Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - 2x^{3} - 13x^{2} + 14x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.170498\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.12311 q^{2} +9.00000 q^{4} +13.4424 q^{5} +31.4219 q^{7} -4.12311 q^{8} +O(q^{10})\) \(q-4.12311 q^{2} +9.00000 q^{4} +13.4424 q^{5} +31.4219 q^{7} -4.12311 q^{8} -55.4243 q^{10} +40.4962 q^{11} -129.556 q^{14} -55.0000 q^{16} +43.1414 q^{17} +26.9779 q^{19} +120.981 q^{20} -166.970 q^{22} +19.0100 q^{23} +55.6971 q^{25} +282.797 q^{28} +154.111 q^{29} -308.270 q^{31} +259.756 q^{32} -177.877 q^{34} +422.384 q^{35} -43.5116 q^{37} -111.233 q^{38} -55.4243 q^{40} +47.8384 q^{41} +342.121 q^{43} +364.466 q^{44} -78.3802 q^{46} +133.468 q^{47} +644.334 q^{49} -229.645 q^{50} +438.454 q^{53} +544.364 q^{55} -129.556 q^{56} -635.418 q^{58} +590.553 q^{59} -541.304 q^{61} +1271.03 q^{62} -631.000 q^{64} +230.345 q^{67} +388.273 q^{68} -1741.54 q^{70} +449.412 q^{71} +389.711 q^{73} +179.403 q^{74} +242.801 q^{76} +1272.47 q^{77} -897.820 q^{79} -739.330 q^{80} -197.243 q^{82} -1300.24 q^{83} +579.923 q^{85} -1410.60 q^{86} -166.970 q^{88} -925.045 q^{89} +171.090 q^{92} -550.301 q^{94} +362.647 q^{95} +1560.49 q^{97} -2656.66 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.12311 −1.45774 −0.728869 0.684653i \(-0.759954\pi\)
−0.728869 + 0.684653i \(0.759954\pi\)
\(3\) 0 0
\(4\) 9.00000 1.12500
\(5\) 13.4424 1.20232 0.601161 0.799128i \(-0.294705\pi\)
0.601161 + 0.799128i \(0.294705\pi\)
\(6\) 0 0
\(7\) 31.4219 1.69662 0.848311 0.529498i \(-0.177620\pi\)
0.848311 + 0.529498i \(0.177620\pi\)
\(8\) −4.12311 −0.182217
\(9\) 0 0
\(10\) −55.4243 −1.75267
\(11\) 40.4962 1.11001 0.555003 0.831849i \(-0.312717\pi\)
0.555003 + 0.831849i \(0.312717\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −129.556 −2.47323
\(15\) 0 0
\(16\) −55.0000 −0.859375
\(17\) 43.1414 0.615490 0.307745 0.951469i \(-0.400426\pi\)
0.307745 + 0.951469i \(0.400426\pi\)
\(18\) 0 0
\(19\) 26.9779 0.325745 0.162873 0.986647i \(-0.447924\pi\)
0.162873 + 0.986647i \(0.447924\pi\)
\(20\) 120.981 1.35261
\(21\) 0 0
\(22\) −166.970 −1.61810
\(23\) 19.0100 0.172342 0.0861709 0.996280i \(-0.472537\pi\)
0.0861709 + 0.996280i \(0.472537\pi\)
\(24\) 0 0
\(25\) 55.6971 0.445577
\(26\) 0 0
\(27\) 0 0
\(28\) 282.797 1.90870
\(29\) 154.111 0.986820 0.493410 0.869797i \(-0.335750\pi\)
0.493410 + 0.869797i \(0.335750\pi\)
\(30\) 0 0
\(31\) −308.270 −1.78603 −0.893016 0.450025i \(-0.851415\pi\)
−0.893016 + 0.450025i \(0.851415\pi\)
\(32\) 259.756 1.43496
\(33\) 0 0
\(34\) −177.877 −0.897223
\(35\) 422.384 2.03988
\(36\) 0 0
\(37\) −43.5116 −0.193331 −0.0966657 0.995317i \(-0.530818\pi\)
−0.0966657 + 0.995317i \(0.530818\pi\)
\(38\) −111.233 −0.474851
\(39\) 0 0
\(40\) −55.4243 −0.219084
\(41\) 47.8384 0.182222 0.0911110 0.995841i \(-0.470958\pi\)
0.0911110 + 0.995841i \(0.470958\pi\)
\(42\) 0 0
\(43\) 342.121 1.21333 0.606663 0.794959i \(-0.292508\pi\)
0.606663 + 0.794959i \(0.292508\pi\)
\(44\) 364.466 1.24876
\(45\) 0 0
\(46\) −78.3802 −0.251229
\(47\) 133.468 0.414218 0.207109 0.978318i \(-0.433594\pi\)
0.207109 + 0.978318i \(0.433594\pi\)
\(48\) 0 0
\(49\) 644.334 1.87853
\(50\) −229.645 −0.649535
\(51\) 0 0
\(52\) 0 0
\(53\) 438.454 1.13635 0.568173 0.822909i \(-0.307650\pi\)
0.568173 + 0.822909i \(0.307650\pi\)
\(54\) 0 0
\(55\) 544.364 1.33458
\(56\) −129.556 −0.309154
\(57\) 0 0
\(58\) −635.418 −1.43852
\(59\) 590.553 1.30311 0.651555 0.758601i \(-0.274117\pi\)
0.651555 + 0.758601i \(0.274117\pi\)
\(60\) 0 0
\(61\) −541.304 −1.13618 −0.568089 0.822967i \(-0.692317\pi\)
−0.568089 + 0.822967i \(0.692317\pi\)
\(62\) 1271.03 2.60357
\(63\) 0 0
\(64\) −631.000 −1.23242
\(65\) 0 0
\(66\) 0 0
\(67\) 230.345 0.420018 0.210009 0.977700i \(-0.432651\pi\)
0.210009 + 0.977700i \(0.432651\pi\)
\(68\) 388.273 0.692426
\(69\) 0 0
\(70\) −1741.54 −2.97362
\(71\) 449.412 0.751203 0.375601 0.926781i \(-0.377436\pi\)
0.375601 + 0.926781i \(0.377436\pi\)
\(72\) 0 0
\(73\) 389.711 0.624826 0.312413 0.949946i \(-0.398863\pi\)
0.312413 + 0.949946i \(0.398863\pi\)
\(74\) 179.403 0.281826
\(75\) 0 0
\(76\) 242.801 0.366464
\(77\) 1272.47 1.88326
\(78\) 0 0
\(79\) −897.820 −1.27864 −0.639321 0.768940i \(-0.720784\pi\)
−0.639321 + 0.768940i \(0.720784\pi\)
\(80\) −739.330 −1.03325
\(81\) 0 0
\(82\) −197.243 −0.265632
\(83\) −1300.24 −1.71952 −0.859759 0.510700i \(-0.829386\pi\)
