Properties

Label 177.6.d.a.176.1
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 176.1
Root \(-0.422380 - 1.67976i\) of defining polynomial
Character \(\chi\) \(=\) 177.176
Dual form 177.6.d.a.176.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-14.7009 - 5.18489i) q^{3} -32.0000 q^{4} -61.6839i q^{5} +145.431 q^{7} +(189.234 + 152.445i) q^{9} +O(q^{10})\) \(q+(-14.7009 - 5.18489i) q^{3} -32.0000 q^{4} -61.6839i q^{5} +145.431 q^{7} +(189.234 + 152.445i) q^{9} +(470.429 + 165.916i) q^{12} +(-319.824 + 906.809i) q^{15} +1024.00 q^{16} +683.622i q^{17} +2164.66 q^{19} +1973.88i q^{20} +(-2137.97 - 754.045i) q^{21} -679.900 q^{25} +(-1991.50 - 3222.24i) q^{27} -4653.80 q^{28} -5700.02i q^{29} -8970.77i q^{35} +(-6055.48 - 4878.25i) q^{36} -18375.2i q^{41} +(9403.41 - 11672.7i) q^{45} +(-15053.7 - 5309.33i) q^{48} +4343.27 q^{49} +(3544.50 - 10049.9i) q^{51} +8353.58i q^{53} +(-31822.4 - 11223.5i) q^{57} +26738.1i q^{59} +(10234.4 - 29017.9i) q^{60} +(27520.5 + 22170.3i) q^{63} -32768.0 q^{64} -21875.9i q^{68} -59459.7i q^{71} +(9995.15 + 3525.21i) q^{75} -69269.0 q^{76} -1559.45 q^{79} -63164.3i q^{80} +(12569.9 + 57695.6i) q^{81} +(68415.2 + 24129.4i) q^{84} +42168.5 q^{85} +(-29553.9 + 83795.5i) q^{87} -133524. i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 192q^{4} + O(q^{10}) \) \( 6q - 192q^{4} + 3153q^{15} + 6144q^{16} - 3075q^{21} - 18750q^{25} - 11949q^{27} + 81147q^{45} + 100842q^{49} - 131001q^{57} - 100896q^{60} + 180732q^{63} - 196608q^{64} - 197304q^{75} + 98400q^{84} + 221799q^{87} + O(q^{100}) \)

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 \(0.5\pi\)
\(3\) −14.7009 5.18489i −0.943064 0.332611i
\(4\) −32.0000 −1.00000
\(5\) 61.6839i 1.10343i −0.834031 0.551717i \(-0.813973\pi\)
0.834031 0.551717i \(-0.186027\pi\)
\(6\) 0 0
\(7\) 145.431 1.12179 0.560897 0.827886i \(-0.310456\pi\)
0.560897 + 0.827886i \(0.310456\pi\)
\(8\) 0 0
\(9\) 189.234 + 152.445i 0.778740 + 0.627347i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 470.429 + 165.916i 0.943064 + 0.332611i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −319.824 + 906.809i −0.367014 + 1.04061i
\(16\) 1024.00 1.00000
\(17\) 683.622i 0.573712i 0.957974 + 0.286856i \(0.0926101\pi\)
−0.957974 + 0.286856i \(0.907390\pi\)
\(18\) 0 0
\(19\) 2164.66 1.37564 0.687821 0.725881i \(-0.258568\pi\)
0.687821 + 0.725881i \(0.258568\pi\)
\(20\) 1973.88i 1.10343i
\(21\) −2137.97 754.045i −1.05792 0.373120i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −679.900 −0.217568
\(26\) 0 0
\(27\) −1991.50 3222.24i −0.525740 0.850645i
\(28\) −4653.80 −1.12179
\(29\) 5700.02i 1.25858i −0.777170 0.629290i \(-0.783346\pi\)
0.777170 0.629290i \(-0.216654\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8970.77i 1.23783i
\(36\) −6055.48 4878.25i −0.778740 0.627347i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18375.2i 1.70716i −0.520965 0.853578i \(-0.674428\pi\)
0.520965 0.853578i \(-0.325572\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 9403.41 11672.7i 0.692236 0.859289i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −15053.7 5309.33i −0.943064 0.332611i
\(49\) 4343.27 0.258420
\(50\) 0 0
\(51\) 3544.50 10049.9i 0.190823 0.541047i
\(52\) 0 0
\(53\) 8353.58i 0.408492i 0.978920 + 0.204246i \(0.0654742\pi\)
−0.978920 + 0.204246i \(0.934526\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −31822.4 11223.5i −1.29732 0.457553i
\(58\) 0 0
\(59\) 26738.1i 1.00000i
\(60\) 10234.4 29017.9i 0.367014 1.04061i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 27520.5 + 22170.3i 0.873586 + 0.703753i
