Properties

Label 1521.4.a.ba.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1520092.1
Defining polynomial: \( x^{4} - x^{3} - 40x^{2} - 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-5.49403\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.85588 q^{2} +6.86783 q^{4} -19.5342 q^{5} +6.26434 q^{7} +4.36551 q^{8} +O(q^{10})\) \(q-3.85588 q^{2} +6.86783 q^{4} -19.5342 q^{5} +6.26434 q^{7} +4.36551 q^{8} +75.3217 q^{10} -27.2460 q^{11} -24.1546 q^{14} -71.7755 q^{16} -30.8471 q^{17} -127.850 q^{19} -134.158 q^{20} +105.057 q^{22} -84.3198 q^{23} +256.586 q^{25} +43.0224 q^{28} -272.460 q^{29} +166.264 q^{31} +241.834 q^{32} +118.943 q^{34} -122.369 q^{35} +198.229 q^{37} +492.976 q^{38} -85.2769 q^{40} +160.385 q^{41} +158.943 q^{43} -187.121 q^{44} +325.127 q^{46} -305.889 q^{47} -303.758 q^{49} -989.366 q^{50} -356.780 q^{53} +532.229 q^{55} +27.3470 q^{56} +1050.57 q^{58} -470.384 q^{59} -171.815 q^{61} -641.096 q^{62} -358.279 q^{64} -1022.49 q^{67} -211.852 q^{68} +471.840 q^{70} -188.616 q^{71} -959.975 q^{73} -764.349 q^{74} -878.055 q^{76} -170.678 q^{77} -1034.80 q^{79} +1402.08 q^{80} -618.424 q^{82} -105.383 q^{83} +602.574 q^{85} -612.864 q^{86} -118.943 q^{88} -649.860 q^{89} -579.094 q^{92} +1179.47 q^{94} +2497.46 q^{95} -707.746 q^{97} +1171.26 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 58 q^{4} - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 58 q^{4} - 36 q^{7} - 4 q^{10} + 354 q^{16} - 84 q^{19} + 176 q^{22} + 660 q^{25} - 988 q^{28} + 604 q^{31} + 720 q^{34} - 184 q^{37} - 2356 q^{40} + 880 q^{43} + 2888 q^{46} - 116 q^{49} + 1152 q^{55} + 1760 q^{58} + 656 q^{61} + 3482 q^{64} - 3052 q^{67} + 4696 q^{70} + 312 q^{73} + 2044 q^{76} - 720 q^{79} + 396 q^{82} - 32 q^{85} - 720 q^{88} + 4840 q^{94} + 344 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.85588 −1.36326 −0.681630 0.731697i \(-0.738729\pi\)
−0.681630 + 0.731697i \(0.738729\pi\)
\(3\) 0 0
\(4\) 6.86783 0.858479
\(5\) −19.5342 −1.74719 −0.873597 0.486650i \(-0.838219\pi\)
−0.873597 + 0.486650i \(0.838219\pi\)
\(6\) 0 0
\(7\) 6.26434 0.338242 0.169121 0.985595i \(-0.445907\pi\)
0.169121 + 0.985595i \(0.445907\pi\)
\(8\) 4.36551 0.192930
\(9\) 0 0
\(10\) 75.3217 2.38188
\(11\) −27.2460 −0.746816 −0.373408 0.927667i \(-0.621811\pi\)
−0.373408 + 0.927667i \(0.621811\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −24.1546 −0.461113
\(15\) 0 0
\(16\) −71.7755 −1.12149
\(17\) −30.8471 −0.440089 −0.220044 0.975490i \(-0.570620\pi\)
−0.220044 + 0.975490i \(0.570620\pi\)
\(18\) 0 0
\(19\) −127.850 −1.54373 −0.771865 0.635786i \(-0.780676\pi\)
−0.771865 + 0.635786i \(0.780676\pi\)
\(20\) −134.158 −1.49993
\(21\) 0 0
\(22\) 105.057 1.01810
\(23\) −84.3198 −0.764430 −0.382215 0.924073i \(-0.624839\pi\)
−0.382215 + 0.924073i \(0.624839\pi\)
\(24\) 0 0
\(25\) 256.586 2.05269
\(26\) 0 0
\(27\) 0 0
\(28\) 43.0224 0.290374
\(29\) −272.460 −1.74464 −0.872320 0.488936i \(-0.837385\pi\)
−0.872320 + 0.488936i \(0.837385\pi\)
\(30\) 0 0
\(31\) 166.264 0.963289 0.481644 0.876367i \(-0.340040\pi\)
0.481644 + 0.876367i \(0.340040\pi\)
\(32\) 241.834 1.33596
\(33\) 0 0
\(34\) 118.943 0.599956
\(35\) −122.369 −0.590975
\(36\) 0 0
\(37\) 198.229 0.880776 0.440388 0.897808i \(-0.354841\pi\)
0.440388 + 0.897808i \(0.354841\pi\)
\(38\) 492.976 2.10451
\(39\) 0 0
\(40\) −85.2769 −0.337086
\(41\) 160.385 0.610923 0.305462 0.952204i \(-0.401189\pi\)
0.305462 + 0.952204i \(0.401189\pi\)
\(42\) 0 0
\(43\) 158.943 0.563687 0.281843 0.959460i \(-0.409054\pi\)
0.281843 + 0.959460i \(0.409054\pi\)
\(44\) −187.121 −0.641126
\(45\) 0 0
\(46\) 325.127 1.04212
\(47\) −305.889 −0.949329 −0.474665 0.880167i \(-0.657431\pi\)
−0.474665 + 0.880167i \(0.657431\pi\)
\(48\) 0 0
\(49\) −303.758 −0.885592
\(50\) −989.366 −2.79835
\(51\) 0 0
\(52\) 0 0
\(53\) −356.780 −0.924669 −0.462335 0.886706i \(-0.652988\pi\)
−0.462335 + 0.886706i \(0.652988\pi\)
\(54\) 0 0
\(55\) 532.229 1.30483
\(56\) 27.3470 0.0652571
\(57\) 0 0
\(58\) 1050.57 2.37840
\(59\) −470.384 −1.03795 −0.518973 0.854791i \(-0.673685\pi\)
−0.518973 + 0.854791i \(0.673685\pi\)
\(60\) 0 0
\(61\) −171.815 −0.360635 −0.180317 0.983608i \(-0.557712\pi\)
−0.180317 + 0.983608i \(0.557712\pi\)
\(62\) −641.096 −1.31321
\(63\) 0 0
\(64\) −358.279 −0.699764
\(65\) 0 0
\(66\) 0 0
\(67\) −1022.49 −1.86444 −0.932220 0.361892i \(-0.882131\pi\)
−0.932220 + 0.361892i \(0.882131\pi\)
