Properties

Label 1104.6.a.r.1.3
Level $1104$
Weight $6$
Character 1104.1
Self dual yes
Analytic conductor $177.064$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(177.063737074\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-7.90234\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.00000 q^{3} +53.3906 q^{5} -89.8688 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +53.3906 q^{5} -89.8688 q^{7} +81.0000 q^{9} -225.013 q^{11} +725.747 q^{13} -480.515 q^{15} +44.9540 q^{17} -1212.50 q^{19} +808.819 q^{21} +529.000 q^{23} -274.446 q^{25} -729.000 q^{27} +2592.49 q^{29} -585.421 q^{31} +2025.11 q^{33} -4798.15 q^{35} -3849.04 q^{37} -6531.73 q^{39} -4299.92 q^{41} +20567.4 q^{43} +4324.64 q^{45} +5221.50 q^{47} -8730.61 q^{49} -404.586 q^{51} -15753.7 q^{53} -12013.6 q^{55} +10912.5 q^{57} +13780.6 q^{59} -18988.5 q^{61} -7279.37 q^{63} +38748.1 q^{65} -1951.43 q^{67} -4761.00 q^{69} -75281.4 q^{71} -4177.11 q^{73} +2470.01 q^{75} +20221.6 q^{77} +95896.7 q^{79} +6561.00 q^{81} -13366.7 q^{83} +2400.12 q^{85} -23332.4 q^{87} +50932.1 q^{89} -65222.0 q^{91} +5268.79 q^{93} -64735.9 q^{95} +44280.7 q^{97} -18226.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) 53.3906 0.955080 0.477540 0.878610i \(-0.341529\pi\)
0.477540 + 0.878610i \(0.341529\pi\)
\(6\) 0 0
\(7\) −89.8688 −0.693208 −0.346604 0.938012i \(-0.612665\pi\)
−0.346604 + 0.938012i \(0.612665\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −225.013 −0.560693 −0.280347 0.959899i \(-0.590449\pi\)
−0.280347 + 0.959899i \(0.590449\pi\)
\(12\) 0 0
\(13\) 725.747 1.19104 0.595521 0.803340i \(-0.296946\pi\)
0.595521 + 0.803340i \(0.296946\pi\)
\(14\) 0 0
\(15\) −480.515 −0.551416
\(16\) 0 0
\(17\) 44.9540 0.0377265 0.0188632 0.999822i \(-0.493995\pi\)
0.0188632 + 0.999822i \(0.493995\pi\)
\(18\) 0 0
\(19\) −1212.50 −0.770542 −0.385271 0.922803i \(-0.625892\pi\)
−0.385271 + 0.922803i \(0.625892\pi\)
\(20\) 0 0
\(21\) 808.819 0.400224
\(22\) 0 0
\(23\) 529.000 0.208514
\(24\) 0 0
\(25\) −274.446 −0.0878226
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 2592.49 0.572430 0.286215 0.958165i \(-0.407603\pi\)
0.286215 + 0.958165i \(0.407603\pi\)
\(30\) 0 0
\(31\) −585.421 −0.109412 −0.0547059 0.998503i \(-0.517422\pi\)
−0.0547059 + 0.998503i \(0.517422\pi\)
\(32\) 0 0
\(33\) 2025.11 0.323716
\(34\) 0 0
\(35\) −4798.15 −0.662069
\(36\) 0 0
\(37\) −3849.04 −0.462219 −0.231109 0.972928i \(-0.574236\pi\)
−0.231109 + 0.972928i \(0.574236\pi\)
\(38\) 0 0
\(39\) −6531.73 −0.687648
\(40\) 0 0
\(41\) −4299.92 −0.399485 −0.199743 0.979848i \(-0.564011\pi\)
−0.199743 + 0.979848i \(0.564011\pi\)
\(42\) 0 0
\(43\) 20567.4 1.69632 0.848162 0.529737i \(-0.177709\pi\)
0.848162 + 0.529737i \(0.177709\pi\)
\(44\) 0 0
\(45\) 4324.64 0.318360
\(46\) 0 0
\(47\) 5221.50 0.344787 0.172393 0.985028i \(-0.444850\pi\)
0.172393 + 0.985028i \(0.444850\pi\)
\(48\) 0 0
\(49\) −8730.61 −0.519462
\(50\) 0 0
\(51\) −404.586 −0.0217814
\(52\) 0 0
\(53\) −15753.7 −0.770360 −0.385180 0.922841i \(-0.625861\pi\)
−0.385180 + 0.922841i \(0.625861\pi\)
\(54\) 0 0
\(55\) −12013.6 −0.535507
\(56\) 0 0
\(57\) 10912.5 0.444873
\(58\) 0 0
\(59\) 13780.6 0.515392 0.257696 0.966226i \(-0.417037\pi\)
0.257696 + 0.966226i \(0.417037\pi\)
\(60\) 0 0
\(61\) −18988.5 −0.653380 −0.326690 0.945131i \(-0.605933\pi\)
−0.326690 + 0.945131i \(0.605933\pi\)
\(62\) 0 0
\(63\) −7279.37 −0.231069
\(64\) 0 0
\(65\) 38748.1 1.13754
\(66\) 0 0
\(67\) −1951.43 −0.0531088 −0.0265544 0.999647i \(-0.508454\pi\)
−0.0265544 + 0.999647i \(0.508454\pi\)
\(68\) 0 0
\(69\) −4761.00 −0.120386
\(70\) 0 0
\(71\) −75281.4 −1.77232 −0.886160 0.463380i \(-0.846637\pi\)
−0.886160 + 0.463380i \(0.846637\pi\)
\(72\) 0 0
\(73\) −4177.11 −0.0917422 −0.0458711 0.998947i \(-0.514606\pi\)
−0.0458711 + 0.998947i \(0.514606\pi\)
\(74\) 0 0
\(75\) 2470.01 0.0507044
\(76\) 0 0
\(77\) 20221.6 0.388677
\(78\) 0 0
\(79\) 95896.7 1.72877 0.864383 0.502835i \(-0.167710\pi\)
0.864383 + 0.502835i \(0.167710\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −13366.7 −0.212976 −0.106488 0.994314i \(-0.533961\pi\)
−0.106488 + 0.994314i \(0.533961\pi\)
\(84\) 0 0
\(85\) 2400.12 0.0360318
