Properties

Label 2898.2.a.ba.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.56155 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.56155 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.56155 q^{10} +3.12311 q^{11} +0.561553 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.12311 q^{17} +3.12311 q^{19} -2.56155 q^{20} +3.12311 q^{22} -1.00000 q^{23} +1.56155 q^{25} +0.561553 q^{26} -1.00000 q^{28} -3.43845 q^{29} -5.12311 q^{31} +1.00000 q^{32} -7.12311 q^{34} +2.56155 q^{35} +2.56155 q^{37} +3.12311 q^{38} -2.56155 q^{40} +0.561553 q^{41} -5.68466 q^{43} +3.12311 q^{44} -1.00000 q^{46} -6.56155 q^{47} +1.00000 q^{49} +1.56155 q^{50} +0.561553 q^{52} +2.24621 q^{53} -8.00000 q^{55} -1.00000 q^{56} -3.43845 q^{58} -13.1231 q^{59} +9.12311 q^{61} -5.12311 q^{62} +1.00000 q^{64} -1.43845 q^{65} +7.12311 q^{67} -7.12311 q^{68} +2.56155 q^{70} -15.3693 q^{71} +0.876894 q^{73} +2.56155 q^{74} +3.12311 q^{76} -3.12311 q^{77} -2.24621 q^{79} -2.56155 q^{80} +0.561553 q^{82} +0.876894 q^{83} +18.2462 q^{85} -5.68466 q^{86} +3.12311 q^{88} -14.0000 q^{89} -0.561553 q^{91} -1.00000 q^{92} -6.56155 q^{94} -8.00000 q^{95} -16.5616 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + 2q^{8} - q^{10} - 2q^{11} - 3q^{13} - 2q^{14} + 2q^{16} - 6q^{17} - 2q^{19} - q^{20} - 2q^{22} - 2q^{23} - q^{25} - 3q^{26} - 2q^{28} - 11q^{29} - 2q^{31} + 2q^{32} - 6q^{34} + q^{35} + q^{37} - 2q^{38} - q^{40} - 3q^{41} + q^{43} - 2q^{44} - 2q^{46} - 9q^{47} + 2q^{49} - q^{50} - 3q^{52} - 12q^{53} - 16q^{55} - 2q^{56} - 11q^{58} - 18q^{59} + 10q^{61} - 2q^{62} + 2q^{64} - 7q^{65} + 6q^{67} - 6q^{68} + q^{70} - 6q^{71} + 10q^{73} + q^{74} - 2q^{76} + 2q^{77} + 12q^{79} - q^{80} - 3q^{82} + 10q^{83} + 20q^{85} + q^{86} - 2q^{88} - 28q^{89} + 3q^{91} - 2q^{92} - 9q^{94} - 16q^{95} - 29q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.56155 −1.14556 −0.572781 0.819709i \(-0.694135\pi\)
−0.572781 + 0.819709i \(0.694135\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.56155 −0.810034
\(11\) 3.12311 0.941652 0.470826 0.882226i \(-0.343956\pi\)
0.470826 + 0.882226i \(0.343956\pi\)
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.12311 −1.72761 −0.863803 0.503829i \(-0.831924\pi\)
−0.863803 + 0.503829i \(0.831924\pi\)
\(18\) 0 0
\(19\) 3.12311 0.716490 0.358245 0.933628i \(-0.383375\pi\)
0.358245 + 0.933628i \(0.383375\pi\)
\(20\) −2.56155 −0.572781
\(21\) 0 0
\(22\) 3.12311 0.665848
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.56155 0.312311
\(26\) 0.561553 0.110130
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −3.43845 −0.638504 −0.319252 0.947670i \(-0.603432\pi\)
−0.319252 + 0.947670i \(0.603432\pi\)
\(30\) 0 0
\(31\) −5.12311 −0.920137 −0.460068 0.887883i \(-0.652175\pi\)
−0.460068 + 0.887883i \(0.652175\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −7.12311 −1.22160
\(35\) 2.56155 0.432981
\(36\) 0 0
\(37\) 2.56155 0.421117 0.210558 0.977581i \(-0.432472\pi\)
0.210558 + 0.977581i \(0.432472\pi\)
\(38\) 3.12311 0.506635
\(39\) 0 0
\(40\) −2.56155 −0.405017
\(41\) 0.561553 0.0876998 0.0438499 0.999038i \(-0.486038\pi\)
0.0438499 + 0.999038i \(0.486038\pi\)
\(42\) 0 0
\(43\) −5.68466 −0.866902 −0.433451 0.901177i \(-0.642704\pi\)
−0.433451 + 0.901177i \(0.642704\pi\)
\(44\) 3.12311 0.470826
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.56155 −0.957101 −0.478550 0.878060i \(-0.658838\pi\)
−0.478550 + 0.878060i \(0.658838\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.56155 0.220837
\(51\) 0 0
\(52\) 0.561553 0.0778734
\(53\) 2.24621 0.308541 0.154270 0.988029i \(-0.450697\pi\)
0.154270 + 0.988029i \(0.450697\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −3.43845 −0.451490
\(59\) −13.1231 −1.70848 −0.854241 0.519877i \(-0.825978\pi\)
−0.854241 + 0.519877i \(0.825978\pi\)
\(60\) 0 0
\(61\) 9.12311 1.16809 0.584047 0.811720i \(-0.301468\pi\)
0.584047 + 0.811720i \(0.301468\pi\)
\(62\) −5.12311 −0.650635
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.43845 −0.178417
\(66\) 0 0
\(67\) 7.12311 0.870226 0.435113 0.900376i \(-0.356708\pi\)
0.435113 + 0.900376i \(0.356708\pi\)
\(68\) −7.12311 −0.863803
\(69\) 0 0
\(70\) 2.56155 0.306164
\(71\) −15.3693 −1.82400 −0.912001 0.410188i \(-0.865463\pi\)
−0.912001 + 0.410188i \(0.865463\pi\)
\(72\) 0 0
\(73\) 0.876894 0.102633 0.0513164 0.998682i \(-0.483658\pi\)
0.0513164 + 0.998682i \(0.483658\pi\)