−0.859759 + 0.510700i \(0.829386\pi\)
\(84\) 0 0
\(85\) 579.923 0.740017
\(86\) −1410.60 −1.76871
\(87\) 0 0
\(88\) −166.970 −0.202262
\(89\) −925.045 −1.10174 −0.550869 0.834592i \(-0.685703\pi\)
−0.550869 + 0.834592i \(0.685703\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 171.090 0.193884
\(93\) 0 0
\(94\) −550.301 −0.603822
\(95\) 362.647 0.391651
\(96\) 0 0
\(97\) 1560.49 1.63344 0.816722 0.577031i \(-0.195789\pi\)
0.816722 + 0.577031i \(0.195789\pi\)
\(98\) −2656.66 −2.73840
\(99\) 0 0
\(100\) 501.274 0.501274
\(101\) −958.840 −0.944635 −0.472318 0.881428i \(-0.656582\pi\)
−0.472318 + 0.881428i \(0.656582\pi\)
\(102\) 0 0
\(103\) 635.153 0.607606 0.303803 0.952735i \(-0.401743\pi\)
0.303803 + 0.952735i \(0.401743\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1807.79 −1.65649
\(107\) 1448.32 1.30855 0.654275 0.756257i \(-0.272974\pi\)
0.654275 + 0.756257i \(0.272974\pi\)
\(108\) 0 0
\(109\) 331.084 0.290937 0.145468 0.989363i \(-0.453531\pi\)
0.145468 + 0.989363i \(0.453531\pi\)
\(110\) −2244.47 −1.94547
\(111\) 0 0
\(112\) −1728.20 −1.45803
\(113\) −695.204 −0.578755 −0.289378 0.957215i \(-0.593448\pi\)
−0.289378 + 0.957215i \(0.593448\pi\)
\(114\) 0 0
\(115\) 255.539 0.207210
\(116\) 1387.00 1.11017
\(117\) 0 0
\(118\) −2434.91 −1.89959
\(119\) 1355.58 1.04425
\(120\) 0 0
\(121\) 308.940 0.232111
\(122\) 2231.85 1.65625
\(123\) 0 0
\(124\) −2774.43 −2.00929
\(125\) −931.594 −0.666595
\(126\) 0 0
\(127\) 247.154 0.172688 0.0863441 0.996265i \(-0.472482\pi\)
0.0863441 + 0.996265i \(0.472482\pi\)
\(128\) 523.634 0.361587
\(129\) 0 0
\(130\) 0 0
\(131\) −472.243 −0.314962 −0.157481 0.987522i \(-0.550337\pi\)
−0.157481 + 0.987522i \(0.550337\pi\)
\(132\) 0 0
\(133\) 847.697 0.552667
\(134\) −949.739 −0.612275
\(135\) 0 0
\(136\) −177.877 −0.112153
\(137\) −1830.70 −1.14166 −0.570829 0.821069i \(-0.693378\pi\)
−0.570829 + 0.821069i \(0.693378\pi\)
\(138\) 0 0
\(139\) −100.000 −0.0610208 −0.0305104 0.999534i \(-0.509713\pi\)
−0.0305104 + 0.999534i \(0.509713\pi\)
\(140\) 3801.46 2.29487
\(141\) 0 0
\(142\) −1852.97 −1.09506
\(143\) 0 0
\(144\) 0 0
\(145\) 2071.62 1.18647
\(146\) −1606.82 −0.910832
\(147\) 0 0
\(148\) −391.604 −0.217498
\(149\) 149.557 0.0822293 0.0411147 0.999154i \(-0.486909\pi\)
0.0411147 + 0.999154i \(0.486909\pi\)
\(150\) 0 0
\(151\) 800.032 0.431163 0.215582 0.976486i \(-0.430835\pi\)
0.215582 + 0.976486i \(0.430835\pi\)
\(152\) −111.233 −0.0593564
\(153\) 0 0
\(154\) −5246.51 −2.74530
\(155\) −4143.88 −2.14739
\(156\) 0 0
\(157\) −2706.16 −1.37564 −0.687818 0.725884i \(-0.741431\pi\)
−0.687818 + 0.725884i \(0.741431\pi\)
\(158\) 3701.81 1.86392
\(159\) 0 0
\(160\) 3491.73 1.72528
\(161\) 597.330 0.292399
\(162\) 0 0
\(163\) −3678.25 −1.76750 −0.883750 0.467959i \(-0.844990\pi\)
−0.883750 + 0.467959i \(0.844990\pi\)
\(164\) 430.546 0.205000
\(165\) 0 0
\(166\) 5361.03 2.50661
\(167\) −3223.11 −1.49348 −0.746742 0.665114i \(-0.768383\pi\)
−0.746742 + 0.665114i \(0.768383\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −2391.08 −1.07875
\(171\) 0 0
\(172\) 3079.09 1.36499
\(173\) 2689.54 1.18198 0.590988 0.806680i \(-0.298738\pi\)
0.590988 + 0.806680i \(0.298738\pi\)
\(174\) 0 0
\(175\) 1750.11 0.755976
\(176\) −2227.29 −0.953911
\(177\) 0 0
\(178\) 3814.06 1.60604
\(179\) 1524.04 0.636381 0.318191 0.948027i \(-0.396925\pi\)
0.318191 + 0.948027i \(0.396925\pi\)
\(180\) 0 0
\(181\) 476.881 0.195836 0.0979180 0.995194i \(-0.468782\pi\)
0.0979180 + 0.995194i \(0.468782\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −78.3802 −0.0314036
\(185\) −584.899 −0.232446
\(186\) 0 0
\(187\) 1747.06 0.683197
\(188\) 1201.21 0.465996
\(189\) 0 0
\(190\) −1495.23 −0.570924
\(191\) 1369.74 0.518906 0.259453 0.965756i \(-0.416458\pi\)
0.259453 + 0.965756i \(0.416458\pi\)
\(192\) 0 0
\(193\) 2144.72 0.799898 0.399949 0.916537i \(-0.369028\pi\)
0.399949 + 0.916537i \(0.369028\pi\)
\(194\) −6434.08 −2.38113
\(195\) 0 0
\(196\) 5799.01 2.11334
\(197\) −239.739 −0.0867040 −0.0433520 0.999060i \(-0.513804\pi\)
−0.0433520 + 0.999060i \(0.513804\pi\)
\(198\) 0 0
\(199\) −1589.94 −0.566371 −0.283185 0.959065i \(-0.591391\pi\)
−0.283185 + 0.959065i \(0.591391\pi\)
\(200\) −229.645 −0.0811918
\(201\) 0 0
\(202\) 3953.40 1.37703
\(203\) 4842.47 1.67426
\(204\) 0 0
\(205\) 643.061 0.219090
\(206\) −2618.80 −0.885731
\(207\) 0 0
\(208\) 0 0
