Properties

Label 8100.2.d.l.649.1
Level $8100$
Weight $2$
Character 8100.649
Analytic conductor $64.679$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1620)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 8100.649
Dual form 8100.2.d.l.649.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.73205i q^{7} +O(q^{10})\) \(q-2.73205i q^{7} -1.73205 q^{11} -5.46410i q^{13} +4.73205i q^{17} +4.46410 q^{19} +3.46410i q^{23} -7.73205 q^{29} +5.92820 q^{31} -6.19615i q^{37} +11.1962 q^{41} -3.26795i q^{43} +1.26795i q^{47} -0.464102 q^{49} -7.26795i q^{53} -7.73205 q^{59} -4.00000 q^{61} +6.39230i q^{67} +11.1962 q^{71} +0.196152i q^{73} +4.73205i q^{77} +14.3923 q^{79} -15.1244i q^{83} +5.19615 q^{89} -14.9282 q^{91} +0.732051i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 4 q^{19} - 24 q^{29} - 4 q^{31} + 24 q^{41} + 12 q^{49} - 24 q^{59} - 16 q^{61} + 24 q^{71} + 16 q^{79} - 32 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.73205i − 1.03262i −0.856402 0.516309i \(-0.827306\pi\)
0.856402 0.516309i \(-0.172694\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.73205 −0.522233 −0.261116 0.965307i \(-0.584091\pi\)
−0.261116 + 0.965307i \(0.584091\pi\)
\(12\) 0 0
\(13\) − 5.46410i − 1.51547i −0.652563 0.757735i \(-0.726306\pi\)
0.652563 0.757735i \(-0.273694\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.73205i 1.14769i 0.818964 + 0.573845i \(0.194549\pi\)
−0.818964 + 0.573845i \(0.805451\pi\)
\(18\) 0 0
\(19\) 4.46410 1.02414 0.512068 0.858945i \(-0.328880\pi\)
0.512068 + 0.858945i \(0.328880\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.73205 −1.43581 −0.717903 0.696143i \(-0.754898\pi\)
−0.717903 + 0.696143i \(0.754898\pi\)
\(30\) 0 0
\(31\) 5.92820 1.06474 0.532368 0.846513i \(-0.321302\pi\)
0.532368 + 0.846513i \(0.321302\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.19615i − 1.01864i −0.860577 0.509321i \(-0.829897\pi\)
0.860577 0.509321i \(-0.170103\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.1962 1.74855 0.874273 0.485435i \(-0.161339\pi\)
0.874273 + 0.485435i \(0.161339\pi\)
\(42\) 0 0
\(43\) − 3.26795i − 0.498358i −0.968458 0.249179i \(-0.919839\pi\)
0.968458 0.249179i \(-0.0801607\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.26795i 0.184949i 0.995715 + 0.0924747i \(0.0294777\pi\)
−0.995715 + 0.0924747i \(0.970522\pi\)
\(48\) 0 0
\(49\) −0.464102 −0.0663002
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.26795i − 0.998330i −0.866507 0.499165i \(-0.833640\pi\)
0.866507 0.499165i \(-0.166360\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.73205 −1.00663 −0.503314 0.864104i \(-0.667886\pi\)
−0.503314 + 0.864104i \(0.667886\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.39230i 0.780944i 0.920615 + 0.390472i \(0.127688\pi\)
−0.920615 + 0.390472i \(0.872312\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.1962 1.32874 0.664369 0.747404i \(-0.268700\pi\)
0.664369 + 0.747404i \(0.268700\pi\)
\(72\) 0 0
\(73\) 0.196152i 0.0229579i 0.999934 + 0.0114790i \(0.00365394\pi\)
−0.999934 + 0.0114790i \(0.996346\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.73205i 0.539267i
\(78\) 0 0
\(79\) 14.3923 1.61926 0.809630 0.586940i \(-0.199668\pi\)
0.809630 + 0.586940i \(0.199668\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 15.1244i − 1.66011i −0.557679 0.830057i \(-0.688308\pi\)
0.557679 0.830057i \(-0.311692\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −14.9282 −1.56490
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.732051i 0.0743285i 0.999309 + 0.0371642i \(0.0118325\pi\)
