Properties

Label 1850.2.b.m.149.3
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.3
Root \(-2.37228i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.m.149.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -4.37228i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -4.37228i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.37228 q^{11} +2.00000i q^{12} -6.74456i q^{13} +4.37228 q^{14} +1.00000 q^{16} +0.372281i q^{17} -1.00000i q^{18} +2.00000 q^{19} -8.74456 q^{21} +2.37228i q^{22} +4.74456i q^{23} -2.00000 q^{24} +6.74456 q^{26} -4.00000i q^{27} +4.37228i q^{28} +9.11684 q^{29} -8.37228 q^{31} +1.00000i q^{32} -4.74456i q^{33} -0.372281 q^{34} +1.00000 q^{36} +1.00000i q^{37} +2.00000i q^{38} -13.4891 q^{39} -0.372281 q^{41} -8.74456i q^{42} -1.62772i q^{43} -2.37228 q^{44} -4.74456 q^{46} +2.74456i q^{47} -2.00000i q^{48} -12.1168 q^{49} +0.744563 q^{51} +6.74456i q^{52} +4.37228i q^{53} +4.00000 q^{54} -4.37228 q^{56} -4.00000i q^{57} +9.11684i q^{58} -1.25544 q^{59} +0.372281 q^{61} -8.37228i q^{62} +4.37228i q^{63} -1.00000 q^{64} +4.74456 q^{66} -6.74456i q^{67} -0.372281i q^{68} +9.48913 q^{69} +4.74456 q^{71} +1.00000i q^{72} +2.74456i q^{73} -1.00000 q^{74} -2.00000 q^{76} -10.3723i q^{77} -13.4891i q^{78} -6.74456 q^{79} -11.0000 q^{81} -0.372281i q^{82} -10.7446i q^{83} +8.74456 q^{84} +1.62772 q^{86} -18.2337i q^{87} -2.37228i q^{88} -10.0000 q^{89} -29.4891 q^{91} -4.74456i q^{92} +16.7446i q^{93} -2.74456 q^{94} +2.00000 q^{96} +17.1168i q^{97} -12.1168i q^{98} -2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 2 q^{11} + 6 q^{14} + 4 q^{16} + 8 q^{19} - 12 q^{21} - 8 q^{24} + 4 q^{26} + 2 q^{29} - 22 q^{31} + 10 q^{34} + 4 q^{36} - 8 q^{39} + 10 q^{41} + 2 q^{44} + 4 q^{46} - 14 q^{49} - 20 q^{51} + 16 q^{54} - 6 q^{56} - 28 q^{59} - 10 q^{61} - 4 q^{64} - 4 q^{66} - 8 q^{69} - 4 q^{71} - 4 q^{74} - 8 q^{76} - 4 q^{79} - 44 q^{81} + 12 q^{84} + 18 q^{86} - 40 q^{89} - 72 q^{91} + 12 q^{94} + 8 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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 \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) − 4.37228i − 1.65257i −0.563254 0.826284i \(-0.690451\pi\)
0.563254 0.826284i \(-0.309549\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.37228 0.715270 0.357635 0.933862i \(-0.383583\pi\)
0.357635 + 0.933862i \(0.383583\pi\)
\(12\) 2.00000i 0.577350i
\(13\) − 6.74456i − 1.87061i −0.353849 0.935303i \(-0.615127\pi\)
0.353849 0.935303i \(-0.384873\pi\)
\(14\) 4.37228 1.16854
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.372281i 0.0902915i 0.998980 + 0.0451457i \(0.0143752\pi\)
−0.998980 + 0.0451457i \(0.985625\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −8.74456 −1.90822
\(22\) 2.37228i 0.505772i
\(23\) 4.74456i 0.989310i 0.869090 + 0.494655i \(0.164706\pi\)
−0.869090 + 0.494655i \(0.835294\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 6.74456 1.32272
\(27\) − 4.00000i − 0.769800i
\(28\) 4.37228i 0.826284i
\(29\) 9.11684 1.69296 0.846478 0.532424i \(-0.178719\pi\)
0.846478 + 0.532424i \(0.178719\pi\)
\(30\) 0 0
\(31\) −8.37228 −1.50371 −0.751853 0.659331i \(-0.770840\pi\)
−0.751853 + 0.659331i \(0.770840\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 4.74456i − 0.825922i
\(34\) −0.372281 −0.0638457
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000i 0.164399i
\(38\) 2.00000i 0.324443i
\(39\) −13.4891 −2.15999
\(40\) 0 0
\(41\) −0.372281 −0.0581406 −0.0290703 0.999577i \(-0.509255\pi\)
−0.0290703 + 0.999577i \(0.509255\pi\)
\(42\) − 8.74456i − 1.34932i
\(43\) − 1.62772i − 0.248225i −0.992268 0.124112i \(-0.960392\pi\)
0.992268 0.124112i \(-0.0396083\pi\)
\(44\) −2.37228 −0.357635
\(45\) 0 0
\(46\) −4.74456 −0.699548
\(47\) 2.74456i 0.400336i 0.979762 + 0.200168i \(0.0641487\pi\)
−0.979762 + 0.200168i \(0.935851\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) −12.1168 −1.73098
\(50\) 0 0
\(51\) 0.744563 0.104260
\(52\) 6.74456i 0.935303i
\(53\) 4.37228i 0.600579i 0.953848 + 0.300290i \(0.0970833\pi\)
−0.953848 + 0.300290i \(0.902917\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −4.37228 −0.584271
\(57\) − 4.00000i − 0.529813i
\(58\) 9.11684i 1.19710i
\(59\) −1.25544 −0.163444 −0.0817220 0.996655i \(-0.526042\pi\)
−0.0817220 + 0.996655i \(0.526042\pi\)
\(60\) 0 0
\(61\) 0.372281 0.0476657 0.0238329 0.999716i \(-0.492413\pi\)
0.0238329 + 0.999716i \(0.492413\pi\)
\(62\) − 8.37228i − 1.06328i
\(63\) 4.37228i 0.550856i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.74456 0.584015
\(67\) − 6.74456i − 0.823979i −0.911188 0.411990i \(-0.864834\pi\)
0.911188 0.411990i \(-0.135166\pi\)
\(68\) − 0.372281i − 0.0451457i
\(69\) 9.48913 1.14236
\(70\) 0 0
\(71\) 4.74456 0.563076 0.281538 0.959550i \(-0.409155\pi\)
