Properties

Label 729.2.a.c.1.4
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.684040\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.684040 q^{2} -1.53209 q^{4} +1.04801 q^{5} -0.120615 q^{7} -2.41609 q^{8} +O(q^{10})\) \(q+0.684040 q^{2} -1.53209 q^{4} +1.04801 q^{5} -0.120615 q^{7} -2.41609 q^{8} +0.716881 q^{10} -5.43372 q^{11} -4.57398 q^{13} -0.0825054 q^{14} +1.41147 q^{16} +4.77833 q^{17} -0.588526 q^{19} -1.60565 q^{20} -3.71688 q^{22} -7.79596 q^{23} -3.90167 q^{25} -3.12879 q^{26} +0.184793 q^{28} +5.06975 q^{29} -8.75877 q^{31} +5.79769 q^{32} +3.26857 q^{34} -0.126406 q^{35} -2.18479 q^{37} -0.402575 q^{38} -2.53209 q^{40} +7.55839 q^{41} -1.30541 q^{43} +8.32494 q^{44} -5.33275 q^{46} +2.41609 q^{47} -6.98545 q^{49} -2.66890 q^{50} +7.00774 q^{52} -3.04628 q^{53} -5.69459 q^{55} +0.291416 q^{56} +3.46791 q^{58} -0.0439002 q^{59} -10.2121 q^{61} -5.99135 q^{62} +1.14290 q^{64} -4.79358 q^{65} -1.85978 q^{67} -7.32083 q^{68} -0.0864665 q^{70} -6.51038 q^{71} +12.2344 q^{73} -1.49449 q^{74} +0.901674 q^{76} +0.655386 q^{77} +0.702333 q^{79} +1.47924 q^{80} +5.17024 q^{82} +6.77660 q^{83} +5.00774 q^{85} -0.892951 q^{86} +13.1284 q^{88} +6.85565 q^{89} +0.551689 q^{91} +11.9441 q^{92} +1.65270 q^{94} -0.616781 q^{95} -9.02734 q^{97} -4.77833 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64} + 6 q^{67} + 30 q^{70} + 12 q^{73} - 18 q^{76} - 48 q^{79} - 12 q^{82} - 18 q^{85} + 42 q^{88} + 12 q^{94} - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.684040 0.483690 0.241845 0.970315i \(-0.422248\pi\)
0.241845 + 0.970315i \(0.422248\pi\)
\(3\) 0 0
\(4\) −1.53209 −0.766044
\(5\) 1.04801 0.468685 0.234342 0.972154i \(-0.424706\pi\)
0.234342 + 0.972154i \(0.424706\pi\)
\(6\) 0 0
\(7\) −0.120615 −0.0455881 −0.0227940 0.999740i \(-0.507256\pi\)
−0.0227940 + 0.999740i \(0.507256\pi\)
\(8\) −2.41609 −0.854217
\(9\) 0 0
\(10\) 0.716881 0.226698
\(11\) −5.43372 −1.63833 −0.819164 0.573560i \(-0.805562\pi\)
−0.819164 + 0.573560i \(0.805562\pi\)
\(12\) 0 0
\(13\) −4.57398 −1.26859 −0.634297 0.773090i \(-0.718710\pi\)
−0.634297 + 0.773090i \(0.718710\pi\)
\(14\) −0.0825054 −0.0220505
\(15\) 0 0
\(16\) 1.41147 0.352869
\(17\) 4.77833 1.15892 0.579458 0.815002i \(-0.303264\pi\)
0.579458 + 0.815002i \(0.303264\pi\)
\(18\) 0 0
\(19\) −0.588526 −0.135017 −0.0675085 0.997719i \(-0.521505\pi\)
−0.0675085 + 0.997719i \(0.521505\pi\)
\(20\) −1.60565 −0.359033
\(21\) 0 0
\(22\) −3.71688 −0.792442
\(23\) −7.79596 −1.62557 −0.812785 0.582564i \(-0.802049\pi\)
−0.812785 + 0.582564i \(0.802049\pi\)
\(24\) 0 0
\(25\) −3.90167 −0.780335
\(26\) −3.12879 −0.613605
\(27\) 0 0
\(28\) 0.184793 0.0349225
\(29\) 5.06975 0.941428 0.470714 0.882286i \(-0.343996\pi\)
0.470714 + 0.882286i \(0.343996\pi\)
\(30\) 0 0
\(31\) −8.75877 −1.57312 −0.786561 0.617513i \(-0.788140\pi\)
−0.786561 + 0.617513i \(0.788140\pi\)
\(32\) 5.79769 1.02490
\(33\) 0 0
\(34\) 3.26857 0.560555
\(35\) −0.126406 −0.0213664
\(36\) 0 0
\(37\) −2.18479 −0.359178 −0.179589 0.983742i \(-0.557477\pi\)
−0.179589 + 0.983742i \(0.557477\pi\)
\(38\) −0.402575 −0.0653064
\(39\) 0 0
\(40\) −2.53209 −0.400358
\(41\) 7.55839 1.18042 0.590211 0.807249i \(-0.299044\pi\)
0.590211 + 0.807249i \(0.299044\pi\)
\(42\) 0 0
\(43\) −1.30541 −0.199073 −0.0995364 0.995034i \(-0.531736\pi\)
−0.0995364 + 0.995034i \(0.531736\pi\)
\(44\) 8.32494 1.25503
\(45\) 0 0
\(46\) −5.33275 −0.786271
\(47\) 2.41609 0.352423 0.176212 0.984352i \(-0.443616\pi\)
0.176212 + 0.984352i \(0.443616\pi\)
\(48\) 0 0
\(49\) −6.98545 −0.997922
\(50\) −2.66890 −0.377440
\(51\) 0 0
\(52\) 7.00774 0.971799
\(53\) −3.04628 −0.418439 −0.209219 0.977869i \(-0.567092\pi\)
−0.209219 + 0.977869i \(0.567092\pi\)
\(54\) 0 0
\(55\) −5.69459 −0.767859
\(56\) 0.291416 0.0389421
\(57\) 0 0
\(58\) 3.46791 0.455359
\(59\) −0.0439002 −0.00571532 −0.00285766 0.999996i \(-0.500910\pi\)
−0.00285766 + 0.999996i \(0.500910\pi\)
\(60\) 0 0
\(61\) −10.2121 −1.30753 −0.653765 0.756698i \(-0.726811\pi\)
−0.653765 + 0.756698i \(0.726811\pi\)
\(62\) −5.99135 −0.760902
\(63\) 0 0
\(64\) 1.14290 0.142863
\(65\) −4.79358 −0.594570
\(66\) 0 0
\(67\) −1.85978 −0.227209 −0.113604 0.993526i \(-0.536240\pi\)
−0.113604 + 0.993526i \(0.536240\pi\)
\(68\) −7.32083 −0.887781
\(69\) 0 0
\(70\) −0.0864665 −0.0103347
\(71\) −6.51038 −0.772640 −0.386320 0.922365i \(-0.626254\pi\)
−0.386320 + 0.922365i \(0.626254\pi\)
\(72\) 0 0
\(73\) 12.2344 1.43193 0.715965 0.698136i \(-0.245987\pi\)
0.715965 + 0.698136i \(0.245987\pi\)
\(74\) −1.49449 −0.173730
\(75\) 0 0
\(76\) 0.901674 0.103429
\(77\) 0.655386 0.0746882
