Properties

Label 507.2.a.h.1.2
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -2.82843 q^{5} +2.41421 q^{6} +2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -2.82843 q^{5} +2.41421 q^{6} +2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} -6.82843 q^{10} +2.00000 q^{11} +3.82843 q^{12} +6.82843 q^{14} -2.82843 q^{15} +3.00000 q^{16} -3.65685 q^{17} +2.41421 q^{18} -2.82843 q^{19} -10.8284 q^{20} +2.82843 q^{21} +4.82843 q^{22} -4.00000 q^{23} +4.41421 q^{24} +3.00000 q^{25} +1.00000 q^{27} +10.8284 q^{28} +2.00000 q^{29} -6.82843 q^{30} +6.82843 q^{31} -1.58579 q^{32} +2.00000 q^{33} -8.82843 q^{34} -8.00000 q^{35} +3.82843 q^{36} -3.65685 q^{37} -6.82843 q^{38} -12.4853 q^{40} -10.8284 q^{41} +6.82843 q^{42} +9.65685 q^{43} +7.65685 q^{44} -2.82843 q^{45} -9.65685 q^{46} +0.343146 q^{47} +3.00000 q^{48} +1.00000 q^{49} +7.24264 q^{50} -3.65685 q^{51} -2.00000 q^{53} +2.41421 q^{54} -5.65685 q^{55} +12.4853 q^{56} -2.82843 q^{57} +4.82843 q^{58} +3.65685 q^{59} -10.8284 q^{60} -9.31371 q^{61} +16.4853 q^{62} +2.82843 q^{63} -9.82843 q^{64} +4.82843 q^{66} -1.17157 q^{67} -14.0000 q^{68} -4.00000 q^{69} -19.3137 q^{70} -2.00000 q^{71} +4.41421 q^{72} -11.6569 q^{73} -8.82843 q^{74} +3.00000 q^{75} -10.8284 q^{76} +5.65685 q^{77} +11.3137 q^{79} -8.48528 q^{80} +1.00000 q^{81} -26.1421 q^{82} +7.65685 q^{83} +10.8284 q^{84} +10.3431 q^{85} +23.3137 q^{86} +2.00000 q^{87} +8.82843 q^{88} -9.17157 q^{89} -6.82843 q^{90} -15.3137 q^{92} +6.82843 q^{93} +0.828427 q^{94} +8.00000 q^{95} -1.58579 q^{96} +7.65685 q^{97} +2.41421 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{8} + 2q^{9} - 8q^{10} + 4q^{11} + 2q^{12} + 8q^{14} + 6q^{16} + 4q^{17} + 2q^{18} - 16q^{20} + 4q^{22} - 8q^{23} + 6q^{24} + 6q^{25} + 2q^{27} + 16q^{28} + 4q^{29} - 8q^{30} + 8q^{31} - 6q^{32} + 4q^{33} - 12q^{34} - 16q^{35} + 2q^{36} + 4q^{37} - 8q^{38} - 8q^{40} - 16q^{41} + 8q^{42} + 8q^{43} + 4q^{44} - 8q^{46} + 12q^{47} + 6q^{48} + 2q^{49} + 6q^{50} + 4q^{51} - 4q^{53} + 2q^{54} + 8q^{56} + 4q^{58} - 4q^{59} - 16q^{60} + 4q^{61} + 16q^{62} - 14q^{64} + 4q^{66} - 8q^{67} - 28q^{68} - 8q^{69} - 16q^{70} - 4q^{71} + 6q^{72} - 12q^{73} - 12q^{74} + 6q^{75} - 16q^{76} + 2q^{81} - 24q^{82} + 4q^{83} + 16q^{84} + 32q^{85} + 24q^{86} + 4q^{87} + 12q^{88} - 24q^{89} - 8q^{90} - 8q^{92} + 8q^{93} - 4q^{94} + 16q^{95} - 6q^{96} + 4q^{97} + 2q^{98} + 4q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.82843 1.91421
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 2.41421 0.985599
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 4.41421 1.56066
\(9\) 1.00000 0.333333
\(10\) −6.82843 −2.15934
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 3.82843 1.10517
\(13\) 0 0
\(14\) 6.82843 1.82497
\(15\) −2.82843 −0.730297
\(16\) 3.00000 0.750000
\(17\) −3.65685 −0.886917 −0.443459 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 2.41421 0.569036
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) −10.8284 −2.42131
\(21\) 2.82843 0.617213
\(22\) 4.82843 1.02942
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 4.41421 0.901048
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 10.8284 2.04638
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −6.82843 −1.24669
\(31\) 6.82843 1.22642 0.613211 0.789919i \(-0.289878\pi\)
0.613211 + 0.789919i \(0.289878\pi\)
\(32\) −1.58579 −0.280330
\(33\) 2.00000 0.348155
\(34\) −8.82843 −1.51406
\(35\) −8.00000 −1.35225
\(36\) 3.82843 0.638071
\(37\) −3.65685 −0.601183 −0.300592 0.953753i \(-0.597184\pi\)
−0.300592 + 0.953753i \(0.597184\pi\)
\(38\) −6.82843 −1.10772
\(39\) 0 0
\(40\) −12.4853 −1.97410
\(41\) −10.8284 −1.69112 −0.845558 0.533883i \(-0.820732\pi\)
−0.845558 + 0.533883i \(0.820732\pi\)
\(42\) 6.82843 1.05365
\(43\) 9.65685 1.47266 0.736328 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) 7.65685 1.15431
\(45\) −2.82843 −0.421637
\(46\) −9.65685 −1.42383
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 3.00000 0.433013
\(49\) 1.00000 0.142857
\(50\) 7.24264 1.02426
\(51\) −3.65685 −0.512062
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 2.41421 0.328533
\(55\) −5.65685 −0.762770
\(56\) 12.4853 1.66842
\(57\) −2.82843 −0.374634
\(58\) 4.82843 0.634004
\(59\) 3.65685 0.476082 0.238041 0.971255i \(-0.423495\pi\)
0.238041 + 0.971255i \(0.423495\pi\)
\(60\) −10.8284 −1.39794
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 16.4853 2.09363
\(63\) 2.82843 0.356348
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 4.82843 0.594338
\(67\) −1.17157 −0.143130 −0.0715652 0.997436i \(-0.522799\pi\)
−0.0715652 + 0.997436i \(0.522799\pi\)
\(68\) −14.0000 −1.69775
\(69\) −4.00000 −0.481543
\(70\) −19.3137 −2.30843
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 4.41421 0.520220
\(73\) −11.6569 −1.36433 −0.682166 0.731198i \(-0.738962\pi\)
−0.682166 + 0.731198i \(0.738962\pi\)
\(74\) −8.82843 −1.02628
\(75\) 3.00000 0.346410
\(76\) −10.8284 −1.24211