\(74\) 2.56155 0.297774
\(75\) 0 0
\(76\) 3.12311 0.358245
\(77\) −3.12311 −0.355911
\(78\) 0 0
\(79\) −2.24621 −0.252719 −0.126359 0.991985i \(-0.540329\pi\)
−0.126359 + 0.991985i \(0.540329\pi\)
\(80\) −2.56155 −0.286390
\(81\) 0 0
\(82\) 0.561553 0.0620131
\(83\) 0.876894 0.0962517 0.0481258 0.998841i \(-0.484675\pi\)
0.0481258 + 0.998841i \(0.484675\pi\)
\(84\) 0 0
\(85\) 18.2462 1.97908
\(86\) −5.68466 −0.612992
\(87\) 0 0
\(88\) 3.12311 0.332924
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −0.561553 −0.0588667
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −6.56155 −0.676772
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −16.5616 −1.68157 −0.840785 0.541368i \(-0.817906\pi\)
−0.840785 + 0.541368i \(0.817906\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.56155 0.156155
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) −9.93087 −0.978518 −0.489259 0.872139i \(-0.662733\pi\)
−0.489259 + 0.872139i \(0.662733\pi\)
\(104\) 0.561553 0.0550648
\(105\) 0 0
\(106\) 2.24621 0.218171
\(107\) −3.12311 −0.301922 −0.150961 0.988540i \(-0.548237\pi\)
−0.150961 + 0.988540i \(0.548237\pi\)
\(108\) 0 0
\(109\) 14.5616 1.39474 0.697372 0.716709i \(-0.254353\pi\)
0.697372 + 0.716709i \(0.254353\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −9.68466 −0.911056 −0.455528 0.890221i \(-0.650550\pi\)
−0.455528 + 0.890221i \(0.650550\pi\)
\(114\) 0 0
\(115\) 2.56155 0.238866
\(116\) −3.43845 −0.319252
\(117\) 0 0
\(118\) −13.1231 −1.20808
\(119\) 7.12311 0.652974
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 9.12311 0.825967
\(123\) 0 0
\(124\) −5.12311 −0.460068
\(125\) 8.80776 0.787790
\(126\) 0 0
\(127\) −4.31534 −0.382925 −0.191462 0.981500i \(-0.561323\pi\)
−0.191462 + 0.981500i \(0.561323\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −1.43845 −0.126160
\(131\) −13.1231 −1.14657 −0.573286 0.819356i \(-0.694331\pi\)
−0.573286 + 0.819356i \(0.694331\pi\)
\(132\) 0 0
\(133\) −3.12311 −0.270808
\(134\) 7.12311 0.615343
\(135\) 0 0
\(136\) −7.12311 −0.610801
\(137\) 4.56155 0.389720 0.194860 0.980831i \(-0.437575\pi\)
0.194860 + 0.980831i \(0.437575\pi\)
\(138\) 0 0
\(139\) −16.8078 −1.42562 −0.712808 0.701359i \(-0.752577\pi\)
−0.712808 + 0.701359i \(0.752577\pi\)
\(140\) 2.56155 0.216491
\(141\) 0 0
\(142\) −15.3693 −1.28976
\(143\) 1.75379 0.146659
\(144\) 0 0
\(145\) 8.80776 0.731445
\(146\) 0.876894 0.0725723
\(147\) 0 0
\(148\) 2.56155 0.210558
\(149\) −10.2462 −0.839402 −0.419701 0.907662i \(-0.637865\pi\)
−0.419701 + 0.907662i \(0.637865\pi\)
\(150\) 0 0
\(151\) 19.6847 1.60191 0.800957 0.598721i \(-0.204324\pi\)
0.800957 + 0.598721i \(0.204324\pi\)
\(152\) 3.12311 0.253317
\(153\) 0 0
\(154\) −3.12311 −0.251667
\(155\) 13.1231 1.05407
\(156\) 0 0
\(157\) −2.24621 −0.179267 −0.0896336 0.995975i \(-0.528570\pi\)
−0.0896336 + 0.995975i \(0.528570\pi\)
\(158\) −2.24621 −0.178699
\(159\) 0 0
\(160\) −2.56155 −0.202509
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0.561553 0.0438499
\(165\) 0 0
\(166\) 0.876894 0.0680602
\(167\) −2.24621 −0.173817 −0.0869085 0.996216i \(-0.527699\pi\)
−0.0869085 + 0.996216i \(0.527699\pi\)
\(168\) 0 0
\(169\) −12.6847 −0.975743
\(170\) 18.2462 1.39942
\(171\) 0 0
\(172\) −5.68466 −0.433451
\(173\) 14.4924 1.10184 0.550919 0.834559i \(-0.314277\pi\)
0.550919 + 0.834559i \(0.314277\pi\)
\(174\) 0 0
\(175\) −1.56155 −0.118042
\(176\) 3.12311 0.235413
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) −16.8078 −1.25627 −0.628136 0.778104i \(-0.716182\pi\)
−0.628136 + 0.778104i \(0.716182\pi\)
\(180\) 0 0
\(181\) −6.87689 −0.511156 −0.255578 0.966789i \(-0.582266\pi\)
−0.255578 + 0.966789i \(0.582266\pi\)
\(182\) −0.561553 −0.0416251
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −6.56155 −0.482415
\(186\) 0 0
\(187\) −22.2462 −1.62680
\(188\) −6.56155 −0.478550
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) −9.68466 −0.697117 −0.348558 0.937287i \(-0.613329\pi\)
−0.348558 + 0.937287i \(0.613329\pi\)
\(194\) −16.5616 −1.18905
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 3.93087 0.280063 0.140031 0.990147i \(-0.455280\pi\)
0.140031 + 0.990147i \(0.455280\pi\)
\(198\) 0 0
\(199\) 24.1771 1.71387 0.856934 0.515426i \(-0.172366\pi\)
0.856934 + 0.515426i \(0.172366\pi\)
\(200\) 1.56155 0.110418
\(201\) 0 0
\(202\) 0.246211 0.0173234
\(203\) 3.43845 0.241332
\(204\) 0 0
