Properties

Label 9.18.a.a.1.1
Level $9$
Weight $18$
Character 9.1
Self dual yes
Analytic conductor $16.490$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,18,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, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
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: \(16.4899878610\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-204.000 q^{2} -89456.0 q^{4} +163554. q^{5} -2.08466e7 q^{7} +4.49877e7 q^{8} +O(q^{10})\) \(q-204.000 q^{2} -89456.0 q^{4} +163554. q^{5} -2.08466e7 q^{7} +4.49877e7 q^{8} -3.33650e7 q^{10} -8.17372e8 q^{11} +2.99590e8 q^{13} +4.25270e9 q^{14} +2.54768e9 q^{16} +4.47756e10 q^{17} +7.87487e10 q^{19} -1.46309e10 q^{20} +1.66744e11 q^{22} +7.04672e11 q^{23} -7.36190e11 q^{25} -6.11163e10 q^{26} +1.86485e12 q^{28} +1.63794e11 q^{29} +1.04986e12 q^{31} -6.41636e12 q^{32} -9.13422e12 q^{34} -3.40954e12 q^{35} -1.98057e13 q^{37} -1.60647e13 q^{38} +7.35792e12 q^{40} -1.46600e13 q^{41} +1.16039e14 q^{43} +7.31189e13 q^{44} -1.43753e14 q^{46} +1.76607e14 q^{47} +2.01949e14 q^{49} +1.50183e14 q^{50} -2.68001e13 q^{52} -1.52863e14 q^{53} -1.33685e14 q^{55} -9.37839e14 q^{56} -3.34139e13 q^{58} +2.62797e14 q^{59} -1.35855e15 q^{61} -2.14172e14 q^{62} +9.75007e14 q^{64} +4.89991e13 q^{65} +4.44864e14 q^{67} -4.00545e15 q^{68} +6.95546e14 q^{70} +4.00327e15 q^{71} +9.24833e14 q^{73} +4.04037e15 q^{74} -7.04454e15 q^{76} +1.70394e16 q^{77} +1.47473e16 q^{79} +4.16684e14 q^{80} +2.99065e15 q^{82} -2.64230e16 q^{83} +7.32323e15 q^{85} -2.36719e16 q^{86} -3.67717e16 q^{88} +3.88837e16 q^{89} -6.24542e15 q^{91} -6.30371e16 q^{92} -3.60277e16 q^{94} +1.28797e16 q^{95} -2.53744e16 q^{97} -4.11975e16 q^{98} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −204.000 −0.563476 −0.281738 0.959491i \(-0.590911\pi\)
−0.281738 + 0.959491i \(0.590911\pi\)
\(3\) 0 0
\(4\) −89456.0 −0.682495
\(5\) 163554. 0.187248 0.0936238 0.995608i \(-0.470155\pi\)
0.0936238 + 0.995608i \(0.470155\pi\)
\(6\) 0 0
\(7\) −2.08466e7 −1.36679 −0.683394 0.730050i \(-0.739497\pi\)
−0.683394 + 0.730050i \(0.739497\pi\)
\(8\) 4.49877e7 0.948045
\(9\) 0 0
\(10\) −3.33650e7 −0.105509
\(11\) −8.17372e8 −1.14969 −0.574847 0.818261i \(-0.694938\pi\)
−0.574847 + 0.818261i \(0.694938\pi\)
\(12\) 0 0
\(13\) 2.99590e8 0.101861 0.0509306 0.998702i \(-0.483781\pi\)
0.0509306 + 0.998702i \(0.483781\pi\)
\(14\) 4.25270e9 0.770152
\(15\) 0 0
\(16\) 2.54768e9 0.148295
\(17\) 4.47756e10 1.55677 0.778387 0.627785i \(-0.216038\pi\)
0.778387 + 0.627785i \(0.216038\pi\)
\(18\) 0 0
\(19\) 7.87487e10 1.06374 0.531872 0.846824i \(-0.321489\pi\)
0.531872 + 0.846824i \(0.321489\pi\)
\(20\) −1.46309e10 −0.127796
\(21\) 0 0
\(22\) 1.66744e11 0.647824
\(23\) 7.04672e11 1.87629 0.938146 0.346240i \(-0.112542\pi\)
0.938146 + 0.346240i \(0.112542\pi\)
\(24\) 0 0
\(25\) −7.36190e11 −0.964938
\(26\) −6.11163e10 −0.0573963
\(27\) 0 0
\(28\) 1.86485e12 0.932826
\(29\) 1.63794e11 0.0608015 0.0304008 0.999538i \(-0.490322\pi\)
0.0304008 + 0.999538i \(0.490322\pi\)
\(30\) 0 0
\(31\) 1.04986e12 0.221084 0.110542 0.993871i \(-0.464741\pi\)
0.110542 + 0.993871i \(0.464741\pi\)
\(32\) −6.41636e12 −1.03161
\(33\) 0 0
\(34\) −9.13422e12 −0.877204
\(35\) −3.40954e12 −0.255928
\(36\) 0 0
\(37\) −1.98057e13 −0.926993 −0.463496 0.886099i \(-0.653405\pi\)
−0.463496 + 0.886099i \(0.653405\pi\)
\(38\) −1.60647e13 −0.599394
\(39\) 0 0
\(40\) 7.35792e12 0.177519
\(41\) −1.46600e13 −0.286729 −0.143365 0.989670i \(-0.545792\pi\)
−0.143365 + 0.989670i \(0.545792\pi\)
\(42\) 0 0
\(43\) 1.16039e14 1.51399 0.756993 0.653424i \(-0.226668\pi\)
0.756993 + 0.653424i \(0.226668\pi\)
\(44\) 7.31189e13 0.784660
\(45\) 0 0
\(46\) −1.43753e14 −1.05724
\(47\) 1.76607e14 1.08187 0.540935 0.841064i \(-0.318070\pi\)
0.540935 + 0.841064i \(0.318070\pi\)
\(48\) 0 0
\(49\) 2.01949e14 0.868109
\(50\) 1.50183e14 0.543719
\(51\) 0 0
\(52\) −2.68001e13 −0.0695197
\(53\) −1.52863e14 −0.337255 −0.168628 0.985680i \(-0.553934\pi\)
−0.168628 + 0.985680i \(0.553934\pi\)
\(54\) 0 0
\(55\) −1.33685e14 −0.215277
\(56\) −9.37839e14 −1.29578
\(57\) 0 0
\(58\) −3.34139e13 −0.0342602
\(59\) 2.62797e14 0.233012 0.116506 0.993190i \(-0.462831\pi\)
0.116506 + 0.993190i \(0.462831\pi\)
\(60\) 0 0
\(61\) −1.35855e15 −0.907346 −0.453673 0.891168i \(-0.649887\pi\)
−0.453673 + 0.891168i \(0.649887\pi\)
\(62\) −2.14172e14 −0.124575
\(63\) 0 0
\(64\) 9.75007e14 0.432990
\(65\) 4.89991e13 0.0190732
\(66\) 0 0
\(67\) 4.44864e14 0.133842 0.0669208 0.997758i \(-0.478683\pi\)
0.0669208 + 0.997758i \(0.478683\pi\)
\(68\) −4.00545e15 −1.06249
\(69\) 0 0
\(70\) 6.95546e14 0.144209
\(71\) 4.00327e15 0.735731 0.367865 0.929879i \(-0.380089\pi\)
0.367865 + 0.929879i \(0.380089\pi\)
\(72\) 0 0
