Properties

Label 684.6.a.e.1.1
Level $684$
Weight $6$
Character 684.1
Self dual yes
Analytic conductor $109.703$
Analytic rank $1$
Dimension $4$
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] = [684,6,Mod(1,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("684.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 684.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: \(109.702532752\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 140x^{2} - 84x + 3103 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(11.0547\) of defining polynomial
Character \(\chi\) \(=\) 684.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-87.4671 q^{5} +76.9277 q^{7} +O(q^{10})\) \(q-87.4671 q^{5} +76.9277 q^{7} +99.2130 q^{11} -79.5656 q^{13} +465.981 q^{17} -361.000 q^{19} -2231.89 q^{23} +4525.49 q^{25} -757.584 q^{29} +2718.02 q^{31} -6728.64 q^{35} +10309.9 q^{37} +6945.22 q^{41} -2107.29 q^{43} +20564.6 q^{47} -10889.1 q^{49} +17567.4 q^{53} -8677.87 q^{55} +34514.7 q^{59} -40975.0 q^{61} +6959.37 q^{65} +68248.5 q^{67} -38876.4 q^{71} -83648.9 q^{73} +7632.23 q^{77} +39001.5 q^{79} -99454.0 q^{83} -40758.0 q^{85} +88908.3 q^{89} -6120.80 q^{91} +31575.6 q^{95} -87484.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 110 q^{5} + 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 110 q^{5} + 30 q^{7} - 706 q^{11} + 788 q^{13} - 240 q^{17} - 1444 q^{19} - 5884 q^{23} + 11774 q^{25} - 5240 q^{29} - 860 q^{31} - 20322 q^{35} - 20732 q^{37} + 10204 q^{41} - 12554 q^{43} + 4826 q^{47} - 21376 q^{49} + 76484 q^{53} - 72914 q^{55} - 23898 q^{59} - 32482 q^{61} + 6076 q^{65} + 5022 q^{67} - 121300 q^{71} - 104700 q^{73} + 10002 q^{77} + 117128 q^{79} - 92832 q^{83} + 80322 q^{85} - 5988 q^{89} + 165618 q^{91} - 39710 q^{95} + 22972 q^{97}+O(q^{100}) \) 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
\(5\) −87.4671 −1.56466 −0.782329 0.622865i \(-0.785969\pi\)
−0.782329 + 0.622865i \(0.785969\pi\)
\(6\) 0 0
\(7\) 76.9277 0.593386 0.296693 0.954973i \(-0.404116\pi\)
0.296693 + 0.954973i \(0.404116\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 99.2130 0.247222 0.123611 0.992331i \(-0.460553\pi\)
0.123611 + 0.992331i \(0.460553\pi\)
\(12\) 0 0
\(13\) −79.5656 −0.130577 −0.0652886 0.997866i \(-0.520797\pi\)
−0.0652886 + 0.997866i \(0.520797\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 465.981 0.391062 0.195531 0.980698i \(-0.437357\pi\)
0.195531 + 0.980698i \(0.437357\pi\)
\(18\) 0 0
\(19\) −361.000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2231.89 −0.879738 −0.439869 0.898062i \(-0.644975\pi\)
−0.439869 + 0.898062i \(0.644975\pi\)
\(24\) 0 0
\(25\) 4525.49 1.44816
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −757.584 −0.167277 −0.0836384 0.996496i \(-0.526654\pi\)
−0.0836384 + 0.996496i \(0.526654\pi\)
\(30\) 0 0
\(31\) 2718.02 0.507983 0.253991 0.967206i \(-0.418257\pi\)
0.253991 + 0.967206i \(0.418257\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6728.64 −0.928447
\(36\) 0 0
\(37\) 10309.9 1.23808 0.619040 0.785360i \(-0.287522\pi\)
0.619040 + 0.785360i \(0.287522\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6945.22 0.645248 0.322624 0.946527i \(-0.395435\pi\)
0.322624 + 0.946527i \(0.395435\pi\)
\(42\) 0 0
\(43\) −2107.29 −0.173802 −0.0869009 0.996217i \(-0.527696\pi\)
−0.0869009 + 0.996217i \(0.527696\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 20564.6 1.35793 0.678963 0.734173i \(-0.262430\pi\)
0.678963 + 0.734173i \(0.262430\pi\)
\(48\) 0 0
\(49\) −10889.1 −0.647893
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 17567.4 0.859050 0.429525 0.903055i \(-0.358681\pi\)
0.429525 + 0.903055i \(0.358681\pi\)
\(54\) 0 0
\(55\) −8677.87 −0.386818
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 34514.7 1.29084 0.645422 0.763827i \(-0.276682\pi\)
0.645422 + 0.763827i \(0.276682\pi\)
\(60\) 0 0
\(61\) −40975.0 −1.40992 −0.704959 0.709248i \(-0.749035\pi\)
−0.704959 + 0.709248i \(0.749035\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6959.37 0.204309
\(66\) 0 0
