Properties

Label 177.6.d.a.176.3
Level $177$
Weight $6$
Character 177.176
Analytic conductor $28.388$
Analytic rank $0$
Dimension $6$
CM discriminant -59
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.149721291.1
Defining polynomial: \(x^{6} - 7 x^{3} + 27\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 176.3
Root \(1.66591 - 0.474089i\) of defining polynomial
Character \(\chi\) \(=\) 177.176
Dual form 177.6.d.a.176.4

$q$-expansion

\(f(q)\) \(=\) \(q+(2.86021 - 15.3238i) q^{3} -32.0000 q^{4} +49.9128i q^{5} -258.614 q^{7} +(-226.638 - 87.6587i) q^{9} +O(q^{10})\) \(q+(2.86021 - 15.3238i) q^{3} -32.0000 q^{4} +49.9128i q^{5} -258.614 q^{7} +(-226.638 - 87.6587i) q^{9} +(-91.5268 + 490.362i) q^{12} +(764.854 + 142.761i) q^{15} +1024.00 q^{16} -683.622i q^{17} +896.052 q^{19} -1597.21i q^{20} +(-739.692 + 3962.96i) q^{21} +633.714 q^{25} +(-1991.50 + 3222.24i) q^{27} +8275.66 q^{28} +3246.37i q^{29} -12908.2i q^{35} +(7252.43 + 2805.08i) q^{36} -18900.1i q^{41} +(4375.29 - 11312.2i) q^{45} +(2928.86 - 15691.6i) q^{48} +50074.3 q^{49} +(-10475.7 - 1955.30i) q^{51} +38850.3i q^{53} +(2562.90 - 13730.9i) q^{57} -26738.1i q^{59} +(-24475.3 - 4568.36i) q^{60} +(58611.9 + 22669.8i) q^{63} -32768.0 q^{64} +21875.9i q^{68} +59459.7i q^{71} +(1812.56 - 9710.91i) q^{75} -28673.7 q^{76} +96849.2 q^{79} +51110.7i q^{80} +(43680.9 + 39733.7i) q^{81} +(23670.1 - 126815. i) q^{84} +34121.5 q^{85} +(49746.7 + 9285.31i) q^{87} +44724.5i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 192q^{4} + O(q^{10}) \) \( 6q - 192q^{4} + 3153q^{15} + 6144q^{16} - 3075q^{21} - 18750q^{25} - 11949q^{27} + 81147q^{45} + 100842q^{49} - 131001q^{57} - 100896q^{60} + 180732q^{63} - 196608q^{64} - 197304q^{75} + 98400q^{84} + 221799q^{87} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(-1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 2.86021 15.3238i 0.183483 0.983023i
\(4\) −32.0000 −1.00000
\(5\) 49.9128i 0.892867i 0.894817 + 0.446434i \(0.147306\pi\)
−0.894817 + 0.446434i \(0.852694\pi\)
\(6\) 0 0
\(7\) −258.614 −1.99484 −0.997418 0.0718079i \(-0.977123\pi\)
−0.997418 + 0.0718079i \(0.977123\pi\)
\(8\) 0 0
\(9\) −226.638 87.6587i −0.932668 0.360736i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −91.5268 + 490.362i −0.183483 + 0.983023i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 764.854 + 142.761i 0.877709 + 0.163826i
\(16\) 1024.00 1.00000
\(17\) 683.622i 0.573712i −0.957974 0.286856i \(-0.907390\pi\)
0.957974 0.286856i \(-0.0926101\pi\)
\(18\) 0 0
\(19\) 896.052 0.569442 0.284721 0.958610i \(-0.408099\pi\)
0.284721 + 0.958610i \(0.408099\pi\)
\(20\) 1597.21i 0.892867i
\(21\) −739.692 + 3962.96i −0.366018 + 1.96097i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 633.714 0.202788
\(26\) 0 0
\(27\) −1991.50 + 3222.24i −0.525740 + 0.850645i
\(28\) 8275.66 1.99484
\(29\) 3246.37i 0.716808i 0.933567 + 0.358404i \(0.116679\pi\)
−0.933567 + 0.358404i \(0.883321\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12908.2i 1.78112i
\(36\) 7252.43 + 2805.08i 0.932668 + 0.360736i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18900.1i 1.75592i −0.478737 0.877958i \(-0.658905\pi\)
