Properties

Label 882.4.a.bi.1.1
Level $882$
Weight $4$
Character 882.1
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 294)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -3.89949 q^{5} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} -3.89949 q^{5} +8.00000 q^{8} -7.79899 q^{10} +61.3970 q^{11} -53.6985 q^{13} +16.0000 q^{16} +32.1005 q^{17} +55.7990 q^{19} -15.5980 q^{20} +122.794 q^{22} +94.6030 q^{23} -109.794 q^{25} -107.397 q^{26} -138.191 q^{29} -132.603 q^{31} +32.0000 q^{32} +64.2010 q^{34} +149.206 q^{37} +111.598 q^{38} -31.1960 q^{40} +427.497 q^{41} +437.588 q^{43} +245.588 q^{44} +189.206 q^{46} +57.0051 q^{47} -219.588 q^{50} -214.794 q^{52} +263.588 q^{53} -239.417 q^{55} -276.382 q^{58} +451.799 q^{59} -579.307 q^{61} -265.206 q^{62} +64.0000 q^{64} +209.397 q^{65} +309.588 q^{67} +128.402 q^{68} +1058.98 q^{71} -1193.66 q^{73} +298.412 q^{74} +223.196 q^{76} +1319.56 q^{79} -62.3919 q^{80} +854.995 q^{82} +1190.33 q^{83} -125.176 q^{85} +875.176 q^{86} +491.176 q^{88} -233.085 q^{89} +378.412 q^{92} +114.010 q^{94} -217.588 q^{95} +1609.44 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 12q^{5} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 12q^{5} + 16q^{8} + 24q^{10} + 4q^{11} - 48q^{13} + 32q^{16} + 84q^{17} + 72q^{19} + 48q^{20} + 8q^{22} + 308q^{23} + 18q^{25} - 96q^{26} + 80q^{29} - 384q^{31} + 64q^{32} + 168q^{34} + 536q^{37} + 144q^{38} + 96q^{40} + 756q^{41} + 400q^{43} + 16q^{44} + 616q^{46} + 312q^{47} + 36q^{50} - 192q^{52} + 52q^{53} - 1152q^{55} + 160q^{58} + 864q^{59} - 1416q^{61} - 768q^{62} + 128q^{64} + 300q^{65} + 144q^{67} + 336q^{68} + 1524q^{71} - 744q^{73} + 1072q^{74} + 288q^{76} + 976q^{79} + 192q^{80} + 1512q^{82} - 312q^{83} + 700q^{85} + 800q^{86} + 32q^{88} + 108q^{89} + 1232q^{92} + 624q^{94} + 40q^{95} + 744q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −3.89949 −0.348781 −0.174391 0.984677i \(-0.555796\pi\)
−0.174391 + 0.984677i \(0.555796\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −7.79899 −0.246626
\(11\) 61.3970 1.68290 0.841449 0.540336i \(-0.181703\pi\)
0.841449 + 0.540336i \(0.181703\pi\)
\(12\) 0 0
\(13\) −53.6985 −1.14564 −0.572818 0.819682i \(-0.694150\pi\)
−0.572818 + 0.819682i \(0.694150\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 32.1005 0.457972 0.228986 0.973430i \(-0.426459\pi\)
0.228986 + 0.973430i \(0.426459\pi\)
\(18\) 0 0
\(19\) 55.7990 0.673746 0.336873 0.941550i \(-0.390631\pi\)
0.336873 + 0.941550i \(0.390631\pi\)
\(20\) −15.5980 −0.174391
\(21\) 0 0
\(22\) 122.794 1.18999
\(23\) 94.6030 0.857656 0.428828 0.903386i \(-0.358927\pi\)
0.428828 + 0.903386i \(0.358927\pi\)
\(24\) 0 0
\(25\) −109.794 −0.878352
\(26\) −107.397 −0.810088
\(27\) 0 0
\(28\) 0 0
\(29\) −138.191 −0.884876 −0.442438 0.896799i \(-0.645886\pi\)
−0.442438 + 0.896799i \(0.645886\pi\)
\(30\) 0 0
\(31\) −132.603 −0.768265 −0.384132 0.923278i \(-0.625499\pi\)
−0.384132 + 0.923278i \(0.625499\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 64.2010 0.323835
\(35\) 0 0
\(36\) 0 0
\(37\) 149.206 0.662955 0.331477 0.943463i \(-0.392453\pi\)
0.331477 + 0.943463i \(0.392453\pi\)
\(38\) 111.598 0.476410
\(39\) 0 0
\(40\) −31.1960 −0.123313
\(41\) 427.497 1.62839 0.814194 0.580593i \(-0.197179\pi\)
0.814194 + 0.580593i \(0.197179\pi\)
\(42\) 0 0
\(43\) 437.588 1.55190 0.775948 0.630797i \(-0.217272\pi\)
0.775948 + 0.630797i \(0.217272\pi\)
\(44\) 245.588 0.841449
\(45\) 0 0
\(46\) 189.206 0.606455
\(47\) 57.0051 0.176916 0.0884579 0.996080i \(-0.471806\pi\)
0.0884579 + 0.996080i \(0.471806\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −219.588 −0.621088
\(51\) 0 0
\(52\) −214.794 −0.572818
\(53\) 263.588 0.683143 0.341572 0.939856i \(-0.389041\pi\)
0.341572 + 0.939856i \(0.389041\pi\)
\(54\) 0 0
\(55\) −239.417 −0.586964
\(56\) 0 0
\(57\) 0 0
\(58\) −276.382 −0.625702
\(59\) 451.799 0.996936 0.498468 0.866908i \(-0.333896\pi\)
0.498468 + 0.866908i \(0.333896\pi\)
\(60\) 0 0
\(61\) −579.307 −1.21594 −0.607972 0.793958i \(-0.708017\pi\)
−0.607972 + 0.793958i \(0.708017\pi\)
\(62\) −265.206 −0.543245
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 209.397 0.399577
\(66\) 0 0
\(67\) 309.588 0.564510 0.282255 0.959339i \(-0.408918\pi\)
0.282255 + 0.959339i \(0.408918\pi\)
\(68\) 128.402 0.228986
\(69\) 0 0
\(70\) 0 0
\(71\) 1058.98 1.77012 0.885059 0.465479i \(-0.154118\pi\)
0.885059 + 0.465479i \(0.154118\pi\)
\(72\) 0 0
