Properties

Label 76.6.a.b.1.4
Level $76$
Weight $6$
Character 76.1
Self dual yes
Analytic conductor $12.189$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1891703058\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 140 x^{2} - 84 x + 3103\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-10.0437\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q+27.9817 q^{3} -64.0791 q^{5} +65.8093 q^{7} +539.976 q^{9} +O(q^{10})\) \(q+27.9817 q^{3} -64.0791 q^{5} +65.8093 q^{7} +539.976 q^{9} +635.790 q^{11} +467.894 q^{13} -1793.04 q^{15} +522.359 q^{17} -361.000 q^{19} +1841.46 q^{21} -3220.28 q^{23} +981.128 q^{25} +8309.91 q^{27} +6979.01 q^{29} -3388.33 q^{31} +17790.5 q^{33} -4217.00 q^{35} -13423.9 q^{37} +13092.5 q^{39} +7104.69 q^{41} -14035.1 q^{43} -34601.2 q^{45} -1444.37 q^{47} -12476.1 q^{49} +14616.5 q^{51} -37171.0 q^{53} -40740.8 q^{55} -10101.4 q^{57} +37865.1 q^{59} +39660.1 q^{61} +35535.5 q^{63} -29982.2 q^{65} -11022.0 q^{67} -90108.9 q^{69} +9099.27 q^{71} +29926.9 q^{73} +27453.7 q^{75} +41840.9 q^{77} -33256.4 q^{79} +101311. q^{81} -50768.6 q^{83} -33472.3 q^{85} +195285. q^{87} -128742. q^{89} +30791.8 q^{91} -94811.3 q^{93} +23132.5 q^{95} -111792. q^{97} +343312. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + O(q^{10}) \) \( 4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + 706q^{11} + 788q^{13} + 2792q^{15} + 240q^{17} - 1444q^{19} + 7128q^{21} + 5884q^{23} + 11774q^{25} + 4426q^{27} + 5240q^{29} - 860q^{31} + 10748q^{33} + 20322q^{35} - 20732q^{37} + 15404q^{39} - 10204q^{41} - 12554q^{43} - 71306q^{45} - 4826q^{47} - 21376q^{49} + 34518q^{51} - 76484q^{53} - 72914q^{55} - 3610q^{57} + 23898q^{59} - 32482q^{61} - 43566q^{63} - 6076q^{65} + 5022q^{67} - 149524q^{69} + 121300q^{71} - 104700q^{73} - 95230q^{75} - 10002q^{77} + 117128q^{79} + 44924q^{81} + 92832q^{83} + 80322q^{85} + 300148q^{87} + 5988q^{89} + 165618q^{91} + 16180q^{93} + 39710q^{95} + 22972q^{97} + 441418q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 27.9817 1.79503 0.897514 0.440986i \(-0.145371\pi\)
0.897514 + 0.440986i \(0.145371\pi\)
\(4\) 0 0
\(5\) −64.0791 −1.14628 −0.573141 0.819457i \(-0.694275\pi\)
−0.573141 + 0.819457i \(0.694275\pi\)
\(6\) 0 0
\(7\) 65.8093 0.507624 0.253812 0.967254i \(-0.418316\pi\)
0.253812 + 0.967254i \(0.418316\pi\)
\(8\) 0 0
\(9\) 539.976 2.22213
\(10\) 0 0
\(11\) 635.790 1.58428 0.792140 0.610340i \(-0.208967\pi\)
0.792140 + 0.610340i \(0.208967\pi\)
\(12\) 0 0
\(13\) 467.894 0.767873 0.383936 0.923360i \(-0.374568\pi\)
0.383936 + 0.923360i \(0.374568\pi\)
\(14\) 0 0
\(15\) −1793.04 −2.05761
\(16\) 0 0
\(17\) 522.359 0.438376 0.219188 0.975683i \(-0.429659\pi\)
0.219188 + 0.975683i \(0.429659\pi\)
\(18\) 0 0
\(19\) −361.000 −0.229416
\(20\) 0 0
\(21\) 1841.46 0.911199
\(22\) 0 0
\(23\) −3220.28 −1.26933 −0.634664 0.772788i \(-0.718861\pi\)
−0.634664 + 0.772788i \(0.718861\pi\)
\(24\) 0 0
\(25\) 981.128 0.313961
\(26\) 0 0
\(27\) 8309.91 2.19375
\(28\) 0 0
\(29\) 6979.01 1.54099 0.770493 0.637448i \(-0.220010\pi\)
0.770493 + 0.637448i \(0.220010\pi\)
\(30\) 0 0
\(31\) −3388.33 −0.633259 −0.316630 0.948549i \(-0.602551\pi\)
−0.316630 + 0.948549i \(0.602551\pi\)
\(32\) 0 0
\(33\) 17790.5 2.84383
\(34\) 0 0
\(35\) −4217.00 −0.581880
\(36\) 0 0
\(37\) −13423.9 −1.61204 −0.806020 0.591888i \(-0.798383\pi\)
−0.806020 + 0.591888i \(0.798383\pi\)
\(38\) 0 0
\(39\) 13092.5 1.37835
\(40\) 0 0
\(41\) 7104.69 0.660063 0.330032 0.943970i \(-0.392941\pi\)
0.330032 + 0.943970i \(0.392941\pi\)
\(42\) 0 0
\(43\) −14035.1 −1.15756 −0.578781 0.815483i \(-0.696471\pi\)
−0.578781 + 0.815483i \(0.696471\pi\)
\(44\) 0 0
\(45\) −34601.2 −2.54718
\(46\) 0 0
\(47\) −1444.37 −0.0953745 −0.0476873 0.998862i \(-0.515185\pi\)
−0.0476873 + 0.998862i \(0.515185\pi\)
\(48\) 0 0
\(49\) −12476.1 −0.742318
\(50\) 0 0
\(51\) 14616.5 0.786898
\(52\) 0 0
\(53\) −37171.0 −1.81767 −0.908834 0.417158i \(-0.863026\pi\)
−0.908834 + 0.417158i \(0.863026\pi\)
\(54\) 0 0
\(55\) −40740.8 −1.81603
\(56\) 0 0
\(57\) −10101.4 −0.411808
\(58\) 0 0
\(59\) 37865.1 1.41615 0.708074 0.706138i \(-0.249564\pi\)
0.708074 + 0.706138i \(0.249564\pi\)
\(60\) 0 0
\(61\) 39660.1 1.36468 0.682338 0.731037i \(-0.260963\pi\)
0.682338 + 0.731037i \(0.260963\pi\)