\(86\) 0 0
\(87\) −23332.4 −0.330492
\(88\) 0 0
\(89\) 50932.1 0.681580 0.340790 0.940139i \(-0.389305\pi\)
0.340790 + 0.940139i \(0.389305\pi\)
\(90\) 0 0
\(91\) −65222.0 −0.825640
\(92\) 0 0
\(93\) 5268.79 0.0631690
\(94\) 0 0
\(95\) −64735.9 −0.735929
\(96\) 0 0
\(97\) 44280.7 0.477843 0.238921 0.971039i \(-0.423206\pi\)
0.238921 + 0.971039i \(0.423206\pi\)
\(98\) 0 0
\(99\) −18226.0 −0.186898
\(100\) 0 0
\(101\) 198402. 1.93527 0.967637 0.252346i \(-0.0812021\pi\)
0.967637 + 0.252346i \(0.0812021\pi\)
\(102\) 0 0
\(103\) −201915. −1.87532 −0.937658 0.347559i \(-0.887011\pi\)
−0.937658 + 0.347559i \(0.887011\pi\)
\(104\) 0 0
\(105\) 43183.3 0.382246
\(106\) 0 0
\(107\) −37688.4 −0.318235 −0.159117 0.987260i \(-0.550865\pi\)
−0.159117 + 0.987260i \(0.550865\pi\)
\(108\) 0 0
\(109\) −86493.8 −0.697298 −0.348649 0.937253i \(-0.613360\pi\)
−0.348649 + 0.937253i \(0.613360\pi\)
\(110\) 0 0
\(111\) 34641.3 0.266862
\(112\) 0 0
\(113\) −150822. −1.11114 −0.555571 0.831469i \(-0.687500\pi\)
−0.555571 + 0.831469i \(0.687500\pi\)
\(114\) 0 0
\(115\) 28243.6 0.199148
\(116\) 0 0
\(117\) 58785.5 0.397014
\(118\) 0 0
\(119\) −4039.96 −0.0261523
\(120\) 0 0
\(121\) −110420. −0.685623
\(122\) 0 0
\(123\) 38699.3 0.230643
\(124\) 0 0
\(125\) −181498. −1.03896
\(126\) 0 0
\(127\) 47605.4 0.261907 0.130953 0.991389i \(-0.458196\pi\)
0.130953 + 0.991389i \(0.458196\pi\)
\(128\) 0 0
\(129\) −185107. −0.979373
\(130\) 0 0
\(131\) 260443. 1.32597 0.662985 0.748632i \(-0.269289\pi\)
0.662985 + 0.748632i \(0.269289\pi\)
\(132\) 0 0
\(133\) 108966. 0.534146
\(134\) 0 0
\(135\) −38921.7 −0.183805
\(136\) 0 0
\(137\) 121613. 0.553580 0.276790 0.960930i \(-0.410729\pi\)
0.276790 + 0.960930i \(0.410729\pi\)
\(138\) 0 0
\(139\) −61341.2 −0.269287 −0.134643 0.990894i \(-0.542989\pi\)
−0.134643 + 0.990894i \(0.542989\pi\)
\(140\) 0 0
\(141\) −46993.5 −0.199063
\(142\) 0 0
\(143\) −163302. −0.667809
\(144\) 0 0
\(145\) 138415. 0.546716
\(146\) 0 0
\(147\) 78575.4 0.299912
\(148\) 0 0
\(149\) −171257. −0.631950 −0.315975 0.948767i \(-0.602332\pi\)
−0.315975 + 0.948767i \(0.602332\pi\)
\(150\) 0 0
\(151\) 459607. 1.64038 0.820189 0.572092i \(-0.193868\pi\)
0.820189 + 0.572092i \(0.193868\pi\)
\(152\) 0 0
\(153\) 3641.27 0.0125755
\(154\) 0 0
\(155\) −31256.0 −0.104497
\(156\) 0 0
\(157\) 387631. 1.25508 0.627538 0.778586i \(-0.284063\pi\)
0.627538 + 0.778586i \(0.284063\pi\)
\(158\) 0 0
\(159\) 141784. 0.444767
\(160\) 0 0
\(161\) −47540.6 −0.144544
\(162\) 0 0
\(163\) 286971. 0.845998 0.422999 0.906130i \(-0.360977\pi\)
0.422999 + 0.906130i \(0.360977\pi\)
\(164\) 0 0
\(165\) 108122. 0.309175
\(166\) 0 0
\(167\) −580711. −1.61127 −0.805636 0.592411i \(-0.798176\pi\)
−0.805636 + 0.592411i \(0.798176\pi\)
\(168\) 0 0
\(169\) 155416. 0.418581
\(170\) 0 0
\(171\) −98212.2 −0.256847
\(172\) 0 0
\(173\) −191878. −0.487427 −0.243714 0.969847i \(-0.578366\pi\)
−0.243714 + 0.969847i \(0.578366\pi\)
\(174\) 0 0
\(175\) 24664.1 0.0608793
\(176\) 0 0
\(177\) −124025. −0.297562
\(178\) 0 0
\(179\) −18757.8 −0.0437571 −0.0218785 0.999761i \(-0.506965\pi\)
−0.0218785 + 0.999761i \(0.506965\pi\)
\(180\) 0 0
\(181\) 47967.6 0.108831 0.0544154 0.998518i \(-0.482670\pi\)
0.0544154 + 0.998518i \(0.482670\pi\)
\(182\) 0 0
\(183\) 170897. 0.377229
\(184\) 0 0
\(185\) −205502. −0.441456
\(186\) 0 0
\(187\) −10115.2 −0.0211530
\(188\) 0 0
\(189\) 65514.3 0.133408
\(190\) 0 0
\(191\) 14294.3 0.0283518 0.0141759 0.999900i \(-0.495488\pi\)
0.0141759 + 0.999900i \(0.495488\pi\)
\(192\) 0 0
\(193\) −422764. −0.816968 −0.408484 0.912765i \(-0.633942\pi\)
−0.408484 + 0.912765i \(0.633942\pi\)
\(194\) 0 0
\(195\) −348733. −0.656759
\(196\) 0 0
\(197\) −311005. −0.570955 −0.285477 0.958385i \(-0.592152\pi\)
−0.285477 + 0.958385i \(0.592152\pi\)
\(198\) 0 0
\(199\) −60066.4 −0.107522 −0.0537612 0.998554i \(-0.517121\pi\)
−0.0537612 + 0.998554i \(0.517121\pi\)
\(200\) 0 0
\(201\) 17562.9 0.0306624
\(202\) 0 0
\(203\) −232984. −0.396813
\(204\) 0 0
\(205\) −229575. −0.381540
\(206\) 0 0
\(207\) 42849.0 0.0695048
\(208\) 0 0
\(209\) 272827. 0.432038
\(210\) 0 0
\(211\) 1.08896e6 1.68385 0.841926 0.539593i \(-0.181422\pi\)
