Properties

Label 45.4.a.e.1.1
Level $45$
Weight $4$
Character 45.1
Self dual yes
Analytic conductor $2.655$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.65508595026\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 45.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{2} +17.0000 q^{4} -5.00000 q^{5} -30.0000 q^{7} +45.0000 q^{8} +O(q^{10})\) \(q+5.00000 q^{2} +17.0000 q^{4} -5.00000 q^{5} -30.0000 q^{7} +45.0000 q^{8} -25.0000 q^{10} +50.0000 q^{11} -20.0000 q^{13} -150.000 q^{14} +89.0000 q^{16} -10.0000 q^{17} -44.0000 q^{19} -85.0000 q^{20} +250.000 q^{22} +120.000 q^{23} +25.0000 q^{25} -100.000 q^{26} -510.000 q^{28} -50.0000 q^{29} +108.000 q^{31} +85.0000 q^{32} -50.0000 q^{34} +150.000 q^{35} -40.0000 q^{37} -220.000 q^{38} -225.000 q^{40} +400.000 q^{41} +280.000 q^{43} +850.000 q^{44} +600.000 q^{46} -280.000 q^{47} +557.000 q^{49} +125.000 q^{50} -340.000 q^{52} -610.000 q^{53} -250.000 q^{55} -1350.00 q^{56} -250.000 q^{58} +50.0000 q^{59} -518.000 q^{61} +540.000 q^{62} -287.000 q^{64} +100.000 q^{65} -180.000 q^{67} -170.000 q^{68} +750.000 q^{70} +700.000 q^{71} -410.000 q^{73} -200.000 q^{74} -748.000 q^{76} -1500.00 q^{77} -516.000 q^{79} -445.000 q^{80} +2000.00 q^{82} +660.000 q^{83} +50.0000 q^{85} +1400.00 q^{86} +2250.00 q^{88} -1500.00 q^{89} +600.000 q^{91} +2040.00 q^{92} -1400.00 q^{94} +220.000 q^{95} -1630.00 q^{97} +2785.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.00000 1.76777 0.883883 0.467707i \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) 0 0
\(4\) 17.0000 2.12500
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −30.0000 −1.61985 −0.809924 0.586535i \(-0.800492\pi\)
−0.809924 + 0.586535i \(0.800492\pi\)
\(8\) 45.0000 1.98874
\(9\) 0 0
\(10\) −25.0000 −0.790569
\(11\) 50.0000 1.37051 0.685253 0.728305i \(-0.259692\pi\)
0.685253 + 0.728305i \(0.259692\pi\)
\(12\) 0 0
\(13\) −20.0000 −0.426692 −0.213346 0.976977i \(-0.568436\pi\)
−0.213346 + 0.976977i \(0.568436\pi\)
\(14\) −150.000 −2.86351
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) −10.0000 −0.142668 −0.0713340 0.997452i \(-0.522726\pi\)
−0.0713340 + 0.997452i \(0.522726\pi\)
\(18\) 0 0
\(19\) −44.0000 −0.531279 −0.265639 0.964072i \(-0.585583\pi\)
−0.265639 + 0.964072i \(0.585583\pi\)
\(20\) −85.0000 −0.950329
\(21\) 0 0
\(22\) 250.000 2.42274
\(23\) 120.000 1.08790 0.543951 0.839117i \(-0.316928\pi\)
0.543951 + 0.839117i \(0.316928\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −100.000 −0.754293
\(27\) 0 0
\(28\) −510.000 −3.44218
\(29\) −50.0000 −0.320164 −0.160082 0.987104i \(-0.551176\pi\)
−0.160082 + 0.987104i \(0.551176\pi\)
\(30\) 0 0
\(31\) 108.000 0.625722 0.312861 0.949799i \(-0.398713\pi\)
0.312861 + 0.949799i \(0.398713\pi\)
\(32\) 85.0000 0.469563
\(33\) 0 0
\(34\) −50.0000 −0.252204
\(35\) 150.000 0.724418
\(36\) 0 0
\(37\) −40.0000 −0.177729 −0.0888643 0.996044i \(-0.528324\pi\)
−0.0888643 + 0.996044i \(0.528324\pi\)
\(38\) −220.000 −0.939177
\(39\) 0 0
\(40\) −225.000 −0.889391
\(41\) 400.000 1.52365 0.761823 0.647785i \(-0.224304\pi\)
0.761823 + 0.647785i \(0.224304\pi\)
\(42\) 0 0
\(43\) 280.000 0.993014 0.496507 0.868033i \(-0.334616\pi\)
0.496507 + 0.868033i \(0.334616\pi\)
\(44\) 850.000 2.91233
\(45\) 0 0
\(46\) 600.000 1.92316
\(47\) −280.000 −0.868983 −0.434491 0.900676i \(-0.643072\pi\)
−0.434491 + 0.900676i \(0.643072\pi\)
\(48\) 0 0
\(49\) 557.000 1.62391
\(50\) 125.000 0.353553
\(51\) 0 0
\(52\) −340.000 −0.906721
\(53\) −610.000 −1.58094 −0.790471 0.612499i \(-0.790164\pi\)
−0.790471 + 0.612499i \(0.790164\pi\)
\(54\) 0 0
\(55\) −250.000 −0.612909
\(56\) −1350.00 −3.22145
\(57\) 0 0
\(58\) −250.000 −0.565976
\(59\) 50.0000 0.110330 0.0551648 0.998477i \(-0.482432\pi\)
0.0551648 + 0.998477i \(0.482432\pi\)
\(60\) 0 0
\(61\) −518.000 −1.08726 −0.543632 0.839324i \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(62\) 540.000 1.10613
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) 100.000 0.190823
\(66\) 0 0
\(67\) −180.000 −0.328216 −0.164108 0.986442i \(-0.552475\pi\)
−0.164108 + 0.986442i \(0.552475\pi\)
\(68\) −170.000 −0.303170
\(69\) 0 0
\(70\) 750.000 1.28060
\(71\) 700.000 1.17007 0.585033 0.811009i \(-0.301081\pi\)
0.585033 + 0.811009i \(0.301081\pi\)
\(72\) 0 0
\(73\) −410.000 −0.657354 −0.328677 0.944442i \(-0.606603\pi\)
−0.328677 + 0.944442i \(0.606603\pi\)
\(74\) −200.000 −0.314183
\(75\) 0 0
\(76\) −748.000 −1.12897
\(77\) −1500.00 −2.22001
\(78\) 0 0
\(79\) −516.000 −0.734868 −0.367434 0.930050i \(-0.619764\pi\)
