Properties

Label 1104.6.a.h.1.1
Level $1104$
Weight $6$
Character 1104.1
Self dual yes
Analytic conductor $177.064$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{29}) \)
Defining polynomial: \(x^{2} - x - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.19258\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.00000 q^{3} +41.6148 q^{5} -0.236813 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+9.00000 q^{3} +41.6148 q^{5} -0.236813 q^{7} +81.0000 q^{9} +421.598 q^{11} +254.502 q^{13} +374.534 q^{15} -975.007 q^{17} -2039.79 q^{19} -2.13132 q^{21} -529.000 q^{23} -1393.21 q^{25} +729.000 q^{27} +2671.55 q^{29} -9039.14 q^{31} +3794.38 q^{33} -9.85493 q^{35} -12665.3 q^{37} +2290.52 q^{39} +10146.4 q^{41} -19523.2 q^{43} +3370.80 q^{45} -27679.7 q^{47} -16806.9 q^{49} -8775.06 q^{51} +10852.8 q^{53} +17544.7 q^{55} -18358.1 q^{57} -11907.7 q^{59} +39861.9 q^{61} -19.1818 q^{63} +10591.1 q^{65} +28550.5 q^{67} -4761.00 q^{69} -52179.8 q^{71} +56918.0 q^{73} -12538.8 q^{75} -99.8398 q^{77} -23178.3 q^{79} +6561.00 q^{81} +18344.6 q^{83} -40574.8 q^{85} +24043.9 q^{87} +47362.4 q^{89} -60.2694 q^{91} -81352.3 q^{93} -84885.4 q^{95} -140379. q^{97} +34149.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} + 94 q^{5} + 118 q^{7} + 162 q^{9} + O(q^{10}) \) \( 2 q + 18 q^{3} + 94 q^{5} + 118 q^{7} + 162 q^{9} - 320 q^{11} - 288 q^{13} + 846 q^{15} - 1810 q^{17} - 730 q^{19} + 1062 q^{21} - 1058 q^{23} - 1774 q^{25} + 1458 q^{27} + 8208 q^{29} - 1772 q^{31} - 2880 q^{33} + 6184 q^{35} - 23112 q^{37} - 2592 q^{39} + 5516 q^{41} - 10322 q^{43} + 7614 q^{45} - 42952 q^{47} - 19634 q^{49} - 16290 q^{51} - 25350 q^{53} - 21304 q^{55} - 6570 q^{57} - 18344 q^{59} + 37224 q^{61} + 9558 q^{63} - 17828 q^{65} + 7482 q^{67} - 9522 q^{69} - 126848 q^{71} + 137660 q^{73} - 15966 q^{75} - 87784 q^{77} - 62286 q^{79} + 13122 q^{81} - 83120 q^{83} - 84316 q^{85} + 73872 q^{87} + 69770 q^{89} - 64204 q^{91} - 15948 q^{93} - 16272 q^{95} - 170104 q^{97} - 25920 q^{99} + O(q^{100}) \)

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
\(3\) 9.00000 0.577350
\(4\) 0 0
\(5\) 41.6148 0.744429 0.372214 0.928147i \(-0.378599\pi\)
0.372214 + 0.928147i \(0.378599\pi\)
\(6\) 0 0
\(7\) −0.236813 −0.00182667 −0.000913335 1.00000i \(-0.500291\pi\)
−0.000913335 1.00000i \(0.500291\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 421.598 1.05055 0.525275 0.850933i \(-0.323963\pi\)
0.525275 + 0.850933i \(0.323963\pi\)
\(12\) 0 0
\(13\) 254.502 0.417670 0.208835 0.977951i \(-0.433033\pi\)
0.208835 + 0.977951i \(0.433033\pi\)
\(14\) 0 0
\(15\) 374.534 0.429796
\(16\) 0 0
\(17\) −975.007 −0.818249 −0.409125 0.912479i \(-0.634166\pi\)
−0.409125 + 0.912479i \(0.634166\pi\)
\(18\) 0 0
\(19\) −2039.79 −1.29629 −0.648143 0.761519i \(-0.724454\pi\)
−0.648143 + 0.761519i \(0.724454\pi\)
\(20\) 0 0
\(21\) −2.13132 −0.00105463
\(22\) 0 0
\(23\) −529.000 −0.208514
\(24\) 0 0
\(25\) −1393.21 −0.445826
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 2671.55 0.589885 0.294943 0.955515i \(-0.404699\pi\)
0.294943 + 0.955515i \(0.404699\pi\)
\(30\) 0 0
\(31\) −9039.14 −1.68936 −0.844681 0.535270i \(-0.820210\pi\)
−0.844681 + 0.535270i \(0.820210\pi\)
\(32\) 0 0
\(33\) 3794.38 0.606535
\(34\) 0 0
\(35\) −9.85493 −0.00135983
\(36\) 0 0
\(37\) −12665.3 −1.52094 −0.760471 0.649372i \(-0.775032\pi\)
−0.760471 + 0.649372i \(0.775032\pi\)
\(38\) 0 0
\(39\) 2290.52 0.241142
\(40\) 0 0
\(41\) 10146.4 0.942658 0.471329 0.881957i \(-0.343774\pi\)
0.471329 + 0.881957i \(0.343774\pi\)
\(42\) 0 0
\(43\) −19523.2 −1.61020 −0.805102 0.593137i \(-0.797889\pi\)
−0.805102 + 0.593137i \(0.797889\pi\)
\(44\) 0 0
\(45\) 3370.80 0.248143
\(46\) 0 0
\(47\) −27679.7 −1.82775 −0.913875 0.405995i \(-0.866925\pi\)
−0.913875 + 0.405995i \(0.866925\pi\)
\(48\) 0 0
\(49\) −16806.9 −0.999997
\(50\) 0 0
\(51\) −8775.06 −0.472416
\(52\) 0 0
\(53\) 10852.8 0.530703 0.265351 0.964152i \(-0.414512\pi\)
0.265351 + 0.964152i \(0.414512\pi\)
\(54\) 0 0
\(55\) 17544.7 0.782059
\(56\) 0 0
\(57\) −18358.1 −0.748411
\(58\) 0 0
\(59\) −11907.7 −0.445345 −0.222672 0.974893i \(-0.571478\pi\)
−0.222672 + 0.974893i \(0.571478\pi\)
\(60\) 0 0
\(61\) 39861.9 1.37162 0.685809 0.727782i \(-0.259449\pi\)
0.685809 + 0.727782i \(0.259449\pi\)
\(62\) 0 0
