Properties

Label 225.10.a.e.1.1
Level $225$
Weight $10$
Character 225.1
Self dual yes
Analytic conductor $115.883$
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] = [225,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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)
 
\(f(q)\) \(=\) \(q+18.0000 q^{2} -188.000 q^{4} -9128.00 q^{7} -12600.0 q^{8} +O(q^{10})\) \(q+18.0000 q^{2} -188.000 q^{4} -9128.00 q^{7} -12600.0 q^{8} -21132.0 q^{11} -31214.0 q^{13} -164304. q^{14} -130544. q^{16} -279342. q^{17} +144020. q^{19} -380376. q^{22} -1.76350e6 q^{23} -561852. q^{26} +1.71606e6 q^{28} -4.69251e6 q^{29} -369088. q^{31} +4.10141e6 q^{32} -5.02816e6 q^{34} -9.34708e6 q^{37} +2.59236e6 q^{38} +7.22684e6 q^{41} +2.31475e7 q^{43} +3.97282e6 q^{44} -3.17429e7 q^{46} +2.29719e7 q^{47} +4.29668e7 q^{49} +5.86823e6 q^{52} +7.84772e7 q^{53} +1.15013e8 q^{56} -8.44652e7 q^{58} +2.03107e7 q^{59} -1.79340e8 q^{61} -6.64358e6 q^{62} +1.40664e8 q^{64} -2.74528e8 q^{67} +5.25163e7 q^{68} +3.63426e7 q^{71} +2.47090e8 q^{73} -1.68247e8 q^{74} -2.70758e7 q^{76} +1.92893e8 q^{77} +1.91875e8 q^{79} +1.30083e8 q^{82} -2.76159e8 q^{83} +4.16655e8 q^{86} +2.66263e8 q^{88} +6.78997e8 q^{89} +2.84921e8 q^{91} +3.31537e8 q^{92} +4.13494e8 q^{94} +5.67658e8 q^{97} +7.73402e8 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.0000 0.795495 0.397748 0.917495i \(-0.369792\pi\)
0.397748 + 0.917495i \(0.369792\pi\)
\(3\) 0 0
\(4\) −188.000 −0.367188
\(5\) 0 0
\(6\) 0 0
\(7\) −9128.00 −1.43693 −0.718463 0.695565i \(-0.755154\pi\)
−0.718463 + 0.695565i \(0.755154\pi\)
\(8\) −12600.0 −1.08759
\(9\) 0 0
\(10\) 0 0
\(11\) −21132.0 −0.435185 −0.217592 0.976040i \(-0.569820\pi\)
−0.217592 + 0.976040i \(0.569820\pi\)
\(12\) 0 0
\(13\) −31214.0 −0.303113 −0.151556 0.988449i \(-0.548429\pi\)
−0.151556 + 0.988449i \(0.548429\pi\)
\(14\) −164304. −1.14307
\(15\) 0 0
\(16\) −130544. −0.497986
\(17\) −279342. −0.811178 −0.405589 0.914056i \(-0.632934\pi\)
−0.405589 + 0.914056i \(0.632934\pi\)
\(18\) 0 0
\(19\) 144020. 0.253531 0.126766 0.991933i \(-0.459540\pi\)
0.126766 + 0.991933i \(0.459540\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −380376. −0.346187
\(23\) −1.76350e6 −1.31401 −0.657006 0.753885i \(-0.728177\pi\)
−0.657006 + 0.753885i \(0.728177\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −561852. −0.241125
\(27\) 0 0
\(28\) 1.71606e6 0.527621
\(29\) −4.69251e6 −1.23201 −0.616005 0.787742i \(-0.711250\pi\)
−0.616005 + 0.787742i \(0.711250\pi\)
\(30\) 0 0
\(31\) −369088. −0.0717798 −0.0358899 0.999356i \(-0.511427\pi\)
−0.0358899 + 0.999356i \(0.511427\pi\)
\(32\) 4.10141e6 0.691446
\(33\) 0 0
\(34\) −5.02816e6 −0.645288
\(35\) 0 0
\(36\) 0 0
\(37\) −9.34708e6 −0.819914 −0.409957 0.912105i \(-0.634456\pi\)
−0.409957 + 0.912105i \(0.634456\pi\)
\(38\) 2.59236e6 0.201683
\(39\) 0 0
\(40\) 0 0
\(41\) 7.22684e6 0.399412 0.199706 0.979856i \(-0.436001\pi\)
0.199706 + 0.979856i \(0.436001\pi\)
\(42\) 0 0
\(43\) 2.31475e7 1.03251 0.516257 0.856434i \(-0.327325\pi\)
0.516257 + 0.856434i \(0.327325\pi\)
\(44\) 3.97282e6 0.159794
\(45\) 0 0
\(46\) −3.17429e7 −1.04529
\(47\) 2.29719e7 0.686683 0.343342 0.939211i \(-0.388441\pi\)
0.343342 + 0.939211i \(0.388441\pi\)
\(48\) 0 0
\(49\) 4.29668e7 1.06476
\(50\) 0 0
\(51\) 0 0
\(52\) 5.86823e6 0.111299
\(53\) 7.84772e7 1.36616 0.683081 0.730343i \(-0.260640\pi\)
0.683081 + 0.730343i \(0.260640\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.15013e8 1.56279
\(57\) 0 0
\(58\) −8.44652e7 −0.980058
\(59\) 2.03107e7 0.218218 0.109109 0.994030i \(-0.465200\pi\)
0.109109 + 0.994030i \(0.465200\pi\)
\(60\) 0 0
\(61\) −1.79340e8 −1.65841 −0.829207 0.558942i \(-0.811207\pi\)
−0.829207 + 0.558942i \(0.811207\pi\)
\(62\) −6.64358e6 −0.0571005
\(63\) 0 0
\(64\) 1.40664e8 1.04803
\(65\) 0 0
\(66\) 0 0
\(67\) −2.74528e8 −1.66437 −0.832186 0.554496i \(-0.812911\pi\)
−0.832186 + 0.554496i \(0.812911\pi\)
\(68\) 5.25163e7 0.297854
\(69\) 0 0
\(70\) 0 0
\(71\) 3.63426e7 0.169728 0.0848641 0.996393i \(-0.472954\pi\)
0.0848641 + 0.996393i \(0.472954\pi\)
\(72\) 0 0
\(73\) 2.47090e8 1.01836 0.509180 0.860660i \(-0.329949\pi\)
0.509180 + 0.860660i \(0.329949\pi\)
\(74\) −1.68247e8 −0.652237
\(75\) 0 0
\(76\) −2.70758e7 −0.0930935
\(77\) 1.92893e8 0.625328
\(78\) 0 0
\(79\) 1.91875e8 0.554238 0.277119 0.960836i \(-0.410620\pi\)
0.277119 + 0.960836i \(0.410620\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.30083e8 0.317730
\(83\) −2.76159e8 −0.638717 −0.319358 0.947634i \(-0.603467\pi\)
