Properties

Label 504.6.a.t.1.1
Level $504$
Weight $6$
Character 504.1
Self dual yes
Analytic conductor $80.833$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,6,Mod(1,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(80.8334451857\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(8.38987\) of defining polynomial
Character \(\chi\) \(=\) 504.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.440532 q^{5} +49.0000 q^{7} +O(q^{10})\) \(q+0.440532 q^{5} +49.0000 q^{7} +203.392 q^{11} -402.951 q^{13} -1761.53 q^{17} +1775.40 q^{19} +2710.72 q^{23} -3124.81 q^{25} -781.595 q^{29} -9655.74 q^{31} +21.5861 q^{35} +3788.80 q^{37} +18034.1 q^{41} -13199.7 q^{43} +26648.6 q^{47} +2401.00 q^{49} -22760.3 q^{53} +89.6008 q^{55} -1579.56 q^{59} -35327.9 q^{61} -177.513 q^{65} -48572.1 q^{67} -12764.0 q^{71} +10152.3 q^{73} +9966.21 q^{77} -8703.14 q^{79} +26277.1 q^{83} -776.012 q^{85} -33598.5 q^{89} -19744.6 q^{91} +782.121 q^{95} +23000.5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{5} + 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 64 q^{5} + 98 q^{7} - 540 q^{11} + 204 q^{13} - 304 q^{17} - 1120 q^{19} + 940 q^{23} - 2210 q^{25} - 932 q^{29} - 16408 q^{31} + 3136 q^{35} - 1764 q^{37} + 2552 q^{41} - 24632 q^{43} + 36760 q^{47} + 4802 q^{49} - 9164 q^{53} - 47160 q^{55} + 39888 q^{59} - 25084 q^{61} + 38400 q^{65} - 2592 q^{67} + 35508 q^{71} - 17188 q^{73} - 26460 q^{77} - 95800 q^{79} - 65352 q^{83} + 91864 q^{85} + 23000 q^{89} + 9996 q^{91} - 183248 q^{95} - 108388 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.440532 0.00788048 0.00394024 0.999992i \(-0.498746\pi\)
0.00394024 + 0.999992i \(0.498746\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 203.392 0.506818 0.253409 0.967359i \(-0.418448\pi\)
0.253409 + 0.967359i \(0.418448\pi\)
\(12\) 0 0
\(13\) −402.951 −0.661294 −0.330647 0.943755i \(-0.607267\pi\)
−0.330647 + 0.943755i \(0.607267\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1761.53 −1.47832 −0.739160 0.673530i \(-0.764777\pi\)
−0.739160 + 0.673530i \(0.764777\pi\)
\(18\) 0 0
\(19\) 1775.40 1.12827 0.564134 0.825683i \(-0.309210\pi\)
0.564134 + 0.825683i \(0.309210\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2710.72 1.06848 0.534239 0.845334i \(-0.320598\pi\)
0.534239 + 0.845334i \(0.320598\pi\)
\(24\) 0 0
\(25\) −3124.81 −0.999938
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −781.595 −0.172578 −0.0862892 0.996270i \(-0.527501\pi\)
−0.0862892 + 0.996270i \(0.527501\pi\)
\(30\) 0 0
\(31\) −9655.74 −1.80460 −0.902300 0.431108i \(-0.858123\pi\)
−0.902300 + 0.431108i \(0.858123\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 21.5861 0.00297854
\(36\) 0 0
\(37\) 3788.80 0.454985 0.227493 0.973780i \(-0.426947\pi\)
0.227493 + 0.973780i \(0.426947\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18034.1 1.67546 0.837730 0.546084i \(-0.183882\pi\)
0.837730 + 0.546084i \(0.183882\pi\)
\(42\) 0 0
\(43\) −13199.7 −1.08866 −0.544329 0.838872i \(-0.683216\pi\)
−0.544329 + 0.838872i \(0.683216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 26648.6 1.75966 0.879831 0.475286i \(-0.157655\pi\)
0.879831 + 0.475286i \(0.157655\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −22760.3 −1.11298 −0.556490 0.830854i \(-0.687852\pi\)
−0.556490 + 0.830854i \(0.687852\pi\)
\(54\) 0 0
\(55\) 89.6008 0.00399397
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1579.56 −0.0590752 −0.0295376 0.999564i \(-0.509403\pi\)
−0.0295376 + 0.999564i \(0.509403\pi\)
\(60\) 0 0
\(61\) −35327.9 −1.21561 −0.607804 0.794087i \(-0.707949\pi\)
−0.607804 + 0.794087i \(0.707949\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −177.513 −0.00521131
\(66\) 0 0
\(67\) −48572.1 −1.32190 −0.660952 0.750428i \(-0.729847\pi\)
−0.660952 + 0.750428i \(0.729847\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12764.0 −0.300498 −0.150249 0.988648i \(-0.548007\pi\)
−0.150249 + 0.988648i \(0.548007\pi\)
\(72\) 0 0
\(73\) 10152.3 0.222976 0.111488 0.993766i \(-0.464438\pi\)
