Properties

Label 1521.4.a.u.1.3
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.20905\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.20905 q^{2} +9.71610 q^{4} -11.4322 q^{5} +11.2543 q^{7} +7.22315 q^{8} +O(q^{10})\) \(q+4.20905 q^{2} +9.71610 q^{4} -11.4322 q^{5} +11.2543 q^{7} +7.22315 q^{8} -48.1187 q^{10} +25.8785 q^{11} +47.3699 q^{14} -47.3262 q^{16} +20.3276 q^{17} -154.712 q^{19} -111.076 q^{20} +108.924 q^{22} +180.418 q^{23} +5.69520 q^{25} +109.348 q^{28} +20.4522 q^{29} -266.424 q^{31} -256.984 q^{32} +85.5599 q^{34} -128.661 q^{35} -115.984 q^{37} -651.190 q^{38} -82.5765 q^{40} +391.184 q^{41} +151.407 q^{43} +251.438 q^{44} +759.390 q^{46} -467.365 q^{47} -216.341 q^{49} +23.9714 q^{50} -79.9842 q^{53} -295.848 q^{55} +81.2915 q^{56} +86.0843 q^{58} -873.710 q^{59} -187.068 q^{61} -1121.39 q^{62} -703.047 q^{64} +609.204 q^{67} +197.505 q^{68} -541.542 q^{70} +248.038 q^{71} -852.765 q^{73} -488.181 q^{74} -1503.20 q^{76} +291.244 q^{77} -331.221 q^{79} +541.043 q^{80} +1646.51 q^{82} -435.432 q^{83} -232.389 q^{85} +637.281 q^{86} +186.924 q^{88} +259.233 q^{89} +1752.96 q^{92} -1967.16 q^{94} +1768.70 q^{95} -1225.17 q^{97} -910.589 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8} - 4 q^{10} - 16 q^{11} + 176 q^{14} - 110 q^{16} + 146 q^{17} - 94 q^{19} - 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 80 q^{28} + 2 q^{29} - 302 q^{31} + 154 q^{32} - 164 q^{34} - 80 q^{35} - 374 q^{37} - 312 q^{38} - 516 q^{40} + 480 q^{41} - 260 q^{43} + 712 q^{44} + 1104 q^{46} - 24 q^{47} + 447 q^{49} + 814 q^{50} + 678 q^{53} - 1552 q^{55} - 96 q^{56} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 74 q^{67} + 460 q^{68} - 1216 q^{70} - 948 q^{71} + 222 q^{73} - 1724 q^{74} - 2392 q^{76} - 112 q^{77} - 24 q^{79} + 1100 q^{80} + 564 q^{82} - 796 q^{83} + 248 q^{85} + 1800 q^{86} + 1608 q^{88} + 1436 q^{89} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} - 3242 q^{97} - 5070 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.20905 1.48812 0.744062 0.668111i \(-0.232897\pi\)
0.744062 + 0.668111i \(0.232897\pi\)
\(3\) 0 0
\(4\) 9.71610 1.21451
\(5\) −11.4322 −1.02253 −0.511264 0.859424i \(-0.670822\pi\)
−0.511264 + 0.859424i \(0.670822\pi\)
\(6\) 0 0
\(7\) 11.2543 0.607675 0.303838 0.952724i \(-0.401732\pi\)
0.303838 + 0.952724i \(0.401732\pi\)
\(8\) 7.22315 0.319221
\(9\) 0 0
\(10\) −48.1187 −1.52165
\(11\) 25.8785 0.709333 0.354666 0.934993i \(-0.384594\pi\)
0.354666 + 0.934993i \(0.384594\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 47.3699 0.904296
\(15\) 0 0
\(16\) −47.3262 −0.739472
\(17\) 20.3276 0.290010 0.145005 0.989431i \(-0.453680\pi\)
0.145005 + 0.989431i \(0.453680\pi\)
\(18\) 0 0
\(19\) −154.712 −1.86807 −0.934035 0.357181i \(-0.883738\pi\)
−0.934035 + 0.357181i \(0.883738\pi\)
\(20\) −111.076 −1.24187
\(21\) 0 0
\(22\) 108.924 1.05558
\(23\) 180.418 1.63565 0.817823 0.575471i \(-0.195181\pi\)
0.817823 + 0.575471i \(0.195181\pi\)
\(24\) 0 0
\(25\) 5.69520 0.0455616
\(26\) 0 0
\(27\) 0 0
\(28\) 109.348 0.738029
\(29\) 20.4522 0.130961 0.0654806 0.997854i \(-0.479142\pi\)
0.0654806 + 0.997854i \(0.479142\pi\)
\(30\) 0 0
\(31\) −266.424 −1.54359 −0.771794 0.635873i \(-0.780640\pi\)
−0.771794 + 0.635873i \(0.780640\pi\)
\(32\) −256.984 −1.41965
\(33\) 0 0
\(34\) 85.5599 0.431571
\(35\) −128.661 −0.621364
\(36\) 0 0
\(37\) −115.984 −0.515340 −0.257670 0.966233i \(-0.582955\pi\)
−0.257670 + 0.966233i \(0.582955\pi\)
\(38\) −651.190 −2.77992
\(39\) 0 0
\(40\) −82.5765 −0.326412
\(41\) 391.184 1.49006 0.745032 0.667029i \(-0.232434\pi\)
0.745032 + 0.667029i \(0.232434\pi\)
\(42\) 0 0
\(43\) 151.407 0.536963 0.268482 0.963285i \(-0.413478\pi\)
0.268482 + 0.963285i \(0.413478\pi\)
\(44\) 251.438 0.861494
\(45\) 0 0
\(46\) 759.390 2.43404
\(47\) −467.365 −1.45047 −0.725236 0.688500i \(-0.758269\pi\)
−0.725236 + 0.688500i \(0.758269\pi\)
\(48\) 0 0
\(49\) −216.341 −0.630731
\(50\) 23.9714 0.0678012
\(51\) 0 0
\(52\) 0 0
\(53\) −79.9842 −0.207296 −0.103648 0.994614i \(-0.533051\pi\)
−0.103648 + 0.994614i \(0.533051\pi\)
\(54\) 0 0
\(55\) −295.848 −0.725312
\(56\) 81.2915 0.193983
\(57\) 0 0
\(58\) 86.0843 0.194887
\(59\) −873.710 −1.92792 −0.963960 0.266045i \(-0.914283\pi\)
−0.963960 + 0.266045i \(0.914283\pi\)
\(60\) 0 0
\(61\) −187.068 −0.392649 −0.196325 0.980539i \(-0.562901\pi\)
−0.196325 + 0.980539i \(0.562901\pi\)
\(62\) −1121.39 −2.29705
\(63\) 0 0
\(64\) −703.047 −1.37314
\(65\) 0 0
\(66\) 0 0
