Properties

Label 261.12.a.a.1.11
Level $261$
Weight $12$
Character 261.1
Self dual yes
Analytic conductor $200.538$
Analytic rank $1$
Dimension $11$
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] = [261,12,Mod(1,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 261.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: \(200.537570126\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + \cdots - 75\!\cdots\!58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-82.7218\) of defining polynomial
Character \(\chi\) \(=\) 261.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+85.7218 q^{2} +5300.23 q^{4} +1506.26 q^{5} -7069.49 q^{7} +278787. q^{8} +O(q^{10})\) \(q+85.7218 q^{2} +5300.23 q^{4} +1506.26 q^{5} -7069.49 q^{7} +278787. q^{8} +129120. q^{10} -598395. q^{11} -284167. q^{13} -606010. q^{14} +1.30433e7 q^{16} -6.02872e6 q^{17} -1.04134e7 q^{19} +7.98356e6 q^{20} -5.12955e7 q^{22} -3.07410e7 q^{23} -4.65593e7 q^{25} -2.43593e7 q^{26} -3.74700e7 q^{28} -2.05111e7 q^{29} +2.00119e7 q^{31} +5.47138e8 q^{32} -5.16793e8 q^{34} -1.06485e7 q^{35} -3.29372e8 q^{37} -8.92657e8 q^{38} +4.19928e8 q^{40} +1.33409e8 q^{41} -1.51972e8 q^{43} -3.17163e9 q^{44} -2.63517e9 q^{46} -5.65222e8 q^{47} -1.92735e9 q^{49} -3.99115e9 q^{50} -1.50615e9 q^{52} +5.83151e9 q^{53} -9.01341e8 q^{55} -1.97089e9 q^{56} -1.75825e9 q^{58} +8.59503e9 q^{59} -1.25836e10 q^{61} +1.71546e9 q^{62} +2.01890e10 q^{64} -4.28030e8 q^{65} +1.31795e10 q^{67} -3.19536e10 q^{68} -9.12811e8 q^{70} +4.99850e9 q^{71} +3.78444e9 q^{73} -2.82344e10 q^{74} -5.51936e10 q^{76} +4.23035e9 q^{77} -3.61053e10 q^{79} +1.96467e10 q^{80} +1.14360e10 q^{82} +3.39624e10 q^{83} -9.08085e9 q^{85} -1.30273e10 q^{86} -1.66825e11 q^{88} -8.21218e10 q^{89} +2.00891e9 q^{91} -1.62934e11 q^{92} -4.84519e10 q^{94} -1.56854e10 q^{95} -4.18025e10 q^{97} -1.65216e11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 32 q^{2} + 9146 q^{4} + 2740 q^{5} - 49432 q^{7} + 150054 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 32 q^{2} + 9146 q^{4} + 2740 q^{5} - 49432 q^{7} + 150054 q^{8} - 685834 q^{10} + 612246 q^{11} + 1510364 q^{13} - 3955400 q^{14} + 3024818 q^{16} + 3291098 q^{17} - 44121388 q^{19} + 49472662 q^{20} - 43435618 q^{22} + 88684076 q^{23} - 44195521 q^{25} + 324999762 q^{26} - 391274848 q^{28} - 225622639 q^{29} - 292235934 q^{31} + 632542514 q^{32} - 1113307936 q^{34} + 1312820120 q^{35} - 1380429338 q^{37} + 1222857284 q^{38} - 2713154106 q^{40} + 1062067494 q^{41} + 74588594 q^{43} - 52891466 q^{44} - 87670324 q^{46} + 1821239394 q^{47} + 4692522003 q^{49} - 9494259926 q^{50} + 3266669866 q^{52} - 7818635688 q^{53} - 191002682 q^{55} - 11263587512 q^{56} - 656356768 q^{58} - 1230002712 q^{59} - 18602654230 q^{61} - 22075953162 q^{62} + 11813658086 q^{64} - 32245789334 q^{65} + 27481284652 q^{67} - 29588811820 q^{68} + 42862666712 q^{70} + 20347168516 q^{71} - 57740010478 q^{73} + 2640709564 q^{74} - 33350650772 q^{76} - 871959792 q^{77} - 120245016462 q^{79} + 84319695274 q^{80} - 111495532412 q^{82} + 142463983824 q^{83} - 181628566552 q^{85} - 47870165542 q^{86} - 180608014462 q^{88} + 96700717270 q^{89} - 355162031176 q^{91} + 22429477796 q^{92} + 172608565078 q^{94} + 195922150708 q^{95} - 303190852014 q^{97} + 123776497136 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 85.7218 1.89420 0.947101 0.320934i \(-0.103997\pi\)
0.947101 + 0.320934i \(0.103997\pi\)
\(3\) 0 0
\(4\) 5300.23 2.58800
\(5\) 1506.26 0.215559 0.107780 0.994175i \(-0.465626\pi\)
0.107780 + 0.994175i \(0.465626\pi\)
\(6\) 0 0
\(7\) −7069.49 −0.158982 −0.0794912 0.996836i \(-0.525330\pi\)
−0.0794912 + 0.996836i \(0.525330\pi\)
\(8\) 278787. 3.00800
\(9\) 0 0
\(10\) 129120. 0.408313
\(11\) −598395. −1.12028 −0.560142 0.828397i \(-0.689253\pi\)
−0.560142 + 0.828397i \(0.689253\pi\)
\(12\) 0 0
\(13\) −284167. −0.212268 −0.106134 0.994352i \(-0.533847\pi\)
−0.106134 + 0.994352i \(0.533847\pi\)
\(14\) −606010. −0.301145
\(15\) 0 0
\(16\) 1.30433e7 3.10976
\(17\) −6.02872e6 −1.02981 −0.514904 0.857248i \(-0.672172\pi\)
−0.514904 + 0.857248i \(0.672172\pi\)
\(18\) 0 0
\(19\) −1.04134e7 −0.964825 −0.482413 0.875944i \(-0.660239\pi\)
−0.482413 + 0.875944i \(0.660239\pi\)
\(20\) 7.98356e6 0.557868
\(21\) 0 0
\(22\) −5.12955e7 −2.12204
\(23\) −3.07410e7 −0.995897 −0.497949 0.867207i \(-0.665913\pi\)
−0.497949 + 0.867207i \(0.665913\pi\)
\(24\) 0 0
\(25\) −4.65593e7 −0.953534
\(26\) −2.43593e7 −0.402079
\(27\) 0 0
\(28\) −3.74700e7 −0.411447
\(29\) −2.05111e7 −0.185695
\(30\) 0 0
\(31\) 2.00119e7 0.125545 0.0627725 0.998028i \(-0.480006\pi\)
0.0627725 + 0.998028i \(0.480006\pi\)
\(32\) 5.47138e8 2.88252
\(33\) 0 0
\(34\) −5.16793e8 −1.95066
\(35\) −1.06485e7 −0.0342701
\(36\) 0 0
\(37\) −3.29372e8 −0.780867 −0.390433 0.920631i \(-0.627675\pi\)
−0.390433 + 0.920631i \(0.627675\pi\)
\(38\) −8.92657e8 −1.82757
\(39\) 0 0
