Properties

Label 1850.2.a.q.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −1.37228 −0.518674 −0.259337 0.965787i \(-0.583504\pi\)
−0.259337 + 0.965787i \(0.583504\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −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 \(-0.271423\pi\)
0.657952 + 0.753059i \(0.271423\pi\)
\(14\) 1.37228 0.366758
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.37228 1.30297 0.651485 0.758662i \(-0.274146\pi\)
0.651485 + 0.758662i \(0.274146\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.74456 0.598913
\(22\) 3.37228 0.718973
\(23\) −6.74456 −1.40634 −0.703169 0.711022i \(-0.748232\pi\)
−0.703169 + 0.711022i \(0.748232\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) −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 \(0.228301\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 2.00000 0.324443
\(39\) −9.48913 −1.51948
\(40\) 0 0
\(41\) 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 \(-0.690018\pi\)
−0.562131 + 0.827048i \(0.690018\pi\)
\(44\) −3.37228 −0.508391
\(45\) 0 0
\(46\) 6.74456 0.994432
\(47\) 8.74456 1.27553 0.637763 0.770233i \(-0.279860\pi\)
0.637763 + 0.770233i \(0.279860\pi\)
\(48\) −2.00000 −0.288675
\(49\) −5.11684 −0.730978
\(50\) 0 0
\(51\) −10.7446 −1.50454
\(52\) 4.74456 0.657952
\(53\) −1.37228 −0.188497 −0.0942487 0.995549i \(-0.530045\pi\)
−0.0942487 + 0.995549i \(0.530045\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 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 \(0.188568\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) −6.74456 −0.830198
\(67\) −4.74456 −0.579641 −0.289820 0.957081i \(-0.593596\pi\)
−0.289820 + 0.957081i \(0.593596\pi\)
\(68\) 5.37228 0.651485
\(69\) 13.4891 1.62390
\(70\) 0 0
\(71\) −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 \(-0.670998\pi\)
−0.511737 + 0.859142i \(0.670998\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −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 \(0.586000\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −5.37228 −0.593269
\(83\) 0.744563 0.0817264 0.0408632 0.999165i \(-0.486989\pi\)
0.0408632 + 0.999165i \(0.486989\pi\)
\(84\) 2.74456 0.299456
\(85\) 0 0
\(86\) 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 \(0.322192\pi\)
\(90\) 0 0
\(91\) −6.51087 −0.682525
\(92\) −6.74456 −0.703169
\(93\) 5.25544 0.544963
\(94\) −8.74456 −0.901933
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 0.116844 0.0118637 0.00593185 0.999982i \(-0.498112\pi\)
0.00593185 + 0.999982i \(0.498112\pi\)
\(98\) 5.11684 0.516879
\(99\) −3.37228 −0.338927
\(100\) 0 0
\(101\) −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 \(-0.345156\pi\)
0.467496 + 0.883995i \(0.345156\pi\)
\(104\) −4.74456 −0.465243
\(105\) 0 0
\(106\) 1.37228 0.133288
\(107\) 3.48913 0.337306 0.168653 0.985675i \(-0.446058\pi\)
0.168653 + 0.985675i \(0.446058\pi\)
\(108\) 4.00000 0.384900
\(109\) 0.116844 0.0111916 0.00559581 0.999984i \(-0.498219\pi\)
0.00559581 + 0.999984i \(0.498219\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.37228 −0.129668
\(113\) −17.3723 −1.63425 −0.817123 0.576463i \(-0.804433\pi\)
−0.817123 + 0.576463i \(0.804433\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 8.11684 0.753630
\(117\) 4.74456 0.438635
\(118\) −12.7446 −1.17323
\(119\) −7.37228 −0.675816
\(120\) 0 0
