Properties

Label 729.2.a.d.1.5
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1397493.1
Defining polynomial: \( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.584534\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.11662 q^{2} +2.48009 q^{4} +2.68310 q^{5} +0.972333 q^{7} +1.01617 q^{8} +O(q^{10})\) \(q+2.11662 q^{2} +2.48009 q^{4} +2.68310 q^{5} +0.972333 q^{7} +1.01617 q^{8} +5.67911 q^{10} +0.316902 q^{11} -1.51403 q^{13} +2.05806 q^{14} -2.80933 q^{16} -1.17468 q^{17} +6.22080 q^{19} +6.65433 q^{20} +0.670762 q^{22} -2.16274 q^{23} +2.19901 q^{25} -3.20463 q^{26} +2.41147 q^{28} +4.40491 q^{29} -8.67323 q^{31} -7.97863 q^{32} -2.48636 q^{34} +2.60886 q^{35} -4.46665 q^{37} +13.1671 q^{38} +2.72649 q^{40} -5.84518 q^{41} +5.59099 q^{43} +0.785946 q^{44} -4.57771 q^{46} +2.47607 q^{47} -6.05457 q^{49} +4.65448 q^{50} -3.75493 q^{52} +10.8920 q^{53} +0.850279 q^{55} +0.988058 q^{56} +9.32353 q^{58} +1.72421 q^{59} +1.01478 q^{61} -18.3580 q^{62} -11.2691 q^{64} -4.06229 q^{65} +0.856551 q^{67} -2.91332 q^{68} +5.52198 q^{70} +9.59577 q^{71} -15.2418 q^{73} -9.45420 q^{74} +15.4282 q^{76} +0.308134 q^{77} -11.2138 q^{79} -7.53771 q^{80} -12.3721 q^{82} -4.68493 q^{83} -3.15179 q^{85} +11.8340 q^{86} +0.322027 q^{88} +15.4995 q^{89} -1.47214 q^{91} -5.36380 q^{92} +5.24090 q^{94} +16.6910 q^{95} -5.54949 q^{97} -12.8152 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35} + 3 q^{37} - 3 q^{38} + 6 q^{40} + 15 q^{41} + 3 q^{44} + 3 q^{46} + 21 q^{47} - 12 q^{49} + 3 q^{50} + 12 q^{52} + 9 q^{53} - 6 q^{55} - 6 q^{56} - 12 q^{58} + 24 q^{59} - 9 q^{61} - 12 q^{62} - 12 q^{64} - 6 q^{65} - 9 q^{67} - 9 q^{68} + 15 q^{70} + 27 q^{71} - 6 q^{73} - 12 q^{74} + 6 q^{76} - 12 q^{77} - 21 q^{80} - 6 q^{82} + 12 q^{83} - 21 q^{86} + 12 q^{88} + 9 q^{89} - 6 q^{91} + 6 q^{92} + 6 q^{94} + 12 q^{95} - 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.11662 1.49668 0.748339 0.663316i \(-0.230852\pi\)
0.748339 + 0.663316i \(0.230852\pi\)
\(3\) 0 0
\(4\) 2.48009 1.24005
\(5\) 2.68310 1.19992 0.599959 0.800031i \(-0.295184\pi\)
0.599959 + 0.800031i \(0.295184\pi\)
\(6\) 0 0
\(7\) 0.972333 0.367507 0.183754 0.982972i \(-0.441175\pi\)
0.183754 + 0.982972i \(0.441175\pi\)
\(8\) 1.01617 0.359271
\(9\) 0 0
\(10\) 5.67911 1.79589
\(11\) 0.316902 0.0955496 0.0477748 0.998858i \(-0.484787\pi\)
0.0477748 + 0.998858i \(0.484787\pi\)
\(12\) 0 0
\(13\) −1.51403 −0.419916 −0.209958 0.977710i \(-0.567333\pi\)
−0.209958 + 0.977710i \(0.567333\pi\)
\(14\) 2.05806 0.550040
\(15\) 0 0
\(16\) −2.80933 −0.702332
\(17\) −1.17468 −0.284903 −0.142451 0.989802i \(-0.545498\pi\)
−0.142451 + 0.989802i \(0.545498\pi\)
\(18\) 0 0
\(19\) 6.22080 1.42715 0.713575 0.700579i \(-0.247075\pi\)
0.713575 + 0.700579i \(0.247075\pi\)
\(20\) 6.65433 1.48795
\(21\) 0 0
\(22\) 0.670762 0.143007
\(23\) −2.16274 −0.450963 −0.225481 0.974247i \(-0.572395\pi\)
−0.225481 + 0.974247i \(0.572395\pi\)
\(24\) 0 0
\(25\) 2.19901 0.439803
\(26\) −3.20463 −0.628480
\(27\) 0 0
\(28\) 2.41147 0.455726
\(29\) 4.40491 0.817971 0.408986 0.912541i \(-0.365883\pi\)
0.408986 + 0.912541i \(0.365883\pi\)
\(30\) 0 0
\(31\) −8.67323 −1.55776 −0.778879 0.627174i \(-0.784211\pi\)
−0.778879 + 0.627174i \(0.784211\pi\)
\(32\) −7.97863 −1.41044
\(33\) 0 0
\(34\) −2.48636 −0.426408
\(35\) 2.60886 0.440978
\(36\) 0 0
\(37\) −4.46665 −0.734312 −0.367156 0.930159i \(-0.619668\pi\)
−0.367156 + 0.930159i \(0.619668\pi\)
\(38\) 13.1671 2.13598
\(39\) 0 0
\(40\) 2.72649 0.431096
\(41\) −5.84518 −0.912864 −0.456432 0.889758i \(-0.650873\pi\)
−0.456432 + 0.889758i \(0.650873\pi\)
\(42\) 0 0
\(43\) 5.59099 0.852618 0.426309 0.904578i \(-0.359814\pi\)
0.426309 + 0.904578i \(0.359814\pi\)
\(44\) 0.785946 0.118486
\(45\) 0 0
\(46\) −4.57771 −0.674946
\(47\) 2.47607 0.361172 0.180586 0.983559i \(-0.442201\pi\)
0.180586 + 0.983559i \(0.442201\pi\)
\(48\) 0 0
\(49\) −6.05457 −0.864938
\(50\) 4.65448 0.658243
\(51\) 0 0
\(52\) −3.75493 −0.520716
\(53\) 10.8920 1.49613 0.748063 0.663628i \(-0.230984\pi\)
0.748063 + 0.663628i \(0.230984\pi\)
\(54\) 0 0
\(55\) 0.850279 0.114652
\(56\) 0.988058 0.132035
\(57\) 0 0
\(58\) 9.32353 1.22424
\(59\) 1.72421 0.224473 0.112237 0.993682i \(-0.464199\pi\)
0.112237 + 0.993682i \(0.464199\pi\)
\(60\) 0 0
\(61\) 1.01478 0.129929 0.0649645 0.997888i \(-0.479307\pi\)
0.0649645 + 0.997888i \(0.479307\pi\)
\(62\) −18.3580 −2.33146
\(63\) 0 0
\(64\) −11.2691 −1.40864
\(65\) −4.06229 −0.503865
\(66\) 0 0
\(67\) 0.856551 0.104644 0.0523222 0.998630i \(-0.483338\pi\)
0.0523222 + 0.998630i \(0.483338\pi\)
\(68\) −2.91332 −0.353292
\(69\) 0 0
\(70\) 5.52198 0.660003
