Properties

Label 360.2.m.a.179.2
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 179.2
Root \(1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 360.179
Dual form 360.2.m.a.179.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +1.16228 q^{7} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +1.16228 q^{7} +2.82843i q^{8} +3.16228 q^{10} -5.88635i q^{11} +7.16228 q^{13} -1.64371i q^{14} +4.00000 q^{16} +6.32456 q^{19} -4.47214i q^{20} -8.32456 q^{22} +4.47214i q^{23} -5.00000 q^{25} -10.1290i q^{26} -2.32456 q^{28} -5.65685i q^{32} +2.59893i q^{35} -4.83772 q^{37} -8.94427i q^{38} -6.32456 q^{40} +7.53006i q^{41} +11.7727i q^{44} +6.32456 q^{46} -2.82843i q^{47} -5.64911 q^{49} +7.07107i q^{50} -14.3246 q^{52} +5.65685i q^{53} +13.1623 q^{55} +3.28742i q^{56} -14.3716i q^{59} -8.00000 q^{64} +16.0153i q^{65} +3.67544 q^{70} +6.84157i q^{74} -12.6491 q^{76} -6.84157i q^{77} +8.94427i q^{80} +10.6491 q^{82} +16.6491 q^{88} -0.955223i q^{89} +8.32456 q^{91} -8.94427i q^{92} -4.00000 q^{94} +14.1421i q^{95} +7.98905i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{4} - 8q^{7} + 16q^{13} + 16q^{16} - 8q^{22} - 20q^{25} + 16q^{28} - 32q^{37} + 28q^{49} - 32q^{52} + 40q^{55} - 32q^{64} + 40q^{70} - 8q^{82} + 16q^{88} + 8q^{91} - 16q^{94} + O(q^{100}) \)

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
\(7\) 1.16228 0.439300 0.219650 0.975579i \(-0.429509\pi\)
0.219650 + 0.975579i \(0.429509\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) 3.16228 1.00000
\(11\) − 5.88635i − 1.77480i −0.460999 0.887401i \(-0.652509\pi\)
0.460999 0.887401i \(-0.347491\pi\)
\(12\) 0 0
\(13\) 7.16228 1.98646 0.993229 0.116171i \(-0.0370621\pi\)
0.993229 + 0.116171i \(0.0370621\pi\)
\(14\) − 1.64371i − 0.439300i
\(15\) 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 \(-0.241620\pi\)
0.725476 + 0.688247i \(0.241620\pi\)
\(20\) − 4.47214i − 1.00000i
\(21\) 0 0
\(22\) −8.32456 −1.77480
\(23\) 4.47214i 0.932505i 0.884652 + 0.466252i \(0.154396\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) − 10.1290i − 1.98646i
\(27\) 0 0
\(28\) −2.32456 −0.439300
\(29\) 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
\(35\) 2.59893i 0.439300i
\(36\) 0 0
\(37\) −4.83772 −0.795317 −0.397658 0.917534i \(-0.630177\pi\)
−0.397658 + 0.917534i \(0.630177\pi\)
\(38\) − 8.94427i − 1.45095i
\(39\) 0 0
\(40\) −6.32456 −1.00000
\(41\) 7.53006i 1.17600i 0.808862 + 0.587999i \(0.200084\pi\)
−0.808862 + 0.587999i \(0.799916\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 11.7727i 1.77480i
\(45\) 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
\(49\) −5.64911 −0.807016
\(50\) 7.07107i 1.00000i
\(51\) 0 0
\(52\) −14.3246 −1.98646
\(53\) 5.65685i 0.777029i 0.921443 + 0.388514i \(0.127012\pi\)
−0.921443 + 0.388514i \(0.872988\pi\)
\(54\) 0 0
\(55\) 13.1623 1.77480
\(56\) 3.28742i 0.439300i
\(57\) 0 0
\(58\) 0 0
\(59\) − 14.3716i − 1.87103i −0.353291 0.935513i \(-0.614937\pi\)
0.353291 0.935513i \(-0.385063\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 16.0153i 1.98646i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 3.67544 0.439300
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 6.84157i 0.795317i
\(75\) 0 0
\(76\) −12.6491 −1.45095
\(77\) − 6.84157i − 0.779670i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.94427i 1.00000i
\(81\) 0 0
\(82\) 10.6491 1.17600
