Properties

Label 567.2.a.c.1.3
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 1.3
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 567.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.879385 q^{2} -1.22668 q^{4} -1.34730 q^{5} +1.00000 q^{7} -2.83750 q^{8} +O(q^{10})\) \(q+0.879385 q^{2} -1.22668 q^{4} -1.34730 q^{5} +1.00000 q^{7} -2.83750 q^{8} -1.18479 q^{10} -1.65270 q^{11} -3.36959 q^{13} +0.879385 q^{14} -0.0418891 q^{16} -0.467911 q^{17} -3.22668 q^{19} +1.65270 q^{20} -1.45336 q^{22} -8.94356 q^{23} -3.18479 q^{25} -2.96316 q^{26} -1.22668 q^{28} -6.26857 q^{29} +9.23442 q^{31} +5.63816 q^{32} -0.411474 q^{34} -1.34730 q^{35} +9.23442 q^{37} -2.83750 q^{38} +3.82295 q^{40} -3.41147 q^{41} -4.41147 q^{43} +2.02734 q^{44} -7.86484 q^{46} -9.35504 q^{47} +1.00000 q^{49} -2.80066 q^{50} +4.13341 q^{52} +0.573978 q^{53} +2.22668 q^{55} -2.83750 q^{56} -5.51249 q^{58} +10.3969 q^{59} +7.63816 q^{61} +8.12061 q^{62} +5.04189 q^{64} +4.53983 q^{65} +0.596267 q^{67} +0.573978 q^{68} -1.18479 q^{70} +0.554378 q^{71} +2.04963 q^{73} +8.12061 q^{74} +3.95811 q^{76} -1.65270 q^{77} -2.40373 q^{79} +0.0564370 q^{80} -3.00000 q^{82} +15.0496 q^{83} +0.630415 q^{85} -3.87939 q^{86} +4.68954 q^{88} -9.08647 q^{89} -3.36959 q^{91} +10.9709 q^{92} -8.22668 q^{94} +4.34730 q^{95} -1.89899 q^{97} +0.879385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} - 6q^{8} - 6q^{11} - 3q^{13} - 3q^{14} + 3q^{16} - 6q^{17} - 3q^{19} + 6q^{20} + 9q^{22} - 12q^{23} - 6q^{25} + 3q^{26} + 3q^{28} - 9q^{29} - 3q^{31} + 9q^{34} - 3q^{35} - 3q^{37} - 6q^{38} - 9q^{40} - 3q^{43} - 15q^{44} - 3q^{47} + 3q^{49} + 6q^{50} - 21q^{52} - 6q^{53} - 6q^{56} - 9q^{58} + 3q^{59} + 6q^{61} + 30q^{62} + 12q^{64} - 15q^{65} - 12q^{67} - 6q^{68} - 9q^{71} - 21q^{73} + 30q^{74} + 15q^{76} - 6q^{77} - 21q^{79} + 15q^{80} - 9q^{82} + 18q^{83} + 9q^{85} - 6q^{86} + 27q^{88} - 12q^{89} - 3q^{91} - 3q^{92} - 18q^{94} + 12q^{95} - 3q^{97} - 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.879385 0.621819 0.310910 0.950439i \(-0.399366\pi\)
0.310910 + 0.950439i \(0.399366\pi\)
\(3\) 0 0
\(4\) −1.22668 −0.613341
\(5\) −1.34730 −0.602529 −0.301265 0.953541i \(-0.597409\pi\)
−0.301265 + 0.953541i \(0.597409\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.83750 −1.00321
\(9\) 0 0
\(10\) −1.18479 −0.374664
\(11\) −1.65270 −0.498309 −0.249154 0.968464i \(-0.580153\pi\)
−0.249154 + 0.968464i \(0.580153\pi\)
\(12\) 0 0
\(13\) −3.36959 −0.934555 −0.467277 0.884111i \(-0.654765\pi\)
−0.467277 + 0.884111i \(0.654765\pi\)
\(14\) 0.879385 0.235026
\(15\) 0 0
\(16\) −0.0418891 −0.0104723
\(17\) −0.467911 −0.113485 −0.0567426 0.998389i \(-0.518071\pi\)
−0.0567426 + 0.998389i \(0.518071\pi\)
\(18\) 0 0
\(19\) −3.22668 −0.740252 −0.370126 0.928982i \(-0.620685\pi\)
−0.370126 + 0.928982i \(0.620685\pi\)
\(20\) 1.65270 0.369556
\(21\) 0 0
\(22\) −1.45336 −0.309858
\(23\) −8.94356 −1.86486 −0.932431 0.361348i \(-0.882317\pi\)
−0.932431 + 0.361348i \(0.882317\pi\)
\(24\) 0 0
\(25\) −3.18479 −0.636959
\(26\) −2.96316 −0.581124
\(27\) 0 0
\(28\) −1.22668 −0.231821
\(29\) −6.26857 −1.16404 −0.582022 0.813173i \(-0.697738\pi\)
−0.582022 + 0.813173i \(0.697738\pi\)
\(30\) 0 0
\(31\) 9.23442 1.65855 0.829276 0.558840i \(-0.188753\pi\)
0.829276 + 0.558840i \(0.188753\pi\)
\(32\) 5.63816 0.996695
\(33\) 0 0
\(34\) −0.411474 −0.0705672
\(35\) −1.34730 −0.227735
\(36\) 0 0
\(37\) 9.23442 1.51813 0.759065 0.651015i \(-0.225657\pi\)
0.759065 + 0.651015i \(0.225657\pi\)
\(38\) −2.83750 −0.460303
\(39\) 0 0
\(40\) 3.82295 0.604461
\(41\) −3.41147 −0.532783 −0.266391 0.963865i \(-0.585831\pi\)
−0.266391 + 0.963865i \(0.585831\pi\)
\(42\) 0 0
\(43\) −4.41147 −0.672743 −0.336372 0.941729i \(-0.609200\pi\)
−0.336372 + 0.941729i \(0.609200\pi\)
\(44\) 2.02734 0.305633
\(45\) 0 0
\(46\) −7.86484 −1.15961
\(47\) −9.35504 −1.36457 −0.682286 0.731085i \(-0.739014\pi\)
−0.682286 + 0.731085i \(0.739014\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.80066 −0.396073
\(51\) 0 0
\(52\) 4.13341 0.573201
\(53\) 0.573978 0.0788419 0.0394210 0.999223i \(-0.487449\pi\)
0.0394210 + 0.999223i \(0.487449\pi\)
\(54\) 0 0
\(55\) 2.22668 0.300246
\(56\) −2.83750 −0.379176
\(57\) 0 0
\(58\) −5.51249 −0.723825
\(59\) 10.3969 1.35356 0.676782 0.736183i \(-0.263374\pi\)
0.676782 + 0.736183i \(0.263374\pi\)
\(60\) 0 0
\(61\) 7.63816 0.977966 0.488983 0.872293i \(-0.337368\pi\)
0.488983 + 0.872293i \(0.337368\pi\)
\(62\) 8.12061 1.03132
\(63\) 0 0
\(64\) 5.04189 0.630236
\(65\) 4.53983 0.563097
\(66\) 0 0
\(67\) 0.596267 0.0728456 0.0364228 0.999336i \(-0.488404\pi\)
0.0364228 + 0.999336i \(0.488404\pi\)
\(68\) 0.573978 0.0696051
\(69\) 0 0
\(70\) −1.18479 −0.141610
\(71\) 0.554378 0.0657925 0.0328963 0.999459i \(-0.489527\pi\)
