Properties

Label 784.6.a.s.1.1
Level $784$
Weight $6$
Character 784.1
Self dual yes
Analytic conductor $125.741$
Analytic rank $0$
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] = [784,6,Mod(1,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, 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("784.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.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: \(125.740914733\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{79}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-8.88819\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-24.7764 q^{3} -36.1056 q^{5} +370.869 q^{9} +O(q^{10})\) \(q-24.7764 q^{3} -36.1056 q^{5} +370.869 q^{9} -155.435 q^{11} +1158.87 q^{13} +894.565 q^{15} +1238.08 q^{17} +280.279 q^{19} +3482.39 q^{23} -1821.39 q^{25} -3168.14 q^{27} -5656.78 q^{29} +2314.70 q^{31} +3851.11 q^{33} -2333.18 q^{37} -28712.6 q^{39} +3812.61 q^{41} -3925.73 q^{43} -13390.4 q^{45} +11116.5 q^{47} -30675.2 q^{51} -11186.2 q^{53} +5612.06 q^{55} -6944.31 q^{57} +6010.35 q^{59} -14838.7 q^{61} -41841.6 q^{65} +42983.9 q^{67} -86281.1 q^{69} +19962.4 q^{71} +45550.6 q^{73} +45127.4 q^{75} -108992. q^{79} -11626.1 q^{81} -55829.0 q^{83} -44701.6 q^{85} +140155. q^{87} -95545.8 q^{89} -57349.9 q^{93} -10119.6 q^{95} +15004.9 q^{97} -57646.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} + 70 q^{5} + 244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} + 70 q^{5} + 244 q^{9} - 62 q^{11} + 1820 q^{13} + 2038 q^{15} + 1694 q^{17} - 826 q^{19} + 2734 q^{23} + 6312 q^{25} - 7154 q^{27} - 2852 q^{29} + 2674 q^{31} + 4858 q^{33} - 9146 q^{37} - 21588 q^{39} + 6132 q^{41} + 16040 q^{43} - 26852 q^{45} + 25326 q^{47} - 25762 q^{51} + 14958 q^{53} + 15526 q^{55} - 18866 q^{57} + 1106 q^{59} - 28042 q^{61} + 28308 q^{65} + 102642 q^{67} - 94346 q^{69} + 11056 q^{71} + 35070 q^{73} + 132776 q^{75} - 101762 q^{79} - 23750 q^{81} + 44632 q^{83} + 3674 q^{85} + 170380 q^{87} - 75474 q^{89} - 53478 q^{93} - 127502 q^{95} - 8316 q^{97} - 69500 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −24.7764 −1.58941 −0.794703 0.606998i \(-0.792373\pi\)
−0.794703 + 0.606998i \(0.792373\pi\)
\(4\) 0 0
\(5\) −36.1056 −0.645876 −0.322938 0.946420i \(-0.604671\pi\)
−0.322938 + 0.946420i \(0.604671\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 370.869 1.52621
\(10\) 0 0
\(11\) −155.435 −0.387317 −0.193658 0.981069i \(-0.562035\pi\)
−0.193658 + 0.981069i \(0.562035\pi\)
\(12\) 0 0
\(13\) 1158.87 1.90185 0.950925 0.309422i \(-0.100136\pi\)
0.950925 + 0.309422i \(0.100136\pi\)
\(14\) 0 0
\(15\) 894.565 1.02656
\(16\) 0 0
\(17\) 1238.08 1.03903 0.519513 0.854462i \(-0.326113\pi\)
0.519513 + 0.854462i \(0.326113\pi\)
\(18\) 0 0
\(19\) 280.279 0.178118 0.0890588 0.996026i \(-0.471614\pi\)
0.0890588 + 0.996026i \(0.471614\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3482.39 1.37264 0.686322 0.727298i \(-0.259224\pi\)
0.686322 + 0.727298i \(0.259224\pi\)
\(24\) 0 0
\(25\) −1821.39 −0.582844
\(26\) 0 0
\(27\) −3168.14 −0.836364
\(28\) 0 0
\(29\) −5656.78 −1.24903 −0.624517 0.781011i \(-0.714704\pi\)
−0.624517 + 0.781011i \(0.714704\pi\)
\(30\) 0 0
\(31\) 2314.70 0.432604 0.216302 0.976326i \(-0.430600\pi\)
0.216302 + 0.976326i \(0.430600\pi\)
\(32\) 0 0
\(33\) 3851.11 0.615604
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2333.18 −0.280184 −0.140092 0.990139i \(-0.544740\pi\)
−0.140092 + 0.990139i \(0.544740\pi\)
\(38\) 0 0
\(39\) −28712.6 −3.02281
\(40\) 0 0
\(41\) 3812.61 0.354211 0.177106 0.984192i \(-0.443327\pi\)
0.177106 + 0.984192i \(0.443327\pi\)
\(42\) 0 0
\(43\) −3925.73 −0.323780 −0.161890 0.986809i \(-0.551759\pi\)
−0.161890 + 0.986809i \(0.551759\pi\)
\(44\) 0 0
\(45\) −13390.4 −0.985743
\(46\) 0 0
\(47\) 11116.5 0.734043 0.367022 0.930212i \(-0.380378\pi\)
0.367022 + 0.930212i \(0.380378\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −30675.2 −1.65143
\(52\) 0 0
\(53\) −11186.2 −0.547007 −0.273504 0.961871i \(-0.588183\pi\)
−0.273504 + 0.961871i \(0.588183\pi\)
\(54\) 0 0
\(55\) 5612.06 0.250159
\(56\) 0 0
\(57\) −6944.31 −0.283101
\(58\) 0 0
\(59\) 6010.35 0.224786 0.112393 0.993664i \(-0.464148\pi\)
0.112393 + 0.993664i \(0.464148\pi\)
\(60\) 0 0
\(61\) −14838.7 −0.510589 −0.255295 0.966863i \(-0.582172\pi\)
