Properties

Label 8100.2.d.l.649.2
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 649.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 8100.649
Dual form 8100.2.d.l.649.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.732051i q^{7} +O(q^{10})\) \(q-0.732051i q^{7} +1.73205 q^{11} -1.46410i q^{13} -1.26795i q^{17} -2.46410 q^{19} +3.46410i q^{23} -4.26795 q^{29} -7.92820 q^{31} -4.19615i q^{37} +0.803848 q^{41} +6.73205i q^{43} -4.73205i q^{47} +6.46410 q^{49} +10.7321i q^{53} -4.26795 q^{59} -4.00000 q^{61} +14.3923i q^{67} +0.803848 q^{71} +10.1962i q^{73} -1.26795i q^{77} -6.39230 q^{79} -9.12436i q^{83} -5.19615 q^{89} -1.07180 q^{91} +2.73205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 4 q^{19} - 24 q^{29} - 4 q^{31} + 24 q^{41} + 12 q^{49} - 24 q^{59} - 16 q^{61} + 24 q^{71} + 16 q^{79} - 32 q^{91} + O(q^{100}) \)

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
\(7\) − 0.732051i − 0.276689i −0.990384 0.138345i \(-0.955822\pi\)
0.990384 0.138345i \(-0.0441781\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.73205 0.522233 0.261116 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(12\) 0 0
\(13\) − 1.46410i − 0.406069i −0.979172 0.203034i \(-0.934920\pi\)
0.979172 0.203034i \(-0.0650803\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.26795i − 0.307523i −0.988108 0.153761i \(-0.950861\pi\)
0.988108 0.153761i \(-0.0491387\pi\)
\(18\) 0 0
\(19\) −2.46410 −0.565304 −0.282652 0.959223i \(-0.591214\pi\)
−0.282652 + 0.959223i \(0.591214\pi\)
\(20\) 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
\(29\) −4.26795 −0.792538 −0.396269 0.918134i \(-0.629695\pi\)
−0.396269 + 0.918134i \(0.629695\pi\)
\(30\) 0 0
\(31\) −7.92820 −1.42395 −0.711974 0.702206i \(-0.752198\pi\)
−0.711974 + 0.702206i \(0.752198\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 4.19615i − 0.689843i −0.938631 0.344922i \(-0.887905\pi\)
0.938631 0.344922i \(-0.112095\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.803848 0.125540 0.0627700 0.998028i \(-0.480007\pi\)
0.0627700 + 0.998028i \(0.480007\pi\)
\(42\) 0 0
\(43\) 6.73205i 1.02663i 0.858201 + 0.513314i \(0.171582\pi\)
−0.858201 + 0.513314i \(0.828418\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4.73205i − 0.690241i −0.938558 0.345120i \(-0.887838\pi\)
0.938558 0.345120i \(-0.112162\pi\)
\(48\) 0 0
\(49\) 6.46410 0.923443
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.7321i 1.47416i 0.675805 + 0.737080i \(0.263796\pi\)
−0.675805 + 0.737080i \(0.736204\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.26795 −0.555640 −0.277820 0.960633i \(-0.589612\pi\)
−0.277820 + 0.960633i \(0.589612\pi\)
\(60\) 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
\(67\) 14.3923i 1.75830i 0.476545 + 0.879150i \(0.341889\pi\)
−0.476545 + 0.879150i \(0.658111\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.803848 0.0953992 0.0476996 0.998862i \(-0.484811\pi\)
0.0476996 + 0.998862i \(0.484811\pi\)
\(72\) 0 0
\(73\) 10.1962i 1.19337i 0.802476 + 0.596685i \(0.203516\pi\)
−0.802476 + 0.596685i \(0.796484\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.26795i − 0.144496i
\(78\) 0 0
\(79\) −6.39230 −0.719190 −0.359595 0.933108i \(-0.617085\pi\)
−0.359595 + 0.933108i \(0.617085\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 9.12436i − 1.00153i −0.865584 0.500764i \(-0.833052\pi\)
0.865584 0.500764i \(-0.166948\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.19615 −0.550791 −0.275396 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 0 0
