Properties

Label 2850.2.d.w.799.4
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.4
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.w.799.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.44949i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.44949i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.44949 q^{11} +1.00000i q^{12} +2.44949i q^{13} -4.44949 q^{14} +1.00000 q^{16} +4.44949i q^{17} -1.00000i q^{18} -1.00000 q^{19} +4.44949 q^{21} +3.44949i q^{22} +1.00000i q^{23} -1.00000 q^{24} -2.44949 q^{26} +1.00000i q^{27} -4.44949i q^{28} +4.34847 q^{29} -3.00000 q^{31} +1.00000i q^{32} -3.44949i q^{33} -4.44949 q^{34} +1.00000 q^{36} -7.79796i q^{37} -1.00000i q^{38} +2.44949 q^{39} -0.898979 q^{41} +4.44949i q^{42} +2.44949i q^{43} -3.44949 q^{44} -1.00000 q^{46} -7.79796i q^{47} -1.00000i q^{48} -12.7980 q^{49} +4.44949 q^{51} -2.44949i q^{52} +7.44949i q^{53} -1.00000 q^{54} +4.44949 q^{56} +1.00000i q^{57} +4.34847i q^{58} -6.44949 q^{59} -9.44949 q^{61} -3.00000i q^{62} -4.44949i q^{63} -1.00000 q^{64} +3.44949 q^{66} +15.2474i q^{67} -4.44949i q^{68} +1.00000 q^{69} -1.55051 q^{71} +1.00000i q^{72} +1.00000i q^{73} +7.79796 q^{74} +1.00000 q^{76} +15.3485i q^{77} +2.44949i q^{78} +5.00000 q^{79} +1.00000 q^{81} -0.898979i q^{82} -8.34847i q^{83} -4.44949 q^{84} -2.44949 q^{86} -4.34847i q^{87} -3.44949i q^{88} -2.10102 q^{89} -10.8990 q^{91} -1.00000i q^{92} +3.00000i q^{93} +7.79796 q^{94} +1.00000 q^{96} +1.55051i q^{97} -12.7980i q^{98} -3.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{11} - 8 q^{14} + 4 q^{16} - 4 q^{19} + 8 q^{21} - 4 q^{24} - 12 q^{29} - 12 q^{31} - 8 q^{34} + 4 q^{36} + 16 q^{41} - 4 q^{44} - 4 q^{46} - 12 q^{49} + 8 q^{51} - 4 q^{54} + 8 q^{56} - 16 q^{59} - 28 q^{61} - 4 q^{64} + 4 q^{66} + 4 q^{69} - 16 q^{71} - 8 q^{74} + 4 q^{76} + 20 q^{79} + 4 q^{81} - 8 q^{84} - 28 q^{89} - 24 q^{91} - 8 q^{94} + 4 q^{96} - 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.44949i 1.68175i 0.541230 + 0.840875i \(0.317959\pi\)
−0.541230 + 0.840875i \(0.682041\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.44949 1.04006 0.520030 0.854148i \(-0.325921\pi\)
0.520030 + 0.854148i \(0.325921\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) −4.44949 −1.18918
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.44949i 1.07916i 0.841934 + 0.539580i \(0.181417\pi\)
−0.841934 + 0.539580i \(0.818583\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 4.44949 0.970958
\(22\) 3.44949i 0.735434i
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.44949 −0.480384
\(27\) 1.00000i 0.192450i
\(28\) − 4.44949i − 0.840875i
\(29\) 4.34847 0.807490 0.403745 0.914871i \(-0.367708\pi\)
0.403745 + 0.914871i \(0.367708\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.44949i − 0.600479i
\(34\) −4.44949 −0.763081
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 7.79796i − 1.28198i −0.767551 0.640988i \(-0.778525\pi\)
0.767551 0.640988i \(-0.221475\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 2.44949 0.392232
\(40\) 0 0
\(41\) −0.898979 −0.140397 −0.0701985 0.997533i \(-0.522363\pi\)
−0.0701985 + 0.997533i \(0.522363\pi\)
\(42\) 4.44949i 0.686571i
\(43\) 2.44949i 0.373544i 0.982403 + 0.186772i \(0.0598025\pi\)
−0.982403 + 0.186772i \(0.940197\pi\)
\(44\) −3.44949 −0.520030
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 7.79796i − 1.13745i −0.822528 0.568725i \(-0.807437\pi\)
0.822528 0.568725i \(-0.192563\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −12.7980 −1.82828
\(50\) 0 0
\(51\) 4.44949 0.623053
\(52\) − 2.44949i − 0.339683i
\(53\) 7.44949i 1.02327i 0.859204 + 0.511633i \(0.170959\pi\)
−0.859204 + 0.511633i \(0.829041\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.44949 0.594588
\(57\) 1.00000i 0.132453i
\(58\) 4.34847i 0.570982i
\(59\) −6.44949 −0.839652 −0.419826 0.907605i \(-0.637909\pi\)
−0.419826 + 0.907605i \(0.637909\pi\)
\(60\) 0 0
\(61\) −9.44949 −1.20988 −0.604942 0.796270i \(-0.706804\pi\)
−0.604942 + 0.796270i \(0.706804\pi\)
\(62\) − 3.00000i − 0.381000i
\(63\) − 4.44949i − 0.560583i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.44949 0.424603
\(67\) 15.2474i 1.86277i 0.364033 + 0.931386i \(0.381400\pi\)
−0.364033 + 0.931386i \(0.618600\pi\)
\(68\) − 4.44949i − 0.539580i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −1.55051 −0.184012 −0.0920059 0.995758i \(-0.529328\pi\)
−0.0920059 + 0.995758i \(0.529328\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 1.00000i 0.117041i 0.998286 + 0.0585206i \(0.0186383\pi\)
−0.998286 + 0.0585206i \(0.981362\pi\)
\(74\) 7.79796 0.906494
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 15.3485i 1.74912i
\(78\) 2.44949i 0.277350i
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 0.898979i − 0.0992757i
