Properties

Label 225.4.b.b.199.1
Level $225$
Weight $4$
Character 225.199
Analytic conductor $13.275$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.4.b.b.199.2

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000i q^{2} -17.0000 q^{4} +30.0000i q^{7} +45.0000i q^{8} +O(q^{10})\) \(q-5.00000i q^{2} -17.0000 q^{4} +30.0000i q^{7} +45.0000i q^{8} +50.0000 q^{11} -20.0000i q^{13} +150.000 q^{14} +89.0000 q^{16} +10.0000i q^{17} +44.0000 q^{19} -250.000i q^{22} +120.000i q^{23} -100.000 q^{26} -510.000i q^{28} +50.0000 q^{29} +108.000 q^{31} -85.0000i q^{32} +50.0000 q^{34} +40.0000i q^{37} -220.000i q^{38} +400.000 q^{41} +280.000i q^{43} -850.000 q^{44} +600.000 q^{46} +280.000i q^{47} -557.000 q^{49} +340.000i q^{52} -610.000i q^{53} -1350.00 q^{56} -250.000i q^{58} -50.0000 q^{59} -518.000 q^{61} -540.000i q^{62} +287.000 q^{64} +180.000i q^{67} -170.000i q^{68} +700.000 q^{71} -410.000i q^{73} +200.000 q^{74} -748.000 q^{76} +1500.00i q^{77} +516.000 q^{79} -2000.00i q^{82} +660.000i q^{83} +1400.00 q^{86} +2250.00i q^{88} +1500.00 q^{89} +600.000 q^{91} -2040.00i q^{92} +1400.00 q^{94} +1630.00i q^{97} +2785.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 34 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 34 q^{4} + 100 q^{11} + 300 q^{14} + 178 q^{16} + 88 q^{19} - 200 q^{26} + 100 q^{29} + 216 q^{31} + 100 q^{34} + 800 q^{41} - 1700 q^{44} + 1200 q^{46} - 1114 q^{49} - 2700 q^{56} - 100 q^{59} - 1036 q^{61} + 574 q^{64} + 1400 q^{71} + 400 q^{74} - 1496 q^{76} + 1032 q^{79} + 2800 q^{86} + 3000 q^{89} + 1200 q^{91} + 2800 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.00000i − 1.76777i −0.467707 0.883883i \(-0.654920\pi\)
0.467707 0.883883i \(-0.345080\pi\)
\(3\) 0 0
\(4\) −17.0000 −2.12500
\(5\) 0 0
\(6\) 0 0
\(7\) 30.0000i 1.61985i 0.586535 + 0.809924i \(0.300492\pi\)
−0.586535 + 0.809924i \(0.699508\pi\)
\(8\) 45.0000i 1.98874i
\(9\) 0 0
\(10\) 0 0
\(11\) 50.0000 1.37051 0.685253 0.728305i \(-0.259692\pi\)
0.685253 + 0.728305i \(0.259692\pi\)
\(12\) 0 0
\(13\) − 20.0000i − 0.426692i −0.976977 0.213346i \(-0.931564\pi\)
0.976977 0.213346i \(-0.0684362\pi\)
\(14\) 150.000 2.86351
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) 10.0000i 0.142668i 0.997452 + 0.0713340i \(0.0227256\pi\)
−0.997452 + 0.0713340i \(0.977274\pi\)
\(18\) 0 0
\(19\) 44.0000 0.531279 0.265639 0.964072i \(-0.414417\pi\)
0.265639 + 0.964072i \(0.414417\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 250.000i − 2.42274i
\(23\) 120.000i 1.08790i 0.839117 + 0.543951i \(0.183072\pi\)
−0.839117 + 0.543951i \(0.816928\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −100.000 −0.754293
\(27\) 0 0
\(28\) − 510.000i − 3.44218i
\(29\) 50.0000 0.320164 0.160082 0.987104i \(-0.448824\pi\)
0.160082 + 0.987104i \(0.448824\pi\)
\(30\) 0 0
\(31\) 108.000 0.625722 0.312861 0.949799i \(-0.398713\pi\)
0.312861 + 0.949799i \(0.398713\pi\)
\(32\) − 85.0000i − 0.469563i
\(33\) 0 0
\(34\) 50.0000 0.252204
\(35\) 0 0
\(36\) 0 0
\(37\) 40.0000i 0.177729i 0.996044 + 0.0888643i \(0.0283238\pi\)
−0.996044 + 0.0888643i \(0.971676\pi\)
\(38\) − 220.000i − 0.939177i
\(39\) 0 0
\(40\) 0 0
\(41\) 400.000 1.52365 0.761823 0.647785i \(-0.224304\pi\)
0.761823 + 0.647785i \(0.224304\pi\)
\(42\) 0 0
\(43\) 280.000i 0.993014i 0.868033 + 0.496507i \(0.165384\pi\)
−0.868033 + 0.496507i \(0.834616\pi\)
\(44\) −850.000 −2.91233
\(45\) 0 0
\(46\) 600.000 1.92316
\(47\) 280.000i 0.868983i 0.900676 + 0.434491i \(0.143072\pi\)
−0.900676 + 0.434491i \(0.856928\pi\)
\(48\) 0 0
\(49\) −557.000 −1.62391
\(50\) 0 0
\(51\) 0 0
\(52\) 340.000i 0.906721i
\(53\) − 610.000i − 1.58094i −0.612499 0.790471i \(-0.709836\pi\)
0.612499 0.790471i \(-0.290164\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1350.00 −3.22145
\(57\) 0 0
\(58\) − 250.000i − 0.565976i
\(59\) −50.0000 −0.110330 −0.0551648 0.998477i \(-0.517568\pi\)
−0.0551648 + 0.998477i \(0.517568\pi\)
\(60\) 0 0
\(61\) −518.000 −1.08726 −0.543632 0.839324i \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(62\) − 540.000i − 1.10613i
\(63\) 0 0
\(64\) 287.000 0.560547
\(65\) 0 0
\(66\) 0 0
\(67\) 180.000i 0.328216i 0.986442 + 0.164108i \(0.0524746\pi\)
−0.986442 + 0.164108i \(0.947525\pi\)
\(68\) − 170.000i − 0.303170i
\(69\) 0 0
\(70\) 0 0
\(71\) 700.000 1.17007 0.585033 0.811009i \(-0.301081\pi\)
0.585033 + 0.811009i \(0.301081\pi\)
\(72\) 0 0
\(73\) − 410.000i − 0.657354i −0.944442 0.328677i \(-0.893397\pi\)
