Properties

Label 225.6.a.h.1.1
Level $225$
Weight $6$
Character 225.1
Self dual yes
Analytic conductor $36.086$
Analytic rank $0$
Dimension $1$
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] = [225,6,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("225.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.00000 q^{2} +17.0000 q^{4} -12.0000 q^{7} -105.000 q^{8} +O(q^{10})\) \(q+7.00000 q^{2} +17.0000 q^{4} -12.0000 q^{7} -105.000 q^{8} -112.000 q^{11} +974.000 q^{13} -84.0000 q^{14} -1279.00 q^{16} +2182.00 q^{17} +1420.00 q^{19} -784.000 q^{22} +3216.00 q^{23} +6818.00 q^{26} -204.000 q^{28} +4150.00 q^{29} -5688.00 q^{31} -5593.00 q^{32} +15274.0 q^{34} -6482.00 q^{37} +9940.00 q^{38} -5402.00 q^{41} +21764.0 q^{43} -1904.00 q^{44} +22512.0 q^{46} -368.000 q^{47} -16663.0 q^{49} +16558.0 q^{52} +12586.0 q^{53} +1260.00 q^{56} +29050.0 q^{58} +25520.0 q^{59} +11782.0 q^{61} -39816.0 q^{62} +1777.00 q^{64} +13188.0 q^{67} +37094.0 q^{68} +35968.0 q^{71} -73186.0 q^{73} -45374.0 q^{74} +24140.0 q^{76} +1344.00 q^{77} -52440.0 q^{79} -37814.0 q^{82} +69036.0 q^{83} +152348. q^{86} +11760.0 q^{88} +33870.0 q^{89} -11688.0 q^{91} +54672.0 q^{92} -2576.00 q^{94} -143042. q^{97} -116641. q^{98} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.00000 1.23744 0.618718 0.785613i \(-0.287652\pi\)
0.618718 + 0.785613i \(0.287652\pi\)
\(3\) 0 0
\(4\) 17.0000 0.531250
\(5\) 0 0
\(6\) 0 0
\(7\) −12.0000 −0.0925627 −0.0462814 0.998928i \(-0.514737\pi\)
−0.0462814 + 0.998928i \(0.514737\pi\)
\(8\) −105.000 −0.580049
\(9\) 0 0
\(10\) 0 0
\(11\) −112.000 −0.279085 −0.139542 0.990216i \(-0.544563\pi\)
−0.139542 + 0.990216i \(0.544563\pi\)
\(12\) 0 0
\(13\) 974.000 1.59846 0.799228 0.601028i \(-0.205242\pi\)
0.799228 + 0.601028i \(0.205242\pi\)
\(14\) −84.0000 −0.114541
\(15\) 0 0
\(16\) −1279.00 −1.24902
\(17\) 2182.00 1.83119 0.915593 0.402106i \(-0.131722\pi\)
0.915593 + 0.402106i \(0.131722\pi\)
\(18\) 0 0
\(19\) 1420.00 0.902411 0.451205 0.892420i \(-0.350994\pi\)
0.451205 + 0.892420i \(0.350994\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −784.000 −0.345350
\(23\) 3216.00 1.26764 0.633821 0.773480i \(-0.281486\pi\)
0.633821 + 0.773480i \(0.281486\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6818.00 1.97799
\(27\) 0 0
\(28\) −204.000 −0.0491739
\(29\) 4150.00 0.916333 0.458166 0.888867i \(-0.348506\pi\)
0.458166 + 0.888867i \(0.348506\pi\)
\(30\) 0 0
\(31\) −5688.00 −1.06305 −0.531527 0.847041i \(-0.678382\pi\)
−0.531527 + 0.847041i \(0.678382\pi\)
\(32\) −5593.00 −0.965539
\(33\) 0 0
\(34\) 15274.0 2.26598
\(35\) 0 0
\(36\) 0 0
\(37\) −6482.00 −0.778403 −0.389202 0.921153i \(-0.627249\pi\)
−0.389202 + 0.921153i \(0.627249\pi\)
\(38\) 9940.00 1.11668
\(39\) 0 0
\(40\) 0 0
\(41\) −5402.00 −0.501874 −0.250937 0.968003i \(-0.580739\pi\)
−0.250937 + 0.968003i \(0.580739\pi\)
\(42\) 0 0
\(43\) 21764.0 1.79501 0.897506 0.441001i \(-0.145377\pi\)
0.897506 + 0.441001i \(0.145377\pi\)
\(44\) −1904.00 −0.148264
\(45\) 0 0
\(46\) 22512.0 1.56863
\(47\) −368.000 −0.0242998 −0.0121499 0.999926i \(-0.503868\pi\)
−0.0121499 + 0.999926i \(0.503868\pi\)
\(48\) 0 0
\(49\) −16663.0 −0.991432
\(50\) 0 0
\(51\) 0 0
\(52\) 16558.0 0.849180
\(53\) 12586.0 0.615457 0.307729 0.951474i \(-0.400431\pi\)
0.307729 + 0.951474i \(0.400431\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1260.00 0.0536909
\(57\) 0 0
\(58\) 29050.0 1.13390
\(59\) 25520.0 0.954444 0.477222 0.878783i \(-0.341644\pi\)
0.477222 + 0.878783i \(0.341644\pi\)
\(60\) 0 0
\(61\) 11782.0 0.405410 0.202705 0.979240i \(-0.435027\pi\)
0.202705 + 0.979240i \(0.435027\pi\)
\(62\) −39816.0 −1.31546
\(63\) 0 0
\(64\) 1777.00 0.0542297
\(65\) 0 0
\(66\) 0 0
\(67\) 13188.0 0.358915 0.179458 0.983766i \(-0.442566\pi\)
0.179458 + 0.983766i \(0.442566\pi\)
\(68\) 37094.0 0.972818
\(69\) 0 0
\(70\) 0 0
\(71\) 35968.0 0.846780 0.423390 0.905948i \(-0.360840\pi\)
0.423390 + 0.905948i \(0.360840\pi\)
\(72\) 0 0
\(73\) −73186.0 −1.60739 −0.803694 0.595042i \(-0.797135\pi\)
−0.803694 + 0.595042i \(0.797135\pi\)
\(74\) −45374.0 −0.963225
\(75\) 0 0
\(76\) 24140.0 0.479406
\(77\) 1344.00 0.0258329
\(78\) 0 0
\(79\) −52440.0 −0.945355 −0.472678 0.881235i \(-0.656712\pi\)
−0.472678 + 0.881235i \(0.656712\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −37814.0 −0.621038
\(83\) 69036.0 1.09997 0.549984 0.835175i \(-0.314634\pi\)