\(77\) 5.65685 0.644658
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) −8.48528 −0.948683
\(81\) 1.00000 0.111111
\(82\) −26.1421 −2.88692
\(83\) 7.65685 0.840449 0.420224 0.907420i \(-0.361951\pi\)
0.420224 + 0.907420i \(0.361951\pi\)
\(84\) 10.8284 1.18148
\(85\) 10.3431 1.12187
\(86\) 23.3137 2.51398
\(87\) 2.00000 0.214423
\(88\) 8.82843 0.941113
\(89\) −9.17157 −0.972185 −0.486092 0.873907i \(-0.661578\pi\)
−0.486092 + 0.873907i \(0.661578\pi\)
\(90\) −6.82843 −0.719779
\(91\) 0 0
\(92\) −15.3137 −1.59656
\(93\) 6.82843 0.708075
\(94\) 0.828427 0.0854457
\(95\) 8.00000 0.820783
\(96\) −1.58579 −0.161849
\(97\) 7.65685 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(98\) 2.41421 0.243872
\(99\) 2.00000 0.201008
\(100\) 11.4853 1.14853
\(101\) −3.65685 −0.363871 −0.181935 0.983311i \(-0.558236\pi\)
−0.181935 + 0.983311i \(0.558236\pi\)
\(102\) −8.82843 −0.874145
\(103\) 13.6569 1.34565 0.672825 0.739802i \(-0.265081\pi\)
0.672825 + 0.739802i \(0.265081\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) −4.82843 −0.468978
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 3.82843 0.368391
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) −13.6569 −1.30213
\(111\) −3.65685 −0.347093
\(112\) 8.48528 0.801784
\(113\) 17.3137 1.62874 0.814368 0.580348i \(-0.197084\pi\)
0.814368 + 0.580348i \(0.197084\pi\)
\(114\) −6.82843 −0.639541
\(115\) 11.3137 1.05501
\(116\) 7.65685 0.710921
\(117\) 0 0
\(118\) 8.82843 0.812723
\(119\) −10.3431 −0.948155
\(120\) −12.4853 −1.13975
\(121\) −7.00000 −0.636364
\(122\) −22.4853 −2.03572
\(123\) −10.8284 −0.976366
\(124\) 26.1421 2.34763
\(125\) 5.65685 0.505964
\(126\) 6.82843 0.608325
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) −20.5563 −1.81694
\(129\) 9.65685 0.850239
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 7.65685 0.666444
\(133\) −8.00000 −0.693688
\(134\) −2.82843 −0.244339
\(135\) −2.82843 −0.243432
\(136\) −16.1421 −1.38418
\(137\) 5.17157 0.441837 0.220919 0.975292i \(-0.429094\pi\)
0.220919 + 0.975292i \(0.429094\pi\)
\(138\) −9.65685 −0.822046
\(139\) 15.3137 1.29889 0.649446 0.760408i \(-0.275001\pi\)
0.649446 + 0.760408i \(0.275001\pi\)
\(140\) −30.6274 −2.58849
\(141\) 0.343146 0.0288981
\(142\) −4.82843 −0.405193
\(143\) 0 0
\(144\) 3.00000 0.250000
\(145\) −5.65685 −0.469776
\(146\) −28.1421 −2.32906
\(147\) 1.00000 0.0824786
\(148\) −14.0000 −1.15079
\(149\) 14.8284 1.21479 0.607396 0.794399i \(-0.292214\pi\)
0.607396 + 0.794399i \(0.292214\pi\)
\(150\) 7.24264 0.591359
\(151\) 20.4853 1.66707 0.833534 0.552468i \(-0.186314\pi\)
0.833534 + 0.552468i \(0.186314\pi\)
\(152\) −12.4853 −1.01269
\(153\) −3.65685 −0.295639
\(154\) 13.6569 1.10050
\(155\) −19.3137 −1.55131
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 27.3137 2.17296
\(159\) −2.00000 −0.158610
\(160\) 4.48528 0.354593
\(161\) −11.3137 −0.891645
\(162\) 2.41421 0.189679
\(163\) −13.1716 −1.03168 −0.515839 0.856686i \(-0.672520\pi\)
−0.515839 + 0.856686i \(0.672520\pi\)
\(164\) −41.4558 −3.23716
\(165\) −5.65685 −0.440386
\(166\) 18.4853 1.43474
\(167\) −7.65685 −0.592505 −0.296253 0.955110i \(-0.595737\pi\)
−0.296253 + 0.955110i \(0.595737\pi\)
\(168\) 12.4853 0.963260
\(169\) 0 0
\(170\) 24.9706 1.91515
\(171\) −2.82843 −0.216295
\(172\) 36.9706 2.81898
\(173\) −0.343146 −0.0260889 −0.0130444 0.999915i \(-0.504152\pi\)
−0.0130444 + 0.999915i \(0.504152\pi\)
\(174\) 4.82843 0.366042
\(175\) 8.48528 0.641427
\(176\) 6.00000 0.452267
\(177\) 3.65685 0.274866
\(178\) −22.1421 −1.65962
\(179\) −0.686292 −0.0512958 −0.0256479 0.999671i \(-0.508165\pi\)
−0.0256479 + 0.999671i \(0.508165\pi\)
\(180\) −10.8284 −0.807103
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −9.31371 −0.688489
\(184\) −17.6569 −1.30168
\(185\) 10.3431 0.760443
\(186\) 16.4853 1.20876
\(187\) −7.31371 −0.534831
\(188\) 1.31371 0.0958120
\(189\) 2.82843 0.205738
\(190\) 19.3137 1.40116
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) −9.82843 −0.709306
\(193\) 17.3137 1.24627 0.623134 0.782115i \(-0.285859\pi\)
0.623134 + 0.782115i \(0.285859\pi\)
\(194\) 18.4853 1.32717
\(195\) 0 0
\(196\) 3.82843 0.273459
\(197\) 16.4853 1.17453 0.587264 0.809396i \(-0.300205\pi\)
0.587264 + 0.809396i \(0.300205\pi\)
\(198\) 4.82843 0.343141
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 13.2426 0.936396
\(201\) −1.17157 −0.0826364
\(202\) −8.82843 −0.621166
\(203\) 5.65685 0.397033
\(204\) −14.0000 −0.980196
\(205\) 30.6274 2.13911
\(206\) 32.9706 2.29717
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) −19.3137 −1.33277
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −7.65685 −0.525875
\(213\) −2.00000 −0.137038
\(214\) 27.3137 1.86713
\(215\) −27.3137 −1.86278
\(216\) 4.41421 0.300349
\(217\) 19.3137 1.31110
\(218\) 41.7990 2.83098
\(219\) −11.6569 −0.787697
\(220\) −21.6569 −1.46010
\(221\) 0 0
\(222\) −8.82843 −0.592525
\(223\) −4.48528 −0.300357 −0.150178 0.988659i \(-0.547985\pi\)
