Properties

Label 5850.2.e.h.5149.1
Level $5850$
Weight $2$
Character 5850.5149
Analytic conductor $46.712$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.7124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5149.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5850.5149
Dual form 5850.2.e.h.5149.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} +2.00000 q^{19} +4.00000i q^{22} -2.00000i q^{23} -1.00000 q^{26} -2.00000i q^{28} +8.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +4.00000 q^{34} -6.00000i q^{37} -2.00000i q^{38} -10.0000 q^{41} +4.00000i q^{43} +4.00000 q^{44} -2.00000 q^{46} +3.00000 q^{49} +1.00000i q^{52} -6.00000i q^{53} -2.00000 q^{56} -8.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +8.00000i q^{67} -4.00000i q^{68} -6.00000 q^{74} -2.00000 q^{76} -8.00000i q^{77} +8.00000 q^{79} +10.0000i q^{82} +12.0000i q^{83} +4.00000 q^{86} -4.00000i q^{88} -10.0000 q^{89} +2.00000 q^{91} +2.00000i q^{92} +8.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 8q^{11} + 4q^{14} + 2q^{16} + 4q^{19} - 2q^{26} + 16q^{29} + 8q^{31} + 8q^{34} - 20q^{41} + 8q^{44} - 4q^{46} + 6q^{49} - 4q^{56} - 24q^{59} - 4q^{61} - 2q^{64} - 12q^{74} - 4q^{76} + 16q^{79} + 8q^{86} - 20q^{89} + 4q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times\).

\(n\) \(2251\) \(3251\) \(3277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) − 2.00000i − 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) − 8.00000i − 1.05045i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) − 8.00000i − 0.911685i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.0000i 1.10432i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) − 4.00000i − 0.426401i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 2.00000i 0.208514i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) −20.0000 −1.99007 −0.995037 0.0995037i \(-0.968274\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000i 0.188982i
\(113\) 16.0000i 1.50515i 0.658505 + 0.752577i \(0.271189\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) 12.0000i 1.10469i
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000i 0.181071i
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 20.0000i − 1.77471i −0.461084 0.887357i \(-0.652539\pi\)
0.461084 0.887357i \(-0.347461\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) − 2.00000i − 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 6.00000i 0.493197i
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 2.00000i 0.162221i
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) 0 0
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) − 4.00000i − 0.304997i
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 10.0000i 0.749532i
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) − 2.00000i − 0.148250i
\(183\) 0 0
\(184\) 2.00000 0.147442
\(185\) 0 0
\(186\) 0 0
\(187\) − 16.0000i − 1.17004i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 12.0000i 0.863779i 0.901927 + 0.431889i \(0.142153\pi\)
−0.901927 + 0.431889i \(0.857847\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 10.0000i 0.712470i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 20.0000i 1.40720i
\(203\) 16.0000i 1.12298i
\(204\) 0 0
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) − 1.00000i − 0.0693375i
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000i 0.543075i
\(218\) 4.00000i 0.270914i
\(219\) 0 0
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) − 22.0000i − 1.47323i −0.676313 0.736614i \(-0.736423\pi\)
0.676313 0.736614i \(-0.263577\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 16.0000 1.06430
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.00000i 0.525226i
\(233\) − 28.0000i − 1.83434i −0.398495 0.917170i \(-0.630467\pi\)
0.398495 0.917170i \(-0.369533\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 8.00000i 0.518563i
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.00000i − 0.127257i
\(248\) 4.00000i 0.254000i
\(249\) 0 0
\(250\) 0 0
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 8.00000i − 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) 14.0000i 0.864923i
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) − 8.00000i − 0.488678i
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) − 16.0000i − 0.959616i
