Properties

Label 9.90.a.b.1.1
Level $9$
Weight $90$
Character 9.1
Self dual yes
Analytic conductor $451.462$
Analytic rank $0$
Dimension $7$
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] = [9,90,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 90, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 90);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 90 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(451.461862736\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3 x^{6} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{43}\cdot 5^{9}\cdot 7^{5}\cdot 11^{2}\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.01484e12\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.42254e13 q^{2} +1.33691e27 q^{4} -3.67006e30 q^{5} -4.35255e37 q^{7} -3.17513e40 q^{8} +O(q^{10})\) \(q-4.42254e13 q^{2} +1.33691e27 q^{4} -3.67006e30 q^{5} -4.35255e37 q^{7} -3.17513e40 q^{8} +1.62310e44 q^{10} +2.45458e46 q^{11} -6.70001e49 q^{13} +1.92493e51 q^{14} +5.76705e53 q^{16} -4.97159e54 q^{17} +4.54728e56 q^{19} -4.90655e57 q^{20} -1.08555e60 q^{22} -2.88267e60 q^{23} -1.48089e62 q^{25} +2.96311e63 q^{26} -5.81898e64 q^{28} +1.57790e65 q^{29} +4.07570e66 q^{31} -5.85187e66 q^{32} +2.19871e68 q^{34} +1.59741e68 q^{35} -7.75996e69 q^{37} -2.01105e70 q^{38} +1.16529e71 q^{40} -2.95976e71 q^{41} -1.81368e72 q^{43} +3.28156e73 q^{44} +1.27487e74 q^{46} -1.54316e73 q^{47} +2.58682e74 q^{49} +6.54931e75 q^{50} -8.95734e76 q^{52} -3.79389e76 q^{53} -9.00846e76 q^{55} +1.38199e78 q^{56} -6.97831e78 q^{58} -4.03039e78 q^{59} -2.95660e79 q^{61} -1.80249e80 q^{62} -9.81620e79 q^{64} +2.45894e80 q^{65} +5.44278e80 q^{67} -6.64659e81 q^{68} -7.06461e81 q^{70} -6.75644e81 q^{71} -1.55795e82 q^{73} +3.43187e83 q^{74} +6.07933e83 q^{76} -1.06837e84 q^{77} -4.89294e84 q^{79} -2.11654e84 q^{80} +1.30897e85 q^{82} +5.34367e84 q^{83} +1.82460e85 q^{85} +8.02105e85 q^{86} -7.79362e86 q^{88} +7.65546e86 q^{89} +2.91621e87 q^{91} -3.85389e87 q^{92} +6.82466e86 q^{94} -1.66888e87 q^{95} -1.46156e87 q^{97} -1.14403e88 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 31407330351408 q^{2} + 22\!\cdots\!04 q^{4}+ \cdots + 17\!\cdots\!20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 31407330351408 q^{2} + 22\!\cdots\!04 q^{4}+ \cdots + 17\!\cdots\!56 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.42254e13 −1.77761 −0.888805 0.458285i \(-0.848464\pi\)
−0.888805 + 0.458285i \(0.848464\pi\)
\(3\) 0 0
\(4\) 1.33691e27 2.15990
\(5\) −3.67006e30 −0.288741 −0.144370 0.989524i \(-0.546116\pi\)
−0.144370 + 0.989524i \(0.546116\pi\)
\(6\) 0 0
\(7\) −4.35255e37 −1.07617 −0.538085 0.842891i \(-0.680852\pi\)
−0.538085 + 0.842891i \(0.680852\pi\)
\(8\) −3.17513e40 −2.06185
\(9\) 0 0
\(10\) 1.62310e44 0.513269
\(11\) 2.45458e46 1.11687 0.558435 0.829548i \(-0.311402\pi\)
0.558435 + 0.829548i \(0.311402\pi\)
\(12\) 0 0
\(13\) −6.70001e49 −1.80134 −0.900672 0.434501i \(-0.856925\pi\)
−0.900672 + 0.434501i \(0.856925\pi\)
\(14\) 1.92493e51 1.91301
\(15\) 0 0
\(16\) 5.76705e53 1.50527
\(17\) −4.97159e54 −0.874014 −0.437007 0.899458i \(-0.643962\pi\)
−0.437007 + 0.899458i \(0.643962\pi\)
\(18\) 0 0
\(19\) 4.54728e56 0.566522 0.283261 0.959043i \(-0.408584\pi\)
0.283261 + 0.959043i \(0.408584\pi\)
\(20\) −4.90655e57 −0.623651
\(21\) 0 0
\(22\) −1.08555e60 −1.98536
\(23\) −2.88267e60 −0.729301 −0.364650 0.931144i \(-0.618812\pi\)
−0.364650 + 0.931144i \(0.618812\pi\)
\(24\) 0 0
\(25\) −1.48089e62 −0.916629
\(26\) 2.96311e63 3.20209
\(27\) 0 0
\(28\) −5.81898e64 −2.32442
\(29\) 1.57790e65 1.32242 0.661208 0.750203i \(-0.270044\pi\)
0.661208 + 0.750203i \(0.270044\pi\)
\(30\) 0 0
\(31\) 4.07570e66 1.75633 0.878166 0.478356i \(-0.158767\pi\)
0.878166 + 0.478356i \(0.158767\pi\)
\(32\) −5.85187e66 −0.613932
\(33\) 0 0
\(34\) 2.19871e68 1.55366
\(35\) 1.59741e68 0.310734
\(36\) 0 0
\(37\) −7.75996e69 −1.27316 −0.636580 0.771211i \(-0.719651\pi\)
−0.636580 + 0.771211i \(0.719651\pi\)
\(38\) −2.01105e70 −1.00706
\(39\) 0 0
\(40\) 1.16529e71 0.595340
\(41\) −2.95976e71 −0.503936 −0.251968 0.967736i \(-0.581078\pi\)
−0.251968 + 0.967736i \(0.581078\pi\)
\(42\) 0 0
\(43\) −1.81368e72 −0.370863 −0.185431 0.982657i \(-0.559368\pi\)
−0.185431 + 0.982657i \(0.559368\pi\)
\(44\) 3.28156e73 2.41233
\(45\) 0 0
\(46\) 1.27487e74 1.29641
\(47\) −1.54316e73 −0.0602636 −0.0301318 0.999546i \(-0.509593\pi\)
−0.0301318 + 0.999546i \(0.509593\pi\)
\(48\) 0 0
\(49\) 2.58682e74 0.158140
\(50\) 6.54931e75 1.62941
\(51\) 0 0
\(52\) −8.95734e76 −3.89072
\(53\) −3.79389e76 −0.706005 −0.353003 0.935622i \(-0.614839\pi\)
−0.353003 + 0.935622i \(0.614839\pi\)
\(54\) 0 0
\(55\) −9.00846e76 −0.322486
\(56\) 1.38199e78 2.21890
\(57\) 0 0
\(58\) −6.97831e78 −2.35074
\(59\) −4.03039e78 −0.634502 −0.317251 0.948342i \(-0.602760\pi\)
−0.317251 + 0.948342i \(0.602760\pi\)