−0.319358 + 0.947634i \(0.603467\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.16655e8 0.821359
\(87\) 0 0
\(88\) 2.66263e8 0.473303
\(89\) 6.78997e8 1.14713 0.573566 0.819160i \(-0.305560\pi\)
0.573566 + 0.819160i \(0.305560\pi\)
\(90\) 0 0
\(91\) 2.84921e8 0.435551
\(92\) 3.31537e8 0.482489
\(93\) 0 0
\(94\) 4.13494e8 0.546253
\(95\) 0 0
\(96\) 0 0
\(97\) 5.67658e8 0.651049 0.325524 0.945534i \(-0.394459\pi\)
0.325524 + 0.945534i \(0.394459\pi\)
\(98\) 7.73402e8 0.847009
\(99\) 0 0
\(100\) 0 0
\(101\) −1.62282e9 −1.55176 −0.775881 0.630879i \(-0.782694\pi\)
−0.775881 + 0.630879i \(0.782694\pi\)
\(102\) 0 0
\(103\) 1.75103e9 1.53294 0.766470 0.642280i \(-0.222011\pi\)
0.766470 + 0.642280i \(0.222011\pi\)
\(104\) 3.93296e8 0.329663
\(105\) 0 0
\(106\) 1.41259e9 1.08677
\(107\) −1.54296e9 −1.13796 −0.568980 0.822352i \(-0.692662\pi\)
−0.568980 + 0.822352i \(0.692662\pi\)
\(108\) 0 0
\(109\) 4.57665e8 0.310548 0.155274 0.987871i \(-0.450374\pi\)
0.155274 + 0.987871i \(0.450374\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.19161e9 0.715569
\(113\) 3.26794e9 1.88548 0.942739 0.333531i \(-0.108240\pi\)
0.942739 + 0.333531i \(0.108240\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.82192e8 0.452379
\(117\) 0 0
\(118\) 3.65592e8 0.173591
\(119\) 2.54983e9 1.16560
\(120\) 0 0
\(121\) −1.91139e9 −0.810614
\(122\) −3.22812e9 −1.31926
\(123\) 0 0
\(124\) 6.93885e7 0.0263566
\(125\) 0 0
\(126\) 0 0
\(127\) −9.28879e8 −0.316842 −0.158421 0.987372i \(-0.550640\pi\)
−0.158421 + 0.987372i \(0.550640\pi\)
\(128\) 4.32029e8 0.142255
\(129\) 0 0
\(130\) 0 0
\(131\) 9.88659e8 0.293309 0.146655 0.989188i \(-0.453149\pi\)
0.146655 + 0.989188i \(0.453149\pi\)
\(132\) 0 0
\(133\) −1.31461e9 −0.364306
\(134\) −4.94151e9 −1.32400
\(135\) 0 0
\(136\) 3.51971e9 0.882230
\(137\) −5.73253e7 −0.0139028 −0.00695142 0.999976i \(-0.502213\pi\)
−0.00695142 + 0.999976i \(0.502213\pi\)
\(138\) 0 0
\(139\) −4.65052e9 −1.05666 −0.528330 0.849039i \(-0.677182\pi\)
−0.528330 + 0.849039i \(0.677182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.54168e8 0.135018
\(143\) 6.59614e8 0.131910
\(144\) 0 0
\(145\) 0 0
\(146\) 4.44761e9 0.810101
\(147\) 0 0
\(148\) 1.75725e9 0.301062
\(149\) 1.40236e9 0.233089 0.116545 0.993185i \(-0.462818\pi\)
0.116545 + 0.993185i \(0.462818\pi\)
\(150\) 0 0
\(151\) 1.01548e10 1.58955 0.794773 0.606907i \(-0.207590\pi\)
0.794773 + 0.606907i \(0.207590\pi\)
\(152\) −1.81465e9 −0.275738
\(153\) 0 0
\(154\) 3.47207e9 0.497445
\(155\) 0 0
\(156\) 0 0
\(157\) −9.36605e9 −1.23029 −0.615146 0.788413i \(-0.710903\pi\)
−0.615146 + 0.788413i \(0.710903\pi\)
\(158\) 3.45375e9 0.440893
\(159\) 0 0
\(160\) 0 0
\(161\) 1.60972e10 1.88814
\(162\) 0 0
\(163\) 7.34780e8 0.0815292 0.0407646 0.999169i \(-0.487021\pi\)
0.0407646 + 0.999169i \(0.487021\pi\)
\(164\) −1.35865e9 −0.146659
\(165\) 0 0
\(166\) −4.97087e9 −0.508096
\(167\) 1.55584e9 0.154789 0.0773947 0.997001i \(-0.475340\pi\)
0.0773947 + 0.997001i \(0.475340\pi\)
\(168\) 0 0
\(169\) −9.63019e9 −0.908123
\(170\) 0 0
\(171\) 0 0
\(172\) −4.35173e9 −0.379126
\(173\) 1.90448e10 1.61648 0.808238 0.588856i \(-0.200421\pi\)
0.808238 + 0.588856i \(0.200421\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.75866e9 0.216716
\(177\) 0 0
\(178\) 1.22220e10 0.912537
\(179\) −4.32852e9 −0.315138 −0.157569 0.987508i \(-0.550366\pi\)
−0.157569 + 0.987508i \(0.550366\pi\)
\(180\) 0 0
\(181\) −9.56757e8 −0.0662595 −0.0331298 0.999451i \(-0.510547\pi\)
−0.0331298 + 0.999451i \(0.510547\pi\)
\(182\) 5.12859e9 0.346479
\(183\) 0 0
\(184\) 2.22200e10 1.42911
\(185\) 0 0
\(186\) 0 0
\(187\) 5.90306e9 0.353012
\(188\) −4.31871e9 −0.252141
\(189\) 0 0
\(190\) 0 0
\(191\) 1.76438e10 0.959274 0.479637 0.877467i \(-0.340768\pi\)
0.479637 + 0.877467i \(0.340768\pi\)
\(192\) 0 0
\(193\) −1.30993e10 −0.679581 −0.339790 0.940501i \(-0.610356\pi\)
−0.339790 + 0.940501i \(0.610356\pi\)
\(194\) 1.02178e10 0.517906
\(195\) 0 0
\(196\) −8.07775e9 −0.390965
\(197\) 4.74589e9 0.224502 0.112251 0.993680i \(-0.464194\pi\)
0.112251 + 0.993680i \(0.464194\pi\)
\(198\) 0 0
\(199\) −1.92102e10 −0.868348 −0.434174 0.900829i \(-0.642960\pi\)
−0.434174 + 0.900829i \(0.642960\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.92108e10 −1.23442
\(203\) 4.28332e10 1.77031
\(204\) 0 0
\(205\) 0 0
\(206\) 3.15185e10 1.21945
\(207\) 0 0
\(208\) 4.07480e9 0.150946
\(209\) −3.04343e9 −0.110333
\(210\) 0 0
\(211\) −1.81273e10 −0.629597 −0.314798 0.949159i \(-0.601937\pi\)
−0.314798 + 0.949159i \(0.601937\pi\)
\(212\) −1.47537e10 −0.501637
\(213\) 0 0