\(67\) 68248.5 1.85740 0.928701 0.370829i \(-0.120926\pi\)
0.928701 + 0.370829i \(0.120926\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −38876.4 −0.915251 −0.457626 0.889145i \(-0.651300\pi\)
−0.457626 + 0.889145i \(0.651300\pi\)
\(72\) 0 0
\(73\) −83648.9 −1.83719 −0.918593 0.395205i \(-0.870674\pi\)
−0.918593 + 0.395205i \(0.870674\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7632.23 0.146698
\(78\) 0 0
\(79\) 39001.5 0.703094 0.351547 0.936170i \(-0.385656\pi\)
0.351547 + 0.936170i \(0.385656\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −99454.0 −1.58463 −0.792314 0.610114i \(-0.791124\pi\)
−0.792314 + 0.610114i \(0.791124\pi\)
\(84\) 0 0
\(85\) −40758.0 −0.611879
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 88908.3 1.18978 0.594891 0.803806i \(-0.297195\pi\)
0.594891 + 0.803806i \(0.297195\pi\)
\(90\) 0 0
\(91\) −6120.80 −0.0774827
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 31575.6 0.358957
\(96\) 0 0
\(97\) −87484.2 −0.944061 −0.472031 0.881582i \(-0.656479\pi\)
−0.472031 + 0.881582i \(0.656479\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −124712. −1.21648 −0.608238 0.793754i \(-0.708124\pi\)
−0.608238 + 0.793754i \(0.708124\pi\)
\(102\) 0 0
\(103\) −28096.0 −0.260947 −0.130473 0.991452i \(-0.541650\pi\)
−0.130473 + 0.991452i \(0.541650\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −106686. −0.900839 −0.450419 0.892817i \(-0.648726\pi\)
−0.450419 + 0.892817i \(0.648726\pi\)
\(108\) 0 0
\(109\) −81738.7 −0.658964 −0.329482 0.944162i \(-0.606874\pi\)
−0.329482 + 0.944162i \(0.606874\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −265828. −1.95841 −0.979206 0.202871i \(-0.934973\pi\)
−0.979206 + 0.202871i \(0.934973\pi\)
\(114\) 0 0
\(115\) 195217. 1.37649
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 35846.8 0.232051
\(120\) 0 0
\(121\) −151208. −0.938881
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −122497. −0.701213
\(126\) 0 0
\(127\) −320548. −1.76353 −0.881766 0.471688i \(-0.843645\pi\)
−0.881766 + 0.471688i \(0.843645\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 104893. 0.534034 0.267017 0.963692i \(-0.413962\pi\)
0.267017 + 0.963692i \(0.413962\pi\)
\(132\) 0 0
\(133\) −27770.9 −0.136132
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −276152. −1.25704 −0.628518 0.777795i \(-0.716338\pi\)
−0.628518 + 0.777795i \(0.716338\pi\)
\(138\) 0 0
\(139\) 51956.7 0.228089 0.114045 0.993476i \(-0.463619\pi\)
0.114045 + 0.993476i \(0.463619\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −7893.94 −0.0322815
\(144\) 0 0
\(145\) 66263.7 0.261731
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 420951. 1.55334 0.776670 0.629908i \(-0.216907\pi\)
0.776670 + 0.629908i \(0.216907\pi\)
\(150\) 0 0
\(151\) −56718.6 −0.202434 −0.101217 0.994864i \(-0.532274\pi\)
−0.101217 + 0.994864i \(0.532274\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −237738. −0.794820
\(156\) 0 0
\(157\) −65573.0 −0.212313 −0.106156 0.994349i \(-0.533854\pi\)
−0.106156 + 0.994349i \(0.533854\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −171694. −0.522025
\(162\) 0 0
\(163\) 445737. 1.31404 0.657022 0.753872i \(-0.271816\pi\)
0.657022 + 0.753872i \(0.271816\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 150553. 0.417734 0.208867 0.977944i \(-0.433022\pi\)
0.208867 + 0.977944i \(0.433022\pi\)
\(168\) 0 0
\(169\) −364962. −0.982950
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 106599. 0.270793 0.135397 0.990791i \(-0.456769\pi\)
0.135397 + 0.990791i \(0.456769\pi\)
\(174\) 0 0
\(175\) 348135. 0.859316
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −416262. −0.971034 −0.485517 0.874227i \(-0.661369\pi\)
−0.485517 + 0.874227i \(0.661369\pi\)
\(180\) 0 0
\(181\) 100405. 0.227803 0.113901 0.993492i \(-0.463665\pi\)
0.113901 + 0.993492i \(0.463665\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −901774. −1.93717
\(186\) 0 0
\(187\) 46231.4 0.0966791
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −597176. −1.18446 −0.592228 0.805770i \(-0.701752\pi\)