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) − 20.0000i − 1.18056i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 22.0000i 1.27443i
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 4.00000i 0.230174i
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 8.00000i 0.455842i
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) −32.0000 −1.79166
\(320\) 0 0
\(321\) 0 0
\(322\) − 4.00000i − 0.222911i
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) − 10.0000i − 0.552158i
\(329\) 0 0
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000i 1.41631i 0.706057 + 0.708155i \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 16.0000i 0.858925i 0.903085 + 0.429463i \(0.141297\pi\)
−0.903085 + 0.429463i \(0.858703\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000i 0.213201i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) − 2.00000i − 0.105703i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 22.0000i 1.15629i
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000i 0.208798i 0.994535 + 0.104399i \(0.0332919\pi\)
−0.994535 + 0.104399i \(0.966708\pi\)
\(368\) − 2.00000i − 0.104257i
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) −16.0000 −0.827340
\(375\) 0 0
\(376\) 0 0
\(377\) − 8.00000i − 0.412021i
\(378\) 0 0
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 8.00000i − 0.409316i
\(383\) 20.0000i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.0000 0.610784
\(387\) 0 0
\(388\) − 8.00000i − 0.406138i
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 3.00000i 0.151523i
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) 0 0
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) − 4.00000i − 0.199254i
\(404\) 20.0000 0.995037
\(405\) 0 0
\(406\) 16.0000 0.794067
\(407\) 24.0000i 1.18964i
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 4.00000i − 0.197066i
\(413\) − 24.0000i − 1.18096i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 8.00000i 0.391293i
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) − 4.00000i − 0.193574i
\(428\) − 4.00000i − 0.193347i
\(429\) 0 0
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) − 4.00000i − 0.191346i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 4.00000i − 0.190261i
\(443\) 16.0000i 0.760183i 0.924949 + 0.380091i \(0.124107\pi\)
−0.924949 + 0.380091i \(0.875893\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −22.0000 −1.04173
\(447\) 0 0
\(448\) − 2.00000i − 0.0944911i
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) − 16.0000i − 0.752577i
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 16.0000i 0.748448i 0.927338 + 0.374224i \(0.122091\pi\)
−0.927338 + 0.374224i \(0.877909\pi\)
\(458\) 28.0000i 1.30835i
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 2.00000i 0.0929479i 0.998920 + 0.0464739i \(0.0147984\pi\)
−0.998920 + 0.0464739i \(0.985202\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) −28.0000 −1.29707
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) − 12.0000i − 0.552345i
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 8.00000i 0.365911i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 22.0000i 1.00207i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) − 38.0000i − 1.72194i −0.508652 0.860972i \(-0.669856\pi\)
0.508652 0.860972i \(-0.330144\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 0 0
\(490\) 0 0
\(491\) −38.0000 −1.71492 −0.857458 0.514554i \(-0.827958\pi\)
−0.857458 + 0.514554i \(0.827958\pi\)
\(492\) 0 0
\(493\) 32.0000i 1.44121i
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 10.0000i 0.446322i
\(503\) 42.0000i 1.87269i 0.351085 + 0.936344i \(0.385813\pi\)
−0.351085 + 0.936344i \(0.614187\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 20.0000i 0.887357i
\(509\) 38.0000 1.68432 0.842160 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −8.00000 −0.352865
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) − 12.0000i − 0.527250i
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 14.0000 0.611593
\(525\) 0 0
\(526\) 18.0000 0.784837
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) − 4.00000i − 0.173422i
\(533\) 10.0000i 0.433148i
\(534\) 0 0
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 4.00000i 0.172452i
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 28.0000i 1.20270i
