Properties

Label 6039.2.a.i.1.12
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 1.12
Root \(-2.37960\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.37960 q^{2} +3.66252 q^{4} -2.55185 q^{5} +1.67712 q^{7} +3.95613 q^{8} +O(q^{10})\) \(q+2.37960 q^{2} +3.66252 q^{4} -2.55185 q^{5} +1.67712 q^{7} +3.95613 q^{8} -6.07239 q^{10} -1.00000 q^{11} +5.65147 q^{13} +3.99087 q^{14} +2.08899 q^{16} +4.01907 q^{17} +2.77230 q^{19} -9.34619 q^{20} -2.37960 q^{22} -6.13877 q^{23} +1.51194 q^{25} +13.4483 q^{26} +6.14247 q^{28} -1.25605 q^{29} +5.37742 q^{31} -2.94129 q^{32} +9.56379 q^{34} -4.27975 q^{35} +2.89012 q^{37} +6.59699 q^{38} -10.0954 q^{40} -6.31673 q^{41} +9.33950 q^{43} -3.66252 q^{44} -14.6078 q^{46} +8.72405 q^{47} -4.18728 q^{49} +3.59781 q^{50} +20.6986 q^{52} +4.90573 q^{53} +2.55185 q^{55} +6.63489 q^{56} -2.98889 q^{58} +4.68512 q^{59} -1.00000 q^{61} +12.7961 q^{62} -11.1771 q^{64} -14.4217 q^{65} +5.67098 q^{67} +14.7199 q^{68} -10.1841 q^{70} +2.71844 q^{71} +15.1095 q^{73} +6.87735 q^{74} +10.1536 q^{76} -1.67712 q^{77} -1.47612 q^{79} -5.33079 q^{80} -15.0313 q^{82} +2.19058 q^{83} -10.2561 q^{85} +22.2243 q^{86} -3.95613 q^{88} +13.1550 q^{89} +9.47818 q^{91} -22.4833 q^{92} +20.7598 q^{94} -7.07450 q^{95} -7.84526 q^{97} -9.96407 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37960 1.68263 0.841317 0.540542i \(-0.181781\pi\)
0.841317 + 0.540542i \(0.181781\pi\)
\(3\) 0 0
\(4\) 3.66252 1.83126
\(5\) −2.55185 −1.14122 −0.570611 0.821221i \(-0.693294\pi\)
−0.570611 + 0.821221i \(0.693294\pi\)
\(6\) 0 0
\(7\) 1.67712 0.633891 0.316945 0.948444i \(-0.397343\pi\)
0.316945 + 0.948444i \(0.397343\pi\)
\(8\) 3.95613 1.39870
\(9\) 0 0
\(10\) −6.07239 −1.92026
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.65147 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(14\) 3.99087 1.06661
\(15\) 0 0
\(16\) 2.08899 0.522248
\(17\) 4.01907 0.974767 0.487383 0.873188i \(-0.337951\pi\)
0.487383 + 0.873188i \(0.337951\pi\)
\(18\) 0 0
\(19\) 2.77230 0.636010 0.318005 0.948089i \(-0.396987\pi\)
0.318005 + 0.948089i \(0.396987\pi\)
\(20\) −9.34619 −2.08987
\(21\) 0 0
\(22\) −2.37960 −0.507333
\(23\) −6.13877 −1.28002 −0.640011 0.768366i \(-0.721070\pi\)
−0.640011 + 0.768366i \(0.721070\pi\)
\(24\) 0 0
\(25\) 1.51194 0.302387
\(26\) 13.4483 2.63742
\(27\) 0 0
\(28\) 6.14247 1.16082
\(29\) −1.25605 −0.233242 −0.116621 0.993177i \(-0.537206\pi\)
−0.116621 + 0.993177i \(0.537206\pi\)
\(30\) 0 0
\(31\) 5.37742 0.965814 0.482907 0.875672i \(-0.339581\pi\)
0.482907 + 0.875672i \(0.339581\pi\)
\(32\) −2.94129 −0.519951
\(33\) 0 0
\(34\) 9.56379 1.64018
\(35\) −4.27975 −0.723410
\(36\) 0 0
\(37\) 2.89012 0.475134 0.237567 0.971371i \(-0.423650\pi\)
0.237567 + 0.971371i \(0.423650\pi\)
\(38\) 6.59699 1.07017
\(39\) 0 0
\(40\) −10.0954 −1.59623
\(41\) −6.31673 −0.986508 −0.493254 0.869885i \(-0.664193\pi\)
−0.493254 + 0.869885i \(0.664193\pi\)
\(42\) 0 0
\(43\) 9.33950 1.42426 0.712130 0.702047i \(-0.247731\pi\)
0.712130 + 0.702047i \(0.247731\pi\)
\(44\) −3.66252 −0.552145
\(45\) 0 0
\(46\) −14.6078 −2.15381
\(47\) 8.72405 1.27253 0.636267 0.771469i \(-0.280478\pi\)
0.636267 + 0.771469i \(0.280478\pi\)
\(48\) 0 0
\(49\) −4.18728 −0.598183
\(50\) 3.59781 0.508807
\(51\) 0 0
\(52\) 20.6986 2.87038
\(53\) 4.90573 0.673854 0.336927 0.941531i \(-0.390612\pi\)
0.336927 + 0.941531i \(0.390612\pi\)
\(54\) 0 0
\(55\) 2.55185 0.344091
\(56\) 6.63489 0.886625
\(57\) 0 0
\(58\) −2.98889 −0.392461
\(59\) 4.68512 0.609951 0.304976 0.952360i \(-0.401352\pi\)
0.304976 + 0.952360i \(0.401352\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 12.7961 1.62511
\(63\) 0 0
\(64\) −11.1771 −1.39714
\(65\) −14.4217 −1.78879
\(66\) 0 0
\(67\) 5.67098 0.692820 0.346410 0.938083i \(-0.387401\pi\)
0.346410 + 0.938083i \(0.387401\pi\)
\(68\) 14.7199 1.78505
\(69\) 0 0
\(70\) −10.1841 −1.21723
\(71\) 2.71844 0.322620 0.161310 0.986904i \(-0.448428\pi\)
0.161310 + 0.986904i \(0.448428\pi\)
\(72\) 0 0
\(73\) 15.1095 1.76843 0.884215 0.467080i \(-0.154694\pi\)
0.884215 + 0.467080i \(0.154694\pi\)
\(74\) 6.87735 0.799476
\(75\) 0 0
\(76\) 10.1536 1.16470
\(77\) −1.67712 −0.191125
\(78\) 0 0
\(79\) −1.47612 −0.166076 −0.0830381 0.996546i \(-0.526462\pi\)
−0.0830381 + 0.996546i \(0.526462\pi\)
\(80\) −5.33079 −0.596001
\(81\) 0 0
\(82\) −15.0313 −1.65993