0.549984 + 0.835175i \(0.314634\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 152348. 2.22122
\(87\) 0 0
\(88\) 11760.0 0.161883
\(89\) 33870.0 0.453252 0.226626 0.973982i \(-0.427230\pi\)
0.226626 + 0.973982i \(0.427230\pi\)
\(90\) 0 0
\(91\) −11688.0 −0.147957
\(92\) 54672.0 0.673435
\(93\) 0 0
\(94\) −2576.00 −0.0300695
\(95\) 0 0
\(96\) 0 0
\(97\) −143042. −1.54360 −0.771799 0.635867i \(-0.780643\pi\)
−0.771799 + 0.635867i \(0.780643\pi\)
\(98\) −116641. −1.22683
\(99\) 0 0
\(100\) 0 0
\(101\) −63042.0 −0.614931 −0.307466 0.951559i \(-0.599481\pi\)
−0.307466 + 0.951559i \(0.599481\pi\)
\(102\) 0 0
\(103\) −38636.0 −0.358839 −0.179419 0.983773i \(-0.557422\pi\)
−0.179419 + 0.983773i \(0.557422\pi\)
\(104\) −102270. −0.927182
\(105\) 0 0
\(106\) 88102.0 0.761590
\(107\) −69228.0 −0.584551 −0.292275 0.956334i \(-0.594412\pi\)
−0.292275 + 0.956334i \(0.594412\pi\)
\(108\) 0 0
\(109\) 51590.0 0.415910 0.207955 0.978138i \(-0.433319\pi\)
0.207955 + 0.978138i \(0.433319\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 15348.0 0.115613
\(113\) 20206.0 0.148862 0.0744311 0.997226i \(-0.476286\pi\)
0.0744311 + 0.997226i \(0.476286\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 70550.0 0.486802
\(117\) 0 0
\(118\) 178640. 1.18106
\(119\) −26184.0 −0.169500
\(120\) 0 0
\(121\) −148507. −0.922112
\(122\) 82474.0 0.501669
\(123\) 0 0
\(124\) −96696.0 −0.564747
\(125\) 0 0
\(126\) 0 0
\(127\) 198908. 1.09432 0.547158 0.837029i \(-0.315710\pi\)
0.547158 + 0.837029i \(0.315710\pi\)
\(128\) 191415. 1.03264
\(129\) 0 0
\(130\) 0 0
\(131\) −150672. −0.767104 −0.383552 0.923519i \(-0.625299\pi\)
−0.383552 + 0.923519i \(0.625299\pi\)
\(132\) 0 0
\(133\) −17040.0 −0.0835296
\(134\) 92316.0 0.444135
\(135\) 0 0
\(136\) −229110. −1.06218
\(137\) −19098.0 −0.0869334 −0.0434667 0.999055i \(-0.513840\pi\)
−0.0434667 + 0.999055i \(0.513840\pi\)
\(138\) 0 0
\(139\) 196460. 0.862456 0.431228 0.902243i \(-0.358080\pi\)
0.431228 + 0.902243i \(0.358080\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 251776. 1.04784
\(143\) −109088. −0.446105
\(144\) 0 0
\(145\) 0 0
\(146\) −512302. −1.98904
\(147\) 0 0
\(148\) −110194. −0.413527
\(149\) 362710. 1.33842 0.669212 0.743071i \(-0.266632\pi\)
0.669212 + 0.743071i \(0.266632\pi\)
\(150\) 0 0
\(151\) 76072.0 0.271508 0.135754 0.990743i \(-0.456654\pi\)
0.135754 + 0.990743i \(0.456654\pi\)
\(152\) −149100. −0.523442
\(153\) 0 0
\(154\) 9408.00 0.0319665
\(155\) 0 0
\(156\) 0 0
\(157\) −252722. −0.818265 −0.409132 0.912475i \(-0.634169\pi\)
−0.409132 + 0.912475i \(0.634169\pi\)
\(158\) −367080. −1.16982
\(159\) 0 0
\(160\) 0 0
\(161\) −38592.0 −0.117336
\(162\) 0 0
\(163\) −85916.0 −0.253282 −0.126641 0.991949i \(-0.540420\pi\)
−0.126641 + 0.991949i \(0.540420\pi\)
\(164\) −91834.0 −0.266621
\(165\) 0 0
\(166\) 483252. 1.36114
\(167\) 316272. 0.877545 0.438773 0.898598i \(-0.355413\pi\)
0.438773 + 0.898598i \(0.355413\pi\)
\(168\) 0 0
\(169\) 577383. 1.55506
\(170\) 0 0
\(171\) 0 0
\(172\) 369988. 0.953601
\(173\) −597534. −1.51791 −0.758957 0.651140i \(-0.774291\pi\)
−0.758957 + 0.651140i \(0.774291\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 143248. 0.348584
\(177\) 0 0
\(178\) 237090. 0.560871
\(179\) 282680. 0.659421 0.329710 0.944082i \(-0.393049\pi\)
0.329710 + 0.944082i \(0.393049\pi\)
\(180\) 0 0
\(181\) −294658. −0.668531 −0.334266 0.942479i \(-0.608488\pi\)
−0.334266 + 0.942479i \(0.608488\pi\)
\(182\) −81816.0 −0.183088
\(183\) 0 0
\(184\) −337680. −0.735294
\(185\) 0 0
\(186\) 0 0
\(187\) −244384. −0.511056
\(188\) −6256.00 −0.0129093
\(189\) 0 0
\(190\) 0 0
\(191\) −723272. −1.43456 −0.717279 0.696786i \(-0.754613\pi\)
−0.717279 + 0.696786i \(0.754613\pi\)
\(192\) 0 0
\(193\) −80426.0 −0.155419 −0.0777093 0.996976i \(-0.524761\pi\)
−0.0777093 + 0.996976i \(0.524761\pi\)
\(194\) −1.00129e6 −1.91011
\(195\) 0 0
\(196\) −283271. −0.526698
\(197\) 509802. 0.935914 0.467957 0.883751i \(-0.344990\pi\)
0.467957 + 0.883751i \(0.344990\pi\)
\(198\) 0 0
\(199\) −435320. −0.779248 −0.389624 0.920974i \(-0.627395\pi\)
−0.389624 + 0.920974i \(0.627395\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −441294. −0.760939
\(203\) −49800.0 −0.0848182
\(204\) 0 0
\(205\) 0 0
\(206\) −270452. −0.444040
\(207\) 0 0
\(208\) −1.24575e6 −1.99651
\(209\) −159040. −0.251849
\(210\) 0 0
\(211\) −1.12275e6 −1.73611 −0.868053 0.496472i \(-0.834629\pi\)