\(73\) 9.24833e14 0.134220 0.0671102 0.997746i \(-0.478622\pi\)
0.0671102 + 0.997746i \(0.478622\pi\)
\(74\) 4.04037e15 0.522338
\(75\) 0 0
\(76\) −7.04454e15 −0.726001
\(77\) 1.70394e16 1.57139
\(78\) 0 0
\(79\) 1.47473e16 1.09366 0.546830 0.837244i \(-0.315834\pi\)
0.546830 + 0.837244i \(0.315834\pi\)
\(80\) 4.16684e14 0.0277678
\(81\) 0 0
\(82\) 2.99065e15 0.161565
\(83\) −2.64230e16 −1.28771 −0.643855 0.765148i \(-0.722666\pi\)
−0.643855 + 0.765148i \(0.722666\pi\)
\(84\) 0 0
\(85\) 7.32323e15 0.291502
\(86\) −2.36719e16 −0.853094
\(87\) 0 0
\(88\) −3.67717e16 −1.08996
\(89\) 3.88837e16 1.04702 0.523508 0.852021i \(-0.324623\pi\)
0.523508 + 0.852021i \(0.324623\pi\)
\(90\) 0 0
\(91\) −6.24542e15 −0.139223
\(92\) −6.30371e16 −1.28056
\(93\) 0 0
\(94\) −3.60277e16 −0.609608
\(95\) 1.28797e16 0.199184
\(96\) 0 0
\(97\) −2.53744e16 −0.328727 −0.164364 0.986400i \(-0.552557\pi\)
−0.164364 + 0.986400i \(0.552557\pi\)
\(98\) −4.11975e16 −0.489158
\(99\) 0 0
\(100\) 6.58566e16 0.658566
\(101\) 7.44216e16 0.683860 0.341930 0.939725i \(-0.388919\pi\)
0.341930 + 0.939725i \(0.388919\pi\)
\(102\) 0 0
\(103\) 2.06558e17 1.60667 0.803333 0.595530i \(-0.203058\pi\)
0.803333 + 0.595530i \(0.203058\pi\)
\(104\) 1.34779e16 0.0965689
\(105\) 0 0
\(106\) 3.11842e16 0.190035
\(107\) 2.23373e17 1.25681 0.628405 0.777887i \(-0.283708\pi\)
0.628405 + 0.777887i \(0.283708\pi\)
\(108\) 0 0
\(109\) −2.66379e17 −1.28049 −0.640244 0.768172i \(-0.721167\pi\)
−0.640244 + 0.768172i \(0.721167\pi\)
\(110\) 2.72716e16 0.121304
\(111\) 0 0
\(112\) −5.31104e16 −0.202687
\(113\) 6.75157e16 0.238912 0.119456 0.992839i \(-0.461885\pi\)
0.119456 + 0.992839i \(0.461885\pi\)
\(114\) 0 0
\(115\) 1.15252e17 0.351331
\(116\) −1.46523e16 −0.0414967
\(117\) 0 0
\(118\) −5.36106e16 −0.131297
\(119\) −9.33417e17 −2.12778
\(120\) 0 0
\(121\) 1.62651e17 0.321795
\(122\) 2.77145e17 0.511267
\(123\) 0 0
\(124\) −9.39164e16 −0.150889
\(125\) −2.45189e17 −0.367930
\(126\) 0 0
\(127\) −2.11177e17 −0.276896 −0.138448 0.990370i \(-0.544211\pi\)
−0.138448 + 0.990370i \(0.544211\pi\)
\(128\) 6.42103e17 0.787626
\(129\) 0 0
\(130\) −9.99582e15 −0.0107473
\(131\) −1.02202e18 −1.02957 −0.514783 0.857321i \(-0.672127\pi\)
−0.514783 + 0.857321i \(0.672127\pi\)
\(132\) 0 0
\(133\) −1.64164e18 −1.45391
\(134\) −9.07522e16 −0.0754165
\(135\) 0 0
\(136\) 2.01435e18 1.47589
\(137\) 1.70778e18 1.17573 0.587865 0.808959i \(-0.299969\pi\)
0.587865 + 0.808959i \(0.299969\pi\)
\(138\) 0 0
\(139\) 1.05313e18 0.640994 0.320497 0.947249i \(-0.396150\pi\)
0.320497 + 0.947249i \(0.396150\pi\)
\(140\) 3.05004e17 0.174669
\(141\) 0 0
\(142\) −8.16667e17 −0.414567
\(143\) −2.44876e17 −0.117109
\(144\) 0 0
\(145\) 2.67891e16 0.0113849
\(146\) −1.88666e17 −0.0756300
\(147\) 0 0
\(148\) 1.77174e18 0.632668
\(149\) 1.41948e18 0.478681 0.239340 0.970936i \(-0.423069\pi\)
0.239340 + 0.970936i \(0.423069\pi\)
\(150\) 0 0
\(151\) −8.64830e17 −0.260391 −0.130196 0.991488i \(-0.541561\pi\)
−0.130196 + 0.991488i \(0.541561\pi\)
\(152\) 3.54272e18 1.00848
\(153\) 0 0
\(154\) −3.47604e18 −0.885438
\(155\) 1.71709e17 0.0413974
\(156\) 0 0
\(157\) 4.23286e18 0.915138 0.457569 0.889174i \(-0.348720\pi\)
0.457569 + 0.889174i \(0.348720\pi\)
\(158\) −3.00845e18 −0.616251
\(159\) 0 0
\(160\) −1.04942e18 −0.193166
\(161\) −1.46900e19 −2.56449
\(162\) 0 0
\(163\) −4.49946e18 −0.707238 −0.353619 0.935390i \(-0.615049\pi\)
−0.353619 + 0.935390i \(0.615049\pi\)
\(164\) 1.31143e18 0.195691
\(165\) 0 0
\(166\) 5.39028e18 0.725593
\(167\) 2.89131e18 0.369832 0.184916 0.982754i \(-0.440799\pi\)
0.184916 + 0.982754i \(0.440799\pi\)
\(168\) 0 0
\(169\) −8.56066e18 −0.989624
\(170\) −1.49394e18 −0.164254
\(171\) 0 0
\(172\) −1.03804e19 −1.03329
\(173\) 5.47551e18 0.518839 0.259420 0.965765i \(-0.416469\pi\)
0.259420 + 0.965765i \(0.416469\pi\)
\(174\) 0 0
\(175\) 1.53470e19 1.31887
\(176\) −2.08241e18 −0.170493
\(177\) 0 0
\(178\) −7.93228e18 −0.589968
\(179\) 2.46059e19 1.74497 0.872485 0.488642i \(-0.162507\pi\)
0.872485 + 0.488642i \(0.162507\pi\)
\(180\) 0 0
\(181\) 2.09528e19 1.35199 0.675997 0.736904i \(-0.263713\pi\)
0.675997 + 0.736904i \(0.263713\pi\)
\(182\) 1.27406e18 0.0784485
\(183\) 0 0
\(184\) 3.17016e19 1.77881
\(185\) −3.23931e18 −0.173577
\(186\) 0 0
\(187\) −3.65983e19 −1.78981
\(188\) −1.57985e19 −0.738371
\(189\) 0 0
\(190\) −2.62745e18 −0.112235
\(191\) 6.03799e18 0.246666 0.123333 0.992365i \(-0.460642\pi\)
0.123333 + 0.992365i \(0.460642\pi\)
\(192\) 0 0
\(193\) 1.32605e19 0.495817 0.247909 0.968783i \(-0.420257\pi\)
0.247909 + 0.968783i \(0.420257\pi\)
\(194\) 5.17638e18 0.185230
\(195\) 0 0
\(196\) −1.80655e19 −0.592480
\(197\) 2.62791e19 0.825368 0.412684 0.910874i \(-0.364591\pi\)
0.412684 + 0.910874i \(0.364591\pi\)