0.478737 0.877958i \(-0.341095\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 4375.29 11312.2i 0.322089 0.832749i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 2928.86 15691.6i 0.183483 0.983023i
\(49\) 50074.3 2.97937
\(50\) 0 0
\(51\) −10475.7 1955.30i −0.563972 0.105266i
\(52\) 0 0
\(53\) 38850.3i 1.89978i 0.312578 + 0.949892i \(0.398808\pi\)
−0.312578 + 0.949892i \(0.601192\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2562.90 13730.9i 0.104483 0.559774i
\(58\) 0 0
\(59\) 26738.1i 1.00000i
\(60\) −24475.3 4568.36i −0.877709 0.163826i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 58611.9 + 22669.8i 1.86052 + 0.719609i
\(64\) −32768.0 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 21875.9i 0.573712i
\(69\) 0 0
\(70\) 0 0
\(71\) 59459.7i 1.39984i 0.714223 + 0.699918i \(0.246780\pi\)
−0.714223 + 0.699918i \(0.753220\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1812.56 9710.91i 0.0372082 0.199346i
\(76\) −28673.7 −0.569442
\(77\) 0 0
\(78\) 0 0
\(79\) 96849.2 1.74594 0.872968 0.487777i \(-0.162192\pi\)
0.872968 + 0.487777i \(0.162192\pi\)
\(80\) 51110.7i 0.892867i
\(81\) 43680.9 + 39733.7i 0.739740 + 0.672893i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 23670.1 126815.i 0.366018 1.96097i
\(85\) 34121.5 0.512248
\(86\) 0 0
\(87\) 49746.7 + 9285.31i 0.704639 + 0.131522i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 44724.5i 0.508436i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −20278.8 −0.202788
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) −197802. 36920.1i −1.75089 0.326806i
\(106\) 0 0
\(107\) 148039.i 1.25002i −0.780618 0.625009i \(-0.785095\pi\)
0.780618 0.625009i \(-0.214905\pi\)
\(108\) 63728.0 103112.i 0.525740 0.850645i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −264821. −1.99484
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 103884.i 0.716808i
\(117\) 0 0
\(118\) 0 0
\(119\) 176794.i 1.14446i
\(120\) 0 0
\(121\) −161051. −1.00000
\(122\) 0 0
\(123\) −289621. 54058.2i −1.72611 0.322180i
\(124\) 0 0
\(125\) 187608.i 1.07393i
\(126\) 0 0
\(127\) −169429. −0.932135 −0.466067 0.884749i \(-0.654330\pi\)
−0.466067 + 0.884749i \(0.654330\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −231732. −1.13594
\(134\) 0 0
\(135\) −160831. 99401.3i −0.759513 0.469416i
\(136\) 0 0
\(137\) 426180.i 1.93995i 0.243195 + 0.969977i \(0.421804\pi\)
−0.243195 + 0.969977i \(0.578196\pi\)
\(138\) 0 0
\(139\) 399275. 1.75281 0.876406 0.481574i \(-0.159935\pi\)
0.876406 + 0.481574i \(0.159935\pi\)
\(140\) 413061.i 1.78112i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −232078. 89762.5i −0.932668 0.360736i
\(145\) −162035. −0.640014
\(146\) 0 0
\(147\) 143223. 767330.i 0.546664 2.92879i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −59925.4 + 154935.i −0.206958 + 0.535083i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 595334. + 111120.i 1.86753 + 0.348578i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 537581. 1.58480 0.792401 0.610001i \(-0.208831\pi\)
0.792401 + 0.610001i \(0.208831\pi\)
\(164\) 604802.i 1.75592i
\(165\) 0 0
\(166\) 0 0
\(167\) 423603.i 1.17535i −0.809096 0.587676i \(-0.800043\pi\)
0.809096 0.587676i \(-0.199957\pi\)
