Properties

Label 1850.2.b.k.149.1
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.k.149.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.44949i q^{3} -1.00000 q^{4} -1.44949 q^{6} -2.44949i q^{7} +1.00000i q^{8} +0.898979 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.44949i q^{3} -1.00000 q^{4} -1.44949 q^{6} -2.44949i q^{7} +1.00000i q^{8} +0.898979 q^{9} +3.44949 q^{11} +1.44949i q^{12} -0.449490i q^{13} -2.44949 q^{14} +1.00000 q^{16} -3.44949i q^{17} -0.898979i q^{18} +5.00000 q^{19} -3.55051 q^{21} -3.44949i q^{22} +2.00000i q^{23} +1.44949 q^{24} -0.449490 q^{26} -5.65153i q^{27} +2.44949i q^{28} +0.898979 q^{29} +4.44949 q^{31} -1.00000i q^{32} -5.00000i q^{33} -3.44949 q^{34} -0.898979 q^{36} +1.00000i q^{37} -5.00000i q^{38} -0.651531 q^{39} +1.00000 q^{41} +3.55051i q^{42} -1.10102i q^{43} -3.44949 q^{44} +2.00000 q^{46} -9.79796i q^{47} -1.44949i q^{48} +1.00000 q^{49} -5.00000 q^{51} +0.449490i q^{52} +6.00000i q^{53} -5.65153 q^{54} +2.44949 q^{56} -7.24745i q^{57} -0.898979i q^{58} +2.00000 q^{59} -6.44949 q^{61} -4.44949i q^{62} -2.20204i q^{63} -1.00000 q^{64} -5.00000 q^{66} -4.55051i q^{67} +3.44949i q^{68} +2.89898 q^{69} -7.55051 q^{71} +0.898979i q^{72} +12.7980i q^{73} +1.00000 q^{74} -5.00000 q^{76} -8.44949i q^{77} +0.651531i q^{78} -7.79796 q^{79} -5.49490 q^{81} -1.00000i q^{82} +3.44949i q^{83} +3.55051 q^{84} -1.10102 q^{86} -1.30306i q^{87} +3.44949i q^{88} -14.3485 q^{89} -1.10102 q^{91} -2.00000i q^{92} -6.44949i q^{93} -9.79796 q^{94} -1.44949 q^{96} -14.0000i q^{97} -1.00000i q^{98} +3.10102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 16 q^{9} + 4 q^{11} + 4 q^{16} + 20 q^{19} - 24 q^{21} - 4 q^{24} + 8 q^{26} - 16 q^{29} + 8 q^{31} - 4 q^{34} + 16 q^{36} - 32 q^{39} + 4 q^{41} - 4 q^{44} + 8 q^{46} + 4 q^{49} - 20 q^{51} - 52 q^{54} + 8 q^{59} - 16 q^{61} - 4 q^{64} - 20 q^{66} - 8 q^{69} - 40 q^{71} + 4 q^{74} - 20 q^{76} + 8 q^{79} + 76 q^{81} + 24 q^{84} - 24 q^{86} - 28 q^{89} - 24 q^{91} + 4 q^{96} + 32 q^{99}+O(q^{100}) \) 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
\(3\) − 1.44949i − 0.836863i −0.908248 0.418432i \(-0.862580\pi\)
0.908248 0.418432i \(-0.137420\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.44949 −0.591752
\(7\) − 2.44949i − 0.925820i −0.886405 0.462910i \(-0.846805\pi\)
0.886405 0.462910i \(-0.153195\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0.898979 0.299660
\(10\) 0 0
\(11\) 3.44949 1.04006 0.520030 0.854148i \(-0.325921\pi\)
0.520030 + 0.854148i \(0.325921\pi\)
\(12\) 1.44949i 0.418432i
\(13\) − 0.449490i − 0.124666i −0.998055 0.0623330i \(-0.980146\pi\)
0.998055 0.0623330i \(-0.0198541\pi\)
\(14\) −2.44949 −0.654654
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.44949i − 0.836624i −0.908303 0.418312i \(-0.862622\pi\)
0.908303 0.418312i \(-0.137378\pi\)
\(18\) − 0.898979i − 0.211891i
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) −3.55051 −0.774785
\(22\) − 3.44949i − 0.735434i
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 1.44949 0.295876
\(25\) 0 0
\(26\) −0.449490 −0.0881522
\(27\) − 5.65153i − 1.08764i
\(28\) 2.44949i 0.462910i
\(29\) 0.898979 0.166936 0.0834681 0.996510i \(-0.473400\pi\)
0.0834681 + 0.996510i \(0.473400\pi\)
\(30\) 0 0
\(31\) 4.44949 0.799152 0.399576 0.916700i \(-0.369157\pi\)
0.399576 + 0.916700i \(0.369157\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 5.00000i − 0.870388i
\(34\) −3.44949 −0.591583
\(35\) 0 0
\(36\) −0.898979 −0.149830
\(37\) 1.00000i 0.164399i
\(38\) − 5.00000i − 0.811107i
\(39\) −0.651531 −0.104328
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 3.55051i 0.547856i
\(43\) − 1.10102i − 0.167904i −0.996470 0.0839520i \(-0.973246\pi\)
0.996470 0.0839520i \(-0.0267543\pi\)
\(44\) −3.44949 −0.520030
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) − 9.79796i − 1.42918i −0.699544 0.714590i \(-0.746613\pi\)
0.699544 0.714590i \(-0.253387\pi\)
\(48\) − 1.44949i − 0.209216i
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0.449490i 0.0623330i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −5.65153 −0.769076
\(55\) 0 0
\(56\) 2.44949 0.327327
\(57\) − 7.24745i − 0.959948i
\(58\) − 0.898979i − 0.118042i
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −6.44949 −0.825773 −0.412886 0.910783i \(-0.635479\pi\)
−0.412886 + 0.910783i \(0.635479\pi\)
\(62\) − 4.44949i − 0.565086i
\(63\) − 2.20204i − 0.277431i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) − 4.55051i − 0.555933i −0.960591 0.277967i \(-0.910340\pi\)
0.960591 0.277967i \(-0.0896605\pi\)
\(68\) 3.44949i 0.418312i
\(69\) 2.89898 0.348996
\(70\) 0 0
\(71\) −7.55051 −0.896081 −0.448040 0.894013i \(-0.647878\pi\)
−0.448040 + 0.894013i \(0.647878\pi\)
\(72\) 0.898979i 0.105946i
\(73\) 12.7980i 1.49789i 0.662633 + 0.748944i \(0.269439\pi\)
