Properties

Label 2151.2.a.k.1.3
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.02621\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.02621 q^{2} +2.10552 q^{4} +3.48073 q^{5} +1.37004 q^{7} -0.213806 q^{8} +O(q^{10})\) \(q-2.02621 q^{2} +2.10552 q^{4} +3.48073 q^{5} +1.37004 q^{7} -0.213806 q^{8} -7.05267 q^{10} +0.191564 q^{11} +4.24334 q^{13} -2.77599 q^{14} -3.77782 q^{16} +4.07863 q^{17} -5.16823 q^{19} +7.32874 q^{20} -0.388149 q^{22} +0.978985 q^{23} +7.11545 q^{25} -8.59789 q^{26} +2.88465 q^{28} +10.0111 q^{29} -4.04813 q^{31} +8.08227 q^{32} -8.26416 q^{34} +4.76874 q^{35} +2.15163 q^{37} +10.4719 q^{38} -0.744200 q^{40} +0.0317692 q^{41} +2.77809 q^{43} +0.403342 q^{44} -1.98363 q^{46} +0.680962 q^{47} -5.12298 q^{49} -14.4174 q^{50} +8.93444 q^{52} +4.91306 q^{53} +0.666782 q^{55} -0.292924 q^{56} -20.2846 q^{58} +3.94941 q^{59} +3.63175 q^{61} +8.20235 q^{62} -8.82072 q^{64} +14.7699 q^{65} +5.37197 q^{67} +8.58765 q^{68} -9.66247 q^{70} -8.95527 q^{71} -7.76915 q^{73} -4.35966 q^{74} -10.8818 q^{76} +0.262451 q^{77} -12.9221 q^{79} -13.1496 q^{80} -0.0643711 q^{82} +2.22931 q^{83} +14.1966 q^{85} -5.62898 q^{86} -0.0409576 q^{88} -9.94855 q^{89} +5.81356 q^{91} +2.06127 q^{92} -1.37977 q^{94} -17.9892 q^{95} -4.84284 q^{97} +10.3802 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.02621 −1.43275 −0.716373 0.697718i \(-0.754199\pi\)
−0.716373 + 0.697718i \(0.754199\pi\)
\(3\) 0 0
\(4\) 2.10552 1.05276
\(5\) 3.48073 1.55663 0.778314 0.627875i \(-0.216075\pi\)
0.778314 + 0.627875i \(0.216075\pi\)
\(6\) 0 0
\(7\) 1.37004 0.517828 0.258914 0.965900i \(-0.416635\pi\)
0.258914 + 0.965900i \(0.416635\pi\)
\(8\) −0.213806 −0.0755918
\(9\) 0 0
\(10\) −7.05267 −2.23025
\(11\) 0.191564 0.0577588 0.0288794 0.999583i \(-0.490806\pi\)
0.0288794 + 0.999583i \(0.490806\pi\)
\(12\) 0 0
\(13\) 4.24334 1.17689 0.588445 0.808537i \(-0.299740\pi\)
0.588445 + 0.808537i \(0.299740\pi\)
\(14\) −2.77599 −0.741915
\(15\) 0 0
\(16\) −3.77782 −0.944456
\(17\) 4.07863 0.989214 0.494607 0.869117i \(-0.335312\pi\)
0.494607 + 0.869117i \(0.335312\pi\)
\(18\) 0 0
\(19\) −5.16823 −1.18567 −0.592837 0.805323i \(-0.701992\pi\)
−0.592837 + 0.805323i \(0.701992\pi\)
\(20\) 7.32874 1.63876
\(21\) 0 0
\(22\) −0.388149 −0.0827536
\(23\) 0.978985 0.204132 0.102066 0.994778i \(-0.467455\pi\)
0.102066 + 0.994778i \(0.467455\pi\)
\(24\) 0 0
\(25\) 7.11545 1.42309
\(26\) −8.59789 −1.68619
\(27\) 0 0
\(28\) 2.88465 0.545148
\(29\) 10.0111 1.85901 0.929507 0.368806i \(-0.120233\pi\)
0.929507 + 0.368806i \(0.120233\pi\)
\(30\) 0 0
\(31\) −4.04813 −0.727065 −0.363533 0.931582i \(-0.618429\pi\)
−0.363533 + 0.931582i \(0.618429\pi\)
\(32\) 8.08227 1.42876
\(33\) 0 0
\(34\) −8.26416 −1.41729
\(35\) 4.76874 0.806065
\(36\) 0 0
\(37\) 2.15163 0.353726 0.176863 0.984235i \(-0.443405\pi\)
0.176863 + 0.984235i \(0.443405\pi\)
\(38\) 10.4719 1.69877
\(39\) 0 0
\(40\) −0.744200 −0.117668
\(41\) 0.0317692 0.00496152 0.00248076 0.999997i \(-0.499210\pi\)
0.00248076 + 0.999997i \(0.499210\pi\)
\(42\) 0 0
\(43\) 2.77809 0.423654 0.211827 0.977307i \(-0.432059\pi\)
0.211827 + 0.977307i \(0.432059\pi\)
\(44\) 0.403342 0.0608061
\(45\) 0 0
\(46\) −1.98363 −0.292470
\(47\) 0.680962 0.0993286 0.0496643 0.998766i \(-0.484185\pi\)
0.0496643 + 0.998766i \(0.484185\pi\)
\(48\) 0 0
\(49\) −5.12298 −0.731854
\(50\) −14.4174 −2.03893
\(51\) 0 0
\(52\) 8.93444 1.23898
\(53\) 4.91306 0.674861 0.337431 0.941350i \(-0.390442\pi\)
0.337431 + 0.941350i \(0.390442\pi\)
\(54\) 0 0
\(55\) 0.666782 0.0899089
\(56\) −0.292924 −0.0391436
\(57\) 0 0
\(58\) −20.2846 −2.66349
\(59\) 3.94941 0.514169 0.257085 0.966389i \(-0.417238\pi\)
0.257085 + 0.966389i \(0.417238\pi\)
\(60\) 0 0
\(61\) 3.63175 0.464997 0.232499 0.972597i \(-0.425310\pi\)
0.232499 + 0.972597i \(0.425310\pi\)
\(62\) 8.20235 1.04170
\(63\) 0 0
\(64\) −8.82072 −1.10259
\(65\) 14.7699 1.83198
\(66\) 0 0
\(67\) 5.37197 0.656290 0.328145 0.944627i \(-0.393576\pi\)
0.328145 + 0.944627i \(0.393576\pi\)
\(68\) 8.58765 1.04141
\(69\) 0 0
\(70\) −9.66247 −1.15489
\(71\) −8.95527 −1.06279 −0.531397 0.847123i \(-0.678333\pi\)
−0.531397 + 0.847123i \(0.678333\pi\)
\(72\) 0 0
\(73\) −7.76915 −0.909310 −0.454655 0.890668i \(-0.650237\pi\)
−0.454655 + 0.890668i \(0.650237\pi\)
\(74\) −4.35966 −0.506800
\(75\) 0 0
\(76\) −10.8818 −1.24823
\(77\) 0.262451 0.0299091
\(78\) 0 0
