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Properties

Label 5775.2.a.a.1.1
Level $5775$
Weight $2$
Character 5775.1
Self dual yes
Analytic conductor $46.114$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5775,2,Mod(1,5775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.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: \(46.1136071673\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5775.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} +1.00000 q^{21} +2.00000 q^{22} -3.00000 q^{23} -4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -3.00000 q^{29} +8.00000 q^{32} -1.00000 q^{33} -6.00000 q^{34} +2.00000 q^{36} +6.00000 q^{37} +10.0000 q^{38} +2.00000 q^{39} -4.00000 q^{41} -2.00000 q^{42} -7.00000 q^{43} -2.00000 q^{44} +6.00000 q^{46} +4.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} +4.00000 q^{52} -9.00000 q^{53} -2.00000 q^{54} -5.00000 q^{57} +6.00000 q^{58} -11.0000 q^{59} -1.00000 q^{61} +1.00000 q^{63} -8.00000 q^{64} +2.00000 q^{66} +2.00000 q^{67} +6.00000 q^{68} -3.00000 q^{69} -8.00000 q^{71} -4.00000 q^{73} -12.0000 q^{74} -10.0000 q^{76} -1.00000 q^{77} -4.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} +8.00000 q^{82} -11.0000 q^{83} +2.00000 q^{84} +14.0000 q^{86} -3.00000 q^{87} -7.00000 q^{89} +2.00000 q^{91} -6.00000 q^{92} -8.00000 q^{94} +8.00000 q^{96} -1.00000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 2.00000 0.577350
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −2.00000 −0.471405
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 2.00000 0.426401
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 8.00000 1.41421
\(33\) −1.00000 −0.174078
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 10.0000 1.62221
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) −2.00000 −0.308607
\(43\) −7.00000 −1.06749 −0.533745 − 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 4.00000 0.554700
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 6.00000 0.787839
\(59\) −11.0000 −1.43208 −0.716039 − 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.00000 0.727607
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −12.0000 −1.39497
\(75\) 0 0
\(76\) −10.0000 −1.14708
\(77\) −1.00000 −0.113961
\(78\) −4.00000 −0.452911
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.00000 0.883452
\(83\) −11.0000 −1.20741 −0.603703 − 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 14.0000 1.50966
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −7.00000 −0.741999 −0.370999 − 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) −2.00000 −0.202031
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −6.00000 −0.594089
\(103\) 3.00000 0.295599 0.147799 − 0.989017i \(-0.452781\pi\)
0.147799 + 0.989017i \(0.452781\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 2.00000 0.192450
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) −4.00000 −0.377964
\(113\) −13.0000 −1.22294 −0.611469 − 0.791269i \(-0.709421\pi\)
−0.611469 + 0.791269i \(0.709421\pi\)
\(114\) 10.0000 0.936586
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 22.0000 2.02526
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −19.0000 −1.68598 −0.842989 − 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) −7.00000 −0.616316
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 − 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −2.00000 −0.174078
\(133\) −5.00000 −0.433555
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 6.00000 0.510754
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 16.0000 1.34269
\(143\) −2.00000 −0.167248
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 1.00000 0.0824786
\(148\) 12.0000 0.986394
\(149\) −14.0000 −1.14692 −0.573462 − 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 − 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 3.00000 0.239426 0.119713 − 0.992809i \(-0.461803\pi\)
0.119713 + 0.992809i \(0.461803\pi\)
\(158\) −20.0000 −1.59111
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) −2.00000 −0.157135
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 22.0000 1.70753
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) −14.0000 −1.06749
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −11.0000 −0.826811
\(178\) 14.0000 1.04934
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 − 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −4.00000 −0.296500
