Properties

Label 360.2.m.a.179.1
Level $360$
Weight $2$
Character 360.179
Analytic conductor $2.875$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 179.1
Root \(-1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 360.179
Dual form 360.2.m.a.179.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -5.16228 q^{7} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -5.16228 q^{7} +2.82843i q^{8} -3.16228 q^{10} +3.05792i q^{11} +0.837722 q^{13} +7.30056i q^{14} +4.00000 q^{16} -6.32456 q^{19} +4.47214i q^{20} +4.32456 q^{22} -4.47214i q^{23} -5.00000 q^{25} -1.18472i q^{26} +10.3246 q^{28} -5.65685i q^{32} +11.5432i q^{35} -11.1623 q^{37} +8.94427i q^{38} +6.32456 q^{40} -10.3585i q^{41} -6.11584i q^{44} -6.32456 q^{46} -2.82843i q^{47} +19.6491 q^{49} +7.07107i q^{50} -1.67544 q^{52} +5.65685i q^{53} +6.83772 q^{55} -14.6011i q^{56} -5.42736i q^{59} -8.00000 q^{64} -1.87320i q^{65} +16.3246 q^{70} +15.7858i q^{74} +12.6491 q^{76} -15.7858i q^{77} -8.94427i q^{80} -14.6491 q^{82} -8.64911 q^{88} -18.8438i q^{89} -4.32456 q^{91} +8.94427i q^{92} -4.00000 q^{94} +14.1421i q^{95} -27.7880i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{4} - 8q^{7} + 16q^{13} + 16q^{16} - 8q^{22} - 20q^{25} + 16q^{28} - 32q^{37} + 28q^{49} - 32q^{52} + 40q^{55} - 32q^{64} + 40q^{70} - 8q^{82} + 16q^{88} + 8q^{91} - 16q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) − 2.23607i − 1.00000i
\(6\) 0 0
\(7\) −5.16228 −1.95116 −0.975579 0.219650i \(-0.929509\pi\)
−0.975579 + 0.219650i \(0.929509\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) −3.16228 −1.00000
\(11\) 3.05792i 0.921998i 0.887401 + 0.460999i \(0.152509\pi\)
−0.887401 + 0.460999i \(0.847491\pi\)
\(12\) 0 0
\(13\) 0.837722 0.232342 0.116171 0.993229i \(-0.462938\pi\)
0.116171 + 0.993229i \(0.462938\pi\)
\(14\) 7.30056i 1.95116i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −6.32456 −1.45095 −0.725476 0.688247i \(-0.758380\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 4.47214i 1.00000i
\(21\) 0 0
\(22\) 4.32456 0.921998
\(23\) − 4.47214i − 0.932505i −0.884652 0.466252i \(-0.845604\pi\)
0.884652 0.466252i \(-0.154396\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) − 1.18472i − 0.232342i
\(27\) 0 0
\(28\) 10.3246 1.95116
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 5.65685i − 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 11.5432i 1.95116i
\(36\) 0 0
\(37\) −11.1623 −1.83507 −0.917534 0.397658i \(-0.869823\pi\)
−0.917534 + 0.397658i \(0.869823\pi\)
\(38\) 8.94427i 1.45095i
\(39\) 0 0
\(40\) 6.32456 1.00000
\(41\) − 10.3585i − 1.61772i −0.587999 0.808862i \(-0.700084\pi\)
0.587999 0.808862i \(-0.299916\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) − 6.11584i − 0.921998i
\(45\) 0 0
\(46\) −6.32456 −0.932505
\(47\) − 2.82843i − 0.412568i −0.978492 0.206284i \(-0.933863\pi\)
0.978492 0.206284i \(-0.0661372\pi\)
\(48\) 0 0
\(49\) 19.6491 2.80702
\(50\) 7.07107i 1.00000i
\(51\) 0 0
\(52\) −1.67544 −0.232342
\(53\) 5.65685i 0.777029i 0.921443 + 0.388514i \(0.127012\pi\)
−0.921443 + 0.388514i \(0.872988\pi\)
\(54\) 0 0
\(55\) 6.83772 0.921998
\(56\) − 14.6011i − 1.95116i
\(57\) 0 0
\(58\) 0 0
\(59\) − 5.42736i − 0.706582i −0.935513 0.353291i \(-0.885063\pi\)
