Properties

Label 567.2.a.c.1.1
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 1.1
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 567.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.53209 q^{2} +4.41147 q^{4} +0.879385 q^{5} +1.00000 q^{7} -6.10607 q^{8} +O(q^{10})\) \(q-2.53209 q^{2} +4.41147 q^{4} +0.879385 q^{5} +1.00000 q^{7} -6.10607 q^{8} -2.22668 q^{10} -3.87939 q^{11} -5.45336 q^{13} -2.53209 q^{14} +6.63816 q^{16} -1.65270 q^{17} +2.41147 q^{19} +3.87939 q^{20} +9.82295 q^{22} -3.16250 q^{23} -4.22668 q^{25} +13.8084 q^{26} +4.41147 q^{28} +6.04963 q^{29} -4.55438 q^{31} -4.59627 q^{32} +4.18479 q^{34} +0.879385 q^{35} -4.55438 q^{37} -6.10607 q^{38} -5.36959 q^{40} +1.18479 q^{41} +0.184793 q^{43} -17.1138 q^{44} +8.00774 q^{46} +1.02229 q^{47} +1.00000 q^{49} +10.7023 q^{50} -24.0574 q^{52} -7.29086 q^{53} -3.41147 q^{55} -6.10607 q^{56} -15.3182 q^{58} -6.66044 q^{59} -2.59627 q^{61} +11.5321 q^{62} -1.63816 q^{64} -4.79561 q^{65} -2.95811 q^{67} -7.29086 q^{68} -2.22668 q^{70} +3.68004 q^{71} -12.7811 q^{73} +11.5321 q^{74} +10.6382 q^{76} -3.87939 q^{77} -5.95811 q^{79} +5.83750 q^{80} -3.00000 q^{82} +0.218941 q^{83} -1.45336 q^{85} -0.467911 q^{86} +23.6878 q^{88} -11.0273 q^{89} -5.45336 q^{91} -13.9513 q^{92} -2.58853 q^{94} +2.12061 q^{95} +12.5030 q^{97} -2.53209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} - 6q^{8} - 6q^{11} - 3q^{13} - 3q^{14} + 3q^{16} - 6q^{17} - 3q^{19} + 6q^{20} + 9q^{22} - 12q^{23} - 6q^{25} + 3q^{26} + 3q^{28} - 9q^{29} - 3q^{31} + 9q^{34} - 3q^{35} - 3q^{37} - 6q^{38} - 9q^{40} - 3q^{43} - 15q^{44} - 3q^{47} + 3q^{49} + 6q^{50} - 21q^{52} - 6q^{53} - 6q^{56} - 9q^{58} + 3q^{59} + 6q^{61} + 30q^{62} + 12q^{64} - 15q^{65} - 12q^{67} - 6q^{68} - 9q^{71} - 21q^{73} + 30q^{74} + 15q^{76} - 6q^{77} - 21q^{79} + 15q^{80} - 9q^{82} + 18q^{83} + 9q^{85} - 6q^{86} + 27q^{88} - 12q^{89} - 3q^{91} - 3q^{92} - 18q^{94} + 12q^{95} - 3q^{97} - 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.53209 −1.79046 −0.895229 0.445607i \(-0.852988\pi\)
−0.895229 + 0.445607i \(0.852988\pi\)
\(3\) 0 0
\(4\) 4.41147 2.20574
\(5\) 0.879385 0.393273 0.196637 0.980476i \(-0.436998\pi\)
0.196637 + 0.980476i \(0.436998\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −6.10607 −2.15882
\(9\) 0 0
\(10\) −2.22668 −0.704139
\(11\) −3.87939 −1.16968 −0.584839 0.811149i \(-0.698842\pi\)
−0.584839 + 0.811149i \(0.698842\pi\)
\(12\) 0 0
\(13\) −5.45336 −1.51249 −0.756245 0.654288i \(-0.772968\pi\)
−0.756245 + 0.654288i \(0.772968\pi\)
\(14\) −2.53209 −0.676729
\(15\) 0 0
\(16\) 6.63816 1.65954
\(17\) −1.65270 −0.400840 −0.200420 0.979710i \(-0.564231\pi\)
−0.200420 + 0.979710i \(0.564231\pi\)
\(18\) 0 0
\(19\) 2.41147 0.553230 0.276615 0.960981i \(-0.410787\pi\)
0.276615 + 0.960981i \(0.410787\pi\)
\(20\) 3.87939 0.867457
\(21\) 0 0
\(22\) 9.82295 2.09426
\(23\) −3.16250 −0.659428 −0.329714 0.944081i \(-0.606952\pi\)
−0.329714 + 0.944081i \(0.606952\pi\)
\(24\) 0 0
\(25\) −4.22668 −0.845336
\(26\) 13.8084 2.70805
\(27\) 0 0
\(28\) 4.41147 0.833690
\(29\) 6.04963 1.12339 0.561694 0.827345i \(-0.310150\pi\)
0.561694 + 0.827345i \(0.310150\pi\)
\(30\) 0 0
\(31\) −4.55438 −0.817990 −0.408995 0.912537i \(-0.634121\pi\)
−0.408995 + 0.912537i \(0.634121\pi\)
\(32\) −4.59627 −0.812513
\(33\) 0 0
\(34\) 4.18479 0.717686
\(35\) 0.879385 0.148643
\(36\) 0 0
\(37\) −4.55438 −0.748735 −0.374368 0.927280i \(-0.622140\pi\)
−0.374368 + 0.927280i \(0.622140\pi\)
\(38\) −6.10607 −0.990535
\(39\) 0 0
\(40\) −5.36959 −0.849006
\(41\) 1.18479 0.185034 0.0925168 0.995711i \(-0.470509\pi\)
0.0925168 + 0.995711i \(0.470509\pi\)
\(42\) 0 0
\(43\) 0.184793 0.0281806 0.0140903 0.999901i \(-0.495515\pi\)
0.0140903 + 0.999901i \(0.495515\pi\)
\(44\) −17.1138 −2.58000
\(45\) 0 0
\(46\) 8.00774 1.18068
\(47\) 1.02229 0.149116 0.0745581 0.997217i \(-0.476245\pi\)
0.0745581 + 0.997217i \(0.476245\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 10.7023 1.51354
\(51\) 0 0
\(52\) −24.0574 −3.33616
\(53\) −7.29086 −1.00148 −0.500738 0.865599i \(-0.666938\pi\)
−0.500738 + 0.865599i \(0.666938\pi\)
\(54\) 0 0
\(55\) −3.41147 −0.460003
\(56\) −6.10607 −0.815958
\(57\) 0 0
\(58\) −15.3182 −2.01138
\(59\) −6.66044 −0.867116 −0.433558 0.901126i \(-0.642742\pi\)
−0.433558 + 0.901126i \(0.642742\pi\)
\(60\) 0 0
\(61\) −2.59627 −0.332418 −0.166209 0.986091i \(-0.553153\pi\)
−0.166209 + 0.986091i \(0.553153\pi\)
\(62\) 11.5321 1.46458
\(63\) 0 0
\(64\) −1.63816 −0.204769
\(65\) −4.79561 −0.594822
\(66\) 0 0
\(67\) −2.95811 −0.361391 −0.180695 0.983539i \(-0.557835\pi\)
−0.180695 + 0.983539i \(0.557835\pi\)
\(68\) −7.29086 −0.884147
\(69\) 0 0
\(70\) −2.22668 −0.266139
