Properties

Label 1850.2.a.r.1.2
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.44949 q^{3} +1.00000 q^{4} -1.44949 q^{6} -2.44949 q^{7} -1.00000 q^{8} -0.898979 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.44949 q^{3} +1.00000 q^{4} -1.44949 q^{6} -2.44949 q^{7} -1.00000 q^{8} -0.898979 q^{9} +3.44949 q^{11} +1.44949 q^{12} +0.449490 q^{13} +2.44949 q^{14} +1.00000 q^{16} -3.44949 q^{17} +0.898979 q^{18} -5.00000 q^{19} -3.55051 q^{21} -3.44949 q^{22} -2.00000 q^{23} -1.44949 q^{24} -0.449490 q^{26} -5.65153 q^{27} -2.44949 q^{28} -0.898979 q^{29} +4.44949 q^{31} -1.00000 q^{32} +5.00000 q^{33} +3.44949 q^{34} -0.898979 q^{36} +1.00000 q^{37} +5.00000 q^{38} +0.651531 q^{39} +1.00000 q^{41} +3.55051 q^{42} +1.10102 q^{43} +3.44949 q^{44} +2.00000 q^{46} -9.79796 q^{47} +1.44949 q^{48} -1.00000 q^{49} -5.00000 q^{51} +0.449490 q^{52} -6.00000 q^{53} +5.65153 q^{54} +2.44949 q^{56} -7.24745 q^{57} +0.898979 q^{58} -2.00000 q^{59} -6.44949 q^{61} -4.44949 q^{62} +2.20204 q^{63} +1.00000 q^{64} -5.00000 q^{66} -4.55051 q^{67} -3.44949 q^{68} -2.89898 q^{69} -7.55051 q^{71} +0.898979 q^{72} -12.7980 q^{73} -1.00000 q^{74} -5.00000 q^{76} -8.44949 q^{77} -0.651531 q^{78} +7.79796 q^{79} -5.49490 q^{81} -1.00000 q^{82} -3.44949 q^{83} -3.55051 q^{84} -1.10102 q^{86} -1.30306 q^{87} -3.44949 q^{88} +14.3485 q^{89} -1.10102 q^{91} -2.00000 q^{92} +6.44949 q^{93} +9.79796 q^{94} -1.44949 q^{96} -14.0000 q^{97} +1.00000 q^{98} -3.10102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{17} - 8 q^{18} - 10 q^{19} - 12 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} + 4 q^{26} - 26 q^{27} + 8 q^{29} + 4 q^{31} - 2 q^{32} + 10 q^{33} + 2 q^{34} + 8 q^{36} + 2 q^{37} + 10 q^{38} + 16 q^{39} + 2 q^{41} + 12 q^{42} + 12 q^{43} + 2 q^{44} + 4 q^{46} - 2 q^{48} - 2 q^{49} - 10 q^{51} - 4 q^{52} - 12 q^{53} + 26 q^{54} + 10 q^{57} - 8 q^{58} - 4 q^{59} - 8 q^{61} - 4 q^{62} + 24 q^{63} + 2 q^{64} - 10 q^{66} - 14 q^{67} - 2 q^{68} + 4 q^{69} - 20 q^{71} - 8 q^{72} - 6 q^{73} - 2 q^{74} - 10 q^{76} - 12 q^{77} - 16 q^{78} - 4 q^{79} + 38 q^{81} - 2 q^{82} - 2 q^{83} - 12 q^{84} - 12 q^{86} - 32 q^{87} - 2 q^{88} + 14 q^{89} - 12 q^{91} - 4 q^{92} + 8 q^{93} + 2 q^{96} - 28 q^{97} + 2 q^{98} - 16 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.44949 0.836863 0.418432 0.908248i \(-0.362580\pi\)
0.418432 + 0.908248i \(0.362580\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.44949 −0.591752
\(7\) −2.44949 −0.925820 −0.462910 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.898979 −0.299660
\(10\) 0 0
\(11\) 3.44949 1.04006 0.520030 0.854148i \(-0.325921\pi\)
0.520030 + 0.854148i \(0.325921\pi\)
\(12\) 1.44949 0.418432
\(13\) 0.449490 0.124666 0.0623330 0.998055i \(-0.480146\pi\)
0.0623330 + 0.998055i \(0.480146\pi\)
\(14\) 2.44949 0.654654
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.44949 −0.836624 −0.418312 0.908303i \(-0.637378\pi\)
−0.418312 + 0.908303i \(0.637378\pi\)
\(18\) 0.898979 0.211891
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) −3.55051 −0.774785
\(22\) −3.44949 −0.735434
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) −1.44949 −0.295876
\(25\) 0 0
\(26\) −0.449490 −0.0881522
\(27\) −5.65153 −1.08764
\(28\) −2.44949 −0.462910
\(29\) −0.898979 −0.166936 −0.0834681 0.996510i \(-0.526600\pi\)
−0.0834681 + 0.996510i \(0.526600\pi\)
\(30\) 0 0
\(31\) 4.44949 0.799152 0.399576 0.916700i \(-0.369157\pi\)
0.399576 + 0.916700i \(0.369157\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.00000 0.870388
\(34\) 3.44949 0.591583
\(35\) 0 0
\(36\) −0.898979 −0.149830
\(37\) 1.00000 0.164399
\(38\) 5.00000 0.811107
\(39\) 0.651531 0.104328
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 3.55051 0.547856
\(43\) 1.10102 0.167904 0.0839520 0.996470i \(-0.473246\pi\)
0.0839520 + 0.996470i \(0.473246\pi\)
\(44\) 3.44949 0.520030
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −9.79796 −1.42918 −0.714590 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 1.44949 0.209216
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0.449490 0.0623330
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.65153 0.769076
\(55\) 0 0
\(56\) 2.44949 0.327327
\(57\) −7.24745 −0.959948
\(58\) 0.898979 0.118042
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −6.44949 −0.825773 −0.412886 0.910783i \(-0.635479\pi\)
−0.412886 + 0.910783i \(0.635479\pi\)
\(62\) −4.44949 −0.565086
\(63\) 2.20204 0.277431
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) −4.55051 −0.555933 −0.277967 0.960591i \(-0.589660\pi\)
−0.277967 + 0.960591i \(0.589660\pi\)
\(68\) −3.44949 −0.418312
\(69\) −2.89898 −0.348996