\(63\) −19.1818 −0.000608890 0
\(64\) 0 0
\(65\) 10591.1 0.310925
\(66\) 0 0
\(67\) 28550.5 0.777009 0.388504 0.921447i \(-0.372992\pi\)
0.388504 + 0.921447i \(0.372992\pi\)
\(68\) 0 0
\(69\) −4761.00 −0.120386
\(70\) 0 0
\(71\) −52179.8 −1.22845 −0.614223 0.789132i \(-0.710531\pi\)
−0.614223 + 0.789132i \(0.710531\pi\)
\(72\) 0 0
\(73\) 56918.0 1.25009 0.625047 0.780587i \(-0.285080\pi\)
0.625047 + 0.780587i \(0.285080\pi\)
\(74\) 0 0
\(75\) −12538.8 −0.257398
\(76\) 0 0
\(77\) −99.8398 −0.00191901
\(78\) 0 0
\(79\) −23178.3 −0.417844 −0.208922 0.977932i \(-0.566996\pi\)
−0.208922 + 0.977932i \(0.566996\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 18344.6 0.292289 0.146144 0.989263i \(-0.453314\pi\)
0.146144 + 0.989263i \(0.453314\pi\)
\(84\) 0 0
\(85\) −40574.8 −0.609128
\(86\) 0 0
\(87\) 24043.9 0.340571
\(88\) 0 0
\(89\) 47362.4 0.633810 0.316905 0.948457i \(-0.397356\pi\)
0.316905 + 0.948457i \(0.397356\pi\)
\(90\) 0 0
\(91\) −60.2694 −0.000762945 0
\(92\) 0 0
\(93\) −81352.3 −0.975354
\(94\) 0 0
\(95\) −84885.4 −0.964992
\(96\) 0 0
\(97\) −140379. −1.51486 −0.757432 0.652915i \(-0.773546\pi\)
−0.757432 + 0.652915i \(0.773546\pi\)
\(98\) 0 0
\(99\) 34149.4 0.350183
\(100\) 0 0
\(101\) 87080.3 0.849408 0.424704 0.905332i \(-0.360378\pi\)
0.424704 + 0.905332i \(0.360378\pi\)
\(102\) 0 0
\(103\) −17988.9 −0.167075 −0.0835375 0.996505i \(-0.526622\pi\)
−0.0835375 + 0.996505i \(0.526622\pi\)
\(104\) 0 0
\(105\) −88.6944 −0.000785096 0
\(106\) 0 0
\(107\) 127485. 1.07646 0.538231 0.842797i \(-0.319093\pi\)
0.538231 + 0.842797i \(0.319093\pi\)
\(108\) 0 0
\(109\) −136418. −1.09978 −0.549890 0.835237i \(-0.685330\pi\)
−0.549890 + 0.835237i \(0.685330\pi\)
\(110\) 0 0
\(111\) −113988. −0.878116
\(112\) 0 0
\(113\) −104012. −0.766279 −0.383139 0.923691i \(-0.625157\pi\)
−0.383139 + 0.923691i \(0.625157\pi\)
\(114\) 0 0
\(115\) −22014.2 −0.155224
\(116\) 0 0
\(117\) 20614.7 0.139223
\(118\) 0 0
\(119\) 230.894 0.00149467
\(120\) 0 0
\(121\) 16693.7 0.103655
\(122\) 0 0
\(123\) 91318.0 0.544244
\(124\) 0 0
\(125\) −188024. −1.07631
\(126\) 0 0
\(127\) −203763. −1.12103 −0.560513 0.828145i \(-0.689396\pi\)
−0.560513 + 0.828145i \(0.689396\pi\)
\(128\) 0 0
\(129\) −175709. −0.929651
\(130\) 0 0
\(131\) −60674.1 −0.308905 −0.154453 0.988000i \(-0.549361\pi\)
−0.154453 + 0.988000i \(0.549361\pi\)
\(132\) 0 0
\(133\) 483.048 0.00236789
\(134\) 0 0
\(135\) 30337.2 0.143265
\(136\) 0 0
\(137\) 94446.5 0.429917 0.214958 0.976623i \(-0.431038\pi\)
0.214958 + 0.976623i \(0.431038\pi\)
\(138\) 0 0
\(139\) −403508. −1.77139 −0.885697 0.464263i \(-0.846319\pi\)
−0.885697 + 0.464263i \(0.846319\pi\)
\(140\) 0 0
\(141\) −249117. −1.05525
\(142\) 0 0
\(143\) 107298. 0.438783
\(144\) 0 0
\(145\) 111176. 0.439128
\(146\) 0 0
\(147\) −151262. −0.577348
\(148\) 0 0
\(149\) 384528. 1.41893 0.709467 0.704738i \(-0.248936\pi\)
0.709467 + 0.704738i \(0.248936\pi\)
\(150\) 0 0
\(151\) 442508. 1.57935 0.789675 0.613525i \(-0.210249\pi\)
0.789675 + 0.613525i \(0.210249\pi\)
\(152\) 0 0
\(153\) −78975.6 −0.272750
\(154\) 0 0
\(155\) −376162. −1.25761
\(156\) 0 0
\(157\) 240724. 0.779416 0.389708 0.920938i \(-0.372576\pi\)
0.389708 + 0.920938i \(0.372576\pi\)
\(158\) 0 0
\(159\) 97675.1 0.306402
\(160\) 0 0
\(161\) 125.274 0.000380887 0
\(162\) 0 0
\(163\) 178188. 0.525301 0.262651 0.964891i \(-0.415403\pi\)
0.262651 + 0.964891i \(0.415403\pi\)
\(164\) 0 0
\(165\) 157903. 0.451522
\(166\) 0 0
\(167\) −25525.2 −0.0708236 −0.0354118 0.999373i \(-0.511274\pi\)
−0.0354118 + 0.999373i \(0.511274\pi\)
\(168\) 0 0
\(169\) −306522. −0.825552
\(170\) 0 0
\(171\) −165223. −0.432095
\(172\) 0 0
\(173\) 701865. 1.78295 0.891474 0.453072i \(-0.149672\pi\)
0.891474 + 0.453072i \(0.149672\pi\)
\(174\) 0 0
\(175\) 329.929 0.000814377 0
\(176\) 0 0
\(177\) −107169. −0.257120
\(178\) 0 0
\(179\) 292917. 0.683302 0.341651 0.939827i \(-0.389014\pi\)
0.341651 + 0.939827i \(0.389014\pi\)
\(180\) 0 0
\(181\) −38932.7 −0.0883321 −0.0441660 0.999024i \(-0.514063\pi\)
−0.0441660 + 0.999024i \(0.514063\pi\)
\(182\) 0 0
\(183\) 358757. 0.791904
\(184\) 0 0
\(185\) −527066. −1.13223
\(186\) 0 0
\(187\) −411061. −0.859611
\(188\) 0 0
\(189\) −172.637 −0.000351543 0
\(190\) 0 0
\(191\) −37244.8 −0.0738724 −0.0369362 0.999318i \(-0.511760\pi\)