0.841926 + 0.539593i \(0.181422\pi\)
\(212\) 0 0
\(213\) 677533. 1.02325
\(214\) 0 0
\(215\) 1.09811e6 1.62012
\(216\) 0 0
\(217\) 52611.1 0.0758452
\(218\) 0 0
\(219\) 37594.0 0.0529674
\(220\) 0 0
\(221\) 32625.2 0.0449338
\(222\) 0 0
\(223\) −365915. −0.492741 −0.246370 0.969176i \(-0.579238\pi\)
−0.246370 + 0.969176i \(0.579238\pi\)
\(224\) 0 0
\(225\) −22230.1 −0.0292742
\(226\) 0 0
\(227\) −1.08830e6 −1.40179 −0.700894 0.713265i \(-0.747216\pi\)
−0.700894 + 0.713265i \(0.747216\pi\)
\(228\) 0 0
\(229\) −1.45107e6 −1.82853 −0.914263 0.405121i \(-0.867229\pi\)
−0.914263 + 0.405121i \(0.867229\pi\)
\(230\) 0 0
\(231\) −181994. −0.224403
\(232\) 0 0
\(233\) −1.27334e6 −1.53657 −0.768286 0.640106i \(-0.778890\pi\)
−0.768286 + 0.640106i \(0.778890\pi\)
\(234\) 0 0
\(235\) 278779. 0.329299
\(236\) 0 0
\(237\) −863070. −0.998103
\(238\) 0 0
\(239\) −799348. −0.905193 −0.452596 0.891715i \(-0.649502\pi\)
−0.452596 + 0.891715i \(0.649502\pi\)
\(240\) 0 0
\(241\) 727560. 0.806913 0.403456 0.914999i \(-0.367809\pi\)
0.403456 + 0.914999i \(0.367809\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) −466132. −0.496128
\(246\) 0 0
\(247\) −879966. −0.917748
\(248\) 0 0
\(249\) 120301. 0.122962
\(250\) 0 0
\(251\) −960924. −0.962730 −0.481365 0.876520i \(-0.659859\pi\)
−0.481365 + 0.876520i \(0.659859\pi\)
\(252\) 0 0
\(253\) −119032. −0.116913
\(254\) 0 0
\(255\) −21601.1 −0.0208030
\(256\) 0 0
\(257\) −1.57691e6 −1.48927 −0.744637 0.667470i \(-0.767377\pi\)
−0.744637 + 0.667470i \(0.767377\pi\)
\(258\) 0 0
\(259\) 345908. 0.320414
\(260\) 0 0
\(261\) 209992. 0.190810
\(262\) 0 0
\(263\) −1.42682e6 −1.27198 −0.635991 0.771696i \(-0.719409\pi\)
−0.635991 + 0.771696i \(0.719409\pi\)
\(264\) 0 0
\(265\) −841101. −0.735755
\(266\) 0 0
\(267\) −458389. −0.393510
\(268\) 0 0
\(269\) −341906. −0.288089 −0.144044 0.989571i \(-0.546011\pi\)
−0.144044 + 0.989571i \(0.546011\pi\)
\(270\) 0 0
\(271\) −723709. −0.598605 −0.299303 0.954158i \(-0.596754\pi\)
−0.299303 + 0.954158i \(0.596754\pi\)
\(272\) 0 0
\(273\) 586998. 0.476683
\(274\) 0 0
\(275\) 61753.7 0.0492415
\(276\) 0 0
\(277\) −1.82928e6 −1.43245 −0.716226 0.697868i \(-0.754132\pi\)
−0.716226 + 0.697868i \(0.754132\pi\)
\(278\) 0 0
\(279\) −47419.1 −0.0364706
\(280\) 0 0
\(281\) 1.14935e6 0.868337 0.434169 0.900832i \(-0.357042\pi\)
0.434169 + 0.900832i \(0.357042\pi\)
\(282\) 0 0
\(283\) 1.83262e6 1.36021 0.680105 0.733115i \(-0.261934\pi\)
0.680105 + 0.733115i \(0.261934\pi\)
\(284\) 0 0
\(285\) 582623. 0.424889
\(286\) 0 0
\(287\) 386429. 0.276926
\(288\) 0 0
\(289\) −1.41784e6 −0.998577
\(290\) 0 0
\(291\) −398526. −0.275883
\(292\) 0 0
\(293\) 1.33169e6 0.906221 0.453111 0.891454i \(-0.350314\pi\)
0.453111 + 0.891454i \(0.350314\pi\)
\(294\) 0 0
\(295\) 735754. 0.492241
\(296\) 0 0
\(297\) 164034. 0.107905
\(298\) 0 0
\(299\) 383920. 0.248349
\(300\) 0 0
\(301\) −1.84837e6 −1.17591
\(302\) 0 0
\(303\) −1.78562e6 −1.11733
\(304\) 0 0
\(305\) −1.01381e6 −0.624031
\(306\) 0 0
\(307\) −3.00299e6 −1.81848 −0.909239 0.416275i \(-0.863335\pi\)
−0.909239 + 0.416275i \(0.863335\pi\)
\(308\) 0 0
\(309\) 1.81723e6 1.08271
\(310\) 0 0
\(311\) −2.75093e6 −1.61279 −0.806397 0.591374i \(-0.798586\pi\)
−0.806397 + 0.591374i \(0.798586\pi\)
\(312\) 0 0
\(313\) −119894. −0.0691727 −0.0345864 0.999402i \(-0.511011\pi\)
−0.0345864 + 0.999402i \(0.511011\pi\)
\(314\) 0 0
\(315\) −388650. −0.220690
\(316\) 0 0
\(317\) 514254. 0.287428 0.143714 0.989619i \(-0.454095\pi\)
0.143714 + 0.989619i \(0.454095\pi\)
\(318\) 0 0
\(319\) −583343. −0.320957
\(320\) 0 0
\(321\) 339195. 0.183733
\(322\) 0 0
\(323\) −54506.6 −0.0290698
\(324\) 0 0
\(325\) −199178. −0.104600
\(326\) 0 0
\(327\) 778444. 0.402585
\(328\) 0 0
\(329\) −469250. −0.239009
\(330\) 0 0
\(331\) −2.46021e6 −1.23425 −0.617124 0.786866i \(-0.711702\pi\)
−0.617124 + 0.786866i \(0.711702\pi\)
\(332\) 0 0
\(333\) −311772. −0.154073
\(334\) 0 0
\(335\) −104188. −0.0507231
\(336\) 0 0
\(337\) −1.42694e6 −0.684432 −0.342216 0.939621i \(-0.611177\pi\)
−0.342216 + 0.939621i \(0.611177\pi\)
\(338\) 0 0
\(339\) 1.35740e6 0.641518
\(340\) 0 0
\(341\) 131727. 0.0613465
\(342\) 0 0
