Properties

Label 6039.2.a.i.1.3
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
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: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.14727\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

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

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 \(-0.774397\pi\)
−0.759175 + 0.650887i \(0.774397\pi\)
\(3\) 0 0
\(4\) 2.61077 1.30539
\(5\) 3.62125 1.61947 0.809736 0.586794i \(-0.199610\pi\)
0.809736 + 0.586794i \(0.199610\pi\)
\(6\) 0 0
\(7\) 2.48742 0.940158 0.470079 0.882624i \(-0.344225\pi\)
0.470079 + 0.882624i \(0.344225\pi\)
\(8\) −1.31149 −0.463681
\(9\) 0 0
\(10\) −7.77580 −2.45892
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.181722 0.0504006 0.0252003 0.999682i \(-0.491978\pi\)
0.0252003 + 0.999682i \(0.491978\pi\)
\(14\) −5.34117 −1.42749
\(15\) 0 0
\(16\) −2.40542 −0.601355
\(17\) 1.02658 0.248982 0.124491 0.992221i \(-0.460270\pi\)
0.124491 + 0.992221i \(0.460270\pi\)
\(18\) 0 0
\(19\) 8.45421 1.93953 0.969765 0.244042i \(-0.0784736\pi\)
0.969765 + 0.244042i \(0.0784736\pi\)
\(20\) 9.45425 2.11403
\(21\) 0 0
\(22\) 2.14727 0.457800
\(23\) 4.42711 0.923116 0.461558 0.887110i \(-0.347291\pi\)
0.461558 + 0.887110i \(0.347291\pi\)
\(24\) 0 0
\(25\) 8.11345 1.62269
\(26\) −0.390206 −0.0765258
\(27\) 0 0
\(28\) 6.49409 1.22727
\(29\) −5.84092 −1.08463 −0.542316 0.840175i \(-0.682452\pi\)
−0.542316 + 0.840175i \(0.682452\pi\)
\(30\) 0 0
\(31\) 9.14893 1.64320 0.821598 0.570067i \(-0.193083\pi\)
0.821598 + 0.570067i \(0.193083\pi\)
\(32\) 7.78806 1.37675
\(33\) 0 0
\(34\) −2.20434 −0.378041
\(35\) 9.00758 1.52256
\(36\) 0 0
\(37\) 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 0
\(40\) −4.74923 −0.750919
\(41\) −6.75705 −1.05527 −0.527637 0.849470i \(-0.676922\pi\)
−0.527637 + 0.849470i \(0.676922\pi\)
\(42\) 0 0
\(43\) 3.54999 0.541369 0.270684 0.962668i \(-0.412750\pi\)
0.270684 + 0.962668i \(0.412750\pi\)
\(44\) −2.61077 −0.393588
\(45\) 0 0
\(46\) −9.50620 −1.40161
\(47\) −3.99461 −0.582674 −0.291337 0.956621i \(-0.594100\pi\)
−0.291337 + 0.956621i \(0.594100\pi\)
\(48\) 0 0
\(49\) −0.812722 −0.116103
\(50\) −17.4218 −2.46381
\(51\) 0 0
\(52\) 0.474435 0.0657922
\(53\) −10.0865 −1.38548 −0.692741 0.721187i \(-0.743597\pi\)
−0.692741 + 0.721187i \(0.743597\pi\)
\(54\) 0 0
\(55\) −3.62125 −0.488289
\(56\) −3.26223 −0.435934
\(57\) 0 0
\(58\) 12.5420 1.64685
\(59\) 1.12062 0.145892 0.0729459 0.997336i \(-0.476760\pi\)
0.0729459 + 0.997336i \(0.476760\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −19.6452 −2.49495
\(63\) 0 0
\(64\) −11.9122 −1.48903
\(65\) 0.658061 0.0816224
\(66\) 0 0
\(67\) −2.55535 −0.312185 −0.156093 0.987742i \(-0.549890\pi\)
−0.156093 + 0.987742i \(0.549890\pi\)
\(68\) 2.68016 0.325017
\(69\) 0 0
\(70\) −19.3417 −2.31178
\(71\) 13.1336 1.55867 0.779336 0.626606i \(-0.215557\pi\)
0.779336 + 0.626606i \(0.215557\pi\)
\(72\) 0 0
\(73\) 8.20991 0.960897 0.480449 0.877023i \(-0.340474\pi\)
0.480449 + 0.877023i \(0.340474\pi\)
\(74\) −2.33869 −0.271867
\(75\) 0 0
\(76\) 22.0720 2.53183
\(77\) −2.48742 −0.283468
\(78\) 0 0
\(79\) 5.35933 0.602972 0.301486 0.953471i \(-0.402517\pi\)
0.301486 + 0.953471i \(0.402517\pi\)
\(80\) −8.71063 −0.973878
\(81\) 0 0
\(82\) 14.5092 1.60228
\(83\) −9.24024 −1.01425 −0.507124 0.861873i \(-0.669291\pi\)
−0.507124 + 0.861873i \(0.669291\pi\)
\(84\) 0 0
\(85\) 3.71749 0.403219
\(86\) −7.62280 −0.821987
\(87\) 0 0
\(88\) 1.31149 0.139805
\(89\) 8.09587 0.858161 0.429080 0.903266i \(-0.358838\pi\)
0.429080 + 0.903266i \(0.358838\pi\)
\(90\) 0 0
\(91\) 0.452020 0.0473846
\(92\) 11.5582 1.20502
\(93\) 0 0
\(94\) 8.57750 0.884702
\(95\) 30.6148 3.14101
\(96\) 0 0
\(97\) 14.9871 1.52171 0.760855 0.648922i \(-0.224780\pi\)
0.760855 + 0.648922i \(0.224780\pi\)
\(98\) 1.74513 0.176285
\(99\) 0 0
\(100\) 21.1824 2.11824
\(101\) −9.37540 −0.932888 −0.466444 0.884551i \(-0.654465\pi\)
−0.466444 + 0.884551i \(0.654465\pi\)
\(102\) 0 0
\(103\) −1.55392 −0.153112 −0.0765561 0.997065i \(-0.524392\pi\)
−0.0765561 + 0.997065i \(0.524392\pi\)
\(104\) −0.238326 −0.0233698
\(105\) 0 0
\(106\) 21.6583 2.10364
