Properties

Label 304.3.e.b.113.1
Level $304$
Weight $3$
Character 304.113
Analytic conductor $8.283$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.28340003655\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-29}) \)
Defining polynomial: \(x^{2} + 29\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.1
Root \(-5.38516i\) of defining polynomial
Character \(\chi\) \(=\) 304.113
Dual form 304.3.e.b.113.2

$q$-expansion

\(f(q)\) \(=\) \(q-5.38516i q^{3} -4.00000 q^{5} +1.00000 q^{7} -20.0000 q^{9} +O(q^{10})\) \(q-5.38516i q^{3} -4.00000 q^{5} +1.00000 q^{7} -20.0000 q^{9} -14.0000 q^{11} +16.1555i q^{13} +21.5407i q^{15} +23.0000 q^{17} +(-10.0000 - 16.1555i) q^{19} -5.38516i q^{21} +1.00000 q^{23} -9.00000 q^{25} +59.2368i q^{27} -48.4665i q^{29} +32.3110i q^{31} +75.3923i q^{33} -4.00000 q^{35} +32.3110i q^{37} +87.0000 q^{39} +32.3110i q^{41} -68.0000 q^{43} +80.0000 q^{45} -26.0000 q^{47} -48.0000 q^{49} -123.859i q^{51} -80.7775i q^{53} +56.0000 q^{55} +(-87.0000 + 53.8516i) q^{57} -16.1555i q^{59} -40.0000 q^{61} -20.0000 q^{63} -64.6220i q^{65} +16.1555i q^{67} -5.38516i q^{69} +32.3110i q^{71} -7.00000 q^{73} +48.4665i q^{75} -14.0000 q^{77} -96.9330i q^{79} +139.000 q^{81} -32.0000 q^{83} -92.0000 q^{85} -261.000 q^{87} -129.244i q^{89} +16.1555i q^{91} +174.000 q^{93} +(40.0000 + 64.6220i) q^{95} -96.9330i q^{97} +280.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{5} + 2q^{7} - 40q^{9} + O(q^{10}) \) \( 2q - 8q^{5} + 2q^{7} - 40q^{9} - 28q^{11} + 46q^{17} - 20q^{19} + 2q^{23} - 18q^{25} - 8q^{35} + 174q^{39} - 136q^{43} + 160q^{45} - 52q^{47} - 96q^{49} + 112q^{55} - 174q^{57} - 80q^{61} - 40q^{63} - 14q^{73} - 28q^{77} + 278q^{81} - 64q^{83} - 184q^{85} - 522q^{87} + 348q^{93} + 80q^{95} + 560q^{99} + O(q^{100}) \)

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
\(3\) 5.38516i 1.79505i −0.440959 0.897527i \(-0.645361\pi\)
0.440959 0.897527i \(-0.354639\pi\)
\(4\) 0 0
\(5\) −4.00000 −0.800000 −0.400000 0.916515i \(-0.630990\pi\)
−0.400000 + 0.916515i \(0.630990\pi\)
\(6\) 0 0
\(7\) 1.00000 0.142857 0.0714286 0.997446i \(-0.477244\pi\)
0.0714286 + 0.997446i \(0.477244\pi\)
\(8\) 0 0
\(9\) −20.0000 −2.22222
\(10\) 0 0
\(11\) −14.0000 −1.27273 −0.636364 0.771389i \(-0.719562\pi\)
−0.636364 + 0.771389i \(0.719562\pi\)
\(12\) 0 0
\(13\) 16.1555i 1.24273i 0.783521 + 0.621365i \(0.213422\pi\)
−0.783521 + 0.621365i \(0.786578\pi\)
\(14\) 0 0
\(15\) 21.5407i 1.43604i
\(16\) 0 0
\(17\) 23.0000 1.35294 0.676471 0.736470i \(-0.263509\pi\)
0.676471 + 0.736470i \(0.263509\pi\)
\(18\) 0 0
\(19\) −10.0000 16.1555i −0.526316 0.850289i
\(20\) 0 0
\(21\) 5.38516i 0.256436i
\(22\) 0 0
\(23\) 1.00000 0.0434783 0.0217391 0.999764i \(-0.493080\pi\)
0.0217391 + 0.999764i \(0.493080\pi\)
\(24\) 0 0
\(25\) −9.00000 −0.360000
\(26\) 0 0
\(27\) 59.2368i 2.19396i
\(28\) 0 0
\(29\) 48.4665i 1.67126i −0.549294 0.835629i \(-0.685103\pi\)
0.549294 0.835629i \(-0.314897\pi\)
\(30\) 0 0
\(31\) 32.3110i 1.04229i 0.853468 + 0.521145i \(0.174495\pi\)
−0.853468 + 0.521145i \(0.825505\pi\)
\(32\) 0 0
\(33\) 75.3923i 2.28462i
\(34\) 0 0
\(35\) −4.00000 −0.114286
\(36\) 0 0
\(37\) 32.3110i 0.873270i 0.899639 + 0.436635i \(0.143830\pi\)
−0.899639 + 0.436635i \(0.856170\pi\)
\(38\) 0 0
\(39\) 87.0000 2.23077
\(40\) 0 0
\(41\) 32.3110i 0.788073i 0.919095 + 0.394036i \(0.128922\pi\)
−0.919095 + 0.394036i \(0.871078\pi\)
\(42\) 0 0
\(43\) −68.0000 −1.58140 −0.790698 0.612207i \(-0.790282\pi\)
