Properties

Label 304.7.e.c.113.1
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 113.1
Root \(-42.5965i\) of defining polynomial
Character \(\chi\) \(=\) 304.113
Dual form 304.7.e.c.113.8

$q$-expansion

\(f(q)\) \(=\) \(q-42.5965i q^{3} +103.439 q^{5} +619.422 q^{7} -1085.46 q^{9} +O(q^{10})\) \(q-42.5965i q^{3} +103.439 q^{5} +619.422 q^{7} -1085.46 q^{9} +57.6072 q^{11} +481.215i q^{13} -4406.15i q^{15} -2339.07 q^{17} +(-4359.73 + 5295.16i) q^{19} -26385.2i q^{21} +8694.38 q^{23} -4925.35 q^{25} +15184.1i q^{27} -46018.1i q^{29} -36728.1i q^{31} -2453.87i q^{33} +64072.4 q^{35} +40458.4i q^{37} +20498.1 q^{39} -61518.9i q^{41} +64375.2 q^{43} -112279. q^{45} +93996.5 q^{47} +266034. q^{49} +99636.2i q^{51} -97805.1i q^{53} +5958.83 q^{55} +(225555. + 185709. i) q^{57} -132484. i q^{59} -1921.32 q^{61} -672360. q^{63} +49776.4i q^{65} -378794. i q^{67} -370350. i q^{69} -451799. i q^{71} -669896. q^{73} +209803. i q^{75} +35683.1 q^{77} -747688. i q^{79} -144512. q^{81} +495260. q^{83} -241951. q^{85} -1.96021e6 q^{87} +1.35076e6i q^{89} +298075. i q^{91} -1.56449e6 q^{93} +(-450966. + 547726. i) q^{95} -75990.3i q^{97} -62530.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{5} - 362q^{7} - 4348q^{9} + O(q^{10}) \) \( 8q + 2q^{5} - 362q^{7} - 4348q^{9} - 902q^{11} + 1550q^{17} - 6232q^{19} + 18820q^{23} - 12158q^{25} + 101762q^{35} - 167028q^{39} + 335042q^{43} - 57230q^{45} + 570394q^{47} + 448182q^{49} - 1089198q^{55} + 341316q^{57} - 632014q^{61} - 328174q^{63} - 852938q^{73} + 1850530q^{77} - 1819456q^{81} - 441200q^{83} - 1828374q^{85} - 1483380q^{87} + 2131176q^{93} - 627950q^{95} + 865394q^{99} + O(q^{100}) \)

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
\(3\) 42.5965i 1.57765i −0.614618 0.788825i \(-0.710690\pi\)
0.614618 0.788825i \(-0.289310\pi\)
\(4\) 0 0
\(5\) 103.439 0.827513 0.413756 0.910388i \(-0.364217\pi\)
0.413756 + 0.910388i \(0.364217\pi\)
\(6\) 0 0
\(7\) 619.422 1.80589 0.902947 0.429752i \(-0.141399\pi\)
0.902947 + 0.429752i \(0.141399\pi\)
\(8\) 0 0
\(9\) −1085.46 −1.48898
\(10\) 0 0
\(11\) 57.6072 0.0432811 0.0216406 0.999766i \(-0.493111\pi\)
0.0216406 + 0.999766i \(0.493111\pi\)
\(12\) 0 0
\(13\) 481.215i 0.219033i 0.993985 + 0.109516i \(0.0349302\pi\)
−0.993985 + 0.109516i \(0.965070\pi\)
\(14\) 0 0
\(15\) 4406.15i 1.30552i
\(16\) 0 0
\(17\) −2339.07 −0.476098 −0.238049 0.971253i \(-0.576508\pi\)
−0.238049 + 0.971253i \(0.576508\pi\)
\(18\) 0 0
\(19\) −4359.73 + 5295.16i −0.635621 + 0.772001i
\(20\) 0 0
\(21\) 26385.2i 2.84907i
\(22\) 0 0
\(23\) 8694.38 0.714587 0.357293 0.933992i \(-0.383700\pi\)
0.357293 + 0.933992i \(0.383700\pi\)
\(24\) 0 0
\(25\) −4925.35 −0.315222
\(26\) 0 0
\(27\) 15184.1i 0.771434i
\(28\) 0 0
\(29\) 46018.1i 1.88684i −0.331603 0.943419i \(-0.607590\pi\)
0.331603 0.943419i \(-0.392410\pi\)
\(30\) 0 0
\(31\) 36728.1i 1.23286i −0.787410 0.616430i \(-0.788579\pi\)
0.787410 0.616430i \(-0.211421\pi\)
\(32\) 0 0
\(33\) 2453.87i 0.0682824i
\(34\) 0 0
\(35\) 64072.4 1.49440
\(36\) 0 0
\(37\) 40458.4i 0.798737i 0.916791 + 0.399368i \(0.130771\pi\)
−0.916791 + 0.399368i \(0.869229\pi\)
\(38\) 0 0
\(39\) 20498.1 0.345557
\(40\) 0 0
\(41\) 61518.9i 0.892600i −0.894883 0.446300i \(-0.852741\pi\)
0.894883 0.446300i \(-0.147259\pi\)
\(42\) 0 0
\(43\) 64375.2 0.809680 0.404840 0.914388i \(-0.367327\pi\)
0.404840 + 0.914388i \(0.367327\pi\)
\(44\) 0 0
\(45\) −112279. −1.23215
\(46\) 0 0
\(47\) 93996.5 0.905353 0.452676 0.891675i \(-0.350469\pi\)
0.452676 + 0.891675i \(0.350469\pi\)