\(121\) 0.372281 0.0338438
\(122\) 5.37228 0.486383
\(123\) −10.7446 −0.968805
\(124\) −2.62772 −0.235976
\(125\) 0 0
\(126\) 1.37228 0.122253
\(127\) 16.7446 1.48584 0.742920 0.669380i \(-0.233440\pi\)
0.742920 + 0.669380i \(0.233440\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 14.7446 1.29819
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) −5.37228 −0.460669
\(137\) −16.7446 −1.43058 −0.715292 0.698825i \(-0.753706\pi\)
−0.715292 + 0.698825i \(0.753706\pi\)
\(138\) −13.4891 −1.14827
\(139\) −1.88316 −0.159727 −0.0798636 0.996806i \(-0.525448\pi\)
−0.0798636 + 0.996806i \(0.525448\pi\)
\(140\) 0 0
\(141\) −17.4891 −1.47285
\(142\) 6.74456 0.565991
\(143\) −16.0000 −1.33799
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 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 \(0.655967\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) −9.48913 −0.759738
\(157\) −17.3723 −1.38646 −0.693229 0.720717i \(-0.743813\pi\)
−0.693229 + 0.720717i \(0.743813\pi\)
\(158\) 4.74456 0.377457
\(159\) 2.74456 0.217658
\(160\) 0 0
\(161\) 9.25544 0.729431
\(162\) 11.0000 0.864242
\(163\) −19.3723 −1.51735 −0.758677 0.651467i \(-0.774154\pi\)
−0.758677 + 0.651467i \(0.774154\pi\)
\(164\) 5.37228 0.419505
\(165\) 0 0
\(166\) −0.744563 −0.0577893
\(167\) −1.48913 −0.115232 −0.0576160 0.998339i \(-0.518350\pi\)
−0.0576160 + 0.998339i \(0.518350\pi\)
\(168\) −2.74456 −0.211748
\(169\) 9.51087 0.731606
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) −7.37228 −0.562131
\(173\) −22.8614 −1.73812 −0.869060 0.494706i \(-0.835276\pi\)
−0.869060 + 0.494706i \(0.835276\pi\)
\(174\) 16.2337 1.23067
\(175\) 0 0
\(176\) −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 \(0.936873\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −5.25544 −0.385347
\(187\) −18.1168 −1.32483
\(188\) 8.74456 0.637763
\(189\) −5.48913 −0.399275
\(190\) 0 0
\(191\) 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 \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −0.116844 −0.00838891
\(195\) 0 0
\(196\) −5.11684 −0.365489
\(197\) −27.4891 −1.95852 −0.979260 0.202610i \(-0.935058\pi\)
−0.979260 + 0.202610i \(0.935058\pi\)
\(198\) 3.37228 0.239658
\(199\) −11.2554 −0.797877 −0.398938 0.916978i \(-0.630621\pi\)
−0.398938 + 0.916978i \(0.630621\pi\)
\(200\) 0 0
\(201\) 9.48913 0.669311
\(202\) 11.4891 0.808372
\(203\) −11.1386 −0.781776
\(204\) −10.7446 −0.752270
\(205\) 0 0
\(206\) −9.48913 −0.661139
\(207\) −6.74456 −0.468780
\(208\) 4.74456 0.328976
\(209\) 6.74456 0.466531
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 3.60597 0.244789
\(218\) −0.116844 −0.00791367
\(219\) 17.4891 1.18181
\(220\) 0 0
\(221\) 25.4891 1.71458
\(222\) −2.00000 −0.134231
\(223\) 1.37228 0.0918948 0.0459474 0.998944i \(-0.485369\pi\)
0.0459474 + 0.998944i \(0.485369\pi\)
\(224\) 1.37228 0.0916894
\(225\) 0 0
\(226\) 17.3723 1.15559
\(227\) 11.3723 0.754805 0.377402 0.926049i \(-0.376817\pi\)
0.377402 + 0.926049i \(0.376817\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −9.25544 −0.608963
\(232\) −8.11684 −0.532897
\(233\) −6.23369 −0.408382 −0.204191 0.978931i \(-0.565456\pi\)
−0.204191 + 0.978931i \(0.565456\pi\)
\(234\) −4.74456 −0.310162
\(235\) 0 0
\(236\) 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 \(0.307169\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) 10.7446 0.685048