\(70\) 0 0
\(71\) −7.55051 −0.896081 −0.448040 0.894013i \(-0.647878\pi\)
−0.448040 + 0.894013i \(0.647878\pi\)
\(72\) 0.898979 0.105946
\(73\) −12.7980 −1.49789 −0.748944 0.662633i \(-0.769439\pi\)
−0.748944 + 0.662633i \(0.769439\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) −8.44949 −0.962909
\(78\) −0.651531 −0.0737713
\(79\) 7.79796 0.877339 0.438669 0.898648i \(-0.355450\pi\)
0.438669 + 0.898648i \(0.355450\pi\)
\(80\) 0 0
\(81\) −5.49490 −0.610544
\(82\) −1.00000 −0.110432
\(83\) −3.44949 −0.378631 −0.189315 0.981916i \(-0.560627\pi\)
−0.189315 + 0.981916i \(0.560627\pi\)
\(84\) −3.55051 −0.387392
\(85\) 0 0
\(86\) −1.10102 −0.118726
\(87\) −1.30306 −0.139703
\(88\) −3.44949 −0.367717
\(89\) 14.3485 1.52093 0.760467 0.649376i \(-0.224970\pi\)
0.760467 + 0.649376i \(0.224970\pi\)
\(90\) 0 0
\(91\) −1.10102 −0.115418
\(92\) −2.00000 −0.208514
\(93\) 6.44949 0.668781
\(94\) 9.79796 1.01058
\(95\) 0 0
\(96\) −1.44949 −0.147938
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) −3.10102 −0.311664
\(100\) 0 0
\(101\) 1.10102 0.109556 0.0547778 0.998499i \(-0.482555\pi\)
0.0547778 + 0.998499i \(0.482555\pi\)
\(102\) 5.00000 0.495074
\(103\) −6.44949 −0.635487 −0.317744 0.948177i \(-0.602925\pi\)
−0.317744 + 0.948177i \(0.602925\pi\)
\(104\) −0.449490 −0.0440761
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 8.55051 0.826609 0.413305 0.910593i \(-0.364375\pi\)
0.413305 + 0.910593i \(0.364375\pi\)
\(108\) −5.65153 −0.543819
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 1.44949 0.137579
\(112\) −2.44949 −0.231455
\(113\) 4.55051 0.428076 0.214038 0.976825i \(-0.431338\pi\)
0.214038 + 0.976825i \(0.431338\pi\)
\(114\) 7.24745 0.678786
\(115\) 0 0
\(116\) −0.898979 −0.0834681
\(117\) −0.404082 −0.0373574
\(118\) 2.00000 0.184115
\(119\) 8.44949 0.774563
\(120\) 0 0
\(121\) 0.898979 0.0817254
\(122\) 6.44949 0.583909
\(123\) 1.44949 0.130696
\(124\) 4.44949 0.399576
\(125\) 0 0
\(126\) −2.20204 −0.196173
\(127\) 16.2474 1.44173 0.720864 0.693077i \(-0.243745\pi\)
0.720864 + 0.693077i \(0.243745\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.59592 0.140513
\(130\) 0 0
\(131\) 8.69694 0.759855 0.379928 0.925016i \(-0.375949\pi\)
0.379928 + 0.925016i \(0.375949\pi\)
\(132\) 5.00000 0.435194
\(133\) 12.2474 1.06199
\(134\) 4.55051 0.393104
\(135\) 0 0
\(136\) 3.44949 0.295791
\(137\) −9.69694 −0.828465 −0.414233 0.910171i \(-0.635950\pi\)
−0.414233 + 0.910171i \(0.635950\pi\)
\(138\) 2.89898 0.246778
\(139\) −12.5505 −1.06452 −0.532260 0.846581i \(-0.678657\pi\)
−0.532260 + 0.846581i \(0.678657\pi\)
\(140\) 0 0
\(141\) −14.2020 −1.19603
\(142\) 7.55051 0.633625
\(143\) 1.55051 0.129660
\(144\) −0.898979 −0.0749150
\(145\) 0 0
\(146\) 12.7980 1.05917
\(147\) −1.44949 −0.119552
\(148\) 1.00000 0.0821995
\(149\) −4.65153 −0.381068 −0.190534 0.981681i \(-0.561022\pi\)
−0.190534 + 0.981681i \(0.561022\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 5.00000 0.405554
\(153\) 3.10102 0.250703
\(154\) 8.44949 0.680879
\(155\) 0 0
\(156\) 0.651531 0.0521642
\(157\) −16.4495 −1.31281 −0.656406 0.754408i \(-0.727924\pi\)
−0.656406 + 0.754408i \(0.727924\pi\)
\(158\) −7.79796 −0.620372
\(159\) −8.69694 −0.689712
\(160\) 0 0
\(161\) 4.89898 0.386094
\(162\) 5.49490 0.431720
\(163\) −10.1010 −0.791173 −0.395586 0.918429i \(-0.629459\pi\)
−0.395586 + 0.918429i \(0.629459\pi\)
\(164\) 1.00000 0.0780869
\(165\) 0 0
\(166\) 3.44949 0.267732
\(167\) 10.4495 0.808606 0.404303 0.914625i \(-0.367514\pi\)
0.404303 + 0.914625i \(0.367514\pi\)
\(168\) 3.55051 0.273928
\(169\) −12.7980 −0.984458
\(170\) 0 0
\(171\) 4.49490 0.343733
\(172\) 1.10102 0.0839520
\(173\) 6.69694 0.509159 0.254579 0.967052i \(-0.418063\pi\)
0.254579 + 0.967052i \(0.418063\pi\)
\(174\) 1.30306 0.0987848
\(175\) 0 0
\(176\) 3.44949 0.260015
\(177\) −2.89898 −0.217901
\(178\) −14.3485 −1.07546
\(179\) −0.797959 −0.0596423 −0.0298211 0.999555i \(-0.509494\pi\)
−0.0298211 + 0.999555i \(0.509494\pi\)
\(180\) 0 0
\(181\) 6.89898 0.512797 0.256399 0.966571i \(-0.417464\pi\)
0.256399 + 0.966571i \(0.417464\pi\)
\(182\) 1.10102 0.0816131
\(183\) −9.34847 −0.691059
\(184\) 2.00000 0.147442
\(185\) 0 0
\(186\) −6.44949 −0.472900
\(187\) −11.8990 −0.870140
\(188\) −9.79796 −0.714590
\(189\) 13.8434 1.00696
\(190\) 0 0
\(191\) −7.79796 −0.564241 −0.282120 0.959379i \(-0.591038\pi\)
−0.282120 + 0.959379i \(0.591038\pi\)
\(192\) 1.44949 0.104608
\(193\) 2.55051 0.183590 0.0917949 0.995778i \(-0.470740\pi\)
0.0917949 + 0.995778i \(0.470740\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) 17.3485 1.23603 0.618014 0.786167i \(-0.287938\pi\)
0.618014 + 0.786167i \(0.287938\pi\)