\(64\) −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 \(-0.753220\pi\)
0.714223 0.699918i \(-0.246780\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 9995.15 + 3525.21i 0.205181 + 0.0723655i
\(76\) −69269.0 −1.37564
\(77\) 0 0
\(78\) 0 0
\(79\) −1559.45 −0.0281127 −0.0140564 0.999901i \(-0.504474\pi\)
−0.0140564 + 0.999901i \(0.504474\pi\)
\(80\) 63164.3i 1.10343i
\(81\) 12569.9 + 57695.6i 0.212873 + 0.977080i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 68415.2 + 24129.4i 1.05792 + 0.373120i
\(85\) 42168.5 0.633053
\(86\) 0 0
\(87\) −29553.9 + 83795.5i −0.418617 + 1.18692i
\(88\) 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
\(95\) 133524.i 1.51793i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 21756.8 0.217568
\(101\) 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
\(105\) −46512.4 + 131878.i −0.411714 + 1.16735i
\(106\) 0 0
\(107\) 234144.i 1.97708i −0.150964 0.988539i \(-0.548238\pi\)
0.150964 0.988539i \(-0.451762\pi\)
\(108\) 63728.0 + 103112.i 0.525740 + 0.850645i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 148922. 1.12179
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 182401.i 1.25858i
\(117\) 0 0
\(118\) 0 0
\(119\) 99420.0i 0.643586i
\(120\) 0 0
\(121\) −161051. −1.00000
\(122\) 0 0
\(123\) −95273.5 + 270133.i −0.567818 + 1.60996i
\(124\) 0 0
\(125\) 150823.i 0.863363i
\(126\) 0 0
\(127\) −193827. −1.06636 −0.533182 0.846001i \(-0.679004\pi\)
−0.533182 + 0.846001i \(0.679004\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 314809. 1.54319
\(134\) 0 0
\(135\) −198760. + 122843.i −0.938632 + 0.580120i
\(136\) 0 0
\(137\) 120553.i 0.548750i 0.961623 + 0.274375i \(0.0884710\pi\)
−0.961623 + 0.274375i \(0.911529\pi\)
\(138\) 0 0
\(139\) 399275. 1.75281 0.876406 0.481574i \(-0.159935\pi\)
0.876406 + 0.481574i \(0.159935\pi\)
\(140\) 287065.i 1.23783i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 193775. + 156104.i 0.778740 + 0.627347i
\(145\) −351599. −1.38876
\(146\) 0 0
\(147\) −63850.1 22519.4i −0.243707 0.0859534i
\(148\) 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
\(153\) −104215. + 129364.i −0.359916 + 0.446772i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 43312.4 122805.i 0.135869 0.385234i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 537581. 1.58480 0.792401 0.610001i \(-0.208831\pi\)
0.792401 + 0.610001i \(0.208831\pi\)
\(164\) 588007.i 1.70716i
\(165\) 0 0
\(166\) 0 0
\(167\) 716872.i 1.98907i −0.104394 0.994536i \(-0.533290\pi\)
0.104394 0.994536i \(-0.466710\pi\)
\(168\) 0 0
\(169\) 371293. 1.00000
\(170\) 0 0
\(171\) 409627. + 329992.i 1.07127 + 0.863004i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −98878.8 −0.244066
\(176\) 0 0
\(177\) 138634. 393074.i 0.332611 0.943064i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −300909. + 373526.i −0.692236 + 0.859289i
\(181\) 201231. 0.456561 0.228281 0.973595i \(-0.426690\pi\)
0.228281 + 0.973595i \(0.426690\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −289626. 468615.i −0.589771 0.954248i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 481720. + 169898.i 0.943064 + 0.332611i
\(193\) −432620. −0.836013 −0.418007 0.908444i \(-0.637271\pi\)
−0.418007 + 0.908444i \(0.637271\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −138985. −0.258420
\(197\) 1.07084e6i 1.96588i −0.183921 0.982941i \(-0.558879\pi\)
0.183921 0.982941i \(-0.441121\pi\)
\(198\) 0 0
\(199\) −923948. −1.65392 −0.826961 0.562260i \(-0.809932\pi\)
