Properties

Label 1521.4.a.bm.1.16
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $18$
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: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \( x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} - 37096332 x^{4} + 27824160 x^{2} - 6889792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 7\cdot 13^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(4.56788\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.56788 q^{2} +12.8655 q^{4} -1.42360 q^{5} -3.82462 q^{7} +22.2252 q^{8} +O(q^{10})\) \(q+4.56788 q^{2} +12.8655 q^{4} -1.42360 q^{5} -3.82462 q^{7} +22.2252 q^{8} -6.50284 q^{10} +18.2903 q^{11} -17.4704 q^{14} -1.40218 q^{16} -93.1475 q^{17} -74.6094 q^{19} -18.3154 q^{20} +83.5478 q^{22} +7.35466 q^{23} -122.973 q^{25} -49.2058 q^{28} -211.812 q^{29} +183.636 q^{31} -184.207 q^{32} -425.487 q^{34} +5.44473 q^{35} -289.095 q^{37} -340.807 q^{38} -31.6398 q^{40} +131.331 q^{41} +394.550 q^{43} +235.314 q^{44} +33.5952 q^{46} +201.122 q^{47} -328.372 q^{49} -561.728 q^{50} -16.7015 q^{53} -26.0380 q^{55} -85.0031 q^{56} -967.531 q^{58} -446.118 q^{59} +475.647 q^{61} +838.830 q^{62} -830.217 q^{64} -252.055 q^{67} -1198.39 q^{68} +24.8709 q^{70} +295.597 q^{71} -892.965 q^{73} -1320.55 q^{74} -959.890 q^{76} -69.9534 q^{77} +170.848 q^{79} +1.99615 q^{80} +599.903 q^{82} +1331.42 q^{83} +132.605 q^{85} +1802.26 q^{86} +406.505 q^{88} -1500.50 q^{89} +94.6217 q^{92} +918.701 q^{94} +106.214 q^{95} -472.304 q^{97} -1499.97 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 56 q^{4} - 94 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55} - 1626 q^{58} - 1784 q^{61} - 4274 q^{64} - 2872 q^{67} - 5772 q^{70} - 3078 q^{73} - 5726 q^{76} + 3326 q^{79} - 1038 q^{82} - 2340 q^{85} + 780 q^{88} + 1220 q^{94} - 4630 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.56788 1.61499 0.807495 0.589874i \(-0.200823\pi\)
0.807495 + 0.589874i \(0.200823\pi\)
\(3\) 0 0
\(4\) 12.8655 1.60819
\(5\) −1.42360 −0.127331 −0.0636653 0.997971i \(-0.520279\pi\)
−0.0636653 + 0.997971i \(0.520279\pi\)
\(6\) 0 0
\(7\) −3.82462 −0.206510 −0.103255 0.994655i \(-0.532926\pi\)
−0.103255 + 0.994655i \(0.532926\pi\)
\(8\) 22.2252 0.982225
\(9\) 0 0
\(10\) −6.50284 −0.205638
\(11\) 18.2903 0.501339 0.250669 0.968073i \(-0.419349\pi\)
0.250669 + 0.968073i \(0.419349\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −17.4704 −0.333512
\(15\) 0 0
\(16\) −1.40218 −0.0219091
\(17\) −93.1475 −1.32892 −0.664459 0.747325i \(-0.731338\pi\)
−0.664459 + 0.747325i \(0.731338\pi\)
\(18\) 0 0
\(19\) −74.6094 −0.900872 −0.450436 0.892809i \(-0.648732\pi\)
−0.450436 + 0.892809i \(0.648732\pi\)
\(20\) −18.3154 −0.204772
\(21\) 0 0
\(22\) 83.5478 0.809657
\(23\) 7.35466 0.0666762 0.0333381 0.999444i \(-0.489386\pi\)
0.0333381 + 0.999444i \(0.489386\pi\)
\(24\) 0 0
\(25\) −122.973 −0.983787
\(26\) 0 0
\(27\) 0 0
\(28\) −49.2058 −0.332108
\(29\) −211.812 −1.35629 −0.678146 0.734927i \(-0.737216\pi\)
−0.678146 + 0.734927i \(0.737216\pi\)
\(30\) 0 0
\(31\) 183.636 1.06394 0.531969 0.846764i \(-0.321452\pi\)
0.531969 + 0.846764i \(0.321452\pi\)
\(32\) −184.207 −1.01761
\(33\) 0 0
\(34\) −425.487 −2.14619
\(35\) 5.44473 0.0262951
\(36\) 0 0
\(37\) −289.095 −1.28451 −0.642255 0.766491i \(-0.722001\pi\)
−0.642255 + 0.766491i \(0.722001\pi\)
\(38\) −340.807 −1.45490
\(39\) 0 0
\(40\) −31.6398 −0.125067
\(41\) 131.331 0.500254 0.250127 0.968213i \(-0.419528\pi\)
0.250127 + 0.968213i \(0.419528\pi\)
\(42\) 0 0
\(43\) 394.550 1.39926 0.699632 0.714503i \(-0.253347\pi\)
0.699632 + 0.714503i \(0.253347\pi\)
\(44\) 235.314 0.806249
\(45\) 0 0
\(46\) 33.5952 0.107681
\(47\) 201.122 0.624184 0.312092 0.950052i \(-0.398970\pi\)
0.312092 + 0.950052i \(0.398970\pi\)
\(48\) 0 0
\(49\) −328.372 −0.957354
\(50\) −561.728 −1.58881
\(51\) 0 0
\(52\) 0 0
\(53\) −16.7015 −0.0432853 −0.0216426 0.999766i \(-0.506890\pi\)
−0.0216426 + 0.999766i \(0.506890\pi\)
\(54\) 0 0
\(55\) −26.0380 −0.0638358
\(56\) −85.0031 −0.202839
\(57\) 0 0
\(58\) −967.531 −2.19040
\(59\) −446.118 −0.984401 −0.492201 0.870482i \(-0.663807\pi\)
−0.492201 + 0.870482i \(0.663807\pi\)
\(60\) 0 0
\(61\) 475.647 0.998366 0.499183 0.866497i \(-0.333634\pi\)
0.499183 + 0.866497i \(0.333634\pi\)
\(62\) 838.830 1.71825
\(63\) 0 0
\(64\) −830.217 −1.62152
\(65\) 0 0
\(66\) 0 0
\(67\) −252.055 −0.459603 −0.229801 0.973238i \(-0.573808\pi\)
