Properties

Label 10.28.a.a.1.1
Level $10$
Weight $28$
Character 10.1
Self dual yes
Analytic conductor $46.186$
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] = [10,28,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, 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("10.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(46.1855574838\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 19551870 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4422.25\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8192.00 q^{2} -3.52746e6 q^{3} +6.71089e7 q^{4} +1.22070e9 q^{5} +2.88969e10 q^{6} -2.10875e11 q^{7} -5.49756e11 q^{8} +4.81735e12 q^{9} -1.00000e13 q^{10} -2.00632e14 q^{11} -2.36724e14 q^{12} +1.66667e15 q^{13} +1.72749e15 q^{14} -4.30598e15 q^{15} +4.50360e15 q^{16} +4.72049e16 q^{17} -3.94638e16 q^{18} +2.80684e17 q^{19} +8.19200e16 q^{20} +7.43854e17 q^{21} +1.64358e18 q^{22} -3.44510e18 q^{23} +1.93924e18 q^{24} +1.49012e18 q^{25} -1.36534e19 q^{26} +9.90596e18 q^{27} -1.41516e19 q^{28} +5.38410e19 q^{29} +3.52746e19 q^{30} -6.02019e19 q^{31} -3.68935e19 q^{32} +7.07721e20 q^{33} -3.86703e20 q^{34} -2.57416e20 q^{35} +3.23287e20 q^{36} +2.15489e21 q^{37} -2.29937e21 q^{38} -5.87910e21 q^{39} -6.71089e20 q^{40} -5.73864e21 q^{41} -6.09365e21 q^{42} +4.73192e20 q^{43} -1.34642e22 q^{44} +5.88056e21 q^{45} +2.82223e22 q^{46} -1.36579e22 q^{47} -1.58863e22 q^{48} -2.12439e22 q^{49} -1.22070e22 q^{50} -1.66513e23 q^{51} +1.11848e23 q^{52} +1.17314e23 q^{53} -8.11496e22 q^{54} -2.44912e23 q^{55} +1.15930e23 q^{56} -9.90101e23 q^{57} -4.41066e23 q^{58} -6.96143e23 q^{59} -2.88969e23 q^{60} -1.66138e24 q^{61} +4.93174e23 q^{62} -1.01586e24 q^{63} +3.02231e23 q^{64} +2.03451e24 q^{65} -5.79765e24 q^{66} -6.87655e22 q^{67} +3.16787e24 q^{68} +1.21524e25 q^{69} +2.10875e24 q^{70} -1.69024e24 q^{71} -2.64837e24 q^{72} -5.53884e24 q^{73} -1.76529e25 q^{74} -5.25632e24 q^{75} +1.88364e25 q^{76} +4.23083e25 q^{77} +4.81616e25 q^{78} -5.02072e24 q^{79} +5.49756e24 q^{80} -7.16780e25 q^{81} +4.70109e25 q^{82} +1.56395e26 q^{83} +4.99192e25 q^{84} +5.76232e25 q^{85} -3.87639e24 q^{86} -1.89922e26 q^{87} +1.10299e26 q^{88} -6.32996e25 q^{89} -4.81735e25 q^{90} -3.51459e26 q^{91} -2.31197e26 q^{92} +2.12360e26 q^{93} +1.11886e26 q^{94} +3.42632e26 q^{95} +1.30140e26 q^{96} -5.95949e26 q^{97} +1.74030e26 q^{98} -9.66515e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16384 q^{2} - 3340644 q^{3} + 134217728 q^{4} + 2441406250 q^{5} + 27366555648 q^{6} - 58706842292 q^{7} - 1099511627776 q^{8} - 2773343978406 q^{9} - 20000000000000 q^{10} - 148584207397296 q^{11}+ \cdots - 13\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8192.00 −0.707107
\(3\) −3.52746e6 −1.27739 −0.638697 0.769458i \(-0.720526\pi\)
−0.638697 + 0.769458i \(0.720526\pi\)
\(4\) 6.71089e7 0.500000
\(5\) 1.22070e9 0.447214
\(6\) 2.88969e10 0.903254
\(7\) −2.10875e11 −0.822626 −0.411313 0.911494i \(-0.634930\pi\)
−0.411313 + 0.911494i \(0.634930\pi\)
