Properties

Label 8004.2.a.e.1.8
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.511914\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.34732 q^{5} +2.91839 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.34732 q^{5} +2.91839 q^{7} +1.00000 q^{9} +3.69746 q^{11} -6.03680 q^{13} -1.34732 q^{15} +5.75034 q^{17} -6.30556 q^{19} -2.91839 q^{21} -1.00000 q^{23} -3.18473 q^{25} -1.00000 q^{27} +1.00000 q^{29} -0.671792 q^{31} -3.69746 q^{33} +3.93200 q^{35} -4.42860 q^{37} +6.03680 q^{39} -2.20091 q^{41} -8.58949 q^{43} +1.34732 q^{45} +0.549485 q^{47} +1.51697 q^{49} -5.75034 q^{51} -2.93706 q^{53} +4.98166 q^{55} +6.30556 q^{57} -4.20516 q^{59} -10.6176 q^{61} +2.91839 q^{63} -8.13349 q^{65} -3.98884 q^{67} +1.00000 q^{69} -9.08197 q^{71} -2.11062 q^{73} +3.18473 q^{75} +10.7906 q^{77} -12.8630 q^{79} +1.00000 q^{81} -6.50563 q^{83} +7.74754 q^{85} -1.00000 q^{87} +4.64297 q^{89} -17.6177 q^{91} +0.671792 q^{93} -8.49560 q^{95} -1.50834 q^{97} +3.69746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.34732 0.602539 0.301270 0.953539i \(-0.402590\pi\)
0.301270 + 0.953539i \(0.402590\pi\)
\(6\) 0 0
\(7\) 2.91839 1.10305 0.551523 0.834160i \(-0.314047\pi\)
0.551523 + 0.834160i \(0.314047\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.69746 1.11483 0.557413 0.830235i \(-0.311794\pi\)
0.557413 + 0.830235i \(0.311794\pi\)
\(12\) 0 0
\(13\) −6.03680 −1.67431 −0.837153 0.546969i \(-0.815782\pi\)
−0.837153 + 0.546969i \(0.815782\pi\)
\(14\) 0 0
\(15\) −1.34732 −0.347876
\(16\) 0 0
\(17\) 5.75034 1.39466 0.697331 0.716750i \(-0.254371\pi\)
0.697331 + 0.716750i \(0.254371\pi\)
\(18\) 0 0
\(19\) −6.30556 −1.44659 −0.723297 0.690537i \(-0.757374\pi\)
−0.723297 + 0.690537i \(0.757374\pi\)
\(20\) 0 0
\(21\) −2.91839 −0.636844
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.18473 −0.636946
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −0.671792 −0.120657 −0.0603287 0.998179i \(-0.519215\pi\)
−0.0603287 + 0.998179i \(0.519215\pi\)
\(32\) 0 0
\(33\) −3.69746 −0.643645
\(34\) 0 0
\(35\) 3.93200 0.664629
\(36\) 0 0
\(37\) −4.42860 −0.728058 −0.364029 0.931388i \(-0.618599\pi\)
−0.364029 + 0.931388i \(0.618599\pi\)
\(38\) 0 0
\(39\) 6.03680 0.966661
\(40\) 0 0
\(41\) −2.20091 −0.343725 −0.171862 0.985121i \(-0.554978\pi\)
−0.171862 + 0.985121i \(0.554978\pi\)
\(42\) 0 0
\(43\) −8.58949 −1.30988 −0.654942 0.755679i \(-0.727307\pi\)
−0.654942 + 0.755679i \(0.727307\pi\)
\(44\) 0 0
\(45\) 1.34732 0.200846
\(46\) 0 0
\(47\) 0.549485 0.0801506 0.0400753 0.999197i \(-0.487240\pi\)
0.0400753 + 0.999197i \(0.487240\pi\)
\(48\) 0 0
\(49\) 1.51697 0.216711
\(50\) 0 0
\(51\) −5.75034 −0.805208
\(52\) 0 0
\(53\) −2.93706 −0.403436 −0.201718 0.979444i \(-0.564652\pi\)
−0.201718 + 0.979444i \(0.564652\pi\)
\(54\) 0 0
\(55\) 4.98166 0.671726
\(56\) 0 0
\(57\) 6.30556 0.835191
\(58\) 0 0
\(59\) −4.20516 −0.547465 −0.273733 0.961806i \(-0.588258\pi\)
−0.273733 + 0.961806i \(0.588258\pi\)
\(60\) 0 0
\(61\) −10.6176 −1.35945 −0.679724 0.733468i \(-0.737900\pi\)
−0.679724 + 0.733468i \(0.737900\pi\)
\(62\) 0 0
\(63\) 2.91839 0.367682
\(64\) 0 0
\(65\) −8.13349 −1.00884
\(66\) 0 0
\(67\) −3.98884 −0.487314 −0.243657 0.969861i \(-0.578347\pi\)
−0.243657 + 0.969861i \(0.578347\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −9.08197 −1.07783 −0.538916 0.842359i \(-0.681166\pi\)
−0.538916 + 0.842359i \(0.681166\pi\)
\(72\) 0 0
\(73\) −2.11062 −0.247029 −0.123515 0.992343i \(-0.539417\pi\)
−0.123515 + 0.992343i \(0.539417\pi\)
\(74\) 0 0
\(75\) 3.18473 0.367741
\(76\) 0 0
\(77\) 10.7906 1.22970
\(78\) 0 0
\(79\) −12.8630 −1.44720 −0.723598 0.690222i \(-0.757513\pi\)
−0.723598 + 0.690222i \(0.757513\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.50563 −0.714085 −0.357043 0.934088i \(-0.616215\pi\)
−0.357043 + 0.934088i \(0.616215\pi\)
\(84\) 0 0
\(85\) 7.74754 0.840338
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 4.64297 0.492154 0.246077 0.969250i \(-0.420858\pi\)
0.246077 + 0.969250i \(0.420858\pi\)
\(90\) 0 0
