Properties

Label 16.22.a.f.1.3
Level $16$
Weight $22$
Character 16.1
Self dual yes
Analytic conductor $44.716$
Analytic rank $1$
Dimension $3$
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] = [16,22,Mod(1,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("16.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 16.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: \(44.7163750859\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4963x + 96223 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-78.2002\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+97085.2 q^{3} -3.39292e7 q^{5} +5.28731e8 q^{7} -1.03483e9 q^{9} +O(q^{10})\) \(q+97085.2 q^{3} -3.39292e7 q^{5} +5.28731e8 q^{7} -1.03483e9 q^{9} +1.21955e11 q^{11} +4.33605e11 q^{13} -3.29402e12 q^{15} -1.31741e13 q^{17} -2.18808e13 q^{19} +5.13320e13 q^{21} -1.40779e14 q^{23} +6.74355e14 q^{25} -1.11601e15 q^{27} +1.17115e15 q^{29} -9.57156e14 q^{31} +1.18400e16 q^{33} -1.79394e16 q^{35} -3.53720e16 q^{37} +4.20966e16 q^{39} -1.71456e17 q^{41} -1.35679e17 q^{43} +3.51109e16 q^{45} +5.75514e17 q^{47} -2.78989e17 q^{49} -1.27901e18 q^{51} -9.62123e17 q^{53} -4.13782e18 q^{55} -2.12430e18 q^{57} -4.87671e18 q^{59} +4.59195e18 q^{61} -5.47145e17 q^{63} -1.47119e19 q^{65} +1.78611e19 q^{67} -1.36675e19 q^{69} -2.93629e19 q^{71} -8.32585e18 q^{73} +6.54699e19 q^{75} +6.44812e19 q^{77} +7.05718e19 q^{79} -9.75235e19 q^{81} -2.11929e20 q^{83} +4.46988e20 q^{85} +1.13702e20 q^{87} +2.62972e20 q^{89} +2.29260e20 q^{91} -9.29257e19 q^{93} +7.42397e20 q^{95} +3.84293e20 q^{97} -1.26202e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 96764 q^{3} - 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 96764 q^{3} - 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9} + 40335108684 q^{11} + 133734425946 q^{13} + 1223136458200 q^{15} + 7797732274422 q^{17} - 35788199781996 q^{19} + 198539224853088 q^{21} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots + 94\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 97085.2 0.949248 0.474624 0.880189i \(-0.342584\pi\)
0.474624 + 0.880189i \(0.342584\pi\)
\(4\) 0 0
\(5\) −3.39292e7 −1.55378 −0.776889 0.629638i \(-0.783203\pi\)
−0.776889 + 0.629638i \(0.783203\pi\)
\(6\) 0 0
\(7\) 5.28731e8 0.707466 0.353733 0.935346i \(-0.384912\pi\)
0.353733 + 0.935346i \(0.384912\pi\)
\(8\) 0 0
\(9\) −1.03483e9 −0.0989284
\(10\) 0 0
\(11\) 1.21955e11 1.41767 0.708835 0.705375i \(-0.249221\pi\)
0.708835 + 0.705375i \(0.249221\pi\)
\(12\) 0 0
\(13\) 4.33605e11 0.872346 0.436173 0.899863i \(-0.356334\pi\)
0.436173 + 0.899863i \(0.356334\pi\)
\(14\) 0 0
\(15\) −3.29402e12 −1.47492
\(16\) 0 0
\(17\) −1.31741e13 −1.58492 −0.792461 0.609922i \(-0.791201\pi\)
−0.792461 + 0.609922i \(0.791201\pi\)
\(18\) 0 0
\(19\) −2.18808e13 −0.818747 −0.409373 0.912367i \(-0.634253\pi\)
−0.409373 + 0.912367i \(0.634253\pi\)
\(20\) 0 0
\(21\) 5.13320e13 0.671561
\(22\) 0 0
\(23\) −1.40779e14 −0.708589 −0.354294 0.935134i \(-0.615279\pi\)
−0.354294 + 0.935134i \(0.615279\pi\)
\(24\) 0 0
\(25\) 6.74355e14 1.41423
\(26\) 0 0
\(27\) −1.11601e15 −1.04316
\(28\) 0 0
\(29\) 1.17115e15 0.516934 0.258467 0.966020i \(-0.416783\pi\)
0.258467 + 0.966020i \(0.416783\pi\)
\(30\) 0 0
\(31\) −9.57156e14 −0.209742 −0.104871 0.994486i \(-0.533443\pi\)
−0.104871 + 0.994486i \(0.533443\pi\)
\(32\) 0 0
\(33\) 1.18400e16 1.34572
\(34\) 0 0
\(35\) −1.79394e16 −1.09925
\(36\) 0 0
\(37\) −3.53720e16 −1.20932 −0.604661 0.796483i \(-0.706691\pi\)
−0.604661 + 0.796483i \(0.706691\pi\)
\(38\) 0 0
\(39\) 4.20966e16 0.828073
\(40\) 0 0
\(41\) −1.71456e17 −1.99490 −0.997451 0.0713504i \(-0.977269\pi\)
−0.997451 + 0.0713504i \(0.977269\pi\)
\(42\) 0 0
\(43\) −1.35679e17 −0.957398 −0.478699 0.877979i \(-0.658892\pi\)
−0.478699 + 0.877979i \(0.658892\pi\)
\(44\) 0 0
\(45\) 3.51109e16 0.153713
\(46\) 0 0
\(47\) 5.75514e17 1.59598 0.797991 0.602669i \(-0.205896\pi\)
0.797991 + 0.602669i \(0.205896\pi\)
\(48\) 0 0
\(49\) −2.78989e17 −0.499492
\(50\) 0 0
\(51\) −1.27901e18 −1.50448
\(52\) 0 0
\(53\) −9.62123e17 −0.755673 −0.377837 0.925872i \(-0.623332\pi\)
−0.377837 + 0.925872i \(0.623332\pi\)
\(54\) 0 0
\(55\) −4.13782e18 −2.20274
\(56\) 0 0
\(57\) −2.12430e18 −0.777194
\(58\) 0 0
\(59\) −4.87671e18 −1.24217 −0.621085 0.783743i \(-0.713308\pi\)
−0.621085 + 0.783743i \(0.713308\pi\)
\(60\) 0 0
\(61\) 4.59195e18 0.824202 0.412101 0.911138i \(-0.364795\pi\)