0.111488 + 0.993766i \(0.464438\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9966.21 0.191559
\(78\) 0 0
\(79\) −8703.14 −0.156895 −0.0784474 0.996918i \(-0.524996\pi\)
−0.0784474 + 0.996918i \(0.524996\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 26277.1 0.418680 0.209340 0.977843i \(-0.432868\pi\)
0.209340 + 0.977843i \(0.432868\pi\)
\(84\) 0 0
\(85\) −776.012 −0.0116499
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −33598.5 −0.449619 −0.224809 0.974403i \(-0.572176\pi\)
−0.224809 + 0.974403i \(0.572176\pi\)
\(90\) 0 0
\(91\) −19744.6 −0.249946
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 782.121 0.00889130
\(96\) 0 0
\(97\) 23000.5 0.248203 0.124102 0.992270i \(-0.460395\pi\)
0.124102 + 0.992270i \(0.460395\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −57527.8 −0.561144 −0.280572 0.959833i \(-0.590524\pi\)
−0.280572 + 0.959833i \(0.590524\pi\)
\(102\) 0 0
\(103\) −167794. −1.55842 −0.779210 0.626763i \(-0.784379\pi\)
−0.779210 + 0.626763i \(0.784379\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −67979.3 −0.574007 −0.287004 0.957929i \(-0.592659\pi\)
−0.287004 + 0.957929i \(0.592659\pi\)
\(108\) 0 0
\(109\) 91585.8 0.738349 0.369175 0.929360i \(-0.379640\pi\)
0.369175 + 0.929360i \(0.379640\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −154355. −1.13717 −0.568584 0.822625i \(-0.692508\pi\)
−0.568584 + 0.822625i \(0.692508\pi\)
\(114\) 0 0
\(115\) 1194.16 0.00842012
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −86315.1 −0.558752
\(120\) 0 0
\(121\) −119683. −0.743135
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2753.24 −0.0157605
\(126\) 0 0
\(127\) −54085.3 −0.297557 −0.148778 0.988871i \(-0.547534\pi\)
−0.148778 + 0.988871i \(0.547534\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −43981.5 −0.223919 −0.111960 0.993713i \(-0.535713\pi\)
−0.111960 + 0.993713i \(0.535713\pi\)
\(132\) 0 0
\(133\) 86994.6 0.426445
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −79578.6 −0.362239 −0.181119 0.983461i \(-0.557972\pi\)
−0.181119 + 0.983461i \(0.557972\pi\)
\(138\) 0 0
\(139\) −207284. −0.909973 −0.454987 0.890498i \(-0.650356\pi\)
−0.454987 + 0.890498i \(0.650356\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −81957.1 −0.335156
\(144\) 0 0
\(145\) −344.318 −0.00136000
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −382884. −1.41287 −0.706434 0.707779i \(-0.749697\pi\)
−0.706434 + 0.707779i \(0.749697\pi\)
\(150\) 0 0
\(151\) 50287.6 0.179481 0.0897406 0.995965i \(-0.471396\pi\)
0.0897406 + 0.995965i \(0.471396\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4253.66 −0.0142211
\(156\) 0 0
\(157\) 367616. 1.19027 0.595135 0.803626i \(-0.297099\pi\)
0.595135 + 0.803626i \(0.297099\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 132825. 0.403847
\(162\) 0 0
\(163\) −206972. −0.610158 −0.305079 0.952327i \(-0.598683\pi\)
−0.305079 + 0.952327i \(0.598683\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −257285. −0.713878 −0.356939 0.934128i \(-0.616180\pi\)
−0.356939 + 0.934128i \(0.616180\pi\)
\(168\) 0 0
\(169\) −208923. −0.562691
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 269032. 0.683421 0.341710 0.939805i \(-0.388994\pi\)
0.341710 + 0.939805i \(0.388994\pi\)
\(174\) 0 0
\(175\) −153115. −0.377941
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −533998. −1.24568 −0.622841 0.782348i \(-0.714022\pi\)
−0.622841 + 0.782348i \(0.714022\pi\)
\(180\) 0 0
\(181\) −629728. −1.42875 −0.714375 0.699763i \(-0.753289\pi\)
−0.714375 + 0.699763i \(0.753289\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1669.09 0.00358550
\(186\) 0 0
\(187\) −358282. −0.749239
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 452199. 0.896903 0.448452 0.893807i \(-0.351976\pi\)
0.448452 + 0.893807i \(0.351976\pi\)
\(192\) 0 0
\(193\) −372356. −0.719556 −0.359778 0.933038i \(-0.617148\pi\)
−0.359778 + 0.933038i \(0.617148\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 250110. 0.459161 0.229581 0.973290i \(-0.426265\pi\)