\(91\) −1.07180 −0.112355
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.73205i 0.277398i 0.990335 + 0.138699i \(0.0442920\pi\)
−0.990335 + 0.138699i \(0.955708\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.1244 1.80344 0.901720 0.432320i \(-0.142305\pi\)
0.901720 + 0.432320i \(0.142305\pi\)
\(102\) 0 0
\(103\) − 2.39230i − 0.235721i −0.993030 0.117860i \(-0.962396\pi\)
0.993030 0.117860i \(-0.0376035\pi\)
\(104\) 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
\(109\) −5.92820 −0.567819 −0.283909 0.958851i \(-0.591632\pi\)
−0.283909 + 0.958851i \(0.591632\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.6603i 1.66134i 0.556767 + 0.830668i \(0.312041\pi\)
−0.556767 + 0.830668i \(0.687959\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.928203 −0.0850883
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.19615i 0.549820i 0.961470 + 0.274910i \(0.0886480\pi\)
−0.961470 + 0.274910i \(0.911352\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.7321 −1.19977 −0.599887 0.800084i \(-0.704788\pi\)
−0.599887 + 0.800084i \(0.704788\pi\)
\(132\) 0 0
\(133\) 1.80385i 0.156413i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 16.3923i − 1.40049i −0.713903 0.700245i \(-0.753074\pi\)
0.713903 0.700245i \(-0.246926\pi\)
\(138\) 0 0
\(139\) 5.39230 0.457369 0.228685 0.973501i \(-0.426558\pi\)
0.228685 + 0.973501i \(0.426558\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 2.53590i − 0.212062i
\(144\) 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
\(151\) 3.39230 0.276062 0.138031 0.990428i \(-0.455923\pi\)
0.138031 + 0.990428i \(0.455923\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 6.73205i − 0.537276i −0.963241 0.268638i \(-0.913426\pi\)
0.963241 0.268638i \(-0.0865736\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.53590 0.199857
\(162\) 0 0
\(163\) 15.2679i 1.19588i 0.801542 + 0.597939i \(0.204014\pi\)
−0.801542 + 0.597939i \(0.795986\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.12436i 0.241770i 0.992667 + 0.120885i \(0.0385732\pi\)
−0.992667 + 0.120885i \(0.961427\pi\)
\(168\) 0 0
\(169\) 10.8564 0.835108
\(170\) 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 \(-0.350315\pi\)
0.453108 + 0.891455i \(0.350315\pi\)
\(180\) 0 0
\(181\) −9.53590 −0.708798 −0.354399 0.935094i \(-0.615314\pi\)
−0.354399 + 0.935094i \(0.615314\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.19615i − 0.160599i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0526 1.37859 0.689297 0.724479i \(-0.257919\pi\)
0.689297 + 0.724479i \(0.257919\pi\)
\(192\) 0 0
\(193\) 20.5885i 1.48199i 0.671511 + 0.740995i \(0.265646\pi\)
−0.671511 + 0.740995i \(0.734354\pi\)
\(194\) 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
\(199\) 11.8564 0.840478 0.420239 0.907413i \(-0.361946\pi\)
0.420239 + 0.907413i \(0.361946\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.12436i 0.219287i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.26795 −0.295220
\(210\) 0 0
\(211\) −6.07180 −0.418000 −0.209000 0.977916i \(-0.567021\pi\)
−0.209000 + 0.977916i \(0.567021\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.80385i 0.393991i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.85641 −0.124875
\(222\) 0 0
\(223\) 21.8564i 1.46361i 0.681512 + 0.731807i \(0.261323\pi\)
−0.681512 + 0.731807i \(0.738677\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.8038i 1.04894i 0.851429 + 0.524469i \(0.175736\pi\)