\(68\) −211.852 −0.377807
\(69\) 0 0
\(70\) 471.840 0.805653
\(71\) −188.616 −0.315276 −0.157638 0.987497i \(-0.550388\pi\)
−0.157638 + 0.987497i \(0.550388\pi\)
\(72\) 0 0
\(73\) −959.975 −1.53913 −0.769566 0.638568i \(-0.779527\pi\)
−0.769566 + 0.638568i \(0.779527\pi\)
\(74\) −764.349 −1.20073
\(75\) 0 0
\(76\) −878.055 −1.32526
\(77\) −170.678 −0.252605
\(78\) 0 0
\(79\) −1034.80 −1.47373 −0.736863 0.676042i \(-0.763694\pi\)
−0.736863 + 0.676042i \(0.763694\pi\)
\(80\) 1402.08 1.95947
\(81\) 0 0
\(82\) −618.424 −0.832847
\(83\) −105.383 −0.139365 −0.0696824 0.997569i \(-0.522199\pi\)
−0.0696824 + 0.997569i \(0.522199\pi\)
\(84\) 0 0
\(85\) 602.574 0.768921
\(86\) −612.864 −0.768452
\(87\) 0 0
\(88\) −118.943 −0.144083
\(89\) −649.860 −0.773990 −0.386995 0.922082i \(-0.626487\pi\)
−0.386995 + 0.922082i \(0.626487\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −579.094 −0.656247
\(93\) 0 0
\(94\) 1179.47 1.29418
\(95\) 2497.46 2.69720
\(96\) 0 0
\(97\) −707.746 −0.740832 −0.370416 0.928866i \(-0.620785\pi\)
−0.370416 + 0.928866i \(0.620785\pi\)
\(98\) 1171.26 1.20729
\(99\) 0 0
\(100\) 1762.19 1.76219
\(101\) −1409.59 −1.38871 −0.694353 0.719634i \(-0.744310\pi\)
−0.694353 + 0.719634i \(0.744310\pi\)
\(102\) 0 0
\(103\) 1519.63 1.45373 0.726863 0.686783i \(-0.240978\pi\)
0.726863 + 0.686783i \(0.240978\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1375.70 1.26056
\(107\) −1957.57 −1.76865 −0.884323 0.466875i \(-0.845380\pi\)
−0.884323 + 0.466875i \(0.845380\pi\)
\(108\) 0 0
\(109\) −1517.63 −1.33360 −0.666801 0.745236i \(-0.732337\pi\)
−0.666801 + 0.745236i \(0.732337\pi\)
\(110\) −2052.21 −1.77883
\(111\) 0 0
\(112\) −449.626 −0.379336
\(113\) 2200.27 1.83171 0.915857 0.401505i \(-0.131513\pi\)
0.915857 + 0.401505i \(0.131513\pi\)
\(114\) 0 0
\(115\) 1647.12 1.33561
\(116\) −1871.21 −1.49774
\(117\) 0 0
\(118\) 1813.75 1.41499
\(119\) −193.236 −0.148857
\(120\) 0 0
\(121\) −588.656 −0.442266
\(122\) 662.500 0.491639
\(123\) 0 0
\(124\) 1141.88 0.826963
\(125\) −2570.43 −1.83925
\(126\) 0 0
\(127\) 1812.16 1.26617 0.633083 0.774084i \(-0.281789\pi\)
0.633083 + 0.774084i \(0.281789\pi\)
\(128\) −553.190 −0.381996
\(129\) 0 0
\(130\) 0 0
\(131\) −1001.31 −0.667822 −0.333911 0.942605i \(-0.608369\pi\)
−0.333911 + 0.942605i \(0.608369\pi\)
\(132\) 0 0
\(133\) −800.898 −0.522155
\(134\) 3942.62 2.54172
\(135\) 0 0
\(136\) −134.663 −0.0849064
\(137\) 1244.11 0.775849 0.387925 0.921691i \(-0.373192\pi\)
0.387925 + 0.921691i \(0.373192\pi\)
\(138\) 0 0
\(139\) 1348.85 0.823077 0.411539 0.911392i \(-0.364991\pi\)
0.411539 + 0.911392i \(0.364991\pi\)
\(140\) −840.410 −0.507340
\(141\) 0 0
\(142\) 727.281 0.429804
\(143\) 0 0
\(144\) 0 0
\(145\) 5322.29 3.04822
\(146\) 3701.55 2.09824
\(147\) 0 0
\(148\) 1361.41 0.756128
\(149\) −1041.63 −0.572710 −0.286355 0.958124i \(-0.592444\pi\)
−0.286355 + 0.958124i \(0.592444\pi\)
\(150\) 0 0
\(151\) −1361.90 −0.733970 −0.366985 0.930227i \(-0.619610\pi\)
−0.366985 + 0.930227i \(0.619610\pi\)
\(152\) −558.132 −0.297832
\(153\) 0 0
\(154\) 658.115 0.344366
\(155\) −3247.85 −1.68305
\(156\) 0 0
\(157\) 86.8978 0.0441733 0.0220866 0.999756i \(-0.492969\pi\)
0.0220866 + 0.999756i \(0.492969\pi\)
\(158\) 3990.08 2.00907
\(159\) 0 0
\(160\) −4724.04 −2.33418
\(161\) −528.208 −0.258563
\(162\) 0 0
\(163\) 28.5386 0.0137136 0.00685679 0.999976i \(-0.497817\pi\)
0.00685679 + 0.999976i \(0.497817\pi\)
\(164\) 1101.49 0.524465
\(165\) 0 0
\(166\) 406.344 0.189990
\(167\) 1289.87 0.597683 0.298842 0.954303i \(-0.403400\pi\)
0.298842 + 0.954303i \(0.403400\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −2323.45 −1.04824
\(171\) 0 0
\(172\) 1091.59 0.483913
\(173\) 603.489 0.265216 0.132608 0.991169i \(-0.457665\pi\)
0.132608 + 0.991169i \(0.457665\pi\)
\(174\) 0 0
\(175\) 1607.34 0.694306
\(176\) 1955.60 0.837549
\(177\) 0 0
\(178\) 2505.79 1.05515
\(179\) −1985.29 −0.828980 −0.414490 0.910054i \(-0.636040\pi\)
−0.414490 + 0.910054i \(0.636040\pi\)
\(180\) 0 0
\(181\) −3945.79 −1.62038 −0.810189 0.586169i \(-0.800635\pi\)
−0.810189 + 0.586169i \(0.800635\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −368.099 −0.147482
\(185\) −3872.26 −1.53889
\(186\) 0 0
\(187\) 840.459 0.328665
\(188\) −2100.79 −0.814979
\(189\) 0 0
\(190\) −9629.91 −3.67698
\(191\) 1446.48 0.547979 0.273989 0.961733i \(-0.411657\pi\)
0.273989 + 0.961733i \(0.411657\pi\)
\(192\) 0 0
