Properties

Label 507.2.a.l.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 $1$

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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: \(3\)
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 \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.69202 q^{2} -1.00000 q^{3} +5.24698 q^{4} -1.04892 q^{5} -2.69202 q^{6} +0.554958 q^{7} +8.74094 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.69202 q^{2} -1.00000 q^{3} +5.24698 q^{4} -1.04892 q^{5} -2.69202 q^{6} +0.554958 q^{7} +8.74094 q^{8} +1.00000 q^{9} -2.82371 q^{10} +2.91185 q^{11} -5.24698 q^{12} +1.49396 q^{14} +1.04892 q^{15} +13.0368 q^{16} -4.85086 q^{17} +2.69202 q^{18} -0.753020 q^{19} -5.50365 q^{20} -0.554958 q^{21} +7.83877 q^{22} +5.76271 q^{23} -8.74094 q^{24} -3.89977 q^{25} -1.00000 q^{27} +2.91185 q^{28} -1.91185 q^{29} +2.82371 q^{30} -9.51573 q^{31} +17.6136 q^{32} -2.91185 q^{33} -13.0586 q^{34} -0.582105 q^{35} +5.24698 q^{36} +5.75302 q^{37} -2.02715 q^{38} -9.16852 q^{40} +4.91185 q^{41} -1.49396 q^{42} -11.0978 q^{43} +15.2784 q^{44} -1.04892 q^{45} +15.5133 q^{46} +0.753020 q^{47} -13.0368 q^{48} -6.69202 q^{49} -10.4983 q^{50} +4.85086 q^{51} -7.58211 q^{53} -2.69202 q^{54} -3.05429 q^{55} +4.85086 q^{56} +0.753020 q^{57} -5.14675 q^{58} +4.09783 q^{59} +5.50365 q^{60} -3.42327 q^{61} -25.6165 q^{62} +0.554958 q^{63} +21.3424 q^{64} -7.83877 q^{66} -1.87263 q^{67} -25.4523 q^{68} -5.76271 q^{69} -1.56704 q^{70} -10.5036 q^{71} +8.74094 q^{72} +10.4765 q^{73} +15.4873 q^{74} +3.89977 q^{75} -3.95108 q^{76} +1.61596 q^{77} +1.33513 q^{79} -13.6746 q^{80} +1.00000 q^{81} +13.2228 q^{82} +2.64310 q^{83} -2.91185 q^{84} +5.08815 q^{85} -29.8756 q^{86} +1.91185 q^{87} +25.4523 q^{88} +9.92692 q^{89} -2.82371 q^{90} +30.2368 q^{92} +9.51573 q^{93} +2.02715 q^{94} +0.789856 q^{95} -17.6136 q^{96} +17.0737 q^{97} -18.0151 q^{98} +2.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 11q^{4} + 6q^{5} - 3q^{6} + 2q^{7} + 12q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 11q^{4} + 6q^{5} - 3q^{6} + 2q^{7} + 12q^{8} + 3q^{9} - q^{10} + 5q^{11} - 11q^{12} - 5q^{14} - 6q^{15} + 11q^{16} - q^{17} + 3q^{18} - 7q^{19} + 15q^{20} - 2q^{21} - 9q^{22} - 12q^{24} + 11q^{25} - 3q^{27} + 5q^{28} - 2q^{29} + q^{30} - 16q^{31} + 22q^{32} - 5q^{33} - 8q^{34} + 4q^{35} + 11q^{36} + 22q^{37} + 3q^{40} + 11q^{41} + 5q^{42} - 15q^{43} + 16q^{44} + 6q^{45} - 7q^{46} + 7q^{47} - 11q^{48} - 15q^{49} - 3q^{50} + q^{51} - 17q^{53} - 3q^{54} + 3q^{55} + q^{56} + 7q^{57} + 12q^{58} - 6q^{59} - 15q^{60} - 13q^{61} - 2q^{62} + 2q^{63} + 9q^{66} + 11q^{67} - 13q^{68} - 24q^{70} + 12q^{72} + 6q^{73} + 15q^{74} - 11q^{75} - 21q^{76} + 15q^{77} + 3q^{79} - 20q^{80} + 3q^{81} - 3q^{82} + 12q^{83} - 5q^{84} + 19q^{85} - 29q^{86} + 2q^{87} + 13q^{88} + q^{89} - q^{90} + 7q^{92} + 16q^{93} - 21q^{95} - 22q^{96} - 5q^{97} - 29q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69202 1.90355 0.951773 0.306802i \(-0.0992590\pi\)
0.951773 + 0.306802i \(0.0992590\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.24698 2.62349
\(5\) −1.04892 −0.469090 −0.234545 0.972105i \(-0.575360\pi\)
−0.234545 + 0.972105i \(0.575360\pi\)
\(6\) −2.69202 −1.09901
\(7\) 0.554958 0.209754 0.104877 0.994485i \(-0.466555\pi\)
0.104877 + 0.994485i \(0.466555\pi\)
\(8\) 8.74094 3.09039
\(9\) 1.00000 0.333333
\(10\) −2.82371 −0.892935
\(11\) 2.91185 0.877957 0.438979 0.898498i \(-0.355340\pi\)
0.438979 + 0.898498i \(0.355340\pi\)
\(12\) −5.24698 −1.51467
\(13\) 0 0
\(14\) 1.49396 0.399277
\(15\) 1.04892 0.270829
\(16\) 13.0368 3.25921
\(17\) −4.85086 −1.17651 −0.588253 0.808677i \(-0.700184\pi\)
−0.588253 + 0.808677i \(0.700184\pi\)
\(18\) 2.69202 0.634516
\(19\) −0.753020 −0.172755 −0.0863774 0.996262i \(-0.527529\pi\)
−0.0863774 + 0.996262i \(0.527529\pi\)
\(20\) −5.50365 −1.23065
\(21\) −0.554958 −0.121102
\(22\) 7.83877 1.67123
\(23\) 5.76271 1.20161 0.600804 0.799396i \(-0.294847\pi\)
0.600804 + 0.799396i \(0.294847\pi\)
\(24\) −8.74094 −1.78424
\(25\) −3.89977 −0.779954
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.91185 0.550289
\(29\) −1.91185 −0.355022 −0.177511 0.984119i \(-0.556805\pi\)
−0.177511 + 0.984119i \(0.556805\pi\)
\(30\) 2.82371 0.515536
\(31\) −9.51573 −1.70908 −0.854538 0.519389i \(-0.826159\pi\)
−0.854538 + 0.519389i \(0.826159\pi\)
\(32\) 17.6136 3.11367
\(33\) −2.91185 −0.506889
\(34\) −13.0586 −2.23953
\(35\) −0.582105 −0.0983937
\(36\) 5.24698 0.874497
\(37\) 5.75302 0.945791 0.472895 0.881119i \(-0.343209\pi\)
0.472895 + 0.881119i \(0.343209\pi\)
\(38\) −2.02715 −0.328847
\(39\) 0 0
\(40\) −9.16852 −1.44967
\(41\) 4.91185 0.767103 0.383551 0.923520i \(-0.374701\pi\)
0.383551 + 0.923520i \(0.374701\pi\)
\(42\) −1.49396 −0.230523
\(43\) −11.0978 −1.69240 −0.846202 0.532862i \(-0.821116\pi\)
−0.846202 + 0.532862i \(0.821116\pi\)
\(44\) 15.2784 2.30331
\(45\) −1.04892 −0.156363
\(46\) 15.5133 2.28732
\(47\) 0.753020 0.109839 0.0549197 0.998491i \(-0.482510\pi\)
0.0549197 + 0.998491i \(0.482510\pi\)
\(48\) −13.0368 −1.88171
\(49\) −6.69202 −0.956003
\(50\) −10.4983 −1.48468
\(51\) 4.85086 0.679256
\(52\) 0 0