\(78\) 0 0
\(79\) 0.702333 0.0790187 0.0395093 0.999219i \(-0.487421\pi\)
0.0395093 + 0.999219i \(0.487421\pi\)
\(80\) 1.47924 0.165384
\(81\) 0 0
\(82\) 5.17024 0.570958
\(83\) 6.77660 0.743828 0.371914 0.928267i \(-0.378702\pi\)
0.371914 + 0.928267i \(0.378702\pi\)
\(84\) 0 0
\(85\) 5.00774 0.543166
\(86\) −0.892951 −0.0962894
\(87\) 0 0
\(88\) 13.1284 1.39949
\(89\) 6.85565 0.726697 0.363349 0.931653i \(-0.381633\pi\)
0.363349 + 0.931653i \(0.381633\pi\)
\(90\) 0 0
\(91\) 0.551689 0.0578327
\(92\) 11.9441 1.24526
\(93\) 0 0
\(94\) 1.65270 0.170463
\(95\) −0.616781 −0.0632804
\(96\) 0 0
\(97\) −9.02734 −0.916588 −0.458294 0.888801i \(-0.651539\pi\)
−0.458294 + 0.888801i \(0.651539\pi\)
\(98\) −4.77833 −0.482684
\(99\) 0 0
\(100\) 5.97771 0.597771
\(101\) 12.9921 1.29276 0.646382 0.763014i \(-0.276281\pi\)
0.646382 + 0.763014i \(0.276281\pi\)
\(102\) 0 0
\(103\) 9.07192 0.893883 0.446941 0.894563i \(-0.352513\pi\)
0.446941 + 0.894563i \(0.352513\pi\)
\(104\) 11.0511 1.08365
\(105\) 0 0
\(106\) −2.08378 −0.202394
\(107\) 11.3865 1.10077 0.550386 0.834911i \(-0.314481\pi\)
0.550386 + 0.834911i \(0.314481\pi\)
\(108\) 0 0
\(109\) 2.89899 0.277672 0.138836 0.990315i \(-0.455664\pi\)
0.138836 + 0.990315i \(0.455664\pi\)
\(110\) −3.89533 −0.371405
\(111\) 0 0
\(112\) −0.170245 −0.0160866
\(113\) −11.5801 −1.08937 −0.544683 0.838642i \(-0.683350\pi\)
−0.544683 + 0.838642i \(0.683350\pi\)
\(114\) 0 0
\(115\) −8.17024 −0.761879
\(116\) −7.76730 −0.721176
\(117\) 0 0
\(118\) −0.0300295 −0.00276444
\(119\) −0.576337 −0.0528327
\(120\) 0 0
\(121\) 18.5253 1.68412
\(122\) −6.98551 −0.632438
\(123\) 0 0
\(124\) 13.4192 1.20508
\(125\) −9.32905 −0.834415
\(126\) 0 0
\(127\) −15.4192 −1.36823 −0.684117 0.729372i \(-0.739812\pi\)
−0.684117 + 0.729372i \(0.739812\pi\)
\(128\) −10.8136 −0.955795
\(129\) 0 0
\(130\) −3.27900 −0.287587
\(131\) 13.3561 1.16693 0.583463 0.812140i \(-0.301697\pi\)
0.583463 + 0.812140i \(0.301697\pi\)
\(132\) 0 0
\(133\) 0.0709849 0.00615517
\(134\) −1.27217 −0.109899
\(135\) 0 0
\(136\) −11.5449 −0.989965
\(137\) −19.6236 −1.67656 −0.838279 0.545242i \(-0.816438\pi\)
−0.838279 + 0.545242i \(0.816438\pi\)
\(138\) 0 0
\(139\) 16.3063 1.38309 0.691543 0.722335i \(-0.256931\pi\)
0.691543 + 0.722335i \(0.256931\pi\)
\(140\) 0.193665 0.0163676
\(141\) 0 0
\(142\) −4.45336 −0.373718
\(143\) 24.8537 2.07837
\(144\) 0 0
\(145\) 5.31315 0.441233
\(146\) 8.36884 0.692610
\(147\) 0 0
\(148\) 3.34730 0.275146
\(149\) 11.0477 0.905062 0.452531 0.891749i \(-0.350521\pi\)
0.452531 + 0.891749i \(0.350521\pi\)
\(150\) 0 0
\(151\) −6.80066 −0.553430 −0.276715 0.960952i \(-0.589246\pi\)
−0.276715 + 0.960952i \(0.589246\pi\)
\(152\) 1.42193 0.115334
\(153\) 0 0
\(154\) 0.448311 0.0361259
\(155\) −9.17928 −0.737298
\(156\) 0 0
\(157\) 16.8871 1.34774 0.673870 0.738850i \(-0.264631\pi\)
0.673870 + 0.738850i \(0.264631\pi\)
\(158\) 0.480424 0.0382205
\(159\) 0 0
\(160\) 6.07604 0.480353
\(161\) 0.940307 0.0741066
\(162\) 0 0
\(163\) 8.41147 0.658838 0.329419 0.944184i \(-0.393147\pi\)
0.329419 + 0.944184i \(0.393147\pi\)
\(164\) −11.5801 −0.904256
\(165\) 0 0
\(166\) 4.63547 0.359782
\(167\) −2.62500 −0.203129 −0.101564 0.994829i \(-0.532385\pi\)
−0.101564 + 0.994829i \(0.532385\pi\)
\(168\) 0 0
\(169\) 7.92127 0.609329
\(170\) 3.42550 0.262724
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 13.1033 0.996223 0.498112 0.867113i \(-0.334027\pi\)
0.498112 + 0.867113i \(0.334027\pi\)
\(174\) 0 0
\(175\) 0.470599 0.0355740
\(176\) −7.66955 −0.578114
\(177\) 0 0
\(178\) 4.68954 0.351496
\(179\) −22.0988 −1.65174 −0.825872 0.563857i \(-0.809317\pi\)
−0.825872 + 0.563857i \(0.809317\pi\)
\(180\) 0 0
\(181\) −2.05913 −0.153054 −0.0765268 0.997068i \(-0.524383\pi\)
−0.0765268 + 0.997068i \(0.524383\pi\)
\(182\) 0.377378 0.0279731
\(183\) 0 0
\(184\) 18.8357 1.38859
\(185\) −2.28969 −0.168341
\(186\) 0 0
\(187\) −25.9641 −1.89868
\(188\) −3.70167 −0.269972
\(189\) 0 0
\(190\) −0.421903 −0.0306081
\(191\) −3.94918 −0.285753 −0.142876 0.989741i \(-0.545635\pi\)
−0.142876 + 0.989741i \(0.545635\pi\)
\(192\) 0 0
\(193\) −8.44656 −0.607996 −0.303998 0.952673i \(-0.598322\pi\)
−0.303998 + 0.952673i \(0.598322\pi\)
\(194\) −6.17507 −0.443344
\(195\) 0 0
\(196\) 10.7023 0.764452
\(197\) −2.29498 −0.163511 −0.0817553 0.996652i \(-0.526053\pi\)
−0.0817553 + 0.996652i \(0.526053\pi\)
\(198\) 0 0
\(199\) −23.5030 −1.66608 −0.833042 0.553210i \(-0.813403\pi\)
−0.833042 + 0.553210i \(0.813403\pi\)
\(200\) 9.42680 0.666575
\(201\) 0 0
\(202\) 8.88713 0.625296
\(203\) −0.611486 −0.0429179
\(204\) 0 0