\(67\) 609.204 1.11084 0.555418 0.831571i \(-0.312558\pi\)
0.555418 + 0.831571i \(0.312558\pi\)
\(68\) 197.505 0.352221
\(69\) 0 0
\(70\) −541.542 −0.924667
\(71\) 248.038 0.414601 0.207301 0.978277i \(-0.433532\pi\)
0.207301 + 0.978277i \(0.433532\pi\)
\(72\) 0 0
\(73\) −852.765 −1.36724 −0.683621 0.729838i \(-0.739596\pi\)
−0.683621 + 0.729838i \(0.739596\pi\)
\(74\) −488.181 −0.766890
\(75\) 0 0
\(76\) −1503.20 −2.26880
\(77\) 291.244 0.431044
\(78\) 0 0
\(79\) −331.221 −0.471712 −0.235856 0.971788i \(-0.575789\pi\)
−0.235856 + 0.971788i \(0.575789\pi\)
\(80\) 541.043 0.756130
\(81\) 0 0
\(82\) 1646.51 2.21740
\(83\) −435.432 −0.575842 −0.287921 0.957654i \(-0.592964\pi\)
−0.287921 + 0.957654i \(0.592964\pi\)
\(84\) 0 0
\(85\) −232.389 −0.296543
\(86\) 637.281 0.799067
\(87\) 0 0
\(88\) 186.924 0.226434
\(89\) 259.233 0.308749 0.154375 0.988012i \(-0.450664\pi\)
0.154375 + 0.988012i \(0.450664\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1752.96 1.98651
\(93\) 0 0
\(94\) −1967.16 −2.15848
\(95\) 1768.70 1.91015
\(96\) 0 0
\(97\) −1225.17 −1.28245 −0.641223 0.767355i \(-0.721572\pi\)
−0.641223 + 0.767355i \(0.721572\pi\)
\(98\) −910.589 −0.938606
\(99\) 0 0
\(100\) 55.3351 0.0553351
\(101\) −645.416 −0.635855 −0.317927 0.948115i \(-0.602987\pi\)
−0.317927 + 0.948115i \(0.602987\pi\)
\(102\) 0 0
\(103\) −511.137 −0.488969 −0.244484 0.969653i \(-0.578619\pi\)
−0.244484 + 0.969653i \(0.578619\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −336.657 −0.308482
\(107\) −608.195 −0.549499 −0.274750 0.961516i \(-0.588595\pi\)
−0.274750 + 0.961516i \(0.588595\pi\)
\(108\) 0 0
\(109\) 1300.04 1.14239 0.571197 0.820813i \(-0.306479\pi\)
0.571197 + 0.820813i \(0.306479\pi\)
\(110\) −1245.24 −1.07935
\(111\) 0 0
\(112\) −532.623 −0.449359
\(113\) −42.1953 −0.0351274 −0.0175637 0.999846i \(-0.505591\pi\)
−0.0175637 + 0.999846i \(0.505591\pi\)
\(114\) 0 0
\(115\) −2062.58 −1.67249
\(116\) 198.716 0.159054
\(117\) 0 0
\(118\) −3677.49 −2.86899
\(119\) 228.773 0.176232
\(120\) 0 0
\(121\) −661.303 −0.496847
\(122\) −787.378 −0.584311
\(123\) 0 0
\(124\) −2588.61 −1.87471
\(125\) 1363.92 0.975939
\(126\) 0 0
\(127\) −311.018 −0.217310 −0.108655 0.994080i \(-0.534654\pi\)
−0.108655 + 0.994080i \(0.534654\pi\)
\(128\) −903.291 −0.623753
\(129\) 0 0
\(130\) 0 0
\(131\) −2000.98 −1.33456 −0.667278 0.744809i \(-0.732541\pi\)
−0.667278 + 0.744809i \(0.732541\pi\)
\(132\) 0 0
\(133\) −1741.17 −1.13518
\(134\) 2564.17 1.65306
\(135\) 0 0
\(136\) 146.829 0.0925773
\(137\) 1038.53 0.647644 0.323822 0.946118i \(-0.395032\pi\)
0.323822 + 0.946118i \(0.395032\pi\)
\(138\) 0 0
\(139\) −2858.46 −1.74426 −0.872128 0.489277i \(-0.837261\pi\)
−0.872128 + 0.489277i \(0.837261\pi\)
\(140\) −1250.09 −0.754655
\(141\) 0 0
\(142\) 1044.00 0.616978
\(143\) 0 0
\(144\) 0 0
\(145\) −233.814 −0.133911
\(146\) −3589.33 −2.03462
\(147\) 0 0
\(148\) −1126.91 −0.625887
\(149\) 743.479 0.408780 0.204390 0.978890i \(-0.434479\pi\)
0.204390 + 0.978890i \(0.434479\pi\)
\(150\) 0 0
\(151\) −2277.24 −1.22728 −0.613640 0.789586i \(-0.710295\pi\)
−0.613640 + 0.789586i \(0.710295\pi\)
\(152\) −1117.51 −0.596328
\(153\) 0 0
\(154\) 1225.86 0.641447
\(155\) 3045.82 1.57836
\(156\) 0 0
\(157\) 3173.51 1.61321 0.806605 0.591091i \(-0.201303\pi\)
0.806605 + 0.591091i \(0.201303\pi\)
\(158\) −1394.12 −0.701966
\(159\) 0 0
\(160\) 2937.89 1.45163
\(161\) 2030.48 0.993941
\(162\) 0 0
\(163\) 2314.65 1.11225 0.556126 0.831098i \(-0.312287\pi\)
0.556126 + 0.831098i \(0.312287\pi\)
\(164\) 3800.78 1.80970
\(165\) 0 0
\(166\) −1832.76 −0.856925
\(167\) −2665.65 −1.23517 −0.617587 0.786502i \(-0.711890\pi\)
−0.617587 + 0.786502i \(0.711890\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −978.138 −0.441293
\(171\) 0 0
\(172\) 1471.09 0.652148
\(173\) 165.243 0.0726198 0.0363099 0.999341i \(-0.488440\pi\)
0.0363099 + 0.999341i \(0.488440\pi\)
\(174\) 0 0
\(175\) 64.0954 0.0276866
\(176\) −1224.73 −0.524532
\(177\) 0 0
\(178\) 1091.13 0.459457
\(179\) −712.339 −0.297446 −0.148723 0.988879i \(-0.547516\pi\)
−0.148723 + 0.988879i \(0.547516\pi\)
\(180\) 0 0
\(181\) 2206.53 0.906133 0.453066 0.891477i \(-0.350330\pi\)
0.453066 + 0.891477i \(0.350330\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1303.19 0.522132
\(185\) 1325.95 0.526949
\(186\) 0 0
\(187\) 526.048 0.205714
\(188\) −4540.96 −1.76162
\(189\) 0 0
\(190\) 7444.54 2.84254
\(191\) −1470.64 −0.557129 −0.278565 0.960417i \(-0.589859\pi\)
−0.278565 + 0.960417i \(0.589859\pi\)