\(193\) −784.523 −0.292597 −0.146299 0.989240i \(-0.546736\pi\)
−0.146299 + 0.989240i \(0.546736\pi\)
\(194\) 2728.98 1.00995
\(195\) 0 0
\(196\) −2086.16 −0.760262
\(197\) −494.336 −0.178782 −0.0893909 0.995997i \(-0.528492\pi\)
−0.0893909 + 0.995997i \(0.528492\pi\)
\(198\) 0 0
\(199\) −3022.59 −1.07671 −0.538357 0.842717i \(-0.680955\pi\)
−0.538357 + 0.842717i \(0.680955\pi\)
\(200\) 1120.13 0.396025
\(201\) 0 0
\(202\) 5435.21 1.89317
\(203\) −1706.78 −0.590111
\(204\) 0 0
\(205\) −3132.99 −1.06740
\(206\) −5859.52 −1.98181
\(207\) 0 0
\(208\) 0 0
\(209\) 3483.41 1.15288
\(210\) 0 0
\(211\) 3185.24 1.03925 0.519623 0.854396i \(-0.326073\pi\)
0.519623 + 0.854396i \(0.326073\pi\)
\(212\) −2450.30 −0.793809
\(213\) 0 0
\(214\) 7548.15 2.41113
\(215\) −3104.82 −0.984870
\(216\) 0 0
\(217\) 1041.54 0.325825
\(218\) 5851.81 1.81805
\(219\) 0 0
\(220\) 3655.26 1.12017
\(221\) 0 0
\(222\) 0 0
\(223\) −997.686 −0.299596 −0.149798 0.988717i \(-0.547862\pi\)
−0.149798 + 0.988717i \(0.547862\pi\)
\(224\) 1514.93 0.451877
\(225\) 0 0
\(226\) −8483.97 −2.49710
\(227\) 6496.57 1.89953 0.949763 0.312969i \(-0.101324\pi\)
0.949763 + 0.312969i \(0.101324\pi\)
\(228\) 0 0
\(229\) −708.434 −0.204431 −0.102215 0.994762i \(-0.532593\pi\)
−0.102215 + 0.994762i \(0.532593\pi\)
\(230\) −6351.11 −1.82078
\(231\) 0 0
\(232\) −1189.43 −0.336593
\(233\) 4919.02 1.38307 0.691536 0.722342i \(-0.256934\pi\)
0.691536 + 0.722342i \(0.256934\pi\)
\(234\) 0 0
\(235\) 5975.30 1.65866
\(236\) −3230.52 −0.891054
\(237\) 0 0
\(238\) 745.097 0.202930
\(239\) −324.370 −0.0877898 −0.0438949 0.999036i \(-0.513977\pi\)
−0.0438949 + 0.999036i \(0.513977\pi\)
\(240\) 0 0
\(241\) 1940.43 0.518649 0.259324 0.965790i \(-0.416500\pi\)
0.259324 + 0.965790i \(0.416500\pi\)
\(242\) 2269.79 0.602924
\(243\) 0 0
\(244\) −1180.00 −0.309597
\(245\) 5933.68 1.54730
\(246\) 0 0
\(247\) 0 0
\(248\) 725.829 0.185847
\(249\) 0 0
\(250\) 9911.28 2.50738
\(251\) −1958.45 −0.492495 −0.246248 0.969207i \(-0.579198\pi\)
−0.246248 + 0.969207i \(0.579198\pi\)
\(252\) 0 0
\(253\) 2297.38 0.570889
\(254\) −6987.47 −1.72611
\(255\) 0 0
\(256\) 4999.27 1.22052
\(257\) −4330.35 −1.05105 −0.525525 0.850778i \(-0.676131\pi\)
−0.525525 + 0.850778i \(0.676131\pi\)
\(258\) 0 0
\(259\) 1241.78 0.297916
\(260\) 0 0
\(261\) 0 0
\(262\) 3860.93 0.910415
\(263\) −2933.60 −0.687807 −0.343904 0.939005i \(-0.611749\pi\)
−0.343904 + 0.939005i \(0.611749\pi\)
\(264\) 0 0
\(265\) 6969.42 1.61558
\(266\) 3088.17 0.711834
\(267\) 0 0
\(268\) −7022.31 −1.60058
\(269\) 2458.25 0.557184 0.278592 0.960410i \(-0.410132\pi\)
0.278592 + 0.960410i \(0.410132\pi\)
\(270\) 0 0
\(271\) −2089.91 −0.468462 −0.234231 0.972181i \(-0.575257\pi\)
−0.234231 + 0.972181i \(0.575257\pi\)
\(272\) 2214.06 0.493557
\(273\) 0 0
\(274\) −4797.14 −1.05768
\(275\) −6990.94 −1.53298
\(276\) 0 0
\(277\) −2462.00 −0.534033 −0.267017 0.963692i \(-0.586038\pi\)
−0.267017 + 0.963692i \(0.586038\pi\)
\(278\) −5201.00 −1.12207
\(279\) 0 0
\(280\) −534.203 −0.114017
\(281\) 5775.91 1.22620 0.613100 0.790005i \(-0.289922\pi\)
0.613100 + 0.790005i \(0.289922\pi\)
\(282\) 0 0
\(283\) 5093.45 1.06987 0.534936 0.844892i \(-0.320336\pi\)
0.534936 + 0.844892i \(0.320336\pi\)
\(284\) −1295.38 −0.270658
\(285\) 0 0
\(286\) 0 0
\(287\) 1004.70 0.206640
\(288\) 0 0
\(289\) −3961.46 −0.806322
\(290\) −20522.1 −4.15552
\(291\) 0 0
\(292\) −6592.95 −1.32131
\(293\) 1503.73 0.299826 0.149913 0.988699i \(-0.452101\pi\)
0.149913 + 0.988699i \(0.452101\pi\)
\(294\) 0 0
\(295\) 9188.59 1.81349
\(296\) 865.372 0.169928
\(297\) 0 0
\(298\) 4016.41 0.780753
\(299\) 0 0
\(300\) 0 0
\(301\) 995.670 0.190663
\(302\) 5251.31 1.00059
\(303\) 0 0
\(304\) 9176.53 1.73128
\(305\) 3356.28 0.630099
\(306\) 0 0
\(307\) 6567.59 1.22095 0.610476 0.792035i \(-0.290978\pi\)
0.610476 + 0.792035i \(0.290978\pi\)
\(308\) −1172.19 −0.216856
\(309\) 0 0
\(310\) 12523.3 2.29444
\(311\) 5714.66 1.04196 0.520978 0.853570i \(-0.325567\pi\)
0.520978 + 0.853570i \(0.325567\pi\)
\(312\) 0 0
\(313\) −4953.59 −0.894548 −0.447274 0.894397i \(-0.647605\pi\)
−0.447274 + 0.894397i \(0.647605\pi\)
\(314\) −335.068 −0.0602197
\(315\) 0 0
\(316\) −7106.85 −1.26516
\(317\) 2526.64 0.447666 0.223833 0.974628i \(-0.428143\pi\)
0.223833 + 0.974628i \(0.428143\pi\)
\(318\) 0 0
\(319\) 7423.44 1.30292
\(320\) 6998.71 1.22262
\(321\) 0 0
\(322\) 2036.71 0.352488
\(323\) 3943.81 0.679379
\(324\) 0 0
\(325\) 0 0