−0.851429 + 0.524469i \(0.824264\pi\)
\(228\) 0 0
\(229\) −9.85641 −0.651330 −0.325665 0.945485i \(-0.605588\pi\)
−0.325665 + 0.945485i \(0.605588\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.58846i 0.431624i 0.976435 + 0.215812i \(0.0692399\pi\)
−0.976435 + 0.215812i \(0.930760\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.4641 1.00029 0.500145 0.865942i \(-0.333280\pi\)
0.500145 + 0.865942i \(0.333280\pi\)
\(240\) 0 0
\(241\) −24.3205 −1.56662 −0.783311 0.621630i \(-0.786471\pi\)
−0.783311 + 0.621630i \(0.786471\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.60770i 0.229552i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −27.4641 −1.73352 −0.866759 0.498727i \(-0.833801\pi\)
−0.866759 + 0.498727i \(0.833801\pi\)
\(252\) 0 0
\(253\) 6.00000i 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.53590i 0.532455i 0.963910 + 0.266227i \(0.0857772\pi\)
−0.963910 + 0.266227i \(0.914223\pi\)
\(258\) 0 0
\(259\) −3.07180 −0.190872
\(260\) 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
\(269\) −19.7321 −1.20308 −0.601542 0.798841i \(-0.705447\pi\)
−0.601542 + 0.798841i \(0.705447\pi\)
\(270\) 0 0
\(271\) −24.7846 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 19.1244i 1.14907i 0.818480 + 0.574536i \(0.194817\pi\)
−0.818480 + 0.574536i \(0.805183\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.32051 0.317395 0.158697 0.987327i \(-0.449271\pi\)
0.158697 + 0.987327i \(0.449271\pi\)
\(282\) 0 0
\(283\) 23.4641i 1.39480i 0.716684 + 0.697398i \(0.245659\pi\)
−0.716684 + 0.697398i \(0.754341\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 0.588457i − 0.0347355i
\(288\) 0 0
\(289\) 15.3923 0.905430
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.58846i 0.384902i 0.981307 + 0.192451i \(0.0616436\pi\)
−0.981307 + 0.192451i \(0.938356\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.07180 0.293310
\(300\) 0 0
\(301\) 4.92820 0.284057
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.1962i 0.696071i 0.937481 + 0.348036i \(0.113151\pi\)
−0.937481 + 0.348036i \(0.886849\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −28.5167 −1.61703 −0.808516 0.588475i \(-0.799729\pi\)
−0.808516 + 0.588475i \(0.799729\pi\)
\(312\) 0 0
\(313\) 1.07180i 0.0605815i 0.999541 + 0.0302908i \(0.00964333\pi\)
−0.999541 + 0.0302908i \(0.990357\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 21.1244i − 1.18646i −0.805032 0.593231i \(-0.797852\pi\)
0.805032 0.593231i \(-0.202148\pi\)
\(318\) 0 0
\(319\) −7.39230 −0.413890
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.12436i 0.173844i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.46410 −0.190982
\(330\) 0 0
\(331\) −8.60770 −0.473122 −0.236561 0.971617i \(-0.576020\pi\)
−0.236561 + 0.971617i \(0.576020\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 14.2487i − 0.776177i −0.921622 0.388088i \(-0.873136\pi\)
0.921622 0.388088i \(-0.126864\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −13.7321 −0.743632
\(342\) 0 0
\(343\) − 9.85641i − 0.532196i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.1962i 1.72838i 0.503166 + 0.864190i \(0.332169\pi\)
−0.503166 + 0.864190i \(0.667831\pi\)
\(348\) 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
\(353\) − 22.9808i − 1.22314i −0.791189 0.611571i \(-0.790538\pi\)
0.791189 0.611571i \(-0.209462\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.6603 1.40707 0.703537 0.710658i \(-0.251603\pi\)
0.703537 + 0.710658i \(0.251603\pi\)
\(360\) 0 0
