Properties

Label 6039.2.a.i.1.5
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 1.5
Root \(1.50067\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.50067 q^{2} +0.252024 q^{4} +0.569898 q^{5} -3.97554 q^{7} +2.62314 q^{8} +O(q^{10})\) \(q-1.50067 q^{2} +0.252024 q^{4} +0.569898 q^{5} -3.97554 q^{7} +2.62314 q^{8} -0.855232 q^{10} -1.00000 q^{11} -3.48724 q^{13} +5.96599 q^{14} -4.44053 q^{16} -7.75664 q^{17} +1.56002 q^{19} +0.143628 q^{20} +1.50067 q^{22} -4.91362 q^{23} -4.67522 q^{25} +5.23321 q^{26} -1.00193 q^{28} -6.29679 q^{29} +2.87665 q^{31} +1.41751 q^{32} +11.6402 q^{34} -2.26565 q^{35} -9.34414 q^{37} -2.34109 q^{38} +1.49492 q^{40} -1.58419 q^{41} +5.31329 q^{43} -0.252024 q^{44} +7.37374 q^{46} -3.46571 q^{47} +8.80489 q^{49} +7.01598 q^{50} -0.878869 q^{52} -10.8254 q^{53} -0.569898 q^{55} -10.4284 q^{56} +9.44943 q^{58} -0.418492 q^{59} -1.00000 q^{61} -4.31692 q^{62} +6.75385 q^{64} -1.98737 q^{65} +9.84375 q^{67} -1.95486 q^{68} +3.40001 q^{70} -9.93167 q^{71} +7.11859 q^{73} +14.0225 q^{74} +0.393164 q^{76} +3.97554 q^{77} -13.1270 q^{79} -2.53065 q^{80} +2.37735 q^{82} -4.04331 q^{83} -4.42050 q^{85} -7.97351 q^{86} -2.62314 q^{88} +13.4410 q^{89} +13.8636 q^{91} -1.23835 q^{92} +5.20090 q^{94} +0.889055 q^{95} -1.77923 q^{97} -13.2133 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.50067 −1.06114 −0.530569 0.847642i \(-0.678022\pi\)
−0.530569 + 0.847642i \(0.678022\pi\)
\(3\) 0 0
\(4\) 0.252024 0.126012
\(5\) 0.569898 0.254866 0.127433 0.991847i \(-0.459326\pi\)
0.127433 + 0.991847i \(0.459326\pi\)
\(6\) 0 0
\(7\) −3.97554 −1.50261 −0.751306 0.659954i \(-0.770576\pi\)
−0.751306 + 0.659954i \(0.770576\pi\)
\(8\) 2.62314 0.927421
\(9\) 0 0
\(10\) −0.855232 −0.270448
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.48724 −0.967186 −0.483593 0.875293i \(-0.660669\pi\)
−0.483593 + 0.875293i \(0.660669\pi\)
\(14\) 5.96599 1.59448
\(15\) 0 0
\(16\) −4.44053 −1.11013
\(17\) −7.75664 −1.88126 −0.940631 0.339430i \(-0.889766\pi\)
−0.940631 + 0.339430i \(0.889766\pi\)
\(18\) 0 0
\(19\) 1.56002 0.357894 0.178947 0.983859i \(-0.442731\pi\)
0.178947 + 0.983859i \(0.442731\pi\)
\(20\) 0.143628 0.0321162
\(21\) 0 0
\(22\) 1.50067 0.319945
\(23\) −4.91362 −1.02456 −0.512280 0.858819i \(-0.671199\pi\)
−0.512280 + 0.858819i \(0.671199\pi\)
\(24\) 0 0
\(25\) −4.67522 −0.935043
\(26\) 5.23321 1.02632
\(27\) 0 0
\(28\) −1.00193 −0.189347
\(29\) −6.29679 −1.16928 −0.584642 0.811291i \(-0.698765\pi\)
−0.584642 + 0.811291i \(0.698765\pi\)
\(30\) 0 0
\(31\) 2.87665 0.516662 0.258331 0.966056i \(-0.416828\pi\)
0.258331 + 0.966056i \(0.416828\pi\)
\(32\) 1.41751 0.250583
\(33\) 0 0
\(34\) 11.6402 1.99628
\(35\) −2.26565 −0.382965
\(36\) 0 0
\(37\) −9.34414 −1.53617 −0.768083 0.640350i \(-0.778789\pi\)
−0.768083 + 0.640350i \(0.778789\pi\)
\(38\) −2.34109 −0.379775
\(39\) 0 0
\(40\) 1.49492 0.236368
\(41\) −1.58419 −0.247409 −0.123704 0.992319i \(-0.539477\pi\)
−0.123704 + 0.992319i \(0.539477\pi\)
\(42\) 0 0
\(43\) 5.31329 0.810269 0.405134 0.914257i \(-0.367225\pi\)
0.405134 + 0.914257i \(0.367225\pi\)
\(44\) −0.252024 −0.0379941
\(45\) 0 0
\(46\) 7.37374 1.08720
\(47\) −3.46571 −0.505526 −0.252763 0.967528i \(-0.581339\pi\)
−0.252763 + 0.967528i \(0.581339\pi\)
\(48\) 0 0
\(49\) 8.80489 1.25784
\(50\) 7.01598 0.992209
\(51\) 0 0
\(52\) −0.878869 −0.121877
\(53\) −10.8254 −1.48699 −0.743493 0.668744i \(-0.766832\pi\)
−0.743493 + 0.668744i \(0.766832\pi\)
\(54\) 0 0
\(55\) −0.569898 −0.0768451
\(56\) −10.4284 −1.39355
\(57\) 0 0
\(58\) 9.44943 1.24077
\(59\) −0.418492 −0.0544831 −0.0272415 0.999629i \(-0.508672\pi\)
−0.0272415 + 0.999629i \(0.508672\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −4.31692 −0.548249
\(63\) 0 0
\(64\) 6.75385 0.844231
\(65\) −1.98737 −0.246503
\(66\) 0 0
\(67\) 9.84375 1.20260 0.601302 0.799022i \(-0.294649\pi\)
0.601302 + 0.799022i \(0.294649\pi\)
\(68\) −1.95486 −0.237062
\(69\) 0 0
\(70\) 3.40001 0.406378
\(71\) −9.93167 −1.17867 −0.589336 0.807888i \(-0.700611\pi\)
−0.589336 + 0.807888i \(0.700611\pi\)
\(72\) 0 0
\(73\) 7.11859 0.833168 0.416584 0.909097i \(-0.363227\pi\)
0.416584 + 0.909097i \(0.363227\pi\)
\(74\) 14.0225 1.63008
\(75\) 0 0
\(76\) 0.393164 0.0450990
\(77\) 3.97554 0.453054
\(78\) 0 0
\(79\) −13.1270 −1.47690 −0.738450 0.674308i \(-0.764442\pi\)
−0.738450 + 0.674308i \(0.764442\pi\)
\(80\) −2.53065 −0.282935
\(81\) 0 0
