Properties

Label 3381.2.a.bc.1.1
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7997584.1
Defining polynomial: \(x^{6} - x^{5} - 7 x^{4} + 5 x^{3} + 12 x^{2} - 4 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.27267\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.27267 q^{2} -1.00000 q^{3} +3.16503 q^{4} +4.15120 q^{5} +2.27267 q^{6} -2.64772 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.27267 q^{2} -1.00000 q^{3} +3.16503 q^{4} +4.15120 q^{5} +2.27267 q^{6} -2.64772 q^{8} +1.00000 q^{9} -9.43432 q^{10} -3.18706 q^{11} -3.16503 q^{12} +0.175661 q^{13} -4.15120 q^{15} -0.312654 q^{16} -1.32687 q^{17} -2.27267 q^{18} +0.0504261 q^{19} +13.1387 q^{20} +7.24313 q^{22} -1.00000 q^{23} +2.64772 q^{24} +12.2325 q^{25} -0.399220 q^{26} -1.00000 q^{27} +1.10049 q^{29} +9.43432 q^{30} -8.71037 q^{31} +6.00601 q^{32} +3.18706 q^{33} +3.01553 q^{34} +3.16503 q^{36} +3.76233 q^{37} -0.114602 q^{38} -0.175661 q^{39} -10.9912 q^{40} +4.84719 q^{41} -10.5958 q^{43} -10.0871 q^{44} +4.15120 q^{45} +2.27267 q^{46} -8.95013 q^{47} +0.312654 q^{48} -27.8004 q^{50} +1.32687 q^{51} +0.555973 q^{52} -9.97629 q^{53} +2.27267 q^{54} -13.2301 q^{55} -0.0504261 q^{57} -2.50105 q^{58} -6.38897 q^{59} -13.1387 q^{60} -7.23261 q^{61} +19.7958 q^{62} -13.0244 q^{64} +0.729206 q^{65} -7.24313 q^{66} +6.02059 q^{67} -4.19957 q^{68} +1.00000 q^{69} +6.71519 q^{71} -2.64772 q^{72} -4.61372 q^{73} -8.55052 q^{74} -12.2325 q^{75} +0.159600 q^{76} +0.399220 q^{78} -15.4854 q^{79} -1.29789 q^{80} +1.00000 q^{81} -11.0161 q^{82} +7.21102 q^{83} -5.50809 q^{85} +24.0807 q^{86} -1.10049 q^{87} +8.43845 q^{88} -1.94147 q^{89} -9.43432 q^{90} -3.16503 q^{92} +8.71037 q^{93} +20.3407 q^{94} +0.209329 q^{95} -6.00601 q^{96} -12.7879 q^{97} -3.18706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} - 3q^{10} - 14q^{11} - 3q^{12} - 3q^{15} - 7q^{16} + 15q^{17} - q^{18} - q^{19} + 17q^{20} - 6q^{22} - 6q^{23} + 3q^{24} + 9q^{25} - 15q^{26} - 6q^{27} - 6q^{29} + 3q^{30} - 11q^{31} + 3q^{32} + 14q^{33} + 15q^{34} + 3q^{36} - 5q^{37} + 14q^{38} - 17q^{40} + 18q^{41} - 37q^{43} - 10q^{44} + 3q^{45} + q^{46} + 3q^{47} + 7q^{48} - 30q^{50} - 15q^{51} - 7q^{52} - 15q^{53} + q^{54} - 2q^{55} + q^{57} + 4q^{58} - 2q^{59} - 17q^{60} - 12q^{61} + 36q^{62} - 23q^{64} - 17q^{65} + 6q^{66} - 10q^{67} - q^{68} + 6q^{69} - 21q^{71} - 3q^{72} - 8q^{73} - 16q^{74} - 9q^{75} + 18q^{76} + 15q^{78} - 17q^{79} + 3q^{80} + 6q^{81} - 48q^{82} + 12q^{83} - 13q^{85} + 22q^{86} + 6q^{87} - 2q^{88} + 18q^{89} - 3q^{90} - 3q^{92} + 11q^{93} + 3q^{94} - 16q^{95} - 3q^{96} - 2q^{97} - 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.27267 −1.60702 −0.803510 0.595291i \(-0.797037\pi\)
−0.803510 + 0.595291i \(0.797037\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.16503 1.58251
\(5\) 4.15120 1.85647 0.928237 0.371988i \(-0.121324\pi\)
0.928237 + 0.371988i \(0.121324\pi\)
\(6\) 2.27267 0.927814
\(7\) 0 0
\(8\) −2.64772 −0.936112
\(9\) 1.00000 0.333333
\(10\) −9.43432 −2.98339
\(11\) −3.18706 −0.960934 −0.480467 0.877013i \(-0.659533\pi\)
−0.480467 + 0.877013i \(0.659533\pi\)
\(12\) −3.16503 −0.913665
\(13\) 0.175661 0.0487197 0.0243598 0.999703i \(-0.492245\pi\)
0.0243598 + 0.999703i \(0.492245\pi\)
\(14\) 0 0
\(15\) −4.15120 −1.07184
\(16\) −0.312654 −0.0781634
\(17\) −1.32687 −0.321812 −0.160906 0.986970i \(-0.551442\pi\)
−0.160906 + 0.986970i \(0.551442\pi\)
\(18\) −2.27267 −0.535673
\(19\) 0.0504261 0.0115685 0.00578427 0.999983i \(-0.498159\pi\)
0.00578427 + 0.999983i \(0.498159\pi\)
\(20\) 13.1387 2.93790
\(21\) 0 0
\(22\) 7.24313 1.54424
\(23\) −1.00000 −0.208514
\(24\) 2.64772 0.540464
\(25\) 12.2325 2.44650
\(26\) −0.399220 −0.0782935
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.10049 0.204356 0.102178 0.994766i \(-0.467419\pi\)
0.102178 + 0.994766i \(0.467419\pi\)
\(30\) 9.43432 1.72246
\(31\) −8.71037 −1.56443 −0.782214 0.623010i \(-0.785910\pi\)
−0.782214 + 0.623010i \(0.785910\pi\)
\(32\) 6.00601 1.06172
\(33\) 3.18706 0.554795
\(34\) 3.01553 0.517159
\(35\) 0 0
\(36\) 3.16503 0.527505
\(37\) 3.76233 0.618523 0.309261 0.950977i \(-0.399918\pi\)
0.309261 + 0.950977i \(0.399918\pi\)
\(38\) −0.114602 −0.0185909
\(39\) −0.175661 −0.0281283
\(40\) −10.9912 −1.73787
\(41\) 4.84719 0.757004 0.378502 0.925601i \(-0.376439\pi\)
0.378502 + 0.925601i \(0.376439\pi\)
\(42\) 0 0
\(43\) −10.5958 −1.61584 −0.807920 0.589293i \(-0.799406\pi\)
−0.807920 + 0.589293i \(0.799406\pi\)
\(44\) −10.0871 −1.52069
\(45\) 4.15120 0.618825
\(46\) 2.27267 0.335087
\(47\) −8.95013 −1.30551 −0.652755 0.757569i \(-0.726387\pi\)
−0.652755 + 0.757569i \(0.726387\pi\)
\(48\) 0.312654 0.0451277
\(49\) 0 0
\(50\) −27.8004 −3.93157
\(51\) 1.32687 0.185798
\(52\) 0.555973 0.0770996
\(53\) −9.97629 −1.37035 −0.685175 0.728379i \(-0.740274\pi\)
−0.685175 + 0.728379i \(0.740274\pi\)
\(54\) 2.27267 0.309271
\(55\) −13.2301 −1.78395
\(56\) 0 0
