Properties

Label 324.6.a.d.1.2
Level 324
Weight 6
Character 324.1
Self dual yes
Analytic conductor 51.964
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.9643576194\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 110 x^{3} + 39 x^{2} + 2214 x - 1944\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{7} \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(10.4099\) of defining polynomial
Character \(\chi\) \(=\) 324.1

$q$-expansion

\(f(q)\) \(=\) \(q-28.1436 q^{5} -151.408 q^{7} +O(q^{10})\) \(q-28.1436 q^{5} -151.408 q^{7} +277.747 q^{11} +583.858 q^{13} +1612.01 q^{17} +1368.76 q^{19} -856.028 q^{23} -2332.94 q^{25} -8534.97 q^{29} +2938.38 q^{31} +4261.17 q^{35} +4036.80 q^{37} -18899.6 q^{41} -20317.1 q^{43} +295.779 q^{47} +6117.35 q^{49} -3039.13 q^{53} -7816.81 q^{55} +17236.6 q^{59} +25652.5 q^{61} -16431.9 q^{65} -26280.3 q^{67} -76665.7 q^{71} +1496.33 q^{73} -42053.1 q^{77} +99274.2 q^{79} -50051.3 q^{83} -45367.7 q^{85} -136635. q^{89} -88400.8 q^{91} -38522.0 q^{95} -66649.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 21q^{5} - 29q^{7} + O(q^{10}) \) \( 5q - 21q^{5} - 29q^{7} + 177q^{11} + 181q^{13} - 1140q^{17} - 416q^{19} + 399q^{23} + 4778q^{25} - 6033q^{29} - 2759q^{31} - 18573q^{35} - 7586q^{37} - 18435q^{41} - 1469q^{43} - 25155q^{47} + 4056q^{49} - 58422q^{53} + 7389q^{55} - 90537q^{59} - 1403q^{61} - 148407q^{65} - 13907q^{67} - 114684q^{71} + 7600q^{73} - 211983q^{77} - 29993q^{79} - 228951q^{83} + 49662q^{85} - 299166q^{89} + 62465q^{91} - 394764q^{95} - 40541q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −28.1436 −0.503449 −0.251724 0.967799i \(-0.580998\pi\)
−0.251724 + 0.967799i \(0.580998\pi\)
\(6\) 0 0
\(7\) −151.408 −1.16789 −0.583947 0.811792i \(-0.698492\pi\)
−0.583947 + 0.811792i \(0.698492\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 277.747 0.692098 0.346049 0.938216i \(-0.387523\pi\)
0.346049 + 0.938216i \(0.387523\pi\)
\(12\) 0 0
\(13\) 583.858 0.958185 0.479092 0.877765i \(-0.340966\pi\)
0.479092 + 0.877765i \(0.340966\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1612.01 1.35283 0.676417 0.736519i \(-0.263532\pi\)
0.676417 + 0.736519i \(0.263532\pi\)
\(18\) 0 0
\(19\) 1368.76 0.869851 0.434925 0.900467i \(-0.356775\pi\)
0.434925 + 0.900467i \(0.356775\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −856.028 −0.337418 −0.168709 0.985666i \(-0.553960\pi\)
−0.168709 + 0.985666i \(0.553960\pi\)
\(24\) 0 0
\(25\) −2332.94 −0.746539
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8534.97 −1.88455 −0.942274 0.334844i \(-0.891317\pi\)
−0.942274 + 0.334844i \(0.891317\pi\)
\(30\) 0 0
\(31\) 2938.38 0.549166 0.274583 0.961563i \(-0.411460\pi\)
0.274583 + 0.961563i \(0.411460\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4261.17 0.587975
\(36\) 0 0
\(37\) 4036.80 0.484767 0.242383 0.970181i \(-0.422071\pi\)
0.242383 + 0.970181i \(0.422071\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −18899.6 −1.75587 −0.877937 0.478776i \(-0.841081\pi\)
−0.877937 + 0.478776i \(0.841081\pi\)
\(42\) 0 0
\(43\) −20317.1 −1.67568 −0.837840 0.545916i \(-0.816182\pi\)
−0.837840 + 0.545916i \(0.816182\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 295.779 0.0195309 0.00976546 0.999952i \(-0.496892\pi\)
0.00976546 + 0.999952i \(0.496892\pi\)
\(48\) 0 0
\(49\) 6117.35 0.363976
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3039.13 −0.148614 −0.0743069 0.997235i \(-0.523674\pi\)
−0.0743069 + 0.997235i \(0.523674\pi\)
\(54\) 0 0
\(55\) −7816.81 −0.348436
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 17236.6 0.644647 0.322324 0.946630i \(-0.395536\pi\)
0.322324 + 0.946630i \(0.395536\pi\)
\(60\) 0 0
\(61\) 25652.5 0.882683 0.441342 0.897339i \(-0.354503\pi\)
0.441342 + 0.897339i \(0.354503\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −16431.9 −0.482397
\(66\) 0 0
\(67\) −26280.3 −0.715225 −0.357613 0.933870i \(-0.616409\pi\)
−0.357613 + 0.933870i \(0.616409\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −76665.7 −1.80491 −0.902454 0.430786i \(-0.858236\pi\)