\(82\) 2.37735 0.262535
\(83\) −4.04331 −0.443811 −0.221906 0.975068i \(-0.571228\pi\)
−0.221906 + 0.975068i \(0.571228\pi\)
\(84\) 0 0
\(85\) −4.42050 −0.479470
\(86\) −7.97351 −0.859806
\(87\) 0 0
\(88\) −2.62314 −0.279628
\(89\) 13.4410 1.42474 0.712371 0.701803i \(-0.247621\pi\)
0.712371 + 0.701803i \(0.247621\pi\)
\(90\) 0 0
\(91\) 13.8636 1.45331
\(92\) −1.23835 −0.129107
\(93\) 0 0
\(94\) 5.20090 0.536432
\(95\) 0.889055 0.0912151
\(96\) 0 0
\(97\) −1.77923 −0.180653 −0.0903267 0.995912i \(-0.528791\pi\)
−0.0903267 + 0.995912i \(0.528791\pi\)
\(98\) −13.2133 −1.33474
\(99\) 0 0
\(100\) −1.17827 −0.117827
\(101\) 18.9035 1.88096 0.940482 0.339844i \(-0.110374\pi\)
0.940482 + 0.339844i \(0.110374\pi\)
\(102\) 0 0
\(103\) 2.41802 0.238254 0.119127 0.992879i \(-0.461990\pi\)
0.119127 + 0.992879i \(0.461990\pi\)
\(104\) −9.14753 −0.896989
\(105\) 0 0
\(106\) 16.2454 1.57790
\(107\) −10.7538 −1.03961 −0.519806 0.854285i \(-0.673996\pi\)
−0.519806 + 0.854285i \(0.673996\pi\)
\(108\) 0 0
\(109\) −14.6227 −1.40060 −0.700298 0.713851i \(-0.746950\pi\)
−0.700298 + 0.713851i \(0.746950\pi\)
\(110\) 0.855232 0.0815432
\(111\) 0 0
\(112\) 17.6535 1.66810
\(113\) 8.88757 0.836072 0.418036 0.908430i \(-0.362719\pi\)
0.418036 + 0.908430i \(0.362719\pi\)
\(114\) 0 0
\(115\) −2.80026 −0.261126
\(116\) −1.58694 −0.147344
\(117\) 0 0
\(118\) 0.628021 0.0578140
\(119\) 30.8368 2.82681
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.50067 0.135865
\(123\) 0 0
\(124\) 0.724986 0.0651057
\(125\) −5.51389 −0.493177
\(126\) 0 0
\(127\) −16.2612 −1.44295 −0.721473 0.692443i \(-0.756535\pi\)
−0.721473 + 0.692443i \(0.756535\pi\)
\(128\) −12.9703 −1.14643
\(129\) 0 0
\(130\) 2.98240 0.261574
\(131\) −16.7549 −1.46389 −0.731943 0.681366i \(-0.761386\pi\)
−0.731943 + 0.681366i \(0.761386\pi\)
\(132\) 0 0
\(133\) −6.20193 −0.537776
\(134\) −14.7723 −1.27613
\(135\) 0 0
\(136\) −20.3468 −1.74472
\(137\) −21.7052 −1.85440 −0.927198 0.374571i \(-0.877790\pi\)
−0.927198 + 0.374571i \(0.877790\pi\)
\(138\) 0 0
\(139\) 9.86673 0.836885 0.418442 0.908243i \(-0.362576\pi\)
0.418442 + 0.908243i \(0.362576\pi\)
\(140\) −0.570999 −0.0482582
\(141\) 0 0
\(142\) 14.9042 1.25073
\(143\) 3.48724 0.291618
\(144\) 0 0
\(145\) −3.58853 −0.298011
\(146\) −10.6827 −0.884106
\(147\) 0 0
\(148\) −2.35495 −0.193576
\(149\) −16.9431 −1.38803 −0.694017 0.719959i \(-0.744161\pi\)
−0.694017 + 0.719959i \(0.744161\pi\)
\(150\) 0 0
\(151\) 16.5384 1.34587 0.672937 0.739700i \(-0.265032\pi\)
0.672937 + 0.739700i \(0.265032\pi\)
\(152\) 4.09217 0.331918
\(153\) 0 0
\(154\) −5.96599 −0.480753
\(155\) 1.63940 0.131680
\(156\) 0 0
\(157\) 17.8806 1.42703 0.713515 0.700640i \(-0.247102\pi\)
0.713515 + 0.700640i \(0.247102\pi\)
\(158\) 19.6993 1.56719
\(159\) 0 0
\(160\) 0.807836 0.0638650
\(161\) 19.5343 1.53952
\(162\) 0 0
\(163\) −0.855017 −0.0669701 −0.0334851 0.999439i \(-0.510661\pi\)
−0.0334851 + 0.999439i \(0.510661\pi\)
\(164\) −0.399254 −0.0311765
\(165\) 0 0
\(166\) 6.06770 0.470945
\(167\) −13.4101 −1.03770 −0.518852 0.854864i \(-0.673640\pi\)
−0.518852 + 0.854864i \(0.673640\pi\)
\(168\) 0 0
\(169\) −0.839160 −0.0645508
\(170\) 6.63373 0.508784
\(171\) 0 0
\(172\) 1.33908 0.102104
\(173\) 9.03817 0.687160 0.343580 0.939123i \(-0.388360\pi\)
0.343580 + 0.939123i \(0.388360\pi\)
\(174\) 0 0
\(175\) 18.5865 1.40501
\(176\) 4.44053 0.334718
\(177\) 0 0
\(178\) −20.1706 −1.51185
\(179\) 26.1042 1.95112 0.975559 0.219736i \(-0.0705196\pi\)
0.975559 + 0.219736i \(0.0705196\pi\)
\(180\) 0 0
\(181\) 3.16435 0.235204 0.117602 0.993061i \(-0.462479\pi\)
0.117602 + 0.993061i \(0.462479\pi\)
\(182\) −20.8048 −1.54216
\(183\) 0 0
\(184\) −12.8891 −0.950198
\(185\) −5.32521 −0.391517
\(186\) 0 0
\(187\) 7.75664 0.567222
\(188\) −0.873443 −0.0637024
\(189\) 0 0
\(190\) −1.33418 −0.0967917
\(191\) 0.339788 0.0245862 0.0122931 0.999924i \(-0.496087\pi\)
0.0122931 + 0.999924i \(0.496087\pi\)
\(192\) 0 0
\(193\) −10.9988 −0.791714 −0.395857 0.918312i \(-0.629552\pi\)
−0.395857 + 0.918312i \(0.629552\pi\)
\(194\) 2.67005 0.191698
\(195\) 0 0
\(196\) 2.21905 0.158503
\(197\) −13.6647 −0.973571 −0.486786 0.873522i \(-0.661831\pi\)
−0.486786 + 0.873522i \(0.661831\pi\)
\(198\) 0 0
\(199\) −5.05352 −0.358235 −0.179117 0.983828i \(-0.557324\pi\)
−0.179117 + 0.983828i \(0.557324\pi\)
\(200\) −12.2638 −0.867179
\(201\) 0 0
\(202\) −28.3679 −1.99596