\(198\) 3.10102 0.220380
\(199\) −15.5959 −1.10557 −0.552783 0.833325i \(-0.686434\pi\)
−0.552783 + 0.833325i \(0.686434\pi\)
\(200\) 0 0
\(201\) −6.59592 −0.465240
\(202\) −1.10102 −0.0774675
\(203\) 2.20204 0.154553
\(204\) −5.00000 −0.350070
\(205\) 0 0
\(206\) 6.44949 0.449357
\(207\) 1.79796 0.124967
\(208\) 0.449490 0.0311665
\(209\) −17.2474 −1.19303
\(210\) 0 0
\(211\) −4.55051 −0.313270 −0.156635 0.987657i \(-0.550065\pi\)
−0.156635 + 0.987657i \(0.550065\pi\)
\(212\) −6.00000 −0.412082
\(213\) −10.9444 −0.749897
\(214\) −8.55051 −0.584501
\(215\) 0 0
\(216\) 5.65153 0.384538
\(217\) −10.8990 −0.739871
\(218\) −14.0000 −0.948200
\(219\) −18.5505 −1.25353
\(220\) 0 0
\(221\) −1.55051 −0.104299
\(222\) −1.44949 −0.0972834
\(223\) −21.7980 −1.45970 −0.729850 0.683608i \(-0.760410\pi\)
−0.729850 + 0.683608i \(0.760410\pi\)
\(224\) 2.44949 0.163663
\(225\) 0 0
\(226\) −4.55051 −0.302695
\(227\) 16.6969 1.10821 0.554107 0.832445i \(-0.313060\pi\)
0.554107 + 0.832445i \(0.313060\pi\)
\(228\) −7.24745 −0.479974
\(229\) 5.79796 0.383140 0.191570 0.981479i \(-0.438642\pi\)
0.191570 + 0.981479i \(0.438642\pi\)
\(230\) 0 0
\(231\) −12.2474 −0.805823
\(232\) 0.898979 0.0590209
\(233\) 28.4949 1.86676 0.933381 0.358886i \(-0.116843\pi\)
0.933381 + 0.358886i \(0.116843\pi\)
\(234\) 0.404082 0.0264157
\(235\) 0 0
\(236\) −2.00000 −0.130189
\(237\) 11.3031 0.734213
\(238\) −8.44949 −0.547699
\(239\) 0.898979 0.0581501 0.0290751 0.999577i \(-0.490744\pi\)
0.0290751 + 0.999577i \(0.490744\pi\)
\(240\) 0 0
\(241\) −2.55051 −0.164293 −0.0821464 0.996620i \(-0.526178\pi\)
−0.0821464 + 0.996620i \(0.526178\pi\)
\(242\) −0.898979 −0.0577886
\(243\) 8.98979 0.576696
\(244\) −6.44949 −0.412886
\(245\) 0 0
\(246\) −1.44949 −0.0924161
\(247\) −2.24745 −0.143002
\(248\) −4.44949 −0.282543
\(249\) −5.00000 −0.316862
\(250\) 0 0
\(251\) −30.7980 −1.94395 −0.971975 0.235084i \(-0.924463\pi\)
−0.971975 + 0.235084i \(0.924463\pi\)
\(252\) 2.20204 0.138716
\(253\) −6.89898 −0.433735
\(254\) −16.2474 −1.01946
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.89898 0.430347 0.215173 0.976576i \(-0.430968\pi\)
0.215173 + 0.976576i \(0.430968\pi\)
\(258\) −1.59592 −0.0993575
\(259\) −2.44949 −0.152204
\(260\) 0 0
\(261\) 0.808164 0.0500241
\(262\) −8.69694 −0.537299
\(263\) −0.202041 −0.0124584 −0.00622919 0.999981i \(-0.501983\pi\)
−0.00622919 + 0.999981i \(0.501983\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) −12.2474 −0.750939
\(267\) 20.7980 1.27281
\(268\) −4.55051 −0.277967
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) 30.0454 1.82513 0.912564 0.408933i \(-0.134099\pi\)
0.912564 + 0.408933i \(0.134099\pi\)
\(272\) −3.44949 −0.209156
\(273\) −1.59592 −0.0965893
\(274\) 9.69694 0.585813
\(275\) 0 0
\(276\) −2.89898 −0.174498
\(277\) −5.79796 −0.348366 −0.174183 0.984713i \(-0.555728\pi\)
−0.174183 + 0.984713i \(0.555728\pi\)
\(278\) 12.5505 0.752730
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 8.89898 0.530869 0.265434 0.964129i \(-0.414485\pi\)
0.265434 + 0.964129i \(0.414485\pi\)
\(282\) 14.2020 0.845719
\(283\) 11.6969 0.695311 0.347655 0.937622i \(-0.386978\pi\)
0.347655 + 0.937622i \(0.386978\pi\)
\(284\) −7.55051 −0.448040
\(285\) 0 0
\(286\) −1.55051 −0.0916836
\(287\) −2.44949 −0.144589
\(288\) 0.898979 0.0529729
\(289\) −5.10102 −0.300060
\(290\) 0 0
\(291\) −20.2929 −1.18959
\(292\) −12.7980 −0.748944
\(293\) 28.0454 1.63843 0.819215 0.573486i \(-0.194409\pi\)
0.819215 + 0.573486i \(0.194409\pi\)
\(294\) 1.44949 0.0845360
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −19.4949 −1.13121
\(298\) 4.65153 0.269456
\(299\) −0.898979 −0.0519893
\(300\) 0 0
\(301\) −2.69694 −0.155449
\(302\) 14.0000 0.805609
\(303\) 1.59592 0.0916831
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) −3.10102 −0.177274
\(307\) −7.65153 −0.436696 −0.218348 0.975871i \(-0.570067\pi\)
−0.218348 + 0.975871i \(0.570067\pi\)
\(308\) −8.44949 −0.481454
\(309\) −9.34847 −0.531816
\(310\) 0 0
\(311\) −8.44949 −0.479127 −0.239563 0.970881i \(-0.577004\pi\)
−0.239563 + 0.970881i \(0.577004\pi\)
\(312\) −0.651531 −0.0368857
\(313\) −8.89898 −0.503000 −0.251500 0.967857i \(-0.580924\pi\)
−0.251500 + 0.967857i \(0.580924\pi\)
\(314\) 16.4495 0.928298
\(315\) 0 0
\(316\) 7.79796 0.438669
\(317\) 9.55051 0.536410 0.268205 0.963362i \(-0.413570\pi\)
0.268205 + 0.963362i \(0.413570\pi\)
\(318\) 8.69694 0.487700
\(319\) −3.10102 −0.173624
\(320\) 0 0
\(321\) 12.3939 0.691759
\(322\) −4.89898 −0.273009
\(323\) 17.2474 0.959674
\(324\) −5.49490 −0.305272
\(325\) 0 0
\(326\) 10.1010 0.559444
\(327\) 20.2929 1.12220
\(328\) −1.00000 −0.0552158
