Properties

Label 507.2.a.k.1.3
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.80194 q^{2} +1.00000 q^{3} +1.24698 q^{4} +1.44504 q^{5} +1.80194 q^{6} +3.44504 q^{7} -1.35690 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.80194 q^{2} +1.00000 q^{3} +1.24698 q^{4} +1.44504 q^{5} +1.80194 q^{6} +3.44504 q^{7} -1.35690 q^{8} +1.00000 q^{9} +2.60388 q^{10} -5.18598 q^{11} +1.24698 q^{12} +6.20775 q^{14} +1.44504 q^{15} -4.93900 q^{16} -0.753020 q^{17} +1.80194 q^{18} +7.96077 q^{19} +1.80194 q^{20} +3.44504 q^{21} -9.34481 q^{22} -2.82908 q^{23} -1.35690 q^{24} -2.91185 q^{25} +1.00000 q^{27} +4.29590 q^{28} -3.91185 q^{29} +2.60388 q^{30} -4.89977 q^{31} -6.18598 q^{32} -5.18598 q^{33} -1.35690 q^{34} +4.97823 q^{35} +1.24698 q^{36} +6.24698 q^{37} +14.3448 q^{38} -1.96077 q^{40} +1.80194 q^{41} +6.20775 q^{42} -7.09783 q^{43} -6.46681 q^{44} +1.44504 q^{45} -5.09783 q^{46} +10.5526 q^{47} -4.93900 q^{48} +4.86831 q^{49} -5.24698 q^{50} -0.753020 q^{51} -3.08815 q^{53} +1.80194 q^{54} -7.49396 q^{55} -4.67456 q^{56} +7.96077 q^{57} -7.04892 q^{58} +1.87800 q^{59} +1.80194 q^{60} +3.34481 q^{61} -8.82908 q^{62} +3.44504 q^{63} -1.26875 q^{64} -9.34481 q^{66} -4.54288 q^{67} -0.939001 q^{68} -2.82908 q^{69} +8.97046 q^{70} -9.11960 q^{71} -1.35690 q^{72} +2.95108 q^{73} +11.2567 q^{74} -2.91185 q^{75} +9.92692 q^{76} -17.8659 q^{77} -9.43296 q^{79} -7.13706 q^{80} +1.00000 q^{81} +3.24698 q^{82} -6.46681 q^{83} +4.29590 q^{84} -1.08815 q^{85} -12.7899 q^{86} -3.91185 q^{87} +7.03684 q^{88} +1.15883 q^{89} +2.60388 q^{90} -3.52781 q^{92} -4.89977 q^{93} +19.0151 q^{94} +11.5036 q^{95} -6.18598 q^{96} +8.65817 q^{97} +8.77240 q^{98} -5.18598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 3q^{3} - q^{4} + 4q^{5} + q^{6} + 10q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} + 3q^{3} - q^{4} + 4q^{5} + q^{6} + 10q^{7} + 3q^{9} - q^{10} - q^{11} - q^{12} + q^{14} + 4q^{15} - 5q^{16} - 7q^{17} + q^{18} + 11q^{19} + q^{20} + 10q^{21} - 5q^{22} + 2q^{23} - 5q^{25} + 3q^{27} - q^{28} - 8q^{29} - q^{30} + 8q^{31} - 4q^{32} - q^{33} + 18q^{35} - q^{36} + 14q^{37} + 20q^{38} + 7q^{40} + q^{41} + q^{42} - 3q^{43} - 16q^{44} + 4q^{45} + 3q^{46} - 9q^{47} - 5q^{48} + 17q^{49} - 11q^{50} - 7q^{51} - 13q^{53} + q^{54} - 13q^{55} + 7q^{56} + 11q^{57} - 12q^{58} - 14q^{59} + q^{60} - 13q^{61} - 16q^{62} + 10q^{63} + 4q^{64} - 5q^{66} + 5q^{67} + 7q^{68} + 2q^{69} - 8q^{70} - 6q^{71} + 18q^{73} + 7q^{74} - 5q^{75} + q^{76} - 15q^{77} - 9q^{79} - 16q^{80} + 3q^{81} + 5q^{82} - 16q^{83} - q^{84} - 7q^{85} - 15q^{86} - 8q^{87} - 7q^{88} - 5q^{89} - q^{90} - 17q^{92} + 8q^{93} + 32q^{94} + 3q^{95} - 4q^{96} + 5q^{97} - 13q^{98} - q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.80194 1.27416 0.637081 0.770797i \(-0.280142\pi\)
0.637081 + 0.770797i \(0.280142\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.24698 0.623490
\(5\) 1.44504 0.646242 0.323121 0.946358i \(-0.395268\pi\)
0.323121 + 0.946358i \(0.395268\pi\)
\(6\) 1.80194 0.735638
\(7\) 3.44504 1.30210 0.651052 0.759033i \(-0.274328\pi\)
0.651052 + 0.759033i \(0.274328\pi\)
\(8\) −1.35690 −0.479735
\(9\) 1.00000 0.333333
\(10\) 2.60388 0.823418
\(11\) −5.18598 −1.56363 −0.781816 0.623509i \(-0.785706\pi\)
−0.781816 + 0.623509i \(0.785706\pi\)
\(12\) 1.24698 0.359972
\(13\) 0 0
\(14\) 6.20775 1.65909
\(15\) 1.44504 0.373108
\(16\) −4.93900 −1.23475
\(17\) −0.753020 −0.182634 −0.0913171 0.995822i \(-0.529108\pi\)
−0.0913171 + 0.995822i \(0.529108\pi\)
\(18\) 1.80194 0.424721
\(19\) 7.96077 1.82633 0.913163 0.407594i \(-0.133632\pi\)
0.913163 + 0.407594i \(0.133632\pi\)
\(20\) 1.80194 0.402926
\(21\) 3.44504 0.751770
\(22\) −9.34481 −1.99232
\(23\) −2.82908 −0.589905 −0.294952 0.955512i \(-0.595304\pi\)
−0.294952 + 0.955512i \(0.595304\pi\)
\(24\) −1.35690 −0.276975
\(25\) −2.91185 −0.582371
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 4.29590 0.811848
\(29\) −3.91185 −0.726413 −0.363207 0.931709i \(-0.618318\pi\)
−0.363207 + 0.931709i \(0.618318\pi\)
\(30\) 2.60388 0.475400
\(31\) −4.89977 −0.880025 −0.440013 0.897992i \(-0.645026\pi\)
−0.440013 + 0.897992i \(0.645026\pi\)
\(32\) −6.18598 −1.09354
\(33\) −5.18598 −0.902763
\(34\) −1.35690 −0.232706
\(35\) 4.97823 0.841474
\(36\) 1.24698 0.207830
\(37\) 6.24698 1.02700 0.513499 0.858090i \(-0.328349\pi\)
0.513499 + 0.858090i \(0.328349\pi\)
\(38\) 14.3448 2.32704
\(39\) 0 0
\(40\) −1.96077 −0.310025
\(41\) 1.80194 0.281415 0.140708 0.990051i \(-0.455062\pi\)
0.140708 + 0.990051i \(0.455062\pi\)
\(42\) 6.20775 0.957877
\(43\) −7.09783 −1.08241 −0.541205 0.840891i \(-0.682032\pi\)
−0.541205 + 0.840891i \(0.682032\pi\)
\(44\) −6.46681 −0.974909
\(45\) 1.44504 0.215414
\(46\) −5.09783 −0.751635
\(47\) 10.5526 1.53925 0.769625 0.638496i \(-0.220443\pi\)
0.769625 + 0.638496i \(0.220443\pi\)
\(48\) −4.93900 −0.712883
\(49\) 4.86831 0.695473
\(50\) −5.24698 −0.742035
\(51\) −0.753020 −0.105444
\(52\) 0 0
\(53\) −3.08815 −0.424189 −0.212095 0.977249i \(-0.568029\pi\)
−0.212095 + 0.977249i \(0.568029\pi\)
\(54\) 1.80194 0.245213
\(55\) −7.49396 −1.01049
\(56\) −4.67456 −0.624665
\(57\) 7.96077 1.05443