0.935513 0.353291i \(-0.114937\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) − 1.87320i − 0.232342i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 16.3246 1.95116
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 15.7858i 1.83507i
\(75\) 0 0
\(76\) 12.6491 1.45095
\(77\) − 15.7858i − 1.79896i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) − 8.94427i − 1.00000i
\(81\) 0 0
\(82\) −14.6491 −1.61772
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −8.64911 −0.921998
\(89\) − 18.8438i − 1.99744i −0.0506267 0.998718i \(-0.516122\pi\)
0.0506267 0.998718i \(-0.483878\pi\)
\(90\) 0 0
\(91\) −4.32456 −0.453337
\(92\) 8.94427i 0.932505i
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 14.1421i 1.45095i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 27.7880i − 2.80702i
\(99\) 0 0
\(100\) 10.0000 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −17.1623 −1.69105 −0.845525 0.533936i \(-0.820712\pi\)
−0.845525 + 0.533936i \(0.820712\pi\)
\(104\) 2.36944i 0.232342i
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) − 9.67000i − 0.921998i
\(111\) 0 0
\(112\) −20.6491 −1.95116
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −10.0000 −0.932505
\(116\) 0 0
\(117\) 0 0
\(118\) −7.67544 −0.706582
\(119\) 0 0
\(120\) 0 0
\(121\) 1.64911 0.149919
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 13.8114 1.22556 0.612781 0.790253i \(-0.290051\pi\)
0.612781 + 0.790253i \(0.290051\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 0 0
\(130\) −2.64911 −0.232342
\(131\) 20.0285i 1.74990i 0.484216 + 0.874948i \(0.339105\pi\)
−0.484216 + 0.874948i \(0.660895\pi\)
\(132\) 0 0
\(133\) 32.6491 2.83104
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) − 23.0864i − 1.95116i
\(141\) 0 0
\(142\) 0 0
\(143\) 2.56169i 0.214219i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 22.3246 1.83507
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) − 17.8885i − 1.45095i
\(153\) 0 0
\(154\) −22.3246 −1.79896
\(155\) 0 0
\(156\) 0 0
\(157\) 7.81139 0.623417 0.311708 0.950178i \(-0.399099\pi\)
0.311708 + 0.950178i \(0.399099\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −12.6491 −1.00000
\(161\) 23.0864i 1.81946i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 20.7170i 1.61772i
\(165\) 0 0
\(166\) 0 0
\(167\) − 4.47214i − 0.346064i −0.984916 0.173032i \(-0.944644\pi\)
0.984916 0.173032i \(-0.0553564\pi\)
\(168\) 0 0
\(169\) −12.2982 −0.946017
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.6274i 1.72033i 0.510015 + 0.860165i \(0.329640\pi\)
−0.510015 + 0.860165i \(0.670360\pi\)
\(174\) 0 0
\(175\) 25.8114 1.95116
\(176\) 12.2317i 0.921998i
\(177\) 0 0
\(178\) −26.6491 −1.99744
\(179\) − 13.9126i − 1.03988i −0.854203 0.519940i \(-0.825954\pi\)
0.854203 0.519940i \(-0.174046\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 6.11584i 0.453337i
\(183\) 0 0
\(184\) 12.6491 0.932505
\(185\) 24.9596i 1.83507i
\(186\) 0 0
\(187\) 0 0
\(188\) 5.65685i 0.412568i
\(189\) 0 0
\(190\) 20.0000 1.45095
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −39.2982 −2.80702
\(197\) 22.3607i 1.59313i 0.604551 + 0.796566i \(0.293352\pi\)
