Properties

Label 76.6.a.b.1.3
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 1.3
Root \(11.0547\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q+17.5489 q^{3} +87.4671 q^{5} +76.9277 q^{7} +64.9636 q^{9} +O(q^{10})\) \(q+17.5489 q^{3} +87.4671 q^{5} +76.9277 q^{7} +64.9636 q^{9} -99.2130 q^{11} -79.5656 q^{13} +1534.95 q^{15} -465.981 q^{17} -361.000 q^{19} +1350.00 q^{21} +2231.89 q^{23} +4525.49 q^{25} -3124.34 q^{27} +757.584 q^{29} +2718.02 q^{31} -1741.08 q^{33} +6728.64 q^{35} +10309.9 q^{37} -1396.29 q^{39} -6945.22 q^{41} -2107.29 q^{43} +5682.17 q^{45} -20564.6 q^{47} -10889.1 q^{49} -8177.45 q^{51} -17567.4 q^{53} -8677.87 q^{55} -6335.15 q^{57} -34514.7 q^{59} -40975.0 q^{61} +4997.50 q^{63} -6959.37 q^{65} +68248.5 q^{67} +39167.2 q^{69} +38876.4 q^{71} -83648.9 q^{73} +79417.3 q^{75} -7632.23 q^{77} +39001.5 q^{79} -70614.9 q^{81} +99454.0 q^{83} -40758.0 q^{85} +13294.8 q^{87} -88908.3 q^{89} -6120.80 q^{91} +47698.3 q^{93} -31575.6 q^{95} -87484.2 q^{97} -6445.23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + O(q^{10}) \) \( 4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + 706q^{11} + 788q^{13} + 2792q^{15} + 240q^{17} - 1444q^{19} + 7128q^{21} + 5884q^{23} + 11774q^{25} + 4426q^{27} + 5240q^{29} - 860q^{31} + 10748q^{33} + 20322q^{35} - 20732q^{37} + 15404q^{39} - 10204q^{41} - 12554q^{43} - 71306q^{45} - 4826q^{47} - 21376q^{49} + 34518q^{51} - 76484q^{53} - 72914q^{55} - 3610q^{57} + 23898q^{59} - 32482q^{61} - 43566q^{63} - 6076q^{65} + 5022q^{67} - 149524q^{69} + 121300q^{71} - 104700q^{73} - 95230q^{75} - 10002q^{77} + 117128q^{79} + 44924q^{81} + 92832q^{83} + 80322q^{85} + 300148q^{87} + 5988q^{89} + 165618q^{91} + 16180q^{93} + 39710q^{95} + 22972q^{97} + 441418q^{99} + O(q^{100}) \)

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
\(3\) 17.5489 1.12576 0.562881 0.826538i \(-0.309693\pi\)
0.562881 + 0.826538i \(0.309693\pi\)
\(4\) 0 0
\(5\) 87.4671 1.56466 0.782329 0.622865i \(-0.214031\pi\)
0.782329 + 0.622865i \(0.214031\pi\)
\(6\) 0 0
\(7\) 76.9277 0.593386 0.296693 0.954973i \(-0.404116\pi\)
0.296693 + 0.954973i \(0.404116\pi\)
\(8\) 0 0
\(9\) 64.9636 0.267340
\(10\) 0 0
\(11\) −99.2130 −0.247222 −0.123611 0.992331i \(-0.539447\pi\)
−0.123611 + 0.992331i \(0.539447\pi\)
\(12\) 0 0
\(13\) −79.5656 −0.130577 −0.0652886 0.997866i \(-0.520797\pi\)
−0.0652886 + 0.997866i \(0.520797\pi\)
\(14\) 0 0
\(15\) 1534.95 1.76143
\(16\) 0 0
\(17\) −465.981 −0.391062 −0.195531 0.980698i \(-0.562643\pi\)
−0.195531 + 0.980698i \(0.562643\pi\)
\(18\) 0 0
\(19\) −361.000 −0.229416
\(20\) 0 0
\(21\) 1350.00 0.668012
\(22\) 0 0
\(23\) 2231.89 0.879738 0.439869 0.898062i \(-0.355025\pi\)
0.439869 + 0.898062i \(0.355025\pi\)
\(24\) 0 0
\(25\) 4525.49 1.44816
\(26\) 0 0
\(27\) −3124.34 −0.824801
\(28\) 0 0
\(29\) 757.584 0.167277 0.0836384 0.996496i \(-0.473346\pi\)
0.0836384 + 0.996496i \(0.473346\pi\)
\(30\) 0 0
\(31\) 2718.02 0.507983 0.253991 0.967206i \(-0.418257\pi\)
0.253991 + 0.967206i \(0.418257\pi\)
\(32\) 0 0
\(33\) −1741.08 −0.278313
\(34\) 0 0
\(35\) 6728.64 0.928447
\(36\) 0 0
\(37\) 10309.9 1.23808 0.619040 0.785360i \(-0.287522\pi\)
0.619040 + 0.785360i \(0.287522\pi\)
\(38\) 0 0
\(39\) −1396.29 −0.146999
\(40\) 0 0
\(41\) −6945.22 −0.645248 −0.322624 0.946527i \(-0.604565\pi\)
−0.322624 + 0.946527i \(0.604565\pi\)
\(42\) 0 0
\(43\) −2107.29 −0.173802 −0.0869009 0.996217i \(-0.527696\pi\)
−0.0869009 + 0.996217i \(0.527696\pi\)
\(44\) 0 0
\(45\) 5682.17 0.418296
\(46\) 0 0
\(47\) −20564.6 −1.35793 −0.678963 0.734173i \(-0.737570\pi\)
−0.678963 + 0.734173i \(0.737570\pi\)
\(48\) 0 0
\(49\) −10889.1 −0.647893
\(50\) 0 0
\(51\) −8177.45 −0.440243
\(52\) 0 0
\(53\) −17567.4 −0.859050 −0.429525 0.903055i \(-0.641319\pi\)
−0.429525 + 0.903055i \(0.641319\pi\)
\(54\) 0 0
\(55\) −8677.87 −0.386818
\(56\) 0 0
\(57\) −6335.15 −0.258267
\(58\) 0 0
\(59\) −34514.7 −1.29084 −0.645422 0.763827i \(-0.723318\pi\)
−0.645422 + 0.763827i \(0.723318\pi\)
\(60\) 0 0
\(61\) −40975.0 −1.40992 −0.704959 0.709248i \(-0.749035\pi\)
−0.704959 + 0.709248i \(0.749035\pi\)
\(62\) 0 0
\(63\) 4997.50 0.158636
\(64\) 0 0
\(65\) −6959.37 −0.204309
\(66\) 0 0
\(67\) 68248.5 1.85740 0.928701 0.370829i \(-0.120926\pi\)