−0.662633 + 0.748944i \(0.730561\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) − 8.44949i − 0.962909i
\(78\) 0.651531i 0.0737713i
\(79\) −7.79796 −0.877339 −0.438669 0.898648i \(-0.644550\pi\)
−0.438669 + 0.898648i \(0.644550\pi\)
\(80\) 0 0
\(81\) −5.49490 −0.610544
\(82\) − 1.00000i − 0.110432i
\(83\) 3.44949i 0.378631i 0.981916 + 0.189315i \(0.0606268\pi\)
−0.981916 + 0.189315i \(0.939373\pi\)
\(84\) 3.55051 0.387392
\(85\) 0 0
\(86\) −1.10102 −0.118726
\(87\) − 1.30306i − 0.139703i
\(88\) 3.44949i 0.367717i
\(89\) −14.3485 −1.52093 −0.760467 0.649376i \(-0.775030\pi\)
−0.760467 + 0.649376i \(0.775030\pi\)
\(90\) 0 0
\(91\) −1.10102 −0.115418
\(92\) − 2.00000i − 0.208514i
\(93\) − 6.44949i − 0.668781i
\(94\) −9.79796 −1.01058
\(95\) 0 0
\(96\) −1.44949 −0.147938
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 3.10102 0.311664
\(100\) 0 0
\(101\) 1.10102 0.109556 0.0547778 0.998499i \(-0.482555\pi\)
0.0547778 + 0.998499i \(0.482555\pi\)
\(102\) 5.00000i 0.495074i
\(103\) 6.44949i 0.635487i 0.948177 + 0.317744i \(0.102925\pi\)
−0.948177 + 0.317744i \(0.897075\pi\)
\(104\) 0.449490 0.0440761
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 8.55051i 0.826609i 0.910593 + 0.413305i \(0.135625\pi\)
−0.910593 + 0.413305i \(0.864375\pi\)
\(108\) 5.65153i 0.543819i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 1.44949 0.137579
\(112\) − 2.44949i − 0.231455i
\(113\) − 4.55051i − 0.428076i −0.976825 0.214038i \(-0.931338\pi\)
0.976825 0.214038i \(-0.0686616\pi\)
\(114\) −7.24745 −0.678786
\(115\) 0 0
\(116\) −0.898979 −0.0834681
\(117\) − 0.404082i − 0.0373574i
\(118\) − 2.00000i − 0.184115i
\(119\) −8.44949 −0.774563
\(120\) 0 0
\(121\) 0.898979 0.0817254
\(122\) 6.44949i 0.583909i
\(123\) − 1.44949i − 0.130696i
\(124\) −4.44949 −0.399576
\(125\) 0 0
\(126\) −2.20204 −0.196173
\(127\) 16.2474i 1.44173i 0.693077 + 0.720864i \(0.256255\pi\)
−0.693077 + 0.720864i \(0.743745\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −1.59592 −0.140513
\(130\) 0 0
\(131\) 8.69694 0.759855 0.379928 0.925016i \(-0.375949\pi\)
0.379928 + 0.925016i \(0.375949\pi\)
\(132\) 5.00000i 0.435194i
\(133\) − 12.2474i − 1.06199i
\(134\) −4.55051 −0.393104
\(135\) 0 0
\(136\) 3.44949 0.295791
\(137\) − 9.69694i − 0.828465i −0.910171 0.414233i \(-0.864050\pi\)
0.910171 0.414233i \(-0.135950\pi\)
\(138\) − 2.89898i − 0.246778i
\(139\) 12.5505 1.06452 0.532260 0.846581i \(-0.321343\pi\)
0.532260 + 0.846581i \(0.321343\pi\)
\(140\) 0 0
\(141\) −14.2020 −1.19603
\(142\) 7.55051i 0.633625i
\(143\) − 1.55051i − 0.129660i
\(144\) 0.898979 0.0749150
\(145\) 0 0
\(146\) 12.7980 1.05917
\(147\) − 1.44949i − 0.119552i
\(148\) − 1.00000i − 0.0821995i
\(149\) 4.65153 0.381068 0.190534 0.981681i \(-0.438978\pi\)
0.190534 + 0.981681i \(0.438978\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 5.00000i 0.405554i
\(153\) − 3.10102i − 0.250703i
\(154\) −8.44949 −0.680879
\(155\) 0 0
\(156\) 0.651531 0.0521642
\(157\) − 16.4495i − 1.31281i −0.754408 0.656406i \(-0.772076\pi\)
0.754408 0.656406i \(-0.227924\pi\)
\(158\) 7.79796i 0.620372i
\(159\) 8.69694 0.689712
\(160\) 0 0
\(161\) 4.89898 0.386094
\(162\) 5.49490i 0.431720i
\(163\) 10.1010i 0.791173i 0.918429 + 0.395586i \(0.129459\pi\)
−0.918429 + 0.395586i \(0.870541\pi\)
\(164\) −1.00000 −0.0780869
\(165\) 0 0
\(166\) 3.44949 0.267732
\(167\) 10.4495i 0.808606i 0.914625 + 0.404303i \(0.132486\pi\)
−0.914625 + 0.404303i \(0.867514\pi\)
\(168\) − 3.55051i − 0.273928i
\(169\) 12.7980 0.984458
\(170\) 0 0
\(171\) 4.49490 0.343733
\(172\) 1.10102i 0.0839520i
\(173\) − 6.69694i − 0.509159i −0.967052 0.254579i \(-0.918063\pi\)
0.967052 0.254579i \(-0.0819370\pi\)
\(174\) −1.30306 −0.0987848
\(175\) 0 0
\(176\) 3.44949 0.260015
\(177\) − 2.89898i − 0.217901i
\(178\) 14.3485i 1.07546i
\(179\) 0.797959 0.0596423 0.0298211 0.999555i \(-0.490506\pi\)
0.0298211 + 0.999555i \(0.490506\pi\)
\(180\) 0 0
\(181\) 6.89898 0.512797 0.256399 0.966571i \(-0.417464\pi\)
0.256399 + 0.966571i \(0.417464\pi\)
\(182\) 1.10102i 0.0816131i
\(183\) 9.34847i 0.691059i
\(184\) −2.00000 −0.147442
\(185\) 0 0
\(186\) −6.44949 −0.472900
\(187\) − 11.8990i − 0.870140i
\(188\) 9.79796i 0.714590i
\(189\) −13.8434 −1.00696
\(190\) 0 0
\(191\) −7.79796 −0.564241 −0.282120 0.959379i \(-0.591038\pi\)
−0.282120 + 0.959379i \(0.591038\pi\)
\(192\) 1.44949i 0.104608i
\(193\) − 2.55051i − 0.183590i −0.995778 0.0917949i \(-0.970740\pi\)
0.995778 0.0917949i \(-0.0292604\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) 17.3485i 1.23603i 0.786167 + 0.618014i \(0.212062\pi\)
−0.786167 + 0.618014i \(0.787938\pi\)
\(198\) − 3.10102i − 0.220380i
\(199\) 15.5959 1.10557 0.552783 0.833325i \(-0.313566\pi\)
0.552783 + 0.833325i \(0.313566\pi\)
\(200\) 0 0
\(201\) −6.59592 −0.465240
\(202\) − 1.10102i − 0.0774675i