\(209\) 1092.50 0.361579
\(210\) 0 0
\(211\) −1872.85 −0.611055 −0.305527 0.952183i \(-0.598833\pi\)
−0.305527 + 0.952183i \(0.598833\pi\)
\(212\) 3946.09 1.27839
\(213\) 0 0
\(214\) −5971.59 −1.90752
\(215\) 4598.92 1.45881
\(216\) 0 0
\(217\) −9686.43 −3.03022
\(218\) −1365.09 −0.424109
\(219\) 0 0
\(220\) 4899.28 1.50141
\(221\) 0 0
\(222\) 0 0
\(223\) 56.1283 0.0168548 0.00842742 0.999964i \(-0.497317\pi\)
0.00842742 + 0.999964i \(0.497317\pi\)
\(224\) 8162.01 2.43459
\(225\) 0 0
\(226\) 2866.40 0.843673
\(227\) −667.390 −0.195137 −0.0975687 0.995229i \(-0.531107\pi\)
−0.0975687 + 0.995229i \(0.531107\pi\)
\(228\) 0 0
\(229\) −723.299 −0.208720 −0.104360 0.994540i \(-0.533279\pi\)
−0.104360 + 0.994540i \(0.533279\pi\)
\(230\) −1053.62 −0.302058
\(231\) 0 0
\(232\) −635.418 −0.179816
\(233\) 275.451 0.0774482 0.0387241 0.999250i \(-0.487671\pi\)
0.0387241 + 0.999250i \(0.487671\pi\)
\(234\) 0 0
\(235\) 1794.12 0.498024
\(236\) 5314.98 1.46600
\(237\) 0 0
\(238\) −5589.22 −1.52225
\(239\) −1529.39 −0.413925 −0.206963 0.978349i \(-0.566358\pi\)
−0.206963 + 0.978349i \(0.566358\pi\)
\(240\) 0 0
\(241\) −975.526 −0.260743 −0.130372 0.991465i \(-0.541617\pi\)
−0.130372 + 0.991465i \(0.541617\pi\)
\(242\) −1273.79 −0.338357
\(243\) 0 0
\(244\) −4871.74 −1.27820
\(245\) 8661.38 2.25859
\(246\) 0 0
\(247\) 0 0
\(248\) 1271.03 0.325446
\(249\) 0 0
\(250\) 3841.06 0.971720
\(251\) 1874.14 0.471294 0.235647 0.971839i \(-0.424279\pi\)
0.235647 + 0.971839i \(0.424279\pi\)
\(252\) 0 0
\(253\) 769.832 0.191300
\(254\) −1019.04 −0.251734
\(255\) 0 0
\(256\) 2889.00 0.705322
\(257\) 1818.19 0.441305 0.220653 0.975352i \(-0.429181\pi\)
0.220653 + 0.975352i \(0.429181\pi\)
\(258\) 0 0
\(259\) −1367.22 −0.328010
\(260\) 0 0
\(261\) 0 0
\(262\) 1947.11 0.459132
\(263\) −673.799 −0.157978 −0.0789890 0.996875i \(-0.525169\pi\)
−0.0789890 + 0.996875i \(0.525169\pi\)
\(264\) 0 0
\(265\) 5893.86 1.36625
\(266\) −3495.14 −0.805643
\(267\) 0 0
\(268\) 2073.11 0.472520
\(269\) 3356.40 0.760756 0.380378 0.924831i \(-0.375794\pi\)
0.380378 + 0.924831i \(0.375794\pi\)
\(270\) 0 0
\(271\) −8915.55 −1.99845 −0.999227 0.0393133i \(-0.987483\pi\)
−0.999227 + 0.0393133i \(0.987483\pi\)
\(272\) −2372.78 −0.528937
\(273\) 0 0
\(274\) 7548.16 1.66424
\(275\) 2255.52 0.494593
\(276\) 0 0
\(277\) 4017.31 0.871396 0.435698 0.900093i \(-0.356502\pi\)
0.435698 + 0.900093i \(0.356502\pi\)
\(278\) 412.311 0.0889523
\(279\) 0 0
\(280\) −1741.54 −0.371702
\(281\) 1841.12 0.390860 0.195430 0.980718i \(-0.437390\pi\)
0.195430 + 0.980718i \(0.437390\pi\)
\(282\) 0 0
\(283\) 4849.40 1.01861 0.509305 0.860586i \(-0.329902\pi\)
0.509305 + 0.860586i \(0.329902\pi\)
\(284\) 4044.71 0.845103
\(285\) 0 0
\(286\) 0 0
\(287\) 1503.17 0.309162
\(288\) 0 0
\(289\) −3051.82 −0.621172
\(290\) −8541.52 −1.72957
\(291\) 0 0
\(292\) 3507.40 0.702929
\(293\) −1413.85 −0.281905 −0.140953 0.990016i \(-0.545017\pi\)
−0.140953 + 0.990016i \(0.545017\pi\)
\(294\) 0 0
\(295\) 7938.43 1.56676
\(296\) 179.403 0.0352283
\(297\) 0 0
\(298\) −616.639 −0.119869
\(299\) 0 0
\(300\) 0 0
\(301\) 10750.1 2.05856
\(302\) −3298.62 −0.628523
\(303\) 0 0
\(304\) −1483.79 −0.279937
\(305\) −7276.41 −1.36605
\(306\) 0 0
\(307\) 4625.64 0.859932 0.429966 0.902845i \(-0.358526\pi\)
0.429966 + 0.902845i \(0.358526\pi\)
\(308\) 11452.2 2.11867
\(309\) 0 0
\(310\) 17085.7 3.13032
\(311\) 6060.79 1.10507 0.552534 0.833490i \(-0.313661\pi\)
0.552534 + 0.833490i \(0.313661\pi\)
\(312\) 0 0
\(313\) 969.946 0.175158 0.0875792 0.996158i \(-0.472087\pi\)
0.0875792 + 0.996158i \(0.472087\pi\)
\(314\) 11157.8 2.00532
\(315\) 0 0
\(316\) −8080.38 −1.43847
\(317\) 8741.63 1.54883 0.774414 0.632679i \(-0.218045\pi\)
0.774414 + 0.632679i \(0.218045\pi\)
\(318\) 0 0
\(319\) 6240.92 1.09537
\(320\) −8482.13 −1.48177
\(321\) 0 0
\(322\) −2462.85 −0.426241
\(323\) 1163.87 0.200493
\(324\) 0 0
\(325\) 0 0
\(326\) 15165.8 2.57655
\(327\) 0 0
\(328\) −197.243 −0.0332040
\(329\) 4193.81 0.702772
\(330\) 0 0
\(331\) −6987.76 −1.16037 −0.580184 0.814485i \(-0.697019\pi\)
−0.580184 + 0.814485i \(0.697019\pi\)
\(332\) −11702.2 −1.93446
\(333\) 0 0
\(334\) 13289.2 2.17711
\(335\) 3096.39 0.504996
\(336\) 0 0
\(337\) −4156.59 −0.671881 −0.335940 0.941883i \(-0.609054\pi\)
−0.335940 + 0.941883i \(0.609054\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 5219.30 0.832519
\(341\) −12483.8 −1.98250
