Properties

Label 48.22.a.f.1.1
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
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] = [48,22,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(134.149125258\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+59049.0 q^{3} +1.29542e7 q^{5} +4.79513e8 q^{7} +3.48678e9 q^{9} +O(q^{10})\) \(q+59049.0 q^{3} +1.29542e7 q^{5} +4.79513e8 q^{7} +3.48678e9 q^{9} -1.15658e11 q^{11} +2.95658e11 q^{13} +7.64931e11 q^{15} +6.62698e12 q^{17} -2.85762e13 q^{19} +2.83148e13 q^{21} -3.35385e14 q^{23} -3.09027e14 q^{25} +2.05891e14 q^{27} -6.99224e14 q^{29} +3.48496e15 q^{31} -6.82948e15 q^{33} +6.21170e15 q^{35} -3.51815e16 q^{37} +1.74583e16 q^{39} +6.13206e15 q^{41} -2.33261e17 q^{43} +4.51684e16 q^{45} +5.80206e17 q^{47} -3.28613e17 q^{49} +3.91317e17 q^{51} -1.39447e18 q^{53} -1.49825e18 q^{55} -1.68740e18 q^{57} -2.35248e18 q^{59} +9.92063e18 q^{61} +1.67196e18 q^{63} +3.83001e18 q^{65} -2.60690e19 q^{67} -1.98042e19 q^{69} +1.33370e19 q^{71} +9.03753e18 q^{73} -1.82477e19 q^{75} -5.54594e19 q^{77} +7.72839e19 q^{79} +1.21577e19 q^{81} +1.55698e20 q^{83} +8.58471e19 q^{85} -4.12885e19 q^{87} +2.53837e20 q^{89} +1.41772e20 q^{91} +2.05783e20 q^{93} -3.70181e20 q^{95} -1.03072e21 q^{97} -4.03274e20 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 59049.0 0.577350
\(4\) 0 0
\(5\) 1.29542e7 0.593232 0.296616 0.954997i \(-0.404142\pi\)
0.296616 + 0.954997i \(0.404142\pi\)
\(6\) 0 0
\(7\) 4.79513e8 0.641610 0.320805 0.947145i \(-0.396047\pi\)
0.320805 + 0.947145i \(0.396047\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −1.15658e11 −1.34447 −0.672236 0.740337i \(-0.734666\pi\)
−0.672236 + 0.740337i \(0.734666\pi\)
\(12\) 0 0
\(13\) 2.95658e11 0.594819 0.297409 0.954750i \(-0.403877\pi\)
0.297409 + 0.954750i \(0.403877\pi\)
\(14\) 0 0
\(15\) 7.64931e11 0.342503
\(16\) 0 0
\(17\) 6.62698e12 0.797264 0.398632 0.917111i \(-0.369485\pi\)
0.398632 + 0.917111i \(0.369485\pi\)
\(18\) 0 0
\(19\) −2.85762e13 −1.06928 −0.534640 0.845080i \(-0.679553\pi\)
−0.534640 + 0.845080i \(0.679553\pi\)
\(20\) 0 0
\(21\) 2.83148e13 0.370434
\(22\) 0 0
\(23\) −3.35385e14 −1.68811 −0.844057 0.536254i \(-0.819839\pi\)
−0.844057 + 0.536254i \(0.819839\pi\)
\(24\) 0 0
\(25\) −3.09027e14 −0.648076
\(26\) 0 0
\(27\) 2.05891e14 0.192450
\(28\) 0 0
\(29\) −6.99224e14 −0.308630 −0.154315 0.988022i \(-0.549317\pi\)
−0.154315 + 0.988022i \(0.549317\pi\)
\(30\) 0 0
\(31\) 3.48496e15 0.763659 0.381830 0.924233i \(-0.375294\pi\)
0.381830 + 0.924233i \(0.375294\pi\)
\(32\) 0 0
\(33\) −6.82948e15 −0.776231
\(34\) 0 0
\(35\) 6.21170e15 0.380624
\(36\) 0 0
\(37\) −3.51815e16 −1.20281 −0.601404 0.798945i \(-0.705392\pi\)
−0.601404 + 0.798945i \(0.705392\pi\)
\(38\) 0 0
\(39\) 1.74583e16 0.343419
\(40\) 0 0
\(41\) 6.13206e15 0.0713470 0.0356735 0.999363i \(-0.488642\pi\)
0.0356735 + 0.999363i \(0.488642\pi\)
\(42\) 0 0
\(43\) −2.33261e17 −1.64597 −0.822987 0.568060i \(-0.807694\pi\)
−0.822987 + 0.568060i \(0.807694\pi\)
\(44\) 0 0
\(45\) 4.51684e16 0.197744
\(46\) 0 0
\(47\) 5.80206e17 1.60899 0.804497 0.593957i \(-0.202435\pi\)
0.804497 + 0.593957i \(0.202435\pi\)
\(48\) 0 0
\(49\) −3.28613e17 −0.588337
\(50\) 0 0
\(51\) 3.91317e17 0.460301
\(52\) 0 0
\(53\) −1.39447e18 −1.09525 −0.547625 0.836724i \(-0.684468\pi\)
−0.547625 + 0.836724i \(0.684468\pi\)
\(54\) 0 0
\(55\) −1.49825e18 −0.797584
\(56\) 0 0
\(57\) −1.68740e18 −0.617349
\(58\) 0 0
\(59\) −2.35248e18 −0.599210 −0.299605 0.954063i \(-0.596855\pi\)
−0.299605 + 0.954063i \(0.596855\pi\)
\(60\) 0 0
\(61\) 9.92063e18 1.78064 0.890320 0.455336i \(-0.150481\pi\)
0.890320 + 0.455336i \(0.150481\pi\)
\(62\) 0 0
\(63\) 1.67196e18 0.213870
\(64\) 0 0
\(65\) 3.83001e18 0.352866
\(66\) 0 0
\(67\) −2.60690e19 −1.74718 −0.873592 0.486659i \(-0.838215\pi\)
−0.873592 + 0.486659i \(0.838215\pi\)
\(68\) 0 0
\(69\) −1.98042e19 −0.974633
\(70\) 0 0
\(71\) 1.33370e19 0.486233 0.243116 0.969997i \(-0.421830\pi\)
0.243116 + 0.969997i \(0.421830\pi\)
\(72\) 0 0
\(73\) 9.03753e18 0.246127 0.123064 0.992399i \(-0.460728\pi\)
0.123064 + 0.992399i \(0.460728\pi\)
\(74\) 0 0
\(75\) −1.82477e19 −0.374167
\(76\) 0 0
\(77\) −5.54594e19 −0.862626
\(78\) 0 0
\(79\) 7.72839e19 0.918342 0.459171 0.888348i \(-0.348147\pi\)