−0.255295 + 0.966863i \(0.582172\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −41841.6 −1.22836
\(66\) 0 0
\(67\) 42983.9 1.16982 0.584909 0.811099i \(-0.301130\pi\)
0.584909 + 0.811099i \(0.301130\pi\)
\(68\) 0 0
\(69\) −86281.1 −2.18169
\(70\) 0 0
\(71\) 19962.4 0.469967 0.234984 0.971999i \(-0.424496\pi\)
0.234984 + 0.971999i \(0.424496\pi\)
\(72\) 0 0
\(73\) 45550.6 1.00043 0.500215 0.865901i \(-0.333254\pi\)
0.500215 + 0.865901i \(0.333254\pi\)
\(74\) 0 0
\(75\) 45127.4 0.926376
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −108992. −1.96484 −0.982419 0.186687i \(-0.940225\pi\)
−0.982419 + 0.186687i \(0.940225\pi\)
\(80\) 0 0
\(81\) −11626.1 −0.196890
\(82\) 0 0
\(83\) −55829.0 −0.889538 −0.444769 0.895645i \(-0.646714\pi\)
−0.444769 + 0.895645i \(0.646714\pi\)
\(84\) 0 0
\(85\) −44701.6 −0.671082
\(86\) 0 0
\(87\) 140155. 1.98522
\(88\) 0 0
\(89\) −95545.8 −1.27861 −0.639303 0.768955i \(-0.720777\pi\)
−0.639303 + 0.768955i \(0.720777\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −57349.9 −0.687584
\(94\) 0 0
\(95\) −10119.6 −0.115042
\(96\) 0 0
\(97\) 15004.9 0.161922 0.0809609 0.996717i \(-0.474201\pi\)
0.0809609 + 0.996717i \(0.474201\pi\)
\(98\) 0 0
\(99\) −57646.0 −0.591127
\(100\) 0 0
\(101\) 26876.1 0.262158 0.131079 0.991372i \(-0.458156\pi\)
0.131079 + 0.991372i \(0.458156\pi\)
\(102\) 0 0
\(103\) 51915.2 0.482171 0.241086 0.970504i \(-0.422497\pi\)
0.241086 + 0.970504i \(0.422497\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 24974.8 0.210883 0.105442 0.994425i \(-0.466374\pi\)
0.105442 + 0.994425i \(0.466374\pi\)
\(108\) 0 0
\(109\) −3636.14 −0.0293140 −0.0146570 0.999893i \(-0.504666\pi\)
−0.0146570 + 0.999893i \(0.504666\pi\)
\(110\) 0 0
\(111\) 57807.7 0.445326
\(112\) 0 0
\(113\) 62175.0 0.458057 0.229028 0.973420i \(-0.426445\pi\)
0.229028 + 0.973420i \(0.426445\pi\)
\(114\) 0 0
\(115\) −125734. −0.886557
\(116\) 0 0
\(117\) 429789. 2.90262
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −136891. −0.849986
\(122\) 0 0
\(123\) −94462.7 −0.562986
\(124\) 0 0
\(125\) 178592. 1.02232
\(126\) 0 0
\(127\) −63550.3 −0.349630 −0.174815 0.984601i \(-0.555933\pi\)
−0.174815 + 0.984601i \(0.555933\pi\)
\(128\) 0 0
\(129\) 97265.5 0.514617
\(130\) 0 0
\(131\) −136297. −0.693918 −0.346959 0.937880i \(-0.612786\pi\)
−0.346959 + 0.937880i \(0.612786\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 114388. 0.540187
\(136\) 0 0
\(137\) 335303. 1.52629 0.763144 0.646228i \(-0.223655\pi\)
0.763144 + 0.646228i \(0.223655\pi\)
\(138\) 0 0
\(139\) −58195.2 −0.255476 −0.127738 0.991808i \(-0.540772\pi\)
−0.127738 + 0.991808i \(0.540772\pi\)
\(140\) 0 0
\(141\) −275426. −1.16669
\(142\) 0 0
\(143\) −180129. −0.736618
\(144\) 0 0
\(145\) 204241. 0.806721
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 283058. 1.04450 0.522251 0.852792i \(-0.325092\pi\)
0.522251 + 0.852792i \(0.325092\pi\)
\(150\) 0 0
\(151\) 241558. 0.862142 0.431071 0.902318i \(-0.358136\pi\)
0.431071 + 0.902318i \(0.358136\pi\)
\(152\) 0 0
\(153\) 459166. 1.58577
\(154\) 0 0
\(155\) −83573.6 −0.279409
\(156\) 0 0
\(157\) −214029. −0.692984 −0.346492 0.938053i \(-0.612627\pi\)
−0.346492 + 0.938053i \(0.612627\pi\)
\(158\) 0 0
\(159\) 277154. 0.869417
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −64466.0 −0.190047 −0.0950237 0.995475i \(-0.530293\pi\)
−0.0950237 + 0.995475i \(0.530293\pi\)
\(164\) 0 0
\(165\) −139047. −0.397604
\(166\) 0 0
\(167\) 442694. 1.22832 0.614162 0.789180i \(-0.289494\pi\)
0.614162 + 0.789180i \(0.289494\pi\)
\(168\) 0 0
\(169\) 971685. 2.61703
\(170\) 0 0
\(171\) 103947. 0.271845
\(172\) 0 0
\(173\) −78599.0 −0.199665 −0.0998325 0.995004i \(-0.531831\pi\)
−0.0998325 + 0.995004i \(0.531831\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −148915. −0.357277
\(178\) 0 0
\(179\) 510427. 1.19070 0.595348 0.803468i \(-0.297014\pi\)
0.595348 + 0.803468i \(0.297014\pi\)
\(180\) 0 0
\(181\) −22051.8 −0.0500319 −0.0250160 0.999687i \(-0.507964\pi\)
−0.0250160 + 0.999687i \(0.507964\pi\)
\(182\) 0 0
\(183\) 367650. 0.811534
\(184\) 0 0
\(185\) 84240.6 0.180964
\(186\) 0 0
\(187\) −192441. −0.402432
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 558152. 1.10706 0.553528 0.832831i \(-0.313281\pi\)