−0.150178 + 0.988659i \(0.547985\pi\)
\(224\) −4.48528 −0.299685
\(225\) 3.00000 0.200000
\(226\) 41.7990 2.78043
\(227\) −5.31371 −0.352683 −0.176342 0.984329i \(-0.556426\pi\)
−0.176342 + 0.984329i \(0.556426\pi\)
\(228\) −10.8284 −0.717130
\(229\) −21.3137 −1.40845 −0.704225 0.709977i \(-0.748705\pi\)
−0.704225 + 0.709977i \(0.748705\pi\)
\(230\) 27.3137 1.80101
\(231\) 5.65685 0.372194
\(232\) 8.82843 0.579615
\(233\) −26.9706 −1.76690 −0.883450 0.468525i \(-0.844786\pi\)
−0.883450 + 0.468525i \(0.844786\pi\)
\(234\) 0 0
\(235\) −0.970563 −0.0633125
\(236\) 14.0000 0.911322
\(237\) 11.3137 0.734904
\(238\) −24.9706 −1.61860
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) −8.48528 −0.547723
\(241\) −11.6569 −0.750884 −0.375442 0.926846i \(-0.622509\pi\)
−0.375442 + 0.926846i \(0.622509\pi\)
\(242\) −16.8995 −1.08634
\(243\) 1.00000 0.0641500
\(244\) −35.6569 −2.28270
\(245\) −2.82843 −0.180702
\(246\) −26.1421 −1.66676
\(247\) 0 0
\(248\) 30.1421 1.91403
\(249\) 7.65685 0.485233
\(250\) 13.6569 0.863735
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 10.8284 0.682127
\(253\) −8.00000 −0.502956
\(254\) −13.6569 −0.856907
\(255\) 10.3431 0.647713
\(256\) −29.9706 −1.87316
\(257\) −15.6569 −0.976648 −0.488324 0.872662i \(-0.662392\pi\)
−0.488324 + 0.872662i \(0.662392\pi\)
\(258\) 23.3137 1.45145
\(259\) −10.3431 −0.642692
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −19.3137 −1.19320
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 8.82843 0.543352
\(265\) 5.65685 0.347498
\(266\) −19.3137 −1.18420
\(267\) −9.17157 −0.561291
\(268\) −4.48528 −0.273982
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −6.82843 −0.415565
\(271\) −11.7990 −0.716738 −0.358369 0.933580i \(-0.616667\pi\)
−0.358369 + 0.933580i \(0.616667\pi\)
\(272\) −10.9706 −0.665188
\(273\) 0 0
\(274\) 12.4853 0.754263
\(275\) 6.00000 0.361814
\(276\) −15.3137 −0.921777
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 36.9706 2.21735
\(279\) 6.82843 0.408807
\(280\) −35.3137 −2.11040
\(281\) −26.8284 −1.60045 −0.800225 0.599700i \(-0.795287\pi\)
−0.800225 + 0.599700i \(0.795287\pi\)
\(282\) 0.828427 0.0493321
\(283\) −4.97056 −0.295469 −0.147735 0.989027i \(-0.547198\pi\)
−0.147735 + 0.989027i \(0.547198\pi\)
\(284\) −7.65685 −0.454351
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −30.6274 −1.80788
\(288\) −1.58579 −0.0934434
\(289\) −3.62742 −0.213377
\(290\) −13.6569 −0.801958
\(291\) 7.65685 0.448853
\(292\) −44.6274 −2.61162
\(293\) 26.1421 1.52724 0.763620 0.645666i \(-0.223420\pi\)
0.763620 + 0.645666i \(0.223420\pi\)
\(294\) 2.41421 0.140800
\(295\) −10.3431 −0.602201
\(296\) −16.1421 −0.938243
\(297\) 2.00000 0.116052
\(298\) 35.7990 2.07378
\(299\) 0 0
\(300\) 11.4853 0.663103
\(301\) 27.3137 1.57434
\(302\) 49.4558 2.84586
\(303\) −3.65685 −0.210081
\(304\) −8.48528 −0.486664
\(305\) 26.3431 1.50840
\(306\) −8.82843 −0.504688
\(307\) 17.1716 0.980033 0.490017 0.871713i \(-0.336991\pi\)
0.490017 + 0.871713i \(0.336991\pi\)
\(308\) 21.6569 1.23401
\(309\) 13.6569 0.776911
\(310\) −46.6274 −2.64826
\(311\) 34.6274 1.96354 0.981770 0.190071i \(-0.0608718\pi\)
0.981770 + 0.190071i \(0.0608718\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −24.1421 −1.36242
\(315\) −8.00000 −0.450749
\(316\) 43.3137 2.43659
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) −4.82843 −0.270765
\(319\) 4.00000 0.223957
\(320\) 27.7990 1.55401
\(321\) 11.3137 0.631470
\(322\) −27.3137 −1.52213
\(323\) 10.3431 0.575508
\(324\) 3.82843 0.212690
\(325\) 0 0
\(326\) −31.7990 −1.76118
\(327\) 17.3137 0.957450
\(328\) −47.7990 −2.63926
\(329\) 0.970563 0.0535089
\(330\) −13.6569 −0.751785
\(331\) 2.14214 0.117742 0.0588712 0.998266i \(-0.481250\pi\)
0.0588712 + 0.998266i \(0.481250\pi\)
\(332\) 29.3137 1.60880
\(333\) −3.65685 −0.200394
\(334\) −18.4853 −1.01147
\(335\) 3.31371 0.181047
\(336\) 8.48528 0.462910
\(337\) −13.3137 −0.725244 −0.362622 0.931936i \(-0.618118\pi\)
−0.362622 + 0.931936i \(0.618118\pi\)
\(338\) 0 0
\(339\) 17.3137 0.940352
\(340\) 39.5980 2.14750
\(341\) 13.6569 0.739560
\(342\) −6.82843 −0.369239
\(343\) −16.9706 −0.916324
\(344\) 42.6274 2.29832
\(345\) 11.3137 0.609110
\(346\) −0.828427 −0.0445365
\(347\) −31.3137 −1.68101 −0.840504 0.541805i \(-0.817741\pi\)
−0.840504 + 0.541805i \(0.817741\pi\)
\(348\) 7.65685 0.410450
\(349\) 7.65685 0.409862 0.204931 0.978776i \(-0.434303\pi\)
0.204931 + 0.978776i \(0.434303\pi\)
\(350\) 20.4853 1.09498
\(351\) 0 0
\(352\) −3.17157 −0.169045
\(353\) −17.4558 −0.929081 −0.464540 0.885552i \(-0.653780\pi\)
−0.464540 + 0.885552i \(0.653780\pi\)
\(354\) 8.82843 0.469226
\(355\) 5.65685 0.300235
\(356\) −35.1127 −1.86097
\(357\) −10.3431 −0.547417
\(358\) −1.65685 −0.0875675
\(359\) −1.02944 −0.0543316 −0.0271658 0.999631i \(-0.508648\pi\)
−0.0271658 + 0.999631i \(0.508648\pi\)
\(360\) −12.4853 −0.658032
\(361\) −11.0000 −0.578947
\(362\) 33.7990 1.77644