\(60\) 0 0
\(61\) −2.95660e79 −1.05588 −0.527940 0.849282i \(-0.677035\pi\)
−0.527940 + 0.849282i \(0.677035\pi\)
\(62\) −1.80249e80 −3.12207
\(63\) 0 0
\(64\) −9.81620e79 −0.413937
\(65\) 2.45894e80 0.520121
\(66\) 0 0
\(67\) 5.44278e80 0.298877 0.149438 0.988771i \(-0.452253\pi\)
0.149438 + 0.988771i \(0.452253\pi\)
\(68\) −6.64659e81 −1.88778
\(69\) 0 0
\(70\) −7.06461e81 −0.552364
\(71\) −6.75644e81 −0.281010 −0.140505 0.990080i \(-0.544873\pi\)
−0.140505 + 0.990080i \(0.544873\pi\)
\(72\) 0 0
\(73\) −1.55795e82 −0.188229 −0.0941146 0.995561i \(-0.530002\pi\)
−0.0941146 + 0.995561i \(0.530002\pi\)
\(74\) 3.43187e83 2.26318
\(75\) 0 0
\(76\) 6.07933e83 1.22363
\(77\) −1.06837e84 −1.20194
\(78\) 0 0
\(79\) −4.89294e84 −1.75859 −0.879294 0.476280i \(-0.841985\pi\)
−0.879294 + 0.476280i \(0.841985\pi\)
\(80\) −2.11654e84 −0.434633
\(81\) 0 0
\(82\) 1.30897e85 0.895802
\(83\) 5.34367e84 0.213238 0.106619 0.994300i \(-0.465997\pi\)
0.106619 + 0.994300i \(0.465997\pi\)
\(84\) 0 0
\(85\) 1.82460e85 0.252363
\(86\) 8.02105e85 0.659250
\(87\) 0 0
\(88\) −7.79362e86 −2.30282
\(89\) 7.65546e86 1.36809 0.684047 0.729438i \(-0.260218\pi\)
0.684047 + 0.729438i \(0.260218\pi\)
\(90\) 0 0
\(91\) 2.91621e87 1.93855
\(92\) −3.85389e87 −1.57522
\(93\) 0 0
\(94\) 6.82466e86 0.107125
\(95\) −1.66888e87 −0.163578
\(96\) 0 0
\(97\) −1.46156e87 −0.0566859 −0.0283430 0.999598i \(-0.509023\pi\)
−0.0283430 + 0.999598i \(0.509023\pi\)
\(98\) −1.14403e88 −0.281111
\(99\) 0 0
\(100\) −1.97983e89 −1.97983
\(101\) −1.76473e89 −1.13338 −0.566691 0.823931i \(-0.691777\pi\)
−0.566691 + 0.823931i \(0.691777\pi\)
\(102\) 0 0
\(103\) 1.57477e89 0.422632 0.211316 0.977418i \(-0.432225\pi\)
0.211316 + 0.977418i \(0.432225\pi\)
\(104\) 2.12734e90 3.71410
\(105\) 0 0
\(106\) 1.67786e90 1.25500
\(107\) −1.57684e90 −0.776621 −0.388311 0.921529i \(-0.626941\pi\)
−0.388311 + 0.921529i \(0.626941\pi\)
\(108\) 0 0
\(109\) 8.09287e90 1.74833 0.874164 0.485630i \(-0.161410\pi\)
0.874164 + 0.485630i \(0.161410\pi\)
\(110\) 3.98402e90 0.573255
\(111\) 0 0
\(112\) −2.51013e91 −1.61992
\(113\) −1.43205e91 −0.622255 −0.311128 0.950368i \(-0.600707\pi\)
−0.311128 + 0.950368i \(0.600707\pi\)
\(114\) 0 0
\(115\) 1.05796e91 0.210579
\(116\) 2.10951e92 2.85629
\(117\) 0 0
\(118\) 1.78246e92 1.12790
\(119\) 2.16391e92 0.940587
\(120\) 0 0
\(121\) 1.19495e92 0.247400
\(122\) 1.30757e93 1.87694
\(123\) 0 0
\(124\) 5.44886e93 3.79350
\(125\) 1.13643e93 0.553409
\(126\) 0 0
\(127\) −4.89751e93 −1.17682 −0.588409 0.808563i \(-0.700246\pi\)
−0.588409 + 0.808563i \(0.700246\pi\)
\(128\) 7.96338e93 1.34975
\(129\) 0 0
\(130\) −1.08748e94 −0.924573
\(131\) 1.69999e94 1.02772 0.513858 0.857875i \(-0.328216\pi\)
0.513858 + 0.857875i \(0.328216\pi\)
\(132\) 0 0
\(133\) −1.97923e94 −0.609674
\(134\) −2.40709e94 −0.531287
\(135\) 0 0
\(136\) 1.57855e95 1.80209
\(137\) −6.04210e94 −0.497878 −0.248939 0.968519i \(-0.580082\pi\)
−0.248939 + 0.968519i \(0.580082\pi\)
\(138\) 0 0
\(139\) −1.73639e95 −0.750739 −0.375370 0.926875i \(-0.622484\pi\)
−0.375370 + 0.926875i \(0.622484\pi\)
\(140\) 2.13560e95 0.671154
\(141\) 0 0
\(142\) 2.98806e95 0.499526
\(143\) −1.64457e96 −2.01187
\(144\) 0 0
\(145\) −5.79098e95 −0.381835
\(146\) 6.89010e95 0.334598
\(147\) 0 0
\(148\) −1.03744e97 −2.74990
\(149\) −8.32082e96 −1.63447 −0.817236 0.576304i \(-0.804494\pi\)
−0.817236 + 0.576304i \(0.804494\pi\)
\(150\) 0 0
\(151\) −1.01165e96 −0.109788 −0.0548942 0.998492i \(-0.517482\pi\)
−0.0548942 + 0.998492i \(0.517482\pi\)
\(152\) −1.44382e97 −1.16808
\(153\) 0 0
\(154\) 4.72490e97 2.13658
\(155\) −1.49581e97 −0.507124
\(156\) 0 0
\(157\) −3.72359e97 −0.713554 −0.356777 0.934190i \(-0.616124\pi\)
−0.356777 + 0.934190i \(0.616124\pi\)
\(158\) 2.16392e98 3.12608
\(159\) 0 0
\(160\) 2.14767e97 0.177267
\(161\) 1.25470e98 0.784851
\(162\) 0 0
\(163\) −1.03319e98 −0.373107 −0.186554 0.982445i \(-0.559732\pi\)
−0.186554 + 0.982445i \(0.559732\pi\)
\(164\) −3.95695e98 −1.08845
\(165\) 0 0
\(166\) −2.36326e98 −0.379054
\(167\) 1.50110e99 1.84301 0.921503 0.388372i \(-0.126962\pi\)
0.921503 + 0.388372i \(0.126962\pi\)
\(168\) 0 0
\(169\) 3.10558e99 2.24484
\(170\) −8.06938e98 −0.448604
\(171\) 0 0
\(172\) −2.42473e99 −0.801027
\(173\) −1.10447e99 −0.281906 −0.140953 0.990016i \(-0.545017\pi\)
−0.140953 + 0.990016i \(0.545017\pi\)
\(174\) 0 0
\(175\) 6.44566e99 0.986448
\(176\) 1.41557e100 1.68119
\(177\) 0 0
\(178\) −3.38566e100 −2.43194
\(179\) −1.69588e100 −0.949372 −0.474686 0.880155i \(-0.657438\pi\)
−0.474686 + 0.880155i \(0.657438\pi\)
\(180\) 0 0
\(181\) −2.52788e100 −0.863099 −0.431549 0.902089i \(-0.642033\pi\)
−0.431549 + 0.902089i \(0.642033\pi\)
\(182\) −1.28971e101 −3.44599
\(183\) 0 0
\(184\) 9.15287e100 1.50371
\(185\) 2.84795e100 0.367613
\(186\) 0 0
\(187\) −1.22032e101 −0.976161
\(188\) −2.06307e100 −0.130163