\(343\) 2.29503e6 1.05330
\(344\) 0 0
\(345\) −254193. −0.114978
\(346\) 0 0
\(347\) −1.63440e6 −0.728676 −0.364338 0.931267i \(-0.618705\pi\)
−0.364338 + 0.931267i \(0.618705\pi\)
\(348\) 0 0
\(349\) −2.65498e6 −1.16680 −0.583402 0.812184i \(-0.698279\pi\)
−0.583402 + 0.812184i \(0.698279\pi\)
\(350\) 0 0
\(351\) −529070. −0.229216
\(352\) 0 0
\(353\) 934019. 0.398950 0.199475 0.979903i \(-0.436076\pi\)
0.199475 + 0.979903i \(0.436076\pi\)
\(354\) 0 0
\(355\) −4.01932e6 −1.69271
\(356\) 0 0
\(357\) 36359.7 0.0150990
\(358\) 0 0
\(359\) −2.98227e6 −1.22127 −0.610634 0.791913i \(-0.709085\pi\)
−0.610634 + 0.791913i \(0.709085\pi\)
\(360\) 0 0
\(361\) −1.00595e6 −0.406265
\(362\) 0 0
\(363\) 993783. 0.395845
\(364\) 0 0
\(365\) −223019. −0.0876211
\(366\) 0 0
\(367\) −4.17728e6 −1.61893 −0.809466 0.587167i \(-0.800243\pi\)
−0.809466 + 0.587167i \(0.800243\pi\)
\(368\) 0 0
\(369\) −348294. −0.133162
\(370\) 0 0
\(371\) 1.41577e6 0.534020
\(372\) 0 0
\(373\) −1.12351e6 −0.418124 −0.209062 0.977902i \(-0.567041\pi\)
−0.209062 + 0.977902i \(0.567041\pi\)
\(374\) 0 0
\(375\) 1.63349e6 0.599842
\(376\) 0 0
\(377\) 1.88149e6 0.681788
\(378\) 0 0
\(379\) −3.45021e6 −1.23381 −0.616904 0.787039i \(-0.711613\pi\)
−0.616904 + 0.787039i \(0.711613\pi\)
\(380\) 0 0
\(381\) −428448. −0.151212
\(382\) 0 0
\(383\) 573250. 0.199686 0.0998430 0.995003i \(-0.468166\pi\)
0.0998430 + 0.995003i \(0.468166\pi\)
\(384\) 0 0
\(385\) 1.07964e6 0.371218
\(386\) 0 0
\(387\) 1.66596e6 0.565441
\(388\) 0 0
\(389\) −5.10842e6 −1.71164 −0.855820 0.517274i \(-0.826947\pi\)
−0.855820 + 0.517274i \(0.826947\pi\)
\(390\) 0 0
\(391\) 23780.7 0.00786651
\(392\) 0 0
\(393\) −2.34398e6 −0.765550
\(394\) 0 0
\(395\) 5.11998e6 1.65111
\(396\) 0 0
\(397\) −417826. −0.133051 −0.0665256 0.997785i \(-0.521191\pi\)
−0.0665256 + 0.997785i \(0.521191\pi\)
\(398\) 0 0
\(399\) −980690. −0.308389
\(400\) 0 0
\(401\) 3.79619e6 1.17893 0.589463 0.807795i \(-0.299339\pi\)
0.589463 + 0.807795i \(0.299339\pi\)
\(402\) 0 0
\(403\) −424868. −0.130314
\(404\) 0 0
\(405\) 350296. 0.106120
\(406\) 0 0
\(407\) 866082. 0.259163
\(408\) 0 0
\(409\) 4.95389e6 1.46433 0.732164 0.681129i \(-0.238511\pi\)
0.732164 + 0.681129i \(0.238511\pi\)
\(410\) 0 0
\(411\) −1.09452e6 −0.319609
\(412\) 0 0
\(413\) −1.23844e6 −0.357274
\(414\) 0 0
\(415\) −713658. −0.203409
\(416\) 0 0
\(417\) 552071. 0.155473
\(418\) 0 0
\(419\) 2.38845e6 0.664631 0.332316 0.943168i \(-0.392170\pi\)
0.332316 + 0.943168i \(0.392170\pi\)
\(420\) 0 0
\(421\) −582907. −0.160285 −0.0801427 0.996783i \(-0.525538\pi\)
−0.0801427 + 0.996783i \(0.525538\pi\)
\(422\) 0 0
\(423\) 422942. 0.114929
\(424\) 0 0
\(425\) −12337.4 −0.00331324
\(426\) 0 0
\(427\) 1.70647e6 0.452929
\(428\) 0 0
\(429\) 1.46972e6 0.385560
\(430\) 0 0
\(431\) 1.17212e6 0.303935 0.151967 0.988386i \(-0.451439\pi\)
0.151967 + 0.988386i \(0.451439\pi\)
\(432\) 0 0
\(433\) −5.34133e6 −1.36908 −0.684542 0.728974i \(-0.739998\pi\)
−0.684542 + 0.728974i \(0.739998\pi\)
\(434\) 0 0
\(435\) −1.24573e6 −0.315647
\(436\) 0 0
\(437\) −641410. −0.160669
\(438\) 0 0
\(439\) −3.68652e6 −0.912967 −0.456483 0.889732i \(-0.650891\pi\)
−0.456483 + 0.889732i \(0.650891\pi\)
\(440\) 0 0
\(441\) −707179. −0.173154
\(442\) 0 0
\(443\) 705619. 0.170829 0.0854143 0.996346i \(-0.472779\pi\)
0.0854143 + 0.996346i \(0.472779\pi\)
\(444\) 0 0
\(445\) 2.71930e6 0.650963
\(446\) 0 0
\(447\) 1.54131e6 0.364857
\(448\) 0 0
\(449\) 442688. 0.103629 0.0518146 0.998657i \(-0.483500\pi\)
0.0518146 + 0.998657i \(0.483500\pi\)
\(450\) 0 0
\(451\) 967537. 0.223989
\(452\) 0 0
\(453\) −4.13646e6 −0.947073
\(454\) 0 0
\(455\) −3.48224e6 −0.788552
\(456\) 0 0
\(457\) 4.85276e6 1.08692 0.543461 0.839434i \(-0.317113\pi\)
0.543461 + 0.839434i \(0.317113\pi\)
\(458\) 0 0
\(459\) −32771.5 −0.00726046
\(460\) 0 0
\(461\) 8.87833e6 1.94571 0.972857 0.231408i \(-0.0743331\pi\)
0.972857 + 0.231408i \(0.0743331\pi\)
\(462\) 0 0
\(463\) 3.58364e6 0.776911 0.388456 0.921467i \(-0.373009\pi\)
0.388456 + 0.921467i \(0.373009\pi\)
\(464\) 0 0
\(465\) 281304. 0.0603314
\(466\) 0 0
\(467\) 6.13857e6 1.30249 0.651246 0.758867i \(-0.274247\pi\)
0.651246 + 0.758867i \(0.274247\pi\)