0.459171 + 0.888348i \(0.348147\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) 1.55698e20 1.10145 0.550724 0.834687i \(-0.314352\pi\)
0.550724 + 0.834687i \(0.314352\pi\)
\(84\) 0 0
\(85\) 8.58471e19 0.472963
\(86\) 0 0
\(87\) −4.12885e19 −0.178187
\(88\) 0 0
\(89\) 2.53837e20 0.862900 0.431450 0.902137i \(-0.358002\pi\)
0.431450 + 0.902137i \(0.358002\pi\)
\(90\) 0 0
\(91\) 1.41772e20 0.381642
\(92\) 0 0
\(93\) 2.05783e20 0.440899
\(94\) 0 0
\(95\) −3.70181e20 −0.634331
\(96\) 0 0
\(97\) −1.03072e21 −1.41918 −0.709591 0.704613i \(-0.751120\pi\)
−0.709591 + 0.704613i \(0.751120\pi\)
\(98\) 0 0
\(99\) −4.03274e20 −0.448157
\(100\) 0 0
\(101\) −7.65513e20 −0.689570 −0.344785 0.938682i \(-0.612048\pi\)
−0.344785 + 0.938682i \(0.612048\pi\)
\(102\) 0 0
\(103\) 8.97524e20 0.658044 0.329022 0.944322i \(-0.393281\pi\)
0.329022 + 0.944322i \(0.393281\pi\)
\(104\) 0 0
\(105\) 3.66794e20 0.219753
\(106\) 0 0
\(107\) −1.71774e21 −0.844166 −0.422083 0.906557i \(-0.638701\pi\)
−0.422083 + 0.906557i \(0.638701\pi\)
\(108\) 0 0
\(109\) 2.99848e21 1.21317 0.606586 0.795018i \(-0.292539\pi\)
0.606586 + 0.795018i \(0.292539\pi\)
\(110\) 0 0
\(111\) −2.07743e21 −0.694442
\(112\) 0 0
\(113\) −5.11711e21 −1.41808 −0.709041 0.705168i \(-0.750872\pi\)
−0.709041 + 0.705168i \(0.750872\pi\)
\(114\) 0 0
\(115\) −4.34464e21 −1.00144
\(116\) 0 0
\(117\) 1.03090e21 0.198273
\(118\) 0 0
\(119\) 3.17773e21 0.511533
\(120\) 0 0
\(121\) 5.97647e21 0.807604
\(122\) 0 0
\(123\) 3.62092e20 0.0411922
\(124\) 0 0
\(125\) −1.01802e22 −0.977691
\(126\) 0 0
\(127\) −4.05103e21 −0.329326 −0.164663 0.986350i \(-0.552654\pi\)
−0.164663 + 0.986350i \(0.552654\pi\)
\(128\) 0 0
\(129\) −1.37738e22 −0.950304
\(130\) 0 0
\(131\) 1.92867e22 1.13217 0.566083 0.824348i \(-0.308458\pi\)
0.566083 + 0.824348i \(0.308458\pi\)
\(132\) 0 0
\(133\) −1.37027e22 −0.686061
\(134\) 0 0
\(135\) 2.66715e21 0.114168
\(136\) 0 0
\(137\) −2.93536e22 −1.07670 −0.538351 0.842721i \(-0.680953\pi\)
−0.538351 + 0.842721i \(0.680953\pi\)
\(138\) 0 0
\(139\) −4.24068e22 −1.33592 −0.667959 0.744198i \(-0.732832\pi\)
−0.667959 + 0.744198i \(0.732832\pi\)
\(140\) 0 0
\(141\) 3.42606e22 0.928953
\(142\) 0 0
\(143\) −3.41952e22 −0.799717
\(144\) 0 0
\(145\) −9.05787e21 −0.183089
\(146\) 0 0
\(147\) −1.94043e22 −0.339676
\(148\) 0 0
\(149\) 5.04492e22 0.766299 0.383150 0.923686i \(-0.374839\pi\)
0.383150 + 0.923686i \(0.374839\pi\)
\(150\) 0 0
\(151\) −8.30789e22 −1.09707 −0.548533 0.836129i \(-0.684814\pi\)
−0.548533 + 0.836129i \(0.684814\pi\)
\(152\) 0 0
\(153\) 2.31069e22 0.265755
\(154\) 0 0
\(155\) 4.51447e22 0.453027
\(156\) 0 0
\(157\) −1.66285e23 −1.45850 −0.729250 0.684247i \(-0.760131\pi\)
−0.729250 + 0.684247i \(0.760131\pi\)
\(158\) 0 0
\(159\) −8.23422e22 −0.632343
\(160\) 0 0
\(161\) −1.60822e23 −1.08311
\(162\) 0 0
\(163\) 1.13921e23 0.673958 0.336979 0.941512i \(-0.390595\pi\)
0.336979 + 0.941512i \(0.390595\pi\)
\(164\) 0 0
\(165\) −8.84702e22 −0.460485
\(166\) 0 0
\(167\) −3.52378e23 −1.61617 −0.808084 0.589068i \(-0.799495\pi\)
−0.808084 + 0.589068i \(0.799495\pi\)
\(168\) 0 0
\(169\) −1.59651e23 −0.646190
\(170\) 0 0
\(171\) −9.96390e22 −0.356427
\(172\) 0 0
\(173\) −5.43709e22 −0.172140 −0.0860702 0.996289i \(-0.527431\pi\)
−0.0860702 + 0.996289i \(0.527431\pi\)
\(174\) 0 0
\(175\) −1.48182e23 −0.415812
\(176\) 0 0
\(177\) −1.38911e23 −0.345954
\(178\) 0 0
\(179\) −5.57843e23 −1.23468 −0.617341 0.786695i \(-0.711790\pi\)
−0.617341 + 0.786695i \(0.711790\pi\)
\(180\) 0 0
\(181\) −5.29021e23 −1.04195 −0.520976 0.853571i \(-0.674432\pi\)
−0.520976 + 0.853571i \(0.674432\pi\)
\(182\) 0 0
\(183\) 5.85803e23 1.02805
\(184\) 0 0
\(185\) −4.55748e23 −0.713545
\(186\) 0 0
\(187\) −7.66462e23 −1.07190
\(188\) 0 0
\(189\) 9.87275e22 0.123478
\(190\) 0 0
\(191\) 8.19262e22 0.0917429 0.0458715 0.998947i \(-0.485394\pi\)
0.0458715 + 0.998947i \(0.485394\pi\)
\(192\) 0 0
\(193\) 6.60174e23 0.662684 0.331342 0.943511i \(-0.392499\pi\)
0.331342 + 0.943511i \(0.392499\pi\)
\(194\) 0 0
\(195\) 2.26158e23 0.203727
\(196\) 0 0
\(197\) −8.79420e23 −0.711707 −0.355853 0.934542i \(-0.615810\pi\)
−0.355853 + 0.934542i \(0.615810\pi\)
\(198\) 0 0
\(199\) −8.82376e23 −0.642239 −0.321119 0.947039i \(-0.604059\pi\)
−0.321119 + 0.947039i \(0.604059\pi\)
\(200\) 0 0
\(201\) −1.53935e24 −1.00874