\(326\) −110.041 −0.0186952
\(327\) 0 0
\(328\) 700.160 0.117865
\(329\) −1916.19 −0.321103
\(330\) 0 0
\(331\) −7436.06 −1.23481 −0.617406 0.786645i \(-0.711816\pi\)
−0.617406 + 0.786645i \(0.711816\pi\)
\(332\) −723.752 −0.119642
\(333\) 0 0
\(334\) −4973.59 −0.814798
\(335\) 19973.6 3.25754
\(336\) 0 0
\(337\) 9467.96 1.53042 0.765212 0.643778i \(-0.222634\pi\)
0.765212 + 0.643778i \(0.222634\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 4138.37 0.660102
\(341\) −4530.04 −0.719400
\(342\) 0 0
\(343\) −4051.51 −0.637787
\(344\) 693.866 0.108752
\(345\) 0 0
\(346\) −2326.98 −0.361559
\(347\) 2794.85 0.432379 0.216189 0.976351i \(-0.430637\pi\)
0.216189 + 0.976351i \(0.430637\pi\)
\(348\) 0 0
\(349\) −2602.91 −0.399228 −0.199614 0.979875i \(-0.563969\pi\)
−0.199614 + 0.979875i \(0.563969\pi\)
\(350\) −6197.72 −0.946520
\(351\) 0 0
\(352\) −6589.01 −0.997714
\(353\) −8875.63 −1.33825 −0.669125 0.743150i \(-0.733331\pi\)
−0.669125 + 0.743150i \(0.733331\pi\)
\(354\) 0 0
\(355\) 3684.47 0.550849
\(356\) −4463.13 −0.664454
\(357\) 0 0
\(358\) 7655.04 1.13012
\(359\) −8946.74 −1.31529 −0.657647 0.753326i \(-0.728448\pi\)
−0.657647 + 0.753326i \(0.728448\pi\)
\(360\) 0 0
\(361\) 9486.72 1.38310
\(362\) 15214.5 2.20900
\(363\) 0 0
\(364\) 0 0
\(365\) 18752.4 2.68916
\(366\) 0 0
\(367\) −3965.30 −0.563997 −0.281998 0.959415i \(-0.590997\pi\)
−0.281998 + 0.959415i \(0.590997\pi\)
\(368\) 6052.10 0.857303
\(369\) 0 0
\(370\) 14931.0 2.09790
\(371\) −2234.99 −0.312762
\(372\) 0 0
\(373\) −1457.44 −0.202315 −0.101157 0.994870i \(-0.532255\pi\)
−0.101157 + 0.994870i \(0.532255\pi\)
\(374\) −3240.71 −0.448057
\(375\) 0 0
\(376\) −1335.36 −0.183154
\(377\) 0 0
\(378\) 0 0
\(379\) −1991.75 −0.269945 −0.134973 0.990849i \(-0.543095\pi\)
−0.134973 + 0.990849i \(0.543095\pi\)
\(380\) 17152.1 2.31549
\(381\) 0 0
\(382\) −5577.48 −0.747038
\(383\) 2504.45 0.334130 0.167065 0.985946i \(-0.446571\pi\)
0.167065 + 0.985946i \(0.446571\pi\)
\(384\) 0 0
\(385\) 3334.06 0.441350
\(386\) 3025.03 0.398886
\(387\) 0 0
\(388\) −4860.68 −0.635988
\(389\) 1448.12 0.188747 0.0943735 0.995537i \(-0.469915\pi\)
0.0943735 + 0.995537i \(0.469915\pi\)
\(390\) 0 0
\(391\) 2601.02 0.336417
\(392\) −1326.06 −0.170857
\(393\) 0 0
\(394\) 1906.10 0.243726
\(395\) 20214.1 2.57489
\(396\) 0 0
\(397\) 5634.14 0.712265 0.356133 0.934435i \(-0.384095\pi\)
0.356133 + 0.934435i \(0.384095\pi\)
\(398\) 11654.8 1.46784
\(399\) 0 0
\(400\) −18416.6 −2.30208
\(401\) 3648.07 0.454304 0.227152 0.973859i \(-0.427058\pi\)
0.227152 + 0.973859i \(0.427058\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −9680.82 −1.19218
\(405\) 0 0
\(406\) 6581.15 0.804475
\(407\) −5400.96 −0.657778
\(408\) 0 0
\(409\) 4459.91 0.539190 0.269595 0.962974i \(-0.413110\pi\)
0.269595 + 0.962974i \(0.413110\pi\)
\(410\) 12080.4 1.45515
\(411\) 0 0
\(412\) 10436.6 1.24799
\(413\) −2946.64 −0.351077
\(414\) 0 0
\(415\) 2058.57 0.243497
\(416\) 0 0
\(417\) 0 0
\(418\) −13431.6 −1.57168
\(419\) 4726.74 0.551114 0.275557 0.961285i \(-0.411138\pi\)
0.275557 + 0.961285i \(0.411138\pi\)
\(420\) 0 0
\(421\) 3090.51 0.357772 0.178886 0.983870i \(-0.442751\pi\)
0.178886 + 0.983870i \(0.442751\pi\)
\(422\) −12281.9 −1.41676
\(423\) 0 0
\(424\) −1557.53 −0.178397
\(425\) −7914.92 −0.903365
\(426\) 0 0
\(427\) −1076.31 −0.121982
\(428\) −13444.2 −1.51835
\(429\) 0 0
\(430\) 11971.8 1.34263
\(431\) 11131.9 1.24409 0.622045 0.782981i \(-0.286302\pi\)
0.622045 + 0.782981i \(0.286302\pi\)
\(432\) 0 0
\(433\) 6773.33 0.751745 0.375872 0.926671i \(-0.377343\pi\)
0.375872 + 0.926671i \(0.377343\pi\)
\(434\) −4016.04 −0.444185
\(435\) 0 0
\(436\) −10422.8 −1.14487
\(437\) 10780.3 1.18007
\(438\) 0 0
\(439\) −5384.95 −0.585443 −0.292722 0.956198i \(-0.594561\pi\)
−0.292722 + 0.956198i \(0.594561\pi\)
\(440\) 2323.45 0.251742
\(441\) 0 0
\(442\) 0 0
\(443\) 921.806 0.0988631 0.0494315 0.998778i \(-0.484259\pi\)
0.0494315 + 0.998778i \(0.484259\pi\)
\(444\) 0 0
\(445\) 12694.5 1.35231
\(446\) 3846.96 0.408428
\(447\) 0 0
\(448\) −2244.38 −0.236690
\(449\) 11902.6 1.25104 0.625521 0.780207i \(-0.284887\pi\)
0.625521 + 0.780207i \(0.284887\pi\)
\(450\) 0 0
\(451\) −4369.84 −0.456247
\(452\) 15111.1 1.57249
\(453\) 0 0
\(454\) −25050.0 −2.58955
\(455\) 0 0
\(456\) 0 0
\(457\) −1420.66 −0.145418 −0.0727088 0.997353i \(-0.523164\pi\)
−0.0727088 + 0.997353i \(0.523164\pi\)
\(458\) 2731.64 0.278692
\(459\) 0 0
