.0757544 0.997127i \(-0.475864\pi\)
0.0757544 + 0.997127i \(0.475864\pi\)
\(614\) −1.22717e7 −1.31366
\(615\) 117647. 0.0125428
\(616\) −1.18782e6 −0.126124
\(617\) 1.51488e7 1.60201 0.801005 0.598658i \(-0.204299\pi\)
0.801005 + 0.598658i \(0.204299\pi\)
\(618\) −1.23878e7 −1.30474
\(619\) 1.58021e7 1.65763 0.828816 0.559521i \(-0.189015\pi\)
0.828816 + 0.559521i \(0.189015\pi\)
\(620\) −258348. −0.0269915
\(621\) −1.55973e6 −0.162301
\(622\) 6.56584e6 0.680478
\(623\) 475430. 0.0490757
\(624\) 1.02714e6 0.105601
\(625\) 8.87414e6 0.908712
\(626\) −6.34988e6 −0.647634
\(627\) 1.84499e7 1.87424
\(628\) 492430. 0.0498247
\(629\) 1.58982e6 0.160222
\(630\) −65840.4 −0.00660908
\(631\) −1.08922e7 −1.08904 −0.544519 0.838749i \(-0.683288\pi\)
−0.544519 + 0.838749i \(0.683288\pi\)
\(632\) −1.09081e7 −1.08631
\(633\) 6.99944e6 0.694311
\(634\) 3.59690e6 0.355390
\(635\) −1.07724e6 −0.106017
\(636\) −554554. −0.0543627
\(637\) −1.79622e6 −0.175392
\(638\) 9.61466e6 0.935152
\(639\) −6.39650e6 −0.619712
\(640\) −1.30296e6 −0.125742
\(641\) 2.41942e6 0.232577 0.116289 0.993215i \(-0.462900\pi\)
0.116289 + 0.993215i \(0.462900\pi\)
\(642\) 9.30260e6 0.890772
\(643\) 1.38769e7 1.32363 0.661814 0.749668i \(-0.269787\pi\)
0.661814 + 0.749668i \(0.269787\pi\)
\(644\) −20621.6 −0.00195933
\(645\) −204223. −0.0193288
\(646\) 5.03985e6 0.475156
\(647\) −7.58249e6 −0.712116 −0.356058 0.934464i \(-0.615880\pi\)
−0.356058 + 0.934464i \(0.615880\pi\)
\(648\) 3.33867e6 0.312346
\(649\) 1.53011e7 1.42597
\(650\) −1.70021e6 −0.157841
\(651\) −684645. −0.0633159
\(652\) −2.13978e6 −0.197129
\(653\) 3.26919e6 0.300025 0.150012 0.988684i \(-0.452069\pi\)
0.150012 + 0.988684i \(0.452069\pi\)
\(654\) 5.04208e6 0.460962
\(655\) 777389. 0.0708003
\(656\) −901819. −0.0818201
\(657\) 1.08079e6 0.0976850
\(658\) 520578. 0.0468728
\(659\) 1.94787e6 0.174721 0.0873606 0.996177i \(-0.472157\pi\)
0.0873606 + 0.996177i \(0.472157\pi\)
\(660\) 296490. 0.0264942
\(661\) −1.04897e7 −0.933810 −0.466905 0.884307i \(-0.654631\pi\)
−0.466905 + 0.884307i \(0.654631\pi\)
\(662\) 1.00798e7 0.893936
\(663\) −399142. −0.0352650
\(664\) 1.12450e6 0.0989784
\(665\) 319785. 0.0280417
\(666\) −2.92312e6 −0.255365
\(667\) 1.27580e6 0.111037
\(668\) 2.25064e6 0.195148
\(669\) 8.78494e6 0.758881
\(670\) 1.34262e6 0.115549
\(671\) 2.04480e7 1.75325
\(672\) 216252. 0.0184730
\(673\) −1.81198e6 −0.154211 −0.0771054 0.997023i \(-0.524568\pi\)
−0.0771054 + 0.997023i \(0.524568\pi\)
\(674\) 1.81134e6 0.153586
\(675\) −1.22538e7 −1.03517
\(676\) 1.73267e6 0.145831
