Properties

Label 1110.2.a.p.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.37228 q^{11} -1.00000 q^{12} +1.37228 q^{13} +3.37228 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.37228 q^{17} -1.00000 q^{18} -1.37228 q^{19} +1.00000 q^{20} +3.37228 q^{21} +1.37228 q^{22} +3.37228 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.37228 q^{26} -1.00000 q^{27} -3.37228 q^{28} +6.00000 q^{29} +1.00000 q^{30} +2.74456 q^{31} -1.00000 q^{32} +1.37228 q^{33} +1.37228 q^{34} -3.37228 q^{35} +1.00000 q^{36} -1.00000 q^{37} +1.37228 q^{38} -1.37228 q^{39} -1.00000 q^{40} +8.74456 q^{41} -3.37228 q^{42} -4.00000 q^{43} -1.37228 q^{44} +1.00000 q^{45} -3.37228 q^{46} -4.74456 q^{47} -1.00000 q^{48} +4.37228 q^{49} -1.00000 q^{50} +1.37228 q^{51} +1.37228 q^{52} +5.37228 q^{53} +1.00000 q^{54} -1.37228 q^{55} +3.37228 q^{56} +1.37228 q^{57} -6.00000 q^{58} +14.7446 q^{59} -1.00000 q^{60} -2.74456 q^{61} -2.74456 q^{62} -3.37228 q^{63} +1.00000 q^{64} +1.37228 q^{65} -1.37228 q^{66} +2.74456 q^{67} -1.37228 q^{68} -3.37228 q^{69} +3.37228 q^{70} +1.25544 q^{71} -1.00000 q^{72} -4.11684 q^{73} +1.00000 q^{74} -1.00000 q^{75} -1.37228 q^{76} +4.62772 q^{77} +1.37228 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.74456 q^{82} +0.627719 q^{83} +3.37228 q^{84} -1.37228 q^{85} +4.00000 q^{86} -6.00000 q^{87} +1.37228 q^{88} +13.3723 q^{89} -1.00000 q^{90} -4.62772 q^{91} +3.37228 q^{92} -2.74456 q^{93} +4.74456 q^{94} -1.37228 q^{95} +1.00000 q^{96} +13.4891 q^{97} -4.37228 q^{98} -1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - q^{7} - 2q^{8} + 2q^{9} - 2q^{10} + 3q^{11} - 2q^{12} - 3q^{13} + q^{14} - 2q^{15} + 2q^{16} + 3q^{17} - 2q^{18} + 3q^{19} + 2q^{20} + q^{21} - 3q^{22} + q^{23} + 2q^{24} + 2q^{25} + 3q^{26} - 2q^{27} - q^{28} + 12q^{29} + 2q^{30} - 6q^{31} - 2q^{32} - 3q^{33} - 3q^{34} - q^{35} + 2q^{36} - 2q^{37} - 3q^{38} + 3q^{39} - 2q^{40} + 6q^{41} - q^{42} - 8q^{43} + 3q^{44} + 2q^{45} - q^{46} + 2q^{47} - 2q^{48} + 3q^{49} - 2q^{50} - 3q^{51} - 3q^{52} + 5q^{53} + 2q^{54} + 3q^{55} + q^{56} - 3q^{57} - 12q^{58} + 18q^{59} - 2q^{60} + 6q^{61} + 6q^{62} - q^{63} + 2q^{64} - 3q^{65} + 3q^{66} - 6q^{67} + 3q^{68} - q^{69} + q^{70} + 14q^{71} - 2q^{72} + 9q^{73} + 2q^{74} - 2q^{75} + 3q^{76} + 15q^{77} - 3q^{78} + 8q^{79} + 2q^{80} + 2q^{81} - 6q^{82} + 7q^{83} + q^{84} + 3q^{85} + 8q^{86} - 12q^{87} - 3q^{88} + 21q^{89} - 2q^{90} - 15q^{91} + q^{92} + 6q^{93} - 2q^{94} + 3q^{95} + 2q^{96} + 4q^{97} - 3q^{98} + 3q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −3.37228 −1.27460 −0.637301 0.770615i \(-0.719949\pi\)
−0.637301 + 0.770615i \(0.719949\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.37228 −0.413758 −0.206879 0.978366i \(-0.566331\pi\)
−0.206879 + 0.978366i \(0.566331\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.37228 0.380602 0.190301 0.981726i \(-0.439054\pi\)
0.190301 + 0.981726i \(0.439054\pi\)
\(14\) 3.37228 0.901280
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.37228 −0.332827 −0.166414 0.986056i \(-0.553219\pi\)
−0.166414 + 0.986056i \(0.553219\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.37228 −0.314823 −0.157411 0.987533i \(-0.550315\pi\)
−0.157411 + 0.987533i \(0.550315\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.37228 0.735892
\(22\) 1.37228 0.292571
\(23\) 3.37228 0.703169 0.351585 0.936156i \(-0.385643\pi\)
0.351585 + 0.936156i \(0.385643\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −1.37228 −0.269127
\(27\) −1.00000 −0.192450
\(28\) −3.37228 −0.637301
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) 2.74456 0.492938 0.246469 0.969151i \(-0.420730\pi\)
0.246469 + 0.969151i \(0.420730\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.37228 0.238884
\(34\) 1.37228 0.235344
\(35\) −3.37228 −0.570020
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 1.37228 0.222613
\(39\) −1.37228 −0.219741
\(40\) −1.00000 −0.158114
\(41\) 8.74456 1.36567 0.682836 0.730572i \(-0.260747\pi\)
0.682836 + 0.730572i \(0.260747\pi\)
\(42\) −3.37228 −0.520354
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.37228 −0.206879
\(45\) 1.00000 0.149071
\(46\) −3.37228 −0.497216
\(47\) −4.74456 −0.692066 −0.346033 0.938222i \(-0.612471\pi\)
−0.346033 + 0.938222i \(0.612471\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.37228 0.624612
\(50\) −1.00000 −0.141421
\(51\) 1.37228 0.192158
\(52\) 1.37228 0.190301
\(53\) 5.37228 0.737940 0.368970 0.929441i \(-0.379711\pi\)
0.368970 + 0.929441i \(0.379711\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.37228 −0.185038
\(56\) 3.37228 0.450640
\(57\) 1.37228 0.181763
\(58\) −6.00000 −0.787839
\(59\) 14.7446 1.91958 0.959789 0.280721i \(-0.0905737\pi\)
0.959789 + 0.280721i \(0.0905737\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.74456 −0.351405 −0.175703 0.984443i \(-0.556220\pi\)
−0.175703 + 0.984443i \(0.556220\pi\)
\(62\) −2.74456 −0.348560
\(63\) −3.37228 −0.424868
\(64\) 1.00000 0.125000
\(65\) 1.37228 0.170211
\(66\) −1.37228 −0.168916