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 32.9706 1.72576
\(366\) −22.4853 −1.17532
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −12.0000 −0.625543
\(369\) −10.8284 −0.563705
\(370\) 24.9706 1.29816
\(371\) −5.65685 −0.293689
\(372\) 26.1421 1.35541
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −17.6569 −0.913014
\(375\) 5.65685 0.292119
\(376\) 1.51472 0.0781156
\(377\) 0 0
\(378\) 6.82843 0.351216
\(379\) 16.4853 0.846792 0.423396 0.905945i \(-0.360838\pi\)
0.423396 + 0.905945i \(0.360838\pi\)
\(380\) 30.6274 1.57115
\(381\) −5.65685 −0.289809
\(382\) −46.6274 −2.38567
\(383\) 2.97056 0.151789 0.0758943 0.997116i \(-0.475819\pi\)
0.0758943 + 0.997116i \(0.475819\pi\)
\(384\) −20.5563 −1.04901
\(385\) −16.0000 −0.815436
\(386\) 41.7990 2.12751
\(387\) 9.65685 0.490885
\(388\) 29.3137 1.48818
\(389\) 6.97056 0.353422 0.176711 0.984263i \(-0.443454\pi\)
0.176711 + 0.984263i \(0.443454\pi\)
\(390\) 0 0
\(391\) 14.6274 0.739740
\(392\) 4.41421 0.222951
\(393\) −8.00000 −0.403547
\(394\) 39.7990 2.00504
\(395\) −32.0000 −1.61009
\(396\) 7.65685 0.384771
\(397\) 2.97056 0.149088 0.0745441 0.997218i \(-0.476250\pi\)
0.0745441 + 0.997218i \(0.476250\pi\)
\(398\) 24.9706 1.25166
\(399\) −8.00000 −0.400501
\(400\) 9.00000 0.450000
\(401\) 2.14214 0.106973 0.0534866 0.998569i \(-0.482967\pi\)
0.0534866 + 0.998569i \(0.482967\pi\)
\(402\) −2.82843 −0.141069
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) −2.82843 −0.140546
\(406\) 13.6569 0.677778
\(407\) −7.31371 −0.362527
\(408\) −16.1421 −0.799155
\(409\) 1.02944 0.0509024 0.0254512 0.999676i \(-0.491898\pi\)
0.0254512 + 0.999676i \(0.491898\pi\)
\(410\) 73.9411 3.65169
\(411\) 5.17157 0.255095
\(412\) 52.2843 2.57586
\(413\) 10.3431 0.508953
\(414\) −9.65685 −0.474608
\(415\) −21.6569 −1.06309
\(416\) 0 0
\(417\) 15.3137 0.749916
\(418\) −13.6569 −0.667979
\(419\) −30.6274 −1.49625 −0.748124 0.663559i \(-0.769045\pi\)
−0.748124 + 0.663559i \(0.769045\pi\)
\(420\) −30.6274 −1.49446
\(421\) −14.6863 −0.715766 −0.357883 0.933766i \(-0.616501\pi\)
−0.357883 + 0.933766i \(0.616501\pi\)
\(422\) −28.9706 −1.41026
\(423\) 0.343146 0.0166843
\(424\) −8.82843 −0.428746
\(425\) −10.9706 −0.532150
\(426\) −4.82843 −0.233938
\(427\) −26.3431 −1.27483
\(428\) 43.3137 2.09365
\(429\) 0 0
\(430\) −65.9411 −3.17996
\(431\) −19.6569 −0.946837 −0.473419 0.880838i \(-0.656980\pi\)
−0.473419 + 0.880838i \(0.656980\pi\)
\(432\) 3.00000 0.144338
\(433\) 1.31371 0.0631328 0.0315664 0.999502i \(-0.489950\pi\)
0.0315664 + 0.999502i \(0.489950\pi\)
\(434\) 46.6274 2.23819
\(435\) −5.65685 −0.271225
\(436\) 66.2843 3.17444
\(437\) 11.3137 0.541208
\(438\) −28.1421 −1.34468
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) −24.9706 −1.19042
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 41.9411 1.99268 0.996342 0.0854611i \(-0.0272364\pi\)
0.996342 + 0.0854611i \(0.0272364\pi\)
\(444\) −14.0000 −0.664411
\(445\) 25.9411 1.22973
\(446\) −10.8284 −0.512741
\(447\) 14.8284 0.701361
\(448\) −27.7990 −1.31338
\(449\) −7.79899 −0.368057 −0.184029 0.982921i \(-0.558914\pi\)
−0.184029 + 0.982921i \(0.558914\pi\)
\(450\) 7.24264 0.341421
\(451\) −21.6569 −1.01978
\(452\) 66.2843 3.11775
\(453\) 20.4853 0.962482
\(454\) −12.8284 −0.602068
\(455\) 0 0
\(456\) −12.4853 −0.584677
\(457\) −3.65685 −0.171060 −0.0855302 0.996336i \(-0.527258\pi\)
−0.0855302 + 0.996336i \(0.527258\pi\)
\(458\) −51.4558 −2.40437
\(459\) −3.65685 −0.170687
\(460\) 43.3137 2.01951
\(461\) −10.8284 −0.504330 −0.252165 0.967684i \(-0.581143\pi\)
−0.252165 + 0.967684i \(0.581143\pi\)
\(462\) 13.6569 0.635374
\(463\) 7.51472 0.349239 0.174619 0.984636i \(-0.444131\pi\)
0.174619 + 0.984636i \(0.444131\pi\)
\(464\) 6.00000 0.278543
\(465\) −19.3137 −0.895652
\(466\) −65.1127 −3.01629
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −3.31371 −0.153013
\(470\) −2.34315 −0.108081
\(471\) −10.0000 −0.460776
\(472\) 16.1421 0.743002
\(473\) 19.3137 0.888045
\(474\) 27.3137 1.25456
\(475\) −8.48528 −0.389331
\(476\) −39.5980 −1.81497
\(477\) −2.00000 −0.0915737
\(478\) −4.82843 −0.220847
\(479\) −2.68629 −0.122740 −0.0613699 0.998115i \(-0.519547\pi\)
−0.0613699 + 0.998115i \(0.519547\pi\)
\(480\) 4.48528 0.204724
\(481\) 0 0
\(482\) −28.1421 −1.28184
\(483\) −11.3137 −0.514792
\(484\) −26.7990 −1.21814
\(485\) −21.6569 −0.983387
\(486\) 2.41421 0.109511
\(487\) −31.7990 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(488\) −41.1127 −1.86108
\(489\) −13.1716 −0.595639
\(490\) −6.82843 −0.308477
\(491\) −14.6274 −0.660126 −0.330063 0.943959i \(-0.607070\pi\)
−0.330063 + 0.943959i \(0.607070\pi\)
\(492\) −41.4558 −1.86897
\(493\) −7.31371 −0.329393
\(494\) 0 0
\(495\) −5.65685 −0.254257
\(496\) 20.4853 0.919816
\(497\) −5.65685 −0.253745
\(498\) 18.4853 0.828345
\(499\) 2.14214 0.0958952 0.0479476 0.998850i \(-0.484732\pi\)
0.0479476 + 0.998850i \(0.484732\pi\)
\(500\) 21.6569 0.968524
\(501\) −7.65685 −0.342083
\(502\) 0 0