0.0328963 + 0.999459i \(0.489527\pi\)
\(72\) 0 0
\(73\) 2.04963 0.239891 0.119946 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(74\) 8.12061 0.944002
\(75\) 0 0
\(76\) 3.95811 0.454026
\(77\) −1.65270 −0.188343
\(78\) 0 0
\(79\) −2.40373 −0.270441 −0.135221 0.990816i \(-0.543174\pi\)
−0.135221 + 0.990816i \(0.543174\pi\)
\(80\) 0.0564370 0.00630985
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 15.0496 1.65191 0.825956 0.563735i \(-0.190636\pi\)
0.825956 + 0.563735i \(0.190636\pi\)
\(84\) 0 0
\(85\) 0.630415 0.0683781
\(86\) −3.87939 −0.418325
\(87\) 0 0
\(88\) 4.68954 0.499907
\(89\) −9.08647 −0.963164 −0.481582 0.876401i \(-0.659938\pi\)
−0.481582 + 0.876401i \(0.659938\pi\)
\(90\) 0 0
\(91\) −3.36959 −0.353228
\(92\) 10.9709 1.14380
\(93\) 0 0
\(94\) −8.22668 −0.848517
\(95\) 4.34730 0.446023
\(96\) 0 0
\(97\) −1.89899 −0.192813 −0.0964064 0.995342i \(-0.530735\pi\)
−0.0964064 + 0.995342i \(0.530735\pi\)
\(98\) 0.879385 0.0888313
\(99\) 0 0
\(100\) 3.90673 0.390673
\(101\) 1.70914 0.170066 0.0850329 0.996378i \(-0.472900\pi\)
0.0850329 + 0.996378i \(0.472900\pi\)
\(102\) 0 0
\(103\) −3.63816 −0.358478 −0.179239 0.983806i \(-0.557364\pi\)
−0.179239 + 0.983806i \(0.557364\pi\)
\(104\) 9.56118 0.937551
\(105\) 0 0
\(106\) 0.504748 0.0490254
\(107\) −7.12836 −0.689124 −0.344562 0.938764i \(-0.611973\pi\)
−0.344562 + 0.938764i \(0.611973\pi\)
\(108\) 0 0
\(109\) 0.403733 0.0386706 0.0193353 0.999813i \(-0.493845\pi\)
0.0193353 + 0.999813i \(0.493845\pi\)
\(110\) 1.95811 0.186699
\(111\) 0 0
\(112\) −0.0418891 −0.00395814
\(113\) −14.3696 −1.35178 −0.675888 0.737004i \(-0.736240\pi\)
−0.675888 + 0.737004i \(0.736240\pi\)
\(114\) 0 0
\(115\) 12.0496 1.12363
\(116\) 7.68954 0.713956
\(117\) 0 0
\(118\) 9.14290 0.841672
\(119\) −0.467911 −0.0428933
\(120\) 0 0
\(121\) −8.26857 −0.751688
\(122\) 6.71688 0.608118
\(123\) 0 0
\(124\) −11.3277 −1.01726
\(125\) 11.0273 0.986315
\(126\) 0 0
\(127\) −20.7716 −1.84318 −0.921589 0.388167i \(-0.873108\pi\)
−0.921589 + 0.388167i \(0.873108\pi\)
\(128\) −6.84255 −0.604802
\(129\) 0 0
\(130\) 3.99226 0.350144
\(131\) 7.16519 0.626026 0.313013 0.949749i \(-0.398662\pi\)
0.313013 + 0.949749i \(0.398662\pi\)
\(132\) 0 0
\(133\) −3.22668 −0.279789
\(134\) 0.524348 0.0452968
\(135\) 0 0
\(136\) 1.32770 0.113849
\(137\) −2.56893 −0.219478 −0.109739 0.993960i \(-0.535002\pi\)
−0.109739 + 0.993960i \(0.535002\pi\)
\(138\) 0 0
\(139\) −6.13341 −0.520229 −0.260114 0.965578i \(-0.583760\pi\)
−0.260114 + 0.965578i \(0.583760\pi\)
\(140\) 1.65270 0.139679
\(141\) 0 0
\(142\) 0.487511 0.0409111
\(143\) 5.56893 0.465697
\(144\) 0 0
\(145\) 8.44562 0.701371
\(146\) 1.80241 0.149169
\(147\) 0 0
\(148\) −11.3277 −0.931131
\(149\) −0.431074 −0.0353150 −0.0176575 0.999844i \(-0.505621\pi\)
−0.0176575 + 0.999844i \(0.505621\pi\)
\(150\) 0 0
\(151\) −2.47060 −0.201055 −0.100527 0.994934i \(-0.532053\pi\)
−0.100527 + 0.994934i \(0.532053\pi\)
\(152\) 9.15570 0.742625
\(153\) 0 0
\(154\) −1.45336 −0.117115
\(155\) −12.4415 −0.999326
\(156\) 0 0
\(157\) 10.1334 0.808734 0.404367 0.914597i \(-0.367492\pi\)
0.404367 + 0.914597i \(0.367492\pi\)
\(158\) −2.11381 −0.168166
\(159\) 0 0
\(160\) −7.59627 −0.600538
\(161\) −8.94356 −0.704852
\(162\) 0 0
\(163\) −2.59627 −0.203355 −0.101678 0.994817i \(-0.532421\pi\)
−0.101678 + 0.994817i \(0.532421\pi\)
\(164\) 4.18479 0.326777
\(165\) 0 0
\(166\) 13.2344 1.02719
\(167\) 23.1830 1.79396 0.896979 0.442074i \(-0.145757\pi\)
0.896979 + 0.442074i \(0.145757\pi\)
\(168\) 0 0
\(169\) −1.64590 −0.126607
\(170\) 0.554378 0.0425188
\(171\) 0 0
\(172\) 5.41147 0.412621
\(173\) 4.75196 0.361285 0.180643 0.983549i \(-0.442182\pi\)
0.180643 + 0.983549i \(0.442182\pi\)
\(174\) 0 0
\(175\) −3.18479 −0.240748
\(176\) 0.0692302 0.00521842
\(177\) 0 0
\(178\) −7.99050 −0.598914
\(179\) 8.53209 0.637718 0.318859 0.947802i \(-0.396700\pi\)
0.318859 + 0.947802i \(0.396700\pi\)
\(180\) 0 0
\(181\) −17.2344 −1.28102 −0.640512 0.767948i \(-0.721278\pi\)
−0.640512 + 0.767948i \(0.721278\pi\)
\(182\) −2.96316 −0.219644
\(183\) 0 0
\(184\) 25.3773 1.87084
\(185\) −12.4415 −0.914718
\(186\) 0 0
\(187\) 0.773318 0.0565506
\(188\) 11.4757 0.836948
\(189\) 0 0
\(190\) 3.82295 0.277346
\(191\) −12.9094 −0.934092 −0.467046 0.884233i \(-0.654682\pi\)
−0.467046 + 0.884233i \(0.654682\pi\)
\(192\) 0 0
\(193\) −0.638156 −0.0459355 −0.0229677 0.999736i \(-0.507311\pi\)
−0.0229677 + 0.999736i \(0.507311\pi\)
\(194\) −1.66994 −0.119895
\(195\) 0 0
\(196\) −1.22668 −0.0876201
\(197\) −11.4456 −0.815467 −0.407733 0.913101i \(-0.633681\pi\)
−0.407733 + 0.913101i \(0.633681\pi\)
\(198\) 0 0
\(199\) −3.63816 −0.257902 −0.128951 0.991651i \(-0.541161\pi\)
−0.128951 + 0.991651i \(0.541161\pi\)