\(202\) 0 0
\(203\) −3.35287e23 −0.198020
\(204\) 0 0
\(205\) 7.94357e22 0.0423253
\(206\) 0 0
\(207\) −1.16942e24 −0.562704
\(208\) 0 0
\(209\) 3.30506e24 1.43762
\(210\) 0 0
\(211\) 2.86260e24 1.12667 0.563333 0.826230i \(-0.309519\pi\)
0.563333 + 0.826230i \(0.309519\pi\)
\(212\) 0 0
\(213\) 7.87534e23 0.280727
\(214\) 0 0
\(215\) −3.02170e24 −0.976445
\(216\) 0 0
\(217\) 1.67108e24 0.489971
\(218\) 0 0
\(219\) 5.33657e23 0.142102
\(220\) 0 0
\(221\) 1.95932e24 0.474228
\(222\) 0 0
\(223\) 1.04892e24 0.230963 0.115481 0.993310i \(-0.463159\pi\)
0.115481 + 0.993310i \(0.463159\pi\)
\(224\) 0 0
\(225\) −1.07751e24 −0.216025
\(226\) 0 0
\(227\) 4.22561e24 0.772002 0.386001 0.922498i \(-0.373856\pi\)
0.386001 + 0.922498i \(0.373856\pi\)
\(228\) 0 0
\(229\) −9.40000e24 −1.56623 −0.783115 0.621877i \(-0.786370\pi\)
−0.783115 + 0.621877i \(0.786370\pi\)
\(230\) 0 0
\(231\) −3.27482e24 −0.498038
\(232\) 0 0
\(233\) −5.59528e24 −0.777293 −0.388646 0.921387i \(-0.627057\pi\)
−0.388646 + 0.921387i \(0.627057\pi\)
\(234\) 0 0
\(235\) 7.51609e24 0.954507
\(236\) 0 0
\(237\) 4.56353e24 0.530205
\(238\) 0 0
\(239\) 4.79661e24 0.510219 0.255109 0.966912i \(-0.417889\pi\)
0.255109 + 0.966912i \(0.417889\pi\)
\(240\) 0 0
\(241\) 1.30240e24 0.126930 0.0634649 0.997984i \(-0.479785\pi\)
0.0634649 + 0.997984i \(0.479785\pi\)
\(242\) 0 0
\(243\) 7.17898e23 0.0641500
\(244\) 0 0
\(245\) −4.25691e24 −0.349020
\(246\) 0 0
\(247\) −8.44878e24 −0.636028
\(248\) 0 0
\(249\) 9.19384e24 0.635922
\(250\) 0 0
\(251\) 1.25489e25 0.798055 0.399027 0.916939i \(-0.369348\pi\)
0.399027 + 0.916939i \(0.369348\pi\)
\(252\) 0 0
\(253\) 3.87899e25 2.26962
\(254\) 0 0
\(255\) 5.06919e24 0.273065
\(256\) 0 0
\(257\) 3.12478e25 1.55068 0.775339 0.631546i \(-0.217579\pi\)
0.775339 + 0.631546i \(0.217579\pi\)
\(258\) 0 0
\(259\) −1.68700e25 −0.771734
\(260\) 0 0
\(261\) −2.43804e24 −0.102877
\(262\) 0 0
\(263\) 2.04345e25 0.795845 0.397923 0.917419i \(-0.369731\pi\)
0.397923 + 0.917419i \(0.369731\pi\)
\(264\) 0 0
\(265\) −1.80642e25 −0.649737
\(266\) 0 0
\(267\) 1.49888e25 0.498195
\(268\) 0 0
\(269\) −6.02651e24 −0.185211 −0.0926055 0.995703i \(-0.529520\pi\)
−0.0926055 + 0.995703i \(0.529520\pi\)
\(270\) 0 0
\(271\) 1.13011e25 0.321324 0.160662 0.987009i \(-0.448637\pi\)
0.160662 + 0.987009i \(0.448637\pi\)
\(272\) 0 0
\(273\) 8.37150e24 0.220341
\(274\) 0 0
\(275\) 3.57413e25 0.871319
\(276\) 0 0
\(277\) 7.51731e25 1.69834 0.849170 0.528119i \(-0.177103\pi\)
0.849170 + 0.528119i \(0.177103\pi\)
\(278\) 0 0
\(279\) 1.21513e25 0.254553
\(280\) 0 0
\(281\) −2.24659e25 −0.436623 −0.218312 0.975879i \(-0.570055\pi\)
−0.218312 + 0.975879i \(0.570055\pi\)
\(282\) 0 0
\(283\) 6.02460e25 1.08685 0.543426 0.839457i \(-0.317127\pi\)
0.543426 + 0.839457i \(0.317127\pi\)
\(284\) 0 0
\(285\) −2.18588e25 −0.366231
\(286\) 0 0
\(287\) 2.94040e24 0.0457769
\(288\) 0 0
\(289\) −2.51750e25 −0.364370
\(290\) 0 0
\(291\) −6.08631e25 −0.819366
\(292\) 0 0
\(293\) 4.65033e24 0.0582604 0.0291302 0.999576i \(-0.490726\pi\)
0.0291302 + 0.999576i \(0.490726\pi\)
\(294\) 0 0
\(295\) −3.04744e25 −0.355471
\(296\) 0 0
\(297\) −2.38129e25 −0.258744
\(298\) 0 0
\(299\) −9.91594e25 −1.00412
\(300\) 0 0
\(301\) −1.11852e26 −1.05607
\(302\) 0 0
\(303\) −4.52028e25 −0.398123
\(304\) 0 0
\(305\) 1.28514e26 1.05633
\(306\) 0 0
\(307\) 6.89098e25 0.528844 0.264422 0.964407i \(-0.414819\pi\)
0.264422 + 0.964407i \(0.414819\pi\)
\(308\) 0 0
\(309\) 5.29979e25 0.379922
\(310\) 0 0
\(311\) 2.75254e25 0.184395 0.0921976 0.995741i \(-0.470611\pi\)
0.0921976 + 0.995741i \(0.470611\pi\)
\(312\) 0 0
\(313\) −1.51255e26 −0.947316 −0.473658 0.880709i \(-0.657067\pi\)
−0.473658 + 0.880709i \(0.657067\pi\)
\(314\) 0 0
\(315\) 2.16588e25 0.126875
\(316\) 0 0
\(317\) −5.45958e25 −0.299252 −0.149626 0.988743i \(-0.547807\pi\)
−0.149626 + 0.988743i \(0.547807\pi\)
\(318\) 0 0
\(319\) 8.08707e25 0.414944
\(320\) 0 0
\(321\) −1.01431e26 −0.487379
\(322\) 0 0
\(323\) −1.89374e26 −0.852499
\(324\) 0 0
\(325\) −9.13662e25 −0.385488
\(326\) 0 0
\(327\) 1.77057e26 0.700425
\(328\) 0 0
\(329\) 2.78216e26 1.03235
\(330\) 0 0
\(331\) −4.57958e26 −1.59453 −0.797263 0.603632i \(-0.793720\pi\)
−0.797263 + 0.603632i \(0.793720\pi\)
\(332\) 0 0
\(333\) −1.22670e26 −0.400936
\(334\) 0 0
\(335\) −3.37702e26 −1.03649