−0.367434 + 0.930050i \(0.619764\pi\)
\(80\) −445.000 −0.621906
\(81\) 0 0
\(82\) 2000.00 2.69345
\(83\) 660.000 0.872824 0.436412 0.899747i \(-0.356249\pi\)
0.436412 + 0.899747i \(0.356249\pi\)
\(84\) 0 0
\(85\) 50.0000 0.0638031
\(86\) 1400.00 1.75542
\(87\) 0 0
\(88\) 2250.00 2.72558
\(89\) −1500.00 −1.78651 −0.893257 0.449547i \(-0.851585\pi\)
−0.893257 + 0.449547i \(0.851585\pi\)
\(90\) 0 0
\(91\) 600.000 0.691177
\(92\) 2040.00 2.31179
\(93\) 0 0
\(94\) −1400.00 −1.53616
\(95\) 220.000 0.237595
\(96\) 0 0
\(97\) −1630.00 −1.70620 −0.853100 0.521747i \(-0.825280\pi\)
−0.853100 + 0.521747i \(0.825280\pi\)
\(98\) 2785.00 2.87069
\(99\) 0 0
\(100\) 425.000 0.425000
\(101\) −450.000 −0.443333 −0.221667 0.975122i \(-0.571150\pi\)
−0.221667 + 0.975122i \(0.571150\pi\)
\(102\) 0 0
\(103\) 770.000 0.736605 0.368303 0.929706i \(-0.379939\pi\)
0.368303 + 0.929706i \(0.379939\pi\)
\(104\) −900.000 −0.848579
\(105\) 0 0
\(106\) −3050.00 −2.79474
\(107\) 660.000 0.596305 0.298152 0.954518i \(-0.403630\pi\)
0.298152 + 0.954518i \(0.403630\pi\)
\(108\) 0 0
\(109\) 1754.00 1.54131 0.770655 0.637253i \(-0.219929\pi\)
0.770655 + 0.637253i \(0.219929\pi\)
\(110\) −1250.00 −1.08348
\(111\) 0 0
\(112\) −2670.00 −2.25260
\(113\) −310.000 −0.258074 −0.129037 0.991640i \(-0.541189\pi\)
−0.129037 + 0.991640i \(0.541189\pi\)
\(114\) 0 0
\(115\) −600.000 −0.486524
\(116\) −850.000 −0.680349
\(117\) 0 0
\(118\) 250.000 0.195037
\(119\) 300.000 0.231100
\(120\) 0 0
\(121\) 1169.00 0.878287
\(122\) −2590.00 −1.92203
\(123\) 0 0
\(124\) 1836.00 1.32966
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1070.00 −0.747615 −0.373808 0.927506i \(-0.621948\pi\)
−0.373808 + 0.927506i \(0.621948\pi\)
\(128\) −2115.00 −1.46048
\(129\) 0 0
\(130\) 500.000 0.337330
\(131\) 1950.00 1.30055 0.650276 0.759698i \(-0.274653\pi\)
0.650276 + 0.759698i \(0.274653\pi\)
\(132\) 0 0
\(133\) 1320.00 0.860590
\(134\) −900.000 −0.580210
\(135\) 0 0
\(136\) −450.000 −0.283729
\(137\) 1050.00 0.654800 0.327400 0.944886i \(-0.393828\pi\)
0.327400 + 0.944886i \(0.393828\pi\)
\(138\) 0 0
\(139\) 1676.00 1.02271 0.511354 0.859370i \(-0.329144\pi\)
0.511354 + 0.859370i \(0.329144\pi\)
\(140\) 2550.00 1.53939
\(141\) 0 0
\(142\) 3500.00 2.06840
\(143\) −1000.00 −0.584785
\(144\) 0 0
\(145\) 250.000 0.143182
\(146\) −2050.00 −1.16205
\(147\) 0 0
\(148\) −680.000 −0.377673
\(149\) 2050.00 1.12713 0.563566 0.826071i \(-0.309429\pi\)
0.563566 + 0.826071i \(0.309429\pi\)
\(150\) 0 0
\(151\) 448.000 0.241442 0.120721 0.992686i \(-0.461479\pi\)
0.120721 + 0.992686i \(0.461479\pi\)
\(152\) −1980.00 −1.05657
\(153\) 0 0
\(154\) −7500.00 −3.92446
\(155\) −540.000 −0.279831
\(156\) 0 0
\(157\) −100.000 −0.0508336 −0.0254168 0.999677i \(-0.508091\pi\)
−0.0254168 + 0.999677i \(0.508091\pi\)
\(158\) −2580.00 −1.29907
\(159\) 0 0
\(160\) −425.000 −0.209995
\(161\) −3600.00 −1.76223
\(162\) 0 0
\(163\) −1900.00 −0.913003 −0.456501 0.889723i \(-0.650898\pi\)
−0.456501 + 0.889723i \(0.650898\pi\)
\(164\) 6800.00 3.23775
\(165\) 0 0
\(166\) 3300.00 1.54295
\(167\) 1920.00 0.889665 0.444833 0.895614i \(-0.353263\pi\)
0.444833 + 0.895614i \(0.353263\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) 250.000 0.112789
\(171\) 0 0
\(172\) 4760.00 2.11015
\(173\) 2550.00 1.12065 0.560326 0.828272i \(-0.310676\pi\)
0.560326 + 0.828272i \(0.310676\pi\)
\(174\) 0 0
\(175\) −750.000 −0.323970
\(176\) 4450.00 1.90586
\(177\) 0 0
\(178\) −7500.00 −3.15814
\(179\) 3650.00 1.52410 0.762050 0.647518i \(-0.224193\pi\)
0.762050 + 0.647518i \(0.224193\pi\)
\(180\) 0 0
\(181\) −4342.00 −1.78308 −0.891542 0.452937i \(-0.850376\pi\)
−0.891542 + 0.452937i \(0.850376\pi\)
\(182\) 3000.00 1.22184
\(183\) 0 0
\(184\) 5400.00 2.16355
\(185\) 200.000 0.0794827
\(186\) 0 0
\(187\) −500.000 −0.195527
\(188\) −4760.00 −1.84659
\(189\) 0 0
\(190\) 1100.00 0.420013
\(191\) −3500.00 −1.32592 −0.662961 0.748654i \(-0.730701\pi\)
−0.662961 + 0.748654i \(0.730701\pi\)
\(192\) 0 0
\(193\) 3350.00 1.24942 0.624711 0.780856i \(-0.285217\pi\)
0.624711 + 0.780856i \(0.285217\pi\)
\(194\) −8150.00 −3.01616
\(195\) 0 0
\(196\) 9469.00 3.45080
\(197\) 90.0000 0.0325494 0.0162747 0.999868i \(-0.494819\pi\)
0.0162747 + 0.999868i \(0.494819\pi\)
\(198\) 0 0
\(199\) 3664.00 1.30520 0.652598 0.757704i \(-0.273679\pi\)
0.652598 + 0.757704i \(0.273679\pi\)
\(200\) 1125.00 0.397748
\(201\) 0 0
\(202\) −2250.00 −0.783710
\(203\) 1500.00 0.518618
\(204\) 0 0
\(205\) −2000.00 −0.681395
\(206\) 3850.00 1.30215
