Properties

Label 567.2.a.c.1.2
Level $567$
Weight $2$
Character 567.1
Self dual yes
Analytic conductor $4.528$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.52751779461\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 567.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.34730 q^{2} -0.184793 q^{4} -2.53209 q^{5} +1.00000 q^{7} +2.94356 q^{8} +O(q^{10})\) \(q-1.34730 q^{2} -0.184793 q^{4} -2.53209 q^{5} +1.00000 q^{7} +2.94356 q^{8} +3.41147 q^{10} -0.467911 q^{11} +5.82295 q^{13} -1.34730 q^{14} -3.59627 q^{16} -3.87939 q^{17} -2.18479 q^{19} +0.467911 q^{20} +0.630415 q^{22} +0.106067 q^{23} +1.41147 q^{25} -7.84524 q^{26} -0.184793 q^{28} -8.78106 q^{29} -7.68004 q^{31} -1.04189 q^{32} +5.22668 q^{34} -2.53209 q^{35} -7.68004 q^{37} +2.94356 q^{38} -7.45336 q^{40} +2.22668 q^{41} +1.22668 q^{43} +0.0864665 q^{44} -0.142903 q^{46} +5.33275 q^{47} +1.00000 q^{49} -1.90167 q^{50} -1.07604 q^{52} +0.716881 q^{53} +1.18479 q^{55} +2.94356 q^{56} +11.8307 q^{58} -0.736482 q^{59} +0.958111 q^{61} +10.3473 q^{62} +8.59627 q^{64} -14.7442 q^{65} -9.63816 q^{67} +0.716881 q^{68} +3.41147 q^{70} -13.2344 q^{71} -10.2686 q^{73} +10.3473 q^{74} +0.403733 q^{76} -0.467911 q^{77} -12.6382 q^{79} +9.10607 q^{80} -3.00000 q^{82} +2.73143 q^{83} +9.82295 q^{85} -1.65270 q^{86} -1.37733 q^{88} +8.11381 q^{89} +5.82295 q^{91} -0.0196004 q^{92} -7.18479 q^{94} +5.53209 q^{95} -13.6040 q^{97} -1.34730 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} - 6q^{8} - 6q^{11} - 3q^{13} - 3q^{14} + 3q^{16} - 6q^{17} - 3q^{19} + 6q^{20} + 9q^{22} - 12q^{23} - 6q^{25} + 3q^{26} + 3q^{28} - 9q^{29} - 3q^{31} + 9q^{34} - 3q^{35} - 3q^{37} - 6q^{38} - 9q^{40} - 3q^{43} - 15q^{44} - 3q^{47} + 3q^{49} + 6q^{50} - 21q^{52} - 6q^{53} - 6q^{56} - 9q^{58} + 3q^{59} + 6q^{61} + 30q^{62} + 12q^{64} - 15q^{65} - 12q^{67} - 6q^{68} - 9q^{71} - 21q^{73} + 30q^{74} + 15q^{76} - 6q^{77} - 21q^{79} + 15q^{80} - 9q^{82} + 18q^{83} + 9q^{85} - 6q^{86} + 27q^{88} - 12q^{89} - 3q^{91} - 3q^{92} - 18q^{94} + 12q^{95} - 3q^{97} - 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34730 −0.952682 −0.476341 0.879261i \(-0.658037\pi\)
−0.476341 + 0.879261i \(0.658037\pi\)
\(3\) 0 0
\(4\) −0.184793 −0.0923963
\(5\) −2.53209 −1.13238 −0.566192 0.824273i \(-0.691584\pi\)
−0.566192 + 0.824273i \(0.691584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.94356 1.04071
\(9\) 0 0
\(10\) 3.41147 1.07880
\(11\) −0.467911 −0.141081 −0.0705403 0.997509i \(-0.522472\pi\)
−0.0705403 + 0.997509i \(0.522472\pi\)
\(12\) 0 0
\(13\) 5.82295 1.61500 0.807498 0.589871i \(-0.200821\pi\)
0.807498 + 0.589871i \(0.200821\pi\)
\(14\) −1.34730 −0.360080
\(15\) 0 0
\(16\) −3.59627 −0.899067
\(17\) −3.87939 −0.940889 −0.470445 0.882430i \(-0.655906\pi\)
−0.470445 + 0.882430i \(0.655906\pi\)
\(18\) 0 0
\(19\) −2.18479 −0.501226 −0.250613 0.968087i \(-0.580632\pi\)
−0.250613 + 0.968087i \(0.580632\pi\)
\(20\) 0.467911 0.104628
\(21\) 0 0
\(22\) 0.630415 0.134405
\(23\) 0.106067 0.0221165 0.0110582 0.999939i \(-0.496480\pi\)
0.0110582 + 0.999939i \(0.496480\pi\)
\(24\) 0 0
\(25\) 1.41147 0.282295
\(26\) −7.84524 −1.53858
\(27\) 0 0
\(28\) −0.184793 −0.0349225
\(29\) −8.78106 −1.63060 −0.815301 0.579038i \(-0.803428\pi\)
−0.815301 + 0.579038i \(0.803428\pi\)
\(30\) 0 0
\(31\) −7.68004 −1.37938 −0.689688 0.724106i \(-0.742252\pi\)
−0.689688 + 0.724106i \(0.742252\pi\)
\(32\) −1.04189 −0.184182
\(33\) 0 0
\(34\) 5.22668 0.896368
\(35\) −2.53209 −0.428001
\(36\) 0 0
\(37\) −7.68004 −1.26259 −0.631296 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) 2.94356 0.477509
\(39\) 0 0
\(40\) −7.45336 −1.17848
\(41\) 2.22668 0.347749 0.173875 0.984768i \(-0.444371\pi\)
0.173875 + 0.984768i \(0.444371\pi\)
\(42\) 0 0
\(43\) 1.22668 0.187067 0.0935336 0.995616i \(-0.470184\pi\)
0.0935336 + 0.995616i \(0.470184\pi\)
\(44\) 0.0864665 0.0130353
\(45\) 0 0
\(46\) −0.142903 −0.0210700
\(47\) 5.33275 0.777861 0.388931 0.921267i \(-0.372845\pi\)
0.388931 + 0.921267i \(0.372845\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.90167 −0.268937
\(51\) 0 0
\(52\) −1.07604 −0.149220
\(53\) 0.716881 0.0984712 0.0492356 0.998787i \(-0.484321\pi\)
0.0492356 + 0.998787i \(0.484321\pi\)
\(54\) 0 0
\(55\) 1.18479 0.159757
\(56\) 2.94356 0.393350
\(57\) 0 0
\(58\) 11.8307 1.55345
\(59\) −0.736482 −0.0958818 −0.0479409 0.998850i \(-0.515266\pi\)
−0.0479409 + 0.998850i \(0.515266\pi\)
\(60\) 0 0
\(61\) 0.958111 0.122674 0.0613368 0.998117i \(-0.480464\pi\)
0.0613368 + 0.998117i \(0.480464\pi\)
\(62\) 10.3473 1.31411
\(63\) 0 0
\(64\) 8.59627 1.07453
\(65\) −14.7442 −1.82880
\(66\) 0 0
\(67\) −9.63816 −1.17749 −0.588744 0.808320i \(-0.700377\pi\)
−0.588744 + 0.808320i \(0.700377\pi\)
\(68\) 0.716881 0.0869346
\(69\) 0 0
