Properties

Label 1323.4.a.ba.1.3
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.0595269376\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{13})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.92081\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.55133 q^{2} -5.59339 q^{4} -9.81086 q^{5} -21.0878 q^{8} +O(q^{10})\) \(q+1.55133 q^{2} -5.59339 q^{4} -9.81086 q^{5} -21.0878 q^{8} -15.2198 q^{10} +70.6023 q^{11} -55.5603 q^{13} +12.0331 q^{16} -13.4164 q^{17} -91.3735 q^{19} +54.8759 q^{20} +109.527 q^{22} -113.784 q^{23} -28.7471 q^{25} -86.1922 q^{26} -12.4959 q^{29} -222.934 q^{31} +187.369 q^{32} -20.8132 q^{34} +257.440 q^{37} -141.750 q^{38} +206.889 q^{40} -286.765 q^{41} +4.81323 q^{43} -394.906 q^{44} -176.516 q^{46} +609.517 q^{47} -44.5961 q^{50} +310.770 q^{52} -691.107 q^{53} -692.669 q^{55} -19.3852 q^{58} -217.936 q^{59} +764.747 q^{61} -345.843 q^{62} +194.407 q^{64} +545.094 q^{65} -98.5487 q^{67} +75.0432 q^{68} +921.274 q^{71} +219.934 q^{73} +399.373 q^{74} +511.088 q^{76} -9.42805 q^{79} -118.055 q^{80} -444.866 q^{82} +800.364 q^{83} +131.626 q^{85} +7.46689 q^{86} -1488.85 q^{88} +253.574 q^{89} +636.436 q^{92} +945.560 q^{94} +896.453 q^{95} +59.2529 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 26q^{4} + O(q^{10}) \) \( 4q + 26q^{4} - 206q^{10} + 68q^{13} + 290q^{16} - 172q^{19} - 94q^{22} + 272q^{25} - 408q^{31} - 180q^{34} + 1320q^{37} - 1446q^{40} + 116q^{43} + 1374q^{46} + 3952q^{52} - 352q^{55} - 1432q^{58} + 2672q^{61} + 826q^{64} + 1444q^{67} + 396q^{73} + 1222q^{76} + 1220q^{79} + 3590q^{82} + 720q^{85} - 6294q^{88} + 3492q^{94} + 624q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.55133 0.548477 0.274238 0.961662i \(-0.411574\pi\)
0.274238 + 0.961662i \(0.411574\pi\)
\(3\) 0 0
\(4\) −5.59339 −0.699173
\(5\) −9.81086 −0.877510 −0.438755 0.898607i \(-0.644580\pi\)
−0.438755 + 0.898607i \(0.644580\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −21.0878 −0.931957
\(9\) 0 0
\(10\) −15.2198 −0.481294
\(11\) 70.6023 1.93522 0.967609 0.252453i \(-0.0812372\pi\)
0.967609 + 0.252453i \(0.0812372\pi\)
\(12\) 0 0
\(13\) −55.5603 −1.18536 −0.592679 0.805439i \(-0.701930\pi\)
−0.592679 + 0.805439i \(0.701930\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 12.0331 0.188017
\(17\) −13.4164 −0.191409 −0.0957046 0.995410i \(-0.530510\pi\)
−0.0957046 + 0.995410i \(0.530510\pi\)
\(18\) 0 0
\(19\) −91.3735 −1.10329 −0.551646 0.834079i \(-0.686000\pi\)
−0.551646 + 0.834079i \(0.686000\pi\)
\(20\) 54.8759 0.613531
\(21\) 0 0
\(22\) 109.527 1.06142
\(23\) −113.784 −1.03155 −0.515773 0.856726i \(-0.672495\pi\)
−0.515773 + 0.856726i \(0.672495\pi\)
\(24\) 0 0
\(25\) −28.7471 −0.229977
\(26\) −86.1922 −0.650141
\(27\) 0 0
\(28\) 0 0
\(29\) −12.4959 −0.0800147 −0.0400073 0.999199i \(-0.512738\pi\)
−0.0400073 + 0.999199i \(0.512738\pi\)
\(30\) 0 0
\(31\) −222.934 −1.29162 −0.645808 0.763500i \(-0.723479\pi\)
−0.645808 + 0.763500i \(0.723479\pi\)
\(32\) 187.369 1.03508
\(33\) 0 0
\(34\) −20.8132 −0.104983
\(35\) 0 0
\(36\) 0 0
\(37\) 257.440 1.14386 0.571930 0.820302i \(-0.306195\pi\)
0.571930 + 0.820302i \(0.306195\pi\)
\(38\) −141.750 −0.605129
\(39\) 0 0
\(40\) 206.889 0.817801
\(41\) −286.765 −1.09232 −0.546160 0.837681i \(-0.683911\pi\)
−0.546160 + 0.837681i \(0.683911\pi\)
\(42\) 0 0
\(43\) 4.81323 0.0170700 0.00853500 0.999964i \(-0.497283\pi\)
0.00853500 + 0.999964i \(0.497283\pi\)
\(44\) −394.906 −1.35305
\(45\) 0 0
\(46\) −176.516 −0.565778
\(47\) 609.517 1.89164 0.945822 0.324686i \(-0.105259\pi\)
0.945822 + 0.324686i \(0.105259\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −44.5961 −0.126137
\(51\) 0 0
\(52\) 310.770 0.828771
\(53\) −691.107 −1.79115 −0.895574 0.444913i \(-0.853235\pi\)
−0.895574 + 0.444913i \(0.853235\pi\)
\(54\) 0 0
\(55\) −692.669 −1.69817
\(56\) 0 0
\(57\) 0 0
\(58\) −19.3852 −0.0438862
\(59\) −217.936 −0.480895 −0.240448 0.970662i \(-0.577294\pi\)
−0.240448 + 0.970662i \(0.577294\pi\)
\(60\) 0 0
\(61\) 764.747 1.60518 0.802589 0.596533i \(-0.203455\pi\)
0.802589 + 0.596533i \(0.203455\pi\)
\(62\) −345.843 −0.708421
\(63\) 0 0
\(64\) 194.407 0.379700
\(65\) 545.094 1.04016
\(66\) 0 0
\(67\) −98.5487 −0.179696 −0.0898481 0.995955i \(-0.528638\pi\)
−0.0898481 + 0.995955i \(0.528638\pi\)
\(68\) 75.0432 0.133828
\(69\) 0 0
\(70\) 0 0
\(71\) 921.274 1.53993 0.769966 0.638085i \(-0.220273\pi\)
