Properties

Label 370.2.a.f.1.2
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
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 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.37228 q^{11} +2.00000 q^{12} -4.74456 q^{13} +1.37228 q^{14} +2.00000 q^{15} +1.00000 q^{16} -5.37228 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.74456 q^{21} -3.37228 q^{22} +6.74456 q^{23} +2.00000 q^{24} +1.00000 q^{25} -4.74456 q^{26} -4.00000 q^{27} +1.37228 q^{28} +8.11684 q^{29} +2.00000 q^{30} -2.62772 q^{31} +1.00000 q^{32} -6.74456 q^{33} -5.37228 q^{34} +1.37228 q^{35} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} -9.48913 q^{39} +1.00000 q^{40} +5.37228 q^{41} +2.74456 q^{42} +7.37228 q^{43} -3.37228 q^{44} +1.00000 q^{45} +6.74456 q^{46} -8.74456 q^{47} +2.00000 q^{48} -5.11684 q^{49} +1.00000 q^{50} -10.7446 q^{51} -4.74456 q^{52} +1.37228 q^{53} -4.00000 q^{54} -3.37228 q^{55} +1.37228 q^{56} -4.00000 q^{57} +8.11684 q^{58} +12.7446 q^{59} +2.00000 q^{60} -5.37228 q^{61} -2.62772 q^{62} +1.37228 q^{63} +1.00000 q^{64} -4.74456 q^{65} -6.74456 q^{66} +4.74456 q^{67} -5.37228 q^{68} +13.4891 q^{69} +1.37228 q^{70} -6.74456 q^{71} +1.00000 q^{72} +8.74456 q^{73} +1.00000 q^{74} +2.00000 q^{75} -2.00000 q^{76} -4.62772 q^{77} -9.48913 q^{78} -4.74456 q^{79} +1.00000 q^{80} -11.0000 q^{81} +5.37228 q^{82} -0.744563 q^{83} +2.74456 q^{84} -5.37228 q^{85} +7.37228 q^{86} +16.2337 q^{87} -3.37228 q^{88} +10.0000 q^{89} +1.00000 q^{90} -6.51087 q^{91} +6.74456 q^{92} -5.25544 q^{93} -8.74456 q^{94} -2.00000 q^{95} +2.00000 q^{96} -0.116844 q^{97} -5.11684 q^{98} -3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 4 q^{12} + 2 q^{13} - 3 q^{14} + 4 q^{15} + 2 q^{16} - 5 q^{17} + 2 q^{18} - 4 q^{19} + 2 q^{20} - 6 q^{21} - q^{22} + 2 q^{23} + 4 q^{24} + 2 q^{25} + 2 q^{26} - 8 q^{27} - 3 q^{28} - q^{29} + 4 q^{30} - 11 q^{31} + 2 q^{32} - 2 q^{33} - 5 q^{34} - 3 q^{35} + 2 q^{36} + 2 q^{37} - 4 q^{38} + 4 q^{39} + 2 q^{40} + 5 q^{41} - 6 q^{42} + 9 q^{43} - q^{44} + 2 q^{45} + 2 q^{46} - 6 q^{47} + 4 q^{48} + 7 q^{49} + 2 q^{50} - 10 q^{51} + 2 q^{52} - 3 q^{53} - 8 q^{54} - q^{55} - 3 q^{56} - 8 q^{57} - q^{58} + 14 q^{59} + 4 q^{60} - 5 q^{61} - 11 q^{62} - 3 q^{63} + 2 q^{64} + 2 q^{65} - 2 q^{66} - 2 q^{67} - 5 q^{68} + 4 q^{69} - 3 q^{70} - 2 q^{71} + 2 q^{72} + 6 q^{73} + 2 q^{74} + 4 q^{75} - 4 q^{76} - 15 q^{77} + 4 q^{78} + 2 q^{79} + 2 q^{80} - 22 q^{81} + 5 q^{82} + 10 q^{83} - 6 q^{84} - 5 q^{85} + 9 q^{86} - 2 q^{87} - q^{88} + 20 q^{89} + 2 q^{90} - 36 q^{91} + 2 q^{92} - 22 q^{93} - 6 q^{94} - 4 q^{95} + 4 q^{96} + 17 q^{97} + 7 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) 1.37228 0.518674 0.259337 0.965787i \(-0.416496\pi\)
0.259337 + 0.965787i \(0.416496\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −3.37228 −1.01678 −0.508391 0.861127i \(-0.669759\pi\)
−0.508391 + 0.861127i \(0.669759\pi\)
\(12\) 2.00000 0.577350
\(13\) −4.74456 −1.31590 −0.657952 0.753059i \(-0.728577\pi\)
−0.657952 + 0.753059i \(0.728577\pi\)
\(14\) 1.37228 0.366758
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −5.37228 −1.30297 −0.651485 0.758662i \(-0.725854\pi\)
−0.651485 + 0.758662i \(0.725854\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.74456 0.598913
\(22\) −3.37228 −0.718973
\(23\) 6.74456 1.40634 0.703169 0.711022i \(-0.251768\pi\)
0.703169 + 0.711022i \(0.251768\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) −4.74456 −0.930485
\(27\) −4.00000 −0.769800
\(28\) 1.37228 0.259337
\(29\) 8.11684 1.50726 0.753630 0.657299i \(-0.228301\pi\)
0.753630 + 0.657299i \(0.228301\pi\)
\(30\) 2.00000 0.365148
\(31\) −2.62772 −0.471952 −0.235976 0.971759i \(-0.575829\pi\)
−0.235976 + 0.971759i \(0.575829\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.74456 −1.17408
\(34\) −5.37228 −0.921339
\(35\) 1.37228 0.231958
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) −9.48913 −1.51948
\(40\) 1.00000 0.158114
\(41\) 5.37228 0.839009 0.419505 0.907753i \(-0.362204\pi\)
0.419505 + 0.907753i \(0.362204\pi\)
\(42\) 2.74456 0.423495
\(43\) 7.37228 1.12426 0.562131 0.827048i \(-0.309982\pi\)
0.562131 + 0.827048i \(0.309982\pi\)
\(44\) −3.37228 −0.508391
\(45\) 1.00000 0.149071
\(46\) 6.74456 0.994432
\(47\) −8.74456 −1.27553 −0.637763 0.770233i \(-0.720140\pi\)
−0.637763 + 0.770233i \(0.720140\pi\)
\(48\) 2.00000 0.288675
\(49\) −5.11684 −0.730978
\(50\) 1.00000 0.141421
\(51\) −10.7446 −1.50454
\(52\) −4.74456 −0.657952
\(53\) 1.37228 0.188497 0.0942487 0.995549i \(-0.469955\pi\)
0.0942487 + 0.995549i \(0.469955\pi\)
\(54\) −4.00000 −0.544331
\(55\) −3.37228 −0.454718
\(56\) 1.37228 0.183379
\(57\) −4.00000 −0.529813
\(58\) 8.11684 1.06579
\(59\) 12.7446 1.65920 0.829600 0.558358i \(-0.188568\pi\)
0.829600 + 0.558358i \(0.188568\pi\)
\(60\) 2.00000 0.258199
\(61\) −5.37228 −0.687850 −0.343925 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) −2.62772 −0.333721
\(63\) 1.37228 0.172891
\(64\) 1.00000 0.125000
\(65\) −4.74456 −0.588491
\(66\) −6.74456 −0.830198