\(70\) 3.41147 0.407749
\(71\) −13.2344 −1.57064 −0.785318 0.619092i \(-0.787501\pi\)
−0.785318 + 0.619092i \(0.787501\pi\)
\(72\) 0 0
\(73\) −10.2686 −1.20185 −0.600923 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(74\) 10.3473 1.20285
\(75\) 0 0
\(76\) 0.403733 0.0463114
\(77\) −0.467911 −0.0533234
\(78\) 0 0
\(79\) −12.6382 −1.42190 −0.710952 0.703241i \(-0.751736\pi\)
−0.710952 + 0.703241i \(0.751736\pi\)
\(80\) 9.10607 1.01809
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 2.73143 0.299813 0.149907 0.988700i \(-0.452103\pi\)
0.149907 + 0.988700i \(0.452103\pi\)
\(84\) 0 0
\(85\) 9.82295 1.06545
\(86\) −1.65270 −0.178216
\(87\) 0 0
\(88\) −1.37733 −0.146823
\(89\) 8.11381 0.860062 0.430031 0.902814i \(-0.358503\pi\)
0.430031 + 0.902814i \(0.358503\pi\)
\(90\) 0 0
\(91\) 5.82295 0.610411
\(92\) −0.0196004 −0.00204348
\(93\) 0 0
\(94\) −7.18479 −0.741055
\(95\) 5.53209 0.567580
\(96\) 0 0
\(97\) −13.6040 −1.38128 −0.690639 0.723200i \(-0.742671\pi\)
−0.690639 + 0.723200i \(0.742671\pi\)
\(98\) −1.34730 −0.136097
\(99\) 0 0
\(100\) −0.260830 −0.0260830
\(101\) 9.57398 0.952646 0.476323 0.879270i \(-0.341969\pi\)
0.476323 + 0.879270i \(0.341969\pi\)
\(102\) 0 0
\(103\) 3.04189 0.299726 0.149863 0.988707i \(-0.452117\pi\)
0.149863 + 0.988707i \(0.452117\pi\)
\(104\) 17.1402 1.68074
\(105\) 0 0
\(106\) −0.965852 −0.0938118
\(107\) 6.51754 0.630074 0.315037 0.949079i \(-0.397983\pi\)
0.315037 + 0.949079i \(0.397983\pi\)
\(108\) 0 0
\(109\) 10.6382 1.01895 0.509475 0.860485i \(-0.329840\pi\)
0.509475 + 0.860485i \(0.329840\pi\)
\(110\) −1.59627 −0.152198
\(111\) 0 0
\(112\) −3.59627 −0.339815
\(113\) −5.17705 −0.487016 −0.243508 0.969899i \(-0.578298\pi\)
−0.243508 + 0.969899i \(0.578298\pi\)
\(114\) 0 0
\(115\) −0.268571 −0.0250443
\(116\) 1.62267 0.150662
\(117\) 0 0
\(118\) 0.992259 0.0913449
\(119\) −3.87939 −0.355623
\(120\) 0 0
\(121\) −10.7811 −0.980096
\(122\) −1.29086 −0.116869
\(123\) 0 0
\(124\) 1.41921 0.127449
\(125\) 9.08647 0.812718
\(126\) 0 0
\(127\) −8.88207 −0.788157 −0.394078 0.919077i \(-0.628936\pi\)
−0.394078 + 0.919077i \(0.628936\pi\)
\(128\) −9.49794 −0.839507
\(129\) 0 0
\(130\) 19.8648 1.74226
\(131\) −11.3628 −0.992771 −0.496385 0.868102i \(-0.665340\pi\)
−0.496385 + 0.868102i \(0.665340\pi\)
\(132\) 0 0
\(133\) −2.18479 −0.189446
\(134\) 12.9855 1.12177
\(135\) 0 0
\(136\) −11.4192 −0.979190
\(137\) 5.72462 0.489087 0.244544 0.969638i \(-0.421362\pi\)
0.244544 + 0.969638i \(0.421362\pi\)
\(138\) 0 0
\(139\) −0.923963 −0.0783695 −0.0391847 0.999232i \(-0.512476\pi\)
−0.0391847 + 0.999232i \(0.512476\pi\)
\(140\) 0.467911 0.0395457
\(141\) 0 0
\(142\) 17.8307 1.49632
\(143\) −2.72462 −0.227844
\(144\) 0 0
\(145\) 22.2344 1.84647
\(146\) 13.8348 1.14498
\(147\) 0 0
\(148\) 1.41921 0.116659
\(149\) −8.72462 −0.714749 −0.357374 0.933961i \(-0.616328\pi\)
−0.357374 + 0.933961i \(0.616328\pi\)
\(150\) 0 0
\(151\) 18.4270 1.49956 0.749782 0.661685i \(-0.230158\pi\)
0.749782 + 0.661685i \(0.230158\pi\)
\(152\) −6.43107 −0.521629
\(153\) 0 0
\(154\) 0.630415 0.0508003
\(155\) 19.4466 1.56198
\(156\) 0 0
\(157\) 4.92396 0.392975 0.196488 0.980506i \(-0.437046\pi\)
0.196488 + 0.980506i \(0.437046\pi\)
\(158\) 17.0273 1.35462
\(159\) 0 0
\(160\) 2.63816 0.208565
\(161\) 0.106067 0.00835924
\(162\) 0 0
\(163\) 7.63816 0.598267 0.299133 0.954211i \(-0.403302\pi\)
0.299133 + 0.954211i \(0.403302\pi\)
\(164\) −0.411474 −0.0321307
\(165\) 0 0
\(166\) −3.68004 −0.285627
\(167\) 5.65539 0.437627 0.218814 0.975767i \(-0.429781\pi\)
0.218814 + 0.975767i \(0.429781\pi\)
\(168\) 0 0
\(169\) 20.9067 1.60821
\(170\) −13.2344 −1.01503
\(171\) 0 0
\(172\) −0.226682 −0.0172843
\(173\) −21.0692 −1.60186 −0.800932 0.598755i \(-0.795662\pi\)
−0.800932 + 0.598755i \(0.795662\pi\)
\(174\) 0 0
\(175\) 1.41147 0.106697
\(176\) 1.68273 0.126841
\(177\) 0 0
\(178\) −10.9317 −0.819366
\(179\) 5.12061 0.382733 0.191366 0.981519i \(-0.438708\pi\)
0.191366 + 0.981519i \(0.438708\pi\)
\(180\) 0 0
\(181\) −0.319955 −0.0237821 −0.0118910 0.999929i \(-0.503785\pi\)
−0.0118910 + 0.999929i \(0.503785\pi\)
\(182\) −7.84524 −0.581528
\(183\) 0 0
\(184\) 0.312214 0.0230168
\(185\) 19.4466 1.42974
\(186\) 0 0
\(187\) 1.81521 0.132741
\(188\) −0.985452 −0.0718715
\(189\) 0 0
\(190\) −7.45336 −0.540724
\(191\) 15.5672 1.12640 0.563200 0.826320i \(-0.309570\pi\)
0.563200 + 0.826320i \(0.309570\pi\)
\(192\) 0 0
\(193\) 6.04189 0.434905 0.217452 0.976071i \(-0.430225\pi\)
0.217452 + 0.976071i \(0.430225\pi\)
\(194\) 18.3286 1.31592
\(195\) 0 0
\(196\) −0.184793 −0.0131995
\(197\) −25.2344 −1.79788 −0.898939 0.438074i \(-0.855661\pi\)
−0.898939 + 0.438074i \(0.855661\pi\)
\(198\) 0 0
\(199\) 3.04189 0.215634 0.107817 0.994171i \(-0.465614\pi\)
0.107817 + 0.994171i \(0.465614\pi\)