\(503\) 15.3137 0.682805 0.341402 0.939917i \(-0.389098\pi\)
0.341402 + 0.939917i \(0.389098\pi\)
\(504\) 12.4853 0.556139
\(505\) 10.3431 0.460264
\(506\) −19.3137 −0.858599
\(507\) 0 0
\(508\) −21.6569 −0.960868
\(509\) 27.7990 1.23217 0.616084 0.787680i \(-0.288718\pi\)
0.616084 + 0.787680i \(0.288718\pi\)
\(510\) 24.9706 1.10572
\(511\) −32.9706 −1.45853
\(512\) −31.2426 −1.38074
\(513\) −2.82843 −0.124878
\(514\) −37.7990 −1.66724
\(515\) −38.6274 −1.70213
\(516\) 36.9706 1.62754
\(517\) 0.686292 0.0301831
\(518\) −24.9706 −1.09714
\(519\) −0.343146 −0.0150624
\(520\) 0 0
\(521\) 2.68629 0.117689 0.0588443 0.998267i \(-0.481258\pi\)
0.0588443 + 0.998267i \(0.481258\pi\)
\(522\) 4.82843 0.211335
\(523\) 7.31371 0.319806 0.159903 0.987133i \(-0.448882\pi\)
0.159903 + 0.987133i \(0.448882\pi\)
\(524\) −30.6274 −1.33796
\(525\) 8.48528 0.370328
\(526\) 28.9706 1.26318
\(527\) −24.9706 −1.08773
\(528\) 6.00000 0.261116
\(529\) −7.00000 −0.304348
\(530\) 13.6569 0.593216
\(531\) 3.65685 0.158694
\(532\) −30.6274 −1.32787
\(533\) 0 0
\(534\) −22.1421 −0.958184
\(535\) −32.0000 −1.38348
\(536\) −5.17157 −0.223378
\(537\) −0.686292 −0.0296157
\(538\) 43.4558 1.87351
\(539\) 2.00000 0.0861461
\(540\) −10.8284 −0.465981
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −28.4853 −1.22355
\(543\) 14.0000 0.600798
\(544\) 5.79899 0.248630
\(545\) −48.9706 −2.09767
\(546\) 0 0
\(547\) 0.686292 0.0293437 0.0146719 0.999892i \(-0.495330\pi\)
0.0146719 + 0.999892i \(0.495330\pi\)
\(548\) 19.7990 0.845771
\(549\) −9.31371 −0.397499
\(550\) 14.4853 0.617654
\(551\) −5.65685 −0.240990
\(552\) −17.6569 −0.751526
\(553\) 32.0000 1.36078
\(554\) −4.82843 −0.205140
\(555\) 10.3431 0.439042
\(556\) 58.6274 2.48636
\(557\) −31.7990 −1.34737 −0.673683 0.739020i \(-0.735289\pi\)
−0.673683 + 0.739020i \(0.735289\pi\)
\(558\) 16.4853 0.697878
\(559\) 0 0
\(560\) −24.0000 −1.01419
\(561\) −7.31371 −0.308785
\(562\) −64.7696 −2.73214
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 1.31371 0.0553171
\(565\) −48.9706 −2.06021
\(566\) −12.0000 −0.504398
\(567\) 2.82843 0.118783
\(568\) −8.82843 −0.370433
\(569\) −9.02944 −0.378534 −0.189267 0.981926i \(-0.560611\pi\)
−0.189267 + 0.981926i \(0.560611\pi\)
\(570\) 19.3137 0.808962
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) 0 0
\(573\) −19.3137 −0.806842
\(574\) −73.9411 −3.08624
\(575\) −12.0000 −0.500435
\(576\) −9.82843 −0.409518
\(577\) −35.9411 −1.49625 −0.748124 0.663559i \(-0.769045\pi\)
−0.748124 + 0.663559i \(0.769045\pi\)
\(578\) −8.75736 −0.364258
\(579\) 17.3137 0.719533
\(580\) −21.6569 −0.899252
\(581\) 21.6569 0.898478
\(582\) 18.4853 0.766240
\(583\) −4.00000 −0.165663
\(584\) −51.4558 −2.12926
\(585\) 0 0
\(586\) 63.1127 2.60716
\(587\) −22.9706 −0.948097 −0.474048 0.880499i \(-0.657208\pi\)
−0.474048 + 0.880499i \(0.657208\pi\)
\(588\) 3.82843 0.157882
\(589\) −19.3137 −0.795807
\(590\) −24.9706 −1.02802
\(591\) 16.4853 0.678114
\(592\) −10.9706 −0.450887
\(593\) 3.51472 0.144332 0.0721661 0.997393i \(-0.477009\pi\)
0.0721661 + 0.997393i \(0.477009\pi\)
\(594\) 4.82843 0.198113
\(595\) 29.2548 1.19933
\(596\) 56.7696 2.32537
\(597\) 10.3431 0.423317
\(598\) 0 0
\(599\) −0.686292 −0.0280411 −0.0140206 0.999902i \(-0.504463\pi\)
−0.0140206 + 0.999902i \(0.504463\pi\)
\(600\) 13.2426 0.540629
\(601\) 44.6274 1.82039 0.910195 0.414180i \(-0.135931\pi\)
0.910195 + 0.414180i \(0.135931\pi\)
\(602\) 65.9411 2.68756
\(603\) −1.17157 −0.0477101
\(604\) 78.4264 3.19113
\(605\) 19.7990 0.804943
\(606\) −8.82843 −0.358630
\(607\) −25.9411 −1.05292 −0.526459 0.850201i \(-0.676481\pi\)
−0.526459 + 0.850201i \(0.676481\pi\)
\(608\) 4.48528 0.181902
\(609\) 5.65685 0.229227
\(610\) 63.5980 2.57501
\(611\) 0 0
\(612\) −14.0000 −0.565916
\(613\) 36.3431 1.46789 0.733943 0.679211i \(-0.237678\pi\)
0.733943 + 0.679211i \(0.237678\pi\)
\(614\) 41.4558 1.67302
\(615\) 30.6274 1.23502
\(616\) 24.9706 1.00609
\(617\) 29.1716 1.17440 0.587202 0.809441i \(-0.300230\pi\)
0.587202 + 0.809441i \(0.300230\pi\)
\(618\) 32.9706 1.32627
\(619\) 15.7990 0.635015 0.317508 0.948256i \(-0.397154\pi\)
0.317508 + 0.948256i \(0.397154\pi\)
\(620\) −73.9411 −2.96955
\(621\) −4.00000 −0.160514
\(622\) 83.5980 3.35197
\(623\) −25.9411 −1.03931
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 14.4853 0.578948
\(627\) −5.65685 −0.225913
\(628\) −38.2843 −1.52771
\(629\) 13.3726 0.533200
\(630\) −19.3137 −0.769477
\(631\) 19.1127 0.760865 0.380432 0.924809i \(-0.375775\pi\)
0.380432 + 0.924809i \(0.375775\pi\)
\(632\) 49.9411 1.98655
\(633\) −12.0000 −0.476957
\(634\) 20.4853 0.813574
\(635\) 16.0000 0.634941
\(636\) −7.65685 −0.303614
\(637\) 0 0
\(638\) 9.65685 0.382319
\(639\) −2.00000 −0.0791188
\(640\) 58.1421 2.29827
\(641\) −26.2843 −1.03817 −0.519083 0.854724i \(-0.673727\pi\)
−0.519083 + 0.854724i \(0.673727\pi\)
\(642\) 27.3137 1.07799
\(643\) −17.1716 −0.677181 −0.338590 0.940934i \(-0.609950\pi\)