−0.790698 + 0.612207i \(0.790282\pi\)
\(44\) 0 0
\(45\) 80.0000 1.77778
\(46\) 0 0
\(47\) −26.0000 −0.553191 −0.276596 0.960986i \(-0.589206\pi\)
−0.276596 + 0.960986i \(0.589206\pi\)
\(48\) 0 0
\(49\) −48.0000 −0.979592
\(50\) 0 0
\(51\) 123.859i 2.42860i
\(52\) 0 0
\(53\) 80.7775i 1.52410i −0.647516 0.762052i \(-0.724192\pi\)
0.647516 0.762052i \(-0.275808\pi\)
\(54\) 0 0
\(55\) 56.0000 1.01818
\(56\) 0 0
\(57\) −87.0000 + 53.8516i −1.52632 + 0.944766i
\(58\) 0 0
\(59\) 16.1555i 0.273822i −0.990583 0.136911i \(-0.956283\pi\)
0.990583 0.136911i \(-0.0437174\pi\)
\(60\) 0 0
\(61\) −40.0000 −0.655738 −0.327869 0.944723i \(-0.606330\pi\)
−0.327869 + 0.944723i \(0.606330\pi\)
\(62\) 0 0
\(63\) −20.0000 −0.317460
\(64\) 0 0
\(65\) 64.6220i 0.994184i
\(66\) 0 0
\(67\) 16.1555i 0.241127i 0.992706 + 0.120563i \(0.0384701\pi\)
−0.992706 + 0.120563i \(0.961530\pi\)
\(68\) 0 0
\(69\) 5.38516i 0.0780459i
\(70\) 0 0
\(71\) 32.3110i 0.455084i 0.973768 + 0.227542i \(0.0730690\pi\)
−0.973768 + 0.227542i \(0.926931\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.0958904 −0.0479452 0.998850i \(-0.515267\pi\)
−0.0479452 + 0.998850i \(0.515267\pi\)
\(74\) 0 0
\(75\) 48.4665i 0.646220i
\(76\) 0 0
\(77\) −14.0000 −0.181818
\(78\) 0 0
\(79\) 96.9330i 1.22700i −0.789695 0.613500i \(-0.789761\pi\)
0.789695 0.613500i \(-0.210239\pi\)
\(80\) 0 0
\(81\) 139.000 1.71605
\(82\) 0 0
\(83\) −32.0000 −0.385542 −0.192771 0.981244i \(-0.561747\pi\)
−0.192771 + 0.981244i \(0.561747\pi\)
\(84\) 0 0
\(85\) −92.0000 −1.08235
\(86\) 0 0
\(87\) −261.000 −3.00000
\(88\) 0 0
\(89\) 129.244i 1.45218i −0.687600 0.726090i \(-0.741336\pi\)
0.687600 0.726090i \(-0.258664\pi\)
\(90\) 0 0
\(91\) 16.1555i 0.177533i
\(92\) 0 0
\(93\) 174.000 1.87097
\(94\) 0 0
\(95\) 40.0000 + 64.6220i 0.421053 + 0.680231i
\(96\) 0 0
\(97\) 96.9330i 0.999309i −0.866225 0.499654i \(-0.833460\pi\)
0.866225 0.499654i \(-0.166540\pi\)
\(98\) 0 0
\(99\) 280.000 2.82828
\(100\) 0 0
\(101\) 14.0000 0.138614 0.0693069 0.997595i \(-0.477921\pi\)
0.0693069 + 0.997595i \(0.477921\pi\)
\(102\) 0 0
\(103\) 129.244i 1.25480i −0.778699 0.627398i \(-0.784120\pi\)
0.778699 0.627398i \(-0.215880\pi\)
\(104\) 0 0
\(105\) 21.5407i 0.205149i
\(106\) 0 0
\(107\) 16.1555i 0.150986i −0.997146 0.0754930i \(-0.975947\pi\)
0.997146 0.0754930i \(-0.0240530\pi\)
\(108\) 0 0
\(109\) 16.1555i 0.148216i 0.997250 + 0.0741078i \(0.0236109\pi\)
−0.997250 + 0.0741078i \(0.976389\pi\)
\(110\) 0 0
\(111\) 174.000 1.56757
\(112\) 0 0
\(113\) 96.9330i 0.857814i 0.903349 + 0.428907i \(0.141101\pi\)
−0.903349 + 0.428907i \(0.858899\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.0347826
\(116\) 0 0
\(117\) 323.110i 2.76162i
\(118\) 0 0
\(119\) 23.0000 0.193277
\(120\) 0 0
\(121\) 75.0000 0.619835
\(122\) 0 0
\(123\) 174.000 1.41463
\(124\) 0 0
\(125\) 136.000 1.08800
\(126\) 0 0
\(127\) 64.6220i 0.508834i 0.967095 + 0.254417i \(0.0818836\pi\)
−0.967095 + 0.254417i \(0.918116\pi\)
\(128\) 0 0
\(129\) 366.191i 2.83869i
\(130\) 0 0
\(131\) −56.0000 −0.427481 −0.213740 0.976890i \(-0.568565\pi\)
−0.213740 + 0.976890i \(0.568565\pi\)
\(132\) 0 0
\(133\) −10.0000 16.1555i −0.0751880 0.121470i
\(134\) 0 0
\(135\) 236.947i 1.75516i
\(136\) 0 0
\(137\) −19.0000 −0.138686 −0.0693431 0.997593i \(-0.522090\pi\)
−0.0693431 + 0.997593i \(0.522090\pi\)
\(138\) 0 0
\(139\) −122.000 −0.877698 −0.438849 0.898561i \(-0.644614\pi\)
−0.438849 + 0.898561i \(0.644614\pi\)
\(140\) 0 0
\(141\) 140.014i 0.993009i
