Properties

Label 507.2.b.g.337.1
Level $507$
Weight $2$
Character 507.337
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.2.b.g.337.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.80194i q^{2} +1.00000 q^{3} -1.24698 q^{4} -1.44504i q^{5} -1.80194i q^{6} +3.44504i q^{7} -1.35690i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.80194i q^{2} +1.00000 q^{3} -1.24698 q^{4} -1.44504i q^{5} -1.80194i q^{6} +3.44504i q^{7} -1.35690i q^{8} +1.00000 q^{9} -2.60388 q^{10} -5.18598i q^{11} -1.24698 q^{12} +6.20775 q^{14} -1.44504i q^{15} -4.93900 q^{16} +0.753020 q^{17} -1.80194i q^{18} -7.96077i q^{19} +1.80194i q^{20} +3.44504i q^{21} -9.34481 q^{22} +2.82908 q^{23} -1.35690i q^{24} +2.91185 q^{25} +1.00000 q^{27} -4.29590i q^{28} -3.91185 q^{29} -2.60388 q^{30} +4.89977i q^{31} +6.18598i q^{32} -5.18598i q^{33} -1.35690i q^{34} +4.97823 q^{35} -1.24698 q^{36} +6.24698i q^{37} -14.3448 q^{38} -1.96077 q^{40} -1.80194i q^{41} +6.20775 q^{42} +7.09783 q^{43} +6.46681i q^{44} -1.44504i q^{45} -5.09783i q^{46} +10.5526i q^{47} -4.93900 q^{48} -4.86831 q^{49} -5.24698i q^{50} +0.753020 q^{51} -3.08815 q^{53} -1.80194i q^{54} -7.49396 q^{55} +4.67456 q^{56} -7.96077i q^{57} +7.04892i q^{58} +1.87800i q^{59} +1.80194i q^{60} +3.34481 q^{61} +8.82908 q^{62} +3.44504i q^{63} +1.26875 q^{64} -9.34481 q^{66} +4.54288i q^{67} -0.939001 q^{68} +2.82908 q^{69} -8.97046i q^{70} +9.11960i q^{71} -1.35690i q^{72} +2.95108i q^{73} +11.2567 q^{74} +2.91185 q^{75} +9.92692i q^{76} +17.8659 q^{77} -9.43296 q^{79} +7.13706i q^{80} +1.00000 q^{81} -3.24698 q^{82} +6.46681i q^{83} -4.29590i q^{84} -1.08815i q^{85} -12.7899i q^{86} -3.91185 q^{87} -7.03684 q^{88} +1.15883i q^{89} -2.60388 q^{90} -3.52781 q^{92} +4.89977i q^{93} +19.0151 q^{94} -11.5036 q^{95} +6.18598i q^{96} -8.65817i q^{97} +8.77240i q^{98} -5.18598i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} + 2q^{4} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} + 2q^{4} + 6q^{9} + 2q^{10} + 2q^{12} + 2q^{14} - 10q^{16} + 14q^{17} - 10q^{22} - 4q^{23} + 10q^{25} + 6q^{27} - 16q^{29} + 2q^{30} + 36q^{35} + 2q^{36} - 40q^{38} + 14q^{40} + 2q^{42} + 6q^{43} - 10q^{48} - 34q^{49} + 14q^{51} - 26q^{53} - 26q^{55} - 14q^{56} - 26q^{61} + 32q^{62} - 8q^{64} - 10q^{66} + 14q^{68} - 4q^{69} + 14q^{74} + 10q^{75} + 30q^{77} - 18q^{79} + 6q^{81} - 10q^{82} - 16q^{87} + 14q^{88} + 2q^{90} - 34q^{92} + 64q^{94} - 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.80194i − 1.27416i −0.770797 0.637081i \(-0.780142\pi\)
0.770797 0.637081i \(-0.219858\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.24698 −0.623490
\(5\) − 1.44504i − 0.646242i −0.946358 0.323121i \(-0.895268\pi\)
0.946358 0.323121i \(-0.104732\pi\)
\(6\) − 1.80194i − 0.735638i
\(7\) 3.44504i 1.30210i 0.759033 + 0.651052i \(0.225672\pi\)
−0.759033 + 0.651052i \(0.774328\pi\)
\(8\) − 1.35690i − 0.479735i
\(9\) 1.00000 0.333333
\(10\) −2.60388 −0.823418
\(11\) − 5.18598i − 1.56363i −0.623509 0.781816i \(-0.714294\pi\)
0.623509 0.781816i \(-0.285706\pi\)
\(12\) −1.24698 −0.359972
\(13\) 0 0
\(14\) 6.20775 1.65909
\(15\) − 1.44504i − 0.373108i
\(16\) −4.93900 −1.23475
\(17\) 0.753020 0.182634 0.0913171 0.995822i \(-0.470892\pi\)
0.0913171 + 0.995822i \(0.470892\pi\)
\(18\) − 1.80194i − 0.424721i
\(19\) − 7.96077i − 1.82633i −0.407594 0.913163i \(-0.633632\pi\)
0.407594 0.913163i \(-0.366368\pi\)
\(20\) 1.80194i 0.402926i
\(21\) 3.44504i 0.751770i
\(22\) −9.34481 −1.99232
\(23\) 2.82908 0.589905 0.294952 0.955512i \(-0.404696\pi\)
0.294952 + 0.955512i \(0.404696\pi\)
\(24\) − 1.35690i − 0.276975i
\(25\) 2.91185 0.582371
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 4.29590i − 0.811848i
\(29\) −3.91185 −0.726413 −0.363207 0.931709i \(-0.618318\pi\)
−0.363207 + 0.931709i \(0.618318\pi\)
\(30\) −2.60388 −0.475400
\(31\) 4.89977i 0.880025i 0.897992 + 0.440013i \(0.145026\pi\)
−0.897992 + 0.440013i \(0.854974\pi\)
\(32\) 6.18598i 1.09354i
\(33\) − 5.18598i − 0.902763i
\(34\) − 1.35690i − 0.232706i
\(35\) 4.97823 0.841474
\(36\) −1.24698 −0.207830
\(37\) 6.24698i 1.02700i 0.858090 + 0.513499i \(0.171651\pi\)
−0.858090 + 0.513499i \(0.828349\pi\)
\(38\) −14.3448 −2.32704
\(39\) 0 0
\(40\) −1.96077 −0.310025
\(41\) − 1.80194i − 0.281415i −0.990051 0.140708i \(-0.955062\pi\)
0.990051 0.140708i \(-0.0449378\pi\)
\(42\) 6.20775 0.957877
\(43\) 7.09783 1.08241 0.541205 0.840891i \(-0.317968\pi\)
0.541205 + 0.840891i \(0.317968\pi\)
\(44\) 6.46681i 0.974909i
\(45\) − 1.44504i − 0.215414i
\(46\) − 5.09783i − 0.751635i
\(47\) 10.5526i 1.53925i 0.638496 + 0.769625i \(0.279557\pi\)
−0.638496 + 0.769625i \(0.720443\pi\)
\(48\) −4.93900 −0.712883
\(49\) −4.86831 −0.695473
\(50\) − 5.24698i − 0.742035i
\(51\) 0.753020 0.105444
\(52\) 0 0
\(53\) −3.08815 −0.424189 −0.212095 0.977249i \(-0.568029\pi\)
−0.212095 + 0.977249i \(0.568029\pi\)
\(54\) − 1.80194i − 0.245213i
\(55\) −7.49396 −1.01049
\(56\) 4.67456 0.624665
\(57\) − 7.96077i − 1.05443i
\(58\) 7.04892i 0.925568i
\(59\) 1.87800i 0.244495i 0.992500 + 0.122248i \(0.0390102\pi\)
−0.992500 + 0.122248i \(0.960990\pi\)
\(60\) 1.80194i 0.232629i
\(61\) 3.34481 0.428260 0.214130 0.976805i \(-0.431308\pi\)
0.214130 + 0.976805i \(0.431308\pi\)
\(62\) 8.82908 1.12129
\(63\) 3.44504i 0.434034i
\(64\) 1.26875 0.158594
