Properties

Label 729.2.a.b.1.3
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: 6.6.7459857.1
Defining polynomial: \( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.45779\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.57840 q^{2} +0.491360 q^{4} +1.67851 q^{5} +2.77928 q^{7} +2.38124 q^{8} +O(q^{10})\) \(q-1.57840 q^{2} +0.491360 q^{4} +1.67851 q^{5} +2.77928 q^{7} +2.38124 q^{8} -2.64936 q^{10} -4.15122 q^{11} +6.87605 q^{13} -4.38683 q^{14} -4.74129 q^{16} +0.976551 q^{17} +2.68529 q^{19} +0.824751 q^{20} +6.55231 q^{22} +1.61317 q^{23} -2.18261 q^{25} -10.8532 q^{26} +1.36563 q^{28} -8.22788 q^{29} +1.04407 q^{31} +2.72118 q^{32} -1.54139 q^{34} +4.66505 q^{35} -1.30834 q^{37} -4.23847 q^{38} +3.99694 q^{40} -4.84817 q^{41} +9.84023 q^{43} -2.03974 q^{44} -2.54623 q^{46} +12.4977 q^{47} +0.724408 q^{49} +3.44504 q^{50} +3.37861 q^{52} +7.34280 q^{53} -6.96786 q^{55} +6.61815 q^{56} +12.9869 q^{58} +9.05188 q^{59} -1.28574 q^{61} -1.64796 q^{62} +5.18745 q^{64} +11.5415 q^{65} -4.64630 q^{67} +0.479838 q^{68} -7.36333 q^{70} +5.62373 q^{71} -4.56144 q^{73} +2.06510 q^{74} +1.31944 q^{76} -11.5374 q^{77} +4.65680 q^{79} -7.95828 q^{80} +7.65237 q^{82} +5.76439 q^{83} +1.63915 q^{85} -15.5319 q^{86} -9.88507 q^{88} +4.54442 q^{89} +19.1105 q^{91} +0.792647 q^{92} -19.7264 q^{94} +4.50728 q^{95} +8.57544 q^{97} -1.14341 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28} - 21 q^{29} + 15 q^{31} - 30 q^{35} + 3 q^{37} - 15 q^{38} + 3 q^{40} + 12 q^{41} + 6 q^{43} + 33 q^{44} - 3 q^{46} + 15 q^{47} + 12 q^{49} + 24 q^{50} + 3 q^{52} + 9 q^{53} + 15 q^{55} - 12 q^{56} - 15 q^{58} - 6 q^{59} + 24 q^{61} + 30 q^{62} + 6 q^{64} + 15 q^{65} + 15 q^{67} - 36 q^{68} - 15 q^{70} + 12 q^{73} - 24 q^{74} + 9 q^{76} - 15 q^{77} + 24 q^{79} + 21 q^{80} - 21 q^{82} + 6 q^{83} - 18 q^{85} + 30 q^{86} - 21 q^{88} + 9 q^{89} + 18 q^{91} - 6 q^{92} - 6 q^{94} + 33 q^{95} - 21 q^{97} - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.57840 −1.11610 −0.558050 0.829807i \(-0.688450\pi\)
−0.558050 + 0.829807i \(0.688450\pi\)
\(3\) 0 0
\(4\) 0.491360 0.245680
\(5\) 1.67851 0.750651 0.375326 0.926893i \(-0.377531\pi\)
0.375326 + 0.926893i \(0.377531\pi\)
\(6\) 0 0
\(7\) 2.77928 1.05047 0.525235 0.850957i \(-0.323977\pi\)
0.525235 + 0.850957i \(0.323977\pi\)
\(8\) 2.38124 0.841897
\(9\) 0 0
\(10\) −2.64936 −0.837802
\(11\) −4.15122 −1.25164 −0.625820 0.779967i \(-0.715236\pi\)
−0.625820 + 0.779967i \(0.715236\pi\)
\(12\) 0 0
\(13\) 6.87605 1.90707 0.953536 0.301279i \(-0.0974137\pi\)
0.953536 + 0.301279i \(0.0974137\pi\)
\(14\) −4.38683 −1.17243
\(15\) 0 0
\(16\) −4.74129 −1.18532
\(17\) 0.976551 0.236848 0.118424 0.992963i \(-0.462216\pi\)
0.118424 + 0.992963i \(0.462216\pi\)
\(18\) 0 0
\(19\) 2.68529 0.616048 0.308024 0.951379i \(-0.400332\pi\)
0.308024 + 0.951379i \(0.400332\pi\)
\(20\) 0.824751 0.184420
\(21\) 0 0
\(22\) 6.55231 1.39696
\(23\) 1.61317 0.336369 0.168185 0.985756i \(-0.446210\pi\)
0.168185 + 0.985756i \(0.446210\pi\)
\(24\) 0 0
\(25\) −2.18261 −0.436522
\(26\) −10.8532 −2.12848
\(27\) 0 0
\(28\) 1.36563 0.258079
\(29\) −8.22788 −1.52788 −0.763939 0.645288i \(-0.776737\pi\)
−0.763939 + 0.645288i \(0.776737\pi\)
\(30\) 0 0
\(31\) 1.04407 0.187520 0.0937602 0.995595i \(-0.470111\pi\)
0.0937602 + 0.995595i \(0.470111\pi\)
\(32\) 2.72118 0.481041
\(33\) 0 0
\(34\) −1.54139 −0.264347
\(35\) 4.66505 0.788537
\(36\) 0 0
\(37\) −1.30834 −0.215091 −0.107545 0.994200i \(-0.534299\pi\)
−0.107545 + 0.994200i \(0.534299\pi\)
\(38\) −4.23847 −0.687571
\(39\) 0 0
\(40\) 3.99694 0.631971
\(41\) −4.84817 −0.757157 −0.378578 0.925569i \(-0.623587\pi\)
−0.378578 + 0.925569i \(0.623587\pi\)
\(42\) 0 0
\(43\) 9.84023 1.50062 0.750310 0.661086i \(-0.229904\pi\)
0.750310 + 0.661086i \(0.229904\pi\)
\(44\) −2.03974 −0.307503
\(45\) 0 0
\(46\) −2.54623 −0.375422
\(47\) 12.4977 1.82298 0.911488 0.411326i \(-0.134934\pi\)
0.911488 + 0.411326i \(0.134934\pi\)
\(48\) 0 0
\(49\) 0.724408 0.103487
\(50\) 3.44504 0.487203
\(51\) 0 0
\(52\) 3.37861 0.468529
\(53\) 7.34280 1.00861 0.504305 0.863525i \(-0.331749\pi\)
0.504305 + 0.863525i \(0.331749\pi\)
\(54\) 0 0
\(55\) −6.96786 −0.939546
\(56\) 6.61815 0.884387
\(57\) 0 0
\(58\) 12.9869 1.70527
\(59\) 9.05188 1.17845 0.589227 0.807968i \(-0.299432\pi\)
0.589227 + 0.807968i \(0.299432\pi\)
\(60\) 0 0
\(61\) −1.28574 −0.164622 −0.0823112 0.996607i \(-0.526230\pi\)
−0.0823112 + 0.996607i \(0.526230\pi\)
\(62\) −1.64796 −0.209292
\(63\) 0 0
\(64\) 5.18745 0.648432
\(65\) 11.5415 1.43155
\(66\) 0 0
\(67\) −4.64630 −0.567636 −0.283818 0.958878i \(-0.591601\pi\)