\(198\) 0 0
\(199\) −2.06774e19 −0.595998 −0.297999 0.954566i \(-0.596319\pi\)
−0.297999 + 0.954566i \(0.596319\pi\)
\(200\) −3.31195e19 −0.914805
\(201\) 0 0
\(202\) −1.51820e19 −0.385338
\(203\) −3.41454e18 −0.0831028
\(204\) 0 0
\(205\) −2.39771e18 −0.0536894
\(206\) −4.21378e19 −0.905317
\(207\) 0 0
\(208\) 7.63260e17 0.0151055
\(209\) −6.43670e19 −1.22298
\(210\) 0 0
\(211\) 1.03175e20 1.80790 0.903948 0.427643i \(-0.140656\pi\)
0.903948 + 0.427643i \(0.140656\pi\)
\(212\) 1.36746e19 0.230175
\(213\) 0 0
\(214\) −4.55682e19 −0.708181
\(215\) 1.89786e19 0.283490
\(216\) 0 0
\(217\) −2.18860e19 −0.302175
\(218\) 5.43414e19 0.721523
\(219\) 0 0
\(220\) 1.19589e19 0.146926
\(221\) 1.34143e19 0.158575
\(222\) 0 0
\(223\) −7.25219e19 −0.794105 −0.397053 0.917796i \(-0.629967\pi\)
−0.397053 + 0.917796i \(0.629967\pi\)
\(224\) 1.33759e20 1.40999
\(225\) 0 0
\(226\) −1.37732e19 −0.134621
\(227\) −9.29766e18 −0.0875294 −0.0437647 0.999042i \(-0.513935\pi\)
−0.0437647 + 0.999042i \(0.513935\pi\)
\(228\) 0 0
\(229\) 9.87595e19 0.862933 0.431467 0.902129i \(-0.357996\pi\)
0.431467 + 0.902129i \(0.357996\pi\)
\(230\) −2.35114e19 −0.197967
\(231\) 0 0
\(232\) 7.36871e18 0.0576426
\(233\) −3.53284e19 −0.266439 −0.133220 0.991087i \(-0.542532\pi\)
−0.133220 + 0.991087i \(0.542532\pi\)
\(234\) 0 0
\(235\) 2.88847e19 0.202578
\(236\) −2.35088e19 −0.159030
\(237\) 0 0
\(238\) 1.90417e20 1.19895
\(239\) 4.68557e19 0.284695 0.142348 0.989817i \(-0.454535\pi\)
0.142348 + 0.989817i \(0.454535\pi\)
\(240\) 0 0
\(241\) −2.55944e20 −1.44877 −0.724385 0.689396i \(-0.757876\pi\)
−0.724385 + 0.689396i \(0.757876\pi\)
\(242\) −3.31807e19 −0.181324
\(243\) 0 0
\(244\) 1.21531e20 0.619259
\(245\) 3.30295e19 0.162551
\(246\) 0 0
\(247\) 2.35923e19 0.108354
\(248\) 4.72308e19 0.209598
\(249\) 0 0
\(250\) 5.00185e19 0.207320
\(251\) −2.06944e20 −0.829135 −0.414567 0.910019i \(-0.636067\pi\)
−0.414567 + 0.910019i \(0.636067\pi\)
\(252\) 0 0
\(253\) −5.75979e20 −2.15716
\(254\) 4.30802e19 0.156024
\(255\) 0 0
\(256\) −2.58785e20 −0.876798
\(257\) −4.67477e20 −1.53225 −0.766124 0.642693i \(-0.777817\pi\)
−0.766124 + 0.642693i \(0.777817\pi\)
\(258\) 0 0
\(259\) 4.12881e20 1.26700
\(260\) −4.38326e18 −0.0130174
\(261\) 0 0
\(262\) 2.08492e20 0.580135
\(263\) −1.26367e20 −0.340416 −0.170208 0.985408i \(-0.554444\pi\)
−0.170208 + 0.985408i \(0.554444\pi\)
\(264\) 0 0
\(265\) −2.50014e19 −0.0631502
\(266\) 3.34894e20 0.819245
\(267\) 0 0
\(268\) −3.97957e19 −0.0913462
\(269\) 1.18589e20 0.263724 0.131862 0.991268i \(-0.457904\pi\)
0.131862 + 0.991268i \(0.457904\pi\)
\(270\) 0 0
\(271\) 4.33191e20 0.904567 0.452284 0.891874i \(-0.350610\pi\)
0.452284 + 0.891874i \(0.350610\pi\)
\(272\) 1.14074e20 0.230861
\(273\) 0 0
\(274\) −3.48388e20 −0.662495
\(275\) 6.01741e20 1.10938
\(276\) 0 0
\(277\) 4.26563e20 0.739444 0.369722 0.929142i \(-0.379453\pi\)
0.369722 + 0.929142i \(0.379453\pi\)
\(278\) −2.14838e20 −0.361185
\(279\) 0 0
\(280\) −1.53387e20 −0.242631
\(281\) 9.97107e20 1.53016 0.765082 0.643933i \(-0.222699\pi\)
0.765082 + 0.643933i \(0.222699\pi\)
\(282\) 0 0
\(283\) −4.85988e20 −0.702168 −0.351084 0.936344i \(-0.614187\pi\)
−0.351084 + 0.936344i \(0.614187\pi\)
\(284\) −3.58117e20 −0.502133
\(285\) 0 0
\(286\) 4.99548e19 0.0659881
\(287\) 3.05611e20 0.391898
\(288\) 0 0
\(289\) 1.17761e21 1.42355
\(290\) −5.46498e18 −0.00641513
\(291\) 0 0
\(292\) −8.27318e19 −0.0916048
\(293\) −1.48194e21 −1.59388 −0.796938 0.604061i \(-0.793548\pi\)
−0.796938 + 0.604061i \(0.793548\pi\)
\(294\) 0 0
\(295\) 4.29815e19 0.0436310
\(296\) −8.91015e20 −0.878831
\(297\) 0 0
\(298\) −2.89574e20 −0.269725
\(299\) 2.11113e20 0.191121
\(300\) 0 0
\(301\) −2.41901e21 −2.06930
\(302\) 1.76425e20 0.146724
\(303\) 0 0
\(304\) 2.00627e20 0.157748
\(305\) −2.22197e20 −0.169898
\(306\) 0 0
\(307\) 3.29698e18 0.00238473 0.00119237 0.999999i \(-0.499620\pi\)
0.00119237 + 0.999999i \(0.499620\pi\)
\(308\) −1.52428e21 −1.07246
\(309\) 0 0
\(310\) −3.50286e19 −0.0233265
\(311\) 2.79172e21 1.80887 0.904437 0.426607i \(-0.140291\pi\)
0.904437 + 0.426607i \(0.140291\pi\)
\(312\) 0 0
\(313\) 1.83126e21 1.12363 0.561815 0.827263i \(-0.310103\pi\)
0.561815 + 0.827263i \(0.310103\pi\)
\(314\) −8.63503e20 −0.515658
\(315\) 0 0
\(316\) −1.31924e21 −0.746418
\(317\) −1.74407e21 −0.960639 −0.480319 0.877094i \(-0.659479\pi\)
−0.480319 + 0.877094i \(0.659479\pi\)
\(318\) 0 0
\(319\) −1.33881e20 −0.0699031
\(320\) 1.59466e20 0.0810763
\(321\) 0 0
\(322\) 2.99676e21 1.44503
\(323\) 3.52602e21 1.65601
\(324\) 0 0
\(325\) −2.20555e20 −0.0982897
\(326\) 9.17889e20 0.398511
\(327\) 0 0
\(328\) −6.59521e20 −0.271832
\(329\) −3.68164e21 −1.47869
\(330\) 0 0
\(331\) 2.03545e21 0.776468 0.388234 0.921561i \(-0.373085\pi\)
