Properties

Label 2013.2.a.e.1.11
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(2.14727\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.14727 q^{2} -1.00000 q^{3} +2.61077 q^{4} -3.62125 q^{5} -2.14727 q^{6} +2.48742 q^{7} +1.31149 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.14727 q^{2} -1.00000 q^{3} +2.61077 q^{4} -3.62125 q^{5} -2.14727 q^{6} +2.48742 q^{7} +1.31149 q^{8} +1.00000 q^{9} -7.77580 q^{10} +1.00000 q^{11} -2.61077 q^{12} +0.181722 q^{13} +5.34117 q^{14} +3.62125 q^{15} -2.40542 q^{16} -1.02658 q^{17} +2.14727 q^{18} +8.45421 q^{19} -9.45425 q^{20} -2.48742 q^{21} +2.14727 q^{22} -4.42711 q^{23} -1.31149 q^{24} +8.11345 q^{25} +0.390206 q^{26} -1.00000 q^{27} +6.49409 q^{28} +5.84092 q^{29} +7.77580 q^{30} +9.14893 q^{31} -7.78806 q^{32} -1.00000 q^{33} -2.20434 q^{34} -9.00758 q^{35} +2.61077 q^{36} +1.08915 q^{37} +18.1535 q^{38} -0.181722 q^{39} -4.74923 q^{40} +6.75705 q^{41} -5.34117 q^{42} +3.54999 q^{43} +2.61077 q^{44} -3.62125 q^{45} -9.50620 q^{46} +3.99461 q^{47} +2.40542 q^{48} -0.812722 q^{49} +17.4218 q^{50} +1.02658 q^{51} +0.474435 q^{52} +10.0865 q^{53} -2.14727 q^{54} -3.62125 q^{55} +3.26223 q^{56} -8.45421 q^{57} +12.5420 q^{58} -1.12062 q^{59} +9.45425 q^{60} -1.00000 q^{61} +19.6452 q^{62} +2.48742 q^{63} -11.9122 q^{64} -0.658061 q^{65} -2.14727 q^{66} -2.55535 q^{67} -2.68016 q^{68} +4.42711 q^{69} -19.3417 q^{70} -13.1336 q^{71} +1.31149 q^{72} +8.20991 q^{73} +2.33869 q^{74} -8.11345 q^{75} +22.0720 q^{76} +2.48742 q^{77} -0.390206 q^{78} +5.35933 q^{79} +8.71063 q^{80} +1.00000 q^{81} +14.5092 q^{82} +9.24024 q^{83} -6.49409 q^{84} +3.71749 q^{85} +7.62280 q^{86} -5.84092 q^{87} +1.31149 q^{88} -8.09587 q^{89} -7.77580 q^{90} +0.452020 q^{91} -11.5582 q^{92} -9.14893 q^{93} +8.57750 q^{94} -30.6148 q^{95} +7.78806 q^{96} +14.9871 q^{97} -1.74513 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 2q^{2} - 13q^{3} + 16q^{4} + 3q^{5} - 2q^{6} + 11q^{7} + 9q^{8} + 13q^{9} + O(q^{10}) \) \( 13q + 2q^{2} - 13q^{3} + 16q^{4} + 3q^{5} - 2q^{6} + 11q^{7} + 9q^{8} + 13q^{9} + 6q^{10} + 13q^{11} - 16q^{12} + 13q^{13} + q^{14} - 3q^{15} + 18q^{16} + 17q^{17} + 2q^{18} + 14q^{19} - 7q^{20} - 11q^{21} + 2q^{22} + 7q^{23} - 9q^{24} + 18q^{25} - 10q^{26} - 13q^{27} + 19q^{28} - 6q^{29} - 6q^{30} + 27q^{31} + 5q^{32} - 13q^{33} + 6q^{34} + 14q^{35} + 16q^{36} + 10q^{37} + 2q^{38} - 13q^{39} + 8q^{40} + 3q^{41} - q^{42} + 29q^{43} + 16q^{44} + 3q^{45} - 24q^{46} + 8q^{47} - 18q^{48} + 8q^{49} - 27q^{50} - 17q^{51} + 37q^{52} - 24q^{53} - 2q^{54} + 3q^{55} + 24q^{56} - 14q^{57} - 5q^{58} + 13q^{59} + 7q^{60} - 13q^{61} + 39q^{62} + 11q^{63} + 47q^{64} - 11q^{65} - 2q^{66} + 44q^{67} - 8q^{68} - 7q^{69} - 12q^{70} + 3q^{71} + 9q^{72} + 48q^{73} - 22q^{74} - 18q^{75} + 47q^{76} + 11q^{77} + 10q^{78} - 17q^{79} - 26q^{80} + 13q^{81} + 56q^{82} + 50q^{83} - 19q^{84} + 8q^{85} + 18q^{86} + 6q^{87} + 9q^{88} - 15q^{89} + 6q^{90} + 47q^{91} + 14q^{92} - 27q^{93} + 45q^{94} - q^{95} - 5q^{96} + 27q^{97} + 47q^{98} + 13q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.14727 1.51835 0.759175 0.650887i \(-0.225603\pi\)
0.759175 + 0.650887i \(0.225603\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.61077 1.30539
\(5\) −3.62125 −1.61947 −0.809736 0.586794i \(-0.800390\pi\)
−0.809736 + 0.586794i \(0.800390\pi\)
\(6\) −2.14727 −0.876619
\(7\) 2.48742 0.940158 0.470079 0.882624i \(-0.344225\pi\)
0.470079 + 0.882624i \(0.344225\pi\)
\(8\) 1.31149 0.463681
\(9\) 1.00000 0.333333
\(10\) −7.77580 −2.45892
\(11\) 1.00000 0.301511
\(12\) −2.61077 −0.753664
\(13\) 0.181722 0.0504006 0.0252003 0.999682i \(-0.491978\pi\)
0.0252003 + 0.999682i \(0.491978\pi\)
\(14\) 5.34117 1.42749
\(15\) 3.62125 0.935003
\(16\) −2.40542 −0.601355
\(17\) −1.02658 −0.248982 −0.124491 0.992221i \(-0.539730\pi\)
−0.124491 + 0.992221i \(0.539730\pi\)
\(18\) 2.14727 0.506116
\(19\) 8.45421 1.93953 0.969765 0.244042i \(-0.0784736\pi\)
0.969765 + 0.244042i \(0.0784736\pi\)
\(20\) −9.45425 −2.11403
\(21\) −2.48742 −0.542800
\(22\) 2.14727 0.457800
\(23\) −4.42711 −0.923116 −0.461558 0.887110i \(-0.652709\pi\)
−0.461558 + 0.887110i \(0.652709\pi\)
\(24\) −1.31149 −0.267706
\(25\) 8.11345 1.62269
\(26\) 0.390206 0.0765258
\(27\) −1.00000 −0.192450
\(28\) 6.49409 1.22727
\(29\) 5.84092 1.08463 0.542316 0.840175i \(-0.317548\pi\)
0.542316 + 0.840175i \(0.317548\pi\)
\(30\) 7.77580 1.41966
\(31\) 9.14893 1.64320 0.821598 0.570067i \(-0.193083\pi\)
0.821598 + 0.570067i \(0.193083\pi\)
\(32\) −7.78806 −1.37675
\(33\) −1.00000 −0.174078
\(34\) −2.20434 −0.378041
\(35\) −9.00758 −1.52256
\(36\) 2.61077 0.435128
\(37\) 1.08915 0.179054 0.0895272 0.995984i \(-0.471464\pi\)
0.0895272 + 0.995984i \(0.471464\pi\)
\(38\) 18.1535 2.94488
\(39\) −0.181722 −0.0290988
\(40\) −4.74923 −0.750919
\(41\) 6.75705 1.05527 0.527637 0.849470i \(-0.323078\pi\)
0.527637 + 0.849470i \(0.323078\pi\)
\(42\) −5.34117 −0.824161
\(43\) 3.54999 0.541369 0.270684 0.962668i \(-0.412750\pi\)
0.270684 + 0.962668i \(0.412750\pi\)
\(44\) 2.61077 0.393588
\(45\) −3.62125 −0.539824
\(46\) −9.50620 −1.40161
\(47\) 3.99461 0.582674 0.291337 0.956621i \(-0.405900\pi\)
0.291337 + 0.956621i \(0.405900\pi\)