\(677\) −1.71538e7 −1.43843 −0.719216 0.694786i \(-0.755499\pi\)
−0.719216 + 0.694786i \(0.755499\pi\)
\(678\) −5.98682e6 −0.500175
\(679\) 2572.50 0.000214132 0
\(680\) 619024. 0.0513376
\(681\) −8.96308e6 −0.740610
\(682\) 1.58979e7 1.30882
\(683\) 1.80610e7 1.48146 0.740732 0.671801i \(-0.234479\pi\)
0.740732 + 0.671801i \(0.234479\pi\)
\(684\) 1.64207e6 0.134200
\(685\) 3.53033e6 0.287467
\(686\) 1.93890e6 0.157306
\(687\) 7.92140e6 0.640339
\(688\) 1.56546e6 0.126087
\(689\) −1.09996e6 −0.0882732
\(690\) −222015. −0.0177525
\(691\) −1.44368e7 −1.15021 −0.575105 0.818080i \(-0.695039\pi\)
−0.575105 + 0.818080i \(0.695039\pi\)
\(692\) −1.42792e6 −0.113354
\(693\) −717966. −0.0567899
\(694\) −2.23513e7 −1.76159
\(695\) −3.24271e6 −0.254651
\(696\) −7.15768e6 −0.560079
\(697\) 350442. 0.0273233
\(698\) 1.34636e7 1.04598
\(699\) 2.47145e6 0.191319
\(700\) −162011. −0.0124968
\(701\) −1.39926e7 −1.07549 −0.537743 0.843109i \(-0.680723\pi\)
−0.537743 + 0.843109i \(0.680723\pi\)
\(702\) 2.27090e6 0.173923
\(703\) 1.41975e7 1.08349
\(704\) −2.01197e7 −1.53000
\(705\) −993165. −0.0752573
\(706\) −1.20894e7 −0.912836
\(707\) 725425. 0.0545813
\(708\) −1.49034e6 −0.111738
\(709\) −1.37678e7 −1.02860 −0.514302 0.857609i \(-0.671949\pi\)
−0.514302 + 0.857609i \(0.671949\pi\)
\(710\) −2.81740e6 −0.209750
\(711\) −6.59331e6 −0.489136
\(712\) 8.21864e6 0.607575
\(713\) 2.10955e6 0.155405
\(714\) 214631. 0.0157561
\(715\) 588090. 0.0430208
\(716\) −1.01916e6 −0.0742947
\(717\) −1.97362e7 −1.43373
\(718\) −7.38120e6 −0.534338
\(719\) −8.26505e6 −0.596243 −0.298121 0.954528i \(-0.596360\pi\)
−0.298121 + 0.954528i \(0.596360\pi\)
\(720\) −962867. −0.0692205
\(721\) −2.34137e6 −0.167738
\(722\) 3.20974e7 2.29154
\(723\) −1.11060e7 −0.790158
\(724\) 904479. 0.0641286
\(725\) 1.00232e7 0.708206
\(726\) −8.78333e6 −0.618469
\(727\) 7.18688e6 0.504318 0.252159 0.967686i \(-0.418859\pi\)
0.252159 + 0.967686i \(0.418859\pi\)
\(728\) 229481. 0.0160479
\(729\) 1.30957e7 0.912663
\(730\) 476043. 0.0330628
\(731\) −608330. −0.0421062
\(732\) −1.99166e6 −0.137384
\(733\) −1.10728e6 −0.0761196 −0.0380598 0.999275i \(-0.512118\pi\)
−0.0380598 + 0.999275i \(0.512118\pi\)
\(734\) 1.37966e7 0.945217
\(735\) −1.84272e6 −0.125818
\(736\) −666322. −0.0453409
\(737\) 1.46408e7 0.992878
\(738\) −644339. −0.0435485
\(739\) 1.98449e7 1.33671 0.668356 0.743841i \(-0.266998\pi\)
0.668356 + 0.743841i \(0.266998\pi\)
\(740\) 228154. 0.0153161
\(741\) −3.56444e6 −0.238477
\(742\) 591484. 0.0394396
\(743\) −2.19204e7 −1.45672 −0.728360 0.685194i \(-0.759717\pi\)