\(71\) 9.59577 1.13881 0.569404 0.822058i \(-0.307174\pi\)
0.569404 + 0.822058i \(0.307174\pi\)
\(72\) 0 0
\(73\) −15.2418 −1.78392 −0.891960 0.452113i \(-0.850670\pi\)
−0.891960 + 0.452113i \(0.850670\pi\)
\(74\) −9.45420 −1.09903
\(75\) 0 0
\(76\) 15.4282 1.76973
\(77\) 0.308134 0.0351152
\(78\) 0 0
\(79\) −11.2138 −1.26165 −0.630824 0.775926i \(-0.717283\pi\)
−0.630824 + 0.775926i \(0.717283\pi\)
\(80\) −7.53771 −0.842741
\(81\) 0 0
\(82\) −12.3721 −1.36626
\(83\) −4.68493 −0.514238 −0.257119 0.966380i \(-0.582773\pi\)
−0.257119 + 0.966380i \(0.582773\pi\)
\(84\) 0 0
\(85\) −3.15179 −0.341860
\(86\) 11.8340 1.27610
\(87\) 0 0
\(88\) 0.322027 0.0343282
\(89\) 15.4995 1.64295 0.821473 0.570248i \(-0.193153\pi\)
0.821473 + 0.570248i \(0.193153\pi\)
\(90\) 0 0
\(91\) −1.47214 −0.154322
\(92\) −5.36380 −0.559215
\(93\) 0 0
\(94\) 5.24090 0.540558
\(95\) 16.6910 1.71246
\(96\) 0 0
\(97\) −5.54949 −0.563465 −0.281732 0.959493i \(-0.590909\pi\)
−0.281732 + 0.959493i \(0.590909\pi\)
\(98\) −12.8152 −1.29453
\(99\) 0 0
\(100\) 5.45376 0.545376
\(101\) 10.1390 1.00887 0.504436 0.863449i \(-0.331700\pi\)
0.504436 + 0.863449i \(0.331700\pi\)
\(102\) 0 0
\(103\) −9.85329 −0.970874 −0.485437 0.874272i \(-0.661339\pi\)
−0.485437 + 0.874272i \(0.661339\pi\)
\(104\) −1.53852 −0.150864
\(105\) 0 0
\(106\) 23.0542 2.23922
\(107\) −5.17080 −0.499880 −0.249940 0.968261i \(-0.580411\pi\)
−0.249940 + 0.968261i \(0.580411\pi\)
\(108\) 0 0
\(109\) −7.31065 −0.700234 −0.350117 0.936706i \(-0.613858\pi\)
−0.350117 + 0.936706i \(0.613858\pi\)
\(110\) 1.79972 0.171597
\(111\) 0 0
\(112\) −2.73160 −0.258112
\(113\) −10.3756 −0.976058 −0.488029 0.872827i \(-0.662284\pi\)
−0.488029 + 0.872827i \(0.662284\pi\)
\(114\) 0 0
\(115\) −5.80285 −0.541118
\(116\) 10.9246 1.01432
\(117\) 0 0
\(118\) 3.64950 0.335964
\(119\) −1.14218 −0.104704
\(120\) 0 0
\(121\) −10.8996 −0.990870
\(122\) 2.14790 0.194462
\(123\) 0 0
\(124\) −21.5104 −1.93169
\(125\) −7.51532 −0.672191
\(126\) 0 0
\(127\) 5.22743 0.463860 0.231930 0.972733i \(-0.425496\pi\)
0.231930 + 0.972733i \(0.425496\pi\)
\(128\) −7.89516 −0.697841
\(129\) 0 0
\(130\) −8.59834 −0.754124
\(131\) 7.23430 0.632064 0.316032 0.948749i \(-0.397649\pi\)
0.316032 + 0.948749i \(0.397649\pi\)
\(132\) 0 0
\(133\) 6.04869 0.524488
\(134\) 1.81300 0.156619
\(135\) 0 0
\(136\) −1.19368 −0.102357
\(137\) −11.2493 −0.961094 −0.480547 0.876969i \(-0.659562\pi\)
−0.480547 + 0.876969i \(0.659562\pi\)
\(138\) 0 0
\(139\) 9.38016 0.795615 0.397808 0.917469i \(-0.369771\pi\)
0.397808 + 0.917469i \(0.369771\pi\)
\(140\) 6.47022 0.546833
\(141\) 0 0
\(142\) 20.3106 1.70443
\(143\) −0.479799 −0.0401228
\(144\) 0 0
\(145\) 11.8188 0.981498
\(146\) −32.2612 −2.66996
\(147\) 0 0
\(148\) −11.0777 −0.910581
\(149\) −19.0512 −1.56074 −0.780369 0.625319i \(-0.784969\pi\)
−0.780369 + 0.625319i \(0.784969\pi\)
\(150\) 0 0
\(151\) 4.01400 0.326655 0.163327 0.986572i \(-0.447777\pi\)
0.163327 + 0.986572i \(0.447777\pi\)
\(152\) 6.32141 0.512734
\(153\) 0 0
\(154\) 0.652204 0.0525561
\(155\) −23.2711 −1.86918
\(156\) 0 0
\(157\) 7.27592 0.580681 0.290341 0.956923i \(-0.406231\pi\)
0.290341 + 0.956923i \(0.406231\pi\)
\(158\) −23.7353 −1.88828
\(159\) 0 0
\(160\) −21.4075 −1.69241
\(161\) −2.10290 −0.165732
\(162\) 0 0
\(163\) 12.4492 0.975094 0.487547 0.873097i \(-0.337892\pi\)
0.487547 + 0.873097i \(0.337892\pi\)
\(164\) −14.4966 −1.13199
\(165\) 0 0
\(166\) −9.91623 −0.769649
\(167\) 2.33189 0.180447 0.0902234 0.995922i \(-0.471242\pi\)
0.0902234 + 0.995922i \(0.471242\pi\)
\(168\) 0 0
\(169\) −10.7077 −0.823670
\(170\) −6.67116 −0.511654
\(171\) 0 0
\(172\) 13.8662 1.05729
\(173\) 3.58227 0.272355 0.136177 0.990684i \(-0.456518\pi\)
0.136177 + 0.990684i \(0.456518\pi\)
\(174\) 0 0
\(175\) 2.13817 0.161631
\(176\) −0.890282 −0.0671076
\(177\) 0 0
\(178\) 32.8066 2.45896
\(179\) 19.9957 1.49455 0.747275 0.664515i \(-0.231362\pi\)
0.747275 + 0.664515i \(0.231362\pi\)
\(180\) 0 0
\(181\) 9.73232 0.723398 0.361699 0.932295i \(-0.382197\pi\)
0.361699 + 0.932295i \(0.382197\pi\)
\(182\) −3.11597 −0.230971
\(183\) 0 0
\(184\) −2.19772 −0.162018
\(185\) −11.9844 −0.881114
\(186\) 0 0
\(187\) −0.372260 −0.0272223
\(188\) 6.14088 0.447869
\(189\) 0 0
\(190\) 35.3286 2.56301
\(191\) 17.7565 1.28482 0.642409 0.766362i \(-0.277935\pi\)
0.642409 + 0.766362i \(0.277935\pi\)
\(192\) 0 0
\(193\) 10.5843 0.761877 0.380939 0.924600i \(-0.375601\pi\)
0.380939 + 0.924600i \(0.375601\pi\)
\(194\) −11.7462 −0.843326
\(195\) 0 0
\(196\) −15.0159 −1.07256
\(197\) −14.1589 −1.00878 −0.504390 0.863476i \(-0.668282\pi\)
−0.504390 + 0.863476i \(0.668282\pi\)
\(198\) 0 0
\(199\) 7.54019 0.534510 0.267255 0.963626i \(-0.413883\pi\)