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 8.00000 0.583460
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −8.00000 −0.577350
\(193\) 18.0000 1.29567 0.647834 − 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −4.00000 −0.284988 −0.142494 − 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 2.00000 0.142134
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 12.0000 0.844317
\(203\) −3.00000 −0.210559
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) −3.00000 −0.208514
\(208\) −8.00000 −0.554700
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 − 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) −18.0000 −1.23625
\(213\) −8.00000 −0.548151
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 8.00000 0.541828
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) −12.0000 −0.805387
\(223\) −11.0000 −0.736614 −0.368307 − 0.929704i \(-0.620063\pi\)
−0.368307 + 0.929704i \(0.620063\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 26.0000 1.72949
\(227\) 21.0000 1.39382 0.696909 − 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) −10.0000 −0.662266
\(229\) −16.0000 −1.05731 −0.528655 − 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −22.0000 −1.43208
\(237\) 10.0000 0.649570
\(238\) −6.00000 −0.388922
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) −10.0000 −0.636285
\(248\) 0 0
\(249\) −11.0000 −0.697097
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 − 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 2.00000 0.125988
\(253\) 3.00000 0.188608
\(254\) 38.0000 2.38433
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 14.0000 0.871602
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) −20.0000 −1.23560
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 10.0000 0.613139
\(267\) −7.00000 −0.428393
\(268\) 4.00000 0.244339
\(269\) −15.0000 −0.914566 −0.457283 − 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) 29.0000 1.76162 0.880812 − 0.473466i \(-0.156997\pi\)
0.880812 + 0.473466i \(0.156997\pi\)
\(272\) −12.0000 −0.727607
\(273\) 2.00000 0.121046
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −32.0000 −1.91923
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) −8.00000 −0.476393
\(283\) 14.0000 0.832214 0.416107 − 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −4.00000 −0.236113
\(288\) 8.00000 0.471405
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) −8.00000 −0.468165
\(293\) −9.00000 −0.525786 −0.262893 − 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 28.0000 1.62200
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) −36.0000 −2.07157
\(303\) −6.00000 −0.344691
\(304\) 20.0000 1.14708
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 26.0000 1.48390 0.741949 − 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) −2.00000 −0.113961
\(309\) 3.00000 0.170664
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 − 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 − 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 20.0000 1.12509
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 18.0000 1.00939
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 6.00000 0.334367
\(323\) −15.0000 −0.834622
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) −28.0000 −1.55078
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −11.0000 −0.604615 −0.302307 − 0.953211i \(-0.597757\pi\)
−0.302307 + 0.953211i \(0.597757\pi\)
\(332\) −22.0000 −1.20741
\(333\) 6.00000 0.328798
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −31.0000 −1.68868 −0.844339 − 0.535810i \(-0.820006\pi\)
−0.844339 + 0.535810i \(0.820006\pi\)
\(338\) 18.0000 0.979071
\(339\) −13.0000 −0.706063
\(340\) 0 0
\(341\) 0 0
\(342\) 10.0000 0.540738
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) 36.0000 1.93258 0.966291 − 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) −6.00000 −0.321634
\(349\) 7.00000 0.374701 0.187351 − 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −8.00000 −0.426401
\(353\) 30.0000 1.59674 0.798369 − 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 22.0000 1.16929
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) −9.00000 −0.475002 −0.237501 − 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 28.0000 1.47165
\(363\) 1.00000 0.0524864
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −13.0000 −0.678594 −0.339297 − 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 12.0000 0.625543