\(48\) 2.40542 0.347192
\(49\) −0.812722 −0.116103
\(50\) 17.4218 2.46381
\(51\) 1.02658 0.143750
\(52\) 0.474435 0.0657922
\(53\) 10.0865 1.38548 0.692741 0.721187i \(-0.256403\pi\)
0.692741 + 0.721187i \(0.256403\pi\)
\(54\) −2.14727 −0.292206
\(55\) −3.62125 −0.488289
\(56\) 3.26223 0.435934
\(57\) −8.45421 −1.11979
\(58\) 12.5420 1.64685
\(59\) −1.12062 −0.145892 −0.0729459 0.997336i \(-0.523240\pi\)
−0.0729459 + 0.997336i \(0.523240\pi\)
\(60\) 9.45425 1.22054
\(61\) −1.00000 −0.128037
\(62\) 19.6452 2.49495
\(63\) 2.48742 0.313386
\(64\) −11.9122 −1.48903
\(65\) −0.658061 −0.0816224
\(66\) −2.14727 −0.264311
\(67\) −2.55535 −0.312185 −0.156093 0.987742i \(-0.549890\pi\)
−0.156093 + 0.987742i \(0.549890\pi\)
\(68\) −2.68016 −0.325017
\(69\) 4.42711 0.532961
\(70\) −19.3417 −2.31178
\(71\) −13.1336 −1.55867 −0.779336 0.626606i \(-0.784443\pi\)
−0.779336 + 0.626606i \(0.784443\pi\)
\(72\) 1.31149 0.154560
\(73\) 8.20991 0.960897 0.480449 0.877023i \(-0.340474\pi\)
0.480449 + 0.877023i \(0.340474\pi\)
\(74\) 2.33869 0.271867
\(75\) −8.11345 −0.936861
\(76\) 22.0720 2.53183
\(77\) 2.48742 0.283468
\(78\) −0.390206 −0.0441822
\(79\) 5.35933 0.602972 0.301486 0.953471i \(-0.402517\pi\)
0.301486 + 0.953471i \(0.402517\pi\)
\(80\) 8.71063 0.973878
\(81\) 1.00000 0.111111
\(82\) 14.5092 1.60228
\(83\) 9.24024 1.01425 0.507124 0.861873i \(-0.330709\pi\)
0.507124 + 0.861873i \(0.330709\pi\)
\(84\) −6.49409 −0.708564
\(85\) 3.71749 0.403219
\(86\) 7.62280 0.821987
\(87\) −5.84092 −0.626213
\(88\) 1.31149 0.139805
\(89\) −8.09587 −0.858161 −0.429080 0.903266i \(-0.641162\pi\)
−0.429080 + 0.903266i \(0.641162\pi\)
\(90\) −7.77580 −0.819642
\(91\) 0.452020 0.0473846
\(92\) −11.5582 −1.20502
\(93\) −9.14893 −0.948700
\(94\) 8.57750 0.884702
\(95\) −30.6148 −3.14101
\(96\) 7.78806 0.794866
\(97\) 14.9871 1.52171 0.760855 0.648922i \(-0.224780\pi\)
0.760855 + 0.648922i \(0.224780\pi\)
\(98\) −1.74513 −0.176285
\(99\) 1.00000 0.100504
\(100\) 21.1824 2.11824
\(101\) 9.37540 0.932888 0.466444 0.884551i \(-0.345535\pi\)
0.466444 + 0.884551i \(0.345535\pi\)
\(102\) 2.20434 0.218262
\(103\) −1.55392 −0.153112 −0.0765561 0.997065i \(-0.524392\pi\)
−0.0765561 + 0.997065i \(0.524392\pi\)
\(104\) 0.238326 0.0233698
\(105\) 9.00758 0.879050
\(106\) 21.6583 2.10364
\(107\) −16.1691 −1.56313 −0.781564 0.623825i \(-0.785578\pi\)
−0.781564 + 0.623825i \(0.785578\pi\)
\(108\) −2.61077 −0.251221
\(109\) −3.55078 −0.340103 −0.170051 0.985435i \(-0.554393\pi\)
−0.170051 + 0.985435i \(0.554393\pi\)
\(110\) −7.77580 −0.741394
\(111\) −1.08915 −0.103377
\(112\) −5.98330 −0.565369
\(113\) −6.78075 −0.637879 −0.318939 0.947775i \(-0.603327\pi\)
−0.318939 + 0.947775i \(0.603327\pi\)
\(114\) −18.1535 −1.70023
\(115\) 16.0317 1.49496
\(116\) 15.2493 1.41586
\(117\) 0.181722 0.0168002
\(118\) −2.40627 −0.221515
\(119\) −2.55353 −0.234082
\(120\) 4.74923 0.433543
\(121\) 1.00000 0.0909091
\(122\) −2.14727 −0.194405
\(123\) −6.75705 −0.609263
\(124\) 23.8857 2.14500
\(125\) −11.2746 −1.00843
\(126\) 5.34117 0.475829
\(127\) 2.97038 0.263579 0.131789 0.991278i \(-0.457928\pi\)
0.131789 + 0.991278i \(0.457928\pi\)
\(128\) −10.0027 −0.884120
\(129\) −3.54999 −0.312560
\(130\) −1.41303 −0.123931
\(131\) 12.2455 1.06989 0.534945 0.844887i \(-0.320332\pi\)
0.534945 + 0.844887i \(0.320332\pi\)
\(132\) −2.61077 −0.227238
\(133\) 21.0292 1.82346
\(134\) −5.48702 −0.474006
\(135\) 3.62125 0.311668
\(136\) −1.34634 −0.115448
\(137\) 4.62204 0.394887 0.197444 0.980314i \(-0.436736\pi\)
0.197444 + 0.980314i \(0.436736\pi\)
\(138\) 9.50620 0.809221
\(139\) −18.2752 −1.55008 −0.775040 0.631912i \(-0.782270\pi\)
−0.775040 + 0.631912i \(0.782270\pi\)
\(140\) −23.5167 −1.98753
\(141\) −3.99461 −0.336407
\(142\) −28.2014 −2.36661
\(143\) 0.181722 0.0151964
\(144\) −2.40542 −0.200452
\(145\) −21.1514 −1.75653
\(146\) 17.6289 1.45898
\(147\) 0.812722 0.0670322
\(148\) 2.84351 0.233735
\(149\) −19.4843 −1.59622 −0.798109 0.602513i \(-0.794166\pi\)
−0.798109 + 0.602513i \(0.794166\pi\)
\(150\) −17.4218 −1.42248
\(151\) −22.0551 −1.79482 −0.897408 0.441202i \(-0.854552\pi\)
−0.897408 + 0.441202i \(0.854552\pi\)
\(152\) 11.0876 0.899323
\(153\) −1.02658 −0.0829939
\(154\) 5.34117 0.430404
\(155\) −33.1306 −2.66111
\(156\) −0.474435 −0.0379852
\(157\) −12.2301 −0.976071 −0.488035 0.872824i \(-0.662286\pi\)
−0.488035 + 0.872824i \(0.662286\pi\)
\(158\) 11.5079 0.915522
\(159\) −10.0865 −0.799908
\(160\) 28.2025 2.22961
\(161\) −11.0121 −0.867875
\(162\) 2.14727 0.168705
\(163\) 20.2586 1.58677 0.793387 0.608718i \(-0.208316\pi\)
0.793387 + 0.608718i \(0.208316\pi\)
\(164\) 17.6411 1.37754
\(165\) 3.62125 0.281914
\(166\) 19.8413 1.53998
\(167\) 1.97347 0.152712 0.0763558 0.997081i \(-0.475672\pi\)
0.0763558 + 0.997081i \(0.475672\pi\)
\(168\) −3.26223 −0.251686
\(169\) −12.9670 −0.997460
\(170\) 7.98246 0.612227
\(171\) 8.45421 0.646510
\(172\) 9.26822 0.706695
\(173\) 15.7980 1.20110 0.600549 0.799588i \(-0.294949\pi\)
0.600549 + 0.799588i \(0.294949\pi\)
\(174\) −12.5420 −0.950810
\(175\) 20.1816 1.52558
\(176\) −2.40542 −0.181315
\(177\) 1.12062 0.0842307
\(178\) −17.3840 −1.30299
\(179\) 3.95408 0.295542 0.147771 0.989022i \(-0.452790\pi\)
