Properties

Label 304.7.e.c.113.3
Level $304$
Weight $7$
Character 304.113
Analytic conductor $69.936$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 5090 x^{6} + 8905881 x^{4} + 5831691048 x^{2} + 827887219200\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.3
Root \(-38.1507i\) of defining polynomial
Character \(\chi\) \(=\) 304.113
Dual form 304.7.e.c.113.6

$q$-expansion

\(f(q)\) \(=\) \(q-38.1507i q^{3} +102.472 q^{5} -512.609 q^{7} -726.473 q^{9} +O(q^{10})\) \(q-38.1507i q^{3} +102.472 q^{5} -512.609 q^{7} -726.473 q^{9} -2416.13 q^{11} -3771.60i q^{13} -3909.38i q^{15} +1567.21 q^{17} +(2736.35 - 6289.54i) q^{19} +19556.4i q^{21} +626.209 q^{23} -5124.47 q^{25} -96.4222i q^{27} +15705.3i q^{29} +26199.8i q^{31} +92177.0i q^{33} -52528.1 q^{35} +49488.8i q^{37} -143889. q^{39} -119758. i q^{41} +13335.4 q^{43} -74443.2 q^{45} +186343. q^{47} +145119. q^{49} -59790.3i q^{51} +204595. i q^{53} -247586. q^{55} +(-239950. - 104393. i) q^{57} +18264.5i q^{59} -353270. q^{61} +372396. q^{63} -386484. i q^{65} +519557. i q^{67} -23890.3i q^{69} +268077. i q^{71} +58589.5 q^{73} +195502. i q^{75} +1.23853e6 q^{77} -473406. i q^{79} -533277. q^{81} -127056. q^{83} +160596. q^{85} +599166. q^{87} +607216. i q^{89} +1.93336e6i q^{91} +999541. q^{93} +(280399. - 644502. i) q^{95} -282195. i q^{97} +1.75525e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{5} - 362q^{7} - 4348q^{9} + O(q^{10}) \) \( 8q + 2q^{5} - 362q^{7} - 4348q^{9} - 902q^{11} + 1550q^{17} - 6232q^{19} + 18820q^{23} - 12158q^{25} + 101762q^{35} - 167028q^{39} + 335042q^{43} - 57230q^{45} + 570394q^{47} + 448182q^{49} - 1089198q^{55} + 341316q^{57} - 632014q^{61} - 328174q^{63} - 852938q^{73} + 1850530q^{77} - 1819456q^{81} - 441200q^{83} - 1828374q^{85} - 1483380q^{87} + 2131176q^{93} - 627950q^{95} + 865394q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 38.1507i 1.41299i −0.707719 0.706494i \(-0.750276\pi\)
0.707719 0.706494i \(-0.249724\pi\)
\(4\) 0 0
\(5\) 102.472 0.819777 0.409888 0.912136i \(-0.365568\pi\)
0.409888 + 0.912136i \(0.365568\pi\)
\(6\) 0 0
\(7\) −512.609 −1.49449 −0.747243 0.664551i \(-0.768623\pi\)
−0.747243 + 0.664551i \(0.768623\pi\)
\(8\) 0 0
\(9\) −726.473 −0.996533
\(10\) 0 0
\(11\) −2416.13 −1.81528 −0.907638 0.419754i \(-0.862116\pi\)
−0.907638 + 0.419754i \(0.862116\pi\)
\(12\) 0 0
\(13\) 3771.60i 1.71671i −0.513059 0.858353i \(-0.671488\pi\)
0.513059 0.858353i \(-0.328512\pi\)
\(14\) 0 0
\(15\) 3909.38i 1.15833i
\(16\) 0 0
\(17\) 1567.21 0.318993 0.159497 0.987198i \(-0.449013\pi\)
0.159497 + 0.987198i \(0.449013\pi\)
\(18\) 0 0
\(19\) 2736.35 6289.54i 0.398942 0.916976i
\(20\) 0 0
\(21\) 19556.4i 2.11169i
\(22\) 0 0
\(23\) 626.209 0.0514678 0.0257339 0.999669i \(-0.491808\pi\)
0.0257339 + 0.999669i \(0.491808\pi\)
\(24\) 0 0
\(25\) −5124.47 −0.327966
\(26\) 0 0
\(27\) 96.4222i 0.00489875i
\(28\) 0 0
\(29\) 15705.3i 0.643948i 0.946748 + 0.321974i \(0.104346\pi\)
−0.946748 + 0.321974i \(0.895654\pi\)
\(30\) 0 0
\(31\) 26199.8i 0.879454i 0.898131 + 0.439727i \(0.144925\pi\)
−0.898131 + 0.439727i \(0.855075\pi\)
\(32\) 0 0
\(33\) 92177.0i 2.56496i
\(34\) 0 0
\(35\) −52528.1 −1.22515
\(36\) 0 0
\(37\) 49488.8i 0.977015i 0.872560 + 0.488508i \(0.162459\pi\)
−0.872560 + 0.488508i \(0.837541\pi\)
\(38\) 0 0
\(39\) −143889. −2.42568
\(40\) 0 0
\(41\) 119758.i 1.73761i −0.495155 0.868805i \(-0.664889\pi\)
0.495155 0.868805i \(-0.335111\pi\)
\(42\) 0 0
\(43\) 13335.4 0.167726 0.0838628 0.996477i \(-0.473274\pi\)
0.0838628 + 0.996477i \(0.473274\pi\)