0.267255 + 0.963626i \(0.413883\pi\)
\(200\) 2.23458 0.158009
\(201\) 0 0
\(202\) 21.4605 1.50996
\(203\) 4.28304 0.300610
\(204\) 0 0
\(205\) −15.6832 −1.09536
\(206\) −20.8557 −1.45309
\(207\) 0 0
\(208\) 4.25341 0.294921
\(209\) 1.97139 0.136364
\(210\) 0 0
\(211\) 5.21364 0.358922 0.179461 0.983765i \(-0.442565\pi\)
0.179461 + 0.983765i \(0.442565\pi\)
\(212\) 27.0131 1.85526
\(213\) 0 0
\(214\) −10.9446 −0.748160
\(215\) 15.0012 1.02307
\(216\) 0 0
\(217\) −8.43326 −0.572487
\(218\) −15.4739 −1.04802
\(219\) 0 0
\(220\) 2.10877 0.142173
\(221\) 1.77851 0.119635
\(222\) 0 0
\(223\) 17.6938 1.18486 0.592432 0.805620i \(-0.298168\pi\)
0.592432 + 0.805620i \(0.298168\pi\)
\(224\) −7.75789 −0.518346
\(225\) 0 0
\(226\) −21.9613 −1.46085
\(227\) 15.7720 1.04682 0.523412 0.852080i \(-0.324659\pi\)
0.523412 + 0.852080i \(0.324659\pi\)
\(228\) 0 0
\(229\) −1.76686 −0.116758 −0.0583788 0.998295i \(-0.518593\pi\)
−0.0583788 + 0.998295i \(0.518593\pi\)
\(230\) −12.2824 −0.809880
\(231\) 0 0
\(232\) 4.47615 0.293874
\(233\) 13.8984 0.910514 0.455257 0.890360i \(-0.349547\pi\)
0.455257 + 0.890360i \(0.349547\pi\)
\(234\) 0 0
\(235\) 6.64354 0.433376
\(236\) 4.27620 0.278357
\(237\) 0 0
\(238\) −2.41757 −0.156708
\(239\) 19.8327 1.28287 0.641435 0.767177i \(-0.278339\pi\)
0.641435 + 0.767177i \(0.278339\pi\)
\(240\) 0 0
\(241\) 19.3747 1.24803 0.624017 0.781411i \(-0.285500\pi\)
0.624017 + 0.781411i \(0.285500\pi\)
\(242\) −23.0703 −1.48301
\(243\) 0 0
\(244\) 2.51674 0.161118
\(245\) −16.2450 −1.03786
\(246\) 0 0
\(247\) −9.41849 −0.599284
\(248\) −8.81349 −0.559657
\(249\) 0 0
\(250\) −15.9071 −1.00605
\(251\) 5.47572 0.345625 0.172812 0.984955i \(-0.444715\pi\)
0.172812 + 0.984955i \(0.444715\pi\)
\(252\) 0 0
\(253\) −0.685377 −0.0430893
\(254\) 11.0645 0.694248
\(255\) 0 0
\(256\) 5.82712 0.364195
\(257\) 11.5652 0.721415 0.360708 0.932679i \(-0.382535\pi\)
0.360708 + 0.932679i \(0.382535\pi\)
\(258\) 0 0
\(259\) −4.34307 −0.269865
\(260\) −10.0749 −0.624816
\(261\) 0 0
\(262\) 15.3123 0.945996
\(263\) −6.47794 −0.399447 −0.199723 0.979852i \(-0.564004\pi\)
−0.199723 + 0.979852i \(0.564004\pi\)
\(264\) 0 0
\(265\) 29.2242 1.79523
\(266\) 12.8028 0.784990
\(267\) 0 0
\(268\) 2.12432 0.129764
\(269\) 13.8387 0.843758 0.421879 0.906652i \(-0.361371\pi\)
0.421879 + 0.906652i \(0.361371\pi\)
\(270\) 0 0
\(271\) 1.94536 0.118172 0.0590860 0.998253i \(-0.481181\pi\)
0.0590860 + 0.998253i \(0.481181\pi\)
\(272\) 3.30007 0.200096
\(273\) 0 0
\(274\) −23.8106 −1.43845
\(275\) 0.696872 0.0420230
\(276\) 0 0
\(277\) 12.4727 0.749411 0.374706 0.927144i \(-0.377744\pi\)
0.374706 + 0.927144i \(0.377744\pi\)
\(278\) 19.8543 1.19078
\(279\) 0 0
\(280\) 2.65106 0.158431
\(281\) −9.75587 −0.581986 −0.290993 0.956725i \(-0.593986\pi\)
−0.290993 + 0.956725i \(0.593986\pi\)
\(282\) 0 0
\(283\) 26.5712 1.57950 0.789748 0.613431i \(-0.210211\pi\)
0.789748 + 0.613431i \(0.210211\pi\)
\(284\) 23.7984 1.41217
\(285\) 0 0
\(286\) −1.01555 −0.0600510
\(287\) −5.68346 −0.335484
\(288\) 0 0
\(289\) −15.6201 −0.918830
\(290\) 25.0160 1.46899
\(291\) 0 0
\(292\) −37.8011 −2.21214
\(293\) −12.2575 −0.716089 −0.358044 0.933705i \(-0.616556\pi\)
−0.358044 + 0.933705i \(0.616556\pi\)
\(294\) 0 0
\(295\) 4.62623 0.269349
\(296\) −4.53888 −0.263817
\(297\) 0 0
\(298\) −40.3243 −2.33592
\(299\) 3.27446 0.189367
\(300\) 0 0
\(301\) 5.43630 0.313343
\(302\) 8.49613 0.488897
\(303\) 0 0
\(304\) −17.4763 −1.00233
\(305\) 2.72275 0.155904
\(306\) 0 0
\(307\) −26.4740 −1.51095 −0.755475 0.655178i \(-0.772594\pi\)
−0.755475 + 0.655178i \(0.772594\pi\)
\(308\) 0.764201 0.0435444
\(309\) 0 0
\(310\) −49.2562 −2.79756
\(311\) 17.6595 1.00138 0.500689 0.865627i \(-0.333080\pi\)
0.500689 + 0.865627i \(0.333080\pi\)
\(312\) 0 0
\(313\) 9.64721 0.545292 0.272646 0.962114i \(-0.412101\pi\)
0.272646 + 0.962114i \(0.412101\pi\)
\(314\) 15.4004 0.869093
\(315\) 0 0
\(316\) −27.8112 −1.56450
\(317\) 3.71049 0.208402 0.104201 0.994556i \(-0.466771\pi\)
0.104201 + 0.994556i \(0.466771\pi\)
\(318\) 0 0
\(319\) 1.39593 0.0781568
\(320\) −30.2361 −1.69025
\(321\) 0 0
\(322\) −4.45106 −0.248048
\(323\) −7.30748 −0.406599
\(324\) 0 0
\(325\) −3.32937 −0.184680
\(326\) 26.3502 1.45940
\(327\) 0 0
\(328\) −5.93972 −0.327966
\(329\) 2.40756 0.132733
\(330\) 0 0
\(331\) −1.41280 −0.0776543 −0.0388271 0.999246i \(-0.512362\pi\)
−0.0388271 + 0.999246i \(0.512362\pi\)
\(332\) −11.6191 −0.637678
\(333\) 0 0
\(334\) 4.93573 0.270071
\(335\) 2.29821 0.125565
\(336\) 0 0
\(337\) −12.9901 −0.707614 −0.353807 0.935318i \(-0.615113\pi\)
−0.353807 + 0.935318i \(0.615113\pi\)
\(338\) −22.6642 −1.23277
\(339\) 0 0