0.281538 + 0.959550i \(0.409155\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 2.74456i 0.321227i 0.987017 + 0.160613i \(0.0513472\pi\)
−0.987017 + 0.160613i \(0.948653\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) − 10.3723i − 1.18203i
\(78\) − 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
\(82\) − 0.372281i − 0.0411116i
\(83\) − 10.7446i − 1.17937i −0.807634 0.589684i \(-0.799252\pi\)
0.807634 0.589684i \(-0.200748\pi\)
\(84\) 8.74456 0.954110
\(85\) 0 0
\(86\) 1.62772 0.175521
\(87\) − 18.2337i − 1.95486i
\(88\) − 2.37228i − 0.252886i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −29.4891 −3.09130
\(92\) − 4.74456i − 0.494655i
\(93\) 16.7446i 1.73633i
\(94\) −2.74456 −0.283080
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 17.1168i 1.73795i 0.494854 + 0.868976i \(0.335222\pi\)
−0.494854 + 0.868976i \(0.664778\pi\)
\(98\) − 12.1168i − 1.22399i
\(99\) −2.37228 −0.238423
\(100\) 0 0
\(101\) 11.4891 1.14321 0.571605 0.820529i \(-0.306321\pi\)
0.571605 + 0.820529i \(0.306321\pi\)
\(102\) 0.744563i 0.0737227i
\(103\) − 13.4891i − 1.32912i −0.747234 0.664562i \(-0.768618\pi\)
0.747234 0.664562i \(-0.231382\pi\)
\(104\) −6.74456 −0.661359
\(105\) 0 0
\(106\) −4.37228 −0.424674
\(107\) 19.4891i 1.88408i 0.335494 + 0.942042i \(0.391097\pi\)
−0.335494 + 0.942042i \(0.608903\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 17.1168 1.63950 0.819748 0.572724i \(-0.194113\pi\)
0.819748 + 0.572724i \(0.194113\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 4.37228i − 0.413142i
\(113\) − 11.6277i − 1.09384i −0.837184 0.546922i \(-0.815799\pi\)
0.837184 0.546922i \(-0.184201\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −9.11684 −0.846478
\(117\) 6.74456i 0.623535i
\(118\) − 1.25544i − 0.115572i
\(119\) 1.62772 0.149213
\(120\) 0 0
\(121\) −5.37228 −0.488389
\(122\) 0.372281i 0.0337048i
\(123\) 0.744563i 0.0671350i
\(124\) 8.37228 0.751853
\(125\) 0 0
\(126\) −4.37228 −0.389514
\(127\) − 5.25544i − 0.466345i −0.972435 0.233172i \(-0.925089\pi\)
0.972435 0.233172i \(-0.0749106\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −3.25544 −0.286625
\(130\) 0 0
\(131\) −14.7446 −1.28824 −0.644119 0.764925i \(-0.722776\pi\)
−0.644119 + 0.764925i \(0.722776\pi\)
\(132\) 4.74456i 0.412961i
\(133\) − 8.74456i − 0.758250i
\(134\) 6.74456 0.582641
\(135\) 0 0
\(136\) 0.372281 0.0319229
\(137\) 5.25544i 0.449002i 0.974474 + 0.224501i \(0.0720753\pi\)
−0.974474 + 0.224501i \(0.927925\pi\)
\(138\) 9.48913i 0.807768i
\(139\) 19.1168 1.62147 0.810735 0.585414i \(-0.199068\pi\)
0.810735 + 0.585414i \(0.199068\pi\)
\(140\) 0 0
\(141\) 5.48913 0.462268
\(142\) 4.74456i 0.398155i
\(143\) − 16.0000i − 1.33799i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.74456 −0.227142
\(147\) 24.2337i 1.99876i
\(148\) − 1.00000i − 0.0821995i
\(149\) −11.4891 −0.941226 −0.470613 0.882340i \(-0.655967\pi\)
−0.470613 + 0.882340i \(0.655967\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) − 2.00000i − 0.162221i
\(153\) − 0.372281i − 0.0300972i
\(154\) 10.3723 0.835822
\(155\) 0 0
\(156\) 13.4891 1.07999
\(157\) 11.6277i 0.927993i 0.885837 + 0.463996i \(0.153585\pi\)
−0.885837 + 0.463996i \(0.846415\pi\)
\(158\) − 6.74456i − 0.536569i
\(159\) 8.74456 0.693489
\(160\) 0 0
\(161\) 20.7446 1.63490
\(162\) − 11.0000i − 0.864242i
\(163\) − 13.6277i − 1.06741i −0.845672 0.533703i \(-0.820800\pi\)
0.845672 0.533703i \(-0.179200\pi\)
\(164\) 0.372281 0.0290703
\(165\) 0 0
\(166\) 10.7446 0.833940
\(167\) − 21.4891i − 1.66288i −0.555616 0.831439i \(-0.687517\pi\)
0.555616 0.831439i \(-0.312483\pi\)
\(168\) 8.74456i 0.674658i
\(169\) −32.4891 −2.49916
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 1.62772i 0.124112i
\(173\) 5.86141i 0.445634i 0.974860 + 0.222817i \(0.0715253\pi\)
−0.974860 + 0.222817i \(0.928475\pi\)
\(174\) 18.2337 1.38229
\(175\) 0 0
\(176\) 2.37228 0.178817
\(177\) 2.51087i 0.188729i
\(178\) − 10.0000i − 0.749532i
\(179\) −8.23369 −0.615415 −0.307707 0.951481i \(-0.599562\pi\)
−0.307707 + 0.951481i \(0.599562\pi\)
\(180\) 0 0
\(181\) −15.4891 −1.15130 −0.575649 0.817697i \(-0.695250\pi\)
−0.575649 + 0.817697i \(0.695250\pi\)
\(182\) − 29.4891i − 2.18588i
\(183\) − 0.744563i − 0.0550397i
\(184\) 4.74456 0.349774
\(185\) 0 0
\(186\) −16.7446 −1.22777
\(187\) 0.883156i 0.0645828i
\(188\) − 2.74456i − 0.200168i
\(189\) −17.4891 −1.27215
\(190\) 0 0
\(191\) 24.3723 1.76352 0.881758 0.471702i \(-0.156360\pi\)
0.881758 + 0.471702i \(0.156360\pi\)
\(192\) 2.00000i 0.144338i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) −17.1168 −1.22892
\(195\) 0 0
\(196\) 12.1168 0.865489
\(197\) 4.51087i 0.321387i 0.987004 + 0.160693i \(0.0513730\pi\)
−0.987004 + 0.160693i \(0.948627\pi\)
\(198\) − 2.37228i − 0.168591i
\(199\) 22.7446 1.61232 0.806160 0.591698i \(-0.201542\pi\)