−0.229801 + 0.973238i \(0.573808\pi\)
\(68\) −1198.39 −2.13715
\(69\) 0 0
\(70\) 24.8709 0.0424663
\(71\) 295.597 0.494098 0.247049 0.969003i \(-0.420539\pi\)
0.247049 + 0.969003i \(0.420539\pi\)
\(72\) 0 0
\(73\) −892.965 −1.43169 −0.715847 0.698257i \(-0.753959\pi\)
−0.715847 + 0.698257i \(0.753959\pi\)
\(74\) −1320.55 −2.07447
\(75\) 0 0
\(76\) −959.890 −1.44878
\(77\) −69.9534 −0.103532
\(78\) 0 0
\(79\) 170.848 0.243315 0.121658 0.992572i \(-0.461179\pi\)
0.121658 + 0.992572i \(0.461179\pi\)
\(80\) 1.99615 0.00278970
\(81\) 0 0
\(82\) 599.903 0.807905
\(83\) 1331.42 1.76075 0.880375 0.474279i \(-0.157291\pi\)
0.880375 + 0.474279i \(0.157291\pi\)
\(84\) 0 0
\(85\) 132.605 0.169212
\(86\) 1802.26 2.25980
\(87\) 0 0
\(88\) 406.505 0.492427
\(89\) −1500.50 −1.78711 −0.893554 0.448957i \(-0.851796\pi\)
−0.893554 + 0.448957i \(0.851796\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 94.6217 0.107228
\(93\) 0 0
\(94\) 918.701 1.00805
\(95\) 106.214 0.114709
\(96\) 0 0
\(97\) −472.304 −0.494384 −0.247192 0.968967i \(-0.579508\pi\)
−0.247192 + 0.968967i \(0.579508\pi\)
\(98\) −1499.97 −1.54612
\(99\) 0 0
\(100\) −1582.12 −1.58212
\(101\) 858.856 0.846133 0.423066 0.906099i \(-0.360954\pi\)
0.423066 + 0.906099i \(0.360954\pi\)
\(102\) 0 0
\(103\) −17.4895 −0.0167310 −0.00836548 0.999965i \(-0.502663\pi\)
−0.00836548 + 0.999965i \(0.502663\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −76.2902 −0.0699053
\(107\) −1674.26 −1.51268 −0.756341 0.654177i \(-0.773015\pi\)
−0.756341 + 0.654177i \(0.773015\pi\)
\(108\) 0 0
\(109\) −749.237 −0.658385 −0.329192 0.944263i \(-0.606776\pi\)
−0.329192 + 0.944263i \(0.606776\pi\)
\(110\) −118.939 −0.103094
\(111\) 0 0
\(112\) 5.36282 0.00452445
\(113\) −935.250 −0.778592 −0.389296 0.921113i \(-0.627282\pi\)
−0.389296 + 0.921113i \(0.627282\pi\)
\(114\) 0 0
\(115\) −10.4701 −0.00848993
\(116\) −2725.07 −2.18118
\(117\) 0 0
\(118\) −2037.82 −1.58980
\(119\) 356.254 0.274435
\(120\) 0 0
\(121\) −996.466 −0.748660
\(122\) 2172.70 1.61235
\(123\) 0 0
\(124\) 2362.58 1.71102
\(125\) 353.015 0.252597
\(126\) 0 0
\(127\) 1383.93 0.966960 0.483480 0.875355i \(-0.339373\pi\)
0.483480 + 0.875355i \(0.339373\pi\)
\(128\) −2318.68 −1.60113
\(129\) 0 0
\(130\) 0 0
\(131\) −1104.68 −0.736764 −0.368382 0.929675i \(-0.620088\pi\)
−0.368382 + 0.929675i \(0.620088\pi\)
\(132\) 0 0
\(133\) 285.353 0.186039
\(134\) −1151.36 −0.742254
\(135\) 0 0
\(136\) −2070.22 −1.30530
\(137\) −1281.58 −0.799215 −0.399607 0.916686i \(-0.630854\pi\)
−0.399607 + 0.916686i \(0.630854\pi\)
\(138\) 0 0
\(139\) −1622.35 −0.989972 −0.494986 0.868901i \(-0.664827\pi\)
−0.494986 + 0.868901i \(0.664827\pi\)
\(140\) 70.0494 0.0422876
\(141\) 0 0
\(142\) 1350.25 0.797964
\(143\) 0 0
\(144\) 0 0
\(145\) 301.535 0.172698
\(146\) −4078.96 −2.31217
\(147\) 0 0
\(148\) −3719.36 −2.06574
\(149\) −3015.76 −1.65812 −0.829062 0.559157i \(-0.811125\pi\)
−0.829062 + 0.559157i \(0.811125\pi\)
\(150\) 0 0
\(151\) 3452.62 1.86073 0.930364 0.366636i \(-0.119491\pi\)
0.930364 + 0.366636i \(0.119491\pi\)
\(152\) −1658.21 −0.884859
\(153\) 0 0
\(154\) −319.539 −0.167202
\(155\) −261.425 −0.135472
\(156\) 0 0
\(157\) −637.489 −0.324059 −0.162029 0.986786i \(-0.551804\pi\)
−0.162029 + 0.986786i \(0.551804\pi\)
\(158\) 780.414 0.392952
\(159\) 0 0
\(160\) 262.237 0.129573
\(161\) −28.1288 −0.0137693
\(162\) 0 0
\(163\) −730.216 −0.350889 −0.175445 0.984489i \(-0.556136\pi\)
−0.175445 + 0.984489i \(0.556136\pi\)
\(164\) 1689.64 0.804504
\(165\) 0 0
\(166\) 6081.76 2.84359
\(167\) 1001.04 0.463850 0.231925 0.972734i \(-0.425498\pi\)
0.231925 + 0.972734i \(0.425498\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 605.723 0.273275
\(171\) 0 0
\(172\) 5076.10 2.25029
\(173\) −1772.12 −0.778796 −0.389398 0.921070i \(-0.627317\pi\)
−0.389398 + 0.921070i \(0.627317\pi\)
\(174\) 0 0
\(175\) 470.327 0.203162
\(176\) −25.6463 −0.0109839
\(177\) 0 0
\(178\) −6854.10 −2.88616
\(179\) 3809.57 1.59073 0.795365 0.606131i \(-0.207279\pi\)
0.795365 + 0.606131i \(0.207279\pi\)
\(180\) 0 0
\(181\) 2025.78 0.831908 0.415954 0.909386i \(-0.363448\pi\)
0.415954 + 0.909386i \(0.363448\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 163.459 0.0654911
\(185\) 411.555 0.163558
\(186\) 0 0
\(187\) −1703.69 −0.666237
\(188\) 2587.54 1.00381
\(189\) 0 0
\(190\) 485.173 0.185253
\(191\) 979.007 0.370882 0.185441 0.982655i \(-0.440629\pi\)
0.185441 + 0.982655i \(0.440629\pi\)