\(67\) 4.74456 0.579641 0.289820 0.957081i \(-0.406404\pi\)
0.289820 + 0.957081i \(0.406404\pi\)
\(68\) −5.37228 −0.651485
\(69\) 13.4891 1.62390
\(70\) 1.37228 0.164019
\(71\) −6.74456 −0.800432 −0.400216 0.916421i \(-0.631065\pi\)
−0.400216 + 0.916421i \(0.631065\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.74456 1.02347 0.511737 0.859142i \(-0.329002\pi\)
0.511737 + 0.859142i \(0.329002\pi\)
\(74\) 1.00000 0.116248
\(75\) 2.00000 0.230940
\(76\) −2.00000 −0.229416
\(77\) −4.62772 −0.527377
\(78\) −9.48913 −1.07443
\(79\) −4.74456 −0.533805 −0.266903 0.963724i \(-0.586000\pi\)
−0.266903 + 0.963724i \(0.586000\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 5.37228 0.593269
\(83\) −0.744563 −0.0817264 −0.0408632 0.999165i \(-0.513011\pi\)
−0.0408632 + 0.999165i \(0.513011\pi\)
\(84\) 2.74456 0.299456
\(85\) −5.37228 −0.582706
\(86\) 7.37228 0.794974
\(87\) 16.2337 1.74043
\(88\) −3.37228 −0.359486
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 1.00000 0.105409
\(91\) −6.51087 −0.682525
\(92\) 6.74456 0.703169
\(93\) −5.25544 −0.544963
\(94\) −8.74456 −0.901933
\(95\) −2.00000 −0.205196
\(96\) 2.00000 0.204124
\(97\) −0.116844 −0.0118637 −0.00593185 0.999982i \(-0.501888\pi\)
−0.00593185 + 0.999982i \(0.501888\pi\)
\(98\) −5.11684 −0.516879
\(99\) −3.37228 −0.338927
\(100\) 1.00000 0.100000
\(101\) −11.4891 −1.14321 −0.571605 0.820529i \(-0.693679\pi\)
−0.571605 + 0.820529i \(0.693679\pi\)
\(102\) −10.7446 −1.06387
\(103\) −9.48913 −0.934991 −0.467496 0.883995i \(-0.654844\pi\)
−0.467496 + 0.883995i \(0.654844\pi\)
\(104\) −4.74456 −0.465243
\(105\) 2.74456 0.267842
\(106\) 1.37228 0.133288
\(107\) −3.48913 −0.337306 −0.168653 0.985675i \(-0.553942\pi\)
−0.168653 + 0.985675i \(0.553942\pi\)
\(108\) −4.00000 −0.384900
\(109\) 0.116844 0.0111916 0.00559581 0.999984i \(-0.498219\pi\)
0.00559581 + 0.999984i \(0.498219\pi\)
\(110\) −3.37228 −0.321534
\(111\) 2.00000 0.189832
\(112\) 1.37228 0.129668
\(113\) 17.3723 1.63425 0.817123 0.576463i \(-0.195567\pi\)
0.817123 + 0.576463i \(0.195567\pi\)
\(114\) −4.00000 −0.374634
\(115\) 6.74456 0.628934
\(116\) 8.11684 0.753630
\(117\) −4.74456 −0.438635
\(118\) 12.7446 1.17323
\(119\) −7.37228 −0.675816
\(120\) 2.00000 0.182574
\(121\) 0.372281 0.0338438
\(122\) −5.37228 −0.486383
\(123\) 10.7446 0.968805
\(124\) −2.62772 −0.235976
\(125\) 1.00000 0.0894427
\(126\) 1.37228 0.122253
\(127\) −16.7446 −1.48584 −0.742920 0.669380i \(-0.766560\pi\)
−0.742920 + 0.669380i \(0.766560\pi\)
\(128\) 1.00000 0.0883883
\(129\) 14.7446 1.29819
\(130\) −4.74456 −0.416126
\(131\) −3.25544 −0.284429 −0.142214 0.989836i \(-0.545422\pi\)
−0.142214 + 0.989836i \(0.545422\pi\)
\(132\) −6.74456 −0.587039
\(133\) −2.74456 −0.237984
\(134\) 4.74456 0.409868
\(135\) −4.00000 −0.344265
\(136\) −5.37228 −0.460669
\(137\) 16.7446 1.43058 0.715292 0.698825i \(-0.246294\pi\)
0.715292 + 0.698825i \(0.246294\pi\)
\(138\) 13.4891 1.14827
\(139\) −1.88316 −0.159727 −0.0798636 0.996806i \(-0.525448\pi\)
−0.0798636 + 0.996806i \(0.525448\pi\)
\(140\) 1.37228 0.115979
\(141\) −17.4891 −1.47285
\(142\) −6.74456 −0.565991
\(143\) 16.0000 1.33799
\(144\) 1.00000 0.0833333
\(145\) 8.11684 0.674067
\(146\) 8.74456 0.723705
\(147\) −10.2337 −0.844060
\(148\) 1.00000 0.0821995
\(149\) −11.4891 −0.941226 −0.470613 0.882340i \(-0.655967\pi\)
−0.470613 + 0.882340i \(0.655967\pi\)
\(150\) 2.00000 0.163299
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) −2.00000 −0.162221
\(153\) −5.37228 −0.434323
\(154\) −4.62772 −0.372912
\(155\) −2.62772 −0.211063
\(156\) −9.48913 −0.759738
\(157\) 17.3723 1.38646 0.693229 0.720717i \(-0.256187\pi\)
0.693229 + 0.720717i \(0.256187\pi\)
\(158\) −4.74456 −0.377457
\(159\) 2.74456 0.217658
\(160\) 1.00000 0.0790569
\(161\) 9.25544 0.729431
\(162\) −11.0000 −0.864242
\(163\) 19.3723 1.51735 0.758677 0.651467i \(-0.225846\pi\)
0.758677 + 0.651467i \(0.225846\pi\)
\(164\) 5.37228 0.419505
\(165\) −6.74456 −0.525063
\(166\) −0.744563 −0.0577893
\(167\) 1.48913 0.115232 0.0576160 0.998339i \(-0.481650\pi\)
0.0576160 + 0.998339i \(0.481650\pi\)
\(168\) 2.74456 0.211748
\(169\) 9.51087 0.731606
\(170\) −5.37228 −0.412035
\(171\) −2.00000 −0.152944
\(172\) 7.37228 0.562131
\(173\) 22.8614 1.73812 0.869060 0.494706i \(-0.164724\pi\)
0.869060 + 0.494706i \(0.164724\pi\)
\(174\) 16.2337 1.23067
\(175\) 1.37228 0.103735
\(176\) −3.37228 −0.254195
\(177\) 25.4891 1.91588
\(178\) 10.0000 0.749532
\(179\) −26.2337 −1.96080 −0.980399 0.197023i \(-0.936873\pi\)
−0.980399 + 0.197023i \(0.936873\pi\)
\(180\) 1.00000 0.0745356
\(181\) 7.48913 0.556662 0.278331 0.960485i \(-0.410219\pi\)
0.278331 + 0.960485i \(0.410219\pi\)
\(182\) −6.51087 −0.482618
\(183\) −10.7446 −0.794261
\(184\) 6.74456 0.497216
\(185\) 1.00000 0.0735215
\(186\) −5.25544 −0.385347
\(187\) 18.1168 1.32483
\(188\) −8.74456 −0.637763
\(189\) −5.48913 −0.399275
\(190\) −2.00000 −0.145095
\(191\) 18.6277 1.34785 0.673927 0.738798i \(-0.264606\pi\)
0.673927 + 0.738798i \(0.264606\pi\)