\(48\) 0 0
\(49\) 266034. 2.26125
\(50\) 0 0
\(51\) 99636.2i 0.751115i
\(52\) 0 0
\(53\) 97805.1i 0.656952i −0.944512 0.328476i \(-0.893465\pi\)
0.944512 0.328476i \(-0.106535\pi\)
\(54\) 0 0
\(55\) 5958.83 0.0358157
\(56\) 0 0
\(57\) 225555. + 185709.i 1.21795 + 1.00279i
\(58\) 0 0
\(59\) 132484.i 0.645070i −0.946558 0.322535i \(-0.895465\pi\)
0.946558 0.322535i \(-0.104535\pi\)
\(60\) 0 0
\(61\) −1921.32 −0.00846469 −0.00423235 0.999991i \(-0.501347\pi\)
−0.00423235 + 0.999991i \(0.501347\pi\)
\(62\) 0 0
\(63\) −672360. −2.68893
\(64\) 0 0
\(65\) 49776.4i 0.181252i
\(66\) 0 0
\(67\) 378794.i 1.25944i −0.776821 0.629721i \(-0.783169\pi\)
0.776821 0.629721i \(-0.216831\pi\)
\(68\) 0 0
\(69\) 370350.i 1.12737i
\(70\) 0 0
\(71\) 451799.i 1.26232i −0.775652 0.631161i \(-0.782579\pi\)
0.775652 0.631161i \(-0.217421\pi\)
\(72\) 0 0
\(73\) −669896. −1.72202 −0.861011 0.508587i \(-0.830168\pi\)
−0.861011 + 0.508587i \(0.830168\pi\)
\(74\) 0 0
\(75\) 209803.i 0.497310i
\(76\) 0 0
\(77\) 35683.1 0.0781611
\(78\) 0 0
\(79\) 747688.i 1.51649i −0.651971 0.758244i \(-0.726058\pi\)
0.651971 0.758244i \(-0.273942\pi\)
\(80\) 0 0
\(81\) −144512. −0.271925
\(82\) 0 0
\(83\) 495260. 0.866161 0.433081 0.901355i \(-0.357427\pi\)
0.433081 + 0.901355i \(0.357427\pi\)
\(84\) 0 0
\(85\) −241951. −0.393977
\(86\) 0 0
\(87\) −1.96021e6 −2.97677
\(88\) 0 0
\(89\) 1.35076e6i 1.91606i 0.286672 + 0.958029i \(0.407451\pi\)
−0.286672 + 0.958029i \(0.592549\pi\)
\(90\) 0 0
\(91\) 298075.i 0.395550i
\(92\) 0 0
\(93\) −1.56449e6 −1.94502
\(94\) 0 0
\(95\) −450966. + 547726.i −0.525985 + 0.638841i
\(96\) 0 0
\(97\) 75990.3i 0.0832612i −0.999133 0.0416306i \(-0.986745\pi\)
0.999133 0.0416306i \(-0.0132553\pi\)
\(98\) 0 0
\(99\) −62530.5 −0.0644446
\(100\) 0 0
\(101\) 1.08825e6 1.05624 0.528121 0.849169i \(-0.322897\pi\)
0.528121 + 0.849169i \(0.322897\pi\)
\(102\) 0 0
\(103\) 1.69273e6i 1.54909i 0.632518 + 0.774545i \(0.282021\pi\)
−0.632518 + 0.774545i \(0.717979\pi\)
\(104\) 0 0
\(105\) 2.72926e6i 2.35764i
\(106\) 0 0
\(107\) 624544.i 0.509814i −0.966966 0.254907i \(-0.917955\pi\)
0.966966 0.254907i \(-0.0820448\pi\)
\(108\) 0 0
\(109\) 1.40732e6i 1.08671i −0.839503 0.543356i \(-0.817154\pi\)
0.839503 0.543356i \(-0.182846\pi\)
\(110\) 0 0
\(111\) 1.72339e6 1.26013
\(112\) 0 0
\(113\) 1.11788e6i 0.774749i 0.921922 + 0.387375i \(0.126618\pi\)
−0.921922 + 0.387375i \(0.873382\pi\)
\(114\) 0 0
\(115\) 899339. 0.591330
\(116\) 0 0
\(117\) 522341.i 0.326135i
\(118\) 0 0
\(119\) −1.44887e6 −0.859782
\(120\) 0 0
\(121\) −1.76824e6 −0.998127
\(122\) 0 0
\(123\) −2.62049e6 −1.40821
\(124\) 0 0
\(125\) −2.12571e6 −1.08836
\(126\) 0 0
\(127\) 2.06130e6i 1.00631i −0.864198 0.503153i \(-0.832173\pi\)
0.864198 0.503153i \(-0.167827\pi\)
\(128\) 0 0
\(129\) 2.74216e6i 1.27739i
\(130\) 0 0
\(131\) 2.33564e6 1.03894 0.519471 0.854488i \(-0.326129\pi\)
0.519471 + 0.854488i \(0.326129\pi\)
\(132\) 0 0
\(133\) −2.70051e6 + 3.27993e6i −1.14786 + 1.39415i
\(134\) 0 0
\(135\) 1.57063e6i 0.638371i
\(136\) 0 0
\(137\) −3.26330e6 −1.26910 −0.634550 0.772882i \(-0.718815\pi\)
−0.634550 + 0.772882i \(0.718815\pi\)
\(138\) 0 0
\(139\) −114357. −0.0425813 −0.0212907 0.999773i \(-0.506778\pi\)
−0.0212907 + 0.999773i \(0.506778\pi\)
\(140\) 0 0
\(141\) 4.00392e6i 1.42833i
\(142\) 0 0
\(143\) 27721.4i 0.00947998i
\(144\) 0 0
\(145\) 4.76007e6i 1.56138i
\(146\) 0 0
\(147\) 1.13321e7i 3.56746i
\(148\) 0 0
\(149\) 4.12978e6 1.24844 0.624221 0.781248i \(-0.285417\pi\)
0.624221 + 0.781248i \(0.285417\pi\)