0.944442 0.328677i \(-0.106603\pi\)
\(74\) 200.000 0.314183
\(75\) 0 0
\(76\) −748.000 −1.12897
\(77\) 1500.00i 2.22001i
\(78\) 0 0
\(79\) 516.000 0.734868 0.367434 0.930050i \(-0.380236\pi\)
0.367434 + 0.930050i \(0.380236\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 2000.00i − 2.69345i
\(83\) 660.000i 0.872824i 0.899747 + 0.436412i \(0.143751\pi\)
−0.899747 + 0.436412i \(0.856249\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1400.00 1.75542
\(87\) 0 0
\(88\) 2250.00i 2.72558i
\(89\) 1500.00 1.78651 0.893257 0.449547i \(-0.148415\pi\)
0.893257 + 0.449547i \(0.148415\pi\)
\(90\) 0 0
\(91\) 600.000 0.691177
\(92\) − 2040.00i − 2.31179i
\(93\) 0 0
\(94\) 1400.00 1.53616
\(95\) 0 0
\(96\) 0 0
\(97\) 1630.00i 1.70620i 0.521747 + 0.853100i \(0.325280\pi\)
−0.521747 + 0.853100i \(0.674720\pi\)
\(98\) 2785.00i 2.87069i
\(99\) 0 0
\(100\) 0 0
\(101\) −450.000 −0.443333 −0.221667 0.975122i \(-0.571150\pi\)
−0.221667 + 0.975122i \(0.571150\pi\)
\(102\) 0 0
\(103\) 770.000i 0.736605i 0.929706 + 0.368303i \(0.120061\pi\)
−0.929706 + 0.368303i \(0.879939\pi\)
\(104\) 900.000 0.848579
\(105\) 0 0
\(106\) −3050.00 −2.79474
\(107\) − 660.000i − 0.596305i −0.954518 0.298152i \(-0.903630\pi\)
0.954518 0.298152i \(-0.0963704\pi\)
\(108\) 0 0
\(109\) −1754.00 −1.54131 −0.770655 0.637253i \(-0.780071\pi\)
−0.770655 + 0.637253i \(0.780071\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2670.00i 2.25260i
\(113\) − 310.000i − 0.258074i −0.991640 0.129037i \(-0.958811\pi\)
0.991640 0.129037i \(-0.0411886\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −850.000 −0.680349
\(117\) 0 0
\(118\) 250.000i 0.195037i
\(119\) −300.000 −0.231100
\(120\) 0 0
\(121\) 1169.00 0.878287
\(122\) 2590.00i 1.92203i
\(123\) 0 0
\(124\) −1836.00 −1.32966
\(125\) 0 0
\(126\) 0 0
\(127\) 1070.00i 0.747615i 0.927506 + 0.373808i \(0.121948\pi\)
−0.927506 + 0.373808i \(0.878052\pi\)
\(128\) − 2115.00i − 1.46048i
\(129\) 0 0
\(130\) 0 0
\(131\) 1950.00 1.30055 0.650276 0.759698i \(-0.274653\pi\)
0.650276 + 0.759698i \(0.274653\pi\)
\(132\) 0 0
\(133\) 1320.00i 0.860590i
\(134\) 900.000 0.580210
\(135\) 0 0
\(136\) −450.000 −0.283729
\(137\) − 1050.00i − 0.654800i −0.944886 0.327400i \(-0.893828\pi\)
0.944886 0.327400i \(-0.106172\pi\)
\(138\) 0 0
\(139\) −1676.00 −1.02271 −0.511354 0.859370i \(-0.670856\pi\)
−0.511354 + 0.859370i \(0.670856\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 3500.00i − 2.06840i
\(143\) − 1000.00i − 0.584785i
\(144\) 0 0
\(145\) 0 0
\(146\) −2050.00 −1.16205
\(147\) 0 0
\(148\) − 680.000i − 0.377673i
\(149\) −2050.00 −1.12713 −0.563566 0.826071i \(-0.690571\pi\)
−0.563566 + 0.826071i \(0.690571\pi\)
\(150\) 0 0
\(151\) 448.000 0.241442 0.120721 0.992686i \(-0.461479\pi\)
0.120721 + 0.992686i \(0.461479\pi\)
\(152\) 1980.00i 1.05657i
\(153\) 0 0
\(154\) 7500.00 3.92446
\(155\) 0 0
\(156\) 0 0
\(157\) 100.000i 0.0508336i 0.999677 + 0.0254168i \(0.00809128\pi\)
−0.999677 + 0.0254168i \(0.991909\pi\)
\(158\) − 2580.00i − 1.29907i
\(159\) 0 0
\(160\) 0 0
\(161\) −3600.00 −1.76223
\(162\) 0 0
\(163\) − 1900.00i − 0.913003i −0.889723 0.456501i \(-0.849102\pi\)
0.889723 0.456501i \(-0.150898\pi\)
\(164\) −6800.00 −3.23775
\(165\) 0 0
\(166\) 3300.00 1.54295
\(167\) − 1920.00i − 0.889665i −0.895614 0.444833i \(-0.853263\pi\)
0.895614 0.444833i \(-0.146737\pi\)
\(168\) 0 0
\(169\) 1797.00 0.817934
\(170\) 0 0
\(171\) 0 0
\(172\) − 4760.00i − 2.11015i
\(173\) 2550.00i 1.12065i 0.828272 + 0.560326i \(0.189324\pi\)
−0.828272 + 0.560326i \(0.810676\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4450.00 1.90586
\(177\) 0 0
\(178\) − 7500.00i − 3.15814i
\(179\) −3650.00 −1.52410 −0.762050 0.647518i \(-0.775807\pi\)
−0.762050 + 0.647518i \(0.775807\pi\)
\(180\) 0 0
\(181\) −4342.00 −1.78308 −0.891542 0.452937i \(-0.850376\pi\)
−0.891542 + 0.452937i \(0.850376\pi\)
\(182\) − 3000.00i − 1.22184i
\(183\) 0 0
\(184\) −5400.00 −2.16355
\(185\) 0 0
\(186\) 0 0
\(187\) 500.000i 0.195527i
\(188\) − 4760.00i − 1.84659i
\(189\) 0 0
\(190\) 0 0
\(191\) −3500.00 −1.32592 −0.662961 0.748654i \(-0.730701\pi\)
−0.662961 + 0.748654i \(0.730701\pi\)
\(192\) 0 0
\(193\) 3350.00i 1.24942i 0.780856 + 0.624711i \(0.214783\pi\)
−0.780856 + 0.624711i \(0.785217\pi\)
\(194\) 8150.00 3.01616
\(195\) 0 0
\(196\) 9469.00 3.45080
\(197\) − 90.0000i − 0.0325494i −0.999868 0.0162747i \(-0.994819\pi\)
0.999868 0.0162747i \(-0.00518063\pi\)
\(198\) 0 0
\(199\) −3664.00 −1.30520 −0.652598 0.757704i \(-0.726321\pi\)