\(192\) 0 0
\(193\) −369.560 −0.137832 −0.0689158 0.997622i \(-0.521954\pi\)
−0.0689158 + 0.997622i \(0.521954\pi\)
\(194\) −5156.80 −1.90844
\(195\) 0 0
\(196\) −2101.99 −0.766031
\(197\) −4273.41 −1.54552 −0.772761 0.634697i \(-0.781125\pi\)
−0.772761 + 0.634697i \(0.781125\pi\)
\(198\) 0 0
\(199\) 4154.31 1.47985 0.739927 0.672687i \(-0.234860\pi\)
0.739927 + 0.672687i \(0.234860\pi\)
\(200\) 41.1373 0.0145442
\(201\) 0 0
\(202\) −2716.59 −0.946230
\(203\) 230.175 0.0795819
\(204\) 0 0
\(205\) −4472.09 −1.52363
\(206\) −2151.40 −0.727646
\(207\) 0 0
\(208\) 0 0
\(209\) −4003.71 −1.32508
\(210\) 0 0
\(211\) 1231.59 0.401830 0.200915 0.979609i \(-0.435608\pi\)
0.200915 + 0.979609i \(0.435608\pi\)
\(212\) −777.134 −0.251763
\(213\) 0 0
\(214\) −2559.92 −0.817723
\(215\) −1730.92 −0.549059
\(216\) 0 0
\(217\) −2998.42 −0.938000
\(218\) 5471.92 1.70002
\(219\) 0 0
\(220\) −2874.49 −0.880901
\(221\) 0 0
\(222\) 0 0
\(223\) 2187.24 0.656809 0.328404 0.944537i \(-0.393489\pi\)
0.328404 + 0.944537i \(0.393489\pi\)
\(224\) −2892.17 −0.862684
\(225\) 0 0
\(226\) −177.602 −0.0522739
\(227\) 4138.67 1.21010 0.605051 0.796187i \(-0.293153\pi\)
0.605051 + 0.796187i \(0.293153\pi\)
\(228\) 0 0
\(229\) 835.354 0.241056 0.120528 0.992710i \(-0.461541\pi\)
0.120528 + 0.992710i \(0.461541\pi\)
\(230\) −8681.50 −2.48887
\(231\) 0 0
\(232\) 147.729 0.0418056
\(233\) −3685.51 −1.03625 −0.518124 0.855305i \(-0.673370\pi\)
−0.518124 + 0.855305i \(0.673370\pi\)
\(234\) 0 0
\(235\) 5343.01 1.48315
\(236\) −8489.05 −2.34148
\(237\) 0 0
\(238\) 962.917 0.262255
\(239\) 3026.21 0.819034 0.409517 0.912303i \(-0.365697\pi\)
0.409517 + 0.912303i \(0.365697\pi\)
\(240\) 0 0
\(241\) −3265.58 −0.872839 −0.436420 0.899743i \(-0.643754\pi\)
−0.436420 + 0.899743i \(0.643754\pi\)
\(242\) −2783.46 −0.739370
\(243\) 0 0
\(244\) −1817.57 −0.476877
\(245\) 2473.25 0.644940
\(246\) 0 0
\(247\) 0 0
\(248\) −1924.42 −0.492746
\(249\) 0 0
\(250\) 5740.79 1.45232
\(251\) 6363.16 1.60016 0.800078 0.599897i \(-0.204792\pi\)
0.800078 + 0.599897i \(0.204792\pi\)
\(252\) 0 0
\(253\) 4668.96 1.16022
\(254\) −1309.09 −0.323385
\(255\) 0 0
\(256\) 1822.38 0.444917
\(257\) 6085.36 1.47702 0.738511 0.674242i \(-0.235529\pi\)
0.738511 + 0.674242i \(0.235529\pi\)
\(258\) 0 0
\(259\) −1305.31 −0.313159
\(260\) 0 0
\(261\) 0 0
\(262\) −8422.24 −1.98598
\(263\) −123.227 −0.0288916 −0.0144458 0.999896i \(-0.504598\pi\)
−0.0144458 + 0.999896i \(0.504598\pi\)
\(264\) 0 0
\(265\) 914.395 0.211965
\(266\) −7328.69 −1.68929
\(267\) 0 0
\(268\) 5919.08 1.34913
\(269\) 1935.79 0.438763 0.219381 0.975639i \(-0.429596\pi\)
0.219381 + 0.975639i \(0.429596\pi\)
\(270\) 0 0
\(271\) 4612.69 1.03395 0.516976 0.856000i \(-0.327058\pi\)
0.516976 + 0.856000i \(0.327058\pi\)
\(272\) −962.028 −0.214454
\(273\) 0 0
\(274\) 4371.20 0.963774
\(275\) 147.383 0.0323183
\(276\) 0 0
\(277\) −5834.30 −1.26552 −0.632761 0.774347i \(-0.718078\pi\)
−0.632761 + 0.774347i \(0.718078\pi\)
\(278\) −12031.4 −2.59567
\(279\) 0 0
\(280\) −929.341 −0.198353
\(281\) 4691.91 0.996071 0.498036 0.867157i \(-0.334055\pi\)
0.498036 + 0.867157i \(0.334055\pi\)
\(282\) 0 0
\(283\) 3465.60 0.727945 0.363973 0.931410i \(-0.381420\pi\)
0.363973 + 0.931410i \(0.381420\pi\)
\(284\) 2409.96 0.503539
\(285\) 0 0
\(286\) 0 0
\(287\) 4402.50 0.905475
\(288\) 0 0
\(289\) −4499.79 −0.915894
\(290\) −984.133 −0.199277
\(291\) 0 0
\(292\) −8285.55 −1.66053
\(293\) 2677.31 0.533822 0.266911 0.963721i \(-0.413997\pi\)
0.266911 + 0.963721i \(0.413997\pi\)
\(294\) 0 0
\(295\) 9988.43 1.97135
\(296\) −837.767 −0.164507
\(297\) 0 0
\(298\) 3129.34 0.608315
\(299\) 0 0
\(300\) 0 0
\(301\) 1703.98 0.326299
\(302\) −9585.02 −1.82634
\(303\) 0 0
\(304\) 7321.93 1.38139
\(305\) 2138.60 0.401494
\(306\) 0 0
\(307\) −471.915 −0.0877316 −0.0438658 0.999037i \(-0.513967\pi\)
−0.0438658 + 0.999037i \(0.513967\pi\)
\(308\) 2829.76 0.523508
\(309\) 0 0
\(310\) 12820.0 2.34880
\(311\) 1518.52 0.276872 0.138436 0.990371i \(-0.455793\pi\)
0.138436 + 0.990371i \(0.455793\pi\)
\(312\) 0 0
\(313\) 4049.86 0.731348 0.365674 0.930743i \(-0.380839\pi\)
0.365674 + 0.930743i \(0.380839\pi\)
\(314\) 13357.5 2.40066
\(315\) 0 0
\(316\) −3218.17 −0.572900
\(317\) 3253.96 0.576532 0.288266 0.957550i \(-0.406921\pi\)
0.288266 + 0.957550i \(0.406921\pi\)
\(318\) 0 0
\(319\) 529.272 0.0928951
\(320\) 8037.37 1.40407
\(321\) 0 0
\(322\) 8546.40 1.47911
\(323\) −3144.92 −0.541759
\(324\) 0 0
\(325\) 0 0
\(326\) 9742.46 1.65517