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 19.8990 1.09375 0.546873 0.837215i \(-0.315818\pi\)
0.546873 + 0.837215i \(0.315818\pi\)
\(332\) −3.44949 −0.189315
\(333\) −0.898979 −0.0492638
\(334\) −10.4495 −0.571771
\(335\) 0 0
\(336\) −3.55051 −0.193696
\(337\) −15.6969 −0.855067 −0.427533 0.904000i \(-0.640617\pi\)
−0.427533 + 0.904000i \(0.640617\pi\)
\(338\) 12.7980 0.696117
\(339\) 6.59592 0.358241
\(340\) 0 0
\(341\) 15.3485 0.831166
\(342\) −4.49490 −0.243056
\(343\) 19.5959 1.05808
\(344\) −1.10102 −0.0593630
\(345\) 0 0
\(346\) −6.69694 −0.360030
\(347\) 18.7980 1.00913 0.504564 0.863374i \(-0.331653\pi\)
0.504564 + 0.863374i \(0.331653\pi\)
\(348\) −1.30306 −0.0698514
\(349\) −12.4495 −0.666406 −0.333203 0.942855i \(-0.608129\pi\)
−0.333203 + 0.942855i \(0.608129\pi\)
\(350\) 0 0
\(351\) −2.54031 −0.135591
\(352\) −3.44949 −0.183858
\(353\) −14.8990 −0.792993 −0.396496 0.918036i \(-0.629774\pi\)
−0.396496 + 0.918036i \(0.629774\pi\)
\(354\) 2.89898 0.154079
\(355\) 0 0
\(356\) 14.3485 0.760467
\(357\) 12.2474 0.648204
\(358\) 0.797959 0.0421734
\(359\) −31.3939 −1.65691 −0.828453 0.560059i \(-0.810778\pi\)
−0.828453 + 0.560059i \(0.810778\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −6.89898 −0.362602
\(363\) 1.30306 0.0683930
\(364\) −1.10102 −0.0577092
\(365\) 0 0
\(366\) 9.34847 0.488652
\(367\) 14.2474 0.743711 0.371855 0.928291i \(-0.378722\pi\)
0.371855 + 0.928291i \(0.378722\pi\)
\(368\) −2.00000 −0.104257
\(369\) −0.898979 −0.0467990
\(370\) 0 0
\(371\) 14.6969 0.763027
\(372\) 6.44949 0.334390
\(373\) 20.0454 1.03791 0.518956 0.854801i \(-0.326321\pi\)
0.518956 + 0.854801i \(0.326321\pi\)
\(374\) 11.8990 0.615282
\(375\) 0 0
\(376\) 9.79796 0.505291
\(377\) −0.404082 −0.0208113
\(378\) −13.8434 −0.712026
\(379\) −21.0454 −1.08103 −0.540515 0.841334i \(-0.681771\pi\)
−0.540515 + 0.841334i \(0.681771\pi\)
\(380\) 0 0
\(381\) 23.5505 1.20653
\(382\) 7.79796 0.398978
\(383\) 5.34847 0.273294 0.136647 0.990620i \(-0.456367\pi\)
0.136647 + 0.990620i \(0.456367\pi\)
\(384\) −1.44949 −0.0739690
\(385\) 0 0
\(386\) −2.55051 −0.129818
\(387\) −0.989795 −0.0503141
\(388\) −14.0000 −0.710742
\(389\) 13.1464 0.666550 0.333275 0.942830i \(-0.391846\pi\)
0.333275 + 0.942830i \(0.391846\pi\)
\(390\) 0 0
\(391\) 6.89898 0.348896
\(392\) 1.00000 0.0505076
\(393\) 12.6061 0.635895
\(394\) −17.3485 −0.874003
\(395\) 0 0
\(396\) −3.10102 −0.155832
\(397\) 15.3485 0.770318 0.385159 0.922850i \(-0.374147\pi\)
0.385159 + 0.922850i \(0.374147\pi\)
\(398\) 15.5959 0.781753
\(399\) 17.7526 0.888739
\(400\) 0 0
\(401\) 15.9444 0.796225 0.398112 0.917337i \(-0.369665\pi\)
0.398112 + 0.917337i \(0.369665\pi\)
\(402\) 6.59592 0.328974
\(403\) 2.00000 0.0996271
\(404\) 1.10102 0.0547778
\(405\) 0 0
\(406\) −2.20204 −0.109285
\(407\) 3.44949 0.170985
\(408\) 5.00000 0.247537
\(409\) −34.1464 −1.68843 −0.844216 0.536003i \(-0.819934\pi\)
−0.844216 + 0.536003i \(0.819934\pi\)
\(410\) 0 0
\(411\) −14.0556 −0.693312
\(412\) −6.44949 −0.317744
\(413\) 4.89898 0.241063
\(414\) −1.79796 −0.0883649
\(415\) 0 0
\(416\) −0.449490 −0.0220380
\(417\) −18.1918 −0.890858
\(418\) 17.2474 0.843600
\(419\) −21.4495 −1.04788 −0.523938 0.851756i \(-0.675538\pi\)
−0.523938 + 0.851756i \(0.675538\pi\)
\(420\) 0 0
\(421\) 32.8990 1.60340 0.801699 0.597728i \(-0.203930\pi\)
0.801699 + 0.597728i \(0.203930\pi\)
\(422\) 4.55051 0.221515
\(423\) 8.80816 0.428268
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 10.9444 0.530257
\(427\) 15.7980 0.764517
\(428\) 8.55051 0.413305
\(429\) 2.24745 0.108508
\(430\) 0 0
\(431\) −20.6969 −0.996936 −0.498468 0.866908i \(-0.666104\pi\)
−0.498468 + 0.866908i \(0.666104\pi\)
\(432\) −5.65153 −0.271909
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 10.8990 0.523168
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 10.0000 0.478365
\(438\) 18.5505 0.886378
\(439\) −19.5959 −0.935262 −0.467631 0.883924i \(-0.654892\pi\)
−0.467631 + 0.883924i \(0.654892\pi\)
\(440\) 0 0
\(441\) 0.898979 0.0428085
\(442\) 1.55051 0.0737503
\(443\) 17.4495 0.829050 0.414525 0.910038i \(-0.363948\pi\)
0.414525 + 0.910038i \(0.363948\pi\)
\(444\) 1.44949 0.0687897
\(445\) 0 0
\(446\) 21.7980 1.03216
\(447\) −6.74235 −0.318902
\(448\) −2.44949 −0.115728
\(449\) 33.2474 1.56904 0.784522 0.620101i \(-0.212908\pi\)
0.784522 + 0.620101i \(0.212908\pi\)
\(450\) 0 0
\(451\) 3.44949 0.162430
\(452\) 4.55051 0.214038
\(453\) −20.2929 −0.953442
\(454\) −16.6969 −0.783626
\(455\) 0 0
\(456\) 7.24745 0.339393
\(457\) −15.2474 −0.713246 −0.356623 0.934248i \(-0.616072\pi\)
−0.356623 + 0.934248i \(0.616072\pi\)
\(458\) −5.79796 −0.270921
\(459\) 19.4949 0.909944