\(192\) 0 0
\(193\) 647.578 0.241522 0.120761 0.992682i \(-0.461467\pi\)
0.120761 + 0.992682i \(0.461467\pi\)
\(194\) −2157.43 −0.798424
\(195\) 0 0
\(196\) −4224.69 −1.53961
\(197\) 3358.09 1.21449 0.607244 0.794515i \(-0.292275\pi\)
0.607244 + 0.794515i \(0.292275\pi\)
\(198\) 0 0
\(199\) −3178.62 −1.13229 −0.566146 0.824305i \(-0.691566\pi\)
−0.566146 + 0.824305i \(0.691566\pi\)
\(200\) −2733.11 −0.966300
\(201\) 0 0
\(202\) 3923.15 1.36650
\(203\) 810.100 0.280088
\(204\) 0 0
\(205\) −186.962 −0.0636976
\(206\) −79.8898 −0.0270203
\(207\) 0 0
\(208\) 0 0
\(209\) −1364.63 −0.451642
\(210\) 0 0
\(211\) 789.477 0.257582 0.128791 0.991672i \(-0.458890\pi\)
0.128791 + 0.991672i \(0.458890\pi\)
\(212\) −214.873 −0.0696111
\(213\) 0 0
\(214\) −7647.83 −2.44297
\(215\) −561.682 −0.178169
\(216\) 0 0
\(217\) −702.340 −0.219714
\(218\) −3422.43 −1.06328
\(219\) 0 0
\(220\) −334.993 −0.102660
\(221\) 0 0
\(222\) 0 0
\(223\) 2349.00 0.705383 0.352691 0.935740i \(-0.385267\pi\)
0.352691 + 0.935740i \(0.385267\pi\)
\(224\) 704.521 0.210146
\(225\) 0 0
\(226\) −4272.11 −1.25742
\(227\) 124.518 0.0364078 0.0182039 0.999834i \(-0.494205\pi\)
0.0182039 + 0.999834i \(0.494205\pi\)
\(228\) 0 0
\(229\) −872.745 −0.251846 −0.125923 0.992040i \(-0.540189\pi\)
−0.125923 + 0.992040i \(0.540189\pi\)
\(230\) −47.8262 −0.0137112
\(231\) 0 0
\(232\) −4707.56 −1.33218
\(233\) 5501.06 1.54672 0.773361 0.633966i \(-0.218574\pi\)
0.773361 + 0.633966i \(0.218574\pi\)
\(234\) 0 0
\(235\) −286.317 −0.0794777
\(236\) −5739.55 −1.58311
\(237\) 0 0
\(238\) 1627.33 0.443210
\(239\) 4035.74 1.09226 0.546130 0.837700i \(-0.316100\pi\)
0.546130 + 0.837700i \(0.316100\pi\)
\(240\) 0 0
\(241\) 3291.83 0.879856 0.439928 0.898033i \(-0.355004\pi\)
0.439928 + 0.898033i \(0.355004\pi\)
\(242\) −4551.74 −1.20908
\(243\) 0 0
\(244\) 6119.45 1.60557
\(245\) 467.471 0.121900
\(246\) 0 0
\(247\) 0 0
\(248\) 4081.36 1.04503
\(249\) 0 0
\(250\) 1612.53 0.407941
\(251\) 6212.66 1.56231 0.781155 0.624337i \(-0.214631\pi\)
0.781155 + 0.624337i \(0.214631\pi\)
\(252\) 0 0
\(253\) 134.519 0.0334274
\(254\) 6321.63 1.56163
\(255\) 0 0
\(256\) −3949.72 −0.964286
\(257\) −1856.15 −0.450519 −0.225260 0.974299i \(-0.572323\pi\)
−0.225260 + 0.974299i \(0.572323\pi\)
\(258\) 0 0
\(259\) 1105.68 0.265265
\(260\) 0 0
\(261\) 0 0
\(262\) −5046.03 −1.18987
\(263\) 6004.40 1.40778 0.703892 0.710307i \(-0.251444\pi\)
0.703892 + 0.710307i \(0.251444\pi\)
\(264\) 0 0
\(265\) 23.7762 0.00551155
\(266\) 1303.46 0.300452
\(267\) 0 0
\(268\) −3242.82 −0.739130
\(269\) 6213.11 1.40825 0.704126 0.710075i \(-0.251339\pi\)
0.704126 + 0.710075i \(0.251339\pi\)
\(270\) 0 0
\(271\) −5678.54 −1.27287 −0.636433 0.771332i \(-0.719591\pi\)
−0.636433 + 0.771332i \(0.719591\pi\)
\(272\) 130.610 0.0291154
\(273\) 0 0
\(274\) −5854.09 −1.29072
\(275\) −2249.22 −0.493210
\(276\) 0 0
\(277\) 7111.40 1.54254 0.771268 0.636510i \(-0.219623\pi\)
0.771268 + 0.636510i \(0.219623\pi\)
\(278\) −7410.71 −1.59879
\(279\) 0 0
\(280\) 121.010 0.0258277
\(281\) −4708.92 −0.999683 −0.499841 0.866117i \(-0.666608\pi\)
−0.499841 + 0.866117i \(0.666608\pi\)
\(282\) 0 0
\(283\) −3538.68 −0.743295 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(284\) 3803.02 0.794605
\(285\) 0 0
\(286\) 0 0
\(287\) −502.290 −0.103308
\(288\) 0 0
\(289\) 3763.46 0.766021
\(290\) 1377.38 0.278905
\(291\) 0 0
\(292\) −11488.5 −2.30244
\(293\) −5172.71 −1.03137 −0.515687 0.856777i \(-0.672463\pi\)
−0.515687 + 0.856777i \(0.672463\pi\)
\(294\) 0 0
\(295\) 635.094 0.125344
\(296\) −6425.20 −1.26168
\(297\) 0 0
\(298\) −13775.6 −2.67785
\(299\) 0 0
\(300\) 0 0
\(301\) −1509.01 −0.288962
\(302\) 15771.1 3.00506
\(303\) 0 0
\(304\) 104.616 0.0197373
\(305\) −677.131 −0.127123
\(306\) 0 0
\(307\) −8903.06 −1.65513 −0.827564 0.561371i \(-0.810274\pi\)
−0.827564 + 0.561371i \(0.810274\pi\)
\(308\) −899.988 −0.166499
\(309\) 0 0
\(310\) −1194.16 −0.218786
\(311\) 9012.95 1.64334 0.821668 0.569967i \(-0.193044\pi\)
0.821668 + 0.569967i \(0.193044\pi\)
\(312\) 0 0
\(313\) 6177.27 1.11553 0.557763 0.830000i \(-0.311660\pi\)
0.557763 + 0.830000i \(0.311660\pi\)
\(314\) −2911.98 −0.523351
\(315\) 0 0
\(316\) 2198.05 0.391298
\(317\) 1524.80 0.270162 0.135081 0.990835i \(-0.456870\pi\)
0.135081 + 0.990835i \(0.456870\pi\)
\(318\) 0 0
\(319\) −3874.09 −0.679961
\(320\) 1181.90 0.206469
\(321\) 0 0
\(322\) −128.489 −0.0222373
\(323\) 6949.68 1.19718