−0.902454 + 0.430786i \(0.858236\pi\)
\(72\) 0 0
\(73\) 1496.33 0.0328640 0.0164320 0.999865i \(-0.494769\pi\)
0.0164320 + 0.999865i \(0.494769\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −42053.1 −0.808297
\(78\) 0 0
\(79\) 99274.2 1.78965 0.894826 0.446414i \(-0.147299\pi\)
0.894826 + 0.446414i \(0.147299\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −50051.3 −0.797481 −0.398740 0.917064i \(-0.630553\pi\)
−0.398740 + 0.917064i \(0.630553\pi\)
\(84\) 0 0
\(85\) −45367.7 −0.681082
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −136635. −1.82847 −0.914235 0.405185i \(-0.867207\pi\)
−0.914235 + 0.405185i \(0.867207\pi\)
\(90\) 0 0
\(91\) −88400.8 −1.11906
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −38522.0 −0.437925
\(96\) 0 0
\(97\) −66649.9 −0.719234 −0.359617 0.933100i \(-0.617093\pi\)
−0.359617 + 0.933100i \(0.617093\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 24677.4 0.240711 0.120355 0.992731i \(-0.461597\pi\)
0.120355 + 0.992731i \(0.461597\pi\)
\(102\) 0 0
\(103\) 115767. 1.07521 0.537603 0.843198i \(-0.319330\pi\)
0.537603 + 0.843198i \(0.319330\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −84364.3 −0.712360 −0.356180 0.934417i \(-0.615921\pi\)
−0.356180 + 0.934417i \(0.615921\pi\)
\(108\) 0 0
\(109\) 198400. 1.59947 0.799735 0.600354i \(-0.204973\pi\)
0.799735 + 0.600354i \(0.204973\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −228905. −1.68640 −0.843199 0.537602i \(-0.819330\pi\)
−0.843199 + 0.537602i \(0.819330\pi\)
\(114\) 0 0
\(115\) 24091.8 0.169873
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −244070. −1.57997
\(120\) 0 0
\(121\) −83907.7 −0.521001
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 153606. 0.879293
\(126\) 0 0
\(127\) −246629. −1.35686 −0.678430 0.734665i \(-0.737339\pi\)
−0.678430 + 0.734665i \(0.737339\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 169225. 0.861563 0.430781 0.902456i \(-0.358238\pi\)
0.430781 + 0.902456i \(0.358238\pi\)
\(132\) 0 0
\(133\) −207242. −1.01589
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −99970.6 −0.455062 −0.227531 0.973771i \(-0.573065\pi\)
−0.227531 + 0.973771i \(0.573065\pi\)
\(138\) 0 0
\(139\) −37398.7 −0.164180 −0.0820899 0.996625i \(-0.526159\pi\)
−0.0820899 + 0.996625i \(0.526159\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 162165. 0.663158
\(144\) 0 0
\(145\) 240205. 0.948773
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −254534. −0.939246 −0.469623 0.882867i \(-0.655610\pi\)
−0.469623 + 0.882867i \(0.655610\pi\)
\(150\) 0 0
\(151\) −236726. −0.844895 −0.422448 0.906387i \(-0.638829\pi\)
−0.422448 + 0.906387i \(0.638829\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −82696.6 −0.276477
\(156\) 0 0
\(157\) 254812. 0.825031 0.412515 0.910951i \(-0.364650\pi\)
0.412515 + 0.910951i \(0.364650\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 129609. 0.394069
\(162\) 0 0
\(163\) 215050. 0.633973 0.316987 0.948430i \(-0.397329\pi\)
0.316987 + 0.948430i \(0.397329\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −192337. −0.533669 −0.266834 0.963742i \(-0.585978\pi\)
−0.266834 + 0.963742i \(0.585978\pi\)
\(168\) 0 0
\(169\) −30402.3 −0.0818823
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8301.15 0.0210874 0.0105437 0.999944i \(-0.496644\pi\)
0.0105437 + 0.999944i \(0.496644\pi\)
\(174\) 0 0
\(175\) 353225. 0.871879
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −574496. −1.34015 −0.670077 0.742292i \(-0.733739\pi\)
−0.670077 + 0.742292i \(0.733739\pi\)
\(180\) 0 0
\(181\) −224707. −0.509823 −0.254912 0.966964i \(-0.582046\pi\)
−0.254912 + 0.966964i \(0.582046\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −113610. −0.244055
\(186\) 0 0
\(187\) 447730. 0.936293
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 478082. 0.948241 0.474121 0.880460i \(-0.342766\pi\)
0.474121 + 0.880460i \(0.342766\pi\)
\(192\) 0 0
\(193\) −526050. −1.01656 −0.508281 0.861191i \(-0.669719\pi\)
−0.508281 + 0.861191i \(0.669719\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 222278. 0.408067 0.204034 0.978964i \(-0.434595\pi\)