\(327\) 0 0
\(328\) 2825.58 0.475660
\(329\) −5259.86 −0.881415
\(330\) 0 0
\(331\) 3422.45 0.568322 0.284161 0.958777i \(-0.408285\pi\)
0.284161 + 0.958777i \(0.408285\pi\)
\(332\) −4230.71 −0.699368
\(333\) 0 0
\(334\) −11219.8 −1.83809
\(335\) −6964.54 −1.13586
\(336\) 0 0
\(337\) −9301.67 −1.50354 −0.751772 0.659423i \(-0.770801\pi\)
−0.751772 + 0.659423i \(0.770801\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −2257.92 −0.360155
\(341\) −6894.66 −1.09492
\(342\) 0 0
\(343\) −6294.99 −0.990955
\(344\) 1093.64 0.171410
\(345\) 0 0
\(346\) 695.518 0.108067
\(347\) −216.898 −0.0335554 −0.0167777 0.999859i \(-0.505341\pi\)
−0.0167777 + 0.999859i \(0.505341\pi\)
\(348\) 0 0
\(349\) 4809.84 0.737721 0.368861 0.929485i \(-0.379748\pi\)
0.368861 + 0.929485i \(0.379748\pi\)
\(350\) 269.781 0.0412011
\(351\) 0 0
\(352\) −6650.35 −1.00700
\(353\) −2859.64 −0.431170 −0.215585 0.976485i \(-0.569166\pi\)
−0.215585 + 0.976485i \(0.569166\pi\)
\(354\) 0 0
\(355\) −2835.62 −0.423941
\(356\) 2518.74 0.374980
\(357\) 0 0
\(358\) −2998.27 −0.442636
\(359\) 3686.04 0.541899 0.270949 0.962594i \(-0.412662\pi\)
0.270949 + 0.962594i \(0.412662\pi\)
\(360\) 0 0
\(361\) 17076.8 2.48969
\(362\) 9287.39 1.34844
\(363\) 0 0
\(364\) 0 0
\(365\) 9748.98 1.39804
\(366\) 0 0
\(367\) −3470.59 −0.493633 −0.246816 0.969062i \(-0.579384\pi\)
−0.246816 + 0.969062i \(0.579384\pi\)
\(368\) −8538.52 −1.20951
\(369\) 0 0
\(370\) 5580.98 0.784166
\(371\) −900.166 −0.125968
\(372\) 0 0
\(373\) −11963.4 −1.66070 −0.830352 0.557240i \(-0.811860\pi\)
−0.830352 + 0.557240i \(0.811860\pi\)
\(374\) 2214.16 0.306127
\(375\) 0 0
\(376\) −3375.85 −0.463021
\(377\) 0 0
\(378\) 0 0
\(379\) −345.604 −0.0468403 −0.0234202 0.999726i \(-0.507456\pi\)
−0.0234202 + 0.999726i \(0.507456\pi\)
\(380\) 17184.8 2.31990
\(381\) 0 0
\(382\) −6189.99 −0.829078
\(383\) −3386.40 −0.451793 −0.225897 0.974151i \(-0.572531\pi\)
−0.225897 + 0.974151i \(0.572531\pi\)
\(384\) 0 0
\(385\) −3329.56 −0.440754
\(386\) −1555.49 −0.205110
\(387\) 0 0
\(388\) −11903.9 −1.55755
\(389\) 1629.88 0.212438 0.106219 0.994343i \(-0.466126\pi\)
0.106219 + 0.994343i \(0.466126\pi\)
\(390\) 0 0
\(391\) 3667.47 0.474353
\(392\) −1562.66 −0.201343
\(393\) 0 0
\(394\) −17987.0 −2.29993
\(395\) 3786.58 0.482338
\(396\) 0 0
\(397\) −7938.94 −1.00364 −0.501819 0.864973i \(-0.667336\pi\)
−0.501819 + 0.864973i \(0.667336\pi\)
\(398\) 17485.7 2.20221
\(399\) 0 0
\(400\) −269.532 −0.0336915
\(401\) 214.402 0.0267001 0.0133500 0.999911i \(-0.495750\pi\)
0.0133500 + 0.999911i \(0.495750\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −6270.93 −0.772253
\(405\) 0 0
\(406\) 968.819 0.118428
\(407\) −3001.48 −0.365548
\(408\) 0 0
\(409\) 4783.73 0.578338 0.289169 0.957278i \(-0.406621\pi\)
0.289169 + 0.957278i \(0.406621\pi\)
\(410\) −18823.2 −2.26735
\(411\) 0 0
\(412\) −4966.25 −0.593859
\(413\) −9832.99 −1.17155
\(414\) 0 0
\(415\) 4977.95 0.588815
\(416\) 0 0
\(417\) 0 0
\(418\) −16851.8 −1.97189
\(419\) 9903.67 1.15472 0.577358 0.816491i \(-0.304084\pi\)
0.577358 + 0.816491i \(0.304084\pi\)
\(420\) 0 0
\(421\) 12120.6 1.40314 0.701572 0.712598i \(-0.252482\pi\)
0.701572 + 0.712598i \(0.252482\pi\)
\(422\) 5183.82 0.597973
\(423\) 0 0
\(424\) −577.738 −0.0661732
\(425\) 115.770 0.0132133
\(426\) 0 0
\(427\) −2105.32 −0.238603
\(428\) −5909.28 −0.667374
\(429\) 0 0
\(430\) −7285.53 −0.817068
\(431\) −13672.6 −1.52805 −0.764023 0.645189i \(-0.776779\pi\)
−0.764023 + 0.645189i \(0.776779\pi\)
\(432\) 0 0
\(433\) 7113.10 0.789455 0.394727 0.918798i \(-0.370839\pi\)
0.394727 + 0.918798i \(0.370839\pi\)
\(434\) −12620.5 −1.39586
\(435\) 0 0
\(436\) 12631.3 1.38745
\(437\) −27912.9 −3.05550
\(438\) 0 0
\(439\) −6022.04 −0.654707 −0.327353 0.944902i \(-0.606157\pi\)
−0.327353 + 0.944902i \(0.606157\pi\)
\(440\) −2136.96 −0.231535
\(441\) 0 0
\(442\) 0 0
\(443\) 12994.4 1.39364 0.696821 0.717245i \(-0.254597\pi\)
0.696821 + 0.717245i \(0.254597\pi\)
\(444\) 0 0
\(445\) −2963.61 −0.315704
\(446\) 9206.20 0.977413
\(447\) 0 0
\(448\) −7912.30 −0.834422
\(449\) 10984.3 1.15452 0.577260 0.816560i \(-0.304122\pi\)
0.577260 + 0.816560i \(0.304122\pi\)
\(450\) 0 0
\(451\) 10123.2 1.05695
\(452\) −409.973 −0.0426627
\(453\) 0 0
\(454\) 17419.9 1.80078
\(455\) 0 0
\(456\) 0 0
\(457\) −9834.10 −1.00661 −0.503304 0.864109i \(-0.667882\pi\)
−0.503304 + 0.864109i \(0.667882\pi\)
\(458\) 3516.05 0.358721
\(459\) 0 0
\(460\) −20040.2 −2.03126
