Properties

Label 1216.3.e.c.1025.2
Level $1216$
Weight $3$
Character 1216.1025
Analytic conductor $33.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1025
Dual form 1216.3.e.c.1025.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.65685i q^{3} -7.00000 q^{5} -11.0000 q^{7} -23.0000 q^{9} +O(q^{10})\) \(q+5.65685i q^{3} -7.00000 q^{5} -11.0000 q^{7} -23.0000 q^{9} +3.00000 q^{11} +11.3137i q^{13} -39.5980i q^{15} -17.0000 q^{17} -19.0000 q^{19} -62.2254i q^{21} -2.00000 q^{23} +24.0000 q^{25} -79.1960i q^{27} +39.5980i q^{29} -5.65685i q^{31} +16.9706i q^{33} +77.0000 q^{35} +39.5980i q^{37} -64.0000 q^{39} +39.5980i q^{41} -21.0000 q^{43} +161.000 q^{45} +5.00000 q^{47} +72.0000 q^{49} -96.1665i q^{51} -5.65685i q^{53} -21.0000 q^{55} -107.480i q^{57} +33.9411i q^{59} -23.0000 q^{61} +253.000 q^{63} -79.1960i q^{65} +39.5980i q^{67} -11.3137i q^{69} -90.5097i q^{71} +39.0000 q^{73} +135.765i q^{75} -33.0000 q^{77} +96.1665i q^{79} +241.000 q^{81} -6.00000 q^{83} +119.000 q^{85} -224.000 q^{87} +118.794i q^{89} -124.451i q^{91} +32.0000 q^{93} +133.000 q^{95} -169.706i q^{97} -69.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{5} - 22q^{7} - 46q^{9} + O(q^{10}) \) \( 2q - 14q^{5} - 22q^{7} - 46q^{9} + 6q^{11} - 34q^{17} - 38q^{19} - 4q^{23} + 48q^{25} + 154q^{35} - 128q^{39} - 42q^{43} + 322q^{45} + 10q^{47} + 144q^{49} - 42q^{55} - 46q^{61} + 506q^{63} + 78q^{73} - 66q^{77} + 482q^{81} - 12q^{83} + 238q^{85} - 448q^{87} + 64q^{93} + 266q^{95} - 138q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 5.65685i 1.88562i 0.333333 + 0.942809i \(0.391827\pi\)
−0.333333 + 0.942809i \(0.608173\pi\)
\(4\) 0 0
\(5\) −7.00000 −1.40000 −0.700000 0.714143i \(-0.746817\pi\)
−0.700000 + 0.714143i \(0.746817\pi\)
\(6\) 0 0
\(7\) −11.0000 −1.57143 −0.785714 0.618590i \(-0.787704\pi\)
−0.785714 + 0.618590i \(0.787704\pi\)
\(8\) 0 0
\(9\) −23.0000 −2.55556
\(10\) 0 0
\(11\) 3.00000 0.272727 0.136364 0.990659i \(-0.456458\pi\)
0.136364 + 0.990659i \(0.456458\pi\)
\(12\) 0 0
\(13\) 11.3137i 0.870285i 0.900362 + 0.435143i \(0.143302\pi\)
−0.900362 + 0.435143i \(0.856698\pi\)
\(14\) 0 0
\(15\) − 39.5980i − 2.63987i
\(16\) 0 0
\(17\) −17.0000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −19.0000 −1.00000
\(20\) 0 0
\(21\) − 62.2254i − 2.96311i
\(22\) 0 0
\(23\) −2.00000 −0.0869565 −0.0434783 0.999054i \(-0.513844\pi\)
−0.0434783 + 0.999054i \(0.513844\pi\)
\(24\) 0 0
\(25\) 24.0000 0.960000
\(26\) 0 0
\(27\) − 79.1960i − 2.93318i
\(28\) 0 0
\(29\) 39.5980i 1.36545i 0.730677 + 0.682724i \(0.239205\pi\)
−0.730677 + 0.682724i \(0.760795\pi\)
\(30\) 0 0
\(31\) − 5.65685i − 0.182479i −0.995829 0.0912396i \(-0.970917\pi\)
0.995829 0.0912396i \(-0.0290829\pi\)
\(32\) 0 0
\(33\) 16.9706i 0.514259i
\(34\) 0 0
\(35\) 77.0000 2.20000
\(36\) 0 0
\(37\) 39.5980i 1.07022i 0.844784 + 0.535108i \(0.179729\pi\)
−0.844784 + 0.535108i \(0.820271\pi\)
\(38\) 0 0
\(39\) −64.0000 −1.64103
\(40\) 0 0
\(41\) 39.5980i 0.965804i 0.875674 + 0.482902i \(0.160417\pi\)
−0.875674 + 0.482902i \(0.839583\pi\)
\(42\) 0 0
\(43\) −21.0000 −0.488372 −0.244186 0.969728i \(-0.578521\pi\)
−0.244186 + 0.969728i \(0.578521\pi\)
\(44\) 0 0
\(45\) 161.000 3.57778
\(46\) 0 0
\(47\) 5.00000 0.106383 0.0531915 0.998584i \(-0.483061\pi\)
0.0531915 + 0.998584i \(0.483061\pi\)
\(48\) 0 0
\(49\) 72.0000 1.46939
\(50\) 0 0
\(51\) − 96.1665i − 1.88562i
\(52\) 0 0
\(53\) − 5.65685i − 0.106733i −0.998575 0.0533665i \(-0.983005\pi\)
0.998575 0.0533665i \(-0.0169952\pi\)
\(54\) 0 0
\(55\) −21.0000 −0.381818
\(56\) 0 0
\(57\) − 107.480i − 1.88562i
\(58\) 0 0
\(59\) 33.9411i 0.575273i 0.957740 + 0.287637i \(0.0928695\pi\)
−0.957740 + 0.287637i \(0.907130\pi\)
\(60\) 0 0
\(61\) −23.0000 −0.377049 −0.188525 0.982068i \(-0.560371\pi\)
−0.188525 + 0.982068i \(0.560371\pi\)
\(62\) 0 0
\(63\) 253.000 4.01587
\(64\) 0 0
\(65\) − 79.1960i − 1.21840i
\(66\) 0 0
\(67\) 39.5980i 0.591015i 0.955340 + 0.295507i \(0.0954887\pi\)
−0.955340 + 0.295507i \(0.904511\pi\)
\(68\) 0 0
\(69\) − 11.3137i − 0.163967i
\(70\) 0 0
\(71\) − 90.5097i − 1.27478i −0.770540 0.637392i \(-0.780013\pi\)
0.770540 0.637392i \(-0.219987\pi\)
\(72\) 0 0
\(73\) 39.0000 0.534247 0.267123 0.963662i \(-0.413927\pi\)
0.267123 + 0.963662i \(0.413927\pi\)
\(74\) 0 0
\(75\) 135.765i 1.81019i