\(53\) −7.58211 −1.04148 −0.520741 0.853715i \(-0.674344\pi\)
−0.520741 + 0.853715i \(0.674344\pi\)
\(54\) −2.69202 −0.366338
\(55\) −3.05429 −0.411841
\(56\) 4.85086 0.648223
\(57\) 0.753020 0.0997400
\(58\) −5.14675 −0.675802
\(59\) 4.09783 0.533493 0.266746 0.963767i \(-0.414051\pi\)
0.266746 + 0.963767i \(0.414051\pi\)
\(60\) 5.50365 0.710518
\(61\) −3.42327 −0.438305 −0.219153 0.975691i \(-0.570329\pi\)
−0.219153 + 0.975691i \(0.570329\pi\)
\(62\) −25.6165 −3.25330
\(63\) 0.554958 0.0699182
\(64\) 21.3424 2.66780
\(65\) 0 0
\(66\) −7.83877 −0.964886
\(67\) −1.87263 −0.228778 −0.114389 0.993436i \(-0.536491\pi\)
−0.114389 + 0.993436i \(0.536491\pi\)
\(68\) −25.4523 −3.08655
\(69\) −5.76271 −0.693749
\(70\) −1.56704 −0.187297
\(71\) −10.5036 −1.24655 −0.623277 0.782001i \(-0.714199\pi\)
−0.623277 + 0.782001i \(0.714199\pi\)
\(72\) 8.74094 1.03013
\(73\) 10.4765 1.22618 0.613091 0.790012i \(-0.289926\pi\)
0.613091 + 0.790012i \(0.289926\pi\)
\(74\) 15.4873 1.80036
\(75\) 3.89977 0.450307
\(76\) −3.95108 −0.453220
\(77\) 1.61596 0.184155
\(78\) 0 0
\(79\) 1.33513 0.150213 0.0751067 0.997176i \(-0.476070\pi\)
0.0751067 + 0.997176i \(0.476070\pi\)
\(80\) −13.6746 −1.52886
\(81\) 1.00000 0.111111
\(82\) 13.2228 1.46022
\(83\) 2.64310 0.290118 0.145059 0.989423i \(-0.453663\pi\)
0.145059 + 0.989423i \(0.453663\pi\)
\(84\) −2.91185 −0.317709
\(85\) 5.08815 0.551887
\(86\) −29.8756 −3.22157
\(87\) 1.91185 0.204972
\(88\) 25.4523 2.71323
\(89\) 9.92692 1.05225 0.526126 0.850407i \(-0.323644\pi\)
0.526126 + 0.850407i \(0.323644\pi\)
\(90\) −2.82371 −0.297645
\(91\) 0 0
\(92\) 30.2368 3.15241
\(93\) 9.51573 0.986735
\(94\) 2.02715 0.209084
\(95\) 0.789856 0.0810375
\(96\) −17.6136 −1.79768
\(97\) 17.0737 1.73357 0.866784 0.498683i \(-0.166183\pi\)
0.866784 + 0.498683i \(0.166183\pi\)
\(98\) −18.0151 −1.81980
\(99\) 2.91185 0.292652
\(100\) −20.4620 −2.04620
\(101\) 7.32304 0.728670 0.364335 0.931268i \(-0.381296\pi\)
0.364335 + 0.931268i \(0.381296\pi\)
\(102\) 13.0586 1.29299
\(103\) −4.21983 −0.415792 −0.207896 0.978151i \(-0.566662\pi\)
−0.207896 + 0.978151i \(0.566662\pi\)
\(104\) 0 0
\(105\) 0.582105 0.0568077
\(106\) −20.4112 −1.98251
\(107\) 6.39373 0.618105 0.309053 0.951045i \(-0.399988\pi\)
0.309053 + 0.951045i \(0.399988\pi\)
\(108\) −5.24698 −0.504891
\(109\) −3.46011 −0.331418 −0.165709 0.986175i \(-0.552991\pi\)
−0.165709 + 0.986175i \(0.552991\pi\)
\(110\) −8.22223 −0.783958
\(111\) −5.75302 −0.546053
\(112\) 7.23490 0.683634
\(113\) 9.35690 0.880223 0.440111 0.897943i \(-0.354939\pi\)
0.440111 + 0.897943i \(0.354939\pi\)
\(114\) 2.02715 0.189860
\(115\) −6.04461 −0.563662
\(116\) −10.0315 −0.931398
\(117\) 0 0
\(118\) 11.0315 1.01553
\(119\) −2.69202 −0.246777
\(120\) 9.16852 0.836968
\(121\) −2.52111 −0.229191
\(122\) −9.21552 −0.834334
\(123\) −4.91185 −0.442887
\(124\) −49.9288 −4.48374
\(125\) 9.33513 0.834959
\(126\) 1.49396 0.133092
\(127\) −4.48188 −0.397702 −0.198851 0.980030i \(-0.563721\pi\)
−0.198851 + 0.980030i \(0.563721\pi\)
\(128\) 22.2271 1.96462
\(129\) 11.0978 0.977110
\(130\) 0 0
\(131\) −9.21744 −0.805331 −0.402666 0.915347i \(-0.631916\pi\)
−0.402666 + 0.915347i \(0.631916\pi\)
\(132\) −15.2784 −1.32982
\(133\) −0.417895 −0.0362361
\(134\) −5.04115 −0.435489
\(135\) 1.04892 0.0902764
\(136\) −42.4010 −3.63586
\(137\) −7.46980 −0.638188 −0.319094 0.947723i \(-0.603379\pi\)
−0.319094 + 0.947723i \(0.603379\pi\)
\(138\) −15.5133 −1.32058
\(139\) −17.9976 −1.52654 −0.763269 0.646081i \(-0.776407\pi\)
−0.763269 + 0.646081i \(0.776407\pi\)
\(140\) −3.05429 −0.258135
\(141\) −0.753020 −0.0634158
\(142\) −28.2760 −2.37287
\(143\) 0 0
\(144\) 13.0368 1.08640
\(145\) 2.00538 0.166537
\(146\) 28.2030 2.33409
\(147\) 6.69202 0.551949
\(148\) 30.1860 2.48127
\(149\) 15.3351 1.25630 0.628151 0.778091i \(-0.283812\pi\)
0.628151 + 0.778091i \(0.283812\pi\)
\(150\) 10.4983 0.857180
\(151\) 2.53079 0.205953 0.102977 0.994684i \(-0.467163\pi\)
0.102977 + 0.994684i \(0.467163\pi\)
\(152\) −6.58211 −0.533879
\(153\) −4.85086 −0.392168
\(154\) 4.35019 0.350548
\(155\) 9.98121 0.801710
\(156\) 0 0
\(157\) 17.2392 1.37584 0.687919 0.725787i \(-0.258524\pi\)
0.687919 + 0.725787i \(0.258524\pi\)
\(158\) 3.59419 0.285938
\(159\) 7.58211 0.601300
\(160\) −18.4752 −1.46059
\(161\) 3.19806 0.252043
\(162\) 2.69202 0.211505
\(163\) 15.7071 1.23027 0.615137 0.788420i \(-0.289101\pi\)
0.615137 + 0.788420i \(0.289101\pi\)
\(164\) 25.7724 2.01249
\(165\) 3.05429 0.237776
\(166\) 7.11529 0.552254
\(167\) 5.39612 0.417565 0.208782 0.977962i \(-0.433050\pi\)
0.208782 + 0.977962i \(0.433050\pi\)
\(168\) −4.85086 −0.374252
\(169\) 0 0
\(170\) 13.6974 1.05054
\(171\) −0.753020 −0.0575849
\(172\) −58.2301 −4.44000
\(173\) 23.9420 1.82028 0.910138 0.414306i \(-0.135976\pi\)
0.910138 + 0.414306i \(0.135976\pi\)
\(174\) 5.14675 0.390174
\(175\) −2.16421 −0.163599
\(176\) 37.9614 2.86145
\(177\) −4.09783 −0.308012
\(178\) 26.7235 2.00301
\(179\) 18.4088 1.37594 0.687969 0.725740i \(-0.258502\pi\)
0.687969 + 0.725740i \(0.258502\pi\)
\(180\) −5.50365 −0.410218
\(181\) −3.63342 −0.270070 −0.135035 0.990841i \(-0.543115\pi\)