\(203\) 25.0331 1.75698
\(204\) 0 0
\(205\) −0.902827 −0.0630561
\(206\) −3.62865 −0.252820
\(207\) 0 0
\(208\) 15.4852 1.07371
\(209\) −1.56002 −0.107909
\(210\) 0 0
\(211\) 0.418379 0.0288024 0.0144012 0.999896i \(-0.495416\pi\)
0.0144012 + 0.999896i \(0.495416\pi\)
\(212\) −2.72827 −0.187378
\(213\) 0 0
\(214\) 16.1380 1.10317
\(215\) 3.02803 0.206510
\(216\) 0 0
\(217\) −11.4362 −0.776342
\(218\) 21.9438 1.48622
\(219\) 0 0
\(220\) −0.143628 −0.00968341
\(221\) 27.0493 1.81953
\(222\) 0 0
\(223\) 8.46953 0.567162 0.283581 0.958948i \(-0.408478\pi\)
0.283581 + 0.958948i \(0.408478\pi\)
\(224\) −5.63536 −0.376528
\(225\) 0 0
\(226\) −13.3373 −0.887187
\(227\) 1.28958 0.0855927 0.0427963 0.999084i \(-0.486373\pi\)
0.0427963 + 0.999084i \(0.486373\pi\)
\(228\) 0 0
\(229\) 18.0318 1.19157 0.595787 0.803143i \(-0.296840\pi\)
0.595787 + 0.803143i \(0.296840\pi\)
\(230\) 4.20228 0.277090
\(231\) 0 0
\(232\) −16.5174 −1.08442
\(233\) −15.1799 −0.994470 −0.497235 0.867616i \(-0.665651\pi\)
−0.497235 + 0.867616i \(0.665651\pi\)
\(234\) 0 0
\(235\) −1.97510 −0.128841
\(236\) −0.105470 −0.00686553
\(237\) 0 0
\(238\) −46.2760 −2.99963
\(239\) −5.61796 −0.363396 −0.181698 0.983354i \(-0.558159\pi\)
−0.181698 + 0.983354i \(0.558159\pi\)
\(240\) 0 0
\(241\) −9.85967 −0.635117 −0.317559 0.948239i \(-0.602863\pi\)
−0.317559 + 0.948239i \(0.602863\pi\)
\(242\) −1.50067 −0.0964670
\(243\) 0 0
\(244\) −0.252024 −0.0161342
\(245\) 5.01789 0.320581
\(246\) 0 0
\(247\) −5.44018 −0.346150
\(248\) 7.54587 0.479163
\(249\) 0 0
\(250\) 8.27455 0.523329
\(251\) −31.0856 −1.96210 −0.981052 0.193743i \(-0.937937\pi\)
−0.981052 + 0.193743i \(0.937937\pi\)
\(252\) 0 0
\(253\) 4.91362 0.308916
\(254\) 24.4027 1.53116
\(255\) 0 0
\(256\) 5.95657 0.372286
\(257\) 10.0493 0.626855 0.313428 0.949612i \(-0.398523\pi\)
0.313428 + 0.949612i \(0.398523\pi\)
\(258\) 0 0
\(259\) 37.1480 2.30826
\(260\) −0.500866 −0.0310624
\(261\) 0 0
\(262\) 25.1437 1.55338
\(263\) −19.8284 −1.22267 −0.611337 0.791370i \(-0.709368\pi\)
−0.611337 + 0.791370i \(0.709368\pi\)
\(264\) 0 0
\(265\) −6.16938 −0.378982
\(266\) 9.30708 0.570654
\(267\) 0 0
\(268\) 2.48086 0.151543
\(269\) 11.7548 0.716702 0.358351 0.933587i \(-0.383339\pi\)
0.358351 + 0.933587i \(0.383339\pi\)
\(270\) 0 0
\(271\) −19.0082 −1.15467 −0.577334 0.816508i \(-0.695907\pi\)
−0.577334 + 0.816508i \(0.695907\pi\)
\(272\) 34.4436 2.08845
\(273\) 0 0
\(274\) 32.5724 1.96777
\(275\) 4.67522 0.281926
\(276\) 0 0
\(277\) −27.7437 −1.66695 −0.833477 0.552553i \(-0.813654\pi\)
−0.833477 + 0.552553i \(0.813654\pi\)
\(278\) −14.8067 −0.888050
\(279\) 0 0
\(280\) −5.94313 −0.355170
\(281\) 26.9692 1.60885 0.804424 0.594056i \(-0.202474\pi\)
0.804424 + 0.594056i \(0.202474\pi\)
\(282\) 0 0
\(283\) 30.2378 1.79745 0.898725 0.438513i \(-0.144495\pi\)
0.898725 + 0.438513i \(0.144495\pi\)
\(284\) −2.50302 −0.148527
\(285\) 0 0
\(286\) −5.23321 −0.309446
\(287\) 6.29800 0.371759
\(288\) 0 0
\(289\) 43.1655 2.53915
\(290\) 5.38521 0.316230
\(291\) 0 0
\(292\) 1.79406 0.104989
\(293\) −6.99534 −0.408672 −0.204336 0.978901i \(-0.565504\pi\)
−0.204336 + 0.978901i \(0.565504\pi\)
\(294\) 0 0
\(295\) −0.238498 −0.0138859
\(296\) −24.5110 −1.42467
\(297\) 0 0
\(298\) 25.4261 1.47289
\(299\) 17.1350 0.990940
\(300\) 0 0
\(301\) −21.1232 −1.21752
\(302\) −24.8187 −1.42816
\(303\) 0 0
\(304\) −6.92734 −0.397310
\(305\) −0.569898 −0.0326323
\(306\) 0 0
\(307\) 7.80207 0.445288 0.222644 0.974900i \(-0.428531\pi\)
0.222644 + 0.974900i \(0.428531\pi\)
\(308\) 1.00193 0.0570904
\(309\) 0 0
\(310\) −2.46020 −0.139730
\(311\) 1.36849 0.0775998 0.0387999 0.999247i \(-0.487647\pi\)
0.0387999 + 0.999247i \(0.487647\pi\)
\(312\) 0 0
\(313\) −5.14485 −0.290804 −0.145402 0.989373i \(-0.546448\pi\)
−0.145402 + 0.989373i \(0.546448\pi\)
\(314\) −26.8330 −1.51427
\(315\) 0 0
\(316\) −3.30832 −0.186107
\(317\) −30.3916 −1.70696 −0.853482 0.521122i \(-0.825514\pi\)
−0.853482 + 0.521122i \(0.825514\pi\)
\(318\) 0 0
\(319\) 6.29679 0.352552
\(320\) 3.84900 0.215166
\(321\) 0 0
\(322\) −29.3146 −1.63364
\(323\) −12.1006 −0.673293
\(324\) 0 0
\(325\) 16.3036 0.904361
\(326\) 1.28310 0.0710645
\(327\) 0 0
\(328\) −4.15555 −0.229452
\(329\) 13.7781 0.759609
\(330\) 0 0
\(331\) 17.3933 0.956024 0.478012 0.878353i \(-0.341357\pi\)
0.478012 + 0.878353i \(0.341357\pi\)
\(332\) −1.01901 −0.0559256
\(333\) 0 0
\(334\) 20.1242 1.10115