0.229581 + 0.973290i \(0.426265\pi\)
\(198\) 0 0
\(199\) −936025. −1.67554 −0.837770 0.546023i \(-0.816141\pi\)
−0.837770 + 0.546023i \(0.816141\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −38298.1 −0.0652285
\(204\) 0 0
\(205\) 7944.59 0.0132034
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 361102. 0.571827
\(210\) 0 0
\(211\) 961698. 1.48707 0.743537 0.668695i \(-0.233147\pi\)
0.743537 + 0.668695i \(0.233147\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5814.88 −0.00857916
\(216\) 0 0
\(217\) −473131. −0.682075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 709812. 0.977604
\(222\) 0 0
\(223\) −624199. −0.840545 −0.420273 0.907398i \(-0.638066\pi\)
−0.420273 + 0.907398i \(0.638066\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −698450. −0.899644 −0.449822 0.893118i \(-0.648513\pi\)
−0.449822 + 0.893118i \(0.648513\pi\)
\(228\) 0 0
\(229\) −987151. −1.24393 −0.621964 0.783046i \(-0.713665\pi\)
−0.621964 + 0.783046i \(0.713665\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.11183e6 1.34168 0.670839 0.741603i \(-0.265934\pi\)
0.670839 + 0.741603i \(0.265934\pi\)
\(234\) 0 0
\(235\) 11739.6 0.0138670
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.50111e6 −1.69988 −0.849941 0.526878i \(-0.823362\pi\)
−0.849941 + 0.526878i \(0.823362\pi\)
\(240\) 0 0
\(241\) −172734. −0.191574 −0.0957869 0.995402i \(-0.530537\pi\)
−0.0957869 + 0.995402i \(0.530537\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1057.72 0.00112578
\(246\) 0 0
\(247\) −715400. −0.746117
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 136981. 0.137238 0.0686192 0.997643i \(-0.478141\pi\)
0.0686192 + 0.997643i \(0.478141\pi\)
\(252\) 0 0
\(253\) 551339. 0.541524
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.97671e6 1.86686 0.933428 0.358765i \(-0.116802\pi\)
0.933428 + 0.358765i \(0.116802\pi\)
\(258\) 0 0
\(259\) 185651. 0.171968
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −414337. −0.369372 −0.184686 0.982798i \(-0.559127\pi\)
−0.184686 + 0.982798i \(0.559127\pi\)
\(264\) 0 0
\(265\) −10026.6 −0.00877082
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −241748. −0.203696 −0.101848 0.994800i \(-0.532476\pi\)
−0.101848 + 0.994800i \(0.532476\pi\)
\(270\) 0 0
\(271\) 558814. 0.462215 0.231107 0.972928i \(-0.425765\pi\)
0.231107 + 0.972928i \(0.425765\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −635561. −0.506787
\(276\) 0 0
\(277\) 232158. 0.181796 0.0908981 0.995860i \(-0.471026\pi\)
0.0908981 + 0.995860i \(0.471026\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.85817e6 −1.40385 −0.701923 0.712253i \(-0.747675\pi\)
−0.701923 + 0.712253i \(0.747675\pi\)
\(282\) 0 0
\(283\) 2.42916e6 1.80297 0.901487 0.432805i \(-0.142476\pi\)
0.901487 + 0.432805i \(0.142476\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 883670. 0.633265
\(288\) 0 0
\(289\) 1.68314e6 1.18543
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.61912e6 1.10182 0.550908 0.834566i \(-0.314281\pi\)
0.550908 + 0.834566i \(0.314281\pi\)
\(294\) 0 0
\(295\) −695.846 −0.000465541 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.09229e6 −0.706577
\(300\) 0 0
\(301\) −646784. −0.411474
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15563.1 −0.00957958
\(306\) 0 0
\(307\) 936495. 0.567100 0.283550 0.958957i \(-0.408488\pi\)
0.283550 + 0.958957i \(0.408488\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −48784.9 −0.0286012 −0.0143006 0.999898i \(-0.504552\pi\)
−0.0143006 + 0.999898i \(0.504552\pi\)
\(312\) 0 0
\(313\) 1.31634e6 0.759466 0.379733 0.925096i \(-0.376016\pi\)
0.379733 + 0.925096i \(0.376016\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.08008e6 1.72153 0.860763 0.509006i \(-0.169987\pi\)
0.860763 + 0.509006i \(0.169987\pi\)
\(318\) 0 0
\(319\) −158970. −0.0874659
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.12743e6 −1.66794
\(324\) 0 0
\(325\) 1.25915e6 0.661253
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.30578e6 0.665090
\(330\) 0 0