\(142\) 0 0
\(143\) 226.177i 1.58166i
\(144\) 0 0
\(145\) 193.866i 1.33701i
\(146\) 0 0
\(147\) 258.488i 1.75842i
\(148\) 0 0
\(149\) −82.0000 −0.550336 −0.275168 0.961396i \(-0.588733\pi\)
−0.275168 + 0.961396i \(0.588733\pi\)
\(150\) 0 0
\(151\) 226.177i 1.49786i 0.662649 + 0.748930i \(0.269432\pi\)
−0.662649 + 0.748930i \(0.730568\pi\)
\(152\) 0 0
\(153\) −460.000 −3.00654
\(154\) 0 0
\(155\) 129.244i 0.833832i
\(156\) 0 0
\(157\) 242.000 1.54140 0.770701 0.637197i \(-0.219906\pi\)
0.770701 + 0.637197i \(0.219906\pi\)
\(158\) 0 0
\(159\) −435.000 −2.73585
\(160\) 0 0
\(161\) 1.00000 0.00621118
\(162\) 0 0
\(163\) 214.000 1.31288 0.656442 0.754377i \(-0.272061\pi\)
0.656442 + 0.754377i \(0.272061\pi\)
\(164\) 0 0
\(165\) 301.569i 1.82769i
\(166\) 0 0
\(167\) 32.3110i 0.193479i −0.995310 0.0967395i \(-0.969159\pi\)
0.995310 0.0967395i \(-0.0308414\pi\)
\(168\) 0 0
\(169\) −92.0000 −0.544379
\(170\) 0 0
\(171\) 200.000 + 323.110i 1.16959 + 1.88953i
\(172\) 0 0
\(173\) 161.555i 0.933844i −0.884299 0.466922i \(-0.845363\pi\)
0.884299 0.466922i \(-0.154637\pi\)
\(174\) 0 0
\(175\) −9.00000 −0.0514286
\(176\) 0 0
\(177\) −87.0000 −0.491525
\(178\) 0 0
\(179\) 290.799i 1.62457i −0.583257 0.812287i \(-0.698222\pi\)
0.583257 0.812287i \(-0.301778\pi\)
\(180\) 0 0
\(181\) 96.9330i 0.535541i 0.963483 + 0.267771i \(0.0862869\pi\)
−0.963483 + 0.267771i \(0.913713\pi\)
\(182\) 0 0
\(183\) 215.407i 1.17709i
\(184\) 0 0
\(185\) 129.244i 0.698616i
\(186\) 0 0
\(187\) −322.000 −1.72193
\(188\) 0 0
\(189\) 59.2368i 0.313422i
\(190\) 0 0
\(191\) 67.0000 0.350785 0.175393 0.984499i \(-0.443881\pi\)
0.175393 + 0.984499i \(0.443881\pi\)
\(192\) 0 0
\(193\) 96.9330i 0.502243i −0.967956 0.251122i \(-0.919201\pi\)
0.967956 0.251122i \(-0.0807994\pi\)
\(194\) 0 0
\(195\) −348.000 −1.78462
\(196\) 0 0
\(197\) −142.000 −0.720812 −0.360406 0.932796i \(-0.617362\pi\)
−0.360406 + 0.932796i \(0.617362\pi\)
\(198\) 0 0
\(199\) −263.000 −1.32161 −0.660804 0.750558i \(-0.729785\pi\)
−0.660804 + 0.750558i \(0.729785\pi\)
\(200\) 0 0
\(201\) 87.0000 0.432836
\(202\) 0 0
\(203\) 48.4665i 0.238751i
\(204\) 0 0
\(205\) 129.244i 0.630458i
\(206\) 0 0
\(207\) −20.0000 −0.0966184
\(208\) 0 0
\(209\) 140.000 + 226.177i 0.669856 + 1.08219i
\(210\) 0 0
\(211\) 80.7775i 0.382832i 0.981509 + 0.191416i \(0.0613079\pi\)
−0.981509 + 0.191416i \(0.938692\pi\)
\(212\) 0 0
\(213\) 174.000 0.816901
\(214\) 0 0
\(215\) 272.000 1.26512
\(216\) 0 0
\(217\) 32.3110i 0.148899i
\(218\) 0 0
\(219\) 37.6962i 0.172129i
\(220\) 0 0
\(221\) 371.576i 1.68134i
\(222\) 0 0
\(223\) 258.488i 1.15914i 0.814923 + 0.579569i \(0.196779\pi\)
−0.814923 + 0.579569i \(0.803221\pi\)
\(224\) 0 0
\(225\) 180.000 0.800000
\(226\) 0 0
\(227\) 80.7775i 0.355848i 0.984044 + 0.177924i \(0.0569381\pi\)
−0.984044 + 0.177924i \(0.943062\pi\)
\(228\) 0 0
\(229\) −304.000 −1.32751 −0.663755 0.747950i \(-0.731038\pi\)
−0.663755 + 0.747950i \(0.731038\pi\)
\(230\) 0 0
\(231\) 75.3923i 0.326374i
\(232\) 0 0
\(233\) 2.00000 0.00858369 0.00429185 0.999991i \(-0.498634\pi\)
0.00429185 + 0.999991i \(0.498634\pi\)
\(234\) 0 0
\(235\) 104.000 0.442553
\(236\) 0 0
\(237\) −522.000 −2.20253
\(238\) 0 0
\(239\) −89.0000 −0.372385 −0.186192 0.982513i \(-0.559615\pi\)
−0.186192 + 0.982513i \(0.559615\pi\)
\(240\) 0 0
\(241\) 161.555i 0.670352i −0.942155 0.335176i \(-0.891204\pi\)
0.942155 0.335176i \(-0.108796\pi\)
\(242\) 0 0
\(243\) 215.407i 0.886447i
\(244\) 0 0
\(245\) 192.000 0.783673