0.412101 + 0.911138i \(0.364795\pi\)
\(62\) 0 0
\(63\) −5.47145e17 −0.0699885
\(64\) 0 0
\(65\) −1.47119e19 −1.35543
\(66\) 0 0
\(67\) 1.78611e19 1.19708 0.598539 0.801094i \(-0.295748\pi\)
0.598539 + 0.801094i \(0.295748\pi\)
\(68\) 0 0
\(69\) −1.36675e19 −0.672626
\(70\) 0 0
\(71\) −2.93629e19 −1.07050 −0.535250 0.844694i \(-0.679783\pi\)
−0.535250 + 0.844694i \(0.679783\pi\)
\(72\) 0 0
\(73\) −8.32585e18 −0.226745 −0.113373 0.993553i \(-0.536165\pi\)
−0.113373 + 0.993553i \(0.536165\pi\)
\(74\) 0 0
\(75\) 6.54699e19 1.34245
\(76\) 0 0
\(77\) 6.44812e19 1.00295
\(78\) 0 0
\(79\) 7.05718e19 0.838584 0.419292 0.907851i \(-0.362278\pi\)
0.419292 + 0.907851i \(0.362278\pi\)
\(80\) 0 0
\(81\) −9.75235e19 −0.891285
\(82\) 0 0
\(83\) −2.11929e20 −1.49924 −0.749620 0.661869i \(-0.769764\pi\)
−0.749620 + 0.661869i \(0.769764\pi\)
\(84\) 0 0
\(85\) 4.46988e20 2.46262
\(86\) 0 0
\(87\) 1.13702e20 0.490698
\(88\) 0 0
\(89\) 2.62972e20 0.893952 0.446976 0.894546i \(-0.352501\pi\)
0.446976 + 0.894546i \(0.352501\pi\)
\(90\) 0 0
\(91\) 2.29260e20 0.617155
\(92\) 0 0
\(93\) −9.29257e19 −0.199097
\(94\) 0 0
\(95\) 7.42397e20 1.27215
\(96\) 0 0
\(97\) 3.84293e20 0.529126 0.264563 0.964368i \(-0.414772\pi\)
0.264563 + 0.964368i \(0.414772\pi\)
\(98\) 0 0
\(99\) −1.26202e20 −0.140248
\(100\) 0 0
\(101\) −1.92132e21 −1.73071 −0.865356 0.501158i \(-0.832908\pi\)
−0.865356 + 0.501158i \(0.832908\pi\)
\(102\) 0 0
\(103\) −6.06205e20 −0.444456 −0.222228 0.974995i \(-0.571333\pi\)
−0.222228 + 0.974995i \(0.571333\pi\)
\(104\) 0 0
\(105\) −1.74165e21 −1.04346
\(106\) 0 0
\(107\) −7.79821e20 −0.383235 −0.191618 0.981470i \(-0.561373\pi\)
−0.191618 + 0.981470i \(0.561373\pi\)
\(108\) 0 0
\(109\) −2.58673e20 −0.104658 −0.0523291 0.998630i \(-0.516664\pi\)
−0.0523291 + 0.998630i \(0.516664\pi\)
\(110\) 0 0
\(111\) −3.43410e21 −1.14795
\(112\) 0 0
\(113\) 1.56041e21 0.432428 0.216214 0.976346i \(-0.430629\pi\)
0.216214 + 0.976346i \(0.430629\pi\)
\(114\) 0 0
\(115\) 4.77651e21 1.10099
\(116\) 0 0
\(117\) −4.48706e20 −0.0862998
\(118\) 0 0
\(119\) −6.96557e21 −1.12128
\(120\) 0 0
\(121\) 7.47267e21 1.00979
\(122\) 0 0
\(123\) −1.66458e22 −1.89366
\(124\) 0 0
\(125\) −6.70164e21 −0.643615
\(126\) 0 0
\(127\) 7.53317e21 0.612405 0.306202 0.951966i \(-0.400942\pi\)
0.306202 + 0.951966i \(0.400942\pi\)
\(128\) 0 0
\(129\) −1.31724e22 −0.908808
\(130\) 0 0
\(131\) 1.41613e22 0.831291 0.415645 0.909527i \(-0.363556\pi\)
0.415645 + 0.909527i \(0.363556\pi\)
\(132\) 0 0
\(133\) −1.15690e22 −0.579236
\(134\) 0 0
\(135\) 3.78654e22 1.62083
\(136\) 0 0
\(137\) 2.76400e22 1.01385 0.506924 0.861991i \(-0.330782\pi\)
0.506924 + 0.861991i \(0.330782\pi\)
\(138\) 0 0
\(139\) −1.98007e21 −0.0623770 −0.0311885 0.999514i \(-0.509929\pi\)
−0.0311885 + 0.999514i \(0.509929\pi\)
\(140\) 0 0
\(141\) 5.58738e22 1.51498
\(142\) 0 0
\(143\) 5.28801e22 1.23670
\(144\) 0 0
\(145\) −3.97363e22 −0.803200
\(146\) 0 0
\(147\) −2.70857e22 −0.474141
\(148\) 0 0
\(149\) 2.76024e22 0.419268 0.209634 0.977780i \(-0.432773\pi\)
0.209634 + 0.977780i \(0.432773\pi\)
\(150\) 0 0
\(151\) −4.72139e22 −0.623465 −0.311732 0.950170i \(-0.600909\pi\)
−0.311732 + 0.950170i \(0.600909\pi\)
\(152\) 0 0
\(153\) 1.36329e22 0.156794
\(154\) 0 0
\(155\) 3.24756e22 0.325892
\(156\) 0 0
\(157\) 1.89837e23 1.66508 0.832540 0.553965i \(-0.186886\pi\)
0.832540 + 0.553965i \(0.186886\pi\)
\(158\) 0 0
\(159\) −9.34079e22 −0.717321
\(160\) 0 0
\(161\) −7.44340e22 −0.501302
\(162\) 0 0
\(163\) −1.59931e23 −0.946158 −0.473079 0.881020i \(-0.656857\pi\)
−0.473079 + 0.881020i \(0.656857\pi\)
\(164\) 0 0
\(165\) −4.01721e23 −2.09095
\(166\) 0 0
\(167\) −3.26974e22 −0.149965 −0.0749825 0.997185i \(-0.523890\pi\)
−0.0749825 + 0.997185i \(0.523890\pi\)
\(168\) 0 0
\(169\) −5.90514e22 −0.239012
\(170\) 0 0
\(171\) 2.26428e22 0.0809973
\(172\) 0 0
\(173\) −5.17436e22 −0.163822 −0.0819111 0.996640i \(-0.526102\pi\)
−0.0819111 + 0.996640i \(0.526102\pi\)
\(174\) 0 0
\(175\) 3.56553e23 1.00052
\(176\) 0 0
\(177\) −4.73457e23 −1.17913
\(178\) 0 0
\(179\) −3.75079e22 −0.0830167 −0.0415084 0.999138i \(-0.513216\pi\)
−0.0415084 + 0.999138i \(0.513216\pi\)
\(180\) 0 0
\(181\) −3.65662e22 −0.0720203 −0.0360102 0.999351i \(-0.511465\pi\)
−0.0360102 + 0.999351i \(0.511465\pi\)
\(182\) 0 0
\(183\) 4.45810e23 0.782372
\(184\) 0 0
\(185\) 1.20015e24 1.87902
\(186\) 0 0
\(187\) −1.60664e24 −2.24690