\(76\) 0 0
\(77\) −33.0000 −0.428571
\(78\) 0 0
\(79\) 96.1665i 1.21730i 0.793440 + 0.608649i \(0.208288\pi\)
−0.793440 + 0.608649i \(0.791712\pi\)
\(80\) 0 0
\(81\) 241.000 2.97531
\(82\) 0 0
\(83\) −6.00000 −0.0722892 −0.0361446 0.999347i \(-0.511508\pi\)
−0.0361446 + 0.999347i \(0.511508\pi\)
\(84\) 0 0
\(85\) 119.000 1.40000
\(86\) 0 0
\(87\) −224.000 −2.57471
\(88\) 0 0
\(89\) 118.794i 1.33476i 0.744716 + 0.667382i \(0.232585\pi\)
−0.744716 + 0.667382i \(0.767415\pi\)
\(90\) 0 0
\(91\) − 124.451i − 1.36759i
\(92\) 0 0
\(93\) 32.0000 0.344086
\(94\) 0 0
\(95\) 133.000 1.40000
\(96\) 0 0
\(97\) − 169.706i − 1.74954i −0.484536 0.874771i \(-0.661012\pi\)
0.484536 0.874771i \(-0.338988\pi\)
\(98\) 0 0
\(99\) −69.0000 −0.696970
\(100\) 0 0
\(101\) −122.000 −1.20792 −0.603960 0.797014i \(-0.706411\pi\)
−0.603960 + 0.797014i \(0.706411\pi\)
\(102\) 0 0
\(103\) − 101.823i − 0.988576i −0.869298 0.494288i \(-0.835429\pi\)
0.869298 0.494288i \(-0.164571\pi\)
\(104\) 0 0
\(105\) 435.578i 4.14836i
\(106\) 0 0
\(107\) 158.392i 1.48030i 0.672443 + 0.740149i \(0.265245\pi\)
−0.672443 + 0.740149i \(0.734755\pi\)
\(108\) 0 0
\(109\) 118.794i 1.08985i 0.838484 + 0.544926i \(0.183442\pi\)
−0.838484 + 0.544926i \(0.816558\pi\)
\(110\) 0 0
\(111\) −224.000 −2.01802
\(112\) 0 0
\(113\) 50.9117i 0.450546i 0.974296 + 0.225273i \(0.0723274\pi\)
−0.974296 + 0.225273i \(0.927673\pi\)
\(114\) 0 0
\(115\) 14.0000 0.121739
\(116\) 0 0
\(117\) − 260.215i − 2.22406i
\(118\) 0 0
\(119\) 187.000 1.57143
\(120\) 0 0
\(121\) −112.000 −0.925620
\(122\) 0 0
\(123\) −224.000 −1.82114
\(124\) 0 0
\(125\) 7.00000 0.0560000
\(126\) 0 0
\(127\) 39.5980i 0.311795i 0.987773 + 0.155898i \(0.0498270\pi\)
−0.987773 + 0.155898i \(0.950173\pi\)
\(128\) 0 0
\(129\) − 118.794i − 0.920883i
\(130\) 0 0
\(131\) −149.000 −1.13740 −0.568702 0.822543i \(-0.692554\pi\)
−0.568702 + 0.822543i \(0.692554\pi\)
\(132\) 0 0
\(133\) 209.000 1.57143
\(134\) 0 0
\(135\) 554.372i 4.10646i
\(136\) 0 0
\(137\) 95.0000 0.693431 0.346715 0.937970i \(-0.387297\pi\)
0.346715 + 0.937970i \(0.387297\pi\)
\(138\) 0 0
\(139\) 155.000 1.11511 0.557554 0.830141i \(-0.311740\pi\)
0.557554 + 0.830141i \(0.311740\pi\)
\(140\) 0 0
\(141\) 28.2843i 0.200598i
\(142\) 0 0
\(143\) 33.9411i 0.237351i
\(144\) 0 0
\(145\) − 277.186i − 1.91163i
\(146\) 0 0
\(147\) 407.294i 2.77070i
\(148\) 0 0
\(149\) −63.0000 −0.422819 −0.211409 0.977398i \(-0.567805\pi\)
−0.211409 + 0.977398i \(0.567805\pi\)
\(150\) 0 0
\(151\) 124.451i 0.824177i 0.911144 + 0.412089i \(0.135201\pi\)
−0.911144 + 0.412089i \(0.864799\pi\)
\(152\) 0 0
\(153\) 391.000 2.55556
\(154\) 0 0
\(155\) 39.5980i 0.255471i
\(156\) 0 0
\(157\) 150.000 0.955414 0.477707 0.878519i \(-0.341468\pi\)
0.477707 + 0.878519i \(0.341468\pi\)
\(158\) 0 0
\(159\) 32.0000 0.201258
\(160\) 0 0
\(161\) 22.0000 0.136646
\(162\) 0 0
\(163\) −166.000 −1.01840 −0.509202 0.860647i \(-0.670060\pi\)
−0.509202 + 0.860647i \(0.670060\pi\)
\(164\) 0 0
\(165\) − 118.794i − 0.719963i
\(166\) 0 0
\(167\) − 209.304i − 1.25332i −0.779295 0.626658i \(-0.784423\pi\)
0.779295 0.626658i \(-0.215577\pi\)
\(168\) 0 0
\(169\) 41.0000 0.242604
\(170\) 0 0
\(171\) 437.000 2.55556
\(172\) 0 0
\(173\) − 56.5685i − 0.326986i −0.986545 0.163493i \(-0.947724\pi\)
0.986545 0.163493i \(-0.0522761\pi\)
\(174\) 0 0
\(175\) −264.000 −1.50857
\(176\) 0 0
\(177\) −192.000 −1.08475
\(178\) 0 0
\(179\) − 73.5391i − 0.410833i −0.978675 0.205416i \(-0.934145\pi\)
0.978675 0.205416i \(-0.0658549\pi\)
\(180\) 0 0
\(181\) 79.1960i 0.437547i 0.975776 + 0.218773i \(0.0702055\pi\)
−0.975776 + 0.218773i \(0.929794\pi\)
\(182\) 0 0
\(183\) − 130.108i − 0.710971i
\(184\) 0 0
\(185\) − 277.186i − 1.49830i
\(186\) 0 0
\(187\) −51.0000 −0.272727
\(188\) 0 0
\(189\) 871.156i 4.60929i
\(190\) 0 0
\(191\) 301.000 1.57592 0.787958 0.615729i \(-0.211138\pi\)
0.787958 + 0.615729i \(0.211138\pi\)
\(192\) 0 0
\(193\) 152.735i 0.791373i 0.918386 + 0.395687i \(0.129493\pi\)
−0.918386 + 0.395687i \(0.870507\pi\)
\(194\) 0 0
\(195\) 448.000 2.29744
\(196\) 0 0
\(197\) −90.0000 −0.456853 −0.228426 0.973561i \(-0.573358\pi\)
−0.228426 + 0.973561i \(0.573358\pi\)
\(198\) 0 0
\(199\) −147.000 −0.738693 −0.369347 0.929292i \(-0.620419\pi\)
−0.369347 + 0.929292i \(0.620419\pi\)