\(79\) −12.9221 −1.45385 −0.726924 0.686718i \(-0.759051\pi\)
−0.726924 + 0.686718i \(0.759051\pi\)
\(80\) −13.1496 −1.47017
\(81\) 0 0
\(82\) −0.0643711 −0.00710860
\(83\) 2.22931 0.244698 0.122349 0.992487i \(-0.460957\pi\)
0.122349 + 0.992487i \(0.460957\pi\)
\(84\) 0 0
\(85\) 14.1966 1.53984
\(86\) −5.62898 −0.606989
\(87\) 0 0
\(88\) −0.0409576 −0.00436609
\(89\) −9.94855 −1.05454 −0.527272 0.849696i \(-0.676785\pi\)
−0.527272 + 0.849696i \(0.676785\pi\)
\(90\) 0 0
\(91\) 5.81356 0.609427
\(92\) 2.06127 0.214903
\(93\) 0 0
\(94\) −1.37977 −0.142313
\(95\) −17.9892 −1.84565
\(96\) 0 0
\(97\) −4.84284 −0.491715 −0.245858 0.969306i \(-0.579070\pi\)
−0.245858 + 0.969306i \(0.579070\pi\)
\(98\) 10.3802 1.04856
\(99\) 0 0
\(100\) 14.9817 1.49817
\(101\) 12.8427 1.27790 0.638950 0.769249i \(-0.279369\pi\)
0.638950 + 0.769249i \(0.279369\pi\)
\(102\) 0 0
\(103\) −9.83159 −0.968736 −0.484368 0.874864i \(-0.660950\pi\)
−0.484368 + 0.874864i \(0.660950\pi\)
\(104\) −0.907252 −0.0889634
\(105\) 0 0
\(106\) −9.95489 −0.966904
\(107\) 6.72004 0.649650 0.324825 0.945774i \(-0.394695\pi\)
0.324825 + 0.945774i \(0.394695\pi\)
\(108\) 0 0
\(109\) 17.0804 1.63601 0.818005 0.575211i \(-0.195080\pi\)
0.818005 + 0.575211i \(0.195080\pi\)
\(110\) −1.35104 −0.128817
\(111\) 0 0
\(112\) −5.17578 −0.489066
\(113\) −12.2581 −1.15315 −0.576574 0.817045i \(-0.695611\pi\)
−0.576574 + 0.817045i \(0.695611\pi\)
\(114\) 0 0
\(115\) 3.40758 0.317758
\(116\) 21.0786 1.95709
\(117\) 0 0
\(118\) −8.00233 −0.736674
\(119\) 5.58791 0.512243
\(120\) 0 0
\(121\) −10.9633 −0.996664
\(122\) −7.35867 −0.666223
\(123\) 0 0
\(124\) −8.52341 −0.765425
\(125\) 7.36330 0.658593
\(126\) 0 0
\(127\) −13.4140 −1.19030 −0.595151 0.803614i \(-0.702908\pi\)
−0.595151 + 0.803614i \(0.702908\pi\)
\(128\) 1.70807 0.150973
\(129\) 0 0
\(130\) −29.9269 −2.62476
\(131\) −15.1828 −1.32652 −0.663262 0.748387i \(-0.730829\pi\)
−0.663262 + 0.748387i \(0.730829\pi\)
\(132\) 0 0
\(133\) −7.08070 −0.613974
\(134\) −10.8847 −0.940297
\(135\) 0 0
\(136\) −0.872037 −0.0747765
\(137\) 6.50914 0.556113 0.278057 0.960565i \(-0.410310\pi\)
0.278057 + 0.960565i \(0.410310\pi\)
\(138\) 0 0
\(139\) −5.06206 −0.429359 −0.214679 0.976685i \(-0.568871\pi\)
−0.214679 + 0.976685i \(0.568871\pi\)
\(140\) 10.0407 0.848593
\(141\) 0 0
\(142\) 18.1452 1.52271
\(143\) 0.812872 0.0679758
\(144\) 0 0
\(145\) 34.8459 2.89379
\(146\) 15.7419 1.30281
\(147\) 0 0
\(148\) 4.53031 0.372389
\(149\) −10.6906 −0.875805 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(150\) 0 0
\(151\) 15.4018 1.25338 0.626691 0.779268i \(-0.284409\pi\)
0.626691 + 0.779268i \(0.284409\pi\)
\(152\) 1.10500 0.0896272
\(153\) 0 0
\(154\) −0.531781 −0.0428521
\(155\) −14.0904 −1.13177
\(156\) 0 0
\(157\) −3.81548 −0.304509 −0.152254 0.988341i \(-0.548653\pi\)
−0.152254 + 0.988341i \(0.548653\pi\)
\(158\) 26.1828 2.08299
\(159\) 0 0
\(160\) 28.1322 2.22404
\(161\) 1.34125 0.105705
\(162\) 0 0
\(163\) 10.4217 0.816292 0.408146 0.912917i \(-0.366175\pi\)
0.408146 + 0.912917i \(0.366175\pi\)
\(164\) 0.0668908 0.00522329
\(165\) 0 0
\(166\) −4.51705 −0.350591
\(167\) 1.09246 0.0845374 0.0422687 0.999106i \(-0.486541\pi\)
0.0422687 + 0.999106i \(0.486541\pi\)
\(168\) 0 0
\(169\) 5.00594 0.385072
\(170\) −28.7653 −2.20620
\(171\) 0 0
\(172\) 5.84932 0.446006
\(173\) 10.4717 0.796148 0.398074 0.917353i \(-0.369679\pi\)
0.398074 + 0.917353i \(0.369679\pi\)
\(174\) 0 0
\(175\) 9.74847 0.736915
\(176\) −0.723696 −0.0545506
\(177\) 0 0
\(178\) 20.1578 1.51089
\(179\) −21.7287 −1.62408 −0.812041 0.583600i \(-0.801643\pi\)
−0.812041 + 0.583600i \(0.801643\pi\)
\(180\) 0 0
\(181\) 12.6108 0.937351 0.468676 0.883370i \(-0.344731\pi\)
0.468676 + 0.883370i \(0.344731\pi\)
\(182\) −11.7795 −0.873153
\(183\) 0 0
\(184\) −0.209313 −0.0154308
\(185\) 7.48924 0.550620
\(186\) 0 0
\(187\) 0.781320 0.0571358
\(188\) 1.43378 0.104569
\(189\) 0 0
\(190\) 36.4498 2.64435
\(191\) 5.93669 0.429564 0.214782 0.976662i \(-0.431096\pi\)
0.214782 + 0.976662i \(0.431096\pi\)
\(192\) 0 0
\(193\) 20.5726 1.48085 0.740426 0.672138i \(-0.234624\pi\)
0.740426 + 0.672138i \(0.234624\pi\)
\(194\) 9.81259 0.704503
\(195\) 0 0
\(196\) −10.7865 −0.770467
\(197\) 3.06174 0.218140 0.109070 0.994034i \(-0.465213\pi\)
0.109070 + 0.994034i \(0.465213\pi\)
\(198\) 0 0
\(199\) 23.1755 1.64286 0.821432 0.570306i \(-0.193175\pi\)
0.821432 + 0.570306i \(0.193175\pi\)