0.553528 + 0.832831i \(0.313281\pi\)
\(192\) 0 0
\(193\) −49316.3 −0.0953009 −0.0476504 0.998864i \(-0.515173\pi\)
−0.0476504 + 0.998864i \(0.515173\pi\)
\(194\) 0 0
\(195\) 1.03668e6 1.95236
\(196\) 0 0
\(197\) −941509. −1.72846 −0.864229 0.503099i \(-0.832193\pi\)
−0.864229 + 0.503099i \(0.832193\pi\)
\(198\) 0 0
\(199\) −640012. −1.14566 −0.572830 0.819675i \(-0.694154\pi\)
−0.572830 + 0.819675i \(0.694154\pi\)
\(200\) 0 0
\(201\) −1.06499e6 −1.85932
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −137656. −0.228777
\(206\) 0 0
\(207\) 1.29151e6 2.09495
\(208\) 0 0
\(209\) −43565.1 −0.0689879
\(210\) 0 0
\(211\) −921869. −1.42549 −0.712743 0.701425i \(-0.752548\pi\)
−0.712743 + 0.701425i \(0.752548\pi\)
\(212\) 0 0
\(213\) −494597. −0.746969
\(214\) 0 0
\(215\) 141741. 0.209121
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.12858e6 −1.59009
\(220\) 0 0
\(221\) 1.43477e6 1.97607
\(222\) 0 0
\(223\) 837700. 1.12805 0.564023 0.825759i \(-0.309253\pi\)
0.564023 + 0.825759i \(0.309253\pi\)
\(224\) 0 0
\(225\) −675497. −0.889544
\(226\) 0 0
\(227\) 1.33069e6 1.71400 0.857002 0.515313i \(-0.172324\pi\)
0.857002 + 0.515313i \(0.172324\pi\)
\(228\) 0 0
\(229\) 1.01063e6 1.27352 0.636759 0.771063i \(-0.280275\pi\)
0.636759 + 0.771063i \(0.280275\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.48610e6 1.79332 0.896661 0.442718i \(-0.145986\pi\)
0.896661 + 0.442718i \(0.145986\pi\)
\(234\) 0 0
\(235\) −401366. −0.474101
\(236\) 0 0
\(237\) 2.70043e6 3.12293
\(238\) 0 0
\(239\) −875637. −0.991584 −0.495792 0.868441i \(-0.665122\pi\)
−0.495792 + 0.868441i \(0.665122\pi\)
\(240\) 0 0
\(241\) 1.45294e6 1.61141 0.805706 0.592316i \(-0.201786\pi\)
0.805706 + 0.592316i \(0.201786\pi\)
\(242\) 0 0
\(243\) 1.05791e6 1.14930
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 324807. 0.338753
\(248\) 0 0
\(249\) 1.38324e6 1.41384
\(250\) 0 0
\(251\) 198384. 0.198757 0.0993786 0.995050i \(-0.468315\pi\)
0.0993786 + 0.995050i \(0.468315\pi\)
\(252\) 0 0
\(253\) −541284. −0.531648
\(254\) 0 0
\(255\) 1.10754e6 1.06662
\(256\) 0 0
\(257\) 479572. 0.452919 0.226460 0.974021i \(-0.427285\pi\)
0.226460 + 0.974021i \(0.427285\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.09793e6 −1.90629
\(262\) 0 0
\(263\) −454943. −0.405572 −0.202786 0.979223i \(-0.565000\pi\)
−0.202786 + 0.979223i \(0.565000\pi\)
\(264\) 0 0
\(265\) 403884. 0.353299
\(266\) 0 0
\(267\) 2.36728e6 2.03222
\(268\) 0 0
\(269\) −860136. −0.724747 −0.362374 0.932033i \(-0.618034\pi\)
−0.362374 + 0.932033i \(0.618034\pi\)
\(270\) 0 0
\(271\) −1.30558e6 −1.07989 −0.539946 0.841700i \(-0.681555\pi\)
−0.539946 + 0.841700i \(0.681555\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 283107. 0.225745
\(276\) 0 0
\(277\) −543109. −0.425292 −0.212646 0.977129i \(-0.568208\pi\)
−0.212646 + 0.977129i \(0.568208\pi\)
\(278\) 0 0
\(279\) 858452. 0.660246
\(280\) 0 0
\(281\) 998089. 0.754055 0.377028 0.926202i \(-0.376946\pi\)
0.377028 + 0.926202i \(0.376946\pi\)
\(282\) 0 0
\(283\) −1.77732e6 −1.31917 −0.659583 0.751632i \(-0.729267\pi\)
−0.659583 + 0.751632i \(0.729267\pi\)
\(284\) 0 0
\(285\) 250728. 0.182848
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 112986. 0.0795759
\(290\) 0 0
\(291\) −371768. −0.257359
\(292\) 0 0
\(293\) −1.63987e6 −1.11594 −0.557969 0.829862i \(-0.688419\pi\)
−0.557969 + 0.829862i \(0.688419\pi\)
\(294\) 0 0
\(295\) −217007. −0.145184
\(296\) 0 0
\(297\) 492439. 0.323938
\(298\) 0 0
\(299\) 4.03564e6 2.61056
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −665894. −0.416676
\(304\) 0 0
\(305\) 535760. 0.329777
\(306\) 0 0
\(307\) −2.38130e6 −1.44201 −0.721006 0.692928i \(-0.756320\pi\)
−0.721006 + 0.692928i \(0.756320\pi\)
\(308\) 0 0
\(309\) −1.28627e6 −0.766366
\(310\) 0 0
\(311\) 1.24513e6 0.729983 0.364992 0.931011i \(-0.381072\pi\)
0.364992 + 0.931011i \(0.381072\pi\)
\(312\) 0 0
\(313\) 1.66299e6 0.959465 0.479732 0.877415i \(-0.340734\pi\)
0.479732 + 0.877415i \(0.340734\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.62486e6 −1.46709 −0.733547 0.679639i \(-0.762136\pi\)
−0.733547 + 0.679639i \(0.762136\pi\)
\(318\) 0 0
\(319\) 879260. 0.483772
\(320\) 0 0
\(321\) −618784. −0.335179
\(322\) 0 0
\(323\) 347008. 0.185069
\(324\) 0 0
\(325\) −2.11075e6 −1.10848
\(326\) 0 0