\(67\) 2.74456 0.335302 0.167651 0.985846i \(-0.446382\pi\)
0.167651 + 0.985846i \(0.446382\pi\)
\(68\) −1.37228 −0.166414
\(69\) −3.37228 −0.405975
\(70\) 3.37228 0.403065
\(71\) 1.25544 0.148993 0.0744965 0.997221i \(-0.476265\pi\)
0.0744965 + 0.997221i \(0.476265\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.11684 −0.481840 −0.240920 0.970545i \(-0.577449\pi\)
−0.240920 + 0.970545i \(0.577449\pi\)
\(74\) 1.00000 0.116248
\(75\) −1.00000 −0.115470
\(76\) −1.37228 −0.157411
\(77\) 4.62772 0.527377
\(78\) 1.37228 0.155380
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −8.74456 −0.965675
\(83\) 0.627719 0.0689011 0.0344505 0.999406i \(-0.489032\pi\)
0.0344505 + 0.999406i \(0.489032\pi\)
\(84\) 3.37228 0.367946
\(85\) −1.37228 −0.148845
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) 1.37228 0.146286
\(89\) 13.3723 1.41746 0.708729 0.705480i \(-0.249269\pi\)
0.708729 + 0.705480i \(0.249269\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.62772 −0.485117
\(92\) 3.37228 0.351585
\(93\) −2.74456 −0.284598
\(94\) 4.74456 0.489364
\(95\) −1.37228 −0.140793
\(96\) 1.00000 0.102062
\(97\) 13.4891 1.36961 0.684807 0.728725i \(-0.259887\pi\)
0.684807 + 0.728725i \(0.259887\pi\)
\(98\) −4.37228 −0.441667
\(99\) −1.37228 −0.137919
\(100\) 1.00000 0.100000
\(101\) 10.7446 1.06912 0.534562 0.845129i \(-0.320477\pi\)
0.534562 + 0.845129i \(0.320477\pi\)
\(102\) −1.37228 −0.135876
\(103\) −0.744563 −0.0733639 −0.0366820 0.999327i \(-0.511679\pi\)
−0.0366820 + 0.999327i \(0.511679\pi\)
\(104\) −1.37228 −0.134563
\(105\) 3.37228 0.329101
\(106\) −5.37228 −0.521802
\(107\) 3.37228 0.326011 0.163005 0.986625i \(-0.447881\pi\)
0.163005 + 0.986625i \(0.447881\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.62772 0.443255 0.221628 0.975131i \(-0.428863\pi\)
0.221628 + 0.975131i \(0.428863\pi\)
\(110\) 1.37228 0.130842
\(111\) 1.00000 0.0949158
\(112\) −3.37228 −0.318651
\(113\) 19.4891 1.83338 0.916691 0.399596i \(-0.130850\pi\)
0.916691 + 0.399596i \(0.130850\pi\)
\(114\) −1.37228 −0.128526
\(115\) 3.37228 0.314467
\(116\) 6.00000 0.557086
\(117\) 1.37228 0.126867
\(118\) −14.7446 −1.35735
\(119\) 4.62772 0.424222
\(120\) 1.00000 0.0912871
\(121\) −9.11684 −0.828804
\(122\) 2.74456 0.248481
\(123\) −8.74456 −0.788471
\(124\) 2.74456 0.246469
\(125\) 1.00000 0.0894427
\(126\) 3.37228 0.300427
\(127\) 3.37228 0.299242 0.149621 0.988743i \(-0.452195\pi\)
0.149621 + 0.988743i \(0.452195\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) −1.37228 −0.120357
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 1.37228 0.119442
\(133\) 4.62772 0.401274
\(134\) −2.74456 −0.237094
\(135\) −1.00000 −0.0860663
\(136\) 1.37228 0.117672
\(137\) 10.7446 0.917970 0.458985 0.888444i \(-0.348213\pi\)
0.458985 + 0.888444i \(0.348213\pi\)
\(138\) 3.37228 0.287068
\(139\) 2.74456 0.232791 0.116395 0.993203i \(-0.462866\pi\)
0.116395 + 0.993203i \(0.462866\pi\)
\(140\) −3.37228 −0.285010
\(141\) 4.74456 0.399564
\(142\) −1.25544 −0.105354
\(143\) −1.88316 −0.157477
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 4.11684 0.340712
\(147\) −4.37228 −0.360620
\(148\) −1.00000 −0.0821995
\(149\) 9.25544 0.758235 0.379117 0.925349i \(-0.376228\pi\)
0.379117 + 0.925349i \(0.376228\pi\)
\(150\) 1.00000 0.0816497
\(151\) −3.37228 −0.274432 −0.137216 0.990541i \(-0.543816\pi\)
−0.137216 + 0.990541i \(0.543816\pi\)
\(152\) 1.37228 0.111307
\(153\) −1.37228 −0.110942
\(154\) −4.62772 −0.372912
\(155\) 2.74456 0.220449
\(156\) −1.37228 −0.109870
\(157\) 3.25544 0.259812 0.129906 0.991526i \(-0.458532\pi\)
0.129906 + 0.991526i \(0.458532\pi\)
\(158\) −4.00000 −0.318223
\(159\) −5.37228 −0.426050
\(160\) −1.00000 −0.0790569
\(161\) −11.3723 −0.896261
\(162\) −1.00000 −0.0785674
\(163\) 4.86141 0.380775 0.190387 0.981709i \(-0.439026\pi\)
0.190387 + 0.981709i \(0.439026\pi\)
\(164\) 8.74456 0.682836
\(165\) 1.37228 0.106832
\(166\) −0.627719 −0.0487204
\(167\) −1.88316 −0.145723 −0.0728615 0.997342i \(-0.523213\pi\)
−0.0728615 + 0.997342i \(0.523213\pi\)
\(168\) −3.37228 −0.260177
\(169\) −11.1168 −0.855142
\(170\) 1.37228 0.105249
\(171\) −1.37228 −0.104941
\(172\) −4.00000 −0.304997
\(173\) −6.86141 −0.521663 −0.260832 0.965384i \(-0.583997\pi\)
−0.260832 + 0.965384i \(0.583997\pi\)
\(174\) 6.00000 0.454859
\(175\) −3.37228 −0.254921
\(176\) −1.37228 −0.103440
\(177\) −14.7446 −1.10827
\(178\) −13.3723 −1.00229
\(179\) 5.48913 0.410276 0.205138 0.978733i \(-0.434236\pi\)
0.205138 + 0.978733i \(0.434236\pi\)
\(180\) 1.00000 0.0745356
\(181\) −20.9783 −1.55930 −0.779651 0.626215i \(-0.784603\pi\)
−0.779651 + 0.626215i \(0.784603\pi\)
\(182\) 4.62772 0.343029
\(183\) 2.74456 0.202884
\(184\) −3.37228 −0.248608
\(185\) −1.00000 −0.0735215
\(186\) 2.74456 0.201241
\(187\) 1.88316 0.137710
\(188\) −4.74456 −0.346033
\(189\) 3.37228 0.245297
\(190\) 1.37228 0.0995558
\(191\) −19.3723 −1.40173 −0.700865 0.713294i \(-0.747202\pi\)
−0.700865 + 0.713294i \(0.747202\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −1.25544 −0.0903684 −0.0451842 0.998979i \(-0.514387\pi\)
