Properties

Label 3850.2.c.y.1849.2
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
Defining polynomial: \(x^{4} + 17 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.2
Root \(-2.37228i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.y.1849.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +6.74456i q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.74456i q^{17} +1.00000i q^{18} +6.74456 q^{19} +2.00000 q^{21} +1.00000i q^{22} -6.74456i q^{23} -2.00000 q^{24} +6.74456 q^{26} +4.00000i q^{27} +1.00000i q^{28} -8.74456 q^{29} +4.74456 q^{31} -1.00000i q^{32} -2.00000i q^{33} +6.74456 q^{34} +1.00000 q^{36} -0.744563i q^{37} -6.74456i q^{38} -13.4891 q^{39} -4.00000 q^{41} -2.00000i q^{42} +4.00000i q^{43} +1.00000 q^{44} -6.74456 q^{46} -4.74456i q^{47} +2.00000i q^{48} -1.00000 q^{49} -13.4891 q^{51} -6.74456i q^{52} -12.7446i q^{53} +4.00000 q^{54} +1.00000 q^{56} +13.4891i q^{57} +8.74456i q^{58} +8.74456 q^{59} +1.25544 q^{61} -4.74456i q^{62} +1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +4.00000i q^{67} -6.74456i q^{68} +13.4891 q^{69} -4.00000 q^{71} -1.00000i q^{72} +10.7446i q^{73} -0.744563 q^{74} -6.74456 q^{76} +1.00000i q^{77} +13.4891i q^{78} -6.74456 q^{79} -11.0000 q^{81} +4.00000i q^{82} +8.00000i q^{83} -2.00000 q^{84} +4.00000 q^{86} -17.4891i q^{87} -1.00000i q^{88} -15.4891 q^{89} +6.74456 q^{91} +6.74456i q^{92} +9.48913i q^{93} -4.74456 q^{94} +2.00000 q^{96} +16.7446i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 4 q^{11} - 4 q^{14} + 4 q^{16} + 4 q^{19} + 8 q^{21} - 8 q^{24} + 4 q^{26} - 12 q^{29} - 4 q^{31} + 4 q^{34} + 4 q^{36} - 8 q^{39} - 16 q^{41} + 4 q^{44} - 4 q^{46} - 4 q^{49} - 8 q^{51} + 16 q^{54} + 4 q^{56} + 12 q^{59} + 28 q^{61} - 4 q^{64} - 8 q^{66} + 8 q^{69} - 16 q^{71} + 20 q^{74} - 4 q^{76} - 4 q^{79} - 44 q^{81} - 8 q^{84} + 16 q^{86} - 16 q^{89} + 4 q^{91} + 4 q^{94} + 8 q^{96} + 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 2.00000i − 0.577350i
\(13\) 6.74456i 1.87061i 0.353849 + 0.935303i \(0.384873\pi\)
−0.353849 + 0.935303i \(0.615127\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.74456i 1.63580i 0.575363 + 0.817898i \(0.304861\pi\)
−0.575363 + 0.817898i \(0.695139\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 6.74456 1.54731 0.773654 0.633608i \(-0.218427\pi\)
0.773654 + 0.633608i \(0.218427\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 1.00000i 0.213201i
\(23\) − 6.74456i − 1.40634i −0.711022 0.703169i \(-0.751768\pi\)
0.711022 0.703169i \(-0.248232\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 6.74456 1.32272
\(27\) 4.00000i 0.769800i
\(28\) 1.00000i 0.188982i
\(29\) −8.74456 −1.62382 −0.811912 0.583779i \(-0.801573\pi\)
−0.811912 + 0.583779i \(0.801573\pi\)
\(30\) 0 0
\(31\) 4.74456 0.852149 0.426074 0.904688i \(-0.359896\pi\)
0.426074 + 0.904688i \(0.359896\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) 6.74456 1.15668
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 0.744563i − 0.122405i −0.998125 0.0612027i \(-0.980506\pi\)
0.998125 0.0612027i \(-0.0194936\pi\)
\(38\) − 6.74456i − 1.09411i
\(39\) −13.4891 −2.15999
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.74456 −0.994432
\(47\) − 4.74456i − 0.692066i −0.938222 0.346033i \(-0.887529\pi\)
0.938222 0.346033i \(-0.112471\pi\)
\(48\) 2.00000i 0.288675i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −13.4891 −1.88886
\(52\) − 6.74456i − 0.935303i
\(53\) − 12.7446i − 1.75060i −0.483580 0.875300i \(-0.660664\pi\)
0.483580 0.875300i \(-0.339336\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 13.4891i 1.78668i
\(58\) 8.74456i 1.14822i
\(59\) 8.74456 1.13845 0.569223 0.822183i \(-0.307244\pi\)
0.569223 + 0.822183i \(0.307244\pi\)
\(60\) 0 0
\(61\) 1.25544 0.160742 0.0803711 0.996765i \(-0.474389\pi\)
0.0803711 + 0.996765i \(0.474389\pi\)
\(62\) − 4.74456i − 0.602560i
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 6.74456i − 0.817898i
\(69\) 13.4891 1.62390
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 10.7446i 1.25756i 0.777585 + 0.628778i \(0.216445\pi\)
−0.777585 + 0.628778i \(0.783555\pi\)
\(74\) −0.744563 −0.0865536
\(75\) 0 0
\(76\) −6.74456 −0.773654
\(77\) 1.00000i 0.113961i
\(78\) 13.4891i 1.52734i
\(79\) −6.74456 −0.758823 −0.379411 0.925228i \(-0.623873\pi\)
−0.379411 + 0.925228i \(0.623873\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 4.00000i 0.441726i
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) − 17.4891i − 1.87503i
\(88\) − 1.00000i − 0.106600i
\(89\) −15.4891 −1.64184 −0.820922 0.571040i \(-0.806540\pi\)