−0.135035 + 0.990841i \(0.543115\pi\)
\(182\) 0 0
\(183\) 3.42327 0.253056
\(184\) 50.3715 3.71344
\(185\) −6.03444 −0.443661
\(186\) 25.6165 1.87830
\(187\) −14.1250 −1.03292
\(188\) 3.95108 0.288162
\(189\) −0.554958 −0.0403673
\(190\) 2.12631 0.154259
\(191\) −21.1782 −1.53240 −0.766201 0.642601i \(-0.777855\pi\)
−0.766201 + 0.642601i \(0.777855\pi\)
\(192\) −21.3424 −1.54026
\(193\) −17.6112 −1.26768 −0.633840 0.773464i \(-0.718522\pi\)
−0.633840 + 0.773464i \(0.718522\pi\)
\(194\) 45.9627 3.29993
\(195\) 0 0
\(196\) −35.1129 −2.50806
\(197\) −4.66248 −0.332188 −0.166094 0.986110i \(-0.553116\pi\)
−0.166094 + 0.986110i \(0.553116\pi\)
\(198\) 7.83877 0.557077
\(199\) 15.0368 1.06593 0.532967 0.846136i \(-0.321077\pi\)
0.532967 + 0.846136i \(0.321077\pi\)
\(200\) −34.0877 −2.41036
\(201\) 1.87263 0.132085
\(202\) 19.7138 1.38706
\(203\) −1.06100 −0.0744675
\(204\) 25.4523 1.78202
\(205\) −5.15213 −0.359840
\(206\) −11.3599 −0.791480
\(207\) 5.76271 0.400536
\(208\) 0 0
\(209\) −2.19269 −0.151671
\(210\) 1.56704 0.108136
\(211\) −0.460107 −0.0316751 −0.0158375 0.999875i \(-0.505041\pi\)
−0.0158375 + 0.999875i \(0.505041\pi\)
\(212\) −39.7832 −2.73232
\(213\) 10.5036 0.719698
\(214\) 17.2121 1.17659
\(215\) 11.6407 0.793890
\(216\) −8.74094 −0.594746
\(217\) −5.28083 −0.358486
\(218\) −9.31468 −0.630870
\(219\) −10.4765 −0.707936
\(220\) −16.0258 −1.08046
\(221\) 0 0
\(222\) −15.4873 −1.03944
\(223\) −16.3502 −1.09489 −0.547445 0.836842i \(-0.684399\pi\)
−0.547445 + 0.836842i \(0.684399\pi\)
\(224\) 9.77479 0.653106
\(225\) −3.89977 −0.259985
\(226\) 25.1890 1.67555
\(227\) 6.56033 0.435425 0.217712 0.976013i \(-0.430141\pi\)
0.217712 + 0.976013i \(0.430141\pi\)
\(228\) 3.95108 0.261667
\(229\) −3.95539 −0.261380 −0.130690 0.991423i \(-0.541719\pi\)
−0.130690 + 0.991423i \(0.541719\pi\)
\(230\) −16.2722 −1.07296
\(231\) −1.61596 −0.106322
\(232\) −16.7114 −1.09716
\(233\) 8.35690 0.547478 0.273739 0.961804i \(-0.411739\pi\)
0.273739 + 0.961804i \(0.411739\pi\)
\(234\) 0 0
\(235\) −0.789856 −0.0515245
\(236\) 21.5013 1.39961
\(237\) −1.33513 −0.0867257
\(238\) −7.24698 −0.469752
\(239\) 20.1008 1.30021 0.650107 0.759843i \(-0.274724\pi\)
0.650107 + 0.759843i \(0.274724\pi\)
\(240\) 13.6746 0.882689
\(241\) −19.0127 −1.22471 −0.612357 0.790581i \(-0.709778\pi\)
−0.612357 + 0.790581i \(0.709778\pi\)
\(242\) −6.78687 −0.436277
\(243\) −1.00000 −0.0641500
\(244\) −17.9618 −1.14989
\(245\) 7.01938 0.448452
\(246\) −13.2228 −0.843056
\(247\) 0 0
\(248\) −83.1764 −5.28171
\(249\) −2.64310 −0.167500
\(250\) 25.1304 1.58938
\(251\) −0.763774 −0.0482090 −0.0241045 0.999709i \(-0.507673\pi\)
−0.0241045 + 0.999709i \(0.507673\pi\)
\(252\) 2.91185 0.183430
\(253\) 16.7802 1.05496
\(254\) −12.0653 −0.757045
\(255\) −5.08815 −0.318632
\(256\) 17.1511 1.07194
\(257\) −13.0911 −0.816602 −0.408301 0.912847i \(-0.633879\pi\)
−0.408301 + 0.912847i \(0.633879\pi\)
\(258\) 29.8756 1.85997
\(259\) 3.19269 0.198384
\(260\) 0 0
\(261\) −1.91185 −0.118341
\(262\) −24.8135 −1.53299
\(263\) 18.3773 1.13320 0.566598 0.823995i \(-0.308259\pi\)
0.566598 + 0.823995i \(0.308259\pi\)
\(264\) −25.4523 −1.56648
\(265\) 7.95300 0.488549
\(266\) −1.12498 −0.0689771
\(267\) −9.92692 −0.607518
\(268\) −9.82563 −0.600196
\(269\) 23.6625 1.44273 0.721363 0.692557i \(-0.243516\pi\)
0.721363 + 0.692557i \(0.243516\pi\)
\(270\) 2.82371 0.171845
\(271\) −19.7530 −1.19991 −0.599955 0.800034i \(-0.704815\pi\)
−0.599955 + 0.800034i \(0.704815\pi\)
\(272\) −63.2398 −3.83448
\(273\) 0 0
\(274\) −20.1089 −1.21482
\(275\) −11.3556 −0.684767
\(276\) −30.2368 −1.82004
\(277\) 1.77777 0.106816 0.0534081 0.998573i \(-0.482992\pi\)
0.0534081 + 0.998573i \(0.482992\pi\)
\(278\) −48.4499 −2.90583
\(279\) −9.51573 −0.569692
\(280\) −5.08815 −0.304075
\(281\) 1.62133 0.0967207 0.0483603 0.998830i \(-0.484600\pi\)
0.0483603 + 0.998830i \(0.484600\pi\)
\(282\) −2.02715 −0.120715
\(283\) 4.98361 0.296245 0.148122 0.988969i \(-0.452677\pi\)
0.148122 + 0.988969i \(0.452677\pi\)
\(284\) −55.1124 −3.27032
\(285\) −0.789856 −0.0467870
\(286\) 0 0
\(287\) 2.72587 0.160903
\(288\) 17.6136 1.03789
\(289\) 6.53079 0.384164
\(290\) 5.39852 0.317012
\(291\) −17.0737 −1.00088
\(292\) 54.9700 3.21688
\(293\) −0.0717525 −0.00419183 −0.00209591 0.999998i \(-0.500667\pi\)
−0.00209591 + 0.999998i \(0.500667\pi\)
\(294\) 18.0151 1.05066
\(295\) −4.29829 −0.250256
\(296\) 50.2868 2.92286
\(297\) −2.91185 −0.168963
\(298\) 41.2825 2.39143
\(299\) 0 0
\(300\) 20.4620 1.18138
\(301\) −6.15883 −0.354989
\(302\) 6.81295 0.392041
\(303\) −7.32304 −0.420698
\(304\) −9.81700 −0.563044
\(305\) 3.59073 0.205605
\(306\) −13.0586 −0.746511
\(307\) −5.19806 −0.296669 −0.148335 0.988937i \(-0.547391\pi\)
−0.148335 + 0.988937i \(0.547391\pi\)
\(308\) 8.47889 0.483130
\(309\) 4.21983 0.240058
\(310\) 26.8696 1.52609
\(311\) −22.5429 −1.27829 −0.639145 0.769087i \(-0.720711\pi\)
−0.639145 + 0.769087i \(0.720711\pi\)
\(312\) 0 0
\(313\) 22.6612 1.28088 0.640442 0.768006i \(-0.278751\pi\)
0.640442 + 0.768006i \(0.278751\pi\)
\(314\) 46.4083 2.61897
\(315\) −0.582105 −0.0327979
\(316\) 7.00538 0.394083
\(317\) 26.3424 1.47954 0.739769 0.672861i \(-0.234935\pi\)