−0.592228 + 0.805770i \(0.701752\pi\)
\(192\) 0 0
\(193\) 714364. 1.38047 0.690234 0.723586i \(-0.257507\pi\)
0.690234 + 0.723586i \(0.257507\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11062.3 0.0203085 0.0101543 0.999948i \(-0.496768\pi\)
0.0101543 + 0.999948i \(0.496768\pi\)
\(198\) 0 0
\(199\) 357858. 0.640587 0.320294 0.947318i \(-0.396218\pi\)
0.320294 + 0.947318i \(0.396218\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −58279.2 −0.0992598
\(204\) 0 0
\(205\) −607478. −1.00959
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −35815.9 −0.0567166
\(210\) 0 0
\(211\) −284130. −0.439350 −0.219675 0.975573i \(-0.570500\pi\)
−0.219675 + 0.975573i \(0.570500\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 184319. 0.271940
\(216\) 0 0
\(217\) 209091. 0.301430
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −37076.1 −0.0510638
\(222\) 0 0
\(223\) 870495. 1.17221 0.586103 0.810236i \(-0.300661\pi\)
0.586103 + 0.810236i \(0.300661\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −484247. −0.623738 −0.311869 0.950125i \(-0.600955\pi\)
−0.311869 + 0.950125i \(0.600955\pi\)
\(228\) 0 0
\(229\) −512807. −0.646197 −0.323099 0.946365i \(-0.604725\pi\)
−0.323099 + 0.946365i \(0.604725\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.14046e6 −1.37623 −0.688115 0.725601i \(-0.741562\pi\)
−0.688115 + 0.725601i \(0.741562\pi\)
\(234\) 0 0
\(235\) −1.79873e6 −2.12469
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 564259. 0.638975 0.319488 0.947590i \(-0.396489\pi\)
0.319488 + 0.947590i \(0.396489\pi\)
\(240\) 0 0
\(241\) −499520. −0.554001 −0.277000 0.960870i \(-0.589340\pi\)
−0.277000 + 0.960870i \(0.589340\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 952441. 1.01373
\(246\) 0 0
\(247\) 28723.2 0.0299564
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.32259e6 −1.32508 −0.662539 0.749027i \(-0.730521\pi\)
−0.662539 + 0.749027i \(0.730521\pi\)
\(252\) 0 0
\(253\) −221433. −0.217491
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −29088.9 −0.0274722 −0.0137361 0.999906i \(-0.504372\pi\)
−0.0137361 + 0.999906i \(0.504372\pi\)
\(258\) 0 0
\(259\) 793114. 0.734660
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.39828e6 −1.24653 −0.623266 0.782010i \(-0.714194\pi\)
−0.623266 + 0.782010i \(0.714194\pi\)
\(264\) 0 0
\(265\) −1.53657e6 −1.34412
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.62054e6 −1.36546 −0.682728 0.730672i \(-0.739207\pi\)
−0.682728 + 0.730672i \(0.739207\pi\)
\(270\) 0 0
\(271\) −311537. −0.257683 −0.128842 0.991665i \(-0.541126\pi\)
−0.128842 + 0.991665i \(0.541126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 448988. 0.358016
\(276\) 0 0
\(277\) −801500. −0.627631 −0.313815 0.949484i \(-0.601607\pi\)
−0.313815 + 0.949484i \(0.601607\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.59535e6 −1.20528 −0.602642 0.798012i \(-0.705885\pi\)
−0.602642 + 0.798012i \(0.705885\pi\)
\(282\) 0 0
\(283\) −1.04144e6 −0.772980 −0.386490 0.922294i \(-0.626313\pi\)
−0.386490 + 0.922294i \(0.626313\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 534280. 0.382881
\(288\) 0 0
\(289\) −1.20272e6 −0.847070
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 391025. 0.266094 0.133047 0.991110i \(-0.457524\pi\)
0.133047 + 0.991110i \(0.457524\pi\)
\(294\) 0 0
\(295\) −3.01890e6 −2.01973
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 177582. 0.114874
\(300\) 0 0
\(301\) −162109. −0.103132
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.58396e6 2.20604
\(306\) 0 0
\(307\) −504246. −0.305349 −0.152675 0.988277i \(-0.548789\pi\)
−0.152675 + 0.988277i \(0.548789\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 80004.3 0.0469042 0.0234521 0.999725i \(-0.492534\pi\)
0.0234521 + 0.999725i \(0.492534\pi\)
\(312\) 0 0
\(313\) −99620.5 −0.0574762 −0.0287381 0.999587i \(-0.509149\pi\)
−0.0287381 + 0.999587i \(0.509149\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 177148. 0.0990121 0.0495060 0.998774i \(-0.484235\pi\)
