Properties

Label 1521.4.a.bk.1.10
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(5.36472\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.36472 q^{2} +20.7803 q^{4} -2.69631 q^{5} +15.2025 q^{7} +68.5626 q^{8} +O(q^{10})\) \(q+5.36472 q^{2} +20.7803 q^{4} -2.69631 q^{5} +15.2025 q^{7} +68.5626 q^{8} -14.4650 q^{10} +66.8848 q^{11} +81.5570 q^{14} +201.577 q^{16} -4.16354 q^{17} +26.0850 q^{19} -56.0301 q^{20} +358.819 q^{22} -47.3242 q^{23} -117.730 q^{25} +315.911 q^{28} -257.007 q^{29} +206.242 q^{31} +532.906 q^{32} -22.3362 q^{34} -40.9906 q^{35} +175.686 q^{37} +139.939 q^{38} -184.866 q^{40} -156.463 q^{41} +51.9845 q^{43} +1389.88 q^{44} -253.881 q^{46} +354.222 q^{47} -111.885 q^{49} -631.588 q^{50} +10.4723 q^{53} -180.342 q^{55} +1042.32 q^{56} -1378.77 q^{58} -445.114 q^{59} +119.696 q^{61} +1106.43 q^{62} +1246.28 q^{64} -22.4078 q^{67} -86.5195 q^{68} -219.903 q^{70} -285.207 q^{71} +740.989 q^{73} +942.507 q^{74} +542.053 q^{76} +1016.81 q^{77} -547.679 q^{79} -543.516 q^{80} -839.378 q^{82} -603.056 q^{83} +11.2262 q^{85} +278.882 q^{86} +4585.80 q^{88} -215.668 q^{89} -983.409 q^{92} +1900.31 q^{94} -70.3333 q^{95} +1447.50 q^{97} -600.233 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.36472 1.89672 0.948358 0.317201i \(-0.102743\pi\)
0.948358 + 0.317201i \(0.102743\pi\)
\(3\) 0 0
\(4\) 20.7803 2.59753
\(5\) −2.69631 −0.241165 −0.120583 0.992703i \(-0.538476\pi\)
−0.120583 + 0.992703i \(0.538476\pi\)
\(6\) 0 0
\(7\) 15.2025 0.820856 0.410428 0.911893i \(-0.365379\pi\)
0.410428 + 0.911893i \(0.365379\pi\)
\(8\) 68.5626 3.03007
\(9\) 0 0
\(10\) −14.4650 −0.457423
\(11\) 66.8848 1.83332 0.916661 0.399666i \(-0.130874\pi\)
0.916661 + 0.399666i \(0.130874\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 81.5570 1.55693
\(15\) 0 0
\(16\) 201.577 3.14965
\(17\) −4.16354 −0.0594004 −0.0297002 0.999559i \(-0.509455\pi\)
−0.0297002 + 0.999559i \(0.509455\pi\)
\(18\) 0 0
\(19\) 26.0850 0.314963 0.157482 0.987522i \(-0.449662\pi\)
0.157482 + 0.987522i \(0.449662\pi\)
\(20\) −56.0301 −0.626436
\(21\) 0 0
\(22\) 358.819 3.47729
\(23\) −47.3242 −0.429034 −0.214517 0.976720i \(-0.568818\pi\)
−0.214517 + 0.976720i \(0.568818\pi\)
\(24\) 0 0
\(25\) −117.730 −0.941839
\(26\) 0 0
\(27\) 0 0
\(28\) 315.911 2.13220
\(29\) −257.007 −1.64569 −0.822845 0.568266i \(-0.807614\pi\)
−0.822845 + 0.568266i \(0.807614\pi\)
\(30\) 0 0
\(31\) 206.242 1.19491 0.597455 0.801903i \(-0.296179\pi\)
0.597455 + 0.801903i \(0.296179\pi\)
\(32\) 532.906 2.94392
\(33\) 0 0
\(34\) −22.3362 −0.112666
\(35\) −40.9906 −0.197962
\(36\) 0 0
\(37\) 175.686 0.780611 0.390305 0.920685i \(-0.372369\pi\)
0.390305 + 0.920685i \(0.372369\pi\)
\(38\) 139.939 0.597396
\(39\) 0 0
\(40\) −184.866 −0.730748
\(41\) −156.463 −0.595984 −0.297992 0.954568i \(-0.596317\pi\)
−0.297992 + 0.954568i \(0.596317\pi\)
\(42\) 0 0
\(43\) 51.9845 0.184362 0.0921809 0.995742i \(-0.470616\pi\)
0.0921809 + 0.995742i \(0.470616\pi\)
\(44\) 1389.88 4.76211
\(45\) 0 0
\(46\) −253.881 −0.813755
\(47\) 354.222 1.09933 0.549666 0.835384i \(-0.314755\pi\)
0.549666 + 0.835384i \(0.314755\pi\)
\(48\) 0 0
\(49\) −111.885 −0.326196
\(50\) −631.588 −1.78640
\(51\) 0 0
\(52\) 0 0
\(53\) 10.4723 0.0271412 0.0135706 0.999908i \(-0.495680\pi\)
0.0135706 + 0.999908i \(0.495680\pi\)
\(54\) 0 0
\(55\) −180.342 −0.442134
\(56\) 1042.32 2.48725
\(57\) 0 0
\(58\) −1378.77 −3.12141
\(59\) −445.114 −0.982185 −0.491092 0.871107i \(-0.663402\pi\)
−0.491092 + 0.871107i \(0.663402\pi\)
\(60\) 0 0
\(61\) 119.696 0.251238 0.125619 0.992079i \(-0.459908\pi\)
0.125619 + 0.992079i \(0.459908\pi\)
\(62\) 1106.43 2.26640
\(63\) 0 0
\(64\) 1246.28 2.43413
\(65\) 0 0
\(66\) 0 0
\(67\) −22.4078 −0.0408589 −0.0204294 0.999791i \(-0.506503\pi\)
−0.0204294 + 0.999791i \(0.506503\pi\)
\(68\) −86.5195 −0.154295
\(69\) 0 0
\(70\) −219.903 −0.375478
\(71\) −285.207 −0.476731 −0.238365 0.971176i \(-0.576612\pi\)
−0.238365 + 0.971176i \(0.576612\pi\)
\(72\) 0 0
\(73\) 740.989 1.18803 0.594015 0.804454i \(-0.297542\pi\)
0.594015 + 0.804454i \(0.297542\pi\)
\(74\) 942.507 1.48060
\(75\) 0 0
\(76\) 542.053 0.818128
\(77\) 1016.81 1.50489
\(78\) 0 0
\(79\) −547.679 −0.779983 −0.389992 0.920818i \(-0.627522\pi\)
−0.389992 + 0.920818i \(0.627522\pi\)
\(80\) −543.516 −0.759586
\(81\) 0 0
\(82\) −839.378 −1.13041
\(83\) −603.056 −0.797518 −0.398759 0.917056i \(-0.630559\pi\)