\(71\) 3.68004 0.436741 0.218370 0.975866i \(-0.429926\pi\)
0.218370 + 0.975866i \(0.429926\pi\)
\(72\) 0 0
\(73\) −12.7811 −1.49591 −0.747955 0.663750i \(-0.768964\pi\)
−0.747955 + 0.663750i \(0.768964\pi\)
\(74\) 11.5321 1.34058
\(75\) 0 0
\(76\) 10.6382 1.22028
\(77\) −3.87939 −0.442097
\(78\) 0 0
\(79\) −5.95811 −0.670340 −0.335170 0.942158i \(-0.608794\pi\)
−0.335170 + 0.942158i \(0.608794\pi\)
\(80\) 5.83750 0.652652
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 0.218941 0.0240319 0.0120159 0.999928i \(-0.496175\pi\)
0.0120159 + 0.999928i \(0.496175\pi\)
\(84\) 0 0
\(85\) −1.45336 −0.157639
\(86\) −0.467911 −0.0504562
\(87\) 0 0
\(88\) 23.6878 2.52513
\(89\) −11.0273 −1.16890 −0.584448 0.811431i \(-0.698689\pi\)
−0.584448 + 0.811431i \(0.698689\pi\)
\(90\) 0 0
\(91\) −5.45336 −0.571668
\(92\) −13.9513 −1.45452
\(93\) 0 0
\(94\) −2.58853 −0.266986
\(95\) 2.12061 0.217570
\(96\) 0 0
\(97\) 12.5030 1.26949 0.634743 0.772723i \(-0.281106\pi\)
0.634743 + 0.772723i \(0.281106\pi\)
\(98\) −2.53209 −0.255780
\(99\) 0 0
\(100\) −18.6459 −1.86459
\(101\) 9.71688 0.966866 0.483433 0.875381i \(-0.339390\pi\)
0.483433 + 0.875381i \(0.339390\pi\)
\(102\) 0 0
\(103\) 6.59627 0.649949 0.324975 0.945723i \(-0.394644\pi\)
0.324975 + 0.945723i \(0.394644\pi\)
\(104\) 33.2986 3.26520
\(105\) 0 0
\(106\) 18.4611 1.79310
\(107\) −2.38919 −0.230971 −0.115486 0.993309i \(-0.536842\pi\)
−0.115486 + 0.993309i \(0.536842\pi\)
\(108\) 0 0
\(109\) 3.95811 0.379118 0.189559 0.981869i \(-0.439294\pi\)
0.189559 + 0.981869i \(0.439294\pi\)
\(110\) 8.63816 0.823616
\(111\) 0 0
\(112\) 6.63816 0.627247
\(113\) −16.4534 −1.54780 −0.773901 0.633306i \(-0.781697\pi\)
−0.773901 + 0.633306i \(0.781697\pi\)
\(114\) 0 0
\(115\) −2.78106 −0.259335
\(116\) 26.6878 2.47790
\(117\) 0 0
\(118\) 16.8648 1.55253
\(119\) −1.65270 −0.151503
\(120\) 0 0
\(121\) 4.04963 0.368148
\(122\) 6.57398 0.595180
\(123\) 0 0
\(124\) −20.0915 −1.80427
\(125\) −8.11381 −0.725721
\(126\) 0 0
\(127\) 17.6536 1.56651 0.783253 0.621702i \(-0.213559\pi\)
0.783253 + 0.621702i \(0.213559\pi\)
\(128\) 13.3405 1.17914
\(129\) 0 0
\(130\) 12.1429 1.06500
\(131\) 19.1976 1.67730 0.838650 0.544670i \(-0.183345\pi\)
0.838650 + 0.544670i \(0.183345\pi\)
\(132\) 0 0
\(133\) 2.41147 0.209101
\(134\) 7.49020 0.647055
\(135\) 0 0
\(136\) 10.0915 0.865341
\(137\) −18.1557 −1.55115 −0.775573 0.631258i \(-0.782539\pi\)
−0.775573 + 0.631258i \(0.782539\pi\)
\(138\) 0 0
\(139\) 22.0574 1.87088 0.935441 0.353483i \(-0.115003\pi\)
0.935441 + 0.353483i \(0.115003\pi\)
\(140\) 3.87939 0.327868
\(141\) 0 0
\(142\) −9.31820 −0.781966
\(143\) 21.1557 1.76913
\(144\) 0 0
\(145\) 5.31996 0.441798
\(146\) 32.3628 2.67836
\(147\) 0 0
\(148\) −20.0915 −1.65151
\(149\) 15.1557 1.24160 0.620802 0.783968i \(-0.286807\pi\)
0.620802 + 0.783968i \(0.286807\pi\)
\(150\) 0 0
\(151\) −18.9564 −1.54265 −0.771323 0.636444i \(-0.780405\pi\)
−0.771323 + 0.636444i \(0.780405\pi\)
\(152\) −14.7246 −1.19432
\(153\) 0 0
\(154\) 9.82295 0.791556
\(155\) −4.00505 −0.321694
\(156\) 0 0
\(157\) −18.0574 −1.44114 −0.720568 0.693385i \(-0.756119\pi\)
−0.720568 + 0.693385i \(0.756119\pi\)
\(158\) 15.0865 1.20021
\(159\) 0 0
\(160\) −4.04189 −0.319539
\(161\) −3.16250 −0.249240
\(162\) 0 0
\(163\) 0.958111 0.0750450 0.0375225 0.999296i \(-0.488053\pi\)
0.0375225 + 0.999296i \(0.488053\pi\)
\(164\) 5.22668 0.408135
\(165\) 0 0
\(166\) −0.554378 −0.0430280
\(167\) −19.8384 −1.53514 −0.767572 0.640963i \(-0.778535\pi\)
−0.767572 + 0.640963i \(0.778535\pi\)
\(168\) 0 0
\(169\) 16.7392 1.28763
\(170\) 3.68004 0.282247
\(171\) 0 0
\(172\) 0.815207 0.0621590
\(173\) −22.6827 −1.72454 −0.862268 0.506452i \(-0.830957\pi\)
−0.862268 + 0.506452i \(0.830957\pi\)
\(174\) 0 0
\(175\) −4.22668 −0.319507
\(176\) −25.7520 −1.94113
\(177\) 0 0
\(178\) 27.9222 2.09286
\(179\) 7.34730 0.549163 0.274581 0.961564i \(-0.411461\pi\)
0.274581 + 0.961564i \(0.411461\pi\)
\(180\) 0 0
\(181\) −3.44562 −0.256111 −0.128056 0.991767i \(-0.540874\pi\)
−0.128056 + 0.991767i \(0.540874\pi\)
\(182\) 13.8084 1.02355
\(183\) 0 0
\(184\) 19.3105 1.42359
\(185\) −4.00505 −0.294457
\(186\) 0 0
\(187\) 6.41147 0.468853
\(188\) 4.50980 0.328911
\(189\) 0 0
\(190\) −5.36959 −0.389551
\(191\) −5.65776 −0.409381 −0.204690 0.978827i \(-0.565619\pi\)
−0.204690 + 0.978827i \(0.565619\pi\)
\(192\) 0 0
\(193\) 9.59627 0.690754 0.345377 0.938464i \(-0.387751\pi\)
0.345377 + 0.938464i \(0.387751\pi\)
\(194\) −31.6587 −2.27296
\(195\) 0 0
\(196\) 4.41147 0.315105
\(197\) −8.31996 −0.592772 −0.296386 0.955068i \(-0.595782\pi\)
−0.296386 + 0.955068i \(0.595782\pi\)
\(198\) 0 0
\(199\) 6.59627 0.467597 0.233798 0.972285i \(-0.424884\pi\)
0.233798 + 0.972285i \(0.424884\pi\)
\(200\) 25.8084 1.82493