\(203\) − 2.20204i − 0.154553i
\(204\) 5.00000 0.350070
\(205\) 0 0
\(206\) 6.44949 0.449357
\(207\) 1.79796i 0.124967i
\(208\) − 0.449490i − 0.0311665i
\(209\) 17.2474 1.19303
\(210\) 0 0
\(211\) −4.55051 −0.313270 −0.156635 0.987657i \(-0.550065\pi\)
−0.156635 + 0.987657i \(0.550065\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 10.9444i 0.749897i
\(214\) 8.55051 0.584501
\(215\) 0 0
\(216\) 5.65153 0.384538
\(217\) − 10.8990i − 0.739871i
\(218\) 14.0000i 0.948200i
\(219\) 18.5505 1.25353
\(220\) 0 0
\(221\) −1.55051 −0.104299
\(222\) − 1.44949i − 0.0972834i
\(223\) 21.7980i 1.45970i 0.683608 + 0.729850i \(0.260410\pi\)
−0.683608 + 0.729850i \(0.739590\pi\)
\(224\) −2.44949 −0.163663
\(225\) 0 0
\(226\) −4.55051 −0.302695
\(227\) 16.6969i 1.10821i 0.832445 + 0.554107i \(0.186940\pi\)
−0.832445 + 0.554107i \(0.813060\pi\)
\(228\) 7.24745i 0.479974i
\(229\) −5.79796 −0.383140 −0.191570 0.981479i \(-0.561358\pi\)
−0.191570 + 0.981479i \(0.561358\pi\)
\(230\) 0 0
\(231\) −12.2474 −0.805823
\(232\) 0.898979i 0.0590209i
\(233\) − 28.4949i − 1.86676i −0.358886 0.933381i \(-0.616843\pi\)
0.358886 0.933381i \(-0.383157\pi\)
\(234\) −0.404082 −0.0264157
\(235\) 0 0
\(236\) −2.00000 −0.130189
\(237\) 11.3031i 0.734213i
\(238\) 8.44949i 0.547699i
\(239\) −0.898979 −0.0581501 −0.0290751 0.999577i \(-0.509256\pi\)
−0.0290751 + 0.999577i \(0.509256\pi\)
\(240\) 0 0
\(241\) −2.55051 −0.164293 −0.0821464 0.996620i \(-0.526178\pi\)
−0.0821464 + 0.996620i \(0.526178\pi\)
\(242\) − 0.898979i − 0.0577886i
\(243\) − 8.98979i − 0.576696i
\(244\) 6.44949 0.412886
\(245\) 0 0
\(246\) −1.44949 −0.0924161
\(247\) − 2.24745i − 0.143002i
\(248\) 4.44949i 0.282543i
\(249\) 5.00000 0.316862
\(250\) 0 0
\(251\) −30.7980 −1.94395 −0.971975 0.235084i \(-0.924463\pi\)
−0.971975 + 0.235084i \(0.924463\pi\)
\(252\) 2.20204i 0.138716i
\(253\) 6.89898i 0.433735i
\(254\) 16.2474 1.01946
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.89898i 0.430347i 0.976576 + 0.215173i \(0.0690316\pi\)
−0.976576 + 0.215173i \(0.930968\pi\)
\(258\) 1.59592i 0.0993575i
\(259\) 2.44949 0.152204
\(260\) 0 0
\(261\) 0.808164 0.0500241
\(262\) − 8.69694i − 0.537299i
\(263\) 0.202041i 0.0124584i 0.999981 + 0.00622919i \(0.00198283\pi\)
−0.999981 + 0.00622919i \(0.998017\pi\)
\(264\) 5.00000 0.307729
\(265\) 0 0
\(266\) −12.2474 −0.750939
\(267\) 20.7980i 1.27281i
\(268\) 4.55051i 0.277967i
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 30.0454 1.82513 0.912564 0.408933i \(-0.134099\pi\)
0.912564 + 0.408933i \(0.134099\pi\)
\(272\) − 3.44949i − 0.209156i
\(273\) 1.59592i 0.0965893i
\(274\) −9.69694 −0.585813
\(275\) 0 0
\(276\) −2.89898 −0.174498
\(277\) − 5.79796i − 0.348366i −0.984713 0.174183i \(-0.944272\pi\)
0.984713 0.174183i \(-0.0557284\pi\)
\(278\) − 12.5505i − 0.752730i
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 8.89898 0.530869 0.265434 0.964129i \(-0.414485\pi\)
0.265434 + 0.964129i \(0.414485\pi\)
\(282\) 14.2020i 0.845719i
\(283\) − 11.6969i − 0.695311i −0.937622 0.347655i \(-0.886978\pi\)
0.937622 0.347655i \(-0.113022\pi\)
\(284\) 7.55051 0.448040
\(285\) 0 0
\(286\) −1.55051 −0.0916836
\(287\) − 2.44949i − 0.144589i
\(288\) − 0.898979i − 0.0529729i
\(289\) 5.10102 0.300060
\(290\) 0 0
\(291\) −20.2929 −1.18959
\(292\) − 12.7980i − 0.748944i
\(293\) − 28.0454i − 1.63843i −0.573486 0.819215i \(-0.694409\pi\)
0.573486 0.819215i \(-0.305591\pi\)
\(294\) −1.44949 −0.0845360
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) − 19.4949i − 1.13121i
\(298\) − 4.65153i − 0.269456i
\(299\) 0.898979 0.0519893
\(300\) 0 0
\(301\) −2.69694 −0.155449
\(302\) 14.0000i 0.805609i
\(303\) − 1.59592i − 0.0916831i
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) −3.10102 −0.177274
\(307\) − 7.65153i − 0.436696i −0.975871 0.218348i \(-0.929933\pi\)
0.975871 0.218348i \(-0.0700668\pi\)
\(308\) 8.44949i 0.481454i
\(309\) 9.34847 0.531816
\(310\) 0 0
\(311\) −8.44949 −0.479127 −0.239563 0.970881i \(-0.577004\pi\)
−0.239563 + 0.970881i \(0.577004\pi\)
\(312\) − 0.651531i − 0.0368857i
\(313\) 8.89898i 0.503000i 0.967857 + 0.251500i \(0.0809239\pi\)
−0.967857 + 0.251500i \(0.919076\pi\)
\(314\) −16.4495 −0.928298
\(315\) 0 0
\(316\) 7.79796 0.438669
\(317\) 9.55051i 0.536410i 0.963362 + 0.268205i \(0.0864305\pi\)
−0.963362 + 0.268205i \(0.913570\pi\)
\(318\) − 8.69694i − 0.487700i
\(319\) 3.10102 0.173624
\(320\) 0 0
\(321\) 12.3939 0.691759
\(322\) − 4.89898i − 0.273009i
\(323\) − 17.2474i − 0.959674i
\(324\) 5.49490 0.305272
\(325\) 0 0
\(326\) 10.1010 0.559444
\(327\) 20.2929i 1.12220i
\(328\) 1.00000i 0.0552158i
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 19.8990 1.09375 0.546873 0.837215i \(-0.315818\pi\)
0.546873 + 0.837215i \(0.315818\pi\)
\(332\) − 3.44949i − 0.189315i
\(333\) 0.898979i 0.0492638i