\(62\) 0 0
\(63\) 35535.5 1.12800
\(64\) 0 0
\(65\) −29982.2 −0.880198
\(66\) 0 0
\(67\) −11022.0 −0.299968 −0.149984 0.988688i \(-0.547922\pi\)
−0.149984 + 0.988688i \(0.547922\pi\)
\(68\) 0 0
\(69\) −90108.9 −2.27848
\(70\) 0 0
\(71\) 9099.27 0.214220 0.107110 0.994247i \(-0.465840\pi\)
0.107110 + 0.994247i \(0.465840\pi\)
\(72\) 0 0
\(73\) 29926.9 0.657286 0.328643 0.944454i \(-0.393409\pi\)
0.328643 + 0.944454i \(0.393409\pi\)
\(74\) 0 0
\(75\) 27453.7 0.563569
\(76\) 0 0
\(77\) 41840.9 0.804218
\(78\) 0 0
\(79\) −33256.4 −0.599525 −0.299763 0.954014i \(-0.596907\pi\)
−0.299763 + 0.954014i \(0.596907\pi\)
\(80\) 0 0
\(81\) 101311. 1.71572
\(82\) 0 0
\(83\) −50768.6 −0.808910 −0.404455 0.914558i \(-0.632539\pi\)
−0.404455 + 0.914558i \(0.632539\pi\)
\(84\) 0 0
\(85\) −33472.3 −0.502503
\(86\) 0 0
\(87\) 195285. 2.76611
\(88\) 0 0
\(89\) −128742. −1.72284 −0.861421 0.507892i \(-0.830425\pi\)
−0.861421 + 0.507892i \(0.830425\pi\)
\(90\) 0 0
\(91\) 30791.8 0.389791
\(92\) 0 0
\(93\) −94811.3 −1.13672
\(94\) 0 0
\(95\) 23132.5 0.262975
\(96\) 0 0
\(97\) −111792. −1.20638 −0.603189 0.797598i \(-0.706103\pi\)
−0.603189 + 0.797598i \(0.706103\pi\)
\(98\) 0 0
\(99\) 343312. 3.52047
\(100\) 0 0
\(101\) −106980. −1.04352 −0.521758 0.853093i \(-0.674724\pi\)
−0.521758 + 0.853093i \(0.674724\pi\)
\(102\) 0 0
\(103\) −2453.53 −0.0227876 −0.0113938 0.999935i \(-0.503627\pi\)
−0.0113938 + 0.999935i \(0.503627\pi\)
\(104\) 0 0
\(105\) −117999. −1.04449
\(106\) 0 0
\(107\) 25637.7 0.216481 0.108241 0.994125i \(-0.465478\pi\)
0.108241 + 0.994125i \(0.465478\pi\)
\(108\) 0 0
\(109\) 97941.0 0.789584 0.394792 0.918771i \(-0.370817\pi\)
0.394792 + 0.918771i \(0.370817\pi\)
\(110\) 0 0
\(111\) −375625. −2.89366
\(112\) 0 0
\(113\) 221831. 1.63428 0.817138 0.576442i \(-0.195559\pi\)
0.817138 + 0.576442i \(0.195559\pi\)
\(114\) 0 0
\(115\) 206352. 1.45501
\(116\) 0 0
\(117\) 252652. 1.70631
\(118\) 0 0
\(119\) 34376.1 0.222530
\(120\) 0 0
\(121\) 243178. 1.50994
\(122\) 0 0
\(123\) 198801. 1.18483
\(124\) 0 0
\(125\) 137377. 0.786394
\(126\) 0 0
\(127\) −308453. −1.69699 −0.848496 0.529202i \(-0.822491\pi\)
−0.848496 + 0.529202i \(0.822491\pi\)
\(128\) 0 0
\(129\) −392726. −2.07786
\(130\) 0 0
\(131\) 262055. 1.33418 0.667090 0.744977i \(-0.267539\pi\)
0.667090 + 0.744977i \(0.267539\pi\)
\(132\) 0 0
\(133\) −23757.2 −0.116457
\(134\) 0 0
\(135\) −532491. −2.51465
\(136\) 0 0
\(137\) 191566. 0.872001 0.436001 0.899946i \(-0.356395\pi\)
0.436001 + 0.899946i \(0.356395\pi\)
\(138\) 0 0
\(139\) −398392. −1.74893 −0.874467 0.485085i \(-0.838789\pi\)
−0.874467 + 0.485085i \(0.838789\pi\)
\(140\) 0 0
\(141\) −40415.8 −0.171200
\(142\) 0 0
\(143\) 297482. 1.21653
\(144\) 0 0
\(145\) −447209. −1.76640
\(146\) 0 0
\(147\) −349104. −1.33248
\(148\) 0 0
\(149\) −91457.4 −0.337484 −0.168742 0.985660i \(-0.553970\pi\)
−0.168742 + 0.985660i \(0.553970\pi\)
\(150\) 0 0
\(151\) 168957. 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(152\) 0 0
\(153\) 282062. 0.974127
\(154\) 0 0
\(155\) 217121. 0.725893
\(156\) 0 0
\(157\) −210494. −0.681539 −0.340770 0.940147i \(-0.610688\pi\)
−0.340770 + 0.940147i \(0.610688\pi\)
\(158\) 0 0
\(159\) −1.04011e6 −3.26277
\(160\) 0 0
\(161\) −211924. −0.644341
\(162\) 0 0
\(163\) 619993. 1.82775 0.913876 0.405992i \(-0.133074\pi\)
0.913876 + 0.405992i \(0.133074\pi\)
\(164\) 0 0
\(165\) −1.14000e6 −3.25983
\(166\) 0 0
\(167\) 49657.8 0.137783 0.0688916 0.997624i \(-0.478054\pi\)
0.0688916 + 0.997624i \(0.478054\pi\)
\(168\) 0 0
\(169\) −152368. −0.410372
\(170\) 0 0
\(171\) −194932. −0.509791
\(172\) 0 0
\(173\) 268048. 0.680923 0.340461 0.940259i \(-0.389417\pi\)
0.340461 + 0.940259i \(0.389417\pi\)
\(174\) 0 0
\(175\) 64567.4 0.159374
\(176\) 0 0
\(177\) 1.05953e6 2.54203
\(178\) 0 0
\(179\) −266354. −0.621337 −0.310668 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(180\) 0 0
\(181\) 276641. 0.627654 0.313827 0.949480i \(-0.398389\pi\)
0.313827 + 0.949480i \(0.398389\pi\)
\(182\) 0 0
\(183\) 1.10976e6 2.44963
\(184\) 0 0
\(185\) 860194. 1.84785
\(186\) 0 0
\(187\) 332111. 0.694511
\(188\) 0 0
\(189\) 546869. 1.11360
\(190\) 0 0
\(191\) 643635. 1.27660 0.638302 0.769786i \(-0.279637\pi\)
0.638302 + 0.769786i \(0.279637\pi\)
\(192\) 0 0