−0.338590 + 0.940934i \(0.609950\pi\)
\(644\) −43.3137 −1.70680
\(645\) −27.3137 −1.07548
\(646\) 24.9706 0.982454
\(647\) −11.3137 −0.444788 −0.222394 0.974957i \(-0.571387\pi\)
−0.222394 + 0.974957i \(0.571387\pi\)
\(648\) 4.41421 0.173407
\(649\) 7.31371 0.287088
\(650\) 0 0
\(651\) 19.3137 0.756964
\(652\) −50.4264 −1.97485
\(653\) −2.68629 −0.105123 −0.0525614 0.998618i \(-0.516739\pi\)
−0.0525614 + 0.998618i \(0.516739\pi\)
\(654\) 41.7990 1.63447
\(655\) 22.6274 0.884126
\(656\) −32.4853 −1.26834
\(657\) −11.6569 −0.454777
\(658\) 2.34315 0.0913453
\(659\) −24.6863 −0.961641 −0.480821 0.876819i \(-0.659661\pi\)
−0.480821 + 0.876819i \(0.659661\pi\)
\(660\) −21.6569 −0.842992
\(661\) 1.02944 0.0400405 0.0200202 0.999800i \(-0.493627\pi\)
0.0200202 + 0.999800i \(0.493627\pi\)
\(662\) 5.17157 0.200999
\(663\) 0 0
\(664\) 33.7990 1.31166
\(665\) 22.6274 0.877454
\(666\) −8.82843 −0.342095
\(667\) −8.00000 −0.309761
\(668\) −29.3137 −1.13418
\(669\) −4.48528 −0.173411
\(670\) 8.00000 0.309067
\(671\) −18.6274 −0.719103
\(672\) −4.48528 −0.173023
\(673\) −28.6274 −1.10351 −0.551753 0.834008i \(-0.686041\pi\)
−0.551753 + 0.834008i \(0.686041\pi\)
\(674\) −32.1421 −1.23807
\(675\) 3.00000 0.115470
\(676\) 0 0
\(677\) 49.3137 1.89528 0.947640 0.319341i \(-0.103462\pi\)
0.947640 + 0.319341i \(0.103462\pi\)
\(678\) 41.7990 1.60528
\(679\) 21.6569 0.831114
\(680\) 45.6569 1.75086
\(681\) −5.31371 −0.203622
\(682\) 32.9706 1.26251
\(683\) 19.9411 0.763026 0.381513 0.924363i \(-0.375403\pi\)
0.381513 + 0.924363i \(0.375403\pi\)
\(684\) −10.8284 −0.414035
\(685\) −14.6274 −0.558885
\(686\) −40.9706 −1.56426
\(687\) −21.3137 −0.813169
\(688\) 28.9706 1.10449
\(689\) 0 0
\(690\) 27.3137 1.03982
\(691\) 34.1421 1.29883 0.649414 0.760435i \(-0.275014\pi\)
0.649414 + 0.760435i \(0.275014\pi\)
\(692\) −1.31371 −0.0499397
\(693\) 5.65685 0.214886
\(694\) −75.5980 −2.86966
\(695\) −43.3137 −1.64298
\(696\) 8.82843 0.334641
\(697\) 39.5980 1.49988
\(698\) 18.4853 0.699678
\(699\) −26.9706 −1.02012
\(700\) 32.4853 1.22783
\(701\) 38.9706 1.47190 0.735949 0.677037i \(-0.236736\pi\)
0.735949 + 0.677037i \(0.236736\pi\)
\(702\) 0 0
\(703\) 10.3431 0.390099
\(704\) −19.6569 −0.740846
\(705\) −0.970563 −0.0365535
\(706\) −42.1421 −1.58604
\(707\) −10.3431 −0.388994
\(708\) 14.0000 0.526152
\(709\) −40.6274 −1.52579 −0.762897 0.646520i \(-0.776224\pi\)
−0.762897 + 0.646520i \(0.776224\pi\)
\(710\) 13.6569 0.512533
\(711\) 11.3137 0.424297
\(712\) −40.4853 −1.51725
\(713\) −27.3137 −1.02291
\(714\) −24.9706 −0.934500
\(715\) 0 0
\(716\) −2.62742 −0.0981912
\(717\) −2.00000 −0.0746914
\(718\) −2.48528 −0.0927499
\(719\) 37.9411 1.41497 0.707483 0.706731i \(-0.249831\pi\)
0.707483 + 0.706731i \(0.249831\pi\)
\(720\) −8.48528 −0.316228
\(721\) 38.6274 1.43856
\(722\) −26.5563 −0.988325
\(723\) −11.6569 −0.433523
\(724\) 53.5980 1.99195
\(725\) 6.00000 0.222834
\(726\) −16.8995 −0.627199
\(727\) −21.6569 −0.803208 −0.401604 0.915813i \(-0.631547\pi\)
−0.401604 + 0.915813i \(0.631547\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 79.5980 2.94605
\(731\) −35.3137 −1.30612
\(732\) −35.6569 −1.31792
\(733\) −8.62742 −0.318661 −0.159330 0.987225i \(-0.550934\pi\)
−0.159330 + 0.987225i \(0.550934\pi\)
\(734\) −57.9411 −2.13865
\(735\) −2.82843 −0.104328
\(736\) 6.34315 0.233811
\(737\) −2.34315 −0.0863109
\(738\) −26.1421 −0.962305
\(739\) −10.1421 −0.373084 −0.186542 0.982447i \(-0.559728\pi\)
−0.186542 + 0.982447i \(0.559728\pi\)
\(740\) 39.5980 1.45565
\(741\) 0 0
\(742\) −13.6569 −0.501359
\(743\) −2.00000 −0.0733729 −0.0366864 0.999327i \(-0.511680\pi\)
−0.0366864 + 0.999327i \(0.511680\pi\)
\(744\) 30.1421 1.10506
\(745\) −41.9411 −1.53660
\(746\) 24.1421 0.883906
\(747\) 7.65685 0.280150
\(748\) −28.0000 −1.02378
\(749\) 32.0000 1.16925
\(750\) 13.6569 0.498678
\(751\) 32.9706 1.20311 0.601556 0.798830i \(-0.294547\pi\)
0.601556 + 0.798830i \(0.294547\pi\)
\(752\) 1.02944 0.0375397
\(753\) 0 0
\(754\) 0 0
\(755\) −57.9411 −2.10869
\(756\) 10.8284 0.393826
\(757\) −15.9411 −0.579390 −0.289695 0.957119i \(-0.593554\pi\)
−0.289695 + 0.957119i \(0.593554\pi\)
\(758\) 39.7990 1.44556
\(759\) −8.00000 −0.290382
\(760\) 35.3137 1.28096
\(761\) −15.5147 −0.562408 −0.281204 0.959648i \(-0.590734\pi\)
−0.281204 + 0.959648i \(0.590734\pi\)
\(762\) −13.6569 −0.494736
\(763\) 48.9706 1.77285
\(764\) −73.9411 −2.67510
\(765\) 10.3431 0.373957
\(766\) 7.17157 0.259119
\(767\) 0 0
\(768\) −29.9706 −1.08147
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) −38.6274 −1.39204
\(771\) −15.6569 −0.563868
\(772\) 66.2843 2.38562
\(773\) −5.85786 −0.210693 −0.105346 0.994436i \(-0.533595\pi\)
−0.105346 + 0.994436i \(0.533595\pi\)
\(774\) 23.3137 0.837994
\(775\) 20.4853 0.735853
\(776\) 33.7990 1.21331
\(777\) −10.3431 −0.371058
\(778\) 16.8284 0.603328
\(779\) 30.6274 1.09734
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 35.3137 1.26282