−0.999309 + 0.0371642i \(0.988168\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.12436 −0.609396 −0.304698 0.952449i \(-0.598556\pi\)
−0.304698 + 0.952449i \(0.598556\pi\)
\(102\) 0 0
\(103\) − 18.3923i − 1.81225i −0.423013 0.906124i \(-0.639027\pi\)
0.423013 0.906124i \(-0.360973\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 3.46410i − 0.334887i −0.985882 0.167444i \(-0.946449\pi\)
0.985882 0.167444i \(-0.0535512\pi\)
\(108\) 0 0
\(109\) 7.92820 0.759384 0.379692 0.925113i \(-0.376030\pi\)
0.379692 + 0.925113i \(0.376030\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 0.339746i − 0.0319606i −0.999872 0.0159803i \(-0.994913\pi\)
0.999872 0.0159803i \(-0.00508691\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.9282 1.18513
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.19615i 0.372348i 0.982517 + 0.186174i \(0.0596089\pi\)
−0.982517 + 0.186174i \(0.940391\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.2679 −0.897115 −0.448557 0.893754i \(-0.648062\pi\)
−0.448557 + 0.893754i \(0.648062\pi\)
\(132\) 0 0
\(133\) − 12.1962i − 1.05754i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.39230i − 0.375260i −0.982240 0.187630i \(-0.939919\pi\)
0.982240 0.187630i \(-0.0600806\pi\)
\(138\) 0 0
\(139\) −15.3923 −1.30556 −0.652779 0.757548i \(-0.726397\pi\)
−0.652779 + 0.757548i \(0.726397\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.46410i 0.791428i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −17.3923 −1.41537 −0.707683 0.706530i \(-0.750259\pi\)
−0.707683 + 0.706530i \(0.750259\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.26795i 0.260811i 0.991461 + 0.130405i \(0.0416279\pi\)
−0.991461 + 0.130405i \(0.958372\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.46410 0.745876
\(162\) 0 0
\(163\) − 18.7321i − 1.46721i −0.679577 0.733604i \(-0.737837\pi\)
0.679577 0.733604i \(-0.262163\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.1244i 1.63465i 0.576176 + 0.817326i \(0.304544\pi\)
−0.576176 + 0.817326i \(0.695456\pi\)
\(168\) 0 0
\(169\) −16.8564 −1.29665
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 24.2487i − 1.84360i −0.387671 0.921798i \(-0.626720\pi\)
0.387671 0.921798i \(-0.373280\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.1244 −0.906217 −0.453108 0.891455i \(-0.649685\pi\)
−0.453108 + 0.891455i \(0.649685\pi\)
\(180\) 0 0
\(181\) −16.4641 −1.22377 −0.611884 0.790948i \(-0.709588\pi\)
−0.611884 + 0.790948i \(0.709588\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.19615i − 0.599362i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.0526 −1.37859 −0.689297 0.724479i \(-0.742081\pi\)
−0.689297 + 0.724479i \(0.742081\pi\)
\(192\) 0 0
\(193\) 10.5885i 0.762174i 0.924539 + 0.381087i \(0.124450\pi\)
−0.924539 + 0.381087i \(0.875550\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.8564i 0.987228i 0.869681 + 0.493614i \(0.164324\pi\)
−0.869681 + 0.493614i \(0.835676\pi\)
\(198\) 0 0
\(199\) −15.8564 −1.12403 −0.562015 0.827127i \(-0.689974\pi\)
−0.562015 + 0.827127i \(0.689974\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 21.1244i 1.48264i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.73205 −0.534837
\(210\) 0 0
\(211\) −19.9282 −1.37191 −0.685957 0.727642i \(-0.740616\pi\)
−0.685957 + 0.727642i \(0.740616\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 16.1962i − 1.09947i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 25.8564 1.73929
\(222\) 0 0
\(223\) 5.85641i 0.392174i 0.980587 + 0.196087i \(0.0628235\pi\)