0.806160 + 0.591698i \(0.201542\pi\)
\(200\) 0 0
\(201\) −13.4891 −0.951450
\(202\) 11.4891i 0.808372i
\(203\) − 39.8614i − 2.79772i
\(204\) −0.744563 −0.0521298
\(205\) 0 0
\(206\) 13.4891 0.939832
\(207\) − 4.74456i − 0.329770i
\(208\) − 6.74456i − 0.467651i
\(209\) 4.74456 0.328188
\(210\) 0 0
\(211\) 3.11684 0.214572 0.107286 0.994228i \(-0.465784\pi\)
0.107286 + 0.994228i \(0.465784\pi\)
\(212\) − 4.37228i − 0.300290i
\(213\) − 9.48913i − 0.650184i
\(214\) −19.4891 −1.33225
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 36.6060i 2.48498i
\(218\) 17.1168i 1.15930i
\(219\) 5.48913 0.370921
\(220\) 0 0
\(221\) 2.51087 0.168900
\(222\) 2.00000i 0.134231i
\(223\) − 4.37228i − 0.292790i −0.989226 0.146395i \(-0.953233\pi\)
0.989226 0.146395i \(-0.0467670\pi\)
\(224\) 4.37228 0.292135
\(225\) 0 0
\(226\) 11.6277 0.773464
\(227\) − 5.62772i − 0.373525i −0.982405 0.186762i \(-0.940201\pi\)
0.982405 0.186762i \(-0.0597995\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −20.7446 −1.36489
\(232\) − 9.11684i − 0.598550i
\(233\) 28.2337i 1.84965i 0.380392 + 0.924825i \(0.375789\pi\)
−0.380392 + 0.924825i \(0.624211\pi\)
\(234\) −6.74456 −0.440906
\(235\) 0 0
\(236\) 1.25544 0.0817220
\(237\) 13.4891i 0.876213i
\(238\) 1.62772i 0.105509i
\(239\) 22.6060 1.46226 0.731129 0.682239i \(-0.238994\pi\)
0.731129 + 0.682239i \(0.238994\pi\)
\(240\) 0 0
\(241\) 24.2337 1.56103 0.780515 0.625138i \(-0.214957\pi\)
0.780515 + 0.625138i \(0.214957\pi\)
\(242\) − 5.37228i − 0.345343i
\(243\) 10.0000i 0.641500i
\(244\) −0.372281 −0.0238329
\(245\) 0 0
\(246\) −0.744563 −0.0474716
\(247\) − 13.4891i − 0.858292i
\(248\) 8.37228i 0.531640i
\(249\) −21.4891 −1.36182
\(250\) 0 0
\(251\) −11.4891 −0.725187 −0.362594 0.931947i \(-0.618109\pi\)
−0.362594 + 0.931947i \(0.618109\pi\)
\(252\) − 4.37228i − 0.275428i
\(253\) 11.2554i 0.707623i
\(254\) 5.25544 0.329755
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.9783i 1.55810i 0.626962 + 0.779050i \(0.284298\pi\)
−0.626962 + 0.779050i \(0.715702\pi\)
\(258\) − 3.25544i − 0.202675i
\(259\) 4.37228 0.271680
\(260\) 0 0
\(261\) −9.11684 −0.564318
\(262\) − 14.7446i − 0.910922i
\(263\) 17.1168i 1.05547i 0.849409 + 0.527735i \(0.176959\pi\)
−0.849409 + 0.527735i \(0.823041\pi\)
\(264\) −4.74456 −0.292008
\(265\) 0 0
\(266\) 8.74456 0.536164
\(267\) 20.0000i 1.22398i
\(268\) 6.74456i 0.411990i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 4.74456 0.288212 0.144106 0.989562i \(-0.453969\pi\)
0.144106 + 0.989562i \(0.453969\pi\)
\(272\) 0.372281i 0.0225729i
\(273\) 58.9783i 3.56953i
\(274\) −5.25544 −0.317493
\(275\) 0 0
\(276\) −9.48913 −0.571178
\(277\) − 10.7446i − 0.645578i −0.946471 0.322789i \(-0.895380\pi\)
0.946471 0.322789i \(-0.104620\pi\)
\(278\) 19.1168i 1.14655i
\(279\) 8.37228 0.501235
\(280\) 0 0
\(281\) 13.2554 0.790753 0.395377 0.918519i \(-0.370614\pi\)
0.395377 + 0.918519i \(0.370614\pi\)
\(282\) 5.48913i 0.326873i
\(283\) 17.4891i 1.03962i 0.854282 + 0.519810i \(0.173997\pi\)
−0.854282 + 0.519810i \(0.826003\pi\)
\(284\) −4.74456 −0.281538
\(285\) 0 0
\(286\) 16.0000 0.946100
\(287\) 1.62772i 0.0960812i
\(288\) − 1.00000i − 0.0589256i
\(289\) 16.8614 0.991847
\(290\) 0 0
\(291\) 34.2337 2.00681
\(292\) − 2.74456i − 0.160613i
\(293\) − 9.86141i − 0.576110i −0.957614 0.288055i \(-0.906991\pi\)
0.957614 0.288055i \(-0.0930085\pi\)
\(294\) −24.2337 −1.41334
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) − 9.48913i − 0.550615i
\(298\) − 11.4891i − 0.665547i
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) −7.11684 −0.410208
\(302\) − 20.0000i − 1.15087i
\(303\) − 22.9783i − 1.32007i
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 0.372281 0.0212819
\(307\) − 0.510875i − 0.0291572i −0.999894 0.0145786i \(-0.995359\pi\)
0.999894 0.0145786i \(-0.00464067\pi\)
\(308\) 10.3723i 0.591016i
\(309\) −26.9783 −1.53474
\(310\) 0 0
\(311\) 4.37228 0.247929 0.123965 0.992287i \(-0.460439\pi\)
0.123965 + 0.992287i \(0.460439\pi\)
\(312\) 13.4891i 0.763671i
\(313\) − 19.4891i − 1.10159i −0.834640 0.550795i \(-0.814325\pi\)
0.834640 0.550795i \(-0.185675\pi\)
\(314\) −11.6277 −0.656190
\(315\) 0 0
\(316\) 6.74456 0.379411
\(317\) 19.6277i 1.10240i 0.834372 + 0.551201i \(0.185830\pi\)
−0.834372 + 0.551201i \(0.814170\pi\)
\(318\) 8.74456i 0.490371i
\(319\) 21.6277 1.21092
\(320\) 0 0
\(321\) 38.9783 2.17555
\(322\) 20.7446i 1.15605i
\(323\) 0.744563i 0.0414286i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 13.6277 0.754770
\(327\) − 34.2337i − 1.89313i
\(328\) 0.372281i 0.0205558i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 0.510875 0.0280802 0.0140401 0.999901i \(-0.495531\pi\)
0.0140401 + 0.999901i \(0.495531\pi\)
\(332\) 10.7446i 0.589684i
\(333\) − 1.00000i − 0.0547997i