\(168\) 0 0
\(169\) 371293. 1.00000
\(170\) 0 0
\(171\) −203080. 78546.8i −0.531100 0.205418i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −163887. −0.404530
\(176\) 0 0
\(177\) −409729. 76476.6i −0.983023 0.183483i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −140009. + 361989.i −0.322089 + 0.832749i
\(181\) −843867. −1.91460 −0.957299 0.289101i \(-0.906644\pi\)
−0.957299 + 0.289101i \(0.906644\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 515030. 833317.i 1.04877 1.69690i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −93723.5 + 502131.i −0.183483 + 0.983023i
\(193\) 1.03055e6 1.99148 0.995739 0.0922178i \(-0.0293956\pi\)
0.995739 + 0.0922178i \(0.0293956\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.60238e6 −2.97937
\(197\) 1.07084e6i 1.96588i 0.183921 + 0.982941i \(0.441121\pi\)
−0.183921 + 0.982941i \(0.558879\pi\)
\(198\) 0 0
\(199\) −82065.4 −0.146902 −0.0734510 0.997299i \(-0.523401\pi\)
−0.0734510 + 0.997299i \(0.523401\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 839557.i 1.42992i
\(204\) 335222. + 62569.8i 0.563972 + 0.105266i
\(205\) 943355. 1.56780
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 1.24321e6i 1.89978i
\(213\) 911150. + 170068.i 1.37607 + 0.256846i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 447431. 0.602510 0.301255 0.953544i \(-0.402595\pi\)
0.301255 + 0.953544i \(0.402595\pi\)
\(224\) 0 0
\(225\) −143624. 55550.6i −0.189134 0.0731530i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −82012.8 + 439390.i −0.104483 + 0.559774i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 855618.i 1.00000i
\(237\) 277009. 1.48410e6i 0.320349 1.71630i
\(238\) 0 0
\(239\) 1.51927e6i 1.72044i −0.509925 0.860219i \(-0.670327\pi\)
0.509925 0.860219i \(-0.329673\pi\)
\(240\) 783211. + 146187.i 0.877709 + 0.163826i
\(241\) 1.79577e6 1.99163 0.995813 0.0914190i \(-0.0291403\pi\)
0.995813 + 0.0914190i \(0.0291403\pi\)
\(242\) 0 0
\(243\) 733808. 555711.i 0.797199 0.603717i
\(244\) 0 0
\(245\) 2.49935e6i 2.66019i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.85585e6i 1.85934i −0.368392 0.929671i \(-0.620091\pi\)
0.368392 0.929671i \(-0.379909\pi\)
\(252\) −1.87558e6 725434.i −1.86052 0.719609i
\(253\) 0 0
\(254\) 0 0
\(255\) 97594.7 522871.i 0.0939887 0.503552i
\(256\) 1.04858e6 1.00000
\(257\) 1.35202e6i 1.27688i 0.769672 + 0.638440i \(0.220420\pi\)
−0.769672 + 0.638440i \(0.779580\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 284573. 735752.i 0.258578 0.668544i
\(262\) 0 0
\(263\) 1.55233e6i 1.38387i 0.721959 + 0.691936i \(0.243242\pi\)
−0.721959 + 0.691936i \(0.756758\pi\)
\(264\) 0 0
\(265\) −1.93912e6 −1.69625
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 137643. 0.113850 0.0569248 0.998378i \(-0.481870\pi\)
0.0569248 + 0.998378i \(0.481870\pi\)
\(272\) 700029.i 0.573712i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 800959. 0.627207 0.313603 0.949554i \(-0.398464\pi\)
0.313603 + 0.949554i \(0.398464\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 125876.i 0.0950996i −0.998869 0.0475498i \(-0.984859\pi\)
0.998869 0.0475498i \(-0.0151413\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 1.90271e6i 1.39984i