0.739769 + 0.672861i \(0.234935\pi\)
\(318\) 20.4112 1.14460
\(319\) −5.56704 −0.311694
\(320\) −22.3864 −1.25144
\(321\) −6.39373 −0.356863
\(322\) 8.60925 0.479775
\(323\) 3.65279 0.203247
\(324\) 5.24698 0.291499
\(325\) 0 0
\(326\) 42.2838 2.34188
\(327\) 3.46011 0.191344
\(328\) 42.9342 2.37065
\(329\) 0.417895 0.0230393
\(330\) 8.22223 0.452619
\(331\) −11.2295 −0.617230 −0.308615 0.951187i \(-0.599866\pi\)
−0.308615 + 0.951187i \(0.599866\pi\)
\(332\) 13.8683 0.761123
\(333\) 5.75302 0.315264
\(334\) 14.5265 0.794854
\(335\) 1.96423 0.107317
\(336\) −7.23490 −0.394696
\(337\) 2.30798 0.125724 0.0628618 0.998022i \(-0.479977\pi\)
0.0628618 + 0.998022i \(0.479977\pi\)
\(338\) 0 0
\(339\) −9.35690 −0.508197
\(340\) 26.6974 1.44787
\(341\) −27.7084 −1.50049
\(342\) −2.02715 −0.109616
\(343\) −7.59850 −0.410280
\(344\) −97.0055 −5.23019
\(345\) 6.04461 0.325431
\(346\) 64.4523 3.46498
\(347\) −9.13706 −0.490503 −0.245252 0.969459i \(-0.578871\pi\)
−0.245252 + 0.969459i \(0.578871\pi\)
\(348\) 10.0315 0.537743
\(349\) 23.9758 1.28340 0.641699 0.766957i \(-0.278230\pi\)
0.641699 + 0.766957i \(0.278230\pi\)
\(350\) −5.82610 −0.311418
\(351\) 0 0
\(352\) 51.2881 2.73367
\(353\) −27.1239 −1.44366 −0.721830 0.692070i \(-0.756699\pi\)
−0.721830 + 0.692070i \(0.756699\pi\)
\(354\) −11.0315 −0.586315
\(355\) 11.0175 0.584746
\(356\) 52.0863 2.76057
\(357\) 2.69202 0.142477
\(358\) 49.5569 2.61916
\(359\) −26.0790 −1.37640 −0.688200 0.725521i \(-0.741599\pi\)
−0.688200 + 0.725521i \(0.741599\pi\)
\(360\) −9.16852 −0.483224
\(361\) −18.4330 −0.970156
\(362\) −9.78123 −0.514090
\(363\) 2.52111 0.132324
\(364\) 0 0
\(365\) −10.9890 −0.575190
\(366\) 9.21552 0.481703
\(367\) 9.57434 0.499776 0.249888 0.968275i \(-0.419606\pi\)
0.249888 + 0.968275i \(0.419606\pi\)
\(368\) 75.1275 3.91629
\(369\) 4.91185 0.255701
\(370\) −16.2448 −0.844530
\(371\) −4.20775 −0.218456
\(372\) 49.9288 2.58869
\(373\) 28.1497 1.45754 0.728769 0.684760i \(-0.240093\pi\)
0.728769 + 0.684760i \(0.240093\pi\)
\(374\) −38.0248 −1.96621
\(375\) −9.33513 −0.482064
\(376\) 6.58211 0.339446
\(377\) 0 0
\(378\) −1.49396 −0.0768410
\(379\) 16.0465 0.824255 0.412127 0.911126i \(-0.364786\pi\)
0.412127 + 0.911126i \(0.364786\pi\)
\(380\) 4.14436 0.212601
\(381\) 4.48188 0.229614
\(382\) −57.0122 −2.91700
\(383\) −24.6165 −1.25785 −0.628923 0.777467i \(-0.716504\pi\)
−0.628923 + 0.777467i \(0.716504\pi\)
\(384\) −22.2271 −1.13427
\(385\) −1.69501 −0.0863855
\(386\) −47.4097 −2.41309
\(387\) −11.0978 −0.564135
\(388\) 89.5852 4.54800
\(389\) −17.2198 −0.873080 −0.436540 0.899685i \(-0.643796\pi\)
−0.436540 + 0.899685i \(0.643796\pi\)
\(390\) 0 0
\(391\) −27.9541 −1.41370
\(392\) −58.4946 −2.95442
\(393\) 9.21744 0.464958
\(394\) −12.5515 −0.632335
\(395\) −1.40044 −0.0704636
\(396\) 15.2784 0.767770
\(397\) −2.03923 −0.102346 −0.0511730 0.998690i \(-0.516296\pi\)
−0.0511730 + 0.998690i \(0.516296\pi\)
\(398\) 40.4795 2.02905
\(399\) 0.417895 0.0209209
\(400\) −50.8407 −2.54203
\(401\) −1.46144 −0.0729806 −0.0364903 0.999334i \(-0.511618\pi\)
−0.0364903 + 0.999334i \(0.511618\pi\)
\(402\) 5.04115 0.251430
\(403\) 0 0
\(404\) 38.4239 1.91166
\(405\) −1.04892 −0.0521211
\(406\) −2.85623 −0.141752
\(407\) 16.7520 0.830364
\(408\) 42.4010 2.09916
\(409\) −29.9390 −1.48039 −0.740194 0.672393i \(-0.765266\pi\)
−0.740194 + 0.672393i \(0.765266\pi\)
\(410\) −13.8696 −0.684973
\(411\) 7.46980 0.368458
\(412\) −22.1414 −1.09083
\(413\) 2.27413 0.111902
\(414\) 15.5133 0.762439
\(415\) −2.77240 −0.136092
\(416\) 0 0
\(417\) 17.9976 0.881347
\(418\) −5.90276 −0.288713
\(419\) 6.64742 0.324748 0.162374 0.986729i \(-0.448085\pi\)
0.162374 + 0.986729i \(0.448085\pi\)
\(420\) 3.05429 0.149034
\(421\) 13.5646 0.661100 0.330550 0.943788i \(-0.392766\pi\)
0.330550 + 0.943788i \(0.392766\pi\)
\(422\) −1.23862 −0.0602950
\(423\) 0.753020 0.0366131
\(424\) −66.2747 −3.21858
\(425\) 18.9172 0.917620
\(426\) 28.2760 1.36998
\(427\) −1.89977 −0.0919364
\(428\) 33.5478 1.62159
\(429\) 0 0
\(430\) 31.3370 1.51121
\(431\) 35.9463 1.73147 0.865736 0.500501i \(-0.166851\pi\)
0.865736 + 0.500501i \(0.166851\pi\)
\(432\) −13.0368 −0.627235
\(433\) −32.4741 −1.56061 −0.780303 0.625402i \(-0.784935\pi\)
−0.780303 + 0.625402i \(0.784935\pi\)
\(434\) −14.2161 −0.682395
\(435\) −2.00538 −0.0961505
\(436\) −18.1551 −0.869472
\(437\) −4.33944 −0.207583
\(438\) −28.2030 −1.34759
\(439\) 12.8321 0.612441 0.306221 0.951961i \(-0.400935\pi\)
0.306221 + 0.951961i \(0.400935\pi\)
\(440\) −26.6974 −1.27275
\(441\) −6.69202 −0.318668
\(442\) 0 0
\(443\) 11.9608 0.568273 0.284137 0.958784i \(-0.408293\pi\)
0.284137 + 0.958784i \(0.408293\pi\)
\(444\) −30.1860 −1.43256
\(445\) −10.4125 −0.493601
\(446\) −44.0151 −2.08417
\(447\) −15.3351 −0.725327
\(448\) 11.8442 0.559584
\(449\) −12.4789 −0.588915 −0.294458 0.955665i \(-0.595139\pi\)
−0.294458 + 0.955665i \(0.595139\pi\)
\(450\) −10.4983 −0.494893
\(451\) 14.3026 0.673483
\(452\) 49.0954 2.30926
\(453\) −2.53079 −0.118907
\(454\) 17.6606 0.828851
\(455\) 0 0
\(456\) 6.58211 0.308235
\(457\) 32.4523 1.51806 0.759028 0.651058i \(-0.225674\pi\)