\(205\) 7.92127 0.553246
\(206\) 6.20556 0.432362
\(207\) 0 0
\(208\) −6.45605 −0.447647
\(209\) 3.19788 0.221202
\(210\) 0 0
\(211\) −1.66725 −0.114778 −0.0573892 0.998352i \(-0.518278\pi\)
−0.0573892 + 0.998352i \(0.518278\pi\)
\(212\) 4.66717 0.320543
\(213\) 0 0
\(214\) 7.78880 0.532431
\(215\) −1.36808 −0.0933023
\(216\) 0 0
\(217\) 1.05644 0.0717156
\(218\) 1.98302 0.134307
\(219\) 0 0
\(220\) 8.72462 0.588214
\(221\) −21.8560 −1.47019
\(222\) 0 0
\(223\) −9.80066 −0.656301 −0.328150 0.944626i \(-0.606425\pi\)
−0.328150 + 0.944626i \(0.606425\pi\)
\(224\) −0.699287 −0.0467231
\(225\) 0 0
\(226\) −7.92127 −0.526915
\(227\) −15.4808 −1.02749 −0.513747 0.857942i \(-0.671743\pi\)
−0.513747 + 0.857942i \(0.671743\pi\)
\(228\) 0 0
\(229\) −11.2540 −0.743687 −0.371843 0.928295i \(-0.621274\pi\)
−0.371843 + 0.928295i \(0.621274\pi\)
\(230\) −5.58878 −0.368513
\(231\) 0 0
\(232\) −12.2490 −0.804184
\(233\) −5.23476 −0.342940 −0.171470 0.985189i \(-0.554852\pi\)
−0.171470 + 0.985189i \(0.554852\pi\)
\(234\) 0 0
\(235\) 2.53209 0.165175
\(236\) 0.0672590 0.00437819
\(237\) 0 0
\(238\) −0.394238 −0.0255546
\(239\) 10.3098 0.666885 0.333443 0.942770i \(-0.391790\pi\)
0.333443 + 0.942770i \(0.391790\pi\)
\(240\) 0 0
\(241\) 10.7442 0.692096 0.346048 0.938217i \(-0.387523\pi\)
0.346048 + 0.938217i \(0.387523\pi\)
\(242\) 12.6720 0.814590
\(243\) 0 0
\(244\) 15.6459 1.00163
\(245\) −7.32083 −0.467710
\(246\) 0 0
\(247\) 2.69190 0.171282
\(248\) 21.1620 1.34379
\(249\) 0 0
\(250\) −6.38144 −0.403598
\(251\) −15.0729 −0.951392 −0.475696 0.879610i \(-0.657804\pi\)
−0.475696 + 0.879610i \(0.657804\pi\)
\(252\) 0 0
\(253\) 42.3610 2.66321
\(254\) −10.5474 −0.661800
\(255\) 0 0
\(256\) −9.68273 −0.605171
\(257\) 3.32245 0.207249 0.103624 0.994617i \(-0.466956\pi\)
0.103624 + 0.994617i \(0.466956\pi\)
\(258\) 0 0
\(259\) 0.263518 0.0163742
\(260\) 7.34419 0.455467
\(261\) 0 0
\(262\) 9.13610 0.564430
\(263\) −8.83867 −0.545016 −0.272508 0.962154i \(-0.587853\pi\)
−0.272508 + 0.962154i \(0.587853\pi\)
\(264\) 0 0
\(265\) −3.19253 −0.196116
\(266\) 0.0485565 0.00297719
\(267\) 0 0
\(268\) 2.84936 0.174052
\(269\) −8.09267 −0.493419 −0.246709 0.969090i \(-0.579349\pi\)
−0.246709 + 0.969090i \(0.579349\pi\)
\(270\) 0 0
\(271\) −19.0000 −1.15417 −0.577084 0.816685i \(-0.695809\pi\)
−0.577084 + 0.816685i \(0.695809\pi\)
\(272\) 6.74449 0.408945
\(273\) 0 0
\(274\) −13.4233 −0.810933
\(275\) 21.2006 1.27844
\(276\) 0 0
\(277\) −4.22937 −0.254118 −0.127059 0.991895i \(-0.540554\pi\)
−0.127059 + 0.991895i \(0.540554\pi\)
\(278\) 11.1542 0.668984
\(279\) 0 0
\(280\) 0.305407 0.0182516
\(281\) −7.30212 −0.435608 −0.217804 0.975993i \(-0.569889\pi\)
−0.217804 + 0.975993i \(0.569889\pi\)
\(282\) 0 0
\(283\) 16.0205 0.952322 0.476161 0.879358i \(-0.342028\pi\)
0.476161 + 0.879358i \(0.342028\pi\)
\(284\) 9.97448 0.591877
\(285\) 0 0
\(286\) 17.0009 1.00529
\(287\) −0.911654 −0.0538132
\(288\) 0 0
\(289\) 5.83244 0.343085
\(290\) 3.63441 0.213420
\(291\) 0 0
\(292\) −18.7442 −1.09692
\(293\) −18.0045 −1.05184 −0.525918 0.850535i \(-0.676278\pi\)
−0.525918 + 0.850535i \(0.676278\pi\)
\(294\) 0 0
\(295\) −0.0460079 −0.00267868
\(296\) 5.27866 0.306816
\(297\) 0 0
\(298\) 7.55707 0.437769
\(299\) 35.6585 2.06219
\(300\) 0 0
\(301\) 0.157451 0.00907535
\(302\) −4.65193 −0.267688
\(303\) 0 0
\(304\) −0.830689 −0.0476433
\(305\) −10.7024 −0.612819
\(306\) 0 0
\(307\) −13.5107 −0.771098 −0.385549 0.922687i \(-0.625988\pi\)
−0.385549 + 0.922687i \(0.625988\pi\)
\(308\) −1.00411 −0.0572145
\(309\) 0 0
\(310\) −6.27900 −0.356623
\(311\) −16.4223 −0.931221 −0.465611 0.884990i \(-0.654165\pi\)
−0.465611 + 0.884990i \(0.654165\pi\)
\(312\) 0 0
\(313\) −19.5648 −1.10587 −0.552934 0.833225i \(-0.686492\pi\)
−0.552934 + 0.833225i \(0.686492\pi\)
\(314\) 11.5515 0.651887
\(315\) 0 0
\(316\) −1.07604 −0.0605318
\(317\) 24.3786 1.36924 0.684619 0.728902i \(-0.259969\pi\)
0.684619 + 0.728902i \(0.259969\pi\)
\(318\) 0 0
\(319\) −27.5476 −1.54237
\(320\) 1.19777 0.0669576
\(321\) 0 0
\(322\) 0.643208 0.0358446
\(323\) −2.81217 −0.156473
\(324\) 0 0
\(325\) 17.8462 0.989927
\(326\) 5.75379 0.318673
\(327\) 0 0
\(328\) −18.2618 −1.00834
\(329\) −0.291416 −0.0160663
\(330\) 0 0
\(331\) 28.4989 1.56644 0.783220 0.621745i \(-0.213576\pi\)
0.783220 + 0.621745i \(0.213576\pi\)
\(332\) −10.3824 −0.569806
\(333\) 0 0
\(334\) −1.79561 −0.0982512
\(335\) −1.94907 −0.106489
\(336\) 0 0
\(337\) −17.4466 −0.950374 −0.475187 0.879885i \(-0.657620\pi\)
−0.475187 + 0.879885i \(0.657620\pi\)
\(338\) 5.41847 0.294726
\(339\) 0 0
