↑

Properties

Label 8550.2.a.bi.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8550,2,Mod(1,8550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2850)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.00000 q^{8} +3.00000 q^{11} +6.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{19} +3.00000 q^{22} -1.00000 q^{23} +6.00000 q^{26} +2.00000 q^{28} +5.00000 q^{29} +7.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{37} -1.00000 q^{38} -2.00000 q^{41} +6.00000 q^{43} +3.00000 q^{44} -1.00000 q^{46} -12.0000 q^{47} -3.00000 q^{49} +6.00000 q^{52} +9.00000 q^{53} +2.00000 q^{56} +5.00000 q^{58} +10.0000 q^{59} -13.0000 q^{61} +7.00000 q^{62} +1.00000 q^{64} +7.00000 q^{67} -2.00000 q^{68} -12.0000 q^{71} -9.00000 q^{73} +2.00000 q^{74} -1.00000 q^{76} +6.00000 q^{77} +5.00000 q^{79} -2.00000 q^{82} -1.00000 q^{83} +6.00000 q^{86} +3.00000 q^{88} -5.00000 q^{89} +12.0000 q^{91} -1.00000 q^{92} -12.0000 q^{94} +12.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 5.00000 0.656532
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 − 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 7.00000 0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 − 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 − 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 − 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −1.00000 −0.109764 −0.0548821 − 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −5.00000 −0.529999 −0.264999 − 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −9.00000 −0.886796 −0.443398 − 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 19.0000 1.78737 0.893685 − 0.448695i \(-0.148111\pi\)
0.893685 + 0.448695i \(0.148111\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −13.0000 −1.17696
\(123\) 0 0
\(124\) 7.00000 0.628619
\(125\) 0 0
\(126\) 0 0
\(127\) −3.00000 −0.266207 −0.133103 − 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −7.00000 −0.611593 −0.305796 − 0.952097i \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 7.00000 0.604708
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 18.0000 1.50524
\(144\) 0 0
\(145\) 0 0
\(146\) −9.00000 −0.744845
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 20.0000 1.63846 0.819232 − 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 − 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 5.00000 0.397779
\(159\) 0 0
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 − 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −1.00000 −0.0776151
\(167\) −2.00000 −0.154765 −0.0773823 − 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) −21.0000 −1.59660 −0.798300 − 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −5.00000 −0.374766
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 12.0000 0.889499
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) −17.0000 −1.23008 −0.615038 − 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 − 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 − 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) 0 0
\(206\) −9.00000 −0.627060
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 − 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 14.0000 0.950382
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 11.0000 0.736614 0.368307 − 0.929704i \(-0.379937\pi\)
0.368307 + 0.929704i \(0.379937\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 19.0000 1.26386
\(227\) −2.00000 −0.132745 −0.0663723 − 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 − 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.00000 0.328266
\(233\) −26.0000 −1.70332 −0.851658 − 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 7.00000 0.444500
\(249\) 0 0
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 − 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) −3.00000 −0.188237
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.0000 −1.68421 −0.842107 − 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) −7.00000 −0.432461
\(263\) −1.00000 −0.0616626 −0.0308313 − 0.999525i \(-0.509815\pi\)
−0.0308313 + 0.999525i \(0.509815\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 7.00000 0.427593
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 − 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) 7.00000 0.420589 0.210295 − 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.00000 −0.417585 −0.208792 − 0.977960i \(-0.566953\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 18.0000 1.06436
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −9.00000 −0.526685
\(293\) −1.00000 −0.0584206 −0.0292103 − 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 20.0000 1.15857
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 12.0000 0.690522
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 − 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 6.00000 0.341882
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −29.0000 −1.63918 −0.819588 − 0.572953i \(-0.805798\pi\)
−0.819588 + 0.572953i \(0.805798\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) 3.00000 0.168497 0.0842484 − 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) 15.0000 0.839839
\(320\) 0 0
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) 0 0
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 − 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) −1.00000 −0.0548821
\(333\) 0 0
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 23.0000 1.25104
\(339\) 0 0
\(340\) 0 0
\(341\) 21.0000 1.13721
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −21.0000 −1.12897
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 25.0000 1.33822 0.669110 − 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −5.00000 −0.264999
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 − 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) 18.0000 0.934513
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 30.0000 1.54508
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −17.0000 −0.869796
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) 0 0
\(388\) 12.0000 0.609208
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) 37.0000 1.85698 0.928488 − 0.371361i \(-0.121109\pi\)