0.147771 + 0.989022i \(0.452790\pi\)
\(180\) −9.45425 −0.704678
\(181\) 21.0521 1.56479 0.782397 0.622780i \(-0.213997\pi\)
0.782397 + 0.622780i \(0.213997\pi\)
\(182\) 0.970609 0.0719463
\(183\) 1.00000 0.0739221
\(184\) −5.80610 −0.428031
\(185\) −3.94407 −0.289974
\(186\) −19.6452 −1.44046
\(187\) −1.02658 −0.0750708
\(188\) 10.4290 0.760613
\(189\) −2.48742 −0.180933
\(190\) −65.7383 −4.76916
\(191\) −16.1292 −1.16707 −0.583536 0.812088i \(-0.698331\pi\)
−0.583536 + 0.812088i \(0.698331\pi\)
\(192\) 11.9122 0.859692
\(193\) 16.0649 1.15638 0.578189 0.815903i \(-0.303760\pi\)
0.578189 + 0.815903i \(0.303760\pi\)
\(194\) 32.1814 2.31049
\(195\) 0.658061 0.0471247
\(196\) −2.12183 −0.151559
\(197\) 10.7311 0.764558 0.382279 0.924047i \(-0.375139\pi\)
0.382279 + 0.924047i \(0.375139\pi\)
\(198\) 2.14727 0.152600
\(199\) −1.75556 −0.124448 −0.0622242 0.998062i \(-0.519819\pi\)
−0.0622242 + 0.998062i \(0.519819\pi\)
\(200\) 10.6407 0.752411
\(201\) 2.55535 0.180240
\(202\) 20.1315 1.41645
\(203\) 14.5288 1.01973
\(204\) 2.68016 0.187649
\(205\) −24.4690 −1.70899
\(206\) −3.33669 −0.232478
\(207\) −4.42711 −0.307705
\(208\) −0.437118 −0.0303087
\(209\) 8.45421 0.584790
\(210\) 19.3417 1.33471
\(211\) −4.40424 −0.303201 −0.151600 0.988442i \(-0.548443\pi\)
−0.151600 + 0.988442i \(0.548443\pi\)
\(212\) 26.3334 1.80859
\(213\) 13.1336 0.899900
\(214\) −34.7195 −2.37337
\(215\) −12.8554 −0.876732
\(216\) −1.31149 −0.0892355
\(217\) 22.7573 1.54486
\(218\) −7.62448 −0.516395
\(219\) −8.20991 −0.554774
\(220\) −9.45425 −0.637405
\(221\) −0.186552 −0.0125488
\(222\) −2.33869 −0.156963
\(223\) −9.81727 −0.657413 −0.328707 0.944432i \(-0.606613\pi\)
−0.328707 + 0.944432i \(0.606613\pi\)
\(224\) −19.3722 −1.29436
\(225\) 8.11345 0.540897
\(226\) −14.5601 −0.968523
\(227\) 18.5665 1.23230 0.616151 0.787628i \(-0.288691\pi\)
0.616151 + 0.787628i \(0.288691\pi\)
\(228\) −22.0720 −1.46175
\(229\) −18.6934 −1.23529 −0.617647 0.786455i \(-0.711914\pi\)
−0.617647 + 0.786455i \(0.711914\pi\)
\(230\) 34.4243 2.26987
\(231\) −2.48742 −0.163660
\(232\) 7.66030 0.502924
\(233\) −13.1659 −0.862524 −0.431262 0.902227i \(-0.641932\pi\)
−0.431262 + 0.902227i \(0.641932\pi\)
\(234\) 0.390206 0.0255086
\(235\) −14.4655 −0.943624
\(236\) −2.92567 −0.190445
\(237\) −5.35933 −0.348126
\(238\) −5.48313 −0.355418
\(239\) −18.0130 −1.16516 −0.582581 0.812772i \(-0.697957\pi\)
−0.582581 + 0.812772i \(0.697957\pi\)
\(240\) −8.71063 −0.562268
\(241\) −28.6918 −1.84820 −0.924102 0.382146i \(-0.875185\pi\)
−0.924102 + 0.382146i \(0.875185\pi\)
\(242\) 2.14727 0.138032
\(243\) −1.00000 −0.0641500
\(244\) −2.61077 −0.167137
\(245\) 2.94307 0.188026
\(246\) −14.5092 −0.925074
\(247\) 1.53632 0.0977535
\(248\) 11.9987 0.761919
\(249\) −9.24024 −0.585576
\(250\) −24.2096 −1.53115
\(251\) −1.25826 −0.0794204 −0.0397102 0.999211i \(-0.512643\pi\)
−0.0397102 + 0.999211i \(0.512643\pi\)
\(252\) 6.49409 0.409089
\(253\) −4.42711 −0.278330
\(254\) 6.37821 0.400204
\(255\) −3.71749 −0.232798
\(256\) 2.34604 0.146627
\(257\) −10.9135 −0.680763 −0.340382 0.940287i \(-0.610556\pi\)
−0.340382 + 0.940287i \(0.610556\pi\)
\(258\) −7.62280 −0.474575
\(259\) 2.70917 0.168339
\(260\) −1.71805 −0.106549
\(261\) 5.84092 0.361544
\(262\) 26.2943 1.62447
\(263\) 21.9507 1.35354 0.676768 0.736197i \(-0.263380\pi\)
0.676768 + 0.736197i \(0.263380\pi\)
\(264\) −1.31149 −0.0807165
\(265\) −36.5256 −2.24375
\(266\) 45.1554 2.76865
\(267\) 8.09587 0.495459
\(268\) −6.67142 −0.407522
\(269\) −1.26259 −0.0769817 −0.0384909 0.999259i \(-0.512255\pi\)
−0.0384909 + 0.999259i \(0.512255\pi\)
\(270\) 7.77580 0.473220
\(271\) 1.70719 0.103704 0.0518522 0.998655i \(-0.483488\pi\)
0.0518522 + 0.998655i \(0.483488\pi\)
\(272\) 2.46935 0.149726
\(273\) −0.452020 −0.0273575
\(274\) 9.92477 0.599577
\(275\) 8.11345 0.489259
\(276\) 11.5582 0.695719
\(277\) −0.510990 −0.0307024 −0.0153512 0.999882i \(-0.504887\pi\)
−0.0153512 + 0.999882i \(0.504887\pi\)
\(278\) −39.2417 −2.35356
\(279\) 9.14893 0.547732
\(280\) −11.8133 −0.705982
\(281\) −16.3798 −0.977135 −0.488567 0.872526i \(-0.662480\pi\)
−0.488567 + 0.872526i \(0.662480\pi\)
\(282\) −8.57750 −0.510783
\(283\) −14.7590 −0.877332 −0.438666 0.898650i \(-0.644549\pi\)
−0.438666 + 0.898650i \(0.644549\pi\)
\(284\) −34.2888 −2.03467
\(285\) 30.6148 1.81346
\(286\) 0.390206 0.0230734
\(287\) 16.8077 0.992125
\(288\) −7.78806 −0.458916
\(289\) −15.9461 −0.938008
\(290\) −45.4179 −2.66703
\(291\) −14.9871 −0.878560
\(292\) 21.4342 1.25434
\(293\) 11.5784 0.676417 0.338208 0.941071i \(-0.390179\pi\)
0.338208 + 0.941071i \(0.390179\pi\)
\(294\) 1.74513 0.101778
\(295\) 4.05803 0.236268
\(296\) 1.42840 0.0830242
\(297\) −1.00000 −0.0580259
\(298\) −41.8381 −2.42362
\(299\) −0.804503 −0.0465256
\(300\) −21.1824 −1.22296
\(301\) 8.83034 0.508972
\(302\) −47.3582 −2.72516
\(303\) −9.37540 −0.538603
\(304\) −20.3359 −1.16635
\(305\) 3.62125 0.207352
\(306\) −2.20434 −0.126014
\(307\) −26.5991 −1.51809 −0.759047 0.651036i \(-0.774335\pi\)
−0.759047 + 0.651036i \(0.774335\pi\)
\(308\) 6.49409 0.370035
\(309\) 1.55392 0.0883994
\(310\) −71.1403 −4.04049
\(311\) −20.9090 −1.18564 −0.592819 0.805336i \(-0.701985\pi\)