−0.398759 + 0.917056i \(0.630559\pi\)
\(84\) 0 0
\(85\) 11.2262 0.0143253
\(86\) 278.882 0.349682
\(87\) 0 0
\(88\) 4585.80 5.55509
\(89\) −215.668 −0.256863 −0.128431 0.991718i \(-0.540994\pi\)
−0.128431 + 0.991718i \(0.540994\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −983.409 −1.11443
\(93\) 0 0
\(94\) 1900.31 2.08512
\(95\) −70.3333 −0.0759583
\(96\) 0 0
\(97\) 1447.50 1.51517 0.757586 0.652735i \(-0.226378\pi\)
0.757586 + 0.652735i \(0.226378\pi\)
\(98\) −600.233 −0.618700
\(99\) 0 0
\(100\) −2446.46 −2.44646
\(101\) 883.450 0.870362 0.435181 0.900343i \(-0.356684\pi\)
0.435181 + 0.900343i \(0.356684\pi\)
\(102\) 0 0
\(103\) 1251.74 1.19745 0.598726 0.800954i \(-0.295674\pi\)
0.598726 + 0.800954i \(0.295674\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 56.1812 0.0514792
\(107\) 341.614 0.308645 0.154323 0.988021i \(-0.450681\pi\)
0.154323 + 0.988021i \(0.450681\pi\)
\(108\) 0 0
\(109\) 775.177 0.681179 0.340589 0.940212i \(-0.389373\pi\)
0.340589 + 0.940212i \(0.389373\pi\)
\(110\) −967.487 −0.838603
\(111\) 0 0
\(112\) 3064.47 2.58541
\(113\) −1279.05 −1.06480 −0.532402 0.846492i \(-0.678710\pi\)
−0.532402 + 0.846492i \(0.678710\pi\)
\(114\) 0 0
\(115\) 127.601 0.103468
\(116\) −5340.67 −4.27473
\(117\) 0 0
\(118\) −2387.91 −1.86293
\(119\) −63.2961 −0.0487592
\(120\) 0 0
\(121\) 3142.58 2.36107
\(122\) 642.137 0.476528
\(123\) 0 0
\(124\) 4285.77 3.10382
\(125\) 654.476 0.468305
\(126\) 0 0
\(127\) −1113.82 −0.778233 −0.389117 0.921188i \(-0.627220\pi\)
−0.389117 + 0.921188i \(0.627220\pi\)
\(128\) 2422.68 1.67294
\(129\) 0 0
\(130\) 0 0
\(131\) −2100.12 −1.40068 −0.700339 0.713811i \(-0.746968\pi\)
−0.700339 + 0.713811i \(0.746968\pi\)
\(132\) 0 0
\(133\) 396.556 0.258540
\(134\) −120.211 −0.0774977
\(135\) 0 0
\(136\) −285.463 −0.179987
\(137\) 1205.02 0.751471 0.375736 0.926727i \(-0.377390\pi\)
0.375736 + 0.926727i \(0.377390\pi\)
\(138\) 0 0
\(139\) 322.890 0.197030 0.0985149 0.995136i \(-0.468591\pi\)
0.0985149 + 0.995136i \(0.468591\pi\)
\(140\) −851.796 −0.514213
\(141\) 0 0
\(142\) −1530.06 −0.904223
\(143\) 0 0
\(144\) 0 0
\(145\) 692.971 0.396884
\(146\) 3975.20 2.25336
\(147\) 0 0
\(148\) 3650.80 2.02766
\(149\) −1128.86 −0.620669 −0.310335 0.950627i \(-0.600441\pi\)
−0.310335 + 0.950627i \(0.600441\pi\)
\(150\) 0 0
\(151\) −2940.44 −1.58470 −0.792350 0.610066i \(-0.791143\pi\)
−0.792350 + 0.610066i \(0.791143\pi\)
\(152\) 1788.46 0.954361
\(153\) 0 0
\(154\) 5454.93 2.85436
\(155\) −556.093 −0.288171
\(156\) 0 0
\(157\) 629.388 0.319940 0.159970 0.987122i \(-0.448860\pi\)
0.159970 + 0.987122i \(0.448860\pi\)
\(158\) −2938.15 −1.47941
\(159\) 0 0
\(160\) −1436.88 −0.709972
\(161\) −719.444 −0.352175
\(162\) 0 0
\(163\) −394.912 −0.189766 −0.0948832 0.995488i \(-0.530248\pi\)
−0.0948832 + 0.995488i \(0.530248\pi\)
\(164\) −3251.33 −1.54809
\(165\) 0 0
\(166\) −3235.23 −1.51267
\(167\) 151.860 0.0703669 0.0351834 0.999381i \(-0.488798\pi\)
0.0351834 + 0.999381i \(0.488798\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 60.2255 0.0271711
\(171\) 0 0
\(172\) 1080.25 0.478886
\(173\) −538.813 −0.236793 −0.118397 0.992966i \(-0.537775\pi\)
−0.118397 + 0.992966i \(0.537775\pi\)
\(174\) 0 0
\(175\) −1789.78 −0.773114
\(176\) 13482.5 5.77432
\(177\) 0 0
\(178\) −1157.00 −0.487196
\(179\) 2220.80 0.927319 0.463659 0.886014i \(-0.346536\pi\)
0.463659 + 0.886014i \(0.346536\pi\)
\(180\) 0 0
\(181\) −3822.78 −1.56986 −0.784932 0.619582i \(-0.787302\pi\)
−0.784932 + 0.619582i \(0.787302\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3244.67 −1.30000
\(185\) −473.704 −0.188256
\(186\) 0 0
\(187\) −278.478 −0.108900
\(188\) 7360.84 2.85555
\(189\) 0 0
\(190\) −377.319 −0.144071
\(191\) −3464.19 −1.31236 −0.656178 0.754606i \(-0.727828\pi\)
−0.656178 + 0.754606i \(0.727828\pi\)
\(192\) 0 0
\(193\) 4697.40 1.75195 0.875975 0.482357i \(-0.160219\pi\)
0.875975 + 0.482357i \(0.160219\pi\)
\(194\) 7765.46 2.87385
\(195\) 0 0
\(196\) −2325.00 −0.847304
\(197\) 2887.89 1.04443 0.522217 0.852813i \(-0.325105\pi\)
0.522217 + 0.852813i \(0.325105\pi\)
\(198\) 0 0
\(199\) 63.0092 0.0224453 0.0112226 0.999937i \(-0.496428\pi\)
0.0112226 + 0.999937i \(0.496428\pi\)
\(200\) −8071.87 −2.85384
\(201\) 0 0
\(202\) 4739.47 1.65083
\(203\) −3907.14 −1.35087
\(204\) 0 0
\(205\) 421.872 0.143731
\(206\) 6715.24 2.27123
\(207\) 0 0
\(208\) 0 0
\(209\) 1744.69 0.577429
\(210\) 0 0