\(200\) 9.03684 0.639001
\(201\) 0 0
\(202\) 1.50299 0.105750
\(203\) −6.26857 −0.439967
\(204\) 0 0
\(205\) 4.59627 0.321017
\(206\) −3.19934 −0.222909
\(207\) 0 0
\(208\) 0.141149 0.00978691
\(209\) 5.33275 0.368874
\(210\) 0 0
\(211\) 5.82295 0.400868 0.200434 0.979707i \(-0.435765\pi\)
0.200434 + 0.979707i \(0.435765\pi\)
\(212\) −0.704088 −0.0483570
\(213\) 0 0
\(214\) −6.26857 −0.428511
\(215\) 5.94356 0.405348
\(216\) 0 0
\(217\) 9.23442 0.626873
\(218\) 0.355037 0.0240461
\(219\) 0 0
\(220\) −2.73143 −0.184153
\(221\) 1.57667 0.106058
\(222\) 0 0
\(223\) 7.08378 0.474365 0.237182 0.971465i \(-0.423776\pi\)
0.237182 + 0.971465i \(0.423776\pi\)
\(224\) 5.63816 0.376715
\(225\) 0 0
\(226\) −12.6364 −0.840561
\(227\) 11.9436 0.792722 0.396361 0.918095i \(-0.370273\pi\)
0.396361 + 0.918095i \(0.370273\pi\)
\(228\) 0 0
\(229\) −17.5526 −1.15991 −0.579955 0.814649i \(-0.696930\pi\)
−0.579955 + 0.814649i \(0.696930\pi\)
\(230\) 10.5963 0.698697
\(231\) 0 0
\(232\) 17.7870 1.16778
\(233\) −16.2540 −1.06484 −0.532418 0.846481i \(-0.678717\pi\)
−0.532418 + 0.846481i \(0.678717\pi\)
\(234\) 0 0
\(235\) 12.6040 0.822195
\(236\) −12.7537 −0.830196
\(237\) 0 0
\(238\) −0.411474 −0.0266719
\(239\) 15.0993 0.976690 0.488345 0.872651i \(-0.337601\pi\)
0.488345 + 0.872651i \(0.337601\pi\)
\(240\) 0 0
\(241\) −15.6382 −1.00734 −0.503671 0.863896i \(-0.668018\pi\)
−0.503671 + 0.863896i \(0.668018\pi\)
\(242\) −7.27126 −0.467414
\(243\) 0 0
\(244\) −9.36959 −0.599826
\(245\) −1.34730 −0.0860756
\(246\) 0 0
\(247\) 10.8726 0.691806
\(248\) −26.2026 −1.66387
\(249\) 0 0
\(250\) 9.69728 0.613310
\(251\) −19.0651 −1.20338 −0.601690 0.798730i \(-0.705506\pi\)
−0.601690 + 0.798730i \(0.705506\pi\)
\(252\) 0 0
\(253\) 14.7811 0.929277
\(254\) −18.2662 −1.14612
\(255\) 0 0
\(256\) −16.1010 −1.00631
\(257\) −26.5817 −1.65812 −0.829061 0.559158i \(-0.811124\pi\)
−0.829061 + 0.559158i \(0.811124\pi\)
\(258\) 0 0
\(259\) 9.23442 0.573799
\(260\) −5.56893 −0.345370
\(261\) 0 0
\(262\) 6.30096 0.389275
\(263\) 0.734118 0.0452676 0.0226338 0.999744i \(-0.492795\pi\)
0.0226338 + 0.999744i \(0.492795\pi\)
\(264\) 0 0
\(265\) −0.773318 −0.0475046
\(266\) −2.83750 −0.173978
\(267\) 0 0
\(268\) −0.731429 −0.0446792
\(269\) 20.8503 1.27126 0.635632 0.771992i \(-0.280739\pi\)
0.635632 + 0.771992i \(0.280739\pi\)
\(270\) 0 0
\(271\) 6.95811 0.422675 0.211338 0.977413i \(-0.432218\pi\)
0.211338 + 0.977413i \(0.432218\pi\)
\(272\) 0.0196004 0.00118845
\(273\) 0 0
\(274\) −2.25908 −0.136476
\(275\) 5.26352 0.317402
\(276\) 0 0
\(277\) 17.8726 1.07386 0.536930 0.843627i \(-0.319584\pi\)
0.536930 + 0.843627i \(0.319584\pi\)
\(278\) −5.39363 −0.323488
\(279\) 0 0
\(280\) 3.82295 0.228465
\(281\) −22.3105 −1.33093 −0.665465 0.746429i \(-0.731767\pi\)
−0.665465 + 0.746429i \(0.731767\pi\)
\(282\) 0 0
\(283\) −18.5945 −1.10533 −0.552665 0.833404i \(-0.686389\pi\)
−0.552665 + 0.833404i \(0.686389\pi\)
\(284\) −0.680045 −0.0403532
\(285\) 0 0
\(286\) 4.89723 0.289579
\(287\) −3.41147 −0.201373
\(288\) 0 0
\(289\) −16.7811 −0.987121
\(290\) 7.42696 0.436126
\(291\) 0 0
\(292\) −2.51424 −0.147135
\(293\) −13.0915 −0.764815 −0.382407 0.923994i \(-0.624905\pi\)
−0.382407 + 0.923994i \(0.624905\pi\)
\(294\) 0 0
\(295\) −14.0077 −0.815562
\(296\) −26.2026 −1.52300
\(297\) 0 0
\(298\) −0.379081 −0.0219595
\(299\) 30.1361 1.74282
\(300\) 0 0
\(301\) −4.41147 −0.254273
\(302\) −2.17261 −0.125020
\(303\) 0 0
\(304\) 0.135163 0.00775211
\(305\) −10.2909 −0.589253
\(306\) 0 0
\(307\) 6.31046 0.360157 0.180078 0.983652i \(-0.442365\pi\)
0.180078 + 0.983652i \(0.442365\pi\)
\(308\) 2.02734 0.115518
\(309\) 0 0
\(310\) −10.9409 −0.621400
\(311\) 9.52435 0.540076 0.270038 0.962850i \(-0.412964\pi\)
0.270038 + 0.962850i \(0.412964\pi\)
\(312\) 0 0
\(313\) −17.6287 −0.996431 −0.498215 0.867053i \(-0.666011\pi\)
−0.498215 + 0.867053i \(0.666011\pi\)
\(314\) 8.91117 0.502886
\(315\) 0 0
\(316\) 2.94862 0.165873
\(317\) −8.07697 −0.453648 −0.226824 0.973936i \(-0.572834\pi\)
−0.226824 + 0.973936i \(0.572834\pi\)
\(318\) 0 0
\(319\) 10.3601 0.580054
\(320\) −6.79292 −0.379736
\(321\) 0 0
\(322\) −7.86484 −0.438290
\(323\) 1.50980 0.0840075
\(324\) 0 0
\(325\) 10.7314 0.595273
\(326\) −2.28312 −0.126450
\(327\) 0 0
\(328\) 9.68004 0.534491
\(329\) −9.35504 −0.515760
\(330\) 0 0
\(331\) 23.0496 1.26692 0.633461 0.773775i \(-0.281634\pi\)
0.633461 + 0.773775i \(0.281634\pi\)
\(332\) −18.4611 −1.01318
\(333\) 0 0
\(334\) 20.3868 1.11552
\(335\) −0.803348 −0.0438916
\(336\) 0 0
\(337\) 29.0232 1.58100 0.790498 0.612465i \(-0.209822\pi\)
0.790498 + 0.612465i \(0.209822\pi\)
\(338\) −1.44738 −0.0787269
\(339\) 0 0
\(340\) −0.773318 −0.0419391
\(341\) −15.2618 −0.826471
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.5175 0.674901