\(150\) 0 0
\(151\) 504349.i 0.146487i 0.997314 + 0.0732437i \(0.0233351\pi\)
−0.997314 + 0.0732437i \(0.976665\pi\)
\(152\) 0 0
\(153\) 2.53898e6 0.708899
\(154\) 0 0
\(155\) 3.79912e6i 1.02021i
\(156\) 0 0
\(157\) 3.09258e6 0.799138 0.399569 0.916703i \(-0.369160\pi\)
0.399569 + 0.916703i \(0.369160\pi\)
\(158\) 0 0
\(159\) −4.16616e6 −1.03644
\(160\) 0 0
\(161\) 5.38549e6 1.29047
\(162\) 0 0
\(163\) 587796. 0.135726 0.0678632 0.997695i \(-0.478382\pi\)
0.0678632 + 0.997695i \(0.478382\pi\)
\(164\) 0 0
\(165\) 253826.i 0.0565046i
\(166\) 0 0
\(167\) 8.10037e6i 1.73922i 0.493736 + 0.869612i \(0.335631\pi\)
−0.493736 + 0.869612i \(0.664369\pi\)
\(168\) 0 0
\(169\) 4.59524e6 0.952025
\(170\) 0 0
\(171\) 4.73233e6 5.74770e6i 0.946425 1.14949i
\(172\) 0 0
\(173\) 1.92729e6i 0.372229i 0.982528 + 0.186114i \(0.0595895\pi\)
−0.982528 + 0.186114i \(0.940411\pi\)
\(174\) 0 0
\(175\) −3.05087e6 −0.569258
\(176\) 0 0
\(177\) −5.64335e6 −1.01769
\(178\) 0 0
\(179\) 4.82927e6i 0.842020i 0.907056 + 0.421010i \(0.138324\pi\)
−0.907056 + 0.421010i \(0.861676\pi\)
\(180\) 0 0
\(181\) 3.36727e6i 0.567862i −0.958845 0.283931i \(-0.908361\pi\)
0.958845 0.283931i \(-0.0916387\pi\)
\(182\) 0 0
\(183\) 81841.7i 0.0133543i
\(184\) 0 0
\(185\) 4.18498e6i 0.660965i
\(186\) 0 0
\(187\) −134747. −0.0206061
\(188\) 0 0
\(189\) 9.40538e6i 1.39313i
\(190\) 0 0
\(191\) −5.65880e6 −0.812128 −0.406064 0.913845i \(-0.633099\pi\)
−0.406064 + 0.913845i \(0.633099\pi\)
\(192\) 0 0
\(193\) 1.02483e7i 1.42554i 0.701400 + 0.712768i \(0.252559\pi\)
−0.701400 + 0.712768i \(0.747441\pi\)
\(194\) 0 0
\(195\) 2.12030e6 0.285953
\(196\) 0 0
\(197\) 1.47915e7 1.93470 0.967351 0.253441i \(-0.0815622\pi\)
0.967351 + 0.253441i \(0.0815622\pi\)
\(198\) 0 0
\(199\) 4.92464e6 0.624907 0.312453 0.949933i \(-0.398849\pi\)
0.312453 + 0.949933i \(0.398849\pi\)
\(200\) 0 0
\(201\) −1.61353e7 −1.98696
\(202\) 0 0
\(203\) 2.85046e7i 3.40743i
\(204\) 0 0
\(205\) 6.36346e6i 0.738638i
\(206\) 0 0
\(207\) −9.43744e6 −1.06400
\(208\) 0 0
\(209\) −251152. + 305039.i −0.0275104 + 0.0334131i
\(210\) 0 0
\(211\) 997042.i 0.106137i −0.998591 0.0530684i \(-0.983100\pi\)
0.998591 0.0530684i \(-0.0169001\pi\)
\(212\) 0 0
\(213\) −1.92451e7 −1.99150
\(214\) 0 0
\(215\) 6.65892e6 0.670021
\(216\) 0 0
\(217\) 2.27502e7i 2.22641i
\(218\) 0 0
\(219\) 2.85352e7i 2.71675i
\(220\) 0 0
\(221\) 1.12559e6i 0.104281i
\(222\) 0 0
\(223\) 1.24527e7i 1.12292i 0.827502 + 0.561462i \(0.189761\pi\)
−0.827502 + 0.561462i \(0.810239\pi\)
\(224\) 0 0
\(225\) 5.34629e6 0.469359
\(226\) 0 0
\(227\) 7.22930e6i 0.618043i 0.951055 + 0.309022i \(0.100002\pi\)
−0.951055 + 0.309022i \(0.899998\pi\)
\(228\) 0 0
\(229\) 4.79865e6 0.399588 0.199794 0.979838i \(-0.435973\pi\)
0.199794 + 0.979838i \(0.435973\pi\)
\(230\) 0 0
\(231\) 1.51998e6i 0.123311i
\(232\) 0 0
\(233\) −2.55746e6 −0.202181 −0.101091 0.994877i \(-0.532233\pi\)
−0.101091 + 0.994877i \(0.532233\pi\)
\(234\) 0 0
\(235\) 9.72291e6 0.749191
\(236\) 0 0
\(237\) −3.18489e7 −2.39249
\(238\) 0 0
\(239\) −2.36214e7 −1.73026 −0.865132 0.501544i \(-0.832765\pi\)
−0.865132 + 0.501544i \(0.832765\pi\)
\(240\) 0 0
\(241\) 1.86092e7i 1.32946i 0.747082 + 0.664732i \(0.231454\pi\)
−0.747082 + 0.664732i \(0.768546\pi\)
\(242\) 0 0
\(243\) 1.72249e7i 1.20044i
\(244\) 0 0
\(245\) 2.75183e7 1.87122
\(246\) 0 0
\(247\) −2.54811e6 2.09796e6i −0.169093 0.139222i
\(248\) 0 0
\(249\) 2.10963e7i 1.36650i
\(250\) 0 0
\(251\) −1.87813e7 −1.18769 −0.593846 0.804578i \(-0.702391\pi\)
−0.593846 + 0.804578i \(0.702391\pi\)