\(340\) −7.67230 −0.416089
\(341\) 47.5927 2.57729
\(342\) 0 0
\(343\) 1.68685 0.0910814
\(344\) 3.15398 0.170051
\(345\) 0 0
\(346\) 8.96316 0.481863
\(347\) −25.1890 −1.35222 −0.676109 0.736802i \(-0.736335\pi\)
−0.676109 + 0.736802i \(0.736335\pi\)
\(348\) 0 0
\(349\) −11.8520 −0.634425 −0.317213 0.948354i \(-0.602747\pi\)
−0.317213 + 0.948354i \(0.602747\pi\)
\(350\) 0.321909 0.0172068
\(351\) 0 0
\(352\) −31.5030 −1.67912
\(353\) 8.39749 0.446953 0.223477 0.974709i \(-0.428259\pi\)
0.223477 + 0.974709i \(0.428259\pi\)
\(354\) 0 0
\(355\) −6.82295 −0.362124
\(356\) −10.5035 −0.556682
\(357\) 0 0
\(358\) −15.1165 −0.798932
\(359\) −2.64025 −0.139347 −0.0696735 0.997570i \(-0.522196\pi\)
−0.0696735 + 0.997570i \(0.522196\pi\)
\(360\) 0 0
\(361\) −18.6536 −0.981770
\(362\) −1.40852 −0.0740304
\(363\) 0 0
\(364\) −0.845237 −0.0443025
\(365\) 12.8218 0.671124
\(366\) 0 0
\(367\) 9.19759 0.480110 0.240055 0.970759i \(-0.422835\pi\)
0.240055 + 0.970759i \(0.422835\pi\)
\(368\) −11.0038 −0.573612
\(369\) 0 0
\(370\) −1.56624 −0.0814248
\(371\) 0.367426 0.0190758
\(372\) 0 0
\(373\) −26.7912 −1.38719 −0.693597 0.720363i \(-0.743975\pi\)
−0.693597 + 0.720363i \(0.743975\pi\)
\(374\) −17.7605 −0.918373
\(375\) 0 0
\(376\) −5.83750 −0.301046
\(377\) −23.1889 −1.19429
\(378\) 0 0
\(379\) −27.1242 −1.39328 −0.696639 0.717422i \(-0.745322\pi\)
−0.696639 + 0.717422i \(0.745322\pi\)
\(380\) 0.944964 0.0484756
\(381\) 0 0
\(382\) −2.70140 −0.138216
\(383\) 34.4268 1.75913 0.879564 0.475781i \(-0.157834\pi\)
0.879564 + 0.475781i \(0.157834\pi\)
\(384\) 0 0
\(385\) 0.686852 0.0350052
\(386\) −5.77778 −0.294081
\(387\) 0 0
\(388\) 13.8307 0.702147
\(389\) 0.756594 0.0383609 0.0191804 0.999816i \(-0.493894\pi\)
0.0191804 + 0.999816i \(0.493894\pi\)
\(390\) 0 0
\(391\) −37.2517 −1.88390
\(392\) 16.8775 0.852442
\(393\) 0 0
\(394\) −1.56986 −0.0790884
\(395\) 0.736053 0.0370348
\(396\) 0 0
\(397\) 7.00774 0.351708 0.175854 0.984416i \(-0.443731\pi\)
0.175854 + 0.984416i \(0.443731\pi\)
\(398\) −16.0770 −0.805867
\(399\) 0 0
\(400\) −5.50711 −0.275356
\(401\) −9.45080 −0.471950 −0.235975 0.971759i \(-0.575828\pi\)
−0.235975 + 0.971759i \(0.575828\pi\)
\(402\) 0 0
\(403\) 40.0624 1.99565
\(404\) −19.9051 −0.990314
\(405\) 0 0
\(406\) −0.418281 −0.0207590
\(407\) 11.8715 0.588451
\(408\) 0 0
\(409\) 5.51754 0.272825 0.136412 0.990652i \(-0.456443\pi\)
0.136412 + 0.990652i \(0.456443\pi\)
\(410\) 5.41847 0.267599
\(411\) 0 0
\(412\) −13.8990 −0.684754
\(413\) 0.00529501 0.000260550 0
\(414\) 0 0
\(415\) 7.10195 0.348621
\(416\) −26.5185 −1.30018
\(417\) 0 0
\(418\) 2.18748 0.106993
\(419\) −0.466378 −0.0227841 −0.0113920 0.999935i \(-0.503626\pi\)
−0.0113920 + 0.999935i \(0.503626\pi\)
\(420\) 0 0
\(421\) −2.41416 −0.117659 −0.0588295 0.998268i \(-0.518737\pi\)
−0.0588295 + 0.998268i \(0.518737\pi\)
\(422\) −1.14047 −0.0555171
\(423\) 0 0
\(424\) 7.36009 0.357438
\(425\) −18.6435 −0.904342
\(426\) 0 0
\(427\) 1.23173 0.0596078
\(428\) −17.4451 −0.843240
\(429\) 0 0
\(430\) −0.935822 −0.0451294
\(431\) 12.0992 0.582796 0.291398 0.956602i \(-0.405880\pi\)
0.291398 + 0.956602i \(0.405880\pi\)
\(432\) 0 0
\(433\) 33.3509 1.60274 0.801371 0.598167i \(-0.204104\pi\)
0.801371 + 0.598167i \(0.204104\pi\)
\(434\) 0.722645 0.0346881
\(435\) 0 0
\(436\) −4.44150 −0.212709
\(437\) 4.58812 0.219480
\(438\) 0 0
\(439\) −9.04458 −0.431674 −0.215837 0.976429i \(-0.569248\pi\)
−0.215837 + 0.976429i \(0.569248\pi\)
\(440\) 13.7587 0.655918
\(441\) 0 0
\(442\) −14.9504 −0.711117
\(443\) 4.67712 0.222217 0.111108 0.993808i \(-0.464560\pi\)
0.111108 + 0.993808i \(0.464560\pi\)
\(444\) 0 0
\(445\) 7.18479 0.340592
\(446\) −6.70405 −0.317446
\(447\) 0 0
\(448\) −0.137851 −0.00651285
\(449\) 26.7069 1.26037 0.630187 0.776443i \(-0.282978\pi\)
0.630187 + 0.776443i \(0.282978\pi\)
\(450\) 0 0
\(451\) −41.0702 −1.93392
\(452\) 17.7418 0.834503
\(453\) 0 0
\(454\) −10.5895 −0.496988
\(455\) 0.578176 0.0271053
\(456\) 0 0
\(457\) −0.657756 −0.0307685 −0.0153843 0.999882i \(-0.504897\pi\)
−0.0153843 + 0.999882i \(0.504897\pi\)
\(458\) −7.69820 −0.359713
\(459\) 0 0
\(460\) 12.5175 0.583633
\(461\) −20.7037 −0.964268 −0.482134 0.876097i \(-0.660138\pi\)
−0.482134 + 0.876097i \(0.660138\pi\)
\(462\) 0 0
\(463\) −24.8239 −1.15366 −0.576832 0.816863i \(-0.695711\pi\)
−0.576832 + 0.816863i \(0.695711\pi\)
\(464\) 7.15582 0.332200
\(465\) 0 0
\(466\) −3.58079 −0.165877
\(467\) −26.6729 −1.23428 −0.617138 0.786855i \(-0.711708\pi\)
−0.617138 + 0.786855i \(0.711708\pi\)
\(468\) 0 0
\(469\) 0.224318 0.0103580
\(470\) 1.73205 0.0798935