\(40\) 4.19928e8 0.648402
\(41\) 1.33409e8 0.179834 0.0899172 0.995949i \(-0.471340\pi\)
0.0899172 + 0.995949i \(0.471340\pi\)
\(42\) 0 0
\(43\) −1.51972e8 −0.157647 −0.0788236 0.996889i \(-0.525116\pi\)
−0.0788236 + 0.996889i \(0.525116\pi\)
\(44\) −3.17163e9 −2.89930
\(45\) 0 0
\(46\) −2.63517e9 −1.88643
\(47\) −5.65222e8 −0.359485 −0.179743 0.983714i \(-0.557526\pi\)
−0.179743 + 0.983714i \(0.557526\pi\)
\(48\) 0 0
\(49\) −1.92735e9 −0.974725
\(50\) −3.99115e9 −1.80619
\(51\) 0 0
\(52\) −1.50615e9 −0.549351
\(53\) 5.83151e9 1.91542 0.957709 0.287738i \(-0.0929032\pi\)
0.957709 + 0.287738i \(0.0929032\pi\)
\(54\) 0 0
\(55\) −9.01341e8 −0.241487
\(56\) −1.97089e9 −0.478220
\(57\) 0 0
\(58\) −1.75825e9 −0.351745
\(59\) 8.59503e9 1.56517 0.782585 0.622544i \(-0.213901\pi\)
0.782585 + 0.622544i \(0.213901\pi\)
\(60\) 0 0
\(61\) −1.25836e10 −1.90761 −0.953807 0.300419i \(-0.902873\pi\)
−0.953807 + 0.300419i \(0.902873\pi\)
\(62\) 1.71546e9 0.237808
\(63\) 0 0
\(64\) 2.01890e10 2.35031
\(65\) −4.28030e8 −0.0457563
\(66\) 0 0
\(67\) 1.31795e10 1.19258 0.596291 0.802768i \(-0.296640\pi\)
0.596291 + 0.802768i \(0.296640\pi\)
\(68\) −3.19536e10 −2.66515
\(69\) 0 0
\(70\) −9.12811e8 −0.0649145
\(71\) 4.99850e9 0.328790 0.164395 0.986395i \(-0.447433\pi\)
0.164395 + 0.986395i \(0.447433\pi\)
\(72\) 0 0
\(73\) 3.78444e9 0.213661 0.106831 0.994277i \(-0.465930\pi\)
0.106831 + 0.994277i \(0.465930\pi\)
\(74\) −2.82344e10 −1.47912
\(75\) 0 0
\(76\) −5.51936e10 −2.49697
\(77\) 4.23035e9 0.178105
\(78\) 0 0
\(79\) −3.61053e10 −1.32015 −0.660074 0.751201i \(-0.729475\pi\)
−0.660074 + 0.751201i \(0.729475\pi\)
\(80\) 1.96467e10 0.670338
\(81\) 0 0
\(82\) 1.14360e10 0.340643
\(83\) 3.39624e10 0.946387 0.473193 0.880959i \(-0.343101\pi\)
0.473193 + 0.880959i \(0.343101\pi\)
\(84\) 0 0
\(85\) −9.08085e9 −0.221984
\(86\) −1.30273e10 −0.298616
\(87\) 0 0
\(88\) −1.66825e11 −3.36982
\(89\) −8.21218e10 −1.55888 −0.779441 0.626475i \(-0.784497\pi\)
−0.779441 + 0.626475i \(0.784497\pi\)
\(90\) 0 0
\(91\) 2.00891e9 0.0337469
\(92\) −1.62934e11 −2.57739
\(93\) 0 0
\(94\) −4.84519e10 −0.680938
\(95\) −1.56854e10 −0.207977
\(96\) 0 0
\(97\) −4.18025e10 −0.494262 −0.247131 0.968982i \(-0.579488\pi\)
−0.247131 + 0.968982i \(0.579488\pi\)
\(98\) −1.65216e11 −1.84633
\(99\) 0 0
\(100\) −2.46775e11 −2.46775
\(101\) 1.93279e11 1.82986 0.914930 0.403613i \(-0.132246\pi\)
0.914930 + 0.403613i \(0.132246\pi\)
\(102\) 0 0
\(103\) 1.18942e11 1.01095 0.505475 0.862841i \(-0.331317\pi\)
0.505475 + 0.862841i \(0.331317\pi\)
\(104\) −7.92221e10 −0.638503
\(105\) 0 0
\(106\) 4.99888e11 3.62819
\(107\) −1.74684e11 −1.20404 −0.602022 0.798480i \(-0.705638\pi\)
−0.602022 + 0.798480i \(0.705638\pi\)
\(108\) 0 0
\(109\) −4.14362e10 −0.257949 −0.128975 0.991648i \(-0.541169\pi\)
−0.128975 + 0.991648i \(0.541169\pi\)
\(110\) −7.72646e10 −0.457426
\(111\) 0 0
\(112\) −9.22095e10 −0.494398
\(113\) 7.24493e10 0.369916 0.184958 0.982746i \(-0.440785\pi\)
0.184958 + 0.982746i \(0.440785\pi\)
\(114\) 0 0
\(115\) −4.63040e10 −0.214675
\(116\) −1.08714e11 −0.480580
\(117\) 0 0
\(118\) 7.36782e11 2.96475
\(119\) 4.26200e10 0.163721
\(120\) 0 0
\(121\) 7.27646e10 0.255036
\(122\) −1.07869e12 −3.61341
\(123\) 0 0
\(124\) 1.06068e11 0.324911
\(125\) −1.43679e11 −0.421102
\(126\) 0 0
\(127\) 1.62132e11 0.435460 0.217730 0.976009i \(-0.430135\pi\)
0.217730 + 0.976009i \(0.430135\pi\)
\(128\) 6.10103e11 1.56945
\(129\) 0 0
\(130\) −3.66915e10 −0.0866718
\(131\) −4.23024e10 −0.0958016 −0.0479008 0.998852i \(-0.515253\pi\)
−0.0479008 + 0.998852i \(0.515253\pi\)
\(132\) 0 0
\(133\) 7.36176e10 0.153390
\(134\) 1.12977e12 2.25899
\(135\) 0 0
\(136\) −1.68073e12 −3.09766
\(137\) 9.90449e11 1.75335 0.876676 0.481082i \(-0.159756\pi\)
0.876676 + 0.481082i \(0.159756\pi\)
\(138\) 0 0
\(139\) 5.58932e10 0.0913646 0.0456823 0.998956i \(-0.485454\pi\)
0.0456823 + 0.998956i \(0.485454\pi\)
\(140\) −5.64397e10 −0.0886912
\(141\) 0 0
\(142\) 4.28480e11 0.622795
\(143\) 1.70044e11 0.237801
\(144\) 0 0
\(145\) −3.08952e10 −0.0400283
\(146\) 3.24409e11 0.404718
\(147\) 0 0
\(148\) −1.74575e12 −2.02089
\(149\) −5.84976e11 −0.652550 −0.326275 0.945275i \(-0.605794\pi\)
−0.326275 + 0.945275i \(0.605794\pi\)
\(150\) 0 0
\(151\) 1.14354e12 1.18543 0.592716 0.805412i \(-0.298056\pi\)
0.592716 + 0.805412i \(0.298056\pi\)
\(152\) −2.90313e12 −2.90220
\(153\) 0 0
\(154\) 3.62633e11 0.337368
\(155\) 3.01433e10 0.0270624
\(156\) 0 0
\(157\) −1.19892e12 −1.00309 −0.501547 0.865130i \(-0.667236\pi\)
−0.501547 + 0.865130i \(0.667236\pi\)
\(158\) −3.09502e12 −2.50063
\(159\) 0 0
\(160\) 8.24135e11 0.621353
\(161\) 2.17323e11 0.158330
\(162\) 0 0
\(163\) −2.10135e12 −1.43043 −0.715216 0.698904i \(-0.753671\pi\)
−0.715216 + 0.698904i \(0.753671\pi\)
\(164\) 7.07097e11 0.465413
\(165\) 0 0
\(166\) 2.91132e12 1.79265
\(167\) −2.49583e12 −1.48688 −0.743438 0.668805i \(-0.766806\pi\)