\(211\) −1049.70 −0.342484 −0.171242 0.985229i \(-0.554778\pi\)
−0.171242 + 0.985229i \(0.554778\pi\)
\(212\) 217.618 0.0705002
\(213\) 0 0
\(214\) 1832.66 0.585412
\(215\) −140.166 −0.0444617
\(216\) 0 0
\(217\) 3135.39 0.980848
\(218\) 4158.61 1.29200
\(219\) 0 0
\(220\) −3747.56 −1.14846
\(221\) 0 0
\(222\) 0 0
\(223\) 2313.49 0.694722 0.347361 0.937732i \(-0.387078\pi\)
0.347361 + 0.937732i \(0.387078\pi\)
\(224\) 8101.49 2.41653
\(225\) 0 0
\(226\) −6861.74 −2.01963
\(227\) −3799.46 −1.11092 −0.555460 0.831543i \(-0.687458\pi\)
−0.555460 + 0.831543i \(0.687458\pi\)
\(228\) 0 0
\(229\) −4321.07 −1.24692 −0.623459 0.781856i \(-0.714273\pi\)
−0.623459 + 0.781856i \(0.714273\pi\)
\(230\) 684.543 0.196250
\(231\) 0 0
\(232\) −17621.1 −4.98655
\(233\) −5279.77 −1.48450 −0.742251 0.670122i \(-0.766242\pi\)
−0.742251 + 0.670122i \(0.766242\pi\)
\(234\) 0 0
\(235\) −955.094 −0.265121
\(236\) −9249.59 −2.55126
\(237\) 0 0
\(238\) −339.566 −0.0924823
\(239\) −1547.92 −0.418939 −0.209469 0.977815i \(-0.567174\pi\)
−0.209469 + 0.977815i \(0.567174\pi\)
\(240\) 0 0
\(241\) −4918.01 −1.31451 −0.657255 0.753669i \(-0.728282\pi\)
−0.657255 + 0.753669i \(0.728282\pi\)
\(242\) 16859.1 4.47828
\(243\) 0 0
\(244\) 2487.32 0.652600
\(245\) 301.677 0.0786671
\(246\) 0 0
\(247\) 0 0
\(248\) 14140.5 3.62066
\(249\) 0 0
\(250\) 3511.08 0.888241
\(251\) −1155.78 −0.290646 −0.145323 0.989384i \(-0.546422\pi\)
−0.145323 + 0.989384i \(0.546422\pi\)
\(252\) 0 0
\(253\) −3165.27 −0.786556
\(254\) −5975.34 −1.47609
\(255\) 0 0
\(256\) 3026.80 0.738965
\(257\) −2351.95 −0.570859 −0.285429 0.958400i \(-0.592136\pi\)
−0.285429 + 0.958400i \(0.592136\pi\)
\(258\) 0 0
\(259\) 2670.86 0.640769
\(260\) 0 0
\(261\) 0 0
\(262\) −11266.6 −2.65669
\(263\) −5521.88 −1.29465 −0.647326 0.762213i \(-0.724113\pi\)
−0.647326 + 0.762213i \(0.724113\pi\)
\(264\) 0 0
\(265\) −28.2367 −0.00654553
\(266\) 2127.41 0.490376
\(267\) 0 0
\(268\) −465.639 −0.106132
\(269\) −3916.95 −0.887810 −0.443905 0.896074i \(-0.646407\pi\)
−0.443905 + 0.896074i \(0.646407\pi\)
\(270\) 0 0
\(271\) −2777.53 −0.622593 −0.311297 0.950313i \(-0.600763\pi\)
−0.311297 + 0.950313i \(0.600763\pi\)
\(272\) −839.276 −0.187090
\(273\) 0 0
\(274\) 6464.58 1.42533
\(275\) −7874.34 −1.72669
\(276\) 0 0
\(277\) 6583.08 1.42794 0.713969 0.700177i \(-0.246896\pi\)
0.713969 + 0.700177i \(0.246896\pi\)
\(278\) 1732.21 0.373710
\(279\) 0 0
\(280\) −2810.42 −0.599839
\(281\) −2871.66 −0.609640 −0.304820 0.952410i \(-0.598596\pi\)
−0.304820 + 0.952410i \(0.598596\pi\)
\(282\) 0 0
\(283\) 7518.04 1.57916 0.789578 0.613651i \(-0.210300\pi\)
0.789578 + 0.613651i \(0.210300\pi\)
\(284\) −5926.69 −1.23832
\(285\) 0 0
\(286\) 0 0
\(287\) −2378.62 −0.489217
\(288\) 0 0
\(289\) −4895.66 −0.996472
\(290\) 3717.60 0.752776
\(291\) 0 0
\(292\) 15397.9 3.08595
\(293\) −4506.57 −0.898555 −0.449278 0.893392i \(-0.648319\pi\)
−0.449278 + 0.893392i \(0.648319\pi\)
\(294\) 0 0
\(295\) 1200.17 0.236869
\(296\) 12045.5 2.36530
\(297\) 0 0
\(298\) −6056.02 −1.17723
\(299\) 0 0
\(300\) 0 0
\(301\) 790.292 0.151335
\(302\) −15774.7 −3.00573
\(303\) 0 0
\(304\) 5258.15 0.992024
\(305\) −322.738 −0.0605900
\(306\) 0 0
\(307\) 9538.89 1.77333 0.886667 0.462409i \(-0.153015\pi\)
0.886667 + 0.462409i \(0.153015\pi\)
\(308\) 21129.7 3.90901
\(309\) 0 0
\(310\) −2983.29 −0.546578
\(311\) 7466.28 1.36133 0.680666 0.732594i \(-0.261691\pi\)
0.680666 + 0.732594i \(0.261691\pi\)
\(312\) 0 0
\(313\) −1821.65 −0.328964 −0.164482 0.986380i \(-0.552595\pi\)
−0.164482 + 0.986380i \(0.552595\pi\)
\(314\) 3376.49 0.606836
\(315\) 0 0
\(316\) −11380.9 −2.02603
\(317\) −3125.14 −0.553708 −0.276854 0.960912i \(-0.589292\pi\)
−0.276854 + 0.960912i \(0.589292\pi\)
\(318\) 0 0
\(319\) −17189.9 −3.01708
\(320\) −3360.35 −0.587029
\(321\) 0 0
\(322\) −3859.62 −0.667976
\(323\) −108.606 −0.0187090
\(324\) 0 0
\(325\) 0 0
\(326\) −2118.60 −0.359933
\(327\) 0 0
\(328\) −10727.5 −1.80587
\(329\) 5385.05 0.902394
\(330\) 0 0
\(331\) −1553.67 −0.257999 −0.128999 0.991645i \(-0.541177\pi\)
−0.128999 + 0.991645i \(0.541177\pi\)
\(332\) −12531.7 −2.07158
\(333\) 0 0
\(334\) 814.686 0.133466
\(335\) 60.4183 0.00985375
\(336\) 0 0
\(337\) −3190.43 −0.515709 −0.257855 0.966184i \(-0.583016\pi\)
−0.257855 + 0.966184i \(0.583016\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 233.283 0.0372105
\(341\) 13794.5 2.19065
\(342\) 0 0