\(73\) −1193.66 −1.91380 −0.956898 0.290424i \(-0.906204\pi\)
−0.956898 + 0.290424i \(0.906204\pi\)
\(74\) 298.412 0.468780
\(75\) 0 0
\(76\) 223.196 0.336873
\(77\) 0 0
\(78\) 0 0
\(79\) 1319.56 1.87926 0.939632 0.342187i \(-0.111168\pi\)
0.939632 + 0.342187i \(0.111168\pi\)
\(80\) −62.3919 −0.0871954
\(81\) 0 0
\(82\) 854.995 1.15144
\(83\) 1190.33 1.57417 0.787083 0.616847i \(-0.211590\pi\)
0.787083 + 0.616847i \(0.211590\pi\)
\(84\) 0 0
\(85\) −125.176 −0.159732
\(86\) 875.176 1.09736
\(87\) 0 0
\(88\) 491.176 0.594994
\(89\) −233.085 −0.277607 −0.138803 0.990320i \(-0.544326\pi\)
−0.138803 + 0.990320i \(0.544326\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 378.412 0.428828
\(93\) 0 0
\(94\) 114.010 0.125098
\(95\) −217.588 −0.234990
\(96\) 0 0
\(97\) 1609.44 1.68468 0.842338 0.538950i \(-0.181179\pi\)
0.842338 + 0.538950i \(0.181179\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −439.176 −0.439176
\(101\) −1479.26 −1.45734 −0.728671 0.684864i \(-0.759861\pi\)
−0.728671 + 0.684864i \(0.759861\pi\)
\(102\) 0 0
\(103\) 1145.35 1.09567 0.547837 0.836585i \(-0.315452\pi\)
0.547837 + 0.836585i \(0.315452\pi\)
\(104\) −429.588 −0.405044
\(105\) 0 0
\(106\) 527.176 0.483055
\(107\) −436.955 −0.394785 −0.197392 0.980325i \(-0.563247\pi\)
−0.197392 + 0.980325i \(0.563247\pi\)
\(108\) 0 0
\(109\) −166.352 −0.146180 −0.0730898 0.997325i \(-0.523286\pi\)
−0.0730898 + 0.997325i \(0.523286\pi\)
\(110\) −478.834 −0.415046
\(111\) 0 0
\(112\) 0 0
\(113\) −490.824 −0.408609 −0.204305 0.978907i \(-0.565493\pi\)
−0.204305 + 0.978907i \(0.565493\pi\)
\(114\) 0 0
\(115\) −368.904 −0.299135
\(116\) −552.764 −0.442438
\(117\) 0 0
\(118\) 903.598 0.704940
\(119\) 0 0
\(120\) 0 0
\(121\) 2438.59 1.83215
\(122\) −1158.61 −0.859802
\(123\) 0 0
\(124\) −530.412 −0.384132
\(125\) 915.578 0.655134
\(126\) 0 0
\(127\) −2616.70 −1.82831 −0.914153 0.405369i \(-0.867143\pi\)
−0.914153 + 0.405369i \(0.867143\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 418.794 0.282544
\(131\) 177.588 0.118442 0.0592211 0.998245i \(-0.481138\pi\)
0.0592211 + 0.998245i \(0.481138\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 619.176 0.399169
\(135\) 0 0
\(136\) 256.804 0.161917
\(137\) −27.0152 −0.0168472 −0.00842358 0.999965i \(-0.502681\pi\)
−0.00842358 + 0.999965i \(0.502681\pi\)
\(138\) 0 0
\(139\) 922.754 0.563071 0.281536 0.959551i \(-0.409156\pi\)
0.281536 + 0.959551i \(0.409156\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2117.97 1.25166
\(143\) −3296.92 −1.92799
\(144\) 0 0
\(145\) 538.875 0.308628
\(146\) −2387.32 −1.35326
\(147\) 0 0
\(148\) 596.824 0.331477
\(149\) −746.703 −0.410552 −0.205276 0.978704i \(-0.565809\pi\)
−0.205276 + 0.978704i \(0.565809\pi\)
\(150\) 0 0
\(151\) 2073.15 1.11729 0.558643 0.829408i \(-0.311322\pi\)
0.558643 + 0.829408i \(0.311322\pi\)
\(152\) 446.392 0.238205
\(153\) 0 0
\(154\) 0 0
\(155\) 517.085 0.267956
\(156\) 0 0
\(157\) −1566.22 −0.796166 −0.398083 0.917349i \(-0.630324\pi\)
−0.398083 + 0.917349i \(0.630324\pi\)
\(158\) 2639.12 1.32884
\(159\) 0 0
\(160\) −124.784 −0.0616564
\(161\) 0 0
\(162\) 0 0
\(163\) 98.7333 0.0474441 0.0237221 0.999719i \(-0.492448\pi\)
0.0237221 + 0.999719i \(0.492448\pi\)
\(164\) 1709.99 0.814194
\(165\) 0 0
\(166\) 2380.66 1.11310
\(167\) −2231.36 −1.03394 −0.516969 0.856004i \(-0.672940\pi\)
−0.516969 + 0.856004i \(0.672940\pi\)
\(168\) 0 0
\(169\) 686.527 0.312484
\(170\) −250.352 −0.112948
\(171\) 0 0
\(172\) 1750.35 0.775948
\(173\) 2100.84 0.923261 0.461631 0.887072i \(-0.347265\pi\)
0.461631 + 0.887072i \(0.347265\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 982.352 0.420725
\(177\) 0 0
\(178\) −466.171 −0.196298
\(179\) −1722.54 −0.719267 −0.359634 0.933094i \(-0.617098\pi\)
−0.359634 + 0.933094i \(0.617098\pi\)
\(180\) 0 0
\(181\) −1655.00 −0.679644 −0.339822 0.940490i \(-0.610367\pi\)
−0.339822 + 0.940490i \(0.610367\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 756.824 0.303227
\(185\) −581.828 −0.231226
\(186\) 0 0
\(187\) 1970.87 0.770720
\(188\) 228.020 0.0884579
\(189\) 0 0
\(190\) −435.176 −0.166163
\(191\) 1007.69 0.381747 0.190874 0.981615i \(-0.438868\pi\)
0.190874 + 0.981615i \(0.438868\pi\)
\(192\) 0 0
\(193\) −7.64849 −0.00285259 −0.00142630 0.999999i \(-0.500454\pi\)
−0.00142630 + 0.999999i \(0.500454\pi\)
\(194\) 3218.87 1.19125
\(195\) 0 0
\(196\) 0 0
\(197\) −2689.88 −0.972822 −0.486411 0.873730i \(-0.661694\pi\)