−0.980587 + 0.196087i \(0.937177\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 26.1962i − 1.73870i −0.494197 0.869350i \(-0.664538\pi\)
0.494197 0.869350i \(-0.335462\pi\)
\(228\) 0 0
\(229\) 17.8564 1.17998 0.589992 0.807409i \(-0.299131\pi\)
0.589992 + 0.807409i \(0.299131\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.5885i 1.61084i 0.592702 + 0.805422i \(0.298061\pi\)
−0.592702 + 0.805422i \(0.701939\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.53590 0.552141 0.276071 0.961137i \(-0.410968\pi\)
0.276071 + 0.961137i \(0.410968\pi\)
\(240\) 0 0
\(241\) 10.3205 0.664802 0.332401 0.943138i \(-0.392141\pi\)
0.332401 + 0.943138i \(0.392141\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 24.3923i − 1.55205i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.5359 −1.29621 −0.648107 0.761549i \(-0.724439\pi\)
−0.648107 + 0.761549i \(0.724439\pi\)
\(252\) 0 0
\(253\) − 6.00000i − 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 15.4641i − 0.964624i −0.875999 0.482312i \(-0.839797\pi\)
0.875999 0.482312i \(-0.160203\pi\)
\(258\) 0 0
\(259\) −16.9282 −1.05187
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 24.2487i − 1.49524i −0.664127 0.747620i \(-0.731197\pi\)
0.664127 0.747620i \(-0.268803\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.2679 −0.991874 −0.495937 0.868358i \(-0.665175\pi\)
−0.495937 + 0.868358i \(0.665175\pi\)
\(270\) 0 0
\(271\) 16.7846 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.12436i 0.307893i 0.988079 + 0.153946i \(0.0491983\pi\)
−0.988079 + 0.153946i \(0.950802\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −29.3205 −1.74911 −0.874557 0.484922i \(-0.838848\pi\)
−0.874557 + 0.484922i \(0.838848\pi\)
\(282\) 0 0
\(283\) − 16.5359i − 0.982957i −0.870890 0.491479i \(-0.836457\pi\)
0.870890 0.491479i \(-0.163543\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 30.5885i − 1.80558i
\(288\) 0 0
\(289\) −5.39230 −0.317194
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.5885i 1.43647i 0.695799 + 0.718237i \(0.255050\pi\)
−0.695799 + 0.718237i \(0.744950\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 18.9282 1.09465
\(300\) 0 0
\(301\) −8.92820 −0.514613
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.80385i − 0.102951i −0.998674 0.0514755i \(-0.983608\pi\)
0.998674 0.0514755i \(-0.0163924\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.5167 0.936574 0.468287 0.883576i \(-0.344871\pi\)
0.468287 + 0.883576i \(0.344871\pi\)
\(312\) 0 0
\(313\) − 14.9282i − 0.843792i −0.906644 0.421896i \(-0.861365\pi\)
0.906644 0.421896i \(-0.138635\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3.12436i − 0.175481i −0.996143 0.0877406i \(-0.972035\pi\)
0.996143 0.0877406i \(-0.0279647\pi\)
\(318\) 0 0
\(319\) 13.3923 0.749825
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 21.1244i 1.17539i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.46410 0.190982
\(330\) 0 0
\(331\) −29.3923 −1.61555 −0.807774 0.589493i \(-0.799328\pi\)
−0.807774 + 0.589493i \(0.799328\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 34.2487i − 1.86565i −0.360335 0.932823i \(-0.617338\pi\)
0.360335 0.932823i \(-0.382662\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.2679 −0.556041
\(342\) 0 0
\(343\) − 17.8564i − 0.964155i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 21.8038i − 1.17049i −0.810856 0.585246i \(-0.800998\pi\)
0.810856 0.585246i \(-0.199002\pi\)
\(348\) 0 0
\(349\) 25.0000 1.33822 0.669110 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 28.9808i − 1.54249i −0.636538 0.771245i \(-0.719634\pi\)