−0.283818 + 0.958878i \(0.591601\pi\)
\(68\) 0.479838 0.0581889
\(69\) 0 0
\(70\) −7.36333 −0.880086
\(71\) 5.62373 0.667414 0.333707 0.942677i \(-0.391700\pi\)
0.333707 + 0.942677i \(0.391700\pi\)
\(72\) 0 0
\(73\) −4.56144 −0.533877 −0.266938 0.963714i \(-0.586012\pi\)
−0.266938 + 0.963714i \(0.586012\pi\)
\(74\) 2.06510 0.240063
\(75\) 0 0
\(76\) 1.31944 0.151351
\(77\) −11.5374 −1.31481
\(78\) 0 0
\(79\) 4.65680 0.523931 0.261966 0.965077i \(-0.415629\pi\)
0.261966 + 0.965077i \(0.415629\pi\)
\(80\) −7.95828 −0.889763
\(81\) 0 0
\(82\) 7.65237 0.845063
\(83\) 5.76439 0.632724 0.316362 0.948639i \(-0.397539\pi\)
0.316362 + 0.948639i \(0.397539\pi\)
\(84\) 0 0
\(85\) 1.63915 0.177791
\(86\) −15.5319 −1.67484
\(87\) 0 0
\(88\) −9.88507 −1.05375
\(89\) 4.54442 0.481707 0.240854 0.970561i \(-0.422573\pi\)
0.240854 + 0.970561i \(0.422573\pi\)
\(90\) 0 0
\(91\) 19.1105 2.00332
\(92\) 0.792647 0.0826391
\(93\) 0 0
\(94\) −19.7264 −2.03463
\(95\) 4.50728 0.462437
\(96\) 0 0
\(97\) 8.57544 0.870704 0.435352 0.900260i \(-0.356624\pi\)
0.435352 + 0.900260i \(0.356624\pi\)
\(98\) −1.14341 −0.115502
\(99\) 0 0
\(100\) −1.07245 −0.107245
\(101\) −7.80570 −0.776696 −0.388348 0.921513i \(-0.626954\pi\)
−0.388348 + 0.921513i \(0.626954\pi\)
\(102\) 0 0
\(103\) −2.16615 −0.213437 −0.106718 0.994289i \(-0.534034\pi\)
−0.106718 + 0.994289i \(0.534034\pi\)
\(104\) 16.3735 1.60556
\(105\) 0 0
\(106\) −11.5899 −1.12571
\(107\) 12.5849 1.21663 0.608317 0.793695i \(-0.291845\pi\)
0.608317 + 0.793695i \(0.291845\pi\)
\(108\) 0 0
\(109\) −12.2140 −1.16989 −0.584945 0.811073i \(-0.698884\pi\)
−0.584945 + 0.811073i \(0.698884\pi\)
\(110\) 10.9981 1.04863
\(111\) 0 0
\(112\) −13.1774 −1.24514
\(113\) 0.450833 0.0424108 0.0212054 0.999775i \(-0.493250\pi\)
0.0212054 + 0.999775i \(0.493250\pi\)
\(114\) 0 0
\(115\) 2.70772 0.252496
\(116\) −4.04285 −0.375369
\(117\) 0 0
\(118\) −14.2875 −1.31527
\(119\) 2.71411 0.248802
\(120\) 0 0
\(121\) 6.23265 0.566604
\(122\) 2.02942 0.183735
\(123\) 0 0
\(124\) 0.513014 0.0460700
\(125\) −12.0561 −1.07833
\(126\) 0 0
\(127\) −0.531069 −0.0471247 −0.0235624 0.999722i \(-0.507501\pi\)
−0.0235624 + 0.999722i \(0.507501\pi\)
\(128\) −13.6303 −1.20476
\(129\) 0 0
\(130\) −18.2171 −1.59775
\(131\) −11.4160 −0.997424 −0.498712 0.866768i \(-0.666193\pi\)
−0.498712 + 0.866768i \(0.666193\pi\)
\(132\) 0 0
\(133\) 7.46318 0.647139
\(134\) 7.33374 0.633539
\(135\) 0 0
\(136\) 2.32541 0.199402
\(137\) −4.22924 −0.361329 −0.180664 0.983545i \(-0.557825\pi\)
−0.180664 + 0.983545i \(0.557825\pi\)
\(138\) 0 0
\(139\) −11.1555 −0.946200 −0.473100 0.881009i \(-0.656865\pi\)
−0.473100 + 0.881009i \(0.656865\pi\)
\(140\) 2.29222 0.193728
\(141\) 0 0
\(142\) −8.87653 −0.744902
\(143\) −28.5440 −2.38697
\(144\) 0 0
\(145\) −13.8106 −1.14690
\(146\) 7.19980 0.595860
\(147\) 0 0
\(148\) −0.642868 −0.0528434
\(149\) 19.4777 1.59567 0.797837 0.602873i \(-0.205977\pi\)
0.797837 + 0.602873i \(0.205977\pi\)
\(150\) 0 0
\(151\) −1.24286 −0.101143 −0.0505713 0.998720i \(-0.516104\pi\)
−0.0505713 + 0.998720i \(0.516104\pi\)
\(152\) 6.39433 0.518649
\(153\) 0 0
\(154\) 18.2107 1.46746
\(155\) 1.75248 0.140762
\(156\) 0 0
\(157\) −3.53314 −0.281975 −0.140988 0.990011i \(-0.545028\pi\)
−0.140988 + 0.990011i \(0.545028\pi\)
\(158\) −7.35031 −0.584760
\(159\) 0 0
\(160\) 4.56752 0.361094
\(161\) 4.48345 0.353346
\(162\) 0 0
\(163\) 15.9509 1.24937 0.624685 0.780877i \(-0.285228\pi\)
0.624685 + 0.780877i \(0.285228\pi\)
\(164\) −2.38220 −0.186018
\(165\) 0 0
\(166\) −9.09854 −0.706184
\(167\) −14.4846 −1.12086 −0.560428 0.828203i \(-0.689363\pi\)
−0.560428 + 0.828203i \(0.689363\pi\)
\(168\) 0 0
\(169\) 34.2800 2.63692
\(170\) −2.58724 −0.198432
\(171\) 0 0
\(172\) 4.83509 0.368672
\(173\) −12.6174 −0.959283 −0.479641 0.877465i \(-0.659233\pi\)
−0.479641 + 0.877465i \(0.659233\pi\)
\(174\) 0 0
\(175\) −6.06609 −0.458554
\(176\) 19.6821 1.48360
\(177\) 0 0
\(178\) −7.17293 −0.537634
\(179\) −0.295899 −0.0221165 −0.0110582 0.999939i \(-0.503520\pi\)
−0.0110582 + 0.999939i \(0.503520\pi\)
\(180\) 0 0
\(181\) 1.42050 0.105585 0.0527925 0.998606i \(-0.483188\pi\)
0.0527925 + 0.998606i \(0.483188\pi\)
\(182\) −30.1640 −2.23591
\(183\) 0 0
\(184\) 3.84135 0.283188
\(185\) −2.19607 −0.161458
\(186\) 0 0
\(187\) −4.05388 −0.296449
\(188\) 6.14087 0.447869
\(189\) 0 0
\(190\) −7.11431 −0.516126
\(191\) −20.6241 −1.49231 −0.746153 0.665774i \(-0.768101\pi\)
−0.746153 + 0.665774i \(0.768101\pi\)
\(192\) 0 0
\(193\) −20.9559 −1.50844 −0.754221 0.656621i \(-0.771985\pi\)
−0.754221 + 0.656621i \(0.771985\pi\)
\(194\) −13.5355 −0.971793