\(189\) 0 0
\(190\) 7.38069e100 0.290778
\(191\) 3.78547e101 1.18069 0.590344 0.807152i \(-0.298992\pi\)
0.590344 + 0.807152i \(0.298992\pi\)
\(192\) 0 0
\(193\) 1.50651e101 0.295578 0.147789 0.989019i \(-0.452784\pi\)
0.147789 + 0.989019i \(0.452784\pi\)
\(194\) 6.46382e100 0.100766
\(195\) 0 0
\(196\) 3.45836e101 0.341566
\(197\) 6.14218e101 0.483700 0.241850 0.970314i \(-0.422246\pi\)
0.241850 + 0.970314i \(0.422246\pi\)
\(198\) 0 0
\(199\) −1.92526e102 −0.967223 −0.483611 0.875283i \(-0.660675\pi\)
−0.483611 + 0.875283i \(0.660675\pi\)
\(200\) 4.70203e102 1.88995
\(201\) 0 0
\(202\) 7.80456e102 2.01471
\(203\) −6.86787e102 −1.42314
\(204\) 0 0
\(205\) 1.08625e102 0.145507
\(206\) −6.96450e102 −0.751275
\(207\) 0 0
\(208\) −3.86393e103 −2.71151
\(209\) 1.11617e103 0.632732
\(210\) 0 0
\(211\) −2.45026e103 −0.909166 −0.454583 0.890704i \(-0.650212\pi\)
−0.454583 + 0.890704i \(0.650212\pi\)
\(212\) −5.07210e103 −1.52490
\(213\) 0 0
\(214\) 6.97363e103 1.38053
\(215\) 6.65630e102 0.107083
\(216\) 0 0
\(217\) −1.77397e104 −1.89011
\(218\) −3.57910e104 −3.10785
\(219\) 0 0
\(220\) −1.20435e104 −0.696538
\(221\) 3.33097e104 1.57440
\(222\) 0 0
\(223\) −4.73510e104 −1.49887 −0.749434 0.662079i \(-0.769674\pi\)
−0.749434 + 0.662079i \(0.769674\pi\)
\(224\) 2.54705e104 0.660695
\(225\) 0 0
\(226\) 6.33331e104 1.10613
\(227\) 1.36699e104 0.196161 0.0980807 0.995178i \(-0.468730\pi\)
0.0980807 + 0.995178i \(0.468730\pi\)
\(228\) 0 0
\(229\) 2.84461e104 0.276276 0.138138 0.990413i \(-0.455888\pi\)
0.138138 + 0.990413i \(0.455888\pi\)
\(230\) −4.67886e104 −0.374327
\(231\) 0 0
\(232\) −5.01003e105 −2.72662
\(233\) 1.32351e104 0.0594823 0.0297411 0.999558i \(-0.490532\pi\)
0.0297411 + 0.999558i \(0.490532\pi\)
\(234\) 0 0
\(235\) 5.66347e103 0.0174005
\(236\) −5.38829e105 −1.37046
\(237\) 0 0
\(238\) −9.56997e105 −1.67200
\(239\) −1.73403e105 −0.251391 −0.125696 0.992069i \(-0.540116\pi\)
−0.125696 + 0.992069i \(0.540116\pi\)
\(240\) 0 0
\(241\) −3.77326e105 −0.377536 −0.188768 0.982022i \(-0.560449\pi\)
−0.188768 + 0.982022i \(0.560449\pi\)
\(242\) −5.28470e105 −0.439781
\(243\) 0 0
\(244\) −3.95271e106 −2.28059
\(245\) −9.49380e104 −0.0456614
\(246\) 0 0
\(247\) −3.04669e106 −1.02050
\(248\) −1.29409e107 −3.62130
\(249\) 0 0
\(250\) −5.02589e106 −0.983745
\(251\) −1.00216e107 −1.64231 −0.821155 0.570705i \(-0.806670\pi\)
−0.821155 + 0.570705i \(0.806670\pi\)
\(252\) 0 0
\(253\) −7.07576e106 −0.814535
\(254\) 2.16594e107 2.09192
\(255\) 0 0
\(256\) −2.91424e107 −1.98540
\(257\) −1.47865e107 −0.846920 −0.423460 0.905915i \(-0.639185\pi\)
−0.423460 + 0.905915i \(0.639185\pi\)
\(258\) 0 0
\(259\) 3.37756e107 1.37014
\(260\) 3.28740e107 1.12341
\(261\) 0 0
\(262\) −7.51826e107 −1.82688
\(263\) −5.33852e106 −0.109494 −0.0547469 0.998500i \(-0.517435\pi\)
−0.0547469 + 0.998500i \(0.517435\pi\)
\(264\) 0 0
\(265\) 1.39238e107 0.203852
\(266\) 8.75320e107 1.08376
\(267\) 0 0
\(268\) 7.27653e107 0.645544
\(269\) −1.15813e108 −0.870524 −0.435262 0.900304i \(-0.643344\pi\)
−0.435262 + 0.900304i \(0.643344\pi\)
\(270\) 0 0
\(271\) −2.64602e108 −1.43041 −0.715203 0.698917i \(-0.753666\pi\)
−0.715203 + 0.698917i \(0.753666\pi\)
\(272\) −2.86714e108 −1.31563
\(273\) 0 0
\(274\) 2.67214e108 0.885033
\(275\) −3.63497e108 −1.02376
\(276\) 0 0
\(277\) −1.07845e108 −0.220015 −0.110007 0.993931i \(-0.535087\pi\)
−0.110007 + 0.993931i \(0.535087\pi\)
\(278\) 7.67926e108 1.33452
\(279\) 0 0
\(280\) −5.07199e108 −0.640687
\(281\) 4.97309e108 0.536037 0.268019 0.963414i \(-0.413631\pi\)
0.268019 + 0.963414i \(0.413631\pi\)
\(282\) 0 0
\(283\) 1.20179e109 0.944785 0.472392 0.881388i \(-0.343391\pi\)
0.472392 + 0.881388i \(0.343391\pi\)
\(284\) −9.03278e108 −0.606953
\(285\) 0 0
\(286\) 7.27318e109 3.57632
\(287\) 1.28825e109 0.542320
\(288\) 0 0
\(289\) −7.63920e108 −0.236099
\(290\) 2.56108e109 0.678754
\(291\) 0 0
\(292\) −2.08285e109 −0.406556
\(293\) −3.43775e109 −0.576322 −0.288161 0.957582i \(-0.593044\pi\)
−0.288161 + 0.957582i \(0.593044\pi\)
\(294\) 0 0
\(295\) 1.47918e109 0.183206
\(296\) 2.46389e110 2.62507
\(297\) 0 0
\(298\) 3.67991e110 2.90545
\(299\) 1.93140e110 1.31372
\(300\) 0 0
\(301\) 7.89410e109 0.399111
\(302\) 4.47407e109 0.195161
\(303\) 0 0
\(304\) 2.62244e110 0.852769
\(305\) 1.08509e110 0.304875
\(306\) 0 0
\(307\) −5.58639e110 −1.17347 −0.586737 0.809778i \(-0.699588\pi\)
−0.586737 + 0.809778i \(0.699588\pi\)
\(308\) −1.42832e111 −2.59607
\(309\) 0 0
\(310\) 6.61525e110 0.901470
\(311\) −9.96934e110 −1.17714 −0.588572 0.808445i \(-0.700309\pi\)
−0.588572 + 0.808445i \(0.700309\pi\)
\(312\) 0 0
\(313\) −7.36331e110 −0.653657 −0.326828 0.945084i \(-0.605980\pi\)
−0.326828 + 0.945084i \(0.605980\pi\)
\(314\) 1.64677e111 1.26842
\(315\) 0 0
\(316\) −6.54143e111 −3.79837
\(317\) −1.40479e111 −0.708721 −0.354361 0.935109i \(-0.615302\pi\)