\(342\) 0 0
\(343\) 9468.49 1.49053
\(344\) −1410.60 −0.221089
\(345\) 0 0
\(346\) −11089.3 −1.72301
\(347\) 312.513 0.0483475 0.0241737 0.999708i \(-0.492305\pi\)
0.0241737 + 0.999708i \(0.492305\pi\)
\(348\) 0 0
\(349\) 4458.75 0.683872 0.341936 0.939723i \(-0.388917\pi\)
0.341936 + 0.939723i \(0.388917\pi\)
\(350\) −7215.88 −1.10201
\(351\) 0 0
\(352\) 10519.1 1.59281
\(353\) −2249.17 −0.339126 −0.169563 0.985519i \(-0.554236\pi\)
−0.169563 + 0.985519i \(0.554236\pi\)
\(354\) 0 0
\(355\) 6041.16 0.903187
\(356\) −8325.41 −1.23945
\(357\) 0 0
\(358\) −6283.78 −0.927677
\(359\) −7842.79 −1.15300 −0.576499 0.817098i \(-0.695582\pi\)
−0.576499 + 0.817098i \(0.695582\pi\)
\(360\) 0 0
\(361\) −6131.19 −0.893890
\(362\) −1966.23 −0.285478
\(363\) 0 0
\(364\) 0 0
\(365\) 5238.64 0.751241
\(366\) 0 0
\(367\) 6660.24 0.947307 0.473653 0.880711i \(-0.342935\pi\)
0.473653 + 0.880711i \(0.342935\pi\)
\(368\) −1045.55 −0.148106
\(369\) 0 0
\(370\) 2411.60 0.338846
\(371\) 13777.1 1.92795
\(372\) 0 0
\(373\) −36.9873 −0.00513439 −0.00256720 0.999997i \(-0.500817\pi\)
−0.00256720 + 0.999997i \(0.500817\pi\)
\(374\) −7203.32 −0.995923
\(375\) 0 0
\(376\) −550.301 −0.0754777
\(377\) 0 0
\(378\) 0 0
\(379\) −12079.9 −1.63721 −0.818603 0.574360i \(-0.805251\pi\)
−0.818603 + 0.574360i \(0.805251\pi\)
\(380\) 3263.82 0.440607
\(381\) 0 0
\(382\) −5647.59 −0.756429
\(383\) −10567.4 −1.40984 −0.704919 0.709287i \(-0.749017\pi\)
−0.704919 + 0.709287i \(0.749017\pi\)
\(384\) 0 0
\(385\) 17104.9 2.26428
\(386\) −8842.90 −1.16604
\(387\) 0 0
\(388\) 14044.4 1.83763
\(389\) 9757.49 1.27179 0.635893 0.771778i \(-0.280632\pi\)
0.635893 + 0.771778i \(0.280632\pi\)
\(390\) 0 0
\(391\) 820.119 0.106075
\(392\) −2656.66 −0.342300
\(393\) 0 0
\(394\) 988.469 0.126392
\(395\) −12068.8 −1.53734
\(396\) 0 0
\(397\) 14200.5 1.79522 0.897612 0.440786i \(-0.145300\pi\)
0.897612 + 0.440786i \(0.145300\pi\)
\(398\) 6555.48 0.825620
\(399\) 0 0
\(400\) −3063.34 −0.382918
\(401\) −12676.4 −1.57863 −0.789313 0.613992i \(-0.789563\pi\)
−0.789313 + 0.613992i \(0.789563\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −8629.56 −1.06271
\(405\) 0 0
\(406\) −19966.0 −2.44063
\(407\) −1762.05 −0.214599
\(408\) 0 0
\(409\) −1533.68 −0.185417 −0.0927083 0.995693i \(-0.529552\pi\)
−0.0927083 + 0.995693i \(0.529552\pi\)
\(410\) −2651.41 −0.319375
\(411\) 0 0
\(412\) 5716.38 0.683557
\(413\) 18556.3 2.21089
\(414\) 0 0
\(415\) −17478.3 −2.06741
\(416\) 0 0
\(417\) 0 0
\(418\) −4504.50 −0.527087
\(419\) 2165.89 0.252532 0.126266 0.991996i \(-0.459701\pi\)
0.126266 + 0.991996i \(0.459701\pi\)
\(420\) 0 0
\(421\) −734.575 −0.0850380 −0.0425190 0.999096i \(-0.513538\pi\)
−0.0425190 + 0.999096i \(0.513538\pi\)
\(422\) 7721.98 0.890758
\(423\) 0 0
\(424\) −1807.79 −0.207062
\(425\) 2402.85 0.274248
\(426\) 0 0
\(427\) −17008.8 −1.92767
\(428\) 13034.9 1.47212
\(429\) 0 0
\(430\) −18961.8 −2.12656
\(431\) −13709.3 −1.53215 −0.766073 0.642754i \(-0.777792\pi\)
−0.766073 + 0.642754i \(0.777792\pi\)
\(432\) 0 0
\(433\) 10049.9 1.11540 0.557701 0.830042i \(-0.311684\pi\)
0.557701 + 0.830042i \(0.311684\pi\)
\(434\) 39938.2 4.41727
\(435\) 0 0
\(436\) 2979.76 0.327304
\(437\) 512.850 0.0561395
\(438\) 0 0
\(439\) −8133.47 −0.884258 −0.442129 0.896951i \(-0.645777\pi\)
−0.442129 + 0.896951i \(0.645777\pi\)
\(440\) −2244.47 −0.243184
\(441\) 0 0
\(442\) 0 0
\(443\) 2370.78 0.254264 0.127132 0.991886i \(-0.459423\pi\)
0.127132 + 0.991886i \(0.459423\pi\)
\(444\) 0 0
\(445\) −12434.8 −1.32464
\(446\) −231.423 −0.0245699
\(447\) 0 0
\(448\) −19827.2 −2.09095
\(449\) 12923.2 1.35832 0.679158 0.733992i \(-0.262345\pi\)
0.679158 + 0.733992i \(0.262345\pi\)
\(450\) 0 0
\(451\) 1937.27 0.202267
\(452\) −6256.84 −0.651099
\(453\) 0 0
\(454\) 2751.72 0.284459
\(455\) 0 0
\(456\) 0 0
\(457\) −8401.26 −0.859944 −0.429972 0.902842i \(-0.641477\pi\)
−0.429972 + 0.902842i \(0.641477\pi\)
\(458\) 2982.24 0.304260
\(459\) 0 0
\(460\) 2299.85 0.233111
\(461\) 17627.3 1.78088 0.890441 0.455098i \(-0.150396\pi\)
0.890441 + 0.455098i \(0.150396\pi\)
\(462\) 0 0
\(463\) 5461.81 0.548233 0.274116 0.961697i \(-0.411615\pi\)
0.274116 + 0.961697i \(0.411615\pi\)
\(464\) −8476.13 −0.848048
\(465\) 0 0
\(466\) −1135.72 −0.112899
\(467\) −8262.19 −0.818691 −0.409345 0.912379i \(-0.634243\pi\)
−0.409345 + 0.912379i \(0.634243\pi\)
\(468\) 0 0
\(469\) 7237.89 0.712611