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 16.6491 1.77480
\(89\) − 0.955223i − 0.101253i −0.998718 0.0506267i \(-0.983878\pi\)
0.998718 0.0506267i \(-0.0161219\pi\)
\(90\) 0 0
\(91\) 8.32456 0.872651
\(92\) − 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 \(\pi\)
\(98\) 7.98905i 0.807016i
\(99\) 0 0
\(100\) 10.0000 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −10.8377 −1.06787 −0.533936 0.845525i \(-0.679288\pi\)
−0.533936 + 0.845525i \(0.679288\pi\)
\(104\) 20.2580i 1.98646i
\(105\) 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 \(\pi\)
\(110\) − 18.6143i − 1.77480i
\(111\) 0 0
\(112\) 4.64911 0.439300
\(113\) 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
\(118\) −20.3246 −1.87103
\(119\) 0 0
\(120\) 0 0
\(121\) −23.6491 −2.14992
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.1803i − 1.00000i
\(126\) 0 0
\(127\) −17.8114 −1.58051 −0.790253 0.612781i \(-0.790051\pi\)
−0.790253 + 0.612781i \(0.790051\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 0 0
\(130\) 22.6491 1.98646
\(131\) 11.0842i 0.968432i 0.874948 + 0.484216i \(0.160895\pi\)
−0.874948 + 0.484216i \(0.839105\pi\)
\(132\) 0 0
\(133\) 7.35089 0.637403
\(134\) 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 \(0.297654\pi\)
\(140\) − 5.19786i − 0.439300i
\(141\) 0 0
\(142\) 0 0
\(143\) − 42.1597i − 3.52557i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 9.67544 0.795317
\(149\) 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
\(154\) −9.67544 −0.779670
\(155\) 0 0
\(156\) 0 0
\(157\) −23.8114 −1.90036 −0.950178 0.311708i \(-0.899099\pi\)
−0.950178 + 0.311708i \(0.899099\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 12.6491 1.00000
\(161\) 5.19786i 0.409649i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) − 15.0601i − 1.17600i
\(165\) 0 0
\(166\) 0 0
\(167\) 4.47214i 0.346064i 0.984916 + 0.173032i \(0.0553564\pi\)
−0.984916 + 0.173032i \(0.944644\pi\)
\(168\) 0 0
\(169\) 38.2982 2.94602
\(170\) 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
\(175\) −5.81139 −0.439300
\(176\) − 23.5454i − 1.77480i
\(177\) 0 0
\(178\) −1.35089 −0.101253
\(179\) − 22.8569i − 1.70841i −0.519940 0.854203i \(-0.674046\pi\)
0.519940 0.854203i \(-0.325954\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) − 11.7727i − 0.872651i
\(183\) 0 0
\(184\) −12.6491 −0.932505
\(185\) − 10.8175i − 0.795317i
\(186\) 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
\(196\) 11.2982 0.807016
\(197\) − 22.3607i − 1.59313i −0.604551 0.796566i \(-0.706648\pi\)
0.604551 0.796566i \(-0.293352\pi\)
\(198\) 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
\(205\) −16.8377 −1.17600
\(206\) 15.3269i 1.06787i
\(207\) 0 0
\(208\) 28.6491 1.98646
\(209\) − 37.2285i − 2.57515i
\(210\) 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
\(220\) −26.3246 −1.77480
\(221\) 0 0
\(222\) 0 0
\(223\) −29.8114 −1.99632 −0.998159 0.0606498i \(-0.980683\pi\)
−0.998159 + 0.0606498i \(0.980683\pi\)
\(224\) − 6.57484i − 0.439300i
\(225\) 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
\(236\) 28.7433i 1.87103i
\(237\) 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 \(-0.196846\pi\)
0.814801 + 0.579741i \(0.196846\pi\)
\(242\) 33.4449i 2.14992i
\(243\) 0 0
\(244\) 0 0
\(245\) − 12.6318i − 0.807016i
\(246\) 0 0
\(247\) 45.2982 2.88226
\(248\) 0 0
\(249\) 0 0
\(250\) −15.8114 −1.00000