−0.868053 + 0.496472i \(0.834629\pi\)
\(212\) 213962. 0.326962
\(213\) 0 0
\(214\) −484596. −0.723345
\(215\) 0 0
\(216\) 0 0
\(217\) 68256.0 0.0983992
\(218\) 361130. 0.514662
\(219\) 0 0
\(220\) 0 0
\(221\) 2.12527e6 2.92707
\(222\) 0 0
\(223\) 325084. 0.437757 0.218879 0.975752i \(-0.429760\pi\)
0.218879 + 0.975752i \(0.429760\pi\)
\(224\) 67116.0 0.0893729
\(225\) 0 0
\(226\) 141442. 0.184207
\(227\) −311308. −0.400983 −0.200491 0.979695i \(-0.564254\pi\)
−0.200491 + 0.979695i \(0.564254\pi\)
\(228\) 0 0
\(229\) −615450. −0.775540 −0.387770 0.921756i \(-0.626754\pi\)
−0.387770 + 0.921756i \(0.626754\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −435750. −0.531517
\(233\) −304434. −0.367370 −0.183685 0.982985i \(-0.558803\pi\)
−0.183685 + 0.982985i \(0.558803\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 433840. 0.507049
\(237\) 0 0
\(238\) −183288. −0.209745
\(239\) −780760. −0.884144 −0.442072 0.896980i \(-0.645756\pi\)
−0.442072 + 0.896980i \(0.645756\pi\)
\(240\) 0 0
\(241\) 635842. 0.705191 0.352595 0.935776i \(-0.385299\pi\)
0.352595 + 0.935776i \(0.385299\pi\)
\(242\) −1.03955e6 −1.14105
\(243\) 0 0
\(244\) 200294. 0.215374
\(245\) 0 0
\(246\) 0 0
\(247\) 1.38308e6 1.44246
\(248\) 597240. 0.616623
\(249\) 0 0
\(250\) 0 0
\(251\) −1.20559e6 −1.20786 −0.603929 0.797038i \(-0.706399\pi\)
−0.603929 + 0.797038i \(0.706399\pi\)
\(252\) 0 0
\(253\) −360192. −0.353780
\(254\) 1.39236e6 1.35415
\(255\) 0 0
\(256\) 1.28304e6 1.22360
\(257\) 1.96702e6 1.85770 0.928852 0.370452i \(-0.120797\pi\)
0.928852 + 0.370452i \(0.120797\pi\)
\(258\) 0 0
\(259\) 77784.0 0.0720511
\(260\) 0 0
\(261\) 0 0
\(262\) −1.05470e6 −0.949243
\(263\) −625264. −0.557409 −0.278705 0.960377i \(-0.589905\pi\)
−0.278705 + 0.960377i \(0.589905\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −119280. −0.103363
\(267\) 0 0
\(268\) 224196. 0.190674
\(269\) −527050. −0.444090 −0.222045 0.975036i \(-0.571273\pi\)
−0.222045 + 0.975036i \(0.571273\pi\)
\(270\) 0 0
\(271\) 2.10923e6 1.74462 0.872311 0.488952i \(-0.162621\pi\)
0.872311 + 0.488952i \(0.162621\pi\)
\(272\) −2.79078e6 −2.28719
\(273\) 0 0
\(274\) −133686. −0.107575
\(275\) 0 0
\(276\) 0 0
\(277\) 267438. 0.209423 0.104711 0.994503i \(-0.466608\pi\)
0.104711 + 0.994503i \(0.466608\pi\)
\(278\) 1.37522e6 1.06724
\(279\) 0 0
\(280\) 0 0
\(281\) −838002. −0.633110 −0.316555 0.948574i \(-0.602526\pi\)
−0.316555 + 0.948574i \(0.602526\pi\)
\(282\) 0 0
\(283\) 2.23772e6 1.66089 0.830444 0.557102i \(-0.188087\pi\)
0.830444 + 0.557102i \(0.188087\pi\)
\(284\) 611456. 0.449852
\(285\) 0 0
\(286\) −763616. −0.552027
\(287\) 64824.0 0.0464549
\(288\) 0 0
\(289\) 3.34127e6 2.35324
\(290\) 0 0
\(291\) 0 0
\(292\) −1.24416e6 −0.853925
\(293\) 785706. 0.534676 0.267338 0.963603i \(-0.413856\pi\)
0.267338 + 0.963603i \(0.413856\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 680610. 0.451512
\(297\) 0 0
\(298\) 2.53897e6 1.65622
\(299\) 3.13238e6 2.02627
\(300\) 0 0
\(301\) −261168. −0.166151
\(302\) 532504. 0.335974
\(303\) 0 0
\(304\) −1.81618e6 −1.12713
\(305\) 0 0
\(306\) 0 0
\(307\) 2.94955e6 1.78612 0.893058 0.449942i \(-0.148555\pi\)
0.893058 + 0.449942i \(0.148555\pi\)
\(308\) 22848.0 0.0137237
\(309\) 0 0
\(310\) 0 0
\(311\) 3.07757e6 1.80429 0.902146 0.431431i \(-0.141991\pi\)
0.902146 + 0.431431i \(0.141991\pi\)
\(312\) 0 0
\(313\) −1.61367e6 −0.931007 −0.465503 0.885046i \(-0.654127\pi\)
−0.465503 + 0.885046i \(0.654127\pi\)
\(314\) −1.76905e6 −1.01255
\(315\) 0 0
\(316\) −891480. −0.502220
\(317\) −2.00496e6 −1.12062 −0.560308 0.828284i \(-0.689317\pi\)
−0.560308 + 0.828284i \(0.689317\pi\)
\(318\) 0 0
\(319\) −464800. −0.255735
\(320\) 0 0
\(321\) 0 0
\(322\) −270144. −0.145196
\(323\) 3.09844e6 1.65248
\(324\) 0 0
\(325\) 0 0
\(326\) −601412. −0.313421
\(327\) 0 0
\(328\) 567210. 0.291111
\(329\) 4416.00 0.00224926
\(330\) 0 0
\(331\) 470772. 0.236179 0.118089 0.993003i \(-0.462323\pi\)
0.118089 + 0.993003i \(0.462323\pi\)
\(332\) 1.17361e6 0.584358
\(333\) 0 0
\(334\) 2.21390e6 1.08591
\(335\) 0 0
\(336\) 0 0
\(337\) −2.31548e6 −1.11062 −0.555311 0.831642i \(-0.687401\pi\)
−0.555311 + 0.831642i \(0.687401\pi\)
\(338\) 4.04168e6 1.92429
\(339\) 0 0
\(340\) 0 0
\(341\) 637056. 0.296682
\(342\) 0 0
\(343\) 401640. 0.184332
\(344\) −2.28522e6 −1.04119
\(345\) 0 0
\(346\) −4.18274e6 −1.87832
\(347\) 1.25393e6 0.559050 0.279525 0.960138i \(-0.409823\pi\)