\(324\) 0 0
\(325\) 0 0
\(326\) −3335.54 −0.566683
\(327\) 0 0
\(328\) 2918.85 0.491362
\(329\) −769.215 −0.128900
\(330\) 0 0
\(331\) −8562.08 −1.42180 −0.710898 0.703295i \(-0.751711\pi\)
−0.710898 + 0.703295i \(0.751711\pi\)
\(332\) 17129.4 2.83162
\(333\) 0 0
\(334\) 4572.64 0.749114
\(335\) 358.825 0.0585215
\(336\) 0 0
\(337\) −4599.39 −0.743457 −0.371728 0.928342i \(-0.621235\pi\)
−0.371728 + 0.928342i \(0.621235\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1706.03 0.272125
\(341\) 3358.76 0.533393
\(342\) 0 0
\(343\) 2567.75 0.404213
\(344\) 8768.97 1.37439
\(345\) 0 0
\(346\) −8094.83 −1.25775
\(347\) 7849.33 1.21433 0.607167 0.794574i \(-0.292306\pi\)
0.607167 + 0.794574i \(0.292306\pi\)
\(348\) 0 0
\(349\) −5989.62 −0.918673 −0.459336 0.888262i \(-0.651913\pi\)
−0.459336 + 0.888262i \(0.651913\pi\)
\(350\) 2148.40 0.328105
\(351\) 0 0
\(352\) −3369.19 −0.510166
\(353\) 6425.03 0.968752 0.484376 0.874860i \(-0.339047\pi\)
0.484376 + 0.874860i \(0.339047\pi\)
\(354\) 0 0
\(355\) −420.813 −0.0629138
\(356\) −19304.7 −2.87401
\(357\) 0 0
\(358\) 17401.7 2.56901
\(359\) −6077.13 −0.893422 −0.446711 0.894678i \(-0.647405\pi\)
−0.446711 + 0.894678i \(0.647405\pi\)
\(360\) 0 0
\(361\) −1292.44 −0.188429
\(362\) 9253.54 1.34352
\(363\) 0 0
\(364\) 0 0
\(365\) 1271.23 0.182299
\(366\) 0 0
\(367\) −7171.22 −1.01998 −0.509992 0.860179i \(-0.670352\pi\)
−0.509992 + 0.860179i \(0.670352\pi\)
\(368\) −10.3126 −0.00146082
\(369\) 0 0
\(370\) 1879.94 0.264144
\(371\) 63.8767 0.00893886
\(372\) 0 0
\(373\) −323.348 −0.0448856 −0.0224428 0.999748i \(-0.507144\pi\)
−0.0224428 + 0.999748i \(0.507144\pi\)
\(374\) −7782.27 −1.07597
\(375\) 0 0
\(376\) 4469.98 0.613089
\(377\) 0 0
\(378\) 0 0
\(379\) −2778.92 −0.376632 −0.188316 0.982108i \(-0.560303\pi\)
−0.188316 + 0.982108i \(0.560303\pi\)
\(380\) 1366.50 0.184474
\(381\) 0 0
\(382\) 4471.99 0.598971
\(383\) −10057.2 −1.34177 −0.670883 0.741563i \(-0.734085\pi\)
−0.670883 + 0.741563i \(0.734085\pi\)
\(384\) 0 0
\(385\) 99.5856 0.0131827
\(386\) 2958.06 0.390055
\(387\) 0 0
\(388\) −6076.45 −0.795064
\(389\) 11319.2 1.47534 0.737670 0.675162i \(-0.235926\pi\)
0.737670 + 0.675162i \(0.235926\pi\)
\(390\) 0 0
\(391\) −685.069 −0.0886072
\(392\) −7298.14 −0.940337
\(393\) 0 0
\(394\) 15339.4 1.96139
\(395\) −243.219 −0.0309815
\(396\) 0 0
\(397\) −12259.5 −1.54984 −0.774921 0.632058i \(-0.782210\pi\)
−0.774921 + 0.632058i \(0.782210\pi\)
\(398\) −14519.5 −1.82864
\(399\) 0 0
\(400\) 172.431 0.0215539
\(401\) 11202.2 1.39504 0.697518 0.716567i \(-0.254288\pi\)
0.697518 + 0.716567i \(0.254288\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 11049.7 1.36074
\(405\) 0 0
\(406\) 3700.44 0.452339
\(407\) −5287.62 −0.643975
\(408\) 0 0
\(409\) −769.899 −0.0930784 −0.0465392 0.998916i \(-0.514819\pi\)
−0.0465392 + 0.998916i \(0.514819\pi\)
\(410\) −854.022 −0.102871
\(411\) 0 0
\(412\) −225.012 −0.0269066
\(413\) 1706.23 0.203289
\(414\) 0 0
\(415\) −1895.41 −0.224197
\(416\) 0 0
\(417\) 0 0
\(418\) −6233.45 −0.729397
\(419\) 8310.48 0.968958 0.484479 0.874803i \(-0.339009\pi\)
0.484479 + 0.874803i \(0.339009\pi\)
\(420\) 0 0
\(421\) −3440.71 −0.398313 −0.199157 0.979968i \(-0.563820\pi\)
−0.199157 + 0.979968i \(0.563820\pi\)
\(422\) 3606.24 0.415993
\(423\) 0 0
\(424\) −371.193 −0.0425159
\(425\) 11454.7 1.30737
\(426\) 0 0
\(427\) −1819.17 −0.206173
\(428\) −21540.3 −2.43269
\(429\) 0 0
\(430\) −2565.70 −0.287742
\(431\) 9740.73 1.08862 0.544309 0.838885i \(-0.316792\pi\)
0.544309 + 0.838885i \(0.316792\pi\)
\(432\) 0 0
\(433\) −16264.5 −1.80513 −0.902563 0.430557i \(-0.858317\pi\)
−0.902563 + 0.430557i \(0.858317\pi\)
\(434\) −3208.21 −0.354836
\(435\) 0 0
\(436\) −9639.34 −1.05881
\(437\) −548.727 −0.0600668
\(438\) 0 0
\(439\) −10934.6 −1.18879 −0.594395 0.804173i \(-0.702608\pi\)
−0.594395 + 0.804173i \(0.702608\pi\)
\(440\) −578.701 −0.0627011
\(441\) 0 0
\(442\) 0 0
\(443\) −16871.8 −1.80949 −0.904744 0.425957i \(-0.859938\pi\)
−0.904744 + 0.425957i \(0.859938\pi\)
\(444\) 0 0
\(445\) 2136.11 0.227554
\(446\) 10729.9 1.13919
\(447\) 0 0
\(448\) 3175.27 0.334860
\(449\) −592.053 −0.0622287 −0.0311144 0.999516i \(-0.509906\pi\)
−0.0311144 + 0.999516i \(0.509906\pi\)
\(450\) 0 0
\(451\) 2402.07 0.250797
\(452\) −12032.5 −1.25213
\(453\) 0 0
\(454\) 568.785 0.0587983
\(455\) 0 0
\(456\) 0 0
\(457\) 14750.9 1.50988 0.754941 0.655792i \(-0.227665\pi\)
