Properties

Label 75.22.a.j.1.1
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $6$
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] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(646.238\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2568.95 q^{2} -59049.0 q^{3} +4.50235e6 q^{4} +1.51694e8 q^{6} +5.58323e8 q^{7} -6.17885e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-2568.95 q^{2} -59049.0 q^{3} +4.50235e6 q^{4} +1.51694e8 q^{6} +5.58323e8 q^{7} -6.17885e9 q^{8} +3.48678e9 q^{9} -3.70933e10 q^{11} -2.65860e11 q^{12} -7.75851e10 q^{13} -1.43430e12 q^{14} +6.43103e12 q^{16} +1.27282e13 q^{17} -8.95738e12 q^{18} +2.24955e12 q^{19} -3.29684e13 q^{21} +9.52909e13 q^{22} +2.53576e14 q^{23} +3.64855e14 q^{24} +1.99312e14 q^{26} -2.05891e14 q^{27} +2.51377e15 q^{28} -4.09365e15 q^{29} +2.85297e15 q^{31} -3.56301e15 q^{32} +2.19032e15 q^{33} -3.26982e16 q^{34} +1.56987e16 q^{36} +1.07806e16 q^{37} -5.77899e15 q^{38} +4.58132e15 q^{39} +4.19762e16 q^{41} +8.46942e16 q^{42} +6.32027e16 q^{43} -1.67007e17 q^{44} -6.51424e17 q^{46} -4.39281e17 q^{47} -3.79746e17 q^{48} -2.46822e17 q^{49} -7.51590e17 q^{51} -3.49316e17 q^{52} -1.77175e18 q^{53} +5.28924e17 q^{54} -3.44979e18 q^{56} -1.32834e17 q^{57} +1.05164e19 q^{58} -2.10312e18 q^{59} -4.93323e18 q^{61} -7.32914e18 q^{62} +1.94675e18 q^{63} -4.33365e18 q^{64} -5.62683e18 q^{66} -5.24702e18 q^{67} +5.73071e19 q^{68} -1.49734e19 q^{69} -4.41600e19 q^{71} -2.15443e19 q^{72} +2.66577e19 q^{73} -2.76949e19 q^{74} +1.01283e19 q^{76} -2.07100e19 q^{77} -1.17692e19 q^{78} +9.60972e19 q^{79} +1.21577e19 q^{81} -1.07835e20 q^{82} -1.56177e20 q^{83} -1.48435e20 q^{84} -1.62364e20 q^{86} +2.41726e20 q^{87} +2.29194e20 q^{88} -3.75254e20 q^{89} -4.33175e19 q^{91} +1.14169e21 q^{92} -1.68465e20 q^{93} +1.12849e21 q^{94} +2.10392e20 q^{96} +9.05983e20 q^{97} +6.34073e20 q^{98} -1.29336e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} - 259251563952 q^{12} + 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} + 5382900513068 q^{17} + 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} - 19701817271864 q^{22} + 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} - 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2568.95 −1.77395 −0.886974 0.461820i \(-0.847197\pi\)
−0.886974 + 0.461820i \(0.847197\pi\)
\(3\) −59049.0 −0.577350
\(4\) 4.50235e6 2.14689
\(5\) 0 0
\(6\) 1.51694e8 1.02419
\(7\) 5.58323e8 0.747061 0.373530 0.927618i \(-0.378147\pi\)
0.373530 + 0.927618i \(0.378147\pi\)
\(8\) −6.17885e9 −2.03452
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −3.70933e10 −0.431194 −0.215597 0.976482i \(-0.569170\pi\)
−0.215597 + 0.976482i \(0.569170\pi\)
\(12\) −2.65860e11 −1.23951
\(13\) −7.75851e10 −0.156089 −0.0780446 0.996950i \(-0.524868\pi\)
−0.0780446 + 0.996950i \(0.524868\pi\)
\(14\) −1.43430e12 −1.32525
\(15\) 0 0
\(16\) 6.43103e12 1.46225
\(17\) 1.27282e13 1.53128 0.765640 0.643269i \(-0.222422\pi\)
0.765640 + 0.643269i \(0.222422\pi\)
\(18\) −8.95738e12 −0.591316
\(19\) 2.24955e12 0.0841751 0.0420875 0.999114i \(-0.486599\pi\)
0.0420875 + 0.999114i \(0.486599\pi\)
\(20\) 0 0
\(21\) −3.29684e13 −0.431316
\(22\) 9.52909e13 0.764915
\(23\) 2.53576e14 1.27634 0.638169 0.769896i \(-0.279692\pi\)
0.638169 + 0.769896i \(0.279692\pi\)
\(24\) 3.64855e14 1.17463
\(25\) 0 0
\(26\) 1.99312e14 0.276894
\(27\) −2.05891e14 −0.192450
\(28\) 2.51377e15 1.60386
\(29\) −4.09365e15 −1.80689 −0.903444 0.428705i \(-0.858970\pi\)
−0.903444 + 0.428705i \(0.858970\pi\)
\(30\) 0 0
\(31\) 2.85297e15 0.625172 0.312586 0.949889i \(-0.398805\pi\)
0.312586 + 0.949889i \(0.398805\pi\)
\(32\) −3.56301e15 −0.559425
\(33\) 2.19032e15 0.248950
\(34\) −3.26982e16 −2.71641
\(35\) 0 0
\(36\) 1.56987e16 0.715630
\(37\) 1.07806e16 0.368575 0.184287 0.982872i \(-0.441002\pi\)
0.184287 + 0.982872i \(0.441002\pi\)
\(38\) −5.77899e15 −0.149322
\(39\) 4.58132e15 0.0901182
\(40\) 0 0
\(41\) 4.19762e16 0.488396 0.244198 0.969725i \(-0.421475\pi\)
0.244198 + 0.969725i \(0.421475\pi\)
\(42\) 8.46942e16 0.765131
\(43\) 6.32027e16 0.445981 0.222991 0.974821i \(-0.428418\pi\)
0.222991 + 0.974821i \(0.428418\pi\)
\(44\) −1.67007e17 −0.925726
\(45\) 0 0
\(46\) −6.51424e17 −2.26416
\(47\) −4.39281e17 −1.21819 −0.609095 0.793097i \(-0.708467\pi\)
−0.609095 + 0.793097i \(0.708467\pi\)
\(48\) −3.79746e17 −0.844228
\(49\) −2.46822e17 −0.441900
\(50\) 0 0
\(51\) −7.51590e17 −0.884085
\(52\) −3.49316e17 −0.335106
\(53\) −1.77175e18 −1.39157 −0.695785 0.718250i \(-0.744943\pi\)
−0.695785 + 0.718250i \(0.744943\pi\)
\(54\) 5.28924e17 0.341396
\(55\) 0 0
\(56\) −3.44979e18 −1.51991
\(57\) −1.32834e17 −0.0485985
\(58\) 1.05164e19 3.20533
\(59\) −2.10312e18 −0.535695 −0.267848 0.963461i \(-0.586312\pi\)
−0.267848 + 0.963461i \(0.586312\pi\)
\(60\) 0 0
\(61\) −4.93323e18 −0.885457 −0.442729 0.896656i \(-0.645990\pi\)
−0.442729 + 0.896656i \(0.645990\pi\)
\(62\) −7.32914e18 −1.10902
\(63\) 1.94675e18 0.249020
\(64\) −4.33365e18 −0.469855
\(65\) 0 0
\(66\) −5.62683e18 −0.441624