\(470\) −7397.35 −0.725988
\(471\) 0 0
\(472\) −2434.91 −0.237449
\(473\) 13854.6 1.34680
\(474\) 0 0
\(475\) 1502.59 0.145145
\(476\) 12200.3 1.17479
\(477\) 0 0
\(478\) 6305.84 0.603395
\(479\) −1575.87 −0.150320 −0.0751601 0.997171i \(-0.523947\pi\)
−0.0751601 + 0.997171i \(0.523947\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 4022.20 0.380095
\(483\) 0 0
\(484\) 2780.46 0.261125
\(485\) 20976.7 1.96393
\(486\) 0 0
\(487\) 12595.7 1.17200 0.586001 0.810310i \(-0.300701\pi\)
0.586001 + 0.810310i \(0.300701\pi\)
\(488\) 2231.85 0.207031
\(489\) 0 0
\(490\) −35711.8 −3.29244
\(491\) 1071.21 0.0984586 0.0492293 0.998788i \(-0.484323\pi\)
0.0492293 + 0.998788i \(0.484323\pi\)
\(492\) 0 0
\(493\) 6648.59 0.607378
\(494\) 0 0
\(495\) 0 0
\(496\) 16954.9 1.53487
\(497\) 14121.4 1.27451
\(498\) 0 0
\(499\) 1422.30 0.127597 0.0637985 0.997963i \(-0.479678\pi\)
0.0637985 + 0.997963i \(0.479678\pi\)
\(500\) −8384.35 −0.749919
\(501\) 0 0
\(502\) −7727.28 −0.687022
\(503\) 9349.34 0.828760 0.414380 0.910104i \(-0.363998\pi\)
0.414380 + 0.910104i \(0.363998\pi\)
\(504\) 0 0
\(505\) −12889.1 −1.13576
\(506\) −3174.10 −0.278866
\(507\) 0 0
\(508\) 2224.39 0.194274
\(509\) 13736.3 1.19617 0.598086 0.801432i \(-0.295928\pi\)
0.598086 + 0.801432i \(0.295928\pi\)
\(510\) 0 0
\(511\) 12245.5 1.06009
\(512\) −16100.7 −1.38976
\(513\) 0 0
\(514\) −7496.58 −0.643307
\(515\) 8537.96 0.730538
\(516\) 0 0
\(517\) 5404.93 0.459785
\(518\) 5637.17 0.478153
\(519\) 0 0
\(520\) 0 0
\(521\) −11052.3 −0.929386 −0.464693 0.885472i \(-0.653835\pi\)
−0.464693 + 0.885472i \(0.653835\pi\)
\(522\) 0 0
\(523\) 6477.04 0.541532 0.270766 0.962645i \(-0.412723\pi\)
0.270766 + 0.962645i \(0.412723\pi\)
\(524\) −4250.19 −0.354332
\(525\) 0 0
\(526\) 2778.14 0.230290
\(527\) −13299.2 −1.09929
\(528\) 0 0
\(529\) −11805.6 −0.970298
\(530\) −24301.0 −1.99164
\(531\) 0 0
\(532\) 7629.27 0.621750
\(533\) 0 0
\(534\) 0 0
\(535\) 19468.9 1.57330
\(536\) −949.739 −0.0765344
\(537\) 0 0
\(538\) −13838.8 −1.10898
\(539\) 26093.1 2.08517
\(540\) 0 0
\(541\) −18341.5 −1.45761 −0.728803 0.684723i \(-0.759923\pi\)
−0.728803 + 0.684723i \(0.759923\pi\)
\(542\) 36759.7 2.91322
\(543\) 0 0
\(544\) 11206.2 0.883204
\(545\) 4450.55 0.349799
\(546\) 0 0
\(547\) −18943.1 −1.48071 −0.740356 0.672215i \(-0.765343\pi\)
−0.740356 + 0.672215i \(0.765343\pi\)
\(548\) −16476.3 −1.28436
\(549\) 0 0
\(550\) −9299.75 −0.720987
\(551\) 4157.61 0.321452
\(552\) 0 0
\(553\) −28211.2 −2.16937
\(554\) −16563.8 −1.27027
\(555\) 0 0
\(556\) −900.000 −0.0686484
\(557\) 415.532 0.0316098 0.0158049 0.999875i \(-0.494969\pi\)
0.0158049 + 0.999875i \(0.494969\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −23231.1 −1.75303
\(561\) 0 0
\(562\) −7591.12 −0.569772
\(563\) −18291.8 −1.36929 −0.684643 0.728879i \(-0.740042\pi\)
−0.684643 + 0.728879i \(0.740042\pi\)
\(564\) 0 0
\(565\) −9345.19 −0.695850
\(566\) −19994.6 −1.48487
\(567\) 0 0
\(568\) −1852.97 −0.136882
\(569\) −4347.47 −0.320308 −0.160154 0.987092i \(-0.551199\pi\)
−0.160154 + 0.987092i \(0.551199\pi\)
\(570\) 0 0
\(571\) −16756.0 −1.22805 −0.614024 0.789288i \(-0.710450\pi\)
−0.614024 + 0.789288i \(0.710450\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6197.74 −0.450677
\(575\) 1058.80 0.0767915
\(576\) 0 0
\(577\) −19974.7 −1.44117 −0.720587 0.693364i \(-0.756128\pi\)
−0.720587 + 0.693364i \(0.756128\pi\)
\(578\) 12583.0 0.905506
\(579\) 0 0
\(580\) 18644.6 1.33478
\(581\) −40856.0 −2.91737
\(582\) 0 0
\(583\) 17755.7 1.26135
\(584\) −1606.82 −0.113854
\(585\) 0 0
\(586\) 5829.47 0.410944
\(587\) −15748.7 −1.10735 −0.553677 0.832732i \(-0.686776\pi\)
−0.553677 + 0.832732i \(0.686776\pi\)
\(588\) 0 0
\(589\) −8316.50 −0.581792
\(590\) −32731.0 −2.28392
\(591\) 0 0
\(592\) 2393.14 0.166144
\(593\) 13318.4 0.922297 0.461148 0.887323i \(-0.347438\pi\)
0.461148 + 0.887323i \(0.347438\pi\)
\(594\) 0 0
\(595\) 18222.3 1.25553
\(596\) 1346.01 0.0925080
\(597\) 0 0
\(598\) 0 0
\(599\) −2970.80 −0.202644 −0.101322 0.994854i \(-0.532307\pi\)
−0.101322 + 0.994854i \(0.532307\pi\)
\(600\) 0 0
\(601\) 10632.6 0.721654 0.360827 0.932633i \(-0.382495\pi\)
0.360827 + 0.932633i \(0.382495\pi\)
\(602\) −44323.8 −3.00083
\(603\) 0 0
\(604\) 7200.29 0.485059
\(605\) 4152.88 0.279072
\(606\) 0 0
\(607\) 11587.9 0.774856 0.387428 0.921900i \(-0.373364\pi\)
0.387428 + 0.921900i \(0.373364\pi\)