0.769966 + 0.638085i \(0.220273\pi\)
\(72\) 0 0
\(73\) 219.934 0.352621 0.176310 0.984335i \(-0.443584\pi\)
0.176310 + 0.984335i \(0.443584\pi\)
\(74\) 399.373 0.627381
\(75\) 0 0
\(76\) 511.088 0.771392
\(77\) 0 0
\(78\) 0 0
\(79\) −9.42805 −0.0134271 −0.00671354 0.999977i \(-0.502137\pi\)
−0.00671354 + 0.999977i \(0.502137\pi\)
\(80\) −118.055 −0.164986
\(81\) 0 0
\(82\) −444.866 −0.599112
\(83\) 800.364 1.05845 0.529225 0.848481i \(-0.322483\pi\)
0.529225 + 0.848481i \(0.322483\pi\)
\(84\) 0 0
\(85\) 131.626 0.167963
\(86\) 7.46689 0.00936250
\(87\) 0 0
\(88\) −1488.85 −1.80354
\(89\) 253.574 0.302008 0.151004 0.988533i \(-0.451749\pi\)
0.151004 + 0.988533i \(0.451749\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 636.436 0.721229
\(93\) 0 0
\(94\) 945.560 1.03752
\(95\) 896.453 0.968149
\(96\) 0 0
\(97\) 59.2529 0.0620229 0.0310114 0.999519i \(-0.490127\pi\)
0.0310114 + 0.999519i \(0.490127\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 160.794 0.160794
\(101\) 683.043 0.672924 0.336462 0.941697i \(-0.390770\pi\)
0.336462 + 0.941697i \(0.390770\pi\)
\(102\) 0 0
\(103\) 520.537 0.497962 0.248981 0.968508i \(-0.419904\pi\)
0.248981 + 0.968508i \(0.419904\pi\)
\(104\) 1171.64 1.10470
\(105\) 0 0
\(106\) −1072.13 −0.982403
\(107\) 714.428 0.645480 0.322740 0.946488i \(-0.395396\pi\)
0.322740 + 0.946488i \(0.395396\pi\)
\(108\) 0 0
\(109\) −182.802 −0.160635 −0.0803175 0.996769i \(-0.525593\pi\)
−0.0803175 + 0.996769i \(0.525593\pi\)
\(110\) −1074.56 −0.931408
\(111\) 0 0
\(112\) 0 0
\(113\) −154.433 −0.128565 −0.0642825 0.997932i \(-0.520476\pi\)
−0.0642825 + 0.997932i \(0.520476\pi\)
\(114\) 0 0
\(115\) 1116.32 0.905191
\(116\) 69.8942 0.0559441
\(117\) 0 0
\(118\) −338.089 −0.263760
\(119\) 0 0
\(120\) 0 0
\(121\) 3653.69 2.74507
\(122\) 1186.37 0.880402
\(123\) 0 0
\(124\) 1246.96 0.903064
\(125\) 1508.39 1.07932
\(126\) 0 0
\(127\) 2072.00 1.44772 0.723858 0.689949i \(-0.242367\pi\)
0.723858 + 0.689949i \(0.242367\pi\)
\(128\) −1197.37 −0.826823
\(129\) 0 0
\(130\) 845.619 0.570505
\(131\) −25.5712 −0.0170547 −0.00852736 0.999964i \(-0.502714\pi\)
−0.00852736 + 0.999964i \(0.502714\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −152.881 −0.0985591
\(135\) 0 0
\(136\) 282.922 0.178385
\(137\) −581.202 −0.362448 −0.181224 0.983442i \(-0.558006\pi\)
−0.181224 + 0.983442i \(0.558006\pi\)
\(138\) 0 0
\(139\) −1185.01 −0.723103 −0.361552 0.932352i \(-0.617753\pi\)
−0.361552 + 0.932352i \(0.617753\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1429.20 0.844616
\(143\) −3922.69 −2.29393
\(144\) 0 0
\(145\) 122.595 0.0702136
\(146\) 341.189 0.193404
\(147\) 0 0
\(148\) −1439.96 −0.799756
\(149\) −471.892 −0.259456 −0.129728 0.991550i \(-0.541410\pi\)
−0.129728 + 0.991550i \(0.541410\pi\)
\(150\) 0 0
\(151\) 2635.18 1.42019 0.710093 0.704108i \(-0.248653\pi\)
0.710093 + 0.704108i \(0.248653\pi\)
\(152\) 1926.87 1.02822
\(153\) 0 0
\(154\) 0 0
\(155\) 2187.17 1.13341
\(156\) 0 0
\(157\) 359.210 0.182599 0.0912996 0.995823i \(-0.470898\pi\)
0.0912996 + 0.995823i \(0.470898\pi\)
\(158\) −14.6260 −0.00736444
\(159\) 0 0
\(160\) −1838.25 −0.908292
\(161\) 0 0
\(162\) 0 0
\(163\) 2659.91 1.27816 0.639081 0.769140i \(-0.279315\pi\)
0.639081 + 0.769140i \(0.279315\pi\)
\(164\) 1603.99 0.763722
\(165\) 0 0
\(166\) 1241.63 0.580536
\(167\) 702.648 0.325584 0.162792 0.986660i \(-0.447950\pi\)
0.162792 + 0.986660i \(0.447950\pi\)
\(168\) 0 0
\(169\) 889.949 0.405075
\(170\) 204.196 0.0921240
\(171\) 0 0
\(172\) −26.9222 −0.0119349
\(173\) −738.814 −0.324688 −0.162344 0.986734i \(-0.551905\pi\)
−0.162344 + 0.986734i \(0.551905\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 849.563 0.363853
\(177\) 0 0
\(178\) 393.375 0.165645
\(179\) −3011.14 −1.25734 −0.628668 0.777674i \(-0.716400\pi\)
−0.628668 + 0.777674i \(0.716400\pi\)
\(180\) 0 0
\(181\) −204.288 −0.0838928 −0.0419464 0.999120i \(-0.513356\pi\)
−0.0419464 + 0.999120i \(0.513356\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2399.45 0.961356
\(185\) −2525.70 −1.00375
\(186\) 0 0
\(187\) −947.230 −0.370419
\(188\) −3409.27 −1.32259
\(189\) 0 0
\(190\) 1390.69 0.531007
\(191\) −3802.34 −1.44046 −0.720229 0.693736i \(-0.755963\pi\)
−0.720229 + 0.693736i \(0.755963\pi\)
\(192\) 0 0
\(193\) −3085.69 −1.15084 −0.575422 0.817857i \(-0.695162\pi\)
