Properties

Label 4998.2.a.y.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
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] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, 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("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.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: \(39.9092309302\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 714)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +3.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -3.00000 q^{19} -3.00000 q^{20} +2.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} +3.00000 q^{30} +1.00000 q^{32} -2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +3.00000 q^{37} -3.00000 q^{38} -3.00000 q^{40} -1.00000 q^{43} +2.00000 q^{44} -3.00000 q^{45} -5.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -1.00000 q^{51} -6.00000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +3.00000 q^{57} +6.00000 q^{58} +5.00000 q^{59} +3.00000 q^{60} +10.0000 q^{61} +1.00000 q^{64} -2.00000 q^{66} -7.00000 q^{67} +1.00000 q^{68} +5.00000 q^{69} +3.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} +3.00000 q^{74} -4.00000 q^{75} -3.00000 q^{76} +16.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -3.00000 q^{85} -1.00000 q^{86} -6.00000 q^{87} +2.00000 q^{88} +5.00000 q^{89} -3.00000 q^{90} -5.00000 q^{92} -6.00000 q^{94} +9.00000 q^{95} -1.00000 q^{96} -4.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 3.00000 0.547723
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −3.00000 −0.486664
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 2.00000 0.301511
\(45\) −3.00000 −0.447214
\(46\) −5.00000 −0.737210
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 6.00000 0.787839
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 3.00000 0.387298
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 1.00000 0.121268
\(69\) 5.00000 0.601929
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 3.00000 0.348743
\(75\) −4.00000 −0.461880
\(76\) −3.00000 −0.344124
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −1.00000 −0.107833
\(87\) −6.00000 −0.643268
\(88\) 2.00000 0.213201
\(89\) 5.00000 0.529999 0.264999 0.964249i \(-0.414628\pi\)
0.264999 + 0.964249i \(0.414628\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −5.00000 −0.521286
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 9.00000 0.923381
\(96\) −1.00000 −0.102062
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 4.00000 0.400000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) −6.00000 −0.572078
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 3.00000 0.280976
\(115\) 15.0000 1.39876
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 5.00000 0.460287
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) −7.00000 −0.636364
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −7.00000 −0.604708
\(135\) 3.00000 0.258199
\(136\) 1.00000 0.0857493
\(137\) 11.0000 0.939793 0.469897 0.882721i \(-0.344291\pi\)
0.469897 + 0.882721i \(0.344291\pi\)
\(138\) 5.00000 0.425628
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 3.00000 0.251754
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −18.0000 −1.49482
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 3.00000 0.246598
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −4.00000 −0.326599
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −3.00000 −0.243332
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 16.0000 1.27289
\(159\) 6.00000 0.475831
\(160\) −3.00000 −0.237171
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) 6.00000 0.467099
\(166\) −12.0000 −0.931381
\(167\) −25.0000 −1.93456 −0.967279 0.253715i \(-0.918347\pi\)
−0.967279 + 0.253715i \(0.918347\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −3.00000 −0.230089
\(171\) −3.00000 −0.229416
\(172\) −1.00000 −0.0762493
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) −5.00000 −0.375823
\(178\) 5.00000 0.374766
\(179\) −7.00000 −0.523205 −0.261602 0.965176i \(-0.584251\pi\)
−0.261602 + 0.965176i \(0.584251\pi\)
\(180\) −3.00000 −0.223607
\(181\) −1.00000 −0.0743294 −0.0371647 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) −5.00000 −0.368605
