Properties

Label 4864.2.a.o.1.1
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -4.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -4.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} -5.00000 q^{13} -12.0000 q^{15} -5.00000 q^{17} +1.00000 q^{19} -3.00000 q^{21} +3.00000 q^{23} +11.0000 q^{25} +9.00000 q^{27} +7.00000 q^{29} +10.0000 q^{31} +4.00000 q^{35} -2.00000 q^{37} -15.0000 q^{39} +6.00000 q^{41} +4.00000 q^{43} -24.0000 q^{45} +8.00000 q^{47} -6.00000 q^{49} -15.0000 q^{51} -9.00000 q^{53} +3.00000 q^{57} -1.00000 q^{59} -2.00000 q^{61} -6.00000 q^{63} +20.0000 q^{65} +7.00000 q^{67} +9.00000 q^{69} +12.0000 q^{71} -11.0000 q^{73} +33.0000 q^{75} +16.0000 q^{79} +9.00000 q^{81} +14.0000 q^{83} +20.0000 q^{85} +21.0000 q^{87} +4.00000 q^{89} +5.00000 q^{91} +30.0000 q^{93} -4.00000 q^{95} -12.0000 q^{97} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) −12.0000 −3.09839
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −15.0000 −2.40192
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −24.0000 −3.57771
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −15.0000 −2.10042
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) 20.0000 2.48069
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 0 0
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 33.0000 3.81051
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 20.0000 2.16930
\(86\) 0 0
\(87\) 21.0000 2.25144
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 0 0
\(93\) 30.0000 3.11086
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 0 0
\(117\) −30.0000 −2.77350
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 18.0000 1.62301
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −36.0000 −3.09839
\(136\) 0 0
\(137\) 15.0000 1.28154 0.640768 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −28.0000 −2.32527
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −30.0000 −2.42536
\(154\) 0 0
\(155\) −40.0000 −3.21288
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) −27.0000 −2.14124
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −11.0000 −0.831522
\(176\) 0 0
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) −5.00000 −0.361787 −0.180894 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) 0 0
\(195\) 60.0000 4.29669
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 0 0
\(201\) 21.0000 1.48123
\(202\) 0 0
\(203\) −7.00000 −0.491304
\(204\) 0 0
\(205\) −24.0000 −1.67623
\(206\) 0 0
\(207\) 18.0000 1.25109
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) 36.0000 2.46668
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) −33.0000 −2.22993
\(220\) 0 0
\(221\) 25.0000 1.68168
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) 66.0000 4.40000
\(226\) 0 0
\(227\) −17.0000 −1.12833 −0.564165 0.825662i \(-0.690802\pi\)
−0.564165 + 0.825662i \(0.690802\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −32.0000 −2.08745
\(236\) 0 0
\(237\) 48.0000 3.11794
\(238\) 0 0
\(239\) 11.0000 0.711531 0.355765 0.934575i \(-0.384220\pi\)
0.355765 + 0.934575i \(0.384220\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 24.0000 1.53330
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) 0 0
\(249\) 42.0000 2.66164
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 60.0000 3.75735
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 42.0000 2.59973
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 36.0000 2.21146
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 0 0
\(273\) 15.0000 0.907841
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 0 0
\(279\) 60.0000 3.59211
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 0 0
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −36.0000 −2.11036
\(292\) 0 0
\(293\) −7.00000 −0.408944 −0.204472 0.978872i \(-0.565548\pi\)
−0.204472 + 0.978872i \(0.565548\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −15.0000 −0.867472
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 35.0000 1.98467 0.992334 0.123585i \(-0.0394392\pi\)
0.992334 + 0.123585i \(0.0394392\pi\)
\(312\) 0 0
\(313\) 9.00000 0.508710 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) 0 0
\(315\) 24.0000 1.35225
\(316\) 0 0
\(317\) −23.0000 −1.29181 −0.645904 0.763418i \(-0.723520\pi\)
−0.645904 + 0.763418i \(0.723520\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 27.0000 1.50699
\(322\) 0 0
\(323\) −5.00000 −0.278207
\(324\) 0 0
\(325\) −55.0000 −3.05085
\(326\) 0 0