\(343\) −6915.37 −1.08862
\(344\) 3564.19 0.558629
\(345\) 0 0
\(346\) −2890.59 −0.449130
\(347\) 5718.68 0.884712 0.442356 0.896840i \(-0.354143\pi\)
0.442356 + 0.896840i \(0.354143\pi\)
\(348\) 0 0
\(349\) −3328.46 −0.510511 −0.255256 0.966874i \(-0.582160\pi\)
−0.255256 + 0.966874i \(0.582160\pi\)
\(350\) −9601.70 −1.46638
\(351\) 0 0
\(352\) 35643.4 5.39715
\(353\) 12306.5 1.85555 0.927774 0.373142i \(-0.121719\pi\)
0.927774 + 0.373142i \(0.121719\pi\)
\(354\) 0 0
\(355\) 769.008 0.114971
\(356\) −4481.64 −0.667210
\(357\) 0 0
\(358\) 11914.0 1.75886
\(359\) −8539.97 −1.25549 −0.627747 0.778418i \(-0.716023\pi\)
−0.627747 + 0.778418i \(0.716023\pi\)
\(360\) 0 0
\(361\) −6178.57 −0.900798
\(362\) −20508.2 −2.97759
\(363\) 0 0
\(364\) 0 0
\(365\) −1997.94 −0.286512
\(366\) 0 0
\(367\) 2496.65 0.355107 0.177553 0.984111i \(-0.443182\pi\)
0.177553 + 0.984111i \(0.443182\pi\)
\(368\) −9539.49 −1.35130
\(369\) 0 0
\(370\) −2541.29 −0.357069
\(371\) 159.205 0.0222790
\(372\) 0 0
\(373\) −1142.91 −0.158653 −0.0793264 0.996849i \(-0.525277\pi\)
−0.0793264 + 0.996849i \(0.525277\pi\)
\(374\) −1493.96 −0.206552
\(375\) 0 0
\(376\) 24286.4 3.33105
\(377\) 0 0
\(378\) 0 0
\(379\) −12181.8 −1.65102 −0.825512 0.564384i \(-0.809114\pi\)
−0.825512 + 0.564384i \(0.809114\pi\)
\(380\) −1461.54 −0.197304
\(381\) 0 0
\(382\) −18584.4 −2.48917
\(383\) 10180.6 1.35824 0.679118 0.734029i \(-0.262362\pi\)
0.679118 + 0.734029i \(0.262362\pi\)
\(384\) 0 0
\(385\) −2741.65 −0.362928
\(386\) 25200.3 3.32295
\(387\) 0 0
\(388\) 30079.5 3.93571
\(389\) −5845.83 −0.761941 −0.380971 0.924587i \(-0.624410\pi\)
−0.380971 + 0.924587i \(0.624410\pi\)
\(390\) 0 0
\(391\) 197.036 0.0254848
\(392\) −7671.13 −0.988395
\(393\) 0 0
\(394\) 15492.7 1.98100
\(395\) 1476.71 0.188105
\(396\) 0 0
\(397\) −2500.92 −0.316166 −0.158083 0.987426i \(-0.550531\pi\)
−0.158083 + 0.987426i \(0.550531\pi\)
\(398\) 338.027 0.0425723
\(399\) 0 0
\(400\) −23731.7 −2.96646
\(401\) 9189.25 1.14436 0.572181 0.820127i \(-0.306098\pi\)
0.572181 + 0.820127i \(0.306098\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 18358.3 2.26080
\(405\) 0 0
\(406\) −20960.7 −2.56223
\(407\) 11750.7 1.43111
\(408\) 0 0
\(409\) 9214.38 1.11399 0.556995 0.830516i \(-0.311954\pi\)
0.556995 + 0.830516i \(0.311954\pi\)
\(410\) 2263.23 0.272617
\(411\) 0 0
\(412\) 26011.5 3.11042
\(413\) −6766.83 −0.806232
\(414\) 0 0
\(415\) 1626.03 0.192334
\(416\) 0 0
\(417\) 0 0
\(418\) 9359.78 1.09522
\(419\) 6494.41 0.757214 0.378607 0.925558i \(-0.376403\pi\)
0.378607 + 0.925558i \(0.376403\pi\)
\(420\) 0 0
\(421\) 3059.56 0.354190 0.177095 0.984194i \(-0.443330\pi\)
0.177095 + 0.984194i \(0.443330\pi\)
\(422\) −5631.33 −0.649595
\(423\) 0 0
\(424\) 718.010 0.0822398
\(425\) 490.173 0.0559456
\(426\) 0 0
\(427\) 1819.68 0.206230
\(428\) 7098.82 0.801716
\(429\) 0 0
\(430\) −751.954 −0.0843313
\(431\) −7937.05 −0.887040 −0.443520 0.896264i \(-0.646271\pi\)
−0.443520 + 0.896264i \(0.646271\pi\)
\(432\) 0 0
\(433\) 7294.37 0.809573 0.404786 0.914411i \(-0.367346\pi\)
0.404786 + 0.914411i \(0.367346\pi\)
\(434\) 16820.5 1.86039
\(435\) 0 0
\(436\) 16108.4 1.76938
\(437\) −1234.45 −0.135130
\(438\) 0 0
\(439\) 15214.7 1.65412 0.827059 0.562115i \(-0.190012\pi\)
0.827059 + 0.562115i \(0.190012\pi\)
\(440\) −12364.7 −1.33970
\(441\) 0 0
\(442\) 0 0
\(443\) −1517.05 −0.162703 −0.0813515 0.996685i \(-0.525924\pi\)
−0.0813515 + 0.996685i \(0.525924\pi\)
\(444\) 0 0
\(445\) 581.509 0.0619464
\(446\) 12411.3 1.31769
\(447\) 0 0
\(448\) 18946.5 1.99807
\(449\) 705.247 0.0741262 0.0370631 0.999313i \(-0.488200\pi\)
0.0370631 + 0.999313i \(0.488200\pi\)
\(450\) 0 0
\(451\) −10465.0 −1.09263
\(452\) −26579.0 −2.76586
\(453\) 0 0
\(454\) −20383.1 −2.10710
\(455\) 0 0
\(456\) 0 0
\(457\) −7277.73 −0.744940 −0.372470 0.928044i \(-0.621489\pi\)
−0.372470 + 0.928044i \(0.621489\pi\)
\(458\) −23181.4 −2.36505
\(459\) 0 0
\(460\) 2651.58 0.268762
\(461\) −1961.88 −0.198208 −0.0991041 0.995077i \(-0.531598\pi\)
−0.0991041 + 0.995077i \(0.531598\pi\)
\(462\) 0 0
\(463\) −10374.1 −1.04131 −0.520653 0.853768i \(-0.674311\pi\)
−0.520653 + 0.853768i \(0.674311\pi\)
\(464\) −51806.8 −5.18334
\(465\) 0 0
\(466\) −28324.5 −2.81568
\(467\) −8788.92 −0.870883 −0.435442 0.900217i \(-0.643408\pi\)
−0.435442 + 0.900217i \(0.643408\pi\)
\(468\) 0 0
\(469\) −340.653 −0.0335392
\(470\) −5123.82 −0.502860
\(471\) 0 0