−0.604551 + 0.796566i \(0.706648\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) − 14.1421i − 1.00000i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −23.1623 −1.61772
\(206\) 24.2711i 1.69105i
\(207\) 0 0
\(208\) 3.35089 0.232342
\(209\) − 19.3400i − 1.33778i
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) − 11.3137i − 0.777029i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −13.6754 −0.921998
\(221\) 0 0
\(222\) 0 0
\(223\) 1.81139 0.121300 0.0606498 0.998159i \(-0.480683\pi\)
0.0606498 + 0.998159i \(0.480683\pi\)
\(224\) 29.2023i 1.95116i
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 14.1421i 0.932505i
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −6.32456 −0.412568
\(236\) 10.8547i 0.706582i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −25.2982 −1.62960 −0.814801 0.579741i \(-0.803154\pi\)
−0.814801 + 0.579741i \(0.803154\pi\)
\(242\) − 2.33219i − 0.149919i
\(243\) 0 0
\(244\) 0 0
\(245\) − 43.9367i − 2.80702i
\(246\) 0 0
\(247\) −5.29822 −0.337118
\(248\) 0 0
\(249\) 0 0
\(250\) 15.8114 1.00000
\(251\) − 23.7749i − 1.50066i −0.661065 0.750329i \(-0.729895\pi\)
0.661065 0.750329i \(-0.270105\pi\)
\(252\) 0 0
\(253\) 13.6754 0.859768
\(254\) − 19.5323i − 1.22556i
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 57.6228 3.58051
\(260\) 3.74641i 0.232342i
\(261\) 0 0
\(262\) 28.3246 1.74990
\(263\) − 31.3050i − 1.93035i −0.261612 0.965173i \(-0.584254\pi\)
0.261612 0.965173i \(-0.415746\pi\)
\(264\) 0 0
\(265\) 12.6491 0.777029
\(266\) − 46.1728i − 2.83104i
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 15.2896i − 0.921998i
\(276\) 0 0
\(277\) −30.1359 −1.81069 −0.905347 0.424673i \(-0.860389\pi\)
−0.905347 + 0.424673i \(0.860389\pi\)
\(278\) − 19.7990i − 1.18746i
\(279\) 0 0
\(280\) −32.6491 −1.95116
\(281\) 33.4449i 1.99516i 0.0695635 + 0.997578i \(0.477839\pi\)
−0.0695635 + 0.997578i \(0.522161\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 3.62278 0.214219
\(287\) 53.4734i 3.15643i
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 4.47214i − 0.261265i −0.991431 0.130632i \(-0.958299\pi\)
0.991431 0.130632i \(-0.0417008\pi\)
\(294\) 0 0
\(295\) −12.1359 −0.706582
\(296\) − 31.5717i − 1.83507i
\(297\) 0 0
\(298\) 0 0
\(299\) − 3.74641i − 0.216660i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −25.2982 −1.45095
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 31.5717i 1.79896i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) − 11.0470i − 0.623417i
\(315\) 0 0
\(316\) 0 0
\(317\) − 31.3050i − 1.75826i −0.476581 0.879131i \(-0.658124\pi\)
0.476581 0.879131i \(-0.341876\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 17.8885i 1.00000i
\(321\) 0 0
\(322\) 32.6491 1.81946
\(323\) 0 0
\(324\) 0 0
\(325\) −4.18861 −0.232342
\(326\) 0 0
\(327\) 0 0
\(328\) 29.2982 1.61772
\(329\) 14.6011i 0.804986i
\(330\) 0 0
\(331\) 31.6228 1.73814 0.869072 0.494685i \(-0.164716\pi\)
0.869072 + 0.494685i \(0.164716\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −6.32456 −0.346064
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 17.3923i 0.946017i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −65.2982 −3.52577
\(344\) 0 0
\(345\) 0 0
\(346\) 32.0000 1.72033
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) − 36.5028i − 1.95116i
\(351\) 0 0