\(207\) 0 0
\(208\) −1780.00 −0.593369
\(209\) −2200.00 −0.728120
\(210\) 0 0
\(211\) −268.000 −0.0874402 −0.0437201 0.999044i \(-0.513921\pi\)
−0.0437201 + 0.999044i \(0.513921\pi\)
\(212\) −10370.0 −3.35950
\(213\) 0 0
\(214\) 3300.00 1.05413
\(215\) −1400.00 −0.444089
\(216\) 0 0
\(217\) −3240.00 −1.01357
\(218\) 8770.00 2.72468
\(219\) 0 0
\(220\) −4250.00 −1.30243
\(221\) 200.000 0.0608754
\(222\) 0 0
\(223\) −3670.00 −1.10207 −0.551034 0.834482i \(-0.685767\pi\)
−0.551034 + 0.834482i \(0.685767\pi\)
\(224\) −2550.00 −0.760621
\(225\) 0 0
\(226\) −1550.00 −0.456214
\(227\) 3760.00 1.09938 0.549692 0.835368i \(-0.314745\pi\)
0.549692 + 0.835368i \(0.314745\pi\)
\(228\) 0 0
\(229\) −1434.00 −0.413805 −0.206903 0.978362i \(-0.566338\pi\)
−0.206903 + 0.978362i \(0.566338\pi\)
\(230\) −3000.00 −0.860061
\(231\) 0 0
\(232\) −2250.00 −0.636723
\(233\) −3450.00 −0.970030 −0.485015 0.874506i \(-0.661186\pi\)
−0.485015 + 0.874506i \(0.661186\pi\)
\(234\) 0 0
\(235\) 1400.00 0.388621
\(236\) 850.000 0.234450
\(237\) 0 0
\(238\) 1500.00 0.408532
\(239\) −4900.00 −1.32617 −0.663085 0.748544i \(-0.730753\pi\)
−0.663085 + 0.748544i \(0.730753\pi\)
\(240\) 0 0
\(241\) 4822.00 1.28885 0.644424 0.764668i \(-0.277097\pi\)
0.644424 + 0.764668i \(0.277097\pi\)
\(242\) 5845.00 1.55261
\(243\) 0 0
\(244\) −8806.00 −2.31044
\(245\) −2785.00 −0.726233
\(246\) 0 0
\(247\) 880.000 0.226693
\(248\) 4860.00 1.24440
\(249\) 0 0
\(250\) −625.000 −0.158114
\(251\) −4650.00 −1.16934 −0.584672 0.811270i \(-0.698777\pi\)
−0.584672 + 0.811270i \(0.698777\pi\)
\(252\) 0 0
\(253\) 6000.00 1.49098
\(254\) −5350.00 −1.32161
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) 5130.00 1.24514 0.622569 0.782565i \(-0.286089\pi\)
0.622569 + 0.782565i \(0.286089\pi\)
\(258\) 0 0
\(259\) 1200.00 0.287893
\(260\) 1700.00 0.405498
\(261\) 0 0
\(262\) 9750.00 2.29907
\(263\) −1280.00 −0.300107 −0.150054 0.988678i \(-0.547945\pi\)
−0.150054 + 0.988678i \(0.547945\pi\)
\(264\) 0 0
\(265\) 3050.00 0.707019
\(266\) 6600.00 1.52132
\(267\) 0 0
\(268\) −3060.00 −0.697460
\(269\) 3350.00 0.759305 0.379653 0.925129i \(-0.376044\pi\)
0.379653 + 0.925129i \(0.376044\pi\)
\(270\) 0 0
\(271\) 5512.00 1.23554 0.617768 0.786361i \(-0.288037\pi\)
0.617768 + 0.786361i \(0.288037\pi\)
\(272\) −890.000 −0.198398
\(273\) 0 0
\(274\) 5250.00 1.15753
\(275\) 1250.00 0.274101
\(276\) 0 0
\(277\) 4920.00 1.06720 0.533600 0.845737i \(-0.320839\pi\)
0.533600 + 0.845737i \(0.320839\pi\)
\(278\) 8380.00 1.80791
\(279\) 0 0
\(280\) 6750.00 1.44068
\(281\) 4500.00 0.955329 0.477665 0.878542i \(-0.341483\pi\)
0.477665 + 0.878542i \(0.341483\pi\)
\(282\) 0 0
\(283\) −6900.00 −1.44934 −0.724669 0.689098i \(-0.758007\pi\)
−0.724669 + 0.689098i \(0.758007\pi\)
\(284\) 11900.0 2.48639
\(285\) 0 0
\(286\) −5000.00 −1.03376
\(287\) −12000.0 −2.46808
\(288\) 0 0
\(289\) −4813.00 −0.979646
\(290\) 1250.00 0.253112
\(291\) 0 0
\(292\) −6970.00 −1.39688
\(293\) 1530.00 0.305063 0.152532 0.988299i \(-0.451257\pi\)
0.152532 + 0.988299i \(0.451257\pi\)
\(294\) 0 0
\(295\) −250.000 −0.0493409
\(296\) −1800.00 −0.353456
\(297\) 0 0
\(298\) 10250.0 1.99251
\(299\) −2400.00 −0.464199
\(300\) 0 0
\(301\) −8400.00 −1.60853
\(302\) 2240.00 0.426813
\(303\) 0 0
\(304\) −3916.00 −0.738809
\(305\) 2590.00 0.486239
\(306\) 0 0
\(307\) 3040.00 0.565153 0.282576 0.959245i \(-0.408811\pi\)
0.282576 + 0.959245i \(0.408811\pi\)
\(308\) −25500.0 −4.71752
\(309\) 0 0
\(310\) −2700.00 −0.494676
\(311\) −5700.00 −1.03928 −0.519642 0.854384i \(-0.673935\pi\)
−0.519642 + 0.854384i \(0.673935\pi\)
\(312\) 0 0
\(313\) 3110.00 0.561622 0.280811 0.959763i \(-0.409397\pi\)
0.280811 + 0.959763i \(0.409397\pi\)
\(314\) −500.000 −0.0898619
\(315\) 0 0
\(316\) −8772.00 −1.56159
\(317\) −950.000 −0.168320 −0.0841598 0.996452i \(-0.526821\pi\)
−0.0841598 + 0.996452i \(0.526821\pi\)
\(318\) 0 0
\(319\) −2500.00 −0.438787
\(320\) 1435.00 0.250684
\(321\) 0 0
\(322\) −18000.0 −3.11522
\(323\) 440.000 0.0757965
\(324\) 0 0
\(325\) −500.000 −0.0853385
\(326\) −9500.00 −1.61398
\(327\) 0 0
\(328\) 18000.0 3.03013
\(329\) 8400.00 1.40762
\(330\) 0 0
\(331\) 2292.00 0.380603 0.190302 0.981726i \(-0.439053\pi\)
0.190302 + 0.981726i \(0.439053\pi\)
\(332\) 11220.0 1.85475
\(333\) 0 0
\(334\) 9600.00 1.57272
\(335\) 900.000 0.146783
\(336\) 0 0
\(337\) −7730.00 −1.24950 −0.624748 0.780827i \(-0.714798\pi\)
−0.624748 + 0.780827i \(0.714798\pi\)
\(338\) −8985.00 −1.44592
\(339\) 0 0
\(340\) 850.000 0.135582