\(345\) 0 0
\(346\) 4.17881 0.224654
\(347\) −12.9463 −0.694991 −0.347496 0.937682i \(-0.612968\pi\)
−0.347496 + 0.937682i \(0.612968\pi\)
\(348\) 0 0
\(349\) 1.46286 0.0783050 0.0391525 0.999233i \(-0.487534\pi\)
0.0391525 + 0.999233i \(0.487534\pi\)
\(350\) −2.80066 −0.149702
\(351\) 0 0
\(352\) −9.31820 −0.496662
\(353\) −14.3327 −0.762855 −0.381428 0.924399i \(-0.624567\pi\)
−0.381428 + 0.924399i \(0.624567\pi\)
\(354\) 0 0
\(355\) −0.746911 −0.0396419
\(356\) 11.1462 0.590747
\(357\) 0 0
\(358\) 7.50299 0.396546
\(359\) 20.9368 1.10500 0.552500 0.833513i \(-0.313674\pi\)
0.552500 + 0.833513i \(0.313674\pi\)
\(360\) 0 0
\(361\) −8.58853 −0.452028
\(362\) −15.1557 −0.796566
\(363\) 0 0
\(364\) 4.13341 0.216649
\(365\) −2.76146 −0.144541
\(366\) 0 0
\(367\) −12.0574 −0.629390 −0.314695 0.949193i \(-0.601902\pi\)
−0.314695 + 0.949193i \(0.601902\pi\)
\(368\) 0.374638 0.0195293
\(369\) 0 0
\(370\) −10.9409 −0.568789
\(371\) 0.573978 0.0297995
\(372\) 0 0
\(373\) −0.781059 −0.0404417 −0.0202209 0.999796i \(-0.506437\pi\)
−0.0202209 + 0.999796i \(0.506437\pi\)
\(374\) 0.680045 0.0351643
\(375\) 0 0
\(376\) 26.5449 1.36895
\(377\) 21.1225 1.08786
\(378\) 0 0
\(379\) −6.92396 −0.355660 −0.177830 0.984061i \(-0.556908\pi\)
−0.177830 + 0.984061i \(0.556908\pi\)
\(380\) −5.33275 −0.273564
\(381\) 0 0
\(382\) −11.3523 −0.580837
\(383\) −7.73236 −0.395105 −0.197553 0.980292i \(-0.563299\pi\)
−0.197553 + 0.980292i \(0.563299\pi\)
\(384\) 0 0
\(385\) 2.22668 0.113482
\(386\) −0.561185 −0.0285636
\(387\) 0 0
\(388\) 2.32945 0.118260
\(389\) −5.39961 −0.273771 −0.136886 0.990587i \(-0.543709\pi\)
−0.136886 + 0.990587i \(0.543709\pi\)
\(390\) 0 0
\(391\) 4.18479 0.211634
\(392\) −2.83750 −0.143315
\(393\) 0 0
\(394\) −10.0651 −0.507073
\(395\) 3.23854 0.162949
\(396\) 0 0
\(397\) −29.2344 −1.46723 −0.733617 0.679563i \(-0.762169\pi\)
−0.733617 + 0.679563i \(0.762169\pi\)
\(398\) −3.19934 −0.160368
\(399\) 0 0
\(400\) 0.133408 0.00667040
\(401\) 27.3979 1.36818 0.684092 0.729396i \(-0.260199\pi\)
0.684092 + 0.729396i \(0.260199\pi\)
\(402\) 0 0
\(403\) −31.1162 −1.55001
\(404\) −2.09657 −0.104308
\(405\) 0 0
\(406\) −5.51249 −0.273580
\(407\) −15.2618 −0.756498
\(408\) 0 0
\(409\) −9.02498 −0.446256 −0.223128 0.974789i \(-0.571627\pi\)
−0.223128 + 0.974789i \(0.571627\pi\)
\(410\) 4.04189 0.199615
\(411\) 0 0
\(412\) 4.46286 0.219869
\(413\) 10.3969 0.511599
\(414\) 0 0
\(415\) −20.2763 −0.995325
\(416\) −18.9982 −0.931466
\(417\) 0 0
\(418\) 4.68954 0.229373
\(419\) −0.175297 −0.00856382 −0.00428191 0.999991i \(-0.501363\pi\)
−0.00428191 + 0.999991i \(0.501363\pi\)
\(420\) 0 0
\(421\) −24.7050 −1.20405 −0.602025 0.798478i \(-0.705639\pi\)
−0.602025 + 0.798478i \(0.705639\pi\)
\(422\) 5.12061 0.249268
\(423\) 0 0
\(424\) −1.62866 −0.0790947
\(425\) 1.49020 0.0722853
\(426\) 0 0
\(427\) 7.63816 0.369636
\(428\) 8.74422 0.422668
\(429\) 0 0
\(430\) 5.22668 0.252053
\(431\) 29.3191 1.41225 0.706126 0.708086i \(-0.250441\pi\)
0.706126 + 0.708086i \(0.250441\pi\)
\(432\) 0 0
\(433\) 19.6554 0.944578 0.472289 0.881444i \(-0.343428\pi\)
0.472289 + 0.881444i \(0.343428\pi\)
\(434\) 8.12061 0.389802
\(435\) 0 0
\(436\) −0.495252 −0.0237183
\(437\) 28.8580 1.38047
\(438\) 0 0
\(439\) −21.9299 −1.04666 −0.523330 0.852130i \(-0.675310\pi\)
−0.523330 + 0.852130i \(0.675310\pi\)
\(440\) −6.31820 −0.301208
\(441\) 0 0
\(442\) 1.38650 0.0659489
\(443\) 18.7101 0.888942 0.444471 0.895793i \(-0.353392\pi\)
0.444471 + 0.895793i \(0.353392\pi\)
\(444\) 0 0
\(445\) 12.2422 0.580334
\(446\) 6.22937 0.294969
\(447\) 0 0
\(448\) 5.04189 0.238207
\(449\) −6.68004 −0.315251 −0.157625 0.987499i \(-0.550384\pi\)
−0.157625 + 0.987499i \(0.550384\pi\)
\(450\) 0 0
\(451\) 5.63816 0.265490
\(452\) 17.6269 0.829100
\(453\) 0 0
\(454\) 10.5030 0.492930
\(455\) 4.53983 0.212830
\(456\) 0 0
\(457\) −19.4287 −0.908837 −0.454418 0.890788i \(-0.650153\pi\)
−0.454418 + 0.890788i \(0.650153\pi\)
\(458\) −15.4355 −0.721254
\(459\) 0 0
\(460\) −14.7811 −0.689170
\(461\) 0.965852 0.0449842 0.0224921 0.999747i \(-0.492840\pi\)
0.0224921 + 0.999747i \(0.492840\pi\)
\(462\) 0 0
\(463\) −0.445622 −0.0207098 −0.0103549 0.999946i \(-0.503296\pi\)
−0.0103549 + 0.999946i \(0.503296\pi\)
\(464\) 0.262585 0.0121902
\(465\) 0 0
\(466\) −14.2935 −0.662136
\(467\) 34.2148 1.58327 0.791637 0.610992i \(-0.209229\pi\)
0.791637 + 0.610992i \(0.209229\pi\)
\(468\) 0 0
\(469\) 0.596267 0.0275330
\(470\) 11.0838 0.511257
\(471\) 0 0
\(472\) −29.5012 −1.35790
\(473\) 7.29086 0.335234
\(474\) 0 0
\(475\) 10.2763 0.471510
\(476\) 0.573978 0.0263082
\(477\) 0 0
\(478\) 13.2781 0.607325
\(479\) 21.7929 0.995744 0.497872 0.867251i \(-0.334115\pi\)
0.497872 + 0.867251i \(0.334115\pi\)
\(480\) 0 0