\(361\) −12.9282 −0.680432
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.60770i 0.292719i 0.989231 + 0.146360i \(0.0467557\pi\)
−0.989231 + 0.146360i \(0.953244\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.85641 0.407884
\(372\) 0 0
\(373\) 24.0526i 1.24539i 0.782463 + 0.622697i \(0.213963\pi\)
−0.782463 + 0.622697i \(0.786037\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.24871i 0.321825i
\(378\) 0 0
\(379\) −11.4641 −0.588871 −0.294436 0.955671i \(-0.595132\pi\)
−0.294436 + 0.955671i \(0.595132\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 30.2487i 1.54564i 0.634627 + 0.772818i \(0.281154\pi\)
−0.634627 + 0.772818i \(0.718846\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.5359 1.04121 0.520606 0.853797i \(-0.325706\pi\)
0.520606 + 0.853797i \(0.325706\pi\)
\(390\) 0 0
\(391\) 4.39230 0.222128
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.67949i − 0.134480i −0.997737 0.0672399i \(-0.978581\pi\)
0.997737 0.0672399i \(-0.0214193\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.7846 −1.33756 −0.668780 0.743461i \(-0.733183\pi\)
−0.668780 + 0.743461i \(0.733183\pi\)
\(402\) 0 0
\(403\) 11.6077i 0.578220i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 7.26795i − 0.360259i
\(408\) 0 0
\(409\) −9.85641 −0.487368 −0.243684 0.969855i \(-0.578356\pi\)
−0.243684 + 0.969855i \(0.578356\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.12436i 0.153739i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.5359 0.710125 0.355063 0.934843i \(-0.384460\pi\)
0.355063 + 0.934843i \(0.384460\pi\)
\(420\) 0 0
\(421\) −27.7846 −1.35414 −0.677070 0.735919i \(-0.736750\pi\)
−0.677070 + 0.735919i \(0.736750\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.92820i 0.141706i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.41154 0.116160 0.0580800 0.998312i \(-0.481502\pi\)
0.0580800 + 0.998312i \(0.481502\pi\)
\(432\) 0 0
\(433\) 4.53590i 0.217981i 0.994043 + 0.108991i \(0.0347619\pi\)
−0.994043 + 0.108991i \(0.965238\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 8.53590i − 0.408327i
\(438\) 0 0
\(439\) −27.3923 −1.30736 −0.653682 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 40.7321i − 1.93524i −0.252415 0.967619i \(-0.581225\pi\)
0.252415 0.967619i \(-0.418775\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.124356 −0.00586871 −0.00293435 0.999996i \(-0.500934\pi\)
−0.00293435 + 0.999996i \(0.500934\pi\)
\(450\) 0 0
\(451\) 1.39230 0.0655611
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 11.8038i − 0.552161i −0.961135 0.276080i \(-0.910964\pi\)
0.961135 0.276080i \(-0.0890356\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.5885 1.00547 0.502737 0.864439i \(-0.332326\pi\)
0.502737 + 0.864439i \(0.332326\pi\)
\(462\) 0 0
\(463\) 18.3923i 0.854763i 0.904071 + 0.427381i \(0.140564\pi\)
−0.904071 + 0.427381i \(0.859436\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15.8038i 0.731315i 0.930750 + 0.365657i \(0.119156\pi\)
−0.930750 + 0.365657i \(0.880844\pi\)
\(468\) 0 0
\(469\) 10.5359 0.486503
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.6603i 0.536139i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.6603 −0.943991 −0.471996 0.881601i \(-0.656466\pi\)
−0.471996 + 0.881601i \(0.656466\pi\)
\(480\) 0 0
\(481\) −6.14359 −0.280124
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 37.4641i 1.69766i 0.528666 + 0.848830i \(0.322693\pi\)
−0.528666 + 0.848830i \(0.677307\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.73205 0.348943 0.174471 0.984662i \(-0.444178\pi\)