−0.820922 + 0.571040i \(0.806540\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) 6.74456i 0.703169i
\(93\) 9.48913i 0.983976i
\(94\) −4.74456 −0.489364
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 16.7446i 1.70015i 0.526659 + 0.850076i \(0.323444\pi\)
−0.526659 + 0.850076i \(0.676556\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 2.74456 0.273094 0.136547 0.990634i \(-0.456399\pi\)
0.136547 + 0.990634i \(0.456399\pi\)
\(102\) 13.4891i 1.33562i
\(103\) − 0.744563i − 0.0733639i −0.999327 0.0366820i \(-0.988321\pi\)
0.999327 0.0366820i \(-0.0116789\pi\)
\(104\) −6.74456 −0.661359
\(105\) 0 0
\(106\) −12.7446 −1.23786
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 4.00000i − 0.384900i
\(109\) 18.2337 1.74647 0.873235 0.487299i \(-0.162018\pi\)
0.873235 + 0.487299i \(0.162018\pi\)
\(110\) 0 0
\(111\) 1.48913 0.141342
\(112\) − 1.00000i − 0.0944911i
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 13.4891 1.26337
\(115\) 0 0
\(116\) 8.74456 0.811912
\(117\) − 6.74456i − 0.623535i
\(118\) − 8.74456i − 0.805002i
\(119\) 6.74456 0.618273
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 1.25544i − 0.113662i
\(123\) − 8.00000i − 0.721336i
\(124\) −4.74456 −0.426074
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −2.74456 −0.239794 −0.119897 0.992786i \(-0.538256\pi\)
−0.119897 + 0.992786i \(0.538256\pi\)
\(132\) 2.00000i 0.174078i
\(133\) − 6.74456i − 0.584828i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.74456 −0.578341
\(137\) − 3.48913i − 0.298096i −0.988830 0.149048i \(-0.952379\pi\)
0.988830 0.149048i \(-0.0476209\pi\)
\(138\) − 13.4891i − 1.14827i
\(139\) −14.7446 −1.25062 −0.625309 0.780377i \(-0.715027\pi\)
−0.625309 + 0.780377i \(0.715027\pi\)
\(140\) 0 0
\(141\) 9.48913 0.799129
\(142\) 4.00000i 0.335673i
\(143\) − 6.74456i − 0.564009i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.7446 0.889226
\(147\) − 2.00000i − 0.164957i
\(148\) 0.744563i 0.0612027i
\(149\) 0.744563 0.0609969 0.0304985 0.999535i \(-0.490291\pi\)
0.0304985 + 0.999535i \(0.490291\pi\)
\(150\) 0 0
\(151\) −9.25544 −0.753197 −0.376598 0.926377i \(-0.622906\pi\)
−0.376598 + 0.926377i \(0.622906\pi\)
\(152\) 6.74456i 0.547056i
\(153\) − 6.74456i − 0.545266i
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 13.4891 1.07999
\(157\) − 0.510875i − 0.0407722i −0.999792 0.0203861i \(-0.993510\pi\)
0.999792 0.0203861i \(-0.00648955\pi\)
\(158\) 6.74456i 0.536569i
\(159\) 25.4891 2.02142
\(160\) 0 0
\(161\) −6.74456 −0.531546
\(162\) 11.0000i 0.864242i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 2.00000i 0.154303i
\(169\) −32.4891 −2.49916
\(170\) 0 0
\(171\) −6.74456 −0.515770
\(172\) − 4.00000i − 0.304997i
\(173\) 20.2337i 1.53834i 0.639045 + 0.769169i \(0.279330\pi\)
−0.639045 + 0.769169i \(0.720670\pi\)
\(174\) −17.4891 −1.32585
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 17.4891i 1.31456i
\(178\) 15.4891i 1.16096i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −19.4891 −1.44862 −0.724308 0.689477i \(-0.757840\pi\)
−0.724308 + 0.689477i \(0.757840\pi\)
\(182\) − 6.74456i − 0.499940i
\(183\) 2.51087i 0.185609i
\(184\) 6.74456 0.497216
\(185\) 0 0
\(186\) 9.48913 0.695776
\(187\) − 6.74456i − 0.493211i
\(188\) 4.74456i 0.346033i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) − 2.00000i − 0.144338i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 16.7446 1.20219
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) − 1.00000i − 0.0710669i
\(199\) −3.25544 −0.230772 −0.115386 0.993321i \(-0.536810\pi\)
−0.115386 + 0.993321i \(0.536810\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) − 2.74456i − 0.193107i
\(203\) 8.74456i 0.613748i
\(204\) 13.4891 0.944428
\(205\) 0 0
\(206\) −0.744563 −0.0518761
\(207\) 6.74456i 0.468780i
\(208\) 6.74456i 0.467651i
\(209\) −6.74456 −0.466531
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 12.7446i 0.875300i
\(213\) − 8.00000i − 0.548151i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) − 4.74456i − 0.322082i
\(218\) − 18.2337i − 1.23494i
\(219\) −21.4891 −1.45210
\(220\) 0 0
\(221\) −45.4891 −3.05993
\(222\) − 1.48913i − 0.0999435i
\(223\) 15.2554i 1.02158i 0.859706 + 0.510790i \(0.170647\pi\)
−0.859706 + 0.510790i \(0.829353\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) − 13.4891i − 0.893339i
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) − 8.74456i − 0.574109i
\(233\) − 24.9783i − 1.63638i −0.574948 0.818190i \(-0.694978\pi\)
0.574948 0.818190i \(-0.305022\pi\)
\(234\) −6.74456 −0.440906
\(235\) 0 0
\(236\) −8.74456 −0.569223