−0.652598 + 0.757704i \(0.726321\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2250.00i 0.783710i
\(203\) 1500.00i 0.518618i
\(204\) 0 0
\(205\) 0 0
\(206\) 3850.00 1.30215
\(207\) 0 0
\(208\) − 1780.00i − 0.593369i
\(209\) 2200.00 0.728120
\(210\) 0 0
\(211\) −268.000 −0.0874402 −0.0437201 0.999044i \(-0.513921\pi\)
−0.0437201 + 0.999044i \(0.513921\pi\)
\(212\) 10370.0i 3.35950i
\(213\) 0 0
\(214\) −3300.00 −1.05413
\(215\) 0 0
\(216\) 0 0
\(217\) 3240.00i 1.01357i
\(218\) 8770.00i 2.72468i
\(219\) 0 0
\(220\) 0 0
\(221\) 200.000 0.0608754
\(222\) 0 0
\(223\) − 3670.00i − 1.10207i −0.834482 0.551034i \(-0.814233\pi\)
0.834482 0.551034i \(-0.185767\pi\)
\(224\) 2550.00 0.760621
\(225\) 0 0
\(226\) −1550.00 −0.456214
\(227\) − 3760.00i − 1.09938i −0.835368 0.549692i \(-0.814745\pi\)
0.835368 0.549692i \(-0.185255\pi\)
\(228\) 0 0
\(229\) 1434.00 0.413805 0.206903 0.978362i \(-0.433662\pi\)
0.206903 + 0.978362i \(0.433662\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2250.00i 0.636723i
\(233\) − 3450.00i − 0.970030i −0.874506 0.485015i \(-0.838814\pi\)
0.874506 0.485015i \(-0.161186\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 850.000 0.234450
\(237\) 0 0
\(238\) 1500.00i 0.408532i
\(239\) 4900.00 1.32617 0.663085 0.748544i \(-0.269247\pi\)
0.663085 + 0.748544i \(0.269247\pi\)
\(240\) 0 0
\(241\) 4822.00 1.28885 0.644424 0.764668i \(-0.277097\pi\)
0.644424 + 0.764668i \(0.277097\pi\)
\(242\) − 5845.00i − 1.55261i
\(243\) 0 0
\(244\) 8806.00 2.31044
\(245\) 0 0
\(246\) 0 0
\(247\) − 880.000i − 0.226693i
\(248\) 4860.00i 1.24440i
\(249\) 0 0
\(250\) 0 0
\(251\) −4650.00 −1.16934 −0.584672 0.811270i \(-0.698777\pi\)
−0.584672 + 0.811270i \(0.698777\pi\)
\(252\) 0 0
\(253\) 6000.00i 1.49098i
\(254\) 5350.00 1.32161
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) − 5130.00i − 1.24514i −0.782565 0.622569i \(-0.786089\pi\)
0.782565 0.622569i \(-0.213911\pi\)
\(258\) 0 0
\(259\) −1200.00 −0.287893
\(260\) 0 0
\(261\) 0 0
\(262\) − 9750.00i − 2.29907i
\(263\) − 1280.00i − 0.300107i −0.988678 0.150054i \(-0.952055\pi\)
0.988678 0.150054i \(-0.0479446\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6600.00 1.52132
\(267\) 0 0
\(268\) − 3060.00i − 0.697460i
\(269\) −3350.00 −0.759305 −0.379653 0.925129i \(-0.623956\pi\)
−0.379653 + 0.925129i \(0.623956\pi\)
\(270\) 0 0
\(271\) 5512.00 1.23554 0.617768 0.786361i \(-0.288037\pi\)
0.617768 + 0.786361i \(0.288037\pi\)
\(272\) 890.000i 0.198398i
\(273\) 0 0
\(274\) −5250.00 −1.15753
\(275\) 0 0
\(276\) 0 0
\(277\) − 4920.00i − 1.06720i −0.845737 0.533600i \(-0.820839\pi\)
0.845737 0.533600i \(-0.179161\pi\)
\(278\) 8380.00i 1.80791i
\(279\) 0 0
\(280\) 0 0
\(281\) 4500.00 0.955329 0.477665 0.878542i \(-0.341483\pi\)
0.477665 + 0.878542i \(0.341483\pi\)
\(282\) 0 0
\(283\) − 6900.00i − 1.44934i −0.689098 0.724669i \(-0.741993\pi\)
0.689098 0.724669i \(-0.258007\pi\)
\(284\) −11900.0 −2.48639
\(285\) 0 0
\(286\) −5000.00 −1.03376
\(287\) 12000.0i 2.46808i
\(288\) 0 0
\(289\) 4813.00 0.979646
\(290\) 0 0
\(291\) 0 0
\(292\) 6970.00i 1.39688i
\(293\) 1530.00i 0.305063i 0.988299 + 0.152532i \(0.0487426\pi\)
−0.988299 + 0.152532i \(0.951257\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1800.00 −0.353456
\(297\) 0 0
\(298\) 10250.0i 1.99251i
\(299\) 2400.00 0.464199
\(300\) 0 0
\(301\) −8400.00 −1.60853
\(302\) − 2240.00i − 0.426813i
\(303\) 0 0
\(304\) 3916.00 0.738809
\(305\) 0 0
\(306\) 0 0
\(307\) − 3040.00i − 0.565153i −0.959245 0.282576i \(-0.908811\pi\)
0.959245 0.282576i \(-0.0911891\pi\)
\(308\) − 25500.0i − 4.71752i
\(309\) 0 0
\(310\) 0 0
\(311\) −5700.00 −1.03928 −0.519642 0.854384i \(-0.673935\pi\)
−0.519642 + 0.854384i \(0.673935\pi\)
\(312\) 0 0
\(313\) 3110.00i 0.561622i 0.959763 + 0.280811i \(0.0906034\pi\)
−0.959763 + 0.280811i \(0.909397\pi\)
\(314\) 500.000 0.0898619
\(315\) 0 0
\(316\) −8772.00 −1.56159
\(317\) 950.000i 0.168320i 0.996452 + 0.0841598i \(0.0268206\pi\)
−0.996452 + 0.0841598i \(0.973179\pi\)
\(318\) 0 0
\(319\) 2500.00 0.438787
\(320\) 0 0
\(321\) 0 0
\(322\) 18000.0i 3.11522i
\(323\) 440.000i 0.0757965i
\(324\) 0 0
\(325\) 0 0
\(326\) −9500.00 −1.61398
\(327\) 0 0
\(328\) 18000.0i 3.03013i
\(329\) −8400.00 −1.40762
\(330\) 0 0
\(331\) 2292.00 0.380603 0.190302 0.981726i \(-0.439053\pi\)
0.190302 + 0.981726i \(0.439053\pi\)
\(332\) − 11220.0i − 1.85475i
\(333\) 0 0
\(334\) −9600.00 −1.57272
\(335\) 0 0
\(336\) 0 0
\(337\) 7730.00i 1.24950i 0.780827 + 0.624748i \(0.214798\pi\)