0.759028 + 0.651058i \(0.225674\pi\)
\(458\) −10.6480 −0.497549
\(459\) 4.85086 0.226419
\(460\) −31.7159 −1.47876
\(461\) 24.4034 1.13658 0.568290 0.822828i \(-0.307605\pi\)
0.568290 + 0.822828i \(0.307605\pi\)
\(462\) −4.35019 −0.202389
\(463\) −33.1836 −1.54217 −0.771086 0.636731i \(-0.780286\pi\)
−0.771086 + 0.636731i \(0.780286\pi\)
\(464\) −24.9245 −1.15709
\(465\) −9.98121 −0.462868
\(466\) 22.4969 1.04215
\(467\) −38.5206 −1.78252 −0.891261 0.453490i \(-0.850179\pi\)
−0.891261 + 0.453490i \(0.850179\pi\)
\(468\) 0 0
\(469\) −1.03923 −0.0479871
\(470\) −2.12631 −0.0980794
\(471\) −17.2392 −0.794341
\(472\) 35.8189 1.64870
\(473\) −32.3153 −1.48586
\(474\) −3.59419 −0.165086
\(475\) 2.93661 0.134741
\(476\) −14.1250 −0.647417
\(477\) −7.58211 −0.347161
\(478\) 54.1118 2.47502
\(479\) −8.34481 −0.381284 −0.190642 0.981660i \(-0.561057\pi\)
−0.190642 + 0.981660i \(0.561057\pi\)
\(480\) 18.4752 0.843272
\(481\) 0 0
\(482\) −51.1825 −2.33130
\(483\) −3.19806 −0.145517
\(484\) −13.2282 −0.601282
\(485\) −17.9089 −0.813200
\(486\) −2.69202 −0.122113
\(487\) −14.8586 −0.673309 −0.336654 0.941628i \(-0.609295\pi\)
−0.336654 + 0.941628i \(0.609295\pi\)
\(488\) −29.9226 −1.35453
\(489\) −15.7071 −0.710299
\(490\) 18.8963 0.853648
\(491\) 4.99894 0.225599 0.112799 0.993618i \(-0.464018\pi\)
0.112799 + 0.993618i \(0.464018\pi\)
\(492\) −25.7724 −1.16191
\(493\) 9.27413 0.417686
\(494\) 0 0
\(495\) −3.05429 −0.137280
\(496\) −124.055 −5.57023
\(497\) −5.82908 −0.261470
\(498\) −7.11529 −0.318844
\(499\) −0.385371 −0.0172516 −0.00862579 0.999963i \(-0.502746\pi\)
−0.00862579 + 0.999963i \(0.502746\pi\)
\(500\) 48.9812 2.19051
\(501\) −5.39612 −0.241081
\(502\) −2.05610 −0.0917681
\(503\) 22.6179 1.00848 0.504241 0.863563i \(-0.331772\pi\)
0.504241 + 0.863563i \(0.331772\pi\)
\(504\) 4.85086 0.216074
\(505\) −7.68127 −0.341812
\(506\) 45.1726 2.00817
\(507\) 0 0
\(508\) −23.5163 −1.04337
\(509\) −11.6039 −0.514333 −0.257166 0.966367i \(-0.582789\pi\)
−0.257166 + 0.966367i \(0.582789\pi\)
\(510\) −13.6974 −0.606531
\(511\) 5.81402 0.257197
\(512\) 1.71678 0.0758715
\(513\) 0.753020 0.0332467
\(514\) −35.2416 −1.55444
\(515\) 4.42626 0.195044
\(516\) 58.2301 2.56344
\(517\) 2.19269 0.0964342
\(518\) 8.59478 0.377633
\(519\) −23.9420 −1.05094
\(520\) 0 0
\(521\) −1.62671 −0.0712675 −0.0356337 0.999365i \(-0.511345\pi\)
−0.0356337 + 0.999365i \(0.511345\pi\)
\(522\) −5.14675 −0.225267
\(523\) 10.0718 0.440407 0.220203 0.975454i \(-0.429328\pi\)
0.220203 + 0.975454i \(0.429328\pi\)
\(524\) −48.3637 −2.11278
\(525\) 2.16421 0.0944539
\(526\) 49.4722 2.15709
\(527\) 46.1594 2.01074
\(528\) −37.9614 −1.65206
\(529\) 10.2088 0.443862
\(530\) 21.4097 0.929976
\(531\) 4.09783 0.177831
\(532\) −2.19269 −0.0950650
\(533\) 0 0
\(534\) −26.7235 −1.15644
\(535\) −6.70650 −0.289947
\(536\) −16.3685 −0.707012
\(537\) −18.4088 −0.794398
\(538\) 63.6999 2.74630
\(539\) −19.4862 −0.839330
\(540\) 5.50365 0.236839
\(541\) −20.4674 −0.879962 −0.439981 0.898007i \(-0.645015\pi\)
−0.439981 + 0.898007i \(0.645015\pi\)
\(542\) −53.1756 −2.28409
\(543\) 3.63342 0.155925
\(544\) −85.4408 −3.66325
\(545\) 3.62937 0.155465
\(546\) 0 0
\(547\) −27.5478 −1.17786 −0.588929 0.808185i \(-0.700450\pi\)
−0.588929 + 0.808185i \(0.700450\pi\)
\(548\) −39.1939 −1.67428
\(549\) −3.42327 −0.146102
\(550\) −30.5694 −1.30348
\(551\) 1.43967 0.0613318
\(552\) −50.3715 −2.14395
\(553\) 0.740939 0.0315079
\(554\) 4.78581 0.203329
\(555\) 6.03444 0.256148
\(556\) −94.4331 −4.00485
\(557\) 37.9855 1.60950 0.804749 0.593615i \(-0.202300\pi\)
0.804749 + 0.593615i \(0.202300\pi\)
\(558\) −25.6165 −1.08443
\(559\) 0 0
\(560\) −7.58881 −0.320686
\(561\) 14.1250 0.596357
\(562\) 4.36467 0.184112
\(563\) −29.7724 −1.25476 −0.627378 0.778714i \(-0.715872\pi\)
−0.627378 + 0.778714i \(0.715872\pi\)
\(564\) −3.95108 −0.166371
\(565\) −9.81461 −0.412904
\(566\) 13.4160 0.563916
\(567\) 0.554958 0.0233061
\(568\) −91.8117 −3.85234
\(569\) −21.9541 −0.920362 −0.460181 0.887825i \(-0.652216\pi\)
−0.460181 + 0.887825i \(0.652216\pi\)
\(570\) −2.12631 −0.0890613
\(571\) 2.46575 0.103188 0.0515942 0.998668i \(-0.483570\pi\)
0.0515942 + 0.998668i \(0.483570\pi\)
\(572\) 0 0
\(573\) 21.1782 0.884732
\(574\) 7.33811 0.306287
\(575\) −22.4733 −0.937199
\(576\) 21.3424 0.889268
\(577\) 17.4547 0.726650 0.363325 0.931662i \(-0.381641\pi\)
0.363325 + 0.931662i \(0.381641\pi\)
\(578\) 17.5810 0.731275
\(579\) 17.6112 0.731895
\(580\) 10.5222 0.436909
\(581\) 1.46681 0.0608536
\(582\) −45.9627 −1.90521
\(583\) −22.0780 −0.914377
\(584\) 91.5745 3.78938
\(585\) 0 0
\(586\) −0.193159 −0.00797934
\(587\) 6.26337 0.258517 0.129259 0.991611i \(-0.458740\pi\)
0.129259 + 0.991611i \(0.458740\pi\)
\(588\) 35.1129 1.44803
\(589\) 7.16554 0.295251
\(590\) −11.5711 −0.476374
\(591\) 4.66248 0.191789
\(592\) 75.0012 3.08253
\(593\) 22.8745 0.939345 0.469672 0.882841i \(-0.344372\pi\)
0.469672 + 0.882841i \(0.344372\pi\)
\(594\) −7.83877 −0.321629
\(595\) 2.82371 0.115761
\(596\) 80.4631 3.29590
\(597\) −15.0368 −0.615417
\(598\) 0 0
\(599\) −1.05621 −0.0431557 −0.0215778 0.999767i \(-0.506869\pi\)
−0.0215778 + 0.999767i \(0.506869\pi\)