\(67\) −5.24702e18 −0.351664 −0.175832 0.984420i \(-0.556261\pi\)
−0.175832 + 0.984420i \(0.556261\pi\)
\(68\) 5.73071e19 3.28749
\(69\) −1.49734e19 −0.736894
\(70\) 0 0
\(71\) −4.41600e19 −1.60996 −0.804982 0.593300i \(-0.797825\pi\)
−0.804982 + 0.593300i \(0.797825\pi\)
\(72\) −2.15443e19 −0.678174
\(73\) 2.66577e19 0.725994 0.362997 0.931790i \(-0.381754\pi\)
0.362997 + 0.931790i \(0.381754\pi\)
\(74\) −2.76949e19 −0.653832
\(75\) 0 0
\(76\) 1.01283e19 0.180715
\(77\) −2.07100e19 −0.322128
\(78\) −1.17692e19 −0.159865
\(79\) 9.60972e19 1.14189 0.570947 0.820987i \(-0.306576\pi\)
0.570947 + 0.820987i \(0.306576\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) −1.07835e20 −0.866389
\(83\) −1.56177e20 −1.10483 −0.552415 0.833569i \(-0.686294\pi\)
−0.552415 + 0.833569i \(0.686294\pi\)
\(84\) −1.48435e20 −0.925987
\(85\) 0 0
\(86\) −1.62364e20 −0.791147
\(87\) 2.41726e20 1.04321
\(88\) 2.29194e20 0.877273
\(89\) −3.75254e20 −1.27564 −0.637822 0.770184i \(-0.720165\pi\)
−0.637822 + 0.770184i \(0.720165\pi\)
\(90\) 0 0
\(91\) −4.33175e19 −0.116608
\(92\) 1.14169e21 2.74016
\(93\) −1.68465e20 −0.360943
\(94\) 1.12849e21 2.16101
\(95\) 0 0
\(96\) 2.10392e20 0.322984
\(97\) 9.05983e20 1.24743 0.623716 0.781651i \(-0.285622\pi\)
0.623716 + 0.781651i \(0.285622\pi\)
\(98\) 6.34073e20 0.783908
\(99\) −1.29336e20 −0.143731
\(100\) 0 0
\(101\) 1.86796e21 1.68265 0.841326 0.540528i \(-0.181776\pi\)
0.841326 + 0.540528i \(0.181776\pi\)
\(102\) 1.93080e21 1.56832
\(103\) 1.38607e21 1.01624 0.508118 0.861287i \(-0.330341\pi\)
0.508118 + 0.861287i \(0.330341\pi\)
\(104\) 4.79386e20 0.317567
\(105\) 0 0
\(106\) 4.55153e21 2.46857
\(107\) 2.83321e20 0.139235 0.0696176 0.997574i \(-0.477822\pi\)
0.0696176 + 0.997574i \(0.477822\pi\)
\(108\) −9.26995e20 −0.413169
\(109\) 4.02076e21 1.62678 0.813391 0.581717i \(-0.197619\pi\)
0.813391 + 0.581717i \(0.197619\pi\)
\(110\) 0 0
\(111\) −6.36584e20 −0.212797
\(112\) 3.59059e21 1.09239
\(113\) −3.17848e21 −0.880838 −0.440419 0.897792i \(-0.645170\pi\)
−0.440419 + 0.897792i \(0.645170\pi\)
\(114\) 3.41244e20 0.0862112
\(115\) 0 0
\(116\) −1.84311e22 −3.87919
\(117\) −2.70522e20 −0.0520298
\(118\) 5.40281e21 0.950296
\(119\) 7.10647e21 1.14396
\(120\) 0 0
\(121\) −6.02434e21 −0.814072
\(122\) 1.26732e22 1.57075
\(123\) −2.47865e21 −0.281976
\(124\) 1.28451e22 1.34218
\(125\) 0 0
\(126\) −5.00111e21 −0.441749
\(127\) 3.71270e21 0.301822 0.150911 0.988547i \(-0.451779\pi\)
0.150911 + 0.988547i \(0.451779\pi\)
\(128\) 1.86051e22 1.39292
\(129\) −3.73205e21 −0.257487
\(130\) 0 0
\(131\) −2.81704e22 −1.65365 −0.826827 0.562456i \(-0.809856\pi\)
−0.826827 + 0.562456i \(0.809856\pi\)
\(132\) 9.86161e21 0.534468
\(133\) 1.25598e21 0.0628839
\(134\) 1.34793e22 0.623833
\(135\) 0 0
\(136\) −7.86458e22 −3.11542
\(137\) 9.97284e21 0.365808 0.182904 0.983131i \(-0.441450\pi\)
0.182904 + 0.983131i \(0.441450\pi\)
\(138\) 3.84659e22 1.30721
\(139\) −4.59334e22 −1.44701 −0.723507 0.690317i \(-0.757471\pi\)
−0.723507 + 0.690317i \(0.757471\pi\)
\(140\) 0 0
\(141\) 2.59391e22 0.703323
\(142\) 1.13445e23 2.85599
\(143\) 2.87789e21 0.0673047
\(144\) 2.24236e22 0.487415
\(145\) 0 0
\(146\) −6.84824e22 −1.28787
\(147\) 1.45746e22 0.255131
\(148\) 4.85381e22 0.791289
\(149\) 1.23009e23 1.86845 0.934227 0.356678i \(-0.116091\pi\)
0.934227 + 0.356678i \(0.116091\pi\)
\(150\) 0 0
\(151\) −5.14951e22 −0.679999 −0.340000 0.940426i \(-0.610427\pi\)
−0.340000 + 0.940426i \(0.610427\pi\)
\(152\) −1.38996e22 −0.171256
\(153\) 4.43806e22 0.510427
\(154\) 5.32031e22 0.571438
\(155\) 0 0
\(156\) 2.06267e22 0.193474
\(157\) 1.04915e23 0.920218 0.460109 0.887862i \(-0.347810\pi\)
0.460109 + 0.887862i \(0.347810\pi\)
\(158\) −2.46869e23 −2.02566
\(159\) 1.04620e23 0.803423
\(160\) 0 0
\(161\) 1.41577e23 0.953502
\(162\) −3.12324e22 −0.197105
\(163\) −2.23011e23 −1.31934 −0.659668 0.751557i \(-0.729303\pi\)
−0.659668 + 0.751557i \(0.729303\pi\)
\(164\) 1.88991e23 1.04853
\(165\) 0 0
\(166\) 4.01210e23 1.95991
\(167\) 9.52097e22 0.436675 0.218337 0.975873i \(-0.429937\pi\)
0.218337 + 0.975873i \(0.429937\pi\)
\(168\) 2.03707e23 0.877521
\(169\) −2.41045e23 −0.975636
\(170\) 0 0
\(171\) 7.84371e21 0.0280584
\(172\) 2.84561e23 0.957472
\(173\) −3.78983e23 −1.19987 −0.599937 0.800048i \(-0.704808\pi\)
−0.599937 + 0.800048i \(0.704808\pi\)
\(174\) −6.20982e23 −1.85060
\(175\) 0 0
\(176\) −2.38548e23 −0.630511
\(177\) 1.24187e23 0.309284
\(178\) 9.64008e23 2.26293
\(179\) 4.80425e23 1.06333 0.531666 0.846954i \(-0.321566\pi\)
0.531666 + 0.846954i \(0.321566\pi\)
\(180\) 0 0
\(181\) 7.81248e23 1.53874 0.769368 0.638806i \(-0.220571\pi\)
0.769368 + 0.638806i \(0.220571\pi\)
\(182\) 1.11281e23 0.206857
\(183\) 2.91302e23 0.511219
\(184\) −1.56681e24 −2.59674
\(185\) 0 0
\(186\) 4.32778e23 0.640294
\(187\) −4.72133e23 −0.660279
\(188\) −1.97780e24 −2.61532
\(189\) −1.14954e23 −0.143772
\(190\) 0 0
\(191\) −6.69659e23 −0.749900 −0.374950 0.927045i \(-0.622340\pi\)
−0.374950 + 0.927045i \(0.622340\pi\)
\(192\) 2.55898e23 0.271271
\(193\) 4.74605e23 0.476410 0.238205 0.971215i \(-0.423441\pi\)