−0.780827 + 0.624748i \(0.785202\pi\)
\(338\) − 8985.00i − 1.44592i
\(339\) 0 0
\(340\) 0 0
\(341\) 5400.00 0.857555
\(342\) 0 0
\(343\) − 6420.00i − 1.01063i
\(344\) −12600.0 −1.97484
\(345\) 0 0
\(346\) 12750.0 1.98105
\(347\) 1120.00i 0.173270i 0.996240 + 0.0866351i \(0.0276114\pi\)
−0.996240 + 0.0866351i \(0.972389\pi\)
\(348\) 0 0
\(349\) −1186.00 −0.181906 −0.0909529 0.995855i \(-0.528991\pi\)
−0.0909529 + 0.995855i \(0.528991\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 4250.00i − 0.643539i
\(353\) 3630.00i 0.547324i 0.961826 + 0.273662i \(0.0882350\pi\)
−0.961826 + 0.273662i \(0.911765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −25500.0 −3.79634
\(357\) 0 0
\(358\) 18250.0i 2.69425i
\(359\) 1800.00 0.264625 0.132312 0.991208i \(-0.457760\pi\)
0.132312 + 0.991208i \(0.457760\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) 21710.0i 3.15208i
\(363\) 0 0
\(364\) −10200.0 −1.46875
\(365\) 0 0
\(366\) 0 0
\(367\) − 8490.00i − 1.20756i −0.797151 0.603780i \(-0.793661\pi\)
0.797151 0.603780i \(-0.206339\pi\)
\(368\) 10680.0i 1.51286i
\(369\) 0 0
\(370\) 0 0
\(371\) 18300.0 2.56089
\(372\) 0 0
\(373\) 100.000i 0.0138815i 0.999976 + 0.00694076i \(0.00220933\pi\)
−0.999976 + 0.00694076i \(0.997791\pi\)
\(374\) 2500.00 0.345647
\(375\) 0 0
\(376\) −12600.0 −1.72818
\(377\) − 1000.00i − 0.136612i
\(378\) 0 0
\(379\) 8084.00 1.09564 0.547820 0.836597i \(-0.315458\pi\)
0.547820 + 0.836597i \(0.315458\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 17500.0i 2.34392i
\(383\) − 9480.00i − 1.26477i −0.774656 0.632383i \(-0.782077\pi\)
0.774656 0.632383i \(-0.217923\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16750.0 2.20869
\(387\) 0 0
\(388\) − 27710.0i − 3.62568i
\(389\) 10950.0 1.42722 0.713608 0.700545i \(-0.247060\pi\)
0.713608 + 0.700545i \(0.247060\pi\)
\(390\) 0 0
\(391\) −1200.00 −0.155209
\(392\) − 25065.0i − 3.22952i
\(393\) 0 0
\(394\) −450.000 −0.0575398
\(395\) 0 0
\(396\) 0 0
\(397\) − 13840.0i − 1.74965i −0.484442 0.874823i \(-0.660977\pi\)
0.484442 0.874823i \(-0.339023\pi\)
\(398\) 18320.0i 2.30728i
\(399\) 0 0
\(400\) 0 0
\(401\) 9300.00 1.15815 0.579077 0.815273i \(-0.303413\pi\)
0.579077 + 0.815273i \(0.303413\pi\)
\(402\) 0 0
\(403\) − 2160.00i − 0.266991i
\(404\) 7650.00 0.942083
\(405\) 0 0
\(406\) 7500.00 0.916795
\(407\) 2000.00i 0.243578i
\(408\) 0 0
\(409\) 2854.00 0.345040 0.172520 0.985006i \(-0.444809\pi\)
0.172520 + 0.985006i \(0.444809\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 13090.0i − 1.56529i
\(413\) − 1500.00i − 0.178717i
\(414\) 0 0
\(415\) 0 0
\(416\) −1700.00 −0.200359
\(417\) 0 0
\(418\) − 11000.0i − 1.28715i
\(419\) −1150.00 −0.134084 −0.0670420 0.997750i \(-0.521356\pi\)
−0.0670420 + 0.997750i \(0.521356\pi\)
\(420\) 0 0
\(421\) −11162.0 −1.29217 −0.646084 0.763266i \(-0.723594\pi\)
−0.646084 + 0.763266i \(0.723594\pi\)
\(422\) 1340.00i 0.154574i
\(423\) 0 0
\(424\) 27450.0 3.14408
\(425\) 0 0
\(426\) 0 0
\(427\) − 15540.0i − 1.76120i
\(428\) 11220.0i 1.26715i
\(429\) 0 0
\(430\) 0 0
\(431\) −1200.00 −0.134111 −0.0670556 0.997749i \(-0.521361\pi\)
−0.0670556 + 0.997749i \(0.521361\pi\)
\(432\) 0 0
\(433\) 1510.00i 0.167589i 0.996483 + 0.0837944i \(0.0267039\pi\)
−0.996483 + 0.0837944i \(0.973296\pi\)
\(434\) 16200.0 1.79176
\(435\) 0 0
\(436\) 29818.0 3.27528
\(437\) 5280.00i 0.577979i
\(438\) 0 0
\(439\) −424.000 −0.0460966 −0.0230483 0.999734i \(-0.507337\pi\)
−0.0230483 + 0.999734i \(0.507337\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 1000.00i − 0.107613i
\(443\) 12360.0i 1.32560i 0.748796 + 0.662801i \(0.230632\pi\)
−0.748796 + 0.662801i \(0.769368\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −18350.0 −1.94820
\(447\) 0 0
\(448\) 8610.00i 0.908001i
\(449\) 1300.00 0.136639 0.0683194 0.997664i \(-0.478236\pi\)
0.0683194 + 0.997664i \(0.478236\pi\)
\(450\) 0 0
\(451\) 20000.0 2.08817
\(452\) 5270.00i 0.548407i
\(453\) 0 0
\(454\) −18800.0 −1.94345
\(455\) 0 0
\(456\) 0 0
\(457\) 7190.00i 0.735961i 0.929834 + 0.367980i \(0.119951\pi\)
−0.929834 + 0.367980i \(0.880049\pi\)
\(458\) − 7170.00i − 0.731511i
\(459\) 0 0
\(460\) 0 0
\(461\) −150.000 −0.0151544 −0.00757722 0.999971i \(-0.502412\pi\)
−0.00757722 + 0.999971i \(0.502412\pi\)
\(462\) 0 0
\(463\) 2670.00i 0.268003i 0.990981 + 0.134002i \(0.0427827\pi\)
−0.990981 + 0.134002i \(0.957217\pi\)
\(464\) 4450.00 0.445229
\(465\) 0 0
\(466\) −17250.0 −1.71479