\(336\) 0 0
\(337\) 1.77477e26 0.511714 0.255857 0.966715i \(-0.417642\pi\)
0.255857 + 0.966715i \(0.417642\pi\)
\(338\) 0 0
\(339\) −3.02160e26 −0.818730
\(340\) 0 0
\(341\) −4.03062e26 −1.02672
\(342\) 0 0
\(343\) −4.25404e26 −1.01909
\(344\) 0 0
\(345\) −2.56547e26 −0.578183
\(346\) 0 0
\(347\) 3.31317e26 0.702724 0.351362 0.936240i \(-0.385719\pi\)
0.351362 + 0.936240i \(0.385719\pi\)
\(348\) 0 0
\(349\) −1.82866e26 −0.365146 −0.182573 0.983192i \(-0.558443\pi\)
−0.182573 + 0.983192i \(0.558443\pi\)
\(350\) 0 0
\(351\) 6.08734e25 0.114473
\(352\) 0 0
\(353\) −5.58761e26 −0.989900 −0.494950 0.868921i \(-0.664814\pi\)
−0.494950 + 0.868921i \(0.664814\pi\)
\(354\) 0 0
\(355\) 1.72769e26 0.288449
\(356\) 0 0
\(357\) 1.87642e26 0.295333
\(358\) 0 0
\(359\) 8.65986e26 1.28534 0.642671 0.766142i \(-0.277826\pi\)
0.642671 + 0.766142i \(0.277826\pi\)
\(360\) 0 0
\(361\) 1.02389e26 0.143360
\(362\) 0 0
\(363\) 3.52905e26 0.466270
\(364\) 0 0
\(365\) 1.17074e26 0.146011
\(366\) 0 0
\(367\) 1.31813e26 0.155227 0.0776133 0.996984i \(-0.475270\pi\)
0.0776133 + 0.996984i \(0.475270\pi\)
\(368\) 0 0
\(369\) 2.13812e25 0.0237823
\(370\) 0 0
\(371\) −6.68667e26 −0.702723
\(372\) 0 0
\(373\) −4.80048e26 −0.476806 −0.238403 0.971166i \(-0.576624\pi\)
−0.238403 + 0.971166i \(0.576624\pi\)
\(374\) 0 0
\(375\) −6.01132e26 −0.564470
\(376\) 0 0
\(377\) −2.06731e26 −0.183579
\(378\) 0 0
\(379\) −3.61041e26 −0.303281 −0.151640 0.988436i \(-0.548456\pi\)
−0.151640 + 0.988436i \(0.548456\pi\)
\(380\) 0 0
\(381\) −2.39209e26 −0.190136
\(382\) 0 0
\(383\) 4.45743e26 0.335349 0.167675 0.985842i \(-0.446374\pi\)
0.167675 + 0.985842i \(0.446374\pi\)
\(384\) 0 0
\(385\) −7.18431e26 −0.511738
\(386\) 0 0
\(387\) −8.13330e26 −0.548658
\(388\) 0 0
\(389\) 2.63616e27 1.68461 0.842307 0.538998i \(-0.181197\pi\)
0.842307 + 0.538998i \(0.181197\pi\)
\(390\) 0 0
\(391\) −2.22259e27 −1.34587
\(392\) 0 0
\(393\) 1.13886e27 0.653657
\(394\) 0 0
\(395\) 1.00115e27 0.544790
\(396\) 0 0
\(397\) −1.91276e27 −0.987100 −0.493550 0.869717i \(-0.664301\pi\)
−0.493550 + 0.869717i \(0.664301\pi\)
\(398\) 0 0
\(399\) −8.09128e26 −0.396097
\(400\) 0 0
\(401\) 2.31621e27 1.07588 0.537939 0.842984i \(-0.319203\pi\)
0.537939 + 0.842984i \(0.319203\pi\)
\(402\) 0 0
\(403\) 1.03036e27 0.454239
\(404\) 0 0
\(405\) 1.57493e26 0.0659147
\(406\) 0 0
\(407\) 4.06902e27 1.61714
\(408\) 0 0
\(409\) 1.77500e27 0.670045 0.335022 0.942210i \(-0.391256\pi\)
0.335022 + 0.942210i \(0.391256\pi\)
\(410\) 0 0
\(411\) −1.73330e27 −0.621634
\(412\) 0 0
\(413\) −1.12804e27 −0.384459
\(414\) 0 0
\(415\) 2.01694e27 0.653415
\(416\) 0 0
\(417\) −2.50408e27 −0.771293
\(418\) 0 0
\(419\) 2.22580e27 0.651988 0.325994 0.945372i \(-0.394301\pi\)
0.325994 + 0.945372i \(0.394301\pi\)
\(420\) 0 0
\(421\) 2.26274e27 0.630482 0.315241 0.949012i \(-0.397915\pi\)
0.315241 + 0.949012i \(0.397915\pi\)
\(422\) 0 0
\(423\) 2.02305e27 0.536331
\(424\) 0 0
\(425\) −2.04791e27 −0.516687
\(426\) 0 0
\(427\) 4.75707e27 1.14248
\(428\) 0 0
\(429\) −2.01919e27 −0.461717
\(430\) 0 0
\(431\) 1.80751e27 0.393612 0.196806 0.980442i \(-0.436943\pi\)
0.196806 + 0.980442i \(0.436943\pi\)
\(432\) 0 0
\(433\) 7.12467e27 1.47789 0.738945 0.673766i \(-0.235324\pi\)
0.738945 + 0.673766i \(0.235324\pi\)
\(434\) 0 0
\(435\) −5.34858e26 −0.105706
\(436\) 0 0
\(437\) 9.58403e27 1.80507
\(438\) 0 0
\(439\) −1.58875e27 −0.285218 −0.142609 0.989779i \(-0.545549\pi\)
−0.142609 + 0.989779i \(0.545549\pi\)
\(440\) 0 0
\(441\) −1.14580e27 −0.196112
\(442\) 0 0
\(443\) −3.52672e26 −0.0575614 −0.0287807 0.999586i \(-0.509162\pi\)
−0.0287807 + 0.999586i \(0.509162\pi\)
\(444\) 0 0
\(445\) 3.28825e27 0.511900
\(446\) 0 0
\(447\) 2.97897e27 0.442423
\(448\) 0 0
\(449\) 7.74724e27 1.09789 0.548947 0.835857i \(-0.315029\pi\)
0.548947 + 0.835857i \(0.315029\pi\)
\(450\) 0 0
\(451\) −7.09220e26 −0.0959240
\(452\) 0 0
\(453\) −4.90573e27 −0.633391
\(454\) 0 0
\(455\) 1.83654e27 0.226402
\(456\) 0 0
\(457\) −1.41977e28 −1.67147 −0.835735 0.549132i \(-0.814958\pi\)
−0.835735 + 0.549132i \(0.814958\pi\)
\(458\) 0 0
\(459\) 1.36444e27 0.153434
\(460\) 0 0
\(461\) −2.10321e27 −0.225956 −0.112978 0.993598i \(-0.536039\pi\)
−0.112978 + 0.993598i \(0.536039\pi\)
\(462\) 0 0
\(463\) −1.66348e28 −1.70772 −0.853860 0.520503i \(-0.825744\pi\)