0.928701 + 0.370829i \(0.120926\pi\)
\(68\) 0 0
\(69\) 39167.2 0.990376
\(70\) 0 0
\(71\) 38876.4 0.915251 0.457626 0.889145i \(-0.348700\pi\)
0.457626 + 0.889145i \(0.348700\pi\)
\(72\) 0 0
\(73\) −83648.9 −1.83719 −0.918593 0.395205i \(-0.870674\pi\)
−0.918593 + 0.395205i \(0.870674\pi\)
\(74\) 0 0
\(75\) 79417.3 1.63028
\(76\) 0 0
\(77\) −7632.23 −0.146698
\(78\) 0 0
\(79\) 39001.5 0.703094 0.351547 0.936170i \(-0.385656\pi\)
0.351547 + 0.936170i \(0.385656\pi\)
\(80\) 0 0
\(81\) −70614.9 −1.19587
\(82\) 0 0
\(83\) 99454.0 1.58463 0.792314 0.610114i \(-0.208876\pi\)
0.792314 + 0.610114i \(0.208876\pi\)
\(84\) 0 0
\(85\) −40758.0 −0.611879
\(86\) 0 0
\(87\) 13294.8 0.188314
\(88\) 0 0
\(89\) −88908.3 −1.18978 −0.594891 0.803806i \(-0.702805\pi\)
−0.594891 + 0.803806i \(0.702805\pi\)
\(90\) 0 0
\(91\) −6120.80 −0.0774827
\(92\) 0 0
\(93\) 47698.3 0.571868
\(94\) 0 0
\(95\) −31575.6 −0.358957
\(96\) 0 0
\(97\) −87484.2 −0.944061 −0.472031 0.881582i \(-0.656479\pi\)
−0.472031 + 0.881582i \(0.656479\pi\)
\(98\) 0 0
\(99\) −6445.23 −0.0660923
\(100\) 0 0
\(101\) 124712. 1.21648 0.608238 0.793754i \(-0.291876\pi\)
0.608238 + 0.793754i \(0.291876\pi\)
\(102\) 0 0
\(103\) −28096.0 −0.260947 −0.130473 0.991452i \(-0.541650\pi\)
−0.130473 + 0.991452i \(0.541650\pi\)
\(104\) 0 0
\(105\) 118080. 1.04521
\(106\) 0 0
\(107\) 106686. 0.900839 0.450419 0.892817i \(-0.351274\pi\)
0.450419 + 0.892817i \(0.351274\pi\)
\(108\) 0 0
\(109\) −81738.7 −0.658964 −0.329482 0.944162i \(-0.606874\pi\)
−0.329482 + 0.944162i \(0.606874\pi\)
\(110\) 0 0
\(111\) 180927. 1.39378
\(112\) 0 0
\(113\) 265828. 1.95841 0.979206 0.202871i \(-0.0650271\pi\)
0.979206 + 0.202871i \(0.0650271\pi\)
\(114\) 0 0
\(115\) 195217. 1.37649
\(116\) 0 0
\(117\) −5168.87 −0.0349085
\(118\) 0 0
\(119\) −35846.8 −0.232051
\(120\) 0 0
\(121\) −151208. −0.938881
\(122\) 0 0
\(123\) −121881. −0.726395
\(124\) 0 0
\(125\) 122497. 0.701213
\(126\) 0 0
\(127\) −320548. −1.76353 −0.881766 0.471688i \(-0.843645\pi\)
−0.881766 + 0.471688i \(0.843645\pi\)
\(128\) 0 0
\(129\) −36980.7 −0.195659
\(130\) 0 0
\(131\) −104893. −0.534034 −0.267017 0.963692i \(-0.586038\pi\)
−0.267017 + 0.963692i \(0.586038\pi\)
\(132\) 0 0
\(133\) −27770.9 −0.136132
\(134\) 0 0
\(135\) −273277. −1.29053
\(136\) 0 0
\(137\) 276152. 1.25704 0.628518 0.777795i \(-0.283662\pi\)
0.628518 + 0.777795i \(0.283662\pi\)
\(138\) 0 0
\(139\) 51956.7 0.228089 0.114045 0.993476i \(-0.463619\pi\)
0.114045 + 0.993476i \(0.463619\pi\)
\(140\) 0 0
\(141\) −360886. −1.52870
\(142\) 0 0
\(143\) 7893.94 0.0322815
\(144\) 0 0
\(145\) 66263.7 0.261731
\(146\) 0 0
\(147\) −191092. −0.729373
\(148\) 0 0
\(149\) −420951. −1.55334 −0.776670 0.629908i \(-0.783093\pi\)
−0.776670 + 0.629908i \(0.783093\pi\)
\(150\) 0 0
\(151\) −56718.6 −0.202434 −0.101217 0.994864i \(-0.532274\pi\)
−0.101217 + 0.994864i \(0.532274\pi\)
\(152\) 0 0
\(153\) −30271.8 −0.104547
\(154\) 0 0
\(155\) 237738. 0.794820
\(156\) 0 0
\(157\) −65573.0 −0.212313 −0.106156 0.994349i \(-0.533854\pi\)
−0.106156 + 0.994349i \(0.533854\pi\)
\(158\) 0 0
\(159\) −308289. −0.967085
\(160\) 0 0
\(161\) 171694. 0.522025
\(162\) 0 0
\(163\) 445737. 1.31404 0.657022 0.753872i \(-0.271816\pi\)
0.657022 + 0.753872i \(0.271816\pi\)
\(164\) 0 0
\(165\) −152287. −0.435465
\(166\) 0 0
\(167\) −150553. −0.417734 −0.208867 0.977944i \(-0.566978\pi\)
−0.208867 + 0.977944i \(0.566978\pi\)
\(168\) 0 0
\(169\) −364962. −0.982950
\(170\) 0 0
\(171\) −23451.9 −0.0613320
\(172\) 0 0
\(173\) −106599. −0.270793 −0.135397 0.990791i \(-0.543231\pi\)
−0.135397 + 0.990791i \(0.543231\pi\)
\(174\) 0 0
\(175\) 348135. 0.859316
\(176\) 0 0
\(177\) −605694. −1.45318
\(178\) 0 0
\(179\) 416262. 0.971034 0.485517 0.874227i \(-0.338631\pi\)
0.485517 + 0.874227i \(0.338631\pi\)
\(180\) 0 0
\(181\) 100405. 0.227803 0.113901 0.993492i \(-0.463665\pi\)
0.113901 + 0.993492i \(0.463665\pi\)
\(182\) 0 0
\(183\) −719065. −1.58723
\(184\) 0 0
\(185\) 901774. 1.93717
\(186\) 0 0
\(187\) 46231.4 0.0966791
\(188\) 0 0
\(189\) −240348. −0.489426
\(190\) 0 0
\(191\) 597176. 1.18446 0.592228 0.805770i \(-0.298248\pi\)
0.592228 + 0.805770i \(0.298248\pi\)
\(192\) 0 0
\(193\) 714364. 1.38047 0.690234 0.723586i \(-0.257507\pi\)
0.690234 + 0.723586i \(0.257507\pi\)