\(467\) − 1180.00i − 0.116925i −0.998290 0.0584624i \(-0.981380\pi\)
0.998290 0.0584624i \(-0.0186198\pi\)
\(468\) 0 0
\(469\) −5400.00 −0.531661
\(470\) 0 0
\(471\) 0 0
\(472\) − 2250.00i − 0.219417i
\(473\) 14000.0i 1.36093i
\(474\) 0 0
\(475\) 0 0
\(476\) 5100.00 0.491088
\(477\) 0 0
\(478\) − 24500.0i − 2.34436i
\(479\) 14100.0 1.34498 0.672490 0.740106i \(-0.265225\pi\)
0.672490 + 0.740106i \(0.265225\pi\)
\(480\) 0 0
\(481\) 800.000 0.0758355
\(482\) − 24110.0i − 2.27838i
\(483\) 0 0
\(484\) −19873.0 −1.86636
\(485\) 0 0
\(486\) 0 0
\(487\) 9850.00i 0.916522i 0.888818 + 0.458261i \(0.151527\pi\)
−0.888818 + 0.458261i \(0.848473\pi\)
\(488\) − 23310.0i − 2.16228i
\(489\) 0 0
\(490\) 0 0
\(491\) −2450.00 −0.225187 −0.112594 0.993641i \(-0.535916\pi\)
−0.112594 + 0.993641i \(0.535916\pi\)
\(492\) 0 0
\(493\) 500.000i 0.0456772i
\(494\) −4400.00 −0.400740
\(495\) 0 0
\(496\) 9612.00 0.870144
\(497\) 21000.0i 1.89533i
\(498\) 0 0
\(499\) 17036.0 1.52833 0.764164 0.645021i \(-0.223152\pi\)
0.764164 + 0.645021i \(0.223152\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 23250.0i 2.06713i
\(503\) − 20600.0i − 1.82606i −0.407891 0.913030i \(-0.633736\pi\)
0.407891 0.913030i \(-0.366264\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 30000.0 2.63570
\(507\) 0 0
\(508\) − 18190.0i − 1.58868i
\(509\) −5750.00 −0.500716 −0.250358 0.968153i \(-0.580548\pi\)
−0.250358 + 0.968153i \(0.580548\pi\)
\(510\) 0 0
\(511\) 12300.0 1.06481
\(512\) 24475.0i 2.11260i
\(513\) 0 0
\(514\) −25650.0 −2.20111
\(515\) 0 0
\(516\) 0 0
\(517\) 14000.0i 1.19095i
\(518\) 6000.00i 0.508928i
\(519\) 0 0
\(520\) 0 0
\(521\) −15500.0 −1.30339 −0.651696 0.758480i \(-0.725942\pi\)
−0.651696 + 0.758480i \(0.725942\pi\)
\(522\) 0 0
\(523\) − 13940.0i − 1.16549i −0.812653 0.582747i \(-0.801978\pi\)
0.812653 0.582747i \(-0.198022\pi\)
\(524\) −33150.0 −2.76367
\(525\) 0 0
\(526\) −6400.00 −0.530520
\(527\) 1080.00i 0.0892705i
\(528\) 0 0
\(529\) −2233.00 −0.183529
\(530\) 0 0
\(531\) 0 0
\(532\) − 22440.0i − 1.82875i
\(533\) − 8000.00i − 0.650128i
\(534\) 0 0
\(535\) 0 0
\(536\) −8100.00 −0.652736
\(537\) 0 0
\(538\) 16750.0i 1.34227i
\(539\) −27850.0 −2.22557
\(540\) 0 0
\(541\) −20478.0 −1.62739 −0.813695 0.581292i \(-0.802547\pi\)
−0.813695 + 0.581292i \(0.802547\pi\)
\(542\) − 27560.0i − 2.18414i
\(543\) 0 0
\(544\) 850.000 0.0669916
\(545\) 0 0
\(546\) 0 0
\(547\) − 12040.0i − 0.941121i −0.882368 0.470561i \(-0.844052\pi\)
0.882368 0.470561i \(-0.155948\pi\)
\(548\) 17850.0i 1.39145i
\(549\) 0 0
\(550\) 0 0
\(551\) 2200.00 0.170096
\(552\) 0 0
\(553\) 15480.0i 1.19037i
\(554\) −24600.0 −1.88656
\(555\) 0 0
\(556\) 28492.0 2.17326
\(557\) − 23550.0i − 1.79146i −0.444594 0.895732i \(-0.646652\pi\)
0.444594 0.895732i \(-0.353348\pi\)
\(558\) 0 0
\(559\) 5600.00 0.423712
\(560\) 0 0
\(561\) 0 0
\(562\) − 22500.0i − 1.68880i
\(563\) − 6120.00i − 0.458130i −0.973411 0.229065i \(-0.926433\pi\)
0.973411 0.229065i \(-0.0735669\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −34500.0 −2.56209
\(567\) 0 0
\(568\) 31500.0i 2.32696i
\(569\) 11700.0 0.862020 0.431010 0.902347i \(-0.358157\pi\)
0.431010 + 0.902347i \(0.358157\pi\)
\(570\) 0 0
\(571\) −8188.00 −0.600100 −0.300050 0.953923i \(-0.597003\pi\)
−0.300050 + 0.953923i \(0.597003\pi\)
\(572\) 17000.0i 1.24267i
\(573\) 0 0
\(574\) 60000.0 4.36298
\(575\) 0 0
\(576\) 0 0
\(577\) − 11690.0i − 0.843433i −0.906728 0.421717i \(-0.861428\pi\)
0.906728 0.421717i \(-0.138572\pi\)
\(578\) − 24065.0i − 1.73179i
\(579\) 0 0
\(580\) 0 0
\(581\) −19800.0 −1.41384
\(582\) 0 0
\(583\) − 30500.0i − 2.16669i
\(584\) 18450.0 1.30731
\(585\) 0 0
\(586\) 7650.00 0.539281
\(587\) − 21060.0i − 1.48082i −0.672157 0.740408i \(-0.734632\pi\)
0.672157 0.740408i \(-0.265368\pi\)
\(588\) 0 0
\(589\) 4752.00 0.332433
\(590\) 0 0
\(591\) 0 0
\(592\) 3560.00i 0.247154i
\(593\) − 22910.0i − 1.58651i −0.608889 0.793255i \(-0.708385\pi\)
0.608889 0.793255i \(-0.291615\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 34850.0 2.39515
\(597\) 0 0
\(598\) − 12000.0i − 0.820596i
\(599\) 1400.00 0.0954966 0.0477483 0.998859i \(-0.484795\pi\)
0.0477483 + 0.998859i \(0.484795\pi\)
\(600\) 0 0
\(601\) −11002.0 −0.746724 −0.373362 0.927686i \(-0.621795\pi\)
−0.373362 + 0.927686i \(0.621795\pi\)
\(602\) 42000.0i 2.84351i
\(603\) 0 0
\(604\) −7616.00 −0.513064
\(605\) 0 0
\(606\) 0 0
\(607\) − 4630.00i − 0.309598i −0.987946 0.154799i \(-0.950527\pi\)
0.987946 0.154799i \(-0.0494730\pi\)