−0.592819 + 0.805336i \(0.701985\pi\)
\(312\) −0.238326 −0.0134926
\(313\) 24.9511 1.41032 0.705160 0.709048i \(-0.250875\pi\)
0.705160 + 0.709048i \(0.250875\pi\)
\(314\) −26.2614 −1.48202
\(315\) −9.00758 −0.507520
\(316\) 13.9920 0.787111
\(317\) 15.8388 0.889596 0.444798 0.895631i \(-0.353276\pi\)
0.444798 + 0.895631i \(0.353276\pi\)
\(318\) −21.6583 −1.21454
\(319\) 5.84092 0.327029
\(320\) 43.1372 2.41144
\(321\) 16.1691 0.902472
\(322\) −23.6459 −1.31774
\(323\) −8.67890 −0.482907
\(324\) 2.61077 0.145043
\(325\) 1.47439 0.0817846
\(326\) 43.5006 2.40928
\(327\) 3.55078 0.196359
\(328\) 8.86180 0.489311
\(329\) 9.93629 0.547805
\(330\) 7.77580 0.428044
\(331\) 16.5911 0.911928 0.455964 0.889998i \(-0.349294\pi\)
0.455964 + 0.889998i \(0.349294\pi\)
\(332\) 24.1241 1.32398
\(333\) 1.08915 0.0596848
\(334\) 4.23757 0.231870
\(335\) 9.25355 0.505575
\(336\) 5.98330 0.326416
\(337\) 15.9284 0.867673 0.433836 0.900992i \(-0.357160\pi\)
0.433836 + 0.900992i \(0.357160\pi\)
\(338\) −27.8436 −1.51449
\(339\) 6.78075 0.368280
\(340\) 9.70552 0.526356
\(341\) 9.14893 0.495442
\(342\) 18.1535 0.981628
\(343\) −19.4336 −1.04931
\(344\) 4.65578 0.251023
\(345\) −16.0317 −0.863116
\(346\) 33.9225 1.82369
\(347\) 10.0215 0.537983 0.268991 0.963143i \(-0.413310\pi\)
0.268991 + 0.963143i \(0.413310\pi\)
\(348\) −15.2493 −0.817449
\(349\) 10.7818 0.577135 0.288567 0.957460i \(-0.406821\pi\)
0.288567 + 0.957460i \(0.406821\pi\)
\(350\) 43.3353 2.31637
\(351\) −0.181722 −0.00969961
\(352\) −7.78806 −0.415105
\(353\) −14.5658 −0.775258 −0.387629 0.921816i \(-0.626706\pi\)
−0.387629 + 0.921816i \(0.626706\pi\)
\(354\) 2.40627 0.127892
\(355\) 47.5601 2.52423
\(356\) −21.1365 −1.12023
\(357\) 2.55353 0.135147
\(358\) 8.49049 0.448736
\(359\) −8.13977 −0.429601 −0.214800 0.976658i \(-0.568910\pi\)
−0.214800 + 0.976658i \(0.568910\pi\)
\(360\) −4.74923 −0.250306
\(361\) 52.4737 2.76177
\(362\) 45.2046 2.37590
\(363\) −1.00000 −0.0524864
\(364\) 1.18012 0.0618551
\(365\) −29.7301 −1.55615
\(366\) 2.14727 0.112240
\(367\) 10.4128 0.543545 0.271773 0.962361i \(-0.412390\pi\)
0.271773 + 0.962361i \(0.412390\pi\)
\(368\) 10.6491 0.555120
\(369\) 6.75705 0.351758
\(370\) −8.46898 −0.440281
\(371\) 25.0893 1.30257
\(372\) −23.8857 −1.23842
\(373\) 14.4160 0.746434 0.373217 0.927744i \(-0.378255\pi\)
0.373217 + 0.927744i \(0.378255\pi\)
\(374\) −2.20434 −0.113984
\(375\) 11.2746 0.582217
\(376\) 5.23888 0.270175
\(377\) 1.06142 0.0546661
\(378\) −5.34117 −0.274720
\(379\) 12.7476 0.654802 0.327401 0.944886i \(-0.393827\pi\)
0.327401 + 0.944886i \(0.393827\pi\)
\(380\) −79.9282 −4.10023
\(381\) −2.97038 −0.152177
\(382\) −34.6339 −1.77202
\(383\) 10.1427 0.518270 0.259135 0.965841i \(-0.416563\pi\)
0.259135 + 0.965841i \(0.416563\pi\)
\(384\) 10.0027 0.510447
\(385\) −9.00758 −0.459069
\(386\) 34.4957 1.75579
\(387\) 3.54999 0.180456
\(388\) 39.1279 1.98642
\(389\) 17.0449 0.864208 0.432104 0.901824i \(-0.357771\pi\)
0.432104 + 0.901824i \(0.357771\pi\)
\(390\) 1.41303 0.0715518
\(391\) 4.54477 0.229839
\(392\) −1.06588 −0.0538348
\(393\) −12.2455 −0.617702
\(394\) 23.0425 1.16087
\(395\) −19.4075 −0.976496
\(396\) 2.61077 0.131196
\(397\) −10.6886 −0.536444 −0.268222 0.963357i \(-0.586436\pi\)
−0.268222 + 0.963357i \(0.586436\pi\)
\(398\) −3.76966 −0.188956
\(399\) −21.0292 −1.05278
\(400\) −19.5163 −0.975813
\(401\) −3.16010 −0.157808 −0.0789039 0.996882i \(-0.525142\pi\)
−0.0789039 + 0.996882i \(0.525142\pi\)
\(402\) 5.48702 0.273668
\(403\) 1.66256 0.0828181
\(404\) 24.4770 1.21778
\(405\) −3.62125 −0.179941
\(406\) 31.1974 1.54830
\(407\) 1.08915 0.0539869
\(408\) 1.34634 0.0666540
\(409\) −13.0980 −0.647655 −0.323828 0.946116i \(-0.604970\pi\)
−0.323828 + 0.946116i \(0.604970\pi\)
\(410\) −52.5415 −2.59484
\(411\) −4.62204 −0.227988
\(412\) −4.05693 −0.199870
\(413\) −2.78745 −0.137161
\(414\) −9.50620 −0.467204
\(415\) −33.4612 −1.64255
\(416\) −1.41526 −0.0693890
\(417\) 18.2752 0.894939
\(418\) 18.1535 0.887916
\(419\) 21.4749 1.04912 0.524558 0.851375i \(-0.324231\pi\)
0.524558 + 0.851375i \(0.324231\pi\)
\(420\) 23.5167 1.14750
\(421\) −14.0457 −0.684546 −0.342273 0.939601i \(-0.611197\pi\)
−0.342273 + 0.939601i \(0.611197\pi\)
\(422\) −9.45710 −0.460364
\(423\) 3.99461 0.194225
\(424\) 13.2283 0.642422
\(425\) −8.32909 −0.404020
\(426\) 28.2014 1.36636
\(427\) −2.48742 −0.120375
\(428\) −42.2139 −2.04048
\(429\) −0.181722 −0.00877362
\(430\) −27.6041 −1.33119
\(431\) 23.1921 1.11712 0.558561 0.829463i \(-0.311354\pi\)
0.558561 + 0.829463i \(0.311354\pi\)
\(432\) 2.40542 0.115731
\(433\) 14.7223 0.707507 0.353753 0.935339i \(-0.384905\pi\)
0.353753 + 0.935339i \(0.384905\pi\)
\(434\) 48.8660 2.34564
\(435\) 21.1514 1.01413
\(436\) −9.27027 −0.443965
\(437\) −37.4277 −1.79041
\(438\) −17.6289 −0.842341
\(439\) −36.6864 −1.75095 −0.875474 0.483266i \(-0.839450\pi\)
−0.875474 + 0.483266i \(0.839450\pi\)
\(440\) −4.74923 −0.226411
\(441\) −0.812722 −0.0387010
\(442\) −0.400577 −0.0190535
\(443\) 41.4579 1.96972 0.984862 0.173338i \(-0.0554553\pi\)
0.984862 + 0.173338i \(0.0554553\pi\)
\(444\) −2.84351 −0.134947
\(445\) 29.3172 1.38977
\(446\) −21.0803 −0.998183