\(327\) 90090.5 0.0465918
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.73886e6 −0.872355 −0.436178 0.899861i \(-0.643668\pi\)
−0.436178 + 0.899861i \(0.643668\pi\)
\(332\) 0 0
\(333\) −865303. −0.427620
\(334\) 0 0
\(335\) −1.55196e6 −0.755558
\(336\) 0 0
\(337\) 853564. 0.409413 0.204706 0.978823i \(-0.434376\pi\)
0.204706 + 0.978823i \(0.434376\pi\)
\(338\) 0 0
\(339\) −1.54047e6 −0.728038
\(340\) 0 0
\(341\) −359785. −0.167555
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 3.11523e6 1.40910
\(346\) 0 0
\(347\) 2.98500e6 1.33082 0.665411 0.746477i \(-0.268256\pi\)
0.665411 + 0.746477i \(0.268256\pi\)
\(348\) 0 0
\(349\) 2.87467e6 1.26335 0.631676 0.775233i \(-0.282367\pi\)
0.631676 + 0.775233i \(0.282367\pi\)
\(350\) 0 0
\(351\) −3.67146e6 −1.59064
\(352\) 0 0
\(353\) 1.47109e6 0.628352 0.314176 0.949365i \(-0.398272\pi\)
0.314176 + 0.949365i \(0.398272\pi\)
\(354\) 0 0
\(355\) −720755. −0.303540
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.07799e6 −1.26047 −0.630233 0.776406i \(-0.717041\pi\)
−0.630233 + 0.776406i \(0.717041\pi\)
\(360\) 0 0
\(361\) −2.39754e6 −0.968274
\(362\) 0 0
\(363\) 3.39167e6 1.35097
\(364\) 0 0
\(365\) −1.64463e6 −0.646154
\(366\) 0 0
\(367\) −1.61670e6 −0.626561 −0.313281 0.949661i \(-0.601428\pi\)
−0.313281 + 0.949661i \(0.601428\pi\)
\(368\) 0 0
\(369\) 1.41398e6 0.540602
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.76119e6 −1.77192 −0.885958 0.463766i \(-0.846498\pi\)
−0.885958 + 0.463766i \(0.846498\pi\)
\(374\) 0 0
\(375\) −4.42487e6 −1.62488
\(376\) 0 0
\(377\) −6.55547e6 −2.37548
\(378\) 0 0
\(379\) −1.00193e6 −0.358294 −0.179147 0.983822i \(-0.557334\pi\)
−0.179147 + 0.983822i \(0.557334\pi\)
\(380\) 0 0
\(381\) 1.57455e6 0.555704
\(382\) 0 0
\(383\) 2.82152e6 0.982849 0.491424 0.870920i \(-0.336476\pi\)
0.491424 + 0.870920i \(0.336476\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.45593e6 −0.494156
\(388\) 0 0
\(389\) −1.24240e6 −0.416281 −0.208140 0.978099i \(-0.566741\pi\)
−0.208140 + 0.978099i \(0.566741\pi\)
\(390\) 0 0
\(391\) 4.31148e6 1.42621
\(392\) 0 0
\(393\) 3.37695e6 1.10292
\(394\) 0 0
\(395\) 3.93522e6 1.26904
\(396\) 0 0
\(397\) 2.74562e6 0.874308 0.437154 0.899387i \(-0.355986\pi\)
0.437154 + 0.899387i \(0.355986\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.71335e6 −1.15320 −0.576601 0.817026i \(-0.695621\pi\)
−0.576601 + 0.817026i \(0.695621\pi\)
\(402\) 0 0
\(403\) 2.68244e6 0.822748
\(404\) 0 0
\(405\) 419768. 0.127166
\(406\) 0 0
\(407\) 362656. 0.108520
\(408\) 0 0
\(409\) 3.75940e6 1.11125 0.555623 0.831434i \(-0.312480\pi\)
0.555623 + 0.831434i \(0.312480\pi\)
\(410\) 0 0
\(411\) −8.30761e6 −2.42589
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 2.01574e6 0.574531
\(416\) 0 0
\(417\) 1.44187e6 0.406055
\(418\) 0 0
\(419\) −4.60027e6 −1.28011 −0.640056 0.768328i \(-0.721089\pi\)
−0.640056 + 0.768328i \(0.721089\pi\)
\(420\) 0 0
\(421\) −404864. −0.111328 −0.0556639 0.998450i \(-0.517728\pi\)
−0.0556639 + 0.998450i \(0.517728\pi\)
\(422\) 0 0
\(423\) 4.12275e6 1.12031
\(424\) 0 0
\(425\) −2.25503e6 −0.605591
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4.46293e6 1.17079
\(430\) 0 0
\(431\) −94758.5 −0.0245711 −0.0122856 0.999925i \(-0.503911\pi\)
−0.0122856 + 0.999925i \(0.503911\pi\)
\(432\) 0 0
\(433\) 4.31727e6 1.10660 0.553298 0.832983i \(-0.313369\pi\)
0.553298 + 0.832983i \(0.313369\pi\)
\(434\) 0 0
\(435\) −5.06036e6 −1.28221
\(436\) 0 0
\(437\) 976041. 0.244492
\(438\) 0 0
\(439\) 2.22367e6 0.550693 0.275346 0.961345i \(-0.411207\pi\)
0.275346 + 0.961345i \(0.411207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.68425e6 1.13405 0.567024 0.823701i \(-0.308095\pi\)
0.567024 + 0.823701i \(0.308095\pi\)
\(444\) 0 0
\(445\) 3.44973e6 0.825821
\(446\) 0 0
\(447\) −7.01315e6 −1.66014
\(448\) 0 0
\(449\) 897932. 0.210198 0.105099 0.994462i \(-0.466484\pi\)
0.105099 + 0.994462i \(0.466484\pi\)
\(450\) 0 0
\(451\) −592612. −0.137192
\(452\) 0 0
\(453\) −5.98493e6 −1.37029
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.98181e6 1.33981 0.669903 0.742448i \(-0.266336\pi\)
0.669903 + 0.742448i \(0.266336\pi\)
\(458\) 0 0
\(459\) −3.92242e6 −0.869004
\(460\) 0 0
\(461\) 1.62507e6 0.356138 0.178069 0.984018i \(-0.443015\pi\)
0.178069 + 0.984018i \(0.443015\pi\)