\(481\) −31.1162 −1.41878
\(482\) −13.7520 −0.626385
\(483\) 0 0
\(484\) 10.1429 0.461041
\(485\) 2.55850 0.116175
\(486\) 0 0
\(487\) 19.3928 0.878772 0.439386 0.898298i \(-0.355196\pi\)
0.439386 + 0.898298i \(0.355196\pi\)
\(488\) −21.6732 −0.981101
\(489\) 0 0
\(490\) −1.18479 −0.0535235
\(491\) −26.1566 −1.18043 −0.590216 0.807245i \(-0.700957\pi\)
−0.590216 + 0.807245i \(0.700957\pi\)
\(492\) 0 0
\(493\) 2.93313 0.132102
\(494\) 9.56118 0.430178
\(495\) 0 0
\(496\) −0.386821 −0.0173688
\(497\) 0.554378 0.0248672
\(498\) 0 0
\(499\) −14.3013 −0.640214 −0.320107 0.947381i \(-0.603719\pi\)
−0.320107 + 0.947381i \(0.603719\pi\)
\(500\) −13.5270 −0.604947
\(501\) 0 0
\(502\) −16.7656 −0.748284
\(503\) −18.7033 −0.833937 −0.416969 0.908921i \(-0.636908\pi\)
−0.416969 + 0.908921i \(0.636908\pi\)
\(504\) 0 0
\(505\) −2.30272 −0.102470
\(506\) 12.9982 0.577842
\(507\) 0 0
\(508\) 25.4801 1.13050
\(509\) 25.6091 1.13510 0.567551 0.823338i \(-0.307891\pi\)
0.567551 + 0.823338i \(0.307891\pi\)
\(510\) 0 0
\(511\) 2.04963 0.0906703
\(512\) −0.473897 −0.0209435
\(513\) 0 0
\(514\) −23.3756 −1.03105
\(515\) 4.90167 0.215994
\(516\) 0 0
\(517\) 15.4611 0.679979
\(518\) 8.12061 0.356799
\(519\) 0 0
\(520\) −12.8817 −0.564902
\(521\) 21.2121 0.929320 0.464660 0.885489i \(-0.346176\pi\)
0.464660 + 0.885489i \(0.346176\pi\)
\(522\) 0 0
\(523\) 20.8057 0.909770 0.454885 0.890550i \(-0.349680\pi\)
0.454885 + 0.890550i \(0.349680\pi\)
\(524\) −8.78941 −0.383967
\(525\) 0 0
\(526\) 0.645572 0.0281483
\(527\) −4.32089 −0.188221
\(528\) 0 0
\(529\) 56.9873 2.47771
\(530\) −0.680045 −0.0295393
\(531\) 0 0
\(532\) 3.95811 0.171606
\(533\) 11.4953 0.497915
\(534\) 0 0
\(535\) 9.60401 0.415217
\(536\) −1.69190 −0.0730791
\(537\) 0 0
\(538\) 18.3354 0.790497
\(539\) −1.65270 −0.0711870
\(540\) 0 0
\(541\) 26.7297 1.14920 0.574599 0.818435i \(-0.305158\pi\)
0.574599 + 0.818435i \(0.305158\pi\)
\(542\) 6.11886 0.262828
\(543\) 0 0
\(544\) −2.63816 −0.113110
\(545\) −0.543948 −0.0233002
\(546\) 0 0
\(547\) 36.7624 1.57185 0.785923 0.618324i \(-0.212188\pi\)
0.785923 + 0.618324i \(0.212188\pi\)
\(548\) 3.15125 0.134615
\(549\) 0 0
\(550\) 4.62866 0.197367
\(551\) 20.2267 0.861686
\(552\) 0 0
\(553\) −2.40373 −0.102217
\(554\) 15.7169 0.667746
\(555\) 0 0
\(556\) 7.52374 0.319078
\(557\) −32.3387 −1.37024 −0.685118 0.728432i \(-0.740249\pi\)
−0.685118 + 0.728432i \(0.740249\pi\)
\(558\) 0 0
\(559\) 14.8648 0.628716
\(560\) 0.0564370 0.00238490
\(561\) 0 0
\(562\) −19.6195 −0.827598
\(563\) 17.7419 0.747730 0.373865 0.927483i \(-0.378032\pi\)
0.373865 + 0.927483i \(0.378032\pi\)
\(564\) 0 0
\(565\) 19.3601 0.814485
\(566\) −16.3517 −0.687315
\(567\) 0 0
\(568\) −1.57304 −0.0660035
\(569\) 26.6013 1.11519 0.557593 0.830115i \(-0.311725\pi\)
0.557593 + 0.830115i \(0.311725\pi\)
\(570\) 0 0
\(571\) −10.0172 −0.419208 −0.209604 0.977786i \(-0.567218\pi\)
−0.209604 + 0.977786i \(0.567218\pi\)
\(572\) −6.83130 −0.285631
\(573\) 0 0
\(574\) −3.00000 −0.125218
\(575\) 28.4834 1.18784
\(576\) 0 0
\(577\) −32.9145 −1.37025 −0.685124 0.728427i \(-0.740252\pi\)
−0.685124 + 0.728427i \(0.740252\pi\)
\(578\) −14.7570 −0.613811
\(579\) 0 0
\(580\) −10.3601 −0.430179
\(581\) 15.0496 0.624364
\(582\) 0 0
\(583\) −0.948615 −0.0392876
\(584\) −5.81582 −0.240660
\(585\) 0 0
\(586\) −11.5125 −0.475577
\(587\) 15.0729 0.622123 0.311062 0.950390i \(-0.399315\pi\)
0.311062 + 0.950390i \(0.399315\pi\)
\(588\) 0 0
\(589\) −29.7965 −1.22775
\(590\) −12.3182 −0.507132
\(591\) 0 0
\(592\) −0.386821 −0.0158983
\(593\) −41.0009 −1.68371 −0.841853 0.539706i \(-0.818535\pi\)
−0.841853 + 0.539706i \(0.818535\pi\)
\(594\) 0 0
\(595\) 0.630415 0.0258445
\(596\) 0.528791 0.0216601
\(597\) 0 0
\(598\) 26.5012 1.08372
\(599\) −6.07367 −0.248164 −0.124082 0.992272i \(-0.539599\pi\)
−0.124082 + 0.992272i \(0.539599\pi\)
\(600\) 0 0
\(601\) −14.1352 −0.576585 −0.288293 0.957542i \(-0.593088\pi\)
−0.288293 + 0.957542i \(0.593088\pi\)
\(602\) −3.87939 −0.158112
\(603\) 0 0
\(604\) 3.03064 0.123315
\(605\) 11.1402 0.452914
\(606\) 0 0
\(607\) 46.0898 1.87073 0.935363 0.353689i \(-0.115073\pi\)
0.935363 + 0.353689i \(0.115073\pi\)
\(608\) −18.1925 −0.737805
\(609\) 0 0
\(610\) −9.04963 −0.366409
\(611\) 31.5226 1.27527
\(612\) 0 0
\(613\) −26.4938 −1.07008 −0.535038 0.844828i \(-0.679703\pi\)
−0.535038 + 0.844828i \(0.679703\pi\)
\(614\) 5.54933 0.223953
\(615\) 0 0
\(616\) 4.68954 0.188947
\(617\) 2.24990 0.0905777 0.0452889 0.998974i \(-0.485579\pi\)
0.0452889 + 0.998974i \(0.485579\pi\)
\(618\) 0 0
\(619\) 6.19078 0.248828 0.124414 0.992230i \(-0.460295\pi\)
0.124414 + 0.992230i \(0.460295\pi\)
\(620\) 15.2618 0.612927
\(621\) 0 0
\(622\) 8.37557 0.335830
\(623\) −9.08647 −0.364042
\(624\) 0 0
\(625\) 1.06687 0.0426746