\(334\) 10.4495 0.571771
\(335\) 0 0
\(336\) −3.55051 −0.193696
\(337\) − 15.6969i − 0.855067i −0.904000 0.427533i \(-0.859383\pi\)
0.904000 0.427533i \(-0.140617\pi\)
\(338\) − 12.7980i − 0.696117i
\(339\) −6.59592 −0.358241
\(340\) 0 0
\(341\) 15.3485 0.831166
\(342\) − 4.49490i − 0.243056i
\(343\) − 19.5959i − 1.05808i
\(344\) 1.10102 0.0593630
\(345\) 0 0
\(346\) −6.69694 −0.360030
\(347\) 18.7980i 1.00913i 0.863374 + 0.504564i \(0.168347\pi\)
−0.863374 + 0.504564i \(0.831653\pi\)
\(348\) 1.30306i 0.0698514i
\(349\) 12.4495 0.666406 0.333203 0.942855i \(-0.391871\pi\)
0.333203 + 0.942855i \(0.391871\pi\)
\(350\) 0 0
\(351\) −2.54031 −0.135591
\(352\) − 3.44949i − 0.183858i
\(353\) 14.8990i 0.792993i 0.918036 + 0.396496i \(0.129774\pi\)
−0.918036 + 0.396496i \(0.870226\pi\)
\(354\) −2.89898 −0.154079
\(355\) 0 0
\(356\) 14.3485 0.760467
\(357\) 12.2474i 0.648204i
\(358\) − 0.797959i − 0.0421734i
\(359\) 31.3939 1.65691 0.828453 0.560059i \(-0.189222\pi\)
0.828453 + 0.560059i \(0.189222\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) − 6.89898i − 0.362602i
\(363\) − 1.30306i − 0.0683930i
\(364\) 1.10102 0.0577092
\(365\) 0 0
\(366\) 9.34847 0.488652
\(367\) 14.2474i 0.743711i 0.928291 + 0.371855i \(0.121278\pi\)
−0.928291 + 0.371855i \(0.878722\pi\)
\(368\) 2.00000i 0.104257i
\(369\) 0.898979 0.0467990
\(370\) 0 0
\(371\) 14.6969 0.763027
\(372\) 6.44949i 0.334390i
\(373\) − 20.0454i − 1.03791i −0.854801 0.518956i \(-0.826321\pi\)
0.854801 0.518956i \(-0.173679\pi\)
\(374\) −11.8990 −0.615282
\(375\) 0 0
\(376\) 9.79796 0.505291
\(377\) − 0.404082i − 0.0208113i
\(378\) 13.8434i 0.712026i
\(379\) 21.0454 1.08103 0.540515 0.841334i \(-0.318229\pi\)
0.540515 + 0.841334i \(0.318229\pi\)
\(380\) 0 0
\(381\) 23.5505 1.20653
\(382\) 7.79796i 0.398978i
\(383\) − 5.34847i − 0.273294i −0.990620 0.136647i \(-0.956367\pi\)
0.990620 0.136647i \(-0.0436326\pi\)
\(384\) 1.44949 0.0739690
\(385\) 0 0
\(386\) −2.55051 −0.129818
\(387\) − 0.989795i − 0.0503141i
\(388\) 14.0000i 0.710742i
\(389\) −13.1464 −0.666550 −0.333275 0.942830i \(-0.608154\pi\)
−0.333275 + 0.942830i \(0.608154\pi\)
\(390\) 0 0
\(391\) 6.89898 0.348896
\(392\) 1.00000i 0.0505076i
\(393\) − 12.6061i − 0.635895i
\(394\) 17.3485 0.874003
\(395\) 0 0
\(396\) −3.10102 −0.155832
\(397\) 15.3485i 0.770318i 0.922850 + 0.385159i \(0.125853\pi\)
−0.922850 + 0.385159i \(0.874147\pi\)
\(398\) − 15.5959i − 0.781753i
\(399\) −17.7526 −0.888739
\(400\) 0 0
\(401\) 15.9444 0.796225 0.398112 0.917337i \(-0.369665\pi\)
0.398112 + 0.917337i \(0.369665\pi\)
\(402\) 6.59592i 0.328974i
\(403\) − 2.00000i − 0.0996271i
\(404\) −1.10102 −0.0547778
\(405\) 0 0
\(406\) −2.20204 −0.109285
\(407\) 3.44949i 0.170985i
\(408\) − 5.00000i − 0.247537i
\(409\) 34.1464 1.68843 0.844216 0.536003i \(-0.180066\pi\)
0.844216 + 0.536003i \(0.180066\pi\)
\(410\) 0 0
\(411\) −14.0556 −0.693312
\(412\) − 6.44949i − 0.317744i
\(413\) − 4.89898i − 0.241063i
\(414\) 1.79796 0.0883649
\(415\) 0 0
\(416\) −0.449490 −0.0220380
\(417\) − 18.1918i − 0.890858i
\(418\) − 17.2474i − 0.843600i
\(419\) 21.4495 1.04788 0.523938 0.851756i \(-0.324462\pi\)
0.523938 + 0.851756i \(0.324462\pi\)
\(420\) 0 0
\(421\) 32.8990 1.60340 0.801699 0.597728i \(-0.203930\pi\)
0.801699 + 0.597728i \(0.203930\pi\)
\(422\) 4.55051i 0.221515i
\(423\) − 8.80816i − 0.428268i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 10.9444 0.530257
\(427\) 15.7980i 0.764517i
\(428\) − 8.55051i − 0.413305i
\(429\) −2.24745 −0.108508
\(430\) 0 0
\(431\) −20.6969 −0.996936 −0.498468 0.866908i \(-0.666104\pi\)
−0.498468 + 0.866908i \(0.666104\pi\)
\(432\) − 5.65153i − 0.271909i
\(433\) 19.0000i 0.913082i 0.889702 + 0.456541i \(0.150912\pi\)
−0.889702 + 0.456541i \(0.849088\pi\)
\(434\) −10.8990 −0.523168
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 10.0000i 0.478365i
\(438\) − 18.5505i − 0.886378i
\(439\) 19.5959 0.935262 0.467631 0.883924i \(-0.345108\pi\)
0.467631 + 0.883924i \(0.345108\pi\)
\(440\) 0 0
\(441\) 0.898979 0.0428085
\(442\) 1.55051i 0.0737503i
\(443\) − 17.4495i − 0.829050i −0.910038 0.414525i \(-0.863948\pi\)
0.910038 0.414525i \(-0.136052\pi\)
\(444\) −1.44949 −0.0687897
\(445\) 0 0
\(446\) 21.7980 1.03216
\(447\) − 6.74235i − 0.318902i
\(448\) 2.44949i 0.115728i
\(449\) −33.2474 −1.56904 −0.784522 0.620101i \(-0.787092\pi\)
−0.784522 + 0.620101i \(0.787092\pi\)
\(450\) 0 0
\(451\) 3.44949 0.162430
\(452\) 4.55051i 0.214038i
\(453\) 20.2929i 0.953442i
\(454\) 16.6969 0.783626
\(455\) 0 0
\(456\) 7.24745 0.339393
\(457\) − 15.2474i − 0.713246i −0.934248 0.356623i \(-0.883928\pi\)
0.934248 0.356623i \(-0.116072\pi\)
\(458\) 5.79796i 0.270921i
\(459\) −19.4949 −0.909944
\(460\) 0 0
\(461\) 9.30306 0.433287 0.216643 0.976251i \(-0.430489\pi\)
0.216643 + 0.976251i \(0.430489\pi\)
\(462\) 12.2474i 0.569803i