\(200\) −1.52133 −0.107574
\(201\) 0 0
\(202\) −26.0220 −1.83090
\(203\) 13.7156 0.962649
\(204\) 0 0
\(205\) 0.110580 0.00772324
\(206\) 19.9209 1.38795
\(207\) 0 0
\(208\) −16.0306 −1.11152
\(209\) −0.990047 −0.0684830
\(210\) 0 0
\(211\) 3.22027 0.221693 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(212\) 10.3446 0.710467
\(213\) 0 0
\(214\) −13.6162 −0.930784
\(215\) 9.66976 0.659472
\(216\) 0 0
\(217\) −5.54611 −0.376494
\(218\) −34.6085 −2.34399
\(219\) 0 0
\(220\) 1.40392 0.0946525
\(221\) 17.3070 1.16420
\(222\) 0 0
\(223\) 2.78961 0.186806 0.0934032 0.995628i \(-0.470225\pi\)
0.0934032 + 0.995628i \(0.470225\pi\)
\(224\) 11.0731 0.739850
\(225\) 0 0
\(226\) 24.8375 1.65217
\(227\) 15.2465 1.01195 0.505975 0.862548i \(-0.331133\pi\)
0.505975 + 0.862548i \(0.331133\pi\)
\(228\) 0 0
\(229\) 17.2752 1.14158 0.570788 0.821097i \(-0.306638\pi\)
0.570788 + 0.821097i \(0.306638\pi\)
\(230\) −6.90446 −0.455267
\(231\) 0 0
\(232\) −2.14043 −0.140526
\(233\) −12.5824 −0.824298 −0.412149 0.911116i \(-0.635222\pi\)
−0.412149 + 0.911116i \(0.635222\pi\)
\(234\) 0 0
\(235\) 2.37024 0.154618
\(236\) 8.31556 0.541297
\(237\) 0 0
\(238\) −11.3223 −0.733913
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −0.658194 −0.0423980 −0.0211990 0.999775i \(-0.506748\pi\)
−0.0211990 + 0.999775i \(0.506748\pi\)
\(242\) 22.2139 1.42797
\(243\) 0 0
\(244\) 7.64671 0.489531
\(245\) −17.8317 −1.13922
\(246\) 0 0
\(247\) −21.9306 −1.39541
\(248\) 0.865514 0.0549602
\(249\) 0 0
\(250\) −14.9196 −0.943597
\(251\) 18.5453 1.17057 0.585283 0.810829i \(-0.300983\pi\)
0.585283 + 0.810829i \(0.300983\pi\)
\(252\) 0 0
\(253\) 0.187538 0.0117904
\(254\) 27.1796 1.70540
\(255\) 0 0
\(256\) 14.1805 0.886283
\(257\) 23.0065 1.43511 0.717555 0.696502i \(-0.245261\pi\)
0.717555 + 0.696502i \(0.245261\pi\)
\(258\) 0 0
\(259\) 2.94783 0.183169
\(260\) 31.0983 1.92864
\(261\) 0 0
\(262\) 30.7634 1.90057
\(263\) 3.78505 0.233396 0.116698 0.993167i \(-0.462769\pi\)
0.116698 + 0.993167i \(0.462769\pi\)
\(264\) 0 0
\(265\) 17.1010 1.05051
\(266\) 14.3470 0.879669
\(267\) 0 0
\(268\) 11.3108 0.690916
\(269\) −2.62526 −0.160065 −0.0800325 0.996792i \(-0.525502\pi\)
−0.0800325 + 0.996792i \(0.525502\pi\)
\(270\) 0 0
\(271\) −11.6427 −0.707242 −0.353621 0.935389i \(-0.615050\pi\)
−0.353621 + 0.935389i \(0.615050\pi\)
\(272\) −15.4084 −0.934270
\(273\) 0 0
\(274\) −13.1889 −0.796769
\(275\) 1.36306 0.0821959
\(276\) 0 0
\(277\) 4.72065 0.283636 0.141818 0.989893i \(-0.454705\pi\)
0.141818 + 0.989893i \(0.454705\pi\)
\(278\) 10.2568 0.615162
\(279\) 0 0
\(280\) −1.01959 −0.0609319
\(281\) −25.0192 −1.49252 −0.746260 0.665654i \(-0.768153\pi\)
−0.746260 + 0.665654i \(0.768153\pi\)
\(282\) 0 0
\(283\) −9.85106 −0.585585 −0.292792 0.956176i \(-0.594584\pi\)
−0.292792 + 0.956176i \(0.594584\pi\)
\(284\) −18.8555 −1.11887
\(285\) 0 0
\(286\) −1.64705 −0.0973920
\(287\) 0.0435252 0.00256921
\(288\) 0 0
\(289\) −0.364740 −0.0214553
\(290\) −70.6050 −4.14607
\(291\) 0 0
\(292\) −16.3581 −0.957285
\(293\) −21.2915 −1.24386 −0.621932 0.783071i \(-0.713652\pi\)
−0.621932 + 0.783071i \(0.713652\pi\)
\(294\) 0 0
\(295\) 13.7468 0.800370
\(296\) −0.460032 −0.0267388
\(297\) 0 0
\(298\) 21.6613 1.25481
\(299\) 4.15417 0.240242
\(300\) 0 0
\(301\) 3.80610 0.219380
\(302\) −31.2073 −1.79578
\(303\) 0 0
\(304\) 19.5247 1.11982
\(305\) 12.6411 0.723828
\(306\) 0 0
\(307\) −17.0413 −0.972597 −0.486299 0.873793i \(-0.661653\pi\)
−0.486299 + 0.873793i \(0.661653\pi\)
\(308\) 0.552596 0.0314871
\(309\) 0 0
\(310\) 28.5501 1.62154
\(311\) 22.1896 1.25826 0.629130 0.777300i \(-0.283411\pi\)
0.629130 + 0.777300i \(0.283411\pi\)
\(312\) 0 0
\(313\) −4.79466 −0.271010 −0.135505 0.990777i \(-0.543266\pi\)
−0.135505 + 0.990777i \(0.543266\pi\)
\(314\) 7.73096 0.436283
\(315\) 0 0
\(316\) −27.2077 −1.53055
\(317\) −10.5786 −0.594152 −0.297076 0.954854i \(-0.596012\pi\)
−0.297076 + 0.954854i \(0.596012\pi\)
\(318\) 0 0
\(319\) 1.91777 0.107374
\(320\) −30.7025 −1.71632
\(321\) 0 0
\(322\) −2.71766 −0.151449
\(323\) −21.0793 −1.17288
\(324\) 0 0
\(325\) 30.1933 1.67482
\(326\) −21.1166 −1.16954
\(327\) 0 0
\(328\) −0.00679245 −0.000375050 0
\(329\) 0.932948 0.0514351
\(330\) 0 0
\(331\) 33.0018 1.81394 0.906972 0.421192i \(-0.138388\pi\)
0.906972 + 0.421192i \(0.138388\pi\)
\(332\) 4.69386 0.257609
\(333\) 0 0
\(334\) −2.21356 −0.121121
\(335\) 18.6983 1.02160