\(91\) −17.6177 −1.84684
\(92\) 0 0
\(93\) 0.671792 0.0696616
\(94\) 0 0
\(95\) −8.49560 −0.871630
\(96\) 0 0
\(97\) −1.50834 −0.153148 −0.0765742 0.997064i \(-0.524398\pi\)
−0.0765742 + 0.997064i \(0.524398\pi\)
\(98\) 0 0
\(99\) 3.69746 0.371609
\(100\) 0 0
\(101\) 3.26567 0.324946 0.162473 0.986713i \(-0.448053\pi\)
0.162473 + 0.986713i \(0.448053\pi\)
\(102\) 0 0
\(103\) −5.93651 −0.584942 −0.292471 0.956274i \(-0.594478\pi\)
−0.292471 + 0.956274i \(0.594478\pi\)
\(104\) 0 0
\(105\) −3.93200 −0.383724
\(106\) 0 0
\(107\) 11.1575 1.07863 0.539317 0.842103i \(-0.318682\pi\)
0.539317 + 0.842103i \(0.318682\pi\)
\(108\) 0 0
\(109\) −1.50930 −0.144564 −0.0722822 0.997384i \(-0.523028\pi\)
−0.0722822 + 0.997384i \(0.523028\pi\)
\(110\) 0 0
\(111\) 4.42860 0.420345
\(112\) 0 0
\(113\) −4.54937 −0.427968 −0.213984 0.976837i \(-0.568644\pi\)
−0.213984 + 0.976837i \(0.568644\pi\)
\(114\) 0 0
\(115\) −1.34732 −0.125638
\(116\) 0 0
\(117\) −6.03680 −0.558102
\(118\) 0 0
\(119\) 16.7817 1.53838
\(120\) 0 0
\(121\) 2.67121 0.242837
\(122\) 0 0
\(123\) 2.20091 0.198450
\(124\) 0 0
\(125\) −11.0274 −0.986325
\(126\) 0 0
\(127\) −3.98492 −0.353605 −0.176802 0.984246i \(-0.556575\pi\)
−0.176802 + 0.984246i \(0.556575\pi\)
\(128\) 0 0
\(129\) 8.58949 0.756262
\(130\) 0 0
\(131\) 1.73658 0.151726 0.0758628 0.997118i \(-0.475829\pi\)
0.0758628 + 0.997118i \(0.475829\pi\)
\(132\) 0 0
\(133\) −18.4020 −1.59566
\(134\) 0 0
\(135\) −1.34732 −0.115959
\(136\) 0 0
\(137\) −8.15181 −0.696456 −0.348228 0.937410i \(-0.613217\pi\)
−0.348228 + 0.937410i \(0.613217\pi\)
\(138\) 0 0
\(139\) −0.954858 −0.0809900 −0.0404950 0.999180i \(-0.512893\pi\)
−0.0404950 + 0.999180i \(0.512893\pi\)
\(140\) 0 0
\(141\) −0.549485 −0.0462750
\(142\) 0 0
\(143\) −22.3208 −1.86656
\(144\) 0 0
\(145\) 1.34732 0.111889
\(146\) 0 0
\(147\) −1.51697 −0.125118
\(148\) 0 0
\(149\) −0.579094 −0.0474412 −0.0237206 0.999719i \(-0.507551\pi\)
−0.0237206 + 0.999719i \(0.507551\pi\)
\(150\) 0 0
\(151\) 16.8135 1.36827 0.684134 0.729357i \(-0.260181\pi\)
0.684134 + 0.729357i \(0.260181\pi\)
\(152\) 0 0
\(153\) 5.75034 0.464887
\(154\) 0 0
\(155\) −0.905118 −0.0727008
\(156\) 0 0
\(157\) 11.7695 0.939307 0.469654 0.882851i \(-0.344379\pi\)
0.469654 + 0.882851i \(0.344379\pi\)
\(158\) 0 0
\(159\) 2.93706 0.232924
\(160\) 0 0
\(161\) −2.91839 −0.230001
\(162\) 0 0
\(163\) −1.97311 −0.154546 −0.0772729 0.997010i \(-0.524621\pi\)
−0.0772729 + 0.997010i \(0.524621\pi\)
\(164\) 0 0
\(165\) −4.98166 −0.387821
\(166\) 0 0
\(167\) 15.3458 1.18750 0.593748 0.804651i \(-0.297648\pi\)
0.593748 + 0.804651i \(0.297648\pi\)
\(168\) 0 0
\(169\) 23.4429 1.80330
\(170\) 0 0
\(171\) −6.30556 −0.482198
\(172\) 0 0
\(173\) 0.507138 0.0385570 0.0192785 0.999814i \(-0.493863\pi\)
0.0192785 + 0.999814i \(0.493863\pi\)
\(174\) 0 0
\(175\) −9.29428 −0.702581
\(176\) 0 0
\(177\) 4.20516 0.316079
\(178\) 0 0
\(179\) 17.0584 1.27501 0.637503 0.770448i \(-0.279967\pi\)
0.637503 + 0.770448i \(0.279967\pi\)
\(180\) 0 0
\(181\) −8.69457 −0.646262 −0.323131 0.946354i \(-0.604735\pi\)
−0.323131 + 0.946354i \(0.604735\pi\)
\(182\) 0 0
\(183\) 10.6176 0.784877
\(184\) 0 0
\(185\) −5.96674 −0.438684
\(186\) 0 0
\(187\) 21.2616 1.55480
\(188\) 0 0
\(189\) −2.91839 −0.212281
\(190\) 0 0
\(191\) 7.97988 0.577404 0.288702 0.957419i \(-0.406776\pi\)
0.288702 + 0.957419i \(0.406776\pi\)
\(192\) 0 0
\(193\) 16.3801 1.17906 0.589531 0.807746i \(-0.299312\pi\)
0.589531 + 0.807746i \(0.299312\pi\)
\(194\) 0 0
\(195\) 8.13349 0.582451
\(196\) 0 0
\(197\) −10.4305 −0.743140 −0.371570 0.928405i \(-0.621180\pi\)
−0.371570 + 0.928405i \(0.621180\pi\)
\(198\) 0 0
\(199\) 21.3102 1.51064 0.755321 0.655355i \(-0.227481\pi\)
0.755321 + 0.655355i \(0.227481\pi\)
\(200\) 0 0
\(201\) 3.98884 0.281351
\(202\) 0 0
\(203\) 2.91839 0.204831
\(204\) 0 0
\(205\) −2.96533 −0.207108
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −23.3145 −1.61270
\(210\) 0 0
\(211\) −22.2304 −1.53040 −0.765201 0.643791i \(-0.777360\pi\)
−0.765201 + 0.643791i \(0.777360\pi\)
