Properties

Label 1850.2.b.m.149.4
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 149.4
Root \(3.37228i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.m.149.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.37228i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.37228i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.37228 q^{11} +2.00000i q^{12} +4.74456i q^{13} -1.37228 q^{14} +1.00000 q^{16} -5.37228i q^{17} -1.00000i q^{18} +2.00000 q^{19} +2.74456 q^{21} -3.37228i q^{22} -6.74456i q^{23} -2.00000 q^{24} -4.74456 q^{26} -4.00000i q^{27} -1.37228i q^{28} -8.11684 q^{29} -2.62772 q^{31} +1.00000i q^{32} +6.74456i q^{33} +5.37228 q^{34} +1.00000 q^{36} +1.00000i q^{37} +2.00000i q^{38} +9.48913 q^{39} +5.37228 q^{41} +2.74456i q^{42} -7.37228i q^{43} +3.37228 q^{44} +6.74456 q^{46} -8.74456i q^{47} -2.00000i q^{48} +5.11684 q^{49} -10.7446 q^{51} -4.74456i q^{52} -1.37228i q^{53} +4.00000 q^{54} +1.37228 q^{56} -4.00000i q^{57} -8.11684i q^{58} -12.7446 q^{59} -5.37228 q^{61} -2.62772i q^{62} -1.37228i q^{63} -1.00000 q^{64} -6.74456 q^{66} +4.74456i q^{67} +5.37228i q^{68} -13.4891 q^{69} -6.74456 q^{71} +1.00000i q^{72} -8.74456i q^{73} -1.00000 q^{74} -2.00000 q^{76} -4.62772i q^{77} +9.48913i q^{78} +4.74456 q^{79} -11.0000 q^{81} +5.37228i q^{82} +0.744563i q^{83} -2.74456 q^{84} +7.37228 q^{86} +16.2337i q^{87} +3.37228i q^{88} -10.0000 q^{89} -6.51087 q^{91} +6.74456i q^{92} +5.25544i q^{93} +8.74456 q^{94} +2.00000 q^{96} -0.116844i q^{97} +5.11684i q^{98} +3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) 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
\(7\) 1.37228i 0.518674i 0.965787 + 0.259337i \(0.0835040\pi\)
−0.965787 + 0.259337i \(0.916496\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.37228 −1.01678 −0.508391 0.861127i \(-0.669759\pi\)
−0.508391 + 0.861127i \(0.669759\pi\)
\(12\) 2.00000i 0.577350i
\(13\) 4.74456i 1.31590i 0.753059 + 0.657952i \(0.228577\pi\)
−0.753059 + 0.657952i \(0.771423\pi\)
\(14\) −1.37228 −0.366758
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 5.37228i − 1.30297i −0.758662 0.651485i \(-0.774146\pi\)
0.758662 0.651485i \(-0.225854\pi\)
\(18\) − 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
\(21\) 2.74456 0.598913
\(22\) − 3.37228i − 0.718973i
\(23\) − 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
\(26\) −4.74456 −0.930485
\(27\) − 4.00000i − 0.769800i
\(28\) − 1.37228i − 0.259337i
\(29\) −8.11684 −1.50726 −0.753630 0.657299i \(-0.771699\pi\)
−0.753630 + 0.657299i \(0.771699\pi\)
\(30\) 0 0
\(31\) −2.62772 −0.471952 −0.235976 0.971759i \(-0.575829\pi\)
−0.235976 + 0.971759i \(0.575829\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 6.74456i 1.17408i
\(34\) 5.37228 0.921339
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000i 0.164399i
\(38\) 2.00000i 0.324443i
\(39\) 9.48913 1.51948
\(40\) 0 0
\(41\) 5.37228 0.839009 0.419505 0.907753i \(-0.362204\pi\)
0.419505 + 0.907753i \(0.362204\pi\)
\(42\) 2.74456i 0.423495i
\(43\) − 7.37228i − 1.12426i −0.827048 0.562131i \(-0.809982\pi\)
0.827048 0.562131i \(-0.190018\pi\)
\(44\) 3.37228 0.508391
\(45\) 0 0
\(46\) 6.74456 0.994432
\(47\) − 8.74456i − 1.27553i −0.770233 0.637763i \(-0.779860\pi\)
0.770233 0.637763i \(-0.220140\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) 5.11684 0.730978
\(50\) 0 0
\(51\) −10.7446 −1.50454
\(52\) − 4.74456i − 0.657952i
\(53\) − 1.37228i − 0.188497i −0.995549 0.0942487i \(-0.969955\pi\)
0.995549 0.0942487i \(-0.0300449\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.37228 0.183379
\(57\) − 4.00000i − 0.529813i
\(58\) − 8.11684i − 1.06579i
\(59\) −12.7446 −1.65920 −0.829600 0.558358i \(-0.811432\pi\)
−0.829600 + 0.558358i \(0.811432\pi\)
\(60\) 0 0
\(61\) −5.37228 −0.687850 −0.343925 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) − 2.62772i − 0.333721i
\(63\) − 1.37228i − 0.172891i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.74456 −0.830198
\(67\) 4.74456i 0.579641i 0.957081 + 0.289820i \(0.0935955\pi\)
−0.957081 + 0.289820i \(0.906404\pi\)
\(68\) 5.37228i 0.651485i
\(69\) −13.4891 −1.62390
\(70\) 0 0
\(71\) −6.74456 −0.800432 −0.400216 0.916421i \(-0.631065\pi\)
−0.400216 + 0.916421i \(0.631065\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 8.74456i − 1.02347i −0.859142 0.511737i \(-0.829002\pi\)
0.859142 0.511737i \(-0.170998\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) − 4.62772i − 0.527377i
\(78\) 9.48913i 1.07443i
\(79\) 4.74456 0.533805 0.266903 0.963724i \(-0.414000\pi\)
0.266903 + 0.963724i \(0.414000\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 5.37228i 0.593269i
\(83\) 0.744563i 0.0817264i 0.999165 + 0.0408632i \(0.0130108\pi\)