\(201\) 0 0
\(202\) −24.6040 −1.73113
\(203\) 6.04963 0.424601
\(204\) 0 0
\(205\) 1.04189 0.0727687
\(206\) −16.7023 −1.16371
\(207\) 0 0
\(208\) −36.2003 −2.51004
\(209\) −9.35504 −0.647101
\(210\) 0 0
\(211\) −3.36959 −0.231972 −0.115986 0.993251i \(-0.537003\pi\)
−0.115986 + 0.993251i \(0.537003\pi\)
\(212\) −32.1634 −2.20899
\(213\) 0 0
\(214\) 6.04963 0.413544
\(215\) 0.162504 0.0110827
\(216\) 0 0
\(217\) −4.55438 −0.309171
\(218\) −10.0223 −0.678795
\(219\) 0 0
\(220\) −15.0496 −1.01465
\(221\) 9.01279 0.606266
\(222\) 0 0
\(223\) −6.27631 −0.420293 −0.210146 0.977670i \(-0.567394\pi\)
−0.210146 + 0.977670i \(0.567394\pi\)
\(224\) −4.59627 −0.307101
\(225\) 0 0
\(226\) 41.6614 2.77127
\(227\) 6.16250 0.409020 0.204510 0.978865i \(-0.434440\pi\)
0.204510 + 0.978865i \(0.434440\pi\)
\(228\) 0 0
\(229\) 23.3851 1.54533 0.772664 0.634815i \(-0.218924\pi\)
0.772664 + 0.634815i \(0.218924\pi\)
\(230\) 7.04189 0.464328
\(231\) 0 0
\(232\) −36.9394 −2.42519
\(233\) 8.52528 0.558510 0.279255 0.960217i \(-0.409913\pi\)
0.279255 + 0.960217i \(0.409913\pi\)
\(234\) 0 0
\(235\) 0.898986 0.0586434
\(236\) −29.3824 −1.91263
\(237\) 0 0
\(238\) 4.18479 0.271260
\(239\) −14.5621 −0.941945 −0.470973 0.882148i \(-0.656097\pi\)
−0.470973 + 0.882148i \(0.656097\pi\)
\(240\) 0 0
\(241\) −5.40373 −0.348085 −0.174043 0.984738i \(-0.555683\pi\)
−0.174043 + 0.984738i \(0.555683\pi\)
\(242\) −10.2540 −0.659154
\(243\) 0 0
\(244\) −11.4534 −0.733226
\(245\) 0.879385 0.0561819
\(246\) 0 0
\(247\) −13.1506 −0.836755
\(248\) 27.8093 1.76589
\(249\) 0 0
\(250\) 20.5449 1.29937
\(251\) 12.0669 0.761654 0.380827 0.924646i \(-0.375639\pi\)
0.380827 + 0.924646i \(0.375639\pi\)
\(252\) 0 0
\(253\) 12.2686 0.771318
\(254\) −44.7006 −2.80476
\(255\) 0 0
\(256\) −30.5030 −1.90644
\(257\) −10.5662 −0.659104 −0.329552 0.944137i \(-0.606898\pi\)
−0.329552 + 0.944137i \(0.606898\pi\)
\(258\) 0 0
\(259\) −4.55438 −0.282995
\(260\) −21.1557 −1.31202
\(261\) 0 0
\(262\) −48.6100 −3.00314
\(263\) 28.3533 1.74834 0.874169 0.485622i \(-0.161407\pi\)
0.874169 + 0.485622i \(0.161407\pi\)
\(264\) 0 0
\(265\) −6.41147 −0.393854
\(266\) −6.10607 −0.374387
\(267\) 0 0
\(268\) −13.0496 −0.797133
\(269\) −7.48339 −0.456271 −0.228135 0.973629i \(-0.573263\pi\)
−0.228135 + 0.973629i \(0.573263\pi\)
\(270\) 0 0
\(271\) 13.6382 0.828459 0.414229 0.910172i \(-0.364051\pi\)
0.414229 + 0.910172i \(0.364051\pi\)
\(272\) −10.9709 −0.665209
\(273\) 0 0
\(274\) 45.9718 2.77726
\(275\) 16.3969 0.988772
\(276\) 0 0
\(277\) −6.15064 −0.369556 −0.184778 0.982780i \(-0.559157\pi\)
−0.184778 + 0.982780i \(0.559157\pi\)
\(278\) −55.8512 −3.34973
\(279\) 0 0
\(280\) −5.36959 −0.320894
\(281\) −3.31221 −0.197590 −0.0987951 0.995108i \(-0.531499\pi\)
−0.0987951 + 0.995108i \(0.531499\pi\)
\(282\) 0 0
\(283\) 29.0232 1.72525 0.862626 0.505843i \(-0.168818\pi\)
0.862626 + 0.505843i \(0.168818\pi\)
\(284\) 16.2344 0.963336
\(285\) 0 0
\(286\) −53.5681 −3.16755
\(287\) 1.18479 0.0699361
\(288\) 0 0
\(289\) −14.2686 −0.839328
\(290\) −13.4706 −0.791021
\(291\) 0 0
\(292\) −56.3833 −3.29958
\(293\) 8.41921 0.491856 0.245928 0.969288i \(-0.420907\pi\)
0.245928 + 0.969288i \(0.420907\pi\)
\(294\) 0 0
\(295\) −5.85710 −0.341013
\(296\) 27.8093 1.61638
\(297\) 0 0
\(298\) −38.3756 −2.22304
\(299\) 17.2463 0.997378
\(300\) 0 0
\(301\) 0.184793 0.0106513
\(302\) 47.9992 2.76204
\(303\) 0 0
\(304\) 16.0077 0.918107
\(305\) −2.28312 −0.130731
\(306\) 0 0
\(307\) −12.6878 −0.724130 −0.362065 0.932153i \(-0.617928\pi\)
−0.362065 + 0.932153i \(0.617928\pi\)
\(308\) −17.1138 −0.975150
\(309\) 0 0
\(310\) 10.1411 0.575979
\(311\) 16.4902 0.935073 0.467537 0.883974i \(-0.345142\pi\)
0.467537 + 0.883974i \(0.345142\pi\)
\(312\) 0 0
\(313\) 28.5185 1.61196 0.805980 0.591943i \(-0.201639\pi\)
0.805980 + 0.591943i \(0.201639\pi\)
\(314\) 45.7229 2.58029
\(315\) 0 0
\(316\) −26.2841 −1.47859
\(317\) 25.8949 1.45440 0.727200 0.686425i \(-0.240821\pi\)
0.727200 + 0.686425i \(0.240821\pi\)
\(318\) 0 0
\(319\) −23.4688 −1.31400
\(320\) −1.44057 −0.0805303
\(321\) 0 0
\(322\) 8.00774 0.446254
\(323\) −3.98545 −0.221756
\(324\) 0 0
\(325\) 23.0496 1.27856
\(326\) −2.42602 −0.134365
\(327\) 0 0
\(328\) −7.23442 −0.399454
\(329\) 1.02229 0.0563606
\(330\) 0 0
\(331\) 8.21894 0.451754 0.225877 0.974156i \(-0.427475\pi\)
0.225877 + 0.974156i \(0.427475\pi\)
\(332\) 0.965852 0.0530080
\(333\) 0 0
\(334\) 50.2327 2.74861
\(335\) −2.60132 −0.142125
\(336\) 0 0
\(337\) 4.57129 0.249014 0.124507 0.992219i \(-0.460265\pi\)
0.124507 + 0.992219i \(0.460265\pi\)
\(338\) −42.3851 −2.30544
\(339\) 0 0
\(340\) −6.41147 −0.347711
\(341\) 17.6682 0.956786
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −1.12836 −0.0608369