\(247\) −9.48913 −0.603779
\(248\) 2.62772 0.166860
\(249\) −1.48913 −0.0943695
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.9783 1.30859 0.654294 0.756241i \(-0.272966\pi\)
0.654294 + 0.756241i \(0.272966\pi\)
\(258\) −14.7446 −0.917956
\(259\) 1.37228 0.0852694
\(260\) 0 0
\(261\) 8.11684 0.502420
\(262\) 3.25544 0.201122
\(263\) −0.116844 −0.00720491 −0.00360245 0.999994i \(-0.501147\pi\)
−0.00360245 + 0.999994i \(0.501147\pi\)
\(264\) −6.74456 −0.415099
\(265\) 0 0
\(266\) −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 \(0.401392\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) 13.4891 0.811950
\(277\) −0.744563 −0.0447364 −0.0223682 0.999750i \(-0.507121\pi\)
−0.0223682 + 0.999750i \(0.507121\pi\)
\(278\) 1.88316 0.112944
\(279\) −2.62772 −0.157317
\(280\) 0 0
\(281\) 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 \(-0.552165\pi\)
−0.163147 + 0.986602i \(0.552165\pi\)
\(284\) −6.74456 −0.400216
\(285\) 0 0
\(286\) 16.0000 0.946100
\(287\) −7.37228 −0.435172
\(288\) −1.00000 −0.0589256
\(289\) 11.8614 0.697730
\(290\) 0 0
\(291\) −0.233688 −0.0136990
\(292\) −8.74456 −0.511737
\(293\) 18.8614 1.10190 0.550948 0.834540i \(-0.314266\pi\)
0.550948 + 0.834540i \(0.314266\pi\)
\(294\) −10.2337 −0.596841
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −13.4891 −0.782718
\(298\) 11.4891 0.665547
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) 10.1168 0.583125
\(302\) 20.0000 1.15087
\(303\) 22.9783 1.32007
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −5.37228 −0.307113
\(307\) 23.4891 1.34060 0.670298 0.742092i \(-0.266166\pi\)
0.670298 + 0.742092i \(0.266166\pi\)
\(308\) 4.62772 0.263689
\(309\) −18.9783 −1.07963
\(310\) 0 0
\(311\) −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 \(-0.468561\pi\)
0.0986085 + 0.995126i \(0.468561\pi\)
\(314\) 17.3723 0.980375
\(315\) 0 0
\(316\) −4.74456 −0.266903
\(317\) −25.3723 −1.42505 −0.712525 0.701647i \(-0.752448\pi\)
−0.712525 + 0.701647i \(0.752448\pi\)
\(318\) −2.74456 −0.153907
\(319\) −27.3723 −1.53255
\(320\) 0 0
\(321\) −6.97825 −0.389488
\(322\) −9.25544 −0.515785
\(323\) −10.7446 −0.597843
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 19.3723 1.07293
\(327\) −0.233688 −0.0129230
\(328\) −5.37228 −0.296635
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 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
\(335\) 0 0
\(336\) 2.74456 0.149728
\(337\) 7.25544 0.395229 0.197614 0.980280i \(-0.436681\pi\)
0.197614 + 0.980280i \(0.436681\pi\)
\(338\) −9.51087 −0.517323
\(339\) 34.7446 1.88707
\(340\) 0 0
\(341\) 8.86141 0.479872
\(342\) 2.00000 0.108148
\(343\) 16.6277 0.897812
\(344\) 7.37228 0.397487
\(345\) 0 0
\(346\) 22.8614 1.22904
\(347\) 22.9783 1.23354 0.616769 0.787145i \(-0.288441\pi\)
0.616769 + 0.787145i \(0.288441\pi\)
\(348\) −16.2337 −0.870217
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 18.9783 1.01298
\(352\) 3.37228 0.179743
\(353\) 22.8614 1.21679 0.608395 0.793634i \(-0.291814\pi\)
0.608395 + 0.793634i \(0.291814\pi\)
\(354\) 25.4891 1.35473
\(355\) 0 0
\(356\) 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 \(0.804632\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −7.48913 −0.393620
\(363\) −0.744563 −0.0390794
\(364\) −6.51087 −0.341263
\(365\) 0 0
\(366\) −10.7446 −0.561627
\(367\) −3.88316 −0.202699 −0.101350 0.994851i \(-0.532316\pi\)