\(212\) 0 0
\(213\) 9.08197 0.622287
\(214\) 0 0
\(215\) −11.5728 −0.789257
\(216\) 0 0
\(217\) −1.96055 −0.133091
\(218\) 0 0
\(219\) 2.11062 0.142622
\(220\) 0 0
\(221\) −34.7136 −2.33509
\(222\) 0 0
\(223\) 3.34435 0.223955 0.111977 0.993711i \(-0.464282\pi\)
0.111977 + 0.993711i \(0.464282\pi\)
\(224\) 0 0
\(225\) −3.18473 −0.212315
\(226\) 0 0
\(227\) −15.7776 −1.04720 −0.523598 0.851965i \(-0.675411\pi\)
−0.523598 + 0.851965i \(0.675411\pi\)
\(228\) 0 0
\(229\) 4.93012 0.325791 0.162896 0.986643i \(-0.447917\pi\)
0.162896 + 0.986643i \(0.447917\pi\)
\(230\) 0 0
\(231\) −10.7906 −0.709970
\(232\) 0 0
\(233\) −2.88297 −0.188869 −0.0944347 0.995531i \(-0.530104\pi\)
−0.0944347 + 0.995531i \(0.530104\pi\)
\(234\) 0 0
\(235\) 0.740332 0.0482939
\(236\) 0 0
\(237\) 12.8630 0.835539
\(238\) 0 0
\(239\) −13.1677 −0.851749 −0.425874 0.904782i \(-0.640033\pi\)
−0.425874 + 0.904782i \(0.640033\pi\)
\(240\) 0 0
\(241\) −1.71060 −0.110189 −0.0550947 0.998481i \(-0.517546\pi\)
−0.0550947 + 0.998481i \(0.517546\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.04385 0.130577
\(246\) 0 0
\(247\) 38.0654 2.42204
\(248\) 0 0
\(249\) 6.50563 0.412277
\(250\) 0 0
\(251\) 8.52631 0.538176 0.269088 0.963116i \(-0.413278\pi\)
0.269088 + 0.963116i \(0.413278\pi\)
\(252\) 0 0
\(253\) −3.69746 −0.232457
\(254\) 0 0
\(255\) −7.74754 −0.485170
\(256\) 0 0
\(257\) 9.31579 0.581103 0.290551 0.956859i \(-0.406161\pi\)
0.290551 + 0.956859i \(0.406161\pi\)
\(258\) 0 0
\(259\) −12.9244 −0.803082
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 31.3704 1.93438 0.967190 0.254054i \(-0.0817642\pi\)
0.967190 + 0.254054i \(0.0817642\pi\)
\(264\) 0 0
\(265\) −3.95715 −0.243086
\(266\) 0 0
\(267\) −4.64297 −0.284145
\(268\) 0 0
\(269\) −4.98936 −0.304206 −0.152103 0.988365i \(-0.548605\pi\)
−0.152103 + 0.988365i \(0.548605\pi\)
\(270\) 0 0
\(271\) 28.3242 1.72057 0.860285 0.509813i \(-0.170286\pi\)
0.860285 + 0.509813i \(0.170286\pi\)
\(272\) 0 0
\(273\) 17.6177 1.06627
\(274\) 0 0
\(275\) −11.7754 −0.710084
\(276\) 0 0
\(277\) −4.33021 −0.260177 −0.130088 0.991502i \(-0.541526\pi\)
−0.130088 + 0.991502i \(0.541526\pi\)
\(278\) 0 0
\(279\) −0.671792 −0.0402191
\(280\) 0 0
\(281\) −9.22411 −0.550264 −0.275132 0.961406i \(-0.588722\pi\)
−0.275132 + 0.961406i \(0.588722\pi\)
\(282\) 0 0
\(283\) 20.4601 1.21623 0.608114 0.793850i \(-0.291927\pi\)
0.608114 + 0.793850i \(0.291927\pi\)
\(284\) 0 0
\(285\) 8.49560 0.503236
\(286\) 0 0
\(287\) −6.42311 −0.379144
\(288\) 0 0
\(289\) 16.0664 0.945080
\(290\) 0 0
\(291\) 1.50834 0.0884203
\(292\) 0 0
\(293\) 4.53864 0.265150 0.132575 0.991173i \(-0.457675\pi\)
0.132575 + 0.991173i \(0.457675\pi\)
\(294\) 0 0
\(295\) −5.66569 −0.329869
\(296\) 0 0
\(297\) −3.69746 −0.214548
\(298\) 0 0
\(299\) 6.03680 0.349117
\(300\) 0 0
\(301\) −25.0674 −1.44486
\(302\) 0 0
\(303\) −3.26567 −0.187608
\(304\) 0 0
\(305\) −14.3053 −0.819120
\(306\) 0 0
\(307\) −17.5891 −1.00387 −0.501933 0.864907i \(-0.667378\pi\)
−0.501933 + 0.864907i \(0.667378\pi\)
\(308\) 0 0
\(309\) 5.93651 0.337717
\(310\) 0 0
\(311\) −25.8397 −1.46524 −0.732618 0.680640i \(-0.761702\pi\)
−0.732618 + 0.680640i \(0.761702\pi\)
\(312\) 0 0
\(313\) −7.28452 −0.411745 −0.205873 0.978579i \(-0.566003\pi\)
−0.205873 + 0.978579i \(0.566003\pi\)
\(314\) 0 0
\(315\) 3.93200 0.221543
\(316\) 0 0
\(317\) −8.21892 −0.461620 −0.230810 0.972999i \(-0.574138\pi\)
−0.230810 + 0.972999i \(0.574138\pi\)
\(318\) 0 0
\(319\) 3.69746 0.207018
\(320\) 0 0
\(321\) −11.1575 −0.622750
\(322\) 0 0
\(323\) −36.2591 −2.01751
\(324\) 0 0
\(325\) 19.2256 1.06644
\(326\) 0 0
\(327\) 1.50930 0.0834643
\(328\) 0 0
\(329\) 1.60361 0.0884098
\(330\) 0 0
\(331\) 3.55431 0.195363 0.0976813 0.995218i \(-0.468857\pi\)
0.0976813 + 0.995218i \(0.468857\pi\)
\(332\) 0 0
\(333\) −4.42860 −0.242686
\(334\) 0 0
\(335\) −5.37424 −0.293626
\(336\) 0 0
\(337\) −20.7733 −1.13160 −0.565798 0.824544i \(-0.691432\pi\)
−0.565798 + 0.824544i \(0.691432\pi\)
\(338\) 0 0
\(339\) 4.54937 0.247088
\(340\) 0 0
\(341\) −2.48392 −0.134512