\(472\) −30518.2 −2.97609
\(473\) 3476.97 0.337995
\(474\) 0 0
\(475\) −3070.98 −0.296645
\(476\) −1315.31 −0.126654
\(477\) 0 0
\(478\) −8304.14 −0.794608
\(479\) 11141.7 1.06279 0.531394 0.847125i \(-0.321668\pi\)
0.531394 + 0.847125i \(0.321668\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −26383.8 −2.49325
\(483\) 0 0
\(484\) 65303.7 6.13295
\(485\) −3902.92 −0.365407
\(486\) 0 0
\(487\) −19640.5 −1.82750 −0.913752 0.406273i \(-0.866828\pi\)
−0.913752 + 0.406273i \(0.866828\pi\)
\(488\) 8206.69 0.761269
\(489\) 0 0
\(490\) 1618.41 0.149209
\(491\) 3410.31 0.313453 0.156726 0.987642i \(-0.449906\pi\)
0.156726 + 0.987642i \(0.449906\pi\)
\(492\) 0 0
\(493\) 1070.06 0.0977546
\(494\) 0 0
\(495\) 0 0
\(496\) 41573.8 3.76354
\(497\) −4335.86 −0.391327
\(498\) 0 0
\(499\) −5032.44 −0.451469 −0.225735 0.974189i \(-0.572478\pi\)
−0.225735 + 0.974189i \(0.572478\pi\)
\(500\) 13600.2 1.21644
\(501\) 0 0
\(502\) −6200.44 −0.551274
\(503\) −17189.4 −1.52373 −0.761866 0.647735i \(-0.775716\pi\)
−0.761866 + 0.647735i \(0.775716\pi\)
\(504\) 0 0
\(505\) −2382.06 −0.209901
\(506\) −16980.8 −1.49187
\(507\) 0 0
\(508\) −23145.5 −2.02149
\(509\) 930.560 0.0810341 0.0405170 0.999179i \(-0.487099\pi\)
0.0405170 + 0.999179i \(0.487099\pi\)
\(510\) 0 0
\(511\) 11264.9 0.975201
\(512\) −3143.50 −0.271337
\(513\) 0 0
\(514\) −12617.6 −1.08276
\(515\) −3375.08 −0.288784
\(516\) 0 0
\(517\) 23692.1 2.01543
\(518\) 14328.4 1.21536
\(519\) 0 0
\(520\) 0 0
\(521\) 9869.60 0.829933 0.414966 0.909837i \(-0.363793\pi\)
0.414966 + 0.909837i \(0.363793\pi\)
\(522\) 0 0
\(523\) −21420.6 −1.79093 −0.895466 0.445129i \(-0.853158\pi\)
−0.895466 + 0.445129i \(0.853158\pi\)
\(524\) −43641.1 −3.63831
\(525\) 0 0
\(526\) −29623.4 −2.45559
\(527\) −858.697 −0.0709781
\(528\) 0 0
\(529\) −9927.42 −0.815930
\(530\) −151.482 −0.0124150
\(531\) 0 0
\(532\) 8240.54 0.671565
\(533\) 0 0
\(534\) 0 0
\(535\) −921.097 −0.0744345
\(536\) −1536.34 −0.123805
\(537\) 0 0
\(538\) −21013.4 −1.68392
\(539\) −7483.41 −0.598021
\(540\) 0 0
\(541\) 7771.50 0.617602 0.308801 0.951127i \(-0.400072\pi\)
0.308801 + 0.951127i \(0.400072\pi\)
\(542\) −14900.7 −1.18088
\(543\) 0 0
\(544\) −2218.78 −0.174870
\(545\) −2090.12 −0.164277
\(546\) 0 0
\(547\) −15577.5 −1.21763 −0.608817 0.793310i \(-0.708356\pi\)
−0.608817 + 0.793310i \(0.708356\pi\)
\(548\) 25040.6 1.95197
\(549\) 0 0
\(550\) −42243.7 −3.27505
\(551\) −6704.02 −0.518332
\(552\) 0 0
\(553\) −8326.07 −0.640254
\(554\) 35316.4 2.70840
\(555\) 0 0
\(556\) 6709.74 0.511792
\(557\) 22804.5 1.73475 0.867377 0.497652i \(-0.165804\pi\)
0.867377 + 0.497652i \(0.165804\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −8262.78 −0.623511
\(561\) 0 0
\(562\) −15405.7 −1.15631
\(563\) 3517.41 0.263306 0.131653 0.991296i \(-0.457972\pi\)
0.131653 + 0.991296i \(0.457972\pi\)
\(564\) 0 0
\(565\) 3448.71 0.256794
\(566\) 40332.2 2.99521
\(567\) 0 0
\(568\) −19554.6 −1.44453
\(569\) 6093.44 0.448946 0.224473 0.974480i \(-0.427934\pi\)
0.224473 + 0.974480i \(0.427934\pi\)
\(570\) 0 0
\(571\) 10460.2 0.766630 0.383315 0.923618i \(-0.374782\pi\)
0.383315 + 0.923618i \(0.374782\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −12760.6 −0.927906
\(575\) 5571.47 0.404081
\(576\) 0 0
\(577\) −9648.19 −0.696117 −0.348058 0.937473i \(-0.613159\pi\)
−0.348058 + 0.937473i \(0.613159\pi\)
\(578\) −26263.9 −1.89002
\(579\) 0 0
\(580\) 14400.1 1.03092
\(581\) −9167.93 −0.654647
\(582\) 0 0
\(583\) 700.440 0.0497586
\(584\) 50804.1 3.59981
\(585\) 0 0
\(586\) −24176.5 −1.70430
\(587\) −2170.73 −0.152633 −0.0763166 0.997084i \(-0.524316\pi\)
−0.0763166 + 0.997084i \(0.524316\pi\)
\(588\) 0 0
\(589\) 5379.82 0.376353
\(590\) 6438.56 0.449274
\(591\) 0 0
\(592\) 35414.3 2.45865
\(593\) −22885.9 −1.58484 −0.792421 0.609975i \(-0.791180\pi\)
−0.792421 + 0.609975i \(0.791180\pi\)
\(594\) 0 0
\(595\) 170.666 0.0117590
\(596\) −23458.0 −1.61221
\(597\) 0 0
\(598\) 0 0
\(599\) 23978.7 1.63563 0.817815 0.575482i \(-0.195185\pi\)
0.817815 + 0.575482i \(0.195185\pi\)
\(600\) 0 0
\(601\) 12873.2 0.873728 0.436864 0.899528i \(-0.356089\pi\)
0.436864 + 0.899528i \(0.356089\pi\)
\(602\) 4239.70 0.287039
\(603\) 0 0
\(604\) −61103.2 −4.11631
\(605\) −8473.38 −0.569408
\(606\) 0 0
\(607\) −7117.15 −0.475908 −0.237954 0.971276i \(-0.576477\pi\)
−0.237954 + 0.971276i \(0.576477\pi\)
\(608\) 13900.9 0.927227
\(609\) 0 0