\(331\) −3.20819e6 −1.60950 −0.804749 0.593615i \(-0.797700\pi\)
−0.804749 + 0.593615i \(0.797700\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −21397.6 −0.0104172
\(336\) 0 0
\(337\) 2.77901e6 1.33295 0.666477 0.745525i \(-0.267801\pi\)
0.666477 + 0.745525i \(0.267801\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.96390e6 −0.914604
\(342\) 0 0
\(343\) 117649. 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.01762e6 −1.34537 −0.672684 0.739930i \(-0.734859\pi\)
−0.672684 + 0.739930i \(0.734859\pi\)
\(348\) 0 0
\(349\) 1.35064e6 0.593574 0.296787 0.954944i \(-0.404085\pi\)
0.296787 + 0.954944i \(0.404085\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −158501. −0.0677011 −0.0338506 0.999427i \(-0.510777\pi\)
−0.0338506 + 0.999427i \(0.510777\pi\)
\(354\) 0 0
\(355\) −5622.96 −0.00236807
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.54504e6 0.632709 0.316354 0.948641i \(-0.397541\pi\)
0.316354 + 0.948641i \(0.397541\pi\)
\(360\) 0 0
\(361\) 675948. 0.272989
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4472.43 0.00175716
\(366\) 0 0
\(367\) 2.51991e6 0.976607 0.488304 0.872674i \(-0.337616\pi\)
0.488304 + 0.872674i \(0.337616\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.11525e6 −0.420667
\(372\) 0 0
\(373\) 2.94012e6 1.09419 0.547096 0.837070i \(-0.315733\pi\)
0.547096 + 0.837070i \(0.315733\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 314945. 0.114125
\(378\) 0 0
\(379\) 4.62230e6 1.65295 0.826475 0.562973i \(-0.190342\pi\)
0.826475 + 0.562973i \(0.190342\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.60273e6 1.60331 0.801656 0.597785i \(-0.203952\pi\)
0.801656 + 0.597785i \(0.203952\pi\)
\(384\) 0 0
\(385\) 4390.44 0.00150958
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.16928e6 1.06191 0.530953 0.847401i \(-0.321834\pi\)
0.530953 + 0.847401i \(0.321834\pi\)
\(390\) 0 0
\(391\) −4.77503e6 −1.57955
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3834.01 −0.00123641
\(396\) 0 0
\(397\) −3.11443e6 −0.991749 −0.495874 0.868394i \(-0.665152\pi\)
−0.495874 + 0.868394i \(0.665152\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.35836e6 −1.97462 −0.987312 0.158795i \(-0.949239\pi\)
−0.987312 + 0.158795i \(0.949239\pi\)
\(402\) 0 0
\(403\) 3.89079e6 1.19337
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 770612. 0.230595
\(408\) 0 0
\(409\) 4.46377e6 1.31945 0.659726 0.751507i \(-0.270673\pi\)
0.659726 + 0.751507i \(0.270673\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −77398.3 −0.0223283
\(414\) 0 0
\(415\) 11575.9 0.00329940
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.96754e6 0.825773 0.412887 0.910782i \(-0.364521\pi\)
0.412887 + 0.910782i \(0.364521\pi\)
\(420\) 0 0
\(421\) 6.81021e6 1.87264 0.936322 0.351143i \(-0.114207\pi\)
0.936322 + 0.351143i \(0.114207\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.50445e6 1.47823
\(426\) 0 0
\(427\) −1.73107e6 −0.459457
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 675836. 0.175246 0.0876230 0.996154i \(-0.472073\pi\)
0.0876230 + 0.996154i \(0.472073\pi\)
\(432\) 0 0
\(433\) −1.24399e6 −0.318859 −0.159429 0.987209i \(-0.550965\pi\)
−0.159429 + 0.987209i \(0.550965\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.81262e6 1.20553
\(438\) 0 0
\(439\) 313929. 0.0777445 0.0388723 0.999244i \(-0.487623\pi\)
0.0388723 + 0.999244i \(0.487623\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.43067e6 0.830557 0.415279 0.909694i \(-0.363684\pi\)
0.415279 + 0.909694i \(0.363684\pi\)
\(444\) 0 0
\(445\) −14801.2 −0.00354321
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −363883. −0.0851816 −0.0425908 0.999093i \(-0.513561\pi\)
−0.0425908 + 0.999093i \(0.513561\pi\)
\(450\) 0 0
\(451\) 3.66799e6 0.849154
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8698.14 −0.00196969
\(456\) 0 0
\(457\) −5.52131e6 −1.23666 −0.618332 0.785917i \(-0.712191\pi\)
−0.618332 + 0.785917i \(0.712191\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 300842. 0.0659306 0.0329653 0.999456i \(-0.489505\pi\)