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 25.0000 1.29445 0.647225 − 0.762299i \(-0.275929\pi\)
0.647225 + 0.762299i \(0.275929\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) −2.00000 −0.102869
\(379\) 5.00000 0.256833 0.128416 − 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −19.0000 −0.973399
\(382\) 24.0000 1.22795
\(383\) −6.00000 −0.306586 −0.153293 − 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −36.0000 −1.83235
\(387\) −7.00000 −0.355830
\(388\) −2.00000 −0.101535
\(389\) −34.0000 −1.72387 −0.861934 − 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) −9.00000 −0.455150
\(392\) 0 0
\(393\) 10.0000 0.504433
\(394\) 8.00000 0.403034
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −10.0000 −0.501886 −0.250943 − 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 8.00000 0.401004
\(399\) −5.00000 −0.250313
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 − 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 − 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 6.00000 0.295599
\(413\) −11.0000 −0.541275
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 16.0000 0.784465
\(417\) 16.0000 0.783523
\(418\) −10.0000 −0.489116
\(419\) 23.0000 1.12362 0.561812 − 0.827265i \(-0.310105\pi\)
0.561812 + 0.827265i \(0.310105\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 − 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) −20.0000 −0.973585
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) −1.00000 −0.0483934
\(428\) 8.00000 0.386695
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −4.00000 −0.192450
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) 15.0000 0.717547
\(438\) 8.00000 0.382255
\(439\) 27.0000 1.28864 0.644320 − 0.764756i \(-0.277141\pi\)
0.644320 + 0.764756i \(0.277141\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 12.0000 0.569495
\(445\) 0 0
\(446\) 22.0000 1.04173
\(447\) −14.0000 −0.662177
\(448\) −8.00000 −0.377964
\(449\) −8.00000 −0.377543 −0.188772 − 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −26.0000 −1.22294
\(453\) 18.0000 0.845714
\(454\) −42.0000 −1.97116
\(455\) 0 0
\(456\) 0 0
\(457\) 37.0000 1.73079 0.865393 − 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) 32.0000 1.49526
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 2.00000 0.0930484
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 12.0000 0.557086
\(465\) 0 0
\(466\) 36.0000 1.66767
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 4.00000 0.184900
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 3.00000 0.138233
\(472\) 0 0
\(473\) 7.00000 0.321860
\(474\) −20.0000 −0.918630
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) −9.00000 −0.412082
\(478\) 42.0000 1.92104
\(479\) −30.0000 −1.37073 −0.685367 − 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 4.00000 0.182195
\(483\) −3.00000 −0.136505
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −2.00000 −0.0907218
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 − 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) −8.00000 −0.360668
\(493\) −9.00000 −0.405340
\(494\) 20.0000 0.899843
\(495\) 0 0
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 22.0000 0.985844
\(499\) 23.0000 1.02962 0.514811 − 0.857304i \(-0.327862\pi\)
0.514811 + 0.857304i \(0.327862\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 8.00000 0.357057
\(503\) −17.0000 −0.757993 −0.378996 − 0.925398i \(-0.623731\pi\)
−0.378996 + 0.925398i \(0.623731\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −9.00000 −0.399704
\(508\) −38.0000 −1.68598
\(509\) 19.0000 0.842160 0.421080 − 0.907023i \(-0.361651\pi\)
0.421080 + 0.907023i \(0.361651\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) −32.0000 −1.41421
\(513\) −5.00000 −0.220755
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) −14.0000 −0.616316
\(517\) −4.00000 −0.175920
\(518\) −12.0000 −0.527250
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −25.0000 −1.09527 −0.547635 − 0.836717i \(-0.684472\pi\)
−0.547635 + 0.836717i \(0.684472\pi\)
\(522\) 6.00000 0.262613
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −48.0000 −2.09290
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −11.0000 −0.477359
\(532\) −10.0000 −0.433555
\(533\) −8.00000 −0.346518
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 − 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) −58.0000 −2.49131
\(543\) −14.0000 −0.600798
\(544\) 24.0000 1.02899