\(285\) 685349. + 127921.i 0.499804 + 0.0932892i
\(286\) 0 0
\(287\) 4.88783e6i 3.50277i
\(288\) 0 0
\(289\) 952518. 0.670855
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.68260e6i 1.82552i 0.408492 + 0.912762i \(0.366055\pi\)
−0.408492 + 0.912762i \(0.633945\pi\)
\(294\) 0 0
\(295\) 1.33457e6 0.892867
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −58001.8 + 310749.i −0.0372082 + 0.199346i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 917557. 0.569442
\(305\) 0 0
\(306\) 0 0
\(307\) 1.49127e6 0.903045 0.451522 0.892260i \(-0.350881\pi\)
0.451522 + 0.892260i \(0.350881\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.58709e6i 0.930469i −0.885188 0.465234i \(-0.845970\pi\)
0.885188 0.465234i \(-0.154030\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −1.13151e6 + 2.92548e6i −0.642515 + 1.66120i
\(316\) −3.09917e6 −1.74594
\(317\) 3.15415e6i 1.76293i −0.472250 0.881465i \(-0.656558\pi\)
0.472250 0.881465i \(-0.343442\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.63554e6i 0.892867i
\(321\) −2.26852e6 423422.i −1.22880 0.229357i
\(322\) 0 0
\(323\) 612561.i 0.326695i
\(324\) −1.39779e6 1.27148e6i −0.739740 0.672893i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.36568e6 −1.18682 −0.593411 0.804900i \(-0.702219\pi\)
−0.593411 + 0.804900i \(0.702219\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −757445. + 4.05807e6i −0.366018 + 1.96097i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −1.09189e6 −0.512248
\(341\) 0 0
\(342\) 0 0
\(343\) −8.60341e6 −3.94853
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −1.59190e6 297130.i −0.704639 0.131522i
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) −2.96780e6 −1.24987
\(356\) 0 0
\(357\) 2.70916e6 + 505670.i 1.12503 + 0.209989i
\(358\) 0 0
\(359\) 4.23515e6i 1.73434i −0.498016 0.867168i \(-0.665938\pi\)
0.498016 0.867168i \(-0.334062\pi\)
\(360\) 0 0
\(361\) −1.67319e6 −0.675736
\(362\) 0 0
\(363\) −460640. + 2.46792e6i −0.183483 + 0.983023i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1.65676e6 + 4.28348e6i −0.633421 + 1.63769i
\(370\) 0 0
\(371\) 1.00472e7i 3.78976i
\(372\) 0 0
\(373\) 5.36607e6 1.99703 0.998514 0.0544950i \(-0.0173549\pi\)
0.998514 + 0.0544950i \(0.0173549\pi\)
\(374\) 0 0
\(375\) 2.87487e6 + 536599.i 1.05570 + 0.197048i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.83372e6 1.37095 0.685475 0.728096i \(-0.259594\pi\)
0.685475 + 0.728096i \(0.259594\pi\)
\(380\) 1.43118e6i 0.508436i
\(381\) −484603. + 2.59630e6i −0.171031 + 0.916310i
\(382\) 0 0
\(383\) 5.18018e6i 1.80446i −0.431252 0.902231i \(-0.641928\pi\)
0.431252 0.902231i \(-0.358072\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.43319e6i 0.815273i −0.913144 0.407636i \(-0.866353\pi\)
0.913144 0.407636i \(-0.133647\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.83401e6i 1.55889i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) −662803. + 3.55102e6i −0.208426 + 1.11666i
\(400\) 648923. 0.202788
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.98322e6 + 2.18023e6i −0.600804 + 0.660489i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 6.53070e6 + 1.21897e6i 1.90702 + 0.355948i