\(462\) 0 0
\(463\) 8.46295e6 1.83472 0.917359 0.398060i \(-0.130317\pi\)
0.917359 + 0.398060i \(0.130317\pi\)
\(464\) 0 0
\(465\) 2.07065e6 0.444094
\(466\) 0 0
\(467\) −7.66594e6 −1.62657 −0.813286 0.581864i \(-0.802324\pi\)
−0.813286 + 0.581864i \(0.802324\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5.30286e6 1.10143
\(472\) 0 0
\(473\) 610195. 0.125405
\(474\) 0 0
\(475\) −510497. −0.103815
\(476\) 0 0
\(477\) −4.14862e6 −0.834849
\(478\) 0 0
\(479\) 2.40292e6 0.478521 0.239261 0.970955i \(-0.423095\pi\)
0.239261 + 0.970955i \(0.423095\pi\)
\(480\) 0 0
\(481\) −2.70385e6 −0.532867
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −541762. −0.104581
\(486\) 0 0
\(487\) 170302. 0.0325384 0.0162692 0.999868i \(-0.494821\pi\)
0.0162692 + 0.999868i \(0.494821\pi\)
\(488\) 0 0
\(489\) 1.59724e6 0.302063
\(490\) 0 0
\(491\) 5.77628e6 1.08130 0.540648 0.841249i \(-0.318179\pi\)
0.540648 + 0.841249i \(0.318179\pi\)
\(492\) 0 0
\(493\) −7.00355e6 −1.29778
\(494\) 0 0
\(495\) 2.08134e6 0.381795
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −543602. −0.0977303 −0.0488652 0.998805i \(-0.515560\pi\)
−0.0488652 + 0.998805i \(0.515560\pi\)
\(500\) 0 0
\(501\) −1.09684e7 −1.95230
\(502\) 0 0
\(503\) 5.40086e6 0.951794 0.475897 0.879501i \(-0.342124\pi\)
0.475897 + 0.879501i \(0.342124\pi\)
\(504\) 0 0
\(505\) −970378. −0.169322
\(506\) 0 0
\(507\) −2.40749e7 −4.15953
\(508\) 0 0
\(509\) 8.34418e6 1.42754 0.713772 0.700378i \(-0.246985\pi\)
0.713772 + 0.700378i \(0.246985\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −887964. −0.148971
\(514\) 0 0
\(515\) −1.87443e6 −0.311423
\(516\) 0 0
\(517\) −1.72788e6 −0.284307
\(518\) 0 0
\(519\) 1.94740e6 0.317349
\(520\) 0 0
\(521\) −1.14784e7 −1.85262 −0.926310 0.376762i \(-0.877037\pi\)
−0.926310 + 0.376762i \(0.877037\pi\)
\(522\) 0 0
\(523\) 4.38734e6 0.701370 0.350685 0.936494i \(-0.385949\pi\)
0.350685 + 0.936494i \(0.385949\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.86579e6 0.449487
\(528\) 0 0
\(529\) 5.69070e6 0.884151
\(530\) 0 0
\(531\) 2.22906e6 0.343071
\(532\) 0 0
\(533\) 4.41832e6 0.673657
\(534\) 0 0
\(535\) −901728. −0.136204
\(536\) 0 0
\(537\) −1.26465e7 −1.89250
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.22306e6 1.06103 0.530515 0.847675i \(-0.321998\pi\)
0.530515 + 0.847675i \(0.321998\pi\)
\(542\) 0 0
\(543\) 546364. 0.0795211
\(544\) 0 0
\(545\) 131285. 0.0189332
\(546\) 0 0
\(547\) −3.54529e6 −0.506622 −0.253311 0.967385i \(-0.581520\pi\)
−0.253311 + 0.967385i \(0.581520\pi\)
\(548\) 0 0
\(549\) −5.50323e6 −0.779267
\(550\) 0 0
\(551\) −1.58548e6 −0.222475
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2.08718e6 −0.287625
\(556\) 0 0
\(557\) 1.03663e7 1.41575 0.707873 0.706340i \(-0.249655\pi\)
0.707873 + 0.706340i \(0.249655\pi\)
\(558\) 0 0
\(559\) −4.54941e6 −0.615780
\(560\) 0 0
\(561\) 4.76799e6 0.639628
\(562\) 0 0
\(563\) 4.48282e6 0.596047 0.298023 0.954559i \(-0.403673\pi\)
0.298023 + 0.954559i \(0.403673\pi\)
\(564\) 0 0
\(565\) −2.24486e6 −0.295848
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.49335e6 0.840792 0.420396 0.907341i \(-0.361891\pi\)
0.420396 + 0.907341i \(0.361891\pi\)
\(570\) 0 0
\(571\) 497508. 0.0638571 0.0319286 0.999490i \(-0.489835\pi\)
0.0319286 + 0.999490i \(0.489835\pi\)
\(572\) 0 0
\(573\) −1.38290e7 −1.75956
\(574\) 0 0
\(575\) −6.34279e6 −0.800038
\(576\) 0 0
\(577\) −6.55123e6 −0.819187 −0.409594 0.912268i \(-0.634330\pi\)
−0.409594 + 0.912268i \(0.634330\pi\)
\(578\) 0 0
\(579\) 1.22188e6 0.151472
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.73873e6 0.211865
\(584\) 0 0
\(585\) −1.55178e7 −1.87474
\(586\) 0 0
\(587\) 30793.5 0.00368862 0.00184431 0.999998i \(-0.499413\pi\)
0.00184431 + 0.999998i \(0.499413\pi\)
\(588\) 0 0
\(589\) 648763. 0.0770544
\(590\) 0 0
\(591\) 2.33272e7 2.74722
\(592\) 0 0
\(593\) 570937. 0.0666732 0.0333366 0.999444i \(-0.489387\pi\)
0.0333366 + 0.999444i \(0.489387\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.58572e7 1.82092
\(598\) 0 0
\(599\) −9.07111e6 −1.03298 −0.516492 0.856292i \(-0.672762\pi\)
−0.516492 + 0.856292i \(0.672762\pi\)
\(600\) 0 0
\(601\) −1.36309e7 −1.53936 −0.769679 0.638431i \(-0.779584\pi\)
−0.769679 + 0.638431i \(0.779584\pi\)
\(602\) 0 0
\(603\) 1.59414e7 1.78539