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) −1.00000 −0.0427569 −0.0213785 − 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 12.0000 0.512615
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 15.0000 0.639021
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 44.0000 1.86938
\(555\) 0 0
\(556\) 32.0000 1.35710
\(557\) 42.0000 1.77960 0.889799 − 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) −14.0000 −0.592137
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 12.0000 0.506189
\(563\) −28.0000 −1.18006 −0.590030 − 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 9.00000 0.377300 0.188650 − 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 − 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) −4.00000 −0.167248
\(573\) −12.0000 −0.501307
\(574\) 8.00000 0.333914
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) −42.0000 −1.74848 −0.874241 − 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 16.0000 0.665512
\(579\) 18.0000 0.748054
\(580\) 0 0
\(581\) −11.0000 −0.456357
\(582\) 2.00000 0.0829027
\(583\) 9.00000 0.372742
\(584\) 0 0
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 42.0000 1.73353 0.866763 − 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) −4.00000 −0.164538
\(592\) −24.0000 −0.986394
\(593\) −42.0000 −1.72473 −0.862367 − 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −28.0000 −1.14692
\(597\) −4.00000 −0.163709
\(598\) 12.0000 0.490716
\(599\) 6.00000 0.245153 0.122577 − 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 − 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 14.0000 0.570597
\(603\) 2.00000 0.0814463
\(604\) 36.0000 1.46482
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) 14.0000 0.568242 0.284121 − 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −40.0000 −1.62221
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 6.00000 0.242536
\(613\) 10.0000 0.403896 0.201948 − 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) −52.0000 −2.09855
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −6.00000 −0.241355
\(619\) 10.0000 0.401934 0.200967 − 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −3.00000 −0.120386
\(622\) −8.00000 −0.320771
\(623\) −7.00000 −0.280449
\(624\) −8.00000 −0.320256
\(625\) 0 0
\(626\) 38.0000 1.51879
\(627\) 5.00000 0.199681
\(628\) 6.00000 0.239426
\(629\) 18.0000 0.717707
\(630\) 0 0
\(631\) −35.0000 −1.39333 −0.696664 − 0.717398i \(-0.745333\pi\)
−0.696664 + 0.717398i \(0.745333\pi\)
\(632\) 0 0
\(633\) 10.0000 0.397464
\(634\) −44.0000 −1.74746
\(635\) 0 0
\(636\) −18.0000 −0.713746
\(637\) 2.00000 0.0792429
\(638\) −6.00000 −0.237542
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −8.00000 −0.315735
\(643\) −9.00000 −0.354925 −0.177463 − 0.984128i \(-0.556789\pi\)
−0.177463 + 0.984128i \(0.556789\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) 30.0000 1.18033
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) 28.0000 1.09656
\(653\) −3.00000 −0.117399 −0.0586995 − 0.998276i \(-0.518695\pi\)
−0.0586995 + 0.998276i \(0.518695\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) 16.0000 0.624695
\(657\) −4.00000 −0.156055
\(658\) −8.00000 −0.311872
\(659\) 41.0000 1.59713 0.798567 − 0.601906i \(-0.205592\pi\)
0.798567 + 0.601906i \(0.205592\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 − 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 22.0000 0.855054
\(663\) 6.00000 0.233021
\(664\) 0 0
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) 9.00000 0.348481
\(668\) −16.0000 −0.619059
\(669\) −11.0000 −0.425285
\(670\) 0 0
\(671\) 1.00000 0.0386046
\(672\) 8.00000 0.308607
\(673\) −29.0000 −1.11787 −0.558934 − 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 62.0000 2.38815
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 15.0000 0.576497 0.288248 − 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 26.0000 0.998524
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 21.0000 0.804722
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) −10.0000 −0.382360
\(685\) 0 0
\(686\) −2.00000 −0.0763604
\(687\) −16.0000 −0.610438
\(688\) 28.0000 1.06749
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −50.0000 −1.90209 −0.951045 − 0.309053i \(-0.899988\pi\)
−0.951045 + 0.309053i \(0.899988\pi\)
\(692\) 12.0000 0.456172
\(693\) −1.00000 −0.0379869
\(694\) −72.0000 −2.73308
\(695\) 0 0
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −14.0000 −0.529908