−0.575422 + 0.817857i \(0.695162\pi\)
\(194\) 91.9206 0.0340181
\(195\) 0 0
\(196\) 0 0
\(197\) −2710.10 −0.980134 −0.490067 0.871685i \(-0.663028\pi\)
−0.490067 + 0.871685i \(0.663028\pi\)
\(198\) 0 0
\(199\) 4807.21 1.71243 0.856217 0.516616i \(-0.172809\pi\)
0.856217 + 0.516616i \(0.172809\pi\)
\(200\) 606.212 0.214328
\(201\) 0 0
\(202\) 1059.62 0.369083
\(203\) 0 0
\(204\) 0 0
\(205\) 2813.41 0.958522
\(206\) 807.523 0.273120
\(207\) 0 0
\(208\) −668.561 −0.222867
\(209\) −6451.19 −2.13511
\(210\) 0 0
\(211\) 5648.81 1.84303 0.921517 0.388338i \(-0.126951\pi\)
0.921517 + 0.388338i \(0.126951\pi\)
\(212\) 3865.63 1.25232
\(213\) 0 0
\(214\) 1108.31 0.354031
\(215\) −47.2219 −0.0149791
\(216\) 0 0
\(217\) 0 0
\(218\) −283.585 −0.0881046
\(219\) 0 0
\(220\) 3874.37 1.18732
\(221\) 745.420 0.226889
\(222\) 0 0
\(223\) −4595.61 −1.38002 −0.690011 0.723799i \(-0.742394\pi\)
−0.690011 + 0.723799i \(0.742394\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −239.576 −0.0705150
\(227\) 1856.53 0.542829 0.271414 0.962463i \(-0.412509\pi\)
0.271414 + 0.962463i \(0.412509\pi\)
\(228\) 0 0
\(229\) 3830.37 1.10532 0.552659 0.833408i \(-0.313613\pi\)
0.552659 + 0.833408i \(0.313613\pi\)
\(230\) 1731.77 0.496476
\(231\) 0 0
\(232\) 263.510 0.0745702
\(233\) 2244.80 0.631166 0.315583 0.948898i \(-0.397800\pi\)
0.315583 + 0.948898i \(0.397800\pi\)
\(234\) 0 0
\(235\) −5979.89 −1.65994
\(236\) 1219.00 0.336229
\(237\) 0 0
\(238\) 0 0
\(239\) −3184.89 −0.861980 −0.430990 0.902357i \(-0.641836\pi\)
−0.430990 + 0.902357i \(0.641836\pi\)
\(240\) 0 0
\(241\) −1607.14 −0.429564 −0.214782 0.976662i \(-0.568904\pi\)
−0.214782 + 0.976662i \(0.568904\pi\)
\(242\) 5668.06 1.50561
\(243\) 0 0
\(244\) −4277.53 −1.12230
\(245\) 0 0
\(246\) 0 0
\(247\) 5076.74 1.30780
\(248\) 4701.18 1.20373
\(249\) 0 0
\(250\) 2340.01 0.591980
\(251\) 3398.32 0.854582 0.427291 0.904114i \(-0.359468\pi\)
0.427291 + 0.904114i \(0.359468\pi\)
\(252\) 0 0
\(253\) −8033.39 −1.99627
\(254\) 3214.34 0.794039
\(255\) 0 0
\(256\) −3412.76 −0.833193
\(257\) −3127.70 −0.759147 −0.379573 0.925162i \(-0.623929\pi\)
−0.379573 + 0.925162i \(0.623929\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −3048.92 −0.727255
\(261\) 0 0
\(262\) −39.6693 −0.00935412
\(263\) 4470.12 1.04806 0.524030 0.851700i \(-0.324428\pi\)
0.524030 + 0.851700i \(0.324428\pi\)
\(264\) 0 0
\(265\) 6780.35 1.57175
\(266\) 0 0
\(267\) 0 0
\(268\) 551.221 0.125639
\(269\) 3021.24 0.684789 0.342394 0.939556i \(-0.388762\pi\)
0.342394 + 0.939556i \(0.388762\pi\)
\(270\) 0 0
\(271\) 6635.95 1.48747 0.743736 0.668473i \(-0.233052\pi\)
0.743736 + 0.668473i \(0.233052\pi\)
\(272\) −161.441 −0.0359881
\(273\) 0 0
\(274\) −901.634 −0.198794
\(275\) −2029.61 −0.445055
\(276\) 0 0
\(277\) −2223.61 −0.482324 −0.241162 0.970485i \(-0.577529\pi\)
−0.241162 + 0.970485i \(0.577529\pi\)
\(278\) −1838.34 −0.396605
\(279\) 0 0
\(280\) 0 0
\(281\) −6027.19 −1.27954 −0.639772 0.768565i \(-0.720971\pi\)
−0.639772 + 0.768565i \(0.720971\pi\)
\(282\) 0 0
\(283\) 769.183 0.161566 0.0807830 0.996732i \(-0.474258\pi\)
0.0807830 + 0.996732i \(0.474258\pi\)
\(284\) −5153.04 −1.07668
\(285\) 0 0
\(286\) −6085.37 −1.25817
\(287\) 0 0
\(288\) 0 0
\(289\) −4733.00 −0.963363
\(290\) 190.185 0.0385105
\(291\) 0 0
\(292\) −1230.18 −0.246543
\(293\) −5232.52 −1.04330 −0.521650 0.853160i \(-0.674683\pi\)
−0.521650 + 0.853160i \(0.674683\pi\)
\(294\) 0 0
\(295\) 2138.14 0.421990
\(296\) −5428.83 −1.06603
\(297\) 0 0
\(298\) −732.059 −0.142305
\(299\) 6321.86 1.22275
\(300\) 0 0
\(301\) 0 0
\(302\) 4088.03 0.778939
\(303\) 0 0
\(304\) −1099.50 −0.207437
\(305\) −7502.82 −1.40856
\(306\) 0 0
\(307\) 1705.68 0.317096 0.158548 0.987351i \(-0.449319\pi\)
0.158548 + 0.987351i \(0.449319\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3393.02 0.621647
\(311\) −7220.87 −1.31659 −0.658293 0.752762i \(-0.728721\pi\)
−0.658293 + 0.752762i \(0.728721\pi\)
\(312\) 0 0
\(313\) 9173.62 1.65663 0.828313 0.560266i \(-0.189301\pi\)
0.828313 + 0.560266i \(0.189301\pi\)
\(314\) 557.252 0.100151
\(315\) 0 0
\(316\) 52.7347 0.00938785
\(317\) 1082.48 0.191793 0.0958965 0.995391i \(-0.469428\pi\)
0.0958965 + 0.995391i \(0.469428\pi\)
\(318\) 0 0
\(319\) −882.238 −0.154846
\(320\) −1907.30 −0.333191
\(321\) 0 0
\(322\) 0 0
\(323\) 1225.90 0.211180
\(324\) 0 0
\(325\) 1597.20 0.272605