\(192\) 2.00000 0.144338
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −0.116844 −0.00838891
\(195\) −9.48913 −0.679530
\(196\) −5.11684 −0.365489
\(197\) 27.4891 1.95852 0.979260 0.202610i \(-0.0649423\pi\)
0.979260 + 0.202610i \(0.0649423\pi\)
\(198\) −3.37228 −0.239658
\(199\) −11.2554 −0.797877 −0.398938 0.916978i \(-0.630621\pi\)
−0.398938 + 0.916978i \(0.630621\pi\)
\(200\) 1.00000 0.0707107
\(201\) 9.48913 0.669311
\(202\) −11.4891 −0.808372
\(203\) 11.1386 0.781776
\(204\) −10.7446 −0.752270
\(205\) 5.37228 0.375216
\(206\) −9.48913 −0.661139
\(207\) 6.74456 0.468780
\(208\) −4.74456 −0.328976
\(209\) 6.74456 0.466531
\(210\) 2.74456 0.189393
\(211\) −14.1168 −0.971844 −0.485922 0.874002i \(-0.661516\pi\)
−0.485922 + 0.874002i \(0.661516\pi\)
\(212\) 1.37228 0.0942487
\(213\) −13.4891 −0.924260
\(214\) −3.48913 −0.238512
\(215\) 7.37228 0.502785
\(216\) −4.00000 −0.272166
\(217\) −3.60597 −0.244789
\(218\) 0.116844 0.00791367
\(219\) 17.4891 1.18181
\(220\) −3.37228 −0.227359
\(221\) 25.4891 1.71458
\(222\) 2.00000 0.134231
\(223\) −1.37228 −0.0918948 −0.0459474 0.998944i \(-0.514631\pi\)
−0.0459474 + 0.998944i \(0.514631\pi\)
\(224\) 1.37228 0.0916894
\(225\) 1.00000 0.0666667
\(226\) 17.3723 1.15559
\(227\) −11.3723 −0.754805 −0.377402 0.926049i \(-0.623183\pi\)
−0.377402 + 0.926049i \(0.623183\pi\)
\(228\) −4.00000 −0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 6.74456 0.444723
\(231\) −9.25544 −0.608963
\(232\) 8.11684 0.532897
\(233\) 6.23369 0.408382 0.204191 0.978931i \(-0.434544\pi\)
0.204191 + 0.978931i \(0.434544\pi\)
\(234\) −4.74456 −0.310162
\(235\) −8.74456 −0.570432
\(236\) 12.7446 0.829600
\(237\) −9.48913 −0.616385
\(238\) −7.37228 −0.477874
\(239\) 17.6060 1.13884 0.569418 0.822048i \(-0.307169\pi\)
0.569418 + 0.822048i \(0.307169\pi\)
\(240\) 2.00000 0.129099
\(241\) −10.2337 −0.659210 −0.329605 0.944119i \(-0.606916\pi\)
−0.329605 + 0.944119i \(0.606916\pi\)
\(242\) 0.372281 0.0239311
\(243\) −10.0000 −0.641500
\(244\) −5.37228 −0.343925
\(245\) −5.11684 −0.326903
\(246\) 10.7446 0.685048
\(247\) 9.48913 0.603779
\(248\) −2.62772 −0.166860
\(249\) −1.48913 −0.0943695
\(250\) 1.00000 0.0632456
\(251\) 11.4891 0.725187 0.362594 0.931947i \(-0.381891\pi\)
0.362594 + 0.931947i \(0.381891\pi\)
\(252\) 1.37228 0.0864456
\(253\) −22.7446 −1.42994
\(254\) −16.7446 −1.05065
\(255\) −10.7446 −0.672851
\(256\) 1.00000 0.0625000
\(257\) −20.9783 −1.30859 −0.654294 0.756241i \(-0.727034\pi\)
−0.654294 + 0.756241i \(0.727034\pi\)
\(258\) 14.7446 0.917956
\(259\) 1.37228 0.0852694
\(260\) −4.74456 −0.294245
\(261\) 8.11684 0.502420
\(262\) −3.25544 −0.201122
\(263\) 0.116844 0.00720491 0.00360245 0.999994i \(-0.498853\pi\)
0.00360245 + 0.999994i \(0.498853\pi\)
\(264\) −6.74456 −0.415099
\(265\) 1.37228 0.0842986
\(266\) −2.74456 −0.168280
\(267\) 20.0000 1.22398
\(268\) 4.74456 0.289820
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −4.00000 −0.243432
\(271\) −6.74456 −0.409703 −0.204852 0.978793i \(-0.565671\pi\)
−0.204852 + 0.978793i \(0.565671\pi\)
\(272\) −5.37228 −0.325742
\(273\) −13.0217 −0.788112
\(274\) 16.7446 1.01158
\(275\) −3.37228 −0.203356
\(276\) 13.4891 0.811950
\(277\) 0.744563 0.0447364 0.0223682 0.999750i \(-0.492879\pi\)
0.0223682 + 0.999750i \(0.492879\pi\)
\(278\) −1.88316 −0.112944
\(279\) −2.62772 −0.157317
\(280\) 1.37228 0.0820095
\(281\) 24.7446 1.47614 0.738068 0.674726i \(-0.235738\pi\)
0.738068 + 0.674726i \(0.235738\pi\)
\(282\) −17.4891 −1.04146
\(283\) 5.48913 0.326295 0.163147 0.986602i \(-0.447835\pi\)
0.163147 + 0.986602i \(0.447835\pi\)
\(284\) −6.74456 −0.400216
\(285\) −4.00000 −0.236940
\(286\) 16.0000 0.946100
\(287\) 7.37228 0.435172
\(288\) 1.00000 0.0589256
\(289\) 11.8614 0.697730
\(290\) 8.11684 0.476637
\(291\) −0.233688 −0.0136990
\(292\) 8.74456 0.511737
\(293\) −18.8614 −1.10190 −0.550948 0.834540i \(-0.685734\pi\)
−0.550948 + 0.834540i \(0.685734\pi\)
\(294\) −10.2337 −0.596841
\(295\) 12.7446 0.742017
\(296\) 1.00000 0.0581238
\(297\) 13.4891 0.782718
\(298\) −11.4891 −0.665547
\(299\) −32.0000 −1.85061
\(300\) 2.00000 0.115470
\(301\) 10.1168 0.583125
\(302\) −20.0000 −1.15087
\(303\) −22.9783 −1.32007
\(304\) −2.00000 −0.114708
\(305\) −5.37228 −0.307616
\(306\) −5.37228 −0.307113
\(307\) −23.4891 −1.34060 −0.670298 0.742092i \(-0.733834\pi\)
−0.670298 + 0.742092i \(0.733834\pi\)
\(308\) −4.62772 −0.263689
\(309\) −18.9783 −1.07963
\(310\) −2.62772 −0.149244
\(311\) −1.37228 −0.0778149 −0.0389075 0.999243i \(-0.512388\pi\)
−0.0389075 + 0.999243i \(0.512388\pi\)
\(312\) −9.48913 −0.537216
\(313\) −3.48913 −0.197217 −0.0986085 0.995126i \(-0.531439\pi\)
−0.0986085 + 0.995126i \(0.531439\pi\)
\(314\) 17.3723 0.980375
\(315\) 1.37228 0.0773193
\(316\) −4.74456 −0.266903
\(317\) 25.3723 1.42505 0.712525 0.701647i \(-0.247552\pi\)
0.712525 + 0.701647i \(0.247552\pi\)
\(318\) 2.74456 0.153907
\(319\) −27.3723 −1.53255
\(320\) 1.00000 0.0559017
\(321\) −6.97825 −0.389488
\(322\) 9.25544 0.515785
\(323\) 10.7446 0.597843