\(195\) 0 0
\(196\) 0.355945 0.0254246
\(197\) 9.59621 0.683702 0.341851 0.939754i \(-0.388946\pi\)
0.341851 + 0.939754i \(0.388946\pi\)
\(198\) 0 0
\(199\) −10.6917 −0.757912 −0.378956 0.925415i \(-0.623717\pi\)
−0.378956 + 0.925415i \(0.623717\pi\)
\(200\) −5.19733 −0.367507
\(201\) 0 0
\(202\) 12.3206 0.866871
\(203\) −22.8676 −1.60499
\(204\) 0 0
\(205\) −8.13769 −0.568361
\(206\) 3.41906 0.238217
\(207\) 0 0
\(208\) −32.6013 −2.26049
\(209\) −11.1472 −0.771070
\(210\) 0 0
\(211\) −15.0502 −1.03610 −0.518051 0.855350i \(-0.673342\pi\)
−0.518051 + 0.855350i \(0.673342\pi\)
\(212\) 3.60796 0.247795
\(213\) 0 0
\(214\) −19.8641 −1.35788
\(215\) 16.5169 1.12644
\(216\) 0 0
\(217\) 2.90176 0.196984
\(218\) 19.2786 1.30571
\(219\) 0 0
\(220\) −3.42373 −0.230828
\(221\) 6.71481 0.451687
\(222\) 0 0
\(223\) −12.2881 −0.822873 −0.411436 0.911439i \(-0.634973\pi\)
−0.411436 + 0.911439i \(0.634973\pi\)
\(224\) 7.56292 0.505319
\(225\) 0 0
\(226\) −0.711597 −0.0473347
\(227\) −3.75256 −0.249066 −0.124533 0.992215i \(-0.539743\pi\)
−0.124533 + 0.992215i \(0.539743\pi\)
\(228\) 0 0
\(229\) −18.6024 −1.22928 −0.614641 0.788807i \(-0.710699\pi\)
−0.614641 + 0.788807i \(0.710699\pi\)
\(230\) −4.27387 −0.281811
\(231\) 0 0
\(232\) −19.5926 −1.28632
\(233\) −0.545784 −0.0357555 −0.0178777 0.999840i \(-0.505691\pi\)
−0.0178777 + 0.999840i \(0.505691\pi\)
\(234\) 0 0
\(235\) 20.9775 1.36842
\(236\) 4.44773 0.289522
\(237\) 0 0
\(238\) −4.28396 −0.277688
\(239\) 20.0947 1.29982 0.649911 0.760011i \(-0.274806\pi\)
0.649911 + 0.760011i \(0.274806\pi\)
\(240\) 0 0
\(241\) 15.2913 0.984999 0.492500 0.870313i \(-0.336083\pi\)
0.492500 + 0.870313i \(0.336083\pi\)
\(242\) −9.83763 −0.632387
\(243\) 0 0
\(244\) −0.631762 −0.0404444
\(245\) 1.21592 0.0776825
\(246\) 0 0
\(247\) 18.4642 1.17485
\(248\) 2.48618 0.157873
\(249\) 0 0
\(250\) 19.0294 1.20352
\(251\) −12.7563 −0.805171 −0.402586 0.915382i \(-0.631888\pi\)
−0.402586 + 0.915382i \(0.631888\pi\)
\(252\) 0 0
\(253\) −6.69663 −0.421013
\(254\) 0.838241 0.0525959
\(255\) 0 0
\(256\) 11.1391 0.696196
\(257\) 13.1047 0.817449 0.408725 0.912658i \(-0.365974\pi\)
0.408725 + 0.912658i \(0.365974\pi\)
\(258\) 0 0
\(259\) −3.63626 −0.225946
\(260\) 5.67103 0.351702
\(261\) 0 0
\(262\) 18.0191 1.11323
\(263\) 9.51418 0.586669 0.293335 0.956010i \(-0.405235\pi\)
0.293335 + 0.956010i \(0.405235\pi\)
\(264\) 0 0
\(265\) 12.3249 0.757115
\(266\) −11.7799 −0.722272
\(267\) 0 0
\(268\) −2.28301 −0.139457
\(269\) −22.1408 −1.34995 −0.674973 0.737842i \(-0.735845\pi\)
−0.674973 + 0.737842i \(0.735845\pi\)
\(270\) 0 0
\(271\) 27.9627 1.69861 0.849307 0.527899i \(-0.177020\pi\)
0.849307 + 0.527899i \(0.177020\pi\)
\(272\) −4.63011 −0.280742
\(273\) 0 0
\(274\) 6.67545 0.403279
\(275\) 9.06051 0.546369
\(276\) 0 0
\(277\) −18.8837 −1.13461 −0.567305 0.823508i \(-0.692014\pi\)
−0.567305 + 0.823508i \(0.692014\pi\)
\(278\) 17.6079 1.05605
\(279\) 0 0
\(280\) 11.1086 0.663867
\(281\) 20.0017 1.19320 0.596602 0.802537i \(-0.296517\pi\)
0.596602 + 0.802537i \(0.296517\pi\)
\(282\) 0 0
\(283\) 16.7450 0.995389 0.497695 0.867352i \(-0.334180\pi\)
0.497695 + 0.867352i \(0.334180\pi\)
\(284\) 2.76328 0.163970
\(285\) 0 0
\(286\) 45.0540 2.66410
\(287\) −13.4744 −0.795371
\(288\) 0 0
\(289\) −16.0463 −0.943903
\(290\) 21.7986 1.28006
\(291\) 0 0
\(292\) −2.24131 −0.131163
\(293\) −19.5720 −1.14341 −0.571704 0.820460i \(-0.693718\pi\)
−0.571704 + 0.820460i \(0.693718\pi\)
\(294\) 0 0
\(295\) 15.1936 0.884608
\(296\) −3.11549 −0.181084
\(297\) 0 0
\(298\) −30.7437 −1.78093
\(299\) 11.0922 0.641480
\(300\) 0 0
\(301\) 27.3488 1.57636
\(302\) 1.96174 0.112885
\(303\) 0 0
\(304\) −12.7317 −0.730214
\(305\) −2.15813 −0.123574
\(306\) 0 0
\(307\) 14.8995 0.850357 0.425179 0.905109i \(-0.360211\pi\)
0.425179 + 0.905109i \(0.360211\pi\)
\(308\) −5.66902 −0.323023
\(309\) 0 0
\(310\) −2.76612 −0.157105
\(311\) −4.59236 −0.260409 −0.130204 0.991487i \(-0.541563\pi\)
−0.130204 + 0.991487i \(0.541563\pi\)
\(312\) 0 0
\(313\) 11.8687 0.670858 0.335429 0.942066i \(-0.391119\pi\)
0.335429 + 0.942066i \(0.391119\pi\)
\(314\) 5.57672 0.314713
\(315\) 0 0
\(316\) 2.28817 0.128719
\(317\) 14.5173 0.815375 0.407688 0.913121i \(-0.366335\pi\)
0.407688 + 0.913121i \(0.366335\pi\)
\(318\) 0 0
\(319\) 34.1557 1.91235
\(320\) 8.70718 0.486746
\(321\) 0 0
\(322\) −7.07670 −0.394369
\(323\) 2.62232 0.145910
\(324\) 0 0
\(325\) −15.0077 −0.832480
\(326\) −25.1769 −1.39442
\(327\) 0 0
\(328\) −11.5447 −0.637448
\(329\) 34.7346 1.91498
\(330\) 0 0
\(331\) −8.10570 −0.445530 −0.222765 0.974872i \(-0.571508\pi\)