\(65\) 0 0
\(66\) −9.34481 −1.15027
\(67\) 4.54288i 0.555001i 0.960726 + 0.277500i \(0.0895060\pi\)
−0.960726 + 0.277500i \(0.910494\pi\)
\(68\) −0.939001 −0.113871
\(69\) 2.82908 0.340582
\(70\) − 8.97046i − 1.07218i
\(71\) 9.11960i 1.08230i 0.840927 + 0.541149i \(0.182011\pi\)
−0.840927 + 0.541149i \(0.817989\pi\)
\(72\) − 1.35690i − 0.159912i
\(73\) 2.95108i 0.345398i 0.984975 + 0.172699i \(0.0552488\pi\)
−0.984975 + 0.172699i \(0.944751\pi\)
\(74\) 11.2567 1.30856
\(75\) 2.91185 0.336232
\(76\) 9.92692i 1.13870i
\(77\) 17.8659 2.03601
\(78\) 0 0
\(79\) −9.43296 −1.06129 −0.530645 0.847594i \(-0.678050\pi\)
−0.530645 + 0.847594i \(0.678050\pi\)
\(80\) 7.13706i 0.797948i
\(81\) 1.00000 0.111111
\(82\) −3.24698 −0.358569
\(83\) 6.46681i 0.709825i 0.934900 + 0.354912i \(0.115489\pi\)
−0.934900 + 0.354912i \(0.884511\pi\)
\(84\) − 4.29590i − 0.468721i
\(85\) − 1.08815i − 0.118026i
\(86\) − 12.7899i − 1.37917i
\(87\) −3.91185 −0.419395
\(88\) −7.03684 −0.750129
\(89\) 1.15883i 0.122836i 0.998112 + 0.0614181i \(0.0195623\pi\)
−0.998112 + 0.0614181i \(0.980438\pi\)
\(90\) −2.60388 −0.274473
\(91\) 0 0
\(92\) −3.52781 −0.367800
\(93\) 4.89977i 0.508083i
\(94\) 19.0151 1.96125
\(95\) −11.5036 −1.18025
\(96\) 6.18598i 0.631354i
\(97\) − 8.65817i − 0.879104i −0.898217 0.439552i \(-0.855137\pi\)
0.898217 0.439552i \(-0.144863\pi\)
\(98\) 8.77240i 0.886146i
\(99\) − 5.18598i − 0.521211i
\(100\) −3.63102 −0.363102
\(101\) 8.47650 0.843443 0.421722 0.906725i \(-0.361426\pi\)
0.421722 + 0.906725i \(0.361426\pi\)
\(102\) − 1.35690i − 0.134353i
\(103\) −5.64742 −0.556456 −0.278228 0.960515i \(-0.589747\pi\)
−0.278228 + 0.960515i \(0.589747\pi\)
\(104\) 0 0
\(105\) 4.97823 0.485825
\(106\) 5.56465i 0.540486i
\(107\) −6.73556 −0.651151 −0.325576 0.945516i \(-0.605558\pi\)
−0.325576 + 0.945516i \(0.605558\pi\)
\(108\) −1.24698 −0.119991
\(109\) 2.07606i 0.198851i 0.995045 + 0.0994255i \(0.0317005\pi\)
−0.995045 + 0.0994255i \(0.968300\pi\)
\(110\) 13.5036i 1.28752i
\(111\) 6.24698i 0.592937i
\(112\) − 17.0151i − 1.60777i
\(113\) 6.16852 0.580286 0.290143 0.956983i \(-0.406297\pi\)
0.290143 + 0.956983i \(0.406297\pi\)
\(114\) −14.3448 −1.34351
\(115\) − 4.08815i − 0.381222i
\(116\) 4.87800 0.452911
\(117\) 0 0
\(118\) 3.38404 0.311526
\(119\) 2.59419i 0.237809i
\(120\) −1.96077 −0.178993
\(121\) −15.8944 −1.44495
\(122\) − 6.02715i − 0.545672i
\(123\) − 1.80194i − 0.162475i
\(124\) − 6.10992i − 0.548687i
\(125\) − 11.4330i − 1.02260i
\(126\) 6.20775 0.553030
\(127\) 14.2620 1.26555 0.632776 0.774335i \(-0.281915\pi\)
0.632776 + 0.774335i \(0.281915\pi\)
\(128\) 10.0858i 0.891463i
\(129\) 7.09783 0.624929
\(130\) 0 0
\(131\) 22.6015 1.97470 0.987350 0.158554i \(-0.0506831\pi\)
0.987350 + 0.158554i \(0.0506831\pi\)
\(132\) 6.46681i 0.562864i
\(133\) 27.4252 2.37807
\(134\) 8.18598 0.707161
\(135\) − 1.44504i − 0.124369i
\(136\) − 1.02177i − 0.0876161i
\(137\) − 13.6353i − 1.16495i −0.812850 0.582473i \(-0.802085\pi\)
0.812850 0.582473i \(-0.197915\pi\)
\(138\) − 5.09783i − 0.433957i
\(139\) 17.6015 1.49294 0.746469 0.665420i \(-0.231748\pi\)
0.746469 + 0.665420i \(0.231748\pi\)
\(140\) −6.20775 −0.524651
\(141\) 10.5526i 0.888686i
\(142\) 16.4330 1.37902
\(143\) 0 0
\(144\) −4.93900 −0.411583
\(145\) 5.65279i 0.469439i
\(146\) 5.31767 0.440093
\(147\) −4.86831 −0.401532
\(148\) − 7.78986i − 0.640322i
\(149\) − 12.7385i − 1.04358i −0.853073 0.521791i \(-0.825264\pi\)
0.853073 0.521791i \(-0.174736\pi\)
\(150\) − 5.24698i − 0.428414i
\(151\) 15.6407i 1.27282i 0.771350 + 0.636412i \(0.219582\pi\)
−0.771350 + 0.636412i \(0.780418\pi\)
\(152\) −10.8019 −0.876153
\(153\) 0.753020 0.0608781
\(154\) − 32.1933i − 2.59421i
\(155\) 7.08038 0.568710
\(156\) 0 0
\(157\) −0.823708 −0.0657391 −0.0328695 0.999460i \(-0.510465\pi\)
−0.0328695 + 0.999460i \(0.510465\pi\)
\(158\) 16.9976i 1.35226i
\(159\) −3.08815 −0.244906
\(160\) 8.93900 0.706690
\(161\) 9.74632i 0.768117i
\(162\) − 1.80194i − 0.141574i
\(163\) 6.26875i 0.491006i 0.969396 + 0.245503i \(0.0789532\pi\)
−0.969396 + 0.245503i \(0.921047\pi\)
\(164\) 2.24698i 0.175460i
\(165\) −7.49396 −0.583404
\(166\) 11.6528 0.904432
\(167\) − 7.45042i − 0.576531i −0.957551 0.288265i \(-0.906921\pi\)
0.957551 0.288265i \(-0.0930785\pi\)
\(168\) 4.67456 0.360650
\(169\) 0 0
\(170\) −1.96077 −0.150384
\(171\) − 7.96077i − 0.608775i
\(172\) −8.85086 −0.674871
\(173\) 2.00969 0.152794 0.0763969 0.997077i \(-0.475658\pi\)
0.0763969 + 0.997077i \(0.475658\pi\)
\(174\) 7.04892i 0.534377i
\(175\) 10.0315i 0.758307i
\(176\) 25.6136i 1.93070i
\(177\) 1.87800i 0.141159i
\(178\) 2.08815 0.156513
\(179\) −20.0368 −1.49762 −0.748812 0.662783i \(-0.769375\pi\)
−0.748812 + 0.662783i \(0.769375\pi\)
\(180\) 1.80194i 0.134309i
\(181\) −24.1226 −1.79302 −0.896509 0.443026i \(-0.853905\pi\)
−0.896509 + 0.443026i \(0.853905\pi\)
\(182\) 0 0
\(183\) 3.34481 0.247256
\(184\) − 3.83877i − 0.282998i
\(185\) 9.02715 0.663689
\(186\) 8.82908 0.647380
\(187\) − 3.90515i − 0.285573i
\(188\) − 13.1588i − 0.959707i
\(189\) 3.44504i 0.250590i
\(190\) 20.7289i 1.50383i
\(191\) −7.08038 −0.512318 −0.256159 0.966635i \(-0.582457\pi\)
−0.256159 + 0.966635i \(0.582457\pi\)
\(192\) 1.26875 0.0915641
\(193\) 9.76809i 0.703122i 0.936165 + 0.351561i \(0.114349\pi\)