\(194\) 0 0
\(195\) −122129. −0.230003
\(196\) 0 0
\(197\) −11062.3 −0.0203085 −0.0101543 0.999948i \(-0.503232\pi\)
−0.0101543 + 0.999948i \(0.503232\pi\)
\(198\) 0 0
\(199\) 357858. 0.640587 0.320294 0.947318i \(-0.396218\pi\)
0.320294 + 0.947318i \(0.396218\pi\)
\(200\) 0 0
\(201\) 1.19769e6 2.09099
\(202\) 0 0
\(203\) 58279.2 0.0992598
\(204\) 0 0
\(205\) −607478. −1.00959
\(206\) 0 0
\(207\) 144992. 0.235189
\(208\) 0 0
\(209\) 35815.9 0.0567166
\(210\) 0 0
\(211\) −284130. −0.439350 −0.219675 0.975573i \(-0.570500\pi\)
−0.219675 + 0.975573i \(0.570500\pi\)
\(212\) 0 0
\(213\) 682238. 1.03035
\(214\) 0 0
\(215\) −184319. −0.271940
\(216\) 0 0
\(217\) 209091. 0.301430
\(218\) 0 0
\(219\) −1.46795e6 −2.06823
\(220\) 0 0
\(221\) 37076.1 0.0510638
\(222\) 0 0
\(223\) 870495. 1.17221 0.586103 0.810236i \(-0.300661\pi\)
0.586103 + 0.810236i \(0.300661\pi\)
\(224\) 0 0
\(225\) 293992. 0.387150
\(226\) 0 0
\(227\) 484247. 0.623738 0.311869 0.950125i \(-0.399045\pi\)
0.311869 + 0.950125i \(0.399045\pi\)
\(228\) 0 0
\(229\) −512807. −0.646197 −0.323099 0.946365i \(-0.604725\pi\)
−0.323099 + 0.946365i \(0.604725\pi\)
\(230\) 0 0
\(231\) −133937. −0.165147
\(232\) 0 0
\(233\) 1.14046e6 1.37623 0.688115 0.725601i \(-0.258438\pi\)
0.688115 + 0.725601i \(0.258438\pi\)
\(234\) 0 0
\(235\) −1.79873e6 −2.12469
\(236\) 0 0
\(237\) 684433. 0.791516
\(238\) 0 0
\(239\) −564259. −0.638975 −0.319488 0.947590i \(-0.603511\pi\)
−0.319488 + 0.947590i \(0.603511\pi\)
\(240\) 0 0
\(241\) −499520. −0.554001 −0.277000 0.960870i \(-0.589340\pi\)
−0.277000 + 0.960870i \(0.589340\pi\)
\(242\) 0 0
\(243\) −479998. −0.521463
\(244\) 0 0
\(245\) −952441. −1.01373
\(246\) 0 0
\(247\) 28723.2 0.0299564
\(248\) 0 0
\(249\) 1.74531e6 1.78391
\(250\) 0 0
\(251\) 1.32259e6 1.32508 0.662539 0.749027i \(-0.269479\pi\)
0.662539 + 0.749027i \(0.269479\pi\)
\(252\) 0 0
\(253\) −221433. −0.217491
\(254\) 0 0
\(255\) −715258. −0.688830
\(256\) 0 0
\(257\) 29088.9 0.0274722 0.0137361 0.999906i \(-0.495628\pi\)
0.0137361 + 0.999906i \(0.495628\pi\)
\(258\) 0 0
\(259\) 793114. 0.734660
\(260\) 0 0
\(261\) 49215.4 0.0447198
\(262\) 0 0
\(263\) 1.39828e6 1.24653 0.623266 0.782010i \(-0.285806\pi\)
0.623266 + 0.782010i \(0.285806\pi\)
\(264\) 0 0
\(265\) −1.53657e6 −1.34412
\(266\) 0 0
\(267\) −1.56024e6 −1.33941
\(268\) 0 0
\(269\) 1.62054e6 1.36546 0.682728 0.730672i \(-0.260793\pi\)
0.682728 + 0.730672i \(0.260793\pi\)
\(270\) 0 0
\(271\) −311537. −0.257683 −0.128842 0.991665i \(-0.541126\pi\)
−0.128842 + 0.991665i \(0.541126\pi\)
\(272\) 0 0
\(273\) −107413. −0.0872270
\(274\) 0 0
\(275\) −448988. −0.358016
\(276\) 0 0
\(277\) −801500. −0.627631 −0.313815 0.949484i \(-0.601607\pi\)
−0.313815 + 0.949484i \(0.601607\pi\)
\(278\) 0 0
\(279\) 176573. 0.135804
\(280\) 0 0
\(281\) 1.59535e6 1.20528 0.602642 0.798012i \(-0.294115\pi\)
0.602642 + 0.798012i \(0.294115\pi\)
\(282\) 0 0
\(283\) −1.04144e6 −0.772980 −0.386490 0.922294i \(-0.626313\pi\)
−0.386490 + 0.922294i \(0.626313\pi\)
\(284\) 0 0
\(285\) −554117. −0.404100
\(286\) 0 0
\(287\) −534280. −0.382881
\(288\) 0 0
\(289\) −1.20272e6 −0.847070
\(290\) 0 0
\(291\) −1.53525e6 −1.06279
\(292\) 0 0
\(293\) −391025. −0.266094 −0.133047 0.991110i \(-0.542476\pi\)
−0.133047 + 0.991110i \(0.542476\pi\)
\(294\) 0 0
\(295\) −3.01890e6 −2.01973
\(296\) 0 0
\(297\) 309975. 0.203909
\(298\) 0 0
\(299\) −177582. −0.114874
\(300\) 0 0
\(301\) −162109. −0.103132
\(302\) 0 0
\(303\) 2.18855e6 1.36946
\(304\) 0 0
\(305\) −3.58396e6 −2.20604
\(306\) 0 0
\(307\) −504246. −0.305349 −0.152675 0.988277i \(-0.548789\pi\)
−0.152675 + 0.988277i \(0.548789\pi\)
\(308\) 0 0
\(309\) −493054. −0.293764
\(310\) 0 0
\(311\) −80004.3 −0.0469042 −0.0234521 0.999725i \(-0.507466\pi\)
−0.0234521 + 0.999725i \(0.507466\pi\)
\(312\) 0 0
\(313\) −99620.5 −0.0574762 −0.0287381 0.999587i \(-0.509149\pi\)
−0.0287381 + 0.999587i \(0.509149\pi\)
\(314\) 0 0
\(315\) 437117. 0.248211
\(316\) 0 0
\(317\) −177148. −0.0990121 −0.0495060 0.998774i \(-0.515765\pi\)
−0.0495060 + 0.998774i \(0.515765\pi\)
\(318\) 0 0
\(319\) −75162.2 −0.0413545
\(320\) 0 0
\(321\) 1.87222e6 1.01413
\(322\) 0 0
\(323\) 168219. 0.0897158
\(324\) 0 0
\(325\) −360073. −0.189096
\(326\) 0 0