−0.222765 + 0.974872i \(0.571508\pi\)
\(332\) 2.83239 0.155448
\(333\) 0 0
\(334\) 22.8626 1.25099
\(335\) −7.79885 −0.426097
\(336\) 0 0
\(337\) −18.8080 −1.02453 −0.512267 0.858826i \(-0.671194\pi\)
−0.512267 + 0.858826i \(0.671194\pi\)
\(338\) −54.1077 −2.94307
\(339\) 0 0
\(340\) 0.805412 0.0436796
\(341\) −4.33416 −0.234708
\(342\) 0 0
\(343\) −17.4416 −0.941760
\(344\) 23.4320 1.26337
\(345\) 0 0
\(346\) 19.9154 1.07066
\(347\) 17.6176 0.945764 0.472882 0.881126i \(-0.343214\pi\)
0.472882 + 0.881126i \(0.343214\pi\)
\(348\) 0 0
\(349\) −16.9893 −0.909416 −0.454708 0.890641i \(-0.650256\pi\)
−0.454708 + 0.890641i \(0.650256\pi\)
\(350\) 9.57475 0.511792
\(351\) 0 0
\(352\) −11.2962 −0.602090
\(353\) 15.1556 0.806651 0.403326 0.915057i \(-0.367854\pi\)
0.403326 + 0.915057i \(0.367854\pi\)
\(354\) 0 0
\(355\) 9.43948 0.500996
\(356\) 2.23294 0.118346
\(357\) 0 0
\(358\) 0.467048 0.0246842
\(359\) −2.45096 −0.129357 −0.0646783 0.997906i \(-0.520602\pi\)
−0.0646783 + 0.997906i \(0.520602\pi\)
\(360\) 0 0
\(361\) −11.7892 −0.620485
\(362\) −2.24213 −0.117844
\(363\) 0 0
\(364\) 9.39012 0.492176
\(365\) −7.65642 −0.400755
\(366\) 0 0
\(367\) −1.31353 −0.0685659 −0.0342829 0.999412i \(-0.510915\pi\)
−0.0342829 + 0.999412i \(0.510915\pi\)
\(368\) −7.64850 −0.398706
\(369\) 0 0
\(370\) 3.46628 0.180203
\(371\) 20.4077 1.05952
\(372\) 0 0
\(373\) −9.62617 −0.498424 −0.249212 0.968449i \(-0.580172\pi\)
−0.249212 + 0.968449i \(0.580172\pi\)
\(374\) 6.39866 0.330867
\(375\) 0 0
\(376\) 29.7601 1.53476
\(377\) −56.5753 −2.91377
\(378\) 0 0
\(379\) −8.56311 −0.439857 −0.219929 0.975516i \(-0.570582\pi\)
−0.219929 + 0.975516i \(0.570582\pi\)
\(380\) 2.21470 0.113611
\(381\) 0 0
\(382\) 32.5531 1.66556
\(383\) −33.6346 −1.71865 −0.859324 0.511431i \(-0.829116\pi\)
−0.859324 + 0.511431i \(0.829116\pi\)
\(384\) 0 0
\(385\) −19.3656 −0.986964
\(386\) 33.0769 1.68357
\(387\) 0 0
\(388\) 4.21363 0.213915
\(389\) 14.9540 0.758197 0.379098 0.925356i \(-0.376234\pi\)
0.379098 + 0.925356i \(0.376234\pi\)
\(390\) 0 0
\(391\) 1.57534 0.0796685
\(392\) 1.72499 0.0871252
\(393\) 0 0
\(394\) −15.1467 −0.763080
\(395\) 7.81648 0.393290
\(396\) 0 0
\(397\) −16.7788 −0.842102 −0.421051 0.907037i \(-0.638339\pi\)
−0.421051 + 0.907037i \(0.638339\pi\)
\(398\) 16.8758 0.845906
\(399\) 0 0
\(400\) 10.3484 0.517419
\(401\) −13.1263 −0.655496 −0.327748 0.944765i \(-0.606290\pi\)
−0.327748 + 0.944765i \(0.606290\pi\)
\(402\) 0 0
\(403\) 7.17907 0.357615
\(404\) −3.83541 −0.190819
\(405\) 0 0
\(406\) 36.0943 1.79133
\(407\) 5.43123 0.269216
\(408\) 0 0
\(409\) 25.4505 1.25845 0.629223 0.777225i \(-0.283373\pi\)
0.629223 + 0.777225i \(0.283373\pi\)
\(410\) 12.8446 0.634348
\(411\) 0 0
\(412\) −1.06436 −0.0524371
\(413\) 25.1577 1.23793
\(414\) 0 0
\(415\) 9.67558 0.474955
\(416\) 18.7109 0.917379
\(417\) 0 0
\(418\) 17.5948 0.860592
\(419\) 12.6701 0.618973 0.309486 0.950904i \(-0.399843\pi\)
0.309486 + 0.950904i \(0.399843\pi\)
\(420\) 0 0
\(421\) −17.6772 −0.861536 −0.430768 0.902463i \(-0.641757\pi\)
−0.430768 + 0.902463i \(0.641757\pi\)
\(422\) 23.7554 1.15639
\(423\) 0 0
\(424\) 17.4850 0.849146
\(425\) −2.13143 −0.103390
\(426\) 0 0
\(427\) −3.57344 −0.172931
\(428\) 6.18374 0.298902
\(429\) 0 0
\(430\) −26.0703 −1.25722
\(431\) −15.6974 −0.756117 −0.378059 0.925782i \(-0.623408\pi\)
−0.378059 + 0.925782i \(0.623408\pi\)
\(432\) 0 0
\(433\) −12.6258 −0.606759 −0.303380 0.952870i \(-0.598115\pi\)
−0.303380 + 0.952870i \(0.598115\pi\)
\(434\) −4.58015 −0.219854
\(435\) 0 0
\(436\) −6.00147 −0.287418
\(437\) 4.33183 0.207219
\(438\) 0 0
\(439\) 26.9369 1.28563 0.642814 0.766022i \(-0.277767\pi\)
0.642814 + 0.766022i \(0.277767\pi\)
\(440\) −16.5922 −0.791001
\(441\) 0 0
\(442\) −10.5987 −0.504128
\(443\) −34.7279 −1.64997 −0.824986 0.565153i \(-0.808817\pi\)
−0.824986 + 0.565153i \(0.808817\pi\)
\(444\) 0 0
\(445\) 7.62784 0.361594
\(446\) 19.3956 0.918408
\(447\) 0 0
\(448\) 14.4174 0.681158
\(449\) 20.7461 0.979070 0.489535 0.871984i \(-0.337166\pi\)
0.489535 + 0.871984i \(0.337166\pi\)
\(450\) 0 0
\(451\) 20.1258 0.947688
\(452\) 0.221521 0.0104195
\(453\) 0 0
\(454\) 5.92306 0.277983
\(455\) 32.0771 1.50380
\(456\) 0 0
\(457\) −7.20773 −0.337164 −0.168582 0.985688i \(-0.553919\pi\)
−0.168582 + 0.985688i \(0.553919\pi\)
\(458\) 29.3621 1.37200
\(459\) 0 0
\(460\) 1.33046 0.0620332
\(461\) −23.1268 −1.07712 −0.538562 0.842586i \(-0.681032\pi\)
−0.538562 + 0.842586i \(0.681032\pi\)
\(462\) 0 0
\(463\) 4.97584 0.231247 0.115623 0.993293i \(-0.463113\pi\)
0.115623 + 0.993293i \(0.463113\pi\)
\(464\) 39.0107 1.81103