\(447\) 19.4843 0.921577
\(448\) −29.6308 −1.39992
\(449\) −7.78679 −0.367481 −0.183741 0.982975i \(-0.558821\pi\)
−0.183741 + 0.982975i \(0.558821\pi\)
\(450\) 17.4218 0.821270
\(451\) 6.75705 0.318177
\(452\) −17.7030 −0.832678
\(453\) 22.0551 1.03624
\(454\) 39.8673 1.87106
\(455\) −1.63688 −0.0767380
\(456\) −11.0876 −0.519225
\(457\) 36.5049 1.70763 0.853814 0.520578i \(-0.174284\pi\)
0.853814 + 0.520578i \(0.174284\pi\)
\(458\) −40.1398 −1.87561
\(459\) 1.02658 0.0479165
\(460\) 41.8550 1.95150
\(461\) 2.24092 0.104370 0.0521850 0.998637i \(-0.483381\pi\)
0.0521850 + 0.998637i \(0.483381\pi\)
\(462\) −5.34117 −0.248494
\(463\) −9.86522 −0.458476 −0.229238 0.973370i \(-0.573623\pi\)
−0.229238 + 0.973370i \(0.573623\pi\)
\(464\) −14.0499 −0.652249
\(465\) 33.1306 1.53639
\(466\) −28.2707 −1.30961
\(467\) −8.97808 −0.415456 −0.207728 0.978187i \(-0.566607\pi\)
−0.207728 + 0.978187i \(0.566607\pi\)
\(468\) 0.474435 0.0219307
\(469\) −6.35623 −0.293503
\(470\) −31.0613 −1.43275
\(471\) 12.2301 0.563535
\(472\) −1.46968 −0.0676473
\(473\) 3.54999 0.163229
\(474\) −11.5079 −0.528577
\(475\) 68.5928 3.14725
\(476\) −6.66669 −0.305567
\(477\) 10.0865 0.461827
\(478\) −38.6787 −1.76912
\(479\) 16.0004 0.731078 0.365539 0.930796i \(-0.380885\pi\)
0.365539 + 0.930796i \(0.380885\pi\)
\(480\) −28.2025 −1.28726
\(481\) 0.197922 0.00902446
\(482\) −61.6092 −2.80622
\(483\) 11.0121 0.501068
\(484\) 2.61077 0.118671
\(485\) −54.2720 −2.46437
\(486\) −2.14727 −0.0974022
\(487\) 3.78986 0.171735 0.0858676 0.996307i \(-0.472634\pi\)
0.0858676 + 0.996307i \(0.472634\pi\)
\(488\) −1.31149 −0.0593683
\(489\) −20.2586 −0.916124
\(490\) 6.31957 0.285489
\(491\) −8.31228 −0.375128 −0.187564 0.982252i \(-0.560059\pi\)
−0.187564 + 0.982252i \(0.560059\pi\)
\(492\) −17.6411 −0.795323
\(493\) −5.99616 −0.270053
\(494\) 3.29889 0.148424
\(495\) −3.62125 −0.162763
\(496\) −22.0070 −0.988144
\(497\) −32.6688 −1.46540
\(498\) −19.8413 −0.889110
\(499\) 32.3218 1.44692 0.723461 0.690365i \(-0.242550\pi\)
0.723461 + 0.690365i \(0.242550\pi\)
\(500\) −29.4353 −1.31639
\(501\) −1.97347 −0.0881681
\(502\) −2.70182 −0.120588
\(503\) −5.08665 −0.226803 −0.113401 0.993549i \(-0.536175\pi\)
−0.113401 + 0.993549i \(0.536175\pi\)
\(504\) 3.26223 0.145311
\(505\) −33.9507 −1.51079
\(506\) −9.50620 −0.422602
\(507\) 12.9670 0.575884
\(508\) 7.75498 0.344071
\(509\) 4.62312 0.204916 0.102458 0.994737i \(-0.467329\pi\)
0.102458 + 0.994737i \(0.467329\pi\)
\(510\) −7.98246 −0.353469
\(511\) 20.4215 0.903395
\(512\) 25.0429 1.10675
\(513\) −8.45421 −0.373263
\(514\) −23.4342 −1.03364
\(515\) 5.62713 0.247961
\(516\) −9.26822 −0.408011
\(517\) 3.99461 0.175683
\(518\) 5.81731 0.255598
\(519\) −15.7980 −0.693454
\(520\) −0.863040 −0.0378468
\(521\) 8.80274 0.385655 0.192828 0.981233i \(-0.438234\pi\)
0.192828 + 0.981233i \(0.438234\pi\)
\(522\) 12.5420 0.548950
\(523\) 28.2658 1.23598 0.617989 0.786186i \(-0.287947\pi\)
0.617989 + 0.786186i \(0.287947\pi\)
\(524\) 31.9701 1.39662
\(525\) −20.1816 −0.880797
\(526\) 47.1340 2.05514
\(527\) −9.39208 −0.409126
\(528\) 2.40542 0.104682
\(529\) −3.40072 −0.147857
\(530\) −78.4303 −3.40679
\(531\) −1.12062 −0.0486306
\(532\) 54.9024 2.38032
\(533\) 1.22791 0.0531865
\(534\) 17.3840 0.752280
\(535\) 58.5524 2.53144
\(536\) −3.35131 −0.144754
\(537\) −3.95408 −0.170631
\(538\) −2.71113 −0.116885
\(539\) −0.812722 −0.0350064
\(540\) 9.45425 0.406846
\(541\) −43.2186 −1.85811 −0.929056 0.369940i \(-0.879378\pi\)
−0.929056 + 0.369940i \(0.879378\pi\)
\(542\) 3.66580 0.157459
\(543\) −21.0521 −0.903434
\(544\) 7.99505 0.342785
\(545\) 12.8583 0.550787
\(546\) −0.970609 −0.0415382
\(547\) −21.7658 −0.930638 −0.465319 0.885143i \(-0.654060\pi\)
−0.465319 + 0.885143i \(0.654060\pi\)
\(548\) 12.0671 0.515480
\(549\) −1.00000 −0.0426790
\(550\) 17.4218 0.742867
\(551\) 49.3804 2.10368
\(552\) 5.80610 0.247124
\(553\) 13.3309 0.566889
\(554\) −1.09723 −0.0466170
\(555\) 3.94407 0.167416
\(556\) −47.7123 −2.02345
\(557\) −33.1891 −1.40627 −0.703134 0.711057i \(-0.748217\pi\)
−0.703134 + 0.711057i \(0.748217\pi\)
\(558\) 19.6452 0.831648
\(559\) 0.645112 0.0272853
\(560\) 21.6670 0.915599
\(561\) 1.02658 0.0433421
\(562\) −35.1718 −1.48363
\(563\) 16.7017 0.703892 0.351946 0.936020i \(-0.385520\pi\)
0.351946 + 0.936020i \(0.385520\pi\)
\(564\) −10.4290 −0.439140
\(565\) 24.5548 1.03303
\(566\) −31.6916 −1.33210
\(567\) 2.48742 0.104462
\(568\) −17.2246 −0.722727
\(569\) −46.4050 −1.94540 −0.972700 0.232068i \(-0.925451\pi\)
−0.972700 + 0.232068i \(0.925451\pi\)
\(570\) 65.7383 2.75347
\(571\) 5.17531 0.216580 0.108290 0.994119i \(-0.465462\pi\)
0.108290 + 0.994119i \(0.465462\pi\)
\(572\) 0.474435 0.0198371
\(573\) 16.1292 0.673809
\(574\) 36.0906 1.50639
\(575\) −35.9191 −1.49793
\(576\) −11.9122 −0.496343
\(577\) 11.8916 0.495055 0.247528 0.968881i \(-0.420382\pi\)
0.247528 + 0.968881i \(0.420382\pi\)
\(578\) −34.2407 −1.42422
\(579\) −16.0649 −0.667635
\(580\) −55.2215 −2.29295
\(581\) 22.9844 0.953553
\(582\) −32.1814 −1.33396
\(583\) 10.0865 0.417738
\(584\) 10.7672 0.445550
\(585\) −0.658061 −0.0272075
\(586\) 24.8619 1.02704
\(587\) 32.9881 1.36157 0.680783 0.732485i \(-0.261640\pi\)