0.636538 0.771245i \(-0.280366\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.33975 0.492933 0.246466 0.969151i \(-0.420730\pi\)
0.246466 + 0.969151i \(0.420730\pi\)
\(360\) 0 0
\(361\) 0.928203 0.0488528
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 26.3923i − 1.37767i −0.724920 0.688834i \(-0.758123\pi\)
0.724920 0.688834i \(-0.241877\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −19.8564 −1.03089
\(372\) 0 0
\(373\) 14.0526i 0.727614i 0.931474 + 0.363807i \(0.118523\pi\)
−0.931474 + 0.363807i \(0.881477\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 42.2487i 2.17592i
\(378\) 0 0
\(379\) −4.53590 −0.232993 −0.116497 0.993191i \(-0.537166\pi\)
−0.116497 + 0.993191i \(0.537166\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 18.2487i 0.932466i 0.884662 + 0.466233i \(0.154389\pi\)
−0.884662 + 0.466233i \(0.845611\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 27.4641 1.39249 0.696243 0.717807i \(-0.254854\pi\)
0.696243 + 0.717807i \(0.254854\pi\)
\(390\) 0 0
\(391\) −16.3923 −0.828994
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 37.3205i 1.87306i 0.350584 + 0.936531i \(0.385983\pi\)
−0.350584 + 0.936531i \(0.614017\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.7846 0.738308 0.369154 0.929368i \(-0.379647\pi\)
0.369154 + 0.929368i \(0.379647\pi\)
\(402\) 0 0
\(403\) − 32.3923i − 1.61358i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.7321i 0.531968i
\(408\) 0 0
\(409\) 17.8564 0.882942 0.441471 0.897275i \(-0.354457\pi\)
0.441471 + 0.897275i \(0.354457\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 21.1244i 1.03946i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.4641 1.04859 0.524295 0.851537i \(-0.324329\pi\)
0.524295 + 0.851537i \(0.324329\pi\)
\(420\) 0 0
\(421\) 13.7846 0.671821 0.335910 0.941894i \(-0.390956\pi\)
0.335910 + 0.941894i \(0.390956\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 10.9282i 0.528853i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 33.5885 1.61790 0.808950 0.587878i \(-0.200037\pi\)
0.808950 + 0.587878i \(0.200037\pi\)
\(432\) 0 0
\(433\) − 11.4641i − 0.550930i −0.961311 0.275465i \(-0.911168\pi\)
0.961311 0.275465i \(-0.0888317\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.4641i 0.739748i
\(438\) 0 0
\(439\) −6.60770 −0.315368 −0.157684 0.987490i \(-0.550403\pi\)
−0.157684 + 0.987490i \(0.550403\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 37.2679i 1.77065i 0.464969 + 0.885327i \(0.346065\pi\)
−0.464969 + 0.885327i \(0.653935\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.1244 1.13850 0.569249 0.822165i \(-0.307234\pi\)
0.569249 + 0.822165i \(0.307234\pi\)
\(450\) 0 0
\(451\) −19.3923 −0.913148
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.1962i 1.03829i 0.854686 + 0.519146i \(0.173750\pi\)
−0.854686 + 0.519146i \(0.826250\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.58846 −0.446579 −0.223289 0.974752i \(-0.571680\pi\)
−0.223289 + 0.974752i \(0.571680\pi\)
\(462\) 0 0
\(463\) 2.39230i 0.111180i 0.998454 + 0.0555899i \(0.0177039\pi\)
−0.998454 + 0.0555899i \(0.982296\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 26.1962i − 1.21221i −0.795383 0.606107i \(-0.792730\pi\)
0.795383 0.606107i \(-0.207270\pi\)
\(468\) 0 0
\(469\) 17.4641 0.806417
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.66025i 0.260259i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.33975 −0.152597 −0.0762984 0.997085i \(-0.524310\pi\)
−0.0762984 + 0.997085i \(0.524310\pi\)
\(480\) 0 0
\(481\) −33.8564 −1.54372