\(600\) 34.0877 1.39162
\(601\) −33.3236 −1.35930 −0.679650 0.733537i \(-0.737868\pi\)
−0.679650 + 0.733537i \(0.737868\pi\)
\(602\) −16.5797 −0.675739
\(603\) −1.87263 −0.0762592
\(604\) 13.2790 0.540316
\(605\) 2.64443 0.107511
\(606\) −19.7138 −0.800818
\(607\) 16.2403 0.659172 0.329586 0.944125i \(-0.393091\pi\)
0.329586 + 0.944125i \(0.393091\pi\)
\(608\) −13.2634 −0.537901
\(609\) 1.06100 0.0429938
\(610\) 9.66632 0.391378
\(611\) 0 0
\(612\) −25.4523 −1.02885
\(613\) 11.0479 0.446219 0.223109 0.974793i \(-0.428379\pi\)
0.223109 + 0.974793i \(0.428379\pi\)
\(614\) −13.9933 −0.564723
\(615\) 5.15213 0.207754
\(616\) 14.1250 0.569112
\(617\) −4.65950 −0.187584 −0.0937922 0.995592i \(-0.529899\pi\)
−0.0937922 + 0.995592i \(0.529899\pi\)
\(618\) 11.3599 0.456961
\(619\) 31.9259 1.28321 0.641604 0.767036i \(-0.278269\pi\)
0.641604 + 0.767036i \(0.278269\pi\)
\(620\) 52.3712 2.10328
\(621\) −5.76271 −0.231250
\(622\) −60.6859 −2.43328
\(623\) 5.50902 0.220714
\(624\) 0 0
\(625\) 9.70709 0.388283
\(626\) 61.0043 2.43822
\(627\) 2.19269 0.0875674
\(628\) 90.4538 3.60950
\(629\) −27.9071 −1.11273
\(630\) −1.56704 −0.0624324
\(631\) −39.4413 −1.57013 −0.785067 0.619411i \(-0.787372\pi\)
−0.785067 + 0.619411i \(0.787372\pi\)
\(632\) 11.6703 0.464218
\(633\) 0.460107 0.0182876
\(634\) 70.9144 2.81637
\(635\) 4.70112 0.186558
\(636\) 39.7832 1.57750
\(637\) 0 0
\(638\) −14.9866 −0.593325
\(639\) −10.5036 −0.415518
\(640\) −23.3144 −0.921583
\(641\) 45.4510 1.79521 0.897603 0.440804i \(-0.145307\pi\)
0.897603 + 0.440804i \(0.145307\pi\)
\(642\) −17.2121 −0.679306
\(643\) −29.7469 −1.17310 −0.586552 0.809912i \(-0.699515\pi\)
−0.586552 + 0.809912i \(0.699515\pi\)
\(644\) 16.7802 0.661231
\(645\) −11.6407 −0.458353
\(646\) 9.83340 0.386890
\(647\) −16.9312 −0.665635 −0.332818 0.942991i \(-0.607999\pi\)
−0.332818 + 0.942991i \(0.607999\pi\)
\(648\) 8.74094 0.343377
\(649\) 11.9323 0.468384
\(650\) 0 0
\(651\) 5.28083 0.206972
\(652\) 82.4148 3.22761
\(653\) −33.1976 −1.29912 −0.649561 0.760309i \(-0.725047\pi\)
−0.649561 + 0.760309i \(0.725047\pi\)
\(654\) 9.31468 0.364233
\(655\) 9.66833 0.377773
\(656\) 64.0350 2.50015
\(657\) 10.4765 0.408727
\(658\) 1.12498 0.0438564
\(659\) −42.6571 −1.66168 −0.830842 0.556508i \(-0.812141\pi\)
−0.830842 + 0.556508i \(0.812141\pi\)
\(660\) 16.0258 0.623804
\(661\) 38.6902 1.50488 0.752438 0.658664i \(-0.228878\pi\)
0.752438 + 0.658664i \(0.228878\pi\)
\(662\) −30.2301 −1.17493
\(663\) 0 0
\(664\) 23.1032 0.896578
\(665\) 0.438337 0.0169980
\(666\) 15.4873 0.600119
\(667\) −11.0175 −0.426598
\(668\) 28.3134 1.09548
\(669\) 16.3502 0.632135
\(670\) 5.28775 0.204283
\(671\) −9.96807 −0.384813
\(672\) −9.77479 −0.377071
\(673\) −2.59419 −0.0999986 −0.0499993 0.998749i \(-0.515922\pi\)
−0.0499993 + 0.998749i \(0.515922\pi\)
\(674\) 6.21313 0.239321
\(675\) 3.89977 0.150102
\(676\) 0 0
\(677\) 1.75302 0.0673740 0.0336870 0.999432i \(-0.489275\pi\)
0.0336870 + 0.999432i \(0.489275\pi\)
\(678\) −25.1890 −0.967376
\(679\) 9.47517 0.363624
\(680\) 44.4752 1.70555
\(681\) −6.56033 −0.251393
\(682\) −74.5916 −2.85626
\(683\) 16.3351 0.625046 0.312523 0.949910i \(-0.398826\pi\)
0.312523 + 0.949910i \(0.398826\pi\)
\(684\) −3.95108 −0.151073
\(685\) 7.83520 0.299368
\(686\) −20.4553 −0.780988
\(687\) 3.95539 0.150908
\(688\) −144.681 −5.51590
\(689\) 0 0
\(690\) 16.2722 0.619472
\(691\) 15.9105 0.605265 0.302632 0.953107i \(-0.402135\pi\)
0.302632 + 0.953107i \(0.402135\pi\)
\(692\) 125.623 4.77547
\(693\) 1.61596 0.0613851
\(694\) −24.5972 −0.933696
\(695\) 18.8780 0.716083
\(696\) 16.7114 0.633444
\(697\) −23.8267 −0.902500
\(698\) 64.5435 2.44301
\(699\) −8.35690 −0.316087
\(700\) −11.3556 −0.429200
\(701\) −20.8635 −0.788005 −0.394002 0.919109i \(-0.628910\pi\)
−0.394002 + 0.919109i \(0.628910\pi\)
\(702\) 0 0
\(703\) −4.33214 −0.163390
\(704\) 62.1460 2.34222
\(705\) 0.789856 0.0297477
\(706\) −73.0182 −2.74807
\(707\) 4.06398 0.152842
\(708\) −21.5013 −0.808067
\(709\) 15.6485 0.587691 0.293846 0.955853i \(-0.405065\pi\)
0.293846 + 0.955853i \(0.405065\pi\)
\(710\) 29.6592 1.11309
\(711\) 1.33513 0.0500711
\(712\) 86.7706 3.25187
\(713\) −54.8364 −2.05364
\(714\) 7.24698 0.271211
\(715\) 0 0
\(716\) 96.5906 3.60976
\(717\) −20.1008 −0.750679
\(718\) −70.2054 −2.62004
\(719\) −27.1594 −1.01288 −0.506438 0.862276i \(-0.669038\pi\)
−0.506438 + 0.862276i \(0.669038\pi\)
\(720\) −13.6746 −0.509621
\(721\) −2.34183 −0.0872143
\(722\) −49.6219 −1.84674
\(723\) 19.0127 0.707089
\(724\) −19.0645 −0.708525
\(725\) 7.45580 0.276901
\(726\) 6.78687 0.251884
\(727\) 31.7784 1.17859 0.589297 0.807916i \(-0.299405\pi\)
0.589297 + 0.807916i \(0.299405\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −29.5826 −1.09490
\(731\) 53.8340 1.99112
\(732\) 17.9618 0.663889
\(733\) 46.8907 1.73195 0.865973 0.500090i \(-0.166700\pi\)
0.865973 + 0.500090i \(0.166700\pi\)
\(734\) 25.7743 0.951347
\(735\) −7.01938 −0.258914
\(736\) 101.502 3.74141
\(737\) −5.45281 −0.200857
\(738\) 13.2228 0.486739
\(739\) 17.0479 0.627115 0.313558 0.949569i \(-0.398479\pi\)
0.313558 + 0.949569i \(0.398479\pi\)
\(740\) −31.6626 −1.16394
