Properties

Label 225.12.a.e.1.1
Level $225$
Weight $12$
Character 225.1
Self dual yes
Analytic conductor $172.877$
Analytic rank $1$
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] = [225,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(172.877215626\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
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)
 
\(f(q)\) \(=\) \(q+34.0000 q^{2} -892.000 q^{4} +17556.0 q^{7} -99960.0 q^{8} +O(q^{10})\) \(q+34.0000 q^{2} -892.000 q^{4} +17556.0 q^{7} -99960.0 q^{8} +468788. q^{11} +374042. q^{13} +596904. q^{14} -1.57182e6 q^{16} -3.72429e6 q^{17} -379460. q^{19} +1.59388e7 q^{22} -3.24581e7 q^{23} +1.27174e7 q^{26} -1.56600e7 q^{28} -6.96967e7 q^{29} +1.71449e8 q^{31} +1.51276e8 q^{32} -1.26626e8 q^{34} +2.91341e8 q^{37} -1.29016e7 q^{38} -1.91343e8 q^{41} +1.75986e9 q^{43} -4.18159e8 q^{44} -1.10358e9 q^{46} +1.62347e9 q^{47} -1.66911e9 q^{49} -3.33645e8 q^{52} -6.44889e8 q^{53} -1.75490e9 q^{56} -2.36969e9 q^{58} -9.25569e8 q^{59} -1.08986e10 q^{61} +5.82925e9 q^{62} +8.36248e9 q^{64} -3.79567e9 q^{67} +3.32206e9 q^{68} +2.29669e10 q^{71} -9.88082e9 q^{73} +9.90558e9 q^{74} +3.38478e8 q^{76} +8.23004e9 q^{77} -2.07689e10 q^{79} -6.50567e9 q^{82} +3.20486e9 q^{83} +5.98352e10 q^{86} -4.68600e10 q^{88} -6.31763e10 q^{89} +6.56668e9 q^{91} +2.89526e10 q^{92} +5.51980e10 q^{94} -1.26494e11 q^{97} -5.67499e10 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 34.0000 0.751301 0.375650 0.926761i \(-0.377419\pi\)
0.375650 + 0.926761i \(0.377419\pi\)
\(3\) 0 0
\(4\) −892.000 −0.435547
\(5\) 0 0
\(6\) 0 0
\(7\) 17556.0 0.394808 0.197404 0.980322i \(-0.436749\pi\)
0.197404 + 0.980322i \(0.436749\pi\)
\(8\) −99960.0 −1.07853
\(9\) 0 0
\(10\) 0 0
\(11\) 468788. 0.877641 0.438820 0.898575i \(-0.355396\pi\)
0.438820 + 0.898575i \(0.355396\pi\)
\(12\) 0 0
\(13\) 374042. 0.279404 0.139702 0.990194i \(-0.455386\pi\)
0.139702 + 0.990194i \(0.455386\pi\)
\(14\) 596904. 0.296620
\(15\) 0 0
\(16\) −1.57182e6 −0.374752
\(17\) −3.72429e6 −0.636171 −0.318086 0.948062i \(-0.603040\pi\)
−0.318086 + 0.948062i \(0.603040\pi\)
\(18\) 0 0
\(19\) −379460. −0.0351578 −0.0175789 0.999845i \(-0.505596\pi\)
−0.0175789 + 0.999845i \(0.505596\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.59388e7 0.659372
\(23\) −3.24581e7 −1.05153 −0.525763 0.850631i \(-0.676220\pi\)
−0.525763 + 0.850631i \(0.676220\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.27174e7 0.209916
\(27\) 0 0
\(28\) −1.56600e7 −0.171958
\(29\) −6.96967e7 −0.630991 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(30\) 0 0
\(31\) 1.71449e8 1.07558 0.537792 0.843077i \(-0.319258\pi\)
0.537792 + 0.843077i \(0.319258\pi\)
\(32\) 1.51276e8 0.796976
\(33\) 0 0
\(34\) −1.26626e8 −0.477956
\(35\) 0 0
\(36\) 0 0
\(37\) 2.91341e8 0.690703 0.345352 0.938473i \(-0.387760\pi\)
0.345352 + 0.938473i \(0.387760\pi\)
\(38\) −1.29016e7 −0.0264141
\(39\) 0 0
\(40\) 0 0
\(41\) −1.91343e8 −0.257930 −0.128965 0.991649i \(-0.541165\pi\)
−0.128965 + 0.991649i \(0.541165\pi\)
\(42\) 0 0
\(43\) 1.75986e9 1.82558 0.912790 0.408429i \(-0.133923\pi\)
0.912790 + 0.408429i \(0.133923\pi\)
\(44\) −4.18159e8 −0.382254
\(45\) 0 0
\(46\) −1.10358e9 −0.790012
\(47\) 1.62347e9 1.03254 0.516269 0.856426i \(-0.327320\pi\)
0.516269 + 0.856426i \(0.327320\pi\)
\(48\) 0 0
\(49\) −1.66911e9 −0.844126
\(50\) 0 0
\(51\) 0 0
\(52\) −3.33645e8 −0.121693
\(53\) −6.44889e8 −0.211820 −0.105910 0.994376i \(-0.533776\pi\)
−0.105910 + 0.994376i \(0.533776\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.75490e9 −0.425812
\(57\) 0 0
\(58\) −2.36969e9 −0.474064
\(59\) −9.25569e8 −0.168548 −0.0842739 0.996443i \(-0.526857\pi\)
−0.0842739 + 0.996443i \(0.526857\pi\)
\(60\) 0 0
\(61\) −1.08986e10 −1.65218 −0.826088 0.563541i \(-0.809439\pi\)
−0.826088 + 0.563541i \(0.809439\pi\)
\(62\) 5.82925e9 0.808088
\(63\) 0 0
\(64\) 8.36248e9 0.973521
\(65\) 0 0
\(66\) 0 0
\(67\) −3.79567e9 −0.343461 −0.171731 0.985144i \(-0.554936\pi\)
−0.171731 + 0.985144i \(0.554936\pi\)
\(68\) 3.32206e9 0.277082
\(69\) 0 0
\(70\) 0 0
\(71\) 2.29669e10 1.51071 0.755357 0.655313i \(-0.227463\pi\)
0.755357 + 0.655313i \(0.227463\pi\)
\(72\) 0 0
\(73\) −9.88082e9 −0.557850 −0.278925 0.960313i \(-0.589978\pi\)
−0.278925 + 0.960313i \(0.589978\pi\)
\(74\) 9.90558e9 0.518926
\(75\) 0 0
\(76\) 3.38478e8 0.0153129
\(77\) 8.23004e9 0.346500
\(78\) 0 0
\(79\) −2.07689e10 −0.759389 −0.379694 0.925112i \(-0.623971\pi\)
−0.379694 + 0.925112i \(0.623971\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.50567e9 −0.193783