0.0495060 + 0.998774i \(0.484235\pi\)
\(318\) 0 0
\(319\) −75162.2 −0.0413545
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −168219. −0.0897158
\(324\) 0 0
\(325\) −360073. −0.189096
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.58199e6 0.805774
\(330\) 0 0
\(331\) 1.05867e6 0.531116 0.265558 0.964095i \(-0.414444\pi\)
0.265558 + 0.964095i \(0.414444\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.96950e6 −2.90620
\(336\) 0 0
\(337\) 3.24019e6 1.55416 0.777081 0.629401i \(-0.216700\pi\)
0.777081 + 0.629401i \(0.216700\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 269663. 0.125584
\(342\) 0 0
\(343\) −2.13060e6 −0.977837
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.67241e6 1.19146 0.595729 0.803185i \(-0.296863\pi\)
0.595729 + 0.803185i \(0.296863\pi\)
\(348\) 0 0
\(349\) 3.02667e6 1.33015 0.665076 0.746775i \(-0.268399\pi\)
0.665076 + 0.746775i \(0.268399\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −790142. −0.337496 −0.168748 0.985659i \(-0.553972\pi\)
−0.168748 + 0.985659i \(0.553972\pi\)
\(354\) 0 0
\(355\) 3.40041e6 1.43206
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.80303e6 −0.738359 −0.369179 0.929358i \(-0.620361\pi\)
−0.369179 + 0.929358i \(0.620361\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.31653e6 2.87457
\(366\) 0 0
\(367\) 1.68799e6 0.654192 0.327096 0.944991i \(-0.393930\pi\)
0.327096 + 0.944991i \(0.393930\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.35142e6 0.509748
\(372\) 0 0
\(373\) 4.48620e6 1.66958 0.834788 0.550571i \(-0.185590\pi\)
0.834788 + 0.550571i \(0.185590\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 60277.6 0.0218425
\(378\) 0 0
\(379\) 103084. 0.0368634 0.0184317 0.999830i \(-0.494133\pi\)
0.0184317 + 0.999830i \(0.494133\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.21030e6 −0.769934 −0.384967 0.922930i \(-0.625787\pi\)
−0.384967 + 0.922930i \(0.625787\pi\)
\(384\) 0 0
\(385\) −667569. −0.229532
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.56273e6 −1.19374 −0.596870 0.802338i \(-0.703589\pi\)
−0.596870 + 0.802338i \(0.703589\pi\)
\(390\) 0 0
\(391\) −1.04002e6 −0.344032
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.41135e6 −1.10010
\(396\) 0 0
\(397\) −4.69992e6 −1.49663 −0.748315 0.663344i \(-0.769137\pi\)
−0.748315 + 0.663344i \(0.769137\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.76391e6 0.858347 0.429173 0.903222i \(-0.358805\pi\)
0.429173 + 0.903222i \(0.358805\pi\)
\(402\) 0 0
\(403\) −216261. −0.0663309
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.02287e6 0.306080
\(408\) 0 0
\(409\) 4.43668e6 1.31144 0.655722 0.755002i \(-0.272364\pi\)
0.655722 + 0.755002i \(0.272364\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.65513e6 0.765969
\(414\) 0 0
\(415\) 8.69895e6 2.47940
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.67747e6 1.02333 0.511664 0.859186i \(-0.329029\pi\)
0.511664 + 0.859186i \(0.329029\pi\)
\(420\) 0 0
\(421\) 2.97195e6 0.817216 0.408608 0.912710i \(-0.366014\pi\)
0.408608 + 0.912710i \(0.366014\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.10879e6 0.566319
\(426\) 0 0
\(427\) −3.15211e6 −0.836626
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.90998e6 0.754565 0.377283 0.926098i \(-0.376859\pi\)
0.377283 + 0.926098i \(0.376859\pi\)
\(432\) 0 0
\(433\) 3.06888e6 0.786611 0.393305 0.919408i \(-0.371332\pi\)
0.393305 + 0.919408i \(0.371332\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 805713. 0.201826
\(438\) 0 0
\(439\) 2.87829e6 0.712809 0.356404 0.934332i \(-0.384003\pi\)
0.356404 + 0.934332i \(0.384003\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.29112e6 −0.312578 −0.156289 0.987711i \(-0.549953\pi\)
−0.156289 + 0.987711i \(0.549953\pi\)
\(444\) 0 0
\(445\) −7.77655e6 −1.86160
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.24274e6 −0.993186 −0.496593 0.867984i \(-0.665416\pi\)
−0.496593 + 0.867984i \(0.665416\pi\)
\(450\) 0 0
\(451\) 689056. 0.159519
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 535368. 0.121234