−0.853860 + 0.520503i \(0.825744\pi\)
\(464\) 0 0
\(465\) 2.66575e27 0.261555
\(466\) 0 0
\(467\) 1.92552e28 1.80601 0.903006 0.429629i \(-0.141356\pi\)
0.903006 + 0.429629i \(0.141356\pi\)
\(468\) 0 0
\(469\) −1.25004e28 −1.12101
\(470\) 0 0
\(471\) −9.81896e27 −0.842065
\(472\) 0 0
\(473\) 2.69784e28 2.21297
\(474\) 0 0
\(475\) 8.83080e27 0.692974
\(476\) 0 0
\(477\) −4.86222e27 −0.365083
\(478\) 0 0
\(479\) −1.98417e28 −1.42579 −0.712894 0.701272i \(-0.752616\pi\)
−0.712894 + 0.701272i \(0.752616\pi\)
\(480\) 0 0
\(481\) −1.04017e28 −0.715454
\(482\) 0 0
\(483\) −9.49635e27 −0.625334
\(484\) 0 0
\(485\) −1.33522e28 −0.841905
\(486\) 0 0
\(487\) 1.91036e26 0.0115361 0.00576807 0.999983i \(-0.498164\pi\)
0.00576807 + 0.999983i \(0.498164\pi\)
\(488\) 0 0
\(489\) 6.72691e27 0.389110
\(490\) 0 0
\(491\) −8.06003e27 −0.446664 −0.223332 0.974742i \(-0.571693\pi\)
−0.223332 + 0.974742i \(0.571693\pi\)
\(492\) 0 0
\(493\) −4.63375e27 −0.246059
\(494\) 0 0
\(495\) −5.22408e27 −0.265861
\(496\) 0 0
\(497\) 6.39525e27 0.311972
\(498\) 0 0
\(499\) −3.12631e28 −1.46210 −0.731048 0.682326i \(-0.760968\pi\)
−0.731048 + 0.682326i \(0.760968\pi\)
\(500\) 0 0
\(501\) −2.08076e28 −0.933095
\(502\) 0 0
\(503\) 4.14717e28 1.78356 0.891782 0.452466i \(-0.149456\pi\)
0.891782 + 0.452466i \(0.149456\pi\)
\(504\) 0 0
\(505\) −9.91659e27 −0.409075
\(506\) 0 0
\(507\) −9.42722e27 −0.373078
\(508\) 0 0
\(509\) 2.54263e28 0.965488 0.482744 0.875762i \(-0.339640\pi\)
0.482744 + 0.875762i \(0.339640\pi\)
\(510\) 0 0
\(511\) 4.33361e27 0.157918
\(512\) 0 0
\(513\) −5.88358e27 −0.205783
\(514\) 0 0
\(515\) 1.16267e28 0.390373
\(516\) 0 0
\(517\) −6.71053e28 −2.16325
\(518\) 0 0
\(519\) −3.21055e27 −0.0993853
\(520\) 0 0
\(521\) −3.24742e28 −0.965479 −0.482739 0.875764i \(-0.660358\pi\)
−0.482739 + 0.875764i \(0.660358\pi\)
\(522\) 0 0
\(523\) −1.45605e28 −0.415822 −0.207911 0.978148i \(-0.566666\pi\)
−0.207911 + 0.978148i \(0.566666\pi\)
\(524\) 0 0
\(525\) −8.75002e27 −0.240069
\(526\) 0 0
\(527\) 2.30948e28 0.608838
\(528\) 0 0
\(529\) 7.30116e28 1.84973
\(530\) 0 0
\(531\) −8.20258e27 −0.199737
\(532\) 0 0
\(533\) 1.81299e27 0.0424385
\(534\) 0 0
\(535\) −2.22519e28 −0.500786
\(536\) 0 0
\(537\) −3.29401e28 −0.712844
\(538\) 0 0
\(539\) 3.80067e28 0.791002
\(540\) 0 0
\(541\) 1.59820e28 0.319933 0.159967 0.987122i \(-0.448861\pi\)
0.159967 + 0.987122i \(0.448861\pi\)
\(542\) 0 0
\(543\) −3.12381e28 −0.601571
\(544\) 0 0
\(545\) 3.88428e28 0.719692
\(546\) 0 0
\(547\) −1.03425e29 −1.84399 −0.921994 0.387203i \(-0.873441\pi\)
−0.921994 + 0.387203i \(0.873441\pi\)
\(548\) 0 0
\(549\) 3.45911e28 0.593546
\(550\) 0 0
\(551\) 1.99812e28 0.330011
\(552\) 0 0
\(553\) 3.70586e28 0.589217
\(554\) 0 0
\(555\) −2.69114e28 −0.411965
\(556\) 0 0
\(557\) −9.81350e28 −1.44659 −0.723293 0.690541i \(-0.757372\pi\)
−0.723293 + 0.690541i \(0.757372\pi\)
\(558\) 0 0
\(559\) −6.89655e28 −0.979057
\(560\) 0 0
\(561\) −4.52588e28 −0.618861
\(562\) 0 0
\(563\) −7.10142e28 −0.935421 −0.467711 0.883882i \(-0.654921\pi\)
−0.467711 + 0.883882i \(0.654921\pi\)
\(564\) 0 0
\(565\) −6.62879e28 −0.841251
\(566\) 0 0
\(567\) 5.82976e27 0.0712900
\(568\) 0 0
\(569\) 1.47837e29 1.74222 0.871110 0.491087i \(-0.163400\pi\)
0.871110 + 0.491087i \(0.163400\pi\)
\(570\) 0 0
\(571\) 1.30988e29 1.48783 0.743914 0.668275i \(-0.232967\pi\)
0.743914 + 0.668275i \(0.232967\pi\)
\(572\) 0 0
\(573\) 4.83766e27 0.0529678
\(574\) 0 0
\(575\) 1.03643e29 1.09403
\(576\) 0 0
\(577\) −1.13007e29 −1.15017 −0.575083 0.818095i \(-0.695030\pi\)
−0.575083 + 0.818095i \(0.695030\pi\)
\(578\) 0 0
\(579\) 3.89826e28 0.382601
\(580\) 0 0
\(581\) 7.46594e28 0.706700
\(582\) 0 0
\(583\) 1.61281e29 1.47253
\(584\) 0 0
\(585\) 1.33544e28 0.117622
\(586\) 0 0
\(587\) 1.05612e29 0.897455 0.448728 0.893669i \(-0.351877\pi\)
0.448728 + 0.893669i \(0.351877\pi\)
\(588\) 0 0
\(589\) −9.95868e28 −0.816566
\(590\) 0 0
\(591\) −5.19289e28 −0.410904
\(592\) 0 0
\(593\) 1.22381e29 0.934630 0.467315 0.884091i \(-0.345221\pi\)
0.467315 + 0.884091i \(0.345221\pi\)
\(594\) 0 0
\(595\) 4.11648e28 0.303458
\(596\) 0 0
\(597\) −5.21034e28 −0.370797
\(598\) 0 0
\(599\) −1.37213e29 −0.942789 −0.471394 0.881922i \(-0.656249\pi\)
−0.471394 + 0.881922i \(0.656249\pi\)
\(600\) 0 0