0.204034 + 0.978964i \(0.434595\pi\)
\(198\) 0 0
\(199\) −109696. −0.196363 −0.0981813 0.995169i \(-0.531303\pi\)
−0.0981813 + 0.995169i \(0.531303\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.29226e6 2.20095
\(204\) 0 0
\(205\) 531904. 0.883992
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 380170. 0.602022
\(210\) 0 0
\(211\) −618873. −0.956963 −0.478482 0.878098i \(-0.658813\pi\)
−0.478482 + 0.878098i \(0.658813\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 571797. 0.843618
\(216\) 0 0
\(217\) −444893. −0.641367
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 941183. 1.29626
\(222\) 0 0
\(223\) 463306. 0.623886 0.311943 0.950101i \(-0.399020\pi\)
0.311943 + 0.950101i \(0.399020\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.14665e6 1.47695 0.738477 0.674279i \(-0.235545\pi\)
0.738477 + 0.674279i \(0.235545\pi\)
\(228\) 0 0
\(229\) −351055. −0.442371 −0.221185 0.975232i \(-0.570993\pi\)
−0.221185 + 0.975232i \(0.570993\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −814275. −0.982610 −0.491305 0.870988i \(-0.663480\pi\)
−0.491305 + 0.870988i \(0.663480\pi\)
\(234\) 0 0
\(235\) −8324.30 −0.00983281
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 714903. 0.809566 0.404783 0.914413i \(-0.367347\pi\)
0.404783 + 0.914413i \(0.367347\pi\)
\(240\) 0 0
\(241\) 45296.9 0.0502373 0.0251186 0.999684i \(-0.492004\pi\)
0.0251186 + 0.999684i \(0.492004\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −172164. −0.183243
\(246\) 0 0
\(247\) 799165. 0.833477
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.35110e6 1.35364 0.676822 0.736147i \(-0.263357\pi\)
0.676822 + 0.736147i \(0.263357\pi\)
\(252\) 0 0
\(253\) −237759. −0.233526
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.20019e6 −1.13349 −0.566747 0.823892i \(-0.691798\pi\)
−0.566747 + 0.823892i \(0.691798\pi\)
\(258\) 0 0
\(259\) −611203. −0.566156
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.31360e6 1.17105 0.585525 0.810654i \(-0.300888\pi\)
0.585525 + 0.810654i \(0.300888\pi\)
\(264\) 0 0
\(265\) 85532.1 0.0748195
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −321363. −0.270779 −0.135389 0.990792i \(-0.543229\pi\)
−0.135389 + 0.990792i \(0.543229\pi\)
\(270\) 0 0
\(271\) 384928. 0.318388 0.159194 0.987247i \(-0.449110\pi\)
0.159194 + 0.987247i \(0.449110\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −647966. −0.516678
\(276\) 0 0
\(277\) −1.69593e6 −1.32803 −0.664015 0.747719i \(-0.731149\pi\)
−0.664015 + 0.747719i \(0.731149\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.06134e6 −0.801841 −0.400920 0.916113i \(-0.631310\pi\)
−0.400920 + 0.916113i \(0.631310\pi\)
\(282\) 0 0
\(283\) −384626. −0.285478 −0.142739 0.989760i \(-0.545591\pi\)
−0.142739 + 0.989760i \(0.545591\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.86155e6 2.05067
\(288\) 0 0
\(289\) 1.17871e6 0.830158
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.33529e6 0.908667 0.454334 0.890832i \(-0.349877\pi\)
0.454334 + 0.890832i \(0.349877\pi\)
\(294\) 0 0
\(295\) −485101. −0.324547
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −499799. −0.323309
\(300\) 0 0
\(301\) 3.07617e6 1.95702
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −721954. −0.444386
\(306\) 0 0
\(307\) 636269. 0.385296 0.192648 0.981268i \(-0.438292\pi\)
0.192648 + 0.981268i \(0.438292\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 539781. 0.316458 0.158229 0.987402i \(-0.449422\pi\)
0.158229 + 0.987402i \(0.449422\pi\)
\(312\) 0 0
\(313\) 1.95321e6 1.12691 0.563454 0.826148i \(-0.309472\pi\)
0.563454 + 0.826148i \(0.309472\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.56397e6 −0.874136 −0.437068 0.899428i \(-0.643983\pi\)
−0.437068 + 0.899428i \(0.643983\pi\)
\(318\) 0 0
\(319\) −2.37056e6 −1.30429
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.20646e6 1.17676
\(324\) 0 0
\(325\) −1.36210e6 −0.715323
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −44783.3 −0.0228100
\(330\) 0 0
\(331\) −2.56000e6 −1.28431 −0.642154 0.766575i \(-0.721959\pi\)