\(610\) −1731.40 −0.114922
\(611\) 0 0
\(612\) 0 0
\(613\) 173.297 0.0114183 0.00570913 0.999984i \(-0.498183\pi\)
0.00570913 + 0.999984i \(0.498183\pi\)
\(614\) 51173.5 3.36351
\(615\) 0 0
\(616\) 69715.5 4.55993
\(617\) 6102.75 0.398197 0.199099 0.979979i \(-0.436199\pi\)
0.199099 + 0.979979i \(0.436199\pi\)
\(618\) 0 0
\(619\) −14867.8 −0.965409 −0.482705 0.875783i \(-0.660346\pi\)
−0.482705 + 0.875783i \(0.660346\pi\)
\(620\) −11555.8 −0.748534
\(621\) 0 0
\(622\) 40054.6 2.58206
\(623\) −3278.69 −0.210847
\(624\) 0 0
\(625\) 12951.6 0.828900
\(626\) −9772.64 −0.623951
\(627\) 0 0
\(628\) 13078.9 0.831056
\(629\) −731.475 −0.0463686
\(630\) 0 0
\(631\) −17210.6 −1.08580 −0.542902 0.839796i \(-0.682675\pi\)
−0.542902 + 0.839796i \(0.682675\pi\)
\(632\) −37550.3 −2.36340
\(633\) 0 0
\(634\) −16765.5 −1.05023
\(635\) 3003.21 0.187683
\(636\) 0 0
\(637\) 0 0
\(638\) −92218.9 −5.72254
\(639\) 0 0
\(640\) −6532.30 −0.403456
\(641\) 12636.0 0.778616 0.389308 0.921108i \(-0.372714\pi\)
0.389308 + 0.921108i \(0.372714\pi\)
\(642\) 0 0
\(643\) −9586.37 −0.587947 −0.293973 0.955814i \(-0.594978\pi\)
−0.293973 + 0.955814i \(0.594978\pi\)
\(644\) −14950.2 −0.914786
\(645\) 0 0
\(646\) −582.641 −0.0354856
\(647\) −5244.32 −0.318664 −0.159332 0.987225i \(-0.550934\pi\)
−0.159332 + 0.987225i \(0.550934\pi\)
\(648\) 0 0
\(649\) −29771.4 −1.80066
\(650\) 0 0
\(651\) 0 0
\(652\) −8206.39 −0.492925
\(653\) −18869.0 −1.13078 −0.565392 0.824822i \(-0.691275\pi\)
−0.565392 + 0.824822i \(0.691275\pi\)
\(654\) 0 0
\(655\) 5662.59 0.337795
\(656\) −31539.3 −1.87714
\(657\) 0 0
\(658\) 28889.3 1.71159
\(659\) 24299.8 1.43639 0.718197 0.695839i \(-0.244967\pi\)
0.718197 + 0.695839i \(0.244967\pi\)
\(660\) 0 0
\(661\) 29915.0 1.76030 0.880151 0.474694i \(-0.157441\pi\)
0.880151 + 0.474694i \(0.157441\pi\)
\(662\) −8335.03 −0.489350
\(663\) 0 0
\(664\) −41347.1 −2.41653
\(665\) −1069.24 −0.0623508
\(666\) 0 0
\(667\) 12162.6 0.706056
\(668\) 3155.69 0.182780
\(669\) 0 0
\(670\) 324.128 0.0186898
\(671\) 8005.86 0.460600
\(672\) 0 0
\(673\) 15493.7 0.887426 0.443713 0.896169i \(-0.353661\pi\)
0.443713 + 0.896169i \(0.353661\pi\)
\(674\) −17115.8 −0.978154
\(675\) 0 0
\(676\) 0 0
\(677\) −11729.7 −0.665891 −0.332945 0.942946i \(-0.608042\pi\)
−0.332945 + 0.942946i \(0.608042\pi\)
\(678\) 0 0
\(679\) 22005.6 1.24374
\(680\) 769.698 0.0434067
\(681\) 0 0
\(682\) 74003.5 4.15505
\(683\) 1168.42 0.0654587 0.0327294 0.999464i \(-0.489580\pi\)
0.0327294 + 0.999464i \(0.489580\pi\)
\(684\) 0 0
\(685\) −3249.10 −0.181229
\(686\) −37099.1 −2.06480
\(687\) 0 0
\(688\) 10478.9 0.580675
\(689\) 0 0
\(690\) 0 0
\(691\) 32992.9 1.81637 0.908183 0.418574i \(-0.137470\pi\)
0.908183 + 0.418574i \(0.137470\pi\)
\(692\) −11196.7 −0.615078
\(693\) 0 0
\(694\) 30679.2 1.67805
\(695\) −870.612 −0.0475168
\(696\) 0 0
\(697\) 651.438 0.0354017
\(698\) −17856.3 −0.968295
\(699\) 0 0
\(700\) −37192.2 −2.00819
\(701\) 14785.8 0.796651 0.398326 0.917244i \(-0.369591\pi\)
0.398326 + 0.917244i \(0.369591\pi\)
\(702\) 0 0
\(703\) 4582.77 0.245864
\(704\) 83357.0 4.46255
\(705\) 0 0
\(706\) 66021.0 3.51945
\(707\) 13430.6 0.714442
\(708\) 0 0
\(709\) 12634.0 0.669225 0.334612 0.942356i \(-0.391395\pi\)
0.334612 + 0.942356i \(0.391395\pi\)
\(710\) 4125.52 0.218067
\(711\) 0 0
\(712\) −14786.8 −0.778312
\(713\) −9760.24 −0.512656
\(714\) 0 0
\(715\) 0 0
\(716\) 46148.7 2.40874
\(717\) 0 0
\(718\) −45814.6 −2.38132
\(719\) 27296.5 1.41584 0.707920 0.706292i \(-0.249634\pi\)
0.707920 + 0.706292i \(0.249634\pi\)
\(720\) 0 0
\(721\) 19029.5 0.982936
\(722\) −33146.3 −1.70856
\(723\) 0 0
\(724\) −79438.5 −4.07777
\(725\) 30257.4 1.54997
\(726\) 0 0
\(727\) 4658.21 0.237639 0.118819 0.992916i \(-0.462089\pi\)
0.118819 + 0.992916i \(0.462089\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −10718.4 −0.543432
\(731\) −216.439 −0.0109512
\(732\) 0 0
\(733\) −166.474 −0.00838864 −0.00419432 0.999991i \(-0.501335\pi\)
−0.00419432 + 0.999991i \(0.501335\pi\)
\(734\) 13393.9 0.673537
\(735\) 0 0
\(736\) −25219.4 −1.26304
\(737\) −1498.74 −0.0749074
\(738\) 0 0
\(739\) 12738.1 0.634069 0.317035 0.948414i \(-0.397313\pi\)
0.317035 + 0.948414i \(0.397313\pi\)
\(740\) −9843.70 −0.489002
\(741\) 0 0
\(742\) 854.092 0.0422570
\(743\) 30724.5 1.51705 0.758527 0.651641i \(-0.225919\pi\)
0.758527 + 0.651641i \(0.225919\pi\)
\(744\) 0 0
\(745\) 3043.76 0.149684