\(340\) −7.81673 −0.423922
\(341\) −2.74856 −0.148843
\(342\) 0 0
\(343\) −12.6934 −0.685378
\(344\) 5.68141 0.306321
\(345\) 0 0
\(346\) 7.58231 0.407628
\(347\) 4.80046 0.257702 0.128851 0.991664i \(-0.458871\pi\)
0.128851 + 0.991664i \(0.458871\pi\)
\(348\) 0 0
\(349\) −22.5776 −1.20855 −0.604275 0.796776i \(-0.706537\pi\)
−0.604275 + 0.796776i \(0.706537\pi\)
\(350\) 4.52571 0.241909
\(351\) 0 0
\(352\) −2.52845 −0.134767
\(353\) 29.7066 1.58112 0.790560 0.612384i \(-0.209789\pi\)
0.790560 + 0.612384i \(0.209789\pi\)
\(354\) 0 0
\(355\) 25.7464 1.36648
\(356\) 38.4402 2.03733
\(357\) 0 0
\(358\) 42.3234 2.23686
\(359\) −13.4198 −0.708271 −0.354136 0.935194i \(-0.615225\pi\)
−0.354136 + 0.935194i \(0.615225\pi\)
\(360\) 0 0
\(361\) 19.6984 1.03676
\(362\) 20.5997 1.08269
\(363\) 0 0
\(364\) −3.65105 −0.191367
\(365\) −40.8953 −2.14056
\(366\) 0 0
\(367\) −7.94998 −0.414985 −0.207493 0.978237i \(-0.566530\pi\)
−0.207493 + 0.978237i \(0.566530\pi\)
\(368\) 6.07585 0.316726
\(369\) 0 0
\(370\) −25.3666 −1.31874
\(371\) 10.5906 0.549837
\(372\) 0 0
\(373\) −11.4205 −0.591332 −0.295666 0.955291i \(-0.595541\pi\)
−0.295666 + 0.955291i \(0.595541\pi\)
\(374\) −0.787934 −0.0407431
\(375\) 0 0
\(376\) 2.51611 0.129759
\(377\) −6.66917 −0.343480
\(378\) 0 0
\(379\) −24.1705 −1.24155 −0.620777 0.783987i \(-0.713183\pi\)
−0.620777 + 0.783987i \(0.713183\pi\)
\(380\) 41.3953 2.12353
\(381\) 0 0
\(382\) 37.5839 1.92296
\(383\) −9.44328 −0.482529 −0.241265 0.970459i \(-0.577562\pi\)
−0.241265 + 0.970459i \(0.577562\pi\)
\(384\) 0 0
\(385\) 0.826754 0.0421353
\(386\) 22.4030 1.14029
\(387\) 0 0
\(388\) −13.7632 −0.698722
\(389\) 2.54771 0.129174 0.0645869 0.997912i \(-0.479427\pi\)
0.0645869 + 0.997912i \(0.479427\pi\)
\(390\) 0 0
\(391\) 2.54054 0.128481
\(392\) −6.15249 −0.310747
\(393\) 0 0
\(394\) −29.9690 −1.50982
\(395\) −30.0876 −1.51387
\(396\) 0 0
\(397\) 3.67517 0.184452 0.0922258 0.995738i \(-0.470602\pi\)
0.0922258 + 0.995738i \(0.470602\pi\)
\(398\) 15.9597 0.799990
\(399\) 0 0
\(400\) −6.17776 −0.308888
\(401\) −16.1487 −0.806429 −0.403214 0.915106i \(-0.632107\pi\)
−0.403214 + 0.915106i \(0.632107\pi\)
\(402\) 0 0
\(403\) 13.1315 0.654128
\(404\) 25.1458 1.25105
\(405\) 0 0
\(406\) 9.06558 0.449917
\(407\) −1.41549 −0.0701632
\(408\) 0 0
\(409\) 9.18197 0.454019 0.227010 0.973893i \(-0.427105\pi\)
0.227010 + 0.973893i \(0.427105\pi\)
\(410\) −33.1954 −1.63941
\(411\) 0 0
\(412\) −24.4371 −1.20393
\(413\) 1.67651 0.0824955
\(414\) 0 0
\(415\) −12.5701 −0.617043
\(416\) 12.0799 0.592266
\(417\) 0 0
\(418\) 4.17268 0.204092
\(419\) 6.97888 0.340940 0.170470 0.985363i \(-0.445471\pi\)
0.170470 + 0.985363i \(0.445471\pi\)
\(420\) 0 0
\(421\) 30.8106 1.50162 0.750809 0.660520i \(-0.229664\pi\)
0.750809 + 0.660520i \(0.229664\pi\)
\(422\) 11.0353 0.537190
\(423\) 0 0
\(424\) 11.0681 0.537515
\(425\) −2.58315 −0.125301
\(426\) 0 0
\(427\) 0.986702 0.0477498
\(428\) −12.8241 −0.619874
\(429\) 0 0
\(430\) 31.7518 1.53121
\(431\) −27.8971 −1.34376 −0.671879 0.740661i \(-0.734513\pi\)
−0.671879 + 0.740661i \(0.734513\pi\)
\(432\) 0 0
\(433\) 19.1706 0.921278 0.460639 0.887588i \(-0.347620\pi\)
0.460639 + 0.887588i \(0.347620\pi\)
\(434\) −17.8500 −0.856829
\(435\) 0 0
\(436\) −18.1311 −0.868322
\(437\) −13.4540 −0.643592
\(438\) 0 0
\(439\) −23.7490 −1.13348 −0.566739 0.823898i \(-0.691795\pi\)
−0.566739 + 0.823898i \(0.691795\pi\)
\(440\) 0.864030 0.0411910
\(441\) 0 0
\(442\) 3.76443 0.179056
\(443\) −23.3583 −1.10978 −0.554892 0.831922i \(-0.687241\pi\)
−0.554892 + 0.831922i \(0.687241\pi\)
\(444\) 0 0
\(445\) 41.5867 1.97140
\(446\) 37.4511 1.77336
\(447\) 0 0
\(448\) −10.9573 −0.517685
\(449\) −4.81906 −0.227426 −0.113713 0.993514i \(-0.536274\pi\)
−0.113713 + 0.993514i \(0.536274\pi\)
\(450\) 0 0
\(451\) −1.85235 −0.0872238
\(452\) −25.7325 −1.21036
\(453\) 0 0
\(454\) 33.3833 1.56676
\(455\) −3.94990 −0.185174
\(456\) 0 0
\(457\) −4.89360 −0.228913 −0.114456 0.993428i \(-0.536513\pi\)
−0.114456 + 0.993428i \(0.536513\pi\)
\(458\) −3.73978 −0.174749
\(459\) 0 0
\(460\) −14.3916 −0.671012
\(461\) −27.9905 −1.30365 −0.651823 0.758371i \(-0.725995\pi\)
−0.651823 + 0.758371i \(0.725995\pi\)
\(462\) 0 0
\(463\) −27.4753 −1.27689 −0.638444 0.769669i \(-0.720421\pi\)
−0.638444 + 0.769669i \(0.720421\pi\)
\(464\) −12.3748 −0.574488
\(465\) 0 0
\(466\) 29.4177 1.36275
\(467\) 21.2465 0.983170 0.491585 0.870830i \(-0.336418\pi\)
0.491585 + 0.870830i \(0.336418\pi\)
\(468\) 0 0
\(469\) 0.832853 0.0384576
\(470\) 14.0619 0.648625
\(471\) 0 0
\(472\) 1.75210 0.0806467
\(473\) 1.77180 0.0814673
\(474\) 0 0
\(475\) 13.6796 0.627665
\(476\) −2.83272 −0.129838