−0.642154 + 0.766575i \(0.721959\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 739622. 0.360079
\(336\) 0 0
\(337\) −599292. −0.287451 −0.143725 0.989618i \(-0.545908\pi\)
−0.143725 + 0.989618i \(0.545908\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 816125. 0.380076
\(342\) 0 0
\(343\) 1.61850e6 0.742808
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.68773e6 −1.19829 −0.599144 0.800641i \(-0.704492\pi\)
−0.599144 + 0.800641i \(0.704492\pi\)
\(348\) 0 0
\(349\) 671614. 0.295159 0.147580 0.989050i \(-0.452852\pi\)
0.147580 + 0.989050i \(0.452852\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.52976e6 0.653412 0.326706 0.945126i \(-0.394061\pi\)
0.326706 + 0.945126i \(0.394061\pi\)
\(354\) 0 0
\(355\) 2.15765e6 0.908678
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.27210e6 −1.33996 −0.669978 0.742381i \(-0.733697\pi\)
−0.669978 + 0.742381i \(0.733697\pi\)
\(360\) 0 0
\(361\) −602583. −0.243360
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −42112.2 −0.0165453
\(366\) 0 0
\(367\) 627142. 0.243053 0.121526 0.992588i \(-0.461221\pi\)
0.121526 + 0.992588i \(0.461221\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 460148. 0.173565
\(372\) 0 0
\(373\) 132374. 0.0492639 0.0246320 0.999697i \(-0.492159\pi\)
0.0246320 + 0.999697i \(0.492159\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.98322e6 −1.80574
\(378\) 0 0
\(379\) −163225. −0.0583700 −0.0291850 0.999574i \(-0.509291\pi\)
−0.0291850 + 0.999574i \(0.509291\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 796601. 0.277488 0.138744 0.990328i \(-0.455693\pi\)
0.138744 + 0.990328i \(0.455693\pi\)
\(384\) 0 0
\(385\) 1.18353e6 0.406936
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.09490e6 −1.37205 −0.686024 0.727579i \(-0.740646\pi\)
−0.686024 + 0.727579i \(0.740646\pi\)
\(390\) 0 0
\(391\) −1.37992e6 −0.456471
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.79394e6 −0.900998
\(396\) 0 0
\(397\) −5.58867e6 −1.77964 −0.889820 0.456312i \(-0.849170\pi\)
−0.889820 + 0.456312i \(0.849170\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.23353e6 −0.693634 −0.346817 0.937933i \(-0.612738\pi\)
−0.346817 + 0.937933i \(0.612738\pi\)
\(402\) 0 0
\(403\) 1.71560e6 0.526202
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.12121e6 0.335506
\(408\) 0 0
\(409\) −4.60512e6 −1.36123 −0.680617 0.732640i \(-0.738288\pi\)
−0.680617 + 0.732640i \(0.738288\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.60976e6 −0.752880
\(414\) 0 0
\(415\) 1.40863e6 0.401491
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.14213e6 −0.317819 −0.158909 0.987293i \(-0.550798\pi\)
−0.158909 + 0.987293i \(0.550798\pi\)
\(420\) 0 0
\(421\) 1.92477e6 0.529264 0.264632 0.964349i \(-0.414749\pi\)
0.264632 + 0.964349i \(0.414749\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.76071e6 −1.00994
\(426\) 0 0
\(427\) −3.88399e6 −1.03088
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.21303e6 −0.573843 −0.286922 0.957954i \(-0.592632\pi\)
−0.286922 + 0.957954i \(0.592632\pi\)
\(432\) 0 0
\(433\) 3.00235e6 0.769558 0.384779 0.923009i \(-0.374278\pi\)
0.384779 + 0.923009i \(0.374278\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.17170e6 −0.293503
\(438\) 0 0
\(439\) 1.92171e6 0.475913 0.237956 0.971276i \(-0.423522\pi\)
0.237956 + 0.971276i \(0.423522\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.84852e6 0.447521 0.223761 0.974644i \(-0.428167\pi\)
0.223761 + 0.974644i \(0.428167\pi\)
\(444\) 0 0
\(445\) 3.84541e6 0.920540
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.45640e6 −0.809110 −0.404555 0.914514i \(-0.632574\pi\)
−0.404555 + 0.914514i \(0.632574\pi\)
\(450\) 0 0
\(451\) −5.24931e6 −1.21524
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.48792e6 0.563388
\(456\) 0 0
\(457\) 3.58637e6 0.803276 0.401638 0.915798i \(-0.368441\pi\)
0.401638 + 0.915798i \(0.368441\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.43550e6 −1.84867 −0.924333 0.381586i \(-0.875378\pi\)
−0.924333 + 0.381586i \(0.875378\pi\)
\(462\) 0 0
\(463\) 4.94474e6 1.07199 0.535995 0.844221i \(-0.319937\pi\)