\(456\) 0 0
\(457\) −6.19374e6 −1.38727 −0.693637 0.720324i \(-0.743993\pi\)
−0.693637 + 0.720324i \(0.743993\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.87032e6 1.28650 0.643250 0.765656i \(-0.277586\pi\)
0.643250 + 0.765656i \(0.277586\pi\)
\(462\) 0 0
\(463\) −8.40538e6 −1.82224 −0.911119 0.412144i \(-0.864780\pi\)
−0.911119 + 0.412144i \(0.864780\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 148549. 0.0315193 0.0157597 0.999876i \(-0.494983\pi\)
0.0157597 + 0.999876i \(0.494983\pi\)
\(468\) 0 0
\(469\) 5.25020e6 1.10216
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −209071. −0.0429676
\(474\) 0 0
\(475\) −1.63370e6 −0.332230
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.35414e6 −1.06623 −0.533115 0.846043i \(-0.678979\pi\)
−0.533115 + 0.846043i \(0.678979\pi\)
\(480\) 0 0
\(481\) −820311. −0.161665
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.65198e6 1.47713
\(486\) 0 0
\(487\) −2.86815e6 −0.547998 −0.273999 0.961730i \(-0.588346\pi\)
−0.273999 + 0.961730i \(0.588346\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.73298e6 −1.44758 −0.723791 0.690019i \(-0.757602\pi\)
−0.723791 + 0.690019i \(0.757602\pi\)
\(492\) 0 0
\(493\) −353020. −0.0654156
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.99067e6 −0.543097
\(498\) 0 0
\(499\) 1.93075e6 0.347116 0.173558 0.984824i \(-0.444474\pi\)
0.173558 + 0.984824i \(0.444474\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.50103e6 0.969447 0.484723 0.874668i \(-0.338920\pi\)
0.484723 + 0.874668i \(0.338920\pi\)
\(504\) 0 0
\(505\) 1.09082e7 1.90337
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.93464e6 0.844230 0.422115 0.906542i \(-0.361288\pi\)
0.422115 + 0.906542i \(0.361288\pi\)
\(510\) 0 0
\(511\) −6.43492e6 −1.09016
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.45748e6 0.408292
\(516\) 0 0
\(517\) 2.04028e6 0.335709
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5.98044e6 −0.965248 −0.482624 0.875828i \(-0.660316\pi\)
−0.482624 + 0.875828i \(0.660316\pi\)
\(522\) 0 0
\(523\) −3.74282e6 −0.598336 −0.299168 0.954200i \(-0.596709\pi\)
−0.299168 + 0.954200i \(0.596709\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.26655e6 0.198653
\(528\) 0 0
\(529\) −1.45500e6 −0.226061
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −552601. −0.0842546
\(534\) 0 0
\(535\) 9.33149e6 1.40951
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.08034e6 −0.160173
\(540\) 0 0
\(541\) 7.66502e6 1.12595 0.562977 0.826473i \(-0.309656\pi\)
0.562977 + 0.826473i \(0.309656\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.14945e6 1.03105
\(546\) 0 0
\(547\) −8.02578e6 −1.14688 −0.573441 0.819247i \(-0.694392\pi\)
−0.573441 + 0.819247i \(0.694392\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 273488. 0.0383759
\(552\) 0 0
\(553\) 3.00029e6 0.417206
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.39685e7 1.90771 0.953855 0.300266i \(-0.0970756\pi\)
0.953855 + 0.300266i \(0.0970756\pi\)
\(558\) 0 0
\(559\) 167668. 0.0226945
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.14218e6 0.683717 0.341858 0.939751i \(-0.388944\pi\)
0.341858 + 0.939751i \(0.388944\pi\)
\(564\) 0 0
\(565\) 2.32512e7 3.06425
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.38063e7 −1.78771 −0.893853 0.448359i \(-0.852009\pi\)
−0.893853 + 0.448359i \(0.852009\pi\)
\(570\) 0 0
\(571\) 1.91974e6 0.246406 0.123203 0.992381i \(-0.460683\pi\)
0.123203 + 0.992381i \(0.460683\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.01004e7 −1.27400
\(576\) 0 0
\(577\) −7.86662e6 −0.983669 −0.491834 0.870689i \(-0.663673\pi\)
−0.491834 + 0.870689i \(0.663673\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.65077e6 −0.940296
\(582\) 0 0
\(583\) 1.74292e6 0.212376
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.61716e6 0.672856 0.336428 0.941709i \(-0.390781\pi\)
0.336428 + 0.941709i \(0.390781\pi\)
\(588\) 0 0
\(589\) −981206. −0.116539
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.37137e6 0.393704 0.196852 0.980433i \(-0.436928\pi\)