\(335\) 5.60993 0.306503
\(336\) 0 0
\(337\) 3.38221 0.184241 0.0921203 0.995748i \(-0.470636\pi\)
0.0921203 + 0.995748i \(0.470636\pi\)
\(338\) 1.25931 0.0684972
\(339\) 0 0
\(340\) −1.11407 −0.0604191
\(341\) −2.87665 −0.155779
\(342\) 0 0
\(343\) −7.17541 −0.387436
\(344\) 13.9375 0.751460
\(345\) 0 0
\(346\) −13.5634 −0.729171
\(347\) −22.0213 −1.18217 −0.591084 0.806610i \(-0.701300\pi\)
−0.591084 + 0.806610i \(0.701300\pi\)
\(348\) 0 0
\(349\) −6.15318 −0.329372 −0.164686 0.986346i \(-0.552661\pi\)
−0.164686 + 0.986346i \(0.552661\pi\)
\(350\) −27.8923 −1.49090
\(351\) 0 0
\(352\) −1.41751 −0.0755535
\(353\) 2.81549 0.149853 0.0749267 0.997189i \(-0.476128\pi\)
0.0749267 + 0.997189i \(0.476128\pi\)
\(354\) 0 0
\(355\) −5.66004 −0.300404
\(356\) 3.38746 0.179535
\(357\) 0 0
\(358\) −39.1739 −2.07040
\(359\) 19.3450 1.02099 0.510494 0.859881i \(-0.329463\pi\)
0.510494 + 0.859881i \(0.329463\pi\)
\(360\) 0 0
\(361\) −16.5663 −0.871912
\(362\) −4.74865 −0.249584
\(363\) 0 0
\(364\) 3.49398 0.183134
\(365\) 4.05687 0.212346
\(366\) 0 0
\(367\) 9.48578 0.495154 0.247577 0.968868i \(-0.420366\pi\)
0.247577 + 0.968868i \(0.420366\pi\)
\(368\) 21.8191 1.13740
\(369\) 0 0
\(370\) 7.99140 0.415453
\(371\) 43.0368 2.23436
\(372\) 0 0
\(373\) −28.1396 −1.45701 −0.728507 0.685038i \(-0.759785\pi\)
−0.728507 + 0.685038i \(0.759785\pi\)
\(374\) −11.6402 −0.601900
\(375\) 0 0
\(376\) −9.09105 −0.468835
\(377\) 21.9584 1.13092
\(378\) 0 0
\(379\) −28.6718 −1.47277 −0.736386 0.676561i \(-0.763469\pi\)
−0.736386 + 0.676561i \(0.763469\pi\)
\(380\) 0.224063 0.0114942
\(381\) 0 0
\(382\) −0.509911 −0.0260893
\(383\) 35.9467 1.83679 0.918395 0.395666i \(-0.129486\pi\)
0.918395 + 0.395666i \(0.129486\pi\)
\(384\) 0 0
\(385\) 2.26565 0.115468
\(386\) 16.5057 0.840118
\(387\) 0 0
\(388\) −0.448409 −0.0227645
\(389\) 36.5774 1.85455 0.927274 0.374383i \(-0.122145\pi\)
0.927274 + 0.374383i \(0.122145\pi\)
\(390\) 0 0
\(391\) 38.1132 1.92747
\(392\) 23.0965 1.16655
\(393\) 0 0
\(394\) 20.5063 1.03309
\(395\) −7.48104 −0.376412
\(396\) 0 0
\(397\) 17.4290 0.874738 0.437369 0.899282i \(-0.355910\pi\)
0.437369 + 0.899282i \(0.355910\pi\)
\(398\) 7.58370 0.380136
\(399\) 0 0
\(400\) 20.7604 1.03802
\(401\) 28.3816 1.41731 0.708655 0.705556i \(-0.249303\pi\)
0.708655 + 0.705556i \(0.249303\pi\)
\(402\) 0 0
\(403\) −10.0316 −0.499708
\(404\) 4.76413 0.237024
\(405\) 0 0
\(406\) −37.5665 −1.86440
\(407\) 9.34414 0.463172
\(408\) 0 0
\(409\) 27.6614 1.36777 0.683883 0.729592i \(-0.260290\pi\)
0.683883 + 0.729592i \(0.260290\pi\)
\(410\) 1.35485 0.0669112
\(411\) 0 0
\(412\) 0.609399 0.0300229
\(413\) 1.66373 0.0818669
\(414\) 0 0
\(415\) −2.30428 −0.113113
\(416\) −4.94319 −0.242360
\(417\) 0 0
\(418\) 2.34109 0.114506
\(419\) 1.14196 0.0557883 0.0278942 0.999611i \(-0.491120\pi\)
0.0278942 + 0.999611i \(0.491120\pi\)
\(420\) 0 0
\(421\) 5.24502 0.255627 0.127813 0.991798i \(-0.459204\pi\)
0.127813 + 0.991798i \(0.459204\pi\)
\(422\) −0.627850 −0.0305633
\(423\) 0 0
\(424\) −28.3966 −1.37906
\(425\) 36.2640 1.75906
\(426\) 0 0
\(427\) 3.97554 0.192390
\(428\) −2.71022 −0.131004
\(429\) 0 0
\(430\) −4.54409 −0.219136
\(431\) −39.7537 −1.91487 −0.957434 0.288651i \(-0.906793\pi\)
−0.957434 + 0.288651i \(0.906793\pi\)
\(432\) 0 0
\(433\) −22.4352 −1.07817 −0.539084 0.842252i \(-0.681229\pi\)
−0.539084 + 0.842252i \(0.681229\pi\)
\(434\) 17.1621 0.823806
\(435\) 0 0
\(436\) −3.68526 −0.176492
\(437\) −7.66536 −0.366684
\(438\) 0 0
\(439\) −5.46960 −0.261050 −0.130525 0.991445i \(-0.541666\pi\)
−0.130525 + 0.991445i \(0.541666\pi\)
\(440\) −1.49492 −0.0712677
\(441\) 0 0
\(442\) −40.5922 −1.93077
\(443\) 4.21494 0.200258 0.100129 0.994974i \(-0.468074\pi\)
0.100129 + 0.994974i \(0.468074\pi\)
\(444\) 0 0
\(445\) 7.66000 0.363119
\(446\) −12.7100 −0.601837
\(447\) 0 0
\(448\) −26.8502 −1.26855
\(449\) −23.8577 −1.12592 −0.562958 0.826485i \(-0.690337\pi\)
−0.562958 + 0.826485i \(0.690337\pi\)
\(450\) 0 0
\(451\) 1.58419 0.0745966
\(452\) 2.23988 0.105355
\(453\) 0 0
\(454\) −1.93525 −0.0908256
\(455\) 7.90087 0.370398
\(456\) 0 0
\(457\) 35.7911 1.67424 0.837119 0.547021i \(-0.184238\pi\)
0.837119 + 0.547021i \(0.184238\pi\)
\(458\) −27.0598 −1.26442
\(459\) 0 0
\(460\) −0.705734 −0.0329050
\(461\) −33.0708 −1.54026 −0.770131 0.637886i \(-0.779809\pi\)
−0.770131 + 0.637886i \(0.779809\pi\)
\(462\) 0 0
\(463\) −30.1684 −1.40205 −0.701023 0.713139i \(-0.747273\pi\)