−0.0451842 + 0.998979i \(0.514387\pi\)
\(194\) −13.4891 −0.968463
\(195\) −1.37228 −0.0982711
\(196\) 4.37228 0.312306
\(197\) −18.8614 −1.34382 −0.671910 0.740633i \(-0.734526\pi\)
−0.671910 + 0.740633i \(0.734526\pi\)
\(198\) 1.37228 0.0975238
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.74456 −0.193587
\(202\) −10.7446 −0.755985
\(203\) −20.2337 −1.42013
\(204\) 1.37228 0.0960789
\(205\) 8.74456 0.610747
\(206\) 0.744563 0.0518761
\(207\) 3.37228 0.234390
\(208\) 1.37228 0.0951506
\(209\) 1.88316 0.130261
\(210\) −3.37228 −0.232710
\(211\) −9.25544 −0.637171 −0.318585 0.947894i \(-0.603208\pi\)
−0.318585 + 0.947894i \(0.603208\pi\)
\(212\) 5.37228 0.368970
\(213\) −1.25544 −0.0860212
\(214\) −3.37228 −0.230524
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) −9.25544 −0.628300
\(218\) −4.62772 −0.313429
\(219\) 4.11684 0.278191
\(220\) −1.37228 −0.0925192
\(221\) −1.88316 −0.126675
\(222\) −1.00000 −0.0671156
\(223\) 18.9783 1.27088 0.635439 0.772151i \(-0.280819\pi\)
0.635439 + 0.772151i \(0.280819\pi\)
\(224\) 3.37228 0.225320
\(225\) 1.00000 0.0666667
\(226\) −19.4891 −1.29640
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 1.37228 0.0908816
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −3.37228 −0.222362
\(231\) −4.62772 −0.304482
\(232\) −6.00000 −0.393919
\(233\) −22.7446 −1.49005 −0.745023 0.667039i \(-0.767561\pi\)
−0.745023 + 0.667039i \(0.767561\pi\)
\(234\) −1.37228 −0.0897088
\(235\) −4.74456 −0.309501
\(236\) 14.7446 0.959789
\(237\) −4.00000 −0.259828
\(238\) −4.62772 −0.299970
\(239\) −5.48913 −0.355062 −0.177531 0.984115i \(-0.556811\pi\)
−0.177531 + 0.984115i \(0.556811\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −0.510875 −0.0329083 −0.0164542 0.999865i \(-0.505238\pi\)
−0.0164542 + 0.999865i \(0.505238\pi\)
\(242\) 9.11684 0.586053
\(243\) −1.00000 −0.0641500
\(244\) −2.74456 −0.175703
\(245\) 4.37228 0.279335
\(246\) 8.74456 0.557533
\(247\) −1.88316 −0.119822
\(248\) −2.74456 −0.174280
\(249\) −0.627719 −0.0397801
\(250\) −1.00000 −0.0632456
\(251\) −26.7446 −1.68810 −0.844051 0.536263i \(-0.819836\pi\)
−0.844051 + 0.536263i \(0.819836\pi\)
\(252\) −3.37228 −0.212434
\(253\) −4.62772 −0.290942
\(254\) −3.37228 −0.211596
\(255\) 1.37228 0.0859356
\(256\) 1.00000 0.0625000
\(257\) −5.37228 −0.335114 −0.167557 0.985862i \(-0.553588\pi\)
−0.167557 + 0.985862i \(0.553588\pi\)
\(258\) −4.00000 −0.249029
\(259\) 3.37228 0.209543
\(260\) 1.37228 0.0851053
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 10.2337 0.631036 0.315518 0.948920i \(-0.397822\pi\)
0.315518 + 0.948920i \(0.397822\pi\)
\(264\) −1.37228 −0.0844581
\(265\) 5.37228 0.330017
\(266\) −4.62772 −0.283744
\(267\) −13.3723 −0.818370
\(268\) 2.74456 0.167651
\(269\) 0.627719 0.0382727 0.0191363 0.999817i \(-0.493908\pi\)
0.0191363 + 0.999817i \(0.493908\pi\)
\(270\) 1.00000 0.0608581
\(271\) 13.4891 0.819406 0.409703 0.912219i \(-0.365632\pi\)
0.409703 + 0.912219i \(0.365632\pi\)
\(272\) −1.37228 −0.0832068
\(273\) 4.62772 0.280082
\(274\) −10.7446 −0.649103
\(275\) −1.37228 −0.0827517
\(276\) −3.37228 −0.202987
\(277\) 25.6060 1.53851 0.769257 0.638940i \(-0.220627\pi\)
0.769257 + 0.638940i \(0.220627\pi\)
\(278\) −2.74456 −0.164608
\(279\) 2.74456 0.164313
\(280\) 3.37228 0.201532
\(281\) 1.37228 0.0818634 0.0409317 0.999162i \(-0.486967\pi\)
0.0409317 + 0.999162i \(0.486967\pi\)
\(282\) −4.74456 −0.282535
\(283\) −11.6060 −0.689903 −0.344952 0.938620i \(-0.612105\pi\)
−0.344952 + 0.938620i \(0.612105\pi\)
\(284\) 1.25544 0.0744965
\(285\) 1.37228 0.0812869
\(286\) 1.88316 0.111353
\(287\) −29.4891 −1.74069
\(288\) −1.00000 −0.0589256
\(289\) −15.1168 −0.889226
\(290\) −6.00000 −0.352332
\(291\) −13.4891 −0.790747
\(292\) −4.11684 −0.240920
\(293\) −4.11684 −0.240509 −0.120254 0.992743i \(-0.538371\pi\)
−0.120254 + 0.992743i \(0.538371\pi\)
\(294\) 4.37228 0.254997
\(295\) 14.7446 0.858462
\(296\) 1.00000 0.0581238
\(297\) 1.37228 0.0796278
\(298\) −9.25544 −0.536153
\(299\) 4.62772 0.267628
\(300\) −1.00000 −0.0577350
\(301\) 13.4891 0.777500
\(302\) 3.37228 0.194053
\(303\) −10.7446 −0.617259
\(304\) −1.37228 −0.0787057
\(305\) −2.74456 −0.157153
\(306\) 1.37228 0.0784481
\(307\) −10.7446 −0.613225 −0.306612 0.951834i \(-0.599195\pi\)
−0.306612 + 0.951834i \(0.599195\pi\)
\(308\) 4.62772 0.263689
\(309\) 0.744563 0.0423567
\(310\) −2.74456 −0.155881
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 1.37228 0.0776901
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −3.25544 −0.183715
\(315\) −3.37228 −0.190007
\(316\) 4.00000 0.225018
\(317\) −16.9783 −0.953594 −0.476797 0.879014i \(-0.658202\pi\)
−0.476797 + 0.879014i \(0.658202\pi\)
\(318\) 5.37228 0.301263
\(319\) −8.23369 −0.460998
\(320\) 1.00000 0.0559017
\(321\) −3.37228 −0.188222
\(322\) 11.3723 0.633752
\(323\) 1.88316 0.104782
\(324\) 1.00000 0.0555556
\(325\) 1.37228 0.0761205
\(326\) −4.86141 −0.269248
\(327\) −4.62772 −0.255913
\(328\) −8.74456 −0.482838
\(329\) 16.0000 0.882109