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 30.5359i − 1.38371i −0.722035 0.691857i \(-0.756793\pi\)
0.722035 0.691857i \(-0.243207\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.26795 0.192610 0.0963049 0.995352i \(-0.469298\pi\)
0.0963049 + 0.995352i \(0.469298\pi\)
\(492\) 0 0
\(493\) − 36.5885i − 1.64786i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 30.5885i − 1.37208i
\(498\) 0 0
\(499\) −27.3923 −1.22625 −0.613124 0.789987i \(-0.710087\pi\)
−0.613124 + 0.789987i \(0.710087\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 35.3205i − 1.57486i −0.616402 0.787432i \(-0.711410\pi\)
0.616402 0.787432i \(-0.288590\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −26.7846 −1.18721 −0.593603 0.804758i \(-0.702295\pi\)
−0.593603 + 0.804758i \(0.702295\pi\)
\(510\) 0 0
\(511\) 0.535898 0.0237067
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.19615i − 0.0965867i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.3923 0.455295 0.227648 0.973744i \(-0.426897\pi\)
0.227648 + 0.973744i \(0.426897\pi\)
\(522\) 0 0
\(523\) − 3.60770i − 0.157753i −0.996884 0.0788767i \(-0.974867\pi\)
0.996884 0.0788767i \(-0.0251334\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.0526i 1.22199i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 61.1769i − 2.64987i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.803848 0.0346242
\(540\) 0 0
\(541\) 2.46410 0.105940 0.0529700 0.998596i \(-0.483131\pi\)
0.0529700 + 0.998596i \(0.483131\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.78461i 0.204575i 0.994755 + 0.102288i \(0.0326162\pi\)
−0.994755 + 0.102288i \(0.967384\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −34.5167 −1.47046
\(552\) 0 0
\(553\) − 39.3205i − 1.67208i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.39230i 0.186108i 0.995661 + 0.0930540i \(0.0296629\pi\)
−0.995661 + 0.0930540i \(0.970337\pi\)
\(558\) 0 0
\(559\) −17.8564 −0.755246
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.7321i 0.958042i 0.877804 + 0.479021i \(0.159008\pi\)
−0.877804 + 0.479021i \(0.840992\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −43.0526 −1.80486 −0.902429 0.430839i \(-0.858218\pi\)
−0.902429 + 0.430839i \(0.858218\pi\)
\(570\) 0 0
\(571\) 0.856406 0.0358395 0.0179197 0.999839i \(-0.494296\pi\)
0.0179197 + 0.999839i \(0.494296\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.8038i 0.491401i 0.969346 + 0.245700i \(0.0790179\pi\)
−0.969346 + 0.245700i \(0.920982\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −41.3205 −1.71426
\(582\) 0 0
\(583\) 12.5885i 0.521361i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 3.80385i − 0.157002i −0.996914 0.0785008i \(-0.974987\pi\)
0.996914 0.0785008i \(-0.0250133\pi\)
\(588\) 0 0
\(589\) 26.4641 1.09043
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 23.0718i − 0.947445i −0.880674 0.473723i \(-0.842910\pi\)
0.880674 0.473723i \(-0.157090\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.4115 −0.588840 −0.294420 0.955676i \(-0.595126\pi\)
−0.294420 + 0.955676i \(0.595126\pi\)
\(600\) 0 0
\(601\) −24.3205 −0.992054 −0.496027 0.868307i \(-0.665208\pi\)
−0.496027 + 0.868307i \(0.665208\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 2.58846i 0.105062i 0.998619 + 0.0525311i \(0.0167289\pi\)
−0.998619 + 0.0525311i \(0.983271\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.92820 0.280285
\(612\) 0 0
\(613\) − 24.3923i − 0.985196i −0.870257 0.492598i \(-0.836047\pi\)
0.870257 0.492598i \(-0.163953\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000i 0.483102i 0.970388 + 0.241551i \(0.0776561\pi\)