\(58\) −7.04892 −0.925568
\(59\) 1.87800 0.244495 0.122248 0.992500i \(-0.460990\pi\)
0.122248 + 0.992500i \(0.460990\pi\)
\(60\) 1.80194 0.232629
\(61\) 3.34481 0.428260 0.214130 0.976805i \(-0.431308\pi\)
0.214130 + 0.976805i \(0.431308\pi\)
\(62\) −8.82908 −1.12129
\(63\) 3.44504 0.434034
\(64\) −1.26875 −0.158594
\(65\) 0 0
\(66\) −9.34481 −1.15027
\(67\) −4.54288 −0.555001 −0.277500 0.960726i \(-0.589506\pi\)
−0.277500 + 0.960726i \(0.589506\pi\)
\(68\) −0.939001 −0.113871
\(69\) −2.82908 −0.340582
\(70\) 8.97046 1.07218
\(71\) −9.11960 −1.08230 −0.541149 0.840927i \(-0.682011\pi\)
−0.541149 + 0.840927i \(0.682011\pi\)
\(72\) −1.35690 −0.159912
\(73\) 2.95108 0.345398 0.172699 0.984975i \(-0.444751\pi\)
0.172699 + 0.984975i \(0.444751\pi\)
\(74\) 11.2567 1.30856
\(75\) −2.91185 −0.336232
\(76\) 9.92692 1.13870
\(77\) −17.8659 −2.03601
\(78\) 0 0
\(79\) −9.43296 −1.06129 −0.530645 0.847594i \(-0.678050\pi\)
−0.530645 + 0.847594i \(0.678050\pi\)
\(80\) −7.13706 −0.797948
\(81\) 1.00000 0.111111
\(82\) 3.24698 0.358569
\(83\) −6.46681 −0.709825 −0.354912 0.934900i \(-0.615489\pi\)
−0.354912 + 0.934900i \(0.615489\pi\)
\(84\) 4.29590 0.468721
\(85\) −1.08815 −0.118026
\(86\) −12.7899 −1.37917
\(87\) −3.91185 −0.419395
\(88\) 7.03684 0.750129
\(89\) 1.15883 0.122836 0.0614181 0.998112i \(-0.480438\pi\)
0.0614181 + 0.998112i \(0.480438\pi\)
\(90\) 2.60388 0.274473
\(91\) 0 0
\(92\) −3.52781 −0.367800
\(93\) −4.89977 −0.508083
\(94\) 19.0151 1.96125
\(95\) 11.5036 1.18025
\(96\) −6.18598 −0.631354
\(97\) 8.65817 0.879104 0.439552 0.898217i \(-0.355137\pi\)
0.439552 + 0.898217i \(0.355137\pi\)
\(98\) 8.77240 0.886146
\(99\) −5.18598 −0.521211
\(100\) −3.63102 −0.363102
\(101\) −8.47650 −0.843443 −0.421722 0.906725i \(-0.638574\pi\)
−0.421722 + 0.906725i \(0.638574\pi\)
\(102\) −1.35690 −0.134353
\(103\) 5.64742 0.556456 0.278228 0.960515i \(-0.410253\pi\)
0.278228 + 0.960515i \(0.410253\pi\)
\(104\) 0 0
\(105\) 4.97823 0.485825
\(106\) −5.56465 −0.540486
\(107\) −6.73556 −0.651151 −0.325576 0.945516i \(-0.605558\pi\)
−0.325576 + 0.945516i \(0.605558\pi\)
\(108\) 1.24698 0.119991
\(109\) −2.07606 −0.198851 −0.0994255 0.995045i \(-0.531700\pi\)
−0.0994255 + 0.995045i \(0.531700\pi\)
\(110\) −13.5036 −1.28752
\(111\) 6.24698 0.592937
\(112\) −17.0151 −1.60777
\(113\) 6.16852 0.580286 0.290143 0.956983i \(-0.406297\pi\)
0.290143 + 0.956983i \(0.406297\pi\)
\(114\) 14.3448 1.34351
\(115\) −4.08815 −0.381222
\(116\) −4.87800 −0.452911
\(117\) 0 0
\(118\) 3.38404 0.311526
\(119\) −2.59419 −0.237809
\(120\) −1.96077 −0.178993
\(121\) 15.8944 1.44495
\(122\) 6.02715 0.545672
\(123\) 1.80194 0.162475
\(124\) −6.10992 −0.548687
\(125\) −11.4330 −1.02260
\(126\) 6.20775 0.553030
\(127\) −14.2620 −1.26555 −0.632776 0.774335i \(-0.718085\pi\)
−0.632776 + 0.774335i \(0.718085\pi\)
\(128\) 10.0858 0.891463
\(129\) −7.09783 −0.624929
\(130\) 0 0
\(131\) 22.6015 1.97470 0.987350 0.158554i \(-0.0506831\pi\)
0.987350 + 0.158554i \(0.0506831\pi\)
\(132\) −6.46681 −0.562864
\(133\) 27.4252 2.37807
\(134\) −8.18598 −0.707161
\(135\) 1.44504 0.124369
\(136\) 1.02177 0.0876161
\(137\) −13.6353 −1.16495 −0.582473 0.812850i \(-0.697915\pi\)
−0.582473 + 0.812850i \(0.697915\pi\)
\(138\) −5.09783 −0.433957
\(139\) 17.6015 1.49294 0.746469 0.665420i \(-0.231748\pi\)
0.746469 + 0.665420i \(0.231748\pi\)
\(140\) 6.20775 0.524651
\(141\) 10.5526 0.888686
\(142\) −16.4330 −1.37902
\(143\) 0 0
\(144\) −4.93900 −0.411583
\(145\) −5.65279 −0.469439
\(146\) 5.31767 0.440093
\(147\) 4.86831 0.401532
\(148\) 7.78986 0.640322
\(149\) 12.7385 1.04358 0.521791 0.853073i \(-0.325264\pi\)
0.521791 + 0.853073i \(0.325264\pi\)
\(150\) −5.24698 −0.428414
\(151\) 15.6407 1.27282 0.636412 0.771350i \(-0.280418\pi\)
0.636412 + 0.771350i \(0.280418\pi\)
\(152\) −10.8019 −0.876153
\(153\) −0.753020 −0.0608781
\(154\) −32.1933 −2.59421
\(155\) −7.08038 −0.568710
\(156\) 0 0
\(157\) −0.823708 −0.0657391 −0.0328695 0.999460i \(-0.510465\pi\)
−0.0328695 + 0.999460i \(0.510465\pi\)
\(158\) −16.9976 −1.35226
\(159\) −3.08815 −0.244906
\(160\) −8.93900 −0.706690
\(161\) −9.74632 −0.768117
\(162\) 1.80194 0.141574
\(163\) 6.26875 0.491006 0.245503 0.969396i \(-0.421047\pi\)
0.245503 + 0.969396i \(0.421047\pi\)
\(164\) 2.24698 0.175460
\(165\) −7.49396 −0.583404
\(166\) −11.6528 −0.904432
\(167\) −7.45042 −0.576531 −0.288265 0.957551i \(-0.593079\pi\)
−0.288265 + 0.957551i \(0.593079\pi\)
\(168\) −4.67456 −0.360650
\(169\) 0 0
\(170\) −1.96077 −0.150384
\(171\) 7.96077 0.608775
\(172\) −8.85086 −0.674871
\(173\) −2.00969 −0.152794 −0.0763969 0.997077i \(-0.524342\pi\)
−0.0763969 + 0.997077i \(0.524342\pi\)
\(174\) −7.04892 −0.534377
\(175\) −10.0315 −0.758307
\(176\) 25.6136 1.93070
\(177\) 1.87800 0.141159
\(178\) 2.08815 0.156513
\(179\) 20.0368 1.49762 0.748812 0.662783i \(-0.230625\pi\)
0.748812 + 0.662783i \(0.230625\pi\)
\(180\) 1.80194 0.134309
\(181\) 24.1226 1.79302 0.896509 0.443026i \(-0.146095\pi\)
0.896509 + 0.443026i \(0.146095\pi\)
\(182\) 0 0
\(183\) 3.34481 0.247256
\(184\) 3.83877 0.282998
\(185\) 9.02715 0.663689
\(186\) −8.82908 −0.647380