\(57\) −0.0504261 −0.00667910
\(58\) −2.50105 −0.328404
\(59\) −6.38897 −0.831774 −0.415887 0.909416i \(-0.636529\pi\)
−0.415887 + 0.909416i \(0.636529\pi\)
\(60\) −13.1387 −1.69620
\(61\) −7.23261 −0.926041 −0.463021 0.886348i \(-0.653234\pi\)
−0.463021 + 0.886348i \(0.653234\pi\)
\(62\) 19.7958 2.51407
\(63\) 0 0
\(64\) −13.0244 −1.62805
\(65\) 0.729206 0.0904468
\(66\) −7.24313 −0.891567
\(67\) 6.02059 0.735533 0.367766 0.929918i \(-0.380123\pi\)
0.367766 + 0.929918i \(0.380123\pi\)
\(68\) −4.19957 −0.509272
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 6.71519 0.796947 0.398473 0.917180i \(-0.369540\pi\)
0.398473 + 0.917180i \(0.369540\pi\)
\(72\) −2.64772 −0.312037
\(73\) −4.61372 −0.539995 −0.269998 0.962861i \(-0.587023\pi\)
−0.269998 + 0.962861i \(0.587023\pi\)
\(74\) −8.55052 −0.993978
\(75\) −12.2325 −1.41249
\(76\) 0.159600 0.0183074
\(77\) 0 0
\(78\) 0.399220 0.0452028
\(79\) −15.4854 −1.74225 −0.871124 0.491064i \(-0.836608\pi\)
−0.871124 + 0.491064i \(0.836608\pi\)
\(80\) −1.29789 −0.145108
\(81\) 1.00000 0.111111
\(82\) −11.0161 −1.21652
\(83\) 7.21102 0.791512 0.395756 0.918356i \(-0.370483\pi\)
0.395756 + 0.918356i \(0.370483\pi\)
\(84\) 0 0
\(85\) −5.50809 −0.597436
\(86\) 24.0807 2.59669
\(87\) −1.10049 −0.117985
\(88\) 8.43845 0.899541
\(89\) −1.94147 −0.205795 −0.102898 0.994692i \(-0.532811\pi\)
−0.102898 + 0.994692i \(0.532811\pi\)
\(90\) −9.43432 −0.994464
\(91\) 0 0
\(92\) −3.16503 −0.329977
\(93\) 8.71037 0.903223
\(94\) 20.3407 2.09798
\(95\) 0.209329 0.0214767
\(96\) −6.00601 −0.612985
\(97\) −12.7879 −1.29841 −0.649207 0.760612i \(-0.724899\pi\)
−0.649207 + 0.760612i \(0.724899\pi\)
\(98\) 0 0
\(99\) −3.18706 −0.320311
\(100\) 38.7162 3.87162
\(101\) −1.64583 −0.163766 −0.0818831 0.996642i \(-0.526093\pi\)
−0.0818831 + 0.996642i \(0.526093\pi\)
\(102\) −3.01553 −0.298582
\(103\) 8.99275 0.886082 0.443041 0.896501i \(-0.353900\pi\)
0.443041 + 0.896501i \(0.353900\pi\)
\(104\) −0.465103 −0.0456071
\(105\) 0 0
\(106\) 22.6728 2.20218
\(107\) −18.9831 −1.83516 −0.917581 0.397549i \(-0.869861\pi\)
−0.917581 + 0.397549i \(0.869861\pi\)
\(108\) −3.16503 −0.304555
\(109\) −11.7183 −1.12241 −0.561203 0.827678i \(-0.689661\pi\)
−0.561203 + 0.827678i \(0.689661\pi\)
\(110\) 30.0677 2.86684
\(111\) −3.76233 −0.357104
\(112\) 0 0
\(113\) 16.6028 1.56186 0.780931 0.624617i \(-0.214745\pi\)
0.780931 + 0.624617i \(0.214745\pi\)
\(114\) 0.114602 0.0107335
\(115\) −4.15120 −0.387102
\(116\) 3.48308 0.323396
\(117\) 0.175661 0.0162399
\(118\) 14.5200 1.33668
\(119\) 0 0
\(120\) 10.9912 1.00336
\(121\) −0.842668 −0.0766062
\(122\) 16.4373 1.48817
\(123\) −4.84719 −0.437056
\(124\) −27.5686 −2.47573
\(125\) 30.0236 2.68539
\(126\) 0 0
\(127\) 17.8466 1.58363 0.791815 0.610761i \(-0.209136\pi\)
0.791815 + 0.610761i \(0.209136\pi\)
\(128\) 17.5881 1.55458
\(129\) 10.5958 0.932905
\(130\) −1.65724 −0.145350
\(131\) 21.5620 1.88388 0.941940 0.335781i \(-0.109000\pi\)
0.941940 + 0.335781i \(0.109000\pi\)
\(132\) 10.0871 0.877971
\(133\) 0 0
\(134\) −13.6828 −1.18202
\(135\) −4.15120 −0.357279
\(136\) 3.51317 0.301252
\(137\) 0.565532 0.0483166 0.0241583 0.999708i \(-0.492309\pi\)
0.0241583 + 0.999708i \(0.492309\pi\)
\(138\) −2.27267 −0.193463
\(139\) −19.5670 −1.65965 −0.829825 0.558024i \(-0.811560\pi\)
−0.829825 + 0.558024i \(0.811560\pi\)
\(140\) 0 0
\(141\) 8.95013 0.753737
\(142\) −15.2614 −1.28071
\(143\) −0.559842 −0.0468164
\(144\) −0.312654 −0.0260545
\(145\) 4.56836 0.379382
\(146\) 10.4855 0.867783
\(147\) 0 0
\(148\) 11.9079 0.978821
\(149\) 7.52726 0.616657 0.308329 0.951280i \(-0.400230\pi\)
0.308329 + 0.951280i \(0.400230\pi\)
\(150\) 27.8004 2.26989
\(151\) −21.3621 −1.73843 −0.869213 0.494437i \(-0.835374\pi\)
−0.869213 + 0.494437i \(0.835374\pi\)
\(152\) −0.133514 −0.0108294
\(153\) −1.32687 −0.107271
\(154\) 0 0
\(155\) −36.1585 −2.90432
\(156\) −0.555973 −0.0445135
\(157\) 11.7594 0.938503 0.469251 0.883065i \(-0.344524\pi\)
0.469251 + 0.883065i \(0.344524\pi\)
\(158\) 35.1933 2.79983
\(159\) 9.97629 0.791171
\(160\) 24.9322 1.97106
\(161\) 0 0
\(162\) −2.27267 −0.178558
\(163\) −15.3219 −1.20010 −0.600051 0.799962i \(-0.704853\pi\)
−0.600051 + 0.799962i \(0.704853\pi\)
\(164\) 15.3415 1.19797
\(165\) 13.2301 1.02996
\(166\) −16.3883 −1.27198
\(167\) −8.02472 −0.620972 −0.310486 0.950578i \(-0.600492\pi\)
−0.310486 + 0.950578i \(0.600492\pi\)
\(168\) 0 0
\(169\) −12.9691 −0.997626
\(170\) 12.5181 0.960092
\(171\) 0.0504261 0.00385618
\(172\) −33.5359 −2.55709
\(173\) 15.5225 1.18016 0.590078 0.807346i \(-0.299097\pi\)
0.590078 + 0.807346i \(0.299097\pi\)
\(174\) 2.50105 0.189604
\(175\) 0 0
\(176\) 0.996445 0.0751099
\(177\) 6.38897 0.480225
\(178\) 4.41232 0.330717
\(179\) 8.28280 0.619086 0.309543 0.950885i \(-0.399824\pi\)
0.309543 + 0.950885i \(0.399824\pi\)
\(180\) 13.1387 0.979299
\(181\) −5.71189 −0.424561 −0.212281 0.977209i \(-0.568089\pi\)
−0.212281 + 0.977209i \(0.568089\pi\)