\(188\) 0 0
\(189\) −5.90070e23 −0.737997
\(190\) 0 0
\(191\) 4.61401e23 0.516688 0.258344 0.966053i \(-0.416823\pi\)
0.258344 + 0.966053i \(0.416823\pi\)
\(192\) 0 0
\(193\) 2.52365e23 0.253324 0.126662 0.991946i \(-0.459574\pi\)
0.126662 + 0.991946i \(0.459574\pi\)
\(194\) 0 0
\(195\) −1.42830e24 −1.28664
\(196\) 0 0
\(197\) −7.91314e23 −0.640403 −0.320202 0.947349i \(-0.603751\pi\)
−0.320202 + 0.947349i \(0.603751\pi\)
\(198\) 0 0
\(199\) 2.46245e24 1.79230 0.896150 0.443752i \(-0.146353\pi\)
0.896150 + 0.443752i \(0.146353\pi\)
\(200\) 0 0
\(201\) 1.73405e24 1.13632
\(202\) 0 0
\(203\) 6.19226e23 0.365713
\(204\) 0 0
\(205\) 5.81736e24 3.09964
\(206\) 0 0
\(207\) 1.45681e23 0.0700995
\(208\) 0 0
\(209\) −2.66846e24 −1.16071
\(210\) 0 0
\(211\) 1.79516e24 0.706542 0.353271 0.935521i \(-0.385069\pi\)
0.353271 + 0.935521i \(0.385069\pi\)
\(212\) 0 0
\(213\) −2.85070e24 −1.01617
\(214\) 0 0
\(215\) 4.60347e24 1.48758
\(216\) 0 0
\(217\) −5.06079e23 −0.148385
\(218\) 0 0
\(219\) −8.08316e23 −0.215238
\(220\) 0 0
\(221\) −5.71236e24 −1.38260
\(222\) 0 0
\(223\) −3.76539e24 −0.829103 −0.414551 0.910026i \(-0.636061\pi\)
−0.414551 + 0.910026i \(0.636061\pi\)
\(224\) 0 0
\(225\) −6.97841e23 −0.139907
\(226\) 0 0
\(227\) −8.37475e24 −1.53003 −0.765016 0.644011i \(-0.777269\pi\)
−0.765016 + 0.644011i \(0.777269\pi\)
\(228\) 0 0
\(229\) 3.14314e24 0.523710 0.261855 0.965107i \(-0.415666\pi\)
0.261855 + 0.965107i \(0.415666\pi\)
\(230\) 0 0
\(231\) 6.26017e24 0.952051
\(232\) 0 0
\(233\) −4.21087e24 −0.584971 −0.292486 0.956270i \(-0.594482\pi\)
−0.292486 + 0.956270i \(0.594482\pi\)
\(234\) 0 0
\(235\) −1.95267e25 −2.47980
\(236\) 0 0
\(237\) 6.85147e24 0.796024
\(238\) 0 0
\(239\) 9.27921e24 0.987036 0.493518 0.869736i \(-0.335711\pi\)
0.493518 + 0.869736i \(0.335711\pi\)
\(240\) 0 0
\(241\) 1.45105e24 0.141418 0.0707090 0.997497i \(-0.477474\pi\)
0.0707090 + 0.997497i \(0.477474\pi\)
\(242\) 0 0
\(243\) 2.20579e24 0.197105
\(244\) 0 0
\(245\) 9.46588e24 0.776099
\(246\) 0 0
\(247\) −9.48760e24 −0.714231
\(248\) 0 0
\(249\) −2.05752e25 −1.42315
\(250\) 0 0
\(251\) −2.04939e25 −1.30332 −0.651658 0.758513i \(-0.725926\pi\)
−0.651658 + 0.758513i \(0.725926\pi\)
\(252\) 0 0
\(253\) −1.71686e25 −1.00454
\(254\) 0 0
\(255\) 4.33959e25 2.33763
\(256\) 0 0
\(257\) 1.77825e25 0.882462 0.441231 0.897394i \(-0.354542\pi\)
0.441231 + 0.897394i \(0.354542\pi\)
\(258\) 0 0
\(259\) −1.87023e25 −0.855554
\(260\) 0 0
\(261\) −1.21194e24 −0.0511394
\(262\) 0 0
\(263\) 3.76203e25 1.46517 0.732583 0.680678i \(-0.238315\pi\)
0.732583 + 0.680678i \(0.238315\pi\)
\(264\) 0 0
\(265\) 3.26441e25 1.17415
\(266\) 0 0
\(267\) 2.55307e25 0.848582
\(268\) 0 0
\(269\) −5.82547e25 −1.79033 −0.895163 0.445739i \(-0.852941\pi\)
−0.895163 + 0.445739i \(0.852941\pi\)
\(270\) 0 0
\(271\) −4.00531e25 −1.13883 −0.569414 0.822051i \(-0.692830\pi\)
−0.569414 + 0.822051i \(0.692830\pi\)
\(272\) 0 0
\(273\) 2.22578e25 0.585834
\(274\) 0 0
\(275\) 8.22407e25 2.00490
\(276\) 0 0
\(277\) 5.12820e25 1.15858 0.579292 0.815120i \(-0.303329\pi\)
0.579292 + 0.815120i \(0.303329\pi\)
\(278\) 0 0
\(279\) 9.90490e23 0.0207494
\(280\) 0 0
\(281\) −1.22313e25 −0.237714 −0.118857 0.992911i \(-0.537923\pi\)
−0.118857 + 0.992911i \(0.537923\pi\)
\(282\) 0 0
\(283\) −8.45355e25 −1.52504 −0.762520 0.646964i \(-0.776038\pi\)
−0.762520 + 0.646964i \(0.776038\pi\)
\(284\) 0 0
\(285\) 7.20758e25 1.20759
\(286\) 0 0
\(287\) −9.06541e25 −1.41133
\(288\) 0 0
\(289\) 1.04466e26 1.51198
\(290\) 0 0
\(291\) 3.73092e25 0.502272
\(292\) 0 0
\(293\) −2.19428e25 −0.274905 −0.137453 0.990508i \(-0.543891\pi\)
−0.137453 + 0.990508i \(0.543891\pi\)
\(294\) 0 0
\(295\) 1.65463e26 1.93006
\(296\) 0 0
\(297\) −1.36103e26 −1.47885
\(298\) 0 0
\(299\) −6.10422e25 −0.618135
\(300\) 0 0
\(301\) −7.17375e25 −0.677327
\(302\) 0 0
\(303\) −1.86531e26 −1.64287
\(304\) 0 0
\(305\) −1.55801e26 −1.28063
\(306\) 0 0
\(307\) −3.79642e25 −0.291354 −0.145677 0.989332i \(-0.546536\pi\)
−0.145677 + 0.989332i \(0.546536\pi\)
\(308\) 0 0
\(309\) −5.88535e25 −0.421899
\(310\) 0 0
\(311\) −7.02783e25 −0.470801 −0.235401 0.971898i \(-0.575640\pi\)
−0.235401 + 0.971898i \(0.575640\pi\)
\(312\) 0 0
\(313\) −9.09673e25 −0.569731 −0.284865 0.958568i \(-0.591949\pi\)
−0.284865 + 0.958568i \(0.591949\pi\)
\(314\) 0 0
\(315\) 1.85642e25 0.108747
\(316\) 0 0
\(317\) −2.99033e25 −0.163907 −0.0819533 0.996636i \(-0.526116\pi\)
