Properties

Label 9.28.a.b.1.2
Level $9$
Weight $28$
Character 9.1
Self dual yes
Analytic conductor $41.567$
Analytic rank $1$
Dimension $2$
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] = [9,28,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, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 28 \)
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: \(41.5670017354\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{30001}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 7500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-86.1040\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+121.102 q^{2} -1.34203e8 q^{4} +1.07803e9 q^{5} -1.13904e10 q^{7} -3.25063e10 q^{8} +O(q^{10})\) \(q+121.102 q^{2} -1.34203e8 q^{4} +1.07803e9 q^{5} -1.13904e10 q^{7} -3.25063e10 q^{8} +1.30551e11 q^{10} +5.05749e13 q^{11} +1.21304e15 q^{13} -1.37940e12 q^{14} +1.80085e16 q^{16} -7.00452e16 q^{17} +2.51027e16 q^{19} -1.44674e17 q^{20} +6.12472e15 q^{22} +3.40370e18 q^{23} -6.28844e18 q^{25} +1.46901e17 q^{26} +1.52863e18 q^{28} -5.74556e19 q^{29} +1.08167e20 q^{31} +6.54378e18 q^{32} -8.48261e18 q^{34} -1.22791e19 q^{35} -1.90073e21 q^{37} +3.03999e18 q^{38} -3.50426e19 q^{40} -6.66148e21 q^{41} +1.03265e21 q^{43} -6.78731e21 q^{44} +4.12194e20 q^{46} -6.43133e22 q^{47} -6.55826e22 q^{49} -7.61542e20 q^{50} -1.62794e23 q^{52} -2.13714e23 q^{53} +5.45211e22 q^{55} +3.70259e20 q^{56} -6.95799e21 q^{58} -1.07026e24 q^{59} +8.16628e23 q^{61} +1.30992e22 q^{62} -2.41627e24 q^{64} +1.30769e24 q^{65} +1.68630e24 q^{67} +9.40028e24 q^{68} -1.48703e21 q^{70} +3.18547e24 q^{71} +2.09853e25 q^{73} -2.30183e23 q^{74} -3.36886e24 q^{76} -5.76068e23 q^{77} -8.88148e22 q^{79} +1.94136e25 q^{80} -8.06718e23 q^{82} +6.50402e25 q^{83} -7.55105e25 q^{85} +1.25056e23 q^{86} -1.64400e24 q^{88} -3.38617e26 q^{89} -1.38170e25 q^{91} -4.56787e26 q^{92} -7.78847e24 q^{94} +2.70614e25 q^{95} -4.16124e26 q^{97} -7.94218e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 21582 q^{2} + 202603844 q^{4} + 1771946100 q^{5} + 369665199904 q^{7} - 4429319872824 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 21582 q^{2} + 202603844 q^{4} + 1771946100 q^{5} + 369665199904 q^{7} - 4429319872824 q^{8} - 14929666656300 q^{10} - 75762335668248 q^{11} - 103021079177588 q^{13} - 82\!\cdots\!36 q^{14}+ \cdots - 17\!\cdots\!78 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 121.102 0.0104531 0.00522656 0.999986i \(-0.498336\pi\)
0.00522656 + 0.999986i \(0.498336\pi\)
\(3\) 0 0
\(4\) −1.34203e8 −0.999891
\(5\) 1.07803e9 0.394943 0.197471 0.980309i \(-0.436727\pi\)
0.197471 + 0.980309i \(0.436727\pi\)
\(6\) 0 0
\(7\) −1.13904e10 −0.0444340 −0.0222170 0.999753i \(-0.507072\pi\)
−0.0222170 + 0.999753i \(0.507072\pi\)
\(8\) −3.25063e10 −0.0209051
\(9\) 0 0
\(10\) 1.30551e11 0.00412839
\(11\) 5.05749e13 0.441707 0.220853 0.975307i \(-0.429116\pi\)
0.220853 + 0.975307i \(0.429116\pi\)
\(12\) 0 0
\(13\) 1.21304e15 1.11081 0.555405 0.831580i \(-0.312563\pi\)
0.555405 + 0.831580i \(0.312563\pi\)
\(14\) −1.37940e12 −0.000464474 0
\(15\) 0 0
\(16\) 1.80085e16 0.999672
\(17\) −7.00452e16 −1.71521 −0.857606 0.514307i \(-0.828049\pi\)
−0.857606 + 0.514307i \(0.828049\pi\)
\(18\) 0 0
\(19\) 2.51027e16 0.136945 0.0684726 0.997653i \(-0.478187\pi\)
0.0684726 + 0.997653i \(0.478187\pi\)
\(20\) −1.44674e17 −0.394900
\(21\) 0 0
\(22\) 6.12472e15 0.00461722
\(23\) 3.40370e18 1.40807 0.704037 0.710164i \(-0.251379\pi\)
0.704037 + 0.710164i \(0.251379\pi\)
\(24\) 0 0
\(25\) −6.28844e18 −0.844020
\(26\) 1.46901e17 0.0116114
\(27\) 0 0
\(28\) 1.52863e18 0.0444291
\(29\) −5.74556e19 −1.03982 −0.519912 0.854220i \(-0.674035\pi\)
−0.519912 + 0.854220i \(0.674035\pi\)
\(30\) 0 0
\(31\) 1.08167e20 0.795632 0.397816 0.917465i \(-0.369768\pi\)
0.397816 + 0.917465i \(0.369768\pi\)
\(32\) 6.54378e18 0.0313548
\(33\) 0 0
\(34\) −8.48261e18 −0.0179293
\(35\) −1.22791e19 −0.0175489
\(36\) 0 0
\(37\) −1.90073e21 −1.28292 −0.641458 0.767159i \(-0.721670\pi\)
−0.641458 + 0.767159i \(0.721670\pi\)
\(38\) 3.03999e18 0.00143151
\(39\) 0 0
\(40\) −3.50426e19 −0.00825632
\(41\) −6.66148e21 −1.12458 −0.562288 0.826942i \(-0.690079\pi\)
−0.562288 + 0.826942i \(0.690079\pi\)
\(42\) 0 0
\(43\) 1.03265e21 0.0916490 0.0458245 0.998950i \(-0.485408\pi\)
0.0458245 + 0.998950i \(0.485408\pi\)
\(44\) −6.78731e21 −0.441659
\(45\) 0 0
\(46\) 4.12194e20 0.0147188
\(47\) −6.43133e22 −1.71783 −0.858914 0.512120i \(-0.828860\pi\)
−0.858914 + 0.512120i \(0.828860\pi\)
\(48\) 0 0
\(49\) −6.55826e22 −0.998026
\(50\) −7.61542e20 −0.00882265
\(51\) 0 0
\(52\) −1.62794e23 −1.11069
\(53\) −2.13714e23 −1.12748 −0.563740 0.825953i \(-0.690638\pi\)
−0.563740 + 0.825953i \(0.690638\pi\)
\(54\) 0 0
\(55\) 5.45211e22 0.174449
\(56\) 3.70259e20 0.000928897 0
\(57\) 0 0
\(58\) −6.95799e21 −0.0108694