0.388234 + 0.921561i \(0.373085\pi\)
\(332\) 2.36369e21 0.878855
\(333\) 0 0
\(334\) −5.89828e20 −0.208392
\(335\) 7.27592e19 0.0250615
\(336\) 0 0
\(337\) −3.47485e21 −1.13784 −0.568921 0.822392i \(-0.692639\pi\)
−0.568921 + 0.822392i \(0.692639\pi\)
\(338\) 1.74638e21 0.557629
\(339\) 0 0
\(340\) −6.55107e20 −0.198949
\(341\) −8.58127e20 −0.254179
\(342\) 0 0
\(343\) 6.39613e20 0.180268
\(344\) 5.22032e21 1.43533
\(345\) 0 0
\(346\) −1.11700e21 −0.292353
\(347\) 1.59958e21 0.408513 0.204256 0.978917i \(-0.434522\pi\)
0.204256 + 0.978917i \(0.434522\pi\)
\(348\) 0 0
\(349\) 1.65498e21 0.402509 0.201255 0.979539i \(-0.435498\pi\)
0.201255 + 0.979539i \(0.435498\pi\)
\(350\) −3.13079e21 −0.743149
\(351\) 0 0
\(352\) 5.24455e21 1.18603
\(353\) −8.34568e21 −1.84237 −0.921185 0.389126i \(-0.872777\pi\)
−0.921185 + 0.389126i \(0.872777\pi\)
\(354\) 0 0
\(355\) 6.54751e20 0.137764
\(356\) −3.47838e21 −0.714583
\(357\) 0 0
\(358\) −5.01960e21 −0.983248
\(359\) 3.48877e20 0.0667375 0.0333688 0.999443i \(-0.489376\pi\)
0.0333688 + 0.999443i \(0.489376\pi\)
\(360\) 0 0
\(361\) 7.20963e20 0.131553
\(362\) −4.27438e21 −0.761816
\(363\) 0 0
\(364\) 5.58690e20 0.0950187
\(365\) 1.51260e20 0.0251325
\(366\) 0 0
\(367\) −6.36726e21 −1.00993 −0.504965 0.863140i \(-0.668495\pi\)
−0.504965 + 0.863140i \(0.668495\pi\)
\(368\) 1.79528e21 0.278244
\(369\) 0 0
\(370\) 6.60819e20 0.0978065
\(371\) 3.18668e21 0.460957
\(372\) 0 0
\(373\) −7.78865e21 −1.07631 −0.538155 0.842846i \(-0.680878\pi\)
−0.538155 + 0.842846i \(0.680878\pi\)
\(374\) 7.46606e21 1.00852
\(375\) 0 0
\(376\) 7.94513e21 1.02566
\(377\) 4.90709e19 0.00619331
\(378\) 0 0
\(379\) −3.17769e21 −0.383423 −0.191711 0.981451i \(-0.561404\pi\)
−0.191711 + 0.981451i \(0.561404\pi\)
\(380\) −1.15216e21 −0.135942
\(381\) 0 0
\(382\) −1.23175e21 −0.138990
\(383\) −6.69797e21 −0.739186 −0.369593 0.929194i \(-0.620503\pi\)
−0.369593 + 0.929194i \(0.620503\pi\)
\(384\) 0 0
\(385\) 2.78686e21 0.294238
\(386\) −2.70514e21 −0.279381
\(387\) 0 0
\(388\) 2.26989e21 0.224355
\(389\) 1.10700e22 1.07048 0.535238 0.844701i \(-0.320222\pi\)
0.535238 + 0.844701i \(0.320222\pi\)
\(390\) 0 0
\(391\) 3.15521e22 2.92096
\(392\) 9.08520e21 0.823006
\(393\) 0 0
\(394\) −5.36094e21 −0.465075
\(395\) 2.41198e21 0.204785
\(396\) 0 0
\(397\) −1.43915e22 −1.17054 −0.585270 0.810839i \(-0.699011\pi\)
−0.585270 + 0.810839i \(0.699011\pi\)
\(398\) 4.21819e21 0.335831
\(399\) 0 0
\(400\) −1.87558e21 −0.143095
\(401\) −8.63300e21 −0.644814 −0.322407 0.946601i \(-0.604492\pi\)
−0.322407 + 0.946601i \(0.604492\pi\)
\(402\) 0 0
\(403\) 3.14528e20 0.0225199
\(404\) −6.65746e21 −0.466731
\(405\) 0 0
\(406\) 6.96566e20 0.0468264
\(407\) 1.61887e22 1.06576
\(408\) 0 0
\(409\) 1.62508e21 0.102619 0.0513094 0.998683i \(-0.483661\pi\)
0.0513094 + 0.998683i \(0.483661\pi\)
\(410\) 4.89132e20 0.0302527
\(411\) 0 0
\(412\) −1.84778e22 −1.09654
\(413\) −5.47842e21 −0.318478
\(414\) 0 0
\(415\) −4.32158e21 −0.241120
\(416\) −1.92227e21 −0.105081
\(417\) 0 0
\(418\) 1.31309e22 0.689120
\(419\) −1.93761e22 −0.996433 −0.498216 0.867053i \(-0.666012\pi\)
−0.498216 + 0.867053i \(0.666012\pi\)
\(420\) 0 0
\(421\) −2.73783e22 −1.35210 −0.676051 0.736855i \(-0.736310\pi\)
−0.676051 + 0.736855i \(0.736310\pi\)
\(422\) −2.10477e22 −1.01871
\(423\) 0 0
\(424\) −6.87698e21 −0.319733
\(425\) −3.29633e22 −1.50219
\(426\) 0 0
\(427\) 2.83211e22 1.24015
\(428\) −1.99821e22 −0.857766
\(429\) 0 0
\(430\) −3.87164e21 −0.159740
\(431\) 3.54083e22 1.43235 0.716174 0.697922i \(-0.245892\pi\)
0.716174 + 0.697922i \(0.245892\pi\)
\(432\) 0 0
\(433\) 1.25200e22 0.486920 0.243460 0.969911i \(-0.421718\pi\)
0.243460 + 0.969911i \(0.421718\pi\)
\(434\) 4.46474e21 0.170268
\(435\) 0 0
\(436\) 2.38292e22 0.873926
\(437\) 5.54920e22 1.99590
\(438\) 0 0
\(439\) −3.05896e22 −1.05834 −0.529170 0.848516i \(-0.677497\pi\)
−0.529170 + 0.848516i \(0.677497\pi\)
\(440\) −6.01416e21 −0.204093
\(441\) 0 0
\(442\) −2.73652e21 −0.0893530
\(443\) 1.49267e22 0.478114 0.239057 0.971005i \(-0.423162\pi\)
0.239057 + 0.971005i \(0.423162\pi\)
\(444\) 0 0
\(445\) 6.35959e21 0.196051
\(446\) 1.47945e22 0.447459
\(447\) 0 0
\(448\) −2.03255e22 −0.591805
\(449\) −4.93173e21 −0.140898 −0.0704491 0.997515i \(-0.522443\pi\)
−0.0704491 + 0.997515i \(0.522443\pi\)
\(450\) 0 0
\(451\) 1.19827e22 0.329651
\(452\) −6.03968e21 −0.163056
\(453\) 0 0
\(454\) 1.89672e21 0.0493207
\(455\) −1.02146e21 −0.0260691
\(456\) 0 0
\(457\) 6.67847e22 1.64206 0.821031 0.570884i \(-0.193400\pi\)
0.821031 + 0.570884i \(0.193400\pi\)
\(458\) −2.01469e22 −0.486242
\(459\) 0 0
\(460\) −1.03100e22 −0.239782
\(461\) 3.58593e22 0.818736 0.409368 0.912369i \(-0.365749\pi\)