\(187\) 3.90515 0.285573
\(188\) 13.1588 0.959707
\(189\) 3.44504 0.250590
\(190\) 20.7289 1.50383
\(191\) −7.08038 −0.512318 −0.256159 0.966635i \(-0.582457\pi\)
−0.256159 + 0.966635i \(0.582457\pi\)
\(192\) −1.26875 −0.0915641
\(193\) 9.76809 0.703122 0.351561 0.936165i \(-0.385651\pi\)
0.351561 + 0.936165i \(0.385651\pi\)
\(194\) 15.6015 1.12012
\(195\) 0 0
\(196\) 6.07069 0.433621
\(197\) 23.4112 1.66798 0.833989 0.551781i \(-0.186052\pi\)
0.833989 + 0.551781i \(0.186052\pi\)
\(198\) −9.34481 −0.664107
\(199\) 4.02475 0.285307 0.142654 0.989773i \(-0.454437\pi\)
0.142654 + 0.989773i \(0.454437\pi\)
\(200\) 3.95108 0.279384
\(201\) −4.54288 −0.320430
\(202\) −15.2741 −1.07468
\(203\) −13.4765 −0.945865
\(204\) −0.939001 −0.0657432
\(205\) 2.60388 0.181863
\(206\) 10.1763 0.709016
\(207\) −2.82908 −0.196635
\(208\) 0 0
\(209\) −41.2844 −2.85570
\(210\) 8.97046 0.619021
\(211\) −3.91185 −0.269303 −0.134652 0.990893i \(-0.542992\pi\)
−0.134652 + 0.990893i \(0.542992\pi\)
\(212\) −3.85086 −0.264478
\(213\) −9.11960 −0.624865
\(214\) −12.1371 −0.829673
\(215\) −10.2567 −0.699499
\(216\) −1.35690 −0.0923251
\(217\) −16.8799 −1.14588
\(218\) −3.74094 −0.253368
\(219\) 2.95108 0.199416
\(220\) −9.34481 −0.630027
\(221\) 0 0
\(222\) 11.2567 0.755498
\(223\) 7.44935 0.498846 0.249423 0.968395i \(-0.419759\pi\)
0.249423 + 0.968395i \(0.419759\pi\)
\(224\) −21.3110 −1.42390
\(225\) −2.91185 −0.194124
\(226\) 11.1153 0.739378
\(227\) −21.2500 −1.41041 −0.705205 0.709004i \(-0.749145\pi\)
−0.705205 + 0.709004i \(0.749145\pi\)
\(228\) 9.92692 0.657426
\(229\) 9.29590 0.614290 0.307145 0.951663i \(-0.400626\pi\)
0.307145 + 0.951663i \(0.400626\pi\)
\(230\) −7.36658 −0.485738
\(231\) −17.8659 −1.17549
\(232\) 5.30798 0.348486
\(233\) 16.2107 1.06200 0.531000 0.847372i \(-0.321816\pi\)
0.531000 + 0.847372i \(0.321816\pi\)
\(234\) 0 0
\(235\) 15.2489 0.994728
\(236\) 2.34183 0.152440
\(237\) −9.43296 −0.612737
\(238\) −4.67456 −0.303007
\(239\) −13.5090 −0.873826 −0.436913 0.899504i \(-0.643928\pi\)
−0.436913 + 0.899504i \(0.643928\pi\)
\(240\) −7.13706 −0.460695
\(241\) 6.26875 0.403806 0.201903 0.979406i \(-0.435287\pi\)
0.201903 + 0.979406i \(0.435287\pi\)
\(242\) 28.6407 1.84109
\(243\) 1.00000 0.0641500
\(244\) 4.17092 0.267015
\(245\) 7.03492 0.449444
\(246\) 3.24698 0.207020
\(247\) 0 0
\(248\) 6.64848 0.422179
\(249\) −6.46681 −0.409818
\(250\) −20.6015 −1.30295
\(251\) −0.753020 −0.0475302 −0.0237651 0.999718i \(-0.507565\pi\)
−0.0237651 + 0.999718i \(0.507565\pi\)
\(252\) 4.29590 0.270616
\(253\) 14.6716 0.922394
\(254\) −25.6993 −1.61252
\(255\) −1.08815 −0.0681423
\(256\) 20.7114 1.29446
\(257\) 19.7265 1.23050 0.615252 0.788331i \(-0.289054\pi\)
0.615252 + 0.788331i \(0.289054\pi\)
\(258\) −12.7899 −0.796262
\(259\) 21.5211 1.33726
\(260\) 0 0
\(261\) −3.91185 −0.242138
\(262\) 40.7265 2.51609
\(263\) −17.6093 −1.08583 −0.542917 0.839787i \(-0.682680\pi\)
−0.542917 + 0.839787i \(0.682680\pi\)
\(264\) 7.03684 0.433087
\(265\) −4.46250 −0.274129
\(266\) 49.4185 3.03004
\(267\) 1.15883 0.0709195
\(268\) −5.66487 −0.346037
\(269\) −16.3870 −0.999135 −0.499567 0.866275i \(-0.666508\pi\)
−0.499567 + 0.866275i \(0.666508\pi\)
\(270\) 2.60388 0.158467
\(271\) −0.795233 −0.0483070 −0.0241535 0.999708i \(-0.507689\pi\)
−0.0241535 + 0.999708i \(0.507689\pi\)
\(272\) 3.71917 0.225508
\(273\) 0 0
\(274\) −24.5700 −1.48433
\(275\) 15.1008 0.910614
\(276\) −3.52781 −0.212349
\(277\) −4.83340 −0.290411 −0.145205 0.989402i \(-0.546384\pi\)
−0.145205 + 0.989402i \(0.546384\pi\)
\(278\) 31.7168 1.90225
\(279\) −4.89977 −0.293342
\(280\) −6.75494 −0.403685
\(281\) 18.7748 1.12001 0.560005 0.828489i \(-0.310799\pi\)
0.560005 + 0.828489i \(0.310799\pi\)
\(282\) 19.0151 1.13233
\(283\) −7.91723 −0.470631 −0.235315 0.971919i \(-0.575612\pi\)
−0.235315 + 0.971919i \(0.575612\pi\)
\(284\) −11.3720 −0.674802
\(285\) 11.5036 0.681417
\(286\) 0 0
\(287\) 6.20775 0.366432
\(288\) −6.18598 −0.364512
\(289\) −16.4330 −0.966645
\(290\) −10.1860 −0.598141
\(291\) 8.65817 0.507551
\(292\) 3.67994 0.215352
\(293\) 6.57912 0.384356 0.192178 0.981360i \(-0.438445\pi\)
0.192178 + 0.981360i \(0.438445\pi\)
\(294\) 8.77240 0.511617
\(295\) 2.71379 0.158003
\(296\) −8.47650 −0.492687
\(297\) −5.18598 −0.300921
\(298\) 22.9541 1.32969
\(299\) 0 0
\(300\) −3.63102 −0.209637
\(301\) −24.4523 −1.40941
\(302\) 28.1836 1.62178
\(303\) −8.47650 −0.486962
\(304\) −39.3183 −2.25506
\(305\) 4.83340 0.276759
\(306\) −1.35690 −0.0775686
\(307\) −24.8649 −1.41911 −0.709556 0.704649i \(-0.751105\pi\)
−0.709556 + 0.704649i \(0.751105\pi\)
\(308\) −22.2784 −1.26943
\(309\) 5.64742 0.321270
\(310\) −12.7584 −0.724628
\(311\) −17.0804 −0.968539 −0.484270 0.874919i \(-0.660915\pi\)
−0.484270 + 0.874919i \(0.660915\pi\)
\(312\) 0 0
\(313\) 15.6974 0.887269 0.443635 0.896208i \(-0.353689\pi\)
0.443635 + 0.896208i \(0.353689\pi\)
\(314\) −1.48427 −0.0837622
\(315\) 4.97823 0.280491
\(316\) −11.7627 −0.661704
\(317\) 32.7821 1.84123 0.920613 0.390477i \(-0.127690\pi\)
0.920613 + 0.390477i \(0.127690\pi\)
\(318\) −5.56465 −0.312050
\(319\) 20.2868 1.13584
\(320\) −1.83340 −0.102490
