Properties

Label 441.4.a.s.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Defining polynomial: \(x^{3} - x^{2} - 24 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.30829\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.30829 q^{2} +20.1780 q^{4} -5.56140 q^{5} -64.6443 q^{8} +O(q^{10})\) \(q-5.30829 q^{2} +20.1780 q^{4} -5.56140 q^{5} -64.6443 q^{8} +29.5215 q^{10} +13.9174 q^{11} +38.6718 q^{13} +181.727 q^{16} -43.4788 q^{17} -109.028 q^{19} -112.218 q^{20} -73.8775 q^{22} +74.8778 q^{23} -94.0708 q^{25} -205.281 q^{26} +72.3589 q^{29} +64.0431 q^{31} -447.507 q^{32} +230.798 q^{34} +188.727 q^{37} +578.751 q^{38} +359.513 q^{40} +24.7923 q^{41} -243.881 q^{43} +280.825 q^{44} -397.474 q^{46} +620.549 q^{47} +499.356 q^{50} +780.319 q^{52} +287.839 q^{53} -77.4001 q^{55} -384.102 q^{58} -525.051 q^{59} -383.436 q^{61} -339.960 q^{62} +921.681 q^{64} -215.069 q^{65} +198.117 q^{67} -877.314 q^{68} -785.432 q^{71} -331.141 q^{73} -1001.82 q^{74} -2199.96 q^{76} +437.647 q^{79} -1010.66 q^{80} -131.605 q^{82} -241.241 q^{83} +241.803 q^{85} +1294.59 q^{86} -899.680 q^{88} -1585.54 q^{89} +1510.88 q^{92} -3294.05 q^{94} +606.347 q^{95} +79.2754 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} + O(q^{10}) \) \( 3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} - 55q^{10} - 35q^{11} + 62q^{13} + 241q^{16} - 48q^{17} - 202q^{19} - 439q^{20} - 7q^{22} - 216q^{23} + 130q^{25} - 274q^{26} - 53q^{29} - 95q^{31} - 683q^{32} - 24q^{34} + 262q^{37} + 398q^{38} + 21q^{40} - 244q^{41} + 360q^{43} + 905q^{44} - 1056q^{46} + 210q^{47} + 1378q^{50} + 324q^{52} - 393q^{53} - 1031q^{55} - 1249q^{58} - 1143q^{59} - 70q^{61} - 1059q^{62} - 399q^{64} + 472q^{65} - 628q^{67} - 1944q^{68} - 318q^{71} + 988q^{73} - 1002q^{74} - 2340q^{76} + 861q^{79} - 175q^{80} + 124q^{82} - 519q^{83} + 1800q^{85} + 3208q^{86} - 891q^{88} - 1766q^{89} + 672q^{92} - 3294q^{94} + 736q^{95} + 19q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.30829 −1.87677 −0.938383 0.345598i \(-0.887676\pi\)
−0.938383 + 0.345598i \(0.887676\pi\)
\(3\) 0 0
\(4\) 20.1780 2.52225
\(5\) −5.56140 −0.497427 −0.248713 0.968577i \(-0.580008\pi\)
−0.248713 + 0.968577i \(0.580008\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −64.6443 −2.85690
\(9\) 0 0
\(10\) 29.5215 0.933553
\(11\) 13.9174 0.381477 0.190738 0.981641i \(-0.438912\pi\)
0.190738 + 0.981641i \(0.438912\pi\)
\(12\) 0 0
\(13\) 38.6718 0.825048 0.412524 0.910947i \(-0.364647\pi\)
0.412524 + 0.910947i \(0.364647\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 181.727 2.83949
\(17\) −43.4788 −0.620303 −0.310152 0.950687i \(-0.600380\pi\)
−0.310152 + 0.950687i \(0.600380\pi\)
\(18\) 0 0
\(19\) −109.028 −1.31646 −0.658228 0.752818i \(-0.728694\pi\)
−0.658228 + 0.752818i \(0.728694\pi\)
\(20\) −112.218 −1.25463
\(21\) 0 0
\(22\) −73.8775 −0.715943
\(23\) 74.8778 0.678831 0.339415 0.940637i \(-0.389771\pi\)
0.339415 + 0.940637i \(0.389771\pi\)
\(24\) 0 0
\(25\) −94.0708 −0.752567
\(26\) −205.281 −1.54842
\(27\) 0 0
\(28\) 0 0
\(29\) 72.3589 0.463335 0.231667 0.972795i \(-0.425582\pi\)
0.231667 + 0.972795i \(0.425582\pi\)
\(30\) 0 0
\(31\) 64.0431 0.371048 0.185524 0.982640i \(-0.440602\pi\)
0.185524 + 0.982640i \(0.440602\pi\)
\(32\) −447.507 −2.47215
\(33\) 0 0
\(34\) 230.798 1.16416
\(35\) 0 0
\(36\) 0 0
\(37\) 188.727 0.838556 0.419278 0.907858i \(-0.362283\pi\)
0.419278 + 0.907858i \(0.362283\pi\)
\(38\) 578.751 2.47068
\(39\) 0 0
\(40\) 359.513 1.42110
\(41\) 24.7923 0.0944367 0.0472184 0.998885i \(-0.484964\pi\)
0.0472184 + 0.998885i \(0.484964\pi\)
\(42\) 0 0
\(43\) −243.881 −0.864920 −0.432460 0.901653i \(-0.642354\pi\)
−0.432460 + 0.901653i \(0.642354\pi\)
\(44\) 280.825 0.962180
\(45\) 0 0
\(46\) −397.474 −1.27401
\(47\) 620.549 1.92588 0.962940 0.269717i \(-0.0869303\pi\)
0.962940 + 0.269717i \(0.0869303\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 499.356 1.41239
\(51\) 0 0
\(52\) 780.319 2.08098
\(53\) 287.839 0.745995 0.372997 0.927832i \(-0.378330\pi\)
0.372997 + 0.927832i \(0.378330\pi\)
\(54\) 0 0
\(55\) −77.4001 −0.189757
\(56\) 0 0
\(57\) 0 0
\(58\) −384.102 −0.869571
\(59\) −525.051 −1.15857 −0.579287 0.815124i \(-0.696669\pi\)
−0.579287 + 0.815124i \(0.696669\pi\)
\(60\) 0 0
\(61\) −383.436 −0.804818 −0.402409 0.915460i \(-0.631827\pi\)
−0.402409 + 0.915460i \(0.631827\pi\)
\(62\) −339.960 −0.696369
\(63\) 0 0
\(64\) 921.681 1.80016
\(65\) −215.069 −0.410401
\(66\) 0 0