\(741\) 0 0
\(742\) −11.3274 −0.415840
\(743\) −11.6324 −0.426750 −0.213375 0.976970i \(-0.568446\pi\)
−0.213375 + 0.976970i \(0.568446\pi\)
\(744\) 83.1764 3.04940
\(745\) −16.0853 −0.589319
\(746\) 75.7797 2.77449
\(747\) 2.64310 0.0967061
\(748\) −74.1135 −2.70986
\(749\) 3.54825 0.129650
\(750\) −25.1304 −0.917631
\(751\) −13.0295 −0.475455 −0.237727 0.971332i \(-0.576402\pi\)
−0.237727 + 0.971332i \(0.576402\pi\)
\(752\) 9.81700 0.357989
\(753\) 0.763774 0.0278335
\(754\) 0 0
\(755\) −2.65459 −0.0966106
\(756\) −2.91185 −0.105903
\(757\) 22.7899 0.828311 0.414156 0.910206i \(-0.364077\pi\)
0.414156 + 0.910206i \(0.364077\pi\)
\(758\) 43.1976 1.56901
\(759\) −16.7802 −0.609081
\(760\) 6.90408 0.250437
\(761\) −38.3424 −1.38991 −0.694956 0.719052i \(-0.744576\pi\)
−0.694956 + 0.719052i \(0.744576\pi\)
\(762\) 12.0653 0.437080
\(763\) −1.92021 −0.0695164
\(764\) −111.122 −4.02024
\(765\) 5.08815 0.183962
\(766\) −66.2683 −2.39437
\(767\) 0 0
\(768\) −17.1511 −0.618886
\(769\) −3.63879 −0.131218 −0.0656091 0.997845i \(-0.520899\pi\)
−0.0656091 + 0.997845i \(0.520899\pi\)
\(770\) −4.56299 −0.164439
\(771\) 13.0911 0.471466
\(772\) −92.4055 −3.32575
\(773\) −39.3424 −1.41505 −0.707524 0.706689i \(-0.750188\pi\)
−0.707524 + 0.706689i \(0.750188\pi\)
\(774\) −29.8756 −1.07386
\(775\) 37.1092 1.33300
\(776\) 149.240 5.35740
\(777\) −3.19269 −0.114537
\(778\) −46.3562 −1.66195
\(779\) −3.69873 −0.132521
\(780\) 0 0
\(781\) −30.5851 −1.09442
\(782\) −75.2529 −2.69104
\(783\) 1.91185 0.0683241
\(784\) −87.2428 −3.11581
\(785\) −18.0825 −0.645392
\(786\) 24.8135 0.885070
\(787\) −25.4252 −0.906310 −0.453155 0.891432i \(-0.649702\pi\)
−0.453155 + 0.891432i \(0.649702\pi\)
\(788\) −24.4639 −0.871492
\(789\) −18.3773 −0.654251
\(790\) −3.77000 −0.134131
\(791\) 5.19269 0.184631
\(792\) 25.4523 0.904409
\(793\) 0 0
\(794\) −5.48965 −0.194820
\(795\) −7.95300 −0.282064
\(796\) 78.8980 2.79646
\(797\) 20.7138 0.733720 0.366860 0.930276i \(-0.380433\pi\)
0.366860 + 0.930276i \(0.380433\pi\)
\(798\) 1.12498 0.0398239
\(799\) −3.65279 −0.129227
\(800\) −68.6889 −2.42852
\(801\) 9.92692 0.350750
\(802\) −3.93422 −0.138922
\(803\) 30.5060 1.07653
\(804\) 9.82563 0.346523
\(805\) −3.35450 −0.118231
\(806\) 0 0
\(807\) −23.6625 −0.832959
\(808\) 64.0103 2.25187
\(809\) 11.0978 0.390179 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(810\) −2.82371 −0.0992150
\(811\) −4.84223 −0.170034 −0.0850169 0.996380i \(-0.527094\pi\)
−0.0850169 + 0.996380i \(0.527094\pi\)
\(812\) −5.56704 −0.195365
\(813\) 19.7530 0.692769
\(814\) 45.0966 1.58064
\(815\) −16.4754 −0.577109
\(816\) 63.2398 2.21384
\(817\) 8.35690 0.292371
\(818\) −80.5964 −2.81799
\(819\) 0 0
\(820\) −27.0331 −0.944037
\(821\) −43.7381 −1.52647 −0.763235 0.646121i \(-0.776390\pi\)
−0.763235 + 0.646121i \(0.776390\pi\)
\(822\) 20.1089 0.701377
\(823\) 12.0998 0.421771 0.210885 0.977511i \(-0.432365\pi\)
0.210885 + 0.977511i \(0.432365\pi\)
\(824\) −36.8853 −1.28496
\(825\) 11.3556 0.395350
\(826\) 6.12200 0.213012
\(827\) −35.3212 −1.22824 −0.614120 0.789213i \(-0.710489\pi\)
−0.614120 + 0.789213i \(0.710489\pi\)
\(828\) 30.2368 1.05080
\(829\) 53.4946 1.85794 0.928971 0.370152i \(-0.120694\pi\)
0.928971 + 0.370152i \(0.120694\pi\)
\(830\) −7.46335 −0.259057
\(831\) −1.77777 −0.0616703
\(832\) 0 0
\(833\) 32.4620 1.12474
\(834\) 48.4499 1.67768
\(835\) −5.66009 −0.195875
\(836\) −11.5050 −0.397908
\(837\) 9.51573 0.328912
\(838\) 17.8950 0.618172
\(839\) 23.1521 0.799300 0.399650 0.916668i \(-0.369132\pi\)
0.399650 + 0.916668i \(0.369132\pi\)
\(840\) 5.08815 0.175558
\(841\) −25.3448 −0.873959
\(842\) 36.5163 1.25844
\(843\) −1.62133 −0.0558417
\(844\) −2.41417 −0.0830993
\(845\) 0 0
\(846\) 2.02715 0.0696948
\(847\) −1.39911 −0.0480739
\(848\) −98.8467 −3.39441
\(849\) −4.98361 −0.171037
\(850\) 50.9256 1.74673
\(851\) 33.1530 1.13647
\(852\) 55.1124 1.88812
\(853\) 26.7265 0.915097 0.457548 0.889185i \(-0.348728\pi\)
0.457548 + 0.889185i \(0.348728\pi\)
\(854\) −5.11423 −0.175005
\(855\) 0.789856 0.0270125
\(856\) 55.8872 1.91019
\(857\) 42.6064 1.45541 0.727703 0.685892i \(-0.240588\pi\)
0.727703 + 0.685892i \(0.240588\pi\)
\(858\) 0 0
\(859\) 33.6079 1.14669 0.573344 0.819315i \(-0.305646\pi\)
0.573344 + 0.819315i \(0.305646\pi\)
\(860\) 61.0786 2.08276
\(861\) −2.72587 −0.0928975
\(862\) 96.7682 3.29594
\(863\) 18.7047 0.636715 0.318358 0.947971i \(-0.396869\pi\)
0.318358 + 0.947971i \(0.396869\pi\)
\(864\) −17.6136 −0.599226
\(865\) −25.1132 −0.853873
\(866\) −87.4210 −2.97069
\(867\) −6.53079 −0.221797
\(868\) −27.7084 −0.940485
\(869\) 3.88769 0.131881
\(870\) −5.39852 −0.183027
\(871\) 0 0
\(872\) −30.2446 −1.02421
\(873\) 17.0737 0.577856
\(874\) −11.6819 −0.395145
\(875\) 5.18060 0.175136
\(876\) −54.9700 −1.85726
\(877\) 36.8237 1.24345 0.621724 0.783236i \(-0.286433\pi\)
0.621724 + 0.783236i \(0.286433\pi\)
\(878\) 34.5442 1.16581
\(879\) 0.0717525 0.00242015
\(880\) −39.8183 −1.34228
\(881\) −41.1250 −1.38554 −0.692768 0.721161i \(-0.743609\pi\)
−0.692768 + 0.721161i \(0.743609\pi\)
\(882\) −18.0151 −0.606599
\(883\) 30.7482 1.03476 0.517380 0.855756i \(-0.326907\pi\)
