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Properties

Label 6450.2.a.y.1.1
Level $6450$
Weight $2$
Character 6450.1
Self dual yes
Analytic conductor $51.504$
Analytic rank $0$
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] = [6450,2,Mod(1,6450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6450, 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("6450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6450 = 2 \cdot 3 \cdot 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6450.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: \(51.5035093037\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1290)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} +10.0000 q^{29} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -8.00000 q^{38} -6.00000 q^{39} -6.00000 q^{41} -1.00000 q^{43} +4.00000 q^{44} +4.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{57} +10.0000 q^{58} -4.00000 q^{59} -6.00000 q^{61} +1.00000 q^{64} -4.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +6.00000 q^{74} -8.00000 q^{76} -6.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -1.00000 q^{86} -10.0000 q^{87} +4.00000 q^{88} +2.00000 q^{89} +4.00000 q^{92} -12.0000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −8.00000 −1.29777
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 6.00000 0.832050
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 10.0000 1.31306
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) −10.0000 −1.07211
\(88\) 4.00000 0.426401
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −7.00000 −0.707107
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −6.00000 −0.594089
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −20.0000 −1.93347 −0.966736 − 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) 6.00000 0.554700
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 − 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −4.00000 −0.340503
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 8.00000 0.671345
\(143\) 24.0000 2.00698
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 7.00000 0.577350
\(148\) 6.00000 0.493197
\(149\) 2.00000 0.163846 0.0819232 − 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −8.00000 −0.648886
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) −1.00000 −0.0762493
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000 0.300658
\(178\) 2.00000 0.149906
\(179\) −16.0000 −1.19590 −0.597948 − 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 4.00000 0.284268
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 4.00000 0.278019
\(208\) 6.00000 0.416025
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) −20.0000 −1.36717
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) −6.00000 −0.402694
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 8.00000 0.529813
\(229\) 22.0000 1.45380 0.726900 − 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) −22.0000 −1.44127 −0.720634 − 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 − 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −48.0000 −3.05417
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 − 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 1.00000 0.0622573
\(259\) 0 0
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) −4.00000 −0.244339
\(269\) −10.0000 −0.609711 −0.304855 − 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 12.0000 0.714590
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 10.0000 0.585206
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) −4.00000 −0.232104
\(298\) 2.00000 0.115857
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 2.00000 0.114897
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 − 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) −6.00000 −0.339683
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −34.0000 −1.90963 −0.954815 − 0.297200i \(-0.903947\pi\)
−0.954815 + 0.297200i \(0.903947\pi\)
\(318\) −6.00000 −0.336463
\(319\) 40.0000 2.23957
\(320\) 0 0
\(321\) 20.0000 1.11629
\(322\) 0 0
\(323\) −48.0000 −2.67079
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 − 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 12.0000 0.658586
\(333\) 6.00000 0.328798
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 − 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 23.0000 1.25104
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 0 0
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −10.0000 −0.536056
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 4.00000 0.213201
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) −16.0000 −0.845626
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 14.0000 0.735824
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) 4.00000 0.208798 0.104399 − 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 4.00000 0.208514
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 60.0000 3.09016
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −20.0000 −1.02463
\(382\) −8.00000 −0.409316
\(383\) 32.0000 1.63512 0.817562 − 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −1.00000 −0.0508329
\(388\) −10.0000 −0.507673
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) −7.00000 −0.353553
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) −6.00000 −0.297044
\(409\) −30.0000 −1.48340 −0.741702 − 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 20.0000 0.979404
\(418\) −32.0000 −1.56517
\(419\) −8.00000 −0.390826 −0.195413 − 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 − 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −12.0000 −0.583460
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −20.0000 −0.966736
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −22.0000 −1.05725 −0.528626 − 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −32.0000 −1.53077
\(438\) −10.0000 −0.477818
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 36.0000 1.71235
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) −2.00000 −0.0945968
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 − 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 18.0000 0.846649
\(453\) −16.0000 −0.751746
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 22.0000 1.02799
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) −4.00000 −0.184115
\(473\) −4.00000 −0.183920
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 16.0000 0.731823
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −20.0000 −0.906287 −0.453143 − 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) −6.00000 −0.271607
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 − 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 6.00000 0.270501
\(493\) 60.0000 2.70226
\(494\) −48.0000 −2.15962
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 4.00000 0.178529
\(503\) 32.0000 1.42681 0.713405 − 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) −23.0000 −1.02147
\(508\) 20.0000 0.887357
\(509\) 6.00000 0.265945 0.132973 − 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) 1.00000 0.0440225
\(517\) −48.0000 −2.11104
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 10.0000 0.437688
\(523\) −4.00000 −0.174908 −0.0874539 − 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) −2.00000 −0.0865485
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 16.0000 0.690451
\(538\) −10.0000 −0.431131
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −16.0000 −0.687259
\(543\) −14.0000 −0.600798
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 18.0000 0.768922
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −80.0000 −3.40811