−0.701023 + 0.713139i \(0.747273\pi\)
\(464\) 27.9611 1.29806
\(465\) 0 0
\(466\) 22.7801 1.05527
\(467\) −13.3041 −0.615642 −0.307821 0.951444i \(-0.599600\pi\)
−0.307821 + 0.951444i \(0.599600\pi\)
\(468\) 0 0
\(469\) −39.1342 −1.80705
\(470\) 2.96399 0.136718
\(471\) 0 0
\(472\) −1.09777 −0.0505287
\(473\) −5.31329 −0.244305
\(474\) 0 0
\(475\) −7.29345 −0.334646
\(476\) 7.77163 0.356212
\(477\) 0 0
\(478\) 8.43073 0.385613
\(479\) −36.2381 −1.65576 −0.827881 0.560904i \(-0.810454\pi\)
−0.827881 + 0.560904i \(0.810454\pi\)
\(480\) 0 0
\(481\) 32.5852 1.48576
\(482\) 14.7962 0.673947
\(483\) 0 0
\(484\) 0.252024 0.0114557
\(485\) −1.01398 −0.0460425
\(486\) 0 0
\(487\) 4.94148 0.223920 0.111960 0.993713i \(-0.464287\pi\)
0.111960 + 0.993713i \(0.464287\pi\)
\(488\) −2.62314 −0.118744
\(489\) 0 0
\(490\) −7.53022 −0.340181
\(491\) 14.0150 0.632489 0.316245 0.948678i \(-0.397578\pi\)
0.316245 + 0.948678i \(0.397578\pi\)
\(492\) 0 0
\(493\) 48.8419 2.19973
\(494\) 8.16394 0.367313
\(495\) 0 0
\(496\) −12.7739 −0.573563
\(497\) 39.4837 1.77109
\(498\) 0 0
\(499\) 15.0317 0.672913 0.336456 0.941699i \(-0.390772\pi\)
0.336456 + 0.941699i \(0.390772\pi\)
\(500\) −1.38963 −0.0621463
\(501\) 0 0
\(502\) 46.6493 2.08206
\(503\) 4.69369 0.209281 0.104641 0.994510i \(-0.466631\pi\)
0.104641 + 0.994510i \(0.466631\pi\)
\(504\) 0 0
\(505\) 10.7730 0.479394
\(506\) −7.37374 −0.327803
\(507\) 0 0
\(508\) −4.09821 −0.181829
\(509\) −2.29316 −0.101643 −0.0508214 0.998708i \(-0.516184\pi\)
−0.0508214 + 0.998708i \(0.516184\pi\)
\(510\) 0 0
\(511\) −28.3002 −1.25193
\(512\) 17.0018 0.751381
\(513\) 0 0
\(514\) −15.0807 −0.665179
\(515\) 1.37802 0.0607229
\(516\) 0 0
\(517\) 3.46571 0.152422
\(518\) −55.7470 −2.44938
\(519\) 0 0
\(520\) −5.21316 −0.228612
\(521\) −6.53621 −0.286356 −0.143178 0.989697i \(-0.545732\pi\)
−0.143178 + 0.989697i \(0.545732\pi\)
\(522\) 0 0
\(523\) 28.4759 1.24516 0.622582 0.782554i \(-0.286084\pi\)
0.622582 + 0.782554i \(0.286084\pi\)
\(524\) −4.22265 −0.184467
\(525\) 0 0
\(526\) 29.7560 1.29742
\(527\) −22.3132 −0.971977
\(528\) 0 0
\(529\) 1.14362 0.0497224
\(530\) 9.25824 0.402152
\(531\) 0 0
\(532\) −1.56304 −0.0677663
\(533\) 5.52445 0.239290
\(534\) 0 0
\(535\) −6.12858 −0.264962
\(536\) 25.8215 1.11532
\(537\) 0 0
\(538\) −17.6401 −0.760519
\(539\) −8.80489 −0.379253
\(540\) 0 0
\(541\) −22.4670 −0.965931 −0.482965 0.875640i \(-0.660440\pi\)
−0.482965 + 0.875640i \(0.660440\pi\)
\(542\) 28.5252 1.22526
\(543\) 0 0
\(544\) −10.9951 −0.471412
\(545\) −8.33342 −0.356965
\(546\) 0 0
\(547\) −21.9796 −0.939779 −0.469889 0.882725i \(-0.655706\pi\)
−0.469889 + 0.882725i \(0.655706\pi\)
\(548\) −5.47023 −0.233677
\(549\) 0 0
\(550\) −7.01598 −0.299162
\(551\) −9.82314 −0.418480
\(552\) 0 0
\(553\) 52.1868 2.21921
\(554\) 41.6342 1.76887
\(555\) 0 0
\(556\) 2.48665 0.105458
\(557\) 0.280608 0.0118897 0.00594487 0.999982i \(-0.498108\pi\)
0.00594487 + 0.999982i \(0.498108\pi\)
\(558\) 0 0
\(559\) −18.5287 −0.783681
\(560\) 10.0607 0.425142
\(561\) 0 0
\(562\) −40.4720 −1.70721
\(563\) −5.01496 −0.211355 −0.105678 0.994400i \(-0.533701\pi\)
−0.105678 + 0.994400i \(0.533701\pi\)
\(564\) 0 0
\(565\) 5.06501 0.213087
\(566\) −45.3771 −1.90734
\(567\) 0 0
\(568\) −26.0522 −1.09313
\(569\) 0.592920 0.0248565 0.0124282 0.999923i \(-0.496044\pi\)
0.0124282 + 0.999923i \(0.496044\pi\)
\(570\) 0 0
\(571\) 0.318940 0.0133472 0.00667360 0.999978i \(-0.497876\pi\)
0.00667360 + 0.999978i \(0.497876\pi\)
\(572\) 0.878869 0.0367474
\(573\) 0 0
\(574\) −9.45125 −0.394488
\(575\) 22.9722 0.958008
\(576\) 0 0
\(577\) 36.7863 1.53143 0.765716 0.643179i \(-0.222385\pi\)
0.765716 + 0.643179i \(0.222385\pi\)
\(578\) −64.7774 −2.69439
\(579\) 0 0
\(580\) −0.904396 −0.0375530
\(581\) 16.0743 0.666876
\(582\) 0 0
\(583\) 10.8254 0.448343
\(584\) 18.6731 0.772698
\(585\) 0 0
\(586\) 10.4977 0.433657
\(587\) −35.9317 −1.48306 −0.741530 0.670920i \(-0.765899\pi\)
−0.741530 + 0.670920i \(0.765899\pi\)
\(588\) 0 0
\(589\) 4.48765 0.184910
\(590\) 0.357908 0.0147348
\(591\) 0 0
\(592\) 41.4929 1.70535
\(593\) −5.63179 −0.231270 −0.115635 0.993292i \(-0.536890\pi\)
−0.115635 + 0.993292i \(0.536890\pi\)
\(594\) 0 0
\(595\) 17.5739 0.720458
\(596\) −4.27008 −0.174909
\(597\) 0 0
\(598\) −25.7140 −1.05152
\(599\) 25.5479 1.04386 0.521929 0.852989i \(-0.325213\pi\)
0.521929 + 0.852989i \(0.325213\pi\)