0.238205 + 0.971215i \(0.423441\pi\)
\(194\) −2.32742e24 −2.21288
\(195\) 0 0
\(196\) −1.11128e24 −0.948712
\(197\) −2.28623e24 −1.85023 −0.925114 0.379689i \(-0.876031\pi\)
−0.925114 + 0.379689i \(0.876031\pi\)
\(198\) 3.32259e23 0.254972
\(199\) 4.11818e23 0.299743 0.149871 0.988706i \(-0.452114\pi\)
0.149871 + 0.988706i \(0.452114\pi\)
\(200\) 0 0
\(201\) 3.09831e23 0.203033
\(202\) −4.79871e24 −2.98494
\(203\) −2.28558e24 −1.34986
\(204\) −3.38392e24 −1.89803
\(205\) 0 0
\(206\) −3.56075e24 −1.80275
\(207\) 8.84165e23 0.425446
\(208\) −4.98952e23 −0.228241
\(209\) −8.34434e22 −0.0362958
\(210\) 0 0
\(211\) 2.31862e24 0.912565 0.456283 0.889835i \(-0.349181\pi\)
0.456283 + 0.889835i \(0.349181\pi\)
\(212\) −7.97703e24 −2.98755
\(213\) 2.60760e24 0.929513
\(214\) −7.27837e23 −0.246996
\(215\) 0 0
\(216\) 1.27217e24 0.391544
\(217\) 1.59288e24 0.467041
\(218\) −1.03291e25 −2.88583
\(219\) −1.57411e24 −0.419153
\(220\) 0 0
\(221\) −9.87522e23 −0.239016
\(222\) 1.63535e24 0.377490
\(223\) 1.79125e24 0.394416 0.197208 0.980362i \(-0.436813\pi\)
0.197208 + 0.980362i \(0.436813\pi\)
\(224\) −1.98931e24 −0.417925
\(225\) 0 0
\(226\) 8.16536e24 1.56256
\(227\) 7.37978e24 1.34826 0.674128 0.738615i \(-0.264520\pi\)
0.674128 + 0.738615i \(0.264520\pi\)
\(228\) −5.98065e23 −0.104336
\(229\) −9.84331e24 −1.64009 −0.820047 0.572297i \(-0.806053\pi\)
−0.820047 + 0.572297i \(0.806053\pi\)
\(230\) 0 0
\(231\) 1.22291e24 0.185981
\(232\) 2.52940e25 3.67616
\(233\) −3.95232e23 −0.0549054 −0.0274527 0.999623i \(-0.508740\pi\)
−0.0274527 + 0.999623i \(0.508740\pi\)
\(234\) 6.94959e23 0.0922981
\(235\) 0 0
\(236\) −9.46899e24 −1.15008
\(237\) −5.67444e24 −0.659273
\(238\) −1.82562e25 −2.02932
\(239\) 8.45833e24 0.899718 0.449859 0.893100i \(-0.351474\pi\)
0.449859 + 0.893100i \(0.351474\pi\)
\(240\) 0 0
\(241\) 4.92775e24 0.480253 0.240126 0.970742i \(-0.422811\pi\)
0.240126 + 0.970742i \(0.422811\pi\)
\(242\) 1.54762e25 1.44412
\(243\) −7.17898e23 −0.0641500
\(244\) −2.22111e25 −1.90098
\(245\) 0 0
\(246\) 6.36753e24 0.500210
\(247\) −1.74532e23 −0.0131388
\(248\) −1.76281e25 −1.27193
\(249\) 9.22207e24 0.637874
\(250\) 0 0
\(251\) −1.27655e25 −0.811825 −0.405912 0.913912i \(-0.633046\pi\)
−0.405912 + 0.913912i \(0.633046\pi\)
\(252\) 8.76496e24 0.534619
\(253\) −9.40597e24 −0.550349
\(254\) −9.53775e24 −0.535417
\(255\) 0 0
\(256\) −3.87072e25 −2.00112
\(257\) −2.25390e24 −0.111850 −0.0559252 0.998435i \(-0.517811\pi\)
−0.0559252 + 0.998435i \(0.517811\pi\)
\(258\) 9.58746e24 0.456769
\(259\) 6.01906e24 0.275348
\(260\) 0 0
\(261\) −1.42737e25 −0.602296
\(262\) 7.23684e25 2.93350
\(263\) 2.95338e25 1.15023 0.575115 0.818073i \(-0.304957\pi\)
0.575115 + 0.818073i \(0.304957\pi\)
\(264\) −1.35337e25 −0.506494
\(265\) 0 0
\(266\) −3.22654e24 −0.111553
\(267\) 2.21583e25 0.736494
\(268\) −2.36239e25 −0.754983
\(269\) −8.12800e24 −0.249795 −0.124898 0.992170i \(-0.539860\pi\)
−0.124898 + 0.992170i \(0.539860\pi\)
\(270\) 0 0
\(271\) −3.49730e25 −0.994388 −0.497194 0.867639i \(-0.665636\pi\)
−0.497194 + 0.867639i \(0.665636\pi\)
\(272\) 8.18557e25 2.23911
\(273\) 2.55786e24 0.0673237
\(274\) −2.56197e25 −0.648924
\(275\) 0 0
\(276\) −6.74156e25 −1.58203
\(277\) −3.52529e25 −0.796448 −0.398224 0.917288i \(-0.630373\pi\)
−0.398224 + 0.917288i \(0.630373\pi\)
\(278\) 1.18001e26 2.56693
\(279\) 9.94769e24 0.208391
\(280\) 0 0
\(281\) −1.13577e25 −0.220736 −0.110368 0.993891i \(-0.535203\pi\)
−0.110368 + 0.993891i \(0.535203\pi\)
\(282\) −6.66363e25 −1.24766
\(283\) 7.93896e25 1.43221 0.716104 0.697994i \(-0.245924\pi\)
0.716104 + 0.697994i \(0.245924\pi\)
\(284\) −1.98824e26 −3.45641
\(285\) 0 0
\(286\) −7.39315e24 −0.119395
\(287\) 2.34362e25 0.364861
\(288\) −1.24234e25 −0.186475
\(289\) 9.29162e25 1.34482
\(290\) 0 0
\(291\) −5.34974e25 −0.720205
\(292\) 1.20022e26 1.55863
\(293\) −1.37769e25 −0.172600 −0.0863001 0.996269i \(-0.527504\pi\)
−0.0863001 + 0.996269i \(0.527504\pi\)
\(294\) −3.74414e25 −0.452590
\(295\) 0 0
\(296\) −6.66117e25 −0.749873
\(297\) 7.63719e24 0.0829833
\(298\) −3.16005e26 −3.31454
\(299\) −1.96737e25 −0.199223
\(300\) 0 0
\(301\) 3.52875e25 0.333175
\(302\) 1.32288e26 1.20628
\(303\) −1.10301e26 −0.971480
\(304\) 1.44669e25 0.123085
\(305\) 0 0
\(306\) −1.14012e26 −0.905470
\(307\) 4.85445e24 0.0372552 0.0186276 0.999826i \(-0.494070\pi\)
0.0186276 + 0.999826i \(0.494070\pi\)
\(308\) −9.32439e25 −0.691573
\(309\) −8.18461e25 −0.586724
\(310\) 0 0
\(311\) 4.06543e25 0.272347 0.136174 0.990685i \(-0.456520\pi\)
0.136174 + 0.990685i \(0.456520\pi\)
\(312\) −2.83073e25 −0.183347
\(313\) −1.76866e26 −1.10772 −0.553859 0.832611i \(-0.686845\pi\)
−0.553859 + 0.832611i \(0.686845\pi\)
\(314\) −2.69521e26 −1.63242
\(315\) 0 0
\(316\) 4.32664e26 2.45152
\(317\) 4.48172e25 0.245653 0.122827 0.992428i \(-0.460804\pi\)
0.122827 + 0.992428i \(0.460804\pi\)
\(318\) −2.68763e26 −1.42523
\(319\) 1.51847e26 0.779119
\(320\) 0 0
\(321\) −1.67298e25 −0.0803875
\(322\) −3.63705e26 −1.69146
\(323\) 2.86329e25 0.128896