−0.101350 + 0.994851i \(0.532316\pi\)
\(368\) −6.74456 −0.351585
\(369\) 5.37228 0.279670
\(370\) 0 0
\(371\) 1.88316 0.0977686
\(372\) 5.25544 0.272482
\(373\) 31.4891 1.63045 0.815223 0.579148i \(-0.196615\pi\)
0.815223 + 0.579148i \(0.196615\pi\)
\(374\) 18.1168 0.936800
\(375\) 0 0
\(376\) −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 \(0.565871\pi\)
\(380\) 0 0
\(381\) −33.4891 −1.71570
\(382\) −18.6277 −0.953077
\(383\) 13.4891 0.689262 0.344631 0.938738i \(-0.388004\pi\)
0.344631 + 0.938738i \(0.388004\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 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 \(0.377041\pi\)
\(390\) 0 0
\(391\) −36.2337 −1.83242
\(392\) 5.11684 0.258440
\(393\) 6.51087 0.328430
\(394\) 27.4891 1.38488
\(395\) 0 0
\(396\) −3.37228 −0.169464
\(397\) −24.9783 −1.25362 −0.626811 0.779171i \(-0.715640\pi\)
−0.626811 + 0.779171i \(0.715640\pi\)
\(398\) 11.2554 0.564184
\(399\) −5.48913 −0.274800
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) 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 \(0.450746\pi\)
\(410\) 0 0
\(411\) 33.4891 1.65190
\(412\) 9.48913 0.467496
\(413\) −17.4891 −0.860584
\(414\) 6.74456 0.331477
\(415\) 0 0
\(416\) −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 \(0.393122\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) −13.4891 −0.653550
\(427\) 7.37228 0.356770
\(428\) 3.48913 0.168653
\(429\) 32.0000 1.54497
\(430\) 0 0
\(431\) −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 \(-0.600948\pi\)
−0.311847 + 0.950132i \(0.600948\pi\)
\(434\) −3.60597 −0.173092
\(435\) 0 0
\(436\) 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 \(0.603385\pi\)
\(440\) 0 0
\(441\) −5.11684 −0.243659
\(442\) −25.4891 −1.21239
\(443\) 16.9783 0.806661 0.403331 0.915054i \(-0.367852\pi\)
0.403331 + 0.915054i \(0.367852\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −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 \(0.360367\pi\)
\(450\) 0 0
\(451\) −18.1168 −0.853089
\(452\) −17.3723 −0.817123
\(453\) 40.0000 1.87936
\(454\) −11.3723 −0.533728
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −18.8614 −0.882299 −0.441150 0.897434i \(-0.645429\pi\)
−0.441150 + 0.897434i \(0.645429\pi\)
\(458\) 10.0000 0.467269
\(459\) 21.4891 1.00303
\(460\) 0 0
\(461\) −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 \(-0.459288\pi\)
0.127551 + 0.991832i \(0.459288\pi\)
\(464\) 8.11684 0.376815
\(465\) 0 0
\(466\) 6.23369 0.288770
\(467\) −30.3505 −1.40446 −0.702228 0.711953i \(-0.747811\pi\)
−0.702228 + 0.711953i \(0.747811\pi\)
\(468\) 4.74456 0.219317
\(469\) 6.51087 0.300644
\(470\) 0 0
\(471\) 34.7446 1.60094
\(472\) −12.7446 −0.586616
\(473\) 24.8614 1.14313
\(474\) −9.48913 −0.435850
\(475\) 0 0
\(476\) −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 \(0.476305\pi\)
\(480\) 0 0
\(481\) −4.74456 −0.216333
\(482\) 10.2337 0.466132
\(483\) −18.5109 −0.842274
\(484\) 0.372281 0.0169219
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 37.7228 1.70938 0.854692 0.519136i \(-0.173746\pi\)
0.854692 + 0.519136i \(0.173746\pi\)
\(488\) 5.37228 0.243192
\(489\) 38.7446 1.75209
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) −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 \(0.688849\pi\)
\(500\) 0 0
\(501\) 2.97825 0.133058
\(502\) −11.4891 −0.512785