\(252\) 0 0
\(253\) 500859. 0.0309281
\(254\) 0 0
\(255\) 1.03063e7i 0.621558i
\(256\) 0 0
\(257\) 1.43001e7i 0.842441i 0.906958 + 0.421220i \(0.138398\pi\)
−0.906958 + 0.421220i \(0.861602\pi\)
\(258\) 0 0
\(259\) 2.50608e7i 1.44243i
\(260\) 0 0
\(261\) 4.99510e7i 2.80946i
\(262\) 0 0
\(263\) 2.68886e7 1.47809 0.739044 0.673657i \(-0.235278\pi\)
0.739044 + 0.673657i \(0.235278\pi\)
\(264\) 0 0
\(265\) 1.01169e7i 0.543636i
\(266\) 0 0
\(267\) 5.75377e7 3.02287
\(268\) 0 0
\(269\) 1.30414e7i 0.669986i 0.942221 + 0.334993i \(0.108734\pi\)
−0.942221 + 0.334993i \(0.891266\pi\)
\(270\) 0 0
\(271\) 1.69825e7 0.853284 0.426642 0.904421i \(-0.359697\pi\)
0.426642 + 0.904421i \(0.359697\pi\)
\(272\) 0 0
\(273\) 1.26970e7 0.624039
\(274\) 0 0
\(275\) −283736. −0.0136432
\(276\) 0 0
\(277\) 6.95844e6 0.327395 0.163698 0.986511i \(-0.447658\pi\)
0.163698 + 0.986511i \(0.447658\pi\)
\(278\) 0 0
\(279\) 3.98670e7i 1.83570i
\(280\) 0 0
\(281\) 7.61455e6i 0.343183i −0.985168 0.171591i \(-0.945109\pi\)
0.985168 0.171591i \(-0.0548908\pi\)
\(282\) 0 0
\(283\) 1.19767e7 0.528418 0.264209 0.964465i \(-0.414889\pi\)
0.264209 + 0.964465i \(0.414889\pi\)
\(284\) 0 0
\(285\) 2.33312e7 + 1.92096e7i 1.00787 + 0.829819i
\(286\) 0 0
\(287\) 3.81061e7i 1.61194i
\(288\) 0 0
\(289\) −1.86663e7 −0.773331
\(290\) 0 0
\(291\) −3.23692e6 −0.131357
\(292\) 0 0
\(293\) 6.84558e6i 0.272149i 0.990699 + 0.136075i \(0.0434487\pi\)
−0.990699 + 0.136075i \(0.956551\pi\)
\(294\) 0 0
\(295\) 1.37040e7i 0.533804i
\(296\) 0 0
\(297\) 874715.i 0.0333885i
\(298\) 0 0
\(299\) 4.18386e6i 0.156518i
\(300\) 0 0
\(301\) 3.98754e7 1.46220
\(302\) 0 0
\(303\) 4.63555e7i 1.66638i
\(304\) 0 0
\(305\) −198740. −0.00700464
\(306\) 0 0
\(307\) 1.30979e7i 0.452675i 0.974049 + 0.226338i \(0.0726753\pi\)
−0.974049 + 0.226338i \(0.927325\pi\)
\(308\) 0 0
\(309\) 7.21046e7 2.44392
\(310\) 0 0
\(311\) 3.13128e6 0.104097 0.0520487 0.998645i \(-0.483425\pi\)
0.0520487 + 0.998645i \(0.483425\pi\)
\(312\) 0 0
\(313\) −2.85162e7 −0.929948 −0.464974 0.885324i \(-0.653936\pi\)
−0.464974 + 0.885324i \(0.653936\pi\)
\(314\) 0 0
\(315\) −6.95483e7 −2.22513
\(316\) 0 0
\(317\) 2.08011e6i 0.0652993i 0.999467 + 0.0326497i \(0.0103946\pi\)
−0.999467 + 0.0326497i \(0.989605\pi\)
\(318\) 0 0
\(319\) 2.65097e6i 0.0816645i
\(320\) 0 0
\(321\) −2.66034e7 −0.804307
\(322\) 0 0
\(323\) 1.01977e7 1.23857e7i 0.302618 0.367548i
\(324\) 0 0
\(325\) 2.37015e6i 0.0690440i
\(326\) 0 0
\(327\) −5.99471e7 −1.71445
\(328\) 0 0
\(329\) 5.82234e7 1.63497
\(330\) 0 0
\(331\) 8.68996e6i 0.239626i −0.992796 0.119813i \(-0.961770\pi\)
0.992796 0.119813i \(-0.0382295\pi\)
\(332\) 0 0
\(333\) 4.39161e7i 1.18930i
\(334\) 0 0
\(335\) 3.91821e7i 1.04220i
\(336\) 0 0
\(337\) 1.62302e7i 0.424066i −0.977262 0.212033i \(-0.931992\pi\)
0.977262 0.212033i \(-0.0680085\pi\)
\(338\) 0 0
\(339\) 4.76179e7 1.22228
\(340\) 0 0
\(341\) 2.11580e6i 0.0533595i
\(342\) 0 0
\(343\) 9.19130e7 2.27769
\(344\) 0 0
\(345\) 3.83087e7i 0.932911i
\(346\) 0 0
\(347\) −6.91081e7 −1.65402 −0.827010 0.562188i \(-0.809960\pi\)
−0.827010 + 0.562188i \(0.809960\pi\)
\(348\) 0 0
\(349\) 1.74773e7 0.411148 0.205574 0.978642i \(-0.434094\pi\)
0.205574 + 0.978642i \(0.434094\pi\)
\(350\) 0 0
\(351\) −7.30683e6 −0.168969
\(352\) 0 0
\(353\) −2.02764e7 −0.460964 −0.230482 0.973077i \(-0.574030\pi\)
−0.230482 + 0.973077i \(0.574030\pi\)
\(354\) 0 0
\(355\) 4.67337e7i 1.04459i
\(356\) 0 0
\(357\) 6.17168e7i 1.35643i
\(358\) 0 0
\(359\) 6.66182e7 1.43982 0.719912 0.694065i \(-0.244182\pi\)