\(327\) −1.43442e6 −0.741837
\(328\) 0 0
\(329\) −1.58199e6 −0.805774
\(330\) 0 0
\(331\) 1.05867e6 0.531116 0.265558 0.964095i \(-0.414444\pi\)
0.265558 + 0.964095i \(0.414444\pi\)
\(332\) 0 0
\(333\) 669766. 0.330988
\(334\) 0 0
\(335\) 5.96950e6 2.90620
\(336\) 0 0
\(337\) 3.24019e6 1.55416 0.777081 0.629401i \(-0.216700\pi\)
0.777081 + 0.629401i \(0.216700\pi\)
\(338\) 0 0
\(339\) 4.66498e6 2.20470
\(340\) 0 0
\(341\) −269663. −0.125584
\(342\) 0 0
\(343\) −2.13060e6 −0.977837
\(344\) 0 0
\(345\) 3.42584e6 1.54960
\(346\) 0 0
\(347\) −2.67241e6 −1.19146 −0.595729 0.803185i \(-0.703137\pi\)
−0.595729 + 0.803185i \(0.703137\pi\)
\(348\) 0 0
\(349\) 3.02667e6 1.33015 0.665076 0.746775i \(-0.268399\pi\)
0.665076 + 0.746775i \(0.268399\pi\)
\(350\) 0 0
\(351\) 248590. 0.107700
\(352\) 0 0
\(353\) 790142. 0.337496 0.168748 0.985659i \(-0.446028\pi\)
0.168748 + 0.985659i \(0.446028\pi\)
\(354\) 0 0
\(355\) 3.40041e6 1.43206
\(356\) 0 0
\(357\) −629072. −0.261234
\(358\) 0 0
\(359\) 1.80303e6 0.738359 0.369179 0.929358i \(-0.379639\pi\)
0.369179 + 0.929358i \(0.379639\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) −2.65353e6 −1.05696
\(364\) 0 0
\(365\) −7.31653e6 −2.87457
\(366\) 0 0
\(367\) 1.68799e6 0.654192 0.327096 0.944991i \(-0.393930\pi\)
0.327096 + 0.944991i \(0.393930\pi\)
\(368\) 0 0
\(369\) −451186. −0.172500
\(370\) 0 0
\(371\) −1.35142e6 −0.509748
\(372\) 0 0
\(373\) 4.48620e6 1.66958 0.834788 0.550571i \(-0.185590\pi\)
0.834788 + 0.550571i \(0.185590\pi\)
\(374\) 0 0
\(375\) 2.14968e6 0.789398
\(376\) 0 0
\(377\) −60277.6 −0.0218425
\(378\) 0 0
\(379\) 103084. 0.0368634 0.0184317 0.999830i \(-0.494133\pi\)
0.0184317 + 0.999830i \(0.494133\pi\)
\(380\) 0 0
\(381\) −5.62525e6 −1.98532
\(382\) 0 0
\(383\) 2.21030e6 0.769934 0.384967 0.922930i \(-0.374213\pi\)
0.384967 + 0.922930i \(0.374213\pi\)
\(384\) 0 0
\(385\) −667569. −0.229532
\(386\) 0 0
\(387\) −136897. −0.0464641
\(388\) 0 0
\(389\) 3.56273e6 1.19374 0.596870 0.802338i \(-0.296411\pi\)
0.596870 + 0.802338i \(0.296411\pi\)
\(390\) 0 0
\(391\) −1.04002e6 −0.344032
\(392\) 0 0
\(393\) −1.84076e6 −0.601196
\(394\) 0 0
\(395\) 3.41135e6 1.10010
\(396\) 0 0
\(397\) −4.69992e6 −1.49663 −0.748315 0.663344i \(-0.769137\pi\)
−0.748315 + 0.663344i \(0.769137\pi\)
\(398\) 0 0
\(399\) −487348. −0.153252
\(400\) 0 0
\(401\) −2.76391e6 −0.858347 −0.429173 0.903222i \(-0.641195\pi\)
−0.429173 + 0.903222i \(0.641195\pi\)
\(402\) 0 0
\(403\) −216261. −0.0663309
\(404\) 0 0
\(405\) −6.17648e6 −1.87113
\(406\) 0 0
\(407\) −1.02287e6 −0.306080
\(408\) 0 0
\(409\) 4.43668e6 1.31144 0.655722 0.755002i \(-0.272364\pi\)
0.655722 + 0.755002i \(0.272364\pi\)
\(410\) 0 0
\(411\) 4.84617e6 1.41512
\(412\) 0 0
\(413\) −2.65513e6 −0.765969
\(414\) 0 0
\(415\) 8.69895e6 2.47940
\(416\) 0 0
\(417\) 911783. 0.256774
\(418\) 0 0
\(419\) −3.67747e6 −1.02333 −0.511664 0.859186i \(-0.670971\pi\)
−0.511664 + 0.859186i \(0.670971\pi\)
\(420\) 0 0
\(421\) 2.97195e6 0.817216 0.408608 0.912710i \(-0.366014\pi\)
0.408608 + 0.912710i \(0.366014\pi\)
\(422\) 0 0
\(423\) −1.33595e6 −0.363028
\(424\) 0 0
\(425\) −2.10879e6 −0.566319
\(426\) 0 0
\(427\) −3.15211e6 −0.836626
\(428\) 0 0
\(429\) 138530. 0.0363413
\(430\) 0 0
\(431\) −2.90998e6 −0.754565 −0.377283 0.926098i \(-0.623141\pi\)
−0.377283 + 0.926098i \(0.623141\pi\)
\(432\) 0 0
\(433\) 3.06888e6 0.786611 0.393305 0.919408i \(-0.371332\pi\)
0.393305 + 0.919408i \(0.371332\pi\)
\(434\) 0 0
\(435\) 1.16285e6 0.294647
\(436\) 0 0
\(437\) −805713. −0.201826
\(438\) 0 0
\(439\) 2.87829e6 0.712809 0.356404 0.934332i \(-0.384003\pi\)
0.356404 + 0.934332i \(0.384003\pi\)
\(440\) 0 0
\(441\) −707397. −0.173208
\(442\) 0 0
\(443\) 1.29112e6 0.312578 0.156289 0.987711i \(-0.450047\pi\)
0.156289 + 0.987711i \(0.450047\pi\)
\(444\) 0 0
\(445\) −7.77655e6 −1.86160
\(446\) 0 0
\(447\) −7.38723e6 −1.74869
\(448\) 0 0
\(449\) 4.24274e6 0.993186 0.496593 0.867984i \(-0.334584\pi\)
0.496593 + 0.867984i \(0.334584\pi\)
\(450\) 0 0
\(451\) 689056. 0.159519
\(452\) 0 0
\(453\) −995349. −0.227892
\(454\) 0 0
\(455\) −535368. −0.121234
\(456\) 0 0
\(457\) −6.19374e6 −1.38727 −0.693637 0.720324i \(-0.743993\pi\)
−0.693637 + 0.720324i \(0.743993\pi\)
\(458\) 0 0
\(459\) 1.45588e6 0.322548