\(200\) 4.15476 0.293786
\(201\) 0 0
\(202\) −12.8990 −0.907569
\(203\) −8.78106 −0.616310
\(204\) 0 0
\(205\) −5.63816 −0.393786
\(206\) −4.09833 −0.285544
\(207\) 0 0
\(208\) −20.9409 −1.45199
\(209\) 1.02229 0.0707132
\(210\) 0 0
\(211\) −5.45336 −0.375425 −0.187713 0.982224i \(-0.560107\pi\)
−0.187713 + 0.982224i \(0.560107\pi\)
\(212\) −0.132474 −0.00909837
\(213\) 0 0
\(214\) −8.78106 −0.600261
\(215\) −3.10607 −0.211832
\(216\) 0 0
\(217\) −7.68004 −0.521355
\(218\) −14.3327 −0.970736
\(219\) 0 0
\(220\) −0.218941 −0.0147610
\(221\) −22.5895 −1.51953
\(222\) 0 0
\(223\) 14.1925 0.950402 0.475201 0.879877i \(-0.342375\pi\)
0.475201 + 0.879877i \(0.342375\pi\)
\(224\) −1.04189 −0.0696141
\(225\) 0 0
\(226\) 6.97502 0.463972
\(227\) 2.89393 0.192077 0.0960385 0.995378i \(-0.469383\pi\)
0.0960385 + 0.995378i \(0.469383\pi\)
\(228\) 0 0
\(229\) 9.16756 0.605809 0.302905 0.953021i \(-0.402044\pi\)
0.302905 + 0.953021i \(0.402044\pi\)
\(230\) 0.361844 0.0238593
\(231\) 0 0
\(232\) −25.8476 −1.69698
\(233\) −13.2713 −0.869429 −0.434715 0.900568i \(-0.643151\pi\)
−0.434715 + 0.900568i \(0.643151\pi\)
\(234\) 0 0
\(235\) −13.5030 −0.880838
\(236\) 0.136096 0.00885912
\(237\) 0 0
\(238\) 5.22668 0.338795
\(239\) −9.53714 −0.616906 −0.308453 0.951240i \(-0.599811\pi\)
−0.308453 + 0.951240i \(0.599811\pi\)
\(240\) 0 0
\(241\) −8.95811 −0.577043 −0.288521 0.957473i \(-0.593164\pi\)
−0.288521 + 0.957473i \(0.593164\pi\)
\(242\) 14.5253 0.933720
\(243\) 0 0
\(244\) −0.177052 −0.0113346
\(245\) −2.53209 −0.161769
\(246\) 0 0
\(247\) −12.7219 −0.809477
\(248\) −22.6067 −1.43553
\(249\) 0 0
\(250\) −12.2422 −0.774262
\(251\) 24.9982 1.57788 0.788938 0.614473i \(-0.210631\pi\)
0.788938 + 0.614473i \(0.210631\pi\)
\(252\) 0 0
\(253\) −0.0496299 −0.00312020
\(254\) 11.9668 0.750863
\(255\) 0 0
\(256\) −4.39599 −0.274750
\(257\) −10.8520 −0.676932 −0.338466 0.940979i \(-0.609908\pi\)
−0.338466 + 0.940979i \(0.609908\pi\)
\(258\) 0 0
\(259\) −7.68004 −0.477215
\(260\) 2.72462 0.168974
\(261\) 0 0
\(262\) 15.3090 0.945795
\(263\) −26.0874 −1.60862 −0.804309 0.594211i \(-0.797464\pi\)
−0.804309 + 0.594211i \(0.797464\pi\)
\(264\) 0 0
\(265\) −1.81521 −0.111507
\(266\) 2.94356 0.180481
\(267\) 0 0
\(268\) 1.78106 0.108796
\(269\) 7.63310 0.465399 0.232699 0.972549i \(-0.425244\pi\)
0.232699 + 0.972549i \(0.425244\pi\)
\(270\) 0 0
\(271\) 3.40373 0.206762 0.103381 0.994642i \(-0.467034\pi\)
0.103381 + 0.994642i \(0.467034\pi\)
\(272\) 13.9513 0.845922
\(273\) 0 0
\(274\) −7.71276 −0.465945
\(275\) −0.660444 −0.0398263
\(276\) 0 0
\(277\) −5.72193 −0.343798 −0.171899 0.985115i \(-0.554990\pi\)
−0.171899 + 0.985115i \(0.554990\pi\)
\(278\) 1.24485 0.0746612
\(279\) 0 0
\(280\) −7.45336 −0.445424
\(281\) −28.3773 −1.69285 −0.846425 0.532508i \(-0.821249\pi\)
−0.846425 + 0.532508i \(0.821249\pi\)
\(282\) 0 0
\(283\) 4.57129 0.271735 0.135867 0.990727i \(-0.456618\pi\)
0.135867 + 0.990727i \(0.456618\pi\)
\(284\) 2.44562 0.145121
\(285\) 0 0
\(286\) 3.67087 0.217063
\(287\) 2.22668 0.131437
\(288\) 0 0
\(289\) −1.95037 −0.114728
\(290\) −29.9564 −1.75910
\(291\) 0 0
\(292\) 1.89756 0.111046
\(293\) −4.32770 −0.252827 −0.126413 0.991978i \(-0.540347\pi\)
−0.126413 + 0.991978i \(0.540347\pi\)
\(294\) 0 0
\(295\) 1.86484 0.108575
\(296\) −22.6067 −1.31399
\(297\) 0 0
\(298\) 11.7547 0.680929
\(299\) 0.617622 0.0357180
\(300\) 0 0
\(301\) 1.22668 0.0707048
\(302\) −24.8266 −1.42861
\(303\) 0 0
\(304\) 7.85710 0.450635
\(305\) −2.42602 −0.138914
\(306\) 0 0
\(307\) 12.3773 0.706411 0.353206 0.935546i \(-0.385092\pi\)
0.353206 + 0.935546i \(0.385092\pi\)
\(308\) 0.0864665 0.00492688
\(309\) 0 0
\(310\) −26.2003 −1.48808
\(311\) 21.9855 1.24668 0.623340 0.781951i \(-0.285775\pi\)
0.623340 + 0.781951i \(0.285775\pi\)
\(312\) 0 0
\(313\) −13.8898 −0.785099 −0.392549 0.919731i \(-0.628407\pi\)
−0.392549 + 0.919731i \(0.628407\pi\)
\(314\) −6.63404 −0.374380
\(315\) 0 0
\(316\) 2.33544 0.131379
\(317\) 6.18210 0.347222 0.173611 0.984814i \(-0.444457\pi\)
0.173611 + 0.984814i \(0.444457\pi\)
\(318\) 0 0
\(319\) 4.10876 0.230046
\(320\) −21.7665 −1.21678
\(321\) 0 0
\(322\) −0.142903 −0.00796370
\(323\) 8.47565 0.471598
\(324\) 0 0
\(325\) 8.21894 0.455905
\(326\) −10.2909 −0.569958
\(327\) 0 0
\(328\) 6.55438 0.361905
\(329\) 5.33275 0.294004
\(330\) 0 0
\(331\) 10.7314 0.589853 0.294926 0.955520i \(-0.404705\pi\)
0.294926 + 0.955520i \(0.404705\pi\)
\(332\) −0.504748 −0.0277016
\(333\) 0 0
\(334\) −7.61949 −0.416920
\(335\) 24.4047 1.33337
\(336\) 0 0
\(337\) −18.5945 −1.01291 −0.506454 0.862267i \(-0.669044\pi\)
−0.506454 + 0.862267i \(0.669044\pi\)
\(338\) −28.1676 −1.53211
\(339\) 0 0
\(340\) −1.81521 −0.0984434
\(341\) 3.59358 0.194603