\(83\) − 8.34847i − 0.916364i −0.888859 0.458182i \(-0.848501\pi\)
0.888859 0.458182i \(-0.151499\pi\)
\(84\) −4.44949 −0.485479
\(85\) 0 0
\(86\) −2.44949 −0.264135
\(87\) − 4.34847i − 0.466205i
\(88\) − 3.44949i − 0.367717i
\(89\) −2.10102 −0.222708 −0.111354 0.993781i \(-0.535519\pi\)
−0.111354 + 0.993781i \(0.535519\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) − 1.00000i − 0.104257i
\(93\) 3.00000i 0.311086i
\(94\) 7.79796 0.804298
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 1.55051i 0.157430i 0.996897 + 0.0787152i \(0.0250818\pi\)
−0.996897 + 0.0787152i \(0.974918\pi\)
\(98\) − 12.7980i − 1.29279i
\(99\) −3.44949 −0.346687
\(100\) 0 0
\(101\) −1.55051 −0.154282 −0.0771408 0.997020i \(-0.524579\pi\)
−0.0771408 + 0.997020i \(0.524579\pi\)
\(102\) 4.44949i 0.440565i
\(103\) − 1.89898i − 0.187112i −0.995614 0.0935560i \(-0.970177\pi\)
0.995614 0.0935560i \(-0.0298234\pi\)
\(104\) 2.44949 0.240192
\(105\) 0 0
\(106\) −7.44949 −0.723558
\(107\) 17.3485i 1.67714i 0.544794 + 0.838570i \(0.316608\pi\)
−0.544794 + 0.838570i \(0.683392\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 2.89898 0.277672 0.138836 0.990315i \(-0.455664\pi\)
0.138836 + 0.990315i \(0.455664\pi\)
\(110\) 0 0
\(111\) −7.79796 −0.740150
\(112\) 4.44949i 0.420437i
\(113\) 16.7980i 1.58022i 0.612966 + 0.790110i \(0.289976\pi\)
−0.612966 + 0.790110i \(0.710024\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −4.34847 −0.403745
\(117\) − 2.44949i − 0.226455i
\(118\) − 6.44949i − 0.593724i
\(119\) −19.7980 −1.81488
\(120\) 0 0
\(121\) 0.898979 0.0817254
\(122\) − 9.44949i − 0.855517i
\(123\) 0.898979i 0.0810583i
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) 4.44949 0.396392
\(127\) 0.101021i 0.00896412i 0.999990 + 0.00448206i \(0.00142669\pi\)
−0.999990 + 0.00448206i \(0.998573\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 2.44949 0.215666
\(130\) 0 0
\(131\) 9.24745 0.807953 0.403977 0.914769i \(-0.367628\pi\)
0.403977 + 0.914769i \(0.367628\pi\)
\(132\) 3.44949i 0.300240i
\(133\) − 4.44949i − 0.385820i
\(134\) −15.2474 −1.31718
\(135\) 0 0
\(136\) 4.44949 0.381541
\(137\) − 4.89898i − 0.418548i −0.977857 0.209274i \(-0.932890\pi\)
0.977857 0.209274i \(-0.0671101\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) 22.2474 1.88700 0.943502 0.331367i \(-0.107510\pi\)
0.943502 + 0.331367i \(0.107510\pi\)
\(140\) 0 0
\(141\) −7.79796 −0.656707
\(142\) − 1.55051i − 0.130116i
\(143\) 8.44949i 0.706582i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) 12.7980i 1.05556i
\(148\) 7.79796i 0.640988i
\(149\) −5.79796 −0.474987 −0.237494 0.971389i \(-0.576326\pi\)
−0.237494 + 0.971389i \(0.576326\pi\)
\(150\) 0 0
\(151\) −23.7980 −1.93665 −0.968325 0.249692i \(-0.919671\pi\)
−0.968325 + 0.249692i \(0.919671\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 4.44949i − 0.359720i
\(154\) −15.3485 −1.23681
\(155\) 0 0
\(156\) −2.44949 −0.196116
\(157\) − 4.89898i − 0.390981i −0.980706 0.195491i \(-0.937370\pi\)
0.980706 0.195491i \(-0.0626299\pi\)
\(158\) 5.00000i 0.397779i
\(159\) 7.44949 0.590783
\(160\) 0 0
\(161\) −4.44949 −0.350669
\(162\) 1.00000i 0.0785674i
\(163\) − 19.7980i − 1.55070i −0.631534 0.775348i \(-0.717575\pi\)
0.631534 0.775348i \(-0.282425\pi\)
\(164\) 0.898979 0.0701985
\(165\) 0 0
\(166\) 8.34847 0.647967
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) − 4.44949i − 0.343286i
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 2.44949i − 0.186772i
\(173\) 1.65153i 0.125564i 0.998027 + 0.0627818i \(0.0199972\pi\)
−0.998027 + 0.0627818i \(0.980003\pi\)
\(174\) 4.34847 0.329657
\(175\) 0 0
\(176\) 3.44949 0.260015
\(177\) 6.44949i 0.484773i
\(178\) − 2.10102i − 0.157478i
\(179\) −25.7980 −1.92823 −0.964115 0.265485i \(-0.914468\pi\)
−0.964115 + 0.265485i \(0.914468\pi\)
\(180\) 0 0
\(181\) −14.4495 −1.07402 −0.537011 0.843575i \(-0.680447\pi\)
−0.537011 + 0.843575i \(0.680447\pi\)
\(182\) − 10.8990i − 0.807886i
\(183\) 9.44949i 0.698526i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −3.00000 −0.219971
\(187\) 15.3485i 1.12239i
\(188\) 7.79796i 0.568725i
\(189\) −4.44949 −0.323653
\(190\) 0 0
\(191\) −0.101021 −0.00730959 −0.00365479 0.999993i \(-0.501163\pi\)
−0.00365479 + 0.999993i \(0.501163\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 15.3485i 1.10481i 0.833577 + 0.552403i \(0.186289\pi\)
−0.833577 + 0.552403i \(0.813711\pi\)
\(194\) −1.55051 −0.111320
\(195\) 0 0
\(196\) 12.7980 0.914140
\(197\) 27.3485i 1.94850i 0.225475 + 0.974249i \(0.427606\pi\)
−0.225475 + 0.974249i \(0.572394\pi\)
\(198\) − 3.44949i − 0.245145i
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 15.2474 1.07547
\(202\) − 1.55051i − 0.109094i
\(203\) 19.3485i 1.35800i
\(204\) −4.44949 −0.311527
\(205\) 0 0
\(206\) 1.89898 0.132308
\(207\) − 1.00000i − 0.0695048i