−0.970388 + 0.241551i \(0.922344\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 14.1962i − 0.568757i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 29.3205 1.16909
\(630\) 0 0
\(631\) −0.0717968 −0.00285818 −0.00142909 0.999999i \(-0.500455\pi\)
−0.00142909 + 0.999999i \(0.500455\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.53590i 0.100476i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.8038 0.505722 0.252861 0.967503i \(-0.418629\pi\)
0.252861 + 0.967503i \(0.418629\pi\)
\(642\) 0 0
\(643\) − 14.5885i − 0.575313i −0.957734 0.287656i \(-0.907124\pi\)
0.957734 0.287656i \(-0.0928761\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 0.248711i − 0.00977785i −0.999988 0.00488893i \(-0.998444\pi\)
0.999988 0.00488893i \(-0.00155620\pi\)
\(648\) 0 0
\(649\) 13.3923 0.525694
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.53590i − 0.0992374i −0.998768 0.0496187i \(-0.984199\pi\)
0.998768 0.0496187i \(-0.0158006\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.53590 −0.0987846 −0.0493923 0.998779i \(-0.515728\pi\)
−0.0493923 + 0.998779i \(0.515728\pi\)
\(660\) 0 0
\(661\) 15.3923 0.598691 0.299346 0.954145i \(-0.403232\pi\)
0.299346 + 0.954145i \(0.403232\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 26.7846i − 1.03710i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.92820 0.267460
\(672\) 0 0
\(673\) 38.3923i 1.47991i 0.672654 + 0.739957i \(0.265154\pi\)
−0.672654 + 0.739957i \(0.734846\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 40.6410i − 1.56196i −0.624555 0.780981i \(-0.714720\pi\)
0.624555 0.780981i \(-0.285280\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 21.4641i 0.821301i 0.911793 + 0.410651i \(0.134698\pi\)
−0.911793 + 0.410651i \(0.865302\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −39.7128 −1.51294
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 52.9808i 2.00679i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.8756 0.675154 0.337577 0.941298i \(-0.390393\pi\)
0.337577 + 0.941298i \(0.390393\pi\)
\(702\) 0 0
\(703\) − 27.6603i − 1.04323i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.7321i 0.629274i
\(708\) 0 0
\(709\) −10.5359 −0.395684 −0.197842 0.980234i \(-0.563393\pi\)
−0.197842 + 0.980234i \(0.563393\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.5359i 0.769075i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17.1962 −0.641308 −0.320654 0.947196i \(-0.603903\pi\)
−0.320654 + 0.947196i \(0.603903\pi\)
\(720\) 0 0
\(721\) −50.2487 −1.87136
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 29.1769i − 1.08211i −0.840987 0.541056i \(-0.818025\pi\)
0.840987 0.541056i \(-0.181975\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15.4641 0.571960
\(732\) 0 0
\(733\) − 43.5692i − 1.60927i −0.593773 0.804633i \(-0.702362\pi\)
0.593773 0.804633i \(-0.297638\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 11.0718i − 0.407835i
\(738\) 0 0
\(739\) 26.1769 0.962933 0.481467 0.876464i \(-0.340104\pi\)
0.481467 + 0.876464i \(0.340104\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 5.41154i − 0.198530i −0.995061 0.0992651i \(-0.968351\pi\)
0.995061 0.0992651i \(-0.0316492\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.46410 −0.345811
\(750\) 0 0
\(751\) 40.7846 1.48825 0.744126 0.668040i \(-0.232866\pi\)
0.744126 + 0.668040i \(0.232866\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 20.3923i − 0.741171i −0.928798 0.370585i \(-0.879157\pi\)
0.928798 0.370585i \(-0.120843\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.1244 1.09201 0.546004 0.837783i \(-0.316149\pi\)