\(107\) 16.1691 1.56313 0.781564 0.623825i \(-0.214422\pi\)
0.781564 + 0.623825i \(0.214422\pi\)
\(108\) 0 0
\(109\) −3.55078 −0.340103 −0.170051 0.985435i \(-0.554393\pi\)
−0.170051 + 0.985435i \(0.554393\pi\)
\(110\) 7.77580 0.741394
\(111\) 0 0
\(112\) −5.98330 −0.565369
\(113\) 6.78075 0.637879 0.318939 0.947775i \(-0.396673\pi\)
0.318939 + 0.947775i \(0.396673\pi\)
\(114\) 0 0
\(115\) 16.0317 1.49496
\(116\) −15.2493 −1.41586
\(117\) 0 0
\(118\) −2.40627 −0.221515
\(119\) 2.55353 0.234082
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.14727 0.194405
\(123\) 0 0
\(124\) 23.8857 2.14500
\(125\) 11.2746 1.00843
\(126\) 0 0
\(127\) 2.97038 0.263579 0.131789 0.991278i \(-0.457928\pi\)
0.131789 + 0.991278i \(0.457928\pi\)
\(128\) 10.0027 0.884120
\(129\) 0 0
\(130\) −1.41303 −0.123931
\(131\) −12.2455 −1.06989 −0.534945 0.844887i \(-0.679668\pi\)
−0.534945 + 0.844887i \(0.679668\pi\)
\(132\) 0 0
\(133\) 21.0292 1.82346
\(134\) 5.48702 0.474006
\(135\) 0 0
\(136\) −1.34634 −0.115448
\(137\) −4.62204 −0.394887 −0.197444 0.980314i \(-0.563264\pi\)
−0.197444 + 0.980314i \(0.563264\pi\)
\(138\) 0 0
\(139\) −18.2752 −1.55008 −0.775040 0.631912i \(-0.782270\pi\)
−0.775040 + 0.631912i \(0.782270\pi\)
\(140\) 23.5167 1.98753
\(141\) 0 0
\(142\) −28.2014 −2.36661
\(143\) −0.181722 −0.0151964
\(144\) 0 0
\(145\) −21.1514 −1.75653
\(146\) −17.6289 −1.45898
\(147\) 0 0
\(148\) 2.84351 0.233735
\(149\) 19.4843 1.59622 0.798109 0.602513i \(-0.205834\pi\)
0.798109 + 0.602513i \(0.205834\pi\)
\(150\) 0 0
\(151\) −22.0551 −1.79482 −0.897408 0.441202i \(-0.854552\pi\)
−0.897408 + 0.441202i \(0.854552\pi\)
\(152\) −11.0876 −0.899323
\(153\) 0 0
\(154\) 5.34117 0.430404
\(155\) 33.1306 2.66111
\(156\) 0 0
\(157\) −12.2301 −0.976071 −0.488035 0.872824i \(-0.662286\pi\)
−0.488035 + 0.872824i \(0.662286\pi\)
\(158\) −11.5079 −0.915522
\(159\) 0 0
\(160\) 28.2025 2.22961
\(161\) 11.0121 0.867875
\(162\) 0 0
\(163\) 20.2586 1.58677 0.793387 0.608718i \(-0.208316\pi\)
0.793387 + 0.608718i \(0.208316\pi\)
\(164\) −17.6411 −1.37754
\(165\) 0 0
\(166\) 19.8413 1.53998
\(167\) −1.97347 −0.152712 −0.0763558 0.997081i \(-0.524328\pi\)
−0.0763558 + 0.997081i \(0.524328\pi\)
\(168\) 0 0
\(169\) −12.9670 −0.997460
\(170\) −7.98246 −0.612227
\(171\) 0 0
\(172\) 9.26822 0.706695
\(173\) −15.7980 −1.20110 −0.600549 0.799588i \(-0.705051\pi\)
−0.600549 + 0.799588i \(0.705051\pi\)
\(174\) 0 0
\(175\) 20.1816 1.52558
\(176\) 2.40542 0.181315
\(177\) 0 0
\(178\) −17.3840 −1.30299
\(179\) −3.95408 −0.295542 −0.147771 0.989022i \(-0.547210\pi\)
−0.147771 + 0.989022i \(0.547210\pi\)
\(180\) 0 0
\(181\) 21.0521 1.56479 0.782397 0.622780i \(-0.213997\pi\)
0.782397 + 0.622780i \(0.213997\pi\)
\(182\) −0.970609 −0.0719463
\(183\) 0 0
\(184\) −5.80610 −0.428031
\(185\) 3.94407 0.289974
\(186\) 0 0
\(187\) −1.02658 −0.0750708
\(188\) −10.4290 −0.760613
\(189\) 0 0
\(190\) −65.7383 −4.76916
\(191\) 16.1292 1.16707 0.583536 0.812088i \(-0.301669\pi\)
0.583536 + 0.812088i \(0.301669\pi\)
\(192\) 0 0
\(193\) 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 0
\(196\) −2.12183 −0.151559
\(197\) −10.7311 −0.764558 −0.382279 0.924047i \(-0.624861\pi\)
−0.382279 + 0.924047i \(0.624861\pi\)
\(198\) 0 0
\(199\) −1.75556 −0.124448 −0.0622242 0.998062i \(-0.519819\pi\)
−0.0622242 + 0.998062i \(0.519819\pi\)
\(200\) −10.6407 −0.752411
\(201\) 0 0
\(202\) 20.1315 1.41645
\(203\) −14.5288 −1.01973
\(204\) 0 0
\(205\) −24.4690 −1.70899
\(206\) 3.33669 0.232478
\(207\) 0 0
\(208\) −0.437118 −0.0303087
\(209\) −8.45421 −0.584790
\(210\) 0 0
\(211\) −4.40424 −0.303201 −0.151600 0.988442i \(-0.548443\pi\)
−0.151600 + 0.988442i \(0.548443\pi\)
\(212\) −26.3334 −1.80859
\(213\) 0 0
\(214\) −34.7195 −2.37337
\(215\) 12.8554 0.876732
\(216\) 0 0
\(217\) 22.7573 1.54486
\(218\) 7.62448 0.516395
\(219\) 0 0
\(220\) −9.45425 −0.637405
\(221\) 0.186552 0.0125488
\(222\) 0 0
\(223\) −9.81727 −0.657413 −0.328707 0.944432i \(-0.606613\pi\)
−0.328707 + 0.944432i \(0.606613\pi\)
\(224\) 19.3722 1.29436
\(225\) 0 0
\(226\) −14.5601 −0.968523
\(227\) −18.5665 −1.23230 −0.616151 0.787628i \(-0.711309\pi\)
−0.616151 + 0.787628i \(0.711309\pi\)
\(228\) 0 0