\(352\) 17.2982 0.921998
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 37.6875i 1.99744i
\(357\) 0 0
\(358\) −19.6754 −1.03988
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 21.0000 1.10526
\(362\) 0 0
\(363\) 0 0
\(364\) 8.64911 0.453337
\(365\) 0 0
\(366\) 0 0
\(367\) 37.8114 1.97374 0.986869 0.161521i \(-0.0516401\pi\)
0.986869 + 0.161521i \(0.0516401\pi\)
\(368\) − 17.8885i − 0.932505i
\(369\) 0 0
\(370\) 35.2982 1.83507
\(371\) − 29.2023i − 1.51611i
\(372\) 0 0
\(373\) −6.13594 −0.317707 −0.158854 0.987302i \(-0.550780\pi\)
−0.158854 + 0.987302i \(0.550780\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) − 28.2843i − 1.45095i
\(381\) 0 0
\(382\) 0 0
\(383\) − 36.7696i − 1.87884i −0.342773 0.939418i \(-0.611366\pi\)
0.342773 0.939418i \(-0.388634\pi\)
\(384\) 0 0
\(385\) −35.2982 −1.79896
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 55.5761i 2.80702i
\(393\) 0 0
\(394\) 31.6228 1.59313
\(395\) 0 0
\(396\) 0 0
\(397\) 24.8377 1.24657 0.623285 0.781995i \(-0.285798\pi\)
0.623285 + 0.781995i \(0.285798\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −20.0000 −1.00000
\(401\) − 35.8143i − 1.78848i −0.447586 0.894241i \(-0.647716\pi\)
0.447586 0.894241i \(-0.352284\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 34.1334i − 1.69193i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 32.7564i 1.61772i
\(411\) 0 0
\(412\) 34.3246 1.69105
\(413\) 28.0175i 1.37865i
\(414\) 0 0
\(415\) 0 0
\(416\) − 4.73887i − 0.232342i
\(417\) 0 0
\(418\) −27.3509 −1.33778
\(419\) − 40.7455i − 1.99055i −0.0971178 0.995273i \(-0.530962\pi\)
0.0971178 0.995273i \(-0.469038\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 31.1127i 1.51454i
\(423\) 0 0
\(424\) −16.0000 −0.777029
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 28.2843i 1.35302i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 19.3400i 0.921998i
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −42.1359 −1.99744
\(446\) − 2.56169i − 0.121300i
\(447\) 0 0
\(448\) 41.2982 1.95116
\(449\) 41.9302i 1.97881i 0.145191 + 0.989404i \(0.453620\pi\)
−0.145191 + 0.989404i \(0.546380\pi\)
\(450\) 0 0
\(451\) 31.6754 1.49154
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.67000i 0.453337i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 20.0000 0.932505
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −10.1886 −0.473505 −0.236752 0.971570i \(-0.576083\pi\)
−0.236752 + 0.971570i \(0.576083\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8.94427i 0.412568i
\(471\) 0 0
\(472\) 15.3509 0.706582
\(473\) 0 0
\(474\) 0 0
\(475\) 31.6228 1.45095
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −9.35089 −0.426364
\(482\) 35.7771i 1.62960i
\(483\) 0 0
\(484\) −3.29822 −0.149919
\(485\) 0 0
\(486\) 0 0
\(487\) −0.135944 −0.00616019 −0.00308010 0.999995i \(-0.500980\pi\)
−0.00308010 + 0.999995i \(0.500980\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −62.1359 −2.80702
\(491\) 29.8907i 1.34895i 0.738298 + 0.674475i \(0.235630\pi\)
−0.738298 + 0.674475i \(0.764370\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 7.49282i 0.337118i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −44.2719 −1.98188 −0.990941 0.134298i \(-0.957122\pi\)
−0.990941 + 0.134298i \(0.957122\pi\)