\(345\) 0 0
\(346\) 57.4347 3.08771
\(347\) −22.4662 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(348\) 0 0
\(349\) 26.0993 1.39706 0.698531 0.715580i \(-0.253838\pi\)
0.698531 + 0.715580i \(0.253838\pi\)
\(350\) 10.7023 0.572064
\(351\) 0 0
\(352\) 17.8307 0.950379
\(353\) 0.355037 0.0188967 0.00944836 0.999955i \(-0.496992\pi\)
0.00944836 + 0.999955i \(0.496992\pi\)
\(354\) 0 0
\(355\) 3.23618 0.171758
\(356\) −48.6468 −2.57828
\(357\) 0 0
\(358\) −18.6040 −0.983252
\(359\) −5.45605 −0.287959 −0.143980 0.989581i \(-0.545990\pi\)
−0.143980 + 0.989581i \(0.545990\pi\)
\(360\) 0 0
\(361\) −13.1848 −0.693936
\(362\) 8.72462 0.458556
\(363\) 0 0
\(364\) −24.0574 −1.26095
\(365\) −11.2395 −0.588301
\(366\) 0 0
\(367\) 10.9240 0.570226 0.285113 0.958494i \(-0.407969\pi\)
0.285113 + 0.958494i \(0.407969\pi\)
\(368\) −20.9932 −1.09435
\(369\) 0 0
\(370\) 10.1411 0.527213
\(371\) −7.29086 −0.378523
\(372\) 0 0
\(373\) 1.73143 0.0896500 0.0448250 0.998995i \(-0.485727\pi\)
0.0448250 + 0.998995i \(0.485727\pi\)
\(374\) −16.2344 −0.839462
\(375\) 0 0
\(376\) −6.24216 −0.321915
\(377\) −32.9908 −1.69911
\(378\) 0 0
\(379\) −12.1334 −0.623251 −0.311626 0.950205i \(-0.600873\pi\)
−0.311626 + 0.950205i \(0.600873\pi\)
\(380\) 9.35504 0.479903
\(381\) 0 0
\(382\) 14.3259 0.732979
\(383\) 8.71183 0.445154 0.222577 0.974915i \(-0.428553\pi\)
0.222577 + 0.974915i \(0.428553\pi\)
\(384\) 0 0
\(385\) −3.41147 −0.173865
\(386\) −24.2986 −1.23677
\(387\) 0 0
\(388\) 55.1566 2.80015
\(389\) −3.64321 −0.184718 −0.0923590 0.995726i \(-0.529441\pi\)
−0.0923590 + 0.995726i \(0.529441\pi\)
\(390\) 0 0
\(391\) 5.22668 0.264325
\(392\) −6.10607 −0.308403
\(393\) 0 0
\(394\) 21.0669 1.06133
\(395\) −5.23947 −0.263627
\(396\) 0 0
\(397\) −15.4456 −0.775194 −0.387597 0.921829i \(-0.626695\pi\)
−0.387597 + 0.921829i \(0.626695\pi\)
\(398\) −16.7023 −0.837212
\(399\) 0 0
\(400\) −28.0574 −1.40287
\(401\) −18.4219 −0.919946 −0.459973 0.887933i \(-0.652141\pi\)
−0.459973 + 0.887933i \(0.652141\pi\)
\(402\) 0 0
\(403\) 24.8367 1.23720
\(404\) 42.8658 2.13265
\(405\) 0 0
\(406\) −15.3182 −0.760230
\(407\) 17.6682 0.875779
\(408\) 0 0
\(409\) −28.6364 −1.41598 −0.707989 0.706223i \(-0.750398\pi\)
−0.707989 + 0.706223i \(0.750398\pi\)
\(410\) −2.63816 −0.130289
\(411\) 0 0
\(412\) 29.0993 1.43362
\(413\) −6.66044 −0.327739
\(414\) 0 0
\(415\) 0.192533 0.00945109
\(416\) 25.0651 1.22892
\(417\) 0 0
\(418\) 23.6878 1.15861
\(419\) 34.6955 1.69499 0.847494 0.530805i \(-0.178111\pi\)
0.847494 + 0.530805i \(0.178111\pi\)
\(420\) 0 0
\(421\) −27.4020 −1.33549 −0.667745 0.744390i \(-0.732740\pi\)
−0.667745 + 0.744390i \(0.732740\pi\)
\(422\) 8.53209 0.415336
\(423\) 0 0
\(424\) 44.5185 2.16201
\(425\) 6.98545 0.338844
\(426\) 0 0
\(427\) −2.59627 −0.125642
\(428\) −10.5398 −0.509462
\(429\) 0 0
\(430\) −0.411474 −0.0198430
\(431\) −26.5921 −1.28090 −0.640449 0.768000i \(-0.721252\pi\)
−0.640449 + 0.768000i \(0.721252\pi\)
\(432\) 0 0
\(433\) 37.1830 1.78690 0.893451 0.449160i \(-0.148277\pi\)
0.893451 + 0.449160i \(0.148277\pi\)
\(434\) 11.5321 0.553558
\(435\) 0 0
\(436\) 17.4611 0.836235
\(437\) −7.62630 −0.364815
\(438\) 0 0
\(439\) 25.0746 1.19675 0.598373 0.801218i \(-0.295814\pi\)
0.598373 + 0.801218i \(0.295814\pi\)
\(440\) 20.8307 0.993064
\(441\) 0 0
\(442\) −22.8212 −1.08549
\(443\) −2.04458 −0.0971408 −0.0485704 0.998820i \(-0.515467\pi\)
−0.0485704 + 0.998820i \(0.515467\pi\)
\(444\) 0 0
\(445\) −9.69728 −0.459695
\(446\) 15.8922 0.752516
\(447\) 0 0
\(448\) −1.63816 −0.0773956
\(449\) 10.2344 0.482992 0.241496 0.970402i \(-0.422362\pi\)
0.241496 + 0.970402i \(0.422362\pi\)
\(450\) 0 0
\(451\) −4.59627 −0.216430
\(452\) −72.5836 −3.41404
\(453\) 0 0
\(454\) −15.6040 −0.732332
\(455\) −4.79561 −0.224822
\(456\) 0 0
\(457\) −42.5945 −1.99249 −0.996244 0.0865948i \(-0.972401\pi\)
−0.996244 + 0.0865948i \(0.972401\pi\)
\(458\) −59.2131 −2.76684
\(459\) 0 0
\(460\) −12.2686 −0.572025
\(461\) −0.504748 −0.0235084 −0.0117542 0.999931i \(-0.503742\pi\)
−0.0117542 + 0.999931i \(0.503742\pi\)
\(462\) 0 0
\(463\) 2.68004 0.124552 0.0622761 0.998059i \(-0.480164\pi\)
0.0622761 + 0.998059i \(0.480164\pi\)
\(464\) 40.1584 1.86431
\(465\) 0 0
\(466\) −21.5868 −0.999988
\(467\) 31.4165 1.45378 0.726892 0.686752i \(-0.240964\pi\)
0.726892 + 0.686752i \(0.240964\pi\)
\(468\) 0 0
\(469\) −2.95811 −0.136593
\(470\) −2.27631 −0.104998
\(471\) 0 0
\(472\) 40.6691 1.87195
\(473\) −0.716881 −0.0329622
\(474\) 0 0
\(475\) −10.1925 −0.467666
\(476\) −7.29086 −0.334176
\(477\) 0 0
\(478\) 36.8726 1.68651
\(479\) 16.4406 0.751189 0.375594 0.926784i \(-0.377439\pi\)
0.375594 + 0.926784i \(0.377439\pi\)
\(480\) 0 0
\(481\) 24.8367 1.13245