\(182\) 0 0
\(183\) 7.23261 0.534650
\(184\) 2.64772 0.195193
\(185\) 15.6182 1.14827
\(186\) −19.7958 −1.45150
\(187\) 4.22880 0.309240
\(188\) −28.3274 −2.06599
\(189\) 0 0
\(190\) −0.475736 −0.0345135
\(191\) −1.55152 −0.112264 −0.0561322 0.998423i \(-0.517877\pi\)
−0.0561322 + 0.998423i \(0.517877\pi\)
\(192\) 13.0244 0.939952
\(193\) 7.32290 0.527114 0.263557 0.964644i \(-0.415104\pi\)
0.263557 + 0.964644i \(0.415104\pi\)
\(194\) 29.0626 2.08658
\(195\) −0.729206 −0.0522195
\(196\) 0 0
\(197\) −6.84137 −0.487427 −0.243714 0.969847i \(-0.578366\pi\)
−0.243714 + 0.969847i \(0.578366\pi\)
\(198\) 7.24313 0.514747
\(199\) 7.64366 0.541845 0.270922 0.962601i \(-0.412671\pi\)
0.270922 + 0.962601i \(0.412671\pi\)
\(200\) −32.3883 −2.29020
\(201\) −6.02059 −0.424660
\(202\) 3.74043 0.263176
\(203\) 0 0
\(204\) 4.19957 0.294028
\(205\) 20.1217 1.40536
\(206\) −20.4376 −1.42395
\(207\) −1.00000 −0.0695048
\(208\) −0.0549212 −0.00380810
\(209\) −0.160711 −0.0111166
\(210\) 0 0
\(211\) −4.69603 −0.323288 −0.161644 0.986849i \(-0.551680\pi\)
−0.161644 + 0.986849i \(0.551680\pi\)
\(212\) −31.5752 −2.16860
\(213\) −6.71519 −0.460117
\(214\) 43.1422 2.94914
\(215\) −43.9852 −2.99976
\(216\) 2.64772 0.180155
\(217\) 0 0
\(218\) 26.6318 1.80373
\(219\) 4.61372 0.311766
\(220\) −41.8737 −2.82313
\(221\) −0.233079 −0.0156786
\(222\) 8.55052 0.573874
\(223\) 24.2891 1.62652 0.813260 0.581900i \(-0.197690\pi\)
0.813260 + 0.581900i \(0.197690\pi\)
\(224\) 0 0
\(225\) 12.2325 0.815500
\(226\) −37.7327 −2.50994
\(227\) 4.33303 0.287593 0.143797 0.989607i \(-0.454069\pi\)
0.143797 + 0.989607i \(0.454069\pi\)
\(228\) −0.159600 −0.0105698
\(229\) −12.4437 −0.822303 −0.411152 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(230\) 9.43432 0.622080
\(231\) 0 0
\(232\) −2.91380 −0.191300
\(233\) 3.24764 0.212760 0.106380 0.994326i \(-0.466074\pi\)
0.106380 + 0.994326i \(0.466074\pi\)
\(234\) −0.399220 −0.0260978
\(235\) −37.1538 −2.42365
\(236\) −20.2213 −1.31629
\(237\) 15.4854 1.00589
\(238\) 0 0
\(239\) 26.4823 1.71300 0.856500 0.516147i \(-0.172634\pi\)
0.856500 + 0.516147i \(0.172634\pi\)
\(240\) 1.29789 0.0837784
\(241\) 2.56149 0.165000 0.0825001 0.996591i \(-0.473710\pi\)
0.0825001 + 0.996591i \(0.473710\pi\)
\(242\) 1.91511 0.123108
\(243\) −1.00000 −0.0641500
\(244\) −22.8914 −1.46547
\(245\) 0 0
\(246\) 11.0161 0.702358
\(247\) 0.00885791 0.000563616 0
\(248\) 23.0626 1.46448
\(249\) −7.21102 −0.456980
\(250\) −68.2336 −4.31547
\(251\) 14.6123 0.922317 0.461159 0.887318i \(-0.347434\pi\)
0.461159 + 0.887318i \(0.347434\pi\)
\(252\) 0 0
\(253\) 3.18706 0.200369
\(254\) −40.5594 −2.54493
\(255\) 5.50809 0.344930
\(256\) −13.9231 −0.870196
\(257\) −5.12857 −0.319911 −0.159956 0.987124i \(-0.551135\pi\)
−0.159956 + 0.987124i \(0.551135\pi\)
\(258\) −24.0807 −1.49920
\(259\) 0 0
\(260\) 2.30796 0.143133
\(261\) 1.10049 0.0681187
\(262\) −49.0033 −3.02743
\(263\) 3.24813 0.200288 0.100144 0.994973i \(-0.468070\pi\)
0.100144 + 0.994973i \(0.468070\pi\)
\(264\) −8.43845 −0.519351
\(265\) −41.4136 −2.54402
\(266\) 0 0
\(267\) 1.94147 0.118816
\(268\) 19.0553 1.16399
\(269\) −4.83101 −0.294552 −0.147276 0.989095i \(-0.547050\pi\)
−0.147276 + 0.989095i \(0.547050\pi\)
\(270\) 9.43432 0.574154
\(271\) −12.1239 −0.736473 −0.368236 0.929732i \(-0.620038\pi\)
−0.368236 + 0.929732i \(0.620038\pi\)
\(272\) 0.414849 0.0251539
\(273\) 0 0
\(274\) −1.28527 −0.0776458
\(275\) −38.9857 −2.35092
\(276\) 3.16503 0.190512
\(277\) −21.9014 −1.31593 −0.657964 0.753050i \(-0.728582\pi\)
−0.657964 + 0.753050i \(0.728582\pi\)
\(278\) 44.4693 2.66709
\(279\) −8.71037 −0.521476
\(280\) 0 0
\(281\) −24.2364 −1.44582 −0.722912 0.690940i \(-0.757197\pi\)
−0.722912 + 0.690940i \(0.757197\pi\)
\(282\) −20.3407 −1.21127
\(283\) 5.14581 0.305887 0.152943 0.988235i \(-0.451125\pi\)
0.152943 + 0.988235i \(0.451125\pi\)
\(284\) 21.2538 1.26118
\(285\) −0.209329 −0.0123996
\(286\) 1.27234 0.0752349
\(287\) 0 0
\(288\) 6.00601 0.353907
\(289\) −15.2394 −0.896437
\(290\) −10.3824 −0.609674
\(291\) 12.7879 0.749639
\(292\) −14.6026 −0.854550
\(293\) −24.5374 −1.43349 −0.716745 0.697335i \(-0.754369\pi\)
−0.716745 + 0.697335i \(0.754369\pi\)
\(294\) 0 0
\(295\) −26.5219 −1.54417
\(296\) −9.96160 −0.579006
\(297\) 3.18706 0.184932
\(298\) −17.1070 −0.990981
\(299\) −0.175661 −0.0101588
\(300\) −38.7162 −2.23528
\(301\) 0 0
\(302\) 48.5491 2.79369
\(303\) 1.64583 0.0945505
\(304\) −0.0157659 −0.000904237 0
\(305\) −30.0241 −1.71917
\(306\) 3.01553 0.172386
\(307\) 7.16659 0.409019 0.204509 0.978865i \(-0.434440\pi\)
0.204509 + 0.978865i \(0.434440\pi\)
\(308\) 0 0
\(309\) −8.99275 −0.511580
\(310\) 82.1764 4.66730
\(311\) 5.87280 0.333016 0.166508 0.986040i \(-0.446751\pi\)
0.166508 + 0.986040i \(0.446751\pi\)
\(312\) 0.465103 0.0263312
\(313\) −3.80364 −0.214994 −0.107497 0.994205i \(-0.534284\pi\)
−0.107497 + 0.994205i \(0.534284\pi\)