\(601\) 1.89598e29 1.25792 0.628958 0.777439i \(-0.283482\pi\)
0.628958 + 0.777439i \(0.283482\pi\)
\(602\) 0 0
\(603\) −9.08969e28 −0.582395
\(604\) 0 0
\(605\) 7.74203e28 0.479097
\(606\) 0 0
\(607\) 1.07832e29 0.644563 0.322281 0.946644i \(-0.395550\pi\)
0.322281 + 0.946644i \(0.395550\pi\)
\(608\) 0 0
\(609\) −1.97984e28 −0.114327
\(610\) 0 0
\(611\) 1.71543e29 0.957060
\(612\) 0 0
\(613\) 4.49900e28 0.242538 0.121269 0.992620i \(-0.461304\pi\)
0.121269 + 0.992620i \(0.461304\pi\)
\(614\) 0 0
\(615\) 4.69060e27 0.0244365
\(616\) 0 0
\(617\) 2.83973e29 1.42982 0.714911 0.699215i \(-0.246467\pi\)
0.714911 + 0.699215i \(0.246467\pi\)
\(618\) 0 0
\(619\) −1.17452e29 −0.571624 −0.285812 0.958286i \(-0.592263\pi\)
−0.285812 + 0.958286i \(0.592263\pi\)
\(620\) 0 0
\(621\) −6.90528e28 −0.324878
\(622\) 0 0
\(623\) 1.21718e29 0.553645
\(624\) 0 0
\(625\) 1.54791e28 0.0680776
\(626\) 0 0
\(627\) 1.95160e29 0.830008
\(628\) 0 0
\(629\) −2.33147e29 −0.958956
\(630\) 0 0
\(631\) −1.11858e29 −0.444999 −0.222500 0.974933i \(-0.571422\pi\)
−0.222500 + 0.974933i \(0.571422\pi\)
\(632\) 0 0
\(633\) 1.69034e29 0.650481
\(634\) 0 0
\(635\) −5.24777e28 −0.195367
\(636\) 0 0
\(637\) −9.71572e28 −0.349954
\(638\) 0 0
\(639\) 4.65031e28 0.162078
\(640\) 0 0
\(641\) 5.02292e29 1.69413 0.847066 0.531488i \(-0.178367\pi\)
0.847066 + 0.531488i \(0.178367\pi\)
\(642\) 0 0
\(643\) −4.27106e29 −1.39419 −0.697093 0.716980i \(-0.745524\pi\)
−0.697093 + 0.716980i \(0.745524\pi\)
\(644\) 0 0
\(645\) −1.78428e29 −0.563751
\(646\) 0 0
\(647\) 4.32869e29 1.32392 0.661960 0.749539i \(-0.269725\pi\)
0.661960 + 0.749539i \(0.269725\pi\)
\(648\) 0 0
\(649\) 2.72082e29 0.805621
\(650\) 0 0
\(651\) 9.86758e28 0.282885
\(652\) 0 0
\(653\) −2.73969e29 −0.760525 −0.380263 0.924879i \(-0.624166\pi\)
−0.380263 + 0.924879i \(0.624166\pi\)
\(654\) 0 0
\(655\) 2.49844e29 0.671637
\(656\) 0 0
\(657\) 3.15119e28 0.0820424
\(658\) 0 0
\(659\) −3.07743e28 −0.0776053 −0.0388026 0.999247i \(-0.512354\pi\)
−0.0388026 + 0.999247i \(0.512354\pi\)
\(660\) 0 0
\(661\) −3.06325e29 −0.748286 −0.374143 0.927371i \(-0.622063\pi\)
−0.374143 + 0.927371i \(0.622063\pi\)
\(662\) 0 0
\(663\) 1.15696e29 0.273796
\(664\) 0 0
\(665\) −1.77507e29 −0.406993
\(666\) 0 0
\(667\) 2.34509e29 0.521002
\(668\) 0 0
\(669\) 6.19378e28 0.133346
\(670\) 0 0
\(671\) −1.14740e30 −2.39402
\(672\) 0 0
\(673\) 5.09843e29 1.03105 0.515523 0.856876i \(-0.327598\pi\)
0.515523 + 0.856876i \(0.327598\pi\)
\(674\) 0 0
\(675\) −6.36258e28 −0.124722
\(676\) 0 0
\(677\) −7.38047e29 −1.40250 −0.701250 0.712915i \(-0.747374\pi\)
−0.701250 + 0.712915i \(0.747374\pi\)
\(678\) 0 0
\(679\) −4.94245e29 −0.910562
\(680\) 0 0
\(681\) 2.49518e29 0.445716
\(682\) 0 0
\(683\) 4.68059e29 0.810743 0.405372 0.914152i \(-0.367142\pi\)
0.405372 + 0.914152i \(0.367142\pi\)
\(684\) 0 0
\(685\) −3.80252e29 −0.638734
\(686\) 0 0
\(687\) −5.55061e29 −0.904263
\(688\) 0 0
\(689\) −4.12287e29 −0.651475
\(690\) 0 0
\(691\) 4.77475e29 0.731865 0.365932 0.930641i \(-0.380750\pi\)
0.365932 + 0.930641i \(0.380750\pi\)
\(692\) 0 0
\(693\) −1.93375e29 −0.287542
\(694\) 0 0
\(695\) −5.49345e29 −0.792510
\(696\) 0 0
\(697\) 4.06370e28 0.0568824
\(698\) 0 0
\(699\) −3.30396e29 −0.448770
\(700\) 0 0
\(701\) −9.76926e29 −1.28772 −0.643862 0.765142i \(-0.722669\pi\)
−0.643862 + 0.765142i \(0.722669\pi\)
\(702\) 0 0
\(703\) 1.00535e30 1.28614
\(704\) 0 0
\(705\) 4.43817e29 0.551085
\(706\) 0 0
\(707\) −3.67073e29 −0.442435
\(708\) 0 0
\(709\) −8.06956e29 −0.944201 −0.472101 0.881545i \(-0.656504\pi\)
−0.472101 + 0.881545i \(0.656504\pi\)
\(710\) 0 0
\(711\) 2.69472e29 0.306114
\(712\) 0 0
\(713\) −1.16880e30 −1.28914
\(714\) 0 0
\(715\) −4.42970e29 −0.474418
\(716\) 0 0
\(717\) 2.83235e29 0.294575
\(718\) 0 0
\(719\) −2.20303e29 −0.222519 −0.111259 0.993791i \(-0.535488\pi\)
−0.111259 + 0.993791i \(0.535488\pi\)
\(720\) 0 0
\(721\) 4.30374e29 0.422208
\(722\) 0 0
\(723\) 7.69051e28 0.0732830
\(724\) 0 0
\(725\) 2.16079e29 0.200015
\(726\) 0 0
\(727\) −1.52128e30 −1.36804 −0.684020 0.729463i \(-0.739770\pi\)
−0.684020 + 0.729463i \(0.739770\pi\)
\(728\) 0 0
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) −1.54582e30 −1.31228
\(732\) 0 0
\(733\) 3.36651e29 0.277708 0.138854 0.990313i \(-0.455658\pi\)