\(468\) 0 0
\(469\) 175373. 0.0368155
\(470\) 0 0
\(471\) −3.48868e6 −0.724618
\(472\) 0 0
\(473\) −4.62793e6 −0.951117
\(474\) 0 0
\(475\) 332764. 0.0676710
\(476\) 0 0
\(477\) −1.27605e6 −0.256787
\(478\) 0 0
\(479\) −505577. −0.100681 −0.0503406 0.998732i \(-0.516031\pi\)
−0.0503406 + 0.998732i \(0.516031\pi\)
\(480\) 0 0
\(481\) −2.79343e6 −0.550522
\(482\) 0 0
\(483\) 427865. 0.0834525
\(484\) 0 0
\(485\) 2.36417e6 0.456378
\(486\) 0 0
\(487\) 751626. 0.143608 0.0718041 0.997419i \(-0.477124\pi\)
0.0718041 + 0.997419i \(0.477124\pi\)
\(488\) 0 0
\(489\) −2.58274e6 −0.488437
\(490\) 0 0
\(491\) 2.19083e6 0.410115 0.205057 0.978750i \(-0.434262\pi\)
0.205057 + 0.978750i \(0.434262\pi\)
\(492\) 0 0
\(493\) 116543. 0.0215957
\(494\) 0 0
\(495\) −973098. −0.178502
\(496\) 0 0
\(497\) 6.76545e6 1.22859
\(498\) 0 0
\(499\) −2.21183e6 −0.397649 −0.198824 0.980035i \(-0.563712\pi\)
−0.198824 + 0.980035i \(0.563712\pi\)
\(500\) 0 0
\(501\) 5.22640e6 0.930268
\(502\) 0 0
\(503\) 1.05756e7 1.86373 0.931866 0.362803i \(-0.118180\pi\)
0.931866 + 0.362803i \(0.118180\pi\)
\(504\) 0 0
\(505\) 1.05928e7 1.84834
\(506\) 0 0
\(507\) −1.39874e6 −0.241668
\(508\) 0 0
\(509\) −1.00676e7 −1.72239 −0.861195 0.508275i \(-0.830283\pi\)
−0.861195 + 0.508275i \(0.830283\pi\)
\(510\) 0 0
\(511\) 375392. 0.0635964
\(512\) 0 0
\(513\) 883910. 0.148291
\(514\) 0 0
\(515\) −1.07803e7 −1.79108
\(516\) 0 0
\(517\) −1.17490e6 −0.193320
\(518\) 0 0
\(519\) 1.72690e6 0.281416
\(520\) 0 0
\(521\) 7.71005e6 1.24441 0.622204 0.782855i \(-0.286237\pi\)
0.622204 + 0.782855i \(0.286237\pi\)
\(522\) 0 0
\(523\) −153540. −0.0245453 −0.0122726 0.999925i \(-0.503907\pi\)
−0.0122726 + 0.999925i \(0.503907\pi\)
\(524\) 0 0
\(525\) −221977. −0.0351487
\(526\) 0 0
\(527\) −26317.0 −0.00412772
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 0 0
\(531\) 1.11623e6 0.171797
\(532\) 0 0
\(533\) −3.12066e6 −0.475804
\(534\) 0 0
\(535\) −2.01220e6 −0.303940
\(536\) 0 0
\(537\) 168820. 0.0252632
\(538\) 0 0
\(539\) 1.96450e6 0.291259
\(540\) 0 0
\(541\) 267983. 0.0393653 0.0196827 0.999806i \(-0.493734\pi\)
0.0196827 + 0.999806i \(0.493734\pi\)
\(542\) 0 0
\(543\) −431708. −0.0628334
\(544\) 0 0
\(545\) −4.61795e6 −0.665976
\(546\) 0 0
\(547\) −6.44942e6 −0.921621 −0.460810 0.887499i \(-0.652441\pi\)
−0.460810 + 0.887499i \(0.652441\pi\)
\(548\) 0 0
\(549\) −1.53807e6 −0.217793
\(550\) 0 0
\(551\) −3.14338e6 −0.441081
\(552\) 0 0
\(553\) −8.61812e6 −1.19839
\(554\) 0 0
\(555\) 1.84952e6 0.254875
\(556\) 0 0
\(557\) −1.36650e7 −1.86626 −0.933132 0.359534i \(-0.882936\pi\)
−0.933132 + 0.359534i \(0.882936\pi\)
\(558\) 0 0
\(559\) 1.49268e7 2.02039
\(560\) 0 0
\(561\) 91037.0 0.0122127
\(562\) 0 0
\(563\) −564712. −0.0750855 −0.0375427 0.999295i \(-0.511953\pi\)
−0.0375427 + 0.999295i \(0.511953\pi\)
\(564\) 0 0
\(565\) −8.05250e6 −1.06123
\(566\) 0 0
\(567\) −589629. −0.0770231
\(568\) 0 0
\(569\) 5.68519e6 0.736147 0.368074 0.929797i \(-0.380017\pi\)
0.368074 + 0.929797i \(0.380017\pi\)
\(570\) 0 0
\(571\) −5.94914e6 −0.763597 −0.381798 0.924246i \(-0.624695\pi\)
−0.381798 + 0.924246i \(0.624695\pi\)
\(572\) 0 0
\(573\) −128649. −0.0163689
\(574\) 0 0
\(575\) −145182. −0.0183123
\(576\) 0 0
\(577\) −7.28162e6 −0.910518 −0.455259 0.890359i \(-0.650453\pi\)
−0.455259 + 0.890359i \(0.650453\pi\)
\(578\) 0 0
\(579\) 3.80488e6 0.471677
\(580\) 0 0
\(581\) 1.20125e6 0.147637
\(582\) 0 0
\(583\) 3.54479e6 0.431935
\(584\) 0 0
\(585\) 3.13859e6 0.379180
\(586\) 0 0
\(587\) −9.14151e6 −1.09502 −0.547511 0.836799i \(-0.684425\pi\)
−0.547511 + 0.836799i \(0.684425\pi\)
\(588\) 0 0
\(589\) 709821. 0.0843064
\(590\) 0 0
\(591\) 2.79904e6 0.329641
\(592\) 0 0
\(593\) 6.14133e6 0.717176 0.358588 0.933496i \(-0.383258\pi\)
0.358588 + 0.933496i \(0.383258\pi\)
\(594\) 0 0
\(595\) −215696. −0.0249775
\(596\) 0 0
\(597\) 540598. 0.0620781
\(598\) 0 0
\(599\) 6.53791e6 0.744512 0.372256 0.928130i \(-0.378584\pi\)
0.372256 + 0.928130i \(0.378584\pi\)
\(600\) 0 0
\(601\) −9.92849e6 −1.12124 −0.560618 0.828075i \(-0.689436\pi\)
−0.560618 + 0.828075i \(0.689436\pi\)
\(602\) 0 0
\(603\) −158066. −0.0177029
\(604\) 0 0