0.279525 + 0.960138i \(0.409823\pi\)
\(348\) 0 0
\(349\) 616390. 0.270889 0.135445 0.990785i \(-0.456754\pi\)
0.135445 + 0.990785i \(0.456754\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 626416. 0.269467
\(353\) −281274. −0.120141 −0.0600707 0.998194i \(-0.519133\pi\)
−0.0600707 + 0.998194i \(0.519133\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 575790. 0.240790
\(357\) 0 0
\(358\) 1.97876e6 0.815991
\(359\) 1.19148e6 0.487922 0.243961 0.969785i \(-0.421553\pi\)
0.243961 + 0.969785i \(0.421553\pi\)
\(360\) 0 0
\(361\) −459699. −0.185655
\(362\) −2.06261e6 −0.827265
\(363\) 0 0
\(364\) −198696. −0.0786024
\(365\) 0 0
\(366\) 0 0
\(367\) 793068. 0.307359 0.153679 0.988121i \(-0.450888\pi\)
0.153679 + 0.988121i \(0.450888\pi\)
\(368\) −4.11326e6 −1.58331
\(369\) 0 0
\(370\) 0 0
\(371\) −151032. −0.0569684
\(372\) 0 0
\(373\) −635626. −0.236554 −0.118277 0.992981i \(-0.537737\pi\)
−0.118277 + 0.992981i \(0.537737\pi\)
\(374\) −1.71069e6 −0.632400
\(375\) 0 0
\(376\) 38640.0 0.0140951
\(377\) 4.04210e6 1.46472
\(378\) 0 0
\(379\) −2.12834e6 −0.761102 −0.380551 0.924760i \(-0.624266\pi\)
−0.380551 + 0.924760i \(0.624266\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5.06290e6 −1.77518
\(383\) −4.88174e6 −1.70051 −0.850253 0.526375i \(-0.823551\pi\)
−0.850253 + 0.526375i \(0.823551\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −562982. −0.192321
\(387\) 0 0
\(388\) −2.43171e6 −0.820036
\(389\) 2.30607e6 0.772678 0.386339 0.922357i \(-0.373740\pi\)
0.386339 + 0.922357i \(0.373740\pi\)
\(390\) 0 0
\(391\) 7.01731e6 2.32129
\(392\) 1.74962e6 0.575079
\(393\) 0 0
\(394\) 3.56861e6 1.15813
\(395\) 0 0
\(396\) 0 0
\(397\) 423398. 0.134826 0.0674128 0.997725i \(-0.478526\pi\)
0.0674128 + 0.997725i \(0.478526\pi\)
\(398\) −3.04724e6 −0.964271
\(399\) 0 0
\(400\) 0 0
\(401\) 2.60756e6 0.809791 0.404896 0.914363i \(-0.367308\pi\)
0.404896 + 0.914363i \(0.367308\pi\)
\(402\) 0 0
\(403\) −5.54011e6 −1.69924
\(404\) −1.07171e6 −0.326682
\(405\) 0 0
\(406\) −348600. −0.104957
\(407\) 725984. 0.217241
\(408\) 0 0
\(409\) −4.80871e6 −1.42141 −0.710707 0.703489i \(-0.751625\pi\)
−0.710707 + 0.703489i \(0.751625\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −656812. −0.190633
\(413\) −306240. −0.0883460
\(414\) 0 0
\(415\) 0 0
\(416\) −5.44758e6 −1.54337
\(417\) 0 0
\(418\) −1.11328e6 −0.311648
\(419\) −139760. −0.0388909 −0.0194454 0.999811i \(-0.506190\pi\)
−0.0194454 + 0.999811i \(0.506190\pi\)
\(420\) 0 0
\(421\) −3.00310e6 −0.825780 −0.412890 0.910781i \(-0.635481\pi\)
−0.412890 + 0.910781i \(0.635481\pi\)
\(422\) −7.85924e6 −2.14832
\(423\) 0 0
\(424\) −1.32153e6 −0.356995
\(425\) 0 0
\(426\) 0 0
\(427\) −141384. −0.0375259
\(428\) −1.17688e6 −0.310543
\(429\) 0 0
\(430\) 0 0
\(431\) −5.97955e6 −1.55051 −0.775257 0.631646i \(-0.782379\pi\)
−0.775257 + 0.631646i \(0.782379\pi\)
\(432\) 0 0
\(433\) −1.75235e6 −0.449159 −0.224580 0.974456i \(-0.572101\pi\)
−0.224580 + 0.974456i \(0.572101\pi\)
\(434\) 477792. 0.121763
\(435\) 0 0
\(436\) 877030. 0.220952
\(437\) 4.56672e6 1.14393
\(438\) 0 0
\(439\) −4.72556e6 −1.17029 −0.585143 0.810930i \(-0.698962\pi\)
−0.585143 + 0.810930i \(0.698962\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.48769e7 3.62206
\(443\) 2.48584e6 0.601815 0.300908 0.953653i \(-0.402710\pi\)
0.300908 + 0.953653i \(0.402710\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.27559e6 0.541697
\(447\) 0 0
\(448\) −21324.0 −0.00501965
\(449\) −1.46233e6 −0.342318 −0.171159 0.985243i \(-0.554751\pi\)
−0.171159 + 0.985243i \(0.554751\pi\)
\(450\) 0 0
\(451\) 605024. 0.140066
\(452\) 343502. 0.0790830
\(453\) 0 0
\(454\) −2.17916e6 −0.496191
\(455\) 0 0
\(456\) 0 0
\(457\) 1.45684e6 0.326303 0.163151 0.986601i \(-0.447834\pi\)
0.163151 + 0.986601i \(0.447834\pi\)
\(458\) −4.30815e6 −0.959681
\(459\) 0 0
\(460\) 0 0
\(461\) −4.32280e6 −0.947356 −0.473678 0.880698i \(-0.657074\pi\)
−0.473678 + 0.880698i \(0.657074\pi\)
\(462\) 0 0
\(463\) −1.07848e6 −0.233807 −0.116904 0.993143i \(-0.537297\pi\)
−0.116904 + 0.993143i \(0.537297\pi\)
\(464\) −5.30785e6 −1.14452
\(465\) 0 0
\(466\) −2.13104e6 −0.454597
\(467\) −3.40023e6 −0.721466 −0.360733 0.932669i \(-0.617473\pi\)
−0.360733 + 0.932669i \(0.617473\pi\)
\(468\) 0 0
\(469\) −158256. −0.0332222
\(470\) 0 0
\(471\) 0 0
\(472\) −2.67960e6 −0.553624
\(473\) −2.43757e6 −0.500961
\(474\) 0 0
\(475\) 0 0