\(604\) 0 0
\(605\) 4.94253e6 0.548985
\(606\) 0 0
\(607\) 3.95651e6 0.435853 0.217927 0.975965i \(-0.430071\pi\)
0.217927 + 0.975965i \(0.430071\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.28825e7 1.39604
\(612\) 0 0
\(613\) 1.32310e6 0.142214 0.0711070 0.997469i \(-0.477347\pi\)
0.0711070 + 0.997469i \(0.477347\pi\)
\(614\) 0 0
\(615\) 3.41063e6 0.363619
\(616\) 0 0
\(617\) 9.26370e6 0.979651 0.489826 0.871820i \(-0.337060\pi\)
0.489826 + 0.871820i \(0.337060\pi\)
\(618\) 0 0
\(619\) −669385. −0.0702182 −0.0351091 0.999383i \(-0.511178\pi\)
−0.0351091 + 0.999383i \(0.511178\pi\)
\(620\) 0 0
\(621\) −1.10327e7 −1.14803
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −756327. −0.0774479
\(626\) 0 0
\(627\) 1.07939e6 0.109650
\(628\) 0 0
\(629\) −2.88866e6 −0.291118
\(630\) 0 0
\(631\) 666246. 0.0666133 0.0333067 0.999445i \(-0.489396\pi\)
0.0333067 + 0.999445i \(0.489396\pi\)
\(632\) 0 0
\(633\) 2.28406e7 2.26568
\(634\) 0 0
\(635\) 2.29452e6 0.225817
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 7.40345e6 0.717269
\(640\) 0 0
\(641\) −2.10877e6 −0.202714 −0.101357 0.994850i \(-0.532318\pi\)
−0.101357 + 0.994850i \(0.532318\pi\)
\(642\) 0 0
\(643\) −1.25977e7 −1.20161 −0.600806 0.799395i \(-0.705154\pi\)
−0.600806 + 0.799395i \(0.705154\pi\)
\(644\) 0 0
\(645\) −3.51182e6 −0.332379
\(646\) 0 0
\(647\) −1.71299e7 −1.60877 −0.804386 0.594107i \(-0.797506\pi\)
−0.804386 + 0.594107i \(0.797506\pi\)
\(648\) 0 0
\(649\) −934217. −0.0870635
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8.25525e6 −0.757613 −0.378806 0.925476i \(-0.623665\pi\)
−0.378806 + 0.925476i \(0.623665\pi\)
\(654\) 0 0
\(655\) 4.92108e6 0.448185
\(656\) 0 0
\(657\) 1.68933e7 1.52687
\(658\) 0 0
\(659\) 1.05417e7 0.945581 0.472790 0.881175i \(-0.343247\pi\)
0.472790 + 0.881175i \(0.343247\pi\)
\(660\) 0 0
\(661\) 4.47640e6 0.398498 0.199249 0.979949i \(-0.436150\pi\)
0.199249 + 0.979949i \(0.436150\pi\)
\(662\) 0 0
\(663\) −3.55485e7 −3.14078
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.96991e7 −1.71448
\(668\) 0 0
\(669\) −2.07552e7 −1.79292
\(670\) 0 0
\(671\) 2.30645e6 0.197760
\(672\) 0 0
\(673\) −1.87165e7 −1.59289 −0.796447 0.604708i \(-0.793290\pi\)
−0.796447 + 0.604708i \(0.793290\pi\)
\(674\) 0 0
\(675\) 5.77042e6 0.487470
\(676\) 0 0
\(677\) 4.16232e6 0.349031 0.174515 0.984654i \(-0.444164\pi\)
0.174515 + 0.984654i \(0.444164\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.29697e7 −2.72425
\(682\) 0 0
\(683\) 1.28691e7 1.05559 0.527796 0.849371i \(-0.323019\pi\)
0.527796 + 0.849371i \(0.323019\pi\)
\(684\) 0 0
\(685\) −1.21063e7 −0.985793
\(686\) 0 0
\(687\) −2.50398e7 −2.02414
\(688\) 0 0
\(689\) −1.29634e7 −1.04033
\(690\) 0 0
\(691\) 9.29037e6 0.740181 0.370090 0.928996i \(-0.379327\pi\)
0.370090 + 0.928996i \(0.379327\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.10117e6 0.165006
\(696\) 0 0
\(697\) 4.72032e6 0.368035
\(698\) 0 0
\(699\) −3.68202e7 −2.85032
\(700\) 0 0
\(701\) 2.50809e7 1.92774 0.963870 0.266375i \(-0.0858258\pi\)
0.963870 + 0.266375i \(0.0858258\pi\)
\(702\) 0 0
\(703\) −653940. −0.0499057
\(704\) 0 0
\(705\) 9.94439e6 0.753539
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 9.93780e6 0.742463 0.371231 0.928540i \(-0.378936\pi\)
0.371231 + 0.928540i \(0.378936\pi\)
\(710\) 0 0
\(711\) −4.04218e7 −2.99876
\(712\) 0 0
\(713\) 8.06069e6 0.593811
\(714\) 0 0
\(715\) 6.50364e6 0.475764
\(716\) 0 0
\(717\) 2.16951e7 1.57603
\(718\) 0 0
\(719\) 1.40009e7 1.01003 0.505016 0.863110i \(-0.331487\pi\)
0.505016 + 0.863110i \(0.331487\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −3.59987e7 −2.56119
\(724\) 0 0
\(725\) 1.03032e7 0.727993
\(726\) 0 0
\(727\) −8.27315e6 −0.580544 −0.290272 0.956944i \(-0.593746\pi\)
−0.290272 + 0.956944i \(0.593746\pi\)
\(728\) 0 0
\(729\) −2.33861e7 −1.62982
\(730\) 0 0
\(731\) −4.86037e6 −0.336416
\(732\) 0 0
\(733\) −3.28591e6 −0.225889 −0.112945 0.993601i \(-0.536028\pi\)
−0.112945 + 0.993601i \(0.536028\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.68119e6 −0.453090
\(738\) 0 0
\(739\) −1.00913e7 −0.679728 −0.339864 0.940475i \(-0.610381\pi\)
−0.339864 + 0.940475i \(0.610381\pi\)
\(740\) 0 0
\(741\) −8.04754e6 −0.538416
\(742\) 0 0
\(743\) −1.73443e7 −1.15261 −0.576307 0.817233i \(-0.695507\pi\)