\(314\) −26.7252 −1.50819
\(315\) 0 0
\(316\) −49.0118 −2.75713
\(317\) 9.80284 0.550582 0.275291 0.961361i \(-0.411226\pi\)
0.275291 + 0.961361i \(0.411226\pi\)
\(318\) −22.6728 −1.27143
\(319\) −3.50733 −0.196373
\(320\) −54.0668 −3.02243
\(321\) 18.9831 1.05953
\(322\) 0 0
\(323\) −0.0669087 −0.00372290
\(324\) 3.16503 0.175835
\(325\) 2.14878 0.119193
\(326\) 34.8216 1.92859
\(327\) 11.7183 0.648022
\(328\) −12.8340 −0.708640
\(329\) 0 0
\(330\) −30.0677 −1.65517
\(331\) −11.1896 −0.615036 −0.307518 0.951542i \(-0.599499\pi\)
−0.307518 + 0.951542i \(0.599499\pi\)
\(332\) 22.8231 1.25258
\(333\) 3.76233 0.206174
\(334\) 18.2375 0.997914
\(335\) 24.9927 1.36550
\(336\) 0 0
\(337\) −31.7991 −1.73221 −0.866103 0.499865i \(-0.833383\pi\)
−0.866103 + 0.499865i \(0.833383\pi\)
\(338\) 29.4746 1.60321
\(339\) −16.6028 −0.901742
\(340\) −17.4333 −0.945451
\(341\) 27.7604 1.50331
\(342\) −0.114602 −0.00619696
\(343\) 0 0
\(344\) 28.0547 1.51261
\(345\) 4.15120 0.223493
\(346\) −35.2776 −1.89653
\(347\) −1.37388 −0.0737539 −0.0368770 0.999320i \(-0.511741\pi\)
−0.0368770 + 0.999320i \(0.511741\pi\)
\(348\) −3.48308 −0.186713
\(349\) −6.43146 −0.344268 −0.172134 0.985074i \(-0.555066\pi\)
−0.172134 + 0.985074i \(0.555066\pi\)
\(350\) 0 0
\(351\) −0.175661 −0.00937611
\(352\) −19.1415 −1.02024
\(353\) 25.7366 1.36982 0.684910 0.728628i \(-0.259842\pi\)
0.684910 + 0.728628i \(0.259842\pi\)
\(354\) −14.5200 −0.771731
\(355\) 27.8761 1.47951
\(356\) −6.14480 −0.325674
\(357\) 0 0
\(358\) −18.8241 −0.994883
\(359\) −23.9634 −1.26474 −0.632369 0.774667i \(-0.717918\pi\)
−0.632369 + 0.774667i \(0.717918\pi\)
\(360\) −10.9912 −0.579289
\(361\) −18.9975 −0.999866
\(362\) 12.9812 0.682278
\(363\) 0.842668 0.0442286
\(364\) 0 0
\(365\) −19.1525 −1.00249
\(366\) −16.4373 −0.859194
\(367\) 25.6783 1.34040 0.670199 0.742181i \(-0.266209\pi\)
0.670199 + 0.742181i \(0.266209\pi\)
\(368\) 0.312654 0.0162982
\(369\) 4.84719 0.252335
\(370\) −35.4950 −1.84530
\(371\) 0 0
\(372\) 27.5686 1.42936
\(373\) 5.80092 0.300360 0.150180 0.988659i \(-0.452015\pi\)
0.150180 + 0.988659i \(0.452015\pi\)
\(374\) −9.61066 −0.496955
\(375\) −30.0236 −1.55041
\(376\) 23.6975 1.22210
\(377\) 0.193314 0.00995616
\(378\) 0 0
\(379\) 21.0504 1.08129 0.540643 0.841252i \(-0.318181\pi\)
0.540643 + 0.841252i \(0.318181\pi\)
\(380\) 0.662532 0.0339872
\(381\) −17.8466 −0.914309
\(382\) 3.52610 0.180411
\(383\) −4.82289 −0.246438 −0.123219 0.992379i \(-0.539322\pi\)
−0.123219 + 0.992379i \(0.539322\pi\)
\(384\) −17.5881 −0.897537
\(385\) 0 0
\(386\) −16.6425 −0.847083
\(387\) −10.5958 −0.538613
\(388\) −40.4740 −2.05476
\(389\) −19.5414 −0.990790 −0.495395 0.868668i \(-0.664977\pi\)
−0.495395 + 0.868668i \(0.664977\pi\)
\(390\) 1.65724 0.0839178
\(391\) 1.32687 0.0671025
\(392\) 0 0
\(393\) −21.5620 −1.08766
\(394\) 15.5482 0.783305
\(395\) −64.2832 −3.23444
\(396\) −10.0871 −0.506897
\(397\) 36.5849 1.83614 0.918072 0.396413i \(-0.129745\pi\)
0.918072 + 0.396413i \(0.129745\pi\)
\(398\) −17.3715 −0.870756
\(399\) 0 0
\(400\) −3.82454 −0.191227
\(401\) −4.67488 −0.233452 −0.116726 0.993164i \(-0.537240\pi\)
−0.116726 + 0.993164i \(0.537240\pi\)
\(402\) 13.6828 0.682437
\(403\) −1.53007 −0.0762184
\(404\) −5.20910 −0.259162
\(405\) 4.15120 0.206275
\(406\) 0 0
\(407\) −11.9907 −0.594359
\(408\) −3.51317 −0.173928
\(409\) 11.0059 0.544208 0.272104 0.962268i \(-0.412280\pi\)
0.272104 + 0.962268i \(0.412280\pi\)
\(410\) −45.7299 −2.25844
\(411\) −0.565532 −0.0278956
\(412\) 28.4623 1.40224
\(413\) 0 0
\(414\) 2.27267 0.111696
\(415\) 29.9344 1.46942
\(416\) 1.05502 0.0517267
\(417\) 19.5670 0.958199
\(418\) 0.365243 0.0178646
\(419\) 18.8771 0.922206 0.461103 0.887347i \(-0.347454\pi\)
0.461103 + 0.887347i \(0.347454\pi\)
\(420\) 0 0
\(421\) −10.1341 −0.493905 −0.246952 0.969028i \(-0.579429\pi\)
−0.246952 + 0.969028i \(0.579429\pi\)
\(422\) 10.6725 0.519531
\(423\) −8.95013 −0.435170
\(424\) 26.4145 1.28280
\(425\) −16.2309 −0.787313
\(426\) 15.2614 0.739418
\(427\) 0 0
\(428\) −60.0819 −2.90417
\(429\) 0.559842 0.0270295
\(430\) 99.9638 4.82068
\(431\) 7.87183 0.379173 0.189586 0.981864i \(-0.439285\pi\)
0.189586 + 0.981864i \(0.439285\pi\)
\(432\) 0.312654 0.0150426
\(433\) −20.5867 −0.989335 −0.494668 0.869082i \(-0.664710\pi\)
−0.494668 + 0.869082i \(0.664710\pi\)
\(434\) 0 0
\(435\) −4.56836 −0.219036
\(436\) −37.0886 −1.77622
\(437\) −0.0504261 −0.00241221
\(438\) −10.4855 −0.501015
\(439\) 33.6678 1.60687 0.803437 0.595389i \(-0.203002\pi\)
0.803437 + 0.595389i \(0.203002\pi\)
\(440\) 35.0297 1.66998
\(441\) 0 0
\(442\) 0.529711 0.0251958
\(443\) 7.33528 0.348510 0.174255 0.984701i \(-0.444248\pi\)
0.174255 + 0.984701i \(0.444248\pi\)
\(444\) −11.9079 −0.565122
\(445\) −8.05943 −0.382054
\(446\) −55.2012 −2.61385
\(447\) −7.52726 −0.356027
\(448\) 0 0
\(449\) −21.1508 −0.998170 −0.499085 0.866553i \(-0.666330\pi\)
−0.499085 + 0.866553i \(0.666330\pi\)