\(477\) 0 0
\(478\) 41.9783 1.92004
\(479\) 41.6788 1.90435 0.952177 0.305546i \(-0.0988389\pi\)
0.952177 + 0.305546i \(0.0988389\pi\)
\(480\) 0 0
\(481\) 6.76264 0.308350
\(482\) 41.0089 1.86791
\(483\) 0 0
\(484\) −27.0319 −1.22872
\(485\) −14.8898 −0.676112
\(486\) 0 0
\(487\) 4.02801 0.182527 0.0912634 0.995827i \(-0.470909\pi\)
0.0912634 + 0.995827i \(0.470909\pi\)
\(488\) 1.03119 0.0466797
\(489\) 0 0
\(490\) −34.3845 −1.55334
\(491\) 38.6214 1.74296 0.871479 0.490433i \(-0.163162\pi\)
0.871479 + 0.490433i \(0.163162\pi\)
\(492\) 0 0
\(493\) −5.17438 −0.233042
\(494\) −19.9354 −0.896935
\(495\) 0 0
\(496\) 24.3660 1.09406
\(497\) 9.33028 0.418520
\(498\) 0 0
\(499\) −4.07186 −0.182281 −0.0911407 0.995838i \(-0.529051\pi\)
−0.0911407 + 0.995838i \(0.529051\pi\)
\(500\) −18.6387 −0.833547
\(501\) 0 0
\(502\) 11.5900 0.517289
\(503\) 3.42594 0.152755 0.0763775 0.997079i \(-0.475665\pi\)
0.0763775 + 0.997079i \(0.475665\pi\)
\(504\) 0 0
\(505\) 27.2041 1.21056
\(506\) −1.45069 −0.0644908
\(507\) 0 0
\(508\) 12.9645 0.575207
\(509\) 12.2634 0.543566 0.271783 0.962359i \(-0.412387\pi\)
0.271783 + 0.962359i \(0.412387\pi\)
\(510\) 0 0
\(511\) −14.8201 −0.655604
\(512\) 28.1241 1.24292
\(513\) 0 0
\(514\) 24.4791 1.07973
\(515\) −26.4373 −1.16497
\(516\) 0 0
\(517\) 0.784671 0.0345098
\(518\) −9.19263 −0.403901
\(519\) 0 0
\(520\) −4.12799 −0.181024
\(521\) −14.0823 −0.616959 −0.308479 0.951231i \(-0.599820\pi\)
−0.308479 + 0.951231i \(0.599820\pi\)
\(522\) 0 0
\(523\) 9.77912 0.427611 0.213806 0.976876i \(-0.431414\pi\)
0.213806 + 0.976876i \(0.431414\pi\)
\(524\) 17.9417 0.783788
\(525\) 0 0
\(526\) −13.7113 −0.597843
\(527\) 10.1883 0.443809
\(528\) 0 0
\(529\) −18.3225 −0.796632
\(530\) 61.8566 2.68688
\(531\) 0 0
\(532\) 15.0013 0.650389
\(533\) 8.84979 0.383327
\(534\) 0 0
\(535\) −13.8738 −0.599815
\(536\) 0.870404 0.0375957
\(537\) 0 0
\(538\) 29.2912 1.26283
\(539\) −1.91871 −0.0826445
\(540\) 0 0
\(541\) 40.9454 1.76038 0.880189 0.474623i \(-0.157416\pi\)
0.880189 + 0.474623i \(0.157416\pi\)
\(542\) 4.11759 0.176866
\(543\) 0 0
\(544\) 9.37238 0.401837
\(545\) −19.6152 −0.840223
\(546\) 0 0
\(547\) 1.11028 0.0474720 0.0237360 0.999718i \(-0.492444\pi\)
0.0237360 + 0.999718i \(0.492444\pi\)
\(548\) −27.8993 −1.19180
\(549\) 0 0
\(550\) 1.47502 0.0628949
\(551\) 27.4021 1.16737
\(552\) 0 0
\(553\) −10.9035 −0.463664
\(554\) 26.4000 1.12163
\(555\) 0 0
\(556\) 23.2637 0.986599
\(557\) −35.0403 −1.48470 −0.742352 0.670010i \(-0.766290\pi\)
−0.742352 + 0.670010i \(0.766290\pi\)
\(558\) 0 0
\(559\) −8.46493 −0.358028
\(560\) −7.32916 −0.309713
\(561\) 0 0
\(562\) −20.6495 −0.871046
\(563\) 38.7712 1.63401 0.817005 0.576630i \(-0.195633\pi\)
0.817005 + 0.576630i \(0.195633\pi\)
\(564\) 0 0
\(565\) −27.8389 −1.17119
\(566\) 56.2413 2.36400
\(567\) 0 0
\(568\) 9.75095 0.409141
\(569\) 33.9683 1.42403 0.712013 0.702166i \(-0.247784\pi\)
0.712013 + 0.702166i \(0.247784\pi\)
\(570\) 0 0
\(571\) 10.0889 0.422206 0.211103 0.977464i \(-0.432294\pi\)
0.211103 + 0.977464i \(0.432294\pi\)
\(572\) −1.18995 −0.0497542
\(573\) 0 0
\(574\) −12.0297 −0.502112
\(575\) −4.75590 −0.198335
\(576\) 0 0
\(577\) −12.1323 −0.505074 −0.252537 0.967587i \(-0.581265\pi\)
−0.252537 + 0.967587i \(0.581265\pi\)
\(578\) −33.0619 −1.37519
\(579\) 0 0
\(580\) 29.3117 1.21710
\(581\) −4.55531 −0.188986
\(582\) 0 0
\(583\) 3.45168 0.142954
\(584\) −15.4883 −0.640911
\(585\) 0 0
\(586\) −25.9444 −1.07175
\(587\) −31.7492 −1.31043 −0.655215 0.755443i \(-0.727422\pi\)
−0.655215 + 0.755443i \(0.727422\pi\)
\(588\) 0 0
\(589\) −53.9544 −2.22315
\(590\) 9.79197 0.403129
\(591\) 0 0
\(592\) 12.5483 0.515731
\(593\) −13.4906 −0.553993 −0.276996 0.960871i \(-0.589339\pi\)
−0.276996 + 0.960871i \(0.589339\pi\)
\(594\) 0 0
\(595\) −3.06459 −0.125636
\(596\) −47.2488 −1.93539
\(597\) 0 0
\(598\) 6.93079 0.283421
\(599\) 42.4304 1.73366 0.866829 0.498606i \(-0.166154\pi\)
0.866829 + 0.498606i \(0.166154\pi\)
\(600\) 0 0
\(601\) −19.8047 −0.807852 −0.403926 0.914792i \(-0.632355\pi\)
−0.403926 + 0.914792i \(0.632355\pi\)
\(602\) 11.5066 0.468974
\(603\) 0 0
\(604\) 9.95509 0.405067
\(605\) −29.2446 −1.18896
\(606\) 0 0
\(607\) −35.9990 −1.46116 −0.730578 0.682830i \(-0.760749\pi\)
−0.730578 + 0.682830i \(0.760749\pi\)
\(608\) −49.6335 −2.01291
\(609\) 0 0
\(610\) 5.76303 0.233338
\(611\) −3.74884 −0.151662
\(612\) 0 0
\(613\) 26.4628 1.06882 0.534411 0.845225i \(-0.320533\pi\)
0.534411 + 0.845225i \(0.320533\pi\)
\(614\) −56.0354 −2.26141
\(615\) 0 0
\(616\) 0.313118 0.0126159
\(617\) −49.1167 −1.97737 −0.988683 0.150022i \(-0.952066\pi\)
−0.988683 + 0.150022i \(0.952066\pi\)