\(324\) 5.47381e25 0.238543
\(325\) 0 0
\(326\) 5.72903e26 2.34043
\(327\) −2.37422e26 −0.939223
\(328\) −2.59364e26 −0.993653
\(329\) −2.45261e26 −0.910062
\(330\) 0 0
\(331\) 2.48182e26 0.864125 0.432063 0.901844i \(-0.357786\pi\)
0.432063 + 0.901844i \(0.357786\pi\)
\(332\) −7.03162e26 −2.37195
\(333\) 3.75897e25 0.122858
\(334\) −2.44589e26 −0.774638
\(335\) 0 0
\(336\) −2.12021e26 −0.630689
\(337\) 5.22985e26 1.50791 0.753955 0.656926i \(-0.228144\pi\)
0.753955 + 0.656926i \(0.228144\pi\)
\(338\) 6.19233e26 1.73073
\(339\) 1.87686e26 0.508552
\(340\) 0 0
\(341\) −1.05826e26 −0.269570
\(342\) −2.01501e25 −0.0497740
\(343\) −4.49655e26 −1.07719
\(344\) −3.90519e26 −0.907359
\(345\) 0 0
\(346\) 9.73588e26 2.12851
\(347\) −2.90390e26 −0.615917 −0.307959 0.951400i \(-0.599646\pi\)
−0.307959 + 0.951400i \(0.599646\pi\)
\(348\) 1.08834e27 2.23965
\(349\) 2.17227e26 0.433757 0.216878 0.976199i \(-0.430412\pi\)
0.216878 + 0.976199i \(0.430412\pi\)
\(350\) 0 0
\(351\) 1.59741e25 0.0300394
\(352\) 1.32164e26 0.241221
\(353\) 2.00928e26 0.355964 0.177982 0.984034i \(-0.443043\pi\)
0.177982 + 0.984034i \(0.443043\pi\)
\(354\) −3.19031e26 −0.548653
\(355\) 0 0
\(356\) −1.68952e27 −2.73867
\(357\) −4.19630e26 −0.660465
\(358\) −1.23419e27 −1.88629
\(359\) 5.85740e26 0.869388 0.434694 0.900578i \(-0.356857\pi\)
0.434694 + 0.900578i \(0.356857\pi\)
\(360\) 0 0
\(361\) −7.09149e26 −0.992915
\(362\) −2.00699e27 −2.72964
\(363\) 3.55731e26 0.470005
\(364\) −1.95031e26 −0.250345
\(365\) 0 0
\(366\) −7.48340e26 −0.906876
\(367\) 6.17342e26 0.726996 0.363498 0.931595i \(-0.381582\pi\)
0.363498 + 0.931595i \(0.381582\pi\)
\(368\) 1.63075e27 1.86632
\(369\) 1.46362e26 0.162799
\(370\) 0 0
\(371\) −9.89206e26 −1.03959
\(372\) −7.58489e26 −0.774905
\(373\) 2.81352e26 0.279452 0.139726 0.990190i \(-0.455378\pi\)
0.139726 + 0.990190i \(0.455378\pi\)
\(374\) 1.21289e27 1.17130
\(375\) 0 0
\(376\) 2.71425e27 2.47844
\(377\) 3.17606e26 0.282036
\(378\) 2.95310e26 0.255044
\(379\) −5.21975e26 −0.438468 −0.219234 0.975672i \(-0.570356\pi\)
−0.219234 + 0.975672i \(0.570356\pi\)
\(380\) 0 0
\(381\) −2.19232e26 −0.174257
\(382\) 1.72032e27 1.33028
\(383\) −1.96111e27 −1.47542 −0.737709 0.675118i \(-0.764093\pi\)
−0.737709 + 0.675118i \(0.764093\pi\)
\(384\) −1.09861e27 −0.804205
\(385\) 0 0
\(386\) −1.21924e27 −0.845126
\(387\) 2.20374e26 0.148660
\(388\) 4.07905e27 2.67810
\(389\) 7.13182e25 0.0455753 0.0227876 0.999740i \(-0.492746\pi\)
0.0227876 + 0.999740i \(0.492746\pi\)
\(390\) 0 0
\(391\) 3.22758e27 1.95443
\(392\) 1.52507e27 0.899056
\(393\) 1.66343e27 0.954737
\(394\) 5.87322e27 3.28221
\(395\) 0 0
\(396\) −5.82318e26 −0.308575
\(397\) 1.96147e27 1.01224 0.506118 0.862464i \(-0.331080\pi\)
0.506118 + 0.862464i \(0.331080\pi\)
\(398\) −1.05794e27 −0.531728
\(399\) −7.41642e25 −0.0363060
\(400\) 0 0
\(401\) −1.16289e27 −0.540158 −0.270079 0.962838i \(-0.587050\pi\)
−0.270079 + 0.962838i \(0.587050\pi\)
\(402\) −7.95941e26 −0.360170
\(403\) −2.21348e26 −0.0975826
\(404\) 8.41024e27 3.61247
\(405\) 0 0
\(406\) 5.87153e27 2.39457
\(407\) −3.99889e26 −0.158927
\(408\) 4.64396e27 1.79869
\(409\) 3.42582e27 1.29321 0.646605 0.762825i \(-0.276188\pi\)
0.646605 + 0.762825i \(0.276188\pi\)
\(410\) 0 0
\(411\) −5.88886e26 −0.211199
\(412\) 6.24058e27 2.18175
\(413\) −1.17422e27 −0.400197
\(414\) −2.27137e27 −0.754719
\(415\) 0 0
\(416\) 2.76436e26 0.0873203
\(417\) 2.71232e27 0.835434
\(418\) 2.14362e26 0.0643868
\(419\) 7.94594e26 0.232754 0.116377 0.993205i \(-0.462872\pi\)
0.116377 + 0.993205i \(0.462872\pi\)
\(420\) 0 0
\(421\) −3.54391e27 −0.987462 −0.493731 0.869615i \(-0.664367\pi\)
−0.493731 + 0.869615i \(0.664367\pi\)
\(422\) −5.95642e27 −1.61884
\(423\) −1.53168e27 −0.406064
\(424\) 1.09473e28 2.83118
\(425\) 0 0
\(426\) −6.69880e27 −1.64891
\(427\) −2.75433e27 −0.661490
\(428\) 1.27561e27 0.298923
\(429\) −1.69936e26 −0.0388584
\(430\) 0 0
\(431\) −7.84829e27 −1.70909 −0.854543 0.519381i \(-0.826163\pi\)
−0.854543 + 0.519381i \(0.826163\pi\)
\(432\) −1.32409e27 −0.281409
\(433\) −9.01390e27 −1.86978 −0.934888 0.354942i \(-0.884501\pi\)
−0.934888 + 0.354942i \(0.884501\pi\)
\(434\) −4.09202e27 −0.828507
\(435\) 0 0
\(436\) 1.81029e28 3.49252
\(437\) 5.70432e26 0.107436
\(438\) 4.04381e27 0.743555
\(439\) 4.74868e27 0.852503 0.426251 0.904605i \(-0.359834\pi\)
0.426251 + 0.904605i \(0.359834\pi\)
\(440\) 0 0
\(441\) −8.60614e26 −0.147300
\(442\) 2.53689e27 0.424003
\(443\) −3.11361e27 −0.508188 −0.254094 0.967179i \(-0.581777\pi\)
−0.254094 + 0.967179i \(0.581777\pi\)
\(444\) −2.86613e27 −0.456851
\(445\) 0 0
\(446\) −4.60163e27 −0.699673
\(447\) −7.26358e27 −1.07875
\(448\) −2.41957e27 −0.351010
\(449\) 2.79453e27 0.396025 0.198013 0.980199i \(-0.436551\pi\)
0.198013 + 0.980199i \(0.436551\pi\)
\(450\) 0 0
\(451\) −1.55703e27 −0.210593
\(452\) −1.43106e28 −1.89106
\(453\) 3.04074e27 0.392598
\(454\) −1.89583e28 −2.39173
\(455\) 0 0
\(456\) 8.20760e26 0.0988747
\(457\) −1.33559e28 −1.57236 −0.786180 0.617998i \(-0.787944\pi\)
−0.786180 + 0.617998i \(0.787944\pi\)