\(205\) −1.43845 −0.100466
\(206\) −9.93087 −0.691916
\(207\) 0 0
\(208\) 0.561553 0.0389367
\(209\) 9.75379 0.674684
\(210\) 0 0
\(211\) 16.4924 1.13539 0.567693 0.823241i \(-0.307836\pi\)
0.567693 + 0.823241i \(0.307836\pi\)
\(212\) 2.24621 0.154270
\(213\) 0 0
\(214\) −3.12311 −0.213491
\(215\) 14.5616 0.993090
\(216\) 0 0
\(217\) 5.12311 0.347779
\(218\) 14.5616 0.986233
\(219\) 0 0
\(220\) −8.00000 −0.539360
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −5.12311 −0.343069 −0.171534 0.985178i \(-0.554872\pi\)
−0.171534 + 0.985178i \(0.554872\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −9.68466 −0.644214
\(227\) 26.8078 1.77929 0.889647 0.456649i \(-0.150951\pi\)
0.889647 + 0.456649i \(0.150951\pi\)
\(228\) 0 0
\(229\) −14.2462 −0.941416 −0.470708 0.882289i \(-0.656001\pi\)
−0.470708 + 0.882289i \(0.656001\pi\)
\(230\) 2.56155 0.168904
\(231\) 0 0
\(232\) −3.43845 −0.225745
\(233\) −14.4924 −0.949430 −0.474715 0.880140i \(-0.657449\pi\)
−0.474715 + 0.880140i \(0.657449\pi\)
\(234\) 0 0
\(235\) 16.8078 1.09642
\(236\) −13.1231 −0.854241
\(237\) 0 0
\(238\) 7.12311 0.461722
\(239\) 28.4924 1.84302 0.921511 0.388353i \(-0.126956\pi\)
0.921511 + 0.388353i \(0.126956\pi\)
\(240\) 0 0
\(241\) 17.0540 1.09854 0.549272 0.835644i \(-0.314905\pi\)
0.549272 + 0.835644i \(0.314905\pi\)
\(242\) −1.24621 −0.0801095
\(243\) 0 0
\(244\) 9.12311 0.584047
\(245\) −2.56155 −0.163652
\(246\) 0 0
\(247\) 1.75379 0.111591
\(248\) −5.12311 −0.325318
\(249\) 0 0
\(250\) 8.80776 0.557052
\(251\) 4.56155 0.287923 0.143961 0.989583i \(-0.454016\pi\)
0.143961 + 0.989583i \(0.454016\pi\)
\(252\) 0 0
\(253\) −3.12311 −0.196348
\(254\) −4.31534 −0.270769
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −2.56155 −0.159167
\(260\) −1.43845 −0.0892087
\(261\) 0 0
\(262\) −13.1231 −0.810748
\(263\) 5.93087 0.365713 0.182857 0.983140i \(-0.441466\pi\)
0.182857 + 0.983140i \(0.441466\pi\)
\(264\) 0 0
\(265\) −5.75379 −0.353452
\(266\) −3.12311 −0.191490
\(267\) 0 0
\(268\) 7.12311 0.435113
\(269\) 4.24621 0.258896 0.129448 0.991586i \(-0.458679\pi\)
0.129448 + 0.991586i \(0.458679\pi\)
\(270\) 0 0
\(271\) −2.24621 −0.136448 −0.0682238 0.997670i \(-0.521733\pi\)
−0.0682238 + 0.997670i \(0.521733\pi\)
\(272\) −7.12311 −0.431902
\(273\) 0 0
\(274\) 4.56155 0.275573
\(275\) 4.87689 0.294088
\(276\) 0 0
\(277\) 4.87689 0.293024 0.146512 0.989209i \(-0.453195\pi\)
0.146512 + 0.989209i \(0.453195\pi\)
\(278\) −16.8078 −1.00806
\(279\) 0 0
\(280\) 2.56155 0.153082
\(281\) 23.9309 1.42760 0.713798 0.700352i \(-0.246973\pi\)
0.713798 + 0.700352i \(0.246973\pi\)
\(282\) 0 0
\(283\) 22.4924 1.33704 0.668518 0.743696i \(-0.266929\pi\)
0.668518 + 0.743696i \(0.266929\pi\)
\(284\) −15.3693 −0.912001
\(285\) 0 0
\(286\) 1.75379 0.103704
\(287\) −0.561553 −0.0331474
\(288\) 0 0
\(289\) 33.7386 1.98463
\(290\) 8.80776 0.517210
\(291\) 0 0
\(292\) 0.876894 0.0513164
\(293\) 13.6155 0.795428 0.397714 0.917510i \(-0.369804\pi\)
0.397714 + 0.917510i \(0.369804\pi\)
\(294\) 0 0
\(295\) 33.6155 1.95717
\(296\) 2.56155 0.148887
\(297\) 0 0
\(298\) −10.2462 −0.593547
\(299\) −0.561553 −0.0324754
\(300\) 0 0
\(301\) 5.68466 0.327658
\(302\) 19.6847 1.13272
\(303\) 0 0
\(304\) 3.12311 0.179122
\(305\) −23.3693 −1.33812
\(306\) 0 0
\(307\) −25.9309 −1.47995 −0.739976 0.672633i \(-0.765163\pi\)
−0.739976 + 0.672633i \(0.765163\pi\)
\(308\) −3.12311 −0.177955
\(309\) 0 0
\(310\) 13.1231 0.745342
\(311\) 30.7386 1.74303 0.871514 0.490371i \(-0.163139\pi\)
0.871514 + 0.490371i \(0.163139\pi\)
\(312\) 0 0
\(313\) −12.2462 −0.692197 −0.346098 0.938198i \(-0.612494\pi\)
−0.346098 + 0.938198i \(0.612494\pi\)
\(314\) −2.24621 −0.126761
\(315\) 0 0
\(316\) −2.24621 −0.126359
\(317\) −2.80776 −0.157700 −0.0788499 0.996887i \(-0.525125\pi\)
−0.0788499 + 0.996887i \(0.525125\pi\)
\(318\) 0 0
\(319\) −10.7386 −0.601248
\(320\) −2.56155 −0.143195
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) −22.2462 −1.23781
\(324\) 0 0
\(325\) 0.876894 0.0486413
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 0.561553 0.0310066
\(329\) 6.56155 0.361750
\(330\) 0 0
\(331\) 8.49242 0.466786 0.233393 0.972383i \(-0.425017\pi\)
0.233393 + 0.972383i \(0.425017\pi\)
\(332\) 0.876894 0.0481258
\(333\) 0 0
\(334\) −2.24621 −0.122907
\(335\) −18.2462 −0.996897
\(336\) 0 0
\(337\) 2.63068 0.143302 0.0716512 0.997430i \(-0.477173\pi\)