\(476\) −445128. −0.0900466
\(477\) 0 0
\(478\) −5.46532e6 −1.09407
\(479\) −2.75268e6 −0.548172 −0.274086 0.961705i \(-0.588375\pi\)
−0.274086 + 0.961705i \(0.588375\pi\)
\(480\) 0 0
\(481\) −6.31347e6 −1.24424
\(482\) 4.45089e6 0.872629
\(483\) 0 0
\(484\) −2.52462e6 −0.489872
\(485\) 0 0
\(486\) 0 0
\(487\) −3.56997e6 −0.682091 −0.341046 0.940047i \(-0.610781\pi\)
−0.341046 + 0.940047i \(0.610781\pi\)
\(488\) −1.23711e6 −0.235157
\(489\) 0 0
\(490\) 0 0
\(491\) −1.96455e6 −0.367756 −0.183878 0.982949i \(-0.558865\pi\)
−0.183878 + 0.982949i \(0.558865\pi\)
\(492\) 0 0
\(493\) 9.05530e6 1.67798
\(494\) 9.68156e6 1.78496
\(495\) 0 0
\(496\) 7.27495e6 1.32778
\(497\) −431616. −0.0783802
\(498\) 0 0
\(499\) 5.14798e6 0.925519 0.462760 0.886484i \(-0.346859\pi\)
0.462760 + 0.886484i \(0.346859\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −8.43914e6 −1.49465
\(503\) 1.97502e6 0.348057 0.174029 0.984741i \(-0.444321\pi\)
0.174029 + 0.984741i \(0.444321\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.52134e6 −0.437780
\(507\) 0 0
\(508\) 3.38144e6 0.581356
\(509\) 3.32447e6 0.568759 0.284379 0.958712i \(-0.408213\pi\)
0.284379 + 0.958712i \(0.408213\pi\)
\(510\) 0 0
\(511\) 878232. 0.148784
\(512\) 2.85601e6 0.481487
\(513\) 0 0
\(514\) 1.37692e7 2.29879
\(515\) 0 0
\(516\) 0 0
\(517\) 41216.0 0.00678171
\(518\) 544488. 0.0891587
\(519\) 0 0
\(520\) 0 0
\(521\) 2.97960e6 0.480910 0.240455 0.970660i \(-0.422703\pi\)
0.240455 + 0.970660i \(0.422703\pi\)
\(522\) 0 0
\(523\) 6.19108e6 0.989720 0.494860 0.868973i \(-0.335219\pi\)
0.494860 + 0.868973i \(0.335219\pi\)
\(524\) −2.56142e6 −0.407524
\(525\) 0 0
\(526\) −4.37685e6 −0.689759
\(527\) −1.24112e7 −1.94665
\(528\) 0 0
\(529\) 3.90631e6 0.606915
\(530\) 0 0
\(531\) 0 0
\(532\) −289680. −0.0443751
\(533\) −5.26155e6 −0.802224
\(534\) 0 0
\(535\) 0 0
\(536\) −1.38474e6 −0.208188
\(537\) 0 0
\(538\) −3.68935e6 −0.549533
\(539\) 1.86626e6 0.276694
\(540\) 0 0
\(541\) −8.55398e6 −1.25654 −0.628268 0.777997i \(-0.716236\pi\)
−0.628268 + 0.777997i \(0.716236\pi\)
\(542\) 1.47646e7 2.15886
\(543\) 0 0
\(544\) −1.22039e7 −1.76808
\(545\) 0 0
\(546\) 0 0
\(547\) 2.54371e6 0.363495 0.181748 0.983345i \(-0.441825\pi\)
0.181748 + 0.983345i \(0.441825\pi\)
\(548\) −324666. −0.0461833
\(549\) 0 0
\(550\) 0 0
\(551\) 5.89300e6 0.826908
\(552\) 0 0
\(553\) 629280. 0.0875046
\(554\) 1.87207e6 0.259147
\(555\) 0 0
\(556\) 3.33982e6 0.458180
\(557\) 7.58704e6 1.03618 0.518089 0.855327i \(-0.326644\pi\)
0.518089 + 0.855327i \(0.326644\pi\)
\(558\) 0 0
\(559\) 2.11981e7 2.86925
\(560\) 0 0
\(561\) 0 0
\(562\) −5.86601e6 −0.783434
\(563\) 7.50940e6 0.998468 0.499234 0.866467i \(-0.333615\pi\)
0.499234 + 0.866467i \(0.333615\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.56641e7 2.05524
\(567\) 0 0
\(568\) −3.77664e6 −0.491173
\(569\) −1.38890e7 −1.79842 −0.899209 0.437519i \(-0.855857\pi\)
−0.899209 + 0.437519i \(0.855857\pi\)
\(570\) 0 0
\(571\) −1.93539e6 −0.248415 −0.124207 0.992256i \(-0.539639\pi\)
−0.124207 + 0.992256i \(0.539639\pi\)
\(572\) −1.85450e6 −0.236993
\(573\) 0 0
\(574\) 453768. 0.0574849
\(575\) 0 0
\(576\) 0 0
\(577\) 4.89408e6 0.611972 0.305986 0.952036i \(-0.401014\pi\)
0.305986 + 0.952036i \(0.401014\pi\)
\(578\) 2.33889e7 2.91199
\(579\) 0 0
\(580\) 0 0
\(581\) −828432. −0.101816
\(582\) 0 0
\(583\) −1.40963e6 −0.171765
\(584\) 7.68453e6 0.932363
\(585\) 0 0
\(586\) 5.49994e6 0.661628
\(587\) −6.43883e6 −0.771279 −0.385640 0.922650i \(-0.626019\pi\)
−0.385640 + 0.922650i \(0.626019\pi\)
\(588\) 0 0
\(589\) −8.07696e6 −0.959312
\(590\) 0 0
\(591\) 0 0
\(592\) 8.29048e6 0.972244
\(593\) 4.30365e6 0.502574 0.251287 0.967913i \(-0.419146\pi\)
0.251287 + 0.967913i \(0.419146\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.16607e6 0.711038
\(597\) 0 0
\(598\) 2.19267e7 2.50738
\(599\) 4.50988e6 0.513568 0.256784 0.966469i \(-0.417337\pi\)
0.256784 + 0.966469i \(0.417337\pi\)
\(600\) 0 0
\(601\) −5.11596e6 −0.577751 −0.288876 0.957367i \(-0.593281\pi\)
−0.288876 + 0.957367i \(0.593281\pi\)
\(602\) −1.82818e6 −0.205602
\(603\) 0 0
\(604\) 1.29322e6 0.144239
\(605\) 0 0
\(606\) 0 0
\(607\) −1.61925e7 −1.78378 −0.891891 0.452250i \(-0.850622\pi\)
−0.891891 + 0.452250i \(0.850622\pi\)
\(608\) −7.94206e6 −0.871313
\(609\) 0 0
\(610\) 0 0
\(611\) −358432. −0.0388422
\(612\) 0 0
\(613\) −1.55525e7 −1.67166 −0.835830 0.548988i \(-0.815013\pi\)