\(44\) 0 0
\(45\) −74443.2 −0.816935
\(46\) 0 0
\(47\) 186343. 1.79482 0.897408 0.441201i \(-0.145447\pi\)
0.897408 + 0.441201i \(0.145447\pi\)
\(48\) 0 0
\(49\) 145119. 1.23349
\(50\) 0 0
\(51\) 59790.3i 0.450734i
\(52\) 0 0
\(53\) 204595.i 1.37425i 0.726537 + 0.687127i \(0.241128\pi\)
−0.726537 + 0.687127i \(0.758872\pi\)
\(54\) 0 0
\(55\) −247586. −1.48812
\(56\) 0 0
\(57\) −239950. 104393.i −1.29568 0.563701i
\(58\) 0 0
\(59\) 18264.5i 0.0889306i 0.999011 + 0.0444653i \(0.0141584\pi\)
−0.999011 + 0.0444653i \(0.985842\pi\)
\(60\) 0 0
\(61\) −353270. −1.55639 −0.778193 0.628025i \(-0.783864\pi\)
−0.778193 + 0.628025i \(0.783864\pi\)
\(62\) 0 0
\(63\) 372396. 1.48930
\(64\) 0 0
\(65\) 386484.i 1.40732i
\(66\) 0 0
\(67\) 519557.i 1.72746i 0.503952 + 0.863732i \(0.331879\pi\)
−0.503952 + 0.863732i \(0.668121\pi\)
\(68\) 0 0
\(69\) 23890.3i 0.0727234i
\(70\) 0 0
\(71\) 268077.i 0.749004i 0.927226 + 0.374502i \(0.122186\pi\)
−0.927226 + 0.374502i \(0.877814\pi\)
\(72\) 0 0
\(73\) 58589.5 0.150609 0.0753045 0.997161i \(-0.476007\pi\)
0.0753045 + 0.997161i \(0.476007\pi\)
\(74\) 0 0
\(75\) 195502.i 0.463412i
\(76\) 0 0
\(77\) 1.23853e6 2.71290
\(78\) 0 0
\(79\) 473406.i 0.960179i −0.877219 0.480090i \(-0.840604\pi\)
0.877219 0.480090i \(-0.159396\pi\)
\(80\) 0 0
\(81\) −533277. −1.00345
\(82\) 0 0
\(83\) −127056. −0.222208 −0.111104 0.993809i \(-0.535439\pi\)
−0.111104 + 0.993809i \(0.535439\pi\)
\(84\) 0 0
\(85\) 160596. 0.261503
\(86\) 0 0
\(87\) 599166. 0.909891
\(88\) 0 0
\(89\) 607216.i 0.861337i 0.902510 + 0.430669i \(0.141722\pi\)
−0.902510 + 0.430669i \(0.858278\pi\)
\(90\) 0 0
\(91\) 1.93336e6i 2.56559i
\(92\) 0 0
\(93\) 999541. 1.24266
\(94\) 0 0
\(95\) 280399. 644502.i 0.327044 0.751716i
\(96\) 0 0
\(97\) 282195.i 0.309196i −0.987977 0.154598i \(-0.950592\pi\)
0.987977 0.154598i \(-0.0494083\pi\)
\(98\) 0 0
\(99\) 1.75525e6 1.80898
\(100\) 0 0
\(101\) 443149. 0.430116 0.215058 0.976601i \(-0.431006\pi\)
0.215058 + 0.976601i \(0.431006\pi\)
\(102\) 0 0
\(103\) 836042.i 0.765097i −0.923935 0.382549i \(-0.875046\pi\)
0.923935 0.382549i \(-0.124954\pi\)
\(104\) 0 0
\(105\) 2.00398e6i 1.73111i
\(106\) 0 0
\(107\) 617387.i 0.503972i 0.967731 + 0.251986i \(0.0810836\pi\)
−0.967731 + 0.251986i \(0.918916\pi\)
\(108\) 0 0
\(109\) 955640.i 0.737929i 0.929443 + 0.368965i \(0.120288\pi\)
−0.929443 + 0.368965i \(0.879712\pi\)
\(110\) 0 0
\(111\) 1.88803e6 1.38051
\(112\) 0 0
\(113\) 262850.i 0.182168i 0.995843 + 0.0910842i \(0.0290333\pi\)
−0.995843 + 0.0910842i \(0.970967\pi\)
\(114\) 0 0
\(115\) 64169.0 0.0421921
\(116\) 0 0
\(117\) 2.73997e6i 1.71075i
\(118\) 0 0
\(119\) −803368. −0.476731
\(120\) 0 0
\(121\) 4.06613e6 2.29523
\(122\) 0 0
\(123\) −4.56884e6 −2.45522
\(124\) 0 0
\(125\) −2.12624e6 −1.08864
\(126\) 0 0
\(127\) 933464.i 0.455708i −0.973695 0.227854i \(-0.926829\pi\)
0.973695 0.227854i \(-0.0731709\pi\)
\(128\) 0 0
\(129\) 508753.i 0.236994i
\(130\) 0 0
\(131\) 1.46381e6 0.651135 0.325568 0.945519i \(-0.394445\pi\)
0.325568 + 0.945519i \(0.394445\pi\)
\(132\) 0 0
\(133\) −1.40267e6 + 3.22407e6i −0.596214 + 1.37041i
\(134\) 0 0
\(135\) 9880.58i 0.00401589i
\(136\) 0 0
\(137\) 1.07931e6 0.419743 0.209871 0.977729i \(-0.432695\pi\)
0.209871 + 0.977729i \(0.432695\pi\)
\(138\) 0 0
\(139\) −2.79160e6 −1.03946 −0.519731 0.854330i \(-0.673968\pi\)
−0.519731 + 0.854330i \(0.673968\pi\)
\(140\) 0 0
\(141\) 7.10912e6i 2.53605i
\(142\) 0 0
\(143\) 9.11270e6i 3.11630i
\(144\) 0 0
\(145\) 1.60935e6i 0.527894i
\(146\) 0 0
\(147\) 5.53638e6i 1.74290i