\(450\) −27.8004 −1.31052
\(451\) −15.4483 −0.727431
\(452\) 52.5484 2.47167
\(453\) 21.3621 1.00368
\(454\) −9.84755 −0.462168
\(455\) 0 0
\(456\) 0.133514 0.00625239
\(457\) 8.96782 0.419497 0.209748 0.977755i \(-0.432736\pi\)
0.209748 + 0.977755i \(0.432736\pi\)
\(458\) 28.2804 1.32146
\(459\) 1.32687 0.0619328
\(460\) −13.1387 −0.612594
\(461\) 6.30195 0.293511 0.146756 0.989173i \(-0.453117\pi\)
0.146756 + 0.989173i \(0.453117\pi\)
\(462\) 0 0
\(463\) −15.1092 −0.702183 −0.351092 0.936341i \(-0.614189\pi\)
−0.351092 + 0.936341i \(0.614189\pi\)
\(464\) −0.344072 −0.0159732
\(465\) 36.1585 1.67681
\(466\) −7.38081 −0.341909
\(467\) −18.9605 −0.877387 −0.438694 0.898637i \(-0.644559\pi\)
−0.438694 + 0.898637i \(0.644559\pi\)
\(468\) 0.555973 0.0256999
\(469\) 0 0
\(470\) 84.4384 3.89485
\(471\) −11.7594 −0.541845
\(472\) 16.9162 0.778633
\(473\) 33.7693 1.55271
\(474\) −35.1933 −1.61648
\(475\) 0.616837 0.0283024
\(476\) 0 0
\(477\) −9.97629 −0.456783
\(478\) −60.1856 −2.75283
\(479\) −28.5772 −1.30572 −0.652862 0.757477i \(-0.726432\pi\)
−0.652862 + 0.757477i \(0.726432\pi\)
\(480\) −24.9322 −1.13799
\(481\) 0.660895 0.0301342
\(482\) −5.82143 −0.265159
\(483\) 0 0
\(484\) −2.66707 −0.121230
\(485\) −53.0851 −2.41047
\(486\) 2.27267 0.103090
\(487\) 24.9927 1.13253 0.566264 0.824224i \(-0.308388\pi\)
0.566264 + 0.824224i \(0.308388\pi\)
\(488\) 19.1500 0.866878
\(489\) 15.3219 0.692879
\(490\) 0 0
\(491\) −22.0515 −0.995170 −0.497585 0.867415i \(-0.665780\pi\)
−0.497585 + 0.867415i \(0.665780\pi\)
\(492\) −15.3415 −0.691648
\(493\) −1.46020 −0.0657642
\(494\) −0.0201311 −0.000905742 0
\(495\) −13.2301 −0.594650
\(496\) 2.72333 0.122281
\(497\) 0 0
\(498\) 16.3883 0.734376
\(499\) −39.4251 −1.76491 −0.882455 0.470397i \(-0.844111\pi\)
−0.882455 + 0.470397i \(0.844111\pi\)
\(500\) 95.0254 4.24967
\(501\) 8.02472 0.358518
\(502\) −33.2088 −1.48218
\(503\) −31.3079 −1.39595 −0.697975 0.716122i \(-0.745915\pi\)
−0.697975 + 0.716122i \(0.745915\pi\)
\(504\) 0 0
\(505\) −6.83218 −0.304028
\(506\) −7.24313 −0.321996
\(507\) 12.9691 0.575980
\(508\) 56.4850 2.50612
\(509\) 15.4660 0.685519 0.342760 0.939423i \(-0.388638\pi\)
0.342760 + 0.939423i \(0.388638\pi\)
\(510\) −12.5181 −0.554309
\(511\) 0 0
\(512\) −3.53344 −0.156158
\(513\) −0.0504261 −0.00222637
\(514\) 11.6555 0.514104
\(515\) 37.3308 1.64499
\(516\) 33.5359 1.47634
\(517\) 28.5246 1.25451
\(518\) 0 0
\(519\) −15.5225 −0.681363
\(520\) −1.93074 −0.0846684
\(521\) −6.81604 −0.298616 −0.149308 0.988791i \(-0.547705\pi\)
−0.149308 + 0.988791i \(0.547705\pi\)
\(522\) −2.50105 −0.109468
\(523\) −8.53560 −0.373236 −0.186618 0.982433i \(-0.559753\pi\)
−0.186618 + 0.982433i \(0.559753\pi\)
\(524\) 68.2443 2.98127
\(525\) 0 0
\(526\) −7.38192 −0.321867
\(527\) 11.5575 0.503452
\(528\) −0.996445 −0.0433647
\(529\) 1.00000 0.0434783
\(530\) 94.1195 4.08829
\(531\) −6.38897 −0.277258
\(532\) 0 0
\(533\) 0.851464 0.0368810
\(534\) −4.41232 −0.190940
\(535\) −78.8026 −3.40693
\(536\) −15.9409 −0.688541
\(537\) −8.28280 −0.357429
\(538\) 10.9793 0.473350
\(539\) 0 0
\(540\) −13.1387 −0.565399
\(541\) 1.23462 0.0530806 0.0265403 0.999648i \(-0.491551\pi\)
0.0265403 + 0.999648i \(0.491551\pi\)
\(542\) 27.5536 1.18353
\(543\) 5.71189 0.245121
\(544\) −7.96916 −0.341675
\(545\) −48.6449 −2.08372
\(546\) 0 0
\(547\) 26.1193 1.11678 0.558390 0.829578i \(-0.311419\pi\)
0.558390 + 0.829578i \(0.311419\pi\)
\(548\) 1.78992 0.0764618
\(549\) −7.23261 −0.308680
\(550\) 88.6015 3.77798
\(551\) 0.0554935 0.00236410
\(552\) −2.64772 −0.112695
\(553\) 0 0
\(554\) 49.7746 2.11472
\(555\) −15.6182 −0.662955
\(556\) −61.9301 −2.62642
\(557\) −25.7744 −1.09210 −0.546049 0.837753i \(-0.683869\pi\)
−0.546049 + 0.837753i \(0.683869\pi\)
\(558\) 19.7958 0.838023
\(559\) −1.86127 −0.0787232
\(560\) 0 0
\(561\) −4.22880 −0.178540
\(562\) 55.0814 2.32347
\(563\) 11.1504 0.469932 0.234966 0.972004i \(-0.424502\pi\)
0.234966 + 0.972004i \(0.424502\pi\)
\(564\) 28.3274 1.19280
\(565\) 68.9217 2.89956
\(566\) −11.6947 −0.491567
\(567\) 0 0
\(568\) −17.7800 −0.746031
\(569\) −9.49752 −0.398157 −0.199078 0.979984i \(-0.563795\pi\)
−0.199078 + 0.979984i \(0.563795\pi\)
\(570\) 0.475736 0.0199264
\(571\) −25.5625 −1.06976 −0.534878 0.844929i \(-0.679643\pi\)
−0.534878 + 0.844929i \(0.679643\pi\)
\(572\) −1.77192 −0.0740876
\(573\) 1.55152 0.0648159
\(574\) 0 0
\(575\) −12.2325 −0.510130
\(576\) −13.0244 −0.542682
\(577\) 15.3566 0.639302 0.319651 0.947535i \(-0.396434\pi\)
0.319651 + 0.947535i \(0.396434\pi\)
\(578\) 34.6342 1.44059
\(579\) −7.32290 −0.304330
\(580\) 14.4590 0.600377
\(581\) 0 0
\(582\) −29.0626 −1.20469
\(583\) 31.7950 1.31681
\(584\) 12.2159 0.505496
\(585\) 0.729206 0.0301489
\(586\) 55.7654 2.30365
\(587\) −2.88985 −0.119277 −0.0596384 0.998220i \(-0.518995\pi\)
−0.0596384 + 0.998220i \(0.518995\pi\)
\(588\) 0 0
\(589\) −0.439230 −0.0180982
\(590\) 60.2756 2.48151