\(326\) 4126.39 0.701042
\(327\) 0 0
\(328\) 6047.23 1.01800
\(329\) 0 0
\(330\) 0 0
\(331\) −1753.27 −0.291143 −0.145572 0.989348i \(-0.546502\pi\)
−0.145572 + 0.989348i \(0.546502\pi\)
\(332\) −4476.75 −0.740041
\(333\) 0 0
\(334\) 1090.04 0.178575
\(335\) 966.847 0.157685
\(336\) 0 0
\(337\) 8918.33 1.44158 0.720790 0.693153i \(-0.243779\pi\)
0.720790 + 0.693153i \(0.243779\pi\)
\(338\) 1380.60 0.222174
\(339\) 0 0
\(340\) −736.238 −0.117436
\(341\) −15739.7 −2.49956
\(342\) 0 0
\(343\) 0 0
\(344\) −101.500 −0.0159085
\(345\) 0 0
\(346\) −1146.14 −0.178084
\(347\) −6811.49 −1.05377 −0.526887 0.849935i \(-0.676641\pi\)
−0.526887 + 0.849935i \(0.676641\pi\)
\(348\) 0 0
\(349\) −9254.60 −1.41945 −0.709724 0.704480i \(-0.751180\pi\)
−0.709724 + 0.704480i \(0.751180\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 13228.7 2.00311
\(353\) 8005.75 1.20709 0.603546 0.797329i \(-0.293754\pi\)
0.603546 + 0.797329i \(0.293754\pi\)
\(354\) 0 0
\(355\) −9038.49 −1.35130
\(356\) −1418.34 −0.211156
\(357\) 0 0
\(358\) −4671.26 −0.689619
\(359\) 6611.34 0.971958 0.485979 0.873970i \(-0.338463\pi\)
0.485979 + 0.873970i \(0.338463\pi\)
\(360\) 0 0
\(361\) 1490.12 0.217251
\(362\) −316.917 −0.0460132
\(363\) 0 0
\(364\) 0 0
\(365\) −2157.74 −0.309428
\(366\) 0 0
\(367\) −1836.21 −0.261170 −0.130585 0.991437i \(-0.541686\pi\)
−0.130585 + 0.991437i \(0.541686\pi\)
\(368\) −1369.17 −0.193948
\(369\) 0 0
\(370\) −3918.19 −0.550533
\(371\) 0 0
\(372\) 0 0
\(373\) −4819.81 −0.669062 −0.334531 0.942385i \(-0.608578\pi\)
−0.334531 + 0.942385i \(0.608578\pi\)
\(374\) −1469.46 −0.203166
\(375\) 0 0
\(376\) −12853.4 −1.76293
\(377\) 694.275 0.0948461
\(378\) 0 0
\(379\) 6710.00 0.909418 0.454709 0.890640i \(-0.349743\pi\)
0.454709 + 0.890640i \(0.349743\pi\)
\(380\) −5014.21 −0.676904
\(381\) 0 0
\(382\) −5898.67 −0.790058
\(383\) 11584.9 1.54559 0.772797 0.634653i \(-0.218857\pi\)
0.772797 + 0.634653i \(0.218857\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4786.91 −0.631211
\(387\) 0 0
\(388\) −331.424 −0.0433648
\(389\) −7523.59 −0.980620 −0.490310 0.871548i \(-0.663116\pi\)
−0.490310 + 0.871548i \(0.663116\pi\)
\(390\) 0 0
\(391\) 1526.57 0.197447
\(392\) 0 0
\(393\) 0 0
\(394\) −4204.24 −0.537581
\(395\) 92.4973 0.0117824
\(396\) 0 0
\(397\) 3302.26 0.417471 0.208735 0.977972i \(-0.433065\pi\)
0.208735 + 0.977972i \(0.433065\pi\)
\(398\) 7457.56 0.939230
\(399\) 0 0
\(400\) −345.916 −0.0432395
\(401\) −6996.80 −0.871331 −0.435665 0.900109i \(-0.643487\pi\)
−0.435665 + 0.900109i \(0.643487\pi\)
\(402\) 0 0
\(403\) 12386.3 1.53103
\(404\) −3820.52 −0.470491
\(405\) 0 0
\(406\) 0 0
\(407\) 18175.8 2.21362
\(408\) 0 0
\(409\) 4534.39 0.548193 0.274097 0.961702i \(-0.411621\pi\)
0.274097 + 0.961702i \(0.411621\pi\)
\(410\) 4364.52 0.525727
\(411\) 0 0
\(412\) −2911.57 −0.348161
\(413\) 0 0
\(414\) 0 0
\(415\) −7852.26 −0.928801
\(416\) −10410.3 −1.22694
\(417\) 0 0
\(418\) −10007.9 −1.17106
\(419\) −1497.50 −0.174601 −0.0873004 0.996182i \(-0.527824\pi\)
−0.0873004 + 0.996182i \(0.527824\pi\)
\(420\) 0 0
\(421\) −4669.50 −0.540564 −0.270282 0.962781i \(-0.587117\pi\)
−0.270282 + 0.962781i \(0.587117\pi\)
\(422\) 8763.15 1.01086
\(423\) 0 0
\(424\) 14573.9 1.66927
\(425\) 385.683 0.0440197
\(426\) 0 0
\(427\) 0 0
\(428\) −3996.07 −0.451302
\(429\) 0 0
\(430\) −73.2565 −0.00821568
\(431\) 11985.2 1.33946 0.669732 0.742603i \(-0.266409\pi\)
0.669732 + 0.742603i \(0.266409\pi\)
\(432\) 0 0
\(433\) 8226.59 0.913036 0.456518 0.889714i \(-0.349096\pi\)
0.456518 + 0.889714i \(0.349096\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1022.48 0.112312
\(437\) 10396.8 1.13809
\(438\) 0 0
\(439\) 8154.12 0.886503 0.443251 0.896397i \(-0.353825\pi\)
0.443251 + 0.896397i \(0.353825\pi\)
\(440\) 14606.9 1.58262
\(441\) 0 0
\(442\) 1156.39 0.124443
\(443\) 11406.0 1.22329 0.611644 0.791133i \(-0.290508\pi\)
0.611644 + 0.791133i \(0.290508\pi\)
\(444\) 0 0
\(445\) −2487.77 −0.265015
\(446\) −7129.29 −0.756910
\(447\) 0 0
\(448\) 0 0
\(449\) −2463.59 −0.258940 −0.129470 0.991583i \(-0.541328\pi\)
−0.129470 + 0.991583i \(0.541328\pi\)
\(450\) 0 0
\(451\) −20246.3 −2.11388
\(452\) 863.805 0.0898893
\(453\) 0 0
\(454\) 2880.08 0.297729
\(455\) 0 0
\(456\) 0 0
\(457\) −16349.9 −1.67356 −0.836778 0.547542i \(-0.815564\pi\)
−0.836778 + 0.547542i \(0.815564\pi\)
\(458\) 5942.15 0.606241