−0.0369362 + 0.999318i \(0.511760\pi\)
\(192\) 0 0
\(193\) −42454.3 −0.0820406 −0.0410203 0.999158i \(-0.513061\pi\)
−0.0410203 + 0.999158i \(0.513061\pi\)
\(194\) 0 0
\(195\) 95319.6 0.179513
\(196\) 0 0
\(197\) 87335.1 0.160333 0.0801666 0.996781i \(-0.474455\pi\)
0.0801666 + 0.996781i \(0.474455\pi\)
\(198\) 0 0
\(199\) −1.08779e6 −1.94721 −0.973604 0.228245i \(-0.926701\pi\)
−0.973604 + 0.228245i \(0.926701\pi\)
\(200\) 0 0
\(201\) 256954. 0.448606
\(202\) 0 0
\(203\) −632.657 −0.00107753
\(204\) 0 0
\(205\) 422243. 0.701742
\(206\) 0 0
\(207\) −42849.0 −0.0695048
\(208\) 0 0
\(209\) −859969. −1.36181
\(210\) 0 0
\(211\) 1.02777e6 1.58923 0.794617 0.607111i \(-0.207672\pi\)
0.794617 + 0.607111i \(0.207672\pi\)
\(212\) 0 0
\(213\) −469618. −0.709244
\(214\) 0 0
\(215\) −812456. −1.19868
\(216\) 0 0
\(217\) 2140.58 0.00308591
\(218\) 0 0
\(219\) 512262. 0.721742
\(220\) 0 0
\(221\) −248141. −0.341758
\(222\) 0 0
\(223\) 209627. 0.282284 0.141142 0.989989i \(-0.454923\pi\)
0.141142 + 0.989989i \(0.454923\pi\)
\(224\) 0 0
\(225\) −112850. −0.148609
\(226\) 0 0
\(227\) 347103. 0.447088 0.223544 0.974694i \(-0.428237\pi\)
0.223544 + 0.974694i \(0.428237\pi\)
\(228\) 0 0
\(229\) −474094. −0.597414 −0.298707 0.954345i \(-0.596555\pi\)
−0.298707 + 0.954345i \(0.596555\pi\)
\(230\) 0 0
\(231\) −898.558 −0.00110794
\(232\) 0 0
\(233\) 67426.2 0.0813652 0.0406826 0.999172i \(-0.487047\pi\)
0.0406826 + 0.999172i \(0.487047\pi\)
\(234\) 0 0
\(235\) −1.15189e6 −1.36063
\(236\) 0 0
\(237\) −208605. −0.241243
\(238\) 0 0
\(239\) 631686. 0.715331 0.357665 0.933850i \(-0.383573\pi\)
0.357665 + 0.933850i \(0.383573\pi\)
\(240\) 0 0
\(241\) −582036. −0.645516 −0.322758 0.946482i \(-0.604610\pi\)
−0.322758 + 0.946482i \(0.604610\pi\)
\(242\) 0 0
\(243\) 59049.0 0.0641500
\(244\) 0 0
\(245\) −699418. −0.744426
\(246\) 0 0
\(247\) −519130. −0.541419
\(248\) 0 0
\(249\) 165101. 0.168753
\(250\) 0 0
\(251\) 1.28835e6 1.29077 0.645386 0.763856i \(-0.276696\pi\)
0.645386 + 0.763856i \(0.276696\pi\)
\(252\) 0 0
\(253\) −223025. −0.219055
\(254\) 0 0
\(255\) −365173. −0.351680
\(256\) 0 0
\(257\) −204908. −0.193520 −0.0967601 0.995308i \(-0.530848\pi\)
−0.0967601 + 0.995308i \(0.530848\pi\)
\(258\) 0 0
\(259\) 2999.32 0.00277826
\(260\) 0 0
\(261\) 216395. 0.196628
\(262\) 0 0
\(263\) −418862. −0.373407 −0.186703 0.982416i \(-0.559780\pi\)
−0.186703 + 0.982416i \(0.559780\pi\)
\(264\) 0 0
\(265\) 451637. 0.395071
\(266\) 0 0
\(267\) 426262. 0.365930
\(268\) 0 0
\(269\) −1.48662e6 −1.25262 −0.626308 0.779576i \(-0.715435\pi\)
−0.626308 + 0.779576i \(0.715435\pi\)
\(270\) 0 0
\(271\) 658640. 0.544785 0.272392 0.962186i \(-0.412185\pi\)
0.272392 + 0.962186i \(0.412185\pi\)
\(272\) 0 0
\(273\) −542.425 −0.000440487 0
\(274\) 0 0
\(275\) −587372. −0.468362
\(276\) 0 0
\(277\) 236526. 0.185216 0.0926082 0.995703i \(-0.470480\pi\)
0.0926082 + 0.995703i \(0.470480\pi\)
\(278\) 0 0
\(279\) −732170. −0.563121
\(280\) 0 0
\(281\) −1.66253e6 −1.25604 −0.628020 0.778197i \(-0.716134\pi\)
−0.628020 + 0.778197i \(0.716134\pi\)
\(282\) 0 0
\(283\) −966812. −0.717589 −0.358795 0.933417i \(-0.616812\pi\)
−0.358795 + 0.933417i \(0.616812\pi\)
\(284\) 0 0
\(285\) −763968. −0.557139
\(286\) 0 0
\(287\) −2402.81 −0.00172193
\(288\) 0 0
\(289\) −469218. −0.330469
\(290\) 0 0
\(291\) −1.26341e6 −0.874607
\(292\) 0 0
\(293\) −82436.0 −0.0560981 −0.0280490 0.999607i \(-0.508929\pi\)
−0.0280490 + 0.999607i \(0.508929\pi\)
\(294\) 0 0
\(295\) −495535. −0.331528
\(296\) 0 0
\(297\) 307345. 0.202178
\(298\) 0 0
\(299\) −134632. −0.0870902
\(300\) 0 0
\(301\) 4623.35 0.00294131
\(302\) 0 0
\(303\) 783722. 0.490406
\(304\) 0 0
\(305\) 1.65884e6 1.02107
\(306\) 0 0
\(307\) −471104. −0.285280 −0.142640 0.989775i \(-0.545559\pi\)
−0.142640 + 0.989775i \(0.545559\pi\)
\(308\) 0 0
\(309\) −161900. −0.0964608
\(310\) 0 0
\(311\) −974141. −0.571112 −0.285556 0.958362i \(-0.592178\pi\)
−0.285556 + 0.958362i \(0.592178\pi\)
\(312\) 0 0
\(313\) 203897. 0.117639 0.0588194 0.998269i \(-0.481266\pi\)
0.0588194 + 0.998269i \(0.481266\pi\)
\(314\) 0 0
\(315\) −798.249 −0.000453275 0
\(316\) 0 0
\(317\) −1.11600e6 −0.623757 −0.311879 0.950122i \(-0.600958\pi\)
−0.311879 + 0.950122i \(0.600958\pi\)
\(318\) 0 0
\(319\) 1.12632e6 0.619704
\(320\) 0 0