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 3.61081 0.194682
\(345\) 0 0
\(346\) 28.3865 1.52607
\(347\) 20.4124 1.09580 0.547898 0.836545i \(-0.315428\pi\)
0.547898 + 0.836545i \(0.315428\pi\)
\(348\) 0 0
\(349\) −3.56212 −0.190676 −0.0953379 0.995445i \(-0.530393\pi\)
−0.0953379 + 0.995445i \(0.530393\pi\)
\(350\) −1.90167 −0.101649
\(351\) 0 0
\(352\) 0.487511 0.0259844
\(353\) −10.0223 −0.533433 −0.266716 0.963775i \(-0.585939\pi\)
−0.266716 + 0.963775i \(0.585939\pi\)
\(354\) 0 0
\(355\) 33.5107 1.77857
\(356\) −1.49937 −0.0794665
\(357\) 0 0
\(358\) −6.89899 −0.364623
\(359\) −9.48070 −0.500372 −0.250186 0.968198i \(-0.580492\pi\)
−0.250186 + 0.968198i \(0.580492\pi\)
\(360\) 0 0
\(361\) −14.2267 −0.748773
\(362\) 0.431074 0.0226568
\(363\) 0 0
\(364\) −1.07604 −0.0563997
\(365\) 26.0009 1.36095
\(366\) 0 0
\(367\) 16.1334 0.842157 0.421079 0.907024i \(-0.361652\pi\)
0.421079 + 0.907024i \(0.361652\pi\)
\(368\) −0.381445 −0.0198842
\(369\) 0 0
\(370\) −26.2003 −1.36209
\(371\) 0.716881 0.0372186
\(372\) 0 0
\(373\) 14.0496 0.727462 0.363731 0.931504i \(-0.381503\pi\)
0.363731 + 0.931504i \(0.381503\pi\)
\(374\) −2.44562 −0.126460
\(375\) 0 0
\(376\) 15.6973 0.809525
\(377\) −51.1317 −2.63341
\(378\) 0 0
\(379\) 16.0574 0.824812 0.412406 0.911000i \(-0.364689\pi\)
0.412406 + 0.911000i \(0.364689\pi\)
\(380\) −1.02229 −0.0524423
\(381\) 0 0
\(382\) −20.9736 −1.07310
\(383\) 32.0205 1.63617 0.818086 0.575095i \(-0.195035\pi\)
0.818086 + 0.575095i \(0.195035\pi\)
\(384\) 0 0
\(385\) 1.18479 0.0603826
\(386\) −8.14022 −0.414326
\(387\) 0 0
\(388\) 2.51392 0.127625
\(389\) 30.0428 1.52323 0.761616 0.648029i \(-0.224406\pi\)
0.761616 + 0.648029i \(0.224406\pi\)
\(390\) 0 0
\(391\) −0.411474 −0.0208091
\(392\) 2.94356 0.148672
\(393\) 0 0
\(394\) 33.9982 1.71281
\(395\) 32.0009 1.61014
\(396\) 0 0
\(397\) −12.3200 −0.618321 −0.309160 0.951010i \(-0.600048\pi\)
−0.309160 + 0.951010i \(0.600048\pi\)
\(398\) −4.09833 −0.205431
\(399\) 0 0
\(400\) −5.07604 −0.253802
\(401\) −20.9760 −1.04749 −0.523745 0.851875i \(-0.675465\pi\)
−0.523745 + 0.851875i \(0.675465\pi\)
\(402\) 0 0
\(403\) −44.7205 −2.22769
\(404\) −1.76920 −0.0880210
\(405\) 0 0
\(406\) 11.8307 0.587147
\(407\) 3.59358 0.178127
\(408\) 0 0
\(409\) 25.6614 1.26887 0.634437 0.772975i \(-0.281232\pi\)
0.634437 + 0.772975i \(0.281232\pi\)
\(410\) 7.59627 0.375153
\(411\) 0 0
\(412\) −0.562118 −0.0276936
\(413\) −0.736482 −0.0362399
\(414\) 0 0
\(415\) −6.91622 −0.339504
\(416\) −6.06687 −0.297453
\(417\) 0 0
\(418\) −1.37733 −0.0673672
\(419\) 1.47977 0.0722915 0.0361458 0.999347i \(-0.488492\pi\)
0.0361458 + 0.999347i \(0.488492\pi\)
\(420\) 0 0
\(421\) 13.1070 0.638796 0.319398 0.947621i \(-0.396519\pi\)
0.319398 + 0.947621i \(0.396519\pi\)
\(422\) 7.34730 0.357661
\(423\) 0 0
\(424\) 2.11019 0.102480
\(425\) −5.47565 −0.265608
\(426\) 0 0
\(427\) 0.958111 0.0463662
\(428\) −1.20439 −0.0582165
\(429\) 0 0
\(430\) 4.18479 0.201809
\(431\) −17.7270 −0.853879 −0.426939 0.904280i \(-0.640408\pi\)
−0.426939 + 0.904280i \(0.640408\pi\)
\(432\) 0 0
\(433\) −5.83843 −0.280577 −0.140289 0.990111i \(-0.544803\pi\)
−0.140289 + 0.990111i \(0.544803\pi\)
\(434\) 10.3473 0.496686
\(435\) 0 0
\(436\) −1.96585 −0.0941472
\(437\) −0.231734 −0.0110853
\(438\) 0 0
\(439\) 29.8553 1.42492 0.712459 0.701714i \(-0.247582\pi\)
0.712459 + 0.701714i \(0.247582\pi\)
\(440\) 3.48751 0.166261
\(441\) 0 0
\(442\) 30.4347 1.44763
\(443\) −10.6655 −0.506733 −0.253367 0.967370i \(-0.581538\pi\)
−0.253367 + 0.967370i \(0.581538\pi\)
\(444\) 0 0
\(445\) −20.5449 −0.973921
\(446\) −19.1215 −0.905432
\(447\) 0 0
\(448\) 8.59627 0.406135
\(449\) −3.55438 −0.167741 −0.0838707 0.996477i \(-0.526728\pi\)
−0.0838707 + 0.996477i \(0.526728\pi\)
\(450\) 0 0
\(451\) −1.04189 −0.0490606
\(452\) 0.956680 0.0449985
\(453\) 0 0
\(454\) −3.89899 −0.182988
\(455\) −14.7442 −0.691220
\(456\) 0 0
\(457\) 5.02322 0.234976 0.117488 0.993074i \(-0.462516\pi\)
0.117488 + 0.993074i \(0.462516\pi\)
\(458\) −12.3514 −0.577144
\(459\) 0 0
\(460\) 0.0496299 0.00231400
\(461\) −18.4611 −0.859819 −0.429910 0.902872i \(-0.641455\pi\)
−0.429910 + 0.902872i \(0.641455\pi\)
\(462\) 0 0
\(463\) −14.2344 −0.661530 −0.330765 0.943713i \(-0.607307\pi\)
−0.330765 + 0.943713i \(0.607307\pi\)
\(464\) 31.5790 1.46602
\(465\) 0 0
\(466\) 17.8803 0.828290
\(467\) 3.36865 0.155883 0.0779413 0.996958i \(-0.475165\pi\)
0.0779413 + 0.996958i \(0.475165\pi\)
\(468\) 0 0
\(469\) −9.63816 −0.445049
\(470\) 18.1925 0.839159
\(471\) 0 0
\(472\) −2.16788 −0.0997848
\(473\) −0.573978 −0.0263915
\(474\) 0 0
\(475\) −3.08378 −0.141493
\(476\) 0.716881 0.0328582
\(477\) 0 0
\(478\) 12.8494 0.587716
\(479\) 36.7665 1.67990 0.839952 0.542660i \(-0.182583\pi\)
0.839952 + 0.542660i \(0.182583\pi\)