\(193\) 59352.1 0.114695 0.0573473 0.998354i \(-0.481736\pi\)
0.0573473 + 0.998354i \(0.481736\pi\)
\(194\) 0 0
\(195\) −838954. −1.57998
\(196\) 0 0
\(197\) 60559.8 0.111178 0.0555890 0.998454i \(-0.482296\pi\)
0.0555890 + 0.998454i \(0.482296\pi\)
\(198\) 0 0
\(199\) −260073. −0.465546 −0.232773 0.972531i \(-0.574780\pi\)
−0.232773 + 0.972531i \(0.574780\pi\)
\(200\) 0 0
\(201\) −308416. −0.538451
\(202\) 0 0
\(203\) 459284. 0.782242
\(204\) 0 0
\(205\) −455262. −0.756618
\(206\) 0 0
\(207\) −1.73887e6 −2.82061
\(208\) 0 0
\(209\) −229520. −0.363459
\(210\) 0 0
\(211\) −501653. −0.775706 −0.387853 0.921721i \(-0.626783\pi\)
−0.387853 + 0.921721i \(0.626783\pi\)
\(212\) 0 0
\(213\) 254613. 0.384531
\(214\) 0 0
\(215\) 899356. 1.32689
\(216\) 0 0
\(217\) −222984. −0.321458
\(218\) 0 0
\(219\) 837406. 1.17985
\(220\) 0 0
\(221\) 244409. 0.336617
\(222\) 0 0
\(223\) −625533. −0.842341 −0.421170 0.906981i \(-0.638381\pi\)
−0.421170 + 0.906981i \(0.638381\pi\)
\(224\) 0 0
\(225\) 529786. 0.697661
\(226\) 0 0
\(227\) −128165. −0.165084 −0.0825422 0.996588i \(-0.526304\pi\)
−0.0825422 + 0.996588i \(0.526304\pi\)
\(228\) 0 0
\(229\) −400845. −0.505112 −0.252556 0.967582i \(-0.581271\pi\)
−0.252556 + 0.967582i \(0.581271\pi\)
\(230\) 0 0
\(231\) 1.17078e6 1.44359
\(232\) 0 0
\(233\) −233608. −0.281902 −0.140951 0.990017i \(-0.545016\pi\)
−0.140951 + 0.990017i \(0.545016\pi\)
\(234\) 0 0
\(235\) 92553.6 0.109326
\(236\) 0 0
\(237\) −930571. −1.07616
\(238\) 0 0
\(239\) 1.47603e6 1.67147 0.835737 0.549130i \(-0.185041\pi\)
0.835737 + 0.549130i \(0.185041\pi\)
\(240\) 0 0
\(241\) 1.50349e6 1.66747 0.833735 0.552164i \(-0.186198\pi\)
0.833735 + 0.552164i \(0.186198\pi\)
\(242\) 0 0
\(243\) 815556. 0.886009
\(244\) 0 0
\(245\) 799459. 0.850905
\(246\) 0 0
\(247\) −168910. −0.176162
\(248\) 0 0
\(249\) −1.42059e6 −1.45202
\(250\) 0 0
\(251\) −1.24638e6 −1.24872 −0.624361 0.781136i \(-0.714641\pi\)
−0.624361 + 0.781136i \(0.714641\pi\)
\(252\) 0 0
\(253\) −2.04742e6 −2.01097
\(254\) 0 0
\(255\) −936613. −0.902006
\(256\) 0 0
\(257\) −185409. −0.175105 −0.0875525 0.996160i \(-0.527905\pi\)
−0.0875525 + 0.996160i \(0.527905\pi\)
\(258\) 0 0
\(259\) −883421. −0.818311
\(260\) 0 0
\(261\) 3.76850e6 3.42427
\(262\) 0 0
\(263\) 685160. 0.610805 0.305403 0.952223i \(-0.401209\pi\)
0.305403 + 0.952223i \(0.401209\pi\)
\(264\) 0 0
\(265\) 2.38188e6 2.08356
\(266\) 0 0
\(267\) −3.60242e6 −3.09255
\(268\) 0 0
\(269\) −46236.6 −0.0389587 −0.0194794 0.999810i \(-0.506201\pi\)
−0.0194794 + 0.999810i \(0.506201\pi\)
\(270\) 0 0
\(271\) 635296. 0.525476 0.262738 0.964867i \(-0.415375\pi\)
0.262738 + 0.964867i \(0.415375\pi\)
\(272\) 0 0
\(273\) 861607. 0.699685
\(274\) 0 0
\(275\) 623791. 0.497402
\(276\) 0 0
\(277\) −912448. −0.714511 −0.357255 0.934007i \(-0.616287\pi\)
−0.357255 + 0.934007i \(0.616287\pi\)
\(278\) 0 0
\(279\) −1.82962e6 −1.40718
\(280\) 0 0
\(281\) −2.37085e6 −1.79118 −0.895589 0.444883i \(-0.853245\pi\)
−0.895589 + 0.444883i \(0.853245\pi\)
\(282\) 0 0
\(283\) −87826.9 −0.0651871 −0.0325936 0.999469i \(-0.510377\pi\)
−0.0325936 + 0.999469i \(0.510377\pi\)
\(284\) 0 0
\(285\) 647288. 0.472047
\(286\) 0 0
\(287\) 467555. 0.335064
\(288\) 0 0
\(289\) −1.14700e6 −0.807826
\(290\) 0 0
\(291\) −3.12815e6 −2.16548
\(292\) 0 0
\(293\) 368613. 0.250842 0.125421 0.992104i \(-0.459972\pi\)
0.125421 + 0.992104i \(0.459972\pi\)
\(294\) 0 0
\(295\) −2.42636e6 −1.62330
\(296\) 0 0
\(297\) 5.28336e6 3.47551
\(298\) 0 0
\(299\) −1.50675e6 −0.974682
\(300\) 0 0
\(301\) −923640. −0.587607
\(302\) 0 0
\(303\) −2.99348e6 −1.87314
\(304\) 0 0
\(305\) −2.54138e6 −1.56430
\(306\) 0 0
\(307\) −915441. −0.554351 −0.277175 0.960819i \(-0.589398\pi\)
−0.277175 + 0.960819i \(0.589398\pi\)
\(308\) 0 0
\(309\) −68654.0 −0.0409044
\(310\) 0 0
\(311\) 1.65178e6 0.968390 0.484195 0.874960i \(-0.339113\pi\)
0.484195 + 0.874960i \(0.339113\pi\)
\(312\) 0 0
\(313\) 2.74796e6 1.58544 0.792720 0.609586i \(-0.208664\pi\)
0.792720 + 0.609586i \(0.208664\pi\)
\(314\) 0 0
\(315\) −2.27708e6 −1.29301
\(316\) 0 0
\(317\) −20962.0 −0.0117161 −0.00585807 0.999983i \(-0.501865\pi\)
−0.00585807 + 0.999983i \(0.501865\pi\)
\(318\) 0 0
\(319\) 4.43718e6 2.44135
\(320\) 0 0
\(321\) 717387. 0.388589
\(322\) 0 0
\(323\) −188572. −0.100570