\(185\) −9.00000 −0.661693
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 9.00000 0.652929
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 0 0
\(197\) −21.0000 −1.49619 −0.748094 0.663593i \(-0.769031\pi\)
−0.748094 + 0.663593i \(0.769031\pi\)
\(198\) 2.00000 0.142134
\(199\) 21.0000 1.48865 0.744325 0.667817i \(-0.232771\pi\)
0.744325 + 0.667817i \(0.232771\pi\)
\(200\) 4.00000 0.282843
\(201\) 7.00000 0.493742
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −5.00000 −0.347524
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) −6.00000 −0.412082
\(213\) −3.00000 −0.205557
\(214\) −16.0000 −1.09374
\(215\) 3.00000 0.204598
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −11.0000 −0.745014
\(219\) 4.00000 0.270295
\(220\) −6.00000 −0.404520
\(221\) 0 0
\(222\) −3.00000 −0.201347
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −14.0000 −0.931266
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 3.00000 0.198680
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 15.0000 0.989071
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) 5.00000 0.325472
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 3.00000 0.193649
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −7.00000 −0.449977
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 3.00000 0.189737
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −10.0000 −0.628695
\(254\) −20.0000 −1.25491
\(255\) 3.00000 0.187867
\(256\) 1.00000 0.0625000
\(257\) −13.0000 −0.810918 −0.405459 0.914113i \(-0.632888\pi\)
−0.405459 + 0.914113i \(0.632888\pi\)
\(258\) 1.00000 0.0622573
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −10.0000 −0.617802
\(263\) −26.0000 −1.60323 −0.801614 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) −2.00000 −0.123091
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) −5.00000 −0.305995
\(268\) −7.00000 −0.427593
\(269\) −19.0000 −1.15845 −0.579225 0.815168i \(-0.696645\pi\)
−0.579225 + 0.815168i \(0.696645\pi\)
\(270\) 3.00000 0.182574
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 11.0000 0.664534
\(275\) 8.00000 0.482418
\(276\) 5.00000 0.300965
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 6.00000 0.357295
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 3.00000 0.178017
\(285\) −9.00000 −0.533114
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) −18.0000 −1.05700
\(291\) 4.00000 0.234484
\(292\) −4.00000 −0.234082
\(293\) 8.00000 0.467365 0.233682 0.972313i \(-0.424922\pi\)
0.233682 + 0.972313i \(0.424922\pi\)
\(294\) 0 0
\(295\) −15.0000 −0.873334
\(296\) 3.00000 0.174371
\(297\) −2.00000 −0.116052
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) −3.00000 −0.172062
\(305\) −30.0000 −1.71780
\(306\) 1.00000 0.0571662
\(307\) 19.0000 1.08439 0.542194 0.840254i \(-0.317594\pi\)
0.542194 + 0.840254i \(0.317594\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 0 0
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) −1.00000 −0.0561656 −0.0280828 0.999606i \(-0.508940\pi\)
−0.0280828 + 0.999606i \(0.508940\pi\)
\(318\) 6.00000 0.336463
\(319\) 12.0000 0.671871
\(320\) −3.00000 −0.167705
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 11.0000 0.608301
\(328\) 0 0
\(329\) 0 0
\(330\) 6.00000 0.330289
\(331\) 9.00000 0.494685 0.247342 0.968928i \(-0.420443\pi\)
0.247342 + 0.968928i \(0.420443\pi\)
\(332\) −12.0000 −0.658586
\(333\) 3.00000 0.164399
\(334\) −25.0000 −1.36794
\(335\) 21.0000 1.14735
\(336\) 0 0
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) −13.0000 −0.707107
\(339\) 14.0000 0.760376
\(340\) −3.00000 −0.162698
\(341\) 0 0
\(342\) −3.00000 −0.162221
\(343\) 0 0
\(344\) −1.00000 −0.0539164
\(345\) −15.0000 −0.807573
\(346\) 9.00000 0.483843
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) −5.00000 −0.265747
\(355\) −9.00000 −0.477670
\(356\) 5.00000 0.264999
\(357\) 0 0
\(358\) −7.00000 −0.369961
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) −3.00000 −0.158114
\(361\) −10.0000 −0.526316
\(362\) −1.00000 −0.0525588
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) −10.0000 −0.522708
\(367\) 29.0000 1.51379 0.756894 0.653538i \(-0.226716\pi\)