−0.743438 + 0.668805i \(0.766806\pi\)
\(168\) 0 0
\(169\) −1.71141e12 −0.954942
\(170\) −7.78427e11 −0.420483
\(171\) 0 0
\(172\) −8.05485e11 −0.407992
\(173\) 1.72047e11 0.0844100 0.0422050 0.999109i \(-0.486562\pi\)
0.0422050 + 0.999109i \(0.486562\pi\)
\(174\) 0 0
\(175\) 3.29151e11 0.151595
\(176\) −7.80504e12 −3.48382
\(177\) 0 0
\(178\) −7.03963e12 −2.95284
\(179\) 3.23099e12 1.31415 0.657074 0.753826i \(-0.271794\pi\)
0.657074 + 0.753826i \(0.271794\pi\)
\(180\) 0 0
\(181\) 2.32931e12 0.891239 0.445620 0.895222i \(-0.352983\pi\)
0.445620 + 0.895222i \(0.352983\pi\)
\(182\) 1.72208e11 0.0639235
\(183\) 0 0
\(184\) −8.57020e12 −2.99566
\(185\) −4.96121e11 −0.168323
\(186\) 0 0
\(187\) 3.60755e12 1.15368
\(188\) −2.99581e12 −0.930349
\(189\) 0 0
\(190\) −1.34458e12 −0.393950
\(191\) −8.46740e11 −0.241028 −0.120514 0.992712i \(-0.538454\pi\)
−0.120514 + 0.992712i \(0.538454\pi\)
\(192\) 0 0
\(193\) −2.76009e12 −0.741921 −0.370960 0.928649i \(-0.620971\pi\)
−0.370960 + 0.928649i \(0.620971\pi\)
\(194\) −3.58339e12 −0.936233
\(195\) 0 0
\(196\) −1.02154e13 −2.52259
\(197\) −4.20436e12 −1.00957 −0.504785 0.863245i \(-0.668428\pi\)
−0.504785 + 0.863245i \(0.668428\pi\)
\(198\) 0 0
\(199\) −7.87735e12 −1.78932 −0.894661 0.446746i \(-0.852583\pi\)
−0.894661 + 0.446746i \(0.852583\pi\)
\(200\) −1.29801e13 −2.86823
\(201\) 0 0
\(202\) 1.65683e13 3.46613
\(203\) 1.45003e11 0.0295223
\(204\) 0 0
\(205\) 2.00949e11 0.0387650
\(206\) 1.01959e13 1.91494
\(207\) 0 0
\(208\) −3.70647e12 −0.660104
\(209\) 6.23134e12 1.08088
\(210\) 0 0
\(211\) 1.03418e13 1.70233 0.851165 0.524899i \(-0.175897\pi\)
0.851165 + 0.524899i \(0.175897\pi\)
\(212\) 3.09084e13 4.95711
\(213\) 0 0
\(214\) −1.49742e13 −2.28070
\(215\) −2.28910e11 −0.0339823
\(216\) 0 0
\(217\) −1.41474e11 −0.0199595
\(218\) −3.55199e12 −0.488608
\(219\) 0 0
\(220\) −4.77732e12 −0.624970
\(221\) 1.71316e12 0.218595
\(222\) 0 0
\(223\) 4.02014e12 0.488162 0.244081 0.969755i \(-0.421514\pi\)
0.244081 + 0.969755i \(0.421514\pi\)
\(224\) −3.86799e12 −0.458270
\(225\) 0 0
\(226\) 6.21049e12 0.700696
\(227\) −1.61832e13 −1.78206 −0.891030 0.453944i \(-0.850017\pi\)
−0.891030 + 0.453944i \(0.850017\pi\)
\(228\) 0 0
\(229\) 1.45521e13 1.52697 0.763485 0.645826i \(-0.223487\pi\)
0.763485 + 0.645826i \(0.223487\pi\)
\(230\) −3.96927e12 −0.406637
\(231\) 0 0
\(232\) −5.71825e12 −0.558572
\(233\) −1.36170e13 −1.29904 −0.649520 0.760344i \(-0.725030\pi\)
−0.649520 + 0.760344i \(0.725030\pi\)
\(234\) 0 0
\(235\) −8.51374e11 −0.0774903
\(236\) 4.55557e13 4.05067
\(237\) 0 0
\(238\) 3.65346e12 0.310121
\(239\) −3.48235e12 −0.288858 −0.144429 0.989515i \(-0.546135\pi\)
−0.144429 + 0.989515i \(0.546135\pi\)
\(240\) 0 0
\(241\) −5.57570e12 −0.441780 −0.220890 0.975299i \(-0.570896\pi\)
−0.220890 + 0.975299i \(0.570896\pi\)
\(242\) 6.23752e12 0.483089
\(243\) 0 0
\(244\) −6.66960e13 −4.93691
\(245\) −2.90310e12 −0.210111
\(246\) 0 0
\(247\) 2.95915e12 0.204802
\(248\) 5.57908e12 0.377640
\(249\) 0 0
\(250\) −1.23164e13 −0.797653
\(251\) 1.23140e13 0.780177 0.390088 0.920777i \(-0.372444\pi\)
0.390088 + 0.920777i \(0.372444\pi\)
\(252\) 0 0
\(253\) 1.83952e13 1.11569
\(254\) 1.38982e13 0.824849
\(255\) 0 0
\(256\) 1.09519e13 0.622546
\(257\) −2.71133e13 −1.50852 −0.754258 0.656578i \(-0.772003\pi\)
−0.754258 + 0.656578i \(0.772003\pi\)
\(258\) 0 0
\(259\) 2.32849e12 0.124144
\(260\) −2.26866e12 −0.118418
\(261\) 0 0
\(262\) −3.62624e12 −0.181468
\(263\) −1.68153e13 −0.824038 −0.412019 0.911175i \(-0.635176\pi\)
−0.412019 + 0.911175i \(0.635176\pi\)
\(264\) 0 0
\(265\) 8.78380e12 0.412886
\(266\) 6.31064e12 0.290552
\(267\) 0 0
\(268\) 6.98546e13 3.08641
\(269\) 3.03478e13 1.31368 0.656840 0.754030i \(-0.271893\pi\)
0.656840 + 0.754030i \(0.271893\pi\)
\(270\) 0 0
\(271\) −1.31818e13 −0.547826 −0.273913 0.961754i \(-0.588318\pi\)
−0.273913 + 0.961754i \(0.588318\pi\)
\(272\) −7.86344e13 −3.20246
\(273\) 0 0
\(274\) 8.49031e13 3.32120
\(275\) 2.78608e13 1.06823
\(276\) 0 0
\(277\) 4.85013e13 1.78696 0.893480 0.449103i \(-0.148256\pi\)
0.893480 + 0.449103i \(0.148256\pi\)
\(278\) 4.79127e12 0.173063
\(279\) 0 0
\(280\) −2.96868e12 −0.103085
\(281\) 1.57905e13 0.537663 0.268832 0.963187i \(-0.413362\pi\)
0.268832 + 0.963187i \(0.413362\pi\)
\(282\) 0 0
\(283\) −3.55960e13 −1.16567 −0.582836 0.812590i \(-0.698057\pi\)
−0.582836 + 0.812590i \(0.698057\pi\)
\(284\) 2.64932e13 0.850910
\(285\) 0 0
\(286\) 1.45765e13 0.450442
\(287\) −9.43132e11 −0.0285905
\(288\) 0 0
\(289\) 2.07357e12 0.0605036
\(290\) −2.64840e12 −0.0758218
\(291\) 0 0
\(292\) 2.00584e13 0.552957
\(293\) 6.08679e13 1.64671 0.823354 0.567528i \(-0.192100\pi\)
0.823354 + 0.567528i \(0.192100\pi\)
\(294\) 0 0
\(295\) 1.29464e13 0.337387
\(296\) −9.18247e13 −2.34885
\(297\) 0 0
\(298\) −5.01452e13 −1.23606
\(299\) 8.73556e12 0.211397
\(300\) 0 0
\(301\) 1.07436e12 0.0250631