\(200\) 0 0
\(201\) −224.000 −1.11443
\(202\) 0 0
\(203\) − 435.578i − 2.14570i
\(204\) 0 0
\(205\) − 277.186i − 1.35213i
\(206\) 0 0
\(207\) 46.0000 0.222222
\(208\) 0 0
\(209\) −57.0000 −0.272727
\(210\) 0 0
\(211\) − 328.098i − 1.55496i −0.628905 0.777482i \(-0.716496\pi\)
0.628905 0.777482i \(-0.283504\pi\)
\(212\) 0 0
\(213\) 512.000 2.40376
\(214\) 0 0
\(215\) 147.000 0.683721
\(216\) 0 0
\(217\) 62.2254i 0.286753i
\(218\) 0 0
\(219\) 220.617i 1.00739i
\(220\) 0 0
\(221\) − 192.333i − 0.870285i
\(222\) 0 0
\(223\) − 356.382i − 1.59812i −0.601248 0.799062i \(-0.705330\pi\)
0.601248 0.799062i \(-0.294670\pi\)
\(224\) 0 0
\(225\) −552.000 −2.45333
\(226\) 0 0
\(227\) − 316.784i − 1.39552i −0.716330 0.697762i \(-0.754179\pi\)
0.716330 0.697762i \(-0.245821\pi\)
\(228\) 0 0
\(229\) 257.000 1.12227 0.561135 0.827724i \(-0.310365\pi\)
0.561135 + 0.827724i \(0.310365\pi\)
\(230\) 0 0
\(231\) − 186.676i − 0.808122i
\(232\) 0 0
\(233\) −177.000 −0.759657 −0.379828 0.925057i \(-0.624017\pi\)
−0.379828 + 0.925057i \(0.624017\pi\)
\(234\) 0 0
\(235\) −35.0000 −0.148936
\(236\) 0 0
\(237\) −544.000 −2.29536
\(238\) 0 0
\(239\) −363.000 −1.51883 −0.759414 0.650607i \(-0.774514\pi\)
−0.759414 + 0.650607i \(0.774514\pi\)
\(240\) 0 0
\(241\) 356.382i 1.47876i 0.673287 + 0.739381i \(0.264882\pi\)
−0.673287 + 0.739381i \(0.735118\pi\)
\(242\) 0 0
\(243\) 650.538i 2.67711i
\(244\) 0 0
\(245\) −504.000 −2.05714
\(246\) 0 0
\(247\) − 214.960i − 0.870285i
\(248\) 0 0
\(249\) − 33.9411i − 0.136310i
\(250\) 0 0
\(251\) −133.000 −0.529880 −0.264940 0.964265i \(-0.585352\pi\)
−0.264940 + 0.964265i \(0.585352\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.0237154
\(254\) 0 0
\(255\) 673.166i 2.63987i
\(256\) 0 0
\(257\) − 135.765i − 0.528267i −0.964486 0.264133i \(-0.914914\pi\)
0.964486 0.264133i \(-0.0850859\pi\)
\(258\) 0 0
\(259\) − 435.578i − 1.68177i
\(260\) 0 0
\(261\) − 910.754i − 3.48948i
\(262\) 0 0
\(263\) 101.000 0.384030 0.192015 0.981392i \(-0.438498\pi\)
0.192015 + 0.981392i \(0.438498\pi\)
\(264\) 0 0
\(265\) 39.5980i 0.149426i
\(266\) 0 0
\(267\) −672.000 −2.51685
\(268\) 0 0
\(269\) − 356.382i − 1.32484i −0.749133 0.662420i \(-0.769530\pi\)
0.749133 0.662420i \(-0.230470\pi\)
\(270\) 0 0
\(271\) 142.000 0.523985 0.261993 0.965070i \(-0.415620\pi\)
0.261993 + 0.965070i \(0.415620\pi\)
\(272\) 0 0
\(273\) 704.000 2.57875
\(274\) 0 0
\(275\) 72.0000 0.261818
\(276\) 0 0
\(277\) −199.000 −0.718412 −0.359206 0.933258i \(-0.616952\pi\)
−0.359206 + 0.933258i \(0.616952\pi\)
\(278\) 0 0
\(279\) 130.108i 0.466336i
\(280\) 0 0
\(281\) 463.862i 1.65075i 0.564582 + 0.825377i \(0.309038\pi\)
−0.564582 + 0.825377i \(0.690962\pi\)
\(282\) 0 0
\(283\) 427.000 1.50883 0.754417 0.656395i \(-0.227920\pi\)
0.754417 + 0.656395i \(0.227920\pi\)
\(284\) 0 0
\(285\) 752.362i 2.63987i
\(286\) 0 0
\(287\) − 435.578i − 1.51769i
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 960.000 3.29897
\(292\) 0 0
\(293\) 237.588i 0.810880i 0.914122 + 0.405440i \(0.132882\pi\)
−0.914122 + 0.405440i \(0.867118\pi\)
\(294\) 0 0
\(295\) − 237.588i − 0.805383i
\(296\) 0 0
\(297\) − 237.588i − 0.799959i
\(298\) 0 0
\(299\) − 22.6274i − 0.0756770i
\(300\) 0 0
\(301\) 231.000 0.767442
\(302\) 0 0
\(303\) − 690.136i − 2.27768i
\(304\) 0 0
\(305\) 161.000 0.527869
\(306\) 0 0
\(307\) − 526.087i − 1.71364i −0.515616 0.856820i \(-0.672437\pi\)
0.515616 0.856820i \(-0.327563\pi\)
\(308\) 0 0
\(309\) 576.000 1.86408
\(310\) 0 0
\(311\) −235.000 −0.755627 −0.377814 0.925882i \(-0.623324\pi\)
−0.377814 + 0.925882i \(0.623324\pi\)
\(312\) 0 0
\(313\) 530.000 1.69329 0.846645 0.532158i \(-0.178619\pi\)
0.846645 + 0.532158i \(0.178619\pi\)
\(314\) 0 0
\(315\) −1771.00 −5.62222
\(316\) 0 0
\(317\) 214.960i 0.678109i 0.940767 + 0.339054i \(0.110107\pi\)
−0.940767 + 0.339054i \(0.889893\pi\)
\(318\) 0 0
\(319\) 118.794i 0.372395i
\(320\) 0 0
\(321\) −896.000 −2.79128
\(322\) 0 0
\(323\) 323.000 1.00000
\(324\) 0 0
\(325\) 271.529i 0.835474i
\(326\) 0 0
\(327\) −672.000 −2.05505
\(328\) 0 0
\(329\) −55.0000 −0.167173
\(330\) 0 0
\(331\) 96.1665i 0.290533i 0.989393 + 0.145267i \(0.0464040\pi\)
−0.989393 + 0.145267i \(0.953596\pi\)
\(332\) 0 0
\(333\) − 910.754i − 2.73500i
\(334\) 0 0
\(335\) − 277.186i − 0.827420i
\(336\) 0 0