\(458\) 2.52870e28 2.90944
\(459\) −2.62063e27 −0.294695
\(460\) 0 0
\(461\) −7.84878e27 −0.843223 −0.421612 0.906777i \(-0.638535\pi\)
−0.421612 + 0.906777i \(0.638535\pi\)
\(462\) −3.14159e27 −0.329920
\(463\) 7.86992e27 0.807922 0.403961 0.914776i \(-0.367633\pi\)
0.403961 + 0.914776i \(0.367633\pi\)
\(464\) −2.63263e28 −2.64212
\(465\) 0 0
\(466\) 1.01533e27 0.0973992
\(467\) −9.65288e26 −0.0905377 −0.0452689 0.998975i \(-0.514414\pi\)
−0.0452689 + 0.998975i \(0.514414\pi\)
\(468\) −1.21799e27 −0.111702
\(469\) −2.92953e27 −0.262714
\(470\) 0 0
\(471\) −6.19512e27 −0.531288
\(472\) 1.29949e28 1.08988
\(473\) −2.34440e27 −0.192304
\(474\) 1.45774e28 1.16952
\(475\) 0 0
\(476\) 3.19958e28 2.45595
\(477\) −6.17770e27 −0.463856
\(478\) −2.17290e28 −1.59605
\(479\) −7.09688e27 −0.509970 −0.254985 0.966945i \(-0.582070\pi\)
−0.254985 + 0.966945i \(0.582070\pi\)
\(480\) 0 0
\(481\) −8.36415e26 −0.0575305
\(482\) −1.26591e28 −0.851943
\(483\) −8.35999e27 −0.550505
\(484\) −2.71237e28 −1.74772
\(485\) 0 0
\(486\) 1.84424e27 0.113799
\(487\) −5.40175e27 −0.326198 −0.163099 0.986610i \(-0.552149\pi\)
−0.163099 + 0.986610i \(0.552149\pi\)
\(488\) 3.04816e28 1.80148
\(489\) 1.31686e28 0.761720
\(490\) 0 0
\(491\) 1.37622e28 0.762665 0.381332 0.924438i \(-0.375465\pi\)
0.381332 + 0.924438i \(0.375465\pi\)
\(492\) −1.11598e28 −0.605370
\(493\) −5.21049e28 −2.76685
\(494\) 4.48363e26 0.0233076
\(495\) 0 0
\(496\) 1.83475e28 0.914155
\(497\) −2.46555e28 −1.20274
\(498\) −2.36910e28 −1.13156
\(499\) 2.91157e28 1.36167 0.680834 0.732437i \(-0.261617\pi\)
0.680834 + 0.732437i \(0.261617\pi\)
\(500\) 0 0
\(501\) −5.62204e27 −0.252114
\(502\) 3.27938e28 1.44013
\(503\) −2.54685e28 −1.09532 −0.547659 0.836702i \(-0.684481\pi\)
−0.547659 + 0.836702i \(0.684481\pi\)
\(504\) −1.20287e28 −0.506637
\(505\) 0 0
\(506\) 2.41635e28 0.976290
\(507\) 1.42335e28 0.563284
\(508\) 1.67159e28 0.647979
\(509\) −1.73800e28 −0.659954 −0.329977 0.943989i \(-0.607041\pi\)
−0.329977 + 0.943989i \(0.607041\pi\)
\(510\) 0 0
\(511\) 1.48836e28 0.542361
\(512\) 6.04193e28 2.15695
\(513\) −4.63163e26 −0.0161995
\(514\) 5.79016e27 0.198417
\(515\) 0 0
\(516\) −1.68030e28 −0.552797
\(517\) 1.62944e28 0.525276
\(518\) −1.54627e28 −0.488452
\(519\) 2.23786e28 0.692747
\(520\) 0 0
\(521\) −1.00200e28 −0.297901 −0.148950 0.988845i \(-0.547589\pi\)
−0.148950 + 0.988845i \(0.547589\pi\)
\(522\) 3.66683e28 1.06844
\(523\) −3.19105e28 −0.911309 −0.455654 0.890157i \(-0.650595\pi\)
−0.455654 + 0.890157i \(0.650595\pi\)
\(524\) −1.26833e29 −3.55021
\(525\) 0 0
\(526\) −7.58709e28 −2.04045
\(527\) 3.63133e28 0.957314
\(528\) 1.40860e28 0.364026
\(529\) 2.48292e28 0.629039
\(530\) 0 0
\(531\) −7.33313e27 −0.178565
\(532\) 5.65485e27 0.135005
\(533\) −3.25672e27 −0.0762334
\(534\) −5.69237e28 −1.30650
\(535\) 0 0
\(536\) 3.24205e28 0.715467
\(537\) −2.83686e28 −0.613915
\(538\) 2.08804e28 0.443124
\(539\) 9.15544e27 0.190545
\(540\) 0 0
\(541\) −5.44256e28 −1.08951 −0.544755 0.838595i \(-0.683378\pi\)
−0.544755 + 0.838595i \(0.683378\pi\)
\(542\) 8.98440e28 1.76399
\(543\) −4.61319e28 −0.888390
\(544\) −4.53508e28 −0.856637
\(545\) 0 0
\(546\) −6.57100e27 −0.119429
\(547\) −1.01579e29 −1.81107 −0.905536 0.424270i \(-0.860531\pi\)
−0.905536 + 0.424270i \(0.860531\pi\)
\(548\) 4.49013e28 0.785349
\(549\) −1.72011e28 −0.295152
\(550\) 0 0
\(551\) −9.20888e27 −0.152095
\(552\) 9.25183e28 1.49923
\(553\) 5.36532e28 0.853065
\(554\) 9.05630e28 1.41286
\(555\) 0 0
\(556\) −2.06808e29 −3.10658
\(557\) 2.16249e28 0.318767 0.159384 0.987217i \(-0.449049\pi\)
0.159384 + 0.987217i \(0.449049\pi\)
\(558\) −2.55551e28 −0.369674
\(559\) −4.90358e27 −0.0696129
\(560\) 0 0
\(561\) 2.78790e28 0.381212
\(562\) 2.91774e28 0.391575
\(563\) −1.31464e29 −1.73168 −0.865840 0.500320i \(-0.833216\pi\)
−0.865840 + 0.500320i \(0.833216\pi\)
\(564\) 1.16787e29 1.50996
\(565\) 0 0
\(566\) −2.03948e29 −2.54066
\(567\) 6.78790e27 0.0830067
\(568\) 2.72858e29 3.27551
\(569\) 1.59225e29 1.87643 0.938215 0.346053i \(-0.112478\pi\)
0.938215 + 0.346053i \(0.112478\pi\)
\(570\) 0 0
\(571\) −1.20057e29 −1.36367 −0.681833 0.731508i \(-0.738817\pi\)
−0.681833 + 0.731508i \(0.738817\pi\)
\(572\) 1.29573e28 0.144496
\(573\) 3.95427e28 0.432955
\(574\) −6.02065e28 −0.647245
\(575\) 0 0
\(576\) −1.51105e28 −0.156618
\(577\) 8.22193e28 0.836812 0.418406 0.908260i \(-0.362589\pi\)
0.418406 + 0.908260i \(0.362589\pi\)
\(578\) −2.38697e29 −2.38564
\(579\) −2.80249e28 −0.275055
\(580\) 0 0
\(581\) −8.71969e28 −0.825376
\(582\) 1.37432e29 1.27761
\(583\) 6.57199e28 0.600036
\(584\) −1.64714e29 −1.47705
\(585\) 0 0
\(586\) 3.53921e28 0.306184
\(587\) −2.00399e29 −1.70292 −0.851461 0.524417i \(-0.824283\pi\)
−0.851461 + 0.524417i \(0.824283\pi\)
\(588\) 6.56199e28 0.547739
\(589\) 6.41791e27 0.0526239
\(590\) 0 0
\(591\) 1.35000e29 1.06823
\(592\) 6.93304e28 0.538947
\(593\) −6.29550e28 −0.480791 −0.240395 0.970675i \(-0.577277\pi\)
−0.240395 + 0.970675i \(0.577277\pi\)
\(594\) −1.96196e28 −0.147208
\(595\) 0 0