−0.354361 + 0.935109i \(0.615302\pi\)
\(318\) 0 0
\(319\) 3.87308e111 1.47697
\(320\) 3.60260e110 0.119520
\(321\) 0 0
\(322\) −5.54894e111 −1.39516
\(323\) −2.26072e111 −0.495148
\(324\) 0 0
\(325\) 9.92201e111 1.65116
\(326\) 4.56934e111 0.663239
\(327\) 0 0
\(328\) 9.39764e111 1.03904
\(329\) 6.71665e110 0.0648538
\(330\) 0 0
\(331\) −1.93674e112 −1.42799 −0.713997 0.700149i \(-0.753117\pi\)
−0.713997 + 0.700149i \(0.753117\pi\)
\(332\) 7.14403e111 0.460573
\(333\) 0 0
\(334\) −6.63867e112 −3.27615
\(335\) −1.99753e111 −0.0862979
\(336\) 0 0
\(337\) −3.04243e112 −1.00853 −0.504266 0.863548i \(-0.668237\pi\)
−0.504266 + 0.863548i \(0.668237\pi\)
\(338\) −1.37346e113 −3.99045
\(339\) 0 0
\(340\) 2.43934e112 0.545080
\(341\) 1.00041e113 1.96160
\(342\) 0 0
\(343\) 5.99389e112 0.905984
\(344\) 5.75866e112 0.764664
\(345\) 0 0
\(346\) 4.88458e112 0.501119
\(347\) −5.62569e112 −0.507591 −0.253796 0.967258i \(-0.581679\pi\)
−0.253796 + 0.967258i \(0.581679\pi\)
\(348\) 0 0
\(349\) −4.27631e111 −0.0298771 −0.0149385 0.999888i \(-0.504755\pi\)
−0.0149385 + 0.999888i \(0.504755\pi\)
\(350\) −2.85062e113 −1.75352
\(351\) 0 0
\(352\) −1.43639e113 −0.685683
\(353\) −1.17997e113 −0.496473 −0.248236 0.968699i \(-0.579851\pi\)
−0.248236 + 0.968699i \(0.579851\pi\)
\(354\) 0 0
\(355\) 2.47965e112 0.0811390
\(356\) 1.02347e114 2.95495
\(357\) 0 0
\(358\) 7.50011e113 1.68761
\(359\) 3.09309e113 0.614738 0.307369 0.951590i \(-0.400551\pi\)
0.307369 + 0.951590i \(0.400551\pi\)
\(360\) 0 0
\(361\) −4.37496e113 −0.679053
\(362\) 1.11797e114 1.53425
\(363\) 0 0
\(364\) 3.89872e114 4.18707
\(365\) 5.71778e112 0.0543494
\(366\) 0 0
\(367\) 8.47042e113 0.631342 0.315671 0.948869i \(-0.397770\pi\)
0.315671 + 0.948869i \(0.397770\pi\)
\(368\) −1.66245e114 −1.09779
\(369\) 0 0
\(370\) −1.25952e114 −0.653473
\(371\) 1.65131e114 0.759781
\(372\) 0 0
\(373\) −1.26483e114 −0.458131 −0.229066 0.973411i \(-0.573567\pi\)
−0.229066 + 0.973411i \(0.573567\pi\)
\(374\) 5.39690e114 1.73523
\(375\) 0 0
\(376\) 4.89972e113 0.124255
\(377\) −1.05719e115 −2.38212
\(378\) 0 0
\(379\) −1.05579e114 −0.187989 −0.0939943 0.995573i \(-0.529964\pi\)
−0.0939943 + 0.995573i \(0.529964\pi\)
\(380\) −2.23115e114 −0.353312
\(381\) 0 0
\(382\) −1.67414e115 −2.09880
\(383\) −1.02001e115 −1.13831 −0.569155 0.822230i \(-0.692730\pi\)
−0.569155 + 0.822230i \(0.692730\pi\)
\(384\) 0 0
\(385\) 3.92097e114 0.347049
\(386\) −6.66260e114 −0.525422
\(387\) 0 0
\(388\) −1.95398e114 −0.122436
\(389\) 2.00350e115 1.11952 0.559760 0.828655i \(-0.310893\pi\)
0.559760 + 0.828655i \(0.310893\pi\)
\(390\) 0 0
\(391\) 1.43315e115 0.637420
\(392\) −8.21351e114 −0.326061
\(393\) 0 0
\(394\) −2.71640e115 −0.859830
\(395\) 1.79574e115 0.507776
\(396\) 0 0
\(397\) 1.43963e115 0.325142 0.162571 0.986697i \(-0.448021\pi\)
0.162571 + 0.986697i \(0.448021\pi\)
\(398\) 8.51452e115 1.71935
\(399\) 0 0
\(400\) −8.54039e115 −1.37977
\(401\) 6.32353e115 0.914187 0.457093 0.889419i \(-0.348890\pi\)
0.457093 + 0.889419i \(0.348890\pi\)
\(402\) 0 0
\(403\) −2.73072e116 −3.16376
\(404\) −2.35929e116 −2.44799
\(405\) 0 0
\(406\) 3.03734e116 2.52979
\(407\) −1.90475e116 −1.42195
\(408\) 0 0
\(409\) 5.70354e114 0.0342339 0.0171170 0.999853i \(-0.494551\pi\)
0.0171170 + 0.999853i \(0.494551\pi\)
\(410\) −4.80399e115 −0.258654
\(411\) 0 0
\(412\) 2.10534e116 0.912842
\(413\) 1.75425e116 0.682831
\(414\) 0 0
\(415\) −1.96116e115 −0.0615705
\(416\) 3.92076e116 1.10590
\(417\) 0 0
\(418\) −4.93629e116 −1.12475
\(419\) −2.82318e116 −0.578382 −0.289191 0.957271i \(-0.593386\pi\)
−0.289191 + 0.957271i \(0.593386\pi\)
\(420\) 0 0
\(421\) 6.04056e116 1.00121 0.500604 0.865676i \(-0.333111\pi\)
0.500604 + 0.865676i \(0.333111\pi\)
\(422\) 1.08364e117 1.61614
\(423\) 0 0
\(424\) 1.20461e117 1.45568
\(425\) 7.36240e116 0.801147
\(426\) 0 0
\(427\) 1.28687e117 1.13630
\(428\) −2.10810e117 −1.67742
\(429\) 0 0
\(430\) −2.94377e116 −0.190352
\(431\) 2.26583e117 1.32126 0.660632 0.750710i \(-0.270288\pi\)
0.660632 + 0.750710i \(0.270288\pi\)
\(432\) 0 0
\(433\) 7.27088e116 0.345045 0.172522 0.985006i \(-0.444808\pi\)
0.172522 + 0.985006i \(0.444808\pi\)
\(434\) 7.84543e117 3.35988
\(435\) 0 0
\(436\) 1.08195e118 3.77622
\(437\) −1.31083e117 −0.413165
\(438\) 0 0
\(439\) 4.51676e116 0.116187 0.0580933 0.998311i \(-0.481498\pi\)
0.0580933 + 0.998311i \(0.481498\pi\)
\(440\) 2.86031e117 0.664918
\(441\) 0 0
\(442\) −1.47314e118 −2.79867
\(443\) −8.00727e117 −1.37568 −0.687842 0.725860i \(-0.741442\pi\)
−0.687842 + 0.725860i \(0.741442\pi\)
\(444\) 0 0
\(445\) −2.80960e117 −0.395024
\(446\) 2.09412e118 2.66440
\(447\) 0 0
\(448\) 4.27254e117 0.445466
\(449\) −1.58652e118 −1.49790 −0.748949 0.662627i \(-0.769441\pi\)
−0.748949 + 0.662627i \(0.769441\pi\)
\(450\) 0 0
\(451\) −7.26498e117 −0.562831