\(327\) 27.0000 1.49310
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −11.0000 −0.604615 −0.302307 0.953211i \(-0.597757\pi\)
−0.302307 + 0.953211i \(0.597757\pi\)
\(332\) 0 0
\(333\) −12.0000 −0.657596
\(334\) 0 0
\(335\) −28.0000 −1.52980
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 24.0000 1.30350
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) −36.0000 −1.93817
\(346\) 0 0
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) −45.0000 −2.40192
\(352\) 0 0
\(353\) 29.0000 1.54351 0.771757 0.635917i \(-0.219378\pi\)
0.771757 + 0.635917i \(0.219378\pi\)
\(354\) 0 0
\(355\) −48.0000 −2.54758
\(356\) 0 0
\(357\) 15.0000 0.793884
\(358\) 0 0
\(359\) −27.0000 −1.42501 −0.712503 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −33.0000 −1.73205
\(364\) 0 0
\(365\) 44.0000 2.30307
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) 36.0000 1.87409
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 0 0
\(373\) −15.0000 −0.776671 −0.388335 0.921518i \(-0.626950\pi\)
−0.388335 + 0.921518i \(0.626950\pi\)
\(374\) 0 0
\(375\) −72.0000 −3.71806
\(376\) 0 0
\(377\) −35.0000 −1.80259
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 24.0000 1.21999
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −15.0000 −0.758583
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −64.0000 −3.22019
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) −4.00000 −0.199750 −0.0998752 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) 0 0
\(403\) −50.0000 −2.49068
\(404\) 0 0
\(405\) −36.0000 −1.78885
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) 45.0000 2.21969
\(412\) 0 0
\(413\) 1.00000 0.0492068
\(414\) 0 0
\(415\) −56.0000 −2.74893
\(416\) 0 0
\(417\) −42.0000 −2.05675
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −21.0000 −1.02348 −0.511739 0.859141i \(-0.670998\pi\)
−0.511739 + 0.859141i \(0.670998\pi\)
\(422\) 0 0
\(423\) 48.0000 2.33384
\(424\) 0 0
\(425\) −55.0000 −2.66789
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.0000 1.05970 0.529851 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −84.0000 −4.02749
\(436\) 0 0
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) 32.0000 1.52037 0.760183 0.649709i \(-0.225109\pi\)
0.760183 + 0.649709i \(0.225109\pi\)
\(444\) 0 0
\(445\) −16.0000 −0.758473
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 24.0000 1.12762
\(454\) 0 0
\(455\) −20.0000 −0.937614
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 0 0
\(459\) −45.0000 −2.10042
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) −120.000 −5.56487
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −7.00000 −0.323230
\(470\) 0 0
\(471\) −60.0000 −2.76465
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 11.0000 0.504715
\(476\) 0 0
\(477\) −54.0000 −2.47249
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) −9.00000 −0.409514
\(484\) 0 0
\(485\) 48.0000 2.17957
\(486\) 0 0
\(487\) 30.0000 1.35943 0.679715 0.733476i \(-0.262104\pi\)
0.679715 + 0.733476i \(0.262104\pi\)
\(488\) 0 0
\(489\) 30.0000 1.35665
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −35.0000 −1.57632
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 0 0
\(503\) −29.0000 −1.29305 −0.646523 0.762894i \(-0.723778\pi\)
−0.646523 + 0.762894i \(0.723778\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 36.0000 1.59882
\(508\) 0 0
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 11.0000 0.486611
\(512\) 0 0
\(513\) 9.00000 0.397360
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 42.0000 1.84360
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 1.00000 0.0437269 0.0218635 0.999761i \(-0.493040\pi\)
0.0218635 + 0.999761i \(0.493040\pi\)
\(524\) 0 0
\(525\) −33.0000 −1.44024
\(526\) 0 0
\(527\) −50.0000 −2.17803
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) −30.0000 −1.29944
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 0 0
\(537\) 36.0000 1.55351
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 0 0
\(545\) −36.0000 −1.54207
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) 7.00000 0.298210
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 24.0000 1.01874
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −32.0000 −1.34625
\(566\) 0 0
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 0 0
\(573\) −15.0000 −0.626634
\(574\) 0 0
\(575\) 33.0000 1.37620
\(576\) 0 0
\(577\) −33.0000 −1.37381 −0.686904 0.726748i \(-0.741031\pi\)