\(618\) 0 0
\(619\) −24.2062 −0.972930 −0.486465 0.873700i \(-0.661714\pi\)
−0.486465 + 0.873700i \(0.661714\pi\)
\(620\) −57.7145 −2.31787
\(621\) 0 0
\(622\) 37.3785 1.49874
\(623\) 15.0707 0.603794
\(624\) 0 0
\(625\) −31.1594 −1.24638
\(626\) 20.4195 0.816127
\(627\) 0 0
\(628\) 18.0449 0.720072
\(629\) 5.24690 0.209208
\(630\) 0 0
\(631\) 17.6968 0.704500 0.352250 0.935906i \(-0.385417\pi\)
0.352250 + 0.935906i \(0.385417\pi\)
\(632\) −11.3951 −0.453274
\(633\) 0 0
\(634\) 7.85371 0.311911
\(635\) 14.0257 0.556593
\(636\) 0 0
\(637\) 9.16680 0.363202
\(638\) 2.95465 0.116976
\(639\) 0 0
\(640\) −21.1835 −0.837351
\(641\) −38.0504 −1.50290 −0.751450 0.659790i \(-0.770645\pi\)
−0.751450 + 0.659790i \(0.770645\pi\)
\(642\) 0 0
\(643\) 46.9425 1.85123 0.925616 0.378465i \(-0.123548\pi\)
0.925616 + 0.378465i \(0.123548\pi\)
\(644\) −5.21540 −0.205515
\(645\) 0 0
\(646\) −15.4672 −0.608548
\(647\) 28.2333 1.10997 0.554983 0.831862i \(-0.312725\pi\)
0.554983 + 0.831862i \(0.312725\pi\)
\(648\) 0 0
\(649\) 0.546406 0.0214483
\(650\) −7.04703 −0.276407
\(651\) 0 0
\(652\) 30.8751 1.20916
\(653\) −35.3072 −1.38168 −0.690839 0.723009i \(-0.742759\pi\)
−0.690839 + 0.723009i \(0.742759\pi\)
\(654\) 0 0
\(655\) 19.4103 0.758425
\(656\) 16.4210 0.641134
\(657\) 0 0
\(658\) 5.09590 0.198659
\(659\) −41.6937 −1.62416 −0.812078 0.583548i \(-0.801664\pi\)
−0.812078 + 0.583548i \(0.801664\pi\)
\(660\) 0 0
\(661\) 1.27391 0.0495495 0.0247747 0.999693i \(-0.492113\pi\)
0.0247747 + 0.999693i \(0.492113\pi\)
\(662\) −2.99036 −0.116223
\(663\) 0 0
\(664\) −4.76070 −0.184751
\(665\) 16.2292 0.629343
\(666\) 0 0
\(667\) −9.52668 −0.368875
\(668\) 5.78329 0.223762
\(669\) 0 0
\(670\) 4.86444 0.187930
\(671\) 0.321585 0.0124147
\(672\) 0 0
\(673\) 35.6328 1.37354 0.686771 0.726874i \(-0.259028\pi\)
0.686771 + 0.726874i \(0.259028\pi\)
\(674\) −27.4951 −1.05907
\(675\) 0 0
\(676\) −26.5561 −1.02139
\(677\) −18.0019 −0.691869 −0.345934 0.938259i \(-0.612438\pi\)
−0.345934 + 0.938259i \(0.612438\pi\)
\(678\) 0 0
\(679\) −5.39595 −0.207077
\(680\) −3.20276 −0.122820
\(681\) 0 0
\(682\) −5.81767 −0.222770
\(683\) −39.7614 −1.52143 −0.760715 0.649087i \(-0.775151\pi\)
−0.760715 + 0.649087i \(0.775151\pi\)
\(684\) 0 0
\(685\) −30.1830 −1.15323
\(686\) −26.8671 −1.02579
\(687\) 0 0
\(688\) −15.7069 −0.598821
\(689\) −16.4908 −0.628248
\(690\) 0 0
\(691\) −16.8407 −0.640651 −0.320325 0.947308i \(-0.603792\pi\)
−0.320325 + 0.947308i \(0.603792\pi\)
\(692\) 8.88436 0.337733
\(693\) 0 0
\(694\) 10.1608 0.385697
\(695\) 25.1679 0.954673
\(696\) 0 0
\(697\) 6.86625 0.260078
\(698\) −47.7882 −1.80881
\(699\) 0 0
\(700\) 5.30287 0.200430
\(701\) 8.96921 0.338762 0.169381 0.985551i \(-0.445823\pi\)
0.169381 + 0.985551i \(0.445823\pi\)
\(702\) 0 0
\(703\) −27.7861 −1.04797
\(704\) −3.57120 −0.134595
\(705\) 0 0
\(706\) 62.8776 2.36643
\(707\) 9.85853 0.370768
\(708\) 0 0
\(709\) 18.1416 0.681323 0.340662 0.940186i \(-0.389349\pi\)
0.340662 + 0.940186i \(0.389349\pi\)
\(710\) 54.4954 2.04518
\(711\) 0 0
\(712\) 15.7502 0.590263
\(713\) 18.7580 0.702491
\(714\) 0 0
\(715\) −1.28735 −0.0481441
\(716\) 49.5912 1.85331
\(717\) 0 0
\(718\) −28.4047 −1.06005
\(719\) −31.5720 −1.17744 −0.588718 0.808339i \(-0.700367\pi\)
−0.588718 + 0.808339i \(0.700367\pi\)
\(720\) 0 0
\(721\) −9.58068 −0.356803
\(722\) 41.6941 1.55169
\(723\) 0 0
\(724\) 24.1370 0.897046
\(725\) 9.68646 0.359746
\(726\) 0 0
\(727\) 38.4093 1.42452 0.712260 0.701915i \(-0.247672\pi\)
0.712260 + 0.701915i \(0.247672\pi\)
\(728\) −1.49595 −0.0554436
\(729\) 0 0
\(730\) −86.5599 −3.20373
\(731\) −6.56765 −0.242913
\(732\) 0 0
\(733\) −30.3303 −1.12028 −0.560138 0.828400i \(-0.689252\pi\)
−0.560138 + 0.828400i \(0.689252\pi\)
\(734\) −16.8271 −0.621100
\(735\) 0 0
\(736\) 17.2557 0.636055
\(737\) 0.271443 0.00999872
\(738\) 0 0
\(739\) 10.0025 0.367949 0.183975 0.982931i \(-0.441104\pi\)
0.183975 + 0.982931i \(0.441104\pi\)
\(740\) −29.7225 −1.09262
\(741\) 0 0
\(742\) 22.4163 0.822929
\(743\) −36.2738 −1.33076 −0.665378 0.746506i \(-0.731730\pi\)
−0.665378 + 0.746506i \(0.731730\pi\)
\(744\) 0 0
\(745\) −51.1164 −1.87276
\(746\) −24.1729 −0.885033
\(747\) 0 0
\(748\) −0.923239 −0.0337569
\(749\) −5.02774 −0.183710
\(750\) 0 0
\(751\) −14.5544 −0.531096 −0.265548 0.964098i \(-0.585553\pi\)
−0.265548 + 0.964098i \(0.585553\pi\)
\(752\) −6.95609 −0.253663
\(753\) 0 0
\(754\) −14.1161 −0.514078
\(755\) 10.7700 0.391959
\(756\) 0 0
\(757\) −45.5754 −1.65646 −0.828232 0.560385i \(-0.810653\pi\)
−0.828232 + 0.560385i \(0.810653\pi\)
\(758\) −51.1598 −1.85821
\(759\) 0 0
\(760\) 16.9610 0.615239
\(761\) −22.2456 −0.806401 −0.403200 0.915112i \(-0.632102\pi\)