\(324\) −11.0000 −0.611111
\(325\) −4.74456 −0.263181
\(326\) 19.3723 1.07293
\(327\) 0.233688 0.0129230
\(328\) 5.37228 0.296635
\(329\) −12.0000 −0.661581
\(330\) −6.74456 −0.371276
\(331\) 23.4891 1.29108 0.645540 0.763727i \(-0.276633\pi\)
0.645540 + 0.763727i \(0.276633\pi\)
\(332\) −0.744563 −0.0408632
\(333\) 1.00000 0.0547997
\(334\) 1.48913 0.0814813
\(335\) 4.74456 0.259223
\(336\) 2.74456 0.149728
\(337\) −7.25544 −0.395229 −0.197614 0.980280i \(-0.563319\pi\)
−0.197614 + 0.980280i \(0.563319\pi\)
\(338\) 9.51087 0.517323
\(339\) 34.7446 1.88707
\(340\) −5.37228 −0.291353
\(341\) 8.86141 0.479872
\(342\) −2.00000 −0.108148
\(343\) −16.6277 −0.897812
\(344\) 7.37228 0.397487
\(345\) 13.4891 0.726230
\(346\) 22.8614 1.22904
\(347\) −22.9783 −1.23354 −0.616769 0.787145i \(-0.711559\pi\)
−0.616769 + 0.787145i \(0.711559\pi\)
\(348\) 16.2337 0.870217
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 1.37228 0.0733515
\(351\) 18.9783 1.01298
\(352\) −3.37228 −0.179743
\(353\) −22.8614 −1.21679 −0.608395 0.793634i \(-0.708186\pi\)
−0.608395 + 0.793634i \(0.708186\pi\)
\(354\) 25.4891 1.35473
\(355\) −6.74456 −0.357964
\(356\) 10.0000 0.529999
\(357\) −14.7446 −0.780365
\(358\) −26.2337 −1.38649
\(359\) −30.9783 −1.63497 −0.817485 0.575950i \(-0.804632\pi\)
−0.817485 + 0.575950i \(0.804632\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) 7.48913 0.393620
\(363\) 0.744563 0.0390794
\(364\) −6.51087 −0.341263
\(365\) 8.74456 0.457711
\(366\) −10.7446 −0.561627
\(367\) 3.88316 0.202699 0.101350 0.994851i \(-0.467684\pi\)
0.101350 + 0.994851i \(0.467684\pi\)
\(368\) 6.74456 0.351585
\(369\) 5.37228 0.279670
\(370\) 1.00000 0.0519875
\(371\) 1.88316 0.0977686
\(372\) −5.25544 −0.272482
\(373\) −31.4891 −1.63045 −0.815223 0.579148i \(-0.803385\pi\)
−0.815223 + 0.579148i \(0.803385\pi\)
\(374\) 18.1168 0.936800
\(375\) 2.00000 0.103280
\(376\) −8.74456 −0.450966
\(377\) −38.5109 −1.98341
\(378\) −5.48913 −0.282330
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) −2.00000 −0.102598
\(381\) −33.4891 −1.71570
\(382\) 18.6277 0.953077
\(383\) −13.4891 −0.689262 −0.344631 0.938738i \(-0.611996\pi\)
−0.344631 + 0.938738i \(0.611996\pi\)
\(384\) 2.00000 0.102062
\(385\) −4.62772 −0.235850
\(386\) 2.00000 0.101797
\(387\) 7.37228 0.374754
\(388\) −0.116844 −0.00593185
\(389\) 14.8614 0.753503 0.376752 0.926314i \(-0.377041\pi\)
0.376752 + 0.926314i \(0.377041\pi\)
\(390\) −9.48913 −0.480501
\(391\) −36.2337 −1.83242
\(392\) −5.11684 −0.258440
\(393\) −6.51087 −0.328430
\(394\) 27.4891 1.38488
\(395\) −4.74456 −0.238725
\(396\) −3.37228 −0.169464
\(397\) 24.9783 1.25362 0.626811 0.779171i \(-0.284360\pi\)
0.626811 + 0.779171i \(0.284360\pi\)
\(398\) −11.2554 −0.564184
\(399\) −5.48913 −0.274800
\(400\) 1.00000 0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 9.48913 0.473275
\(403\) 12.4674 0.621044
\(404\) −11.4891 −0.571605
\(405\) −11.0000 −0.546594
\(406\) 11.1386 0.552799
\(407\) −3.37228 −0.167158
\(408\) −10.7446 −0.531935
\(409\) 6.23369 0.308236 0.154118 0.988052i \(-0.450746\pi\)
0.154118 + 0.988052i \(0.450746\pi\)
\(410\) 5.37228 0.265318
\(411\) 33.4891 1.65190
\(412\) −9.48913 −0.467496
\(413\) 17.4891 0.860584
\(414\) 6.74456 0.331477
\(415\) −0.744563 −0.0365491
\(416\) −4.74456 −0.232621
\(417\) −3.76631 −0.184437
\(418\) 6.74456 0.329887
\(419\) 13.4891 0.658987 0.329493 0.944158i \(-0.393122\pi\)
0.329493 + 0.944158i \(0.393122\pi\)
\(420\) 2.74456 0.133921
\(421\) −7.48913 −0.364998 −0.182499 0.983206i \(-0.558419\pi\)
−0.182499 + 0.983206i \(0.558419\pi\)
\(422\) −14.1168 −0.687197
\(423\) −8.74456 −0.425175
\(424\) 1.37228 0.0666439
\(425\) −5.37228 −0.260594
\(426\) −13.4891 −0.653550
\(427\) −7.37228 −0.356770
\(428\) −3.48913 −0.168653
\(429\) 32.0000 1.54497
\(430\) 7.37228 0.355523
\(431\) −1.37228 −0.0661005 −0.0330502 0.999454i \(-0.510522\pi\)
−0.0330502 + 0.999454i \(0.510522\pi\)
\(432\) −4.00000 −0.192450
\(433\) 12.9783 0.623695 0.311847 0.950132i \(-0.399052\pi\)
0.311847 + 0.950132i \(0.399052\pi\)
\(434\) −3.60597 −0.173092
\(435\) 16.2337 0.778346
\(436\) 0.116844 0.00559581
\(437\) −13.4891 −0.645272
\(438\) 17.4891 0.835663
\(439\) −13.3723 −0.638224 −0.319112 0.947717i \(-0.603385\pi\)
−0.319112 + 0.947717i \(0.603385\pi\)
\(440\) −3.37228 −0.160767
\(441\) −5.11684 −0.243659
\(442\) 25.4891 1.21239
\(443\) −16.9783 −0.806661 −0.403331 0.915054i \(-0.632148\pi\)
−0.403331 + 0.915054i \(0.632148\pi\)
\(444\) 2.00000 0.0949158
\(445\) 10.0000 0.474045
\(446\) −1.37228 −0.0649794
\(447\) −22.9783 −1.08683
\(448\) 1.37228 0.0648342
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) −18.1168 −0.853089
\(452\) 17.3723 0.817123
\(453\) −40.0000 −1.87936
\(454\) −11.3723 −0.533728
\(455\) −6.51087 −0.305235
\(456\) −4.00000 −0.187317
\(457\) 18.8614 0.882299 0.441150 0.897434i \(-0.354571\pi\)
0.441150 + 0.897434i \(0.354571\pi\)
\(458\) −10.0000 −0.467269
\(459\) 21.4891 1.00303
\(460\) 6.74456 0.314467
\(461\) −39.0951 −1.82084 −0.910420 0.413685i \(-0.864241\pi\)