0.174471 + 0.984662i \(0.444178\pi\)
\(492\) 0 0
\(493\) 5.41154i 0.243724i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 0.588457i − 0.0263959i
\(498\) 0 0
\(499\) −6.60770 −0.295801 −0.147901 0.989002i \(-0.547252\pi\)
−0.147901 + 0.989002i \(0.547252\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.679492i 0.0302970i 0.999885 + 0.0151485i \(0.00482211\pi\)
−0.999885 + 0.0151485i \(0.995178\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.7846 0.655316 0.327658 0.944796i \(-0.393741\pi\)
0.327658 + 0.944796i \(0.393741\pi\)
\(510\) 0 0
\(511\) 7.46410 0.330192
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 8.19615i − 0.360466i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) 0 0
\(523\) 24.3923i 1.06660i 0.845926 + 0.533301i \(0.179048\pi\)
−0.845926 + 0.533301i \(0.820952\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.0526i 0.437896i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 1.17691i − 0.0509778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11.1962 0.482252
\(540\) 0 0
\(541\) −4.46410 −0.191927 −0.0959634 0.995385i \(-0.530593\pi\)
−0.0959634 + 0.995385i \(0.530593\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 36.7846i 1.57280i 0.617720 + 0.786398i \(0.288057\pi\)
−0.617720 + 0.786398i \(0.711943\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.5167 0.448025
\(552\) 0 0
\(553\) 4.67949i 0.198992i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.3923i 0.694564i 0.937761 + 0.347282i \(0.112895\pi\)
−0.937761 + 0.347282i \(0.887105\pi\)
\(558\) 0 0
\(559\) 9.85641 0.416882
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 19.2679i − 0.812047i −0.913863 0.406024i \(-0.866915\pi\)
0.913863 0.406024i \(-0.133085\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.94744 −0.207408 −0.103704 0.994608i \(-0.533069\pi\)
−0.103704 + 0.994608i \(0.533069\pi\)
\(570\) 0 0
\(571\) −26.8564 −1.12391 −0.561953 0.827169i \(-0.689950\pi\)
−0.561953 + 0.827169i \(0.689950\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 22.1962i − 0.924038i −0.886870 0.462019i \(-0.847125\pi\)
0.886870 0.462019i \(-0.152875\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.67949 −0.277112
\(582\) 0 0
\(583\) 18.5885i 0.769855i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.1962i 0.585938i 0.956122 + 0.292969i \(0.0946433\pi\)
−0.956122 + 0.292969i \(0.905357\pi\)
\(588\) 0 0
\(589\) 19.5359 0.804963
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 36.9282i 1.51646i 0.651987 + 0.758230i \(0.273935\pi\)
−0.651987 + 0.758230i \(0.726065\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −45.5885 −1.86269 −0.931347 0.364133i \(-0.881365\pi\)
−0.931347 + 0.364133i \(0.881365\pi\)
\(600\) 0 0
\(601\) 10.3205 0.420982 0.210491 0.977596i \(-0.432494\pi\)
0.210491 + 0.977596i \(0.432494\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 28.5885i 1.16037i 0.814485 + 0.580185i \(0.197020\pi\)
−0.814485 + 0.580185i \(0.802980\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6.92820 −0.280285
\(612\) 0 0
\(613\) 3.60770i 0.145713i 0.997342 + 0.0728567i \(0.0232116\pi\)
−0.997342 + 0.0728567i \(0.976788\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12.0000i − 0.483102i −0.970388 0.241551i \(-0.922344\pi\)
0.970388 0.241551i \(-0.0776561\pi\)
\(618\) 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
\(623\) 3.80385i 0.152398i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.32051 −0.212143