\(214\) −2.77732e10 −0.905241
\(215\) 0 0
\(216\) 0 0
\(217\) 3.36904e9 0.103142
\(218\) 8.23797e9 0.247039
\(219\) 0 0
\(220\) 0 0
\(221\) 8.71938e9 0.245878
\(222\) 0 0
\(223\) −3.91081e9 −0.105900 −0.0529499 0.998597i \(-0.516862\pi\)
−0.0529499 + 0.998597i \(0.516862\pi\)
\(224\) −3.74377e10 −0.993556
\(225\) 0 0
\(226\) 5.88230e10 1.49989
\(227\) −3.58126e10 −0.895199 −0.447599 0.894234i \(-0.647721\pi\)
−0.447599 + 0.894234i \(0.647721\pi\)
\(228\) 0 0
\(229\) 5.30445e9 0.127462 0.0637310 0.997967i \(-0.479700\pi\)
0.0637310 + 0.997967i \(0.479700\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.91256e10 1.33992
\(233\) −2.54172e10 −0.564970 −0.282485 0.959272i \(-0.591159\pi\)
−0.282485 + 0.959272i \(0.591159\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.81840e9 −0.0801268
\(237\) 0 0
\(238\) 4.58970e10 0.927231
\(239\) 6.33494e10 1.25589 0.627946 0.778257i \(-0.283896\pi\)
0.627946 + 0.778257i \(0.283896\pi\)
\(240\) 0 0
\(241\) 1.01929e11 1.94635 0.973173 0.230073i \(-0.0738965\pi\)
0.973173 + 0.230073i \(0.0738965\pi\)
\(242\) −3.44050e10 −0.644840
\(243\) 0 0
\(244\) 3.37159e10 0.608949
\(245\) 0 0
\(246\) 0 0
\(247\) −4.49544e9 −0.0768486
\(248\) 4.65051e9 0.0780671
\(249\) 0 0
\(250\) 0 0
\(251\) −9.64745e10 −1.53420 −0.767098 0.641530i \(-0.778300\pi\)
−0.767098 + 0.641530i \(0.778300\pi\)
\(252\) 0 0
\(253\) 3.72662e10 0.571838
\(254\) −1.67198e10 −0.252046
\(255\) 0 0
\(256\) −6.42434e10 −0.934864
\(257\) −7.99591e9 −0.114332 −0.0571661 0.998365i \(-0.518206\pi\)
−0.0571661 + 0.998365i \(0.518206\pi\)
\(258\) 0 0
\(259\) 8.53201e10 1.17816
\(260\) 0 0
\(261\) 0 0
\(262\) 1.77959e10 0.233326
\(263\) 1.06269e11 1.36964 0.684821 0.728712i \(-0.259881\pi\)
0.684821 + 0.728712i \(0.259881\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.36631e10 −0.289803
\(267\) 0 0
\(268\) 5.16113e10 0.611137
\(269\) 1.00886e11 1.17475 0.587377 0.809313i \(-0.300160\pi\)
0.587377 + 0.809313i \(0.300160\pi\)
\(270\) 0 0
\(271\) −2.86345e10 −0.322498 −0.161249 0.986914i \(-0.551552\pi\)
−0.161249 + 0.986914i \(0.551552\pi\)
\(272\) 3.64664e10 0.403955
\(273\) 0 0
\(274\) −1.03186e9 −0.0110596
\(275\) 0 0
\(276\) 0 0
\(277\) −4.12251e10 −0.420729 −0.210365 0.977623i \(-0.567465\pi\)
−0.210365 + 0.977623i \(0.567465\pi\)
\(278\) −8.37094e10 −0.840568
\(279\) 0 0
\(280\) 0 0
\(281\) −1.21800e11 −1.16538 −0.582692 0.812693i \(-0.698001\pi\)
−0.582692 + 0.812693i \(0.698001\pi\)
\(282\) 0 0
\(283\) −2.32820e10 −0.215765 −0.107883 0.994164i \(-0.534407\pi\)
−0.107883 + 0.994164i \(0.534407\pi\)
\(284\) −6.83242e9 −0.0623221
\(285\) 0 0
\(286\) 1.18731e10 0.104934
\(287\) −6.59666e10 −0.573925
\(288\) 0 0
\(289\) −4.05559e10 −0.341990
\(290\) 0 0
\(291\) 0 0
\(292\) −4.64528e10 −0.373929
\(293\) −9.12801e10 −0.723555 −0.361778 0.932264i \(-0.617830\pi\)
−0.361778 + 0.932264i \(0.617830\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.17773e11 0.891731
\(297\) 0 0
\(298\) 2.52425e10 0.185421
\(299\) 5.50458e10 0.398294
\(300\) 0 0
\(301\) −2.11290e11 −1.48365
\(302\) 1.82786e11 1.26448
\(303\) 0 0
\(304\) −1.88009e10 −0.126255
\(305\) 0 0
\(306\) 0 0
\(307\) 4.39070e10 0.282105 0.141053 0.990002i \(-0.454951\pi\)
0.141053 + 0.990002i \(0.454951\pi\)
\(308\) −3.62639e10 −0.229613
\(309\) 0 0
\(310\) 0 0
\(311\) 7.64835e10 0.463603 0.231801 0.972763i \(-0.425538\pi\)
0.231801 + 0.972763i \(0.425538\pi\)
\(312\) 0 0
\(313\) −2.68488e11 −1.58116 −0.790580 0.612359i \(-0.790221\pi\)
−0.790580 + 0.612359i \(0.790221\pi\)
\(314\) −1.68589e11 −0.978691
\(315\) 0 0
\(316\) −3.60725e10 −0.203509
\(317\) 1.52056e11 0.845738 0.422869 0.906191i \(-0.361023\pi\)
0.422869 + 0.906191i \(0.361023\pi\)
\(318\) 0 0
\(319\) 9.91621e10 0.536152
\(320\) 0 0
\(321\) 0 0
\(322\) 2.89749e11 1.50200
\(323\) −4.02308e10 −0.205659
\(324\) 0 0
\(325\) 0 0
\(326\) 1.32260e10 0.0648561
\(327\) 0 0
\(328\) −9.10582e10 −0.434397
\(329\) −2.09687e11 −0.986713
\(330\) 0 0
\(331\) −2.63057e11 −1.20455 −0.602274 0.798289i \(-0.705739\pi\)
−0.602274 + 0.798289i \(0.705739\pi\)
\(332\) 5.19179e10 0.234529
\(333\) 0 0
\(334\) 2.80051e10 0.123134
\(335\) 0 0
\(336\) 0 0
\(337\) −2.97183e11 −1.25513 −0.627566 0.778564i \(-0.715949\pi\)
−0.627566 + 0.778564i \(0.715949\pi\)
\(338\) −1.73343e11 −0.722407
\(339\) 0 0
\(340\) 0 0
\(341\) 7.79957e9 0.0312375
\(342\) 0 0
\(343\) −2.38530e10 −0.0930507
\(344\) −2.91658e11 −1.12295
\(345\) 0 0
\(346\) 3.42807e11 1.28590
\(347\) 1.40219e11 0.519186 0.259593 0.965718i \(-0.416412\pi\)
0.259593 + 0.965718i \(0.416412\pi\)