\(302\) 9.80260e13 2.24545
\(303\) 0 0
\(304\) −1.35825e14 −3.00038
\(305\) −1.89542e13 −0.411204
\(306\) 0 0
\(307\) 1.27841e12 0.0267552 0.0133776 0.999911i \(-0.495742\pi\)
0.0133776 + 0.999911i \(0.495742\pi\)
\(308\) 2.24218e13 0.460938
\(309\) 0 0
\(310\) 2.58394e12 0.0512616
\(311\) 5.67748e13 1.10656 0.553278 0.832996i \(-0.313377\pi\)
0.553278 + 0.832996i \(0.313377\pi\)
\(312\) 0 0
\(313\) −3.35116e13 −0.630524 −0.315262 0.949005i \(-0.602092\pi\)
−0.315262 + 0.949005i \(0.602092\pi\)
\(314\) −1.02774e14 −1.90006
\(315\) 0 0
\(316\) −1.91367e14 −3.41655
\(317\) 1.98376e13 0.348066 0.174033 0.984740i \(-0.444320\pi\)
0.174033 + 0.984740i \(0.444320\pi\)
\(318\) 0 0
\(319\) 1.22738e13 0.208031
\(320\) 3.04100e13 0.506632
\(321\) 0 0
\(322\) 1.86293e13 0.299909
\(323\) 6.27796e13 0.993584
\(324\) 0 0
\(325\) 1.32306e13 0.202405
\(326\) −1.80132e14 −2.70953
\(327\) 0 0
\(328\) 3.71927e13 0.540943
\(329\) 3.99583e12 0.0571518
\(330\) 0 0
\(331\) −4.54179e13 −0.628309 −0.314155 0.949372i \(-0.601721\pi\)
−0.314155 + 0.949372i \(0.601721\pi\)
\(332\) 1.80009e14 2.44925
\(333\) 0 0
\(334\) −2.13947e14 −2.81644
\(335\) 1.98519e13 0.257072
\(336\) 0 0
\(337\) 7.95830e13 0.997369 0.498685 0.866784i \(-0.333817\pi\)
0.498685 + 0.866784i \(0.333817\pi\)
\(338\) −1.46705e14 −1.80885
\(339\) 0 0
\(340\) −4.81306e13 −0.574497
\(341\) −1.19750e13 −0.140646
\(342\) 0 0
\(343\) 2.76041e13 0.313946
\(344\) −4.23678e13 −0.474203
\(345\) 0 0
\(346\) 1.47482e13 0.159890
\(347\) 5.46578e13 0.583230 0.291615 0.956536i \(-0.405807\pi\)
0.291615 + 0.956536i \(0.405807\pi\)
\(348\) 0 0
\(349\) −9.78873e13 −1.01201 −0.506007 0.862530i \(-0.668879\pi\)
−0.506007 + 0.862530i \(0.668879\pi\)
\(350\) 2.82154e13 0.287152
\(351\) 0 0
\(352\) −3.27405e14 −3.22924
\(353\) −1.31032e14 −1.27238 −0.636189 0.771533i \(-0.719490\pi\)
−0.636189 + 0.771533i \(0.719490\pi\)
\(354\) 0 0
\(355\) 7.52906e12 0.0708736
\(356\) −4.35265e14 −4.03440
\(357\) 0 0
\(358\) 2.76967e14 2.48926
\(359\) −2.77646e13 −0.245738 −0.122869 0.992423i \(-0.539210\pi\)
−0.122869 + 0.992423i \(0.539210\pi\)
\(360\) 0 0
\(361\) −8.05095e12 −0.0691127
\(362\) 1.99672e14 1.68819
\(363\) 0 0
\(364\) 1.06477e13 0.0873371
\(365\) 5.70037e12 0.0460567
\(366\) 0 0
\(367\) −1.08509e14 −0.850754 −0.425377 0.905016i \(-0.639858\pi\)
−0.425377 + 0.905016i \(0.639858\pi\)
\(368\) −4.00964e14 −3.09700
\(369\) 0 0
\(370\) −4.25284e13 −0.318838
\(371\) −4.12258e13 −0.304518
\(372\) 0 0
\(373\) −4.02186e13 −0.288422 −0.144211 0.989547i \(-0.546064\pi\)
−0.144211 + 0.989547i \(0.546064\pi\)
\(374\) 3.09246e14 2.18530
\(375\) 0 0
\(376\) −1.57577e14 −1.08133
\(377\) 5.82859e12 0.0394172
\(378\) 0 0
\(379\) −1.80219e14 −1.18382 −0.591908 0.806005i \(-0.701625\pi\)
−0.591908 + 0.806005i \(0.701625\pi\)
\(380\) −8.31361e13 −0.538245
\(381\) 0 0
\(382\) −7.25841e13 −0.456555
\(383\) 1.41856e14 0.879535 0.439768 0.898112i \(-0.355061\pi\)
0.439768 + 0.898112i \(0.355061\pi\)
\(384\) 0 0
\(385\) 6.37202e12 0.0383922
\(386\) −2.36600e14 −1.40535
\(387\) 0 0
\(388\) −2.21563e14 −1.27915
\(389\) −2.91132e14 −1.65717 −0.828586 0.559862i \(-0.810854\pi\)
−0.828586 + 0.559862i \(0.810854\pi\)
\(390\) 0 0
\(391\) 1.85329e14 1.02558
\(392\) −5.37321e14 −2.93197
\(393\) 0 0
\(394\) −3.60406e14 −1.91233
\(395\) −5.43842e13 −0.284570
\(396\) 0 0
\(397\) 1.35401e14 0.689086 0.344543 0.938771i \(-0.388034\pi\)
0.344543 + 0.938771i \(0.388034\pi\)
\(398\) −6.75261e14 −3.38934
\(399\) 0 0
\(400\) −6.07286e14 −2.96527
\(401\) −9.06673e12 −0.0436673 −0.0218337 0.999762i \(-0.506950\pi\)
−0.0218337 + 0.999762i \(0.506950\pi\)
\(402\) 0 0
\(403\) −5.68673e12 −0.0266492
\(404\) 1.02443e15 4.73569
\(405\) 0 0
\(406\) 1.24300e13 0.0559212
\(407\) 1.97094e14 0.874792
\(408\) 0 0
\(409\) 8.87571e13 0.383464 0.191732 0.981447i \(-0.438589\pi\)
0.191732 + 0.981447i \(0.438589\pi\)
\(410\) 1.72257e13 0.0734287
\(411\) 0 0
\(412\) 6.30419e14 2.61634
\(413\) −6.07625e13 −0.248834
\(414\) 0 0
\(415\) 5.11563e13 0.204002
\(416\) −1.55479e14 −0.611867
\(417\) 0 0
\(418\) 5.34162e14 2.04740
\(419\) 2.74320e13 0.103772 0.0518860 0.998653i \(-0.483477\pi\)
0.0518860 + 0.998653i \(0.483477\pi\)
\(420\) 0 0
\(421\) 4.33872e14 1.59886 0.799429 0.600760i \(-0.205135\pi\)
0.799429 + 0.600760i \(0.205135\pi\)
\(422\) 8.86520e14 3.22456
\(423\) 0 0
\(424\) 1.62575e15 5.76158
\(425\) 2.80693e14 0.981957
\(426\) 0 0
\(427\) 8.89596e13 0.303277
\(428\) −9.25865e14 −3.11607
\(429\) 0 0
\(430\) −1.96225e13 −0.0643693
\(431\) −3.83089e14 −1.24072 −0.620361 0.784316i \(-0.713014\pi\)
−0.620361 + 0.784316i \(0.713014\pi\)
\(432\) 0 0
\(433\) 9.06861e13 0.286324 0.143162 0.989699i \(-0.454273\pi\)
0.143162 + 0.989699i \(0.454273\pi\)
\(434\) −1.21274e13 −0.0378073
\(435\) 0 0
\(436\) −2.19621e14 −0.667573
\(437\) 3.20119e14 0.960867
\(438\) 0 0