\(630\) 0 0
\(631\) −13.9282 −0.554473 −0.277237 0.960802i \(-0.589419\pi\)
−0.277237 + 0.960802i \(0.589419\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 9.46410i − 0.374981i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 23.1962 0.916193 0.458096 0.888902i \(-0.348531\pi\)
0.458096 + 0.888902i \(0.348531\pi\)
\(642\) 0 0
\(643\) − 16.5885i − 0.654185i −0.944992 0.327092i \(-0.893931\pi\)
0.944992 0.327092i \(-0.106069\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 48.2487i − 1.89685i −0.316998 0.948426i \(-0.602675\pi\)
0.316998 0.948426i \(-0.397325\pi\)
\(648\) 0 0
\(649\) −7.39230 −0.290173
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.46410i 0.370359i 0.982705 + 0.185179i \(0.0592866\pi\)
−0.982705 + 0.185179i \(0.940713\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.46410 −0.368669 −0.184335 0.982864i \(-0.559013\pi\)
−0.184335 + 0.982864i \(0.559013\pi\)
\(660\) 0 0
\(661\) −5.39230 −0.209736 −0.104868 0.994486i \(-0.533442\pi\)
−0.104868 + 0.994486i \(0.533442\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 14.7846i − 0.572462i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) − 17.6077i − 0.678727i −0.940655 0.339363i \(-0.889788\pi\)
0.940655 0.339363i \(-0.110212\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 28.6410i − 1.10076i −0.834913 0.550382i \(-0.814482\pi\)
0.834913 0.550382i \(-0.185518\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 14.5359i − 0.556201i −0.960552 0.278100i \(-0.910295\pi\)
0.960552 0.278100i \(-0.0897048\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.7128 0.598610
\(690\) 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
\(697\) − 1.01924i − 0.0386064i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.1244 1.59101 0.795507 0.605944i \(-0.207204\pi\)
0.795507 + 0.605944i \(0.207204\pi\)
\(702\) 0 0
\(703\) 10.3397i 0.389971i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 13.2679i − 0.498993i
\(708\) 0 0
\(709\) −17.4641 −0.655878 −0.327939 0.944699i \(-0.606354\pi\)
−0.327939 + 0.944699i \(0.606354\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 27.4641i − 1.02854i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.80385 −0.253741 −0.126870 0.991919i \(-0.540493\pi\)
−0.126870 + 0.991919i \(0.540493\pi\)
\(720\) 0 0
\(721\) −1.75129 −0.0652214
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 33.1769i − 1.23046i −0.788346 0.615232i \(-0.789062\pi\)
0.788346 0.615232i \(-0.210938\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.53590 0.315712
\(732\) 0 0
\(733\) − 39.5692i − 1.46152i −0.682633 0.730761i \(-0.739165\pi\)
0.682633 0.730761i \(-0.260835\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.9282i 0.918242i
\(738\) 0 0
\(739\) −36.1769 −1.33079 −0.665395 0.746492i \(-0.731737\pi\)
−0.665395 + 0.746492i \(0.731737\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.5885i 1.34230i 0.741321 + 0.671150i \(0.234199\pi\)
−0.741321 + 0.671150i \(0.765801\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.53590 −0.0926597
\(750\) 0 0
\(751\) −0.784610 −0.0286308 −0.0143154 0.999898i \(-0.504557\pi\)
−0.0143154 + 0.999898i \(0.504557\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 0.392305i − 0.0142586i −0.999975 0.00712928i \(-0.997731\pi\)
0.999975 0.00712928i \(-0.00226934\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.87564 0.212992 0.106496 0.994313i \(-0.466037\pi\)
0.106496 + 0.994313i \(0.466037\pi\)
\(762\) 0 0