\(342\) 0 0
\(343\) −16.0016 −0.864004
\(344\) 0 0
\(345\) 1.34732 0.0725372
\(346\) 0 0
\(347\) −7.56583 −0.406155 −0.203078 0.979163i \(-0.565094\pi\)
−0.203078 + 0.979163i \(0.565094\pi\)
\(348\) 0 0
\(349\) 9.14262 0.489393 0.244697 0.969600i \(-0.421312\pi\)
0.244697 + 0.969600i \(0.421312\pi\)
\(350\) 0 0
\(351\) 6.03680 0.322220
\(352\) 0 0
\(353\) 21.0243 1.11901 0.559506 0.828826i \(-0.310991\pi\)
0.559506 + 0.828826i \(0.310991\pi\)
\(354\) 0 0
\(355\) −12.2363 −0.649436
\(356\) 0 0
\(357\) −16.7817 −0.888182
\(358\) 0 0
\(359\) 18.8797 0.996433 0.498216 0.867053i \(-0.333989\pi\)
0.498216 + 0.867053i \(0.333989\pi\)
\(360\) 0 0
\(361\) 20.7600 1.09263
\(362\) 0 0
\(363\) −2.67121 −0.140202
\(364\) 0 0
\(365\) −2.84368 −0.148845
\(366\) 0 0
\(367\) 20.2455 1.05681 0.528404 0.848993i \(-0.322791\pi\)
0.528404 + 0.848993i \(0.322791\pi\)
\(368\) 0 0
\(369\) −2.20091 −0.114575
\(370\) 0 0
\(371\) −8.57147 −0.445008
\(372\) 0 0
\(373\) −8.30473 −0.430003 −0.215001 0.976614i \(-0.568976\pi\)
−0.215001 + 0.976614i \(0.568976\pi\)
\(374\) 0 0
\(375\) 11.0274 0.569455
\(376\) 0 0
\(377\) −6.03680 −0.310911
\(378\) 0 0
\(379\) −23.1184 −1.18751 −0.593756 0.804645i \(-0.702356\pi\)
−0.593756 + 0.804645i \(0.702356\pi\)
\(380\) 0 0
\(381\) 3.98492 0.204154
\(382\) 0 0
\(383\) −17.7677 −0.907889 −0.453945 0.891030i \(-0.649984\pi\)
−0.453945 + 0.891030i \(0.649984\pi\)
\(384\) 0 0
\(385\) 14.5384 0.740945
\(386\) 0 0
\(387\) −8.58949 −0.436628
\(388\) 0 0
\(389\) −4.16020 −0.210930 −0.105465 0.994423i \(-0.533633\pi\)
−0.105465 + 0.994423i \(0.533633\pi\)
\(390\) 0 0
\(391\) −5.75034 −0.290807
\(392\) 0 0
\(393\) −1.73658 −0.0875988
\(394\) 0 0
\(395\) −17.3305 −0.871992
\(396\) 0 0
\(397\) −1.81500 −0.0910922 −0.0455461 0.998962i \(-0.514503\pi\)
−0.0455461 + 0.998962i \(0.514503\pi\)
\(398\) 0 0
\(399\) 18.4020 0.921255
\(400\) 0 0
\(401\) −4.76103 −0.237754 −0.118877 0.992909i \(-0.537929\pi\)
−0.118877 + 0.992909i \(0.537929\pi\)
\(402\) 0 0
\(403\) 4.05547 0.202017
\(404\) 0 0
\(405\) 1.34732 0.0669488
\(406\) 0 0
\(407\) −16.3746 −0.811658
\(408\) 0 0
\(409\) 3.00888 0.148780 0.0743898 0.997229i \(-0.476299\pi\)
0.0743898 + 0.997229i \(0.476299\pi\)
\(410\) 0 0
\(411\) 8.15181 0.402099
\(412\) 0 0
\(413\) −12.2723 −0.603879
\(414\) 0 0
\(415\) −8.76515 −0.430265
\(416\) 0 0
\(417\) 0.954858 0.0467596
\(418\) 0 0
\(419\) 6.26848 0.306236 0.153118 0.988208i \(-0.451069\pi\)
0.153118 + 0.988208i \(0.451069\pi\)
\(420\) 0 0
\(421\) 3.78433 0.184437 0.0922184 0.995739i \(-0.470604\pi\)
0.0922184 + 0.995739i \(0.470604\pi\)
\(422\) 0 0
\(423\) 0.549485 0.0267169
\(424\) 0 0
\(425\) −18.3133 −0.888324
\(426\) 0 0
\(427\) −30.9863 −1.49953
\(428\) 0 0
\(429\) 22.3208 1.07766
\(430\) 0 0
\(431\) 29.6641 1.42887 0.714435 0.699702i \(-0.246684\pi\)
0.714435 + 0.699702i \(0.246684\pi\)
\(432\) 0 0
\(433\) −27.8508 −1.33843 −0.669213 0.743071i \(-0.733369\pi\)
−0.669213 + 0.743071i \(0.733369\pi\)
\(434\) 0 0
\(435\) −1.34732 −0.0645990
\(436\) 0 0
\(437\) 6.30556 0.301636
\(438\) 0 0
\(439\) −12.0457 −0.574910 −0.287455 0.957794i \(-0.592809\pi\)
−0.287455 + 0.957794i \(0.592809\pi\)
\(440\) 0 0
\(441\) 1.51697 0.0722369
\(442\) 0 0
\(443\) 1.13889 0.0541102 0.0270551 0.999634i \(-0.491387\pi\)
0.0270551 + 0.999634i \(0.491387\pi\)
\(444\) 0 0
\(445\) 6.25556 0.296542
\(446\) 0 0
\(447\) 0.579094 0.0273902
\(448\) 0 0
\(449\) −16.5017 −0.778764 −0.389382 0.921076i \(-0.627311\pi\)
−0.389382 + 0.921076i \(0.627311\pi\)
\(450\) 0 0
\(451\) −8.13778 −0.383193
\(452\) 0 0
\(453\) −16.8135 −0.789969
\(454\) 0 0
\(455\) −23.7367 −1.11279
\(456\) 0 0
\(457\) 0.941855 0.0440581 0.0220291 0.999757i \(-0.492987\pi\)
0.0220291 + 0.999757i \(0.492987\pi\)
\(458\) 0 0
\(459\) −5.75034 −0.268403
\(460\) 0 0
\(461\) −7.61417 −0.354627 −0.177314 0.984154i \(-0.556741\pi\)
−0.177314 + 0.984154i \(0.556741\pi\)
\(462\) 0 0
\(463\) 3.38347 0.157243 0.0786217 0.996905i \(-0.474948\pi\)
0.0786217 + 0.996905i \(0.474948\pi\)
\(464\) 0 0
\(465\) 0.905118 0.0419739