−0.910420 + 0.413685i \(0.864241\pi\)
\(462\) −9.25544 −0.430602
\(463\) −5.48913 −0.255101 −0.127551 0.991832i \(-0.540712\pi\)
−0.127551 + 0.991832i \(0.540712\pi\)
\(464\) 8.11684 0.376815
\(465\) −5.25544 −0.243715
\(466\) 6.23369 0.288770
\(467\) 30.3505 1.40446 0.702228 0.711953i \(-0.252189\pi\)
0.702228 + 0.711953i \(0.252189\pi\)
\(468\) −4.74456 −0.219317
\(469\) 6.51087 0.300644
\(470\) −8.74456 −0.403357
\(471\) 34.7446 1.60094
\(472\) 12.7446 0.586616
\(473\) −24.8614 −1.14313
\(474\) −9.48913 −0.435850
\(475\) −2.00000 −0.0917663
\(476\) −7.37228 −0.337908
\(477\) 1.37228 0.0628324
\(478\) 17.6060 0.805278
\(479\) 3.25544 0.148745 0.0743724 0.997231i \(-0.476305\pi\)
0.0743724 + 0.997231i \(0.476305\pi\)
\(480\) 2.00000 0.0912871
\(481\) −4.74456 −0.216333
\(482\) −10.2337 −0.466132
\(483\) 18.5109 0.842274
\(484\) 0.372281 0.0169219
\(485\) −0.116844 −0.00530561
\(486\) −10.0000 −0.453609
\(487\) −37.7228 −1.70938 −0.854692 0.519136i \(-0.826254\pi\)
−0.854692 + 0.519136i \(0.826254\pi\)
\(488\) −5.37228 −0.243192
\(489\) 38.7446 1.75209
\(490\) −5.11684 −0.231155
\(491\) 14.9783 0.675959 0.337979 0.941153i \(-0.390257\pi\)
0.337979 + 0.941153i \(0.390257\pi\)
\(492\) 10.7446 0.484402
\(493\) −43.6060 −1.96391
\(494\) 9.48913 0.426936
\(495\) −3.37228 −0.151573
\(496\) −2.62772 −0.117988
\(497\) −9.25544 −0.415163
\(498\) −1.48913 −0.0667293
\(499\) −24.9783 −1.11818 −0.559090 0.829107i \(-0.688849\pi\)
−0.559090 + 0.829107i \(0.688849\pi\)
\(500\) 1.00000 0.0447214
\(501\) 2.97825 0.133058
\(502\) 11.4891 0.512785
\(503\) 29.4891 1.31486 0.657428 0.753518i \(-0.271645\pi\)
0.657428 + 0.753518i \(0.271645\pi\)
\(504\) 1.37228 0.0611263
\(505\) −11.4891 −0.511259
\(506\) −22.7446 −1.01112
\(507\) 19.0217 0.844786
\(508\) −16.7446 −0.742920
\(509\) −11.2554 −0.498888 −0.249444 0.968389i \(-0.580248\pi\)
−0.249444 + 0.968389i \(0.580248\pi\)
\(510\) −10.7446 −0.475777
\(511\) 12.0000 0.530849
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −20.9783 −0.925311
\(515\) −9.48913 −0.418141
\(516\) 14.7446 0.649093
\(517\) 29.4891 1.29693
\(518\) 1.37228 0.0602946
\(519\) 45.7228 2.00701
\(520\) −4.74456 −0.208063
\(521\) 27.0951 1.18706 0.593529 0.804813i \(-0.297734\pi\)
0.593529 + 0.804813i \(0.297734\pi\)
\(522\) 8.11684 0.355265
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −3.25544 −0.142214
\(525\) 2.74456 0.119783
\(526\) 0.116844 0.00509464
\(527\) 14.1168 0.614939
\(528\) −6.74456 −0.293519
\(529\) 22.4891 0.977788
\(530\) 1.37228 0.0596081
\(531\) 12.7446 0.553067
\(532\) −2.74456 −0.118992
\(533\) −25.4891 −1.10406
\(534\) 20.0000 0.865485
\(535\) −3.48913 −0.150848
\(536\) 4.74456 0.204934
\(537\) −52.4674 −2.26413
\(538\) 10.0000 0.431131
\(539\) 17.2554 0.743244
\(540\) −4.00000 −0.172133
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −6.74456 −0.289704
\(543\) 14.9783 0.642778
\(544\) −5.37228 −0.230335
\(545\) 0.116844 0.00500505
\(546\) −13.0217 −0.557279
\(547\) 39.3723 1.68344 0.841719 0.539916i \(-0.181544\pi\)
0.841719 + 0.539916i \(0.181544\pi\)
\(548\) 16.7446 0.715292
\(549\) −5.37228 −0.229283
\(550\) −3.37228 −0.143795
\(551\) −16.2337 −0.691578
\(552\) 13.4891 0.574135
\(553\) −6.51087 −0.276871
\(554\) 0.744563 0.0316334
\(555\) 2.00000 0.0848953
\(556\) −1.88316 −0.0798636
\(557\) 8.74456 0.370519 0.185260 0.982690i \(-0.440687\pi\)
0.185260 + 0.982690i \(0.440687\pi\)
\(558\) −2.62772 −0.111240
\(559\) −34.9783 −1.47942
\(560\) 1.37228 0.0579895
\(561\) 36.2337 1.52979
\(562\) 24.7446 1.04379
\(563\) 33.0951 1.39479 0.697396 0.716686i \(-0.254342\pi\)
0.697396 + 0.716686i \(0.254342\pi\)
\(564\) −17.4891 −0.736425
\(565\) 17.3723 0.730857
\(566\) 5.48913 0.230725
\(567\) −15.0951 −0.633934
\(568\) −6.74456 −0.282996
\(569\) −32.9783 −1.38252 −0.691260 0.722606i \(-0.742944\pi\)
−0.691260 + 0.722606i \(0.742944\pi\)
\(570\) −4.00000 −0.167542
\(571\) −27.6060 −1.15527 −0.577637 0.816294i \(-0.696025\pi\)
−0.577637 + 0.816294i \(0.696025\pi\)
\(572\) 16.0000 0.668994
\(573\) 37.2554 1.55637
\(574\) 7.37228 0.307713
\(575\) 6.74456 0.281268
\(576\) 1.00000 0.0416667
\(577\) −7.48913 −0.311776 −0.155888 0.987775i \(-0.549824\pi\)
−0.155888 + 0.987775i \(0.549824\pi\)
\(578\) 11.8614 0.493369
\(579\) 4.00000 0.166234
\(580\) 8.11684 0.337034
\(581\) −1.02175 −0.0423893
\(582\) −0.233688 −0.00968668
\(583\) −4.62772 −0.191661
\(584\) 8.74456 0.361853
\(585\) −4.74456 −0.196164
\(586\) −18.8614 −0.779158
\(587\) 47.8397 1.97455 0.987277 0.159010i \(-0.0508302\pi\)
0.987277 + 0.159010i \(0.0508302\pi\)
\(588\) −10.2337 −0.422030
\(589\) 5.25544 0.216547
\(590\) 12.7446 0.524685
\(591\) 54.9783 2.26150
\(592\) 1.00000 0.0410997
\(593\) −10.2337 −0.420247 −0.210124 0.977675i \(-0.567387\pi\)
−0.210124 + 0.977675i \(0.567387\pi\)
\(594\) 13.4891 0.553466
\(595\) −7.37228 −0.302234
\(596\) −11.4891 −0.470613
\(597\) −22.5109 −0.921309
\(598\) −32.0000 −1.30858
\(599\) 17.4891 0.714586 0.357293 0.933992i \(-0.383700\pi\)
0.357293 + 0.933992i \(0.383700\pi\)