\(321\) 1.14736e6 0.621496
\(322\) 0 0
\(323\) 1.98881e6 1.06068
\(324\) 0 0
\(325\) −354574. −0.186208
\(326\) 0 0
\(327\) −1.22776e6 −0.634958
\(328\) 0 0
\(329\) 6554.91 0.00333870
\(330\) 0 0
\(331\) −2.08998e6 −1.04851 −0.524254 0.851562i \(-0.675656\pi\)
−0.524254 + 0.851562i \(0.675656\pi\)
\(332\) 0 0
\(333\) −1.02589e6 −0.506981
\(334\) 0 0
\(335\) 1.18812e6 0.578428
\(336\) 0 0
\(337\) 901699. 0.432501 0.216250 0.976338i \(-0.430617\pi\)
0.216250 + 0.976338i \(0.430617\pi\)
\(338\) 0 0
\(339\) −936107. −0.442411
\(340\) 0 0
\(341\) −3.81088e6 −1.77476
\(342\) 0 0
\(343\) 7960.21 0.00365333
\(344\) 0 0
\(345\) −198128. −0.0896187
\(346\) 0 0
\(347\) 1.53207e6 0.683053 0.341526 0.939872i \(-0.389056\pi\)
0.341526 + 0.939872i \(0.389056\pi\)
\(348\) 0 0
\(349\) 1.28353e6 0.564084 0.282042 0.959402i \(-0.408988\pi\)
0.282042 + 0.959402i \(0.408988\pi\)
\(350\) 0 0
\(351\) 185532. 0.0803806
\(352\) 0 0
\(353\) −4.45978e6 −1.90492 −0.952460 0.304663i \(-0.901456\pi\)
−0.952460 + 0.304663i \(0.901456\pi\)
\(354\) 0 0
\(355\) −2.17145e6 −0.914491
\(356\) 0 0
\(357\) 2078.05 0.000862949 0
\(358\) 0 0
\(359\) −4.67506e6 −1.91448 −0.957240 0.289296i \(-0.906579\pi\)
−0.957240 + 0.289296i \(0.906579\pi\)
\(360\) 0 0
\(361\) 1.68463e6 0.680356
\(362\) 0 0
\(363\) 150243. 0.0598451
\(364\) 0 0
\(365\) 2.36863e6 0.930606
\(366\) 0 0
\(367\) −2.43229e6 −0.942650 −0.471325 0.881960i \(-0.656224\pi\)
−0.471325 + 0.881960i \(0.656224\pi\)
\(368\) 0 0
\(369\) 821862. 0.314219
\(370\) 0 0
\(371\) −2570.08 −0.000969419 0
\(372\) 0 0
\(373\) 1.37458e6 0.511561 0.255780 0.966735i \(-0.417668\pi\)
0.255780 + 0.966735i \(0.417668\pi\)
\(374\) 0 0
\(375\) −1.69222e6 −0.621410
\(376\) 0 0
\(377\) 679914. 0.246377
\(378\) 0 0
\(379\) −2.01997e6 −0.722350 −0.361175 0.932498i \(-0.617624\pi\)
−0.361175 + 0.932498i \(0.617624\pi\)
\(380\) 0 0
\(381\) −1.83387e6 −0.647225
\(382\) 0 0
\(383\) −2.00834e6 −0.699584 −0.349792 0.936827i \(-0.613748\pi\)
−0.349792 + 0.936827i \(0.613748\pi\)
\(384\) 0 0
\(385\) −4154.82 −0.00142857
\(386\) 0 0
\(387\) −1.58138e6 −0.536734
\(388\) 0 0
\(389\) 3.25055e6 1.08914 0.544569 0.838716i \(-0.316693\pi\)
0.544569 + 0.838716i \(0.316693\pi\)
\(390\) 0 0
\(391\) 515779. 0.170617
\(392\) 0 0
\(393\) −546067. −0.178346
\(394\) 0 0
\(395\) −964563. −0.311055
\(396\) 0 0
\(397\) 5.43707e6 1.73137 0.865683 0.500592i \(-0.166884\pi\)
0.865683 + 0.500592i \(0.166884\pi\)
\(398\) 0 0
\(399\) 4347.43 0.00136710
\(400\) 0 0
\(401\) 3.27208e6 1.01616 0.508081 0.861309i \(-0.330355\pi\)
0.508081 + 0.861309i \(0.330355\pi\)
\(402\) 0 0
\(403\) −2.30048e6 −0.705596
\(404\) 0 0
\(405\) 273035. 0.0827143
\(406\) 0 0
\(407\) −5.33968e6 −1.59783
\(408\) 0 0
\(409\) −1.56299e6 −0.462007 −0.231003 0.972953i \(-0.574201\pi\)
−0.231003 + 0.972953i \(0.574201\pi\)
\(410\) 0 0
\(411\) 850019. 0.248213
\(412\) 0 0
\(413\) 2819.89 0.000813498 0
\(414\) 0 0
\(415\) 763406. 0.217588
\(416\) 0 0
\(417\) −3.63157e6 −1.02272
\(418\) 0 0
\(419\) 4.76111e6 1.32487 0.662434 0.749120i \(-0.269523\pi\)
0.662434 + 0.749120i \(0.269523\pi\)
\(420\) 0 0
\(421\) −4.56989e6 −1.25661 −0.628305 0.777967i \(-0.716251\pi\)
−0.628305 + 0.777967i \(0.716251\pi\)
\(422\) 0 0
\(423\) −2.24206e6 −0.609250
\(424\) 0 0
\(425\) 1.35839e6 0.364796
\(426\) 0 0
\(427\) −9439.80 −0.00250549
\(428\) 0 0
\(429\) 965678. 0.253331
\(430\) 0 0
\(431\) 3.28891e6 0.852824 0.426412 0.904529i \(-0.359777\pi\)
0.426412 + 0.904529i \(0.359777\pi\)
\(432\) 0 0
\(433\) 2.11805e6 0.542895 0.271447 0.962453i \(-0.412498\pi\)
0.271447 + 0.962453i \(0.412498\pi\)
\(434\) 0 0
\(435\) 1.00058e6 0.253531
\(436\) 0 0
\(437\) 1.07905e6 0.270294
\(438\) 0 0
\(439\) −758443. −0.187829 −0.0939143 0.995580i \(-0.529938\pi\)
−0.0939143 + 0.995580i \(0.529938\pi\)
\(440\) 0 0
\(441\) −1.36136e6 −0.333332
\(442\) 0 0
\(443\) −650377. −0.157455 −0.0787274 0.996896i \(-0.525086\pi\)
−0.0787274 + 0.996896i \(0.525086\pi\)
\(444\) 0 0
\(445\) 1.97098e6 0.471826
\(446\) 0 0
\(447\) 3.46075e6 0.819222
\(448\) 0 0
\(449\) −7.58111e6 −1.77467 −0.887333 0.461129i \(-0.847445\pi\)
−0.887333 + 0.461129i \(0.847445\pi\)
\(450\) 0 0
\(451\) 4.27772e6 0.990309
\(452\) 0 0
\(453\) 3.98257e6 0.911838
\(454\) 0 0
\(455\) −2508.10 −0.000567958 0
\(456\) 0 0