−0.728360 + 0.685194i \(0.759717\pi\)
\(744\) −1.18353e7 −0.783875
\(745\) −237006. −0.0156448
\(746\) 1.82098e7 1.19800
\(747\) 679697. 0.0445671
\(748\) 883172. 0.0577153
\(749\) 1.75825e6 0.114518
\(750\) −3.54379e6 −0.230046
\(751\) −1.62135e7 −1.04900 −0.524502 0.851409i \(-0.675748\pi\)
−0.524502 + 0.851409i \(0.675748\pi\)
\(752\) 7.61307e6 0.490925
\(753\) −1.42140e6 −0.0913545
\(754\) −1.85751e6 −0.118988
\(755\) −2.73146e6 −0.174392
\(756\) 216392. 0.0137701
\(757\) 1.31863e7 0.836342 0.418171 0.908368i \(-0.362671\pi\)
0.418171 + 0.908368i \(0.362671\pi\)
\(758\) 7.41412e6 0.468691
\(759\) −2.42099e6 −0.152542
\(760\) 5.52805e6 0.347166
\(761\) −1.92620e7 −1.20570 −0.602851 0.797854i \(-0.705969\pi\)
−0.602851 + 0.797854i \(0.705969\pi\)
\(762\) −6.45670e6 −0.402831
\(763\) 952983. 0.0592617
\(764\) 3.15973e6 0.195847
\(765\) 374165. 0.0231158
\(766\) −1.46736e7 −0.903577
\(767\) −2.95610e6 −0.181439
\(768\) 5.20888e6 0.318670
\(769\) −456438. −0.0278334 −0.0139167 0.999903i \(-0.504430\pi\)
−0.0139167 + 0.999903i \(0.504430\pi\)
\(770\) −316235. −0.0192213
\(771\) −1.75710e7 −1.06454
\(772\) −2.16953e6 −0.131016
\(773\) −6.14461e6 −0.369867 −0.184933 0.982751i \(-0.559207\pi\)
−0.184933 + 0.982751i \(0.559207\pi\)
\(774\) 1.11850e6 0.0671097
\(775\) 1.65734e7 0.991191
\(776\) 44470.2 0.00265103
\(777\) 604628. 0.0359282
\(778\) −861961. −0.0510551
\(779\) 3.12953e6 0.184772
\(780\) −57280.6 −0.00337110
\(781\) −3.07227e7 −1.80232
\(782\) −661328. −0.0386723
\(783\) −1.33875e7 −0.780361
\(784\) 1.41253e7 0.820745
\(785\) 1.00203e6 0.0580369
\(786\) 4.65949e6 0.269018
\(787\) 8.50558e6 0.489516 0.244758 0.969584i \(-0.421291\pi\)
0.244758 + 0.969584i \(0.421291\pi\)
\(788\) 4.68755e6 0.268925
\(789\) 1.20897e6 0.0691389
\(790\) −2.90408e6 −0.165554
\(791\) −1.13154e6 −0.0643029
\(792\) −1.24113e7 −0.703080
\(793\) −3.95047e6 −0.223082
\(794\) −1.97833e7 −1.11365
\(795\) −1.12844e6 −0.0633228
\(796\) −1.25164e6 −0.0700160
\(797\) 1.28439e7 0.716227 0.358113 0.933678i \(-0.383420\pi\)
0.358113 + 0.933678i \(0.383420\pi\)
\(798\) 1.91671e6 0.106549
\(799\) −2.95840e6 −0.163942
\(800\) −5.23488e6 −0.289189
\(801\) 4.96770e6 0.273573
\(802\) 2.39517e7 1.31492
\(803\) 5.19108e6 0.284099
\(804\) −1.42603e6 −0.0778016
\(805\) −41962.1 −0.00228227
\(806\) −3.07141e6 −0.166533
\(807\) −9.34670e6 −0.505213
\(808\) 1.25403e7 0.675738
\(809\) 2.98801e7 1.60513 0.802567 0.596562i \(-0.203467\pi\)
0.802567 + 0.596562i \(0.203467\pi\)
\(810\) 888861. 0.0476016