\(600\) 2.00000 0.0816497
\(601\) −5.37228 −0.219140 −0.109570 0.993979i \(-0.534947\pi\)
−0.109570 + 0.993979i \(0.534947\pi\)
\(602\) 10.1168 0.412332
\(603\) 4.74456 0.193214
\(604\) −20.0000 −0.813788
\(605\) 0.372281 0.0151354
\(606\) −22.9783 −0.933428
\(607\) 17.7228 0.719347 0.359673 0.933078i \(-0.382888\pi\)
0.359673 + 0.933078i \(0.382888\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 22.2772 0.902717
\(610\) −5.37228 −0.217517
\(611\) 41.4891 1.67847
\(612\) −5.37228 −0.217162
\(613\) 43.0951 1.74059 0.870297 0.492527i \(-0.163927\pi\)
0.870297 + 0.492527i \(0.163927\pi\)
\(614\) −23.4891 −0.947944
\(615\) 10.7446 0.433263
\(616\) −4.62772 −0.186456
\(617\) −31.7228 −1.27711 −0.638556 0.769575i \(-0.720468\pi\)
−0.638556 + 0.769575i \(0.720468\pi\)
\(618\) −18.9783 −0.763417
\(619\) 30.1168 1.21050 0.605249 0.796036i \(-0.293074\pi\)
0.605249 + 0.796036i \(0.293074\pi\)
\(620\) −2.62772 −0.105532
\(621\) −26.9783 −1.08260
\(622\) −1.37228 −0.0550235
\(623\) 13.7228 0.549793
\(624\) −9.48913 −0.379869
\(625\) 1.00000 0.0400000
\(626\) −3.48913 −0.139453
\(627\) 13.4891 0.538704
\(628\) 17.3723 0.693229
\(629\) −5.37228 −0.214207
\(630\) 1.37228 0.0546730
\(631\) −19.0951 −0.760164 −0.380082 0.924953i \(-0.624104\pi\)
−0.380082 + 0.924953i \(0.624104\pi\)
\(632\) −4.74456 −0.188729
\(633\) −28.2337 −1.12219
\(634\) 25.3723 1.00766
\(635\) −16.7446 −0.664488
\(636\) 2.74456 0.108829
\(637\) 24.2772 0.961897
\(638\) −27.3723 −1.08368
\(639\) −6.74456 −0.266811
\(640\) 1.00000 0.0395285
\(641\) −49.6060 −1.95932 −0.979659 0.200670i \(-0.935688\pi\)
−0.979659 + 0.200670i \(0.935688\pi\)
\(642\) −6.97825 −0.275410
\(643\) 3.37228 0.132990 0.0664949 0.997787i \(-0.478818\pi\)
0.0664949 + 0.997787i \(0.478818\pi\)
\(644\) 9.25544 0.364715
\(645\) 14.7446 0.580567
\(646\) 10.7446 0.422739
\(647\) −13.4891 −0.530312 −0.265156 0.964205i \(-0.585423\pi\)
−0.265156 + 0.964205i \(0.585423\pi\)
\(648\) −11.0000 −0.432121
\(649\) −42.9783 −1.68704
\(650\) −4.74456 −0.186097
\(651\) −7.21194 −0.282658
\(652\) 19.3723 0.758677
\(653\) 0.510875 0.0199921 0.00999604 0.999950i \(-0.496818\pi\)
0.00999604 + 0.999950i \(0.496818\pi\)
\(654\) 0.233688 0.00913792
\(655\) −3.25544 −0.127200
\(656\) 5.37228 0.209752
\(657\) 8.74456 0.341158
\(658\) −12.0000 −0.467809
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) −6.74456 −0.262532
\(661\) 4.35053 0.169216 0.0846080 0.996414i \(-0.473036\pi\)
0.0846080 + 0.996414i \(0.473036\pi\)
\(662\) 23.4891 0.912931
\(663\) 50.9783 1.97983
\(664\) −0.744563 −0.0288946
\(665\) −2.74456 −0.106430
\(666\) 1.00000 0.0387492
\(667\) 54.7446 2.11972
\(668\) 1.48913 0.0576160
\(669\) −2.74456 −0.106111
\(670\) 4.74456 0.183298
\(671\) 18.1168 0.699393
\(672\) 2.74456 0.105874
\(673\) −8.51087 −0.328070 −0.164035 0.986455i \(-0.552451\pi\)
−0.164035 + 0.986455i \(0.552451\pi\)
\(674\) −7.25544 −0.279469
\(675\) −4.00000 −0.153960
\(676\) 9.51087 0.365803
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 34.7446 1.33436
\(679\) −0.160343 −0.00615339
\(680\) −5.37228 −0.206018
\(681\) −22.7446 −0.871574
\(682\) 8.86141 0.339321
\(683\) 8.62772 0.330130 0.165065 0.986283i \(-0.447217\pi\)
0.165065 + 0.986283i \(0.447217\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 16.7446 0.639777
\(686\) −16.6277 −0.634849
\(687\) −20.0000 −0.763048
\(688\) 7.37228 0.281066
\(689\) −6.51087 −0.248045
\(690\) 13.4891 0.513522
\(691\) 22.3505 0.850254 0.425127 0.905134i \(-0.360229\pi\)
0.425127 + 0.905134i \(0.360229\pi\)
\(692\) 22.8614 0.869060
\(693\) −4.62772 −0.175792
\(694\) −22.9783 −0.872242
\(695\) −1.88316 −0.0714322
\(696\) 16.2337 0.615336
\(697\) −28.8614 −1.09320
\(698\) −22.0000 −0.832712
\(699\) 12.4674 0.471559
\(700\) 1.37228 0.0518674
\(701\) −26.4674 −0.999659 −0.499829 0.866124i \(-0.666604\pi\)
−0.499829 + 0.866124i \(0.666604\pi\)
\(702\) 18.9783 0.716288
\(703\) −2.00000 −0.0754314
\(704\) −3.37228 −0.127098
\(705\) −17.4891 −0.658679
\(706\) −22.8614 −0.860400
\(707\) −15.7663 −0.592953
\(708\) 25.4891 0.957940
\(709\) 40.1168 1.50662 0.753310 0.657666i \(-0.228456\pi\)
0.753310 + 0.657666i \(0.228456\pi\)
\(710\) −6.74456 −0.253119
\(711\) −4.74456 −0.177935
\(712\) 10.0000 0.374766
\(713\) −17.7228 −0.663725
\(714\) −14.7446 −0.551801
\(715\) 16.0000 0.598366
\(716\) −26.2337 −0.980399
\(717\) 35.2119 1.31501
\(718\) −30.9783 −1.15610
\(719\) −34.7446 −1.29575 −0.647877 0.761745i \(-0.724343\pi\)
−0.647877 + 0.761745i \(0.724343\pi\)
\(720\) 1.00000 0.0372678
\(721\) −13.0217 −0.484955
\(722\) −15.0000 −0.558242
\(723\) −20.4674 −0.761190
\(724\) 7.48913 0.278331
\(725\) 8.11684 0.301452
\(726\) 0.744563 0.0276333
\(727\) −48.0000 −1.78022 −0.890111 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) −6.51087 −0.241309
\(729\) 13.0000 0.481481
\(730\) 8.74456 0.323651
\(731\) −39.6060 −1.46488
\(732\) −10.7446 −0.397130
\(733\) −29.3723 −1.08489 −0.542445 0.840091i \(-0.682501\pi\)
−0.542445 + 0.840091i \(0.682501\pi\)