\(208\) 2.44949i 0.169842i
\(209\) −3.44949 −0.238606
\(210\) 0 0
\(211\) −3.65153 −0.251382 −0.125691 0.992069i \(-0.540115\pi\)
−0.125691 + 0.992069i \(0.540115\pi\)
\(212\) − 7.44949i − 0.511633i
\(213\) 1.55051i 0.106239i
\(214\) −17.3485 −1.18592
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 13.3485i − 0.906153i
\(218\) 2.89898i 0.196344i
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) −10.8990 −0.733145
\(222\) − 7.79796i − 0.523365i
\(223\) − 16.1010i − 1.07820i −0.842240 0.539102i \(-0.818764\pi\)
0.842240 0.539102i \(-0.181236\pi\)
\(224\) −4.44949 −0.297294
\(225\) 0 0
\(226\) −16.7980 −1.11738
\(227\) 29.5959i 1.96435i 0.187969 + 0.982175i \(0.439810\pi\)
−0.187969 + 0.982175i \(0.560190\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) −7.24745 −0.478925 −0.239462 0.970906i \(-0.576971\pi\)
−0.239462 + 0.970906i \(0.576971\pi\)
\(230\) 0 0
\(231\) 15.3485 1.00986
\(232\) − 4.34847i − 0.285491i
\(233\) − 6.24745i − 0.409284i −0.978837 0.204642i \(-0.934397\pi\)
0.978837 0.204642i \(-0.0656030\pi\)
\(234\) 2.44949 0.160128
\(235\) 0 0
\(236\) 6.44949 0.419826
\(237\) − 5.00000i − 0.324785i
\(238\) − 19.7980i − 1.28331i
\(239\) 8.69694 0.562558 0.281279 0.959626i \(-0.409241\pi\)
0.281279 + 0.959626i \(0.409241\pi\)
\(240\) 0 0
\(241\) −4.44949 −0.286617 −0.143308 0.989678i \(-0.545774\pi\)
−0.143308 + 0.989678i \(0.545774\pi\)
\(242\) 0.898979i 0.0577886i
\(243\) − 1.00000i − 0.0641500i
\(244\) 9.44949 0.604942
\(245\) 0 0
\(246\) −0.898979 −0.0573168
\(247\) − 2.44949i − 0.155857i
\(248\) 3.00000i 0.190500i
\(249\) −8.34847 −0.529063
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 4.44949i 0.280292i
\(253\) 3.44949i 0.216868i
\(254\) −0.101021 −0.00633859
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 21.4949i − 1.34081i −0.741993 0.670407i \(-0.766119\pi\)
0.741993 0.670407i \(-0.233881\pi\)
\(258\) 2.44949i 0.152499i
\(259\) 34.6969 2.15596
\(260\) 0 0
\(261\) −4.34847 −0.269163
\(262\) 9.24745i 0.571309i
\(263\) 6.79796i 0.419180i 0.977789 + 0.209590i \(0.0672129\pi\)
−0.977789 + 0.209590i \(0.932787\pi\)
\(264\) −3.44949 −0.212301
\(265\) 0 0
\(266\) 4.44949 0.272816
\(267\) 2.10102i 0.128580i
\(268\) − 15.2474i − 0.931386i
\(269\) 2.89898 0.176754 0.0883769 0.996087i \(-0.471832\pi\)
0.0883769 + 0.996087i \(0.471832\pi\)
\(270\) 0 0
\(271\) 22.9444 1.39377 0.696886 0.717182i \(-0.254568\pi\)
0.696886 + 0.717182i \(0.254568\pi\)
\(272\) 4.44949i 0.269790i
\(273\) 10.8990i 0.659636i
\(274\) 4.89898 0.295958
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 21.0454i 1.26450i 0.774766 + 0.632248i \(0.217868\pi\)
−0.774766 + 0.632248i \(0.782132\pi\)
\(278\) 22.2474i 1.33431i
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) 7.00000 0.417585 0.208792 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) − 7.79796i − 0.464362i
\(283\) − 29.7980i − 1.77130i −0.464349 0.885652i \(-0.653712\pi\)
0.464349 0.885652i \(-0.346288\pi\)
\(284\) 1.55051 0.0920059
\(285\) 0 0
\(286\) −8.44949 −0.499629
\(287\) − 4.00000i − 0.236113i
\(288\) − 1.00000i − 0.0589256i
\(289\) −2.79796 −0.164586
\(290\) 0 0
\(291\) 1.55051 0.0908925
\(292\) − 1.00000i − 0.0585206i
\(293\) 23.2474i 1.35813i 0.734078 + 0.679065i \(0.237615\pi\)
−0.734078 + 0.679065i \(0.762385\pi\)
\(294\) −12.7980 −0.746392
\(295\) 0 0
\(296\) −7.79796 −0.453247
\(297\) 3.44949i 0.200160i
\(298\) − 5.79796i − 0.335867i
\(299\) −2.44949 −0.141658
\(300\) 0 0
\(301\) −10.8990 −0.628207
\(302\) − 23.7980i − 1.36942i
\(303\) 1.55051i 0.0890745i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 4.44949 0.254360
\(307\) − 16.3485i − 0.933056i −0.884506 0.466528i \(-0.845505\pi\)
0.884506 0.466528i \(-0.154495\pi\)
\(308\) − 15.3485i − 0.874560i
\(309\) −1.89898 −0.108029
\(310\) 0 0
\(311\) −23.7980 −1.34946 −0.674729 0.738065i \(-0.735740\pi\)
−0.674729 + 0.738065i \(0.735740\pi\)
\(312\) − 2.44949i − 0.138675i
\(313\) − 31.8990i − 1.80304i −0.432741 0.901518i \(-0.642453\pi\)
0.432741 0.901518i \(-0.357547\pi\)
\(314\) 4.89898 0.276465
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 15.2474i 0.856382i 0.903688 + 0.428191i \(0.140849\pi\)
−0.903688 + 0.428191i \(0.859151\pi\)
\(318\) 7.44949i 0.417747i
\(319\) 15.0000 0.839839
\(320\) 0 0
\(321\) 17.3485 0.968297
\(322\) − 4.44949i − 0.247960i
\(323\) − 4.44949i − 0.247576i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 19.7980 1.09651
\(327\) − 2.89898i − 0.160314i
\(328\) 0.898979i 0.0496378i
\(329\) 34.6969 1.91290
\(330\) 0 0
\(331\) −22.3485 −1.22838 −0.614191 0.789157i \(-0.710518\pi\)
−0.614191 + 0.789157i \(0.710518\pi\)
\(332\) 8.34847i 0.458182i
\(333\) 7.79796i 0.427326i
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 4.44949 0.242740
\(337\) − 24.8990i − 1.35633i −0.734908 0.678167i \(-0.762775\pi\)