\(229\) −18.6934 −1.23529 −0.617647 0.786455i \(-0.711914\pi\)
−0.617647 + 0.786455i \(0.711914\pi\)
\(230\) −34.4243 −2.26987
\(231\) 0 0
\(232\) 7.66030 0.502924
\(233\) 13.1659 0.862524 0.431262 0.902227i \(-0.358068\pi\)
0.431262 + 0.902227i \(0.358068\pi\)
\(234\) 0 0
\(235\) −14.4655 −0.943624
\(236\) 2.92567 0.190445
\(237\) 0 0
\(238\) −5.48313 −0.355418
\(239\) 18.0130 1.16516 0.582581 0.812772i \(-0.302043\pi\)
0.582581 + 0.812772i \(0.302043\pi\)
\(240\) 0 0
\(241\) −28.6918 −1.84820 −0.924102 0.382146i \(-0.875185\pi\)
−0.924102 + 0.382146i \(0.875185\pi\)
\(242\) −2.14727 −0.138032
\(243\) 0 0
\(244\) −2.61077 −0.167137
\(245\) −2.94307 −0.188026
\(246\) 0 0
\(247\) 1.53632 0.0977535
\(248\) −11.9987 −0.761919
\(249\) 0 0
\(250\) −24.2096 −1.53115
\(251\) 1.25826 0.0794204 0.0397102 0.999211i \(-0.487357\pi\)
0.0397102 + 0.999211i \(0.487357\pi\)
\(252\) 0 0
\(253\) −4.42711 −0.278330
\(254\) −6.37821 −0.400204
\(255\) 0 0
\(256\) 2.34604 0.146627
\(257\) 10.9135 0.680763 0.340382 0.940287i \(-0.389444\pi\)
0.340382 + 0.940287i \(0.389444\pi\)
\(258\) 0 0
\(259\) 2.70917 0.168339
\(260\) 1.71805 0.106549
\(261\) 0 0
\(262\) 26.2943 1.62447
\(263\) −21.9507 −1.35354 −0.676768 0.736197i \(-0.736620\pi\)
−0.676768 + 0.736197i \(0.736620\pi\)
\(264\) 0 0
\(265\) −36.5256 −2.24375
\(266\) −45.1554 −2.76865
\(267\) 0 0
\(268\) −6.67142 −0.407522
\(269\) 1.26259 0.0769817 0.0384909 0.999259i \(-0.487745\pi\)
0.0384909 + 0.999259i \(0.487745\pi\)
\(270\) 0 0
\(271\) 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 0
\(274\) 9.92477 0.599577
\(275\) −8.11345 −0.489259
\(276\) 0 0
\(277\) −0.510990 −0.0307024 −0.0153512 0.999882i \(-0.504887\pi\)
−0.0153512 + 0.999882i \(0.504887\pi\)
\(278\) 39.2417 2.35356
\(279\) 0 0
\(280\) −11.8133 −0.705982
\(281\) 16.3798 0.977135 0.488567 0.872526i \(-0.337520\pi\)
0.488567 + 0.872526i \(0.337520\pi\)
\(282\) 0 0
\(283\) −14.7590 −0.877332 −0.438666 0.898650i \(-0.644549\pi\)
−0.438666 + 0.898650i \(0.644549\pi\)
\(284\) 34.2888 2.03467
\(285\) 0 0
\(286\) 0.390206 0.0230734
\(287\) −16.8077 −0.992125
\(288\) 0 0
\(289\) −15.9461 −0.938008
\(290\) 45.4179 2.66703
\(291\) 0 0
\(292\) 21.4342 1.25434
\(293\) −11.5784 −0.676417 −0.338208 0.941071i \(-0.609821\pi\)
−0.338208 + 0.941071i \(0.609821\pi\)
\(294\) 0 0
\(295\) 4.05803 0.236268
\(296\) −1.42840 −0.0830242
\(297\) 0 0
\(298\) −41.8381 −2.42362
\(299\) 0.804503 0.0465256
\(300\) 0 0
\(301\) 8.83034 0.508972
\(302\) 47.3582 2.72516
\(303\) 0 0
\(304\) −20.3359 −1.16635
\(305\) −3.62125 −0.207352
\(306\) 0 0
\(307\) −26.5991 −1.51809 −0.759047 0.651036i \(-0.774335\pi\)
−0.759047 + 0.651036i \(0.774335\pi\)
\(308\) −6.49409 −0.370035
\(309\) 0 0
\(310\) −71.1403 −4.04049
\(311\) 20.9090 1.18564 0.592819 0.805336i \(-0.298015\pi\)
0.592819 + 0.805336i \(0.298015\pi\)
\(312\) 0 0
\(313\) 24.9511 1.41032 0.705160 0.709048i \(-0.250875\pi\)
0.705160 + 0.709048i \(0.250875\pi\)
\(314\) 26.2614 1.48202
\(315\) 0 0
\(316\) 13.9920 0.787111
\(317\) −15.8388 −0.889596 −0.444798 0.895631i \(-0.646724\pi\)
−0.444798 + 0.895631i \(0.646724\pi\)
\(318\) 0 0
\(319\) 5.84092 0.327029
\(320\) −43.1372 −2.41144
\(321\) 0 0
\(322\) −23.6459 −1.31774
\(323\) 8.67890 0.482907
\(324\) 0 0
\(325\) 1.47439 0.0817846
\(326\) −43.5006 −2.40928
\(327\) 0 0
\(328\) 8.86180 0.489311
\(329\) −9.93629 −0.547805
\(330\) 0 0
\(331\) 16.5911 0.911928 0.455964 0.889998i \(-0.349294\pi\)
0.455964 + 0.889998i \(0.349294\pi\)
\(332\) −24.1241 −1.32398
\(333\) 0 0
\(334\) 4.23757 0.231870
\(335\) −9.25355 −0.505575
\(336\) 0 0
\(337\) 15.9284 0.867673 0.433836 0.900992i \(-0.357160\pi\)
0.433836 + 0.900992i \(0.357160\pi\)
\(338\) 27.8436 1.51449
\(339\) 0 0
\(340\) 9.70552 0.526356
\(341\) −9.14893 −0.495442
\(342\) 0 0
\(343\) −19.4336 −1.04931
\(344\) −4.65578 −0.251023
\(345\) 0 0
\(346\) 33.9225 1.82369
\(347\) −10.0215 −0.537983 −0.268991 0.963143i \(-0.586690\pi\)
−0.268991 + 0.963143i \(0.586690\pi\)
\(348\) 0 0
\(349\) 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 0
\(352\) −7.78806 −0.415105
\(353\) 14.5658 0.775258 0.387629 0.921816i \(-0.373294\pi\)
0.387629 + 0.921816i \(0.373294\pi\)
\(354\) 0 0
\(355\) 47.5601 2.52423