\(608\) 7007.67 0.467432
\(609\) 0 0
\(610\) 30001.4 1.99135
\(611\) 0 0
\(612\) 0 0
\(613\) −20792.3 −1.36998 −0.684988 0.728555i \(-0.740192\pi\)
−0.684988 + 0.728555i \(0.740192\pi\)
\(614\) −19072.0 −1.25356
\(615\) 0 0
\(616\) −5246.51 −0.343162
\(617\) −1562.78 −0.101969 −0.0509846 0.998699i \(-0.516236\pi\)
−0.0509846 + 0.998699i \(0.516236\pi\)
\(618\) 0 0
\(619\) −758.406 −0.0492454 −0.0246227 0.999697i \(-0.507838\pi\)
−0.0246227 + 0.999697i \(0.507838\pi\)
\(620\) −37294.9 −2.41581
\(621\) 0 0
\(622\) −24989.3 −1.61090
\(623\) −29066.7 −1.86923
\(624\) 0 0
\(625\) −19485.0 −1.24704
\(626\) −3999.19 −0.255335
\(627\) 0 0
\(628\) −24355.4 −1.54759
\(629\) −1877.15 −0.118994
\(630\) 0 0
\(631\) 14265.2 0.899981 0.449990 0.893033i \(-0.351427\pi\)
0.449990 + 0.893033i \(0.351427\pi\)
\(632\) 3701.81 0.232990
\(633\) 0 0
\(634\) −36042.7 −2.25779
\(635\) 3322.34 0.207627
\(636\) 0 0
\(637\) 0 0
\(638\) −25732.0 −1.59677
\(639\) 0 0
\(640\) 7038.88 0.434744
\(641\) −3985.64 −0.245590 −0.122795 0.992432i \(-0.539186\pi\)
−0.122795 + 0.992432i \(0.539186\pi\)
\(642\) 0 0
\(643\) 8156.55 0.500254 0.250127 0.968213i \(-0.419528\pi\)
0.250127 + 0.968213i \(0.419528\pi\)
\(644\) 5375.97 0.328949
\(645\) 0 0
\(646\) −4798.74 −0.292266
\(647\) −11279.2 −0.685368 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(648\) 0 0
\(649\) 23915.2 1.44646
\(650\) 0 0
\(651\) 0 0
\(652\) −33104.2 −1.98844
\(653\) −6565.75 −0.393473 −0.196736 0.980456i \(-0.563034\pi\)
−0.196736 + 0.980456i \(0.563034\pi\)
\(654\) 0 0
\(655\) −6348.06 −0.378686
\(656\) −2631.11 −0.156597
\(657\) 0 0
\(658\) −17291.5 −1.02446
\(659\) −4799.35 −0.283696 −0.141848 0.989888i \(-0.545304\pi\)
−0.141848 + 0.989888i \(0.545304\pi\)
\(660\) 0 0
\(661\) −15593.6 −0.917581 −0.458790 0.888545i \(-0.651717\pi\)
−0.458790 + 0.888545i \(0.651717\pi\)
\(662\) 28811.3 1.69151
\(663\) 0 0
\(664\) 5361.03 0.313326
\(665\) 11395.1 0.664483
\(666\) 0 0
\(667\) 2929.66 0.170070
\(668\) −29008.0 −1.68017
\(669\) 0 0
\(670\) −12766.7 −0.736152
\(671\) −21920.8 −1.26116
\(672\) 0 0
\(673\) −2205.54 −0.126326 −0.0631630 0.998003i \(-0.520119\pi\)
−0.0631630 + 0.998003i \(0.520119\pi\)
\(674\) 17138.1 0.979426
\(675\) 0 0
\(676\) 0 0
\(677\) 15046.4 0.854182 0.427091 0.904209i \(-0.359538\pi\)
0.427091 + 0.904209i \(0.359538\pi\)
\(678\) 0 0
\(679\) 49033.6 2.77134
\(680\) −2391.08 −0.134844
\(681\) 0 0
\(682\) 51471.9 2.88997
\(683\) 30632.5 1.71614 0.858068 0.513537i \(-0.171665\pi\)
0.858068 + 0.513537i \(0.171665\pi\)
\(684\) 0 0
\(685\) −24608.9 −1.37264
\(686\) −39039.6 −2.17280
\(687\) 0 0
\(688\) −18816.7 −1.04270
\(689\) 0 0
\(690\) 0 0
\(691\) −2175.72 −0.119780 −0.0598901 0.998205i \(-0.519075\pi\)
−0.0598901 + 0.998205i \(0.519075\pi\)
\(692\) 24205.9 1.32972
\(693\) 0 0
\(694\) −1288.52 −0.0704780
\(695\) −1344.24 −0.0733666
\(696\) 0 0
\(697\) 2063.82 0.112156
\(698\) −18383.9 −0.996906
\(699\) 0 0
\(700\) 15751.0 0.850473
\(701\) 32718.2 1.76284 0.881419 0.472335i \(-0.156589\pi\)
0.881419 + 0.472335i \(0.156589\pi\)
\(702\) 0 0
\(703\) −1173.85 −0.0629768
\(704\) −25553.1 −1.36799
\(705\) 0 0
\(706\) 9273.58 0.494356
\(707\) −30128.6 −1.60269
\(708\) 0 0
\(709\) 25219.7 1.33589 0.667945 0.744211i \(-0.267174\pi\)
0.667945 + 0.744211i \(0.267174\pi\)
\(710\) −24908.3 −1.31661
\(711\) 0 0
\(712\) 3814.06 0.200756
\(713\) −5860.22 −0.307808
\(714\) 0 0
\(715\) 0 0
\(716\) 13716.4 0.715929
\(717\) 0 0
\(718\) 32336.6 1.68077
\(719\) 35466.2 1.83959 0.919796 0.392398i \(-0.128354\pi\)
0.919796 + 0.392398i \(0.128354\pi\)
\(720\) 0 0
\(721\) 19957.7 1.03088
\(722\) 25279.5 1.30306
\(723\) 0 0
\(724\) 4291.93 0.220315
\(725\) 8583.57 0.439704
\(726\) 0 0
\(727\) −14262.2 −0.727588 −0.363794 0.931479i \(-0.618519\pi\)
−0.363794 + 0.931479i \(0.618519\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −21599.5 −1.09511
\(731\) 14759.6 0.746790
\(732\) 0 0
\(733\) −16022.5 −0.807371 −0.403685 0.914898i \(-0.632271\pi\)
−0.403685 + 0.914898i \(0.632271\pi\)
\(734\) −27460.9 −1.38093
\(735\) 0 0
\(736\) 4937.96 0.247304
\(737\) 9328.11 0.466222
\(738\) 0 0
\(739\) −3796.21 −0.188966 −0.0944830 0.995526i \(-0.530120\pi\)
−0.0944830 + 0.995526i \(0.530120\pi\)
\(740\) −5264.09 −0.261502
\(741\) 0 0
\(742\) −56804.3 −2.81044
\(743\) −30329.3 −1.49754 −0.748772 0.662827i \(-0.769356\pi\)
−0.748772 + 0.662827i \(0.769356\pi\)
\(744\) 0 0