\(763\) 4.33975i 0.157109i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.24871i 0.225628i
\(768\) 0 0
\(769\) 13.2487 0.477761 0.238880 0.971049i \(-0.423220\pi\)
0.238880 + 0.971049i \(0.423220\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 0.339746i − 0.0122198i −0.999981 0.00610991i \(-0.998055\pi\)
0.999981 0.00610991i \(-0.00194486\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.98076 −0.0709682
\(780\) 0 0
\(781\) 1.39230 0.0498206
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 25.8038i 0.919808i 0.887969 + 0.459904i \(0.152116\pi\)
−0.887969 + 0.459904i \(0.847884\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.9282 0.459674
\(792\) 0 0
\(793\) 5.85641i 0.207967i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 11.3205i − 0.400993i −0.979694 0.200496i \(-0.935744\pi\)
0.979694 0.200496i \(-0.0642555\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 17.6603i 0.623217i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −39.5885 −1.39186 −0.695928 0.718112i \(-0.745007\pi\)
−0.695928 + 0.718112i \(0.745007\pi\)
\(810\) 0 0
\(811\) 23.2487 0.816373 0.408186 0.912899i \(-0.366161\pi\)
0.408186 + 0.912899i \(0.366161\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.5885i − 0.580357i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.6603 −0.721048 −0.360524 0.932750i \(-0.617402\pi\)
−0.360524 + 0.932750i \(0.617402\pi\)
\(822\) 0 0
\(823\) − 3.07180i − 0.107076i −0.998566 0.0535381i \(-0.982950\pi\)
0.998566 0.0535381i \(-0.0170498\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 38.5359i − 1.34002i −0.742350 0.670012i \(-0.766289\pi\)
0.742350 0.670012i \(-0.233711\pi\)
\(828\) 0 0
\(829\) 10.2154 0.354795 0.177398 0.984139i \(-0.443232\pi\)
0.177398 + 0.984139i \(0.443232\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 8.19615i − 0.283980i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −39.5885 −1.36675 −0.683373 0.730070i \(-0.739488\pi\)
−0.683373 + 0.730070i \(0.739488\pi\)
\(840\) 0 0
\(841\) −10.7846 −0.371883
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.85641i 0.201229i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 14.5359 0.498284
\(852\) 0 0
\(853\) 10.1962i 0.349110i 0.984647 + 0.174555i \(0.0558486\pi\)
−0.984647 + 0.174555i \(0.944151\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 42.9282i − 1.46640i −0.680013 0.733200i \(-0.738026\pi\)
0.680013 0.733200i \(-0.261974\pi\)
\(858\) 0 0
\(859\) 33.7846 1.15272 0.576358 0.817197i \(-0.304473\pi\)
0.576358 + 0.817197i \(0.304473\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.48334i 0.152615i 0.997084 + 0.0763073i \(0.0243130\pi\)
−0.997084 + 0.0763073i \(0.975687\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −11.0718 −0.375585
\(870\) 0 0
\(871\) 21.0718 0.713991
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.2487i 1.15650i 0.815861 + 0.578248i \(0.196264\pi\)
−0.815861 + 0.578248i \(0.803736\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −17.1962 −0.579353 −0.289677 0.957125i \(-0.593548\pi\)
−0.289677 + 0.957125i \(0.593548\pi\)
\(882\) 0 0
\(883\) 32.8372i 1.10506i 0.833493 + 0.552529i \(0.186337\pi\)
−0.833493 + 0.552529i \(0.813663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.7321i 0.561807i 0.959736 + 0.280904i \(0.0906341\pi\)
−0.959736 + 0.280904i \(0.909366\pi\)
\(888\) 0 0
\(889\) 4.53590 0.152129
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.6603i 0.390196i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 33.8372 1.12853