\(251\) 20.9465i 1.32213i 0.750329 + 0.661065i \(0.229895\pi\)
−0.750329 + 0.661065i \(0.770105\pi\)
\(252\) 0 0
\(253\) 26.3246 1.65501
\(254\) 25.1891i 1.58051i
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −5.62278 −0.349382
\(260\) − 32.0307i − 1.98646i
\(261\) 0 0
\(262\) 15.6754 0.968432
\(263\) 31.3050i 1.93035i 0.261612 + 0.965173i \(0.415746\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(264\) 0 0
\(265\) −12.6491 −0.777029
\(266\) − 10.3957i − 0.637403i
\(267\) 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
\(275\) 29.4317i 1.77480i
\(276\) 0 0
\(277\) 14.1359 0.849347 0.424673 0.905347i \(-0.360389\pi\)
0.424673 + 0.905347i \(0.360389\pi\)
\(278\) − 19.7990i − 1.18746i
\(279\) 0 0
\(280\) −7.35089 −0.439300
\(281\) − 2.33219i − 0.139127i −0.997578 0.0695635i \(-0.977839\pi\)
0.997578 0.0695635i \(-0.0221607\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −59.6228 −3.52557
\(287\) 8.75202i 0.516615i
\(288\) 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 \(0.0417008\pi\)
−0.991431 + 0.130632i \(0.958299\pi\)
\(294\) 0 0
\(295\) 32.1359 1.87103
\(296\) − 13.6831i − 0.795317i
\(297\) 0 0
\(298\) 0 0
\(299\) 32.0307i 1.85238i
\(300\) 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 \(\pi\)
\(308\) 13.6831i 0.779670i
\(309\) 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 \(\pi\)
\(314\) 33.6744i 1.90036i
\(315\) 0 0
\(316\) 0 0
\(317\) 31.3050i 1.75826i 0.476581 + 0.879131i \(0.341876\pi\)
−0.476581 + 0.879131i \(0.658124\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 17.8885i − 1.00000i
\(321\) 0 0
\(322\) 7.35089 0.409649
\(323\) 0 0
\(324\) 0 0
\(325\) −35.8114 −1.98646
\(326\) 0 0
\(327\) 0 0
\(328\) −21.2982 −1.17600
\(329\) − 3.28742i − 0.181241i
\(330\) 0 0
\(331\) −31.6228 −1.73814 −0.869072 0.494685i \(-0.835284\pi\)
−0.869072 + 0.494685i \(0.835284\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 6.32456 0.346064
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) − 54.1619i − 2.94602i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −14.7018 −0.793821
\(344\) 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 \(\pi\)
\(350\) 8.21854i 0.439300i
\(351\) 0 0
\(352\) −33.2982 −1.77480
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.91045i 0.101253i
\(357\) 0 0
\(358\) −32.3246 −1.70841
\(359\) 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
\(364\) −16.6491 −0.872651
\(365\) 0 0
\(366\) 0 0
\(367\) 6.18861 0.323043 0.161521 0.986869i \(-0.448360\pi\)
0.161521 + 0.986869i \(0.448360\pi\)
\(368\) 17.8885i 0.932505i
\(369\) 0 0
\(370\) −15.2982 −0.795317
\(371\) 6.57484i 0.341348i
\(372\) 0 0
\(373\) 38.1359 1.97460 0.987302 0.158854i \(-0.0507798\pi\)
0.987302 + 0.158854i \(0.0507798\pi\)
\(374\) 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
\(385\) 15.2982 0.779670
\(386\) 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
\(392\) − 15.9781i − 0.807016i
\(393\) 0 0
\(394\) −31.6228 −1.59313
\(395\) 0 0
\(396\) 0 0
\(397\) 31.1623 1.56399 0.781995 0.623285i \(-0.214202\pi\)
0.781995 + 0.623285i \(0.214202\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −20.0000 −1.00000
\(401\) − 17.9258i − 0.895171i −0.894241 0.447586i \(-0.852284\pi\)
0.894241 0.447586i \(-0.147716\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 28.4765i 1.41153i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 23.8121i 1.17600i