\(246\) 0 0
\(247\) 261.000 161.555i 1.05668 0.654069i
\(248\) 0 0
\(249\) 172.325i 0.692069i
\(250\) 0 0
\(251\) 370.000 1.47410 0.737052 0.675836i \(-0.236217\pi\)
0.737052 + 0.675836i \(0.236217\pi\)
\(252\) 0 0
\(253\) −14.0000 −0.0553360
\(254\) 0 0
\(255\) 495.435i 1.94288i
\(256\) 0 0
\(257\) 193.866i 0.754342i −0.926144 0.377171i \(-0.876897\pi\)
0.926144 0.377171i \(-0.123103\pi\)
\(258\) 0 0
\(259\) 32.3110i 0.124753i
\(260\) 0 0
\(261\) 969.330i 3.71391i
\(262\) 0 0
\(263\) 394.000 1.49810 0.749049 0.662514i \(-0.230511\pi\)
0.749049 + 0.662514i \(0.230511\pi\)
\(264\) 0 0
\(265\) 323.110i 1.21928i
\(266\) 0 0
\(267\) −696.000 −2.60674
\(268\) 0 0
\(269\) 96.9330i 0.360346i 0.983635 + 0.180173i \(0.0576657\pi\)
−0.983635 + 0.180173i \(0.942334\pi\)
\(270\) 0 0
\(271\) 403.000 1.48708 0.743542 0.668689i \(-0.233144\pi\)
0.743542 + 0.668689i \(0.233144\pi\)
\(272\) 0 0
\(273\) 87.0000 0.318681
\(274\) 0 0
\(275\) 126.000 0.458182
\(276\) 0 0
\(277\) 206.000 0.743682 0.371841 0.928296i \(-0.378727\pi\)
0.371841 + 0.928296i \(0.378727\pi\)
\(278\) 0 0
\(279\) 646.220i 2.31620i
\(280\) 0 0
\(281\) 258.488i 0.919886i −0.887948 0.459943i \(-0.847870\pi\)
0.887948 0.459943i \(-0.152130\pi\)
\(282\) 0 0
\(283\) −56.0000 −0.197880 −0.0989399 0.995093i \(-0.531545\pi\)
−0.0989399 + 0.995093i \(0.531545\pi\)
\(284\) 0 0
\(285\) 348.000 215.407i 1.22105 0.755813i
\(286\) 0 0
\(287\) 32.3110i 0.112582i
\(288\) 0 0
\(289\) 240.000 0.830450
\(290\) 0 0
\(291\) −522.000 −1.79381
\(292\) 0 0
\(293\) 80.7775i 0.275691i −0.990454 0.137846i \(-0.955982\pi\)
0.990454 0.137846i \(-0.0440177\pi\)
\(294\) 0 0
\(295\) 64.6220i 0.219058i
\(296\) 0 0
\(297\) 829.315i 2.79231i
\(298\) 0 0
\(299\) 16.1555i 0.0540318i
\(300\) 0 0
\(301\) −68.0000 −0.225914
\(302\) 0 0
\(303\) 75.3923i 0.248819i
\(304\) 0 0
\(305\) 160.000 0.524590
\(306\) 0 0
\(307\) 420.043i 1.36822i −0.729380 0.684109i \(-0.760191\pi\)
0.729380 0.684109i \(-0.239809\pi\)
\(308\) 0 0
\(309\) −696.000 −2.25243
\(310\) 0 0
\(311\) 265.000 0.852090 0.426045 0.904702i \(-0.359907\pi\)
0.426045 + 0.904702i \(0.359907\pi\)
\(312\) 0 0
\(313\) 77.0000 0.246006 0.123003 0.992406i \(-0.460747\pi\)
0.123003 + 0.992406i \(0.460747\pi\)
\(314\) 0 0
\(315\) 80.0000 0.253968
\(316\) 0 0
\(317\) 403.887i 1.27409i 0.770826 + 0.637046i \(0.219844\pi\)
−0.770826 + 0.637046i \(0.780156\pi\)
\(318\) 0 0
\(319\) 678.531i 2.12706i
\(320\) 0 0
\(321\) −87.0000 −0.271028
\(322\) 0 0
\(323\) −230.000 371.576i −0.712074 1.15039i
\(324\) 0 0
\(325\) 145.399i 0.447383i
\(326\) 0 0
\(327\) 87.0000 0.266055
\(328\) 0 0
\(329\) −26.0000 −0.0790274
\(330\) 0 0
\(331\) 436.198i 1.31782i 0.752222 + 0.658910i \(0.228982\pi\)
−0.752222 + 0.658910i \(0.771018\pi\)
\(332\) 0 0
\(333\) 646.220i 1.94060i
\(334\) 0 0
\(335\) 64.6220i 0.192901i
\(336\) 0 0
\(337\) 387.732i 1.15054i −0.817964 0.575270i \(-0.804897\pi\)
0.817964 0.575270i \(-0.195103\pi\)
\(338\) 0 0
\(339\) 522.000 1.53982
\(340\) 0 0
\(341\) 452.354i 1.32655i
\(342\) 0 0
\(343\) −97.0000 −0.282799
\(344\) 0 0
\(345\) 21.5407i 0.0624367i
\(346\) 0 0
\(347\) −392.000 −1.12968 −0.564841 0.825199i \(-0.691063\pi\)
−0.564841 + 0.825199i \(0.691063\pi\)
\(348\) 0 0
\(349\) 410.000 1.17479 0.587393 0.809302i \(-0.300154\pi\)
0.587393 + 0.809302i \(0.300154\pi\)
\(350\) 0 0
\(351\) −957.000 −2.72650
\(352\) 0 0
\(353\) −481.000 −1.36261 −0.681303 0.732001i \(-0.738586\pi\)
−0.681303 + 0.732001i \(0.738586\pi\)
\(354\) 0 0