\(59\) −1.07026e24 −1.32736 −0.663679 0.748017i \(-0.731006\pi\)
−0.663679 + 0.748017i \(0.731006\pi\)
\(60\) 0 0
\(61\) 8.16628e23 0.645761 0.322880 0.946440i \(-0.395349\pi\)
0.322880 + 0.946440i \(0.395349\pi\)
\(62\) 1.30992e22 0.00831684
\(63\) 0 0
\(64\) −2.41627e24 −0.999344
\(65\) 1.30769e24 0.438706
\(66\) 0 0
\(67\) 1.68630e24 0.375773 0.187886 0.982191i \(-0.439836\pi\)
0.187886 + 0.982191i \(0.439836\pi\)
\(68\) 9.40028e24 1.71502
\(69\) 0 0
\(70\) −1.48703e21 −0.000183441 0
\(71\) 3.18547e24 0.324478 0.162239 0.986751i \(-0.448128\pi\)
0.162239 + 0.986751i \(0.448128\pi\)
\(72\) 0 0
\(73\) 2.09853e25 1.46912 0.734559 0.678545i \(-0.237389\pi\)
0.734559 + 0.678545i \(0.237389\pi\)
\(74\) −2.30183e23 −0.0134105
\(75\) 0 0
\(76\) −3.36886e24 −0.136930
\(77\) −5.76068e23 −0.0196268
\(78\) 0 0
\(79\) −8.88148e22 −0.00214052 −0.00107026 0.999999i \(-0.500341\pi\)
−0.00107026 + 0.999999i \(0.500341\pi\)
\(80\) 1.94136e25 0.394813
\(81\) 0 0
\(82\) −8.06718e23 −0.0117553
\(83\) 6.50402e25 0.804689 0.402344 0.915488i \(-0.368195\pi\)
0.402344 + 0.915488i \(0.368195\pi\)
\(84\) 0 0
\(85\) −7.55105e25 −0.677411
\(86\) 1.25056e23 0.000958019 0
\(87\) 0 0
\(88\) −1.64400e24 −0.00923393
\(89\) −3.38617e26 −1.63284 −0.816420 0.577459i \(-0.804044\pi\)
−0.816420 + 0.577459i \(0.804044\pi\)
\(90\) 0 0
\(91\) −1.38170e25 −0.0493577
\(92\) −4.56787e26 −1.40792
\(93\) 0 0
\(94\) −7.78847e24 −0.0179567
\(95\) 2.70614e25 0.0540855
\(96\) 0 0
\(97\) −4.16124e26 −0.627775 −0.313888 0.949460i \(-0.601632\pi\)
−0.313888 + 0.949460i \(0.601632\pi\)
\(98\) −7.94218e24 −0.0104325
\(99\) 0 0
\(100\) 8.43928e26 0.843928
\(101\) 9.90798e26 0.866256 0.433128 0.901332i \(-0.357410\pi\)
0.433128 + 0.901332i \(0.357410\pi\)
\(102\) 0 0
\(103\) −2.01221e27 −1.35012 −0.675059 0.737764i \(-0.735882\pi\)
−0.675059 + 0.737764i \(0.735882\pi\)
\(104\) −3.94314e25 −0.0232216
\(105\) 0 0
\(106\) −2.58812e25 −0.0117857
\(107\) −2.23219e25 −0.00895470 −0.00447735 0.999990i \(-0.501425\pi\)
−0.00447735 + 0.999990i \(0.501425\pi\)
\(108\) 0 0
\(109\) 5.40319e27 1.68808 0.844039 0.536281i \(-0.180171\pi\)
0.844039 + 0.536281i \(0.180171\pi\)
\(110\) 6.60261e24 0.00182354
\(111\) 0 0
\(112\) −2.05124e26 −0.0444194
\(113\) 3.71413e27 0.713342 0.356671 0.934230i \(-0.383912\pi\)
0.356671 + 0.934230i \(0.383912\pi\)
\(114\) 0 0
\(115\) 3.66927e27 0.556108
\(116\) 7.71072e27 1.03971
\(117\) 0 0
\(118\) −1.29611e26 −0.0138750
\(119\) 7.97842e26 0.0762137
\(120\) 0 0
\(121\) −1.05522e28 −0.804895
\(122\) 9.88953e25 0.00675022
\(123\) 0 0
\(124\) −1.45164e28 −0.795545
\(125\) −1.48110e28 −0.728282
\(126\) 0 0
\(127\) 2.13925e28 0.849007 0.424503 0.905426i \(-0.360449\pi\)
0.424503 + 0.905426i \(0.360449\pi\)
\(128\) −1.17091e27 −0.0418011
\(129\) 0 0
\(130\) 1.58364e26 0.00458585
\(131\) −5.14435e28 −1.34328 −0.671642 0.740876i \(-0.734411\pi\)
−0.671642 + 0.740876i \(0.734411\pi\)
\(132\) 0 0
\(133\) −2.85930e26 −0.00608502
\(134\) 2.04214e26 0.00392800
\(135\) 0 0
\(136\) 2.27691e27 0.0358567
\(137\) −3.71968e28 −0.530612 −0.265306 0.964164i \(-0.585473\pi\)
−0.265306 + 0.964164i \(0.585473\pi\)
\(138\) 0 0
\(139\) −1.00529e29 −1.17921 −0.589605 0.807692i \(-0.700717\pi\)
−0.589605 + 0.807692i \(0.700717\pi\)
\(140\) 1.64790e27 0.0175470
\(141\) 0 0
\(142\) 3.85766e26 0.00339181
\(143\) 6.13494e28 0.490652
\(144\) 0 0
\(145\) −6.19387e28 −0.410671
\(146\) 2.54136e27 0.0153569
\(147\) 0 0
\(148\) 2.55084e29 1.28277
\(149\) 2.06193e29 0.946801 0.473400 0.880847i \(-0.343026\pi\)
0.473400 + 0.880847i \(0.343026\pi\)
\(150\) 0 0
\(151\) 8.73857e28 0.335160 0.167580 0.985859i \(-0.446405\pi\)
0.167580 + 0.985859i \(0.446405\pi\)
\(152\) −8.15995e26 −0.00286285
\(153\) 0 0
\(154\) −6.97630e25 −0.000205161 0
\(155\) 1.16607e29 0.314229
\(156\) 0 0
\(157\) 3.66647e29 0.831004 0.415502 0.909592i \(-0.363606\pi\)
0.415502 + 0.909592i \(0.363606\pi\)
\(158\) −1.07556e25 −2.23751e−5 0
\(159\) 0 0
\(160\) 7.05437e27 0.0123834
\(161\) −3.87694e28 −0.0625663
\(162\) 0 0
\(163\) −9.33235e29 −1.27485 −0.637424 0.770513i \(-0.720000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(164\) 8.93991e29 1.12445
\(165\) 0 0
\(166\) 7.87650e27 0.00841151
\(167\) −3.46815e29 −0.341527 −0.170764 0.985312i \(-0.554623\pi\)
−0.170764 + 0.985312i \(0.554623\pi\)
\(168\) 0 0
\(169\) 2.78930e29 0.233897
\(170\) −9.14447e27 −0.00708106
\(171\) 0 0
\(172\) −1.38584e29 −0.0916390
\(173\) −2.65803e30 −1.62532 −0.812659 0.582740i \(-0.801981\pi\)
−0.812659 + 0.582740i \(0.801981\pi\)
\(174\) 0 0
\(175\) 7.16278e28 0.0375032
\(176\) 9.10778e29 0.441562
\(177\) 0 0
\(178\) −4.10072e28 −0.0170683
\(179\) −4.30907e30 −1.66291 −0.831454 0.555594i \(-0.812491\pi\)
−0.831454 + 0.555594i \(0.812491\pi\)
\(180\) 0 0
\(181\) −5.56401e29 −0.184811 −0.0924053 0.995721i \(-0.529456\pi\)
−0.0924053 + 0.995721i \(0.529456\pi\)
\(182\) −1.67326e27 −0.000515942 0