\(466\) 0 0
\(467\) 22.0548 1.02057 0.510286 0.860005i \(-0.329540\pi\)
0.510286 + 0.860005i \(0.329540\pi\)
\(468\) 0 0
\(469\) −11.6410 −0.537530
\(470\) 0 0
\(471\) −11.7695 −0.542309
\(472\) 0 0
\(473\) −31.7593 −1.46029
\(474\) 0 0
\(475\) 20.0815 0.921403
\(476\) 0 0
\(477\) −2.93706 −0.134479
\(478\) 0 0
\(479\) 22.8852 1.04565 0.522825 0.852440i \(-0.324878\pi\)
0.522825 + 0.852440i \(0.324878\pi\)
\(480\) 0 0
\(481\) 26.7346 1.21899
\(482\) 0 0
\(483\) 2.91839 0.132791
\(484\) 0 0
\(485\) −2.03221 −0.0922780
\(486\) 0 0
\(487\) 32.7383 1.48351 0.741756 0.670670i \(-0.233993\pi\)
0.741756 + 0.670670i \(0.233993\pi\)
\(488\) 0 0
\(489\) 1.97311 0.0892271
\(490\) 0 0
\(491\) −11.7397 −0.529805 −0.264902 0.964275i \(-0.585340\pi\)
−0.264902 + 0.964275i \(0.585340\pi\)
\(492\) 0 0
\(493\) 5.75034 0.258982
\(494\) 0 0
\(495\) 4.98166 0.223909
\(496\) 0 0
\(497\) −26.5047 −1.18890
\(498\) 0 0
\(499\) −33.6706 −1.50730 −0.753651 0.657275i \(-0.771709\pi\)
−0.753651 + 0.657275i \(0.771709\pi\)
\(500\) 0 0
\(501\) −15.3458 −0.685601
\(502\) 0 0
\(503\) −29.0415 −1.29490 −0.647448 0.762110i \(-0.724164\pi\)
−0.647448 + 0.762110i \(0.724164\pi\)
\(504\) 0 0
\(505\) 4.39990 0.195793
\(506\) 0 0
\(507\) −23.4429 −1.04114
\(508\) 0 0
\(509\) −34.1695 −1.51454 −0.757269 0.653103i \(-0.773467\pi\)
−0.757269 + 0.653103i \(0.773467\pi\)
\(510\) 0 0
\(511\) −6.15960 −0.272485
\(512\) 0 0
\(513\) 6.30556 0.278397
\(514\) 0 0
\(515\) −7.99838 −0.352451
\(516\) 0 0
\(517\) 2.03170 0.0893540
\(518\) 0 0
\(519\) −0.507138 −0.0222609
\(520\) 0 0
\(521\) −17.6254 −0.772185 −0.386092 0.922460i \(-0.626175\pi\)
−0.386092 + 0.922460i \(0.626175\pi\)
\(522\) 0 0
\(523\) 2.67968 0.117174 0.0585870 0.998282i \(-0.481340\pi\)
0.0585870 + 0.998282i \(0.481340\pi\)
\(524\) 0 0
\(525\) 9.29428 0.405635
\(526\) 0 0
\(527\) −3.86303 −0.168276
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.20516 −0.182488
\(532\) 0 0
\(533\) 13.2865 0.575500
\(534\) 0 0
\(535\) 15.0327 0.649920
\(536\) 0 0
\(537\) −17.0584 −0.736125
\(538\) 0 0
\(539\) 5.60895 0.241595
\(540\) 0 0
\(541\) −0.691956 −0.0297495 −0.0148748 0.999889i \(-0.504735\pi\)
−0.0148748 + 0.999889i \(0.504735\pi\)
\(542\) 0 0
\(543\) 8.69457 0.373120
\(544\) 0 0
\(545\) −2.03350 −0.0871058
\(546\) 0 0
\(547\) 14.2353 0.608659 0.304330 0.952567i \(-0.401568\pi\)
0.304330 + 0.952567i \(0.401568\pi\)
\(548\) 0 0
\(549\) −10.6176 −0.453149
\(550\) 0 0
\(551\) −6.30556 −0.268626
\(552\) 0 0
\(553\) −37.5391 −1.59632
\(554\) 0 0
\(555\) 5.96674 0.253274
\(556\) 0 0
\(557\) −5.54928 −0.235130 −0.117565 0.993065i \(-0.537509\pi\)
−0.117565 + 0.993065i \(0.537509\pi\)
\(558\) 0 0
\(559\) 51.8530 2.19315
\(560\) 0 0
\(561\) −21.2616 −0.897667
\(562\) 0 0
\(563\) 26.6802 1.12444 0.562219 0.826989i \(-0.309948\pi\)
0.562219 + 0.826989i \(0.309948\pi\)
\(564\) 0 0
\(565\) −6.12945 −0.257868
\(566\) 0 0
\(567\) 2.91839 0.122561
\(568\) 0 0
\(569\) −8.81271 −0.369448 −0.184724 0.982790i \(-0.559139\pi\)
−0.184724 + 0.982790i \(0.559139\pi\)
\(570\) 0 0
\(571\) −41.6835 −1.74440 −0.872200 0.489150i \(-0.837307\pi\)
−0.872200 + 0.489150i \(0.837307\pi\)
\(572\) 0 0
\(573\) −7.97988 −0.333364
\(574\) 0 0
\(575\) 3.18473 0.132812
\(576\) 0 0
\(577\) −10.8443 −0.451452 −0.225726 0.974191i \(-0.572475\pi\)
−0.225726 + 0.974191i \(0.572475\pi\)
\(578\) 0 0
\(579\) −16.3801 −0.680732
\(580\) 0 0
\(581\) −18.9859 −0.787669
\(582\) 0 0
\(583\) −10.8597 −0.449761
\(584\) 0 0
\(585\) −8.13349 −0.336278
\(586\) 0 0
\(587\) 0.0198519 0.000819377 0 0.000409689 1.00000i \(-0.499870\pi\)
0.000409689 1.00000i \(0.499870\pi\)
\(588\) 0 0
\(589\) 4.23602 0.174542
\(590\) 0 0
\(591\) 10.4305 0.429052
\(592\) 0 0
\(593\) 6.59076 0.270650 0.135325 0.990801i \(-0.456792\pi\)
0.135325 + 0.990801i \(0.456792\pi\)
\(594\) 0 0
\(595\) 22.6103 0.926932
\(596\) 0 0
\(597\) −21.3102 −0.872169
\(598\) 0 0
\(599\) 31.4616 1.28549 0.642743 0.766082i \(-0.277796\pi\)
0.642743 + 0.766082i \(0.277796\pi\)
\(600\) 0 0