\(336\) 0 0
\(337\) −31.0856 −1.69334 −0.846671 0.532116i \(-0.821397\pi\)
−0.846671 + 0.532116i \(0.821397\pi\)
\(338\) −10.1431 −0.551711
\(339\) 0 0
\(340\) 29.8912 1.62108
\(341\) −0.775476 −0.0419944
\(342\) 0 0
\(343\) −16.6090 −0.896802
\(344\) −0.593972 −0.0320248
\(345\) 0 0
\(346\) −21.2178 −1.14068
\(347\) −22.9937 −1.23436 −0.617182 0.786820i \(-0.711726\pi\)
−0.617182 + 0.786820i \(0.711726\pi\)
\(348\) 0 0
\(349\) 17.7700 0.951208 0.475604 0.879659i \(-0.342230\pi\)
0.475604 + 0.879659i \(0.342230\pi\)
\(350\) −19.7524 −1.05581
\(351\) 0 0
\(352\) 1.54827 0.0825233
\(353\) −13.9632 −0.743188 −0.371594 0.928395i \(-0.621189\pi\)
−0.371594 + 0.928395i \(0.621189\pi\)
\(354\) 0 0
\(355\) −31.1708 −1.65438
\(356\) −20.9469 −1.11018
\(357\) 0 0
\(358\) 44.0269 2.32690
\(359\) 21.7844 1.14974 0.574869 0.818245i \(-0.305053\pi\)
0.574869 + 0.818245i \(0.305053\pi\)
\(360\) 0 0
\(361\) 7.71060 0.405821
\(362\) −25.5521 −1.34299
\(363\) 0 0
\(364\) 12.2406 0.641580
\(365\) −27.0423 −1.41546
\(366\) 0 0
\(367\) 5.31876 0.277637 0.138818 0.990318i \(-0.455670\pi\)
0.138818 + 0.990318i \(0.455670\pi\)
\(368\) −3.69843 −0.192794
\(369\) 0 0
\(370\) −15.1748 −0.788898
\(371\) 6.73111 0.349462
\(372\) 0 0
\(373\) −23.1559 −1.19897 −0.599484 0.800387i \(-0.704628\pi\)
−0.599484 + 0.800387i \(0.704628\pi\)
\(374\) −1.58312 −0.0818610
\(375\) 0 0
\(376\) −0.145594 −0.00750843
\(377\) 42.4805 2.18786
\(378\) 0 0
\(379\) −5.92833 −0.304518 −0.152259 0.988341i \(-0.548655\pi\)
−0.152259 + 0.988341i \(0.548655\pi\)
\(380\) −37.8766 −1.94303
\(381\) 0 0
\(382\) −12.0290 −0.615456
\(383\) 34.8364 1.78006 0.890029 0.455903i \(-0.150684\pi\)
0.890029 + 0.455903i \(0.150684\pi\)
\(384\) 0 0
\(385\) 0.913520 0.0465573
\(386\) −41.6845 −2.12168
\(387\) 0 0
\(388\) −10.1967 −0.517658
\(389\) 1.77761 0.0901285 0.0450643 0.998984i \(-0.485651\pi\)
0.0450643 + 0.998984i \(0.485651\pi\)
\(390\) 0 0
\(391\) 3.99292 0.201931
\(392\) 1.09532 0.0553222
\(393\) 0 0
\(394\) −6.20373 −0.312540
\(395\) −44.9782 −2.26310
\(396\) 0 0
\(397\) −12.8987 −0.647365 −0.323682 0.946166i \(-0.604921\pi\)
−0.323682 + 0.946166i \(0.604921\pi\)
\(398\) −46.9583 −2.35381
\(399\) 0 0
\(400\) −26.8809 −1.34405
\(401\) 25.4445 1.27064 0.635319 0.772250i \(-0.280869\pi\)
0.635319 + 0.772250i \(0.280869\pi\)
\(402\) 0 0
\(403\) −17.1776 −0.855676
\(404\) 27.0406 1.34532
\(405\) 0 0
\(406\) −27.7907 −1.37923
\(407\) 0.412176 0.0204308
\(408\) 0 0
\(409\) −33.1140 −1.63738 −0.818690 0.574235i \(-0.805299\pi\)
−0.818690 + 0.574235i \(0.805299\pi\)
\(410\) −0.224058 −0.0110654
\(411\) 0 0
\(412\) −20.7006 −1.01985
\(413\) 5.41086 0.266251
\(414\) 0 0
\(415\) 7.75961 0.380904
\(416\) 34.2958 1.68149
\(417\) 0 0
\(418\) 2.00604 0.0981187
\(419\) 33.1423 1.61911 0.809553 0.587047i \(-0.199710\pi\)
0.809553 + 0.587047i \(0.199710\pi\)
\(420\) 0 0
\(421\) −14.3129 −0.697570 −0.348785 0.937203i \(-0.613406\pi\)
−0.348785 + 0.937203i \(0.613406\pi\)
\(422\) −6.52495 −0.317629
\(423\) 0 0
\(424\) −1.05044 −0.0510140
\(425\) 29.0213 1.40774
\(426\) 0 0
\(427\) 4.97565 0.240789
\(428\) 14.1492 0.683926
\(429\) 0 0
\(430\) −19.5929 −0.944856
\(431\) 24.1655 1.16401 0.582006 0.813185i \(-0.302268\pi\)
0.582006 + 0.813185i \(0.302268\pi\)
\(432\) 0 0
\(433\) −4.95559 −0.238150 −0.119075 0.992885i \(-0.537993\pi\)
−0.119075 + 0.992885i \(0.537993\pi\)
\(434\) 11.2376 0.539421
\(435\) 0 0
\(436\) 35.9632 1.72233
\(437\) −5.05962 −0.242034
\(438\) 0 0
\(439\) −20.9535 −1.00005 −0.500027 0.866010i \(-0.666677\pi\)
−0.500027 + 0.866010i \(0.666677\pi\)
\(440\) −0.142562 −0.00679638
\(441\) 0 0
\(442\) −35.0677 −1.66800
\(443\) 3.09007 0.146814 0.0734069 0.997302i \(-0.476613\pi\)
0.0734069 + 0.997302i \(0.476613\pi\)
\(444\) 0 0
\(445\) −34.6282 −1.64153
\(446\) −5.65234 −0.267646
\(447\) 0 0
\(448\) −12.0848 −0.570952
\(449\) 7.59148 0.358264 0.179132 0.983825i \(-0.442671\pi\)
0.179132 + 0.983825i \(0.442671\pi\)
\(450\) 0 0
\(451\) 0.00608585 0.000286571 0
\(452\) −25.8097 −1.21399
\(453\) 0 0
\(454\) −30.8927 −1.44987
\(455\) 20.2354 0.948650
\(456\) 0 0
\(457\) −24.8554 −1.16269 −0.581344 0.813658i \(-0.697473\pi\)
−0.581344 + 0.813658i \(0.697473\pi\)
\(458\) −35.0031 −1.63559
\(459\) 0 0
\(460\) 7.17472 0.334523
\(461\) −6.55243 −0.305177 −0.152589 0.988290i \(-0.548761\pi\)
−0.152589 + 0.988290i \(0.548761\pi\)
\(462\) 0 0