−0.0819533 + 0.996636i \(0.526116\pi\)
\(318\) 0 0
\(319\) 1.42828e26 0.732841
\(320\) 0 0
\(321\) −7.57091e25 −0.363785
\(322\) 0 0
\(323\) 2.88260e26 1.29765
\(324\) 0 0
\(325\) 2.92404e26 1.23369
\(326\) 0 0
\(327\) −2.51133e25 −0.0993465
\(328\) 0 0
\(329\) 3.04292e26 1.12910
\(330\) 0 0
\(331\) 6.61150e25 0.230200 0.115100 0.993354i \(-0.463281\pi\)
0.115100 + 0.993354i \(0.463281\pi\)
\(332\) 0 0
\(333\) 3.66039e25 0.119636
\(334\) 0 0
\(335\) −6.06013e26 −1.85999
\(336\) 0 0
\(337\) −1.11607e26 −0.321793 −0.160896 0.986971i \(-0.551438\pi\)
−0.160896 + 0.986971i \(0.551438\pi\)
\(338\) 0 0
\(339\) 1.51492e26 0.410482
\(340\) 0 0
\(341\) −1.16730e26 −0.297345
\(342\) 0 0
\(343\) −4.42831e26 −1.06084
\(344\) 0 0
\(345\) 4.63728e26 1.04511
\(346\) 0 0
\(347\) 5.94900e26 1.26178 0.630891 0.775872i \(-0.282690\pi\)
0.630891 + 0.775872i \(0.282690\pi\)
\(348\) 0 0
\(349\) 4.45113e26 0.888799 0.444400 0.895829i \(-0.353417\pi\)
0.444400 + 0.895829i \(0.353417\pi\)
\(350\) 0 0
\(351\) −4.83908e26 −0.909993
\(352\) 0 0
\(353\) 2.37403e26 0.420584 0.210292 0.977639i \(-0.432559\pi\)
0.210292 + 0.977639i \(0.432559\pi\)
\(354\) 0 0
\(355\) 9.96262e26 1.66332
\(356\) 0 0
\(357\) −6.76254e26 −1.06437
\(358\) 0 0
\(359\) 3.45652e26 0.513036 0.256518 0.966539i \(-0.417425\pi\)
0.256518 + 0.966539i \(0.417425\pi\)
\(360\) 0 0
\(361\) −2.35442e26 −0.329654
\(362\) 0 0
\(363\) 7.25485e26 0.958537
\(364\) 0 0
\(365\) 2.82490e26 0.352312
\(366\) 0 0
\(367\) 8.76689e26 1.03241 0.516205 0.856465i \(-0.327344\pi\)
0.516205 + 0.856465i \(0.327344\pi\)
\(368\) 0 0
\(369\) 1.77427e26 0.197353
\(370\) 0 0
\(371\) −5.08705e26 −0.534613
\(372\) 0 0
\(373\) −1.54618e27 −1.53573 −0.767867 0.640609i \(-0.778682\pi\)
−0.767867 + 0.640609i \(0.778682\pi\)
\(374\) 0 0
\(375\) −6.50630e26 −0.610950
\(376\) 0 0
\(377\) 5.07818e26 0.450945
\(378\) 0 0
\(379\) −1.48367e27 −1.24631 −0.623156 0.782097i \(-0.714150\pi\)
−0.623156 + 0.782097i \(0.714150\pi\)
\(380\) 0 0
\(381\) 7.31359e26 0.581324
\(382\) 0 0
\(383\) −1.86137e27 −1.40038 −0.700190 0.713956i \(-0.746901\pi\)
−0.700190 + 0.713956i \(0.746901\pi\)
\(384\) 0 0
\(385\) −2.18780e27 −1.55837
\(386\) 0 0
\(387\) 1.40404e26 0.0947139
\(388\) 0 0
\(389\) −4.43791e26 −0.283601 −0.141801 0.989895i \(-0.545289\pi\)
−0.141801 + 0.989895i \(0.545289\pi\)
\(390\) 0 0
\(391\) 1.85463e27 1.12306
\(392\) 0 0
\(393\) 1.37485e27 0.789101
\(394\) 0 0
\(395\) −2.39445e27 −1.30297
\(396\) 0 0
\(397\) −3.85978e26 −0.199188 −0.0995939 0.995028i \(-0.531754\pi\)
−0.0995939 + 0.995028i \(0.531754\pi\)
\(398\) 0 0
\(399\) −1.12318e27 −0.549838
\(400\) 0 0
\(401\) −8.65938e26 −0.402227 −0.201113 0.979568i \(-0.564456\pi\)
−0.201113 + 0.979568i \(0.564456\pi\)
\(402\) 0 0
\(403\) −4.15028e26 −0.182967
\(404\) 0 0
\(405\) 3.30890e27 1.38486
\(406\) 0 0
\(407\) −4.31378e27 −1.71442
\(408\) 0 0
\(409\) 6.19682e26 0.233924 0.116962 0.993136i \(-0.462684\pi\)
0.116962 + 0.993136i \(0.462684\pi\)
\(410\) 0 0
\(411\) 2.68344e27 0.962393
\(412\) 0 0
\(413\) −2.57847e27 −0.878793
\(414\) 0 0
\(415\) 7.19060e27 2.32948
\(416\) 0 0
\(417\) −1.92235e26 −0.0592112
\(418\) 0 0
\(419\) 2.87299e27 0.841562 0.420781 0.907162i \(-0.361756\pi\)
0.420781 + 0.907162i \(0.361756\pi\)
\(420\) 0 0
\(421\) 2.25957e27 0.629598 0.314799 0.949158i \(-0.398063\pi\)
0.314799 + 0.949158i \(0.398063\pi\)
\(422\) 0 0
\(423\) −5.95557e26 −0.157888
\(424\) 0 0
\(425\) −8.88404e27 −2.24144
\(426\) 0 0
\(427\) 2.42791e27 0.583095
\(428\) 0 0
\(429\) 5.13387e27 1.17393
\(430\) 0 0
\(431\) 3.01840e27 0.657303 0.328652 0.944451i \(-0.393406\pi\)
0.328652 + 0.944451i \(0.393406\pi\)
\(432\) 0 0
\(433\) 6.44418e27 1.33673 0.668367 0.743832i \(-0.266994\pi\)
0.668367 + 0.743832i \(0.266994\pi\)
\(434\) 0 0
\(435\) −3.85781e27 −0.762436
\(436\) 0 0
\(437\) 3.08034e27 0.580155
\(438\) 0 0
\(439\) 8.50049e27 1.52604 0.763021 0.646374i \(-0.223715\pi\)
0.763021 + 0.646374i \(0.223715\pi\)
\(440\) 0 0
\(441\) 2.88705e26 0.0494139
\(442\) 0 0
\(443\) −6.00246e27 −0.979694 −0.489847 0.871809i \(-0.662947\pi\)
−0.489847 + 0.871809i \(0.662947\pi\)
\(444\) 0 0
\(445\) −8.92243e27 −1.38900
\(446\) 0 0
\(447\) 2.67979e27 0.397989
\(448\) 0 0
\(449\) 8.58205e27 1.21620 0.608100 0.793861i \(-0.291932\pi\)
0.608100 + 0.793861i \(0.291932\pi\)
\(450\) 0 0
\(451\) −2.09098e28 −2.82811
\(452\) 0 0
\(453\) −4.58377e27 −0.591823
\(454\) 0 0