−0.486411 + 0.873730i \(0.661694\pi\)
\(198\) 0 0
\(199\) −867.497 −0.309021 −0.154511 0.987991i \(-0.549380\pi\)
−0.154511 + 0.987991i \(0.549380\pi\)
\(200\) −878.352 −0.310544
\(201\) 0 0
\(202\) −2958.51 −1.03050
\(203\) 0 0
\(204\) 0 0
\(205\) −1667.02 −0.567951
\(206\) 2290.69 0.774758
\(207\) 0 0
\(208\) −859.176 −0.286409
\(209\) 3425.89 1.13385
\(210\) 0 0
\(211\) 162.030 0.0528655 0.0264328 0.999651i \(-0.491585\pi\)
0.0264328 + 0.999651i \(0.491585\pi\)
\(212\) 1054.35 0.341572
\(213\) 0 0
\(214\) −873.909 −0.279155
\(215\) −1706.37 −0.541272
\(216\) 0 0
\(217\) 0 0
\(218\) −332.703 −0.103365
\(219\) 0 0
\(220\) −957.669 −0.293482
\(221\) −1723.75 −0.524669
\(222\) 0 0
\(223\) 4577.85 1.37469 0.687344 0.726332i \(-0.258777\pi\)
0.687344 + 0.726332i \(0.258777\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −981.648 −0.288930
\(227\) −2218.19 −0.648575 −0.324287 0.945959i \(-0.605125\pi\)
−0.324287 + 0.945959i \(0.605125\pi\)
\(228\) 0 0
\(229\) −785.217 −0.226588 −0.113294 0.993562i \(-0.536140\pi\)
−0.113294 + 0.993562i \(0.536140\pi\)
\(230\) −737.808 −0.211520
\(231\) 0 0
\(232\) −1105.53 −0.312851
\(233\) 5369.12 1.50963 0.754813 0.655940i \(-0.227727\pi\)
0.754813 + 0.655940i \(0.227727\pi\)
\(234\) 0 0
\(235\) −222.291 −0.0617049
\(236\) 1807.20 0.498468
\(237\) 0 0
\(238\) 0 0
\(239\) −3713.28 −1.00499 −0.502493 0.864581i \(-0.667584\pi\)
−0.502493 + 0.864581i \(0.667584\pi\)
\(240\) 0 0
\(241\) −6998.62 −1.87063 −0.935313 0.353821i \(-0.884882\pi\)
−0.935313 + 0.353821i \(0.884882\pi\)
\(242\) 4877.18 1.29552
\(243\) 0 0
\(244\) −2317.23 −0.607972
\(245\) 0 0
\(246\) 0 0
\(247\) −2996.32 −0.771868
\(248\) −1060.82 −0.271623
\(249\) 0 0
\(250\) 1831.16 0.463250
\(251\) 3722.75 0.936168 0.468084 0.883684i \(-0.344945\pi\)
0.468084 + 0.883684i \(0.344945\pi\)
\(252\) 0 0
\(253\) 5808.34 1.44335
\(254\) −5233.41 −1.29281
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 1230.15 0.298578 0.149289 0.988794i \(-0.452301\pi\)
0.149289 + 0.988794i \(0.452301\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 837.588 0.199788
\(261\) 0 0
\(262\) 355.176 0.0837513
\(263\) −2388.63 −0.560036 −0.280018 0.959995i \(-0.590340\pi\)
−0.280018 + 0.959995i \(0.590340\pi\)
\(264\) 0 0
\(265\) −1027.86 −0.238268
\(266\) 0 0
\(267\) 0 0
\(268\) 1238.35 0.282255
\(269\) 6702.31 1.51913 0.759567 0.650429i \(-0.225411\pi\)
0.759567 + 0.650429i \(0.225411\pi\)
\(270\) 0 0
\(271\) −4950.37 −1.10964 −0.554822 0.831969i \(-0.687214\pi\)
−0.554822 + 0.831969i \(0.687214\pi\)
\(272\) 513.608 0.114493
\(273\) 0 0
\(274\) −54.0303 −0.0119127
\(275\) −6741.02 −1.47818
\(276\) 0 0
\(277\) −3705.18 −0.803691 −0.401846 0.915707i \(-0.631631\pi\)
−0.401846 + 0.915707i \(0.631631\pi\)
\(278\) 1845.51 0.398152
\(279\) 0 0
\(280\) 0 0
\(281\) −9324.74 −1.97960 −0.989800 0.142465i \(-0.954497\pi\)
−0.989800 + 0.142465i \(0.954497\pi\)
\(282\) 0 0
\(283\) 5569.49 1.16986 0.584932 0.811082i \(-0.301121\pi\)
0.584932 + 0.811082i \(0.301121\pi\)
\(284\) 4235.94 0.885059
\(285\) 0 0
\(286\) −6593.85 −1.36330
\(287\) 0 0
\(288\) 0 0
\(289\) −3882.56 −0.790262
\(290\) 1077.75 0.218233
\(291\) 0 0
\(292\) −4774.63 −0.956898
\(293\) −1665.31 −0.332042 −0.166021 0.986122i \(-0.553092\pi\)
−0.166021 + 0.986122i \(0.553092\pi\)
\(294\) 0 0
\(295\) −1761.79 −0.347713
\(296\) 1193.65 0.234390
\(297\) 0 0
\(298\) −1493.41 −0.290304
\(299\) −5080.04 −0.982563
\(300\) 0 0
\(301\) 0 0
\(302\) 4146.29 0.790040
\(303\) 0 0
\(304\) 892.784 0.168436
\(305\) 2259.00 0.424099
\(306\) 0 0
\(307\) 5303.32 0.985916 0.492958 0.870053i \(-0.335916\pi\)
0.492958 + 0.870053i \(0.335916\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1034.17 0.189474
\(311\) 1125.99 0.205302 0.102651 0.994717i \(-0.467267\pi\)
0.102651 + 0.994717i \(0.467267\pi\)
\(312\) 0 0
\(313\) 8299.51 1.49877 0.749386 0.662133i \(-0.230349\pi\)
0.749386 + 0.662133i \(0.230349\pi\)
\(314\) −3132.44 −0.562974
\(315\) 0 0
\(316\) 5278.23 0.939632
\(317\) 4278.76 0.758105 0.379053 0.925375i \(-0.376250\pi\)
0.379053 + 0.925375i \(0.376250\pi\)
\(318\) 0 0
\(319\) −8484.50 −1.48916
\(320\) −249.568 −0.0435977
\(321\) 0 0
\(322\) 0 0
\(323\) 1791.18 0.308556
\(324\) 0 0
\(325\) 5895.77 1.00627
\(326\) 197.467 0.0335481
\(327\) 0 0
\(328\) 3419.98 0.575722
\(329\) 0 0
\(330\) 0 0
\(331\) −1707.12 −0.283479 −0.141739 0.989904i \(-0.545270\pi\)
−0.141739 + 0.989904i \(0.545270\pi\)