−0.999165 + 0.0408632i \(0.986989\pi\)
\(84\) −2.74456 −0.299456
\(85\) 0 0
\(86\) 7.37228 0.794974
\(87\) 16.2337i 1.74043i
\(88\) 3.37228i 0.359486i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −6.51087 −0.682525
\(92\) 6.74456i 0.703169i
\(93\) 5.25544i 0.544963i
\(94\) 8.74456 0.901933
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) − 0.116844i − 0.0118637i −0.999982 0.00593185i \(-0.998112\pi\)
0.999982 0.00593185i \(-0.00188818\pi\)
\(98\) 5.11684i 0.516879i
\(99\) 3.37228 0.338927
\(100\) 0 0
\(101\) −11.4891 −1.14321 −0.571605 0.820529i \(-0.693679\pi\)
−0.571605 + 0.820529i \(0.693679\pi\)
\(102\) − 10.7446i − 1.06387i
\(103\) 9.48913i 0.934991i 0.883995 + 0.467496i \(0.154844\pi\)
−0.883995 + 0.467496i \(0.845156\pi\)
\(104\) 4.74456 0.465243
\(105\) 0 0
\(106\) 1.37228 0.133288
\(107\) − 3.48913i − 0.337306i −0.985675 0.168653i \(-0.946058\pi\)
0.985675 0.168653i \(-0.0539418\pi\)
\(108\) 4.00000i 0.384900i
\(109\) −0.116844 −0.0111916 −0.00559581 0.999984i \(-0.501781\pi\)
−0.00559581 + 0.999984i \(0.501781\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 1.37228i 0.129668i
\(113\) − 17.3723i − 1.63425i −0.576463 0.817123i \(-0.695567\pi\)
0.576463 0.817123i \(-0.304433\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 8.11684 0.753630
\(117\) − 4.74456i − 0.438635i
\(118\) − 12.7446i − 1.17323i
\(119\) 7.37228 0.675816
\(120\) 0 0
\(121\) 0.372281 0.0338438
\(122\) − 5.37228i − 0.486383i
\(123\) − 10.7446i − 0.968805i
\(124\) 2.62772 0.235976
\(125\) 0 0
\(126\) 1.37228 0.122253
\(127\) − 16.7446i − 1.48584i −0.669380 0.742920i \(-0.733440\pi\)
0.669380 0.742920i \(-0.266560\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −14.7446 −1.29819
\(130\) 0 0
\(131\) −3.25544 −0.284429 −0.142214 0.989836i \(-0.545422\pi\)
−0.142214 + 0.989836i \(0.545422\pi\)
\(132\) − 6.74456i − 0.587039i
\(133\) 2.74456i 0.237984i
\(134\) −4.74456 −0.409868
\(135\) 0 0
\(136\) −5.37228 −0.460669
\(137\) 16.7446i 1.43058i 0.698825 + 0.715292i \(0.253706\pi\)
−0.698825 + 0.715292i \(0.746294\pi\)
\(138\) − 13.4891i − 1.14827i
\(139\) 1.88316 0.159727 0.0798636 0.996806i \(-0.474552\pi\)
0.0798636 + 0.996806i \(0.474552\pi\)
\(140\) 0 0
\(141\) −17.4891 −1.47285
\(142\) − 6.74456i − 0.565991i
\(143\) − 16.0000i − 1.33799i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 8.74456 0.723705
\(147\) − 10.2337i − 0.844060i
\(148\) − 1.00000i − 0.0821995i
\(149\) 11.4891 0.941226 0.470613 0.882340i \(-0.344033\pi\)
0.470613 + 0.882340i \(0.344033\pi\)
\(150\) 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
\(153\) 5.37228i 0.434323i
\(154\) 4.62772 0.372912
\(155\) 0 0
\(156\) −9.48913 −0.759738
\(157\) 17.3723i 1.38646i 0.720717 + 0.693229i \(0.243813\pi\)
−0.720717 + 0.693229i \(0.756187\pi\)
\(158\) 4.74456i 0.377457i
\(159\) −2.74456 −0.217658
\(160\) 0 0
\(161\) 9.25544 0.729431
\(162\) − 11.0000i − 0.864242i
\(163\) − 19.3723i − 1.51735i −0.651467 0.758677i \(-0.725846\pi\)
0.651467 0.758677i \(-0.274154\pi\)
\(164\) −5.37228 −0.419505
\(165\) 0 0
\(166\) −0.744563 −0.0577893
\(167\) 1.48913i 0.115232i 0.998339 + 0.0576160i \(0.0183499\pi\)
−0.998339 + 0.0576160i \(0.981650\pi\)
\(168\) − 2.74456i − 0.211748i
\(169\) −9.51087 −0.731606
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 7.37228i 0.562131i
\(173\) − 22.8614i − 1.73812i −0.494706 0.869060i \(-0.664724\pi\)
0.494706 0.869060i \(-0.335276\pi\)
\(174\) −16.2337 −1.23067
\(175\) 0 0
\(176\) −3.37228 −0.254195
\(177\) 25.4891i 1.91588i
\(178\) − 10.0000i − 0.749532i
\(179\) 26.2337 1.96080 0.980399 0.197023i \(-0.0631272\pi\)
0.980399 + 0.197023i \(0.0631272\pi\)
\(180\) 0 0
\(181\) 7.48913 0.556662 0.278331 0.960485i \(-0.410219\pi\)
0.278331 + 0.960485i \(0.410219\pi\)
\(182\) − 6.51087i − 0.482618i
\(183\) 10.7446i 0.794261i
\(184\) −6.74456 −0.497216
\(185\) 0 0
\(186\) −5.25544 −0.385347
\(187\) 18.1168i 1.32483i
\(188\) 8.74456i 0.637763i
\(189\) 5.48913 0.399275
\(190\) 0 0
\(191\) 18.6277 1.34785 0.673927 0.738798i \(-0.264606\pi\)
0.673927 + 0.738798i \(0.264606\pi\)
\(192\) 2.00000i 0.144338i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 0.116844 0.00838891
\(195\) 0 0
\(196\) −5.11684 −0.365489
\(197\) 27.4891i 1.95852i 0.202610 + 0.979260i \(0.435058\pi\)
−0.202610 + 0.979260i \(0.564942\pi\)
\(198\) 3.37228i 0.239658i
\(199\) 11.2554 0.797877 0.398938 0.916978i \(-0.369379\pi\)
0.398938 + 0.916978i \(0.369379\pi\)
\(200\) 0 0
\(201\) 9.48913 0.669311
\(202\) − 11.4891i − 0.808372i
\(203\) − 11.1386i − 0.781776i
\(204\) 10.7446 0.752270
\(205\) 0 0
\(206\) −9.48913 −0.661139
\(207\) 6.74456i 0.468780i
\(208\) 4.74456i 0.328976i
\(209\) −6.74456 −0.466531
\(210\) 0 0
\(211\) −14.1168 −0.971844 −0.485922 0.874002i \(-0.661516\pi\)