\(452\) −1.91453e118 −1.34401
\(453\) 0 0
\(454\) −6.04555e117 −0.348699
\(455\) −1.07027e118 −0.559738
\(456\) 0 0
\(457\) −1.01717e117 −0.0437644 −0.0218822 0.999761i \(-0.506966\pi\)
−0.0218822 + 0.999761i \(0.506966\pi\)
\(458\) −1.25804e118 −0.491111
\(459\) 0 0
\(460\) 1.41440e118 0.454829
\(461\) 5.67592e118 1.65709 0.828543 0.559926i \(-0.189170\pi\)
0.828543 + 0.559926i \(0.189170\pi\)
\(462\) 0 0
\(463\) −3.42212e118 −0.824028 −0.412014 0.911178i \(-0.635174\pi\)
−0.412014 + 0.911178i \(0.635174\pi\)
\(464\) 9.09981e118 1.99059
\(465\) 0 0
\(466\) −5.85326e117 −0.105736
\(467\) 5.74060e118 0.942662 0.471331 0.881956i \(-0.343774\pi\)
0.471331 + 0.881956i \(0.343774\pi\)
\(468\) 0 0
\(469\) −2.36900e118 −0.321642
\(470\) −2.50469e117 −0.0309314
\(471\) 0 0
\(472\) 1.27970e119 1.30825
\(473\) −4.45181e118 −0.414206
\(474\) 0 0
\(475\) −6.73405e118 −0.519291
\(476\) 2.89296e119 2.03157
\(477\) 0 0
\(478\) 7.66882e118 0.446876
\(479\) 7.35495e118 0.390526 0.195263 0.980751i \(-0.437444\pi\)
0.195263 + 0.980751i \(0.437444\pi\)
\(480\) 0 0
\(481\) 5.19918e119 2.29340
\(482\) 1.66874e119 0.671112
\(483\) 0 0
\(484\) 1.59754e119 0.534360
\(485\) 5.36402e117 0.0163675
\(486\) 0 0
\(487\) 1.73116e119 0.439843 0.219922 0.975518i \(-0.429420\pi\)
0.219922 + 0.975518i \(0.429420\pi\)
\(488\) 9.38758e119 2.17707
\(489\) 0 0
\(490\) 4.19867e118 0.0811682
\(491\) −9.48426e119 −1.67447 −0.837234 0.546845i \(-0.815829\pi\)
−0.837234 + 0.546845i \(0.815829\pi\)
\(492\) 0 0
\(493\) −7.84466e119 −1.15581
\(494\) 1.34741e120 1.81405
\(495\) 0 0
\(496\) 2.35047e120 2.64375
\(497\) 2.94077e119 0.302414
\(498\) 0 0
\(499\) 2.96564e119 0.255061 0.127530 0.991835i \(-0.459295\pi\)
0.127530 + 0.991835i \(0.459295\pi\)
\(500\) 1.51930e120 1.19531
\(501\) 0 0
\(502\) 4.43208e120 2.91939
\(503\) 1.72800e120 1.04176 0.520881 0.853630i \(-0.325604\pi\)
0.520881 + 0.853630i \(0.325604\pi\)
\(504\) 0 0
\(505\) 6.47665e119 0.327253
\(506\) 3.12928e120 1.44793
\(507\) 0 0
\(508\) −6.54754e120 −2.54181
\(509\) 1.24174e120 0.441661 0.220830 0.975312i \(-0.429123\pi\)
0.220830 + 0.975312i \(0.429123\pi\)
\(510\) 0 0
\(511\) 6.78106e119 0.202566
\(512\) 7.95925e120 2.17951
\(513\) 0 0
\(514\) 6.53940e120 1.50549
\(515\) −5.77952e119 −0.122031
\(516\) 0 0
\(517\) −3.78780e119 −0.0673066
\(518\) −1.49374e121 −2.43557
\(519\) 0 0
\(520\) −7.80748e120 −1.07241
\(521\) −6.02844e120 −0.760197 −0.380099 0.924946i \(-0.624110\pi\)
−0.380099 + 0.924946i \(0.624110\pi\)
\(522\) 0 0
\(523\) −1.30371e121 −1.38630 −0.693148 0.720796i \(-0.743777\pi\)
−0.693148 + 0.720796i \(0.743777\pi\)
\(524\) 2.27274e121 2.21976
\(525\) 0 0
\(526\) 2.36098e120 0.194638
\(527\) −2.02627e121 −1.53506
\(528\) 0 0
\(529\) −7.31366e120 −0.468120
\(530\) −6.15785e120 −0.362370
\(531\) 0 0
\(532\) −2.64605e121 −1.31683
\(533\) 1.98305e121 0.907762
\(534\) 0 0
\(535\) 5.78709e120 0.224242
\(536\) −1.72816e121 −0.616240
\(537\) 0 0
\(538\) 5.12187e121 1.54745
\(539\) 6.34957e120 0.176622
\(540\) 0 0
\(541\) −3.02509e121 −0.713609 −0.356804 0.934179i \(-0.616134\pi\)
−0.356804 + 0.934179i \(0.616134\pi\)
\(542\) 1.17021e122 2.54270
\(543\) 0 0
\(544\) 2.90931e121 0.536586
\(545\) −2.97013e121 −0.504814
\(546\) 0 0
\(547\) 6.07810e120 0.0877671 0.0438836 0.999037i \(-0.486027\pi\)
0.0438836 + 0.999037i \(0.486027\pi\)
\(548\) −8.07777e121 −1.07537
\(549\) 0 0
\(550\) 1.60758e122 1.81984
\(551\) 7.17515e121 0.749178
\(552\) 0 0
\(553\) 2.12967e122 1.89254
\(554\) 4.76950e121 0.391101
\(555\) 0 0
\(556\) −2.32141e122 −1.62152
\(557\) 9.48715e121 0.611758 0.305879 0.952070i \(-0.401050\pi\)
0.305879 + 0.952070i \(0.401050\pi\)
\(558\) 0 0
\(559\) 1.21516e122 0.668051
\(560\) 9.21234e121 0.467738
\(561\) 0 0
\(562\) −2.19937e122 −0.952866
\(563\) −3.99353e122 −1.59858 −0.799288 0.600949i \(-0.794790\pi\)
−0.799288 + 0.600949i \(0.794790\pi\)
\(564\) 0 0
\(565\) 5.25572e121 0.179670
\(566\) −5.31497e122 −1.67946
\(567\) 0 0
\(568\) 2.14526e122 0.579401
\(569\) 4.99506e122 1.24751 0.623756 0.781619i \(-0.285606\pi\)
0.623756 + 0.781619i \(0.285606\pi\)
\(570\) 0 0
\(571\) −7.98755e121 −0.170650 −0.0853251 0.996353i \(-0.527193\pi\)
−0.0853251 + 0.996353i \(0.527193\pi\)
\(572\) −2.19865e123 −4.34543
\(573\) 0 0
\(574\) −5.69734e122 −0.964035
\(575\) 4.26893e122 0.668498
\(576\) 0 0
\(577\) −1.20737e123 −1.62000 −0.810001 0.586429i \(-0.800533\pi\)
−0.810001 + 0.586429i \(0.800533\pi\)
\(578\) 3.37847e122 0.419692
\(579\) 0 0
\(580\) −7.74204e122 −0.824726
\(581\) −2.32586e122 −0.229480
\(582\) 0 0
\(583\) −9.31240e122 −0.788517
\(584\) 4.94671e122 0.388101
\(585\) 0 0
\(586\) 1.52036e123 1.02448
\(587\) −6.63777e122 −0.414597 −0.207299 0.978278i \(-0.566467\pi\)
−0.207299 + 0.978278i \(0.566467\pi\)
\(588\) 0 0
\(589\) 1.85334e123 0.995001
\(590\) −6.54172e122 −0.325670
\(591\) 0 0