\(460\) 0 0
\(461\) −5.87032e6 −1.28650 −0.643250 0.765656i \(-0.722414\pi\)
−0.643250 + 0.765656i \(0.722414\pi\)
\(462\) 0 0
\(463\) −8.40538e6 −1.82224 −0.911119 0.412144i \(-0.864780\pi\)
−0.911119 + 0.412144i \(0.864780\pi\)
\(464\) 0 0
\(465\) 4.17203e6 0.894778
\(466\) 0 0
\(467\) −148549. −0.0315193 −0.0157597 0.999876i \(-0.505017\pi\)
−0.0157597 + 0.999876i \(0.505017\pi\)
\(468\) 0 0
\(469\) 5.25020e6 1.10216
\(470\) 0 0
\(471\) −1.15073e6 −0.239013
\(472\) 0 0
\(473\) 209071. 0.0429676
\(474\) 0 0
\(475\) −1.63370e6 −0.332230
\(476\) 0 0
\(477\) −1.14124e6 −0.229658
\(478\) 0 0
\(479\) 5.35414e6 1.06623 0.533115 0.846043i \(-0.321021\pi\)
0.533115 + 0.846043i \(0.321021\pi\)
\(480\) 0 0
\(481\) −820311. −0.161665
\(482\) 0 0
\(483\) 3.01304e6 0.587675
\(484\) 0 0
\(485\) −7.65198e6 −1.47713
\(486\) 0 0
\(487\) −2.86815e6 −0.547998 −0.273999 0.961730i \(-0.588346\pi\)
−0.273999 + 0.961730i \(0.588346\pi\)
\(488\) 0 0
\(489\) 7.82219e6 1.47930
\(490\) 0 0
\(491\) 7.73298e6 1.44758 0.723791 0.690019i \(-0.242398\pi\)
0.723791 + 0.690019i \(0.242398\pi\)
\(492\) 0 0
\(493\) −353020. −0.0654156
\(494\) 0 0
\(495\) −563746. −0.103412
\(496\) 0 0
\(497\) 2.99067e6 0.543097
\(498\) 0 0
\(499\) 1.93075e6 0.347116 0.173558 0.984824i \(-0.444474\pi\)
0.173558 + 0.984824i \(0.444474\pi\)
\(500\) 0 0
\(501\) −2.64205e6 −0.470269
\(502\) 0 0
\(503\) −5.50103e6 −0.969447 −0.484723 0.874668i \(-0.661080\pi\)
−0.484723 + 0.874668i \(0.661080\pi\)
\(504\) 0 0
\(505\) 1.09082e7 1.90337
\(506\) 0 0
\(507\) −6.40468e6 −1.10657
\(508\) 0 0
\(509\) −4.93464e6 −0.844230 −0.422115 0.906542i \(-0.638712\pi\)
−0.422115 + 0.906542i \(0.638712\pi\)
\(510\) 0 0
\(511\) −6.43492e6 −1.09016
\(512\) 0 0
\(513\) 1.12789e6 0.189222
\(514\) 0 0
\(515\) −2.45748e6 −0.408292
\(516\) 0 0
\(517\) 2.04028e6 0.335709
\(518\) 0 0
\(519\) −1.87069e6 −0.304849
\(520\) 0 0
\(521\) 5.98044e6 0.965248 0.482624 0.875828i \(-0.339684\pi\)
0.482624 + 0.875828i \(0.339684\pi\)
\(522\) 0 0
\(523\) −3.74282e6 −0.598336 −0.299168 0.954200i \(-0.596709\pi\)
−0.299168 + 0.954200i \(0.596709\pi\)
\(524\) 0 0
\(525\) 6.10939e6 0.967386
\(526\) 0 0
\(527\) −1.26655e6 −0.198653
\(528\) 0 0
\(529\) −1.45500e6 −0.226061
\(530\) 0 0
\(531\) −2.24220e6 −0.345094
\(532\) 0 0
\(533\) 552601. 0.0842546
\(534\) 0 0
\(535\) 9.33149e6 1.40951
\(536\) 0 0
\(537\) 7.30494e6 1.09315
\(538\) 0 0
\(539\) 1.08034e6 0.160173
\(540\) 0 0
\(541\) 7.66502e6 1.12595 0.562977 0.826473i \(-0.309656\pi\)
0.562977 + 0.826473i \(0.309656\pi\)
\(542\) 0 0
\(543\) 1.76200e6 0.256452
\(544\) 0 0
\(545\) −7.14945e6 −1.03105
\(546\) 0 0
\(547\) −8.02578e6 −1.14688 −0.573441 0.819247i \(-0.694392\pi\)
−0.573441 + 0.819247i \(0.694392\pi\)
\(548\) 0 0
\(549\) −2.66188e6 −0.376927
\(550\) 0 0
\(551\) −273488. −0.0383759
\(552\) 0 0
\(553\) 3.00029e6 0.417206
\(554\) 0 0
\(555\) 1.58251e7 2.18079
\(556\) 0 0
\(557\) −1.39685e7 −1.90771 −0.953855 0.300266i \(-0.902924\pi\)
−0.953855 + 0.300266i \(0.902924\pi\)
\(558\) 0 0
\(559\) 167668. 0.0226945
\(560\) 0 0
\(561\) 811309. 0.108838
\(562\) 0 0
\(563\) −5.14218e6 −0.683717 −0.341858 0.939751i \(-0.611056\pi\)
−0.341858 + 0.939751i \(0.611056\pi\)
\(564\) 0 0
\(565\) 2.32512e7 3.06425
\(566\) 0 0
\(567\) −5.43224e6 −0.709612
\(568\) 0 0
\(569\) 1.38063e7 1.78771 0.893853 0.448359i \(-0.147991\pi\)
0.893853 + 0.448359i \(0.147991\pi\)
\(570\) 0 0
\(571\) 1.91974e6 0.246406 0.123203 0.992381i \(-0.460683\pi\)
0.123203 + 0.992381i \(0.460683\pi\)
\(572\) 0 0
\(573\) 1.04798e7 1.33342
\(574\) 0 0
\(575\) 1.01004e7 1.27400
\(576\) 0 0
\(577\) −7.86662e6 −0.983669 −0.491834 0.870689i \(-0.663673\pi\)
−0.491834 + 0.870689i \(0.663673\pi\)
\(578\) 0 0
\(579\) 1.25363e7 1.55408
\(580\) 0 0
\(581\) 7.65077e6 0.940296
\(582\) 0 0
\(583\) 1.74292e6 0.212376
\(584\) 0 0
\(585\) −452106. −0.0546198
\(586\) 0 0
\(587\) −5.61716e6 −0.672856 −0.336428 0.941709i \(-0.609219\pi\)
−0.336428 + 0.941709i \(0.609219\pi\)
\(588\) 0 0
\(589\) −981206. −0.116539
\(590\) 0 0
\(591\) −194130. −0.0228625
\(592\) 0 0
\(593\) −3.37137e6 −0.393704 −0.196852 0.980433i \(-0.563072\pi\)
−0.196852 + 0.980433i \(0.563072\pi\)
\(594\) 0 0
\(595\) −3.13542e6 −0.363081
\(596\) 0 0
\(597\) 6.28002e6 0.721149