0.0716512 + 0.997430i \(0.477173\pi\)
\(338\) −12.6847 −0.689954
\(339\) 0 0
\(340\) 18.2462 0.989540
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −5.68466 −0.306496
\(345\) 0 0
\(346\) 14.4924 0.779117
\(347\) 13.9309 0.747848 0.373924 0.927459i \(-0.378012\pi\)
0.373924 + 0.927459i \(0.378012\pi\)
\(348\) 0 0
\(349\) 0.246211 0.0131794 0.00658969 0.999978i \(-0.497902\pi\)
0.00658969 + 0.999978i \(0.497902\pi\)
\(350\) −1.56155 −0.0834685
\(351\) 0 0
\(352\) 3.12311 0.166462
\(353\) 13.1922 0.702152 0.351076 0.936347i \(-0.385816\pi\)
0.351076 + 0.936347i \(0.385816\pi\)
\(354\) 0 0
\(355\) 39.3693 2.08951
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −16.8078 −0.888318
\(359\) 10.5616 0.557417 0.278709 0.960376i \(-0.410094\pi\)
0.278709 + 0.960376i \(0.410094\pi\)
\(360\) 0 0
\(361\) −9.24621 −0.486643
\(362\) −6.87689 −0.361442
\(363\) 0 0
\(364\) −0.561553 −0.0294334
\(365\) −2.24621 −0.117572
\(366\) 0 0
\(367\) −22.5616 −1.17770 −0.588852 0.808241i \(-0.700420\pi\)
−0.588852 + 0.808241i \(0.700420\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −6.56155 −0.341119
\(371\) −2.24621 −0.116617
\(372\) 0 0
\(373\) 6.87689 0.356072 0.178036 0.984024i \(-0.443026\pi\)
0.178036 + 0.984024i \(0.443026\pi\)
\(374\) −22.2462 −1.15032
\(375\) 0 0
\(376\) −6.56155 −0.338386
\(377\) −1.93087 −0.0994448
\(378\) 0 0
\(379\) −17.0540 −0.876004 −0.438002 0.898974i \(-0.644314\pi\)
−0.438002 + 0.898974i \(0.644314\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 20.0000 1.02329
\(383\) −10.2462 −0.523557 −0.261778 0.965128i \(-0.584309\pi\)
−0.261778 + 0.965128i \(0.584309\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −9.68466 −0.492936
\(387\) 0 0
\(388\) −16.5616 −0.840785
\(389\) 4.63068 0.234785 0.117392 0.993086i \(-0.462546\pi\)
0.117392 + 0.993086i \(0.462546\pi\)
\(390\) 0 0
\(391\) 7.12311 0.360231
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 3.93087 0.198034
\(395\) 5.75379 0.289505
\(396\) 0 0
\(397\) −24.2462 −1.21688 −0.608441 0.793599i \(-0.708205\pi\)
−0.608441 + 0.793599i \(0.708205\pi\)
\(398\) 24.1771 1.21189
\(399\) 0 0
\(400\) 1.56155 0.0780776
\(401\) 0.246211 0.0122952 0.00614760 0.999981i \(-0.498043\pi\)
0.00614760 + 0.999981i \(0.498043\pi\)
\(402\) 0 0
\(403\) −2.87689 −0.143308
\(404\) 0.246211 0.0122495
\(405\) 0 0
\(406\) 3.43845 0.170647
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −7.61553 −0.376564 −0.188282 0.982115i \(-0.560292\pi\)
−0.188282 + 0.982115i \(0.560292\pi\)
\(410\) −1.43845 −0.0710398
\(411\) 0 0
\(412\) −9.93087 −0.489259
\(413\) 13.1231 0.645746
\(414\) 0 0
\(415\) −2.24621 −0.110262
\(416\) 0.561553 0.0275324
\(417\) 0 0
\(418\) 9.75379 0.477073
\(419\) 33.8617 1.65425 0.827127 0.562015i \(-0.189974\pi\)
0.827127 + 0.562015i \(0.189974\pi\)
\(420\) 0 0
\(421\) −12.1771 −0.593475 −0.296737 0.954959i \(-0.595899\pi\)
−0.296737 + 0.954959i \(0.595899\pi\)
\(422\) 16.4924 0.802839
\(423\) 0 0
\(424\) 2.24621 0.109086
\(425\) −11.1231 −0.539550
\(426\) 0 0
\(427\) −9.12311 −0.441498
\(428\) −3.12311 −0.150961
\(429\) 0 0
\(430\) 14.5616 0.702220
\(431\) 15.6847 0.755503 0.377752 0.925907i \(-0.376697\pi\)
0.377752 + 0.925907i \(0.376697\pi\)
\(432\) 0 0
\(433\) 2.31534 0.111268 0.0556341 0.998451i \(-0.482282\pi\)
0.0556341 + 0.998451i \(0.482282\pi\)
\(434\) 5.12311 0.245917
\(435\) 0 0
\(436\) 14.5616 0.697372
\(437\) −3.12311 −0.149398
\(438\) 0 0
\(439\) 17.6155 0.840743 0.420372 0.907352i \(-0.361900\pi\)
0.420372 + 0.907352i \(0.361900\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −37.9309 −1.80215 −0.901075 0.433663i \(-0.857221\pi\)
−0.901075 + 0.433663i \(0.857221\pi\)
\(444\) 0 0
\(445\) 35.8617 1.70001
\(446\) −5.12311 −0.242586
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −16.2462 −0.766706 −0.383353 0.923602i \(-0.625231\pi\)
−0.383353 + 0.923602i \(0.625231\pi\)
\(450\) 0 0
\(451\) 1.75379 0.0825827
\(452\) −9.68466 −0.455528
\(453\) 0 0
\(454\) 26.8078 1.25815
\(455\) 1.43845 0.0674354
\(456\) 0 0
\(457\) −17.3693 −0.812502 −0.406251 0.913761i \(-0.633164\pi\)
−0.406251 + 0.913761i \(0.633164\pi\)
\(458\) −14.2462 −0.665682
\(459\) 0 0
\(460\) 2.56155 0.119433
\(461\) 8.73863 0.406999 0.203499 0.979075i \(-0.434769\pi\)
0.203499 + 0.979075i \(0.434769\pi\)
\(462\) 0 0
\(463\) −9.43845 −0.438642 −0.219321 0.975653i \(-0.570384\pi\)
−0.219321 + 0.975653i \(0.570384\pi\)