\(471\) 0 0
\(472\) 0.106067 0.00488212
\(473\) 7.09321 0.326146
\(474\) 0 0
\(475\) 2.29624 0.105359
\(476\) 0.883000 0.0404722
\(477\) 0 0
\(478\) 7.05232 0.322566
\(479\) 3.22654 0.147424 0.0737121 0.997280i \(-0.476515\pi\)
0.0737121 + 0.997280i \(0.476515\pi\)
\(480\) 0 0
\(481\) 9.99319 0.455650
\(482\) 7.34948 0.334760
\(483\) 0 0
\(484\) −28.3824 −1.29011
\(485\) −9.46075 −0.429590
\(486\) 0 0
\(487\) −7.77930 −0.352514 −0.176257 0.984344i \(-0.556399\pi\)
−0.176257 + 0.984344i \(0.556399\pi\)
\(488\) 24.6734 1.11691
\(489\) 0 0
\(490\) −5.00774 −0.226227
\(491\) −17.0460 −0.769273 −0.384637 0.923068i \(-0.625673\pi\)
−0.384637 + 0.923068i \(0.625673\pi\)
\(492\) 0 0
\(493\) 24.2249 1.09104
\(494\) 1.84137 0.0828472
\(495\) 0 0
\(496\) −12.3628 −0.555105
\(497\) 0.785248 0.0352232
\(498\) 0 0
\(499\) 7.93077 0.355030 0.177515 0.984118i \(-0.443194\pi\)
0.177515 + 0.984118i \(0.443194\pi\)
\(500\) 14.2929 0.639199
\(501\) 0 0
\(502\) −10.3105 −0.460178
\(503\) −24.9496 −1.11245 −0.556224 0.831032i \(-0.687750\pi\)
−0.556224 + 0.831032i \(0.687750\pi\)
\(504\) 0 0
\(505\) 13.6159 0.605898
\(506\) 28.9766 1.28817
\(507\) 0 0
\(508\) 23.6236 1.04813
\(509\) 40.4690 1.79376 0.896878 0.442278i \(-0.145830\pi\)
0.896878 + 0.442278i \(0.145830\pi\)
\(510\) 0 0
\(511\) −1.47565 −0.0652790
\(512\) 15.0038 0.663080
\(513\) 0 0
\(514\) 2.27269 0.100244
\(515\) 9.50747 0.418949
\(516\) 0 0
\(517\) −13.1284 −0.577384
\(518\) 0.180257 0.00792004
\(519\) 0 0
\(520\) 11.5817 0.507892
\(521\) −25.3674 −1.11137 −0.555684 0.831394i \(-0.687543\pi\)
−0.555684 + 0.831394i \(0.687543\pi\)
\(522\) 0 0
\(523\) 12.7219 0.556291 0.278146 0.960539i \(-0.410280\pi\)
0.278146 + 0.960539i \(0.410280\pi\)
\(524\) −20.4627 −0.893917
\(525\) 0 0
\(526\) −6.04601 −0.263618
\(527\) −41.8523 −1.82311
\(528\) 0 0
\(529\) 37.7769 1.64248
\(530\) −2.18382 −0.0948591
\(531\) 0 0
\(532\) −0.108755 −0.00471514
\(533\) −34.5719 −1.49748
\(534\) 0 0
\(535\) 11.9331 0.515914
\(536\) 4.49341 0.194086
\(537\) 0 0
\(538\) −5.53571 −0.238661
\(539\) 37.9570 1.63492
\(540\) 0 0
\(541\) 2.09327 0.0899969 0.0449984 0.998987i \(-0.485672\pi\)
0.0449984 + 0.998987i \(0.485672\pi\)
\(542\) −12.9968 −0.558259
\(543\) 0 0
\(544\) 27.7033 1.18777
\(545\) 3.03817 0.130141
\(546\) 0 0
\(547\) 3.45100 0.147554 0.0737770 0.997275i \(-0.476495\pi\)
0.0737770 + 0.997275i \(0.476495\pi\)
\(548\) 30.0651 1.28432
\(549\) 0 0
\(550\) 14.5021 0.618370
\(551\) −2.98368 −0.127109
\(552\) 0 0
\(553\) −0.0847118 −0.00360231
\(554\) −2.89306 −0.122914
\(555\) 0 0
\(556\) −24.9828 −1.05951
\(557\) 11.1003 0.470337 0.235168 0.971955i \(-0.424436\pi\)
0.235168 + 0.971955i \(0.424436\pi\)
\(558\) 0 0
\(559\) 5.97090 0.252542
\(560\) −0.178418 −0.00753954
\(561\) 0 0
\(562\) −4.99495 −0.210699
\(563\) −24.3107 −1.02457 −0.512286 0.858815i \(-0.671201\pi\)
−0.512286 + 0.858815i \(0.671201\pi\)
\(564\) 0 0
\(565\) −12.1361 −0.510569
\(566\) 10.9587 0.460628
\(567\) 0 0
\(568\) 15.7297 0.660002
\(569\) −42.7640 −1.79276 −0.896379 0.443288i \(-0.853812\pi\)
−0.896379 + 0.443288i \(0.853812\pi\)
\(570\) 0 0
\(571\) 18.0283 0.754460 0.377230 0.926120i \(-0.376877\pi\)
0.377230 + 0.926120i \(0.376877\pi\)
\(572\) −38.0781 −1.59212
\(573\) 0 0
\(574\) −0.623608 −0.0260289
\(575\) 30.4173 1.26849
\(576\) 0 0
\(577\) 25.2763 1.05227 0.526133 0.850402i \(-0.323641\pi\)
0.526133 + 0.850402i \(0.323641\pi\)
\(578\) 3.98963 0.165947
\(579\) 0 0
\(580\) −8.14022 −0.338004
\(581\) −0.817358 −0.0339097
\(582\) 0 0
\(583\) 16.5526 0.685540
\(584\) −29.5595 −1.22318
\(585\) 0 0
\(586\) −12.3158 −0.508763
\(587\) −28.5653 −1.17902 −0.589508 0.807762i \(-0.700679\pi\)
−0.589508 + 0.807762i \(0.700679\pi\)
\(588\) 0 0
\(589\) 5.15476 0.212398
\(590\) −0.0314712 −0.00129565
\(591\) 0 0
\(592\) −3.08378 −0.126743
\(593\) −20.6009 −0.845977 −0.422989 0.906135i \(-0.639019\pi\)
−0.422989 + 0.906135i \(0.639019\pi\)
\(594\) 0 0
\(595\) −0.604007 −0.0247619
\(596\) −16.9260 −0.693318
\(597\) 0 0
\(598\) 24.3919 0.997458
\(599\) 14.4748 0.591424 0.295712 0.955277i \(-0.404443\pi\)
0.295712 + 0.955277i \(0.404443\pi\)
\(600\) 0 0
\(601\) −13.3550 −0.544763 −0.272382 0.962189i \(-0.587811\pi\)
−0.272382 + 0.962189i \(0.587811\pi\)
\(602\) 0.107703 0.00438965
\(603\) 0 0
\(604\) 10.4192 0.423952
\(605\) 19.4147 0.789319
\(606\) 0 0
\(607\) 31.2131 1.26690 0.633450 0.773784i \(-0.281638\pi\)
0.633450 + 0.773784i \(0.281638\pi\)
\(608\) −3.41209 −0.138378
\(609\) 0 0
\(610\) −7.32089 −0.296414
\(611\) −11.0511 −0.447082
\(612\) 0 0
\(613\) 30.0651 1.21432 0.607159 0.794580i \(-0.292309\pi\)