0.719912 + 0.694065i \(0.244182\pi\)
\(360\) 0 0
\(361\) −9.03146e6 4.61709e7i −0.191971 0.981401i
\(362\) 0 0
\(363\) 7.53210e7i 1.57469i
\(364\) 0 0
\(365\) −6.92934e7 −1.42499
\(366\) 0 0
\(367\) 2.30816e7 0.466948 0.233474 0.972363i \(-0.424991\pi\)
0.233474 + 0.972363i \(0.424991\pi\)
\(368\) 0 0
\(369\) 6.67766e7i 1.32906i
\(370\) 0 0
\(371\) 6.05826e7i 1.18639i
\(372\) 0 0
\(373\) 8.39871e7i 1.61840i 0.587532 + 0.809201i \(0.300100\pi\)
−0.587532 + 0.809201i \(0.699900\pi\)
\(374\) 0 0
\(375\) 9.05479e7i 1.71706i
\(376\) 0 0
\(377\) 2.21446e7 0.413279
\(378\) 0 0
\(379\) 9.57993e7i 1.75973i 0.475228 + 0.879863i \(0.342366\pi\)
−0.475228 + 0.879863i \(0.657634\pi\)
\(380\) 0 0
\(381\) −8.78042e7 −1.58760
\(382\) 0 0
\(383\) 5.84087e7i 1.03964i −0.854277 0.519818i \(-0.826000\pi\)
0.854277 0.519818i \(-0.174000\pi\)
\(384\) 0 0
\(385\) 3.69103e6 0.0646793
\(386\) 0 0
\(387\) −6.98770e7 −1.20559
\(388\) 0 0
\(389\) 4.66058e7 0.791755 0.395878 0.918303i \(-0.370440\pi\)
0.395878 + 0.918303i \(0.370440\pi\)
\(390\) 0 0
\(391\) −2.03368e7 −0.340213
\(392\) 0 0
\(393\) 9.94900e7i 1.63909i
\(394\) 0 0
\(395\) 7.73401e7i 1.25491i
\(396\) 0 0
\(397\) −3.42232e6 −0.0546952 −0.0273476 0.999626i \(-0.508706\pi\)
−0.0273476 + 0.999626i \(0.508706\pi\)
\(398\) 0 0
\(399\) 1.39714e8 + 1.15032e8i 2.19948 + 1.81093i
\(400\) 0 0
\(401\) 3.36234e7i 0.521444i 0.965414 + 0.260722i \(0.0839607\pi\)
−0.965414 + 0.260722i \(0.916039\pi\)
\(402\) 0 0
\(403\) 1.76741e7 0.270036
\(404\) 0 0
\(405\) −1.49482e7 −0.225021
\(406\) 0 0
\(407\) 2.33069e6i 0.0345702i
\(408\) 0 0
\(409\) 3.29912e7i 0.482201i 0.970500 + 0.241100i \(0.0775083\pi\)
−0.970500 + 0.241100i \(0.922492\pi\)
\(410\) 0 0
\(411\) 1.39005e8i 2.00219i
\(412\) 0 0
\(413\) 8.20633e7i 1.16493i
\(414\) 0 0
\(415\) 5.12292e7 0.716760
\(416\) 0 0
\(417\) 4.87122e6i 0.0671784i
\(418\) 0 0
\(419\) −7.94603e7 −1.08021 −0.540105 0.841598i \(-0.681615\pi\)
−0.540105 + 0.841598i \(0.681615\pi\)
\(420\) 0 0
\(421\) 1.20635e8i 1.61669i −0.588706 0.808347i \(-0.700362\pi\)
0.588706 0.808347i \(-0.299638\pi\)
\(422\) 0 0
\(423\) −1.02030e8 −1.34805
\(424\) 0 0
\(425\) 1.15207e7 0.150077
\(426\) 0 0
\(427\) −1.19011e6 −0.0152863
\(428\) 0 0
\(429\) 1.18084e6 0.0149561
\(430\) 0 0
\(431\) 1.08169e8i 1.35105i 0.737337 + 0.675525i \(0.236083\pi\)
−0.737337 + 0.675525i \(0.763917\pi\)
\(432\) 0 0
\(433\) 5.91296e7i 0.728352i −0.931330 0.364176i \(-0.881351\pi\)
0.931330 0.364176i \(-0.118649\pi\)
\(434\) 0 0
\(435\) −2.02762e8 −2.46331
\(436\) 0 0
\(437\) −3.79051e7 + 4.60381e7i −0.454207 + 0.551662i
\(438\) 0 0
\(439\) 1.06497e8i 1.25877i −0.777094 0.629384i \(-0.783307\pi\)
0.777094 0.629384i \(-0.216693\pi\)
\(440\) 0 0
\(441\) −2.88771e8 −3.36695
\(442\) 0 0
\(443\) −1.08889e7 −0.125248 −0.0626242 0.998037i \(-0.519947\pi\)
−0.0626242 + 0.998037i \(0.519947\pi\)
\(444\) 0 0
\(445\) 1.39722e8i 1.58556i
\(446\) 0 0
\(447\) 1.75914e8i 1.96960i
\(448\) 0 0
\(449\) 1.25704e7i 0.138870i 0.997586 + 0.0694351i \(0.0221197\pi\)
−0.997586 + 0.0694351i \(0.977880\pi\)
\(450\) 0 0
\(451\) 3.54393e6i 0.0386327i
\(452\) 0 0
\(453\) 2.14835e7 0.231106
\(454\) 0 0
\(455\) 3.08326e7i 0.327323i
\(456\) 0 0
\(457\) 3.27105e7 0.342719 0.171359 0.985209i \(-0.445184\pi\)
0.171359 + 0.985209i \(0.445184\pi\)
\(458\) 0 0
\(459\) 3.55167e7i 0.367278i
\(460\) 0 0
\(461\) 6.30874e7 0.643932 0.321966 0.946751i \(-0.395656\pi\)
0.321966 + 0.946751i \(0.395656\pi\)
\(462\) 0 0
\(463\) −1.48175e8 −1.49290 −0.746450 0.665441i \(-0.768243\pi\)