0.680783 + 0.732485i \(0.261640\pi\)
\(588\) 2.12183 0.0875028
\(589\) 77.3470 3.18703
\(590\) 8.71369 0.358737
\(591\) −10.7311 −0.441418
\(592\) −2.61985 −0.107675
\(593\) 2.08016 0.0854221 0.0427111 0.999087i \(-0.486401\pi\)
0.0427111 + 0.999087i \(0.486401\pi\)
\(594\) −2.14727 −0.0881036
\(595\) 9.24698 0.379089
\(596\) −50.8691 −2.08368
\(597\) 1.75556 0.0718503
\(598\) −1.72749 −0.0706421
\(599\) 3.43085 0.140181 0.0700903 0.997541i \(-0.477671\pi\)
0.0700903 + 0.997541i \(0.477671\pi\)
\(600\) −10.6407 −0.434405
\(601\) −11.5256 −0.470141 −0.235070 0.971978i \(-0.575532\pi\)
−0.235070 + 0.971978i \(0.575532\pi\)
\(602\) 18.9611 0.772798
\(603\) −2.55535 −0.104062
\(604\) −57.5807 −2.34293
\(605\) −3.62125 −0.147225
\(606\) −20.1315 −0.817787
\(607\) −29.6459 −1.20329 −0.601645 0.798764i \(-0.705488\pi\)
−0.601645 + 0.798764i \(0.705488\pi\)
\(608\) −65.8419 −2.67024
\(609\) −14.5288 −0.588739
\(610\) 7.77580 0.314833
\(611\) 0.725909 0.0293671
\(612\) −2.68016 −0.108339
\(613\) −32.1201 −1.29732 −0.648659 0.761079i \(-0.724670\pi\)
−0.648659 + 0.761079i \(0.724670\pi\)
\(614\) −57.1156 −2.30500
\(615\) 24.4690 0.986684
\(616\) 3.26223 0.131439
\(617\) 3.29006 0.132453 0.0662265 0.997805i \(-0.478904\pi\)
0.0662265 + 0.997805i \(0.478904\pi\)
\(618\) 3.33669 0.134221
\(619\) 30.1376 1.21133 0.605666 0.795719i \(-0.292907\pi\)
0.605666 + 0.795719i \(0.292907\pi\)
\(620\) −86.4963 −3.47377
\(621\) 4.42711 0.177654
\(622\) −44.8972 −1.80021
\(623\) −20.1379 −0.806806
\(624\) 0.437118 0.0174987
\(625\) 0.260823 0.0104329
\(626\) 53.5768 2.14136
\(627\) −8.45421 −0.337629
\(628\) −31.9301 −1.27415
\(629\) −1.11809 −0.0445813
\(630\) −19.3417 −0.770592
\(631\) −12.1776 −0.484782 −0.242391 0.970179i \(-0.577932\pi\)
−0.242391 + 0.970179i \(0.577932\pi\)
\(632\) 7.02870 0.279587
\(633\) 4.40424 0.175053
\(634\) 34.0102 1.35072
\(635\) −10.7565 −0.426858
\(636\) −26.3334 −1.04419
\(637\) −0.147690 −0.00585167
\(638\) 12.5420 0.496544
\(639\) −13.1336 −0.519557
\(640\) 36.2222 1.43181
\(641\) 16.9243 0.668469 0.334234 0.942490i \(-0.391522\pi\)
0.334234 + 0.942490i \(0.391522\pi\)
\(642\) 34.7195 1.37027
\(643\) −19.6269 −0.774009 −0.387004 0.922078i \(-0.626490\pi\)
−0.387004 + 0.922078i \(0.626490\pi\)
\(644\) −28.7500 −1.13291
\(645\) 12.8554 0.506181
\(646\) −18.6360 −0.733222
\(647\) 24.6067 0.967389 0.483695 0.875237i \(-0.339295\pi\)
0.483695 + 0.875237i \(0.339295\pi\)
\(648\) 1.31149 0.0515201
\(649\) −1.12062 −0.0439880
\(650\) 3.16592 0.124178
\(651\) −22.7573 −0.891927
\(652\) 52.8905 2.07135
\(653\) 23.1376 0.905446 0.452723 0.891651i \(-0.350453\pi\)
0.452723 + 0.891651i \(0.350453\pi\)
\(654\) 7.62448 0.298141
\(655\) −44.3439 −1.73266
\(656\) −16.2535 −0.634594
\(657\) 8.20991 0.320299
\(658\) 21.3359 0.831760
\(659\) −9.65499 −0.376105 −0.188053 0.982159i \(-0.560218\pi\)
−0.188053 + 0.982159i \(0.560218\pi\)
\(660\) 9.45425 0.368006
\(661\) 42.9636 1.67109 0.835545 0.549421i \(-0.185152\pi\)
0.835545 + 0.549421i \(0.185152\pi\)
\(662\) 35.6255 1.38463
\(663\) 0.186552 0.00724507
\(664\) 12.1185 0.470288
\(665\) −76.1520 −2.95305
\(666\) 2.33869 0.0906224
\(667\) −25.8584 −1.00124
\(668\) 5.15227 0.199347
\(669\) 9.81727 0.379558
\(670\) 19.8699 0.767640
\(671\) −1.00000 −0.0386046
\(672\) 19.3722 0.747299
\(673\) 39.0023 1.50343 0.751715 0.659489i \(-0.229227\pi\)
0.751715 + 0.659489i \(0.229227\pi\)
\(674\) 34.2025 1.31743
\(675\) −8.11345 −0.312287
\(676\) −33.8538 −1.30207
\(677\) −49.2672 −1.89349 −0.946746 0.321981i \(-0.895651\pi\)
−0.946746 + 0.321981i \(0.895651\pi\)
\(678\) 14.5601 0.559177
\(679\) 37.2793 1.43065
\(680\) 4.87545 0.186965
\(681\) −18.5665 −0.711470
\(682\) 19.6452 0.752254
\(683\) −18.1420 −0.694183 −0.347092 0.937831i \(-0.612831\pi\)
−0.347092 + 0.937831i \(0.612831\pi\)
\(684\) 22.0720 0.843944
\(685\) −16.7376 −0.639509
\(686\) −41.7291 −1.59322
\(687\) 18.6934 0.713198
\(688\) −8.53922 −0.325555
\(689\) 1.83293 0.0698291
\(690\) −34.4243 −1.31051
\(691\) 39.8244 1.51499 0.757496 0.652840i \(-0.226423\pi\)
0.757496 + 0.652840i \(0.226423\pi\)
\(692\) 41.2449 1.56790
\(693\) 2.48742 0.0944894
\(694\) 21.5189 0.816846
\(695\) 66.1789 2.51031
\(696\) −7.66030 −0.290363
\(697\) −6.93664 −0.262744
\(698\) 23.1514 0.876292
\(699\) 13.1659 0.497979
\(700\) 52.6895 1.99148
\(701\) −1.05971 −0.0400247 −0.0200123 0.999800i \(-0.506371\pi\)
−0.0200123 + 0.999800i \(0.506371\pi\)
\(702\) −0.390206 −0.0147274
\(703\) 9.20787 0.347281
\(704\) −11.9122 −0.448959
\(705\) 14.4655 0.544801
\(706\) −31.2766 −1.17711
\(707\) 23.3206 0.877062
\(708\) 2.92567 0.109953
\(709\) −16.3607 −0.614440 −0.307220 0.951639i \(-0.599399\pi\)
−0.307220 + 0.951639i \(0.599399\pi\)
\(710\) 102.124 3.83266
\(711\) 5.35933 0.200991
\(712\) −10.6176 −0.397913
\(713\) −40.5033 −1.51686
\(714\) 5.48313 0.205201
\(715\) −0.658061 −0.0246101
\(716\) 10.3232 0.385796
\(717\) 18.0130 0.672707
\(718\) −17.4783 −0.652284
\(719\) −19.9404 −0.743651 −0.371826 0.928303i \(-0.621268\pi\)
−0.371826 + 0.928303i \(0.621268\pi\)
\(720\) 8.71063 0.324626
\(721\) −3.86526 −0.143950
\(722\) 112.675 4.19334
\(723\) 28.6918 1.06706
\(724\) 54.9623 2.04266
\(725\) 47.3900 1.76002