0.734908 0.678167i \(-0.237225\pi\)
\(338\) 7.00000i 0.380750i
\(339\) 16.7980 0.912340
\(340\) 0 0
\(341\) −10.3485 −0.560401
\(342\) 1.00000i 0.0540738i
\(343\) − 25.7980i − 1.39296i
\(344\) 2.44949 0.132068
\(345\) 0 0
\(346\) −1.65153 −0.0887868
\(347\) − 23.5959i − 1.26670i −0.773867 0.633348i \(-0.781680\pi\)
0.773867 0.633348i \(-0.218320\pi\)
\(348\) 4.34847i 0.233102i
\(349\) 25.9444 1.38877 0.694386 0.719603i \(-0.255676\pi\)
0.694386 + 0.719603i \(0.255676\pi\)
\(350\) 0 0
\(351\) −2.44949 −0.130744
\(352\) 3.44949i 0.183858i
\(353\) 18.2474i 0.971214i 0.874177 + 0.485607i \(0.161401\pi\)
−0.874177 + 0.485607i \(0.838599\pi\)
\(354\) −6.44949 −0.342787
\(355\) 0 0
\(356\) 2.10102 0.111354
\(357\) 19.7980i 1.04782i
\(358\) − 25.7980i − 1.36346i
\(359\) 22.8990 1.20856 0.604281 0.796771i \(-0.293460\pi\)
0.604281 + 0.796771i \(0.293460\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 14.4495i − 0.759448i
\(363\) − 0.898979i − 0.0471842i
\(364\) 10.8990 0.571262
\(365\) 0 0
\(366\) −9.44949 −0.493933
\(367\) − 5.55051i − 0.289734i −0.989451 0.144867i \(-0.953725\pi\)
0.989451 0.144867i \(-0.0462755\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 0.898979 0.0467990
\(370\) 0 0
\(371\) −33.1464 −1.72088
\(372\) − 3.00000i − 0.155543i
\(373\) 22.4495i 1.16239i 0.813764 + 0.581195i \(0.197415\pi\)
−0.813764 + 0.581195i \(0.802585\pi\)
\(374\) −15.3485 −0.793650
\(375\) 0 0
\(376\) −7.79796 −0.402149
\(377\) 10.6515i 0.548582i
\(378\) − 4.44949i − 0.228857i
\(379\) 1.30306 0.0669338 0.0334669 0.999440i \(-0.489345\pi\)
0.0334669 + 0.999440i \(0.489345\pi\)
\(380\) 0 0
\(381\) 0.101021 0.00517544
\(382\) − 0.101021i − 0.00516866i
\(383\) − 16.2474i − 0.830206i −0.909774 0.415103i \(-0.863746\pi\)
0.909774 0.415103i \(-0.136254\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −15.3485 −0.781217
\(387\) − 2.44949i − 0.124515i
\(388\) − 1.55051i − 0.0787152i
\(389\) −21.5959 −1.09496 −0.547478 0.836820i \(-0.684412\pi\)
−0.547478 + 0.836820i \(0.684412\pi\)
\(390\) 0 0
\(391\) −4.44949 −0.225020
\(392\) 12.7980i 0.646395i
\(393\) − 9.24745i − 0.466472i
\(394\) −27.3485 −1.37780
\(395\) 0 0
\(396\) 3.44949 0.173343
\(397\) 23.9444i 1.20173i 0.799349 + 0.600867i \(0.205178\pi\)
−0.799349 + 0.600867i \(0.794822\pi\)
\(398\) 10.0000i 0.501255i
\(399\) −4.44949 −0.222753
\(400\) 0 0
\(401\) 11.2020 0.559403 0.279702 0.960087i \(-0.409764\pi\)
0.279702 + 0.960087i \(0.409764\pi\)
\(402\) 15.2474i 0.760474i
\(403\) − 7.34847i − 0.366053i
\(404\) 1.55051 0.0771408
\(405\) 0 0
\(406\) −19.3485 −0.960248
\(407\) − 26.8990i − 1.33333i
\(408\) − 4.44949i − 0.220283i
\(409\) 22.8990 1.13228 0.566141 0.824309i \(-0.308436\pi\)
0.566141 + 0.824309i \(0.308436\pi\)
\(410\) 0 0
\(411\) −4.89898 −0.241649
\(412\) 1.89898i 0.0935560i
\(413\) − 28.6969i − 1.41208i
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −2.44949 −0.120096
\(417\) − 22.2474i − 1.08946i
\(418\) − 3.44949i − 0.168720i
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 3.65153i − 0.177754i
\(423\) 7.79796i 0.379150i
\(424\) 7.44949 0.361779
\(425\) 0 0
\(426\) −1.55051 −0.0751225
\(427\) − 42.0454i − 2.03472i
\(428\) − 17.3485i − 0.838570i
\(429\) 8.44949 0.407945
\(430\) 0 0
\(431\) −5.10102 −0.245708 −0.122854 0.992425i \(-0.539205\pi\)
−0.122854 + 0.992425i \(0.539205\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 4.65153i − 0.223538i −0.993734 0.111769i \(-0.964348\pi\)
0.993734 0.111769i \(-0.0356517\pi\)
\(434\) 13.3485 0.640747
\(435\) 0 0
\(436\) −2.89898 −0.138836
\(437\) − 1.00000i − 0.0478365i
\(438\) 1.00000i 0.0477818i
\(439\) 12.1010 0.577550 0.288775 0.957397i \(-0.406752\pi\)
0.288775 + 0.957397i \(0.406752\pi\)
\(440\) 0 0
\(441\) 12.7980 0.609427
\(442\) − 10.8990i − 0.518412i
\(443\) − 21.2474i − 1.00950i −0.863267 0.504748i \(-0.831585\pi\)
0.863267 0.504748i \(-0.168415\pi\)
\(444\) 7.79796 0.370075
\(445\) 0 0
\(446\) 16.1010 0.762405
\(447\) 5.79796i 0.274234i
\(448\) − 4.44949i − 0.210219i
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −3.10102 −0.146021
\(452\) − 16.7980i − 0.790110i
\(453\) 23.7980i 1.11813i
\(454\) −29.5959 −1.38901
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) − 24.8990i − 1.16473i −0.812929 0.582363i \(-0.802128\pi\)
0.812929 0.582363i \(-0.197872\pi\)
\(458\) − 7.24745i − 0.338651i
\(459\) −4.44949 −0.207684
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 15.3485i 0.714075i
\(463\) 14.6969i 0.683025i 0.939877 + 0.341512i \(0.110939\pi\)
−0.939877 + 0.341512i \(0.889061\pi\)
\(464\) 4.34847 0.201873
\(465\) 0 0
\(466\) 6.24745 0.289407
\(467\) − 6.34847i − 0.293772i −0.989153 0.146886i \(-0.953075\pi\)
0.989153 0.146886i \(-0.0469251\pi\)
\(468\) 2.44949i 0.113228i
\(469\) −67.8434 −3.13272
\(470\) 0 0