\(439\) −2.09425e14 −0.613019 −0.306510 0.951868i \(-0.599161\pi\)
−0.306510 + 0.951868i \(0.599161\pi\)
\(440\) −2.51283e14 −0.726395
\(441\) 0 0
\(442\) 1.46855e14 0.414064
\(443\) 1.56421e14 0.435585 0.217793 0.975995i \(-0.430114\pi\)
0.217793 + 0.975995i \(0.430114\pi\)
\(444\) 0 0
\(445\) −1.23697e14 −0.336031
\(446\) 3.44614e14 0.924678
\(447\) 0 0
\(448\) −1.42726e14 −0.373659
\(449\) 4.22292e14 1.09209 0.546045 0.837756i \(-0.316133\pi\)
0.546045 + 0.837756i \(0.316133\pi\)
\(450\) 0 0
\(451\) −7.98311e13 −0.201466
\(452\) 3.83998e14 0.957344
\(453\) 0 0
\(454\) −1.38725e15 −3.37558
\(455\) 3.02596e12 0.00727445
\(456\) 0 0
\(457\) 4.05203e14 0.950898 0.475449 0.879743i \(-0.342286\pi\)
0.475449 + 0.879743i \(0.342286\pi\)
\(458\) 1.24743e15 2.89239
\(459\) 0 0
\(460\) −2.45422e14 −0.555579
\(461\) 5.53814e14 1.23882 0.619410 0.785067i \(-0.287372\pi\)
0.619410 + 0.785067i \(0.287372\pi\)
\(462\) 0 0
\(463\) −2.46478e14 −0.538372 −0.269186 0.963088i \(-0.586755\pi\)
−0.269186 + 0.963088i \(0.586755\pi\)
\(464\) −2.67533e14 −0.577469
\(465\) 0 0
\(466\) −1.16727e15 −2.46065
\(467\) −3.13903e14 −0.653961 −0.326981 0.945031i \(-0.606031\pi\)
−0.326981 + 0.945031i \(0.606031\pi\)
\(468\) 0 0
\(469\) −9.31726e13 −0.189600
\(470\) −7.29814e13 −0.146782
\(471\) 0 0
\(472\) 2.39619e15 4.70803
\(473\) 9.09390e13 0.176610
\(474\) 0 0
\(475\) 4.84841e14 0.919994
\(476\) 2.25896e14 0.423711
\(477\) 0 0
\(478\) −2.98514e14 −0.547156
\(479\) −2.29175e14 −0.415262 −0.207631 0.978207i \(-0.566575\pi\)
−0.207631 + 0.978207i \(0.566575\pi\)
\(480\) 0 0
\(481\) 9.35965e13 0.165753
\(482\) −4.77959e14 −0.836821
\(483\) 0 0
\(484\) 3.85669e14 0.660033
\(485\) −6.29656e13 −0.106543
\(486\) 0 0
\(487\) −2.16926e14 −0.358841 −0.179420 0.983773i \(-0.557422\pi\)
−0.179420 + 0.983773i \(0.557422\pi\)
\(488\) −3.50815e15 −5.73811
\(489\) 0 0
\(490\) −2.48859e14 −0.397992
\(491\) 1.19339e15 1.88727 0.943637 0.330982i \(-0.107380\pi\)
0.943637 + 0.330982i \(0.107380\pi\)
\(492\) 0 0
\(493\) 1.23656e14 0.191230
\(494\) 2.53664e14 0.387936
\(495\) 0 0
\(496\) 2.61022e14 0.390416
\(497\) −3.53368e13 −0.0522718
\(498\) 0 0
\(499\) 2.45455e14 0.355156 0.177578 0.984107i \(-0.443174\pi\)
0.177578 + 0.984107i \(0.443174\pi\)
\(500\) −7.61531e14 −1.08981
\(501\) 0 0
\(502\) 1.05558e15 1.47781
\(503\) −2.39448e13 −0.0331579 −0.0165790 0.999863i \(-0.505277\pi\)
−0.0165790 + 0.999863i \(0.505277\pi\)
\(504\) 0 0
\(505\) 2.91130e14 0.394443
\(506\) 1.57687e15 2.11334
\(507\) 0 0
\(508\) 8.59337e14 1.12697
\(509\) −7.32323e14 −0.950069 −0.475034 0.879967i \(-0.657564\pi\)
−0.475034 + 0.879967i \(0.657564\pi\)
\(510\) 0 0
\(511\) −2.67541e13 −0.0339684
\(512\) −3.10669e14 −0.390223
\(513\) 0 0
\(514\) −2.32420e15 −2.85744
\(515\) 1.79158e14 0.217919
\(516\) 0 0
\(517\) 3.38226e14 0.402725
\(518\) 1.99603e14 0.235154
\(519\) 0 0
\(520\) −1.19329e14 −0.137635
\(521\) −3.34135e14 −0.381341 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(522\) 0 0
\(523\) −4.01114e13 −0.0448238 −0.0224119 0.999749i \(-0.507135\pi\)
−0.0224119 + 0.999749i \(0.507135\pi\)
\(524\) −2.24213e14 −0.247935
\(525\) 0 0
\(526\) −1.44144e15 −1.56090
\(527\) −1.20646e14 −0.129287
\(528\) 0 0
\(529\) −7.80235e12 −0.00818878
\(530\) 7.52963e14 0.782090
\(531\) 0 0
\(532\) 3.90190e14 0.396975
\(533\) −3.79103e13 −0.0381731
\(534\) 0 0
\(535\) −2.63120e14 −0.259542
\(536\) 3.67429e15 3.58729
\(537\) 0 0
\(538\) 2.60147e15 2.48838
\(539\) 1.15332e15 1.09197
\(540\) 0 0
\(541\) 3.82729e14 0.355063 0.177532 0.984115i \(-0.443189\pi\)
0.177532 + 0.984115i \(0.443189\pi\)
\(542\) −1.12997e15 −1.03769
\(543\) 0 0
\(544\) −3.29854e15 −2.96844
\(545\) −6.24139e13 −0.0556033
\(546\) 0 0
\(547\) 1.10953e15 0.968742 0.484371 0.874863i \(-0.339048\pi\)
0.484371 + 0.874863i \(0.339048\pi\)
\(548\) 5.24961e15 4.53768
\(549\) 0 0
\(550\) 2.38828e15 2.02344
\(551\) 2.13591e14 0.179164
\(552\) 0 0
\(553\) 2.55246e14 0.209880
\(554\) 4.15762e15 3.38486
\(555\) 0 0
\(556\) 2.96247e14 0.236452
\(557\) −9.17117e14 −0.724805 −0.362402 0.932022i \(-0.618043\pi\)
−0.362402 + 0.932022i \(0.618043\pi\)
\(558\) 0 0
\(559\) 4.31853e13 0.0334635
\(560\) −1.38892e14 −0.106572
\(561\) 0 0
\(562\) 1.35359e15 1.01844
\(563\) 1.33076e15 0.991522 0.495761 0.868459i \(-0.334889\pi\)
0.495761 + 0.868459i \(0.334889\pi\)
\(564\) 0 0
\(565\) 1.09128e14 0.0797387
\(566\) −3.05136e15 −2.20802
\(567\) 0 0
\(568\) 1.39352e15 0.989001
\(569\) 1.00669e15 0.707584 0.353792 0.935324i \(-0.384892\pi\)
0.353792 + 0.935324i \(0.384892\pi\)
\(570\) 0 0
\(571\) −2.06731e15 −1.42530 −0.712652 0.701518i \(-0.752506\pi\)
−0.712652 + 0.701518i \(0.752506\pi\)
\(572\) 9.01272e14 0.615429
\(573\) 0 0
\(574\) −8.08470e13 −0.0541562
\(575\) 1.43128e15 0.949622
\(576\) 0 0
\(577\) 7.80901e14 0.508310 0.254155 0.967163i \(-0.418203\pi\)
0.254155 + 0.967163i \(0.418203\pi\)
\(578\) 1.77750e14 0.114606
\(579\) 0 0