\(900\) 0 0
\(901\) 13.6077 0.453338
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 42.7846i 1.42064i 0.703879 + 0.710320i \(0.251450\pi\)
−0.703879 + 0.710320i \(0.748550\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 17.1962 0.569734 0.284867 0.958567i \(-0.408051\pi\)
0.284867 + 0.958567i \(0.408051\pi\)
\(912\) 0 0
\(913\) − 15.8038i − 0.523031i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.0526i 0.331965i
\(918\) 0 0
\(919\) −30.6077 −1.00965 −0.504827 0.863220i \(-0.668444\pi\)
−0.504827 + 0.863220i \(0.668444\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 1.17691i − 0.0387386i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46.5167 1.52616 0.763081 0.646303i \(-0.223686\pi\)
0.763081 + 0.646303i \(0.223686\pi\)
\(930\) 0 0
\(931\) −15.9282 −0.522026
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 32.9282i − 1.07572i −0.843035 0.537859i \(-0.819233\pi\)
0.843035 0.537859i \(-0.180767\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 15.4641 0.504115 0.252058 0.967712i \(-0.418893\pi\)
0.252058 + 0.967712i \(0.418893\pi\)
\(942\) 0 0
\(943\) 2.78461i 0.0906794i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.6410i 0.540760i 0.962754 + 0.270380i \(0.0871494\pi\)
−0.962754 + 0.270380i \(0.912851\pi\)
\(948\) 0 0
\(949\) 14.9282 0.484590
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.46410i 0.306572i 0.988182 + 0.153286i \(0.0489856\pi\)
−0.988182 + 0.153286i \(0.951014\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 31.8564 1.02763
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 38.5885i − 1.24092i −0.784238 0.620461i \(-0.786946\pi\)
0.784238 0.620461i \(-0.213054\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.7321 −0.825781 −0.412890 0.910781i \(-0.635481\pi\)
−0.412890 + 0.910781i \(0.635481\pi\)
\(972\) 0 0
\(973\) − 3.94744i − 0.126549i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 16.3923i − 0.524436i −0.965009 0.262218i \(-0.915546\pi\)
0.965009 0.262218i \(-0.0844540\pi\)
\(978\) 0 0
\(979\) −9.00000 −0.287641
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 49.2679i 1.57140i 0.618605 + 0.785702i \(0.287698\pi\)
−0.618605 + 0.785702i \(0.712302\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −23.3205 −0.741549
\(990\) 0 0
\(991\) 14.2154 0.451567 0.225783 0.974178i \(-0.427506\pi\)
0.225783 + 0.974178i \(0.427506\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 41.8038i − 1.32394i −0.749530 0.661971i \(-0.769720\pi\)
0.749530 0.661971i \(-0.230280\pi\)
\(998\) 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.649.2 4
3.2 odd 2 8100.2.d.m.649.2 4
5.2 odd 4 1620.2.a.h.1.2 yes 2
5.3 odd 4 8100.2.a.s.1.1 2
5.4 even 2 inner 8100.2.d.l.649.3 4
15.2 even 4 1620.2.a.g.1.2 2
15.8 even 4 8100.2.a.t.1.1 2
15.14 odd 2 8100.2.d.m.649.3 4
20.7 even 4 6480.2.a.bp.1.1 2
45.2 even 12 1620.2.i.n.1081.1 4
45.7 odd 12 1620.2.i.m.1081.1 4
45.22 odd 12 1620.2.i.m.541.1 4
45.32 even 12 1620.2.i.n.541.1 4
60.47 odd 4 6480.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.a.g.1.2 2 15.2 even 4
1620.2.a.h.1.2 yes 2 5.2 odd 4
1620.2.i.m.541.1 4 45.22 odd 12
1620.2.i.m.1081.1 4 45.7 odd 12
1620.2.i.n.541.1 4 45.32 even 12
1620.2.i.n.1081.1 4 45.2 even 12
6480.2.a.bh.1.1 2 60.47 odd 4
6480.2.a.bp.1.1 2 20.7 even 4
8100.2.a.s.1.1 2 5.3 odd 4
8100.2.a.t.1.1 2 15.8 even 4
8100.2.d.l.649.2 4 1.1 even 1 trivial
8100.2.d.l.649.3 4 5.4 even 2 inner
8100.2.d.m.649.2 4 3.2 odd 2
8100.2.d.m.649.3 4 15.14 odd 2