−0.746450 + 0.665441i \(0.768243\pi\)
\(464\) 0 0
\(465\) −1.61829e8 −1.60953
\(466\) 0 0
\(467\) −6.45801e7 −0.634086 −0.317043 0.948411i \(-0.602690\pi\)
−0.317043 + 0.948411i \(0.602690\pi\)
\(468\) 0 0
\(469\) 2.34633e8i 2.27442i
\(470\) 0 0
\(471\) 1.31733e8i 1.26076i
\(472\) 0 0
\(473\) 3.70847e6 0.0350439
\(474\) 0 0
\(475\) 2.14732e7 2.60805e7i 0.200362 0.243352i
\(476\) 0 0
\(477\) 1.06164e8i 0.978187i
\(478\) 0 0
\(479\) 1.70533e8 1.55168 0.775838 0.630932i \(-0.217328\pi\)
0.775838 + 0.630932i \(0.217328\pi\)
\(480\) 0 0
\(481\) −1.94692e7 −0.174949
\(482\) 0 0
\(483\) 2.29403e8i 2.03591i
\(484\) 0 0
\(485\) 7.86037e6i 0.0688997i
\(486\) 0 0
\(487\) 1.79460e8i 1.55375i −0.629657 0.776873i \(-0.716805\pi\)
0.629657 0.776873i \(-0.283195\pi\)
\(488\) 0 0
\(489\) 2.50381e7i 0.214129i
\(490\) 0 0
\(491\) 1.53625e8 1.29783 0.648913 0.760863i \(-0.275224\pi\)
0.648913 + 0.760863i \(0.275224\pi\)
\(492\) 0 0
\(493\) 1.07639e8i 0.898320i
\(494\) 0 0
\(495\) −6.46810e6 −0.0533287
\(496\) 0 0
\(497\) 2.79854e8i 2.27962i
\(498\) 0 0
\(499\) 1.34585e7 0.108317 0.0541584 0.998532i \(-0.482752\pi\)
0.0541584 + 0.998532i \(0.482752\pi\)
\(500\) 0 0
\(501\) 3.45048e8 2.74388
\(502\) 0 0
\(503\) −5.76561e7 −0.453045 −0.226523 0.974006i \(-0.572736\pi\)
−0.226523 + 0.974006i \(0.572736\pi\)
\(504\) 0 0
\(505\) 1.12567e8 0.874054
\(506\) 0 0
\(507\) 1.95741e8i 1.50196i
\(508\) 0 0
\(509\) 2.15748e8i 1.63604i 0.575193 + 0.818018i \(0.304927\pi\)
−0.575193 + 0.818018i \(0.695073\pi\)
\(510\) 0 0
\(511\) −4.14948e8 −3.10979
\(512\) 0 0
\(513\) −8.04023e7 6.61987e7i −0.595548 0.490340i
\(514\) 0 0
\(515\) 1.75095e8i 1.28189i
\(516\) 0 0
\(517\) 5.41487e6 0.0391847
\(518\) 0 0
\(519\) 8.20961e7 0.587246
\(520\) 0 0
\(521\) 8.48064e7i 0.599674i −0.953990 0.299837i \(-0.903068\pi\)
0.953990 0.299837i \(-0.0969323\pi\)
\(522\) 0 0
\(523\) 2.36947e7i 0.165633i 0.996565 + 0.0828165i \(0.0263915\pi\)
−0.996565 + 0.0828165i \(0.973608\pi\)
\(524\) 0 0
\(525\) 1.29956e8i 0.898090i
\(526\) 0 0
\(527\) 8.59096e7i 0.586962i
\(528\) 0 0
\(529\) −7.24437e7 −0.489366
\(530\) 0 0
\(531\) 1.43806e8i 0.960494i
\(532\) 0 0
\(533\) 2.96038e7 0.195509
\(534\) 0 0
\(535\) 6.46022e7i 0.421877i
\(536\) 0 0
\(537\) 2.05710e8 1.32841
\(538\) 0 0
\(539\) 1.53255e7 0.0978696
\(540\) 0 0
\(541\) −2.29774e8 −1.45114 −0.725569 0.688149i \(-0.758423\pi\)
−0.725569 + 0.688149i \(0.758423\pi\)
\(542\) 0 0
\(543\) −1.43434e8 −0.895887
\(544\) 0 0
\(545\) 1.45572e8i 0.899268i
\(546\) 0 0
\(547\) 1.04309e8i 0.637326i −0.947868 0.318663i \(-0.896766\pi\)
0.947868 0.318663i \(-0.103234\pi\)
\(548\) 0 0
\(549\) 2.08553e6 0.0126037
\(550\) 0 0
\(551\) 2.43673e8 + 2.00626e8i 1.45664 + 1.19931i
\(552\) 0 0
\(553\) 4.63134e8i 2.73862i
\(554\) 0 0
\(555\) 1.78266e8 1.04277
\(556\) 0 0
\(557\) 1.31503e8 0.760973 0.380486 0.924787i \(-0.375757\pi\)
0.380486 + 0.924787i \(0.375757\pi\)
\(558\) 0 0
\(559\) 3.09783e7i 0.177346i
\(560\) 0 0
\(561\) 5.73976e6i 0.0325091i
\(562\) 0 0
\(563\) 7.60307e7i 0.426053i −0.977046 0.213027i \(-0.931668\pi\)
0.977046 0.213027i \(-0.0683321\pi\)
\(564\) 0 0
\(565\) 1.15633e8i 0.641115i
\(566\) 0 0
\(567\) −8.95138e7 −0.491067
\(568\) 0 0
\(569\) 1.84138e8i 0.999553i 0.866155 + 0.499776i \(0.166584\pi\)
−0.866155 + 0.499776i \(0.833416\pi\)
\(570\) 0 0
\(571\) 1.94513e8 1.04482 0.522409 0.852695i \(-0.325033\pi\)
0.522409 + 0.852695i \(0.325033\pi\)
\(572\) 0 0
\(573\) 2.41045e8i 1.28125i
\(574\) 0 0
\(575\) −4.28229e7 −0.225254
\(576\) 0 0