\(482\) 13.6827 0.623231
\(483\) 0 0
\(484\) 17.8648 0.812038
\(485\) 10.9949 0.499255
\(486\) 0 0
\(487\) −2.97535 −0.134826 −0.0674129 0.997725i \(-0.521474\pi\)
−0.0674129 + 0.997725i \(0.521474\pi\)
\(488\) 15.8530 0.717631
\(489\) 0 0
\(490\) −2.22668 −0.100591
\(491\) 26.4861 1.19530 0.597650 0.801757i \(-0.296101\pi\)
0.597650 + 0.801757i \(0.296101\pi\)
\(492\) 0 0
\(493\) −9.99825 −0.450298
\(494\) 33.2986 1.49817
\(495\) 0 0
\(496\) −30.2327 −1.35749
\(497\) 3.68004 0.165073
\(498\) 0 0
\(499\) −13.4439 −0.601830 −0.300915 0.953651i \(-0.597292\pi\)
−0.300915 + 0.953651i \(0.597292\pi\)
\(500\) −35.7939 −1.60075
\(501\) 0 0
\(502\) −30.5544 −1.36371
\(503\) 22.6631 1.01050 0.505250 0.862973i \(-0.331400\pi\)
0.505250 + 0.862973i \(0.331400\pi\)
\(504\) 0 0
\(505\) 8.54488 0.380242
\(506\) −31.0651 −1.38101
\(507\) 0 0
\(508\) 77.8786 3.45530
\(509\) −9.54757 −0.423189 −0.211594 0.977358i \(-0.567866\pi\)
−0.211594 + 0.977358i \(0.567866\pi\)
\(510\) 0 0
\(511\) −12.7811 −0.565401
\(512\) 50.5553 2.23425
\(513\) 0 0
\(514\) 26.7547 1.18010
\(515\) 5.80066 0.255608
\(516\) 0 0
\(517\) −3.96585 −0.174418
\(518\) 11.5321 0.506691
\(519\) 0 0
\(520\) 29.2823 1.28411
\(521\) 3.11287 0.136377 0.0681887 0.997672i \(-0.478278\pi\)
0.0681887 + 0.997672i \(0.478278\pi\)
\(522\) 0 0
\(523\) −16.1489 −0.706142 −0.353071 0.935597i \(-0.614863\pi\)
−0.353071 + 0.935597i \(0.614863\pi\)
\(524\) 84.6897 3.69968
\(525\) 0 0
\(526\) −71.7930 −3.13032
\(527\) 7.52704 0.327883
\(528\) 0 0
\(529\) −12.9986 −0.565155
\(530\) 16.2344 0.705178
\(531\) 0 0
\(532\) 10.6382 0.461223
\(533\) −6.46110 −0.279861
\(534\) 0 0
\(535\) −2.10101 −0.0908348
\(536\) 18.0624 0.780178
\(537\) 0 0
\(538\) 18.9486 0.816933
\(539\) −3.87939 −0.167097
\(540\) 0 0
\(541\) −5.01548 −0.215632 −0.107816 0.994171i \(-0.534386\pi\)
−0.107816 + 0.994171i \(0.534386\pi\)
\(542\) −34.5330 −1.48332
\(543\) 0 0
\(544\) 7.59627 0.325687
\(545\) 3.48070 0.149097
\(546\) 0 0
\(547\) 16.4780 0.704549 0.352275 0.935897i \(-0.385408\pi\)
0.352275 + 0.935897i \(0.385408\pi\)
\(548\) −80.0934 −3.42142
\(549\) 0 0
\(550\) −41.5185 −1.77035
\(551\) 14.5885 0.621492
\(552\) 0 0
\(553\) −5.95811 −0.253365
\(554\) 15.5740 0.661675
\(555\) 0 0
\(556\) 97.3055 4.12667
\(557\) 34.5631 1.46448 0.732242 0.681045i \(-0.238474\pi\)
0.732242 + 0.681045i \(0.238474\pi\)
\(558\) 0 0
\(559\) −1.00774 −0.0426229
\(560\) 5.83750 0.246679
\(561\) 0 0
\(562\) 8.38682 0.353777
\(563\) 37.2104 1.56823 0.784115 0.620615i \(-0.213117\pi\)
0.784115 + 0.620615i \(0.213117\pi\)
\(564\) 0 0
\(565\) −14.4688 −0.608709
\(566\) −73.4894 −3.08899
\(567\) 0 0
\(568\) −22.4706 −0.942845
\(569\) −0.404667 −0.0169645 −0.00848226 0.999964i \(-0.502700\pi\)
−0.00848226 + 0.999964i \(0.502700\pi\)
\(570\) 0 0
\(571\) −37.7793 −1.58101 −0.790507 0.612453i \(-0.790183\pi\)
−0.790507 + 0.612453i \(0.790183\pi\)
\(572\) 93.3278 3.90223
\(573\) 0 0
\(574\) −3.00000 −0.125218
\(575\) 13.3669 0.557438
\(576\) 0 0
\(577\) −2.21120 −0.0920535 −0.0460267 0.998940i \(-0.514656\pi\)
−0.0460267 + 0.998940i \(0.514656\pi\)
\(578\) 36.1293 1.50278
\(579\) 0 0
\(580\) 23.4688 0.974491
\(581\) 0.218941 0.00908320
\(582\) 0 0
\(583\) 28.2841 1.17141
\(584\) 78.0420 3.22940
\(585\) 0 0
\(586\) −21.3182 −0.880647
\(587\) −24.2098 −0.999244 −0.499622 0.866243i \(-0.666528\pi\)
−0.499622 + 0.866243i \(0.666528\pi\)
\(588\) 0 0
\(589\) −10.9828 −0.452537
\(590\) 14.8307 0.610570
\(591\) 0 0
\(592\) −30.2327 −1.24255
\(593\) −12.2385 −0.502577 −0.251288 0.967912i \(-0.580854\pi\)
−0.251288 + 0.967912i \(0.580854\pi\)
\(594\) 0 0
\(595\) −1.45336 −0.0595821
\(596\) 66.8590 2.73865
\(597\) 0 0
\(598\) −43.6691 −1.78576
\(599\) −39.6168 −1.61870 −0.809349 0.587328i \(-0.800180\pi\)
−0.809349 + 0.587328i \(0.800180\pi\)
\(600\) 0 0
\(601\) −30.0077 −1.22404 −0.612021 0.790842i \(-0.709643\pi\)
−0.612021 + 0.790842i \(0.709643\pi\)
\(602\) −0.467911 −0.0190706
\(603\) 0 0
\(604\) −83.6255 −3.40267
\(605\) 3.56118 0.144783
\(606\) 0 0
\(607\) −19.4843 −0.790844 −0.395422 0.918499i \(-0.629402\pi\)
−0.395422 + 0.918499i \(0.629402\pi\)
\(608\) −11.0838 −0.449507
\(609\) 0 0
\(610\) 5.78106 0.234068
\(611\) −5.57491 −0.225537
\(612\) 0 0
\(613\) −18.5276 −0.748325 −0.374162 0.927363i \(-0.622070\pi\)
−0.374162 + 0.927363i \(0.622070\pi\)
\(614\) 32.1266 1.29652
\(615\) 0 0
\(616\) 23.6878 0.954408
\(617\) −27.8402 −1.12080 −0.560402 0.828221i \(-0.689353\pi\)
−0.560402 + 0.828221i \(0.689353\pi\)
\(618\) 0 0
\(619\) −44.9813 −1.80795 −0.903976 0.427583i \(-0.859365\pi\)
−0.903976 + 0.427583i \(0.859365\pi\)
\(620\) −17.6682 −0.709571
\(621\) 0 0
\(622\) −41.7547 −1.67421
\(623\) −11.0273 −0.441801
\(624\) 0 0