\(330\) −1.37228 −0.0755416
\(331\) −8.74456 −0.480645 −0.240322 0.970693i \(-0.577253\pi\)
−0.240322 + 0.970693i \(0.577253\pi\)
\(332\) 0.627719 0.0344505
\(333\) −1.00000 −0.0547997
\(334\) 1.88316 0.103042
\(335\) 2.74456 0.149951
\(336\) 3.37228 0.183973
\(337\) −4.11684 −0.224259 −0.112129 0.993694i \(-0.535767\pi\)
−0.112129 + 0.993694i \(0.535767\pi\)
\(338\) 11.1168 0.604677
\(339\) −19.4891 −1.05850
\(340\) −1.37228 −0.0744224
\(341\) −3.76631 −0.203957
\(342\) 1.37228 0.0742045
\(343\) 8.86141 0.478471
\(344\) 4.00000 0.215666
\(345\) −3.37228 −0.181558
\(346\) 6.86141 0.368872
\(347\) 1.48913 0.0799404 0.0399702 0.999201i \(-0.487274\pi\)
0.0399702 + 0.999201i \(0.487274\pi\)
\(348\) −6.00000 −0.321634
\(349\) 16.7446 0.896316 0.448158 0.893954i \(-0.352080\pi\)
0.448158 + 0.893954i \(0.352080\pi\)
\(350\) 3.37228 0.180256
\(351\) −1.37228 −0.0732470
\(352\) 1.37228 0.0731428
\(353\) 3.48913 0.185707 0.0928537 0.995680i \(-0.470401\pi\)
0.0928537 + 0.995680i \(0.470401\pi\)
\(354\) 14.7446 0.783665
\(355\) 1.25544 0.0666317
\(356\) 13.3723 0.708729
\(357\) −4.62772 −0.244925
\(358\) −5.48913 −0.290109
\(359\) 32.4674 1.71356 0.856781 0.515680i \(-0.172461\pi\)
0.856781 + 0.515680i \(0.172461\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −17.1168 −0.900887
\(362\) 20.9783 1.10259
\(363\) 9.11684 0.478510
\(364\) −4.62772 −0.242558
\(365\) −4.11684 −0.215485
\(366\) −2.74456 −0.143461
\(367\) 20.6277 1.07676 0.538379 0.842703i \(-0.319037\pi\)
0.538379 + 0.842703i \(0.319037\pi\)
\(368\) 3.37228 0.175792
\(369\) 8.74456 0.455224
\(370\) 1.00000 0.0519875
\(371\) −18.1168 −0.940580
\(372\) −2.74456 −0.142299
\(373\) 3.48913 0.180660 0.0903300 0.995912i \(-0.471208\pi\)
0.0903300 + 0.995912i \(0.471208\pi\)
\(374\) −1.88316 −0.0973757
\(375\) −1.00000 −0.0516398
\(376\) 4.74456 0.244682
\(377\) 8.23369 0.424057
\(378\) −3.37228 −0.173451
\(379\) 24.2337 1.24480 0.622400 0.782699i \(-0.286158\pi\)
0.622400 + 0.782699i \(0.286158\pi\)
\(380\) −1.37228 −0.0703965
\(381\) −3.37228 −0.172767
\(382\) 19.3723 0.991172
\(383\) 27.6060 1.41060 0.705300 0.708909i \(-0.250812\pi\)
0.705300 + 0.708909i \(0.250812\pi\)
\(384\) 1.00000 0.0510310
\(385\) 4.62772 0.235850
\(386\) 1.25544 0.0639001
\(387\) −4.00000 −0.203331
\(388\) 13.4891 0.684807
\(389\) 23.4891 1.19095 0.595473 0.803375i \(-0.296965\pi\)
0.595473 + 0.803375i \(0.296965\pi\)
\(390\) 1.37228 0.0694882
\(391\) −4.62772 −0.234034
\(392\) −4.37228 −0.220834
\(393\) 0 0
\(394\) 18.8614 0.950224
\(395\) 4.00000 0.201262
\(396\) −1.37228 −0.0689597
\(397\) 20.7446 1.04114 0.520570 0.853819i \(-0.325720\pi\)
0.520570 + 0.853819i \(0.325720\pi\)
\(398\) −8.00000 −0.401004
\(399\) −4.62772 −0.231676
\(400\) 1.00000 0.0500000
\(401\) −36.1168 −1.80359 −0.901795 0.432165i \(-0.857750\pi\)
−0.901795 + 0.432165i \(0.857750\pi\)
\(402\) 2.74456 0.136886
\(403\) 3.76631 0.187613
\(404\) 10.7446 0.534562
\(405\) 1.00000 0.0496904
\(406\) 20.2337 1.00418
\(407\) 1.37228 0.0680215
\(408\) −1.37228 −0.0679380
\(409\) −8.74456 −0.432391 −0.216195 0.976350i \(-0.569365\pi\)
−0.216195 + 0.976350i \(0.569365\pi\)
\(410\) −8.74456 −0.431863
\(411\) −10.7446 −0.529990
\(412\) −0.744563 −0.0366820
\(413\) −49.7228 −2.44670
\(414\) −3.37228 −0.165739
\(415\) 0.627719 0.0308135
\(416\) −1.37228 −0.0672816
\(417\) −2.74456 −0.134402
\(418\) −1.88316 −0.0921082
\(419\) 7.88316 0.385117 0.192559 0.981285i \(-0.438321\pi\)
0.192559 + 0.981285i \(0.438321\pi\)
\(420\) 3.37228 0.164550
\(421\) 24.2337 1.18108 0.590539 0.807009i \(-0.298915\pi\)
0.590539 + 0.807009i \(0.298915\pi\)
\(422\) 9.25544 0.450548
\(423\) −4.74456 −0.230689
\(424\) −5.37228 −0.260901
\(425\) −1.37228 −0.0665654
\(426\) 1.25544 0.0608261
\(427\) 9.25544 0.447902
\(428\) 3.37228 0.163005
\(429\) 1.88316 0.0909196
\(430\) 4.00000 0.192897
\(431\) −31.6060 −1.52241 −0.761203 0.648514i \(-0.775391\pi\)
−0.761203 + 0.648514i \(0.775391\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −10.8614 −0.521966 −0.260983 0.965343i \(-0.584047\pi\)
−0.260983 + 0.965343i \(0.584047\pi\)
\(434\) 9.25544 0.444275
\(435\) −6.00000 −0.287678
\(436\) 4.62772 0.221628
\(437\) −4.62772 −0.221374
\(438\) −4.11684 −0.196710
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 1.37228 0.0654209
\(441\) 4.37228 0.208204
\(442\) 1.88316 0.0895726
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 1.00000 0.0474579
\(445\) 13.3723 0.633907
\(446\) −18.9783 −0.898646
\(447\) −9.25544 −0.437767
\(448\) −3.37228 −0.159325
\(449\) 36.9783 1.74511 0.872556 0.488515i \(-0.162461\pi\)
0.872556 + 0.488515i \(0.162461\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −12.0000 −0.565058
\(452\) 19.4891 0.916691
\(453\) 3.37228 0.158444
\(454\) −4.00000 −0.187729
\(455\) −4.62772 −0.216951
\(456\) −1.37228 −0.0642630
\(457\) −40.4674 −1.89298 −0.946492 0.322727i \(-0.895400\pi\)
−0.946492 + 0.322727i \(0.895400\pi\)
\(458\) 6.00000 0.280362
\(459\) 1.37228 0.0640526
\(460\) 3.37228 0.157233