\(626\) −15.5024 −0.619600
\(627\) 0 0
\(628\) −12.4305 −0.496030
\(629\) −4.32089 −0.172285
\(630\) 0 0
\(631\) 26.1661 1.04166 0.520829 0.853661i \(-0.325623\pi\)
0.520829 + 0.853661i \(0.325623\pi\)
\(632\) 6.82058 0.271308
\(633\) 0 0
\(634\) −7.10277 −0.282087
\(635\) 27.9855 1.11057
\(636\) 0 0
\(637\) −3.36959 −0.133508
\(638\) 9.11051 0.360689
\(639\) 0 0
\(640\) 9.21894 0.364411
\(641\) −4.88888 −0.193099 −0.0965496 0.995328i \(-0.530781\pi\)
−0.0965496 + 0.995328i \(0.530781\pi\)
\(642\) 0 0
\(643\) −40.3678 −1.59195 −0.795976 0.605328i \(-0.793042\pi\)
−0.795976 + 0.605328i \(0.793042\pi\)
\(644\) 10.9709 0.432314
\(645\) 0 0
\(646\) 1.32770 0.0522375
\(647\) 2.28075 0.0896657 0.0448329 0.998995i \(-0.485724\pi\)
0.0448329 + 0.998995i \(0.485724\pi\)
\(648\) 0 0
\(649\) −17.1830 −0.674493
\(650\) 9.43706 0.370152
\(651\) 0 0
\(652\) 3.18479 0.124726
\(653\) −23.4793 −0.918815 −0.459407 0.888226i \(-0.651938\pi\)
−0.459407 + 0.888226i \(0.651938\pi\)
\(654\) 0 0
\(655\) −9.65364 −0.377199
\(656\) 0.142903 0.00557944
\(657\) 0 0
\(658\) −8.22668 −0.320709
\(659\) 47.9623 1.86835 0.934174 0.356818i \(-0.116138\pi\)
0.934174 + 0.356818i \(0.116138\pi\)
\(660\) 0 0
\(661\) 29.3090 1.13999 0.569995 0.821648i \(-0.306945\pi\)
0.569995 + 0.821648i \(0.306945\pi\)
\(662\) 20.2695 0.787797
\(663\) 0 0
\(664\) −42.7033 −1.65721
\(665\) 4.34730 0.168581
\(666\) 0 0
\(667\) 56.0634 2.17078
\(668\) −28.4382 −1.10031
\(669\) 0 0
\(670\) −0.706452 −0.0272926
\(671\) −12.6236 −0.487329
\(672\) 0 0
\(673\) 26.3182 1.01449 0.507246 0.861801i \(-0.330664\pi\)
0.507246 + 0.861801i \(0.330664\pi\)
\(674\) 25.5226 0.983094
\(675\) 0 0
\(676\) 2.01899 0.0776535
\(677\) −35.8907 −1.37939 −0.689697 0.724098i \(-0.742256\pi\)
−0.689697 + 0.724098i \(0.742256\pi\)
\(678\) 0 0
\(679\) −1.89899 −0.0728764
\(680\) −1.78880 −0.0685973
\(681\) 0 0
\(682\) −13.4210 −0.513915
\(683\) −35.0642 −1.34169 −0.670847 0.741596i \(-0.734069\pi\)
−0.670847 + 0.741596i \(0.734069\pi\)
\(684\) 0 0
\(685\) 3.46110 0.132242
\(686\) 0.879385 0.0335751
\(687\) 0 0
\(688\) 0.184793 0.00704515
\(689\) −1.93407 −0.0736821
\(690\) 0 0
\(691\) 2.06687 0.0786273 0.0393136 0.999227i \(-0.487483\pi\)
0.0393136 + 0.999227i \(0.487483\pi\)
\(692\) −5.82915 −0.221591
\(693\) 0 0
\(694\) −11.3847 −0.432159
\(695\) 8.26352 0.313453
\(696\) 0 0
\(697\) 1.59627 0.0604629
\(698\) 1.28642 0.0486916
\(699\) 0 0
\(700\) 3.90673 0.147660
\(701\) 7.36009 0.277987 0.138993 0.990293i \(-0.455613\pi\)
0.138993 + 0.990293i \(0.455613\pi\)
\(702\) 0 0
\(703\) −29.7965 −1.12380
\(704\) −8.33275 −0.314052
\(705\) 0 0
\(706\) −12.6040 −0.474358
\(707\) 1.70914 0.0642788
\(708\) 0 0
\(709\) 9.10876 0.342086 0.171043 0.985264i \(-0.445286\pi\)
0.171043 + 0.985264i \(0.445286\pi\)
\(710\) −0.656822 −0.0246501
\(711\) 0 0
\(712\) 25.7828 0.966252
\(713\) −82.5886 −3.09297
\(714\) 0 0
\(715\) −7.50299 −0.280596
\(716\) −10.4662 −0.391139
\(717\) 0 0
\(718\) 18.4115 0.687110
\(719\) 25.9537 0.967908 0.483954 0.875093i \(-0.339200\pi\)
0.483954 + 0.875093i \(0.339200\pi\)
\(720\) 0 0
\(721\) −3.63816 −0.135492
\(722\) −7.55262 −0.281080
\(723\) 0 0
\(724\) 21.1411 0.785705
\(725\) 19.9641 0.741448
\(726\) 0 0
\(727\) −10.1601 −0.376819 −0.188409 0.982091i \(-0.560333\pi\)
−0.188409 + 0.982091i \(0.560333\pi\)
\(728\) 9.56118 0.354361
\(729\) 0 0
\(730\) −2.42839 −0.0898786
\(731\) 2.06418 0.0763464
\(732\) 0 0
\(733\) 40.6614 1.50186 0.750931 0.660381i \(-0.229605\pi\)
0.750931 + 0.660381i \(0.229605\pi\)
\(734\) −10.6031 −0.391367
\(735\) 0 0
\(736\) −50.4252 −1.85870
\(737\) −0.985452 −0.0362996
\(738\) 0 0
\(739\) −25.3618 −0.932951 −0.466475 0.884534i \(-0.654476\pi\)
−0.466475 + 0.884534i \(0.654476\pi\)
\(740\) 15.2618 0.561034
\(741\) 0 0
\(742\) 0.504748 0.0185299
\(743\) 22.4442 0.823398 0.411699 0.911320i \(-0.364936\pi\)
0.411699 + 0.911320i \(0.364936\pi\)
\(744\) 0 0
\(745\) 0.580785 0.0212783
\(746\) −0.686852 −0.0251474
\(747\) 0 0
\(748\) −0.948615 −0.0346848
\(749\) −7.12836 −0.260464
\(750\) 0 0
\(751\) 24.2172 0.883698 0.441849 0.897090i \(-0.354323\pi\)
0.441849 + 0.897090i \(0.354323\pi\)
\(752\) 0.391874 0.0142902
\(753\) 0 0
\(754\) 18.5748 0.676454
\(755\) 3.32863 0.121141
\(756\) 0 0
\(757\) 9.11793 0.331397 0.165698 0.986176i \(-0.447012\pi\)
0.165698 + 0.986176i \(0.447012\pi\)
\(758\) −6.08883 −0.221156
\(759\) 0 0
\(760\) −12.3354 −0.447453
\(761\) 18.2722 0.662366 0.331183 0.943566i \(-0.392552\pi\)
0.331183 + 0.943566i \(0.392552\pi\)
\(762\) 0 0
\(763\) 0.403733 0.0146161
\(764\) 15.8357 0.572917
\(765\) 0 0
\(766\) −6.79973 −0.245684
\(767\) −35.0333 −1.26498
\(768\) 0 0
\(769\) 18.5294 0.668187 0.334094 0.942540i \(-0.391570\pi\)
0.334094 + 0.942540i \(0.391570\pi\)
