Properties

Label 8470.2.a.bo.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{9} +1.00000 q^{10} +2.82843 q^{12} -2.00000 q^{13} -1.00000 q^{14} -2.82843 q^{15} +1.00000 q^{16} -7.65685 q^{17} -5.00000 q^{18} +2.82843 q^{19} -1.00000 q^{20} +2.82843 q^{21} -2.82843 q^{23} -2.82843 q^{24} +1.00000 q^{25} +2.00000 q^{26} +5.65685 q^{27} +1.00000 q^{28} -3.17157 q^{29} +2.82843 q^{30} -5.65685 q^{31} -1.00000 q^{32} +7.65685 q^{34} -1.00000 q^{35} +5.00000 q^{36} +3.17157 q^{37} -2.82843 q^{38} -5.65685 q^{39} +1.00000 q^{40} +0.828427 q^{41} -2.82843 q^{42} +9.65685 q^{43} -5.00000 q^{45} +2.82843 q^{46} -8.00000 q^{47} +2.82843 q^{48} +1.00000 q^{49} -1.00000 q^{50} -21.6569 q^{51} -2.00000 q^{52} -10.4853 q^{53} -5.65685 q^{54} -1.00000 q^{56} +8.00000 q^{57} +3.17157 q^{58} -8.00000 q^{59} -2.82843 q^{60} +9.31371 q^{61} +5.65685 q^{62} +5.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +15.3137 q^{67} -7.65685 q^{68} -8.00000 q^{69} +1.00000 q^{70} -5.65685 q^{71} -5.00000 q^{72} -7.65685 q^{73} -3.17157 q^{74} +2.82843 q^{75} +2.82843 q^{76} +5.65685 q^{78} +5.17157 q^{79} -1.00000 q^{80} +1.00000 q^{81} -0.828427 q^{82} +2.34315 q^{83} +2.82843 q^{84} +7.65685 q^{85} -9.65685 q^{86} -8.97056 q^{87} -11.6569 q^{89} +5.00000 q^{90} -2.00000 q^{91} -2.82843 q^{92} -16.0000 q^{93} +8.00000 q^{94} -2.82843 q^{95} -2.82843 q^{96} -0.828427 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 10 q^{18} - 2 q^{20} + 2 q^{25} + 4 q^{26} + 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 10 q^{36} + 12 q^{37} + 2 q^{40} - 4 q^{41} + 8 q^{43} - 10 q^{45} - 16 q^{47} + 2 q^{49} - 2 q^{50} - 32 q^{51} - 4 q^{52} - 4 q^{53} - 2 q^{56} + 16 q^{57} + 12 q^{58} - 16 q^{59} - 4 q^{61} + 10 q^{63} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} - 16 q^{69} + 2 q^{70} - 10 q^{72} - 4 q^{73} - 12 q^{74} + 16 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 16 q^{83} + 4 q^{85} - 8 q^{86} + 16 q^{87} - 12 q^{89} + 10 q^{90} - 4 q^{91} - 32 q^{93} + 16 q^{94} + 4 q^{97} - 2 q^{98} + O(q^{100}) \)

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
\(3\) 2.82843 1.63299 0.816497 0.577350i \(-0.195913\pi\)
0.816497 + 0.577350i \(0.195913\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.82843 −1.15470
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 5.00000 1.66667
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 2.82843 0.816497
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.82843 −0.730297
\(16\) 1.00000 0.250000
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) −5.00000 −1.17851
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) −2.82843 −0.577350
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 5.65685 1.08866
\(28\) 1.00000 0.188982
\(29\) −3.17157 −0.588946 −0.294473 0.955660i \(-0.595144\pi\)
−0.294473 + 0.955660i \(0.595144\pi\)
\(30\) 2.82843 0.516398
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.65685 1.31314
\(35\) −1.00000 −0.169031
\(36\) 5.00000 0.833333
\(37\) 3.17157 0.521403 0.260702 0.965419i \(-0.416046\pi\)
0.260702 + 0.965419i \(0.416046\pi\)
\(38\) −2.82843 −0.458831
\(39\) −5.65685 −0.905822
\(40\) 1.00000 0.158114
\(41\) 0.828427 0.129379 0.0646893 0.997905i \(-0.479394\pi\)
0.0646893 + 0.997905i \(0.479394\pi\)
\(42\) −2.82843 −0.436436
\(43\) 9.65685 1.47266 0.736328 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) 0 0
\(45\) −5.00000 −0.745356
\(46\) 2.82843 0.417029
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 2.82843 0.408248
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −21.6569 −3.03257
\(52\) −2.00000 −0.277350
\(53\) −10.4853 −1.44026 −0.720132 0.693837i \(-0.755919\pi\)
−0.720132 + 0.693837i \(0.755919\pi\)
\(54\) −5.65685 −0.769800
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 8.00000 1.05963
\(58\) 3.17157 0.416448
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −2.82843 −0.365148
\(61\) 9.31371 1.19250 0.596249 0.802799i \(-0.296657\pi\)
0.596249 + 0.802799i \(0.296657\pi\)
\(62\) 5.65685 0.718421
\(63\) 5.00000 0.629941
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 15.3137 1.87087 0.935434 0.353502i \(-0.115009\pi\)
0.935434 + 0.353502i \(0.115009\pi\)
\(68\) −7.65685 −0.928530
\(69\) −8.00000 −0.963087
\(70\) 1.00000 0.119523
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) −5.00000 −0.589256
\(73\) −7.65685 −0.896167 −0.448084 0.893992i \(-0.647893\pi\)
−0.448084 + 0.893992i \(0.647893\pi\)
\(74\) −3.17157 −0.368688
\(75\) 2.82843 0.326599
\(76\) 2.82843 0.324443
\(77\) 0 0
\(78\) 5.65685 0.640513
\(79\) 5.17157 0.581847 0.290924 0.956746i \(-0.406037\pi\)
0.290924 + 0.956746i \(0.406037\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −0.828427 −0.0914845