−0.403200 + 0.915112i \(0.632102\pi\)
\(762\) 0 0
\(763\) −7.10839 −0.257341
\(764\) 44.0378 1.59323
\(765\) 0 0
\(766\) −19.9879 −0.722191
\(767\) −2.61051 −0.0942600
\(768\) 0 0
\(769\) 13.6649 0.492770 0.246385 0.969172i \(-0.420757\pi\)
0.246385 + 0.969172i \(0.420757\pi\)
\(770\) 1.74993 0.0630630
\(771\) 0 0
\(772\) 26.2501 0.944763
\(773\) 20.6540 0.742871 0.371436 0.928459i \(-0.378866\pi\)
0.371436 + 0.928459i \(0.378866\pi\)
\(774\) 0 0
\(775\) −19.0726 −0.685106
\(776\) −5.63923 −0.202437
\(777\) 0 0
\(778\) 5.39253 0.193332
\(779\) −36.3617 −1.30279
\(780\) 0 0
\(781\) 3.04092 0.108813
\(782\) 5.37736 0.192294
\(783\) 0 0
\(784\) 17.0093 0.607474
\(785\) 19.5220 0.696770
\(786\) 0 0
\(787\) −24.7034 −0.880580 −0.440290 0.897856i \(-0.645124\pi\)
−0.440290 + 0.897856i \(0.645124\pi\)
\(788\) −35.1153 −1.25093
\(789\) 0 0
\(790\) −63.6842 −2.26578
\(791\) −10.0886 −0.358708
\(792\) 0 0
\(793\) −1.53640 −0.0545593
\(794\) 7.77895 0.276065
\(795\) 0 0
\(796\) 18.7004 0.662817
\(797\) 30.5658 1.08270 0.541348 0.840799i \(-0.317914\pi\)
0.541348 + 0.840799i \(0.317914\pi\)
\(798\) 0 0
\(799\) −2.90860 −0.102899
\(800\) −17.5451 −0.620314
\(801\) 0 0
\(802\) −34.1808 −1.20696
\(803\) −4.83017 −0.170453
\(804\) 0 0
\(805\) −5.64230 −0.198865
\(806\) 27.7945 0.979019
\(807\) 0 0
\(808\) 10.3030 0.362459
\(809\) −46.8599 −1.64751 −0.823753 0.566949i \(-0.808124\pi\)
−0.823753 + 0.566949i \(0.808124\pi\)
\(810\) 0 0
\(811\) 10.9984 0.386206 0.193103 0.981178i \(-0.438145\pi\)
0.193103 + 0.981178i \(0.438145\pi\)
\(812\) 10.6223 0.372771
\(813\) 0 0
\(814\) −2.99606 −0.105012
\(815\) 33.4023 1.17003
\(816\) 0 0
\(817\) 34.7805 1.21681
\(818\) 19.4348 0.679521
\(819\) 0 0
\(820\) −38.8958 −1.35830
\(821\) 35.9833 1.25583 0.627913 0.778283i \(-0.283909\pi\)
0.627913 + 0.778283i \(0.283909\pi\)
\(822\) 0 0
\(823\) −48.9866 −1.70757 −0.853783 0.520628i \(-0.825698\pi\)
−0.853783 + 0.520628i \(0.825698\pi\)
\(824\) −10.0126 −0.348807
\(825\) 0 0
\(826\) 3.54853 0.123469
\(827\) 15.6107 0.542836 0.271418 0.962462i \(-0.412507\pi\)
0.271418 + 0.962462i \(0.412507\pi\)
\(828\) 0 0
\(829\) 11.4708 0.398398 0.199199 0.979959i \(-0.436166\pi\)
0.199199 + 0.979959i \(0.436166\pi\)
\(830\) −26.6062 −0.923515
\(831\) 0 0
\(832\) 17.0618 0.591510
\(833\) 7.11221 0.246423
\(834\) 0 0
\(835\) 6.25668 0.216521
\(836\) 4.88922 0.169097
\(837\) 0 0
\(838\) 14.7716 0.510278
\(839\) 0.675324 0.0233148 0.0116574 0.999932i \(-0.496289\pi\)
0.0116574 + 0.999932i \(0.496289\pi\)
\(840\) 0 0
\(841\) −9.59676 −0.330923
\(842\) 65.2145 2.24744
\(843\) 0 0
\(844\) 12.9303 0.445079
\(845\) −28.7298 −0.988337
\(846\) 0 0
\(847\) −10.5980 −0.364152
\(848\) −30.5991 −1.05078
\(849\) 0 0
\(850\) −5.46755 −0.187535
\(851\) 9.66020 0.331147
\(852\) 0 0
\(853\) 39.3558 1.34752 0.673758 0.738952i \(-0.264679\pi\)
0.673758 + 0.738952i \(0.264679\pi\)
\(854\) 2.08848 0.0714662
\(855\) 0 0
\(856\) −5.25443 −0.179593
\(857\) −32.6621 −1.11572 −0.557858 0.829936i \(-0.688377\pi\)
−0.557858 + 0.829936i \(0.688377\pi\)
\(858\) 0 0
\(859\) −33.1966 −1.13265 −0.566327 0.824180i \(-0.691636\pi\)
−0.566327 + 0.824180i \(0.691636\pi\)
\(860\) 37.2043 1.26866
\(861\) 0 0
\(862\) −59.0477 −2.01117
\(863\) −22.6796 −0.772024 −0.386012 0.922494i \(-0.626148\pi\)
−0.386012 + 0.922494i \(0.626148\pi\)
\(864\) 0 0
\(865\) 9.61158 0.326804
\(866\) 40.5768 1.37886
\(867\) 0 0
\(868\) −20.9153 −0.709910
\(869\) −3.55367 −0.120550
\(870\) 0 0
\(871\) −1.29684 −0.0439419
\(872\) −7.42888 −0.251574
\(873\) 0 0
\(874\) −28.4770 −0.963250
\(875\) −7.30739 −0.247035
\(876\) 0 0
\(877\) 9.24376 0.312140 0.156070 0.987746i \(-0.450118\pi\)
0.156070 + 0.987746i \(0.450118\pi\)
\(878\) −50.2676 −1.69645
\(879\) 0 0
\(880\) −2.38871 −0.0805236
\(881\) −7.78755 −0.262369 −0.131185 0.991358i \(-0.541878\pi\)
−0.131185 + 0.991358i \(0.541878\pi\)
\(882\) 0 0
\(883\) −32.4618 −1.09243 −0.546213 0.837647i \(-0.683931\pi\)
−0.546213 + 0.837647i \(0.683931\pi\)
\(884\) 4.41086 0.148353
\(885\) 0 0
\(886\) −49.4406 −1.66099
\(887\) 33.8477 1.13649 0.568247 0.822858i \(-0.307622\pi\)
0.568247 + 0.822858i \(0.307622\pi\)
\(888\) 0 0
\(889\) 5.08280 0.170472
\(890\) 88.0234 2.95055
\(891\) 0 0
\(892\) 43.8822 1.46929
\(893\) 15.4031 0.515446
\(894\) 0 0
\(895\) 53.6504 1.79334
\(896\) −7.67673 −0.256461
\(897\) 0 0
\(898\) −10.2001 −0.340383
\(899\) −38.2048 −1.27420
\(900\) 0 0
\(901\) −12.7946 −0.426250
\(902\) −3.92073 −0.130546
\(903\) 0 0
\(904\) −10.5434 −0.350670
\(905\) 26.1128 0.868018
\(906\) 0 0
\(907\) −11.0598 −0.367236 −0.183618 0.982998i \(-0.558781\pi\)
−0.183618 + 0.982998i \(0.558781\pi\)