\(67\) 198.117 0.361251 0.180625 0.983552i \(-0.442188\pi\)
0.180625 + 0.983552i \(0.442188\pi\)
\(68\) −877.314 −1.56456
\(69\) 0 0
\(70\) 0 0
\(71\) −785.432 −1.31287 −0.656434 0.754384i \(-0.727936\pi\)
−0.656434 + 0.754384i \(0.727936\pi\)
\(72\) 0 0
\(73\) −331.141 −0.530919 −0.265459 0.964122i \(-0.585524\pi\)
−0.265459 + 0.964122i \(0.585524\pi\)
\(74\) −1001.82 −1.57377
\(75\) 0 0
\(76\) −2199.96 −3.32043
\(77\) 0 0
\(78\) 0 0
\(79\) 437.647 0.623280 0.311640 0.950200i \(-0.399122\pi\)
0.311640 + 0.950200i \(0.399122\pi\)
\(80\) −1010.66 −1.41244
\(81\) 0 0
\(82\) −131.605 −0.177236
\(83\) −241.241 −0.319032 −0.159516 0.987195i \(-0.550993\pi\)
−0.159516 + 0.987195i \(0.550993\pi\)
\(84\) 0 0
\(85\) 241.803 0.308555
\(86\) 1294.59 1.62325
\(87\) 0 0
\(88\) −899.680 −1.08984
\(89\) −1585.54 −1.88840 −0.944198 0.329378i \(-0.893161\pi\)
−0.944198 + 0.329378i \(0.893161\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1510.88 1.71218
\(93\) 0 0
\(94\) −3294.05 −3.61442
\(95\) 606.347 0.654841
\(96\) 0 0
\(97\) 79.2754 0.0829814 0.0414907 0.999139i \(-0.486789\pi\)
0.0414907 + 0.999139i \(0.486789\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1898.16 −1.89816
\(101\) −1154.97 −1.13786 −0.568931 0.822385i \(-0.692643\pi\)
−0.568931 + 0.822385i \(0.692643\pi\)
\(102\) 0 0
\(103\) −1444.86 −1.38220 −0.691098 0.722761i \(-0.742873\pi\)
−0.691098 + 0.722761i \(0.742873\pi\)
\(104\) −2499.91 −2.35708
\(105\) 0 0
\(106\) −1527.93 −1.40006
\(107\) −990.960 −0.895325 −0.447662 0.894203i \(-0.647743\pi\)
−0.447662 + 0.894203i \(0.647743\pi\)
\(108\) 0 0
\(109\) 1953.17 1.71633 0.858164 0.513376i \(-0.171605\pi\)
0.858164 + 0.513376i \(0.171605\pi\)
\(110\) 410.862 0.356129
\(111\) 0 0
\(112\) 0 0
\(113\) −672.882 −0.560172 −0.280086 0.959975i \(-0.590363\pi\)
−0.280086 + 0.959975i \(0.590363\pi\)
\(114\) 0 0
\(115\) −416.426 −0.337669
\(116\) 1460.06 1.16865
\(117\) 0 0
\(118\) 2787.13 2.17437
\(119\) 0 0
\(120\) 0 0
\(121\) −1137.31 −0.854475
\(122\) 2035.39 1.51045
\(123\) 0 0
\(124\) 1292.26 0.935874
\(125\) 1218.34 0.871773
\(126\) 0 0
\(127\) 175.815 0.122843 0.0614216 0.998112i \(-0.480437\pi\)
0.0614216 + 0.998112i \(0.480437\pi\)
\(128\) −1312.50 −0.906325
\(129\) 0 0
\(130\) 1141.65 0.770226
\(131\) −1125.93 −0.750939 −0.375470 0.926835i \(-0.622519\pi\)
−0.375470 + 0.926835i \(0.622519\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1051.66 −0.677983
\(135\) 0 0
\(136\) 2810.66 1.77215
\(137\) −1868.70 −1.16536 −0.582678 0.812703i \(-0.697995\pi\)
−0.582678 + 0.812703i \(0.697995\pi\)
\(138\) 0 0
\(139\) −2817.19 −1.71907 −0.859537 0.511074i \(-0.829248\pi\)
−0.859537 + 0.511074i \(0.829248\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4169.30 2.46394
\(143\) 538.210 0.314737
\(144\) 0 0
\(145\) −402.417 −0.230475
\(146\) 1757.79 0.996410
\(147\) 0 0
\(148\) 3808.14 2.11505
\(149\) −1800.33 −0.989855 −0.494928 0.868934i \(-0.664805\pi\)
−0.494928 + 0.868934i \(0.664805\pi\)
\(150\) 0 0
\(151\) −452.984 −0.244128 −0.122064 0.992522i \(-0.538951\pi\)
−0.122064 + 0.992522i \(0.538951\pi\)
\(152\) 7048.03 3.76099
\(153\) 0 0
\(154\) 0 0
\(155\) −356.169 −0.184569
\(156\) 0 0
\(157\) −1863.66 −0.947364 −0.473682 0.880696i \(-0.657075\pi\)
−0.473682 + 0.880696i \(0.657075\pi\)
\(158\) −2323.16 −1.16975
\(159\) 0 0
\(160\) 2488.77 1.22971
\(161\) 0 0
\(162\) 0 0
\(163\) 2321.14 1.11537 0.557686 0.830052i \(-0.311689\pi\)
0.557686 + 0.830052i \(0.311689\pi\)
\(164\) 500.259 0.238193
\(165\) 0 0
\(166\) 1280.58 0.598749
\(167\) −3211.62 −1.48816 −0.744079 0.668092i \(-0.767111\pi\)
−0.744079 + 0.668092i \(0.767111\pi\)
\(168\) 0 0
\(169\) −701.494 −0.319296
\(170\) −1283.56 −0.579086
\(171\) 0 0
\(172\) −4921.04 −2.18154
\(173\) −214.277 −0.0941687 −0.0470844 0.998891i \(-0.514993\pi\)
−0.0470844 + 0.998891i \(0.514993\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2529.17 1.08320
\(177\) 0 0
\(178\) 8416.53 3.54408
\(179\) 2437.22 1.01769 0.508845 0.860858i \(-0.330073\pi\)
0.508845 + 0.860858i \(0.330073\pi\)
\(180\) 0 0
\(181\) −248.631 −0.102103 −0.0510514 0.998696i \(-0.516257\pi\)
−0.0510514 + 0.998696i \(0.516257\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4840.43 −1.93935
\(185\) −1049.59 −0.417120
\(186\) 0 0
\(187\) −605.110 −0.236631
\(188\) 12521.4 4.85755
\(189\) 0 0
\(190\) −3218.67 −1.22898
\(191\) 4313.08 1.63394 0.816972 0.576677i \(-0.195651\pi\)
0.816972 + 0.576677i \(0.195651\pi\)
\(192\) 0 0