\(321\) −6.73556 −0.375942
\(322\) −17.5623 −0.978706
\(323\) −5.99462 −0.333550
\(324\) 1.24698 0.0692766
\(325\) 0 0
\(326\) 11.2959 0.625622
\(327\) −2.07606 −0.114807
\(328\) −2.44504 −0.135005
\(329\) 36.3540 2.00426
\(330\) −13.5036 −0.743351
\(331\) −29.1618 −1.60288 −0.801439 0.598076i \(-0.795932\pi\)
−0.801439 + 0.598076i \(0.795932\pi\)
\(332\) −8.06398 −0.442569
\(333\) 6.24698 0.342332
\(334\) −13.4252 −0.734594
\(335\) −6.56465 −0.358665
\(336\) −17.0151 −0.928248
\(337\) −33.2911 −1.81348 −0.906741 0.421688i \(-0.861438\pi\)
−0.906741 + 0.421688i \(0.861438\pi\)
\(338\) 0 0
\(339\) 6.16852 0.335028
\(340\) −1.35690 −0.0735880
\(341\) 25.4101 1.37604
\(342\) 14.3448 0.775679
\(343\) −7.34375 −0.396525
\(344\) 9.63102 0.519270
\(345\) −4.08815 −0.220098
\(346\) −3.62133 −0.194684
\(347\) −0.873690 −0.0469022 −0.0234511 0.999725i \(-0.507465\pi\)
−0.0234511 + 0.999725i \(0.507465\pi\)
\(348\) −4.87800 −0.261488
\(349\) −3.23191 −0.173000 −0.0865002 0.996252i \(-0.527568\pi\)
−0.0865002 + 0.996252i \(0.527568\pi\)
\(350\) −18.0761 −0.966206
\(351\) 0 0
\(352\) 32.0804 1.70989
\(353\) 8.14675 0.433608 0.216804 0.976215i \(-0.430437\pi\)
0.216804 + 0.976215i \(0.430437\pi\)
\(354\) 3.38404 0.179860
\(355\) −13.1782 −0.699427
\(356\) 1.44504 0.0765871
\(357\) −2.59419 −0.137299
\(358\) 36.1051 1.90822
\(359\) 2.64071 0.139371 0.0696857 0.997569i \(-0.477800\pi\)
0.0696857 + 0.997569i \(0.477800\pi\)
\(360\) −1.96077 −0.103342
\(361\) 44.3739 2.33547
\(362\) 43.4674 2.28460
\(363\) 15.8944 0.834239
\(364\) 0 0
\(365\) 4.26444 0.223211
\(366\) 6.02715 0.315044
\(367\) −2.90408 −0.151592 −0.0757960 0.997123i \(-0.524150\pi\)
−0.0757960 + 0.997123i \(0.524150\pi\)
\(368\) 13.9729 0.728385
\(369\) 1.80194 0.0938051
\(370\) 16.2664 0.845648
\(371\) −10.6388 −0.552339
\(372\) −6.10992 −0.316784
\(373\) −8.39852 −0.434859 −0.217429 0.976076i \(-0.569767\pi\)
−0.217429 + 0.976076i \(0.569767\pi\)
\(374\) 7.03684 0.363866
\(375\) −11.4330 −0.590396
\(376\) −14.3187 −0.738432
\(377\) 0 0
\(378\) 6.20775 0.319292
\(379\) 15.7482 0.808932 0.404466 0.914553i \(-0.367457\pi\)
0.404466 + 0.914553i \(0.367457\pi\)
\(380\) 14.3448 0.735873
\(381\) −14.2620 −0.730667
\(382\) −12.7584 −0.652776
\(383\) 12.7385 0.650909 0.325455 0.945558i \(-0.394483\pi\)
0.325455 + 0.945558i \(0.394483\pi\)
\(384\) 10.0858 0.514686
\(385\) −25.8170 −1.31576
\(386\) 17.6015 0.895892
\(387\) −7.09783 −0.360803
\(388\) 10.7966 0.548112
\(389\) −0.310371 −0.0157365 −0.00786823 0.999969i \(-0.502505\pi\)
−0.00786823 + 0.999969i \(0.502505\pi\)
\(390\) 0 0
\(391\) 2.13036 0.107737
\(392\) −6.60579 −0.333643
\(393\) 22.6015 1.14009
\(394\) 42.1855 2.12528
\(395\) −13.6310 −0.685851
\(396\) −6.46681 −0.324970
\(397\) 1.49098 0.0748299 0.0374150 0.999300i \(-0.488088\pi\)
0.0374150 + 0.999300i \(0.488088\pi\)
\(398\) 7.25236 0.363528
\(399\) 27.4252 1.37298
\(400\) 14.3817 0.719083
\(401\) −23.8334 −1.19018 −0.595092 0.803658i \(-0.702884\pi\)
−0.595092 + 0.803658i \(0.702884\pi\)
\(402\) −8.18598 −0.408280
\(403\) 0 0
\(404\) −10.5700 −0.525878
\(405\) 1.44504 0.0718047
\(406\) −24.2838 −1.20519
\(407\) −32.3967 −1.60585
\(408\) 1.02177 0.0505852
\(409\) 4.26742 0.211010 0.105505 0.994419i \(-0.466354\pi\)
0.105505 + 0.994419i \(0.466354\pi\)
\(410\) 4.69202 0.231722
\(411\) −13.6353 −0.672581
\(412\) 7.04221 0.346945
\(413\) 6.46980 0.318358
\(414\) −5.09783 −0.250545
\(415\) −9.34481 −0.458719
\(416\) 0 0
\(417\) 17.6015 0.861948
\(418\) −74.3919 −3.63863
\(419\) 29.6896 1.45043 0.725217 0.688521i \(-0.241740\pi\)
0.725217 + 0.688521i \(0.241740\pi\)
\(420\) 6.20775 0.302907
\(421\) 29.3991 1.43282 0.716412 0.697677i \(-0.245783\pi\)
0.716412 + 0.697677i \(0.245783\pi\)
\(422\) −7.04892 −0.343136
\(423\) 10.5526 0.513083
\(424\) 4.19029 0.203499
\(425\) 2.19269 0.106361
\(426\) −16.4330 −0.796180
\(427\) 11.5230 0.557638
\(428\) −8.39911 −0.405986
\(429\) 0 0
\(430\) −18.4819 −0.891275
\(431\) −33.0562 −1.59226 −0.796131 0.605124i \(-0.793123\pi\)
−0.796131 + 0.605124i \(0.793123\pi\)
\(432\) −4.93900 −0.237628
\(433\) −29.2664 −1.40645 −0.703226 0.710967i \(-0.748258\pi\)
−0.703226 + 0.710967i \(0.748258\pi\)
\(434\) −30.4166 −1.46004
\(435\) −5.65279 −0.271031
\(436\) −2.58881 −0.123982
\(437\) −22.5217 −1.07736
\(438\) 5.31767 0.254088
\(439\) −2.13169 −0.101740 −0.0508699 0.998705i \(-0.516199\pi\)
−0.0508699 + 0.998705i \(0.516199\pi\)
\(440\) 10.1685 0.484765
\(441\) 4.86831 0.231824
\(442\) 0 0
\(443\) −22.9922 −1.09239 −0.546197 0.837657i \(-0.683925\pi\)
−0.546197 + 0.837657i \(0.683925\pi\)
\(444\) 7.78986 0.369690
\(445\) 1.67456 0.0793819
\(446\) 13.4233 0.635610
\(447\) 12.7385 0.602513
\(448\) −4.37090 −0.206505
\(449\) −12.9379 −0.610579 −0.305289 0.952260i \(-0.598753\pi\)
−0.305289 + 0.952260i \(0.598753\pi\)
\(450\) −5.24698 −0.247345
\(451\) −9.34481 −0.440030
\(452\) 7.69202 0.361802
\(453\) 15.6407 0.734865
\(454\) −38.2911 −1.79709
\(455\) 0 0
\(456\) −10.8019 −0.505847
\(457\) −4.85325 −0.227025 −0.113513 0.993537i \(-0.536210\pi\)
−0.113513 + 0.993537i \(0.536210\pi\)
\(458\) 16.7506 0.782705
\(459\) −0.753020 −0.0351480