\(455\) −7.77863e27 −0.958922
\(456\) 0 0
\(457\) 1.09251e28 1.28619 0.643093 0.765788i \(-0.277651\pi\)
0.643093 + 0.765788i \(0.277651\pi\)
\(458\) 0 0
\(459\) 1.47025e28 1.65332
\(460\) 0 0
\(461\) −7.29500e27 −0.783728 −0.391864 0.920023i \(-0.628170\pi\)
−0.391864 + 0.920023i \(0.628170\pi\)
\(462\) 0 0
\(463\) 1.13930e28 1.16960 0.584800 0.811178i \(-0.301173\pi\)
0.584800 + 0.811178i \(0.301173\pi\)
\(464\) 0 0
\(465\) 3.15290e27 0.309353
\(466\) 0 0
\(467\) −1.51114e28 −1.41735 −0.708677 0.705533i \(-0.750708\pi\)
−0.708677 + 0.705533i \(0.750708\pi\)
\(468\) 0 0
\(469\) 9.44371e27 0.846892
\(470\) 0 0
\(471\) 1.84304e28 1.58057
\(472\) 0 0
\(473\) −1.65466e28 −1.35727
\(474\) 0 0
\(475\) −1.47554e28 −1.15789
\(476\) 0 0
\(477\) 9.95630e26 0.0747576
\(478\) 0 0
\(479\) −3.89042e27 −0.279559 −0.139779 0.990183i \(-0.544639\pi\)
−0.139779 + 0.990183i \(0.544639\pi\)
\(480\) 0 0
\(481\) −1.53375e28 −1.05495
\(482\) 0 0
\(483\) −7.22644e27 −0.475860
\(484\) 0 0
\(485\) −1.30388e28 −0.822145
\(486\) 0 0
\(487\) −1.89704e28 −1.14557 −0.572787 0.819704i \(-0.694138\pi\)
−0.572787 + 0.819704i \(0.694138\pi\)
\(488\) 0 0
\(489\) −1.55270e28 −0.898138
\(490\) 0 0
\(491\) 1.33589e28 0.740313 0.370157 0.928969i \(-0.379304\pi\)
0.370157 + 0.928969i \(0.379304\pi\)
\(492\) 0 0
\(493\) −1.54289e28 −0.819300
\(494\) 0 0
\(495\) 4.28193e27 0.217914
\(496\) 0 0
\(497\) −1.55251e28 −0.757343
\(498\) 0 0
\(499\) 2.86135e28 1.33818 0.669090 0.743181i \(-0.266684\pi\)
0.669090 + 0.743181i \(0.266684\pi\)
\(500\) 0 0
\(501\) −3.17443e27 −0.142354
\(502\) 0 0
\(503\) −9.44655e27 −0.406265 −0.203133 0.979151i \(-0.565112\pi\)
−0.203133 + 0.979151i \(0.565112\pi\)
\(504\) 0 0
\(505\) 6.51888e28 2.68914
\(506\) 0 0
\(507\) −5.73302e27 −0.226882
\(508\) 0 0
\(509\) 2.06060e28 0.782452 0.391226 0.920295i \(-0.372051\pi\)
0.391226 + 0.920295i \(0.372051\pi\)
\(510\) 0 0
\(511\) −4.40214e27 −0.160415
\(512\) 0 0
\(513\) 2.44192e28 0.854080
\(514\) 0 0
\(515\) 2.05681e28 0.690586
\(516\) 0 0
\(517\) 7.01865e28 2.26258
\(518\) 0 0
\(519\) −5.02354e27 −0.155508
\(520\) 0 0
\(521\) −4.30969e28 −1.28130 −0.640649 0.767834i \(-0.721335\pi\)
−0.640649 + 0.767834i \(0.721335\pi\)
\(522\) 0 0
\(523\) −1.08864e28 −0.310897 −0.155448 0.987844i \(-0.549682\pi\)
−0.155448 + 0.987844i \(0.549682\pi\)
\(524\) 0 0
\(525\) 3.46160e28 0.949739
\(526\) 0 0
\(527\) 1.26097e28 0.332425
\(528\) 0 0
\(529\) −1.96530e28 −0.497902
\(530\) 0 0
\(531\) 5.04655e27 0.122886
\(532\) 0 0
\(533\) −7.43441e28 −1.74025
\(534\) 0 0
\(535\) 2.64587e28 0.595462
\(536\) 0 0
\(537\) −3.64146e27 −0.0788035
\(538\) 0 0
\(539\) −3.40240e28 −0.708114
\(540\) 0 0
\(541\) 3.99516e28 0.799765 0.399883 0.916566i \(-0.369051\pi\)
0.399883 + 0.916566i \(0.369051\pi\)
\(542\) 0 0
\(543\) −3.55004e27 −0.0683651
\(544\) 0 0
\(545\) 8.77658e27 0.162616
\(546\) 0 0
\(547\) 5.58498e28 0.995760 0.497880 0.867246i \(-0.334112\pi\)
0.497880 + 0.867246i \(0.334112\pi\)
\(548\) 0 0
\(549\) −4.75187e27 −0.0815370
\(550\) 0 0
\(551\) −2.56257e28 −0.423238
\(552\) 0 0
\(553\) 3.73135e28 0.593270
\(554\) 0 0
\(555\) 1.16516e29 1.78365
\(556\) 0 0
\(557\) −8.87710e28 −1.30855 −0.654277 0.756255i \(-0.727027\pi\)
−0.654277 + 0.756255i \(0.727027\pi\)
\(558\) 0 0
\(559\) −5.88309e28 −0.835183
\(560\) 0 0
\(561\) −1.55981e29 −2.13286
\(562\) 0 0
\(563\) −7.63115e28 −1.00520 −0.502599 0.864520i \(-0.667623\pi\)
−0.502599 + 0.864520i \(0.667623\pi\)
\(564\) 0 0
\(565\) −5.29434e28 −0.671898
\(566\) 0 0
\(567\) −5.15637e28 −0.630554
\(568\) 0 0
\(569\) 1.48073e29 1.74500 0.872502 0.488610i \(-0.162496\pi\)
0.872502 + 0.488610i \(0.162496\pi\)
\(570\) 0 0
\(571\) 7.51419e28 0.853499 0.426749 0.904370i \(-0.359659\pi\)
0.426749 + 0.904370i \(0.359659\pi\)
\(572\) 0 0
\(573\) 4.47952e28 0.490465
\(574\) 0 0
\(575\) −9.49348e28 −1.00210
\(576\) 0 0
\(577\) 1.56608e28 0.159392 0.0796961 0.996819i \(-0.474605\pi\)
0.0796961 + 0.996819i \(0.474605\pi\)
\(578\) 0 0
\(579\) 2.45009e28 0.240468
\(580\) 0 0
\(581\) −1.12054e29 −1.06066
\(582\) 0 0
\(583\) −1.17335e29 −1.07129
\(584\) 0 0
\(585\) 1.52242e28 0.134091
\(586\) 0 0
\(587\) −1.19163e29 −1.01260 −0.506302 0.862356i \(-0.668988\pi\)
−0.506302 + 0.862356i \(0.668988\pi\)
\(588\) 0 0
\(589\) 2.09433e28 0.171725
\(590\) 0 0
\(591\) −7.68248e28 −0.607901
\(592\) 0 0
\(593\) 3.34770e28 0.255666 0.127833 0.991796i \(-0.459198\pi\)