−0.686904 + 0.726748i \(0.741031\pi\)
\(578\) 0 0
\(579\) 60.0000 2.49351
\(580\) 0 0
\(581\) −14.0000 −0.580818
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 120.000 4.96139
\(586\) 0 0
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) −20.0000 −0.819920
\(596\) 0 0
\(597\) −15.0000 −0.613909
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) 42.0000 1.71037
\(604\) 0 0
\(605\) 44.0000 1.78885
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) −21.0000 −0.850963
\(610\) 0 0
\(611\) −40.0000 −1.61823
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) −72.0000 −2.90332
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 34.0000 1.36658 0.683288 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) 27.0000 1.08347
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) −3.00000 −0.119239
\(634\) 0 0
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 30.0000 1.18864
\(638\) 0 0
\(639\) 72.0000 2.84828
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) −48.0000 −1.89000
\(646\) 0 0
\(647\) −25.0000 −0.982851 −0.491426 0.870919i \(-0.663524\pi\)
−0.491426 + 0.870919i \(0.663524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −30.0000 −1.17579
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −66.0000 −2.57491
\(658\) 0 0
\(659\) 21.0000 0.818044 0.409022 0.912525i \(-0.365870\pi\)
0.409022 + 0.912525i \(0.365870\pi\)
\(660\) 0 0
\(661\) 13.0000 0.505641 0.252821 0.967513i \(-0.418642\pi\)
0.252821 + 0.967513i \(0.418642\pi\)
\(662\) 0 0
\(663\) 75.0000 2.91276
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 21.0000 0.813123
\(668\) 0 0
\(669\) −42.0000 −1.62381
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 99.0000 3.81051
\(676\) 0 0
\(677\) 1.00000 0.0384331 0.0192166 0.999815i \(-0.493883\pi\)
0.0192166 + 0.999815i \(0.493883\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) −51.0000 −1.95432
\(682\) 0 0
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) 0 0
\(685\) −60.0000 −2.29248
\(686\) 0 0
\(687\) −66.0000 −2.51806
\(688\) 0 0
\(689\) 45.0000 1.71436
\(690\) 0 0
\(691\) −6.00000 −0.228251 −0.114125 0.993466i \(-0.536407\pi\)
−0.114125 + 0.993466i \(0.536407\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 56.0000 2.12420
\(696\) 0 0
\(697\) −30.0000 −1.13633
\(698\) 0 0
\(699\) −54.0000 −2.04247
\(700\) 0 0
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) −96.0000 −3.61557
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 96.0000 3.60028
\(712\) 0 0
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 33.0000 1.23241
\(718\) 0 0
\(719\) 39.0000 1.45445 0.727227 0.686397i \(-0.240809\pi\)
0.727227 + 0.686397i \(0.240809\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) 0 0
\(723\) −42.0000 −1.56200
\(724\) 0 0
\(725\) 77.0000 2.85971
\(726\) 0 0
\(727\) −25.0000 −0.927199 −0.463599 0.886045i \(-0.653442\pi\)
−0.463599 + 0.886045i \(0.653442\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) 40.0000 1.47743 0.738717 0.674016i \(-0.235432\pi\)
0.738717 + 0.674016i \(0.235432\pi\)
\(734\) 0 0
\(735\) 72.0000 2.65576
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) −15.0000 −0.551039
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) 0 0
\(747\) 84.0000 3.07340
\(748\) 0 0
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 0 0
\(753\) −60.0000 −2.18652
\(754\) 0 0
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.00000 −0.0362500 −0.0181250 0.999836i \(-0.505770\pi\)
−0.0181250 + 0.999836i \(0.505770\pi\)
\(762\) 0 0
\(763\) −9.00000 −0.325822
\(764\) 0 0
\(765\) 120.000 4.33861
\(766\) 0 0
\(767\) 5.00000 0.180540
\(768\) 0 0
\(769\) −7.00000 −0.252426 −0.126213 0.992003i \(-0.540282\pi\)
−0.126213 + 0.992003i \(0.540282\pi\)
\(770\) 0 0
\(771\) 36.0000 1.29651
\(772\) 0 0
\(773\) 17.0000 0.611448 0.305724 0.952120i \(-0.401102\pi\)
0.305724 + 0.952120i \(0.401102\pi\)
\(774\) 0 0
\(775\) 110.000 3.95132
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 63.0000 2.25144
\(784\) 0 0
\(785\) 80.0000 2.85532
\(786\) 0 0
\(787\) −13.0000 −0.463400 −0.231700 0.972787i \(-0.574429\pi\)
−0.231700 + 0.972787i \(0.574429\pi\)
\(788\) 0 0
\(789\) −72.0000 −2.56327
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) 108.000 3.83037
\(796\) 0 0
\(797\) 27.0000 0.956389 0.478195 0.878254i \(-0.341291\pi\)
0.478195 + 0.878254i \(0.341291\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) −42.0000 −1.47847
\(808\) 0 0
\(809\) 29.0000 1.01959 0.509793 0.860297i \(-0.329722\pi\)