\(580\) −1.63752e14 −0.103593
\(581\) −2.40097e14 −0.150459
\(582\) 0 0
\(583\) −3.48954e15 −2.14581
\(584\) 1.05506e15 0.642694
\(585\) 0 0
\(586\) 5.21771e15 3.11920
\(587\) −2.95350e15 −1.74915 −0.874574 0.484891i \(-0.838859\pi\)
−0.874574 + 0.484891i \(0.838859\pi\)
\(588\) 0 0
\(589\) −2.08393e14 −0.121129
\(590\) 1.10979e15 0.639079
\(591\) 0 0
\(592\) −4.29609e15 −2.42831
\(593\) 1.10716e14 0.0620026 0.0310013 0.999519i \(-0.490130\pi\)
0.0310013 + 0.999519i \(0.490130\pi\)
\(594\) 0 0
\(595\) 6.41970e13 0.0352916
\(596\) −3.10051e15 −1.68880
\(597\) 0 0
\(598\) 7.48828e14 0.400429
\(599\) 2.23003e14 0.118158 0.0590791 0.998253i \(-0.481184\pi\)
0.0590791 + 0.998253i \(0.481184\pi\)
\(600\) 0 0
\(601\) 2.38409e15 1.24026 0.620130 0.784499i \(-0.287080\pi\)
0.620130 + 0.784499i \(0.287080\pi\)
\(602\) 9.20963e13 0.0474746
\(603\) 0 0
\(604\) 6.06101e15 3.06790
\(605\) 1.09603e14 0.0549752
\(606\) 0 0
\(607\) 9.55050e14 0.470423 0.235211 0.971944i \(-0.424422\pi\)
0.235211 + 0.971944i \(0.424422\pi\)
\(608\) −5.69758e15 −2.78113
\(609\) 0 0
\(610\) −1.62479e15 −0.778903
\(611\) 1.60617e14 0.0763073
\(612\) 0 0
\(613\) 3.30769e15 1.54345 0.771725 0.635956i \(-0.219394\pi\)
0.771725 + 0.635956i \(0.219394\pi\)
\(614\) 1.09587e14 0.0506798
\(615\) 0 0
\(616\) 1.17937e15 0.535742
\(617\) 4.36956e15 1.96730 0.983648 0.180104i \(-0.0576434\pi\)
0.983648 + 0.180104i \(0.0576434\pi\)
\(618\) 0 0
\(619\) −2.19128e13 −0.00969169 −0.00484585 0.999988i \(-0.501542\pi\)
−0.00484585 + 0.999988i \(0.501542\pi\)
\(620\) 1.59766e14 0.0700376
\(621\) 0 0
\(622\) 4.86684e15 2.09604
\(623\) 5.80560e14 0.247835
\(624\) 0 0
\(625\) 2.05698e15 0.862762
\(626\) −2.87268e15 −1.19434
\(627\) 0 0
\(628\) −6.35455e15 −2.59601
\(629\) 1.98569e15 0.804143
\(630\) 0 0
\(631\) −2.46904e15 −0.982577 −0.491288 0.870997i \(-0.663474\pi\)
−0.491288 + 0.870997i \(0.663474\pi\)
\(632\) −1.00657e16 −3.97101
\(633\) 0 0
\(634\) 1.70051e15 0.659308
\(635\) 2.44214e14 0.0938673
\(636\) 0 0
\(637\) 5.47688e14 0.206903
\(638\) 1.05213e15 0.394054
\(639\) 0 0
\(640\) 9.18976e14 0.338310
\(641\) −4.93142e14 −0.179992 −0.0899959 0.995942i \(-0.528685\pi\)
−0.0899959 + 0.995942i \(0.528685\pi\)
\(642\) 0 0
\(643\) −3.23186e15 −1.15956 −0.579779 0.814774i \(-0.696861\pi\)
−0.579779 + 0.814774i \(0.696861\pi\)
\(644\) 1.15186e15 0.409759
\(645\) 0 0
\(646\) 5.38158e15 1.88205
\(647\) 3.57806e15 1.24072 0.620361 0.784317i \(-0.286986\pi\)
0.620361 + 0.784317i \(0.286986\pi\)
\(648\) 0 0
\(649\) −5.14322e15 −1.75343
\(650\) 1.13415e15 0.383396
\(651\) 0 0
\(652\) −1.11377e16 −3.70196
\(653\) −1.86179e15 −0.613631 −0.306816 0.951769i \(-0.599264\pi\)
−0.306816 + 0.951769i \(0.599264\pi\)
\(654\) 0 0
\(655\) −6.37186e13 −0.0206509
\(656\) 1.74009e15 0.559243
\(657\) 0 0
\(658\) 3.42530e14 0.108257
\(659\) −2.15375e15 −0.675034 −0.337517 0.941319i \(-0.609587\pi\)
−0.337517 + 0.941319i \(0.609587\pi\)
\(660\) 0 0
\(661\) 9.87662e14 0.304439 0.152219 0.988347i \(-0.451358\pi\)
0.152219 + 0.988347i \(0.451358\pi\)
\(662\) −3.89331e15 −1.19014
\(663\) 0 0
\(664\) 9.46829e15 2.84673
\(665\) 1.10888e14 0.0330646
\(666\) 0 0
\(667\) 6.30533e14 0.184933
\(668\) −1.32285e16 −3.84804
\(669\) 0 0
\(670\) 1.70174e15 0.486947
\(671\) 7.52996e15 2.13707
\(672\) 0 0
\(673\) −3.49944e15 −0.977048 −0.488524 0.872550i \(-0.662465\pi\)
−0.488524 + 0.872550i \(0.662465\pi\)
\(674\) 6.82200e15 1.88922
\(675\) 0 0
\(676\) −9.07087e15 −2.47139
\(677\) 1.40384e14 0.0379386 0.0189693 0.999820i \(-0.493962\pi\)
0.0189693 + 0.999820i \(0.493962\pi\)
\(678\) 0 0
\(679\) 2.95522e14 0.0785790
\(680\) −2.53163e15 −0.667730
\(681\) 0 0
\(682\) −1.02652e15 −0.266412
\(683\) 4.41379e15 1.13631 0.568157 0.822920i \(-0.307657\pi\)
0.568157 + 0.822920i \(0.307657\pi\)
\(684\) 0 0
\(685\) 1.49188e15 0.377951
\(686\) 2.36627e15 0.594678
\(687\) 0 0
\(688\) −1.98221e15 −0.490245
\(689\) −1.65712e15 −0.406582
\(690\) 0 0
\(691\) −4.16695e15 −1.00621 −0.503105 0.864225i \(-0.667809\pi\)
−0.503105 + 0.864225i \(0.667809\pi\)
\(692\) 9.11891e14 0.218454
\(693\) 0 0
\(694\) 4.68536e15 1.10476
\(695\) 8.41900e13 0.0196945
\(696\) 0 0
\(697\) −8.04284e14 −0.185195
\(698\) −8.39108e15 −1.91696
\(699\) 0 0
\(700\) 1.74457e15 0.392329
\(701\) −1.15347e14 −0.0257369 −0.0128685 0.999917i \(-0.504096\pi\)
−0.0128685 + 0.999917i \(0.504096\pi\)
\(702\) 0 0
\(703\) 3.42989e15 0.753400
\(704\) −1.20810e16 −2.63302
\(705\) 0 0
\(706\) −1.12323e16 −2.41014
\(707\) −1.36639e15 −0.290916
\(708\) 0 0
\(709\) 8.73974e15 1.83208 0.916039 0.401088i \(-0.131368\pi\)
0.916039 + 0.401088i \(0.131368\pi\)
\(710\) 6.45405e14 0.134249
\(711\) 0 0
\(712\) −2.28945e16 −4.68912
\(713\) −6.15186e14 −0.125030
\(714\) 0 0
\(715\) 2.56131e14 0.0512601
\(716\) 1.71250e16 3.40102
\(717\) 0 0
\(718\) −2.38004e15 −0.465478
\(719\) 7.62293e14 0.147949 0.0739747 0.997260i \(-0.476432\pi\)