\(83\) 3.20486e9 0.0893058 0.0446529 0.999003i \(-0.485782\pi\)
0.0446529 + 0.999003i \(0.485782\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.98352e10 1.37156
\(87\) 0 0
\(88\) −4.68600e10 −0.946560
\(89\) −6.31763e10 −1.19925 −0.599624 0.800282i \(-0.704683\pi\)
−0.599624 + 0.800282i \(0.704683\pi\)
\(90\) 0 0
\(91\) 6.56668e9 0.110311
\(92\) 2.89526e10 0.457989
\(93\) 0 0
\(94\) 5.51980e10 0.775747
\(95\) 0 0
\(96\) 0 0
\(97\) −1.26494e11 −1.49564 −0.747820 0.663902i \(-0.768899\pi\)
−0.747820 + 0.663902i \(0.768899\pi\)
\(98\) −5.67499e10 −0.634193
\(99\) 0 0
\(100\) 0 0
\(101\) −3.28905e10 −0.311389 −0.155694 0.987805i \(-0.549762\pi\)
−0.155694 + 0.987805i \(0.549762\pi\)
\(102\) 0 0
\(103\) −7.62224e10 −0.647855 −0.323928 0.946082i \(-0.605003\pi\)
−0.323928 + 0.946082i \(0.605003\pi\)
\(104\) −3.73892e10 −0.301344
\(105\) 0 0
\(106\) −2.19262e10 −0.159141
\(107\) 5.58566e10 0.385002 0.192501 0.981297i \(-0.438340\pi\)
0.192501 + 0.981297i \(0.438340\pi\)
\(108\) 0 0
\(109\) −1.66137e11 −1.03424 −0.517119 0.855914i \(-0.672995\pi\)
−0.517119 + 0.855914i \(0.672995\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.75949e10 −0.147955
\(113\) −1.79452e11 −0.916254 −0.458127 0.888887i \(-0.651480\pi\)
−0.458127 + 0.888887i \(0.651480\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.21695e10 0.274826
\(117\) 0 0
\(118\) −3.14694e10 −0.126630
\(119\) −6.53836e10 −0.251166
\(120\) 0 0
\(121\) −6.55495e10 −0.229747
\(122\) −3.70552e11 −1.24128
\(123\) 0 0
\(124\) −1.52932e11 −0.468468
\(125\) 0 0
\(126\) 0 0
\(127\) 5.37949e10 0.144484 0.0722421 0.997387i \(-0.476985\pi\)
0.0722421 + 0.997387i \(0.476985\pi\)
\(128\) −2.54890e10 −0.0655689
\(129\) 0 0
\(130\) 0 0
\(131\) −5.31507e11 −1.20370 −0.601848 0.798610i \(-0.705569\pi\)
−0.601848 + 0.798610i \(0.705569\pi\)
\(132\) 0 0
\(133\) −6.66180e9 −0.0138806
\(134\) −1.29053e11 −0.258043
\(135\) 0 0
\(136\) 3.72280e11 0.686128
\(137\) 3.92828e11 0.695407 0.347703 0.937605i \(-0.386962\pi\)
0.347703 + 0.937605i \(0.386962\pi\)
\(138\) 0 0
\(139\) −1.04604e12 −1.70988 −0.854940 0.518727i \(-0.826406\pi\)
−0.854940 + 0.518727i \(0.826406\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.80876e11 1.13500
\(143\) 1.75346e11 0.245216
\(144\) 0 0
\(145\) 0 0
\(146\) −3.35948e11 −0.419113
\(147\) 0 0
\(148\) −2.59876e11 −0.300834
\(149\) −4.45289e11 −0.496727 −0.248363 0.968667i \(-0.579893\pi\)
−0.248363 + 0.968667i \(0.579893\pi\)
\(150\) 0 0
\(151\) 2.66615e11 0.276383 0.138191 0.990406i \(-0.455871\pi\)
0.138191 + 0.990406i \(0.455871\pi\)
\(152\) 3.79308e10 0.0379186
\(153\) 0 0
\(154\) 2.79821e11 0.260326
\(155\) 0 0
\(156\) 0 0
\(157\) 1.61959e12 1.35505 0.677526 0.735498i \(-0.263052\pi\)
0.677526 + 0.735498i \(0.263052\pi\)
\(158\) −7.06142e11 −0.570530
\(159\) 0 0
\(160\) 0 0
\(161\) −5.69834e11 −0.415151
\(162\) 0 0
\(163\) −1.55712e12 −1.05996 −0.529980 0.848010i \(-0.677801\pi\)
−0.529980 + 0.848010i \(0.677801\pi\)
\(164\) 1.70678e11 0.112341
\(165\) 0 0
\(166\) 1.08965e11 0.0670956
\(167\) −8.10194e11 −0.482668 −0.241334 0.970442i \(-0.577585\pi\)
−0.241334 + 0.970442i \(0.577585\pi\)
\(168\) 0 0
\(169\) −1.65225e12 −0.921934
\(170\) 0 0
\(171\) 0 0
\(172\) −1.56979e12 −0.795126
\(173\) −3.62920e12 −1.78056 −0.890281 0.455412i \(-0.849492\pi\)
−0.890281 + 0.455412i \(0.849492\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −7.36852e11 −0.328898
\(177\) 0 0
\(178\) −2.14799e12 −0.900997
\(179\) 3.78383e12 1.53900 0.769502 0.638644i \(-0.220504\pi\)
0.769502 + 0.638644i \(0.220504\pi\)
\(180\) 0 0
\(181\) −2.63828e10 −0.0100946 −0.00504729 0.999987i \(-0.501607\pi\)
−0.00504729 + 0.999987i \(0.501607\pi\)
\(182\) 2.23267e11 0.0828767
\(183\) 0 0
\(184\) 3.24451e12 1.13410
\(185\) 0 0
\(186\) 0 0
\(187\) −1.74590e12 −0.558330
\(188\) −1.44814e12 −0.449719
\(189\) 0 0
\(190\) 0 0
\(191\) −3.18849e12 −0.907615 −0.453807 0.891100i \(-0.649935\pi\)
−0.453807 + 0.891100i \(0.649935\pi\)
\(192\) 0 0
\(193\) 2.17255e12 0.583990 0.291995 0.956420i \(-0.405681\pi\)
0.291995 + 0.956420i \(0.405681\pi\)
\(194\) −4.30081e12 −1.12368
\(195\) 0 0
\(196\) 1.48885e12 0.367657
\(197\) 3.09042e12 0.742085 0.371043 0.928616i \(-0.379000\pi\)
0.371043 + 0.928616i \(0.379000\pi\)
\(198\) 0 0
\(199\) 4.50627e12 1.02359 0.511794 0.859108i \(-0.328981\pi\)
0.511794 + 0.859108i \(0.328981\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1.11828e12 −0.233947
\(203\) −1.22360e12 −0.249121
\(204\) 0 0