−0.936165 + 0.351561i \(0.885651\pi\)
\(194\) −15.6015 −1.12012
\(195\) 0 0
\(196\) 6.07069 0.433621
\(197\) − 23.4112i − 1.66798i −0.551781 0.833989i \(-0.686052\pi\)
0.551781 0.833989i \(-0.313948\pi\)
\(198\) −9.34481 −0.664107
\(199\) −4.02475 −0.285307 −0.142654 0.989773i \(-0.545563\pi\)
−0.142654 + 0.989773i \(0.545563\pi\)
\(200\) − 3.95108i − 0.279384i
\(201\) 4.54288i 0.320430i
\(202\) − 15.2741i − 1.07468i
\(203\) − 13.4765i − 0.945865i
\(204\) −0.939001 −0.0657432
\(205\) −2.60388 −0.181863
\(206\) 10.1763i 0.709016i
\(207\) 2.82908 0.196635
\(208\) 0 0
\(209\) −41.2844 −2.85570
\(210\) − 8.97046i − 0.619021i
\(211\) −3.91185 −0.269303 −0.134652 0.990893i \(-0.542992\pi\)
−0.134652 + 0.990893i \(0.542992\pi\)
\(212\) 3.85086 0.264478
\(213\) 9.11960i 0.624865i
\(214\) 12.1371i 0.829673i
\(215\) − 10.2567i − 0.699499i
\(216\) − 1.35690i − 0.0923251i
\(217\) −16.8799 −1.14588
\(218\) 3.74094 0.253368
\(219\) 2.95108i 0.199416i
\(220\) 9.34481 0.630027
\(221\) 0 0
\(222\) 11.2567 0.755498
\(223\) − 7.44935i − 0.498846i −0.968395 0.249423i \(-0.919759\pi\)
0.968395 0.249423i \(-0.0802409\pi\)
\(224\) −21.3110 −1.42390
\(225\) 2.91185 0.194124
\(226\) − 11.1153i − 0.739378i
\(227\) 21.2500i 1.41041i 0.709004 + 0.705205i \(0.249145\pi\)
−0.709004 + 0.705205i \(0.750855\pi\)
\(228\) 9.92692i 0.657426i
\(229\) 9.29590i 0.614290i 0.951663 + 0.307145i \(0.0993737\pi\)
−0.951663 + 0.307145i \(0.900626\pi\)
\(230\) −7.36658 −0.485738
\(231\) 17.8659 1.17549
\(232\) 5.30798i 0.348486i
\(233\) −16.2107 −1.06200 −0.531000 0.847372i \(-0.678184\pi\)
−0.531000 + 0.847372i \(0.678184\pi\)
\(234\) 0 0
\(235\) 15.2489 0.994728
\(236\) − 2.34183i − 0.152440i
\(237\) −9.43296 −0.612737
\(238\) 4.67456 0.303007
\(239\) 13.5090i 0.873826i 0.899504 + 0.436913i \(0.143928\pi\)
−0.899504 + 0.436913i \(0.856072\pi\)
\(240\) 7.13706i 0.460695i
\(241\) 6.26875i 0.403806i 0.979406 + 0.201903i \(0.0647125\pi\)
−0.979406 + 0.201903i \(0.935287\pi\)
\(242\) 28.6407i 1.84109i
\(243\) 1.00000 0.0641500
\(244\) −4.17092 −0.267015
\(245\) 7.03492i 0.449444i
\(246\) −3.24698 −0.207020
\(247\) 0 0
\(248\) 6.64848 0.422179
\(249\) 6.46681i 0.409818i
\(250\) −20.6015 −1.30295
\(251\) 0.753020 0.0475302 0.0237651 0.999718i \(-0.492435\pi\)
0.0237651 + 0.999718i \(0.492435\pi\)
\(252\) − 4.29590i − 0.270616i
\(253\) − 14.6716i − 0.922394i
\(254\) − 25.6993i − 1.61252i
\(255\) − 1.08815i − 0.0681423i
\(256\) 20.7114 1.29446
\(257\) −19.7265 −1.23050 −0.615252 0.788331i \(-0.710946\pi\)
−0.615252 + 0.788331i \(0.710946\pi\)
\(258\) − 12.7899i − 0.796262i
\(259\) −21.5211 −1.33726
\(260\) 0 0
\(261\) −3.91185 −0.242138
\(262\) − 40.7265i − 2.51609i
\(263\) −17.6093 −1.08583 −0.542917 0.839787i \(-0.682680\pi\)
−0.542917 + 0.839787i \(0.682680\pi\)
\(264\) −7.03684 −0.433087
\(265\) 4.46250i 0.274129i
\(266\) − 49.4185i − 3.03004i
\(267\) 1.15883i 0.0709195i
\(268\) − 5.66487i − 0.346037i
\(269\) −16.3870 −0.999135 −0.499567 0.866275i \(-0.666508\pi\)
−0.499567 + 0.866275i \(0.666508\pi\)
\(270\) −2.60388 −0.158467
\(271\) − 0.795233i − 0.0483070i −0.999708 0.0241535i \(-0.992311\pi\)
0.999708 0.0241535i \(-0.00768904\pi\)
\(272\) −3.71917 −0.225508
\(273\) 0 0
\(274\) −24.5700 −1.48433
\(275\) − 15.1008i − 0.910614i
\(276\) −3.52781 −0.212349
\(277\) 4.83340 0.290411 0.145205 0.989402i \(-0.453616\pi\)
0.145205 + 0.989402i \(0.453616\pi\)
\(278\) − 31.7168i − 1.90225i
\(279\) 4.89977i 0.293342i
\(280\) − 6.75494i − 0.403685i
\(281\) 18.7748i 1.12001i 0.828489 + 0.560005i \(0.189201\pi\)
−0.828489 + 0.560005i \(0.810799\pi\)
\(282\) 19.0151 1.13233
\(283\) 7.91723 0.470631 0.235315 0.971919i \(-0.424388\pi\)
0.235315 + 0.971919i \(0.424388\pi\)
\(284\) − 11.3720i − 0.674802i
\(285\) −11.5036 −0.681417
\(286\) 0 0
\(287\) 6.20775 0.366432
\(288\) 6.18598i 0.364512i
\(289\) −16.4330 −0.966645
\(290\) 10.1860 0.598141
\(291\) − 8.65817i − 0.507551i
\(292\) − 3.67994i − 0.215352i
\(293\) 6.57912i 0.384356i 0.981360 + 0.192178i \(0.0615552\pi\)
−0.981360 + 0.192178i \(0.938445\pi\)
\(294\) 8.77240i 0.511617i
\(295\) 2.71379 0.158003
\(296\) 8.47650 0.492687
\(297\) − 5.18598i − 0.300921i
\(298\) −22.9541 −1.32969
\(299\) 0 0
\(300\) −3.63102 −0.209637
\(301\) 24.4523i 1.40941i
\(302\) 28.1836 1.62178
\(303\) 8.47650 0.486962
\(304\) 39.3183i 2.25506i
\(305\) − 4.83340i − 0.276759i
\(306\) − 1.35690i − 0.0775686i
\(307\) − 24.8649i − 1.41911i −0.704649 0.709556i \(-0.748895\pi\)
0.704649 0.709556i \(-0.251105\pi\)
\(308\) −22.2784 −1.26943
\(309\) −5.64742 −0.321270
\(310\) − 12.7584i − 0.724628i
\(311\) 17.0804 0.968539 0.484270 0.874919i \(-0.339085\pi\)
0.484270 + 0.874919i \(0.339085\pi\)
\(312\) 0 0
\(313\) 15.6974 0.887269 0.443635 0.896208i \(-0.353689\pi\)
0.443635 + 0.896208i \(0.353689\pi\)
\(314\) 1.48427i 0.0837622i
\(315\) 4.97823 0.280491
\(316\) 11.7627 0.661704
\(317\) − 32.7821i − 1.84123i −0.390477 0.920613i \(-0.627690\pi\)
0.390477 0.920613i \(-0.372310\pi\)
\(318\) 5.56465i 0.312050i
\(319\) 20.2868i 1.13584i
\(320\) − 1.83340i − 0.102490i
\(321\) −6.73556 −0.375942
\(322\) 17.5623 0.978706
\(323\) − 5.99462i − 0.333550i
\(324\) −1.24698 −0.0692766
\(325\) 0 0
\(326\) 11.2959 0.625622
\(327\) 2.07606i 0.114807i
\(328\) −2.44504 −0.135005
\(329\) −36.3540 −2.00426
\(330\) 13.5036i 0.743351i