\(592\) −4.47521e123 −1.91645
\(593\) 3.67636e123 1.46044 0.730221 0.683211i \(-0.239417\pi\)
0.730221 + 0.683211i \(0.239417\pi\)
\(594\) 0 0
\(595\) −7.94167e122 −0.271586
\(596\) −1.11242e124 −3.53029
\(597\) 0 0
\(598\) −8.54167e123 −2.33529
\(599\) −7.52276e123 −1.90934 −0.954671 0.297663i \(-0.903793\pi\)
−0.954671 + 0.297663i \(0.903793\pi\)
\(600\) 0 0
\(601\) 2.62225e123 0.573798 0.286899 0.957961i \(-0.407376\pi\)
0.286899 + 0.957961i \(0.407376\pi\)
\(602\) −3.49120e123 −0.709464
\(603\) 0 0
\(604\) −1.35249e123 −0.237132
\(605\) −4.38553e122 −0.0714345
\(606\) 0 0
\(607\) −5.96123e123 −0.838377 −0.419188 0.907899i \(-0.637685\pi\)
−0.419188 + 0.907899i \(0.637685\pi\)
\(608\) −2.66101e123 −0.347806
\(609\) 0 0
\(610\) −4.79884e123 −0.541950
\(611\) 1.03392e123 0.108555
\(612\) 0 0
\(613\) 1.49430e124 1.35659 0.678293 0.734792i \(-0.262720\pi\)
0.678293 + 0.734792i \(0.262720\pi\)
\(614\) 2.47060e124 2.08598
\(615\) 0 0
\(616\) 3.39221e124 2.47823
\(617\) 1.42286e124 0.967101 0.483551 0.875316i \(-0.339347\pi\)
0.483551 + 0.875316i \(0.339347\pi\)
\(618\) 0 0
\(619\) −1.75124e124 −1.03065 −0.515327 0.856994i \(-0.672329\pi\)
−0.515327 + 0.856994i \(0.672329\pi\)
\(620\) −1.99976e124 −1.09534
\(621\) 0 0
\(622\) 4.40898e124 2.09250
\(623\) −3.33207e124 −1.47230
\(624\) 0 0
\(625\) 1.97544e124 0.756837
\(626\) 3.25645e124 1.16195
\(627\) 0 0
\(628\) −4.97811e124 −1.54121
\(629\) 3.85794e124 1.11276
\(630\) 0 0
\(631\) 2.10999e124 0.528413 0.264207 0.964466i \(-0.414890\pi\)
0.264207 + 0.964466i \(0.414890\pi\)
\(632\) 1.55357e125 3.62595
\(633\) 0 0
\(634\) 6.21276e124 1.25983
\(635\) 1.79741e124 0.339795
\(636\) 0 0
\(637\) −1.73317e124 −0.284864
\(638\) −1.71288e125 −2.62547
\(639\) 0 0
\(640\) −2.92261e124 −0.389728
\(641\) −9.40134e124 −1.16952 −0.584759 0.811207i \(-0.698811\pi\)
−0.584759 + 0.811207i \(0.698811\pi\)
\(642\) 0 0
\(643\) −5.35874e124 −0.580330 −0.290165 0.956977i \(-0.593710\pi\)
−0.290165 + 0.956977i \(0.593710\pi\)
\(644\) 1.67742e125 1.69520
\(645\) 0 0
\(646\) 9.99814e124 0.880181
\(647\) 1.76192e125 1.44792 0.723962 0.689840i \(-0.242319\pi\)
0.723962 + 0.689840i \(0.242319\pi\)
\(648\) 0 0
\(649\) −9.89293e124 −0.708657
\(650\) −4.38804e125 −2.93513
\(651\) 0 0
\(652\) −1.38129e125 −0.805874
\(653\) 2.70255e125 1.47278 0.736388 0.676560i \(-0.236530\pi\)
0.736388 + 0.676560i \(0.236530\pi\)
\(654\) 0 0
\(655\) −6.23906e124 −0.296743
\(656\) −1.70691e125 −0.758560
\(657\) 0 0
\(658\) −2.97047e124 −0.115285
\(659\) −4.92460e125 −1.78636 −0.893182 0.449696i \(-0.851532\pi\)
−0.893182 + 0.449696i \(0.851532\pi\)
\(660\) 0 0
\(661\) 1.53102e123 0.00485306 0.00242653 0.999997i \(-0.499228\pi\)
0.00242653 + 0.999997i \(0.499228\pi\)
\(662\) 8.56532e125 2.53842
\(663\) 0 0
\(664\) −1.69669e125 −0.439665
\(665\) 7.26388e124 0.176038
\(666\) 0 0
\(667\) −4.54856e125 −0.964439
\(668\) 2.00684e126 3.98071
\(669\) 0 0
\(670\) 8.83417e124 0.153404
\(671\) −7.25720e125 −1.17928
\(672\) 0 0
\(673\) 1.14314e126 1.62716 0.813579 0.581455i \(-0.197516\pi\)
0.813579 + 0.581455i \(0.197516\pi\)
\(674\) 1.34553e126 1.79278
\(675\) 0 0
\(676\) 4.15190e126 4.84862
\(677\) −8.57614e125 −0.937770 −0.468885 0.883259i \(-0.655344\pi\)
−0.468885 + 0.883259i \(0.655344\pi\)
\(678\) 0 0
\(679\) 6.36152e124 0.0610036
\(680\) −5.79336e125 −0.520336
\(681\) 0 0
\(682\) −4.42436e126 −3.48695
\(683\) 8.28357e125 0.611641 0.305820 0.952089i \(-0.401069\pi\)
0.305820 + 0.952089i \(0.401069\pi\)
\(684\) 0 0
\(685\) 2.21749e125 0.143758
\(686\) −2.65082e126 −1.61049
\(687\) 0 0
\(688\) −1.04596e126 −0.558249
\(689\) 2.54191e126 1.27176
\(690\) 0 0
\(691\) 4.99715e125 0.219760 0.109880 0.993945i \(-0.464953\pi\)
0.109880 + 0.993945i \(0.464953\pi\)
\(692\) −1.47659e126 −0.608889
\(693\) 0 0
\(694\) 2.48798e126 0.902299
\(695\) 6.37266e125 0.216769
\(696\) 0 0
\(697\) 1.47147e126 0.440447
\(698\) 1.89121e125 0.0531098
\(699\) 0 0
\(700\) 8.61729e126 2.13063
\(701\) −1.20764e126 −0.280212 −0.140106 0.990137i \(-0.544744\pi\)
−0.140106 + 0.990137i \(0.544744\pi\)
\(702\) 0 0
\(703\) −3.52868e126 −0.721273
\(704\) −2.40947e126 −0.462314
\(705\) 0 0
\(706\) 5.21845e126 0.882535
\(707\) 7.68105e126 1.21971
\(708\) 0 0
\(709\) −4.71841e126 −0.660751 −0.330376 0.943850i \(-0.607175\pi\)
−0.330376 + 0.943850i \(0.607175\pi\)
\(710\) −1.09664e126 −0.144234
\(711\) 0 0
\(712\) −2.43071e127 −2.82081
\(713\) −1.17489e127 −1.28089
\(714\) 0 0
\(715\) 6.03568e126 0.580908
\(716\) −2.26725e127 −2.05055
\(717\) 0 0
\(718\) −1.36793e127 −1.09276
\(719\) 1.53894e127 1.15554 0.577770 0.816200i \(-0.303923\pi\)
0.577770 + 0.816200i \(0.303923\pi\)
\(720\) 0 0
\(721\) −6.85428e126 −0.454823
\(722\) 1.93484e127 1.20709
\(723\) 0 0
\(724\) −3.37956e127 −1.86421
\(725\) −2.33670e127 −1.21216
\(726\) 0 0
\(727\) −9.29984e126 −0.426770 −0.213385 0.976968i \(-0.568449\pi\)