−0.485922 + 0.874002i \(0.661516\pi\)
\(212\) 1.37228i 0.0942487i
\(213\) 13.4891i 0.924260i
\(214\) 3.48913 0.238512
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) − 3.60597i − 0.244789i
\(218\) − 0.116844i − 0.00791367i
\(219\) −17.4891 −1.18181
\(220\) 0 0
\(221\) 25.4891 1.71458
\(222\) 2.00000i 0.134231i
\(223\) 1.37228i 0.0918948i 0.998944 + 0.0459474i \(0.0146307\pi\)
−0.998944 + 0.0459474i \(0.985369\pi\)
\(224\) −1.37228 −0.0916894
\(225\) 0 0
\(226\) 17.3723 1.15559
\(227\) − 11.3723i − 0.754805i −0.926049 0.377402i \(-0.876817\pi\)
0.926049 0.377402i \(-0.123183\pi\)
\(228\) 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
\(231\) −9.25544 −0.608963
\(232\) 8.11684i 0.532897i
\(233\) − 6.23369i − 0.408382i −0.978931 0.204191i \(-0.934544\pi\)
0.978931 0.204191i \(-0.0654564\pi\)
\(234\) 4.74456 0.310162
\(235\) 0 0
\(236\) 12.7446 0.829600
\(237\) − 9.48913i − 0.616385i
\(238\) 7.37228i 0.477874i
\(239\) −17.6060 −1.13884 −0.569418 0.822048i \(-0.692831\pi\)
−0.569418 + 0.822048i \(0.692831\pi\)
\(240\) 0 0
\(241\) −10.2337 −0.659210 −0.329605 0.944119i \(-0.606916\pi\)
−0.329605 + 0.944119i \(0.606916\pi\)
\(242\) 0.372281i 0.0239311i
\(243\) 10.0000i 0.641500i
\(244\) 5.37228 0.343925
\(245\) 0 0
\(246\) 10.7446 0.685048
\(247\) 9.48913i 0.603779i
\(248\) 2.62772i 0.166860i
\(249\) 1.48913 0.0943695
\(250\) 0 0
\(251\) 11.4891 0.725187 0.362594 0.931947i \(-0.381891\pi\)
0.362594 + 0.931947i \(0.381891\pi\)
\(252\) 1.37228i 0.0864456i
\(253\) 22.7446i 1.42994i
\(254\) 16.7446 1.05065
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 20.9783i − 1.30859i −0.756241 0.654294i \(-0.772966\pi\)
0.756241 0.654294i \(-0.227034\pi\)
\(258\) − 14.7446i − 0.917956i
\(259\) −1.37228 −0.0852694
\(260\) 0 0
\(261\) 8.11684 0.502420
\(262\) − 3.25544i − 0.201122i
\(263\) − 0.116844i − 0.00720491i −0.999994 0.00360245i \(-0.998853\pi\)
0.999994 0.00360245i \(-0.00114670\pi\)
\(264\) 6.74456 0.415099
\(265\) 0 0
\(266\) −2.74456 −0.168280
\(267\) 20.0000i 1.22398i
\(268\) − 4.74456i − 0.289820i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −6.74456 −0.409703 −0.204852 0.978793i \(-0.565671\pi\)
−0.204852 + 0.978793i \(0.565671\pi\)
\(272\) − 5.37228i − 0.325742i
\(273\) 13.0217i 0.788112i
\(274\) −16.7446 −1.01158
\(275\) 0 0
\(276\) 13.4891 0.811950
\(277\) 0.744563i 0.0447364i 0.999750 + 0.0223682i \(0.00712062\pi\)
−0.999750 + 0.0223682i \(0.992879\pi\)
\(278\) 1.88316i 0.112944i
\(279\) 2.62772 0.157317
\(280\) 0 0
\(281\) 24.7446 1.47614 0.738068 0.674726i \(-0.235738\pi\)
0.738068 + 0.674726i \(0.235738\pi\)
\(282\) − 17.4891i − 1.04146i
\(283\) − 5.48913i − 0.326295i −0.986602 0.163147i \(-0.947835\pi\)
0.986602 0.163147i \(-0.0521646\pi\)
\(284\) 6.74456 0.400216
\(285\) 0 0
\(286\) 16.0000 0.946100
\(287\) 7.37228i 0.435172i
\(288\) − 1.00000i − 0.0589256i
\(289\) −11.8614 −0.697730
\(290\) 0 0
\(291\) −0.233688 −0.0136990
\(292\) 8.74456i 0.511737i
\(293\) 18.8614i 1.10190i 0.834540 + 0.550948i \(0.185734\pi\)
−0.834540 + 0.550948i \(0.814266\pi\)
\(294\) 10.2337 0.596841
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 13.4891i 0.782718i
\(298\) 11.4891i 0.665547i
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) 10.1168 0.583125
\(302\) − 20.0000i − 1.15087i
\(303\) 22.9783i 1.32007i
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) −5.37228 −0.307113
\(307\) − 23.4891i − 1.34060i −0.742092 0.670298i \(-0.766166\pi\)
0.742092 0.670298i \(-0.233834\pi\)
\(308\) 4.62772i 0.263689i
\(309\) 18.9783 1.07963
\(310\) 0 0
\(311\) −1.37228 −0.0778149 −0.0389075 0.999243i \(-0.512388\pi\)
−0.0389075 + 0.999243i \(0.512388\pi\)
\(312\) − 9.48913i − 0.537216i
\(313\) 3.48913i 0.197217i 0.995126 + 0.0986085i \(0.0314392\pi\)
−0.995126 + 0.0986085i \(0.968561\pi\)
\(314\) −17.3723 −0.980375
\(315\) 0 0
\(316\) −4.74456 −0.266903
\(317\) 25.3723i 1.42505i 0.701647 + 0.712525i \(0.252448\pi\)
−0.701647 + 0.712525i \(0.747552\pi\)
\(318\) − 2.74456i − 0.153907i
\(319\) 27.3723 1.53255
\(320\) 0 0
\(321\) −6.97825 −0.389488
\(322\) 9.25544i 0.515785i
\(323\) − 10.7446i − 0.597843i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 19.3723 1.07293
\(327\) 0.233688i 0.0129230i
\(328\) − 5.37228i − 0.296635i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 23.4891 1.29108 0.645540 0.763727i \(-0.276633\pi\)
0.645540 + 0.763727i \(0.276633\pi\)
\(332\) − 0.744563i − 0.0408632i
\(333\) − 1.00000i − 0.0547997i
\(334\) −1.48913 −0.0814813
\(335\) 0 0
\(336\) 2.74456 0.149728
\(337\) − 7.25544i − 0.395229i −0.980280 0.197614i \(-0.936681\pi\)
0.980280 0.197614i \(-0.0633194\pi\)
\(338\) − 9.51087i − 0.517323i
\(339\) −34.7446 −1.88707
\(340\) 0 0