\(8\) −5.49756e11 −0.353553
\(9\) 4.81735e12 0.631735
\(10\) −1.00000e13 −0.316228
\(11\) −2.00632e14 −1.75226 −0.876131 0.482074i \(-0.839884\pi\)
−0.876131 + 0.482074i \(0.839884\pi\)
\(12\) −2.36724e14 −0.638697
\(13\) 1.66667e15 1.52621 0.763105 0.646275i \(-0.223674\pi\)
0.763105 + 0.646275i \(0.223674\pi\)
\(14\) 1.72749e15 0.581684
\(15\) −4.30598e15 −0.571268
\(16\) 4.50360e15 0.250000
\(17\) 4.72049e16 1.15592 0.577959 0.816066i \(-0.303849\pi\)
0.577959 + 0.816066i \(0.303849\pi\)
\(18\) −3.94638e16 −0.446704
\(19\) 2.80684e17 1.53124 0.765622 0.643291i \(-0.222431\pi\)
0.765622 + 0.643291i \(0.222431\pi\)
\(20\) 8.19200e16 0.223607
\(21\) 7.43854e17 1.05082
\(22\) 1.64358e18 1.23904
\(23\) −3.44510e18 −1.42520 −0.712601 0.701570i \(-0.752483\pi\)
−0.712601 + 0.701570i \(0.752483\pi\)
\(24\) 1.93924e18 0.451627
\(25\) 1.49012e18 0.200000
\(26\) −1.36534e19 −1.07919
\(27\) 9.90596e18 0.470420
\(28\) −1.41516e19 −0.411313
\(29\) 5.38410e19 0.974407 0.487204 0.873288i \(-0.338017\pi\)
0.487204 + 0.873288i \(0.338017\pi\)
\(30\) 3.52746e19 0.403947
\(31\) −6.02019e19 −0.442820 −0.221410 0.975181i \(-0.571066\pi\)
−0.221410 + 0.975181i \(0.571066\pi\)
\(32\) −3.68935e19 −0.176777
\(33\) 7.07721e20 2.23833
\(34\) −3.86703e20 −0.817357
\(35\) −2.57416e20 −0.367889
\(36\) 3.23287e20 0.315867
\(37\) 2.15489e21 1.45446 0.727231 0.686393i \(-0.240807\pi\)
0.727231 + 0.686393i \(0.240807\pi\)
\(38\) −2.29937e21 −1.08275
\(39\) −5.87910e21 −1.94957
\(40\) −6.71089e20 −0.158114
\(41\) −5.73864e21 −0.968784 −0.484392 0.874851i \(-0.660959\pi\)
−0.484392 + 0.874851i \(0.660959\pi\)
\(42\) −6.09365e21 −0.743040
\(43\) 4.73192e20 0.0419965 0.0209983 0.999780i \(-0.493316\pi\)
0.0209983 + 0.999780i \(0.493316\pi\)
\(44\) −1.34642e22 −0.876131
\(45\) 5.88056e21 0.282520
\(46\) 2.82223e22 1.00777
\(47\) −1.36579e22 −0.364808 −0.182404 0.983224i \(-0.558388\pi\)
−0.182404 + 0.983224i \(0.558388\pi\)
\(48\) −1.58863e22 −0.319348
\(49\) −2.12439e22 −0.323287
\(50\) −1.22070e22 −0.141421
\(51\) −1.66513e23 −1.47656
\(52\) 1.11848e23 0.763105
\(53\) 1.17314e23 0.618907 0.309454 0.950915i \(-0.399854\pi\)
0.309454 + 0.950915i \(0.399854\pi\)
\(54\) −8.11496e22 −0.332637
\(55\) −2.44912e23 −0.783635
\(56\) 1.15930e23 0.290842
\(57\) −9.90101e23 −1.95600
\(58\) −4.41066e23 −0.689010
\(59\) −6.96143e23 −0.863369 −0.431684 0.902025i \(-0.642081\pi\)
−0.431684 + 0.902025i \(0.642081\pi\)
\(60\) −2.88969e23 −0.285634
\(61\) −1.66138e24 −1.31376 −0.656882 0.753993i \(-0.728125\pi\)
−0.656882 + 0.753993i \(0.728125\pi\)
\(62\) 4.93174e23 0.313121
\(63\) −1.01586e24 −0.519681
\(64\) 3.02231e23 0.125000
\(65\) 2.03451e24 0.682541
\(66\) −5.79765e24 −1.58274
\(67\) −6.87655e22 −0.0153236 −0.00766180 0.999971i \(-0.502439\pi\)
−0.00766180 + 0.999971i \(0.502439\pi\)
\(68\) 3.16787e24 0.577959
\(69\) 1.21524e25 1.82054
\(70\) 2.10875e24 0.260137
\(71\) −1.69024e24 −0.172171 −0.0860856 0.996288i \(-0.527436\pi\)
−0.0860856 + 0.996288i \(0.527436\pi\)