\(237\) − 13.4891i − 0.876213i
\(238\) − 6.74456i − 0.437185i
\(239\) 14.7446 0.953746 0.476873 0.878972i \(-0.341770\pi\)
0.476873 + 0.878972i \(0.341770\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) − 10.0000i − 0.641500i
\(244\) −1.25544 −0.0803711
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) 45.4891i 2.89440i
\(248\) 4.74456i 0.301280i
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) −26.2337 −1.65586 −0.827928 0.560835i \(-0.810480\pi\)
−0.827928 + 0.560835i \(0.810480\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) 6.74456i 0.424027i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.2337i 0.638360i 0.947694 + 0.319180i \(0.103407\pi\)
−0.947694 + 0.319180i \(0.896593\pi\)
\(258\) 8.00000i 0.498058i
\(259\) −0.744563 −0.0462649
\(260\) 0 0
\(261\) 8.74456 0.541275
\(262\) 2.74456i 0.169560i
\(263\) 26.9783i 1.66355i 0.555113 + 0.831775i \(0.312675\pi\)
−0.555113 + 0.831775i \(0.687325\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −6.74456 −0.413536
\(267\) − 30.9783i − 1.89584i
\(268\) − 4.00000i − 0.244339i
\(269\) −20.9783 −1.27907 −0.639533 0.768763i \(-0.720872\pi\)
−0.639533 + 0.768763i \(0.720872\pi\)
\(270\) 0 0
\(271\) 14.9783 0.909864 0.454932 0.890526i \(-0.349664\pi\)
0.454932 + 0.890526i \(0.349664\pi\)
\(272\) 6.74456i 0.408949i
\(273\) 13.4891i 0.816399i
\(274\) −3.48913 −0.210786
\(275\) 0 0
\(276\) −13.4891 −0.811950
\(277\) − 15.4891i − 0.930651i −0.885140 0.465326i \(-0.845937\pi\)
0.885140 0.465326i \(-0.154063\pi\)
\(278\) 14.7446i 0.884320i
\(279\) −4.74456 −0.284050
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) − 9.48913i − 0.565069i
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) −6.74456 −0.398814
\(287\) 4.00000i 0.236113i
\(288\) 1.00000i 0.0589256i
\(289\) −28.4891 −1.67583
\(290\) 0 0
\(291\) −33.4891 −1.96317
\(292\) − 10.7446i − 0.628778i
\(293\) 13.2554i 0.774391i 0.921998 + 0.387195i \(0.126556\pi\)
−0.921998 + 0.387195i \(0.873444\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 0.744563 0.0432768
\(297\) − 4.00000i − 0.232104i
\(298\) − 0.744563i − 0.0431314i
\(299\) 45.4891 2.63070
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 9.25544i 0.532591i
\(303\) 5.48913i 0.315342i
\(304\) 6.74456 0.386827
\(305\) 0 0
\(306\) −6.74456 −0.385561
\(307\) 1.48913i 0.0849889i 0.999097 + 0.0424944i \(0.0135305\pi\)
−0.999097 + 0.0424944i \(0.986470\pi\)
\(308\) − 1.00000i − 0.0569803i
\(309\) 1.48913 0.0847134
\(310\) 0 0
\(311\) −20.7446 −1.17632 −0.588158 0.808746i \(-0.700147\pi\)
−0.588158 + 0.808746i \(0.700147\pi\)
\(312\) − 13.4891i − 0.763671i
\(313\) − 2.23369i − 0.126256i −0.998005 0.0631278i \(-0.979892\pi\)
0.998005 0.0631278i \(-0.0201076\pi\)
\(314\) −0.510875 −0.0288303
\(315\) 0 0
\(316\) 6.74456 0.379411
\(317\) − 2.23369i − 0.125456i −0.998031 0.0627282i \(-0.980020\pi\)
0.998031 0.0627282i \(-0.0199801\pi\)
\(318\) − 25.4891i − 1.42936i
\(319\) 8.74456 0.489602
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 6.74456i 0.375860i
\(323\) 45.4891i 2.53108i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 36.4674i 2.01665i
\(328\) − 4.00000i − 0.220863i
\(329\) −4.74456 −0.261576
\(330\) 0 0
\(331\) 14.9783 0.823279 0.411640 0.911347i \(-0.364956\pi\)
0.411640 + 0.911347i \(0.364956\pi\)
\(332\) − 8.00000i − 0.439057i
\(333\) 0.744563i 0.0408018i
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 26.0000i − 1.41631i −0.706057 0.708155i \(-0.749528\pi\)
0.706057 0.708155i \(-0.250472\pi\)
\(338\) 32.4891i 1.76718i
\(339\) −20.0000 −1.08625
\(340\) 0 0
\(341\) −4.74456 −0.256932
\(342\) 6.74456i 0.364704i
\(343\) 1.00000i 0.0539949i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 20.2337 1.08777
\(347\) 22.9783i 1.23354i 0.787145 + 0.616769i \(0.211559\pi\)
−0.787145 + 0.616769i \(0.788441\pi\)
\(348\) 17.4891i 0.937516i
\(349\) −30.7446 −1.64572 −0.822859 0.568245i \(-0.807623\pi\)
−0.822859 + 0.568245i \(0.807623\pi\)
\(350\) 0 0
\(351\) −26.9783 −1.43999
\(352\) 1.00000i 0.0533002i
\(353\) 8.74456i 0.465426i 0.972545 + 0.232713i \(0.0747603\pi\)
−0.972545 + 0.232713i \(0.925240\pi\)
\(354\) 17.4891 0.929537
\(355\) 0 0
\(356\) 15.4891 0.820922
\(357\) 13.4891i 0.713920i
\(358\) − 4.00000i − 0.211407i
\(359\) 1.25544 0.0662594 0.0331297 0.999451i \(-0.489453\pi\)
0.0331297 + 0.999451i \(0.489453\pi\)
\(360\) 0 0
\(361\) 26.4891 1.39416
\(362\) 19.4891i 1.02433i
\(363\) 2.00000i 0.104973i
\(364\) −6.74456 −0.353511
\(365\) 0 0
\(366\) 2.51087 0.131246
\(367\) 8.74456i 0.456462i 0.973607 + 0.228231i \(0.0732942\pi\)
−0.973607 + 0.228231i \(0.926706\pi\)