\(355\) 129.244i 0.364067i
\(356\) 0 0
\(357\) 123.859i 0.346943i
\(358\) 0 0
\(359\) 109.000 0.303621 0.151811 0.988410i \(-0.451490\pi\)
0.151811 + 0.988410i \(0.451490\pi\)
\(360\) 0 0
\(361\) −161.000 + 323.110i −0.445983 + 0.895041i
\(362\) 0 0
\(363\) 403.887i 1.11264i
\(364\) 0 0
\(365\) 28.0000 0.0767123
\(366\) 0 0
\(367\) −50.0000 −0.136240 −0.0681199 0.997677i \(-0.521700\pi\)
−0.0681199 + 0.997677i \(0.521700\pi\)
\(368\) 0 0
\(369\) 646.220i 1.75127i
\(370\) 0 0
\(371\) 80.7775i 0.217729i
\(372\) 0 0
\(373\) 500.820i 1.34268i 0.741149 + 0.671341i \(0.234281\pi\)
−0.741149 + 0.671341i \(0.765719\pi\)
\(374\) 0 0
\(375\) 732.382i 1.95302i
\(376\) 0 0
\(377\) 783.000 2.07692
\(378\) 0 0
\(379\) 210.021i 0.554146i −0.960849 0.277073i \(-0.910636\pi\)
0.960849 0.277073i \(-0.0893644\pi\)
\(380\) 0 0
\(381\) 348.000 0.913386
\(382\) 0 0
\(383\) 258.488i 0.674903i −0.941343 0.337452i \(-0.890435\pi\)
0.941343 0.337452i \(-0.109565\pi\)
\(384\) 0 0
\(385\) 56.0000 0.145455
\(386\) 0 0
\(387\) 1360.00 3.51421
\(388\) 0 0
\(389\) 578.000 1.48586 0.742931 0.669368i \(-0.233435\pi\)
0.742931 + 0.669368i \(0.233435\pi\)
\(390\) 0 0
\(391\) 23.0000 0.0588235
\(392\) 0 0
\(393\) 301.569i 0.767352i
\(394\) 0 0
\(395\) 387.732i 0.981600i
\(396\) 0 0
\(397\) −658.000 −1.65743 −0.828715 0.559670i \(-0.810928\pi\)
−0.828715 + 0.559670i \(0.810928\pi\)
\(398\) 0 0
\(399\) −87.0000 + 53.8516i −0.218045 + 0.134967i
\(400\) 0 0
\(401\) 775.464i 1.93382i −0.255109 0.966912i \(-0.582111\pi\)
0.255109 0.966912i \(-0.417889\pi\)
\(402\) 0 0
\(403\) −522.000 −1.29529
\(404\) 0 0
\(405\) −556.000 −1.37284
\(406\) 0 0
\(407\) 452.354i 1.11143i
\(408\) 0 0
\(409\) 355.421i 0.869000i 0.900672 + 0.434500i \(0.143075\pi\)
−0.900672 + 0.434500i \(0.856925\pi\)
\(410\) 0 0
\(411\) 102.318i 0.248949i
\(412\) 0 0
\(413\) 16.1555i 0.0391174i
\(414\) 0 0
\(415\) 128.000 0.308434
\(416\) 0 0
\(417\) 656.990i 1.57552i
\(418\) 0 0
\(419\) −8.00000 −0.0190931 −0.00954654 0.999954i \(-0.503039\pi\)
−0.00954654 + 0.999954i \(0.503039\pi\)
\(420\) 0 0
\(421\) 500.820i 1.18960i 0.803875 + 0.594798i \(0.202768\pi\)
−0.803875 + 0.594798i \(0.797232\pi\)
\(422\) 0 0
\(423\) 520.000 1.22931
\(424\) 0 0
\(425\) −207.000 −0.487059
\(426\) 0 0
\(427\) −40.0000 −0.0936768
\(428\) 0 0
\(429\) −1218.00 −2.83916
\(430\) 0 0
\(431\) 646.220i 1.49935i 0.661806 + 0.749675i \(0.269790\pi\)
−0.661806 + 0.749675i \(0.730210\pi\)
\(432\) 0 0
\(433\) 193.866i 0.447727i 0.974620 + 0.223864i \(0.0718670\pi\)
−0.974620 + 0.223864i \(0.928133\pi\)
\(434\) 0 0
\(435\) 1044.00 2.40000
\(436\) 0 0
\(437\) −10.0000 16.1555i −0.0228833 0.0369691i
\(438\) 0 0
\(439\) 323.110i 0.736013i 0.929823 + 0.368007i \(0.119960\pi\)
−0.929823 + 0.368007i \(0.880040\pi\)
\(440\) 0 0
\(441\) 960.000 2.17687
\(442\) 0 0
\(443\) −182.000 −0.410835 −0.205418 0.978674i \(-0.565855\pi\)
−0.205418 + 0.978674i \(0.565855\pi\)
\(444\) 0 0
\(445\) 516.976i 1.16174i
\(446\) 0 0
\(447\) 441.584i 0.987883i
\(448\) 0 0
\(449\) 420.043i 0.935507i −0.883859 0.467754i \(-0.845063\pi\)
0.883859 0.467754i \(-0.154937\pi\)
\(450\) 0 0
\(451\) 452.354i 1.00300i
\(452\) 0 0
\(453\) 1218.00 2.68874
\(454\) 0 0
\(455\) 64.6220i 0.142026i
\(456\) 0 0
\(457\) 317.000 0.693654 0.346827 0.937929i \(-0.387259\pi\)
0.346827 + 0.937929i \(0.387259\pi\)
\(458\) 0 0
\(459\) 1362.45i 2.96829i
\(460\) 0 0
\(461\) −172.000 −0.373102 −0.186551 0.982445i \(-0.559731\pi\)
−0.186551 + 0.982445i \(0.559731\pi\)