0.409368 + 0.912369i \(0.365749\pi\)
\(462\) 0 0
\(463\) −2.17954e22 −0.479652 −0.239826 0.970816i \(-0.577090\pi\)
−0.239826 + 0.970816i \(0.577090\pi\)
\(464\) 4.17295e20 0.00901654
\(465\) 0 0
\(466\) 7.20699e21 0.150132
\(467\) 1.01862e22 0.208362 0.104181 0.994558i \(-0.466778\pi\)
0.104181 + 0.994558i \(0.466778\pi\)
\(468\) 0 0
\(469\) −9.27388e21 −0.182933
\(470\) −5.89248e21 −0.114148
\(471\) 0 0
\(472\) 1.18226e22 0.220906
\(473\) −9.48470e22 −1.74062
\(474\) 0 0
\(475\) −5.79739e22 −1.02645
\(476\) 8.34998e22 1.45220
\(477\) 0 0
\(478\) −9.55856e21 −0.160419
\(479\) −6.74899e22 −1.11272 −0.556361 0.830940i \(-0.687803\pi\)
−0.556361 + 0.830940i \(0.687803\pi\)
\(480\) 0 0
\(481\) −5.93360e21 −0.0944245
\(482\) 5.22125e22 0.816347
\(483\) 0 0
\(484\) −1.45501e22 −0.219624
\(485\) −4.15008e21 −0.0615534
\(486\) 0 0
\(487\) 2.15075e22 0.308031 0.154016 0.988068i \(-0.450779\pi\)
0.154016 + 0.988068i \(0.450779\pi\)
\(488\) −6.11182e22 −0.860205
\(489\) 0 0
\(490\) −6.73802e21 −0.0915937
\(491\) −1.36833e22 −0.182809 −0.0914044 0.995814i \(-0.529136\pi\)
−0.0914044 + 0.995814i \(0.529136\pi\)
\(492\) 0 0
\(493\) 7.33397e21 0.0946542
\(494\) −4.81283e21 −0.0610550
\(495\) 0 0
\(496\) 2.67471e21 0.0327856
\(497\) −8.34544e22 −1.00559
\(498\) 0 0
\(499\) −1.36348e22 −0.158779 −0.0793894 0.996844i \(-0.525297\pi\)
−0.0793894 + 0.996844i \(0.525297\pi\)
\(500\) 2.19336e22 0.251110
\(501\) 0 0
\(502\) 4.22165e22 0.467197
\(503\) −6.17808e22 −0.672242 −0.336121 0.941819i \(-0.609115\pi\)
−0.336121 + 0.941819i \(0.609115\pi\)
\(504\) 0 0
\(505\) 1.21719e22 0.128051
\(506\) 1.17500e23 1.21551
\(507\) 0 0
\(508\) 1.88911e22 0.188980
\(509\) 9.19553e22 0.904639 0.452320 0.891856i \(-0.350597\pi\)
0.452320 + 0.891856i \(0.350597\pi\)
\(510\) 0 0
\(511\) −1.92796e22 −0.183451
\(512\) −3.13696e22 −0.293572
\(513\) 0 0
\(514\) 9.53654e22 0.863384
\(515\) 3.37834e22 0.300844
\(516\) 0 0
\(517\) −1.44353e23 −1.24382
\(518\) −8.42278e22 −0.713925
\(519\) 0 0
\(520\) 2.20436e21 0.0180823
\(521\) 1.76716e23 1.42611 0.713057 0.701106i \(-0.247310\pi\)
0.713057 + 0.701106i \(0.247310\pi\)
\(522\) 0 0
\(523\) −2.39601e23 −1.87165 −0.935823 0.352470i \(-0.885342\pi\)
−0.935823 + 0.352470i \(0.885342\pi\)
\(524\) 9.14260e22 0.702674
\(525\) 0 0
\(526\) 2.57789e22 0.191816
\(527\) 4.70082e22 0.344178
\(528\) 0 0
\(529\) 3.55513e23 2.52047
\(530\) 5.10029e21 0.0355836
\(531\) 0 0
\(532\) 1.46854e23 0.992289
\(533\) −4.39200e21 −0.0292066
\(534\) 0 0
\(535\) 3.65336e22 0.235334
\(536\) 2.00134e22 0.126888
\(537\) 0 0
\(538\) −2.41921e22 −0.148602
\(539\) −1.65067e23 −0.998059
\(540\) 0 0
\(541\) 1.61766e22 0.0947784 0.0473892 0.998877i \(-0.484910\pi\)
0.0473892 + 0.998877i \(0.484910\pi\)
\(542\) −8.83710e22 −0.509702
\(543\) 0 0
\(544\) −2.87296e23 −1.60598
\(545\) −4.35674e22 −0.239768
\(546\) 0 0
\(547\) 2.46037e23 1.31253 0.656263 0.754532i \(-0.272136\pi\)
0.656263 + 0.754532i \(0.272136\pi\)
\(548\) −1.52771e23 −0.802430
\(549\) 0 0
\(550\) −1.22755e23 −0.625111
\(551\) 1.28985e22 0.0646773
\(552\) 0 0
\(553\) −3.07431e23 −1.49480
\(554\) −8.70189e22 −0.416659
\(555\) 0 0
\(556\) −9.42084e22 −0.437475
\(557\) 8.61414e22 0.393951 0.196976 0.980408i \(-0.436888\pi\)
0.196976 + 0.980408i \(0.436888\pi\)
\(558\) 0 0
\(559\) 3.47641e22 0.154216
\(560\) −8.68642e21 −0.0379527
\(561\) 0 0
\(562\) −2.03410e23 −0.862210
\(563\) −2.78426e23 −1.16249 −0.581244 0.813730i \(-0.697434\pi\)
−0.581244 + 0.813730i \(0.697434\pi\)
\(564\) 0 0
\(565\) 1.10425e22 0.0447357
\(566\) 9.91416e22 0.395655
\(567\) 0 0
\(568\) 1.80098e23 0.697506
\(569\) 4.52461e23 1.72634 0.863172 0.504910i \(-0.168474\pi\)
0.863172 + 0.504910i \(0.168474\pi\)
\(570\) 0 0
\(571\) 7.63957e22 0.282919 0.141459 0.989944i \(-0.454821\pi\)
0.141459 + 0.989944i \(0.454821\pi\)
\(572\) 2.19057e22 0.0799264
\(573\) 0 0
\(574\) −6.23447e22 −0.220825
\(575\) −5.18772e23 −1.81051
\(576\) 0 0
\(577\) 2.40302e23 0.814261 0.407131 0.913370i \(-0.366529\pi\)
0.407131 + 0.913370i \(0.366529\pi\)
\(578\) −2.40233e23 −0.802134
\(579\) 0 0
\(580\) −2.39645e21 −0.00777016
\(581\) 5.50828e23 1.76002
\(582\) 0 0
\(583\) 1.24946e23 0.387740
\(584\) 4.16061e22 0.127247
\(585\) 0 0
\(586\) 3.02315e23 0.898111
\(587\) 1.19663e23 0.350377 0.175188 0.984535i \(-0.443947\pi\)
0.175188 + 0.984535i \(0.443947\pi\)
\(588\) 0 0
\(589\) 8.26751e22 0.235177
\(590\) −8.76824e21 −0.0245850
\(591\) 0 0
\(592\) −5.04587e22 −0.137468
\(593\) 2.98534e23 0.801731 0.400866 0.916137i \(-0.368709\pi\)
0.400866 + 0.916137i \(0.368709\pi\)
\(594\) 0 0
\(595\) −1.52664e23 −0.398422
\(596\) −1.26981e23 −0.326697
\(597\) 0 0
\(598\) −4.30670e22 −0.107692
\(599\) 2.69094e23 0.663401 0.331700 0.943385i \(-0.392378\pi\)
0.331700 + 0.943385i \(0.392378\pi\)