\(463\) 6.94066 0.322560 0.161280 0.986909i \(-0.448438\pi\)
0.161280 + 0.986909i \(0.448438\pi\)
\(464\) −37.8202 −1.75576
\(465\) 0 0
\(466\) 25.4945 1.18101
\(467\) 39.0558 1.80729 0.903644 0.428285i \(-0.140882\pi\)
0.903644 + 0.428285i \(0.140882\pi\)
\(468\) 0 0
\(469\) 7.35983 0.339845
\(470\) −4.80261 −0.221528
\(471\) 0 0
\(472\) −0.844407 −0.0388670
\(473\) 0.532182 0.0244697
\(474\) 0 0
\(475\) −36.7743 −1.68732
\(476\) 11.7654 0.539268
\(477\) 0 0
\(478\) −2.02621 −0.0926766
\(479\) −18.4393 −0.842513 −0.421257 0.906942i \(-0.638411\pi\)
−0.421257 + 0.906942i \(0.638411\pi\)
\(480\) 0 0
\(481\) 9.13011 0.416297
\(482\) 1.33364 0.0607456
\(483\) 0 0
\(484\) −23.0835 −1.04925
\(485\) −16.8566 −0.765418
\(486\) 0 0
\(487\) 37.3485 1.69242 0.846212 0.532846i \(-0.178878\pi\)
0.846212 + 0.532846i \(0.178878\pi\)
\(488\) −0.776489 −0.0351500
\(489\) 0 0
\(490\) 36.1307 1.63222
\(491\) −33.4616 −1.51010 −0.755050 0.655668i \(-0.772387\pi\)
−0.755050 + 0.655668i \(0.772387\pi\)
\(492\) 0 0
\(493\) 40.8316 1.83896
\(494\) 44.4359 1.99926
\(495\) 0 0
\(496\) 15.2931 0.686681
\(497\) −12.2691 −0.550345
\(498\) 0 0
\(499\) −32.5621 −1.45768 −0.728841 0.684684i \(-0.759940\pi\)
−0.728841 + 0.684684i \(0.759940\pi\)
\(500\) 15.5036 0.693341
\(501\) 0 0
\(502\) −37.5766 −1.67712
\(503\) 4.28613 0.191109 0.0955545 0.995424i \(-0.469538\pi\)
0.0955545 + 0.995424i \(0.469538\pi\)
\(504\) 0 0
\(505\) 44.7020 1.98921
\(506\) −0.379992 −0.0168927
\(507\) 0 0
\(508\) −28.2435 −1.25310
\(509\) 28.3767 1.25777 0.628887 0.777497i \(-0.283511\pi\)
0.628887 + 0.777497i \(0.283511\pi\)
\(510\) 0 0
\(511\) −10.6441 −0.470866
\(512\) −32.1489 −1.42079
\(513\) 0 0
\(514\) −46.6161 −2.05615
\(515\) −34.2211 −1.50796
\(516\) 0 0
\(517\) 0.130448 0.00573710
\(518\) −5.97292 −0.262435
\(519\) 0 0
\(520\) −3.15789 −0.138483
\(521\) −33.0835 −1.44941 −0.724706 0.689058i \(-0.758024\pi\)
−0.724706 + 0.689058i \(0.758024\pi\)
\(522\) 0 0
\(523\) 36.8785 1.61258 0.806292 0.591517i \(-0.201471\pi\)
0.806292 + 0.591517i \(0.201471\pi\)
\(524\) −31.9676 −1.39651
\(525\) 0 0
\(526\) −7.66931 −0.334398
\(527\) −16.5108 −0.719223
\(528\) 0 0
\(529\) −22.0416 −0.958330
\(530\) −34.6502 −1.50511
\(531\) 0 0
\(532\) −14.9086 −0.646368
\(533\) 0.134808 0.00583917
\(534\) 0 0
\(535\) 23.3906 1.01126
\(536\) −1.14856 −0.0496102
\(537\) 0 0
\(538\) 5.31932 0.229332
\(539\) −0.981380 −0.0422710
\(540\) 0 0
\(541\) 27.7829 1.19448 0.597241 0.802062i \(-0.296264\pi\)
0.597241 + 0.802062i \(0.296264\pi\)
\(542\) 23.5905 1.01330
\(543\) 0 0
\(544\) 32.9646 1.41335
\(545\) 59.4523 2.54666
\(546\) 0 0
\(547\) −23.6920 −1.01300 −0.506498 0.862241i \(-0.669060\pi\)
−0.506498 + 0.862241i \(0.669060\pi\)
\(548\) 13.7051 0.585454
\(549\) 0 0
\(550\) −2.76185 −0.117766
\(551\) −51.7396 −2.20418
\(552\) 0 0
\(553\) −17.7038 −0.752843
\(554\) −9.56502 −0.406379
\(555\) 0 0
\(556\) −10.6583 −0.452012
\(557\) 16.2888 0.690178 0.345089 0.938570i \(-0.387849\pi\)
0.345089 + 0.938570i \(0.387849\pi\)
\(558\) 0 0
\(559\) 11.7884 0.498595
\(560\) −18.0155 −0.761293
\(561\) 0 0
\(562\) 50.6941 2.13840
\(563\) −0.216197 −0.00911160 −0.00455580 0.999990i \(-0.501450\pi\)
−0.00455580 + 0.999990i \(0.501450\pi\)
\(564\) 0 0
\(565\) −42.6672 −1.79502
\(566\) 19.9603 0.838994
\(567\) 0 0
\(568\) 1.91469 0.0803386
\(569\) 13.6370 0.571691 0.285845 0.958276i \(-0.407726\pi\)
0.285845 + 0.958276i \(0.407726\pi\)
\(570\) 0 0
\(571\) −17.2641 −0.722482 −0.361241 0.932473i \(-0.617647\pi\)
−0.361241 + 0.932473i \(0.617647\pi\)
\(572\) 1.71152 0.0715622
\(573\) 0 0
\(574\) −0.0881912 −0.00368103
\(575\) 6.96592 0.290499
\(576\) 0 0
\(577\) −45.0478 −1.87537 −0.937683 0.347493i \(-0.887033\pi\)
−0.937683 + 0.347493i \(0.887033\pi\)
\(578\) 0.739039 0.0307400
\(579\) 0 0
\(580\) 73.3687 3.04647
\(581\) 3.05425 0.126712
\(582\) 0 0
\(583\) 0.941167 0.0389791
\(584\) 1.66109 0.0687364
\(585\) 0 0
\(586\) 43.1411 1.78214
\(587\) 6.87236 0.283653 0.141826 0.989892i \(-0.454703\pi\)
0.141826 + 0.989892i \(0.454703\pi\)
\(588\) 0 0
\(589\) 20.9216 0.862062
\(590\) −27.8539 −1.14673
\(591\) 0 0
\(592\) −8.12849 −0.334079
\(593\) −11.7147 −0.481067 −0.240534 0.970641i \(-0.577322\pi\)
−0.240534 + 0.970641i \(0.577322\pi\)
\(594\) 0 0
\(595\) 19.4500 0.797371
\(596\) −22.5092 −0.922013
\(597\) 0 0
\(598\) −8.41721 −0.344205
\(599\) 29.8786 1.22081 0.610403 0.792091i \(-0.291008\pi\)