0.0329653 + 0.999456i \(0.489505\pi\)
\(462\) 0 0
\(463\) −6.80343e6 −1.47494 −0.737472 0.675378i \(-0.763981\pi\)
−0.737472 + 0.675378i \(0.763981\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.40398e6 −0.510080 −0.255040 0.966931i \(-0.582089\pi\)
−0.255040 + 0.966931i \(0.582089\pi\)
\(468\) 0 0
\(469\) −2.38003e6 −0.499633
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.68471e6 −0.551752
\(474\) 0 0
\(475\) −5.54778e6 −1.12820
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.09265e6 −1.01416 −0.507078 0.861900i \(-0.669274\pi\)
−0.507078 + 0.861900i \(0.669274\pi\)
\(480\) 0 0
\(481\) −1.52670e6 −0.300879
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10132.4 0.00195596
\(486\) 0 0
\(487\) −7.33002e6 −1.40050 −0.700249 0.713899i \(-0.746928\pi\)
−0.700249 + 0.713899i \(0.746928\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.36290e6 −0.255130 −0.127565 0.991830i \(-0.540716\pi\)
−0.127565 + 0.991830i \(0.540716\pi\)
\(492\) 0 0
\(493\) 1.37680e6 0.255126
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −625436. −0.113577
\(498\) 0 0
\(499\) 5.05087e6 0.908060 0.454030 0.890986i \(-0.349986\pi\)
0.454030 + 0.890986i \(0.349986\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.93004e6 1.04505 0.522526 0.852623i \(-0.324990\pi\)
0.522526 + 0.852623i \(0.324990\pi\)
\(504\) 0 0
\(505\) −25342.9 −0.00442208
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.01042e6 1.19936 0.599680 0.800240i \(-0.295294\pi\)
0.599680 + 0.800240i \(0.295294\pi\)
\(510\) 0 0
\(511\) 497464. 0.0842770
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −73918.9 −0.0122811
\(516\) 0 0
\(517\) 5.42011e6 0.891829
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.40461e6 0.710908 0.355454 0.934694i \(-0.384326\pi\)
0.355454 + 0.934694i \(0.384326\pi\)
\(522\) 0 0
\(523\) −7.18270e6 −1.14824 −0.574121 0.818770i \(-0.694656\pi\)
−0.574121 + 0.818770i \(0.694656\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.70089e7 2.66778
\(528\) 0 0
\(529\) 911672. 0.141644
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.26686e6 −1.10797
\(534\) 0 0
\(535\) −29947.1 −0.00452345
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 488344. 0.0724026
\(540\) 0 0
\(541\) 9.23327e6 1.35632 0.678161 0.734914i \(-0.262777\pi\)
0.678161 + 0.734914i \(0.262777\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 40346.5 0.00581855
\(546\) 0 0
\(547\) −4.61993e6 −0.660187 −0.330093 0.943948i \(-0.607080\pi\)
−0.330093 + 0.943948i \(0.607080\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.38764e6 −0.194715
\(552\) 0 0
\(553\) −426454. −0.0593006
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.71536e6 −1.32685 −0.663424 0.748244i \(-0.730897\pi\)
−0.663424 + 0.748244i \(0.730897\pi\)
\(558\) 0 0
\(559\) 5.31882e6 0.719923
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.70410e6 −1.29028 −0.645141 0.764064i \(-0.723201\pi\)
−0.645141 + 0.764064i \(0.723201\pi\)
\(564\) 0 0
\(565\) −67998.3 −0.00896143
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.91242e6 −1.02454 −0.512270 0.858825i \(-0.671195\pi\)
−0.512270 + 0.858825i \(0.671195\pi\)
\(570\) 0 0
\(571\) 7.91956e6 1.01651 0.508254 0.861207i \(-0.330291\pi\)
0.508254 + 0.861207i \(0.330291\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.47048e6 −1.06841
\(576\) 0 0
\(577\) −6.81761e6 −0.852497 −0.426248 0.904606i \(-0.640165\pi\)
−0.426248 + 0.904606i \(0.640165\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.28758e6 0.158246
\(582\) 0 0
\(583\) −4.62925e6 −0.564079
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.75763e6 −0.689682 −0.344841 0.938661i \(-0.612067\pi\)
−0.344841 + 0.938661i \(0.612067\pi\)
\(588\) 0 0
\(589\) −1.71428e7 −2.03607
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.22350e6 −0.142879 −0.0714393 0.997445i \(-0.522759\pi\)
−0.0714393 + 0.997445i \(0.522759\pi\)
\(594\) 0 0
\(595\) −38024.6 −0.00440324
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 590873. 0.0672864 0.0336432 0.999434i \(-0.489289\pi\)