\(324\) 0 0
\(325\) 459064. 0.241082
\(326\) 0 0
\(327\) 2.74056e6 1.41733
\(328\) 0 0
\(329\) −95052.7 −0.0484144
\(330\) 0 0
\(331\) 1.41911e6 0.711947 0.355973 0.934496i \(-0.384149\pi\)
0.355973 + 0.934496i \(0.384149\pi\)
\(332\) 0 0
\(333\) −7.24862e6 −3.58216
\(334\) 0 0
\(335\) 706282. 0.343848
\(336\) 0 0
\(337\) −2.30351e6 −1.10488 −0.552441 0.833552i \(-0.686303\pi\)
−0.552441 + 0.833552i \(0.686303\pi\)
\(338\) 0 0
\(339\) 6.20720e6 2.93357
\(340\) 0 0
\(341\) −2.15427e6 −1.00326
\(342\) 0 0
\(343\) −1.92710e6 −0.884442
\(344\) 0 0
\(345\) 5.77410e6 2.61178
\(346\) 0 0
\(347\) 2.93674e6 1.30931 0.654653 0.755929i \(-0.272815\pi\)
0.654653 + 0.755929i \(0.272815\pi\)
\(348\) 0 0
\(349\) −3.80733e6 −1.67323 −0.836617 0.547788i \(-0.815470\pi\)
−0.836617 + 0.547788i \(0.815470\pi\)
\(350\) 0 0
\(351\) 3.88816e6 1.68452
\(352\) 0 0
\(353\) 679178. 0.290100 0.145050 0.989424i \(-0.453666\pi\)
0.145050 + 0.989424i \(0.453666\pi\)
\(354\) 0 0
\(355\) −583073. −0.245557
\(356\) 0 0
\(357\) 961902. 0.399448
\(358\) 0 0
\(359\) 3.29854e6 1.35078 0.675392 0.737459i \(-0.263974\pi\)
0.675392 + 0.737459i \(0.263974\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) 6.80453e6 2.71039
\(364\) 0 0
\(365\) −1.91769e6 −0.753435
\(366\) 0 0
\(367\) −1.36424e6 −0.528720 −0.264360 0.964424i \(-0.585161\pi\)
−0.264360 + 0.964424i \(0.585161\pi\)
\(368\) 0 0
\(369\) 3.83637e6 1.46674
\(370\) 0 0
\(371\) −2.44620e6 −0.922692
\(372\) 0 0
\(373\) −359656. −0.133849 −0.0669246 0.997758i \(-0.521319\pi\)
−0.0669246 + 0.997758i \(0.521319\pi\)
\(374\) 0 0
\(375\) 3.84405e6 1.41160
\(376\) 0 0
\(377\) 3.26544e6 1.18328
\(378\) 0 0
\(379\) 2.90865e6 1.04014 0.520072 0.854123i \(-0.325905\pi\)
0.520072 + 0.854123i \(0.325905\pi\)
\(380\) 0 0
\(381\) −8.63105e6 −3.04615
\(382\) 0 0
\(383\) 2.04074e6 0.710871 0.355436 0.934701i \(-0.384333\pi\)
0.355436 + 0.934701i \(0.384333\pi\)
\(384\) 0 0
\(385\) −2.68112e6 −0.921861
\(386\) 0 0
\(387\) −7.57863e6 −2.57225
\(388\) 0 0
\(389\) −3.37257e6 −1.13002 −0.565011 0.825083i \(-0.691128\pi\)
−0.565011 + 0.825083i \(0.691128\pi\)
\(390\) 0 0
\(391\) −1.68214e6 −0.556443
\(392\) 0 0
\(393\) 7.33276e6 2.39489
\(394\) 0 0
\(395\) 2.13104e6 0.687225
\(396\) 0 0
\(397\) 850462. 0.270819 0.135409 0.990790i \(-0.456765\pi\)
0.135409 + 0.990790i \(0.456765\pi\)
\(398\) 0 0
\(399\) −664766. −0.209043
\(400\) 0 0
\(401\) −805071. −0.250019 −0.125010 0.992156i \(-0.539896\pi\)
−0.125010 + 0.992156i \(0.539896\pi\)
\(402\) 0 0
\(403\) −1.58538e6 −0.486262
\(404\) 0 0
\(405\) −6.49193e6 −1.96669
\(406\) 0 0
\(407\) −8.53481e6 −2.55392
\(408\) 0 0
\(409\) −1.41286e6 −0.417630 −0.208815 0.977955i \(-0.566961\pi\)
−0.208815 + 0.977955i \(0.566961\pi\)
\(410\) 0 0
\(411\) 5.36035e6 1.56527
\(412\) 0 0
\(413\) 2.49187e6 0.718871
\(414\) 0 0
\(415\) 3.25321e6 0.927238
\(416\) 0 0
\(417\) −1.11477e7 −3.13939
\(418\) 0 0
\(419\) −1.55942e6 −0.433938 −0.216969 0.976179i \(-0.569617\pi\)
−0.216969 + 0.976179i \(0.569617\pi\)
\(420\) 0 0
\(421\) 1.59717e6 0.439183 0.219592 0.975592i \(-0.429528\pi\)
0.219592 + 0.975592i \(0.429528\pi\)
\(422\) 0 0
\(423\) −779923. −0.211934
\(424\) 0 0
\(425\) 512502. 0.137633
\(426\) 0 0
\(427\) 2.61000e6 0.692742
\(428\) 0 0
\(429\) 8.32407e6 2.18370
\(430\) 0 0
\(431\) 4.28243e6 1.11045 0.555223 0.831701i \(-0.312633\pi\)
0.555223 + 0.831701i \(0.312633\pi\)
\(432\) 0 0
\(433\) −3.66079e6 −0.938328 −0.469164 0.883111i \(-0.655445\pi\)
−0.469164 + 0.883111i \(0.655445\pi\)
\(434\) 0 0
\(435\) −1.25137e7 −3.17075
\(436\) 0 0
\(437\) 1.16252e6 0.291204
\(438\) 0 0
\(439\) 5.49744e6 1.36144 0.680722 0.732542i \(-0.261666\pi\)
0.680722 + 0.732542i \(0.261666\pi\)
\(440\) 0 0
\(441\) −6.73682e6 −1.64952
\(442\) 0 0
\(443\) 1.82338e6 0.441436 0.220718 0.975338i \(-0.429160\pi\)
0.220718 + 0.975338i \(0.429160\pi\)
\(444\) 0 0
\(445\) 8.24967e6 1.97486
\(446\) 0 0
\(447\) −2.55913e6 −0.605793
\(448\) 0 0
\(449\) 1.53215e6 0.358662 0.179331 0.983789i \(-0.442607\pi\)
0.179331 + 0.983789i \(0.442607\pi\)
\(450\) 0 0
\(451\) 4.51709e6 1.04572
\(452\) 0 0
\(453\) 4.72771e6 1.08244
\(454\) 0 0
\(455\) −1.97311e6 −0.446810
\(456\) 0 0
\(457\) 5.28539e6 1.18382 0.591912 0.806003i \(-0.298373\pi\)
0.591912 + 0.806003i \(0.298373\pi\)