\(726\) −2.14727 −0.0796927
\(727\) −42.7616 −1.58594 −0.792969 0.609262i \(-0.791466\pi\)
−0.792969 + 0.609262i \(0.791466\pi\)
\(728\) 0.592819 0.0219713
\(729\) 1.00000 0.0370370
\(730\) −63.8386 −2.36277
\(731\) −3.64434 −0.134791
\(732\) 2.61077 0.0964968
\(733\) −47.7154 −1.76241 −0.881204 0.472736i \(-0.843267\pi\)
−0.881204 + 0.472736i \(0.843267\pi\)
\(734\) 22.3592 0.825292
\(735\) −2.94307 −0.108557
\(736\) 34.4786 1.27090
\(737\) −2.55535 −0.0941274
\(738\) 14.5092 0.534092
\(739\) −43.5566 −1.60226 −0.801128 0.598494i \(-0.795766\pi\)
−0.801128 + 0.598494i \(0.795766\pi\)
\(740\) −10.2971 −0.378527
\(741\) −1.53632 −0.0564380
\(742\) 53.8735 1.97776
\(743\) 10.8289 0.397275 0.198637 0.980073i \(-0.436348\pi\)
0.198637 + 0.980073i \(0.436348\pi\)
\(744\) −11.9987 −0.439894
\(745\) 70.5576 2.58503
\(746\) 30.9551 1.13335
\(747\) 9.24024 0.338083
\(748\) −2.68016 −0.0979963
\(749\) −40.2195 −1.46959
\(750\) 24.2096 0.884009
\(751\) 40.1160 1.46385 0.731926 0.681384i \(-0.238621\pi\)
0.731926 + 0.681384i \(0.238621\pi\)
\(752\) −9.60871 −0.350394
\(753\) 1.25826 0.0458534
\(754\) 2.27917 0.0830023
\(755\) 79.8669 2.90665
\(756\) −6.49409 −0.236188
\(757\) −27.3129 −0.992705 −0.496353 0.868121i \(-0.665328\pi\)
−0.496353 + 0.868121i \(0.665328\pi\)
\(758\) 27.3726 0.994218
\(759\) 4.42711 0.160694
\(760\) −40.1510 −1.45643
\(761\) 1.27815 0.0463331 0.0231665 0.999732i \(-0.492625\pi\)
0.0231665 + 0.999732i \(0.492625\pi\)
\(762\) −6.37821 −0.231058
\(763\) −8.83229 −0.319750
\(764\) −42.1098 −1.52348
\(765\) 3.71749 0.134406
\(766\) 21.7792 0.786914
\(767\) −0.203641 −0.00735304
\(768\) −2.34604 −0.0846554
\(769\) 16.2719 0.586781 0.293391 0.955993i \(-0.405216\pi\)
0.293391 + 0.955993i \(0.405216\pi\)
\(770\) −19.3417 −0.697027
\(771\) 10.9135 0.393039
\(772\) 41.9418 1.50952
\(773\) −43.2949 −1.55721 −0.778604 0.627515i \(-0.784072\pi\)
−0.778604 + 0.627515i \(0.784072\pi\)
\(774\) 7.62280 0.273996
\(775\) 74.2294 2.66640
\(776\) 19.6554 0.705588
\(777\) −2.70917 −0.0971908
\(778\) 36.5999 1.31217
\(779\) 57.1256 2.04674
\(780\) 1.71805 0.0615159
\(781\) −13.1336 −0.469957
\(782\) 9.75885 0.348976
\(783\) −5.84092 −0.208738
\(784\) 1.95494 0.0698192
\(785\) 44.2884 1.58072
\(786\) −26.2943 −0.937887
\(787\) 1.65622 0.0590380 0.0295190 0.999564i \(-0.490602\pi\)
0.0295190 + 0.999564i \(0.490602\pi\)
\(788\) 28.0164 0.998042
\(789\) −21.9507 −0.781464
\(790\) −41.6731 −1.48266
\(791\) −16.8666 −0.599707
\(792\) 1.31149 0.0466017
\(793\) −0.181722 −0.00645314
\(794\) −22.9513 −0.814510
\(795\) 36.5256 1.29543
\(796\) −4.58336 −0.162453
\(797\) 3.43162 0.121554 0.0607772 0.998151i \(-0.480642\pi\)
0.0607772 + 0.998151i \(0.480642\pi\)
\(798\) −45.1554 −1.59848
\(799\) −4.10078 −0.145075
\(800\) −63.1881 −2.23404
\(801\) −8.09587 −0.286054
\(802\) −6.78558 −0.239607
\(803\) 8.20991 0.289721
\(804\) 6.67142 0.235283
\(805\) 39.8775 1.40550
\(806\) 3.56997 0.125747
\(807\) 1.26259 0.0444454
\(808\) 12.2957 0.432562
\(809\) −17.6124 −0.619221 −0.309610 0.950864i \(-0.600199\pi\)
−0.309610 + 0.950864i \(0.600199\pi\)
\(810\) −7.77580 −0.273214
\(811\) −33.2624 −1.16800 −0.584001 0.811753i \(-0.698514\pi\)
−0.584001 + 0.811753i \(0.698514\pi\)
\(812\) 37.9315 1.33113
\(813\) −1.70719 −0.0598737
\(814\) 2.33869 0.0819710
\(815\) −73.3614 −2.56974
\(816\) −2.46935 −0.0864445
\(817\) 30.0124 1.05000
\(818\) −28.1250 −0.983367
\(819\) 0.452020 0.0157949
\(820\) −63.8829 −2.23089
\(821\) −38.6262 −1.34806 −0.674032 0.738702i \(-0.735439\pi\)
−0.674032 + 0.738702i \(0.735439\pi\)
\(822\) −9.92477 −0.346166
\(823\) 33.8042 1.17834 0.589170 0.808009i \(-0.299455\pi\)
0.589170 + 0.808009i \(0.299455\pi\)
\(824\) −2.03795 −0.0709953
\(825\) −8.11345 −0.282474
\(826\) −5.98540 −0.208259
\(827\) 0.557114 0.0193727 0.00968637 0.999953i \(-0.496917\pi\)
0.00968637 + 0.999953i \(0.496917\pi\)
\(828\) −11.5582 −0.401674
\(829\) 5.19937 0.180581 0.0902907 0.995915i \(-0.471220\pi\)
0.0902907 + 0.995915i \(0.471220\pi\)
\(830\) −71.8503 −2.49396
\(831\) 0.510990 0.0177260
\(832\) −2.16472 −0.0750481
\(833\) 0.834322 0.0289075
\(834\) 39.2417 1.35883
\(835\) −7.14642 −0.247312
\(836\) 22.0720 0.763376
\(837\) −9.14893 −0.316233
\(838\) 46.1123 1.59292
\(839\) −11.9950 −0.414112 −0.207056 0.978329i \(-0.566388\pi\)
−0.207056 + 0.978329i \(0.566388\pi\)
\(840\) 11.8133 0.407599
\(841\) 5.11637 0.176427
\(842\) −30.1599 −1.03938
\(843\) 16.3798 0.564149
\(844\) −11.4985 −0.395793
\(845\) 46.9567 1.61536
\(846\) 8.57750 0.294901
\(847\) 2.48742 0.0854689
\(848\) −24.2622 −0.833166
\(849\) 14.7590 0.506528
\(850\) −17.8848 −0.613444
\(851\) −4.82176 −0.165288
\(852\) 34.2888 1.17472
\(853\) −22.8770 −0.783294 −0.391647 0.920115i \(-0.628095\pi\)
−0.391647 + 0.920115i \(0.628095\pi\)
\(854\) −5.34117 −0.182771
\(855\) −30.6148 −1.04700
\(856\) −21.2056 −0.724793
\(857\) 42.2589 1.44354 0.721768 0.692135i \(-0.243330\pi\)
0.721768 + 0.692135i \(0.243330\pi\)
\(858\) −0.390206 −0.0133214
\(859\) 14.4661 0.493576 0.246788 0.969070i \(-0.420625\pi\)
0.246788 + 0.969070i \(0.420625\pi\)
\(860\) −33.5625 −1.14447
\(861\) −16.8077 −0.572803
\(862\) 49.7996 1.69618