\(193\) 2060.85 0.768618 0.384309 0.923205i \(-0.374440\pi\)
0.384309 + 0.923205i \(0.374440\pi\)
\(194\) −420.817 −0.155737
\(195\) 0 0
\(196\) 0 0
\(197\) 1666.09 0.602557 0.301279 0.953536i \(-0.402587\pi\)
0.301279 + 0.953536i \(0.402587\pi\)
\(198\) 0 0
\(199\) −1087.53 −0.387403 −0.193702 0.981061i \(-0.562049\pi\)
−0.193702 + 0.981061i \(0.562049\pi\)
\(200\) 6081.15 2.15001
\(201\) 0 0
\(202\) 6130.94 2.13550
\(203\) 0 0
\(204\) 0 0
\(205\) −137.880 −0.0469754
\(206\) 7669.74 2.59406
\(207\) 0 0
\(208\) 7027.72 2.34271
\(209\) −1517.38 −0.502198
\(210\) 0 0
\(211\) −4676.47 −1.52579 −0.762895 0.646522i \(-0.776223\pi\)
−0.762895 + 0.646522i \(0.776223\pi\)
\(212\) 5808.01 1.88158
\(213\) 0 0
\(214\) 5260.31 1.68031
\(215\) 1356.32 0.430234
\(216\) 0 0
\(217\) 0 0
\(218\) −10368.0 −3.22114
\(219\) 0 0
\(220\) −1561.78 −0.478614
\(221\) −1681.40 −0.511780
\(222\) 0 0
\(223\) 3246.03 0.974754 0.487377 0.873192i \(-0.337954\pi\)
0.487377 + 0.873192i \(0.337954\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3571.86 1.05131
\(227\) −5138.16 −1.50234 −0.751171 0.660108i \(-0.770510\pi\)
−0.751171 + 0.660108i \(0.770510\pi\)
\(228\) 0 0
\(229\) 614.806 0.177413 0.0887064 0.996058i \(-0.471727\pi\)
0.0887064 + 0.996058i \(0.471727\pi\)
\(230\) 2210.51 0.633725
\(231\) 0 0
\(232\) −4677.59 −1.32370
\(233\) −2827.42 −0.794979 −0.397490 0.917607i \(-0.630119\pi\)
−0.397490 + 0.917607i \(0.630119\pi\)
\(234\) 0 0
\(235\) −3451.12 −0.957984
\(236\) −10594.5 −2.92221
\(237\) 0 0
\(238\) 0 0
\(239\) 3432.45 0.928983 0.464491 0.885578i \(-0.346237\pi\)
0.464491 + 0.885578i \(0.346237\pi\)
\(240\) 0 0
\(241\) −2636.11 −0.704593 −0.352296 0.935888i \(-0.614599\pi\)
−0.352296 + 0.935888i \(0.614599\pi\)
\(242\) 6037.16 1.60365
\(243\) 0 0
\(244\) −7736.96 −2.02995
\(245\) 0 0
\(246\) 0 0
\(247\) −4216.30 −1.08614
\(248\) −4140.02 −1.06005
\(249\) 0 0
\(250\) −6467.31 −1.63611
\(251\) −2057.57 −0.517422 −0.258711 0.965955i \(-0.583298\pi\)
−0.258711 + 0.965955i \(0.583298\pi\)
\(252\) 0 0
\(253\) 1042.10 0.258958
\(254\) −933.279 −0.230548
\(255\) 0 0
\(256\) −406.321 −0.0991996
\(257\) −2150.21 −0.521892 −0.260946 0.965353i \(-0.584034\pi\)
−0.260946 + 0.965353i \(0.584034\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4339.66 −1.03513
\(261\) 0 0
\(262\) 5976.77 1.40934
\(263\) 4590.15 1.07620 0.538100 0.842881i \(-0.319142\pi\)
0.538100 + 0.842881i \(0.319142\pi\)
\(264\) 0 0
\(265\) −1600.79 −0.371078
\(266\) 0 0
\(267\) 0 0
\(268\) 3997.59 0.911164
\(269\) 379.378 0.0859891 0.0429945 0.999075i \(-0.486310\pi\)
0.0429945 + 0.999075i \(0.486310\pi\)
\(270\) 0 0
\(271\) −5368.84 −1.20345 −0.601723 0.798705i \(-0.705519\pi\)
−0.601723 + 0.798705i \(0.705519\pi\)
\(272\) −7901.28 −1.76134
\(273\) 0 0
\(274\) 9919.61 2.18710
\(275\) −1309.22 −0.287087
\(276\) 0 0
\(277\) −4781.60 −1.03718 −0.518590 0.855023i \(-0.673543\pi\)
−0.518590 + 0.855023i \(0.673543\pi\)
\(278\) 14954.5 3.22630
\(279\) 0 0
\(280\) 0 0
\(281\) 2076.57 0.440845 0.220423 0.975404i \(-0.429256\pi\)
0.220423 + 0.975404i \(0.429256\pi\)
\(282\) 0 0
\(283\) 2557.62 0.537224 0.268612 0.963248i \(-0.413435\pi\)
0.268612 + 0.963248i \(0.413435\pi\)
\(284\) −15848.4 −3.31138
\(285\) 0 0
\(286\) −2856.97 −0.590687
\(287\) 0 0
\(288\) 0 0
\(289\) −3022.60 −0.615224
\(290\) 2136.15 0.432548
\(291\) 0 0
\(292\) −6681.75 −1.33911
\(293\) −560.049 −0.111667 −0.0558335 0.998440i \(-0.517782\pi\)
−0.0558335 + 0.998440i \(0.517782\pi\)
\(294\) 0 0
\(295\) 2920.02 0.576305
\(296\) −12200.2 −2.39567
\(297\) 0 0
\(298\) 9556.66 1.85773
\(299\) 2895.66 0.560068
\(300\) 0 0
\(301\) 0 0
\(302\) 2404.57 0.458171
\(303\) 0 0
\(304\) −19813.3 −3.73806
\(305\) 2132.44 0.400338
\(306\) 0 0
\(307\) 3653.02 0.679117 0.339558 0.940585i \(-0.389722\pi\)
0.339558 + 0.940585i \(0.389722\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1890.65 0.346393
\(311\) −3492.27 −0.636747 −0.318374 0.947965i \(-0.603137\pi\)
−0.318374 + 0.947965i \(0.603137\pi\)
\(312\) 0 0
\(313\) 8712.09 1.57328 0.786640 0.617412i \(-0.211819\pi\)
0.786640 + 0.617412i \(0.211819\pi\)
\(314\) 9892.85 1.77798
\(315\) 0 0
\(316\) 8830.83 1.57207
\(317\) −1940.33 −0.343785 −0.171892 0.985116i \(-0.554988\pi\)
−0.171892 + 0.985116i \(0.554988\pi\)
\(318\) 0 0
\(319\) 1007.05 0.176752
\(320\) −5125.84 −0.895447
\(321\) 0 0
\(322\) 0 0
\(323\) 4740.39 0.816602
\(324\) 0 0
\(325\) −3637.89 −0.620904
\(326\) −12321.3 −2.09329
\(327\) 0 0
\(328\) −1602.68 −0.269797