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) 0 0
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) − 10.0000i − 0.423714i −0.977301 0.211857i \(-0.932049\pi\)
0.977301 0.211857i \(-0.0679510\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) − 10.0000i − 0.421825i
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) 0 0
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) − 4.00000i − 0.167248i
\(573\) 0 0
\(574\) −20.0000 −0.834784
\(575\) 0 0
\(576\) 0 0
\(577\) 4.00000i 0.166522i 0.996528 + 0.0832611i \(0.0265335\pi\)
−0.996528 + 0.0832611i \(0.973466\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 0 0
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) 24.0000i 0.993978i
\(584\) 0 0
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) − 6.00000i − 0.246598i
\(593\) − 34.0000i − 1.39621i −0.715994 0.698106i \(-0.754026\pi\)
0.715994 0.698106i \(-0.245974\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) 2.00000i 0.0817861i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) 12.0000i 0.487065i 0.969893 + 0.243532i \(0.0783062\pi\)
−0.969893 + 0.243532i \(0.921694\pi\)
\(608\) − 2.00000i − 0.0811107i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 38.0000i 1.52982i 0.644136 + 0.764911i \(0.277217\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 16.0000i − 0.641542i
\(623\) − 20.0000i − 0.801283i
\(624\) 0 0
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) − 10.0000i − 0.399043i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.00000i − 0.118864i
\(638\) 32.0000i 1.26689i
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) − 18.0000i − 0.707653i −0.935311 0.353827i \(-0.884880\pi\)
0.935311 0.353827i \(-0.115120\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) − 12.0000i − 0.469956i
\(653\) 38.0000i 1.48705i 0.668705 + 0.743527i \(0.266849\pi\)
−0.668705 + 0.743527i \(0.733151\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 10.0000i 0.388661i
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) − 16.0000i − 0.619522i
\(668\) 12.0000i 0.464294i
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) − 46.0000i − 1.77317i −0.462566 0.886585i \(-0.653071\pi\)
0.462566 0.886585i \(-0.346929\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 0 0
\(682\) 16.0000i 0.612672i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 4.00000i 0.152499i
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) 14.0000i 0.532200i
\(693\) 0 0
\(694\) 16.0000 0.607352
\(695\) 0 0
\(696\) 0 0
\(697\) − 40.0000i − 1.51511i
\(698\) − 28.0000i − 1.05982i
\(699\) 0 0
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) − 12.0000i − 0.452589i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) − 40.0000i − 1.50435i
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 10.0000i − 0.374766i
\(713\) − 8.00000i − 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 0 0
\(718\) − 24.0000i − 0.895672i
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 15.0000i 0.558242i
\(723\) 0 0
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) 0 0
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 2.00000i 0.0741249i
\(729\) 0 0
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 6.00000i 0.221615i 0.993842 + 0.110808i \(0.0353437\pi\)
−0.993842 + 0.110808i \(0.964656\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) − 32.0000i − 1.17874i
\(738\) 0 0
\(739\) −50.0000 −1.83928 −0.919640 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 12.0000i − 0.440534i
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 16.0000i 0.585018i
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) − 14.0000i − 0.508503i
\(759\) 0 0
\(760\) 0 0
\(761\) 46.0000 1.66750 0.833749 0.552143i \(-0.186190\pi\)
0.833749 + 0.552143i \(0.186190\pi\)
\(762\) 0 0
\(763\) − 8.00000i − 0.289619i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 12.0000i − 0.431889i
\(773\) − 26.0000i − 0.935155i −0.883952 0.467578i \(-0.845127\pi\)
0.883952 0.467578i \(-0.154873\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) − 20.0000i − 0.717035i
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) − 8.00000i − 0.286079i
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) − 10.0000i − 0.356235i
\(789\) 0 0
\(790\) 0 0
\(791\) −32.0000 −1.13779
\(792\) 0 0
\(793\) 2.00000i 0.0710221i