0.0336432 + 0.999434i \(0.489289\pi\)
\(600\) 0 0
\(601\) −1.27472e7 −1.43955 −0.719776 0.694207i \(-0.755755\pi\)
−0.719776 + 0.694207i \(0.755755\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −52724.1 −0.00585626
\(606\) 0 0
\(607\) 7.36918e6 0.811797 0.405899 0.913918i \(-0.366959\pi\)
0.405899 + 0.913918i \(0.366959\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.07381e7 −1.16365
\(612\) 0 0
\(613\) −5.05204e6 −0.543020 −0.271510 0.962436i \(-0.587523\pi\)
−0.271510 + 0.962436i \(0.587523\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.04265e7 −1.10262 −0.551309 0.834301i \(-0.685871\pi\)
−0.551309 + 0.834301i \(0.685871\pi\)
\(618\) 0 0
\(619\) 1.84513e7 1.93553 0.967765 0.251856i \(-0.0810410\pi\)
0.967765 + 0.251856i \(0.0810410\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.64633e6 −0.169940
\(624\) 0 0
\(625\) 9.76381e6 0.999814
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.67410e6 −0.672614
\(630\) 0 0
\(631\) 2.28762e6 0.228723 0.114362 0.993439i \(-0.463518\pi\)
0.114362 + 0.993439i \(0.463518\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −23826.3 −0.00234489
\(636\) 0 0
\(637\) −967487. −0.0944705
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.98845e6 0.671793 0.335897 0.941899i \(-0.390961\pi\)
0.335897 + 0.941899i \(0.390961\pi\)
\(642\) 0 0
\(643\) −1.47461e7 −1.40653 −0.703265 0.710928i \(-0.748275\pi\)
−0.703265 + 0.710928i \(0.748275\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.20477e6 0.300979 0.150490 0.988612i \(-0.451915\pi\)
0.150490 + 0.988612i \(0.451915\pi\)
\(648\) 0 0
\(649\) −321269. −0.0299404
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.10963e6 0.193608 0.0968041 0.995303i \(-0.469138\pi\)
0.0968041 + 0.995303i \(0.469138\pi\)
\(654\) 0 0
\(655\) −19375.3 −0.00176459
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.74852e7 1.56840 0.784202 0.620505i \(-0.213072\pi\)
0.784202 + 0.620505i \(0.213072\pi\)
\(660\) 0 0
\(661\) −973131. −0.0866299 −0.0433149 0.999061i \(-0.513792\pi\)
−0.0433149 + 0.999061i \(0.513792\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 38323.9 0.00336059
\(666\) 0 0
\(667\) −2.11869e6 −0.184396
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.18542e6 −0.616092
\(672\) 0 0
\(673\) 651051. 0.0554086 0.0277043 0.999616i \(-0.491180\pi\)
0.0277043 + 0.999616i \(0.491180\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.62839e6 0.388113 0.194057 0.980990i \(-0.437835\pi\)
0.194057 + 0.980990i \(0.437835\pi\)
\(678\) 0 0
\(679\) 1.12702e6 0.0938119
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.77222e7 −1.45367 −0.726836 0.686811i \(-0.759010\pi\)
−0.726836 + 0.686811i \(0.759010\pi\)
\(684\) 0 0
\(685\) −35057.0 −0.00285462
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.17128e6 0.736007
\(690\) 0 0
\(691\) 1.41125e7 1.12437 0.562186 0.827011i \(-0.309960\pi\)
0.562186 + 0.827011i \(0.309960\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −91315.3 −0.00717103
\(696\) 0 0
\(697\) −3.17676e7 −2.47687
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.02386e6 −0.0786947 −0.0393474 0.999226i \(-0.512528\pi\)
−0.0393474 + 0.999226i \(0.512528\pi\)
\(702\) 0 0
\(703\) 6.72664e6 0.513346
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.81886e6 −0.212092
\(708\) 0 0
\(709\) −7.14313e6 −0.533670 −0.266835 0.963742i \(-0.585978\pi\)
−0.266835 + 0.963742i \(0.585978\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.61740e7 −1.92818
\(714\) 0 0
\(715\) −36104.8 −0.00264119
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.74905e7 −1.26177 −0.630884 0.775878i \(-0.717307\pi\)
−0.630884 + 0.775878i \(0.717307\pi\)
\(720\) 0 0
\(721\) −8.22193e6 −0.589027
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.44233e6 0.172568
\(726\) 0 0
\(727\) −8.07868e6 −0.566897 −0.283449 0.958987i \(-0.591479\pi\)
−0.283449 + 0.958987i \(0.591479\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.32516e7 1.60939
\(732\) 0 0
\(733\) −6.42261e6 −0.441521 −0.220761 0.975328i \(-0.570854\pi\)