\(600\) 0 0
\(601\) 39.1627 1.59748 0.798741 0.601675i \(-0.205500\pi\)
0.798741 + 0.601675i \(0.205500\pi\)
\(602\) 31.6990 1.29195
\(603\) 0 0
\(604\) 4.16807 0.169597
\(605\) 0.569898 0.0231697
\(606\) 0 0
\(607\) 8.19398 0.332583 0.166292 0.986077i \(-0.446821\pi\)
0.166292 + 0.986077i \(0.446821\pi\)
\(608\) 2.21135 0.0896820
\(609\) 0 0
\(610\) 0.855232 0.0346273
\(611\) 12.0858 0.488938
\(612\) 0 0
\(613\) 29.6138 1.19609 0.598046 0.801462i \(-0.295944\pi\)
0.598046 + 0.801462i \(0.295944\pi\)
\(614\) −11.7084 −0.472511
\(615\) 0 0
\(616\) 10.4284 0.420172
\(617\) −24.1413 −0.971894 −0.485947 0.873988i \(-0.661525\pi\)
−0.485947 + 0.873988i \(0.661525\pi\)
\(618\) 0 0
\(619\) −15.2803 −0.614168 −0.307084 0.951682i \(-0.599353\pi\)
−0.307084 + 0.951682i \(0.599353\pi\)
\(620\) 0.413168 0.0165932
\(621\) 0 0
\(622\) −2.05365 −0.0823440
\(623\) −53.4352 −2.14084
\(624\) 0 0
\(625\) 20.2337 0.809349
\(626\) 7.72075 0.308583
\(627\) 0 0
\(628\) 4.50635 0.179823
\(629\) 72.4791 2.88993
\(630\) 0 0
\(631\) −25.2273 −1.00428 −0.502142 0.864785i \(-0.667454\pi\)
−0.502142 + 0.864785i \(0.667454\pi\)
\(632\) −34.4339 −1.36971
\(633\) 0 0
\(634\) 45.6080 1.81132
\(635\) −9.26722 −0.367758
\(636\) 0 0
\(637\) −30.7048 −1.21657
\(638\) −9.44943 −0.374106
\(639\) 0 0
\(640\) −7.39177 −0.292186
\(641\) 11.1758 0.441417 0.220709 0.975340i \(-0.429163\pi\)
0.220709 + 0.975340i \(0.429163\pi\)
\(642\) 0 0
\(643\) 33.0083 1.30172 0.650861 0.759197i \(-0.274408\pi\)
0.650861 + 0.759197i \(0.274408\pi\)
\(644\) 4.92311 0.193998
\(645\) 0 0
\(646\) 18.1590 0.714456
\(647\) 10.2919 0.404616 0.202308 0.979322i \(-0.435156\pi\)
0.202308 + 0.979322i \(0.435156\pi\)
\(648\) 0 0
\(649\) 0.418492 0.0164273
\(650\) −24.4664 −0.959651
\(651\) 0 0
\(652\) −0.215485 −0.00843905
\(653\) 8.56890 0.335327 0.167664 0.985844i \(-0.446378\pi\)
0.167664 + 0.985844i \(0.446378\pi\)
\(654\) 0 0
\(655\) −9.54861 −0.373095
\(656\) 7.03464 0.274657
\(657\) 0 0
\(658\) −20.6764 −0.806049
\(659\) −0.185337 −0.00721972 −0.00360986 0.999993i \(-0.501149\pi\)
−0.00360986 + 0.999993i \(0.501149\pi\)
\(660\) 0 0
\(661\) 6.58896 0.256281 0.128141 0.991756i \(-0.459099\pi\)
0.128141 + 0.991756i \(0.459099\pi\)
\(662\) −26.1017 −1.01447
\(663\) 0 0
\(664\) −10.6062 −0.411600
\(665\) −3.53447 −0.137061
\(666\) 0 0
\(667\) 30.9400 1.19800
\(668\) −3.37967 −0.130763
\(669\) 0 0
\(670\) −8.41868 −0.325242
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 32.8732 1.26717 0.633584 0.773674i \(-0.281583\pi\)
0.633584 + 0.773674i \(0.281583\pi\)
\(674\) −5.07559 −0.195505
\(675\) 0 0
\(676\) −0.211489 −0.00813418
\(677\) 49.0282 1.88431 0.942154 0.335181i \(-0.108797\pi\)
0.942154 + 0.335181i \(0.108797\pi\)
\(678\) 0 0
\(679\) 7.07339 0.271452
\(680\) −11.5956 −0.444671
\(681\) 0 0
\(682\) 4.31692 0.165303
\(683\) 24.9821 0.955912 0.477956 0.878384i \(-0.341378\pi\)
0.477956 + 0.878384i \(0.341378\pi\)
\(684\) 0 0
\(685\) −12.3697 −0.472623
\(686\) 10.7680 0.411122
\(687\) 0 0
\(688\) −23.5938 −0.899506
\(689\) 37.7508 1.43819
\(690\) 0 0
\(691\) 8.91705 0.339221 0.169610 0.985511i \(-0.445749\pi\)
0.169610 + 0.985511i \(0.445749\pi\)
\(692\) 2.27784 0.0865905
\(693\) 0 0
\(694\) 33.0469 1.25444
\(695\) 5.62303 0.213294
\(696\) 0 0
\(697\) 12.2880 0.465441
\(698\) 9.23392 0.349509
\(699\) 0 0
\(700\) 4.68425 0.177048
\(701\) 19.3770 0.731861 0.365930 0.930642i \(-0.380751\pi\)
0.365930 + 0.930642i \(0.380751\pi\)
\(702\) 0 0
\(703\) −14.5771 −0.549785
\(704\) −6.75385 −0.254545
\(705\) 0 0
\(706\) −4.22513 −0.159015
\(707\) −75.1514 −2.82636
\(708\) 0 0
\(709\) −31.7098 −1.19089 −0.595444 0.803397i \(-0.703024\pi\)
−0.595444 + 0.803397i \(0.703024\pi\)
\(710\) 8.49388 0.318770
\(711\) 0 0
\(712\) 35.2577 1.32134
\(713\) −14.1348 −0.529351
\(714\) 0 0
\(715\) 1.98737 0.0743235
\(716\) 6.57889 0.245865
\(717\) 0 0
\(718\) −29.0305 −1.08341
\(719\) 29.9033 1.11521 0.557603 0.830108i \(-0.311721\pi\)
0.557603 + 0.830108i \(0.311721\pi\)
\(720\) 0 0
\(721\) −9.61291 −0.358003
\(722\) 24.8607 0.925218
\(723\) 0 0
\(724\) 0.797492 0.0296386
\(725\) 29.4388 1.09333
\(726\) 0 0
\(727\) −11.3206 −0.419859 −0.209930 0.977717i \(-0.567323\pi\)
−0.209930 + 0.977717i \(0.567323\pi\)
\(728\) 36.3663 1.34783
\(729\) 0 0
\(730\) −6.08805 −0.225329
\(731\) −41.2133 −1.52433
\(732\) 0 0
\(733\) 46.5548 1.71954 0.859771 0.510680i \(-0.170606\pi\)