\(341\) 5400.00 0.857555
\(342\) 0 0
\(343\) −6420.00 −1.01063
\(344\) 12600.0 1.97484
\(345\) 0 0
\(346\) 12750.0 1.98105
\(347\) −1120.00 −0.173270 −0.0866351 0.996240i \(-0.527611\pi\)
−0.0866351 + 0.996240i \(0.527611\pi\)
\(348\) 0 0
\(349\) 1186.00 0.181906 0.0909529 0.995855i \(-0.471009\pi\)
0.0909529 + 0.995855i \(0.471009\pi\)
\(350\) −3750.00 −0.572703
\(351\) 0 0
\(352\) 4250.00 0.643539
\(353\) 3630.00 0.547324 0.273662 0.961826i \(-0.411765\pi\)
0.273662 + 0.961826i \(0.411765\pi\)
\(354\) 0 0
\(355\) −3500.00 −0.523270
\(356\) −25500.0 −3.79634
\(357\) 0 0
\(358\) 18250.0 2.69425
\(359\) −1800.00 −0.264625 −0.132312 0.991208i \(-0.542240\pi\)
−0.132312 + 0.991208i \(0.542240\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) −21710.0 −3.15208
\(363\) 0 0
\(364\) 10200.0 1.46875
\(365\) 2050.00 0.293978
\(366\) 0 0
\(367\) 8490.00 1.20756 0.603780 0.797151i \(-0.293661\pi\)
0.603780 + 0.797151i \(0.293661\pi\)
\(368\) 10680.0 1.51286
\(369\) 0 0
\(370\) 1000.00 0.140507
\(371\) 18300.0 2.56089
\(372\) 0 0
\(373\) 100.000 0.0138815 0.00694076 0.999976i \(-0.497791\pi\)
0.00694076 + 0.999976i \(0.497791\pi\)
\(374\) −2500.00 −0.345647
\(375\) 0 0
\(376\) −12600.0 −1.72818
\(377\) 1000.00 0.136612
\(378\) 0 0
\(379\) −8084.00 −1.09564 −0.547820 0.836597i \(-0.684542\pi\)
−0.547820 + 0.836597i \(0.684542\pi\)
\(380\) 3740.00 0.504889
\(381\) 0 0
\(382\) −17500.0 −2.34392
\(383\) −9480.00 −1.26477 −0.632383 0.774656i \(-0.717923\pi\)
−0.632383 + 0.774656i \(0.717923\pi\)
\(384\) 0 0
\(385\) 7500.00 0.992819
\(386\) 16750.0 2.20869
\(387\) 0 0
\(388\) −27710.0 −3.62568
\(389\) −10950.0 −1.42722 −0.713608 0.700545i \(-0.752940\pi\)
−0.713608 + 0.700545i \(0.752940\pi\)
\(390\) 0 0
\(391\) −1200.00 −0.155209
\(392\) 25065.0 3.22952
\(393\) 0 0
\(394\) 450.000 0.0575398
\(395\) 2580.00 0.328643
\(396\) 0 0
\(397\) 13840.0 1.74965 0.874823 0.484442i \(-0.160977\pi\)
0.874823 + 0.484442i \(0.160977\pi\)
\(398\) 18320.0 2.30728
\(399\) 0 0
\(400\) 2225.00 0.278125
\(401\) 9300.00 1.15815 0.579077 0.815273i \(-0.303413\pi\)
0.579077 + 0.815273i \(0.303413\pi\)
\(402\) 0 0
\(403\) −2160.00 −0.266991
\(404\) −7650.00 −0.942083
\(405\) 0 0
\(406\) 7500.00 0.916795
\(407\) −2000.00 −0.243578
\(408\) 0 0
\(409\) −2854.00 −0.345040 −0.172520 0.985006i \(-0.555191\pi\)
−0.172520 + 0.985006i \(0.555191\pi\)
\(410\) −10000.0 −1.20455
\(411\) 0 0
\(412\) 13090.0 1.56529
\(413\) −1500.00 −0.178717
\(414\) 0 0
\(415\) −3300.00 −0.390339
\(416\) −1700.00 −0.200359
\(417\) 0 0
\(418\) −11000.0 −1.28715
\(419\) 1150.00 0.134084 0.0670420 0.997750i \(-0.478644\pi\)
0.0670420 + 0.997750i \(0.478644\pi\)
\(420\) 0 0
\(421\) −11162.0 −1.29217 −0.646084 0.763266i \(-0.723594\pi\)
−0.646084 + 0.763266i \(0.723594\pi\)
\(422\) −1340.00 −0.154574
\(423\) 0 0
\(424\) −27450.0 −3.14408
\(425\) −250.000 −0.0285336
\(426\) 0 0
\(427\) 15540.0 1.76120
\(428\) 11220.0 1.26715
\(429\) 0 0
\(430\) −7000.00 −0.785046
\(431\) −1200.00 −0.134111 −0.0670556 0.997749i \(-0.521361\pi\)
−0.0670556 + 0.997749i \(0.521361\pi\)
\(432\) 0 0
\(433\) 1510.00 0.167589 0.0837944 0.996483i \(-0.473296\pi\)
0.0837944 + 0.996483i \(0.473296\pi\)
\(434\) −16200.0 −1.79176
\(435\) 0 0
\(436\) 29818.0 3.27528
\(437\) −5280.00 −0.577979
\(438\) 0 0
\(439\) 424.000 0.0460966 0.0230483 0.999734i \(-0.492663\pi\)
0.0230483 + 0.999734i \(0.492663\pi\)
\(440\) −11250.0 −1.21892
\(441\) 0 0
\(442\) 1000.00 0.107613
\(443\) 12360.0 1.32560 0.662801 0.748796i \(-0.269368\pi\)
0.662801 + 0.748796i \(0.269368\pi\)
\(444\) 0 0
\(445\) 7500.00 0.798953
\(446\) −18350.0 −1.94820
\(447\) 0 0
\(448\) 8610.00 0.908001
\(449\) −1300.00 −0.136639 −0.0683194 0.997664i \(-0.521764\pi\)
−0.0683194 + 0.997664i \(0.521764\pi\)
\(450\) 0 0
\(451\) 20000.0 2.08817
\(452\) −5270.00 −0.548407
\(453\) 0 0
\(454\) 18800.0 1.94345
\(455\) −3000.00 −0.309104
\(456\) 0 0
\(457\) −7190.00 −0.735961 −0.367980 0.929834i \(-0.619951\pi\)
−0.367980 + 0.929834i \(0.619951\pi\)
\(458\) −7170.00 −0.731511
\(459\) 0 0
\(460\) −10200.0 −1.03386
\(461\) −150.000 −0.0151544 −0.00757722 0.999971i \(-0.502412\pi\)
−0.00757722 + 0.999971i \(0.502412\pi\)
\(462\) 0 0
\(463\) 2670.00 0.268003 0.134002 0.990981i \(-0.457217\pi\)
0.134002 + 0.990981i \(0.457217\pi\)
\(464\) −4450.00 −0.445229
\(465\) 0 0
\(466\) −17250.0 −1.71479
\(467\) 1180.00 0.116925 0.0584624 0.998290i \(-0.481380\pi\)
0.0584624 + 0.998290i \(0.481380\pi\)
\(468\) 0 0
\(469\) 5400.00 0.531661
\(470\) 7000.00 0.686991