\(148\) 0 0
\(149\) −1.47423e6 −0.445662 −0.222831 0.974857i \(-0.571530\pi\)
−0.222831 + 0.974857i \(0.571530\pi\)
\(150\) 0 0
\(151\) 3.70643e6i 1.07653i −0.842776 0.538264i \(-0.819080\pi\)
0.842776 0.538264i \(-0.180920\pi\)
\(152\) 0 0
\(153\) −1.13854e6 −0.317887
\(154\) 0 0
\(155\) 2.68475e6i 0.720956i
\(156\) 0 0
\(157\) 517298. 0.133673 0.0668363 0.997764i \(-0.478709\pi\)
0.0668363 + 0.997764i \(0.478709\pi\)
\(158\) 0 0
\(159\) 7.80543e6 1.94180
\(160\) 0 0
\(161\) −321000. −0.0769180
\(162\) 0 0
\(163\) −3.07611e6 −0.710296 −0.355148 0.934810i \(-0.615570\pi\)
−0.355148 + 0.934810i \(0.615570\pi\)
\(164\) 0 0
\(165\) 9.44557e6i 2.10270i
\(166\) 0 0
\(167\) 3.20302e6i 0.687717i 0.939021 + 0.343859i \(0.111734\pi\)
−0.939021 + 0.343859i \(0.888266\pi\)
\(168\) 0 0
\(169\) −9.39819e6 −1.94708
\(170\) 0 0
\(171\) −1.98788e6 + 4.56918e6i −0.397559 + 0.913797i
\(172\) 0 0
\(173\) 2.52360e6i 0.487395i −0.969851 0.243698i \(-0.921640\pi\)
0.969851 0.243698i \(-0.0783605\pi\)
\(174\) 0 0
\(175\) 2.62685e6 0.490141
\(176\) 0 0
\(177\) 696802. 0.125658
\(178\) 0 0
\(179\) 7.68706e6i 1.34030i 0.742227 + 0.670149i \(0.233770\pi\)
−0.742227 + 0.670149i \(0.766230\pi\)
\(180\) 0 0
\(181\) 3.05055e6i 0.514449i −0.966352 0.257225i \(-0.917192\pi\)
0.966352 0.257225i \(-0.0828081\pi\)
\(182\) 0 0
\(183\) 1.34775e7i 2.19915i
\(184\) 0 0
\(185\) 5.07122e6i 0.800934i
\(186\) 0 0
\(187\) −3.78660e6 −0.579061
\(188\) 0 0
\(189\) 49426.9i 0.00732112i
\(190\) 0 0
\(191\) 1.01067e7 1.45047 0.725234 0.688503i \(-0.241732\pi\)
0.725234 + 0.688503i \(0.241732\pi\)
\(192\) 0 0
\(193\) 2.57566e6i 0.358275i −0.983824 0.179138i \(-0.942669\pi\)
0.983824 0.179138i \(-0.0573307\pi\)
\(194\) 0 0
\(195\) −1.47446e7 −1.98852
\(196\) 0 0
\(197\) −333309. −0.0435962 −0.0217981 0.999762i \(-0.506939\pi\)
−0.0217981 + 0.999762i \(0.506939\pi\)
\(198\) 0 0
\(199\) 9.52584e6 1.20877 0.604385 0.796692i \(-0.293419\pi\)
0.604385 + 0.796692i \(0.293419\pi\)
\(200\) 0 0
\(201\) 1.98214e7 2.44088
\(202\) 0 0
\(203\) 8.05065e6i 0.962372i
\(204\) 0 0
\(205\) 1.22718e7i 1.42445i
\(206\) 0 0
\(207\) −454924. −0.0512894
\(208\) 0 0
\(209\) −6.61137e6 + 1.51964e7i −0.724191 + 1.66456i
\(210\) 0 0
\(211\) 7.87152e6i 0.837937i −0.908001 0.418969i \(-0.862392\pi\)
0.908001 0.418969i \(-0.137608\pi\)
\(212\) 0 0
\(213\) 1.02273e7 1.05833
\(214\) 0 0
\(215\) 1.36650e6 0.137498
\(216\) 0 0
\(217\) 1.34303e7i 1.31433i
\(218\) 0 0
\(219\) 2.23523e6i 0.212809i
\(220\) 0 0
\(221\) 5.91091e6i 0.547618i
\(222\) 0 0
\(223\) 1.47262e7i 1.32793i −0.747762 0.663967i \(-0.768872\pi\)
0.747762 0.663967i \(-0.231128\pi\)
\(224\) 0 0
\(225\) 3.72279e6 0.326829
\(226\) 0 0
\(227\) 7.49958e6i 0.641150i 0.947223 + 0.320575i \(0.103876\pi\)
−0.947223 + 0.320575i \(0.896124\pi\)
\(228\) 0 0
\(229\) 5.68862e6 0.473697 0.236848 0.971547i \(-0.423886\pi\)
0.236848 + 0.971547i \(0.423886\pi\)
\(230\) 0 0
\(231\) 4.72508e7i 3.83330i
\(232\) 0 0
\(233\) −1.88368e7 −1.48916 −0.744578 0.667536i \(-0.767349\pi\)
−0.744578 + 0.667536i \(0.767349\pi\)
\(234\) 0 0
\(235\) 1.90950e7 1.47135
\(236\) 0 0
\(237\) −1.80607e7 −1.35672
\(238\) 0 0
\(239\) −8.34352e6 −0.611161 −0.305581 0.952166i \(-0.598851\pi\)
−0.305581 + 0.952166i \(0.598851\pi\)
\(240\) 0 0
\(241\) 1.05537e7i 0.753970i 0.926219 + 0.376985i \(0.123039\pi\)
−0.926219 + 0.376985i \(0.876961\pi\)
\(242\) 0 0
\(243\) 2.02746e7i 1.41297i
\(244\) 0 0
\(245\) 1.48706e7 1.01119
\(246\) 0 0
\(247\) −2.37217e7 1.03204e7i −1.57418 0.684867i
\(248\) 0 0
\(249\) 4.84726e6i 0.313977i