\(459\) 0 0
\(460\) −6243.98 −0.632885
\(461\) −2872.86 −0.290244 −0.145122 0.989414i \(-0.546357\pi\)
−0.145122 + 0.989414i \(0.546357\pi\)
\(462\) 0 0
\(463\) 2984.20 0.299541 0.149771 0.988721i \(-0.452146\pi\)
0.149771 + 0.988721i \(0.452146\pi\)
\(464\) −150.364 −0.0150441
\(465\) 0 0
\(466\) 3482.42 0.346180
\(467\) 19419.5 1.92426 0.962129 0.272595i \(-0.0878821\pi\)
0.962129 + 0.272595i \(0.0878821\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −9276.76 −0.910436
\(471\) 0 0
\(472\) 4595.78 0.448174
\(473\) 339.825 0.0330342
\(474\) 0 0
\(475\) 2626.72 0.253731
\(476\) 0 0
\(477\) 0 0
\(478\) −4940.80 −0.472776
\(479\) 18229.3 1.73887 0.869435 0.494047i \(-0.164483\pi\)
0.869435 + 0.494047i \(0.164483\pi\)
\(480\) 0 0
\(481\) −14303.4 −1.35588
\(482\) −2493.20 −0.235606
\(483\) 0 0
\(484\) −20436.5 −1.91928
\(485\) −581.322 −0.0544257
\(486\) 0 0
\(487\) 17079.6 1.58922 0.794609 0.607122i \(-0.207676\pi\)
0.794609 + 0.607122i \(0.207676\pi\)
\(488\) −16126.8 −1.49596
\(489\) 0 0
\(490\) 0 0
\(491\) −1087.13 −0.0999214 −0.0499607 0.998751i \(-0.515910\pi\)
−0.0499607 + 0.998751i \(0.515910\pi\)
\(492\) 0 0
\(493\) 167.650 0.0153155
\(494\) 7875.69 0.717295
\(495\) 0 0
\(496\) −2682.58 −0.242845
\(497\) 0 0
\(498\) 0 0
\(499\) 18785.8 1.68530 0.842652 0.538459i \(-0.180993\pi\)
0.842652 + 0.538459i \(0.180993\pi\)
\(500\) −8437.01 −0.754629
\(501\) 0 0
\(502\) 5271.91 0.468719
\(503\) −12278.7 −1.08843 −0.544214 0.838947i \(-0.683172\pi\)
−0.544214 + 0.838947i \(0.683172\pi\)
\(504\) 0 0
\(505\) −6701.24 −0.590497
\(506\) −12462.4 −1.09490
\(507\) 0 0
\(508\) −11589.5 −1.01220
\(509\) 1226.83 0.106834 0.0534168 0.998572i \(-0.482989\pi\)
0.0534168 + 0.998572i \(0.482989\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 4284.63 0.369836
\(513\) 0 0
\(514\) −4852.09 −0.416374
\(515\) −5106.91 −0.436966
\(516\) 0 0
\(517\) 43033.3 3.66074
\(518\) 0 0
\(519\) 0 0
\(520\) −11494.8 −0.969388
\(521\) −12571.3 −1.05712 −0.528559 0.848897i \(-0.677267\pi\)
−0.528559 + 0.848897i \(0.677267\pi\)
\(522\) 0 0
\(523\) 7828.18 0.654498 0.327249 0.944938i \(-0.393878\pi\)
0.327249 + 0.944938i \(0.393878\pi\)
\(524\) 143.030 0.0119242
\(525\) 0 0
\(526\) 6934.62 0.574836
\(527\) 2990.97 0.247227
\(528\) 0 0
\(529\) 779.727 0.0640854
\(530\) 10518.5 0.862068
\(531\) 0 0
\(532\) 0 0
\(533\) 15932.7 1.29479
\(534\) 0 0
\(535\) −7009.15 −0.566415
\(536\) 2078.17 0.167469
\(537\) 0 0
\(538\) 4686.93 0.375591
\(539\) 0 0
\(540\) 0 0
\(541\) −5726.90 −0.455118 −0.227559 0.973764i \(-0.573074\pi\)
−0.227559 + 0.973764i \(0.573074\pi\)
\(542\) 10294.5 0.815844
\(543\) 0 0
\(544\) −2513.82 −0.198124
\(545\) 1793.44 0.140959
\(546\) 0 0
\(547\) −7738.41 −0.604882 −0.302441 0.953168i \(-0.597801\pi\)
−0.302441 + 0.953168i \(0.597801\pi\)
\(548\) 3250.89 0.253414
\(549\) 0 0
\(550\) −3148.59 −0.244102
\(551\) 1141.79 0.0882795
\(552\) 0 0
\(553\) 0 0
\(554\) −3449.55 −0.264544
\(555\) 0 0
\(556\) 6628.23 0.505575
\(557\) −6714.94 −0.510810 −0.255405 0.966834i \(-0.582209\pi\)
−0.255405 + 0.966834i \(0.582209\pi\)
\(558\) 0 0
\(559\) −267.424 −0.0202341
\(560\) 0 0
\(561\) 0 0
\(562\) −9350.13 −0.701800
\(563\) −25016.6 −1.87269 −0.936343 0.351087i \(-0.885812\pi\)
−0.936343 + 0.351087i \(0.885812\pi\)
\(564\) 0 0
\(565\) 1515.12 0.112817
\(566\) 1193.25 0.0886152
\(567\) 0 0
\(568\) −19427.6 −1.43515
\(569\) −11748.3 −0.865578 −0.432789 0.901495i \(-0.642470\pi\)
−0.432789 + 0.901495i \(0.642470\pi\)
\(570\) 0 0
\(571\) 2168.65 0.158941 0.0794705 0.996837i \(-0.474677\pi\)
0.0794705 + 0.996837i \(0.474677\pi\)
\(572\) 21941.1 1.60385
\(573\) 0 0
\(574\) 0 0
\(575\) 3270.95 0.237231
\(576\) 0 0
\(577\) −3941.26 −0.284362 −0.142181 0.989841i \(-0.545411\pi\)
−0.142181 + 0.989841i \(0.545411\pi\)
\(578\) −7342.43 −0.528382
\(579\) 0 0
\(580\) −685.722 −0.0490915
\(581\) 0 0
\(582\) 0 0
\(583\) −48793.8 −3.46626
\(584\) −4637.92 −0.328627
\(585\) 0 0
\(586\) −8117.34 −0.572226
\(587\) −13873.7 −0.975521 −0.487760 0.872978i \(-0.662186\pi\)
−0.487760 + 0.872978i \(0.662186\pi\)
\(588\) 0 0
\(589\) 20370.3 1.42503
\(590\) 3316.95 0.231452
\(591\) 0 0
\(592\) 3097.79 0.215065
\(593\) 9754.07 0.675467 0.337733 0.941242i \(-0.390340\pi\)
0.337733 + 0.941242i \(0.390340\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2639.48 0.181405
\(597\) 0 0
\(598\) 9807.26 0.670650