\(461\) 7.48913 0.348803 0.174402 0.984675i \(-0.444201\pi\)
0.174402 + 0.984675i \(0.444201\pi\)
\(462\) 4.62772 0.215301
\(463\) −19.7228 −0.916597 −0.458298 0.888798i \(-0.651541\pi\)
−0.458298 + 0.888798i \(0.651541\pi\)
\(464\) 6.00000 0.278543
\(465\) −2.74456 −0.127276
\(466\) 22.7446 1.05362
\(467\) 1.48913 0.0689085 0.0344543 0.999406i \(-0.489031\pi\)
0.0344543 + 0.999406i \(0.489031\pi\)
\(468\) 1.37228 0.0634337
\(469\) −9.25544 −0.427376
\(470\) 4.74456 0.218850
\(471\) −3.25544 −0.150003
\(472\) −14.7446 −0.678674
\(473\) 5.48913 0.252390
\(474\) 4.00000 0.183726
\(475\) −1.37228 −0.0629646
\(476\) 4.62772 0.212111
\(477\) 5.37228 0.245980
\(478\) 5.48913 0.251067
\(479\) −18.1168 −0.827780 −0.413890 0.910327i \(-0.635830\pi\)
−0.413890 + 0.910327i \(0.635830\pi\)
\(480\) 1.00000 0.0456435
\(481\) −1.37228 −0.0625706
\(482\) 0.510875 0.0232697
\(483\) 11.3723 0.517457
\(484\) −9.11684 −0.414402
\(485\) 13.4891 0.612510
\(486\) 1.00000 0.0453609
\(487\) −30.4674 −1.38061 −0.690304 0.723519i \(-0.742523\pi\)
−0.690304 + 0.723519i \(0.742523\pi\)
\(488\) 2.74456 0.124241
\(489\) −4.86141 −0.219840
\(490\) −4.37228 −0.197520
\(491\) 2.62772 0.118587 0.0592936 0.998241i \(-0.481115\pi\)
0.0592936 + 0.998241i \(0.481115\pi\)
\(492\) −8.74456 −0.394235
\(493\) −8.23369 −0.370827
\(494\) 1.88316 0.0847272
\(495\) −1.37228 −0.0616795
\(496\) 2.74456 0.123235
\(497\) −4.23369 −0.189907
\(498\) 0.627719 0.0281287
\(499\) 20.3505 0.911015 0.455507 0.890232i \(-0.349458\pi\)
0.455507 + 0.890232i \(0.349458\pi\)
\(500\) 1.00000 0.0447214
\(501\) 1.88316 0.0841332
\(502\) 26.7446 1.19367
\(503\) 5.48913 0.244748 0.122374 0.992484i \(-0.460949\pi\)
0.122374 + 0.992484i \(0.460949\pi\)
\(504\) 3.37228 0.150213
\(505\) 10.7446 0.478127
\(506\) 4.62772 0.205727
\(507\) 11.1168 0.493716
\(508\) 3.37228 0.149621
\(509\) 18.3505 0.813373 0.406687 0.913568i \(-0.366684\pi\)
0.406687 + 0.913568i \(0.366684\pi\)
\(510\) −1.37228 −0.0607656
\(511\) 13.8832 0.614155
\(512\) −1.00000 −0.0441942
\(513\) 1.37228 0.0605877
\(514\) 5.37228 0.236961
\(515\) −0.744563 −0.0328094
\(516\) 4.00000 0.176090
\(517\) 6.51087 0.286348
\(518\) −3.37228 −0.148170
\(519\) 6.86141 0.301182
\(520\) −1.37228 −0.0601785
\(521\) −5.76631 −0.252627 −0.126313 0.991990i \(-0.540314\pi\)
−0.126313 + 0.991990i \(0.540314\pi\)
\(522\) −6.00000 −0.262613
\(523\) −6.97825 −0.305138 −0.152569 0.988293i \(-0.548755\pi\)
−0.152569 + 0.988293i \(0.548755\pi\)
\(524\) 0 0
\(525\) 3.37228 0.147178
\(526\) −10.2337 −0.446210
\(527\) −3.76631 −0.164063
\(528\) 1.37228 0.0597209
\(529\) −11.6277 −0.505553
\(530\) −5.37228 −0.233357
\(531\) 14.7446 0.639860
\(532\) 4.62772 0.200637
\(533\) 12.0000 0.519778
\(534\) 13.3723 0.578675
\(535\) 3.37228 0.145796
\(536\) −2.74456 −0.118547
\(537\) −5.48913 −0.236873
\(538\) −0.627719 −0.0270629
\(539\) −6.00000 −0.258438
\(540\) −1.00000 −0.0430331
\(541\) 4.86141 0.209008 0.104504 0.994524i \(-0.466674\pi\)
0.104504 + 0.994524i \(0.466674\pi\)
\(542\) −13.4891 −0.579408
\(543\) 20.9783 0.900263
\(544\) 1.37228 0.0588361
\(545\) 4.62772 0.198230
\(546\) −4.62772 −0.198048
\(547\) −2.11684 −0.0905097 −0.0452549 0.998975i \(-0.514410\pi\)
−0.0452549 + 0.998975i \(0.514410\pi\)
\(548\) 10.7446 0.458985
\(549\) −2.74456 −0.117135
\(550\) 1.37228 0.0585143
\(551\) −8.23369 −0.350767
\(552\) 3.37228 0.143534
\(553\) −13.4891 −0.573616
\(554\) −25.6060 −1.08789
\(555\) 1.00000 0.0424476
\(556\) 2.74456 0.116395
\(557\) 40.7446 1.72640 0.863201 0.504860i \(-0.168456\pi\)
0.863201 + 0.504860i \(0.168456\pi\)
\(558\) −2.74456 −0.116187
\(559\) −5.48913 −0.232165
\(560\) −3.37228 −0.142505
\(561\) −1.88316 −0.0795069
\(562\) −1.37228 −0.0578862
\(563\) 13.2554 0.558650 0.279325 0.960197i \(-0.409889\pi\)
0.279325 + 0.960197i \(0.409889\pi\)
\(564\) 4.74456 0.199782
\(565\) 19.4891 0.819914
\(566\) 11.6060 0.487835
\(567\) −3.37228 −0.141623
\(568\) −1.25544 −0.0526770
\(569\) −32.3505 −1.35620 −0.678102 0.734967i \(-0.737197\pi\)
−0.678102 + 0.734967i \(0.737197\pi\)
\(570\) −1.37228 −0.0574785
\(571\) −1.25544 −0.0525384 −0.0262692 0.999655i \(-0.508363\pi\)
−0.0262692 + 0.999655i \(0.508363\pi\)
\(572\) −1.88316 −0.0787387
\(573\) 19.3723 0.809289
\(574\) 29.4891 1.23085
\(575\) 3.37228 0.140634
\(576\) 1.00000 0.0416667
\(577\) 26.7446 1.11339 0.556695 0.830717i \(-0.312069\pi\)
0.556695 + 0.830717i \(0.312069\pi\)
\(578\) 15.1168 0.628778
\(579\) 1.25544 0.0521742
\(580\) 6.00000 0.249136
\(581\) −2.11684 −0.0878215
\(582\) 13.4891 0.559142
\(583\) −7.37228 −0.305329
\(584\) 4.11684 0.170356
\(585\) 1.37228 0.0567368
\(586\) 4.11684 0.170065
\(587\) 1.48913 0.0614628 0.0307314 0.999528i \(-0.490216\pi\)
0.0307314 + 0.999528i \(0.490216\pi\)
\(588\) −4.37228 −0.180310
\(589\) −3.76631 −0.155188
\(590\) −14.7446 −0.607024
\(591\) 18.8614 0.775855
\(592\) −1.00000 −0.0410997
\(593\) −22.9783 −0.943604 −0.471802 0.881705i \(-0.656396\pi\)
−0.471802 + 0.881705i \(0.656396\pi\)
\(594\) −1.37228 −0.0563054
\(595\) 4.62772 0.189718
\(596\) 9.25544 0.379117