\(605\) −5.89540e6 −0.654825
\(606\) 0 0
\(607\) 2.59619e6 0.285999 0.143000 0.989723i \(-0.454325\pi\)
0.143000 + 0.989723i \(0.454325\pi\)
\(608\) 0 0
\(609\) 2.09685e6 0.229100
\(610\) 0 0
\(611\) 3.78949e6 0.410656
\(612\) 0 0
\(613\) −4.12434e6 −0.443306 −0.221653 0.975126i \(-0.571145\pi\)
−0.221653 + 0.975126i \(0.571145\pi\)
\(614\) 0 0
\(615\) 2.06618e6 0.220282
\(616\) 0 0
\(617\) 1.17375e7 1.24126 0.620631 0.784103i \(-0.286876\pi\)
0.620631 + 0.784103i \(0.286876\pi\)
\(618\) 0 0
\(619\) −6.90815e6 −0.724662 −0.362331 0.932050i \(-0.618019\pi\)
−0.362331 + 0.932050i \(0.618019\pi\)
\(620\) 0 0
\(621\) −385641. −0.0401286
\(622\) 0 0
\(623\) −4.57721e6 −0.472477
\(624\) 0 0
\(625\) −8.83266e6 −0.904465
\(626\) 0 0
\(627\) −2.45544e6 −0.249437
\(628\) 0 0
\(629\) −173030. −0.0174379
\(630\) 0 0
\(631\) −793455. −0.0793321 −0.0396660 0.999213i \(-0.512629\pi\)
−0.0396660 + 0.999213i \(0.512629\pi\)
\(632\) 0 0
\(633\) −9.80060e6 −0.972173
\(634\) 0 0
\(635\) 2.54168e6 0.250142
\(636\) 0 0
\(637\) −6.33621e6 −0.618702
\(638\) 0 0
\(639\) −6.09780e6 −0.590773
\(640\) 0 0
\(641\) 3.78009e6 0.363377 0.181689 0.983356i \(-0.441844\pi\)
0.181689 + 0.983356i \(0.441844\pi\)
\(642\) 0 0
\(643\) −1.21266e7 −1.15667 −0.578336 0.815799i \(-0.696298\pi\)
−0.578336 + 0.815799i \(0.696298\pi\)
\(644\) 0 0
\(645\) −9.88296e6 −0.935380
\(646\) 0 0
\(647\) 6.56812e6 0.616851 0.308425 0.951249i \(-0.400198\pi\)
0.308425 + 0.951249i \(0.400198\pi\)
\(648\) 0 0
\(649\) −3.10081e6 −0.288977
\(650\) 0 0
\(651\) −473500. −0.0437892
\(652\) 0 0
\(653\) −9.97904e6 −0.915811 −0.457905 0.889001i \(-0.651400\pi\)
−0.457905 + 0.889001i \(0.651400\pi\)
\(654\) 0 0
\(655\) 1.39052e7 1.26641
\(656\) 0 0
\(657\) −338346. −0.0305807
\(658\) 0 0
\(659\) −1.22411e7 −1.09801 −0.549006 0.835818i \(-0.684994\pi\)
−0.549006 + 0.835818i \(0.684994\pi\)
\(660\) 0 0
\(661\) −7.26433e6 −0.646684 −0.323342 0.946282i \(-0.604806\pi\)
−0.323342 + 0.946282i \(0.604806\pi\)
\(662\) 0 0
\(663\) −293627. −0.0259425
\(664\) 0 0
\(665\) 5.81773e6 0.510152
\(666\) 0 0
\(667\) 1.37143e6 0.119360
\(668\) 0 0
\(669\) 3.29324e6 0.284484
\(670\) 0 0
\(671\) 4.27265e6 0.366346
\(672\) 0 0
\(673\) 1.76906e7 1.50558 0.752791 0.658260i \(-0.228707\pi\)
0.752791 + 0.658260i \(0.228707\pi\)
\(674\) 0 0
\(675\) 200071. 0.0169015
\(676\) 0 0
\(677\) 8.88669e6 0.745193 0.372596 0.927994i \(-0.378468\pi\)
0.372596 + 0.927994i \(0.378468\pi\)
\(678\) 0 0
\(679\) −3.97945e6 −0.331244
\(680\) 0 0
\(681\) 9.79467e6 0.809323
\(682\) 0 0
\(683\) 3.93961e6 0.323148 0.161574 0.986861i \(-0.448343\pi\)
0.161574 + 0.986861i \(0.448343\pi\)
\(684\) 0 0
\(685\) 6.49301e6 0.528713
\(686\) 0 0
\(687\) 1.30597e7 1.05570
\(688\) 0 0
\(689\) −1.14332e7 −0.917531
\(690\) 0 0
\(691\) −2.19400e7 −1.74800 −0.873999 0.485927i \(-0.838482\pi\)
−0.873999 + 0.485927i \(0.838482\pi\)
\(692\) 0 0
\(693\) 1.63795e6 0.129559
\(694\) 0 0
\(695\) −3.27504e6 −0.257190
\(696\) 0 0
\(697\) −193299. −0.0150712
\(698\) 0 0
\(699\) 1.14600e7 0.887141
\(700\) 0 0
\(701\) 1.20994e7 0.929969 0.464984 0.885319i \(-0.346060\pi\)
0.464984 + 0.885319i \(0.346060\pi\)
\(702\) 0 0
\(703\) 4.66694e6 0.356159
\(704\) 0 0
\(705\) −2.50901e6 −0.190121
\(706\) 0 0
\(707\) −1.78301e7 −1.34155
\(708\) 0 0
\(709\) 1.95715e7 1.46221 0.731103 0.682267i \(-0.239006\pi\)
0.731103 + 0.682267i \(0.239006\pi\)
\(710\) 0 0
\(711\) 7.76763e6 0.576255
\(712\) 0 0
\(713\) −309688. −0.0228139
\(714\) 0 0
\(715\) −8.71881e6 −0.637811
\(716\) 0 0
\(717\) 7.19413e6 0.522613
\(718\) 0 0
\(719\) 1.98434e7 1.43151 0.715753 0.698354i \(-0.246084\pi\)
0.715753 + 0.698354i \(0.246084\pi\)
\(720\) 0 0
\(721\) 1.81458e7 1.29998
\(722\) 0 0
\(723\) −6.54804e6 −0.465871
\(724\) 0 0
\(725\) −711497. −0.0502722
\(726\) 0 0
\(727\) −2.50056e7 −1.75469 −0.877345 0.479860i \(-0.840688\pi\)
−0.877345 + 0.479860i \(0.840688\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 924588. 0.0639963
\(732\) 0 0
\(733\) 2.53798e7 1.74473 0.872365 0.488855i \(-0.162585\pi\)
0.872365 + 0.488855i \(0.162585\pi\)
\(734\) 0 0
\(735\) 4.19519e6 0.286440
\(736\) 0 0
\(737\) 439097. 0.0297777
\(738\) 0 0