\(332\) 4761.33 0.787083
\(333\) 0 0
\(334\) −4462.71 −0.731104
\(335\) −1207.24 −0.196891
\(336\) 0 0
\(337\) 1710.67 0.276517 0.138259 0.990396i \(-0.455849\pi\)
0.138259 + 0.990396i \(0.455849\pi\)
\(338\) 1373.05 0.220960
\(339\) 0 0
\(340\) −500.703 −0.0798660
\(341\) −8141.42 −1.29291
\(342\) 0 0
\(343\) 0 0
\(344\) 3500.70 0.548678
\(345\) 0 0
\(346\) 4201.69 0.652844
\(347\) −8910.30 −1.37847 −0.689236 0.724537i \(-0.742054\pi\)
−0.689236 + 0.724537i \(0.742054\pi\)
\(348\) 0 0
\(349\) −5378.68 −0.824969 −0.412485 0.910965i \(-0.635339\pi\)
−0.412485 + 0.910965i \(0.635339\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1964.70 0.297497
\(353\) −4252.56 −0.641193 −0.320596 0.947216i \(-0.603883\pi\)
−0.320596 + 0.947216i \(0.603883\pi\)
\(354\) 0 0
\(355\) −4129.51 −0.617384
\(356\) −932.341 −0.138803
\(357\) 0 0
\(358\) −3445.08 −0.508599
\(359\) −4903.89 −0.720940 −0.360470 0.932771i \(-0.617384\pi\)
−0.360470 + 0.932771i \(0.617384\pi\)
\(360\) 0 0
\(361\) −3745.47 −0.546067
\(362\) −3310.01 −0.480581
\(363\) 0 0
\(364\) 0 0
\(365\) 4654.66 0.667497
\(366\) 0 0
\(367\) 4041.57 0.574845 0.287423 0.957804i \(-0.407202\pi\)
0.287423 + 0.957804i \(0.407202\pi\)
\(368\) 1513.65 0.214414
\(369\) 0 0
\(370\) −1163.66 −0.163502
\(371\) 0 0
\(372\) 0 0
\(373\) −7451.35 −1.03436 −0.517180 0.855877i \(-0.673018\pi\)
−0.517180 + 0.855877i \(0.673018\pi\)
\(374\) 3941.75 0.544981
\(375\) 0 0
\(376\) 456.040 0.0625492
\(377\) 7420.64 1.01375
\(378\) 0 0
\(379\) −12564.4 −1.70288 −0.851438 0.524456i \(-0.824269\pi\)
−0.851438 + 0.524456i \(0.824269\pi\)
\(380\) −870.352 −0.117495
\(381\) 0 0
\(382\) 2015.38 0.269936
\(383\) −4289.93 −0.572337 −0.286169 0.958179i \(-0.592382\pi\)
−0.286169 + 0.958179i \(0.592382\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −15.2970 −0.00201709
\(387\) 0 0
\(388\) 6437.75 0.842338
\(389\) −5662.77 −0.738082 −0.369041 0.929413i \(-0.620314\pi\)
−0.369041 + 0.929413i \(0.620314\pi\)
\(390\) 0 0
\(391\) 3036.81 0.392782
\(392\) 0 0
\(393\) 0 0
\(394\) −5379.76 −0.687889
\(395\) −5145.61 −0.655452
\(396\) 0 0
\(397\) −14561.4 −1.84084 −0.920421 0.390928i \(-0.872154\pi\)
−0.920421 + 0.390928i \(0.872154\pi\)
\(398\) −1734.99 −0.218511
\(399\) 0 0
\(400\) −1756.70 −0.219588
\(401\) 3742.19 0.466025 0.233013 0.972474i \(-0.425142\pi\)
0.233013 + 0.972474i \(0.425142\pi\)
\(402\) 0 0
\(403\) 7120.58 0.880152
\(404\) −5917.02 −0.728671
\(405\) 0 0
\(406\) 0 0
\(407\) 9160.80 1.11569
\(408\) 0 0
\(409\) 3517.17 0.425215 0.212608 0.977138i \(-0.431804\pi\)
0.212608 + 0.977138i \(0.431804\pi\)
\(410\) −3334.05 −0.401602
\(411\) 0 0
\(412\) 4581.39 0.547837
\(413\) 0 0
\(414\) 0 0
\(415\) −4641.69 −0.549040
\(416\) −1718.35 −0.202522
\(417\) 0 0
\(418\) 6851.78 0.801750
\(419\) −7579.52 −0.883732 −0.441866 0.897081i \(-0.645683\pi\)
−0.441866 + 0.897081i \(0.645683\pi\)
\(420\) 0 0
\(421\) −4980.87 −0.576610 −0.288305 0.957539i \(-0.593092\pi\)
−0.288305 + 0.957539i \(0.593092\pi\)
\(422\) 324.061 0.0373816
\(423\) 0 0
\(424\) 2108.70 0.241528
\(425\) −3524.44 −0.402260
\(426\) 0 0
\(427\) 0 0
\(428\) −1747.82 −0.197392
\(429\) 0 0
\(430\) −3412.74 −0.382737
\(431\) −14203.3 −1.58736 −0.793678 0.608339i \(-0.791836\pi\)
−0.793678 + 0.608339i \(0.791836\pi\)
\(432\) 0 0
\(433\) −3874.82 −0.430051 −0.215026 0.976608i \(-0.568983\pi\)
−0.215026 + 0.976608i \(0.568983\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −665.406 −0.0730898
\(437\) 5278.75 0.577842
\(438\) 0 0
\(439\) −7763.82 −0.844070 −0.422035 0.906579i \(-0.638684\pi\)
−0.422035 + 0.906579i \(0.638684\pi\)
\(440\) −1915.34 −0.207523
\(441\) 0 0
\(442\) −3447.50 −0.370997
\(443\) −9662.24 −1.03627 −0.518134 0.855299i \(-0.673373\pi\)
−0.518134 + 0.855299i \(0.673373\pi\)
\(444\) 0 0
\(445\) 908.915 0.0968241
\(446\) 9155.70 0.972051
\(447\) 0 0
\(448\) 0 0
\(449\) 10942.6 1.15014 0.575069 0.818105i \(-0.304975\pi\)
0.575069 + 0.818105i \(0.304975\pi\)
\(450\) 0 0
\(451\) 26247.0 2.74041
\(452\) −1963.30 −0.204305
\(453\) 0 0
\(454\) −4436.38 −0.458612
\(455\) 0 0
\(456\) 0 0
\(457\) 13618.3 1.39396 0.696979 0.717091i \(-0.254527\pi\)
0.696979 + 0.717091i \(0.254527\pi\)
\(458\) −1570.43 −0.160222
\(459\) 0 0
\(460\) −1475.62 −0.149567
\(461\) 11955.8 1.20789 0.603947 0.797025i \(-0.293594\pi\)
0.603947 + 0.797025i \(0.293594\pi\)
\(462\) 0 0
\(463\) 648.503 0.0650939 0.0325470 0.999470i \(-0.489638\pi\)
0.0325470 + 0.999470i \(0.489638\pi\)