\(770\) 1.95811 0.0705654
\(771\) 0 0
\(772\) 0.782814 0.0281741
\(773\) 2.96080 0.106493 0.0532463 0.998581i \(-0.483043\pi\)
0.0532463 + 0.998581i \(0.483043\pi\)
\(774\) 0 0
\(775\) −29.4097 −1.05643
\(776\) 5.38836 0.193431
\(777\) 0 0
\(778\) −4.74834 −0.170236
\(779\) 11.0077 0.394393
\(780\) 0 0
\(781\) −0.916222 −0.0327850
\(782\) 3.68004 0.131598
\(783\) 0 0
\(784\) −0.0418891 −0.00149604
\(785\) −13.6527 −0.487286
\(786\) 0 0
\(787\) −33.4020 −1.19065 −0.595326 0.803484i \(-0.702977\pi\)
−0.595326 + 0.803484i \(0.702977\pi\)
\(788\) 14.0401 0.500159
\(789\) 0 0
\(790\) 2.84793 0.101325
\(791\) −14.3696 −0.510924
\(792\) 0 0
\(793\) −25.7374 −0.913962
\(794\) −25.7083 −0.912354
\(795\) 0 0
\(796\) 4.46286 0.158182
\(797\) −49.3509 −1.74810 −0.874050 0.485837i \(-0.838515\pi\)
−0.874050 + 0.485837i \(0.838515\pi\)
\(798\) 0 0
\(799\) 4.37733 0.154859
\(800\) −17.9564 −0.634853
\(801\) 0 0
\(802\) 24.0933 0.850763
\(803\) −3.38743 −0.119540
\(804\) 0 0
\(805\) 12.0496 0.424694
\(806\) −27.3631 −0.963824
\(807\) 0 0
\(808\) −4.84968 −0.170611
\(809\) −19.8280 −0.697115 −0.348558 0.937287i \(-0.613328\pi\)
−0.348558 + 0.937287i \(0.613328\pi\)
\(810\) 0 0
\(811\) −23.8557 −0.837686 −0.418843 0.908059i \(-0.637564\pi\)
−0.418843 + 0.908059i \(0.637564\pi\)
\(812\) 7.68954 0.269850
\(813\) 0 0
\(814\) −13.4210 −0.470405
\(815\) 3.49794 0.122528
\(816\) 0 0
\(817\) 14.2344 0.497999
\(818\) −7.93643 −0.277491
\(819\) 0 0
\(820\) −5.63816 −0.196893
\(821\) 50.9427 1.77791 0.888957 0.457991i \(-0.151431\pi\)
0.888957 + 0.457991i \(0.151431\pi\)
\(822\) 0 0
\(823\) 13.6149 0.474587 0.237293 0.971438i \(-0.423740\pi\)
0.237293 + 0.971438i \(0.423740\pi\)
\(824\) 10.3233 0.359628
\(825\) 0 0
\(826\) 9.14290 0.318122
\(827\) −36.2158 −1.25935 −0.629673 0.776861i \(-0.716811\pi\)
−0.629673 + 0.776861i \(0.716811\pi\)
\(828\) 0 0
\(829\) 25.3259 0.879606 0.439803 0.898094i \(-0.355048\pi\)
0.439803 + 0.898094i \(0.355048\pi\)
\(830\) −17.8307 −0.618912
\(831\) 0 0
\(832\) −16.9891 −0.588990
\(833\) −0.467911 −0.0162122
\(834\) 0 0
\(835\) −31.2344 −1.08091
\(836\) −6.54158 −0.226245
\(837\) 0 0
\(838\) −0.154154 −0.00532515
\(839\) −8.71419 −0.300847 −0.150424 0.988622i \(-0.548064\pi\)
−0.150424 + 0.988622i \(0.548064\pi\)
\(840\) 0 0
\(841\) 10.2950 0.354999
\(842\) −21.7252 −0.748701
\(843\) 0 0
\(844\) −7.14290 −0.245869
\(845\) 2.21751 0.0762847
\(846\) 0 0
\(847\) −8.26857 −0.284111
\(848\) −0.0240434 −0.000825654 0
\(849\) 0 0
\(850\) 1.31046 0.0449484
\(851\) −82.5886 −2.83110
\(852\) 0 0
\(853\) −11.9813 −0.410233 −0.205117 0.978738i \(-0.565757\pi\)
−0.205117 + 0.978738i \(0.565757\pi\)
\(854\) 6.71688 0.229847
\(855\) 0 0
\(856\) 20.2267 0.691334
\(857\) −6.50030 −0.222046 −0.111023 0.993818i \(-0.535413\pi\)
−0.111023 + 0.993818i \(0.535413\pi\)
\(858\) 0 0
\(859\) −53.5526 −1.82719 −0.913596 0.406623i \(-0.866706\pi\)
−0.913596 + 0.406623i \(0.866706\pi\)
\(860\) −7.29086 −0.248616
\(861\) 0 0
\(862\) 25.7828 0.878166
\(863\) −3.69965 −0.125937 −0.0629687 0.998016i \(-0.520057\pi\)
−0.0629687 + 0.998016i \(0.520057\pi\)
\(864\) 0 0
\(865\) −6.40230 −0.217685
\(866\) 17.2847 0.587357
\(867\) 0 0
\(868\) −11.3277 −0.384487
\(869\) 3.97266 0.134763
\(870\) 0 0
\(871\) −2.00917 −0.0680782
\(872\) −1.14559 −0.0387946
\(873\) 0 0
\(874\) 25.3773 0.858401
\(875\) 11.0273 0.372792
\(876\) 0 0
\(877\) −11.7888 −0.398079 −0.199040 0.979991i \(-0.563782\pi\)
−0.199040 + 0.979991i \(0.563782\pi\)
\(878\) −19.2849 −0.650833
\(879\) 0 0
\(880\) −0.0932736 −0.00314425
\(881\) −49.4858 −1.66722 −0.833609 0.552355i \(-0.813729\pi\)
−0.833609 + 0.552355i \(0.813729\pi\)
\(882\) 0 0
\(883\) −21.5357 −0.724734 −0.362367 0.932035i \(-0.618031\pi\)
−0.362367 + 0.932035i \(0.618031\pi\)
\(884\) −1.93407 −0.0650497
\(885\) 0 0
\(886\) 16.4534 0.552762
\(887\) −11.8848 −0.399051 −0.199526 0.979893i \(-0.563940\pi\)
−0.199526 + 0.979893i \(0.563940\pi\)
\(888\) 0 0
\(889\) −20.7716 −0.696656
\(890\) 10.7656 0.360863
\(891\) 0 0
\(892\) −8.68954 −0.290947
\(893\) 30.1857 1.01013
\(894\) 0 0
\(895\) −11.4953 −0.384244
\(896\) −6.84255 −0.228594
\(897\) 0 0
\(898\) −5.87433 −0.196029
\(899\) −57.8866 −1.93063
\(900\) 0 0
\(901\) −0.268571 −0.00894739
\(902\) 4.95811 0.165087
\(903\) 0 0
\(904\) 40.7736 1.35611
\(905\) 23.2199 0.771855
\(906\) 0 0
\(907\) −26.0215 −0.864029 −0.432014 0.901867i \(-0.642197\pi\)
−0.432014 + 0.901867i \(0.642197\pi\)
\(908\) −14.6509 −0.486209
\(909\) 0 0
\(910\) 3.99226 0.132342
\(911\) 4.03272 0.133610 0.0668050 0.997766i \(-0.478719\pi\)
0.0668050 + 0.997766i \(0.478719\pi\)
\(912\) 0 0
\(913\) −24.8726 −0.823162
\(914\) −17.0853 −0.565132
\(915\) 0 0
\(916\) 21.5315 0.711420
\(917\) 7.16519 0.236615
\(918\) 0 0