\(183\) 0 0
\(184\) −1.10642e29 −0.0294359
\(185\) −2.04904e30 −0.506678
\(186\) 0 0
\(187\) −3.54253e30 −0.757621
\(188\) 8.63104e30 1.71764
\(189\) 0 0
\(190\) 3.27718e27 0.000565363 0
\(191\) 4.19469e30 0.674139 0.337070 0.941480i \(-0.390564\pi\)
0.337070 + 0.941480i \(0.390564\pi\)
\(192\) 0 0
\(193\) 4.85628e28 0.00678078 0.00339039 0.999994i \(-0.498921\pi\)
0.00339039 + 0.999994i \(0.498921\pi\)
\(194\) −5.03934e28 −0.00656221
\(195\) 0 0
\(196\) 8.80139e30 0.997917
\(197\) 2.40097e29 0.0254151 0.0127075 0.999919i \(-0.495955\pi\)
0.0127075 + 0.999919i \(0.495955\pi\)
\(198\) 0 0
\(199\) 7.40632e30 0.684047 0.342024 0.939691i \(-0.388888\pi\)
0.342024 + 0.939691i \(0.388888\pi\)
\(200\) 2.04414e29 0.0176443
\(201\) 0 0
\(202\) 1.19988e29 0.00905509
\(203\) 6.54442e29 0.0462035
\(204\) 0 0
\(205\) −7.18125e30 −0.444143
\(206\) −2.43683e29 −0.0141129
\(207\) 0 0
\(208\) 2.18450e31 1.11045
\(209\) 1.26957e30 0.0604896
\(210\) 0 0
\(211\) −3.11843e31 −1.30654 −0.653271 0.757124i \(-0.726604\pi\)
−0.653271 + 0.757124i \(0.726604\pi\)
\(212\) 2.86811e31 1.12736
\(213\) 0 0
\(214\) −2.70323e27 −9.36046e−5 0
\(215\) 1.11322e30 0.0361961
\(216\) 0 0
\(217\) −1.23207e30 −0.0353531
\(218\) 6.54337e29 0.0176457
\(219\) 0 0
\(220\) −7.31690e30 −0.174430
\(221\) −8.49675e31 −1.90527
\(222\) 0 0
\(223\) −2.26940e31 −0.450605 −0.225302 0.974289i \(-0.572337\pi\)
−0.225302 + 0.974289i \(0.572337\pi\)
\(224\) −7.45362e28 −0.00139322
\(225\) 0 0
\(226\) 4.49788e29 0.00745665
\(227\) −8.41908e31 −1.31497 −0.657484 0.753468i \(-0.728379\pi\)
−0.657484 + 0.753468i \(0.728379\pi\)
\(228\) 0 0
\(229\) 4.09888e31 0.568703 0.284352 0.958720i \(-0.408222\pi\)
0.284352 + 0.958720i \(0.408222\pi\)
\(230\) 4.44356e29 0.00581307
\(231\) 0 0
\(232\) 1.86767e30 0.0217376
\(233\) −6.99046e31 −0.767717 −0.383858 0.923392i \(-0.625405\pi\)
−0.383858 + 0.923392i \(0.625405\pi\)
\(234\) 0 0
\(235\) −6.93314e31 −0.678444
\(236\) 1.43633e32 1.32721
\(237\) 0 0
\(238\) 9.66202e28 0.000796671 0
\(239\) 1.89074e32 1.47320 0.736598 0.676330i \(-0.236431\pi\)
0.736598 + 0.676330i \(0.236431\pi\)
\(240\) 0 0
\(241\) 1.01767e32 0.708563 0.354281 0.935139i \(-0.384725\pi\)
0.354281 + 0.935139i \(0.384725\pi\)
\(242\) −1.27789e30 −0.00841367
\(243\) 0 0
\(244\) −1.09594e32 −0.645690
\(245\) −7.06998e31 −0.394163
\(246\) 0 0
\(247\) 3.04506e31 0.152120
\(248\) −3.51611e30 −0.0166328
\(249\) 0 0
\(250\) −1.79364e30 −0.00761283
\(251\) 2.25120e32 0.905356 0.452678 0.891674i \(-0.350469\pi\)
0.452678 + 0.891674i \(0.350469\pi\)
\(252\) 0 0
\(253\) 1.72142e32 0.621955
\(254\) 2.59067e30 0.00887478
\(255\) 0 0
\(256\) 3.24164e32 0.998908
\(257\) −1.28310e32 −0.375113 −0.187557 0.982254i \(-0.560057\pi\)
−0.187557 + 0.982254i \(0.560057\pi\)
\(258\) 0 0
\(259\) 2.16501e31 0.0570050
\(260\) −1.75496e32 −0.438658
\(261\) 0 0
\(262\) −6.22990e30 −0.0140415
\(263\) 6.17345e32 1.32168 0.660838 0.750528i \(-0.270201\pi\)
0.660838 + 0.750528i \(0.270201\pi\)
\(264\) 0 0
\(265\) −2.30389e32 −0.445290
\(266\) −3.46266e28 −6.36075e−5 0
\(267\) 0 0
\(268\) −2.26307e32 −0.375732
\(269\) 1.70730e31 0.0269559 0.0134779 0.999909i \(-0.495710\pi\)
0.0134779 + 0.999909i \(0.495710\pi\)
\(270\) 0 0
\(271\) −1.06709e33 −1.52446 −0.762229 0.647307i \(-0.775895\pi\)
−0.762229 + 0.647307i \(0.775895\pi\)
\(272\) −1.26141e33 −1.71465
\(273\) 0 0
\(274\) −4.50460e30 −0.00554656
\(275\) −3.18037e32 −0.372809
\(276\) 0 0
\(277\) 5.90772e32 0.627976 0.313988 0.949427i \(-0.398335\pi\)
0.313988 + 0.949427i \(0.398335\pi\)
\(278\) −1.21742e31 −0.0123264
\(279\) 0 0
\(280\) 3.99149e29 0.000366861 0
\(281\) −3.49531e32 −0.306161 −0.153081 0.988214i \(-0.548919\pi\)
−0.153081 + 0.988214i \(0.548919\pi\)
\(282\) 0 0
\(283\) −1.04098e32 −0.0828565 −0.0414282 0.999141i \(-0.513191\pi\)
−0.0414282 + 0.999141i \(0.513191\pi\)
\(284\) −4.27499e32 −0.324443
\(285\) 0 0
\(286\) 7.42953e30 0.00512885
\(287\) 7.58768e31 0.0499693
\(288\) 0 0
\(289\) 3.23862e33 1.94195
\(290\) −7.50089e30 −0.00429279
\(291\) 0 0
\(292\) −2.81629e33 −1.46896
\(293\) 4.57358e32 0.227794 0.113897 0.993493i \(-0.463667\pi\)
0.113897 + 0.993493i \(0.463667\pi\)
\(294\) 0 0
\(295\) −1.15377e33 −0.524231
\(296\) 6.17858e31 0.0268195
\(297\) 0 0
\(298\) 2.49703e31 0.00989703
\(299\) 4.12882e33 1.56410
\(300\) 0 0
\(301\) −1.17623e31 −0.00407233
\(302\) 1.05826e31 0.00350346
\(303\) 0 0
\(304\) 4.52062e32 0.136900
\(305\) 8.80347e32 0.255039
\(306\) 0 0
\(307\) 2.50586e33 0.664642 0.332321 0.943166i \(-0.392168\pi\)
0.332321 + 0.943166i \(0.392168\pi\)
\(308\) 7.73101e31 0.0196246
\(309\) 0 0
\(310\) 1.41213e31 0.00328468
\(311\) 4.09104e33 0.911104 0.455552 0.890209i \(-0.349442\pi\)
0.455552 + 0.890209i \(0.349442\pi\)
\(312\) 0 0
\(313\) 5.57324e33 1.13831 0.569154 0.822231i \(-0.307271\pi\)
0.569154 + 0.822231i \(0.307271\pi\)
\(314\) 4.44017e31 0.00868659