0.546004 + 0.837783i \(0.316149\pi\)
\(762\) 0 0
\(763\) − 21.6603i − 0.784154i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 42.2487i 1.52551i
\(768\) 0 0
\(769\) −35.2487 −1.27110 −0.635551 0.772059i \(-0.719227\pi\)
−0.635551 + 0.772059i \(0.719227\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.6603i 0.635195i 0.948226 + 0.317598i \(0.102876\pi\)
−0.948226 + 0.317598i \(0.897124\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 49.9808 1.79075
\(780\) 0 0
\(781\) −19.3923 −0.693911
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 36.1962i − 1.29025i −0.764076 0.645127i \(-0.776805\pi\)
0.764076 0.645127i \(-0.223195\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.928203 −0.0330031
\(792\) 0 0
\(793\) 21.8564i 0.776144i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 23.3205i − 0.826055i −0.910719 0.413027i \(-0.864471\pi\)
0.910719 0.413027i \(-0.135529\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 0.339746i − 0.0119894i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8.41154 −0.295734 −0.147867 0.989007i \(-0.547241\pi\)
−0.147867 + 0.989007i \(0.547241\pi\)
\(810\) 0 0
\(811\) −25.2487 −0.886602 −0.443301 0.896373i \(-0.646193\pi\)
−0.443301 + 0.896373i \(0.646193\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 14.5885i − 0.510386i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.33975 −0.116558 −0.0582790 0.998300i \(-0.518561\pi\)
−0.0582790 + 0.998300i \(0.518561\pi\)
\(822\) 0 0
\(823\) 16.9282i 0.590080i 0.955485 + 0.295040i \(0.0953330\pi\)
−0.955485 + 0.295040i \(0.904667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 45.4641i 1.58094i 0.612500 + 0.790471i \(0.290164\pi\)
−0.612500 + 0.790471i \(0.709836\pi\)
\(828\) 0 0
\(829\) 51.7846 1.79855 0.899277 0.437380i \(-0.144093\pi\)
0.899277 + 0.437380i \(0.144093\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2.19615i − 0.0760922i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8.41154 −0.290399 −0.145199 0.989402i \(-0.546382\pi\)
−0.145199 + 0.989402i \(0.546382\pi\)
\(840\) 0 0
\(841\) 30.7846 1.06154
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 21.8564i 0.750995i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 21.4641 0.735780
\(852\) 0 0
\(853\) 0.196152i 0.00671613i 0.999994 + 0.00335807i \(0.00106891\pi\)
−0.999994 + 0.00335807i \(0.998931\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 29.0718i 0.993074i 0.868016 + 0.496537i \(0.165395\pi\)
−0.868016 + 0.496537i \(0.834605\pi\)
\(858\) 0 0
\(859\) −7.78461 −0.265607 −0.132804 0.991142i \(-0.542398\pi\)
−0.132804 + 0.991142i \(0.542398\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 49.5167i − 1.68557i −0.538253 0.842783i \(-0.680915\pi\)
0.538253 0.842783i \(-0.319085\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −24.9282 −0.845631
\(870\) 0 0
\(871\) 34.9282 1.18350
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.2487i 0.481145i 0.970631 + 0.240572i \(0.0773351\pi\)
−0.970631 + 0.240572i \(0.922665\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.80385 −0.229227 −0.114614 0.993410i \(-0.536563\pi\)
−0.114614 + 0.993410i \(0.536563\pi\)
\(882\) 0 0
\(883\) 46.8372i 1.57620i 0.615550 + 0.788098i \(0.288934\pi\)
−0.615550 + 0.788098i \(0.711066\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 13.2679i − 0.445494i −0.974876 0.222747i \(-0.928498\pi\)
0.974876 0.222747i \(-0.0715024\pi\)
\(888\) 0 0
\(889\) 11.4641 0.384494
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.66025i 0.189413i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −45.8372 −1.52876