0.535995 + 0.844221i \(0.319937\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.12589e6 0.875438 0.437719 0.899112i \(-0.355787\pi\)
0.437719 + 0.899112i \(0.355787\pi\)
\(468\) 0 0
\(469\) 3.97904e6 0.835307
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.64302e6 −1.15973
\(474\) 0 0
\(475\) −3.19324e6 −0.649378
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.12260e6 0.223557 0.111778 0.993733i \(-0.464345\pi\)
0.111778 + 0.993733i \(0.464345\pi\)
\(480\) 0 0
\(481\) 2.35692e6 0.464496
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.87577e6 0.362097
\(486\) 0 0
\(487\) 8.11380e6 1.55025 0.775126 0.631807i \(-0.217687\pi\)
0.775126 + 0.631807i \(0.217687\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.32962e6 0.623291 0.311645 0.950199i \(-0.399120\pi\)
0.311645 + 0.950199i \(0.399120\pi\)
\(492\) 0 0
\(493\) −1.37584e7 −2.54948
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.16078e7 2.10794
\(498\) 0 0
\(499\) 1.88126e6 0.338218 0.169109 0.985597i \(-0.445911\pi\)
0.169109 + 0.985597i \(0.445911\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.96749e6 1.40411 0.702056 0.712122i \(-0.252266\pi\)
0.702056 + 0.712122i \(0.252266\pi\)
\(504\) 0 0
\(505\) −694511. −0.121186
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.08112e7 −1.84961 −0.924805 0.380441i \(-0.875772\pi\)
−0.924805 + 0.380441i \(0.875772\pi\)
\(510\) 0 0
\(511\) −226556. −0.0383817
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.25810e6 −0.541311
\(516\) 0 0
\(517\) 82151.7 0.0135173
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.19670e6 0.193148 0.0965740 0.995326i \(-0.469212\pi\)
0.0965740 + 0.995326i \(0.469212\pi\)
\(522\) 0 0
\(523\) −6.43371e6 −1.02851 −0.514254 0.857638i \(-0.671931\pi\)
−0.514254 + 0.857638i \(0.671931\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.73668e6 0.742929
\(528\) 0 0
\(529\) −5.70356e6 −0.886149
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.10347e7 −1.68245
\(534\) 0 0
\(535\) 2.37432e6 0.358637
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.69907e6 0.251907
\(540\) 0 0
\(541\) 5.85989e6 0.860788 0.430394 0.902641i \(-0.358375\pi\)
0.430394 + 0.902641i \(0.358375\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.58370e6 −0.805251
\(546\) 0 0
\(547\) 5.18872e6 0.741468 0.370734 0.928739i \(-0.379106\pi\)
0.370734 + 0.928739i \(0.379106\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.16824e7 −1.63927
\(552\) 0 0
\(553\) −1.50309e7 −2.09012
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.07405e6 −0.692973 −0.346487 0.938055i \(-0.612625\pi\)
−0.346487 + 0.938055i \(0.612625\pi\)
\(558\) 0 0
\(559\) −1.18623e7 −1.60561
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.10878e6 0.812239 0.406119 0.913820i \(-0.366882\pi\)
0.406119 + 0.913820i \(0.366882\pi\)
\(564\) 0 0
\(565\) 6.44223e6 0.849014
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.08097e7 1.39969 0.699844 0.714296i \(-0.253253\pi\)
0.699844 + 0.714296i \(0.253253\pi\)
\(570\) 0 0
\(571\) 1.20559e7 1.54743 0.773714 0.633535i \(-0.218397\pi\)
0.773714 + 0.633535i \(0.218397\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.99706e6 0.251896
\(576\) 0 0
\(577\) 1.22323e7 1.52957 0.764786 0.644285i \(-0.222845\pi\)
0.764786 + 0.644285i \(0.222845\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.57816e6 0.931373
\(582\) 0 0
\(583\) −844109. −0.102855
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.59634e6 1.02972 0.514859 0.857275i \(-0.327844\pi\)
0.514859 + 0.857275i \(0.327844\pi\)
\(588\) 0 0
\(589\) 4.02195e6 0.477692
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.78455e6 0.675511 0.337756 0.941234i \(-0.390332\pi\)
0.337756 + 0.941234i \(0.390332\pi\)
\(594\) 0 0
\(595\) 6.86903e6 0.795432
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.25113e6 0.597979 0.298989 0.954256i \(-0.403350\pi\)
0.298989 + 0.954256i \(0.403350\pi\)
\(600\) 0 0
\(601\) −6.22338e6 −0.702813 −0.351407 0.936223i \(-0.614297\pi\)
−0.351407 + 0.936223i \(0.614297\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.36147e6 0.262297