\(739\) −1.50131e7 −1.01125 −0.505625 0.862753i \(-0.668738\pi\)
−0.505625 + 0.862753i \(0.668738\pi\)
\(740\) 0 0
\(741\) 7.91969e6 0.529862
\(742\) 0 0
\(743\) 2.53859e7 1.68702 0.843512 0.537111i \(-0.180484\pi\)
0.843512 + 0.537111i \(0.180484\pi\)
\(744\) 0 0
\(745\) −9.14351e6 −0.603563
\(746\) 0 0
\(747\) −1.08271e6 −0.0709919
\(748\) 0 0
\(749\) 3.38701e6 0.220603
\(750\) 0 0
\(751\) −1.25539e7 −0.812230 −0.406115 0.913822i \(-0.633117\pi\)
−0.406115 + 0.913822i \(0.633117\pi\)
\(752\) 0 0
\(753\) 8.64831e6 0.555833
\(754\) 0 0
\(755\) 2.45387e7 1.56669
\(756\) 0 0
\(757\) 6.15033e6 0.390085 0.195042 0.980795i \(-0.437516\pi\)
0.195042 + 0.980795i \(0.437516\pi\)
\(758\) 0 0
\(759\) 1.07129e6 0.0674995
\(760\) 0 0
\(761\) −6.22830e6 −0.389859 −0.194930 0.980817i \(-0.562448\pi\)
−0.194930 + 0.980817i \(0.562448\pi\)
\(762\) 0 0
\(763\) 7.77309e6 0.483373
\(764\) 0 0
\(765\) 194410. 0.0120106
\(766\) 0 0
\(767\) 1.00012e7 0.613853
\(768\) 0 0
\(769\) 8.72415e6 0.531995 0.265997 0.963974i \(-0.414299\pi\)
0.265997 + 0.963974i \(0.414299\pi\)
\(770\) 0 0
\(771\) 1.41922e7 0.859832
\(772\) 0 0
\(773\) −7.23571e6 −0.435545 −0.217772 0.976000i \(-0.569879\pi\)
−0.217772 + 0.976000i \(0.569879\pi\)
\(774\) 0 0
\(775\) 160666. 0.00960883
\(776\) 0 0
\(777\) −3.11317e6 −0.184991
\(778\) 0 0
\(779\) 5.21364e6 0.307820
\(780\) 0 0
\(781\) 1.69393e7 0.993728
\(782\) 0 0
\(783\) −1.88992e6 −0.110164
\(784\) 0 0
\(785\) 2.06959e7 1.19870
\(786\) 0 0
\(787\) 1.35431e7 0.779440 0.389720 0.920933i \(-0.372572\pi\)
0.389720 + 0.920933i \(0.372572\pi\)
\(788\) 0 0
\(789\) 1.28414e7 0.734379
\(790\) 0 0
\(791\) 1.35542e7 0.770253
\(792\) 0 0
\(793\) −1.37809e7 −0.778203
\(794\) 0 0
\(795\) 7.56991e6 0.424788
\(796\) 0 0
\(797\) 1.01044e7 0.563461 0.281731 0.959494i \(-0.409092\pi\)
0.281731 + 0.959494i \(0.409092\pi\)
\(798\) 0 0
\(799\) 234727. 0.0130076
\(800\) 0 0
\(801\) 4.12550e6 0.227193
\(802\) 0 0
\(803\) 939904. 0.0514392
\(804\) 0 0
\(805\) −2.53822e6 −0.138051
\(806\) 0 0
\(807\) 3.07716e6 0.166328
\(808\) 0 0
\(809\) −3.23912e6 −0.174003 −0.0870013 0.996208i \(-0.527728\pi\)
−0.0870013 + 0.996208i \(0.527728\pi\)
\(810\) 0 0
\(811\) −6.20038e6 −0.331029 −0.165515 0.986207i \(-0.552928\pi\)
−0.165515 + 0.986207i \(0.552928\pi\)
\(812\) 0 0
\(813\) 6.51338e6 0.345605
\(814\) 0 0
\(815\) 1.53216e7 0.807995
\(816\) 0 0
\(817\) −2.49379e7 −1.30709
\(818\) 0 0
\(819\) −5.28298e6 −0.275213
\(820\) 0 0
\(821\) 1.75836e7 0.910439 0.455219 0.890379i \(-0.349561\pi\)
0.455219 + 0.890379i \(0.349561\pi\)
\(822\) 0 0
\(823\) 3.80649e7 1.95896 0.979478 0.201549i \(-0.0645975\pi\)
0.979478 + 0.201549i \(0.0645975\pi\)
\(824\) 0 0
\(825\) −555784. −0.0284296
\(826\) 0 0
\(827\) −8.39081e6 −0.426619 −0.213310 0.976985i \(-0.568424\pi\)
−0.213310 + 0.976985i \(0.568424\pi\)
\(828\) 0 0
\(829\) 394283. 0.0199261 0.00996305 0.999950i \(-0.496829\pi\)
0.00996305 + 0.999950i \(0.496829\pi\)
\(830\) 0 0
\(831\) 1.64635e7 0.827027
\(832\) 0 0
\(833\) −392476. −0.0195975
\(834\) 0 0
\(835\) −3.10045e7 −1.53889
\(836\) 0 0
\(837\) 426772. 0.0210563
\(838\) 0 0
\(839\) −1.63376e7 −0.801276 −0.400638 0.916236i \(-0.631212\pi\)
−0.400638 + 0.916236i \(0.631212\pi\)
\(840\) 0 0
\(841\) −1.37901e7 −0.672324
\(842\) 0 0
\(843\) −1.03442e7 −0.501335
\(844\) 0 0
\(845\) 8.29775e6 0.399778
\(846\) 0 0
\(847\) 9.92334e6 0.475280
\(848\) 0 0
\(849\) −1.64936e7 −0.785317
\(850\) 0 0
\(851\) −2.03614e6 −0.0963793
\(852\) 0 0
\(853\) 2.38816e6 0.112380 0.0561901 0.998420i \(-0.482105\pi\)
0.0561901 + 0.998420i \(0.482105\pi\)
\(854\) 0 0
\(855\) −5.24361e6 −0.245310
\(856\) 0 0
\(857\) −1.41990e7 −0.660398 −0.330199 0.943911i \(-0.607116\pi\)
−0.330199 + 0.943911i \(0.607116\pi\)
\(858\) 0 0
\(859\) −1.13234e7 −0.523595 −0.261797 0.965123i \(-0.584315\pi\)
−0.261797 + 0.965123i \(0.584315\pi\)
\(860\) 0 0
\(861\) −3.47786e6 −0.159884
\(862\) 0 0
\(863\) −2.55589e7 −1.16819 −0.584097 0.811684i \(-0.698551\pi\)
−0.584097 + 0.811684i \(0.698551\pi\)
\(864\) 0 0
\(865\) −1.02445e7 −0.465532
\(866\) 0 0
\(867\) 1.27605e7 0.576529
\(868\) 0 0
\(869\) −2.15780e7 −0.969307
\(870\) 0 0
\(871\) −1.41625e6 −0.0632548