\(471\) −4.89898 −0.225733
\(472\) 6.44949i 0.296862i
\(473\) 8.44949i 0.388508i
\(474\) 5.00000 0.229658
\(475\) 0 0
\(476\) 19.7980 0.907438
\(477\) − 7.44949i − 0.341089i
\(478\) 8.69694i 0.397789i
\(479\) −16.5959 −0.758287 −0.379143 0.925338i \(-0.623781\pi\)
−0.379143 + 0.925338i \(0.623781\pi\)
\(480\) 0 0
\(481\) 19.1010 0.870932
\(482\) − 4.44949i − 0.202669i
\(483\) 4.44949i 0.202459i
\(484\) −0.898979 −0.0408627
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 32.0000i − 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 9.44949i 0.427758i
\(489\) −19.7980 −0.895295
\(490\) 0 0
\(491\) 43.5959 1.96746 0.983728 0.179664i \(-0.0575009\pi\)
0.983728 + 0.179664i \(0.0575009\pi\)
\(492\) − 0.898979i − 0.0405291i
\(493\) 19.3485i 0.871411i
\(494\) 2.44949 0.110208
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) − 6.89898i − 0.309462i
\(498\) − 8.34847i − 0.374104i
\(499\) 23.8434 1.06738 0.533688 0.845682i \(-0.320806\pi\)
0.533688 + 0.845682i \(0.320806\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) − 18.0000i − 0.803379i
\(503\) − 29.7980i − 1.32863i −0.747455 0.664313i \(-0.768724\pi\)
0.747455 0.664313i \(-0.231276\pi\)
\(504\) −4.44949 −0.198196
\(505\) 0 0
\(506\) −3.44949 −0.153349
\(507\) − 7.00000i − 0.310881i
\(508\) − 0.101021i − 0.00448206i
\(509\) 30.1464 1.33622 0.668108 0.744064i \(-0.267104\pi\)
0.668108 + 0.744064i \(0.267104\pi\)
\(510\) 0 0
\(511\) −4.44949 −0.196834
\(512\) 1.00000i 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) 21.4949 0.948099
\(515\) 0 0
\(516\) −2.44949 −0.107833
\(517\) − 26.8990i − 1.18302i
\(518\) 34.6969i 1.52450i
\(519\) 1.65153 0.0724942
\(520\) 0 0
\(521\) 15.6969 0.687695 0.343848 0.939025i \(-0.388270\pi\)
0.343848 + 0.939025i \(0.388270\pi\)
\(522\) − 4.34847i − 0.190327i
\(523\) 33.3939i 1.46021i 0.683334 + 0.730106i \(0.260529\pi\)
−0.683334 + 0.730106i \(0.739471\pi\)
\(524\) −9.24745 −0.403977
\(525\) 0 0
\(526\) −6.79796 −0.296405
\(527\) − 13.3485i − 0.581468i
\(528\) − 3.44949i − 0.150120i
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 6.44949 0.279884
\(532\) 4.44949i 0.192910i
\(533\) − 2.20204i − 0.0953810i
\(534\) −2.10102 −0.0909200
\(535\) 0 0
\(536\) 15.2474 0.658589
\(537\) 25.7980i 1.11326i
\(538\) 2.89898i 0.124984i
\(539\) −44.1464 −1.90152
\(540\) 0 0
\(541\) 22.1464 0.952149 0.476075 0.879405i \(-0.342059\pi\)
0.476075 + 0.879405i \(0.342059\pi\)
\(542\) 22.9444i 0.985546i
\(543\) 14.4495i 0.620087i
\(544\) −4.44949 −0.190770
\(545\) 0 0
\(546\) −10.8990 −0.466433
\(547\) − 46.6413i − 1.99424i −0.0758461 0.997120i \(-0.524166\pi\)
0.0758461 0.997120i \(-0.475834\pi\)
\(548\) 4.89898i 0.209274i
\(549\) 9.44949 0.403294
\(550\) 0 0
\(551\) −4.34847 −0.185251
\(552\) − 1.00000i − 0.0425628i
\(553\) 22.2474i 0.946058i
\(554\) −21.0454 −0.894134
\(555\) 0 0
\(556\) −22.2474 −0.943502
\(557\) 27.3485i 1.15879i 0.815046 + 0.579396i \(0.196711\pi\)
−0.815046 + 0.579396i \(0.803289\pi\)
\(558\) 3.00000i 0.127000i
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 15.3485 0.648013
\(562\) 7.00000i 0.295277i
\(563\) − 6.24745i − 0.263299i −0.991296 0.131649i \(-0.957973\pi\)
0.991296 0.131649i \(-0.0420273\pi\)
\(564\) 7.79796 0.328353
\(565\) 0 0
\(566\) 29.7980 1.25250
\(567\) 4.44949i 0.186861i
\(568\) 1.55051i 0.0650580i
\(569\) 33.1918 1.39147 0.695737 0.718297i \(-0.255078\pi\)
0.695737 + 0.718297i \(0.255078\pi\)
\(570\) 0 0
\(571\) −10.2474 −0.428842 −0.214421 0.976741i \(-0.568787\pi\)
−0.214421 + 0.976741i \(0.568787\pi\)
\(572\) − 8.44949i − 0.353291i
\(573\) 0.101021i 0.00422019i
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 0.101021i 0.00420554i 0.999998 + 0.00210277i \(0.000669333\pi\)
−0.999998 + 0.00210277i \(0.999331\pi\)
\(578\) − 2.79796i − 0.116380i
\(579\) 15.3485 0.637861
\(580\) 0 0
\(581\) 37.1464 1.54109
\(582\) 1.55051i 0.0642707i
\(583\) 25.6969i 1.06426i
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) −23.2474 −0.960343
\(587\) − 33.4495i − 1.38061i −0.723519 0.690304i \(-0.757477\pi\)
0.723519 0.690304i \(-0.242523\pi\)
\(588\) − 12.7980i − 0.527779i
\(589\) 3.00000 0.123613
\(590\) 0 0
\(591\) 27.3485 1.12497
\(592\) − 7.79796i − 0.320494i
\(593\) − 8.49490i − 0.348844i −0.984671 0.174422i \(-0.944194\pi\)
0.984671 0.174422i \(-0.0558056\pi\)
\(594\) −3.44949 −0.141534
\(595\) 0 0
\(596\) 5.79796 0.237494
\(597\) − 10.0000i − 0.409273i
\(598\) − 2.44949i − 0.100167i
\(599\) −16.4495 −0.672108 −0.336054 0.941843i \(-0.609092\pi\)
−0.336054 + 0.941843i \(0.609092\pi\)
\(600\) 0 0
\(601\) 17.1464 0.699417 0.349709 0.936858i \(-0.386281\pi\)
0.349709 + 0.936858i \(0.386281\pi\)
\(602\) − 10.8990i − 0.444209i
\(603\) − 15.2474i − 0.620924i
\(604\) 23.7980 0.968325
\(605\) 0 0
\(606\) −1.55051 −0.0629852
\(607\) 46.1918i 1.87487i 0.348162 + 0.937434i \(0.386806\pi\)