\(458\) 0 0
\(459\) 4.34076e6 0.961688
\(460\) 0 0
\(461\) −6.69418e6 −1.46705 −0.733525 0.679662i \(-0.762126\pi\)
−0.733525 + 0.679662i \(0.762126\pi\)
\(462\) 0 0
\(463\) 812541. 0.176154 0.0880770 0.996114i \(-0.471928\pi\)
0.0880770 + 0.996114i \(0.471928\pi\)
\(464\) 0 0
\(465\) 6.07542e6 1.30300
\(466\) 0 0
\(467\) −3.46112e6 −0.734387 −0.367193 0.930145i \(-0.619681\pi\)
−0.367193 + 0.930145i \(0.619681\pi\)
\(468\) 0 0
\(469\) −725353. −0.152271
\(470\) 0 0
\(471\) −5.88999e6 −1.22338
\(472\) 0 0
\(473\) −8.92337e6 −1.83390
\(474\) 0 0
\(475\) −354187. −0.0720276
\(476\) 0 0
\(477\) −2.00715e7 −4.03909
\(478\) 0 0
\(479\) 2.38975e6 0.475898 0.237949 0.971278i \(-0.423525\pi\)
0.237949 + 0.971278i \(0.423525\pi\)
\(480\) 0 0
\(481\) −6.28099e6 −1.23784
\(482\) 0 0
\(483\) −5.93000e6 −1.15661
\(484\) 0 0
\(485\) 7.16356e6 1.38285
\(486\) 0 0
\(487\) 1.58853e6 0.303511 0.151755 0.988418i \(-0.451507\pi\)
0.151755 + 0.988418i \(0.451507\pi\)
\(488\) 0 0
\(489\) 1.73485e7 3.28087
\(490\) 0 0
\(491\) −949072. −0.177662 −0.0888312 0.996047i \(-0.528313\pi\)
−0.0888312 + 0.996047i \(0.528313\pi\)
\(492\) 0 0
\(493\) 3.64555e6 0.675532
\(494\) 0 0
\(495\) −2.19991e7 −4.03545
\(496\) 0 0
\(497\) 598816. 0.108743
\(498\) 0 0
\(499\) 5.93440e6 1.06690 0.533452 0.845830i \(-0.320894\pi\)
0.533452 + 0.845830i \(0.320894\pi\)
\(500\) 0 0
\(501\) 1.38951e6 0.247325
\(502\) 0 0
\(503\) −6.26101e6 −1.10338 −0.551689 0.834050i \(-0.686016\pi\)
−0.551689 + 0.834050i \(0.686016\pi\)
\(504\) 0 0
\(505\) 6.85518e6 1.19616
\(506\) 0 0
\(507\) −4.26352e6 −0.736628
\(508\) 0 0
\(509\) 3.36180e6 0.575145 0.287572 0.957759i \(-0.407152\pi\)
0.287572 + 0.957759i \(0.407152\pi\)
\(510\) 0 0
\(511\) 1.96947e6 0.333654
\(512\) 0 0
\(513\) −2.99988e6 −0.503281
\(514\) 0 0
\(515\) 157220. 0.0261210
\(516\) 0 0
\(517\) −918313. −0.151100
\(518\) 0 0
\(519\) 7.50045e6 1.22228
\(520\) 0 0
\(521\) −4.54384e6 −0.733380 −0.366690 0.930343i \(-0.619509\pi\)
−0.366690 + 0.930343i \(0.619509\pi\)
\(522\) 0 0
\(523\) −9.28544e6 −1.48439 −0.742195 0.670184i \(-0.766215\pi\)
−0.742195 + 0.670184i \(0.766215\pi\)
\(524\) 0 0
\(525\) 1.80671e6 0.286081
\(526\) 0 0
\(527\) −1.76993e6 −0.277606
\(528\) 0 0
\(529\) 3.93385e6 0.611193
\(530\) 0 0
\(531\) 2.04463e7 3.14686
\(532\) 0 0
\(533\) 3.32424e6 0.506845
\(534\) 0 0
\(535\) −1.64284e6 −0.248148
\(536\) 0 0
\(537\) −7.45305e6 −1.11532
\(538\) 0 0
\(539\) −7.93220e6 −1.17604
\(540\) 0 0
\(541\) −875434. −0.128597 −0.0642984 0.997931i \(-0.520481\pi\)
−0.0642984 + 0.997931i \(0.520481\pi\)
\(542\) 0 0
\(543\) 7.74089e6 1.12666
\(544\) 0 0
\(545\) −6.27597e6 −0.905085
\(546\) 0 0
\(547\) 1.16991e7 1.67180 0.835901 0.548881i \(-0.184946\pi\)
0.835901 + 0.548881i \(0.184946\pi\)
\(548\) 0 0
\(549\) 2.14155e7 3.03248
\(550\) 0 0
\(551\) −2.51942e6 −0.353527
\(552\) 0 0
\(553\) −2.18858e6 −0.304333
\(554\) 0 0
\(555\) 2.40697e7 3.31695
\(556\) 0 0
\(557\) −3.93911e6 −0.537972 −0.268986 0.963144i \(-0.586689\pi\)
−0.268986 + 0.963144i \(0.586689\pi\)
\(558\) 0 0
\(559\) −6.56694e6 −0.888861
\(560\) 0 0
\(561\) 9.29303e6 1.24667
\(562\) 0 0
\(563\) 9.82008e6 1.30570 0.652851 0.757486i \(-0.273573\pi\)
0.652851 + 0.757486i \(0.273573\pi\)
\(564\) 0 0
\(565\) −1.42147e7 −1.87334
\(566\) 0 0
\(567\) 6.66723e6 0.870938
\(568\) 0 0
\(569\) 8.83108e6 1.14349 0.571746 0.820431i \(-0.306266\pi\)
0.571746 + 0.820431i \(0.306266\pi\)
\(570\) 0 0
\(571\) 7.13743e6 0.916118 0.458059 0.888922i \(-0.348545\pi\)
0.458059 + 0.888922i \(0.348545\pi\)
\(572\) 0 0
\(573\) 1.80100e7 2.29154
\(574\) 0 0
\(575\) −3.15951e6 −0.398519
\(576\) 0 0
\(577\) 1.30687e7 1.63415 0.817077 0.576528i \(-0.195593\pi\)
0.817077 + 0.576528i \(0.195593\pi\)
\(578\) 0 0
\(579\) 1.66077e6 0.205880
\(580\) 0 0
\(581\) −3.34105e6 −0.410622
\(582\) 0 0
\(583\) −2.36329e7 −2.87969
\(584\) 0 0
\(585\) −1.61897e7 −1.95591
\(586\) 0 0
\(587\) −9.11604e6 −1.09197 −0.545985 0.837795i \(-0.683845\pi\)
−0.545985 + 0.837795i \(0.683845\pi\)
\(588\) 0 0
\(589\) 1.22319e6 0.145280
\(590\) 0 0
\(591\) 1.69457e6 0.199568
\(592\) 0 0
\(593\) −6.76736e6 −0.790283 −0.395141 0.918620i \(-0.629304\pi\)
−0.395141 + 0.918620i \(0.629304\pi\)
\(594\) 0 0
\(595\) −2.20279e6 −0.255082
\(596\) 0 0