\(597\) −8.00000 −0.327418
\(598\) −4.62772 −0.189241
\(599\) −22.9783 −0.938866 −0.469433 0.882968i \(-0.655542\pi\)
−0.469433 + 0.882968i \(0.655542\pi\)
\(600\) 1.00000 0.0408248
\(601\) 30.8614 1.25886 0.629432 0.777056i \(-0.283288\pi\)
0.629432 + 0.777056i \(0.283288\pi\)
\(602\) −13.4891 −0.549776
\(603\) 2.74456 0.111767
\(604\) −3.37228 −0.137216
\(605\) −9.11684 −0.370652
\(606\) 10.7446 0.436468
\(607\) 38.4674 1.56134 0.780671 0.624942i \(-0.214877\pi\)
0.780671 + 0.624942i \(0.214877\pi\)
\(608\) 1.37228 0.0556534
\(609\) 20.2337 0.819910
\(610\) 2.74456 0.111124
\(611\) −6.51087 −0.263402
\(612\) −1.37228 −0.0554712
\(613\) 30.4674 1.23057 0.615283 0.788306i \(-0.289042\pi\)
0.615283 + 0.788306i \(0.289042\pi\)
\(614\) 10.7446 0.433615
\(615\) −8.74456 −0.352615
\(616\) −4.62772 −0.186456
\(617\) 32.2337 1.29768 0.648840 0.760925i \(-0.275255\pi\)
0.648840 + 0.760925i \(0.275255\pi\)
\(618\) −0.744563 −0.0299507
\(619\) −26.9783 −1.08435 −0.542174 0.840266i \(-0.682399\pi\)
−0.542174 + 0.840266i \(0.682399\pi\)
\(620\) 2.74456 0.110224
\(621\) −3.37228 −0.135325
\(622\) 8.00000 0.320771
\(623\) −45.0951 −1.80670
\(624\) −1.37228 −0.0549352
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) −1.88316 −0.0752060
\(628\) 3.25544 0.129906
\(629\) 1.37228 0.0547164
\(630\) 3.37228 0.134355
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −4.00000 −0.159111
\(633\) 9.25544 0.367871
\(634\) 16.9783 0.674292
\(635\) 3.37228 0.133825
\(636\) −5.37228 −0.213025
\(637\) 6.00000 0.237729
\(638\) 8.23369 0.325975
\(639\) 1.25544 0.0496643
\(640\) −1.00000 −0.0395285
\(641\) 43.7228 1.72695 0.863474 0.504394i \(-0.168284\pi\)
0.863474 + 0.504394i \(0.168284\pi\)
\(642\) 3.37228 0.133093
\(643\) 23.6060 0.930929 0.465464 0.885067i \(-0.345887\pi\)
0.465464 + 0.885067i \(0.345887\pi\)
\(644\) −11.3723 −0.448131
\(645\) 4.00000 0.157500
\(646\) −1.88316 −0.0740918
\(647\) −39.6060 −1.55707 −0.778536 0.627600i \(-0.784037\pi\)
−0.778536 + 0.627600i \(0.784037\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −20.2337 −0.794242
\(650\) −1.37228 −0.0538253
\(651\) 9.25544 0.362749
\(652\) 4.86141 0.190387
\(653\) 19.7228 0.771813 0.385907 0.922538i \(-0.373889\pi\)
0.385907 + 0.922538i \(0.373889\pi\)
\(654\) 4.62772 0.180958
\(655\) 0 0
\(656\) 8.74456 0.341418
\(657\) −4.11684 −0.160613
\(658\) −16.0000 −0.623745
\(659\) −2.23369 −0.0870121 −0.0435061 0.999053i \(-0.513853\pi\)
−0.0435061 + 0.999053i \(0.513853\pi\)
\(660\) 1.37228 0.0534160
\(661\) −26.1168 −1.01583 −0.507914 0.861408i \(-0.669583\pi\)
−0.507914 + 0.861408i \(0.669583\pi\)
\(662\) 8.74456 0.339867
\(663\) 1.88316 0.0731357
\(664\) −0.627719 −0.0243602
\(665\) 4.62772 0.179455
\(666\) 1.00000 0.0387492
\(667\) 20.2337 0.783452
\(668\) −1.88316 −0.0728615
\(669\) −18.9783 −0.733742
\(670\) −2.74456 −0.106032
\(671\) 3.76631 0.145397
\(672\) −3.37228 −0.130089
\(673\) 2.86141 0.110299 0.0551496 0.998478i \(-0.482436\pi\)
0.0551496 + 0.998478i \(0.482436\pi\)
\(674\) 4.11684 0.158575
\(675\) −1.00000 −0.0384900
\(676\) −11.1168 −0.427571
\(677\) −2.86141 −0.109973 −0.0549864 0.998487i \(-0.517512\pi\)
−0.0549864 + 0.998487i \(0.517512\pi\)
\(678\) 19.4891 0.748475
\(679\) −45.4891 −1.74571
\(680\) 1.37228 0.0526246
\(681\) −4.00000 −0.153280
\(682\) 3.76631 0.144220
\(683\) 30.9783 1.18535 0.592675 0.805442i \(-0.298072\pi\)
0.592675 + 0.805442i \(0.298072\pi\)
\(684\) −1.37228 −0.0524705
\(685\) 10.7446 0.410529
\(686\) −8.86141 −0.338330
\(687\) 6.00000 0.228914
\(688\) −4.00000 −0.152499
\(689\) 7.37228 0.280862
\(690\) 3.37228 0.128381
\(691\) 22.7446 0.865244 0.432622 0.901575i \(-0.357588\pi\)
0.432622 + 0.901575i \(0.357588\pi\)
\(692\) −6.86141 −0.260832
\(693\) 4.62772 0.175792
\(694\) −1.48913 −0.0565264
\(695\) 2.74456 0.104107
\(696\) 6.00000 0.227429
\(697\) −12.0000 −0.454532
\(698\) −16.7446 −0.633791
\(699\) 22.7446 0.860278
\(700\) −3.37228 −0.127460
\(701\) −47.7228 −1.80247 −0.901233 0.433335i \(-0.857337\pi\)
−0.901233 + 0.433335i \(0.857337\pi\)
\(702\) 1.37228 0.0517934
\(703\) 1.37228 0.0517566
\(704\) −1.37228 −0.0517198
\(705\) 4.74456 0.178691
\(706\) −3.48913 −0.131315
\(707\) −36.2337 −1.36271
\(708\) −14.7446 −0.554135
\(709\) −20.6277 −0.774690 −0.387345 0.921935i \(-0.626608\pi\)
−0.387345 + 0.921935i \(0.626608\pi\)
\(710\) −1.25544 −0.0471157
\(711\) 4.00000 0.150012
\(712\) −13.3723 −0.501147
\(713\) 9.25544 0.346619
\(714\) 4.62772 0.173188
\(715\) −1.88316 −0.0704260
\(716\) 5.48913 0.205138
\(717\) 5.48913 0.204995
\(718\) −32.4674 −1.21167
\(719\) −13.7228 −0.511775 −0.255887 0.966707i \(-0.582368\pi\)
−0.255887 + 0.966707i \(0.582368\pi\)
\(720\) 1.00000 0.0372678
\(721\) 2.51087 0.0935099
\(722\) 17.1168 0.637023
\(723\) 0.510875 0.0189996
\(724\) −20.9783 −0.779651
\(725\) 6.00000 0.222834
\(726\) −9.11684 −0.338358
\(727\) 8.97825 0.332985 0.166492 0.986043i \(-0.446756\pi\)
0.166492 + 0.986043i \(0.446756\pi\)
\(728\) 4.62772 0.171515
\(729\) 1.00000 0.0370370
\(730\) 4.11684 0.152371