\(811\) −2.05001e7 −1.09447 −0.547234 0.836980i \(-0.684319\pi\)
−0.547234 + 0.836980i \(0.684319\pi\)
\(812\) −177000. −0.00942070
\(813\) −2.51098e7 −1.33234
\(814\) −1.40399e7 −0.742682
\(815\) −4.35415e6 −0.229620
\(816\) 3.13882e6 0.165022
\(817\) −5.43254e6 −0.284740
\(818\) 2.00185e7 1.04604
\(819\) 138708. 0.00722589
\(820\) 50291.7 0.00261193
\(821\) 2.20698e7 1.14272 0.571360 0.820699i \(-0.306416\pi\)
0.571360 + 0.820699i \(0.306416\pi\)
\(822\) 2.11599e7 1.09228
\(823\) 1.59817e6 0.0822476 0.0411238 0.999154i \(-0.486906\pi\)
0.0411238 + 0.999154i \(0.486906\pi\)
\(824\) −4.04747e7 −2.07666
\(825\) −1.90202e7 −0.972929
\(826\) 1.58959e6 0.0810653
\(827\) −1.49218e7 −0.758680 −0.379340 0.925257i \(-0.623849\pi\)
−0.379340 + 0.925257i \(0.623849\pi\)
\(828\) −215473. −0.0109223
\(829\) 7.71986e6 0.390142 0.195071 0.980789i \(-0.437506\pi\)
0.195071 + 0.980789i \(0.437506\pi\)
\(830\) 299379. 0.0150843
\(831\) 1.77439e7 0.891344
\(832\) 3.88704e6 0.194675
\(833\) −5.48902e6 −0.274083
\(834\) −1.94360e7 −0.967592
\(835\) 4.57973e6 0.227313
\(836\) 7.88695e6 0.390295
\(837\) −2.21364e7 −1.09218
\(838\) 1.50115e7 0.738439
\(839\) 3.50556e7 1.71930 0.859652 0.510880i \(-0.170680\pi\)
0.859652 + 0.510880i \(0.170680\pi\)
\(840\) 235422. 0.0115120
\(841\) −9.56068e6 −0.466121
\(842\) 7.40258e6 0.359834
\(843\) 2.54692e7 1.23437
\(844\) 2.99211e6 0.144585
\(845\) 3.52575e6 0.169867
\(846\) 5.43945e6 0.261294
\(847\) −1.66010e6 −0.0795108
\(848\) 8.65001e6 0.413073
\(849\) 2.04284e7 0.972668
\(850\) −5.19565e6 −0.246656
\(851\) −1.86300e6 −0.0881837
\(852\) 2.99243e6 0.141229
\(853\) −2.21072e7 −1.04031 −0.520153 0.854073i \(-0.674125\pi\)
−0.520153 + 0.854073i \(0.674125\pi\)
\(854\) 2.12429e6 0.0996712
\(855\) 3.34139e6 0.156319
\(856\) 3.03944e7 1.41778
\(857\) 1.60053e7 0.744410 0.372205 0.928151i \(-0.378602\pi\)
0.372205 + 0.928151i \(0.378602\pi\)
\(858\) 3.52487e6 0.163465
\(859\) −2.09057e7 −0.966677 −0.483338 0.875434i \(-0.660576\pi\)
−0.483338 + 0.875434i \(0.660576\pi\)
\(860\) −87301.0 −0.00402507
\(861\) 133277. 0.00612700
\(862\) −2.49200e7 −1.14230
\(863\) 1.41874e7 0.648449 0.324224 0.945980i \(-0.394897\pi\)
0.324224 + 0.945980i \(0.394897\pi\)
\(864\) 6.99201e6 0.318653
\(865\) −2.90561e6 −0.132037
\(866\) 1.11322e6 0.0504412
\(867\) 1.47797e7 0.667755
\(868\) −292671. −0.0131850
\(869\) −3.16680e7 −1.42256
\(870\) −1.90560e6 −0.0853561
\(871\) −2.82854e6 −0.126333
\(872\) 1.64740e7 0.733682
\(873\) 26879.7 0.00119368
\(874\) −5.90583e6 −0.261519
\(875\) −669798. −0.0295749
\(876\) −505618. −0.0222619