0.756894 + 0.653538i \(0.226716\pi\)
\(368\) −5.00000 −0.260643
\(369\) 0 0
\(370\) −9.00000 −0.467888
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 2.00000 0.103418
\(375\) −3.00000 −0.154919
\(376\) −6.00000 −0.309426
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 9.00000 0.461690
\(381\) 20.0000 1.02463
\(382\) 12.0000 0.613973
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −1.00000 −0.0508329
\(388\) −4.00000 −0.203069
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) 0 0
\(393\) 10.0000 0.504433
\(394\) −21.0000 −1.05796
\(395\) −48.0000 −2.41514
\(396\) 2.00000 0.100504
\(397\) 23.0000 1.15434 0.577168 0.816625i \(-0.304158\pi\)
0.577168 + 0.816625i \(0.304158\pi\)
\(398\) 21.0000 1.05263
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 7.00000 0.349128
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) −1.00000 −0.0495074
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −11.0000 −0.542590
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) −5.00000 −0.245737
\(415\) 36.0000 1.76717
\(416\) 0 0
\(417\) −16.0000 −0.783523
\(418\) −6.00000 −0.293470
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) −14.0000 −0.681509
\(423\) −6.00000 −0.291730
\(424\) −6.00000 −0.291386
\(425\) 4.00000 0.194029
\(426\) −3.00000 −0.145350
\(427\) 0 0
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 3.00000 0.144673
\(431\) −37.0000 −1.78223 −0.891114 0.453780i \(-0.850075\pi\)
−0.891114 + 0.453780i \(0.850075\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 37.0000 1.77811 0.889053 0.457804i \(-0.151364\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) 18.0000 0.863034
\(436\) −11.0000 −0.526804
\(437\) 15.0000 0.717547
\(438\) 4.00000 0.191127
\(439\) −41.0000 −1.95682 −0.978412 0.206666i \(-0.933739\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) −6.00000 −0.286039
\(441\) 0 0
\(442\) 0 0
\(443\) 35.0000 1.66290 0.831450 0.555599i \(-0.187511\pi\)
0.831450 + 0.555599i \(0.187511\pi\)
\(444\) −3.00000 −0.142374
\(445\) −15.0000 −0.711068
\(446\) −4.00000 −0.189405
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 4.00000 0.188562
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 21.0000 0.982339 0.491169 0.871064i \(-0.336570\pi\)
0.491169 + 0.871064i \(0.336570\pi\)
\(458\) −12.0000 −0.560723
\(459\) −1.00000 −0.0466760
\(460\) 15.0000 0.699379
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 18.0000 0.830278
\(471\) −2.00000 −0.0921551
\(472\) 5.00000 0.230144
\(473\) −2.00000 −0.0919601
\(474\) −16.0000 −0.734904
\(475\) −12.0000 −0.550598
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −20.0000 −0.914779
\(479\) 27.0000 1.23366 0.616831 0.787096i \(-0.288416\pi\)
0.616831 + 0.787096i \(0.288416\pi\)
\(480\) 3.00000 0.136931
\(481\) 0 0
\(482\) −8.00000 −0.364390
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 12.0000 0.544892
\(486\) −1.00000 −0.0453609
\(487\) 17.0000 0.770344 0.385172 0.922845i \(-0.374142\pi\)
0.385172 + 0.922845i \(0.374142\pi\)
\(488\) 10.0000 0.452679
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −27.0000 −1.21849 −0.609246 0.792981i \(-0.708528\pi\)
−0.609246 + 0.792981i \(0.708528\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 3.00000 0.134164
\(501\) 25.0000 1.11692
\(502\) −12.0000 −0.535586
\(503\) 3.00000 0.133763 0.0668817 0.997761i \(-0.478695\pi\)
0.0668817 + 0.997761i \(0.478695\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) −10.0000 −0.444554
\(507\) 13.0000 0.577350
\(508\) −20.0000 −0.887357
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 3.00000 0.132842
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 3.00000 0.132453
\(514\) −13.0000 −0.573405
\(515\) 42.0000 1.85074
\(516\) 1.00000 0.0440225
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) −9.00000 −0.395056
\(520\) 0 0
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 6.00000 0.262613
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) −26.0000 −1.13365
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 2.00000 0.0869565
\(530\) 18.0000 0.781870
\(531\) 5.00000 0.216982
\(532\) 0 0
\(533\) 0 0
\(534\) −5.00000 −0.216371
\(535\) 48.0000 2.07522