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) −6.00000 −0.253095
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 − 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 24.0000 1.00349
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 19.0000 0.790296
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) 24.0000 0.993978
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 7.00000 0.288675
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 6.00000 0.246598
\(593\) −46.0000 −1.88899 −0.944497 − 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) −8.00000 −0.327418
\(598\) 24.0000 0.981433
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 0 0
\(611\) −72.0000 −2.91281
\(612\) 6.00000 0.242536
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 4.00000 0.160904
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 16.0000 0.641542
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 32.0000 1.27796
\(628\) −2.00000 −0.0798087
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −34.0000 −1.35031
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −42.0000 −1.66410
\(638\) 40.0000 1.58362
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 − 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) 20.0000 0.789337
\(643\) 12.0000 0.473234 0.236617 − 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 30.0000 1.17399 0.586995 − 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 8.00000 0.310929
\(663\) −36.0000 −1.39812
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 40.0000 1.54881
\(668\) −12.0000 −0.464294
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 − 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 38.0000 1.46046 0.730229 − 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) −18.0000 −0.691286
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 − 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) −1.00000 −0.0381246
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 24.0000 0.913003 0.456502 − 0.889723i \(-0.349102\pi\)
0.456502 + 0.889723i \(0.349102\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) −36.0000 −1.36360
\(698\) −14.0000 −0.529908
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 − 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −6.00000 −0.226455
\(703\) −48.0000 −1.81035
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 2.00000 0.0749532
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) −16.0000 −0.597531
\(718\) 32.0000 1.19423
\(719\) 32.0000 1.19340 0.596699 − 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 45.0000 1.67473
\(723\) −18.0000 −0.669427
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.00000 −0.221918
\(732\) 6.00000 0.221766
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −16.0000 −0.589368
\(738\) −6.00000 −0.220863
\(739\) 8.00000 0.294285 0.147142 − 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 48.0000 1.76332
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6.00000 0.219676
\(747\) 12.0000 0.439057
\(748\) 24.0000 0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 − 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) −12.0000 −0.437595
\(753\) −4.00000 −0.145768
\(754\) 60.0000 2.18507
\(755\) 0 0
\(756\) 0 0
\(757\) 54.0000 1.96266 0.981332 − 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) −20.0000 −0.726433
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 − 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) −20.0000 −0.724524
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) 34.0000 1.22607 0.613036 − 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −10.0000 −0.359908
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 24.0000 0.858238
\(783\) −10.0000 −0.357371
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 − 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −36.0000 −1.27840
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −34.0000 −1.20434 −0.602171 − 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 2.00000 0.0706225
\(803\) 40.0000 1.41157
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) −2.00000 −0.0703598
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 8.00000 0.279885
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) 0 0
\(821\) 14.0000 0.488603 0.244302 − 0.969699i \(-0.421441\pi\)
0.244302 + 0.969699i \(0.421441\pi\)
\(822\) −18.0000 −0.627822
\(823\) 12.0000 0.418294 0.209147 − 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 − 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 4.00000 0.139010
\(829\) 18.0000 0.625166 0.312583 − 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 6.00000 0.208013
\(833\) −42.0000 −1.45521
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) −32.0000 −1.10674
\(837\) 0 0
\(838\) −8.00000 −0.276355
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 10.0000 0.344623
\(843\) 6.00000 0.206651
\(844\) 0 0
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) −8.00000 −0.274075
\(853\) −42.0000 −1.43805 −0.719026 − 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −24.0000 −0.819346
\(859\) −8.00000 −0.272956 −0.136478 − 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −22.0000 −0.747590
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) −2.00000 −0.0677285
\(873\) −10.0000 −0.338449
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 16.0000 0.539974
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 − 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) −7.00000 −0.235702
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −8.00000 −0.268614 −0.134307 − 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 8.00000 0.267860
\(893\) 96.0000 3.21252
\(894\) −2.00000 −0.0668900
\(895\) 0 0
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) −22.0000 −0.734150
\(899\) 0 0
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) −44.0000 −1.46100 −0.730498 − 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 20.0000 0.663723
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 8.00000 0.264906
\(913\) 48.0000 1.58857
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) −18.0000 −0.592798
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) −4.00000 −0.131377
\(928\) 10.0000 0.328266
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 56.0000 1.83533
\(932\) −22.0000 −0.720634
\(933\) −16.0000 −0.523816
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 2.00000 0.0651635
\(943\) −24.0000 −0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 8.00000 0.259828
\(949\) 60.0000 1.94768
\(950\) 0 0
\(951\) 34.0000 1.10253
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) −40.0000 −1.29302
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 36.0000 1.16069
\(963\) −20.0000 −0.644491
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 − 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 5.00000 0.160706
\(969\) 48.0000 1.54198
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −4.00000 −0.127906
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 8.00000 0.255290
\(983\) −8.00000 −0.255160 −0.127580 − 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 60.0000 1.91079
\(987\) 0 0
\(988\) −48.0000 −1.52708
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) −26.0000 −0.823428 −0.411714 − 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −32.0000 −1.01294
\(999\) −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 6450.2.a.y.1.1 1
5.4 even 2 1290.2.a.c.1.1 ✓ 1
15.14 odd 2 3870.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1290.2.a.c.1.1 ✓ 1 5.4 even 2
3870.2.a.w.1.1 1 15.14 odd 2
6450.2.a.y.1.1 1 1.1 even 1 trivial