Properties

Label 2151.2.a.j.1.18
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
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: \(1\)
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 1.18
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 \(-0.245801\pi\)
0.716373 + 0.697718i \(0.245801\pi\)
\(3\) 0 0
\(4\) 2.10552 1.05276
\(5\) −3.48073 −1.55663 −0.778314 0.627875i \(-0.783925\pi\)
−0.778314 + 0.627875i \(0.783925\pi\)
\(6\) 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 \(-0.509194\pi\)
−0.0288794 + 0.999583i \(0.509194\pi\)
\(12\) 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 \(-0.664688\pi\)
−0.494607 + 0.869117i \(0.664688\pi\)
\(18\) 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 \(-0.532545\pi\)
−0.102066 + 0.994778i \(0.532545\pi\)
\(24\) 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 \(-0.879767\pi\)
−0.929507 + 0.368806i \(0.879767\pi\)
\(30\) 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 \(-0.500790\pi\)
−0.00248076 + 0.999997i \(0.500790\pi\)
\(42\) 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 \(-0.515815\pi\)
−0.0496643 + 0.998766i \(0.515815\pi\)
\(48\) 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 \(-0.609558\pi\)
−0.337431 + 0.941350i \(0.609558\pi\)
\(54\) 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 \(-0.582762\pi\)
−0.257085 + 0.966389i \(0.582762\pi\)
\(60\) 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 \(-0.321667\pi\)
0.531397 + 0.847123i \(0.321667\pi\)
\(72\) 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 \(-0.539043\pi\)
−0.122349 + 0.992487i \(0.539043\pi\)
\(84\) 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 \(-0.323215\pi\)
0.527272 + 0.849696i \(0.323215\pi\)
\(90\) 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 \(-0.720631\pi\)
−0.638950 + 0.769249i \(0.720631\pi\)
\(102\) 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 \(-0.605305\pi\)
−0.324825 + 0.945774i \(0.605305\pi\)
\(108\) 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 \(-0.304389\pi\)
0.576574 + 0.817045i \(0.304389\pi\)
\(114\) 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 \(-0.269171\pi\)
0.663262 + 0.748387i \(0.269171\pi\)
\(132\) 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 \(-0.589690\pi\)
−0.278057 + 0.960565i \(0.589690\pi\)
\(138\) 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 \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(150\) 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 \(-0.513459\pi\)
−0.0422687 + 0.999106i \(0.513459\pi\)
\(168\) 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 \(-0.630321\pi\)
−0.398074 + 0.917353i \(0.630321\pi\)
\(174\) 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 \(-0.198357\pi\)
0.812041 + 0.583600i \(0.198357\pi\)
\(180\) 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 \(-0.568904\pi\)
−0.214782 + 0.976662i \(0.568904\pi\)
\(192\) 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 \(-0.534787\pi\)
−0.109070 + 0.994034i \(0.534787\pi\)
\(198\) 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 \(-0.668867\pi\)
−0.505975 + 0.862548i \(0.668867\pi\)
\(228\) 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 \(-0.364778\pi\)
0.412149 + 0.911116i \(0.364778\pi\)
\(234\) 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 \(-0.699017\pi\)
−0.585283 + 0.810829i \(0.699017\pi\)
\(252\) 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 \(-0.754739\pi\)
−0.717555 + 0.696502i \(0.754739\pi\)
\(258\) 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 \(-0.537231\pi\)
−0.116698 + 0.993167i \(0.537231\pi\)
\(264\) 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 \(-0.474498\pi\)
0.0800325 + 0.996792i \(0.474498\pi\)
\(270\) 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 \(-0.231847\pi\)
0.746260 + 0.665654i \(0.231847\pi\)
\(282\) 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 \(-0.286348\pi\)
0.621932 + 0.783071i \(0.286348\pi\)
\(294\) 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 \(-0.716589\pi\)
−0.629130 + 0.777300i \(0.716589\pi\)
\(312\) 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 \(-0.403988\pi\)
0.297076 + 0.954854i \(0.403988\pi\)
\(318\) 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 \(-0.288274\pi\)
0.617182 + 0.786820i \(0.288274\pi\)
\(348\) 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 \(-0.378811\pi\)
0.371594 + 0.928395i \(0.378811\pi\)
\(354\) 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 \(-0.694947\pi\)
−0.574869 + 0.818245i \(0.694947\pi\)
\(360\) 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 \(-0.849316\pi\)
−0.890029 + 0.455903i \(0.849316\pi\)
\(384\) 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 \(-0.514349\pi\)
−0.0450643 + 0.998984i \(0.514349\pi\)
\(390\) 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 \(-0.719131\pi\)
−0.635319 + 0.772250i \(0.719131\pi\)
\(402\) 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 \(-0.800290\pi\)
−0.809553 + 0.587047i \(0.800290\pi\)
\(420\) 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 \(-0.697732\pi\)
−0.582006 + 0.813185i \(0.697732\pi\)
\(432\) 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 \(-0.523387\pi\)
−0.0734069 + 0.997302i \(0.523387\pi\)