\(337\) − 169.706i − 0.503578i −0.967782 0.251789i \(-0.918981\pi\)
0.967782 0.251789i \(-0.0810188\pi\)
\(338\) 0 0
\(339\) −288.000 −0.849558
\(340\) 0 0
\(341\) − 16.9706i − 0.0497670i
\(342\) 0 0
\(343\) −253.000 −0.737609
\(344\) 0 0
\(345\) 79.1960i 0.229554i
\(346\) 0 0
\(347\) −253.000 −0.729107 −0.364553 0.931183i \(-0.618778\pi\)
−0.364553 + 0.931183i \(0.618778\pi\)
\(348\) 0 0
\(349\) −351.000 −1.00573 −0.502865 0.864365i \(-0.667721\pi\)
−0.502865 + 0.864365i \(0.667721\pi\)
\(350\) 0 0
\(351\) 896.000 2.55271
\(352\) 0 0
\(353\) 98.0000 0.277620 0.138810 0.990319i \(-0.455672\pi\)
0.138810 + 0.990319i \(0.455672\pi\)
\(354\) 0 0
\(355\) 633.568i 1.78470i
\(356\) 0 0
\(357\) 1057.83i 2.96311i
\(358\) 0 0
\(359\) 61.0000 0.169916 0.0849582 0.996385i \(-0.472924\pi\)
0.0849582 + 0.996385i \(0.472924\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) − 633.568i − 1.74537i
\(364\) 0 0
\(365\) −273.000 −0.747945
\(366\) 0 0
\(367\) 686.000 1.86921 0.934605 0.355688i \(-0.115753\pi\)
0.934605 + 0.355688i \(0.115753\pi\)
\(368\) 0 0
\(369\) − 910.754i − 2.46817i
\(370\) 0 0
\(371\) 62.2254i 0.167723i
\(372\) 0 0
\(373\) − 135.765i − 0.363980i −0.983300 0.181990i \(-0.941746\pi\)
0.983300 0.181990i \(-0.0582538\pi\)
\(374\) 0 0
\(375\) 39.5980i 0.105595i
\(376\) 0 0
\(377\) −448.000 −1.18833
\(378\) 0 0
\(379\) 79.1960i 0.208960i 0.994527 + 0.104480i \(0.0333179\pi\)
−0.994527 + 0.104480i \(0.966682\pi\)
\(380\) 0 0
\(381\) −224.000 −0.587927
\(382\) 0 0
\(383\) 16.9706i 0.0443096i 0.999755 + 0.0221548i \(0.00705266\pi\)
−0.999755 + 0.0221548i \(0.992947\pi\)
\(384\) 0 0
\(385\) 231.000 0.600000
\(386\) 0 0
\(387\) 483.000 1.24806
\(388\) 0 0
\(389\) −599.000 −1.53985 −0.769923 0.638137i \(-0.779705\pi\)
−0.769923 + 0.638137i \(0.779705\pi\)
\(390\) 0 0
\(391\) 34.0000 0.0869565
\(392\) 0 0
\(393\) − 842.871i − 2.14471i
\(394\) 0 0
\(395\) − 673.166i − 1.70422i
\(396\) 0 0
\(397\) 569.000 1.43325 0.716625 0.697459i \(-0.245686\pi\)
0.716625 + 0.697459i \(0.245686\pi\)
\(398\) 0 0
\(399\) 1182.28i 2.96311i
\(400\) 0 0
\(401\) − 16.9706i − 0.0423206i −0.999776 0.0211603i \(-0.993264\pi\)
0.999776 0.0211603i \(-0.00673604\pi\)
\(402\) 0 0
\(403\) 64.0000 0.158809
\(404\) 0 0
\(405\) −1687.00 −4.16543
\(406\) 0 0
\(407\) 118.794i 0.291877i
\(408\) 0 0
\(409\) 593.970i 1.45225i 0.687563 + 0.726124i \(0.258680\pi\)
−0.687563 + 0.726124i \(0.741320\pi\)
\(410\) 0 0
\(411\) 537.401i 1.30755i
\(412\) 0 0
\(413\) − 373.352i − 0.904001i
\(414\) 0 0
\(415\) 42.0000 0.101205
\(416\) 0 0
\(417\) 876.812i 2.10267i
\(418\) 0 0
\(419\) −230.000 −0.548926 −0.274463 0.961598i \(-0.588500\pi\)
−0.274463 + 0.961598i \(0.588500\pi\)
\(420\) 0 0
\(421\) 593.970i 1.41085i 0.708782 + 0.705427i \(0.249245\pi\)
−0.708782 + 0.705427i \(0.750755\pi\)
\(422\) 0 0
\(423\) −115.000 −0.271868
\(424\) 0 0
\(425\) −408.000 −0.960000
\(426\) 0 0
\(427\) 253.000 0.592506
\(428\) 0 0
\(429\) −192.000 −0.447552
\(430\) 0 0
\(431\) 475.176i 1.10250i 0.834341 + 0.551248i \(0.185848\pi\)
−0.834341 + 0.551248i \(0.814152\pi\)
\(432\) 0 0
\(433\) 593.970i 1.37175i 0.727717 + 0.685877i \(0.240581\pi\)
−0.727717 + 0.685877i \(0.759419\pi\)
\(434\) 0 0
\(435\) 1568.00 3.60460
\(436\) 0 0
\(437\) 38.0000 0.0869565
\(438\) 0 0
\(439\) − 118.794i − 0.270601i −0.990805 0.135301i \(-0.956800\pi\)
0.990805 0.135301i \(-0.0432000\pi\)
\(440\) 0 0
\(441\) −1656.00 −3.75510
\(442\) 0 0
\(443\) −653.000 −1.47404 −0.737020 0.675871i \(-0.763768\pi\)
−0.737020 + 0.675871i \(0.763768\pi\)
\(444\) 0 0
\(445\) − 831.558i − 1.86867i
\(446\) 0 0
\(447\) − 356.382i − 0.797275i
\(448\) 0 0
\(449\) − 147.078i − 0.327568i −0.986496 0.163784i \(-0.947630\pi\)
0.986496 0.163784i \(-0.0523701\pi\)
\(450\) 0 0
\(451\) 118.794i 0.263401i
\(452\) 0 0
\(453\) −704.000 −1.55408
\(454\) 0 0
\(455\) 871.156i 1.91463i
\(456\) 0 0
\(457\) −817.000 −1.78775 −0.893873 0.448320i \(-0.852022\pi\)
−0.893873 + 0.448320i \(0.852022\pi\)
\(458\) 0 0
\(459\) 1346.33i 2.93318i
\(460\) 0 0
\(461\) −463.000 −1.00434 −0.502169 0.864769i \(-0.667465\pi\)
−0.502169 + 0.864769i \(0.667465\pi\)
\(462\) 0 0
\(463\) 29.0000 0.0626350 0.0313175 0.999509i \(-0.490030\pi\)
0.0313175 + 0.999509i \(0.490030\pi\)
\(464\) 0 0
\(465\) −224.000 −0.481720
\(466\) 0 0