0.127833 + 0.991796i \(0.459198\pi\)
\(594\) 0 0
\(595\) 2.36337e29 1.74222
\(596\) 0 0
\(597\) 2.39068e29 1.70134
\(598\) 0 0
\(599\) −1.39785e29 −0.960461 −0.480231 0.877142i \(-0.659447\pi\)
−0.480231 + 0.877142i \(0.659447\pi\)
\(600\) 0 0
\(601\) −1.02954e27 −0.00683061 −0.00341531 0.999994i \(-0.501087\pi\)
−0.00341531 + 0.999994i \(0.501087\pi\)
\(602\) 0 0
\(603\) −1.84831e28 −0.118425
\(604\) 0 0
\(605\) −2.53542e29 −1.56898
\(606\) 0 0
\(607\) −1.61045e29 −0.962647 −0.481323 0.876543i \(-0.659844\pi\)
−0.481323 + 0.876543i \(0.659844\pi\)
\(608\) 0 0
\(609\) 6.01176e28 0.347152
\(610\) 0 0
\(611\) 2.49546e29 1.39225
\(612\) 0 0
\(613\) −1.08394e29 −0.584345 −0.292173 0.956366i \(-0.594378\pi\)
−0.292173 + 0.956366i \(0.594378\pi\)
\(614\) 0 0
\(615\) 5.64780e29 2.94232
\(616\) 0 0
\(617\) −1.69220e29 −0.852033 −0.426017 0.904715i \(-0.640084\pi\)
−0.426017 + 0.904715i \(0.640084\pi\)
\(618\) 0 0
\(619\) −4.72000e28 −0.229715 −0.114858 0.993382i \(-0.536641\pi\)
−0.114858 + 0.993382i \(0.536641\pi\)
\(620\) 0 0
\(621\) 1.57110e29 0.739168
\(622\) 0 0
\(623\) 1.39041e29 0.632441
\(624\) 0 0
\(625\) −9.41762e28 −0.414191
\(626\) 0 0
\(627\) −2.59068e29 −1.10180
\(628\) 0 0
\(629\) 4.65995e29 1.91668
\(630\) 0 0
\(631\) 4.15217e29 1.65184 0.825918 0.563790i \(-0.190657\pi\)
0.825918 + 0.563790i \(0.190657\pi\)
\(632\) 0 0
\(633\) 1.74284e29 0.670684
\(634\) 0 0
\(635\) −2.55595e29 −0.951541
\(636\) 0 0
\(637\) −1.20971e29 −0.435730
\(638\) 0 0
\(639\) 3.03855e28 0.105903
\(640\) 0 0
\(641\) 9.51731e28 0.321000 0.160500 0.987036i \(-0.448689\pi\)
0.160500 + 0.987036i \(0.448689\pi\)
\(642\) 0 0
\(643\) 3.25707e29 1.06319 0.531596 0.846998i \(-0.321592\pi\)
0.531596 + 0.846998i \(0.321592\pi\)
\(644\) 0 0
\(645\) 4.46929e29 1.41209
\(646\) 0 0
\(647\) 3.06325e28 0.0936888 0.0468444 0.998902i \(-0.485084\pi\)
0.0468444 + 0.998902i \(0.485084\pi\)
\(648\) 0 0
\(649\) −5.94737e29 −1.76099
\(650\) 0 0
\(651\) −4.91327e28 −0.140854
\(652\) 0 0
\(653\) −8.66491e28 −0.240534 −0.120267 0.992742i \(-0.538375\pi\)
−0.120267 + 0.992742i \(0.538375\pi\)
\(654\) 0 0
\(655\) −4.80480e29 −1.29164
\(656\) 0 0
\(657\) 8.61580e27 0.0224316
\(658\) 0 0
\(659\) −7.32944e29 −1.84831 −0.924154 0.382021i \(-0.875228\pi\)
−0.924154 + 0.382021i \(0.875228\pi\)
\(660\) 0 0
\(661\) −5.06204e29 −1.23655 −0.618273 0.785964i \(-0.712167\pi\)
−0.618273 + 0.785964i \(0.712167\pi\)
\(662\) 0 0
\(663\) −5.54586e29 −1.31243
\(664\) 0 0
\(665\) 3.92529e29 0.900004
\(666\) 0 0
\(667\) −1.64873e29 −0.366293
\(668\) 0 0
\(669\) −3.65563e29 −0.787024
\(670\) 0 0
\(671\) 5.60009e29 1.16845
\(672\) 0 0
\(673\) 4.74688e29 0.959953 0.479977 0.877281i \(-0.340645\pi\)
0.479977 + 0.877281i \(0.340645\pi\)
\(674\) 0 0
\(675\) −7.52588e29 −1.47526
\(676\) 0 0
\(677\) 6.45182e29 1.22603 0.613015 0.790071i \(-0.289956\pi\)
0.613015 + 0.790071i \(0.289956\pi\)
\(678\) 0 0
\(679\) 2.03188e29 0.374339
\(680\) 0 0
\(681\) −8.13064e29 −1.45238
\(682\) 0 0
\(683\) 5.14621e29 0.891394 0.445697 0.895184i \(-0.352956\pi\)
0.445697 + 0.895184i \(0.352956\pi\)
\(684\) 0 0
\(685\) −9.37805e29 −1.57529
\(686\) 0 0
\(687\) 3.05152e29 0.497131
\(688\) 0 0
\(689\) −4.17181e29 −0.659209
\(690\) 0 0
\(691\) 1.06801e30 1.63702 0.818512 0.574489i \(-0.194799\pi\)
0.818512 + 0.574489i \(0.194799\pi\)
\(692\) 0 0
\(693\) −6.67268e28 −0.0992205
\(694\) 0 0
\(695\) 6.71821e28 0.0969199
\(696\) 0 0
\(697\) 2.25878e30 3.16177
\(698\) 0 0
\(699\) −4.08813e29 −0.555283
\(700\) 0 0
\(701\) −1.04529e30 −1.37784 −0.688919 0.724839i \(-0.741914\pi\)
−0.688919 + 0.724839i \(0.741914\pi\)
\(702\) 0 0
\(703\) 7.73967e29 0.990128
\(704\) 0 0
\(705\) −1.89576e30 −2.35395
\(706\) 0 0
\(707\) −1.01586e30 −1.22442
\(708\) 0 0
\(709\) 8.68030e28 0.101566 0.0507831 0.998710i \(-0.483828\pi\)
0.0507831 + 0.998710i \(0.483828\pi\)
\(710\) 0 0
\(711\) −7.30295e28 −0.0829598
\(712\) 0 0
\(713\) 1.34747e29 0.148621
\(714\) 0 0
\(715\) −1.79418e30 −1.92155
\(716\) 0 0
\(717\) 9.00873e29 0.936942
\(718\) 0 0
\(719\) 1.16156e30 1.17325 0.586624 0.809859i \(-0.300457\pi\)
0.586624 + 0.809859i \(0.300457\pi\)
\(720\) 0 0
\(721\) −3.20520e29 −0.314438
\(722\) 0 0
\(723\) 1.40876e29 0.134241
\(724\) 0 0
\(725\) 7.89774e29 0.731061
\(726\) 0 0
\(727\) −2.77822e29 −0.249836 −0.124918 0.992167i \(-0.539867\pi\)
−0.124918 + 0.992167i \(0.539867\pi\)
\(728\) 0 0
\(729\) 1.23428e30 1.07839
\(730\) 0 0