\(863\) −34.2785 −1.16685 −0.583427 0.812166i \(-0.698289\pi\)
−0.583427 + 0.812166i \(0.698289\pi\)
\(864\) 7.78806 0.264955
\(865\) −57.2084 −1.94515
\(866\) 31.6127 1.07424
\(867\) 15.9461 0.541559
\(868\) 59.4140 2.01664
\(869\) 5.35933 0.181803
\(870\) 45.4179 1.53981
\(871\) −0.464363 −0.0157343
\(872\) −4.65681 −0.157699
\(873\) 14.9871 0.507237
\(874\) −80.3674 −2.71847
\(875\) −28.0447 −0.948083
\(876\) −21.4342 −0.724194
\(877\) 50.0011 1.68842 0.844209 0.536014i \(-0.180070\pi\)
0.844209 + 0.536014i \(0.180070\pi\)
\(878\) −78.7757 −2.65855
\(879\) −11.5784 −0.390529
\(880\) 8.71063 0.293635
\(881\) −5.16237 −0.173925 −0.0869624 0.996212i \(-0.527716\pi\)
−0.0869624 + 0.996212i \(0.527716\pi\)
\(882\) −1.74513 −0.0587617
\(883\) 54.9858 1.85042 0.925209 0.379457i \(-0.123889\pi\)
0.925209 + 0.379457i \(0.123889\pi\)
\(884\) −0.487044 −0.0163811
\(885\) −4.05803 −0.136409
\(886\) 89.0213 2.99073
\(887\) −19.7988 −0.664779 −0.332389 0.943142i \(-0.607855\pi\)
−0.332389 + 0.943142i \(0.607855\pi\)
\(888\) −1.42840 −0.0479340
\(889\) 7.38859 0.247805
\(890\) 62.9519 2.11015
\(891\) 1.00000 0.0335013
\(892\) −25.6306 −0.858177
\(893\) 33.7713 1.13011
\(894\) 41.8381 1.39928
\(895\) −14.3187 −0.478622
\(896\) −24.8809 −0.831212
\(897\) 0.804503 0.0268616
\(898\) −16.7203 −0.557965
\(899\) 53.4382 1.78226
\(900\) 21.1824 0.706078
\(901\) −10.3545 −0.344959
\(902\) 14.5092 0.483104
\(903\) −8.83034 −0.293855
\(904\) −8.89287 −0.295772
\(905\) −76.2351 −2.53414
\(906\) 47.3582 1.57337
\(907\) −46.3300 −1.53836 −0.769181 0.639031i \(-0.779336\pi\)
−0.769181 + 0.639031i \(0.779336\pi\)
\(908\) 48.4729 1.60863
\(909\) 9.37540 0.310963
\(910\) −3.51482 −0.116515
\(911\) −44.9211 −1.48830 −0.744151 0.668011i \(-0.767146\pi\)
−0.744151 + 0.668011i \(0.767146\pi\)
\(912\) 20.3359 0.673390
\(913\) 9.24024 0.305807
\(914\) 78.3859 2.59278
\(915\) −3.62125 −0.119715
\(916\) −48.8042 −1.61254
\(917\) 30.4596 1.00587
\(918\) 2.20434 0.0727540
\(919\) −30.9670 −1.02151 −0.510754 0.859727i \(-0.670634\pi\)
−0.510754 + 0.859727i \(0.670634\pi\)
\(920\) 21.0253 0.693185
\(921\) 26.5991 0.876472
\(922\) 4.81186 0.158470
\(923\) −2.38667 −0.0785581
\(924\) −6.49409 −0.213640
\(925\) 8.83673 0.290550
\(926\) −21.1833 −0.696126
\(927\) −1.55392 −0.0510374
\(928\) −45.4895 −1.49327
\(929\) −34.3813 −1.12801 −0.564007 0.825770i \(-0.690741\pi\)
−0.564007 + 0.825770i \(0.690741\pi\)
\(930\) 71.1403 2.33278
\(931\) −6.87092 −0.225185
\(932\) −34.3730 −1.12593
\(933\) 20.9090 0.684528
\(934\) −19.2784 −0.630807
\(935\) 3.71749 0.121575
\(936\) 0.238326 0.00778994
\(937\) −8.57984 −0.280291 −0.140146 0.990131i \(-0.544757\pi\)
−0.140146 + 0.990131i \(0.544757\pi\)
\(938\) −13.6485 −0.445641
\(939\) −24.9511 −0.814248
\(940\) −37.7660 −1.23179
\(941\) −42.1060 −1.37262 −0.686309 0.727310i \(-0.740770\pi\)
−0.686309 + 0.727310i \(0.740770\pi\)
\(942\) 26.2614 0.855642
\(943\) −29.9142 −0.974140
\(944\) 2.69555 0.0877328
\(945\) 9.00758 0.293017
\(946\) 7.62280 0.247838
\(947\) −54.0758 −1.75723 −0.878614 0.477532i \(-0.841531\pi\)
−0.878614 + 0.477532i \(0.841531\pi\)
\(948\) −13.9920 −0.454438
\(949\) 1.49192 0.0484298
\(950\) 147.287 4.77863
\(951\) −15.8388 −0.513608
\(952\) −3.34893 −0.108539
\(953\) −23.6590 −0.766392 −0.383196 0.923667i \(-0.625177\pi\)
−0.383196 + 0.923667i \(0.625177\pi\)
\(954\) 21.6583 0.701215
\(955\) 58.4080 1.89004
\(956\) −47.0277 −1.52099
\(957\) −5.84092 −0.188810
\(958\) 34.3572 1.11003
\(959\) 11.4970 0.371257
\(960\) −43.1372 −1.39225
\(961\) 52.7029 1.70009
\(962\) 0.424992 0.0137023
\(963\) −16.1691 −0.521043
\(964\) −74.9078 −2.41262
\(965\) −58.1751 −1.87272
\(966\) 23.6459 0.760796
\(967\) −36.9302 −1.18759 −0.593797 0.804615i \(-0.702372\pi\)
−0.593797 + 0.804615i \(0.702372\pi\)
\(968\) 1.31149 0.0421528
\(969\) 8.67890 0.278807
\(970\) −116.537 −3.74177
\(971\) 9.30267 0.298537 0.149268 0.988797i \(-0.452308\pi\)
0.149268 + 0.988797i \(0.452308\pi\)
\(972\) −2.61077 −0.0837405
\(973\) −45.4581 −1.45732
\(974\) 8.13786 0.260754
\(975\) −1.47439 −0.0472184
\(976\) 2.40542 0.0769956
\(977\) −5.96172 −0.190732 −0.0953662 0.995442i \(-0.530402\pi\)
−0.0953662 + 0.995442i \(0.530402\pi\)
\(978\) −43.5006 −1.39100
\(979\) −8.09587 −0.258745
\(980\) 7.68368 0.245446
\(981\) −3.55078 −0.113368
\(982\) −17.8487 −0.569575
\(983\) 33.8665 1.08018 0.540088 0.841609i \(-0.318391\pi\)
0.540088 + 0.841609i \(0.318391\pi\)
\(984\) −8.86180 −0.282504
\(985\) −38.8599 −1.23818
\(986\) −12.8754 −0.410035
\(987\) −9.93629 −0.316275
\(988\) 4.01097 0.127606
\(989\) −15.7162 −0.499746
\(990\) −7.77580 −0.247131
\(991\) −7.02387 −0.223121 −0.111560 0.993758i \(-0.535585\pi\)
−0.111560 + 0.993758i \(0.535585\pi\)
\(992\) −71.2524 −2.26227
\(993\) −16.5911 −0.526502
\(994\) −70.1488 −2.22499
\(995\) 6.35732 0.201541
\(996\) −24.1241 −0.764403
\(997\) 28.9360 0.916412 0.458206 0.888846i \(-0.348492\pi\)
0.458206 + 0.888846i \(0.348492\pi\)
\(998\) 69.4036 2.19693
\(999\) −1.08915 −0.0344590
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 2013.2.a.e.1.11 13
3.2 odd 2 6039.2.a.i.1.3 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.11 13 1.1 even 1 trivial
6039.2.a.i.1.3 13 3.2 odd 2