\(460\) −5.09783 −0.237688
\(461\) 18.8345 0.877208 0.438604 0.898680i \(-0.355473\pi\)
0.438604 + 0.898680i \(0.355473\pi\)
\(462\) −32.1933 −1.49777
\(463\) −22.8767 −1.06317 −0.531585 0.847005i \(-0.678403\pi\)
−0.531585 + 0.847005i \(0.678403\pi\)
\(464\) 19.3207 0.896939
\(465\) −7.08038 −0.328345
\(466\) 29.2107 1.35316
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) −15.6504 −0.722668
\(470\) 27.4776 1.26745
\(471\) −0.823708 −0.0379545
\(472\) −2.54825 −0.117293
\(473\) 36.8092 1.69249
\(474\) −16.9976 −0.780726
\(475\) −23.1806 −1.06360
\(476\) −3.23490 −0.148271
\(477\) −3.08815 −0.141396
\(478\) −24.3424 −1.11340
\(479\) −38.0901 −1.74038 −0.870190 0.492717i \(-0.836004\pi\)
−0.870190 + 0.492717i \(0.836004\pi\)
\(480\) −8.93900 −0.408008
\(481\) 0 0
\(482\) 11.2959 0.514514
\(483\) −9.74632 −0.443473
\(484\) 19.8200 0.900909
\(485\) 12.5114 0.568114
\(486\) 1.80194 0.0817376
\(487\) −21.2500 −0.962928 −0.481464 0.876466i \(-0.659895\pi\)
−0.481464 + 0.876466i \(0.659895\pi\)
\(488\) −4.53856 −0.205451
\(489\) 6.26875 0.283483
\(490\) 12.6765 0.572665
\(491\) −6.35019 −0.286580 −0.143290 0.989681i \(-0.545768\pi\)
−0.143290 + 0.989681i \(0.545768\pi\)
\(492\) 2.24698 0.101302
\(493\) 2.94571 0.132668
\(494\) 0 0
\(495\) −7.49396 −0.336828
\(496\) 24.2000 1.08661
\(497\) −31.4174 −1.40926
\(498\) −11.6528 −0.522174
\(499\) −4.65087 −0.208202 −0.104101 0.994567i \(-0.533196\pi\)
−0.104101 + 0.994567i \(0.533196\pi\)
\(500\) −14.2567 −0.637578
\(501\) −7.45042 −0.332860
\(502\) −1.35690 −0.0605612
\(503\) 15.4752 0.690004 0.345002 0.938602i \(-0.387878\pi\)
0.345002 + 0.938602i \(0.387878\pi\)
\(504\) −4.67456 −0.208222
\(505\) −12.2489 −0.545069
\(506\) 26.4373 1.17528
\(507\) 0 0
\(508\) −17.7845 −0.789059
\(509\) −20.5047 −0.908855 −0.454428 0.890784i \(-0.650156\pi\)
−0.454428 + 0.890784i \(0.650156\pi\)
\(510\) −1.96077 −0.0868244
\(511\) 10.1666 0.449744
\(512\) 17.1491 0.757892
\(513\) 7.96077 0.351477
\(514\) 35.5459 1.56786
\(515\) 8.16075 0.359606
\(516\) −8.85086 −0.389637
\(517\) −54.7254 −2.40682
\(518\) 38.7797 1.70388
\(519\) −2.00969 −0.0882155
\(520\) 0 0
\(521\) −42.0267 −1.84122 −0.920611 0.390481i \(-0.872309\pi\)
−0.920611 + 0.390481i \(0.872309\pi\)
\(522\) −7.04892 −0.308523
\(523\) −29.9885 −1.31131 −0.655653 0.755062i \(-0.727607\pi\)
−0.655653 + 0.755062i \(0.727607\pi\)
\(524\) 28.1836 1.23121
\(525\) −10.0315 −0.437809
\(526\) −31.7308 −1.38353
\(527\) 3.68963 0.160723
\(528\) 25.6136 1.11469
\(529\) −14.9963 −0.652012
\(530\) −8.04115 −0.349285
\(531\) 1.87800 0.0814984
\(532\) 34.1987 1.48270
\(533\) 0 0
\(534\) 2.08815 0.0903629
\(535\) −9.73317 −0.420802
\(536\) 6.16421 0.266253
\(537\) 20.0368 0.864653
\(538\) −29.5284 −1.27306
\(539\) −25.2470 −1.08746
\(540\) 1.80194 0.0775431
\(541\) 36.3803 1.56411 0.782056 0.623208i \(-0.214171\pi\)
0.782056 + 0.623208i \(0.214171\pi\)
\(542\) −1.43296 −0.0615509
\(543\) 24.1226 1.03520
\(544\) 4.65817 0.199717
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) −25.8159 −1.10381 −0.551905 0.833907i \(-0.686099\pi\)
−0.551905 + 0.833907i \(0.686099\pi\)
\(548\) −17.0030 −0.726331
\(549\) 3.34481 0.142753
\(550\) 27.2107 1.16027
\(551\) −31.1414 −1.32667
\(552\) 3.83877 0.163389
\(553\) −32.4969 −1.38191
\(554\) −8.70948 −0.370030
\(555\) 9.02715 0.383181
\(556\) 21.9487 0.930832
\(557\) 17.9903 0.762274 0.381137 0.924519i \(-0.375533\pi\)
0.381137 + 0.924519i \(0.375533\pi\)
\(558\) −8.82908 −0.373765
\(559\) 0 0
\(560\) −24.5875 −1.03901
\(561\) 3.90515 0.164876
\(562\) 33.8310 1.42707
\(563\) −39.1323 −1.64923 −0.824614 0.565695i \(-0.808608\pi\)
−0.824614 + 0.565695i \(0.808608\pi\)
\(564\) 13.1588 0.554087
\(565\) 8.91377 0.375005
\(566\) −14.2664 −0.599660
\(567\) 3.44504 0.144678
\(568\) 12.3744 0.519216
\(569\) 30.6002 1.28283 0.641413 0.767196i \(-0.278349\pi\)
0.641413 + 0.767196i \(0.278349\pi\)
\(570\) 20.7289 0.868236
\(571\) −2.96184 −0.123949 −0.0619745 0.998078i \(-0.519740\pi\)
−0.0619745 + 0.998078i \(0.519740\pi\)
\(572\) 0 0
\(573\) −7.08038 −0.295787
\(574\) 11.1860 0.466894
\(575\) 8.23788 0.343543
\(576\) −1.26875 −0.0528646
\(577\) −0.819396 −0.0341119 −0.0170560 0.999855i \(-0.505429\pi\)
−0.0170560 + 0.999855i \(0.505429\pi\)
\(578\) −29.6112 −1.23166
\(579\) 9.76809 0.405948
\(580\) −7.04892 −0.292690
\(581\) −22.2784 −0.924265
\(582\) 15.6015 0.646702
\(583\) 16.0151 0.663276
\(584\) −4.00431 −0.165700
\(585\) 0 0
\(586\) 11.8552 0.489732
\(587\) −31.7995 −1.31251 −0.656254 0.754540i \(-0.727860\pi\)
−0.656254 + 0.754540i \(0.727860\pi\)
\(588\) 6.07069 0.250351
\(589\) −39.0060 −1.60721
\(590\) 4.89008 0.201322
\(591\) 23.4112 0.963008
\(592\) −30.8538 −1.26808
\(593\) −4.26337 −0.175076 −0.0875379 0.996161i \(-0.527900\pi\)
−0.0875379 + 0.996161i \(0.527900\pi\)
\(594\) −9.34481 −0.383422
\(595\) −3.74871 −0.153682
\(596\) 15.8847 0.650663
\(597\) 4.02475 0.164722
\(598\) 0 0
\(599\) 24.7278 1.01035 0.505175 0.863017i \(-0.331428\pi\)
0.505175 + 0.863017i \(0.331428\pi\)
\(600\) 3.95108 0.161302
\(601\) 6.82371 0.278345 0.139172 0.990268i \(-0.455556\pi\)
0.139172 + 0.990268i \(0.455556\pi\)