\(412\) 0 0
\(413\) 6.91485e6i 1.99484i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.14201e6 6.11841e6i 0.321611 1.72305i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 6.32967e6 + 1.18144e6i 1.75089 + 0.326806i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 433221.i 0.116342i
\(426\) 0 0
\(427\) 0 0
\(428\) 4.73724e6i 1.25002i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −2.03930e6 + 3.29957e6i −0.525740 + 0.850645i
\(433\) −3.85408e6 −0.987874 −0.493937 0.869498i \(-0.664443\pi\)
−0.493937 + 0.869498i \(0.664443\pi\)
\(434\) 0 0
\(435\) −463456. + 2.48300e6i −0.117432 + 0.629149i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −7.86257e6 −1.94717 −0.973584 0.228328i \(-0.926674\pi\)
−0.973584 + 0.228328i \(0.926674\pi\)
\(440\) 0 0
\(441\) −1.13488e7 4.38945e6i −2.77877 1.07477i
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 8.47427e6 1.99484
\(449\) 6.48642e6i 1.51841i 0.650851 + 0.759205i \(0.274412\pi\)
−0.650851 + 0.759205i \(0.725588\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 2.20279e6 + 1.36143e6i 0.488025 + 0.301623i
\(460\) 0 0
\(461\) 6.82636e6i 1.49602i −0.663689 0.748009i \(-0.731010\pi\)
0.663689 0.748009i \(-0.268990\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 3.32428e6i 0.716808i
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 567841. 0.115476
\(476\) 5.65742e6i 1.14446i
\(477\) 3.40557e6 8.80496e6i 0.685320 1.77187i
\(478\) 0 0
\(479\) 8.05733e6i 1.60455i 0.596957 + 0.802273i \(0.296376\pi\)
−0.596957 + 0.802273i \(0.703624\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 5.15363e6 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 8.12788e6 1.55294 0.776470 0.630154i \(-0.217008\pi\)
0.776470 + 0.630154i \(0.217008\pi\)
\(488\) 0 0
\(489\) 1.53760e6 8.23779e6i 0.290784 1.55790i
\(490\) 0 0
\(491\) 2.70444e6i 0.506259i −0.967432 0.253130i \(-0.918540\pi\)
0.967432 0.253130i \(-0.0814600\pi\)
\(492\) 9.26788e6 + 1.72986e6i 1.72611 + 0.322180i
\(493\) 2.21929e6 0.411241
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.53771e7i 2.79244i
\(498\) 0 0
\(499\) 3.42120e6 0.615073 0.307536 0.951536i \(-0.400495\pi\)
0.307536 + 0.951536i \(0.400495\pi\)
\(500\) 6.00345e6i 1.07393i
\(501\) −6.49121e6 1.21160e6i −1.15540 0.215657i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.06198e6 5.68962e6i 0.183483 0.983023i
\(508\) 5.42173e6 0.932135
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.78449e6 + 2.88730e6i −0.299378 + 0.484393i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.27107e6i 1.49636i 0.663497 + 0.748179i \(0.269072\pi\)
−0.663497 + 0.748179i \(0.730928\pi\)
\(522\) 0 0
\(523\) 1.24612e7 1.99208 0.996041 0.0888983i \(-0.0283346\pi\)
0.996041 + 0.0888983i \(0.0283346\pi\)
\(524\) 0 0
\(525\) −468753. + 2.51138e6i −0.0742243 + 0.397662i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.43634e6 −1.00000
\(530\) 0 0
\(531\) −2.34383e6 + 6.05987e6i −0.360736 + 0.932668i
\(532\) 7.41542e6 1.13594
\(533\) 0 0
\(534\) 0 0
\(535\) 7.38903e6 1.11610