\(596\) 5.53832e29 4.01137
\(597\) −2.43175e28 −0.173056
\(598\) 5.05408e28 0.353411
\(599\) 6.47634e28 0.444988 0.222494 0.974934i \(-0.428580\pi\)
0.222494 + 0.974934i \(0.428580\pi\)
\(600\) 0 0
\(601\) 8.96436e26 0.00594754 0.00297377 0.999996i \(-0.499053\pi\)
0.00297377 + 0.999996i \(0.499053\pi\)
\(602\) −9.06518e28 −0.591035
\(603\) −1.82952e28 −0.117221
\(604\) −2.31849e29 −1.45988
\(605\) 0 0
\(606\) 2.83359e29 1.72335
\(607\) −2.88429e28 −0.172408 −0.0862042 0.996277i \(-0.527474\pi\)
−0.0862042 + 0.996277i \(0.527474\pi\)
\(608\) −8.01518e27 −0.0470897
\(609\) 1.34961e29 0.779339
\(610\) 0 0
\(611\) 3.40817e28 0.190146
\(612\) 1.99817e29 1.09583
\(613\) −2.23267e29 −1.20362 −0.601811 0.798638i \(-0.705554\pi\)
−0.601811 + 0.798638i \(0.705554\pi\)
\(614\) −1.24708e28 −0.0660887
\(615\) 0 0
\(616\) 1.27964e29 0.655376
\(617\) −8.48914e28 −0.427435 −0.213717 0.976896i \(-0.568557\pi\)
−0.213717 + 0.976896i \(0.568557\pi\)
\(618\) 2.10259e29 1.04082
\(619\) −2.50545e29 −1.21937 −0.609684 0.792645i \(-0.708704\pi\)
−0.609684 + 0.792645i \(0.708704\pi\)
\(620\) 0 0
\(621\) −5.22090e28 −0.245631
\(622\) −1.04439e29 −0.483129
\(623\) −2.09513e29 −0.952984
\(624\) 2.94626e28 0.131775
\(625\) 0 0
\(626\) 4.54360e29 1.96503
\(627\) 4.92725e27 0.0209554
\(628\) 4.72364e29 1.97561
\(629\) 1.37218e29 0.564391
\(630\) 0 0
\(631\) −3.77252e29 −1.50080 −0.750400 0.660984i \(-0.770139\pi\)
−0.750400 + 0.660984i \(0.770139\pi\)
\(632\) −5.93770e29 −2.32321
\(633\) −1.36912e29 −0.526870
\(634\) −1.15133e29 −0.435776
\(635\) 0 0
\(636\) 4.71036e29 1.72486
\(637\) 1.91497e28 0.0689759
\(638\) −3.90087e29 −1.38212
\(639\) −1.53976e29 −0.536654
\(640\) 0 0
\(641\) −2.07562e29 −0.700064 −0.350032 0.936738i \(-0.613829\pi\)
−0.350032 + 0.936738i \(0.613829\pi\)
\(642\) 4.29781e28 0.142603
\(643\) −2.75939e29 −0.900738 −0.450369 0.892843i \(-0.648708\pi\)
−0.450369 + 0.892843i \(0.648708\pi\)
\(644\) 6.37431e29 2.04706
\(645\) 0 0
\(646\) −7.35564e28 −0.228654
\(647\) −1.30012e29 −0.397638 −0.198819 0.980036i \(-0.563711\pi\)
−0.198819 + 0.980036i \(0.563711\pi\)
\(648\) −7.51203e28 −0.226058
\(649\) 7.80117e28 0.230989
\(650\) 0 0
\(651\) −9.40578e28 −0.269646
\(652\) −1.00407e30 −2.83247
\(653\) −1.20201e29 −0.333672 −0.166836 0.985985i \(-0.553355\pi\)
−0.166836 + 0.985985i \(0.553355\pi\)
\(654\) 6.09924e29 1.66613
\(655\) 0 0
\(656\) 2.69950e29 0.714155
\(657\) 9.29497e28 0.241998
\(658\) 6.30063e29 1.61440
\(659\) −3.25641e29 −0.821186 −0.410593 0.911819i \(-0.634678\pi\)
−0.410593 + 0.911819i \(0.634678\pi\)
\(660\) 0 0
\(661\) −3.48891e28 −0.0852264 −0.0426132 0.999092i \(-0.513568\pi\)
−0.0426132 + 0.999092i \(0.513568\pi\)
\(662\) −6.37568e29 −1.53291
\(663\) 5.83122e28 0.137996
\(664\) 9.64991e29 2.24780
\(665\) 0 0
\(666\) −9.65660e28 −0.217944
\(667\) −1.03805e30 −2.30620
\(668\) 4.28668e29 0.937493
\(669\) −1.05771e29 −0.227716
\(670\) 0 0
\(671\) 1.82990e29 0.381804
\(672\) 1.17467e29 0.241289
\(673\) 9.26788e29 1.87423 0.937113 0.349025i \(-0.113487\pi\)
0.937113 + 0.349025i \(0.113487\pi\)
\(674\) −1.34352e30 −2.67495
\(675\) 0 0
\(676\) −1.08527e30 −2.09458
\(677\) 3.95810e29 0.752152 0.376076 0.926589i \(-0.377273\pi\)
0.376076 + 0.926589i \(0.377273\pi\)
\(678\) −4.82156e29 −0.902145
\(679\) 5.05831e29 0.931907
\(680\) 0 0
\(681\) −4.35769e29 −0.778416
\(682\) 2.71862e29 0.478203
\(683\) 8.69191e29 1.50556 0.752779 0.658273i \(-0.228713\pi\)
0.752779 + 0.658273i \(0.228713\pi\)
\(684\) 3.53151e28 0.0602382
\(685\) 0 0
\(686\) 1.15514e30 1.91087
\(687\) 5.81238e29 0.946908
\(688\) 4.06458e29 0.652134
\(689\) 1.37461e29 0.217209
\(690\) 0 0
\(691\) 5.25953e29 0.806172 0.403086 0.915162i \(-0.367938\pi\)
0.403086 + 0.915162i \(0.367938\pi\)
\(692\) −1.70631e30 −2.57600
\(693\) −7.22114e28 −0.107376
\(694\) 7.45998e29 1.09260
\(695\) 0 0
\(696\) −1.49359e30 −2.12243
\(697\) 5.34283e29 0.747871
\(698\) −5.58045e29 −0.769462
\(699\) 2.33380e28 0.0316996
\(700\) 0 0
\(701\) −7.49111e29 −0.987431 −0.493716 0.869623i \(-0.664362\pi\)
−0.493716 + 0.869623i \(0.664362\pi\)
\(702\) −4.10366e28 −0.0532883
\(703\) 2.42516e28 0.0310248
\(704\) 1.60749e29 0.202599
\(705\) 0 0
\(706\) −5.16174e29 −0.631461
\(707\) 1.04293e30 1.25704
\(708\) 5.59135e29 0.663999
\(709\) −4.05321e29 −0.474257 −0.237128 0.971478i \(-0.576206\pi\)
−0.237128 + 0.971478i \(0.576206\pi\)
\(710\) 0 0
\(711\) 3.35070e29 0.380632
\(712\) 2.31863e30 2.59533
\(713\) 7.23444e29 0.797931
\(714\) 1.07801e30 1.17163
\(715\) 0 0
\(716\) 2.16304e30 2.28286
\(717\) −4.99456e29 −0.519453
\(718\) −1.50474e30 −1.54225
\(719\) 1.84533e28 0.0186389 0.00931945 0.999957i \(-0.497033\pi\)
0.00931945 + 0.999957i \(0.497033\pi\)
\(720\) 0 0
\(721\) 7.73875e29 0.759190
\(722\) 1.82177e30 1.76138
\(723\) −2.90979e29 −0.277274
\(724\) 3.51746e30 3.30350
\(725\) 0 0
\(726\) −9.13855e29 −0.833764
\(727\) 5.77932e29 0.519715 0.259858 0.965647i \(-0.416324\pi\)
0.259858 + 0.965647i \(0.416324\pi\)
\(728\) 2.67652e29 0.237242
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 8.04459e29 0.682922