\(608\) − 3740.00i − 0.249469i
\(609\) 0 0
\(610\) 0 0
\(611\) 5600.00 0.370788
\(612\) 0 0
\(613\) 24040.0i 1.58396i 0.610548 + 0.791979i \(0.290949\pi\)
−0.610548 + 0.791979i \(0.709051\pi\)
\(614\) −15200.0 −0.999059
\(615\) 0 0
\(616\) −67500.0 −4.41502
\(617\) 1890.00i 0.123320i 0.998097 + 0.0616601i \(0.0196395\pi\)
−0.998097 + 0.0616601i \(0.980361\pi\)
\(618\) 0 0
\(619\) −19244.0 −1.24957 −0.624783 0.780798i \(-0.714813\pi\)
−0.624783 + 0.780798i \(0.714813\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 28500.0i 1.83721i
\(623\) 45000.0i 2.89388i
\(624\) 0 0
\(625\) 0 0
\(626\) 15550.0 0.992816
\(627\) 0 0
\(628\) − 1700.00i − 0.108021i
\(629\) −400.000 −0.0253562
\(630\) 0 0
\(631\) 15892.0 1.00262 0.501308 0.865269i \(-0.332852\pi\)
0.501308 + 0.865269i \(0.332852\pi\)
\(632\) 23220.0i 1.46146i
\(633\) 0 0
\(634\) 4750.00 0.297550
\(635\) 0 0
\(636\) 0 0
\(637\) 11140.0i 0.692909i
\(638\) − 12500.0i − 0.775674i
\(639\) 0 0
\(640\) 0 0
\(641\) −12600.0 −0.776396 −0.388198 0.921576i \(-0.626902\pi\)
−0.388198 + 0.921576i \(0.626902\pi\)
\(642\) 0 0
\(643\) 7260.00i 0.445267i 0.974902 + 0.222633i \(0.0714653\pi\)
−0.974902 + 0.222633i \(0.928535\pi\)
\(644\) 61200.0 3.74475
\(645\) 0 0
\(646\) 2200.00 0.133990
\(647\) − 7400.00i − 0.449651i −0.974399 0.224825i \(-0.927819\pi\)
0.974399 0.224825i \(-0.0721812\pi\)
\(648\) 0 0
\(649\) −2500.00 −0.151207
\(650\) 0 0
\(651\) 0 0
\(652\) 32300.0i 1.94013i
\(653\) − 4790.00i − 0.287055i −0.989646 0.143528i \(-0.954155\pi\)
0.989646 0.143528i \(-0.0458446\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 35600.0 2.11882
\(657\) 0 0
\(658\) 42000.0i 2.48834i
\(659\) 1450.00 0.0857117 0.0428558 0.999081i \(-0.486354\pi\)
0.0428558 + 0.999081i \(0.486354\pi\)
\(660\) 0 0
\(661\) 11818.0 0.695411 0.347706 0.937604i \(-0.386961\pi\)
0.347706 + 0.937604i \(0.386961\pi\)
\(662\) − 11460.0i − 0.672818i
\(663\) 0 0
\(664\) −29700.0 −1.73582
\(665\) 0 0
\(666\) 0 0
\(667\) 6000.00i 0.348307i
\(668\) 32640.0i 1.89054i
\(669\) 0 0
\(670\) 0 0
\(671\) −25900.0 −1.49010
\(672\) 0 0
\(673\) 5550.00i 0.317885i 0.987288 + 0.158943i \(0.0508085\pi\)
−0.987288 + 0.158943i \(0.949192\pi\)
\(674\) 38650.0 2.20882
\(675\) 0 0
\(676\) −30549.0 −1.73811
\(677\) − 12930.0i − 0.734033i −0.930214 0.367016i \(-0.880379\pi\)
0.930214 0.367016i \(-0.119621\pi\)
\(678\) 0 0
\(679\) −48900.0 −2.76378
\(680\) 0 0
\(681\) 0 0
\(682\) − 27000.0i − 1.51596i
\(683\) 32580.0i 1.82524i 0.408809 + 0.912620i \(0.365944\pi\)
−0.408809 + 0.912620i \(0.634056\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −32100.0 −1.78657
\(687\) 0 0
\(688\) 24920.0i 1.38091i
\(689\) −12200.0 −0.674576
\(690\) 0 0
\(691\) 10228.0 0.563085 0.281542 0.959549i \(-0.409154\pi\)
0.281542 + 0.959549i \(0.409154\pi\)
\(692\) − 43350.0i − 2.38139i
\(693\) 0 0
\(694\) 5600.00 0.306301
\(695\) 0 0
\(696\) 0 0
\(697\) 4000.00i 0.217376i
\(698\) 5930.00i 0.321567i
\(699\) 0 0
\(700\) 0 0
\(701\) 8350.00 0.449893 0.224947 0.974371i \(-0.427779\pi\)
0.224947 + 0.974371i \(0.427779\pi\)
\(702\) 0 0
\(703\) 1760.00i 0.0944234i
\(704\) 14350.0 0.768233
\(705\) 0 0
\(706\) 18150.0 0.967541
\(707\) − 13500.0i − 0.718133i
\(708\) 0 0
\(709\) 14954.0 0.792115 0.396057 0.918226i \(-0.370378\pi\)
0.396057 + 0.918226i \(0.370378\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 67500.0i 3.55291i
\(713\) 12960.0i 0.680723i
\(714\) 0 0
\(715\) 0 0
\(716\) 62050.0 3.23871
\(717\) 0 0
\(718\) − 9000.00i − 0.467795i
\(719\) −29400.0 −1.52494 −0.762472 0.647021i \(-0.776015\pi\)
−0.762472 + 0.647021i \(0.776015\pi\)
\(720\) 0 0
\(721\) −23100.0 −1.19319
\(722\) 24615.0i 1.26880i
\(723\) 0 0
\(724\) 73814.0 3.78905
\(725\) 0 0
\(726\) 0 0
\(727\) 16330.0i 0.833076i 0.909118 + 0.416538i \(0.136757\pi\)
−0.909118 + 0.416538i \(0.863243\pi\)
\(728\) 27000.0i 1.37457i
\(729\) 0 0
\(730\) 0 0
\(731\) −2800.00 −0.141671
\(732\) 0 0
\(733\) 30800.0i 1.55201i 0.630726 + 0.776005i \(0.282757\pi\)
−0.630726 + 0.776005i \(0.717243\pi\)
\(734\) −42450.0 −2.13468
\(735\) 0 0
\(736\) 10200.0 0.510838
\(737\) 9000.00i 0.449823i
\(738\) 0 0
\(739\) 9524.00 0.474081 0.237041 0.971500i \(-0.423823\pi\)
0.237041 + 0.971500i \(0.423823\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 91500.0i − 4.52705i
\(743\) − 28600.0i − 1.41216i −0.708134 0.706078i \(-0.750463\pi\)
0.708134 0.706078i \(-0.249537\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 500.000 0.0245393
\(747\) 0 0
\(748\) − 8500.00i − 0.415496i