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 5.14659e6 + 3.18084e6i 0.759513 + 0.469416i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −2.41364e6 + 1.29313e7i −0.351296 + 1.88209i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.06008e7 1.51485 0.757427 0.652920i \(-0.226456\pi\)
0.757427 + 0.652920i \(0.226456\pi\)
\(548\) 1.36378e7i 1.93995i
\(549\) 0 0
\(550\) 0 0
\(551\) 2.90892e6i 0.408180i
\(552\) 0 0
\(553\) −2.50466e7 −3.48286
\(554\) 0 0
\(555\) 0 0
\(556\) −1.27768e7 −1.75281
\(557\) 1.46439e7i 1.99994i 0.00752886 + 0.999972i \(0.497603\pi\)
−0.00752886 + 0.999972i \(0.502397\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.32180e7i 1.78112i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.12965e7 1.02757e7i −1.47566 1.34231i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 7.42649e6 + 2.87240e6i 0.932668 + 0.360736i
\(577\) −8.86441e6 −1.10844 −0.554218 0.832372i \(-0.686982\pi\)
−0.554218 + 0.832372i \(0.686982\pi\)
\(578\) 0 0
\(579\) 2.94759e6 1.57919e7i 0.365402 1.95767i
\(580\) 5.18513e6 0.640014
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −4.58315e6 + 2.45546e7i −0.546664 + 2.92879i
\(589\) 0 0
\(590\) 0 0
\(591\) 1.64093e7 + 3.06282e6i 1.93251 + 0.360706i
\(592\) 0 0
\(593\) 1.69832e7i 1.98327i 0.129057 + 0.991637i \(0.458805\pi\)
−0.129057 + 0.991637i \(0.541195\pi\)
\(594\) 0 0
\(595\) −8.82430e6 −1.02185
\(596\) 0 0
\(597\) −234725. + 1.25756e6i −0.0269540 + 0.144408i
\(598\) 0 0
\(599\) 9.88518e6i 1.12569i 0.826564 + 0.562843i \(0.190293\pi\)
−0.826564 + 0.562843i \(0.809707\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.03850e6i 0.892867i
\(606\) 0 0
\(607\) 1.07713e7 1.18657 0.593287 0.804991i \(-0.297830\pi\)
0.593287 + 0.804991i \(0.297830\pi\)
\(608\) 0 0
\(609\) −1.28652e7 2.40131e6i −1.40564 0.262365i
\(610\) 0 0
\(611\) 0 0
\(612\) 1.91761e6 4.95792e6i 0.206958 0.535083i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 2.69820e6 1.44558e7i 0.287664 1.54118i
\(616\) 0 0
\(617\) 1.51272e7i 1.59973i 0.600180 + 0.799865i \(0.295096\pi\)
−0.600180 + 0.799865i \(0.704904\pi\)
\(618\) 0 0
\(619\) −1.31704e7 −1.38157 −0.690785 0.723060i \(-0.742735\pi\)
−0.690785 + 0.723060i \(0.742735\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.38368e6 −0.756088
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.77761e7 1.77731 0.888655 0.458577i \(-0.151641\pi\)
0.888655 + 0.458577i \(0.151641\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.45668e6i 0.832273i
\(636\) −1.90507e7 3.55584e6i −1.86753 0.348578i
\(637\) 0 0
\(638\) 0 0
\(639\) 5.21217e6 1.34759e7i 0.504971 1.30558i
\(640\) 0 0
\(641\) 1.91009e7i 1.83615i 0.396404 + 0.918076i \(0.370258\pi\)
−0.396404 + 0.918076i \(0.629742\pi\)
\(642\) 0 0
\(643\) −1.83283e7 −1.74822 −0.874108 0.485733i \(-0.838553\pi\)
−0.874108 + 0.485733i \(0.838553\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.12667e7i 1.99728i 0.0520897 + 0.998642i \(0.483412\pi\)
−0.0520897 + 0.998642i \(0.516588\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.72026e7 −1.58480
\(653\) 7.59642e6i 0.697150i −0.937281 0.348575i \(-0.886666\pi\)
0.937281 0.348575i \(-0.113334\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.93537e7i 1.75592i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 1.91517e6 0.170492 0.0852460 0.996360i \(-0.472832\pi\)