\(732\) 1.31154e30 1.09753
\(733\) −1.71345e30 −1.41344 −0.706722 0.707491i \(-0.749827\pi\)
−0.706722 + 0.707491i \(0.749827\pi\)
\(734\) −1.58592e30 −1.28965
\(735\) 0 0
\(736\) −9.03493e29 −0.714016
\(737\) 1.94629e29 0.151635
\(738\) −3.75996e29 −0.288796
\(739\) 1.47473e29 0.111672 0.0558362 0.998440i \(-0.482218\pi\)
0.0558362 + 0.998440i \(0.482218\pi\)
\(740\) 0 0
\(741\) 1.03059e28 0.00758570
\(742\) 2.54122e30 1.84417
\(743\) 1.29857e30 0.929144 0.464572 0.885535i \(-0.346208\pi\)
0.464572 + 0.885535i \(0.346208\pi\)
\(744\) 1.04092e30 0.734347
\(745\) 0 0
\(746\) −7.22778e29 −0.495733
\(747\) −5.44554e29 −0.368277
\(748\) −2.12571e30 −1.41755
\(749\) 1.58184e29 0.104017
\(750\) 0 0
\(751\) −2.64216e30 −1.68943 −0.844713 0.535220i \(-0.820229\pi\)
−0.844713 + 0.535220i \(0.820229\pi\)
\(752\) −2.82503e30 −1.78129
\(753\) 7.53788e29 0.468707
\(754\) −8.15914e29 −0.500317
\(755\) 0 0
\(756\) −5.17562e29 −0.308662
\(757\) 1.84730e30 1.08650 0.543252 0.839570i \(-0.317193\pi\)
0.543252 + 0.839570i \(0.317193\pi\)
\(758\) 1.34093e30 0.777819
\(759\) 5.55413e29 0.317744
\(760\) 0 0
\(761\) 2.82617e30 1.57275 0.786376 0.617749i \(-0.211955\pi\)
0.786376 + 0.617749i \(0.211955\pi\)
\(762\) 5.63195e29 0.309123
\(763\) 2.24488e30 1.21530
\(764\) −3.01504e30 −1.60995
\(765\) 0 0
\(766\) 5.03800e30 2.61732
\(767\) 1.63171e29 0.0836163
\(768\) 2.28562e30 1.15535
\(769\) −1.57374e30 −0.784705 −0.392352 0.919815i \(-0.628339\pi\)
−0.392352 + 0.919815i \(0.628339\pi\)
\(770\) 0 0
\(771\) 1.33091e29 0.0645768
\(772\) 2.13684e30 1.02280
\(773\) 3.55240e30 1.67740 0.838702 0.544591i \(-0.183315\pi\)
0.838702 + 0.544591i \(0.183315\pi\)
\(774\) −5.66130e29 −0.263716
\(775\) 0 0
\(776\) −5.59793e30 −2.53793
\(777\) −3.55420e29 −0.158972
\(778\) −1.83213e29 −0.0808481
\(779\) 9.44276e28 0.0411108
\(780\) 0 0
\(781\) 1.63804e30 0.694206
\(782\) −8.29148e30 −3.46706
\(783\) 8.42846e29 0.347736
\(784\) −1.58732e30 −0.646167
\(785\) 0 0
\(786\) −4.27328e30 −1.69365
\(787\) 1.46623e30 0.573414 0.286707 0.958018i \(-0.407439\pi\)
0.286707 + 0.958018i \(0.407439\pi\)
\(788\) −1.02934e31 −3.97224
\(789\) −1.74394e30 −0.664085
\(790\) 0 0
\(791\) −1.77462e30 −0.658039
\(792\) 7.99150e29 0.292424
\(793\) 3.82745e29 0.138210
\(794\) −5.03892e30 −1.79565
\(795\) 0 0
\(796\) 1.85415e30 0.643514
\(797\) −3.89960e30 −1.33569 −0.667847 0.744298i \(-0.732784\pi\)
−0.667847 + 0.744298i \(0.732784\pi\)
\(798\) 1.90524e29 0.0644050
\(799\) −5.59128e30 −1.86539
\(800\) 0 0
\(801\) −1.30843e30 −0.425215
\(802\) 2.98740e30 0.958213
\(803\) −9.88823e29 −0.313044
\(804\) 1.39497e30 0.435890
\(805\) 0 0
\(806\) 5.68632e29 0.173106
\(807\) 4.79950e29 0.144219
\(808\) −1.15419e31 −3.42339
\(809\) −6.55857e30 −1.92021 −0.960107 0.279633i \(-0.909787\pi\)
−0.960107 + 0.279633i \(0.909787\pi\)
\(810\) 0 0
\(811\) 5.52522e28 0.0157627 0.00788135 0.999969i \(-0.497491\pi\)
0.00788135 + 0.999969i \(0.497491\pi\)
\(812\) −1.02905e31 −2.89799
\(813\) 2.06512e30 0.574110
\(814\) 1.02729e30 0.281928
\(815\) 0 0
\(816\) −4.83349e30 −1.29275
\(817\) 1.42178e29 0.0375405
\(818\) −8.80075e30 −2.29409
\(819\) −1.51039e29 −0.0388694
\(820\) 0 0
\(821\) −5.51600e30 −1.38363 −0.691817 0.722073i \(-0.743189\pi\)
−0.691817 + 0.722073i \(0.743189\pi\)
\(822\) 1.51282e30 0.374656
\(823\) 4.67949e30 1.14419 0.572097 0.820186i \(-0.306130\pi\)
0.572097 + 0.820186i \(0.306130\pi\)
\(824\) −8.56432e30 −2.06755
\(825\) 0 0
\(826\) 3.01651e30 0.709928
\(827\) −4.73472e30 −1.10024 −0.550119 0.835086i \(-0.685418\pi\)
−0.550119 + 0.835086i \(0.685418\pi\)
\(828\) 3.98082e30 0.913386
\(829\) 7.07617e30 1.60316 0.801578 0.597890i \(-0.203994\pi\)
0.801578 + 0.597890i \(0.203994\pi\)
\(830\) 0 0
\(831\) 2.08165e30 0.459829
\(832\) 3.36226e29 0.0733393
\(833\) −3.14161e30 −0.676674
\(834\) −6.96781e30 −1.48202
\(835\) 0 0
\(836\) −3.75692e29 −0.0779230
\(837\) −5.87401e29 −0.120314
\(838\) −2.04127e30 −0.412894
\(839\) 5.63558e30 1.12574 0.562870 0.826545i \(-0.309697\pi\)
0.562870 + 0.826545i \(0.309697\pi\)
\(840\) 0 0
\(841\) 1.16251e31 2.26485
\(842\) 9.10413e30 1.75171
\(843\) 6.70661e29 0.127442
\(844\) 1.04392e31 1.95918
\(845\) 0 0
\(846\) 3.93481e30 0.720335
\(847\) −3.36352e30 −0.608161
\(848\) −1.13941e31 −2.03482
\(849\) −4.68787e30 −0.826885
\(850\) 0 0
\(851\) 2.73370e30 0.470426
\(852\) 1.17403e31 1.99556
\(853\) 3.93728e30 0.661045 0.330523 0.943798i \(-0.392775\pi\)
0.330523 + 0.943798i \(0.392775\pi\)
\(854\) 7.07574e30 1.17345
\(855\) 0 0
\(856\) −1.75060e30 −0.283277
\(857\) 3.31208e30 0.529422 0.264711 0.964328i \(-0.414723\pi\)
0.264711 + 0.964328i \(0.414723\pi\)
\(858\) 4.36558e29 0.0689328
\(859\) −7.48047e30 −1.16681 −0.583406 0.812181i \(-0.698280\pi\)
−0.583406 + 0.812181i \(0.698280\pi\)
\(860\) 0 0
\(861\) −1.38389e30 −0.210653
\(862\) 2.01619e31 3.03183
\(863\) 8.74392e30 1.29895 0.649476 0.760382i \(-0.274988\pi\)
0.649476 + 0.760382i \(0.274988\pi\)
\(864\) 7.33592e29 0.107661
\(865\) 0 0
\(866\) 2.31563e31 3.31689
\(867\) −5.48661e30 −0.776432