\(83\) 2.19058 0.240448 0.120224 0.992747i \(-0.461639\pi\)
0.120224 + 0.992747i \(0.461639\pi\)
\(84\) 0 0
\(85\) −10.2561 −1.11243
\(86\) 22.2243 2.39651
\(87\) 0 0
\(88\) −3.95613 −0.421725
\(89\) 13.1550 1.39442 0.697212 0.716865i \(-0.254424\pi\)
0.697212 + 0.716865i \(0.254424\pi\)
\(90\) 0 0
\(91\) 9.47818 0.993583
\(92\) −22.4833 −2.34405
\(93\) 0 0
\(94\) 20.7598 2.14121
\(95\) −7.07450 −0.725829
\(96\) 0 0
\(97\) −7.84526 −0.796565 −0.398283 0.917263i \(-0.630394\pi\)
−0.398283 + 0.917263i \(0.630394\pi\)
\(98\) −9.96407 −1.00652
\(99\) 0 0
\(100\) 5.53749 0.553749
\(101\) 3.23484 0.321879 0.160939 0.986964i \(-0.448548\pi\)
0.160939 + 0.986964i \(0.448548\pi\)
\(102\) 0 0
\(103\) −7.86347 −0.774811 −0.387405 0.921909i \(-0.626629\pi\)
−0.387405 + 0.921909i \(0.626629\pi\)
\(104\) 22.3580 2.19238
\(105\) 0 0
\(106\) 11.6737 1.13385
\(107\) 18.7522 1.81284 0.906420 0.422378i \(-0.138805\pi\)
0.906420 + 0.422378i \(0.138805\pi\)
\(108\) 0 0
\(109\) −19.5666 −1.87414 −0.937070 0.349141i \(-0.886473\pi\)
−0.937070 + 0.349141i \(0.886473\pi\)
\(110\) 6.07239 0.578980
\(111\) 0 0
\(112\) 3.50348 0.331048
\(113\) −11.4456 −1.07671 −0.538356 0.842718i \(-0.680954\pi\)
−0.538356 + 0.842718i \(0.680954\pi\)
\(114\) 0 0
\(115\) 15.6652 1.46079
\(116\) −4.60029 −0.427126
\(117\) 0 0
\(118\) 11.1487 1.02632
\(119\) 6.74045 0.617896
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.37960 −0.215439
\(123\) 0 0
\(124\) 19.6949 1.76865
\(125\) 8.90101 0.796131
\(126\) 0 0
\(127\) 1.63206 0.144822 0.0724109 0.997375i \(-0.476931\pi\)
0.0724109 + 0.997375i \(0.476931\pi\)
\(128\) −20.7145 −1.83092
\(129\) 0 0
\(130\) −34.3179 −3.00988
\(131\) −5.81437 −0.508004 −0.254002 0.967204i \(-0.581747\pi\)
−0.254002 + 0.967204i \(0.581747\pi\)
\(132\) 0 0
\(133\) 4.64948 0.403161
\(134\) 13.4947 1.16576
\(135\) 0 0
\(136\) 15.9000 1.36341
\(137\) 0.124547 0.0106407 0.00532037 0.999986i \(-0.498306\pi\)
0.00532037 + 0.999986i \(0.498306\pi\)
\(138\) 0 0
\(139\) 5.70439 0.483840 0.241920 0.970296i \(-0.422223\pi\)
0.241920 + 0.970296i \(0.422223\pi\)
\(140\) −15.6747 −1.32475
\(141\) 0 0
\(142\) 6.46881 0.542851
\(143\) −5.65147 −0.472600
\(144\) 0 0
\(145\) 3.20524 0.266181
\(146\) 35.9546 2.97562
\(147\) 0 0
\(148\) 10.5851 0.870092
\(149\) 20.0285 1.64080 0.820399 0.571792i \(-0.193751\pi\)
0.820399 + 0.571792i \(0.193751\pi\)
\(150\) 0 0
\(151\) 8.05186 0.655251 0.327626 0.944808i \(-0.393752\pi\)
0.327626 + 0.944808i \(0.393752\pi\)
\(152\) 10.9676 0.889589
\(153\) 0 0
\(154\) −3.99087 −0.321594
\(155\) −13.7224 −1.10221
\(156\) 0 0
\(157\) 10.2781 0.820281 0.410141 0.912022i \(-0.365480\pi\)
0.410141 + 0.912022i \(0.365480\pi\)
\(158\) −3.51258 −0.279446
\(159\) 0 0
\(160\) 7.50572 0.593379
\(161\) −10.2954 −0.811394
\(162\) 0 0
\(163\) −6.68649 −0.523727 −0.261863 0.965105i \(-0.584337\pi\)
−0.261863 + 0.965105i \(0.584337\pi\)
\(164\) −23.1351 −1.80655
\(165\) 0 0
\(166\) 5.21272 0.404585
\(167\) −14.0284 −1.08555 −0.542773 0.839879i \(-0.682626\pi\)
−0.542773 + 0.839879i \(0.682626\pi\)
\(168\) 0 0
\(169\) 18.9391 1.45686
\(170\) −24.4054 −1.87180
\(171\) 0 0
\(172\) 34.2061 2.60819
\(173\) −10.3570 −0.787430 −0.393715 0.919233i \(-0.628810\pi\)
−0.393715 + 0.919233i \(0.628810\pi\)
\(174\) 0 0
\(175\) 2.53569 0.191680
\(176\) −2.08899 −0.157464
\(177\) 0 0
\(178\) 31.3036 2.34630
\(179\) 7.34446 0.548951 0.274476 0.961594i \(-0.411496\pi\)
0.274476 + 0.961594i \(0.411496\pi\)
\(180\) 0 0
\(181\) −0.531235 −0.0394864 −0.0197432 0.999805i \(-0.506285\pi\)
−0.0197432 + 0.999805i \(0.506285\pi\)
\(182\) 22.5543 1.67184
\(183\) 0 0
\(184\) −24.2858 −1.79037
\(185\) −7.37516 −0.542233
\(186\) 0 0
\(187\) −4.01907 −0.293903
\(188\) 31.9520 2.33034
\(189\) 0 0
\(190\) −16.8345 −1.22130
\(191\) −16.5231 −1.19557 −0.597785 0.801656i \(-0.703952\pi\)
−0.597785 + 0.801656i \(0.703952\pi\)
\(192\) 0 0
\(193\) 17.9043 1.28878 0.644391 0.764697i \(-0.277111\pi\)
0.644391 + 0.764697i \(0.277111\pi\)
\(194\) −18.6686 −1.34033
\(195\) 0 0
\(196\) −15.3360 −1.09543
\(197\) −9.54976 −0.680392 −0.340196 0.940355i \(-0.610493\pi\)
−0.340196 + 0.940355i \(0.610493\pi\)
\(198\) 0 0
\(199\) 7.91086 0.560786 0.280393 0.959885i \(-0.409535\pi\)
0.280393 + 0.959885i \(0.409535\pi\)
\(200\) 5.98142 0.422950
\(201\) 0 0
\(202\) 7.69764 0.541604
\(203\) −2.10654 −0.147850
\(204\) 0 0
\(205\) 16.1194 1.12582