\(464\) −3.43845 −0.159626
\(465\) 0 0
\(466\) −14.4924 −0.671349
\(467\) 9.68466 0.448153 0.224076 0.974572i \(-0.428064\pi\)
0.224076 + 0.974572i \(0.428064\pi\)
\(468\) 0 0
\(469\) −7.12311 −0.328914
\(470\) 16.8078 0.775284
\(471\) 0 0
\(472\) −13.1231 −0.604040
\(473\) −17.7538 −0.816320
\(474\) 0 0
\(475\) 4.87689 0.223767
\(476\) 7.12311 0.326487
\(477\) 0 0
\(478\) 28.4924 1.30321
\(479\) 9.61553 0.439345 0.219672 0.975574i \(-0.429501\pi\)
0.219672 + 0.975574i \(0.429501\pi\)
\(480\) 0 0
\(481\) 1.43845 0.0655875
\(482\) 17.0540 0.776787
\(483\) 0 0
\(484\) −1.24621 −0.0566460
\(485\) 42.4233 1.92634
\(486\) 0 0
\(487\) −1.43845 −0.0651823 −0.0325911 0.999469i \(-0.510376\pi\)
−0.0325911 + 0.999469i \(0.510376\pi\)
\(488\) 9.12311 0.412984
\(489\) 0 0
\(490\) −2.56155 −0.115719
\(491\) −22.7386 −1.02618 −0.513090 0.858335i \(-0.671499\pi\)
−0.513090 + 0.858335i \(0.671499\pi\)
\(492\) 0 0
\(493\) 24.4924 1.10308
\(494\) 1.75379 0.0789067
\(495\) 0 0
\(496\) −5.12311 −0.230034
\(497\) 15.3693 0.689408
\(498\) 0 0
\(499\) 16.6307 0.744492 0.372246 0.928134i \(-0.378588\pi\)
0.372246 + 0.928134i \(0.378588\pi\)
\(500\) 8.80776 0.393895
\(501\) 0 0
\(502\) 4.56155 0.203592
\(503\) −42.1080 −1.87750 −0.938750 0.344598i \(-0.888015\pi\)
−0.938750 + 0.344598i \(0.888015\pi\)
\(504\) 0 0
\(505\) −0.630683 −0.0280650
\(506\) −3.12311 −0.138839
\(507\) 0 0
\(508\) −4.31534 −0.191462
\(509\) −21.3693 −0.947178 −0.473589 0.880746i \(-0.657042\pi\)
−0.473589 + 0.880746i \(0.657042\pi\)
\(510\) 0 0
\(511\) −0.876894 −0.0387915
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 25.4384 1.12095
\(516\) 0 0
\(517\) −20.4924 −0.901256
\(518\) −2.56155 −0.112548
\(519\) 0 0
\(520\) −1.43845 −0.0630801
\(521\) 16.2462 0.711759 0.355880 0.934532i \(-0.384181\pi\)
0.355880 + 0.934532i \(0.384181\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) −13.1231 −0.573286
\(525\) 0 0
\(526\) 5.93087 0.258598
\(527\) 36.4924 1.58963
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −5.75379 −0.249929
\(531\) 0 0
\(532\) −3.12311 −0.135404
\(533\) 0.315342 0.0136590
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 7.12311 0.307671
\(537\) 0 0
\(538\) 4.24621 0.183067
\(539\) 3.12311 0.134522
\(540\) 0 0
\(541\) 9.36932 0.402818 0.201409 0.979507i \(-0.435448\pi\)
0.201409 + 0.979507i \(0.435448\pi\)
\(542\) −2.24621 −0.0964830
\(543\) 0 0
\(544\) −7.12311 −0.305401
\(545\) −37.3002 −1.59776
\(546\) 0 0
\(547\) 37.6155 1.60832 0.804162 0.594410i \(-0.202614\pi\)
0.804162 + 0.594410i \(0.202614\pi\)
\(548\) 4.56155 0.194860
\(549\) 0 0
\(550\) 4.87689 0.207951
\(551\) −10.7386 −0.457481
\(552\) 0 0
\(553\) 2.24621 0.0955186
\(554\) 4.87689 0.207199
\(555\) 0 0
\(556\) −16.8078 −0.712808
\(557\) −36.9848 −1.56710 −0.783549 0.621330i \(-0.786593\pi\)
−0.783549 + 0.621330i \(0.786593\pi\)
\(558\) 0 0
\(559\) −3.19224 −0.135017
\(560\) 2.56155 0.108245
\(561\) 0 0
\(562\) 23.9309 1.00946
\(563\) −15.4384 −0.650653 −0.325326 0.945602i \(-0.605474\pi\)
−0.325326 + 0.945602i \(0.605474\pi\)
\(564\) 0 0
\(565\) 24.8078 1.04367
\(566\) 22.4924 0.945427
\(567\) 0 0
\(568\) −15.3693 −0.644882
\(569\) −25.5464 −1.07096 −0.535480 0.844548i \(-0.679869\pi\)
−0.535480 + 0.844548i \(0.679869\pi\)
\(570\) 0 0
\(571\) 18.6307 0.779670 0.389835 0.920885i \(-0.372532\pi\)
0.389835 + 0.920885i \(0.372532\pi\)
\(572\) 1.75379 0.0733296
\(573\) 0 0
\(574\) −0.561553 −0.0234388
\(575\) −1.56155 −0.0651213
\(576\) 0 0
\(577\) −36.2462 −1.50895 −0.754475 0.656329i \(-0.772108\pi\)
−0.754475 + 0.656329i \(0.772108\pi\)
\(578\) 33.7386 1.40334
\(579\) 0 0
\(580\) 8.80776 0.365722
\(581\) −0.876894 −0.0363797
\(582\) 0 0
\(583\) 7.01515 0.290538
\(584\) 0.876894 0.0362861
\(585\) 0 0
\(586\) 13.6155 0.562452
\(587\) −40.9848 −1.69163 −0.845813 0.533480i \(-0.820884\pi\)
−0.845813 + 0.533480i \(0.820884\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 33.6155 1.38393
\(591\) 0 0
\(592\) 2.56155 0.105279
\(593\) 23.3002 0.956824 0.478412 0.878136i \(-0.341213\pi\)
0.478412 + 0.878136i \(0.341213\pi\)
\(594\) 0 0
\(595\) −18.2462 −0.748022
\(596\) −10.2462 −0.419701
\(597\) 0 0
\(598\) −0.561553 −0.0229636
\(599\) −38.7386 −1.58282 −0.791409 0.611287i \(-0.790652\pi\)
−0.791409 + 0.611287i \(0.790652\pi\)
\(600\) 0 0
\(601\) 21.3693 0.871673 0.435836 0.900026i \(-0.356453\pi\)
0.435836 + 0.900026i \(0.356453\pi\)