\(467\) 19.0000 0.0406852 0.0203426 0.999793i \(-0.493524\pi\)
0.0203426 + 0.999793i \(0.493524\pi\)
\(468\) 0 0
\(469\) − 435.578i − 0.928737i
\(470\) 0 0
\(471\) 848.528i 1.80155i
\(472\) 0 0
\(473\) −63.0000 −0.133192
\(474\) 0 0
\(475\) −456.000 −0.960000
\(476\) 0 0
\(477\) 130.108i 0.272762i
\(478\) 0 0
\(479\) 206.000 0.430063 0.215031 0.976607i \(-0.431015\pi\)
0.215031 + 0.976607i \(0.431015\pi\)
\(480\) 0 0
\(481\) −448.000 −0.931393
\(482\) 0 0
\(483\) 124.451i 0.257662i
\(484\) 0 0
\(485\) 1187.94i 2.44936i
\(486\) 0 0
\(487\) − 45.2548i − 0.0929257i −0.998920 0.0464629i \(-0.985205\pi\)
0.998920 0.0464629i \(-0.0147949\pi\)
\(488\) 0 0
\(489\) − 939.038i − 1.92032i
\(490\) 0 0
\(491\) −246.000 −0.501018 −0.250509 0.968114i \(-0.580598\pi\)
−0.250509 + 0.968114i \(0.580598\pi\)
\(492\) 0 0
\(493\) − 673.166i − 1.36545i
\(494\) 0 0
\(495\) 483.000 0.975758
\(496\) 0 0
\(497\) 995.606i 2.00323i
\(498\) 0 0
\(499\) −21.0000 −0.0420842 −0.0210421 0.999779i \(-0.506698\pi\)
−0.0210421 + 0.999779i \(0.506698\pi\)
\(500\) 0 0
\(501\) 1184.00 2.36327
\(502\) 0 0
\(503\) −386.000 −0.767396 −0.383698 0.923459i \(-0.625350\pi\)
−0.383698 + 0.923459i \(0.625350\pi\)
\(504\) 0 0
\(505\) 854.000 1.69109
\(506\) 0 0
\(507\) 231.931i 0.457458i
\(508\) 0 0
\(509\) 475.176i 0.933548i 0.884377 + 0.466774i \(0.154584\pi\)
−0.884377 + 0.466774i \(0.845416\pi\)
\(510\) 0 0
\(511\) −429.000 −0.839530
\(512\) 0 0
\(513\) 1504.72i 2.93318i
\(514\) 0 0
\(515\) 712.764i 1.38401i
\(516\) 0 0
\(517\) 15.0000 0.0290135
\(518\) 0 0
\(519\) 320.000 0.616570
\(520\) 0 0
\(521\) − 786.303i − 1.50922i −0.656175 0.754609i \(-0.727826\pi\)
0.656175 0.754609i \(-0.272174\pi\)
\(522\) 0 0
\(523\) 395.980i 0.757132i 0.925574 + 0.378566i \(0.123583\pi\)
−0.925574 + 0.378566i \(0.876417\pi\)
\(524\) 0 0
\(525\) − 1493.41i − 2.84459i
\(526\) 0 0
\(527\) 96.1665i 0.182479i
\(528\) 0 0
\(529\) −525.000 −0.992439
\(530\) 0 0
\(531\) − 780.646i − 1.47014i
\(532\) 0 0
\(533\) −448.000 −0.840525
\(534\) 0 0
\(535\) − 1108.74i − 2.07242i
\(536\) 0 0
\(537\) 416.000 0.774674
\(538\) 0 0
\(539\) 216.000 0.400742
\(540\) 0 0
\(541\) −7.00000 −0.0129390 −0.00646950 0.999979i \(-0.502059\pi\)
−0.00646950 + 0.999979i \(0.502059\pi\)
\(542\) 0 0
\(543\) −448.000 −0.825046
\(544\) 0 0
\(545\) − 831.558i − 1.52579i
\(546\) 0 0
\(547\) 712.764i 1.30304i 0.758631 + 0.651521i \(0.225869\pi\)
−0.758631 + 0.651521i \(0.774131\pi\)
\(548\) 0 0
\(549\) 529.000 0.963570
\(550\) 0 0
\(551\) − 752.362i − 1.36545i
\(552\) 0 0
\(553\) − 1057.83i − 1.91290i
\(554\) 0 0
\(555\) 1568.00 2.82523
\(556\) 0 0
\(557\) 1001.00 1.79713 0.898564 0.438843i \(-0.144612\pi\)
0.898564 + 0.438843i \(0.144612\pi\)
\(558\) 0 0
\(559\) − 237.588i − 0.425023i
\(560\) 0 0
\(561\) − 288.500i − 0.514259i
\(562\) 0 0
\(563\) − 322.441i − 0.572719i −0.958122 0.286359i \(-0.907555\pi\)
0.958122 0.286359i \(-0.0924451\pi\)
\(564\) 0 0
\(565\) − 356.382i − 0.630764i
\(566\) 0 0
\(567\) −2651.00 −4.67549
\(568\) 0 0
\(569\) − 712.764i − 1.25266i −0.779558 0.626330i \(-0.784556\pi\)
0.779558 0.626330i \(-0.215444\pi\)
\(570\) 0 0
\(571\) 746.000 1.30648 0.653240 0.757151i \(-0.273409\pi\)
0.653240 + 0.757151i \(0.273409\pi\)
\(572\) 0 0
\(573\) 1702.71i 2.97158i
\(574\) 0 0
\(575\) −48.0000 −0.0834783
\(576\) 0 0
\(577\) −25.0000 −0.0433276 −0.0216638 0.999765i \(-0.506896\pi\)
−0.0216638 + 0.999765i \(0.506896\pi\)
\(578\) 0 0
\(579\) −864.000 −1.49223
\(580\) 0 0
\(581\) 66.0000 0.113597
\(582\) 0 0
\(583\) − 16.9706i − 0.0291090i
\(584\) 0 0
\(585\) 1821.51i 3.11369i
\(586\) 0 0
\(587\) −869.000 −1.48041 −0.740204 0.672382i \(-0.765271\pi\)
−0.740204 + 0.672382i \(0.765271\pi\)
\(588\) 0 0
\(589\) 107.480i 0.182479i
\(590\) 0 0
\(591\) − 509.117i − 0.861450i
\(592\) 0 0
\(593\) 898.000 1.51433 0.757167 0.653221i \(-0.226583\pi\)
0.757167 + 0.653221i \(0.226583\pi\)
\(594\) 0 0
\(595\) −1309.00 −2.20000
\(596\) 0 0
\(597\) − 831.558i − 1.39289i
\(598\) 0 0
\(599\) 277.186i 0.462748i 0.972865 + 0.231374i \(0.0743220\pi\)
−0.972865 + 0.231374i \(0.925678\pi\)
\(600\) 0 0
\(601\) − 475.176i − 0.790642i −0.918543 0.395321i \(-0.870633\pi\)
0.918543 0.395321i \(-0.129367\pi\)
\(602\) 0 0
\(603\) − 910.754i − 1.51037i
\(604\) 0 0
\(605\) 784.000 1.29587
\(606\) 0 0