\(206\) −18.7120 −1.30372
\(207\) 0 0
\(208\) 11.8059 0.818590
\(209\) −2.77230 −0.191764
\(210\) 0 0
\(211\) −8.64980 −0.595477 −0.297738 0.954648i \(-0.596232\pi\)
−0.297738 + 0.954648i \(0.596232\pi\)
\(212\) 17.9673 1.23400
\(213\) 0 0
\(214\) 44.6227 3.05035
\(215\) −23.8330 −1.62540
\(216\) 0 0
\(217\) 9.01857 0.612220
\(218\) −46.5608 −3.15349
\(219\) 0 0
\(220\) 9.34619 0.630120
\(221\) 22.7136 1.52788
\(222\) 0 0
\(223\) 8.83430 0.591588 0.295794 0.955252i \(-0.404416\pi\)
0.295794 + 0.955252i \(0.404416\pi\)
\(224\) −4.93288 −0.329592
\(225\) 0 0
\(226\) −27.2360 −1.81171
\(227\) −15.6307 −1.03744 −0.518722 0.854943i \(-0.673592\pi\)
−0.518722 + 0.854943i \(0.673592\pi\)
\(228\) 0 0
\(229\) −13.1402 −0.868329 −0.434164 0.900834i \(-0.642956\pi\)
−0.434164 + 0.900834i \(0.642956\pi\)
\(230\) 37.2770 2.45797
\(231\) 0 0
\(232\) −4.96908 −0.326236
\(233\) 19.2138 1.25874 0.629368 0.777108i \(-0.283314\pi\)
0.629368 + 0.777108i \(0.283314\pi\)
\(234\) 0 0
\(235\) −22.2625 −1.45224
\(236\) 17.1593 1.11698
\(237\) 0 0
\(238\) 16.0396 1.03969
\(239\) −13.3971 −0.866585 −0.433293 0.901253i \(-0.642648\pi\)
−0.433293 + 0.901253i \(0.642648\pi\)
\(240\) 0 0
\(241\) −25.2756 −1.62814 −0.814072 0.580765i \(-0.802754\pi\)
−0.814072 + 0.580765i \(0.802754\pi\)
\(242\) 2.37960 0.152967
\(243\) 0 0
\(244\) −3.66252 −0.234469
\(245\) 10.6853 0.682659
\(246\) 0 0
\(247\) 15.6676 0.996905
\(248\) 21.2738 1.35089
\(249\) 0 0
\(250\) 21.1809 1.33960
\(251\) −16.2067 −1.02296 −0.511480 0.859295i \(-0.670903\pi\)
−0.511480 + 0.859295i \(0.670903\pi\)
\(252\) 0 0
\(253\) 6.13877 0.385941
\(254\) 3.88366 0.243682
\(255\) 0 0
\(256\) −26.9380 −1.68363
\(257\) 16.7068 1.04214 0.521072 0.853513i \(-0.325532\pi\)
0.521072 + 0.853513i \(0.325532\pi\)
\(258\) 0 0
\(259\) 4.84708 0.301183
\(260\) −52.8197 −3.27574
\(261\) 0 0
\(262\) −13.8359 −0.854785
\(263\) −10.1887 −0.628263 −0.314132 0.949379i \(-0.601713\pi\)
−0.314132 + 0.949379i \(0.601713\pi\)
\(264\) 0 0
\(265\) −12.5187 −0.769017
\(266\) 11.0639 0.678372
\(267\) 0 0
\(268\) 20.7700 1.26873
\(269\) −1.61445 −0.0984350 −0.0492175 0.998788i \(-0.515673\pi\)
−0.0492175 + 0.998788i \(0.515673\pi\)
\(270\) 0 0
\(271\) 25.4804 1.54782 0.773912 0.633294i \(-0.218297\pi\)
0.773912 + 0.633294i \(0.218297\pi\)
\(272\) 8.39580 0.509070
\(273\) 0 0
\(274\) 0.296372 0.0179045
\(275\) −1.51194 −0.0911732
\(276\) 0 0
\(277\) 17.4548 1.04876 0.524378 0.851486i \(-0.324298\pi\)
0.524378 + 0.851486i \(0.324298\pi\)
\(278\) 13.5742 0.814125
\(279\) 0 0
\(280\) −16.9312 −1.01184
\(281\) 16.5498 0.987275 0.493638 0.869668i \(-0.335667\pi\)
0.493638 + 0.869668i \(0.335667\pi\)
\(282\) 0 0
\(283\) 21.5787 1.28272 0.641359 0.767241i \(-0.278371\pi\)
0.641359 + 0.767241i \(0.278371\pi\)
\(284\) 9.95633 0.590800
\(285\) 0 0
\(286\) −13.4483 −0.795213
\(287\) −10.5939 −0.625338
\(288\) 0 0
\(289\) −0.847100 −0.0498294
\(290\) 7.62720 0.447885
\(291\) 0 0
\(292\) 55.3387 3.23845
\(293\) 3.17063 0.185230 0.0926150 0.995702i \(-0.470477\pi\)
0.0926150 + 0.995702i \(0.470477\pi\)
\(294\) 0 0
\(295\) −11.9557 −0.696090
\(296\) 11.4337 0.664571
\(297\) 0 0
\(298\) 47.6599 2.76086
\(299\) −34.6931 −2.00635
\(300\) 0 0
\(301\) 15.6634 0.902825
\(302\) 19.1602 1.10255
\(303\) 0 0
\(304\) 5.79132 0.332155
\(305\) 2.55185 0.146118
\(306\) 0 0
\(307\) 11.3935 0.650259 0.325130 0.945669i \(-0.394592\pi\)
0.325130 + 0.945669i \(0.394592\pi\)
\(308\) −6.14247 −0.350000
\(309\) 0 0
\(310\) −32.6538 −1.85461
\(311\) −13.0228 −0.738456 −0.369228 0.929339i \(-0.620378\pi\)
−0.369228 + 0.929339i \(0.620378\pi\)
\(312\) 0 0
\(313\) −31.9422 −1.80548 −0.902740 0.430186i \(-0.858448\pi\)
−0.902740 + 0.430186i \(0.858448\pi\)
\(314\) 24.4578 1.38023
\(315\) 0 0
\(316\) −5.40631 −0.304129
\(317\) −4.27975 −0.240375 −0.120187 0.992751i \(-0.538350\pi\)
−0.120187 + 0.992751i \(0.538350\pi\)
\(318\) 0 0
\(319\) 1.25605 0.0703250
\(320\) 28.5222 1.59444
\(321\) 0 0
\(322\) −24.4991 −1.36528
\(323\) 11.1421 0.619962
\(324\) 0 0
\(325\) 8.54466 0.473973
\(326\) −15.9112 −0.881240
\(327\) 0 0
\(328\) −24.9898 −1.37983
\(329\) 14.6313 0.806647
\(330\) 0 0
\(331\) −30.6153 −1.68277 −0.841385 0.540436i \(-0.818259\pi\)
−0.841385 + 0.540436i \(0.818259\pi\)
\(332\) 8.02304 0.440321
\(333\) 0 0
\(334\) −33.3819 −1.82658
\(335\) −14.4715 −0.790662
\(336\) 0 0
\(337\) 20.1363 1.09689 0.548447 0.836185i \(-0.315219\pi\)