\(341\) 8.86141 0.479872
\(342\) − 2.00000i − 0.108148i
\(343\) 16.6277i 0.897812i
\(344\) −7.37228 −0.397487
\(345\) 0 0
\(346\) 22.8614 1.22904
\(347\) − 22.9783i − 1.23354i −0.787145 0.616769i \(-0.788441\pi\)
0.787145 0.616769i \(-0.211559\pi\)
\(348\) − 16.2337i − 0.870217i
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 18.9783 1.01298
\(352\) − 3.37228i − 0.179743i
\(353\) 22.8614i 1.21679i 0.793634 + 0.608395i \(0.208186\pi\)
−0.793634 + 0.608395i \(0.791814\pi\)
\(354\) −25.4891 −1.35473
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) − 14.7446i − 0.780365i
\(358\) 26.2337i 1.38649i
\(359\) 30.9783 1.63497 0.817485 0.575950i \(-0.195368\pi\)
0.817485 + 0.575950i \(0.195368\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 7.48913i 0.393620i
\(363\) − 0.744563i − 0.0390794i
\(364\) 6.51087 0.341263
\(365\) 0 0
\(366\) −10.7446 −0.561627
\(367\) 3.88316i 0.202699i 0.994851 + 0.101350i \(0.0323160\pi\)
−0.994851 + 0.101350i \(0.967684\pi\)
\(368\) − 6.74456i − 0.351585i
\(369\) −5.37228 −0.279670
\(370\) 0 0
\(371\) 1.88316 0.0977686
\(372\) − 5.25544i − 0.272482i
\(373\) 31.4891i 1.63045i 0.579148 + 0.815223i \(0.303385\pi\)
−0.579148 + 0.815223i \(0.696615\pi\)
\(374\) −18.1168 −0.936800
\(375\) 0 0
\(376\) −8.74456 −0.450966
\(377\) − 38.5109i − 1.98341i
\(378\) 5.48913i 0.282330i
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −33.4891 −1.71570
\(382\) 18.6277i 0.953077i
\(383\) 13.4891i 0.689262i 0.938738 + 0.344631i \(0.111996\pi\)
−0.938738 + 0.344631i \(0.888004\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 7.37228i 0.374754i
\(388\) 0.116844i 0.00593185i
\(389\) −14.8614 −0.753503 −0.376752 0.926314i \(-0.622959\pi\)
−0.376752 + 0.926314i \(0.622959\pi\)
\(390\) 0 0
\(391\) −36.2337 −1.83242
\(392\) − 5.11684i − 0.258440i
\(393\) 6.51087i 0.328430i
\(394\) −27.4891 −1.38488
\(395\) 0 0
\(396\) −3.37228 −0.169464
\(397\) 24.9783i 1.25362i 0.779171 + 0.626811i \(0.215640\pi\)
−0.779171 + 0.626811i \(0.784360\pi\)
\(398\) 11.2554i 0.564184i
\(399\) 5.48913 0.274800
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 9.48913i 0.473275i
\(403\) − 12.4674i − 0.621044i
\(404\) 11.4891 0.571605
\(405\) 0 0
\(406\) 11.1386 0.552799
\(407\) − 3.37228i − 0.167158i
\(408\) 10.7446i 0.531935i
\(409\) −6.23369 −0.308236 −0.154118 0.988052i \(-0.549254\pi\)
−0.154118 + 0.988052i \(0.549254\pi\)
\(410\) 0 0
\(411\) 33.4891 1.65190
\(412\) − 9.48913i − 0.467496i
\(413\) − 17.4891i − 0.860584i
\(414\) −6.74456 −0.331477
\(415\) 0 0
\(416\) −4.74456 −0.232621
\(417\) − 3.76631i − 0.184437i
\(418\) − 6.74456i − 0.329887i
\(419\) −13.4891 −0.658987 −0.329493 0.944158i \(-0.606878\pi\)
−0.329493 + 0.944158i \(0.606878\pi\)
\(420\) 0 0
\(421\) −7.48913 −0.364998 −0.182499 0.983206i \(-0.558419\pi\)
−0.182499 + 0.983206i \(0.558419\pi\)
\(422\) − 14.1168i − 0.687197i
\(423\) 8.74456i 0.425175i
\(424\) −1.37228 −0.0666439
\(425\) 0 0
\(426\) −13.4891 −0.653550
\(427\) − 7.37228i − 0.356770i
\(428\) 3.48913i 0.168653i
\(429\) −32.0000 −1.54497
\(430\) 0 0
\(431\) −1.37228 −0.0661005 −0.0330502 0.999454i \(-0.510522\pi\)
−0.0330502 + 0.999454i \(0.510522\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) − 12.9783i − 0.623695i −0.950132 0.311847i \(-0.899052\pi\)
0.950132 0.311847i \(-0.100948\pi\)
\(434\) 3.60597 0.173092
\(435\) 0 0
\(436\) 0.116844 0.00559581
\(437\) − 13.4891i − 0.645272i
\(438\) − 17.4891i − 0.835663i
\(439\) 13.3723 0.638224 0.319112 0.947717i \(-0.396615\pi\)
0.319112 + 0.947717i \(0.396615\pi\)
\(440\) 0 0
\(441\) −5.11684 −0.243659
\(442\) 25.4891i 1.21239i
\(443\) 16.9783i 0.806661i 0.915054 + 0.403331i \(0.132148\pi\)
−0.915054 + 0.403331i \(0.867852\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −1.37228 −0.0649794
\(447\) − 22.9783i − 1.08683i
\(448\) − 1.37228i − 0.0648342i
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −18.1168 −0.853089
\(452\) 17.3723i 0.817123i
\(453\) 40.0000i 1.87936i
\(454\) 11.3723 0.533728
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 18.8614i 0.882299i 0.897434 + 0.441150i \(0.145429\pi\)
−0.897434 + 0.441150i \(0.854571\pi\)
\(458\) 10.0000i 0.467269i
\(459\) −21.4891 −1.00303
\(460\) 0 0
\(461\) −39.0951 −1.82084 −0.910420 0.413685i \(-0.864241\pi\)
−0.910420 + 0.413685i \(0.864241\pi\)
\(462\) − 9.25544i − 0.430602i
\(463\) 5.48913i 0.255101i 0.991832 + 0.127551i \(0.0407115\pi\)
−0.991832 + 0.127551i \(0.959288\pi\)
\(464\) −8.11684 −0.376815
\(465\) 0 0
\(466\) 6.23369 0.288770
\(467\) 30.3505i 1.40446i 0.711953 + 0.702228i \(0.247811\pi\)
−0.711953 + 0.702228i \(0.752189\pi\)
\(468\) 4.74456i 0.219317i
\(469\) −6.51087 −0.300644
\(470\) 0 0
\(471\) 34.7446 1.60094
\(472\) 12.7446i 0.586616i