−0.835830 + 0.548988i \(0.815013\pi\)
\(614\) 2.06468e7 2.21021
\(615\) 0 0
\(616\) −141120. −0.0149843
\(617\) −1.46710e7 −1.55148 −0.775740 0.631053i \(-0.782623\pi\)
−0.775740 + 0.631053i \(0.782623\pi\)
\(618\) 0 0
\(619\) −9.66826e6 −1.01420 −0.507098 0.861889i \(-0.669282\pi\)
−0.507098 + 0.861889i \(0.669282\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2.15430e7 2.23270
\(623\) −406440. −0.0419543
\(624\) 0 0
\(625\) 0 0
\(626\) −1.12957e7 −1.15206
\(627\) 0 0
\(628\) −4.29627e6 −0.434703
\(629\) −1.41437e7 −1.42540
\(630\) 0 0
\(631\) −1.16557e6 −0.116537 −0.0582686 0.998301i \(-0.518558\pi\)
−0.0582686 + 0.998301i \(0.518558\pi\)
\(632\) 5.50620e6 0.548352
\(633\) 0 0
\(634\) −1.40347e7 −1.38669
\(635\) 0 0
\(636\) 0 0
\(637\) −1.62298e7 −1.58476
\(638\) −3.25360e6 −0.316455
\(639\) 0 0
\(640\) 0 0
\(641\) −1.95088e6 −0.187537 −0.0937683 0.995594i \(-0.529891\pi\)
−0.0937683 + 0.995594i \(0.529891\pi\)
\(642\) 0 0
\(643\) 1.17387e7 1.11968 0.559839 0.828601i \(-0.310863\pi\)
0.559839 + 0.828601i \(0.310863\pi\)
\(644\) −656064. −0.0623349
\(645\) 0 0
\(646\) 2.16891e7 2.04484
\(647\) −1.01369e7 −0.952015 −0.476008 0.879441i \(-0.657917\pi\)
−0.476008 + 0.879441i \(0.657917\pi\)
\(648\) 0 0
\(649\) −2.85824e6 −0.266371
\(650\) 0 0
\(651\) 0 0
\(652\) −1.46057e6 −0.134556
\(653\) 2.47095e6 0.226767 0.113384 0.993551i \(-0.463831\pi\)
0.113384 + 0.993551i \(0.463831\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.90916e6 0.626853
\(657\) 0 0
\(658\) 30912.0 0.00278332
\(659\) −1.62242e7 −1.45529 −0.727644 0.685955i \(-0.759384\pi\)
−0.727644 + 0.685955i \(0.759384\pi\)
\(660\) 0 0
\(661\) 1.54679e7 1.37698 0.688490 0.725246i \(-0.258274\pi\)
0.688490 + 0.725246i \(0.258274\pi\)
\(662\) 3.29540e6 0.292256
\(663\) 0 0
\(664\) −7.24878e6 −0.638035
\(665\) 0 0
\(666\) 0 0
\(667\) 1.33464e7 1.16158
\(668\) 5.37662e6 0.466196
\(669\) 0 0
\(670\) 0 0
\(671\) −1.31958e6 −0.113144
\(672\) 0 0
\(673\) 1.94441e7 1.65482 0.827410 0.561598i \(-0.189813\pi\)
0.827410 + 0.561598i \(0.189813\pi\)
\(674\) −1.62084e7 −1.37433
\(675\) 0 0
\(676\) 9.81551e6 0.826126
\(677\) 643242. 0.0539390 0.0269695 0.999636i \(-0.491414\pi\)
0.0269695 + 0.999636i \(0.491414\pi\)
\(678\) 0 0
\(679\) 1.71650e6 0.142880
\(680\) 0 0
\(681\) 0 0
\(682\) 4.45939e6 0.367126
\(683\) 1.14412e6 0.0938465 0.0469233 0.998898i \(-0.485058\pi\)
0.0469233 + 0.998898i \(0.485058\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.81148e6 0.228100
\(687\) 0 0
\(688\) −2.78362e7 −2.24201
\(689\) 1.22588e7 0.983781
\(690\) 0 0
\(691\) 1.63625e6 0.130363 0.0651816 0.997873i \(-0.479237\pi\)
0.0651816 + 0.997873i \(0.479237\pi\)
\(692\) −1.01581e7 −0.806392
\(693\) 0 0
\(694\) 8.77752e6 0.691789
\(695\) 0 0
\(696\) 0 0
\(697\) −1.17872e7 −0.919025
\(698\) 4.31473e6 0.335209
\(699\) 0 0
\(700\) 0 0
\(701\) −1.58303e7 −1.21673 −0.608364 0.793658i \(-0.708174\pi\)
−0.608364 + 0.793658i \(0.708174\pi\)
\(702\) 0 0
\(703\) −9.20444e6 −0.702440
\(704\) −199024. −0.0151347
\(705\) 0 0
\(706\) −1.96892e6 −0.148667
\(707\) 756504. 0.0569197
\(708\) 0 0
\(709\) 910870. 0.0680520 0.0340260 0.999421i \(-0.489167\pi\)
0.0340260 + 0.999421i \(0.489167\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.55635e6 −0.262908
\(713\) −1.82926e7 −1.34757
\(714\) 0 0
\(715\) 0 0
\(716\) 4.80556e6 0.350317
\(717\) 0 0
\(718\) 8.34036e6 0.603773
\(719\) 1.52246e7 1.09831 0.549155 0.835721i \(-0.314950\pi\)
0.549155 + 0.835721i \(0.314950\pi\)
\(720\) 0 0
\(721\) 463632. 0.0332151
\(722\) −3.21789e6 −0.229736
\(723\) 0 0
\(724\) −5.00919e6 −0.355157
\(725\) 0 0
\(726\) 0 0
\(727\) −1.81793e7 −1.27567 −0.637837 0.770171i \(-0.720171\pi\)
−0.637837 + 0.770171i \(0.720171\pi\)
\(728\) 1.22724e6 0.0858225
\(729\) 0 0
\(730\) 0 0
\(731\) 4.74890e7 3.28700
\(732\) 0 0
\(733\) −2.08512e7 −1.43341 −0.716707 0.697374i \(-0.754352\pi\)
−0.716707 + 0.697374i \(0.754352\pi\)
\(734\) 5.55148e6 0.380337
\(735\) 0 0
\(736\) −1.79871e7 −1.22396
\(737\) −1.47706e6 −0.100168
\(738\) 0 0
\(739\) −2.12513e7 −1.43144 −0.715722 0.698385i \(-0.753902\pi\)
−0.715722 + 0.698385i \(0.753902\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.05722e6 −0.0704948
\(743\) 7.12262e6 0.473334 0.236667 0.971591i \(-0.423945\pi\)
0.236667 + 0.971591i \(0.423945\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.44938e6 −0.292720
\(747\) 0 0
\(748\) −4.15453e6 −0.271499