\(460\) 0 0
\(461\) 9.30306 0.433287 0.216643 0.976251i \(-0.430489\pi\)
0.216643 + 0.976251i \(0.430489\pi\)
\(462\) 12.2474 0.569803
\(463\) −9.55051 −0.443850 −0.221925 0.975064i \(-0.571234\pi\)
−0.221925 + 0.975064i \(0.571234\pi\)
\(464\) −0.898979 −0.0417341
\(465\) 0 0
\(466\) −28.4949 −1.32000
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −0.404082 −0.0186787
\(469\) 11.1464 0.514694
\(470\) 0 0
\(471\) −23.8434 −1.09864
\(472\) 2.00000 0.0920575
\(473\) 3.79796 0.174630
\(474\) −11.3031 −0.519167
\(475\) 0 0
\(476\) 8.44949 0.387282
\(477\) 5.39388 0.246969
\(478\) −0.898979 −0.0411184
\(479\) 6.24745 0.285453 0.142727 0.989762i \(-0.454413\pi\)
0.142727 + 0.989762i \(0.454413\pi\)
\(480\) 0 0
\(481\) 0.449490 0.0204950
\(482\) 2.55051 0.116173
\(483\) 7.10102 0.323108
\(484\) 0.898979 0.0408627
\(485\) 0 0
\(486\) −8.98979 −0.407785
\(487\) 30.7423 1.39307 0.696534 0.717523i \(-0.254724\pi\)
0.696534 + 0.717523i \(0.254724\pi\)
\(488\) 6.44949 0.291955
\(489\) −14.6413 −0.662104
\(490\) 0 0
\(491\) −41.7980 −1.88632 −0.943158 0.332345i \(-0.892160\pi\)
−0.943158 + 0.332345i \(0.892160\pi\)
\(492\) 1.44949 0.0653480
\(493\) 3.10102 0.139663
\(494\) 2.24745 0.101117
\(495\) 0 0
\(496\) 4.44949 0.199788
\(497\) 18.4949 0.829610
\(498\) 5.00000 0.224055
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) 15.1464 0.676693
\(502\) 30.7980 1.37458
\(503\) −9.79796 −0.436869 −0.218435 0.975852i \(-0.570095\pi\)
−0.218435 + 0.975852i \(0.570095\pi\)
\(504\) −2.20204 −0.0980867
\(505\) 0 0
\(506\) 6.89898 0.306697
\(507\) −18.5505 −0.823857
\(508\) 16.2474 0.720864
\(509\) 29.3485 1.30085 0.650424 0.759571i \(-0.274591\pi\)
0.650424 + 0.759571i \(0.274591\pi\)
\(510\) 0 0
\(511\) 31.3485 1.38677
\(512\) −1.00000 −0.0441942
\(513\) 28.2577 1.24761
\(514\) −6.89898 −0.304301
\(515\) 0 0
\(516\) 1.59592 0.0702564
\(517\) −33.7980 −1.48643
\(518\) 2.44949 0.107624
\(519\) 9.70714 0.426096
\(520\) 0 0
\(521\) 10.1010 0.442534 0.221267 0.975213i \(-0.428981\pi\)
0.221267 + 0.975213i \(0.428981\pi\)
\(522\) −0.808164 −0.0353724
\(523\) 18.3939 0.804308 0.402154 0.915572i \(-0.368262\pi\)
0.402154 + 0.915572i \(0.368262\pi\)
\(524\) 8.69694 0.379928
\(525\) 0 0
\(526\) 0.202041 0.00880941
\(527\) −15.3485 −0.668590
\(528\) 5.00000 0.217597
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 1.79796 0.0780248
\(532\) 12.2474 0.530994
\(533\) 0.449490 0.0194696
\(534\) −20.7980 −0.900016
\(535\) 0 0
\(536\) 4.55051 0.196552
\(537\) −1.15663 −0.0499124
\(538\) −4.00000 −0.172452
\(539\) −3.44949 −0.148580
\(540\) 0 0
\(541\) 25.3485 1.08982 0.544908 0.838496i \(-0.316565\pi\)
0.544908 + 0.838496i \(0.316565\pi\)
\(542\) −30.0454 −1.29056
\(543\) 10.0000 0.429141
\(544\) 3.44949 0.147896
\(545\) 0 0
\(546\) 1.59592 0.0682990
\(547\) −3.69694 −0.158070 −0.0790348 0.996872i \(-0.525184\pi\)
−0.0790348 + 0.996872i \(0.525184\pi\)
\(548\) −9.69694 −0.414233
\(549\) 5.79796 0.247451
\(550\) 0 0
\(551\) 4.49490 0.191489
\(552\) 2.89898 0.123389
\(553\) −19.1010 −0.812258
\(554\) 5.79796 0.246332
\(555\) 0 0
\(556\) −12.5505 −0.532260
\(557\) 21.7980 0.923609 0.461805 0.886982i \(-0.347202\pi\)
0.461805 + 0.886982i \(0.347202\pi\)
\(558\) 4.00000 0.169334
\(559\) 0.494897 0.0209319
\(560\) 0 0
\(561\) −17.2474 −0.728188
\(562\) −8.89898 −0.375381
\(563\) −37.5959 −1.58448 −0.792240 0.610210i \(-0.791085\pi\)
−0.792240 + 0.610210i \(0.791085\pi\)
\(564\) −14.2020 −0.598014
\(565\) 0 0
\(566\) −11.6969 −0.491659
\(567\) 13.4597 0.565254
\(568\) 7.55051 0.316812
\(569\) 27.0454 1.13380 0.566901 0.823786i \(-0.308142\pi\)
0.566901 + 0.823786i \(0.308142\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 1.55051 0.0648301
\(573\) −11.3031 −0.472192
\(574\) 2.44949 0.102240
\(575\) 0 0
\(576\) −0.898979 −0.0374575
\(577\) −21.0454 −0.876132 −0.438066 0.898943i \(-0.644336\pi\)
−0.438066 + 0.898943i \(0.644336\pi\)
\(578\) 5.10102 0.212174
\(579\) 3.69694 0.153640
\(580\) 0 0
\(581\) 8.44949 0.350544
\(582\) 20.2929 0.841166
\(583\) −20.6969 −0.857180
\(584\) 12.7980 0.529583
\(585\) 0 0
\(586\) −28.0454 −1.15855
\(587\) 6.30306 0.260155 0.130078 0.991504i \(-0.458477\pi\)
0.130078 + 0.991504i \(0.458477\pi\)
\(588\) −1.44949 −0.0597759
\(589\) −22.2474 −0.916690
\(590\) 0 0
\(591\) 25.1464 1.03439
\(592\) 1.00000 0.0410997
\(593\) −46.3939 −1.90517 −0.952584 0.304275i \(-0.901586\pi\)
−0.952584 + 0.304275i \(0.901586\pi\)
\(594\) 19.4949 0.799885
\(595\) 0 0
\(596\) −4.65153 −0.190534
\(597\) −22.6061 −0.925207
\(598\) 0.898979 0.0367620
\(599\) 38.9444 1.59122 0.795612 0.605806i \(-0.207149\pi\)