\(872\) 0 0
\(873\) 3.58673e6 0.159281
\(874\) 0 0
\(875\) 1.63110e7 0.720214
\(876\) 0 0
\(877\) 1.94757e7 0.855056 0.427528 0.904002i \(-0.359385\pi\)
0.427528 + 0.904002i \(0.359385\pi\)
\(878\) 0 0
\(879\) −1.19852e7 −0.523207
\(880\) 0 0
\(881\) 9.76547e6 0.423890 0.211945 0.977282i \(-0.432020\pi\)
0.211945 + 0.977282i \(0.432020\pi\)
\(882\) 0 0
\(883\) −4.86226e6 −0.209863 −0.104932 0.994479i \(-0.533462\pi\)
−0.104932 + 0.994479i \(0.533462\pi\)
\(884\) 0 0
\(885\) −6.62178e6 −0.284195
\(886\) 0 0
\(887\) 3.00831e7 1.28385 0.641925 0.766767i \(-0.278136\pi\)
0.641925 + 0.766767i \(0.278136\pi\)
\(888\) 0 0
\(889\) −4.27824e6 −0.181556
\(890\) 0 0
\(891\) −1.47631e6 −0.0622992
\(892\) 0 0
\(893\) −6.33105e6 −0.265673
\(894\) 0 0
\(895\) −1.00149e6 −0.0417915
\(896\) 0 0
\(897\) −3.45528e6 −0.143385
\(898\) 0 0
\(899\) −1.51770e6 −0.0626306
\(900\) 0 0
\(901\) −708193. −0.0290630
\(902\) 0 0
\(903\) 1.66353e7 0.678909
\(904\) 0 0
\(905\) 2.56102e6 0.103942
\(906\) 0 0
\(907\) 3.27277e7 1.32098 0.660491 0.750834i \(-0.270348\pi\)
0.660491 + 0.750834i \(0.270348\pi\)
\(908\) 0 0
\(909\) 1.60706e7 0.645091
\(910\) 0 0
\(911\) 1.44550e6 0.0577062 0.0288531 0.999584i \(-0.490814\pi\)
0.0288531 + 0.999584i \(0.490814\pi\)
\(912\) 0 0
\(913\) 3.00769e6 0.119414
\(914\) 0 0
\(915\) 9.12427e6 0.360284
\(916\) 0 0
\(917\) −2.34057e7 −0.919174
\(918\) 0 0
\(919\) −1.02478e7 −0.400258 −0.200129 0.979770i \(-0.564136\pi\)
−0.200129 + 0.979770i \(0.564136\pi\)
\(920\) 0 0
\(921\) 2.70269e7 1.04990
\(922\) 0 0
\(923\) −5.46353e7 −2.11091
\(924\) 0 0
\(925\) 1.05635e6 0.0405933
\(926\) 0 0
\(927\) −1.63551e7 −0.625105
\(928\) 0 0
\(929\) 4.79655e7 1.82343 0.911716 0.410821i \(-0.134758\pi\)
0.911716 + 0.410821i \(0.134758\pi\)
\(930\) 0 0
\(931\) 1.05858e7 0.400268
\(932\) 0 0
\(933\) 2.47584e7 0.931147
\(934\) 0 0
\(935\) −540058. −0.0202028
\(936\) 0 0
\(937\) −1.18919e7 −0.442489 −0.221244 0.975218i \(-0.571012\pi\)
−0.221244 + 0.975218i \(0.571012\pi\)
\(938\) 0 0
\(939\) 1.07904e6 0.0399369
\(940\) 0 0
\(941\) 3.72086e7 1.36984 0.684919 0.728619i \(-0.259838\pi\)
0.684919 + 0.728619i \(0.259838\pi\)
\(942\) 0 0
\(943\) −2.27466e6 −0.0832985
\(944\) 0 0
\(945\) 3.49785e6 0.127415
\(946\) 0 0
\(947\) −2.24159e7 −0.812235 −0.406118 0.913821i \(-0.633118\pi\)
−0.406118 + 0.913821i \(0.633118\pi\)
\(948\) 0 0
\(949\) −3.03153e6 −0.109269
\(950\) 0 0
\(951\) −4.62829e6 −0.165947
\(952\) 0 0
\(953\) 1.19242e7 0.425302 0.212651 0.977128i \(-0.431790\pi\)
0.212651 + 0.977128i \(0.431790\pi\)
\(954\) 0 0
\(955\) 763182. 0.0270782
\(956\) 0 0
\(957\) 5.25009e6 0.185305
\(958\) 0 0
\(959\) −1.09292e7 −0.383746
\(960\) 0 0
\(961\) −2.82864e7 −0.988029
\(962\) 0 0
\(963\) −3.05276e6 −0.106078
\(964\) 0 0
\(965\) −2.25716e7 −0.780270
\(966\) 0 0
\(967\) −1.40918e7 −0.484619 −0.242309 0.970199i \(-0.577905\pi\)
−0.242309 + 0.970199i \(0.577905\pi\)
\(968\) 0 0
\(969\) 490559. 0.0167835
\(970\) 0 0
\(971\) 2.06293e7 0.702160 0.351080 0.936346i \(-0.385815\pi\)
0.351080 + 0.936346i \(0.385815\pi\)
\(972\) 0 0
\(973\) 5.51265e6 0.186672
\(974\) 0 0
\(975\) 1.79260e6 0.0603911
\(976\) 0 0
\(977\) 2.77805e7 0.931116 0.465558 0.885017i \(-0.345854\pi\)
0.465558 + 0.885017i \(0.345854\pi\)
\(978\) 0 0
\(979\) −1.14604e7 −0.382157
\(980\) 0 0
\(981\) −7.00600e6 −0.232433
\(982\) 0 0
\(983\) 8.68777e6 0.286764 0.143382 0.989667i \(-0.454202\pi\)
0.143382 + 0.989667i \(0.454202\pi\)
\(984\) 0 0
\(985\) −1.66047e7 −0.545307
\(986\) 0 0
\(987\) 4.22325e6 0.137992
\(988\) 0 0
\(989\) 1.08802e7 0.353708
\(990\) 0 0
\(991\) 5.74667e7 1.85880 0.929399 0.369077i \(-0.120326\pi\)
0.929399 + 0.369077i \(0.120326\pi\)
\(992\) 0 0
\(993\) 2.21419e7 0.712594
\(994\) 0 0
\(995\) −3.20698e6 −0.102692
\(996\) 0 0
\(997\) −3.52375e7 −1.12271 −0.561354 0.827576i \(-0.689719\pi\)
−0.561354 + 0.827576i \(0.689719\pi\)
\(998\) 0 0
\(999\) 2.80595e6 0.0889541
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 1104.6.a.r.1.3 5
4.3 odd 2 69.6.a.e.1.3 5
12.11 even 2 207.6.a.f.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.e.1.3 5 4.3 odd 2
207.6.a.f.1.3 5 12.11 even 2
1104.6.a.r.1.3 5 1.1 even 1 trivial