\(463\) 9.55051i 0.443850i 0.975064 + 0.221925i \(0.0712340\pi\)
−0.975064 + 0.221925i \(0.928766\pi\)
\(464\) 0.898979 0.0417341
\(465\) 0 0
\(466\) −28.4949 −1.32000
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0.404082i 0.0186787i
\(469\) −11.1464 −0.514694
\(470\) 0 0
\(471\) −23.8434 −1.09864
\(472\) 2.00000i 0.0920575i
\(473\) − 3.79796i − 0.174630i
\(474\) 11.3031 0.519167
\(475\) 0 0
\(476\) 8.44949 0.387282
\(477\) 5.39388i 0.246969i
\(478\) 0.898979i 0.0411184i
\(479\) −6.24745 −0.285453 −0.142727 0.989762i \(-0.545587\pi\)
−0.142727 + 0.989762i \(0.545587\pi\)
\(480\) 0 0
\(481\) 0.449490 0.0204950
\(482\) 2.55051i 0.116173i
\(483\) − 7.10102i − 0.323108i
\(484\) −0.898979 −0.0408627
\(485\) 0 0
\(486\) −8.98979 −0.407785
\(487\) 30.7423i 1.39307i 0.717523 + 0.696534i \(0.245276\pi\)
−0.717523 + 0.696534i \(0.754724\pi\)
\(488\) − 6.44949i − 0.291955i
\(489\) 14.6413 0.662104
\(490\) 0 0
\(491\) −41.7980 −1.88632 −0.943158 0.332345i \(-0.892160\pi\)
−0.943158 + 0.332345i \(0.892160\pi\)
\(492\) 1.44949i 0.0653480i
\(493\) − 3.10102i − 0.139663i
\(494\) −2.24745 −0.101117
\(495\) 0 0
\(496\) 4.44949 0.199788
\(497\) 18.4949i 0.829610i
\(498\) − 5.00000i − 0.224055i
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) 0 0
\(501\) 15.1464 0.676693
\(502\) 30.7980i 1.37458i
\(503\) 9.79796i 0.436869i 0.975852 + 0.218435i \(0.0700951\pi\)
−0.975852 + 0.218435i \(0.929905\pi\)
\(504\) 2.20204 0.0980867
\(505\) 0 0
\(506\) 6.89898 0.306697
\(507\) − 18.5505i − 0.823857i
\(508\) − 16.2474i − 0.720864i
\(509\) −29.3485 −1.30085 −0.650424 0.759571i \(-0.725409\pi\)
−0.650424 + 0.759571i \(0.725409\pi\)
\(510\) 0 0
\(511\) 31.3485 1.38677
\(512\) − 1.00000i − 0.0441942i
\(513\) − 28.2577i − 1.24761i
\(514\) 6.89898 0.304301
\(515\) 0 0
\(516\) 1.59592 0.0702564
\(517\) − 33.7980i − 1.48643i
\(518\) − 2.44949i − 0.107624i
\(519\) −9.70714 −0.426096
\(520\) 0 0
\(521\) 10.1010 0.442534 0.221267 0.975213i \(-0.428981\pi\)
0.221267 + 0.975213i \(0.428981\pi\)
\(522\) − 0.808164i − 0.0353724i
\(523\) − 18.3939i − 0.804308i −0.915572 0.402154i \(-0.868262\pi\)
0.915572 0.402154i \(-0.131738\pi\)
\(524\) −8.69694 −0.379928
\(525\) 0 0
\(526\) 0.202041 0.00880941
\(527\) − 15.3485i − 0.668590i
\(528\) − 5.00000i − 0.217597i
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 1.79796 0.0780248
\(532\) 12.2474i 0.530994i
\(533\) − 0.449490i − 0.0194696i
\(534\) 20.7980 0.900016
\(535\) 0 0
\(536\) 4.55051 0.196552
\(537\) − 1.15663i − 0.0499124i
\(538\) 4.00000i 0.172452i
\(539\) 3.44949 0.148580
\(540\) 0 0
\(541\) 25.3485 1.08982 0.544908 0.838496i \(-0.316565\pi\)
0.544908 + 0.838496i \(0.316565\pi\)
\(542\) − 30.0454i − 1.29056i
\(543\) − 10.0000i − 0.429141i
\(544\) −3.44949 −0.147896
\(545\) 0 0
\(546\) 1.59592 0.0682990
\(547\) − 3.69694i − 0.158070i −0.996872 0.0790348i \(-0.974816\pi\)
0.996872 0.0790348i \(-0.0251838\pi\)
\(548\) 9.69694i 0.414233i
\(549\) −5.79796 −0.247451
\(550\) 0 0
\(551\) 4.49490 0.191489
\(552\) 2.89898i 0.123389i
\(553\) 19.1010i 0.812258i
\(554\) −5.79796 −0.246332
\(555\) 0 0
\(556\) −12.5505 −0.532260
\(557\) 21.7980i 0.923609i 0.886982 + 0.461805i \(0.152798\pi\)
−0.886982 + 0.461805i \(0.847202\pi\)
\(558\) − 4.00000i − 0.169334i
\(559\) −0.494897 −0.0209319
\(560\) 0 0
\(561\) −17.2474 −0.728188
\(562\) − 8.89898i − 0.375381i
\(563\) 37.5959i 1.58448i 0.610210 + 0.792240i \(0.291085\pi\)
−0.610210 + 0.792240i \(0.708915\pi\)
\(564\) 14.2020 0.598014
\(565\) 0 0
\(566\) −11.6969 −0.491659
\(567\) 13.4597i 0.565254i
\(568\) − 7.55051i − 0.316812i
\(569\) −27.0454 −1.13380 −0.566901 0.823786i \(-0.691858\pi\)
−0.566901 + 0.823786i \(0.691858\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 1.55051i 0.0648301i
\(573\) 11.3031i 0.472192i
\(574\) −2.44949 −0.102240
\(575\) 0 0
\(576\) −0.898979 −0.0374575
\(577\) − 21.0454i − 0.876132i −0.898943 0.438066i \(-0.855664\pi\)
0.898943 0.438066i \(-0.144336\pi\)
\(578\) − 5.10102i − 0.212174i
\(579\) −3.69694 −0.153640
\(580\) 0 0
\(581\) 8.44949 0.350544
\(582\) 20.2929i 0.841166i
\(583\) 20.6969i 0.857180i
\(584\) −12.7980 −0.529583
\(585\) 0 0
\(586\) −28.0454 −1.15855
\(587\) 6.30306i 0.260155i 0.991504 + 0.130078i \(0.0415226\pi\)
−0.991504 + 0.130078i \(0.958477\pi\)
\(588\) 1.44949i 0.0597759i
\(589\) 22.2474 0.916690
\(590\) 0 0
\(591\) 25.1464 1.03439
\(592\) 1.00000i 0.0410997i
\(593\) 46.3939i 1.90517i 0.304275 + 0.952584i \(0.401586\pi\)
−0.304275 + 0.952584i \(0.598414\pi\)
\(594\) −19.4949 −0.799885
\(595\) 0 0
\(596\) −4.65153 −0.190534
\(597\) − 22.6061i − 0.925207i
\(598\) − 0.898979i − 0.0367620i
\(599\) −38.9444 −1.59122 −0.795612 0.605806i \(-0.792851\pi\)
−0.795612 + 0.605806i \(0.792851\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 2.69694i 0.109919i
\(603\) − 4.09082i − 0.166591i