\(602\) 5.68466 0.231689
\(603\) 0 0
\(604\) 19.6847 0.800957
\(605\) 3.19224 0.129783
\(606\) 0 0
\(607\) −6.38447 −0.259138 −0.129569 0.991570i \(-0.541359\pi\)
−0.129569 + 0.991570i \(0.541359\pi\)
\(608\) 3.12311 0.126659
\(609\) 0 0
\(610\) −23.3693 −0.946196
\(611\) −3.68466 −0.149065
\(612\) 0 0
\(613\) −16.8078 −0.678859 −0.339430 0.940631i \(-0.610234\pi\)
−0.339430 + 0.940631i \(0.610234\pi\)
\(614\) −25.9309 −1.04648
\(615\) 0 0
\(616\) −3.12311 −0.125834
\(617\) −48.2462 −1.94232 −0.971160 0.238430i \(-0.923367\pi\)
−0.971160 + 0.238430i \(0.923367\pi\)
\(618\) 0 0
\(619\) −8.24621 −0.331443 −0.165722 0.986173i \(-0.552995\pi\)
−0.165722 + 0.986173i \(0.552995\pi\)
\(620\) 13.1231 0.527037
\(621\) 0 0
\(622\) 30.7386 1.23251
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −30.3693 −1.21477
\(626\) −12.2462 −0.489457
\(627\) 0 0
\(628\) −2.24621 −0.0896336
\(629\) −18.2462 −0.727524
\(630\) 0 0
\(631\) −24.4924 −0.975028 −0.487514 0.873115i \(-0.662096\pi\)
−0.487514 + 0.873115i \(0.662096\pi\)
\(632\) −2.24621 −0.0893495
\(633\) 0 0
\(634\) −2.80776 −0.111511
\(635\) 11.0540 0.438664
\(636\) 0 0
\(637\) 0.561553 0.0222495
\(638\) −10.7386 −0.425147
\(639\) 0 0
\(640\) −2.56155 −0.101254
\(641\) 25.6847 1.01448 0.507242 0.861804i \(-0.330665\pi\)
0.507242 + 0.861804i \(0.330665\pi\)
\(642\) 0 0
\(643\) −38.0000 −1.49857 −0.749287 0.662246i \(-0.769604\pi\)
−0.749287 + 0.662246i \(0.769604\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) −22.2462 −0.875265
\(647\) 36.4924 1.43467 0.717333 0.696731i \(-0.245363\pi\)
0.717333 + 0.696731i \(0.245363\pi\)
\(648\) 0 0
\(649\) −40.9848 −1.60880
\(650\) 0.876894 0.0343946
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 33.5464 1.31277 0.656386 0.754425i \(-0.272084\pi\)
0.656386 + 0.754425i \(0.272084\pi\)
\(654\) 0 0
\(655\) 33.6155 1.31347
\(656\) 0.561553 0.0219250
\(657\) 0 0
\(658\) 6.56155 0.255796
\(659\) 35.6155 1.38738 0.693692 0.720272i \(-0.255983\pi\)
0.693692 + 0.720272i \(0.255983\pi\)
\(660\) 0 0
\(661\) 41.1231 1.59950 0.799752 0.600331i \(-0.204964\pi\)
0.799752 + 0.600331i \(0.204964\pi\)
\(662\) 8.49242 0.330067
\(663\) 0 0
\(664\) 0.876894 0.0340301
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 3.43845 0.133137
\(668\) −2.24621 −0.0869085
\(669\) 0 0
\(670\) −18.2462 −0.704913
\(671\) 28.4924 1.09994
\(672\) 0 0
\(673\) 10.3153 0.397627 0.198814 0.980037i \(-0.436291\pi\)
0.198814 + 0.980037i \(0.436291\pi\)
\(674\) 2.63068 0.101330
\(675\) 0 0
\(676\) −12.6847 −0.487871
\(677\) −27.3693 −1.05189 −0.525944 0.850519i \(-0.676288\pi\)
−0.525944 + 0.850519i \(0.676288\pi\)
\(678\) 0 0
\(679\) 16.5616 0.635574
\(680\) 18.2462 0.699710
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) 40.4924 1.54940 0.774700 0.632329i \(-0.217901\pi\)
0.774700 + 0.632329i \(0.217901\pi\)
\(684\) 0 0
\(685\) −11.6847 −0.446448
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −5.68466 −0.216726
\(689\) 1.26137 0.0480542
\(690\) 0 0
\(691\) −12.9460 −0.492490 −0.246245 0.969208i \(-0.579197\pi\)
−0.246245 + 0.969208i \(0.579197\pi\)
\(692\) 14.4924 0.550919
\(693\) 0 0
\(694\) 13.9309 0.528809
\(695\) 43.0540 1.63313
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0.246211 0.00931923
\(699\) 0 0
\(700\) −1.56155 −0.0590211
\(701\) −17.7538 −0.670551 −0.335276 0.942120i \(-0.608829\pi\)
−0.335276 + 0.942120i \(0.608829\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 3.12311 0.117706
\(705\) 0 0
\(706\) 13.1922 0.496496
\(707\) −0.246211 −0.00925973
\(708\) 0 0
\(709\) 35.3693 1.32832 0.664161 0.747589i \(-0.268789\pi\)
0.664161 + 0.747589i \(0.268789\pi\)
\(710\) 39.3693 1.47750
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) 5.12311 0.191862
\(714\) 0 0
\(715\) −4.49242 −0.168007
\(716\) −16.8078 −0.628136
\(717\) 0 0
\(718\) 10.5616 0.394154
\(719\) −14.5616 −0.543054 −0.271527 0.962431i \(-0.587529\pi\)
−0.271527 + 0.962431i \(0.587529\pi\)
\(720\) 0 0
\(721\) 9.93087 0.369845
\(722\) −9.24621 −0.344108
\(723\) 0 0
\(724\) −6.87689 −0.255578
\(725\) −5.36932 −0.199411
\(726\) 0 0
\(727\) −18.2462 −0.676715 −0.338357 0.941018i \(-0.609871\pi\)
−0.338357 + 0.941018i \(0.609871\pi\)
\(728\) −0.561553 −0.0208125
\(729\) 0 0
\(730\) −2.24621 −0.0831360
\(731\) 40.4924 1.49767
\(732\) 0 0
\(733\) 42.2462 1.56040 0.780200 0.625531i \(-0.215117\pi\)
0.780200 + 0.625531i \(0.215117\pi\)