0.928488 + 0.371361i \(0.121109\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 − 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 0 0
\(403\) 42.0000 2.09217
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 10.0000 0.496292
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −9.00000 −0.443398
\(413\) 20.0000 0.984136
\(414\) 0 0
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) −3.00000 −0.146735
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 17.0000 0.827547
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) −26.0000 −1.25823
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 36.0000 1.73005 0.865025 − 0.501729i \(-0.167303\pi\)
0.865025 + 0.501729i \(0.167303\pi\)
\(434\) 14.0000 0.672022
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 1.00000 0.0478365
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 − 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −21.0000 −0.997740 −0.498870 − 0.866677i \(-0.666252\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 11.0000 0.520865
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −35.0000 −1.65175 −0.825876 − 0.563852i \(-0.809319\pi\)
−0.825876 + 0.563852i \(0.809319\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 19.0000 0.893685
\(453\) 0 0
\(454\) −2.00000 −0.0938647
\(455\) 0 0
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 15.0000 0.700904
\(459\) 0 0
\(460\) 0 0
\(461\) 28.0000 1.30409 0.652045 − 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) −7.00000 −0.323921 −0.161961 − 0.986797i \(-0.551782\pi\)
−0.161961 + 0.986797i \(0.551782\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) 18.0000 0.827641
\(474\) 0 0
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 0 0
\(479\) 15.0000 0.685367 0.342684 − 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −13.0000 −0.588482
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 − 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −10.0000 −0.450377
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) 30.0000 1.34298 0.671492 − 0.741012i \(-0.265654\pi\)
0.671492 + 0.741012i \(0.265654\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 8.00000 0.357057
\(503\) 44.0000 1.96186 0.980932 − 0.194354i \(-0.0622609\pi\)
0.980932 + 0.194354i \(0.0622609\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.00000 −0.133366
\(507\) 0 0
\(508\) −3.00000 −0.133103
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −18.0000 −0.796273
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −27.0000 −1.19092
\(515\) 0 0
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) 4.00000 0.175750
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 − 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 − 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −7.00000 −0.305796
\(525\) 0 0
\(526\) −1.00000 −0.0436021
\(527\) −14.0000 −0.609850
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 7.00000 0.302354
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 − 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 32.0000 1.37452
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −23.0000 −0.983409 −0.491704 − 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 7.00000 0.297402
\(555\) 0 0
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) −7.00000 −0.295277
\(563\) −16.0000 −0.674320 −0.337160 − 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −10.0000 −0.419222 −0.209611 − 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 − 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 18.0000 0.752618
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 0 0
\(577\) 7.00000 0.291414 0.145707 − 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) −1.00000 −0.0413096
\(587\) 3.00000 0.123823 0.0619116 − 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 4.00000 0.164260 0.0821302 − 0.996622i \(-0.473828\pi\)
0.0821302 + 0.996622i \(0.473828\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) −30.0000 −1.22577 −0.612883 − 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 − 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) 0 0
\(607\) −23.0000 −0.933541 −0.466771 − 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −72.0000 −2.91281
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) −22.0000 −0.885687 −0.442843 − 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 − 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 0 0
\(626\) −29.0000 −1.15907
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −48.0000 −1.91085 −0.955425 − 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) 5.00000 0.198889
\(633\) 0 0
\(634\) 3.00000 0.119145
\(635\) 0 0
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 15.0000 0.593856
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) −27.0000 −1.06148 −0.530740 − 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) 24.0000 0.939193 0.469596 − 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) −24.0000 −0.935617
\(659\) −30.0000 −1.16863 −0.584317 − 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 − 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 7.00000 0.272063
\(663\) 0 0
\(664\) −1.00000 −0.0388075
\(665\) 0 0
\(666\) 0 0
\(667\) −5.00000 −0.193601
\(668\) −2.00000 −0.0773823
\(669\) 0 0
\(670\) 0 0
\(671\) −39.0000 −1.50558
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −7.00000 −0.269032 −0.134516 − 0.990911i \(-0.542948\pi\)
−0.134516 + 0.990911i \(0.542948\pi\)
\(678\) 0 0
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) 0 0
\(682\) 21.0000 0.804132
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) 54.0000 2.05724
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −21.0000 −0.798300
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 25.0000 0.946264
\(699\) 0 0
\(700\) 0 0
\(701\) 48.0000 1.81293 0.906467 − 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 − 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −5.00000 −0.187383
\(713\) −7.00000 −0.262152
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5.00000 −0.186469 −0.0932343 − 0.995644i \(-0.529721\pi\)
−0.0932343 + 0.995644i \(0.529721\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 − 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 12.0000 0.444750