\(461\) 3401.42 0.343644 0.171822 0.985128i \(-0.445035\pi\)
0.171822 + 0.985128i \(0.445035\pi\)
\(462\) 0 0
\(463\) −1739.42 −0.174596 −0.0872979 0.996182i \(-0.527823\pi\)
−0.0872979 + 0.996182i \(0.527823\pi\)
\(464\) −967.925 −0.0968422
\(465\) 0 0
\(466\) −15512.5 −1.54207
\(467\) 7958.82 0.788630 0.394315 0.918975i \(-0.370982\pi\)
0.394315 + 0.918975i \(0.370982\pi\)
\(468\) 0 0
\(469\) 6856.16 0.675028
\(470\) 22489.0 2.20711
\(471\) 0 0
\(472\) −6310.94 −0.615433
\(473\) 3918.20 0.380886
\(474\) 0 0
\(475\) −881.114 −0.0851122
\(476\) 2222.78 0.214036
\(477\) 0 0
\(478\) 12737.5 1.21882
\(479\) 8431.98 0.804315 0.402158 0.915570i \(-0.368260\pi\)
0.402158 + 0.915570i \(0.368260\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −13745.0 −1.29889
\(483\) 0 0
\(484\) −6425.29 −0.603427
\(485\) 14006.4 1.31133
\(486\) 0 0
\(487\) 11684.7 1.08723 0.543617 0.839334i \(-0.317055\pi\)
0.543617 + 0.839334i \(0.317055\pi\)
\(488\) −1351.22 −0.125342
\(489\) 0 0
\(490\) 10410.0 0.959750
\(491\) 3954.70 0.363489 0.181745 0.983346i \(-0.441826\pi\)
0.181745 + 0.983346i \(0.441826\pi\)
\(492\) 0 0
\(493\) 415.744 0.0379801
\(494\) 0 0
\(495\) 0 0
\(496\) 12608.8 1.14144
\(497\) 2791.49 0.251943
\(498\) 0 0
\(499\) 5690.37 0.510493 0.255246 0.966876i \(-0.417843\pi\)
0.255246 + 0.966876i \(0.417843\pi\)
\(500\) 13251.9 1.18529
\(501\) 0 0
\(502\) 26782.8 2.38123
\(503\) −10859.1 −0.962595 −0.481298 0.876557i \(-0.659834\pi\)
−0.481298 + 0.876557i \(0.659834\pi\)
\(504\) 0 0
\(505\) 7378.53 0.650178
\(506\) 19651.9 1.72655
\(507\) 0 0
\(508\) −3021.88 −0.263926
\(509\) −18558.6 −1.61610 −0.808049 0.589115i \(-0.799476\pi\)
−0.808049 + 0.589115i \(0.799476\pi\)
\(510\) 0 0
\(511\) −9597.27 −0.830838
\(512\) 14896.8 1.28584
\(513\) 0 0
\(514\) 25613.6 2.19799
\(515\) 5843.42 0.499984
\(516\) 0 0
\(517\) −12094.7 −1.02887
\(518\) −5494.13 −0.466020
\(519\) 0 0
\(520\) 0 0
\(521\) −17297.5 −1.45454 −0.727271 0.686350i \(-0.759212\pi\)
−0.727271 + 0.686350i \(0.759212\pi\)
\(522\) 0 0
\(523\) −5016.11 −0.419386 −0.209693 0.977767i \(-0.567247\pi\)
−0.209693 + 0.977767i \(0.567247\pi\)
\(524\) −19441.8 −1.62084
\(525\) 0 0
\(526\) −518.667 −0.0429942
\(527\) −5415.77 −0.447656
\(528\) 0 0
\(529\) 20383.8 1.67533
\(530\) 3848.73 0.315431
\(531\) 0 0
\(532\) −16917.4 −1.37869
\(533\) 0 0
\(534\) 0 0
\(535\) 6953.01 0.561878
\(536\) 4400.37 0.354603
\(537\) 0 0
\(538\) 8147.83 0.652933
\(539\) −5598.57 −0.447398
\(540\) 0 0
\(541\) −17642.3 −1.40204 −0.701018 0.713144i \(-0.747271\pi\)
−0.701018 + 0.713144i \(0.747271\pi\)
\(542\) 19415.0 1.53865
\(543\) 0 0
\(544\) −5223.86 −0.411712
\(545\) −14862.3 −1.16813
\(546\) 0 0
\(547\) −18414.9 −1.43943 −0.719713 0.694271i \(-0.755727\pi\)
−0.719713 + 0.694271i \(0.755727\pi\)
\(548\) 10090.4 0.786571
\(549\) 0 0
\(550\) 620.343 0.0480937
\(551\) −3164.20 −0.244645
\(552\) 0 0
\(553\) −3727.66 −0.286648
\(554\) −24556.9 −1.88325
\(555\) 0 0
\(556\) −27773.1 −2.11842
\(557\) 8179.15 0.622193 0.311096 0.950378i \(-0.399304\pi\)
0.311096 + 0.950378i \(0.399304\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 6089.05 0.459481
\(561\) 0 0
\(562\) 19748.5 1.48228
\(563\) 1880.07 0.140738 0.0703690 0.997521i \(-0.477582\pi\)
0.0703690 + 0.997521i \(0.477582\pi\)
\(564\) 0 0
\(565\) 482.385 0.0359187
\(566\) 14586.9 1.08327
\(567\) 0 0
\(568\) 1791.62 0.132350
\(569\) −10118.3 −0.745485 −0.372743 0.927935i \(-0.621583\pi\)
−0.372743 + 0.927935i \(0.621583\pi\)
\(570\) 0 0
\(571\) 23428.9 1.71711 0.858555 0.512721i \(-0.171362\pi\)
0.858555 + 0.512721i \(0.171362\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 18530.3 1.34746
\(575\) 1027.52 0.0745225
\(576\) 0 0
\(577\) −20508.1 −1.47966 −0.739831 0.672793i \(-0.765094\pi\)
−0.739831 + 0.672793i \(0.765094\pi\)
\(578\) −18939.8 −1.36296
\(579\) 0 0
\(580\) −2271.76 −0.162637
\(581\) −4900.49 −0.349925
\(582\) 0 0
\(583\) −2069.87 −0.147042
\(584\) −6159.65 −0.436452
\(585\) 0 0
\(586\) 11268.9 0.794394
\(587\) −5968.43 −0.419665 −0.209833 0.977737i \(-0.567292\pi\)
−0.209833 + 0.977737i \(0.567292\pi\)
\(588\) 0 0
\(589\) 41219.0 2.88353
\(590\) 42041.8 2.93361
\(591\) 0 0
\(592\) 5489.06 0.381080
\(593\) −14659.5 −1.01517 −0.507584 0.861602i \(-0.669461\pi\)
−0.507584 + 0.861602i \(0.669461\pi\)
\(594\) 0 0
\(595\) −2615.38 −0.180202
\(596\) 7223.72 0.496468
\(597\) 0 0
\(598\) 0 0
\(599\) −23635.9 −1.61225 −0.806125 0.591746i \(-0.798439\pi\)
−0.806125 + 0.591746i \(0.798439\pi\)
\(600\) 0 0