\(473\) 24.8614i 1.14313i
\(474\) 9.48913 0.435850
\(475\) 0 0
\(476\) −7.37228 −0.337908
\(477\) 1.37228i 0.0628324i
\(478\) − 17.6060i − 0.805278i
\(479\) −3.25544 −0.148745 −0.0743724 0.997231i \(-0.523695\pi\)
−0.0743724 + 0.997231i \(0.523695\pi\)
\(480\) 0 0
\(481\) −4.74456 −0.216333
\(482\) − 10.2337i − 0.466132i
\(483\) − 18.5109i − 0.842274i
\(484\) −0.372281 −0.0169219
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) − 37.7228i − 1.70938i −0.519136 0.854692i \(-0.673746\pi\)
0.519136 0.854692i \(-0.326254\pi\)
\(488\) 5.37228i 0.243192i
\(489\) −38.7446 −1.75209
\(490\) 0 0
\(491\) 14.9783 0.675959 0.337979 0.941153i \(-0.390257\pi\)
0.337979 + 0.941153i \(0.390257\pi\)
\(492\) 10.7446i 0.484402i
\(493\) 43.6060i 1.96391i
\(494\) −9.48913 −0.426936
\(495\) 0 0
\(496\) −2.62772 −0.117988
\(497\) − 9.25544i − 0.415163i
\(498\) 1.48913i 0.0667293i
\(499\) 24.9783 1.11818 0.559090 0.829107i \(-0.311151\pi\)
0.559090 + 0.829107i \(0.311151\pi\)
\(500\) 0 0
\(501\) 2.97825 0.133058
\(502\) 11.4891i 0.512785i
\(503\) − 29.4891i − 1.31486i −0.753518 0.657428i \(-0.771645\pi\)
0.753518 0.657428i \(-0.228355\pi\)
\(504\) −1.37228 −0.0611263
\(505\) 0 0
\(506\) −22.7446 −1.01112
\(507\) 19.0217i 0.844786i
\(508\) 16.7446i 0.742920i
\(509\) 11.2554 0.498888 0.249444 0.968389i \(-0.419752\pi\)
0.249444 + 0.968389i \(0.419752\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 1.00000i 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) 20.9783 0.925311
\(515\) 0 0
\(516\) 14.7446 0.649093
\(517\) 29.4891i 1.29693i
\(518\) − 1.37228i − 0.0602946i
\(519\) −45.7228 −2.00701
\(520\) 0 0
\(521\) 27.0951 1.18706 0.593529 0.804813i \(-0.297734\pi\)
0.593529 + 0.804813i \(0.297734\pi\)
\(522\) 8.11684i 0.355265i
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 3.25544 0.142214
\(525\) 0 0
\(526\) 0.116844 0.00509464
\(527\) 14.1168i 0.614939i
\(528\) 6.74456i 0.293519i
\(529\) −22.4891 −0.977788
\(530\) 0 0
\(531\) 12.7446 0.553067
\(532\) − 2.74456i − 0.118992i
\(533\) 25.4891i 1.10406i
\(534\) −20.0000 −0.865485
\(535\) 0 0
\(536\) 4.74456 0.204934
\(537\) − 52.4674i − 2.26413i
\(538\) − 10.0000i − 0.431131i
\(539\) −17.2554 −0.743244
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) − 6.74456i − 0.289704i
\(543\) − 14.9783i − 0.642778i
\(544\) 5.37228 0.230335
\(545\) 0 0
\(546\) −13.0217 −0.557279
\(547\) 39.3723i 1.68344i 0.539916 + 0.841719i \(0.318456\pi\)
−0.539916 + 0.841719i \(0.681544\pi\)
\(548\) − 16.7446i − 0.715292i
\(549\) 5.37228 0.229283
\(550\) 0 0
\(551\) −16.2337 −0.691578
\(552\) 13.4891i 0.574135i
\(553\) 6.51087i 0.276871i
\(554\) −0.744563 −0.0316334
\(555\) 0 0
\(556\) −1.88316 −0.0798636
\(557\) 8.74456i 0.370519i 0.982690 + 0.185260i \(0.0593126\pi\)
−0.982690 + 0.185260i \(0.940687\pi\)
\(558\) 2.62772i 0.111240i
\(559\) 34.9783 1.47942
\(560\) 0 0
\(561\) 36.2337 1.52979
\(562\) 24.7446i 1.04379i
\(563\) − 33.0951i − 1.39479i −0.716686 0.697396i \(-0.754342\pi\)
0.716686 0.697396i \(-0.245658\pi\)
\(564\) 17.4891 0.736425
\(565\) 0 0
\(566\) 5.48913 0.230725
\(567\) − 15.0951i − 0.633934i
\(568\) 6.74456i 0.282996i
\(569\) 32.9783 1.38252 0.691260 0.722606i \(-0.257056\pi\)
0.691260 + 0.722606i \(0.257056\pi\)
\(570\) 0 0
\(571\) −27.6060 −1.15527 −0.577637 0.816294i \(-0.696025\pi\)
−0.577637 + 0.816294i \(0.696025\pi\)
\(572\) 16.0000i 0.668994i
\(573\) − 37.2554i − 1.55637i
\(574\) −7.37228 −0.307713
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 7.48913i − 0.311776i −0.987775 0.155888i \(-0.950176\pi\)
0.987775 0.155888i \(-0.0498240\pi\)
\(578\) − 11.8614i − 0.493369i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −1.02175 −0.0423893
\(582\) − 0.233688i − 0.00968668i
\(583\) 4.62772i 0.191661i
\(584\) −8.74456 −0.361853
\(585\) 0 0
\(586\) −18.8614 −0.779158
\(587\) 47.8397i 1.97455i 0.159010 + 0.987277i \(0.449170\pi\)
−0.159010 + 0.987277i \(0.550830\pi\)
\(588\) 10.2337i 0.422030i
\(589\) −5.25544 −0.216547
\(590\) 0 0
\(591\) 54.9783 2.26150
\(592\) 1.00000i 0.0410997i
\(593\) 10.2337i 0.420247i 0.977675 + 0.210124i \(0.0673866\pi\)
−0.977675 + 0.210124i \(0.932613\pi\)
\(594\) −13.4891 −0.553466
\(595\) 0 0
\(596\) −11.4891 −0.470613
\(597\) − 22.5109i − 0.921309i
\(598\) 32.0000i 1.30858i
\(599\) −17.4891 −0.714586 −0.357293 0.933992i \(-0.616300\pi\)
−0.357293 + 0.933992i \(0.616300\pi\)
\(600\) 0 0
\(601\) −5.37228 −0.219140 −0.109570 0.993979i \(-0.534947\pi\)
−0.109570 + 0.993979i \(0.534947\pi\)
\(602\) 10.1168i 0.412332i
\(603\) − 4.74456i − 0.193214i
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) −22.9783 −0.933428
\(607\) 17.7228i 0.719347i 0.933078 + 0.359673i \(0.117112\pi\)