\(591\) 6.84137 0.281416
\(592\) −1.17631 −0.0483458
\(593\) 20.9778 0.861454 0.430727 0.902482i \(-0.358257\pi\)
0.430727 + 0.902482i \(0.358257\pi\)
\(594\) −7.24313 −0.297189
\(595\) 0 0
\(596\) 23.8240 0.975869
\(597\) −7.64366 −0.312834
\(598\) 0.399220 0.0163253
\(599\) 8.10812 0.331289 0.165644 0.986186i \(-0.447030\pi\)
0.165644 + 0.986186i \(0.447030\pi\)
\(600\) 32.3883 1.32225
\(601\) 31.3113 1.27721 0.638607 0.769533i \(-0.279511\pi\)
0.638607 + 0.769533i \(0.279511\pi\)
\(602\) 0 0
\(603\) 6.02059 0.245178
\(604\) −67.6118 −2.75108
\(605\) −3.49809 −0.142217
\(606\) −3.74043 −0.151945
\(607\) −25.6566 −1.04137 −0.520685 0.853749i \(-0.674323\pi\)
−0.520685 + 0.853749i \(0.674323\pi\)
\(608\) 0.302860 0.0122826
\(609\) 0 0
\(610\) 68.2348 2.76274
\(611\) −1.57219 −0.0636041
\(612\) −4.19957 −0.169757
\(613\) 29.3962 1.18730 0.593651 0.804722i \(-0.297686\pi\)
0.593651 + 0.804722i \(0.297686\pi\)
\(614\) −16.2873 −0.657301
\(615\) −20.1217 −0.811384
\(616\) 0 0
\(617\) 36.1711 1.45619 0.728097 0.685474i \(-0.240405\pi\)
0.728097 + 0.685474i \(0.240405\pi\)
\(618\) 20.4376 0.822119
\(619\) −22.9578 −0.922752 −0.461376 0.887205i \(-0.652644\pi\)
−0.461376 + 0.887205i \(0.652644\pi\)
\(620\) −114.443 −4.59613
\(621\) 1.00000 0.0401286
\(622\) −13.3469 −0.535163
\(623\) 0 0
\(624\) 0.0549212 0.00219861
\(625\) 63.4715 2.53886
\(626\) 8.64441 0.345500
\(627\) 0.160711 0.00641817
\(628\) 37.2188 1.48519
\(629\) −4.99210 −0.199048
\(630\) 0 0
\(631\) −13.3609 −0.531890 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(632\) 41.0012 1.63094
\(633\) 4.69603 0.186651
\(634\) −22.2786 −0.884797
\(635\) 74.0849 2.93997
\(636\) 31.5752 1.25204
\(637\) 0 0
\(638\) 7.97099 0.315575
\(639\) 6.71519 0.265649
\(640\) 73.0116 2.88604
\(641\) −35.0959 −1.38620 −0.693102 0.720840i \(-0.743756\pi\)
−0.693102 + 0.720840i \(0.743756\pi\)
\(642\) −43.1422 −1.70269
\(643\) 15.1822 0.598726 0.299363 0.954139i \(-0.403226\pi\)
0.299363 + 0.954139i \(0.403226\pi\)
\(644\) 0 0
\(645\) 43.9852 1.73191
\(646\) 0.152061 0.00598277
\(647\) 20.0656 0.788861 0.394430 0.918926i \(-0.370942\pi\)
0.394430 + 0.918926i \(0.370942\pi\)
\(648\) −2.64772 −0.104012
\(649\) 20.3620 0.799279
\(650\) −4.88346 −0.191545
\(651\) 0 0
\(652\) −48.4942 −1.89918
\(653\) −4.13765 −0.161919 −0.0809594 0.996717i \(-0.525798\pi\)
−0.0809594 + 0.996717i \(0.525798\pi\)
\(654\) −26.6318 −1.04138
\(655\) 89.5083 3.49738
\(656\) −1.51549 −0.0591700
\(657\) −4.61372 −0.179998
\(658\) 0 0
\(659\) 14.7450 0.574383 0.287192 0.957873i \(-0.407278\pi\)
0.287192 + 0.957873i \(0.407278\pi\)
\(660\) 41.8737 1.62993
\(661\) 25.9992 1.01125 0.505625 0.862753i \(-0.331262\pi\)
0.505625 + 0.862753i \(0.331262\pi\)
\(662\) 25.4303 0.988376
\(663\) 0.233079 0.00905203
\(664\) −19.0928 −0.740944
\(665\) 0 0
\(666\) −8.55052 −0.331326
\(667\) −1.10049 −0.0426112
\(668\) −25.3985 −0.982697
\(669\) −24.2891 −0.939072
\(670\) −56.8002 −2.19438
\(671\) 23.0507 0.889864
\(672\) 0 0
\(673\) 8.46300 0.326225 0.163112 0.986608i \(-0.447847\pi\)
0.163112 + 0.986608i \(0.447847\pi\)
\(674\) 72.2688 2.78369
\(675\) −12.2325 −0.470829
\(676\) −41.0477 −1.57876
\(677\) 18.7230 0.719583 0.359791 0.933033i \(-0.382848\pi\)
0.359791 + 0.933033i \(0.382848\pi\)
\(678\) 37.7327 1.44912
\(679\) 0 0
\(680\) 14.5839 0.559267
\(681\) −4.33303 −0.166042
\(682\) −63.0903 −2.41585
\(683\) −0.301855 −0.0115502 −0.00577508 0.999983i \(-0.501838\pi\)
−0.00577508 + 0.999983i \(0.501838\pi\)
\(684\) 0.159600 0.00610246
\(685\) 2.34764 0.0896986
\(686\) 0 0
\(687\) 12.4437 0.474757
\(688\) 3.31281 0.126300
\(689\) −1.75245 −0.0667630
\(690\) −9.43432 −0.359158
\(691\) −31.5237 −1.19922 −0.599610 0.800293i \(-0.704677\pi\)
−0.599610 + 0.800293i \(0.704677\pi\)
\(692\) 49.1292 1.86761
\(693\) 0 0
\(694\) 3.12238 0.118524
\(695\) −81.2266 −3.08110
\(696\) 2.91380 0.110447
\(697\) −6.43157 −0.243613
\(698\) 14.6166 0.553246
\(699\) −3.24764 −0.122837
\(700\) 0 0
\(701\) −17.6579 −0.666928 −0.333464 0.942763i \(-0.608218\pi\)
−0.333464 + 0.942763i \(0.608218\pi\)
\(702\) 0.399220 0.0150676
\(703\) 0.189719 0.00715540
\(704\) 41.5094 1.56444
\(705\) 37.1538 1.39929
\(706\) −58.4907 −2.20133
\(707\) 0 0
\(708\) 20.2213 0.759962
\(709\) 7.73921 0.290652 0.145326 0.989384i \(-0.453577\pi\)
0.145326 + 0.989384i \(0.453577\pi\)
\(710\) −63.3532 −2.37760
\(711\) −15.4854 −0.580749
\(712\) 5.14047 0.192647
\(713\) 8.71037 0.326206
\(714\) 0 0
\(715\) −2.32402 −0.0869134
\(716\) 26.2153 0.979712
\(717\) −26.4823 −0.989001
\(718\) 54.4608 2.03246
\(719\) −23.4142 −0.873203 −0.436602 0.899655i \(-0.643818\pi\)
−0.436602 + 0.899655i \(0.643818\pi\)
\(720\) −1.29789 −0.0483695
\(721\) 0 0
\(722\) 43.1749 1.60681
\(723\) −2.56149 −0.0952630
\(724\) −18.0783 −0.671874
\(725\) 13.4617 0.499957
\(726\) −1.91511 −0.0710763
\(727\) −43.6574 −1.61916 −0.809582 0.587007i \(-0.800306\pi\)