\(464\) −2211.05 −0.221219
\(465\) 0 0
\(466\) 10738.2 1.06747
\(467\) 2784.74 0.275937 0.137969 0.990437i \(-0.455943\pi\)
0.137969 + 0.990437i \(0.455943\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −444.582 −0.0436320
\(471\) 0 0
\(472\) 3614.39 0.352470
\(473\) 26866.6 2.61168
\(474\) 0 0
\(475\) −6126.39 −0.591785
\(476\) 0 0
\(477\) 0 0
\(478\) −7426.55 −0.710633
\(479\) 11113.4 1.06009 0.530046 0.847969i \(-0.322175\pi\)
0.530046 + 0.847969i \(0.322175\pi\)
\(480\) 0 0
\(481\) −8012.14 −0.759505
\(482\) −13997.2 −1.32273
\(483\) 0 0
\(484\) 9754.35 0.916074
\(485\) −6275.99 −0.587584
\(486\) 0 0
\(487\) −3786.27 −0.352305 −0.176152 0.984363i \(-0.556365\pi\)
−0.176152 + 0.984363i \(0.556365\pi\)
\(488\) −4634.45 −0.429901
\(489\) 0 0
\(490\) 0 0
\(491\) 9582.12 0.880723 0.440361 0.897821i \(-0.354850\pi\)
0.440361 + 0.897821i \(0.354850\pi\)
\(492\) 0 0
\(493\) −4436.00 −0.405248
\(494\) −5992.64 −0.545793
\(495\) 0 0
\(496\) −2121.65 −0.192066
\(497\) 0 0
\(498\) 0 0
\(499\) 5581.55 0.500730 0.250365 0.968152i \(-0.419449\pi\)
0.250365 + 0.968152i \(0.419449\pi\)
\(500\) 3662.31 0.327567
\(501\) 0 0
\(502\) 7445.51 0.661970
\(503\) −14116.3 −1.25132 −0.625661 0.780095i \(-0.715171\pi\)
−0.625661 + 0.780095i \(0.715171\pi\)
\(504\) 0 0
\(505\) 5768.35 0.508294
\(506\) 11616.7 1.02060
\(507\) 0 0
\(508\) −10466.8 −0.914153
\(509\) 16787.6 1.46188 0.730941 0.682441i \(-0.239081\pi\)
0.730941 + 0.682441i \(0.239081\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 2460.30 0.211127
\(515\) −4466.27 −0.382150
\(516\) 0 0
\(517\) 3499.94 0.297731
\(518\) 0 0
\(519\) 0 0
\(520\) 1675.18 0.141272
\(521\) −5598.61 −0.470786 −0.235393 0.971900i \(-0.575638\pi\)
−0.235393 + 0.971900i \(0.575638\pi\)
\(522\) 0 0
\(523\) 13270.7 1.10954 0.554769 0.832004i \(-0.312807\pi\)
0.554769 + 0.832004i \(0.312807\pi\)
\(524\) 710.352 0.0592211
\(525\) 0 0
\(526\) −4777.27 −0.396005
\(527\) −4256.62 −0.351843
\(528\) 0 0
\(529\) −3217.27 −0.264426
\(530\) −2055.72 −0.168481
\(531\) 0 0
\(532\) 0 0
\(533\) −22956.0 −1.86554
\(534\) 0 0
\(535\) 1703.90 0.137694
\(536\) 2476.70 0.199584
\(537\) 0 0
\(538\) 13404.6 1.07419
\(539\) 0 0
\(540\) 0 0
\(541\) −23017.5 −1.82921 −0.914603 0.404353i \(-0.867497\pi\)
−0.914603 + 0.404353i \(0.867497\pi\)
\(542\) −9900.74 −0.784637
\(543\) 0 0
\(544\) 1027.22 0.0809587
\(545\) 648.687 0.0509848
\(546\) 0 0
\(547\) 4475.84 0.349859 0.174930 0.984581i \(-0.444030\pi\)
0.174930 + 0.984581i \(0.444030\pi\)
\(548\) −108.061 −0.00842358
\(549\) 0 0
\(550\) −13482.0 −1.04523
\(551\) −7710.91 −0.596181
\(552\) 0 0
\(553\) 0 0
\(554\) −7410.35 −0.568295
\(555\) 0 0
\(556\) 3691.01 0.281536
\(557\) −1541.70 −0.117278 −0.0586390 0.998279i \(-0.518676\pi\)
−0.0586390 + 0.998279i \(0.518676\pi\)
\(558\) 0 0
\(559\) −23497.8 −1.77791
\(560\) 0 0
\(561\) 0 0
\(562\) −18649.5 −1.39979
\(563\) 13079.7 0.979115 0.489557 0.871971i \(-0.337158\pi\)
0.489557 + 0.871971i \(0.337158\pi\)
\(564\) 0 0
\(565\) 1913.97 0.142515
\(566\) 11139.0 0.827219
\(567\) 0 0
\(568\) 8471.88 0.625831
\(569\) −11411.2 −0.840740 −0.420370 0.907353i \(-0.638100\pi\)
−0.420370 + 0.907353i \(0.638100\pi\)
\(570\) 0 0
\(571\) 2311.74 0.169428 0.0847139 0.996405i \(-0.473002\pi\)
0.0847139 + 0.996405i \(0.473002\pi\)
\(572\) −13187.7 −0.963995
\(573\) 0 0
\(574\) 0 0
\(575\) −10386.8 −0.753324
\(576\) 0 0
\(577\) −25097.0 −1.81075 −0.905373 0.424617i \(-0.860409\pi\)
−0.905373 + 0.424617i \(0.860409\pi\)
\(578\) −7765.12 −0.558800
\(579\) 0 0
\(580\) 2155.50 0.154314
\(581\) 0 0
\(582\) 0 0
\(583\) 16183.5 1.14966
\(584\) −9549.26 −0.676629
\(585\) 0 0
\(586\) −3330.61 −0.234789
\(587\) 19789.1 1.39145 0.695726 0.718307i \(-0.255083\pi\)
0.695726 + 0.718307i \(0.255083\pi\)
\(588\) 0 0
\(589\) −7399.12 −0.517615
\(590\) −3523.58 −0.245870
\(591\) 0 0
\(592\) 2387.30 0.165739
\(593\) −2774.13 −0.192108 −0.0960540 0.995376i \(-0.530622\pi\)
−0.0960540 + 0.995376i \(0.530622\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2986.81 −0.205276
\(597\) 0 0
\(598\) −10160.1 −0.694777
\(599\) 26190.3 1.78649 0.893245 0.449570i \(-0.148423\pi\)
0.893245 + 0.449570i \(0.148423\pi\)
\(600\) 0 0
\(601\) 11038.6 0.749211 0.374605 0.927184i \(-0.377778\pi\)
0.374605 + 0.927184i \(0.377778\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8292.58 0.558643
\(605\) −9509.26 −0.639019
\(606\) 0 0
\(607\) 1854.39 0.123999 0.0619996 0.998076i \(-0.480252\pi\)