\(601\) −29.9067 −1.21992 −0.609961 0.792432i \(-0.708815\pi\)
−0.609961 + 0.792432i \(0.708815\pi\)
\(602\) 0 0
\(603\) −3.98884 −0.162438
\(604\) 0 0
\(605\) 3.59897 0.146319
\(606\) 0 0
\(607\) −5.08762 −0.206500 −0.103250 0.994655i \(-0.532924\pi\)
−0.103250 + 0.994655i \(0.532924\pi\)
\(608\) 0 0
\(609\) −2.91839 −0.118259
\(610\) 0 0
\(611\) −3.31713 −0.134197
\(612\) 0 0
\(613\) 35.8235 1.44690 0.723449 0.690378i \(-0.242556\pi\)
0.723449 + 0.690378i \(0.242556\pi\)
\(614\) 0 0
\(615\) 2.96533 0.119574
\(616\) 0 0
\(617\) 15.2128 0.612442 0.306221 0.951960i \(-0.400935\pi\)
0.306221 + 0.951960i \(0.400935\pi\)
\(618\) 0 0
\(619\) 30.5644 1.22849 0.614244 0.789116i \(-0.289461\pi\)
0.614244 + 0.789116i \(0.289461\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 13.5500 0.542869
\(624\) 0 0
\(625\) 1.06617 0.0426470
\(626\) 0 0
\(627\) 23.3145 0.931093
\(628\) 0 0
\(629\) −25.4660 −1.01539
\(630\) 0 0
\(631\) −9.73458 −0.387527 −0.193764 0.981048i \(-0.562070\pi\)
−0.193764 + 0.981048i \(0.562070\pi\)
\(632\) 0 0
\(633\) 22.2304 0.883578
\(634\) 0 0
\(635\) −5.36896 −0.213061
\(636\) 0 0
\(637\) −9.15767 −0.362840
\(638\) 0 0
\(639\) −9.08197 −0.359277
\(640\) 0 0
\(641\) −38.9947 −1.54020 −0.770099 0.637924i \(-0.779793\pi\)
−0.770099 + 0.637924i \(0.779793\pi\)
\(642\) 0 0
\(643\) 4.49902 0.177424 0.0887120 0.996057i \(-0.471725\pi\)
0.0887120 + 0.996057i \(0.471725\pi\)
\(644\) 0 0
\(645\) 11.5728 0.455678
\(646\) 0 0
\(647\) −5.20597 −0.204668 −0.102334 0.994750i \(-0.532631\pi\)
−0.102334 + 0.994750i \(0.532631\pi\)
\(648\) 0 0
\(649\) −15.5484 −0.610328
\(650\) 0 0
\(651\) 1.96055 0.0768400
\(652\) 0 0
\(653\) −1.19073 −0.0465968 −0.0232984 0.999729i \(-0.507417\pi\)
−0.0232984 + 0.999729i \(0.507417\pi\)
\(654\) 0 0
\(655\) 2.33972 0.0914206
\(656\) 0 0
\(657\) −2.11062 −0.0823431
\(658\) 0 0
\(659\) 42.9964 1.67490 0.837450 0.546514i \(-0.184045\pi\)
0.837450 + 0.546514i \(0.184045\pi\)
\(660\) 0 0
\(661\) −42.7967 −1.66460 −0.832300 0.554325i \(-0.812976\pi\)
−0.832300 + 0.554325i \(0.812976\pi\)
\(662\) 0 0
\(663\) 34.7136 1.34816
\(664\) 0 0
\(665\) −24.7934 −0.961448
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −3.34435 −0.129300
\(670\) 0 0
\(671\) −39.2582 −1.51555
\(672\) 0 0
\(673\) −24.7493 −0.954015 −0.477007 0.878899i \(-0.658279\pi\)
−0.477007 + 0.878899i \(0.658279\pi\)
\(674\) 0 0
\(675\) 3.18473 0.122580
\(676\) 0 0
\(677\) −24.9081 −0.957296 −0.478648 0.878007i \(-0.658873\pi\)
−0.478648 + 0.878007i \(0.658873\pi\)
\(678\) 0 0
\(679\) −4.40191 −0.168930
\(680\) 0 0
\(681\) 15.7776 0.604599
\(682\) 0 0
\(683\) 16.7745 0.641860 0.320930 0.947103i \(-0.396005\pi\)
0.320930 + 0.947103i \(0.396005\pi\)
\(684\) 0 0
\(685\) −10.9831 −0.419642
\(686\) 0 0
\(687\) −4.93012 −0.188096
\(688\) 0 0
\(689\) 17.7304 0.675475
\(690\) 0 0
\(691\) −2.32052 −0.0882768 −0.0441384 0.999025i \(-0.514054\pi\)
−0.0441384 + 0.999025i \(0.514054\pi\)
\(692\) 0 0
\(693\) 10.7906 0.409901
\(694\) 0 0
\(695\) −1.28650 −0.0487997
\(696\) 0 0
\(697\) −12.6560 −0.479380
\(698\) 0 0
\(699\) 2.88297 0.109044
\(700\) 0 0
\(701\) 5.11535 0.193204 0.0966020 0.995323i \(-0.469203\pi\)
0.0966020 + 0.995323i \(0.469203\pi\)
\(702\) 0 0
\(703\) 27.9248 1.05320
\(704\) 0 0
\(705\) −0.740332 −0.0278825
\(706\) 0 0
\(707\) 9.53049 0.358431
\(708\) 0 0
\(709\) −30.4887 −1.14503 −0.572514 0.819895i \(-0.694032\pi\)
−0.572514 + 0.819895i \(0.694032\pi\)
\(710\) 0 0
\(711\) −12.8630 −0.482399
\(712\) 0 0
\(713\) 0.671792 0.0251588
\(714\) 0 0
\(715\) −30.0733 −1.12468
\(716\) 0 0
\(717\) 13.1677 0.491757
\(718\) 0 0
\(719\) −7.50338 −0.279829 −0.139914 0.990164i \(-0.544683\pi\)
−0.139914 + 0.990164i \(0.544683\pi\)
\(720\) 0 0
\(721\) −17.3250 −0.645218
\(722\) 0 0
\(723\) 1.71060 0.0636179
\(724\) 0 0
\(725\) −3.18473 −0.118278
\(726\) 0 0
\(727\) −25.1529 −0.932870 −0.466435 0.884555i \(-0.654462\pi\)
−0.466435 + 0.884555i \(0.654462\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −49.3925 −1.82685
\(732\) 0 0
\(733\) −18.5387 −0.684742 −0.342371 0.939565i \(-0.611230\pi\)