\(734\) −22.5616 −0.832762
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 22.2462 0.819450
\(738\) 0 0
\(739\) −17.6155 −0.647998 −0.323999 0.946057i \(-0.605027\pi\)
−0.323999 + 0.946057i \(0.605027\pi\)
\(740\) −6.56155 −0.241207
\(741\) 0 0
\(742\) −2.24621 −0.0824610
\(743\) −40.9848 −1.50359 −0.751794 0.659398i \(-0.770811\pi\)
−0.751794 + 0.659398i \(0.770811\pi\)
\(744\) 0 0
\(745\) 26.2462 0.961587
\(746\) 6.87689 0.251781
\(747\) 0 0
\(748\) −22.2462 −0.813402
\(749\) 3.12311 0.114116
\(750\) 0 0
\(751\) −8.49242 −0.309893 −0.154946 0.987923i \(-0.549521\pi\)
−0.154946 + 0.987923i \(0.549521\pi\)
\(752\) −6.56155 −0.239275
\(753\) 0 0
\(754\) −1.93087 −0.0703181
\(755\) −50.4233 −1.83509
\(756\) 0 0
\(757\) −0.630683 −0.0229226 −0.0114613 0.999934i \(-0.503648\pi\)
−0.0114613 + 0.999934i \(0.503648\pi\)
\(758\) −17.0540 −0.619428
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 52.7386 1.91177 0.955887 0.293735i \(-0.0948982\pi\)
0.955887 + 0.293735i \(0.0948982\pi\)
\(762\) 0 0
\(763\) −14.5616 −0.527164
\(764\) 20.0000 0.723575
\(765\) 0 0
\(766\) −10.2462 −0.370211
\(767\) −7.36932 −0.266091
\(768\) 0 0
\(769\) −1.82292 −0.0657361 −0.0328681 0.999460i \(-0.510464\pi\)
−0.0328681 + 0.999460i \(0.510464\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) −9.68466 −0.348558
\(773\) −21.3002 −0.766114 −0.383057 0.923725i \(-0.625129\pi\)
−0.383057 + 0.923725i \(0.625129\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) −16.5616 −0.594525
\(777\) 0 0
\(778\) 4.63068 0.166018
\(779\) 1.75379 0.0628360
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 7.12311 0.254722
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 5.75379 0.205362
\(786\) 0 0
\(787\) −19.1231 −0.681665 −0.340833 0.940124i \(-0.610709\pi\)
−0.340833 + 0.940124i \(0.610709\pi\)
\(788\) 3.93087 0.140031
\(789\) 0 0
\(790\) 5.75379 0.204711
\(791\) 9.68466 0.344347
\(792\) 0 0
\(793\) 5.12311 0.181927
\(794\) −24.2462 −0.860466
\(795\) 0 0
\(796\) 24.1771 0.856934
\(797\) 34.4233 1.21934 0.609668 0.792657i \(-0.291303\pi\)
0.609668 + 0.792657i \(0.291303\pi\)
\(798\) 0 0
\(799\) 46.7386 1.65349
\(800\) 1.56155 0.0552092
\(801\) 0 0
\(802\) 0.246211 0.00869402
\(803\) 2.73863 0.0966443
\(804\) 0 0
\(805\) −2.56155 −0.0902829
\(806\) −2.87689 −0.101334
\(807\) 0 0
\(808\) 0.246211 0.00866168
\(809\) 33.8617 1.19052 0.595258 0.803535i \(-0.297050\pi\)
0.595258 + 0.803535i \(0.297050\pi\)
\(810\) 0 0
\(811\) −12.3153 −0.432450 −0.216225 0.976344i \(-0.569374\pi\)
−0.216225 + 0.976344i \(0.569374\pi\)
\(812\) 3.43845 0.120666
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 20.4924 0.717818
\(816\) 0 0
\(817\) −17.7538 −0.621126
\(818\) −7.61553 −0.266271
\(819\) 0 0
\(820\) −1.43845 −0.0502328
\(821\) −43.4773 −1.51737 −0.758684 0.651459i \(-0.774157\pi\)
−0.758684 + 0.651459i \(0.774157\pi\)
\(822\) 0 0
\(823\) 35.6847 1.24389 0.621944 0.783061i \(-0.286343\pi\)
0.621944 + 0.783061i \(0.286343\pi\)
\(824\) −9.93087 −0.345958
\(825\) 0 0
\(826\) 13.1231 0.456611
\(827\) 0.738634 0.0256848 0.0128424 0.999918i \(-0.495912\pi\)
0.0128424 + 0.999918i \(0.495912\pi\)
\(828\) 0 0
\(829\) −24.7386 −0.859208 −0.429604 0.903017i \(-0.641347\pi\)
−0.429604 + 0.903017i \(0.641347\pi\)
\(830\) −2.24621 −0.0779671
\(831\) 0 0
\(832\) 0.561553 0.0194683
\(833\) −7.12311 −0.246801
\(834\) 0 0
\(835\) 5.75379 0.199118
\(836\) 9.75379 0.337342
\(837\) 0 0
\(838\) 33.8617 1.16973
\(839\) 33.6155 1.16054 0.580268 0.814425i \(-0.302948\pi\)
0.580268 + 0.814425i \(0.302948\pi\)
\(840\) 0 0
\(841\) −17.1771 −0.592313
\(842\) −12.1771 −0.419650
\(843\) 0 0
\(844\) 16.4924 0.567693
\(845\) 32.4924 1.11777
\(846\) 0 0
\(847\) 1.24621 0.0428203
\(848\) 2.24621 0.0771352
\(849\) 0 0
\(850\) −11.1231 −0.381519
\(851\) −2.56155 −0.0878089
\(852\) 0 0
\(853\) −4.56155 −0.156185 −0.0780923 0.996946i \(-0.524883\pi\)
−0.0780923 + 0.996946i \(0.524883\pi\)
\(854\) −9.12311 −0.312186
\(855\) 0 0
\(856\) −3.12311 −0.106746
\(857\) −13.6847 −0.467459 −0.233730 0.972302i \(-0.575093\pi\)
−0.233730 + 0.972302i \(0.575093\pi\)
\(858\) 0 0
\(859\) 12.1771 0.415477 0.207738 0.978184i \(-0.433390\pi\)
0.207738 + 0.978184i \(0.433390\pi\)
\(860\) 14.5616 0.496545
\(861\) 0 0
\(862\) 15.6847 0.534222
\(863\) 38.7386 1.31868 0.659339 0.751846i \(-0.270836\pi\)
0.659339 + 0.751846i \(0.270836\pi\)
\(864\) 0 0
\(865\) −37.1231 −1.26222