\(348\) 0 0
\(349\) −3.14891e11 −1.13618 −0.568088 0.822968i \(-0.692317\pi\)
−0.568088 + 0.822968i \(0.692317\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −8.66710e10 −0.300907
\(353\) −5.31195e11 −1.82082 −0.910411 0.413704i \(-0.864235\pi\)
−0.910411 + 0.413704i \(0.864235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.27652e11 −0.421212
\(357\) 0 0
\(358\) −7.79133e10 −0.250691
\(359\) 4.97684e11 1.58135 0.790676 0.612235i \(-0.209729\pi\)
0.790676 + 0.612235i \(0.209729\pi\)
\(360\) 0 0
\(361\) −3.01946e11 −0.935722
\(362\) −1.72216e10 −0.0527091
\(363\) 0 0
\(364\) −5.35652e10 −0.159929
\(365\) 0 0
\(366\) 0 0
\(367\) −2.77743e11 −0.799183 −0.399591 0.916693i \(-0.630848\pi\)
−0.399591 + 0.916693i \(0.630848\pi\)
\(368\) 2.30214e11 0.654359
\(369\) 0 0
\(370\) 0 0
\(371\) −7.16340e11 −1.96307
\(372\) 0 0
\(373\) −2.55319e11 −0.682957 −0.341478 0.939890i \(-0.610928\pi\)
−0.341478 + 0.939890i \(0.610928\pi\)
\(374\) 1.06255e11 0.280819
\(375\) 0 0
\(376\) −2.89446e11 −0.746830
\(377\) 1.46472e11 0.373438
\(378\) 0 0
\(379\) 8.12103e10 0.202178 0.101089 0.994877i \(-0.467767\pi\)
0.101089 + 0.994877i \(0.467767\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3.17589e11 0.763097
\(383\) −4.13123e11 −0.981036 −0.490518 0.871431i \(-0.663192\pi\)
−0.490518 + 0.871431i \(0.663192\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.35788e11 −0.540603
\(387\) 0 0
\(388\) −1.06720e11 −0.239057
\(389\) −2.53052e11 −0.560321 −0.280160 0.959953i \(-0.590388\pi\)
−0.280160 + 0.959953i \(0.590388\pi\)
\(390\) 0 0
\(391\) 4.92618e11 1.06590
\(392\) −5.41381e11 −1.15802
\(393\) 0 0
\(394\) 8.54259e10 0.178590
\(395\) 0 0
\(396\) 0 0
\(397\) 6.53562e10 0.132047 0.0660236 0.997818i \(-0.478969\pi\)
0.0660236 + 0.997818i \(0.478969\pi\)
\(398\) −3.45784e11 −0.690766
\(399\) 0 0
\(400\) 0 0
\(401\) 2.40886e11 0.465225 0.232612 0.972570i \(-0.425273\pi\)
0.232612 + 0.972570i \(0.425273\pi\)
\(402\) 0 0
\(403\) 1.15207e10 0.0217574
\(404\) 3.05091e11 0.569788
\(405\) 0 0
\(406\) 7.70998e11 1.40827
\(407\) 1.97522e11 0.356814
\(408\) 0 0
\(409\) −3.78340e11 −0.668540 −0.334270 0.942477i \(-0.608490\pi\)
−0.334270 + 0.942477i \(0.608490\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.29193e11 −0.562876
\(413\) −1.85396e11 −0.313563
\(414\) 0 0
\(415\) 0 0
\(416\) −1.28021e11 −0.209586
\(417\) 0 0
\(418\) −5.47818e10 −0.0877693
\(419\) −9.68766e11 −1.53552 −0.767760 0.640737i \(-0.778629\pi\)
−0.767760 + 0.640737i \(0.778629\pi\)
\(420\) 0 0
\(421\) −1.43789e11 −0.223077 −0.111539 0.993760i \(-0.535578\pi\)
−0.111539 + 0.993760i \(0.535578\pi\)
\(422\) −3.26292e11 −0.500841
\(423\) 0 0
\(424\) −9.88812e11 −1.48582
\(425\) 0 0
\(426\) 0 0
\(427\) 1.63701e12 2.38302
\(428\) 2.90076e11 0.417845
\(429\) 0 0
\(430\) 0 0
\(431\) −2.06890e11 −0.288796 −0.144398 0.989520i \(-0.546125\pi\)
−0.144398 + 0.989520i \(0.546125\pi\)
\(432\) 0 0
\(433\) 4.81883e10 0.0658789 0.0329394 0.999457i \(-0.489513\pi\)
0.0329394 + 0.999457i \(0.489513\pi\)
\(434\) 6.06426e10 0.0820492
\(435\) 0 0
\(436\) −8.60411e10 −0.114029
\(437\) −2.53979e11 −0.333143
\(438\) 0 0
\(439\) 4.37941e11 0.562763 0.281381 0.959596i \(-0.409207\pi\)
0.281381 + 0.959596i \(0.409207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.56949e11 0.195595
\(443\) 7.58665e11 0.935908 0.467954 0.883753i \(-0.344991\pi\)
0.467954 + 0.883753i \(0.344991\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −7.03946e10 −0.0842427
\(447\) 0 0
\(448\) −1.28398e12 −1.50594
\(449\) 1.37280e11 0.159403 0.0797017 0.996819i \(-0.474603\pi\)
0.0797017 + 0.996819i \(0.474603\pi\)
\(450\) 0 0
\(451\) −1.52718e11 −0.173818
\(452\) −6.14373e11 −0.692324
\(453\) 0 0
\(454\) −6.44627e11 −0.712126
\(455\) 0 0
\(456\) 0 0
\(457\) −1.31091e11 −0.140589 −0.0702944 0.997526i \(-0.522394\pi\)
−0.0702944 + 0.997526i \(0.522394\pi\)
\(458\) 9.54800e10 0.101395
\(459\) 0 0
\(460\) 0 0
\(461\) 8.03508e11 0.828583 0.414291 0.910144i \(-0.364029\pi\)
0.414291 + 0.910144i \(0.364029\pi\)
\(462\) 0 0
\(463\) 1.78582e12 1.80602 0.903009 0.429621i \(-0.141353\pi\)
0.903009 + 0.429621i \(0.141353\pi\)
\(464\) 6.12579e11 0.613524
\(465\) 0 0
\(466\) −4.57509e11 −0.449431
\(467\) 6.44366e11 0.626912 0.313456 0.949603i \(-0.398513\pi\)
0.313456 + 0.949603i \(0.398513\pi\)
\(468\) 0 0
\(469\) 2.50590e12 2.39158
\(470\) 0 0
\(471\) 0 0
\(472\) −2.55914e11 −0.237332
\(473\) −4.89152e11 −0.449334
\(474\) 0 0
\(475\) 0 0
\(476\) −4.79369e11 −0.427995
\(477\) 0 0
\(478\) 1.14029e12 0.999056