\(457\) −8.04537e6 −1.80200 −0.901002 0.433815i \(-0.857167\pi\)
−0.901002 + 0.433815i \(0.857167\pi\)
\(458\) 0 0
\(459\) −710780. −0.157472
\(460\) 0 0
\(461\) 5.63578e6 1.23510 0.617550 0.786532i \(-0.288125\pi\)
0.617550 + 0.786532i \(0.288125\pi\)
\(462\) 0 0
\(463\) −1.39156e6 −0.301682 −0.150841 0.988558i \(-0.548198\pi\)
−0.150841 + 0.988558i \(0.548198\pi\)
\(464\) 0 0
\(465\) −3.38546e6 −0.726082
\(466\) 0 0
\(467\) −4.05063e6 −0.859469 −0.429734 0.902955i \(-0.641393\pi\)
−0.429734 + 0.902955i \(0.641393\pi\)
\(468\) 0 0
\(469\) −6761.12 −0.00141934
\(470\) 0 0
\(471\) 2.16651e6 0.449996
\(472\) 0 0
\(473\) −8.23095e6 −1.69160
\(474\) 0 0
\(475\) 2.84184e6 0.577917
\(476\) 0 0
\(477\) 879076. 0.176901
\(478\) 0 0
\(479\) −6.55529e6 −1.30543 −0.652714 0.757605i \(-0.726370\pi\)
−0.652714 + 0.757605i \(0.726370\pi\)
\(480\) 0 0
\(481\) −3.22336e6 −0.635252
\(482\) 0 0
\(483\) 1127.47 0.000219905 0
\(484\) 0 0
\(485\) −5.84186e6 −1.12771
\(486\) 0 0
\(487\) 1.98924e6 0.380071 0.190035 0.981777i \(-0.439140\pi\)
0.190035 + 0.981777i \(0.439140\pi\)
\(488\) 0 0
\(489\) 1.60369e6 0.303283
\(490\) 0 0
\(491\) −7.99205e6 −1.49608 −0.748040 0.663654i \(-0.769005\pi\)
−0.748040 + 0.663654i \(0.769005\pi\)
\(492\) 0 0
\(493\) −2.60478e6 −0.482673
\(494\) 0 0
\(495\) 1.42112e6 0.260686
\(496\) 0 0
\(497\) 12356.8 0.00224397
\(498\) 0 0
\(499\) 1.00308e7 1.80338 0.901688 0.432387i \(-0.142329\pi\)
0.901688 + 0.432387i \(0.142329\pi\)
\(500\) 0 0
\(501\) −229727. −0.0408900
\(502\) 0 0
\(503\) 4.82601e6 0.850488 0.425244 0.905079i \(-0.360188\pi\)
0.425244 + 0.905079i \(0.360188\pi\)
\(504\) 0 0
\(505\) 3.62383e6 0.632324
\(506\) 0 0
\(507\) −2.75869e6 −0.476633
\(508\) 0 0
\(509\) 138289. 0.0236589 0.0118294 0.999930i \(-0.496234\pi\)
0.0118294 + 0.999930i \(0.496234\pi\)
\(510\) 0 0
\(511\) −13478.9 −0.00228351
\(512\) 0 0
\(513\) −1.48700e6 −0.249470
\(514\) 0 0
\(515\) −748605. −0.124375
\(516\) 0 0
\(517\) −1.16697e7 −1.92014
\(518\) 0 0
\(519\) 6.31679e6 1.02939
\(520\) 0 0
\(521\) 4.62831e6 0.747012 0.373506 0.927628i \(-0.378155\pi\)
0.373506 + 0.927628i \(0.378155\pi\)
\(522\) 0 0
\(523\) 1.54914e6 0.247649 0.123825 0.992304i \(-0.460484\pi\)
0.123825 + 0.992304i \(0.460484\pi\)
\(524\) 0 0
\(525\) 2969.36 0.000470181 0
\(526\) 0 0
\(527\) 8.81323e6 1.38232
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 0 0
\(531\) −964521. −0.148448
\(532\) 0 0
\(533\) 2.58229e6 0.393720
\(534\) 0 0
\(535\) 5.30526e6 0.801349
\(536\) 0 0
\(537\) 2.63626e6 0.394505
\(538\) 0 0
\(539\) −7.08577e6 −1.05055
\(540\) 0 0
\(541\) −6.52702e6 −0.958787 −0.479393 0.877600i \(-0.659143\pi\)
−0.479393 + 0.877600i \(0.659143\pi\)
\(542\) 0 0
\(543\) −350395. −0.0509985
\(544\) 0 0
\(545\) −5.67701e6 −0.818707
\(546\) 0 0
\(547\) −1.06849e7 −1.52686 −0.763432 0.645888i \(-0.776487\pi\)
−0.763432 + 0.645888i \(0.776487\pi\)
\(548\) 0 0
\(549\) 3.22881e6 0.457206
\(550\) 0 0
\(551\) −5.44938e6 −0.764660
\(552\) 0 0
\(553\) 5488.93 0.000763264 0
\(554\) 0 0
\(555\) −4.74360e6 −0.653695
\(556\) 0 0
\(557\) −3.07804e6 −0.420374 −0.210187 0.977661i \(-0.567407\pi\)
−0.210187 + 0.977661i \(0.567407\pi\)
\(558\) 0 0
\(559\) −4.96871e6 −0.672533
\(560\) 0 0
\(561\) −3.69955e6 −0.496297
\(562\) 0 0
\(563\) −1.10366e7 −1.46745 −0.733724 0.679448i \(-0.762219\pi\)
−0.733724 + 0.679448i \(0.762219\pi\)
\(564\) 0 0
\(565\) −4.32844e6 −0.570440
\(566\) 0 0
\(567\) −1553.73 −0.000202963 0
\(568\) 0 0
\(569\) −9.26308e6 −1.19943 −0.599715 0.800214i \(-0.704719\pi\)
−0.599715 + 0.800214i \(0.704719\pi\)
\(570\) 0 0
\(571\) 1.31913e7 1.69315 0.846576 0.532267i \(-0.178660\pi\)
0.846576 + 0.532267i \(0.178660\pi\)
\(572\) 0 0
\(573\) −335203. −0.0426503
\(574\) 0 0
\(575\) 737006. 0.0929611
\(576\) 0 0
\(577\) −1.29741e6 −0.162232 −0.0811160 0.996705i \(-0.525848\pi\)
−0.0811160 + 0.996705i \(0.525848\pi\)
\(578\) 0 0
\(579\) −382089. −0.0473661
\(580\) 0 0
\(581\) −4344.23 −0.000533916 0
\(582\) 0 0
\(583\) 4.57551e6 0.557530
\(584\) 0 0
\(585\) 857876. 0.103642
\(586\) 0 0
\(587\) −4.51203e6 −0.540477 −0.270238 0.962793i \(-0.587103\pi\)
−0.270238 + 0.962793i \(0.587103\pi\)
\(588\) 0 0
\(589\) 1.84379e7 2.18990
\(590\) 0 0
\(591\) 786016. 0.0925684
\(592\) 0 0
\(593\) 9.27288e6 1.08287 0.541437 0.840741i \(-0.317880\pi\)