\(331\) 29.1618i 1.60288i 0.598076 + 0.801439i \(0.295932\pi\)
−0.598076 + 0.801439i \(0.704068\pi\)
\(332\) − 8.06398i − 0.442569i
\(333\) 6.24698i 0.342332i
\(334\) −13.4252 −0.734594
\(335\) 6.56465 0.358665
\(336\) − 17.0151i − 0.928248i
\(337\) 33.2911 1.81348 0.906741 0.421688i \(-0.138562\pi\)
0.906741 + 0.421688i \(0.138562\pi\)
\(338\) 0 0
\(339\) 6.16852 0.335028
\(340\) 1.35690i 0.0735880i
\(341\) 25.4101 1.37604
\(342\) −14.3448 −0.775679
\(343\) 7.34375i 0.396525i
\(344\) − 9.63102i − 0.519270i
\(345\) − 4.08815i − 0.220098i
\(346\) − 3.62133i − 0.194684i
\(347\) −0.873690 −0.0469022 −0.0234511 0.999725i \(-0.507465\pi\)
−0.0234511 + 0.999725i \(0.507465\pi\)
\(348\) 4.87800 0.261488
\(349\) − 3.23191i − 0.173000i −0.996252 0.0865002i \(-0.972432\pi\)
0.996252 0.0865002i \(-0.0275683\pi\)
\(350\) 18.0761 0.966206
\(351\) 0 0
\(352\) 32.0804 1.70989
\(353\) − 8.14675i − 0.433608i −0.976215 0.216804i \(-0.930437\pi\)
0.976215 0.216804i \(-0.0695632\pi\)
\(354\) 3.38404 0.179860
\(355\) 13.1782 0.699427
\(356\) − 1.44504i − 0.0765871i
\(357\) 2.59419i 0.137299i
\(358\) 36.1051i 1.90822i
\(359\) 2.64071i 0.139371i 0.997569 + 0.0696857i \(0.0221996\pi\)
−0.997569 + 0.0696857i \(0.977800\pi\)
\(360\) −1.96077 −0.103342
\(361\) −44.3739 −2.33547
\(362\) 43.4674i 2.28460i
\(363\) −15.8944 −0.834239
\(364\) 0 0
\(365\) 4.26444 0.223211
\(366\) − 6.02715i − 0.315044i
\(367\) −2.90408 −0.151592 −0.0757960 0.997123i \(-0.524150\pi\)
−0.0757960 + 0.997123i \(0.524150\pi\)
\(368\) −13.9729 −0.728385
\(369\) − 1.80194i − 0.0938051i
\(370\) − 16.2664i − 0.845648i
\(371\) − 10.6388i − 0.552339i
\(372\) − 6.10992i − 0.316784i
\(373\) −8.39852 −0.434859 −0.217429 0.976076i \(-0.569767\pi\)
−0.217429 + 0.976076i \(0.569767\pi\)
\(374\) −7.03684 −0.363866
\(375\) − 11.4330i − 0.590396i
\(376\) 14.3187 0.738432
\(377\) 0 0
\(378\) 6.20775 0.319292
\(379\) − 15.7482i − 0.808932i −0.914553 0.404466i \(-0.867457\pi\)
0.914553 0.404466i \(-0.132543\pi\)
\(380\) 14.3448 0.735873
\(381\) 14.2620 0.730667
\(382\) 12.7584i 0.652776i
\(383\) − 12.7385i − 0.650909i −0.945558 0.325455i \(-0.894483\pi\)
0.945558 0.325455i \(-0.105517\pi\)
\(384\) 10.0858i 0.514686i
\(385\) − 25.8170i − 1.31576i
\(386\) 17.6015 0.895892
\(387\) 7.09783 0.360803
\(388\) 10.7966i 0.548112i
\(389\) 0.310371 0.0157365 0.00786823 0.999969i \(-0.497495\pi\)
0.00786823 + 0.999969i \(0.497495\pi\)
\(390\) 0 0
\(391\) 2.13036 0.107737
\(392\) 6.60579i 0.333643i
\(393\) 22.6015 1.14009
\(394\) −42.1855 −2.12528
\(395\) 13.6310i 0.685851i
\(396\) 6.46681i 0.324970i
\(397\) 1.49098i 0.0748299i 0.999300 + 0.0374150i \(0.0119123\pi\)
−0.999300 + 0.0374150i \(0.988088\pi\)
\(398\) 7.25236i 0.363528i
\(399\) 27.4252 1.37298
\(400\) −14.3817 −0.719083
\(401\) − 23.8334i − 1.19018i −0.803658 0.595092i \(-0.797116\pi\)
0.803658 0.595092i \(-0.202884\pi\)
\(402\) 8.18598 0.408280
\(403\) 0 0
\(404\) −10.5700 −0.525878
\(405\) − 1.44504i − 0.0718047i
\(406\) −24.2838 −1.20519
\(407\) 32.3967 1.60585
\(408\) − 1.02177i − 0.0505852i
\(409\) − 4.26742i − 0.211010i −0.994419 0.105505i \(-0.966354\pi\)
0.994419 0.105505i \(-0.0336460\pi\)
\(410\) 4.69202i 0.231722i
\(411\) − 13.6353i − 0.672581i
\(412\) 7.04221 0.346945
\(413\) −6.46980 −0.318358
\(414\) − 5.09783i − 0.250545i
\(415\) 9.34481 0.458719
\(416\) 0 0
\(417\) 17.6015 0.861948
\(418\) 74.3919i 3.63863i
\(419\) 29.6896 1.45043 0.725217 0.688521i \(-0.241740\pi\)
0.725217 + 0.688521i \(0.241740\pi\)
\(420\) −6.20775 −0.302907
\(421\) − 29.3991i − 1.43282i −0.697677 0.716412i \(-0.745783\pi\)
0.697677 0.716412i \(-0.254217\pi\)
\(422\) 7.04892i 0.343136i
\(423\) 10.5526i 0.513083i
\(424\) 4.19029i 0.203499i
\(425\) 2.19269 0.106361
\(426\) 16.4330 0.796180
\(427\) 11.5230i 0.557638i
\(428\) 8.39911 0.405986
\(429\) 0 0
\(430\) −18.4819 −0.891275
\(431\) 33.0562i 1.59226i 0.605124 + 0.796131i \(0.293123\pi\)
−0.605124 + 0.796131i \(0.706877\pi\)
\(432\) −4.93900 −0.237628
\(433\) 29.2664 1.40645 0.703226 0.710967i \(-0.251742\pi\)
0.703226 + 0.710967i \(0.251742\pi\)
\(434\) 30.4166i 1.46004i
\(435\) 5.65279i 0.271031i
\(436\) − 2.58881i − 0.123982i
\(437\) − 22.5217i − 1.07736i
\(438\) 5.31767 0.254088
\(439\) 2.13169 0.101740 0.0508699 0.998705i \(-0.483801\pi\)
0.0508699 + 0.998705i \(0.483801\pi\)
\(440\) 10.1685i 0.484765i
\(441\) −4.86831 −0.231824
\(442\) 0 0
\(443\) −22.9922 −1.09239 −0.546197 0.837657i \(-0.683925\pi\)
−0.546197 + 0.837657i \(0.683925\pi\)
\(444\) − 7.78986i − 0.369690i
\(445\) 1.67456 0.0793819
\(446\) −13.4233 −0.635610
\(447\) − 12.7385i − 0.602513i
\(448\) 4.37090i 0.206505i
\(449\) − 12.9379i − 0.610579i −0.952260 0.305289i \(-0.901247\pi\)
0.952260 0.305289i \(-0.0987532\pi\)
\(450\) − 5.24698i − 0.247345i
\(451\) −9.34481 −0.440030
\(452\) −7.69202 −0.361802
\(453\) 15.6407i 0.734865i
\(454\) 38.2911 1.79709
\(455\) 0 0
\(456\) −10.8019 −0.505847
\(457\) 4.85325i 0.227025i 0.993537 + 0.113513i \(0.0362103\pi\)
−0.993537 + 0.113513i \(0.963790\pi\)
\(458\) 16.7506 0.782705
\(459\) 0.753020 0.0351480
\(460\) 5.09783i 0.237688i
\(461\) − 18.8345i − 0.877208i −0.898680 0.438604i \(-0.855473\pi\)
0.898680 0.438604i \(-0.144527\pi\)
\(462\) − 32.1933i − 1.49777i
\(463\) − 22.8767i − 1.06317i −0.847005 0.531585i \(-0.821597\pi\)