\(329\) 0 0
\(330\) 0 0
\(331\) −5731.51 −0.951759 −0.475879 0.879510i \(-0.657870\pi\)
−0.475879 + 0.879510i \(0.657870\pi\)
\(332\) −4867.76 −0.804678
\(333\) 0 0
\(334\) 17048.2 2.79292
\(335\) −1101.81 −0.179696
\(336\) 0 0
\(337\) 2403.74 0.388547 0.194273 0.980947i \(-0.437765\pi\)
0.194273 + 0.980947i \(0.437765\pi\)
\(338\) 3723.73 0.599244
\(339\) 0 0
\(340\) 4879.09 0.778253
\(341\) 891.312 0.141546
\(342\) 0 0
\(343\) 0 0
\(344\) 15765.6 2.47099
\(345\) 0 0
\(346\) 1137.45 0.176733
\(347\) 3336.44 0.516166 0.258083 0.966123i \(-0.416909\pi\)
0.258083 + 0.966123i \(0.416909\pi\)
\(348\) 0 0
\(349\) −2424.54 −0.371870 −0.185935 0.982562i \(-0.559531\pi\)
−0.185935 + 0.982562i \(0.559531\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6228.12 −0.943069
\(353\) 12403.1 1.87012 0.935059 0.354491i \(-0.115346\pi\)
0.935059 + 0.354491i \(0.115346\pi\)
\(354\) 0 0
\(355\) 4368.10 0.653055
\(356\) −31993.1 −4.76300
\(357\) 0 0
\(358\) −12937.5 −1.90997
\(359\) −1353.84 −0.199034 −0.0995168 0.995036i \(-0.531730\pi\)
−0.0995168 + 0.995036i \(0.531730\pi\)
\(360\) 0 0
\(361\) 5028.05 0.733059
\(362\) 1319.81 0.191623
\(363\) 0 0
\(364\) 0 0
\(365\) 1841.61 0.264093
\(366\) 0 0
\(367\) −1378.06 −0.196006 −0.0980031 0.995186i \(-0.531246\pi\)
−0.0980031 + 0.995186i \(0.531246\pi\)
\(368\) 13607.3 1.92753
\(369\) 0 0
\(370\) 5571.52 0.782837
\(371\) 0 0
\(372\) 0 0
\(373\) 5456.92 0.757503 0.378752 0.925498i \(-0.376353\pi\)
0.378752 + 0.925498i \(0.376353\pi\)
\(374\) 3212.10 0.444101
\(375\) 0 0
\(376\) −40115.0 −5.50205
\(377\) 2798.25 0.382273
\(378\) 0 0
\(379\) 554.675 0.0751761 0.0375881 0.999293i \(-0.488033\pi\)
0.0375881 + 0.999293i \(0.488033\pi\)
\(380\) 12234.9 1.65167
\(381\) 0 0
\(382\) −22895.1 −3.06653
\(383\) −5860.66 −0.781895 −0.390948 0.920413i \(-0.627853\pi\)
−0.390948 + 0.920413i \(0.627853\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10939.6 −1.44252
\(387\) 0 0
\(388\) 1599.62 0.209300
\(389\) −7778.86 −1.01389 −0.506946 0.861978i \(-0.669226\pi\)
−0.506946 + 0.861978i \(0.669226\pi\)
\(390\) 0 0
\(391\) −3255.60 −0.421081
\(392\) 0 0
\(393\) 0 0
\(394\) −8844.08 −1.13086
\(395\) −2433.93 −0.310036
\(396\) 0 0
\(397\) 8027.88 1.01488 0.507440 0.861687i \(-0.330592\pi\)
0.507440 + 0.861687i \(0.330592\pi\)
\(398\) 5772.95 0.727065
\(399\) 0 0
\(400\) −17095.2 −2.13690
\(401\) −779.980 −0.0971330 −0.0485665 0.998820i \(-0.515465\pi\)
−0.0485665 + 0.998820i \(0.515465\pi\)
\(402\) 0 0
\(403\) 2476.66 0.306132
\(404\) −23305.0 −2.86997
\(405\) 0 0
\(406\) 0 0
\(407\) 2626.59 0.319890
\(408\) 0 0
\(409\) 14692.5 1.77628 0.888139 0.459575i \(-0.151998\pi\)
0.888139 + 0.459575i \(0.151998\pi\)
\(410\) 731.907 0.0881617
\(411\) 0 0
\(412\) −29154.4 −3.48624
\(413\) 0 0
\(414\) 0 0
\(415\) 1341.64 0.158695
\(416\) −17305.9 −2.03964
\(417\) 0 0
\(418\) 8054.70 0.942508
\(419\) −3370.31 −0.392960 −0.196480 0.980508i \(-0.562951\pi\)
−0.196480 + 0.980508i \(0.562951\pi\)
\(420\) 0 0
\(421\) 15651.0 1.81184 0.905919 0.423450i \(-0.139181\pi\)
0.905919 + 0.423450i \(0.139181\pi\)
\(422\) 24824.1 2.86355
\(423\) 0 0
\(424\) −18607.2 −2.13123
\(425\) 4090.08 0.466819
\(426\) 0 0
\(427\) 0 0
\(428\) −19995.6 −2.25823
\(429\) 0 0
\(430\) −7199.75 −0.807449
\(431\) −4888.12 −0.546294 −0.273147 0.961972i \(-0.588064\pi\)
−0.273147 + 0.961972i \(0.588064\pi\)
\(432\) 0 0
\(433\) −5255.73 −0.583313 −0.291656 0.956523i \(-0.594206\pi\)
−0.291656 + 0.956523i \(0.594206\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 39411.0 4.32901
\(437\) −8163.76 −0.893651
\(438\) 0 0
\(439\) −824.977 −0.0896902 −0.0448451 0.998994i \(-0.514279\pi\)
−0.0448451 + 0.998994i \(0.514279\pi\)
\(440\) 5003.48 0.542117
\(441\) 0 0
\(442\) 8925.37 0.960490
\(443\) −13027.3 −1.39717 −0.698583 0.715529i \(-0.746186\pi\)
−0.698583 + 0.715529i \(0.746186\pi\)
\(444\) 0 0
\(445\) 8817.84 0.939339
\(446\) −17230.9 −1.82939
\(447\) 0 0
\(448\) 0 0
\(449\) −16526.1 −1.73700 −0.868500 0.495689i \(-0.834916\pi\)
−0.868500 + 0.495689i \(0.834916\pi\)
\(450\) 0 0
\(451\) 345.044 0.0360254
\(452\) −13577.4 −1.41289
\(453\) 0 0
\(454\) 27274.9 2.81954
\(455\) 0 0
\(456\) 0 0
\(457\) −3710.82 −0.379836 −0.189918 0.981800i \(-0.560822\pi\)
−0.189918 + 0.981800i \(0.560822\pi\)
\(458\) −3263.57 −0.332962
\(459\) 0 0
\(460\) −8402.63 −0.851684
\(461\) 9714.00 0.981401 0.490701 0.871328i \(-0.336741\pi\)
0.490701 + 0.871328i \(0.336741\pi\)
\(462\) 0 0