−0.220761 + 0.975328i \(0.570854\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.87917e6 −0.669965
\(738\) 0 0
\(739\) −2.45407e7 −1.65301 −0.826506 0.562927i \(-0.809675\pi\)
−0.826506 + 0.562927i \(0.809675\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.53250e7 1.01842 0.509212 0.860641i \(-0.329937\pi\)
0.509212 + 0.860641i \(0.329937\pi\)
\(744\) 0 0
\(745\) −168673. −0.0111341
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.33099e6 −0.216954
\(750\) 0 0
\(751\) −1.24971e7 −0.808554 −0.404277 0.914637i \(-0.632477\pi\)
−0.404277 + 0.914637i \(0.632477\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22153.3 0.00141440
\(756\) 0 0
\(757\) −1.56124e7 −0.990217 −0.495108 0.868831i \(-0.664872\pi\)
−0.495108 + 0.868831i \(0.664872\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.82820e7 −1.14436 −0.572179 0.820128i \(-0.693902\pi\)
−0.572179 + 0.820128i \(0.693902\pi\)
\(762\) 0 0
\(763\) 4.48770e6 0.279070
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 636485. 0.0390661
\(768\) 0 0
\(769\) 1.32020e7 0.805051 0.402526 0.915409i \(-0.368132\pi\)
0.402526 + 0.915409i \(0.368132\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.45645e7 0.876689 0.438344 0.898807i \(-0.355565\pi\)
0.438344 + 0.898807i \(0.355565\pi\)
\(774\) 0 0
\(775\) 3.01723e7 1.80449
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.20177e7 1.89037
\(780\) 0 0
\(781\) −2.59610e6 −0.152298
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 161947. 0.00937990
\(786\) 0 0
\(787\) 1.78163e7 1.02537 0.512685 0.858577i \(-0.328651\pi\)
0.512685 + 0.858577i \(0.328651\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7.56339e6 −0.429809
\(792\) 0 0
\(793\) 1.42354e7 0.803874
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.29429e7 1.27939 0.639696 0.768628i \(-0.279060\pi\)
0.639696 + 0.768628i \(0.279060\pi\)
\(798\) 0 0
\(799\) −4.69424e7 −2.60134
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.06490e6 0.113008
\(804\) 0 0
\(805\) 58513.9 0.00318251
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.92525e6 −0.157142 −0.0785710 0.996909i \(-0.525036\pi\)
−0.0785710 + 0.996909i \(0.525036\pi\)
\(810\) 0 0
\(811\) 7.22818e6 0.385902 0.192951 0.981208i \(-0.438194\pi\)
0.192951 + 0.981208i \(0.438194\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −91177.8 −0.00480834
\(816\) 0 0
\(817\) −2.34347e7 −1.22830
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.85795e7 −1.47978 −0.739890 0.672728i \(-0.765122\pi\)
−0.739890 + 0.672728i \(0.765122\pi\)
\(822\) 0 0
\(823\) 1.31774e6 0.0678157 0.0339078 0.999425i \(-0.489205\pi\)
0.0339078 + 0.999425i \(0.489205\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.02202e7 1.53650 0.768252 0.640147i \(-0.221127\pi\)
0.768252 + 0.640147i \(0.221127\pi\)
\(828\) 0 0
\(829\) −8.04061e6 −0.406352 −0.203176 0.979142i \(-0.565126\pi\)
−0.203176 + 0.979142i \(0.565126\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.22944e6 −0.211189
\(834\) 0 0
\(835\) −113342. −0.00562570
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.62114e6 0.226644 0.113322 0.993558i \(-0.463851\pi\)
0.113322 + 0.993558i \(0.463851\pi\)
\(840\) 0 0
\(841\) −1.99003e7 −0.970217
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −92037.4 −0.00443427
\(846\) 0 0
\(847\) −5.86445e6 −0.280879
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.02704e7 0.486142
\(852\) 0 0
\(853\) 2.20395e7 1.03712 0.518559 0.855042i \(-0.326468\pi\)
0.518559 + 0.855042i \(0.326468\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.48477e7 0.690569 0.345284 0.938498i \(-0.387782\pi\)
0.345284 + 0.938498i \(0.387782\pi\)
\(858\) 0 0
\(859\) −668744. −0.0309227 −0.0154613 0.999880i \(-0.504922\pi\)
−0.0154613 + 0.999880i \(0.504922\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.09838e7 0.502024 0.251012 0.967984i \(-0.419237\pi\)
0.251012 + 0.967984i \(0.419237\pi\)
\(864\) 0 0
\(865\) 118517. 0.00538569
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.77015e6 −0.0795171