\(205\) 0 0
\(206\) −2.59156e12 −0.486734
\(207\) 0 0
\(208\) −5.87928e11 −0.104707
\(209\) −1.77886e11 −0.0308559
\(210\) 0 0
\(211\) −1.04739e13 −1.72408 −0.862038 0.506843i \(-0.830812\pi\)
−0.862038 + 0.506843i \(0.830812\pi\)
\(212\) 5.75241e11 0.0922576
\(213\) 0 0
\(214\) 1.89912e12 0.289253
\(215\) 0 0
\(216\) 0 0
\(217\) 3.00995e12 0.424650
\(218\) −5.64865e12 −0.777023
\(219\) 0 0
\(220\) 0 0
\(221\) −1.39304e12 −0.177748
\(222\) 0 0
\(223\) 3.63798e12 0.441757 0.220879 0.975301i \(-0.429108\pi\)
0.220879 + 0.975301i \(0.429108\pi\)
\(224\) 2.65580e12 0.314653
\(225\) 0 0
\(226\) −6.10136e12 −0.688383
\(227\) 1.65864e13 1.82646 0.913229 0.407446i \(-0.133581\pi\)
0.913229 + 0.407446i \(0.133581\pi\)
\(228\) 0 0
\(229\) −1.02798e13 −1.07867 −0.539336 0.842091i \(-0.681325\pi\)
−0.539336 + 0.842091i \(0.681325\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.96688e12 0.680541
\(233\) 2.92760e12 0.279289 0.139645 0.990202i \(-0.455404\pi\)
0.139645 + 0.990202i \(0.455404\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 8.25608e11 0.0734104
\(237\) 0 0
\(238\) −2.22304e12 −0.188701
\(239\) −1.33960e13 −1.11119 −0.555593 0.831454i \(-0.687509\pi\)
−0.555593 + 0.831454i \(0.687509\pi\)
\(240\) 0 0
\(241\) −1.67282e13 −1.32543 −0.662713 0.748874i \(-0.730595\pi\)
−0.662713 + 0.748874i \(0.730595\pi\)
\(242\) −2.22868e12 −0.172609
\(243\) 0 0
\(244\) 9.72154e12 0.719600
\(245\) 0 0
\(246\) 0 0
\(247\) −1.41934e11 −0.00982320
\(248\) −1.71380e13 −1.16005
\(249\) 0 0
\(250\) 0 0
\(251\) −1.88339e13 −1.19326 −0.596629 0.802517i \(-0.703494\pi\)
−0.596629 + 0.802517i \(0.703494\pi\)
\(252\) 0 0
\(253\) −1.52160e13 −0.922862
\(254\) 1.82903e12 0.108551
\(255\) 0 0
\(256\) −1.79930e13 −1.02278
\(257\) −1.20156e12 −0.0668520 −0.0334260 0.999441i \(-0.510642\pi\)
−0.0334260 + 0.999441i \(0.510642\pi\)
\(258\) 0 0
\(259\) 5.11477e12 0.272695
\(260\) 0 0
\(261\) 0 0
\(262\) −1.80712e13 −0.904339
\(263\) 3.79660e12 0.186054 0.0930268 0.995664i \(-0.470346\pi\)
0.0930268 + 0.995664i \(0.470346\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.26501e11 −0.0104285
\(267\) 0 0
\(268\) 3.38574e12 0.149593
\(269\) 2.57422e13 1.11432 0.557159 0.830406i \(-0.311892\pi\)
0.557159 + 0.830406i \(0.311892\pi\)
\(270\) 0 0
\(271\) 1.60144e13 0.665550 0.332775 0.943006i \(-0.392015\pi\)
0.332775 + 0.943006i \(0.392015\pi\)
\(272\) 5.85392e12 0.238406
\(273\) 0 0
\(274\) 1.33561e13 0.522460
\(275\) 0 0
\(276\) 0 0
\(277\) 2.31620e13 0.853369 0.426684 0.904401i \(-0.359681\pi\)
0.426684 + 0.904401i \(0.359681\pi\)
\(278\) −3.55653e13 −1.28463
\(279\) 0 0
\(280\) 0 0
\(281\) −3.71797e13 −1.26596 −0.632981 0.774167i \(-0.718169\pi\)
−0.632981 + 0.774167i \(0.718169\pi\)
\(282\) 0 0
\(283\) −9.57806e12 −0.313655 −0.156827 0.987626i \(-0.550127\pi\)
−0.156827 + 0.987626i \(0.550127\pi\)
\(284\) −2.04865e13 −0.657987
\(285\) 0 0
\(286\) 5.96178e12 0.184231
\(287\) −3.35922e12 −0.101833
\(288\) 0 0
\(289\) −2.04016e13 −0.595286
\(290\) 0 0
\(291\) 0 0
\(292\) 8.81369e12 0.242970
\(293\) −6.51052e13 −1.76134 −0.880672 0.473727i \(-0.842908\pi\)
−0.880672 + 0.473727i \(0.842908\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.91224e13 −0.744943
\(297\) 0 0
\(298\) −1.51398e13 −0.373191
\(299\) −1.21407e13 −0.293800
\(300\) 0 0
\(301\) 3.08961e13 0.720754
\(302\) 9.06490e12 0.207647
\(303\) 0 0
\(304\) 5.96444e11 0.0131754
\(305\) 0 0
\(306\) 0 0
\(307\) 3.37872e13 0.707116 0.353558 0.935413i \(-0.384972\pi\)
0.353558 + 0.935413i \(0.384972\pi\)
\(308\) −7.34120e12 −0.150917
\(309\) 0 0
\(310\) 0 0
\(311\) −3.13268e13 −0.610568 −0.305284 0.952261i \(-0.598751\pi\)
−0.305284 + 0.952261i \(0.598751\pi\)
\(312\) 0 0
\(313\) −6.39027e13 −1.20234 −0.601168 0.799123i \(-0.705298\pi\)
−0.601168 + 0.799123i \(0.705298\pi\)
\(314\) 5.50660e13 1.01805
\(315\) 0 0
\(316\) 1.85258e13 0.330749
\(317\) 1.15517e13 0.202685 0.101342 0.994852i \(-0.467686\pi\)
0.101342 + 0.994852i \(0.467686\pi\)
\(318\) 0 0
\(319\) −3.26730e13 −0.553783
\(320\) 0 0
\(321\) 0 0
\(322\) −1.93744e13 −0.311903
\(323\) 1.41322e12 0.0223664
\(324\) 0 0
\(325\) 0 0
\(326\) −5.29420e13 −0.796349
\(327\) 0 0
\(328\) 1.91267e13 0.278185
\(329\) 2.85016e13 0.407655
\(330\) 0 0
\(331\) 1.17396e14 1.62406 0.812028 0.583618i \(-0.198363\pi\)
0.812028 + 0.583618i \(0.198363\pi\)
\(332\) −2.85874e12 −0.0388969
\(333\) 0 0
\(334\) −2.75466e13 −0.362629
\(335\) 0 0
\(336\) 0 0
\(337\) −3.67400e12 −0.0460441 −0.0230221 0.999735i \(-0.507329\pi\)
−0.0230221 + 0.999735i \(0.507329\pi\)