−0.826961 + 0.562260i \(0.809932\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 828961.i 1.41187i
\(204\) −113424. + 321596.i −0.190823 + 0.541047i
\(205\) −1.13346e6 −1.88373
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 267315.i 0.408492i
\(213\) −308292. + 874113.i −0.465601 + 1.32014i
\(214\) 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
\(225\) −128660. 103648.i −0.169429 0.136491i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 1.01832e6 + 359152.i 1.29732 + 0.457553i
\(229\) 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
\(237\) 22925.3 + 8085.56i 0.0265121 + 0.00935060i
\(238\) 0 0
\(239\) 1.53957e6i 1.74344i −0.490009 0.871718i \(-0.663006\pi\)
0.490009 0.871718i \(-0.336994\pi\)
\(240\) −327500. + 928573.i −0.367014 + 1.04061i
\(241\) −755113. −0.837470 −0.418735 0.908108i \(-0.637526\pi\)
−0.418735 + 0.908108i \(0.637526\pi\)
\(242\) 0 0
\(243\) 114356. 913352.i 0.124235 0.992253i
\(244\) 0 0
\(245\) 267910.i 0.285150i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.56480e6i 1.56774i −0.620923 0.783872i \(-0.713242\pi\)
0.620923 0.783872i \(-0.286758\pi\)
\(252\) −880657. 709450.i −0.873586 0.703753i
\(253\) 0 0
\(254\) 0 0
\(255\) −619915. 218639.i −0.597010 0.210560i
\(256\) 1.04858e6 1.00000
\(257\) 735549.i 0.694671i −0.937741 0.347335i \(-0.887087\pi\)
0.937741 0.347335i \(-0.112913\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 868940. 1.07864e6i 0.789566 0.980107i
\(262\) 0 0
\(263\) 626525.i 0.558533i −0.960214 0.279267i \(-0.909909\pi\)
0.960214 0.279267i \(-0.0900913\pi\)
\(264\) 0 0
\(265\) 515281. 0.450744
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −2.15946e6 −1.78617 −0.893083 0.449891i \(-0.851463\pi\)
−0.893083 + 0.449891i \(0.851463\pi\)
\(272\) 700029.i 0.573712i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.69981e6 1.33107 0.665536 0.746365i \(-0.268203\pi\)
0.665536 + 0.746365i \(0.268203\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.22706e6i 1.68254i 0.540614 + 0.841271i \(0.318192\pi\)
−0.540614 + 0.841271i \(0.681808\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 1.90271e6i 1.39984i
\(285\) −692309. + 1.96293e6i −0.504880 + 1.43151i
\(286\) 0 0
\(287\) 2.67233e6i 1.91508i
\(288\) 0 0
\(289\) 952518. 0.670855
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.38101e6i 1.62029i 0.586229 + 0.810145i \(0.300612\pi\)
−0.586229 + 0.810145i \(0.699388\pi\)
\(294\) 0 0
\(295\) 1.64931e6 1.10343
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −319845. 112807.i −0.205181 0.0723655i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 2.21661e6 1.37564
\(305\) 0 0
\(306\) 0 0
\(307\) −3.29773e6 −1.99696 −0.998481 0.0551001i \(-0.982452\pi\)
−0.998481 + 0.0551001i \(0.982452\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.82160e6i 1.06796i 0.845499 + 0.533978i \(0.179303\pi\)
−0.845499 + 0.533978i \(0.820697\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 1.36755e6 1.69757e6i 0.776546 0.963945i
\(316\) 49902.3 0.0281127
\(317\) 3.15415e6i 1.76293i 0.472250 + 0.881465i \(0.343442\pi\)
−0.472250 + 0.881465i \(0.656558\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.02126e6i 1.10343i
\(321\) −1.21401e6 + 3.44213e6i −0.657597 + 1.86451i
\(322\) 0 0
\(323\) 1.47981e6i 0.789221i
\(324\) −402237. 1.84626e6i −0.212873 0.977080i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.96174e6 1.98754 0.993769 0.111459i \(-0.0355523\pi\)
0.993769 + 0.111459i \(0.0355523\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −2.18929e6 772142.i −1.05792 0.373120i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −1.34939e6 −0.633053