0.795612 + 0.605806i \(0.207149\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 2.69694 0.109919
\(603\) 4.09082 0.166591
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) −1.59592 −0.0648297
\(607\) 28.7423 1.16662 0.583308 0.812251i \(-0.301758\pi\)
0.583308 + 0.812251i \(0.301758\pi\)
\(608\) 5.00000 0.202777
\(609\) 3.19184 0.129340
\(610\) 0 0
\(611\) −4.40408 −0.178170
\(612\) 3.10102 0.125351
\(613\) 34.4949 1.39324 0.696618 0.717442i \(-0.254687\pi\)
0.696618 + 0.717442i \(0.254687\pi\)
\(614\) 7.65153 0.308791
\(615\) 0 0
\(616\) 8.44949 0.340440
\(617\) 4.89898 0.197225 0.0986127 0.995126i \(-0.468559\pi\)
0.0986127 + 0.995126i \(0.468559\pi\)
\(618\) 9.34847 0.376051
\(619\) −32.8990 −1.32232 −0.661161 0.750244i \(-0.729936\pi\)
−0.661161 + 0.750244i \(0.729936\pi\)
\(620\) 0 0
\(621\) 11.3031 0.453576
\(622\) 8.44949 0.338794
\(623\) −35.1464 −1.40811
\(624\) 0.651531 0.0260821
\(625\) 0 0
\(626\) 8.89898 0.355675
\(627\) −25.0000 −0.998404
\(628\) −16.4495 −0.656406
\(629\) −3.44949 −0.137540
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) −7.79796 −0.310186
\(633\) −6.59592 −0.262164
\(634\) −9.55051 −0.379299
\(635\) 0 0
\(636\) −8.69694 −0.344856
\(637\) −0.449490 −0.0178094
\(638\) 3.10102 0.122771
\(639\) 6.78775 0.268519
\(640\) 0 0
\(641\) 21.5959 0.852987 0.426494 0.904491i \(-0.359749\pi\)
0.426494 + 0.904491i \(0.359749\pi\)
\(642\) −12.3939 −0.489147
\(643\) −32.2929 −1.27351 −0.636753 0.771068i \(-0.719723\pi\)
−0.636753 + 0.771068i \(0.719723\pi\)
\(644\) 4.89898 0.193047
\(645\) 0 0
\(646\) −17.2474 −0.678592
\(647\) 40.2474 1.58229 0.791145 0.611628i \(-0.209485\pi\)
0.791145 + 0.611628i \(0.209485\pi\)
\(648\) 5.49490 0.215860
\(649\) −6.89898 −0.270809
\(650\) 0 0
\(651\) −15.7980 −0.619171
\(652\) −10.1010 −0.395586
\(653\) 10.0454 0.393107 0.196554 0.980493i \(-0.437025\pi\)
0.196554 + 0.980493i \(0.437025\pi\)
\(654\) −20.2929 −0.793513
\(655\) 0 0
\(656\) 1.00000 0.0390434
\(657\) 11.5051 0.448857
\(658\) −24.0000 −0.935617
\(659\) 6.55051 0.255172 0.127586 0.991828i \(-0.459277\pi\)
0.127586 + 0.991828i \(0.459277\pi\)
\(660\) 0 0
\(661\) 25.7980 1.00342 0.501712 0.865035i \(-0.332704\pi\)
0.501712 + 0.865035i \(0.332704\pi\)
\(662\) −19.8990 −0.773396
\(663\) −2.24745 −0.0872837
\(664\) 3.44949 0.133866
\(665\) 0 0
\(666\) 0.898979 0.0348347
\(667\) 1.79796 0.0696172
\(668\) 10.4495 0.404303
\(669\) −31.5959 −1.22157
\(670\) 0 0
\(671\) −22.2474 −0.858853
\(672\) 3.55051 0.136964
\(673\) −5.79796 −0.223495 −0.111747 0.993737i \(-0.535645\pi\)
−0.111747 + 0.993737i \(0.535645\pi\)
\(674\) 15.6969 0.604623
\(675\) 0 0
\(676\) −12.7980 −0.492229
\(677\) 43.5959 1.67553 0.837764 0.546033i \(-0.183863\pi\)
0.837764 + 0.546033i \(0.183863\pi\)
\(678\) −6.59592 −0.253315
\(679\) 34.2929 1.31604
\(680\) 0 0
\(681\) 24.2020 0.927424
\(682\) −15.3485 −0.587723
\(683\) 7.00000 0.267848 0.133924 0.990992i \(-0.457242\pi\)
0.133924 + 0.990992i \(0.457242\pi\)
\(684\) 4.49490 0.171867
\(685\) 0 0
\(686\) −19.5959 −0.748176
\(687\) 8.40408 0.320636
\(688\) 1.10102 0.0419760
\(689\) −2.69694 −0.102745
\(690\) 0 0
\(691\) 29.6515 1.12800 0.563999 0.825776i \(-0.309262\pi\)
0.563999 + 0.825776i \(0.309262\pi\)
\(692\) 6.69694 0.254579
\(693\) 7.59592 0.288545
\(694\) −18.7980 −0.713561
\(695\) 0 0
\(696\) 1.30306 0.0493924
\(697\) −3.44949 −0.130659
\(698\) 12.4495 0.471220
\(699\) 41.3031 1.56223
\(700\) 0 0
\(701\) −34.2474 −1.29351 −0.646754 0.762699i \(-0.723874\pi\)
−0.646754 + 0.762699i \(0.723874\pi\)
\(702\) 2.54031 0.0958776
\(703\) −5.00000 −0.188579
\(704\) 3.44949 0.130008
\(705\) 0 0
\(706\) 14.8990 0.560730
\(707\) −2.69694 −0.101429
\(708\) −2.89898 −0.108950
\(709\) 16.8990 0.634654 0.317327 0.948316i \(-0.397215\pi\)
0.317327 + 0.948316i \(0.397215\pi\)
\(710\) 0 0
\(711\) −7.01021 −0.262903
\(712\) −14.3485 −0.537732
\(713\) −8.89898 −0.333269
\(714\) −12.2474 −0.458349
\(715\) 0 0
\(716\) −0.797959 −0.0298211
\(717\) 1.30306 0.0486637
\(718\) 31.3939 1.17161
\(719\) −0.651531 −0.0242980 −0.0121490 0.999926i \(-0.503867\pi\)
−0.0121490 + 0.999926i \(0.503867\pi\)
\(720\) 0 0
\(721\) 15.7980 0.588347
\(722\) −6.00000 −0.223297
\(723\) −3.69694 −0.137491
\(724\) 6.89898 0.256399
\(725\) 0 0
\(726\) −1.30306 −0.0483611
\(727\) −36.0000 −1.33517 −0.667583 0.744535i \(-0.732671\pi\)
−0.667583 + 0.744535i \(0.732671\pi\)
\(728\) 1.10102 0.0408065
\(729\) 29.5153 1.09316
\(730\) 0 0
\(731\) −3.79796 −0.140473
\(732\) −9.34847 −0.345529
\(733\) 15.5959 0.576048 0.288024 0.957623i \(-0.407002\pi\)
0.288024 + 0.957623i \(0.407002\pi\)
\(734\) −14.2474 −0.525883
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) −15.6969 −0.578204