\(411\) 0 0
\(412\) 21.6754 1.06787
\(413\) − 16.7038i − 0.821942i
\(414\) 0 0
\(415\) 0 0
\(416\) − 40.5160i − 1.98646i
\(417\) 0 0
\(418\) −52.6491 −2.57515
\(419\) 3.97590i 0.194236i 0.995273 + 0.0971178i \(0.0309624\pi\)
−0.995273 + 0.0971178i \(0.969038\pi\)
\(420\) 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 \(\pi\)
\(440\) 37.2285i 1.77480i
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 2.13594 0.101253
\(446\) 42.1597i 1.99632i
\(447\) 0 0
\(448\) −9.29822 −0.439300
\(449\) 6.15309i 0.290382i 0.989404 + 0.145191i \(0.0463797\pi\)
−0.989404 + 0.145191i \(0.953620\pi\)
\(450\) 0 0
\(451\) 44.3246 2.08716
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 18.6143i 0.872651i
\(456\) 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
\(463\) −41.8114 −1.94314 −0.971570 0.236752i \(-0.923917\pi\)
−0.971570 + 0.236752i \(0.923917\pi\)
\(464\) 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
\(472\) 40.6491 1.87103
\(473\) 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
\(481\) −34.6491 −1.57986
\(482\) − 35.7771i − 1.62960i
\(483\) 0 0
\(484\) 47.2982 2.14992
\(485\) 0 0
\(486\) 0 0
\(487\) 44.1359 1.99999 0.999995 0.00308010i \(-0.000980427\pi\)
0.999995 + 0.00308010i \(0.000980427\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −17.8641 −0.807016
\(491\) − 32.7192i − 1.47660i −0.674475 0.738298i \(-0.735630\pi\)
0.674475 0.738298i \(-0.264370\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) − 64.0614i − 2.88226i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 44.2719 1.98188 0.990941 0.134298i \(-0.0428781\pi\)
0.990941 + 0.134298i \(0.0428781\pi\)
\(500\) 22.3607i 1.00000i
\(501\) 0 0
\(502\) 29.6228 1.32213
\(503\) − 19.7990i − 0.882793i −0.897312 0.441397i \(-0.854483\pi\)
0.897312 0.441397i \(-0.145517\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) − 37.2285i − 1.65501i
\(507\) 0 0
\(508\) 35.6228 1.58051
\(509\) 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
\(515\) − 24.2339i − 1.06787i
\(516\) 0 0
\(517\) −16.6491 −0.732227
\(518\) 7.95181i 0.349382i
\(519\) 0 0
\(520\) −45.2982 −1.98646
\(521\) − 44.7586i − 1.96091i −0.196744 0.980455i \(-0.563037\pi\)
0.196744 0.980455i \(-0.436963\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) − 22.1684i − 0.968432i
\(525\) 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
\(532\) −14.7018 −0.637403
\(533\) 53.9324i 2.33607i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 33.2526i 1.43229i
\(540\) 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
\(550\) 41.6228 1.77480
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) − 19.9912i − 0.849347i
\(555\) 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
\(560\) 10.3957i 0.439300i
\(561\) 0 0
\(562\) −3.29822 −0.139127
\(563\) 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
\(569\) 41.4712i 1.73856i 0.494318 + 0.869281i \(0.335418\pi\)
−0.494318 + 0.869281i \(0.664582\pi\)
\(570\) 0 0
\(571\) 44.2719 1.85272 0.926360 0.376638i \(-0.122920\pi\)
0.926360 + 0.376638i \(0.122920\pi\)
\(572\) 84.3193i 3.52557i
\(573\) 0 0
\(574\) 12.3772 0.516615
\(575\) − 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
\(583\) 33.2982 1.37907
\(584\) 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
\(590\) − 45.4471i − 1.87103i
\(591\) 0 0
\(592\) −19.3509 −0.795317