\(503\) −29.4891 −1.31486 −0.657428 0.753518i \(-0.728355\pi\)
−0.657428 + 0.753518i \(0.728355\pi\)
\(504\) 1.37228 0.0611263
\(505\) 0 0
\(506\) −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 \(0.580248\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) −20.9783 −0.925311
\(515\) 0 0
\(516\) 14.7446 0.649093
\(517\) −29.4891 −1.29693
\(518\) −1.37228 −0.0602946
\(519\) 45.7228 2.00701
\(520\) 0 0
\(521\) 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 \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −3.25544 −0.142214
\(525\) 0 0
\(526\) 0.116844 0.00509464
\(527\) −14.1168 −0.614939
\(528\) 6.74456 0.293519
\(529\) 22.4891 0.977788
\(530\) 0 0
\(531\) 12.7446 0.553067
\(532\) 2.74456 0.118992
\(533\) 25.4891 1.10406
\(534\) 20.0000 0.865485
\(535\) 0 0
\(536\) 4.74456 0.204934
\(537\) 52.4674 2.26413
\(538\) −10.0000 −0.431131
\(539\) 17.2554 0.743244
\(540\) 0 0
\(541\) 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 0
\(546\) −13.0217 −0.557279
\(547\) −39.3723 −1.68344 −0.841719 0.539916i \(-0.818456\pi\)
−0.841719 + 0.539916i \(0.818456\pi\)
\(548\) −16.7446 −0.715292
\(549\) −5.37228 −0.229283
\(550\) 0 0
\(551\) −16.2337 −0.691578
\(552\) −13.4891 −0.574135
\(553\) 6.51087 0.276871
\(554\) 0.744563 0.0316334
\(555\) 0 0
\(556\) −1.88316 −0.0798636
\(557\) −8.74456 −0.370519 −0.185260 0.982690i \(-0.559313\pi\)
−0.185260 + 0.982690i \(0.559313\pi\)
\(558\) 2.62772 0.111240
\(559\) −34.9783 −1.47942
\(560\) 0 0
\(561\) 36.2337 1.52979
\(562\) −24.7446 −1.04379
\(563\) −33.0951 −1.39479 −0.697396 0.716686i \(-0.745658\pi\)
−0.697396 + 0.716686i \(0.745658\pi\)
\(564\) −17.4891 −0.736425
\(565\) 0 0
\(566\) 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 \(0.742944\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 7.48913 0.311776 0.155888 0.987775i \(-0.450176\pi\)
0.155888 + 0.987775i \(0.450176\pi\)
\(578\) −11.8614 −0.493369
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −1.02175 −0.0423893
\(582\) 0.233688 0.00968668
\(583\) 4.62772 0.191661
\(584\) 8.74456 0.361853
\(585\) 0 0
\(586\) −18.8614 −0.779158
\(587\) −47.8397 −1.97455 −0.987277 0.159010i \(-0.949170\pi\)
−0.987277 + 0.159010i \(0.949170\pi\)
\(588\) 10.2337 0.422030
\(589\) 5.25544 0.216547
\(590\) 0 0
\(591\) 54.9783 2.26150
\(592\) −1.00000 −0.0410997
\(593\) 10.2337 0.420247 0.210124 0.977675i \(-0.432613\pi\)
0.210124 + 0.977675i \(0.432613\pi\)
\(594\) 13.4891 0.553466
\(595\) 0 0
\(596\) −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 \(0.383700\pi\)
\(600\) 0 0
\(601\) −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 0
\(606\) −22.9783 −0.933428
\(607\) −17.7228 −0.719347 −0.359673 0.933078i \(-0.617112\pi\)
−0.359673 + 0.933078i \(0.617112\pi\)
\(608\) 2.00000 0.0811107
\(609\) 22.2772 0.902717
\(610\) 0 0
\(611\) 41.4891 1.67847
\(612\) 5.37228 0.217162
\(613\) −43.0951 −1.74059 −0.870297 0.492527i \(-0.836073\pi\)
−0.870297 + 0.492527i \(0.836073\pi\)
\(614\) −23.4891 −0.947944
\(615\) 0 0
\(616\) −4.62772 −0.186456
\(617\) 31.7228 1.27711 0.638556 0.769575i \(-0.279532\pi\)
0.638556 + 0.769575i \(0.279532\pi\)
\(618\) 18.9783 0.763417
\(619\) 30.1168 1.21050 0.605249 0.796036i \(-0.293074\pi\)
0.605249 + 0.796036i \(0.293074\pi\)
\(620\) 0 0
\(621\) −26.9783 −1.08260
\(622\) 1.37228 0.0550235
\(623\) −13.7228 −0.549793