0.754941 + 0.655792i \(0.227665\pi\)
\(458\) −3986.60 −0.406728
\(459\) 0 0
\(460\) −134.704 −0.0136534
\(461\) −2208.61 −0.223135 −0.111568 0.993757i \(-0.535587\pi\)
−0.111568 + 0.993757i \(0.535587\pi\)
\(462\) 0 0
\(463\) 3946.11 0.396093 0.198047 0.980193i \(-0.436540\pi\)
0.198047 + 0.980193i \(0.436540\pi\)
\(464\) 296.999 0.0297151
\(465\) 0 0
\(466\) 25128.2 2.49794
\(467\) 13673.7 1.35491 0.677455 0.735564i \(-0.263083\pi\)
0.677455 + 0.735564i \(0.263083\pi\)
\(468\) 0 0
\(469\) 964.014 0.0949126
\(470\) −1307.86 −0.128356
\(471\) 0 0
\(472\) −9915.08 −0.966903
\(473\) 7216.43 0.701505
\(474\) 0 0
\(475\) 9174.97 0.886266
\(476\) 4583.40 0.441344
\(477\) 0 0
\(478\) 18434.8 1.76399
\(479\) −13890.4 −1.32499 −0.662495 0.749066i \(-0.730503\pi\)
−0.662495 + 0.749066i \(0.730503\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 15036.7 1.42096
\(483\) 0 0
\(484\) −12820.1 −1.20399
\(485\) 672.372 0.0629502
\(486\) 0 0
\(487\) 10512.6 0.978180 0.489090 0.872233i \(-0.337329\pi\)
0.489090 + 0.872233i \(0.337329\pi\)
\(488\) 10571.4 0.980620
\(489\) 0 0
\(490\) 2135.35 0.196868
\(491\) 11387.5 1.04667 0.523333 0.852128i \(-0.324688\pi\)
0.523333 + 0.852128i \(0.324688\pi\)
\(492\) 0 0
\(493\) 19729.7 1.80240
\(494\) 0 0
\(495\) 0 0
\(496\) −257.492 −0.0233099
\(497\) −1130.55 −0.102036
\(498\) 0 0
\(499\) −12664.5 −1.13615 −0.568076 0.822976i \(-0.692312\pi\)
−0.568076 + 0.822976i \(0.692312\pi\)
\(500\) 4541.73 0.406224
\(501\) 0 0
\(502\) 28378.7 2.52312
\(503\) −9908.07 −0.878288 −0.439144 0.898417i \(-0.644718\pi\)
−0.439144 + 0.898417i \(0.644718\pi\)
\(504\) 0 0
\(505\) −1222.67 −0.107739
\(506\) 614.466 0.0539849
\(507\) 0 0
\(508\) 17805.0 1.55506
\(509\) −1801.99 −0.156919 −0.0784596 0.996917i \(-0.525000\pi\)
−0.0784596 + 0.996917i \(0.525000\pi\)
\(510\) 0 0
\(511\) 3415.26 0.295660
\(512\) 507.602 0.0438146
\(513\) 0 0
\(514\) −8478.67 −0.727584
\(515\) 24.8980 0.00213036
\(516\) 0 0
\(517\) 3678.57 0.312927
\(518\) 5050.61 0.428400
\(519\) 0 0
\(520\) 0 0
\(521\) −15680.2 −1.31855 −0.659273 0.751903i \(-0.729136\pi\)
−0.659273 + 0.751903i \(0.729136\pi\)
\(522\) 0 0
\(523\) 17866.4 1.49377 0.746884 0.664954i \(-0.231549\pi\)
0.746884 + 0.664954i \(0.231549\pi\)
\(524\) −14212.3 −1.18486
\(525\) 0 0
\(526\) 27427.4 2.27356
\(527\) −17105.3 −1.41389
\(528\) 0 0
\(529\) −12112.9 −0.995554
\(530\) 108.607 0.00890109
\(531\) 0 0
\(532\) 3671.22 0.299187
\(533\) 0 0
\(534\) 0 0
\(535\) 2383.48 0.192611
\(536\) −5601.97 −0.451433
\(537\) 0 0
\(538\) 28380.7 2.27431
\(539\) −6006.02 −0.479958
\(540\) 0 0
\(541\) 9640.89 0.766163 0.383082 0.923714i \(-0.374863\pi\)
0.383082 + 0.923714i \(0.374863\pi\)
\(542\) −25938.9 −2.05567
\(543\) 0 0
\(544\) 17158.4 1.35232
\(545\) 1066.61 0.0838325
\(546\) 0 0
\(547\) 2378.17 0.185892 0.0929461 0.995671i \(-0.470372\pi\)
0.0929461 + 0.995671i \(0.470372\pi\)
\(548\) −16488.2 −1.28529
\(549\) 0 0
\(550\) −10274.2 −0.796530
\(551\) 15803.2 1.22185
\(552\) 0 0
\(553\) −653.430 −0.0502471
\(554\) 32484.0 2.49118
\(555\) 0 0
\(556\) −20872.4 −1.59207
\(557\) −6200.98 −0.471712 −0.235856 0.971788i \(-0.575789\pi\)
−0.235856 + 0.971788i \(0.575789\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −7.63451 −0.000576102 0
\(561\) 0 0
\(562\) −21509.8 −1.61448
\(563\) 14559.7 1.08991 0.544953 0.838467i \(-0.316548\pi\)
0.544953 + 0.838467i \(0.316548\pi\)
\(564\) 0 0
\(565\) 1331.42 0.0991387
\(566\) −16164.3 −1.20041
\(567\) 0 0
\(568\) 6569.72 0.485316
\(569\) 15706.3 1.15719 0.578595 0.815615i \(-0.303601\pi\)
0.578595 + 0.815615i \(0.303601\pi\)
\(570\) 0 0
\(571\) 17788.6 1.30373 0.651864 0.758336i \(-0.273987\pi\)
0.651864 + 0.758336i \(0.273987\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2294.40 −0.166841
\(575\) −904.428 −0.0655952
\(576\) 0 0
\(577\) −16136.7 −1.16426 −0.582132 0.813095i \(-0.697781\pi\)
−0.582132 + 0.813095i \(0.697781\pi\)
\(578\) 17191.0 1.23712
\(579\) 0 0
\(580\) 3879.41 0.277731
\(581\) −5092.17 −0.363613
\(582\) 0 0
\(583\) −305.474 −0.0217006
\(584\) −19846.3 −1.40625
\(585\) 0 0
\(586\) −23628.3 −1.66566
\(587\) 2446.59 0.172030 0.0860150 0.996294i \(-0.472587\pi\)
0.0860150 + 0.996294i \(0.472587\pi\)
\(588\) 0 0
\(589\) −13701.0 −0.958473
\(590\) 2901.03 0.202430
\(591\) 0 0
\(592\) 405.364 0.0281425
\(593\) 1730.14 0.119811 0.0599057 0.998204i \(-0.480920\pi\)
0.0599057 + 0.998204i \(0.480920\pi\)
\(594\) 0 0
\(595\) −507.163 −0.0349440