\(900\) 0 0
\(901\) 34.3923 1.14577
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1.21539i − 0.0403564i −0.999796 0.0201782i \(-0.993577\pi\)
0.999796 0.0201782i \(-0.00642335\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.80385 0.225422 0.112711 0.993628i \(-0.464047\pi\)
0.112711 + 0.993628i \(0.464047\pi\)
\(912\) 0 0
\(913\) 26.1962i 0.866966i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 28.0526i 0.926377i
\(918\) 0 0
\(919\) −51.3923 −1.69528 −0.847638 0.530575i \(-0.821976\pi\)
−0.847638 + 0.530575i \(0.821976\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 61.1769i − 2.01366i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.48334 0.0486668 0.0243334 0.999704i \(-0.492254\pi\)
0.0243334 + 0.999704i \(0.492254\pi\)
\(930\) 0 0
\(931\) −2.07180 −0.0679004
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 19.0718i 0.623048i 0.950238 + 0.311524i \(0.100840\pi\)
−0.950238 + 0.311524i \(0.899160\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.53590 0.278262 0.139131 0.990274i \(-0.455569\pi\)
0.139131 + 0.990274i \(0.455569\pi\)
\(942\) 0 0
\(943\) 38.7846i 1.26300i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 52.6410i 1.71060i 0.518130 + 0.855302i \(0.326628\pi\)
−0.518130 + 0.855302i \(0.673372\pi\)
\(948\) 0 0
\(949\) 1.07180 0.0347920
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 2.53590i − 0.0821458i −0.999156 0.0410729i \(-0.986922\pi\)
0.999156 0.0410729i \(-0.0130776\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 4.14359 0.133664
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.41154i 0.238339i 0.992874 + 0.119170i \(0.0380232\pi\)
−0.992874 + 0.119170i \(0.961977\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −22.2679 −0.714612 −0.357306 0.933987i \(-0.616305\pi\)
−0.357306 + 0.933987i \(0.616305\pi\)
\(972\) 0 0
\(973\) 42.0526i 1.34814i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 4.39230i − 0.140522i −0.997529 0.0702611i \(-0.977617\pi\)
0.997529 0.0702611i \(-0.0223833\pi\)
\(978\) 0 0
\(979\) −9.00000 −0.287641
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 52.7321i − 1.68189i −0.541120 0.840946i \(-0.681999\pi\)
0.541120 0.840946i \(-0.318001\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.3205 0.359971
\(990\) 0 0
\(991\) 55.7846 1.77206 0.886028 0.463631i \(-0.153454\pi\)
0.886028 + 0.463631i \(0.153454\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 52.1962i 1.65307i 0.562886 + 0.826534i \(0.309691\pi\)
−0.562886 + 0.826534i \(0.690309\pi\)
\(998\) 0 0
\(999\) 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 8100.2.d.l.649.1 4
3.2 odd 2 8100.2.d.m.649.1 4
5.2 odd 4 8100.2.a.s.1.2 2
5.3 odd 4 1620.2.a.h.1.1 yes 2
5.4 even 2 inner 8100.2.d.l.649.4 4
15.2 even 4 8100.2.a.t.1.2 2
15.8 even 4 1620.2.a.g.1.1 2
15.14 odd 2 8100.2.d.m.649.4 4
20.3 even 4 6480.2.a.bp.1.2 2
45.13 odd 12 1620.2.i.m.541.2 4
45.23 even 12 1620.2.i.n.541.2 4
45.38 even 12 1620.2.i.n.1081.2 4
45.43 odd 12 1620.2.i.m.1081.2 4
60.23 odd 4 6480.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.a.g.1.1 2 15.8 even 4
1620.2.a.h.1.1 yes 2 5.3 odd 4
1620.2.i.m.541.2 4 45.13 odd 12
1620.2.i.m.1081.2 4 45.43 odd 12
1620.2.i.n.541.2 4 45.23 even 12
1620.2.i.n.1081.2 4 45.38 even 12
6480.2.a.bh.1.2 2 60.23 odd 4
6480.2.a.bp.1.2 2 20.3 even 4
8100.2.a.s.1.2 2 5.2 odd 4
8100.2.a.t.1.2 2 15.2 even 4
8100.2.d.l.649.1 4 1.1 even 1 trivial
8100.2.d.l.649.4 4 5.4 even 2 inner
8100.2.d.m.649.1 4 3.2 odd 2
8100.2.d.m.649.4 4 15.14 odd 2