\(877\) 2.06566e7 0.906901 0.453450 0.891282i \(-0.350193\pi\)
0.453450 + 0.891282i \(0.350193\pi\)
\(878\) 2.89759e6 0.126853
\(879\) 2.17477e7 0.949382
\(880\) −4.62470e6 −0.201315
\(881\) −1.35902e7 −0.589909 −0.294955 0.955511i \(-0.595305\pi\)
−0.294955 + 0.955511i \(0.595305\pi\)
\(882\) 1.00924e7 0.436839
\(883\) 1.20900e7 0.521824 0.260912 0.965363i \(-0.415977\pi\)
0.260912 + 0.965363i \(0.415977\pi\)
\(884\) −170625. −0.00734365
\(885\) −3.03264e6 −0.130155
\(886\) −6.44115e6 −0.275663
\(887\) 1.83559e7 0.783371 0.391685 0.920099i \(-0.371892\pi\)
0.391685 + 0.920099i \(0.371892\pi\)
\(888\) 1.04521e7 0.444805
\(889\) −1.22035e6 −0.0517883
\(890\) 2.18807e6 0.0925946
\(891\) 9.69271e6 0.409026
\(892\) 3.75538e6 0.158031
\(893\) −2.64192e7 −1.10864
\(894\) −1.42056e6 −0.0594451
\(895\) −2.07384e6 −0.0865401
\(896\) −1.47607e6 −0.0614238
\(897\) 467725. 0.0194093
\(898\) 8.39635e6 0.347456
\(899\) 1.81067e7 0.747206
\(900\) −1.69283e6 −0.0696639
\(901\) −3.36134e6 −0.137943
\(902\) −3.09479e6 −0.126653
\(903\) −231355. −0.00944190
\(904\) −1.95608e7 −0.796094
\(905\) 1.84049e6 0.0746984
\(906\) −1.63717e7 −0.662634
\(907\) 1.78761e7 0.721531 0.360765 0.932657i \(-0.382515\pi\)
0.360765 + 0.932657i \(0.382515\pi\)
\(908\) −3.83153e6 −0.154226
\(909\) 7.57987e6 0.304265
\(910\) 61095.1 0.00244570
\(911\) −1.47994e7 −0.590809 −0.295405 0.955372i \(-0.595454\pi\)
−0.295405 + 0.955372i \(0.595454\pi\)
\(912\) 2.80305e7 1.11595
\(913\) 3.26462e6 0.129615
\(914\) 2.31588e7 0.916961
\(915\) −4.05275e6 −0.160028
\(916\) 3.38623e6 0.133345
\(917\) 880670. 0.0345852
\(918\) 6.93961e6 0.271787
\(919\) −2.50852e6 −0.0979780 −0.0489890 0.998799i \(-0.515600\pi\)
−0.0489890 + 0.998799i \(0.515600\pi\)
\(920\) −725390. −0.0282554
\(921\) 2.65225e7 1.03030
\(922\) −4.51315e7 −1.74845
\(923\) 5.93549e6 0.229325
\(924\) 335881. 0.0129421
\(925\) −1.46364e7 −0.562445
\(926\) −4.43626e7 −1.70016
\(927\) −2.44646e7 −0.935060
\(928\) −5.71919e6 −0.218004
\(929\) −9.51733e6 −0.361806 −0.180903 0.983501i \(-0.557902\pi\)
−0.180903 + 0.983501i \(0.557902\pi\)
\(930\) −3.15094e6 −0.119463
\(931\) −4.90183e7 −1.85347
\(932\) 1.05649e6 0.0398407
\(933\) −1.41906e7 −0.533698
\(934\) −3.61025e6 −0.135416
\(935\) 1.79713e6 0.0672281
\(936\) 2.39781e6 0.0894593
\(937\) 4.27838e6 0.159195 0.0795977 0.996827i \(-0.474636\pi\)
0.0795977 + 0.996827i \(0.474636\pi\)
\(938\) 1.52099e6 0.0564444
\(939\) 1.37238e7 0.507939
\(940\) −424557. −0.0156717
\(941\) −3.20489e7 −1.17988 −0.589941 0.807446i \(-0.700849\pi\)