0.847005 0.531585i \(-0.178403\pi\)
\(464\) 19.3207 0.896939
\(465\) 7.08038 0.328345
\(466\) 29.2107i 1.35316i
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) −15.6504 −0.722668
\(470\) − 27.4776i − 1.26745i
\(471\) −0.823708 −0.0379545
\(472\) 2.54825 0.117293
\(473\) − 36.8092i − 1.69249i
\(474\) 16.9976i 0.780726i
\(475\) − 23.1806i − 1.06360i
\(476\) − 3.23490i − 0.148271i
\(477\) −3.08815 −0.141396
\(478\) 24.3424 1.11340
\(479\) − 38.0901i − 1.74038i −0.492717 0.870190i \(-0.663996\pi\)
0.492717 0.870190i \(-0.336004\pi\)
\(480\) 8.93900 0.408008
\(481\) 0 0
\(482\) 11.2959 0.514514
\(483\) 9.74632i 0.443473i
\(484\) 19.8200 0.900909
\(485\) −12.5114 −0.568114
\(486\) − 1.80194i − 0.0817376i
\(487\) 21.2500i 0.962928i 0.876466 + 0.481464i \(0.159895\pi\)
−0.876466 + 0.481464i \(0.840105\pi\)
\(488\) − 4.53856i − 0.205451i
\(489\) 6.26875i 0.283483i
\(490\) 12.6765 0.572665
\(491\) 6.35019 0.286580 0.143290 0.989681i \(-0.454232\pi\)
0.143290 + 0.989681i \(0.454232\pi\)
\(492\) 2.24698i 0.101302i
\(493\) −2.94571 −0.132668
\(494\) 0 0
\(495\) −7.49396 −0.336828
\(496\) − 24.2000i − 1.08661i
\(497\) −31.4174 −1.40926
\(498\) 11.6528 0.522174
\(499\) 4.65087i 0.208202i 0.994567 + 0.104101i \(0.0331965\pi\)
−0.994567 + 0.104101i \(0.966804\pi\)
\(500\) 14.2567i 0.637578i
\(501\) − 7.45042i − 0.332860i
\(502\) − 1.35690i − 0.0605612i
\(503\) 15.4752 0.690004 0.345002 0.938602i \(-0.387878\pi\)
0.345002 + 0.938602i \(0.387878\pi\)
\(504\) 4.67456 0.208222
\(505\) − 12.2489i − 0.545069i
\(506\) −26.4373 −1.17528
\(507\) 0 0
\(508\) −17.7845 −0.789059
\(509\) 20.5047i 0.908855i 0.890784 + 0.454428i \(0.150156\pi\)
−0.890784 + 0.454428i \(0.849844\pi\)
\(510\) −1.96077 −0.0868244
\(511\) −10.1666 −0.449744
\(512\) − 17.1491i − 0.757892i
\(513\) − 7.96077i − 0.351477i
\(514\) 35.5459i 1.56786i
\(515\) 8.16075i 0.359606i
\(516\) −8.85086 −0.389637
\(517\) 54.7254 2.40682
\(518\) 38.7797i 1.70388i
\(519\) 2.00969 0.0882155
\(520\) 0 0
\(521\) −42.0267 −1.84122 −0.920611 0.390481i \(-0.872309\pi\)
−0.920611 + 0.390481i \(0.872309\pi\)
\(522\) 7.04892i 0.308523i
\(523\) −29.9885 −1.31131 −0.655653 0.755062i \(-0.727607\pi\)
−0.655653 + 0.755062i \(0.727607\pi\)
\(524\) −28.1836 −1.23121
\(525\) 10.0315i 0.437809i
\(526\) 31.7308i 1.38353i
\(527\) 3.68963i 0.160723i
\(528\) 25.6136i 1.11469i
\(529\) −14.9963 −0.652012
\(530\) 8.04115 0.349285
\(531\) 1.87800i 0.0814984i
\(532\) −34.1987 −1.48270
\(533\) 0 0
\(534\) 2.08815 0.0903629
\(535\) 9.73317i 0.420802i
\(536\) 6.16421 0.266253
\(537\) −20.0368 −0.864653
\(538\) 29.5284i 1.27306i
\(539\) 25.2470i 1.08746i
\(540\) 1.80194i 0.0775431i
\(541\) 36.3803i 1.56411i 0.623208 + 0.782056i \(0.285829\pi\)
−0.623208 + 0.782056i \(0.714171\pi\)
\(542\) −1.43296 −0.0615509
\(543\) −24.1226 −1.03520
\(544\) 4.65817i 0.199717i
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −25.8159 −1.10381 −0.551905 0.833907i \(-0.686099\pi\)
−0.551905 + 0.833907i \(0.686099\pi\)
\(548\) 17.0030i 0.726331i
\(549\) 3.34481 0.142753
\(550\) −27.2107 −1.16027
\(551\) 31.1414i 1.32667i
\(552\) − 3.83877i − 0.163389i
\(553\) − 32.4969i − 1.38191i
\(554\) − 8.70948i − 0.370030i
\(555\) 9.02715 0.383181
\(556\) −21.9487 −0.930832
\(557\) 17.9903i 0.762274i 0.924519 + 0.381137i \(0.124467\pi\)
−0.924519 + 0.381137i \(0.875533\pi\)
\(558\) 8.82908 0.373765
\(559\) 0 0
\(560\) −24.5875 −1.03901
\(561\) − 3.90515i − 0.164876i
\(562\) 33.8310 1.42707
\(563\) 39.1323 1.64923 0.824614 0.565695i \(-0.191392\pi\)
0.824614 + 0.565695i \(0.191392\pi\)
\(564\) − 13.1588i − 0.554087i
\(565\) − 8.91377i − 0.375005i
\(566\) − 14.2664i − 0.599660i
\(567\) 3.44504i 0.144678i
\(568\) 12.3744 0.519216
\(569\) −30.6002 −1.28283 −0.641413 0.767196i \(-0.721651\pi\)
−0.641413 + 0.767196i \(0.721651\pi\)
\(570\) 20.7289i 0.868236i
\(571\) 2.96184 0.123949 0.0619745 0.998078i \(-0.480260\pi\)
0.0619745 + 0.998078i \(0.480260\pi\)
\(572\) 0 0
\(573\) −7.08038 −0.295787
\(574\) − 11.1860i − 0.466894i
\(575\) 8.23788 0.343543
\(576\) 1.26875 0.0528646
\(577\) 0.819396i 0.0341119i 0.999855 + 0.0170560i \(0.00542934\pi\)
−0.999855 + 0.0170560i \(0.994571\pi\)
\(578\) 29.6112i 1.23166i
\(579\) 9.76809i 0.405948i
\(580\) − 7.04892i − 0.292690i
\(581\) −22.2784 −0.924265
\(582\) −15.6015 −0.646702
\(583\) 16.0151i 0.663276i
\(584\) 4.00431 0.165700
\(585\) 0 0
\(586\) 11.8552 0.489732
\(587\) 31.7995i 1.31251i 0.754540 + 0.656254i \(0.227860\pi\)
−0.754540 + 0.656254i \(0.772140\pi\)
\(588\) 6.07069 0.250351
\(589\) 39.0060 1.60721
\(590\) − 4.89008i − 0.201322i
\(591\) − 23.4112i − 0.963008i
\(592\) − 30.8538i − 1.26808i
\(593\) − 4.26337i − 0.175076i −0.996161 0.0875379i \(-0.972100\pi\)
0.996161 0.0875379i \(-0.0278999\pi\)
\(594\) −9.34481 −0.383422
\(595\) 3.74871 0.153682
\(596\) 15.8847i 0.650663i
\(597\) −4.02475 −0.164722
\(598\) 0 0
\(599\) 24.7278 1.01035 0.505175 0.863017i \(-0.331428\pi\)
0.505175 + 0.863017i \(0.331428\pi\)
\(600\) − 3.95108i − 0.161302i
\(601\) 6.82371 0.278345 0.139172 0.990268i \(-0.455556\pi\)
0.139172 + 0.990268i \(0.455556\pi\)
\(602\) 44.0616 1.79582
\(603\) 4.54288i 0.185000i
\(604\) − 19.5036i − 0.793592i
\(605\) 22.9681i 0.933785i
\(606\) − 15.2741i − 0.620469i
\(607\) 31.9963 1.29869 0.649344 0.760494i \(-0.275043\pi\)