−0.809582 + 0.587007i \(0.800306\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 43.5273 1.61102
\(731\) 14.0592 0.519997
\(732\) 22.8914 0.846091
\(733\) −42.2476 −1.56045 −0.780226 0.625497i \(-0.784896\pi\)
−0.780226 + 0.625497i \(0.784896\pi\)
\(734\) −58.3584 −2.15405
\(735\) 0 0
\(736\) −6.00601 −0.221384
\(737\) −19.1880 −0.706798
\(738\) −11.0161 −0.405507
\(739\) −39.2506 −1.44386 −0.721929 0.691967i \(-0.756744\pi\)
−0.721929 + 0.691967i \(0.756744\pi\)
\(740\) 49.4320 1.81716
\(741\) −0.00885791 −0.000325404 0
\(742\) 0 0
\(743\) 20.7027 0.759507 0.379754 0.925088i \(-0.376009\pi\)
0.379754 + 0.925088i \(0.376009\pi\)
\(744\) −23.0626 −0.845518
\(745\) 31.2472 1.14481
\(746\) −13.1836 −0.482685
\(747\) 7.21102 0.263837
\(748\) 13.3843 0.489377
\(749\) 0 0
\(750\) 68.2336 2.49154
\(751\) −0.0631626 −0.00230484 −0.00115242 0.999999i \(-0.500367\pi\)
−0.00115242 + 0.999999i \(0.500367\pi\)
\(752\) 2.79829 0.102043
\(753\) −14.6123 −0.532500
\(754\) −0.439338 −0.0159997
\(755\) −88.6786 −3.22735
\(756\) 0 0
\(757\) 37.1805 1.35135 0.675673 0.737201i \(-0.263853\pi\)
0.675673 + 0.737201i \(0.263853\pi\)
\(758\) −47.8406 −1.73765
\(759\) −3.18706 −0.115683
\(760\) −0.554246 −0.0201046
\(761\) −28.3067 −1.02612 −0.513059 0.858353i \(-0.671488\pi\)
−0.513059 + 0.858353i \(0.671488\pi\)
\(762\) 40.5594 1.46931
\(763\) 0 0
\(764\) −4.91062 −0.177660
\(765\) −5.50809 −0.199145
\(766\) 10.9608 0.396031
\(767\) −1.12230 −0.0405237
\(768\) 13.9231 0.502408
\(769\) 7.54568 0.272104 0.136052 0.990702i \(-0.456559\pi\)
0.136052 + 0.990702i \(0.456559\pi\)
\(770\) 0 0
\(771\) 5.12857 0.184701
\(772\) 23.1772 0.834166
\(773\) −18.3951 −0.661626 −0.330813 0.943696i \(-0.607323\pi\)
−0.330813 + 0.943696i \(0.607323\pi\)
\(774\) 24.0807 0.865562
\(775\) −106.550 −3.82737
\(776\) 33.8588 1.21546
\(777\) 0 0
\(778\) 44.4112 1.59222
\(779\) 0.244425 0.00875743
\(780\) −2.30796 −0.0826381
\(781\) −21.4017 −0.765813
\(782\) −3.01553 −0.107835
\(783\) −1.10049 −0.0393283
\(784\) 0 0
\(785\) 48.8157 1.74231
\(786\) 49.0033 1.74789
\(787\) 17.2390 0.614502 0.307251 0.951628i \(-0.400591\pi\)
0.307251 + 0.951628i \(0.400591\pi\)
\(788\) −21.6531 −0.771360
\(789\) −3.24813 −0.115636
\(790\) 146.094 5.19781
\(791\) 0 0
\(792\) 8.43845 0.299847
\(793\) −1.27049 −0.0451164
\(794\) −83.1455 −2.95072
\(795\) 41.4136 1.46879
\(796\) 24.1924 0.857477
\(797\) −52.7371 −1.86804 −0.934021 0.357217i \(-0.883726\pi\)
−0.934021 + 0.357217i \(0.883726\pi\)
\(798\) 0 0
\(799\) 11.8756 0.420129
\(800\) 73.4684 2.59750
\(801\) −1.94147 −0.0685984
\(802\) 10.6245 0.375163
\(803\) 14.7042 0.518900
\(804\) −19.0553 −0.672030
\(805\) 0 0
\(806\) 3.47735 0.122485
\(807\) 4.83101 0.170059
\(808\) 4.35770 0.153303
\(809\) 6.01310 0.211409 0.105705 0.994398i \(-0.466290\pi\)
0.105705 + 0.994398i \(0.466290\pi\)
\(810\) −9.43432 −0.331488
\(811\) 24.3539 0.855181 0.427591 0.903972i \(-0.359362\pi\)
0.427591 + 0.903972i \(0.359362\pi\)
\(812\) 0 0
\(813\) 12.1239 0.425203
\(814\) 27.2510 0.955147
\(815\) −63.6042 −2.22796
\(816\) −0.414849 −0.0145226
\(817\) −0.534303 −0.0186929
\(818\) −25.0128 −0.874554
\(819\) 0 0
\(820\) 63.6857 2.22400
\(821\) 0.111893 0.00390508 0.00195254 0.999998i \(-0.499378\pi\)
0.00195254 + 0.999998i \(0.499378\pi\)
\(822\) 1.28527 0.0448288
\(823\) 19.9376 0.694983 0.347491 0.937683i \(-0.387034\pi\)
0.347491 + 0.937683i \(0.387034\pi\)
\(824\) −23.8103 −0.829472
\(825\) 38.9857 1.35731
\(826\) 0 0
\(827\) −48.7394 −1.69483 −0.847417 0.530928i \(-0.821843\pi\)
−0.847417 + 0.530928i \(0.821843\pi\)
\(828\) −3.16503 −0.109992
\(829\) 11.3338 0.393638 0.196819 0.980440i \(-0.436939\pi\)
0.196819 + 0.980440i \(0.436939\pi\)
\(830\) −68.0310 −2.36139
\(831\) 21.9014 0.759751
\(832\) −2.28788 −0.0793178
\(833\) 0 0
\(834\) −44.4693 −1.53985
\(835\) −33.3123 −1.15282
\(836\) −0.508654 −0.0175922
\(837\) 8.71037 0.301074
\(838\) −42.9014 −1.48200
\(839\) 42.1263 1.45436 0.727182 0.686445i \(-0.240830\pi\)
0.727182 + 0.686445i \(0.240830\pi\)
\(840\) 0 0
\(841\) −27.7889 −0.958239
\(842\) 23.0314 0.793715
\(843\) 24.2364 0.834747
\(844\) −14.8631 −0.511608
\(845\) −53.8376 −1.85207
\(846\) 20.3407 0.699327
\(847\) 0 0
\(848\) 3.11912 0.107111
\(849\) −5.14581 −0.176604
\(850\) 36.8874 1.26523
\(851\) −3.76233 −0.128971
\(852\) −21.2538 −0.728142
\(853\) −37.5528 −1.28578 −0.642891 0.765957i \(-0.722265\pi\)
−0.642891 + 0.765957i \(0.722265\pi\)
\(854\) 0 0
\(855\) 0.209329 0.00715890
\(856\) 50.2619 1.71792
\(857\) 58.0721 1.98370 0.991852 0.127397i \(-0.0406621\pi\)
0.991852 + 0.127397i \(0.0406621\pi\)
\(858\) −1.27234 −0.0434369
\(859\) −24.9857 −0.852501 −0.426250 0.904605i \(-0.640166\pi\)
−0.426250 + 0.904605i \(0.640166\pi\)
\(860\) −139.214 −4.74717
\(861\) 0 0
\(862\) −17.8901 −0.609338
\(863\) −13.9209 −0.473875 −0.236937 0.971525i \(-0.576144\pi\)
−0.236937 + 0.971525i \(0.576144\pi\)
\(864\) −6.00601 −0.204328
\(865\) 64.4372 2.19093