\(598\) 0 0
\(599\) −4.63713e6 −0.528059 −0.264030 0.964515i \(-0.585052\pi\)
−0.264030 + 0.964515i \(0.585052\pi\)
\(600\) 0 0
\(601\) −1.32857e7 −1.50037 −0.750184 0.661229i \(-0.770035\pi\)
−0.750184 + 0.661229i \(0.770035\pi\)
\(602\) 0 0
\(603\) 4.43367e6 0.496558
\(604\) 0 0
\(605\) −1.32257e7 −1.46903
\(606\) 0 0
\(607\) −5.36627e6 −0.591154 −0.295577 0.955319i \(-0.595512\pi\)
−0.295577 + 0.955319i \(0.595512\pi\)
\(608\) 0 0
\(609\) 1.02273e6 0.111743
\(610\) 0 0
\(611\) 1.63624e6 0.177314
\(612\) 0 0
\(613\) −1.19912e7 −1.28888 −0.644440 0.764655i \(-0.722909\pi\)
−0.644440 + 0.764655i \(0.722909\pi\)
\(614\) 0 0
\(615\) −1.06606e7 −1.13656
\(616\) 0 0
\(617\) −5.57116e6 −0.589159 −0.294579 0.955627i \(-0.595180\pi\)
−0.294579 + 0.955627i \(0.595180\pi\)
\(618\) 0 0
\(619\) 1.06940e7 1.12179 0.560897 0.827885i \(-0.310456\pi\)
0.560897 + 0.827885i \(0.310456\pi\)
\(620\) 0 0
\(621\) −6.97319e6 −0.725609
\(622\) 0 0
\(623\) −6.83951e6 −0.706000
\(624\) 0 0
\(625\) −3.42772e6 −0.350999
\(626\) 0 0
\(627\) 628529. 0.0638494
\(628\) 0 0
\(629\) −4.80420e6 −0.484166
\(630\) 0 0
\(631\) 1.27937e7 1.27915 0.639576 0.768728i \(-0.279110\pi\)
0.639576 + 0.768728i \(0.279110\pi\)
\(632\) 0 0
\(633\) −4.98617e6 −0.494604
\(634\) 0 0
\(635\) −2.80374e7 −2.75932
\(636\) 0 0
\(637\) 866401. 0.0846000
\(638\) 0 0
\(639\) 2.52555e6 0.244683
\(640\) 0 0
\(641\) −3.64712e6 −0.350594 −0.175297 0.984516i \(-0.556089\pi\)
−0.175297 + 0.984516i \(0.556089\pi\)
\(642\) 0 0
\(643\) 9.41881e6 0.898398 0.449199 0.893432i \(-0.351709\pi\)
0.449199 + 0.893432i \(0.351709\pi\)
\(644\) 0 0
\(645\) −3.23459e6 −0.306140
\(646\) 0 0
\(647\) −8.34880e6 −0.784085 −0.392043 0.919947i \(-0.628231\pi\)
−0.392043 + 0.919947i \(0.628231\pi\)
\(648\) 0 0
\(649\) 3.42430e6 0.319125
\(650\) 0 0
\(651\) 3.66932e6 0.339338
\(652\) 0 0
\(653\) 7.48058e6 0.686518 0.343259 0.939241i \(-0.388469\pi\)
0.343259 + 0.939241i \(0.388469\pi\)
\(654\) 0 0
\(655\) −9.17471e6 −0.835582
\(656\) 0 0
\(657\) −5.43413e6 −0.491153
\(658\) 0 0
\(659\) −1.39427e7 −1.25064 −0.625320 0.780368i \(-0.715032\pi\)
−0.625320 + 0.780368i \(0.715032\pi\)
\(660\) 0 0
\(661\) −1.06152e7 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(662\) 0 0
\(663\) 650644. 0.0574856
\(664\) 0 0
\(665\) −2.42904e6 −0.213000
\(666\) 0 0
\(667\) 1.69085e6 0.147160
\(668\) 0 0
\(669\) 1.52762e7 1.31963
\(670\) 0 0
\(671\) 4.06525e6 0.348563
\(672\) 0 0
\(673\) 9.42379e6 0.802025 0.401012 0.916073i \(-0.368658\pi\)
0.401012 + 0.916073i \(0.368658\pi\)
\(674\) 0 0
\(675\) −1.41392e7 −1.19444
\(676\) 0 0
\(677\) 1.80040e6 0.150972 0.0754862 0.997147i \(-0.475949\pi\)
0.0754862 + 0.997147i \(0.475949\pi\)
\(678\) 0 0
\(679\) −6.72995e6 −0.560193
\(680\) 0 0
\(681\) 8.49799e6 0.702180
\(682\) 0 0
\(683\) 1.73202e7 1.42070 0.710348 0.703850i \(-0.248537\pi\)
0.710348 + 0.703850i \(0.248537\pi\)
\(684\) 0 0
\(685\) 2.41543e7 1.96683
\(686\) 0 0
\(687\) −8.99919e6 −0.727464
\(688\) 0 0
\(689\) 1.39776e6 0.112172
\(690\) 0 0
\(691\) −1.68765e7 −1.34458 −0.672292 0.740286i \(-0.734690\pi\)
−0.672292 + 0.740286i \(0.734690\pi\)
\(692\) 0 0
\(693\) −495817. −0.0392182
\(694\) 0 0
\(695\) 4.54450e6 0.356882
\(696\) 0 0
\(697\) 3.23634e6 0.252332
\(698\) 0 0
\(699\) 2.00138e7 1.54931
\(700\) 0 0
\(701\) 5.40330e6 0.415302 0.207651 0.978203i \(-0.433418\pi\)
0.207651 + 0.978203i \(0.433418\pi\)
\(702\) 0 0
\(703\) −3.72186e6 −0.284035
\(704\) 0 0
\(705\) −3.15657e7 −2.39189
\(706\) 0 0
\(707\) 9.59378e6 0.721841
\(708\) 0 0
\(709\) 2.05223e7 1.53324 0.766619 0.642103i \(-0.221938\pi\)
0.766619 + 0.642103i \(0.221938\pi\)
\(710\) 0 0
\(711\) 2.53368e6 0.187965
\(712\) 0 0
\(713\) 6.06633e6 0.446892
\(714\) 0 0
\(715\) 690460. 0.0505096
\(716\) 0 0
\(717\) −9.90213e6 −0.719334
\(718\) 0 0
\(719\) −4.37778e6 −0.315815 −0.157907 0.987454i \(-0.550475\pi\)
−0.157907 + 0.987454i \(0.550475\pi\)
\(720\) 0 0
\(721\) −2.16136e6 −0.154842
\(722\) 0 0
\(723\) −8.76602e6 −0.623673
\(724\) 0 0
\(725\) 3.42844e6 0.242243
\(726\) 0 0
\(727\) −1.75812e7 −1.23371 −0.616853 0.787078i \(-0.711593\pi\)
−0.616853 + 0.787078i \(0.711593\pi\)
\(728\) 0 0
\(729\) 8.73599e6 0.608826
\(730\) 0 0
\(731\) 981959. 0.0679673
\(732\) 0 0