0.610403 + 0.792091i \(0.291008\pi\)
\(600\) 0 0
\(601\) 21.7040 0.885324 0.442662 0.896688i \(-0.354034\pi\)
0.442662 + 0.896688i \(0.354034\pi\)
\(602\) −7.71195 −0.314316
\(603\) 0 0
\(604\) 32.4288 1.31951
\(605\) −38.1602 −1.55143
\(606\) 0 0
\(607\) −12.5826 −0.510713 −0.255357 0.966847i \(-0.582193\pi\)
−0.255357 + 0.966847i \(0.582193\pi\)
\(608\) −41.7710 −1.69404
\(609\) 0 0
\(610\) −25.6135 −1.03706
\(611\) 2.88956 0.116899
\(612\) 0 0
\(613\) −12.3210 −0.497642 −0.248821 0.968549i \(-0.580043\pi\)
−0.248821 + 0.968549i \(0.580043\pi\)
\(614\) 34.5292 1.39348
\(615\) 0 0
\(616\) −0.0561136 −0.00226088
\(617\) −48.6690 −1.95934 −0.979670 0.200617i \(-0.935705\pi\)
−0.979670 + 0.200617i \(0.935705\pi\)
\(618\) 0 0
\(619\) −40.9465 −1.64578 −0.822890 0.568201i \(-0.807640\pi\)
−0.822890 + 0.568201i \(0.807640\pi\)
\(620\) −29.6677 −1.19148
\(621\) 0 0
\(622\) −44.9608 −1.80277
\(623\) −13.6299 −0.546072
\(624\) 0 0
\(625\) −9.94763 −0.397905
\(626\) 9.71497 0.388288
\(627\) 0 0
\(628\) −8.03357 −0.320574
\(629\) 8.77572 0.349911
\(630\) 0 0
\(631\) −22.4459 −0.893558 −0.446779 0.894644i \(-0.647429\pi\)
−0.446779 + 0.894644i \(0.647429\pi\)
\(632\) 2.76282 0.109899
\(633\) 0 0
\(634\) 21.4344 0.851269
\(635\) −46.6905 −1.85286
\(636\) 0 0
\(637\) −21.7386 −0.861313
\(638\) −3.88579 −0.153840
\(639\) 0 0
\(640\) 5.94531 0.235009
\(641\) 11.9975 0.473872 0.236936 0.971525i \(-0.423857\pi\)
0.236936 + 0.971525i \(0.423857\pi\)
\(642\) 0 0
\(643\) −40.5278 −1.59826 −0.799130 0.601158i \(-0.794706\pi\)
−0.799130 + 0.601158i \(0.794706\pi\)
\(644\) 2.82403 0.111282
\(645\) 0 0
\(646\) 42.7111 1.68045
\(647\) −10.4001 −0.408871 −0.204435 0.978880i \(-0.565536\pi\)
−0.204435 + 0.978880i \(0.565536\pi\)
\(648\) 0 0
\(649\) 0.756565 0.0296978
\(650\) −61.1779 −2.39959
\(651\) 0 0
\(652\) 21.9431 0.859359
\(653\) 2.80015 0.109578 0.0547891 0.998498i \(-0.482551\pi\)
0.0547891 + 0.998498i \(0.482551\pi\)
\(654\) 0 0
\(655\) −52.8470 −2.06490
\(656\) −0.120019 −0.00468594
\(657\) 0 0
\(658\) −1.89035 −0.0736934
\(659\) −28.1624 −1.09705 −0.548525 0.836134i \(-0.684811\pi\)
−0.548525 + 0.836134i \(0.684811\pi\)
\(660\) 0 0
\(661\) 15.9663 0.621016 0.310508 0.950571i \(-0.399501\pi\)
0.310508 + 0.950571i \(0.399501\pi\)
\(662\) −66.8685 −2.59892
\(663\) 0 0
\(664\) −0.476640 −0.0184972
\(665\) −24.6460 −0.955730
\(666\) 0 0
\(667\) 9.80071 0.379485
\(668\) 2.30020 0.0889976
\(669\) 0 0
\(670\) −37.8867 −1.46369
\(671\) 0.695712 0.0268577
\(672\) 0 0
\(673\) 20.0409 0.772519 0.386260 0.922390i \(-0.373767\pi\)
0.386260 + 0.922390i \(0.373767\pi\)
\(674\) 62.9860 2.42613
\(675\) 0 0
\(676\) 10.5401 0.405389
\(677\) 36.8324 1.41558 0.707791 0.706422i \(-0.249692\pi\)
0.707791 + 0.706422i \(0.249692\pi\)
\(678\) 0 0
\(679\) −6.63489 −0.254624
\(680\) −3.03532 −0.116399
\(681\) 0 0
\(682\) 1.57128 0.0601673
\(683\) −25.9817 −0.994161 −0.497080 0.867705i \(-0.665595\pi\)
−0.497080 + 0.867705i \(0.665595\pi\)
\(684\) 0 0
\(685\) 22.6565 0.865661
\(686\) 33.6533 1.28489
\(687\) 0 0
\(688\) −10.4951 −0.400123
\(689\) 20.8478 0.794238
\(690\) 0 0
\(691\) −7.84083 −0.298279 −0.149140 0.988816i \(-0.547650\pi\)
−0.149140 + 0.988816i \(0.547650\pi\)
\(692\) 22.0483 0.838152
\(693\) 0 0
\(694\) 46.5900 1.76853
\(695\) −17.6196 −0.668351
\(696\) 0 0
\(697\) 0.129575 0.00490801
\(698\) −36.0058 −1.36284
\(699\) 0 0
\(700\) 20.5256 0.775795
\(701\) 28.2197 1.06584 0.532921 0.846165i \(-0.321094\pi\)
0.532921 + 0.846165i \(0.321094\pi\)
\(702\) 0 0
\(703\) −11.1201 −0.419404
\(704\) −1.68973 −0.0636842
\(705\) 0 0
\(706\) 28.2924 1.06480
\(707\) 17.5951 0.661732
\(708\) 0 0
\(709\) 21.9221 0.823303 0.411652 0.911341i \(-0.364952\pi\)
0.411652 + 0.911341i \(0.364952\pi\)
\(710\) 63.1586 2.37030
\(711\) 0 0
\(712\) 2.12706 0.0797150
\(713\) −3.96306 −0.148418
\(714\) 0 0
\(715\) 2.82938 0.105813
\(716\) −45.7503 −1.70977
\(717\) 0 0
\(718\) −44.1398 −1.64728
\(719\) −51.4814 −1.91993 −0.959966 0.280118i \(-0.909626\pi\)
−0.959966 + 0.280118i \(0.909626\pi\)
\(720\) 0 0
\(721\) −13.4697 −0.501638
\(722\) −15.6233 −0.581438
\(723\) 0 0
\(724\) 26.5522 0.986806
\(725\) 71.2334 2.64554
\(726\) 0 0
\(727\) −4.61082 −0.171006 −0.0855030 0.996338i \(-0.527250\pi\)
−0.0855030 + 0.996338i \(0.527250\pi\)
\(728\) −1.24297 −0.0460677
\(729\) 0 0
\(730\) 54.7933 2.02799
\(731\) 11.3308 0.419085
\(732\) 0 0
\(733\) −3.26928 −0.120754 −0.0603768 0.998176i \(-0.519230\pi\)