\(356\) 21.1365 1.12023
\(357\) 0 0
\(358\) 8.49049 0.448736
\(359\) 8.13977 0.429601 0.214800 0.976658i \(-0.431090\pi\)
0.214800 + 0.976658i \(0.431090\pi\)
\(360\) 0 0
\(361\) 52.4737 2.76177
\(362\) −45.2046 −2.37590
\(363\) 0 0
\(364\) 1.18012 0.0618551
\(365\) 29.7301 1.55615
\(366\) 0 0
\(367\) 10.4128 0.543545 0.271773 0.962361i \(-0.412390\pi\)
0.271773 + 0.962361i \(0.412390\pi\)
\(368\) −10.6491 −0.555120
\(369\) 0 0
\(370\) −8.46898 −0.440281
\(371\) −25.0893 −1.30257
\(372\) 0 0
\(373\) 14.4160 0.746434 0.373217 0.927744i \(-0.378255\pi\)
0.373217 + 0.927744i \(0.378255\pi\)
\(374\) 2.20434 0.113984
\(375\) 0 0
\(376\) 5.23888 0.270175
\(377\) −1.06142 −0.0546661
\(378\) 0 0
\(379\) 12.7476 0.654802 0.327401 0.944886i \(-0.393827\pi\)
0.327401 + 0.944886i \(0.393827\pi\)
\(380\) 79.9282 4.10023
\(381\) 0 0
\(382\) −34.6339 −1.77202
\(383\) −10.1427 −0.518270 −0.259135 0.965841i \(-0.583437\pi\)
−0.259135 + 0.965841i \(0.583437\pi\)
\(384\) 0 0
\(385\) −9.00758 −0.459069
\(386\) −34.4957 −1.75579
\(387\) 0 0
\(388\) 39.1279 1.98642
\(389\) −17.0449 −0.864208 −0.432104 0.901824i \(-0.642229\pi\)
−0.432104 + 0.901824i \(0.642229\pi\)
\(390\) 0 0
\(391\) 4.54477 0.229839
\(392\) 1.06588 0.0538348
\(393\) 0 0
\(394\) 23.0425 1.16087
\(395\) 19.4075 0.976496
\(396\) 0 0
\(397\) −10.6886 −0.536444 −0.268222 0.963357i \(-0.586436\pi\)
−0.268222 + 0.963357i \(0.586436\pi\)
\(398\) 3.76966 0.188956
\(399\) 0 0
\(400\) −19.5163 −0.975813
\(401\) 3.16010 0.157808 0.0789039 0.996882i \(-0.474858\pi\)
0.0789039 + 0.996882i \(0.474858\pi\)
\(402\) 0 0
\(403\) 1.66256 0.0828181
\(404\) −24.4770 −1.21778
\(405\) 0 0
\(406\) 31.1974 1.54830
\(407\) −1.08915 −0.0539869
\(408\) 0 0
\(409\) −13.0980 −0.647655 −0.323828 0.946116i \(-0.604970\pi\)
−0.323828 + 0.946116i \(0.604970\pi\)
\(410\) 52.5415 2.59484
\(411\) 0 0
\(412\) −4.05693 −0.199870
\(413\) 2.78745 0.137161
\(414\) 0 0
\(415\) −33.4612 −1.64255
\(416\) 1.41526 0.0693890
\(417\) 0 0
\(418\) 18.1535 0.887916
\(419\) −21.4749 −1.04912 −0.524558 0.851375i \(-0.675769\pi\)
−0.524558 + 0.851375i \(0.675769\pi\)
\(420\) 0 0
\(421\) −14.0457 −0.684546 −0.342273 0.939601i \(-0.611197\pi\)
−0.342273 + 0.939601i \(0.611197\pi\)
\(422\) 9.45710 0.460364
\(423\) 0 0
\(424\) 13.2283 0.642422
\(425\) 8.32909 0.404020
\(426\) 0 0
\(427\) −2.48742 −0.120375
\(428\) 42.2139 2.04048
\(429\) 0 0
\(430\) −27.6041 −1.33119
\(431\) −23.1921 −1.11712 −0.558561 0.829463i \(-0.688646\pi\)
−0.558561 + 0.829463i \(0.688646\pi\)
\(432\) 0 0
\(433\) 14.7223 0.707507 0.353753 0.935339i \(-0.384905\pi\)
0.353753 + 0.935339i \(0.384905\pi\)
\(434\) −48.8660 −2.34564
\(435\) 0 0
\(436\) −9.27027 −0.443965
\(437\) 37.4277 1.79041
\(438\) 0 0
\(439\) −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 0
\(442\) −0.400577 −0.0190535
\(443\) −41.4579 −1.96972 −0.984862 0.173338i \(-0.944545\pi\)
−0.984862 + 0.173338i \(0.944545\pi\)
\(444\) 0 0
\(445\) 29.3172 1.38977
\(446\) 21.0803 0.998183
\(447\) 0 0
\(448\) −29.6308 −1.39992
\(449\) 7.78679 0.367481 0.183741 0.982975i \(-0.441179\pi\)
0.183741 + 0.982975i \(0.441179\pi\)
\(450\) 0 0
\(451\) 6.75705 0.318177
\(452\) 17.7030 0.832678
\(453\) 0 0
\(454\) 39.8673 1.87106
\(455\) 1.63688 0.0767380
\(456\) 0 0
\(457\) 36.5049 1.70763 0.853814 0.520578i \(-0.174284\pi\)
0.853814 + 0.520578i \(0.174284\pi\)
\(458\) 40.1398 1.87561
\(459\) 0 0
\(460\) 41.8550 1.95150
\(461\) −2.24092 −0.104370 −0.0521850 0.998637i \(-0.516619\pi\)
−0.0521850 + 0.998637i \(0.516619\pi\)
\(462\) 0 0
\(463\) −9.86522 −0.458476 −0.229238 0.973370i \(-0.573623\pi\)
−0.229238 + 0.973370i \(0.573623\pi\)
\(464\) 14.0499 0.652249
\(465\) 0 0
\(466\) −28.2707 −1.30961
\(467\) 8.97808 0.415456 0.207728 0.978187i \(-0.433393\pi\)
0.207728 + 0.978187i \(0.433393\pi\)
\(468\) 0 0
\(469\) −6.35623 −0.293503
\(470\) 31.0613 1.43275
\(471\) 0 0
\(472\) −1.46968 −0.0676473
\(473\) −3.54999 −0.163229
\(474\) 0 0
\(475\) 68.5928 3.14725
\(476\) 6.66669 0.305567
\(477\) 0 0
\(478\) −38.6787 −1.76912
\(479\) −16.0004 −0.731078 −0.365539 0.930796i \(-0.619115\pi\)
−0.365539 + 0.930796i \(0.619115\pi\)
\(480\) 0 0
\(481\) 0.197922 0.00902446
\(482\) 61.6092 2.80622