0.138854 + 0.990313i \(0.455658\pi\)
\(734\) 0 0
\(735\) −2.51366e29 −0.201507
\(736\) 0 0
\(737\) 3.01508e30 2.34904
\(738\) 0 0
\(739\) 2.59622e29 0.196596 0.0982982 0.995157i \(-0.468660\pi\)
0.0982982 + 0.995157i \(0.468660\pi\)
\(740\) 0 0
\(741\) −4.98892e29 −0.367211
\(742\) 0 0
\(743\) 1.31717e30 0.942455 0.471227 0.882012i \(-0.343811\pi\)
0.471227 + 0.882012i \(0.343811\pi\)
\(744\) 0 0
\(745\) 6.53528e29 0.454593
\(746\) 0 0
\(747\) 5.42887e29 0.367150
\(748\) 0 0
\(749\) −8.23679e29 −0.541625
\(750\) 0 0
\(751\) −1.46730e29 −0.0938209 −0.0469104 0.998899i \(-0.514938\pi\)
−0.0469104 + 0.998899i \(0.514938\pi\)
\(752\) 0 0
\(753\) 7.41002e29 0.460757
\(754\) 0 0
\(755\) −1.07622e30 −0.650815
\(756\) 0 0
\(757\) −3.16010e29 −0.185864 −0.0929318 0.995672i \(-0.529624\pi\)
−0.0929318 + 0.995672i \(0.529624\pi\)
\(758\) 0 0
\(759\) 2.29051e30 1.31037
\(760\) 0 0
\(761\) −2.02639e30 −1.12767 −0.563837 0.825886i \(-0.690676\pi\)
−0.563837 + 0.825886i \(0.690676\pi\)
\(762\) 0 0
\(763\) 1.43781e30 0.778383
\(764\) 0 0
\(765\) 2.99330e29 0.157654
\(766\) 0 0
\(767\) −6.95529e29 −0.356422
\(768\) 0 0
\(769\) 1.55510e30 0.775410 0.387705 0.921783i \(-0.373268\pi\)
0.387705 + 0.921783i \(0.373268\pi\)
\(770\) 0 0
\(771\) 1.84515e30 0.895284
\(772\) 0 0
\(773\) −3.90527e30 −1.84402 −0.922012 0.387161i \(-0.873456\pi\)
−0.922012 + 0.387161i \(0.873456\pi\)
\(774\) 0 0
\(775\) −1.07694e30 −0.494909
\(776\) 0 0
\(777\) −9.96157e29 −0.445561
\(778\) 0 0
\(779\) −1.75231e29 −0.0762899
\(780\) 0 0
\(781\) −1.54252e30 −0.653726
\(782\) 0 0
\(783\) −1.43964e29 −0.0593958
\(784\) 0 0
\(785\) −2.15408e30 −0.865229
\(786\) 0 0
\(787\) 6.91303e29 0.270355 0.135177 0.990821i \(-0.456840\pi\)
0.135177 + 0.990821i \(0.456840\pi\)
\(788\) 0 0
\(789\) 1.20664e30 0.459481
\(790\) 0 0
\(791\) −2.45372e30 −0.909855
\(792\) 0 0
\(793\) 2.93312e30 1.05916
\(794\) 0 0
\(795\) −1.06667e30 −0.375126
\(796\) 0 0
\(797\) 3.05715e30 1.04714 0.523569 0.851983i \(-0.324600\pi\)
0.523569 + 0.851983i \(0.324600\pi\)
\(798\) 0 0
\(799\) 3.84501e30 1.28279
\(800\) 0 0
\(801\) 8.85076e29 0.287633
\(802\) 0 0
\(803\) −1.04526e30 −0.330911
\(804\) 0 0
\(805\) −2.08331e30 −0.642536
\(806\) 0 0
\(807\) −3.55859e29 −0.106932
\(808\) 0 0
\(809\) 5.62302e30 1.64630 0.823152 0.567821i \(-0.192213\pi\)
0.823152 + 0.567821i \(0.192213\pi\)
\(810\) 0 0
\(811\) 3.35752e30 0.957854 0.478927 0.877855i \(-0.341026\pi\)
0.478927 + 0.877855i \(0.341026\pi\)
\(812\) 0 0
\(813\) 6.67319e29 0.185517
\(814\) 0 0
\(815\) 1.47575e30 0.399814
\(816\) 0 0
\(817\) 6.66571e30 1.76001
\(818\) 0 0
\(819\) 4.94328e29 0.127214
\(820\) 0 0
\(821\) 1.66494e30 0.417634 0.208817 0.977955i \(-0.433039\pi\)
0.208817 + 0.977955i \(0.433039\pi\)
\(822\) 0 0
\(823\) 4.79393e30 1.17218 0.586088 0.810247i \(-0.300667\pi\)
0.586088 + 0.810247i \(0.300667\pi\)
\(824\) 0 0
\(825\) 2.11049e30 0.503056
\(826\) 0 0
\(827\) −6.92737e30 −1.60976 −0.804879 0.593439i \(-0.797770\pi\)
−0.804879 + 0.593439i \(0.797770\pi\)
\(828\) 0 0
\(829\) −2.55745e30 −0.579408 −0.289704 0.957116i \(-0.593557\pi\)
−0.289704 + 0.957116i \(0.593557\pi\)
\(830\) 0 0
\(831\) 4.43890e30 0.980537
\(832\) 0 0
\(833\) −2.17771e30 −0.469060
\(834\) 0 0
\(835\) −4.56477e30 −0.958763
\(836\) 0 0
\(837\) 7.17522e29 0.146966
\(838\) 0 0
\(839\) 4.16860e29 0.0832702 0.0416351 0.999133i \(-0.486743\pi\)
0.0416351 + 0.999133i \(0.486743\pi\)
\(840\) 0 0
\(841\) −4.64393e30 −0.904748
\(842\) 0 0
\(843\) −1.32659e30 −0.252085
\(844\) 0 0
\(845\) −2.06814e30 −0.383341
\(846\) 0 0
\(847\) 2.86580e30 0.518167
\(848\) 0 0
\(849\) 3.55747e30 0.627495
\(850\) 0 0
\(851\) 1.17994e31 2.03048
\(852\) 0 0
\(853\) 7.40771e30 1.24371 0.621855 0.783132i \(-0.286379\pi\)
0.621855 + 0.783132i \(0.286379\pi\)
\(854\) 0 0
\(855\) −1.29074e30 −0.211444
\(856\) 0 0
\(857\) −3.64448e30 −0.582555 −0.291277 0.956639i \(-0.594080\pi\)
−0.291277 + 0.956639i \(0.594080\pi\)
\(858\) 0 0
\(859\) 3.70416e30 0.577778 0.288889 0.957363i \(-0.406714\pi\)
0.288889 + 0.957363i \(0.406714\pi\)
\(860\) 0 0
\(861\) 1.73628e29 0.0264293
\(862\) 0 0
\(863\) −6.86612e29 −0.101999 −0.0509997 0.998699i \(-0.516241\pi\)
−0.0509997 + 0.998699i \(0.516241\pi\)
\(864\) 0 0
\(865\) −7.04330e29 −0.102119
\(866\) 0 0
\(867\) −1.48656e30 −0.210369
\(868\) 0 0