0.859771 + 0.510680i \(0.170606\pi\)
\(734\) −14.2351 −0.525426
\(735\) 0 0
\(736\) −6.96509 −0.256737
\(737\) −9.84375 −0.362599
\(738\) 0 0
\(739\) 3.40428 0.125229 0.0626143 0.998038i \(-0.480056\pi\)
0.0626143 + 0.998038i \(0.480056\pi\)
\(740\) −1.34208 −0.0493359
\(741\) 0 0
\(742\) −64.5843 −2.37096
\(743\) −3.46368 −0.127070 −0.0635350 0.997980i \(-0.520237\pi\)
−0.0635350 + 0.997980i \(0.520237\pi\)
\(744\) 0 0
\(745\) −9.65585 −0.353763
\(746\) 42.2284 1.54609
\(747\) 0 0
\(748\) 1.95486 0.0714769
\(749\) 42.7522 1.56213
\(750\) 0 0
\(751\) −1.89930 −0.0693065 −0.0346532 0.999399i \(-0.511033\pi\)
−0.0346532 + 0.999399i \(0.511033\pi\)
\(752\) 15.3896 0.561201
\(753\) 0 0
\(754\) −32.9524 −1.20006
\(755\) 9.42519 0.343018
\(756\) 0 0
\(757\) 8.67210 0.315193 0.157596 0.987504i \(-0.449626\pi\)
0.157596 + 0.987504i \(0.449626\pi\)
\(758\) 43.0271 1.56281
\(759\) 0 0
\(760\) 2.33212 0.0845948
\(761\) −37.5110 −1.35977 −0.679887 0.733317i \(-0.737971\pi\)
−0.679887 + 0.733317i \(0.737971\pi\)
\(762\) 0 0
\(763\) 58.1329 2.10455
\(764\) 0.0856348 0.00309816
\(765\) 0 0
\(766\) −53.9443 −1.94909
\(767\) 1.45938 0.0526953
\(768\) 0 0
\(769\) −30.1552 −1.08743 −0.543713 0.839271i \(-0.682982\pi\)
−0.543713 + 0.839271i \(0.682982\pi\)
\(770\) −3.40001 −0.122528
\(771\) 0 0
\(772\) −2.77198 −0.0997656
\(773\) 23.2997 0.838031 0.419015 0.907979i \(-0.362375\pi\)
0.419015 + 0.907979i \(0.362375\pi\)
\(774\) 0 0
\(775\) −13.4490 −0.483101
\(776\) −4.66717 −0.167542
\(777\) 0 0
\(778\) −54.8908 −1.96793
\(779\) −2.47137 −0.0885461
\(780\) 0 0
\(781\) 9.93167 0.355383
\(782\) −57.1955 −2.04531
\(783\) 0 0
\(784\) −39.0984 −1.39637
\(785\) 10.1901 0.363702
\(786\) 0 0
\(787\) 51.3754 1.83134 0.915668 0.401936i \(-0.131663\pi\)
0.915668 + 0.401936i \(0.131663\pi\)
\(788\) −3.44384 −0.122682
\(789\) 0 0
\(790\) 11.2266 0.399425
\(791\) −35.3329 −1.25629
\(792\) 0 0
\(793\) 3.48724 0.123836
\(794\) −26.1553 −0.928217
\(795\) 0 0
\(796\) −1.27361 −0.0451419
\(797\) −17.7177 −0.627593 −0.313796 0.949490i \(-0.601601\pi\)
−0.313796 + 0.949490i \(0.601601\pi\)
\(798\) 0 0
\(799\) 26.8823 0.951027
\(800\) −6.62716 −0.234305
\(801\) 0 0
\(802\) −42.5915 −1.50396
\(803\) −7.11859 −0.251210
\(804\) 0 0
\(805\) 11.1325 0.392370
\(806\) 15.0541 0.530259
\(807\) 0 0
\(808\) 49.5864 1.74445
\(809\) −16.0592 −0.564612 −0.282306 0.959324i \(-0.591099\pi\)
−0.282306 + 0.959324i \(0.591099\pi\)
\(810\) 0 0
\(811\) −12.0185 −0.422028 −0.211014 0.977483i \(-0.567677\pi\)
−0.211014 + 0.977483i \(0.567677\pi\)
\(812\) 6.30895 0.221401
\(813\) 0 0
\(814\) −14.0225 −0.491489
\(815\) −0.487273 −0.0170684
\(816\) 0 0
\(817\) 8.28885 0.289990
\(818\) −41.5107 −1.45139
\(819\) 0 0
\(820\) −0.227534 −0.00794584
\(821\) 12.6542 0.441636 0.220818 0.975315i \(-0.429127\pi\)
0.220818 + 0.975315i \(0.429127\pi\)
\(822\) 0 0
\(823\) −0.0537888 −0.00187496 −0.000937480 1.00000i \(-0.500298\pi\)
−0.000937480 1.00000i \(0.500298\pi\)
\(824\) 6.34280 0.220962
\(825\) 0 0
\(826\) −2.49672 −0.0868720
\(827\) 31.9496 1.11100 0.555498 0.831518i \(-0.312528\pi\)
0.555498 + 0.831518i \(0.312528\pi\)
\(828\) 0 0
\(829\) 35.7815 1.24274 0.621372 0.783516i \(-0.286576\pi\)
0.621372 + 0.783516i \(0.286576\pi\)
\(830\) 3.45797 0.120028
\(831\) 0 0
\(832\) −23.5523 −0.816528
\(833\) −68.2964 −2.36633
\(834\) 0 0
\(835\) −7.64238 −0.264476
\(836\) −0.393164 −0.0135979
\(837\) 0 0
\(838\) −1.71371 −0.0591990
\(839\) −6.85181 −0.236551 −0.118275 0.992981i \(-0.537737\pi\)
−0.118275 + 0.992981i \(0.537737\pi\)
\(840\) 0 0
\(841\) 10.6495 0.367224
\(842\) −7.87107 −0.271255
\(843\) 0 0
\(844\) 0.105442 0.00362945
\(845\) −0.478236 −0.0164518
\(846\) 0 0
\(847\) −3.97554 −0.136601
\(848\) 48.0706 1.65075
\(849\) 0 0
\(850\) −54.4204 −1.86661
\(851\) 45.9135 1.57389
\(852\) 0 0
\(853\) −42.9971 −1.47219 −0.736096 0.676877i \(-0.763333\pi\)
−0.736096 + 0.676877i \(0.763333\pi\)
\(854\) −5.96599 −0.204152
\(855\) 0 0
\(856\) −28.2088 −0.964157
\(857\) 3.32058 0.113429 0.0567145 0.998390i \(-0.481938\pi\)
0.0567145 + 0.998390i \(0.481938\pi\)
\(858\) 0 0
\(859\) 46.4438 1.58464 0.792321 0.610105i \(-0.208873\pi\)
0.792321 + 0.610105i \(0.208873\pi\)
\(860\) 0.763138 0.0260228
\(861\) 0 0
\(862\) 59.6574 2.03194
\(863\) −11.1536 −0.379672 −0.189836 0.981816i \(-0.560796\pi\)
−0.189836 + 0.981816i \(0.560796\pi\)
\(864\) 0 0
\(865\) 5.15084 0.175134