\(783\) 2.00000 0.0714742
\(784\) 3.00000 0.107143
\(785\) 28.2843 1.00951
\(786\) −19.3137 −0.688897
\(787\) −32.7696 −1.16811 −0.584054 0.811715i \(-0.698534\pi\)
−0.584054 + 0.811715i \(0.698534\pi\)
\(788\) 63.1127 2.24830
\(789\) 12.0000 0.427211
\(790\) −77.2548 −2.74860
\(791\) 48.9706 1.74119
\(792\) 8.82843 0.313704
\(793\) 0 0
\(794\) 7.17157 0.254510
\(795\) 5.65685 0.200628
\(796\) 39.5980 1.40351
\(797\) 35.6569 1.26303 0.631515 0.775363i \(-0.282433\pi\)
0.631515 + 0.775363i \(0.282433\pi\)
\(798\) −19.3137 −0.683698
\(799\) −1.25483 −0.0443928
\(800\) −4.75736 −0.168198
\(801\) −9.17157 −0.324062
\(802\) 5.17157 0.182615
\(803\) −23.3137 −0.822723
\(804\) −4.48528 −0.158184
\(805\) 32.0000 1.12785
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) −16.1421 −0.567878
\(809\) 41.3137 1.45251 0.726256 0.687424i \(-0.241259\pi\)
0.726256 + 0.687424i \(0.241259\pi\)
\(810\) −6.82843 −0.239926
\(811\) 1.85786 0.0652384 0.0326192 0.999468i \(-0.489615\pi\)
0.0326192 + 0.999468i \(0.489615\pi\)
\(812\) 21.6569 0.760007
\(813\) −11.7990 −0.413809
\(814\) −17.6569 −0.618872
\(815\) 37.2548 1.30498
\(816\) −10.9706 −0.384047
\(817\) −27.3137 −0.955586
\(818\) 2.48528 0.0868958
\(819\) 0 0
\(820\) 117.255 4.09472
\(821\) −15.7990 −0.551389 −0.275694 0.961245i \(-0.588908\pi\)
−0.275694 + 0.961245i \(0.588908\pi\)
\(822\) 12.4853 0.435474
\(823\) 48.9706 1.70701 0.853503 0.521088i \(-0.174473\pi\)
0.853503 + 0.521088i \(0.174473\pi\)
\(824\) 60.2843 2.10010
\(825\) 6.00000 0.208893
\(826\) 24.9706 0.868837
\(827\) 26.0000 0.904109 0.452054 0.891990i \(-0.350691\pi\)
0.452054 + 0.891990i \(0.350691\pi\)
\(828\) −15.3137 −0.532188
\(829\) 5.31371 0.184553 0.0922764 0.995733i \(-0.470586\pi\)
0.0922764 + 0.995733i \(0.470586\pi\)
\(830\) −52.2843 −1.81481
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −3.65685 −0.126702
\(834\) 36.9706 1.28019
\(835\) 21.6569 0.749466
\(836\) −21.6569 −0.749018
\(837\) 6.82843 0.236025
\(838\) −73.9411 −2.55425
\(839\) 47.2548 1.63142 0.815709 0.578462i \(-0.196347\pi\)
0.815709 + 0.578462i \(0.196347\pi\)
\(840\) −35.3137 −1.21844
\(841\) −25.0000 −0.862069
\(842\) −35.4558 −1.22189
\(843\) −26.8284 −0.924020
\(844\) −45.9411 −1.58136
\(845\) 0 0
\(846\) 0.828427 0.0284819
\(847\) −19.7990 −0.680301
\(848\) −6.00000 −0.206041
\(849\) −4.97056 −0.170589
\(850\) −26.4853 −0.908438
\(851\) 14.6274 0.501421
\(852\) −7.65685 −0.262320
\(853\) 7.65685 0.262166 0.131083 0.991371i \(-0.458155\pi\)
0.131083 + 0.991371i \(0.458155\pi\)
\(854\) −63.5980 −2.17628
\(855\) 8.00000 0.273594
\(856\) 49.9411 1.70695
\(857\) 29.5980 1.01105 0.505524 0.862813i \(-0.331299\pi\)
0.505524 + 0.862813i \(0.331299\pi\)
\(858\) 0 0
\(859\) −23.3137 −0.795453 −0.397727 0.917504i \(-0.630201\pi\)
−0.397727 + 0.917504i \(0.630201\pi\)
\(860\) −104.569 −3.56576
\(861\) −30.6274 −1.04378
\(862\) −47.4558 −1.61635
\(863\) 39.6569 1.34994 0.674968 0.737847i \(-0.264158\pi\)
0.674968 + 0.737847i \(0.264158\pi\)
\(864\) −1.58579 −0.0539496
\(865\) 0.970563 0.0330001
\(866\) 3.17157 0.107774
\(867\) −3.62742 −0.123194
\(868\) 73.9411 2.50973
\(869\) 22.6274 0.767583
\(870\) −13.6569 −0.463011
\(871\) 0 0
\(872\) 76.4264 2.58812
\(873\) 7.65685 0.259145
\(874\) 27.3137 0.923900
\(875\) 16.0000 0.540899
\(876\) −44.6274 −1.50782
\(877\) 14.2843 0.482346 0.241173 0.970482i \(-0.422468\pi\)
0.241173 + 0.970482i \(0.422468\pi\)
\(878\) −40.9706 −1.38269
\(879\) 26.1421 0.881752
\(880\) −16.9706 −0.572078
\(881\) 53.5980 1.80576 0.902881 0.429891i \(-0.141448\pi\)
0.902881 + 0.429891i \(0.141448\pi\)
\(882\) 2.41421 0.0812908
\(883\) −51.5980 −1.73641 −0.868205 0.496205i \(-0.834726\pi\)
−0.868205 + 0.496205i \(0.834726\pi\)
\(884\) 0 0
\(885\) −10.3431 −0.347681
\(886\) 101.255 3.40172
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −16.1421 −0.541695
\(889\) −16.0000 −0.536623
\(890\) 62.6274 2.09928
\(891\) 2.00000 0.0670025
\(892\) −17.1716 −0.574947
\(893\) −0.970563 −0.0324786
\(894\) 35.7990 1.19730
\(895\) 1.94113 0.0648847
\(896\) −58.1421 −1.94239
\(897\) 0 0
\(898\) −18.8284 −0.628313
\(899\) 13.6569 0.455482
\(900\) 11.4853 0.382843
\(901\) 7.31371 0.243655
\(902\) −52.2843 −1.74088
\(903\) 27.3137 0.908943
\(904\) 76.4264 2.54190
\(905\) −39.5980 −1.31628
\(906\) 49.4558 1.64306
\(907\) 20.9706 0.696316 0.348158 0.937436i \(-0.386807\pi\)
0.348158 + 0.937436i \(0.386807\pi\)
\(908\) −20.3431 −0.675111
\(909\) −3.65685 −0.121290
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −8.48528 −0.280976
\(913\) 15.3137 0.506810
\(914\) −8.82843 −0.292018
\(915\) 26.3431 0.870878
\(916\) −81.5980 −2.69607
\(917\) −22.6274 −0.747223
\(918\) −8.82843 −0.291382
\(919\) −19.3137 −0.637100 −0.318550 0.947906i \(-0.603196\pi\)
−0.318550 + 0.947906i \(0.603196\pi\)
\(920\) 49.9411 1.64651
\(921\) 17.1716 0.565823
\(922\) −26.1421 −0.860945
\(923\) 0 0
\(924\) 21.6569 0.712458
\(925\) −10.9706 −0.360710
\(926\) 18.1421 0.596188