\(465\) 0 0
\(466\) 0.861467 0.0399067
\(467\) 12.4814 0.577569 0.288784 0.957394i \(-0.406749\pi\)
0.288784 + 0.957394i \(0.406749\pi\)
\(468\) 0 0
\(469\) −12.9134 −0.596284
\(470\) −33.1110 −1.52729
\(471\) 0 0
\(472\) 21.5547 0.992137
\(473\) −40.8490 −1.87824
\(474\) 0 0
\(475\) −5.86094 −0.268919
\(476\) 1.33361 0.0611257
\(477\) 0 0
\(478\) −31.7176 −1.45073
\(479\) −28.4713 −1.30089 −0.650443 0.759555i \(-0.725417\pi\)
−0.650443 + 0.759555i \(0.725417\pi\)
\(480\) 0 0
\(481\) −8.99624 −0.410193
\(482\) −24.1359 −1.09936
\(483\) 0 0
\(484\) 3.06247 0.139203
\(485\) 14.3939 0.653595
\(486\) 0 0
\(487\) 29.6841 1.34511 0.672557 0.740045i \(-0.265196\pi\)
0.672557 + 0.740045i \(0.265196\pi\)
\(488\) −3.06166 −0.138595
\(489\) 0 0
\(490\) −1.91922 −0.0867015
\(491\) −12.4050 −0.559828 −0.279914 0.960025i \(-0.590306\pi\)
−0.279914 + 0.960025i \(0.590306\pi\)
\(492\) 0 0
\(493\) −8.03494 −0.361876
\(494\) −29.1439 −1.31125
\(495\) 0 0
\(496\) −4.95023 −0.222272
\(497\) 15.6299 0.701099
\(498\) 0 0
\(499\) −2.27982 −0.102059 −0.0510294 0.998697i \(-0.516250\pi\)
−0.0510294 + 0.998697i \(0.516250\pi\)
\(500\) −5.92387 −0.264923
\(501\) 0 0
\(502\) 20.1346 0.898652
\(503\) −41.2812 −1.84064 −0.920320 0.391167i \(-0.872071\pi\)
−0.920320 + 0.391167i \(0.872071\pi\)
\(504\) 0 0
\(505\) −13.1019 −0.583028
\(506\) 10.5700 0.469893
\(507\) 0 0
\(508\) −0.260946 −0.0115776
\(509\) −16.4242 −0.727988 −0.363994 0.931401i \(-0.618587\pi\)
−0.363994 + 0.931401i \(0.618587\pi\)
\(510\) 0 0
\(511\) −12.6775 −0.560821
\(512\) 9.67844 0.427731
\(513\) 0 0
\(514\) −20.6845 −0.912355
\(515\) −3.63589 −0.160217
\(516\) 0 0
\(517\) −51.8807 −2.28171
\(518\) 5.73949 0.252179
\(519\) 0 0
\(520\) 27.4831 1.20521
\(521\) 9.29672 0.407297 0.203648 0.979044i \(-0.434720\pi\)
0.203648 + 0.979044i \(0.434720\pi\)
\(522\) 0 0
\(523\) −22.7471 −0.994662 −0.497331 0.867561i \(-0.665686\pi\)
−0.497331 + 0.867561i \(0.665686\pi\)
\(524\) −5.60938 −0.245047
\(525\) 0 0
\(526\) −15.0172 −0.654782
\(527\) 1.01959 0.0444139
\(528\) 0 0
\(529\) −20.3977 −0.886856
\(530\) −19.4537 −0.845016
\(531\) 0 0
\(532\) 3.66710 0.158989
\(533\) −33.3362 −1.44395
\(534\) 0 0
\(535\) 21.1239 0.913267
\(536\) −11.0640 −0.477891
\(537\) 0 0
\(538\) 34.9471 1.50668
\(539\) −3.00718 −0.129528
\(540\) 0 0
\(541\) 2.38959 0.102737 0.0513683 0.998680i \(-0.483642\pi\)
0.0513683 + 0.998680i \(0.483642\pi\)
\(542\) −44.1365 −1.89582
\(543\) 0 0
\(544\) 2.65737 0.113934
\(545\) −20.5013 −0.878179
\(546\) 0 0
\(547\) 29.6668 1.26846 0.634230 0.773144i \(-0.281317\pi\)
0.634230 + 0.773144i \(0.281317\pi\)
\(548\) −2.07808 −0.0887712
\(549\) 0 0
\(550\) −14.3011 −0.609803
\(551\) −22.0942 −0.941246
\(552\) 0 0
\(553\) 12.9426 0.550374
\(554\) 29.8061 1.26634
\(555\) 0 0
\(556\) −5.48138 −0.232462
\(557\) −8.41413 −0.356518 −0.178259 0.983984i \(-0.557046\pi\)
−0.178259 + 0.983984i \(0.557046\pi\)
\(558\) 0 0
\(559\) 67.6619 2.86179
\(560\) −22.1183 −0.934669
\(561\) 0 0
\(562\) −31.5708 −1.33174
\(563\) 27.2265 1.14746 0.573730 0.819044i \(-0.305496\pi\)
0.573730 + 0.819044i \(0.305496\pi\)
\(564\) 0 0
\(565\) 0.756727 0.0318357
\(566\) −26.4304 −1.11095
\(567\) 0 0
\(568\) 13.3915 0.561894
\(569\) 21.9493 0.920163 0.460082 0.887877i \(-0.347820\pi\)
0.460082 + 0.887877i \(0.347820\pi\)
\(570\) 0 0
\(571\) −44.8535 −1.87706 −0.938530 0.345199i \(-0.887811\pi\)
−0.938530 + 0.345199i \(0.887811\pi\)
\(572\) −14.0254 −0.586430
\(573\) 0 0
\(574\) 21.2681 0.887713
\(575\) −3.52092 −0.146833
\(576\) 0 0
\(577\) 12.0191 0.500361 0.250181 0.968199i \(-0.419510\pi\)
0.250181 + 0.968199i \(0.419510\pi\)
\(578\) 25.3276 1.05349
\(579\) 0 0
\(580\) −6.78595 −0.281771
\(581\) 16.0209 0.664658
\(582\) 0 0
\(583\) −30.4816 −1.26242
\(584\) −10.8619 −0.449469
\(585\) 0 0
\(586\) 30.8925 1.27616
\(587\) 17.0299 0.702898 0.351449 0.936207i \(-0.385689\pi\)
0.351449 + 0.936207i \(0.385689\pi\)
\(588\) 0 0
\(589\) 2.80363 0.115521
\(590\) −23.9817 −0.987311
\(591\) 0 0
\(592\) 6.20324 0.254951
\(593\) 14.9284 0.613037 0.306519 0.951865i \(-0.400836\pi\)
0.306519 + 0.951865i \(0.400836\pi\)
\(594\) 0 0
\(595\) 4.55566 0.186764
\(596\) 9.57056 0.392025
\(597\) 0 0
\(598\) −17.5080 −0.715956
\(599\) −13.5348 −0.553017 −0.276508 0.961011i \(-0.589177\pi\)
−0.276508 + 0.961011i \(0.589177\pi\)
\(600\) 0 0
\(601\) −45.3173 −1.84853 −0.924265 0.381752i \(-0.875321\pi\)
−0.924265 + 0.381752i \(0.875321\pi\)
\(602\) −43.1674 −1.75937
\(603\) 0 0
\(604\) −0.610691 −0.0248487
\(605\) 10.4615 0.425322
\(606\) 0 0
\(607\) 24.4368 0.991859 0.495929 0.868363i \(-0.334827\pi\)