\(368\) − 6.74456i − 0.351585i
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −12.7446 −0.661665
\(372\) − 9.48913i − 0.491988i
\(373\) − 16.9783i − 0.879100i −0.898218 0.439550i \(-0.855138\pi\)
0.898218 0.439550i \(-0.144862\pi\)
\(374\) −6.74456 −0.348753
\(375\) 0 0
\(376\) 4.74456 0.244682
\(377\) − 58.9783i − 3.03753i
\(378\) − 4.00000i − 0.205738i
\(379\) 37.4891 1.92569 0.962844 0.270060i \(-0.0870435\pi\)
0.962844 + 0.270060i \(0.0870435\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.0000i 0.818631i
\(383\) 19.7228i 1.00779i 0.863765 + 0.503894i \(0.168100\pi\)
−0.863765 + 0.503894i \(0.831900\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) − 4.00000i − 0.203331i
\(388\) − 16.7446i − 0.850076i
\(389\) 7.48913 0.379714 0.189857 0.981812i \(-0.439198\pi\)
0.189857 + 0.981812i \(0.439198\pi\)
\(390\) 0 0
\(391\) 45.4891 2.30048
\(392\) − 1.00000i − 0.0505076i
\(393\) − 5.48913i − 0.276890i
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) − 35.4891i − 1.78115i −0.454838 0.890574i \(-0.650303\pi\)
0.454838 0.890574i \(-0.349697\pi\)
\(398\) 3.25544i 0.163180i
\(399\) 13.4891 0.675301
\(400\) 0 0
\(401\) 23.4891 1.17299 0.586495 0.809953i \(-0.300507\pi\)
0.586495 + 0.809953i \(0.300507\pi\)
\(402\) 8.00000i 0.399004i
\(403\) 32.0000i 1.59403i
\(404\) −2.74456 −0.136547
\(405\) 0 0
\(406\) 8.74456 0.433985
\(407\) 0.744563i 0.0369066i
\(408\) − 13.4891i − 0.667811i
\(409\) 21.4891 1.06257 0.531284 0.847194i \(-0.321710\pi\)
0.531284 + 0.847194i \(0.321710\pi\)
\(410\) 0 0
\(411\) 6.97825 0.344212
\(412\) 0.744563i 0.0366820i
\(413\) − 8.74456i − 0.430292i
\(414\) 6.74456 0.331477
\(415\) 0 0
\(416\) 6.74456 0.330679
\(417\) − 29.4891i − 1.44409i
\(418\) 6.74456i 0.329887i
\(419\) 0.744563 0.0363743 0.0181871 0.999835i \(-0.494211\pi\)
0.0181871 + 0.999835i \(0.494211\pi\)
\(420\) 0 0
\(421\) 4.51087 0.219847 0.109923 0.993940i \(-0.464939\pi\)
0.109923 + 0.993940i \(0.464939\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 4.74456i 0.230689i
\(424\) 12.7446 0.618931
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) − 1.25544i − 0.0607549i
\(428\) − 12.0000i − 0.580042i
\(429\) 13.4891 0.651261
\(430\) 0 0
\(431\) −17.2554 −0.831165 −0.415583 0.909555i \(-0.636422\pi\)
−0.415583 + 0.909555i \(0.636422\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 36.7446i 1.76583i 0.469532 + 0.882915i \(0.344423\pi\)
−0.469532 + 0.882915i \(0.655577\pi\)
\(434\) −4.74456 −0.227746
\(435\) 0 0
\(436\) −18.2337 −0.873235
\(437\) − 45.4891i − 2.17604i
\(438\) 21.4891i 1.02679i
\(439\) −14.9783 −0.714873 −0.357436 0.933937i \(-0.616349\pi\)
−0.357436 + 0.933937i \(0.616349\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 45.4891i 2.16370i
\(443\) − 29.4891i − 1.40107i −0.713618 0.700535i \(-0.752945\pi\)
0.713618 0.700535i \(-0.247055\pi\)
\(444\) −1.48913 −0.0706708
\(445\) 0 0
\(446\) 15.2554 0.722366
\(447\) 1.48913i 0.0704332i
\(448\) 1.00000i 0.0472456i
\(449\) 4.97825 0.234938 0.117469 0.993077i \(-0.462522\pi\)
0.117469 + 0.993077i \(0.462522\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) − 10.0000i − 0.470360i
\(453\) − 18.5109i − 0.869717i
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −13.4891 −0.631686
\(457\) − 3.48913i − 0.163214i −0.996665 0.0816072i \(-0.973995\pi\)
0.996665 0.0816072i \(-0.0260053\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) −26.9783 −1.25924
\(460\) 0 0
\(461\) 33.7228 1.57063 0.785314 0.619098i \(-0.212502\pi\)
0.785314 + 0.619098i \(0.212502\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) 1.25544i 0.0583451i 0.999574 + 0.0291726i \(0.00928723\pi\)
−0.999574 + 0.0291726i \(0.990713\pi\)
\(464\) −8.74456 −0.405956
\(465\) 0 0
\(466\) −24.9783 −1.15710
\(467\) − 16.9783i − 0.785660i −0.919611 0.392830i \(-0.871496\pi\)
0.919611 0.392830i \(-0.128504\pi\)
\(468\) 6.74456i 0.311768i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 1.02175 0.0470797
\(472\) 8.74456i 0.402501i
\(473\) − 4.00000i − 0.183920i
\(474\) −13.4891 −0.619576
\(475\) 0 0
\(476\) −6.74456 −0.309137
\(477\) 12.7446i 0.583533i
\(478\) − 14.7446i − 0.674401i
\(479\) 41.4891 1.89569 0.947843 0.318737i \(-0.103259\pi\)
0.947843 + 0.318737i \(0.103259\pi\)
\(480\) 0 0
\(481\) 5.02175 0.228972
\(482\) − 20.0000i − 0.910975i
\(483\) − 13.4891i − 0.613776i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) − 9.25544i − 0.419404i −0.977765 0.209702i \(-0.932751\pi\)
0.977765 0.209702i \(-0.0672494\pi\)
\(488\) 1.25544i 0.0568310i
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 30.9783 1.39803 0.699014 0.715108i \(-0.253622\pi\)