\(536\) −7.00000 −0.302354
\(537\) 7.00000 0.302072
\(538\) −19.0000 −0.819148
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 1.00000 0.0429141
\(544\) 1.00000 0.0428746
\(545\) 33.0000 1.41356
\(546\) 0 0
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) 11.0000 0.469897
\(549\) 10.0000 0.426790
\(550\) 8.00000 0.341121
\(551\) −18.0000 −0.766826
\(552\) 5.00000 0.212814
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 9.00000 0.382029
\(556\) 16.0000 0.678551
\(557\) 4.00000 0.169485 0.0847427 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) −15.0000 −0.632737
\(563\) −41.0000 −1.72794 −0.863972 0.503540i \(-0.832031\pi\)
−0.863972 + 0.503540i \(0.832031\pi\)
\(564\) 6.00000 0.252646
\(565\) 42.0000 1.76695
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) 3.00000 0.125877
\(569\) −17.0000 −0.712677 −0.356339 0.934357i \(-0.615975\pi\)
−0.356339 + 0.934357i \(0.615975\pi\)
\(570\) −9.00000 −0.376969
\(571\) 14.0000 0.585882 0.292941 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) −20.0000 −0.834058
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 1.00000 0.0415945
\(579\) 14.0000 0.581820
\(580\) −18.0000 −0.747409
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) −12.0000 −0.496989
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 8.00000 0.330477
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −15.0000 −0.617540
\(591\) 21.0000 0.863825
\(592\) 3.00000 0.123299
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −21.0000 −0.859473
\(598\) 0 0
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) −4.00000 −0.163299
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −7.00000 −0.285062
\(604\) 0 0
\(605\) 21.0000 0.853771
\(606\) 12.0000 0.487467
\(607\) −47.0000 −1.90767 −0.953836 0.300329i \(-0.902903\pi\)
−0.953836 + 0.300329i \(0.902903\pi\)
\(608\) −3.00000 −0.121666
\(609\) 0 0
\(610\) −30.0000 −1.21466
\(611\) 0 0
\(612\) 1.00000 0.0404226
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 19.0000 0.766778
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 14.0000 0.563163
\(619\) 42.0000 1.68812 0.844061 0.536247i \(-0.180158\pi\)
0.844061 + 0.536247i \(0.180158\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 15.0000 0.601445
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −2.00000 −0.0799361
\(627\) 6.00000 0.239617
\(628\) 2.00000 0.0798087
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) 16.0000 0.636446
\(633\) 14.0000 0.556450
\(634\) −1.00000 −0.0397151
\(635\) 60.0000 2.38103
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 12.0000 0.475085
\(639\) 3.00000 0.118678
\(640\) −3.00000 −0.118585
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 16.0000 0.631470
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) −3.00000 −0.118125
\(646\) −3.00000 −0.118033
\(647\) −2.00000 −0.0786281 −0.0393141 0.999227i \(-0.512517\pi\)
−0.0393141 + 0.999227i \(0.512517\pi\)
\(648\) 1.00000 0.0392837
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 45.0000 1.76099 0.880493 0.474059i \(-0.157212\pi\)
0.880493 + 0.474059i \(0.157212\pi\)
\(654\) 11.0000 0.430134
\(655\) 30.0000 1.17220
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 6.00000 0.233550
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 9.00000 0.349795
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) −30.0000 −1.16160
\(668\) −25.0000 −0.967279
\(669\) 4.00000 0.154649
\(670\) 21.0000 0.811301
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 12.0000 0.462223
\(675\) −4.00000 −0.153960
\(676\) −13.0000 −0.500000
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) −3.00000 −0.115045
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) −32.0000 −1.22445 −0.612223 0.790685i \(-0.709725\pi\)
−0.612223 + 0.790685i \(0.709725\pi\)
\(684\) −3.00000 −0.114708
\(685\) −33.0000 −1.26087
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) −1.00000 −0.0381246
\(689\) 0 0
\(690\) −15.0000 −0.571040
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 9.00000 0.342129
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −48.0000 −1.82074
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) 16.0000 0.605609