\(83\) 2.34315 0.257194 0.128597 0.991697i \(-0.458953\pi\)
0.128597 + 0.991697i \(0.458953\pi\)
\(84\) 2.82843 0.308607
\(85\) 7.65685 0.830502
\(86\) −9.65685 −1.04133
\(87\) −8.97056 −0.961745
\(88\) 0 0
\(89\) −11.6569 −1.23562 −0.617812 0.786326i \(-0.711981\pi\)
−0.617812 + 0.786326i \(0.711981\pi\)
\(90\) 5.00000 0.527046
\(91\) −2.00000 −0.209657
\(92\) −2.82843 −0.294884
\(93\) −16.0000 −1.65912
\(94\) 8.00000 0.825137
\(95\) −2.82843 −0.290191
\(96\) −2.82843 −0.288675
\(97\) −0.828427 −0.0841140 −0.0420570 0.999115i \(-0.513391\pi\)
−0.0420570 + 0.999115i \(0.513391\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 21.6569 2.14435
\(103\) −5.65685 −0.557386 −0.278693 0.960380i \(-0.589901\pi\)
−0.278693 + 0.960380i \(0.589901\pi\)
\(104\) 2.00000 0.196116
\(105\) −2.82843 −0.276026
\(106\) 10.4853 1.01842
\(107\) 1.65685 0.160174 0.0800871 0.996788i \(-0.474480\pi\)
0.0800871 + 0.996788i \(0.474480\pi\)
\(108\) 5.65685 0.544331
\(109\) −14.4853 −1.38744 −0.693719 0.720246i \(-0.744029\pi\)
−0.693719 + 0.720246i \(0.744029\pi\)
\(110\) 0 0
\(111\) 8.97056 0.851448
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −8.00000 −0.749269
\(115\) 2.82843 0.263752
\(116\) −3.17157 −0.294473
\(117\) −10.0000 −0.924500
\(118\) 8.00000 0.736460
\(119\) −7.65685 −0.701903
\(120\) 2.82843 0.258199
\(121\) 0 0
\(122\) −9.31371 −0.843224
\(123\) 2.34315 0.211274
\(124\) −5.65685 −0.508001
\(125\) −1.00000 −0.0894427
\(126\) −5.00000 −0.445435
\(127\) −19.3137 −1.71381 −0.856907 0.515471i \(-0.827617\pi\)
−0.856907 + 0.515471i \(0.827617\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 27.3137 2.40484
\(130\) −2.00000 −0.175412
\(131\) −18.8284 −1.64505 −0.822524 0.568731i \(-0.807435\pi\)
−0.822524 + 0.568731i \(0.807435\pi\)
\(132\) 0 0
\(133\) 2.82843 0.245256
\(134\) −15.3137 −1.32290
\(135\) −5.65685 −0.486864
\(136\) 7.65685 0.656570
\(137\) 5.31371 0.453981 0.226990 0.973897i \(-0.427111\pi\)
0.226990 + 0.973897i \(0.427111\pi\)
\(138\) 8.00000 0.681005
\(139\) −18.8284 −1.59701 −0.798503 0.601991i \(-0.794374\pi\)
−0.798503 + 0.601991i \(0.794374\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −22.6274 −1.90557
\(142\) 5.65685 0.474713
\(143\) 0 0
\(144\) 5.00000 0.416667
\(145\) 3.17157 0.263385
\(146\) 7.65685 0.633686
\(147\) 2.82843 0.233285
\(148\) 3.17157 0.260702
\(149\) 1.51472 0.124091 0.0620453 0.998073i \(-0.480238\pi\)
0.0620453 + 0.998073i \(0.480238\pi\)
\(150\) −2.82843 −0.230940
\(151\) −10.8284 −0.881205 −0.440602 0.897702i \(-0.645235\pi\)
−0.440602 + 0.897702i \(0.645235\pi\)
\(152\) −2.82843 −0.229416
\(153\) −38.2843 −3.09510
\(154\) 0 0
\(155\) 5.65685 0.454369
\(156\) −5.65685 −0.452911
\(157\) 15.6569 1.24955 0.624777 0.780804i \(-0.285190\pi\)
0.624777 + 0.780804i \(0.285190\pi\)
\(158\) −5.17157 −0.411428
\(159\) −29.6569 −2.35194
\(160\) 1.00000 0.0790569
\(161\) −2.82843 −0.222911
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0.828427 0.0646893
\(165\) 0 0
\(166\) −2.34315 −0.181863
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −2.82843 −0.218218
\(169\) −9.00000 −0.692308
\(170\) −7.65685 −0.587254
\(171\) 14.1421 1.08148
\(172\) 9.65685 0.736328
\(173\) 1.31371 0.0998794 0.0499397 0.998752i \(-0.484097\pi\)
0.0499397 + 0.998752i \(0.484097\pi\)
\(174\) 8.97056 0.680057
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −22.6274 −1.70078
\(178\) 11.6569 0.873718
\(179\) 6.34315 0.474109 0.237054 0.971496i \(-0.423818\pi\)
0.237054 + 0.971496i \(0.423818\pi\)
\(180\) −5.00000 −0.372678
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 2.00000 0.148250
\(183\) 26.3431 1.94734
\(184\) 2.82843 0.208514
\(185\) −3.17157 −0.233179
\(186\) 16.0000 1.17318
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 5.65685 0.411476
\(190\) 2.82843 0.205196
\(191\) 19.3137 1.39749 0.698745 0.715370i \(-0.253742\pi\)
0.698745 + 0.715370i \(0.253742\pi\)
\(192\) 2.82843 0.204124
\(193\) −21.3137 −1.53419 −0.767097 0.641531i \(-0.778300\pi\)
−0.767097 + 0.641531i \(0.778300\pi\)
\(194\) 0.828427 0.0594776
\(195\) 5.65685 0.405096
\(196\) 1.00000 0.0714286
\(197\) 12.3431 0.879413 0.439706 0.898142i \(-0.355083\pi\)
0.439706 + 0.898142i \(0.355083\pi\)
\(198\) 0 0
\(199\) −11.3137 −0.802008 −0.401004 0.916076i \(-0.631339\pi\)
−0.401004 + 0.916076i \(0.631339\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 43.3137 3.05511
\(202\) 10.0000 0.703598
\(203\) −3.17157 −0.222601
\(204\) −21.6569 −1.51628
\(205\) −0.828427 −0.0578599
\(206\) 5.65685 0.394132
\(207\) −14.1421 −0.982946
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 2.82843 0.195180
\(211\) −15.3137 −1.05424 −0.527120 0.849791i \(-0.676728\pi\)
−0.527120 + 0.849791i \(0.676728\pi\)