\(250\) 0 0
\(251\) 1.91869e6 0.121335 0.0606673 0.998158i \(-0.480677\pi\)
0.0606673 + 0.998158i \(0.480677\pi\)
\(252\) 0 0
\(253\) −1.51300e6 −0.0934283
\(254\) 0 0
\(255\) 6.12683e6i 0.369501i
\(256\) 0 0
\(257\) 1.62596e7i 0.957877i −0.877848 0.478938i \(-0.841022\pi\)
0.877848 0.478938i \(-0.158978\pi\)
\(258\) 0 0
\(259\) 2.53684e7i 1.46014i
\(260\) 0 0
\(261\) 1.14094e7i 0.641716i
\(262\) 0 0
\(263\) 6.66631e6 0.366453 0.183226 0.983071i \(-0.441346\pi\)
0.183226 + 0.983071i \(0.441346\pi\)
\(264\) 0 0
\(265\) 2.09653e7i 1.12658i
\(266\) 0 0
\(267\) 2.31657e7 1.21706
\(268\) 0 0
\(269\) 2.25023e7i 1.15603i −0.816025 0.578016i \(-0.803827\pi\)
0.816025 0.578016i \(-0.196173\pi\)
\(270\) 0 0
\(271\) −2.63688e7 −1.32490 −0.662448 0.749108i \(-0.730483\pi\)
−0.662448 + 0.749108i \(0.730483\pi\)
\(272\) 0 0
\(273\) 7.37589e7 3.62515
\(274\) 0 0
\(275\) 1.23814e7 0.595349
\(276\) 0 0
\(277\) −2.95860e7 −1.39203 −0.696013 0.718030i \(-0.745044\pi\)
−0.696013 + 0.718030i \(0.745044\pi\)
\(278\) 0 0
\(279\) 1.90335e7i 0.876405i
\(280\) 0 0
\(281\) 1.14941e7i 0.518029i −0.965873 0.259015i \(-0.916602\pi\)
0.965873 0.259015i \(-0.0833978\pi\)
\(282\) 0 0
\(283\) −3.92088e7 −1.72991 −0.864957 0.501846i \(-0.832655\pi\)
−0.864957 + 0.501846i \(0.832655\pi\)
\(284\) 0 0
\(285\) −2.45882e7 1.06974e7i −1.06216 0.462109i
\(286\) 0 0
\(287\) 6.13889e7i 2.59683i
\(288\) 0 0
\(289\) −2.16814e7 −0.898243
\(290\) 0 0
\(291\) −1.07659e7 −0.436891
\(292\) 0 0
\(293\) 5.81821e6i 0.231306i −0.993290 0.115653i \(-0.963104\pi\)
0.993290 0.115653i \(-0.0368960\pi\)
\(294\) 0 0
\(295\) 1.87160e6i 0.0729033i
\(296\) 0 0
\(297\) 232969.i 0.00889259i
\(298\) 0 0
\(299\) 2.36181e6i 0.0883552i
\(300\) 0 0
\(301\) −6.83583e6 −0.250664
\(302\) 0 0
\(303\) 1.69064e7i 0.607748i
\(304\) 0 0
\(305\) −3.62003e7 −1.27589
\(306\) 0 0
\(307\) 2.09554e7i 0.724236i 0.932132 + 0.362118i \(0.117946\pi\)
−0.932132 + 0.362118i \(0.882054\pi\)
\(308\) 0 0
\(309\) −3.18956e7 −1.08107
\(310\) 0 0
\(311\) −2.79502e7 −0.929189 −0.464594 0.885524i \(-0.653800\pi\)
−0.464594 + 0.885524i \(0.653800\pi\)
\(312\) 0 0
\(313\) 1.47351e7 0.480528 0.240264 0.970708i \(-0.422766\pi\)
0.240264 + 0.970708i \(0.422766\pi\)
\(314\) 0 0
\(315\) 3.81602e7 1.22090
\(316\) 0 0
\(317\) 3.55764e7i 1.11682i −0.829564 0.558411i \(-0.811411\pi\)
0.829564 0.558411i \(-0.188589\pi\)
\(318\) 0 0
\(319\) 3.79460e7i 1.16894i
\(320\) 0 0
\(321\) 2.35537e7 0.712106
\(322\) 0 0
\(323\) 4.28844e6 9.85706e6i 0.127260 0.292509i
\(324\) 0 0
\(325\) 1.93275e7i 0.563021i
\(326\) 0 0
\(327\) 3.64583e7 1.04268
\(328\) 0 0
\(329\) −9.55212e7 −2.68233
\(330\) 0 0
\(331\) 4.77022e7i 1.31539i 0.753285 + 0.657694i \(0.228468\pi\)
−0.753285 + 0.657694i \(0.771532\pi\)
\(332\) 0 0
\(333\) 3.59522e7i 0.973628i
\(334\) 0 0
\(335\) 5.32401e7i 1.41613i
\(336\) 0 0
\(337\) 4.64803e7i 1.21445i 0.794530 + 0.607225i \(0.207717\pi\)
−0.794530 + 0.607225i \(0.792283\pi\)
\(338\) 0 0
\(339\) 1.00279e7 0.257402
\(340\) 0 0
\(341\) 6.33022e7i 1.59645i
\(342\) 0 0
\(343\) −1.40812e7 −0.348946
\(344\) 0 0
\(345\) 2.44809e6i 0.0596170i
\(346\) 0 0
\(347\) −5.43540e7 −1.30090 −0.650448 0.759550i \(-0.725419\pi\)
−0.650448 + 0.759550i \(0.725419\pi\)
\(348\) 0 0
\(349\) 3.19672e7 0.752018 0.376009 0.926616i \(-0.377296\pi\)
0.376009 + 0.926616i \(0.377296\pi\)
\(350\) 0 0
\(351\) −363666. −0.00840972
\(352\) 0 0
\(353\) 1.94618e7 0.442444 0.221222 0.975224i \(-0.428995\pi\)
0.221222 + 0.975224i \(0.428995\pi\)
\(354\) 0 0
\(355\) 2.74704e7i 0.614016i
\(356\) 0 0