−0.348162 + 0.937434i \(0.613194\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 19.3485 0.784040
\(610\) 0 0
\(611\) 19.1010 0.772745
\(612\) 4.44949i 0.179860i
\(613\) − 37.1918i − 1.50216i −0.660209 0.751082i \(-0.729532\pi\)
0.660209 0.751082i \(-0.270468\pi\)
\(614\) 16.3485 0.659771
\(615\) 0 0
\(616\) 15.3485 0.618407
\(617\) 28.2929i 1.13903i 0.821982 + 0.569514i \(0.192868\pi\)
−0.821982 + 0.569514i \(0.807132\pi\)
\(618\) − 1.89898i − 0.0763882i
\(619\) −45.1464 −1.81459 −0.907294 0.420497i \(-0.861856\pi\)
−0.907294 + 0.420497i \(0.861856\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 23.7980i − 0.954211i
\(623\) − 9.34847i − 0.374539i
\(624\) 2.44949 0.0980581
\(625\) 0 0
\(626\) 31.8990 1.27494
\(627\) 3.44949i 0.137759i
\(628\) 4.89898i 0.195491i
\(629\) 34.6969 1.38346
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) − 5.00000i − 0.198889i
\(633\) 3.65153i 0.145135i
\(634\) −15.2474 −0.605554
\(635\) 0 0
\(636\) −7.44949 −0.295391
\(637\) − 31.3485i − 1.24207i
\(638\) 15.0000i 0.593856i
\(639\) 1.55051 0.0613372
\(640\) 0 0
\(641\) 37.7980 1.49293 0.746465 0.665425i \(-0.231750\pi\)
0.746465 + 0.665425i \(0.231750\pi\)
\(642\) 17.3485i 0.684689i
\(643\) 0.202041i 0.00796772i 0.999992 + 0.00398386i \(0.00126811\pi\)
−0.999992 + 0.00398386i \(0.998732\pi\)
\(644\) 4.44949 0.175334
\(645\) 0 0
\(646\) 4.44949 0.175063
\(647\) 25.8990i 1.01819i 0.860709 + 0.509097i \(0.170021\pi\)
−0.860709 + 0.509097i \(0.829979\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −22.2474 −0.873289
\(650\) 0 0
\(651\) −13.3485 −0.523168
\(652\) 19.7980i 0.775348i
\(653\) 14.6969i 0.575136i 0.957760 + 0.287568i \(0.0928467\pi\)
−0.957760 + 0.287568i \(0.907153\pi\)
\(654\) 2.89898 0.113359
\(655\) 0 0
\(656\) −0.898979 −0.0350993
\(657\) − 1.00000i − 0.0390137i
\(658\) 34.6969i 1.35263i
\(659\) 18.0454 0.702949 0.351475 0.936197i \(-0.385680\pi\)
0.351475 + 0.936197i \(0.385680\pi\)
\(660\) 0 0
\(661\) 23.5959 0.917775 0.458887 0.888494i \(-0.348248\pi\)
0.458887 + 0.888494i \(0.348248\pi\)
\(662\) − 22.3485i − 0.868598i
\(663\) 10.8990i 0.423281i
\(664\) −8.34847 −0.323983
\(665\) 0 0
\(666\) −7.79796 −0.302165
\(667\) 4.34847i 0.168373i
\(668\) − 18.0000i − 0.696441i
\(669\) −16.1010 −0.622501
\(670\) 0 0
\(671\) −32.5959 −1.25835
\(672\) 4.44949i 0.171643i
\(673\) 45.3485i 1.74806i 0.485876 + 0.874028i \(0.338501\pi\)
−0.485876 + 0.874028i \(0.661499\pi\)
\(674\) 24.8990 0.959073
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) 22.3485i 0.858921i 0.903086 + 0.429461i \(0.141296\pi\)
−0.903086 + 0.429461i \(0.858704\pi\)
\(678\) 16.7980i 0.645122i
\(679\) −6.89898 −0.264759
\(680\) 0 0
\(681\) 29.5959 1.13412
\(682\) − 10.3485i − 0.396263i
\(683\) 35.6413i 1.36378i 0.731456 + 0.681889i \(0.238841\pi\)
−0.731456 + 0.681889i \(0.761159\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 25.7980 0.984971
\(687\) 7.24745i 0.276507i
\(688\) 2.44949i 0.0933859i
\(689\) −18.2474 −0.695172
\(690\) 0 0
\(691\) 9.75255 0.371005 0.185502 0.982644i \(-0.440609\pi\)
0.185502 + 0.982644i \(0.440609\pi\)
\(692\) − 1.65153i − 0.0627818i
\(693\) − 15.3485i − 0.583040i
\(694\) 23.5959 0.895689
\(695\) 0 0
\(696\) −4.34847 −0.164828
\(697\) − 4.00000i − 0.151511i
\(698\) 25.9444i 0.982010i
\(699\) −6.24745 −0.236300
\(700\) 0 0
\(701\) −32.4949 −1.22732 −0.613658 0.789572i \(-0.710302\pi\)
−0.613658 + 0.789572i \(0.710302\pi\)
\(702\) − 2.44949i − 0.0924500i
\(703\) 7.79796i 0.294106i
\(704\) −3.44949 −0.130008
\(705\) 0 0
\(706\) −18.2474 −0.686752
\(707\) − 6.89898i − 0.259463i
\(708\) − 6.44949i − 0.242387i
\(709\) 12.7526 0.478932 0.239466 0.970905i \(-0.423028\pi\)
0.239466 + 0.970905i \(0.423028\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 2.10102i 0.0787391i
\(713\) − 3.00000i − 0.112351i
\(714\) −19.7980 −0.740920
\(715\) 0 0
\(716\) 25.7980 0.964115
\(717\) − 8.69694i − 0.324793i
\(718\) 22.8990i 0.854582i
\(719\) 19.2020 0.716115 0.358058 0.933699i \(-0.383439\pi\)
0.358058 + 0.933699i \(0.383439\pi\)
\(720\) 0 0
\(721\) 8.44949 0.314675
\(722\) 1.00000i 0.0372161i
\(723\) 4.44949i 0.165478i
\(724\) 14.4495 0.537011
\(725\) 0 0
\(726\) 0.898979 0.0333643
\(727\) 22.4949i 0.834290i 0.908840 + 0.417145i \(0.136969\pi\)
−0.908840 + 0.417145i \(0.863031\pi\)
\(728\) 10.8990i 0.403943i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −10.8990 −0.403113
\(732\) − 9.44949i − 0.349263i
\(733\) − 4.14643i − 0.153152i −0.997064 0.0765759i \(-0.975601\pi\)
0.997064 0.0765759i \(-0.0243988\pi\)
\(734\) 5.55051 0.204873
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 52.5959i 1.93740i
\(738\) 0.898979i 0.0330919i
\(739\) −32.8990 −1.21021 −0.605104 0.796146i \(-0.706869\pi\)
−0.605104 + 0.796146i \(0.706869\pi\)
\(740\) 0 0