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −39.0000 −1.44050 −0.720249 − 0.693716i \(-0.755972\pi\)
−0.720249 + 0.693716i \(0.755972\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 21.0000 0.773545
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 − 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 18.0000 0.660801
\(743\) −16.0000 −0.586983 −0.293492 − 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6.00000 0.219676
\(747\) 0 0
\(748\) −6.00000 −0.219382
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) 52.0000 1.89751 0.948753 − 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 30.0000 1.09254
\(755\) 0 0
\(756\) 0 0
\(757\) −23.0000 −0.835949 −0.417975 − 0.908459i \(-0.637260\pi\)
−0.417975 + 0.908459i \(0.637260\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 − 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) −17.0000 −0.615038
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 60.0000 2.16647
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 − 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.0000 0.575853
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.0000 0.430775
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 2.00000 0.0715199
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) −43.0000 −1.53278 −0.766392 − 0.642373i \(-0.777950\pi\)
−0.766392 + 0.642373i \(0.777950\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) 38.0000 1.35112
\(792\) 0 0
\(793\) −78.0000 −2.76986
\(794\) 37.0000 1.31308
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 3.00000 0.105934
\(803\) −27.0000 −0.952809
\(804\) 0 0
\(805\) 0 0
\(806\) 42.0000 1.47939
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −13.0000 −0.456492 −0.228246 − 0.973604i \(-0.573299\pi\)
−0.228246 + 0.973604i \(0.573299\pi\)
\(812\) 10.0000 0.350931
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) −6.00000 −0.209913
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 − 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 0 0
\(823\) −34.0000 −1.18517 −0.592583 − 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) −9.00000 −0.313530
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) 18.0000 0.625921 0.312961 − 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) 0 0
\(829\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.00000 0.208013
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) −3.00000 −0.103757
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) 30.0000 1.03572 0.517858 − 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 22.0000 0.758170
\(843\) 0 0
\(844\) 17.0000 0.585164
\(845\) 0 0
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) 9.00000 0.309061
\(849\) 0 0
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −26.0000 −0.889702
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 − 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) −36.0000 −1.22545 −0.612727 − 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 36.0000 1.22333
\(867\) 0 0
\(868\) 14.0000 0.475191
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 42.0000 1.42312
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) 1.00000 0.0338255
\(875\) 0 0
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 − 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −35.0000 −1.18119
\(879\) 0 0
\(880\) 0 0
\(881\) −32.0000 −1.07811 −0.539054 − 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −21.0000 −0.705509
\(887\) −52.0000 −1.74599 −0.872995 − 0.487730i \(-0.837825\pi\)
−0.872995 + 0.487730i \(0.837825\pi\)
\(888\) 0 0
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) 0 0
\(892\) 11.0000 0.368307
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −35.0000 −1.16797
\(899\) 35.0000 1.16732
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) 19.0000 0.631931
\(905\) 0 0
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 − 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) −2.00000 −0.0663723
\(909\) 0 0
\(910\) 0 0
\(911\) −22.0000 −0.728893 −0.364446 − 0.931224i \(-0.618742\pi\)
−0.364446 + 0.931224i \(0.618742\pi\)
\(912\) 0 0
\(913\) −3.00000 −0.0992855
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) 15.0000 0.495614
\(917\) −14.0000 −0.462321
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 28.0000 0.922131
\(923\) −72.0000 −2.36991
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) 5.00000 0.164133
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) −26.0000 −0.851658
\(933\) 0 0
\(934\) −7.00000 −0.229047
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 14.0000 0.457116
\(939\) 0 0
\(940\) 0 0
\(941\) 43.0000 1.40176 0.700880 − 0.713279i \(-0.252791\pi\)
0.700880 + 0.713279i \(0.252791\pi\)
\(942\) 0 0
\(943\) 2.00000 0.0651290
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 18.0000 0.585230
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) −54.0000 −1.75291
\(950\) 0 0
\(951\) 0 0
\(952\) −4.00000 −0.129641
\(953\) −11.0000 −0.356325 −0.178162 − 0.984001i \(-0.557015\pi\)
−0.178162 + 0.984001i \(0.557015\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 15.0000 0.484628
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 42.0000 1.35063 0.675314 − 0.737530i \(-0.264008\pi\)
0.675314 + 0.737530i \(0.264008\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 − 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −13.0000 −0.416120
\(977\) −42.0000 −1.34370 −0.671850 − 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −10.0000 −0.318465
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 17.0000 0.540023 0.270011 − 0.962857i \(-0.412973\pi\)
0.270011 + 0.962857i \(0.412973\pi\)
\(992\) 7.00000 0.222250
\(993\) 0 0
\(994\) −24.0000 −0.761234
\(995\) 0 0
\(996\) 0 0
\(997\) 17.0000 0.538395 0.269198 − 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) 30.0000 0.949633
\(999\) 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 8550.2.a.bi.1.1 1
3.2 odd 2 2850.2.a.d.1.1 ✓ 1
5.4 even 2 8550.2.a.f.1.1 1
15.2 even 4 2850.2.d.m.799.1 2
15.8 even 4 2850.2.d.m.799.2 2
15.14 odd 2 2850.2.a.z.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.d.1.1 ✓ 1 3.2 odd 2
2850.2.a.z.1.1 yes 1 15.14 odd 2
2850.2.d.m.799.1 2 15.2 even 4
2850.2.d.m.799.2 2 15.8 even 4
8550.2.a.f.1.1 1 5.4 even 2
8550.2.a.bi.1.1 1 1.1 even 1 trivial