\(463\) −43.2780 −0.00434406 −0.00217203 0.999998i \(-0.500691\pi\)
−0.00217203 + 0.999998i \(0.500691\pi\)
\(464\) 13149.6 1.31563
\(465\) 0 0
\(466\) 15008.8 1.49199
\(467\) 1533.24 0.151927 0.0759633 0.997111i \(-0.475797\pi\)
0.0759633 + 0.997111i \(0.475797\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 18319.6 1.79791
\(471\) 0 0
\(472\) 33941.6 3.30993
\(473\) −3394.19 −0.329947
\(474\) 0 0
\(475\) 10256.3 0.990722
\(476\) 0 0
\(477\) 0 0
\(478\) −18220.5 −1.74348
\(479\) 7035.37 0.671095 0.335547 0.942023i \(-0.391079\pi\)
0.335547 + 0.942023i \(0.391079\pi\)
\(480\) 0 0
\(481\) 7298.42 0.691849
\(482\) 13993.3 1.32236
\(483\) 0 0
\(484\) −22948.6 −2.15520
\(485\) −440.882 −0.0412772
\(486\) 0 0
\(487\) −15371.3 −1.43026 −0.715132 0.698989i \(-0.753634\pi\)
−0.715132 + 0.698989i \(0.753634\pi\)
\(488\) 24786.9 2.29929
\(489\) 0 0
\(490\) 0 0
\(491\) 2393.35 0.219980 0.109990 0.993933i \(-0.464918\pi\)
0.109990 + 0.993933i \(0.464918\pi\)
\(492\) 0 0
\(493\) −3146.08 −0.287408
\(494\) 22381.3 2.03843
\(495\) 0 0
\(496\) 11638.4 1.05359
\(497\) 0 0
\(498\) 0 0
\(499\) 693.520 0.0622169 0.0311084 0.999516i \(-0.490096\pi\)
0.0311084 + 0.999516i \(0.490096\pi\)
\(500\) 24583.7 2.19883
\(501\) 0 0
\(502\) 10922.2 0.971079
\(503\) 8646.95 0.766498 0.383249 0.923645i \(-0.374805\pi\)
0.383249 + 0.923645i \(0.374805\pi\)
\(504\) 0 0
\(505\) 6423.27 0.566003
\(506\) −5531.79 −0.486004
\(507\) 0 0
\(508\) 3547.60 0.309841
\(509\) 15500.9 1.34983 0.674916 0.737895i \(-0.264180\pi\)
0.674916 + 0.737895i \(0.264180\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 12656.9 1.09250
\(513\) 0 0
\(514\) 11413.9 0.979469
\(515\) 8035.44 0.687541
\(516\) 0 0
\(517\) 8636.41 0.734678
\(518\) 0 0
\(519\) 0 0
\(520\) 13903.0 1.17248
\(521\) −864.707 −0.0727131 −0.0363565 0.999339i \(-0.511575\pi\)
−0.0363565 + 0.999339i \(0.511575\pi\)
\(522\) 0 0
\(523\) −6255.61 −0.523019 −0.261509 0.965201i \(-0.584220\pi\)
−0.261509 + 0.965201i \(0.584220\pi\)
\(524\) −22719.0 −1.89406
\(525\) 0 0
\(526\) −24365.9 −2.01978
\(527\) −2784.51 −0.230162
\(528\) 0 0
\(529\) −6560.31 −0.539189
\(530\) 8497.45 0.696426
\(531\) 0 0
\(532\) 0 0
\(533\) 958.762 0.0779148
\(534\) 0 0
\(535\) 5511.13 0.445358
\(536\) −12807.1 −1.03206
\(537\) 0 0
\(538\) −2013.85 −0.161381
\(539\) 0 0
\(540\) 0 0
\(541\) 143.871 0.0114334 0.00571671 0.999984i \(-0.498180\pi\)
0.00571671 + 0.999984i \(0.498180\pi\)
\(542\) 28499.4 2.25858
\(543\) 0 0
\(544\) 19457.1 1.53348
\(545\) −10862.4 −0.853747
\(546\) 0 0
\(547\) 5455.65 0.426448 0.213224 0.977003i \(-0.431604\pi\)
0.213224 + 0.977003i \(0.431604\pi\)
\(548\) −37706.6 −2.93932
\(549\) 0 0
\(550\) 6949.72 0.538795
\(551\) −7889.13 −0.609960
\(552\) 0 0
\(553\) 0 0
\(554\) 25382.2 1.94654
\(555\) 0 0
\(556\) −56845.3 −4.33593
\(557\) −24809.9 −1.88730 −0.943652 0.330940i \(-0.892634\pi\)
−0.943652 + 0.330940i \(0.892634\pi\)
\(558\) 0 0
\(559\) −9431.33 −0.713600
\(560\) 0 0
\(561\) 0 0
\(562\) −11023.0 −0.827363
\(563\) −16369.8 −1.22541 −0.612705 0.790312i \(-0.709919\pi\)
−0.612705 + 0.790312i \(0.709919\pi\)
\(564\) 0 0
\(565\) 3742.17 0.278645
\(566\) −13576.6 −1.00824
\(567\) 0 0
\(568\) 50773.7 3.75074
\(569\) 18450.6 1.35938 0.679691 0.733498i \(-0.262114\pi\)
0.679691 + 0.733498i \(0.262114\pi\)
\(570\) 0 0
\(571\) −7108.69 −0.520997 −0.260499 0.965474i \(-0.583887\pi\)
−0.260499 + 0.965474i \(0.583887\pi\)
\(572\) 10860.0 0.793844
\(573\) 0 0
\(574\) 0 0
\(575\) −7043.82 −0.510865
\(576\) 0 0
\(577\) 7594.17 0.547919 0.273960 0.961741i \(-0.411667\pi\)
0.273960 + 0.961741i \(0.411667\pi\)
\(578\) 16044.8 1.15463
\(579\) 0 0
\(580\) −8119.96 −0.581315
\(581\) 0 0
\(582\) 0 0
\(583\) 4005.96 0.284580
\(584\) 21406.4 1.51678
\(585\) 0 0
\(586\) 2972.91 0.209573
\(587\) 1763.34 0.123988 0.0619939 0.998077i \(-0.480254\pi\)
0.0619939 + 0.998077i \(0.480254\pi\)
\(588\) 0 0
\(589\) −6982.47 −0.488468
\(590\) −15500.3 −1.08159
\(591\) 0 0
\(592\) 34296.9 2.38107
\(593\) 12316.1 0.852889 0.426445 0.904514i \(-0.359766\pi\)
0.426445 + 0.904514i \(0.359766\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −36327.0 −2.49666
\(597\) 0 0
\(598\) −15371.0 −1.05112
\(599\) 8903.40 0.607317 0.303659 0.952781i \(-0.401792\pi\)
0.303659 + 0.952781i \(0.401792\pi\)
\(600\) 0 0
\(601\) −19157.1 −1.30022 −0.650112 0.759838i \(-0.725278\pi\)
−0.650112 + 0.759838i \(0.725278\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −9140.31 −0.615751
\(605\) 6325.02 0.425039