\(338\) −5.61766e13 −0.692650
\(339\) 0 0
\(340\) 0 0
\(341\) 8.03731e13 0.943977
\(342\) 0 0
\(343\) −6.40169e13 −0.728077
\(344\) −1.75915e14 −1.96894
\(345\) 0 0
\(346\) −1.23393e14 −1.33774
\(347\) −5.92966e13 −0.632729 −0.316364 0.948638i \(-0.602462\pi\)
−0.316364 + 0.948638i \(0.602462\pi\)
\(348\) 0 0
\(349\) −9.69156e13 −1.00197 −0.500984 0.865457i \(-0.667028\pi\)
−0.500984 + 0.865457i \(0.667028\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 7.09164e13 0.699459
\(353\) 3.93594e13 0.382197 0.191099 0.981571i \(-0.438795\pi\)
0.191099 + 0.981571i \(0.438795\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.63533e13 0.522329
\(357\) 0 0
\(358\) 1.28650e14 1.15626
\(359\) 5.38495e13 0.476609 0.238305 0.971190i \(-0.423408\pi\)
0.238305 + 0.971190i \(0.423408\pi\)
\(360\) 0 0
\(361\) −1.16346e14 −0.998764
\(362\) −8.97014e11 −0.00758406
\(363\) 0 0
\(364\) −5.85748e12 −0.0480456
\(365\) 0 0
\(366\) 0 0
\(367\) −4.55671e13 −0.357263 −0.178632 0.983916i \(-0.557167\pi\)
−0.178632 + 0.983916i \(0.557167\pi\)
\(368\) 5.10184e13 0.394061
\(369\) 0 0
\(370\) 0 0
\(371\) −1.13217e13 −0.0836284
\(372\) 0 0
\(373\) −1.69341e14 −1.21441 −0.607203 0.794547i \(-0.707709\pi\)
−0.607203 + 0.794547i \(0.707709\pi\)
\(374\) −5.93606e13 −0.419474
\(375\) 0 0
\(376\) −1.62282e14 −1.11362
\(377\) −2.60695e13 −0.176301
\(378\) 0 0
\(379\) −5.93802e13 −0.390055 −0.195028 0.980798i \(-0.562480\pi\)
−0.195028 + 0.980798i \(0.562480\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.08409e14 −0.681892
\(383\) 1.21003e14 0.750242 0.375121 0.926976i \(-0.377601\pi\)
0.375121 + 0.926976i \(0.377601\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.38668e13 0.438752
\(387\) 0 0
\(388\) 1.12833e14 0.651421
\(389\) −1.53362e14 −0.872960 −0.436480 0.899714i \(-0.643775\pi\)
−0.436480 + 0.899714i \(0.643775\pi\)
\(390\) 0 0
\(391\) 1.20883e14 0.668950
\(392\) 1.66845e14 0.910414
\(393\) 0 0
\(394\) 1.05074e14 0.557529
\(395\) 0 0
\(396\) 0 0
\(397\) −1.33061e14 −0.677180 −0.338590 0.940934i \(-0.609950\pi\)
−0.338590 + 0.940934i \(0.609950\pi\)
\(398\) 1.53213e14 0.769023
\(399\) 0 0
\(400\) 0 0
\(401\) 1.25080e14 0.602411 0.301205 0.953559i \(-0.402611\pi\)
0.301205 + 0.953559i \(0.402611\pi\)
\(402\) 0 0
\(403\) 6.41290e13 0.300522
\(404\) 2.93383e13 0.135624
\(405\) 0 0
\(406\) −4.16022e13 −0.187165
\(407\) 1.36577e14 0.606189
\(408\) 0 0
\(409\) 1.91902e14 0.829087 0.414544 0.910029i \(-0.363941\pi\)
0.414544 + 0.910029i \(0.363941\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.79904e13 0.282171
\(413\) −1.62493e13 −0.0665440
\(414\) 0 0
\(415\) 0 0
\(416\) 5.65836e13 0.222678
\(417\) 0 0
\(418\) −6.04813e12 −0.0231821
\(419\) 3.39088e14 1.28273 0.641366 0.767235i \(-0.278368\pi\)
0.641366 + 0.767235i \(0.278368\pi\)
\(420\) 0 0
\(421\) 4.09835e14 1.51028 0.755139 0.655564i \(-0.227569\pi\)
0.755139 + 0.655564i \(0.227569\pi\)
\(422\) −3.56114e14 −1.29530
\(423\) 0 0
\(424\) 6.44631e13 0.228454
\(425\) 0 0
\(426\) 0 0
\(427\) −1.91336e14 −0.652293
\(428\) −4.98241e13 −0.167687
\(429\) 0 0
\(430\) 0 0
\(431\) 5.58968e14 1.81035 0.905173 0.425043i \(-0.139741\pi\)
0.905173 + 0.425043i \(0.139741\pi\)
\(432\) 0 0
\(433\) 3.26000e14 1.02928 0.514640 0.857406i \(-0.327925\pi\)
0.514640 + 0.857406i \(0.327925\pi\)
\(434\) 1.02338e14 0.319040
\(435\) 0 0
\(436\) 1.48194e14 0.450459
\(437\) 1.23165e13 0.0369693
\(438\) 0 0
\(439\) 8.91336e13 0.260908 0.130454 0.991454i \(-0.458357\pi\)
0.130454 + 0.991454i \(0.458357\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.73633e13 −0.133543
\(443\) 5.55014e14 1.54555 0.772776 0.634679i \(-0.218868\pi\)
0.772776 + 0.634679i \(0.218868\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.23691e14 0.331893
\(447\) 0 0
\(448\) 1.46812e14 0.384354
\(449\) 2.41405e14 0.624296 0.312148 0.950033i \(-0.398952\pi\)
0.312148 + 0.950033i \(0.398952\pi\)
\(450\) 0 0
\(451\) −8.96994e13 −0.226370
\(452\) 1.60071e14 0.399072
\(453\) 0 0
\(454\) 5.63937e14 1.37222
\(455\) 0 0
\(456\) 0 0
\(457\) −3.60954e13 −0.0847057 −0.0423529 0.999103i \(-0.513485\pi\)
−0.0423529 + 0.999103i \(0.513485\pi\)
\(458\) −3.49513e14 −0.810407
\(459\) 0 0
\(460\) 0 0
\(461\) −2.58827e14 −0.578968 −0.289484 0.957183i \(-0.593484\pi\)
−0.289484 + 0.957183i \(0.593484\pi\)
\(462\) 0 0
\(463\) 5.64823e14 1.23372 0.616860 0.787073i \(-0.288404\pi\)
0.616860 + 0.787073i \(0.288404\pi\)
\(464\) 1.09551e14 0.236465
\(465\) 0 0
\(466\) 9.95384e13 0.209830