0.196852 + 0.980433i \(0.436928\pi\)
\(594\) 0 0
\(595\) −3.13542e6 −0.363081
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.63713e6 0.528059 0.264030 0.964515i \(-0.414948\pi\)
0.264030 + 0.964515i \(0.414948\pi\)
\(600\) 0 0
\(601\) −1.32857e7 −1.50037 −0.750184 0.661229i \(-0.770035\pi\)
−0.750184 + 0.661229i \(0.770035\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.32257e7 1.46903
\(606\) 0 0
\(607\) −5.36627e6 −0.591154 −0.295577 0.955319i \(-0.595512\pi\)
−0.295577 + 0.955319i \(0.595512\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.63624e6 −0.177314
\(612\) 0 0
\(613\) −1.19912e7 −1.28888 −0.644440 0.764655i \(-0.722909\pi\)
−0.644440 + 0.764655i \(0.722909\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.57116e6 0.589159 0.294579 0.955627i \(-0.404820\pi\)
0.294579 + 0.955627i \(0.404820\pi\)
\(618\) 0 0
\(619\) 1.06940e7 1.12179 0.560897 0.827885i \(-0.310456\pi\)
0.560897 + 0.827885i \(0.310456\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.83951e6 0.706000
\(624\) 0 0
\(625\) −3.42772e6 −0.350999
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.80420e6 0.484166
\(630\) 0 0
\(631\) 1.27937e7 1.27915 0.639576 0.768728i \(-0.279110\pi\)
0.639576 + 0.768728i \(0.279110\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.80374e7 2.75932
\(636\) 0 0
\(637\) 866401. 0.0846000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.64712e6 0.350594 0.175297 0.984516i \(-0.443911\pi\)
0.175297 + 0.984516i \(0.443911\pi\)
\(642\) 0 0
\(643\) 9.41881e6 0.898398 0.449199 0.893432i \(-0.351709\pi\)
0.449199 + 0.893432i \(0.351709\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.34880e6 0.784085 0.392043 0.919947i \(-0.371769\pi\)
0.392043 + 0.919947i \(0.371769\pi\)
\(648\) 0 0
\(649\) 3.42430e6 0.319125
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.48058e6 −0.686518 −0.343259 0.939241i \(-0.611531\pi\)
−0.343259 + 0.939241i \(0.611531\pi\)
\(654\) 0 0
\(655\) −9.17471e6 −0.835582
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.39427e7 1.25064 0.625320 0.780368i \(-0.284968\pi\)
0.625320 + 0.780368i \(0.284968\pi\)
\(660\) 0 0
\(661\) −1.06152e7 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.42904e6 0.213000
\(666\) 0 0
\(667\) 1.69085e6 0.147160
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.06525e6 −0.348563
\(672\) 0 0
\(673\) 9.42379e6 0.802025 0.401012 0.916073i \(-0.368658\pi\)
0.401012 + 0.916073i \(0.368658\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.80040e6 −0.150972 −0.0754862 0.997147i \(-0.524051\pi\)
−0.0754862 + 0.997147i \(0.524051\pi\)
\(678\) 0 0
\(679\) −6.72995e6 −0.560193
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.73202e7 −1.42070 −0.710348 0.703850i \(-0.751463\pi\)
−0.710348 + 0.703850i \(0.751463\pi\)
\(684\) 0 0
\(685\) 2.41543e7 1.96683
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.39776e6 −0.112172
\(690\) 0 0
\(691\) −1.68765e7 −1.34458 −0.672292 0.740286i \(-0.734690\pi\)
−0.672292 + 0.740286i \(0.734690\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.54450e6 −0.356882
\(696\) 0 0
\(697\) 3.23634e6 0.252332
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −5.40330e6 −0.415302 −0.207651 0.978203i \(-0.566582\pi\)
−0.207651 + 0.978203i \(0.566582\pi\)
\(702\) 0 0
\(703\) −3.72186e6 −0.284035
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.59378e6 −0.721841
\(708\) 0 0
\(709\) 2.05223e7 1.53324 0.766619 0.642103i \(-0.221938\pi\)
0.766619 + 0.642103i \(0.221938\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.06633e6 −0.446892
\(714\) 0 0
\(715\) 690460. 0.0505096
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.37778e6 0.315815 0.157907 0.987454i \(-0.449525\pi\)
0.157907 + 0.987454i \(0.449525\pi\)
\(720\) 0 0
\(721\) −2.16136e6 −0.154842
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.42844e6 −0.242243
\(726\) 0 0
\(727\) −1.75812e7 −1.23371 −0.616853 0.787078i \(-0.711593\pi\)
−0.616853 + 0.787078i \(0.711593\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −981959. −0.0679673