\(738\) 0.898979 0.0330919
\(739\) −31.5959 −1.16227 −0.581137 0.813806i \(-0.697392\pi\)
−0.581137 + 0.813806i \(0.697392\pi\)
\(740\) 0 0
\(741\) −3.25765 −0.119673
\(742\) −14.6969 −0.539542
\(743\) 6.40408 0.234943 0.117471 0.993076i \(-0.462521\pi\)
0.117471 + 0.993076i \(0.462521\pi\)
\(744\) −6.44949 −0.236450
\(745\) 0 0
\(746\) −20.0454 −0.733915
\(747\) 3.10102 0.113460
\(748\) −11.8990 −0.435070
\(749\) −20.9444 −0.765291
\(750\) 0 0
\(751\) −30.6969 −1.12015 −0.560074 0.828443i \(-0.689227\pi\)
−0.560074 + 0.828443i \(0.689227\pi\)
\(752\) −9.79796 −0.357295
\(753\) −44.6413 −1.62682
\(754\) 0.404082 0.0147158
\(755\) 0 0
\(756\) 13.8434 0.503478
\(757\) 13.7980 0.501495 0.250748 0.968052i \(-0.419324\pi\)
0.250748 + 0.968052i \(0.419324\pi\)
\(758\) 21.0454 0.764404
\(759\) −10.0000 −0.362977
\(760\) 0 0
\(761\) −14.3031 −0.518486 −0.259243 0.965812i \(-0.583473\pi\)
−0.259243 + 0.965812i \(0.583473\pi\)
\(762\) −23.5505 −0.853145
\(763\) −34.2929 −1.24148
\(764\) −7.79796 −0.282120
\(765\) 0 0
\(766\) −5.34847 −0.193248
\(767\) −0.898979 −0.0324603
\(768\) 1.44949 0.0523040
\(769\) 53.2474 1.92015 0.960076 0.279739i \(-0.0902480\pi\)
0.960076 + 0.279739i \(0.0902480\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) 2.55051 0.0917949
\(773\) −22.0454 −0.792918 −0.396459 0.918052i \(-0.629761\pi\)
−0.396459 + 0.918052i \(0.629761\pi\)
\(774\) 0.989795 0.0355774
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) −3.55051 −0.127374
\(778\) −13.1464 −0.471322
\(779\) −5.00000 −0.179144
\(780\) 0 0
\(781\) −26.0454 −0.931978
\(782\) −6.89898 −0.246707
\(783\) 5.08061 0.181566
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −12.6061 −0.449646
\(787\) −30.6969 −1.09423 −0.547114 0.837058i \(-0.684274\pi\)
−0.547114 + 0.837058i \(0.684274\pi\)
\(788\) 17.3485 0.618014
\(789\) −0.292856 −0.0104260
\(790\) 0 0
\(791\) −11.1464 −0.396321
\(792\) 3.10102 0.110190
\(793\) −2.89898 −0.102946
\(794\) −15.3485 −0.544697
\(795\) 0 0
\(796\) −15.5959 −0.552783
\(797\) −4.24745 −0.150452 −0.0752262 0.997166i \(-0.523968\pi\)
−0.0752262 + 0.997166i \(0.523968\pi\)
\(798\) −17.7526 −0.628434
\(799\) 33.7980 1.19569
\(800\) 0 0
\(801\) −12.8990 −0.455763
\(802\) −15.9444 −0.563016
\(803\) −44.1464 −1.55789
\(804\) −6.59592 −0.232620
\(805\) 0 0
\(806\) −2.00000 −0.0704470
\(807\) 5.79796 0.204098
\(808\) −1.10102 −0.0387338
\(809\) −39.3939 −1.38501 −0.692507 0.721411i \(-0.743494\pi\)
−0.692507 + 0.721411i \(0.743494\pi\)
\(810\) 0 0
\(811\) 5.79796 0.203594 0.101797 0.994805i \(-0.467541\pi\)
0.101797 + 0.994805i \(0.467541\pi\)
\(812\) 2.20204 0.0772765
\(813\) 43.5505 1.52738
\(814\) −3.44949 −0.120905
\(815\) 0 0
\(816\) −5.00000 −0.175035
\(817\) −5.50510 −0.192599
\(818\) 34.1464 1.19390
\(819\) 0.989795 0.0345862
\(820\) 0 0
\(821\) 4.04541 0.141186 0.0705929 0.997505i \(-0.477511\pi\)
0.0705929 + 0.997505i \(0.477511\pi\)
\(822\) 14.0556 0.490246
\(823\) −33.8434 −1.17971 −0.589853 0.807511i \(-0.700814\pi\)
−0.589853 + 0.807511i \(0.700814\pi\)
\(824\) 6.44949 0.224679
\(825\) 0 0
\(826\) −4.89898 −0.170457
\(827\) −27.0000 −0.938882 −0.469441 0.882964i \(-0.655545\pi\)
−0.469441 + 0.882964i \(0.655545\pi\)
\(828\) 1.79796 0.0624834
\(829\) 21.3485 0.741463 0.370731 0.928740i \(-0.379107\pi\)
0.370731 + 0.928740i \(0.379107\pi\)
\(830\) 0 0
\(831\) −8.40408 −0.291534
\(832\) 0.449490 0.0155833
\(833\) 3.44949 0.119518
\(834\) 18.1918 0.629932
\(835\) 0 0
\(836\) −17.2474 −0.596515
\(837\) −25.1464 −0.869188
\(838\) 21.4495 0.740960
\(839\) −1.34847 −0.0465543 −0.0232772 0.999729i \(-0.507410\pi\)
−0.0232772 + 0.999729i \(0.507410\pi\)
\(840\) 0 0
\(841\) −28.1918 −0.972132
\(842\) −32.8990 −1.13377
\(843\) 12.8990 0.444264
\(844\) −4.55051 −0.156635
\(845\) 0 0
\(846\) −8.80816 −0.302831
\(847\) −2.20204 −0.0756630
\(848\) −6.00000 −0.206041
\(849\) 16.9546 0.581880
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) −10.9444 −0.374949
\(853\) 19.3485 0.662479 0.331239 0.943547i \(-0.392533\pi\)
0.331239 + 0.943547i \(0.392533\pi\)
\(854\) −15.7980 −0.540595
\(855\) 0 0
\(856\) −8.55051 −0.292250
\(857\) 32.1464 1.09810 0.549051 0.835789i \(-0.314989\pi\)
0.549051 + 0.835789i \(0.314989\pi\)
\(858\) −2.24745 −0.0767266
\(859\) −9.49490 −0.323962 −0.161981 0.986794i \(-0.551788\pi\)
−0.161981 + 0.986794i \(0.551788\pi\)
\(860\) 0 0
\(861\) −3.55051 −0.121001
\(862\) 20.6969 0.704941
\(863\) −18.2474 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(864\) 5.65153 0.192269
\(865\) 0 0
\(866\) 19.0000 0.645646
\(867\) −7.39388 −0.251109
\(868\) −10.8990 −0.369935
\(869\) 26.8990 0.912485
\(870\) 0 0
\(871\) −2.04541 −0.0693060