\(500\) − 22.3607i − 1.00000i
\(501\) 0 0
\(502\) −33.6228 −1.50066
\(503\) − 19.7990i − 0.882793i −0.897312 0.441397i \(-0.854483\pi\)
0.897312 0.441397i \(-0.145517\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) − 19.3400i − 0.859768i
\(507\) 0 0
\(508\) −27.6228 −1.22556
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.6274i − 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) 38.3760i 1.69105i
\(516\) 0 0
\(517\) 8.64911 0.380387
\(518\) − 81.4909i − 3.58051i
\(519\) 0 0
\(520\) 5.29822 0.232342
\(521\) − 8.98151i − 0.393487i −0.980455 0.196744i \(-0.936963\pi\)
0.980455 0.196744i \(-0.0630367\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) − 40.0570i − 1.74990i
\(525\) 0 0
\(526\) −44.2719 −1.93035
\(527\) 0 0
\(528\) 0 0
\(529\) 3.00000 0.130435
\(530\) − 17.8885i − 0.777029i
\(531\) 0 0
\(532\) −65.2982 −2.83104
\(533\) − 8.67753i − 0.375866i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 60.0855i 2.58806i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −21.6228 −0.921998
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 42.6187i 1.81069i
\(555\) 0 0
\(556\) −28.0000 −1.18746
\(557\) − 45.2548i − 1.91751i −0.284236 0.958754i \(-0.591740\pi\)
0.284236 0.958754i \(-0.408260\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 46.1728i 1.95116i
\(561\) 0 0
\(562\) 47.2982 1.99516
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.5826i 0.988636i 0.869281 + 0.494318i \(0.164582\pi\)
−0.869281 + 0.494318i \(0.835418\pi\)
\(570\) 0 0
\(571\) −44.2719 −1.85272 −0.926360 0.376638i \(-0.877080\pi\)
−0.926360 + 0.376638i \(0.877080\pi\)
\(572\) − 5.12338i − 0.214219i
\(573\) 0 0
\(574\) 75.6228 3.15643
\(575\) 22.3607i 0.932505i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 24.0416i 1.00000i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −17.2982 −0.716419
\(584\) 0 0
\(585\) 0 0
\(586\) −6.32456 −0.261265
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 17.1628i 0.706582i
\(591\) 0 0
\(592\) −44.6491 −1.83507
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −5.29822 −0.216660
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −25.2982 −1.03194 −0.515968 0.856608i \(-0.672568\pi\)
−0.515968 + 0.856608i \(0.672568\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 3.68752i − 0.149919i
\(606\) 0 0
\(607\) 32.7851 1.33070 0.665352 0.746530i \(-0.268281\pi\)
0.665352 + 0.746530i \(0.268281\pi\)
\(608\) 35.7771i 1.45095i
\(609\) 0 0
\(610\) 0 0
\(611\) − 2.36944i − 0.0958571i
\(612\) 0 0
\(613\) 38.7851 1.56651 0.783257 0.621698i \(-0.213557\pi\)
0.783257 + 0.621698i \(0.213557\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 44.6491 1.79896
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −46.0000 −1.84890 −0.924448 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 97.2768i 3.89731i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −15.6228 −0.623417
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −44.2719 −1.75826
\(635\) − 30.8832i − 1.22556i
\(636\) 0 0
\(637\) 16.4605 0.652189
\(638\) 0 0
\(639\) 0 0
\(640\) 25.2982 1.00000
\(641\) − 37.1913i − 1.46897i −0.678626 0.734484i \(-0.737424\pi\)
0.678626 0.734484i \(-0.262576\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) − 46.1728i − 1.81946i