\(749\) 19800.0 0.965923
\(750\) 0 0
\(751\) −8252.00 −0.400958 −0.200479 0.979698i \(-0.564250\pi\)
−0.200479 + 0.979698i \(0.564250\pi\)
\(752\) 24920.0i 1.20843i
\(753\) 0 0
\(754\) −5000.00 −0.241498
\(755\) 0 0
\(756\) 0 0
\(757\) 24920.0i 1.19648i 0.801318 + 0.598238i \(0.204132\pi\)
−0.801318 + 0.598238i \(0.795868\pi\)
\(758\) − 40420.0i − 1.93683i
\(759\) 0 0
\(760\) 0 0
\(761\) 27900.0 1.32901 0.664503 0.747285i \(-0.268643\pi\)
0.664503 + 0.747285i \(0.268643\pi\)
\(762\) 0 0
\(763\) − 52620.0i − 2.49669i
\(764\) 59500.0 2.81758
\(765\) 0 0
\(766\) −47400.0 −2.23581
\(767\) 1000.00i 0.0470768i
\(768\) 0 0
\(769\) 11506.0 0.539554 0.269777 0.962923i \(-0.413050\pi\)
0.269777 + 0.962923i \(0.413050\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 56950.0i − 2.65502i
\(773\) − 12510.0i − 0.582087i −0.956710 0.291044i \(-0.905998\pi\)
0.956710 0.291044i \(-0.0940025\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −73350.0 −3.39318
\(777\) 0 0
\(778\) − 54750.0i − 2.52299i
\(779\) 17600.0 0.809481
\(780\) 0 0
\(781\) 35000.0 1.60358
\(782\) 6000.00i 0.274373i
\(783\) 0 0
\(784\) −49573.0 −2.25825
\(785\) 0 0
\(786\) 0 0
\(787\) 1100.00i 0.0498231i 0.999690 + 0.0249115i \(0.00793041\pi\)
−0.999690 + 0.0249115i \(0.992070\pi\)
\(788\) 1530.00i 0.0691675i
\(789\) 0 0
\(790\) 0 0
\(791\) 9300.00 0.418040
\(792\) 0 0
\(793\) 10360.0i 0.463927i
\(794\) −69200.0 −3.09297
\(795\) 0 0
\(796\) 62288.0 2.77354
\(797\) 4490.00i 0.199553i 0.995010 + 0.0997766i \(0.0318128\pi\)
−0.995010 + 0.0997766i \(0.968187\pi\)
\(798\) 0 0
\(799\) −2800.00 −0.123976
\(800\) 0 0
\(801\) 0 0
\(802\) − 46500.0i − 2.04735i
\(803\) − 20500.0i − 0.900908i
\(804\) 0 0
\(805\) 0 0
\(806\) −10800.0 −0.471977
\(807\) 0 0
\(808\) − 20250.0i − 0.881674i
\(809\) −28600.0 −1.24292 −0.621460 0.783446i \(-0.713460\pi\)
−0.621460 + 0.783446i \(0.713460\pi\)
\(810\) 0 0
\(811\) 10068.0 0.435925 0.217963 0.975957i \(-0.430059\pi\)
0.217963 + 0.975957i \(0.430059\pi\)
\(812\) − 25500.0i − 1.10206i
\(813\) 0 0
\(814\) 10000.0 0.430589
\(815\) 0 0
\(816\) 0 0
\(817\) 12320.0i 0.527567i
\(818\) − 14270.0i − 0.609950i
\(819\) 0 0
\(820\) 0 0
\(821\) −14250.0 −0.605759 −0.302880 0.953029i \(-0.597948\pi\)
−0.302880 + 0.953029i \(0.597948\pi\)
\(822\) 0 0
\(823\) 6830.00i 0.289282i 0.989484 + 0.144641i \(0.0462027\pi\)
−0.989484 + 0.144641i \(0.953797\pi\)
\(824\) −34650.0 −1.46491
\(825\) 0 0
\(826\) −7500.00 −0.315930
\(827\) 8920.00i 0.375065i 0.982258 + 0.187533i \(0.0600490\pi\)
−0.982258 + 0.187533i \(0.939951\pi\)
\(828\) 0 0
\(829\) 3534.00 0.148059 0.0740295 0.997256i \(-0.476414\pi\)
0.0740295 + 0.997256i \(0.476414\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 5740.00i − 0.239181i
\(833\) − 5570.00i − 0.231680i
\(834\) 0 0
\(835\) 0 0
\(836\) −37400.0 −1.54726
\(837\) 0 0
\(838\) 5750.00i 0.237029i
\(839\) 8000.00 0.329190 0.164595 0.986361i \(-0.447368\pi\)
0.164595 + 0.986361i \(0.447368\pi\)
\(840\) 0 0
\(841\) −21889.0 −0.897495
\(842\) 55810.0i 2.28425i
\(843\) 0 0
\(844\) 4556.00 0.185810
\(845\) 0 0
\(846\) 0 0
\(847\) 35070.0i 1.42269i
\(848\) − 54290.0i − 2.19850i
\(849\) 0 0
\(850\) 0 0
\(851\) −4800.00 −0.193351
\(852\) 0 0
\(853\) 5160.00i 0.207122i 0.994623 + 0.103561i \(0.0330237\pi\)
−0.994623 + 0.103561i \(0.966976\pi\)
\(854\) −77700.0 −3.11339
\(855\) 0 0
\(856\) 29700.0 1.18589
\(857\) − 7670.00i − 0.305720i −0.988248 0.152860i \(-0.951152\pi\)
0.988248 0.152860i \(-0.0488484\pi\)
\(858\) 0 0
\(859\) −25804.0 −1.02494 −0.512469 0.858706i \(-0.671269\pi\)
−0.512469 + 0.858706i \(0.671269\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6000.00i 0.237078i
\(863\) 400.000i 0.0157777i 0.999969 + 0.00788885i \(0.00251113\pi\)
−0.999969 + 0.00788885i \(0.997489\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 7550.00 0.296258
\(867\) 0 0
\(868\) − 55080.0i − 2.15384i
\(869\) 25800.0 1.00714
\(870\) 0 0
\(871\) 3600.00 0.140047
\(872\) − 78930.0i − 3.06526i
\(873\) 0 0
\(874\) 26400.0 1.02173
\(875\) 0 0
\(876\) 0 0
\(877\) 35100.0i 1.35147i 0.737143 + 0.675737i \(0.236175\pi\)
−0.737143 + 0.675737i \(0.763825\pi\)
\(878\) 2120.00i 0.0814881i
\(879\) 0 0
\(880\) 0 0
\(881\) −18700.0 −0.715118 −0.357559 0.933891i \(-0.616391\pi\)
−0.357559 + 0.933891i \(0.616391\pi\)
\(882\) 0 0
\(883\) − 2980.00i − 0.113573i −0.998386 0.0567865i \(-0.981915\pi\)
0.998386 0.0567865i \(-0.0180854\pi\)
\(884\) −3400.00 −0.129360
\(885\) 0 0
\(886\) 61800.0 2.34335