\(341\) 0 0
\(342\) 0 0
\(343\) −1.81262e6 −0.831899
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 945726. 2.68145e6i 0.418617 1.18692i
\(349\) 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
\(355\) −3.66771e6 −1.54463
\(356\) 0 0
\(357\) 515482. 1.46157e6i 0.214064 0.606943i
\(358\) 0 0
\(359\) 4.22397e6i 1.72976i −0.501981 0.864878i \(-0.667395\pi\)
0.501981 0.864878i \(-0.332605\pi\)
\(360\) 0 0
\(361\) 2.20964e6 0.892389
\(362\) 0 0
\(363\) 2.36760e6 + 835031.i 0.943064 + 0.332611i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 2.80122e6 3.47722e6i 1.07098 1.32943i
\(370\) 0 0
\(371\) 1.21487e6i 0.458243i
\(372\) 0 0
\(373\) 5.36607e6 1.99703 0.998514 0.0544950i \(-0.0173549\pi\)
0.998514 + 0.0544950i \(0.0173549\pi\)
\(374\) 0 0
\(375\) −782002. + 2.21724e6i −0.287164 + 0.814206i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −5.44338e6 −1.94657 −0.973287 0.229591i \(-0.926261\pi\)
−0.973287 + 0.229591i \(0.926261\pi\)
\(380\) 4.27278e6i 1.51793i
\(381\) 2.84944e6 + 1.00497e6i 1.00565 + 0.354684i
\(382\) 0 0
\(383\) 5.18018e6i 1.80446i 0.431252 + 0.902231i \(0.358072\pi\)
−0.431252 + 0.902231i \(0.641928\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.43319e6i 0.815273i 0.913144 + 0.407636i \(0.133647\pi\)
−0.913144 + 0.407636i \(0.866353\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 96192.8i 0.0310206i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) −4.62798e6 1.63225e6i −1.45532 0.513280i
\(400\) −696218. −0.217568
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.55889e6 775361.i 1.07814 0.234891i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 625051. 1.77223e6i 0.182520 0.517507i
\(412\) 0 0
\(413\) 3.88855e6i 1.12179i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.86971e6 2.07020e6i −1.65301 0.583004i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 1.48840e6 4.22011e6i 0.411714 1.16735i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 464795.i 0.124821i
\(426\) 0 0
\(427\) 0 0
\(428\) 7.49261e6i 1.97708i
\(429\) 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
\(433\) 7.80260e6 1.99995 0.999976 0.00698713i \(-0.00222409\pi\)
0.999976 + 0.00698713i \(0.00222409\pi\)
\(434\) 0 0
\(435\) 5.16883e6 + 1.82300e6i 1.30969 + 0.461917i
\(436\) 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
\(441\) 821894. + 662111.i 0.201242 + 0.162119i
\(442\) 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
\(448\) −4.76549e6 −1.12179
\(449\) 1.57248e6i 0.368102i −0.982917 0.184051i \(-0.941079\pi\)
0.982917 0.184051i \(-0.0589212\pi\)
\(450\) 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 \(0.268990\pi\)
−0.663689 + 0.748009i \(0.731010\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 5.83682e6i 1.25858i
\(465\) 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
\(475\) −1.47175e6 −0.299296
\(476\) 3.18144e6i 0.643586i
\(477\) −1.27346e6 + 1.58078e6i −0.256266 + 0.318109i
\(478\) 0 0
\(479\) 8.05733e6i 1.60455i −0.596957 0.802273i \(-0.703624\pi\)
0.596957 0.802273i \(-0.296376\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 5.15363e6 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 1.64860e6 0.314988 0.157494 0.987520i \(-0.449658\pi\)
0.157494 + 0.987520i \(0.449658\pi\)
\(488\) 0 0
\(489\) −7.90293e6 2.78730e6i −1.49457 0.527122i
\(490\) 0 0
\(491\) 1.03035e7i 1.92877i −0.264499 0.964386i \(-0.585207\pi\)
0.264499 0.964386i \(-0.414793\pi\)
\(492\) 3.04875e6 8.64425e6i 0.567818 1.60996i
\(493\) 3.89666e6 0.722063