\(479\) 1.87701e12 1.62913 0.814565 0.580073i \(-0.196976\pi\)
0.814565 + 0.580073i \(0.196976\pi\)
\(480\) 0 0
\(481\) 2.91760e11 0.248526
\(482\) 1.83472e12 1.54831
\(483\) 0 0
\(484\) 3.59341e11 0.297647
\(485\) 0 0
\(486\) 0 0
\(487\) −1.49355e12 −1.20321 −0.601603 0.798795i \(-0.705471\pi\)
−0.601603 + 0.798795i \(0.705471\pi\)
\(488\) 2.25968e12 1.80368
\(489\) 0 0
\(490\) 0 0
\(491\) 7.47313e11 0.580278 0.290139 0.956985i \(-0.406298\pi\)
0.290139 + 0.956985i \(0.406298\pi\)
\(492\) 0 0
\(493\) 1.31082e12 0.999379
\(494\) −8.09179e10 −0.0611327
\(495\) 0 0
\(496\) 4.81822e10 0.0357453
\(497\) −3.31736e11 −0.243887
\(498\) 0 0
\(499\) 1.01300e12 0.731402 0.365701 0.930732i \(-0.380829\pi\)
0.365701 + 0.930732i \(0.380829\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −1.73654e12 −1.22044
\(503\) 2.03399e11 0.141675 0.0708373 0.997488i \(-0.477433\pi\)
0.0708373 + 0.997488i \(0.477433\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.70792e11 0.454894
\(507\) 0 0
\(508\) 1.74629e11 0.116340
\(509\) −5.93699e11 −0.392045 −0.196023 0.980599i \(-0.562803\pi\)
−0.196023 + 0.980599i \(0.562803\pi\)
\(510\) 0 0
\(511\) −2.25543e12 −1.46331
\(512\) −1.37758e12 −0.885935
\(513\) 0 0
\(514\) −1.43926e11 −0.0909508
\(515\) 0 0
\(516\) 0 0
\(517\) −4.85442e11 −0.298834
\(518\) 1.53576e12 0.937217
\(519\) 0 0
\(520\) 0 0
\(521\) 1.74950e12 1.04027 0.520133 0.854085i \(-0.325882\pi\)
0.520133 + 0.854085i \(0.325882\pi\)
\(522\) 0 0
\(523\) −3.20555e11 −0.187346 −0.0936731 0.995603i \(-0.529861\pi\)
−0.0936731 + 0.995603i \(0.529861\pi\)
\(524\) −1.85868e11 −0.107699
\(525\) 0 0
\(526\) 1.91285e12 1.08954
\(527\) 1.03102e11 0.0582262
\(528\) 0 0
\(529\) 1.30877e12 0.726627
\(530\) 0 0
\(531\) 0 0
\(532\) 2.47148e11 0.133769
\(533\) −2.25579e11 −0.121067
\(534\) 0 0
\(535\) 0 0
\(536\) 3.45906e12 1.81016
\(537\) 0 0
\(538\) 1.81595e12 0.934512
\(539\) −9.07974e11 −0.463366
\(540\) 0 0
\(541\) 3.50229e11 0.175778 0.0878889 0.996130i \(-0.471988\pi\)
0.0878889 + 0.996130i \(0.471988\pi\)
\(542\) −5.15421e11 −0.256546
\(543\) 0 0
\(544\) −1.14570e12 −0.560885
\(545\) 0 0
\(546\) 0 0
\(547\) 2.46037e12 1.17505 0.587527 0.809205i \(-0.300102\pi\)
0.587527 + 0.809205i \(0.300102\pi\)
\(548\) 1.07772e10 0.00510495
\(549\) 0 0
\(550\) 0 0
\(551\) −6.75815e11 −0.312353
\(552\) 0 0
\(553\) −1.75143e12 −0.796399
\(554\) −7.42051e11 −0.334688
\(555\) 0 0
\(556\) 8.74299e11 0.387992
\(557\) 1.04356e12 0.459376 0.229688 0.973264i \(-0.426229\pi\)
0.229688 + 0.973264i \(0.426229\pi\)
\(558\) 0 0
\(559\) −7.22525e11 −0.312968
\(560\) 0 0
\(561\) 0 0
\(562\) −2.19240e12 −0.927057
\(563\) 3.81849e12 1.60179 0.800893 0.598808i \(-0.204359\pi\)
0.800893 + 0.598808i \(0.204359\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.19076e11 −0.171640
\(567\) 0 0
\(568\) −4.57917e11 −0.184595
\(569\) −4.84062e11 −0.193596 −0.0967979 0.995304i \(-0.530860\pi\)
−0.0967979 + 0.995304i \(0.530860\pi\)
\(570\) 0 0
\(571\) 1.77044e10 0.00696978 0.00348489 0.999994i \(-0.498891\pi\)
0.00348489 + 0.999994i \(0.498891\pi\)
\(572\) −1.24007e11 −0.0484357
\(573\) 0 0
\(574\) −1.18740e12 −0.456555
\(575\) 0 0
\(576\) 0 0
\(577\) 7.46985e11 0.280557 0.140278 0.990112i \(-0.455200\pi\)
0.140278 + 0.990112i \(0.455200\pi\)
\(578\) −7.30007e11 −0.272052
\(579\) 0 0
\(580\) 0 0
\(581\) 2.52078e12 0.917789
\(582\) 0 0
\(583\) −1.65838e12 −0.594532
\(584\) −3.11333e12 −1.10756
\(585\) 0 0
\(586\) −1.64304e12 −0.575585
\(587\) −4.33948e12 −1.50857 −0.754286 0.656546i \(-0.772017\pi\)
−0.754286 + 0.656546i \(0.772017\pi\)
\(588\) 0 0
\(589\) −5.31561e10 −0.0181984
\(590\) 0 0
\(591\) 0 0
\(592\) 1.22020e12 0.408305
\(593\) 1.90120e12 0.631366 0.315683 0.948865i \(-0.397766\pi\)
0.315683 + 0.948865i \(0.397766\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.63644e11 −0.0855874
\(597\) 0 0
\(598\) 9.90824e11 0.316841
\(599\) 5.49649e12 1.74447 0.872237 0.489083i \(-0.162668\pi\)
0.872237 + 0.489083i \(0.162668\pi\)
\(600\) 0 0
\(601\) 4.77698e12 1.49355 0.746773 0.665079i \(-0.231602\pi\)
0.746773 + 0.665079i \(0.231602\pi\)
\(602\) −3.80322e12 −1.18023
\(603\) 0 0
\(604\) −1.90909e12 −0.583661
\(605\) 0 0
\(606\) 0 0
\(607\) −8.25000e11 −0.246663 −0.123332 0.992366i \(-0.539358\pi\)
−0.123332 + 0.992366i \(0.539358\pi\)
\(608\) 5.90685e11 0.175303
\(609\) 0 0
\(610\) 0 0
\(611\) −7.17045e11 −0.208142
\(612\) 0 0
\(613\) 1.46985e12 0.420438 0.210219 0.977654i \(-0.432582\pi\)
0.210219 + 0.977654i \(0.432582\pi\)
\(614\) 7.90326e11 0.224413
\(615\) 0 0
\(616\) −2.43045e12 −0.680101