\(745\) 2010.40 0.0988661
\(746\) 152.502 0.00748460
\(747\) 0 0
\(748\) 15723.6 0.768597
\(749\) 45509.1 2.22011
\(750\) 0 0
\(751\) 21551.9 1.04719 0.523595 0.851967i \(-0.324591\pi\)
0.523595 + 0.851967i \(0.324591\pi\)
\(752\) −7340.72 −0.355969
\(753\) 0 0
\(754\) 0 0
\(755\) 10754.3 0.518397
\(756\) 0 0
\(757\) −20417.6 −0.980306 −0.490153 0.871637i \(-0.663059\pi\)
−0.490153 + 0.871637i \(0.663059\pi\)
\(758\) 49806.5 2.38662
\(759\) 0 0
\(760\) −1495.23 −0.0713655
\(761\) −31375.4 −1.49456 −0.747278 0.664512i \(-0.768640\pi\)
−0.747278 + 0.664512i \(0.768640\pi\)
\(762\) 0 0
\(763\) 10403.3 0.493609
\(764\) 12327.7 0.583770
\(765\) 0 0
\(766\) 43570.5 2.05518
\(767\) 0 0
\(768\) 0 0
\(769\) −12452.7 −0.583946 −0.291973 0.956427i \(-0.594312\pi\)
−0.291973 + 0.956427i \(0.594312\pi\)
\(770\) −70525.5 −3.30073
\(771\) 0 0
\(772\) 19302.5 0.899885
\(773\) −37449.8 −1.74253 −0.871264 0.490815i \(-0.836699\pi\)
−0.871264 + 0.490815i \(0.836699\pi\)
\(774\) 0 0
\(775\) −17169.8 −0.795815
\(776\) −6434.08 −0.297642
\(777\) 0 0
\(778\) −40231.2 −1.85393
\(779\) 1290.58 0.0593580
\(780\) 0 0
\(781\) 18199.5 0.833839
\(782\) −3381.44 −0.154629
\(783\) 0 0
\(784\) −35438.4 −1.61436
\(785\) −36377.1 −1.65396
\(786\) 0 0
\(787\) 26460.2 1.19848 0.599240 0.800570i \(-0.295470\pi\)
0.599240 + 0.800570i \(0.295470\pi\)
\(788\) −2157.65 −0.0975420
\(789\) 0 0
\(790\) 49761.0 2.24104
\(791\) −21844.6 −0.981928
\(792\) 0 0
\(793\) 0 0
\(794\) −58550.3 −2.61697
\(795\) 0 0
\(796\) −14309.4 −0.637167
\(797\) −4749.47 −0.211085 −0.105543 0.994415i \(-0.533658\pi\)
−0.105543 + 0.994415i \(0.533658\pi\)
\(798\) 0 0
\(799\) 5757.99 0.254947
\(800\) 14467.6 0.639386
\(801\) 0 0
\(802\) 52266.1 2.30122
\(803\) 15781.8 0.693560
\(804\) 0 0
\(805\) 8029.53 0.351557
\(806\) 0 0
\(807\) 0 0
\(808\) 3953.40 0.172129
\(809\) 4464.04 0.194002 0.0970009 0.995284i \(-0.469075\pi\)
0.0970009 + 0.995284i \(0.469075\pi\)
\(810\) 0 0
\(811\) 20774.6 0.899499 0.449749 0.893155i \(-0.351513\pi\)
0.449749 + 0.893155i \(0.351513\pi\)
\(812\) 43582.2 1.88354
\(813\) 0 0
\(814\) 7265.13 0.312829
\(815\) −49444.3 −2.12510
\(816\) 0 0
\(817\) 9229.73 0.395235
\(818\) 6323.51 0.270289
\(819\) 0 0
\(820\) 5787.55 0.246476
\(821\) 35769.2 1.52053 0.760264 0.649614i \(-0.225070\pi\)
0.760264 + 0.649614i \(0.225070\pi\)
\(822\) 0 0
\(823\) 2945.44 0.124753 0.0623764 0.998053i \(-0.480132\pi\)
0.0623764 + 0.998053i \(0.480132\pi\)
\(824\) −2618.80 −0.110716
\(825\) 0 0
\(826\) −76509.6 −3.22289
\(827\) 17878.6 0.751753 0.375877 0.926670i \(-0.377342\pi\)
0.375877 + 0.926670i \(0.377342\pi\)
\(828\) 0 0
\(829\) 13423.8 0.562398 0.281199 0.959649i \(-0.409268\pi\)
0.281199 + 0.959649i \(0.409268\pi\)
\(830\) 72065.0 3.01375
\(831\) 0 0
\(832\) 0 0
\(833\) 27797.5 1.15621
\(834\) 0 0
\(835\) −43326.2 −1.79565
\(836\) 9832.53 0.406776
\(837\) 0 0
\(838\) −8930.21 −0.368125
\(839\) 31542.7 1.29794 0.648971 0.760813i \(-0.275200\pi\)
0.648971 + 0.760813i \(0.275200\pi\)
\(840\) 0 0
\(841\) −638.669 −0.0261867
\(842\) 3028.73 0.123963
\(843\) 0 0
\(844\) −16855.7 −0.687437
\(845\) 0 0
\(846\) 0 0
\(847\) 9707.47 0.393805
\(848\) −24115.0 −0.976547
\(849\) 0 0
\(850\) −9907.22 −0.399782
\(851\) −827.155 −0.0333191
\(852\) 0 0
\(853\) 21810.6 0.875476 0.437738 0.899103i \(-0.355780\pi\)
0.437738 + 0.899103i \(0.355780\pi\)
\(854\) 70129.1 2.81003
\(855\) 0 0
\(856\) −5971.59 −0.238440
\(857\) −33234.6 −1.32470 −0.662352 0.749193i \(-0.730442\pi\)
−0.662352 + 0.749193i \(0.730442\pi\)
\(858\) 0 0
\(859\) 42697.7 1.69596 0.847978 0.530032i \(-0.177820\pi\)
0.847978 + 0.530032i \(0.177820\pi\)
\(860\) 41390.3 1.64116
\(861\) 0 0
\(862\) 56525.0 2.23347
\(863\) 27419.2 1.08153 0.540765 0.841174i \(-0.318135\pi\)
0.540765 + 0.841174i \(0.318135\pi\)
\(864\) 0 0
\(865\) 36153.8 1.42112
\(866\) −41437.0 −1.62596
\(867\) 0 0
\(868\) −87177.9 −3.40900
\(869\) −36358.3 −1.41930
\(870\) 0 0
\(871\) 0 0
\(872\) −1365.09 −0.0530137
\(873\) 0 0
\(874\) −2114.54 −0.0818367
\(875\) −29272.4 −1.13096
\(876\) 0 0
\(877\) −42604.2 −1.64041 −0.820206 0.572068i \(-0.806141\pi\)
−0.820206 + 0.572068i \(0.806141\pi\)
\(878\) 33535.2 1.28902
\(879\) 0 0
\(880\) −29940.0 −1.14691
\(881\) 4699.40 0.179712 0.0898562 0.995955i \(-0.471359\pi\)
0.0898562 + 0.995955i \(0.471359\pi\)
\(882\) 0 0
\(883\) 22233.5 0.847358 0.423679 0.905812i \(-0.360738\pi\)