\(334\) 21.4891 1.17583
\(335\) 0 0
\(336\) −8.74456 −0.477055
\(337\) − 18.7446i − 1.02108i −0.859854 0.510541i \(-0.829445\pi\)
0.859854 0.510541i \(-0.170555\pi\)
\(338\) − 32.4891i − 1.76718i
\(339\) −23.2554 −1.26306
\(340\) 0 0
\(341\) −19.8614 −1.07556
\(342\) − 2.00000i − 0.108148i
\(343\) 22.3723i 1.20799i
\(344\) −1.62772 −0.0877607
\(345\) 0 0
\(346\) −5.86141 −0.315111
\(347\) 22.9783i 1.23354i 0.787145 + 0.616769i \(0.211559\pi\)
−0.787145 + 0.616769i \(0.788441\pi\)
\(348\) 18.2337i 0.977428i
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) −26.9783 −1.43999
\(352\) 2.37228i 0.126443i
\(353\) − 5.86141i − 0.311971i −0.987759 0.155986i \(-0.950145\pi\)
0.987759 0.155986i \(-0.0498553\pi\)
\(354\) −2.51087 −0.133451
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) − 3.25544i − 0.172296i
\(358\) − 8.23369i − 0.435164i
\(359\) −14.9783 −0.790522 −0.395261 0.918569i \(-0.629346\pi\)
−0.395261 + 0.918569i \(0.629346\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) − 15.4891i − 0.814090i
\(363\) 10.7446i 0.563943i
\(364\) 29.4891 1.54565
\(365\) 0 0
\(366\) 0.744563 0.0389189
\(367\) 21.1168i 1.10229i 0.834409 + 0.551145i \(0.185809\pi\)
−0.834409 + 0.551145i \(0.814191\pi\)
\(368\) 4.74456i 0.247327i
\(369\) 0.372281 0.0193802
\(370\) 0 0
\(371\) 19.1168 0.992497
\(372\) − 16.7446i − 0.868165i
\(373\) 8.51087i 0.440676i 0.975424 + 0.220338i \(0.0707161\pi\)
−0.975424 + 0.220338i \(0.929284\pi\)
\(374\) −0.883156 −0.0456669
\(375\) 0 0
\(376\) 2.74456 0.141540
\(377\) − 61.4891i − 3.16685i
\(378\) − 17.4891i − 0.899544i
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −10.5109 −0.538488
\(382\) 24.3723i 1.24699i
\(383\) − 9.48913i − 0.484872i −0.970167 0.242436i \(-0.922054\pi\)
0.970167 0.242436i \(-0.0779464\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 1.62772i 0.0827416i
\(388\) − 17.1168i − 0.868976i
\(389\) 13.8614 0.702801 0.351401 0.936225i \(-0.385706\pi\)
0.351401 + 0.936225i \(0.385706\pi\)
\(390\) 0 0
\(391\) −1.76631 −0.0893262
\(392\) 12.1168i 0.611993i
\(393\) 29.4891i 1.48753i
\(394\) −4.51087 −0.227255
\(395\) 0 0
\(396\) 2.37228 0.119212
\(397\) − 20.9783i − 1.05287i −0.850216 0.526434i \(-0.823529\pi\)
0.850216 0.526434i \(-0.176471\pi\)
\(398\) 22.7446i 1.14008i
\(399\) −17.4891 −0.875551
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) − 13.4891i − 0.672776i
\(403\) 56.4674i 2.81284i
\(404\) −11.4891 −0.571605
\(405\) 0 0
\(406\) 39.8614 1.97829
\(407\) 2.37228i 0.117590i
\(408\) − 0.744563i − 0.0368613i
\(409\) 28.2337 1.39607 0.698033 0.716066i \(-0.254059\pi\)
0.698033 + 0.716066i \(0.254059\pi\)
\(410\) 0 0
\(411\) 10.5109 0.518463
\(412\) 13.4891i 0.664562i
\(413\) 5.48913i 0.270102i
\(414\) 4.74456 0.233183
\(415\) 0 0
\(416\) 6.74456 0.330679
\(417\) − 38.2337i − 1.87231i
\(418\) 4.74456i 0.232064i
\(419\) 9.48913 0.463574 0.231787 0.972767i \(-0.425543\pi\)
0.231787 + 0.972767i \(0.425543\pi\)
\(420\) 0 0
\(421\) 15.4891 0.754894 0.377447 0.926031i \(-0.376802\pi\)
0.377447 + 0.926031i \(0.376802\pi\)
\(422\) 3.11684i 0.151726i
\(423\) − 2.74456i − 0.133445i
\(424\) 4.37228 0.212337
\(425\) 0 0
\(426\) 9.48913 0.459750
\(427\) − 1.62772i − 0.0787708i
\(428\) − 19.4891i − 0.942042i
\(429\) −32.0000 −1.54497
\(430\) 0 0
\(431\) 4.37228 0.210605 0.105303 0.994440i \(-0.466419\pi\)
0.105303 + 0.994440i \(0.466419\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 32.9783i 1.58483i 0.609980 + 0.792417i \(0.291177\pi\)
−0.609980 + 0.792417i \(0.708823\pi\)
\(434\) −36.6060 −1.75714
\(435\) 0 0
\(436\) −17.1168 −0.819748
\(437\) 9.48913i 0.453926i
\(438\) 5.48913i 0.262281i
\(439\) 7.62772 0.364051 0.182026 0.983294i \(-0.441735\pi\)
0.182026 + 0.983294i \(0.441735\pi\)
\(440\) 0 0
\(441\) 12.1168 0.576993
\(442\) 2.51087i 0.119430i
\(443\) − 28.9783i − 1.37680i −0.725332 0.688399i \(-0.758314\pi\)
0.725332 0.688399i \(-0.241686\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 4.37228 0.207034
\(447\) 22.9783i 1.08683i
\(448\) 4.37228i 0.206571i
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −0.883156 −0.0415862
\(452\) 11.6277i 0.546922i
\(453\) 40.0000i 1.87936i
\(454\) 5.62772 0.264122
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) − 9.86141i − 0.461297i −0.973037 0.230649i \(-0.925915\pi\)
0.973037 0.230649i \(-0.0740848\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 1.48913 0.0695064
\(460\) 0 0
\(461\) 24.0951 1.12222 0.561110 0.827741i \(-0.310374\pi\)
0.561110 + 0.827741i \(0.310374\pi\)
\(462\) − 20.7446i − 0.965124i
\(463\) − 17.4891i − 0.812789i −0.913698 0.406394i \(-0.866786\pi\)
0.913698 0.406394i \(-0.133214\pi\)
\(464\) 9.11684 0.423239
\(465\) 0 0
\(466\) −28.2337 −1.30790
\(467\) − 21.3505i − 0.987985i −0.869466 0.493992i \(-0.835537\pi\)