\(597\) −7.27729e6 −0.835668
\(598\) 0 0
\(599\) −3.98188e6 −0.453441 −0.226720 0.973960i \(-0.572800\pi\)
−0.226720 + 0.973960i \(0.572800\pi\)
\(600\) 0 0
\(601\) 1.55761e7 1.75903 0.879516 0.475870i \(-0.157867\pi\)
0.879516 + 0.475870i \(0.157867\pi\)
\(602\) 0 0
\(603\) −5.95164e6 −0.666567
\(604\) 0 0
\(605\) −1.55826e7 −1.73082
\(606\) 0 0
\(607\) 3.12960e6 0.344760 0.172380 0.985031i \(-0.444854\pi\)
0.172380 + 0.985031i \(0.444854\pi\)
\(608\) 0 0
\(609\) 1.28515e7 1.40415
\(610\) 0 0
\(611\) −675810. −0.0732355
\(612\) 0 0
\(613\) 1.40524e7 1.51043 0.755215 0.655477i \(-0.227532\pi\)
0.755215 + 0.655477i \(0.227532\pi\)
\(614\) 0 0
\(615\) −1.27390e7 −1.35815
\(616\) 0 0
\(617\) −5.27324e6 −0.557654 −0.278827 0.960341i \(-0.589946\pi\)
−0.278827 + 0.960341i \(0.589946\pi\)
\(618\) 0 0
\(619\) −7.91873e6 −0.830670 −0.415335 0.909668i \(-0.636336\pi\)
−0.415335 + 0.909668i \(0.636336\pi\)
\(620\) 0 0
\(621\) −2.67602e7 −2.78459
\(622\) 0 0
\(623\) −8.47242e6 −0.874556
\(624\) 0 0
\(625\) −1.18690e7 −1.21539
\(626\) 0 0
\(627\) −6.42237e6 −0.652418
\(628\) 0 0
\(629\) −7.01213e6 −0.706680
\(630\) 0 0
\(631\) 1.62979e7 1.62951 0.814755 0.579805i \(-0.196871\pi\)
0.814755 + 0.579805i \(0.196871\pi\)
\(632\) 0 0
\(633\) −1.40371e7 −1.39241
\(634\) 0 0
\(635\) 1.97654e7 1.94523
\(636\) 0 0
\(637\) −5.83751e6 −0.570006
\(638\) 0 0
\(639\) 4.91339e6 0.476024
\(640\) 0 0
\(641\) 1.00004e7 0.961330 0.480665 0.876904i \(-0.340395\pi\)
0.480665 + 0.876904i \(0.340395\pi\)
\(642\) 0 0
\(643\) −1.66096e7 −1.58428 −0.792138 0.610342i \(-0.791032\pi\)
−0.792138 + 0.610342i \(0.791032\pi\)
\(644\) 0 0
\(645\) 2.51655e7 2.38181
\(646\) 0 0
\(647\) −3.48062e6 −0.326886 −0.163443 0.986553i \(-0.552260\pi\)
−0.163443 + 0.986553i \(0.552260\pi\)
\(648\) 0 0
\(649\) 2.40742e7 2.24358
\(650\) 0 0
\(651\) −6.23947e6 −0.577025
\(652\) 0 0
\(653\) −7.11534e6 −0.652999 −0.326500 0.945197i \(-0.605869\pi\)
−0.326500 + 0.945197i \(0.605869\pi\)
\(654\) 0 0
\(655\) −1.67923e7 −1.52935
\(656\) 0 0
\(657\) 1.61598e7 1.46057
\(658\) 0 0
\(659\) −6.01255e6 −0.539318 −0.269659 0.962956i \(-0.586911\pi\)
−0.269659 + 0.962956i \(0.586911\pi\)
\(660\) 0 0
\(661\) −1.66126e6 −0.147889 −0.0739444 0.997262i \(-0.523559\pi\)
−0.0739444 + 0.997262i \(0.523559\pi\)
\(662\) 0 0
\(663\) 6.83898e6 0.604237
\(664\) 0 0
\(665\) 1.52234e6 0.133492
\(666\) 0 0
\(667\) −2.24744e7 −1.95602
\(668\) 0 0
\(669\) −1.75035e7 −1.51203
\(670\) 0 0
\(671\) 2.52155e7 2.16203
\(672\) 0 0
\(673\) 2.09343e6 0.178164 0.0890821 0.996024i \(-0.471607\pi\)
0.0890821 + 0.996024i \(0.471607\pi\)
\(674\) 0 0
\(675\) 8.15309e6 0.688752
\(676\) 0 0
\(677\) −1.41602e7 −1.18740 −0.593700 0.804686i \(-0.702333\pi\)
−0.593700 + 0.804686i \(0.702333\pi\)
\(678\) 0 0
\(679\) −7.35698e6 −0.612386
\(680\) 0 0
\(681\) −3.58629e6 −0.296331
\(682\) 0 0
\(683\) −4.94105e6 −0.405292 −0.202646 0.979252i \(-0.564954\pi\)
−0.202646 + 0.979252i \(0.564954\pi\)
\(684\) 0 0
\(685\) −1.22754e7 −0.999559
\(686\) 0 0
\(687\) −1.12163e7 −0.906689
\(688\) 0 0
\(689\) −1.73921e7 −1.39574
\(690\) 0 0
\(691\) −9.44784e6 −0.752727 −0.376363 0.926472i \(-0.622826\pi\)
−0.376363 + 0.926472i \(0.622826\pi\)
\(692\) 0 0
\(693\) 2.25931e7 1.78707
\(694\) 0 0
\(695\) 2.55286e7 2.00477
\(696\) 0 0
\(697\) 3.71120e6 0.289356
\(698\) 0 0
\(699\) −6.53675e6 −0.506022
\(700\) 0 0
\(701\) −6.83585e6 −0.525409 −0.262705 0.964876i \(-0.584614\pi\)
−0.262705 + 0.964876i \(0.584614\pi\)
\(702\) 0 0
\(703\) 4.84605e6 0.369827
\(704\) 0 0
\(705\) 2.58981e6 0.196243
\(706\) 0 0
\(707\) −7.04028e6 −0.529714
\(708\) 0 0
\(709\) 2.04148e6 0.152521 0.0762604 0.997088i \(-0.475702\pi\)
0.0762604 + 0.997088i \(0.475702\pi\)
\(710\) 0 0
\(711\) −1.79577e7 −1.33222
\(712\) 0 0
\(713\) 1.09114e7 0.803814
\(714\) 0 0
\(715\) −1.90624e7 −1.39448
\(716\) 0 0
\(717\) 4.13018e7 3.00034
\(718\) 0 0
\(719\) 1.41812e7 1.02303 0.511516 0.859274i \(-0.329084\pi\)
0.511516 + 0.859274i \(0.329084\pi\)
\(720\) 0 0
\(721\) −161465. −0.0115675
\(722\) 0 0
\(723\) 4.20703e7 2.99316
\(724\) 0 0
\(725\) 6.84730e6 0.483810
\(726\) 0 0
\(727\) 7.23601e6 0.507766 0.253883 0.967235i \(-0.418292\pi\)
0.253883 + 0.967235i \(0.418292\pi\)
\(728\) 0 0
\(729\) −1.79800e6 −0.125306
\(730\) 0 0