\(212\) −10.4853 −0.720132
\(213\) −16.0000 −1.09630
\(214\) −1.65685 −0.113260
\(215\) −9.65685 −0.658592
\(216\) −5.65685 −0.384900
\(217\) −5.65685 −0.384012
\(218\) 14.4853 0.981067
\(219\) −21.6569 −1.46343
\(220\) 0 0
\(221\) 15.3137 1.03011
\(222\) −8.97056 −0.602065
\(223\) −28.2843 −1.89405 −0.947027 0.321153i \(-0.895930\pi\)
−0.947027 + 0.321153i \(0.895930\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 5.00000 0.333333
\(226\) −2.00000 −0.133038
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 8.00000 0.529813
\(229\) −1.31371 −0.0868123 −0.0434062 0.999058i \(-0.513821\pi\)
−0.0434062 + 0.999058i \(0.513821\pi\)
\(230\) −2.82843 −0.186501
\(231\) 0 0
\(232\) 3.17157 0.208224
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 10.0000 0.653720
\(235\) 8.00000 0.521862
\(236\) −8.00000 −0.520756
\(237\) 14.6274 0.950153
\(238\) 7.65685 0.496320
\(239\) 7.51472 0.486087 0.243043 0.970015i \(-0.421854\pi\)
0.243043 + 0.970015i \(0.421854\pi\)
\(240\) −2.82843 −0.182574
\(241\) 8.82843 0.568689 0.284344 0.958722i \(-0.408224\pi\)
0.284344 + 0.958722i \(0.408224\pi\)
\(242\) 0 0
\(243\) −14.1421 −0.907218
\(244\) 9.31371 0.596249
\(245\) −1.00000 −0.0638877
\(246\) −2.34315 −0.149394
\(247\) −5.65685 −0.359937
\(248\) 5.65685 0.359211
\(249\) 6.62742 0.419995
\(250\) 1.00000 0.0632456
\(251\) −11.3137 −0.714115 −0.357057 0.934082i \(-0.616220\pi\)
−0.357057 + 0.934082i \(0.616220\pi\)
\(252\) 5.00000 0.314970
\(253\) 0 0
\(254\) 19.3137 1.21185
\(255\) 21.6569 1.35620
\(256\) 1.00000 0.0625000
\(257\) 23.1716 1.44540 0.722702 0.691160i \(-0.242900\pi\)
0.722702 + 0.691160i \(0.242900\pi\)
\(258\) −27.3137 −1.70048
\(259\) 3.17157 0.197072
\(260\) 2.00000 0.124035
\(261\) −15.8579 −0.981577
\(262\) 18.8284 1.16322
\(263\) 14.6274 0.901965 0.450983 0.892533i \(-0.351074\pi\)
0.450983 + 0.892533i \(0.351074\pi\)
\(264\) 0 0
\(265\) 10.4853 0.644106
\(266\) −2.82843 −0.173422
\(267\) −32.9706 −2.01777
\(268\) 15.3137 0.935434
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 5.65685 0.344265
\(271\) −16.9706 −1.03089 −0.515444 0.856923i \(-0.672373\pi\)
−0.515444 + 0.856923i \(0.672373\pi\)
\(272\) −7.65685 −0.464265
\(273\) −5.65685 −0.342368
\(274\) −5.31371 −0.321013
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 13.3137 0.799943 0.399972 0.916528i \(-0.369020\pi\)
0.399972 + 0.916528i \(0.369020\pi\)
\(278\) 18.8284 1.12925
\(279\) −28.2843 −1.69334
\(280\) 1.00000 0.0597614
\(281\) 6.97056 0.415829 0.207914 0.978147i \(-0.433332\pi\)
0.207914 + 0.978147i \(0.433332\pi\)
\(282\) 22.6274 1.34744
\(283\) 22.6274 1.34506 0.672530 0.740070i \(-0.265208\pi\)
0.672530 + 0.740070i \(0.265208\pi\)
\(284\) −5.65685 −0.335673
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 0.828427 0.0489005
\(288\) −5.00000 −0.294628
\(289\) 41.6274 2.44867
\(290\) −3.17157 −0.186241
\(291\) −2.34315 −0.137358
\(292\) −7.65685 −0.448084
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) −2.82843 −0.164957
\(295\) 8.00000 0.465778
\(296\) −3.17157 −0.184344
\(297\) 0 0
\(298\) −1.51472 −0.0877453
\(299\) 5.65685 0.327144
\(300\) 2.82843 0.163299
\(301\) 9.65685 0.556612
\(302\) 10.8284 0.623106
\(303\) −28.2843 −1.62489
\(304\) 2.82843 0.162221
\(305\) −9.31371 −0.533301
\(306\) 38.2843 2.18857
\(307\) 8.97056 0.511977 0.255989 0.966680i \(-0.417599\pi\)
0.255989 + 0.966680i \(0.417599\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) −5.65685 −0.321288
\(311\) 0.970563 0.0550356 0.0275178 0.999621i \(-0.491240\pi\)
0.0275178 + 0.999621i \(0.491240\pi\)
\(312\) 5.65685 0.320256
\(313\) −8.82843 −0.499012 −0.249506 0.968373i \(-0.580268\pi\)
−0.249506 + 0.968373i \(0.580268\pi\)
\(314\) −15.6569 −0.883567
\(315\) −5.00000 −0.281718
\(316\) 5.17157 0.290924
\(317\) 3.17157 0.178133 0.0890666 0.996026i \(-0.471612\pi\)
0.0890666 + 0.996026i \(0.471612\pi\)
\(318\) 29.6569 1.66307
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 4.68629 0.261563
\(322\) 2.82843 0.157622
\(323\) −21.6569 −1.20502
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −12.0000 −0.664619
\(327\) −40.9706 −2.26568
\(328\) −0.828427 −0.0457422
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 28.9706 1.59237 0.796183 0.605056i \(-0.206849\pi\)
0.796183 + 0.605056i \(0.206849\pi\)
\(332\) 2.34315 0.128597
\(333\) 15.8579 0.869006
\(334\) −16.0000 −0.875481
\(335\) −15.3137 −0.836677
\(336\) 2.82843 0.154303
\(337\) −29.3137 −1.59682 −0.798410 0.602115i \(-0.794325\pi\)
−0.798410 + 0.602115i \(0.794325\pi\)
\(338\) 9.00000 0.489535
\(339\) 5.65685 0.307238
\(340\) 7.65685 0.415251
\(341\) 0 0
\(342\) −14.1421 −0.764719
\(343\) 1.00000 0.0539949
\(344\) −9.65685 −0.520663
\(345\) 8.00000 0.430706
\(346\) −1.31371 −0.0706254