0.649344 + 0.760494i \(0.275043\pi\)
\(608\) 49.2452 1.99716
\(609\) − 13.4765i − 0.546095i
\(610\) −8.70948 −0.352637
\(611\) 0 0
\(612\) −0.939001 −0.0379569
\(613\) − 33.5875i − 1.35659i −0.734792 0.678293i \(-0.762720\pi\)
0.734792 0.678293i \(-0.237280\pi\)
\(614\) −44.8049 −1.80818
\(615\) −2.60388 −0.104998
\(616\) − 24.2422i − 0.976746i
\(617\) 26.5870i 1.07035i 0.844740 + 0.535176i \(0.179755\pi\)
−0.844740 + 0.535176i \(0.820245\pi\)
\(618\) 10.1763i 0.409350i
\(619\) − 9.17928i − 0.368946i −0.982838 0.184473i \(-0.940942\pi\)
0.982838 0.184473i \(-0.0590579\pi\)
\(620\) −8.82908 −0.354585
\(621\) 2.82908 0.113527
\(622\) − 30.7778i − 1.23408i
\(623\) −3.99223 −0.159945
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) − 28.2857i − 1.13053i
\(627\) −41.2844 −1.64874
\(628\) 1.02715 0.0409876
\(629\) 4.70410i 0.187565i
\(630\) − 8.97046i − 0.357392i
\(631\) 17.0043i 0.676931i 0.940979 + 0.338465i \(0.109908\pi\)
−0.940979 + 0.338465i \(0.890092\pi\)
\(632\) 12.7995i 0.509139i
\(633\) −3.91185 −0.155482
\(634\) −59.0713 −2.34602
\(635\) − 20.6093i − 0.817853i
\(636\) 3.85086 0.152696
\(637\) 0 0
\(638\) 36.5555 1.44725
\(639\) 9.11960i 0.360766i
\(640\) 14.5743 0.576101
\(641\) 21.6649 0.855711 0.427856 0.903847i \(-0.359269\pi\)
0.427856 + 0.903847i \(0.359269\pi\)
\(642\) 12.1371i 0.479012i
\(643\) − 9.35557i − 0.368948i −0.982837 0.184474i \(-0.940942\pi\)
0.982837 0.184474i \(-0.0590581\pi\)
\(644\) − 12.1535i − 0.478913i
\(645\) − 10.2567i − 0.403856i
\(646\) −10.8019 −0.424997
\(647\) 0.702775 0.0276289 0.0138145 0.999905i \(-0.495603\pi\)
0.0138145 + 0.999905i \(0.495603\pi\)
\(648\) − 1.35690i − 0.0533039i
\(649\) 9.73928 0.382300
\(650\) 0 0
\(651\) −16.8799 −0.661576
\(652\) − 7.81700i − 0.306137i
\(653\) −37.3411 −1.46127 −0.730635 0.682768i \(-0.760776\pi\)
−0.730635 + 0.682768i \(0.760776\pi\)
\(654\) 3.74094 0.146282
\(655\) − 32.6601i − 1.27614i
\(656\) 8.89977i 0.347478i
\(657\) 2.95108i 0.115133i
\(658\) 65.5077i 2.55376i
\(659\) −0.735562 −0.0286534 −0.0143267 0.999897i \(-0.504560\pi\)
−0.0143267 + 0.999897i \(0.504560\pi\)
\(660\) 9.34481 0.363746
\(661\) 13.8485i 0.538643i 0.963050 + 0.269321i \(0.0867994\pi\)
−0.963050 + 0.269321i \(0.913201\pi\)
\(662\) 52.5478 2.04233
\(663\) 0 0
\(664\) 8.77479 0.340528
\(665\) − 39.6305i − 1.53681i
\(666\) 11.2567 0.436187
\(667\) −11.0670 −0.428515
\(668\) 9.29052i 0.359461i
\(669\) − 7.44935i − 0.288009i
\(670\) − 11.8291i − 0.456997i
\(671\) − 17.3461i − 0.669640i
\(672\) −21.3110 −0.822088
\(673\) 6.35019 0.244782 0.122391 0.992482i \(-0.460944\pi\)
0.122391 + 0.992482i \(0.460944\pi\)
\(674\) − 59.9885i − 2.31067i
\(675\) 2.91185 0.112077
\(676\) 0 0
\(677\) 33.7241 1.29612 0.648061 0.761589i \(-0.275580\pi\)
0.648061 + 0.761589i \(0.275580\pi\)
\(678\) − 11.1153i − 0.426880i
\(679\) 29.8278 1.14468
\(680\) −1.47650 −0.0566212
\(681\) 21.2500i 0.814300i
\(682\) − 45.7875i − 1.75329i
\(683\) 19.2687i 0.737298i 0.929569 + 0.368649i \(0.120180\pi\)
−0.929569 + 0.368649i \(0.879820\pi\)
\(684\) 9.92692i 0.379565i
\(685\) −19.7036 −0.752837
\(686\) 13.2330 0.505237
\(687\) 9.29590i 0.354661i
\(688\) −35.0562 −1.33651
\(689\) 0 0
\(690\) −7.36658 −0.280441
\(691\) 39.4010i 1.49889i 0.662069 + 0.749443i \(0.269679\pi\)
−0.662069 + 0.749443i \(0.730321\pi\)
\(692\) −2.50604 −0.0952654
\(693\) 17.8659 0.678670
\(694\) 1.57434i 0.0597610i
\(695\) − 25.4349i − 0.964800i
\(696\) 5.30798i 0.201198i
\(697\) − 1.35690i − 0.0513961i
\(698\) −5.82371 −0.220431
\(699\) −16.2107 −0.613146
\(700\) − 12.5090i − 0.472797i
\(701\) −18.3985 −0.694902 −0.347451 0.937698i \(-0.612953\pi\)
−0.347451 + 0.937698i \(0.612953\pi\)
\(702\) 0 0
\(703\) 49.7308 1.87563
\(704\) − 6.57971i − 0.247982i
\(705\) 15.2489 0.574307
\(706\) −14.6799 −0.552487
\(707\) 29.2019i 1.09825i
\(708\) − 2.34183i − 0.0880114i
\(709\) − 38.4553i − 1.44422i −0.691778 0.722110i \(-0.743172\pi\)
0.691778 0.722110i \(-0.256828\pi\)
\(710\) − 23.7463i − 0.891183i
\(711\) −9.43296 −0.353764
\(712\) 1.57242 0.0589288
\(713\) 13.8619i 0.519131i
\(714\) 4.67456 0.174941
\(715\) 0 0
\(716\) 24.9855 0.933753
\(717\) 13.5090i 0.504504i
\(718\) 4.75840 0.177582
\(719\) −48.4999 −1.80874 −0.904371 0.426747i \(-0.859659\pi\)
−0.904371 + 0.426747i \(0.859659\pi\)
\(720\) 7.13706i 0.265983i
\(721\) − 19.4556i − 0.724564i
\(722\) 79.9590i 2.97576i
\(723\) 6.26875i 0.233137i
\(724\) 30.0804 1.11793
\(725\) −11.3907 −0.423042
\(726\) 28.6407i 1.06296i
\(727\) −19.0344 −0.705948 −0.352974 0.935633i \(-0.614830\pi\)
−0.352974 + 0.935633i \(0.614830\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 7.68425i − 0.284407i
\(731\) 5.34481 0.197685
\(732\) −4.17092 −0.154161
\(733\) − 24.6213i − 0.909410i −0.890642 0.454705i \(-0.849745\pi\)
0.890642 0.454705i \(-0.150255\pi\)
\(734\) 5.23298i 0.193153i
\(735\) 7.03492i 0.259487i
\(736\) 17.5007i 0.645083i
\(737\) 23.5593 0.867817
\(738\) −3.24698 −0.119523
\(739\) − 44.5115i − 1.63738i −0.574233 0.818692i \(-0.694700\pi\)
0.574233 0.818692i \(-0.305300\pi\)
\(740\) −11.2567 −0.413803
\(741\) 0 0
\(742\) −19.1704 −0.703769
\(743\) − 10.4112i − 0.381950i −0.981595 0.190975i \(-0.938835\pi\)
0.981595 0.190975i \(-0.0611649\pi\)
\(744\) 6.64848 0.243745
\(745\) −18.4077 −0.674407
\(746\) 15.1336i 0.554081i