0.548447 + 0.836185i \(0.315219\pi\)
\(338\) 45.0676 2.45136
\(339\) 0 0
\(340\) −37.5630 −2.03714
\(341\) −5.37742 −0.291204
\(342\) 0 0
\(343\) −18.7624 −1.01307
\(344\) 36.9483 1.99212
\(345\) 0 0
\(346\) −24.6456 −1.32496
\(347\) 31.7798 1.70603 0.853015 0.521887i \(-0.174772\pi\)
0.853015 + 0.521887i \(0.174772\pi\)
\(348\) 0 0
\(349\) −28.4164 −1.52110 −0.760548 0.649282i \(-0.775069\pi\)
−0.760548 + 0.649282i \(0.775069\pi\)
\(350\) 6.03395 0.322528
\(351\) 0 0
\(352\) 2.94129 0.156771
\(353\) −13.9859 −0.744392 −0.372196 0.928154i \(-0.621395\pi\)
−0.372196 + 0.928154i \(0.621395\pi\)
\(354\) 0 0
\(355\) −6.93705 −0.368180
\(356\) 48.1803 2.55355
\(357\) 0 0
\(358\) 17.4769 0.923684
\(359\) −9.57884 −0.505552 −0.252776 0.967525i \(-0.581344\pi\)
−0.252776 + 0.967525i \(0.581344\pi\)
\(360\) 0 0
\(361\) −11.3143 −0.595491
\(362\) −1.26413 −0.0664412
\(363\) 0 0
\(364\) 34.7140 1.81951
\(365\) −38.5571 −2.01817
\(366\) 0 0
\(367\) −20.5751 −1.07401 −0.537006 0.843578i \(-0.680445\pi\)
−0.537006 + 0.843578i \(0.680445\pi\)
\(368\) −12.8238 −0.668489
\(369\) 0 0
\(370\) −17.5500 −0.912379
\(371\) 8.22749 0.427150
\(372\) 0 0
\(373\) 27.0493 1.40056 0.700280 0.713868i \(-0.253058\pi\)
0.700280 + 0.713868i \(0.253058\pi\)
\(374\) −9.56379 −0.494532
\(375\) 0 0
\(376\) 34.5135 1.77990
\(377\) −7.09850 −0.365592
\(378\) 0 0
\(379\) −18.5659 −0.953665 −0.476832 0.878994i \(-0.658215\pi\)
−0.476832 + 0.878994i \(0.658215\pi\)
\(380\) −25.9105 −1.32918
\(381\) 0 0
\(382\) −39.3185 −2.01171
\(383\) −3.36015 −0.171696 −0.0858478 0.996308i \(-0.527360\pi\)
−0.0858478 + 0.996308i \(0.527360\pi\)
\(384\) 0 0
\(385\) 4.27975 0.218116
\(386\) 42.6052 2.16855
\(387\) 0 0
\(388\) −28.7334 −1.45872
\(389\) −10.0729 −0.510716 −0.255358 0.966847i \(-0.582193\pi\)
−0.255358 + 0.966847i \(0.582193\pi\)
\(390\) 0 0
\(391\) −24.6721 −1.24772
\(392\) −16.5654 −0.836680
\(393\) 0 0
\(394\) −22.7246 −1.14485
\(395\) 3.76683 0.189530
\(396\) 0 0
\(397\) −11.9242 −0.598461 −0.299230 0.954181i \(-0.596730\pi\)
−0.299230 + 0.954181i \(0.596730\pi\)
\(398\) 18.8247 0.943598
\(399\) 0 0
\(400\) 3.15842 0.157921
\(401\) −11.2101 −0.559804 −0.279902 0.960029i \(-0.590302\pi\)
−0.279902 + 0.960029i \(0.590302\pi\)
\(402\) 0 0
\(403\) 30.3904 1.51385
\(404\) 11.8477 0.589443
\(405\) 0 0
\(406\) −5.01272 −0.248777
\(407\) −2.89012 −0.143258
\(408\) 0 0
\(409\) −29.5649 −1.46189 −0.730944 0.682437i \(-0.760920\pi\)
−0.730944 + 0.682437i \(0.760920\pi\)
\(410\) 38.3577 1.89435
\(411\) 0 0
\(412\) −28.8001 −1.41888
\(413\) 7.85750 0.386642
\(414\) 0 0
\(415\) −5.59003 −0.274404
\(416\) −16.6226 −0.814990
\(417\) 0 0
\(418\) −6.59699 −0.322669
\(419\) −13.1249 −0.641194 −0.320597 0.947216i \(-0.603884\pi\)
−0.320597 + 0.947216i \(0.603884\pi\)
\(420\) 0 0
\(421\) −39.4833 −1.92430 −0.962149 0.272524i \(-0.912142\pi\)
−0.962149 + 0.272524i \(0.912142\pi\)
\(422\) −20.5831 −1.00197
\(423\) 0 0
\(424\) 19.4077 0.942522
\(425\) 6.07657 0.294757
\(426\) 0 0
\(427\) −1.67712 −0.0811614
\(428\) 68.6801 3.31978
\(429\) 0 0
\(430\) −56.7131 −2.73495
\(431\) −8.99935 −0.433483 −0.216742 0.976229i \(-0.569543\pi\)
−0.216742 + 0.976229i \(0.569543\pi\)
\(432\) 0 0
\(433\) −19.8497 −0.953916 −0.476958 0.878926i \(-0.658261\pi\)
−0.476958 + 0.878926i \(0.658261\pi\)
\(434\) 21.4606 1.03014
\(435\) 0 0
\(436\) −71.6630 −3.43203
\(437\) −17.0185 −0.814107
\(438\) 0 0
\(439\) 24.7912 1.18322 0.591609 0.806225i \(-0.298493\pi\)
0.591609 + 0.806225i \(0.298493\pi\)
\(440\) 10.0954 0.481282
\(441\) 0 0
\(442\) 54.0495 2.57087
\(443\) −11.8657 −0.563758 −0.281879 0.959450i \(-0.590958\pi\)
−0.281879 + 0.959450i \(0.590958\pi\)
\(444\) 0 0
\(445\) −33.5695 −1.59135
\(446\) 21.0221 0.995427
\(447\) 0 0
\(448\) −18.7453 −0.885631
\(449\) −5.23662 −0.247132 −0.123566 0.992336i \(-0.539433\pi\)
−0.123566 + 0.992336i \(0.539433\pi\)
\(450\) 0 0
\(451\) 6.31673 0.297443
\(452\) −41.9197 −1.97174
\(453\) 0 0
\(454\) −37.1948 −1.74564
\(455\) −24.1869 −1.13390
\(456\) 0 0
\(457\) 9.12526 0.426862 0.213431 0.976958i \(-0.431536\pi\)
0.213431 + 0.976958i \(0.431536\pi\)
\(458\) −31.2685 −1.46108
\(459\) 0 0
\(460\) 57.3741 2.67508
\(461\) −2.08183 −0.0969604 −0.0484802 0.998824i \(-0.515438\pi\)
−0.0484802 + 0.998824i \(0.515438\pi\)
\(462\) 0 0
\(463\) 27.5553 1.28060 0.640302 0.768124i \(-0.278809\pi\)