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) − 30.0000i − 1.05934i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) − 20.0000i − 0.703598i
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) − 16.0000i − 0.561490i
\(813\) 0 0
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) 8.00000i 0.279885i
\(818\) − 18.0000i − 0.629355i
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) − 20.0000i − 0.697156i −0.937280 0.348578i \(-0.886665\pi\)
0.937280 0.348578i \(-0.113335\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) − 44.0000i − 1.53003i −0.644013 0.765015i \(-0.722732\pi\)
0.644013 0.765015i \(-0.277268\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000i 0.0346688i
\(833\) 12.0000i 0.415775i
\(834\) 0 0
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) − 14.0000i − 0.483622i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 16.0000i 0.551396i
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000i 0.343604i
\(848\) − 6.00000i − 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) − 16.0000i − 0.546550i −0.961936 0.273275i \(-0.911893\pi\)
0.961936 0.273275i \(-0.0881068\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 16.0000i 0.544962i
\(863\) 36.0000i 1.22545i 0.790295 + 0.612727i \(0.209928\pi\)
−0.790295 + 0.612727i \(0.790072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) − 8.00000i − 0.271538i
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) − 4.00000i − 0.135457i
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) − 6.00000i − 0.201460i −0.994914 0.100730i \(-0.967882\pi\)
0.994914 0.100730i \(-0.0321179\pi\)
\(888\) 0 0
\(889\) 40.0000 1.34156
\(890\) 0 0
\(891\) 0 0
\(892\) 22.0000i 0.736614i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 18.0000i 0.600668i
\(899\) 32.0000 1.06726
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) − 40.0000i − 1.33185i
\(903\) 0 0
\(904\) −16.0000 −0.532152
\(905\) 0 0
\(906\) 0 0
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) − 48.0000i − 1.58857i
\(914\) 16.0000 0.529233
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) − 28.0000i − 0.924641i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 30.0000i 0.987997i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 2.00000 0.0657241
\(927\) 0 0
\(928\) − 8.00000i − 0.262613i
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 28.0000i 0.917170i
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 16.0000i 0.522419i
\(939\) 0 0
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) 20.0000i 0.651290i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) − 8.00000i − 0.259281i
\(953\) 36.0000i 1.16615i 0.812417 + 0.583077i \(0.198151\pi\)
−0.812417 + 0.583077i \(0.801849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) − 24.0000i − 0.775405i
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 6.00000i 0.193448i
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) − 14.0000i − 0.450210i −0.974335 0.225105i \(-0.927728\pi\)
0.974335 0.225105i \(-0.0722725\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 0 0
\(973\) 32.0000i 1.02587i
\(974\) −38.0000 −1.21760
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 0 0
\(982\) 38.0000i 1.21263i
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 32.0000 1.01909
\(987\) 0 0
\(988\) 2.00000i 0.0636285i
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) − 14.0000i − 0.443162i
\(999\) 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 5850.2.e.h.5149.1 2
3.2 odd 2 1950.2.e.m.1249.2 2
5.2 odd 4 1170.2.a.j.1.1 1
5.3 odd 4 5850.2.a.s.1.1 1
5.4 even 2 inner 5850.2.e.h.5149.2 2
15.2 even 4 390.2.a.b.1.1 1
15.8 even 4 1950.2.a.ba.1.1 1
15.14 odd 2 1950.2.e.m.1249.1 2
20.7 even 4 9360.2.a.v.1.1 1
60.47 odd 4 3120.2.a.y.1.1 1
195.47 odd 4 5070.2.b.f.1351.1 2
195.77 even 4 5070.2.a.n.1.1 1
195.122 odd 4 5070.2.b.f.1351.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.b.1.1 1 15.2 even 4
1170.2.a.j.1.1 1 5.2 odd 4
1950.2.a.ba.1.1 1 15.8 even 4
1950.2.e.m.1249.1 2 15.14 odd 2
1950.2.e.m.1249.2 2 3.2 odd 2
3120.2.a.y.1.1 1 60.47 odd 4
5070.2.a.n.1.1 1 195.77 even 4
5070.2.b.f.1351.1 2 195.47 odd 4
5070.2.b.f.1351.2 2 195.122 odd 4
5850.2.a.s.1.1 1 5.3 odd 4
5850.2.e.h.5149.1 2 1.1 even 1 trivial
5850.2.e.h.5149.2 2 5.4 even 2 inner
9360.2.a.v.1.1 1 20.7 even 4