0.509793 + 0.860297i \(0.329722\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) 33.0000 1.15736
\(814\) 0 0
\(815\) −40.0000 −1.40114
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 30.0000 1.04828
\(820\) 0 0
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) −51.0000 −1.77775 −0.888874 0.458151i \(-0.848512\pi\)
−0.888874 + 0.458151i \(0.848512\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 43.0000 1.49526 0.747628 0.664117i \(-0.231193\pi\)
0.747628 + 0.664117i \(0.231193\pi\)
\(828\) 0 0
\(829\) 5.00000 0.173657 0.0868286 0.996223i \(-0.472327\pi\)
0.0868286 + 0.996223i \(0.472327\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 90.0000 3.11086
\(838\) 0 0
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) 48.0000 1.65321
\(844\) 0 0
\(845\) −48.0000 −1.65125
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 18.0000 0.617758
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) −24.0000 −0.820783
\(856\) 0 0
\(857\) −20.0000 −0.683187 −0.341593 0.939848i \(-0.610967\pi\)
−0.341593 + 0.939848i \(0.610967\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −18.0000 −0.613438
\(862\) 0 0
\(863\) 10.0000 0.340404 0.170202 0.985409i \(-0.445558\pi\)
0.170202 + 0.985409i \(0.445558\pi\)
\(864\) 0 0
\(865\) −56.0000 −1.90406
\(866\) 0 0
\(867\) 24.0000 0.815083
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −35.0000 −1.18593
\(872\) 0 0
\(873\) −72.0000 −2.43683
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) 37.0000 1.24940 0.624701 0.780864i \(-0.285221\pi\)
0.624701 + 0.780864i \(0.285221\pi\)
\(878\) 0 0
\(879\) −21.0000 −0.708312
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 0 0
\(887\) −26.0000 −0.872995 −0.436497 0.899706i \(-0.643781\pi\)
−0.436497 + 0.899706i \(0.643781\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) −48.0000 −1.60446
\(896\) 0 0
\(897\) −45.0000 −1.50251
\(898\) 0 0
\(899\) 70.0000 2.33463
\(900\) 0 0
\(901\) 45.0000 1.49917
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) −24.0000 −0.797787
\(906\) 0 0
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 24.0000 0.793416
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) 0 0
\(921\) −36.0000 −1.18624
\(922\) 0 0
\(923\) −60.0000 −1.97492
\(924\) 0 0
\(925\) −22.0000 −0.723356
\(926\) 0 0
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) 29.0000 0.951459 0.475730 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 105.000 3.43755
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −31.0000 −1.01273 −0.506363 0.862320i \(-0.669010\pi\)
−0.506363 + 0.862320i \(0.669010\pi\)
\(938\) 0 0
\(939\) 27.0000 0.881112
\(940\) 0 0
\(941\) 51.0000 1.66255 0.831276 0.555860i \(-0.187611\pi\)
0.831276 + 0.555860i \(0.187611\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) 36.0000 1.17108
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 55.0000 1.78538
\(950\) 0 0
\(951\) −69.0000 −2.23748
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 20.0000 0.647185
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15.0000 −0.484375
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 54.0000 1.74013
\(964\) 0 0
\(965\) −80.0000 −2.57529
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) −15.0000 −0.481869
\(970\) 0 0
\(971\) −32.0000 −1.02693 −0.513464 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) 0 0
\(973\) 14.0000 0.448819
\(974\) 0 0
\(975\) −165.000 −5.28423
\(976\) 0 0
\(977\) 26.0000 0.831814 0.415907 0.909407i \(-0.363464\pi\)
0.415907 + 0.909407i \(0.363464\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 54.0000 1.72409
\(982\) 0 0
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) −8.00000 −0.254901
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) −33.0000 −1.04722
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 0 0
\(999\) −18.0000 −0.569495
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 4864.2.a.o.1.1 1
4.3 odd 2 4864.2.a.a.1.1 1
8.3 odd 2 4864.2.a.p.1.1 1
8.5 even 2 4864.2.a.b.1.1 1
16.3 odd 4 1216.2.c.b.609.1 2
16.5 even 4 1216.2.c.c.609.1 yes 2
16.11 odd 4 1216.2.c.b.609.2 yes 2
16.13 even 4 1216.2.c.c.609.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.b.609.1 2 16.3 odd 4
1216.2.c.b.609.2 yes 2 16.11 odd 4
1216.2.c.c.609.1 yes 2 16.5 even 4
1216.2.c.c.609.2 yes 2 16.13 even 4
4864.2.a.a.1.1 1 4.3 odd 2
4864.2.a.b.1.1 1 8.5 even 2
4864.2.a.o.1.1 1 1.1 even 1 trivial
4864.2.a.p.1.1 1 8.3 odd 2