0.495929 + 0.868363i \(0.334827\pi\)
\(608\) 7.30715 0.296344
\(609\) 0 0
\(610\) 3.40640 0.137921
\(611\) 85.9348 3.47655
\(612\) 0 0
\(613\) −25.1996 −1.01780 −0.508901 0.860825i \(-0.669948\pi\)
−0.508901 + 0.860825i \(0.669948\pi\)
\(614\) −23.5174 −0.949084
\(615\) 0 0
\(616\) −27.4734 −1.10693
\(617\) −4.04185 −0.162719 −0.0813594 0.996685i \(-0.525926\pi\)
−0.0813594 + 0.996685i \(0.525926\pi\)
\(618\) 0 0
\(619\) −44.7700 −1.79946 −0.899729 0.436449i \(-0.856236\pi\)
−0.899729 + 0.436449i \(0.856236\pi\)
\(620\) 0.861097 0.0345825
\(621\) 0 0
\(622\) 7.24860 0.290642
\(623\) 12.6302 0.506019
\(624\) 0 0
\(625\) −9.32314 −0.372926
\(626\) −18.7336 −0.748745
\(627\) 0 0
\(628\) −1.73604 −0.0692757
\(629\) −1.27767 −0.0509439
\(630\) 0 0
\(631\) 31.4116 1.25048 0.625238 0.780434i \(-0.285002\pi\)
0.625238 + 0.780434i \(0.285002\pi\)
\(632\) 11.0890 0.441096
\(633\) 0 0
\(634\) −22.9142 −0.910041
\(635\) −0.891403 −0.0353742
\(636\) 0 0
\(637\) 4.98106 0.197357
\(638\) −53.9116 −2.13438
\(639\) 0 0
\(640\) −22.8785 −0.904351
\(641\) −48.6406 −1.92119 −0.960594 0.277954i \(-0.910344\pi\)
−0.960594 + 0.277954i \(0.910344\pi\)
\(642\) 0 0
\(643\) −28.0324 −1.10549 −0.552744 0.833351i \(-0.686419\pi\)
−0.552744 + 0.833351i \(0.686419\pi\)
\(644\) 2.20299 0.0868099
\(645\) 0 0
\(646\) −4.13908 −0.162850
\(647\) 37.5519 1.47632 0.738159 0.674627i \(-0.235696\pi\)
0.738159 + 0.674627i \(0.235696\pi\)
\(648\) 0 0
\(649\) −37.5763 −1.47500
\(650\) 23.6883 0.929131
\(651\) 0 0
\(652\) 7.83762 0.306945
\(653\) 5.11222 0.200057 0.100028 0.994985i \(-0.468107\pi\)
0.100028 + 0.994985i \(0.468107\pi\)
\(654\) 0 0
\(655\) −19.1619 −0.748718
\(656\) 22.9866 0.897474
\(657\) 0 0
\(658\) −54.8253 −2.13731
\(659\) −35.0059 −1.36364 −0.681818 0.731522i \(-0.738811\pi\)
−0.681818 + 0.731522i \(0.738811\pi\)
\(660\) 0 0
\(661\) 1.33346 0.0518656 0.0259328 0.999664i \(-0.491744\pi\)
0.0259328 + 0.999664i \(0.491744\pi\)
\(662\) 12.7941 0.497256
\(663\) 0 0
\(664\) 13.7264 0.532689
\(665\) 12.5270 0.485776
\(666\) 0 0
\(667\) −13.2730 −0.513931
\(668\) −7.11717 −0.275372
\(669\) 0 0
\(670\) 12.3097 0.475567
\(671\) 5.33740 0.206048
\(672\) 0 0
\(673\) −3.57953 −0.137981 −0.0689904 0.997617i \(-0.521978\pi\)
−0.0689904 + 0.997617i \(0.521978\pi\)
\(674\) 29.6866 1.14348
\(675\) 0 0
\(676\) 16.8438 0.647839
\(677\) 36.9389 1.41968 0.709839 0.704364i \(-0.248768\pi\)
0.709839 + 0.704364i \(0.248768\pi\)
\(678\) 0 0
\(679\) 23.8336 0.914648
\(680\) 3.90321 0.149681
\(681\) 0 0
\(682\) 6.84106 0.261958
\(683\) −38.0166 −1.45466 −0.727332 0.686286i \(-0.759240\pi\)
−0.727332 + 0.686286i \(0.759240\pi\)
\(684\) 0 0
\(685\) −7.09881 −0.271232
\(686\) 27.5300 1.05110
\(687\) 0 0
\(688\) −46.6553 −1.77872
\(689\) 50.4894 1.92349
\(690\) 0 0
\(691\) 0.550464 0.0209406 0.0104703 0.999945i \(-0.496667\pi\)
0.0104703 + 0.999945i \(0.496667\pi\)
\(692\) −6.19968 −0.235677
\(693\) 0 0
\(694\) −27.8077 −1.05557
\(695\) −18.7246 −0.710266
\(696\) 0 0
\(697\) −4.73449 −0.179331
\(698\) 26.8160 1.01500
\(699\) 0 0
\(700\) −2.98064 −0.112657
\(701\) −19.0242 −0.718534 −0.359267 0.933235i \(-0.616973\pi\)
−0.359267 + 0.933235i \(0.616973\pi\)
\(702\) 0 0
\(703\) −3.51328 −0.132506
\(704\) −21.5343 −0.811603
\(705\) 0 0
\(706\) −23.9217 −0.900304
\(707\) −21.6942 −0.815896
\(708\) 0 0
\(709\) 12.1568 0.456558 0.228279 0.973596i \(-0.426690\pi\)
0.228279 + 0.973596i \(0.426690\pi\)
\(710\) −14.8993 −0.559161
\(711\) 0 0
\(712\) 10.8214 0.405548
\(713\) 1.68426 0.0630761
\(714\) 0 0
\(715\) −47.9113 −1.79178
\(716\) −0.145393 −0.00543358
\(717\) 0 0
\(718\) 3.86860 0.144375
\(719\) −9.77667 −0.364608 −0.182304 0.983242i \(-0.558356\pi\)
−0.182304 + 0.983242i \(0.558356\pi\)
\(720\) 0 0
\(721\) −6.02033 −0.224209
\(722\) 18.6082 0.692524
\(723\) 0 0
\(724\) 0.697978 0.0259401
\(725\) 17.9583 0.666953
\(726\) 0 0
\(727\) 4.33493 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(728\) 45.5067 1.68659
\(729\) 0 0
\(730\) 12.0849 0.447283
\(731\) 9.60949 0.355420
\(732\) 0 0
\(733\) 29.1937 1.07829 0.539146 0.842212i \(-0.318747\pi\)
0.539146 + 0.842212i \(0.318747\pi\)
\(734\) 2.07329 0.0765264
\(735\) 0 0
\(736\) 4.38972 0.161807
\(737\) 19.2878 0.710476
\(738\) 0 0
\(739\) 41.5553 1.52864 0.764319 0.644838i \(-0.223075\pi\)
0.764319 + 0.644838i \(0.223075\pi\)
\(740\) −1.07906 −0.0396670
\(741\) 0 0
\(742\) −32.2116 −1.18253
\(743\) −21.3774 −0.784259 −0.392130 0.919910i \(-0.628262\pi\)
−0.392130 + 0.919910i \(0.628262\pi\)
\(744\) 0 0
\(745\) 32.6935 1.19780
\(746\) 15.1940 0.556291
\(747\) 0 0
\(748\) −1.99191 −0.0728316
\(749\) 34.9771 1.27804