\(606\) 0 0
\(607\) −7569.93 −0.506184 −0.253092 0.967442i \(-0.581448\pi\)
−0.253092 + 0.967442i \(0.581448\pi\)
\(608\) 48790.7 3.25448
\(609\) 0 0
\(610\) −11319.6 −0.751340
\(611\) 23997.7 1.58894
\(612\) 0 0
\(613\) 2907.13 0.191546 0.0957730 0.995403i \(-0.469468\pi\)
0.0957730 + 0.995403i \(0.469468\pi\)
\(614\) −19391.3 −1.27454
\(615\) 0 0
\(616\) 0 0
\(617\) 12510.9 0.816320 0.408160 0.912910i \(-0.366171\pi\)
0.408160 + 0.912910i \(0.366171\pi\)
\(618\) 0 0
\(619\) 10065.6 0.653585 0.326792 0.945096i \(-0.394032\pi\)
0.326792 + 0.945096i \(0.394032\pi\)
\(620\) −7186.78 −0.465529
\(621\) 0 0
\(622\) 18538.0 1.19503
\(623\) 0 0
\(624\) 0 0
\(625\) 4983.18 0.318923
\(626\) −46246.4 −2.95268
\(627\) 0 0
\(628\) −37604.9 −2.38949
\(629\) −8205.63 −0.520159
\(630\) 0 0
\(631\) −25146.6 −1.58648 −0.793242 0.608907i \(-0.791608\pi\)
−0.793242 + 0.608907i \(0.791608\pi\)
\(632\) −28291.4 −1.78065
\(633\) 0 0
\(634\) 10299.8 0.645203
\(635\) −977.779 −0.0611055
\(636\) 0 0
\(637\) 0 0
\(638\) −5345.70 −0.331721
\(639\) 0 0
\(640\) 7299.33 0.450830
\(641\) 28958.9 1.78441 0.892206 0.451629i \(-0.149157\pi\)
0.892206 + 0.451629i \(0.149157\pi\)
\(642\) 0 0
\(643\) 7341.90 0.450290 0.225145 0.974325i \(-0.427714\pi\)
0.225145 + 0.974325i \(0.427714\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −25163.4 −1.53257
\(647\) −6071.98 −0.368955 −0.184478 0.982837i \(-0.559059\pi\)
−0.184478 + 0.982837i \(0.559059\pi\)
\(648\) 0 0
\(649\) −7307.33 −0.441969
\(650\) 19311.0 1.16529
\(651\) 0 0
\(652\) 46835.9 2.81324
\(653\) −26262.1 −1.57384 −0.786920 0.617056i \(-0.788325\pi\)
−0.786920 + 0.617056i \(0.788325\pi\)
\(654\) 0 0
\(655\) 6261.75 0.373537
\(656\) 4505.44 0.268152
\(657\) 0 0
\(658\) 0 0
\(659\) −26130.1 −1.54459 −0.772296 0.635263i \(-0.780892\pi\)
−0.772296 + 0.635263i \(0.780892\pi\)
\(660\) 0 0
\(661\) −11925.5 −0.701737 −0.350868 0.936425i \(-0.614114\pi\)
−0.350868 + 0.936425i \(0.614114\pi\)
\(662\) 30424.5 1.78623
\(663\) 0 0
\(664\) 15594.9 0.911444
\(665\) 0 0
\(666\) 0 0
\(667\) 5418.08 0.314526
\(668\) −64804.0 −3.75350
\(669\) 0 0
\(670\) 5848.71 0.337247
\(671\) −5336.42 −0.307019
\(672\) 0 0
\(673\) −6359.85 −0.364271 −0.182135 0.983273i \(-0.558301\pi\)
−0.182135 + 0.983273i \(0.558301\pi\)
\(674\) −12759.8 −0.729211
\(675\) 0 0
\(676\) −14154.7 −0.805344
\(677\) −8561.61 −0.486041 −0.243020 0.970021i \(-0.578138\pi\)
−0.243020 + 0.970021i \(0.578138\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −15631.2 −0.881513
\(681\) 0 0
\(682\) −4731.34 −0.265649
\(683\) 6705.88 0.375686 0.187843 0.982199i \(-0.439850\pi\)
0.187843 + 0.982199i \(0.439850\pi\)
\(684\) 0 0
\(685\) 10392.6 0.579679
\(686\) 0 0
\(687\) 0 0
\(688\) −44319.9 −2.45593
\(689\) 11131.2 0.615481
\(690\) 0 0
\(691\) −25330.6 −1.39453 −0.697267 0.716811i \(-0.745601\pi\)
−0.697267 + 0.716811i \(0.745601\pi\)
\(692\) −4323.68 −0.237517
\(693\) 0 0
\(694\) −17710.8 −0.968723
\(695\) 15667.5 0.855113
\(696\) 0 0
\(697\) −1077.94 −0.0585794
\(698\) 12870.2 0.697912
\(699\) 0 0
\(700\) 0 0
\(701\) 27184.1 1.46467 0.732333 0.680947i \(-0.238432\pi\)
0.732333 + 0.680947i \(0.238432\pi\)
\(702\) 0 0
\(703\) −20576.5 −1.10392
\(704\) 12827.4 0.686719
\(705\) 0 0
\(706\) −65839.5 −3.50977
\(707\) 0 0
\(708\) 0 0
\(709\) 16145.5 0.855228 0.427614 0.903961i \(-0.359354\pi\)
0.427614 + 0.903961i \(0.359354\pi\)
\(710\) −23187.2 −1.22563
\(711\) 0 0
\(712\) 102496. 5.39496
\(713\) 4795.41 0.251879
\(714\) 0 0
\(715\) −2993.20 −0.156558
\(716\) 49178.2 2.56687
\(717\) 0 0
\(718\) 7186.59 0.373539
\(719\) −17297.5 −0.897200 −0.448600 0.893733i \(-0.648077\pi\)
−0.448600 + 0.893733i \(0.648077\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −26690.4 −1.37578
\(723\) 0 0
\(724\) −5016.87 −0.257528
\(725\) −6806.86 −0.348690
\(726\) 0 0
\(727\) 3514.71 0.179303 0.0896516 0.995973i \(-0.471425\pi\)
0.0896516 + 0.995973i \(0.471425\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −9775.78 −0.495641
\(731\) 10603.7 0.536513
\(732\) 0 0
\(733\) −27511.2 −1.38629 −0.693144 0.720799i \(-0.743775\pi\)
−0.693144 + 0.720799i \(0.743775\pi\)
\(734\) 7315.16 0.367858
\(735\) 0 0
\(736\) −33508.4 −1.67817
\(737\) 2757.26 0.137809
\(738\) 0 0
\(739\) 16101.6 0.801497 0.400749 0.916188i \(-0.368750\pi\)
0.400749 + 0.916188i \(0.368750\pi\)
\(740\) −21178.6 −1.05208
\(741\) 0 0
\(742\) 0 0
\(743\) −14682.4 −0.724961 −0.362480 0.931991i \(-0.618070\pi\)
−0.362480 + 0.931991i \(0.618070\pi\)