\(444\) 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 \(-0.557329\pi\)
−0.179132 + 0.983825i \(0.557329\pi\)
\(450\) 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 \(-0.451239\pi\)
0.152589 + 0.988290i \(0.451239\pi\)
\(462\) 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 \(-0.859118\pi\)
−0.903644 + 0.428285i \(0.859118\pi\)
\(468\) 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 \(-0.361589\pi\)
0.421257 + 0.906942i \(0.361589\pi\)
\(480\) 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 \(-0.227613\pi\)
0.755050 + 0.655668i \(0.227613\pi\)
\(492\) 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 \(-0.530462\pi\)
−0.0955545 + 0.995424i \(0.530462\pi\)
\(504\) 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 \(-0.716489\pi\)
−0.628887 + 0.777497i \(0.716489\pi\)
\(510\) 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 \(-0.241976\pi\)
0.724706 + 0.689058i \(0.241976\pi\)
\(522\) 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 \(-0.612151\pi\)
−0.345089 + 0.938570i \(0.612151\pi\)
\(558\) 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 \(-0.498550\pi\)
0.00455580 + 0.999990i \(0.498550\pi\)
\(564\) 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 \(-0.592274\pi\)
−0.285845 + 0.958276i \(0.592274\pi\)
\(570\) 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 \(-0.545297\pi\)
−0.141826 + 0.989892i \(0.545297\pi\)
\(588\) 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 \(-0.422678\pi\)
0.240534 + 0.970641i \(0.422678\pi\)
\(594\) 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 \(-0.708992\pi\)
−0.610403 + 0.792091i \(0.708992\pi\)
\(600\) 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 \(-0.0642947\pi\)
0.979670 + 0.200617i \(0.0642947\pi\)
\(618\) 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 \(-0.576143\pi\)
−0.236936 + 0.971525i \(0.576143\pi\)
\(642\) 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 \(-0.434464\pi\)
0.204435 + 0.978880i \(0.434464\pi\)
\(648\) 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 \(-0.517449\pi\)
−0.0547891 + 0.998498i \(0.517449\pi\)
\(654\) 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 \(-0.315189\pi\)
0.548525 + 0.836134i \(0.315189\pi\)
\(660\) 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 \(-0.750308\pi\)
−0.707791 + 0.706422i \(0.750308\pi\)
\(678\) 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 \(-0.334405\pi\)
0.497080 + 0.867705i \(0.334405\pi\)
\(684\) 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 \(-0.678906\pi\)
−0.532921 + 0.846165i \(0.678906\pi\)
\(702\) 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 \(-0.0903735\pi\)
0.959966 + 0.280118i \(0.0903735\pi\)
\(720\) 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 \(-0.185608\pi\)
0.834757 + 0.550619i \(0.185608\pi\)
\(744\) 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 \(-0.783696\pi\)
−0.777862 + 0.628435i \(0.783696\pi\)
\(762\) 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 \(-0.599860\pi\)
−0.308600 + 0.951192i \(0.599860\pi\)
\(774\) 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 \(-0.148415\pi\)
0.893257 + 0.449547i \(0.148415\pi\)
\(798\) 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 \(-0.271546\pi\)
0.657661 + 0.753314i \(0.271546\pi\)
\(810\) 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 \(-0.857499\pi\)
−0.901453 + 0.432877i \(0.857499\pi\)
\(822\) 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 \(-0.500771\pi\)
−0.00242370 + 0.999997i \(0.500771\pi\)
\(828\) 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 \(-0.541323\pi\)
−0.129456 + 0.991585i \(0.541323\pi\)
\(840\) 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 \(-0.447021\pi\)
0.165671 + 0.986181i \(0.447021\pi\)
\(858\) 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 \(-0.723370\pi\)
−0.645545 + 0.763722i \(0.723370\pi\)
\(864\) 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 \(-0.682414\pi\)
−0.542216 + 0.840239i \(0.682414\pi\)
\(882\) 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 \(-0.314552\pi\)
0.550198 + 0.835034i \(0.314552\pi\)
\(888\) 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 \(-0.433204\pi\)
0.208309 + 0.978063i \(0.433204\pi\)
\(912\) 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 \(-0.423574\pi\)
0.237799 + 0.971314i \(0.423574\pi\)
\(930\) 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 \(-0.689279\pi\)
−0.560209 + 0.828351i \(0.689279\pi\)
\(942\) 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 \(-0.472836\pi\)
0.0852343 + 0.996361i \(0.472836\pi\)
\(948\) 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 \(-0.348542\pi\)
0.458068 + 0.888917i \(0.348542\pi\)
\(954\) 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 \(-0.334313\pi\)
0.497333 + 0.867560i \(0.334313\pi\)
\(972\) 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 \(-0.751551\pi\)
−0.710543 + 0.703653i \(0.751551\pi\)
\(978\) 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 \(-0.348464\pi\)
0.458284 + 0.888806i \(0.348464\pi\)
\(984\) 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.j.1.18 20
3.2 odd 2 2151.2.a.k.1.3 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.18 20 1.1 even 1 trivial
2151.2.a.k.1.3 yes 20 3.2 odd 2