\(577\) 230318. 0.00119895 0.000599475 1.00000i \(-0.499809\pi\)
0.000599475 1.00000i \(0.499809\pi\)
\(578\) 0 0
\(579\) 4.36540e8 2.24900
\(580\) 0 0
\(581\) 3.06775e8 1.56420
\(582\) 0 0
\(583\) 5.63427e6i 0.0284336i
\(584\) 0 0
\(585\) 5.40305e7i 0.269881i
\(586\) 0 0
\(587\) 2.41690e8 1.19493 0.597466 0.801894i \(-0.296174\pi\)
0.597466 + 0.801894i \(0.296174\pi\)
\(588\) 0 0
\(589\) 1.94481e8 + 1.60124e8i 0.951768 + 0.783631i
\(590\) 0 0
\(591\) 6.30067e8i 3.05228i
\(592\) 0 0
\(593\) −2.16180e8 −1.03669 −0.518347 0.855170i \(-0.673453\pi\)
−0.518347 + 0.855170i \(0.673453\pi\)
\(594\) 0 0
\(595\) −1.49870e8 −0.711481
\(596\) 0 0
\(597\) 2.09772e8i 0.985883i
\(598\) 0 0
\(599\) 1.80059e8i 0.837788i 0.908035 + 0.418894i \(0.137582\pi\)
−0.908035 + 0.418894i \(0.862418\pi\)
\(600\) 0 0
\(601\) 2.65025e8i 1.22085i −0.792072 0.610427i \(-0.790998\pi\)
0.792072 0.610427i \(-0.209002\pi\)
\(602\) 0 0
\(603\) 4.11167e8i 1.87528i
\(604\) 0 0
\(605\) −1.82905e8 −0.825963
\(606\) 0 0
\(607\) 7.33490e7i 0.327965i 0.986463 + 0.163983i \(0.0524341\pi\)
−0.986463 + 0.163983i \(0.947566\pi\)
\(608\) 0 0
\(609\) −1.21420e9 −5.37573
\(610\) 0 0
\(611\) 4.52325e7i 0.198302i
\(612\) 0 0
\(613\) 2.14007e8 0.929064 0.464532 0.885556i \(-0.346223\pi\)
0.464532 + 0.885556i \(0.346223\pi\)
\(614\) 0 0
\(615\) −2.71061e8 −1.16531
\(616\) 0 0
\(617\) 3.01185e8 1.28226 0.641132 0.767430i \(-0.278465\pi\)
0.641132 + 0.767430i \(0.278465\pi\)
\(618\) 0 0
\(619\) 3.15883e8 1.33185 0.665924 0.746019i \(-0.268037\pi\)
0.665924 + 0.746019i \(0.268037\pi\)
\(620\) 0 0
\(621\) 1.32017e8i 0.551257i
\(622\) 0 0
\(623\) 8.36691e8i 3.46020i
\(624\) 0 0
\(625\) −1.42923e8 −0.585412
\(626\) 0 0
\(627\) 1.29936e7 + 1.06982e7i 0.0527141 + 0.0434018i
\(628\) 0 0
\(629\) 9.46350e7i 0.380277i
\(630\) 0 0
\(631\) −1.94706e8 −0.774980 −0.387490 0.921874i \(-0.626658\pi\)
−0.387490 + 0.921874i \(0.626658\pi\)
\(632\) 0 0
\(633\) −4.24705e7 −0.167447
\(634\) 0 0
\(635\) 2.13219e8i 0.832731i
\(636\) 0 0
\(637\) 1.28020e8i 0.495288i
\(638\) 0 0
\(639\) 4.90411e8i 1.87957i
\(640\) 0 0
\(641\) 4.53818e8i 1.72309i 0.507684 + 0.861543i \(0.330502\pi\)
−0.507684 + 0.861543i \(0.669498\pi\)
\(642\) 0 0
\(643\) −1.37859e8 −0.518564 −0.259282 0.965802i \(-0.583486\pi\)
−0.259282 + 0.965802i \(0.583486\pi\)
\(644\) 0 0
\(645\) 2.83647e8i 1.05706i
\(646\) 0 0
\(647\) 9.74047e7 0.359639 0.179820 0.983700i \(-0.442449\pi\)
0.179820 + 0.983700i \(0.442449\pi\)
\(648\) 0 0
\(649\) 7.63202e6i 0.0279193i
\(650\) 0 0
\(651\) −9.69079e8 −3.51250
\(652\) 0 0
\(653\) −2.31439e7 −0.0831184 −0.0415592 0.999136i \(-0.513233\pi\)
−0.0415592 + 0.999136i \(0.513233\pi\)
\(654\) 0 0
\(655\) 2.41596e8 0.859738
\(656\) 0 0
\(657\) 7.27148e8 2.56405
\(658\) 0 0
\(659\) 2.21467e8i 0.773844i −0.922112 0.386922i \(-0.873538\pi\)
0.922112 0.386922i \(-0.126462\pi\)
\(660\) 0 0
\(661\) 2.13083e8i 0.737811i 0.929467 + 0.368906i \(0.120267\pi\)
−0.929467 + 0.368906i \(0.879733\pi\)
\(662\) 0 0
\(663\) −4.79464e7 −0.164519
\(664\) 0 0
\(665\) −2.79338e8 + 3.39273e8i −0.949873 + 1.15368i
\(666\) 0 0
\(667\) 4.00099e8i 1.34831i
\(668\) 0 0
\(669\) 5.30444e8 1.77158
\(670\) 0 0
\(671\) −110682. −0.000366361
\(672\) 0 0
\(673\) 1.52414e8i 0.500011i −0.968244 0.250006i \(-0.919568\pi\)
0.968244 0.250006i \(-0.0804325\pi\)
\(674\) 0 0
\(675\) 7.47872e7i 0.243173i
\(676\) 0 0
\(677\) 9.86309e7i 0.317868i 0.987289 + 0.158934i \(0.0508057\pi\)
−0.987289 + 0.158934i \(0.949194\pi\)
\(678\) 0 0
\(679\) 4.70700e7i 0.150361i
\(680\) 0 0