0.517380 + 0.855756i \(0.326907\pi\)
\(884\) 0 0
\(885\) 4.29829 0.144485
\(886\) 32.1987 1.08173
\(887\) 7.58940 0.254827 0.127414 0.991850i \(-0.459332\pi\)
0.127414 + 0.991850i \(0.459332\pi\)
\(888\) −50.2868 −1.68751
\(889\) −2.48725 −0.0834198
\(890\) −28.0307 −0.939592
\(891\) 2.91185 0.0975508
\(892\) −85.7891 −2.87243
\(893\) −0.567040 −0.0189753
\(894\) −41.2825 −1.38069
\(895\) −19.3093 −0.645439
\(896\) 12.3351 0.412088
\(897\) 0 0
\(898\) −33.5934 −1.12103
\(899\) 18.1927 0.606760
\(900\) −20.4620 −0.682068
\(901\) 36.7797 1.22531
\(902\) 38.5029 1.28201
\(903\) 6.15883 0.204953
\(904\) 81.7881 2.72023
\(905\) 3.81115 0.126687
\(906\) −6.81295 −0.226345
\(907\) 19.3333 0.641952 0.320976 0.947087i \(-0.395989\pi\)
0.320976 + 0.947087i \(0.395989\pi\)
\(908\) 34.4219 1.14233
\(909\) 7.32304 0.242890
\(910\) 0 0
\(911\) 22.7149 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(912\) 9.81700 0.325073
\(913\) 7.69633 0.254711
\(914\) 87.3624 2.88969
\(915\) −3.59073 −0.118706
\(916\) −20.7539 −0.685727
\(917\) −5.11529 −0.168922
\(918\) 13.0586 0.430998
\(919\) 14.2911 0.471420 0.235710 0.971823i \(-0.424258\pi\)
0.235710 + 0.971823i \(0.424258\pi\)
\(920\) −52.8355 −1.74194
\(921\) 5.19806 0.171282
\(922\) 65.6945 2.16353
\(923\) 0 0
\(924\) −8.47889 −0.278935
\(925\) −22.4355 −0.737674
\(926\) −89.3309 −2.93560
\(927\) −4.21983 −0.138597
\(928\) −33.6746 −1.10542
\(929\) 32.7211 1.07354 0.536772 0.843727i \(-0.319644\pi\)
0.536772 + 0.843727i \(0.319644\pi\)
\(930\) −26.8696 −0.881090
\(931\) 5.03923 0.165154
\(932\) 43.8485 1.43630
\(933\) 22.5429 0.738021
\(934\) −103.698 −3.39311
\(935\) 14.8159 0.484533
\(936\) 0 0
\(937\) −4.01400 −0.131132 −0.0655658 0.997848i \(-0.520885\pi\)
−0.0655658 + 0.997848i \(0.520885\pi\)
\(938\) −2.79763 −0.0913457
\(939\) −22.6612 −0.739519
\(940\) −4.14436 −0.135174
\(941\) −28.2669 −0.921476 −0.460738 0.887536i \(-0.652415\pi\)
−0.460738 + 0.887536i \(0.652415\pi\)
\(942\) −46.4083 −1.51206
\(943\) 28.3056 0.921757
\(944\) 53.4228 1.73876
\(945\) 0.582105 0.0189359
\(946\) −86.9934 −2.82840
\(947\) −37.8455 −1.22981 −0.614906 0.788600i \(-0.710806\pi\)
−0.614906 + 0.788600i \(0.710806\pi\)
\(948\) −7.00538 −0.227524
\(949\) 0 0
\(950\) 7.90541 0.256485
\(951\) −26.3424 −0.854212
\(952\) −23.5308 −0.762637
\(953\) −40.8256 −1.32247 −0.661236 0.750178i \(-0.729968\pi\)
−0.661236 + 0.750178i \(0.729968\pi\)
\(954\) −20.4112 −0.660837
\(955\) 22.2142 0.718834
\(956\) 105.469 3.41110
\(957\) 5.56704 0.179957
\(958\) −22.4644 −0.725792
\(959\) −4.14542 −0.133863
\(960\) 22.3864 0.722519
\(961\) 59.5491 1.92094
\(962\) 0 0
\(963\) 6.39373 0.206035
\(964\) −99.7591 −3.21302
\(965\) 18.4727 0.594656
\(966\) −8.60925 −0.276998
\(967\) 10.2798 0.330575 0.165288 0.986245i \(-0.447145\pi\)
0.165288 + 0.986245i \(0.447145\pi\)
\(968\) −22.0368 −0.708291
\(969\) −3.65279 −0.117345
\(970\) −48.2111 −1.54796
\(971\) −19.4805 −0.625161 −0.312580 0.949891i \(-0.601193\pi\)
−0.312580 + 0.949891i \(0.601193\pi\)
\(972\) −5.24698 −0.168297
\(973\) −9.98792 −0.320198
\(974\) −39.9997 −1.28167
\(975\) 0 0
\(976\) −44.6286 −1.42853
\(977\) 47.0538 1.50539 0.752693 0.658372i \(-0.228755\pi\)
0.752693 + 0.658372i \(0.228755\pi\)
\(978\) −42.2838 −1.35209
\(979\) 28.9057 0.923831
\(980\) 36.8305 1.17651
\(981\) −3.46011 −0.110473
\(982\) 13.4572 0.429438
\(983\) 34.4295 1.09813 0.549065 0.835779i \(-0.314984\pi\)
0.549065 + 0.835779i \(0.314984\pi\)
\(984\) −42.9342 −1.36869
\(985\) 4.89056 0.155826
\(986\) 24.9661 0.795084
\(987\) −0.417895 −0.0133017
\(988\) 0 0
\(989\) −63.9536 −2.03361
\(990\) −8.22223 −0.261319
\(991\) 7.30798 0.232146 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(992\) −167.606 −5.32149
\(993\) 11.2295 0.356358
\(994\) −15.6920 −0.497721
\(995\) −15.7724 −0.500019
\(996\) −13.8683 −0.439434
\(997\) −35.6915 −1.13036 −0.565181 0.824967i \(-0.691194\pi\)
−0.565181 + 0.824967i \(0.691194\pi\)
\(998\) −1.03743 −0.0328392
\(999\) −5.75302 −0.182018
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.l.1.3 yes 3
3.2 odd 2 1521.2.a.n.1.1 3
4.3 odd 2 8112.2.a.cp.1.1 3
13.2 odd 12 507.2.j.i.316.6 12
13.3 even 3 507.2.e.i.22.1 6
13.4 even 6 507.2.e.l.484.3 6
13.5 odd 4 507.2.b.f.337.1 6
13.6 odd 12 507.2.j.i.361.1 12
13.7 odd 12 507.2.j.i.361.6 12
13.8 odd 4 507.2.b.f.337.6 6
13.9 even 3 507.2.e.i.484.1 6
13.10 even 6 507.2.e.l.22.3 6
13.11 odd 12 507.2.j.i.316.1 12
13.12 even 2 507.2.a.i.1.1 3
39.5 even 4 1521.2.b.k.1351.6 6
39.8 even 4 1521.2.b.k.1351.1 6
39.38 odd 2 1521.2.a.s.1.3 3
52.51 odd 2 8112.2.a.cg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.1 3 13.12 even 2
507.2.a.l.1.3 yes 3 1.1 even 1 trivial
507.2.b.f.337.1 6 13.5 odd 4
507.2.b.f.337.6 6 13.8 odd 4
507.2.e.i.22.1 6 13.3 even 3
507.2.e.i.484.1 6 13.9 even 3
507.2.e.l.22.3 6 13.10 even 6
507.2.e.l.484.3 6 13.4 even 6
507.2.j.i.316.1 12 13.11 odd 12
507.2.j.i.316.6 12 13.2 odd 12
507.2.j.i.361.1 12 13.6 odd 12
507.2.j.i.361.6 12 13.7 odd 12
1521.2.a.n.1.1 3 3.2 odd 2
1521.2.a.s.1.3 3 39.38 odd 2
1521.2.b.k.1351.1 6 39.8 even 4
1521.2.b.k.1351.6 6 39.5 even 4
8112.2.a.cg.1.3 3 52.51 odd 2
8112.2.a.cp.1.1 3 4.3 odd 2