0.699014 + 0.715108i \(0.253622\pi\)
\(492\) 8.00000i 0.360668i
\(493\) − 58.9783i − 2.65625i
\(494\) 45.4891 2.04665
\(495\) 0 0
\(496\) 4.74456 0.213037
\(497\) 4.00000i 0.179425i
\(498\) 16.0000i 0.716977i
\(499\) 13.4891 0.603856 0.301928 0.953331i \(-0.402370\pi\)
0.301928 + 0.953331i \(0.402370\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 26.2337i 1.17087i
\(503\) 6.51087i 0.290306i 0.989409 + 0.145153i \(0.0463674\pi\)
−0.989409 + 0.145153i \(0.953633\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 6.74456 0.299832
\(507\) − 64.9783i − 2.88579i
\(508\) 0 0
\(509\) 35.4891 1.57303 0.786514 0.617573i \(-0.211884\pi\)
0.786514 + 0.617573i \(0.211884\pi\)
\(510\) 0 0
\(511\) 10.7446 0.475311
\(512\) − 1.00000i − 0.0441942i
\(513\) 26.9783i 1.19112i
\(514\) 10.2337 0.451389
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 4.74456i 0.208666i
\(518\) 0.744563i 0.0327142i
\(519\) −40.4674 −1.77632
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) − 8.74456i − 0.382739i
\(523\) − 17.4891i − 0.764746i −0.924008 0.382373i \(-0.875107\pi\)
0.924008 0.382373i \(-0.124893\pi\)
\(524\) 2.74456 0.119897
\(525\) 0 0
\(526\) 26.9783 1.17631
\(527\) 32.0000i 1.39394i
\(528\) − 2.00000i − 0.0870388i
\(529\) −22.4891 −0.977788
\(530\) 0 0
\(531\) −8.74456 −0.379482
\(532\) 6.74456i 0.292414i
\(533\) − 26.9783i − 1.16856i
\(534\) −30.9783 −1.34056
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 8.00000i 0.345225i
\(538\) 20.9783i 0.904437i
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 1.76631 0.0759397 0.0379698 0.999279i \(-0.487911\pi\)
0.0379698 + 0.999279i \(0.487911\pi\)
\(542\) − 14.9783i − 0.643371i
\(543\) − 38.9783i − 1.67272i
\(544\) 6.74456 0.289171
\(545\) 0 0
\(546\) 13.4891 0.577281
\(547\) − 14.9783i − 0.640424i −0.947346 0.320212i \(-0.896246\pi\)
0.947346 0.320212i \(-0.103754\pi\)
\(548\) 3.48913i 0.149048i
\(549\) −1.25544 −0.0535808
\(550\) 0 0
\(551\) −58.9783 −2.51256
\(552\) 13.4891i 0.574135i
\(553\) 6.74456i 0.286808i
\(554\) −15.4891 −0.658070
\(555\) 0 0
\(556\) 14.7446 0.625309
\(557\) 0.978251i 0.0414498i 0.999785 + 0.0207249i \(0.00659741\pi\)
−0.999785 + 0.0207249i \(0.993403\pi\)
\(558\) 4.74456i 0.200853i
\(559\) −26.9783 −1.14106
\(560\) 0 0
\(561\) 13.4891 0.569511
\(562\) − 14.0000i − 0.590554i
\(563\) 5.48913i 0.231339i 0.993288 + 0.115670i \(0.0369014\pi\)
−0.993288 + 0.115670i \(0.963099\pi\)
\(564\) −9.48913 −0.399564
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) 11.0000i 0.461957i
\(568\) − 4.00000i − 0.167836i
\(569\) −16.5109 −0.692172 −0.346086 0.938203i \(-0.612489\pi\)
−0.346086 + 0.938203i \(0.612489\pi\)
\(570\) 0 0
\(571\) −17.4891 −0.731897 −0.365949 0.930635i \(-0.619255\pi\)
−0.365949 + 0.930635i \(0.619255\pi\)
\(572\) 6.74456i 0.282004i
\(573\) − 32.0000i − 1.33682i
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 8.74456i 0.364041i 0.983295 + 0.182020i \(0.0582637\pi\)
−0.983295 + 0.182020i \(0.941736\pi\)
\(578\) 28.4891i 1.18499i
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 33.4891i 1.38817i
\(583\) 12.7446i 0.527826i
\(584\) −10.7446 −0.444613
\(585\) 0 0
\(586\) 13.2554 0.547577
\(587\) 4.97825i 0.205474i 0.994709 + 0.102737i \(0.0327601\pi\)
−0.994709 + 0.102737i \(0.967240\pi\)
\(588\) 2.00000i 0.0824786i
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 20.0000 0.822690
\(592\) − 0.744563i − 0.0306013i
\(593\) − 40.2337i − 1.65220i −0.563524 0.826100i \(-0.690555\pi\)
0.563524 0.826100i \(-0.309445\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −0.744563 −0.0304985
\(597\) − 6.51087i − 0.266472i
\(598\) − 45.4891i − 1.86019i
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −33.4891 −1.36605 −0.683025 0.730395i \(-0.739336\pi\)
−0.683025 + 0.730395i \(0.739336\pi\)
\(602\) − 4.00000i − 0.163028i
\(603\) − 4.00000i − 0.162893i
\(604\) 9.25544 0.376598
\(605\) 0 0
\(606\) 5.48913 0.222980
\(607\) − 8.00000i − 0.324710i −0.986732 0.162355i \(-0.948091\pi\)
0.986732 0.162355i \(-0.0519090\pi\)
\(608\) − 6.74456i − 0.273528i
\(609\) −17.4891 −0.708695
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 6.74456i 0.272633i
\(613\) 11.4891i 0.464041i 0.972711 + 0.232021i \(0.0745337\pi\)
−0.972711 + 0.232021i \(0.925466\pi\)
\(614\) 1.48913 0.0600962
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 10.0000i 0.402585i 0.979531 + 0.201292i \(0.0645141\pi\)
−0.979531 + 0.201292i \(0.935486\pi\)
\(618\) − 1.48913i − 0.0599014i
\(619\) −0.744563 −0.0299265 −0.0149632 0.999888i \(-0.504763\pi\)