\(357\) 3.06490e7i 0.673615i
\(358\) 0 0
\(359\) −8.12905e6 −0.175694 −0.0878469 0.996134i \(-0.527999\pi\)
−0.0878469 + 0.996134i \(0.527999\pi\)
\(360\) 0 0
\(361\) −3.20707e7 3.44207e7i −0.681690 0.731641i
\(362\) 0 0
\(363\) 1.55126e8i 3.24313i
\(364\) 0 0
\(365\) 6.00379e6 0.123466
\(366\) 0 0
\(367\) −9.45918e7 −1.91362 −0.956809 0.290716i \(-0.906107\pi\)
−0.956809 + 0.290716i \(0.906107\pi\)
\(368\) 0 0
\(369\) 8.70008e7i 1.73159i
\(370\) 0 0
\(371\) 1.04877e8i 2.05380i
\(372\) 0 0
\(373\) 1.41683e7i 0.273018i 0.990639 + 0.136509i \(0.0435882\pi\)
−0.990639 + 0.136509i \(0.956412\pi\)
\(374\) 0 0
\(375\) 8.11175e7i 1.53823i
\(376\) 0 0
\(377\) 5.92340e7 1.10547
\(378\) 0 0
\(379\) 5.66625e6i 0.104083i 0.998645 + 0.0520413i \(0.0165727\pi\)
−0.998645 + 0.0520413i \(0.983427\pi\)
\(380\) 0 0
\(381\) −3.56123e7 −0.643909
\(382\) 0 0
\(383\) 8.42138e7i 1.49895i 0.662034 + 0.749474i \(0.269694\pi\)
−0.662034 + 0.749474i \(0.730306\pi\)
\(384\) 0 0
\(385\) 1.26915e8 2.22398
\(386\) 0 0
\(387\) −9.68778e6 −0.167144
\(388\) 0 0
\(389\) 1.89837e7 0.322501 0.161251 0.986913i \(-0.448447\pi\)
0.161251 + 0.986913i \(0.448447\pi\)
\(390\) 0 0
\(391\) 981404. 0.0164179
\(392\) 0 0
\(393\) 5.58454e7i 0.920046i
\(394\) 0 0
\(395\) 4.85109e7i 0.787133i
\(396\) 0 0
\(397\) −2.79727e7 −0.447057 −0.223528 0.974697i \(-0.571758\pi\)
−0.223528 + 0.974697i \(0.571758\pi\)
\(398\) 0 0
\(399\) 1.23000e8 + 5.35130e7i 1.93637 + 0.842443i
\(400\) 0 0
\(401\) 9.08568e7i 1.40904i −0.709682 0.704522i \(-0.751161\pi\)
0.709682 0.704522i \(-0.248839\pi\)
\(402\) 0 0
\(403\) 9.88154e7 1.50977
\(404\) 0 0
\(405\) −5.46460e7 −0.822609
\(406\) 0 0
\(407\) 1.19571e8i 1.77355i
\(408\) 0 0
\(409\) 9.73970e7i 1.42356i 0.702403 + 0.711780i \(0.252111\pi\)
−0.702403 + 0.711780i \(0.747889\pi\)
\(410\) 0 0
\(411\) 4.11763e7i 0.593091i
\(412\) 0 0
\(413\) 9.36253e6i 0.132906i
\(414\) 0 0
\(415\) −1.30197e7 −0.182161
\(416\) 0 0
\(417\) 1.06501e8i 1.46875i
\(418\) 0 0
\(419\) 4.44767e7 0.604631 0.302316 0.953208i \(-0.402240\pi\)
0.302316 + 0.953208i \(0.402240\pi\)
\(420\) 0 0
\(421\) 8.49965e7i 1.13908i 0.821963 + 0.569541i \(0.192879\pi\)
−0.821963 + 0.569541i \(0.807121\pi\)
\(422\) 0 0
\(423\) −1.35373e8 −1.78859
\(424\) 0 0
\(425\) −8.03114e6 −0.104619
\(426\) 0 0
\(427\) 1.81089e8 2.32600
\(428\) 0 0
\(429\) 3.47655e8 4.40329
\(430\) 0 0
\(431\) 1.20089e8i 1.49993i −0.661476 0.749966i \(-0.730070\pi\)
0.661476 0.749966i \(-0.269930\pi\)
\(432\) 0 0
\(433\) 5.72220e7i 0.704854i 0.935839 + 0.352427i \(0.114644\pi\)
−0.935839 + 0.352427i \(0.885356\pi\)
\(434\) 0 0
\(435\) 6.13978e7 0.745907
\(436\) 0 0
\(437\) 1.71352e6 3.93857e6i 0.0205327 0.0471948i
\(438\) 0 0
\(439\) 4.39003e7i 0.518889i 0.965758 + 0.259444i \(0.0835394\pi\)
−0.965758 + 0.259444i \(0.916461\pi\)
\(440\) 0 0
\(441\) −1.05425e8 −1.22921
\(442\) 0 0
\(443\) −7.04729e7 −0.810608 −0.405304 0.914182i \(-0.632834\pi\)
−0.405304 + 0.914182i \(0.632834\pi\)
\(444\) 0 0
\(445\) 6.22227e7i 0.706105i
\(446\) 0 0
\(447\) 5.62428e7i 0.629715i
\(448\) 0 0
\(449\) 1.26837e8i 1.40122i 0.713543 + 0.700611i \(0.247089\pi\)
−0.713543 + 0.700611i \(0.752911\pi\)
\(450\) 0 0
\(451\) 2.89351e8i 3.15424i
\(452\) 0 0
\(453\) −1.41403e8 −1.52112
\(454\) 0 0
\(455\) 1.98115e8i 2.10321i
\(456\) 0 0
\(457\) −1.93303e7 −0.202530 −0.101265 0.994859i \(-0.532289\pi\)
−0.101265 + 0.994859i \(0.532289\pi\)
\(458\) 0 0
\(459\) 151114.i 0.00156267i
\(460\) 0 0
\(461\) −6.62945e7 −0.676667 −0.338333 0.941026i \(-0.609863\pi\)
−0.338333 + 0.941026i \(0.609863\pi\)