0.0739747 + 0.997260i \(0.476432\pi\)
\(720\) 0 0
\(721\) −8.40858e14 −0.160723
\(722\) −6.90143e14 −0.130913
\(723\) 0 0
\(724\) 1.23459e16 2.30653
\(725\) 9.54985e14 0.177067
\(726\) 0 0
\(727\) −5.14284e15 −0.939212 −0.469606 0.882876i \(-0.655604\pi\)
−0.469606 + 0.882876i \(0.655604\pi\)
\(728\) 5.60060e14 0.101511
\(729\) 0 0
\(730\) 4.88646e14 0.0872407
\(731\) 9.16195e14 0.162346
\(732\) 0 0
\(733\) −4.01404e15 −0.700663 −0.350332 0.936626i \(-0.613931\pi\)
−0.350332 + 0.936626i \(0.613931\pi\)
\(734\) −9.30162e15 −1.61150
\(735\) 0 0
\(736\) −1.68196e16 −2.87069
\(737\) −7.88656e15 −1.33603
\(738\) 0 0
\(739\) 3.88766e15 0.648850 0.324425 0.945911i \(-0.394829\pi\)
0.324425 + 0.945911i \(0.394829\pi\)
\(740\) −2.62956e15 −0.435621
\(741\) 0 0
\(742\) −3.53395e15 −0.576819
\(743\) −2.10387e15 −0.340863 −0.170432 0.985370i \(-0.554516\pi\)
−0.170432 + 0.985370i \(0.554516\pi\)
\(744\) 0 0
\(745\) −8.81129e14 −0.140663
\(746\) −3.44761e15 −0.546329
\(747\) 0 0
\(748\) 1.91209e16 2.98572
\(749\) 1.23493e15 0.191422
\(750\) 0 0
\(751\) 4.83508e15 0.738557 0.369279 0.929319i \(-0.379605\pi\)
0.369279 + 0.929319i \(0.379605\pi\)
\(752\) −7.37236e15 −1.11791
\(753\) 0 0
\(754\) 4.99637e14 0.0746642
\(755\) 1.72247e15 0.255531
\(756\) 0 0
\(757\) −8.12261e15 −1.18759 −0.593797 0.804615i \(-0.702372\pi\)
−0.593797 + 0.804615i \(0.702372\pi\)
\(758\) −1.54487e16 −2.24239
\(759\) 0 0
\(760\) −4.37288e15 −0.625595
\(761\) −3.35823e15 −0.476974 −0.238487 0.971146i \(-0.576651\pi\)
−0.238487 + 0.971146i \(0.576651\pi\)
\(762\) 0 0
\(763\) 2.92933e14 0.0410094
\(764\) −4.48792e15 −0.623780
\(765\) 0 0
\(766\) 1.21601e16 1.66602
\(767\) −2.44242e15 −0.332236
\(768\) 0 0
\(769\) −1.21660e14 −0.0163137 −0.00815686 0.999967i \(-0.502596\pi\)
−0.00815686 + 0.999967i \(0.502596\pi\)
\(770\) 5.46222e14 0.0727227
\(771\) 0 0
\(772\) −1.46291e16 −1.92009
\(773\) 3.05158e15 0.397684 0.198842 0.980032i \(-0.436282\pi\)
0.198842 + 0.980032i \(0.436282\pi\)
\(774\) 0 0
\(775\) −9.31742e14 −0.119712
\(776\) −1.16540e16 −1.48674
\(777\) 0 0
\(778\) −2.49564e16 −3.13902
\(779\) −1.38924e15 −0.173509
\(780\) 0 0
\(781\) −2.99107e15 −0.368338
\(782\) 1.58867e16 1.94266
\(783\) 0 0
\(784\) −2.51390e16 −3.03116
\(785\) −1.80589e15 −0.216226
\(786\) 0 0
\(787\) 9.93889e15 1.17348 0.586742 0.809774i \(-0.300410\pi\)
0.586742 + 0.809774i \(0.300410\pi\)
\(788\) −2.22841e16 −2.61277
\(789\) 0 0
\(790\) −4.66191e15 −0.539033
\(791\) −5.12180e14 −0.0588101
\(792\) 0 0
\(793\) 3.57584e15 0.404926
\(794\) 1.16068e16 1.30527
\(795\) 0 0
\(796\) −4.17518e16 −4.63077
\(797\) −6.66631e15 −0.734285 −0.367143 0.930165i \(-0.619664\pi\)
−0.367143 + 0.930165i \(0.619664\pi\)
\(798\) 0 0
\(799\) 3.40757e15 0.370201
\(800\) −2.54744e16 −2.74858
\(801\) 0 0
\(802\) −7.77217e14 −0.0827148
\(803\) −2.26459e15 −0.239361
\(804\) 0 0
\(805\) 3.27346e14 0.0341295
\(806\) −4.87477e14 −0.0504790
\(807\) 0 0
\(808\) 5.38839e16 5.50422
\(809\) 1.50060e16 1.52246 0.761232 0.648479i \(-0.224595\pi\)
0.761232 + 0.648479i \(0.224595\pi\)
\(810\) 0 0
\(811\) 9.15416e15 0.916229 0.458115 0.888893i \(-0.348525\pi\)
0.458115 + 0.888893i \(0.348525\pi\)
\(812\) 7.68552e14 0.0764038
\(813\) 0 0
\(814\) 1.68953e16 1.65703
\(815\) −3.16519e15 −0.308342
\(816\) 0 0
\(817\) 1.58254e15 0.152102
\(818\) 7.60843e15 0.726360
\(819\) 0 0
\(820\) 1.06508e15 0.100324
\(821\) −1.78031e16 −1.66574 −0.832872 0.553466i \(-0.813305\pi\)
−0.832872 + 0.553466i \(0.813305\pi\)
\(822\) 0 0
\(823\) 1.32521e16 1.22345 0.611724 0.791071i \(-0.290476\pi\)
0.611724 + 0.791071i \(0.290476\pi\)
\(824\) 3.31595e16 3.04094
\(825\) 0 0
\(826\) −5.20867e15 −0.471343
\(827\) 1.40611e16 1.26398 0.631989 0.774977i \(-0.282239\pi\)
0.631989 + 0.774977i \(0.282239\pi\)
\(828\) 0 0
\(829\) −1.03566e16 −0.918686 −0.459343 0.888259i \(-0.651915\pi\)
−0.459343 + 0.888259i \(0.651915\pi\)
\(830\) 4.38522e15 0.386422
\(831\) 0 0
\(832\) −5.73705e15 −0.498897
\(833\) 1.16194e16 1.00378
\(834\) 0 0
\(835\) −3.75938e15 −0.320510
\(836\) 3.30275e16 2.79732
\(837\) 0 0
\(838\) 2.35152e15 0.196565
\(839\) −9.68827e15 −0.804554 −0.402277 0.915518i \(-0.631781\pi\)
−0.402277 + 0.915518i \(0.631781\pi\)
\(840\) 0 0
\(841\) 4.20707e14 0.0344828
\(842\) 3.71923e16 3.02856
\(843\) 0 0
\(844\) 5.48141e16 4.40564
\(845\) −2.57784e15 −0.205846
\(846\) 0 0
\(847\) −5.14409e14 −0.0405462
\(848\) 7.60621e16 5.95650
\(849\) 0 0
\(850\) 2.40615e16 1.86003
\(851\) 1.01252e16 0.777663
\(852\) 0 0
\(853\) 1.46435e16 1.11026 0.555132 0.831762i \(-0.312668\pi\)
0.555132 + 0.831762i \(0.312668\pi\)
\(854\) 7.62578e15 0.574468
\(855\) 0 0
\(856\) −4.86997e16 −3.62177
\(857\) 9.45909e15 0.698964 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(858\) 0 0
\(859\) 4.86873e15 0.355184 0.177592 0.984104i \(-0.443169\pi\)
0.177592 + 0.984104i \(0.443169\pi\)
\(860\) −1.21327e15 −0.0879463