\(731\) 1.78745e30 1.51740
\(732\) 0 0
\(733\) 2.16010e30 1.78189 0.890947 0.454107i \(-0.150042\pi\)
0.890947 + 0.454107i \(0.150042\pi\)
\(734\) 0 0
\(735\) 9.18996e29 0.736710
\(736\) 0 0
\(737\) 2.17824e30 1.69706
\(738\) 0 0
\(739\) −1.74302e30 −1.31988 −0.659942 0.751317i \(-0.729419\pi\)
−0.659942 + 0.751317i \(0.729419\pi\)
\(740\) 0 0
\(741\) −9.21105e29 −0.677982
\(742\) 0 0
\(743\) 5.89639e29 0.421895 0.210947 0.977497i \(-0.432345\pi\)
0.210947 + 0.977497i \(0.432345\pi\)
\(744\) 0 0
\(745\) −9.36530e29 −0.651449
\(746\) 0 0
\(747\) 2.19310e29 0.148317
\(748\) 0 0
\(749\) −4.12316e29 −0.271126
\(750\) 0 0
\(751\) 8.31754e28 0.0531833 0.0265917 0.999646i \(-0.491535\pi\)
0.0265917 + 0.999646i \(0.491535\pi\)
\(752\) 0 0
\(753\) −1.98965e30 −1.23717
\(754\) 0 0
\(755\) 1.60193e30 0.968726
\(756\) 0 0
\(757\) −2.67582e29 −0.157380 −0.0786902 0.996899i \(-0.525074\pi\)
−0.0786902 + 0.996899i \(0.525074\pi\)
\(758\) 0 0
\(759\) −1.66681e30 −0.953561
\(760\) 0 0
\(761\) 2.95107e30 1.64225 0.821127 0.570745i \(-0.193346\pi\)
0.821127 + 0.570745i \(0.193346\pi\)
\(762\) 0 0
\(763\) −1.36769e29 −0.0740421
\(764\) 0 0
\(765\) −4.62555e29 −0.243623
\(766\) 0 0
\(767\) −2.11457e30 −1.08360
\(768\) 0 0
\(769\) −2.62362e30 −1.30820 −0.654100 0.756408i \(-0.726953\pi\)
−0.654100 + 0.756408i \(0.726953\pi\)
\(770\) 0 0
\(771\) 1.72642e30 0.837675
\(772\) 0 0
\(773\) 9.36482e29 0.442196 0.221098 0.975252i \(-0.429036\pi\)
0.221098 + 0.975252i \(0.429036\pi\)
\(774\) 0 0
\(775\) −6.45464e29 −0.296622
\(776\) 0 0
\(777\) −1.81572e30 −0.812133
\(778\) 0 0
\(779\) 3.75158e30 1.63332
\(780\) 0 0
\(781\) −3.58094e30 −1.51762
\(782\) 0 0
\(783\) −1.30702e30 −0.539242
\(784\) 0 0
\(785\) −6.44103e30 −2.58716
\(786\) 0 0
\(787\) −2.42850e30 −0.949736 −0.474868 0.880057i \(-0.657504\pi\)
−0.474868 + 0.880057i \(0.657504\pi\)
\(788\) 0 0
\(789\) 3.65237e30 1.39081
\(790\) 0 0
\(791\) 8.25036e29 0.305929
\(792\) 0 0
\(793\) 1.99109e30 0.718990
\(794\) 0 0
\(795\) 3.16926e30 1.11456
\(796\) 0 0
\(797\) −1.99411e29 −0.0683025 −0.0341513 0.999417i \(-0.510873\pi\)
−0.0341513 + 0.999417i \(0.510873\pi\)
\(798\) 0 0
\(799\) −7.58189e30 −2.52951
\(800\) 0 0
\(801\) −2.72130e29 −0.0884373
\(802\) 0 0
\(803\) −1.01537e30 −0.321450
\(804\) 0 0
\(805\) 2.52549e30 0.778913
\(806\) 0 0
\(807\) −5.65567e30 −1.69946
\(808\) 0 0
\(809\) 3.77765e30 1.10602 0.553010 0.833175i \(-0.313479\pi\)
0.553010 + 0.833175i \(0.313479\pi\)
\(810\) 0 0
\(811\) −2.72778e30 −0.778199 −0.389099 0.921196i \(-0.627214\pi\)
−0.389099 + 0.921196i \(0.627214\pi\)
\(812\) 0 0
\(813\) −3.88856e30 −1.08103
\(814\) 0 0
\(815\) 5.42635e30 1.47012
\(816\) 0 0
\(817\) 2.96875e30 0.783867
\(818\) 0 0
\(819\) −2.37245e29 −0.0610542
\(820\) 0 0
\(821\) −1.90653e30 −0.478234 −0.239117 0.970991i \(-0.576858\pi\)
−0.239117 + 0.970991i \(0.576858\pi\)
\(822\) 0 0
\(823\) 3.39146e30 0.829255 0.414628 0.909991i \(-0.363912\pi\)
0.414628 + 0.909991i \(0.363912\pi\)
\(824\) 0 0
\(825\) 7.98435e30 1.90315
\(826\) 0 0
\(827\) 3.99616e30 0.928613 0.464307 0.885675i \(-0.346304\pi\)
0.464307 + 0.885675i \(0.346304\pi\)
\(828\) 0 0
\(829\) −2.80835e30 −0.636251 −0.318126 0.948049i \(-0.603053\pi\)
−0.318126 + 0.948049i \(0.603053\pi\)
\(830\) 0 0
\(831\) 4.97872e30 1.09978
\(832\) 0 0
\(833\) 3.67544e30 0.791656
\(834\) 0 0
\(835\) 1.10940e30 0.233012
\(836\) 0 0
\(837\) 1.06820e30 0.218793
\(838\) 0 0
\(839\) −5.63065e30 −1.12475 −0.562377 0.826881i \(-0.690113\pi\)
−0.562377 + 0.826881i \(0.690113\pi\)
\(840\) 0 0
\(841\) −3.76124e30 −0.732779
\(842\) 0 0
\(843\) −1.18748e30 −0.225650
\(844\) 0 0
\(845\) 2.00357e30 0.371372
\(846\) 0 0
\(847\) 3.95103e30 0.714389
\(848\) 0 0
\(849\) −8.20714e30 −1.44764
\(850\) 0 0
\(851\) 4.97962e30 0.856911
\(852\) 0 0
\(853\) 4.25429e30 0.714269 0.357135 0.934053i \(-0.383754\pi\)
0.357135 + 0.934053i \(0.383754\pi\)
\(854\) 0 0
\(855\) −7.68252e29 −0.125852
\(856\) 0 0
\(857\) −5.42388e29 −0.0866984 −0.0433492 0.999060i \(-0.513803\pi\)
−0.0433492 + 0.999060i \(0.513803\pi\)
\(858\) 0 0
\(859\) 3.77718e30 0.589169 0.294585 0.955625i \(-0.404819\pi\)
0.294585 + 0.955625i \(0.404819\pi\)
\(860\) 0 0
\(861\) −8.80116e30 −1.33970
\(862\) 0 0
\(863\) −9.65328e30 −1.43404 −0.717021 0.697052i \(-0.754495\pi\)
−0.717021 + 0.697052i \(0.754495\pi\)
\(864\) 0 0
\(865\) 1.75562e30 0.254543
\(866\) 0 0
\(867\) 1.01421e31 1.43524
\(868\) 0 0