\(471\) 0 0
\(472\) 2250.00 0.219417
\(473\) 14000.0 1.36093
\(474\) 0 0
\(475\) −1100.00 −0.106256
\(476\) 5100.00 0.491088
\(477\) 0 0
\(478\) −24500.0 −2.34436
\(479\) −14100.0 −1.34498 −0.672490 0.740106i \(-0.734775\pi\)
−0.672490 + 0.740106i \(0.734775\pi\)
\(480\) 0 0
\(481\) 800.000 0.0758355
\(482\) 24110.0 2.27838
\(483\) 0 0
\(484\) 19873.0 1.86636
\(485\) 8150.00 0.763036
\(486\) 0 0
\(487\) −9850.00 −0.916522 −0.458261 0.888818i \(-0.651527\pi\)
−0.458261 + 0.888818i \(0.651527\pi\)
\(488\) −23310.0 −2.16228
\(489\) 0 0
\(490\) −13925.0 −1.28381
\(491\) −2450.00 −0.225187 −0.112594 0.993641i \(-0.535916\pi\)
−0.112594 + 0.993641i \(0.535916\pi\)
\(492\) 0 0
\(493\) 500.000 0.0456772
\(494\) 4400.00 0.400740
\(495\) 0 0
\(496\) 9612.00 0.870144
\(497\) −21000.0 −1.89533
\(498\) 0 0
\(499\) −17036.0 −1.52833 −0.764164 0.645021i \(-0.776848\pi\)
−0.764164 + 0.645021i \(0.776848\pi\)
\(500\) −2125.00 −0.190066
\(501\) 0 0
\(502\) −23250.0 −2.06713
\(503\) −20600.0 −1.82606 −0.913030 0.407891i \(-0.866264\pi\)
−0.913030 + 0.407891i \(0.866264\pi\)
\(504\) 0 0
\(505\) 2250.00 0.198265
\(506\) 30000.0 2.63570
\(507\) 0 0
\(508\) −18190.0 −1.58868
\(509\) 5750.00 0.500716 0.250358 0.968153i \(-0.419452\pi\)
0.250358 + 0.968153i \(0.419452\pi\)
\(510\) 0 0
\(511\) 12300.0 1.06481
\(512\) −24475.0 −2.11260
\(513\) 0 0
\(514\) 25650.0 2.20111
\(515\) −3850.00 −0.329420
\(516\) 0 0
\(517\) −14000.0 −1.19095
\(518\) 6000.00 0.508928
\(519\) 0 0
\(520\) 4500.00 0.379496
\(521\) −15500.0 −1.30339 −0.651696 0.758480i \(-0.725942\pi\)
−0.651696 + 0.758480i \(0.725942\pi\)
\(522\) 0 0
\(523\) −13940.0 −1.16549 −0.582747 0.812653i \(-0.698022\pi\)
−0.582747 + 0.812653i \(0.698022\pi\)
\(524\) 33150.0 2.76367
\(525\) 0 0
\(526\) −6400.00 −0.530520
\(527\) −1080.00 −0.0892705
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 15250.0 1.24984
\(531\) 0 0
\(532\) 22440.0 1.82875
\(533\) −8000.00 −0.650128
\(534\) 0 0
\(535\) −3300.00 −0.266676
\(536\) −8100.00 −0.652736
\(537\) 0 0
\(538\) 16750.0 1.34227
\(539\) 27850.0 2.22557
\(540\) 0 0
\(541\) −20478.0 −1.62739 −0.813695 0.581292i \(-0.802547\pi\)
−0.813695 + 0.581292i \(0.802547\pi\)
\(542\) 27560.0 2.18414
\(543\) 0 0
\(544\) −850.000 −0.0669916
\(545\) −8770.00 −0.689295
\(546\) 0 0
\(547\) 12040.0 0.941121 0.470561 0.882368i \(-0.344052\pi\)
0.470561 + 0.882368i \(0.344052\pi\)
\(548\) 17850.0 1.39145
\(549\) 0 0
\(550\) 6250.00 0.484547
\(551\) 2200.00 0.170096
\(552\) 0 0
\(553\) 15480.0 1.19037
\(554\) 24600.0 1.88656
\(555\) 0 0
\(556\) 28492.0 2.17326
\(557\) 23550.0 1.79146 0.895732 0.444594i \(-0.146652\pi\)
0.895732 + 0.444594i \(0.146652\pi\)
\(558\) 0 0
\(559\) −5600.00 −0.423712
\(560\) 13350.0 1.00739
\(561\) 0 0
\(562\) 22500.0 1.68880
\(563\) −6120.00 −0.458130 −0.229065 0.973411i \(-0.573567\pi\)
−0.229065 + 0.973411i \(0.573567\pi\)
\(564\) 0 0
\(565\) 1550.00 0.115414
\(566\) −34500.0 −2.56209
\(567\) 0 0
\(568\) 31500.0 2.32696
\(569\) −11700.0 −0.862020 −0.431010 0.902347i \(-0.641843\pi\)
−0.431010 + 0.902347i \(0.641843\pi\)
\(570\) 0 0
\(571\) −8188.00 −0.600100 −0.300050 0.953923i \(-0.597003\pi\)
−0.300050 + 0.953923i \(0.597003\pi\)
\(572\) −17000.0 −1.24267
\(573\) 0 0
\(574\) −60000.0 −4.36298
\(575\) 3000.00 0.217580
\(576\) 0 0
\(577\) 11690.0 0.843433 0.421717 0.906728i \(-0.361428\pi\)
0.421717 + 0.906728i \(0.361428\pi\)
\(578\) −24065.0 −1.73179
\(579\) 0 0
\(580\) 4250.00 0.304261
\(581\) −19800.0 −1.41384
\(582\) 0 0
\(583\) −30500.0 −2.16669
\(584\) −18450.0 −1.30731
\(585\) 0 0
\(586\) 7650.00 0.539281
\(587\) 21060.0 1.48082 0.740408 0.672157i \(-0.234632\pi\)
0.740408 + 0.672157i \(0.234632\pi\)
\(588\) 0 0
\(589\) −4752.00 −0.332433
\(590\) −1250.00 −0.0872232
\(591\) 0 0
\(592\) −3560.00 −0.247154
\(593\) −22910.0 −1.58651 −0.793255 0.608889i \(-0.791615\pi\)
−0.793255 + 0.608889i \(0.791615\pi\)
\(594\) 0 0
\(595\) −1500.00 −0.103351
\(596\) 34850.0 2.39515
\(597\) 0 0
\(598\) −12000.0 −0.820596
\(599\) −1400.00 −0.0954966 −0.0477483 0.998859i \(-0.515205\pi\)
−0.0477483 + 0.998859i \(0.515205\pi\)
\(600\) 0 0
\(601\) −11002.0 −0.746724 −0.373362 0.927686i \(-0.621795\pi\)
−0.373362 + 0.927686i \(0.621795\pi\)
\(602\) −42000.0 −2.84351
\(603\) 0 0
\(604\) 7616.00 0.513064
\(605\) −5845.00 −0.392782
\(606\) 0 0
\(607\) 4630.00 0.309598 0.154799 0.987946i \(-0.450527\pi\)
0.154799 + 0.987946i \(0.450527\pi\)
\(608\) −3740.00 −0.249469
\(609\) 0 0
\(610\) 12950.0 0.859558
\(611\) 5600.00 0.370788