\(601\) −11527.0 −0.782356 −0.391178 0.920315i \(-0.627932\pi\)
−0.391178 + 0.920315i \(0.627932\pi\)
\(602\) 7172.15 0.485573
\(603\) 0 0
\(604\) −22125.9 −1.49055
\(605\) 7560.15 0.508039
\(606\) 0 0
\(607\) −5098.56 −0.340930 −0.170465 0.985364i \(-0.554527\pi\)
−0.170465 + 0.985364i \(0.554527\pi\)
\(608\) 39758.4 2.65200
\(609\) 0 0
\(610\) 9001.47 0.597473
\(611\) 0 0
\(612\) 0 0
\(613\) −1516.39 −0.0999128 −0.0499564 0.998751i \(-0.515908\pi\)
−0.0499564 + 0.998751i \(0.515908\pi\)
\(614\) −1986.31 −0.130556
\(615\) 0 0
\(616\) 2103.70 0.137598
\(617\) 18539.3 1.20966 0.604832 0.796353i \(-0.293240\pi\)
0.604832 + 0.796353i \(0.293240\pi\)
\(618\) 0 0
\(619\) −25684.9 −1.66779 −0.833897 0.551920i \(-0.813895\pi\)
−0.833897 + 0.551920i \(0.813895\pi\)
\(620\) 29593.5 1.91694
\(621\) 0 0
\(622\) 6391.51 0.412020
\(623\) 2917.49 0.187619
\(624\) 0 0
\(625\) −16304.5 −1.04349
\(626\) 17046.1 1.08834
\(627\) 0 0
\(628\) 30834.2 1.95926
\(629\) −2357.67 −0.149454
\(630\) 0 0
\(631\) 22410.9 1.41389 0.706945 0.707269i \(-0.250073\pi\)
0.706945 + 0.707269i \(0.250073\pi\)
\(632\) −2392.46 −0.150580
\(633\) 0 0
\(634\) 13696.1 0.857950
\(635\) 3555.62 0.222206
\(636\) 0 0
\(637\) 0 0
\(638\) 2227.73 0.138239
\(639\) 0 0
\(640\) 10326.6 0.637804
\(641\) −6827.81 −0.420721 −0.210361 0.977624i \(-0.567464\pi\)
−0.210361 + 0.977624i \(0.567464\pi\)
\(642\) 0 0
\(643\) 23264.3 1.42684 0.713418 0.700738i \(-0.247146\pi\)
0.713418 + 0.700738i \(0.247146\pi\)
\(644\) 19728.4 1.20715
\(645\) 0 0
\(646\) −13237.1 −0.806204
\(647\) −14745.9 −0.896014 −0.448007 0.894030i \(-0.647866\pi\)
−0.448007 + 0.894030i \(0.647866\pi\)
\(648\) 0 0
\(649\) −22610.3 −1.36754
\(650\) 0 0
\(651\) 0 0
\(652\) 22489.3 1.35084
\(653\) −10909.0 −0.653755 −0.326878 0.945067i \(-0.605997\pi\)
−0.326878 + 0.945067i \(0.605997\pi\)
\(654\) 0 0
\(655\) 22875.7 1.36462
\(656\) −18513.2 −1.10186
\(657\) 0 0
\(658\) −22139.0 −1.31166
\(659\) 4182.99 0.247263 0.123631 0.992328i \(-0.460546\pi\)
0.123631 + 0.992328i \(0.460546\pi\)
\(660\) 0 0
\(661\) −2224.23 −0.130881 −0.0654406 0.997856i \(-0.520845\pi\)
−0.0654406 + 0.997856i \(0.520845\pi\)
\(662\) 14405.2 0.845734
\(663\) 0 0
\(664\) −3145.19 −0.183821
\(665\) 19905.4 1.16075
\(666\) 0 0
\(667\) 3689.95 0.214206
\(668\) −25899.7 −1.50013
\(669\) 0 0
\(670\) −29314.1 −1.69030
\(671\) −4841.04 −0.278519
\(672\) 0 0
\(673\) −24152.5 −1.38337 −0.691687 0.722197i \(-0.743132\pi\)
−0.691687 + 0.722197i \(0.743132\pi\)
\(674\) −39151.2 −2.23746
\(675\) 0 0
\(676\) 0 0
\(677\) 15310.7 0.869187 0.434593 0.900627i \(-0.356892\pi\)
0.434593 + 0.900627i \(0.356892\pi\)
\(678\) 0 0
\(679\) −13788.4 −0.779310
\(680\) −1678.58 −0.0946628
\(681\) 0 0
\(682\) −29020.0 −1.62937
\(683\) 11399.6 0.638646 0.319323 0.947646i \(-0.396545\pi\)
0.319323 + 0.947646i \(0.396545\pi\)
\(684\) 0 0
\(685\) −11872.6 −0.662233
\(686\) −26495.9 −1.47466
\(687\) 0 0
\(688\) −7165.54 −0.397069
\(689\) 0 0
\(690\) 0 0
\(691\) 3323.23 0.182955 0.0914773 0.995807i \(-0.470841\pi\)
0.0914773 + 0.995807i \(0.470841\pi\)
\(692\) 1605.52 0.0881976
\(693\) 0 0
\(694\) −912.936 −0.0499345
\(695\) 32678.5 1.78355
\(696\) 0 0
\(697\) 7951.82 0.432133
\(698\) 20244.8 1.09782
\(699\) 0 0
\(700\) 622.758 0.0336257
\(701\) 12670.4 0.682673 0.341336 0.939941i \(-0.389120\pi\)
0.341336 + 0.939941i \(0.389120\pi\)
\(702\) 0 0
\(703\) 17944.0 0.962692
\(704\) −18193.8 −0.974012
\(705\) 0 0
\(706\) −12036.4 −0.641635
\(707\) −7263.71 −0.386393
\(708\) 0 0
\(709\) −13075.2 −0.692594 −0.346297 0.938125i \(-0.612561\pi\)
−0.346297 + 0.938125i \(0.612561\pi\)
\(710\) −11935.3 −0.630877
\(711\) 0 0
\(712\) 1872.48 0.0985592
\(713\) −48067.8 −2.52476
\(714\) 0 0
\(715\) 0 0
\(716\) −6921.16 −0.361251
\(717\) 0 0
\(718\) 15514.7 0.806412
\(719\) 2988.41 0.155005 0.0775026 0.996992i \(-0.475305\pi\)
0.0775026 + 0.996992i \(0.475305\pi\)
\(720\) 0 0
\(721\) −5752.48 −0.297134
\(722\) 71877.0 3.70496
\(723\) 0 0
\(724\) 21438.9 1.10051
\(725\) 116.479 0.00596680
\(726\) 0 0
\(727\) −5507.46 −0.280963 −0.140482 0.990083i \(-0.544865\pi\)
−0.140482 + 0.990083i \(0.544865\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 41033.9 2.08046
\(731\) 3077.75 0.155725
\(732\) 0 0
\(733\) 36585.2 1.84353 0.921764 0.387751i \(-0.126748\pi\)
0.921764 + 0.387751i \(0.126748\pi\)
\(734\) −14607.9 −0.734587
\(735\) 0 0
\(736\) −46364.6 −2.32204
\(737\) 15765.3 0.787953
\(738\) 0 0
\(739\) −6425.89 −0.319865 −0.159933 0.987128i \(-0.551128\pi\)