\(602\) −44.0616 −1.79582
\(603\) −4.54288 −0.185000
\(604\) 19.5036 0.793592
\(605\) 22.9681 0.933785
\(606\) −15.2741 −0.620469
\(607\) 31.9963 1.29869 0.649344 0.760494i \(-0.275043\pi\)
0.649344 + 0.760494i \(0.275043\pi\)
\(608\) −49.2452 −1.99716
\(609\) −13.4765 −0.546095
\(610\) 8.70948 0.352637
\(611\) 0 0
\(612\) −0.939001 −0.0379569
\(613\) 33.5875 1.35659 0.678293 0.734792i \(-0.262720\pi\)
0.678293 + 0.734792i \(0.262720\pi\)
\(614\) −44.8049 −1.80818
\(615\) 2.60388 0.104998
\(616\) 24.2422 0.976746
\(617\) −26.5870 −1.07035 −0.535176 0.844740i \(-0.679755\pi\)
−0.535176 + 0.844740i \(0.679755\pi\)
\(618\) 10.1763 0.409350
\(619\) −9.17928 −0.368946 −0.184473 0.982838i \(-0.559058\pi\)
−0.184473 + 0.982838i \(0.559058\pi\)
\(620\) −8.82908 −0.354585
\(621\) −2.82908 −0.113527
\(622\) −30.7778 −1.23408
\(623\) 3.99223 0.159945
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) 28.2857 1.13053
\(627\) −41.2844 −1.64874
\(628\) −1.02715 −0.0409876
\(629\) −4.70410 −0.187565
\(630\) 8.97046 0.357392
\(631\) 17.0043 0.676931 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(632\) 12.7995 0.509139
\(633\) −3.91185 −0.155482
\(634\) 59.0713 2.34602
\(635\) −20.6093 −0.817853
\(636\) −3.85086 −0.152696
\(637\) 0 0
\(638\) 36.5555 1.44725
\(639\) −9.11960 −0.360766
\(640\) 14.5743 0.576101
\(641\) −21.6649 −0.855711 −0.427856 0.903847i \(-0.640731\pi\)
−0.427856 + 0.903847i \(0.640731\pi\)
\(642\) −12.1371 −0.479012
\(643\) 9.35557 0.368948 0.184474 0.982837i \(-0.440942\pi\)
0.184474 + 0.982837i \(0.440942\pi\)
\(644\) −12.1535 −0.478913
\(645\) −10.2567 −0.403856
\(646\) −10.8019 −0.424997
\(647\) −0.702775 −0.0276289 −0.0138145 0.999905i \(-0.504397\pi\)
−0.0138145 + 0.999905i \(0.504397\pi\)
\(648\) −1.35690 −0.0533039
\(649\) −9.73928 −0.382300
\(650\) 0 0
\(651\) −16.8799 −0.661576
\(652\) 7.81700 0.306137
\(653\) −37.3411 −1.46127 −0.730635 0.682768i \(-0.760776\pi\)
−0.730635 + 0.682768i \(0.760776\pi\)
\(654\) −3.74094 −0.146282
\(655\) 32.6601 1.27614
\(656\) −8.89977 −0.347478
\(657\) 2.95108 0.115133
\(658\) 65.5077 2.55376
\(659\) −0.735562 −0.0286534 −0.0143267 0.999897i \(-0.504560\pi\)
−0.0143267 + 0.999897i \(0.504560\pi\)
\(660\) −9.34481 −0.363746
\(661\) 13.8485 0.538643 0.269321 0.963050i \(-0.413201\pi\)
0.269321 + 0.963050i \(0.413201\pi\)
\(662\) −52.5478 −2.04233
\(663\) 0 0
\(664\) 8.77479 0.340528
\(665\) 39.6305 1.53681
\(666\) 11.2567 0.436187
\(667\) 11.0670 0.428515
\(668\) −9.29052 −0.359461
\(669\) 7.44935 0.288009
\(670\) −11.8291 −0.456997
\(671\) −17.3461 −0.669640
\(672\) −21.3110 −0.822088
\(673\) −6.35019 −0.244782 −0.122391 0.992482i \(-0.539056\pi\)
−0.122391 + 0.992482i \(0.539056\pi\)
\(674\) −59.9885 −2.31067
\(675\) −2.91185 −0.112077
\(676\) 0 0
\(677\) 33.7241 1.29612 0.648061 0.761589i \(-0.275580\pi\)
0.648061 + 0.761589i \(0.275580\pi\)
\(678\) 11.1153 0.426880
\(679\) 29.8278 1.14468
\(680\) 1.47650 0.0566212
\(681\) −21.2500 −0.814300
\(682\) 45.7875 1.75329
\(683\) 19.2687 0.737298 0.368649 0.929569i \(-0.379820\pi\)
0.368649 + 0.929569i \(0.379820\pi\)
\(684\) 9.92692 0.379565
\(685\) −19.7036 −0.752837
\(686\) −13.2330 −0.505237
\(687\) 9.29590 0.354661
\(688\) 35.0562 1.33651
\(689\) 0 0
\(690\) −7.36658 −0.280441
\(691\) −39.4010 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(692\) −2.50604 −0.0952654
\(693\) −17.8659 −0.678670
\(694\) −1.57434 −0.0597610
\(695\) 25.4349 0.964800
\(696\) 5.30798 0.201198
\(697\) −1.35690 −0.0513961
\(698\) −5.82371 −0.220431
\(699\) 16.2107 0.613146
\(700\) −12.5090 −0.472797
\(701\) 18.3985 0.694902 0.347451 0.937698i \(-0.387047\pi\)
0.347451 + 0.937698i \(0.387047\pi\)
\(702\) 0 0
\(703\) 49.7308 1.87563
\(704\) 6.57971 0.247982
\(705\) 15.2489 0.574307
\(706\) 14.6799 0.552487
\(707\) −29.2019 −1.09825
\(708\) 2.34183 0.0880114
\(709\) −38.4553 −1.44422 −0.722110 0.691778i \(-0.756828\pi\)
−0.722110 + 0.691778i \(0.756828\pi\)
\(710\) −23.7463 −0.891183
\(711\) −9.43296 −0.353764
\(712\) −1.57242 −0.0589288
\(713\) 13.8619 0.519131
\(714\) −4.67456 −0.174941
\(715\) 0 0
\(716\) 24.9855 0.933753
\(717\) −13.5090 −0.504504
\(718\) 4.75840 0.177582
\(719\) 48.4999 1.80874 0.904371 0.426747i \(-0.140341\pi\)
0.904371 + 0.426747i \(0.140341\pi\)
\(720\) −7.13706 −0.265983
\(721\) 19.4556 0.724564
\(722\) 79.9590 2.97576
\(723\) 6.26875 0.233137
\(724\) 30.0804 1.11793
\(725\) 11.3907 0.423042
\(726\) 28.6407 1.06296
\(727\) 19.0344 0.705948 0.352974 0.935633i \(-0.385170\pi\)
0.352974 + 0.935633i \(0.385170\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 7.68425 0.284407
\(731\) 5.34481 0.197685
\(732\) 4.17092 0.154161
\(733\) 24.6213 0.909410 0.454705 0.890642i \(-0.349745\pi\)
0.454705 + 0.890642i \(0.349745\pi\)
\(734\) −5.23298 −0.193153
\(735\) 7.03492 0.259487
\(736\) 17.5007 0.645083
\(737\) 23.5593 0.867817
\(738\) 3.24698 0.119523
\(739\) −44.5115 −1.63738 −0.818692 0.574233i \(-0.805300\pi\)
−0.818692 + 0.574233i \(0.805300\pi\)
\(740\) 11.2567 0.413803
\(741\) 0 0
\(742\) −19.1704 −0.703769
\(743\) 10.4112 0.381950 0.190975 0.981595i \(-0.438835\pi\)
0.190975 + 0.981595i \(0.438835\pi\)