0.640302 + 0.768124i \(0.278809\pi\)
\(464\) −2.62387 −0.121810
\(465\) 0 0
\(466\) 45.7211 2.11799
\(467\) −27.9889 −1.29517 −0.647587 0.761992i \(-0.724222\pi\)
−0.647587 + 0.761992i \(0.724222\pi\)
\(468\) 0 0
\(469\) 9.51089 0.439172
\(470\) −52.9759 −2.44359
\(471\) 0 0
\(472\) 18.5350 0.853141
\(473\) −9.33950 −0.429431
\(474\) 0 0
\(475\) 4.19155 0.192321
\(476\) 24.6870 1.13153
\(477\) 0 0
\(478\) −31.8798 −1.45815
\(479\) −0.953001 −0.0435437 −0.0217719 0.999763i \(-0.506931\pi\)
−0.0217719 + 0.999763i \(0.506931\pi\)
\(480\) 0 0
\(481\) 16.3335 0.744741
\(482\) −60.1459 −2.73957
\(483\) 0 0
\(484\) 3.66252 0.166478
\(485\) 20.0199 0.909058
\(486\) 0 0
\(487\) −12.8346 −0.581591 −0.290796 0.956785i \(-0.593920\pi\)
−0.290796 + 0.956785i \(0.593920\pi\)
\(488\) −3.95613 −0.179086
\(489\) 0 0
\(490\) 25.4268 1.14867
\(491\) −39.6400 −1.78893 −0.894465 0.447138i \(-0.852443\pi\)
−0.894465 + 0.447138i \(0.852443\pi\)
\(492\) 0 0
\(493\) −5.04813 −0.227356
\(494\) 37.2827 1.67743
\(495\) 0 0
\(496\) 11.2334 0.504394
\(497\) 4.55914 0.204505
\(498\) 0 0
\(499\) −35.1235 −1.57234 −0.786172 0.618007i \(-0.787940\pi\)
−0.786172 + 0.618007i \(0.787940\pi\)
\(500\) 32.6001 1.45792
\(501\) 0 0
\(502\) −38.5656 −1.72127
\(503\) −2.69005 −0.119943 −0.0599717 0.998200i \(-0.519101\pi\)
−0.0599717 + 0.998200i \(0.519101\pi\)
\(504\) 0 0
\(505\) −8.25482 −0.367335
\(506\) 14.6078 0.649398
\(507\) 0 0
\(508\) 5.97745 0.265206
\(509\) −22.4595 −0.995500 −0.497750 0.867321i \(-0.665840\pi\)
−0.497750 + 0.867321i \(0.665840\pi\)
\(510\) 0 0
\(511\) 25.3403 1.12099
\(512\) −22.6730 −1.00201
\(513\) 0 0
\(514\) 39.7557 1.75355
\(515\) 20.0664 0.884231
\(516\) 0 0
\(517\) −8.72405 −0.383683
\(518\) 11.5341 0.506780
\(519\) 0 0
\(520\) −57.0541 −2.50199
\(521\) −24.9298 −1.09219 −0.546096 0.837722i \(-0.683887\pi\)
−0.546096 + 0.837722i \(0.683887\pi\)
\(522\) 0 0
\(523\) −22.1551 −0.968775 −0.484387 0.874854i \(-0.660957\pi\)
−0.484387 + 0.874854i \(0.660957\pi\)
\(524\) −21.2952 −0.930287
\(525\) 0 0
\(526\) −24.2451 −1.05714
\(527\) 21.6122 0.941444
\(528\) 0 0
\(529\) 14.6845 0.638456
\(530\) −29.7895 −1.29397
\(531\) 0 0
\(532\) 17.0288 0.738292
\(533\) −35.6988 −1.54629
\(534\) 0 0
\(535\) −47.8527 −2.06885
\(536\) 22.4351 0.969050
\(537\) 0 0
\(538\) −3.84176 −0.165630
\(539\) 4.18728 0.180359
\(540\) 0 0
\(541\) 34.9763 1.50375 0.751874 0.659307i \(-0.229150\pi\)
0.751874 + 0.659307i \(0.229150\pi\)
\(542\) 60.6332 2.60442
\(543\) 0 0
\(544\) −11.8212 −0.506831
\(545\) 49.9310 2.13881
\(546\) 0 0
\(547\) 41.5082 1.77476 0.887382 0.461035i \(-0.152522\pi\)
0.887382 + 0.461035i \(0.152522\pi\)
\(548\) 0.456154 0.0194859
\(549\) 0 0
\(550\) −3.59781 −0.153411
\(551\) −3.48214 −0.148344
\(552\) 0 0
\(553\) −2.47562 −0.105274
\(554\) 41.5354 1.76467
\(555\) 0 0
\(556\) 20.8924 0.886036
\(557\) 13.4014 0.567836 0.283918 0.958849i \(-0.408366\pi\)
0.283918 + 0.958849i \(0.408366\pi\)
\(558\) 0 0
\(559\) 52.7819 2.23244
\(560\) −8.94036 −0.377799
\(561\) 0 0
\(562\) 39.3819 1.66122
\(563\) 10.6144 0.447343 0.223671 0.974665i \(-0.428196\pi\)
0.223671 + 0.974665i \(0.428196\pi\)
\(564\) 0 0
\(565\) 29.2074 1.22877
\(566\) 51.3486 2.15834
\(567\) 0 0
\(568\) 10.7545 0.451249
\(569\) 25.0719 1.05107 0.525535 0.850772i \(-0.323865\pi\)
0.525535 + 0.850772i \(0.323865\pi\)
\(570\) 0 0
\(571\) −1.81795 −0.0760790 −0.0380395 0.999276i \(-0.512111\pi\)
−0.0380395 + 0.999276i \(0.512111\pi\)
\(572\) −20.6986 −0.865452
\(573\) 0 0
\(574\) −25.2093 −1.05222
\(575\) −9.28143 −0.387062
\(576\) 0 0
\(577\) 32.7010 1.36136 0.680681 0.732580i \(-0.261684\pi\)
0.680681 + 0.732580i \(0.261684\pi\)
\(578\) −2.01576 −0.0838447
\(579\) 0 0
\(580\) 11.7392 0.487445
\(581\) 3.67386 0.152417
\(582\) 0 0
\(583\) −4.90573 −0.203175
\(584\) 59.7750 2.47351
\(585\) 0 0
\(586\) 7.54484 0.311674
\(587\) −24.0218 −0.991488 −0.495744 0.868469i \(-0.665104\pi\)
−0.495744 + 0.868469i \(0.665104\pi\)
\(588\) 0 0
\(589\) 14.9079 0.614268
\(590\) −28.4499 −1.17126
\(591\) 0 0
\(592\) 6.03745 0.248138
\(593\) 3.62291 0.148775 0.0743875 0.997229i \(-0.476300\pi\)
0.0743875 + 0.997229i \(0.476300\pi\)
\(594\) 0 0
\(595\) −17.2006 −0.705156
\(596\) 73.3547 3.00472
\(597\) 0 0
\(598\) −82.5558 −3.37596
\(599\) −19.0751 −0.779389 −0.389695 0.920944i \(-0.627419\pi\)
−0.389695 + 0.920944i \(0.627419\pi\)
\(600\) 0 0