\(462\) 0 0
\(463\) −146.000 −0.315335 −0.157667 0.987492i \(-0.550397\pi\)
−0.157667 + 0.987492i \(0.550397\pi\)
\(464\) 0 0
\(465\) −696.000 −1.49677
\(466\) 0 0
\(467\) −662.000 −1.41756 −0.708779 0.705430i \(-0.750754\pi\)
−0.708779 + 0.705430i \(0.750754\pi\)
\(468\) 0 0
\(469\) 16.1555i 0.0344467i
\(470\) 0 0
\(471\) 1303.21i 2.76690i
\(472\) 0 0
\(473\) 952.000 2.01268
\(474\) 0 0
\(475\) 90.0000 + 145.399i 0.189474 + 0.306104i
\(476\) 0 0
\(477\) 1615.55i 3.38690i
\(478\) 0 0
\(479\) 130.000 0.271399 0.135699 0.990750i \(-0.456672\pi\)
0.135699 + 0.990750i \(0.456672\pi\)
\(480\) 0 0
\(481\) −522.000 −1.08524
\(482\) 0 0
\(483\) 5.38516i 0.0111494i
\(484\) 0 0
\(485\) 387.732i 0.799447i
\(486\) 0 0
\(487\) 387.732i 0.796164i −0.917350 0.398082i \(-0.869676\pi\)
0.917350 0.398082i \(-0.130324\pi\)
\(488\) 0 0
\(489\) 1152.43i 2.35670i
\(490\) 0 0
\(491\) 280.000 0.570265 0.285132 0.958488i \(-0.407962\pi\)
0.285132 + 0.958488i \(0.407962\pi\)
\(492\) 0 0
\(493\) 1114.73i 2.26111i
\(494\) 0 0
\(495\) −1120.00 −2.26263
\(496\) 0 0
\(497\) 32.3110i 0.0650120i
\(498\) 0 0
\(499\) 460.000 0.921844 0.460922 0.887441i \(-0.347519\pi\)
0.460922 + 0.887441i \(0.347519\pi\)
\(500\) 0 0
\(501\) −174.000 −0.347305
\(502\) 0 0
\(503\) −263.000 −0.522863 −0.261431 0.965222i \(-0.584195\pi\)
−0.261431 + 0.965222i \(0.584195\pi\)
\(504\) 0 0
\(505\) −56.0000 −0.110891
\(506\) 0 0
\(507\) 495.435i 0.977190i
\(508\) 0 0
\(509\) 290.799i 0.571314i −0.958332 0.285657i \(-0.907788\pi\)
0.958332 0.285657i \(-0.0922118\pi\)
\(510\) 0 0
\(511\) −7.00000 −0.0136986
\(512\) 0 0
\(513\) 957.000 592.368i 1.86550 1.15471i
\(514\) 0 0
\(515\) 516.976i 1.00384i
\(516\) 0 0
\(517\) 364.000 0.704062
\(518\) 0 0
\(519\) −870.000 −1.67630
\(520\) 0 0
\(521\) 226.177i 0.434121i −0.976158 0.217060i \(-0.930353\pi\)
0.976158 0.217060i \(-0.0696469\pi\)
\(522\) 0 0
\(523\) 80.7775i 0.154450i −0.997014 0.0772251i \(-0.975394\pi\)
0.997014 0.0772251i \(-0.0246060\pi\)
\(524\) 0 0
\(525\) 48.4665i 0.0923171i
\(526\) 0 0
\(527\) 743.153i 1.41016i
\(528\) 0 0
\(529\) −528.000 −0.998110
\(530\) 0 0
\(531\) 323.110i 0.608493i
\(532\) 0 0
\(533\) −522.000 −0.979362
\(534\) 0 0
\(535\) 64.6220i 0.120789i
\(536\) 0 0
\(537\) −1566.00 −2.91620
\(538\) 0 0
\(539\) 672.000 1.24675
\(540\) 0 0
\(541\) −640.000 −1.18299 −0.591497 0.806307i \(-0.701463\pi\)
−0.591497 + 0.806307i \(0.701463\pi\)
\(542\) 0 0
\(543\) 522.000 0.961326
\(544\) 0 0
\(545\) 64.6220i 0.118572i
\(546\) 0 0
\(547\) 32.3110i 0.0590694i −0.999564 0.0295347i \(-0.990597\pi\)
0.999564 0.0295347i \(-0.00940256\pi\)
\(548\) 0 0
\(549\) 800.000 1.45719
\(550\) 0 0
\(551\) −783.000 + 484.665i −1.42105 + 0.879609i
\(552\) 0 0
\(553\) 96.9330i 0.175286i
\(554\) 0 0
\(555\) −696.000 −1.25405
\(556\) 0 0
\(557\) −844.000 −1.51526 −0.757630 0.652684i \(-0.773643\pi\)
−0.757630 + 0.652684i \(0.773643\pi\)
\(558\) 0 0
\(559\) 1098.57i 1.96525i
\(560\) 0 0
\(561\) 1734.02i 3.09095i
\(562\) 0 0
\(563\) 420.043i 0.746080i −0.927815 0.373040i \(-0.878315\pi\)
0.927815 0.373040i \(-0.121685\pi\)
\(564\) 0 0
\(565\) 387.732i 0.686251i
\(566\) 0 0
\(567\) 139.000 0.245150
\(568\) 0 0
\(569\) 290.799i 0.511070i 0.966800 + 0.255535i \(0.0822516\pi\)
−0.966800 + 0.255535i \(0.917748\pi\)
\(570\) 0 0
\(571\) −458.000 −0.802102 −0.401051 0.916056i \(-0.631355\pi\)
−0.401051 + 0.916056i \(0.631355\pi\)
\(572\) 0 0
\(573\) 360.806i 0.629679i
\(574\) 0 0
\(575\) −9.00000 −0.0156522
\(576\) 0 0