\(596\) −38799.3 −2.66658
\(597\) 0 0
\(598\) 0 0
\(599\) 23597.1 1.60960 0.804800 0.593546i \(-0.202273\pi\)
0.804800 + 0.593546i \(0.202273\pi\)
\(600\) 0 0
\(601\) −20907.8 −1.41905 −0.709525 0.704681i \(-0.751090\pi\)
−0.709525 + 0.704681i \(0.751090\pi\)
\(602\) −6892.96 −0.466671
\(603\) 0 0
\(604\) 44419.8 2.99241
\(605\) 1418.57 0.0953273
\(606\) 0 0
\(607\) 1496.15 0.100045 0.0500223 0.998748i \(-0.484071\pi\)
0.0500223 + 0.998748i \(0.484071\pi\)
\(608\) 13743.6 0.916735
\(609\) 0 0
\(610\) −3093.05 −0.205302
\(611\) 0 0
\(612\) 0 0
\(613\) 9923.66 0.653855 0.326927 0.945049i \(-0.393987\pi\)
0.326927 + 0.945049i \(0.393987\pi\)
\(614\) −40668.1 −2.67302
\(615\) 0 0
\(616\) −1554.73 −0.101691
\(617\) −19544.5 −1.27525 −0.637626 0.770346i \(-0.720083\pi\)
−0.637626 + 0.770346i \(0.720083\pi\)
\(618\) 0 0
\(619\) 13948.1 0.905691 0.452846 0.891589i \(-0.350409\pi\)
0.452846 + 0.891589i \(0.350409\pi\)
\(620\) −3363.37 −0.217865
\(621\) 0 0
\(622\) 41170.1 2.65397
\(623\) 5738.84 0.369056
\(624\) 0 0
\(625\) 14869.1 0.951624
\(626\) 28217.0 1.80156
\(627\) 0 0
\(628\) −8201.64 −0.521149
\(629\) 26928.5 1.70701
\(630\) 0 0
\(631\) −25953.1 −1.63736 −0.818681 0.574249i \(-0.805294\pi\)
−0.818681 + 0.574249i \(0.805294\pi\)
\(632\) 3797.14 0.238991
\(633\) 0 0
\(634\) 6965.12 0.436309
\(635\) −1970.16 −0.123124
\(636\) 0 0
\(637\) 0 0
\(638\) −17696.4 −1.09813
\(639\) 0 0
\(640\) 3300.87 0.203873
\(641\) −3214.66 −0.198083 −0.0990417 0.995083i \(-0.531578\pi\)
−0.0990417 + 0.995083i \(0.531578\pi\)
\(642\) 0 0
\(643\) −1996.07 −0.122422 −0.0612111 0.998125i \(-0.519496\pi\)
−0.0612111 + 0.998125i \(0.519496\pi\)
\(644\) −361.892 −0.0221437
\(645\) 0 0
\(646\) 31745.3 1.93344
\(647\) −5539.40 −0.336594 −0.168297 0.985736i \(-0.553827\pi\)
−0.168297 + 0.985736i \(0.553827\pi\)
\(648\) 0 0
\(649\) −8159.63 −0.493518
\(650\) 0 0
\(651\) 0 0
\(652\) −9394.63 −0.564298
\(653\) −15628.0 −0.936553 −0.468277 0.883582i \(-0.655125\pi\)
−0.468277 + 0.883582i \(0.655125\pi\)
\(654\) 0 0
\(655\) 1572.62 0.0938126
\(656\) −184.150 −0.0109601
\(657\) 0 0
\(658\) −3513.68 −0.208173
\(659\) 17229.9 1.01849 0.509244 0.860622i \(-0.329925\pi\)
0.509244 + 0.860622i \(0.329925\pi\)
\(660\) 0 0
\(661\) −19010.9 −1.11867 −0.559333 0.828943i \(-0.688943\pi\)
−0.559333 + 0.828943i \(0.688943\pi\)
\(662\) −39110.6 −2.29619
\(663\) 0 0
\(664\) 29591.1 1.72945
\(665\) −406.228 −0.0236885
\(666\) 0 0
\(667\) −1557.80 −0.0904324
\(668\) 12879.0 0.745961
\(669\) 0 0
\(670\) 1639.07 0.0945117
\(671\) 8699.71 0.500519
\(672\) 0 0
\(673\) −18524.2 −1.06100 −0.530501 0.847684i \(-0.677996\pi\)
−0.530501 + 0.847684i \(0.677996\pi\)
\(674\) −21009.5 −1.20067
\(675\) 0 0
\(676\) 0 0
\(677\) −27599.1 −1.56679 −0.783397 0.621522i \(-0.786515\pi\)
−0.783397 + 0.621522i \(0.786515\pi\)
\(678\) 0 0
\(679\) 1806.38 0.102095
\(680\) 2947.17 0.166204
\(681\) 0 0
\(682\) 15342.4 0.861425
\(683\) −26301.5 −1.47350 −0.736749 0.676167i \(-0.763640\pi\)
−0.736749 + 0.676167i \(0.763640\pi\)
\(684\) 0 0
\(685\) 1824.45 0.101765
\(686\) 11729.2 0.652801
\(687\) 0 0
\(688\) −553.232 −0.0306566
\(689\) 0 0
\(690\) 0 0
\(691\) −131.528 −0.00724106 −0.00362053 0.999993i \(-0.501152\pi\)
−0.00362053 + 0.999993i \(0.501152\pi\)
\(692\) −22799.3 −1.25245
\(693\) 0 0
\(694\) 35854.8 1.96114
\(695\) 2309.58 0.126054
\(696\) 0 0
\(697\) −12233.1 −0.664796
\(698\) −27359.9 −1.48365
\(699\) 0 0
\(700\) 6051.01 0.326724
\(701\) −11604.4 −0.625239 −0.312620 0.949878i \(-0.601207\pi\)
−0.312620 + 0.949878i \(0.601207\pi\)
\(702\) 0 0
\(703\) 21569.2 1.15718
\(704\) −15184.9 −0.812929
\(705\) 0 0
\(706\) 29348.8 1.56453
\(707\) −3284.80 −0.174735
\(708\) 0 0
\(709\) 2046.79 0.108419 0.0542094 0.998530i \(-0.482736\pi\)
0.0542094 + 0.998530i \(0.482736\pi\)
\(710\) −1922.22 −0.101605
\(711\) 0 0
\(712\) −33348.9 −1.75534
\(713\) 1350.58 0.0709394
\(714\) 0 0
\(715\) 0 0
\(716\) 49012.2 2.55820
\(717\) 0 0
\(718\) −27759.6 −1.44287
\(719\) −34019.7 −1.76456 −0.882282 0.470721i \(-0.843994\pi\)
−0.882282 + 0.470721i \(0.843994\pi\)
\(720\) 0 0
\(721\) 66.8906 0.00345511
\(722\) −5903.69 −0.304311
\(723\) 0 0
\(724\) 26062.8 1.33787
\(725\) 26047.2 1.33430
\(726\) 0 0
\(727\) 10854.7 0.553754 0.276877 0.960905i \(-0.410701\pi\)
0.276877 + 0.960905i \(0.410701\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 5806.81 0.294410
\(731\) −36751.4 −1.85951
\(732\) 0 0