\(601\) −9.56199 −0.390042 −0.195021 0.980799i \(-0.562477\pi\)
−0.195021 + 0.980799i \(0.562477\pi\)
\(602\) 37.2728 1.51912
\(603\) 0 0
\(604\) 29.4901 1.19993
\(605\) −2.55185 −0.103747
\(606\) 0 0
\(607\) −33.3670 −1.35432 −0.677162 0.735834i \(-0.736790\pi\)
−0.677162 + 0.735834i \(0.736790\pi\)
\(608\) −8.15414 −0.330694
\(609\) 0 0
\(610\) 6.07239 0.245864
\(611\) 49.3037 1.99462
\(612\) 0 0
\(613\) 21.0516 0.850268 0.425134 0.905131i \(-0.360227\pi\)
0.425134 + 0.905131i \(0.360227\pi\)
\(614\) 27.1119 1.09415
\(615\) 0 0
\(616\) −6.63489 −0.267327
\(617\) −3.39567 −0.136705 −0.0683523 0.997661i \(-0.521774\pi\)
−0.0683523 + 0.997661i \(0.521774\pi\)
\(618\) 0 0
\(619\) −11.3446 −0.455980 −0.227990 0.973664i \(-0.573215\pi\)
−0.227990 + 0.973664i \(0.573215\pi\)
\(620\) −50.2584 −2.01843
\(621\) 0 0
\(622\) −30.9892 −1.24255
\(623\) 22.0624 0.883912
\(624\) 0 0
\(625\) −30.2737 −1.21095
\(626\) −76.0098 −3.03796
\(627\) 0 0
\(628\) 37.6437 1.50215
\(629\) 11.6156 0.463144
\(630\) 0 0
\(631\) 44.0848 1.75499 0.877494 0.479587i \(-0.159214\pi\)
0.877494 + 0.479587i \(0.159214\pi\)
\(632\) −5.83972 −0.232291
\(633\) 0 0
\(634\) −10.1841 −0.404463
\(635\) −4.16477 −0.165274
\(636\) 0 0
\(637\) −23.6643 −0.937613
\(638\) 2.98889 0.118331
\(639\) 0 0
\(640\) 52.8602 2.08948
\(641\) 5.41854 0.214019 0.107010 0.994258i \(-0.465872\pi\)
0.107010 + 0.994258i \(0.465872\pi\)
\(642\) 0 0
\(643\) 19.3418 0.762768 0.381384 0.924417i \(-0.375448\pi\)
0.381384 + 0.924417i \(0.375448\pi\)
\(644\) −37.7072 −1.48587
\(645\) 0 0
\(646\) 26.5137 1.04317
\(647\) −21.8968 −0.860852 −0.430426 0.902626i \(-0.641637\pi\)
−0.430426 + 0.902626i \(0.641637\pi\)
\(648\) 0 0
\(649\) −4.68512 −0.183907
\(650\) 20.3329 0.797523
\(651\) 0 0
\(652\) −24.4894 −0.959078
\(653\) 4.80308 0.187959 0.0939795 0.995574i \(-0.470041\pi\)
0.0939795 + 0.995574i \(0.470041\pi\)
\(654\) 0 0
\(655\) 14.8374 0.579746
\(656\) −13.1956 −0.515202
\(657\) 0 0
\(658\) 34.8166 1.35729
\(659\) 29.9093 1.16510 0.582550 0.812795i \(-0.302055\pi\)
0.582550 + 0.812795i \(0.302055\pi\)
\(660\) 0 0
\(661\) −33.4154 −1.29971 −0.649854 0.760059i \(-0.725170\pi\)
−0.649854 + 0.760059i \(0.725170\pi\)
\(662\) −72.8523 −2.83149
\(663\) 0 0
\(664\) 8.66623 0.336315
\(665\) −11.8648 −0.460096
\(666\) 0 0
\(667\) 7.71057 0.298555
\(668\) −51.3791 −1.98792
\(669\) 0 0
\(670\) −34.4364 −1.33039
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −7.76575 −0.299348 −0.149674 0.988735i \(-0.547822\pi\)
−0.149674 + 0.988735i \(0.547822\pi\)
\(674\) 47.9165 1.84567
\(675\) 0 0
\(676\) 69.3649 2.66788
\(677\) −32.9643 −1.26692 −0.633460 0.773775i \(-0.718366\pi\)
−0.633460 + 0.773775i \(0.718366\pi\)
\(678\) 0 0
\(679\) −13.1574 −0.504935
\(680\) −40.5743 −1.55595
\(681\) 0 0
\(682\) −12.7961 −0.489990
\(683\) 16.7372 0.640429 0.320215 0.947345i \(-0.396245\pi\)
0.320215 + 0.947345i \(0.396245\pi\)
\(684\) 0 0
\(685\) −0.317824 −0.0121434
\(686\) −44.6470 −1.70463
\(687\) 0 0
\(688\) 19.5101 0.743817
\(689\) 27.7246 1.05622
\(690\) 0 0
\(691\) −17.4790 −0.664931 −0.332465 0.943115i \(-0.607880\pi\)
−0.332465 + 0.943115i \(0.607880\pi\)
\(692\) −37.9328 −1.44199
\(693\) 0 0
\(694\) 75.6233 2.87062
\(695\) −14.5567 −0.552169
\(696\) 0 0
\(697\) −25.3874 −0.961616
\(698\) −67.6198 −2.55945
\(699\) 0 0
\(700\) 9.28702 0.351016
\(701\) −20.9278 −0.790434 −0.395217 0.918588i \(-0.629331\pi\)
−0.395217 + 0.918588i \(0.629331\pi\)
\(702\) 0 0
\(703\) 8.01230 0.302190
\(704\) 11.1771 0.421252
\(705\) 0 0
\(706\) −33.2808 −1.25254
\(707\) 5.42520 0.204036
\(708\) 0 0
\(709\) −24.1466 −0.906845 −0.453423 0.891296i \(-0.649797\pi\)
−0.453423 + 0.891296i \(0.649797\pi\)
\(710\) −16.5074 −0.619513
\(711\) 0 0
\(712\) 52.0427 1.95038
\(713\) −33.0108 −1.23626
\(714\) 0 0
\(715\) 14.4217 0.539341
\(716\) 26.8992 1.00527
\(717\) 0 0
\(718\) −22.7938 −0.850659
\(719\) 32.0699 1.19601 0.598003 0.801494i \(-0.295961\pi\)
0.598003 + 0.801494i \(0.295961\pi\)
\(720\) 0 0
\(721\) −13.1880 −0.491145
\(722\) −26.9236 −1.00199
\(723\) 0 0
\(724\) −1.94566 −0.0723098
\(725\) −1.89906 −0.0705293
\(726\) 0 0
\(727\) −22.5250 −0.835406 −0.417703 0.908584i \(-0.637165\pi\)
−0.417703 + 0.908584i \(0.637165\pi\)
\(728\) 37.4969 1.38973
\(729\) 0 0
\(730\) −91.7506 −3.39584
\(731\) 37.5361 1.38832
\(732\) 0 0
\(733\) 45.0196 1.66284 0.831418 0.555648i \(-0.187530\pi\)