\(480\) 0 0
\(481\) −44.7205 −2.03908
\(482\) 12.0692 0.549738
\(483\) 0 0
\(484\) 1.99226 0.0905572
\(485\) 34.4466 1.56414
\(486\) 0 0
\(487\) −37.4175 −1.69555 −0.847773 0.530358i \(-0.822057\pi\)
−0.847773 + 0.530358i \(0.822057\pi\)
\(488\) 2.82026 0.127667
\(489\) 0 0
\(490\) 3.41147 0.154115
\(491\) 26.6705 1.20363 0.601813 0.798637i \(-0.294445\pi\)
0.601813 + 0.798637i \(0.294445\pi\)
\(492\) 0 0
\(493\) 34.0651 1.53422
\(494\) 17.1402 0.771175
\(495\) 0 0
\(496\) 27.6195 1.24015
\(497\) −13.2344 −0.593645
\(498\) 0 0
\(499\) 33.7452 1.51064 0.755320 0.655356i \(-0.227481\pi\)
0.755320 + 0.655356i \(0.227481\pi\)
\(500\) −1.67911 −0.0750921
\(501\) 0 0
\(502\) −33.6800 −1.50321
\(503\) 32.0401 1.42860 0.714299 0.699840i \(-0.246745\pi\)
0.714299 + 0.699840i \(0.246745\pi\)
\(504\) 0 0
\(505\) −24.2422 −1.07876
\(506\) 0.0668661 0.00297256
\(507\) 0 0
\(508\) 1.64134 0.0728227
\(509\) 7.93851 0.351868 0.175934 0.984402i \(-0.443705\pi\)
0.175934 + 0.984402i \(0.443705\pi\)
\(510\) 0 0
\(511\) −10.2686 −0.454255
\(512\) 24.9186 1.10126
\(513\) 0 0
\(514\) 14.6209 0.644901
\(515\) −7.70233 −0.339405
\(516\) 0 0
\(517\) −2.49525 −0.109741
\(518\) 10.3473 0.454634
\(519\) 0 0
\(520\) −43.4005 −1.90324
\(521\) 14.6750 0.642923 0.321462 0.946923i \(-0.395826\pi\)
0.321462 + 0.946923i \(0.395826\pi\)
\(522\) 0 0
\(523\) 28.3432 1.23936 0.619680 0.784854i \(-0.287262\pi\)
0.619680 + 0.784854i \(0.287262\pi\)
\(524\) 2.09976 0.0917283
\(525\) 0 0
\(526\) 35.1475 1.53250
\(527\) 29.7939 1.29784
\(528\) 0 0
\(529\) −22.9887 −0.999511
\(530\) 2.44562 0.106231
\(531\) 0 0
\(532\) 0.403733 0.0175041
\(533\) 12.9659 0.561613
\(534\) 0 0
\(535\) −16.5030 −0.713487
\(536\) −28.3705 −1.22542
\(537\) 0 0
\(538\) −10.2841 −0.443377
\(539\) −0.467911 −0.0201544
\(540\) 0 0
\(541\) 11.2858 0.485215 0.242607 0.970125i \(-0.421997\pi\)
0.242607 + 0.970125i \(0.421997\pi\)
\(542\) −4.58584 −0.196979
\(543\) 0 0
\(544\) 4.04189 0.173295
\(545\) −26.9368 −1.15384
\(546\) 0 0
\(547\) −29.2404 −1.25023 −0.625115 0.780533i \(-0.714948\pi\)
−0.625115 + 0.780533i \(0.714948\pi\)
\(548\) −1.05787 −0.0451899
\(549\) 0 0
\(550\) 0.889814 0.0379418
\(551\) 19.1848 0.817300
\(552\) 0 0
\(553\) −12.6382 −0.537429
\(554\) 7.70914 0.327530
\(555\) 0 0
\(556\) 0.170741 0.00724105
\(557\) 0.775682 0.0328667 0.0164334 0.999865i \(-0.494769\pi\)
0.0164334 + 0.999865i \(0.494769\pi\)
\(558\) 0 0
\(559\) 7.14290 0.302113
\(560\) 9.10607 0.384802
\(561\) 0 0
\(562\) 38.2327 1.61275
\(563\) −24.9522 −1.05161 −0.525806 0.850605i \(-0.676236\pi\)
−0.525806 + 0.850605i \(0.676236\pi\)
\(564\) 0 0
\(565\) 13.1088 0.551489
\(566\) −6.15888 −0.258877
\(567\) 0 0
\(568\) −38.9564 −1.63457
\(569\) 24.8033 1.03981 0.519905 0.854224i \(-0.325967\pi\)
0.519905 + 0.854224i \(0.325967\pi\)
\(570\) 0 0
\(571\) 8.79654 0.368124 0.184062 0.982915i \(-0.441075\pi\)
0.184062 + 0.982915i \(0.441075\pi\)
\(572\) 0.503490 0.0210520
\(573\) 0 0
\(574\) −3.00000 −0.125218
\(575\) 0.149711 0.00624336
\(576\) 0 0
\(577\) −12.8743 −0.535965 −0.267983 0.963424i \(-0.586357\pi\)
−0.267983 + 0.963424i \(0.586357\pi\)
\(578\) 2.62773 0.109299
\(579\) 0 0
\(580\) −4.10876 −0.170607
\(581\) 2.73143 0.113319
\(582\) 0 0
\(583\) −0.335437 −0.0138924
\(584\) −30.2262 −1.25077
\(585\) 0 0
\(586\) 5.83069 0.240864
\(587\) −44.8631 −1.85170 −0.925849 0.377894i \(-0.876648\pi\)
−0.925849 + 0.377894i \(0.876648\pi\)
\(588\) 0 0
\(589\) 16.7793 0.691379
\(590\) −2.51249 −0.103438
\(591\) 0 0
\(592\) 27.6195 1.13515
\(593\) −3.76053 −0.154426 −0.0772131 0.997015i \(-0.524602\pi\)
−0.0772131 + 0.997015i \(0.524602\pi\)
\(594\) 0 0
\(595\) 9.82295 0.402702
\(596\) 1.61225 0.0660401
\(597\) 0 0
\(598\) −0.832119 −0.0340279
\(599\) 3.69047 0.150789 0.0753943 0.997154i \(-0.475978\pi\)
0.0753943 + 0.997154i \(0.475978\pi\)
\(600\) 0 0
\(601\) −21.8571 −0.891570 −0.445785 0.895140i \(-0.647075\pi\)
−0.445785 + 0.895140i \(0.647075\pi\)
\(602\) −1.65270 −0.0673592
\(603\) 0 0
\(604\) −3.40516 −0.138554
\(605\) 27.2986 1.10985
\(606\) 0 0
\(607\) 24.3946 0.990145 0.495072 0.868852i \(-0.335142\pi\)
0.495072 + 0.868852i \(0.335142\pi\)
\(608\) 2.27631 0.0923166
\(609\) 0 0
\(610\) 3.26857 0.132341
\(611\) 31.0523 1.25624
\(612\) 0 0
\(613\) 42.0215 1.69723 0.848616 0.529010i \(-0.177437\pi\)
0.848616 + 0.529010i \(0.177437\pi\)
\(614\) −16.6759 −0.672986
\(615\) 0 0
\(616\) −1.37733 −0.0554940
\(617\) −46.4097 −1.86838 −0.934192 0.356769i \(-0.883878\pi\)
−0.934192 + 0.356769i \(0.883878\pi\)
\(618\) 0 0
\(619\) −27.2094 −1.09364 −0.546820 0.837250i \(-0.684162\pi\)
−0.546820 + 0.837250i \(0.684162\pi\)
\(620\) −3.59358 −0.144322
\(621\) 0 0
\(622\) −29.6209 −1.18769
\(623\) 8.11381 0.325073
\(624\) 0 0
\(625\) −30.0651 −1.20260