\(315\) 0 0
\(316\) 1.19192e31 0.00214029
\(317\) −6.22580e33 −1.07126 −0.535631 0.844452i \(-0.679926\pi\)
−0.535631 + 0.844452i \(0.679926\pi\)
\(318\) 0 0
\(319\) −2.90581e33 −0.459297
\(320\) −2.60480e33 −0.394684
\(321\) 0 0
\(322\) −4.69505e30 −0.000654013 0
\(323\) −1.75832e33 −0.234890
\(324\) 0 0
\(325\) −7.62812e33 −0.937545
\(326\) −1.13017e32 −0.0133261
\(327\) 0 0
\(328\) 2.16540e32 0.0235094
\(329\) 7.32553e32 0.0763299
\(330\) 0 0
\(331\) −6.69049e33 −0.642363 −0.321181 0.947018i \(-0.604080\pi\)
−0.321181 + 0.947018i \(0.604080\pi\)
\(332\) −8.72860e33 −0.804601
\(333\) 0 0
\(334\) −4.20000e31 −0.00357003
\(335\) 1.81788e33 0.148409
\(336\) 0 0
\(337\) −8.45330e33 −0.636829 −0.318414 0.947952i \(-0.603150\pi\)
−0.318414 + 0.947952i \(0.603150\pi\)
\(338\) 3.37790e31 0.00244496
\(339\) 0 0
\(340\) 1.01337e34 0.677337
\(341\) 5.47054e33 0.351436
\(342\) 0 0
\(343\) 1.49550e33 0.0887802
\(344\) −3.35675e31 −0.00191593
\(345\) 0 0
\(346\) −3.21893e32 −0.0169896
\(347\) −3.54455e34 −1.79934 −0.899671 0.436568i \(-0.856194\pi\)
−0.899671 + 0.436568i \(0.856194\pi\)
\(348\) 0 0
\(349\) 1.62742e34 0.764462 0.382231 0.924067i \(-0.375156\pi\)
0.382231 + 0.924067i \(0.375156\pi\)
\(350\) 8.67426e30 0.000392025 0
\(351\) 0 0
\(352\) 3.30951e32 0.0138496
\(353\) 6.68547e33 0.269261 0.134630 0.990896i \(-0.457015\pi\)
0.134630 + 0.990896i \(0.457015\pi\)
\(354\) 0 0
\(355\) 3.43402e33 0.128150
\(356\) 4.54434e34 1.63266
\(357\) 0 0
\(358\) −5.21837e32 −0.0173826
\(359\) 7.80470e33 0.250369 0.125185 0.992133i \(-0.460048\pi\)
0.125185 + 0.992133i \(0.460048\pi\)
\(360\) 0 0
\(361\) −3.29705e34 −0.981246
\(362\) −6.73812e31 −0.00193185
\(363\) 0 0
\(364\) 1.85428e33 0.0493523
\(365\) 2.26227e34 0.580218
\(366\) 0 0
\(367\) 6.61615e34 1.57621 0.788104 0.615542i \(-0.211063\pi\)
0.788104 + 0.615542i \(0.211063\pi\)
\(368\) 6.12955e34 1.40761
\(369\) 0 0
\(370\) −2.48143e32 −0.00529637
\(371\) 2.43428e33 0.0500984
\(372\) 0 0
\(373\) −6.79473e34 −1.30048 −0.650239 0.759729i \(-0.725331\pi\)
−0.650239 + 0.759729i \(0.725331\pi\)
\(374\) −4.29007e32 −0.00791951
\(375\) 0 0
\(376\) 2.09059e33 0.0359114
\(377\) −6.96959e34 −1.15505
\(378\) 0 0
\(379\) 7.98595e34 1.23224 0.616122 0.787650i \(-0.288703\pi\)
0.616122 + 0.787650i \(0.288703\pi\)
\(380\) −3.63172e33 −0.0540796
\(381\) 0 0
\(382\) 5.07985e32 0.00704686
\(383\) −1.19033e35 −1.59399 −0.796994 0.603987i \(-0.793578\pi\)
−0.796994 + 0.603987i \(0.793578\pi\)
\(384\) 0 0
\(385\) −6.21016e32 −0.00775146
\(386\) 5.88105e30 7.08804e−5 0
\(387\) 0 0
\(388\) 5.58451e34 0.627707
\(389\) −1.48935e34 −0.161688 −0.0808441 0.996727i \(-0.525762\pi\)
−0.0808441 + 0.996727i \(0.525762\pi\)
\(390\) 0 0
\(391\) −2.38413e35 −2.41514
\(392\) 2.13185e33 0.0208638
\(393\) 0 0
\(394\) 2.90762e31 0.000265667 0
\(395\) −9.57447e31 −0.000845384 0
\(396\) 0 0
\(397\) 4.07374e34 0.335986 0.167993 0.985788i \(-0.446271\pi\)
0.167993 + 0.985788i \(0.446271\pi\)
\(398\) 8.96920e32 0.00715043
\(399\) 0 0
\(400\) −1.13245e35 −0.843744
\(401\) 1.64933e35 1.18812 0.594058 0.804422i \(-0.297525\pi\)
0.594058 + 0.804422i \(0.297525\pi\)
\(402\) 0 0
\(403\) 1.31211e35 0.883796
\(404\) −1.32968e35 −0.866162
\(405\) 0 0
\(406\) 7.92542e31 0.000482971 0
\(407\) −9.61295e34 −0.566672
\(408\) 0 0
\(409\) −2.13655e35 −1.17882 −0.589411 0.807833i \(-0.700640\pi\)
−0.589411 + 0.807833i \(0.700640\pi\)
\(410\) −8.69663e32 −0.00464268
\(411\) 0 0
\(412\) 2.70045e35 1.34997
\(413\) 1.21907e34 0.0589798
\(414\) 0 0
\(415\) 7.01151e34 0.317806
\(416\) 7.93786e33 0.0348292
\(417\) 0 0
\(418\) 1.53747e32 0.000632306 0
\(419\) −6.25908e34 −0.249242 −0.124621 0.992204i \(-0.539772\pi\)
−0.124621 + 0.992204i \(0.539772\pi\)
\(420\) 0 0
\(421\) 1.62873e35 0.608191 0.304096 0.952641i \(-0.401646\pi\)
0.304096 + 0.952641i \(0.401646\pi\)
\(422\) −3.77648e33 −0.0136574
\(423\) 0 0
\(424\) 6.94704e33 0.0235701
\(425\) 4.40475e35 1.44767
\(426\) 0 0
\(427\) −9.30171e33 −0.0286937
\(428\) 2.99567e33 0.00895372
\(429\) 0 0
\(430\) 1.34813e32 0.000378363 0
\(431\) 6.21025e35 1.68914 0.844571 0.535444i \(-0.179856\pi\)
0.844571 + 0.535444i \(0.179856\pi\)
\(432\) 0 0
\(433\) 1.04792e35 0.267757 0.133878 0.990998i \(-0.457257\pi\)
0.133878 + 0.990998i \(0.457257\pi\)
\(434\) −1.49206e32 −0.000369550 0
\(435\) 0 0
\(436\) −7.25125e35 −1.68789
\(437\) 8.54420e34 0.192829
\(438\) 0 0
\(439\) 6.93788e35 1.47216 0.736081 0.676893i \(-0.236674\pi\)
0.736081 + 0.676893i \(0.236674\pi\)
\(440\) −1.77228e33 −0.00364687
\(441\) 0 0
\(442\) −1.02897e34 −0.0199161
\(443\) −5.15844e35 −0.968430 −0.484215 0.874949i \(-0.660895\pi\)
−0.484215 + 0.874949i \(0.660895\pi\)
\(444\) 0 0
\(445\) −3.65038e35 −0.644878
\(446\) −2.74829e33 −0.00471023
\(447\) 0 0
\(448\) 2.75222e34 0.0444048
\(449\) −9.16926e35 −1.43552 −0.717759 0.696292i \(-0.754832\pi\)