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 45.2982 1.85238
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 25.2982 1.03194 0.515968 0.856608i \(-0.327432\pi\)
0.515968 + 0.856608i \(0.327432\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 52.8810i − 2.14992i
\(606\) 0 0
\(607\) −36.7851 −1.49306 −0.746530 0.665352i \(-0.768281\pi\)
−0.746530 + 0.665352i \(0.768281\pi\)
\(608\) − 35.7771i − 1.45095i
\(609\) 0 0
\(610\) 0 0
\(611\) − 20.2580i − 0.819550i
\(612\) 0 0
\(613\) −30.7851 −1.24340 −0.621698 0.783257i \(-0.713557\pi\)
−0.621698 + 0.783257i \(0.713557\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 19.3509 0.779670
\(617\) 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
\(623\) − 1.11023i − 0.0444806i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 47.6228 1.90036
\(629\) 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
\(635\) − 39.8275i − 1.58051i
\(636\) 0 0
\(637\) −40.4605 −1.60310
\(638\) 0 0
\(639\) 0 0
\(640\) −25.2982 −1.00000
\(641\) 34.3629i 1.35725i 0.734484 + 0.678626i \(0.237424\pi\)
−0.734484 + 0.678626i \(0.762576\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) − 10.3957i − 0.409649i
\(645\) 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
\(649\) −84.5964 −3.32070
\(650\) 50.6450i 1.98646i
\(651\) 0 0
\(652\) 0 0
\(653\) 4.47214i 0.175008i 0.996164 + 0.0875041i \(0.0278891\pi\)
−0.996164 + 0.0875041i \(0.972111\pi\)
\(654\) 0 0
\(655\) −24.7851 −0.968432
\(656\) 30.1202i 1.17600i
\(657\) 0 0
\(658\) −4.64911 −0.181241
\(659\) − 49.6897i − 1.93564i −0.251649 0.967818i \(-0.580973\pi\)
0.251649 0.967818i \(-0.419027\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 44.7214i 1.73814i
\(663\) 0 0
\(664\) 0 0
\(665\) 16.4371i 0.637403i
\(666\) 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
\(676\) −76.5964 −2.94602
\(677\) − 49.1935i − 1.89066i −0.326116 0.945330i \(-0.605740\pi\)
0.326116 0.945330i \(-0.394260\pi\)
\(678\) 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
\(686\) 20.7915i 0.793821i
\(687\) 0 0
\(688\) 0 0
\(689\) 40.5160i 1.54354i
\(690\) 0 0
\(691\) −31.6228 −1.20299 −0.601494 0.798878i \(-0.705427\pi\)
−0.601494 + 0.798878i \(0.705427\pi\)
\(692\) − 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
\(700\) 11.6228 0.439300
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −30.5964 −1.15397
\(704\) 47.0908i 1.77480i
\(705\) 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
\(712\) 2.70178 0.101253
\(713\) 0 0
\(714\) 0 0
\(715\) 94.2719 3.52557
\(716\) 45.7138i 1.70841i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −12.5964 −0.469116
\(722\) − 29.6985i − 1.10526i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −34.8377 −1.29206 −0.646030 0.763312i \(-0.723572\pi\)
−0.646030 + 0.763312i \(0.723572\pi\)
\(728\) 23.5454i 0.872651i
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −18.7851 −0.693842 −0.346921 0.937894i \(-0.612773\pi\)
−0.346921 + 0.937894i \(0.612773\pi\)
\(734\) − 8.75202i − 0.323043i
\(735\) 0 0
\(736\) 25.2982 0.932505
\(737\) 0 0
\(738\) 0 0
\(739\) 6.32456 0.232653 0.116326 0.993211i \(-0.462888\pi\)
0.116326 + 0.993211i \(0.462888\pi\)
\(740\) 21.6350i 0.795317i
\(741\) 0 0
\(742\) 9.29822 0.341348
\(743\) − 36.7696i − 1.34894i −0.738300 0.674472i \(-0.764371\pi\)
0.738300 0.674472i \(-0.235629\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) − 53.9324i − 1.97460i