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −1.00000 −0.0377695 −0.0188847 − 0.999822i \(-0.506012\pi\)
−0.0188847 + 0.999822i \(0.506012\pi\)
\(702\) −4.00000 −0.150970
\(703\) −30.0000 −1.13147
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) −60.0000 −2.25813
\(707\) −6.00000 −0.225653
\(708\) −22.0000 −0.826811
\(709\) −35.0000 −1.31445 −0.657226 − 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) 0 0
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) −21.0000 −0.784259
\(718\) 18.0000 0.671754
\(719\) −15.0000 −0.559406 −0.279703 − 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) 3.00000 0.111726
\(722\) −12.0000 −0.446594
\(723\) −2.00000 −0.0743808
\(724\) −28.0000 −1.04061
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) 37.0000 1.37225 0.686127 − 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −21.0000 −0.776713
\(732\) −2.00000 −0.0739221
\(733\) −46.0000 −1.69905 −0.849524 − 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) 26.0000 0.959678
\(735\) 0 0
\(736\) −24.0000 −0.884652
\(737\) −2.00000 −0.0736709
\(738\) 8.00000 0.294484
\(739\) −2.00000 −0.0735712 −0.0367856 − 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) −10.0000 −0.367359
\(742\) 18.0000 0.660801
\(743\) −26.0000 −0.953847 −0.476924 − 0.878945i \(-0.658248\pi\)
−0.476924 + 0.878945i \(0.658248\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −50.0000 −1.83063
\(747\) −11.0000 −0.402469
\(748\) −6.00000 −0.219382
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 9.00000 0.328415 0.164207 − 0.986426i \(-0.447493\pi\)
0.164207 + 0.986426i \(0.447493\pi\)
\(752\) −16.0000 −0.583460
\(753\) −4.00000 −0.145768
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 22.0000 0.799604 0.399802 − 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −10.0000 −0.363216
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 − 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 38.0000 1.37659
\(763\) −4.00000 −0.144810
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) −22.0000 −0.794374
\(768\) 16.0000 0.577350
\(769\) −45.0000 −1.62274 −0.811371 − 0.584532i \(-0.801278\pi\)
−0.811371 + 0.584532i \(0.801278\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 36.0000 1.29567
\(773\) −14.0000 −0.503545 −0.251773 − 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 14.0000 0.503220
\(775\) 0 0
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 68.0000 2.43792
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 18.0000 0.643679
\(783\) −3.00000 −0.107211
\(784\) −4.00000 −0.142857
\(785\) 0 0
\(786\) −20.0000 −0.713376
\(787\) −24.0000 −0.855508 −0.427754 − 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) −8.00000 −0.284988
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −13.0000 −0.462227
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 20.0000 0.709773
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −38.0000 −1.34603 −0.673015 − 0.739629i \(-0.735001\pi\)
−0.673015 + 0.739629i \(0.735001\pi\)
\(798\) 10.0000 0.353996
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) −7.00000 −0.247333
\(802\) 56.0000 1.97743
\(803\) 4.00000 0.141157
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −15.0000 −0.528025
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 − 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −6.00000 −0.210559
\(813\) 29.0000 1.01707
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) 35.0000 1.22449
\(818\) 4.00000 0.139857
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −53.0000 −1.84971 −0.924856 − 0.380317i \(-0.875815\pi\)
−0.924856 + 0.380317i \(0.875815\pi\)
\(822\) −12.0000 −0.418548
\(823\) −2.00000 −0.0697156 −0.0348578 − 0.999392i \(-0.511098\pi\)
−0.0348578 + 0.999392i \(0.511098\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 22.0000 0.765478
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) −6.00000 −0.208514
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) −16.0000 −0.554700
\(833\) 3.00000 0.103944
\(834\) −32.0000 −1.10807
\(835\) 0 0
\(836\) 10.0000 0.345857
\(837\) 0 0
\(838\) −46.0000 −1.58904
\(839\) −15.0000 −0.517858 −0.258929 − 0.965896i \(-0.583369\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −2.00000 −0.0689246
\(843\) −6.00000 −0.206651
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 1.00000 0.0343604
\(848\) 36.0000 1.23625
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) −18.0000 −0.617032