\(612\) 0 0
\(613\) 24040.0 1.58396 0.791979 0.610548i \(-0.209051\pi\)
0.791979 + 0.610548i \(0.209051\pi\)
\(614\) 15200.0 0.999059
\(615\) 0 0
\(616\) −67500.0 −4.41502
\(617\) −1890.00 −0.123320 −0.0616601 0.998097i \(-0.519639\pi\)
−0.0616601 + 0.998097i \(0.519639\pi\)
\(618\) 0 0
\(619\) 19244.0 1.24957 0.624783 0.780798i \(-0.285187\pi\)
0.624783 + 0.780798i \(0.285187\pi\)
\(620\) −9180.00 −0.594641
\(621\) 0 0
\(622\) −28500.0 −1.83721
\(623\) 45000.0 2.89388
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 15550.0 0.992816
\(627\) 0 0
\(628\) −1700.00 −0.108021
\(629\) 400.000 0.0253562
\(630\) 0 0
\(631\) 15892.0 1.00262 0.501308 0.865269i \(-0.332852\pi\)
0.501308 + 0.865269i \(0.332852\pi\)
\(632\) −23220.0 −1.46146
\(633\) 0 0
\(634\) −4750.00 −0.297550
\(635\) 5350.00 0.334344
\(636\) 0 0
\(637\) −11140.0 −0.692909
\(638\) −12500.0 −0.775674
\(639\) 0 0
\(640\) 10575.0 0.653146
\(641\) −12600.0 −0.776396 −0.388198 0.921576i \(-0.626902\pi\)
−0.388198 + 0.921576i \(0.626902\pi\)
\(642\) 0 0
\(643\) 7260.00 0.445267 0.222633 0.974902i \(-0.428535\pi\)
0.222633 + 0.974902i \(0.428535\pi\)
\(644\) −61200.0 −3.74475
\(645\) 0 0
\(646\) 2200.00 0.133990
\(647\) 7400.00 0.449651 0.224825 0.974399i \(-0.427819\pi\)
0.224825 + 0.974399i \(0.427819\pi\)
\(648\) 0 0
\(649\) 2500.00 0.151207
\(650\) −2500.00 −0.150859
\(651\) 0 0
\(652\) −32300.0 −1.94013
\(653\) −4790.00 −0.287055 −0.143528 0.989646i \(-0.545845\pi\)
−0.143528 + 0.989646i \(0.545845\pi\)
\(654\) 0 0
\(655\) −9750.00 −0.581624
\(656\) 35600.0 2.11882
\(657\) 0 0
\(658\) 42000.0 2.48834
\(659\) −1450.00 −0.0857117 −0.0428558 0.999081i \(-0.513646\pi\)
−0.0428558 + 0.999081i \(0.513646\pi\)
\(660\) 0 0
\(661\) 11818.0 0.695411 0.347706 0.937604i \(-0.386961\pi\)
0.347706 + 0.937604i \(0.386961\pi\)
\(662\) 11460.0 0.672818
\(663\) 0 0
\(664\) 29700.0 1.73582
\(665\) −6600.00 −0.384868
\(666\) 0 0
\(667\) −6000.00 −0.348307
\(668\) 32640.0 1.89054
\(669\) 0 0
\(670\) 4500.00 0.259478
\(671\) −25900.0 −1.49010
\(672\) 0 0
\(673\) 5550.00 0.317885 0.158943 0.987288i \(-0.449192\pi\)
0.158943 + 0.987288i \(0.449192\pi\)
\(674\) −38650.0 −2.20882
\(675\) 0 0
\(676\) −30549.0 −1.73811
\(677\) 12930.0 0.734033 0.367016 0.930214i \(-0.380379\pi\)
0.367016 + 0.930214i \(0.380379\pi\)
\(678\) 0 0
\(679\) 48900.0 2.76378
\(680\) 2250.00 0.126888
\(681\) 0 0
\(682\) 27000.0 1.51596
\(683\) 32580.0 1.82524 0.912620 0.408809i \(-0.134056\pi\)
0.912620 + 0.408809i \(0.134056\pi\)
\(684\) 0 0
\(685\) −5250.00 −0.292835
\(686\) −32100.0 −1.78657
\(687\) 0 0
\(688\) 24920.0 1.38091
\(689\) 12200.0 0.674576
\(690\) 0 0
\(691\) 10228.0 0.563085 0.281542 0.959549i \(-0.409154\pi\)
0.281542 + 0.959549i \(0.409154\pi\)
\(692\) 43350.0 2.38139
\(693\) 0 0
\(694\) −5600.00 −0.306301
\(695\) −8380.00 −0.457369
\(696\) 0 0
\(697\) −4000.00 −0.217376
\(698\) 5930.00 0.321567
\(699\) 0 0
\(700\) −12750.0 −0.688435
\(701\) 8350.00 0.449893 0.224947 0.974371i \(-0.427779\pi\)
0.224947 + 0.974371i \(0.427779\pi\)
\(702\) 0 0
\(703\) 1760.00 0.0944234
\(704\) −14350.0 −0.768233
\(705\) 0 0
\(706\) 18150.0 0.967541
\(707\) 13500.0 0.718133
\(708\) 0 0
\(709\) −14954.0 −0.792115 −0.396057 0.918226i \(-0.629622\pi\)
−0.396057 + 0.918226i \(0.629622\pi\)
\(710\) −17500.0 −0.925019
\(711\) 0 0
\(712\) −67500.0 −3.55291
\(713\) 12960.0 0.680723
\(714\) 0 0
\(715\) 5000.00 0.261524
\(716\) 62050.0 3.23871
\(717\) 0 0
\(718\) −9000.00 −0.467795
\(719\) 29400.0 1.52494 0.762472 0.647021i \(-0.223985\pi\)
0.762472 + 0.647021i \(0.223985\pi\)
\(720\) 0 0
\(721\) −23100.0 −1.19319
\(722\) −24615.0 −1.26880
\(723\) 0 0
\(724\) −73814.0 −3.78905
\(725\) −1250.00 −0.0640329
\(726\) 0 0
\(727\) −16330.0 −0.833076 −0.416538 0.909118i \(-0.636757\pi\)
−0.416538 + 0.909118i \(0.636757\pi\)
\(728\) 27000.0 1.37457
\(729\) 0 0
\(730\) 10250.0 0.519684
\(731\) −2800.00 −0.141671
\(732\) 0 0
\(733\) 30800.0 1.55201 0.776005 0.630726i \(-0.217243\pi\)
0.776005 + 0.630726i \(0.217243\pi\)
\(734\) 42450.0 2.13468
\(735\) 0 0
\(736\) 10200.0 0.510838
\(737\) −9000.00 −0.449823
\(738\) 0 0
\(739\) −9524.00 −0.474081 −0.237041 0.971500i \(-0.576177\pi\)
−0.237041 + 0.971500i \(0.576177\pi\)
\(740\) 3400.00 0.168901
\(741\) 0 0
\(742\) 91500.0 4.52705
\(743\) −28600.0 −1.41216 −0.706078 0.708134i \(-0.749537\pi\)
−0.706078 + 0.708134i \(0.749537\pi\)
\(744\) 0 0
\(745\) −10250.0 −0.504068
\(746\) 500.000 0.0245393
\(747\) 0 0
\(748\) −8500.00 −0.415496
\(749\) −19800.0 −0.965923