\(747\) 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
\(757\) −9.86406 −0.358515 −0.179258 0.983802i \(-0.557370\pi\)
−0.179258 + 0.983802i \(0.557370\pi\)
\(758\) 48.0833i 1.74646i
\(759\) 0 0
\(760\) −40.0000 −1.45095
\(761\) 51.3334i 1.86084i 0.366501 + 0.930418i \(0.380556\pi\)
−0.366501 + 0.930418i \(0.619444\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −52.0000 −1.87884
\(767\) − 102.934i − 3.71672i
\(768\) 0 0
\(769\) −12.6491 −0.456139 −0.228069 0.973645i \(-0.573241\pi\)
−0.228069 + 0.973645i \(0.573241\pi\)
\(770\) − 21.6350i − 0.779670i
\(771\) 0 0
\(772\) 0 0
\(773\) − 22.3607i − 0.804258i −0.915583 0.402129i \(-0.868270\pi\)
0.915583 0.402129i \(-0.131730\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 47.6243i 1.70632i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −22.5964 −0.807016
\(785\) − 53.2439i − 1.90036i
\(786\) 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
\(794\) − 44.0701i − 1.56399i
\(795\) 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
\(802\) −25.3509 −0.895171
\(803\) 0 0
\(804\) 0 0
\(805\) −11.6228 −0.409649
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17.3923i 0.611481i 0.952115 + 0.305741i \(0.0989040\pi\)
−0.952115 + 0.305741i \(0.901096\pi\)
\(810\) 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
\(814\) 40.2719 1.41153
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) − 19.7990i − 0.692255i
\(819\) 0 0
\(820\) 33.6754 1.17600
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 56.1359 1.95678 0.978388 0.206778i \(-0.0662976\pi\)
0.978388 + 0.206778i \(0.0662976\pi\)
\(824\) − 30.6537i − 1.06787i
\(825\) 0 0
\(826\) −23.6228 −0.821942
\(827\) 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
\(832\) −57.2982 −1.98646
\(833\) 0 0
\(834\) 0 0
\(835\) −10.0000 −0.346064
\(836\) 74.4571i 2.57515i
\(837\) 0 0
\(838\) 5.62278 0.194236
\(839\) 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
\(845\) 85.6374i 2.94602i
\(846\) 0 0
\(847\) −27.4868 −0.944459
\(848\) 22.6274i 0.777029i
\(849\) 0 0
\(850\) 0 0
\(851\) − 21.6350i − 0.741637i
\(852\) 0 0
\(853\) 45.1096 1.54452 0.772262 0.635304i \(-0.219125\pi\)
0.772262 + 0.635304i \(0.219125\pi\)
\(854\) 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 \(0.453868\pi\)
−0.144421 + 0.989516i \(0.546132\pi\)
\(864\) 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
\(875\) − 12.9947i − 0.439300i
\(876\) 0 0
\(877\) 33.1096 1.11803 0.559016 0.829157i \(-0.311179\pi\)
0.559016 + 0.829157i \(0.311179\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 52.6491 1.77480
\(881\) 49.9565i 1.68308i 0.540198 + 0.841538i \(0.318349\pi\)
−0.540198 + 0.841538i \(0.681651\pi\)
\(882\) 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 \(0.0239212\pi\)
−0.997178 + 0.0750798i \(0.976079\pi\)
\(888\) 0 0
\(889\) −20.7018 −0.694315
\(890\) − 3.02068i − 0.101253i
\(891\) 0 0
\(892\) 59.6228 1.99632
\(893\) − 17.8885i − 0.598617i
\(894\) 0 0
\(895\) 51.1096 1.70841
\(896\) 13.1497i 0.439300i
\(897\) 0 0
\(898\) 8.70178 0.290382
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) − 62.6844i − 2.08716i
\(903\) 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
\(910\) 26.3246 0.872651
\(911\) 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
\(917\) 12.8829i 0.425432i
\(918\) 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