0.541437 + 0.840741i \(0.317880\pi\)
\(594\) 0 0
\(595\) 9608.63 0.00111268
\(596\) 0 0
\(597\) −9.79011e6 −1.12422
\(598\) 0 0
\(599\) 6.32225e6 0.719954 0.359977 0.932961i \(-0.382785\pi\)
0.359977 + 0.932961i \(0.382785\pi\)
\(600\) 0 0
\(601\) 3.18785e6 0.360008 0.180004 0.983666i \(-0.442389\pi\)
0.180004 + 0.983666i \(0.442389\pi\)
\(602\) 0 0
\(603\) 2.31259e6 0.259003
\(604\) 0 0
\(605\) 694706. 0.0771636
\(606\) 0 0
\(607\) 5.70410e6 0.628369 0.314185 0.949362i \(-0.398269\pi\)
0.314185 + 0.949362i \(0.398269\pi\)
\(608\) 0 0
\(609\) −5693.91 −0.000622110 0
\(610\) 0 0
\(611\) −7.04455e6 −0.763396
\(612\) 0 0
\(613\) 1.17835e7 1.26655 0.633276 0.773926i \(-0.281710\pi\)
0.633276 + 0.773926i \(0.281710\pi\)
\(614\) 0 0
\(615\) 3.80018e6 0.405151
\(616\) 0 0
\(617\) 1.41628e6 0.149774 0.0748868 0.997192i \(-0.476140\pi\)
0.0748868 + 0.997192i \(0.476140\pi\)
\(618\) 0 0
\(619\) 4.37752e6 0.459200 0.229600 0.973285i \(-0.426258\pi\)
0.229600 + 0.973285i \(0.426258\pi\)
\(620\) 0 0
\(621\) −385641. −0.0401286
\(622\) 0 0
\(623\) −11216.0 −0.00115776
\(624\) 0 0
\(625\) −3.47084e6 −0.355414
\(626\) 0 0
\(627\) −7.73972e6 −0.786243
\(628\) 0 0
\(629\) 1.23488e7 1.24451
\(630\) 0 0
\(631\) 1.61975e7 1.61948 0.809739 0.586790i \(-0.199609\pi\)
0.809739 + 0.586790i \(0.199609\pi\)
\(632\) 0 0
\(633\) 9.24989e6 0.917545
\(634\) 0 0
\(635\) −8.47956e6 −0.834525
\(636\) 0 0
\(637\) −4.27740e6 −0.417668
\(638\) 0 0
\(639\) −4.22656e6 −0.409482
\(640\) 0 0
\(641\) −1.56439e7 −1.50383 −0.751915 0.659260i \(-0.770870\pi\)
−0.751915 + 0.659260i \(0.770870\pi\)
\(642\) 0 0
\(643\) 1.57570e7 1.50295 0.751476 0.659760i \(-0.229342\pi\)
0.751476 + 0.659760i \(0.229342\pi\)
\(644\) 0 0
\(645\) −7.31211e6 −0.692059
\(646\) 0 0
\(647\) 660291. 0.0620119 0.0310059 0.999519i \(-0.490129\pi\)
0.0310059 + 0.999519i \(0.490129\pi\)
\(648\) 0 0
\(649\) −5.02024e6 −0.467857
\(650\) 0 0
\(651\) 19265.3 0.00178165
\(652\) 0 0
\(653\) 1.24310e7 1.14084 0.570419 0.821354i \(-0.306781\pi\)
0.570419 + 0.821354i \(0.306781\pi\)
\(654\) 0 0
\(655\) −2.52494e6 −0.229958
\(656\) 0 0
\(657\) 4.61036e6 0.416698
\(658\) 0 0
\(659\) 13723.9 0.00123101 0.000615507 1.00000i \(-0.499804\pi\)
0.000615507 1.00000i \(0.499804\pi\)
\(660\) 0 0
\(661\) 6.54194e6 0.582375 0.291187 0.956666i \(-0.405950\pi\)
0.291187 + 0.956666i \(0.405950\pi\)
\(662\) 0 0
\(663\) −2.23327e6 −0.197314
\(664\) 0 0
\(665\) 20101.9 0.00176272
\(666\) 0 0
\(667\) −1.41325e6 −0.123000
\(668\) 0 0
\(669\) 1.88665e6 0.162977
\(670\) 0 0
\(671\) 1.68057e7 1.44095
\(672\) 0 0
\(673\) 1.54266e7 1.31290 0.656451 0.754369i \(-0.272057\pi\)
0.656451 + 0.754369i \(0.272057\pi\)
\(674\) 0 0
\(675\) −1.01565e6 −0.0857992
\(676\) 0 0
\(677\) 5.30462e6 0.444818 0.222409 0.974953i \(-0.428608\pi\)
0.222409 + 0.974953i \(0.428608\pi\)
\(678\) 0 0
\(679\) 33243.6 0.00276716
\(680\) 0 0
\(681\) 3.12392e6 0.258127
\(682\) 0 0
\(683\) −1.52230e7 −1.24867 −0.624337 0.781155i \(-0.714631\pi\)
−0.624337 + 0.781155i \(0.714631\pi\)
\(684\) 0 0
\(685\) 3.93038e6 0.320043
\(686\) 0 0
\(687\) −4.26684e6 −0.344917
\(688\) 0 0
\(689\) 2.76206e6 0.221659
\(690\) 0 0
\(691\) −1.04711e7 −0.834248 −0.417124 0.908850i \(-0.636962\pi\)
−0.417124 + 0.908850i \(0.636962\pi\)
\(692\) 0 0
\(693\) −8087.02 −0.000639669 0
\(694\) 0 0
\(695\) −1.67919e7 −1.31868
\(696\) 0 0
\(697\) −9.89286e6 −0.771329
\(698\) 0 0
\(699\) 606836. 0.0469762
\(700\) 0 0
\(701\) −2.28874e7 −1.75914 −0.879572 0.475765i \(-0.842171\pi\)
−0.879572 + 0.475765i \(0.842171\pi\)
\(702\) 0 0
\(703\) 2.58346e7 1.97158
\(704\) 0 0
\(705\) −1.03670e7 −0.785560
\(706\) 0 0
\(707\) −20621.7 −0.00155159
\(708\) 0 0
\(709\) 1.85127e7 1.38310 0.691550 0.722329i \(-0.256928\pi\)
0.691550 + 0.722329i \(0.256928\pi\)
\(710\) 0 0
\(711\) −1.87745e6 −0.139281
\(712\) 0 0
\(713\) 4.78170e6 0.352256
\(714\) 0 0
\(715\) 4.46517e6 0.326643
\(716\) 0 0
\(717\) 5.68518e6 0.412996
\(718\) 0 0
\(719\) −1.25679e7 −0.906652 −0.453326 0.891345i \(-0.649763\pi\)
−0.453326 + 0.891345i \(0.649763\pi\)
\(720\) 0 0
\(721\) 4260.00 0.000305191 0
\(722\) 0 0
\(723\) −5.23832e6 −0.372689
\(724\) 0 0
\(725\) −3.72201e6 −0.262986
\(726\) 0 0
\(727\) −3.64050e6 −0.255462 −0.127731 0.991809i \(-0.540769\pi\)
−0.127731 + 0.991809i \(0.540769\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 1.90353e7 1.31755