\(750\) 0 0
\(751\) −8252.00 −0.400958 −0.200479 0.979698i \(-0.564250\pi\)
−0.200479 + 0.979698i \(0.564250\pi\)
\(752\) −24920.0 −1.20843
\(753\) 0 0
\(754\) 5000.00 0.241498
\(755\) −2240.00 −0.107976
\(756\) 0 0
\(757\) −24920.0 −1.19648 −0.598238 0.801318i \(-0.704132\pi\)
−0.598238 + 0.801318i \(0.704132\pi\)
\(758\) −40420.0 −1.93683
\(759\) 0 0
\(760\) 9900.00 0.472514
\(761\) 27900.0 1.32901 0.664503 0.747285i \(-0.268643\pi\)
0.664503 + 0.747285i \(0.268643\pi\)
\(762\) 0 0
\(763\) −52620.0 −2.49669
\(764\) −59500.0 −2.81758
\(765\) 0 0
\(766\) −47400.0 −2.23581
\(767\) −1000.00 −0.0470768
\(768\) 0 0
\(769\) −11506.0 −0.539554 −0.269777 0.962923i \(-0.586950\pi\)
−0.269777 + 0.962923i \(0.586950\pi\)
\(770\) 37500.0 1.75507
\(771\) 0 0
\(772\) 56950.0 2.65502
\(773\) −12510.0 −0.582087 −0.291044 0.956710i \(-0.594002\pi\)
−0.291044 + 0.956710i \(0.594002\pi\)
\(774\) 0 0
\(775\) 2700.00 0.125144
\(776\) −73350.0 −3.39318
\(777\) 0 0
\(778\) −54750.0 −2.52299
\(779\) −17600.0 −0.809481
\(780\) 0 0
\(781\) 35000.0 1.60358
\(782\) −6000.00 −0.274373
\(783\) 0 0
\(784\) 49573.0 2.25825
\(785\) 500.000 0.0227335
\(786\) 0 0
\(787\) −1100.00 −0.0498231 −0.0249115 0.999690i \(-0.507930\pi\)
−0.0249115 + 0.999690i \(0.507930\pi\)
\(788\) 1530.00 0.0691675
\(789\) 0 0
\(790\) 12900.0 0.580964
\(791\) 9300.00 0.418040
\(792\) 0 0
\(793\) 10360.0 0.463927
\(794\) 69200.0 3.09297
\(795\) 0 0
\(796\) 62288.0 2.77354
\(797\) −4490.00 −0.199553 −0.0997766 0.995010i \(-0.531813\pi\)
−0.0997766 + 0.995010i \(0.531813\pi\)
\(798\) 0 0
\(799\) 2800.00 0.123976
\(800\) 2125.00 0.0939126
\(801\) 0 0
\(802\) 46500.0 2.04735
\(803\) −20500.0 −0.900908
\(804\) 0 0
\(805\) 18000.0 0.788095
\(806\) −10800.0 −0.471977
\(807\) 0 0
\(808\) −20250.0 −0.881674
\(809\) 28600.0 1.24292 0.621460 0.783446i \(-0.286540\pi\)
0.621460 + 0.783446i \(0.286540\pi\)
\(810\) 0 0
\(811\) 10068.0 0.435925 0.217963 0.975957i \(-0.430059\pi\)
0.217963 + 0.975957i \(0.430059\pi\)
\(812\) 25500.0 1.10206
\(813\) 0 0
\(814\) −10000.0 −0.430589
\(815\) 9500.00 0.408307
\(816\) 0 0
\(817\) −12320.0 −0.527567
\(818\) −14270.0 −0.609950
\(819\) 0 0
\(820\) −34000.0 −1.44797
\(821\) −14250.0 −0.605759 −0.302880 0.953029i \(-0.597948\pi\)
−0.302880 + 0.953029i \(0.597948\pi\)
\(822\) 0 0
\(823\) 6830.00 0.289282 0.144641 0.989484i \(-0.453797\pi\)
0.144641 + 0.989484i \(0.453797\pi\)
\(824\) 34650.0 1.46491
\(825\) 0 0
\(826\) −7500.00 −0.315930
\(827\) −8920.00 −0.375065 −0.187533 0.982258i \(-0.560049\pi\)
−0.187533 + 0.982258i \(0.560049\pi\)
\(828\) 0 0
\(829\) −3534.00 −0.148059 −0.0740295 0.997256i \(-0.523586\pi\)
−0.0740295 + 0.997256i \(0.523586\pi\)
\(830\) −16500.0 −0.690028
\(831\) 0 0
\(832\) 5740.00 0.239181
\(833\) −5570.00 −0.231680
\(834\) 0 0
\(835\) −9600.00 −0.397870
\(836\) −37400.0 −1.54726
\(837\) 0 0
\(838\) 5750.00 0.237029
\(839\) −8000.00 −0.329190 −0.164595 0.986361i \(-0.552632\pi\)
−0.164595 + 0.986361i \(0.552632\pi\)
\(840\) 0 0
\(841\) −21889.0 −0.897495
\(842\) −55810.0 −2.28425
\(843\) 0 0
\(844\) −4556.00 −0.185810
\(845\) 8985.00 0.365791
\(846\) 0 0
\(847\) −35070.0 −1.42269
\(848\) −54290.0 −2.19850
\(849\) 0 0
\(850\) −1250.00 −0.0504408
\(851\) −4800.00 −0.193351
\(852\) 0 0
\(853\) 5160.00 0.207122 0.103561 0.994623i \(-0.466976\pi\)
0.103561 + 0.994623i \(0.466976\pi\)
\(854\) 77700.0 3.11339
\(855\) 0 0
\(856\) 29700.0 1.18589
\(857\) 7670.00 0.305720 0.152860 0.988248i \(-0.451152\pi\)
0.152860 + 0.988248i \(0.451152\pi\)
\(858\) 0 0
\(859\) 25804.0 1.02494 0.512469 0.858706i \(-0.328731\pi\)
0.512469 + 0.858706i \(0.328731\pi\)
\(860\) −23800.0 −0.943690
\(861\) 0 0
\(862\) −6000.00 −0.237078
\(863\) 400.000 0.0157777 0.00788885 0.999969i \(-0.497489\pi\)
0.00788885 + 0.999969i \(0.497489\pi\)
\(864\) 0 0
\(865\) −12750.0 −0.501171
\(866\) 7550.00 0.296258
\(867\) 0 0
\(868\) −55080.0 −2.15384
\(869\) −25800.0 −1.00714
\(870\) 0 0
\(871\) 3600.00 0.140047
\(872\) 78930.0 3.06526
\(873\) 0 0
\(874\) −26400.0 −1.02173
\(875\) 3750.00 0.144884
\(876\) 0 0
\(877\) −35100.0 −1.35147 −0.675737 0.737143i \(-0.736175\pi\)
−0.675737 + 0.737143i \(0.736175\pi\)
\(878\) 2120.00 0.0814881
\(879\) 0 0
\(880\) −22250.0 −0.852327
\(881\) −18700.0 −0.715118 −0.357559 0.933891i \(-0.616391\pi\)
−0.357559 + 0.933891i \(0.616391\pi\)
\(882\) 0 0
\(883\) −2980.00 −0.113573 −0.0567865 0.998386i \(-0.518085\pi\)
−0.0567865 + 0.998386i \(0.518085\pi\)
\(884\) 3400.00 0.129360
\(885\) 0 0
\(886\) 61800.0 2.34335