−0.159933 + 0.987128i \(0.551128\pi\)
\(740\) 12883.0 0.639986
\(741\) 0 0
\(742\) −3788.84 −0.187457
\(743\) 20411.0 1.00782 0.503908 0.863757i \(-0.331895\pi\)
0.503908 + 0.863757i \(0.331895\pi\)
\(744\) 0 0
\(745\) −8499.60 −0.417988
\(746\) −50354.6 −2.47133
\(747\) 0 0
\(748\) 5111.13 0.249842
\(749\) −6844.81 −0.333917
\(750\) 0 0
\(751\) −24259.5 −1.17875 −0.589375 0.807860i \(-0.700626\pi\)
−0.589375 + 0.807860i \(0.700626\pi\)
\(752\) 22118.6 1.07258
\(753\) 0 0
\(754\) 0 0
\(755\) 26033.9 1.25493
\(756\) 0 0
\(757\) 9295.39 0.446297 0.223148 0.974785i \(-0.428367\pi\)
0.223148 + 0.974785i \(0.428367\pi\)
\(758\) −1454.66 −0.0697042
\(759\) 0 0
\(760\) 12775.6 0.609761
\(761\) −21974.7 −1.04676 −0.523378 0.852101i \(-0.675328\pi\)
−0.523378 + 0.852101i \(0.675328\pi\)
\(762\) 0 0
\(763\) 14631.0 0.694204
\(764\) −14288.9 −0.676641
\(765\) 0 0
\(766\) −14253.5 −0.672324
\(767\) 0 0
\(768\) 0 0
\(769\) −22987.4 −1.07795 −0.538977 0.842320i \(-0.681189\pi\)
−0.538977 + 0.842320i \(0.681189\pi\)
\(770\) −14014.3 −0.655897
\(771\) 0 0
\(772\) −3590.68 −0.167398
\(773\) 31970.9 1.48760 0.743799 0.668404i \(-0.233022\pi\)
0.743799 + 0.668404i \(0.233022\pi\)
\(774\) 0 0
\(775\) −1517.34 −0.0703283
\(776\) −8849.59 −0.409384
\(777\) 0 0
\(778\) 6860.25 0.316133
\(779\) −60520.8 −2.78354
\(780\) 0 0
\(781\) 6418.85 0.294090
\(782\) 15436.6 0.705896
\(783\) 0 0
\(784\) 10238.6 0.466408
\(785\) −36280.2 −1.64955
\(786\) 0 0
\(787\) −6087.26 −0.275715 −0.137857 0.990452i \(-0.544022\pi\)
−0.137857 + 0.990452i \(0.544022\pi\)
\(788\) −41520.9 −1.87706
\(789\) 0 0
\(790\) 15937.9 0.717779
\(791\) −474.878 −0.0213460
\(792\) 0 0
\(793\) 0 0
\(794\) −33415.4 −1.49354
\(795\) 0 0
\(796\) 40363.6 1.79730
\(797\) −23080.0 −1.02577 −0.512883 0.858458i \(-0.671423\pi\)
−0.512883 + 0.858458i \(0.671423\pi\)
\(798\) 0 0
\(799\) −9500.41 −0.420651
\(800\) −1463.57 −0.0646813
\(801\) 0 0
\(802\) 902.429 0.0397330
\(803\) −22068.3 −0.969829
\(804\) 0 0
\(805\) −23212.9 −1.01633
\(806\) 0 0
\(807\) 0 0
\(808\) −4661.94 −0.202978
\(809\) 32377.8 1.40710 0.703550 0.710646i \(-0.251597\pi\)
0.703550 + 0.710646i \(0.251597\pi\)
\(810\) 0 0
\(811\) −26352.8 −1.14103 −0.570513 0.821288i \(-0.693256\pi\)
−0.570513 + 0.821288i \(0.693256\pi\)
\(812\) 2236.40 0.0966532
\(813\) 0 0
\(814\) −12633.4 −0.543980
\(815\) −26461.5 −1.13731
\(816\) 0 0
\(817\) −23424.5 −1.00308
\(818\) 20135.0 0.860638
\(819\) 0 0
\(820\) −43451.3 −1.85047
\(821\) 35355.3 1.50294 0.751468 0.659770i \(-0.229346\pi\)
0.751468 + 0.659770i \(0.229346\pi\)
\(822\) 0 0
\(823\) −12663.3 −0.536347 −0.268173 0.963371i \(-0.586420\pi\)
−0.268173 + 0.963371i \(0.586420\pi\)
\(824\) −3692.02 −0.156089
\(825\) 0 0
\(826\) −41387.6 −1.74341
\(827\) −16295.2 −0.685176 −0.342588 0.939486i \(-0.611303\pi\)
−0.342588 + 0.939486i \(0.611303\pi\)
\(828\) 0 0
\(829\) 13638.9 0.571411 0.285705 0.958318i \(-0.407772\pi\)
0.285705 + 0.958318i \(0.407772\pi\)
\(830\) 20952.4 0.876229
\(831\) 0 0
\(832\) 0 0
\(833\) −4397.69 −0.182918
\(834\) 0 0
\(835\) 30474.2 1.26300
\(836\) −38900.5 −1.60933
\(837\) 0 0
\(838\) 41685.0 1.71836
\(839\) −1890.31 −0.0777838 −0.0388919 0.999243i \(-0.512383\pi\)
−0.0388919 + 0.999243i \(0.512383\pi\)
\(840\) 0 0
\(841\) −23970.7 −0.982849
\(842\) 51016.4 2.08805
\(843\) 0 0
\(844\) 11966.3 0.488028
\(845\) 0 0
\(846\) 0 0
\(847\) −7442.50 −0.301921
\(848\) 3785.35 0.153289
\(849\) 0 0
\(850\) 487.280 0.0196630
\(851\) −20925.6 −0.842914
\(852\) 0 0
\(853\) −1620.21 −0.0650351 −0.0325175 0.999471i \(-0.510352\pi\)
−0.0325175 + 0.999471i \(0.510352\pi\)
\(854\) −8861.39 −0.355071
\(855\) 0 0
\(856\) −4393.08 −0.175412
\(857\) 14508.4 0.578292 0.289146 0.957285i \(-0.406629\pi\)
0.289146 + 0.957285i \(0.406629\pi\)
\(858\) 0 0
\(859\) 29639.8 1.17730 0.588648 0.808389i \(-0.299660\pi\)
0.588648 + 0.808389i \(0.299660\pi\)
\(860\) −16817.8 −0.666839
\(861\) 0 0
\(862\) −57548.8 −2.27392
\(863\) 21528.8 0.849186 0.424593 0.905384i \(-0.360417\pi\)
0.424593 + 0.905384i \(0.360417\pi\)
\(864\) 0 0
\(865\) −1889.10 −0.0742557
\(866\) 29939.4 1.17481
\(867\) 0 0
\(868\) −29132.9 −1.13921
\(869\) −8571.50 −0.334601
\(870\) 0 0
\(871\) 0 0
\(872\) 9390.36 0.364676
\(873\) 0 0
\(874\) −117487. −4.54696
\(875\) 15349.9 0.593054
\(876\) 0 0
\(877\) −14865.3 −0.572366 −0.286183 0.958175i \(-0.592387\pi\)
−0.286183 + 0.958175i \(0.592387\pi\)
\(878\) −25347.1 −0.974285
\(879\) 0 0
\(880\) 14001.4 0.536348