\(460\) 11312.2 1.14659
\(461\) 1768.23 0.178644 0.0893218 0.996003i \(-0.471530\pi\)
0.0893218 + 0.996003i \(0.471530\pi\)
\(462\) 0 0
\(463\) −18637.4 −1.87074 −0.935370 0.353670i \(-0.884933\pi\)
−0.935370 + 0.353670i \(0.884933\pi\)
\(464\) 19556.0 1.95660
\(465\) 0 0
\(466\) −18967.2 −1.88549
\(467\) −9042.01 −0.895962 −0.447981 0.894043i \(-0.647857\pi\)
−0.447981 + 0.894043i \(0.647857\pi\)
\(468\) 0 0
\(469\) −6405.25 −0.630633
\(470\) −23040.1 −2.26119
\(471\) 0 0
\(472\) −2053.47 −0.200251
\(473\) −4330.55 −0.420970
\(474\) 0 0
\(475\) −32804.6 −3.16880
\(476\) −1327.12 −0.127790
\(477\) 0 0
\(478\) 1250.73 0.119680
\(479\) 9547.64 0.910737 0.455368 0.890303i \(-0.349508\pi\)
0.455368 + 0.890303i \(0.349508\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −7482.08 −0.707053
\(483\) 0 0
\(484\) −4042.79 −0.379676
\(485\) 13825.3 1.29438
\(486\) 0 0
\(487\) −3692.81 −0.343608 −0.171804 0.985131i \(-0.554960\pi\)
−0.171804 + 0.985131i \(0.554960\pi\)
\(488\) −750.062 −0.0695773
\(489\) 0 0
\(490\) −22879.6 −2.10937
\(491\) 3487.77 0.320572 0.160286 0.987071i \(-0.448758\pi\)
0.160286 + 0.987071i \(0.448758\pi\)
\(492\) 0 0
\(493\) 8404.59 0.767796
\(494\) 0 0
\(495\) 0 0
\(496\) −11933.7 −1.08032
\(497\) −1181.55 −0.106640
\(498\) 0 0
\(499\) −15089.0 −1.35366 −0.676830 0.736139i \(-0.736647\pi\)
−0.676830 + 0.736139i \(0.736647\pi\)
\(500\) −17653.3 −1.57896
\(501\) 0 0
\(502\) 7551.56 0.671400
\(503\) −8815.48 −0.781437 −0.390718 0.920510i \(-0.627773\pi\)
−0.390718 + 0.920510i \(0.627773\pi\)
\(504\) 0 0
\(505\) 27535.2 2.42634
\(506\) −8858.41 −0.778270
\(507\) 0 0
\(508\) 12445.6 1.08698
\(509\) 19491.2 1.69731 0.848655 0.528946i \(-0.177413\pi\)
0.848655 + 0.528946i \(0.177413\pi\)
\(510\) 0 0
\(511\) −6013.61 −0.520599
\(512\) −14851.1 −1.28190
\(513\) 0 0
\(514\) 16697.3 1.43285
\(515\) −29684.8 −2.53994
\(516\) 0 0
\(517\) 8334.24 0.708974
\(518\) −4788.14 −0.406137
\(519\) 0 0
\(520\) 0 0
\(521\) 6128.73 0.515364 0.257682 0.966230i \(-0.417041\pi\)
0.257682 + 0.966230i \(0.417041\pi\)
\(522\) 0 0
\(523\) −22618.6 −1.89109 −0.945546 0.325489i \(-0.894471\pi\)
−0.945546 + 0.325489i \(0.894471\pi\)
\(524\) −6876.82 −0.573311
\(525\) 0 0
\(526\) 11311.6 0.937660
\(527\) −5128.77 −0.423933
\(528\) 0 0
\(529\) −5057.17 −0.415647
\(530\) −26873.3 −2.20245
\(531\) 0 0
\(532\) −5500.43 −0.448259
\(533\) 0 0
\(534\) 0 0
\(535\) 38239.6 3.09017
\(536\) −4463.71 −0.359707
\(537\) 0 0
\(538\) −9478.74 −0.759587
\(539\) 8276.19 0.661374
\(540\) 0 0
\(541\) 15950.5 1.26759 0.633796 0.773500i \(-0.281496\pi\)
0.633796 + 0.773500i \(0.281496\pi\)
\(542\) 8058.47 0.638636
\(543\) 0 0
\(544\) −7459.87 −0.587940
\(545\) 29645.7 2.33006
\(546\) 0 0
\(547\) −1972.83 −0.154208 −0.0771042 0.997023i \(-0.524567\pi\)
−0.0771042 + 0.997023i \(0.524567\pi\)
\(548\) 8544.33 0.666050
\(549\) 0 0
\(550\) 26956.2 2.08985
\(551\) 34834.1 2.69325
\(552\) 0 0
\(553\) −6482.35 −0.498477
\(554\) 9493.18 0.728027
\(555\) 0 0
\(556\) 9263.66 0.706595
\(557\) −19974.8 −1.51950 −0.759749 0.650217i \(-0.774678\pi\)
−0.759749 + 0.650217i \(0.774678\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 8783.10 0.662775
\(561\) 0 0
\(562\) −22271.2 −1.67163
\(563\) 13153.4 0.984637 0.492319 0.870415i \(-0.336149\pi\)
0.492319 + 0.870415i \(0.336149\pi\)
\(564\) 0 0
\(565\) −42980.5 −3.20036
\(566\) −19639.7 −1.45851
\(567\) 0 0
\(568\) −823.405 −0.0608263
\(569\) −4133.86 −0.304570 −0.152285 0.988337i \(-0.548663\pi\)
−0.152285 + 0.988337i \(0.548663\pi\)
\(570\) 0 0
\(571\) −3394.69 −0.248798 −0.124399 0.992232i \(-0.539700\pi\)
−0.124399 + 0.992232i \(0.539700\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −3874.02 −0.281704
\(575\) −21635.3 −1.56914
\(576\) 0 0
\(577\) −4210.56 −0.303792 −0.151896 0.988396i \(-0.548538\pi\)
−0.151896 + 0.988396i \(0.548538\pi\)
\(578\) 15274.9 1.09923
\(579\) 0 0
\(580\) 36552.6 2.61684
\(581\) −660.154 −0.0471391
\(582\) 0 0
\(583\) 9720.82 0.690558
\(584\) −4190.78 −0.296945
\(585\) 0 0
\(586\) −5798.21 −0.408740
\(587\) −7252.27 −0.509937 −0.254969 0.966949i \(-0.582065\pi\)
−0.254969 + 0.966949i \(0.582065\pi\)
\(588\) 0 0
\(589\) −21257.0 −1.48706
\(590\) −35430.1 −2.47226
\(591\) 0 0
\(592\) −14228.0 −0.987784
\(593\) −13899.4 −0.962531 −0.481266 0.876575i \(-0.659823\pi\)
−0.481266 + 0.876575i \(0.659823\pi\)
\(594\) 0 0
\(595\) 3774.72 0.260082
\(596\) −7153.75 −0.491659
\(597\) 0 0