\(861\) 0 0
\(862\) −3.28391e16 −2.35018
\(863\) 1.48573e15 0.105652 0.0528262 0.998604i \(-0.483177\pi\)
0.0528262 + 0.998604i \(0.483177\pi\)
\(864\) 0 0
\(865\) 2.59149e14 0.0181954
\(866\) 7.77378e15 0.542355
\(867\) 0 0
\(868\) −7.49846e14 −0.0516552
\(869\) 2.16052e16 1.47894
\(870\) 0 0
\(871\) −3.74518e15 −0.253147
\(872\) −1.15519e16 −0.775912
\(873\) 0 0
\(874\) 2.74412e16 1.82008
\(875\) 1.01574e15 0.0669478
\(876\) 0 0
\(877\) −1.05639e16 −0.687583 −0.343791 0.939046i \(-0.611711\pi\)
−0.343791 + 0.939046i \(0.611711\pi\)
\(878\) −1.79523e16 −1.16118
\(879\) 0 0
\(880\) −1.17565e16 −0.750968
\(881\) 8.44930e15 0.536356 0.268178 0.963369i \(-0.413578\pi\)
0.268178 + 0.963369i \(0.413578\pi\)
\(882\) 0 0
\(883\) −2.60928e16 −1.63583 −0.817913 0.575342i \(-0.804869\pi\)
−0.817913 + 0.575342i \(0.804869\pi\)
\(884\) 9.08016e15 0.565726
\(885\) 0 0
\(886\) 1.34087e16 0.825087
\(887\) −2.69034e16 −1.64523 −0.822615 0.568598i \(-0.807486\pi\)
−0.822615 + 0.568598i \(0.807486\pi\)
\(888\) 0 0
\(889\) −1.14619e15 −0.0692304
\(890\) −1.06036e16 −0.636512
\(891\) 0 0
\(892\) 2.13077e16 1.26337
\(893\) 5.88590e15 0.346840
\(894\) 0 0
\(895\) 4.86673e15 0.283277
\(896\) −4.31312e15 −0.249515
\(897\) 0 0
\(898\) 3.61997e16 2.06864
\(899\) −4.10468e14 −0.0233131
\(900\) 0 0
\(901\) −3.51565e16 −1.97251
\(902\) −6.84327e15 −0.381617
\(903\) 0 0
\(904\) 2.01980e16 1.11271
\(905\) 3.50855e15 0.192115
\(906\) 0 0
\(907\) 2.07428e15 0.112209 0.0561045 0.998425i \(-0.482132\pi\)
0.0561045 + 0.998425i \(0.482132\pi\)
\(908\) −8.57748e16 −4.61198
\(909\) 0 0
\(910\) 2.59391e14 0.0137793
\(911\) −1.42172e16 −0.750696 −0.375348 0.926884i \(-0.622477\pi\)
−0.375348 + 0.926884i \(0.622477\pi\)
\(912\) 0 0
\(913\) −2.03229e16 −1.06022
\(914\) 3.47348e16 1.80119
\(915\) 0 0
\(916\) 7.71295e16 3.95180
\(917\) 2.99056e14 0.0152308
\(918\) 0 0
\(919\) −3.08777e16 −1.55385 −0.776925 0.629593i \(-0.783222\pi\)
−0.776925 + 0.629593i \(0.783222\pi\)
\(920\) −1.29090e16 −0.645742
\(921\) 0 0
\(922\) 4.74739e16 2.34658
\(923\) −1.42041e15 −0.0697916
\(924\) 0 0
\(925\) 1.53353e16 0.744583
\(926\) −2.11285e16 −1.01979
\(927\) 0 0
\(928\) −1.12224e16 −0.535270
\(929\) −3.68373e16 −1.74663 −0.873315 0.487156i \(-0.838034\pi\)
−0.873315 + 0.487156i \(0.838034\pi\)
\(930\) 0 0
\(931\) 2.00703e16 0.940439
\(932\) −7.21731e16 −3.36192
\(933\) 0 0
\(934\) −2.69083e16 −1.23874
\(935\) 5.43393e15 0.248685
\(936\) 0 0
\(937\) −4.33816e16 −1.96217 −0.981087 0.193565i \(-0.937995\pi\)
−0.981087 + 0.193565i \(0.937995\pi\)
\(938\) −7.98693e15 −0.359140
\(939\) 0 0
\(940\) −4.51248e15 −0.200545
\(941\) −1.78920e16 −0.790528 −0.395264 0.918568i \(-0.629347\pi\)
−0.395264 + 0.918568i \(0.629347\pi\)
\(942\) 0 0
\(943\) −4.10111e15 −0.179097
\(944\) 1.12108e17 4.86731
\(945\) 0 0
\(946\) 7.79546e15 0.334534
\(947\) 9.07033e15 0.386989 0.193494 0.981101i \(-0.438018\pi\)
0.193494 + 0.981101i \(0.438018\pi\)
\(948\) 0 0
\(949\) −1.07541e15 −0.0453535
\(950\) 4.15615e16 1.74265
\(951\) 0 0
\(952\) 1.18819e16 0.492474
\(953\) 1.41782e16 0.584266 0.292133 0.956378i \(-0.405635\pi\)
0.292133 + 0.956378i \(0.405635\pi\)
\(954\) 0 0
\(955\) −1.27542e15 −0.0519557
\(956\) −1.84573e16 −0.747566
\(957\) 0 0
\(958\) −1.96453e16 −0.786590
\(959\) −7.00197e15 −0.278752
\(960\) 0 0
\(961\) −2.50080e16 −0.984238
\(962\) 8.02327e15 0.313970
\(963\) 0 0
\(964\) −2.95525e16 −1.14333
\(965\) −4.15742e15 −0.159928
\(966\) 0 0
\(967\) −1.91949e16 −0.730028 −0.365014 0.931002i \(-0.618936\pi\)
−0.365014 + 0.931002i \(0.618936\pi\)
\(968\) 2.02859e16 0.767148
\(969\) 0 0
\(970\) −5.39753e15 −0.201814
\(971\) −3.56830e16 −1.32665 −0.663324 0.748332i \(-0.730855\pi\)
−0.663324 + 0.748332i \(0.730855\pi\)
\(972\) 0 0
\(973\) −3.95137e14 −0.0145254
\(974\) −1.85953e16 −0.679717
\(975\) 0 0
\(976\) −1.64131e17 −5.93223
\(977\) −3.12998e16 −1.12492 −0.562460 0.826824i \(-0.690145\pi\)
−0.562460 + 0.826824i \(0.690145\pi\)
\(978\) 0 0
\(979\) 4.91413e16 1.74639
\(980\) −1.53871e16 −0.543768
\(981\) 0 0
\(982\) 1.02300e17 3.57488
\(983\) −4.86496e16 −1.69058 −0.845289 0.534310i \(-0.820572\pi\)
−0.845289 + 0.534310i \(0.820572\pi\)
\(984\) 0 0
\(985\) −6.33289e15 −0.217622
\(986\) 1.06000e16 0.362229
\(987\) 0 0
\(988\) 1.56842e16 0.530027
\(989\) 4.67176e15 0.157000
\(990\) 0 0
\(991\) 1.35365e16 0.449885 0.224942 0.974372i \(-0.427781\pi\)
0.224942 + 0.974372i \(0.427781\pi\)
\(992\) 1.09493e16 0.361886
\(993\) 0 0
\(994\) −3.02914e15 −0.0990134
\(995\) −1.18654e16 −0.385704
\(996\) 0 0
\(997\) 3.72028e15 0.119606 0.0598028 0.998210i \(-0.480953\pi\)
0.0598028 + 0.998210i \(0.480953\pi\)
\(998\) 2.10409e16 0.672738
\(999\) 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 261.12.a.a.1.11 11
3.2 odd 2 29.12.a.a.1.1 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.12.a.a.1.1 11 3.2 odd 2
261.12.a.a.1.11 11 1.1 even 1 trivial