\(731\) 5.48913 0.203023
\(732\) 2.74456 0.101442
\(733\) 3.48913 0.128874 0.0644369 0.997922i \(-0.479475\pi\)
0.0644369 + 0.997922i \(0.479475\pi\)
\(734\) −20.6277 −0.761383
\(735\) −4.37228 −0.161274
\(736\) −3.37228 −0.124304
\(737\) −3.76631 −0.138734
\(738\) −8.74456 −0.321892
\(739\) 25.7228 0.946229 0.473114 0.881001i \(-0.343130\pi\)
0.473114 + 0.881001i \(0.343130\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 1.88316 0.0691795
\(742\) 18.1168 0.665090
\(743\) 22.4674 0.824248 0.412124 0.911128i \(-0.364787\pi\)
0.412124 + 0.911128i \(0.364787\pi\)
\(744\) 2.74456 0.100621
\(745\) 9.25544 0.339093
\(746\) −3.48913 −0.127746
\(747\) 0.627719 0.0229670
\(748\) 1.88316 0.0688550
\(749\) −11.3723 −0.415534
\(750\) 1.00000 0.0365148
\(751\) −10.5109 −0.383547 −0.191774 0.981439i \(-0.561424\pi\)
−0.191774 + 0.981439i \(0.561424\pi\)
\(752\) −4.74456 −0.173016
\(753\) 26.7446 0.974626
\(754\) −8.23369 −0.299853
\(755\) −3.37228 −0.122730
\(756\) 3.37228 0.122649
\(757\) 44.1168 1.60345 0.801727 0.597690i \(-0.203915\pi\)
0.801727 + 0.597690i \(0.203915\pi\)
\(758\) −24.2337 −0.880207
\(759\) 4.62772 0.167976
\(760\) 1.37228 0.0497779
\(761\) −12.5109 −0.453519 −0.226759 0.973951i \(-0.572813\pi\)
−0.226759 + 0.973951i \(0.572813\pi\)
\(762\) 3.37228 0.122165
\(763\) −15.6060 −0.564974
\(764\) −19.3723 −0.700865
\(765\) −1.37228 −0.0496149
\(766\) −27.6060 −0.997444
\(767\) 20.2337 0.730596
\(768\) −1.00000 −0.0360844
\(769\) −51.4891 −1.85675 −0.928373 0.371651i \(-0.878792\pi\)
−0.928373 + 0.371651i \(0.878792\pi\)
\(770\) −4.62772 −0.166771
\(771\) 5.37228 0.193478
\(772\) −1.25544 −0.0451842
\(773\) 20.1168 0.723553 0.361776 0.932265i \(-0.382170\pi\)
0.361776 + 0.932265i \(0.382170\pi\)
\(774\) 4.00000 0.143777
\(775\) 2.74456 0.0985876
\(776\) −13.4891 −0.484231
\(777\) −3.37228 −0.120980
\(778\) −23.4891 −0.842126
\(779\) −12.0000 −0.429945
\(780\) −1.37228 −0.0491356
\(781\) −1.72281 −0.0616471
\(782\) 4.62772 0.165487
\(783\) −6.00000 −0.214423
\(784\) 4.37228 0.156153
\(785\) 3.25544 0.116192
\(786\) 0 0
\(787\) 10.7446 0.383002 0.191501 0.981492i \(-0.438664\pi\)
0.191501 + 0.981492i \(0.438664\pi\)
\(788\) −18.8614 −0.671910
\(789\) −10.2337 −0.364329
\(790\) −4.00000 −0.142314
\(791\) −65.7228 −2.33683
\(792\) 1.37228 0.0487619
\(793\) −3.76631 −0.133746
\(794\) −20.7446 −0.736197
\(795\) −5.37228 −0.190535
\(796\) 8.00000 0.283552
\(797\) 4.51087 0.159783 0.0798917 0.996804i \(-0.474543\pi\)
0.0798917 + 0.996804i \(0.474543\pi\)
\(798\) 4.62772 0.163819
\(799\) 6.51087 0.230338
\(800\) −1.00000 −0.0353553
\(801\) 13.3723 0.472486
\(802\) 36.1168 1.27533
\(803\) 5.64947 0.199365
\(804\) −2.74456 −0.0967933
\(805\) −11.3723 −0.400820
\(806\) −3.76631 −0.132663
\(807\) −0.627719 −0.0220967
\(808\) −10.7446 −0.377992
\(809\) −24.1168 −0.847903 −0.423952 0.905685i \(-0.639357\pi\)
−0.423952 + 0.905685i \(0.639357\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −20.2337 −0.710063
\(813\) −13.4891 −0.473084
\(814\) −1.37228 −0.0480984
\(815\) 4.86141 0.170288
\(816\) 1.37228 0.0480395
\(817\) 5.48913 0.192040
\(818\) 8.74456 0.305746
\(819\) −4.62772 −0.161706
\(820\) 8.74456 0.305373
\(821\) 26.3505 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(822\) 10.7446 0.374760
\(823\) 25.8832 0.902230 0.451115 0.892466i \(-0.351026\pi\)
0.451115 + 0.892466i \(0.351026\pi\)
\(824\) 0.744563 0.0259381
\(825\) 1.37228 0.0477767
\(826\) 49.7228 1.73008
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 3.37228 0.117195
\(829\) −35.8397 −1.24476 −0.622381 0.782714i \(-0.713835\pi\)
−0.622381 + 0.782714i \(0.713835\pi\)
\(830\) −0.627719 −0.0217884
\(831\) −25.6060 −0.888261
\(832\) 1.37228 0.0475753
\(833\) −6.00000 −0.207888
\(834\) 2.74456 0.0950364
\(835\) −1.88316 −0.0651693
\(836\) 1.88316 0.0651303
\(837\) −2.74456 −0.0948660
\(838\) −7.88316 −0.272319
\(839\) −9.25544 −0.319533 −0.159767 0.987155i \(-0.551074\pi\)
−0.159767 + 0.987155i \(0.551074\pi\)
\(840\) −3.37228 −0.116355
\(841\) 7.00000 0.241379
\(842\) −24.2337 −0.835148
\(843\) −1.37228 −0.0472639
\(844\) −9.25544 −0.318585
\(845\) −11.1168 −0.382431
\(846\) 4.74456 0.163121
\(847\) 30.7446 1.05640
\(848\) 5.37228 0.184485
\(849\) 11.6060 0.398316
\(850\) 1.37228 0.0470689
\(851\) −3.37228 −0.115600
\(852\) −1.25544 −0.0430106
\(853\) 40.3505 1.38158 0.690788 0.723057i \(-0.257264\pi\)
0.690788 + 0.723057i \(0.257264\pi\)
\(854\) −9.25544 −0.316715
\(855\) −1.37228 −0.0469310
\(856\) −3.37228 −0.115262
\(857\) 17.8397 0.609391 0.304696 0.952450i \(-0.401445\pi\)
0.304696 + 0.952450i \(0.401445\pi\)
\(858\) −1.88316 −0.0642899
\(859\) 17.8397 0.608681 0.304341 0.952563i \(-0.401564\pi\)
0.304341 + 0.952563i \(0.401564\pi\)
\(860\) −4.00000 −0.136399
\(861\) 29.4891 1.00499
\(862\) 31.6060 1.07650
\(863\) 3.25544 0.110816 0.0554082 0.998464i \(-0.482354\pi\)
0.0554082 + 0.998464i \(0.482354\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.86141 −0.233295
\(866\) 10.8614 0.369086
\(867\) 15.1168 0.513395