−0.0149632 + 0.999888i \(0.504763\pi\)
\(620\) 0 0
\(621\) 26.9783 1.08260
\(622\) 20.7446i 0.831781i
\(623\) 15.4891i 0.620559i
\(624\) −13.4891 −0.539997
\(625\) 0 0
\(626\) −2.23369 −0.0892761
\(627\) − 13.4891i − 0.538704i
\(628\) 0.510875i 0.0203861i
\(629\) 5.02175 0.200230
\(630\) 0 0
\(631\) −2.97825 −0.118562 −0.0592811 0.998241i \(-0.518881\pi\)
−0.0592811 + 0.998241i \(0.518881\pi\)
\(632\) − 6.74456i − 0.268284i
\(633\) − 24.0000i − 0.953914i
\(634\) −2.23369 −0.0887111
\(635\) 0 0
\(636\) −25.4891 −1.01071
\(637\) − 6.74456i − 0.267229i
\(638\) − 8.74456i − 0.346201i
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 11.4891 0.453793 0.226897 0.973919i \(-0.427142\pi\)
0.226897 + 0.973919i \(0.427142\pi\)
\(642\) 24.0000i 0.947204i
\(643\) 46.4674i 1.83249i 0.400613 + 0.916247i \(0.368797\pi\)
−0.400613 + 0.916247i \(0.631203\pi\)
\(644\) 6.74456 0.265773
\(645\) 0 0
\(646\) 45.4891 1.78975
\(647\) − 8.74456i − 0.343784i −0.985116 0.171892i \(-0.945012\pi\)
0.985116 0.171892i \(-0.0549880\pi\)
\(648\) − 11.0000i − 0.432121i
\(649\) −8.74456 −0.343254
\(650\) 0 0
\(651\) 9.48913 0.371908
\(652\) 4.00000i 0.156652i
\(653\) 15.7228i 0.615281i 0.951503 + 0.307641i \(0.0995394\pi\)
−0.951503 + 0.307641i \(0.900461\pi\)
\(654\) 36.4674 1.42599
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) − 10.7446i − 0.419185i
\(658\) 4.74456i 0.184962i
\(659\) 41.4891 1.61619 0.808093 0.589054i \(-0.200500\pi\)
0.808093 + 0.589054i \(0.200500\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) − 14.9783i − 0.582146i
\(663\) − 90.9783i − 3.53330i
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 0.744563 0.0288512
\(667\) 58.9783i 2.28365i
\(668\) 0 0
\(669\) −30.5109 −1.17962
\(670\) 0 0
\(671\) −1.25544 −0.0484656
\(672\) − 2.00000i − 0.0771517i
\(673\) 20.9783i 0.808652i 0.914615 + 0.404326i \(0.132494\pi\)
−0.914615 + 0.404326i \(0.867506\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 32.4891 1.24958
\(677\) − 36.2337i − 1.39257i −0.717764 0.696287i \(-0.754834\pi\)
0.717764 0.696287i \(-0.245166\pi\)
\(678\) 20.0000i 0.768095i
\(679\) 16.7446 0.642597
\(680\) 0 0
\(681\) 40.0000 1.53280
\(682\) 4.74456i 0.181679i
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 6.74456 0.257885
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 12.0000i 0.457829i
\(688\) 4.00000i 0.152499i
\(689\) 85.9565 3.27468
\(690\) 0 0
\(691\) 1.76631 0.0671937 0.0335968 0.999435i \(-0.489304\pi\)
0.0335968 + 0.999435i \(0.489304\pi\)
\(692\) − 20.2337i − 0.769169i
\(693\) − 1.00000i − 0.0379869i
\(694\) 22.9783 0.872242
\(695\) 0 0
\(696\) 17.4891 0.662924
\(697\) − 26.9783i − 1.02187i
\(698\) 30.7446i 1.16370i
\(699\) 49.9565 1.88953
\(700\) 0 0
\(701\) 22.2337 0.839755 0.419877 0.907581i \(-0.362073\pi\)
0.419877 + 0.907581i \(0.362073\pi\)
\(702\) 26.9783i 1.01823i
\(703\) − 5.02175i − 0.189399i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 8.74456 0.329106
\(707\) − 2.74456i − 0.103220i
\(708\) − 17.4891i − 0.657282i
\(709\) −0.510875 −0.0191863 −0.00959315 0.999954i \(-0.503054\pi\)
−0.00959315 + 0.999954i \(0.503054\pi\)
\(710\) 0 0
\(711\) 6.74456 0.252941
\(712\) − 15.4891i − 0.580480i
\(713\) − 32.0000i − 1.19841i
\(714\) 13.4891 0.504818
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 29.4891i 1.10129i
\(718\) − 1.25544i − 0.0468525i
\(719\) −7.72281 −0.288012 −0.144006 0.989577i \(-0.545999\pi\)
−0.144006 + 0.989577i \(0.545999\pi\)
\(720\) 0 0
\(721\) −0.744563 −0.0277290
\(722\) − 26.4891i − 0.985823i
\(723\) 40.0000i 1.48762i
\(724\) 19.4891 0.724308
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) − 14.2337i − 0.527898i −0.964537 0.263949i \(-0.914975\pi\)
0.964537 0.263949i \(-0.0850251\pi\)
\(728\) 6.74456i 0.249970i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −26.9783 −0.997827
\(732\) − 2.51087i − 0.0928046i
\(733\) − 16.2337i − 0.599605i −0.954001 0.299802i \(-0.903079\pi\)
0.954001 0.299802i \(-0.0969208\pi\)
\(734\) 8.74456 0.322768
\(735\) 0 0
\(736\) −6.74456 −0.248608
\(737\) − 4.00000i − 0.147342i
\(738\) − 4.00000i − 0.147242i
\(739\) 30.9783 1.13955 0.569777 0.821800i \(-0.307030\pi\)
0.569777 + 0.821800i \(0.307030\pi\)
\(740\) 0 0
\(741\) −90.9783 −3.34217
\(742\) 12.7446i 0.467868i
\(743\) 26.9783i 0.989736i 0.868968 + 0.494868i \(0.164784\pi\)
−0.868968 + 0.494868i \(0.835216\pi\)
\(744\) −9.48913 −0.347888
\(745\) 0 0
\(746\) −16.9783 −0.621618
\(747\) − 8.00000i − 0.292705i
\(748\) 6.74456i 0.246606i
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) − 4.74456i − 0.173016i
\(753\) − 52.4674i − 1.91202i