\(606\) 0 0
\(607\) −1.58810e7 −1.74947 −0.874733 0.484604i \(-0.838964\pi\)
−0.874733 + 0.484604i \(0.838964\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 172693. 0.0187142
\(612\) 0 0
\(613\) −1.28661e7 −1.38292 −0.691458 0.722417i \(-0.743031\pi\)
−0.691458 + 0.722417i \(0.743031\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.87594e6 0.727141 0.363571 0.931567i \(-0.381558\pi\)
0.363571 + 0.931567i \(0.381558\pi\)
\(618\) 0 0
\(619\) 1.30482e7 1.36875 0.684373 0.729132i \(-0.260076\pi\)
0.684373 + 0.729132i \(0.260076\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.06877e7 2.13546
\(624\) 0 0
\(625\) 2.96739e6 0.303861
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.50734e6 0.655809
\(630\) 0 0
\(631\) −1.52784e7 −1.52758 −0.763790 0.645464i \(-0.776664\pi\)
−0.763790 + 0.645464i \(0.776664\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.94103e6 0.683109
\(636\) 0 0
\(637\) 3.57166e6 0.348756
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.19231e7 1.14616 0.573080 0.819499i \(-0.305748\pi\)
0.573080 + 0.819499i \(0.305748\pi\)
\(642\) 0 0
\(643\) −6.00603e6 −0.572875 −0.286438 0.958099i \(-0.592471\pi\)
−0.286438 + 0.958099i \(0.592471\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.19440e7 −1.12173 −0.560864 0.827908i \(-0.689531\pi\)
−0.560864 + 0.827908i \(0.689531\pi\)
\(648\) 0 0
\(649\) 4.78742e6 0.446159
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.06593e6 0.281371 0.140686 0.990054i \(-0.455069\pi\)
0.140686 + 0.990054i \(0.455069\pi\)
\(654\) 0 0
\(655\) −4.76261e6 −0.433753
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.10130e6 0.278183 0.139092 0.990280i \(-0.455582\pi\)
0.139092 + 0.990280i \(0.455582\pi\)
\(660\) 0 0
\(661\) 1.29771e7 1.15524 0.577621 0.816305i \(-0.303981\pi\)
0.577621 + 0.816305i \(0.303981\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.83253e6 0.511450
\(666\) 0 0
\(667\) 7.30618e6 0.635881
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.12490e6 0.610903
\(672\) 0 0
\(673\) 2.21406e7 1.88431 0.942154 0.335182i \(-0.108798\pi\)
0.942154 + 0.335182i \(0.108798\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.59970e6 −0.134143 −0.0670713 0.997748i \(-0.521366\pi\)
−0.0670713 + 0.997748i \(0.521366\pi\)
\(678\) 0 0
\(679\) 1.00913e7 0.839989
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.36986e7 −1.12363 −0.561816 0.827262i \(-0.689897\pi\)
−0.561816 + 0.827262i \(0.689897\pi\)
\(684\) 0 0
\(685\) 2.81353e6 0.229100
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.77442e6 −0.142400
\(690\) 0 0
\(691\) −4.18213e6 −0.333198 −0.166599 0.986025i \(-0.553279\pi\)
−0.166599 + 0.986025i \(0.553279\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.05254e6 0.0826561
\(696\) 0 0
\(697\) −3.04663e7 −2.37540
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.47134e7 1.13089 0.565443 0.824788i \(-0.308705\pi\)
0.565443 + 0.824788i \(0.308705\pi\)
\(702\) 0 0
\(703\) 5.52543e6 0.421675
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.73635e6 −0.281125
\(708\) 0 0
\(709\) −8.51816e6 −0.636400 −0.318200 0.948024i \(-0.603078\pi\)
−0.318200 + 0.948024i \(0.603078\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.51533e6 −0.185299
\(714\) 0 0
\(715\) −4.56391e6 −0.333866
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.20183e7 0.867003 0.433502 0.901153i \(-0.357278\pi\)
0.433502 + 0.901153i \(0.357278\pi\)
\(720\) 0 0
\(721\) −1.75280e7 −1.25573
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.99115e7 1.40689
\(726\) 0 0
\(727\) 2.98514e6 0.209473 0.104737 0.994500i \(-0.466600\pi\)
0.104737 + 0.994500i \(0.466600\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.27513e7 −2.26691
\(732\) 0 0
\(733\) 2.11351e7 1.45293 0.726464 0.687205i \(-0.241162\pi\)
0.726464 + 0.687205i \(0.241162\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.29926e6 −0.495006
\(738\) 0 0
\(739\) 1.33961e7 0.902337 0.451168 0.892439i \(-0.351007\pi\)
0.451168 + 0.892439i \(0.351007\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.19084e7 −0.791371 −0.395685 0.918386i \(-0.629493\pi\)