\(600\) 0 0
\(601\) 3.04922e22 0.0730728 0.0365364 0.999332i \(-0.488368\pi\)
0.0365364 + 0.999332i \(0.488368\pi\)
\(602\) 4.93478e23 1.16600
\(603\) 0 0
\(604\) 7.73642e22 0.177716
\(605\) 2.66021e22 0.0602554
\(606\) 0 0
\(607\) −5.58818e23 −1.23074 −0.615371 0.788238i \(-0.710994\pi\)
−0.615371 + 0.788238i \(0.710994\pi\)
\(608\) −5.05279e23 −1.09737
\(609\) 0 0
\(610\) 4.53281e22 0.0957335
\(611\) 5.29095e22 0.110201
\(612\) 0 0
\(613\) −6.34890e23 −1.28613 −0.643064 0.765812i \(-0.722337\pi\)
−0.643064 + 0.765812i \(0.722337\pi\)
\(614\) −6.72584e20 −0.00134374
\(615\) 0 0
\(616\) 7.66564e23 1.48975
\(617\) −2.36658e23 −0.453625 −0.226813 0.973938i \(-0.572830\pi\)
−0.226813 + 0.973938i \(0.572830\pi\)
\(618\) 0 0
\(619\) 3.23770e23 0.603763 0.301882 0.953345i \(-0.402385\pi\)
0.301882 + 0.953345i \(0.402385\pi\)
\(620\) −1.53604e22 −0.0282535
\(621\) 0 0
\(622\) −5.69511e23 −1.01926
\(623\) −8.10592e23 −1.43105
\(624\) 0 0
\(625\) 5.21566e23 0.896044
\(626\) −3.73577e23 −0.633138
\(627\) 0 0
\(628\) −3.78654e23 −0.624577
\(629\) −8.86814e23 −1.44312
\(630\) 0 0
\(631\) −1.43355e23 −0.227073 −0.113536 0.993534i \(-0.536218\pi\)
−0.113536 + 0.993534i \(0.536218\pi\)
\(632\) 6.63448e23 1.03684
\(633\) 0 0
\(634\) 3.55790e23 0.541297
\(635\) −3.45389e22 −0.0518480
\(636\) 0 0
\(637\) 6.05017e22 0.0884265
\(638\) 2.73116e22 0.0393887
\(639\) 0 0
\(640\) 1.05019e23 0.147481
\(641\) −4.66875e23 −0.647004 −0.323502 0.946227i \(-0.604860\pi\)
−0.323502 + 0.946227i \(0.604860\pi\)
\(642\) 0 0
\(643\) −4.89782e23 −0.661013 −0.330506 0.943804i \(-0.607220\pi\)
−0.330506 + 0.943804i \(0.607220\pi\)
\(644\) 1.31411e24 1.75025
\(645\) 0 0
\(646\) −7.19308e23 −0.933122
\(647\) 1.09424e24 1.40096 0.700482 0.713670i \(-0.252968\pi\)
0.700482 + 0.713670i \(0.252968\pi\)
\(648\) 0 0
\(649\) −2.14803e23 −0.267893
\(650\) 4.49932e22 0.0553839
\(651\) 0 0
\(652\) 4.02503e23 0.482686
\(653\) −5.18955e23 −0.614282 −0.307141 0.951664i \(-0.599372\pi\)
−0.307141 + 0.951664i \(0.599372\pi\)
\(654\) 0 0
\(655\) −1.67156e23 −0.192784
\(656\) −3.73491e22 −0.0425205
\(657\) 0 0
\(658\) 7.51055e23 0.833204
\(659\) −4.01367e23 −0.439557 −0.219779 0.975550i \(-0.570534\pi\)
−0.219779 + 0.975550i \(0.570534\pi\)
\(660\) 0 0
\(661\) 1.72655e24 1.84275 0.921375 0.388674i \(-0.127067\pi\)
0.921375 + 0.388674i \(0.127067\pi\)
\(662\) −4.15233e23 −0.437521
\(663\) 0 0
\(664\) −1.18871e24 −1.22081
\(665\) −2.68497e23 −0.272242
\(666\) 0 0
\(667\) 1.15421e23 0.114081
\(668\) −2.58645e23 −0.252409
\(669\) 0 0
\(670\) −1.48429e22 −0.0141216
\(671\) 1.11044e24 1.04317
\(672\) 0 0
\(673\) −9.61197e23 −0.880409 −0.440205 0.897898i \(-0.645094\pi\)
−0.440205 + 0.897898i \(0.645094\pi\)
\(674\) 7.08869e23 0.641146
\(675\) 0 0
\(676\) 7.65803e23 0.675414
\(677\) 6.20817e23 0.540705 0.270352 0.962761i \(-0.412860\pi\)
0.270352 + 0.962761i \(0.412860\pi\)
\(678\) 0 0
\(679\) 5.28969e23 0.449301
\(680\) 3.29455e23 0.276357
\(681\) 0 0
\(682\) 1.75058e23 0.143224
\(683\) 2.70132e23 0.218273 0.109137 0.994027i \(-0.465191\pi\)
0.109137 + 0.994027i \(0.465191\pi\)
\(684\) 0 0
\(685\) 2.79315e23 0.220153
\(686\) −1.30481e23 −0.101576
\(687\) 0 0
\(688\) 2.95630e23 0.224516
\(689\) −4.57963e22 −0.0343532
\(690\) 0 0
\(691\) −5.16537e23 −0.378040 −0.189020 0.981973i \(-0.560531\pi\)
−0.189020 + 0.981973i \(0.560531\pi\)
\(692\) −4.89817e23 −0.354105
\(693\) 0 0
\(694\) −3.26315e23 −0.230187
\(695\) 1.72243e23 0.120025
\(696\) 0 0
\(697\) −6.56412e23 −0.446373
\(698\) −3.37615e23 −0.226804
\(699\) 0 0
\(700\) −1.37288e24 −0.900120
\(701\) −5.95182e23 −0.385520 −0.192760 0.981246i \(-0.561744\pi\)
−0.192760 + 0.981246i \(0.561744\pi\)
\(702\) 0 0
\(703\) −1.55967e24 −0.986084
\(704\) −7.96944e23 −0.497806
\(705\) 0 0
\(706\) 1.70252e24 1.03813
\(707\) −1.55143e24 −0.934691
\(708\) 0 0
\(709\) 1.04141e24 0.612533 0.306266 0.951946i \(-0.400920\pi\)
0.306266 + 0.951946i \(0.400920\pi\)
\(710\) −1.33569e23 −0.0776266
\(711\) 0 0
\(712\) 1.74929e24 0.992618
\(713\) 7.39808e23 0.414818
\(714\) 0 0
\(715\) −4.00505e22 −0.0219284
\(716\) −2.20114e24 −1.19093
\(717\) 0 0
\(718\) −7.11710e22 −0.0376050
\(719\) −1.57223e24 −0.820958 −0.410479 0.911870i \(-0.634638\pi\)
−0.410479 + 0.911870i \(0.634638\pi\)
\(720\) 0 0
\(721\) −4.30602e24 −2.19597
\(722\) −1.47077e23 −0.0741271
\(723\) 0 0
\(724\) −1.87436e24 −0.922730
\(725\) −1.20583e23 −0.0586697
\(726\) 0 0
\(727\) 3.53143e24 1.67845 0.839224 0.543785i \(-0.183009\pi\)
0.839224 + 0.543785i \(0.183009\pi\)
\(728\) −2.80967e23 −0.131989
\(729\) 0 0
\(730\) −3.08571e22 −0.0141615
\(731\) 5.19571e24 2.35693
\(732\) 0 0
\(733\) 2.94036e24 1.30322 0.651609 0.758555i \(-0.274094\pi\)
0.651609 + 0.758555i \(0.274094\pi\)