\(747\) 6.46681i 0.236608i
\(748\) 4.86964i 0.178052i
\(749\) − 23.2043i − 0.847866i
\(750\) −20.6015 −0.752260
\(751\) −1.69979 −0.0620263 −0.0310131 0.999519i \(-0.509873\pi\)
−0.0310131 + 0.999519i \(0.509873\pi\)
\(752\) − 52.1191i − 1.90059i
\(753\) 0.753020 0.0274416
\(754\) 0 0
\(755\) 22.6015 0.822552
\(756\) − 4.29590i − 0.156240i
\(757\) 27.4252 0.996786 0.498393 0.866951i \(-0.333924\pi\)
0.498393 + 0.866951i \(0.333924\pi\)
\(758\) −28.3773 −1.03071
\(759\) − 14.6716i − 0.532545i
\(760\) 15.6093i 0.566207i
\(761\) − 5.02608i − 0.182195i −0.995842 0.0910977i \(-0.970962\pi\)
0.995842 0.0910977i \(-0.0290375\pi\)
\(762\) − 25.6993i − 0.930988i
\(763\) −7.15213 −0.258924
\(764\) 8.82908 0.319425
\(765\) − 1.08815i − 0.0393420i
\(766\) −22.9541 −0.829364
\(767\) 0 0
\(768\) 20.7114 0.747358
\(769\) − 42.4456i − 1.53063i −0.643657 0.765314i \(-0.722584\pi\)
0.643657 0.765314i \(-0.277416\pi\)
\(770\) −46.5206 −1.67649
\(771\) −19.7265 −0.710431
\(772\) − 12.1806i − 0.438390i
\(773\) − 26.3593i − 0.948078i −0.880504 0.474039i \(-0.842796\pi\)
0.880504 0.474039i \(-0.157204\pi\)
\(774\) − 12.7899i − 0.459722i
\(775\) 14.2674i 0.512501i
\(776\) −11.7482 −0.421737
\(777\) −21.5211 −0.772065
\(778\) − 0.559270i − 0.0200508i
\(779\) −14.3448 −0.513956
\(780\) 0 0
\(781\) 47.2941 1.69232
\(782\) − 3.83877i − 0.137274i
\(783\) −3.91185 −0.139798
\(784\) 24.0446 0.858736
\(785\) 1.19029i 0.0424834i
\(786\) − 40.7265i − 1.45266i
\(787\) 17.1424i 0.611062i 0.952182 + 0.305531i \(0.0988340\pi\)
−0.952182 + 0.305531i \(0.901166\pi\)
\(788\) 29.1933i 1.03997i
\(789\) −17.6093 −0.626906
\(790\) 24.5623 0.873886
\(791\) 21.2508i 0.755592i
\(792\) −7.03684 −0.250043
\(793\) 0 0
\(794\) 2.68664 0.0953455
\(795\) 4.46250i 0.158269i
\(796\) 5.01879 0.177886
\(797\) −30.1629 −1.06842 −0.534212 0.845351i \(-0.679392\pi\)
−0.534212 + 0.845351i \(0.679392\pi\)
\(798\) − 49.4185i − 1.74940i
\(799\) 7.94630i 0.281120i
\(800\) 18.0127i 0.636844i
\(801\) 1.15883i 0.0409454i
\(802\) −42.9463 −1.51649
\(803\) 15.3043 0.540076
\(804\) − 5.66487i − 0.199785i
\(805\) 14.0838 0.496390
\(806\) 0 0
\(807\) −16.3870 −0.576851
\(808\) − 11.5017i − 0.404629i
\(809\) −29.8504 −1.04948 −0.524742 0.851261i \(-0.675838\pi\)
−0.524742 + 0.851261i \(0.675838\pi\)
\(810\) −2.60388 −0.0914909
\(811\) − 47.7362i − 1.67624i −0.545484 0.838122i \(-0.683654\pi\)
0.545484 0.838122i \(-0.316346\pi\)
\(812\) 16.8049i 0.589737i
\(813\) − 0.795233i − 0.0278900i
\(814\) − 58.3769i − 2.04611i
\(815\) 9.05861 0.317309
\(816\) −3.71917 −0.130197
\(817\) − 56.5042i − 1.97683i
\(818\) −7.68963 −0.268862
\(819\) 0 0
\(820\) 3.24698 0.113389
\(821\) 17.9299i 0.625758i 0.949793 + 0.312879i \(0.101293\pi\)
−0.949793 + 0.312879i \(0.898707\pi\)
\(822\) −24.5700 −0.856978
\(823\) −54.3196 −1.89346 −0.946731 0.322026i \(-0.895636\pi\)
−0.946731 + 0.322026i \(0.895636\pi\)
\(824\) 7.66296i 0.266952i
\(825\) − 15.1008i − 0.525743i
\(826\) 11.6582i 0.405640i
\(827\) 49.1041i 1.70752i 0.520670 + 0.853758i \(0.325682\pi\)
−0.520670 + 0.853758i \(0.674318\pi\)
\(828\) −3.52781 −0.122600
\(829\) 7.35796 0.255553 0.127776 0.991803i \(-0.459216\pi\)
0.127776 + 0.991803i \(0.459216\pi\)
\(830\) − 16.8388i − 0.584482i
\(831\) 4.83340 0.167669
\(832\) 0 0
\(833\) −3.66594 −0.127017
\(834\) − 31.7168i − 1.09826i
\(835\) −10.7662 −0.372579
\(836\) 51.4808 1.78050
\(837\) 4.89977i 0.169361i
\(838\) − 53.4989i − 1.84809i
\(839\) − 36.5013i − 1.26016i −0.776529 0.630082i \(-0.783021\pi\)
0.776529 0.630082i \(-0.216979\pi\)
\(840\) − 6.75494i − 0.233068i
\(841\) −13.6974 −0.472324
\(842\) −52.9754 −1.82565
\(843\) 18.7748i 0.646638i
\(844\) 4.87800 0.167908
\(845\) 0 0
\(846\) 19.0151 0.653751
\(847\) − 54.7569i − 1.88147i
\(848\) 15.2524 0.523768
\(849\) 7.91723 0.271719
\(850\) − 3.95108i − 0.135521i
\(851\) 17.6732i 0.605831i
\(852\) − 11.3720i − 0.389597i
\(853\) − 9.73855i − 0.333441i −0.986004 0.166721i \(-0.946682\pi\)
0.986004 0.166721i \(-0.0533178\pi\)
\(854\) 20.7638 0.710522
\(855\) −11.5036 −0.393416
\(856\) 9.13946i 0.312380i
\(857\) 15.2030 0.519323 0.259662 0.965700i \(-0.416389\pi\)
0.259662 + 0.965700i \(0.416389\pi\)
\(858\) 0 0
\(859\) −31.9885 −1.09143 −0.545717 0.837970i \(-0.683743\pi\)
−0.545717 + 0.837970i \(0.683743\pi\)
\(860\) 12.7899i 0.436130i
\(861\) 6.20775 0.211560
\(862\) 59.5652 2.02880
\(863\) 35.2905i 1.20130i 0.799511 + 0.600652i \(0.205092\pi\)
−0.799511 + 0.600652i \(0.794908\pi\)
\(864\) 6.18598i 0.210451i
\(865\) − 2.90408i − 0.0987418i
\(866\) − 52.7362i − 1.79205i
\(867\) −16.4330 −0.558093
\(868\) 21.0489 0.714447
\(869\) 48.9191i 1.65947i
\(870\) 10.1860 0.345337
\(871\) 0 0
\(872\) 2.81700 0.0953958
\(873\) − 8.65817i − 0.293035i
\(874\) −40.5827 −1.37273
\(875\) 39.3870 1.33152
\(876\) − 3.67994i − 0.124334i
\(877\) 15.3263i 0.517532i 0.965940 + 0.258766i \(0.0833159\pi\)
−0.965940 + 0.258766i \(0.916684\pi\)
\(878\) − 3.84117i − 0.129633i
\(879\) 6.57912i 0.221908i
\(880\) 37.0127 1.24770
\(881\) 36.4306 1.22738 0.613689 0.789548i \(-0.289685\pi\)
0.613689 + 0.789548i \(0.289685\pi\)
\(882\) 8.77240i 0.295382i
\(883\) 37.6819 1.26810 0.634048 0.773294i \(-0.281392\pi\)
0.634048 + 0.773294i \(0.281392\pi\)
\(884\) 0 0
\(885\) 2.71379 0.0912231
\(886\) 41.4306i 1.39189i