\(866\) 46.7868 1.58988
\(867\) 15.2394 0.517558
\(868\) 0 0
\(869\) 49.3530 1.67418
\(870\) 10.3824 0.351996
\(871\) 1.05759 0.0358349
\(872\) 31.0267 1.05070
\(873\) −12.7879 −0.432804
\(874\) 0.114602 0.00387647
\(875\) 0 0
\(876\) 14.6026 0.493375
\(877\) −7.97539 −0.269310 −0.134655 0.990893i \(-0.542993\pi\)
−0.134655 + 0.990893i \(0.542993\pi\)
\(878\) −76.5157 −2.58228
\(879\) 24.5374 0.827626
\(880\) 4.13645 0.139440
\(881\) −33.6800 −1.13471 −0.567354 0.823474i \(-0.692033\pi\)
−0.567354 + 0.823474i \(0.692033\pi\)
\(882\) 0 0
\(883\) 32.7638 1.10259 0.551294 0.834311i \(-0.314134\pi\)
0.551294 + 0.834311i \(0.314134\pi\)
\(884\) −0.737701 −0.0248116
\(885\) 26.5219 0.891525
\(886\) −16.6707 −0.560062
\(887\) 9.33788 0.313535 0.156768 0.987635i \(-0.449893\pi\)
0.156768 + 0.987635i \(0.449893\pi\)
\(888\) 9.96160 0.334289
\(889\) 0 0
\(890\) 18.3164 0.613968
\(891\) −3.18706 −0.106770
\(892\) 76.8758 2.57399
\(893\) −0.451320 −0.0151029
\(894\) 17.1070 0.572143
\(895\) 34.3836 1.14932
\(896\) 0 0
\(897\) 0.175661 0.00586516
\(898\) 48.0689 1.60408
\(899\) −9.58568 −0.319700
\(900\) 38.7162 1.29054
\(901\) 13.2372 0.440995
\(902\) 35.1088 1.16900
\(903\) 0 0
\(904\) −43.9597 −1.46208
\(905\) −23.7112 −0.788187
\(906\) −48.5491 −1.61294
\(907\) 19.1938 0.637320 0.318660 0.947869i \(-0.396767\pi\)
0.318660 + 0.947869i \(0.396767\pi\)
\(908\) 13.7142 0.455121
\(909\) −1.64583 −0.0545887
\(910\) 0 0
\(911\) −18.4406 −0.610963 −0.305481 0.952198i \(-0.598817\pi\)
−0.305481 + 0.952198i \(0.598817\pi\)
\(912\) 0.0157659 0.000522061 0
\(913\) −22.9819 −0.760591
\(914\) −20.3809 −0.674140
\(915\) 30.0241 0.992565
\(916\) −39.3847 −1.30131
\(917\) 0 0
\(918\) −3.01553 −0.0995272
\(919\) −30.6905 −1.01239 −0.506194 0.862420i \(-0.668948\pi\)
−0.506194 + 0.862420i \(0.668948\pi\)
\(920\) 10.9912 0.362371
\(921\) −7.16659 −0.236147
\(922\) −14.3223 −0.471679
\(923\) 1.17960 0.0388270
\(924\) 0 0
\(925\) 46.0226 1.51321
\(926\) 34.3382 1.12842
\(927\) 8.99275 0.295361
\(928\) 6.60955 0.216969
\(929\) −4.18918 −0.137443 −0.0687213 0.997636i \(-0.521892\pi\)
−0.0687213 + 0.997636i \(0.521892\pi\)
\(930\) −82.1764 −2.69467
\(931\) 0 0
\(932\) 10.2789 0.336696
\(933\) −5.87280 −0.192267
\(934\) 43.0910 1.40998
\(935\) 17.5546 0.574097
\(936\) −0.465103 −0.0152024
\(937\) 31.8927 1.04189 0.520945 0.853590i \(-0.325580\pi\)
0.520945 + 0.853590i \(0.325580\pi\)
\(938\) 0 0
\(939\) 3.80364 0.124127
\(940\) −117.593 −3.83546
\(941\) −30.3580 −0.989643 −0.494822 0.868995i \(-0.664767\pi\)
−0.494822 + 0.868995i \(0.664767\pi\)
\(942\) 26.7252 0.870756
\(943\) −4.84719 −0.157846
\(944\) 1.99754 0.0650143
\(945\) 0 0
\(946\) −76.7465 −2.49524
\(947\) 33.7676 1.09730 0.548650 0.836052i \(-0.315142\pi\)
0.548650 + 0.836052i \(0.315142\pi\)
\(948\) 49.0118 1.59183
\(949\) −0.810452 −0.0263084
\(950\) −1.40187 −0.0454826
\(951\) −9.80284 −0.317879
\(952\) 0 0
\(953\) 6.62261 0.214527 0.107264 0.994231i \(-0.465791\pi\)
0.107264 + 0.994231i \(0.465791\pi\)
\(954\) 22.6728 0.734060
\(955\) −6.44070 −0.208416
\(956\) 83.8173 2.71085
\(957\) 3.50733 0.113376
\(958\) 64.9465 2.09833
\(959\) 0 0
\(960\) 54.0668 1.74500
\(961\) 44.8705 1.44744
\(962\) −1.50200 −0.0484263
\(963\) −18.9831 −0.611721
\(964\) 8.10720 0.261115
\(965\) 30.3989 0.978574
\(966\) 0 0
\(967\) −49.1558 −1.58074 −0.790372 0.612628i \(-0.790113\pi\)
−0.790372 + 0.612628i \(0.790113\pi\)
\(968\) 2.23115 0.0717120
\(969\) 0.0669087 0.00214942
\(970\) 120.645 3.87368
\(971\) −8.76315 −0.281223 −0.140611 0.990065i \(-0.544907\pi\)
−0.140611 + 0.990065i \(0.544907\pi\)
\(972\) −3.16503 −0.101518
\(973\) 0 0
\(974\) −56.8002 −1.82000
\(975\) −2.14878 −0.0688159
\(976\) 2.26130 0.0723826
\(977\) −0.347523 −0.0111183 −0.00555913 0.999985i \(-0.501770\pi\)
−0.00555913 + 0.999985i \(0.501770\pi\)
\(978\) −34.8216 −1.11347
\(979\) 6.18757 0.197756
\(980\) 0 0
\(981\) −11.7183 −0.374136
\(982\) 50.1158 1.59926
\(983\) −53.3283 −1.70091 −0.850454 0.526050i \(-0.823672\pi\)
−0.850454 + 0.526050i \(0.823672\pi\)
\(984\) 12.8340 0.409134
\(985\) −28.3999 −0.904896
\(986\) 3.31856 0.105684
\(987\) 0 0
\(988\) 0.0280355 0.000891930 0
\(989\) 10.5958 0.336926
\(990\) 30.0677 0.955614
\(991\) −5.82924 −0.185172 −0.0925860 0.995705i \(-0.529513\pi\)
−0.0925860 + 0.995705i \(0.529513\pi\)
\(992\) −52.3145 −1.66099
\(993\) 11.1896 0.355091
\(994\) 0 0
\(995\) 31.7304 1.00592
\(996\) −22.8231 −0.723177
\(997\) −31.9358 −1.01142 −0.505708 0.862705i \(-0.668769\pi\)
−0.505708 + 0.862705i \(0.668769\pi\)
\(998\) 89.6002 2.83625
\(999\) −3.76233 −0.119035
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 3381.2.a.bc.1.1 6
7.2 even 3 483.2.i.f.277.6 12
7.4 even 3 483.2.i.f.415.6 yes 12
7.6 odd 2 3381.2.a.bd.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.f.277.6 12 7.2 even 3
483.2.i.f.415.6 yes 12 7.4 even 3
3381.2.a.bc.1.1 6 1.1 even 1 trivial
3381.2.a.bd.1.1 6 7.6 odd 2