\(347\) −28.9706 −1.55522 −0.777611 0.628746i \(-0.783568\pi\)
−0.777611 + 0.628746i \(0.783568\pi\)
\(348\) −8.97056 −0.480873
\(349\) −26.9706 −1.44370 −0.721851 0.692049i \(-0.756708\pi\)
−0.721851 + 0.692049i \(0.756708\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −11.3137 −0.603881
\(352\) 0 0
\(353\) −24.8284 −1.32148 −0.660742 0.750613i \(-0.729758\pi\)
−0.660742 + 0.750613i \(0.729758\pi\)
\(354\) 22.6274 1.20263
\(355\) 5.65685 0.300235
\(356\) −11.6569 −0.617812
\(357\) −21.6569 −1.14620
\(358\) −6.34315 −0.335246
\(359\) −33.4558 −1.76573 −0.882866 0.469625i \(-0.844389\pi\)
−0.882866 + 0.469625i \(0.844389\pi\)
\(360\) 5.00000 0.263523
\(361\) −11.0000 −0.578947
\(362\) 14.0000 0.735824
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 7.65685 0.400778
\(366\) −26.3431 −1.37698
\(367\) 28.2843 1.47643 0.738213 0.674567i \(-0.235670\pi\)
0.738213 + 0.674567i \(0.235670\pi\)
\(368\) −2.82843 −0.147442
\(369\) 4.14214 0.215631
\(370\) 3.17157 0.164882
\(371\) −10.4853 −0.544369
\(372\) −16.0000 −0.829561
\(373\) −3.65685 −0.189345 −0.0946724 0.995508i \(-0.530180\pi\)
−0.0946724 + 0.995508i \(0.530180\pi\)
\(374\) 0 0
\(375\) −2.82843 −0.146059
\(376\) 8.00000 0.412568
\(377\) 6.34315 0.326689
\(378\) −5.65685 −0.290957
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −2.82843 −0.145095
\(381\) −54.6274 −2.79865
\(382\) −19.3137 −0.988175
\(383\) −29.6569 −1.51539 −0.757697 0.652606i \(-0.773676\pi\)
−0.757697 + 0.652606i \(0.773676\pi\)
\(384\) −2.82843 −0.144338
\(385\) 0 0
\(386\) 21.3137 1.08484
\(387\) 48.2843 2.45443
\(388\) −0.828427 −0.0420570
\(389\) 30.9706 1.57027 0.785135 0.619325i \(-0.212594\pi\)
0.785135 + 0.619325i \(0.212594\pi\)
\(390\) −5.65685 −0.286446
\(391\) 21.6569 1.09523
\(392\) −1.00000 −0.0505076
\(393\) −53.2548 −2.68635
\(394\) −12.3431 −0.621839
\(395\) −5.17157 −0.260210
\(396\) 0 0
\(397\) −22.9706 −1.15286 −0.576430 0.817147i \(-0.695555\pi\)
−0.576430 + 0.817147i \(0.695555\pi\)
\(398\) 11.3137 0.567105
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −43.3137 −2.16029
\(403\) 11.3137 0.563576
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) 3.17157 0.157403
\(407\) 0 0
\(408\) 21.6569 1.07217
\(409\) 12.1421 0.600390 0.300195 0.953878i \(-0.402948\pi\)
0.300195 + 0.953878i \(0.402948\pi\)
\(410\) 0.828427 0.0409131
\(411\) 15.0294 0.741347
\(412\) −5.65685 −0.278693
\(413\) −8.00000 −0.393654
\(414\) 14.1421 0.695048
\(415\) −2.34315 −0.115021
\(416\) 2.00000 0.0980581
\(417\) −53.2548 −2.60790
\(418\) 0 0
\(419\) −30.6274 −1.49625 −0.748124 0.663559i \(-0.769045\pi\)
−0.748124 + 0.663559i \(0.769045\pi\)
\(420\) −2.82843 −0.138013
\(421\) 28.6274 1.39521 0.697607 0.716480i \(-0.254248\pi\)
0.697607 + 0.716480i \(0.254248\pi\)
\(422\) 15.3137 0.745460
\(423\) −40.0000 −1.94487
\(424\) 10.4853 0.509210
\(425\) −7.65685 −0.371412
\(426\) 16.0000 0.775203
\(427\) 9.31371 0.450722
\(428\) 1.65685 0.0800871
\(429\) 0 0
\(430\) 9.65685 0.465695
\(431\) 19.7990 0.953684 0.476842 0.878989i \(-0.341781\pi\)
0.476842 + 0.878989i \(0.341781\pi\)
\(432\) 5.65685 0.272166
\(433\) 20.8284 1.00095 0.500475 0.865751i \(-0.333159\pi\)
0.500475 + 0.865751i \(0.333159\pi\)
\(434\) 5.65685 0.271538
\(435\) 8.97056 0.430106
\(436\) −14.4853 −0.693719
\(437\) −8.00000 −0.382692
\(438\) 21.6569 1.03480
\(439\) −24.9706 −1.19178 −0.595890 0.803066i \(-0.703201\pi\)
−0.595890 + 0.803066i \(0.703201\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) −15.3137 −0.728399
\(443\) −8.68629 −0.412698 −0.206349 0.978478i \(-0.566158\pi\)
−0.206349 + 0.978478i \(0.566158\pi\)
\(444\) 8.97056 0.425724
\(445\) 11.6569 0.552588
\(446\) 28.2843 1.33930
\(447\) 4.28427 0.202639
\(448\) 1.00000 0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) −5.00000 −0.235702
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) −30.6274 −1.43900
\(454\) −24.0000 −1.12638
\(455\) 2.00000 0.0937614
\(456\) −8.00000 −0.374634
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 1.31371 0.0613856
\(459\) −43.3137 −2.02171
\(460\) 2.82843 0.131876
\(461\) 11.6569 0.542914 0.271457 0.962451i \(-0.412495\pi\)
0.271457 + 0.962451i \(0.412495\pi\)
\(462\) 0 0
\(463\) −9.45584 −0.439450 −0.219725 0.975562i \(-0.570516\pi\)
−0.219725 + 0.975562i \(0.570516\pi\)
\(464\) −3.17157 −0.147237
\(465\) 16.0000 0.741982
\(466\) −6.00000 −0.277945
\(467\) 11.7990 0.545992 0.272996 0.962015i \(-0.411985\pi\)
0.272996 + 0.962015i \(0.411985\pi\)
\(468\) −10.0000 −0.462250
\(469\) 15.3137 0.707121
\(470\) −8.00000 −0.369012
\(471\) 44.2843 2.04051
\(472\) 8.00000 0.368230
\(473\) 0 0
\(474\) −14.6274 −0.671860
\(475\) 2.82843 0.129777
\(476\) −7.65685 −0.350951
\(477\) −52.4264 −2.40044