\(483\) 0 0
\(484\) 2.61077 0.118671
\(485\) 54.2720 2.46437
\(486\) 0 0
\(487\) 3.78986 0.171735 0.0858676 0.996307i \(-0.472634\pi\)
0.0858676 + 0.996307i \(0.472634\pi\)
\(488\) 1.31149 0.0593683
\(489\) 0 0
\(490\) 6.31957 0.285489
\(491\) 8.31228 0.375128 0.187564 0.982252i \(-0.439941\pi\)
0.187564 + 0.982252i \(0.439941\pi\)
\(492\) 0 0
\(493\) −5.99616 −0.270053
\(494\) −3.29889 −0.148424
\(495\) 0 0
\(496\) −22.0070 −0.988144
\(497\) 32.6688 1.46540
\(498\) 0 0
\(499\) 32.3218 1.44692 0.723461 0.690365i \(-0.242550\pi\)
0.723461 + 0.690365i \(0.242550\pi\)
\(500\) 29.4353 1.31639
\(501\) 0 0
\(502\) −2.70182 −0.120588
\(503\) 5.08665 0.226803 0.113401 0.993549i \(-0.463825\pi\)
0.113401 + 0.993549i \(0.463825\pi\)
\(504\) 0 0
\(505\) −33.9507 −1.51079
\(506\) 9.50620 0.422602
\(507\) 0 0
\(508\) 7.75498 0.344071
\(509\) −4.62312 −0.204916 −0.102458 0.994737i \(-0.532671\pi\)
−0.102458 + 0.994737i \(0.532671\pi\)
\(510\) 0 0
\(511\) 20.4215 0.903395
\(512\) −25.0429 −1.10675
\(513\) 0 0
\(514\) −23.4342 −1.03364
\(515\) −5.62713 −0.247961
\(516\) 0 0
\(517\) 3.99461 0.175683
\(518\) −5.81731 −0.255598
\(519\) 0 0
\(520\) −0.863040 −0.0378468
\(521\) −8.80274 −0.385655 −0.192828 0.981233i \(-0.561766\pi\)
−0.192828 + 0.981233i \(0.561766\pi\)
\(522\) 0 0
\(523\) 28.2658 1.23598 0.617989 0.786186i \(-0.287947\pi\)
0.617989 + 0.786186i \(0.287947\pi\)
\(524\) −31.9701 −1.39662
\(525\) 0 0
\(526\) 47.1340 2.05514
\(527\) 9.39208 0.409126
\(528\) 0 0
\(529\) −3.40072 −0.147857
\(530\) 78.4303 3.40679
\(531\) 0 0
\(532\) 54.9024 2.38032
\(533\) −1.22791 −0.0531865
\(534\) 0 0
\(535\) 58.5524 2.53144
\(536\) 3.35131 0.144754
\(537\) 0 0
\(538\) −2.71113 −0.116885
\(539\) 0.812722 0.0350064
\(540\) 0 0
\(541\) −43.2186 −1.85811 −0.929056 0.369940i \(-0.879378\pi\)
−0.929056 + 0.369940i \(0.879378\pi\)
\(542\) −3.66580 −0.157459
\(543\) 0 0
\(544\) 7.99505 0.342785
\(545\) −12.8583 −0.550787
\(546\) 0 0
\(547\) −21.7658 −0.930638 −0.465319 0.885143i \(-0.654060\pi\)
−0.465319 + 0.885143i \(0.654060\pi\)
\(548\) −12.0671 −0.515480
\(549\) 0 0
\(550\) 17.4218 0.742867
\(551\) −49.3804 −2.10368
\(552\) 0 0
\(553\) 13.3309 0.566889
\(554\) 1.09723 0.0466170
\(555\) 0 0
\(556\) −47.7123 −2.02345
\(557\) 33.1891 1.40627 0.703134 0.711057i \(-0.251783\pi\)
0.703134 + 0.711057i \(0.251783\pi\)
\(558\) 0 0
\(559\) 0.645112 0.0272853
\(560\) −21.6670 −0.915599
\(561\) 0 0
\(562\) −35.1718 −1.48363
\(563\) −16.7017 −0.703892 −0.351946 0.936020i \(-0.614480\pi\)
−0.351946 + 0.936020i \(0.614480\pi\)
\(564\) 0 0
\(565\) 24.5548 1.03303
\(566\) 31.6916 1.33210
\(567\) 0 0
\(568\) −17.2246 −0.722727
\(569\) 46.4050 1.94540 0.972700 0.232068i \(-0.0745493\pi\)
0.972700 + 0.232068i \(0.0745493\pi\)
\(570\) 0 0
\(571\) 5.17531 0.216580 0.108290 0.994119i \(-0.465462\pi\)
0.108290 + 0.994119i \(0.465462\pi\)
\(572\) −0.474435 −0.0198371
\(573\) 0 0
\(574\) 36.0906 1.50639
\(575\) 35.9191 1.49793
\(576\) 0 0
\(577\) 11.8916 0.495055 0.247528 0.968881i \(-0.420382\pi\)
0.247528 + 0.968881i \(0.420382\pi\)
\(578\) 34.2407 1.42422
\(579\) 0 0
\(580\) −55.2215 −2.29295
\(581\) −22.9844 −0.953553
\(582\) 0 0
\(583\) 10.0865 0.417738
\(584\) −10.7672 −0.445550
\(585\) 0 0
\(586\) 24.8619 1.02704
\(587\) −32.9881 −1.36157 −0.680783 0.732485i \(-0.738360\pi\)
−0.680783 + 0.732485i \(0.738360\pi\)
\(588\) 0 0
\(589\) 77.3470 3.18703
\(590\) −8.71369 −0.358737
\(591\) 0 0
\(592\) −2.61985 −0.107675
\(593\) −2.08016 −0.0854221 −0.0427111 0.999087i \(-0.513599\pi\)
−0.0427111 + 0.999087i \(0.513599\pi\)
\(594\) 0 0
\(595\) 9.24698 0.379089
\(596\) 50.8691 2.08368
\(597\) 0 0
\(598\) −1.72749 −0.0706421
\(599\) −3.43085 −0.140181 −0.0700903 0.997541i \(-0.522329\pi\)
−0.0700903 + 0.997541i \(0.522329\pi\)
\(600\) 0 0
\(601\) −11.5256 −0.470141 −0.235070 0.971978i \(-0.575532\pi\)
−0.235070 + 0.971978i \(0.575532\pi\)
\(602\) −18.9611 −0.772798
\(603\) 0 0
\(604\) −57.5807 −2.34293
\(605\) 3.62125 0.147225
\(606\) 0 0
\(607\) −29.6459 −1.20329 −0.601645 0.798764i \(-0.705488\pi\)
−0.601645 + 0.798764i \(0.705488\pi\)
\(608\) 65.8419 2.67024
\(609\) 0 0
\(610\) 7.77580 0.314833
\(611\) −0.725909 −0.0293671
\(612\) 0 0