\(749\) 830736. 0.0541076
\(750\) 0 0
\(751\) −1.00277e7 −0.648785 −0.324393 0.945923i \(-0.605160\pi\)
−0.324393 + 0.945923i \(0.605160\pi\)
\(752\) 470672. 0.0303511
\(753\) 0 0
\(754\) 2.82947e7 1.81249
\(755\) 0 0
\(756\) 0 0
\(757\) 2.18303e7 1.38459 0.692294 0.721616i \(-0.256600\pi\)
0.692294 + 0.721616i \(0.256600\pi\)
\(758\) −1.48984e7 −0.941816
\(759\) 0 0
\(760\) 0 0
\(761\) −2.56780e7 −1.60731 −0.803655 0.595096i \(-0.797114\pi\)
−0.803655 + 0.595096i \(0.797114\pi\)
\(762\) 0 0
\(763\) −619080. −0.0384978
\(764\) −1.22956e7 −0.762109
\(765\) 0 0
\(766\) −3.41722e7 −2.10427
\(767\) 2.48565e7 1.52564
\(768\) 0 0
\(769\) 5.81453e6 0.354567 0.177284 0.984160i \(-0.443269\pi\)
0.177284 + 0.984160i \(0.443269\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.36724e6 −0.0825662
\(773\) 1.55507e7 0.936057 0.468029 0.883713i \(-0.344964\pi\)
0.468029 + 0.883713i \(0.344964\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.50194e7 0.895362
\(777\) 0 0
\(778\) 1.61425e7 0.956140
\(779\) −7.67084e6 −0.452897
\(780\) 0 0
\(781\) −4.02842e6 −0.236323
\(782\) 4.91212e7 2.87245
\(783\) 0 0
\(784\) 2.13120e7 1.23832
\(785\) 0 0
\(786\) 0 0
\(787\) −2.35987e7 −1.35816 −0.679079 0.734065i \(-0.737621\pi\)
−0.679079 + 0.734065i \(0.737621\pi\)
\(788\) 8.66663e6 0.497204
\(789\) 0 0
\(790\) 0 0
\(791\) −242472. −0.0137791
\(792\) 0 0
\(793\) 1.14757e7 0.648030
\(794\) 2.96379e6 0.166838
\(795\) 0 0
\(796\) −7.40044e6 −0.413976
\(797\) −2.01127e7 −1.12157 −0.560783 0.827963i \(-0.689500\pi\)
−0.560783 + 0.827963i \(0.689500\pi\)
\(798\) 0 0
\(799\) −802976. −0.0444975
\(800\) 0 0
\(801\) 0 0
\(802\) 1.82529e7 1.00207
\(803\) 8.19683e6 0.448598
\(804\) 0 0
\(805\) 0 0
\(806\) −3.87808e7 −2.10271
\(807\) 0 0
\(808\) 6.61941e6 0.356690
\(809\) 4.24111e6 0.227829 0.113914 0.993491i \(-0.463661\pi\)
0.113914 + 0.993491i \(0.463661\pi\)
\(810\) 0 0
\(811\) 6.04321e6 0.322638 0.161319 0.986902i \(-0.448425\pi\)
0.161319 + 0.986902i \(0.448425\pi\)
\(812\) −846600. −0.0450597
\(813\) 0 0
\(814\) 5.08189e6 0.268822
\(815\) 0 0
\(816\) 0 0
\(817\) 3.09049e7 1.61984
\(818\) −3.36610e7 −1.75891
\(819\) 0 0
\(820\) 0 0
\(821\) −1.66230e7 −0.860702 −0.430351 0.902662i \(-0.641610\pi\)
−0.430351 + 0.902662i \(0.641610\pi\)
\(822\) 0 0
\(823\) 2.59172e7 1.33380 0.666898 0.745149i \(-0.267622\pi\)
0.666898 + 0.745149i \(0.267622\pi\)
\(824\) 4.05678e6 0.208144
\(825\) 0 0
\(826\) −2.14368e6 −0.109323
\(827\) 1.67704e7 0.852668 0.426334 0.904566i \(-0.359805\pi\)
0.426334 + 0.904566i \(0.359805\pi\)
\(828\) 0 0
\(829\) 6.15999e6 0.311310 0.155655 0.987811i \(-0.450251\pi\)
0.155655 + 0.987811i \(0.450251\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.73080e6 0.0866838
\(833\) −3.63587e7 −1.81550
\(834\) 0 0
\(835\) 0 0
\(836\) −2.70368e6 −0.133795
\(837\) 0 0
\(838\) −978320. −0.0481250
\(839\) −2.80172e6 −0.137410 −0.0687052 0.997637i \(-0.521887\pi\)
−0.0687052 + 0.997637i \(0.521887\pi\)
\(840\) 0 0
\(841\) −3.28865e6 −0.160335
\(842\) −2.10217e7 −1.02185
\(843\) 0 0
\(844\) −1.90867e7 −0.922306
\(845\) 0 0
\(846\) 0 0
\(847\) 1.78208e6 0.0853532
\(848\) −1.60975e7 −0.768721
\(849\) 0 0
\(850\) 0 0
\(851\) −2.08461e7 −0.986736
\(852\) 0 0
\(853\) −2.54991e7 −1.19992 −0.599960 0.800030i \(-0.704817\pi\)
−0.599960 + 0.800030i \(0.704817\pi\)
\(854\) −989688. −0.0464359
\(855\) 0 0
\(856\) 7.26894e6 0.339068
\(857\) −1.19499e7 −0.555794 −0.277897 0.960611i \(-0.589637\pi\)
−0.277897 + 0.960611i \(0.589637\pi\)
\(858\) 0 0
\(859\) −1.01568e7 −0.469651 −0.234825 0.972038i \(-0.575452\pi\)
−0.234825 + 0.972038i \(0.575452\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −4.18569e7 −1.91866
\(863\) 3.66497e7 1.67511 0.837556 0.546351i \(-0.183984\pi\)
0.837556 + 0.546351i \(0.183984\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.22664e7 −0.555806
\(867\) 0 0
\(868\) 1.16035e6 0.0522746
\(869\) 5.87328e6 0.263834
\(870\) 0 0
\(871\) 1.28451e7 0.573710
\(872\) −5.41695e6 −0.241248
\(873\) 0 0
\(874\) 3.19670e7 1.41555
\(875\) 0 0
\(876\) 0 0
\(877\) 8.08232e6 0.354844 0.177422 0.984135i \(-0.443224\pi\)
0.177422 + 0.984135i \(0.443224\pi\)
\(878\) −3.30789e7 −1.44815
\(879\) 0 0
\(880\) 0 0
\(881\) −288202. −0.0125100 −0.00625500 0.999980i \(-0.501991\pi\)
−0.00625500 + 0.999980i \(0.501991\pi\)
\(882\) 0 0
\(883\) −6.20688e6 −0.267899 −0.133950 0.990988i \(-0.542766\pi\)
−0.133950 + 0.990988i \(0.542766\pi\)