\(744\) 6.64848 0.243745
\(745\) 18.4077 0.674407
\(746\) −15.1336 −0.554081
\(747\) −6.46681 −0.236608
\(748\) 4.86964 0.178052
\(749\) −23.2043 −0.847866
\(750\) −20.6015 −0.752260
\(751\) 1.69979 0.0620263 0.0310131 0.999519i \(-0.490127\pi\)
0.0310131 + 0.999519i \(0.490127\pi\)
\(752\) −52.1191 −1.90059
\(753\) −0.753020 −0.0274416
\(754\) 0 0
\(755\) 22.6015 0.822552
\(756\) 4.29590 0.156240
\(757\) 27.4252 0.996786 0.498393 0.866951i \(-0.333924\pi\)
0.498393 + 0.866951i \(0.333924\pi\)
\(758\) 28.3773 1.03071
\(759\) 14.6716 0.532545
\(760\) −15.6093 −0.566207
\(761\) −5.02608 −0.182195 −0.0910977 0.995842i \(-0.529038\pi\)
−0.0910977 + 0.995842i \(0.529038\pi\)
\(762\) −25.6993 −0.930988
\(763\) −7.15213 −0.258924
\(764\) −8.82908 −0.319425
\(765\) −1.08815 −0.0393420
\(766\) 22.9541 0.829364
\(767\) 0 0
\(768\) 20.7114 0.747358
\(769\) 42.4456 1.53063 0.765314 0.643657i \(-0.222584\pi\)
0.765314 + 0.643657i \(0.222584\pi\)
\(770\) −46.5206 −1.67649
\(771\) 19.7265 0.710431
\(772\) 12.1806 0.438390
\(773\) 26.3593 0.948078 0.474039 0.880504i \(-0.342796\pi\)
0.474039 + 0.880504i \(0.342796\pi\)
\(774\) −12.7899 −0.459722
\(775\) 14.2674 0.512501
\(776\) −11.7482 −0.421737
\(777\) 21.5211 0.772065
\(778\) −0.559270 −0.0200508
\(779\) 14.3448 0.513956
\(780\) 0 0
\(781\) 47.2941 1.69232
\(782\) 3.83877 0.137274
\(783\) −3.91185 −0.139798
\(784\) −24.0446 −0.858736
\(785\) −1.19029 −0.0424834
\(786\) 40.7265 1.45266
\(787\) 17.1424 0.611062 0.305531 0.952182i \(-0.401166\pi\)
0.305531 + 0.952182i \(0.401166\pi\)
\(788\) 29.1933 1.03997
\(789\) −17.6093 −0.626906
\(790\) −24.5623 −0.873886
\(791\) 21.2508 0.755592
\(792\) 7.03684 0.250043
\(793\) 0 0
\(794\) 2.68664 0.0953455
\(795\) −4.46250 −0.158269
\(796\) 5.01879 0.177886
\(797\) 30.1629 1.06842 0.534212 0.845351i \(-0.320608\pi\)
0.534212 + 0.845351i \(0.320608\pi\)
\(798\) 49.4185 1.74940
\(799\) −7.94630 −0.281120
\(800\) 18.0127 0.636844
\(801\) 1.15883 0.0409454
\(802\) −42.9463 −1.51649
\(803\) −15.3043 −0.540076
\(804\) −5.66487 −0.199785
\(805\) −14.0838 −0.496390
\(806\) 0 0
\(807\) −16.3870 −0.576851
\(808\) 11.5017 0.404629
\(809\) −29.8504 −1.04948 −0.524742 0.851261i \(-0.675838\pi\)
−0.524742 + 0.851261i \(0.675838\pi\)
\(810\) 2.60388 0.0914909
\(811\) 47.7362 1.67624 0.838122 0.545484i \(-0.183654\pi\)
0.838122 + 0.545484i \(0.183654\pi\)
\(812\) −16.8049 −0.589737
\(813\) −0.795233 −0.0278900
\(814\) −58.3769 −2.04611
\(815\) 9.05861 0.317309
\(816\) 3.71917 0.130197
\(817\) −56.5042 −1.97683
\(818\) 7.68963 0.268862
\(819\) 0 0
\(820\) 3.24698 0.113389
\(821\) −17.9299 −0.625758 −0.312879 0.949793i \(-0.601293\pi\)
−0.312879 + 0.949793i \(0.601293\pi\)
\(822\) −24.5700 −0.856978
\(823\) 54.3196 1.89346 0.946731 0.322026i \(-0.104364\pi\)
0.946731 + 0.322026i \(0.104364\pi\)
\(824\) −7.66296 −0.266952
\(825\) 15.1008 0.525743
\(826\) 11.6582 0.405640
\(827\) 49.1041 1.70752 0.853758 0.520670i \(-0.174318\pi\)
0.853758 + 0.520670i \(0.174318\pi\)
\(828\) −3.52781 −0.122600
\(829\) −7.35796 −0.255553 −0.127776 0.991803i \(-0.540784\pi\)
−0.127776 + 0.991803i \(0.540784\pi\)
\(830\) −16.8388 −0.584482
\(831\) −4.83340 −0.167669
\(832\) 0 0
\(833\) −3.66594 −0.127017
\(834\) 31.7168 1.09826
\(835\) −10.7662 −0.372579
\(836\) −51.4808 −1.78050
\(837\) −4.89977 −0.169361
\(838\) 53.4989 1.84809
\(839\) −36.5013 −1.26016 −0.630082 0.776529i \(-0.716979\pi\)
−0.630082 + 0.776529i \(0.716979\pi\)
\(840\) −6.75494 −0.233068
\(841\) −13.6974 −0.472324
\(842\) 52.9754 1.82565
\(843\) 18.7748 0.646638
\(844\) −4.87800 −0.167908
\(845\) 0 0
\(846\) 19.0151 0.653751
\(847\) 54.7569 1.88147
\(848\) 15.2524 0.523768
\(849\) −7.91723 −0.271719
\(850\) 3.95108 0.135521
\(851\) −17.6732 −0.605831
\(852\) −11.3720 −0.389597
\(853\) −9.73855 −0.333441 −0.166721 0.986004i \(-0.553318\pi\)
−0.166721 + 0.986004i \(0.553318\pi\)
\(854\) 20.7638 0.710522
\(855\) 11.5036 0.393416
\(856\) 9.13946 0.312380
\(857\) −15.2030 −0.519323 −0.259662 0.965700i \(-0.583611\pi\)
−0.259662 + 0.965700i \(0.583611\pi\)
\(858\) 0 0
\(859\) −31.9885 −1.09143 −0.545717 0.837970i \(-0.683743\pi\)
−0.545717 + 0.837970i \(0.683743\pi\)
\(860\) −12.7899 −0.436130
\(861\) 6.20775 0.211560
\(862\) −59.5652 −2.02880
\(863\) −35.2905 −1.20130 −0.600652 0.799511i \(-0.705092\pi\)
−0.600652 + 0.799511i \(0.705092\pi\)
\(864\) −6.18598 −0.210451
\(865\) −2.90408 −0.0987418
\(866\) −52.7362 −1.79205
\(867\) −16.4330 −0.558093
\(868\) −21.0489 −0.714447
\(869\) 48.9191 1.65947
\(870\) −10.1860 −0.345337
\(871\) 0 0
\(872\) 2.81700 0.0953958
\(873\) 8.65817 0.293035
\(874\) −40.5827 −1.37273
\(875\) −39.3870 −1.33152
\(876\) 3.67994 0.124334
\(877\) −15.3263 −0.517532 −0.258766 0.965940i \(-0.583316\pi\)
−0.258766 + 0.965940i \(0.583316\pi\)
\(878\) −3.84117 −0.129633
\(879\) 6.57912 0.221908
\(880\) 37.0127 1.24770
\(881\) −36.4306 −1.22738 −0.613689 0.789548i \(-0.710315\pi\)
−0.613689 + 0.789548i \(0.710315\pi\)
\(882\) 8.77240 0.295382
\(883\) −37.6819 −1.26810 −0.634048 0.773294i \(-0.718608\pi\)
−0.634048 + 0.773294i \(0.718608\pi\)
\(884\) 0 0
\(885\) 2.71379 0.0912231
\(886\) −41.4306 −1.39189