0.0619996 + 0.998076i \(0.480252\pi\)
\(608\) 1785.57 0.119103
\(609\) 0 0
\(610\) 4518.01 0.299883
\(611\) −3061.08 −0.202681
\(612\) 0 0
\(613\) −14856.8 −0.978894 −0.489447 0.872033i \(-0.662801\pi\)
−0.489447 + 0.872033i \(0.662801\pi\)
\(614\) 10606.6 0.697148
\(615\) 0 0
\(616\) 0 0
\(617\) −7600.00 −0.495890 −0.247945 0.968774i \(-0.579755\pi\)
−0.247945 + 0.968774i \(0.579755\pi\)
\(618\) 0 0
\(619\) −22685.3 −1.47302 −0.736509 0.676427i \(-0.763527\pi\)
−0.736509 + 0.676427i \(0.763527\pi\)
\(620\) 2068.34 0.133978
\(621\) 0 0
\(622\) 2251.98 0.145171
\(623\) 0 0
\(624\) 0 0
\(625\) 10154.0 0.649853
\(626\) 16599.0 1.05979
\(627\) 0 0
\(628\) −6264.88 −0.398083
\(629\) 4789.59 0.303614
\(630\) 0 0
\(631\) −12024.1 −0.758595 −0.379297 0.925275i \(-0.623834\pi\)
−0.379297 + 0.925275i \(0.623834\pi\)
\(632\) 10556.5 0.664420
\(633\) 0 0
\(634\) 8557.53 0.536061
\(635\) 10203.8 0.637679
\(636\) 0 0
\(637\) 0 0
\(638\) −16969.0 −1.05299
\(639\) 0 0
\(640\) −499.135 −0.0308282
\(641\) −11320.6 −0.697559 −0.348779 0.937205i \(-0.613404\pi\)
−0.348779 + 0.937205i \(0.613404\pi\)
\(642\) 0 0
\(643\) −16843.6 −1.03304 −0.516521 0.856275i \(-0.672773\pi\)
−0.516521 + 0.856275i \(0.672773\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3582.35 0.218182
\(647\) 7719.32 0.469054 0.234527 0.972110i \(-0.424646\pi\)
0.234527 + 0.972110i \(0.424646\pi\)
\(648\) 0 0
\(649\) 27739.1 1.67774
\(650\) 11791.5 0.711542
\(651\) 0 0
\(652\) 394.933 0.0237221
\(653\) 30803.2 1.84597 0.922987 0.384832i \(-0.125741\pi\)
0.922987 + 0.384832i \(0.125741\pi\)
\(654\) 0 0
\(655\) −692.503 −0.0413104
\(656\) 6839.96 0.407097
\(657\) 0 0
\(658\) 0 0
\(659\) −9760.68 −0.576968 −0.288484 0.957485i \(-0.593151\pi\)
−0.288484 + 0.957485i \(0.593151\pi\)
\(660\) 0 0
\(661\) −18071.1 −1.06336 −0.531682 0.846944i \(-0.678440\pi\)
−0.531682 + 0.846944i \(0.678440\pi\)
\(662\) −3414.23 −0.200450
\(663\) 0 0
\(664\) 9522.65 0.556552
\(665\) 0 0
\(666\) 0 0
\(667\) −13073.3 −0.758920
\(668\) −8925.43 −0.516969
\(669\) 0 0
\(670\) −2414.47 −0.139223
\(671\) −35567.7 −2.04631
\(672\) 0 0
\(673\) 26591.1 1.52305 0.761524 0.648137i \(-0.224451\pi\)
0.761524 + 0.648137i \(0.224451\pi\)
\(674\) 3421.35 0.195527
\(675\) 0 0
\(676\) 2746.11 0.156242
\(677\) 33080.3 1.87796 0.938979 0.343975i \(-0.111773\pi\)
0.938979 + 0.343975i \(0.111773\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1001.41 −0.0564738
\(681\) 0 0
\(682\) −16282.8 −0.914227
\(683\) 10466.1 0.586344 0.293172 0.956060i \(-0.405289\pi\)
0.293172 + 0.956060i \(0.405289\pi\)
\(684\) 0 0
\(685\) 105.345 0.00587597
\(686\) 0 0
\(687\) 0 0
\(688\) 7001.41 0.387974
\(689\) −14154.3 −0.782634
\(690\) 0 0
\(691\) −13269.9 −0.730551 −0.365275 0.930900i \(-0.619025\pi\)
−0.365275 + 0.930900i \(0.619025\pi\)
\(692\) 8403.38 0.461631
\(693\) 0 0
\(694\) −17820.6 −0.974727
\(695\) −3598.27 −0.196389
\(696\) 0 0
\(697\) 13722.9 0.745755
\(698\) −10757.4 −0.583341
\(699\) 0 0
\(700\) 0 0
\(701\) 15169.9 0.817344 0.408672 0.912681i \(-0.365992\pi\)
0.408672 + 0.912681i \(0.365992\pi\)
\(702\) 0 0
\(703\) 8325.55 0.446663
\(704\) 3929.41 0.210362
\(705\) 0 0
\(706\) −8505.13 −0.453392
\(707\) 0 0
\(708\) 0 0
\(709\) 26317.7 1.39405 0.697026 0.717046i \(-0.254506\pi\)
0.697026 + 0.717046i \(0.254506\pi\)
\(710\) −8259.01 −0.436557
\(711\) 0 0
\(712\) −1864.68 −0.0981488
\(713\) −12544.6 −0.658907
\(714\) 0 0
\(715\) 12856.3 0.672447
\(716\) −6890.17 −0.359634
\(717\) 0 0
\(718\) −9807.78 −0.509781
\(719\) 23013.1 1.19366 0.596831 0.802367i \(-0.296426\pi\)
0.596831 + 0.802367i \(0.296426\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −7490.95 −0.386128
\(723\) 0 0
\(724\) −6620.02 −0.339822
\(725\) 15172.5 0.777232
\(726\) 0 0
\(727\) 16265.4 0.829780 0.414890 0.909872i \(-0.363820\pi\)
0.414890 + 0.909872i \(0.363820\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 9309.33 0.471991
\(731\) 14046.8 0.710724
\(732\) 0 0
\(733\) 866.444 0.0436601 0.0218300 0.999762i \(-0.493051\pi\)
0.0218300 + 0.999762i \(0.493051\pi\)
\(734\) 8083.14 0.406477
\(735\) 0 0
\(736\) 3027.30 0.151614
\(737\) 19007.8 0.950013
\(738\) 0 0
\(739\) 10990.1 0.547058 0.273529 0.961864i \(-0.411809\pi\)
0.273529 + 0.961864i \(0.411809\pi\)
\(740\) −2327.31 −0.115613
\(741\) 0 0
\(742\) 0 0
\(743\) 22416.4 1.10683 0.553417 0.832905i \(-0.313324\pi\)
0.553417 + 0.832905i \(0.313324\pi\)
\(744\) 0 0