−0.933078 + 0.359673i \(0.882888\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) −22.2772 −0.902717
\(610\) 0 0
\(611\) 41.4891 1.67847
\(612\) − 5.37228i − 0.217162i
\(613\) − 43.0951i − 1.74059i −0.492527 0.870297i \(-0.663927\pi\)
0.492527 0.870297i \(-0.336073\pi\)
\(614\) 23.4891 0.947944
\(615\) 0 0
\(616\) −4.62772 −0.186456
\(617\) − 31.7228i − 1.27711i −0.769575 0.638556i \(-0.779532\pi\)
0.769575 0.638556i \(-0.220468\pi\)
\(618\) 18.9783i 0.763417i
\(619\) −30.1168 −1.21050 −0.605249 0.796036i \(-0.706926\pi\)
−0.605249 + 0.796036i \(0.706926\pi\)
\(620\) 0 0
\(621\) −26.9783 −1.08260
\(622\) − 1.37228i − 0.0550235i
\(623\) − 13.7228i − 0.549793i
\(624\) 9.48913 0.379869
\(625\) 0 0
\(626\) −3.48913 −0.139453
\(627\) 13.4891i 0.538704i
\(628\) − 17.3723i − 0.693229i
\(629\) 5.37228 0.214207
\(630\) 0 0
\(631\) −19.0951 −0.760164 −0.380082 0.924953i \(-0.624104\pi\)
−0.380082 + 0.924953i \(0.624104\pi\)
\(632\) − 4.74456i − 0.188729i
\(633\) 28.2337i 1.12219i
\(634\) −25.3723 −1.00766
\(635\) 0 0
\(636\) 2.74456 0.108829
\(637\) 24.2772i 0.961897i
\(638\) 27.3723i 1.08368i
\(639\) 6.74456 0.266811
\(640\) 0 0
\(641\) −49.6060 −1.95932 −0.979659 0.200670i \(-0.935688\pi\)
−0.979659 + 0.200670i \(0.935688\pi\)
\(642\) − 6.97825i − 0.275410i
\(643\) − 3.37228i − 0.132990i −0.997787 0.0664949i \(-0.978818\pi\)
0.997787 0.0664949i \(-0.0211816\pi\)
\(644\) −9.25544 −0.364715
\(645\) 0 0
\(646\) 10.7446 0.422739
\(647\) − 13.4891i − 0.530312i −0.964205 0.265156i \(-0.914577\pi\)
0.964205 0.265156i \(-0.0854235\pi\)
\(648\) 11.0000i 0.432121i
\(649\) 42.9783 1.68704
\(650\) 0 0
\(651\) −7.21194 −0.282658
\(652\) 19.3723i 0.758677i
\(653\) − 0.510875i − 0.0199921i −0.999950 0.00999604i \(-0.996818\pi\)
0.999950 0.00999604i \(-0.00318189\pi\)
\(654\) −0.233688 −0.00913792
\(655\) 0 0
\(656\) 5.37228 0.209752
\(657\) 8.74456i 0.341158i
\(658\) 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
\(661\) 4.35053 0.169216 0.0846080 0.996414i \(-0.473036\pi\)
0.0846080 + 0.996414i \(0.473036\pi\)
\(662\) 23.4891i 0.912931i
\(663\) − 50.9783i − 1.97983i
\(664\) 0.744563 0.0288946
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 54.7446i 2.11972i
\(668\) − 1.48913i − 0.0576160i
\(669\) 2.74456 0.106111
\(670\) 0 0
\(671\) 18.1168 0.699393
\(672\) 2.74456i 0.105874i
\(673\) 8.51087i 0.328070i 0.986455 + 0.164035i \(0.0524510\pi\)
−0.986455 + 0.164035i \(0.947549\pi\)
\(674\) 7.25544 0.279469
\(675\) 0 0
\(676\) 9.51087 0.365803
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) − 34.7446i − 1.33436i
\(679\) 0.160343 0.00615339
\(680\) 0 0
\(681\) −22.7446 −0.871574
\(682\) 8.86141i 0.339321i
\(683\) − 8.62772i − 0.330130i −0.986283 0.165065i \(-0.947217\pi\)
0.986283 0.165065i \(-0.0527835\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −16.6277 −0.634849
\(687\) − 20.0000i − 0.763048i
\(688\) − 7.37228i − 0.281066i
\(689\) 6.51087 0.248045
\(690\) 0 0
\(691\) 22.3505 0.850254 0.425127 0.905134i \(-0.360229\pi\)
0.425127 + 0.905134i \(0.360229\pi\)
\(692\) 22.8614i 0.869060i
\(693\) 4.62772i 0.175792i
\(694\) 22.9783 0.872242
\(695\) 0 0
\(696\) 16.2337 0.615336
\(697\) − 28.8614i − 1.09320i
\(698\) 22.0000i 0.832712i
\(699\) −12.4674 −0.471559
\(700\) 0 0
\(701\) −26.4674 −0.999659 −0.499829 0.866124i \(-0.666604\pi\)
−0.499829 + 0.866124i \(0.666604\pi\)
\(702\) 18.9783i 0.716288i
\(703\) 2.00000i 0.0754314i
\(704\) 3.37228 0.127098
\(705\) 0 0
\(706\) −22.8614 −0.860400
\(707\) − 15.7663i − 0.592953i
\(708\) − 25.4891i − 0.957940i
\(709\) −40.1168 −1.50662 −0.753310 0.657666i \(-0.771544\pi\)
−0.753310 + 0.657666i \(0.771544\pi\)
\(710\) 0 0
\(711\) −4.74456 −0.177935
\(712\) 10.0000i 0.374766i
\(713\) 17.7228i 0.663725i
\(714\) 14.7446 0.551801
\(715\) 0 0
\(716\) −26.2337 −0.980399
\(717\) 35.2119i 1.31501i
\(718\) 30.9783i 1.15610i
\(719\) 34.7446 1.29575 0.647877 0.761745i \(-0.275657\pi\)
0.647877 + 0.761745i \(0.275657\pi\)
\(720\) 0 0
\(721\) −13.0217 −0.484955
\(722\) − 15.0000i − 0.558242i
\(723\) 20.4674i 0.761190i
\(724\) −7.48913 −0.278331
\(725\) 0 0
\(726\) 0.744563 0.0276333
\(727\) − 48.0000i − 1.78022i −0.455744 0.890111i \(-0.650627\pi\)
0.455744 0.890111i \(-0.349373\pi\)
\(728\) 6.51087i 0.241309i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −39.6060 −1.46488
\(732\) − 10.7446i − 0.397130i
\(733\) 29.3723i 1.08489i 0.840091 + 0.542445i \(0.182501\pi\)
−0.840091 + 0.542445i \(0.817499\pi\)
\(734\) −3.88316 −0.143330
\(735\) 0 0
\(736\) 6.74456 0.248608
\(737\) − 16.0000i − 0.589368i
\(738\) − 5.37228i − 0.197756i
\(739\) 36.8614 1.35597 0.677984 0.735076i \(-0.262854\pi\)
0.677984 + 0.735076i \(0.262854\pi\)
\(740\) 0 0
\(741\) 18.9783 0.697183
\(742\) 1.88316i 0.0691328i