\(734\) 3.88316 0.143330
\(735\) −10.2337 −0.377475
\(736\) 6.74456 0.248608
\(737\) −16.0000 −0.589368
\(738\) 5.37228 0.197756
\(739\) −36.8614 −1.35597 −0.677984 0.735076i \(-0.737146\pi\)
−0.677984 + 0.735076i \(0.737146\pi\)
\(740\) 1.00000 0.0367607
\(741\) 18.9783 0.697183
\(742\) 1.88316 0.0691328
\(743\) −5.37228 −0.197090 −0.0985449 0.995133i \(-0.531419\pi\)
−0.0985449 + 0.995133i \(0.531419\pi\)
\(744\) −5.25544 −0.192674
\(745\) −11.4891 −0.420929
\(746\) −31.4891 −1.15290
\(747\) −0.744563 −0.0272421
\(748\) 18.1168 0.662417
\(749\) −4.78806 −0.174952
\(750\) 2.00000 0.0730297
\(751\) 10.7446 0.392075 0.196037 0.980596i \(-0.437193\pi\)
0.196037 + 0.980596i \(0.437193\pi\)
\(752\) −8.74456 −0.318881
\(753\) 22.9783 0.837374
\(754\) −38.5109 −1.40248
\(755\) −20.0000 −0.727875
\(756\) −5.48913 −0.199638
\(757\) 43.9565 1.59763 0.798813 0.601579i \(-0.205462\pi\)
0.798813 + 0.601579i \(0.205462\pi\)
\(758\) −8.00000 −0.290573
\(759\) −45.4891 −1.65115
\(760\) −2.00000 −0.0725476
\(761\) 5.37228 0.194745 0.0973725 0.995248i \(-0.468956\pi\)
0.0973725 + 0.995248i \(0.468956\pi\)
\(762\) −33.4891 −1.21318
\(763\) 0.160343 0.00580480
\(764\) 18.6277 0.673927
\(765\) −5.37228 −0.194235
\(766\) −13.4891 −0.487382
\(767\) −60.4674 −2.18335
\(768\) 2.00000 0.0721688
\(769\) 46.2337 1.66723 0.833615 0.552346i \(-0.186267\pi\)
0.833615 + 0.552346i \(0.186267\pi\)
\(770\) −4.62772 −0.166771
\(771\) −41.9565 −1.51103
\(772\) 2.00000 0.0719816
\(773\) −24.3505 −0.875828 −0.437914 0.899017i \(-0.644283\pi\)
−0.437914 + 0.899017i \(0.644283\pi\)
\(774\) 7.37228 0.264991
\(775\) −2.62772 −0.0943904
\(776\) −0.116844 −0.00419445
\(777\) 2.74456 0.0984606
\(778\) 14.8614 0.532807
\(779\) −10.7446 −0.384964
\(780\) −9.48913 −0.339765
\(781\) 22.7446 0.813864
\(782\) −36.2337 −1.29571
\(783\) −32.4674 −1.16029
\(784\) −5.11684 −0.182744
\(785\) 17.3723 0.620043
\(786\) −6.51087 −0.232235
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 27.4891 0.979260
\(789\) 0.233688 0.00831951
\(790\) −4.74456 −0.168804
\(791\) 23.8397 0.847641
\(792\) −3.37228 −0.119829
\(793\) 25.4891 0.905145
\(794\) 24.9783 0.886445
\(795\) 2.74456 0.0973396
\(796\) −11.2554 −0.398938
\(797\) −27.4891 −0.973715 −0.486857 0.873481i \(-0.661857\pi\)
−0.486857 + 0.873481i \(0.661857\pi\)
\(798\) −5.48913 −0.194313
\(799\) 46.9783 1.66197
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) 10.0000 0.353112
\(803\) −29.4891 −1.04065
\(804\) 9.48913 0.334656
\(805\) 9.25544 0.326211
\(806\) 12.4674 0.439145
\(807\) 20.0000 0.704033
\(808\) −11.4891 −0.404186
\(809\) −51.4891 −1.81026 −0.905131 0.425134i \(-0.860227\pi\)
−0.905131 + 0.425134i \(0.860227\pi\)
\(810\) −11.0000 −0.386501
\(811\) 49.4891 1.73780 0.868899 0.494989i \(-0.164828\pi\)
0.868899 + 0.494989i \(0.164828\pi\)
\(812\) 11.1386 0.390888
\(813\) −13.4891 −0.473084
\(814\) −3.37228 −0.118198
\(815\) 19.3723 0.678581
\(816\) −10.7446 −0.376135
\(817\) −14.7446 −0.515847
\(818\) 6.23369 0.217956
\(819\) −6.51087 −0.227508
\(820\) 5.37228 0.187608
\(821\) −32.7446 −1.14279 −0.571397 0.820674i \(-0.693598\pi\)
−0.571397 + 0.820674i \(0.693598\pi\)
\(822\) 33.4891 1.16807
\(823\) −39.7228 −1.38465 −0.692325 0.721586i \(-0.743414\pi\)
−0.692325 + 0.721586i \(0.743414\pi\)
\(824\) −9.48913 −0.330569
\(825\) −6.74456 −0.234816
\(826\) 17.4891 0.608524
\(827\) −54.3505 −1.88995 −0.944977 0.327138i \(-0.893916\pi\)
−0.944977 + 0.327138i \(0.893916\pi\)
\(828\) 6.74456 0.234390
\(829\) −40.3505 −1.40143 −0.700716 0.713440i \(-0.747136\pi\)
−0.700716 + 0.713440i \(0.747136\pi\)
\(830\) −0.744563 −0.0258441
\(831\) 1.48913 0.0516572
\(832\) −4.74456 −0.164488
\(833\) 27.4891 0.952442
\(834\) −3.76631 −0.130417
\(835\) 1.48913 0.0515333
\(836\) 6.74456 0.233266
\(837\) 10.5109 0.363309
\(838\) 13.4891 0.465974
\(839\) −29.4891 −1.01808 −0.509039 0.860744i \(-0.669999\pi\)
−0.509039 + 0.860744i \(0.669999\pi\)
\(840\) 2.74456 0.0946964
\(841\) 36.8832 1.27183
\(842\) −7.48913 −0.258092
\(843\) 49.4891 1.70450
\(844\) −14.1168 −0.485922
\(845\) 9.51087 0.327184
\(846\) −8.74456 −0.300644
\(847\) 0.510875 0.0175539
\(848\) 1.37228 0.0471243
\(849\) 10.9783 0.376773
\(850\) −5.37228 −0.184268
\(851\) 6.74456 0.231201
\(852\) −13.4891 −0.462130
\(853\) 35.4891 1.21512 0.607562 0.794272i \(-0.292148\pi\)
0.607562 + 0.794272i \(0.292148\pi\)
\(854\) −7.37228 −0.252274
\(855\) −2.00000 −0.0683986
\(856\) −3.48913 −0.119256
\(857\) −29.1386 −0.995355 −0.497678 0.867362i \(-0.665814\pi\)
−0.497678 + 0.867362i \(0.665814\pi\)
\(858\) 32.0000 1.09246
\(859\) −19.7228 −0.672934 −0.336467 0.941695i \(-0.609232\pi\)
−0.336467 + 0.941695i \(0.609232\pi\)
\(860\) 7.37228 0.251393
\(861\) 14.7446 0.502493
\(862\) −1.37228 −0.0467401
\(863\) 53.8397 1.83272 0.916362 0.400352i \(-0.131112\pi\)
0.916362 + 0.400352i \(0.131112\pi\)
\(864\) −4.00000 −0.136083
\(865\) 22.8614 0.777311
\(866\) 12.9783 0.441019
\(867\) 23.7228 0.805669
\(868\) −3.60597 −0.122395
\(869\) 16.0000 0.542763
\(870\) 16.2337 0.550374