\(599\) 15994.0 1.09098 0.545491 0.838116i \(-0.316343\pi\)
0.545491 + 0.838116i \(0.316343\pi\)
\(600\) 0 0
\(601\) −15183.7 −1.03054 −0.515272 0.857027i \(-0.672309\pi\)
−0.515272 + 0.857027i \(0.672309\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −14739.6 −0.992957
\(605\) −35845.8 −2.40883
\(606\) 0 0
\(607\) −18747.2 −1.25358 −0.626792 0.779187i \(-0.715632\pi\)
−0.626792 + 0.779187i \(0.715632\pi\)
\(608\) −17120.6 −1.14199
\(609\) 0 0
\(610\) −11639.3 −0.772562
\(611\) −33865.0 −2.24228
\(612\) 0 0
\(613\) −460.802 −0.0303615 −0.0151808 0.999885i \(-0.504832\pi\)
−0.0151808 + 0.999885i \(0.504832\pi\)
\(614\) 2646.07 0.173920
\(615\) 0 0
\(616\) 0 0
\(617\) 27174.7 1.77311 0.886557 0.462620i \(-0.153090\pi\)
0.886557 + 0.462620i \(0.153090\pi\)
\(618\) 0 0
\(619\) −29663.1 −1.92611 −0.963055 0.269305i \(-0.913206\pi\)
−0.963055 + 0.269305i \(0.913206\pi\)
\(620\) −12233.7 −0.792447
\(621\) 0 0
\(622\) −11201.9 −0.722117
\(623\) 0 0
\(624\) 0 0
\(625\) −11205.2 −0.717134
\(626\) 14231.3 0.908621
\(627\) 0 0
\(628\) −2009.20 −0.127669
\(629\) −3453.92 −0.218945
\(630\) 0 0
\(631\) 8710.14 0.549517 0.274758 0.961513i \(-0.411402\pi\)
0.274758 + 0.961513i \(0.411402\pi\)
\(632\) 198.817 0.0125135
\(633\) 0 0
\(634\) 1679.29 0.105194
\(635\) −20328.1 −1.27038
\(636\) 0 0
\(637\) 0 0
\(638\) −1368.64 −0.0849293
\(639\) 0 0
\(640\) 11747.2 0.725545
\(641\) 14429.7 0.889138 0.444569 0.895745i \(-0.353357\pi\)
0.444569 + 0.895745i \(0.353357\pi\)
\(642\) 0 0
\(643\) 25865.5 1.58637 0.793184 0.608982i \(-0.208422\pi\)
0.793184 + 0.608982i \(0.208422\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1901.78 0.115827
\(647\) 5059.78 0.307451 0.153725 0.988114i \(-0.450873\pi\)
0.153725 + 0.988114i \(0.450873\pi\)
\(648\) 0 0
\(649\) −15386.8 −0.930637
\(650\) 2477.77 0.149517
\(651\) 0 0
\(652\) −14877.9 −0.893656
\(653\) −15389.9 −0.922287 −0.461143 0.887326i \(-0.652561\pi\)
−0.461143 + 0.887326i \(0.652561\pi\)
\(654\) 0 0
\(655\) 250.876 0.0149657
\(656\) −3450.66 −0.205375
\(657\) 0 0
\(658\) 0 0
\(659\) −30375.3 −1.79553 −0.897765 0.440474i \(-0.854810\pi\)
−0.897765 + 0.440474i \(0.854810\pi\)
\(660\) 0 0
\(661\) −12410.4 −0.730270 −0.365135 0.930955i \(-0.618977\pi\)
−0.365135 + 0.930955i \(0.618977\pi\)
\(662\) −2719.89 −0.159685
\(663\) 0 0
\(664\) −16877.9 −0.986431
\(665\) 0 0
\(666\) 0 0
\(667\) 1421.83 0.0825387
\(668\) −3930.18 −0.227640
\(669\) 0 0
\(670\) 1499.90 0.0864866
\(671\) 53992.9 3.10637
\(672\) 0 0
\(673\) 23632.8 1.35361 0.676804 0.736163i \(-0.263364\pi\)
0.676804 + 0.736163i \(0.263364\pi\)
\(674\) 13835.2 0.790673
\(675\) 0 0
\(676\) −4977.83 −0.283217
\(677\) 7460.16 0.423511 0.211756 0.977323i \(-0.432082\pi\)
0.211756 + 0.977323i \(0.432082\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2775.71 −0.156535
\(681\) 0 0
\(682\) −24417.3 −1.37095
\(683\) −7798.73 −0.436911 −0.218455 0.975847i \(-0.570102\pi\)
−0.218455 + 0.975847i \(0.570102\pi\)
\(684\) 0 0
\(685\) 5702.09 0.318052
\(686\) 0 0
\(687\) 0 0
\(688\) 57.9179 0.00320945
\(689\) 38398.1 2.12315
\(690\) 0 0
\(691\) −4748.63 −0.261427 −0.130714 0.991420i \(-0.541727\pi\)
−0.130714 + 0.991420i \(0.541727\pi\)
\(692\) 4132.47 0.227013
\(693\) 0 0
\(694\) −10566.8 −0.577971
\(695\) 11626.0 0.634530
\(696\) 0 0
\(697\) 3847.35 0.209080
\(698\) −14356.9 −0.778534
\(699\) 0 0
\(700\) 0 0
\(701\) 13784.1 0.742681 0.371341 0.928497i \(-0.378898\pi\)
0.371341 + 0.928497i \(0.378898\pi\)
\(702\) 0 0
\(703\) −23523.2 −1.26201
\(704\) 13725.6 0.734803
\(705\) 0 0
\(706\) 12419.5 0.662061
\(707\) 0 0
\(708\) 0 0
\(709\) 577.799 0.0306060 0.0153030 0.999883i \(-0.495129\pi\)
0.0153030 + 0.999883i \(0.495129\pi\)
\(710\) −14021.6 −0.741159
\(711\) 0 0
\(712\) −5347.30 −0.281459
\(713\) 25366.2 1.33236
\(714\) 0 0
\(715\) 38484.9 2.01294
\(716\) 16842.5 0.879096
\(717\) 0 0
\(718\) 10256.3 0.533097
\(719\) −20274.7 −1.05162 −0.525812 0.850601i \(-0.676239\pi\)
−0.525812 + 0.850601i \(0.676239\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2311.67 0.119157
\(723\) 0 0
\(724\) 1142.66 0.0586556
\(725\) 359.220 0.0184015
\(726\) 0 0
\(727\) −28861.8 −1.47239 −0.736193 0.676772i \(-0.763378\pi\)
−0.736193 + 0.676772i \(0.763378\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −3347.36 −0.169714
\(731\) −64.5762 −0.00326736
\(732\) 0 0