\(745\) 2911.76 0.143193
\(746\) −14902.7 −0.731403
\(747\) 0 0
\(748\) 7883.49 0.385360
\(749\) 0 0
\(750\) 0 0
\(751\) −19717.9 −0.958080 −0.479040 0.877793i \(-0.659015\pi\)
−0.479040 + 0.877793i \(0.659015\pi\)
\(752\) 912.081 0.0442289
\(753\) 0 0
\(754\) 14841.3 0.716827
\(755\) −8084.22 −0.389689
\(756\) 0 0
\(757\) 839.321 0.0402981 0.0201490 0.999797i \(-0.493586\pi\)
0.0201490 + 0.999797i \(0.493586\pi\)
\(758\) −25128.8 −1.20411
\(759\) 0 0
\(760\) −1740.70 −0.0830815
\(761\) 18364.5 0.874786 0.437393 0.899270i \(-0.355902\pi\)
0.437393 + 0.899270i \(0.355902\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 4030.75 0.190874
\(765\) 0 0
\(766\) −8579.86 −0.404704
\(767\) −24260.9 −1.14213
\(768\) 0 0
\(769\) 14890.8 0.698277 0.349138 0.937071i \(-0.386474\pi\)
0.349138 + 0.937071i \(0.386474\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −30.5939 −0.00142630
\(773\) −16606.2 −0.772683 −0.386341 0.922356i \(-0.626261\pi\)
−0.386341 + 0.922356i \(0.626261\pi\)
\(774\) 0 0
\(775\) 14559.0 0.674807
\(776\) 12875.5 0.595623
\(777\) 0 0
\(778\) −11325.5 −0.521903
\(779\) 23853.9 1.09712
\(780\) 0 0
\(781\) 65018.5 2.97893
\(782\) 6073.61 0.277739
\(783\) 0 0
\(784\) 0 0
\(785\) 6107.47 0.277688
\(786\) 0 0
\(787\) −3156.20 −0.142956 −0.0714781 0.997442i \(-0.522772\pi\)
−0.0714781 + 0.997442i \(0.522772\pi\)
\(788\) −10759.5 −0.486411
\(789\) 0 0
\(790\) −10291.2 −0.463475
\(791\) 0 0
\(792\) 0 0
\(793\) 31107.9 1.39303
\(794\) −29122.8 −1.30167
\(795\) 0 0
\(796\) −3469.99 −0.154511
\(797\) −13514.8 −0.600652 −0.300326 0.953837i \(-0.597096\pi\)
−0.300326 + 0.953837i \(0.597096\pi\)
\(798\) 0 0
\(799\) 1829.89 0.0810224
\(800\) −3513.41 −0.155272
\(801\) 0 0
\(802\) 7484.38 0.329530
\(803\) −73287.0 −3.22072
\(804\) 0 0
\(805\) 0 0
\(806\) 14241.2 0.622362
\(807\) 0 0
\(808\) −11834.0 −0.515248
\(809\) 26758.1 1.16287 0.581437 0.813591i \(-0.302491\pi\)
0.581437 + 0.813591i \(0.302491\pi\)
\(810\) 0 0
\(811\) 15920.7 0.689338 0.344669 0.938724i \(-0.387991\pi\)
0.344669 + 0.938724i \(0.387991\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 18321.6 0.788909
\(815\) −385.010 −0.0165476
\(816\) 0 0
\(817\) 24417.0 1.04558
\(818\) 7034.35 0.300673
\(819\) 0 0
\(820\) −6668.10 −0.283976
\(821\) 19306.2 0.820696 0.410348 0.911929i \(-0.365407\pi\)
0.410348 + 0.911929i \(0.365407\pi\)
\(822\) 0 0
\(823\) 791.000 0.0335025 0.0167512 0.999860i \(-0.494668\pi\)
0.0167512 + 0.999860i \(0.494668\pi\)
\(824\) 9162.77 0.387379
\(825\) 0 0
\(826\) 0 0
\(827\) −29537.6 −1.24199 −0.620993 0.783816i \(-0.713270\pi\)
−0.620993 + 0.783816i \(0.713270\pi\)
\(828\) 0 0
\(829\) 5766.52 0.241592 0.120796 0.992677i \(-0.461455\pi\)
0.120796 + 0.992677i \(0.461455\pi\)
\(830\) −9283.38 −0.388230
\(831\) 0 0
\(832\) −3436.70 −0.143205
\(833\) 0 0
\(834\) 0 0
\(835\) 8701.16 0.360618
\(836\) 13703.6 0.566923
\(837\) 0 0
\(838\) −15159.0 −0.624893
\(839\) 29726.4 1.22321 0.611603 0.791165i \(-0.290525\pi\)
0.611603 + 0.791165i \(0.290525\pi\)
\(840\) 0 0
\(841\) −5292.27 −0.216994
\(842\) −9961.75 −0.407725
\(843\) 0 0
\(844\) 648.121 0.0264328
\(845\) −2677.11 −0.108989
\(846\) 0 0
\(847\) 0 0
\(848\) 4217.41 0.170786
\(849\) 0 0
\(850\) −7048.88 −0.284441
\(851\) 14115.3 0.568587
\(852\) 0 0
\(853\) 17829.6 0.715677 0.357838 0.933784i \(-0.383514\pi\)
0.357838 + 0.933784i \(0.383514\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3495.64 −0.139578
\(857\) −39682.4 −1.58171 −0.790856 0.612003i \(-0.790364\pi\)
−0.790856 + 0.612003i \(0.790364\pi\)
\(858\) 0 0
\(859\) 2195.13 0.0871909 0.0435955 0.999049i \(-0.486119\pi\)
0.0435955 + 0.999049i \(0.486119\pi\)
\(860\) −6825.49 −0.270636
\(861\) 0 0
\(862\) −28406.6 −1.12243
\(863\) −31917.1 −1.25894 −0.629472 0.777023i \(-0.716729\pi\)
−0.629472 + 0.777023i \(0.716729\pi\)
\(864\) 0 0
\(865\) −8192.23 −0.322016
\(866\) −7749.65 −0.304092
\(867\) 0 0
\(868\) 0 0
\(869\) 81016.8 3.16261
\(870\) 0 0
\(871\) −16624.4 −0.646724
\(872\) −1330.81 −0.0516823
\(873\) 0 0
\(874\) 10557.5 0.408596
\(875\) 0 0
\(876\) 0 0
\(877\) 28842.9 1.11055 0.555276 0.831666i \(-0.312613\pi\)
0.555276 + 0.831666i \(0.312613\pi\)
\(878\) −15527.6 −0.596848
\(879\) 0 0
\(880\) −3830.67 −0.146741
\(881\) −15350.2 −0.587015 −0.293508 0.955957i \(-0.594823\pi\)
−0.293508 + 0.955957i \(0.594823\pi\)
\(882\) 0 0
\(883\) 6089.64 0.232087 0.116043 0.993244i \(-0.462979\pi\)
0.116043 + 0.993244i \(0.462979\pi\)