\(871\) −22.5109 −0.762752
\(872\) 0.116844 0.00395684
\(873\) −0.116844 −0.00395457
\(874\) −13.4891 −0.456276
\(875\) 1.37228 0.0463916
\(876\) 17.4891 0.590903
\(877\) 49.3723 1.66718 0.833592 0.552381i \(-0.186281\pi\)
0.833592 + 0.552381i \(0.186281\pi\)
\(878\) −13.3723 −0.451293
\(879\) −37.7228 −1.27236
\(880\) −3.37228 −0.113680
\(881\) 1.37228 0.0462333 0.0231167 0.999733i \(-0.492641\pi\)
0.0231167 + 0.999733i \(0.492641\pi\)
\(882\) −5.11684 −0.172293
\(883\) −11.3723 −0.382708 −0.191354 0.981521i \(-0.561288\pi\)
−0.191354 + 0.981521i \(0.561288\pi\)
\(884\) 25.4891 0.857292
\(885\) 25.4891 0.856808
\(886\) −16.9783 −0.570395
\(887\) −25.3723 −0.851918 −0.425959 0.904743i \(-0.640063\pi\)
−0.425959 + 0.904743i \(0.640063\pi\)
\(888\) 2.00000 0.0671156
\(889\) −22.9783 −0.770666
\(890\) 10.0000 0.335201
\(891\) 37.0951 1.24273
\(892\) −1.37228 −0.0459474
\(893\) 17.4891 0.585251
\(894\) −22.9783 −0.768508
\(895\) −26.2337 −0.876895
\(896\) 1.37228 0.0458447
\(897\) −64.0000 −2.13690
\(898\) 18.0000 0.600668
\(899\) −21.3288 −0.711355
\(900\) 1.00000 0.0333333
\(901\) −7.37228 −0.245606
\(902\) −18.1168 −0.603225
\(903\) 20.2337 0.673335
\(904\) 17.3723 0.577793
\(905\) 7.48913 0.248947
\(906\) −40.0000 −1.32891
\(907\) 40.4674 1.34370 0.671849 0.740689i \(-0.265501\pi\)
0.671849 + 0.740689i \(0.265501\pi\)
\(908\) −11.3723 −0.377402
\(909\) −11.4891 −0.381070
\(910\) −6.51087 −0.215833
\(911\) 17.7663 0.588624 0.294312 0.955709i \(-0.404909\pi\)
0.294312 + 0.955709i \(0.404909\pi\)
\(912\) −4.00000 −0.132453
\(913\) 2.51087 0.0830978
\(914\) 18.8614 0.623880
\(915\) −10.7446 −0.355204
\(916\) −10.0000 −0.330409
\(917\) −4.46738 −0.147526
\(918\) 21.4891 0.709247
\(919\) 7.72281 0.254752 0.127376 0.991854i \(-0.459344\pi\)
0.127376 + 0.991854i \(0.459344\pi\)
\(920\) 6.74456 0.222362
\(921\) −46.9783 −1.54799
\(922\) −39.0951 −1.28753
\(923\) 32.0000 1.05329
\(924\) −9.25544 −0.304482
\(925\) 1.00000 0.0328798
\(926\) −5.48913 −0.180384
\(927\) −9.48913 −0.311664
\(928\) 8.11684 0.266448
\(929\) 31.0951 1.02020 0.510098 0.860116i \(-0.329609\pi\)
0.510098 + 0.860116i \(0.329609\pi\)
\(930\) −5.25544 −0.172333
\(931\) 10.2337 0.335396
\(932\) 6.23369 0.204191
\(933\) −2.74456 −0.0898529
\(934\) 30.3505 0.993100
\(935\) 18.1168 0.592484
\(936\) −4.74456 −0.155081
\(937\) 36.9783 1.20803 0.604013 0.796974i \(-0.293567\pi\)
0.604013 + 0.796974i \(0.293567\pi\)
\(938\) 6.51087 0.212588
\(939\) −6.97825 −0.227727
\(940\) −8.74456 −0.285216
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 34.7446 1.13204
\(943\) 36.2337 1.17993
\(944\) 12.7446 0.414800
\(945\) −5.48913 −0.178561
\(946\) −24.8614 −0.808314
\(947\) 37.0951 1.20543 0.602714 0.797957i \(-0.294086\pi\)
0.602714 + 0.797957i \(0.294086\pi\)
\(948\) −9.48913 −0.308192
\(949\) −41.4891 −1.34679
\(950\) −2.00000 −0.0648886
\(951\) 50.7446 1.64551
\(952\) −7.37228 −0.238937
\(953\) 46.2337 1.49766 0.748828 0.662764i \(-0.230617\pi\)
0.748828 + 0.662764i \(0.230617\pi\)
\(954\) 1.37228 0.0444292
\(955\) 18.6277 0.602779
\(956\) 17.6060 0.569418
\(957\) −54.7446 −1.76964
\(958\) 3.25544 0.105178
\(959\) 22.9783 0.742006
\(960\) 2.00000 0.0645497
\(961\) −24.0951 −0.777261
\(962\) −4.74456 −0.152971
\(963\) −3.48913 −0.112435
\(964\) −10.2337 −0.329605
\(965\) 2.00000 0.0643823
\(966\) 18.5109 0.595578
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) 0.372281 0.0119656
\(969\) 21.4891 0.690330
\(970\) −0.116844 −0.00375163
\(971\) −19.6060 −0.629185 −0.314593 0.949227i \(-0.601868\pi\)
−0.314593 + 0.949227i \(0.601868\pi\)
\(972\) −10.0000 −0.320750
\(973\) −2.58422 −0.0828463
\(974\) −37.7228 −1.20872
\(975\) −9.48913 −0.303895
\(976\) −5.37228 −0.171963
\(977\) −11.8832 −0.380176 −0.190088 0.981767i \(-0.560877\pi\)
−0.190088 + 0.981767i \(0.560877\pi\)
\(978\) 38.7446 1.23891
\(979\) −33.7228 −1.07779
\(980\) −5.11684 −0.163452
\(981\) 0.116844 0.00373054
\(982\) 14.9783 0.477975
\(983\) 30.8614 0.984326 0.492163 0.870503i \(-0.336206\pi\)
0.492163 + 0.870503i \(0.336206\pi\)
\(984\) 10.7446 0.342524
\(985\) 27.4891 0.875876
\(986\) −43.6060 −1.38870
\(987\) −24.0000 −0.763928
\(988\) 9.48913 0.301889
\(989\) 49.7228 1.58109
\(990\) −3.37228 −0.107178
\(991\) 25.6060 0.813400 0.406700 0.913562i \(-0.366679\pi\)
0.406700 + 0.913562i \(0.366679\pi\)
\(992\) −2.62772 −0.0834302
\(993\) 46.9783 1.49081
\(994\) −9.25544 −0.293565
\(995\) −11.2554 −0.356821
\(996\) −1.48913 −0.0471847
\(997\) 26.2337 0.830829 0.415415 0.909632i \(-0.363637\pi\)
0.415415 + 0.909632i \(0.363637\pi\)
\(998\) −24.9783 −0.790673
\(999\) −4.00000 −0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 370.2.a.f.1.2 2
3.2 odd 2 3330.2.a.bb.1.2 2
4.3 odd 2 2960.2.a.o.1.1 2
5.2 odd 4 1850.2.b.m.149.4 4
5.3 odd 4 1850.2.b.m.149.1 4
5.4 even 2 1850.2.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.2 2 1.1 even 1 trivial
1850.2.a.q.1.1 2 5.4 even 2
1850.2.b.m.149.1 4 5.3 odd 4
1850.2.b.m.149.4 4 5.2 odd 4
2960.2.a.o.1.1 2 4.3 odd 2
3330.2.a.bb.1.2 2 3.2 odd 2