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) −9.00000 −0.339441
\(704\) 2.00000 0.0753778
\(705\) −18.0000 −0.677919
\(706\) −9.00000 −0.338719
\(707\) 0 0
\(708\) −5.00000 −0.187912
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) −9.00000 −0.337764
\(711\) 16.0000 0.600047
\(712\) 5.00000 0.187383
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −7.00000 −0.261602
\(717\) 20.0000 0.746914
\(718\) −6.00000 −0.223918
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) −10.0000 −0.372161
\(723\) 8.00000 0.297523
\(724\) −1.00000 −0.0371647
\(725\) 24.0000 0.891338
\(726\) 7.00000 0.259794
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) −1.00000 −0.0369863
\(732\) −10.0000 −0.369611
\(733\) 24.0000 0.886460 0.443230 0.896408i \(-0.353832\pi\)
0.443230 + 0.896408i \(0.353832\pi\)
\(734\) 29.0000 1.07041
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) −14.0000 −0.515697
\(738\) 0 0
\(739\) −3.00000 −0.110357 −0.0551784 0.998477i \(-0.517573\pi\)
−0.0551784 + 0.998477i \(0.517573\pi\)
\(740\) −9.00000 −0.330847
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 30.0000 1.09911
\(746\) 26.0000 0.951928
\(747\) −12.0000 −0.439057
\(748\) 2.00000 0.0731272
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) −43.0000 −1.56909 −0.784546 0.620070i \(-0.787104\pi\)
−0.784546 + 0.620070i \(0.787104\pi\)
\(752\) −6.00000 −0.218797
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 4.00000 0.145382 0.0726912 0.997354i \(-0.476841\pi\)
0.0726912 + 0.997354i \(0.476841\pi\)
\(758\) 0 0
\(759\) 10.0000 0.362977
\(760\) 9.00000 0.326464
\(761\) 47.0000 1.70375 0.851874 0.523746i \(-0.175466\pi\)
0.851874 + 0.523746i \(0.175466\pi\)
\(762\) 20.0000 0.724524
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) −3.00000 −0.108465
\(766\) 14.0000 0.505841
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 19.0000 0.685158 0.342579 0.939489i \(-0.388700\pi\)
0.342579 + 0.939489i \(0.388700\pi\)
\(770\) 0 0
\(771\) 13.0000 0.468184
\(772\) −14.0000 −0.503871
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) −4.00000 −0.143592
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) −5.00000 −0.178800
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) −6.00000 −0.214149
\(786\) 10.0000 0.356688
\(787\) −30.0000 −1.06938 −0.534692 0.845047i \(-0.679572\pi\)
−0.534692 + 0.845047i \(0.679572\pi\)
\(788\) −21.0000 −0.748094
\(789\) 26.0000 0.925625
\(790\) −48.0000 −1.70776
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) 0 0
\(794\) 23.0000 0.816239
\(795\) −18.0000 −0.638394
\(796\) 21.0000 0.744325
\(797\) 44.0000 1.55856 0.779280 0.626676i \(-0.215585\pi\)
0.779280 + 0.626676i \(0.215585\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 4.00000 0.141421
\(801\) 5.00000 0.176666
\(802\) −6.00000 −0.211867
\(803\) −8.00000 −0.282314
\(804\) 7.00000 0.246871
\(805\) 0 0
\(806\) 0 0
\(807\) 19.0000 0.668832
\(808\) −12.0000 −0.422159
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) −3.00000 −0.105409
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 6.00000 0.210300
\(815\) −60.0000 −2.10171
\(816\) −1.00000 −0.0350070
\(817\) 3.00000 0.104957
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) −53.0000 −1.84971 −0.924856 0.380317i \(-0.875815\pi\)
−0.924856 + 0.380317i \(0.875815\pi\)
\(822\) −11.0000 −0.383669
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −14.0000 −0.487713
\(825\) −8.00000 −0.278524
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) −5.00000 −0.173762
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 36.0000 1.24958
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) 75.0000 2.59548
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 24.0000 0.829066
\(839\) −51.0000 −1.76072 −0.880358 0.474310i \(-0.842698\pi\)
−0.880358 + 0.474310i \(0.842698\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) 15.0000 0.516627
\(844\) −14.0000 −0.481900
\(845\) 39.0000 1.34164
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 8.00000 0.274559
\(850\) 4.00000 0.137199
\(851\) −15.0000 −0.514193
\(852\) −3.00000 −0.102778
\(853\) −18.0000 −0.616308 −0.308154 0.951336i \(-0.599711\pi\)
−0.308154 + 0.951336i \(0.599711\pi\)