\(577\) −85.0000 −0.147314 −0.0736568 0.997284i \(-0.523467\pi\)
−0.0736568 + 0.997284i \(0.523467\pi\)
\(578\) 0 0
\(579\) −522.000 −0.901554
\(580\) 0 0
\(581\) −32.0000 −0.0550775
\(582\) 0 0
\(583\) 1130.88i 1.93977i
\(584\) 0 0
\(585\) 1292.44i 2.20930i
\(586\) 0 0
\(587\) −488.000 −0.831346 −0.415673 0.909514i \(-0.636454\pi\)
−0.415673 + 0.909514i \(0.636454\pi\)
\(588\) 0 0
\(589\) 522.000 323.110i 0.886248 0.548574i
\(590\) 0 0
\(591\) 764.693i 1.29390i
\(592\) 0 0
\(593\) −610.000 −1.02867 −0.514334 0.857590i \(-0.671961\pi\)
−0.514334 + 0.857590i \(0.671961\pi\)
\(594\) 0 0
\(595\) −92.0000 −0.154622
\(596\) 0 0
\(597\) 1416.30i 2.37236i
\(598\) 0 0
\(599\) 323.110i 0.539416i 0.962942 + 0.269708i \(0.0869271\pi\)
−0.962942 + 0.269708i \(0.913073\pi\)
\(600\) 0 0
\(601\) 96.9330i 0.161286i −0.996743 0.0806431i \(-0.974303\pi\)
0.996743 0.0806431i \(-0.0256974\pi\)
\(602\) 0 0
\(603\) 323.110i 0.535837i
\(604\) 0 0
\(605\) −300.000 −0.495868
\(606\) 0 0
\(607\) 1001.64i 1.65015i −0.565024 0.825075i \(-0.691133\pi\)
0.565024 0.825075i \(-0.308867\pi\)
\(608\) 0 0
\(609\) −261.000 −0.428571
\(610\) 0 0
\(611\) 420.043i 0.687468i
\(612\) 0 0
\(613\) 200.000 0.326264 0.163132 0.986604i \(-0.447840\pi\)
0.163132 + 0.986604i \(0.447840\pi\)
\(614\) 0 0
\(615\) −696.000 −1.13171
\(616\) 0 0
\(617\) 530.000 0.858995 0.429498 0.903068i \(-0.358691\pi\)
0.429498 + 0.903068i \(0.358691\pi\)
\(618\) 0 0
\(619\) 256.000 0.413570 0.206785 0.978386i \(-0.433700\pi\)
0.206785 + 0.978386i \(0.433700\pi\)
\(620\) 0 0
\(621\) 59.2368i 0.0953894i
\(622\) 0 0
\(623\) 129.244i 0.207454i
\(624\) 0 0
\(625\) −319.000 −0.510400
\(626\) 0 0
\(627\) 1218.00 753.923i 1.94258 1.20243i
\(628\) 0 0
\(629\) 743.153i 1.18148i
\(630\) 0 0
\(631\) 826.000 1.30903 0.654517 0.756048i \(-0.272872\pi\)
0.654517 + 0.756048i \(0.272872\pi\)
\(632\) 0 0
\(633\) 435.000 0.687204
\(634\) 0 0
\(635\) 258.488i 0.407068i
\(636\) 0 0
\(637\) 775.464i 1.21737i
\(638\) 0 0
\(639\) 646.220i 1.01130i
\(640\) 0 0
\(641\) 581.598i 0.907329i −0.891173 0.453664i \(-0.850116\pi\)
0.891173 0.453664i \(-0.149884\pi\)
\(642\) 0 0
\(643\) −170.000 −0.264386 −0.132193 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(644\) 0 0
\(645\) 1464.76i 2.27095i
\(646\) 0 0
\(647\) 25.0000 0.0386399 0.0193199 0.999813i \(-0.493850\pi\)
0.0193199 + 0.999813i \(0.493850\pi\)
\(648\) 0 0
\(649\) 226.177i 0.348501i
\(650\) 0 0
\(651\) 174.000 0.267281
\(652\) 0 0
\(653\) −1054.00 −1.61409 −0.807044 0.590491i \(-0.798934\pi\)
−0.807044 + 0.590491i \(0.798934\pi\)
\(654\) 0 0
\(655\) 224.000 0.341985
\(656\) 0 0
\(657\) 140.000 0.213090
\(658\) 0 0
\(659\) 1243.97i 1.88767i 0.330420 + 0.943834i \(0.392810\pi\)
−0.330420 + 0.943834i \(0.607190\pi\)
\(660\) 0 0
\(661\) 533.131i 0.806553i −0.915078 0.403276i \(-0.867871\pi\)
0.915078 0.403276i \(-0.132129\pi\)
\(662\) 0 0
\(663\) 2001.00 3.01810
\(664\) 0 0
\(665\) 40.0000 + 64.6220i 0.0601504 + 0.0971759i
\(666\) 0 0
\(667\) 48.4665i 0.0726634i
\(668\) 0 0
\(669\) 1392.00 2.08072
\(670\) 0 0
\(671\) 560.000 0.834575
\(672\) 0 0
\(673\) 290.799i 0.432093i −0.976383 0.216047i \(-0.930684\pi\)
0.976383 0.216047i \(-0.0693164\pi\)
\(674\) 0 0
\(675\) 533.131i 0.789824i
\(676\) 0 0
\(677\) 953.174i 1.40794i 0.710231 + 0.703969i \(0.248591\pi\)
−0.710231 + 0.703969i \(0.751409\pi\)
\(678\) 0 0
\(679\) 96.9330i 0.142758i
\(680\) 0 0
\(681\) 435.000 0.638767
\(682\) 0 0
\(683\) 872.397i 1.27730i −0.769497 0.638651i \(-0.779493\pi\)