−0.213385 + 0.976968i \(0.568449\pi\)
\(728\) −9.25936e127 −3.99700
\(729\) 0 0
\(730\) −2.52871e126 −0.0966121
\(731\) 9.01685e126 0.324139
\(732\) 0 0
\(733\) 2.90574e127 0.924974 0.462487 0.886626i \(-0.346957\pi\)
0.462487 + 0.886626i \(0.346957\pi\)
\(734\) −3.74607e127 −1.12228
\(735\) 0 0
\(736\) 1.68690e127 0.447741
\(737\) 1.33598e127 0.333807
\(738\) 0 0
\(739\) −3.49478e127 −0.774002 −0.387001 0.922079i \(-0.626489\pi\)
−0.387001 + 0.922079i \(0.626489\pi\)
\(740\) 3.80747e127 0.794008
\(741\) 0 0
\(742\) −7.30296e127 −1.35060
\(743\) −8.90897e127 −1.55176 −0.775882 0.630879i \(-0.782695\pi\)
−0.775882 + 0.630879i \(0.782695\pi\)
\(744\) 0 0
\(745\) 3.05379e127 0.471938
\(746\) 5.59377e127 0.814379
\(747\) 0 0
\(748\) −1.63146e128 −2.10841
\(749\) 6.86326e127 0.835776
\(750\) 0 0
\(751\) 5.71982e127 0.618594 0.309297 0.950966i \(-0.399906\pi\)
0.309297 + 0.950966i \(0.399906\pi\)
\(752\) −8.89945e126 −0.0907130
\(753\) 0 0
\(754\) 4.67548e128 4.23449
\(755\) 3.71282e126 0.0317004
\(756\) 0 0
\(757\) 2.03779e128 1.54665 0.773325 0.634010i \(-0.218592\pi\)
0.773325 + 0.634010i \(0.218592\pi\)
\(758\) 4.66926e127 0.334171
\(759\) 0 0
\(760\) 5.29892e127 0.337274
\(761\) 1.17204e126 0.00703599 0.00351799 0.999994i \(-0.498880\pi\)
0.00351799 + 0.999994i \(0.498880\pi\)
\(762\) 0 0
\(763\) −3.52246e128 −1.88150
\(764\) 5.06085e128 2.55017
\(765\) 0 0
\(766\) 4.51105e128 2.02347
\(767\) 2.70037e128 1.14296
\(768\) 0 0
\(769\) 3.19825e127 0.120556 0.0602782 0.998182i \(-0.480801\pi\)
0.0602782 + 0.998182i \(0.480801\pi\)
\(770\) −1.73406e128 −0.616919
\(771\) 0 0
\(772\) 2.01407e128 0.638419
\(773\) −3.28146e128 −0.981929 −0.490965 0.871179i \(-0.663356\pi\)
−0.490965 + 0.871179i \(0.663356\pi\)
\(774\) 0 0
\(775\) −6.03567e128 −1.60990
\(776\) 4.64066e127 0.116878
\(777\) 0 0
\(778\) −8.86055e128 −1.99007
\(779\) −1.34589e128 −0.285491
\(780\) 0 0
\(781\) −1.65842e128 −0.313852
\(782\) −6.33815e128 −1.13308
\(783\) 0 0
\(784\) 1.49183e128 0.238043
\(785\) 1.36658e128 0.206032
\(786\) 0 0
\(787\) 1.37748e129 1.85443 0.927214 0.374532i \(-0.122197\pi\)
0.927214 + 0.374532i \(0.122197\pi\)
\(788\) 8.21157e128 1.04474
\(789\) 0 0
\(790\) −7.94171e128 −0.902628
\(791\) 6.23308e128 0.669652
\(792\) 0 0
\(793\) 1.98092e129 1.90200
\(794\) −6.36681e128 −0.577976
\(795\) 0 0
\(796\) −2.57390e129 −2.08910
\(797\) 4.39500e128 0.337337 0.168668 0.985673i \(-0.446053\pi\)
0.168668 + 0.985673i \(0.446053\pi\)
\(798\) 0 0
\(799\) 7.67194e127 0.0526712
\(800\) 8.66599e128 0.562748
\(801\) 0 0
\(802\) −2.79660e129 −1.62507
\(803\) −3.82412e128 −0.210228
\(804\) 0 0
\(805\) −4.60481e128 −0.226618
\(806\) 1.20767e130 5.62393
\(807\) 0 0
\(808\) 5.60324e129 2.33686
\(809\) −3.31962e129 −1.31032 −0.655162 0.755488i \(-0.727400\pi\)
−0.655162 + 0.755488i \(0.727400\pi\)
\(810\) 0 0
\(811\) −3.97411e129 −1.40544 −0.702720 0.711467i \(-0.748031\pi\)
−0.702720 + 0.711467i \(0.748031\pi\)
\(812\) −9.18175e129 −3.07385
\(813\) 0 0
\(814\) 8.42381e129 2.52768
\(815\) 3.79188e128 0.107731
\(816\) 0 0
\(817\) −8.24730e128 −0.210102
\(818\) −2.52241e128 −0.0608546
\(819\) 0 0
\(820\) 1.45222e129 0.314280
\(821\) 1.35805e129 0.278386 0.139193 0.990265i \(-0.455549\pi\)
0.139193 + 0.990265i \(0.455549\pi\)
\(822\) 0 0
\(823\) 1.64457e129 0.302524 0.151262 0.988494i \(-0.451666\pi\)
0.151262 + 0.988494i \(0.451666\pi\)
\(824\) −5.00012e129 −0.871404
\(825\) 0 0
\(826\) −7.75822e129 −1.21381
\(827\) −6.57180e127 −0.00974292 −0.00487146 0.999988i \(-0.501551\pi\)
−0.00487146 + 0.999988i \(0.501551\pi\)
\(828\) 0 0
\(829\) 4.74673e129 0.632001 0.316000 0.948759i \(-0.397660\pi\)
0.316000 + 0.948759i \(0.397660\pi\)
\(830\) 8.67330e128 0.109448
\(831\) 0 0
\(832\) 6.57686e129 0.745643
\(833\) −1.28606e129 −0.138216
\(834\) 0 0
\(835\) −5.50913e129 −0.532151
\(836\) 1.49222e130 1.36664
\(837\) 0 0
\(838\) 1.24856e130 1.02814
\(839\) 1.79965e130 1.40533 0.702667 0.711518i \(-0.251992\pi\)
0.702667 + 0.711518i \(0.251992\pi\)
\(840\) 0 0
\(841\) 1.06605e130 0.748782
\(842\) −2.67146e130 −1.77976
\(843\) 0 0
\(844\) −3.27579e130 −1.96371
\(845\) −1.13977e130 −0.648176
\(846\) 0 0
\(847\) −5.20106e129 −0.266244
\(848\) −2.18795e130 −1.06273
\(849\) 0 0
\(850\) −3.25605e130 −1.42413
\(851\) 2.23694e130 0.928517
\(852\) 0 0
\(853\) 3.67965e130 1.37586 0.687930 0.725777i \(-0.258519\pi\)
0.687930 + 0.725777i \(0.258519\pi\)
\(854\) −5.69124e130 −2.01991
\(855\) 0 0
\(856\) 5.00667e130 1.60128
\(857\) −7.78814e129 −0.236476 −0.118238 0.992985i \(-0.537725\pi\)
−0.118238 + 0.992985i \(0.537725\pi\)
\(858\) 0 0
\(859\) −9.94984e129 −0.272346 −0.136173 0.990685i \(-0.543480\pi\)
−0.136173 + 0.990685i \(0.543480\pi\)
\(860\) 8.89889e129 0.231289
\(861\) 0 0
\(862\) −1.00207e131 −2.34869
\(863\) 8.18207e130 1.82131 0.910653 0.413172i \(-0.135579\pi\)