\(604\) 14.0000 0.569652
\(605\) 0 0
\(606\) −1.59592 −0.0648297
\(607\) 28.7423i 1.16662i 0.812251 + 0.583308i \(0.198242\pi\)
−0.812251 + 0.583308i \(0.801758\pi\)
\(608\) − 5.00000i − 0.202777i
\(609\) −3.19184 −0.129340
\(610\) 0 0
\(611\) −4.40408 −0.178170
\(612\) 3.10102i 0.125351i
\(613\) − 34.4949i − 1.39324i −0.717442 0.696618i \(-0.754687\pi\)
0.717442 0.696618i \(-0.245313\pi\)
\(614\) −7.65153 −0.308791
\(615\) 0 0
\(616\) 8.44949 0.340440
\(617\) 4.89898i 0.197225i 0.995126 + 0.0986127i \(0.0314405\pi\)
−0.995126 + 0.0986127i \(0.968559\pi\)
\(618\) − 9.34847i − 0.376051i
\(619\) 32.8990 1.32232 0.661161 0.750244i \(-0.270064\pi\)
0.661161 + 0.750244i \(0.270064\pi\)
\(620\) 0 0
\(621\) 11.3031 0.453576
\(622\) 8.44949i 0.338794i
\(623\) 35.1464i 1.40811i
\(624\) −0.651531 −0.0260821
\(625\) 0 0
\(626\) 8.89898 0.355675
\(627\) − 25.0000i − 0.998404i
\(628\) 16.4495i 0.656406i
\(629\) 3.44949 0.137540
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) − 7.79796i − 0.310186i
\(633\) 6.59592i 0.262164i
\(634\) 9.55051 0.379299
\(635\) 0 0
\(636\) −8.69694 −0.344856
\(637\) − 0.449490i − 0.0178094i
\(638\) − 3.10102i − 0.122771i
\(639\) −6.78775 −0.268519
\(640\) 0 0
\(641\) 21.5959 0.852987 0.426494 0.904491i \(-0.359749\pi\)
0.426494 + 0.904491i \(0.359749\pi\)
\(642\) − 12.3939i − 0.489147i
\(643\) 32.2929i 1.27351i 0.771068 + 0.636753i \(0.219723\pi\)
−0.771068 + 0.636753i \(0.780277\pi\)
\(644\) −4.89898 −0.193047
\(645\) 0 0
\(646\) −17.2474 −0.678592
\(647\) 40.2474i 1.58229i 0.611628 + 0.791145i \(0.290515\pi\)
−0.611628 + 0.791145i \(0.709485\pi\)
\(648\) − 5.49490i − 0.215860i
\(649\) 6.89898 0.270809
\(650\) 0 0
\(651\) −15.7980 −0.619171
\(652\) − 10.1010i − 0.395586i
\(653\) − 10.0454i − 0.393107i −0.980493 0.196554i \(-0.937025\pi\)
0.980493 0.196554i \(-0.0629750\pi\)
\(654\) 20.2929 0.793513
\(655\) 0 0
\(656\) 1.00000 0.0390434
\(657\) 11.5051i 0.448857i
\(658\) 24.0000i 0.935617i
\(659\) −6.55051 −0.255172 −0.127586 0.991828i \(-0.540723\pi\)
−0.127586 + 0.991828i \(0.540723\pi\)
\(660\) 0 0
\(661\) 25.7980 1.00342 0.501712 0.865035i \(-0.332704\pi\)
0.501712 + 0.865035i \(0.332704\pi\)
\(662\) − 19.8990i − 0.773396i
\(663\) 2.24745i 0.0872837i
\(664\) −3.44949 −0.133866
\(665\) 0 0
\(666\) 0.898979 0.0348347
\(667\) 1.79796i 0.0696172i
\(668\) − 10.4495i − 0.404303i
\(669\) 31.5959 1.22157
\(670\) 0 0
\(671\) −22.2474 −0.858853
\(672\) 3.55051i 0.136964i
\(673\) 5.79796i 0.223495i 0.993737 + 0.111747i \(0.0356448\pi\)
−0.993737 + 0.111747i \(0.964355\pi\)
\(674\) −15.6969 −0.604623
\(675\) 0 0
\(676\) −12.7980 −0.492229
\(677\) 43.5959i 1.67553i 0.546033 + 0.837764i \(0.316137\pi\)
−0.546033 + 0.837764i \(0.683863\pi\)
\(678\) 6.59592i 0.253315i
\(679\) −34.2929 −1.31604
\(680\) 0 0
\(681\) 24.2020 0.927424
\(682\) − 15.3485i − 0.587723i
\(683\) − 7.00000i − 0.267848i −0.990992 0.133924i \(-0.957242\pi\)
0.990992 0.133924i \(-0.0427577\pi\)
\(684\) −4.49490 −0.171867
\(685\) 0 0
\(686\) −19.5959 −0.748176
\(687\) 8.40408i 0.320636i
\(688\) − 1.10102i − 0.0419760i
\(689\) 2.69694 0.102745
\(690\) 0 0
\(691\) 29.6515 1.12800 0.563999 0.825776i \(-0.309262\pi\)
0.563999 + 0.825776i \(0.309262\pi\)
\(692\) 6.69694i 0.254579i
\(693\) − 7.59592i − 0.288545i
\(694\) 18.7980 0.713561
\(695\) 0 0
\(696\) 1.30306 0.0493924
\(697\) − 3.44949i − 0.130659i
\(698\) − 12.4495i − 0.471220i
\(699\) −41.3031 −1.56223
\(700\) 0 0
\(701\) −34.2474 −1.29351 −0.646754 0.762699i \(-0.723874\pi\)
−0.646754 + 0.762699i \(0.723874\pi\)
\(702\) 2.54031i 0.0958776i
\(703\) 5.00000i 0.188579i
\(704\) −3.44949 −0.130008
\(705\) 0 0
\(706\) 14.8990 0.560730
\(707\) − 2.69694i − 0.101429i
\(708\) 2.89898i 0.108950i
\(709\) −16.8990 −0.634654 −0.317327 0.948316i \(-0.602785\pi\)
−0.317327 + 0.948316i \(0.602785\pi\)
\(710\) 0 0
\(711\) −7.01021 −0.262903
\(712\) − 14.3485i − 0.537732i
\(713\) 8.89898i 0.333269i
\(714\) 12.2474 0.458349
\(715\) 0 0
\(716\) −0.797959 −0.0298211
\(717\) 1.30306i 0.0486637i
\(718\) − 31.3939i − 1.17161i
\(719\) 0.651531 0.0242980 0.0121490 0.999926i \(-0.496133\pi\)
0.0121490 + 0.999926i \(0.496133\pi\)
\(720\) 0 0
\(721\) 15.7980 0.588347
\(722\) − 6.00000i − 0.223297i
\(723\) 3.69694i 0.137491i
\(724\) −6.89898 −0.256399
\(725\) 0 0
\(726\) −1.30306 −0.0483611
\(727\) − 36.0000i − 1.33517i −0.744535 0.667583i \(-0.767329\pi\)
0.744535 0.667583i \(-0.232671\pi\)
\(728\) − 1.10102i − 0.0408065i
\(729\) −29.5153 −1.09316
\(730\) 0 0
\(731\) −3.79796 −0.140473
\(732\) − 9.34847i − 0.345529i
\(733\) − 15.5959i − 0.576048i −0.957623 0.288024i \(-0.907002\pi\)
0.957623 0.288024i \(-0.0929984\pi\)
\(734\) 14.2474 0.525883
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) − 15.6969i − 0.578204i
\(738\) − 0.898979i − 0.0330919i
\(739\) 31.5959 1.16227 0.581137 0.813806i \(-0.302608\pi\)