\(884\) 3.61296e7 1.55501
\(885\) 0 0
\(886\) 1.74009e7 0.744708
\(887\) −1.49976e7 −0.640050 −0.320025 0.947409i \(-0.603691\pi\)
−0.320025 + 0.947409i \(0.603691\pi\)
\(888\) 0 0
\(889\) −2.38690e6 −0.101293
\(890\) 0 0
\(891\) 0 0
\(892\) 5.52643e6 0.232559
\(893\) −522560. −0.0219284
\(894\) 0 0
\(895\) 0 0
\(896\) −2.29698e6 −0.0955844
\(897\) 0 0
\(898\) −1.02363e7 −0.423597
\(899\) −2.36052e7 −0.974111
\(900\) 0 0
\(901\) 2.74627e7 1.12702
\(902\) 4.23517e6 0.173322
\(903\) 0 0
\(904\) −2.12163e6 −0.0863473
\(905\) 0 0
\(906\) 0 0
\(907\) 3.92150e7 1.58283 0.791415 0.611279i \(-0.209345\pi\)
0.791415 + 0.611279i \(0.209345\pi\)
\(908\) −5.29224e6 −0.213022
\(909\) 0 0
\(910\) 0 0
\(911\) 3.72997e7 1.48905 0.744526 0.667594i \(-0.232676\pi\)
0.744526 + 0.667594i \(0.232676\pi\)
\(912\) 0 0
\(913\) −7.73203e6 −0.306985
\(914\) 1.01979e7 0.403779
\(915\) 0 0
\(916\) −1.04626e7 −0.412005
\(917\) 1.80806e6 0.0710052
\(918\) 0 0
\(919\) 5.78776e6 0.226059 0.113029 0.993592i \(-0.463945\pi\)
0.113029 + 0.993592i \(0.463945\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3.02596e7 −1.17229
\(923\) 3.50328e7 1.35354
\(924\) 0 0
\(925\) 0 0
\(926\) −7.54933e6 −0.289322
\(927\) 0 0
\(928\) −2.32110e7 −0.884755
\(929\) −1.62700e7 −0.618513 −0.309256 0.950979i \(-0.600080\pi\)
−0.309256 + 0.950979i \(0.600080\pi\)
\(930\) 0 0
\(931\) −2.36615e7 −0.894679
\(932\) −5.17538e6 −0.195165
\(933\) 0 0
\(934\) −2.38016e7 −0.892769
\(935\) 0 0
\(936\) 0 0
\(937\) 1.20396e7 0.447983 0.223992 0.974591i \(-0.428091\pi\)
0.223992 + 0.974591i \(0.428091\pi\)
\(938\) −1.10779e6 −0.0411103
\(939\) 0 0
\(940\) 0 0
\(941\) −3.10171e7 −1.14190 −0.570949 0.820985i \(-0.693425\pi\)
−0.570949 + 0.820985i \(0.693425\pi\)
\(942\) 0 0
\(943\) −1.73728e7 −0.636197
\(944\) −3.26401e7 −1.19212
\(945\) 0 0
\(946\) −1.70630e7 −0.619908
\(947\) 3.27325e6 0.118605 0.0593027 0.998240i \(-0.481112\pi\)
0.0593027 + 0.998240i \(0.481112\pi\)
\(948\) 0 0
\(949\) −7.12832e7 −2.56934
\(950\) 0 0
\(951\) 0 0
\(952\) 2.74932e6 0.0983180
\(953\) −2.62021e7 −0.934552 −0.467276 0.884112i \(-0.654765\pi\)
−0.467276 + 0.884112i \(0.654765\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.32729e7 −0.469701
\(957\) 0 0
\(958\) −1.92688e7 −0.678328
\(959\) 229176. 0.00804679
\(960\) 0 0
\(961\) 3.72419e6 0.130084
\(962\) −4.41943e7 −1.53967
\(963\) 0 0
\(964\) 1.08093e7 0.374633
\(965\) 0 0
\(966\) 0 0
\(967\) 2.01481e7 0.692897 0.346449 0.938069i \(-0.387387\pi\)
0.346449 + 0.938069i \(0.387387\pi\)
\(968\) 1.55932e7 0.534869
\(969\) 0 0
\(970\) 0 0
\(971\) 1.57046e7 0.534537 0.267269 0.963622i \(-0.413879\pi\)
0.267269 + 0.963622i \(0.413879\pi\)
\(972\) 0 0
\(973\) −2.35752e6 −0.0798313
\(974\) −2.49898e7 −0.844045
\(975\) 0 0
\(976\) −1.50692e7 −0.506367
\(977\) 2.84554e7 0.953736 0.476868 0.878975i \(-0.341772\pi\)
0.476868 + 0.878975i \(0.341772\pi\)
\(978\) 0 0
\(979\) −3.79344e6 −0.126496
\(980\) 0 0
\(981\) 0 0
\(982\) −1.37519e7 −0.455075
\(983\) −7.50074e6 −0.247583 −0.123791 0.992308i \(-0.539505\pi\)
−0.123791 + 0.992308i \(0.539505\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 6.33871e7 2.07639
\(987\) 0 0
\(988\) 2.35124e7 0.766309
\(989\) 6.99930e7 2.27543
\(990\) 0 0
\(991\) 3.22184e7 1.04212 0.521062 0.853519i \(-0.325536\pi\)
0.521062 + 0.853519i \(0.325536\pi\)
\(992\) 3.18130e7 1.02642
\(993\) 0 0
\(994\) −3.02131e6 −0.0969906
\(995\) 0 0
\(996\) 0 0
\(997\) −1.22112e7 −0.389065 −0.194532 0.980896i \(-0.562319\pi\)
−0.194532 + 0.980896i \(0.562319\pi\)
\(998\) 3.60359e7 1.14527
\(999\) 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 225.6.a.h.1.1 1
3.2 odd 2 75.6.a.a.1.1 1
5.2 odd 4 225.6.b.a.199.2 2
5.3 odd 4 225.6.b.a.199.1 2
5.4 even 2 45.6.a.a.1.1 1
15.2 even 4 75.6.b.a.49.1 2
15.8 even 4 75.6.b.a.49.2 2
15.14 odd 2 15.6.a.b.1.1 1
20.19 odd 2 720.6.a.q.1.1 1
60.59 even 2 240.6.a.b.1.1 1
105.104 even 2 735.6.a.b.1.1 1
120.29 odd 2 960.6.a.k.1.1 1
120.59 even 2 960.6.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.6.a.b.1.1 1 15.14 odd 2
45.6.a.a.1.1 1 5.4 even 2
75.6.a.a.1.1 1 3.2 odd 2
75.6.b.a.49.1 2 15.2 even 4
75.6.b.a.49.2 2 15.8 even 4
225.6.a.h.1.1 1 1.1 even 1 trivial
225.6.b.a.199.1 2 5.3 odd 4
225.6.b.a.199.2 2 5.2 odd 4
240.6.a.b.1.1 1 60.59 even 2
720.6.a.q.1.1 1 20.19 odd 2
735.6.a.b.1.1 1 105.104 even 2
960.6.a.k.1.1 1 120.29 odd 2
960.6.a.x.1.1 1 120.59 even 2