\(732\) 0 0
\(733\) −4.38735e6 −0.301608 −0.150804 0.988564i \(-0.548186\pi\)
−0.150804 + 0.988564i \(0.548186\pi\)
\(734\) 0 0
\(735\) −6.29476e6 −0.429795
\(736\) 0 0
\(737\) 1.20368e7 0.816287
\(738\) 0 0
\(739\) −1.95857e7 −1.31925 −0.659625 0.751595i \(-0.729285\pi\)
−0.659625 + 0.751595i \(0.729285\pi\)
\(740\) 0 0
\(741\) −4.67217e6 −0.312589
\(742\) 0 0
\(743\) 2.94833e7 1.95931 0.979657 0.200678i \(-0.0643146\pi\)
0.979657 + 0.200678i \(0.0643146\pi\)
\(744\) 0 0
\(745\) 1.60021e7 1.05630
\(746\) 0 0
\(747\) 1.48591e6 0.0974296
\(748\) 0 0
\(749\) −30190.0 −0.00196634
\(750\) 0 0
\(751\) −1.27147e7 −0.822634 −0.411317 0.911492i \(-0.634931\pi\)
−0.411317 + 0.911492i \(0.634931\pi\)
\(752\) 0 0
\(753\) 1.15952e7 0.745228
\(754\) 0 0
\(755\) 1.84149e7 1.17571
\(756\) 0 0
\(757\) 1.63980e7 1.04004 0.520022 0.854153i \(-0.325924\pi\)
0.520022 + 0.854153i \(0.325924\pi\)
\(758\) 0 0
\(759\) −2.00723e6 −0.126471
\(760\) 0 0
\(761\) 8.50863e6 0.532596 0.266298 0.963891i \(-0.414199\pi\)
0.266298 + 0.963891i \(0.414199\pi\)
\(762\) 0 0
\(763\) 32305.5 0.00200893
\(764\) 0 0
\(765\) −3.28656e6 −0.203043
\(766\) 0 0
\(767\) −3.03053e6 −0.186007
\(768\) 0 0
\(769\) 1.70141e7 1.03751 0.518755 0.854923i \(-0.326396\pi\)
0.518755 + 0.854923i \(0.326396\pi\)
\(770\) 0 0
\(771\) −1.84417e6 −0.111729
\(772\) 0 0
\(773\) 3.63916e6 0.219055 0.109527 0.993984i \(-0.465066\pi\)
0.109527 + 0.993984i \(0.465066\pi\)
\(774\) 0 0
\(775\) 1.25934e7 0.753161
\(776\) 0 0
\(777\) 26993.8 0.00160403
\(778\) 0 0
\(779\) −2.06966e7 −1.22195
\(780\) 0 0
\(781\) −2.19989e7 −1.29054
\(782\) 0 0
\(783\) 1.94756e6 0.113524
\(784\) 0 0
\(785\) 1.00177e7 0.580220
\(786\) 0 0
\(787\) 5.30127e6 0.305101 0.152550 0.988296i \(-0.451251\pi\)
0.152550 + 0.988296i \(0.451251\pi\)
\(788\) 0 0
\(789\) −3.76976e6 −0.215586
\(790\) 0 0
\(791\) 24631.3 0.00139974
\(792\) 0 0
\(793\) 1.01449e7 0.572883
\(794\) 0 0
\(795\) 4.06473e6 0.228094
\(796\) 0 0
\(797\) 1.41955e7 0.791598 0.395799 0.918337i \(-0.370468\pi\)
0.395799 + 0.918337i \(0.370468\pi\)
\(798\) 0 0
\(799\) 2.69879e7 1.49555
\(800\) 0 0
\(801\) 3.83636e6 0.211270
\(802\) 0 0
\(803\) 2.39965e7 1.31329
\(804\) 0 0
\(805\) 5213.26 0.000283543 0
\(806\) 0 0
\(807\) −1.33795e7 −0.723198
\(808\) 0 0
\(809\) −1.58542e7 −0.851675 −0.425838 0.904800i \(-0.640021\pi\)
−0.425838 + 0.904800i \(0.640021\pi\)
\(810\) 0 0
\(811\) 1.40129e7 0.748127 0.374063 0.927403i \(-0.377964\pi\)
0.374063 + 0.927403i \(0.377964\pi\)
\(812\) 0 0
\(813\) 5.92776e6 0.314532
\(814\) 0 0
\(815\) 7.41524e6 0.391049
\(816\) 0 0
\(817\) 3.98232e7 2.08728
\(818\) 0 0
\(819\) −4881.82 −0.000254315 0
\(820\) 0 0
\(821\) 1.16279e7 0.602065 0.301033 0.953614i \(-0.402669\pi\)
0.301033 + 0.953614i \(0.402669\pi\)
\(822\) 0 0
\(823\) 2.76237e7 1.42161 0.710807 0.703387i \(-0.248330\pi\)
0.710807 + 0.703387i \(0.248330\pi\)
\(824\) 0 0
\(825\) −5.28635e6 −0.270409
\(826\) 0 0
\(827\) 2.78146e7 1.41420 0.707098 0.707116i \(-0.250004\pi\)
0.707098 + 0.707116i \(0.250004\pi\)
\(828\) 0 0
\(829\) −1.23659e7 −0.624943 −0.312471 0.949927i \(-0.601157\pi\)
−0.312471 + 0.949927i \(0.601157\pi\)
\(830\) 0 0
\(831\) 2.12873e6 0.106935
\(832\) 0 0
\(833\) 1.63869e7 0.818246
\(834\) 0 0
\(835\) −1.06223e6 −0.0527231
\(836\) 0 0
\(837\) −6.58953e6 −0.325118
\(838\) 0 0
\(839\) 3.49502e7 1.71413 0.857067 0.515205i \(-0.172284\pi\)
0.857067 + 0.515205i \(0.172284\pi\)
\(840\) 0 0
\(841\) −1.33740e7 −0.652035
\(842\) 0 0
\(843\) −1.49628e7 −0.725175
\(844\) 0 0
\(845\) −1.27558e7 −0.614565
\(846\) 0 0
\(847\) −3953.28 −0.000189343 0
\(848\) 0 0
\(849\) −8.70131e6 −0.414300
\(850\) 0 0
\(851\) 6.69997e6 0.317138
\(852\) 0 0
\(853\) −9.99950e6 −0.470550 −0.235275 0.971929i \(-0.575599\pi\)
−0.235275 + 0.971929i \(0.575599\pi\)
\(854\) 0 0
\(855\) −6.87571e6 −0.321664
\(856\) 0 0
\(857\) 9.71240e6 0.451725 0.225863 0.974159i \(-0.427480\pi\)
0.225863 + 0.974159i \(0.427480\pi\)
\(858\) 0 0
\(859\) −3.69393e7 −1.70807 −0.854035 0.520216i \(-0.825851\pi\)
−0.854035 + 0.520216i \(0.825851\pi\)
\(860\) 0 0
\(861\) −21625.3 −0.000994155 0
\(862\) 0 0
\(863\) 3.31284e7 1.51416 0.757082 0.653319i \(-0.226624\pi\)
0.757082 + 0.653319i \(0.226624\pi\)
\(864\) 0 0
\(865\) 2.92080e7 1.32728