−0.342371 + 0.939565i \(0.611230\pi\)
\(734\) 0 0
\(735\) −2.04385 −0.0753885
\(736\) 0 0
\(737\) −14.7486 −0.543271
\(738\) 0 0
\(739\) 17.1856 0.632184 0.316092 0.948729i \(-0.397629\pi\)
0.316092 + 0.948729i \(0.397629\pi\)
\(740\) 0 0
\(741\) −38.0654 −1.39837
\(742\) 0 0
\(743\) −27.9275 −1.02456 −0.512280 0.858818i \(-0.671199\pi\)
−0.512280 + 0.858818i \(0.671199\pi\)
\(744\) 0 0
\(745\) −0.780224 −0.0285852
\(746\) 0 0
\(747\) −6.50563 −0.238028
\(748\) 0 0
\(749\) 32.5618 1.18978
\(750\) 0 0
\(751\) −44.7691 −1.63365 −0.816824 0.576888i \(-0.804267\pi\)
−0.816824 + 0.576888i \(0.804267\pi\)
\(752\) 0 0
\(753\) −8.52631 −0.310716
\(754\) 0 0
\(755\) 22.6532 0.824435
\(756\) 0 0
\(757\) 38.2317 1.38955 0.694777 0.719225i \(-0.255503\pi\)
0.694777 + 0.719225i \(0.255503\pi\)
\(758\) 0 0
\(759\) 3.69746 0.134209
\(760\) 0 0
\(761\) −40.0194 −1.45070 −0.725350 0.688380i \(-0.758322\pi\)
−0.725350 + 0.688380i \(0.758322\pi\)
\(762\) 0 0
\(763\) −4.40471 −0.159461
\(764\) 0 0
\(765\) 7.74754 0.280113
\(766\) 0 0
\(767\) 25.3857 0.916624
\(768\) 0 0
\(769\) −33.6601 −1.21381 −0.606906 0.794773i \(-0.707590\pi\)
−0.606906 + 0.794773i \(0.707590\pi\)
\(770\) 0 0
\(771\) −9.31579 −0.335500
\(772\) 0 0
\(773\) 2.00330 0.0720536 0.0360268 0.999351i \(-0.488530\pi\)
0.0360268 + 0.999351i \(0.488530\pi\)
\(774\) 0 0
\(775\) 2.13948 0.0768523
\(776\) 0 0
\(777\) 12.9244 0.463659
\(778\) 0 0
\(779\) 13.8780 0.497230
\(780\) 0 0
\(781\) −33.5802 −1.20160
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 15.8572 0.565969
\(786\) 0 0
\(787\) −0.0531304 −0.00189389 −0.000946946 1.00000i \(-0.500301\pi\)
−0.000946946 1.00000i \(0.500301\pi\)
\(788\) 0 0
\(789\) −31.3704 −1.11681
\(790\) 0 0
\(791\) −13.2768 −0.472069
\(792\) 0 0
\(793\) 64.0964 2.27613
\(794\) 0 0
\(795\) 3.95715 0.140346
\(796\) 0 0
\(797\) 31.2146 1.10568 0.552839 0.833288i \(-0.313545\pi\)
0.552839 + 0.833288i \(0.313545\pi\)
\(798\) 0 0
\(799\) 3.15972 0.111783
\(800\) 0 0
\(801\) 4.64297 0.164051
\(802\) 0 0
\(803\) −7.80393 −0.275395
\(804\) 0 0
\(805\) −3.93200 −0.138585
\(806\) 0 0
\(807\) 4.98936 0.175634
\(808\) 0 0
\(809\) −0.153360 −0.00539185 −0.00269592 0.999996i \(-0.500858\pi\)
−0.00269592 + 0.999996i \(0.500858\pi\)
\(810\) 0 0
\(811\) 11.3984 0.400253 0.200126 0.979770i \(-0.435865\pi\)
0.200126 + 0.979770i \(0.435865\pi\)
\(812\) 0 0
\(813\) −28.3242 −0.993371
\(814\) 0 0
\(815\) −2.65841 −0.0931199
\(816\) 0 0
\(817\) 54.1615 1.89487
\(818\) 0 0
\(819\) −17.6177 −0.615612
\(820\) 0 0
\(821\) −18.9338 −0.660795 −0.330397 0.943842i \(-0.607183\pi\)
−0.330397 + 0.943842i \(0.607183\pi\)
\(822\) 0 0
\(823\) 42.4815 1.48081 0.740405 0.672161i \(-0.234634\pi\)
0.740405 + 0.672161i \(0.234634\pi\)
\(824\) 0 0
\(825\) 11.7754 0.409967
\(826\) 0 0
\(827\) 45.4445 1.58026 0.790129 0.612940i \(-0.210013\pi\)
0.790129 + 0.612940i \(0.210013\pi\)
\(828\) 0 0
\(829\) 49.6222 1.72345 0.861725 0.507375i \(-0.169384\pi\)
0.861725 + 0.507375i \(0.169384\pi\)
\(830\) 0 0
\(831\) 4.33021 0.150213
\(832\) 0 0
\(833\) 8.72311 0.302238
\(834\) 0 0
\(835\) 20.6757 0.715513
\(836\) 0 0
\(837\) 0.671792 0.0232205
\(838\) 0 0
\(839\) 53.2719 1.83915 0.919576 0.392913i \(-0.128533\pi\)
0.919576 + 0.392913i \(0.128533\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 9.22411 0.317695
\(844\) 0 0
\(845\) 31.5851 1.08656
\(846\) 0 0
\(847\) 7.79561 0.267860
\(848\) 0 0
\(849\) −20.4601 −0.702189
\(850\) 0 0
\(851\) 4.42860 0.151811
\(852\) 0 0
\(853\) 24.3344 0.833194 0.416597 0.909091i \(-0.363223\pi\)
0.416597 + 0.909091i \(0.363223\pi\)
\(854\) 0 0
\(855\) −8.49560 −0.290543
\(856\) 0 0
\(857\) 4.88014 0.166702 0.0833512 0.996520i \(-0.473438\pi\)
0.0833512 + 0.996520i \(0.473438\pi\)
\(858\) 0 0
\(859\) −42.3449 −1.44479 −0.722394 0.691481i \(-0.756958\pi\)
−0.722394 + 0.691481i \(0.756958\pi\)
\(860\) 0 0
\(861\) 6.42311 0.218899
\(862\) 0 0
\(863\) −55.7214 −1.89678 −0.948389 0.317110i \(-0.897287\pi\)
−0.948389 + 0.317110i \(0.897287\pi\)
\(864\) 0 0