\(854\) 0 0
\(855\) 9.00000 0.307794
\(856\) −16.0000 −0.546869
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) −15.0000 −0.511793 −0.255897 0.966704i \(-0.582371\pi\)
−0.255897 + 0.966704i \(0.582371\pi\)
\(860\) 3.00000 0.102299
\(861\) 0 0
\(862\) −37.0000 −1.26023
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −27.0000 −0.918028
\(866\) 37.0000 1.25731
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 18.0000 0.610257
\(871\) 0 0
\(872\) −11.0000 −0.372507
\(873\) −4.00000 −0.135379
\(874\) 15.0000 0.507383
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) −41.0000 −1.38368
\(879\) −8.00000 −0.269833
\(880\) −6.00000 −0.202260
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 15.0000 0.504219
\(886\) 35.0000 1.17585
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −3.00000 −0.100673
\(889\) 0 0
\(890\) −15.0000 −0.502801
\(891\) 2.00000 0.0670025
\(892\) −4.00000 −0.133930
\(893\) 18.0000 0.602347
\(894\) 10.0000 0.334450
\(895\) 21.0000 0.701953
\(896\) 0 0
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) 0 0
\(900\) 4.00000 0.133333
\(901\) −6.00000 −0.199889
\(902\) 0 0
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 3.00000 0.0997234
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) −18.0000 −0.597351
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −33.0000 −1.09334 −0.546669 0.837349i \(-0.684105\pi\)
−0.546669 + 0.837349i \(0.684105\pi\)
\(912\) 3.00000 0.0993399
\(913\) −24.0000 −0.794284
\(914\) 21.0000 0.694618
\(915\) 30.0000 0.991769
\(916\) −12.0000 −0.396491
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 15.0000 0.494535
\(921\) −19.0000 −0.626071
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 20.0000 0.657241
\(927\) −14.0000 −0.459820
\(928\) 6.00000 0.196960
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −15.0000 −0.491078
\(934\) −21.0000 −0.687141
\(935\) −6.00000 −0.196221
\(936\) 0 0
\(937\) −39.0000 −1.27407 −0.637037 0.770833i \(-0.719840\pi\)
−0.637037 + 0.770833i \(0.719840\pi\)
\(938\) 0 0
\(939\) 2.00000 0.0652675
\(940\) 18.0000 0.587095
\(941\) 27.0000 0.880175 0.440087 0.897955i \(-0.354947\pi\)
0.440087 + 0.897955i \(0.354947\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) 5.00000 0.162736
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) −30.0000 −0.974869 −0.487435 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) −16.0000 −0.519656
\(949\) 0 0
\(950\) −12.0000 −0.389331
\(951\) 1.00000 0.0324272
\(952\) 0 0
\(953\) −41.0000 −1.32812 −0.664060 0.747679i \(-0.731168\pi\)
−0.664060 + 0.747679i \(0.731168\pi\)
\(954\) −6.00000 −0.194257
\(955\) −36.0000 −1.16493
\(956\) −20.0000 −0.646846
\(957\) −12.0000 −0.387905
\(958\) 27.0000 0.872330
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −16.0000 −0.515593
\(964\) −8.00000 −0.257663
\(965\) 42.0000 1.35203
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −7.00000 −0.224989
\(969\) 3.00000 0.0963739
\(970\) 12.0000 0.385297
\(971\) −1.00000 −0.0320915 −0.0160458 0.999871i \(-0.505108\pi\)
−0.0160458 + 0.999871i \(0.505108\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 17.0000 0.544715
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) −20.0000 −0.639529
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) −27.0000 −0.861605
\(983\) 19.0000 0.606006 0.303003 0.952990i \(-0.402011\pi\)
0.303003 + 0.952990i \(0.402011\pi\)
\(984\) 0 0
\(985\) 63.0000 2.00735
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 0 0
\(989\) 5.00000 0.158991
\(990\) −6.00000 −0.190693
\(991\) −3.00000 −0.0952981 −0.0476491 0.998864i \(-0.515173\pi\)
−0.0476491 + 0.998864i \(0.515173\pi\)
\(992\) 0 0
\(993\) −9.00000 −0.285606
\(994\) 0 0
\(995\) −63.0000 −1.99723
\(996\) 12.0000 0.380235
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 16.0000 0.506471
\(999\) −3.00000 −0.0949158
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 4998.2.a.y.1.1 1
7.2 even 3 714.2.i.f.613.1 yes 2
7.4 even 3 714.2.i.f.205.1 2
7.6 odd 2 4998.2.a.br.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.i.f.205.1 2 7.4 even 3
714.2.i.f.613.1 yes 2 7.2 even 3
4998.2.a.y.1.1 1 1.1 even 1 trivial
4998.2.a.br.1.1 1 7.6 odd 2