\(852\) −16.0000 −0.548151
\(853\) 44.0000 1.50653 0.753266 − 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 0 0
\(857\) 2.00000 0.0683187 0.0341593 − 0.999416i \(-0.489125\pi\)
0.0341593 + 0.999416i \(0.489125\pi\)
\(858\) 4.00000 0.136558
\(859\) −28.0000 −0.955348 −0.477674 − 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 48.0000 1.63489
\(863\) 21.0000 0.714848 0.357424 − 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 8.00000 0.272166
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) −30.0000 −1.01477
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) 19.0000 0.641584 0.320792 − 0.947150i \(-0.396051\pi\)
0.320792 + 0.947150i \(0.396051\pi\)
\(878\) −54.0000 −1.82241
\(879\) −9.00000 −0.303562
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 − 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −8.00000 −0.269221 −0.134611 − 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 33.0000 1.10803 0.554016 − 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(888\) 0 0
\(889\) −19.0000 −0.637240
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −22.0000 −0.736614
\(893\) −20.0000 −0.669274
\(894\) 28.0000 0.936460
\(895\) 0 0
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 16.0000 0.533927
\(899\) 0 0
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) −8.00000 −0.266371
\(903\) −7.00000 −0.232945
\(904\) 0 0
\(905\) 0 0
\(906\) −36.0000 −1.19602
\(907\) −58.0000 −1.92586 −0.962929 − 0.269754i \(-0.913058\pi\)
−0.962929 + 0.269754i \(0.913058\pi\)
\(908\) 42.0000 1.39382
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 20.0000 0.662266
\(913\) 11.0000 0.364047
\(914\) −74.0000 −2.44770
\(915\) 0 0
\(916\) −32.0000 −1.05731
\(917\) 10.0000 0.330229
\(918\) −6.00000 −0.198030
\(919\) 10.0000 0.329870 0.164935 − 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) 26.0000 0.856729
\(922\) 60.0000 1.97599
\(923\) −16.0000 −0.526646
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 44.0000 1.44593
\(927\) 3.00000 0.0985329
\(928\) −24.0000 −0.787839
\(929\) 50.0000 1.64045 0.820223 − 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) −36.0000 −1.17922
\(933\) 4.00000 0.130954
\(934\) 40.0000 1.30884
\(935\) 0 0
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 − 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) −4.00000 −0.130605
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 − 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) −6.00000 −0.195491
\(943\) 12.0000 0.390774
\(944\) 44.0000 1.43208
\(945\) 0 0
\(946\) −14.0000 −0.455179
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 20.0000 0.649570
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 − 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 18.0000 0.582772
\(955\) 0 0
\(956\) −42.0000 −1.35838
\(957\) 3.00000 0.0969762
\(958\) 60.0000 1.93851
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −24.0000 −0.773791
\(963\) 4.00000 0.128898
\(964\) −4.00000 −0.128831
\(965\) 0 0
\(966\) 6.00000 0.193047
\(967\) 5.00000 0.160789 0.0803946 − 0.996763i \(-0.474382\pi\)
0.0803946 + 0.996763i \(0.474382\pi\)
\(968\) 0 0
\(969\) −15.0000 −0.481869
\(970\) 0 0
\(971\) 29.0000 0.930654 0.465327 − 0.885139i \(-0.345937\pi\)
0.465327 + 0.885139i \(0.345937\pi\)
\(972\) 2.00000 0.0641500
\(973\) 16.0000 0.512936
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −45.0000 −1.43968 −0.719839 − 0.694141i \(-0.755784\pi\)
−0.719839 + 0.694141i \(0.755784\pi\)
\(978\) −28.0000 −0.895341
\(979\) 7.00000 0.223721
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 6.00000 0.191468
\(983\) −54.0000 −1.72233 −0.861166 − 0.508323i \(-0.830265\pi\)
−0.861166 + 0.508323i \(0.830265\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 4.00000 0.127321
\(988\) −20.0000 −0.636285
\(989\) 21.0000 0.667761
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 − 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 0 0
\(993\) −11.0000 −0.349074
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) −22.0000 −0.697097
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −46.0000 −1.45610
\(999\) 6.00000 0.189832
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 5775.2.a.a.1.1 1
5.4 even 2 1155.2.a.n.1.1 ✓ 1
15.14 odd 2 3465.2.a.a.1.1 1
35.34 odd 2 8085.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.n.1.1 ✓ 1 5.4 even 2
3465.2.a.a.1.1 1 15.14 odd 2
5775.2.a.a.1.1 1 1.1 even 1 trivial
8085.2.a.z.1.1 1 35.34 odd 2