0.769497 0.638651i \(-0.220507\pi\)
\(684\) 0 0
\(685\) 76.0000 0.110949
\(686\) 0 0
\(687\) 1637.09i 2.38296i
\(688\) 0 0
\(689\) 1305.00 1.89405
\(690\) 0 0
\(691\) −668.000 −0.966715 −0.483357 0.875423i \(-0.660583\pi\)
−0.483357 + 0.875423i \(0.660583\pi\)
\(692\) 0 0
\(693\) 280.000 0.404040
\(694\) 0 0
\(695\) 488.000 0.702158
\(696\) 0 0
\(697\) 743.153i 1.06622i
\(698\) 0 0
\(699\) 10.7703i 0.0154082i
\(700\) 0 0
\(701\) −700.000 −0.998573 −0.499287 0.866437i \(-0.666405\pi\)
−0.499287 + 0.866437i \(0.666405\pi\)
\(702\) 0 0
\(703\) 522.000 323.110i 0.742532 0.459616i
\(704\) 0 0
\(705\) 560.057i 0.794407i
\(706\) 0 0
\(707\) 14.0000 0.0198020
\(708\) 0 0
\(709\) 1292.00 1.82228 0.911142 0.412092i \(-0.135202\pi\)
0.911142 + 0.412092i \(0.135202\pi\)
\(710\) 0 0
\(711\) 1938.66i 2.72667i
\(712\) 0 0
\(713\) 32.3110i 0.0453170i
\(714\) 0 0
\(715\) 904.708i 1.26533i
\(716\) 0 0
\(717\) 479.280i 0.668451i
\(718\) 0 0
\(719\) −1061.00 −1.47566 −0.737830 0.674986i \(-0.764150\pi\)
−0.737830 + 0.674986i \(0.764150\pi\)
\(720\) 0 0
\(721\) 129.244i 0.179257i
\(722\) 0 0
\(723\) −870.000 −1.20332
\(724\) 0 0
\(725\) 436.198i 0.601653i
\(726\) 0 0
\(727\) −83.0000 −0.114168 −0.0570839 0.998369i \(-0.518180\pi\)
−0.0570839 + 0.998369i \(0.518180\pi\)
\(728\) 0 0
\(729\) 91.0000 0.124829
\(730\) 0 0
\(731\) −1564.00 −2.13953
\(732\) 0 0
\(733\) 296.000 0.403820 0.201910 0.979404i \(-0.435285\pi\)
0.201910 + 0.979404i \(0.435285\pi\)
\(734\) 0 0
\(735\) 1033.95i 1.40674i
\(736\) 0 0
\(737\) 226.177i 0.306889i
\(738\) 0 0
\(739\) −548.000 −0.741543 −0.370771 0.928724i \(-0.620907\pi\)
−0.370771 + 0.928724i \(0.620907\pi\)
\(740\) 0 0
\(741\) −870.000 1405.53i −1.17409 1.89680i
\(742\) 0 0
\(743\) 613.909i 0.826257i −0.910673 0.413128i \(-0.864436\pi\)
0.910673 0.413128i \(-0.135564\pi\)
\(744\) 0 0
\(745\) 328.000 0.440268
\(746\) 0 0
\(747\) 640.000 0.856760
\(748\) 0 0
\(749\) 16.1555i 0.0215694i
\(750\) 0 0
\(751\) 420.043i 0.559311i −0.960100 0.279656i \(-0.909780\pi\)
0.960100 0.279656i \(-0.0902203\pi\)
\(752\) 0 0
\(753\) 1992.51i 2.64610i
\(754\) 0 0
\(755\) 904.708i 1.19829i
\(756\) 0 0
\(757\) 188.000 0.248349 0.124174 0.992260i \(-0.460372\pi\)
0.124174 + 0.992260i \(0.460372\pi\)
\(758\) 0 0
\(759\) 75.3923i 0.0993311i
\(760\) 0 0
\(761\) −1183.00 −1.55453 −0.777267 0.629171i \(-0.783394\pi\)
−0.777267 + 0.629171i \(0.783394\pi\)
\(762\) 0 0
\(763\) 16.1555i 0.0211736i
\(764\) 0 0
\(765\) 1840.00 2.40523
\(766\) 0 0
\(767\) 261.000 0.340287
\(768\) 0 0
\(769\) −1177.00 −1.53056 −0.765280 0.643698i \(-0.777399\pi\)
−0.765280 + 0.643698i \(0.777399\pi\)
\(770\) 0 0
\(771\) −1044.00 −1.35409
\(772\) 0 0
\(773\) 1017.80i 1.31668i 0.752719 + 0.658342i \(0.228742\pi\)
−0.752719 + 0.658342i \(0.771258\pi\)
\(774\) 0 0
\(775\) 290.799i 0.375224i
\(776\) 0 0
\(777\) 174.000 0.223938
\(778\) 0 0
\(779\) 522.000 323.110i 0.670090 0.414775i
\(780\) 0 0
\(781\) 452.354i 0.579198i
\(782\) 0 0
\(783\) 2871.00 3.66667
\(784\) 0 0
\(785\) −968.000 −1.23312
\(786\) 0 0
\(787\) 113.088i 0.143696i −0.997416 0.0718478i \(-0.977110\pi\)
0.997416 0.0718478i \(-0.0228896\pi\)
\(788\) 0 0
\(789\) 2121.75i 2.68917i
\(790\) 0 0
\(791\) 96.9330i 0.122545i
\(792\) 0 0
\(793\) 646.220i 0.814905i
\(794\) 0 0
\(795\) 1740.00 2.18868
\(796\) 0 0
\(797\) 177.710i 0.222974i −0.993766 0.111487i \(-0.964439\pi\)
0.993766 0.111487i \(-0.0355614\pi\)
\(798\) 0 0
\(799\) −598.000 −0.748436
\(800\) 0