\(731\) −7.33137e6 −0.507448
\(732\) 0 0
\(733\) 1.34837e7 0.926933 0.463467 0.886114i \(-0.346605\pi\)
0.463467 + 0.886114i \(0.346605\pi\)
\(734\) 0 0
\(735\) 2.23702e7 1.52740
\(736\) 0 0
\(737\) −7.00770e6 −0.475233
\(738\) 0 0
\(739\) 1.52779e6 0.102909 0.0514544 0.998675i \(-0.483614\pi\)
0.0514544 + 0.998675i \(0.483614\pi\)
\(740\) 0 0
\(741\) −4.72639e6 −0.316216
\(742\) 0 0
\(743\) 6.01918e6 0.400005 0.200003 0.979795i \(-0.435905\pi\)
0.200003 + 0.979795i \(0.435905\pi\)
\(744\) 0 0
\(745\) 5.86050e6 0.386852
\(746\) 0 0
\(747\) −2.74139e7 −1.79750
\(748\) 0 0
\(749\) 1.68720e6 0.109891
\(750\) 0 0
\(751\) −2.06310e6 −0.133481 −0.0667407 0.997770i \(-0.521260\pi\)
−0.0667407 + 0.997770i \(0.521260\pi\)
\(752\) 0 0
\(753\) −3.48758e7 −2.24149
\(754\) 0 0
\(755\) −1.08266e7 −0.691234
\(756\) 0 0
\(757\) −2.73849e7 −1.73689 −0.868444 0.495787i \(-0.834880\pi\)
−0.868444 + 0.495787i \(0.834880\pi\)
\(758\) 0 0
\(759\) −5.72903e7 −3.60975
\(760\) 0 0
\(761\) 3.55235e6 0.222359 0.111179 0.993800i \(-0.464537\pi\)
0.111179 + 0.993800i \(0.464537\pi\)
\(762\) 0 0
\(763\) 6.44543e6 0.400812
\(764\) 0 0
\(765\) −1.80743e7 −1.11662
\(766\) 0 0
\(767\) 1.77168e7 1.08742
\(768\) 0 0
\(769\) 9.74838e6 0.594452 0.297226 0.954807i \(-0.403939\pi\)
0.297226 + 0.954807i \(0.403939\pi\)
\(770\) 0 0
\(771\) −5.18807e6 −0.314318
\(772\) 0 0
\(773\) 3.11310e7 1.87389 0.936945 0.349477i \(-0.113641\pi\)
0.936945 + 0.349477i \(0.113641\pi\)
\(774\) 0 0
\(775\) −3.32439e6 −0.198819
\(776\) 0 0
\(777\) −2.47196e7 −1.46889
\(778\) 0 0
\(779\) −2.56479e6 −0.151429
\(780\) 0 0
\(781\) 5.78522e6 0.339385
\(782\) 0 0
\(783\) 5.79950e7 3.38054
\(784\) 0 0
\(785\) 1.34883e7 0.781236
\(786\) 0 0
\(787\) −1.96579e7 −1.13136 −0.565680 0.824625i \(-0.691386\pi\)
−0.565680 + 0.824625i \(0.691386\pi\)
\(788\) 0 0
\(789\) 1.91720e7 1.09641
\(790\) 0 0
\(791\) 1.45985e7 0.829597
\(792\) 0 0
\(793\) 1.85567e7 1.04790
\(794\) 0 0
\(795\) 6.66492e7 3.74005
\(796\) 0 0
\(797\) 1.97340e7 1.10045 0.550224 0.835017i \(-0.314542\pi\)
0.550224 + 0.835017i \(0.314542\pi\)
\(798\) 0 0
\(799\) −754478. −0.0418099
\(800\) 0 0
\(801\) −6.95177e7 −3.82837
\(802\) 0 0
\(803\) 1.90272e7 1.04133
\(804\) 0 0
\(805\) 1.35799e7 0.738596
\(806\) 0 0
\(807\) −1.29378e6 −0.0699320
\(808\) 0 0
\(809\) −2.73904e7 −1.47139 −0.735695 0.677313i \(-0.763144\pi\)
−0.735695 + 0.677313i \(0.763144\pi\)
\(810\) 0 0
\(811\) 9.78395e6 0.522351 0.261175 0.965291i \(-0.415890\pi\)
0.261175 + 0.965291i \(0.415890\pi\)
\(812\) 0 0
\(813\) 1.77767e7 0.943244
\(814\) 0 0
\(815\) −3.97286e7 −2.09512
\(816\) 0 0
\(817\) 5.06667e6 0.265563
\(818\) 0 0
\(819\) 1.66268e7 0.866164
\(820\) 0 0
\(821\) −2.07144e6 −0.107254 −0.0536272 0.998561i \(-0.517078\pi\)
−0.0536272 + 0.998561i \(0.517078\pi\)
\(822\) 0 0
\(823\) 2.10882e7 1.08528 0.542638 0.839967i \(-0.317426\pi\)
0.542638 + 0.839967i \(0.317426\pi\)
\(824\) 0 0
\(825\) 1.74548e7 0.892851
\(826\) 0 0
\(827\) −7.73194e6 −0.393119 −0.196560 0.980492i \(-0.562977\pi\)
−0.196560 + 0.980492i \(0.562977\pi\)
\(828\) 0 0
\(829\) 2.77388e7 1.40185 0.700924 0.713236i \(-0.252771\pi\)
0.700924 + 0.713236i \(0.252771\pi\)
\(830\) 0 0
\(831\) −2.55319e7 −1.28257
\(832\) 0 0
\(833\) −6.51703e6 −0.325415
\(834\) 0 0
\(835\) −3.18203e6 −0.157938
\(836\) 0 0
\(837\) −2.81567e7 −1.38921
\(838\) 0 0
\(839\) 2.08334e7 1.02177 0.510887 0.859648i \(-0.329317\pi\)
0.510887 + 0.859648i \(0.329317\pi\)
\(840\) 0 0
\(841\) 2.81954e7 1.37464
\(842\) 0 0
\(843\) −6.63405e7 −3.21521
\(844\) 0 0
\(845\) 9.76361e6 0.470401
\(846\) 0 0
\(847\) 1.60033e7 0.766483
\(848\) 0 0
\(849\) −2.45755e6 −0.117013
\(850\) 0 0
\(851\) 4.32288e7 2.04621
\(852\) 0 0
\(853\) 3.04500e7 1.43290 0.716449 0.697639i \(-0.245766\pi\)
0.716449 + 0.697639i \(0.245766\pi\)
\(854\) 0 0
\(855\) 1.24910e7 0.584363
\(856\) 0 0
\(857\) −1.74993e6 −0.0813895 −0.0406947 0.999172i \(-0.512957\pi\)
−0.0406947 + 0.999172i \(0.512957\pi\)
\(858\) 0 0
\(859\) −2.31559e7 −1.07073 −0.535364 0.844622i \(-0.679825\pi\)
−0.535364 + 0.844622i \(0.679825\pi\)
\(860\) 0 0
\(861\) 1.30830e7 0.601449
\(862\) 0 0
\(863\) 1.92908e7 0.881704 0.440852 0.897580i \(-0.354676\pi\)
0.440852 + 0.897580i \(0.354676\pi\)
\(864\) 0 0
\(865\) −1.71763e7 −0.780529
\(866\) 0 0