\(919\) 27.4270 0.904732 0.452366 0.891832i \(-0.350580\pi\)
0.452366 + 0.891832i \(0.350580\pi\)
\(920\) −34.1908 −1.12724
\(921\) 0 0
\(922\) 0.849356 0.0279720
\(923\) −1.86802 −0.0614867
\(924\) 0 0
\(925\) −29.4097 −0.966986
\(926\) −0.391874 −0.0128778
\(927\) 0 0
\(928\) −35.3432 −1.16020
\(929\) −7.67675 −0.251866 −0.125933 0.992039i \(-0.540192\pi\)
−0.125933 + 0.992039i \(0.540192\pi\)
\(930\) 0 0
\(931\) −3.22668 −0.105750
\(932\) 19.9385 0.653108
\(933\) 0 0
\(934\) 30.0880 0.984510
\(935\) −1.04189 −0.0340734
\(936\) 0 0
\(937\) −2.02465 −0.0661425 −0.0330713 0.999453i \(-0.510529\pi\)
−0.0330713 + 0.999453i \(0.510529\pi\)
\(938\) 0.524348 0.0171206
\(939\) 0 0
\(940\) −15.4611 −0.504286
\(941\) 6.13928 0.200135 0.100067 0.994981i \(-0.468094\pi\)
0.100067 + 0.994981i \(0.468094\pi\)
\(942\) 0 0
\(943\) 30.5107 0.993566
\(944\) −0.435518 −0.0141749
\(945\) 0 0
\(946\) 6.41147 0.208455
\(947\) −5.56448 −0.180821 −0.0904107 0.995905i \(-0.528818\pi\)
−0.0904107 + 0.995905i \(0.528818\pi\)
\(948\) 0 0
\(949\) −6.90640 −0.224191
\(950\) 9.03684 0.293194
\(951\) 0 0
\(952\) 1.32770 0.0430309
\(953\) 8.72018 0.282474 0.141237 0.989976i \(-0.454892\pi\)
0.141237 + 0.989976i \(0.454892\pi\)
\(954\) 0 0
\(955\) 17.3928 0.562818
\(956\) −18.5220 −0.599044
\(957\) 0 0
\(958\) 19.1644 0.619173
\(959\) −2.56893 −0.0829549
\(960\) 0 0
\(961\) 54.2746 1.75079
\(962\) −27.3631 −0.882222
\(963\) 0 0
\(964\) 19.1830 0.617844
\(965\) 0.859785 0.0276775
\(966\) 0 0
\(967\) −57.7698 −1.85775 −0.928876 0.370391i \(-0.879224\pi\)
−0.928876 + 0.370391i \(0.879224\pi\)
\(968\) 23.4620 0.754098
\(969\) 0 0
\(970\) 2.24990 0.0722401
\(971\) 30.7192 0.985828 0.492914 0.870078i \(-0.335932\pi\)
0.492914 + 0.870078i \(0.335932\pi\)
\(972\) 0 0
\(973\) −6.13341 −0.196628
\(974\) 17.0537 0.546437
\(975\) 0 0
\(976\) −0.319955 −0.0102415
\(977\) −10.3000 −0.329527 −0.164764 0.986333i \(-0.552686\pi\)
−0.164764 + 0.986333i \(0.552686\pi\)
\(978\) 0 0
\(979\) 15.0172 0.479953
\(980\) 1.65270 0.0527937
\(981\) 0 0
\(982\) −23.0018 −0.734015
\(983\) −13.6963 −0.436846 −0.218423 0.975854i \(-0.570091\pi\)
−0.218423 + 0.975854i \(0.570091\pi\)
\(984\) 0 0
\(985\) 15.4206 0.491343
\(986\) 2.57935 0.0821434
\(987\) 0 0
\(988\) −13.3372 −0.424313
\(989\) 39.4543 1.25457
\(990\) 0 0
\(991\) 57.9813 1.84184 0.920919 0.389754i \(-0.127440\pi\)
0.920919 + 0.389754i \(0.127440\pi\)
\(992\) 52.0651 1.65307
\(993\) 0 0
\(994\) 0.487511 0.0154629
\(995\) 4.90167 0.155394
\(996\) 0 0
\(997\) 16.2175 0.513614 0.256807 0.966463i \(-0.417330\pi\)
0.256807 + 0.966463i \(0.417330\pi\)
\(998\) −12.5763 −0.398097
\(999\) 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.1.3 3
3.2 odd 2 567.2.a.h.1.1 3
4.3 odd 2 9072.2.a.bs.1.2 3
7.6 odd 2 3969.2.a.l.1.3 3
9.2 odd 6 63.2.f.a.22.3 6
9.4 even 3 189.2.f.b.127.1 6
9.5 odd 6 63.2.f.a.43.3 yes 6
9.7 even 3 189.2.f.b.64.1 6
12.11 even 2 9072.2.a.ca.1.2 3
21.20 even 2 3969.2.a.q.1.1 3
36.7 odd 6 3024.2.r.k.1009.2 6
36.11 even 6 1008.2.r.h.337.2 6
36.23 even 6 1008.2.r.h.673.2 6
36.31 odd 6 3024.2.r.k.2017.2 6
63.2 odd 6 441.2.g.c.67.3 6
63.4 even 3 1323.2.g.d.667.1 6
63.5 even 6 441.2.h.e.214.1 6
63.11 odd 6 441.2.h.d.373.1 6
63.13 odd 6 1323.2.f.d.883.1 6
63.16 even 3 1323.2.g.d.361.1 6
63.20 even 6 441.2.f.c.148.3 6
63.23 odd 6 441.2.h.d.214.1 6
63.25 even 3 1323.2.h.c.226.3 6
63.31 odd 6 1323.2.g.e.667.1 6
63.32 odd 6 441.2.g.c.79.3 6
63.34 odd 6 1323.2.f.d.442.1 6
63.38 even 6 441.2.h.e.373.1 6
63.40 odd 6 1323.2.h.b.802.3 6
63.41 even 6 441.2.f.c.295.3 6
63.47 even 6 441.2.g.b.67.3 6
63.52 odd 6 1323.2.h.b.226.3 6
63.58 even 3 1323.2.h.c.802.3 6
63.59 even 6 441.2.g.b.79.3 6
63.61 odd 6 1323.2.g.e.361.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 6 9.2 odd 6
63.2.f.a.43.3 yes 6 9.5 odd 6
189.2.f.b.64.1 6 9.7 even 3
189.2.f.b.127.1 6 9.4 even 3
441.2.f.c.148.3 6 63.20 even 6
441.2.f.c.295.3 6 63.41 even 6
441.2.g.b.67.3 6 63.47 even 6
441.2.g.b.79.3 6 63.59 even 6
441.2.g.c.67.3 6 63.2 odd 6
441.2.g.c.79.3 6 63.32 odd 6
441.2.h.d.214.1 6 63.23 odd 6
441.2.h.d.373.1 6 63.11 odd 6
441.2.h.e.214.1 6 63.5 even 6
441.2.h.e.373.1 6 63.38 even 6
567.2.a.c.1.3 3 1.1 even 1 trivial
567.2.a.h.1.1 3 3.2 odd 2
1008.2.r.h.337.2 6 36.11 even 6
1008.2.r.h.673.2 6 36.23 even 6
1323.2.f.d.442.1 6 63.34 odd 6
1323.2.f.d.883.1 6 63.13 odd 6
1323.2.g.d.361.1 6 63.16 even 3
1323.2.g.d.667.1 6 63.4 even 3
1323.2.g.e.361.1 6 63.61 odd 6
1323.2.g.e.667.1 6 63.31 odd 6
1323.2.h.b.226.3 6 63.52 odd 6
1323.2.h.b.802.3 6 63.40 odd 6
1323.2.h.c.226.3 6 63.25 even 3
1323.2.h.c.802.3 6 63.58 even 3
3024.2.r.k.1009.2 6 36.7 odd 6
3024.2.r.k.2017.2 6 36.31 odd 6
3969.2.a.l.1.3 3 7.6 odd 2
3969.2.a.q.1.1 3 21.20 even 2
9072.2.a.bs.1.2 3 4.3 odd 2
9072.2.a.ca.1.2 3 12.11 even 2