\(734\) 1.29892e24 0.569071
\(735\) 0 0
\(736\) −4.52143e24 −1.93559
\(737\) −3.63619e23 −0.153877
\(738\) 0 0
\(739\) 9.01116e23 0.372651 0.186326 0.982488i \(-0.440342\pi\)
0.186326 + 0.982488i \(0.440342\pi\)
\(740\) 2.89775e23 0.118466
\(741\) 0 0
\(742\) −6.50082e23 −0.259738
\(743\) 3.55572e24 1.40450 0.702252 0.711929i \(-0.252178\pi\)
0.702252 + 0.711929i \(0.252178\pi\)
\(744\) 0 0
\(745\) 2.32162e23 0.0896318
\(746\) 1.58888e24 0.606474
\(747\) 0 0
\(748\) 3.27394e24 1.22154
\(749\) −4.65657e24 −1.71779
\(750\) 0 0
\(751\) 1.88476e24 0.679697 0.339849 0.940480i \(-0.389624\pi\)
0.339849 + 0.940480i \(0.389624\pi\)
\(752\) 4.49938e23 0.160436
\(753\) 0 0
\(754\) −1.00105e22 −0.00348978
\(755\) −1.41446e23 −0.0487576
\(756\) 0 0
\(757\) −1.47031e24 −0.495558 −0.247779 0.968817i \(-0.579701\pi\)
−0.247779 + 0.968817i \(0.579701\pi\)
\(758\) 6.48248e23 0.216049
\(759\) 0 0
\(760\) 5.79426e23 0.188835
\(761\) 1.01146e24 0.325972 0.162986 0.986628i \(-0.447888\pi\)
0.162986 + 0.986628i \(0.447888\pi\)
\(762\) 0 0
\(763\) 5.55309e24 1.75015
\(764\) −5.40134e23 −0.168348
\(765\) 0 0
\(766\) 1.36639e24 0.416513
\(767\) 7.87314e22 0.0237349
\(768\) 0 0
\(769\) −4.17674e24 −1.23158 −0.615791 0.787910i \(-0.711163\pi\)
−0.615791 + 0.787910i \(0.711163\pi\)
\(770\) −5.68520e23 −0.165796
\(771\) 0 0
\(772\) −1.18623e24 −0.338393
\(773\) 3.39452e24 0.957750 0.478875 0.877883i \(-0.341045\pi\)
0.478875 + 0.877883i \(0.341045\pi\)
\(774\) 0 0
\(775\) −7.72897e23 −0.213332
\(776\) −1.14154e24 −0.311648
\(777\) 0 0
\(778\) −2.25828e24 −0.603187
\(779\) −1.15446e24 −0.305007
\(780\) 0 0
\(781\) −3.27216e24 −0.845865
\(782\) −6.43663e24 −1.64589
\(783\) 0 0
\(784\) 5.14501e23 0.128736
\(785\) 6.92301e23 0.171357
\(786\) 0 0
\(787\) −7.12654e24 −1.72621 −0.863105 0.505025i \(-0.831483\pi\)
−0.863105 + 0.505025i \(0.831483\pi\)
\(788\) −2.35082e24 −0.563310
\(789\) 0 0
\(790\) −4.92044e23 −0.115391
\(791\) −1.40747e24 −0.326542
\(792\) 0 0
\(793\) −4.07008e23 −0.0924232
\(794\) 2.93586e24 0.659571
\(795\) 0 0
\(796\) 1.84972e24 0.406766
\(797\) −5.98812e24 −1.30285 −0.651426 0.758712i \(-0.725829\pi\)
−0.651426 + 0.758712i \(0.725829\pi\)
\(798\) 0 0
\(799\) 7.90767e24 1.68423
\(800\) 4.72365e24 0.995436
\(801\) 0 0
\(802\) 1.76113e24 0.363337
\(803\) −7.55933e23 −0.154312
\(804\) 0 0
\(805\) −2.40261e24 −0.480195
\(806\) −6.41636e22 −0.0126894
\(807\) 0 0
\(808\) 3.34806e24 0.648330
\(809\) 2.16052e24 0.413996 0.206998 0.978341i \(-0.433631\pi\)
0.206998 + 0.978341i \(0.433631\pi\)
\(810\) 0 0
\(811\) −7.22229e23 −0.135518 −0.0677591 0.997702i \(-0.521585\pi\)
−0.0677591 + 0.997702i \(0.521585\pi\)
\(812\) 3.05451e23 0.0567172
\(813\) 0 0
\(814\) −3.30249e24 −0.600529
\(815\) −7.35904e23 −0.132428
\(816\) 0 0
\(817\) 9.13790e24 1.61049
\(818\) −3.31516e23 −0.0578232
\(819\) 0 0
\(820\) 2.14489e23 0.0366427
\(821\) −1.07145e25 −1.81157 −0.905786 0.423736i \(-0.860719\pi\)
−0.905786 + 0.423736i \(0.860719\pi\)
\(822\) 0 0
\(823\) −8.71705e24 −1.44368 −0.721840 0.692060i \(-0.756703\pi\)
−0.721840 + 0.692060i \(0.756703\pi\)
\(824\) 9.29257e24 1.52319
\(825\) 0 0
\(826\) 1.11760e24 0.179455
\(827\) 3.38564e24 0.538077 0.269038 0.963129i \(-0.413294\pi\)
0.269038 + 0.963129i \(0.413294\pi\)
\(828\) 0 0
\(829\) 1.77079e24 0.275711 0.137855 0.990452i \(-0.455979\pi\)
0.137855 + 0.990452i \(0.455979\pi\)
\(830\) 8.81603e23 0.135865
\(831\) 0 0
\(832\) 2.92102e23 0.0441048
\(833\) 9.04237e24 1.35145
\(834\) 0 0
\(835\) 4.72886e23 0.0692502
\(836\) 5.75801e24 0.834678
\(837\) 0 0
\(838\) 3.95273e24 0.561466
\(839\) 6.16728e23 0.0867196 0.0433598 0.999060i \(-0.486194\pi\)
0.0433598 + 0.999060i \(0.486194\pi\)
\(840\) 0 0
\(841\) −7.23032e24 −0.996303
\(842\) 5.58518e24 0.761876
\(843\) 0 0
\(844\) −9.22962e24 −1.23388
\(845\) −1.40013e24 −0.185305
\(846\) 0 0
\(847\) −3.39070e24 −0.439826
\(848\) −3.89448e23 −0.0500132
\(849\) 0 0
\(850\) 6.72452e24 0.846448
\(851\) −1.39565e25 −1.73931
\(852\) 0 0
\(853\) 8.04241e24 0.982470 0.491235 0.871027i \(-0.336546\pi\)
0.491235 + 0.871027i \(0.336546\pi\)
\(854\) −5.77751e24 −0.698794
\(855\) 0 0
\(856\) 1.00491e25 1.19151
\(857\) −1.71827e23 −0.0201722 −0.0100861 0.999949i \(-0.503211\pi\)
−0.0100861 + 0.999949i \(0.503211\pi\)
\(858\) 0 0
\(859\) 5.30613e24 0.610711 0.305356 0.952238i \(-0.401225\pi\)
0.305356 + 0.952238i \(0.401225\pi\)
\(860\) −1.69775e24 −0.193481
\(861\) 0 0
\(862\) −7.22330e24 −0.807093
\(863\) 1.56974e25 1.73675 0.868375 0.495909i \(-0.165165\pi\)
0.868375 + 0.495909i \(0.165165\pi\)
\(864\) 0 0
\(865\) 8.95541e23 0.0971513
\(866\) −2.55408e24 −0.274367
\(867\) 0 0
\(868\) 1.95783e24 0.206233
\(869\) −1.20540e25 −1.25737
\(870\) 0 0
\(871\) 1.33277e23 0.0136333
\(872\) −1.19838e25 −1.21396