\(617\) 5.17373e12 1.43721 0.718606 0.695418i \(-0.244781\pi\)
0.718606 + 0.695418i \(0.244781\pi\)
\(618\) 0 0
\(619\) −4.80274e12 −1.31487 −0.657433 0.753513i \(-0.728358\pi\)
−0.657433 + 0.753513i \(0.728358\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1.37670e12 0.368794
\(623\) −6.19789e12 −1.64834
\(624\) 0 0
\(625\) 0 0
\(626\) −4.83279e12 −1.25780
\(627\) 0 0
\(628\) 1.76082e12 0.451748
\(629\) 2.61103e12 0.665096
\(630\) 0 0
\(631\) 5.93992e12 1.49159 0.745794 0.666177i \(-0.232071\pi\)
0.745794 + 0.666177i \(0.232071\pi\)
\(632\) −2.41762e12 −0.602784
\(633\) 0 0
\(634\) 2.73700e12 0.672780
\(635\) 0 0
\(636\) 0 0
\(637\) −1.34116e12 −0.322741
\(638\) 1.78492e12 0.426506
\(639\) 0 0
\(640\) 0 0
\(641\) −6.47738e12 −1.51544 −0.757720 0.652580i \(-0.773687\pi\)
−0.757720 + 0.652580i \(0.773687\pi\)
\(642\) 0 0
\(643\) 1.74519e12 0.402619 0.201309 0.979528i \(-0.435480\pi\)
0.201309 + 0.979528i \(0.435480\pi\)
\(644\) −3.02627e12 −0.693301
\(645\) 0 0
\(646\) −7.24155e11 −0.163601
\(647\) 4.97244e12 1.11558 0.557790 0.829982i \(-0.311650\pi\)
0.557790 + 0.829982i \(0.311650\pi\)
\(648\) 0 0
\(649\) −4.29205e11 −0.0949650
\(650\) 0 0
\(651\) 0 0
\(652\) −1.38139e11 −0.0299365
\(653\) −1.45057e12 −0.312198 −0.156099 0.987741i \(-0.549892\pi\)
−0.156099 + 0.987741i \(0.549892\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.43420e11 −0.198901
\(657\) 0 0
\(658\) −3.77437e12 −0.784925
\(659\) 3.05175e12 0.630325 0.315162 0.949038i \(-0.397941\pi\)
0.315162 + 0.949038i \(0.397941\pi\)
\(660\) 0 0
\(661\) −1.88315e12 −0.383688 −0.191844 0.981425i \(-0.561447\pi\)
−0.191844 + 0.981425i \(0.561447\pi\)
\(662\) −4.73503e12 −0.958213
\(663\) 0 0
\(664\) 3.47961e12 0.694662
\(665\) 0 0
\(666\) 0 0
\(667\) 8.27522e12 1.61888
\(668\) −2.92498e11 −0.0568367
\(669\) 0 0
\(670\) 0 0
\(671\) 3.78981e12 0.721716
\(672\) 0 0
\(673\) −7.84359e12 −1.47383 −0.736914 0.675986i \(-0.763718\pi\)
−0.736914 + 0.675986i \(0.763718\pi\)
\(674\) −5.34929e12 −0.998451
\(675\) 0 0
\(676\) 1.81047e12 0.333451
\(677\) 2.32420e12 0.425230 0.212615 0.977136i \(-0.431802\pi\)
0.212615 + 0.977136i \(0.431802\pi\)
\(678\) 0 0
\(679\) −5.18158e12 −0.935509
\(680\) 0 0
\(681\) 0 0
\(682\) 1.40392e11 0.0248493
\(683\) −6.89265e12 −1.21197 −0.605986 0.795475i \(-0.707221\pi\)
−0.605986 + 0.795475i \(0.707221\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −4.29354e11 −0.0740214
\(687\) 0 0
\(688\) −3.02176e12 −0.514177
\(689\) −2.44959e12 −0.414101
\(690\) 0 0
\(691\) 1.11430e13 1.85931 0.929654 0.368434i \(-0.120106\pi\)
0.929654 + 0.368434i \(0.120106\pi\)
\(692\) −3.58043e12 −0.593550
\(693\) 0 0
\(694\) 2.52393e12 0.413010
\(695\) 0 0
\(696\) 0 0
\(697\) −2.01876e12 −0.323994
\(698\) −5.66803e12 −0.903822
\(699\) 0 0
\(700\) 0 0
\(701\) 5.78015e12 0.904082 0.452041 0.891997i \(-0.350696\pi\)
0.452041 + 0.891997i \(0.350696\pi\)
\(702\) 0 0
\(703\) −1.34617e12 −0.207874
\(704\) −2.97251e12 −0.456085
\(705\) 0 0
\(706\) −9.56151e12 −1.44846
\(707\) 1.48131e13 2.22977
\(708\) 0 0
\(709\) −3.39902e12 −0.505180 −0.252590 0.967573i \(-0.581282\pi\)
−0.252590 + 0.967573i \(0.581282\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −8.55537e12 −1.24761
\(713\) 6.50885e11 0.0943195
\(714\) 0 0
\(715\) 0 0
\(716\) 8.13761e11 0.115715
\(717\) 0 0
\(718\) 8.95831e12 1.25796
\(719\) −2.71272e11 −0.0378551 −0.0189276 0.999821i \(-0.506025\pi\)
−0.0189276 + 0.999821i \(0.506025\pi\)
\(720\) 0 0
\(721\) −1.59834e13 −2.20272
\(722\) −5.43503e12 −0.744362
\(723\) 0 0
\(724\) 1.79870e11 0.0243297
\(725\) 0 0
\(726\) 0 0
\(727\) 2.13863e12 0.283943 0.141972 0.989871i \(-0.454656\pi\)
0.141972 + 0.989871i \(0.454656\pi\)
\(728\) −3.59001e12 −0.473701
\(729\) 0 0
\(730\) 0 0
\(731\) −6.46606e12 −0.837552
\(732\) 0 0
\(733\) −4.60033e12 −0.588601 −0.294301 0.955713i \(-0.595087\pi\)
−0.294301 + 0.955713i \(0.595087\pi\)
\(734\) −4.99938e12 −0.635746
\(735\) 0 0
\(736\) −7.23282e12 −0.908568
\(737\) 5.80133e12 0.724309
\(738\) 0 0
\(739\) 8.60245e12 1.06102 0.530508 0.847680i \(-0.322001\pi\)
0.530508 + 0.847680i \(0.322001\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.28941e13 −1.56161
\(743\) 1.31407e12 0.158187 0.0790934 0.996867i \(-0.474797\pi\)
0.0790934 + 0.996867i \(0.474797\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.59574e12 −0.543289
\(747\) 0 0
\(748\) −1.10977e12 −0.129622
\(749\) 1.40841e13 1.63516
\(750\) 0 0
\(751\) −7.95810e12 −0.912913 −0.456457 0.889746i \(-0.650882\pi\)
−0.456457 + 0.889746i \(0.650882\pi\)
\(752\) −2.99884e12 −0.341958