\(741\) −2.44949 −0.0899843
\(742\) − 33.1464i − 1.21684i
\(743\) 53.3939i 1.95883i 0.201854 + 0.979416i \(0.435303\pi\)
−0.201854 + 0.979416i \(0.564697\pi\)
\(744\) 3.00000 0.109985
\(745\) 0 0
\(746\) −22.4495 −0.821934
\(747\) 8.34847i 0.305455i
\(748\) − 15.3485i − 0.561196i
\(749\) −77.1918 −2.82053
\(750\) 0 0
\(751\) −23.7980 −0.868400 −0.434200 0.900817i \(-0.642969\pi\)
−0.434200 + 0.900817i \(0.642969\pi\)
\(752\) − 7.79796i − 0.284362i
\(753\) 18.0000i 0.655956i
\(754\) −10.6515 −0.387906
\(755\) 0 0
\(756\) 4.44949 0.161826
\(757\) 48.1464i 1.74991i 0.484203 + 0.874956i \(0.339110\pi\)
−0.484203 + 0.874956i \(0.660890\pi\)
\(758\) 1.30306i 0.0473293i
\(759\) 3.44949 0.125209
\(760\) 0 0
\(761\) 51.3485 1.86138 0.930690 0.365808i \(-0.119207\pi\)
0.930690 + 0.365808i \(0.119207\pi\)
\(762\) 0.101021i 0.00365959i
\(763\) 12.8990i 0.466974i
\(764\) 0.101021 0.00365479
\(765\) 0 0
\(766\) 16.2474 0.587044
\(767\) − 15.7980i − 0.570431i
\(768\) − 1.00000i − 0.0360844i
\(769\) 7.89898 0.284844 0.142422 0.989806i \(-0.454511\pi\)
0.142422 + 0.989806i \(0.454511\pi\)
\(770\) 0 0
\(771\) −21.4949 −0.774120
\(772\) − 15.3485i − 0.552403i
\(773\) − 18.4949i − 0.665215i −0.943065 0.332608i \(-0.892072\pi\)
0.943065 0.332608i \(-0.107928\pi\)
\(774\) 2.44949 0.0880451
\(775\) 0 0
\(776\) 1.55051 0.0556601
\(777\) − 34.6969i − 1.24475i
\(778\) − 21.5959i − 0.774251i
\(779\) 0.898979 0.0322093
\(780\) 0 0
\(781\) −5.34847 −0.191383
\(782\) − 4.44949i − 0.159113i
\(783\) 4.34847i 0.155402i
\(784\) −12.7980 −0.457070
\(785\) 0 0
\(786\) 9.24745 0.329846
\(787\) − 33.4495i − 1.19235i −0.802856 0.596173i \(-0.796687\pi\)
0.802856 0.596173i \(-0.203313\pi\)
\(788\) − 27.3485i − 0.974249i
\(789\) 6.79796 0.242014
\(790\) 0 0
\(791\) −74.7423 −2.65753
\(792\) 3.44949i 0.122572i
\(793\) − 23.1464i − 0.821954i
\(794\) −23.9444 −0.849755
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 3.50510i 0.124157i 0.998071 + 0.0620786i \(0.0197729\pi\)
−0.998071 + 0.0620786i \(0.980227\pi\)
\(798\) − 4.44949i − 0.157510i
\(799\) 34.6969 1.22749
\(800\) 0 0
\(801\) 2.10102 0.0742359
\(802\) 11.2020i 0.395558i
\(803\) 3.44949i 0.121730i
\(804\) −15.2474 −0.537736
\(805\) 0 0
\(806\) 7.34847 0.258839
\(807\) − 2.89898i − 0.102049i
\(808\) 1.55051i 0.0545468i
\(809\) 3.55051 0.124829 0.0624146 0.998050i \(-0.480120\pi\)
0.0624146 + 0.998050i \(0.480120\pi\)
\(810\) 0 0
\(811\) −19.4495 −0.682964 −0.341482 0.939888i \(-0.610929\pi\)
−0.341482 + 0.939888i \(0.610929\pi\)
\(812\) − 19.3485i − 0.678998i
\(813\) − 22.9444i − 0.804695i
\(814\) 26.8990 0.942809
\(815\) 0 0
\(816\) 4.44949 0.155763
\(817\) − 2.44949i − 0.0856968i
\(818\) 22.8990i 0.800644i
\(819\) 10.8990 0.380841
\(820\) 0 0
\(821\) −23.1464 −0.807816 −0.403908 0.914800i \(-0.632348\pi\)
−0.403908 + 0.914800i \(0.632348\pi\)
\(822\) − 4.89898i − 0.170872i
\(823\) 31.7980i 1.10841i 0.832381 + 0.554204i \(0.186977\pi\)
−0.832381 + 0.554204i \(0.813023\pi\)
\(824\) −1.89898 −0.0661541
\(825\) 0 0
\(826\) 28.6969 0.998494
\(827\) 0.247449i 0.00860463i 0.999991 + 0.00430232i \(0.00136947\pi\)
−0.999991 + 0.00430232i \(0.998631\pi\)
\(828\) 1.00000i 0.0347524i
\(829\) 39.6413 1.37680 0.688400 0.725331i \(-0.258313\pi\)
0.688400 + 0.725331i \(0.258313\pi\)
\(830\) 0 0
\(831\) 21.0454 0.730057
\(832\) − 2.44949i − 0.0849208i
\(833\) − 56.9444i − 1.97301i
\(834\) 22.2474 0.770366
\(835\) 0 0
\(836\) 3.44949 0.119303
\(837\) − 3.00000i − 0.103695i
\(838\) 0 0
\(839\) −10.6515 −0.367732 −0.183866 0.982951i \(-0.558861\pi\)
−0.183866 + 0.982951i \(0.558861\pi\)
\(840\) 0 0
\(841\) −10.0908 −0.347959
\(842\) 22.0000i 0.758170i
\(843\) − 7.00000i − 0.241093i
\(844\) 3.65153 0.125691
\(845\) 0 0
\(846\) −7.79796 −0.268099
\(847\) 4.00000i 0.137442i
\(848\) 7.44949i 0.255817i
\(849\) −29.7980 −1.02266
\(850\) 0 0
\(851\) 7.79796 0.267311
\(852\) − 1.55051i − 0.0531196i
\(853\) − 1.10102i − 0.0376982i −0.999822 0.0188491i \(-0.994000\pi\)
0.999822 0.0188491i \(-0.00600021\pi\)
\(854\) 42.0454 1.43876
\(855\) 0 0
\(856\) 17.3485 0.592958
\(857\) − 36.4949i − 1.24664i −0.781966 0.623321i \(-0.785783\pi\)
0.781966 0.623321i \(-0.214217\pi\)
\(858\) 8.44949i 0.288461i
\(859\) 4.20204 0.143372 0.0716859 0.997427i \(-0.477162\pi\)
0.0716859 + 0.997427i \(0.477162\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) − 5.10102i − 0.173741i
\(863\) 17.3031i 0.589003i 0.955651 + 0.294502i \(0.0951536\pi\)
−0.955651 + 0.294502i \(0.904846\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 4.65153 0.158065
\(867\) 2.79796i 0.0950237i
\(868\) 13.3485i 0.453077i
\(869\) 17.2474 0.585080
\(870\) 0 0
\(871\) −37.3485 −1.26550
\(872\) − 2.89898i − 0.0981718i
\(873\) − 1.55051i − 0.0524768i
\(874\) 1.00000 0.0338255
\(875\) 0 0
\(876\) −1.00000 −0.0337869