\(887\) 35880.0i 1.35821i 0.734041 + 0.679105i \(0.237632\pi\)
−0.734041 + 0.679105i \(0.762368\pi\)
\(888\) 0 0
\(889\) −32100.0 −1.21102
\(890\) 0 0
\(891\) 0 0
\(892\) 62390.0i 2.34190i
\(893\) 12320.0i 0.461672i
\(894\) 0 0
\(895\) 0 0
\(896\) 63450.0 2.36575
\(897\) 0 0
\(898\) − 6500.00i − 0.241545i
\(899\) 5400.00 0.200334
\(900\) 0 0
\(901\) 6100.00 0.225550
\(902\) − 100000.i − 3.69139i
\(903\) 0 0
\(904\) 13950.0 0.513241
\(905\) 0 0
\(906\) 0 0
\(907\) − 45240.0i − 1.65620i −0.560584 0.828098i \(-0.689423\pi\)
0.560584 0.828098i \(-0.310577\pi\)
\(908\) 63920.0i 2.33619i
\(909\) 0 0
\(910\) 0 0
\(911\) 33200.0 1.20743 0.603713 0.797202i \(-0.293687\pi\)
0.603713 + 0.797202i \(0.293687\pi\)
\(912\) 0 0
\(913\) 33000.0i 1.19621i
\(914\) 35950.0 1.30101
\(915\) 0 0
\(916\) −24378.0 −0.879336
\(917\) 58500.0i 2.10670i
\(918\) 0 0
\(919\) −35356.0 −1.26908 −0.634541 0.772889i \(-0.718811\pi\)
−0.634541 + 0.772889i \(0.718811\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 750.000i 0.0267895i
\(923\) − 14000.0i − 0.499259i
\(924\) 0 0
\(925\) 0 0
\(926\) 13350.0 0.473767
\(927\) 0 0
\(928\) − 4250.00i − 0.150337i
\(929\) 25700.0 0.907631 0.453816 0.891096i \(-0.350062\pi\)
0.453816 + 0.891096i \(0.350062\pi\)
\(930\) 0 0
\(931\) −24508.0 −0.862747
\(932\) 58650.0i 2.06131i
\(933\) 0 0
\(934\) −5900.00 −0.206696
\(935\) 0 0
\(936\) 0 0
\(937\) − 52890.0i − 1.84401i −0.387173 0.922007i \(-0.626549\pi\)
0.387173 0.922007i \(-0.373451\pi\)
\(938\) 27000.0i 0.939852i
\(939\) 0 0
\(940\) 0 0
\(941\) 38050.0 1.31817 0.659083 0.752070i \(-0.270945\pi\)
0.659083 + 0.752070i \(0.270945\pi\)
\(942\) 0 0
\(943\) 48000.0i 1.65758i
\(944\) −4450.00 −0.153427
\(945\) 0 0
\(946\) 70000.0 2.40581
\(947\) 29640.0i 1.01708i 0.861040 + 0.508538i \(0.169814\pi\)
−0.861040 + 0.508538i \(0.830186\pi\)
\(948\) 0 0
\(949\) −8200.00 −0.280488
\(950\) 0 0
\(951\) 0 0
\(952\) − 13500.0i − 0.459598i
\(953\) 15170.0i 0.515640i 0.966193 + 0.257820i \(0.0830041\pi\)
−0.966193 + 0.257820i \(0.916996\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −83300.0 −2.81811
\(957\) 0 0
\(958\) − 70500.0i − 2.37761i
\(959\) 31500.0 1.06068
\(960\) 0 0
\(961\) −18127.0 −0.608472
\(962\) − 4000.00i − 0.134059i
\(963\) 0 0
\(964\) −81974.0 −2.73880
\(965\) 0 0
\(966\) 0 0
\(967\) 5470.00i 0.181906i 0.995855 + 0.0909531i \(0.0289913\pi\)
−0.995855 + 0.0909531i \(0.971009\pi\)
\(968\) 52605.0i 1.74668i
\(969\) 0 0
\(970\) 0 0
\(971\) 15150.0 0.500707 0.250354 0.968154i \(-0.419453\pi\)
0.250354 + 0.968154i \(0.419453\pi\)
\(972\) 0 0
\(973\) − 50280.0i − 1.65663i
\(974\) 49250.0 1.62020
\(975\) 0 0
\(976\) −46102.0 −1.51198
\(977\) − 31190.0i − 1.02135i −0.859775 0.510674i \(-0.829396\pi\)
0.859775 0.510674i \(-0.170604\pi\)
\(978\) 0 0
\(979\) 75000.0 2.44843
\(980\) 0 0
\(981\) 0 0
\(982\) 12250.0i 0.398079i
\(983\) − 7560.00i − 0.245297i −0.992450 0.122648i \(-0.960861\pi\)
0.992450 0.122648i \(-0.0391387\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2500.00 0.0807467
\(987\) 0 0
\(988\) 14960.0i 0.481722i
\(989\) −33600.0 −1.08030
\(990\) 0 0
\(991\) 32672.0 1.04729 0.523643 0.851938i \(-0.324573\pi\)
0.523643 + 0.851938i \(0.324573\pi\)
\(992\) − 9180.00i − 0.293816i
\(993\) 0 0
\(994\) 105000. 3.35050
\(995\) 0 0
\(996\) 0 0
\(997\) 4740.00i 0.150569i 0.997162 + 0.0752845i \(0.0239865\pi\)
−0.997162 + 0.0752845i \(0.976014\pi\)
\(998\) − 85180.0i − 2.70173i
\(999\) 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 225.4.b.b.199.1 2
3.2 odd 2 225.4.b.a.199.2 2
5.2 odd 4 45.4.a.e.1.1 yes 1
5.3 odd 4 225.4.a.a.1.1 1
5.4 even 2 inner 225.4.b.b.199.2 2
15.2 even 4 45.4.a.a.1.1 1
15.8 even 4 225.4.a.h.1.1 1
15.14 odd 2 225.4.b.a.199.1 2
20.7 even 4 720.4.a.o.1.1 1
35.27 even 4 2205.4.a.t.1.1 1
45.2 even 12 405.4.e.n.271.1 2
45.7 odd 12 405.4.e.b.271.1 2
45.22 odd 12 405.4.e.b.136.1 2
45.32 even 12 405.4.e.n.136.1 2
60.47 odd 4 720.4.a.bc.1.1 1
105.62 odd 4 2205.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.a.a.1.1 1 15.2 even 4
45.4.a.e.1.1 yes 1 5.2 odd 4
225.4.a.a.1.1 1 5.3 odd 4
225.4.a.h.1.1 1 15.8 even 4
225.4.b.a.199.1 2 15.14 odd 2
225.4.b.a.199.2 2 3.2 odd 2
225.4.b.b.199.1 2 1.1 even 1 trivial
225.4.b.b.199.2 2 5.4 even 2 inner
405.4.e.b.136.1 2 45.22 odd 12
405.4.e.b.271.1 2 45.7 odd 12
405.4.e.n.136.1 2 45.32 even 12
405.4.e.n.271.1 2 45.2 even 12
720.4.a.o.1.1 1 20.7 even 4
720.4.a.bc.1.1 1 60.47 odd 4
2205.4.a.a.1.1 1 105.62 odd 4
2205.4.a.t.1.1 1 35.27 even 4