0.869466 0.493992i \(-0.164463\pi\)
\(468\) − 6.74456i − 0.311768i
\(469\) −29.4891 −1.36168
\(470\) 0 0
\(471\) 23.2554 1.07155
\(472\) 1.25544i 0.0577862i
\(473\) − 3.86141i − 0.177548i
\(474\) −13.4891 −0.619576
\(475\) 0 0
\(476\) −1.62772 −0.0746064
\(477\) − 4.37228i − 0.200193i
\(478\) 22.6060i 1.03397i
\(479\) −14.7446 −0.673696 −0.336848 0.941559i \(-0.609361\pi\)
−0.336848 + 0.941559i \(0.609361\pi\)
\(480\) 0 0
\(481\) 6.74456 0.307526
\(482\) 24.2337i 1.10381i
\(483\) − 41.4891i − 1.88782i
\(484\) 5.37228 0.244195
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 19.7228i 0.893726i 0.894602 + 0.446863i \(0.147459\pi\)
−0.894602 + 0.446863i \(0.852541\pi\)
\(488\) − 0.372281i − 0.0168524i
\(489\) −27.2554 −1.23253
\(490\) 0 0
\(491\) −30.9783 −1.39803 −0.699014 0.715108i \(-0.746378\pi\)
−0.699014 + 0.715108i \(0.746378\pi\)
\(492\) − 0.744563i − 0.0335675i
\(493\) 3.39403i 0.152859i
\(494\) 13.4891 0.606904
\(495\) 0 0
\(496\) −8.37228 −0.375927
\(497\) − 20.7446i − 0.930521i
\(498\) − 21.4891i − 0.962951i
\(499\) −20.9783 −0.939115 −0.469558 0.882902i \(-0.655587\pi\)
−0.469558 + 0.882902i \(0.655587\pi\)
\(500\) 0 0
\(501\) −42.9783 −1.92013
\(502\) − 11.4891i − 0.512785i
\(503\) − 6.51087i − 0.290306i −0.989409 0.145153i \(-0.953633\pi\)
0.989409 0.145153i \(-0.0463674\pi\)
\(504\) 4.37228 0.194757
\(505\) 0 0
\(506\) −11.2554 −0.500365
\(507\) 64.9783i 2.88579i
\(508\) 5.25544i 0.233172i
\(509\) 22.7446 1.00814 0.504068 0.863664i \(-0.331836\pi\)
0.504068 + 0.863664i \(0.331836\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 1.00000i 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) −24.9783 −1.10174
\(515\) 0 0
\(516\) 3.25544 0.143313
\(517\) 6.51087i 0.286348i
\(518\) 4.37228i 0.192107i
\(519\) 11.7228 0.514574
\(520\) 0 0
\(521\) −36.0951 −1.58135 −0.790677 0.612233i \(-0.790271\pi\)
−0.790677 + 0.612233i \(0.790271\pi\)
\(522\) − 9.11684i − 0.399033i
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 14.7446 0.644119
\(525\) 0 0
\(526\) −17.1168 −0.746330
\(527\) − 3.11684i − 0.135772i
\(528\) − 4.74456i − 0.206481i
\(529\) 0.489125 0.0212663
\(530\) 0 0
\(531\) 1.25544 0.0544813
\(532\) 8.74456i 0.379125i
\(533\) 2.51087i 0.108758i
\(534\) −20.0000 −0.865485
\(535\) 0 0
\(536\) −6.74456 −0.291321
\(537\) 16.4674i 0.710620i
\(538\) − 10.0000i − 0.431131i
\(539\) −28.7446 −1.23812
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 4.74456i 0.203796i
\(543\) 30.9783i 1.32940i
\(544\) −0.372281 −0.0159614
\(545\) 0 0
\(546\) −58.9783 −2.52404
\(547\) 33.6277i 1.43782i 0.695104 + 0.718909i \(0.255358\pi\)
−0.695104 + 0.718909i \(0.744642\pi\)
\(548\) − 5.25544i − 0.224501i
\(549\) −0.372281 −0.0158886
\(550\) 0 0
\(551\) 18.2337 0.776781
\(552\) − 9.48913i − 0.403884i
\(553\) 29.4891i 1.25401i
\(554\) 10.7446 0.456493
\(555\) 0 0
\(556\) −19.1168 −0.810735
\(557\) − 2.74456i − 0.116291i −0.998308 0.0581454i \(-0.981481\pi\)
0.998308 0.0581454i \(-0.0185187\pi\)
\(558\) 8.37228i 0.354427i
\(559\) −10.9783 −0.464331
\(560\) 0 0
\(561\) 1.76631 0.0745738
\(562\) 13.2554i 0.559147i
\(563\) 30.0951i 1.26836i 0.773187 + 0.634179i \(0.218662\pi\)
−0.773187 + 0.634179i \(0.781338\pi\)
\(564\) −5.48913 −0.231134
\(565\) 0 0
\(566\) −17.4891 −0.735123
\(567\) 48.0951i 2.01980i
\(568\) − 4.74456i − 0.199077i
\(569\) −12.9783 −0.544077 −0.272038 0.962286i \(-0.587698\pi\)
−0.272038 + 0.962286i \(0.587698\pi\)
\(570\) 0 0
\(571\) 12.6060 0.527543 0.263772 0.964585i \(-0.415033\pi\)
0.263772 + 0.964585i \(0.415033\pi\)
\(572\) 16.0000i 0.668994i
\(573\) − 48.7446i − 2.03633i
\(574\) −1.62772 −0.0679397
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 15.4891i 0.644821i 0.946600 + 0.322410i \(0.104493\pi\)
−0.946600 + 0.322410i \(0.895507\pi\)
\(578\) 16.8614i 0.701342i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −46.9783 −1.94899
\(582\) 34.2337i 1.41903i
\(583\) 10.3723i 0.429576i
\(584\) 2.74456 0.113571
\(585\) 0 0
\(586\) 9.86141 0.407371
\(587\) − 26.8397i − 1.10779i −0.832586 0.553896i \(-0.813141\pi\)
0.832586 0.553896i \(-0.186859\pi\)
\(588\) − 24.2337i − 0.999380i
\(589\) −16.7446 −0.689948
\(590\) 0 0
\(591\) 9.02175 0.371105
\(592\) 1.00000i 0.0410997i
\(593\) − 24.2337i − 0.995158i −0.867419 0.497579i \(-0.834222\pi\)
0.867419 0.497579i \(-0.165778\pi\)
\(594\) 9.48913 0.389344
\(595\) 0 0
\(596\) 11.4891 0.470613
\(597\) − 45.4891i − 1.86175i
\(598\) 32.0000i 1.30858i
\(599\) 5.48913 0.224280 0.112140 0.993692i \(-0.464230\pi\)
0.112140 + 0.993692i \(0.464230\pi\)
\(600\) 0 0
\(601\) 0.372281 0.0151857 0.00759284 0.999971i \(-0.497583\pi\)
0.00759284 + 0.999971i \(0.497583\pi\)
\(602\) − 7.11684i − 0.290061i
\(603\) 6.74456i 0.274660i
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 22.9783 0.933428