\(626\) 18.7137 0.747950
\(627\) 0 0
\(628\) −0.909912 −0.0363094
\(629\) 29.7939 1.18796
\(630\) 0 0
\(631\) −29.6023 −1.17845 −0.589224 0.807970i \(-0.700566\pi\)
−0.589224 + 0.807970i \(0.700566\pi\)
\(632\) −37.2012 −1.47978
\(633\) 0 0
\(634\) −8.32913 −0.330792
\(635\) 22.4902 0.892496
\(636\) 0 0
\(637\) 5.82295 0.230714
\(638\) −5.53571 −0.219161
\(639\) 0 0
\(640\) 24.0496 0.950645
\(641\) 0.279000 0.0110198 0.00550991 0.999985i \(-0.498246\pi\)
0.00550991 + 0.999985i \(0.498246\pi\)
\(642\) 0 0
\(643\) −18.2439 −0.719470 −0.359735 0.933055i \(-0.617133\pi\)
−0.359735 + 0.933055i \(0.617133\pi\)
\(644\) −0.0196004 −0.000772362 0
\(645\) 0 0
\(646\) −11.4192 −0.449283
\(647\) −22.4570 −0.882875 −0.441438 0.897292i \(-0.645531\pi\)
−0.441438 + 0.897292i \(0.645531\pi\)
\(648\) 0 0
\(649\) 0.344608 0.0135270
\(650\) −11.0733 −0.434332
\(651\) 0 0
\(652\) −1.41147 −0.0552776
\(653\) 50.5313 1.97744 0.988721 0.149771i \(-0.0478538\pi\)
0.988721 + 0.149771i \(0.0478538\pi\)
\(654\) 0 0
\(655\) 28.7716 1.12420
\(656\) −8.00774 −0.312650
\(657\) 0 0
\(658\) −7.18479 −0.280092
\(659\) 2.67263 0.104111 0.0520554 0.998644i \(-0.483423\pi\)
0.0520554 + 0.998644i \(0.483423\pi\)
\(660\) 0 0
\(661\) −34.6100 −1.34617 −0.673086 0.739564i \(-0.735032\pi\)
−0.673086 + 0.739564i \(0.735032\pi\)
\(662\) −14.4584 −0.561942
\(663\) 0 0
\(664\) 8.04013 0.312018
\(665\) 5.53209 0.214525
\(666\) 0 0
\(667\) −0.931379 −0.0360631
\(668\) −1.04507 −0.0404351
\(669\) 0 0
\(670\) −32.8803 −1.27028
\(671\) −0.448311 −0.0173068
\(672\) 0 0
\(673\) 16.5125 0.636510 0.318255 0.948005i \(-0.396903\pi\)
0.318255 + 0.948005i \(0.396903\pi\)
\(674\) 25.0523 0.964979
\(675\) 0 0
\(676\) −3.86341 −0.148593
\(677\) 43.7579 1.68175 0.840877 0.541226i \(-0.182040\pi\)
0.840877 + 0.541226i \(0.182040\pi\)
\(678\) 0 0
\(679\) −13.6040 −0.522074
\(680\) 28.9145 1.10882
\(681\) 0 0
\(682\) −4.84161 −0.185395
\(683\) −28.2412 −1.08062 −0.540310 0.841466i \(-0.681693\pi\)
−0.540310 + 0.841466i \(0.681693\pi\)
\(684\) 0 0
\(685\) −14.4953 −0.553835
\(686\) −1.34730 −0.0514400
\(687\) 0 0
\(688\) −4.41147 −0.168186
\(689\) 4.17436 0.159031
\(690\) 0 0
\(691\) −29.0651 −1.10569 −0.552844 0.833284i \(-0.686458\pi\)
−0.552844 + 0.833284i \(0.686458\pi\)
\(692\) 3.89344 0.148006
\(693\) 0 0
\(694\) −27.5016 −1.04395
\(695\) 2.33956 0.0887444
\(696\) 0 0
\(697\) −8.63816 −0.327193
\(698\) 4.79923 0.181654
\(699\) 0 0
\(700\) −0.260830 −0.00985844
\(701\) 1.10876 0.0418771 0.0209386 0.999781i \(-0.493335\pi\)
0.0209386 + 0.999781i \(0.493335\pi\)
\(702\) 0 0
\(703\) 16.7793 0.632843
\(704\) −4.02229 −0.151596
\(705\) 0 0
\(706\) 13.5030 0.508192
\(707\) 9.57398 0.360066
\(708\) 0 0
\(709\) −18.4688 −0.693612 −0.346806 0.937937i \(-0.612734\pi\)
−0.346806 + 0.937937i \(0.612734\pi\)
\(710\) −45.1489 −1.69441
\(711\) 0 0
\(712\) 23.8835 0.895072
\(713\) −0.814598 −0.0305069
\(714\) 0 0
\(715\) 6.89899 0.258007
\(716\) −0.946251 −0.0353631
\(717\) 0 0
\(718\) 12.7733 0.476696
\(719\) 33.7769 1.25967 0.629834 0.776730i \(-0.283123\pi\)
0.629834 + 0.776730i \(0.283123\pi\)
\(720\) 0 0
\(721\) 3.04189 0.113286
\(722\) 19.1676 0.713343
\(723\) 0 0
\(724\) 0.0591253 0.00219738
\(725\) −12.3942 −0.460310
\(726\) 0 0
\(727\) 16.8043 0.623236 0.311618 0.950207i \(-0.399129\pi\)
0.311618 + 0.950207i \(0.399129\pi\)
\(728\) 17.1402 0.635259
\(729\) 0 0
\(730\) −35.0310 −1.29655
\(731\) −4.75877 −0.176009
\(732\) 0 0
\(733\) −13.6364 −0.503672 −0.251836 0.967770i \(-0.581034\pi\)
−0.251836 + 0.967770i \(0.581034\pi\)
\(734\) −21.7365 −0.802308
\(735\) 0 0
\(736\) −0.110510 −0.00407345
\(737\) 4.50980 0.166121
\(738\) 0 0
\(739\) −32.0419 −1.17868 −0.589340 0.807885i \(-0.700612\pi\)
−0.589340 + 0.807885i \(0.700612\pi\)
\(740\) −3.59358 −0.132103
\(741\) 0 0
\(742\) −0.965852 −0.0354575
\(743\) −33.7529 −1.23827 −0.619137 0.785283i \(-0.712517\pi\)
−0.619137 + 0.785283i \(0.712517\pi\)
\(744\) 0 0
\(745\) 22.0915 0.809371
\(746\) −18.9290 −0.693040
\(747\) 0 0
\(748\) −0.335437 −0.0122648
\(749\) 6.51754 0.238146
\(750\) 0 0
\(751\) 26.1165 0.953004 0.476502 0.879173i \(-0.341904\pi\)
0.476502 + 0.879173i \(0.341904\pi\)
\(752\) −19.1780 −0.699349
\(753\) 0 0
\(754\) 68.8895 2.50881
\(755\) −46.6587 −1.69808
\(756\) 0 0
\(757\) 35.6536 1.29585 0.647927 0.761703i \(-0.275636\pi\)
0.647927 + 0.761703i \(0.275636\pi\)
\(758\) −21.6340 −0.785784
\(759\) 0 0
\(760\) 16.2841 0.590685
\(761\) −40.7648 −1.47772 −0.738861 0.673858i \(-0.764636\pi\)
−0.738861 + 0.673858i \(0.764636\pi\)
\(762\) 0 0
\(763\) 10.6382 0.385127
\(764\) −2.87670 −0.104075
\(765\) 0 0
\(766\) −43.1411 −1.55875
\(767\) −4.28850 −0.154849
\(768\) 0 0
\(769\) 39.4270 1.42177 0.710886 0.703307i \(-0.248294\pi\)
0.710886 + 0.703307i \(0.248294\pi\)