\(478\) −7.51472 −0.343715
\(479\) −35.3137 −1.61352 −0.806762 0.590876i \(-0.798782\pi\)
−0.806762 + 0.590876i \(0.798782\pi\)
\(480\) 2.82843 0.129099
\(481\) −6.34315 −0.289223
\(482\) −8.82843 −0.402124
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) 0.828427 0.0376169
\(486\) 14.1421 0.641500
\(487\) −36.7696 −1.66619 −0.833094 0.553132i \(-0.813433\pi\)
−0.833094 + 0.553132i \(0.813433\pi\)
\(488\) −9.31371 −0.421612
\(489\) 33.9411 1.53487
\(490\) 1.00000 0.0451754
\(491\) 8.68629 0.392007 0.196003 0.980603i \(-0.437204\pi\)
0.196003 + 0.980603i \(0.437204\pi\)
\(492\) 2.34315 0.105637
\(493\) 24.2843 1.09371
\(494\) 5.65685 0.254514
\(495\) 0 0
\(496\) −5.65685 −0.254000
\(497\) −5.65685 −0.253745
\(498\) −6.62742 −0.296982
\(499\) 28.9706 1.29690 0.648450 0.761257i \(-0.275417\pi\)
0.648450 + 0.761257i \(0.275417\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 45.2548 2.02184
\(502\) 11.3137 0.504956
\(503\) 3.31371 0.147751 0.0738755 0.997267i \(-0.476463\pi\)
0.0738755 + 0.997267i \(0.476463\pi\)
\(504\) −5.00000 −0.222718
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) −25.4558 −1.13053
\(508\) −19.3137 −0.856907
\(509\) −10.6863 −0.473662 −0.236831 0.971551i \(-0.576109\pi\)
−0.236831 + 0.971551i \(0.576109\pi\)
\(510\) −21.6569 −0.958982
\(511\) −7.65685 −0.338719
\(512\) −1.00000 −0.0441942
\(513\) 16.0000 0.706417
\(514\) −23.1716 −1.02205
\(515\) 5.65685 0.249271
\(516\) 27.3137 1.20242
\(517\) 0 0
\(518\) −3.17157 −0.139351
\(519\) 3.71573 0.163102
\(520\) −2.00000 −0.0877058
\(521\) 41.5980 1.82244 0.911220 0.411919i \(-0.135141\pi\)
0.911220 + 0.411919i \(0.135141\pi\)
\(522\) 15.8579 0.694080
\(523\) 10.3431 0.452274 0.226137 0.974095i \(-0.427390\pi\)
0.226137 + 0.974095i \(0.427390\pi\)
\(524\) −18.8284 −0.822524
\(525\) 2.82843 0.123443
\(526\) −14.6274 −0.637786
\(527\) 43.3137 1.88677
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) −10.4853 −0.455452
\(531\) −40.0000 −1.73585
\(532\) 2.82843 0.122628
\(533\) −1.65685 −0.0717663
\(534\) 32.9706 1.42678
\(535\) −1.65685 −0.0716321
\(536\) −15.3137 −0.661451
\(537\) 17.9411 0.774217
\(538\) −2.00000 −0.0862261
\(539\) 0 0
\(540\) −5.65685 −0.243432
\(541\) 33.1127 1.42363 0.711813 0.702369i \(-0.247874\pi\)
0.711813 + 0.702369i \(0.247874\pi\)
\(542\) 16.9706 0.728948
\(543\) −39.5980 −1.69931
\(544\) 7.65685 0.328285
\(545\) 14.4853 0.620481
\(546\) 5.65685 0.242091
\(547\) −23.3137 −0.996822 −0.498411 0.866941i \(-0.666083\pi\)
−0.498411 + 0.866941i \(0.666083\pi\)
\(548\) 5.31371 0.226990
\(549\) 46.5685 1.98750
\(550\) 0 0
\(551\) −8.97056 −0.382159
\(552\) 8.00000 0.340503
\(553\) 5.17157 0.219918
\(554\) −13.3137 −0.565645
\(555\) −8.97056 −0.380779
\(556\) −18.8284 −0.798503
\(557\) 26.9706 1.14278 0.571390 0.820679i \(-0.306404\pi\)
0.571390 + 0.820679i \(0.306404\pi\)
\(558\) 28.2843 1.19737
\(559\) −19.3137 −0.816883
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −6.97056 −0.294035
\(563\) 22.6274 0.953632 0.476816 0.879003i \(-0.341791\pi\)
0.476816 + 0.879003i \(0.341791\pi\)
\(564\) −22.6274 −0.952786
\(565\) −2.00000 −0.0841406
\(566\) −22.6274 −0.951101
\(567\) 1.00000 0.0419961
\(568\) 5.65685 0.237356
\(569\) −16.6274 −0.697058 −0.348529 0.937298i \(-0.613319\pi\)
−0.348529 + 0.937298i \(0.613319\pi\)
\(570\) 8.00000 0.335083
\(571\) 35.5980 1.48973 0.744865 0.667216i \(-0.232514\pi\)
0.744865 + 0.667216i \(0.232514\pi\)
\(572\) 0 0
\(573\) 54.6274 2.28209
\(574\) −0.828427 −0.0345779
\(575\) −2.82843 −0.117954
\(576\) 5.00000 0.208333
\(577\) 25.1127 1.04546 0.522728 0.852500i \(-0.324914\pi\)
0.522728 + 0.852500i \(0.324914\pi\)
\(578\) −41.6274 −1.73147
\(579\) −60.2843 −2.50533
\(580\) 3.17157 0.131692
\(581\) 2.34315 0.0972101
\(582\) 2.34315 0.0971265
\(583\) 0 0
\(584\) 7.65685 0.316843
\(585\) 10.0000 0.413449
\(586\) 2.00000 0.0826192
\(587\) −38.1421 −1.57429 −0.787147 0.616765i \(-0.788443\pi\)
−0.787147 + 0.616765i \(0.788443\pi\)
\(588\) 2.82843 0.116642
\(589\) −16.0000 −0.659269
\(590\) −8.00000 −0.329355
\(591\) 34.9117 1.43607
\(592\) 3.17157 0.130351
\(593\) −20.3431 −0.835393 −0.417696 0.908587i \(-0.637162\pi\)
−0.417696 + 0.908587i \(0.637162\pi\)
\(594\) 0 0
\(595\) 7.65685 0.313900
\(596\) 1.51472 0.0620453
\(597\) −32.0000 −1.30967
\(598\) −5.65685 −0.231326
\(599\) 21.6569 0.884875 0.442438 0.896799i \(-0.354114\pi\)
0.442438 + 0.896799i \(0.354114\pi\)
\(600\) −2.82843 −0.115470
\(601\) −12.8284 −0.523282 −0.261641 0.965165i \(-0.584264\pi\)
−0.261641 + 0.965165i \(0.584264\pi\)
\(602\) −9.65685 −0.393584
\(603\) 76.5685 3.11811
\(604\) −10.8284 −0.440602
\(605\) 0 0
\(606\) 28.2843 1.14897
\(607\) −29.6569 −1.20373 −0.601867 0.798596i \(-0.705576\pi\)