0.423679 + 0.905812i \(0.360738\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −9774.98 −0.370651
\(887\) 8852.46 0.335103 0.167552 0.985863i \(-0.446414\pi\)
0.167552 + 0.985863i \(0.446414\pi\)
\(888\) 0 0
\(889\) 7766.05 0.292986
\(890\) 51270.0 1.93098
\(891\) 0 0
\(892\) 505.155 0.0189617
\(893\) 3600.68 0.134930
\(894\) 0 0
\(895\) 20486.7 0.765135
\(896\) 16453.6 0.613477
\(897\) 0 0
\(898\) −53283.7 −1.98007
\(899\) −47508.0 −1.76249
\(900\) 0 0
\(901\) 18915.5 0.699410
\(902\) −7987.58 −0.294853
\(903\) 0 0
\(904\) 2866.40 0.105459
\(905\) 6410.41 0.235458
\(906\) 0 0
\(907\) 37172.4 1.36085 0.680424 0.732818i \(-0.261795\pi\)
0.680424 + 0.732818i \(0.261795\pi\)
\(908\) −6006.51 −0.219530
\(909\) 0 0
\(910\) 0 0
\(911\) −38035.1 −1.38327 −0.691635 0.722247i \(-0.743109\pi\)
−0.691635 + 0.722247i \(0.743109\pi\)
\(912\) 0 0
\(913\) −52654.8 −1.90867
\(914\) 34639.3 1.25357
\(915\) 0 0
\(916\) −6509.70 −0.234810
\(917\) −14838.8 −0.534372
\(918\) 0 0
\(919\) −8352.27 −0.299800 −0.149900 0.988701i \(-0.547895\pi\)
−0.149900 + 0.988701i \(0.547895\pi\)
\(920\) −1053.62 −0.0377573
\(921\) 0 0
\(922\) −72679.4 −2.59606
\(923\) 0 0
\(924\) 0 0
\(925\) −2423.47 −0.0861440
\(926\) −22519.6 −0.799179
\(927\) 0 0
\(928\) 40031.3 1.41605
\(929\) 20232.1 0.714524 0.357262 0.934004i \(-0.383710\pi\)
0.357262 + 0.934004i \(0.383710\pi\)
\(930\) 0 0
\(931\) 17382.8 0.611921
\(932\) 2479.06 0.0871292
\(933\) 0 0
\(934\) 34065.9 1.19344
\(935\) 23484.7 0.821423
\(936\) 0 0
\(937\) −27766.9 −0.968095 −0.484048 0.875042i \(-0.660834\pi\)
−0.484048 + 0.875042i \(0.660834\pi\)
\(938\) −29842.6 −1.03880
\(939\) 0 0
\(940\) 16147.1 0.560277
\(941\) −400.765 −0.0138837 −0.00694185 0.999976i \(-0.502210\pi\)
−0.00694185 + 0.999976i \(0.502210\pi\)
\(942\) 0 0
\(943\) 909.408 0.0314045
\(944\) −32480.4 −1.11986
\(945\) 0 0
\(946\) −57124.0 −1.96328
\(947\) 21804.4 0.748201 0.374101 0.927388i \(-0.377951\pi\)
0.374101 + 0.927388i \(0.377951\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −6195.35 −0.211583
\(951\) 0 0
\(952\) −5589.22 −0.190281
\(953\) −30480.4 −1.03605 −0.518026 0.855365i \(-0.673333\pi\)
−0.518026 + 0.855365i \(0.673333\pi\)
\(954\) 0 0
\(955\) 18412.6 0.623892
\(956\) −13764.5 −0.465666
\(957\) 0 0
\(958\) 6497.48 0.219127
\(959\) −57524.0 −1.93696
\(960\) 0 0
\(961\) 65239.6 2.18991
\(962\) 0 0
\(963\) 0 0
\(964\) −8779.73 −0.293336
\(965\) 28830.1 0.961734
\(966\) 0 0
\(967\) −23864.1 −0.793608 −0.396804 0.917903i \(-0.629881\pi\)
−0.396804 + 0.917903i \(0.629881\pi\)
\(968\) −1273.79 −0.0422947
\(969\) 0 0
\(970\) −86489.2 −2.86289
\(971\) 14010.3 0.463040 0.231520 0.972830i \(-0.425630\pi\)
0.231520 + 0.972830i \(0.425630\pi\)
\(972\) 0 0
\(973\) −3142.19 −0.103529
\(974\) −51933.3 −1.70847
\(975\) 0 0
\(976\) 29771.7 0.976404
\(977\) −25448.2 −0.833326 −0.416663 0.909061i \(-0.636800\pi\)
−0.416663 + 0.909061i \(0.636800\pi\)
\(978\) 0 0
\(979\) −37460.8 −1.22293
\(980\) 77952.4 2.54092
\(981\) 0 0
\(982\) −4416.72 −0.143527
\(983\) 52479.9 1.70280 0.851399 0.524519i \(-0.175755\pi\)
0.851399 + 0.524519i \(0.175755\pi\)
\(984\) 0 0
\(985\) −3222.66 −0.104246
\(986\) −27412.8 −0.885398
\(987\) 0 0
\(988\) 0 0
\(989\) 6503.73 0.209107
\(990\) 0 0
\(991\) 38048.5 1.21963 0.609814 0.792545i \(-0.291244\pi\)
0.609814 + 0.792545i \(0.291244\pi\)
\(992\) −80075.0 −2.56289
\(993\) 0 0
\(994\) −58223.9 −1.85790
\(995\) −21372.5 −0.680960
\(996\) 0 0
\(997\) −31236.6 −0.992251 −0.496125 0.868251i \(-0.665244\pi\)
−0.496125 + 0.868251i \(0.665244\pi\)
\(998\) −5864.30 −0.186003
\(999\) 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.z.1.2 4
3.2 odd 2 507.4.a.k.1.3 4
13.6 odd 12 117.4.q.d.10.2 4
13.11 odd 12 117.4.q.d.82.2 4
13.12 even 2 inner 1521.4.a.z.1.3 4
39.5 even 4 507.4.b.e.337.2 4
39.8 even 4 507.4.b.e.337.3 4
39.11 even 12 39.4.j.b.4.1 4
39.32 even 12 39.4.j.b.10.1 yes 4
39.38 odd 2 507.4.a.k.1.2 4
156.11 odd 12 624.4.bv.c.433.1 4
156.71 odd 12 624.4.bv.c.49.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.1 4 39.11 even 12
39.4.j.b.10.1 yes 4 39.32 even 12
117.4.q.d.10.2 4 13.6 odd 12
117.4.q.d.82.2 4 13.11 odd 12
507.4.a.k.1.2 4 39.38 odd 2
507.4.a.k.1.3 4 3.2 odd 2
507.4.b.e.337.2 4 39.5 even 4
507.4.b.e.337.3 4 39.8 even 4
624.4.bv.c.49.2 4 156.71 odd 12
624.4.bv.c.433.1 4 156.11 odd 12
1521.4.a.z.1.2 4 1.1 even 1 trivial
1521.4.a.z.1.3 4 13.12 even 2 inner