−0.717759 + 0.696292i \(0.754832\pi\)
\(450\) 0 0
\(451\) −3.36904e35 −0.496732
\(452\) −4.98447e35 −0.713264
\(453\) 0 0
\(454\) −1.01957e34 −0.0137455
\(455\) −1.48951e34 −0.0194934
\(456\) 0 0
\(457\) 1.55254e36 1.91502 0.957510 0.288400i \(-0.0931232\pi\)
0.957510 + 0.288400i \(0.0931232\pi\)
\(458\) 4.96382e33 0.00594473
\(459\) 0 0
\(460\) −4.92428e35 −0.556048
\(461\) 5.06840e35 0.555787 0.277893 0.960612i \(-0.410364\pi\)
0.277893 + 0.960612i \(0.410364\pi\)
\(462\) 0 0
\(463\) −1.45476e36 −1.50469 −0.752347 0.658767i \(-0.771078\pi\)
−0.752347 + 0.658767i \(0.771078\pi\)
\(464\) −1.03469e36 −1.03948
\(465\) 0 0
\(466\) −8.46559e33 −0.00802504
\(467\) 6.89399e35 0.634881 0.317441 0.948278i \(-0.397177\pi\)
0.317441 + 0.948278i \(0.397177\pi\)
\(468\) 0 0
\(469\) −1.92076e34 −0.0166971
\(470\) −8.39617e33 −0.00709186
\(471\) 0 0
\(472\) 3.47903e34 0.0277486
\(473\) 5.22261e34 0.0404820
\(474\) 0 0
\(475\) −1.57857e35 −0.115585
\(476\) −1.07073e35 −0.0762053
\(477\) 0 0
\(478\) 2.28973e34 0.0153995
\(479\) 6.34595e35 0.414923 0.207461 0.978243i \(-0.433480\pi\)
0.207461 + 0.978243i \(0.433480\pi\)
\(480\) 0 0
\(481\) −2.30566e36 −1.42507
\(482\) 1.23242e34 0.00740670
\(483\) 0 0
\(484\) 1.41613e36 0.804807
\(485\) −4.48592e35 −0.247935
\(486\) 0 0
\(487\) 2.48287e36 1.29812 0.649058 0.760739i \(-0.275163\pi\)
0.649058 + 0.760739i \(0.275163\pi\)
\(488\) −2.65456e34 −0.0134997
\(489\) 0 0
\(490\) −8.56188e33 −0.00412024
\(491\) 2.37415e36 1.11150 0.555748 0.831351i \(-0.312432\pi\)
0.555748 + 0.831351i \(0.312432\pi\)
\(492\) 0 0
\(493\) 4.02449e36 1.78352
\(494\) 3.68762e33 0.00159013
\(495\) 0 0
\(496\) 1.94793e36 0.795371
\(497\) −3.62837e34 −0.0144178
\(498\) 0 0
\(499\) −5.00732e36 −1.88472 −0.942362 0.334595i \(-0.891400\pi\)
−0.942362 + 0.334595i \(0.891400\pi\)
\(500\) 1.98768e36 0.728203
\(501\) 0 0
\(502\) 2.72625e34 0.00946380
\(503\) 1.61560e36 0.545966 0.272983 0.962019i \(-0.411990\pi\)
0.272983 + 0.962019i \(0.411990\pi\)
\(504\) 0 0
\(505\) 1.06811e36 0.342122
\(506\) 2.08467e34 0.00650138
\(507\) 0 0
\(508\) −2.87094e36 −0.848914
\(509\) −1.80638e36 −0.520139 −0.260069 0.965590i \(-0.583745\pi\)
−0.260069 + 0.965590i \(0.583745\pi\)
\(510\) 0 0
\(511\) −2.39031e35 −0.0652787
\(512\) 1.96413e35 0.0522428
\(513\) 0 0
\(514\) −1.55386e34 −0.00392111
\(515\) −2.16922e36 −0.533219
\(516\) 0 0
\(517\) −3.25264e36 −0.758776
\(518\) 2.62187e33 0.000595880 0
\(519\) 0 0
\(520\) −4.25081e34 −0.00917120
\(521\) −4.59028e36 −0.965006 −0.482503 0.875894i \(-0.660272\pi\)
−0.482503 + 0.875894i \(0.660272\pi\)
\(522\) 0 0
\(523\) 5.82915e35 0.116368 0.0581838 0.998306i \(-0.481469\pi\)
0.0581838 + 0.998306i \(0.481469\pi\)
\(524\) 6.90387e36 1.34314
\(525\) 0 0
\(526\) 7.47617e34 0.0138157
\(527\) −7.57658e36 −1.36468
\(528\) 0 0
\(529\) 5.74195e36 0.982670
\(530\) −2.79006e34 −0.00465467
\(531\) 0 0
\(532\) 3.83726e34 0.00608435
\(533\) −8.08063e36 −1.24919
\(534\) 0 0
\(535\) −2.40636e34 −0.00353659
\(536\) −5.48154e34 −0.00785558
\(537\) 0 0
\(538\) 2.06757e33 0.000281773 0
\(539\) −3.31684e36 −0.440835
\(540\) 0 0
\(541\) −6.32438e36 −0.799567 −0.399783 0.916610i \(-0.630915\pi\)
−0.399783 + 0.916610i \(0.630915\pi\)
\(542\) −1.29227e35 −0.0159354
\(543\) 0 0
\(544\) −4.58360e35 −0.0537801
\(545\) 5.82478e36 0.666695
\(546\) 0 0
\(547\) −9.05384e36 −0.986289 −0.493144 0.869948i \(-0.664153\pi\)
−0.493144 + 0.869948i \(0.664153\pi\)
\(548\) 4.99192e36 0.530554
\(549\) 0 0
\(550\) −3.85150e34 −0.00389702
\(551\) −1.44229e36 −0.142399
\(552\) 0 0
\(553\) 1.01164e33 9.51118e−5 0
\(554\) 7.15436e34 0.00656431
\(555\) 0 0
\(556\) 1.34913e37 1.17908
\(557\) 1.74285e37 1.48668 0.743338 0.668917i \(-0.233242\pi\)
0.743338 + 0.668917i \(0.233242\pi\)
\(558\) 0 0
\(559\) 1.25264e36 0.101805
\(560\) −2.21129e35 −0.0175431
\(561\) 0 0
\(562\) −4.23289e34 −0.00320034
\(563\) 7.81650e36 0.576964 0.288482 0.957485i \(-0.406849\pi\)
0.288482 + 0.957485i \(0.406849\pi\)
\(564\) 0 0
\(565\) 4.00393e36 0.281729
\(566\) −1.26065e34 −0.000866109 0
\(567\) 0 0
\(568\) −1.03548e35 −0.00678325
\(569\) 1.42845e37 0.913797 0.456899 0.889519i \(-0.348960\pi\)
0.456899 + 0.889519i \(0.348960\pi\)
\(570\) 0 0
\(571\) 2.26975e37 1.38481 0.692406 0.721508i \(-0.256551\pi\)
0.692406 + 0.721508i \(0.256551\pi\)
\(572\) −8.23327e36 −0.490598
\(573\) 0 0
\(574\) 9.18883e33 0.000522336 0
\(575\) −2.14040e37 −1.18844
\(576\) 0 0
\(577\) −1.49672e37 −0.792987 −0.396494 0.918038i \(-0.629773\pi\)
−0.396494 + 0.918038i \(0.629773\pi\)
\(578\) 3.92203e35 0.0202995
\(579\) 0 0
\(580\) 8.31236e36 0.410626
\(581\) −7.40834e35 −0.0357555
\(582\) 0 0
\(583\) −1.08086e37 −0.498015
\(584\) −6.82154e35 −0.0307121
\(585\) 0 0
\(586\) 5.53869e34 0.00238116
\(587\) 7.65028e36 0.321413 0.160706 0.987002i \(-0.448623\pi\)
0.160706 + 0.987002i \(0.448623\pi\)
\(588\) 0 0
\(589\) 2.71529e36 0.108958