\(462\) 0 0
\(463\) 2.57206e7 0.259142 0.129571 0.991570i \(-0.458640\pi\)
0.129571 + 0.991570i \(0.458640\pi\)
\(464\) 0 0
\(465\) 1.02425e8 1.01870
\(466\) 0 0
\(467\) −6.14641e7 −0.603491 −0.301745 0.953389i \(-0.597569\pi\)
−0.301745 + 0.953389i \(0.597569\pi\)
\(468\) 0 0
\(469\) 2.66330e8i 2.58167i
\(470\) 0 0
\(471\) 1.97353e7i 0.188878i
\(472\) 0 0
\(473\) −3.22200e7 −0.304468
\(474\) 0 0
\(475\) −1.40223e7 + 3.22305e7i −0.130840 + 0.300737i
\(476\) 0 0
\(477\) 1.48633e8i 1.36949i
\(478\) 0 0
\(479\) −1.28933e8 −1.17316 −0.586581 0.809890i \(-0.699527\pi\)
−0.586581 + 0.809890i \(0.699527\pi\)
\(480\) 0 0
\(481\) 1.86652e8 1.67725
\(482\) 0 0
\(483\) 1.22464e7i 0.108684i
\(484\) 0 0
\(485\) 2.89171e7i 0.253472i
\(486\) 0 0
\(487\) 1.30778e8i 1.13227i −0.824314 0.566133i \(-0.808439\pi\)
0.824314 0.566133i \(-0.191561\pi\)
\(488\) 0 0
\(489\) 1.17356e8i 1.00364i
\(490\) 0 0
\(491\) 1.59776e8 1.34980 0.674898 0.737911i \(-0.264188\pi\)
0.674898 + 0.737911i \(0.264188\pi\)
\(492\) 0 0
\(493\) 2.46135e7i 0.205415i
\(494\) 0 0
\(495\) 1.79865e8 1.48296
\(496\) 0 0
\(497\) 1.37419e8i 1.11938i
\(498\) 0 0
\(499\) −9.75200e7 −0.784860 −0.392430 0.919782i \(-0.628365\pi\)
−0.392430 + 0.919782i \(0.628365\pi\)
\(500\) 0 0
\(501\) 1.22197e8 0.971736
\(502\) 0 0
\(503\) 1.63836e8 1.28738 0.643690 0.765286i \(-0.277403\pi\)
0.643690 + 0.765286i \(0.277403\pi\)
\(504\) 0 0
\(505\) 4.54104e7 0.352599
\(506\) 0 0
\(507\) 3.58547e8i 2.75120i
\(508\) 0 0
\(509\) 1.33348e8i 1.01119i 0.862771 + 0.505594i \(0.168727\pi\)
−0.862771 + 0.505594i \(0.831273\pi\)
\(510\) 0 0
\(511\) −3.00335e7 −0.225083
\(512\) 0 0
\(513\) −606451. 263844.i −0.00449204 0.00195432i
\(514\) 0 0
\(515\) 8.56710e7i 0.627209i
\(516\) 0 0
\(517\) −4.50230e8 −3.25809
\(518\) 0 0
\(519\) −9.62768e7 −0.688684
\(520\) 0 0
\(521\) 1.78512e8i 1.26228i 0.775669 + 0.631139i \(0.217412\pi\)
−0.775669 + 0.631139i \(0.782588\pi\)
\(522\) 0 0
\(523\) 1.67427e8i 1.17036i −0.810902 0.585181i \(-0.801023\pi\)
0.810902 0.585181i \(-0.198977\pi\)
\(524\) 0 0
\(525\) 1.00216e8i 0.692562i
\(526\) 0 0
\(527\) 4.10607e7i 0.280540i
\(528\) 0 0
\(529\) −1.47644e8 −0.997351
\(530\) 0 0
\(531\) 1.32686e7i 0.0886223i
\(532\) 0 0
\(533\) −4.51679e8 −2.98297
\(534\) 0 0
\(535\) 6.32649e7i 0.413144i
\(536\) 0 0
\(537\) 2.93266e8 1.89382
\(538\) 0 0
\(539\) −3.50626e8 −2.23912
\(540\) 0 0
\(541\) −1.01782e8 −0.642808 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(542\) 0 0
\(543\) −1.16381e8 −0.726911
\(544\) 0 0
\(545\) 9.79264e7i 0.604937i
\(546\) 0 0
\(547\) 2.32682e8i 1.42168i 0.703354 + 0.710840i \(0.251685\pi\)
−0.703354 + 0.710840i \(0.748315\pi\)
\(548\) 0 0
\(549\) 2.56641e8 1.55099
\(550\) 0 0
\(551\) 9.87788e7 + 4.29750e7i 0.590485 + 0.256898i
\(552\) 0 0
\(553\) 2.42672e8i 1.43497i
\(554\) 0 0
\(555\) 1.93470e8 1.13171
\(556\) 0 0
\(557\) −9.88543e7 −0.572045 −0.286022 0.958223i \(-0.592333\pi\)
−0.286022 + 0.958223i \(0.592333\pi\)
\(558\) 0 0
\(559\) 5.02957e7i 0.287936i
\(560\) 0 0
\(561\) 1.44461e8i 0.818206i
\(562\) 0 0
\(563\) 2.96849e8i 1.66345i 0.555187 + 0.831726i \(0.312647\pi\)
−0.555187 + 0.831726i \(0.687353\pi\)
\(564\) 0 0
\(565\) 2.69348e7i 0.149338i
\(566\) 0 0
\(567\) 2.73363e8 1.49965
\(568\) 0 0
\(569\) 2.25877e8i 1.22613i 0.790034 + 0.613063i \(0.210063\pi\)
−0.790034 + 0.613063i \(0.789937\pi\)
\(570\) 0 0
\(571\) −1.34657e8 −0.723303 −0.361652 0.932313i \(-0.617787\pi\)
−0.361652 + 0.932313i \(0.617787\pi\)
\(572\) 0 0
\(573\) 3.85576e8i 2.04949i
\(574\) 0 0