\(866\) 33.6679 1.14408
\(867\) 0 0
\(868\) −2.88221 −0.0978286
\(869\) 13.1270 0.445302
\(870\) 0 0
\(871\) −34.3275 −1.16314
\(872\) −38.3573 −1.29894
\(873\) 0 0
\(874\) 11.5032 0.389102
\(875\) 21.9207 0.741054
\(876\) 0 0
\(877\) −29.0494 −0.980929 −0.490464 0.871461i \(-0.663173\pi\)
−0.490464 + 0.871461i \(0.663173\pi\)
\(878\) 8.20810 0.277010
\(879\) 0 0
\(880\) 2.53065 0.0853082
\(881\) 21.4381 0.722269 0.361134 0.932514i \(-0.382390\pi\)
0.361134 + 0.932514i \(0.382390\pi\)
\(882\) 0 0
\(883\) −35.2149 −1.18508 −0.592538 0.805542i \(-0.701874\pi\)
−0.592538 + 0.805542i \(0.701874\pi\)
\(884\) 6.81708 0.229283
\(885\) 0 0
\(886\) −6.32525 −0.212501
\(887\) −41.4951 −1.39327 −0.696635 0.717426i \(-0.745320\pi\)
−0.696635 + 0.717426i \(0.745320\pi\)
\(888\) 0 0
\(889\) 64.6469 2.16819
\(890\) −11.4952 −0.385319
\(891\) 0 0
\(892\) 2.13453 0.0714693
\(893\) −5.40659 −0.180925
\(894\) 0 0
\(895\) 14.8767 0.497274
\(896\) 51.5641 1.72263
\(897\) 0 0
\(898\) 35.8027 1.19475
\(899\) −18.1137 −0.604124
\(900\) 0 0
\(901\) 83.9689 2.79741
\(902\) −2.37735 −0.0791572
\(903\) 0 0
\(904\) 23.3134 0.775391
\(905\) 1.80336 0.0599456
\(906\) 0 0
\(907\) −21.7745 −0.723009 −0.361505 0.932370i \(-0.617737\pi\)
−0.361505 + 0.932370i \(0.617737\pi\)
\(908\) 0.325007 0.0107857
\(909\) 0 0
\(910\) −11.8566 −0.393044
\(911\) −26.9675 −0.893472 −0.446736 0.894666i \(-0.647414\pi\)
−0.446736 + 0.894666i \(0.647414\pi\)
\(912\) 0 0
\(913\) 4.04331 0.133814
\(914\) −53.7108 −1.77660
\(915\) 0 0
\(916\) 4.54445 0.150153
\(917\) 66.6099 2.19965
\(918\) 0 0
\(919\) 25.4015 0.837917 0.418958 0.908005i \(-0.362395\pi\)
0.418958 + 0.908005i \(0.362395\pi\)
\(920\) −7.34548 −0.242173
\(921\) 0 0
\(922\) 49.6285 1.63443
\(923\) 34.6341 1.14000
\(924\) 0 0
\(925\) 43.6859 1.43638
\(926\) 45.2730 1.48776
\(927\) 0 0
\(928\) −8.92575 −0.293002
\(929\) 29.0759 0.953949 0.476975 0.878917i \(-0.341733\pi\)
0.476975 + 0.878917i \(0.341733\pi\)
\(930\) 0 0
\(931\) 13.7358 0.450174
\(932\) −3.82571 −0.125315
\(933\) 0 0
\(934\) 19.9652 0.653280
\(935\) 4.42050 0.144566
\(936\) 0 0
\(937\) −21.2285 −0.693506 −0.346753 0.937957i \(-0.612716\pi\)
−0.346753 + 0.937957i \(0.612716\pi\)
\(938\) 58.7277 1.91753
\(939\) 0 0
\(940\) −0.497774 −0.0162356
\(941\) −53.6582 −1.74921 −0.874604 0.484839i \(-0.838878\pi\)
−0.874604 + 0.484839i \(0.838878\pi\)
\(942\) 0 0
\(943\) 7.78410 0.253485
\(944\) 1.85833 0.0604835
\(945\) 0 0
\(946\) 7.97351 0.259241
\(947\) −42.9691 −1.39631 −0.698154 0.715947i \(-0.745995\pi\)
−0.698154 + 0.715947i \(0.745995\pi\)
\(948\) 0 0
\(949\) −24.8242 −0.805829
\(950\) 10.9451 0.355106
\(951\) 0 0
\(952\) 80.8894 2.62164
\(953\) −20.7043 −0.670678 −0.335339 0.942098i \(-0.608851\pi\)
−0.335339 + 0.942098i \(0.608851\pi\)
\(954\) 0 0
\(955\) 0.193644 0.00626619
\(956\) −1.41586 −0.0457923
\(957\) 0 0
\(958\) 54.3816 1.75699
\(959\) 86.2896 2.78644
\(960\) 0 0
\(961\) −22.7249 −0.733060
\(962\) −48.8998 −1.57659
\(963\) 0 0
\(964\) −2.48488 −0.0800325
\(965\) −6.26822 −0.201781
\(966\) 0 0
\(967\) −28.9634 −0.931401 −0.465701 0.884942i \(-0.654198\pi\)
−0.465701 + 0.884942i \(0.654198\pi\)
\(968\) 2.62314 0.0843110
\(969\) 0 0
\(970\) 1.52165 0.0488574
\(971\) −41.1027 −1.31905 −0.659524 0.751684i \(-0.729242\pi\)
−0.659524 + 0.751684i \(0.729242\pi\)
\(972\) 0 0
\(973\) −39.2255 −1.25751
\(974\) −7.41556 −0.237610
\(975\) 0 0
\(976\) 4.44053 0.142138
\(977\) −0.544347 −0.0174152 −0.00870761 0.999962i \(-0.502772\pi\)
−0.00870761 + 0.999962i \(0.502772\pi\)
\(978\) 0 0
\(979\) −13.4410 −0.429576
\(980\) 1.26463 0.0403971
\(981\) 0 0
\(982\) −21.0320 −0.671158
\(983\) −19.5398 −0.623224 −0.311612 0.950209i \(-0.600869\pi\)
−0.311612 + 0.950209i \(0.600869\pi\)
\(984\) 0 0
\(985\) −7.78750 −0.248130
\(986\) −73.2958 −2.33422
\(987\) 0 0
\(988\) −1.37106 −0.0436191
\(989\) −26.1074 −0.830169
\(990\) 0 0
\(991\) −45.5206 −1.44601 −0.723004 0.690843i \(-0.757239\pi\)
−0.723004 + 0.690843i \(0.757239\pi\)
\(992\) 4.07768 0.129466
\(993\) 0 0
\(994\) −59.2522 −1.87937
\(995\) −2.87999 −0.0913020
\(996\) 0 0
\(997\) 8.42755 0.266903 0.133452 0.991055i \(-0.457394\pi\)
0.133452 + 0.991055i \(0.457394\pi\)
\(998\) −22.5577 −0.714052
\(999\) 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.1.5 13
3.2 odd 2 2013.2.a.e.1.9 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.9 13 3.2 odd 2
6039.2.a.i.1.5 13 1.1 even 1 trivial