\(866\) 2.31534 0.0786785
\(867\) 0 0
\(868\) 5.12311 0.173890
\(869\) −7.01515 −0.237973
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 14.5616 0.493116
\(873\) 0 0
\(874\) −3.12311 −0.105641
\(875\) −8.80776 −0.297757
\(876\) 0 0
\(877\) 21.8617 0.738218 0.369109 0.929386i \(-0.379663\pi\)
0.369109 + 0.929386i \(0.379663\pi\)
\(878\) 17.6155 0.594495
\(879\) 0 0
\(880\) −8.00000 −0.269680
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −17.1231 −0.576238 −0.288119 0.957595i \(-0.593030\pi\)
−0.288119 + 0.957595i \(0.593030\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −37.9309 −1.27431
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 4.31534 0.144732
\(890\) 35.8617 1.20209
\(891\) 0 0
\(892\) −5.12311 −0.171534
\(893\) −20.4924 −0.685753
\(894\) 0 0
\(895\) 43.0540 1.43914
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −16.2462 −0.542143
\(899\) 17.6155 0.587511
\(900\) 0 0
\(901\) −16.0000 −0.533037
\(902\) 1.75379 0.0583948
\(903\) 0 0
\(904\) −9.68466 −0.322107
\(905\) 17.6155 0.585560
\(906\) 0 0
\(907\) −17.5464 −0.582619 −0.291309 0.956629i \(-0.594091\pi\)
−0.291309 + 0.956629i \(0.594091\pi\)
\(908\) 26.8078 0.889647
\(909\) 0 0
\(910\) 1.43845 0.0476841
\(911\) 39.0540 1.29392 0.646958 0.762526i \(-0.276041\pi\)
0.646958 + 0.762526i \(0.276041\pi\)
\(912\) 0 0
\(913\) 2.73863 0.0906355
\(914\) −17.3693 −0.574526
\(915\) 0 0
\(916\) −14.2462 −0.470708
\(917\) 13.1231 0.433363
\(918\) 0 0
\(919\) −1.26137 −0.0416086 −0.0208043 0.999784i \(-0.506623\pi\)
−0.0208043 + 0.999784i \(0.506623\pi\)
\(920\) 2.56155 0.0844519
\(921\) 0 0
\(922\) 8.73863 0.287792
\(923\) −8.63068 −0.284082
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) −9.43845 −0.310167
\(927\) 0 0
\(928\) −3.43845 −0.112873
\(929\) 40.5616 1.33078 0.665391 0.746495i \(-0.268265\pi\)
0.665391 + 0.746495i \(0.268265\pi\)
\(930\) 0 0
\(931\) 3.12311 0.102356
\(932\) −14.4924 −0.474715
\(933\) 0 0
\(934\) 9.68466 0.316892
\(935\) 56.9848 1.86360
\(936\) 0 0
\(937\) 18.1771 0.593819 0.296910 0.954906i \(-0.404044\pi\)
0.296910 + 0.954906i \(0.404044\pi\)
\(938\) −7.12311 −0.232578
\(939\) 0 0
\(940\) 16.8078 0.548209
\(941\) −39.0540 −1.27312 −0.636562 0.771226i \(-0.719644\pi\)
−0.636562 + 0.771226i \(0.719644\pi\)
\(942\) 0 0
\(943\) −0.561553 −0.0182867
\(944\) −13.1231 −0.427121
\(945\) 0 0
\(946\) −17.7538 −0.577225
\(947\) 26.0691 0.847133 0.423566 0.905865i \(-0.360778\pi\)
0.423566 + 0.905865i \(0.360778\pi\)
\(948\) 0 0
\(949\) 0.492423 0.0159847
\(950\) 4.87689 0.158227
\(951\) 0 0
\(952\) 7.12311 0.230861
\(953\) −50.0000 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) 0 0
\(955\) −51.2311 −1.65780
\(956\) 28.4924 0.921511
\(957\) 0 0
\(958\) 9.61553 0.310664
\(959\) −4.56155 −0.147300
\(960\) 0 0
\(961\) −4.75379 −0.153348
\(962\) 1.43845 0.0463774
\(963\) 0 0
\(964\) 17.0540 0.549272
\(965\) 24.8078 0.798590
\(966\) 0 0
\(967\) −26.2462 −0.844021 −0.422011 0.906591i \(-0.638676\pi\)
−0.422011 + 0.906591i \(0.638676\pi\)
\(968\) −1.24621 −0.0400547
\(969\) 0 0
\(970\) 42.4233 1.36213
\(971\) −39.6155 −1.27132 −0.635661 0.771968i \(-0.719273\pi\)
−0.635661 + 0.771968i \(0.719273\pi\)
\(972\) 0 0
\(973\) 16.8078 0.538832
\(974\) −1.43845 −0.0460908
\(975\) 0 0
\(976\) 9.12311 0.292023
\(977\) 46.8078 1.49751 0.748757 0.662845i \(-0.230651\pi\)
0.748757 + 0.662845i \(0.230651\pi\)
\(978\) 0 0
\(979\) −43.7235 −1.39741
\(980\) −2.56155 −0.0818258
\(981\) 0 0
\(982\) −22.7386 −0.725619
\(983\) 23.8617 0.761071 0.380536 0.924766i \(-0.375740\pi\)
0.380536 + 0.924766i \(0.375740\pi\)
\(984\) 0 0
\(985\) −10.0691 −0.320829
\(986\) 24.4924 0.779998
\(987\) 0 0
\(988\) 1.75379 0.0557955
\(989\) 5.68466 0.180762
\(990\) 0 0
\(991\) 10.2462 0.325482 0.162741 0.986669i \(-0.447967\pi\)
0.162741 + 0.986669i \(0.447967\pi\)
\(992\) −5.12311 −0.162659
\(993\) 0 0
\(994\) 15.3693 0.487485
\(995\) −61.9309 −1.96334
\(996\) 0 0
\(997\) −47.4773 −1.50362 −0.751810 0.659380i \(-0.770819\pi\)
−0.751810 + 0.659380i \(0.770819\pi\)
\(998\) 16.6307 0.526435
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2898.2.a.ba.1.1 2
3.2 odd 2 966.2.a.l.1.2 2
12.11 even 2 7728.2.a.bm.1.2 2
21.20 even 2 6762.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.l.1.2 2 3.2 odd 2
2898.2.a.ba.1.1 2 1.1 even 1 trivial
6762.2.a.bw.1.1 2 21.20 even 2
7728.2.a.bm.1.2 2 12.11 even 2