−0.395685 + 0.918386i \(0.629493\pi\)
\(744\) 0 0
\(745\) 7.16350e6 0.472862
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.27734e7 0.831961
\(750\) 0 0
\(751\) 4.66367e6 0.301737 0.150868 0.988554i \(-0.451793\pi\)
0.150868 + 0.988554i \(0.451793\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.66232e6 0.425361
\(756\) 0 0
\(757\) 138599. 0.00879065 0.00439532 0.999990i \(-0.498601\pi\)
0.00439532 + 0.999990i \(0.498601\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.39712e6 −0.150047 −0.0750237 0.997182i \(-0.523903\pi\)
−0.0750237 + 0.997182i \(0.523903\pi\)
\(762\) 0 0
\(763\) −3.00394e7 −1.86801
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.00638e7 0.617691
\(768\) 0 0
\(769\) 799479. 0.0487519 0.0243759 0.999703i \(-0.492240\pi\)
0.0243759 + 0.999703i \(0.492240\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.07743e6 −0.0648546 −0.0324273 0.999474i \(-0.510324\pi\)
−0.0324273 + 0.999474i \(0.510324\pi\)
\(774\) 0 0
\(775\) −6.85505e6 −0.409974
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.58691e7 −1.52735
\(780\) 0 0
\(781\) −2.12937e7 −1.24917
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.17132e6 −0.415361
\(786\) 0 0
\(787\) 3.07091e6 0.176738 0.0883692 0.996088i \(-0.471834\pi\)
0.0883692 + 0.996088i \(0.471834\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.46581e7 1.96953
\(792\) 0 0
\(793\) 1.49774e7 0.845774
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.49848e7 −0.835615 −0.417808 0.908536i \(-0.637201\pi\)
−0.417808 + 0.908536i \(0.637201\pi\)
\(798\) 0 0
\(799\) 476797. 0.0264221
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 415601. 0.0227451
\(804\) 0 0
\(805\) −3.64768e6 −0.198393
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.11069e6 0.167103 0.0835516 0.996503i \(-0.473374\pi\)
0.0835516 + 0.996503i \(0.473374\pi\)
\(810\) 0 0
\(811\) 1.12694e7 0.601659 0.300830 0.953678i \(-0.402736\pi\)
0.300830 + 0.953678i \(0.402736\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.05229e6 −0.319173
\(816\) 0 0
\(817\) −2.78093e7 −1.45759
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.09537e7 −1.08493 −0.542467 0.840077i \(-0.682510\pi\)
−0.542467 + 0.840077i \(0.682510\pi\)
\(822\) 0 0
\(823\) 1.94816e6 0.100259 0.0501296 0.998743i \(-0.484037\pi\)
0.0501296 + 0.998743i \(0.484037\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.11807e7 −1.58534 −0.792671 0.609650i \(-0.791310\pi\)
−0.792671 + 0.609650i \(0.791310\pi\)
\(828\) 0 0
\(829\) −1.67293e7 −0.845455 −0.422727 0.906257i \(-0.638927\pi\)
−0.422727 + 0.906257i \(0.638927\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.86120e6 0.492399
\(834\) 0 0
\(835\) 5.41306e6 0.268675
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 8.27321e6 0.405760 0.202880 0.979204i \(-0.434970\pi\)
0.202880 + 0.979204i \(0.434970\pi\)
\(840\) 0 0
\(841\) 5.23346e7 2.55152
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 855632. 0.0412235
\(846\) 0 0
\(847\) 1.27043e7 0.608473
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.45562e6 −0.163569
\(852\) 0 0
\(853\) 3.46938e7 1.63260 0.816300 0.577628i \(-0.196022\pi\)
0.816300 + 0.577628i \(0.196022\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.03725e6 −0.234284 −0.117142 0.993115i \(-0.537373\pi\)
−0.117142 + 0.993115i \(0.537373\pi\)
\(858\) 0 0
\(859\) −4.27819e7 −1.97823 −0.989116 0.147139i \(-0.952994\pi\)
−0.989116 + 0.147139i \(0.952994\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.50904e6 −0.0689721 −0.0344861 0.999405i \(-0.510979\pi\)
−0.0344861 + 0.999405i \(0.510979\pi\)
\(864\) 0 0
\(865\) −233624. −0.0106164
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.75731e7 1.23861
\(870\) 0 0
\(871\) −1.53440e7 −0.685318
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.32572e7 −1.02692
\(876\) 0 0
\(877\) −3.29399e7 −1.44618 −0.723092 0.690752i \(-0.757280\pi\)
−0.723092 + 0.690752i \(0.757280\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.62885e7 −1.14110 −0.570552 0.821261i \(-0.693271\pi\)