\(750\) 0 0
\(751\) −39.9676 −1.45844 −0.729220 0.684280i \(-0.760117\pi\)
−0.729220 + 0.684280i \(0.760117\pi\)
\(752\) −59.2552 −2.16081
\(753\) 0 0
\(754\) 89.2986 3.25206
\(755\) −2.08615 −0.0759228
\(756\) 0 0
\(757\) 6.68348 0.242915 0.121458 0.992597i \(-0.461243\pi\)
0.121458 + 0.992597i \(0.461243\pi\)
\(758\) 13.5161 0.490925
\(759\) 0 0
\(760\) 10.7329 0.389324
\(761\) −42.0250 −1.52341 −0.761703 0.647927i \(-0.775636\pi\)
−0.761703 + 0.647927i \(0.775636\pi\)
\(762\) 0 0
\(763\) −33.9461 −1.22893
\(764\) −10.1338 −0.366630
\(765\) 0 0
\(766\) 53.0890 1.91818
\(767\) 62.2411 2.24740
\(768\) 0 0
\(769\) 2.41323 0.0870233 0.0435117 0.999053i \(-0.486145\pi\)
0.0435117 + 0.999053i \(0.486145\pi\)
\(770\) 30.5668 1.10155
\(771\) 0 0
\(772\) −10.2969 −0.370594
\(773\) −1.39780 −0.0502754 −0.0251377 0.999684i \(-0.508002\pi\)
−0.0251377 + 0.999684i \(0.508002\pi\)
\(774\) 0 0
\(775\) −2.27880 −0.0818569
\(776\) 20.4202 0.733043
\(777\) 0 0
\(778\) −23.6034 −0.846224
\(779\) −13.0187 −0.466445
\(780\) 0 0
\(781\) −23.3454 −0.835363
\(782\) −2.48653 −0.0889181
\(783\) 0 0
\(784\) −3.43462 −0.122665
\(785\) −5.93040 −0.211665
\(786\) 0 0
\(787\) −39.6515 −1.41342 −0.706712 0.707501i \(-0.749822\pi\)
−0.706712 + 0.707501i \(0.749822\pi\)
\(788\) 4.71519 0.167972
\(789\) 0 0
\(790\) −12.3376 −0.438951
\(791\) 1.25299 0.0445513
\(792\) 0 0
\(793\) −8.84082 −0.313947
\(794\) 26.4837 0.939871
\(795\) 0 0
\(796\) −5.25345 −0.186204
\(797\) 3.29387 0.116675 0.0583374 0.998297i \(-0.481420\pi\)
0.0583374 + 0.998297i \(0.481420\pi\)
\(798\) 0 0
\(799\) 12.2046 0.431769
\(800\) −5.93927 −0.209985
\(801\) 0 0
\(802\) 20.7186 0.731599
\(803\) 18.9356 0.668222
\(804\) 0 0
\(805\) 7.52551 0.265239
\(806\) −11.3315 −0.399134
\(807\) 0 0
\(808\) −18.5873 −0.653898
\(809\) 6.54436 0.230087 0.115044 0.993360i \(-0.463299\pi\)
0.115044 + 0.993360i \(0.463299\pi\)
\(810\) 0 0
\(811\) −44.7516 −1.57144 −0.785721 0.618581i \(-0.787708\pi\)
−0.785721 + 0.618581i \(0.787708\pi\)
\(812\) −11.2362 −0.394314
\(813\) 0 0
\(814\) −8.57268 −0.300472
\(815\) 26.7737 0.937841
\(816\) 0 0
\(817\) 26.4239 0.924454
\(818\) −40.1712 −1.40455
\(819\) 0 0
\(820\) −3.99853 −0.139635
\(821\) 49.6840 1.73398 0.866991 0.498324i \(-0.166051\pi\)
0.866991 + 0.498324i \(0.166051\pi\)
\(822\) 0 0
\(823\) 10.6206 0.370211 0.185106 0.982719i \(-0.440737\pi\)
0.185106 + 0.982719i \(0.440737\pi\)
\(824\) −5.15812 −0.179692
\(825\) 0 0
\(826\) −39.7090 −1.38165
\(827\) 16.4008 0.570311 0.285156 0.958481i \(-0.407955\pi\)
0.285156 + 0.958481i \(0.407955\pi\)
\(828\) 0 0
\(829\) −2.95645 −0.102682 −0.0513409 0.998681i \(-0.516350\pi\)
−0.0513409 + 0.998681i \(0.516350\pi\)
\(830\) −15.2720 −0.530098
\(831\) 0 0
\(832\) 35.6692 1.23661
\(833\) 0.707421 0.0245107
\(834\) 0 0
\(835\) −24.3126 −0.841372
\(836\) −5.47730 −0.189436
\(837\) 0 0
\(838\) −19.9985 −0.690835
\(839\) 32.4464 1.12018 0.560088 0.828433i \(-0.310767\pi\)
0.560088 + 0.828433i \(0.310767\pi\)
\(840\) 0 0
\(841\) 38.6980 1.33441
\(842\) 27.9018 0.961561
\(843\) 0 0
\(844\) −7.39508 −0.254549
\(845\) 57.5392 1.97941
\(846\) 0 0
\(847\) 17.3223 0.595201
\(848\) −34.8143 −1.19553
\(849\) 0 0
\(850\) 3.36426 0.115393
\(851\) −2.11058 −0.0723498
\(852\) 0 0
\(853\) 28.3331 0.970107 0.485053 0.874485i \(-0.338800\pi\)
0.485053 + 0.874485i \(0.338800\pi\)
\(854\) 5.64033 0.193008
\(855\) 0 0
\(856\) 29.9678 1.02428
\(857\) −14.0035 −0.478349 −0.239175 0.970977i \(-0.576877\pi\)
−0.239175 + 0.970977i \(0.576877\pi\)
\(858\) 0 0
\(859\) −2.15977 −0.0736905 −0.0368452 0.999321i \(-0.511731\pi\)
−0.0368452 + 0.999321i \(0.511731\pi\)
\(860\) 8.11574 0.276744
\(861\) 0 0
\(862\) 24.7768 0.843903
\(863\) 14.9487 0.508859 0.254430 0.967091i \(-0.418112\pi\)
0.254430 + 0.967091i \(0.418112\pi\)
\(864\) 0 0
\(865\) −21.1784 −0.720087
\(866\) 19.9287 0.677204
\(867\) 0 0
\(868\) 1.42581 0.0483951
\(869\) −19.3314 −0.655773
\(870\) 0 0
\(871\) −31.9482 −1.08252
\(872\) −29.0845 −0.984926
\(873\) 0 0
\(874\) −6.83737 −0.231278
\(875\) −33.5072 −1.13275
\(876\) 0 0
\(877\) −1.88708 −0.0637221 −0.0318611 0.999492i \(-0.510143\pi\)
−0.0318611 + 0.999492i \(0.510143\pi\)
\(878\) −42.5173 −1.43489
\(879\) 0 0
\(880\) 33.0366 1.11366
\(881\) −46.8258 −1.57760 −0.788800 0.614649i \(-0.789297\pi\)
−0.788800 + 0.614649i \(0.789297\pi\)
\(882\) 0 0
\(883\) −30.0635 −1.01172 −0.505858 0.862617i \(-0.668824\pi\)
−0.505858 + 0.862617i \(0.668824\pi\)
\(884\) 3.29939 0.110970
\(885\) 0 0
\(886\) 54.8147 1.84153
\(887\) 29.9139 1.00441 0.502205 0.864749i \(-0.332522\pi\)
0.502205 + 0.864749i \(0.332522\pi\)