\(607\) 452.548i 0.745549i 0.927922 + 0.372775i \(0.121594\pi\)
−0.927922 + 0.372775i \(0.878406\pi\)
\(608\) 0 0
\(609\) 2464.00 4.04598
\(610\) 0 0
\(611\) 56.5685i 0.0925835i
\(612\) 0 0
\(613\) 585.000 0.954323 0.477162 0.878816i \(-0.341666\pi\)
0.477162 + 0.878816i \(0.341666\pi\)
\(614\) 0 0
\(615\) 1568.00 2.54959
\(616\) 0 0
\(617\) −873.000 −1.41491 −0.707455 0.706758i \(-0.750157\pi\)
−0.707455 + 0.706758i \(0.750157\pi\)
\(618\) 0 0
\(619\) 970.000 1.56704 0.783522 0.621364i \(-0.213421\pi\)
0.783522 + 0.621364i \(0.213421\pi\)
\(620\) 0 0
\(621\) 158.392i 0.255059i
\(622\) 0 0
\(623\) − 1306.73i − 2.09749i
\(624\) 0 0
\(625\) −649.000 −1.03840
\(626\) 0 0
\(627\) − 322.441i − 0.514259i
\(628\) 0 0
\(629\) − 673.166i − 1.07022i
\(630\) 0 0
\(631\) −259.000 −0.410460 −0.205230 0.978714i \(-0.565794\pi\)
−0.205230 + 0.978714i \(0.565794\pi\)
\(632\) 0 0
\(633\) 1856.00 2.93207
\(634\) 0 0
\(635\) − 277.186i − 0.436513i
\(636\) 0 0
\(637\) 814.587i 1.27879i
\(638\) 0 0
\(639\) 2081.72i 3.25778i
\(640\) 0 0
\(641\) 1063.49i 1.65911i 0.558426 + 0.829554i \(0.311405\pi\)
−0.558426 + 0.829554i \(0.688595\pi\)
\(642\) 0 0
\(643\) −645.000 −1.00311 −0.501555 0.865126i \(-0.667239\pi\)
−0.501555 + 0.865126i \(0.667239\pi\)
\(644\) 0 0
\(645\) 831.558i 1.28924i
\(646\) 0 0
\(647\) 29.0000 0.0448223 0.0224111 0.999749i \(-0.492866\pi\)
0.0224111 + 0.999749i \(0.492866\pi\)
\(648\) 0 0
\(649\) 101.823i 0.156893i
\(650\) 0 0
\(651\) −352.000 −0.540707
\(652\) 0 0
\(653\) −135.000 −0.206738 −0.103369 0.994643i \(-0.532962\pi\)
−0.103369 + 0.994643i \(0.532962\pi\)
\(654\) 0 0
\(655\) 1043.00 1.59237
\(656\) 0 0
\(657\) −897.000 −1.36530
\(658\) 0 0
\(659\) 780.646i 1.18459i 0.805721 + 0.592296i \(0.201778\pi\)
−0.805721 + 0.592296i \(0.798222\pi\)
\(660\) 0 0
\(661\) 995.606i 1.50621i 0.657899 + 0.753106i \(0.271445\pi\)
−0.657899 + 0.753106i \(0.728555\pi\)
\(662\) 0 0
\(663\) 1088.00 1.64103
\(664\) 0 0
\(665\) −1463.00 −2.20000
\(666\) 0 0
\(667\) − 79.1960i − 0.118735i
\(668\) 0 0
\(669\) 2016.00 3.01345
\(670\) 0 0
\(671\) −69.0000 −0.102832
\(672\) 0 0
\(673\) − 395.980i − 0.588380i −0.955747 0.294190i \(-0.904950\pi\)
0.955747 0.294190i \(-0.0950499\pi\)
\(674\) 0 0
\(675\) − 1900.70i − 2.81586i
\(676\) 0 0
\(677\) − 356.382i − 0.526413i −0.964739 0.263207i \(-0.915220\pi\)
0.964739 0.263207i \(-0.0847801\pi\)
\(678\) 0 0
\(679\) 1866.76i 2.74928i
\(680\) 0 0
\(681\) 1792.00 2.63142
\(682\) 0 0
\(683\) 520.431i 0.761977i 0.924580 + 0.380989i \(0.124416\pi\)
−0.924580 + 0.380989i \(0.875584\pi\)
\(684\) 0 0
\(685\) −665.000 −0.970803
\(686\) 0 0
\(687\) 1453.81i 2.11617i
\(688\) 0 0
\(689\) 64.0000 0.0928882
\(690\) 0 0
\(691\) 835.000 1.20839 0.604197 0.796835i \(-0.293494\pi\)
0.604197 + 0.796835i \(0.293494\pi\)
\(692\) 0 0
\(693\) 759.000 1.09524
\(694\) 0 0
\(695\) −1085.00 −1.56115
\(696\) 0 0
\(697\) − 673.166i − 0.965804i
\(698\) 0 0
\(699\) − 1001.26i − 1.43242i
\(700\) 0 0
\(701\) −1002.00 −1.42939 −0.714693 0.699438i \(-0.753434\pi\)
−0.714693 + 0.699438i \(0.753434\pi\)
\(702\) 0 0
\(703\) − 752.362i − 1.07022i
\(704\) 0 0
\(705\) − 197.990i − 0.280837i
\(706\) 0 0
\(707\) 1342.00 1.89816
\(708\) 0 0
\(709\) −250.000 −0.352609 −0.176305 0.984336i \(-0.556414\pi\)
−0.176305 + 0.984336i \(0.556414\pi\)
\(710\) 0 0
\(711\) − 2211.83i − 3.11087i
\(712\) 0 0
\(713\) 11.3137i 0.0158678i
\(714\) 0 0
\(715\) − 237.588i − 0.332291i
\(716\) 0 0
\(717\) − 2053.44i − 2.86393i
\(718\) 0 0
\(719\) −1171.00 −1.62865 −0.814325 0.580409i \(-0.802893\pi\)
−0.814325 + 0.580409i \(0.802893\pi\)
\(720\) 0 0
\(721\) 1120.06i 1.55348i
\(722\) 0 0
\(723\) −2016.00 −2.78838
\(724\) 0 0
\(725\) 950.352i 1.31083i
\(726\) 0 0
\(727\) −91.0000 −0.125172 −0.0625860 0.998040i \(-0.519935\pi\)
−0.0625860 + 0.998040i \(0.519935\pi\)
\(728\) 0 0
\(729\) −1511.00 −2.07270
\(730\) 0 0
\(731\) 357.000 0.488372
\(732\) 0 0
\(733\) −682.000 −0.930423 −0.465211 0.885200i \(-0.654022\pi\)
−0.465211 + 0.885200i \(0.654022\pi\)
\(734\) 0 0
\(735\) − 2851.05i − 3.87899i
\(736\) 0 0
\(737\) 118.794i 0.161186i
\(738\) 0 0
\(739\) −221.000 −0.299053 −0.149526 0.988758i \(-0.547775\pi\)
−0.149526 + 0.988758i \(0.547775\pi\)
\(740\) 0 0
\(741\) 1216.00 1.64103
\(742\) 0 0
\(743\) 288.500i 0.388290i 0.980973 + 0.194145i \(0.0621933\pi\)