\(743\) 5.37228i 0.197090i 0.995133 + 0.0985449i \(0.0314188\pi\)
−0.995133 + 0.0985449i \(0.968581\pi\)
\(744\) 5.25544 0.192674
\(745\) 0 0
\(746\) −31.4891 −1.15290
\(747\) − 0.744563i − 0.0272421i
\(748\) − 18.1168i − 0.662417i
\(749\) 4.78806 0.174952
\(750\) 0 0
\(751\) 10.7446 0.392075 0.196037 0.980596i \(-0.437193\pi\)
0.196037 + 0.980596i \(0.437193\pi\)
\(752\) − 8.74456i − 0.318881i
\(753\) − 22.9783i − 0.837374i
\(754\) 38.5109 1.40248
\(755\) 0 0
\(756\) −5.48913 −0.199638
\(757\) 43.9565i 1.59763i 0.601579 + 0.798813i \(0.294538\pi\)
−0.601579 + 0.798813i \(0.705462\pi\)
\(758\) 8.00000i 0.290573i
\(759\) 45.4891 1.65115
\(760\) 0 0
\(761\) 5.37228 0.194745 0.0973725 0.995248i \(-0.468956\pi\)
0.0973725 + 0.995248i \(0.468956\pi\)
\(762\) − 33.4891i − 1.21318i
\(763\) − 0.160343i − 0.00580480i
\(764\) −18.6277 −0.673927
\(765\) 0 0
\(766\) −13.4891 −0.487382
\(767\) − 60.4674i − 2.18335i
\(768\) − 2.00000i − 0.0721688i
\(769\) −46.2337 −1.66723 −0.833615 0.552346i \(-0.813733\pi\)
−0.833615 + 0.552346i \(0.813733\pi\)
\(770\) 0 0
\(771\) −41.9565 −1.51103
\(772\) 2.00000i 0.0719816i
\(773\) 24.3505i 0.875828i 0.899017 + 0.437914i \(0.144283\pi\)
−0.899017 + 0.437914i \(0.855717\pi\)
\(774\) −7.37228 −0.264991
\(775\) 0 0
\(776\) −0.116844 −0.00419445
\(777\) 2.74456i 0.0984606i
\(778\) − 14.8614i − 0.532807i
\(779\) 10.7446 0.384964
\(780\) 0 0
\(781\) 22.7446 0.813864
\(782\) − 36.2337i − 1.29571i
\(783\) 32.4674i 1.16029i
\(784\) 5.11684 0.182744
\(785\) 0 0
\(786\) −6.51087 −0.232235
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) − 27.4891i − 0.979260i
\(789\) −0.233688 −0.00831951
\(790\) 0 0
\(791\) 23.8397 0.847641
\(792\) − 3.37228i − 0.119829i
\(793\) − 25.4891i − 0.905145i
\(794\) −24.9783 −0.886445
\(795\) 0 0
\(796\) −11.2554 −0.398938
\(797\) − 27.4891i − 0.973715i −0.873481 0.486857i \(-0.838143\pi\)
0.873481 0.486857i \(-0.161857\pi\)
\(798\) 5.48913i 0.194313i
\(799\) −46.9783 −1.66197
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 10.0000i 0.353112i
\(803\) 29.4891i 1.04065i
\(804\) −9.48913 −0.334656
\(805\) 0 0
\(806\) 12.4674 0.439145
\(807\) 20.0000i 0.704033i
\(808\) 11.4891i 0.404186i
\(809\) 51.4891 1.81026 0.905131 0.425134i \(-0.139773\pi\)
0.905131 + 0.425134i \(0.139773\pi\)
\(810\) 0 0
\(811\) 49.4891 1.73780 0.868899 0.494989i \(-0.164828\pi\)
0.868899 + 0.494989i \(0.164828\pi\)
\(812\) 11.1386i 0.390888i
\(813\) 13.4891i 0.473084i
\(814\) 3.37228 0.118198
\(815\) 0 0
\(816\) −10.7446 −0.376135
\(817\) − 14.7446i − 0.515847i
\(818\) − 6.23369i − 0.217956i
\(819\) 6.51087 0.227508
\(820\) 0 0
\(821\) −32.7446 −1.14279 −0.571397 0.820674i \(-0.693598\pi\)
−0.571397 + 0.820674i \(0.693598\pi\)
\(822\) 33.4891i 1.16807i
\(823\) 39.7228i 1.38465i 0.721586 + 0.692325i \(0.243414\pi\)
−0.721586 + 0.692325i \(0.756586\pi\)
\(824\) 9.48913 0.330569
\(825\) 0 0
\(826\) 17.4891 0.608524
\(827\) − 54.3505i − 1.88995i −0.327138 0.944977i \(-0.606084\pi\)
0.327138 0.944977i \(-0.393916\pi\)
\(828\) − 6.74456i − 0.234390i
\(829\) 40.3505 1.40143 0.700716 0.713440i \(-0.252864\pi\)
0.700716 + 0.713440i \(0.252864\pi\)
\(830\) 0 0
\(831\) 1.48913 0.0516572
\(832\) − 4.74456i − 0.164488i
\(833\) − 27.4891i − 0.952442i
\(834\) 3.76631 0.130417
\(835\) 0 0
\(836\) 6.74456 0.233266
\(837\) 10.5109i 0.363309i
\(838\) − 13.4891i − 0.465974i
\(839\) 29.4891 1.01808 0.509039 0.860744i \(-0.330001\pi\)
0.509039 + 0.860744i \(0.330001\pi\)
\(840\) 0 0
\(841\) 36.8832 1.27183
\(842\) − 7.48913i − 0.258092i
\(843\) − 49.4891i − 1.70450i
\(844\) 14.1168 0.485922
\(845\) 0 0
\(846\) −8.74456 −0.300644
\(847\) 0.510875i 0.0175539i
\(848\) − 1.37228i − 0.0471243i
\(849\) −10.9783 −0.376773
\(850\) 0 0
\(851\) 6.74456 0.231201
\(852\) − 13.4891i − 0.462130i
\(853\) − 35.4891i − 1.21512i −0.794272 0.607562i \(-0.792148\pi\)
0.794272 0.607562i \(-0.207852\pi\)
\(854\) 7.37228 0.252274
\(855\) 0 0
\(856\) −3.48913 −0.119256
\(857\) − 29.1386i − 0.995355i −0.867362 0.497678i \(-0.834186\pi\)
0.867362 0.497678i \(-0.165814\pi\)
\(858\) − 32.0000i − 1.09246i
\(859\) 19.7228 0.672934 0.336467 0.941695i \(-0.390768\pi\)
0.336467 + 0.941695i \(0.390768\pi\)
\(860\) 0 0
\(861\) 14.7446 0.502493
\(862\) − 1.37228i − 0.0467401i
\(863\) − 53.8397i − 1.83272i −0.400352 0.916362i \(-0.631112\pi\)
0.400352 0.916362i \(-0.368888\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 12.9783 0.441019
\(867\) 23.7228i 0.805669i
\(868\) 3.60597i 0.122395i
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −22.5109 −0.762752
\(872\) 0.116844i 0.00395684i
\(873\) 0.116844i 0.00395457i
\(874\) 13.4891 0.456276
\(875\) 0 0
\(876\) 17.4891 0.590903
\(877\) 49.3723i 1.66718i 0.552381 + 0.833592i \(0.313719\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(878\) 13.3723i 0.451293i