\(869\) 8.60655e30 1.18883
\(870\) 0 0
\(871\) 7.74465e30 1.04427
\(872\) 0 0
\(873\) −3.97677e29 −0.0523456
\(874\) 0 0
\(875\) −3.54337e30 −0.455336
\(876\) 0 0
\(877\) −8.26757e30 −1.03725 −0.518623 0.855003i \(-0.673555\pi\)
−0.518623 + 0.855003i \(0.673555\pi\)
\(878\) 0 0
\(879\) −2.13032e30 −0.260953
\(880\) 0 0
\(881\) 3.71573e30 0.444423 0.222212 0.974998i \(-0.428672\pi\)
0.222212 + 0.974998i \(0.428672\pi\)
\(882\) 0 0
\(883\) −3.53555e29 −0.0412923 −0.0206462 0.999787i \(-0.506572\pi\)
−0.0206462 + 0.999787i \(0.506572\pi\)
\(884\) 0 0
\(885\) 1.60640e31 1.83210
\(886\) 0 0
\(887\) −7.78376e28 −0.00866944 −0.00433472 0.999991i \(-0.501380\pi\)
−0.00433472 + 0.999991i \(0.501380\pi\)
\(888\) 0 0
\(889\) 3.98302e30 0.433256
\(890\) 0 0
\(891\) −1.18934e31 −1.26355
\(892\) 0 0
\(893\) −1.25927e31 −1.30671
\(894\) 0 0
\(895\) 1.27261e30 0.128990
\(896\) 0 0
\(897\) −5.92630e30 −0.586763
\(898\) 0 0
\(899\) −1.12098e30 −0.108423
\(900\) 0 0
\(901\) 1.26751e31 1.19768
\(902\) 0 0
\(903\) −6.96465e30 −0.642951
\(904\) 0 0
\(905\) 1.24066e30 0.111904
\(906\) 0 0
\(907\) −7.49286e30 −0.660345 −0.330173 0.943921i \(-0.607107\pi\)
−0.330173 + 0.943921i \(0.607107\pi\)
\(908\) 0 0
\(909\) 1.98823e30 0.171217
\(910\) 0 0
\(911\) −1.85880e31 −1.56419 −0.782096 0.623159i \(-0.785849\pi\)
−0.782096 + 0.623159i \(0.785849\pi\)
\(912\) 0 0
\(913\) −2.58457e31 −2.12543
\(914\) 0 0
\(915\) −1.51260e31 −1.21563
\(916\) 0 0
\(917\) 7.48750e30 0.588110
\(918\) 0 0
\(919\) 1.21998e31 0.936573 0.468286 0.883577i \(-0.344872\pi\)
0.468286 + 0.883577i \(0.344872\pi\)
\(920\) 0 0
\(921\) −3.68576e30 −0.276567
\(922\) 0 0
\(923\) −1.27319e31 −0.933847
\(924\) 0 0
\(925\) −2.38533e31 −1.71025
\(926\) 0 0
\(927\) 6.27317e29 0.0439693
\(928\) 0 0
\(929\) −1.53725e31 −1.05337 −0.526684 0.850061i \(-0.676565\pi\)
−0.526684 + 0.850061i \(0.676565\pi\)
\(930\) 0 0
\(931\) 6.10449e30 0.408957
\(932\) 0 0
\(933\) −6.82298e30 −0.446907
\(934\) 0 0
\(935\) 5.45122e31 3.49118
\(936\) 0 0
\(937\) 1.67237e31 1.04729 0.523645 0.851937i \(-0.324572\pi\)
0.523645 + 0.851937i \(0.324572\pi\)
\(938\) 0 0
\(939\) −8.83157e30 −0.540816
\(940\) 0 0
\(941\) 2.77704e30 0.166299 0.0831497 0.996537i \(-0.473502\pi\)
0.0831497 + 0.996537i \(0.473502\pi\)
\(942\) 0 0
\(943\) 2.41373e31 1.41357
\(944\) 0 0
\(945\) 2.00206e31 1.14668
\(946\) 0 0
\(947\) −1.06419e30 −0.0596132 −0.0298066 0.999556i \(-0.509489\pi\)
−0.0298066 + 0.999556i \(0.509489\pi\)
\(948\) 0 0
\(949\) −3.61013e30 −0.197800
\(950\) 0 0
\(951\) −2.90316e30 −0.155588
\(952\) 0 0
\(953\) 2.76399e31 1.44897 0.724487 0.689289i \(-0.242077\pi\)
0.724487 + 0.689289i \(0.242077\pi\)
\(954\) 0 0
\(955\) −1.56550e31 −0.802818
\(956\) 0 0
\(957\) 1.38664e31 0.695648
\(958\) 0 0
\(959\) 1.46142e31 0.717263
\(960\) 0 0
\(961\) −1.99094e31 −0.956008
\(962\) 0 0
\(963\) 8.06980e29 0.0379129
\(964\) 0 0
\(965\) −8.56254e30 −0.393610
\(966\) 0 0
\(967\) −1.28234e31 −0.576800 −0.288400 0.957510i \(-0.593123\pi\)
−0.288400 + 0.957510i \(0.593123\pi\)
\(968\) 0 0
\(969\) 2.79858e31 1.23179
\(970\) 0 0
\(971\) −7.42766e30 −0.319926 −0.159963 0.987123i \(-0.551138\pi\)
−0.159963 + 0.987123i \(0.551138\pi\)
\(972\) 0 0
\(973\) −1.04692e30 −0.0441296
\(974\) 0 0
\(975\) 2.83881e31 1.17108
\(976\) 0 0
\(977\) −2.93176e31 −1.18368 −0.591842 0.806054i \(-0.701599\pi\)
−0.591842 + 0.806054i \(0.701599\pi\)
\(978\) 0 0
\(979\) 3.20706e31 1.26733
\(980\) 0 0
\(981\) 2.67682e29 0.0103537
\(982\) 0 0
\(983\) −4.75461e31 −1.80013 −0.900064 0.435758i \(-0.856480\pi\)
−0.900064 + 0.435758i \(0.856480\pi\)
\(984\) 0 0
\(985\) 2.68487e31 0.995044
\(986\) 0 0
\(987\) 2.95423e31 1.07180
\(988\) 0 0
\(989\) 1.91006e31 0.678401
\(990\) 0 0
\(991\) −4.88614e31 −1.69900 −0.849498 0.527592i \(-0.823095\pi\)
−0.849498 + 0.527592i \(0.823095\pi\)
\(992\) 0 0
\(993\) 6.41878e30 0.218517
\(994\) 0 0
\(995\) −8.35491e31 −2.78484
\(996\) 0 0
\(997\) 5.22919e31 1.70661 0.853306 0.521410i \(-0.174594\pi\)
0.853306 + 0.521410i \(0.174594\pi\)
\(998\) 0 0
\(999\) 3.94756e31 1.26151
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 16.22.a.f.1.3 3
4.3 odd 2 8.22.a.b.1.1 3
8.3 odd 2 64.22.a.l.1.3 3
8.5 even 2 64.22.a.m.1.1 3
12.11 even 2 72.22.a.f.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.1 3 4.3 odd 2
16.22.a.f.1.3 3 1.1 even 1 trivial
64.22.a.l.1.3 3 8.3 odd 2
64.22.a.m.1.1 3 8.5 even 2
72.22.a.f.1.3 3 12.11 even 2