\(881\) 21336.0 0.815921 0.407961 0.913000i \(-0.366240\pi\)
0.407961 + 0.913000i \(0.366240\pi\)
\(882\) 0 0
\(883\) 37538.2 1.43065 0.715323 0.698794i \(-0.246280\pi\)
0.715323 + 0.698794i \(0.246280\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 54694.1 2.07391
\(887\) −34575.0 −1.30881 −0.654406 0.756144i \(-0.727081\pi\)
−0.654406 + 0.756144i \(0.727081\pi\)
\(888\) 0 0
\(889\) −3500.29 −0.132054
\(890\) −12474.0 −0.469807
\(891\) 0 0
\(892\) 21251.4 0.797703
\(893\) 72306.9 2.70958
\(894\) 0 0
\(895\) 8143.61 0.304146
\(896\) −10165.9 −0.379039
\(897\) 0 0
\(898\) 46233.3 1.71807
\(899\) −5448.96 −0.202150
\(900\) 0 0
\(901\) −1625.89 −0.0601178
\(902\) 42609.2 1.57287
\(903\) 0 0
\(904\) −304.783 −0.0112134
\(905\) −25225.5 −0.926545
\(906\) 0 0
\(907\) −10424.8 −0.381641 −0.190820 0.981625i \(-0.561115\pi\)
−0.190820 + 0.981625i \(0.561115\pi\)
\(908\) 40211.7 1.46968
\(909\) 0 0
\(910\) 0 0
\(911\) −10961.8 −0.398661 −0.199331 0.979932i \(-0.563877\pi\)
−0.199331 + 0.979932i \(0.563877\pi\)
\(912\) 0 0
\(913\) −11268.3 −0.408464
\(914\) −41392.2 −1.49796
\(915\) 0 0
\(916\) 8116.38 0.292765
\(917\) −22519.7 −0.810976
\(918\) 0 0
\(919\) −10779.2 −0.386914 −0.193457 0.981109i \(-0.561970\pi\)
−0.193457 + 0.981109i \(0.561970\pi\)
\(920\) −14898.3 −0.533895
\(921\) 0 0
\(922\) 14316.7 0.511384
\(923\) 0 0
\(924\) 0 0
\(925\) −660.549 −0.0234797
\(926\) −7321.32 −0.259820
\(927\) 0 0
\(928\) −5255.88 −0.185919
\(929\) 5429.07 0.191735 0.0958675 0.995394i \(-0.469437\pi\)
0.0958675 + 0.995394i \(0.469437\pi\)
\(930\) 0 0
\(931\) 33470.5 1.17825
\(932\) −35808.8 −1.25854
\(933\) 0 0
\(934\) 33499.1 1.17358
\(935\) −6013.88 −0.210348
\(936\) 0 0
\(937\) −21300.1 −0.742631 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(938\) 28857.9 1.00453
\(939\) 0 0
\(940\) 51913.2 1.80130
\(941\) 26851.2 0.930207 0.465103 0.885256i \(-0.346017\pi\)
0.465103 + 0.885256i \(0.346017\pi\)
\(942\) 0 0
\(943\) 70576.7 2.43722
\(944\) 41349.4 1.42564
\(945\) 0 0
\(946\) 16491.9 0.566805
\(947\) −8021.68 −0.275258 −0.137629 0.990484i \(-0.543948\pi\)
−0.137629 + 0.990484i \(0.543948\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −3708.65 −0.126658
\(951\) 0 0
\(952\) 1652.46 0.0562569
\(953\) −35715.0 −1.21398 −0.606990 0.794709i \(-0.707623\pi\)
−0.606990 + 0.794709i \(0.707623\pi\)
\(954\) 0 0
\(955\) 16812.6 0.569680
\(956\) 29402.9 0.994726
\(957\) 0 0
\(958\) 35490.6 1.19692
\(959\) 11687.9 0.393557
\(960\) 0 0
\(961\) 41190.9 1.38266
\(962\) 0 0
\(963\) 0 0
\(964\) −31728.7 −1.06007
\(965\) 4224.88 0.140936
\(966\) 0 0
\(967\) −53338.8 −1.77380 −0.886898 0.461965i \(-0.847145\pi\)
−0.886898 + 0.461965i \(0.847145\pi\)
\(968\) −4776.69 −0.158604
\(969\) 0 0
\(970\) 58953.6 1.95143
\(971\) −23112.9 −0.763882 −0.381941 0.924187i \(-0.624744\pi\)
−0.381941 + 0.924187i \(0.624744\pi\)
\(972\) 0 0
\(973\) −32170.0 −1.05994
\(974\) 49181.3 1.61794
\(975\) 0 0
\(976\) 8853.22 0.290353
\(977\) −52874.6 −1.73143 −0.865715 0.500538i \(-0.833136\pi\)
−0.865715 + 0.500538i \(0.833136\pi\)
\(978\) 0 0
\(979\) 6708.57 0.219006
\(980\) 24030.4 0.783287
\(981\) 0 0
\(982\) 16645.5 0.540917
\(983\) 45173.1 1.46572 0.732858 0.680381i \(-0.238186\pi\)
0.732858 + 0.680381i \(0.238186\pi\)
\(984\) 0 0
\(985\) 48854.5 1.58034
\(986\) 1749.89 0.0565190
\(987\) 0 0
\(988\) 0 0
\(989\) 27316.7 0.878281
\(990\) 0 0
\(991\) 60485.6 1.93884 0.969418 0.245414i \(-0.0789239\pi\)
0.969418 + 0.245414i \(0.0789239\pi\)
\(992\) 68466.7 2.19135
\(993\) 0 0
\(994\) 11749.5 0.374922
\(995\) −47492.8 −1.51319
\(996\) 0 0
\(997\) −18108.1 −0.575214 −0.287607 0.957749i \(-0.592860\pi\)
−0.287607 + 0.957749i \(0.592860\pi\)
\(998\) 23951.1 0.759677
\(999\) 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.u.1.3 3
3.2 odd 2 507.4.a.h.1.1 3
13.12 even 2 117.4.a.f.1.1 3
39.5 even 4 507.4.b.g.337.6 6
39.8 even 4 507.4.b.g.337.1 6
39.38 odd 2 39.4.a.c.1.3 3
52.51 odd 2 1872.4.a.bk.1.3 3
156.155 even 2 624.4.a.t.1.1 3
195.194 odd 2 975.4.a.l.1.1 3
273.272 even 2 1911.4.a.k.1.3 3
312.77 odd 2 2496.4.a.bl.1.3 3
312.155 even 2 2496.4.a.bp.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 39.38 odd 2
117.4.a.f.1.1 3 13.12 even 2
507.4.a.h.1.1 3 3.2 odd 2
507.4.b.g.337.1 6 39.8 even 4
507.4.b.g.337.6 6 39.5 even 4
624.4.a.t.1.1 3 156.155 even 2
975.4.a.l.1.1 3 195.194 odd 2
1521.4.a.u.1.3 3 1.1 even 1 trivial
1872.4.a.bk.1.3 3 52.51 odd 2
1911.4.a.k.1.3 3 273.272 even 2
2496.4.a.bl.1.3 3 312.77 odd 2
2496.4.a.bp.1.3 3 312.155 even 2