\(598\) 0 0
\(599\) −18140.7 −1.23741 −0.618704 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(600\) 0 0
\(601\) −26808.3 −1.81952 −0.909761 0.415133i \(-0.863735\pi\)
−0.909761 + 0.415133i \(0.863735\pi\)
\(602\) −3839.19 −0.259923
\(603\) 0 0
\(604\) −9353.27 −0.630098
\(605\) 11498.9 0.772724
\(606\) 0 0
\(607\) 3769.98 0.252091 0.126045 0.992024i \(-0.459772\pi\)
0.126045 + 0.992024i \(0.459772\pi\)
\(608\) −30918.6 −2.06236
\(609\) 0 0
\(610\) −12941.4 −0.858989
\(611\) 0 0
\(612\) 0 0
\(613\) −2722.99 −0.179413 −0.0897067 0.995968i \(-0.528593\pi\)
−0.0897067 + 0.995968i \(0.528593\pi\)
\(614\) −25323.9 −1.66448
\(615\) 0 0
\(616\) −745.097 −0.0487351
\(617\) 11947.1 0.779533 0.389767 0.920914i \(-0.372556\pi\)
0.389767 + 0.920914i \(0.372556\pi\)
\(618\) 0 0
\(619\) 18386.8 1.19391 0.596953 0.802276i \(-0.296378\pi\)
0.596953 + 0.802276i \(0.296378\pi\)
\(620\) −22305.7 −1.44487
\(621\) 0 0
\(622\) −22035.1 −1.42046
\(623\) −4070.95 −0.261796
\(624\) 0 0
\(625\) 18138.1 1.16084
\(626\) 19100.5 1.21950
\(627\) 0 0
\(628\) 596.800 0.0379218
\(629\) −6114.79 −0.387620
\(630\) 0 0
\(631\) 25308.3 1.59669 0.798343 0.602202i \(-0.205710\pi\)
0.798343 + 0.602202i \(0.205710\pi\)
\(632\) −4517.44 −0.284326
\(633\) 0 0
\(634\) −9742.42 −0.610285
\(635\) −35399.1 −2.21224
\(636\) 0 0
\(637\) 0 0
\(638\) −28623.9 −1.77623
\(639\) 0 0
\(640\) 10806.1 0.667422
\(641\) 9670.76 0.595901 0.297950 0.954581i \(-0.403697\pi\)
0.297950 + 0.954581i \(0.403697\pi\)
\(642\) 0 0
\(643\) −19673.0 −1.20657 −0.603286 0.797525i \(-0.706142\pi\)
−0.603286 + 0.797525i \(0.706142\pi\)
\(644\) −3627.64 −0.221971
\(645\) 0 0
\(646\) −15206.9 −0.926170
\(647\) 13369.3 0.812367 0.406183 0.913792i \(-0.366859\pi\)
0.406183 + 0.913792i \(0.366859\pi\)
\(648\) 0 0
\(649\) 12816.1 0.775154
\(650\) 0 0
\(651\) 0 0
\(652\) 195.998 0.0117728
\(653\) 3088.52 0.185089 0.0925446 0.995709i \(-0.470500\pi\)
0.0925446 + 0.995709i \(0.470500\pi\)
\(654\) 0 0
\(655\) 19559.8 1.16682
\(656\) −11511.7 −0.685146
\(657\) 0 0
\(658\) 7388.61 0.437748
\(659\) −31326.8 −1.85177 −0.925886 0.377802i \(-0.876680\pi\)
−0.925886 + 0.377802i \(0.876680\pi\)
\(660\) 0 0
\(661\) −146.828 −0.00863986 −0.00431993 0.999991i \(-0.501375\pi\)
−0.00431993 + 0.999991i \(0.501375\pi\)
\(662\) 28672.6 1.68337
\(663\) 0 0
\(664\) −460.050 −0.0268877
\(665\) 15644.9 0.912307
\(666\) 0 0
\(667\) 22973.8 1.33365
\(668\) 8858.61 0.513098
\(669\) 0 0
\(670\) −77016.0 −4.44087
\(671\) 4681.28 0.269328
\(672\) 0 0
\(673\) −3463.75 −0.198392 −0.0991961 0.995068i \(-0.531627\pi\)
−0.0991961 + 0.995068i \(0.531627\pi\)
\(674\) −36507.4 −2.08637
\(675\) 0 0
\(676\) 0 0
\(677\) 534.786 0.0303597 0.0151798 0.999885i \(-0.495168\pi\)
0.0151798 + 0.999885i \(0.495168\pi\)
\(678\) 0 0
\(679\) −4433.56 −0.250581
\(680\) 2630.54 0.148348
\(681\) 0 0
\(682\) 17467.3 0.980729
\(683\) −22369.0 −1.25318 −0.626592 0.779348i \(-0.715551\pi\)
−0.626592 + 0.779348i \(0.715551\pi\)
\(684\) 0 0
\(685\) −24302.7 −1.35556
\(686\) 15622.2 0.869470
\(687\) 0 0
\(688\) −11408.2 −0.632171
\(689\) 0 0
\(690\) 0 0
\(691\) −25007.2 −1.37673 −0.688363 0.725366i \(-0.741671\pi\)
−0.688363 + 0.725366i \(0.741671\pi\)
\(692\) 4144.66 0.227683
\(693\) 0 0
\(694\) −10776.6 −0.589445
\(695\) −26348.7 −1.43808
\(696\) 0 0
\(697\) −4947.39 −0.268861
\(698\) 10036.5 0.544251
\(699\) 0 0
\(700\) 11038.9 0.596047
\(701\) 27977.1 1.50739 0.753696 0.657223i \(-0.228269\pi\)
0.753696 + 0.657223i \(0.228269\pi\)
\(702\) 0 0
\(703\) −25343.7 −1.35968
\(704\) 9761.67 0.522595
\(705\) 0 0
\(706\) 34223.4 1.82438
\(707\) −8830.14 −0.469720
\(708\) 0 0
\(709\) −6374.31 −0.337648 −0.168824 0.985646i \(-0.553997\pi\)
−0.168824 + 0.985646i \(0.553997\pi\)
\(710\) −14206.9 −0.750950
\(711\) 0 0
\(712\) −2836.97 −0.149326
\(713\) −14019.4 −0.736367
\(714\) 0 0
\(715\) 0 0
\(716\) −13634.6 −0.711662
\(717\) 0 0
\(718\) 34497.6 1.79309
\(719\) 25433.3 1.31919 0.659597 0.751619i \(-0.270727\pi\)
0.659597 + 0.751619i \(0.270727\pi\)
\(720\) 0 0
\(721\) 9519.48 0.491711
\(722\) −36579.7 −1.88553
\(723\) 0 0
\(724\) −27099.0 −1.39106
\(725\) −69909.4 −3.58120
\(726\) 0 0
\(727\) −15847.8 −0.808476 −0.404238 0.914654i \(-0.632463\pi\)
−0.404238 + 0.914654i \(0.632463\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −72306.9 −3.66603
\(731\) −4902.91 −0.248072
\(732\) 0 0
\(733\) −9378.34 −0.472574 −0.236287 0.971683i \(-0.575931\pi\)