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.64731e6i 1.57033i
\(498\) 0 0
\(499\) 7.45662e6 1.34057 0.670286 0.742102i \(-0.266171\pi\)
0.670286 + 0.742102i \(0.266171\pi\)
\(500\) 4.82634e6i 0.863363i
\(501\) −3.71690e6 + 1.05387e7i −0.661587 + 1.87582i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5.45835e6 1.92511e6i −0.943064 0.332611i
\(508\) 6.20247e6 1.06636
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.31092e6 6.97505e6i −0.723229 1.17018i
\(514\) 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 \(-0.730928\pi\)
0.663497 0.748179i \(-0.269072\pi\)
\(522\) 0 0
\(523\) −7.19380e6 −1.15002 −0.575009 0.818147i \(-0.695001\pi\)
−0.575009 + 0.818147i \(0.695001\pi\)
\(524\) 0 0
\(525\) 1.45361e6 + 512675.i 0.230170 + 0.0811791i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.43634e6 −1.00000
\(530\) 0 0
\(531\) −4.07609e6 + 5.05975e6i −0.627347 + 0.778740i
\(532\) −1.00739e7 −1.54319
\(533\) 0 0
\(534\) 0 0
\(535\) −1.44429e7 −2.18158
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 6.36033e6 3.93099e6i 0.938632 0.580120i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −2.95828e6 1.04336e6i −0.430566 0.151857i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.06008e7 1.51485 0.757427 0.652920i \(-0.226456\pi\)
0.757427 + 0.652920i \(0.226456\pi\)
\(548\) 3.85768e6i 0.548750i
\(549\) 0 0
\(550\) 0 0
\(551\) 1.23386e7i 1.73136i
\(552\) 0 0
\(553\) −226793. −0.0315367
\(554\) 0 0
\(555\) 0 0
\(556\) −1.27768e7 −1.75281
\(557\) 7.22645e6i 0.986931i 0.869765 + 0.493466i \(0.164270\pi\)
−0.869765 + 0.493466i \(0.835730\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 9.18607e6i 1.23783i
\(561\) 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
\(567\) 1.82806e6 + 8.39075e6i 0.238799 + 1.09608i
\(568\) 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
\(576\) −6.20082e6 4.99532e6i −0.778740 0.627347i
\(577\) 1.59619e7 1.99593 0.997964 0.0637806i \(-0.0203158\pi\)
0.997964 + 0.0637806i \(0.0203158\pi\)
\(578\) 0 0
\(579\) 6.35991e6 + 2.24308e6i 0.788414 + 0.278067i
\(580\) 1.12512e7 1.38876
\(581\) 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 \(0.5\pi\)
\(588\) 2.04320e6 + 720620.i 0.243707 + 0.0859534i
\(589\) 0 0
\(590\) 0 0
\(591\) −5.55217e6 + 1.57423e7i −0.653873 + 1.85395i
\(592\) 0 0
\(593\) 1.04058e7i 1.21517i 0.794254 + 0.607586i \(0.207862\pi\)
−0.794254 + 0.607586i \(0.792138\pi\)
\(594\) 0 0
\(595\) 6.13261e6 0.710155
\(596\) 0 0
\(597\) 1.35829e7 + 4.79057e6i 1.55975 + 0.550112i
\(598\) 0 0
\(599\) 1.75146e7i 1.99449i 0.0741549 + 0.997247i \(0.476374\pi\)
−0.0741549 + 0.997247i \(0.523626\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.93425e6i 1.10343i
\(606\) 0 0
\(607\) −1.80424e7 −1.98757 −0.993786 0.111307i \(-0.964496\pi\)
−0.993786 + 0.111307i \(0.964496\pi\)
\(608\) 0 0
\(609\) −4.29807e6 + 1.21865e7i −0.469602 + 1.33148i
\(610\) 0 0
\(611\) 0 0
\(612\) 3.33488e6 4.13966e6i 0.359916 0.446772i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 1.66628e7 + 5.87684e6i 1.77648 + 0.626550i
\(616\) 0 0
\(617\) 1.73937e7i 1.83941i 0.392613 + 0.919704i \(0.371571\pi\)
−0.392613 + 0.919704i \(0.628429\pi\)
\(618\) 0 0
\(619\) 1.85240e7 1.94316 0.971581 0.236707i \(-0.0760681\pi\)
0.971581 + 0.236707i \(0.0760681\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.14280e7 −1.17023
\(626\) 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
\(635\) 1.19560e7i 1.17666i
\(636\) −1.38600e6 + 3.92977e6i −0.135869 + 0.385234i
\(637\) 0 0
\(638\) 0 0
\(639\) 9.06435e6 1.12518e7i 0.878182 1.09011i
\(640\) 0 0