\(733\) 26169.5 1.31868 0.659340 0.751844i \(-0.270836\pi\)
0.659340 + 0.751844i \(0.270836\pi\)
\(734\) −2848.56 −0.143246
\(735\) 0 0
\(736\) −21319.6 −1.06773
\(737\) −6957.77 −0.347751
\(738\) 0 0
\(739\) −8351.16 −0.415700 −0.207850 0.978161i \(-0.566647\pi\)
−0.207850 + 0.978161i \(0.566647\pi\)
\(740\) 14127.2 0.701794
\(741\) 0 0
\(742\) 0 0
\(743\) −20732.7 −1.02370 −0.511849 0.859075i \(-0.671039\pi\)
−0.511849 + 0.859075i \(0.671039\pi\)
\(744\) 0 0
\(745\) 4629.67 0.227675
\(746\) −7477.09 −0.366965
\(747\) 0 0
\(748\) 5298.22 0.258987
\(749\) 0 0
\(750\) 0 0
\(751\) 13055.4 0.634351 0.317176 0.948367i \(-0.397265\pi\)
0.317176 + 0.948367i \(0.397265\pi\)
\(752\) 7334.36 0.355661
\(753\) 0 0
\(754\) 1077.05 0.0520208
\(755\) −25853.4 −1.24623
\(756\) 0 0
\(757\) 14425.9 0.692628 0.346314 0.938119i \(-0.387433\pi\)
0.346314 + 0.938119i \(0.387433\pi\)
\(758\) 10409.4 0.498794
\(759\) 0 0
\(760\) −18904.2 −0.902273
\(761\) −5232.01 −0.249225 −0.124613 0.992205i \(-0.539769\pi\)
−0.124613 + 0.992205i \(0.539769\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 21268.0 1.00713
\(765\) 0 0
\(766\) 17972.0 0.847723
\(767\) 12108.6 0.570033
\(768\) 0 0
\(769\) 18570.2 0.870818 0.435409 0.900233i \(-0.356604\pi\)
0.435409 + 0.900233i \(0.356604\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 17259.5 0.804639
\(773\) 23462.7 1.09171 0.545857 0.837878i \(-0.316204\pi\)
0.545857 + 0.837878i \(0.316204\pi\)
\(774\) 0 0
\(775\) 6408.70 0.297042
\(776\) −1249.51 −0.0578027
\(777\) 0 0
\(778\) −11671.5 −0.537847
\(779\) 26202.7 1.20515
\(780\) 0 0
\(781\) 65044.1 2.98010
\(782\) 2368.21 0.108295
\(783\) 0 0
\(784\) 0 0
\(785\) −3524.16 −0.160233
\(786\) 0 0
\(787\) −25626.0 −1.16070 −0.580348 0.814369i \(-0.697083\pi\)
−0.580348 + 0.814369i \(0.697083\pi\)
\(788\) 15158.6 0.685284
\(789\) 0 0
\(790\) 143.493 0.00646236
\(791\) 0 0
\(792\) 0 0
\(793\) −42489.6 −1.90271
\(794\) 5122.89 0.228973
\(795\) 0 0
\(796\) −26888.6 −1.19729
\(797\) 12209.4 0.542633 0.271316 0.962490i \(-0.412541\pi\)
0.271316 + 0.962490i \(0.412541\pi\)
\(798\) 0 0
\(799\) −8177.53 −0.362078
\(800\) −5386.33 −0.238044
\(801\) 0 0
\(802\) −10854.3 −0.477905
\(803\) 15527.8 0.682398
\(804\) 0 0
\(805\) 0 0
\(806\) 19215.2 0.839733
\(807\) 0 0
\(808\) −14403.9 −0.627136
\(809\) −7844.50 −0.340912 −0.170456 0.985365i \(-0.554524\pi\)
−0.170456 + 0.985365i \(0.554524\pi\)
\(810\) 0 0
\(811\) −7602.91 −0.329192 −0.164596 0.986361i \(-0.552632\pi\)
−0.164596 + 0.986361i \(0.552632\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 28196.7 1.21412
\(815\) −26096.0 −1.12160
\(816\) 0 0
\(817\) −439.802 −0.0188332
\(818\) 7034.31 0.300671
\(819\) 0 0
\(820\) −15736.5 −0.670173
\(821\) 29146.5 1.23900 0.619501 0.784996i \(-0.287335\pi\)
0.619501 + 0.784996i \(0.287335\pi\)
\(822\) 0 0
\(823\) −22062.1 −0.934432 −0.467216 0.884143i \(-0.654743\pi\)
−0.467216 + 0.884143i \(0.654743\pi\)
\(824\) −10977.0 −0.464079
\(825\) 0 0
\(826\) 0 0
\(827\) 4571.98 0.192241 0.0961206 0.995370i \(-0.469357\pi\)
0.0961206 + 0.995370i \(0.469357\pi\)
\(828\) 0 0
\(829\) −10311.6 −0.432012 −0.216006 0.976392i \(-0.569303\pi\)
−0.216006 + 0.976392i \(0.569303\pi\)
\(830\) −12181.4 −0.509426
\(831\) 0 0
\(832\) −10801.3 −0.450081
\(833\) 0 0
\(834\) 0 0
\(835\) −6893.57 −0.285703
\(836\) 36084.0 1.49281
\(837\) 0 0
\(838\) −2323.11 −0.0957645
\(839\) 3414.34 0.140496 0.0702480 0.997530i \(-0.477621\pi\)
0.0702480 + 0.997530i \(0.477621\pi\)
\(840\) 0 0
\(841\) −24232.9 −0.993598
\(842\) −7243.92 −0.296487
\(843\) 0 0
\(844\) −31596.0 −1.28860
\(845\) −8731.16 −0.355457
\(846\) 0 0
\(847\) 0 0
\(848\) −8316.14 −0.336766
\(849\) 0 0
\(850\) 598.320 0.0241438
\(851\) −29292.4 −1.17994
\(852\) 0 0
\(853\) −12093.2 −0.485419 −0.242709 0.970099i \(-0.578036\pi\)
−0.242709 + 0.970099i \(0.578036\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −15065.7 −0.601560
\(857\) −35003.7 −1.39522 −0.697610 0.716478i \(-0.745753\pi\)
−0.697610 + 0.716478i \(0.745753\pi\)
\(858\) 0 0
\(859\) 1669.49 0.0663123 0.0331562 0.999450i \(-0.489444\pi\)
0.0331562 + 0.999450i \(0.489444\pi\)
\(860\) 264.130 0.0104730
\(861\) 0 0
\(862\) 18593.0 0.734664
\(863\) 1620.80 0.0639312 0.0319656 0.999489i \(-0.489823\pi\)
0.0319656 + 0.999489i \(0.489823\pi\)
\(864\) 0 0
\(865\) 7248.40 0.284917
\(866\) 12762.1 0.500779
\(867\) 0 0
\(868\) 0 0