\(887\) 4.24890 0.142664 0.0713320 0.997453i \(-0.477275\pi\)
0.0713320 + 0.997453i \(0.477275\pi\)
\(888\) 8.47650 0.284453
\(889\) 49.1333i 1.64788i
\(890\) − 3.01746i − 0.101145i
\(891\) − 5.18598i − 0.173737i
\(892\) 9.28919i 0.311025i
\(893\) 84.0066 2.81117
\(894\) −22.9541 −0.767699
\(895\) 28.9541i 0.967828i
\(896\) −34.7458 −1.16078
\(897\) 0 0
\(898\) −23.3134 −0.777977
\(899\) − 19.1672i − 0.639262i
\(900\) −3.63102 −0.121034
\(901\) −2.32544 −0.0774715
\(902\) 16.8388i 0.560670i
\(903\) 24.4523i 0.813723i
\(904\) − 8.37004i − 0.278383i
\(905\) 34.8582i 1.15872i
\(906\) 28.1836 0.936337
\(907\) 12.6183 0.418985 0.209493 0.977810i \(-0.432819\pi\)
0.209493 + 0.977810i \(0.432819\pi\)
\(908\) − 26.4983i − 0.879376i
\(909\) 8.47650 0.281148
\(910\) 0 0
\(911\) 6.77777 0.224558 0.112279 0.993677i \(-0.464185\pi\)
0.112279 + 0.993677i \(0.464185\pi\)
\(912\) 39.3183i 1.30196i
\(913\) 33.5368 1.10990
\(914\) 8.74525 0.289267
\(915\) − 4.83340i − 0.159787i
\(916\) − 11.5918i − 0.383004i
\(917\) 77.8631i 2.57126i
\(918\) − 1.35690i − 0.0447842i
\(919\) −20.4674 −0.675157 −0.337579 0.941297i \(-0.609608\pi\)
−0.337579 + 0.941297i \(0.609608\pi\)
\(920\) −5.54719 −0.182885
\(921\) − 24.8649i − 0.819325i
\(922\) −33.9385 −1.11771
\(923\) 0 0
\(924\) −22.2784 −0.732907
\(925\) 18.1903i 0.598093i
\(926\) −41.2223 −1.35465
\(927\) −5.64742 −0.185485
\(928\) − 24.1987i − 0.794360i
\(929\) 4.65220i 0.152634i 0.997084 + 0.0763169i \(0.0243161\pi\)
−0.997084 + 0.0763169i \(0.975684\pi\)
\(930\) − 12.7584i − 0.418364i
\(931\) 38.7555i 1.27016i
\(932\) 20.2145 0.662147
\(933\) 17.0804 0.559186
\(934\) 23.4252i 0.766496i
\(935\) −5.64310 −0.184549
\(936\) 0 0
\(937\) −41.8544 −1.36732 −0.683662 0.729798i \(-0.739614\pi\)
−0.683662 + 0.729798i \(0.739614\pi\)
\(938\) 28.2010i 0.920797i
\(939\) 15.6974 0.512265
\(940\) −19.0151 −0.620203
\(941\) − 30.3454i − 0.989232i −0.869112 0.494616i \(-0.835309\pi\)
0.869112 0.494616i \(-0.164691\pi\)
\(942\) 1.48427i 0.0483601i
\(943\) − 5.09783i − 0.166008i
\(944\) − 9.27545i − 0.301890i
\(945\) 4.97823 0.161942
\(946\) −66.3279 −2.15651
\(947\) 12.0325i 0.391004i 0.980703 + 0.195502i \(0.0626337\pi\)
−0.980703 + 0.195502i \(0.937366\pi\)
\(948\) 11.7627 0.382035
\(949\) 0 0
\(950\) −41.7700 −1.35520
\(951\) − 32.7821i − 1.06303i
\(952\) 3.52004 0.114085
\(953\) −22.9825 −0.744478 −0.372239 0.928137i \(-0.621410\pi\)
−0.372239 + 0.928137i \(0.621410\pi\)
\(954\) 5.56465i 0.180162i
\(955\) 10.2314i 0.331082i
\(956\) − 16.8455i − 0.544822i
\(957\) 20.2868i 0.655779i
\(958\) −68.6359 −2.21753
\(959\) 46.9743 1.51688
\(960\) − 1.83340i − 0.0591726i
\(961\) 6.99223 0.225556
\(962\) 0 0
\(963\) −6.73556 −0.217050
\(964\) − 7.81700i − 0.251769i
\(965\) 14.1153 0.454387
\(966\) 17.5623 0.565056
\(967\) 38.8883i 1.25056i 0.780399 + 0.625281i \(0.215016\pi\)
−0.780399 + 0.625281i \(0.784984\pi\)
\(968\) 21.5670i 0.693191i
\(969\) − 5.99462i − 0.192575i
\(970\) 22.5448i 0.723870i
\(971\) −57.5133 −1.84569 −0.922845 0.385171i \(-0.874143\pi\)
−0.922845 + 0.385171i \(0.874143\pi\)
\(972\) −1.24698 −0.0399969
\(973\) 60.6378i 1.94396i
\(974\) 38.2911 1.22693
\(975\) 0 0
\(976\) −16.5200 −0.528794
\(977\) − 16.3690i − 0.523690i −0.965110 0.261845i \(-0.915669\pi\)
0.965110 0.261845i \(-0.0843309\pi\)
\(978\) 11.2959 0.361203
\(979\) 6.00969 0.192070
\(980\) − 8.77240i − 0.280224i
\(981\) 2.07606i 0.0662836i
\(982\) − 11.4426i − 0.365150i
\(983\) − 15.6963i − 0.500635i −0.968164 0.250318i \(-0.919465\pi\)
0.968164 0.250318i \(-0.0805351\pi\)
\(984\) −2.44504 −0.0779451
\(985\) −33.8301 −1.07792
\(986\) 5.30798i 0.169040i
\(987\) −36.3540 −1.15716
\(988\) 0 0
\(989\) 20.0804 0.638519
\(990\) 13.5036i 0.429174i
\(991\) −11.2644 −0.357827 −0.178913 0.983865i \(-0.557258\pi\)
−0.178913 + 0.983865i \(0.557258\pi\)
\(992\) −30.3099 −0.962340
\(993\) 29.1618i 0.925422i
\(994\) 56.6122i 1.79563i
\(995\) 5.81594i 0.184378i
\(996\) − 8.06398i − 0.255517i
\(997\) −7.70112 −0.243897 −0.121948 0.992536i \(-0.538914\pi\)
−0.121948 + 0.992536i \(0.538914\pi\)
\(998\) 8.38059 0.265283
\(999\) 6.24698i 0.197646i
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.b.g.337.1 6
3.2 odd 2 1521.2.b.m.1351.6 6
13.2 odd 12 507.2.e.k.22.3 6
13.3 even 3 507.2.j.h.316.6 12
13.4 even 6 507.2.j.h.361.6 12
13.5 odd 4 507.2.a.j.1.1 3
13.6 odd 12 507.2.e.k.484.3 6
13.7 odd 12 507.2.e.j.484.1 6
13.8 odd 4 507.2.a.k.1.3 yes 3
13.9 even 3 507.2.j.h.361.1 12
13.10 even 6 507.2.j.h.316.1 12
13.11 odd 12 507.2.e.j.22.1 6
13.12 even 2 inner 507.2.b.g.337.6 6
39.5 even 4 1521.2.a.q.1.3 3
39.8 even 4 1521.2.a.p.1.1 3
39.38 odd 2 1521.2.b.m.1351.1 6
52.31 even 4 8112.2.a.by.1.2 3
52.47 even 4 8112.2.a.cf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.1 3 13.5 odd 4
507.2.a.k.1.3 yes 3 13.8 odd 4
507.2.b.g.337.1 6 1.1 even 1 trivial
507.2.b.g.337.6 6 13.12 even 2 inner
507.2.e.j.22.1 6 13.11 odd 12
507.2.e.j.484.1 6 13.7 odd 12
507.2.e.k.22.3 6 13.2 odd 12
507.2.e.k.484.3 6 13.6 odd 12
507.2.j.h.316.1 12 13.10 even 6
507.2.j.h.316.6 12 13.3 even 3
507.2.j.h.361.1 12 13.9 even 3
507.2.j.h.361.6 12 13.4 even 6
1521.2.a.p.1.1 3 39.8 even 4
1521.2.a.q.1.3 3 39.5 even 4
1521.2.b.m.1351.1 6 39.38 odd 2
1521.2.b.m.1351.6 6 3.2 odd 2
8112.2.a.by.1.2 3 52.31 even 4
8112.2.a.cf.1.2 3 52.47 even 4