\(607\) − 39.7228i − 1.61230i −0.591712 0.806150i \(-0.701548\pi\)
0.591712 0.806150i \(-0.298452\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) −79.7228 −3.23053
\(610\) 0 0
\(611\) 18.5109 0.748870
\(612\) 0.372281i 0.0150486i
\(613\) 20.0951i 0.811633i 0.913955 + 0.405817i \(0.133013\pi\)
−0.913955 + 0.405817i \(0.866987\pi\)
\(614\) 0.510875 0.0206172
\(615\) 0 0
\(616\) −10.3723 −0.417911
\(617\) 25.7228i 1.03556i 0.855513 + 0.517781i \(0.173242\pi\)
−0.855513 + 0.517781i \(0.826758\pi\)
\(618\) − 26.9783i − 1.08522i
\(619\) −12.8832 −0.517818 −0.258909 0.965902i \(-0.583363\pi\)
−0.258909 + 0.965902i \(0.583363\pi\)
\(620\) 0 0
\(621\) 18.9783 0.761571
\(622\) 4.37228i 0.175313i
\(623\) 43.7228i 1.75172i
\(624\) −13.4891 −0.539997
\(625\) 0 0
\(626\) 19.4891 0.778942
\(627\) − 9.48913i − 0.378959i
\(628\) − 11.6277i − 0.463996i
\(629\) −0.372281 −0.0148438
\(630\) 0 0
\(631\) 44.0951 1.75540 0.877699 0.479212i \(-0.159078\pi\)
0.877699 + 0.479212i \(0.159078\pi\)
\(632\) 6.74456i 0.268284i
\(633\) − 6.23369i − 0.247767i
\(634\) −19.6277 −0.779516
\(635\) 0 0
\(636\) −8.74456 −0.346744
\(637\) 81.7228i 3.23798i
\(638\) 21.6277i 0.856250i
\(639\) −4.74456 −0.187692
\(640\) 0 0
\(641\) −9.39403 −0.371042 −0.185521 0.982640i \(-0.559397\pi\)
−0.185521 + 0.982640i \(0.559397\pi\)
\(642\) 38.9783i 1.53835i
\(643\) 2.37228i 0.0935536i 0.998905 + 0.0467768i \(0.0148950\pi\)
−0.998905 + 0.0467768i \(0.985105\pi\)
\(644\) −20.7446 −0.817450
\(645\) 0 0
\(646\) −0.744563 −0.0292944
\(647\) 9.48913i 0.373056i 0.982450 + 0.186528i \(0.0597235\pi\)
−0.982450 + 0.186528i \(0.940276\pi\)
\(648\) 11.0000i 0.432121i
\(649\) −2.97825 −0.116907
\(650\) 0 0
\(651\) 73.2119 2.86940
\(652\) 13.6277i 0.533703i
\(653\) − 23.4891i − 0.919201i −0.888126 0.459600i \(-0.847993\pi\)
0.888126 0.459600i \(-0.152007\pi\)
\(654\) 34.2337 1.33864
\(655\) 0 0
\(656\) −0.372281 −0.0145351
\(657\) − 2.74456i − 0.107076i
\(658\) 12.0000i 0.467809i
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −47.3505 −1.84172 −0.920861 0.389891i \(-0.872513\pi\)
−0.920861 + 0.389891i \(0.872513\pi\)
\(662\) 0.510875i 0.0198557i
\(663\) − 5.02175i − 0.195029i
\(664\) −10.7446 −0.416970
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 43.2554i 1.67486i
\(668\) 21.4891i 0.831439i
\(669\) −8.74456 −0.338084
\(670\) 0 0
\(671\) 0.883156 0.0340939
\(672\) − 8.74456i − 0.337329i
\(673\) 31.4891i 1.21382i 0.794772 + 0.606908i \(0.207590\pi\)
−0.794772 + 0.606908i \(0.792410\pi\)
\(674\) 18.7446 0.722014
\(675\) 0 0
\(676\) 32.4891 1.24958
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) − 23.2554i − 0.893120i
\(679\) 74.8397 2.87208
\(680\) 0 0
\(681\) −11.2554 −0.431309
\(682\) − 19.8614i − 0.760533i
\(683\) − 14.3723i − 0.549940i −0.961453 0.274970i \(-0.911332\pi\)
0.961453 0.274970i \(-0.0886679\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −22.3723 −0.854178
\(687\) − 20.0000i − 0.763048i
\(688\) − 1.62772i − 0.0620562i
\(689\) 29.4891 1.12345
\(690\) 0 0
\(691\) −29.3505 −1.11655 −0.558273 0.829657i \(-0.688536\pi\)
−0.558273 + 0.829657i \(0.688536\pi\)
\(692\) − 5.86141i − 0.222817i
\(693\) 10.3723i 0.394010i
\(694\) −22.9783 −0.872242
\(695\) 0 0
\(696\) −18.2337 −0.691146
\(697\) − 0.138593i − 0.00524960i
\(698\) 22.0000i 0.832712i
\(699\) 56.4674 2.13579
\(700\) 0 0
\(701\) 42.4674 1.60397 0.801985 0.597344i \(-0.203777\pi\)
0.801985 + 0.597344i \(0.203777\pi\)
\(702\) − 26.9783i − 1.01823i
\(703\) 2.00000i 0.0754314i
\(704\) −2.37228 −0.0894087
\(705\) 0 0
\(706\) 5.86141 0.220597
\(707\) − 50.2337i − 1.88923i
\(708\) − 2.51087i − 0.0943645i
\(709\) −22.8832 −0.859395 −0.429697 0.902973i \(-0.641380\pi\)
−0.429697 + 0.902973i \(0.641380\pi\)
\(710\) 0 0
\(711\) 6.74456 0.252941
\(712\) 10.0000i 0.374766i
\(713\) − 39.7228i − 1.48763i
\(714\) 3.25544 0.121832
\(715\) 0 0
\(716\) 8.23369 0.307707
\(717\) − 45.2119i − 1.68847i
\(718\) − 14.9783i − 0.558983i
\(719\) 23.2554 0.867281 0.433641 0.901086i \(-0.357229\pi\)
0.433641 + 0.901086i \(0.357229\pi\)
\(720\) 0 0
\(721\) −58.9783 −2.19646
\(722\) − 15.0000i − 0.558242i
\(723\) − 48.4674i − 1.80252i
\(724\) 15.4891 0.575649
\(725\) 0 0
\(726\) −10.7446 −0.398768
\(727\) − 48.0000i − 1.78022i −0.455744 0.890111i \(-0.650627\pi\)
0.455744 0.890111i \(-0.349373\pi\)
\(728\) 29.4891i 1.09294i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0.605969 0.0224126
\(732\) 0.744563i 0.0275198i
\(733\) 23.6277i 0.872710i 0.899775 + 0.436355i \(0.143731\pi\)
−0.899775 + 0.436355i \(0.856269\pi\)
\(734\) −21.1168 −0.779437
\(735\) 0 0
\(736\) −4.74456 −0.174887
\(737\) − 16.0000i − 0.589368i
\(738\) 0.372281i 0.0137039i
\(739\) 8.13859 0.299383 0.149691 0.988733i \(-0.452172\pi\)
0.149691 + 0.988733i \(0.452172\pi\)
\(740\) 0 0