\(624\) −9.48913 −0.379869
\(625\) 0 0
\(626\) −3.48913 −0.139453
\(627\) −13.4891 −0.538704
\(628\) −17.3723 −0.693229
\(629\) −5.37228 −0.214207
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) 2.74456 0.108829
\(637\) −24.2772 −0.961897
\(638\) 27.3723 1.08368
\(639\) −6.74456 −0.266811
\(640\) 0 0
\(641\) −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 \(-0.521182\pi\)
−0.0664949 + 0.997787i \(0.521182\pi\)
\(644\) 9.25544 0.364715
\(645\) 0 0
\(646\) 10.7446 0.422739
\(647\) 13.4891 0.530312 0.265156 0.964205i \(-0.414577\pi\)
0.265156 + 0.964205i \(0.414577\pi\)
\(648\) 11.0000 0.432121
\(649\) −42.9783 −1.68704
\(650\) 0 0
\(651\) −7.21194 −0.282658
\(652\) −19.3723 −0.758677
\(653\) −0.510875 −0.0199921 −0.00999604 0.999950i \(-0.503182\pi\)
−0.00999604 + 0.999950i \(0.503182\pi\)
\(654\) 0.233688 0.00913792
\(655\) 0 0
\(656\) 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 \(0.424908\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) −54.7446 −2.11972
\(668\) −1.48913 −0.0576160
\(669\) −2.74456 −0.106111
\(670\) 0 0
\(671\) 18.1168 0.699393
\(672\) −2.74456 −0.105874
\(673\) 8.51087 0.328070 0.164035 0.986455i \(-0.447549\pi\)
0.164035 + 0.986455i \(0.447549\pi\)
\(674\) −7.25544 −0.279469
\(675\) 0 0
\(676\) 9.51087 0.365803
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) −34.7446 −1.33436
\(679\) −0.160343 −0.00615339
\(680\) 0 0
\(681\) −22.7446 −0.871574
\(682\) −8.86141 −0.339321
\(683\) −8.62772 −0.330130 −0.165065 0.986283i \(-0.552783\pi\)
−0.165065 + 0.986283i \(0.552783\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −16.6277 −0.634849
\(687\) 20.0000 0.763048
\(688\) −7.37228 −0.281066
\(689\) −6.51087 −0.248045
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) 16.2337 0.615336
\(697\) 28.8614 1.09320
\(698\) 22.0000 0.832712
\(699\) 12.4674 0.471559
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) −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 \(0.228456\pi\)
\(710\) 0 0
\(711\) −4.74456 −0.177935
\(712\) −10.0000 −0.374766
\(713\) 17.7228 0.663725
\(714\) −14.7446 −0.551801
\(715\) 0 0
\(716\) −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 \(0.724343\pi\)
\(720\) 0 0
\(721\) −13.0217 −0.484955
\(722\) 15.0000 0.558242
\(723\) 20.4674 0.761190
\(724\) 7.48913 0.278331
\(725\) 0 0
\(726\) 0.744563 0.0276333
\(727\) 48.0000 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) 6.51087 0.241309
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −39.6060 −1.46488
\(732\) 10.7446 0.397130
\(733\) 29.3723 1.08489 0.542445 0.840091i \(-0.317499\pi\)
0.542445 + 0.840091i \(0.317499\pi\)
\(734\) 3.88316 0.143330
\(735\) 0 0
\(736\) 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 \(0.737146\pi\)
\(740\) 0 0
\(741\) 18.9783 0.697183
\(742\) −1.88316 −0.0691328
\(743\) 5.37228 0.197090 0.0985449 0.995133i \(-0.468581\pi\)
0.0985449 + 0.995133i \(0.468581\pi\)
\(744\) −5.25544 −0.192674
\(745\) 0 0
\(746\) −31.4891 −1.15290
\(747\) 0.744563 0.0272421
\(748\) −18.1168 −0.662417
\(749\) −4.78806 −0.174952
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) −5.48913 −0.199638
\(757\) −43.9565 −1.59763 −0.798813 0.601579i \(-0.794538\pi\)
−0.798813 + 0.601579i \(0.794538\pi\)
\(758\) 8.00000 0.290573
\(759\) −45.4891 −1.65115