0.581137 + 0.813806i \(0.302608\pi\)
\(740\) 0 0
\(741\) −3.25765 −0.119673
\(742\) − 14.6969i − 0.539542i
\(743\) − 6.40408i − 0.234943i −0.993076 0.117471i \(-0.962521\pi\)
0.993076 0.117471i \(-0.0374789\pi\)
\(744\) 6.44949 0.236450
\(745\) 0 0
\(746\) −20.0454 −0.733915
\(747\) 3.10102i 0.113460i
\(748\) 11.8990i 0.435070i
\(749\) 20.9444 0.765291
\(750\) 0 0
\(751\) −30.6969 −1.12015 −0.560074 0.828443i \(-0.689227\pi\)
−0.560074 + 0.828443i \(0.689227\pi\)
\(752\) − 9.79796i − 0.357295i
\(753\) 44.6413i 1.62682i
\(754\) −0.404082 −0.0147158
\(755\) 0 0
\(756\) 13.8434 0.503478
\(757\) 13.7980i 0.501495i 0.968052 + 0.250748i \(0.0806764\pi\)
−0.968052 + 0.250748i \(0.919324\pi\)
\(758\) − 21.0454i − 0.764404i
\(759\) 10.0000 0.362977
\(760\) 0 0
\(761\) −14.3031 −0.518486 −0.259243 0.965812i \(-0.583473\pi\)
−0.259243 + 0.965812i \(0.583473\pi\)
\(762\) − 23.5505i − 0.853145i
\(763\) 34.2929i 1.24148i
\(764\) 7.79796 0.282120
\(765\) 0 0
\(766\) −5.34847 −0.193248
\(767\) − 0.898979i − 0.0324603i
\(768\) − 1.44949i − 0.0523040i
\(769\) −53.2474 −1.92015 −0.960076 0.279739i \(-0.909752\pi\)
−0.960076 + 0.279739i \(0.909752\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) 2.55051i 0.0917949i
\(773\) 22.0454i 0.792918i 0.918052 + 0.396459i \(0.129761\pi\)
−0.918052 + 0.396459i \(0.870239\pi\)
\(774\) −0.989795 −0.0355774
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) − 3.55051i − 0.127374i
\(778\) 13.1464i 0.471322i
\(779\) 5.00000 0.179144
\(780\) 0 0
\(781\) −26.0454 −0.931978
\(782\) − 6.89898i − 0.246707i
\(783\) − 5.08061i − 0.181566i
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −12.6061 −0.449646
\(787\) − 30.6969i − 1.09423i −0.837058 0.547114i \(-0.815726\pi\)
0.837058 0.547114i \(-0.184274\pi\)
\(788\) − 17.3485i − 0.618014i
\(789\) 0.292856 0.0104260
\(790\) 0 0
\(791\) −11.1464 −0.396321
\(792\) 3.10102i 0.110190i
\(793\) 2.89898i 0.102946i
\(794\) 15.3485 0.544697
\(795\) 0 0
\(796\) −15.5959 −0.552783
\(797\) − 4.24745i − 0.150452i −0.997166 0.0752262i \(-0.976032\pi\)
0.997166 0.0752262i \(-0.0239679\pi\)
\(798\) 17.7526i 0.628434i
\(799\) −33.7980 −1.19569
\(800\) 0 0
\(801\) −12.8990 −0.455763
\(802\) − 15.9444i − 0.563016i
\(803\) 44.1464i 1.55789i
\(804\) 6.59592 0.232620
\(805\) 0 0
\(806\) −2.00000 −0.0704470
\(807\) 5.79796i 0.204098i
\(808\) 1.10102i 0.0387338i
\(809\) 39.3939 1.38501 0.692507 0.721411i \(-0.256506\pi\)
0.692507 + 0.721411i \(0.256506\pi\)
\(810\) 0 0
\(811\) 5.79796 0.203594 0.101797 0.994805i \(-0.467541\pi\)
0.101797 + 0.994805i \(0.467541\pi\)
\(812\) 2.20204i 0.0772765i
\(813\) − 43.5505i − 1.52738i
\(814\) 3.44949 0.120905
\(815\) 0 0
\(816\) −5.00000 −0.175035
\(817\) − 5.50510i − 0.192599i
\(818\) − 34.1464i − 1.19390i
\(819\) −0.989795 −0.0345862
\(820\) 0 0
\(821\) 4.04541 0.141186 0.0705929 0.997505i \(-0.477511\pi\)
0.0705929 + 0.997505i \(0.477511\pi\)
\(822\) 14.0556i 0.490246i
\(823\) 33.8434i 1.17971i 0.807511 + 0.589853i \(0.200814\pi\)
−0.807511 + 0.589853i \(0.799186\pi\)
\(824\) −6.44949 −0.224679
\(825\) 0 0
\(826\) −4.89898 −0.170457
\(827\) − 27.0000i − 0.938882i −0.882964 0.469441i \(-0.844455\pi\)
0.882964 0.469441i \(-0.155545\pi\)
\(828\) − 1.79796i − 0.0624834i
\(829\) −21.3485 −0.741463 −0.370731 0.928740i \(-0.620893\pi\)
−0.370731 + 0.928740i \(0.620893\pi\)
\(830\) 0 0
\(831\) −8.40408 −0.291534
\(832\) 0.449490i 0.0155833i
\(833\) − 3.44949i − 0.119518i
\(834\) −18.1918 −0.629932
\(835\) 0 0
\(836\) −17.2474 −0.596515
\(837\) − 25.1464i − 0.869188i
\(838\) − 21.4495i − 0.740960i
\(839\) 1.34847 0.0465543 0.0232772 0.999729i \(-0.492590\pi\)
0.0232772 + 0.999729i \(0.492590\pi\)
\(840\) 0 0
\(841\) −28.1918 −0.972132
\(842\) − 32.8990i − 1.13377i
\(843\) − 12.8990i − 0.444264i
\(844\) 4.55051 0.156635
\(845\) 0 0
\(846\) −8.80816 −0.302831
\(847\) − 2.20204i − 0.0756630i
\(848\) 6.00000i 0.206041i
\(849\) −16.9546 −0.581880
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) − 10.9444i − 0.374949i
\(853\) − 19.3485i − 0.662479i −0.943547 0.331239i \(-0.892533\pi\)
0.943547 0.331239i \(-0.107467\pi\)
\(854\) 15.7980 0.540595
\(855\) 0 0
\(856\) −8.55051 −0.292250
\(857\) 32.1464i 1.09810i 0.835789 + 0.549051i \(0.185011\pi\)
−0.835789 + 0.549051i \(0.814989\pi\)
\(858\) 2.24745i 0.0767266i
\(859\) 9.49490 0.323962 0.161981 0.986794i \(-0.448212\pi\)
0.161981 + 0.986794i \(0.448212\pi\)
\(860\) 0 0
\(861\) −3.55051 −0.121001
\(862\) 20.6969i 0.704941i
\(863\) 18.2474i 0.621150i 0.950549 + 0.310575i \(0.100522\pi\)
−0.950549 + 0.310575i \(0.899478\pi\)
\(864\) −5.65153 −0.192269
\(865\) 0 0
\(866\) 19.0000 0.645646
\(867\) − 7.39388i − 0.251109i
\(868\) 10.8990i 0.369935i
\(869\) −26.8990 −0.912485
\(870\) 0 0
\(871\) −2.04541 −0.0693060
\(872\) − 14.0000i − 0.474100i
\(873\) − 12.5857i − 0.425962i