−0.576307 + 0.817233i \(0.695507\pi\)
\(744\) 0 0
\(745\) −1.02200e7 −0.674619
\(746\) 0 0
\(747\) −2.07053e7 −1.35762
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.75971e7 1.13852 0.569262 0.822156i \(-0.307229\pi\)
0.569262 + 0.822156i \(0.307229\pi\)
\(752\) 0 0
\(753\) −4.91525e6 −0.315906
\(754\) 0 0
\(755\) −8.72158e6 −0.556836
\(756\) 0 0
\(757\) −4.66480e6 −0.295865 −0.147932 0.988997i \(-0.547262\pi\)
−0.147932 + 0.988997i \(0.547262\pi\)
\(758\) 0 0
\(759\) 1.34111e7 0.845005
\(760\) 0 0
\(761\) 1.82717e7 1.14371 0.571857 0.820353i \(-0.306223\pi\)
0.571857 + 0.820353i \(0.306223\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.65785e7 −1.02421
\(766\) 0 0
\(767\) 6.96521e6 0.427510
\(768\) 0 0
\(769\) −2.31895e7 −1.41409 −0.707043 0.707171i \(-0.749971\pi\)
−0.707043 + 0.707171i \(0.749971\pi\)
\(770\) 0 0
\(771\) −1.18821e7 −0.719873
\(772\) 0 0
\(773\) 2.02149e6 0.121681 0.0608405 0.998148i \(-0.480622\pi\)
0.0608405 + 0.998148i \(0.480622\pi\)
\(774\) 0 0
\(775\) −4.21597e6 −0.252141
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.06859e6 0.0630913
\(780\) 0 0
\(781\) −3.10285e6 −0.182026
\(782\) 0 0
\(783\) 1.79215e7 1.04465
\(784\) 0 0
\(785\) 7.72763e6 0.447582
\(786\) 0 0
\(787\) −1.57744e7 −0.907856 −0.453928 0.891038i \(-0.649978\pi\)
−0.453928 + 0.891038i \(0.649978\pi\)
\(788\) 0 0
\(789\) 1.12718e7 0.644618
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.71961e7 −0.971064
\(794\) 0 0
\(795\) −1.00068e7 −0.561535
\(796\) 0 0
\(797\) 1.22436e6 0.0682750 0.0341375 0.999417i \(-0.489132\pi\)
0.0341375 + 0.999417i \(0.489132\pi\)
\(798\) 0 0
\(799\) 1.37631e7 0.762690
\(800\) 0 0
\(801\) −3.54350e7 −1.95142
\(802\) 0 0
\(803\) −7.08014e6 −0.387483
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.13111e7 1.15192
\(808\) 0 0
\(809\) −2.77325e7 −1.48976 −0.744882 0.667196i \(-0.767494\pi\)
−0.744882 + 0.667196i \(0.767494\pi\)
\(810\) 0 0
\(811\) 1.19246e7 0.636636 0.318318 0.947984i \(-0.396882\pi\)
0.318318 + 0.947984i \(0.396882\pi\)
\(812\) 0 0
\(813\) 3.23476e7 1.71639
\(814\) 0 0
\(815\) 2.32758e6 0.122747
\(816\) 0 0
\(817\) −1.10030e6 −0.0576709
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.60462e7 0.830836 0.415418 0.909631i \(-0.363635\pi\)
0.415418 + 0.909631i \(0.363635\pi\)
\(822\) 0 0
\(823\) 1.12307e7 0.577970 0.288985 0.957334i \(-0.406682\pi\)
0.288985 + 0.957334i \(0.406682\pi\)
\(824\) 0 0
\(825\) −7.01437e6 −0.358801
\(826\) 0 0
\(827\) 2.68427e7 1.36478 0.682389 0.730989i \(-0.260941\pi\)
0.682389 + 0.730989i \(0.260941\pi\)
\(828\) 0 0
\(829\) 2.80419e7 1.41717 0.708584 0.705626i \(-0.249334\pi\)
0.708584 + 0.705626i \(0.249334\pi\)
\(830\) 0 0
\(831\) 1.34563e7 0.675962
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1.59837e7 −0.793344
\(836\) 0 0
\(837\) −7.33331e6 −0.361815
\(838\) 0 0
\(839\) 3.30603e7 1.62145 0.810723 0.585430i \(-0.199074\pi\)
0.810723 + 0.585430i \(0.199074\pi\)
\(840\) 0 0
\(841\) 1.14880e7 0.560086
\(842\) 0 0
\(843\) −2.47290e7 −1.19850
\(844\) 0 0
\(845\) −3.50832e7 −1.69028
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 4.40356e7 2.09669
\(850\) 0 0
\(851\) −8.12503e6 −0.384593
\(852\) 0 0
\(853\) 2.73897e7 1.28889 0.644444 0.764652i \(-0.277089\pi\)
0.644444 + 0.764652i \(0.277089\pi\)
\(854\) 0 0
\(855\) −3.75306e6 −0.175578
\(856\) 0 0
\(857\) −2.96721e6 −0.138005 −0.0690027 0.997616i \(-0.521982\pi\)
−0.0690027 + 0.997616i \(0.521982\pi\)
\(858\) 0 0
\(859\) 3.97960e7 1.84016 0.920081 0.391729i \(-0.128123\pi\)
0.920081 + 0.391729i \(0.128123\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7.19597e6 −0.328899 −0.164450 0.986385i \(-0.552585\pi\)
−0.164450 + 0.986385i \(0.552585\pi\)
\(864\) 0 0
\(865\) 2.83786e6 0.128959
\(866\) 0 0
\(867\) −2.79940e6 −0.126478
\(868\) 0 0
\(869\) 1.69411e7 0.761015
\(870\) 0 0
\(871\) 4.98127e7 2.22482
\(872\) 0 0
\(873\) 5.56488e6 0.247127
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.51242e7 1.10305 0.551523 0.834160i \(-0.314047\pi\)
0.551523 + 0.834160i \(0.314047\pi\)
\(878\) 0 0
\(879\) 4.06300e7 1.77368
\(880\) 0 0
\(881\) 120262. 0.00522022 0.00261011 0.999997i \(-0.499169\pi\)
0.00261011 + 0.999997i \(0.499169\pi\)
\(882\) 0 0
\(883\) −2.43497e7 −1.05097 −0.525487 0.850802i \(-0.676117\pi\)