−0.0603768 + 0.998176i \(0.519230\pi\)
\(734\) −10.7769 −0.397783
\(735\) 0 0
\(736\) 7.91242 0.291656
\(737\) 1.02908 0.0379065
\(738\) 0 0
\(739\) 35.3375 1.29991 0.649956 0.759972i \(-0.274787\pi\)
0.649956 + 0.759972i \(0.274787\pi\)
\(740\) 15.7688 0.579671
\(741\) 0 0
\(742\) −13.6386 −0.500690
\(743\) −45.5076 −1.66951 −0.834757 0.550619i \(-0.814392\pi\)
−0.834757 + 0.550619i \(0.814392\pi\)
\(744\) 0 0
\(745\) −37.2109 −1.36330
\(746\) 46.9187 1.71782
\(747\) 0 0
\(748\) 1.64509 0.0601503
\(749\) 9.20674 0.336407
\(750\) 0 0
\(751\) 38.6081 1.40883 0.704415 0.709789i \(-0.251210\pi\)
0.704415 + 0.709789i \(0.251210\pi\)
\(752\) −2.57256 −0.0938115
\(753\) 0 0
\(754\) −86.0743 −3.13464
\(755\) 53.6095 1.95105
\(756\) 0 0
\(757\) 12.4991 0.454288 0.227144 0.973861i \(-0.427061\pi\)
0.227144 + 0.973861i \(0.427061\pi\)
\(758\) 12.0120 0.436297
\(759\) 0 0
\(760\) 3.84620 0.139516
\(761\) 42.9165 1.55572 0.777862 0.628435i \(-0.216304\pi\)
0.777862 + 0.628435i \(0.216304\pi\)
\(762\) 0 0
\(763\) 23.4010 0.847171
\(764\) 12.4998 0.452228
\(765\) 0 0
\(766\) −70.5859 −2.55037
\(767\) 16.7587 0.605121
\(768\) 0 0
\(769\) −41.9328 −1.51214 −0.756068 0.654493i \(-0.772882\pi\)
−0.756068 + 0.654493i \(0.772882\pi\)
\(770\) −1.85098 −0.0667048
\(771\) 0 0
\(772\) 43.3161 1.55898
\(773\) 17.1599 0.617200 0.308600 0.951192i \(-0.400140\pi\)
0.308600 + 0.951192i \(0.400140\pi\)
\(774\) 0 0
\(775\) −28.8042 −1.03468
\(776\) 1.03543 0.0371697
\(777\) 0 0
\(778\) −3.60181 −0.129131
\(779\) −0.164191 −0.00588274
\(780\) 0 0
\(781\) −1.71551 −0.0613857
\(782\) −8.09049 −0.289315
\(783\) 0 0
\(784\) 19.3537 0.691205
\(785\) −13.2806 −0.474006
\(786\) 0 0
\(787\) 8.35629 0.297870 0.148935 0.988847i \(-0.452416\pi\)
0.148935 + 0.988847i \(0.452416\pi\)
\(788\) 6.44657 0.229649
\(789\) 0 0
\(790\) 91.1353 3.24245
\(791\) −16.7942 −0.597132
\(792\) 0 0
\(793\) 15.4107 0.547251
\(794\) 26.1354 0.927509
\(795\) 0 0
\(796\) 48.7964 1.72954
\(797\) −50.4354 −1.78651 −0.893257 0.449547i \(-0.851585\pi\)
−0.893257 + 0.449547i \(0.851585\pi\)
\(798\) 0 0
\(799\) 2.77740 0.0982572
\(800\) 57.5090 2.03325
\(801\) 0 0
\(802\) −51.5559 −1.82050
\(803\) −1.48829 −0.0525206
\(804\) 0 0
\(805\) 4.66853 0.164544
\(806\) 34.8054 1.22597
\(807\) 0 0
\(808\) −2.74585 −0.0965988
\(809\) −37.4116 −1.31532 −0.657661 0.753314i \(-0.728454\pi\)
−0.657661 + 0.753314i \(0.728454\pi\)
\(810\) 0 0
\(811\) 34.9470 1.22716 0.613578 0.789634i \(-0.289730\pi\)
0.613578 + 0.789634i \(0.289730\pi\)
\(812\) 28.8785 1.01344
\(813\) 0 0
\(814\) −0.835154 −0.0292721
\(815\) 36.2751 1.27066
\(816\) 0 0
\(817\) −14.3578 −0.502316
\(818\) 67.0958 2.34595
\(819\) 0 0
\(820\) 0.232828 0.00813072
\(821\) 51.6588 1.80291 0.901453 0.432877i \(-0.142501\pi\)
0.901453 + 0.432877i \(0.142501\pi\)
\(822\) 0 0
\(823\) −23.3085 −0.812484 −0.406242 0.913765i \(-0.633161\pi\)
−0.406242 + 0.913765i \(0.633161\pi\)
\(824\) 2.10205 0.0732285
\(825\) 0 0
\(826\) −10.9635 −0.381470
\(827\) 0.139400 0.00484741 0.00242370 0.999997i \(-0.499229\pi\)
0.00242370 + 0.999997i \(0.499229\pi\)
\(828\) 0 0
\(829\) −8.55617 −0.297168 −0.148584 0.988900i \(-0.547472\pi\)
−0.148584 + 0.988900i \(0.547472\pi\)
\(830\) −15.7226 −0.545739
\(831\) 0 0
\(832\) −37.4293 −1.29763
\(833\) −20.8948 −0.723961
\(834\) 0 0
\(835\) 3.80257 0.131593
\(836\) −2.08456 −0.0720962
\(837\) 0 0
\(838\) −67.1532 −2.31977
\(839\) 7.49954 0.258913 0.129456 0.991585i \(-0.458677\pi\)
0.129456 + 0.991585i \(0.458677\pi\)
\(840\) 0 0
\(841\) 71.2220 2.45593
\(842\) 29.0010 0.999440
\(843\) 0 0
\(844\) 6.78035 0.233389
\(845\) 17.4243 0.599414
\(846\) 0 0
\(847\) −15.0202 −0.516100
\(848\) −18.5607 −0.637377
\(849\) 0 0
\(850\) −58.8032 −2.01693
\(851\) 2.10642 0.0722070
\(852\) 0 0
\(853\) 7.83193 0.268160 0.134080 0.990971i \(-0.457192\pi\)
0.134080 + 0.990971i \(0.457192\pi\)
\(854\) −10.0817 −0.344989
\(855\) 0 0
\(856\) −1.43678 −0.0491083
\(857\) −9.69988 −0.331342 −0.165671 0.986181i \(-0.552979\pi\)
−0.165671 + 0.986181i \(0.552979\pi\)
\(858\) 0 0
\(859\) −3.41295 −0.116448 −0.0582242 0.998304i \(-0.518544\pi\)
−0.0582242 + 0.998304i \(0.518544\pi\)
\(860\) 20.3599 0.694266
\(861\) 0 0
\(862\) −48.9643 −1.66773
\(863\) 37.9282 1.29109 0.645545 0.763722i \(-0.276630\pi\)
0.645545 + 0.763722i \(0.276630\pi\)
\(864\) 0 0
\(865\) 36.4491 1.23931
\(866\) 10.0411 0.341209