0.0852460 + 0.996360i \(0.472832\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.15664e7i 1.01425i
\(666\) 0 0
\(667\) 0 0
\(668\) 1.35553e7i 1.17535i
\(669\) 1.27975e6 6.85635e6i 0.110550 0.592281i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.26204e6 + 2.04198e6i −0.106614 + 0.172501i
\(676\) −1.18814e7 −1.00000
\(677\) 388942.i 0.0326147i −0.999867 0.0163074i \(-0.994809\pi\)
0.999867 0.0163074i \(-0.00519102\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 6.49855e6 + 2.51350e6i 0.531100 + 0.205418i
\(685\) −2.12718e7 −1.73212
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.99289e7i 1.56503i
\(696\) 0 0
\(697\) −1.29205e7 −1.00739
\(698\) 0 0
\(699\) 0 0
\(700\) 5.24440e6 0.404530
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 1.31113e7 + 2.44725e6i 0.983023 + 0.183483i
\(709\) 1.67805e7 1.25369 0.626845 0.779144i \(-0.284346\pi\)
0.626845 + 0.779144i \(0.284346\pi\)
\(710\) 0 0
\(711\) −2.19497e7 8.48968e6i −1.62838 0.629821i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −2.32809e7 4.34542e6i −1.69123 0.315671i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 4.48030e6 1.15836e7i 0.322089 0.832749i
\(721\) 0 0
\(722\) 0 0
\(723\) 5.13628e6 2.75180e7i 0.365429 1.95781i
\(724\) 2.70037e7 1.91460
\(725\) 2.05727e6i 0.145360i
\(726\) 0 0
\(727\) −1.14089e7 −0.800586 −0.400293 0.916387i \(-0.631092\pi\)
−0.400293 + 0.916387i \(0.631092\pi\)
\(728\) 0 0
\(729\) −6.41676e6 1.28342e7i −0.447195 0.894436i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.70725e7 1.17365 0.586824 0.809714i \(-0.300378\pi\)
0.586824 + 0.809714i \(0.300378\pi\)
\(734\) 0 0
\(735\) 3.82996e7 + 7.14867e6i 2.61502 + 0.488098i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.03276e7i 1.35087i 0.737419 + 0.675436i \(0.236044\pi\)
−0.737419 + 0.675436i \(0.763956\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.82849e7i 2.49358i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −2.84387e7 5.30813e6i −1.82778 0.341157i
\(754\) 0 0
\(755\) 0 0
\(756\) −1.64810e7 + 2.66662e7i −1.04877 + 1.69690i
\(757\) 6.84658e6 0.434244 0.217122 0.976144i \(-0.430333\pi\)
0.217122 + 0.976144i \(0.430333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.19363e7i 1.99905i −0.0308916 0.999523i \(-0.509835\pi\)
0.0308916 0.999523i \(-0.490165\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −7.73324e6 2.99105e6i −0.477758 0.184786i
\(766\) 0 0
\(767\) 0 0
\(768\) 2.99915e6 1.60682e7i 0.183483 0.983023i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 2.07181e7 + 3.86706e6i 1.25520 + 0.234285i
\(772\) −3.29776e7 −1.99148
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.69354e7i 0.999892i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.04606e7 6.46514e6i −0.609749 0.376855i
\(784\) 5.12761e7 2.97937
\(785\) 0 0
\(786\) 0 0
\(787\) 3.71139e6 0.213599 0.106800 0.994281i \(-0.465940\pi\)
0.106800 + 0.994281i \(0.465940\pi\)
\(788\) 3.42668e7i 1.96588i
\(789\) 2.37877e7 + 4.44001e6i 1.36038 + 0.253917i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −5.54631e6 + 2.97148e7i −0.311234 + 1.66746i
\(796\) 2.62609e6 0.146902
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0