\(645\) 0 0
\(646\) 0 0
\(647\) 31.1127i 1.22317i 0.791180 + 0.611583i \(0.209467\pi\)
−0.791180 + 0.611583i \(0.790533\pi\)
\(648\) 0 0
\(649\) 16.5964 0.651467
\(650\) 5.92359i 0.232342i
\(651\) 0 0
\(652\) 0 0
\(653\) − 4.47214i − 0.175008i −0.996164 0.0875041i \(-0.972111\pi\)
0.996164 0.0875041i \(-0.0278891\pi\)
\(654\) 0 0
\(655\) 44.7851 1.74990
\(656\) − 41.4339i − 1.61772i
\(657\) 0 0
\(658\) 20.6491 0.804986
\(659\) 12.9202i 0.503299i 0.967818 + 0.251649i \(0.0809729\pi\)
−0.967818 + 0.251649i \(0.919027\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) − 44.7214i − 1.73814i
\(663\) 0 0
\(664\) 0 0
\(665\) − 73.0056i − 2.83104i
\(666\) 0 0
\(667\) 0 0
\(668\) 8.94427i 0.346064i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 24.5964 0.946017
\(677\) 49.1935i 1.89066i 0.326116 + 0.945330i \(0.394260\pi\)
−0.326116 + 0.945330i \(0.605740\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 92.3456i 3.52577i
\(687\) 0 0
\(688\) 0 0
\(689\) 4.73887i 0.180537i
\(690\) 0 0
\(691\) 31.6228 1.20299 0.601494 0.798878i \(-0.294573\pi\)
0.601494 + 0.798878i \(0.294573\pi\)
\(692\) − 45.2548i − 1.72033i
\(693\) 0 0
\(694\) 0 0
\(695\) − 31.3050i − 1.18746i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −51.6228 −1.95116
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 70.5964 2.66260
\(704\) − 24.4634i − 0.921998i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 53.2982 1.99744
\(713\) 0 0
\(714\) 0 0
\(715\) 5.72811 0.214219
\(716\) 27.8253i 1.03988i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 88.5964 3.29950
\(722\) − 29.6985i − 1.10526i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −41.1623 −1.52662 −0.763312 0.646030i \(-0.776428\pi\)
−0.763312 + 0.646030i \(0.776428\pi\)
\(728\) − 12.2317i − 0.453337i
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 50.7851 1.87579 0.937894 0.346921i \(-0.112773\pi\)
0.937894 + 0.346921i \(0.112773\pi\)
\(734\) − 53.4734i − 1.97374i
\(735\) 0 0
\(736\) −25.2982 −0.932505
\(737\) 0 0
\(738\) 0 0
\(739\) −6.32456 −0.232653 −0.116326 0.993211i \(-0.537112\pi\)
−0.116326 + 0.993211i \(0.537112\pi\)
\(740\) − 49.9192i − 1.83507i
\(741\) 0 0
\(742\) −41.2982 −1.51611
\(743\) − 36.7696i − 1.34894i −0.738300 0.674472i \(-0.764371\pi\)
0.738300 0.674472i \(-0.235629\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.67753i 0.317707i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) − 11.3137i − 0.412568i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −54.1359 −1.96760 −0.983802 0.179258i \(-0.942630\pi\)
−0.983802 + 0.179258i \(0.942630\pi\)
\(758\) 48.0833i 1.74646i
\(759\) 0 0
\(760\) −40.0000 −1.45095
\(761\) − 20.2207i − 0.733001i −0.930418 0.366501i \(-0.880556\pi\)
0.930418 0.366501i \(-0.119444\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −52.0000 −1.87884
\(767\) − 4.54662i − 0.164169i
\(768\) 0 0
\(769\) 12.6491 0.456139 0.228069 0.973645i \(-0.426759\pi\)
0.228069 + 0.973645i \(0.426759\pi\)
\(770\) 49.9192i 1.79896i
\(771\) 0 0
\(772\) 0 0
\(773\) 22.3607i 0.804258i 0.915583 + 0.402129i \(0.131730\pi\)
−0.915583 + 0.402129i \(0.868270\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 65.5128i 2.34724i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 78.5964 2.80702