\(754\) −58.9783 −2.14786
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 19.2554i 0.699851i 0.936778 + 0.349925i \(0.113793\pi\)
−0.936778 + 0.349925i \(0.886207\pi\)
\(758\) − 37.4891i − 1.36167i
\(759\) −13.4891 −0.489624
\(760\) 0 0
\(761\) −29.4891 −1.06898 −0.534490 0.845175i \(-0.679496\pi\)
−0.534490 + 0.845175i \(0.679496\pi\)
\(762\) 0 0
\(763\) − 18.2337i − 0.660104i
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 19.7228 0.712614
\(767\) 58.9783i 2.12958i
\(768\) 2.00000i 0.0721688i
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 0 0
\(771\) −20.4674 −0.737115
\(772\) 2.00000i 0.0719816i
\(773\) − 51.4891i − 1.85194i −0.377603 0.925968i \(-0.623252\pi\)
0.377603 0.925968i \(-0.376748\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −16.7446 −0.601095
\(777\) − 1.48913i − 0.0534221i
\(778\) − 7.48913i − 0.268498i
\(779\) −26.9783 −0.966596
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) − 45.4891i − 1.62669i
\(783\) − 34.9783i − 1.25002i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −5.48913 −0.195791
\(787\) − 5.48913i − 0.195666i −0.995203 0.0978331i \(-0.968809\pi\)
0.995203 0.0978331i \(-0.0311911\pi\)
\(788\) 10.0000i 0.356235i
\(789\) −53.9565 −1.92090
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 1.00000i 0.0355335i
\(793\) 8.46738i 0.300685i
\(794\) −35.4891 −1.25946
\(795\) 0 0
\(796\) 3.25544 0.115386
\(797\) 42.4674i 1.50427i 0.659008 + 0.752136i \(0.270976\pi\)
−0.659008 + 0.752136i \(0.729024\pi\)
\(798\) − 13.4891i − 0.477510i
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) 15.4891 0.547281
\(802\) − 23.4891i − 0.829430i
\(803\) − 10.7446i − 0.379167i
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 32.0000 1.12715
\(807\) − 41.9565i − 1.47694i
\(808\) 2.74456i 0.0965534i
\(809\) −34.4674 −1.21181 −0.605904 0.795538i \(-0.707189\pi\)
−0.605904 + 0.795538i \(0.707189\pi\)
\(810\) 0 0
\(811\) −18.7446 −0.658211 −0.329105 0.944293i \(-0.606747\pi\)
−0.329105 + 0.944293i \(0.606747\pi\)
\(812\) − 8.74456i − 0.306874i
\(813\) 29.9565i 1.05062i
\(814\) 0.744563 0.0260969
\(815\) 0 0
\(816\) −13.4891 −0.472214
\(817\) 26.9783i 0.943850i
\(818\) − 21.4891i − 0.751350i
\(819\) −6.74456 −0.235674
\(820\) 0 0
\(821\) 7.72281 0.269528 0.134764 0.990878i \(-0.456972\pi\)
0.134764 + 0.990878i \(0.456972\pi\)
\(822\) − 6.97825i − 0.243394i
\(823\) 7.76631i 0.270717i 0.990797 + 0.135358i \(0.0432186\pi\)
−0.990797 + 0.135358i \(0.956781\pi\)
\(824\) 0.744563 0.0259381
\(825\) 0 0
\(826\) −8.74456 −0.304262
\(827\) − 4.00000i − 0.139094i −0.997579 0.0695468i \(-0.977845\pi\)
0.997579 0.0695468i \(-0.0221553\pi\)
\(828\) − 6.74456i − 0.234390i
\(829\) −14.4674 −0.502473 −0.251236 0.967926i \(-0.580837\pi\)
−0.251236 + 0.967926i \(0.580837\pi\)
\(830\) 0 0
\(831\) 30.9783 1.07462
\(832\) − 6.74456i − 0.233826i
\(833\) − 6.74456i − 0.233685i
\(834\) −29.4891 −1.02112
\(835\) 0 0
\(836\) 6.74456 0.233266
\(837\) 18.9783i 0.655984i
\(838\) − 0.744563i − 0.0257205i
\(839\) 3.25544 0.112390 0.0561951 0.998420i \(-0.482103\pi\)
0.0561951 + 0.998420i \(0.482103\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) − 4.51087i − 0.155455i
\(843\) 28.0000i 0.964371i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 4.74456 0.163121
\(847\) − 1.00000i − 0.0343604i
\(848\) − 12.7446i − 0.437650i
\(849\) −56.0000 −1.92192
\(850\) 0 0
\(851\) −5.02175 −0.172143
\(852\) 8.00000i 0.274075i
\(853\) − 22.7446i − 0.778759i −0.921077 0.389379i \(-0.872689\pi\)
0.921077 0.389379i \(-0.127311\pi\)
\(854\) −1.25544 −0.0429602
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 43.2119i 1.47609i 0.674751 + 0.738046i \(0.264251\pi\)
−0.674751 + 0.738046i \(0.735749\pi\)
\(858\) − 13.4891i − 0.460511i
\(859\) 29.2119 0.996698 0.498349 0.866976i \(-0.333940\pi\)
0.498349 + 0.866976i \(0.333940\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 17.2554i 0.587723i
\(863\) − 40.2337i − 1.36957i −0.728745 0.684785i \(-0.759896\pi\)
0.728745 0.684785i \(-0.240104\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 36.7446 1.24863
\(867\) − 56.9783i − 1.93508i
\(868\) 4.74456i 0.161041i
\(869\) 6.74456 0.228794
\(870\) 0 0
\(871\) −26.9783 −0.914123
\(872\) 18.2337i 0.617471i
\(873\) − 16.7446i − 0.566718i
\(874\) −45.4891 −1.53869
\(875\) 0 0
\(876\) 21.4891 0.726050
\(877\) 26.4674i 0.893740i 0.894599 + 0.446870i \(0.147461\pi\)
−0.894599 + 0.446870i \(0.852539\pi\)
\(878\) 14.9783i 0.505491i
\(879\) −26.5109 −0.894190
\(880\) 0 0
\(881\) −55.4891 −1.86948 −0.934738 0.355338i \(-0.884366\pi\)