0.910653 + 0.413172i \(0.135579\pi\)
\(864\) 0 0
\(865\) 4.05349e129 0.0813977
\(866\) −3.21557e130 −0.613356
\(867\) 0 0
\(868\) −2.37164e131 −4.08245
\(869\) −1.20101e131 −1.96411
\(870\) 0 0
\(871\) −3.64667e130 −0.538380
\(872\) −2.56959e131 −3.60480
\(873\) 0 0
\(874\) 5.79721e130 0.734447
\(875\) −4.94635e130 −0.595561
\(876\) 0 0
\(877\) 2.51544e130 0.273610 0.136805 0.990598i \(-0.456317\pi\)
0.136805 + 0.990598i \(0.456317\pi\)
\(878\) −1.99756e130 −0.206535
\(879\) 0 0
\(880\) −5.19522e130 −0.485428
\(881\) −9.16844e130 −0.814456 −0.407228 0.913327i \(-0.633505\pi\)
−0.407228 + 0.913327i \(0.633505\pi\)
\(882\) 0 0
\(883\) 8.89867e130 0.714618 0.357309 0.933986i \(-0.383694\pi\)
0.357309 + 0.933986i \(0.383694\pi\)
\(884\) 4.45322e131 3.40055
\(885\) 0 0
\(886\) 3.54125e131 2.44543
\(887\) 2.22905e131 1.46393 0.731963 0.681344i \(-0.238604\pi\)
0.731963 + 0.681344i \(0.238604\pi\)
\(888\) 0 0
\(889\) 2.13166e131 1.26646
\(890\) 1.24256e131 0.702200
\(891\) 0 0
\(892\) −6.33043e131 −3.23740
\(893\) −7.01717e129 −0.0341407
\(894\) 0 0
\(895\) 6.22400e130 0.274122
\(896\) −3.46610e131 −1.45256
\(897\) 0 0
\(898\) 7.01643e131 2.66268
\(899\) 6.43103e131 2.32260
\(900\) 0 0
\(901\) 1.88617e131 0.617059
\(902\) 3.21296e131 1.00050
\(903\) 0 0
\(904\) 4.54696e131 1.28300
\(905\) 9.27749e130 0.249212
\(906\) 0 0
\(907\) −3.83857e131 −0.934642 −0.467321 0.884088i \(-0.654781\pi\)
−0.467321 + 0.884088i \(0.654781\pi\)
\(908\) 1.82754e131 0.423689
\(909\) 0 0
\(910\) 4.73329e131 0.994997
\(911\) 1.57548e131 0.315387 0.157694 0.987488i \(-0.449594\pi\)
0.157694 + 0.987488i \(0.449594\pi\)
\(912\) 0 0
\(913\) 1.31165e131 0.238159
\(914\) 4.49845e130 0.0777960
\(915\) 0 0
\(916\) 3.80299e131 0.596729
\(917\) −7.39928e131 −1.10600
\(918\) 0 0
\(919\) 3.76660e131 0.510987 0.255493 0.966811i \(-0.417762\pi\)
0.255493 + 0.966811i \(0.417762\pi\)
\(920\) −3.35916e131 −0.434182
\(921\) 0 0
\(922\) −2.51020e132 −2.94565
\(923\) 4.52682e131 0.506195
\(924\) 0 0
\(925\) 1.14917e132 1.16701
\(926\) 1.51345e132 1.46480
\(927\) 0 0
\(928\) −9.23365e131 −0.811873
\(929\) −1.74031e132 −1.45857 −0.729286 0.684209i \(-0.760147\pi\)
−0.729286 + 0.684209i \(0.760147\pi\)
\(930\) 0 0
\(931\) 1.17630e131 0.0895897
\(932\) 1.76941e131 0.128476
\(933\) 0 0
\(934\) −2.53880e132 −1.67569
\(935\) 4.47864e131 0.281857
\(936\) 0 0
\(937\) 1.51226e132 0.865397 0.432698 0.901539i \(-0.357562\pi\)
0.432698 + 0.901539i \(0.357562\pi\)
\(938\) 1.04770e132 0.571754
\(939\) 0 0
\(940\) 7.57157e130 0.0375835
\(941\) 6.31705e131 0.299072 0.149536 0.988756i \(-0.452222\pi\)
0.149536 + 0.988756i \(0.452222\pi\)
\(942\) 0 0
\(943\) 8.53203e131 0.367521
\(944\) −2.32435e132 −0.955097
\(945\) 0 0
\(946\) 1.96883e132 0.736297
\(947\) 1.15806e132 0.413196 0.206598 0.978426i \(-0.433761\pi\)
0.206598 + 0.978426i \(0.433761\pi\)
\(948\) 0 0
\(949\) 1.04383e132 0.339065
\(950\) 2.97816e132 0.923097
\(951\) 0 0
\(952\) −6.87070e132 −1.93935
\(953\) 6.23693e132 1.68011 0.840053 0.542504i \(-0.182524\pi\)
0.840053 + 0.542504i \(0.182524\pi\)
\(954\) 0 0
\(955\) −1.38929e132 −0.340913
\(956\) −2.31825e132 −0.542980
\(957\) 0 0
\(958\) −3.25276e132 −0.694203
\(959\) 2.62985e132 0.535801
\(960\) 0 0
\(961\) 1.12262e133 2.08470
\(962\) −2.29936e133 −4.07677
\(963\) 0 0
\(964\) −5.04452e132 −0.815440
\(965\) −5.52898e131 −0.0853454
\(966\) 0 0
\(967\) −2.82355e132 −0.397482 −0.198741 0.980052i \(-0.563685\pi\)
−0.198741 + 0.980052i \(0.563685\pi\)
\(968\) −3.79412e132 −0.510102
\(969\) 0 0
\(970\) −2.37226e131 −0.0290951
\(971\) −1.37580e132 −0.161176 −0.0805879 0.996748i \(-0.525680\pi\)
−0.0805879 + 0.996748i \(0.525680\pi\)
\(972\) 0 0
\(973\) 7.55772e132 0.807922
\(974\) −7.65611e132 −0.781870
\(975\) 0 0
\(976\) −1.70508e133 −1.58938
\(977\) 1.36516e133 1.21584 0.607920 0.793998i \(-0.292004\pi\)
0.607920 + 0.793998i \(0.292004\pi\)
\(978\) 0 0
\(979\) 1.87909e133 1.52798
\(980\) −1.26924e132 −0.0986241
\(981\) 0 0
\(982\) 4.19445e133 2.97655
\(983\) −1.70343e133 −1.15530 −0.577649 0.816285i \(-0.696030\pi\)
−0.577649 + 0.816285i \(0.696030\pi\)
\(984\) 0 0
\(985\) −2.25422e132 −0.139664
\(986\) 3.46933e133 2.05458
\(987\) 0 0
\(988\) −4.07316e133 −2.20418
\(989\) 5.22823e132 0.270471
\(990\) 0 0
\(991\) 1.06033e133 0.501375 0.250687 0.968068i \(-0.419343\pi\)
0.250687 + 0.968068i \(0.419343\pi\)
\(992\) −2.38504e133 −1.07827
\(993\) 0 0
\(994\) −1.30057e133 −0.537575
\(995\) 7.06580e132 0.279277
\(996\) 0 0
\(997\) 1.46285e133 0.528768 0.264384 0.964417i \(-0.414831\pi\)
0.264384 + 0.964417i \(0.414831\pi\)
\(998\) −1.31156e133 −0.453399
\(999\) 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 9.90.a.b.1.1 7
3.2 odd 2 1.90.a.a.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.90.a.a.1.7 7 3.2 odd 2
9.90.a.b.1.1 7 1.1 even 1 trivial