\(613\) −32.1201 −1.29732 −0.648659 0.761079i \(-0.724670\pi\)
−0.648659 + 0.761079i \(0.724670\pi\)
\(614\) 57.1156 2.30500
\(615\) 0 0
\(616\) 3.26223 0.131439
\(617\) −3.29006 −0.132453 −0.0662265 0.997805i \(-0.521096\pi\)
−0.0662265 + 0.997805i \(0.521096\pi\)
\(618\) 0 0
\(619\) 30.1376 1.21133 0.605666 0.795719i \(-0.292907\pi\)
0.605666 + 0.795719i \(0.292907\pi\)
\(620\) 86.4963 3.47377
\(621\) 0 0
\(622\) −44.8972 −1.80021
\(623\) 20.1379 0.806806
\(624\) 0 0
\(625\) 0.260823 0.0104329
\(626\) −53.5768 −2.14136
\(627\) 0 0
\(628\) −31.9301 −1.27415
\(629\) 1.11809 0.0445813
\(630\) 0 0
\(631\) −12.1776 −0.484782 −0.242391 0.970179i \(-0.577932\pi\)
−0.242391 + 0.970179i \(0.577932\pi\)
\(632\) −7.02870 −0.279587
\(633\) 0 0
\(634\) 34.0102 1.35072
\(635\) 10.7565 0.426858
\(636\) 0 0
\(637\) −0.147690 −0.00585167
\(638\) −12.5420 −0.496544
\(639\) 0 0
\(640\) 36.2222 1.43181
\(641\) −16.9243 −0.668469 −0.334234 0.942490i \(-0.608478\pi\)
−0.334234 + 0.942490i \(0.608478\pi\)
\(642\) 0 0
\(643\) −19.6269 −0.774009 −0.387004 0.922078i \(-0.626490\pi\)
−0.387004 + 0.922078i \(0.626490\pi\)
\(644\) 28.7500 1.13291
\(645\) 0 0
\(646\) −18.6360 −0.733222
\(647\) −24.6067 −0.967389 −0.483695 0.875237i \(-0.660705\pi\)
−0.483695 + 0.875237i \(0.660705\pi\)
\(648\) 0 0
\(649\) −1.12062 −0.0439880
\(650\) −3.16592 −0.124178
\(651\) 0 0
\(652\) 52.8905 2.07135
\(653\) −23.1376 −0.905446 −0.452723 0.891651i \(-0.649547\pi\)
−0.452723 + 0.891651i \(0.649547\pi\)
\(654\) 0 0
\(655\) −44.3439 −1.73266
\(656\) 16.2535 0.634594
\(657\) 0 0
\(658\) 21.3359 0.831760
\(659\) 9.65499 0.376105 0.188053 0.982159i \(-0.439782\pi\)
0.188053 + 0.982159i \(0.439782\pi\)
\(660\) 0 0
\(661\) 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 0
\(664\) 12.1185 0.470288
\(665\) 76.1520 2.95305
\(666\) 0 0
\(667\) −25.8584 −1.00124
\(668\) −5.15227 −0.199347
\(669\) 0 0
\(670\) 19.8699 0.767640
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 39.0023 1.50343 0.751715 0.659489i \(-0.229227\pi\)
0.751715 + 0.659489i \(0.229227\pi\)
\(674\) −34.2025 −1.31743
\(675\) 0 0
\(676\) −33.8538 −1.30207
\(677\) 49.2672 1.89349 0.946746 0.321981i \(-0.104349\pi\)
0.946746 + 0.321981i \(0.104349\pi\)
\(678\) 0 0
\(679\) 37.2793 1.43065
\(680\) −4.87545 −0.186965
\(681\) 0 0
\(682\) 19.6452 0.752254
\(683\) 18.1420 0.694183 0.347092 0.937831i \(-0.387169\pi\)
0.347092 + 0.937831i \(0.387169\pi\)
\(684\) 0 0
\(685\) −16.7376 −0.639509
\(686\) 41.7291 1.59322
\(687\) 0 0
\(688\) −8.53922 −0.325555
\(689\) −1.83293 −0.0698291
\(690\) 0 0
\(691\) 39.8244 1.51499 0.757496 0.652840i \(-0.226423\pi\)
0.757496 + 0.652840i \(0.226423\pi\)
\(692\) −41.2449 −1.56790
\(693\) 0 0
\(694\) 21.5189 0.816846
\(695\) −66.1789 −2.51031
\(696\) 0 0
\(697\) −6.93664 −0.262744
\(698\) −23.1514 −0.876292
\(699\) 0 0
\(700\) 52.6895 1.99148
\(701\) 1.05971 0.0400247 0.0200123 0.999800i \(-0.493629\pi\)
0.0200123 + 0.999800i \(0.493629\pi\)
\(702\) 0 0
\(703\) 9.20787 0.347281
\(704\) 11.9122 0.448959
\(705\) 0 0
\(706\) −31.2766 −1.17711
\(707\) −23.3206 −0.877062
\(708\) 0 0
\(709\) −16.3607 −0.614440 −0.307220 0.951639i \(-0.599399\pi\)
−0.307220 + 0.951639i \(0.599399\pi\)
\(710\) −102.124 −3.83266
\(711\) 0 0
\(712\) −10.6176 −0.397913
\(713\) 40.5033 1.51686
\(714\) 0 0
\(715\) −0.658061 −0.0246101
\(716\) −10.3232 −0.385796
\(717\) 0 0
\(718\) −17.4783 −0.652284
\(719\) 19.9404 0.743651 0.371826 0.928303i \(-0.378732\pi\)
0.371826 + 0.928303i \(0.378732\pi\)
\(720\) 0 0
\(721\) −3.86526 −0.143950
\(722\) −112.675 −4.19334
\(723\) 0 0
\(724\) 54.9623 2.04266
\(725\) −47.3900 −1.76002
\(726\) 0 0
\(727\) −42.7616 −1.58594 −0.792969 0.609262i \(-0.791466\pi\)
−0.792969 + 0.609262i \(0.791466\pi\)
\(728\) −0.592819 −0.0219713
\(729\) 0 0
\(730\) −63.8386 −2.36277
\(731\) 3.64434 0.134791
\(732\) 0 0
\(733\) −47.7154 −1.76241 −0.881204 0.472736i \(-0.843267\pi\)
−0.881204 + 0.472736i \(0.843267\pi\)
\(734\) −22.3592 −0.825292
\(735\) 0 0
\(736\) 34.4786 1.27090
\(737\) 2.55535 0.0941274
\(738\) 0 0
\(739\) −43.5566 −1.60226 −0.801128 0.598494i \(-0.795766\pi\)
−0.801128 + 0.598494i \(0.795766\pi\)
\(740\) 10.2971 0.378527
\(741\) 0 0
\(742\) 53.8735 1.97776