\(873\) 0 0
\(874\) −1.13204e25 −1.12464
\(875\) 5.11134e24 0.502882
\(876\) 0 0
\(877\) −5.17245e24 −0.499114 −0.249557 0.968360i \(-0.580285\pi\)
−0.249557 + 0.968360i \(0.580285\pi\)
\(878\) 6.24027e24 0.596349
\(879\) 0 0
\(880\) −3.40586e23 −0.0319245
\(881\) −1.81020e25 −1.68048 −0.840238 0.542217i \(-0.817585\pi\)
−0.840238 + 0.542217i \(0.817585\pi\)
\(882\) 0 0
\(883\) −1.22106e25 −1.11192 −0.555959 0.831210i \(-0.687649\pi\)
−0.555959 + 0.831210i \(0.687649\pi\)
\(884\) −1.19999e24 −0.108227
\(885\) 0 0
\(886\) −3.04504e24 −0.269406
\(887\) 1.59175e25 1.39484 0.697420 0.716663i \(-0.254331\pi\)
0.697420 + 0.716663i \(0.254331\pi\)
\(888\) 0 0
\(889\) 4.40232e24 0.378457
\(890\) −1.29736e24 −0.110470
\(891\) 0 0
\(892\) 6.48752e24 0.541973
\(893\) 1.39075e25 1.15083
\(894\) 0 0
\(895\) 4.02439e24 0.326741
\(896\) −1.33856e25 −1.07652
\(897\) 0 0
\(898\) 1.00607e24 0.0793927
\(899\) 1.71961e23 0.0134422
\(900\) 0 0
\(901\) −6.84456e24 −0.525031
\(902\) −2.44447e24 −0.185750
\(903\) 0 0
\(904\) 3.03738e24 0.226499
\(905\) 3.42692e24 0.253158
\(906\) 0 0
\(907\) −5.23010e24 −0.379182 −0.189591 0.981863i \(-0.560716\pi\)
−0.189591 + 0.981863i \(0.560716\pi\)
\(908\) 8.31732e23 0.0597384
\(909\) 0 0
\(910\) 2.08378e23 0.0146893
\(911\) −1.46926e25 −1.02611 −0.513054 0.858356i \(-0.671486\pi\)
−0.513054 + 0.858356i \(0.671486\pi\)
\(912\) 0 0
\(913\) 2.15974e25 1.48047
\(914\) −1.36241e25 −0.925262
\(915\) 0 0
\(916\) −8.83463e24 −0.588948
\(917\) 2.13056e25 1.40720
\(918\) 0 0
\(919\) 3.02532e24 0.196151 0.0980753 0.995179i \(-0.468731\pi\)
0.0980753 + 0.995179i \(0.468731\pi\)
\(920\) 5.18492e24 0.333078
\(921\) 0 0
\(922\) −7.31530e24 −0.461338
\(923\) 1.19934e24 0.0749424
\(924\) 0 0
\(925\) 1.45808e25 0.894491
\(926\) 4.44626e24 0.270272
\(927\) 0 0
\(928\) −1.05096e24 −0.0627232
\(929\) 3.31849e25 1.96249 0.981244 0.192771i \(-0.0617474\pi\)
0.981244 + 0.192771i \(0.0617474\pi\)
\(930\) 0 0
\(931\) 1.59032e25 0.923446
\(932\) 3.16033e24 0.181843
\(933\) 0 0
\(934\) −2.07798e24 −0.117407
\(935\) −5.98581e24 −0.335138
\(936\) 0 0
\(937\) −4.73059e24 −0.260093 −0.130047 0.991508i \(-0.541513\pi\)
−0.130047 + 0.991508i \(0.541513\pi\)
\(938\) 1.89187e24 0.103078
\(939\) 0 0
\(940\) −2.58391e24 −0.138258
\(941\) 2.93059e25 1.55397 0.776986 0.629518i \(-0.216748\pi\)
0.776986 + 0.629518i \(0.216748\pi\)
\(942\) 0 0
\(943\) −1.03305e25 −0.537988
\(944\) 6.69524e23 0.0345545
\(945\) 0 0
\(946\) 1.93488e25 0.980797
\(947\) −2.12220e25 −1.06613 −0.533067 0.846073i \(-0.678960\pi\)
−0.533067 + 0.846073i \(0.678960\pi\)
\(948\) 0 0
\(949\) 2.77070e23 0.0136718
\(950\) 1.18267e25 0.578379
\(951\) 0 0
\(952\) −4.19923e25 −2.01723
\(953\) −3.62846e25 −1.72756 −0.863779 0.503870i \(-0.831909\pi\)
−0.863779 + 0.503870i \(0.831909\pi\)
\(954\) 0 0
\(955\) 9.87537e23 0.0461875
\(956\) −4.19152e24 −0.194303
\(957\) 0 0
\(958\) 1.37679e25 0.626992
\(959\) −3.56014e25 −1.60697
\(960\) 0 0
\(961\) −2.14479e25 −0.951122
\(962\) 1.21045e24 0.0532059
\(963\) 0 0
\(964\) 2.28957e25 0.988778
\(965\) 2.16880e24 0.0928406
\(966\) 0 0
\(967\) 2.27508e25 0.956910 0.478455 0.878112i \(-0.341197\pi\)
0.478455 + 0.878112i \(0.341197\pi\)
\(968\) 7.31728e24 0.305077
\(969\) 0 0
\(970\) 8.46617e23 0.0346838
\(971\) −2.55385e25 −1.03713 −0.518563 0.855039i \(-0.673533\pi\)
−0.518563 + 0.855039i \(0.673533\pi\)
\(972\) 0 0
\(973\) −2.19540e25 −0.876103
\(974\) −4.38754e24 −0.173568
\(975\) 0 0
\(976\) −3.46116e24 −0.134555
\(977\) −3.25195e25 −1.25326 −0.626629 0.779318i \(-0.715566\pi\)
−0.626629 + 0.779318i \(0.715566\pi\)
\(978\) 0 0
\(979\) −3.17825e25 −1.20375
\(980\) −2.95469e24 −0.110940
\(981\) 0 0
\(982\) 2.79139e24 0.103008
\(983\) 1.00301e25 0.366946 0.183473 0.983025i \(-0.441266\pi\)
0.183473 + 0.983025i \(0.441266\pi\)
\(984\) 0 0
\(985\) 4.29805e24 0.154548
\(986\) −1.49613e24 −0.0533354
\(987\) 0 0
\(988\) −2.11047e24 −0.0739513
\(989\) 8.17693e25 2.84068
\(990\) 0 0
\(991\) −3.21765e25 −1.09878 −0.549392 0.835564i \(-0.685141\pi\)
−0.549392 + 0.835564i \(0.685141\pi\)
\(992\) −6.73628e24 −0.228072
\(993\) 0 0
\(994\) 1.70247e25 0.566624
\(995\) −3.38187e24 −0.111599
\(996\) 0 0
\(997\) 2.48213e25 0.805222 0.402611 0.915371i \(-0.368103\pi\)
0.402611 + 0.915371i \(0.368103\pi\)
\(998\) 2.78149e24 0.0894680
\(999\) 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 9.18.a.a.1.1 1
3.2 odd 2 3.18.a.a.1.1 1
12.11 even 2 48.18.a.e.1.1 1
15.2 even 4 75.18.b.a.49.2 2
15.8 even 4 75.18.b.a.49.1 2
15.14 odd 2 75.18.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.a.1.1 1 3.2 odd 2
9.18.a.a.1.1 1 1.1 even 1 trivial
48.18.a.e.1.1 1 12.11 even 2
75.18.a.a.1.1 1 15.14 odd 2
75.18.b.a.49.1 2 15.8 even 4
75.18.b.a.49.2 2 15.2 even 4