\(867\) −3.20950e7 −1.45007
\(868\) 0 0
\(869\) −2.11441e7 −0.949816
\(870\) 0 0
\(871\) −5.15715e6 −0.230337
\(872\) 0 0
\(873\) −6.03653e7 −2.68072
\(874\) 0 0
\(875\) 9.04071e6 0.399192
\(876\) 0 0
\(877\) −6.13591e6 −0.269389 −0.134695 0.990887i \(-0.543005\pi\)
−0.134695 + 0.990887i \(0.543005\pi\)
\(878\) 0 0
\(879\) 1.03144e7 0.450269
\(880\) 0 0
\(881\) −2.50035e6 −0.108533 −0.0542665 0.998526i \(-0.517282\pi\)
−0.0542665 + 0.998526i \(0.517282\pi\)
\(882\) 0 0
\(883\) 8.48262e6 0.366124 0.183062 0.983101i \(-0.441399\pi\)
0.183062 + 0.983101i \(0.441399\pi\)
\(884\) 0 0
\(885\) −6.78937e7 −2.91388
\(886\) 0 0
\(887\) 2.42625e7 1.03544 0.517722 0.855549i \(-0.326780\pi\)
0.517722 + 0.855549i \(0.326780\pi\)
\(888\) 0 0
\(889\) −2.02991e7 −0.861434
\(890\) 0 0
\(891\) 6.44127e7 2.71817
\(892\) 0 0
\(893\) 521416. 0.0218804
\(894\) 0 0
\(895\) 1.70677e7 0.712227
\(896\) 0 0
\(897\) −4.21614e7 −1.74958
\(898\) 0 0
\(899\) −2.36472e7 −0.975844
\(900\) 0 0
\(901\) −1.94166e7 −0.796823
\(902\) 0 0
\(903\) −2.58450e7 −1.05477
\(904\) 0 0
\(905\) −1.77269e7 −0.719468
\(906\) 0 0
\(907\) −4.47763e7 −1.80730 −0.903649 0.428274i \(-0.859122\pi\)
−0.903649 + 0.428274i \(0.859122\pi\)
\(908\) 0 0
\(909\) −5.77667e7 −2.31882
\(910\) 0 0
\(911\) 9.62280e6 0.384154 0.192077 0.981380i \(-0.438478\pi\)
0.192077 + 0.981380i \(0.438478\pi\)
\(912\) 0 0
\(913\) −3.22782e7 −1.28154
\(914\) 0 0
\(915\) −7.11123e7 −2.80797
\(916\) 0 0
\(917\) 1.72457e7 0.677262
\(918\) 0 0
\(919\) 4.09421e7 1.59912 0.799561 0.600585i \(-0.205066\pi\)
0.799561 + 0.600585i \(0.205066\pi\)
\(920\) 0 0
\(921\) −2.56156e7 −0.995075
\(922\) 0 0
\(923\) 4.25749e6 0.164494
\(924\) 0 0
\(925\) −1.31706e7 −0.506118
\(926\) 0 0
\(927\) −1.32485e6 −0.0506369
\(928\) 0 0
\(929\) 1.50754e7 0.573099 0.286550 0.958065i \(-0.407492\pi\)
0.286550 + 0.958065i \(0.407492\pi\)
\(930\) 0 0
\(931\) 4.50389e6 0.170299
\(932\) 0 0
\(933\) 4.62195e7 1.73829
\(934\) 0 0
\(935\) −2.12814e7 −0.796105
\(936\) 0 0
\(937\) −2.25249e7 −0.838135 −0.419068 0.907955i \(-0.637643\pi\)
−0.419068 + 0.907955i \(0.637643\pi\)
\(938\) 0 0
\(939\) 7.68927e7 2.84591
\(940\) 0 0
\(941\) −1.04598e7 −0.385078 −0.192539 0.981289i \(-0.561672\pi\)
−0.192539 + 0.981289i \(0.561672\pi\)
\(942\) 0 0
\(943\) −2.28791e7 −0.837837
\(944\) 0 0
\(945\) −3.50429e7 −1.27650
\(946\) 0 0
\(947\) 1.91938e7 0.695481 0.347741 0.937591i \(-0.386949\pi\)
0.347741 + 0.937591i \(0.386949\pi\)
\(948\) 0 0
\(949\) 1.40026e7 0.504712
\(950\) 0 0
\(951\) −586553. −0.0210308
\(952\) 0 0
\(953\) 4.15274e7 1.48116 0.740581 0.671967i \(-0.234550\pi\)
0.740581 + 0.671967i \(0.234550\pi\)
\(954\) 0 0
\(955\) −4.12435e7 −1.46335
\(956\) 0 0
\(957\) 1.24160e8 4.38230
\(958\) 0 0
\(959\) 1.26068e7 0.442649
\(960\) 0 0
\(961\) −1.71484e7 −0.598983
\(962\) 0 0
\(963\) 1.38438e7 0.481048
\(964\) 0 0
\(965\) −3.80323e6 −0.131472
\(966\) 0 0
\(967\) 5.33594e7 1.83504 0.917518 0.397694i \(-0.130189\pi\)
0.917518 + 0.397694i \(0.130189\pi\)
\(968\) 0 0
\(969\) −5.27656e6 −0.180527
\(970\) 0 0
\(971\) 4.51646e7 1.53727 0.768635 0.639688i \(-0.220936\pi\)
0.768635 + 0.639688i \(0.220936\pi\)
\(972\) 0 0
\(973\) −2.62179e7 −0.887801
\(974\) 0 0
\(975\) 1.28454e7 0.432749
\(976\) 0 0
\(977\) 2.79834e6 0.0937916 0.0468958 0.998900i \(-0.485067\pi\)
0.0468958 + 0.998900i \(0.485067\pi\)
\(978\) 0 0
\(979\) −8.18529e7 −2.72946
\(980\) 0 0
\(981\) 5.28858e7 1.75455
\(982\) 0 0
\(983\) 2.61378e7 0.862750 0.431375 0.902173i \(-0.358029\pi\)
0.431375 + 0.902173i \(0.358029\pi\)
\(984\) 0 0
\(985\) −3.88062e6 −0.127441
\(986\) 0 0
\(987\) −2.65974e6 −0.0869052
\(988\) 0 0
\(989\) 4.51969e7 1.46933
\(990\) 0 0
\(991\) 4.58748e7 1.48385 0.741925 0.670483i \(-0.233913\pi\)
0.741925 + 0.670483i \(0.233913\pi\)
\(992\) 0 0
\(993\) 3.97093e7 1.27796
\(994\) 0 0
\(995\) 1.66652e7 0.533647
\(996\) 0 0
\(997\) −4.97107e7 −1.58384 −0.791921 0.610623i \(-0.790919\pi\)
−0.791921 + 0.610623i \(0.790919\pi\)
\(998\) 0 0
\(999\) −1.11552e8 −3.53641
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 76.6.a.b.1.4 4
3.2 odd 2 684.6.a.e.1.3 4
4.3 odd 2 304.6.a.k.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.a.b.1.4 4 1.1 even 1 trivial
304.6.a.k.1.1 4 4.3 odd 2
684.6.a.e.1.3 4 3.2 odd 2