\(770\) −1.59627 −0.0575255
\(771\) 0 0
\(772\) −1.11650 −0.0401836
\(773\) −24.9026 −0.895685 −0.447842 0.894113i \(-0.647807\pi\)
−0.447842 + 0.894113i \(0.647807\pi\)
\(774\) 0 0
\(775\) −10.8402 −0.389391
\(776\) −40.0443 −1.43750
\(777\) 0 0
\(778\) −40.4766 −1.45116
\(779\) −4.86484 −0.174301
\(780\) 0 0
\(781\) 6.19253 0.221586
\(782\) 0.554378 0.0198245
\(783\) 0 0
\(784\) −3.59627 −0.128438
\(785\) −12.4679 −0.444999
\(786\) 0 0
\(787\) −30.7050 −1.09452 −0.547258 0.836964i \(-0.684328\pi\)
−0.547258 + 0.836964i \(0.684328\pi\)
\(788\) 4.66313 0.166117
\(789\) 0 0
\(790\) −43.1147 −1.53395
\(791\) −5.17705 −0.184075
\(792\) 0 0
\(793\) 5.57903 0.198117
\(794\) 16.5986 0.589063
\(795\) 0 0
\(796\) −0.562118 −0.0199238
\(797\) 11.0137 0.390126 0.195063 0.980791i \(-0.437509\pi\)
0.195063 + 0.980791i \(0.437509\pi\)
\(798\) 0 0
\(799\) −20.6878 −0.731881
\(800\) −1.47060 −0.0519935
\(801\) 0 0
\(802\) 28.2608 0.997925
\(803\) 4.80478 0.169557
\(804\) 0 0
\(805\) −0.268571 −0.00946587
\(806\) 60.2518 2.12228
\(807\) 0 0
\(808\) 28.1816 0.991425
\(809\) −16.9881 −0.597271 −0.298636 0.954367i \(-0.596532\pi\)
−0.298636 + 0.954367i \(0.596532\pi\)
\(810\) 0 0
\(811\) 37.9796 1.33364 0.666822 0.745217i \(-0.267654\pi\)
0.666822 + 0.745217i \(0.267654\pi\)
\(812\) 1.62267 0.0569447
\(813\) 0 0
\(814\) −4.84161 −0.169699
\(815\) −19.3405 −0.677468
\(816\) 0 0
\(817\) −2.68004 −0.0937629
\(818\) −34.5735 −1.20883
\(819\) 0 0
\(820\) 1.04189 0.0363843
\(821\) −8.27868 −0.288928 −0.144464 0.989510i \(-0.546146\pi\)
−0.144464 + 0.989510i \(0.546146\pi\)
\(822\) 0 0
\(823\) 54.5526 1.90158 0.950792 0.309829i \(-0.100272\pi\)
0.950792 + 0.309829i \(0.100272\pi\)
\(824\) 8.95399 0.311927
\(825\) 0 0
\(826\) 0.992259 0.0345251
\(827\) 31.8708 1.10826 0.554129 0.832431i \(-0.313052\pi\)
0.554129 + 0.832431i \(0.313052\pi\)
\(828\) 0 0
\(829\) −0.352349 −0.0122376 −0.00611879 0.999981i \(-0.501948\pi\)
−0.00611879 + 0.999981i \(0.501948\pi\)
\(830\) 9.31820 0.323439
\(831\) 0 0
\(832\) 50.0556 1.73537
\(833\) −3.87939 −0.134413
\(834\) 0 0
\(835\) −14.3200 −0.495562
\(836\) −0.188911 −0.00653363
\(837\) 0 0
\(838\) −1.99369 −0.0688709
\(839\) −25.0155 −0.863630 −0.431815 0.901962i \(-0.642127\pi\)
−0.431815 + 0.901962i \(0.642127\pi\)
\(840\) 0 0
\(841\) 48.1070 1.65886
\(842\) −17.6590 −0.608570
\(843\) 0 0
\(844\) 1.00774 0.0346879
\(845\) −52.9377 −1.82111
\(846\) 0 0
\(847\) −10.7811 −0.370442
\(848\) −2.57810 −0.0885322
\(849\) 0 0
\(850\) 7.37733 0.253040
\(851\) −0.814598 −0.0279241
\(852\) 0 0
\(853\) 39.1908 1.34187 0.670933 0.741518i \(-0.265894\pi\)
0.670933 + 0.741518i \(0.265894\pi\)
\(854\) −1.29086 −0.0441723
\(855\) 0 0
\(856\) 19.1848 0.655723
\(857\) −16.4074 −0.560465 −0.280232 0.959932i \(-0.590411\pi\)
−0.280232 + 0.959932i \(0.590411\pi\)
\(858\) 0 0
\(859\) −26.8324 −0.915511 −0.457756 0.889078i \(-0.651347\pi\)
−0.457756 + 0.889078i \(0.651347\pi\)
\(860\) 0.573978 0.0195725
\(861\) 0 0
\(862\) 23.8835 0.813475
\(863\) −14.5057 −0.493779 −0.246890 0.969044i \(-0.579408\pi\)
−0.246890 + 0.969044i \(0.579408\pi\)
\(864\) 0 0
\(865\) 53.3492 1.81393
\(866\) 7.86610 0.267301
\(867\) 0 0
\(868\) 1.41921 0.0481713
\(869\) 5.91353 0.200603
\(870\) 0 0
\(871\) −56.1225 −1.90164
\(872\) 31.3141 1.06043
\(873\) 0 0
\(874\) 0.312214 0.0105608
\(875\) 9.08647 0.307179
\(876\) 0 0
\(877\) 18.9145 0.638696 0.319348 0.947637i \(-0.396536\pi\)
0.319348 + 0.947637i \(0.396536\pi\)
\(878\) −40.2240 −1.35749
\(879\) 0 0
\(880\) −4.26083 −0.143633
\(881\) −53.8976 −1.81585 −0.907927 0.419128i \(-0.862336\pi\)
−0.907927 + 0.419128i \(0.862336\pi\)
\(882\) 0 0
\(883\) 43.4252 1.46137 0.730687 0.682712i \(-0.239200\pi\)
0.730687 + 0.682712i \(0.239200\pi\)
\(884\) 4.17436 0.140399
\(885\) 0 0
\(886\) 14.3696 0.482756
\(887\) 38.9600 1.30815 0.654074 0.756431i \(-0.273058\pi\)
0.654074 + 0.756431i \(0.273058\pi\)
\(888\) 0 0
\(889\) −8.88207 −0.297895
\(890\) 27.6800 0.927837
\(891\) 0 0
\(892\) −2.62267 −0.0878136
\(893\) −11.6509 −0.389884
\(894\) 0 0
\(895\) −12.9659 −0.433401
\(896\) −9.49794 −0.317304
\(897\) 0 0
\(898\) 4.78880 0.159804
\(899\) 67.4389 2.24921
\(900\) 0 0
\(901\) −2.78106 −0.0926505
\(902\) 1.40373 0.0467392
\(903\) 0 0
\(904\) −15.2390 −0.506841
\(905\) 0.810155 0.0269305
\(906\) 0 0
\(907\) 34.5276 1.14647 0.573236 0.819390i \(-0.305688\pi\)
0.573236 + 0.819390i \(0.305688\pi\)
\(908\) −0.534777 −0.0177472
\(909\) 0 0
\(910\) 19.8648 0.658513
\(911\) −46.5262 −1.54148 −0.770741 0.637148i \(-0.780114\pi\)
−0.770741 + 0.637148i \(0.780114\pi\)
\(912\) 0 0
\(913\) −1.27807 −0.0422978
\(914\) −6.76777 −0.223858
\(915\) 0 0
\(916\) −1.69410 −0.0559745
\(917\) −11.3628 −0.375232
\(918\) 0 0