\(467\) −3.88270e14 −0.808892 −0.404446 0.914562i \(-0.632536\pi\)
−0.404446 + 0.914562i \(0.632536\pi\)
\(468\) 0 0
\(469\) −6.66369e13 −0.135601
\(470\) 0 0
\(471\) 0 0
\(472\) 9.25199e13 0.181783
\(473\) 8.25000e14 1.60220
\(474\) 0 0
\(475\) 0 0
\(476\) 5.83221e13 0.109394
\(477\) 0 0
\(478\) −4.55464e14 −0.834836
\(479\) −2.64819e14 −0.479848 −0.239924 0.970792i \(-0.577122\pi\)
−0.239924 + 0.970792i \(0.577122\pi\)
\(480\) 0 0
\(481\) 1.08974e14 0.192985
\(482\) −5.68759e14 −0.995794
\(483\) 0 0
\(484\) 5.84701e13 0.100066
\(485\) 0 0
\(486\) 0 0
\(487\) 9.60424e14 1.58874 0.794372 0.607432i \(-0.207800\pi\)
0.794372 + 0.607432i \(0.207800\pi\)
\(488\) 1.08942e15 1.78192
\(489\) 0 0
\(490\) 0 0
\(491\) −2.37920e14 −0.376255 −0.188127 0.982145i \(-0.560242\pi\)
−0.188127 + 0.982145i \(0.560242\pi\)
\(492\) 0 0
\(493\) 2.59570e14 0.401418
\(494\) −4.82576e12 −0.00738018
\(495\) 0 0
\(496\) −2.69487e14 −0.403078
\(497\) 4.03208e14 0.596443
\(498\) 0 0
\(499\) −9.26509e14 −1.34059 −0.670296 0.742094i \(-0.733833\pi\)
−0.670296 + 0.742094i \(0.733833\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −6.40352e14 −0.896496
\(503\) −2.21172e14 −0.306272 −0.153136 0.988205i \(-0.548937\pi\)
−0.153136 + 0.988205i \(0.548937\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.17343e14 −0.693347
\(507\) 0 0
\(508\) −4.79850e13 −0.0629297
\(509\) 2.22667e14 0.288873 0.144437 0.989514i \(-0.453863\pi\)
0.144437 + 0.989514i \(0.453863\pi\)
\(510\) 0 0
\(511\) −1.73468e14 −0.220244
\(512\) −5.59560e14 −0.702849
\(513\) 0 0
\(514\) −4.08531e13 −0.0502260
\(515\) 0 0
\(516\) 0 0
\(517\) 7.61063e14 0.906197
\(518\) 1.73902e14 0.204876
\(519\) 0 0
\(520\) 0 0
\(521\) 1.00449e15 1.14641 0.573203 0.819413i \(-0.305701\pi\)
0.573203 + 0.819413i \(0.305701\pi\)
\(522\) 0 0
\(523\) 1.01954e15 1.13932 0.569658 0.821882i \(-0.307076\pi\)
0.569658 + 0.821882i \(0.307076\pi\)
\(524\) 4.74104e14 0.524266
\(525\) 0 0
\(526\) 1.29084e14 0.139782
\(527\) −6.38524e14 −0.684256
\(528\) 0 0
\(529\) 1.00718e14 0.105706
\(530\) 0 0
\(531\) 0 0
\(532\) 5.94233e12 0.00604564
\(533\) −7.15704e13 −0.0720666
\(534\) 0 0
\(535\) 0 0
\(536\) 3.79416e14 0.370432
\(537\) 0 0
\(538\) 8.75236e14 0.837188
\(539\) −7.82460e14 −0.740840
\(540\) 0 0
\(541\) 1.57672e15 1.46275 0.731376 0.681975i \(-0.238879\pi\)
0.731376 + 0.681975i \(0.238879\pi\)
\(542\) 5.44491e14 0.500028
\(543\) 0 0
\(544\) −5.63395e14 −0.507013
\(545\) 0 0
\(546\) 0 0
\(547\) 2.73964e14 0.239201 0.119600 0.992822i \(-0.461839\pi\)
0.119600 + 0.992822i \(0.461839\pi\)
\(548\) −3.50402e14 −0.302882
\(549\) 0 0
\(550\) 0 0
\(551\) 2.64471e13 0.0221842
\(552\) 0 0
\(553\) −3.64619e14 −0.299813
\(554\) 7.87507e14 0.641137
\(555\) 0 0
\(556\) 9.33065e14 0.744733
\(557\) 9.10005e13 0.0719184 0.0359592 0.999353i \(-0.488551\pi\)
0.0359592 + 0.999353i \(0.488551\pi\)
\(558\) 0 0
\(559\) 6.58261e14 0.510074
\(560\) 0 0
\(561\) 0 0
\(562\) −1.26411e15 −0.951119
\(563\) 1.02926e14 0.0766886 0.0383443 0.999265i \(-0.487792\pi\)
0.0383443 + 0.999265i \(0.487792\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −3.25654e14 −0.235649
\(567\) 0 0
\(568\) −2.29578e15 −1.62935
\(569\) −2.33956e15 −1.64443 −0.822216 0.569175i \(-0.807263\pi\)
−0.822216 + 0.569175i \(0.807263\pi\)
\(570\) 0 0
\(571\) −2.26931e14 −0.156457 −0.0782286 0.996935i \(-0.524926\pi\)
−0.0782286 + 0.996935i \(0.524926\pi\)
\(572\) −1.56409e14 −0.106803
\(573\) 0 0
\(574\) −1.14214e14 −0.0765072
\(575\) 0 0
\(576\) 0 0
\(577\) −2.42151e15 −1.57623 −0.788115 0.615528i \(-0.788943\pi\)
−0.788115 + 0.615528i \(0.788943\pi\)
\(578\) −6.93654e14 −0.447239
\(579\) 0 0
\(580\) 0 0
\(581\) 5.62646e13 0.0352587
\(582\) 0 0
\(583\) −3.02316e14 −0.185902
\(584\) 9.87687e14 0.601656
\(585\) 0 0
\(586\) −2.21358e15 −1.32330
\(587\) 1.60425e15 0.950087 0.475044 0.879962i \(-0.342432\pi\)
0.475044 + 0.879962i \(0.342432\pi\)
\(588\) 0 0
\(589\) −6.50579e13 −0.0378152
\(590\) 0 0
\(591\) 0 0
\(592\) −4.57936e14 −0.258842
\(593\) −2.02053e15 −1.13153 −0.565763 0.824568i \(-0.691418\pi\)
−0.565763 + 0.824568i \(0.691418\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.97198e14 0.216348
\(597\) 0 0
\(598\) −4.12783e14 −0.220732
\(599\) −1.25788e15 −0.666486 −0.333243 0.942841i \(-0.608143\pi\)
−0.333243 + 0.942841i \(0.608143\pi\)
\(600\) 0 0
\(601\) −8.50070e14 −0.442227 −0.221113 0.975248i \(-0.570969\pi\)
−0.221113 + 0.975248i \(0.570969\pi\)
\(602\) 1.05047e15 0.541503
\(603\) 0 0
\(604\) −2.37820e14 −0.120378
\(605\) 0 0
\(606\) 0 0