\(733\) 32912.5 1.65846 0.829229 0.558910i \(-0.188780\pi\)
0.829229 + 0.558910i \(0.188780\pi\)
\(734\) −32757.3 −1.64727
\(735\) 0 0
\(736\) −1354.78 −0.0678503
\(737\) −4610.15 −0.230417
\(738\) 0 0
\(739\) −26680.8 −1.32810 −0.664052 0.747687i \(-0.731164\pi\)
−0.664052 + 0.747687i \(0.731164\pi\)
\(740\) 5294.88 0.263032
\(741\) 0 0
\(742\) 291.781 0.0144362
\(743\) 31128.8 1.53702 0.768509 0.639838i \(-0.220999\pi\)
0.768509 + 0.639838i \(0.220999\pi\)
\(744\) 0 0
\(745\) 4293.23 0.211130
\(746\) −1477.02 −0.0724898
\(747\) 0 0
\(748\) −21918.9 −1.07144
\(749\) 6403.42 0.312384
\(750\) 0 0
\(751\) −26246.3 −1.27529 −0.637644 0.770331i \(-0.720091\pi\)
−0.637644 + 0.770331i \(0.720091\pi\)
\(752\) −282.010 −0.0136753
\(753\) 0 0
\(754\) 0 0
\(755\) −4915.15 −0.236928
\(756\) 0 0
\(757\) 7414.10 0.355971 0.177986 0.984033i \(-0.443042\pi\)
0.177986 + 0.984033i \(0.443042\pi\)
\(758\) −12693.8 −0.608257
\(759\) 0 0
\(760\) 2360.63 0.112670
\(761\) −13424.0 −0.639448 −0.319724 0.947511i \(-0.603590\pi\)
−0.319724 + 0.947511i \(0.603590\pi\)
\(762\) 0 0
\(763\) 2865.55 0.135963
\(764\) 12595.5 0.596450
\(765\) 0 0
\(766\) −45939.9 −2.16694
\(767\) 0 0
\(768\) 0 0
\(769\) −11201.4 −0.525269 −0.262634 0.964895i \(-0.584591\pi\)
−0.262634 + 0.964895i \(0.584591\pi\)
\(770\) 454.895 0.0212900
\(771\) 0 0
\(772\) 8331.44 0.388413
\(773\) 7544.68 0.351052 0.175526 0.984475i \(-0.443837\pi\)
0.175526 + 0.984475i \(0.443837\pi\)
\(774\) 0 0
\(775\) −22582.4 −1.04669
\(776\) −10497.1 −0.485596
\(777\) 0 0
\(778\) 51704.9 2.38266
\(779\) −9798.50 −0.450665
\(780\) 0 0
\(781\) 5406.56 0.247710
\(782\) −3129.31 −0.143100
\(783\) 0 0
\(784\) 460.438 0.0209748
\(785\) 907.530 0.0412626
\(786\) 0 0
\(787\) −21988.8 −0.995952 −0.497976 0.867191i \(-0.665923\pi\)
−0.497976 + 0.867191i \(0.665923\pi\)
\(788\) 43203.7 1.95313
\(789\) 0 0
\(790\) −1111.00 −0.0500348
\(791\) 3576.98 0.160787
\(792\) 0 0
\(793\) 0 0
\(794\) −56000.0 −2.50298
\(795\) 0 0
\(796\) −40894.6 −1.82094
\(797\) 18599.4 0.826632 0.413316 0.910588i \(-0.364370\pi\)
0.413316 + 0.910588i \(0.364370\pi\)
\(798\) 0 0
\(799\) −18734.0 −0.829488
\(800\) 22652.5 1.00111
\(801\) 0 0
\(802\) 51170.2 2.25297
\(803\) −16332.6 −0.717764
\(804\) 0 0
\(805\) 40.0442 0.00175326
\(806\) 0 0
\(807\) 0 0
\(808\) 19088.3 0.831093
\(809\) 17943.9 0.779818 0.389909 0.920853i \(-0.372506\pi\)
0.389909 + 0.920853i \(0.372506\pi\)
\(810\) 0 0
\(811\) −26483.5 −1.14668 −0.573342 0.819316i \(-0.694353\pi\)
−0.573342 + 0.819316i \(0.694353\pi\)
\(812\) 10422.4 0.450436
\(813\) 0 0
\(814\) −24153.2 −1.04001
\(815\) 1039.54 0.0446790
\(816\) 0 0
\(817\) −29437.2 −1.26056
\(818\) −3516.81 −0.150321
\(819\) 0 0
\(820\) −2405.37 −0.102438
\(821\) 5337.82 0.226907 0.113454 0.993543i \(-0.463809\pi\)
0.113454 + 0.993543i \(0.463809\pi\)
\(822\) 0 0
\(823\) −18329.3 −0.776330 −0.388165 0.921590i \(-0.626891\pi\)
−0.388165 + 0.921590i \(0.626891\pi\)
\(824\) −388.707 −0.0164336
\(825\) 0 0
\(826\) 7793.88 0.328309
\(827\) 22310.2 0.938092 0.469046 0.883174i \(-0.344598\pi\)
0.469046 + 0.883174i \(0.344598\pi\)
\(828\) 0 0
\(829\) −15181.1 −0.636022 −0.318011 0.948087i \(-0.603015\pi\)
−0.318011 + 0.948087i \(0.603015\pi\)
\(830\) −8658.00 −0.362076
\(831\) 0 0
\(832\) 0 0
\(833\) 30587.1 1.27224
\(834\) 0 0
\(835\) −1425.08 −0.0590624
\(836\) −17556.7 −0.726327
\(837\) 0 0
\(838\) 37961.3 1.56486
\(839\) 20959.7 0.862467 0.431234 0.902240i \(-0.358078\pi\)
0.431234 + 0.902240i \(0.358078\pi\)
\(840\) 0 0
\(841\) 20475.2 0.839527
\(842\) −15716.7 −0.643272
\(843\) 0 0
\(844\) 10157.1 0.414242
\(845\) 0 0
\(846\) 0 0
\(847\) 3811.11 0.154606
\(848\) 23.4185 0.000948342 0
\(849\) 0 0
\(850\) 52323.5 2.11139
\(851\) −2126.20 −0.0856464
\(852\) 0 0
\(853\) −44482.9 −1.78554 −0.892769 0.450515i \(-0.851240\pi\)
−0.892769 + 0.450515i \(0.851240\pi\)
\(854\) −8309.75 −0.332967
\(855\) 0 0
\(856\) −37210.8 −1.48580
\(857\) 17206.5 0.685836 0.342918 0.939365i \(-0.388585\pi\)
0.342918 + 0.939365i \(0.388585\pi\)
\(858\) 0 0
\(859\) 3736.65 0.148420 0.0742101 0.997243i \(-0.476356\pi\)
0.0742101 + 0.997243i \(0.476356\pi\)
\(860\) −7226.34 −0.286530
\(861\) 0 0
\(862\) 44494.5 1.75811
\(863\) 7263.00 0.286483 0.143242 0.989688i \(-0.454247\pi\)
0.143242 + 0.989688i \(0.454247\pi\)
\(864\) 0 0
\(865\) 2522.79 0.0991646
\(866\) −74294.1 −2.91526
\(867\) 0 0
\(868\) −9035.99 −0.353343
\(869\) 3124.86 0.121983