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) −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 \(0.186267\pi\)
\(770\) 0 0
\(771\) −41.9565 −1.51103
\(772\) −2.00000 −0.0719816
\(773\) 24.3505 0.875828 0.437914 0.899017i \(-0.355717\pi\)
0.437914 + 0.899017i \(0.355717\pi\)
\(774\) 7.37228 0.264991
\(775\) 0 0
\(776\) −0.116844 −0.00419445
\(777\) −2.74456 −0.0984606
\(778\) −14.8614 −0.532807
\(779\) −10.7446 −0.384964
\(780\) 0 0
\(781\) 22.7446 0.813864
\(782\) 36.2337 1.29571
\(783\) 32.4674 1.16029
\(784\) −5.11684 −0.182744
\(785\) 0 0
\(786\) −6.51087 −0.232235
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −27.4891 −0.979260
\(789\) 0.233688 0.00831951
\(790\) 0 0
\(791\) 23.8397 0.847641
\(792\) 3.37228 0.119829
\(793\) −25.4891 −0.905145
\(794\) 24.9783 0.886445
\(795\) 0 0
\(796\) −11.2554 −0.398938
\(797\) 27.4891 0.973715 0.486857 0.873481i \(-0.338143\pi\)
0.486857 + 0.873481i \(0.338143\pi\)
\(798\) 5.48913 0.194313
\(799\) 46.9783 1.66197
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) −10.0000 −0.353112
\(803\) 29.4891 1.04065
\(804\) 9.48913 0.334656
\(805\) 0 0
\(806\) 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 \(0.860227\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) −10.7446 −0.376135
\(817\) 14.7446 0.515847
\(818\) −6.23369 −0.217956
\(819\) −6.51087 −0.227508
\(820\) 0 0
\(821\) −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 \(-0.256586\pi\)
0.692325 + 0.721586i \(0.256586\pi\)
\(824\) −9.48913 −0.330569
\(825\) 0 0
\(826\) 17.4891 0.608524
\(827\) 54.3505 1.88995 0.944977 0.327138i \(-0.106084\pi\)
0.944977 + 0.327138i \(0.106084\pi\)
\(828\) −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 0
\(831\) 1.48913 0.0516572
\(832\) 4.74456 0.164488
\(833\) −27.4891 −0.952442
\(834\) −3.76631 −0.130417
\(835\) 0 0
\(836\) 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 \(0.669999\pi\)
\(840\) 0 0
\(841\) 36.8832 1.27183
\(842\) 7.48913 0.258092
\(843\) −49.4891 −1.70450
\(844\) −14.1168 −0.485922
\(845\) 0 0
\(846\) −8.74456 −0.300644
\(847\) −0.510875 −0.0175539
\(848\) −1.37228 −0.0471243
\(849\) 10.9783 0.376773
\(850\) 0 0
\(851\) 6.74456 0.231201
\(852\) 13.4891 0.462130
\(853\) −35.4891 −1.21512 −0.607562 0.794272i \(-0.707852\pi\)
−0.607562 + 0.794272i \(0.707852\pi\)
\(854\) −7.37228 −0.252274
\(855\) 0 0
\(856\) −3.48913 −0.119256
\(857\) 29.1386 0.995355 0.497678 0.867362i \(-0.334186\pi\)
0.497678 + 0.867362i \(0.334186\pi\)
\(858\) −32.0000 −1.09246
\(859\) −19.7228 −0.672934 −0.336467 0.941695i \(-0.609232\pi\)
−0.336467 + 0.941695i \(0.609232\pi\)
\(860\) 0 0
\(861\) 14.7446 0.502493
\(862\) 1.37228 0.0467401
\(863\) −53.8397 −1.83272 −0.916362 0.400352i \(-0.868888\pi\)
−0.916362 + 0.400352i \(0.868888\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 12.9783 0.441019
\(867\) −23.7228 −0.805669
\(868\) 3.60597 0.122395
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −22.5109 −0.762752
\(872\) −0.116844 −0.00395684
\(873\) 0.116844 0.00395457
\(874\) −13.4891 −0.456276
\(875\) 0 0
\(876\) 17.4891 0.590903
\(877\) −49.3723 −1.66718 −0.833592 0.552381i \(-0.813719\pi\)
−0.833592 + 0.552381i \(0.813719\pi\)
\(878\) 13.3723 0.451293
\(879\) −37.7228 −1.27236
\(880\) 0 0
\(881\) 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 \(-0.438712\pi\)
0.191354 + 0.981521i \(0.438712\pi\)
\(884\) 25.4891 0.857292
\(885\) 0 0