\(625\) 13.9982 0.559930
\(626\) −72.2113 −2.88614
\(627\) 0 0
\(628\) −79.6596 −3.17877
\(629\) 7.52704 0.300123
\(630\) 0 0
\(631\) 9.43613 0.375646 0.187823 0.982203i \(-0.439857\pi\)
0.187823 + 0.982203i \(0.439857\pi\)
\(632\) 36.3806 1.44714
\(633\) 0 0
\(634\) −65.5681 −2.60404
\(635\) 15.5243 0.616065
\(636\) 0 0
\(637\) −5.45336 −0.216070
\(638\) 59.4252 2.35267
\(639\) 0 0
\(640\) 11.7314 0.463725
\(641\) −37.3901 −1.47682 −0.738410 0.674352i \(-0.764423\pi\)
−0.738410 + 0.674352i \(0.764423\pi\)
\(642\) 0 0
\(643\) 1.61175 0.0635611 0.0317806 0.999495i \(-0.489882\pi\)
0.0317806 + 0.999495i \(0.489882\pi\)
\(644\) −13.9513 −0.549758
\(645\) 0 0
\(646\) 10.0915 0.397046
\(647\) 41.1762 1.61880 0.809402 0.587255i \(-0.199791\pi\)
0.809402 + 0.587255i \(0.199791\pi\)
\(648\) 0 0
\(649\) 25.8384 1.01425
\(650\) −58.3637 −2.28921
\(651\) 0 0
\(652\) 4.22668 0.165530
\(653\) −3.05199 −0.119434 −0.0597169 0.998215i \(-0.519020\pi\)
−0.0597169 + 0.998215i \(0.519020\pi\)
\(654\) 0 0
\(655\) 16.8821 0.659637
\(656\) 7.86484 0.307070
\(657\) 0 0
\(658\) −2.58853 −0.100911
\(659\) −41.6350 −1.62187 −0.810934 0.585138i \(-0.801041\pi\)
−0.810934 + 0.585138i \(0.801041\pi\)
\(660\) 0 0
\(661\) 20.3010 0.789616 0.394808 0.918764i \(-0.370811\pi\)
0.394808 + 0.918764i \(0.370811\pi\)
\(662\) −20.8111 −0.808846
\(663\) 0 0
\(664\) −1.33687 −0.0518805
\(665\) 2.12061 0.0822339
\(666\) 0 0
\(667\) −19.1320 −0.740793
\(668\) −87.5167 −3.38612
\(669\) 0 0
\(670\) 6.58677 0.254469
\(671\) 10.0719 0.388822
\(672\) 0 0
\(673\) −0.830689 −0.0320207 −0.0160104 0.999872i \(-0.505096\pi\)
−0.0160104 + 0.999872i \(0.505096\pi\)
\(674\) −11.5749 −0.445849
\(675\) 0 0
\(676\) 73.8444 2.84017
\(677\) −10.8672 −0.417660 −0.208830 0.977952i \(-0.566966\pi\)
−0.208830 + 0.977952i \(0.566966\pi\)
\(678\) 0 0
\(679\) 12.5030 0.479821
\(680\) 8.87433 0.340315
\(681\) 0 0
\(682\) −44.7374 −1.71308
\(683\) −32.6946 −1.25102 −0.625512 0.780215i \(-0.715110\pi\)
−0.625512 + 0.780215i \(0.715110\pi\)
\(684\) 0 0
\(685\) −15.9659 −0.610024
\(686\) −2.53209 −0.0966756
\(687\) 0 0
\(688\) 1.22668 0.0467668
\(689\) 39.7597 1.51472
\(690\) 0 0
\(691\) 14.9982 0.570560 0.285280 0.958444i \(-0.407913\pi\)
0.285280 + 0.958444i \(0.407913\pi\)
\(692\) −100.064 −3.80387
\(693\) 0 0
\(694\) 56.8863 2.15937
\(695\) 19.3969 0.735767
\(696\) 0 0
\(697\) −1.95811 −0.0741687
\(698\) −66.0856 −2.50138
\(699\) 0 0
\(700\) −18.6459 −0.704749
\(701\) −26.4688 −0.999714 −0.499857 0.866108i \(-0.666614\pi\)
−0.499857 + 0.866108i \(0.666614\pi\)
\(702\) 0 0
\(703\) −10.9828 −0.414223
\(704\) 6.35504 0.239514
\(705\) 0 0
\(706\) −0.898986 −0.0338338
\(707\) 9.71688 0.365441
\(708\) 0 0
\(709\) 15.3601 0.576860 0.288430 0.957501i \(-0.406867\pi\)
0.288430 + 0.957501i \(0.406867\pi\)
\(710\) −8.19429 −0.307526
\(711\) 0 0
\(712\) 67.3337 2.52344
\(713\) 14.4032 0.539405
\(714\) 0 0
\(715\) 18.6040 0.695750
\(716\) 32.4124 1.21131
\(717\) 0 0
\(718\) 13.8152 0.515579
\(719\) −26.7306 −0.996883 −0.498442 0.866923i \(-0.666094\pi\)
−0.498442 + 0.866923i \(0.666094\pi\)
\(720\) 0 0
\(721\) 6.59627 0.245658
\(722\) 33.3851 1.24246
\(723\) 0 0
\(724\) −15.2003 −0.564914
\(725\) −25.5699 −0.949641
\(726\) 0 0
\(727\) −45.6441 −1.69285 −0.846424 0.532510i \(-0.821249\pi\)
−0.846424 + 0.532510i \(0.821249\pi\)
\(728\) 33.2986 1.23413
\(729\) 0 0
\(730\) 28.4593 1.05333
\(731\) −0.305407 −0.0112959
\(732\) 0 0
\(733\) 5.97502 0.220693 0.110346 0.993893i \(-0.464804\pi\)
0.110346 + 0.993893i \(0.464804\pi\)
\(734\) −27.6604 −1.02097
\(735\) 0 0
\(736\) 14.5357 0.535793
\(737\) 11.4757 0.422711
\(738\) 0 0
\(739\) −35.5963 −1.30943 −0.654715 0.755876i \(-0.727211\pi\)
−0.654715 + 0.755876i \(0.727211\pi\)
\(740\) −17.6682 −0.649495
\(741\) 0 0
\(742\) 18.4611 0.677728
\(743\) 29.3087 1.07523 0.537616 0.843190i \(-0.319325\pi\)
0.537616 + 0.843190i \(0.319325\pi\)
\(744\) 0 0
\(745\) 13.3277 0.488289
\(746\) −4.38413 −0.160515
\(747\) 0 0
\(748\) 28.2841 1.03417
\(749\) −2.38919 −0.0872989
\(750\) 0 0
\(751\) −17.3337 −0.632515 −0.316258 0.948673i \(-0.602426\pi\)
−0.316258 + 0.948673i \(0.602426\pi\)
\(752\) 6.78611 0.247464
\(753\) 0 0
\(754\) 83.5357 3.04219
\(755\) −16.6699 −0.606681
\(756\) 0 0
\(757\) −2.77156 −0.100734 −0.0503671 0.998731i \(-0.516039\pi\)
−0.0503671 + 0.998731i \(0.516039\pi\)
\(758\) 30.7229 1.11590
\(759\) 0 0
\(760\) −12.9486 −0.469696
\(761\) −7.50744 −0.272144 −0.136072 0.990699i \(-0.543448\pi\)
−0.136072 + 0.990699i \(0.543448\pi\)
\(762\) 0 0
\(763\) 3.95811 0.143293
\(764\) −24.9590 −0.902987
\(765\) 0 0
\(766\) −22.0591 −0.797029
\(767\) 36.3218 1.31150
\(768\) 0 0
\(769\) 2.04364 0.0736957 0.0368478 0.999321i \(-0.488268\pi\)