\(681\) 3.07943e8 0.975055
\(682\) 0 0
\(683\) 1.83544e8i 0.576074i 0.957619 + 0.288037i \(0.0930026\pi\)
−0.957619 + 0.288037i \(0.906997\pi\)
\(684\) 0 0
\(685\) −3.37553e8 −1.05020
\(686\) 0 0
\(687\) 2.04406e8i 0.630410i
\(688\) 0 0
\(689\) 4.70653e7 0.143894
\(690\) 0 0
\(691\) −9.11147e7 −0.276156 −0.138078 0.990421i \(-0.544092\pi\)
−0.138078 + 0.990421i \(0.544092\pi\)
\(692\) 0 0
\(693\) −3.87328e7 −0.116380
\(694\) 0 0
\(695\) −1.18290e7 −0.0352366
\(696\) 0 0
\(697\) 1.43897e8i 0.424965i
\(698\) 0 0
\(699\) 1.08939e8i 0.318971i
\(700\) 0 0
\(701\) −4.14272e8 −1.20263 −0.601315 0.799012i \(-0.705356\pi\)
−0.601315 + 0.799012i \(0.705356\pi\)
\(702\) 0 0
\(703\) −2.14234e8 1.76388e8i −0.616625 0.507694i
\(704\) 0 0
\(705\) 4.14162e8i 1.18196i
\(706\) 0 0
\(707\) 6.74084e8 1.90746
\(708\) 0 0
\(709\) −3.21198e8 −0.901226 −0.450613 0.892719i \(-0.648795\pi\)
−0.450613 + 0.892719i \(0.648795\pi\)
\(710\) 0 0
\(711\) 8.11588e8i 2.25802i
\(712\) 0 0
\(713\) 3.19328e8i 0.880985i
\(714\) 0 0
\(715\) 2.86748e6i 0.00784481i
\(716\) 0 0
\(717\) 1.00619e9i 2.72975i
\(718\) 0 0
\(719\) 3.98125e8 1.07111 0.535553 0.844502i \(-0.320103\pi\)
0.535553 + 0.844502i \(0.320103\pi\)
\(720\) 0 0
\(721\) 1.04852e9i 2.79749i
\(722\) 0 0
\(723\) 7.92687e8 2.09743
\(724\) 0 0
\(725\) 2.26655e8i 0.594774i
\(726\) 0 0
\(727\) −2.93307e8 −0.763343 −0.381672 0.924298i \(-0.624651\pi\)
−0.381672 + 0.924298i \(0.624651\pi\)
\(728\) 0 0
\(729\) 6.28373e8 1.62194
\(730\) 0 0
\(731\) −1.50578e8 −0.385487
\(732\) 0 0
\(733\) 4.75660e8 1.20777 0.603886 0.797071i \(-0.293618\pi\)
0.603886 + 0.797071i \(0.293618\pi\)
\(734\) 0 0
\(735\) 1.17219e9i 2.95212i
\(736\) 0 0
\(737\) 2.18212e7i 0.0545101i
\(738\) 0 0
\(739\) −2.86348e8 −0.709515 −0.354757 0.934958i \(-0.615437\pi\)
−0.354757 + 0.934958i \(0.615437\pi\)
\(740\) 0 0
\(741\) −8.93660e7 + 1.08541e8i −0.219643 + 0.266770i
\(742\) 0 0
\(743\) 6.29617e8i 1.53501i 0.641045 + 0.767503i \(0.278501\pi\)
−0.641045 + 0.767503i \(0.721499\pi\)
\(744\) 0 0
\(745\) 4.27181e8 1.03310
\(746\) 0 0
\(747\) −5.37587e8 −1.28969
\(748\) 0 0
\(749\) 3.86856e8i 0.920669i
\(750\) 0 0
\(751\) 5.36049e8i 1.26557i 0.774329 + 0.632783i \(0.218087\pi\)
−0.774329 + 0.632783i \(0.781913\pi\)
\(752\) 0 0
\(753\) 8.00018e8i 1.87376i
\(754\) 0 0
\(755\) 5.21694e7i 0.121220i
\(756\) 0 0
\(757\) −4.72181e7 −0.108848 −0.0544240 0.998518i \(-0.517332\pi\)
−0.0544240 + 0.998518i \(0.517332\pi\)
\(758\) 0 0
\(759\) 2.13348e7i 0.0487937i
\(760\) 0 0
\(761\) 3.01530e8 0.684189 0.342095 0.939666i \(-0.388864\pi\)
0.342095 + 0.939666i \(0.388864\pi\)
\(762\) 0 0
\(763\) 8.71726e8i 1.96249i
\(764\) 0 0
\(765\) 2.62629e8 0.586623
\(766\) 0 0
\(767\) 6.37532e7 0.141291
\(768\) 0 0
\(769\) −7.09285e8 −1.55970 −0.779851 0.625965i \(-0.784705\pi\)
−0.779851 + 0.625965i \(0.784705\pi\)
\(770\) 0 0
\(771\) 6.09134e8 1.32908
\(772\) 0 0
\(773\) 4.38166e8i 0.948638i −0.880353 0.474319i \(-0.842694\pi\)
0.880353 0.474319i \(-0.157306\pi\)
\(774\) 0 0
\(775\) 1.80899e8i 0.388625i
\(776\) 0 0
\(777\) 1.06750e9 2.27565
\(778\) 0 0
\(779\) 3.25752e8 + 2.68206e8i 0.689088 + 0.567356i
\(780\) 0 0
\(781\) 2.60269e7i 0.0546347i
\(782\) 0 0
\(783\) 6.98745e8 1.45557
\(784\) 0 0
\(785\) 3.19893e8 0.661297
\(786\) 0 0
\(787\) 4.08238e8i 0.837509i 0.908099 + 0.418755i \(0.137533\pi\)
−0.908099 + 0.418755i \(0.862467\pi\)
\(788\) 0 0
\(789\) 1.14536e9i 2.33190i
\(790\) 0 0
\(791\) 6.92441e8i 1.39911i
\(792\) 0 0
\(793\) 924569.i 0.00185404i
\(794\) 0 0
\(795\) −4.30944e8 −0.857668
\(796\) 0 0