\(733\) 5.07428e6 0.348831 0.174415 0.984672i \(-0.444196\pi\)
0.174415 + 0.984672i \(0.444196\pi\)
\(734\) 0 0
\(735\) −1.67143e7 −1.14122
\(736\) 0 0
\(737\) −6.77114e6 −0.459191
\(738\) 0 0
\(739\) −2.35559e7 −1.58668 −0.793339 0.608780i \(-0.791659\pi\)
−0.793339 + 0.608780i \(0.791659\pi\)
\(740\) 0 0
\(741\) 504060. 0.0337238
\(742\) 0 0
\(743\) 2.16206e7 1.43680 0.718399 0.695631i \(-0.244875\pi\)
0.718399 + 0.695631i \(0.244875\pi\)
\(744\) 0 0
\(745\) −3.68194e7 −2.43045
\(746\) 0 0
\(747\) 6.46089e6 0.423634
\(748\) 0 0
\(749\) 8.20709e6 0.534545
\(750\) 0 0
\(751\) 2.87245e7 1.85845 0.929227 0.369508i \(-0.120474\pi\)
0.929227 + 0.369508i \(0.120474\pi\)
\(752\) 0 0
\(753\) 2.32100e7 1.49172
\(754\) 0 0
\(755\) −4.96101e6 −0.316740
\(756\) 0 0
\(757\) 1.24115e7 0.787199 0.393599 0.919282i \(-0.371230\pi\)
0.393599 + 0.919282i \(0.371230\pi\)
\(758\) 0 0
\(759\) −3.88590e6 −0.244843
\(760\) 0 0
\(761\) 766287. 0.0479656 0.0239828 0.999712i \(-0.492365\pi\)
0.0239828 + 0.999712i \(0.492365\pi\)
\(762\) 0 0
\(763\) −6.28797e6 −0.391020
\(764\) 0 0
\(765\) −2.64779e6 −0.163580
\(766\) 0 0
\(767\) 2.74618e6 0.168555
\(768\) 0 0
\(769\) −1.49872e7 −0.913914 −0.456957 0.889489i \(-0.651061\pi\)
−0.456957 + 0.889489i \(0.651061\pi\)
\(770\) 0 0
\(771\) 510477. 0.0309272
\(772\) 0 0
\(773\) 1.16435e7 0.700867 0.350434 0.936588i \(-0.386034\pi\)
0.350434 + 0.936588i \(0.386034\pi\)
\(774\) 0 0
\(775\) 1.23004e7 0.735639
\(776\) 0 0
\(777\) 1.39183e7 0.827052
\(778\) 0 0
\(779\) 2.50722e6 0.148030
\(780\) 0 0
\(781\) −3.85705e6 −0.226270
\(782\) 0 0
\(783\) −2.36695e6 −0.137970
\(784\) 0 0
\(785\) −5.73547e6 −0.332197
\(786\) 0 0
\(787\) −1.80280e7 −1.03756 −0.518778 0.854909i \(-0.673613\pi\)
−0.518778 + 0.854909i \(0.673613\pi\)
\(788\) 0 0
\(789\) 2.45382e7 1.40330
\(790\) 0 0
\(791\) 2.04495e7 1.16209
\(792\) 0 0
\(793\) 3.26020e6 0.184103
\(794\) 0 0
\(795\) −2.69651e7 −1.51316
\(796\) 0 0
\(797\) 1.97016e7 1.09864 0.549319 0.835612i \(-0.314887\pi\)
0.549319 + 0.835612i \(0.314887\pi\)
\(798\) 0 0
\(799\) 9.58272e6 0.531033
\(800\) 0 0
\(801\) −5.77580e6 −0.318076
\(802\) 0 0
\(803\) 8.29906e6 0.454193
\(804\) 0 0
\(805\) 1.50176e7 0.816790
\(806\) 0 0
\(807\) 2.84386e7 1.53718
\(808\) 0 0
\(809\) −6.78469e6 −0.364468 −0.182234 0.983255i \(-0.558333\pi\)
−0.182234 + 0.983255i \(0.558333\pi\)
\(810\) 0 0
\(811\) 3.30503e7 1.76450 0.882252 0.470777i \(-0.156026\pi\)
0.882252 + 0.470777i \(0.156026\pi\)
\(812\) 0 0
\(813\) −5.46713e6 −0.290090
\(814\) 0 0
\(815\) 3.89873e7 2.05603
\(816\) 0 0
\(817\) 760733. 0.0398729
\(818\) 0 0
\(819\) −397629. −0.0207142
\(820\) 0 0
\(821\) −2.99319e7 −1.54980 −0.774901 0.632083i \(-0.782200\pi\)
−0.774901 + 0.632083i \(0.782200\pi\)
\(822\) 0 0
\(823\) 1.11456e7 0.573591 0.286796 0.957992i \(-0.407410\pi\)
0.286796 + 0.957992i \(0.407410\pi\)
\(824\) 0 0
\(825\) −7.87923e6 −0.403041
\(826\) 0 0
\(827\) 1.66420e7 0.846141 0.423070 0.906097i \(-0.360952\pi\)
0.423070 + 0.906097i \(0.360952\pi\)
\(828\) 0 0
\(829\) −1.67963e7 −0.848844 −0.424422 0.905465i \(-0.639523\pi\)
−0.424422 + 0.905465i \(0.639523\pi\)
\(830\) 0 0
\(831\) −1.40654e7 −0.706563
\(832\) 0 0
\(833\) 5.07413e6 0.253366
\(834\) 0 0
\(835\) −1.31685e7 −0.653611
\(836\) 0 0
\(837\) −8.49203e6 −0.418985
\(838\) 0 0
\(839\) 2.44304e7 1.19819 0.599095 0.800678i \(-0.295527\pi\)
0.599095 + 0.800678i \(0.295527\pi\)
\(840\) 0 0
\(841\) −1.99372e7 −0.972018
\(842\) 0 0
\(843\) 2.79966e7 1.35686
\(844\) 0 0
\(845\) −3.19222e7 −1.53798
\(846\) 0 0
\(847\) −1.16321e7 −0.557119
\(848\) 0 0
\(849\) −1.82761e7 −0.870191
\(850\) 0 0
\(851\) 2.30105e7 1.08919
\(852\) 0 0
\(853\) −3.68577e6 −0.173442 −0.0867212 0.996233i \(-0.527639\pi\)
−0.0867212 + 0.996233i \(0.527639\pi\)
\(854\) 0 0
\(855\) −2.05127e6 −0.0959636
\(856\) 0 0
\(857\) −3.18729e7 −1.48241 −0.741207 0.671276i \(-0.765746\pi\)
−0.741207 + 0.671276i \(0.765746\pi\)
\(858\) 0 0
\(859\) −3.56678e7 −1.64928 −0.824638 0.565661i \(-0.808621\pi\)
−0.824638 + 0.565661i \(0.808621\pi\)
\(860\) 0 0
\(861\) −9.37602e6 −0.431033
\(862\) 0 0
\(863\) 2.38642e7 1.09074 0.545369 0.838196i \(-0.316390\pi\)
0.545369 + 0.838196i \(0.316390\pi\)
\(864\) 0 0
\(865\) −9.32391e6 −0.423699
\(866\) 0 0
\(867\) −2.11064e7 −0.953600