\(927\) 13.6569 0.448550
\(928\) −3.17157 −0.104112
\(929\) 27.7990 0.912055 0.456028 0.889966i \(-0.349272\pi\)
0.456028 + 0.889966i \(0.349272\pi\)
\(930\) −46.6274 −1.52897
\(931\) −2.82843 −0.0926980
\(932\) −103.255 −3.38222
\(933\) 34.6274 1.13365
\(934\) −19.3137 −0.631964
\(935\) 20.6863 0.676514
\(936\) 0 0
\(937\) 1.31371 0.0429170 0.0214585 0.999770i \(-0.493169\pi\)
0.0214585 + 0.999770i \(0.493169\pi\)
\(938\) −8.00000 −0.261209
\(939\) 6.00000 0.195803
\(940\) −3.71573 −0.121194
\(941\) −5.85786 −0.190961 −0.0954805 0.995431i \(-0.530439\pi\)
−0.0954805 + 0.995431i \(0.530439\pi\)
\(942\) −24.1421 −0.786593
\(943\) 43.3137 1.41049
\(944\) 10.9706 0.357061
\(945\) −8.00000 −0.260240
\(946\) 46.6274 1.51599
\(947\) 54.9706 1.78630 0.893152 0.449756i \(-0.148489\pi\)
0.893152 + 0.449756i \(0.148489\pi\)
\(948\) 43.3137 1.40676
\(949\) 0 0
\(950\) −20.4853 −0.664630
\(951\) 8.48528 0.275154
\(952\) −45.6569 −1.47975
\(953\) 51.6569 1.67333 0.836665 0.547715i \(-0.184502\pi\)
0.836665 + 0.547715i \(0.184502\pi\)
\(954\) −4.82843 −0.156326
\(955\) 54.6274 1.76770
\(956\) −7.65685 −0.247640
\(957\) 4.00000 0.129302
\(958\) −6.48528 −0.209530
\(959\) 14.6274 0.472344
\(960\) 27.7990 0.897209
\(961\) 15.6274 0.504110
\(962\) 0 0
\(963\) 11.3137 0.364579
\(964\) −44.6274 −1.43735
\(965\) −48.9706 −1.57642
\(966\) −27.3137 −0.878804
\(967\) 10.1421 0.326149 0.163075 0.986614i \(-0.447859\pi\)
0.163075 + 0.986614i \(0.447859\pi\)
\(968\) −30.8995 −0.993147
\(969\) 10.3431 0.332270
\(970\) −52.2843 −1.67875
\(971\) 7.31371 0.234708 0.117354 0.993090i \(-0.462559\pi\)
0.117354 + 0.993090i \(0.462559\pi\)
\(972\) 3.82843 0.122797
\(973\) 43.3137 1.38857
\(974\) −76.7696 −2.45986
\(975\) 0 0
\(976\) −27.9411 −0.894374
\(977\) −13.8579 −0.443352 −0.221676 0.975120i \(-0.571153\pi\)
−0.221676 + 0.975120i \(0.571153\pi\)
\(978\) −31.7990 −1.01682
\(979\) −18.3431 −0.586249
\(980\) −10.8284 −0.345901
\(981\) 17.3137 0.552784
\(982\) −35.3137 −1.12691
\(983\) −2.68629 −0.0856794 −0.0428397 0.999082i \(-0.513640\pi\)
−0.0428397 + 0.999082i \(0.513640\pi\)
\(984\) −47.7990 −1.52378
\(985\) −46.6274 −1.48567
\(986\) −17.6569 −0.562309
\(987\) 0.970563 0.0308934
\(988\) 0 0
\(989\) −38.6274 −1.22828
\(990\) −13.6569 −0.434043
\(991\) 27.3137 0.867649 0.433824 0.900998i \(-0.357164\pi\)
0.433824 + 0.900998i \(0.357164\pi\)
\(992\) −10.8284 −0.343803
\(993\) 2.14214 0.0679786
\(994\) −13.6569 −0.433169
\(995\) −29.2548 −0.927441
\(996\) 29.3137 0.928840
\(997\) 51.2548 1.62326 0.811628 0.584174i \(-0.198581\pi\)
0.811628 + 0.584174i \(0.198581\pi\)
\(998\) 5.17157 0.163703
\(999\) −3.65685 −0.115698
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 507.2.a.h.1.2 2
3.2 odd 2 1521.2.a.f.1.1 2
4.3 odd 2 8112.2.a.bm.1.1 2
13.2 odd 12 507.2.j.f.316.4 8
13.3 even 3 507.2.e.d.22.1 4
13.4 even 6 507.2.e.h.484.2 4
13.5 odd 4 507.2.b.e.337.1 4
13.6 odd 12 507.2.j.f.361.1 8
13.7 odd 12 507.2.j.f.361.4 8
13.8 odd 4 507.2.b.e.337.4 4
13.9 even 3 507.2.e.d.484.1 4
13.10 even 6 507.2.e.h.22.2 4
13.11 odd 12 507.2.j.f.316.1 8
13.12 even 2 39.2.a.b.1.1 2
39.5 even 4 1521.2.b.j.1351.4 4
39.8 even 4 1521.2.b.j.1351.1 4
39.38 odd 2 117.2.a.c.1.2 2
52.51 odd 2 624.2.a.k.1.2 2
65.12 odd 4 975.2.c.h.274.1 4
65.38 odd 4 975.2.c.h.274.4 4
65.64 even 2 975.2.a.l.1.2 2
91.90 odd 2 1911.2.a.h.1.1 2
104.51 odd 2 2496.2.a.bi.1.1 2
104.77 even 2 2496.2.a.bf.1.1 2
117.25 even 6 1053.2.e.m.352.2 4
117.38 odd 6 1053.2.e.e.352.1 4
117.77 odd 6 1053.2.e.e.703.1 4
117.103 even 6 1053.2.e.m.703.2 4
143.142 odd 2 4719.2.a.p.1.2 2
156.155 even 2 1872.2.a.w.1.1 2
195.38 even 4 2925.2.c.u.2224.1 4
195.77 even 4 2925.2.c.u.2224.4 4
195.194 odd 2 2925.2.a.v.1.1 2
273.272 even 2 5733.2.a.u.1.2 2
312.77 odd 2 7488.2.a.cl.1.2 2
312.155 even 2 7488.2.a.co.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.1 2 13.12 even 2
117.2.a.c.1.2 2 39.38 odd 2
507.2.a.h.1.2 2 1.1 even 1 trivial
507.2.b.e.337.1 4 13.5 odd 4
507.2.b.e.337.4 4 13.8 odd 4
507.2.e.d.22.1 4 13.3 even 3
507.2.e.d.484.1 4 13.9 even 3
507.2.e.h.22.2 4 13.10 even 6
507.2.e.h.484.2 4 13.4 even 6
507.2.j.f.316.1 8 13.11 odd 12
507.2.j.f.316.4 8 13.2 odd 12
507.2.j.f.361.1 8 13.6 odd 12
507.2.j.f.361.4 8 13.7 odd 12
624.2.a.k.1.2 2 52.51 odd 2
975.2.a.l.1.2 2 65.64 even 2
975.2.c.h.274.1 4 65.12 odd 4
975.2.c.h.274.4 4 65.38 odd 4
1053.2.e.e.352.1 4 117.38 odd 6
1053.2.e.e.703.1 4 117.77 odd 6
1053.2.e.m.352.2 4 117.25 even 6
1053.2.e.m.703.2 4 117.103 even 6
1521.2.a.f.1.1 2 3.2 odd 2
1521.2.b.j.1351.1 4 39.8 even 4
1521.2.b.j.1351.4 4 39.5 even 4
1872.2.a.w.1.1 2 156.155 even 2
1911.2.a.h.1.1 2 91.90 odd 2
2496.2.a.bf.1.1 2 104.77 even 2
2496.2.a.bi.1.1 2 104.51 odd 2
2925.2.a.v.1.1 2 195.194 odd 2
2925.2.c.u.2224.1 4 195.38 even 4
2925.2.c.u.2224.4 4 195.77 even 4
4719.2.a.p.1.2 2 143.142 odd 2
5733.2.a.u.1.2 2 273.272 even 2
7488.2.a.cl.1.2 2 312.77 odd 2
7488.2.a.co.1.2 2 312.155 even 2
8112.2.a.bm.1.1 2 4.3 odd 2