\(732\) 0 0
\(733\) 5.07428e6 0.348831 0.174415 0.984672i \(-0.444196\pi\)
0.174415 + 0.984672i \(0.444196\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.77114e6 0.459191
\(738\) 0 0
\(739\) −2.35559e7 −1.58668 −0.793339 0.608780i \(-0.791659\pi\)
−0.793339 + 0.608780i \(0.791659\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.16206e7 −1.43680 −0.718399 0.695631i \(-0.755125\pi\)
−0.718399 + 0.695631i \(0.755125\pi\)
\(744\) 0 0
\(745\) −3.68194e7 −2.43045
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −8.20709e6 −0.534545
\(750\) 0 0
\(751\) 2.87245e7 1.85845 0.929227 0.369508i \(-0.120474\pi\)
0.929227 + 0.369508i \(0.120474\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.96101e6 0.316740
\(756\) 0 0
\(757\) 1.24115e7 0.787199 0.393599 0.919282i \(-0.371230\pi\)
0.393599 + 0.919282i \(0.371230\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −766287. −0.0479656 −0.0239828 0.999712i \(-0.507635\pi\)
−0.0239828 + 0.999712i \(0.507635\pi\)
\(762\) 0 0
\(763\) −6.28797e6 −0.391020
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.74618e6 −0.168555
\(768\) 0 0
\(769\) −1.49872e7 −0.913914 −0.456957 0.889489i \(-0.651061\pi\)
−0.456957 + 0.889489i \(0.651061\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.16435e7 −0.700867 −0.350434 0.936588i \(-0.613966\pi\)
−0.350434 + 0.936588i \(0.613966\pi\)
\(774\) 0 0
\(775\) 1.23004e7 0.735639
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.50722e6 −0.148030
\(780\) 0 0
\(781\) −3.85705e6 −0.226270
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.73547e6 0.332197
\(786\) 0 0
\(787\) −1.80280e7 −1.03756 −0.518778 0.854909i \(-0.673613\pi\)
−0.518778 + 0.854909i \(0.673613\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.04495e7 −1.16209
\(792\) 0 0
\(793\) 3.26020e6 0.184103
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.97016e7 −1.09864 −0.549319 0.835612i \(-0.685113\pi\)
−0.549319 + 0.835612i \(0.685113\pi\)
\(798\) 0 0
\(799\) 9.58272e6 0.531033
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −8.29906e6 −0.454193
\(804\) 0 0
\(805\) 1.50176e7 0.816790
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.78469e6 0.364468 0.182234 0.983255i \(-0.441667\pi\)
0.182234 + 0.983255i \(0.441667\pi\)
\(810\) 0 0
\(811\) 3.30503e7 1.76450 0.882252 0.470777i \(-0.156026\pi\)
0.882252 + 0.470777i \(0.156026\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.89873e7 −2.05603
\(816\) 0 0
\(817\) 760733. 0.0398729
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.99319e7 1.54980 0.774901 0.632083i \(-0.217800\pi\)
0.774901 + 0.632083i \(0.217800\pi\)
\(822\) 0 0
\(823\) 1.11456e7 0.573591 0.286796 0.957992i \(-0.407410\pi\)
0.286796 + 0.957992i \(0.407410\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.66420e7 −0.846141 −0.423070 0.906097i \(-0.639048\pi\)
−0.423070 + 0.906097i \(0.639048\pi\)
\(828\) 0 0
\(829\) −1.67963e7 −0.848844 −0.424422 0.905465i \(-0.639523\pi\)
−0.424422 + 0.905465i \(0.639523\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.07413e6 −0.253366
\(834\) 0 0
\(835\) −1.31685e7 −0.653611
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2.44304e7 −1.19819 −0.599095 0.800678i \(-0.704473\pi\)
−0.599095 + 0.800678i \(0.704473\pi\)
\(840\) 0 0
\(841\) −1.99372e7 −0.972018
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.19222e7 1.53798
\(846\) 0 0
\(847\) −1.16321e7 −0.557119
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.30105e7 −1.08919
\(852\) 0 0
\(853\) −3.68577e6 −0.173442 −0.0867212 0.996233i \(-0.527639\pi\)
−0.0867212 + 0.996233i \(0.527639\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.18729e7 1.48241 0.741207 0.671276i \(-0.234254\pi\)
0.741207 + 0.671276i \(0.234254\pi\)
\(858\) 0 0
\(859\) −3.56678e7 −1.64928 −0.824638 0.565661i \(-0.808621\pi\)
−0.824638 + 0.565661i \(0.808621\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.38642e7 −1.09074 −0.545369 0.838196i \(-0.683610\pi\)
−0.545369 + 0.838196i \(0.683610\pi\)
\(864\) 0 0
\(865\) −9.32391e6 −0.423699