\(868\) 0 0
\(869\) −3.86945e6 −0.173820
\(870\) 0 0
\(871\) −5.43023e6 −0.242534
\(872\) 0 0
\(873\) −5.68328e6 −0.252385
\(874\) 0 0
\(875\) 9.42339e6 0.416090
\(876\) 0 0
\(877\) −1.78983e7 −0.785800 −0.392900 0.919581i \(-0.628528\pi\)
−0.392900 + 0.919581i \(0.628528\pi\)
\(878\) 0 0
\(879\) −6.86206e6 −0.299559
\(880\) 0 0
\(881\) 4.19383e7 1.82042 0.910209 0.414148i \(-0.135921\pi\)
0.910209 + 0.414148i \(0.135921\pi\)
\(882\) 0 0
\(883\) −5.72425e6 −0.247068 −0.123534 0.992340i \(-0.539423\pi\)
−0.123534 + 0.992340i \(0.539423\pi\)
\(884\) 0 0
\(885\) −5.29783e7 −2.27373
\(886\) 0 0
\(887\) 5.28478e6 0.225537 0.112769 0.993621i \(-0.464028\pi\)
0.112769 + 0.993621i \(0.464028\pi\)
\(888\) 0 0
\(889\) −2.46590e7 −1.04646
\(890\) 0 0
\(891\) 7.00592e6 0.295645
\(892\) 0 0
\(893\) 7.42383e6 0.311529
\(894\) 0 0
\(895\) 3.64092e7 1.51934
\(896\) 0 0
\(897\) −3.11636e6 −0.129320
\(898\) 0 0
\(899\) 2.05913e6 0.0849737
\(900\) 0 0
\(901\) 8.18608e6 0.335942
\(902\) 0 0
\(903\) −2.84484e6 −0.116102
\(904\) 0 0
\(905\) 8.78213e6 0.356433
\(906\) 0 0
\(907\) −1.88055e6 −0.0759045 −0.0379523 0.999280i \(-0.512083\pi\)
−0.0379523 + 0.999280i \(0.512083\pi\)
\(908\) 0 0
\(909\) 8.10172e6 0.325213
\(910\) 0 0
\(911\) −1.81410e7 −0.724210 −0.362105 0.932137i \(-0.617942\pi\)
−0.362105 + 0.932137i \(0.617942\pi\)
\(912\) 0 0
\(913\) −9.86713e6 −0.391755
\(914\) 0 0
\(915\) −6.28945e7 −2.48348
\(916\) 0 0
\(917\) −8.06919e6 −0.316889
\(918\) 0 0
\(919\) −3.25793e7 −1.27249 −0.636243 0.771489i \(-0.719512\pi\)
−0.636243 + 0.771489i \(0.719512\pi\)
\(920\) 0 0
\(921\) −8.84897e6 −0.343751
\(922\) 0 0
\(923\) −3.09323e6 −0.119511
\(924\) 0 0
\(925\) 4.66572e7 1.79293
\(926\) 0 0
\(927\) −1.82522e6 −0.0697614
\(928\) 0 0
\(929\) 5.80658e6 0.220740 0.110370 0.993891i \(-0.464796\pi\)
0.110370 + 0.993891i \(0.464796\pi\)
\(930\) 0 0
\(931\) 3.93098e6 0.148637
\(932\) 0 0
\(933\) −1.40399e6 −0.0528030
\(934\) 0 0
\(935\) 4.04372e6 0.151270
\(936\) 0 0
\(937\) 3.50025e7 1.30242 0.651209 0.758899i \(-0.274262\pi\)
0.651209 + 0.758899i \(0.274262\pi\)
\(938\) 0 0
\(939\) −1.74823e6 −0.0647045
\(940\) 0 0
\(941\) −3.99448e7 −1.47057 −0.735285 0.677758i \(-0.762952\pi\)
−0.735285 + 0.677758i \(0.762952\pi\)
\(942\) 0 0
\(943\) −1.55010e7 −0.567649
\(944\) 0 0
\(945\) −2.10226e7 −0.765784
\(946\) 0 0
\(947\) −6.80996e6 −0.246757 −0.123379 0.992360i \(-0.539373\pi\)
−0.123379 + 0.992360i \(0.539373\pi\)
\(948\) 0 0
\(949\) 6.65558e6 0.239894
\(950\) 0 0
\(951\) −3.10875e6 −0.111464
\(952\) 0 0
\(953\) −3.48768e7 −1.24395 −0.621976 0.783036i \(-0.713670\pi\)
−0.621976 + 0.783036i \(0.713670\pi\)
\(954\) 0 0
\(955\) 5.22332e7 1.85327
\(956\) 0 0
\(957\) −1.31901e6 −0.0465553
\(958\) 0 0
\(959\) 2.12438e7 0.745908
\(960\) 0 0
\(961\) −2.12415e7 −0.741954
\(962\) 0 0
\(963\) 6.93069e6 0.240830
\(964\) 0 0
\(965\) 6.24834e7 2.15996
\(966\) 0 0
\(967\) −4.78004e7 −1.64386 −0.821931 0.569587i \(-0.807103\pi\)
−0.821931 + 0.569587i \(0.807103\pi\)
\(968\) 0 0
\(969\) 2.95206e6 0.100999
\(970\) 0 0
\(971\) 3.82374e7 1.30149 0.650744 0.759298i \(-0.274457\pi\)
0.650744 + 0.759298i \(0.274457\pi\)
\(972\) 0 0
\(973\) 3.99691e6 0.135345
\(974\) 0 0
\(975\) −6.31889e6 −0.212877
\(976\) 0 0
\(977\) 1.14899e7 0.385107 0.192553 0.981287i \(-0.438323\pi\)
0.192553 + 0.981287i \(0.438323\pi\)
\(978\) 0 0
\(979\) 8.82086e6 0.294140
\(980\) 0 0
\(981\) −5.31004e6 −0.176167
\(982\) 0 0
\(983\) −5.75149e7 −1.89844 −0.949220 0.314614i \(-0.898125\pi\)
−0.949220 + 0.314614i \(0.898125\pi\)
\(984\) 0 0
\(985\) −967583. −0.0317759
\(986\) 0 0
\(987\) −2.77621e7 −0.907110
\(988\) 0 0
\(989\) −4.70325e6 −0.152900
\(990\) 0 0
\(991\) −1.07593e7 −0.348016 −0.174008 0.984744i \(-0.555672\pi\)
−0.174008 + 0.984744i \(0.555672\pi\)
\(992\) 0 0
\(993\) 1.85784e7 0.597910
\(994\) 0 0
\(995\) 3.13008e7 1.00230
\(996\) 0 0
\(997\) −9.88575e6 −0.314972 −0.157486 0.987521i \(-0.550339\pi\)
−0.157486 + 0.987521i \(0.550339\pi\)
\(998\) 0 0
\(999\) −3.22115e7 −1.02117
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.1.3 4
3.2 odd 2 684.6.a.e.1.1 4
4.3 odd 2 304.6.a.k.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.a.b.1.3 4 1.1 even 1 trivial
304.6.a.k.1.2 4 4.3 odd 2
684.6.a.e.1.1 4 3.2 odd 2