\(887\) 4.24890 0.142664 0.0713320 0.997453i \(-0.477275\pi\)
0.0713320 + 0.997453i \(0.477275\pi\)
\(888\) −8.47650 −0.284453
\(889\) −49.1333 −1.64788
\(890\) 3.01746 0.101145
\(891\) −5.18598 −0.173737
\(892\) 9.28919 0.311025
\(893\) 84.0066 2.81117
\(894\) 22.9541 0.767699
\(895\) 28.9541 0.967828
\(896\) 34.7458 1.16078
\(897\) 0 0
\(898\) −23.3134 −0.777977
\(899\) 19.1672 0.639262
\(900\) −3.63102 −0.121034
\(901\) 2.32544 0.0774715
\(902\) −16.8388 −0.560670
\(903\) −24.4523 −0.813723
\(904\) −8.37004 −0.278383
\(905\) 34.8582 1.15872
\(906\) 28.1836 0.936337
\(907\) −12.6183 −0.418985 −0.209493 0.977810i \(-0.567181\pi\)
−0.209493 + 0.977810i \(0.567181\pi\)
\(908\) −26.4983 −0.879376
\(909\) −8.47650 −0.281148
\(910\) 0 0
\(911\) 6.77777 0.224558 0.112279 0.993677i \(-0.464185\pi\)
0.112279 + 0.993677i \(0.464185\pi\)
\(912\) −39.3183 −1.30196
\(913\) 33.5368 1.10990
\(914\) −8.74525 −0.289267
\(915\) 4.83340 0.159787
\(916\) 11.5918 0.383004
\(917\) 77.8631 2.57126
\(918\) −1.35690 −0.0447842
\(919\) −20.4674 −0.675157 −0.337579 0.941297i \(-0.609608\pi\)
−0.337579 + 0.941297i \(0.609608\pi\)
\(920\) 5.54719 0.182885
\(921\) −24.8649 −0.819325
\(922\) 33.9385 1.11771
\(923\) 0 0
\(924\) −22.2784 −0.732907
\(925\) −18.1903 −0.598093
\(926\) −41.2223 −1.35465
\(927\) 5.64742 0.185485
\(928\) 24.1987 0.794360
\(929\) −4.65220 −0.152634 −0.0763169 0.997084i \(-0.524316\pi\)
−0.0763169 + 0.997084i \(0.524316\pi\)
\(930\) −12.7584 −0.418364
\(931\) 38.7555 1.27016
\(932\) 20.2145 0.662147
\(933\) −17.0804 −0.559186
\(934\) 23.4252 0.766496
\(935\) 5.64310 0.184549
\(936\) 0 0
\(937\) −41.8544 −1.36732 −0.683662 0.729798i \(-0.739614\pi\)
−0.683662 + 0.729798i \(0.739614\pi\)
\(938\) −28.2010 −0.920797
\(939\) 15.6974 0.512265
\(940\) 19.0151 0.620203
\(941\) 30.3454 0.989232 0.494616 0.869112i \(-0.335309\pi\)
0.494616 + 0.869112i \(0.335309\pi\)
\(942\) −1.48427 −0.0483601
\(943\) −5.09783 −0.166008
\(944\) −9.27545 −0.301890
\(945\) 4.97823 0.161942
\(946\) 66.3279 2.15651
\(947\) 12.0325 0.391004 0.195502 0.980703i \(-0.437366\pi\)
0.195502 + 0.980703i \(0.437366\pi\)
\(948\) −11.7627 −0.382035
\(949\) 0 0
\(950\) −41.7700 −1.35520
\(951\) 32.7821 1.06303
\(952\) 3.52004 0.114085
\(953\) 22.9825 0.744478 0.372239 0.928137i \(-0.378590\pi\)
0.372239 + 0.928137i \(0.378590\pi\)
\(954\) −5.56465 −0.180162
\(955\) −10.2314 −0.331082
\(956\) −16.8455 −0.544822
\(957\) 20.2868 0.655779
\(958\) −68.6359 −2.21753
\(959\) −46.9743 −1.51688
\(960\) −1.83340 −0.0591726
\(961\) −6.99223 −0.225556
\(962\) 0 0
\(963\) −6.73556 −0.217050
\(964\) 7.81700 0.251769
\(965\) 14.1153 0.454387
\(966\) −17.5623 −0.565056
\(967\) −38.8883 −1.25056 −0.625281 0.780399i \(-0.715016\pi\)
−0.625281 + 0.780399i \(0.715016\pi\)
\(968\) −21.5670 −0.693191
\(969\) −5.99462 −0.192575
\(970\) 22.5448 0.723870
\(971\) −57.5133 −1.84569 −0.922845 0.385171i \(-0.874143\pi\)
−0.922845 + 0.385171i \(0.874143\pi\)
\(972\) 1.24698 0.0399969
\(973\) 60.6378 1.94396
\(974\) −38.2911 −1.22693
\(975\) 0 0
\(976\) −16.5200 −0.528794
\(977\) 16.3690 0.523690 0.261845 0.965110i \(-0.415669\pi\)
0.261845 + 0.965110i \(0.415669\pi\)
\(978\) 11.2959 0.361203
\(979\) −6.00969 −0.192070
\(980\) 8.77240 0.280224
\(981\) −2.07606 −0.0662836
\(982\) −11.4426 −0.365150
\(983\) −15.6963 −0.500635 −0.250318 0.968164i \(-0.580535\pi\)
−0.250318 + 0.968164i \(0.580535\pi\)
\(984\) −2.44504 −0.0779451
\(985\) 33.8301 1.07792
\(986\) 5.30798 0.169040
\(987\) 36.3540 1.15716
\(988\) 0 0
\(989\) 20.0804 0.638519
\(990\) −13.5036 −0.429174
\(991\) −11.2644 −0.357827 −0.178913 0.983865i \(-0.557258\pi\)
−0.178913 + 0.983865i \(0.557258\pi\)
\(992\) 30.3099 0.962340
\(993\) −29.1618 −0.925422
\(994\) −56.6122 −1.79563
\(995\) 5.81594 0.184378
\(996\) −8.06398 −0.255517
\(997\) −7.70112 −0.243897 −0.121948 0.992536i \(-0.538914\pi\)
−0.121948 + 0.992536i \(0.538914\pi\)
\(998\) −8.38059 −0.265283
\(999\) 6.24698 0.197646
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.k.1.3 yes 3
3.2 odd 2 1521.2.a.p.1.1 3
4.3 odd 2 8112.2.a.cf.1.2 3
13.2 odd 12 507.2.j.h.316.6 12
13.3 even 3 507.2.e.j.22.1 6
13.4 even 6 507.2.e.k.484.3 6
13.5 odd 4 507.2.b.g.337.1 6
13.6 odd 12 507.2.j.h.361.1 12
13.7 odd 12 507.2.j.h.361.6 12
13.8 odd 4 507.2.b.g.337.6 6
13.9 even 3 507.2.e.j.484.1 6
13.10 even 6 507.2.e.k.22.3 6
13.11 odd 12 507.2.j.h.316.1 12
13.12 even 2 507.2.a.j.1.1 3
39.5 even 4 1521.2.b.m.1351.6 6
39.8 even 4 1521.2.b.m.1351.1 6
39.38 odd 2 1521.2.a.q.1.3 3
52.51 odd 2 8112.2.a.by.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.1 3 13.12 even 2
507.2.a.k.1.3 yes 3 1.1 even 1 trivial
507.2.b.g.337.1 6 13.5 odd 4
507.2.b.g.337.6 6 13.8 odd 4
507.2.e.j.22.1 6 13.3 even 3
507.2.e.j.484.1 6 13.9 even 3
507.2.e.k.22.3 6 13.10 even 6
507.2.e.k.484.3 6 13.4 even 6
507.2.j.h.316.1 12 13.11 odd 12
507.2.j.h.316.6 12 13.2 odd 12
507.2.j.h.361.1 12 13.6 odd 12
507.2.j.h.361.6 12 13.7 odd 12
1521.2.a.p.1.1 3 3.2 odd 2
1521.2.a.q.1.3 3 39.38 odd 2
1521.2.b.m.1351.1 6 39.8 even 4
1521.2.b.m.1351.6 6 39.5 even 4
8112.2.a.by.1.2 3 52.51 odd 2
8112.2.a.cf.1.2 3 4.3 odd 2