0.607159 + 0.794580i \(0.292309\pi\)
\(614\) −9.24189 −0.372972
\(615\) 0 0
\(616\) −1.58347 −0.0638000
\(617\) 42.1537 1.69704 0.848521 0.529161i \(-0.177493\pi\)
0.848521 + 0.529161i \(0.177493\pi\)
\(618\) 0 0
\(619\) 11.6108 0.466678 0.233339 0.972395i \(-0.425035\pi\)
0.233339 + 0.972395i \(0.425035\pi\)
\(620\) 14.0635 0.564803
\(621\) 0 0
\(622\) −11.2335 −0.450422
\(623\) −0.826892 −0.0331287
\(624\) 0 0
\(625\) 9.73143 0.389257
\(626\) −13.3831 −0.534897
\(627\) 0 0
\(628\) −25.8726 −1.03243
\(629\) −10.4397 −0.416257
\(630\) 0 0
\(631\) −29.3105 −1.16683 −0.583415 0.812174i \(-0.698284\pi\)
−0.583415 + 0.812174i \(0.698284\pi\)
\(632\) −1.69690 −0.0674991
\(633\) 0 0
\(634\) 16.6759 0.662286
\(635\) −16.1595 −0.641270
\(636\) 0 0
\(637\) 31.9513 1.26596
\(638\) −18.8436 −0.746027
\(639\) 0 0
\(640\) −11.3327 −0.447966
\(641\) 31.0979 1.22829 0.614146 0.789192i \(-0.289501\pi\)
0.614146 + 0.789192i \(0.289501\pi\)
\(642\) 0 0
\(643\) 42.0719 1.65915 0.829577 0.558392i \(-0.188581\pi\)
0.829577 + 0.558392i \(0.188581\pi\)
\(644\) −1.44063 −0.0567690
\(645\) 0 0
\(646\) −1.92364 −0.0756845
\(647\) −4.66717 −0.183485 −0.0917427 0.995783i \(-0.529244\pi\)
−0.0917427 + 0.995783i \(0.529244\pi\)
\(648\) 0 0
\(649\) 0.238541 0.00936356
\(650\) 12.2075 0.478818
\(651\) 0 0
\(652\) −12.8871 −0.504699
\(653\) −3.12413 −0.122257 −0.0611283 0.998130i \(-0.519470\pi\)
−0.0611283 + 0.998130i \(0.519470\pi\)
\(654\) 0 0
\(655\) 13.9973 0.546920
\(656\) 10.6685 0.416534
\(657\) 0 0
\(658\) −0.199340 −0.00777110
\(659\) 37.4292 1.45803 0.729017 0.684495i \(-0.239977\pi\)
0.729017 + 0.684495i \(0.239977\pi\)
\(660\) 0 0
\(661\) 26.5003 1.03074 0.515371 0.856967i \(-0.327654\pi\)
0.515371 + 0.856967i \(0.327654\pi\)
\(662\) 19.4944 0.757671
\(663\) 0 0
\(664\) −16.3729 −0.635391
\(665\) 0.0743929 0.00288483
\(666\) 0 0
\(667\) −39.5235 −1.53036
\(668\) 4.02174 0.155606
\(669\) 0 0
\(670\) −1.33325 −0.0515078
\(671\) 55.4898 2.14216
\(672\) 0 0
\(673\) 1.72100 0.0663397 0.0331698 0.999450i \(-0.489440\pi\)
0.0331698 + 0.999450i \(0.489440\pi\)
\(674\) −11.9341 −0.459686
\(675\) 0 0
\(676\) −12.1361 −0.466773
\(677\) −28.2792 −1.08686 −0.543429 0.839455i \(-0.682874\pi\)
−0.543429 + 0.839455i \(0.682874\pi\)
\(678\) 0 0
\(679\) 1.08883 0.0417855
\(680\) −12.0992 −0.463982
\(681\) 0 0
\(682\) 32.5553 1.24661
\(683\) 24.7139 0.945651 0.472825 0.881156i \(-0.343234\pi\)
0.472825 + 0.881156i \(0.343234\pi\)
\(684\) 0 0
\(685\) −20.5657 −0.785777
\(686\) 1.15387 0.0440551
\(687\) 0 0
\(688\) −1.84255 −0.0702465
\(689\) 13.9336 0.530829
\(690\) 0 0
\(691\) −43.7725 −1.66518 −0.832592 0.553887i \(-0.813144\pi\)
−0.832592 + 0.553887i \(0.813144\pi\)
\(692\) −20.0754 −0.763151
\(693\) 0 0
\(694\) −17.2303 −0.654053
\(695\) 17.0892 0.648231
\(696\) 0 0
\(697\) 36.1165 1.36801
\(698\) −8.10728 −0.306865
\(699\) 0 0
\(700\) −0.721000 −0.0272512
\(701\) 14.6504 0.553338 0.276669 0.960965i \(-0.410769\pi\)
0.276669 + 0.960965i \(0.410769\pi\)
\(702\) 0 0
\(703\) 1.28581 0.0484951
\(704\) −6.21021 −0.234056
\(705\) 0 0
\(706\) 5.74422 0.216187
\(707\) −1.56704 −0.0589346
\(708\) 0 0
\(709\) 5.07367 0.190546 0.0952729 0.995451i \(-0.469628\pi\)
0.0952729 + 0.995451i \(0.469628\pi\)
\(710\) −4.66717 −0.175156
\(711\) 0 0
\(712\) −16.5639 −0.620757
\(713\) 68.2830 2.55722
\(714\) 0 0
\(715\) 26.0469 0.974100
\(716\) 33.8574 1.26531
\(717\) 0 0
\(718\) −1.80604 −0.0674007
\(719\) 5.33717 0.199043 0.0995213 0.995035i \(-0.468269\pi\)
0.0995213 + 0.995035i \(0.468269\pi\)
\(720\) 0 0
\(721\) −1.09421 −0.0407504
\(722\) −12.7598 −0.474872
\(723\) 0 0
\(724\) 3.15476 0.117246
\(725\) −19.7805 −0.734629
\(726\) 0 0
\(727\) 37.1198 1.37670 0.688348 0.725380i \(-0.258336\pi\)
0.688348 + 0.725380i \(0.258336\pi\)
\(728\) −1.33293 −0.0494017
\(729\) 0 0
\(730\) 8.77063 0.324616
\(731\) −6.23767 −0.230708
\(732\) 0 0
\(733\) 29.9463 1.10609 0.553045 0.833151i \(-0.313466\pi\)
0.553045 + 0.833151i \(0.313466\pi\)
\(734\) 6.29152 0.232224
\(735\) 0 0
\(736\) −45.1985 −1.66604
\(737\) 10.1055 0.372243
\(738\) 0 0
\(739\) −28.6100 −1.05244 −0.526218 0.850350i \(-0.676390\pi\)
−0.526218 + 0.850350i \(0.676390\pi\)
\(740\) 3.50800 0.128957
\(741\) 0 0
\(742\) 0.251334 0.00922678
\(743\) −49.6436 −1.82125 −0.910624 0.413237i \(-0.864398\pi\)
−0.910624 + 0.413237i \(0.864398\pi\)
\(744\) 0 0
\(745\) 11.5781 0.424189
\(746\) −18.3262 −0.670971
\(747\) 0 0
\(748\) 39.7793 1.45448
\(749\) −1.37338 −0.0501821
\(750\) 0 0
\(751\) −31.6236 −1.15396 −0.576981 0.816758i \(-0.695769\pi\)
−0.576981 + 0.816758i \(0.695769\pi\)