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.86945e6 0.173820
\(870\) 0 0
\(871\) −5.43023e6 −0.242534
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.42339e6 −0.416090
\(876\) 0 0
\(877\) −1.78983e7 −0.785800 −0.392900 0.919581i \(-0.628528\pi\)
−0.392900 + 0.919581i \(0.628528\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.19383e7 −1.82042 −0.910209 0.414148i \(-0.864079\pi\)
−0.910209 + 0.414148i \(0.864079\pi\)
\(882\) 0 0
\(883\) −5.72425e6 −0.247068 −0.123534 0.992340i \(-0.539423\pi\)
−0.123534 + 0.992340i \(0.539423\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.28478e6 −0.225537 −0.112769 0.993621i \(-0.535972\pi\)
−0.112769 + 0.993621i \(0.535972\pi\)
\(888\) 0 0
\(889\) −2.46590e7 −1.04646
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −7.42383e6 −0.311529
\(894\) 0 0
\(895\) 3.64092e7 1.51934
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.05913e6 −0.0849737
\(900\) 0 0
\(901\) 8.18608e6 0.335942
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.78213e6 −0.356433
\(906\) 0 0
\(907\) −1.88055e6 −0.0759045 −0.0379523 0.999280i \(-0.512083\pi\)
−0.0379523 + 0.999280i \(0.512083\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.81410e7 0.724210 0.362105 0.932137i \(-0.382058\pi\)
0.362105 + 0.932137i \(0.382058\pi\)
\(912\) 0 0
\(913\) −9.86713e6 −0.391755
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.06919e6 0.316889
\(918\) 0 0
\(919\) −3.25793e7 −1.27249 −0.636243 0.771489i \(-0.719512\pi\)
−0.636243 + 0.771489i \(0.719512\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.09323e6 0.119511
\(924\) 0 0
\(925\) 4.66572e7 1.79293
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −5.80658e6 −0.220740 −0.110370 0.993891i \(-0.535204\pi\)
−0.110370 + 0.993891i \(0.535204\pi\)
\(930\) 0 0
\(931\) 3.93098e6 0.148637
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.04372e6 −0.151270
\(936\) 0 0
\(937\) 3.50025e7 1.30242 0.651209 0.758899i \(-0.274262\pi\)
0.651209 + 0.758899i \(0.274262\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.99448e7 1.47057 0.735285 0.677758i \(-0.237048\pi\)
0.735285 + 0.677758i \(0.237048\pi\)
\(942\) 0 0
\(943\) −1.55010e7 −0.567649
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.80996e6 0.246757 0.123379 0.992360i \(-0.460627\pi\)
0.123379 + 0.992360i \(0.460627\pi\)
\(948\) 0 0
\(949\) 6.65558e6 0.239894
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.48768e7 1.24395 0.621976 0.783036i \(-0.286330\pi\)
0.621976 + 0.783036i \(0.286330\pi\)
\(954\) 0 0
\(955\) 5.22332e7 1.85327
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.12438e7 −0.745908
\(960\) 0 0
\(961\) −2.12415e7 −0.741954
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6.24834e7 −2.15996
\(966\) 0 0
\(967\) −4.78004e7 −1.64386 −0.821931 0.569587i \(-0.807103\pi\)
−0.821931 + 0.569587i \(0.807103\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.82374e7 −1.30149 −0.650744 0.759298i \(-0.725543\pi\)
−0.650744 + 0.759298i \(0.725543\pi\)
\(972\) 0 0
\(973\) 3.99691e6 0.135345
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.14899e7 −0.385107 −0.192553 0.981287i \(-0.561677\pi\)
−0.192553 + 0.981287i \(0.561677\pi\)
\(978\) 0 0
\(979\) 8.82086e6 0.294140
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.75149e7 1.89844 0.949220 0.314614i \(-0.101875\pi\)
0.949220 + 0.314614i \(0.101875\pi\)
\(984\) 0 0
\(985\) −967583. −0.0317759
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.70325e6 0.152900
\(990\) 0 0
\(991\) −1.07593e7 −0.348016 −0.174008 0.984744i \(-0.555672\pi\)
−0.174008 + 0.984744i \(0.555672\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.13008e7 −1.00230
\(996\) 0 0
\(997\) −9.88575e6 −0.314972 −0.157486 0.987521i \(-0.550339\pi\)
−0.157486 + 0.987521i \(0.550339\pi\)
\(998\) 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 684.6.a.e.1.1 4
3.2 odd 2 76.6.a.b.1.3 4
12.11 even 2 304.6.a.k.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.a.b.1.3 4 3.2 odd 2
304.6.a.k.1.2 4 12.11 even 2
684.6.a.e.1.1 4 1.1 even 1 trivial