\(869\) −8.93848e30 −1.23468
\(870\) 0 0
\(871\) −7.70751e30 −1.03926
\(872\) 0 0
\(873\) −3.59391e30 −0.473061
\(874\) 0 0
\(875\) −4.88155e30 −0.627297
\(876\) 0 0
\(877\) 1.41886e31 1.78010 0.890049 0.455865i \(-0.150670\pi\)
0.890049 + 0.455865i \(0.150670\pi\)
\(878\) 0 0
\(879\) 2.74597e29 0.0336367
\(880\) 0 0
\(881\) 9.83090e30 1.17583 0.587917 0.808921i \(-0.299948\pi\)
0.587917 + 0.808921i \(0.299948\pi\)
\(882\) 0 0
\(883\) −1.06031e30 −0.123835 −0.0619176 0.998081i \(-0.519722\pi\)
−0.0619176 + 0.998081i \(0.519722\pi\)
\(884\) 0 0
\(885\) −1.79948e30 −0.205231
\(886\) 0 0
\(887\) 8.96894e30 0.998949 0.499474 0.866329i \(-0.333527\pi\)
0.499474 + 0.866329i \(0.333527\pi\)
\(888\) 0 0
\(889\) −1.94252e30 −0.211299
\(890\) 0 0
\(891\) −1.40613e30 −0.149386
\(892\) 0 0
\(893\) −1.65801e31 −1.72047
\(894\) 0 0
\(895\) −7.22640e30 −0.732453
\(896\) 0 0
\(897\) −5.85526e30 −0.579730
\(898\) 0 0
\(899\) −2.43677e30 −0.235688
\(900\) 0 0
\(901\) −9.24114e30 −0.873203
\(902\) 0 0
\(903\) −6.60473e30 −0.609724
\(904\) 0 0
\(905\) −6.85303e30 −0.618119
\(906\) 0 0
\(907\) −3.04599e30 −0.268443 −0.134221 0.990951i \(-0.542853\pi\)
−0.134221 + 0.990951i \(0.542853\pi\)
\(908\) 0 0
\(909\) −2.66918e30 −0.229857
\(910\) 0 0
\(911\) −9.54796e30 −0.803466 −0.401733 0.915757i \(-0.631592\pi\)
−0.401733 + 0.915757i \(0.631592\pi\)
\(912\) 0 0
\(913\) −1.80077e31 −1.48087
\(914\) 0 0
\(915\) 7.58860e30 0.609874
\(916\) 0 0
\(917\) 9.24824e30 0.726409
\(918\) 0 0
\(919\) −1.56370e31 −1.20044 −0.600219 0.799835i \(-0.704920\pi\)
−0.600219 + 0.799835i \(0.704920\pi\)
\(920\) 0 0
\(921\) 4.06905e30 0.305328
\(922\) 0 0
\(923\) 3.94318e30 0.289220
\(924\) 0 0
\(925\) 1.08720e31 0.779511
\(926\) 0 0
\(927\) 3.12947e30 0.219348
\(928\) 0 0
\(929\) −4.04044e30 −0.276862 −0.138431 0.990372i \(-0.544206\pi\)
−0.138431 + 0.990372i \(0.544206\pi\)
\(930\) 0 0
\(931\) 9.39051e30 0.629097
\(932\) 0 0
\(933\) 1.62535e30 0.106461
\(934\) 0 0
\(935\) −9.92888e30 −0.635885
\(936\) 0 0
\(937\) −2.09838e31 −1.31407 −0.657035 0.753860i \(-0.728190\pi\)
−0.657035 + 0.753860i \(0.728190\pi\)
\(938\) 0 0
\(939\) −8.93147e30 −0.546933
\(940\) 0 0
\(941\) 2.17744e31 1.30393 0.651967 0.758247i \(-0.273944\pi\)
0.651967 + 0.758247i \(0.273944\pi\)
\(942\) 0 0
\(943\) −2.05660e30 −0.120442
\(944\) 0 0
\(945\) 1.27893e30 0.0732511
\(946\) 0 0
\(947\) −1.21468e31 −0.680435 −0.340218 0.940347i \(-0.610501\pi\)
−0.340218 + 0.940347i \(0.610501\pi\)
\(948\) 0 0
\(949\) 2.67202e30 0.146401
\(950\) 0 0
\(951\) −3.22383e30 −0.172773
\(952\) 0 0
\(953\) 6.95212e30 0.364453 0.182227 0.983257i \(-0.441670\pi\)
0.182227 + 0.983257i \(0.441670\pi\)
\(954\) 0 0
\(955\) 1.06129e30 0.0544248
\(956\) 0 0
\(957\) 4.77534e30 0.239568
\(958\) 0 0
\(959\) −1.40754e31 −0.690823
\(960\) 0 0
\(961\) −8.68058e30 −0.416824
\(962\) 0 0
\(963\) −5.98939e30 −0.281389
\(964\) 0 0
\(965\) 8.55201e30 0.393126
\(966\) 0 0
\(967\) 8.85795e30 0.398433 0.199216 0.979956i \(-0.436160\pi\)
0.199216 + 0.979956i \(0.436160\pi\)
\(968\) 0 0
\(969\) −1.11823e31 −0.492190
\(970\) 0 0
\(971\) 3.05382e31 1.31535 0.657676 0.753301i \(-0.271540\pi\)
0.657676 + 0.753301i \(0.271540\pi\)
\(972\) 0 0
\(973\) −2.03346e31 −0.857138
\(974\) 0 0
\(975\) −5.39509e30 −0.222561
\(976\) 0 0
\(977\) −2.38019e31 −0.960989 −0.480495 0.876998i \(-0.659543\pi\)
−0.480495 + 0.876998i \(0.659543\pi\)
\(978\) 0 0
\(979\) −2.93583e31 −1.16014
\(980\) 0 0
\(981\) 1.04550e31 0.404391
\(982\) 0 0
\(983\) 3.40761e31 1.29014 0.645072 0.764122i \(-0.276827\pi\)
0.645072 + 0.764122i \(0.276827\pi\)
\(984\) 0 0
\(985\) −1.13922e31 −0.422207
\(986\) 0 0
\(987\) 1.64284e31 0.596026
\(988\) 0 0
\(989\) 7.82322e31 2.77859
\(990\) 0 0
\(991\) 3.53546e31 1.22934 0.614671 0.788784i \(-0.289289\pi\)
0.614671 + 0.788784i \(0.289289\pi\)
\(992\) 0 0
\(993\) −2.70420e31 −0.920600
\(994\) 0 0
\(995\) −1.14305e31 −0.380997
\(996\) 0 0
\(997\) −4.23577e31 −1.38240 −0.691199 0.722664i \(-0.742917\pi\)
−0.691199 + 0.722664i \(0.742917\pi\)
\(998\) 0 0
\(999\) −7.24357e30 −0.231481
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 48.22.a.f.1.1 1
4.3 odd 2 6.22.a.b.1.1 1
12.11 even 2 18.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.22.a.b.1.1 1 4.3 odd 2
18.22.a.a.1.1 1 12.11 even 2
48.22.a.f.1.1 1 1.1 even 1 trivial