\(590\) −1.39724e35 −0.00547985
\(591\) 0 0
\(592\) −3.42293e37 −1.28249
\(593\) 1.64866e37 0.603799 0.301900 0.953340i \(-0.402379\pi\)
0.301900 + 0.953340i \(0.402379\pi\)
\(594\) 0 0
\(595\) 8.60094e35 0.0301000
\(596\) −2.76717e37 −0.946698
\(597\) 0 0
\(598\) 5.00008e35 0.0163497
\(599\) 5.74500e36 0.183665 0.0918327 0.995774i \(-0.470727\pi\)
0.0918327 + 0.995774i \(0.470727\pi\)
\(600\) 0 0
\(601\) −5.09841e36 −0.155822 −0.0779112 0.996960i \(-0.524825\pi\)
−0.0779112 + 0.996960i \(0.524825\pi\)
\(602\) −1.42443e33 −4.25686e−5 0
\(603\) 0 0
\(604\) −1.17274e37 −0.335123
\(605\) −1.13755e37 −0.317888
\(606\) 0 0
\(607\) 1.21769e37 0.325453 0.162727 0.986671i \(-0.447971\pi\)
0.162727 + 0.986671i \(0.447971\pi\)
\(608\) 1.64267e35 0.00429389
\(609\) 0 0
\(610\) 1.06612e35 0.00266595
\(611\) −7.80145e37 −1.90818
\(612\) 0 0
\(613\) 2.55443e37 0.597829 0.298915 0.954280i \(-0.403375\pi\)
0.298915 + 0.954280i \(0.403375\pi\)
\(614\) 3.03464e35 0.00694759
\(615\) 0 0
\(616\) 1.87258e34 0.000410300 0
\(617\) −1.30077e37 −0.278838 −0.139419 0.990234i \(-0.544523\pi\)
−0.139419 + 0.990234i \(0.544523\pi\)
\(618\) 0 0
\(619\) −2.04833e37 −0.420316 −0.210158 0.977667i \(-0.567398\pi\)
−0.210158 + 0.977667i \(0.567398\pi\)
\(620\) −1.56490e37 −0.314195
\(621\) 0 0
\(622\) 4.95433e35 0.00952389
\(623\) 3.85698e36 0.0725535
\(624\) 0 0
\(625\) 3.08859e37 0.556390
\(626\) 6.74931e35 0.0118989
\(627\) 0 0
\(628\) −4.92052e37 −0.830913
\(629\) 1.33137e38 2.20047
\(630\) 0 0
\(631\) 7.16988e37 1.13531 0.567656 0.823266i \(-0.307850\pi\)
0.567656 + 0.823266i \(0.307850\pi\)
\(632\) 2.88704e33 4.47478e−5 0
\(633\) 0 0
\(634\) −7.53957e35 −0.0111980
\(635\) 2.30617e37 0.335309
\(636\) 0 0
\(637\) −7.95543e37 −1.10862
\(638\) −3.51900e35 −0.00480109
\(639\) 0 0
\(640\) −1.26227e36 −0.0165090
\(641\) 1.00079e38 1.28161 0.640807 0.767702i \(-0.278600\pi\)
0.640807 + 0.767702i \(0.278600\pi\)
\(642\) 0 0
\(643\) 7.58172e37 0.930936 0.465468 0.885065i \(-0.345886\pi\)
0.465468 + 0.885065i \(0.345886\pi\)
\(644\) 5.20298e36 0.0625594
\(645\) 0 0
\(646\) −2.12936e35 −0.00245534
\(647\) 7.13094e37 0.805265 0.402632 0.915362i \(-0.368095\pi\)
0.402632 + 0.915362i \(0.368095\pi\)
\(648\) 0 0
\(649\) −5.41285e37 −0.586303
\(650\) −9.23781e35 −0.00980028
\(651\) 0 0
\(652\) 1.25243e38 1.27471
\(653\) −2.47590e37 −0.246835 −0.123417 0.992355i \(-0.539385\pi\)
−0.123417 + 0.992355i \(0.539385\pi\)
\(654\) 0 0
\(655\) −5.54574e37 −0.530520
\(656\) −1.19963e38 −1.12421
\(657\) 0 0
\(658\) 8.87136e34 0.000797886 0
\(659\) −1.51836e38 −1.33789 −0.668946 0.743311i \(-0.733254\pi\)
−0.668946 + 0.743311i \(0.733254\pi\)
\(660\) 0 0
\(661\) 1.92148e38 1.62523 0.812617 0.582798i \(-0.198042\pi\)
0.812617 + 0.582798i \(0.198042\pi\)
\(662\) −8.10232e35 −0.00671470
\(663\) 0 0
\(664\) −2.11422e36 −0.0168221
\(665\) −3.08239e35 −0.00240323
\(666\) 0 0
\(667\) −1.95562e38 −1.46415
\(668\) 4.65436e37 0.341490
\(669\) 0 0
\(670\) 2.20149e35 0.00155134
\(671\) 4.13009e37 0.285237
\(672\) 0 0
\(673\) −9.37859e37 −0.622206 −0.311103 0.950376i \(-0.600698\pi\)
−0.311103 + 0.950376i \(0.600698\pi\)
\(674\) −1.02371e36 −0.00665685
\(675\) 0 0
\(676\) −3.74333e37 −0.233872
\(677\) 2.93781e37 0.179919 0.0899595 0.995945i \(-0.471326\pi\)
0.0899595 + 0.995945i \(0.471326\pi\)
\(678\) 0 0
\(679\) 4.73981e36 0.0278945
\(680\) 2.45457e36 0.0141613
\(681\) 0 0
\(682\) 6.62494e35 0.00367361
\(683\) −1.68599e38 −0.916591 −0.458295 0.888800i \(-0.651540\pi\)
−0.458295 + 0.888800i \(0.651540\pi\)
\(684\) 0 0
\(685\) −4.00991e37 −0.209562
\(686\) 1.81108e35 0.000928031 0
\(687\) 0 0
\(688\) 1.85964e37 0.0916190
\(689\) −2.59243e38 −1.25241
\(690\) 0 0
\(691\) −3.72820e38 −1.73199 −0.865996 0.500051i \(-0.833314\pi\)
−0.865996 + 0.500051i \(0.833314\pi\)
\(692\) 3.56716e38 1.62514
\(693\) 0 0
\(694\) −4.29252e36 −0.0188088
\(695\) −1.08373e38 −0.465720
\(696\) 0 0
\(697\) 4.66604e38 1.92888
\(698\) 1.97084e36 0.00799102
\(699\) 0 0
\(700\) −9.61267e36 −0.0374991
\(701\) −3.38534e38 −1.29541 −0.647707 0.761890i \(-0.724272\pi\)
−0.647707 + 0.761890i \(0.724272\pi\)
\(702\) 0 0
\(703\) −4.77135e37 −0.175689
\(704\) −1.22203e38 −0.441417
\(705\) 0 0
\(706\) 8.09623e35 0.00281462
\(707\) −1.12856e37 −0.0384912
\(708\) 0 0
\(709\) 2.34891e38 0.771155 0.385578 0.922675i \(-0.374002\pi\)
0.385578 + 0.922675i \(0.374002\pi\)
\(710\) 4.15866e35 0.00133957
\(711\) 0 0
\(712\) 1.10072e37 0.0341347
\(713\) 3.68168e38 1.12031
\(714\) 0 0
\(715\) 6.61362e37 0.193779
\(716\) 5.78291e38 1.66273
\(717\) 0 0
\(718\) 9.45165e35 0.00261714
\(719\) 1.57531e38 0.428082 0.214041 0.976825i \(-0.431337\pi\)
0.214041 + 0.976825i \(0.431337\pi\)
\(720\) 0 0
\(721\) 2.29199e37 0.0599911
\(722\) −3.99279e36 −0.0102571
\(723\) 0 0
\(724\) 7.46707e37 0.184790
\(725\) 3.61306e38 0.877632
\(726\) 0 0