\(741\) −26.9783 −0.991071
\(742\) 19.1168i 0.701801i
\(743\) − 0.372281i − 0.0136577i −0.999977 0.00682884i \(-0.997826\pi\)
0.999977 0.00682884i \(-0.00217370\pi\)
\(744\) 16.7446 0.613885
\(745\) 0 0
\(746\) −8.51087 −0.311605
\(747\) 10.7446i 0.393123i
\(748\) − 0.883156i − 0.0322914i
\(749\) 85.2119 3.11358
\(750\) 0 0
\(751\) −0.744563 −0.0271695 −0.0135847 0.999908i \(-0.504324\pi\)
−0.0135847 + 0.999908i \(0.504324\pi\)
\(752\) 2.74456i 0.100084i
\(753\) 22.9783i 0.837374i
\(754\) 61.4891 2.23930
\(755\) 0 0
\(756\) 17.4891 0.636073
\(757\) − 47.9565i − 1.74301i −0.490388 0.871504i \(-0.663145\pi\)
0.490388 0.871504i \(-0.336855\pi\)
\(758\) 8.00000i 0.290573i
\(759\) 22.5109 0.817093
\(760\) 0 0
\(761\) −0.372281 −0.0134952 −0.00674759 0.999977i \(-0.502148\pi\)
−0.00674759 + 0.999977i \(0.502148\pi\)
\(762\) − 10.5109i − 0.380769i
\(763\) − 74.8397i − 2.70938i
\(764\) −24.3723 −0.881758
\(765\) 0 0
\(766\) 9.48913 0.342856
\(767\) 8.46738i 0.305739i
\(768\) − 2.00000i − 0.0721688i
\(769\) −11.7663 −0.424304 −0.212152 0.977237i \(-0.568047\pi\)
−0.212152 + 0.977237i \(0.568047\pi\)
\(770\) 0 0
\(771\) 49.9565 1.79914
\(772\) 2.00000i 0.0719816i
\(773\) − 27.3505i − 0.983730i −0.870671 0.491865i \(-0.836315\pi\)
0.870671 0.491865i \(-0.163685\pi\)
\(774\) −1.62772 −0.0585071
\(775\) 0 0
\(776\) 17.1168 0.614459
\(777\) − 8.74456i − 0.313709i
\(778\) 13.8614i 0.496956i
\(779\) −0.744563 −0.0266767
\(780\) 0 0
\(781\) 11.2554 0.402751
\(782\) − 1.76631i − 0.0631632i
\(783\) − 36.4674i − 1.30324i
\(784\) −12.1168 −0.432744
\(785\) 0 0
\(786\) −29.4891 −1.05184
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) − 4.51087i − 0.160693i
\(789\) 34.2337 1.21875
\(790\) 0 0
\(791\) −50.8397 −1.80765
\(792\) 2.37228i 0.0842953i
\(793\) − 2.51087i − 0.0891638i
\(794\) 20.9783 0.744490
\(795\) 0 0
\(796\) −22.7446 −0.806160
\(797\) − 4.51087i − 0.159783i −0.996804 0.0798917i \(-0.974543\pi\)
0.996804 0.0798917i \(-0.0254574\pi\)
\(798\) − 17.4891i − 0.619108i
\(799\) −1.02175 −0.0361469
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 10.0000i 0.353112i
\(803\) 6.51087i 0.229764i
\(804\) 13.4891 0.475725
\(805\) 0 0
\(806\) −56.4674 −1.98898
\(807\) 20.0000i 0.704033i
\(808\) − 11.4891i − 0.404186i
\(809\) 28.5109 1.00239 0.501194 0.865335i \(-0.332894\pi\)
0.501194 + 0.865335i \(0.332894\pi\)
\(810\) 0 0
\(811\) 26.5109 0.930923 0.465461 0.885068i \(-0.345888\pi\)
0.465461 + 0.885068i \(0.345888\pi\)
\(812\) 39.8614i 1.39886i
\(813\) − 9.48913i − 0.332798i
\(814\) −2.37228 −0.0831484
\(815\) 0 0
\(816\) 0.744563 0.0260649
\(817\) − 3.25544i − 0.113893i
\(818\) 28.2337i 0.987168i
\(819\) 29.4891 1.03043
\(820\) 0 0
\(821\) −21.2554 −0.741820 −0.370910 0.928669i \(-0.620954\pi\)
−0.370910 + 0.928669i \(0.620954\pi\)
\(822\) 10.5109i 0.366609i
\(823\) − 17.7228i − 0.617778i −0.951098 0.308889i \(-0.900043\pi\)
0.951098 0.308889i \(-0.0999572\pi\)
\(824\) −13.4891 −0.469916
\(825\) 0 0
\(826\) −5.48913 −0.190991
\(827\) − 2.64947i − 0.0921310i −0.998938 0.0460655i \(-0.985332\pi\)
0.998938 0.0460655i \(-0.0146683\pi\)
\(828\) 4.74456i 0.164885i
\(829\) −11.3505 −0.394220 −0.197110 0.980381i \(-0.563156\pi\)
−0.197110 + 0.980381i \(0.563156\pi\)
\(830\) 0 0
\(831\) −21.4891 −0.745449
\(832\) 6.74456i 0.233826i
\(833\) − 4.51087i − 0.156293i
\(834\) 38.2337 1.32392
\(835\) 0 0
\(836\) −4.74456 −0.164094
\(837\) 33.4891i 1.15755i
\(838\) 9.48913i 0.327796i
\(839\) 6.51087 0.224780 0.112390 0.993664i \(-0.464149\pi\)
0.112390 + 0.993664i \(0.464149\pi\)
\(840\) 0 0
\(841\) 54.1168 1.86610
\(842\) 15.4891i 0.533791i
\(843\) − 26.5109i − 0.913083i
\(844\) −3.11684 −0.107286
\(845\) 0 0
\(846\) 2.74456 0.0943600
\(847\) 23.4891i 0.807096i
\(848\) 4.37228i 0.150145i
\(849\) 34.9783 1.20045
\(850\) 0 0
\(851\) −4.74456 −0.162642
\(852\) 9.48913i 0.325092i
\(853\) − 12.5109i − 0.428364i −0.976794 0.214182i \(-0.931291\pi\)
0.976794 0.214182i \(-0.0687086\pi\)
\(854\) 1.62772 0.0556994
\(855\) 0 0
\(856\) 19.4891 0.666125
\(857\) − 57.8614i − 1.97651i −0.152820 0.988254i \(-0.548836\pi\)
0.152820 0.988254i \(-0.451164\pi\)
\(858\) − 32.0000i − 1.09246i
\(859\) −37.7228 −1.28709 −0.643543 0.765410i \(-0.722536\pi\)
−0.643543 + 0.765410i \(0.722536\pi\)
\(860\) 0 0
\(861\) 3.25544 0.110945
\(862\) 4.37228i 0.148920i
\(863\) 20.8397i 0.709390i 0.934982 + 0.354695i \(0.115415\pi\)
−0.934982 + 0.354695i \(0.884585\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −32.9783 −1.12065
\(867\) − 33.7228i − 1.14529i
\(868\) − 36.6060i − 1.24249i
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −45.4891 −1.54134
\(872\) − 17.1168i − 0.579649i
\(873\) − 17.1168i − 0.579317i
\(874\) −9.48913 −0.320974
\(875\) 0 0
\(876\) −5.48913 −0.185460
\(877\) 43.6277i 1.47320i 0.676327 + 0.736602i \(0.263571\pi\)