0.0368478 + 0.999321i \(0.488268\pi\)
\(770\) 8.63816 0.311298
\(771\) 0 0
\(772\) 42.3337 1.52362
\(773\) 24.9418 0.897094 0.448547 0.893759i \(-0.351942\pi\)
0.448547 + 0.893759i \(0.351942\pi\)
\(774\) 0 0
\(775\) 19.2499 0.691477
\(776\) −76.3441 −2.74059
\(777\) 0 0
\(778\) 9.22493 0.330730
\(779\) 2.85710 0.102366
\(780\) 0 0
\(781\) −14.2763 −0.510847
\(782\) −13.2344 −0.473262
\(783\) 0 0
\(784\) 6.63816 0.237077
\(785\) −15.8794 −0.566760
\(786\) 0 0
\(787\) 7.10700 0.253337 0.126669 0.991945i \(-0.459572\pi\)
0.126669 + 0.991945i \(0.459572\pi\)
\(788\) −36.7033 −1.30750
\(789\) 0 0
\(790\) 13.2668 0.472012
\(791\) −16.4534 −0.585014
\(792\) 0 0
\(793\) 14.1584 0.502779
\(794\) 39.1097 1.38795
\(795\) 0 0
\(796\) 29.0993 1.03140
\(797\) −33.6628 −1.19240 −0.596199 0.802837i \(-0.703323\pi\)
−0.596199 + 0.802837i \(0.703323\pi\)
\(798\) 0 0
\(799\) −1.68954 −0.0597716
\(800\) 19.4270 0.686847
\(801\) 0 0
\(802\) 46.6459 1.64712
\(803\) 49.5827 1.74973
\(804\) 0 0
\(805\) −2.78106 −0.0980195
\(806\) −62.8887 −2.21516
\(807\) 0 0
\(808\) −59.3319 −2.08729
\(809\) 12.8161 0.450592 0.225296 0.974290i \(-0.427665\pi\)
0.225296 + 0.974290i \(0.427665\pi\)
\(810\) 0 0
\(811\) −26.1239 −0.917335 −0.458667 0.888608i \(-0.651673\pi\)
−0.458667 + 0.888608i \(0.651673\pi\)
\(812\) 26.6878 0.936558
\(813\) 0 0
\(814\) −44.7374 −1.56805
\(815\) 0.842549 0.0295132
\(816\) 0 0
\(817\) 0.445622 0.0155904
\(818\) 72.5099 2.53525
\(819\) 0 0
\(820\) 4.59627 0.160509
\(821\) −27.6641 −0.965483 −0.482741 0.875763i \(-0.660359\pi\)
−0.482741 + 0.875763i \(0.660359\pi\)
\(822\) 0 0
\(823\) 27.8324 0.970178 0.485089 0.874465i \(-0.338787\pi\)
0.485089 + 0.874465i \(0.338787\pi\)
\(824\) −40.2772 −1.40312
\(825\) 0 0
\(826\) 16.8648 0.586803
\(827\) −4.65507 −0.161873 −0.0809363 0.996719i \(-0.525791\pi\)
−0.0809363 + 0.996719i \(0.525791\pi\)
\(828\) 0 0
\(829\) −9.97359 −0.346397 −0.173199 0.984887i \(-0.555410\pi\)
−0.173199 + 0.984887i \(0.555410\pi\)
\(830\) −0.487511 −0.0169218
\(831\) 0 0
\(832\) 8.93346 0.309712
\(833\) −1.65270 −0.0572628
\(834\) 0 0
\(835\) −17.4456 −0.603731
\(836\) −41.2695 −1.42734
\(837\) 0 0
\(838\) −87.8522 −3.03480
\(839\) 6.72967 0.232334 0.116167 0.993230i \(-0.462939\pi\)
0.116167 + 0.993230i \(0.462939\pi\)
\(840\) 0 0
\(841\) 7.59802 0.262001
\(842\) 69.3842 2.39114
\(843\) 0 0
\(844\) −14.8648 −0.511669
\(845\) 14.7202 0.506390
\(846\) 0 0
\(847\) 4.04963 0.139147
\(848\) −48.3979 −1.66199
\(849\) 0 0
\(850\) −17.6878 −0.606686
\(851\) 14.4032 0.493737
\(852\) 0 0
\(853\) 5.79055 0.198265 0.0991324 0.995074i \(-0.468393\pi\)
0.0991324 + 0.995074i \(0.468393\pi\)
\(854\) 6.57398 0.224957
\(855\) 0 0
\(856\) 14.5885 0.498626
\(857\) 34.9077 1.19242 0.596211 0.802827i \(-0.296672\pi\)
0.596211 + 0.802827i \(0.296672\pi\)
\(858\) 0 0
\(859\) −12.6149 −0.430416 −0.215208 0.976568i \(-0.569043\pi\)
−0.215208 + 0.976568i \(0.569043\pi\)
\(860\) 0.716881 0.0244455
\(861\) 0 0
\(862\) 67.3337 2.29339
\(863\) 24.2053 0.823959 0.411979 0.911193i \(-0.364838\pi\)
0.411979 + 0.911193i \(0.364838\pi\)
\(864\) 0 0
\(865\) −19.9469 −0.678214
\(866\) −94.1508 −3.19937
\(867\) 0 0
\(868\) −20.0915 −0.681951
\(869\) 23.1138 0.784082
\(870\) 0 0
\(871\) 16.1317 0.546600
\(872\) −24.1685 −0.818448
\(873\) 0 0
\(874\) 19.3105 0.653186
\(875\) −8.11381 −0.274297
\(876\) 0 0
\(877\) −1.12567 −0.0380111 −0.0190055 0.999819i \(-0.506050\pi\)
−0.0190055 + 0.999819i \(0.506050\pi\)
\(878\) −63.4911 −2.14272
\(879\) 0 0
\(880\) −22.6459 −0.763393
\(881\) 4.38331 0.147678 0.0738388 0.997270i \(-0.476475\pi\)
0.0738388 + 0.997270i \(0.476475\pi\)
\(882\) 0 0
\(883\) −6.88949 −0.231850 −0.115925 0.993258i \(-0.536983\pi\)
−0.115925 + 0.993258i \(0.536983\pi\)
\(884\) 39.7597 1.33726
\(885\) 0 0
\(886\) 5.17705 0.173926
\(887\) −39.0752 −1.31202 −0.656009 0.754753i \(-0.727757\pi\)
−0.656009 + 0.754753i \(0.727757\pi\)
\(888\) 0 0
\(889\) 17.6536 0.592084
\(890\) 24.5544 0.823065
\(891\) 0 0
\(892\) −27.6878 −0.927056
\(893\) 2.46522 0.0824955
\(894\) 0 0
\(895\) 6.46110 0.215971
\(896\) 13.3405 0.445674
\(897\) 0 0
\(898\) −25.9145 −0.864777
\(899\) −27.5523 −0.918921
\(900\) 0 0
\(901\) 12.0496 0.401431
\(902\) 11.6382 0.387508
\(903\) 0 0
\(904\) 100.465 3.34143
\(905\) −3.03003 −0.100722
\(906\) 0 0
\(907\) 42.4938 1.41098 0.705492 0.708718i \(-0.250726\pi\)
0.705492 + 0.708718i \(0.250726\pi\)
\(908\) 27.1857 0.902190
\(909\) 0 0
\(910\) 12.1429 0.402533
\(911\) 15.4935 0.513322 0.256661 0.966501i \(-0.417378\pi\)
0.256661 + 0.966501i \(0.417378\pi\)
\(912\) 0 0
\(913\) −0.849356 −0.0281096
\(914\) 107.853 3.56746
\(915\) 0 0
\(916\) 103.163 3.40859
\(917\) 19.1976 0.633960
\(918\) 0 0