−0.236287 + 0.971683i \(0.575931\pi\)
\(734\) 15289.7 0.768874
\(735\) 0 0
\(736\) −20391.4 −1.02125
\(737\) 27858.9 1.39239
\(738\) 0 0
\(739\) −12956.2 −0.644928 −0.322464 0.946582i \(-0.604511\pi\)
−0.322464 + 0.946582i \(0.604511\pi\)
\(740\) −26594.0 −1.32110
\(741\) 0 0
\(742\) 8617.85 0.426376
\(743\) 16776.6 0.828363 0.414181 0.910194i \(-0.364068\pi\)
0.414181 + 0.910194i \(0.364068\pi\)
\(744\) 0 0
\(745\) 20347.5 1.00064
\(746\) 5619.72 0.275808
\(747\) 0 0
\(748\) 5772.13 0.282152
\(749\) −12262.9 −0.598231
\(750\) 0 0
\(751\) 11213.4 0.544849 0.272425 0.962177i \(-0.412174\pi\)
0.272425 + 0.962177i \(0.412174\pi\)
\(752\) 21955.3 1.06467
\(753\) 0 0
\(754\) 0 0
\(755\) 26603.6 1.28239
\(756\) 0 0
\(757\) 852.253 0.0409190 0.0204595 0.999791i \(-0.493487\pi\)
0.0204595 + 0.999791i \(0.493487\pi\)
\(758\) 7679.95 0.368006
\(759\) 0 0
\(760\) 10902.7 0.520371
\(761\) 35706.1 1.70085 0.850425 0.526097i \(-0.176345\pi\)
0.850425 + 0.526097i \(0.176345\pi\)
\(762\) 0 0
\(763\) −9506.95 −0.451081
\(764\) 9934.21 0.470428
\(765\) 0 0
\(766\) −9656.88 −0.455506
\(767\) 0 0
\(768\) 0 0
\(769\) 3663.63 0.171800 0.0858999 0.996304i \(-0.472623\pi\)
0.0858999 + 0.996304i \(0.472623\pi\)
\(770\) −12855.8 −0.601675
\(771\) 0 0
\(772\) −5387.97 −0.251188
\(773\) 17985.6 0.836865 0.418432 0.908248i \(-0.362580\pi\)
0.418432 + 0.908248i \(0.362580\pi\)
\(774\) 0 0
\(775\) 42661.1 1.97733
\(776\) −3089.67 −0.142929
\(777\) 0 0
\(778\) −5583.78 −0.257311
\(779\) −20505.2 −0.943101
\(780\) 0 0
\(781\) 5139.03 0.235453
\(782\) −10029.2 −0.458624
\(783\) 0 0
\(784\) 21802.4 0.993185
\(785\) −1697.48 −0.0771793
\(786\) 0 0
\(787\) 19322.0 0.875167 0.437584 0.899178i \(-0.355834\pi\)
0.437584 + 0.899178i \(0.355834\pi\)
\(788\) −3395.02 −0.153480
\(789\) 0 0
\(790\) −77943.1 −3.51024
\(791\) 13783.2 0.619563
\(792\) 0 0
\(793\) 0 0
\(794\) −21724.6 −0.971003
\(795\) 0 0
\(796\) −20758.7 −0.924336
\(797\) 10689.5 0.475082 0.237541 0.971378i \(-0.423659\pi\)
0.237541 + 0.971378i \(0.423659\pi\)
\(798\) 0 0
\(799\) 9435.77 0.417789
\(800\) 62051.2 2.74230
\(801\) 0 0
\(802\) −14066.5 −0.619335
\(803\) 26155.5 1.14945
\(804\) 0 0
\(805\) 10318.1 0.451759
\(806\) 0 0
\(807\) 0 0
\(808\) −6153.58 −0.267923
\(809\) −16756.4 −0.728213 −0.364107 0.931357i \(-0.618626\pi\)
−0.364107 + 0.931357i \(0.618626\pi\)
\(810\) 0 0
\(811\) 33333.7 1.44329 0.721643 0.692266i \(-0.243387\pi\)
0.721643 + 0.692266i \(0.243387\pi\)
\(812\) −11721.9 −0.506598
\(813\) 0 0
\(814\) 20825.5 0.896722
\(815\) −557.479 −0.0239603
\(816\) 0 0
\(817\) −20320.9 −0.870180
\(818\) −17196.9 −0.735056
\(819\) 0 0
\(820\) −21516.8 −0.916342
\(821\) 21242.1 0.902988 0.451494 0.892274i \(-0.350891\pi\)
0.451494 + 0.892274i \(0.350891\pi\)
\(822\) 0 0
\(823\) 3994.06 0.169167 0.0845834 0.996416i \(-0.473044\pi\)
0.0845834 + 0.996416i \(0.473044\pi\)
\(824\) 6633.96 0.280467
\(825\) 0 0
\(826\) 11361.9 0.478610
\(827\) −16952.5 −0.712814 −0.356407 0.934331i \(-0.615998\pi\)
−0.356407 + 0.934331i \(0.615998\pi\)
\(828\) 0 0
\(829\) 38917.5 1.63047 0.815237 0.579128i \(-0.196607\pi\)
0.815237 + 0.579128i \(0.196607\pi\)
\(830\) −7937.62 −0.331950
\(831\) 0 0
\(832\) 0 0
\(833\) 9370.04 0.389739
\(834\) 0 0
\(835\) −25196.6 −1.04427
\(836\) 23923.5 0.989726
\(837\) 0 0
\(838\) −18225.8 −0.751311
\(839\) −25671.3 −1.05634 −0.528172 0.849138i \(-0.677122\pi\)
−0.528172 + 0.849138i \(0.677122\pi\)
\(840\) 0 0
\(841\) 49845.4 2.04377
\(842\) −11916.6 −0.487737
\(843\) 0 0
\(844\) 21875.7 0.892170
\(845\) 0 0
\(846\) 0 0
\(847\) −3687.54 −0.149593
\(848\) 25608.1 1.03701
\(849\) 0 0
\(850\) 30519.0 1.23152
\(851\) −16714.7 −0.673292
\(852\) 0 0
\(853\) 37444.6 1.50302 0.751511 0.659720i \(-0.229325\pi\)
0.751511 + 0.659720i \(0.229325\pi\)
\(854\) 4150.12 0.166293
\(855\) 0 0
\(856\) −8545.78 −0.341225
\(857\) −36610.6 −1.45927 −0.729636 0.683836i \(-0.760310\pi\)
−0.729636 + 0.683836i \(0.760310\pi\)
\(858\) 0 0
\(859\) 18043.5 0.716688 0.358344 0.933590i \(-0.383341\pi\)
0.358344 + 0.933590i \(0.383341\pi\)
\(860\) −21323.4 −0.845490
\(861\) 0 0
\(862\) −42923.1 −1.69602
\(863\) 8697.94 0.343084 0.171542 0.985177i \(-0.445125\pi\)
0.171542 + 0.985177i \(0.445125\pi\)
\(864\) 0 0
\(865\) −11788.7 −0.463384
\(866\) −26117.2 −1.02482
\(867\) 0 0
\(868\) 7153.09 0.279714
\(869\) 28194.2 1.10060
\(870\) 0 0
\(871\) 0 0
\(872\) −6625.23 −0.257292
\(873\) 0 0