\(753\) 0 0
\(754\) 2.63650e12 0.297068
\(755\) 0 0
\(756\) 0 0
\(757\) 6.95936e12 0.770261 0.385131 0.922862i \(-0.374156\pi\)
0.385131 + 0.922862i \(0.374156\pi\)
\(758\) 1.46179e12 0.160832
\(759\) 0 0
\(760\) 0 0
\(761\) 1.38632e13 1.49841 0.749207 0.662336i \(-0.230435\pi\)
0.749207 + 0.662336i \(0.230435\pi\)
\(762\) 0 0
\(763\) −4.17757e12 −0.446234
\(764\) −3.31704e12 −0.352233
\(765\) 0 0
\(766\) −7.43621e12 −0.780409
\(767\) −6.33977e11 −0.0661446
\(768\) 0 0
\(769\) 1.35604e13 1.39832 0.699158 0.714967i \(-0.253558\pi\)
0.699158 + 0.714967i \(0.253558\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.46267e12 0.249534
\(773\) 4.96740e12 0.500405 0.250203 0.968194i \(-0.419503\pi\)
0.250203 + 0.968194i \(0.419503\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −7.15248e12 −0.708075
\(777\) 0 0
\(778\) −4.55494e12 −0.445732
\(779\) 1.04081e12 0.101263
\(780\) 0 0
\(781\) −7.67993e11 −0.0738631
\(782\) 8.86713e12 0.847916
\(783\) 0 0
\(784\) −5.60905e12 −0.530234
\(785\) 0 0
\(786\) 0 0
\(787\) 5.25809e12 0.488587 0.244293 0.969701i \(-0.421444\pi\)
0.244293 + 0.969701i \(0.421444\pi\)
\(788\) −8.92227e11 −0.0824342
\(789\) 0 0
\(790\) 0 0
\(791\) −2.98298e13 −2.70929
\(792\) 0 0
\(793\) 5.59792e12 0.502686
\(794\) 1.17641e12 0.105043
\(795\) 0 0
\(796\) 3.61152e12 0.318846
\(797\) 1.61816e13 1.42056 0.710281 0.703918i \(-0.248568\pi\)
0.710281 + 0.703918i \(0.248568\pi\)
\(798\) 0 0
\(799\) −6.41701e12 −0.557022
\(800\) 0 0
\(801\) 0 0
\(802\) 4.33596e12 0.370084
\(803\) −5.22150e12 −0.443175
\(804\) 0 0
\(805\) 0 0
\(806\) 2.07373e11 0.0173079
\(807\) 0 0
\(808\) 2.04476e13 1.68768
\(809\) 1.39276e13 1.14317 0.571583 0.820544i \(-0.306329\pi\)
0.571583 + 0.820544i \(0.306329\pi\)
\(810\) 0 0
\(811\) 1.38190e13 1.12171 0.560857 0.827913i \(-0.310472\pi\)
0.560857 + 0.827913i \(0.310472\pi\)
\(812\) −8.05265e12 −0.650035
\(813\) 0 0
\(814\) 3.55540e12 0.283844
\(815\) 0 0
\(816\) 0 0
\(817\) 3.33370e12 0.261774
\(818\) −6.81012e12 −0.531820
\(819\) 0 0
\(820\) 0 0
\(821\) −3.48578e12 −0.267766 −0.133883 0.990997i \(-0.542745\pi\)
−0.133883 + 0.990997i \(0.542745\pi\)
\(822\) 0 0
\(823\) 1.21472e13 0.922948 0.461474 0.887154i \(-0.347321\pi\)
0.461474 + 0.887154i \(0.347321\pi\)
\(824\) −2.20629e13 −1.66721
\(825\) 0 0
\(826\) −3.33712e12 −0.249438
\(827\) −1.70702e13 −1.26901 −0.634504 0.772919i \(-0.718796\pi\)
−0.634504 + 0.772919i \(0.718796\pi\)
\(828\) 0 0
\(829\) 5.58071e12 0.410387 0.205194 0.978721i \(-0.434218\pi\)
0.205194 + 0.978721i \(0.434218\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.39068e12 −0.317671
\(833\) −1.20024e13 −0.863707
\(834\) 0 0
\(835\) 0 0
\(836\) 5.72165e11 0.0405129
\(837\) 0 0
\(838\) −1.74378e13 −1.22150
\(839\) −1.28642e13 −0.896304 −0.448152 0.893957i \(-0.647918\pi\)
−0.448152 + 0.893957i \(0.647918\pi\)
\(840\) 0 0
\(841\) 7.51250e12 0.517849
\(842\) −2.58819e12 −0.177457
\(843\) 0 0
\(844\) 3.40793e12 0.231180
\(845\) 0 0
\(846\) 0 0
\(847\) 1.74471e13 1.16479
\(848\) −1.02447e13 −0.680329
\(849\) 0 0
\(850\) 0 0
\(851\) 1.64835e13 1.07738
\(852\) 0 0
\(853\) −1.03465e13 −0.669147 −0.334574 0.942370i \(-0.608592\pi\)
−0.334574 + 0.942370i \(0.608592\pi\)
\(854\) 2.94663e13 1.89568
\(855\) 0 0
\(856\) 1.94413e13 1.23763
\(857\) 5.63216e12 0.356666 0.178333 0.983970i \(-0.442930\pi\)
0.178333 + 0.983970i \(0.442930\pi\)
\(858\) 0 0
\(859\) −2.28289e13 −1.43059 −0.715295 0.698822i \(-0.753708\pi\)
−0.715295 + 0.698822i \(0.753708\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −3.72401e12 −0.229736
\(863\) 4.62918e12 0.284090 0.142045 0.989860i \(-0.454632\pi\)
0.142045 + 0.989860i \(0.454632\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8.67389e11 0.0524063
\(867\) 0 0
\(868\) −6.33379e11 −0.0378726
\(869\) −4.05470e12 −0.241196
\(870\) 0 0
\(871\) 8.56913e12 0.504493
\(872\) −5.76658e12 −0.337749
\(873\) 0 0
\(874\) −4.57162e12 −0.265014
\(875\) 0 0
\(876\) 0 0
\(877\) −5.33611e12 −0.304598 −0.152299 0.988334i \(-0.548668\pi\)
−0.152299 + 0.988334i \(0.548668\pi\)
\(878\) 7.88294e12 0.447675
\(879\) 0 0
\(880\) 0 0
\(881\) −7.87890e12 −0.440630 −0.220315 0.975429i \(-0.570709\pi\)
−0.220315 + 0.975429i \(0.570709\pi\)
\(882\) 0 0
\(883\) −1.38973e13 −0.769320 −0.384660 0.923058i \(-0.625681\pi\)
−0.384660 + 0.923058i \(0.625681\pi\)
\(884\) −1.63924e12 −0.0902835
\(885\) 0 0
\(886\) 1.36560e13 0.744510
\(887\) −5.86607e12 −0.318193 −0.159097 0.987263i \(-0.550858\pi\)
−0.159097 + 0.987263i \(0.550858\pi\)
\(888\) 0 0
\(889\) 8.47881e12 0.455278