\(886\) −16.9783 −0.570395
\(887\) 25.3723 0.851918 0.425959 0.904743i \(-0.359937\pi\)
0.425959 + 0.904743i \(0.359937\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −22.9783 −0.770666
\(890\) 0 0
\(891\) 37.0951 1.24273
\(892\) 1.37228 0.0459474
\(893\) −17.4891 −0.585251
\(894\) −22.9783 −0.768508
\(895\) 0 0
\(896\) 1.37228 0.0458447
\(897\) 64.0000 2.13690
\(898\) −18.0000 −0.600668
\(899\) −21.3288 −0.711355
\(900\) 0 0
\(901\) −7.37228 −0.245606
\(902\) 18.1168 0.603225
\(903\) −20.2337 −0.673335
\(904\) 17.3723 0.577793
\(905\) 0 0
\(906\) −40.0000 −1.32891
\(907\) −40.4674 −1.34370 −0.671849 0.740689i \(-0.734499\pi\)
−0.671849 + 0.740689i \(0.734499\pi\)
\(908\) 11.3723 0.377402
\(909\) −11.4891 −0.381070
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) −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 \(0.459344\pi\)
\(920\) 0 0
\(921\) −46.9783 −1.54799
\(922\) 39.0951 1.28753
\(923\) −32.0000 −1.05329
\(924\) −9.25544 −0.304482
\(925\) 0 0
\(926\) −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 \(0.329609\pi\)
\(930\) 0 0
\(931\) 10.2337 0.335396
\(932\) −6.23369 −0.204191
\(933\) 2.74456 0.0898529
\(934\) 30.3505 0.993100
\(935\) 0 0
\(936\) −4.74456 −0.155081
\(937\) −36.9783 −1.20803 −0.604013 0.796974i \(-0.706433\pi\)
−0.604013 + 0.796974i \(0.706433\pi\)
\(938\) −6.51087 −0.212588
\(939\) −6.97825 −0.227727
\(940\) 0 0
\(941\) 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
\(945\) 0 0
\(946\) −24.8614 −0.808314
\(947\) −37.0951 −1.20543 −0.602714 0.797957i \(-0.705914\pi\)
−0.602714 + 0.797957i \(0.705914\pi\)
\(948\) 9.48913 0.308192
\(949\) −41.4891 −1.34679
\(950\) 0 0
\(951\) 50.7446 1.64551
\(952\) 7.37228 0.238937
\(953\) −46.2337 −1.49766 −0.748828 0.662764i \(-0.769383\pi\)
−0.748828 + 0.662764i \(0.769383\pi\)
\(954\) 1.37228 0.0444292
\(955\) 0 0
\(956\) 17.6060 0.569418
\(957\) 54.7446 1.76964
\(958\) −3.25544 −0.105178
\(959\) 22.9783 0.742006
\(960\) 0 0
\(961\) −24.0951 −0.777261
\(962\) 4.74456 0.152971
\(963\) 3.48913 0.112435
\(964\) −10.2337 −0.329605
\(965\) 0 0
\(966\) 18.5109 0.595578
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) −0.372281 −0.0119656
\(969\) 21.4891 0.690330
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −5.37228 −0.171963
\(977\) 11.8832 0.380176 0.190088 0.981767i \(-0.439123\pi\)
0.190088 + 0.981767i \(0.439123\pi\)
\(978\) −38.7446 −1.23891
\(979\) −33.7228 −1.07779
\(980\) 0 0
\(981\) 0.116844 0.00373054
\(982\) −14.9783 −0.477975
\(983\) −30.8614 −0.984326 −0.492163 0.870503i \(-0.663794\pi\)
−0.492163 + 0.870503i \(0.663794\pi\)
\(984\) 10.7446 0.342524
\(985\) 0 0
\(986\) −43.6060 −1.38870
\(987\) 24.0000 0.763928
\(988\) −9.48913 −0.301889
\(989\) 49.7228 1.58109
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) −1.48913 −0.0471847
\(997\) −26.2337 −0.830829 −0.415415 0.909632i \(-0.636363\pi\)
−0.415415 + 0.909632i \(0.636363\pi\)
\(998\) 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 1850.2.a.q.1.1 2
5.2 odd 4 1850.2.b.m.149.1 4
5.3 odd 4 1850.2.b.m.149.4 4
5.4 even 2 370.2.a.f.1.2 2
15.14 odd 2 3330.2.a.bb.1.2 2
20.19 odd 2 2960.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.2 2 5.4 even 2
1850.2.a.q.1.1 2 1.1 even 1 trivial
1850.2.b.m.149.1 4 5.2 odd 4
1850.2.b.m.149.4 4 5.3 odd 4
2960.2.a.o.1.1 2 20.19 odd 2
3330.2.a.bb.1.2 2 15.14 odd 2