\(752\) 3.41025 0.124359
\(753\) 0 0
\(754\) −15.8621 −0.577665
\(755\) −7.12716 −0.259384
\(756\) 0 0
\(757\) 3.04189 0.110559 0.0552797 0.998471i \(-0.482395\pi\)
0.0552797 + 0.998471i \(0.482395\pi\)
\(758\) −18.5541 −0.673914
\(759\) 0 0
\(760\) 1.49020 0.0540552
\(761\) −37.9283 −1.37490 −0.687450 0.726232i \(-0.741270\pi\)
−0.687450 + 0.726232i \(0.741270\pi\)
\(762\) 0 0
\(763\) −0.349660 −0.0126586
\(764\) 6.05050 0.218899
\(765\) 0 0
\(766\) 23.5493 0.850872
\(767\) 0.200798 0.00725041
\(768\) 0 0
\(769\) −20.3878 −0.735201 −0.367601 0.929984i \(-0.619821\pi\)
−0.367601 + 0.929984i \(0.619821\pi\)
\(770\) 0.469834 0.0169317
\(771\) 0 0
\(772\) 12.9409 0.465752
\(773\) −24.4664 −0.879994 −0.439997 0.897999i \(-0.645021\pi\)
−0.439997 + 0.897999i \(0.645021\pi\)
\(774\) 0 0
\(775\) 34.1739 1.22756
\(776\) 21.8109 0.782965
\(777\) 0 0
\(778\) 0.517541 0.0185547
\(779\) −4.44831 −0.159377
\(780\) 0 0
\(781\) 35.3756 1.26584
\(782\) −25.4816 −0.911221
\(783\) 0 0
\(784\) −9.85978 −0.352135
\(785\) 17.6979 0.631665
\(786\) 0 0
\(787\) 37.5749 1.33940 0.669700 0.742631i \(-0.266422\pi\)
0.669700 + 0.742631i \(0.266422\pi\)
\(788\) 3.51611 0.125256
\(789\) 0 0
\(790\) 0.503490 0.0179134
\(791\) 1.39673 0.0496622
\(792\) 0 0
\(793\) 46.7101 1.65872
\(794\) 4.79358 0.170118
\(795\) 0 0
\(796\) 36.0087 1.27629
\(797\) −2.30493 −0.0816449 −0.0408224 0.999166i \(-0.512998\pi\)
−0.0408224 + 0.999166i \(0.512998\pi\)
\(798\) 0 0
\(799\) 11.5449 0.408429
\(800\) −22.6207 −0.799762
\(801\) 0 0
\(802\) −6.46473 −0.228277
\(803\) −66.4784 −2.34597
\(804\) 0 0
\(805\) 0.985452 0.0347326
\(806\) 27.4043 0.965276
\(807\) 0 0
\(808\) −31.3901 −1.10430
\(809\) 28.8614 1.01471 0.507356 0.861736i \(-0.330623\pi\)
0.507356 + 0.861736i \(0.330623\pi\)
\(810\) 0 0
\(811\) −51.8631 −1.82116 −0.910580 0.413334i \(-0.864364\pi\)
−0.910580 + 0.413334i \(0.864364\pi\)
\(812\) 0.936851 0.0328770
\(813\) 0 0
\(814\) 8.12061 0.284627
\(815\) 8.81531 0.308787
\(816\) 0 0
\(817\) 0.768266 0.0268782
\(818\) 3.77422 0.131963
\(819\) 0 0
\(820\) −12.1361 −0.423811
\(821\) −18.4411 −0.643598 −0.321799 0.946808i \(-0.604288\pi\)
−0.321799 + 0.946808i \(0.604288\pi\)
\(822\) 0 0
\(823\) −34.7665 −1.21188 −0.605942 0.795509i \(-0.707204\pi\)
−0.605942 + 0.795509i \(0.707204\pi\)
\(824\) −21.9186 −0.763570
\(825\) 0 0
\(826\) 0.00362200 0.000126025 0
\(827\) 32.7773 1.13978 0.569889 0.821722i \(-0.306986\pi\)
0.569889 + 0.821722i \(0.306986\pi\)
\(828\) 0 0
\(829\) 5.35267 0.185906 0.0929530 0.995670i \(-0.470369\pi\)
0.0929530 + 0.995670i \(0.470369\pi\)
\(830\) 4.85802 0.168624
\(831\) 0 0
\(832\) −5.22762 −0.181235
\(833\) −33.3788 −1.15651
\(834\) 0 0
\(835\) −2.75103 −0.0952033
\(836\) −4.89944 −0.169451
\(837\) 0 0
\(838\) −0.319022 −0.0110204
\(839\) −5.43837 −0.187754 −0.0938768 0.995584i \(-0.529926\pi\)
−0.0938768 + 0.995584i \(0.529926\pi\)
\(840\) 0 0
\(841\) −3.29767 −0.113713
\(842\) −1.65138 −0.0569105
\(843\) 0 0
\(844\) 2.55438 0.0879253
\(845\) 8.30158 0.285583
\(846\) 0 0
\(847\) −2.23442 −0.0767757
\(848\) −4.29975 −0.147654
\(849\) 0 0
\(850\) −12.7529 −0.437421
\(851\) 17.0325 0.583868
\(852\) 0 0
\(853\) −47.1171 −1.61326 −0.806629 0.591057i \(-0.798711\pi\)
−0.806629 + 0.591057i \(0.798711\pi\)
\(854\) 0.842556 0.0288317
\(855\) 0 0
\(856\) −27.5107 −0.940298
\(857\) 46.9560 1.60399 0.801993 0.597333i \(-0.203773\pi\)
0.801993 + 0.597333i \(0.203773\pi\)
\(858\) 0 0
\(859\) 7.16344 0.244413 0.122207 0.992505i \(-0.461003\pi\)
0.122207 + 0.992505i \(0.461003\pi\)
\(860\) 2.09602 0.0714737
\(861\) 0 0
\(862\) 8.27631 0.281892
\(863\) 35.4309 1.20608 0.603041 0.797710i \(-0.293955\pi\)
0.603041 + 0.797710i \(0.293955\pi\)
\(864\) 0 0
\(865\) 13.7324 0.466914
\(866\) 22.8134 0.775230
\(867\) 0 0
\(868\) −1.61856 −0.0549373
\(869\) −3.81628 −0.129458
\(870\) 0 0
\(871\) 8.50662 0.288236
\(872\) −7.00421 −0.237193
\(873\) 0 0
\(874\) 3.13846 0.106160
\(875\) 1.12522 0.0380394
\(876\) 0 0
\(877\) 8.17293 0.275980 0.137990 0.990434i \(-0.455936\pi\)
0.137990 + 0.990434i \(0.455936\pi\)
\(878\) −6.18686 −0.208796
\(879\) 0 0
\(880\) −8.03777 −0.270953
\(881\) 33.2307 1.11957 0.559785 0.828638i \(-0.310884\pi\)
0.559785 + 0.828638i \(0.310884\pi\)
\(882\) 0 0
\(883\) 33.0479 1.11215 0.556075 0.831132i \(-0.312307\pi\)
0.556075 + 0.831132i \(0.312307\pi\)
\(884\) 33.4853 1.12623
\(885\) 0 0
\(886\) 3.19934 0.107484
\(887\) 49.7472 1.67035 0.835174 0.549986i \(-0.185367\pi\)
0.835174 + 0.549986i \(0.185367\pi\)
\(888\) 0 0
\(889\) 1.85978 0.0623752
\(890\) 4.91469 0.164741
\(891\) 0 0