\(607\) 2.25771e15 1.11207 0.556033 0.831160i \(-0.312323\pi\)
0.556033 + 0.831160i \(0.312323\pi\)
\(608\) −5.74032e13 −0.0280199
\(609\) 0 0
\(610\) 0 0
\(611\) 6.07246e14 0.288495
\(612\) 0 0
\(613\) −3.37583e15 −1.57525 −0.787623 0.616157i \(-0.788689\pi\)
−0.787623 + 0.616157i \(0.788689\pi\)
\(614\) 1.14876e15 0.531257
\(615\) 0 0
\(616\) −8.22675e14 −0.373710
\(617\) −1.76114e15 −0.792915 −0.396457 0.918053i \(-0.629761\pi\)
−0.396457 + 0.918053i \(0.629761\pi\)
\(618\) 0 0
\(619\) −5.36923e14 −0.237472 −0.118736 0.992926i \(-0.537884\pi\)
−0.118736 + 0.992926i \(0.537884\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −1.06511e15 −0.458720
\(623\) −1.10912e15 −0.473473
\(624\) 0 0
\(625\) 0 0
\(626\) −2.17269e15 −0.903316
\(627\) 0 0
\(628\) −1.44467e15 −0.590189
\(629\) −1.08504e15 −0.439405
\(630\) 0 0
\(631\) −1.36152e15 −0.541830 −0.270915 0.962603i \(-0.587326\pi\)
−0.270915 + 0.962603i \(0.587326\pi\)
\(632\) 2.07606e15 0.819022
\(633\) 0 0
\(634\) 3.92758e14 0.152277
\(635\) 0 0
\(636\) 0 0
\(637\) −6.24319e14 −0.235852
\(638\) −1.11088e15 −0.416058
\(639\) 0 0
\(640\) 0 0
\(641\) 2.14896e15 0.784348 0.392174 0.919891i \(-0.371723\pi\)
0.392174 + 0.919891i \(0.371723\pi\)
\(642\) 0 0
\(643\) 5.57356e14 0.199973 0.0999867 0.994989i \(-0.468120\pi\)
0.0999867 + 0.994989i \(0.468120\pi\)
\(644\) 5.08292e14 0.180818
\(645\) 0 0
\(646\) 4.80494e13 0.0168039
\(647\) 2.41894e14 0.0838787 0.0419394 0.999120i \(-0.486646\pi\)
0.0419394 + 0.999120i \(0.486646\pi\)
\(648\) 0 0
\(649\) −4.33896e14 −0.147924
\(650\) 0 0
\(651\) 0 0
\(652\) 1.38895e15 0.461663
\(653\) −3.13959e15 −1.03478 −0.517392 0.855748i \(-0.673097\pi\)
−0.517392 + 0.855748i \(0.673097\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.00758e14 0.0966598
\(657\) 0 0
\(658\) 9.69056e14 0.306271
\(659\) −1.06199e15 −0.332851 −0.166425 0.986054i \(-0.553223\pi\)
−0.166425 + 0.986054i \(0.553223\pi\)
\(660\) 0 0
\(661\) −8.87576e14 −0.273588 −0.136794 0.990599i \(-0.543680\pi\)
−0.136794 + 0.990599i \(0.543680\pi\)
\(662\) 3.99148e15 1.22016
\(663\) 0 0
\(664\) −3.20358e14 −0.0963188
\(665\) 0 0
\(666\) 0 0
\(667\) 2.26222e15 0.663503
\(668\) 7.22693e14 0.210224
\(669\) 0 0
\(670\) 0 0
\(671\) −5.10913e15 −1.45002
\(672\) 0 0
\(673\) 2.92399e15 0.816380 0.408190 0.912897i \(-0.366160\pi\)
0.408190 + 0.912897i \(0.366160\pi\)
\(674\) −1.24916e14 −0.0345930
\(675\) 0 0
\(676\) 1.47381e15 0.401545
\(677\) −3.20144e15 −0.865181 −0.432591 0.901590i \(-0.642400\pi\)
−0.432591 + 0.901590i \(0.642400\pi\)
\(678\) 0 0
\(679\) −2.22074e15 −0.590491
\(680\) 0 0
\(681\) 0 0
\(682\) 2.73268e15 0.709211
\(683\) 2.37436e15 0.611269 0.305634 0.952149i \(-0.401131\pi\)
0.305634 + 0.952149i \(0.401131\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.17657e15 −0.547005
\(687\) 0 0
\(688\) −2.76619e15 −0.684140
\(689\) −2.41215e14 −0.0591833
\(690\) 0 0
\(691\) −4.46382e14 −0.107790 −0.0538949 0.998547i \(-0.517164\pi\)
−0.0538949 + 0.998547i \(0.517164\pi\)
\(692\) 3.23724e15 0.775518
\(693\) 0 0
\(694\) −2.01608e15 −0.475370
\(695\) 0 0
\(696\) 0 0
\(697\) 7.12617e14 0.164088
\(698\) −3.29513e15 −0.752780
\(699\) 0 0
\(700\) 0 0
\(701\) −5.79930e15 −1.29398 −0.646988 0.762500i \(-0.723972\pi\)
−0.646988 + 0.762500i \(0.723972\pi\)
\(702\) 0 0
\(703\) −1.10552e14 −0.0242836
\(704\) 3.92023e15 0.854402
\(705\) 0 0
\(706\) 1.33822e15 0.287145
\(707\) −5.77426e14 −0.122939
\(708\) 0 0
\(709\) 3.71150e15 0.778027 0.389014 0.921232i \(-0.372816\pi\)
0.389014 + 0.921232i \(0.372816\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.31511e15 1.29342
\(713\) −5.56490e15 −1.13101
\(714\) 0 0
\(715\) 0 0
\(716\) −3.37518e15 −0.670309
\(717\) 0 0
\(718\) 1.83088e15 0.358077
\(719\) −2.54074e14 −0.0493118 −0.0246559 0.999696i \(-0.507849\pi\)
−0.0246559 + 0.999696i \(0.507849\pi\)
\(720\) 0 0
\(721\) −1.33816e15 −0.255779
\(722\) −3.95577e15 −0.750372
\(723\) 0 0
\(724\) 2.35334e13 0.00439666
\(725\) 0 0
\(726\) 0 0
\(727\) −3.40726e15 −0.622251 −0.311126 0.950369i \(-0.600706\pi\)
−0.311126 + 0.950369i \(0.600706\pi\)
\(728\) −6.56405e14 −0.118973
\(729\) 0 0
\(730\) 0 0
\(731\) −6.55421e15 −1.16138
\(732\) 0 0
\(733\) 3.16708e15 0.552825 0.276412 0.961039i \(-0.410854\pi\)
0.276412 + 0.961039i \(0.410854\pi\)
\(734\) −1.54928e15 −0.268412
\(735\) 0 0
\(736\) −4.91013e15 −0.838041
\(737\) −1.77937e15 −0.301435
\(738\) 0 0
\(739\) 3.82593e15 0.638547 0.319273 0.947663i \(-0.396561\pi\)
0.319273 + 0.947663i \(0.396561\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −3.84937e14 −0.0628301