−0.980973 + 0.194145i \(0.937807\pi\)
\(744\) 0 0
\(745\) 441.000 0.591946
\(746\) 0 0
\(747\) 138.000 0.184739
\(748\) 0 0
\(749\) − 1742.31i − 2.32618i
\(750\) 0 0
\(751\) − 927.724i − 1.23532i −0.786446 0.617659i \(-0.788081\pi\)
0.786446 0.617659i \(-0.211919\pi\)
\(752\) 0 0
\(753\) − 752.362i − 0.999152i
\(754\) 0 0
\(755\) − 871.156i − 1.15385i
\(756\) 0 0
\(757\) −895.000 −1.18230 −0.591149 0.806562i \(-0.701326\pi\)
−0.591149 + 0.806562i \(0.701326\pi\)
\(758\) 0 0
\(759\) − 33.9411i − 0.0447182i
\(760\) 0 0
\(761\) 95.0000 0.124836 0.0624179 0.998050i \(-0.480119\pi\)
0.0624179 + 0.998050i \(0.480119\pi\)
\(762\) 0 0
\(763\) − 1306.73i − 1.71263i
\(764\) 0 0
\(765\) −2737.00 −3.57778
\(766\) 0 0
\(767\) −384.000 −0.500652
\(768\) 0 0
\(769\) 679.000 0.882965 0.441482 0.897270i \(-0.354453\pi\)
0.441482 + 0.897270i \(0.354453\pi\)
\(770\) 0 0
\(771\) 768.000 0.996109
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) − 135.765i − 0.175180i
\(776\) 0 0
\(777\) 2464.00 3.17117
\(778\) 0 0
\(779\) − 752.362i − 0.965804i
\(780\) 0 0
\(781\) − 271.529i − 0.347668i
\(782\) 0 0
\(783\) 3136.00 4.00511
\(784\) 0 0
\(785\) −1050.00 −1.33758
\(786\) 0 0
\(787\) − 1391.59i − 1.76822i −0.467282 0.884108i \(-0.654767\pi\)
0.467282 0.884108i \(-0.345233\pi\)
\(788\) 0 0
\(789\) 571.342i 0.724135i
\(790\) 0 0
\(791\) − 560.029i − 0.708001i
\(792\) 0 0
\(793\) − 260.215i − 0.328140i
\(794\) 0 0
\(795\) −224.000 −0.281761
\(796\) 0 0
\(797\) 356.382i 0.447154i 0.974686 + 0.223577i \(0.0717734\pi\)
−0.974686 + 0.223577i \(0.928227\pi\)
\(798\) 0 0
\(799\) −85.0000 −0.106383
\(800\) 0 0
\(801\) − 2732.26i − 3.41106i
\(802\) 0 0
\(803\) 117.000 0.145704
\(804\) 0 0
\(805\) −154.000 −0.191304
\(806\) 0 0
\(807\) 2016.00 2.49814
\(808\) 0 0
\(809\) 455.000 0.562423 0.281211 0.959646i \(-0.409264\pi\)
0.281211 + 0.959646i \(0.409264\pi\)
\(810\) 0 0
\(811\) 475.176i 0.585913i 0.956126 + 0.292957i \(0.0946392\pi\)
−0.956126 + 0.292957i \(0.905361\pi\)
\(812\) 0 0
\(813\) 803.273i 0.988036i
\(814\) 0 0
\(815\) 1162.00 1.42577
\(816\) 0 0
\(817\) 399.000 0.488372
\(818\) 0 0
\(819\) 2862.37i 3.49496i
\(820\) 0 0
\(821\) −831.000 −1.01218 −0.506090 0.862481i \(-0.668910\pi\)
−0.506090 + 0.862481i \(0.668910\pi\)
\(822\) 0 0
\(823\) 109.000 0.132442 0.0662211 0.997805i \(-0.478906\pi\)
0.0662211 + 0.997805i \(0.478906\pi\)
\(824\) 0 0
\(825\) 407.294i 0.493689i
\(826\) 0 0
\(827\) 1238.85i 1.49801i 0.662567 + 0.749003i \(0.269467\pi\)
−0.662567 + 0.749003i \(0.730533\pi\)
\(828\) 0 0
\(829\) 667.509i 0.805198i 0.915377 + 0.402599i \(0.131893\pi\)
−0.915377 + 0.402599i \(0.868107\pi\)
\(830\) 0 0
\(831\) − 1125.71i − 1.35465i
\(832\) 0 0
\(833\) −1224.00 −1.46939
\(834\) 0 0
\(835\) 1465.13i 1.75464i
\(836\) 0 0
\(837\) −448.000 −0.535245
\(838\) 0 0
\(839\) − 316.784i − 0.377573i −0.982018 0.188787i \(-0.939545\pi\)
0.982018 0.188787i \(-0.0604554\pi\)
\(840\) 0 0
\(841\) −727.000 −0.864447
\(842\) 0 0
\(843\) −2624.00 −3.11269
\(844\) 0 0
\(845\) −287.000 −0.339645
\(846\) 0 0
\(847\) 1232.00 1.45455
\(848\) 0 0
\(849\) 2415.48i 2.84508i
\(850\) 0 0
\(851\) − 79.1960i − 0.0930622i
\(852\) 0 0
\(853\) 1126.00 1.32005 0.660023 0.751245i \(-0.270546\pi\)
0.660023 + 0.751245i \(0.270546\pi\)
\(854\) 0 0
\(855\) −3059.00 −3.57778
\(856\) 0 0
\(857\) − 203.647i − 0.237627i −0.992917 0.118814i \(-0.962091\pi\)
0.992917 0.118814i \(-0.0379091\pi\)
\(858\) 0 0
\(859\) 651.000 0.757858 0.378929 0.925426i \(-0.376292\pi\)
0.378929 + 0.925426i \(0.376292\pi\)
\(860\) 0 0
\(861\) 2464.00 2.86179
\(862\) 0 0
\(863\) − 39.5980i − 0.0458841i −0.999737 0.0229421i \(-0.992697\pi\)
0.999737 0.0229421i \(-0.00730332\pi\)
\(864\) 0 0
\(865\) 395.980i 0.457780i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 288.500i 0.331990i
\(870\) 0 0
\(871\) −448.000 −0.514351
\(872\) 0 0
\(873\) 3903.23i 4.47105i
\(874\) 0 0
\(875\) −77.0000 −0.0880000
\(876\) 0 0
\(877\) − 1029.55i − 1.17394i −0.809608 0.586971i \(-0.800320\pi\)
0.809608 0.586971i \(-0.199680\pi\)
\(878\) 0 0
\(879\) −1344.00 −1.52901
\(880\) 0 0
\(881\) 391.000 0.443814 0.221907 0.975068i \(-0.428772\pi\)
0.221907 + 0.975068i \(0.428772\pi\)
\(882\) 0 0
\(883\) 995.000 1.12684 0.563420 0.826171i \(-0.309485\pi\)
0.563420 + 0.826171i \(0.309485\pi\)