\(575\) −3.20899e6 −0.0168797
\(576\) 0 0
\(577\) 1.22386e8 0.637097 0.318549 0.947907i \(-0.396805\pi\)
0.318549 + 0.947907i \(0.396805\pi\)
\(578\) 0 0
\(579\) −9.82631e7 −0.506238
\(580\) 0 0
\(581\) 6.51298e7 0.332087
\(582\) 0 0
\(583\) 4.94328e8i 2.49465i
\(584\) 0 0
\(585\) 2.80770e8i 1.40244i
\(586\) 0 0
\(587\) −3.54883e8 −1.75457 −0.877285 0.479970i \(-0.840647\pi\)
−0.877285 + 0.479970i \(0.840647\pi\)
\(588\) 0 0
\(589\) 1.64785e8 + 7.16918e7i 0.806439 + 0.350852i
\(590\) 0 0
\(591\) 1.27160e7i 0.0616009i
\(592\) 0 0
\(593\) 3.36642e8 1.61437 0.807186 0.590297i \(-0.200989\pi\)
0.807186 + 0.590297i \(0.200989\pi\)
\(594\) 0 0
\(595\) −8.23228e7 −0.390813
\(596\) 0 0
\(597\) 3.63417e8i 1.70798i
\(598\) 0 0
\(599\) 5.74144e7i 0.267141i 0.991039 + 0.133571i \(0.0426443\pi\)
−0.991039 + 0.133571i \(0.957356\pi\)
\(600\) 0 0
\(601\) 1.19611e7i 0.0550994i 0.999620 + 0.0275497i \(0.00877045\pi\)
−0.999620 + 0.0275497i \(0.991230\pi\)
\(602\) 0 0
\(603\) 3.77444e8i 1.72147i
\(604\) 0 0
\(605\) 4.16665e8 1.88157
\(606\) 0 0
\(607\) 2.53084e8i 1.13161i 0.824538 + 0.565807i \(0.191435\pi\)
−0.824538 + 0.565807i \(0.808565\pi\)
\(608\) 0 0
\(609\) −3.07138e8 −1.35982
\(610\) 0 0
\(611\) 7.02813e8i 3.08117i
\(612\) 0 0
\(613\) 4.80487e7 0.208593 0.104297 0.994546i \(-0.466741\pi\)
0.104297 + 0.994546i \(0.466741\pi\)
\(614\) 0 0
\(615\) −4.68179e8 −2.01273
\(616\) 0 0
\(617\) −1.50367e8 −0.640174 −0.320087 0.947388i \(-0.603712\pi\)
−0.320087 + 0.947388i \(0.603712\pi\)
\(618\) 0 0
\(619\) 2.30454e8 0.971657 0.485828 0.874054i \(-0.338518\pi\)
0.485828 + 0.874054i \(0.338518\pi\)
\(620\) 0 0
\(621\) 60380.5i 0.000252128i
\(622\) 0 0
\(623\) 3.11264e8i 1.28726i
\(624\) 0 0
\(625\) −1.37811e8 −0.564472
\(626\) 0 0
\(627\) 5.79751e8 + 2.52228e8i 2.35201 + 1.02327i
\(628\) 0 0
\(629\) 7.75595e7i 0.311661i
\(630\) 0 0
\(631\) 2.64740e8 1.05374 0.526868 0.849947i \(-0.323366\pi\)
0.526868 + 0.849947i \(0.323366\pi\)
\(632\) 0 0
\(633\) −3.00304e8 −1.18399
\(634\) 0 0
\(635\) 9.56540e7i 0.373579i
\(636\) 0 0
\(637\) 5.47331e8i 2.11754i
\(638\) 0 0
\(639\) 1.94751e8i 0.746408i
\(640\) 0 0
\(641\) 2.39436e8i 0.909108i −0.890719 0.454554i \(-0.849799\pi\)
0.890719 0.454554i \(-0.150201\pi\)
\(642\) 0 0
\(643\) −3.28913e8 −1.23722 −0.618611 0.785697i \(-0.712304\pi\)
−0.618611 + 0.785697i \(0.712304\pi\)
\(644\) 0 0
\(645\) 5.21330e7i 0.194282i
\(646\) 0 0
\(647\) 2.24926e8 0.830476 0.415238 0.909713i \(-0.363698\pi\)
0.415238 + 0.909713i \(0.363698\pi\)
\(648\) 0 0
\(649\) 4.41294e7i 0.161434i
\(650\) 0 0
\(651\) −5.12373e8 −1.85714
\(652\) 0 0
\(653\) 2.25975e8 0.811560 0.405780 0.913971i \(-0.367000\pi\)
0.405780 + 0.913971i \(0.367000\pi\)
\(654\) 0 0
\(655\) 1.50000e8 0.533786
\(656\) 0 0
\(657\) −4.25637e7 −0.150087
\(658\) 0 0
\(659\) 3.30068e8i 1.15331i −0.816987 0.576656i \(-0.804357\pi\)
0.816987 0.576656i \(-0.195643\pi\)
\(660\) 0 0
\(661\) 3.67503e8i 1.27250i −0.771484 0.636248i \(-0.780485\pi\)
0.771484 0.636248i \(-0.219515\pi\)
\(662\) 0 0
\(663\) −2.25505e8 −0.773777
\(664\) 0 0
\(665\) −1.43735e8 + 3.30378e8i −0.488762 + 1.12343i
\(666\) 0 0
\(667\) 9.83477e6i 0.0331426i
\(668\) 0 0
\(669\) −5.61815e8 −1.87635
\(670\) 0 0
\(671\) 8.53547e8 2.82527
\(672\) 0 0
\(673\) 3.21489e8i 1.05468i −0.849654 0.527340i \(-0.823189\pi\)
0.849654 0.527340i \(-0.176811\pi\)
\(674\) 0 0
\(675\) 494112.i 0.00160662i
\(676\) 0 0
\(677\) 3.83480e8i 1.23588i 0.786225 + 0.617941i \(0.212033\pi\)
−0.786225 + 0.617941i \(0.787967\pi\)
\(678\) 0 0
\(679\) 1.44656e8i 0.462090i