\(919\) 6.52940 0.215385 0.107693 0.994184i \(-0.465654\pi\)
0.107693 + 0.994184i \(0.465654\pi\)
\(920\) 16.9813 0.559858
\(921\) 0 0
\(922\) 1.27807 0.0420909
\(923\) −20.0686 −0.660567
\(924\) 0 0
\(925\) 19.2499 0.632933
\(926\) −6.78611 −0.223005
\(927\) 0 0
\(928\) −27.8057 −0.912767
\(929\) −58.2772 −1.91201 −0.956007 0.293343i \(-0.905232\pi\)
−0.956007 + 0.293343i \(0.905232\pi\)
\(930\) 0 0
\(931\) 2.41147 0.0790329
\(932\) 37.6091 1.23193
\(933\) 0 0
\(934\) −79.5494 −2.60294
\(935\) 5.63816 0.184387
\(936\) 0 0
\(937\) 32.4175 1.05903 0.529516 0.848300i \(-0.322374\pi\)
0.529516 + 0.848300i \(0.322374\pi\)
\(938\) 7.49020 0.244564
\(939\) 0 0
\(940\) 3.96585 0.129352
\(941\) 27.3226 0.890693 0.445346 0.895358i \(-0.353081\pi\)
0.445346 + 0.895358i \(0.353081\pi\)
\(942\) 0 0
\(943\) −3.74691 −0.122016
\(944\) −44.2131 −1.43901
\(945\) 0 0
\(946\) 1.81521 0.0590175
\(947\) 38.2131 1.24176 0.620879 0.783906i \(-0.286776\pi\)
0.620879 + 0.783906i \(0.286776\pi\)
\(948\) 0 0
\(949\) 69.6998 2.26255
\(950\) 25.8084 0.837335
\(951\) 0 0
\(952\) 10.0915 0.327068
\(953\) −58.9377 −1.90918 −0.954590 0.297924i \(-0.903706\pi\)
−0.954590 + 0.297924i \(0.903706\pi\)
\(954\) 0 0
\(955\) −4.97535 −0.160998
\(956\) −64.2404 −2.07768
\(957\) 0 0
\(958\) −41.6290 −1.34497
\(959\) −18.1557 −0.586278
\(960\) 0 0
\(961\) −10.2576 −0.330892
\(962\) −62.8887 −2.02761
\(963\) 0 0
\(964\) −23.8384 −0.767784
\(965\) 8.43882 0.271655
\(966\) 0 0
\(967\) 24.7187 0.794901 0.397451 0.917624i \(-0.369895\pi\)
0.397451 + 0.917624i \(0.369895\pi\)
\(968\) −24.7273 −0.794766
\(969\) 0 0
\(970\) −27.8402 −0.893894
\(971\) −8.17623 −0.262388 −0.131194 0.991357i \(-0.541881\pi\)
−0.131194 + 0.991357i \(0.541881\pi\)
\(972\) 0 0
\(973\) 22.0574 0.707127
\(974\) 7.53384 0.241400
\(975\) 0 0
\(976\) −17.2344 −0.551660
\(977\) 15.8485 0.507040 0.253520 0.967330i \(-0.418412\pi\)
0.253520 + 0.967330i \(0.418412\pi\)
\(978\) 0 0
\(979\) 42.7793 1.36723
\(980\) 3.87939 0.123922
\(981\) 0 0
\(982\) −67.0651 −2.14013
\(983\) −53.3063 −1.70021 −0.850104 0.526615i \(-0.823461\pi\)
−0.850104 + 0.526615i \(0.823461\pi\)
\(984\) 0 0
\(985\) −7.31645 −0.233121
\(986\) 25.3164 0.806240
\(987\) 0 0
\(988\) −58.0137 −1.84566
\(989\) −0.584407 −0.0185831
\(990\) 0 0
\(991\) 40.2094 1.27730 0.638648 0.769499i \(-0.279494\pi\)
0.638648 + 0.769499i \(0.279494\pi\)
\(992\) 20.9331 0.664628
\(993\) 0 0
\(994\) −9.31820 −0.295555
\(995\) 5.80066 0.183893
\(996\) 0 0
\(997\) 28.7202 0.909577 0.454789 0.890599i \(-0.349715\pi\)
0.454789 + 0.890599i \(0.349715\pi\)
\(998\) 34.0411 1.07755
\(999\) 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.1.1 3
3.2 odd 2 567.2.a.h.1.3 3
4.3 odd 2 9072.2.a.bs.1.3 3
7.6 odd 2 3969.2.a.l.1.1 3
9.2 odd 6 63.2.f.a.22.1 6
9.4 even 3 189.2.f.b.127.3 6
9.5 odd 6 63.2.f.a.43.1 yes 6
9.7 even 3 189.2.f.b.64.3 6
12.11 even 2 9072.2.a.ca.1.1 3
21.20 even 2 3969.2.a.q.1.3 3
36.7 odd 6 3024.2.r.k.1009.1 6
36.11 even 6 1008.2.r.h.337.3 6
36.23 even 6 1008.2.r.h.673.3 6
36.31 odd 6 3024.2.r.k.2017.1 6
63.2 odd 6 441.2.g.c.67.1 6
63.4 even 3 1323.2.g.d.667.3 6
63.5 even 6 441.2.h.e.214.3 6
63.11 odd 6 441.2.h.d.373.3 6
63.13 odd 6 1323.2.f.d.883.3 6
63.16 even 3 1323.2.g.d.361.3 6
63.20 even 6 441.2.f.c.148.1 6
63.23 odd 6 441.2.h.d.214.3 6
63.25 even 3 1323.2.h.c.226.1 6
63.31 odd 6 1323.2.g.e.667.3 6
63.32 odd 6 441.2.g.c.79.1 6
63.34 odd 6 1323.2.f.d.442.3 6
63.38 even 6 441.2.h.e.373.3 6
63.40 odd 6 1323.2.h.b.802.1 6
63.41 even 6 441.2.f.c.295.1 6
63.47 even 6 441.2.g.b.67.1 6
63.52 odd 6 1323.2.h.b.226.1 6
63.58 even 3 1323.2.h.c.802.1 6
63.59 even 6 441.2.g.b.79.1 6
63.61 odd 6 1323.2.g.e.361.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.1 6 9.2 odd 6
63.2.f.a.43.1 yes 6 9.5 odd 6
189.2.f.b.64.3 6 9.7 even 3
189.2.f.b.127.3 6 9.4 even 3
441.2.f.c.148.1 6 63.20 even 6
441.2.f.c.295.1 6 63.41 even 6
441.2.g.b.67.1 6 63.47 even 6
441.2.g.b.79.1 6 63.59 even 6
441.2.g.c.67.1 6 63.2 odd 6
441.2.g.c.79.1 6 63.32 odd 6
441.2.h.d.214.3 6 63.23 odd 6
441.2.h.d.373.3 6 63.11 odd 6
441.2.h.e.214.3 6 63.5 even 6
441.2.h.e.373.3 6 63.38 even 6
567.2.a.c.1.1 3 1.1 even 1 trivial
567.2.a.h.1.3 3 3.2 odd 2
1008.2.r.h.337.3 6 36.11 even 6
1008.2.r.h.673.3 6 36.23 even 6
1323.2.f.d.442.3 6 63.34 odd 6
1323.2.f.d.883.3 6 63.13 odd 6
1323.2.g.d.361.3 6 63.16 even 3
1323.2.g.d.667.3 6 63.4 even 3
1323.2.g.e.361.3 6 63.61 odd 6
1323.2.g.e.667.3 6 63.31 odd 6
1323.2.h.b.226.1 6 63.52 odd 6
1323.2.h.b.802.1 6 63.40 odd 6
1323.2.h.c.226.1 6 63.25 even 3
1323.2.h.c.802.1 6 63.58 even 3
3024.2.r.k.1009.1 6 36.7 odd 6
3024.2.r.k.2017.1 6 36.31 odd 6
3969.2.a.l.1.1 3 7.6 odd 2
3969.2.a.q.1.3 3 21.20 even 2
9072.2.a.bs.1.3 3 4.3 odd 2
9072.2.a.ca.1.1 3 12.11 even 2