\(727\) −2.38862e38 −0.559027 −0.279513 0.960142i \(-0.590173\pi\)
−0.279513 + 0.960142i \(0.590173\pi\)
\(728\) 4.49139e35 0.00103183
\(729\) 0 0
\(730\) 2.73965e36 0.00606509
\(731\) −7.23320e37 −0.157198
\(732\) 0 0
\(733\) −1.39015e38 −0.291177 −0.145589 0.989345i \(-0.546508\pi\)
−0.145589 + 0.989345i \(0.546508\pi\)
\(734\) 8.01229e36 0.0164763
\(735\) 0 0
\(736\) 2.22731e37 0.0441499
\(737\) 8.52846e37 0.165981
\(738\) 0 0
\(739\) 3.01753e37 0.0566176 0.0283088 0.999599i \(-0.490988\pi\)
0.0283088 + 0.999599i \(0.490988\pi\)
\(740\) 2.74987e38 0.506623
\(741\) 0 0
\(742\) 2.94797e35 0.000523685 0
\(743\) −8.48049e38 −1.47935 −0.739677 0.672962i \(-0.765022\pi\)
−0.739677 + 0.672962i \(0.765022\pi\)
\(744\) 0 0
\(745\) 2.22281e38 0.373932
\(746\) −8.22856e36 −0.0135941
\(747\) 0 0
\(748\) 4.75418e38 0.757538
\(749\) 2.54256e35 0.000397893 0
\(750\) 0 0
\(751\) −1.00379e39 −1.51532 −0.757658 0.652651i \(-0.773657\pi\)
−0.757658 + 0.652651i \(0.773657\pi\)
\(752\) −1.15819e39 −1.71726
\(753\) 0 0
\(754\) −8.44031e36 −0.0120738
\(755\) 9.42041e37 0.132369
\(756\) 0 0
\(757\) 1.01436e39 1.37530 0.687652 0.726040i \(-0.258641\pi\)
0.687652 + 0.726040i \(0.258641\pi\)
\(758\) 9.67114e36 0.0128808
\(759\) 0 0
\(760\) −8.79664e35 −0.00113066
\(761\) 1.18512e39 1.49648 0.748240 0.663428i \(-0.230899\pi\)
0.748240 + 0.663428i \(0.230899\pi\)
\(762\) 0 0
\(763\) −6.15445e37 −0.0750080
\(764\) −5.62940e38 −0.674065
\(765\) 0 0
\(766\) −1.44152e37 −0.0166622
\(767\) −1.29827e39 −1.47444
\(768\) 0 0
\(769\) −1.38650e39 −1.52024 −0.760122 0.649780i \(-0.774861\pi\)
−0.760122 + 0.649780i \(0.774861\pi\)
\(770\) −7.52063e34 −8.10269e−5 0
\(771\) 0 0
\(772\) −6.51727e36 −0.00678004
\(773\) 3.84501e38 0.393074 0.196537 0.980496i \(-0.437030\pi\)
0.196537 + 0.980496i \(0.437030\pi\)
\(774\) 0 0
\(775\) −6.80202e38 −0.671530
\(776\) 1.35266e37 0.0131237
\(777\) 0 0
\(778\) −1.80364e36 −0.00169015
\(779\) −1.67221e38 −0.154005
\(780\) 0 0
\(781\) 1.61105e38 0.143324
\(782\) −2.88722e37 −0.0252458
\(783\) 0 0
\(784\) −1.18104e39 −0.997698
\(785\) 3.95255e38 0.328199
\(786\) 0 0
\(787\) 8.31177e38 0.666860 0.333430 0.942775i \(-0.391794\pi\)
0.333430 + 0.942775i \(0.391794\pi\)
\(788\) −3.22217e37 −0.0254123
\(789\) 0 0
\(790\) −1.15949e34 −8.83690e−6 0
\(791\) −4.23054e37 −0.0316966
\(792\) 0 0
\(793\) 9.90602e38 0.717317
\(794\) 4.93337e36 0.00351210
\(795\) 0 0
\(796\) −9.93951e38 −0.683973
\(797\) 5.86889e38 0.397071 0.198536 0.980094i \(-0.436381\pi\)
0.198536 + 0.980094i \(0.436381\pi\)
\(798\) 0 0
\(799\) 4.50484e39 2.94644
\(800\) −4.11502e37 −0.0264641
\(801\) 0 0
\(802\) 1.99737e37 0.0124195
\(803\) 1.06133e39 0.648919
\(804\) 0 0
\(805\) −4.17945e37 −0.0247101
\(806\) 1.58899e37 0.00923843
\(807\) 0 0
\(808\) −3.22072e37 −0.0181092
\(809\) 2.02827e39 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(810\) 0 0
\(811\) −1.61063e39 −0.861421 −0.430710 0.902490i \(-0.641737\pi\)
−0.430710 + 0.902490i \(0.641737\pi\)
\(812\) −8.78281e37 −0.0461984
\(813\) 0 0
\(814\) −1.16415e37 −0.00592350
\(815\) −1.00605e39 −0.503492
\(816\) 0 0
\(817\) 2.59222e37 0.0125509
\(818\) −2.58740e37 −0.0123224
\(819\) 0 0
\(820\) 9.63745e38 0.444094
\(821\) 2.30031e39 1.04268 0.521342 0.853348i \(-0.325431\pi\)
0.521342 + 0.853348i \(0.325431\pi\)
\(822\) 0 0
\(823\) −2.08776e39 −0.915764 −0.457882 0.889013i \(-0.651392\pi\)
−0.457882 + 0.889013i \(0.651392\pi\)
\(824\) 6.54096e37 0.0282244
\(825\) 0 0
\(826\) 1.47632e36 0.000616523 0
\(827\) −3.43290e39 −1.41038 −0.705192 0.709017i \(-0.749139\pi\)
−0.705192 + 0.709017i \(0.749139\pi\)
\(828\) 0 0
\(829\) 8.03456e38 0.319504 0.159752 0.987157i \(-0.448931\pi\)
0.159752 + 0.987157i \(0.448931\pi\)
\(830\) 8.49107e36 0.00332207
\(831\) 0 0
\(832\) −2.93103e39 −1.11008
\(833\) 4.59375e39 1.71183
\(834\) 0 0
\(835\) −3.73875e38 −0.134884
\(836\) −1.70380e38 −0.0604830
\(837\) 0 0
\(838\) −7.57987e36 −0.00260536
\(839\) −2.54669e39 −0.861369 −0.430685 0.902502i \(-0.641728\pi\)
−0.430685 + 0.902502i \(0.641728\pi\)
\(840\) 0 0
\(841\) 2.48015e38 0.0812329
\(842\) 1.97242e37 0.00635750
\(843\) 0 0
\(844\) 4.18503e39 1.30640
\(845\) 3.00694e38 0.0923760
\(846\) 0 0
\(847\) 1.20193e38 0.0357647
\(848\) −3.84867e39 −1.12711
\(849\) 0 0
\(850\) 5.33424e37 0.0151327
\(851\) −6.46952e39 −1.80644
\(852\) 0 0
\(853\) 1.61849e39 0.437823 0.218912 0.975745i \(-0.429749\pi\)
0.218912 + 0.975745i \(0.429749\pi\)
\(854\) −1.12646e36 −0.000299939 0
\(855\) 0 0
\(856\) 7.25603e35 0.000187199 0
\(857\) −1.79678e38 −0.0456304 −0.0228152 0.999740i \(-0.507263\pi\)
−0.0228152 + 0.999740i \(0.507263\pi\)
\(858\) 0 0
\(859\) −6.83644e39 −1.68237 −0.841187 0.540744i \(-0.818143\pi\)
−0.841187 + 0.540744i \(0.818143\pi\)
\(860\) −1.49398e38 −0.0361922
\(861\) 0 0
\(862\) 7.52074e37 0.0176568
\(863\) 8.93182e37 0.0206440 0.0103220 0.999947i \(-0.496714\pi\)