\(884\) 0 0
\(885\) 1344.00 1.51864
\(886\) 0 0
\(887\) 475.176i 0.535711i 0.963459 + 0.267856i \(0.0863150\pi\)
−0.963459 + 0.267856i \(0.913685\pi\)
\(888\) 0 0
\(889\) − 435.578i − 0.489964i
\(890\) 0 0
\(891\) 723.000 0.811448
\(892\) 0 0
\(893\) −95.0000 −0.106383
\(894\) 0 0
\(895\) 514.774i 0.575166i
\(896\) 0 0
\(897\) 128.000 0.142698
\(898\) 0 0
\(899\) 224.000 0.249166
\(900\) 0 0
\(901\) 96.1665i 0.106733i
\(902\) 0 0
\(903\) 1306.73i 1.44710i
\(904\) 0 0
\(905\) − 554.372i − 0.612565i
\(906\) 0 0
\(907\) − 627.911i − 0.692294i −0.938180 0.346147i \(-0.887490\pi\)
0.938180 0.346147i \(-0.112510\pi\)
\(908\) 0 0
\(909\) 2806.00 3.08691
\(910\) 0 0
\(911\) 803.273i 0.881749i 0.897569 + 0.440874i \(0.145332\pi\)
−0.897569 + 0.440874i \(0.854668\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.0197152
\(914\) 0 0
\(915\) 910.754i 0.995359i
\(916\) 0 0
\(917\) 1639.00 1.78735
\(918\) 0 0
\(919\) −1090.00 −1.18607 −0.593036 0.805176i \(-0.702071\pi\)
−0.593036 + 0.805176i \(0.702071\pi\)
\(920\) 0 0
\(921\) 2976.00 3.23127
\(922\) 0 0
\(923\) 1024.00 1.10943
\(924\) 0 0
\(925\) 950.352i 1.02741i
\(926\) 0 0
\(927\) 2341.94i 2.52636i
\(928\) 0 0
\(929\) 226.000 0.243272 0.121636 0.992575i \(-0.461186\pi\)
0.121636 + 0.992575i \(0.461186\pi\)
\(930\) 0 0
\(931\) −1368.00 −1.46939
\(932\) 0 0
\(933\) − 1329.36i − 1.42482i
\(934\) 0 0
\(935\) 357.000 0.381818
\(936\) 0 0
\(937\) 623.000 0.664888 0.332444 0.943123i \(-0.392127\pi\)
0.332444 + 0.943123i \(0.392127\pi\)
\(938\) 0 0
\(939\) 2998.13i 3.19290i
\(940\) 0 0
\(941\) 1549.98i 1.64716i 0.567200 + 0.823580i \(0.308027\pi\)
−0.567200 + 0.823580i \(0.691973\pi\)
\(942\) 0 0
\(943\) − 79.1960i − 0.0839830i
\(944\) 0 0
\(945\) − 6098.09i − 6.45300i
\(946\) 0 0
\(947\) 602.000 0.635692 0.317846 0.948142i \(-0.397041\pi\)
0.317846 + 0.948142i \(0.397041\pi\)
\(948\) 0 0
\(949\) 441.235i 0.464947i
\(950\) 0 0
\(951\) −1216.00 −1.27865
\(952\) 0 0
\(953\) 277.186i 0.290856i 0.989369 + 0.145428i \(0.0464559\pi\)
−0.989369 + 0.145428i \(0.953544\pi\)
\(954\) 0 0
\(955\) −2107.00 −2.20628
\(956\) 0 0
\(957\) −672.000 −0.702194
\(958\) 0 0
\(959\) −1045.00 −1.08968
\(960\) 0 0
\(961\) 929.000 0.966701
\(962\) 0 0
\(963\) − 3643.01i − 3.78298i
\(964\) 0 0
\(965\) − 1069.15i − 1.10792i
\(966\) 0 0
\(967\) −770.000 −0.796277 −0.398139 0.917325i \(-0.630344\pi\)
−0.398139 + 0.917325i \(0.630344\pi\)
\(968\) 0 0
\(969\) 1827.16i 1.88562i
\(970\) 0 0
\(971\) − 712.764i − 0.734051i −0.930211 0.367026i \(-0.880376\pi\)
0.930211 0.367026i \(-0.119624\pi\)
\(972\) 0 0
\(973\) −1705.00 −1.75231
\(974\) 0 0
\(975\) −1536.00 −1.57538
\(976\) 0 0
\(977\) 1340.67i 1.37224i 0.727490 + 0.686118i \(0.240687\pi\)
−0.727490 + 0.686118i \(0.759313\pi\)
\(978\) 0 0
\(979\) 356.382i 0.364026i
\(980\) 0 0
\(981\) − 2732.26i − 2.78518i
\(982\) 0 0
\(983\) − 554.372i − 0.563959i −0.959420 0.281980i \(-0.909009\pi\)
0.959420 0.281980i \(-0.0909910\pi\)
\(984\) 0 0
\(985\) 630.000 0.639594
\(986\) 0 0
\(987\) − 311.127i − 0.315225i
\(988\) 0 0
\(989\) 42.0000 0.0424671
\(990\) 0 0
\(991\) 639.225i 0.645030i 0.946564 + 0.322515i \(0.104528\pi\)
−0.946564 + 0.322515i \(0.895472\pi\)
\(992\) 0 0
\(993\) −544.000 −0.547835
\(994\) 0 0
\(995\) 1029.00 1.03417
\(996\) 0 0
\(997\) 473.000 0.474423 0.237212 0.971458i \(-0.423767\pi\)
0.237212 + 0.971458i \(0.423767\pi\)
\(998\) 0 0
\(999\) 3136.00 3.13914
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 1216.3.e.c.1025.2 2
4.3 odd 2 1216.3.e.d.1025.1 2
8.3 odd 2 152.3.e.a.113.2 yes 2
8.5 even 2 304.3.e.f.113.1 2
19.18 odd 2 inner 1216.3.e.c.1025.1 2
24.5 odd 2 2736.3.o.b.721.1 2
24.11 even 2 1368.3.o.a.721.1 2
76.75 even 2 1216.3.e.d.1025.2 2
152.37 odd 2 304.3.e.f.113.2 2
152.75 even 2 152.3.e.a.113.1 2
456.227 odd 2 1368.3.o.a.721.2 2
456.341 even 2 2736.3.o.b.721.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.3.e.a.113.1 2 152.75 even 2
152.3.e.a.113.2 yes 2 8.3 odd 2
304.3.e.f.113.1 2 8.5 even 2
304.3.e.f.113.2 2 152.37 odd 2
1216.3.e.c.1025.1 2 19.18 odd 2 inner
1216.3.e.c.1025.2 2 1.1 even 1 trivial
1216.3.e.d.1025.1 2 4.3 odd 2
1216.3.e.d.1025.2 2 76.75 even 2
1368.3.o.a.721.1 2 24.11 even 2
1368.3.o.a.721.2 2 456.227 odd 2
2736.3.o.b.721.1 2 24.5 odd 2
2736.3.o.b.721.2 2 456.341 even 2