\(743\) −10.8289 −0.397275 −0.198637 0.980073i \(-0.563652\pi\)
−0.198637 + 0.980073i \(0.563652\pi\)
\(744\) 0 0
\(745\) 70.5576 2.58503
\(746\) −30.9551 −1.13335
\(747\) 0 0
\(748\) −2.68016 −0.0979963
\(749\) 40.2195 1.46959
\(750\) 0 0
\(751\) 40.1160 1.46385 0.731926 0.681384i \(-0.238621\pi\)
0.731926 + 0.681384i \(0.238621\pi\)
\(752\) 9.60871 0.350394
\(753\) 0 0
\(754\) 2.27917 0.0830023
\(755\) −79.8669 −2.90665
\(756\) 0 0
\(757\) −27.3129 −0.992705 −0.496353 0.868121i \(-0.665328\pi\)
−0.496353 + 0.868121i \(0.665328\pi\)
\(758\) −27.3726 −0.994218
\(759\) 0 0
\(760\) −40.1510 −1.45643
\(761\) −1.27815 −0.0463331 −0.0231665 0.999732i \(-0.507375\pi\)
−0.0231665 + 0.999732i \(0.507375\pi\)
\(762\) 0 0
\(763\) −8.83229 −0.319750
\(764\) 42.1098 1.52348
\(765\) 0 0
\(766\) 21.7792 0.786914
\(767\) 0.203641 0.00735304
\(768\) 0 0
\(769\) 16.2719 0.586781 0.293391 0.955993i \(-0.405216\pi\)
0.293391 + 0.955993i \(0.405216\pi\)
\(770\) 19.3417 0.697027
\(771\) 0 0
\(772\) 41.9418 1.50952
\(773\) 43.2949 1.55721 0.778604 0.627515i \(-0.215928\pi\)
0.778604 + 0.627515i \(0.215928\pi\)
\(774\) 0 0
\(775\) 74.2294 2.66640
\(776\) −19.6554 −0.705588
\(777\) 0 0
\(778\) 36.5999 1.31217
\(779\) −57.1256 −2.04674
\(780\) 0 0
\(781\) −13.1336 −0.469957
\(782\) −9.75885 −0.348976
\(783\) 0 0
\(784\) 1.95494 0.0698192
\(785\) −44.2884 −1.58072
\(786\) 0 0
\(787\) 1.65622 0.0590380 0.0295190 0.999564i \(-0.490602\pi\)
0.0295190 + 0.999564i \(0.490602\pi\)
\(788\) −28.0164 −0.998042
\(789\) 0 0
\(790\) −41.6731 −1.48266
\(791\) 16.8666 0.599707
\(792\) 0 0
\(793\) −0.181722 −0.00645314
\(794\) 22.9513 0.814510
\(795\) 0 0
\(796\) −4.58336 −0.162453
\(797\) −3.43162 −0.121554 −0.0607772 0.998151i \(-0.519358\pi\)
−0.0607772 + 0.998151i \(0.519358\pi\)
\(798\) 0 0
\(799\) −4.10078 −0.145075
\(800\) 63.1881 2.23404
\(801\) 0 0
\(802\) −6.78558 −0.239607
\(803\) −8.20991 −0.289721
\(804\) 0 0
\(805\) 39.8775 1.40550
\(806\) −3.56997 −0.125747
\(807\) 0 0
\(808\) 12.2957 0.432562
\(809\) 17.6124 0.619221 0.309610 0.950864i \(-0.399801\pi\)
0.309610 + 0.950864i \(0.399801\pi\)
\(810\) 0 0
\(811\) −33.2624 −1.16800 −0.584001 0.811753i \(-0.698514\pi\)
−0.584001 + 0.811753i \(0.698514\pi\)
\(812\) −37.9315 −1.33113
\(813\) 0 0
\(814\) 2.33869 0.0819710
\(815\) 73.3614 2.56974
\(816\) 0 0
\(817\) 30.0124 1.05000
\(818\) 28.1250 0.983367
\(819\) 0 0
\(820\) −63.8829 −2.23089
\(821\) 38.6262 1.34806 0.674032 0.738702i \(-0.264561\pi\)
0.674032 + 0.738702i \(0.264561\pi\)
\(822\) 0 0
\(823\) 33.8042 1.17834 0.589170 0.808009i \(-0.299455\pi\)
0.589170 + 0.808009i \(0.299455\pi\)
\(824\) 2.03795 0.0709953
\(825\) 0 0
\(826\) −5.98540 −0.208259
\(827\) −0.557114 −0.0193727 −0.00968637 0.999953i \(-0.503083\pi\)
−0.00968637 + 0.999953i \(0.503083\pi\)
\(828\) 0 0
\(829\) 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 0
\(832\) −2.16472 −0.0750481
\(833\) −0.834322 −0.0289075
\(834\) 0 0
\(835\) −7.14642 −0.247312
\(836\) −22.0720 −0.763376
\(837\) 0 0
\(838\) 46.1123 1.59292
\(839\) 11.9950 0.414112 0.207056 0.978329i \(-0.433612\pi\)
0.207056 + 0.978329i \(0.433612\pi\)
\(840\) 0 0
\(841\) 5.11637 0.176427
\(842\) 30.1599 1.03938
\(843\) 0 0
\(844\) −11.4985 −0.395793
\(845\) −46.9567 −1.61536
\(846\) 0 0
\(847\) 2.48742 0.0854689
\(848\) 24.2622 0.833166
\(849\) 0 0
\(850\) −17.8848 −0.613444
\(851\) 4.82176 0.165288
\(852\) 0 0
\(853\) −22.8770 −0.783294 −0.391647 0.920115i \(-0.628095\pi\)
−0.391647 + 0.920115i \(0.628095\pi\)
\(854\) 5.34117 0.182771
\(855\) 0 0
\(856\) −21.2056 −0.724793
\(857\) −42.2589 −1.44354 −0.721768 0.692135i \(-0.756670\pi\)
−0.721768 + 0.692135i \(0.756670\pi\)
\(858\) 0 0
\(859\) 14.4661 0.493576 0.246788 0.969070i \(-0.420625\pi\)
0.246788 + 0.969070i \(0.420625\pi\)
\(860\) 33.5625 1.14447
\(861\) 0 0
\(862\) 49.7996 1.69618
\(863\) 34.2785 1.16685 0.583427 0.812166i \(-0.301711\pi\)
0.583427 + 0.812166i \(0.301711\pi\)
\(864\) 0 0
\(865\) −57.2084 −1.94515
\(866\) −31.6127 −1.07424
\(867\) 0 0
\(868\) 59.4140 2.01664
\(869\) −5.35933 −0.181803
\(870\) 0 0
\(871\) −0.464363 −0.0157343
\(872\) 4.65681 0.157699
\(873\) 0 0
\(874\) −80.3674 −2.71847