\(877\) − 9.10102i − 0.307320i −0.988124 0.153660i \(-0.950894\pi\)
0.988124 0.153660i \(-0.0491060\pi\)
\(878\) 12.1010i 0.408390i
\(879\) 23.2474 0.784117
\(880\) 0 0
\(881\) 50.6969 1.70802 0.854012 0.520254i \(-0.174163\pi\)
0.854012 + 0.520254i \(0.174163\pi\)
\(882\) 12.7980i 0.430930i
\(883\) − 34.6515i − 1.16612i −0.812430 0.583058i \(-0.801856\pi\)
0.812430 0.583058i \(-0.198144\pi\)
\(884\) 10.8990 0.366572
\(885\) 0 0
\(886\) 21.2474 0.713822
\(887\) − 4.89898i − 0.164492i −0.996612 0.0822458i \(-0.973791\pi\)
0.996612 0.0822458i \(-0.0262093\pi\)
\(888\) 7.79796i 0.261682i
\(889\) −0.449490 −0.0150754
\(890\) 0 0
\(891\) 3.44949 0.115562
\(892\) 16.1010i 0.539102i
\(893\) 7.79796i 0.260949i
\(894\) −5.79796 −0.193913
\(895\) 0 0
\(896\) 4.44949 0.148647
\(897\) 2.44949i 0.0817861i
\(898\) 15.0000i 0.500556i
\(899\) −13.0454 −0.435089
\(900\) 0 0
\(901\) −33.1464 −1.10427
\(902\) − 3.10102i − 0.103253i
\(903\) 10.8990i 0.362695i
\(904\) 16.7980 0.558692
\(905\) 0 0
\(906\) −23.7980 −0.790634
\(907\) − 22.0000i − 0.730498i −0.930910 0.365249i \(-0.880984\pi\)
0.930910 0.365249i \(-0.119016\pi\)
\(908\) − 29.5959i − 0.982175i
\(909\) 1.55051 0.0514272
\(910\) 0 0
\(911\) 6.49490 0.215186 0.107593 0.994195i \(-0.465686\pi\)
0.107593 + 0.994195i \(0.465686\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) − 28.7980i − 0.953073i
\(914\) 24.8990 0.823585
\(915\) 0 0
\(916\) 7.24745 0.239462
\(917\) 41.1464i 1.35877i
\(918\) − 4.44949i − 0.146855i
\(919\) −23.8434 −0.786520 −0.393260 0.919427i \(-0.628653\pi\)
−0.393260 + 0.919427i \(0.628653\pi\)
\(920\) 0 0
\(921\) −16.3485 −0.538700
\(922\) 12.0000i 0.395199i
\(923\) − 3.79796i − 0.125011i
\(924\) −15.3485 −0.504928
\(925\) 0 0
\(926\) −14.6969 −0.482971
\(927\) 1.89898i 0.0623707i
\(928\) 4.34847i 0.142745i
\(929\) 31.5959 1.03663 0.518314 0.855190i \(-0.326560\pi\)
0.518314 + 0.855190i \(0.326560\pi\)
\(930\) 0 0
\(931\) 12.7980 0.419436
\(932\) 6.24745i 0.204642i
\(933\) 23.7980i 0.779110i
\(934\) 6.34847 0.207728
\(935\) 0 0
\(936\) −2.44949 −0.0800641
\(937\) − 19.3939i − 0.633570i −0.948497 0.316785i \(-0.897397\pi\)
0.948497 0.316785i \(-0.102603\pi\)
\(938\) − 67.8434i − 2.21516i
\(939\) −31.8990 −1.04098
\(940\) 0 0
\(941\) 36.6413 1.19447 0.597237 0.802065i \(-0.296265\pi\)
0.597237 + 0.802065i \(0.296265\pi\)
\(942\) − 4.89898i − 0.159617i
\(943\) − 0.898979i − 0.0292748i
\(944\) −6.44949 −0.209913
\(945\) 0 0
\(946\) −8.44949 −0.274717
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) 5.00000i 0.162392i
\(949\) −2.44949 −0.0795138
\(950\) 0 0
\(951\) 15.2474 0.494432
\(952\) 19.7980i 0.641656i
\(953\) 15.4949i 0.501929i 0.967996 + 0.250964i \(0.0807477\pi\)
−0.967996 + 0.250964i \(0.919252\pi\)
\(954\) 7.44949 0.241186
\(955\) 0 0
\(956\) −8.69694 −0.281279
\(957\) − 15.0000i − 0.484881i
\(958\) − 16.5959i − 0.536190i
\(959\) 21.7980 0.703893
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 19.1010i 0.615842i
\(963\) − 17.3485i − 0.559047i
\(964\) 4.44949 0.143308
\(965\) 0 0
\(966\) −4.44949 −0.143160
\(967\) − 34.8990i − 1.12228i −0.827722 0.561138i \(-0.810364\pi\)
0.827722 0.561138i \(-0.189636\pi\)
\(968\) − 0.898979i − 0.0288943i
\(969\) −4.44949 −0.142938
\(970\) 0 0
\(971\) 41.6413 1.33633 0.668167 0.744011i \(-0.267079\pi\)
0.668167 + 0.744011i \(0.267079\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 98.9898i 3.17347i
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −9.44949 −0.302471
\(977\) 19.5959i 0.626929i 0.949600 + 0.313464i \(0.101490\pi\)
−0.949600 + 0.313464i \(0.898510\pi\)
\(978\) − 19.7980i − 0.633069i
\(979\) −7.24745 −0.231629
\(980\) 0 0
\(981\) −2.89898 −0.0925573
\(982\) 43.5959i 1.39120i
\(983\) − 50.4495i − 1.60909i −0.593892 0.804544i \(-0.702410\pi\)
0.593892 0.804544i \(-0.297590\pi\)
\(984\) 0.898979 0.0286584
\(985\) 0 0
\(986\) −19.3485 −0.616181
\(987\) − 34.6969i − 1.10442i
\(988\) 2.44949i 0.0779287i
\(989\) −2.44949 −0.0778892
\(990\) 0 0
\(991\) 44.1010 1.40092 0.700458 0.713694i \(-0.252979\pi\)
0.700458 + 0.713694i \(0.252979\pi\)
\(992\) − 3.00000i − 0.0952501i
\(993\) 22.3485i 0.709207i
\(994\) 6.89898 0.218822
\(995\) 0 0
\(996\) 8.34847 0.264531
\(997\) 39.4495i 1.24938i 0.780874 + 0.624689i \(0.214774\pi\)
−0.780874 + 0.624689i \(0.785226\pi\)
\(998\) 23.8434i 0.754749i
\(999\) 7.79796 0.246717
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 2850.2.d.w.799.4 4
5.2 odd 4 2850.2.a.bc.1.1 2
5.3 odd 4 2850.2.a.bj.1.2 yes 2
5.4 even 2 inner 2850.2.d.w.799.1 4
15.2 even 4 8550.2.a.bv.1.1 2
15.8 even 4 8550.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bc.1.1 2 5.2 odd 4
2850.2.a.bj.1.2 yes 2 5.3 odd 4
2850.2.d.w.799.1 4 5.4 even 2 inner
2850.2.d.w.799.4 4 1.1 even 1 trivial
8550.2.a.bu.1.2 2 15.8 even 4
8550.2.a.bv.1.1 2 15.2 even 4