\(865\) 0.683277 0.0232321
\(866\) 0 0
\(867\) −16.0664 −0.545642
\(868\) 0 0
\(869\) −47.5603 −1.61337
\(870\) 0 0
\(871\) 24.0798 0.815913
\(872\) 0 0
\(873\) −1.50834 −0.0510495
\(874\) 0 0
\(875\) −32.1823 −1.08796
\(876\) 0 0
\(877\) −41.5958 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(878\) 0 0
\(879\) −4.53864 −0.153084
\(880\) 0 0
\(881\) 51.9100 1.74889 0.874446 0.485122i \(-0.161225\pi\)
0.874446 + 0.485122i \(0.161225\pi\)
\(882\) 0 0
\(883\) −8.84338 −0.297604 −0.148802 0.988867i \(-0.547542\pi\)
−0.148802 + 0.988867i \(0.547542\pi\)
\(884\) 0 0
\(885\) 5.66569 0.190450
\(886\) 0 0
\(887\) 14.7401 0.494925 0.247462 0.968897i \(-0.420403\pi\)
0.247462 + 0.968897i \(0.420403\pi\)
\(888\) 0 0
\(889\) −11.6295 −0.390042
\(890\) 0 0
\(891\) 3.69746 0.123870
\(892\) 0 0
\(893\) −3.46481 −0.115945
\(894\) 0 0
\(895\) 22.9831 0.768241
\(896\) 0 0
\(897\) −6.03680 −0.201563
\(898\) 0 0
\(899\) −0.671792 −0.0224055
\(900\) 0 0
\(901\) −16.8891 −0.562657
\(902\) 0 0
\(903\) 25.0674 0.834192
\(904\) 0 0
\(905\) −11.7144 −0.389398
\(906\) 0 0
\(907\) −20.0429 −0.665515 −0.332757 0.943012i \(-0.607979\pi\)
−0.332757 + 0.943012i \(0.607979\pi\)
\(908\) 0 0
\(909\) 3.26567 0.108315
\(910\) 0 0
\(911\) 4.47857 0.148382 0.0741908 0.997244i \(-0.476363\pi\)
0.0741908 + 0.997244i \(0.476363\pi\)
\(912\) 0 0
\(913\) −24.0543 −0.796081
\(914\) 0 0
\(915\) 14.3053 0.472919
\(916\) 0 0
\(917\) 5.06800 0.167360
\(918\) 0 0
\(919\) 20.4388 0.674214 0.337107 0.941466i \(-0.390552\pi\)
0.337107 + 0.941466i \(0.390552\pi\)
\(920\) 0 0
\(921\) 17.5891 0.579582
\(922\) 0 0
\(923\) 54.8260 1.80462
\(924\) 0 0
\(925\) 14.1039 0.463734
\(926\) 0 0
\(927\) −5.93651 −0.194981
\(928\) 0 0
\(929\) −31.2202 −1.02430 −0.512151 0.858895i \(-0.671151\pi\)
−0.512151 + 0.858895i \(0.671151\pi\)
\(930\) 0 0
\(931\) −9.56537 −0.313492
\(932\) 0 0
\(933\) 25.8397 0.845955
\(934\) 0 0
\(935\) 28.6462 0.936831
\(936\) 0 0
\(937\) −35.5696 −1.16201 −0.581005 0.813900i \(-0.697340\pi\)
−0.581005 + 0.813900i \(0.697340\pi\)
\(938\) 0 0
\(939\) 7.28452 0.237721
\(940\) 0 0
\(941\) 0.409490 0.0133490 0.00667449 0.999978i \(-0.497875\pi\)
0.00667449 + 0.999978i \(0.497875\pi\)
\(942\) 0 0
\(943\) 2.20091 0.0716716
\(944\) 0 0
\(945\) −3.93200 −0.127908
\(946\) 0 0
\(947\) 43.0270 1.39819 0.699095 0.715029i \(-0.253586\pi\)
0.699095 + 0.715029i \(0.253586\pi\)
\(948\) 0 0
\(949\) 12.7414 0.413603
\(950\) 0 0
\(951\) 8.21892 0.266517
\(952\) 0 0
\(953\) 17.1891 0.556808 0.278404 0.960464i \(-0.410195\pi\)
0.278404 + 0.960464i \(0.410195\pi\)
\(954\) 0 0
\(955\) 10.7514 0.347908
\(956\) 0 0
\(957\) −3.69746 −0.119522
\(958\) 0 0
\(959\) −23.7901 −0.768223
\(960\) 0 0
\(961\) −30.5487 −0.985442
\(962\) 0 0
\(963\) 11.1575 0.359545
\(964\) 0 0
\(965\) 22.0692 0.710432
\(966\) 0 0
\(967\) 8.03839 0.258497 0.129249 0.991612i \(-0.458743\pi\)
0.129249 + 0.991612i \(0.458743\pi\)
\(968\) 0 0
\(969\) 36.2591 1.16481
\(970\) 0 0
\(971\) 45.4945 1.45999 0.729995 0.683453i \(-0.239522\pi\)
0.729995 + 0.683453i \(0.239522\pi\)
\(972\) 0 0
\(973\) −2.78664 −0.0893357
\(974\) 0 0
\(975\) −19.2256 −0.615711
\(976\) 0 0
\(977\) −15.6237 −0.499846 −0.249923 0.968266i \(-0.580405\pi\)
−0.249923 + 0.968266i \(0.580405\pi\)
\(978\) 0 0
\(979\) 17.1672 0.548666
\(980\) 0 0
\(981\) −1.50930 −0.0481882
\(982\) 0 0
\(983\) 13.8606 0.442084 0.221042 0.975264i \(-0.429054\pi\)
0.221042 + 0.975264i \(0.429054\pi\)
\(984\) 0 0
\(985\) −14.0532 −0.447771
\(986\) 0 0
\(987\) −1.60361 −0.0510434
\(988\) 0 0
\(989\) 8.58949 0.273130
\(990\) 0 0
\(991\) −32.4027 −1.02930 −0.514652 0.857399i \(-0.672079\pi\)
−0.514652 + 0.857399i \(0.672079\pi\)
\(992\) 0 0
\(993\) −3.55431 −0.112793
\(994\) 0 0
\(995\) 28.7117 0.910221
\(996\) 0 0
\(997\) −32.7250 −1.03641 −0.518206 0.855256i \(-0.673400\pi\)
−0.518206 + 0.855256i \(0.673400\pi\)
\(998\) 0 0
\(999\) 4.42860 0.140115
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 8004.2.a.e.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.8 9 1.1 even 1 trivial