−0.601867 + 0.798596i \(0.705576\pi\)
\(608\) −2.82843 −0.114708
\(609\) −8.97056 −0.363506
\(610\) 9.31371 0.377101
\(611\) 16.0000 0.647291
\(612\) −38.2843 −1.54755
\(613\) −22.9706 −0.927772 −0.463886 0.885895i \(-0.653545\pi\)
−0.463886 + 0.885895i \(0.653545\pi\)
\(614\) −8.97056 −0.362022
\(615\) −2.34315 −0.0944848
\(616\) 0 0
\(617\) −32.3431 −1.30209 −0.651043 0.759041i \(-0.725668\pi\)
−0.651043 + 0.759041i \(0.725668\pi\)
\(618\) 16.0000 0.643614
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 5.65685 0.227185
\(621\) −16.0000 −0.642058
\(622\) −0.970563 −0.0389160
\(623\) −11.6569 −0.467022
\(624\) −5.65685 −0.226455
\(625\) 1.00000 0.0400000
\(626\) 8.82843 0.352855
\(627\) 0 0
\(628\) 15.6569 0.624777
\(629\) −24.2843 −0.968277
\(630\) 5.00000 0.199205
\(631\) −22.6274 −0.900783 −0.450392 0.892831i \(-0.648716\pi\)
−0.450392 + 0.892831i \(0.648716\pi\)
\(632\) −5.17157 −0.205714
\(633\) −43.3137 −1.72157
\(634\) −3.17157 −0.125959
\(635\) 19.3137 0.766441
\(636\) −29.6569 −1.17597
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −28.2843 −1.11891
\(640\) 1.00000 0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −4.68629 −0.184953
\(643\) 14.1421 0.557711 0.278856 0.960333i \(-0.410045\pi\)
0.278856 + 0.960333i \(0.410045\pi\)
\(644\) −2.82843 −0.111456
\(645\) −27.3137 −1.07548
\(646\) 21.6569 0.852078
\(647\) 12.2843 0.482945 0.241472 0.970408i \(-0.422370\pi\)
0.241472 + 0.970408i \(0.422370\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) −16.0000 −0.627089
\(652\) 12.0000 0.469956
\(653\) 7.45584 0.291770 0.145885 0.989302i \(-0.453397\pi\)
0.145885 + 0.989302i \(0.453397\pi\)
\(654\) 40.9706 1.60208
\(655\) 18.8284 0.735688
\(656\) 0.828427 0.0323446
\(657\) −38.2843 −1.49361
\(658\) 8.00000 0.311872
\(659\) 9.65685 0.376178 0.188089 0.982152i \(-0.439771\pi\)
0.188089 + 0.982152i \(0.439771\pi\)
\(660\) 0 0
\(661\) −6.00000 −0.233373 −0.116686 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) −28.9706 −1.12597
\(663\) 43.3137 1.68217
\(664\) −2.34315 −0.0909317
\(665\) −2.82843 −0.109682
\(666\) −15.8579 −0.614480
\(667\) 8.97056 0.347342
\(668\) 16.0000 0.619059
\(669\) −80.0000 −3.09298
\(670\) 15.3137 0.591620
\(671\) 0 0
\(672\) −2.82843 −0.109109
\(673\) 36.6274 1.41188 0.705942 0.708270i \(-0.250524\pi\)
0.705942 + 0.708270i \(0.250524\pi\)
\(674\) 29.3137 1.12912
\(675\) 5.65685 0.217732
\(676\) −9.00000 −0.346154
\(677\) 14.0000 0.538064 0.269032 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) −5.65685 −0.217250
\(679\) −0.828427 −0.0317921
\(680\) −7.65685 −0.293627
\(681\) 67.8823 2.60125
\(682\) 0 0
\(683\) 22.3431 0.854937 0.427468 0.904030i \(-0.359406\pi\)
0.427468 + 0.904030i \(0.359406\pi\)
\(684\) 14.1421 0.540738
\(685\) −5.31371 −0.203026
\(686\) −1.00000 −0.0381802
\(687\) −3.71573 −0.141764
\(688\) 9.65685 0.368164
\(689\) 20.9706 0.798915
\(690\) −8.00000 −0.304555
\(691\) 0.970563 0.0369219 0.0184610 0.999830i \(-0.494123\pi\)
0.0184610 + 0.999830i \(0.494123\pi\)
\(692\) 1.31371 0.0499397
\(693\) 0 0
\(694\) 28.9706 1.09971
\(695\) 18.8284 0.714203
\(696\) 8.97056 0.340028
\(697\) −6.34315 −0.240264
\(698\) 26.9706 1.02085
\(699\) 16.9706 0.641886
\(700\) 1.00000 0.0377964
\(701\) −40.8284 −1.54207 −0.771034 0.636794i \(-0.780260\pi\)
−0.771034 + 0.636794i \(0.780260\pi\)
\(702\) 11.3137 0.427008
\(703\) 8.97056 0.338331
\(704\) 0 0
\(705\) 22.6274 0.852198
\(706\) 24.8284 0.934430
\(707\) −10.0000 −0.376089
\(708\) −22.6274 −0.850390
\(709\) 26.2843 0.987127 0.493563 0.869710i \(-0.335694\pi\)
0.493563 + 0.869710i \(0.335694\pi\)
\(710\) −5.65685 −0.212298
\(711\) 25.8579 0.969746
\(712\) 11.6569 0.436859
\(713\) 16.0000 0.599205
\(714\) 21.6569 0.810487
\(715\) 0 0
\(716\) 6.34315 0.237054
\(717\) 21.2548 0.793776
\(718\) 33.4558 1.24856
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) −5.00000 −0.186339
\(721\) −5.65685 −0.210672
\(722\) 11.0000 0.409378
\(723\) 24.9706 0.928665
\(724\) −14.0000 −0.520306
\(725\) −3.17157 −0.117789
\(726\) 0 0
\(727\) 4.68629 0.173805 0.0869025 0.996217i \(-0.472303\pi\)
0.0869025 + 0.996217i \(0.472303\pi\)
\(728\) 2.00000 0.0741249
\(729\) −43.0000 −1.59259
\(730\) −7.65685 −0.283393
\(731\) −73.9411 −2.73481
\(732\) 26.3431 0.973671
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) −28.2843 −1.04399
\(735\) −2.82843 −0.104328
\(736\) 2.82843 0.104257
\(737\) 0 0
\(738\) −4.14214 −0.152474
\(739\) −41.2548 −1.51758 −0.758792 0.651333i \(-0.774210\pi\)
−0.758792 + 0.651333i \(0.774210\pi\)
\(740\) −3.17157 −0.116589
\(741\) −16.0000 −0.587775
\(742\) 10.4853 0.384927
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 16.0000 0.586588
\(745\) −1.51472 −0.0554950
\(746\) 3.65685 0.133887