−0.570552 + 0.821261i \(0.693271\pi\)
\(882\) 0 0
\(883\) −2.07499e7 −0.895602 −0.447801 0.894133i \(-0.647793\pi\)
−0.447801 + 0.894133i \(0.647793\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.28582e7 −1.40228 −0.701140 0.713023i \(-0.747325\pi\)
−0.701140 + 0.713023i \(0.747325\pi\)
\(888\) 0 0
\(889\) 3.73416e7 1.58467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 404852. 0.0169890
\(894\) 0 0
\(895\) 1.61684e7 0.674698
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.50790e7 −1.03493
\(900\) 0 0
\(901\) −4.89909e6 −0.201050
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.32407e6 0.256670
\(906\) 0 0
\(907\) 4.71672e7 1.90380 0.951900 0.306408i \(-0.0991270\pi\)
0.951900 + 0.306408i \(0.0991270\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.80492e7 −1.51897 −0.759486 0.650524i \(-0.774549\pi\)
−0.759486 + 0.650524i \(0.774549\pi\)
\(912\) 0 0
\(913\) −1.39016e7 −0.551935
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.56220e7 −1.00621
\(918\) 0 0
\(919\) 2.54292e7 0.993215 0.496607 0.867975i \(-0.334579\pi\)
0.496607 + 0.867975i \(0.334579\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.47619e7 −1.72943
\(924\) 0 0
\(925\) −9.41759e6 −0.361898
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.40094e7 0.532576 0.266288 0.963894i \(-0.414203\pi\)
0.266288 + 0.963894i \(0.414203\pi\)
\(930\) 0 0
\(931\) 8.37320e6 0.316605
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.26007e7 −0.471375
\(936\) 0 0
\(937\) −1.29313e7 −0.481164 −0.240582 0.970629i \(-0.577338\pi\)
−0.240582 + 0.970629i \(0.577338\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.59945e6 0.353405 0.176702 0.984264i \(-0.443457\pi\)
0.176702 + 0.984264i \(0.443457\pi\)
\(942\) 0 0
\(943\) 1.61786e7 0.592464
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.43337e7 0.519379 0.259690 0.965692i \(-0.416380\pi\)
0.259690 + 0.965692i \(0.416380\pi\)
\(948\) 0 0
\(949\) 873646. 0.0314898
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.14184e7 −1.12060 −0.560302 0.828289i \(-0.689315\pi\)
−0.560302 + 0.828289i \(0.689315\pi\)
\(954\) 0 0
\(955\) −1.34550e7 −0.477391
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.51363e7 0.531464
\(960\) 0 0
\(961\) −1.99951e7 −0.698417
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.48050e7 0.511786
\(966\) 0 0
\(967\) −1.35264e7 −0.465174 −0.232587 0.972576i \(-0.574719\pi\)
−0.232587 + 0.972576i \(0.574719\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.90534e7 −1.32926 −0.664632 0.747171i \(-0.731412\pi\)
−0.664632 + 0.747171i \(0.731412\pi\)
\(972\) 0 0
\(973\) 5.66246e6 0.191744
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.89049e7 −0.633632 −0.316816 0.948487i \(-0.602614\pi\)
−0.316816 + 0.948487i \(0.602614\pi\)
\(978\) 0 0
\(979\) −3.79500e7 −1.26548
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.90025e6 0.194754 0.0973771 0.995248i \(-0.468955\pi\)
0.0973771 + 0.995248i \(0.468955\pi\)
\(984\) 0 0
\(985\) −6.25572e6 −0.205441
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.73920e7 0.565405
\(990\) 0 0
\(991\) −3.67515e7 −1.18875 −0.594375 0.804188i \(-0.702601\pi\)
−0.594375 + 0.804188i \(0.702601\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.08725e6 0.0988585
\(996\) 0 0
\(997\) 2.92149e7 0.930820 0.465410 0.885095i \(-0.345907\pi\)
0.465410 + 0.885095i \(0.345907\pi\)
\(998\) 0 0
\(999\) 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 324.6.a.d.1.2 5
3.2 odd 2 324.6.a.e.1.4 5
9.2 odd 6 36.6.e.a.13.1 10
9.4 even 3 108.6.e.a.73.4 10
9.5 odd 6 36.6.e.a.25.1 yes 10
9.7 even 3 108.6.e.a.37.4 10
36.7 odd 6 432.6.i.d.145.4 10
36.11 even 6 144.6.i.d.49.5 10
36.23 even 6 144.6.i.d.97.5 10
36.31 odd 6 432.6.i.d.289.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.6.e.a.13.1 10 9.2 odd 6
36.6.e.a.25.1 yes 10 9.5 odd 6
108.6.e.a.37.4 10 9.7 even 3
108.6.e.a.73.4 10 9.4 even 3
144.6.i.d.49.5 10 36.11 even 6
144.6.i.d.97.5 10 36.23 even 6
324.6.a.d.1.2 5 1.1 even 1 trivial
324.6.a.e.1.4 5 3.2 odd 2
432.6.i.d.145.4 10 36.7 odd 6
432.6.i.d.289.4 10 36.31 odd 6