\(746\) −6131.38 −0.300919
\(747\) 0 0
\(748\) −5786.84 −0.282871
\(749\) 5193.37 0.253353
\(750\) 0 0
\(751\) −39538.6 −1.92115 −0.960575 0.278021i \(-0.910322\pi\)
−0.960575 + 0.278021i \(0.910322\pi\)
\(752\) 71403.2 3.46251
\(753\) 0 0
\(754\) 0 0
\(755\) 7928.35 0.382175
\(756\) 0 0
\(757\) 23035.1 1.10598 0.552990 0.833188i \(-0.313487\pi\)
0.552990 + 0.833188i \(0.313487\pi\)
\(758\) −65352.1 −3.13153
\(759\) 0 0
\(760\) −4822.23 −0.230159
\(761\) −32454.0 −1.54594 −0.772968 0.634445i \(-0.781229\pi\)
−0.772968 + 0.634445i \(0.781229\pi\)
\(762\) 0 0
\(763\) 11784.6 0.559150
\(764\) −71986.8 −3.40889
\(765\) 0 0
\(766\) 54616.1 2.57619
\(767\) 0 0
\(768\) 0 0
\(769\) −32216.2 −1.51072 −0.755362 0.655307i \(-0.772539\pi\)
−0.755362 + 0.655307i \(0.772539\pi\)
\(770\) −14708.2 −0.688372
\(771\) 0 0
\(772\) 97613.2 4.55075
\(773\) −2924.60 −0.136081 −0.0680404 0.997683i \(-0.521675\pi\)
−0.0680404 + 0.997683i \(0.521675\pi\)
\(774\) 0 0
\(775\) −24280.9 −1.12541
\(776\) 99244.7 4.59108
\(777\) 0 0
\(778\) −31361.2 −1.44519
\(779\) −4081.32 −0.187713
\(780\) 0 0
\(781\) −19076.1 −0.874001
\(782\) 1057.04 0.0483374
\(783\) 0 0
\(784\) −22553.5 −1.02740
\(785\) −1697.03 −0.0771586
\(786\) 0 0
\(787\) 25507.1 1.15531 0.577656 0.816280i \(-0.303967\pi\)
0.577656 + 0.816280i \(0.303967\pi\)
\(788\) 60011.1 2.71295
\(789\) 0 0
\(790\) 7922.16 0.356782
\(791\) −19444.7 −0.874050
\(792\) 0 0
\(793\) 0 0
\(794\) −13416.8 −0.599677
\(795\) 0 0
\(796\) 1309.35 0.0583023
\(797\) −5448.98 −0.242174 −0.121087 0.992642i \(-0.538638\pi\)
−0.121087 + 0.992642i \(0.538638\pi\)
\(798\) 0 0
\(799\) −1474.82 −0.0653008
\(800\) −62739.0 −2.77270
\(801\) 0 0
\(802\) 49297.8 2.17053
\(803\) 49560.9 2.17804
\(804\) 0 0
\(805\) 1939.85 0.0849324
\(806\) 0 0
\(807\) 0 0
\(808\) 60571.7 2.63726
\(809\) 2453.12 0.106610 0.0533048 0.998578i \(-0.483025\pi\)
0.0533048 + 0.998578i \(0.483025\pi\)
\(810\) 0 0
\(811\) −5133.85 −0.222286 −0.111143 0.993804i \(-0.535451\pi\)
−0.111143 + 0.993804i \(0.535451\pi\)
\(812\) −81191.4 −3.50894
\(813\) 0 0
\(814\) 63039.4 2.71441
\(815\) 1064.81 0.0457651
\(816\) 0 0
\(817\) 1356.01 0.0580673
\(818\) 49432.6 2.11292
\(819\) 0 0
\(820\) 8766.61 0.373345
\(821\) 23854.0 1.01402 0.507009 0.861941i \(-0.330751\pi\)
0.507009 + 0.861941i \(0.330751\pi\)
\(822\) 0 0
\(823\) 757.156 0.0320690 0.0160345 0.999871i \(-0.494896\pi\)
0.0160345 + 0.999871i \(0.494896\pi\)
\(824\) 85822.6 3.62836
\(825\) 0 0
\(826\) −36302.2 −1.52919
\(827\) −28621.2 −1.20345 −0.601726 0.798702i \(-0.705520\pi\)
−0.601726 + 0.798702i \(0.705520\pi\)
\(828\) 0 0
\(829\) 27429.2 1.14916 0.574582 0.818447i \(-0.305165\pi\)
0.574582 + 0.818447i \(0.305165\pi\)
\(830\) 8723.19 0.364803
\(831\) 0 0
\(832\) 0 0
\(833\) 465.838 0.0193761
\(834\) 0 0
\(835\) −409.462 −0.0169701
\(836\) 36255.1 1.49989
\(837\) 0 0
\(838\) 34840.7 1.43622
\(839\) 4633.62 0.190668 0.0953339 0.995445i \(-0.469608\pi\)
0.0953339 + 0.995445i \(0.469608\pi\)
\(840\) 0 0
\(841\) 41663.6 1.70829
\(842\) 16413.7 0.671798
\(843\) 0 0
\(844\) −21813.0 −0.889614
\(845\) 0 0
\(846\) 0 0
\(847\) 47775.0 1.93810
\(848\) 2110.99 0.0854853
\(849\) 0 0
\(850\) 2629.64 0.106113
\(851\) −8314.19 −0.334908
\(852\) 0 0
\(853\) −14854.6 −0.596261 −0.298131 0.954525i \(-0.596363\pi\)
−0.298131 + 0.954525i \(0.596363\pi\)
\(854\) 9762.07 0.391161
\(855\) 0 0
\(856\) 23421.9 0.935216
\(857\) 42799.5 1.70595 0.852977 0.521948i \(-0.174794\pi\)
0.852977 + 0.521948i \(0.174794\pi\)
\(858\) 0 0
\(859\) −8246.47 −0.327551 −0.163775 0.986498i \(-0.552367\pi\)
−0.163775 + 0.986498i \(0.552367\pi\)
\(860\) −2912.70 −0.115491
\(861\) 0 0
\(862\) −42580.1 −1.68246
\(863\) −17695.4 −0.697983 −0.348991 0.937126i \(-0.613476\pi\)
−0.348991 + 0.937126i \(0.613476\pi\)
\(864\) 0 0
\(865\) 1452.81 0.0571064
\(866\) 39132.3 1.53553
\(867\) 0 0
\(868\) 65154.2 2.54779
\(869\) −36631.4 −1.42996
\(870\) 0 0
\(871\) 0 0
\(872\) 53148.2 2.06402
\(873\) 0 0
\(874\) −6622.49 −0.256303
\(875\) 9949.64 0.384411
\(876\) 0 0
\(877\) −14346.8 −0.552401 −0.276200 0.961100i \(-0.589075\pi\)
−0.276200 + 0.961100i \(0.589075\pi\)
\(878\) 81622.7 3.13739
\(879\) 0 0
\(880\) −36353.0 −1.39257
\(881\) 2063.51 0.0789121 0.0394560 0.999221i \(-0.487437\pi\)
0.0394560 + 0.999221i \(0.487437\pi\)
\(882\) 0 0
\(883\) −34137.1 −1.30103 −0.650513 0.759495i \(-0.725446\pi\)