\(872\) −14.0000 −0.474100
\(873\) 12.5857 0.425962
\(874\) −10.0000 −0.338255
\(875\) 0 0
\(876\) −18.5505 −0.626764
\(877\) −4.65153 −0.157071 −0.0785355 0.996911i \(-0.525024\pi\)
−0.0785355 + 0.996911i \(0.525024\pi\)
\(878\) 19.5959 0.661330
\(879\) 40.6515 1.37114
\(880\) 0 0
\(881\) 34.2929 1.15536 0.577678 0.816265i \(-0.303959\pi\)
0.577678 + 0.816265i \(0.303959\pi\)
\(882\) −0.898979 −0.0302702
\(883\) −26.1010 −0.878369 −0.439185 0.898397i \(-0.644733\pi\)
−0.439185 + 0.898397i \(0.644733\pi\)
\(884\) −1.55051 −0.0521493
\(885\) 0 0
\(886\) −17.4495 −0.586227
\(887\) −50.6969 −1.70224 −0.851118 0.524974i \(-0.824075\pi\)
−0.851118 + 0.524974i \(0.824075\pi\)
\(888\) −1.44949 −0.0486417
\(889\) −39.7980 −1.33478
\(890\) 0 0
\(891\) −18.9546 −0.635003
\(892\) −21.7980 −0.729850
\(893\) 48.9898 1.63938
\(894\) 6.74235 0.225498
\(895\) 0 0
\(896\) 2.44949 0.0818317
\(897\) −1.30306 −0.0435080
\(898\) −33.2474 −1.10948
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 20.6969 0.689515
\(902\) −3.44949 −0.114855
\(903\) −3.90918 −0.130090
\(904\) −4.55051 −0.151348
\(905\) 0 0
\(906\) 20.2929 0.674185
\(907\) 34.8990 1.15880 0.579401 0.815043i \(-0.303287\pi\)
0.579401 + 0.815043i \(0.303287\pi\)
\(908\) 16.6969 0.554107
\(909\) −0.989795 −0.0328294
\(910\) 0 0
\(911\) −16.2474 −0.538302 −0.269151 0.963098i \(-0.586743\pi\)
−0.269151 + 0.963098i \(0.586743\pi\)
\(912\) −7.24745 −0.239987
\(913\) −11.8990 −0.393799
\(914\) 15.2474 0.504341
\(915\) 0 0
\(916\) 5.79796 0.191570
\(917\) −21.3031 −0.703489
\(918\) −19.4949 −0.643427
\(919\) −51.7980 −1.70866 −0.854329 0.519733i \(-0.826031\pi\)
−0.854329 + 0.519733i \(0.826031\pi\)
\(920\) 0 0
\(921\) −11.0908 −0.365455
\(922\) −9.30306 −0.306380
\(923\) −3.39388 −0.111711
\(924\) −12.2474 −0.402911
\(925\) 0 0
\(926\) 9.55051 0.313849
\(927\) 5.79796 0.190430
\(928\) 0.898979 0.0295104
\(929\) −5.79796 −0.190225 −0.0951124 0.995467i \(-0.530321\pi\)
−0.0951124 + 0.995467i \(0.530321\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) 28.4949 0.933381
\(933\) −12.2474 −0.400963
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0.404082 0.0132078
\(937\) −14.5959 −0.476828 −0.238414 0.971164i \(-0.576627\pi\)
−0.238414 + 0.971164i \(0.576627\pi\)
\(938\) −11.1464 −0.363944
\(939\) −12.8990 −0.420942
\(940\) 0 0
\(941\) −41.3939 −1.34940 −0.674701 0.738091i \(-0.735727\pi\)
−0.674701 + 0.738091i \(0.735727\pi\)
\(942\) 23.8434 0.776859
\(943\) −2.00000 −0.0651290
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) −3.79796 −0.123482
\(947\) −32.0000 −1.03986 −0.519930 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 11.3031 0.367106
\(949\) −5.75255 −0.186736
\(950\) 0 0
\(951\) 13.8434 0.448902
\(952\) −8.44949 −0.273850
\(953\) −25.4949 −0.825861 −0.412930 0.910763i \(-0.635495\pi\)
−0.412930 + 0.910763i \(0.635495\pi\)
\(954\) −5.39388 −0.174633
\(955\) 0 0
\(956\) 0.898979 0.0290751
\(957\) −4.49490 −0.145299
\(958\) −6.24745 −0.201846
\(959\) 23.7526 0.767010
\(960\) 0 0
\(961\) −11.2020 −0.361356
\(962\) −0.449490 −0.0144921
\(963\) −7.68673 −0.247702
\(964\) −2.55051 −0.0821464
\(965\) 0 0
\(966\) −7.10102 −0.228472
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) −0.898979 −0.0288943
\(969\) 25.0000 0.803116
\(970\) 0 0
\(971\) −53.5403 −1.71819 −0.859095 0.511816i \(-0.828973\pi\)
−0.859095 + 0.511816i \(0.828973\pi\)
\(972\) 8.98979 0.288348
\(973\) 30.7423 0.985554
\(974\) −30.7423 −0.985048
\(975\) 0 0
\(976\) −6.44949 −0.206443
\(977\) 23.7423 0.759585 0.379792 0.925072i \(-0.375995\pi\)
0.379792 + 0.925072i \(0.375995\pi\)
\(978\) 14.6413 0.468178
\(979\) 49.4949 1.58186
\(980\) 0 0
\(981\) −12.5857 −0.401831
\(982\) 41.7980 1.33383
\(983\) −18.4949 −0.589896 −0.294948 0.955513i \(-0.595302\pi\)
−0.294948 + 0.955513i \(0.595302\pi\)
\(984\) −1.44949 −0.0462080
\(985\) 0 0
\(986\) −3.10102 −0.0987566
\(987\) 34.7878 1.10731
\(988\) −2.24745 −0.0715009
\(989\) −2.20204 −0.0700208
\(990\) 0 0
\(991\) −16.4495 −0.522535 −0.261268 0.965266i \(-0.584141\pi\)
−0.261268 + 0.965266i \(0.584141\pi\)
\(992\) −4.44949 −0.141271
\(993\) 28.8434 0.915317
\(994\) −18.4949 −0.586623
\(995\) 0 0
\(996\) −5.00000 −0.158431
\(997\) −39.5505 −1.25258 −0.626289 0.779591i \(-0.715427\pi\)
−0.626289 + 0.779591i \(0.715427\pi\)
\(998\) −22.0000 −0.696398
\(999\) −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.a.r.1.2 2
5.2 odd 4 1850.2.b.k.149.1 4
5.3 odd 4 1850.2.b.k.149.4 4
5.4 even 2 1850.2.a.w.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.r.1.2 2 1.1 even 1 trivial
1850.2.a.w.1.1 yes 2 5.4 even 2
1850.2.b.k.149.1 4 5.2 odd 4
1850.2.b.k.149.4 4 5.3 odd 4