\(744\) 0 0
\(745\) 10012.3 0.492380
\(746\) −28966.9 −1.42166
\(747\) 0 0
\(748\) −12209.9 −0.596843
\(749\) 0 0
\(750\) 0 0
\(751\) −7273.06 −0.353393 −0.176696 0.984265i \(-0.556541\pi\)
−0.176696 + 0.984265i \(0.556541\pi\)
\(752\) 112771. 5.46851
\(753\) 0 0
\(754\) −14853.9 −0.717437
\(755\) 2519.23 0.121436
\(756\) 0 0
\(757\) 8505.93 0.408393 0.204196 0.978930i \(-0.434542\pi\)
0.204196 + 0.978930i \(0.434542\pi\)
\(758\) −2944.38 −0.141088
\(759\) 0 0
\(760\) −39196.9 −1.87082
\(761\) −14217.7 −0.677256 −0.338628 0.940920i \(-0.609963\pi\)
−0.338628 + 0.940920i \(0.609963\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 87029.3 4.12121
\(765\) 0 0
\(766\) 31110.1 1.46743
\(767\) −20304.7 −0.955879
\(768\) 0 0
\(769\) 16379.1 0.768068 0.384034 0.923319i \(-0.374534\pi\)
0.384034 + 0.923319i \(0.374534\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 41583.8 1.93865
\(773\) 39896.7 1.85638 0.928192 0.372102i \(-0.121363\pi\)
0.928192 + 0.372102i \(0.121363\pi\)
\(774\) 0 0
\(775\) −6024.59 −0.279238
\(776\) −5124.70 −0.237070
\(777\) 0 0
\(778\) 41292.5 1.90284
\(779\) −2703.05 −0.124322
\(780\) 0 0
\(781\) −10931.1 −0.500829
\(782\) 17281.7 0.790270
\(783\) 0 0
\(784\) 0 0
\(785\) 10364.5 0.471244
\(786\) 0 0
\(787\) −33128.5 −1.50051 −0.750257 0.661146i \(-0.770070\pi\)
−0.750257 + 0.661146i \(0.770070\pi\)
\(788\) 33618.3 1.51980
\(789\) 0 0
\(790\) 12920.0 0.581865
\(791\) 0 0
\(792\) 0 0
\(793\) −14828.1 −0.664013
\(794\) −42614.3 −1.90469
\(795\) 0 0
\(796\) −21944.2 −0.977127
\(797\) 17851.5 0.793390 0.396695 0.917951i \(-0.370157\pi\)
0.396695 + 0.917951i \(0.370157\pi\)
\(798\) 0 0
\(799\) −26980.7 −1.19463
\(800\) 42097.4 1.86046
\(801\) 0 0
\(802\) 4140.36 0.182296
\(803\) −4608.61 −0.202533
\(804\) 0 0
\(805\) 0 0
\(806\) −13146.8 −0.574538
\(807\) 0 0
\(808\) 74662.5 3.25076
\(809\) 5057.03 0.219772 0.109886 0.993944i \(-0.464951\pi\)
0.109886 + 0.993944i \(0.464951\pi\)
\(810\) 0 0
\(811\) −17535.4 −0.759251 −0.379626 0.925140i \(-0.623947\pi\)
−0.379626 + 0.925140i \(0.623947\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −13942.7 −0.600358
\(815\) −12908.8 −0.554815
\(816\) 0 0
\(817\) 26589.8 1.13863
\(818\) −77992.1 −3.33366
\(819\) 0 0
\(820\) −2782.14 −0.118484
\(821\) 18700.8 0.794959 0.397480 0.917611i \(-0.369885\pi\)
0.397480 + 0.917611i \(0.369885\pi\)
\(822\) 0 0
\(823\) 22222.6 0.941230 0.470615 0.882339i \(-0.344032\pi\)
0.470615 + 0.882339i \(0.344032\pi\)
\(824\) 93402.0 3.94880
\(825\) 0 0
\(826\) 0 0
\(827\) −25178.9 −1.05872 −0.529358 0.848399i \(-0.677567\pi\)
−0.529358 + 0.848399i \(0.677567\pi\)
\(828\) 0 0
\(829\) 12278.9 0.514432 0.257216 0.966354i \(-0.417195\pi\)
0.257216 + 0.966354i \(0.417195\pi\)
\(830\) −7121.81 −0.297834
\(831\) 0 0
\(832\) 35643.1 1.48522
\(833\) 0 0
\(834\) 0 0
\(835\) 17861.1 0.740249
\(836\) −30617.7 −1.26667
\(837\) 0 0
\(838\) 17890.6 0.737494
\(839\) 25765.0 1.06020 0.530098 0.847936i \(-0.322155\pi\)
0.530098 + 0.847936i \(0.322155\pi\)
\(840\) 0 0
\(841\) −19153.2 −0.785321
\(842\) −83080.2 −3.40040
\(843\) 0 0
\(844\) −94361.8 −3.84842
\(845\) 3901.29 0.158826
\(846\) 0 0
\(847\) 0 0
\(848\) 52308.2 2.11824
\(849\) 0 0
\(850\) −21711.4 −0.876111
\(851\) 14131.5 0.569238
\(852\) 0 0
\(853\) 37864.5 1.51988 0.759939 0.649995i \(-0.225229\pi\)
0.759939 + 0.649995i \(0.225229\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 64060.0 2.55786
\(857\) 29208.7 1.16424 0.582118 0.813104i \(-0.302224\pi\)
0.582118 + 0.813104i \(0.302224\pi\)
\(858\) 0 0
\(859\) 34902.9 1.38635 0.693173 0.720771i \(-0.256212\pi\)
0.693173 + 0.720771i \(0.256212\pi\)
\(860\) 27367.8 1.08516
\(861\) 0 0
\(862\) 25947.6 1.02527
\(863\) 13589.3 0.536021 0.268011 0.963416i \(-0.413634\pi\)
0.268011 + 0.963416i \(0.413634\pi\)
\(864\) 0 0
\(865\) 1191.68 0.0468420
\(866\) 27899.0 1.09474
\(867\) 0 0
\(868\) 0 0
\(869\) 6090.89 0.237767
\(870\) 0 0
\(871\) 7661.52 0.298049
\(872\) −126261. −4.90338
\(873\) 0 0
\(874\) 43335.7 1.67717
\(875\) 0 0
\(876\) 0 0
\(877\) −2379.75 −0.0916288 −0.0458144 0.998950i \(-0.514588\pi\)
−0.0458144 + 0.998950i \(0.514588\pi\)
\(878\) 4379.22 0.168327
\(879\) 0 0
\(880\) −14065.7 −0.538812
\(881\) 24235.5 0.926803 0.463401 0.886148i \(-0.346629\pi\)
0.463401 + 0.886148i \(0.346629\pi\)
\(882\) 0 0
\(883\) −9844.13 −0.375177 −0.187589 0.982248i \(-0.560067\pi\)
−0.187589 + 0.982248i \(0.560067\pi\)
\(884\) −33927.3 −1.29084
\(885\) 0 0