−0.676327 + 0.736602i \(0.736429\pi\)
\(878\) 7.62772i 0.257423i
\(879\) −19.7228 −0.665234
\(880\) 0 0
\(881\) −4.37228 −0.147306 −0.0736530 0.997284i \(-0.523466\pi\)
−0.0736530 + 0.997284i \(0.523466\pi\)
\(882\) 12.1168i 0.407995i
\(883\) 5.62772i 0.189388i 0.995506 + 0.0946939i \(0.0301872\pi\)
−0.995506 + 0.0946939i \(0.969813\pi\)
\(884\) −2.51087 −0.0844499
\(885\) 0 0
\(886\) 28.9783 0.973543
\(887\) − 19.6277i − 0.659034i −0.944150 0.329517i \(-0.893114\pi\)
0.944150 0.329517i \(-0.106886\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) −22.9783 −0.770666
\(890\) 0 0
\(891\) −26.0951 −0.874219
\(892\) 4.37228i 0.146395i
\(893\) 5.48913i 0.183687i
\(894\) −22.9783 −0.768508
\(895\) 0 0
\(896\) −4.37228 −0.146068
\(897\) − 64.0000i − 2.13690i
\(898\) − 18.0000i − 0.600668i
\(899\) −76.3288 −2.54571
\(900\) 0 0
\(901\) −1.62772 −0.0542272
\(902\) − 0.883156i − 0.0294059i
\(903\) 14.2337i 0.473667i
\(904\) −11.6277 −0.386732
\(905\) 0 0
\(906\) −40.0000 −1.32891
\(907\) − 28.4674i − 0.945244i −0.881265 0.472622i \(-0.843308\pi\)
0.881265 0.472622i \(-0.156692\pi\)
\(908\) 5.62772i 0.186762i
\(909\) −11.4891 −0.381070
\(910\) 0 0
\(911\) 52.2337 1.73058 0.865290 0.501272i \(-0.167134\pi\)
0.865290 + 0.501272i \(0.167134\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 25.4891i − 0.843567i
\(914\) 9.86141 0.326186
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 64.4674i 2.12890i
\(918\) 1.48913i 0.0491485i
\(919\) 49.7228 1.64020 0.820102 0.572217i \(-0.193917\pi\)
0.820102 + 0.572217i \(0.193917\pi\)
\(920\) 0 0
\(921\) −1.02175 −0.0336678
\(922\) 24.0951i 0.793530i
\(923\) − 32.0000i − 1.05329i
\(924\) 20.7446 0.682446
\(925\) 0 0
\(926\) 17.4891 0.574728
\(927\) 13.4891i 0.443041i
\(928\) 9.11684i 0.299275i
\(929\) 32.0951 1.05301 0.526503 0.850173i \(-0.323503\pi\)
0.526503 + 0.850173i \(0.323503\pi\)
\(930\) 0 0
\(931\) −24.2337 −0.794227
\(932\) − 28.2337i − 0.924825i
\(933\) − 8.74456i − 0.286284i
\(934\) 21.3505 0.698611
\(935\) 0 0
\(936\) 6.74456 0.220453
\(937\) − 8.97825i − 0.293307i −0.989188 0.146653i \(-0.953150\pi\)
0.989188 0.146653i \(-0.0468502\pi\)
\(938\) − 29.4891i − 0.962854i
\(939\) −38.9783 −1.27201
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 23.2554i 0.757703i
\(943\) − 1.76631i − 0.0575190i
\(944\) −1.25544 −0.0408610
\(945\) 0 0
\(946\) 3.86141 0.125545
\(947\) − 26.0951i − 0.847977i −0.905668 0.423988i \(-0.860630\pi\)
0.905668 0.423988i \(-0.139370\pi\)
\(948\) − 13.4891i − 0.438106i
\(949\) 18.5109 0.600888
\(950\) 0 0
\(951\) 39.2554 1.27294
\(952\) − 1.62772i − 0.0527547i
\(953\) − 11.7663i − 0.381148i −0.981673 0.190574i \(-0.938965\pi\)
0.981673 0.190574i \(-0.0610349\pi\)
\(954\) 4.37228 0.141558
\(955\) 0 0
\(956\) −22.6060 −0.731129
\(957\) − 43.2554i − 1.39825i
\(958\) − 14.7446i − 0.476375i
\(959\) 22.9783 0.742006
\(960\) 0 0
\(961\) 39.0951 1.26113
\(962\) 6.74456i 0.217453i
\(963\) − 19.4891i − 0.628028i
\(964\) −24.2337 −0.780515
\(965\) 0 0
\(966\) 41.4891 1.33489
\(967\) − 28.0000i − 0.900419i −0.892923 0.450210i \(-0.851349\pi\)
0.892923 0.450210i \(-0.148651\pi\)
\(968\) 5.37228i 0.172672i
\(969\) 1.48913 0.0478376
\(970\) 0 0
\(971\) 20.6060 0.661277 0.330639 0.943757i \(-0.392736\pi\)
0.330639 + 0.943757i \(0.392736\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) − 83.5842i − 2.67959i
\(974\) −19.7228 −0.631960
\(975\) 0 0
\(976\) 0.372281 0.0119164
\(977\) − 29.1168i − 0.931530i −0.884908 0.465765i \(-0.845779\pi\)
0.884908 0.465765i \(-0.154221\pi\)
\(978\) − 27.2554i − 0.871533i
\(979\) −23.7228 −0.758184
\(980\) 0 0
\(981\) −17.1168 −0.546499
\(982\) − 30.9783i − 0.988556i
\(983\) − 2.13859i − 0.0682105i −0.999418 0.0341053i \(-0.989142\pi\)
0.999418 0.0341053i \(-0.0108581\pi\)
\(984\) 0.744563 0.0237358
\(985\) 0 0
\(986\) −3.39403 −0.108088
\(987\) − 24.0000i − 0.763928i
\(988\) 13.4891i 0.429146i
\(989\) 7.72281 0.245571
\(990\) 0 0
\(991\) −14.6060 −0.463974 −0.231987 0.972719i \(-0.574523\pi\)
−0.231987 + 0.972719i \(0.574523\pi\)
\(992\) − 8.37228i − 0.265820i
\(993\) − 1.02175i − 0.0324242i
\(994\) 20.7446 0.657978
\(995\) 0 0
\(996\) 21.4891 0.680909
\(997\) − 8.23369i − 0.260764i −0.991464 0.130382i \(-0.958380\pi\)
0.991464 0.130382i \(-0.0416203\pi\)
\(998\) − 20.9783i − 0.664055i
\(999\) 4.00000 0.126554
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 1850.2.b.m.149.3 4
5.2 odd 4 1850.2.a.q.1.2 2
5.3 odd 4 370.2.a.f.1.1 2
5.4 even 2 inner 1850.2.b.m.149.2 4
15.8 even 4 3330.2.a.bb.1.1 2
20.3 even 4 2960.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.1 2 5.3 odd 4
1850.2.a.q.1.2 2 5.2 odd 4
1850.2.b.m.149.2 4 5.4 even 2 inner
1850.2.b.m.149.3 4 1.1 even 1 trivial
2960.2.a.o.1.2 2 20.3 even 4
3330.2.a.bb.1.1 2 15.8 even 4