\(747\) 11.7157 0.428656
\(748\) 0 0
\(749\) 1.65685 0.0605401
\(750\) 2.82843 0.103280
\(751\) 39.5980 1.44495 0.722475 0.691397i \(-0.243004\pi\)
0.722475 + 0.691397i \(0.243004\pi\)
\(752\) −8.00000 −0.291730
\(753\) −32.0000 −1.16614
\(754\) −6.34315 −0.231004
\(755\) 10.8284 0.394087
\(756\) 5.65685 0.205738
\(757\) −25.1127 −0.912737 −0.456368 0.889791i \(-0.650850\pi\)
−0.456368 + 0.889791i \(0.650850\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 2.82843 0.102598
\(761\) 7.45584 0.270274 0.135137 0.990827i \(-0.456853\pi\)
0.135137 + 0.990827i \(0.456853\pi\)
\(762\) 54.6274 1.97894
\(763\) −14.4853 −0.524402
\(764\) 19.3137 0.698745
\(765\) 38.2843 1.38417
\(766\) 29.6569 1.07155
\(767\) 16.0000 0.577727
\(768\) 2.82843 0.102062
\(769\) 14.4853 0.522353 0.261176 0.965291i \(-0.415890\pi\)
0.261176 + 0.965291i \(0.415890\pi\)
\(770\) 0 0
\(771\) 65.5391 2.36033
\(772\) −21.3137 −0.767097
\(773\) −10.6863 −0.384359 −0.192180 0.981360i \(-0.561556\pi\)
−0.192180 + 0.981360i \(0.561556\pi\)
\(774\) −48.2843 −1.73554
\(775\) −5.65685 −0.203200
\(776\) 0.828427 0.0297388
\(777\) 8.97056 0.321817
\(778\) −30.9706 −1.11035
\(779\) 2.34315 0.0839519
\(780\) 5.65685 0.202548
\(781\) 0 0
\(782\) −21.6569 −0.774448
\(783\) −17.9411 −0.641164
\(784\) 1.00000 0.0357143
\(785\) −15.6569 −0.558817
\(786\) 53.2548 1.89954
\(787\) −30.6274 −1.09175 −0.545875 0.837867i \(-0.683803\pi\)
−0.545875 + 0.837867i \(0.683803\pi\)
\(788\) 12.3431 0.439706
\(789\) 41.3726 1.47290
\(790\) 5.17157 0.183996
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) −18.6274 −0.661479
\(794\) 22.9706 0.815195
\(795\) 29.6569 1.05182
\(796\) −11.3137 −0.401004
\(797\) 6.28427 0.222600 0.111300 0.993787i \(-0.464499\pi\)
0.111300 + 0.993787i \(0.464499\pi\)
\(798\) −8.00000 −0.283197
\(799\) 61.2548 2.16704
\(800\) −1.00000 −0.0353553
\(801\) −58.2843 −2.05937
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) 43.3137 1.52756
\(805\) 2.82843 0.0996890
\(806\) −11.3137 −0.398508
\(807\) 5.65685 0.199131
\(808\) 10.0000 0.351799
\(809\) −22.2843 −0.783473 −0.391737 0.920077i \(-0.628126\pi\)
−0.391737 + 0.920077i \(0.628126\pi\)
\(810\) 1.00000 0.0351364
\(811\) −22.1421 −0.777516 −0.388758 0.921340i \(-0.627096\pi\)
−0.388758 + 0.921340i \(0.627096\pi\)
\(812\) −3.17157 −0.111300
\(813\) −48.0000 −1.68343
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) −21.6569 −0.758142
\(817\) 27.3137 0.955586
\(818\) −12.1421 −0.424540
\(819\) −10.0000 −0.349428
\(820\) −0.828427 −0.0289299
\(821\) 29.7990 1.03999 0.519996 0.854169i \(-0.325933\pi\)
0.519996 + 0.854169i \(0.325933\pi\)
\(822\) −15.0294 −0.524212
\(823\) 21.1716 0.737995 0.368997 0.929430i \(-0.379701\pi\)
0.368997 + 0.929430i \(0.379701\pi\)
\(824\) 5.65685 0.197066
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 49.2548 1.71276 0.856379 0.516347i \(-0.172709\pi\)
0.856379 + 0.516347i \(0.172709\pi\)
\(828\) −14.1421 −0.491473
\(829\) 39.2548 1.36338 0.681688 0.731643i \(-0.261246\pi\)
0.681688 + 0.731643i \(0.261246\pi\)
\(830\) 2.34315 0.0813318
\(831\) 37.6569 1.30630
\(832\) −2.00000 −0.0693375
\(833\) −7.65685 −0.265294
\(834\) 53.2548 1.84406
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −32.0000 −1.10608
\(838\) 30.6274 1.05801
\(839\) −34.3431 −1.18566 −0.592829 0.805329i \(-0.701989\pi\)
−0.592829 + 0.805329i \(0.701989\pi\)
\(840\) 2.82843 0.0975900
\(841\) −18.9411 −0.653142
\(842\) −28.6274 −0.986566
\(843\) 19.7157 0.679046
\(844\) −15.3137 −0.527120
\(845\) 9.00000 0.309609
\(846\) 40.0000 1.37523
\(847\) 0 0
\(848\) −10.4853 −0.360066
\(849\) 64.0000 2.19647
\(850\) 7.65685 0.262628
\(851\) −8.97056 −0.307507
\(852\) −16.0000 −0.548151
\(853\) 25.3137 0.866725 0.433362 0.901220i \(-0.357327\pi\)
0.433362 + 0.901220i \(0.357327\pi\)
\(854\) −9.31371 −0.318709
\(855\) −14.1421 −0.483651
\(856\) −1.65685 −0.0566301
\(857\) 6.97056 0.238110 0.119055 0.992888i \(-0.462014\pi\)
0.119055 + 0.992888i \(0.462014\pi\)
\(858\) 0 0
\(859\) 4.68629 0.159894 0.0799471 0.996799i \(-0.474525\pi\)
0.0799471 + 0.996799i \(0.474525\pi\)
\(860\) −9.65685 −0.329296
\(861\) 2.34315 0.0798542
\(862\) −19.7990 −0.674356
\(863\) −36.7696 −1.25165 −0.625825 0.779963i \(-0.715238\pi\)
−0.625825 + 0.779963i \(0.715238\pi\)
\(864\) −5.65685 −0.192450
\(865\) −1.31371 −0.0446674
\(866\) −20.8284 −0.707779
\(867\) 117.740 3.99866
\(868\) −5.65685 −0.192006
\(869\) 0 0
\(870\) −8.97056 −0.304131
\(871\) −30.6274 −1.03777
\(872\) 14.4853 0.490534
\(873\) −4.14214 −0.140190
\(874\) 8.00000 0.270604
\(875\) −1.00000 −0.0338062
\(876\) −21.6569 −0.731717
\(877\) 2.97056 0.100309 0.0501544 0.998741i \(-0.484029\pi\)
0.0501544 + 0.998741i \(0.484029\pi\)
\(878\) 24.9706 0.842716