\(870\) 0 0
\(871\) 1.95722e7 0.874166
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −134909. −0.00595690
\(876\) 0 0
\(877\) 4.15400e7 1.82376 0.911879 0.410460i \(-0.134632\pi\)
0.911879 + 0.410460i \(0.134632\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.03497e7 −1.75146 −0.875730 0.482801i \(-0.839619\pi\)
−0.875730 + 0.482801i \(0.839619\pi\)
\(882\) 0 0
\(883\) −2.10878e7 −0.910183 −0.455092 0.890445i \(-0.650394\pi\)
−0.455092 + 0.890445i \(0.650394\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.52443e6 −0.150411 −0.0752056 0.997168i \(-0.523961\pi\)
−0.0752056 + 0.997168i \(0.523961\pi\)
\(888\) 0 0
\(889\) −2.65018e6 −0.112466
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.73119e7 1.98537
\(894\) 0 0
\(895\) −235244. −0.00981658
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7.54687e6 0.311435
\(900\) 0 0
\(901\) 4.00929e7 1.64534
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −277415. −0.0112592
\(906\) 0 0
\(907\) −2.95093e7 −1.19108 −0.595540 0.803325i \(-0.703062\pi\)
−0.595540 + 0.803325i \(0.703062\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.32643e7 −0.529527 −0.264763 0.964313i \(-0.585294\pi\)
−0.264763 + 0.964313i \(0.585294\pi\)
\(912\) 0 0
\(913\) 5.34455e6 0.212195
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.15509e6 −0.0846336
\(918\) 0 0
\(919\) 1.89419e7 0.739834 0.369917 0.929065i \(-0.379386\pi\)
0.369917 + 0.929065i \(0.379386\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.14327e6 0.198717
\(924\) 0 0
\(925\) −1.18393e7 −0.454957
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.92211e7 −1.11086 −0.555428 0.831565i \(-0.687445\pi\)
−0.555428 + 0.831565i \(0.687445\pi\)
\(930\) 0 0
\(931\) 4.26274e6 0.161181
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −157835. −0.00590437
\(936\) 0 0
\(937\) −6.46944e6 −0.240723 −0.120362 0.992730i \(-0.538405\pi\)
−0.120362 + 0.992730i \(0.538405\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.11564e7 1.51518 0.757588 0.652733i \(-0.226378\pi\)
0.757588 + 0.652733i \(0.226378\pi\)
\(942\) 0 0
\(943\) 4.88854e7 1.79019
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.00657e7 1.08942 0.544711 0.838624i \(-0.316639\pi\)
0.544711 + 0.838624i \(0.316639\pi\)
\(948\) 0 0
\(949\) −4.09089e6 −0.147453
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.95724e7 −1.05476 −0.527381 0.849629i \(-0.676826\pi\)
−0.527381 + 0.849629i \(0.676826\pi\)
\(954\) 0 0
\(955\) 199208. 0.00706803
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.89935e6 −0.136913
\(960\) 0 0
\(961\) 6.46041e7 2.25658
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −164035. −0.00567045
\(966\) 0 0
\(967\) 3.49436e7 1.20172 0.600858 0.799356i \(-0.294826\pi\)
0.600858 + 0.799356i \(0.294826\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.13425e7 0.386065 0.193032 0.981192i \(-0.438168\pi\)
0.193032 + 0.981192i \(0.438168\pi\)
\(972\) 0 0
\(973\) −1.01569e7 −0.343938
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.24082e7 −1.42139 −0.710694 0.703501i \(-0.751619\pi\)
−0.710694 + 0.703501i \(0.751619\pi\)
\(978\) 0 0
\(979\) −6.83366e6 −0.227875
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.98560e7 1.31556 0.657779 0.753211i \(-0.271496\pi\)
0.657779 + 0.753211i \(0.271496\pi\)
\(984\) 0 0
\(985\) 110182. 0.00361841
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.57806e7 −1.16321
\(990\) 0 0
\(991\) 2.81338e7 0.910007 0.455004 0.890490i \(-0.349638\pi\)
0.455004 + 0.890490i \(0.349638\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −412349. −0.0132041
\(996\) 0 0
\(997\) 5.67938e7 1.80952 0.904760 0.425922i \(-0.140050\pi\)
0.904760 + 0.425922i \(0.140050\pi\)
\(998\) 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 504.6.a.t.1.1 2
3.2 odd 2 168.6.a.h.1.2 2
4.3 odd 2 1008.6.a.bu.1.1 2
12.11 even 2 336.6.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.6.a.h.1.2 2 3.2 odd 2
336.6.a.w.1.2 2 12.11 even 2
504.6.a.t.1.1 2 1.1 even 1 trivial
1008.6.a.bu.1.1 2 4.3 odd 2