\(680\) 0 0
\(681\) 2.86114e8 0.905936
\(682\) 0 0
\(683\) 8.83834e7i 0.277401i −0.990334 0.138701i \(-0.955707\pi\)
0.990334 0.138701i \(-0.0442926\pi\)
\(684\) 0 0
\(685\) 1.10599e8 0.344095
\(686\) 0 0
\(687\) 2.17025e8i 0.669328i
\(688\) 0 0
\(689\) 7.71651e8 2.35919
\(690\) 0 0
\(691\) 1.24144e8 0.376262 0.188131 0.982144i \(-0.439757\pi\)
0.188131 + 0.982144i \(0.439757\pi\)
\(692\) 0 0
\(693\) −8.99759e8 −2.70350
\(694\) 0 0
\(695\) −2.86061e8 −0.852127
\(696\) 0 0
\(697\) 1.87686e8i 0.554286i
\(698\) 0 0
\(699\) 7.18637e8i 2.10416i
\(700\) 0 0
\(701\) 2.26176e8 0.656588 0.328294 0.944576i \(-0.393526\pi\)
0.328294 + 0.944576i \(0.393526\pi\)
\(702\) 0 0
\(703\) 3.11261e8 + 1.35418e8i 0.895900 + 0.389773i
\(704\) 0 0
\(705\) 7.28486e8i 2.07900i
\(706\) 0 0
\(707\) −2.27162e8 −0.642802
\(708\) 0 0
\(709\) 4.83198e8 1.35577 0.677886 0.735167i \(-0.262896\pi\)
0.677886 + 0.735167i \(0.262896\pi\)
\(710\) 0 0
\(711\) 3.43916e8i 0.956851i
\(712\) 0 0
\(713\) 1.64066e7i 0.0452636i
\(714\) 0 0
\(715\) 9.33797e8i 2.55467i
\(716\) 0 0
\(717\) 3.18311e8i 0.863563i
\(718\) 0 0
\(719\) −3.11946e8 −0.839253 −0.419627 0.907697i \(-0.637839\pi\)
−0.419627 + 0.907697i \(0.637839\pi\)
\(720\) 0 0
\(721\) 4.28563e8i 1.14343i
\(722\) 0 0
\(723\) 4.02631e8 1.06535
\(724\) 0 0
\(725\) 8.04811e7i 0.211193i
\(726\) 0 0
\(727\) 1.30234e8 0.338938 0.169469 0.985535i \(-0.445795\pi\)
0.169469 + 0.985535i \(0.445795\pi\)
\(728\) 0 0
\(729\) 3.84730e8 0.993054
\(730\) 0 0
\(731\) 2.08994e7 0.0535034
\(732\) 0 0
\(733\) 1.90792e8 0.484448 0.242224 0.970220i \(-0.422123\pi\)
0.242224 + 0.970220i \(0.422123\pi\)
\(734\) 0 0
\(735\) 5.67324e8i 1.42879i
\(736\) 0 0
\(737\) 1.25532e9i 3.13582i
\(738\) 0 0
\(739\) −4.48582e8 −1.11150 −0.555749 0.831350i \(-0.687568\pi\)
−0.555749 + 0.831350i \(0.687568\pi\)
\(740\) 0 0
\(741\) −3.93731e8 + 9.04997e8i −0.967708 + 2.22429i
\(742\) 0 0
\(743\) 2.16076e8i 0.526794i −0.964687 0.263397i \(-0.915157\pi\)
0.964687 0.263397i \(-0.0848430\pi\)
\(744\) 0 0
\(745\) −1.51067e8 −0.365344
\(746\) 0 0
\(747\) 9.23024e7 0.221438
\(748\) 0 0
\(749\) 3.16478e8i 0.753179i
\(750\) 0 0
\(751\) 1.56034e8i 0.368383i 0.982890 + 0.184191i \(0.0589667\pi\)
−0.982890 + 0.184191i \(0.941033\pi\)
\(752\) 0 0
\(753\) 7.31994e7i 0.171444i
\(754\) 0 0
\(755\) 3.79806e8i 0.882512i
\(756\) 0 0
\(757\) −4.85982e8 −1.12030 −0.560148 0.828393i \(-0.689256\pi\)
−0.560148 + 0.828393i \(0.689256\pi\)
\(758\) 0 0
\(759\) 5.77221e7i 0.132013i
\(760\) 0 0
\(761\) −4.11783e8 −0.934360 −0.467180 0.884162i \(-0.654730\pi\)
−0.467180 + 0.884162i \(0.654730\pi\)
\(762\) 0 0
\(763\) 4.89869e8i 1.10282i
\(764\) 0 0
\(765\) −1.16668e8 −0.260597
\(766\) 0 0
\(767\) 6.88864e7 0.152668
\(768\) 0 0
\(769\) 2.85308e7 0.0627386 0.0313693 0.999508i \(-0.490013\pi\)
0.0313693 + 0.999508i \(0.490013\pi\)
\(770\) 0 0
\(771\) −6.20313e8 −1.35347
\(772\) 0 0
\(773\) 6.67394e8i 1.44492i −0.691413 0.722460i \(-0.743011\pi\)
0.691413 0.722460i \(-0.256989\pi\)
\(774\) 0 0
\(775\) 1.34260e8i 0.288431i
\(776\) 0 0
\(777\) −9.67820e8 −2.06315
\(778\) 0 0
\(779\) −7.53221e8 3.27699e8i −1.59335 0.693206i
\(780\) 0 0
\(781\) 6.47709e8i 1.35965i
\(782\) 0 0
\(783\) 1.51433e6 0.00315454
\(784\) 0 0
\(785\) 5.30087e7 0.109582
\(786\) 0 0
\(787\) 3.31663e8i 0.680413i −0.940351 0.340206i \(-0.889503\pi\)
0.940351 0.340206i \(-0.110497\pi\)
\(788\) 0 0
\(789\) 2.54324e8i 0.517793i
\(790\) 0 0
\(791\) 1.34739e8i 0.272248i
\(792\) 0 0
\(793\) 1.33240e9i 2.67186i
\(794\) 0 0
\(795\) 7.99839e8 1.59185
\(796\) 0 0