−0.934738 + 0.355338i \(0.884366\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) − 8.00000i − 0.269221i −0.990899 0.134611i \(-0.957022\pi\)
0.990899 0.134611i \(-0.0429784\pi\)
\(884\) 45.4891 1.52996
\(885\) 0 0
\(886\) −29.4891 −0.990707
\(887\) 41.4891i 1.39307i 0.717524 + 0.696534i \(0.245276\pi\)
−0.717524 + 0.696534i \(0.754724\pi\)
\(888\) 1.48913i 0.0499718i
\(889\) 0 0
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) − 15.2554i − 0.510790i
\(893\) − 32.0000i − 1.07084i
\(894\) 1.48913 0.0498038
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 90.9783i 3.03768i
\(898\) − 4.97825i − 0.166126i
\(899\) −41.4891 −1.38374
\(900\) 0 0
\(901\) 85.9565 2.86363
\(902\) − 4.00000i − 0.133185i
\(903\) 8.00000i 0.266223i
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) −18.5109 −0.614983
\(907\) 14.5109i 0.481826i 0.970547 + 0.240913i \(0.0774468\pi\)
−0.970547 + 0.240913i \(0.922553\pi\)
\(908\) 20.0000i 0.663723i
\(909\) −2.74456 −0.0910314
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 13.4891i 0.446670i
\(913\) − 8.00000i − 0.264761i
\(914\) −3.48913 −0.115410
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 2.74456i 0.0906334i
\(918\) 26.9783i 0.890415i
\(919\) 55.2119 1.82127 0.910637 0.413207i \(-0.135592\pi\)
0.910637 + 0.413207i \(0.135592\pi\)
\(920\) 0 0
\(921\) −2.97825 −0.0981367
\(922\) − 33.7228i − 1.11060i
\(923\) − 26.9783i − 0.888000i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 1.25544 0.0412562
\(927\) 0.744563i 0.0244546i
\(928\) 8.74456i 0.287054i
\(929\) 28.9783 0.950746 0.475373 0.879784i \(-0.342313\pi\)
0.475373 + 0.879784i \(0.342313\pi\)
\(930\) 0 0
\(931\) −6.74456 −0.221044
\(932\) 24.9783i 0.818190i
\(933\) − 41.4891i − 1.35829i
\(934\) −16.9783 −0.555545
\(935\) 0 0
\(936\) 6.74456 0.220453
\(937\) 18.7446i 0.612358i 0.951974 + 0.306179i \(0.0990506\pi\)
−0.951974 + 0.306179i \(0.900949\pi\)
\(938\) − 4.00000i − 0.130605i
\(939\) 4.46738 0.145787
\(940\) 0 0
\(941\) 12.2337 0.398807 0.199403 0.979917i \(-0.436100\pi\)
0.199403 + 0.979917i \(0.436100\pi\)
\(942\) − 1.02175i − 0.0332904i
\(943\) 26.9783i 0.878533i
\(944\) 8.74456 0.284611
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) 13.4891i 0.438106i
\(949\) −72.4674 −2.35239
\(950\) 0 0
\(951\) 4.46738 0.144865
\(952\) 6.74456i 0.218593i
\(953\) 6.00000i 0.194359i 0.995267 + 0.0971795i \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) 12.7446 0.412620
\(955\) 0 0
\(956\) −14.7446 −0.476873
\(957\) 17.4891i 0.565343i
\(958\) − 41.4891i − 1.34045i
\(959\) −3.48913 −0.112670
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) − 5.02175i − 0.161908i
\(963\) − 12.0000i − 0.386695i
\(964\) −20.0000 −0.644157
\(965\) 0 0
\(966\) −13.4891 −0.434005
\(967\) 50.9783i 1.63935i 0.572829 + 0.819675i \(0.305846\pi\)
−0.572829 + 0.819675i \(0.694154\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) −90.9783 −2.92264
\(970\) 0 0
\(971\) 19.7228 0.632935 0.316468 0.948603i \(-0.397503\pi\)
0.316468 + 0.948603i \(0.397503\pi\)
\(972\) 10.0000i 0.320750i
\(973\) 14.7446i 0.472689i
\(974\) −9.25544 −0.296563
\(975\) 0 0
\(976\) 1.25544 0.0401856
\(977\) − 12.9783i − 0.415211i −0.978213 0.207606i \(-0.933433\pi\)
0.978213 0.207606i \(-0.0665670\pi\)
\(978\) − 8.00000i − 0.255812i
\(979\) 15.4891 0.495035
\(980\) 0 0
\(981\) −18.2337 −0.582157
\(982\) − 30.9783i − 0.988556i
\(983\) 2.23369i 0.0712436i 0.999365 + 0.0356218i \(0.0113412\pi\)
−0.999365 + 0.0356218i \(0.988659\pi\)
\(984\) 8.00000 0.255031
\(985\) 0 0
\(986\) −58.9783 −1.87825
\(987\) − 9.48913i − 0.302042i
\(988\) − 45.4891i − 1.44720i
\(989\) 26.9783 0.857858
\(990\) 0 0
\(991\) 1.48913 0.0473036 0.0236518 0.999720i \(-0.492471\pi\)
0.0236518 + 0.999720i \(0.492471\pi\)
\(992\) − 4.74456i − 0.150640i
\(993\) 29.9565i 0.950641i
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) − 39.2119i − 1.24185i −0.783868 0.620927i \(-0.786756\pi\)
0.783868 0.620927i \(-0.213244\pi\)
\(998\) − 13.4891i − 0.426991i
\(999\) 2.97825 0.0942277
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 3850.2.c.y.1849.2 4
5.2 odd 4 770.2.a.k.1.2 2
5.3 odd 4 3850.2.a.bc.1.1 2
5.4 even 2 inner 3850.2.c.y.1849.3 4
15.2 even 4 6930.2.a.bo.1.2 2
20.7 even 4 6160.2.a.r.1.2 2
35.27 even 4 5390.2.a.bq.1.1 2
55.32 even 4 8470.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.2 2 5.2 odd 4
3850.2.a.bc.1.1 2 5.3 odd 4
3850.2.c.y.1849.2 4 1.1 even 1 trivial
3850.2.c.y.1849.3 4 5.4 even 2 inner
5390.2.a.bq.1.1 2 35.27 even 4
6160.2.a.r.1.2 2 20.7 even 4
6930.2.a.bo.1.2 2 15.2 even 4
8470.2.a.bu.1.1 2 55.32 even 4