−0.650513 + 0.759495i \(0.725446\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −8138.58 −0.308602
\(887\) 22238.5 0.841821 0.420910 0.907102i \(-0.361711\pi\)
0.420910 + 0.907102i \(0.361711\pi\)
\(888\) 0 0
\(889\) −16932.8 −0.638817
\(890\) 3119.63 0.117495
\(891\) 0 0
\(892\) 48075.0 1.80456
\(893\) 9239.89 0.346250
\(894\) 0 0
\(895\) −5987.96 −0.223637
\(896\) 36830.7 1.37325
\(897\) 0 0
\(898\) 3783.45 0.140596
\(899\) −53005.7 −1.96645
\(900\) 0 0
\(901\) −43.6019 −0.00161220
\(902\) −56141.7 −2.07241
\(903\) 0 0
\(904\) −87694.9 −3.22643
\(905\) 10307.4 0.378597
\(906\) 0 0
\(907\) −44160.3 −1.61667 −0.808335 0.588723i \(-0.799631\pi\)
−0.808335 + 0.588723i \(0.799631\pi\)
\(908\) −78953.8 −2.88565
\(909\) 0 0
\(910\) 0 0
\(911\) −11916.3 −0.433376 −0.216688 0.976241i \(-0.569525\pi\)
−0.216688 + 0.976241i \(0.569525\pi\)
\(912\) 0 0
\(913\) −40335.3 −1.46211
\(914\) −39043.0 −1.41294
\(915\) 0 0
\(916\) −89793.0 −3.23891
\(917\) −31927.1 −1.14975
\(918\) 0 0
\(919\) −20128.9 −0.722516 −0.361258 0.932466i \(-0.617653\pi\)
−0.361258 + 0.932466i \(0.617653\pi\)
\(920\) 8748.64 0.313515
\(921\) 0 0
\(922\) −10525.0 −0.375945
\(923\) 0 0
\(924\) 0 0
\(925\) −20683.5 −0.735210
\(926\) −55654.1 −1.97506
\(927\) 0 0
\(928\) −136961. −4.84478
\(929\) 31428.4 1.10994 0.554969 0.831871i \(-0.312730\pi\)
0.554969 + 0.831871i \(0.312730\pi\)
\(930\) 0 0
\(931\) −2918.52 −0.102740
\(932\) −109715. −3.85605
\(933\) 0 0
\(934\) −47150.1 −1.65182
\(935\) 750.863 0.0262629
\(936\) 0 0
\(937\) 42473.2 1.48083 0.740416 0.672149i \(-0.234629\pi\)
0.740416 + 0.672149i \(0.234629\pi\)
\(938\) −1827.51 −0.0636144
\(939\) 0 0
\(940\) −19847.1 −0.688661
\(941\) 42644.0 1.47732 0.738659 0.674079i \(-0.235459\pi\)
0.738659 + 0.674079i \(0.235459\pi\)
\(942\) 0 0
\(943\) 7404.46 0.255697
\(944\) −89725.0 −3.09354
\(945\) 0 0
\(946\) 18653.0 0.641080
\(947\) −4282.95 −0.146966 −0.0734831 0.997296i \(-0.523412\pi\)
−0.0734831 + 0.997296i \(0.523412\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −16475.0 −0.562651
\(951\) 0 0
\(952\) −4339.74 −0.147744
\(953\) −40593.9 −1.37982 −0.689908 0.723897i \(-0.742349\pi\)
−0.689908 + 0.723897i \(0.742349\pi\)
\(954\) 0 0
\(955\) 9340.54 0.316495
\(956\) −32166.1 −1.08821
\(957\) 0 0
\(958\) 59771.9 2.01581
\(959\) 18319.2 0.616850
\(960\) 0 0
\(961\) 12744.8 0.427808
\(962\) 0 0
\(963\) 0 0
\(964\) −102198. −3.41448
\(965\) −12665.7 −0.422510
\(966\) 0 0
\(967\) 17709.4 0.588931 0.294466 0.955662i \(-0.404858\pi\)
0.294466 + 0.955662i \(0.404858\pi\)
\(968\) 215464. 7.15420
\(969\) 0 0
\(970\) −20938.1 −0.693074
\(971\) −4038.97 −0.133488 −0.0667440 0.997770i \(-0.521261\pi\)
−0.0667440 + 0.997770i \(0.521261\pi\)
\(972\) 0 0
\(973\) 4908.72 0.161733
\(974\) −105366. −3.46626
\(975\) 0 0
\(976\) 24128.1 0.791312
\(977\) 2764.30 0.0905197 0.0452598 0.998975i \(-0.485588\pi\)
0.0452598 + 0.998975i \(0.485588\pi\)
\(978\) 0 0
\(979\) −14424.9 −0.470912
\(980\) 6268.93 0.204340
\(981\) 0 0
\(982\) 18295.4 0.594531
\(983\) 17804.5 0.577695 0.288848 0.957375i \(-0.406728\pi\)
0.288848 + 0.957375i \(0.406728\pi\)
\(984\) 0 0
\(985\) −7786.65 −0.251881
\(986\) 5740.57 0.185413
\(987\) 0 0
\(988\) 0 0
\(989\) −2460.12 −0.0790974
\(990\) 0 0
\(991\) −8129.69 −0.260593 −0.130297 0.991475i \(-0.541593\pi\)
−0.130297 + 0.991475i \(0.541593\pi\)
\(992\) 109908. 3.51772
\(993\) 0 0
\(994\) −23260.7 −0.742237
\(995\) −169.893 −0.00541302
\(996\) 0 0
\(997\) −29452.9 −0.935589 −0.467795 0.883837i \(-0.654951\pi\)
−0.467795 + 0.883837i \(0.654951\pi\)
\(998\) −26997.7 −0.856309
\(999\) 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.bk.1.10 10
3.2 odd 2 507.4.a.r.1.1 10
13.2 odd 12 117.4.q.e.82.5 10
13.7 odd 12 117.4.q.e.10.5 10
13.12 even 2 inner 1521.4.a.bk.1.1 10
39.2 even 12 39.4.j.c.4.1 10
39.5 even 4 507.4.b.i.337.10 10
39.8 even 4 507.4.b.i.337.1 10
39.20 even 12 39.4.j.c.10.1 yes 10
39.38 odd 2 507.4.a.r.1.10 10
156.59 odd 12 624.4.bv.h.49.3 10
156.119 odd 12 624.4.bv.h.433.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.1 10 39.2 even 12
39.4.j.c.10.1 yes 10 39.20 even 12
117.4.q.e.10.5 10 13.7 odd 12
117.4.q.e.82.5 10 13.2 odd 12
507.4.a.r.1.1 10 3.2 odd 2
507.4.a.r.1.10 10 39.38 odd 2
507.4.b.i.337.1 10 39.8 even 4
507.4.b.i.337.10 10 39.5 even 4
624.4.bv.h.49.3 10 156.59 odd 12
624.4.bv.h.433.3 10 156.119 odd 12
1521.4.a.bk.1.1 10 13.12 even 2 inner
1521.4.a.bk.1.10 10 1.1 even 1 trivial