\(867\) 0 0
\(868\) −11.6774 −0.396358
\(869\) −2.47541 −0.0839725
\(870\) 0 0
\(871\) 22.7951 0.772382
\(872\) −3.65190 −0.123669
\(873\) 0 0
\(874\) 10.2518 0.346774
\(875\) 10.0880 0.341038
\(876\) 0 0
\(877\) 44.6999 1.50941 0.754704 0.656066i \(-0.227781\pi\)
0.754704 + 0.656066i \(0.227781\pi\)
\(878\) 42.4561 1.43282
\(879\) 0 0
\(880\) −2.51899 −0.0849150
\(881\) 32.1877 1.08443 0.542216 0.840239i \(-0.317586\pi\)
0.542216 + 0.840239i \(0.317586\pi\)
\(882\) 0 0
\(883\) 54.4253 1.83156 0.915778 0.401685i \(-0.131575\pi\)
0.915778 + 0.401685i \(0.131575\pi\)
\(884\) 36.4403 1.22562
\(885\) 0 0
\(886\) −6.26113 −0.210347
\(887\) −32.7726 −1.10040 −0.550198 0.835034i \(-0.685448\pi\)
−0.550198 + 0.835034i \(0.685448\pi\)
\(888\) 0 0
\(889\) −18.3778 −0.616371
\(890\) 70.1639 2.35190
\(891\) 0 0
\(892\) 5.87359 0.196662
\(893\) −3.51937 −0.117771
\(894\) 0 0
\(895\) −75.6318 −2.52809
\(896\) 2.34013 0.0781781
\(897\) 0 0
\(898\) −15.3819 −0.513301
\(899\) −40.5262 −1.35162
\(900\) 0 0
\(901\) 20.0386 0.667582
\(902\) −0.0123312 −0.000410584 0
\(903\) 0 0
\(904\) 2.62086 0.0871685
\(905\) 43.8946 1.45911
\(906\) 0 0
\(907\) 19.5122 0.647890 0.323945 0.946076i \(-0.394991\pi\)
0.323945 + 0.946076i \(0.394991\pi\)
\(908\) 32.1019 1.06534
\(909\) 0 0
\(910\) −41.0011 −1.35917
\(911\) −12.5747 −0.416618 −0.208309 0.978063i \(-0.566796\pi\)
−0.208309 + 0.978063i \(0.566796\pi\)
\(912\) 0 0
\(913\) 0.427056 0.0141335
\(914\) 50.3623 1.66584
\(915\) 0 0
\(916\) 36.3733 1.20181
\(917\) −20.8010 −0.686911
\(918\) 0 0
\(919\) −25.2431 −0.832693 −0.416347 0.909206i \(-0.636690\pi\)
−0.416347 + 0.909206i \(0.636690\pi\)
\(920\) −0.728561 −0.0240199
\(921\) 0 0
\(922\) 13.2766 0.437241
\(923\) −38.0003 −1.25079
\(924\) 0 0
\(925\) 15.3098 0.503384
\(926\) −14.0632 −0.462146
\(927\) 0 0
\(928\) 80.9124 2.65608
\(929\) −14.4960 −0.475599 −0.237799 0.971314i \(-0.576426\pi\)
−0.237799 + 0.971314i \(0.576426\pi\)
\(930\) 0 0
\(931\) 26.4767 0.867740
\(932\) −26.4924 −0.867789
\(933\) 0 0
\(934\) −79.1352 −2.58938
\(935\) 2.71956 0.0889391
\(936\) 0 0
\(937\) −13.7602 −0.449525 −0.224762 0.974414i \(-0.572161\pi\)
−0.224762 + 0.974414i \(0.572161\pi\)
\(938\) −14.9125 −0.486912
\(939\) 0 0
\(940\) 4.99059 0.162775
\(941\) 34.3696 1.12042 0.560209 0.828351i \(-0.310721\pi\)
0.560209 + 0.828351i \(0.310721\pi\)
\(942\) 0 0
\(943\) 0.0311016 0.00101281
\(944\) −14.9202 −0.485610
\(945\) 0 0
\(946\) −1.07831 −0.0350589
\(947\) −5.24589 −0.170469 −0.0852343 0.996361i \(-0.527164\pi\)
−0.0852343 + 0.996361i \(0.527164\pi\)
\(948\) 0 0
\(949\) −32.9671 −1.07016
\(950\) 74.5123 2.41750
\(951\) 0 0
\(952\) −1.19473 −0.0387214
\(953\) −28.2817 −0.916135 −0.458068 0.888917i \(-0.651458\pi\)
−0.458068 + 0.888917i \(0.651458\pi\)
\(954\) 0 0
\(955\) 20.6640 0.668671
\(956\) 2.10552 0.0680974
\(957\) 0 0
\(958\) 37.3619 1.20711
\(959\) 8.91780 0.287971
\(960\) 0 0
\(961\) −14.6127 −0.471376
\(962\) −18.4995 −0.596448
\(963\) 0 0
\(964\) −1.38584 −0.0446349
\(965\) 71.6077 2.30513
\(966\) 0 0
\(967\) 3.52394 0.113322 0.0566611 0.998393i \(-0.481955\pi\)
0.0566611 + 0.998393i \(0.481955\pi\)
\(968\) 2.34402 0.0753397
\(969\) 0 0
\(970\) 34.1549 1.09665
\(971\) −30.9946 −0.994665 −0.497333 0.867560i \(-0.665687\pi\)
−0.497333 + 0.867560i \(0.665687\pi\)
\(972\) 0 0
\(973\) −6.93525 −0.222334
\(974\) −75.6759 −2.42481
\(975\) 0 0
\(976\) −13.7201 −0.439170
\(977\) 44.4189 1.42109 0.710543 0.703653i \(-0.248449\pi\)
0.710543 + 0.703653i \(0.248449\pi\)
\(978\) 0 0
\(979\) −1.90579 −0.0609092
\(980\) −37.5450 −1.19933
\(981\) 0 0
\(982\) 67.8001 2.16359
\(983\) −28.7370 −0.916568 −0.458284 0.888806i \(-0.651536\pi\)
−0.458284 + 0.888806i \(0.651536\pi\)
\(984\) 0 0
\(985\) 10.6571 0.339563
\(986\) −82.7333 −2.63477
\(987\) 0 0
\(988\) −46.1752 −1.46903
\(989\) 2.71971 0.0864816
\(990\) 0 0
\(991\) 18.3353 0.582441 0.291220 0.956656i \(-0.405939\pi\)
0.291220 + 0.956656i \(0.405939\pi\)
\(992\) −32.7181 −1.03880
\(993\) 0 0
\(994\) 24.8598 0.788504
\(995\) 80.6674 2.55733
\(996\) 0 0
\(997\) 18.7618 0.594192 0.297096 0.954848i \(-0.403982\pi\)
0.297096 + 0.954848i \(0.403982\pi\)
\(998\) 65.9777 2.08849
\(999\) 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 2151.2.a.k.1.3 yes 20
3.2 odd 2 2151.2.a.j.1.18 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.18 20 3.2 odd 2
2151.2.a.k.1.3 yes 20 1.1 even 1 trivial