−0.589941 + 0.807446i \(0.700849\pi\)
\(942\) 6.00590e6 0.220521
\(943\) −410657. −0.0150384
\(944\) 2.32466e7 0.849041
\(945\) 440327. 0.0160397
\(946\) 5.37223e6 0.195176
\(947\) 1.34081e7 0.485841 0.242920 0.970046i \(-0.421895\pi\)
0.242920 + 0.970046i \(0.421895\pi\)
\(948\) 3.08450e6 0.111471
\(949\) −1.00289e6 −0.0361485
\(950\) −4.63985e7 −1.66799
\(951\) −7.77389e6 −0.278732
\(952\) 701265. 0.0250778
\(953\) 4.23589e7 1.51082 0.755409 0.655254i \(-0.227438\pi\)
0.755409 + 0.655254i \(0.227438\pi\)
\(954\) 6.18033e6 0.219857
\(955\) 6.42961e6 0.228127
\(956\) −8.43683e6 −0.298562
\(957\) −2.07799e7 −0.733440
\(958\) −3.98267e7 −1.40204
\(959\) 3.99935e6 0.140425
\(960\) 3.98768e6 0.139650
\(961\) 1.31048e6 0.0457745
\(962\) 2.71245e6 0.0944982
\(963\) 1.83717e7 0.638385
\(964\) −4.74760e6 −0.164544
\(965\) −4.41470e6 −0.152610
\(966\) −251511. −0.00867190
\(967\) −4.12202e7 −1.41757 −0.708785 0.705425i \(-0.750756\pi\)
−0.708785 + 0.705425i \(0.750756\pi\)
\(968\) −2.86978e7 −0.984373
\(969\) −1.08925e7 −0.372665
\(970\) 11839.4 0.000404018 0
\(971\) −1.13026e7 −0.384708 −0.192354 0.981326i \(-0.561612\pi\)
−0.192354 + 0.981326i \(0.561612\pi\)
\(972\) 3.79140e6 0.128716
\(973\) −3.67352e6 −0.124394
\(974\) 2.51312e7 0.848820
\(975\) 3.67463e6 0.123795
\(976\) 3.10662e7 1.04391
\(977\) −2.79073e7 −0.935367 −0.467684 0.883896i \(-0.654911\pi\)
−0.467684 + 0.883896i \(0.654911\pi\)
\(978\) −2.60977e7 −0.872480
\(979\) 2.38601e7 0.795638
\(980\) −787725. −0.0262005
\(981\) 9.95758e6 0.330355
\(982\) 347568. 0.0115017
\(983\) −1.22161e7 −0.403226 −0.201613 0.979465i \(-0.564618\pi\)
−0.201613 + 0.979465i \(0.564618\pi\)
\(984\) 2.30393e6 0.0758546
\(985\) 9.53852e6 0.313249
\(986\) −5.67633e6 −0.185941
\(987\) −1.12511e6 −0.0367623
\(988\) −1.52372e6 −0.0496608
\(989\) 712858. 0.0231746
\(990\) −3.30429e6 −0.107150
\(991\) 2.47723e7 0.801277 0.400638 0.916236i \(-0.368788\pi\)
0.400638 + 0.916236i \(0.368788\pi\)
\(992\) −9.45675e6 −0.305114
\(993\) −2.17852e7 −0.701113
\(994\) −3.19170e6 −0.102461
\(995\) −2.54691e6 −0.0815561
\(996\) −317978. −0.0101566
\(997\) 2.12381e7 0.676670 0.338335 0.941026i \(-0.390136\pi\)
0.338335 + 0.941026i \(0.390136\pi\)
\(998\) −1.52873e7 −0.485851
\(999\) 1.95493e7 0.619750
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 43.6.a.a.1.7 8
3.2 odd 2 387.6.a.c.1.2 8
4.3 odd 2 688.6.a.e.1.6 8
5.4 even 2 1075.6.a.a.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.7 8 1.1 even 1 trivial
387.6.a.c.1.2 8 3.2 odd 2
688.6.a.e.1.6 8 4.3 odd 2
1075.6.a.a.1.2 8 5.4 even 2