\(908\) 39.1160 1.29811
\(909\) 0 0
\(910\) −8.36045 −0.277146
\(911\) −12.5452 −0.415640 −0.207820 0.978167i \(-0.566637\pi\)
−0.207820 + 0.978167i \(0.566637\pi\)
\(912\) 0 0
\(913\) −1.48466 −0.0491352
\(914\) −10.3579 −0.342609
\(915\) 0 0
\(916\) −4.38198 −0.144785
\(917\) 7.03415 0.232288
\(918\) 0 0
\(919\) −5.92909 −0.195583 −0.0977913 0.995207i \(-0.531178\pi\)
−0.0977913 + 0.995207i \(0.531178\pi\)
\(920\) −5.89669 −0.194408
\(921\) 0 0
\(922\) −59.2452 −1.95114
\(923\) −14.5283 −0.478204
\(924\) 0 0
\(925\) −9.82222 −0.322953
\(926\) −58.1549 −1.91109
\(927\) 0 0
\(928\) −35.1452 −1.15370
\(929\) 10.0328 0.329165 0.164582 0.986363i \(-0.447372\pi\)
0.164582 + 0.986363i \(0.447372\pi\)
\(930\) 0 0
\(931\) −37.6643 −1.23440
\(932\) 34.4693 1.12908
\(933\) 0 0
\(934\) 44.9708 1.47149
\(935\) −0.998810 −0.0326646
\(936\) 0 0
\(937\) −23.5341 −0.768826 −0.384413 0.923161i \(-0.625596\pi\)
−0.384413 + 0.923161i \(0.625596\pi\)
\(938\) 1.76283 0.0575586
\(939\) 0 0
\(940\) 16.4766 0.537407
\(941\) 29.2996 0.955138 0.477569 0.878594i \(-0.341518\pi\)
0.477569 + 0.878594i \(0.341518\pi\)
\(942\) 0 0
\(943\) 12.6416 0.411668
\(944\) −4.84388 −0.157655
\(945\) 0 0
\(946\) 3.75023 0.121930
\(947\) −53.6619 −1.74378 −0.871889 0.489704i \(-0.837105\pi\)
−0.871889 + 0.489704i \(0.837105\pi\)
\(948\) 0 0
\(949\) 23.0766 0.749098
\(950\) 28.9546 0.939412
\(951\) 0 0
\(952\) −1.16066 −0.0376171
\(953\) 4.89656 0.158615 0.0793076 0.996850i \(-0.474729\pi\)
0.0793076 + 0.996850i \(0.474729\pi\)
\(954\) 0 0
\(955\) 47.6425 1.54167
\(956\) 49.1869 1.59082
\(957\) 0 0
\(958\) 88.2184 2.85021
\(959\) −10.9381 −0.353209
\(960\) 0 0
\(961\) 44.2249 1.42661
\(962\) 14.3140 0.461500
\(963\) 0 0
\(964\) 48.0510 1.54762
\(965\) 28.3988 0.914190
\(966\) 0 0
\(967\) 16.9931 0.546462 0.273231 0.961948i \(-0.411908\pi\)
0.273231 + 0.961948i \(0.411908\pi\)
\(968\) −11.0758 −0.355991
\(969\) 0 0
\(970\) −31.5161 −1.01192
\(971\) 2.68374 0.0861253 0.0430627 0.999072i \(-0.486288\pi\)
0.0430627 + 0.999072i \(0.486288\pi\)
\(972\) 0 0
\(973\) 9.12064 0.292394
\(974\) 8.52579 0.273184
\(975\) 0 0
\(976\) −2.85085 −0.0912533
\(977\) −10.9930 −0.351698 −0.175849 0.984417i \(-0.556267\pi\)
−0.175849 + 0.984417i \(0.556267\pi\)
\(978\) 0 0
\(979\) 4.91183 0.156983
\(980\) −40.2891 −1.28699
\(981\) 0 0
\(982\) 81.7468 2.60865
\(983\) 47.7756 1.52381 0.761903 0.647691i \(-0.224265\pi\)
0.761903 + 0.647691i \(0.224265\pi\)
\(984\) 0 0
\(985\) −37.9897 −1.21045
\(986\) −10.9522 −0.348789
\(987\) 0 0
\(988\) −23.3587 −0.743139
\(989\) −12.0919 −0.384499
\(990\) 0 0
\(991\) 55.5006 1.76303 0.881517 0.472152i \(-0.156523\pi\)
0.881517 + 0.472152i \(0.156523\pi\)
\(992\) 69.2005 2.19712
\(993\) 0 0
\(994\) 19.7487 0.626390
\(995\) 20.2311 0.641368
\(996\) 0 0
\(997\) −45.0467 −1.42664 −0.713322 0.700837i \(-0.752810\pi\)
−0.713322 + 0.700837i \(0.752810\pi\)
\(998\) −8.61859 −0.272817
\(999\) 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 729.2.a.d.1.5 6
3.2 odd 2 729.2.a.a.1.2 6
9.2 odd 6 729.2.c.e.244.5 12
9.4 even 3 729.2.c.b.487.2 12
9.5 odd 6 729.2.c.e.487.5 12
9.7 even 3 729.2.c.b.244.2 12
27.2 odd 18 243.2.e.c.190.2 12
27.4 even 9 81.2.e.a.46.2 12
27.5 odd 18 243.2.e.d.217.1 12
27.7 even 9 81.2.e.a.37.2 12
27.11 odd 18 243.2.e.d.28.1 12
27.13 even 9 243.2.e.b.55.1 12
27.14 odd 18 243.2.e.c.55.2 12
27.16 even 9 243.2.e.a.28.2 12
27.20 odd 18 27.2.e.a.22.1 yes 12
27.22 even 9 243.2.e.a.217.2 12
27.23 odd 18 27.2.e.a.16.1 12
27.25 even 9 243.2.e.b.190.1 12
108.23 even 18 432.2.u.c.97.2 12
108.47 even 18 432.2.u.c.49.2 12
135.23 even 36 675.2.u.b.124.1 24
135.47 even 36 675.2.u.b.49.1 24
135.74 odd 18 675.2.l.c.76.2 12
135.77 even 36 675.2.u.b.124.4 24
135.104 odd 18 675.2.l.c.151.2 12
135.128 even 36 675.2.u.b.49.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.16.1 12 27.23 odd 18
27.2.e.a.22.1 yes 12 27.20 odd 18
81.2.e.a.37.2 12 27.7 even 9
81.2.e.a.46.2 12 27.4 even 9
243.2.e.a.28.2 12 27.16 even 9
243.2.e.a.217.2 12 27.22 even 9
243.2.e.b.55.1 12 27.13 even 9
243.2.e.b.190.1 12 27.25 even 9
243.2.e.c.55.2 12 27.14 odd 18
243.2.e.c.190.2 12 27.2 odd 18
243.2.e.d.28.1 12 27.11 odd 18
243.2.e.d.217.1 12 27.5 odd 18
432.2.u.c.49.2 12 108.47 even 18
432.2.u.c.97.2 12 108.23 even 18
675.2.l.c.76.2 12 135.74 odd 18
675.2.l.c.151.2 12 135.104 odd 18
675.2.u.b.49.1 24 135.47 even 36
675.2.u.b.49.4 24 135.128 even 36
675.2.u.b.124.1 24 135.23 even 36
675.2.u.b.124.4 24 135.77 even 36
729.2.a.a.1.2 6 3.2 odd 2
729.2.a.d.1.5 6 1.1 even 1 trivial
729.2.c.b.244.2 12 9.7 even 3
729.2.c.b.487.2 12 9.4 even 3
729.2.c.e.244.5 12 9.2 odd 6
729.2.c.e.487.5 12 9.5 odd 6