\(785\) − 17.4668i − 0.623417i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) − 44.7214i − 1.59313i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) − 35.1258i − 1.24657i
\(795\) 0 0
\(796\) 0 0
\(797\) 39.5980i 1.40263i 0.712850 + 0.701316i \(0.247404\pi\)
−0.712850 + 0.701316i \(0.752596\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 28.2843i 1.00000i
\(801\) 0 0
\(802\) −50.6491 −1.78848
\(803\) 0 0
\(804\) 0 0
\(805\) 51.6228 1.81946
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 54.1619i − 1.90423i −0.305741 0.952115i \(-0.598904\pi\)
0.305741 0.952115i \(-0.401096\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −48.2719 −1.69193
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) − 19.7990i − 0.692255i
\(819\) 0 0
\(820\) 46.3246 1.61772
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 11.8641 0.413555 0.206778 0.978388i \(-0.433702\pi\)
0.206778 + 0.978388i \(0.433702\pi\)
\(824\) − 48.5423i − 1.69105i
\(825\) 0 0
\(826\) 39.6228 1.37865
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6.70178 −0.232342
\(833\) 0 0
\(834\) 0 0
\(835\) −10.0000 −0.346064
\(836\) 38.6800i 1.33778i
\(837\) 0 0
\(838\) −57.6228 −1.99055
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 44.0000 1.51454
\(845\) 27.4997i 0.946017i
\(846\) 0 0
\(847\) −8.51317 −0.292516
\(848\) 22.6274i 0.777029i
\(849\) 0 0
\(850\) 0 0
\(851\) 49.9192i 1.71121i
\(852\) 0 0
\(853\) −37.1096 −1.27061 −0.635304 0.772262i \(-0.719125\pi\)
−0.635304 + 0.772262i \(0.719125\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 58.1378i − 1.97903i −0.144421 0.989516i \(-0.546132\pi\)
0.144421 0.989516i \(-0.453868\pi\)
\(864\) 0 0
\(865\) 50.5964 1.72033
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 40.0000 1.35302
\(875\) − 57.7160i − 1.95116i
\(876\) 0 0
\(877\) −49.1096 −1.65831 −0.829157 0.559016i \(-0.811179\pi\)
−0.829157 + 0.559016i \(0.811179\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 27.3509 0.921998
\(881\) 32.0679i 1.08040i 0.841538 + 0.540198i \(0.181651\pi\)
−0.841538 + 0.540198i \(0.818349\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 4.47214i − 0.150160i −0.997178 0.0750798i \(-0.976079\pi\)
0.997178 0.0750798i \(-0.0239212\pi\)
\(888\) 0 0
\(889\) −71.2982 −2.39127
\(890\) 59.5892i 1.99744i
\(891\) 0 0
\(892\) −3.62278 −0.121300
\(893\) 17.8885i 0.598617i
\(894\) 0 0
\(895\) −31.1096 −1.03988
\(896\) − 58.4045i − 1.95116i
\(897\) 0 0
\(898\) 59.2982 1.97881
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) − 44.7958i − 1.49154i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 13.6754 0.453337
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 103.393i − 3.41432i
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) − 28.2843i − 0.932505i
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 55.8114 1.83507
\(926\) 14.4089i 0.473505i
\(927\) 0 0
\(928\) 0 0
\(929\) − 11.7355i − 0.385028i −0.981294 0.192514i \(-0.938336\pi\)
0.981294 0.192514i \(-0.0616641\pi\)
\(930\) 0 0
\(931\) −124.272 −4.07285
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 12.6491 0.412568
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −46.3246 −1.50854