\(743\) −2.20995e15 −0.358051 −0.179025 0.983844i \(-0.557294\pi\)
−0.179025 + 0.983844i \(0.557294\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5.75760e15 −0.912385
\(747\) 0 0
\(748\) 1.55734e15 0.243179
\(749\) 9.80618e14 0.152002
\(750\) 0 0
\(751\) 1.22606e15 0.187281 0.0936405 0.995606i \(-0.470150\pi\)
0.0936405 + 0.995606i \(0.470150\pi\)
\(752\) −2.55181e15 −0.386946
\(753\) 0 0
\(754\) −8.86363e14 −0.132455
\(755\) 0 0
\(756\) 0 0
\(757\) −6.70929e15 −0.980955 −0.490478 0.871454i \(-0.663178\pi\)
−0.490478 + 0.871454i \(0.663178\pi\)
\(758\) −2.01893e15 −0.293049
\(759\) 0 0
\(760\) 0 0
\(761\) 5.22698e15 0.742396 0.371198 0.928554i \(-0.378947\pi\)
0.371198 + 0.928554i \(0.378947\pi\)
\(762\) 0 0
\(763\) −2.91670e15 −0.408325
\(764\) 2.84413e15 0.395309
\(765\) 0 0
\(766\) 4.11409e15 0.563657
\(767\) −3.46202e14 −0.0470928
\(768\) 0 0
\(769\) 9.72252e15 1.30372 0.651859 0.758340i \(-0.273989\pi\)
0.651859 + 0.758340i \(0.273989\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.93792e15 −0.254355
\(773\) 9.82829e15 1.28083 0.640414 0.768030i \(-0.278763\pi\)
0.640414 + 0.768030i \(0.278763\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.26444e16 1.61309
\(777\) 0 0
\(778\) −5.21430e15 −0.655855
\(779\) 7.26071e13 0.00906824
\(780\) 0 0
\(781\) 1.07666e16 1.32586
\(782\) 4.11003e15 0.502583
\(783\) 0 0
\(784\) 2.62355e15 0.316338
\(785\) 0 0
\(786\) 0 0
\(787\) −2.54031e15 −0.299934 −0.149967 0.988691i \(-0.547917\pi\)
−0.149967 + 0.988691i \(0.547917\pi\)
\(788\) −2.75666e15 −0.323213
\(789\) 0 0
\(790\) 0 0
\(791\) −3.15045e15 −0.361745
\(792\) 0 0
\(793\) −4.07653e15 −0.461624
\(794\) −4.52408e15 −0.508766
\(795\) 0 0
\(796\) −4.01960e15 −0.445821
\(797\) −3.63405e15 −0.400285 −0.200143 0.979767i \(-0.564141\pi\)
−0.200143 + 0.979767i \(0.564141\pi\)
\(798\) 0 0
\(799\) −6.04627e15 −0.656871
\(800\) 0 0
\(801\) 0 0
\(802\) 4.25271e15 0.452592
\(803\) −4.63201e15 −0.489592
\(804\) 0 0
\(805\) 0 0
\(806\) 2.18039e15 0.225783
\(807\) 0 0
\(808\) 3.28774e15 0.335842
\(809\) 4.02489e15 0.408355 0.204177 0.978934i \(-0.434548\pi\)
0.204177 + 0.978934i \(0.434548\pi\)
\(810\) 0 0
\(811\) −1.12095e16 −1.12194 −0.560970 0.827836i \(-0.689572\pi\)
−0.560970 + 0.827836i \(0.689572\pi\)
\(812\) 1.09145e15 0.108504
\(813\) 0 0
\(814\) 4.64362e15 0.455430
\(815\) 0 0
\(816\) 0 0
\(817\) −6.67795e14 −0.0641833
\(818\) 6.52465e15 0.622894
\(819\) 0 0
\(820\) 0 0
\(821\) 2.11427e16 1.97821 0.989106 0.147204i \(-0.0470274\pi\)
0.989106 + 0.147204i \(0.0470274\pi\)
\(822\) 0 0
\(823\) 3.76589e15 0.347671 0.173836 0.984775i \(-0.444384\pi\)
0.173836 + 0.984775i \(0.444384\pi\)
\(824\) 7.61919e15 0.698730
\(825\) 0 0
\(826\) −5.52476e14 −0.0499946
\(827\) 1.79419e15 0.161282 0.0806412 0.996743i \(-0.474303\pi\)
0.0806412 + 0.996743i \(0.474303\pi\)
\(828\) 0 0
\(829\) −1.11615e16 −0.990084 −0.495042 0.868869i \(-0.664847\pi\)
−0.495042 + 0.868869i \(0.664847\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.12792e15 0.272005
\(833\) 6.21626e15 0.537009
\(834\) 0 0
\(835\) 0 0
\(836\) 1.58675e14 0.0134392
\(837\) 0 0
\(838\) 1.15290e16 0.963717
\(839\) −3.47514e15 −0.288590 −0.144295 0.989535i \(-0.546091\pi\)
−0.144295 + 0.989535i \(0.546091\pi\)
\(840\) 0 0
\(841\) −7.34288e15 −0.601850
\(842\) 1.39344e16 1.13467
\(843\) 0 0
\(844\) 9.34275e15 0.750916
\(845\) 0 0
\(846\) 0 0
\(847\) −1.15079e15 −0.0907060
\(848\) 1.01365e15 0.0793801
\(849\) 0 0
\(850\) 0 0
\(851\) −9.45636e15 −0.726292
\(852\) 0 0
\(853\) −1.64122e16 −1.24436 −0.622182 0.782873i \(-0.713754\pi\)
−0.622182 + 0.782873i \(0.713754\pi\)
\(854\) −6.50541e15 −0.490068
\(855\) 0 0
\(856\) −5.58342e15 −0.415236
\(857\) 4.91921e15 0.363497 0.181748 0.983345i \(-0.441824\pi\)
0.181748 + 0.983345i \(0.441824\pi\)
\(858\) 0 0
\(859\) −1.96890e16 −1.43635 −0.718176 0.695861i \(-0.755023\pi\)
−0.718176 + 0.695861i \(0.755023\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.90049e16 1.36012
\(863\) 2.44068e16 1.73560 0.867802 0.496910i \(-0.165532\pi\)
0.867802 + 0.496910i \(0.165532\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.10840e16 0.773299
\(867\) 0 0
\(868\) −2.68488e15 −0.184955
\(869\) −9.73620e15 −0.666470
\(870\) 0 0
\(871\) −1.41974e15 −0.0959642
\(872\) 1.66070e16 1.11545
\(873\) 0 0
\(874\) 4.18763e14 0.0277751
\(875\) 0 0
\(876\) 0 0
\(877\) −2.88744e16 −1.87938 −0.939691 0.342025i \(-0.888887\pi\)
−0.939691 + 0.342025i \(0.888887\pi\)
\(878\) 3.03054e15 0.196020
\(879\) 0 0