\(925\) 24.1886 0.795317
\(926\) 59.1302i 1.94314i
\(927\) 0 0
\(928\) 0 0
\(929\) 59.8187i 1.96259i 0.192514 + 0.981294i \(0.438336\pi\)
−0.192514 + 0.981294i \(0.561664\pi\)
\(930\) 0 0
\(931\) −35.7281 −1.17094
\(932\) 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
\(943\) −33.6754 −1.09662
\(944\) − 57.4865i − 1.87103i
\(945\) 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
\(962\) 49.0012i 1.57986i
\(963\) 0 0
\(964\) −50.5964 −1.62960
\(965\) 0 0
\(966\) 0 0
\(967\) −15.8641 −0.510154 −0.255077 0.966921i \(-0.582101\pi\)
−0.255077 + 0.966921i \(0.582101\pi\)
\(968\) − 66.8898i − 2.14992i
\(969\) 0 0
\(970\) 0 0
\(971\) − 48.3128i − 1.55043i −0.631697 0.775215i \(-0.717641\pi\)
0.631697 0.775215i \(-0.282359\pi\)
\(972\) 0 0
\(973\) 16.2719 0.521653
\(974\) − 62.4177i − 1.99999i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) −5.62278 −0.179705
\(980\) 25.2636i 0.807016i
\(981\) 0 0
\(982\) −46.2719 −1.47660
\(983\) 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
\(988\) −90.5964 −2.88226
\(989\) 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
\(997\) −6.78505 −0.214885 −0.107442 0.994211i \(-0.534266\pi\)
−0.107442 + 0.994211i \(0.534266\pi\)
\(998\) − 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.179.2 4
3.2 odd 2 inner 360.2.m.a.179.3 yes 4
4.3 odd 2 1440.2.m.b.719.3 4
5.2 odd 4 1800.2.b.f.251.7 8
5.3 odd 4 1800.2.b.f.251.2 8
5.4 even 2 360.2.m.b.179.3 yes 4
8.3 odd 2 360.2.m.b.179.3 yes 4
8.5 even 2 1440.2.m.a.719.2 4
12.11 even 2 1440.2.m.b.719.1 4
15.2 even 4 1800.2.b.f.251.3 8
15.8 even 4 1800.2.b.f.251.6 8
15.14 odd 2 360.2.m.b.179.2 yes 4
20.3 even 4 7200.2.b.f.4751.6 8
20.7 even 4 7200.2.b.f.4751.4 8
20.19 odd 2 1440.2.m.a.719.2 4
24.5 odd 2 1440.2.m.a.719.4 4
24.11 even 2 360.2.m.b.179.2 yes 4
40.3 even 4 1800.2.b.f.251.7 8
40.13 odd 4 7200.2.b.f.4751.4 8
40.19 odd 2 CM 360.2.m.a.179.2 4
40.27 even 4 1800.2.b.f.251.2 8
40.29 even 2 1440.2.m.b.719.3 4
40.37 odd 4 7200.2.b.f.4751.6 8
60.23 odd 4 7200.2.b.f.4751.5 8
60.47 odd 4 7200.2.b.f.4751.3 8
60.59 even 2 1440.2.m.a.719.4 4
120.29 odd 2 1440.2.m.b.719.1 4
120.53 even 4 7200.2.b.f.4751.3 8
120.59 even 2 inner 360.2.m.a.179.3 yes 4
120.77 even 4 7200.2.b.f.4751.5 8
120.83 odd 4 1800.2.b.f.251.3 8
120.107 odd 4 1800.2.b.f.251.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.m.a.179.2 4 1.1 even 1 trivial
360.2.m.a.179.2 4 40.19 odd 2 CM
360.2.m.a.179.3 yes 4 3.2 odd 2 inner
360.2.m.a.179.3 yes 4 120.59 even 2 inner
360.2.m.b.179.2 yes 4 15.14 odd 2
360.2.m.b.179.2 yes 4 24.11 even 2
360.2.m.b.179.3 yes 4 5.4 even 2
360.2.m.b.179.3 yes 4 8.3 odd 2
1440.2.m.a.719.2 4 8.5 even 2
1440.2.m.a.719.2 4 20.19 odd 2
1440.2.m.a.719.4 4 24.5 odd 2
1440.2.m.a.719.4 4 60.59 even 2
1440.2.m.b.719.1 4 12.11 even 2
1440.2.m.b.719.1 4 120.29 odd 2
1440.2.m.b.719.3 4 4.3 odd 2
1440.2.m.b.719.3 4 40.29 even 2
1800.2.b.f.251.2 8 5.3 odd 4
1800.2.b.f.251.2 8 40.27 even 4
1800.2.b.f.251.3 8 15.2 even 4
1800.2.b.f.251.3 8 120.83 odd 4
1800.2.b.f.251.6 8 15.8 even 4
1800.2.b.f.251.6 8 120.107 odd 4
1800.2.b.f.251.7 8 5.2 odd 4
1800.2.b.f.251.7 8 40.3 even 4
7200.2.b.f.4751.3 8 60.47 odd 4
7200.2.b.f.4751.3 8 120.53 even 4
7200.2.b.f.4751.4 8 20.7 even 4
7200.2.b.f.4751.4 8 40.13 odd 4
7200.2.b.f.4751.5 8 60.23 odd 4
7200.2.b.f.4751.5 8 120.77 even 4
7200.2.b.f.4751.6 8 20.3 even 4
7200.2.b.f.4751.6 8 40.37 odd 4