\(879\) 37.7228 1.27236
\(880\) 0 0
\(881\) 1.37228 0.0462333 0.0231167 0.999733i \(-0.492641\pi\)
0.0231167 + 0.999733i \(0.492641\pi\)
\(882\) − 5.11684i − 0.172293i
\(883\) 11.3723i 0.382708i 0.981521 + 0.191354i \(0.0612878\pi\)
−0.981521 + 0.191354i \(0.938712\pi\)
\(884\) −25.4891 −0.857292
\(885\) 0 0
\(886\) −16.9783 −0.570395
\(887\) − 25.3723i − 0.851918i −0.904743 0.425959i \(-0.859937\pi\)
0.904743 0.425959i \(-0.140063\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) 22.9783 0.770666
\(890\) 0 0
\(891\) 37.0951 1.24273
\(892\) − 1.37228i − 0.0459474i
\(893\) − 17.4891i − 0.585251i
\(894\) 22.9783 0.768508
\(895\) 0 0
\(896\) 1.37228 0.0458447
\(897\) − 64.0000i − 2.13690i
\(898\) − 18.0000i − 0.600668i
\(899\) 21.3288 0.711355
\(900\) 0 0
\(901\) −7.37228 −0.245606
\(902\) − 18.1168i − 0.603225i
\(903\) − 20.2337i − 0.673335i
\(904\) −17.3723 −0.577793
\(905\) 0 0
\(906\) −40.0000 −1.32891
\(907\) 40.4674i 1.34370i 0.740689 + 0.671849i \(0.234499\pi\)
−0.740689 + 0.671849i \(0.765501\pi\)
\(908\) 11.3723i 0.377402i
\(909\) 11.4891 0.381070
\(910\) 0 0
\(911\) 17.7663 0.588624 0.294312 0.955709i \(-0.404909\pi\)
0.294312 + 0.955709i \(0.404909\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 2.51087i − 0.0830978i
\(914\) −18.8614 −0.623880
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 4.46738i − 0.147526i
\(918\) − 21.4891i − 0.709247i
\(919\) −7.72281 −0.254752 −0.127376 0.991854i \(-0.540656\pi\)
−0.127376 + 0.991854i \(0.540656\pi\)
\(920\) 0 0
\(921\) −46.9783 −1.54799
\(922\) − 39.0951i − 1.28753i
\(923\) − 32.0000i − 1.05329i
\(924\) 9.25544 0.304482
\(925\) 0 0
\(926\) −5.48913 −0.180384
\(927\) − 9.48913i − 0.311664i
\(928\) − 8.11684i − 0.266448i
\(929\) −31.0951 −1.02020 −0.510098 0.860116i \(-0.670391\pi\)
−0.510098 + 0.860116i \(0.670391\pi\)
\(930\) 0 0
\(931\) 10.2337 0.335396
\(932\) 6.23369i 0.204191i
\(933\) 2.74456i 0.0898529i
\(934\) −30.3505 −0.993100
\(935\) 0 0
\(936\) −4.74456 −0.155081
\(937\) 36.9783i 1.20803i 0.796974 + 0.604013i \(0.206433\pi\)
−0.796974 + 0.604013i \(0.793567\pi\)
\(938\) − 6.51087i − 0.212588i
\(939\) 6.97825 0.227727
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 34.7446i 1.13204i
\(943\) − 36.2337i − 1.17993i
\(944\) −12.7446 −0.414800
\(945\) 0 0
\(946\) −24.8614 −0.808314
\(947\) 37.0951i 1.20543i 0.797957 + 0.602714i \(0.205914\pi\)
−0.797957 + 0.602714i \(0.794086\pi\)
\(948\) 9.48913i 0.308192i
\(949\) 41.4891 1.34679
\(950\) 0 0
\(951\) 50.7446 1.64551
\(952\) − 7.37228i − 0.238937i
\(953\) − 46.2337i − 1.49766i −0.662764 0.748828i \(-0.730617\pi\)
0.662764 0.748828i \(-0.269383\pi\)
\(954\) −1.37228 −0.0444292
\(955\) 0 0
\(956\) 17.6060 0.569418
\(957\) − 54.7446i − 1.76964i
\(958\) − 3.25544i − 0.105178i
\(959\) −22.9783 −0.742006
\(960\) 0 0
\(961\) −24.0951 −0.777261
\(962\) − 4.74456i − 0.152971i
\(963\) 3.48913i 0.112435i
\(964\) 10.2337 0.329605
\(965\) 0 0
\(966\) 18.5109 0.595578
\(967\) − 28.0000i − 0.900419i −0.892923 0.450210i \(-0.851349\pi\)
0.892923 0.450210i \(-0.148651\pi\)
\(968\) − 0.372281i − 0.0119656i
\(969\) −21.4891 −0.690330
\(970\) 0 0
\(971\) −19.6060 −0.629185 −0.314593 0.949227i \(-0.601868\pi\)
−0.314593 + 0.949227i \(0.601868\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) 2.58422i 0.0828463i
\(974\) 37.7228 1.20872
\(975\) 0 0
\(976\) −5.37228 −0.171963
\(977\) − 11.8832i − 0.380176i −0.981767 0.190088i \(-0.939123\pi\)
0.981767 0.190088i \(-0.0608773\pi\)
\(978\) − 38.7446i − 1.23891i
\(979\) 33.7228 1.07779
\(980\) 0 0
\(981\) 0.116844 0.00373054
\(982\) 14.9783i 0.477975i
\(983\) − 30.8614i − 0.984326i −0.870503 0.492163i \(-0.836206\pi\)
0.870503 0.492163i \(-0.163794\pi\)
\(984\) −10.7446 −0.342524
\(985\) 0 0
\(986\) −43.6060 −1.38870
\(987\) − 24.0000i − 0.763928i
\(988\) − 9.48913i − 0.301889i
\(989\) −49.7228 −1.58109
\(990\) 0 0
\(991\) 25.6060 0.813400 0.406700 0.913562i \(-0.366679\pi\)
0.406700 + 0.913562i \(0.366679\pi\)
\(992\) − 2.62772i − 0.0834302i
\(993\) − 46.9783i − 1.49081i
\(994\) 9.25544 0.293565
\(995\) 0 0
\(996\) −1.48913 −0.0471847
\(997\) 26.2337i 0.830829i 0.909632 + 0.415415i \(0.136363\pi\)
−0.909632 + 0.415415i \(0.863637\pi\)
\(998\) 24.9783i 0.790673i
\(999\) 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.149.4 4
5.2 odd 4 1850.2.a.q.1.1 2
5.3 odd 4 370.2.a.f.1.2 2
5.4 even 2 inner 1850.2.b.m.149.1 4
15.8 even 4 3330.2.a.bb.1.2 2
20.3 even 4 2960.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.2 2 5.3 odd 4
1850.2.a.q.1.1 2 5.2 odd 4
1850.2.b.m.149.1 4 5.4 even 2 inner
1850.2.b.m.149.4 4 1.1 even 1 trivial
2960.2.a.o.1.1 2 20.3 even 4
3330.2.a.bb.1.2 2 15.8 even 4