0.831418 + 0.555648i \(0.187530\pi\)
\(734\) −48.9606 −1.80717
\(735\) 0 0
\(736\) 18.0559 0.665549
\(737\) −5.67098 −0.208893
\(738\) 0 0
\(739\) −38.6313 −1.42108 −0.710539 0.703658i \(-0.751549\pi\)
−0.710539 + 0.703658i \(0.751549\pi\)
\(740\) −27.0117 −0.992968
\(741\) 0 0
\(742\) 19.5782 0.718737
\(743\) 40.6081 1.48977 0.744884 0.667194i \(-0.232505\pi\)
0.744884 + 0.667194i \(0.232505\pi\)
\(744\) 0 0
\(745\) −51.1097 −1.87251
\(746\) 64.3666 2.35663
\(747\) 0 0
\(748\) −14.7199 −0.538213
\(749\) 31.4496 1.14914
\(750\) 0 0
\(751\) 24.7212 0.902088 0.451044 0.892502i \(-0.351052\pi\)
0.451044 + 0.892502i \(0.351052\pi\)
\(752\) 18.2245 0.664578
\(753\) 0 0
\(754\) −16.8916 −0.615157
\(755\) −20.5471 −0.747787
\(756\) 0 0
\(757\) 22.0864 0.802743 0.401371 0.915915i \(-0.368534\pi\)
0.401371 + 0.915915i \(0.368534\pi\)
\(758\) −44.1794 −1.60467
\(759\) 0 0
\(760\) −27.9877 −1.01522
\(761\) 4.10116 0.148667 0.0743334 0.997233i \(-0.476317\pi\)
0.0743334 + 0.997233i \(0.476317\pi\)
\(762\) 0 0
\(763\) −32.8155 −1.18800
\(764\) −60.5161 −2.18940
\(765\) 0 0
\(766\) −7.99583 −0.288901
\(767\) 26.4778 0.956060
\(768\) 0 0
\(769\) −1.84340 −0.0664747 −0.0332374 0.999447i \(-0.510582\pi\)
−0.0332374 + 0.999447i \(0.510582\pi\)
\(770\) 10.1841 0.367010
\(771\) 0 0
\(772\) 65.5749 2.36009
\(773\) 46.4688 1.67137 0.835684 0.549211i \(-0.185072\pi\)
0.835684 + 0.549211i \(0.185072\pi\)
\(774\) 0 0
\(775\) 8.13032 0.292050
\(776\) −31.0369 −1.11416
\(777\) 0 0
\(778\) −23.9695 −0.859348
\(779\) −17.5119 −0.627429
\(780\) 0 0
\(781\) −2.71844 −0.0972734
\(782\) −58.7099 −2.09946
\(783\) 0 0
\(784\) −8.74719 −0.312400
\(785\) −26.2281 −0.936123
\(786\) 0 0
\(787\) −26.6731 −0.950795 −0.475397 0.879771i \(-0.657696\pi\)
−0.475397 + 0.879771i \(0.657696\pi\)
\(788\) −34.9761 −1.24597
\(789\) 0 0
\(790\) 8.96357 0.318909
\(791\) −19.1956 −0.682517
\(792\) 0 0
\(793\) −5.65147 −0.200690
\(794\) −28.3750 −1.00699
\(795\) 0 0
\(796\) 28.9737 1.02694
\(797\) 22.8323 0.808763 0.404381 0.914590i \(-0.367487\pi\)
0.404381 + 0.914590i \(0.367487\pi\)
\(798\) 0 0
\(799\) 35.0625 1.24042
\(800\) −4.44704 −0.157227
\(801\) 0 0
\(802\) −26.6755 −0.941945
\(803\) −15.1095 −0.533202
\(804\) 0 0
\(805\) 26.2724 0.925980
\(806\) 72.3170 2.54726
\(807\) 0 0
\(808\) 12.7974 0.450213
\(809\) −5.39701 −0.189749 −0.0948744 0.995489i \(-0.530245\pi\)
−0.0948744 + 0.995489i \(0.530245\pi\)
\(810\) 0 0
\(811\) −49.8393 −1.75009 −0.875047 0.484037i \(-0.839170\pi\)
−0.875047 + 0.484037i \(0.839170\pi\)
\(812\) −7.71522 −0.270751
\(813\) 0 0
\(814\) −6.87735 −0.241051
\(815\) 17.0629 0.597688
\(816\) 0 0
\(817\) 25.8919 0.905844
\(818\) −70.3527 −2.45982
\(819\) 0 0
\(820\) 59.0374 2.06168
\(821\) −17.3451 −0.605347 −0.302674 0.953094i \(-0.597879\pi\)
−0.302674 + 0.953094i \(0.597879\pi\)
\(822\) 0 0
\(823\) −30.8810 −1.07644 −0.538221 0.842804i \(-0.680904\pi\)
−0.538221 + 0.842804i \(0.680904\pi\)
\(824\) −31.1089 −1.08373
\(825\) 0 0
\(826\) 18.6977 0.650578
\(827\) −52.0029 −1.80832 −0.904160 0.427195i \(-0.859502\pi\)
−0.904160 + 0.427195i \(0.859502\pi\)
\(828\) 0 0
\(829\) 48.0581 1.66913 0.834563 0.550912i \(-0.185720\pi\)
0.834563 + 0.550912i \(0.185720\pi\)
\(830\) −13.3021 −0.461722
\(831\) 0 0
\(832\) −63.1670 −2.18992
\(833\) −16.8290 −0.583089
\(834\) 0 0
\(835\) 35.7983 1.23885
\(836\) −10.1536 −0.351170
\(837\) 0 0
\(838\) −31.2321 −1.07890
\(839\) 34.1026 1.17735 0.588676 0.808369i \(-0.299649\pi\)
0.588676 + 0.808369i \(0.299649\pi\)
\(840\) 0 0
\(841\) −27.4223 −0.945598
\(842\) −93.9546 −3.23789
\(843\) 0 0
\(844\) −31.6800 −1.09047
\(845\) −48.3298 −1.66260
\(846\) 0 0
\(847\) 1.67712 0.0576264
\(848\) 10.2480 0.351919
\(849\) 0 0
\(850\) 14.4598 0.495968
\(851\) −17.7418 −0.608181
\(852\) 0 0
\(853\) −1.66881 −0.0571391 −0.0285695 0.999592i \(-0.509095\pi\)
−0.0285695 + 0.999592i \(0.509095\pi\)
\(854\) −3.99087 −0.136565
\(855\) 0 0
\(856\) 74.1860 2.53562
\(857\) 52.5109 1.79374 0.896869 0.442297i \(-0.145836\pi\)
0.896869 + 0.442297i \(0.145836\pi\)
\(858\) 0 0
\(859\) 7.22377 0.246472 0.123236 0.992377i \(-0.460673\pi\)
0.123236 + 0.992377i \(0.460673\pi\)
\(860\) −87.2887 −2.97652
\(861\) 0 0
\(862\) −21.4149 −0.729394
\(863\) −47.7485 −1.62538 −0.812688 0.582698i \(-0.801997\pi\)
−0.812688 + 0.582698i \(0.801997\pi\)
\(864\) 0 0
\(865\) 26.4296 0.898632