\(884\) −6894.99 −0.262335
\(885\) 0 0
\(886\) −19324.5 −0.732752
\(887\) −384.441 −0.0145527 −0.00727637 0.999974i \(-0.502316\pi\)
−0.00727637 + 0.999974i \(0.502316\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1817.83 0.0684650
\(891\) 0 0
\(892\) 18311.4 0.687344
\(893\) 3180.82 0.119196
\(894\) 0 0
\(895\) 6717.05 0.250867
\(896\) 0 0
\(897\) 0 0
\(898\) 21885.2 0.813271
\(899\) 18324.5 0.679819
\(900\) 0 0
\(901\) 8461.30 0.312860
\(902\) 52494.1 1.93776
\(903\) 0 0
\(904\) −3926.59 −0.144465
\(905\) 6453.68 0.237047
\(906\) 0 0
\(907\) −7267.93 −0.266073 −0.133036 0.991111i \(-0.542473\pi\)
−0.133036 + 0.991111i \(0.542473\pi\)
\(908\) −8872.76 −0.324287
\(909\) 0 0
\(910\) 0 0
\(911\) 8535.12 0.310408 0.155204 0.987882i \(-0.450397\pi\)
0.155204 + 0.987882i \(0.450397\pi\)
\(912\) 0 0
\(913\) 73082.7 2.64916
\(914\) 27236.7 0.985678
\(915\) 0 0
\(916\) −3140.87 −0.113294
\(917\) 0 0
\(918\) 0 0
\(919\) −10851.7 −0.389516 −0.194758 0.980851i \(-0.562392\pi\)
−0.194758 + 0.980851i \(0.562392\pi\)
\(920\) −2951.23 −0.105760
\(921\) 0 0
\(922\) 23911.7 0.854110
\(923\) −56865.9 −2.02791
\(924\) 0 0
\(925\) −16381.9 −0.582307
\(926\) 1297.01 0.0460284
\(927\) 0 0
\(928\) −4422.11 −0.156425
\(929\) −1560.64 −0.0551161 −0.0275581 0.999620i \(-0.508773\pi\)
−0.0275581 + 0.999620i \(0.508773\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 21476.5 0.754813
\(933\) 0 0
\(934\) 5569.49 0.195117
\(935\) −7685.41 −0.268813
\(936\) 0 0
\(937\) −11978.4 −0.417627 −0.208813 0.977956i \(-0.566960\pi\)
−0.208813 + 0.977956i \(0.566960\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −889.164 −0.0308525
\(941\) 24597.7 0.852137 0.426068 0.904691i \(-0.359898\pi\)
0.426068 + 0.904691i \(0.359898\pi\)
\(942\) 0 0
\(943\) 40442.6 1.39660
\(944\) 7228.78 0.249234
\(945\) 0 0
\(946\) 53733.1 1.84674
\(947\) 10834.6 0.371783 0.185891 0.982570i \(-0.440483\pi\)
0.185891 + 0.982570i \(0.440483\pi\)
\(948\) 0 0
\(949\) 64097.6 2.19252
\(950\) −12252.8 −0.418456
\(951\) 0 0
\(952\) 0 0
\(953\) −701.418 −0.0238417 −0.0119209 0.999929i \(-0.503795\pi\)
−0.0119209 + 0.999929i \(0.503795\pi\)
\(954\) 0 0
\(955\) −3929.47 −0.133146
\(956\) −14853.1 −0.502493
\(957\) 0 0
\(958\) 22226.8 0.749599
\(959\) 0 0
\(960\) 0 0
\(961\) −12207.4 −0.409769
\(962\) −16024.3 −0.537051
\(963\) 0 0
\(964\) −27994.5 −0.935313
\(965\) 29.8252 0.000994931 0
\(966\) 0 0
\(967\) 42402.5 1.41011 0.705053 0.709154i \(-0.250923\pi\)
0.705053 + 0.709154i \(0.250923\pi\)
\(968\) 19508.7 0.647762
\(969\) 0 0
\(970\) −12552.0 −0.415484
\(971\) 15463.4 0.511064 0.255532 0.966801i \(-0.417749\pi\)
0.255532 + 0.966801i \(0.417749\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −7572.55 −0.249117
\(975\) 0 0
\(976\) −9268.91 −0.303986
\(977\) 34484.1 1.12922 0.564609 0.825359i \(-0.309027\pi\)
0.564609 + 0.825359i \(0.309027\pi\)
\(978\) 0 0
\(979\) −14310.7 −0.467184
\(980\) 0 0
\(981\) 0 0
\(982\) 19164.2 0.622765
\(983\) 7335.36 0.238008 0.119004 0.992894i \(-0.462030\pi\)
0.119004 + 0.992894i \(0.462030\pi\)
\(984\) 0 0
\(985\) 10489.2 0.339302
\(986\) −8872.00 −0.286554
\(987\) 0 0
\(988\) −11985.3 −0.385934
\(989\) 41397.1 1.33099
\(990\) 0 0
\(991\) −10123.0 −0.324487 −0.162243 0.986751i \(-0.551873\pi\)
−0.162243 + 0.986751i \(0.551873\pi\)
\(992\) −4243.30 −0.135811
\(993\) 0 0
\(994\) 0 0
\(995\) 3382.80 0.107781
\(996\) 0 0
\(997\) −56669.2 −1.80013 −0.900066 0.435755i \(-0.856482\pi\)
−0.900066 + 0.435755i \(0.856482\pi\)
\(998\) 11163.1 0.354070
\(999\) 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 882.4.a.bi.1.1 2
3.2 odd 2 294.4.a.k.1.2 yes 2
7.2 even 3 882.4.g.y.361.2 4
7.3 odd 6 882.4.g.bd.667.1 4
7.4 even 3 882.4.g.y.667.2 4
7.5 odd 6 882.4.g.bd.361.1 4
7.6 odd 2 882.4.a.bc.1.2 2
12.11 even 2 2352.4.a.bn.1.2 2
21.2 odd 6 294.4.e.n.67.1 4
21.5 even 6 294.4.e.o.67.2 4
21.11 odd 6 294.4.e.n.79.1 4
21.17 even 6 294.4.e.o.79.2 4
21.20 even 2 294.4.a.j.1.1 2
84.83 odd 2 2352.4.a.cd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.1 2 21.20 even 2
294.4.a.k.1.2 yes 2 3.2 odd 2
294.4.e.n.67.1 4 21.2 odd 6
294.4.e.n.79.1 4 21.11 odd 6
294.4.e.o.67.2 4 21.5 even 6
294.4.e.o.79.2 4 21.17 even 6
882.4.a.bc.1.2 2 7.6 odd 2
882.4.a.bi.1.1 2 1.1 even 1 trivial
882.4.g.y.361.2 4 7.2 even 3
882.4.g.y.667.2 4 7.4 even 3
882.4.g.bd.361.1 4 7.5 odd 6
882.4.g.bd.667.1 4 7.3 odd 6
2352.4.a.bn.1.2 2 12.11 even 2
2352.4.a.cd.1.1 2 84.83 odd 2