\(868\) 7.17170e30 1.00269
\(869\) −3.56456e30 −0.492378
\(870\) 0 0
\(871\) 4.07091e29 0.0548909
\(872\) −2.48436e31 −3.30972
\(873\) 3.15897e30 0.415810
\(874\) −1.46541e30 −0.190586
\(875\) 0 0
\(876\) −7.08721e30 −0.899875
\(877\) −6.82497e30 −0.856259 −0.428130 0.903717i \(-0.640827\pi\)
−0.428130 + 0.903717i \(0.640827\pi\)
\(878\) −1.21991e31 −1.51229
\(879\) 8.13512e29 0.0996507
\(880\) 0 0
\(881\) −1.33104e31 −1.59200 −0.796002 0.605294i \(-0.793056\pi\)
−0.796002 + 0.605294i \(0.793056\pi\)
\(882\) 2.21087e30 0.261303
\(883\) 2.94784e30 0.344284 0.172142 0.985072i \(-0.444931\pi\)
0.172142 + 0.985072i \(0.444931\pi\)
\(884\) −4.44617e30 −0.513142
\(885\) 0 0
\(886\) 7.99870e30 0.901499
\(887\) 1.22822e31 1.36797 0.683987 0.729494i \(-0.260244\pi\)
0.683987 + 0.729494i \(0.260244\pi\)
\(888\) 3.93336e30 0.432939
\(889\) 2.07289e30 0.225480
\(890\) 0 0
\(891\) −4.50968e29 −0.0479104
\(892\) 8.06483e30 0.846768
\(893\) −9.88187e29 −0.102541
\(894\) 1.86598e31 1.91365
\(895\) 0 0
\(896\) 1.03876e31 1.04060
\(897\) 1.16171e30 0.115021
\(898\) −7.17902e30 −0.702528
\(899\) −1.16791e31 −1.12962
\(900\) 0 0
\(901\) −2.25512e31 −2.13088
\(902\) 3.99995e30 0.373582
\(903\) −2.08369e30 −0.192359
\(904\) 1.96393e31 1.79208
\(905\) 0 0
\(906\) −7.81150e30 −0.696448
\(907\) 9.64171e30 0.849724 0.424862 0.905258i \(-0.360323\pi\)
0.424862 + 0.905258i \(0.360323\pi\)
\(908\) 3.32264e31 2.89456
\(909\) 6.51319e30 0.560884
\(910\) 0 0
\(911\) −1.47519e30 −0.124138 −0.0620690 0.998072i \(-0.519770\pi\)
−0.0620690 + 0.998072i \(0.519770\pi\)
\(912\) −8.54258e29 −0.0710630
\(913\) 5.79311e30 0.476396
\(914\) 3.43106e31 2.78928
\(915\) 0 0
\(916\) −4.43181e31 −3.52110
\(917\) −1.57282e31 −1.23538
\(918\) 6.73227e30 0.522774
\(919\) −1.21235e31 −0.930714 −0.465357 0.885123i \(-0.654074\pi\)
−0.465357 + 0.885123i \(0.654074\pi\)
\(920\) 0 0
\(921\) −2.86650e29 −0.0215093
\(922\) 2.01631e31 1.49583
\(923\) 3.42615e30 0.251298
\(924\) 5.50596e30 0.399280
\(925\) 0 0
\(926\) −2.02174e31 −1.43321
\(927\) 4.83293e30 0.338745
\(928\) 1.45857e31 1.01082
\(929\) −1.25133e31 −0.857447 −0.428724 0.903436i \(-0.641037\pi\)
−0.428724 + 0.903436i \(0.641037\pi\)
\(930\) 0 0
\(931\) −5.55239e29 −0.0371970
\(932\) −1.77947e30 −0.117876
\(933\) −2.40060e30 −0.157240
\(934\) 2.47978e30 0.160609
\(935\) 0 0
\(936\) 1.67152e30 0.105856
\(937\) −6.38356e30 −0.399758 −0.199879 0.979821i \(-0.564055\pi\)
−0.199879 + 0.979821i \(0.564055\pi\)
\(938\) 7.52582e30 0.466041
\(939\) 1.04438e31 0.639541
\(940\) 0 0
\(941\) 2.28359e31 1.36750 0.683751 0.729716i \(-0.260348\pi\)
0.683751 + 0.729716i \(0.260348\pi\)
\(942\) 1.59150e31 0.942478
\(943\) 1.06441e31 0.623358
\(944\) −1.35252e31 −0.783319
\(945\) 0 0
\(946\) 6.02264e30 0.341138
\(947\) −6.01017e30 −0.336676 −0.168338 0.985729i \(-0.553840\pi\)
−0.168338 + 0.985729i \(0.553840\pi\)
\(948\) −2.55483e31 −1.41539
\(949\) −2.06824e30 −0.113320
\(950\) 0 0
\(951\) −2.64641e30 −0.141828
\(952\) −4.39098e31 −2.32741
\(953\) 3.54334e31 1.85754 0.928769 0.370660i \(-0.120868\pi\)
0.928769 + 0.370660i \(0.120868\pi\)
\(954\) 1.58702e31 0.822857
\(955\) 0 0
\(956\) 3.80824e31 1.93160
\(957\) −8.96641e30 −0.449825
\(958\) 1.82315e31 0.904660
\(959\) 5.56806e30 0.273281
\(960\) 0 0
\(961\) −1.26861e31 −0.609160
\(962\) 2.14871e30 0.102056
\(963\) 9.87879e29 0.0464117
\(964\) 2.21865e31 1.03105
\(965\) 0 0
\(966\) 2.14764e31 0.976566
\(967\) −2.99312e31 −1.34631 −0.673157 0.739499i \(-0.735062\pi\)
−0.673157 + 0.739499i \(0.735062\pi\)
\(968\) 3.72234e31 1.65625
\(969\) −1.69074e30 −0.0744179
\(970\) 0 0
\(971\) −1.42001e31 −0.611632 −0.305816 0.952091i \(-0.598929\pi\)
−0.305816 + 0.952091i \(0.598929\pi\)
\(972\) −3.23223e30 −0.137723
\(973\) −2.56456e31 −1.08101
\(974\) 1.38768e31 0.578657
\(975\) 0 0
\(976\) −3.17257e31 −1.29476
\(977\) 2.56243e31 1.03457 0.517283 0.855814i \(-0.326943\pi\)
0.517283 + 0.855814i \(0.326943\pi\)
\(978\) −3.38294e31 −1.35125
\(979\) 1.39194e31 0.550050
\(980\) 0 0
\(981\) 1.40195e31 0.542261
\(982\) −3.53545e31 −1.35293
\(983\) −3.29184e31 −1.24631 −0.623156 0.782098i \(-0.714150\pi\)
−0.623156 + 0.782098i \(0.714150\pi\)
\(984\) 1.53152e31 0.573686
\(985\) 0 0
\(986\) 1.33855e32 4.90825
\(987\) 1.44824e31 0.525425
\(988\) −7.85804e29 −0.0282076
\(989\) 1.60267e31 0.569223
\(990\) 0 0
\(991\) −2.22756e31 −0.774560 −0.387280 0.921962i \(-0.626585\pi\)
−0.387280 + 0.921962i \(0.626585\pi\)
\(992\) −1.01652e31 −0.349737
\(993\) −1.46549e31 −0.498903
\(994\) 6.33388e31 2.13360
\(995\) 0 0
\(996\) 4.15210e31 1.36945
\(997\) −2.49754e31 −0.815105 −0.407553 0.913182i \(-0.633618\pi\)
−0.407553 + 0.913182i \(0.633618\pi\)
\(998\) −7.47968e31 −2.41553
\(999\) −2.21963e30 −0.0709322
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 75.22.a.j.1.1 yes 6
5.2 odd 4 75.22.b.i.49.1 12
5.3 odd 4 75.22.b.i.49.12 12
5.4 even 2 75.22.a.i.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.22.a.i.1.6 6 5.4 even 2
75.22.a.j.1.1 yes 6 1.1 even 1 trivial
75.22.b.i.49.1 12 5.2 odd 4
75.22.b.i.49.12 12 5.3 odd 4