\(869\) −665.642 −0.0259843
\(870\) 0 0
\(871\) 5475.40 0.213004
\(872\) 3854.88 0.149705
\(873\) 0 0
\(874\) 16128.9 0.624218
\(875\) 0 0
\(876\) 0 0
\(877\) −14449.2 −0.556346 −0.278173 0.960531i \(-0.589729\pi\)
−0.278173 + 0.960531i \(0.589729\pi\)
\(878\) 12649.7 0.486226
\(879\) 0 0
\(880\) −8334.94 −0.319285
\(881\) −40111.6 −1.53393 −0.766966 0.641688i \(-0.778234\pi\)
−0.766966 + 0.641688i \(0.778234\pi\)
\(882\) 0 0
\(883\) 10152.0 0.386911 0.193456 0.981109i \(-0.438030\pi\)
0.193456 + 0.981109i \(0.438030\pi\)
\(884\) −4169.42 −0.158634
\(885\) 0 0
\(886\) 17694.5 0.670945
\(887\) 18907.0 0.715711 0.357856 0.933777i \(-0.383508\pi\)
0.357856 + 0.933777i \(0.383508\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −3859.35 −0.145355
\(891\) 0 0
\(892\) 25705.0 0.964875
\(893\) −55693.8 −2.08703
\(894\) 0 0
\(895\) 29541.9 1.10332
\(896\) 0 0
\(897\) 0 0
\(898\) −3821.83 −0.142022
\(899\) 2785.75 0.103348
\(900\) 0 0
\(901\) 9272.17 0.342842
\(902\) −31408.6 −1.15941
\(903\) 0 0
\(904\) 3256.65 0.119817
\(905\) 2004.24 0.0736167
\(906\) 0 0
\(907\) −6789.41 −0.248554 −0.124277 0.992248i \(-0.539661\pi\)
−0.124277 + 0.992248i \(0.539661\pi\)
\(908\) −10384.3 −0.379531
\(909\) 0 0
\(910\) 0 0
\(911\) −45716.8 −1.66264 −0.831320 0.555794i \(-0.812414\pi\)
−0.831320 + 0.555794i \(0.812414\pi\)
\(912\) 0 0
\(913\) 56507.6 2.04833
\(914\) −25364.0 −0.917907
\(915\) 0 0
\(916\) −21424.7 −0.772809
\(917\) 0 0
\(918\) 0 0
\(919\) 19232.7 0.690346 0.345173 0.938539i \(-0.387820\pi\)
0.345173 + 0.938539i \(0.387820\pi\)
\(920\) −23540.6 −0.843599
\(921\) 0 0
\(922\) −4456.74 −0.159192
\(923\) −51186.3 −1.82537
\(924\) 0 0
\(925\) −7400.64 −0.263061
\(926\) 4629.47 0.164291
\(927\) 0 0
\(928\) −2341.34 −0.0828215
\(929\) −45139.4 −1.59416 −0.797081 0.603873i \(-0.793623\pi\)
−0.797081 + 0.603873i \(0.793623\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −12556.0 −0.441295
\(933\) 0 0
\(934\) 30126.0 1.05541
\(935\) 9293.13 0.325046
\(936\) 0 0
\(937\) 3684.00 0.128443 0.0642215 0.997936i \(-0.479544\pi\)
0.0642215 + 0.997936i \(0.479544\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 33447.8 1.16058
\(941\) 3082.72 0.106795 0.0533973 0.998573i \(-0.482995\pi\)
0.0533973 + 0.998573i \(0.482995\pi\)
\(942\) 0 0
\(943\) 32629.2 1.12678
\(944\) −2622.43 −0.0904163
\(945\) 0 0
\(946\) 527.180 0.0181185
\(947\) −6475.69 −0.222209 −0.111104 0.993809i \(-0.535439\pi\)
−0.111104 + 0.993809i \(0.535439\pi\)
\(948\) 0 0
\(949\) −12219.6 −0.417982
\(950\) 4074.91 0.139166
\(951\) 0 0
\(952\) 0 0
\(953\) 7643.58 0.259811 0.129906 0.991526i \(-0.458533\pi\)
0.129906 + 0.991526i \(0.458533\pi\)
\(954\) 0 0
\(955\) 37304.2 1.26402
\(956\) 17814.3 0.602674
\(957\) 0 0
\(958\) 28279.6 0.953730
\(959\) 0 0
\(960\) 0 0
\(961\) 19908.5 0.668273
\(962\) −22189.3 −0.743671
\(963\) 0 0
\(964\) 8989.36 0.300340
\(965\) 30273.3 1.00988
\(966\) 0 0
\(967\) 5636.41 0.187440 0.0937202 0.995599i \(-0.470124\pi\)
0.0937202 + 0.995599i \(0.470124\pi\)
\(968\) −77048.2 −2.55829
\(969\) 0 0
\(970\) −901.820 −0.0298512
\(971\) 12541.0 0.414479 0.207240 0.978290i \(-0.433552\pi\)
0.207240 + 0.978290i \(0.433552\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 26496.0 0.871649
\(975\) 0 0
\(976\) 9202.25 0.301800
\(977\) 45250.1 1.48176 0.740879 0.671638i \(-0.234409\pi\)
0.740879 + 0.671638i \(0.234409\pi\)
\(978\) 0 0
\(979\) 17902.9 0.584452
\(980\) 0 0
\(981\) 0 0
\(982\) −1686.49 −0.0548045
\(983\) 48847.0 1.58492 0.792460 0.609923i \(-0.208800\pi\)
0.792460 + 0.609923i \(0.208800\pi\)
\(984\) 0 0
\(985\) 26588.4 0.860077
\(986\) 260.079 0.00840022
\(987\) 0 0
\(988\) −28396.2 −0.914376
\(989\) −547.667 −0.0176085
\(990\) 0 0
\(991\) −31801.6 −1.01939 −0.509693 0.860357i \(-0.670241\pi\)
−0.509693 + 0.860357i \(0.670241\pi\)
\(992\) −41771.0 −1.33693
\(993\) 0 0
\(994\) 0 0
\(995\) −47162.9 −1.50268
\(996\) 0 0
\(997\) 18382.5 0.583930 0.291965 0.956429i \(-0.405691\pi\)
0.291965 + 0.956429i \(0.405691\pi\)
\(998\) 29142.8 0.924349
\(999\) 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 1323.4.a.ba.1.3 4
3.2 odd 2 inner 1323.4.a.ba.1.2 4
7.6 odd 2 189.4.a.l.1.3 yes 4
21.20 even 2 189.4.a.l.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.l.1.2 4 21.20 even 2
189.4.a.l.1.3 yes 4 7.6 odd 2
1323.4.a.ba.1.2 4 3.2 odd 2 inner
1323.4.a.ba.1.3 4 1.1 even 1 trivial