0.0103220 + 0.999947i \(0.496714\pi\)
\(864\) 0 0
\(865\) −2.86543e39 −0.641907
\(866\) 1.26905e37 0.00279890
\(867\) 0 0
\(868\) 1.65347e38 0.0353492
\(869\) −4.49180e36 −0.000945483 0
\(870\) 0 0
\(871\) 2.04555e39 0.417412
\(872\) −1.75638e38 −0.0352895
\(873\) 0 0
\(874\) 1.03472e37 0.00201566
\(875\) 1.68703e38 0.0323605
\(876\) 0 0
\(877\) −4.74244e39 −0.882078 −0.441039 0.897488i \(-0.645390\pi\)
−0.441039 + 0.897488i \(0.645390\pi\)
\(878\) 8.40191e37 0.0153887
\(879\) 0 0
\(880\) 9.81843e38 0.174392
\(881\) 4.04555e39 0.707624 0.353812 0.935317i \(-0.384885\pi\)
0.353812 + 0.935317i \(0.384885\pi\)
\(882\) 0 0
\(883\) 7.15911e39 1.21448 0.607239 0.794519i \(-0.292277\pi\)
0.607239 + 0.794519i \(0.292277\pi\)
\(884\) 1.14029e40 1.90507
\(885\) 0 0
\(886\) −6.24697e37 −0.0101231
\(887\) 3.43934e39 0.548916 0.274458 0.961599i \(-0.411502\pi\)
0.274458 + 0.961599i \(0.411502\pi\)
\(888\) 0 0
\(889\) −2.43669e38 −0.0377247
\(890\) −4.42068e37 −0.00674099
\(891\) 0 0
\(892\) 3.04561e39 0.450556
\(893\) −1.61444e39 −0.235248
\(894\) 0 0
\(895\) −4.64529e39 −0.656753
\(896\) 1.33371e37 0.00185739
\(897\) 0 0
\(898\) −1.11041e38 −0.0150056
\(899\) −6.21481e39 −0.827317
\(900\) 0 0
\(901\) 1.49696e40 1.93387
\(902\) −4.07997e37 −0.00519241
\(903\) 0 0
\(904\) −1.20732e38 −0.0149125
\(905\) −5.99814e38 −0.0729896
\(906\) 0 0
\(907\) 2.71200e38 0.0320325 0.0160163 0.999872i \(-0.494902\pi\)
0.0160163 + 0.999872i \(0.494902\pi\)
\(908\) 1.12987e40 1.31482
\(909\) 0 0
\(910\) −1.80382e36 −0.000203767 0
\(911\) −1.18591e40 −1.31994 −0.659970 0.751292i \(-0.729431\pi\)
−0.659970 + 0.751292i \(0.729431\pi\)
\(912\) 0 0
\(913\) 3.28941e39 0.355436
\(914\) 1.88015e38 0.0200179
\(915\) 0 0
\(916\) −5.50082e39 −0.568641
\(917\) 5.85961e38 0.0596874
\(918\) 0 0
\(919\) −1.70613e40 −1.68753 −0.843766 0.536711i \(-0.819667\pi\)
−0.843766 + 0.536711i \(0.819667\pi\)
\(920\) −1.19274e38 −0.0116255
\(921\) 0 0
\(922\) 6.13793e37 0.00580971
\(923\) 3.86409e39 0.360433
\(924\) 0 0
\(925\) 1.19526e40 1.08281
\(926\) −1.76174e38 −0.0157287
\(927\) 0 0
\(928\) −3.75977e38 −0.0326035
\(929\) 8.89239e39 0.759987 0.379994 0.924989i \(-0.375926\pi\)
0.379994 + 0.924989i \(0.375926\pi\)
\(930\) 0 0
\(931\) −1.64630e39 −0.136675
\(932\) 9.38141e39 0.767633
\(933\) 0 0
\(934\) 8.34875e37 0.00663649
\(935\) −3.81894e39 −0.299217
\(936\) 0 0
\(937\) −1.80674e40 −1.37534 −0.687670 0.726023i \(-0.741367\pi\)
−0.687670 + 0.726023i \(0.741367\pi\)
\(938\) −2.32608e36 −0.000174537 0
\(939\) 0 0
\(940\) 9.30449e39 0.678370
\(941\) −2.62451e40 −1.88620 −0.943101 0.332507i \(-0.892105\pi\)
−0.943101 + 0.332507i \(0.892105\pi\)
\(942\) 0 0
\(943\) −2.26737e40 −1.58348
\(944\) −1.92738e40 −1.32692
\(945\) 0 0
\(946\) 6.32468e36 0.000423163 0
\(947\) 6.71744e39 0.443077 0.221539 0.975152i \(-0.428892\pi\)
0.221539 + 0.975152i \(0.428892\pi\)
\(948\) 0 0
\(949\) 2.54560e40 1.63191
\(950\) −1.91168e37 −0.00120822
\(951\) 0 0
\(952\) −2.59349e37 −0.00159325
\(953\) −1.95322e39 −0.118303 −0.0591516 0.998249i \(-0.518840\pi\)
−0.0591516 + 0.998249i \(0.518840\pi\)
\(954\) 0 0
\(955\) 4.52199e39 0.266246
\(956\) −2.53743e40 −1.47304
\(957\) 0 0
\(958\) 7.68507e37 0.00433724
\(959\) 4.23686e38 0.0235772
\(960\) 0 0
\(961\) −6.78259e39 −0.366969
\(962\) −2.79220e38 −0.0148965
\(963\) 0 0
\(964\) −1.36575e40 −0.708486
\(965\) 5.23519e37 0.00267802
\(966\) 0 0
\(967\) 1.48699e40 0.739690 0.369845 0.929093i \(-0.379411\pi\)
0.369845 + 0.929093i \(0.379411\pi\)
\(968\) 3.43012e38 0.0168264
\(969\) 0 0
\(970\) −5.43254e37 −0.00259170
\(971\) 1.17711e40 0.553804 0.276902 0.960898i \(-0.410692\pi\)
0.276902 + 0.960898i \(0.410692\pi\)
\(972\) 0 0
\(973\) 1.14506e39 0.0523970
\(974\) 3.00681e38 0.0135694
\(975\) 0 0
\(976\) 1.47062e40 0.645549
\(977\) 3.52154e40 1.52460 0.762300 0.647224i \(-0.224070\pi\)
0.762300 + 0.647224i \(0.224070\pi\)
\(978\) 0 0
\(979\) −1.71255e40 −0.721236
\(980\) 9.48813e39 0.394120
\(981\) 0 0
\(982\) 2.87514e38 0.0116186
\(983\) 3.23914e40 1.29109 0.645547 0.763721i \(-0.276629\pi\)
0.645547 + 0.763721i \(0.276629\pi\)
\(984\) 0 0
\(985\) 2.58830e38 0.0100375
\(986\) 4.87374e38 0.0186433
\(987\) 0 0
\(988\) −4.08656e39 −0.152103
\(989\) 3.51482e39 0.129049
\(990\) 0 0
\(991\) 5.97175e38 0.0213357 0.0106679 0.999943i \(-0.496604\pi\)
0.0106679 + 0.999943i \(0.496604\pi\)
\(992\) 7.07822e38 0.0249469
\(993\) 0 0
\(994\) −4.39403e36 −0.000150712 0
\(995\) 7.98421e39 0.270160
\(996\) 0 0
\(997\) −1.51823e40 −0.499980 −0.249990 0.968248i \(-0.580427\pi\)
−0.249990 + 0.968248i \(0.580427\pi\)
\(998\) −6.06396e38 −0.0197013
\(999\) 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 9.28.a.b.1.2 2
3.2 odd 2 3.28.a.b.1.1 2
12.11 even 2 48.28.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.28.a.b.1.1 2 3.2 odd 2
9.28.a.b.1.2 2 1.1 even 1 trivial
48.28.a.e.1.1 2 12.11 even 2