\(886\) 69152.6 2.62215
\(887\) −28609.9 −1.08300 −0.541502 0.840699i \(-0.682144\pi\)
−0.541502 + 0.840699i \(0.682144\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −46807.7 −1.76292
\(891\) 0 0
\(892\) 65498.4 2.45857
\(893\) −67657.0 −2.53534
\(894\) 0 0
\(895\) −13554.4 −0.506226
\(896\) 0 0
\(897\) 0 0
\(898\) 87725.2 3.25994
\(899\) 4634.09 0.171919
\(900\) 0 0
\(901\) −12514.9 −0.462743
\(902\) −1831.59 −0.0676113
\(903\) 0 0
\(904\) 43498.0 1.60036
\(905\) 1382.74 0.0507886
\(906\) 0 0
\(907\) 44578.3 1.63197 0.815986 0.578071i \(-0.196194\pi\)
0.815986 + 0.578071i \(0.196194\pi\)
\(908\) −103678. −3.78928
\(909\) 0 0
\(910\) 0 0
\(911\) −45870.6 −1.66823 −0.834116 0.551589i \(-0.814022\pi\)
−0.834116 + 0.551589i \(0.814022\pi\)
\(912\) 0 0
\(913\) −3357.45 −0.121703
\(914\) 19698.1 0.712863
\(915\) 0 0
\(916\) 12405.6 0.447479
\(917\) 0 0
\(918\) 0 0
\(919\) 31088.3 1.11590 0.557948 0.829876i \(-0.311589\pi\)
0.557948 + 0.829876i \(0.311589\pi\)
\(920\) 26919.6 0.964686
\(921\) 0 0
\(922\) −51564.8 −1.84186
\(923\) −30374.0 −1.08318
\(924\) 0 0
\(925\) −17753.7 −0.631069
\(926\) 229.732 0.00815278
\(927\) 0 0
\(928\) −32381.1 −1.14543
\(929\) 42094.5 1.48663 0.743313 0.668943i \(-0.233253\pi\)
0.743313 + 0.668943i \(0.233253\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −57051.5 −2.00513
\(933\) 0 0
\(934\) −8138.87 −0.285130
\(935\) 3365.26 0.117707
\(936\) 0 0
\(937\) −44385.1 −1.54749 −0.773745 0.633497i \(-0.781619\pi\)
−0.773745 + 0.633497i \(0.781619\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −69636.6 −2.41627
\(941\) −40991.6 −1.42007 −0.710036 0.704165i \(-0.751321\pi\)
−0.710036 + 0.704165i \(0.751321\pi\)
\(942\) 0 0
\(943\) 1856.39 0.0641066
\(944\) −95416.1 −3.28976
\(945\) 0 0
\(946\) 18017.4 0.619233
\(947\) 52622.4 1.80570 0.902851 0.429955i \(-0.141470\pi\)
0.902851 + 0.429955i \(0.141470\pi\)
\(948\) 0 0
\(949\) −12805.8 −0.438034
\(950\) −54443.6 −1.85935
\(951\) 0 0
\(952\) 0 0
\(953\) 10798.1 0.367035 0.183517 0.983016i \(-0.441252\pi\)
0.183517 + 0.983016i \(0.441252\pi\)
\(954\) 0 0
\(955\) −23986.8 −0.812768
\(956\) 69260.0 2.34313
\(957\) 0 0
\(958\) −37345.8 −1.25949
\(959\) 0 0
\(960\) 0 0
\(961\) −25689.5 −0.862324
\(962\) −38742.2 −1.29844
\(963\) 0 0
\(964\) −53191.4 −1.77716
\(965\) −11461.2 −0.382331
\(966\) 0 0
\(967\) 15648.9 0.520408 0.260204 0.965554i \(-0.416210\pi\)
0.260204 + 0.965554i \(0.416210\pi\)
\(968\) 73520.4 2.44115
\(969\) 0 0
\(970\) 2340.33 0.0774675
\(971\) −47259.8 −1.56194 −0.780968 0.624571i \(-0.785274\pi\)
−0.780968 + 0.624571i \(0.785274\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 81595.2 2.68427
\(975\) 0 0
\(976\) −69680.7 −2.28527
\(977\) 48966.5 1.60346 0.801728 0.597689i \(-0.203914\pi\)
0.801728 + 0.597689i \(0.203914\pi\)
\(978\) 0 0
\(979\) −22066.6 −0.720380
\(980\) 0 0
\(981\) 0 0
\(982\) −12704.6 −0.412851
\(983\) 19111.3 0.620097 0.310049 0.950721i \(-0.399655\pi\)
0.310049 + 0.950721i \(0.399655\pi\)
\(984\) 0 0
\(985\) −9265.77 −0.299728
\(986\) 16700.3 0.539397
\(987\) 0 0
\(988\) −85076.4 −2.73951
\(989\) −18261.3 −0.587134
\(990\) 0 0
\(991\) −54102.5 −1.73423 −0.867115 0.498107i \(-0.834029\pi\)
−0.867115 + 0.498107i \(0.834029\pi\)
\(992\) −28659.7 −0.917286
\(993\) 0 0
\(994\) 0 0
\(995\) 6048.21 0.192705
\(996\) 0 0
\(997\) 9192.80 0.292015 0.146008 0.989283i \(-0.453358\pi\)
0.146008 + 0.989283i \(0.453358\pi\)
\(998\) −3681.41 −0.116766
\(999\) 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 441.4.a.s.1.1 3
3.2 odd 2 147.4.a.l.1.3 3
7.2 even 3 63.4.e.c.46.3 6
7.3 odd 6 441.4.e.w.226.3 6
7.4 even 3 63.4.e.c.37.3 6
7.5 odd 6 441.4.e.w.361.3 6
7.6 odd 2 441.4.a.t.1.1 3
12.11 even 2 2352.4.a.ci.1.2 3
21.2 odd 6 21.4.e.b.4.1 6
21.5 even 6 147.4.e.n.67.1 6
21.11 odd 6 21.4.e.b.16.1 yes 6
21.17 even 6 147.4.e.n.79.1 6
21.20 even 2 147.4.a.m.1.3 3
84.11 even 6 336.4.q.k.289.2 6
84.23 even 6 336.4.q.k.193.2 6
84.83 odd 2 2352.4.a.cg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.1 6 21.2 odd 6
21.4.e.b.16.1 yes 6 21.11 odd 6
63.4.e.c.37.3 6 7.4 even 3
63.4.e.c.46.3 6 7.2 even 3
147.4.a.l.1.3 3 3.2 odd 2
147.4.a.m.1.3 3 21.20 even 2
147.4.e.n.67.1 6 21.5 even 6
147.4.e.n.79.1 6 21.17 even 6
336.4.q.k.193.2 6 84.23 even 6
336.4.q.k.289.2 6 84.11 even 6
441.4.a.s.1.1 3 1.1 even 1 trivial
441.4.a.t.1.1 3 7.6 odd 2
441.4.e.w.226.3 6 7.3 odd 6
441.4.e.w.361.3 6 7.5 odd 6
2352.4.a.cg.1.2 3 84.83 odd 2
2352.4.a.ci.1.2 3 12.11 even 2