\(880\) 0 0
\(881\) 2.51928e15 0.159922 0.0799610 0.996798i \(-0.474520\pi\)
0.0799610 + 0.996798i \(0.474520\pi\)
\(882\) 0 0
\(883\) 2.29479e16 1.43866 0.719330 0.694668i \(-0.244449\pi\)
0.719330 + 0.694668i \(0.244449\pi\)
\(884\) 1.24259e15 0.0774178
\(885\) 0 0
\(886\) 1.88705e16 1.16117
\(887\) −3.12029e16 −1.90816 −0.954080 0.299553i \(-0.903163\pi\)
−0.954080 + 0.299553i \(0.903163\pi\)
\(888\) 0 0
\(889\) 9.44423e14 0.0570436
\(890\) 0 0
\(891\) 0 0
\(892\) −3.24508e15 −0.192406
\(893\) −6.16042e14 −0.0363017
\(894\) 0 0
\(895\) 0 0
\(896\) −4.47485e14 −0.0258872
\(897\) 0 0
\(898\) 8.20775e15 0.469034
\(899\) −1.19494e16 −0.678685
\(900\) 0 0
\(901\) 2.40175e15 0.134754
\(902\) −3.04978e15 −0.170072
\(903\) 0 0
\(904\) 1.79380e16 0.988206
\(905\) 0 0
\(906\) 0 0
\(907\) −7.97131e15 −0.431211 −0.215605 0.976481i \(-0.569173\pi\)
−0.215605 + 0.976481i \(0.569173\pi\)
\(908\) −1.47951e16 −0.795508
\(909\) 0 0
\(910\) 0 0
\(911\) 2.15018e16 1.13533 0.567666 0.823259i \(-0.307846\pi\)
0.567666 + 0.823259i \(0.307846\pi\)
\(912\) 0 0
\(913\) 1.50240e15 0.0783784
\(914\) −1.22724e15 −0.0636395
\(915\) 0 0
\(916\) 9.16958e15 0.469812
\(917\) −9.33114e15 −0.475230
\(918\) 0 0
\(919\) −1.93638e16 −0.974442 −0.487221 0.873279i \(-0.661989\pi\)
−0.487221 + 0.873279i \(0.661989\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −8.80012e15 −0.434979
\(923\) 8.59060e15 0.422099
\(924\) 0 0
\(925\) 0 0
\(926\) 1.92040e16 0.926896
\(927\) 0 0
\(928\) −1.05434e16 −0.502885
\(929\) 1.78505e16 0.846379 0.423189 0.906041i \(-0.360911\pi\)
0.423189 + 0.906041i \(0.360911\pi\)
\(930\) 0 0
\(931\) 6.33362e14 0.0296776
\(932\) −2.61142e15 −0.121644
\(933\) 0 0
\(934\) −1.32012e16 −0.607722
\(935\) 0 0
\(936\) 0 0
\(937\) −7.46505e15 −0.337649 −0.168824 0.985646i \(-0.553997\pi\)
−0.168824 + 0.985646i \(0.553997\pi\)
\(938\) −2.26565e15 −0.101877
\(939\) 0 0
\(940\) 0 0
\(941\) −3.05984e16 −1.35194 −0.675968 0.736931i \(-0.736274\pi\)
−0.675968 + 0.736931i \(0.736274\pi\)
\(942\) 0 0
\(943\) 6.21064e15 0.271220
\(944\) 1.45483e15 0.0631636
\(945\) 0 0
\(946\) 2.80500e16 1.20374
\(947\) −2.35138e16 −1.00322 −0.501611 0.865093i \(-0.667259\pi\)
−0.501611 + 0.865093i \(0.667259\pi\)
\(948\) 0 0
\(949\) −3.69584e15 −0.155865
\(950\) 0 0
\(951\) 0 0
\(952\) 6.53574e15 0.270889
\(953\) 1.48744e16 0.612956 0.306478 0.951878i \(-0.400849\pi\)
0.306478 + 0.951878i \(0.400849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.19492e16 0.483974
\(957\) 0 0
\(958\) −9.00384e15 −0.360510
\(959\) 6.89648e15 0.274552
\(960\) 0 0
\(961\) 3.98616e15 0.156883
\(962\) 3.70510e15 0.144990
\(963\) 0 0
\(964\) 1.49216e16 0.577285
\(965\) 0 0
\(966\) 0 0
\(967\) 4.74039e16 1.80289 0.901444 0.432896i \(-0.142508\pi\)
0.901444 + 0.432896i \(0.142508\pi\)
\(968\) 6.55233e15 0.247788
\(969\) 0 0
\(970\) 0 0
\(971\) 1.21083e16 0.450171 0.225086 0.974339i \(-0.427734\pi\)
0.225086 + 0.974339i \(0.427734\pi\)
\(972\) 0 0
\(973\) −1.83642e16 −0.675075
\(974\) 3.26544e16 1.19362
\(975\) 0 0
\(976\) 1.71307e16 0.619156
\(977\) −5.23952e16 −1.88309 −0.941546 0.336885i \(-0.890627\pi\)
−0.941546 + 0.336885i \(0.890627\pi\)
\(978\) 0 0
\(979\) −2.96163e16 −1.05251
\(980\) 0 0
\(981\) 0 0
\(982\) −8.08927e15 −0.282681
\(983\) 3.64254e16 1.26578 0.632892 0.774240i \(-0.281868\pi\)
0.632892 + 0.774240i \(0.281868\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 8.82540e15 0.301586
\(987\) 0 0
\(988\) 1.26605e14 0.00427847
\(989\) −5.71216e16 −1.91964
\(990\) 0 0
\(991\) 4.21088e16 1.39948 0.699742 0.714396i \(-0.253298\pi\)
0.699742 + 0.714396i \(0.253298\pi\)
\(992\) 2.59361e16 0.857216
\(993\) 0 0
\(994\) 1.37091e16 0.448108
\(995\) 0 0
\(996\) 0 0
\(997\) −2.75209e16 −0.884789 −0.442394 0.896821i \(-0.645871\pi\)
−0.442394 + 0.896821i \(0.645871\pi\)
\(998\) −3.15013e16 −1.00719
\(999\) 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 225.12.a.e.1.1 1
3.2 odd 2 25.12.a.a.1.1 1
5.2 odd 4 225.12.b.c.199.2 2
5.3 odd 4 225.12.b.c.199.1 2
5.4 even 2 45.12.a.a.1.1 1
15.2 even 4 25.12.b.a.24.1 2
15.8 even 4 25.12.b.a.24.2 2
15.14 odd 2 5.12.a.a.1.1 1
60.59 even 2 80.12.a.f.1.1 1
105.104 even 2 245.12.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.12.a.a.1.1 1 15.14 odd 2
25.12.a.a.1.1 1 3.2 odd 2
25.12.b.a.24.1 2 15.2 even 4
25.12.b.a.24.2 2 15.8 even 4
45.12.a.a.1.1 1 5.4 even 2
80.12.a.f.1.1 1 60.59 even 2
225.12.a.e.1.1 1 1.1 even 1 trivial
225.12.b.c.199.1 2 5.3 odd 4
225.12.b.c.199.2 2 5.2 odd 4
245.12.a.a.1.1 1 105.104 even 2