\(888\) 0 0
\(889\) −1.47599 −0.0495031
\(890\) −12.0398 −0.403576
\(891\) 0 0
\(892\) −6.03788 −0.202163
\(893\) 33.5599 1.12304
\(894\) 0 0
\(895\) −0.496668 −0.0166018
\(896\) −37.8823 −1.26556
\(897\) 0 0
\(898\) −32.7458 −1.09274
\(899\) −8.59047 −0.286508
\(900\) 0 0
\(901\) 7.17062 0.238888
\(902\) −31.7667 −1.05772
\(903\) 0 0
\(904\) 1.07354 0.0357055
\(905\) 2.38432 0.0792576
\(906\) 0 0
\(907\) 26.7652 0.888725 0.444363 0.895847i \(-0.353430\pi\)
0.444363 + 0.895847i \(0.353430\pi\)
\(908\) −1.84386 −0.0611905
\(909\) 0 0
\(910\) −50.6306 −1.67839
\(911\) −0.441137 −0.0146155 −0.00730776 0.999973i \(-0.502326\pi\)
−0.00730776 + 0.999973i \(0.502326\pi\)
\(912\) 0 0
\(913\) −23.9293 −0.791943
\(914\) 11.3767 0.376308
\(915\) 0 0
\(916\) −9.14048 −0.302010
\(917\) −31.7284 −1.04776
\(918\) 0 0
\(919\) 49.0749 1.61883 0.809416 0.587236i \(-0.199784\pi\)
0.809416 + 0.587236i \(0.199784\pi\)
\(920\) 6.44774 0.212576
\(921\) 0 0
\(922\) 36.5035 1.20218
\(923\) 38.6691 1.27281
\(924\) 0 0
\(925\) 2.85561 0.0938919
\(926\) −7.85389 −0.258095
\(927\) 0 0
\(928\) −22.3895 −0.734972
\(929\) −32.4312 −1.06403 −0.532017 0.846734i \(-0.678566\pi\)
−0.532017 + 0.846734i \(0.678566\pi\)
\(930\) 0 0
\(931\) 1.94524 0.0637528
\(932\) −0.268176 −0.00878441
\(933\) 0 0
\(934\) −19.7006 −0.644625
\(935\) −6.80447 −0.222530
\(936\) 0 0
\(937\) 48.6157 1.58821 0.794103 0.607783i \(-0.207941\pi\)
0.794103 + 0.607783i \(0.207941\pi\)
\(938\) 20.3825 0.665513
\(939\) 0 0
\(940\) 10.3075 0.336193
\(941\) 0.710434 0.0231595 0.0115797 0.999933i \(-0.496314\pi\)
0.0115797 + 0.999933i \(0.496314\pi\)
\(942\) 0 0
\(943\) −7.82092 −0.254684
\(944\) −42.9175 −1.39685
\(945\) 0 0
\(946\) 64.4762 2.09630
\(947\) −26.1077 −0.848387 −0.424194 0.905571i \(-0.639442\pi\)
−0.424194 + 0.905571i \(0.639442\pi\)
\(948\) 0 0
\(949\) −31.3647 −1.01814
\(950\) 9.25094 0.300140
\(951\) 0 0
\(952\) 6.46296 0.209466
\(953\) 51.0054 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(954\) 0 0
\(955\) −34.6177 −1.12020
\(956\) 9.87375 0.319340
\(957\) 0 0
\(958\) 44.9392 1.45192
\(959\) −11.7543 −0.379565
\(960\) 0 0
\(961\) −29.9099 −0.964836
\(962\) 14.1997 0.457817
\(963\) 0 0
\(964\) 7.51353 0.241995
\(965\) −35.1747 −1.13231
\(966\) 0 0
\(967\) 10.0048 0.321733 0.160867 0.986976i \(-0.448571\pi\)
0.160867 + 0.986976i \(0.448571\pi\)
\(968\) 14.8414 0.477022
\(969\) 0 0
\(970\) −22.7195 −0.729478
\(971\) −44.6269 −1.43215 −0.716073 0.698025i \(-0.754062\pi\)
−0.716073 + 0.698025i \(0.754062\pi\)
\(972\) 0 0
\(973\) −31.0044 −0.993954
\(974\) −46.8535 −1.50128
\(975\) 0 0
\(976\) 6.09607 0.195130
\(977\) 39.0856 1.25046 0.625230 0.780440i \(-0.285005\pi\)
0.625230 + 0.780440i \(0.285005\pi\)
\(978\) 0 0
\(979\) −18.8649 −0.602925
\(980\) 0.597456 0.0190850
\(981\) 0 0
\(982\) 19.5800 0.624824
\(983\) −6.79459 −0.216714 −0.108357 0.994112i \(-0.534559\pi\)
−0.108357 + 0.994112i \(0.534559\pi\)
\(984\) 0 0
\(985\) 16.1073 0.513222
\(986\) 12.6824 0.403890
\(987\) 0 0
\(988\) 9.07255 0.288636
\(989\) 15.8740 0.504763
\(990\) 0 0
\(991\) 17.2046 0.546522 0.273261 0.961940i \(-0.411898\pi\)
0.273261 + 0.961940i \(0.411898\pi\)
\(992\) 2.84110 0.0902049
\(993\) 0 0
\(994\) −24.6704 −0.782497
\(995\) −17.9460 −0.568928
\(996\) 0 0
\(997\) 7.33368 0.232260 0.116130 0.993234i \(-0.462951\pi\)
0.116130 + 0.993234i \(0.462951\pi\)
\(998\) 3.59848 0.113908
\(999\) 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.b.1.3 6
3.2 odd 2 729.2.a.e.1.4 yes 6
9.2 odd 6 729.2.c.a.244.3 12
9.4 even 3 729.2.c.d.487.4 12
9.5 odd 6 729.2.c.a.487.3 12
9.7 even 3 729.2.c.d.244.4 12
27.2 odd 18 729.2.e.u.568.1 12
27.4 even 9 729.2.e.s.406.1 12
27.5 odd 18 729.2.e.k.649.2 12
27.7 even 9 729.2.e.s.325.1 12
27.11 odd 18 729.2.e.k.82.2 12
27.13 even 9 729.2.e.j.163.2 12
27.14 odd 18 729.2.e.u.163.1 12
27.16 even 9 729.2.e.t.82.1 12
27.20 odd 18 729.2.e.l.325.2 12
27.22 even 9 729.2.e.t.649.1 12
27.23 odd 18 729.2.e.l.406.2 12
27.25 even 9 729.2.e.j.568.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.3 6 1.1 even 1 trivial
729.2.a.e.1.4 yes 6 3.2 odd 2
729.2.c.a.244.3 12 9.2 odd 6
729.2.c.a.487.3 12 9.5 odd 6
729.2.c.d.244.4 12 9.7 even 3
729.2.c.d.487.4 12 9.4 even 3
729.2.e.j.163.2 12 27.13 even 9
729.2.e.j.568.2 12 27.25 even 9
729.2.e.k.82.2 12 27.11 odd 18
729.2.e.k.649.2 12 27.5 odd 18
729.2.e.l.325.2 12 27.20 odd 18
729.2.e.l.406.2 12 27.23 odd 18
729.2.e.s.325.1 12 27.7 even 9
729.2.e.s.406.1 12 27.4 even 9
729.2.e.t.82.1 12 27.16 even 9
729.2.e.t.649.1 12 27.22 even 9
729.2.e.u.163.1 12 27.14 odd 18
729.2.e.u.568.1 12 27.2 odd 18