\(874\) 10.0000 0.338255
\(875\) 0 0
\(876\) −18.5505 −0.626764
\(877\) − 4.65153i − 0.157071i −0.996911 0.0785355i \(-0.974976\pi\)
0.996911 0.0785355i \(-0.0250244\pi\)
\(878\) − 19.5959i − 0.661330i
\(879\) −40.6515 −1.37114
\(880\) 0 0
\(881\) 34.2929 1.15536 0.577678 0.816265i \(-0.303959\pi\)
0.577678 + 0.816265i \(0.303959\pi\)
\(882\) − 0.898979i − 0.0302702i
\(883\) 26.1010i 0.878369i 0.898397 + 0.439185i \(0.144733\pi\)
−0.898397 + 0.439185i \(0.855267\pi\)
\(884\) 1.55051 0.0521493
\(885\) 0 0
\(886\) −17.4495 −0.586227
\(887\) − 50.6969i − 1.70224i −0.524974 0.851118i \(-0.675925\pi\)
0.524974 0.851118i \(-0.324075\pi\)
\(888\) 1.44949i 0.0486417i
\(889\) 39.7980 1.33478
\(890\) 0 0
\(891\) −18.9546 −0.635003
\(892\) − 21.7980i − 0.729850i
\(893\) − 48.9898i − 1.63938i
\(894\) −6.74235 −0.225498
\(895\) 0 0
\(896\) 2.44949 0.0818317
\(897\) − 1.30306i − 0.0435080i
\(898\) 33.2474i 1.10948i
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 20.6969 0.689515
\(902\) − 3.44949i − 0.114855i
\(903\) 3.90918i 0.130090i
\(904\) 4.55051 0.151348
\(905\) 0 0
\(906\) 20.2929 0.674185
\(907\) 34.8990i 1.15880i 0.815043 + 0.579401i \(0.196713\pi\)
−0.815043 + 0.579401i \(0.803287\pi\)
\(908\) − 16.6969i − 0.554107i
\(909\) 0.989795 0.0328294
\(910\) 0 0
\(911\) −16.2474 −0.538302 −0.269151 0.963098i \(-0.586743\pi\)
−0.269151 + 0.963098i \(0.586743\pi\)
\(912\) − 7.24745i − 0.239987i
\(913\) 11.8990i 0.393799i
\(914\) −15.2474 −0.504341
\(915\) 0 0
\(916\) 5.79796 0.191570
\(917\) − 21.3031i − 0.703489i
\(918\) 19.4949i 0.643427i
\(919\) 51.7980 1.70866 0.854329 0.519733i \(-0.173969\pi\)
0.854329 + 0.519733i \(0.173969\pi\)
\(920\) 0 0
\(921\) −11.0908 −0.365455
\(922\) − 9.30306i − 0.306380i
\(923\) 3.39388i 0.111711i
\(924\) 12.2474 0.402911
\(925\) 0 0
\(926\) 9.55051 0.313849
\(927\) 5.79796i 0.190430i
\(928\) − 0.898979i − 0.0295104i
\(929\) 5.79796 0.190225 0.0951124 0.995467i \(-0.469679\pi\)
0.0951124 + 0.995467i \(0.469679\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) 28.4949i 0.933381i
\(933\) 12.2474i 0.400963i
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0.404082 0.0132078
\(937\) − 14.5959i − 0.476828i −0.971164 0.238414i \(-0.923373\pi\)
0.971164 0.238414i \(-0.0766275\pi\)
\(938\) 11.1464i 0.363944i
\(939\) 12.8990 0.420942
\(940\) 0 0
\(941\) −41.3939 −1.34940 −0.674701 0.738091i \(-0.735727\pi\)
−0.674701 + 0.738091i \(0.735727\pi\)
\(942\) 23.8434i 0.776859i
\(943\) 2.00000i 0.0651290i
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) −3.79796 −0.123482
\(947\) − 32.0000i − 1.03986i −0.854209 0.519930i \(-0.825958\pi\)
0.854209 0.519930i \(-0.174042\pi\)
\(948\) − 11.3031i − 0.367106i
\(949\) 5.75255 0.186736
\(950\) 0 0
\(951\) 13.8434 0.448902
\(952\) − 8.44949i − 0.273850i
\(953\) 25.4949i 0.825861i 0.910763 + 0.412930i \(0.135495\pi\)
−0.910763 + 0.412930i \(0.864505\pi\)
\(954\) 5.39388 0.174633
\(955\) 0 0
\(956\) 0.898979 0.0290751
\(957\) − 4.49490i − 0.145299i
\(958\) 6.24745i 0.201846i
\(959\) −23.7526 −0.767010
\(960\) 0 0
\(961\) −11.2020 −0.361356
\(962\) − 0.449490i − 0.0144921i
\(963\) 7.68673i 0.247702i
\(964\) 2.55051 0.0821464
\(965\) 0 0
\(966\) −7.10102 −0.228472
\(967\) 44.0000i 1.41494i 0.706741 + 0.707472i \(0.250165\pi\)
−0.706741 + 0.707472i \(0.749835\pi\)
\(968\) 0.898979i 0.0288943i
\(969\) −25.0000 −0.803116
\(970\) 0 0
\(971\) −53.5403 −1.71819 −0.859095 0.511816i \(-0.828973\pi\)
−0.859095 + 0.511816i \(0.828973\pi\)
\(972\) 8.98979i 0.288348i
\(973\) − 30.7423i − 0.985554i
\(974\) 30.7423 0.985048
\(975\) 0 0
\(976\) −6.44949 −0.206443
\(977\) 23.7423i 0.759585i 0.925072 + 0.379792i \(0.124005\pi\)
−0.925072 + 0.379792i \(0.875995\pi\)
\(978\) − 14.6413i − 0.468178i
\(979\) −49.4949 −1.58186
\(980\) 0 0
\(981\) −12.5857 −0.401831
\(982\) 41.7980i 1.33383i
\(983\) 18.4949i 0.589896i 0.955513 + 0.294948i \(0.0953023\pi\)
−0.955513 + 0.294948i \(0.904698\pi\)
\(984\) 1.44949 0.0462080
\(985\) 0 0
\(986\) −3.10102 −0.0987566
\(987\) 34.7878i 1.10731i
\(988\) 2.24745i 0.0715009i
\(989\) 2.20204 0.0700208
\(990\) 0 0
\(991\) −16.4495 −0.522535 −0.261268 0.965266i \(-0.584141\pi\)
−0.261268 + 0.965266i \(0.584141\pi\)
\(992\) − 4.44949i − 0.141271i
\(993\) − 28.8434i − 0.915317i
\(994\) 18.4949 0.586623
\(995\) 0 0
\(996\) −5.00000 −0.158431
\(997\) − 39.5505i − 1.25258i −0.779591 0.626289i \(-0.784573\pi\)
0.779591 0.626289i \(-0.215427\pi\)
\(998\) 22.0000i 0.696398i
\(999\) 5.65153 0.178807
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.k.149.1 4
5.2 odd 4 1850.2.a.w.1.1 yes 2
5.3 odd 4 1850.2.a.r.1.2 2
5.4 even 2 inner 1850.2.b.k.149.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.r.1.2 2 5.3 odd 4
1850.2.a.w.1.1 yes 2 5.2 odd 4
1850.2.b.k.149.1 4 1.1 even 1 trivial
1850.2.b.k.149.4 4 5.4 even 2 inner