\(887\) −35880.0 −1.35821 −0.679105 0.734041i \(-0.737632\pi\)
−0.679105 + 0.734041i \(0.737632\pi\)
\(888\) 0 0
\(889\) 32100.0 1.21102
\(890\) 37500.0 1.41236
\(891\) 0 0
\(892\) −62390.0 −2.34190
\(893\) 12320.0 0.461672
\(894\) 0 0
\(895\) −18250.0 −0.681598
\(896\) 63450.0 2.36575
\(897\) 0 0
\(898\) −6500.00 −0.241545
\(899\) −5400.00 −0.200334
\(900\) 0 0
\(901\) 6100.00 0.225550
\(902\) 100000. 3.69139
\(903\) 0 0
\(904\) −13950.0 −0.513241
\(905\) 21710.0 0.797420
\(906\) 0 0
\(907\) 45240.0 1.65620 0.828098 0.560584i \(-0.189423\pi\)
0.828098 + 0.560584i \(0.189423\pi\)
\(908\) 63920.0 2.33619
\(909\) 0 0
\(910\) −15000.0 −0.546423
\(911\) 33200.0 1.20743 0.603713 0.797202i \(-0.293687\pi\)
0.603713 + 0.797202i \(0.293687\pi\)
\(912\) 0 0
\(913\) 33000.0 1.19621
\(914\) −35950.0 −1.30101
\(915\) 0 0
\(916\) −24378.0 −0.879336
\(917\) −58500.0 −2.10670
\(918\) 0 0
\(919\) 35356.0 1.26908 0.634541 0.772889i \(-0.281189\pi\)
0.634541 + 0.772889i \(0.281189\pi\)
\(920\) −27000.0 −0.967569
\(921\) 0 0
\(922\) −750.000 −0.0267895
\(923\) −14000.0 −0.499259
\(924\) 0 0
\(925\) −1000.00 −0.0355457
\(926\) 13350.0 0.473767
\(927\) 0 0
\(928\) −4250.00 −0.150337
\(929\) −25700.0 −0.907631 −0.453816 0.891096i \(-0.649938\pi\)
−0.453816 + 0.891096i \(0.649938\pi\)
\(930\) 0 0
\(931\) −24508.0 −0.862747
\(932\) −58650.0 −2.06131
\(933\) 0 0
\(934\) 5900.00 0.206696
\(935\) 2500.00 0.0874425
\(936\) 0 0
\(937\) 52890.0 1.84401 0.922007 0.387173i \(-0.126549\pi\)
0.922007 + 0.387173i \(0.126549\pi\)
\(938\) 27000.0 0.939852
\(939\) 0 0
\(940\) 23800.0 0.825820
\(941\) 38050.0 1.31817 0.659083 0.752070i \(-0.270945\pi\)
0.659083 + 0.752070i \(0.270945\pi\)
\(942\) 0 0
\(943\) 48000.0 1.65758
\(944\) 4450.00 0.153427
\(945\) 0 0
\(946\) 70000.0 2.40581
\(947\) −29640.0 −1.01708 −0.508538 0.861040i \(-0.669814\pi\)
−0.508538 + 0.861040i \(0.669814\pi\)
\(948\) 0 0
\(949\) 8200.00 0.280488
\(950\) −5500.00 −0.187835
\(951\) 0 0
\(952\) 13500.0 0.459598
\(953\) 15170.0 0.515640 0.257820 0.966193i \(-0.416996\pi\)
0.257820 + 0.966193i \(0.416996\pi\)
\(954\) 0 0
\(955\) 17500.0 0.592970
\(956\) −83300.0 −2.81811
\(957\) 0 0
\(958\) −70500.0 −2.37761
\(959\) −31500.0 −1.06068
\(960\) 0 0
\(961\) −18127.0 −0.608472
\(962\) 4000.00 0.134059
\(963\) 0 0
\(964\) 81974.0 2.73880
\(965\) −16750.0 −0.558758
\(966\) 0 0
\(967\) −5470.00 −0.181906 −0.0909531 0.995855i \(-0.528991\pi\)
−0.0909531 + 0.995855i \(0.528991\pi\)
\(968\) 52605.0 1.74668
\(969\) 0 0
\(970\) 40750.0 1.34887
\(971\) 15150.0 0.500707 0.250354 0.968154i \(-0.419453\pi\)
0.250354 + 0.968154i \(0.419453\pi\)
\(972\) 0 0
\(973\) −50280.0 −1.65663
\(974\) −49250.0 −1.62020
\(975\) 0 0
\(976\) −46102.0 −1.51198
\(977\) 31190.0 1.02135 0.510674 0.859775i \(-0.329396\pi\)
0.510674 + 0.859775i \(0.329396\pi\)
\(978\) 0 0
\(979\) −75000.0 −2.44843
\(980\) −47345.0 −1.54325
\(981\) 0 0
\(982\) −12250.0 −0.398079
\(983\) −7560.00 −0.245297 −0.122648 0.992450i \(-0.539139\pi\)
−0.122648 + 0.992450i \(0.539139\pi\)
\(984\) 0 0
\(985\) −450.000 −0.0145565
\(986\) 2500.00 0.0807467
\(987\) 0 0
\(988\) 14960.0 0.481722
\(989\) 33600.0 1.08030
\(990\) 0 0
\(991\) 32672.0 1.04729 0.523643 0.851938i \(-0.324573\pi\)
0.523643 + 0.851938i \(0.324573\pi\)
\(992\) 9180.00 0.293816
\(993\) 0 0
\(994\) −105000. −3.35050
\(995\) −18320.0 −0.583702
\(996\) 0 0
\(997\) −4740.00 −0.150569 −0.0752845 0.997162i \(-0.523986\pi\)
−0.0752845 + 0.997162i \(0.523986\pi\)
\(998\) −85180.0 −2.70173
\(999\) 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 45.4.a.e.1.1 yes 1
3.2 odd 2 45.4.a.a.1.1 1
4.3 odd 2 720.4.a.o.1.1 1
5.2 odd 4 225.4.b.b.199.2 2
5.3 odd 4 225.4.b.b.199.1 2
5.4 even 2 225.4.a.a.1.1 1
7.6 odd 2 2205.4.a.t.1.1 1
9.2 odd 6 405.4.e.n.271.1 2
9.4 even 3 405.4.e.b.136.1 2
9.5 odd 6 405.4.e.n.136.1 2
9.7 even 3 405.4.e.b.271.1 2
12.11 even 2 720.4.a.bc.1.1 1
15.2 even 4 225.4.b.a.199.1 2
15.8 even 4 225.4.b.a.199.2 2
15.14 odd 2 225.4.a.h.1.1 1
21.20 even 2 2205.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.a.a.1.1 1 3.2 odd 2
45.4.a.e.1.1 yes 1 1.1 even 1 trivial
225.4.a.a.1.1 1 5.4 even 2
225.4.a.h.1.1 1 15.14 odd 2
225.4.b.a.199.1 2 15.2 even 4
225.4.b.a.199.2 2 15.8 even 4
225.4.b.b.199.1 2 5.3 odd 4
225.4.b.b.199.2 2 5.2 odd 4
405.4.e.b.136.1 2 9.4 even 3
405.4.e.b.271.1 2 9.7 even 3
405.4.e.n.136.1 2 9.5 odd 6
405.4.e.n.271.1 2 9.2 odd 6
720.4.a.o.1.1 1 4.3 odd 2
720.4.a.bc.1.1 1 12.11 even 2
2205.4.a.a.1.1 1 21.20 even 2
2205.4.a.t.1.1 1 7.6 odd 2