\(866\) −47.2344 −1.60509
\(867\) 0 0
\(868\) 33.0307 1.12113
\(869\) 1.47612 0.0500739
\(870\) 0 0
\(871\) 32.0494 1.08595
\(872\) −77.4080 −2.62137
\(873\) 0 0
\(874\) −40.4974 −1.36984
\(875\) 14.9280 0.504660
\(876\) 0 0
\(877\) −17.6923 −0.597427 −0.298714 0.954343i \(-0.596558\pi\)
−0.298714 + 0.954343i \(0.596558\pi\)
\(878\) 58.9932 1.99092
\(879\) 0 0
\(880\) 5.33079 0.179701
\(881\) −18.9137 −0.637219 −0.318610 0.947886i \(-0.603216\pi\)
−0.318610 + 0.947886i \(0.603216\pi\)
\(882\) 0 0
\(883\) 59.1263 1.98976 0.994878 0.101079i \(-0.0322294\pi\)
0.994878 + 0.101079i \(0.0322294\pi\)
\(884\) 83.1891 2.79795
\(885\) 0 0
\(886\) −28.2357 −0.948599
\(887\) 34.1377 1.14623 0.573117 0.819474i \(-0.305734\pi\)
0.573117 + 0.819474i \(0.305734\pi\)
\(888\) 0 0
\(889\) 2.73716 0.0918012
\(890\) −79.8821 −2.67765
\(891\) 0 0
\(892\) 32.3558 1.08335
\(893\) 24.1857 0.809344
\(894\) 0 0
\(895\) −18.7420 −0.626475
\(896\) −34.7406 −1.16060
\(897\) 0 0
\(898\) −12.4611 −0.415832
\(899\) −6.75429 −0.225268
\(900\) 0 0
\(901\) 19.7165 0.656851
\(902\) 15.0313 0.500488
\(903\) 0 0
\(904\) −45.2803 −1.50600
\(905\) 1.35563 0.0450628
\(906\) 0 0
\(907\) 8.40299 0.279017 0.139508 0.990221i \(-0.455448\pi\)
0.139508 + 0.990221i \(0.455448\pi\)
\(908\) −57.2476 −1.89983
\(909\) 0 0
\(910\) −57.5552 −1.90794
\(911\) 12.5927 0.417214 0.208607 0.978000i \(-0.433107\pi\)
0.208607 + 0.978000i \(0.433107\pi\)
\(912\) 0 0
\(913\) −2.19058 −0.0724977
\(914\) 21.7145 0.718252
\(915\) 0 0
\(916\) −48.1262 −1.59013
\(917\) −9.75138 −0.322019
\(918\) 0 0
\(919\) 51.7102 1.70576 0.852881 0.522105i \(-0.174853\pi\)
0.852881 + 0.522105i \(0.174853\pi\)
\(920\) 61.9736 2.04321
\(921\) 0 0
\(922\) −4.95393 −0.163149
\(923\) 15.3632 0.505685
\(924\) 0 0
\(925\) 4.36968 0.143674
\(926\) 65.5707 2.15479
\(927\) 0 0
\(928\) 3.69439 0.121274
\(929\) 43.9138 1.44076 0.720382 0.693577i \(-0.243966\pi\)
0.720382 + 0.693577i \(0.243966\pi\)
\(930\) 0 0
\(931\) −11.6084 −0.380450
\(932\) 70.3707 2.30507
\(933\) 0 0
\(934\) −66.6026 −2.17930
\(935\) 10.2561 0.335409
\(936\) 0 0
\(937\) 14.6324 0.478021 0.239010 0.971017i \(-0.423177\pi\)
0.239010 + 0.971017i \(0.423177\pi\)
\(938\) 22.6322 0.738966
\(939\) 0 0
\(940\) −81.5366 −2.65943
\(941\) −46.6148 −1.51960 −0.759799 0.650158i \(-0.774703\pi\)
−0.759799 + 0.650158i \(0.774703\pi\)
\(942\) 0 0
\(943\) 38.7770 1.26275
\(944\) 9.78719 0.318546
\(945\) 0 0
\(946\) −22.2243 −0.722575
\(947\) −6.99474 −0.227299 −0.113649 0.993521i \(-0.536254\pi\)
−0.113649 + 0.993521i \(0.536254\pi\)
\(948\) 0 0
\(949\) 85.3907 2.77190
\(950\) 9.97422 0.323607
\(951\) 0 0
\(952\) 26.6661 0.864253
\(953\) −39.6795 −1.28534 −0.642672 0.766142i \(-0.722174\pi\)
−0.642672 + 0.766142i \(0.722174\pi\)
\(954\) 0 0
\(955\) 42.1645 1.36441
\(956\) −49.0670 −1.58694
\(957\) 0 0
\(958\) −2.26777 −0.0732682
\(959\) 0.208879 0.00674506
\(960\) 0 0
\(961\) −2.08330 −0.0672033
\(962\) 38.8672 1.25313
\(963\) 0 0
\(964\) −92.5722 −2.98155
\(965\) −45.6891 −1.47079
\(966\) 0 0
\(967\) −56.5820 −1.81956 −0.909778 0.415096i \(-0.863748\pi\)
−0.909778 + 0.415096i \(0.863748\pi\)
\(968\) 3.95613 0.127155
\(969\) 0 0
\(970\) 47.6395 1.52961
\(971\) −23.7722 −0.762885 −0.381443 0.924392i \(-0.624573\pi\)
−0.381443 + 0.924392i \(0.624573\pi\)
\(972\) 0 0
\(973\) 9.56692 0.306702
\(974\) −30.5413 −0.978605
\(975\) 0 0
\(976\) −2.08899 −0.0668670
\(977\) 43.0284 1.37660 0.688300 0.725426i \(-0.258357\pi\)
0.688300 + 0.725426i \(0.258357\pi\)
\(978\) 0 0
\(979\) −13.1550 −0.420434
\(980\) 39.1351 1.25013
\(981\) 0 0
\(982\) −94.3276 −3.01011
\(983\) −28.8215 −0.919263 −0.459631 0.888110i \(-0.652018\pi\)
−0.459631 + 0.888110i \(0.652018\pi\)
\(984\) 0 0
\(985\) 24.3695 0.776478
\(986\) −12.0126 −0.382558
\(987\) 0 0
\(988\) 57.3828 1.82559
\(989\) −57.3330 −1.82308
\(990\) 0 0
\(991\) 5.66098 0.179827 0.0899135 0.995950i \(-0.471341\pi\)
0.0899135 + 0.995950i \(0.471341\pi\)
\(992\) −15.8165 −0.502176
\(993\) 0 0
\(994\) 10.8490 0.344108
\(995\) −20.1873 −0.639981
\(996\) 0 0
\(997\) −50.8029 −1.60894 −0.804472 0.593991i \(-0.797552\pi\)
−0.804472 + 0.593991i \(0.797552\pi\)
\(998\) −83.5801 −2.64568
\(999\) 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.1.12 13
3.2 odd 2 2013.2.a.e.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.2 13 3.2 odd 2
6039.2.a.i.1.12 13 1.1 even 1 trivial