\(641\) 1.91009e7i 1.83615i −0.396404 0.918076i \(-0.629742\pi\)
0.396404 0.918076i \(-0.370258\pi\)
\(642\) 0 0
\(643\) 1.79845e7 1.71542 0.857710 0.514133i \(-0.171886\pi\)
0.857710 + 0.514133i \(0.171886\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.67270e6i 0.908420i 0.890895 + 0.454210i \(0.150078\pi\)
−0.890895 + 0.454210i \(0.849922\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.72026e7 −1.58480
\(653\) 1.38912e7i 1.27484i 0.770515 + 0.637422i \(0.219999\pi\)
−0.770515 + 0.637422i \(0.780001\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.88162e7i 1.70716i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −2.03432e7 −1.81099 −0.905496 0.424355i \(-0.860501\pi\)
−0.905496 + 0.424355i \(0.860501\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.94186e7i 1.70280i
\(666\) 0 0
\(667\) 0 0
\(668\) 2.29399e7i 1.98907i
\(669\) −6.57765e6 2.31988e6i −0.568205 0.200401i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 1.35402e6 + 2.19080e6i 0.114384 + 0.185073i
\(676\) −1.18814e7 −1.00000
\(677\) 388942.i 0.0326147i 0.999867 + 0.0163074i \(0.00519102\pi\)
−0.999867 + 0.0163074i \(0.994809\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −1.31080e7 1.05597e7i −1.07127 0.863004i
\(685\) 7.43615e6 0.605510
\(686\) 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
\(695\) 2.46288e7i 1.93411i
\(696\) 0 0
\(697\) 1.25617e7 0.979415
\(698\) 0 0
\(699\) 0 0
\(700\) 3.16412e6 0.244066
\(701\) 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
\(708\) −4.43628e6 + 1.25784e7i −0.332611 + 0.943064i
\(709\) −2.64534e7 −1.97636 −0.988181 0.153291i \(-0.951013\pi\)
−0.988181 + 0.153291i \(0.951013\pi\)
\(710\) 0 0
\(711\) −295100. 237730.i −0.0218925 0.0176364i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.98252e6 + 2.26331e7i −0.579885 + 1.64417i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 9.62909e6 1.19528e7i 0.692236 0.859289i
\(721\) 0 0
\(722\) 0 0
\(723\) 1.11009e7 + 3.91518e6i 0.789788 + 0.278552i
\(724\) −6.43940e6 −0.456561
\(725\) 3.87544e6i 0.273827i
\(726\) 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
\(735\) −1.38908e6 + 3.93852e6i −0.0948439 + 0.268915i
\(736\) 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 \(-0.763956\pi\)
0.737419 0.675436i \(-0.236044\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.40519e7i 2.21787i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −8.11332e6 + 2.30040e7i −0.521448 + 1.47848i
\(754\) 0 0
\(755\) 0 0
\(756\) 9.26805e6 + 1.49957e7i 0.589771 + 0.954248i
\(757\) 2.32339e7 1.47361 0.736805 0.676105i \(-0.236334\pi\)
0.736805 + 0.676105i \(0.236334\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.51133e7i 0.946017i −0.881058 0.473008i \(-0.843168\pi\)
0.881058 0.473008i \(-0.156832\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 7.97970e6 + 6.42838e6i 0.492984 + 0.397144i
\(766\) 0 0
\(767\) 0 0
\(768\) −1.54150e7 5.43675e6i −0.943064 0.332611i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −3.81374e6 + 1.08132e7i −0.231055 + 0.655119i
\(772\) 1.38438e7 0.836013
\(773\) 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
\(779\) 3.97761e7i 2.34843i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.83668e7 + 1.13516e7i −1.07061 + 0.661686i
\(784\) 4.44751e6 0.258420
\(785\) 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
\(789\) −3.24846e6 + 9.21049e6i −0.185774 + 0.526733i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −7.57511e6 2.67168e6i −0.425080 0.149922i
\(796\) 2.95663e7 1.65392
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0