\(866\) 0 0
\(867\) −4.22296e6 −0.190796
\(868\) 0 0
\(869\) −9.77194e6 −0.438966
\(870\) 0 0
\(871\) 7.26615e6 0.324533
\(872\) 0 0
\(873\) −1.13707e7 −0.504954
\(874\) 0 0
\(875\) 44526.6 0.00196607
\(876\) 0 0
\(877\) 1.38007e7 0.605902 0.302951 0.953006i \(-0.402028\pi\)
0.302951 + 0.953006i \(0.402028\pi\)
\(878\) 0 0
\(879\) −741924. −0.0323882
\(880\) 0 0
\(881\) −7.74868e6 −0.336347 −0.168174 0.985757i \(-0.553787\pi\)
−0.168174 + 0.985757i \(0.553787\pi\)
\(882\) 0 0
\(883\) −3.79838e7 −1.63944 −0.819722 0.572761i \(-0.805872\pi\)
−0.819722 + 0.572761i \(0.805872\pi\)
\(884\) 0 0
\(885\) −4.45982e6 −0.191408
\(886\) 0 0
\(887\) 1.51721e7 0.647495 0.323747 0.946144i \(-0.395057\pi\)
0.323747 + 0.946144i \(0.395057\pi\)
\(888\) 0 0
\(889\) 48253.7 0.00204775
\(890\) 0 0
\(891\) 2.76610e6 0.116728
\(892\) 0 0
\(893\) 5.64607e7 2.36929
\(894\) 0 0
\(895\) 1.21897e7 0.508670
\(896\) 0 0
\(897\) −1.21168e6 −0.0502815
\(898\) 0 0
\(899\) −2.41485e7 −0.996530
\(900\) 0 0
\(901\) −1.05815e7 −0.434247
\(902\) 0 0
\(903\) 41610.2 0.00169817
\(904\) 0 0
\(905\) −1.62018e6 −0.0657569
\(906\) 0 0
\(907\) −8.48315e6 −0.342404 −0.171202 0.985236i \(-0.554765\pi\)
−0.171202 + 0.985236i \(0.554765\pi\)
\(908\) 0 0
\(909\) 7.05350e6 0.283136
\(910\) 0 0
\(911\) −2.96302e7 −1.18288 −0.591438 0.806351i \(-0.701439\pi\)
−0.591438 + 0.806351i \(0.701439\pi\)
\(912\) 0 0
\(913\) 7.73403e6 0.307064
\(914\) 0 0
\(915\) 1.49296e7 0.589516
\(916\) 0 0
\(917\) 14368.4 0.000564268 0
\(918\) 0 0
\(919\) 1.27080e7 0.496352 0.248176 0.968715i \(-0.420169\pi\)
0.248176 + 0.968715i \(0.420169\pi\)
\(920\) 0 0
\(921\) −4.23993e6 −0.164706
\(922\) 0 0
\(923\) −1.32799e7 −0.513085
\(924\) 0 0
\(925\) 1.76454e7 0.678075
\(926\) 0 0
\(927\) −1.45710e6 −0.0556916
\(928\) 0 0
\(929\) −4.50533e7 −1.71272 −0.856362 0.516376i \(-0.827281\pi\)
−0.856362 + 0.516376i \(0.827281\pi\)
\(930\) 0 0
\(931\) 3.42826e7 1.29628
\(932\) 0 0
\(933\) −8.76727e6 −0.329731
\(934\) 0 0
\(935\) −1.71062e7 −0.639919
\(936\) 0 0
\(937\) −2.49632e7 −0.928864 −0.464432 0.885609i \(-0.653741\pi\)
−0.464432 + 0.885609i \(0.653741\pi\)
\(938\) 0 0
\(939\) 1.83508e6 0.0679188
\(940\) 0 0
\(941\) −3.36141e7 −1.23751 −0.618754 0.785585i \(-0.712362\pi\)
−0.618754 + 0.785585i \(0.712362\pi\)
\(942\) 0 0
\(943\) −5.36747e6 −0.196558
\(944\) 0 0
\(945\) −7184.24 −0.000261699 0
\(946\) 0 0
\(947\) 5.33926e6 0.193467 0.0967333 0.995310i \(-0.469161\pi\)
0.0967333 + 0.995310i \(0.469161\pi\)
\(948\) 0 0
\(949\) 1.44858e7 0.522127
\(950\) 0 0
\(951\) −1.00440e7 −0.360126
\(952\) 0 0
\(953\) −3.69391e7 −1.31751 −0.658755 0.752358i \(-0.728916\pi\)
−0.658755 + 0.752358i \(0.728916\pi\)
\(954\) 0 0
\(955\) −1.54994e6 −0.0549928
\(956\) 0 0
\(957\) 1.01369e7 0.357786
\(958\) 0 0
\(959\) −22366.2 −0.000785317 0
\(960\) 0 0
\(961\) 5.30769e7 1.85395
\(962\) 0 0
\(963\) 1.03263e7 0.358821
\(964\) 0 0
\(965\) −1.76673e6 −0.0610734
\(966\) 0 0
\(967\) 4.47688e7 1.53960 0.769802 0.638283i \(-0.220355\pi\)
0.769802 + 0.638283i \(0.220355\pi\)
\(968\) 0 0
\(969\) 1.78993e7 0.612386
\(970\) 0 0
\(971\) 4.07416e7 1.38672 0.693361 0.720590i \(-0.256129\pi\)
0.693361 + 0.720590i \(0.256129\pi\)
\(972\) 0 0
\(973\) 95555.9 0.00323575
\(974\) 0 0
\(975\) −3.19116e6 −0.107507
\(976\) 0 0
\(977\) 1.07576e7 0.360561 0.180281 0.983615i \(-0.442299\pi\)
0.180281 + 0.983615i \(0.442299\pi\)
\(978\) 0 0
\(979\) 1.99679e7 0.665849
\(980\) 0 0
\(981\) −1.10499e7 −0.366593
\(982\) 0 0
\(983\) 2.44766e7 0.807917 0.403958 0.914777i \(-0.367634\pi\)
0.403958 + 0.914777i \(0.367634\pi\)
\(984\) 0 0
\(985\) 3.63444e6 0.119357
\(986\) 0 0
\(987\) 58994.2 0.00192760
\(988\) 0 0
\(989\) 1.03278e7 0.335751
\(990\) 0 0
\(991\) 2.09086e6 0.0676301 0.0338151 0.999428i \(-0.489234\pi\)
0.0338151 + 0.999428i \(0.489234\pi\)
\(992\) 0 0
\(993\) −1.88098e7 −0.605357
\(994\) 0 0
\(995\) −4.52682e7 −1.44956
\(996\) 0 0
\(997\) 1.49624e7 0.476721 0.238360 0.971177i \(-0.423390\pi\)
0.238360 + 0.971177i \(0.423390\pi\)
\(998\) 0 0
\(999\) −9.23304e6 −0.292705
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.h.1.1 2
4.3 odd 2 69.6.a.a.1.1 2
12.11 even 2 207.6.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.a.1.1 2 4.3 odd 2
207.6.a.a.1.2 2 12.11 even 2
1104.6.a.h.1.1 2 1.1 even 1 trivial