\(919\) −9.95636 −0.328430 −0.164215 0.986425i \(-0.552509\pi\)
−0.164215 + 0.986425i \(0.552509\pi\)
\(920\) −0.790555 −0.0260638
\(921\) 0 0
\(922\) 24.8726 0.819135
\(923\) −77.0634 −2.53657
\(924\) 0 0
\(925\) −10.8402 −0.356423
\(926\) 19.1780 0.630228
\(927\) 0 0
\(928\) 9.14889 0.300327
\(929\) −9.04601 −0.296790 −0.148395 0.988928i \(-0.547411\pi\)
−0.148395 + 0.988928i \(0.547411\pi\)
\(930\) 0 0
\(931\) −2.18479 −0.0716037
\(932\) 2.45243 0.0803320
\(933\) 0 0
\(934\) −4.53857 −0.148507
\(935\) −4.59627 −0.150314
\(936\) 0 0
\(937\) −24.3928 −0.796878 −0.398439 0.917195i \(-0.630448\pi\)
−0.398439 + 0.917195i \(0.630448\pi\)
\(938\) 12.9855 0.423990
\(939\) 0 0
\(940\) 2.49525 0.0813862
\(941\) 59.5381 1.94089 0.970443 0.241331i \(-0.0775839\pi\)
0.970443 + 0.241331i \(0.0775839\pi\)
\(942\) 0 0
\(943\) 0.236177 0.00769098
\(944\) 2.64858 0.0862041
\(945\) 0 0
\(946\) 0.773318 0.0251428
\(947\) −8.64858 −0.281041 −0.140521 0.990078i \(-0.544878\pi\)
−0.140521 + 0.990078i \(0.544878\pi\)
\(948\) 0 0
\(949\) −59.7934 −1.94097
\(950\) 4.15476 0.134798
\(951\) 0 0
\(952\) −11.4192 −0.370099
\(953\) −3.78249 −0.122527 −0.0612634 0.998122i \(-0.519513\pi\)
−0.0612634 + 0.998122i \(0.519513\pi\)
\(954\) 0 0
\(955\) −39.4175 −1.27552
\(956\) 1.76239 0.0569998
\(957\) 0 0
\(958\) −49.5354 −1.60042
\(959\) 5.72462 0.184858
\(960\) 0 0
\(961\) 27.9831 0.902680
\(962\) 60.2518 1.94260
\(963\) 0 0
\(964\) 1.65539 0.0533166
\(965\) −15.2986 −0.492479
\(966\) 0 0
\(967\) −32.9489 −1.05957 −0.529783 0.848133i \(-0.677727\pi\)
−0.529783 + 0.848133i \(0.677727\pi\)
\(968\) −31.7347 −1.01999
\(969\) 0 0
\(970\) −46.4097 −1.49013
\(971\) 55.4570 1.77970 0.889850 0.456254i \(-0.150809\pi\)
0.889850 + 0.456254i \(0.150809\pi\)
\(972\) 0 0
\(973\) −0.923963 −0.0296209
\(974\) 50.4124 1.61532
\(975\) 0 0
\(976\) −3.44562 −0.110292
\(977\) −56.5485 −1.80915 −0.904573 0.426318i \(-0.859811\pi\)
−0.904573 + 0.426318i \(0.859811\pi\)
\(978\) 0 0
\(979\) −3.79654 −0.121338
\(980\) 0.467911 0.0149469
\(981\) 0 0
\(982\) −35.9331 −1.14667
\(983\) −28.9973 −0.924871 −0.462435 0.886653i \(-0.653024\pi\)
−0.462435 + 0.886653i \(0.653024\pi\)
\(984\) 0 0
\(985\) 63.8958 2.03589
\(986\) −45.8958 −1.46162
\(987\) 0 0
\(988\) 2.35092 0.0747927
\(989\) 0.130110 0.00413726
\(990\) 0 0
\(991\) 6.80922 0.216302 0.108151 0.994134i \(-0.465507\pi\)
0.108151 + 0.994134i \(0.465507\pi\)
\(992\) 8.00175 0.254056
\(993\) 0 0
\(994\) 17.8307 0.565555
\(995\) −7.70233 −0.244180
\(996\) 0 0
\(997\) −38.9377 −1.23317 −0.616585 0.787289i \(-0.711484\pi\)
−0.616585 + 0.787289i \(0.711484\pi\)
\(998\) −45.4647 −1.43916
\(999\) 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 567.2.a.c.1.2 3
3.2 odd 2 567.2.a.h.1.2 3
4.3 odd 2 9072.2.a.bs.1.1 3
7.6 odd 2 3969.2.a.l.1.2 3
9.2 odd 6 63.2.f.a.22.2 6
9.4 even 3 189.2.f.b.127.2 6
9.5 odd 6 63.2.f.a.43.2 yes 6
9.7 even 3 189.2.f.b.64.2 6
12.11 even 2 9072.2.a.ca.1.3 3
21.20 even 2 3969.2.a.q.1.2 3
36.7 odd 6 3024.2.r.k.1009.3 6
36.11 even 6 1008.2.r.h.337.1 6
36.23 even 6 1008.2.r.h.673.1 6
36.31 odd 6 3024.2.r.k.2017.3 6
63.2 odd 6 441.2.g.c.67.2 6
63.4 even 3 1323.2.g.d.667.2 6
63.5 even 6 441.2.h.e.214.2 6
63.11 odd 6 441.2.h.d.373.2 6
63.13 odd 6 1323.2.f.d.883.2 6
63.16 even 3 1323.2.g.d.361.2 6
63.20 even 6 441.2.f.c.148.2 6
63.23 odd 6 441.2.h.d.214.2 6
63.25 even 3 1323.2.h.c.226.2 6
63.31 odd 6 1323.2.g.e.667.2 6
63.32 odd 6 441.2.g.c.79.2 6
63.34 odd 6 1323.2.f.d.442.2 6
63.38 even 6 441.2.h.e.373.2 6
63.40 odd 6 1323.2.h.b.802.2 6
63.41 even 6 441.2.f.c.295.2 6
63.47 even 6 441.2.g.b.67.2 6
63.52 odd 6 1323.2.h.b.226.2 6
63.58 even 3 1323.2.h.c.802.2 6
63.59 even 6 441.2.g.b.79.2 6
63.61 odd 6 1323.2.g.e.361.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 6 9.2 odd 6
63.2.f.a.43.2 yes 6 9.5 odd 6
189.2.f.b.64.2 6 9.7 even 3
189.2.f.b.127.2 6 9.4 even 3
441.2.f.c.148.2 6 63.20 even 6
441.2.f.c.295.2 6 63.41 even 6
441.2.g.b.67.2 6 63.47 even 6
441.2.g.b.79.2 6 63.59 even 6
441.2.g.c.67.2 6 63.2 odd 6
441.2.g.c.79.2 6 63.32 odd 6
441.2.h.d.214.2 6 63.23 odd 6
441.2.h.d.373.2 6 63.11 odd 6
441.2.h.e.214.2 6 63.5 even 6
441.2.h.e.373.2 6 63.38 even 6
567.2.a.c.1.2 3 1.1 even 1 trivial
567.2.a.h.1.2 3 3.2 odd 2
1008.2.r.h.337.1 6 36.11 even 6
1008.2.r.h.673.1 6 36.23 even 6
1323.2.f.d.442.2 6 63.34 odd 6
1323.2.f.d.883.2 6 63.13 odd 6
1323.2.g.d.361.2 6 63.16 even 3
1323.2.g.d.667.2 6 63.4 even 3
1323.2.g.e.361.2 6 63.61 odd 6
1323.2.g.e.667.2 6 63.31 odd 6
1323.2.h.b.226.2 6 63.52 odd 6
1323.2.h.b.802.2 6 63.40 odd 6
1323.2.h.c.226.2 6 63.25 even 3
1323.2.h.c.802.2 6 63.58 even 3
3024.2.r.k.1009.3 6 36.7 odd 6
3024.2.r.k.2017.3 6 36.31 odd 6
3969.2.a.l.1.2 3 7.6 odd 2
3969.2.a.q.1.2 3 21.20 even 2
9072.2.a.bs.1.1 3 4.3 odd 2
9072.2.a.ca.1.3 3 12.11 even 2