\(868\) −9.25544 −0.314150
\(869\) −5.48913 −0.186206
\(870\) 6.00000 0.203419
\(871\) 3.76631 0.127617
\(872\) −4.62772 −0.156714
\(873\) 13.4891 0.456538
\(874\) 4.62772 0.156535
\(875\) −3.37228 −0.114004
\(876\) 4.11684 0.139095
\(877\) −54.2337 −1.83134 −0.915671 0.401929i \(-0.868340\pi\)
−0.915671 + 0.401929i \(0.868340\pi\)
\(878\) 32.0000 1.07995
\(879\) 4.11684 0.138858
\(880\) −1.37228 −0.0462596
\(881\) −14.2337 −0.479545 −0.239773 0.970829i \(-0.577073\pi\)
−0.239773 + 0.970829i \(0.577073\pi\)
\(882\) −4.37228 −0.147222
\(883\) −50.1168 −1.68657 −0.843283 0.537470i \(-0.819380\pi\)
−0.843283 + 0.537470i \(0.819380\pi\)
\(884\) −1.88316 −0.0633374
\(885\) −14.7446 −0.495633
\(886\) 4.00000 0.134383
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −11.3723 −0.381414
\(890\) −13.3723 −0.448240
\(891\) −1.37228 −0.0459732
\(892\) 18.9783 0.635439
\(893\) 6.51087 0.217878
\(894\) 9.25544 0.309548
\(895\) 5.48913 0.183481
\(896\) 3.37228 0.112660
\(897\) −4.62772 −0.154515
\(898\) −36.9783 −1.23398
\(899\) 16.4674 0.549218
\(900\) 1.00000 0.0333333
\(901\) −7.37228 −0.245606
\(902\) 12.0000 0.399556
\(903\) −13.4891 −0.448890
\(904\) −19.4891 −0.648199
\(905\) −20.9783 −0.697341
\(906\) −3.37228 −0.112037
\(907\) −3.60597 −0.119734 −0.0598671 0.998206i \(-0.519068\pi\)
−0.0598671 + 0.998206i \(0.519068\pi\)
\(908\) 4.00000 0.132745
\(909\) 10.7446 0.356375
\(910\) 4.62772 0.153407
\(911\) 34.9783 1.15888 0.579441 0.815014i \(-0.303271\pi\)
0.579441 + 0.815014i \(0.303271\pi\)
\(912\) 1.37228 0.0454408
\(913\) −0.861407 −0.0285084
\(914\) 40.4674 1.33854
\(915\) 2.74456 0.0907324
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) −1.37228 −0.0452920
\(919\) −34.5109 −1.13841 −0.569204 0.822196i \(-0.692749\pi\)
−0.569204 + 0.822196i \(0.692749\pi\)
\(920\) −3.37228 −0.111181
\(921\) 10.7446 0.354045
\(922\) −7.48913 −0.246641
\(923\) 1.72281 0.0567071
\(924\) −4.62772 −0.152241
\(925\) −1.00000 −0.0328798
\(926\) 19.7228 0.648132
\(927\) −0.744563 −0.0244546
\(928\) −6.00000 −0.196960
\(929\) −12.5109 −0.410468 −0.205234 0.978713i \(-0.565796\pi\)
−0.205234 + 0.978713i \(0.565796\pi\)
\(930\) 2.74456 0.0899978
\(931\) −6.00000 −0.196642
\(932\) −22.7446 −0.745023
\(933\) 8.00000 0.261908
\(934\) −1.48913 −0.0487257
\(935\) 1.88316 0.0615858
\(936\) −1.37228 −0.0448544
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 9.25544 0.302201
\(939\) −8.00000 −0.261070
\(940\) −4.74456 −0.154751
\(941\) 15.7663 0.513967 0.256984 0.966416i \(-0.417271\pi\)
0.256984 + 0.966416i \(0.417271\pi\)
\(942\) 3.25544 0.106068
\(943\) 29.4891 0.960298
\(944\) 14.7446 0.479895
\(945\) 3.37228 0.109700
\(946\) −5.48913 −0.178467
\(947\) −28.4674 −0.925065 −0.462533 0.886602i \(-0.653059\pi\)
−0.462533 + 0.886602i \(0.653059\pi\)
\(948\) −4.00000 −0.129914
\(949\) −5.64947 −0.183389
\(950\) 1.37228 0.0445227
\(951\) 16.9783 0.550557
\(952\) −4.62772 −0.149985
\(953\) −50.7446 −1.64378 −0.821889 0.569648i \(-0.807080\pi\)
−0.821889 + 0.569648i \(0.807080\pi\)
\(954\) −5.37228 −0.173934
\(955\) −19.3723 −0.626872
\(956\) −5.48913 −0.177531
\(957\) 8.23369 0.266157
\(958\) 18.1168 0.585329
\(959\) −36.2337 −1.17005
\(960\) −1.00000 −0.0322749
\(961\) −23.4674 −0.757012
\(962\) 1.37228 0.0442441
\(963\) 3.37228 0.108670
\(964\) −0.510875 −0.0164542
\(965\) −1.25544 −0.0404140
\(966\) −11.3723 −0.365897
\(967\) −16.9783 −0.545984 −0.272992 0.962016i \(-0.588013\pi\)
−0.272992 + 0.962016i \(0.588013\pi\)
\(968\) 9.11684 0.293026
\(969\) −1.88316 −0.0604957
\(970\) −13.4891 −0.433110
\(971\) 50.2337 1.61208 0.806038 0.591864i \(-0.201608\pi\)
0.806038 + 0.591864i \(0.201608\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −9.25544 −0.296716
\(974\) 30.4674 0.976238
\(975\) −1.37228 −0.0439482
\(976\) −2.74456 −0.0878513
\(977\) 25.1386 0.804255 0.402127 0.915584i \(-0.368271\pi\)
0.402127 + 0.915584i \(0.368271\pi\)
\(978\) 4.86141 0.155451
\(979\) −18.3505 −0.586486
\(980\) 4.37228 0.139667
\(981\) 4.62772 0.147752
\(982\) −2.62772 −0.0838539
\(983\) −12.5109 −0.399035 −0.199517 0.979894i \(-0.563937\pi\)
−0.199517 + 0.979894i \(0.563937\pi\)
\(984\) 8.74456 0.278766
\(985\) −18.8614 −0.600974
\(986\) 8.23369 0.262214
\(987\) −16.0000 −0.509286
\(988\) −1.88316 −0.0599112
\(989\) −13.4891 −0.428929
\(990\) 1.37228 0.0436140
\(991\) 22.9783 0.729928 0.364964 0.931022i \(-0.381081\pi\)
0.364964 + 0.931022i \(0.381081\pi\)
\(992\) −2.74456 −0.0871400
\(993\) 8.74456 0.277500
\(994\) 4.23369 0.134284
\(995\) 8.00000 0.253617
\(996\) −0.627719 −0.0198900
\(997\) 16.1168 0.510426 0.255213 0.966885i \(-0.417854\pi\)
0.255213 + 0.966885i \(0.417854\pi\)
\(998\) −20.3505 −0.644185
\(999\) 1.00000 0.0316386
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 1110.2.a.p.1.1 2
3.2 odd 2 3330.2.a.be.1.1 2
4.3 odd 2 8880.2.a.br.1.2 2
5.4 even 2 5550.2.a.ca.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.p.1.1 2 1.1 even 1 trivial
3330.2.a.be.1.1 2 3.2 odd 2
5550.2.a.ca.1.2 2 5.4 even 2
8880.2.a.br.1.2 2 4.3 odd 2