\(890\) 0 0
\(891\) 0 0
\(892\) 7.35232e11 0.0388851
\(893\) 3.30841e12 0.174096
\(894\) 0 0
\(895\) 0 0
\(896\) −3.94356e12 −0.204410
\(897\) 0 0
\(898\) 2.47103e12 0.126805
\(899\) 1.73195e12 0.0884334
\(900\) 0 0
\(901\) −2.19220e13 −1.10820
\(902\) −2.74892e12 −0.138271
\(903\) 0 0
\(904\) −4.11761e13 −2.05063
\(905\) 0 0
\(906\) 0 0
\(907\) −8.52250e12 −0.418152 −0.209076 0.977899i \(-0.567046\pi\)
−0.209076 + 0.977899i \(0.567046\pi\)
\(908\) 6.73277e12 0.328706
\(909\) 0 0
\(910\) 0 0
\(911\) −1.63369e13 −0.785843 −0.392921 0.919572i \(-0.628536\pi\)
−0.392921 + 0.919572i \(0.628536\pi\)
\(912\) 0 0
\(913\) 5.83580e12 0.277960
\(914\) −2.35964e12 −0.111838
\(915\) 0 0
\(916\) −9.97236e11 −0.0468024
\(917\) −9.02448e12 −0.421464
\(918\) 0 0
\(919\) 3.24712e13 1.50169 0.750843 0.660481i \(-0.229648\pi\)
0.750843 + 0.660481i \(0.229648\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.44631e13 0.659134
\(923\) −1.13440e12 −0.0514468
\(924\) 0 0
\(925\) 0 0
\(926\) 3.21447e13 1.43668
\(927\) 0 0
\(928\) −1.92459e13 −0.851868
\(929\) 1.24571e13 0.548716 0.274358 0.961628i \(-0.411535\pi\)
0.274358 + 0.961628i \(0.411535\pi\)
\(930\) 0 0
\(931\) 6.18808e12 0.269949
\(932\) 4.77843e12 0.207450
\(933\) 0 0
\(934\) 1.15986e13 0.498706
\(935\) 0 0
\(936\) 0 0
\(937\) −9.29012e12 −0.393725 −0.196863 0.980431i \(-0.563075\pi\)
−0.196863 + 0.980431i \(0.563075\pi\)
\(938\) 4.51061e13 1.90249
\(939\) 0 0
\(940\) 0 0
\(941\) 3.24831e13 1.35053 0.675265 0.737575i \(-0.264029\pi\)
0.675265 + 0.737575i \(0.264029\pi\)
\(942\) 0 0
\(943\) −1.27445e13 −0.524832
\(944\) −2.65143e12 −0.108669
\(945\) 0 0
\(946\) −8.80474e12 −0.357443
\(947\) −4.16258e13 −1.68185 −0.840926 0.541151i \(-0.817989\pi\)
−0.840926 + 0.541151i \(0.817989\pi\)
\(948\) 0 0
\(949\) −7.71265e12 −0.308678
\(950\) 0 0
\(951\) 0 0
\(952\) −3.21279e13 −1.26770
\(953\) 1.86479e13 0.732339 0.366169 0.930548i \(-0.380669\pi\)
0.366169 + 0.930548i \(0.380669\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.19097e13 −0.461148
\(957\) 0 0
\(958\) 3.37861e13 1.29596
\(959\) 5.23265e11 0.0199774
\(960\) 0 0
\(961\) −2.63034e13 −0.994848
\(962\) 5.25167e12 0.197702
\(963\) 0 0
\(964\) −1.91626e13 −0.714674
\(965\) 0 0
\(966\) 0 0
\(967\) 2.32990e13 0.856875 0.428437 0.903572i \(-0.359064\pi\)
0.428437 + 0.903572i \(0.359064\pi\)
\(968\) 2.40835e13 0.881617
\(969\) 0 0
\(970\) 0 0
\(971\) 3.63628e13 1.31272 0.656358 0.754450i \(-0.272096\pi\)
0.656358 + 0.754450i \(0.272096\pi\)
\(972\) 0 0
\(973\) 4.24500e13 1.51834
\(974\) −2.68839e13 −0.957145
\(975\) 0 0
\(976\) 2.34118e13 0.825866
\(977\) 5.18141e13 1.81938 0.909688 0.415291i \(-0.136320\pi\)
0.909688 + 0.415291i \(0.136320\pi\)
\(978\) 0 0
\(979\) −1.43486e13 −0.499214
\(980\) 0 0
\(981\) 0 0
\(982\) 1.34516e13 0.461608
\(983\) −2.54862e13 −0.870592 −0.435296 0.900287i \(-0.643356\pi\)
−0.435296 + 0.900287i \(0.643356\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.35947e13 0.795001
\(987\) 0 0
\(988\) 8.45143e11 0.0282178
\(989\) −4.08205e13 −1.35673
\(990\) 0 0
\(991\) 3.92830e13 1.29382 0.646910 0.762567i \(-0.276061\pi\)
0.646910 + 0.762567i \(0.276061\pi\)
\(992\) −1.51378e12 −0.0496318
\(993\) 0 0
\(994\) −5.97124e12 −0.194011
\(995\) 0 0
\(996\) 0 0
\(997\) −4.98389e13 −1.59750 −0.798748 0.601665i \(-0.794504\pi\)
−0.798748 + 0.601665i \(0.794504\pi\)
\(998\) 1.82340e13 0.581827
\(999\) 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 225.10.a.e.1.1 1
3.2 odd 2 75.10.a.b.1.1 1
5.2 odd 4 225.10.b.c.199.2 2
5.3 odd 4 225.10.b.c.199.1 2
5.4 even 2 9.10.a.a.1.1 1
15.2 even 4 75.10.b.c.49.1 2
15.8 even 4 75.10.b.c.49.2 2
15.14 odd 2 3.10.a.b.1.1 1
20.19 odd 2 144.10.a.m.1.1 1
45.4 even 6 81.10.c.d.55.1 2
45.14 odd 6 81.10.c.b.55.1 2
45.29 odd 6 81.10.c.b.28.1 2
45.34 even 6 81.10.c.d.28.1 2
60.59 even 2 48.10.a.a.1.1 1
105.104 even 2 147.10.a.c.1.1 1
120.29 odd 2 192.10.a.g.1.1 1
120.59 even 2 192.10.a.n.1.1 1
165.164 even 2 363.10.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.10.a.b.1.1 1 15.14 odd 2
9.10.a.a.1.1 1 5.4 even 2
48.10.a.a.1.1 1 60.59 even 2
75.10.a.b.1.1 1 3.2 odd 2
75.10.b.c.49.1 2 15.2 even 4
75.10.b.c.49.2 2 15.8 even 4
81.10.c.b.28.1 2 45.29 odd 6
81.10.c.b.55.1 2 45.14 odd 6
81.10.c.d.28.1 2 45.34 even 6
81.10.c.d.55.1 2 45.4 even 6
144.10.a.m.1.1 1 20.19 odd 2
147.10.a.c.1.1 1 105.104 even 2
192.10.a.g.1.1 1 120.29 odd 2
192.10.a.n.1.1 1 120.59 even 2
225.10.a.e.1.1 1 1.1 even 1 trivial
225.10.b.c.199.1 2 5.3 odd 4
225.10.b.c.199.2 2 5.2 odd 4
363.10.a.a.1.1 1 165.164 even 2