\(870\) 0 0
\(871\) 0 0
\(872\) −16652.0 −0.646682
\(873\) 0 0
\(874\) −2506.52 −0.0970073
\(875\) −1350.15 −0.0521638
\(876\) 0 0
\(877\) −9024.17 −0.347463 −0.173731 0.984793i \(-0.555582\pi\)
−0.173731 + 0.984793i \(0.555582\pi\)
\(878\) −49947.8 −1.91988
\(879\) 0 0
\(880\) 36.5101 0.00139858
\(881\) −9585.56 −0.366567 −0.183284 0.983060i \(-0.558673\pi\)
−0.183284 + 0.983060i \(0.558673\pi\)
\(882\) 0 0
\(883\) −7138.33 −0.272054 −0.136027 0.990705i \(-0.543433\pi\)
−0.136027 + 0.990705i \(0.543433\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −77068.3 −2.92230
\(887\) 34358.1 1.30060 0.650299 0.759678i \(-0.274644\pi\)
0.650299 + 0.759678i \(0.274644\pi\)
\(888\) 0 0
\(889\) −5293.01 −0.199687
\(890\) 9757.50 0.367497
\(891\) 0 0
\(892\) 30221.1 1.13439
\(893\) −15005.6 −0.562310
\(894\) 0 0
\(895\) −5423.30 −0.202549
\(896\) 8868.07 0.330649
\(897\) 0 0
\(898\) −2704.43 −0.100499
\(899\) −38896.4 −1.44301
\(900\) 0 0
\(901\) 1555.70 0.0575226
\(902\) 10972.4 0.405034
\(903\) 0 0
\(904\) −20786.1 −0.764753
\(905\) −2883.91 −0.105927
\(906\) 0 0
\(907\) −13184.2 −0.482661 −0.241330 0.970443i \(-0.577584\pi\)
−0.241330 + 0.970443i \(0.577584\pi\)
\(908\) 1602.00 0.0585508
\(909\) 0 0
\(910\) 0 0
\(911\) 10953.7 0.398368 0.199184 0.979962i \(-0.436171\pi\)
0.199184 + 0.979962i \(0.436171\pi\)
\(912\) 0 0
\(913\) 24352.0 0.882732
\(914\) 67380.2 2.43844
\(915\) 0 0
\(916\) −11228.3 −0.405016
\(917\) 4224.97 0.152149
\(918\) 0 0
\(919\) 16852.4 0.604907 0.302453 0.953164i \(-0.402194\pi\)
0.302453 + 0.953164i \(0.402194\pi\)
\(920\) −232.700 −0.00833902
\(921\) 0 0
\(922\) −10088.7 −0.360361
\(923\) 0 0
\(924\) 0 0
\(925\) 35551.0 1.26369
\(926\) 18025.3 0.639687
\(927\) 0 0
\(928\) 39017.2 1.38017
\(929\) 20981.5 0.740993 0.370496 0.928834i \(-0.379188\pi\)
0.370496 + 0.928834i \(0.379188\pi\)
\(930\) 0 0
\(931\) 24499.7 0.862453
\(932\) 70774.1 2.48743
\(933\) 0 0
\(934\) 62459.8 2.18817
\(935\) 2425.38 0.0848324
\(936\) 0 0
\(937\) 21492.1 0.749323 0.374662 0.927162i \(-0.377759\pi\)
0.374662 + 0.927162i \(0.377759\pi\)
\(938\) 4403.50 0.153283
\(939\) 0 0
\(940\) −3683.62 −0.127815
\(941\) −26577.9 −0.920740 −0.460370 0.887727i \(-0.652283\pi\)
−0.460370 + 0.887727i \(0.652283\pi\)
\(942\) 0 0
\(943\) 965.893 0.0333550
\(944\) 625.539 0.0215673
\(945\) 0 0
\(946\) 32963.8 1.13292
\(947\) −7432.76 −0.255050 −0.127525 0.991835i \(-0.540703\pi\)
−0.127525 + 0.991835i \(0.540703\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 41910.2 1.43131
\(951\) 0 0
\(952\) 7917.82 0.269557
\(953\) 27981.8 0.951121 0.475561 0.879683i \(-0.342245\pi\)
0.475561 + 0.879683i \(0.342245\pi\)
\(954\) 0 0
\(955\) −1393.71 −0.0472246
\(956\) 51922.0 1.75657
\(957\) 0 0
\(958\) −63449.9 −2.13985
\(959\) 4901.54 0.165046
\(960\) 0 0
\(961\) 3931.36 0.131965
\(962\) 0 0
\(963\) 0 0
\(964\) 42351.2 1.41498
\(965\) −921.892 −0.0307531
\(966\) 0 0
\(967\) −43777.1 −1.45582 −0.727910 0.685673i \(-0.759508\pi\)
−0.727910 + 0.685673i \(0.759508\pi\)
\(968\) −22146.7 −0.735352
\(969\) 0 0
\(970\) 3071.32 0.101664
\(971\) −4722.08 −0.156065 −0.0780323 0.996951i \(-0.524864\pi\)
−0.0780323 + 0.996951i \(0.524864\pi\)
\(972\) 0 0
\(973\) 6204.88 0.204439
\(974\) 48020.5 1.57975
\(975\) 0 0
\(976\) −666.944 −0.0218733
\(977\) −22867.5 −0.748821 −0.374410 0.927263i \(-0.622155\pi\)
−0.374410 + 0.927263i \(0.622155\pi\)
\(978\) 0 0
\(979\) −27444.5 −0.895946
\(980\) 6014.26 0.196039
\(981\) 0 0
\(982\) 52017.0 1.69035
\(983\) −51887.9 −1.68359 −0.841794 0.539799i \(-0.818500\pi\)
−0.841794 + 0.539799i \(0.818500\pi\)
\(984\) 0 0
\(985\) −4780.58 −0.154642
\(986\) 90123.1 2.91086
\(987\) 0 0
\(988\) 0 0
\(989\) 2901.79 0.0932977
\(990\) 0 0
\(991\) −26676.4 −0.855099 −0.427550 0.903992i \(-0.640623\pi\)
−0.427550 + 0.903992i \(0.640623\pi\)
\(992\) −33827.1 −1.08267
\(993\) 0 0
\(994\) −5164.21 −0.164788
\(995\) 4525.08 0.144176
\(996\) 0 0
\(997\) −20542.0 −0.652529 −0.326264 0.945279i \(-0.605790\pi\)
−0.326264 + 0.945279i \(0.605790\pi\)
\(998\) −57849.8 −1.83487
\(999\) 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.bm.1.16 yes 18
3.2 odd 2 inner 1521.4.a.bm.1.3 18
13.12 even 2 1521.4.a.bn.1.3 yes 18
39.38 odd 2 1521.4.a.bn.1.16 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1521.4.a.bm.1.3 18 3.2 odd 2 inner
1521.4.a.bm.1.16 yes 18 1.1 even 1 trivial
1521.4.a.bn.1.3 yes 18 13.12 even 2
1521.4.a.bn.1.16 yes 18 39.38 odd 2