\(944\) − 21.7094i − 0.706582i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) − 44.7214i − 1.45095i
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 13.2242i 0.426364i
\(963\) 0 0
\(964\) 50.5964 1.62960
\(965\) 0 0
\(966\) 0 0
\(967\) −60.1359 −1.93384 −0.966921 0.255077i \(-0.917899\pi\)
−0.966921 + 0.255077i \(0.917899\pi\)
\(968\) 4.66439i 0.149919i
\(969\) 0 0
\(970\) 0 0
\(971\) − 39.3685i − 1.26339i −0.775215 0.631697i \(-0.782359\pi\)
0.775215 0.631697i \(-0.217641\pi\)
\(972\) 0 0
\(973\) −72.2719 −2.31693
\(974\) 0.192253i 0.00616019i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 57.6228 1.84163
\(980\) 87.8735i 2.80702i
\(981\) 0 0
\(982\) 42.2719 1.34895
\(983\) 48.0833i 1.53362i 0.641875 + 0.766809i \(0.278157\pi\)
−0.641875 + 0.766809i \(0.721843\pi\)
\(984\) 0 0
\(985\) 50.0000 1.59313
\(986\) 0 0
\(987\) 0 0
\(988\) 10.5964 0.337118
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 62.7851 1.98842 0.994211 0.107442i \(-0.0342661\pi\)
0.994211 + 0.107442i \(0.0342661\pi\)
\(998\) 62.6099i 1.98188i
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 360.2.m.a.179.1 4
3.2 odd 2 inner 360.2.m.a.179.4 yes 4
4.3 odd 2 1440.2.m.b.719.2 4
5.2 odd 4 1800.2.b.f.251.5 8
5.3 odd 4 1800.2.b.f.251.4 8
5.4 even 2 360.2.m.b.179.4 yes 4
8.3 odd 2 360.2.m.b.179.4 yes 4
8.5 even 2 1440.2.m.a.719.3 4
12.11 even 2 1440.2.m.b.719.4 4
15.2 even 4 1800.2.b.f.251.1 8
15.8 even 4 1800.2.b.f.251.8 8
15.14 odd 2 360.2.m.b.179.1 yes 4
20.3 even 4 7200.2.b.f.4751.1 8
20.7 even 4 7200.2.b.f.4751.7 8
20.19 odd 2 1440.2.m.a.719.3 4
24.5 odd 2 1440.2.m.a.719.1 4
24.11 even 2 360.2.m.b.179.1 yes 4
40.3 even 4 1800.2.b.f.251.5 8
40.13 odd 4 7200.2.b.f.4751.7 8
40.19 odd 2 CM 360.2.m.a.179.1 4
40.27 even 4 1800.2.b.f.251.4 8
40.29 even 2 1440.2.m.b.719.2 4
40.37 odd 4 7200.2.b.f.4751.1 8
60.23 odd 4 7200.2.b.f.4751.2 8
60.47 odd 4 7200.2.b.f.4751.8 8
60.59 even 2 1440.2.m.a.719.1 4
120.29 odd 2 1440.2.m.b.719.4 4
120.53 even 4 7200.2.b.f.4751.8 8
120.59 even 2 inner 360.2.m.a.179.4 yes 4
120.77 even 4 7200.2.b.f.4751.2 8
120.83 odd 4 1800.2.b.f.251.1 8
120.107 odd 4 1800.2.b.f.251.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.m.a.179.1 4 1.1 even 1 trivial
360.2.m.a.179.1 4 40.19 odd 2 CM
360.2.m.a.179.4 yes 4 3.2 odd 2 inner
360.2.m.a.179.4 yes 4 120.59 even 2 inner
360.2.m.b.179.1 yes 4 15.14 odd 2
360.2.m.b.179.1 yes 4 24.11 even 2
360.2.m.b.179.4 yes 4 5.4 even 2
360.2.m.b.179.4 yes 4 8.3 odd 2
1440.2.m.a.719.1 4 24.5 odd 2
1440.2.m.a.719.1 4 60.59 even 2
1440.2.m.a.719.3 4 8.5 even 2
1440.2.m.a.719.3 4 20.19 odd 2
1440.2.m.b.719.2 4 4.3 odd 2
1440.2.m.b.719.2 4 40.29 even 2
1440.2.m.b.719.4 4 12.11 even 2
1440.2.m.b.719.4 4 120.29 odd 2
1800.2.b.f.251.1 8 15.2 even 4
1800.2.b.f.251.1 8 120.83 odd 4
1800.2.b.f.251.4 8 5.3 odd 4
1800.2.b.f.251.4 8 40.27 even 4
1800.2.b.f.251.5 8 5.2 odd 4
1800.2.b.f.251.5 8 40.3 even 4
1800.2.b.f.251.8 8 15.8 even 4
1800.2.b.f.251.8 8 120.107 odd 4
7200.2.b.f.4751.1 8 20.3 even 4
7200.2.b.f.4751.1 8 40.37 odd 4
7200.2.b.f.4751.2 8 60.23 odd 4
7200.2.b.f.4751.2 8 120.77 even 4
7200.2.b.f.4751.7 8 20.7 even 4
7200.2.b.f.4751.7 8 40.13 odd 4
7200.2.b.f.4751.8 8 60.47 odd 4
7200.2.b.f.4751.8 8 120.53 even 4