\(874\) −41567.6 −1.60875
\(875\) −16102.0 −0.622113
\(876\) 0 0
\(877\) −26657.4 −1.02640 −0.513202 0.858268i \(-0.671541\pi\)
−0.513202 + 0.858268i \(0.671541\pi\)
\(878\) 20763.7 0.798111
\(879\) 0 0
\(880\) −38201.1 −1.46336
\(881\) 13896.5 0.531424 0.265712 0.964053i \(-0.414393\pi\)
0.265712 + 0.964053i \(0.414393\pi\)
\(882\) 0 0
\(883\) 24343.3 0.927767 0.463884 0.885896i \(-0.346456\pi\)
0.463884 + 0.885896i \(0.346456\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3554.38 −0.134776
\(887\) 4592.33 0.173839 0.0869196 0.996215i \(-0.472298\pi\)
0.0869196 + 0.996215i \(0.472298\pi\)
\(888\) 0 0
\(889\) 11352.0 0.428271
\(890\) −48948.6 −1.84355
\(891\) 0 0
\(892\) −6851.94 −0.257197
\(893\) 39108.0 1.46551
\(894\) 0 0
\(895\) 38781.1 1.44839
\(896\) −3465.37 −0.129207
\(897\) 0 0
\(898\) −45895.0 −1.70550
\(899\) −45300.4 −1.68059
\(900\) 0 0
\(901\) 11005.6 0.406937
\(902\) 16849.6 0.621984
\(903\) 0 0
\(904\) 9605.29 0.353393
\(905\) 77078.0 2.83111
\(906\) 0 0
\(907\) 31406.7 1.14977 0.574885 0.818234i \(-0.305047\pi\)
0.574885 + 0.818234i \(0.305047\pi\)
\(908\) 44617.4 1.63070
\(909\) 0 0
\(910\) 0 0
\(911\) −44825.0 −1.63021 −0.815103 0.579317i \(-0.803319\pi\)
−0.815103 + 0.579317i \(0.803319\pi\)
\(912\) 0 0
\(913\) 2871.26 0.104080
\(914\) 5477.91 0.198242
\(915\) 0 0
\(916\) −4865.40 −0.175499
\(917\) −6272.53 −0.225886
\(918\) 0 0
\(919\) 22241.1 0.798332 0.399166 0.916879i \(-0.369300\pi\)
0.399166 + 0.916879i \(0.369300\pi\)
\(920\) 7190.53 0.257679
\(921\) 0 0
\(922\) −6818.09 −0.243538
\(923\) 0 0
\(924\) 0 0
\(925\) 50862.9 1.80796
\(926\) 71863.6 2.55031
\(927\) 0 0
\(928\) −65890.1 −2.33076
\(929\) −16292.5 −0.575393 −0.287697 0.957722i \(-0.592889\pi\)
−0.287697 + 0.957722i \(0.592889\pi\)
\(930\) 0 0
\(931\) 38835.6 1.36712
\(932\) 33783.0 1.18734
\(933\) 0 0
\(934\) 34864.9 1.22143
\(935\) −16417.7 −0.574242
\(936\) 0 0
\(937\) 24400.8 0.850734 0.425367 0.905021i \(-0.360145\pi\)
0.425367 + 0.905021i \(0.360145\pi\)
\(938\) 24697.9 0.859717
\(939\) 0 0
\(940\) 41037.4 1.42393
\(941\) −41529.9 −1.43872 −0.719360 0.694638i \(-0.755565\pi\)
−0.719360 + 0.694638i \(0.755565\pi\)
\(942\) 0 0
\(943\) −13523.6 −0.467008
\(944\) 33762.1 1.16405
\(945\) 0 0
\(946\) 16698.1 0.573892
\(947\) 29981.8 1.02880 0.514402 0.857549i \(-0.328014\pi\)
0.514402 + 0.857549i \(0.328014\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 126491. 4.31990
\(951\) 0 0
\(952\) −843.575 −0.0287189
\(953\) −28594.4 −0.971943 −0.485972 0.873975i \(-0.661534\pi\)
−0.485972 + 0.873975i \(0.661534\pi\)
\(954\) 0 0
\(955\) −28256.0 −0.957426
\(956\) −2227.72 −0.0753657
\(957\) 0 0
\(958\) −36814.6 −1.24157
\(959\) 7793.52 0.262425
\(960\) 0 0
\(961\) −2147.17 −0.0720745
\(962\) 0 0
\(963\) 0 0
\(964\) 13326.6 0.445249
\(965\) 15325.1 0.511224
\(966\) 0 0
\(967\) 28729.1 0.955393 0.477696 0.878525i \(-0.341472\pi\)
0.477696 + 0.878525i \(0.341472\pi\)
\(968\) −2569.78 −0.0853264
\(969\) 0 0
\(970\) −53308.6 −1.76457
\(971\) −27550.7 −0.910549 −0.455274 0.890351i \(-0.650459\pi\)
−0.455274 + 0.890351i \(0.650459\pi\)
\(972\) 0 0
\(973\) 8449.64 0.278400
\(974\) 14239.0 0.468427
\(975\) 0 0
\(976\) 12332.1 0.404449
\(977\) −23812.0 −0.779746 −0.389873 0.920869i \(-0.627481\pi\)
−0.389873 + 0.920869i \(0.627481\pi\)
\(978\) 0 0
\(979\) 17706.1 0.578028
\(980\) 40751.5 1.32833
\(981\) 0 0
\(982\) −13448.4 −0.437023
\(983\) −32668.3 −1.05998 −0.529988 0.848005i \(-0.677804\pi\)
−0.529988 + 0.848005i \(0.677804\pi\)
\(984\) 0 0
\(985\) 9656.48 0.312367
\(986\) −32407.1 −1.04671
\(987\) 0 0
\(988\) 0 0
\(989\) −13402.0 −0.430899
\(990\) 0 0
\(991\) −19654.5 −0.630015 −0.315008 0.949089i \(-0.602007\pi\)
−0.315008 + 0.949089i \(0.602007\pi\)
\(992\) 40208.4 1.28691
\(993\) 0 0
\(994\) 4555.94 0.145378
\(995\) 59044.0 1.88123
\(996\) 0 0
\(997\) 17666.5 0.561188 0.280594 0.959827i \(-0.409469\pi\)
0.280594 + 0.959827i \(0.409469\pi\)
\(998\) 58181.4 1.84539
\(999\) 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 1521.4.a.ba.1.2 4
3.2 odd 2 inner 1521.4.a.ba.1.3 4
13.12 even 2 117.4.a.g.1.3 yes 4
39.38 odd 2 117.4.a.g.1.2 4
52.51 odd 2 1872.4.a.bo.1.4 4
156.155 even 2 1872.4.a.bo.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.a.g.1.2 4 39.38 odd 2
117.4.a.g.1.3 yes 4 13.12 even 2
1521.4.a.ba.1.2 4 1.1 even 1 trivial
1521.4.a.ba.1.3 4 3.2 odd 2 inner
1872.4.a.bo.1.1 4 156.155 even 2
1872.4.a.bo.1.4 4 52.51 odd 2