−0.525487 + 0.850802i \(0.676117\pi\)
\(884\) 0 0
\(885\) 5.37665e6 0.230756
\(886\) 0 0
\(887\) 2.82766e7 1.20675 0.603377 0.797456i \(-0.293821\pi\)
0.603377 + 0.797456i \(0.293821\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.80710e6 0.0762586
\(892\) 0 0
\(893\) 3.11571e6 0.130746
\(894\) 0 0
\(895\) −1.84293e7 −0.769042
\(896\) 0 0
\(897\) −9.99885e7 −4.14924
\(898\) 0 0
\(899\) −1.30938e7 −0.540337
\(900\) 0 0
\(901\) −1.38494e7 −0.568355
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 796192. 0.0323144
\(906\) 0 0
\(907\) 2.43168e7 0.981496 0.490748 0.871302i \(-0.336724\pi\)
0.490748 + 0.871302i \(0.336724\pi\)
\(908\) 0 0
\(909\) 9.96754e6 0.400109
\(910\) 0 0
\(911\) 3.36224e7 1.34225 0.671123 0.741346i \(-0.265812\pi\)
0.671123 + 0.741346i \(0.265812\pi\)
\(912\) 0 0
\(913\) 8.67777e6 0.344533
\(914\) 0 0
\(915\) −1.32742e7 −0.524150
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.13749e6 0.317835 0.158917 0.987292i \(-0.449200\pi\)
0.158917 + 0.987292i \(0.449200\pi\)
\(920\) 0 0
\(921\) 5.90001e7 2.29194
\(922\) 0 0
\(923\) 2.31338e7 0.893807
\(924\) 0 0
\(925\) 4.24962e6 0.163304
\(926\) 0 0
\(927\) 1.92538e7 0.735896
\(928\) 0 0
\(929\) 3.58054e7 1.36116 0.680580 0.732674i \(-0.261728\pi\)
0.680580 + 0.732674i \(0.261728\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3.08498e7 −1.16024
\(934\) 0 0
\(935\) 6.94818e6 0.259921
\(936\) 0 0
\(937\) 4.65849e7 1.73339 0.866694 0.498840i \(-0.166241\pi\)
0.866694 + 0.498840i \(0.166241\pi\)
\(938\) 0 0
\(939\) −4.12029e7 −1.52498
\(940\) 0 0
\(941\) −2.06725e6 −0.0761059 −0.0380529 0.999276i \(-0.512116\pi\)
−0.0380529 + 0.999276i \(0.512116\pi\)
\(942\) 0 0
\(943\) 1.32770e7 0.486206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.36773e7 −0.857940 −0.428970 0.903319i \(-0.641123\pi\)
−0.428970 + 0.903319i \(0.641123\pi\)
\(948\) 0 0
\(949\) 5.27872e7 1.90267
\(950\) 0 0
\(951\) 6.50345e7 2.33181
\(952\) 0 0
\(953\) −3.40521e6 −0.121454 −0.0607270 0.998154i \(-0.519342\pi\)
−0.0607270 + 0.998154i \(0.519342\pi\)
\(954\) 0 0
\(955\) −2.01524e7 −0.715020
\(956\) 0 0
\(957\) −2.17849e7 −0.768910
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.32713e7 −0.812854
\(962\) 0 0
\(963\) 9.26238e6 0.321852
\(964\) 0 0
\(965\) 1.78059e6 0.0615525
\(966\) 0 0
\(967\) −1.44238e6 −0.0496036 −0.0248018 0.999692i \(-0.507895\pi\)
−0.0248018 + 0.999692i \(0.507895\pi\)
\(968\) 0 0
\(969\) −8.59761e6 −0.294150
\(970\) 0 0
\(971\) 5.15954e6 0.175616 0.0878078 0.996137i \(-0.472014\pi\)
0.0878078 + 0.996137i \(0.472014\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 5.22968e7 1.76183
\(976\) 0 0
\(977\) −6.97650e6 −0.233831 −0.116915 0.993142i \(-0.537301\pi\)
−0.116915 + 0.993142i \(0.537301\pi\)
\(978\) 0 0
\(979\) 1.48511e7 0.495225
\(980\) 0 0
\(981\) −1.34853e6 −0.0447394
\(982\) 0 0
\(983\) 2.81568e7 0.929393 0.464696 0.885470i \(-0.346163\pi\)
0.464696 + 0.885470i \(0.346163\pi\)
\(984\) 0 0
\(985\) 3.39937e7 1.11637
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.36709e7 −0.444434
\(990\) 0 0
\(991\) 2.34043e7 0.757028 0.378514 0.925596i \(-0.376435\pi\)
0.378514 + 0.925596i \(0.376435\pi\)
\(992\) 0 0
\(993\) 4.30825e7 1.38653
\(994\) 0 0
\(995\) 2.31080e7 0.739954
\(996\) 0 0
\(997\) −4.30284e7 −1.37094 −0.685468 0.728103i \(-0.740402\pi\)
−0.685468 + 0.728103i \(0.740402\pi\)
\(998\) 0 0
\(999\) 7.39183e6 0.234336
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 784.6.a.s.1.1 2
4.3 odd 2 98.6.a.h.1.2 2
7.2 even 3 112.6.i.d.81.2 4
7.4 even 3 112.6.i.d.65.2 4
7.6 odd 2 784.6.a.bb.1.2 2
12.11 even 2 882.6.a.ba.1.2 2
28.3 even 6 98.6.c.e.79.2 4
28.11 odd 6 14.6.c.a.9.1 4
28.19 even 6 98.6.c.e.67.2 4
28.23 odd 6 14.6.c.a.11.1 yes 4
28.27 even 2 98.6.a.g.1.1 2
84.11 even 6 126.6.g.j.37.1 4
84.23 even 6 126.6.g.j.109.1 4
84.83 odd 2 882.6.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.c.a.9.1 4 28.11 odd 6
14.6.c.a.11.1 yes 4 28.23 odd 6
98.6.a.g.1.1 2 28.27 even 2
98.6.a.h.1.2 2 4.3 odd 2
98.6.c.e.67.2 4 28.19 even 6
98.6.c.e.79.2 4 28.3 even 6
112.6.i.d.65.2 4 7.4 even 3
112.6.i.d.81.2 4 7.2 even 3
126.6.g.j.37.1 4 84.11 even 6
126.6.g.j.109.1 4 84.23 even 6
784.6.a.s.1.1 2 1.1 even 1 trivial
784.6.a.bb.1.2 2 7.6 odd 2
882.6.a.ba.1.2 2 12.11 even 2
882.6.a.bi.1.1 2 84.83 odd 2