\(875\) 28.0447 0.948083
\(876\) 0 0
\(877\) 50.0011 1.68842 0.844209 0.536014i \(-0.180070\pi\)
0.844209 + 0.536014i \(0.180070\pi\)
\(878\) 78.7757 2.65855
\(879\) 0 0
\(880\) 8.71063 0.293635
\(881\) 5.16237 0.173925 0.0869624 0.996212i \(-0.472284\pi\)
0.0869624 + 0.996212i \(0.472284\pi\)
\(882\) 0 0
\(883\) 54.9858 1.85042 0.925209 0.379457i \(-0.123889\pi\)
0.925209 + 0.379457i \(0.123889\pi\)
\(884\) 0.487044 0.0163811
\(885\) 0 0
\(886\) 89.0213 2.99073
\(887\) 19.7988 0.664779 0.332389 0.943142i \(-0.392145\pi\)
0.332389 + 0.943142i \(0.392145\pi\)
\(888\) 0 0
\(889\) 7.38859 0.247805
\(890\) −62.9519 −2.11015
\(891\) 0 0
\(892\) −25.6306 −0.858177
\(893\) −33.7713 −1.13011
\(894\) 0 0
\(895\) −14.3187 −0.478622
\(896\) 24.8809 0.831212
\(897\) 0 0
\(898\) −16.7203 −0.557965
\(899\) −53.4382 −1.78226
\(900\) 0 0
\(901\) −10.3545 −0.344959
\(902\) −14.5092 −0.483104
\(903\) 0 0
\(904\) −8.89287 −0.295772
\(905\) 76.2351 2.53414
\(906\) 0 0
\(907\) −46.3300 −1.53836 −0.769181 0.639031i \(-0.779336\pi\)
−0.769181 + 0.639031i \(0.779336\pi\)
\(908\) −48.4729 −1.60863
\(909\) 0 0
\(910\) −3.51482 −0.116515
\(911\) 44.9211 1.48830 0.744151 0.668011i \(-0.232854\pi\)
0.744151 + 0.668011i \(0.232854\pi\)
\(912\) 0 0
\(913\) 9.24024 0.305807
\(914\) −78.3859 −2.59278
\(915\) 0 0
\(916\) −48.8042 −1.61254
\(917\) −30.4596 −1.00587
\(918\) 0 0
\(919\) −30.9670 −1.02151 −0.510754 0.859727i \(-0.670634\pi\)
−0.510754 + 0.859727i \(0.670634\pi\)
\(920\) −21.0253 −0.693185
\(921\) 0 0
\(922\) 4.81186 0.158470
\(923\) 2.38667 0.0785581
\(924\) 0 0
\(925\) 8.83673 0.290550
\(926\) 21.1833 0.696126
\(927\) 0 0
\(928\) −45.4895 −1.49327
\(929\) 34.3813 1.12801 0.564007 0.825770i \(-0.309259\pi\)
0.564007 + 0.825770i \(0.309259\pi\)
\(930\) 0 0
\(931\) −6.87092 −0.225185
\(932\) 34.3730 1.12593
\(933\) 0 0
\(934\) −19.2784 −0.630807
\(935\) −3.71749 −0.121575
\(936\) 0 0
\(937\) −8.57984 −0.280291 −0.140146 0.990131i \(-0.544757\pi\)
−0.140146 + 0.990131i \(0.544757\pi\)
\(938\) 13.6485 0.445641
\(939\) 0 0
\(940\) −37.7660 −1.23179
\(941\) 42.1060 1.37262 0.686309 0.727310i \(-0.259230\pi\)
0.686309 + 0.727310i \(0.259230\pi\)
\(942\) 0 0
\(943\) −29.9142 −0.974140
\(944\) −2.69555 −0.0877328
\(945\) 0 0
\(946\) 7.62280 0.247838
\(947\) 54.0758 1.75723 0.878614 0.477532i \(-0.158469\pi\)
0.878614 + 0.477532i \(0.158469\pi\)
\(948\) 0 0
\(949\) 1.49192 0.0484298
\(950\) −147.287 −4.77863
\(951\) 0 0
\(952\) −3.34893 −0.108539
\(953\) 23.6590 0.766392 0.383196 0.923667i \(-0.374823\pi\)
0.383196 + 0.923667i \(0.374823\pi\)
\(954\) 0 0
\(955\) 58.4080 1.89004
\(956\) 47.0277 1.52099
\(957\) 0 0
\(958\) 34.3572 1.11003
\(959\) −11.4970 −0.371257
\(960\) 0 0
\(961\) 52.7029 1.70009
\(962\) −0.424992 −0.0137023
\(963\) 0 0
\(964\) −74.9078 −2.41262
\(965\) 58.1751 1.87272
\(966\) 0 0
\(967\) −36.9302 −1.18759 −0.593797 0.804615i \(-0.702372\pi\)
−0.593797 + 0.804615i \(0.702372\pi\)
\(968\) −1.31149 −0.0421528
\(969\) 0 0
\(970\) −116.537 −3.74177
\(971\) −9.30267 −0.298537 −0.149268 0.988797i \(-0.547692\pi\)
−0.149268 + 0.988797i \(0.547692\pi\)
\(972\) 0 0
\(973\) −45.4581 −1.45732
\(974\) −8.13786 −0.260754
\(975\) 0 0
\(976\) 2.40542 0.0769956
\(977\) 5.96172 0.190732 0.0953662 0.995442i \(-0.469598\pi\)
0.0953662 + 0.995442i \(0.469598\pi\)
\(978\) 0 0
\(979\) −8.09587 −0.258745
\(980\) −7.68368 −0.245446
\(981\) 0 0
\(982\) −17.8487 −0.569575
\(983\) −33.8665 −1.08018 −0.540088 0.841609i \(-0.681609\pi\)
−0.540088 + 0.841609i \(0.681609\pi\)
\(984\) 0 0
\(985\) −38.8599 −1.23818
\(986\) 12.8754 0.410035
\(987\) 0 0
\(988\) 4.01097 0.127606
\(989\) 15.7162 0.499746
\(990\) 0 0
\(991\) −7.02387 −0.223121 −0.111560 0.993758i \(-0.535585\pi\)
−0.111560 + 0.993758i \(0.535585\pi\)
\(992\) 71.2524 2.26227
\(993\) 0 0
\(994\) −70.1488 −2.22499
\(995\) −6.35732 −0.201541
\(996\) 0 0
\(997\) 28.9360 0.916412 0.458206 0.888846i \(-0.348492\pi\)
0.458206 + 0.888846i \(0.348492\pi\)
\(998\) −69.4036 −2.19693
\(999\) 0 0
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 6039.2.a.i.1.3 13
3.2 odd 2 2013.2.a.e.1.11 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.11 13 3.2 odd 2
6039.2.a.i.1.3 13 1.1 even 1 trivial