\(879\) −5.65685 −0.190801
\(880\) 0 0
\(881\) −2.68629 −0.0905035 −0.0452517 0.998976i \(-0.514409\pi\)
−0.0452517 + 0.998976i \(0.514409\pi\)
\(882\) −5.00000 −0.168359
\(883\) 41.6569 1.40186 0.700932 0.713228i \(-0.252767\pi\)
0.700932 + 0.713228i \(0.252767\pi\)
\(884\) 15.3137 0.515056
\(885\) 22.6274 0.760612
\(886\) 8.68629 0.291822
\(887\) 22.6274 0.759754 0.379877 0.925037i \(-0.375966\pi\)
0.379877 + 0.925037i \(0.375966\pi\)
\(888\) −8.97056 −0.301032
\(889\) −19.3137 −0.647761
\(890\) −11.6569 −0.390739
\(891\) 0 0
\(892\) −28.2843 −0.947027
\(893\) −22.6274 −0.757198
\(894\) −4.28427 −0.143287
\(895\) −6.34315 −0.212028
\(896\) −1.00000 −0.0334077
\(897\) 16.0000 0.534224
\(898\) 6.00000 0.200223
\(899\) 17.9411 0.598370
\(900\) 5.00000 0.166667
\(901\) 80.2843 2.67466
\(902\) 0 0
\(903\) 27.3137 0.908943
\(904\) −2.00000 −0.0665190
\(905\) 14.0000 0.465376
\(906\) 30.6274 1.01753
\(907\) 17.6569 0.586286 0.293143 0.956069i \(-0.405299\pi\)
0.293143 + 0.956069i \(0.405299\pi\)
\(908\) 24.0000 0.796468
\(909\) −50.0000 −1.65840
\(910\) −2.00000 −0.0662994
\(911\) −21.6569 −0.717524 −0.358762 0.933429i \(-0.616801\pi\)
−0.358762 + 0.933429i \(0.616801\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −26.3431 −0.870878
\(916\) −1.31371 −0.0434062
\(917\) −18.8284 −0.621769
\(918\) 43.3137 1.42957
\(919\) −20.7696 −0.685124 −0.342562 0.939495i \(-0.611295\pi\)
−0.342562 + 0.939495i \(0.611295\pi\)
\(920\) −2.82843 −0.0932505
\(921\) 25.3726 0.836055
\(922\) −11.6569 −0.383898
\(923\) 11.3137 0.372395
\(924\) 0 0
\(925\) 3.17157 0.104281
\(926\) 9.45584 0.310738
\(927\) −28.2843 −0.928977
\(928\) 3.17157 0.104112
\(929\) −38.9706 −1.27858 −0.639291 0.768965i \(-0.720772\pi\)
−0.639291 + 0.768965i \(0.720772\pi\)
\(930\) −16.0000 −0.524661
\(931\) 2.82843 0.0926980
\(932\) 6.00000 0.196537
\(933\) 2.74517 0.0898727
\(934\) −11.7990 −0.386075
\(935\) 0 0
\(936\) 10.0000 0.326860
\(937\) −24.6274 −0.804543 −0.402271 0.915520i \(-0.631779\pi\)
−0.402271 + 0.915520i \(0.631779\pi\)
\(938\) −15.3137 −0.500010
\(939\) −24.9706 −0.814884
\(940\) 8.00000 0.260931
\(941\) −13.3137 −0.434014 −0.217007 0.976170i \(-0.569630\pi\)
−0.217007 + 0.976170i \(0.569630\pi\)
\(942\) −44.2843 −1.44286
\(943\) −2.34315 −0.0763033
\(944\) −8.00000 −0.260378
\(945\) −5.65685 −0.184017
\(946\) 0 0
\(947\) 35.5980 1.15678 0.578389 0.815761i \(-0.303681\pi\)
0.578389 + 0.815761i \(0.303681\pi\)
\(948\) 14.6274 0.475076
\(949\) 15.3137 0.497104
\(950\) −2.82843 −0.0917663
\(951\) 8.97056 0.290890
\(952\) 7.65685 0.248160
\(953\) 43.2548 1.40116 0.700581 0.713573i \(-0.252924\pi\)
0.700581 + 0.713573i \(0.252924\pi\)
\(954\) 52.4264 1.69737
\(955\) −19.3137 −0.624977
\(956\) 7.51472 0.243043
\(957\) 0 0
\(958\) 35.3137 1.14093
\(959\) 5.31371 0.171589
\(960\) −2.82843 −0.0912871
\(961\) 1.00000 0.0322581
\(962\) 6.34315 0.204511
\(963\) 8.28427 0.266957
\(964\) 8.82843 0.284344
\(965\) 21.3137 0.686113
\(966\) 8.00000 0.257396
\(967\) −59.3137 −1.90740 −0.953700 0.300759i \(-0.902760\pi\)
−0.953700 + 0.300759i \(0.902760\pi\)
\(968\) 0 0
\(969\) −61.2548 −1.96779
\(970\) −0.828427 −0.0265992
\(971\) −41.9411 −1.34595 −0.672977 0.739663i \(-0.734985\pi\)
−0.672977 + 0.739663i \(0.734985\pi\)
\(972\) −14.1421 −0.453609
\(973\) −18.8284 −0.603612
\(974\) 36.7696 1.17817
\(975\) −5.65685 −0.181164
\(976\) 9.31371 0.298125
\(977\) −20.6274 −0.659930 −0.329965 0.943993i \(-0.607037\pi\)
−0.329965 + 0.943993i \(0.607037\pi\)
\(978\) −33.9411 −1.08532
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −72.4264 −2.31240
\(982\) −8.68629 −0.277191
\(983\) 39.5980 1.26298 0.631490 0.775384i \(-0.282444\pi\)
0.631490 + 0.775384i \(0.282444\pi\)
\(984\) −2.34315 −0.0746968
\(985\) −12.3431 −0.393285
\(986\) −24.2843 −0.773369
\(987\) −22.6274 −0.720239
\(988\) −5.65685 −0.179969
\(989\) −27.3137 −0.868525
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 5.65685 0.179605
\(993\) 81.9411 2.60032
\(994\) 5.65685 0.179425
\(995\) 11.3137 0.358669
\(996\) 6.62742 0.209998
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −28.9706 −0.917047
\(999\) 17.9411 0.567632
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 8470.2.a.bo.1.2 2
11.10 odd 2 770.2.a.i.1.2 2
33.32 even 2 6930.2.a.br.1.1 2
44.43 even 2 6160.2.a.ba.1.1 2
55.32 even 4 3850.2.c.u.1849.3 4
55.43 even 4 3850.2.c.u.1849.2 4
55.54 odd 2 3850.2.a.bi.1.1 2
77.76 even 2 5390.2.a.bt.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.i.1.2 2 11.10 odd 2
3850.2.a.bi.1.1 2 55.54 odd 2
3850.2.c.u.1849.2 4 55.43 even 4
3850.2.c.u.1849.3 4 55.32 even 4
5390.2.a.bt.1.1 2 77.76 even 2
6160.2.a.ba.1.1 2 44.43 even 2
6930.2.a.br.1.1 2 33.32 even 2
8470.2.a.bo.1.2 2 1.1 even 1 trivial