\(892\) 15.0155 0.502756
\(893\) −1.42193 −0.0475831
\(894\) 0 0
\(895\) −23.1598 −0.774147
\(896\) 1.30428 0.0435729
\(897\) 0 0
\(898\) 18.2686 0.609630
\(899\) −44.4047 −1.48098
\(900\) 0 0
\(901\) −14.5561 −0.484935
\(902\) −28.0936 −0.935416
\(903\) 0 0
\(904\) 27.9786 0.930556
\(905\) −2.15799 −0.0717338
\(906\) 0 0
\(907\) 34.5485 1.14716 0.573582 0.819148i \(-0.305553\pi\)
0.573582 + 0.819148i \(0.305553\pi\)
\(908\) 23.7179 0.787106
\(909\) 0 0
\(910\) 0.395496 0.0131106
\(911\) 13.6369 0.451811 0.225905 0.974149i \(-0.427466\pi\)
0.225905 + 0.974149i \(0.427466\pi\)
\(912\) 0 0
\(913\) −36.8221 −1.21863
\(914\) −0.449932 −0.0148824
\(915\) 0 0
\(916\) 17.2422 0.569697
\(917\) −1.61094 −0.0531979
\(918\) 0 0
\(919\) 32.7701 1.08099 0.540493 0.841348i \(-0.318238\pi\)
0.540493 + 0.841348i \(0.318238\pi\)
\(920\) 19.7401 0.650810
\(921\) 0 0
\(922\) −14.1622 −0.466407
\(923\) 29.7783 0.980166
\(924\) 0 0
\(925\) 8.52435 0.280279
\(926\) −16.9805 −0.558015
\(927\) 0 0
\(928\) 29.3928 0.964866
\(929\) −5.53147 −0.181482 −0.0907408 0.995875i \(-0.528923\pi\)
−0.0907408 + 0.995875i \(0.528923\pi\)
\(930\) 0 0
\(931\) 4.11112 0.134736
\(932\) 8.02011 0.262708
\(933\) 0 0
\(934\) −18.2453 −0.597006
\(935\) −27.2106 −0.889883
\(936\) 0 0
\(937\) −0.994014 −0.0324730 −0.0162365 0.999868i \(-0.505168\pi\)
−0.0162365 + 0.999868i \(0.505168\pi\)
\(938\) 0.153442 0.00501007
\(939\) 0 0
\(940\) −3.87939 −0.126532
\(941\) −11.4572 −0.373493 −0.186747 0.982408i \(-0.559794\pi\)
−0.186747 + 0.982408i \(0.559794\pi\)
\(942\) 0 0
\(943\) −58.9249 −1.91886
\(944\) −0.0619640 −0.00201676
\(945\) 0 0
\(946\) 4.85204 0.157754
\(947\) 45.6850 1.48456 0.742282 0.670088i \(-0.233743\pi\)
0.742282 + 0.670088i \(0.233743\pi\)
\(948\) 0 0
\(949\) −55.9600 −1.81654
\(950\) 1.57072 0.0509608
\(951\) 0 0
\(952\) 1.39248 0.0451306
\(953\) 14.5053 0.469873 0.234936 0.972011i \(-0.424512\pi\)
0.234936 + 0.972011i \(0.424512\pi\)
\(954\) 0 0
\(955\) −4.13878 −0.133928
\(956\) −15.7955 −0.510864
\(957\) 0 0
\(958\) 2.20708 0.0713076
\(959\) 2.36690 0.0764311
\(960\) 0 0
\(961\) 45.7161 1.47471
\(962\) 6.83575 0.220393
\(963\) 0 0
\(964\) −16.4611 −0.530176
\(965\) −8.85208 −0.284959
\(966\) 0 0
\(967\) −25.0368 −0.805130 −0.402565 0.915391i \(-0.631881\pi\)
−0.402565 + 0.915391i \(0.631881\pi\)
\(968\) −44.7588 −1.43860
\(969\) 0 0
\(970\) −6.47153 −0.207788
\(971\) −27.0907 −0.869383 −0.434692 0.900579i \(-0.643143\pi\)
−0.434692 + 0.900579i \(0.643143\pi\)
\(972\) 0 0
\(973\) −1.96679 −0.0630522
\(974\) −5.32136 −0.170507
\(975\) 0 0
\(976\) −14.4142 −0.461386
\(977\) 2.47340 0.0791310 0.0395655 0.999217i \(-0.487403\pi\)
0.0395655 + 0.999217i \(0.487403\pi\)
\(978\) 0 0
\(979\) −37.2517 −1.19057
\(980\) 11.2162 0.358287
\(981\) 0 0
\(982\) −11.6601 −0.372089
\(983\) −19.1080 −0.609451 −0.304726 0.952440i \(-0.598565\pi\)
−0.304726 + 0.952440i \(0.598565\pi\)
\(984\) 0 0
\(985\) −2.40516 −0.0766349
\(986\) 16.5708 0.527723
\(987\) 0 0
\(988\) −4.12424 −0.131209
\(989\) 10.1769 0.323607
\(990\) 0 0
\(991\) −38.3164 −1.21716 −0.608581 0.793492i \(-0.708261\pi\)
−0.608581 + 0.793492i \(0.708261\pi\)
\(992\) −50.7806 −1.61229
\(993\) 0 0
\(994\) 0.537141 0.0170371
\(995\) −24.6314 −0.780867
\(996\) 0 0
\(997\) 41.3746 1.31035 0.655174 0.755478i \(-0.272595\pi\)
0.655174 + 0.755478i \(0.272595\pi\)
\(998\) 5.42497 0.171724
\(999\) 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.c.1.4 yes 6
3.2 odd 2 inner 729.2.a.c.1.3 6
9.2 odd 6 729.2.c.c.244.4 12
9.4 even 3 729.2.c.c.487.3 12
9.5 odd 6 729.2.c.c.487.4 12
9.7 even 3 729.2.c.c.244.3 12
27.2 odd 18 729.2.e.m.568.2 12
27.4 even 9 729.2.e.r.406.2 12
27.5 odd 18 729.2.e.q.649.1 12
27.7 even 9 729.2.e.r.325.2 12
27.11 odd 18 729.2.e.q.82.1 12
27.13 even 9 729.2.e.m.163.1 12
27.14 odd 18 729.2.e.m.163.2 12
27.16 even 9 729.2.e.q.82.2 12
27.20 odd 18 729.2.e.r.325.1 12
27.22 even 9 729.2.e.q.649.2 12
27.23 odd 18 729.2.e.r.406.1 12
27.25 even 9 729.2.e.m.568.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.3 6 3.2 odd 2 inner
729.2.a.c.1.4 yes 6 1.1 even 1 trivial
729.2.c.c.244.3 12 9.7 even 3
729.2.c.c.244.4 12 9.2 odd 6
729.2.c.c.487.3 12 9.4 even 3
729.2.c.c.487.4 12 9.5 odd 6
729.2.e.m.163.1 12 27.13 even 9
729.2.e.m.163.2 12 27.14 odd 18
729.2.e.m.568.1 12 27.25 even 9
729.2.e.m.568.2 12 27.2 odd 18
729.2.e.q.82.1 12 27.11 odd 18
729.2.e.q.82.2 12 27.16 even 9
729.2.e.q.649.1 12 27.5 odd 18
729.2.e.q.649.2 12 27.22 even 9
729.2.e.r.325.1 12 27.20 odd 18
729.2.e.r.325.2 12 27.7 even 9
729.2.e.r.406.1 12 27.23 odd 18
729.2.e.r.406.2 12 27.4 even 9