\(72\) −2.64837e24 −0.223352
\(73\) −5.53884e24 −0.387758 −0.193879 0.981025i \(-0.562107\pi\)
−0.193879 + 0.981025i \(0.562107\pi\)
\(74\) −1.76529e25 −1.02846
\(75\) −5.25632e24 −0.255479
\(76\) 1.88364e25 0.765622
\(77\) 4.23083e25 1.44146
\(78\) 4.81616e25 1.37855
\(79\) −5.02072e24 −0.121004 −0.0605021 0.998168i \(-0.519270\pi\)
−0.0605021 + 0.998168i \(0.519270\pi\)
\(80\) 5.49756e24 0.111803
\(81\) −7.16780e25 −1.23265
\(82\) 4.70109e25 0.685034
\(83\) 1.56395e26 1.93494 0.967470 0.252988i \(-0.0814132\pi\)
0.967470 + 0.252988i \(0.0814132\pi\)
\(84\) 4.99192e25 0.525408
\(85\) 5.76232e25 0.516942
\(86\) −3.87639e24 −0.0296960
\(87\) −1.89922e26 −1.24470
\(88\) 1.10299e26 0.619518
\(89\) −6.32996e25 −0.305236 −0.152618 0.988285i \(-0.548770\pi\)
−0.152618 + 0.988285i \(0.548770\pi\)
\(90\) −4.81735e25 −0.199772
\(91\) −3.51459e26 −1.25550
\(92\) −2.31197e26 −0.712601
\(93\) 2.12360e26 0.565656
\(94\) 1.11886e26 0.257958
\(95\) 3.42632e26 0.684793
\(96\) 1.30140e26 0.225813
\(97\) −5.95949e26 −0.899063 −0.449532 0.893264i \(-0.648409\pi\)
−0.449532 + 0.893264i \(0.648409\pi\)
\(98\) 1.74030e26 0.228598
\(99\) −9.66515e26 −1.10696
\(100\) 1.00000e26 0.100000
\(101\) −6.01756e26 −0.526117 −0.263058 0.964780i \(-0.584731\pi\)
−0.263058 + 0.964780i \(0.584731\pi\)
\(102\) 1.36408e27 1.04409
\(103\) 5.34812e26 0.358838 0.179419 0.983773i \(-0.442578\pi\)
0.179419 + 0.983773i \(0.442578\pi\)
\(104\) −9.16261e26 −0.539596
\(105\) 9.08025e26 0.469940
\(106\) −9.61036e26 −0.437633
\(107\) −2.27447e27 −0.912429 −0.456214 0.889870i \(-0.650795\pi\)
−0.456214 + 0.889870i \(0.650795\pi\)
\(108\) 6.64777e26 0.235210
\(109\) −2.61656e27 −0.817472 −0.408736 0.912653i \(-0.634030\pi\)
−0.408736 + 0.912653i \(0.634030\pi\)
\(110\) 2.00632e27 0.554114
\(111\) −7.60129e27 −1.85792
\(112\) −9.49698e26 −0.205656
\(113\) −9.28514e27 −1.78332 −0.891660 0.452705i \(-0.850459\pi\)
−0.891660 + 0.452705i \(0.850459\pi\)
\(114\) 8.11091e27 1.38310
\(115\) −4.20545e27 −0.637369
\(116\) 3.61321e27 0.487204
\(117\) 8.02893e27 0.964159
\(118\) 5.70280e27 0.610494
\(119\) −9.95436e27 −0.950888
\(120\) 2.36724e27 0.201974
\(121\) 2.71432e28 2.07042
\(122\) 1.36101e28 0.928972
\(123\) 2.02428e28 1.23752
\(124\) −4.04008e27 −0.221410
\(125\) 1.81899e27 0.0894427
\(126\) 8.32194e27 0.367470
\(127\) 1.34819e28 0.535058 0.267529 0.963550i \(-0.413793\pi\)
0.267529 + 0.963550i \(0.413793\pi\)
\(128\) −2.47588e27 −0.0883883
\(129\) −1.66916e27 −0.0536461
\(130\) −1.66667e28 −0.482630
\(131\) −1.22819e28 −0.320703 −0.160351 0.987060i \(-0.551263\pi\)
−0.160351 + 0.987060i \(0.551263\pi\)
\(132\) 4.74943e28 1.11916
\(133\) −5.91894e28 −1.25964
\(134\) 5.63327e26 0.0108354
\(135\) 1.20922e28 0.210378
\(136\) −2.59512e28 −0.408679
\(137\) 7.77523e28 1.10914 0.554569 0.832138i \(-0.312883\pi\)
0.554569 + 0.832138i \(0.312883\pi\)
\(138\) −9.95528e28 −1.28732
\(139\) −6.09274e28 −0.714682 −0.357341 0.933974i \(-0.616317\pi\)
−0.357341 + 0.933974i \(0.616317\pi\)
\(140\) −1.72749e28 −0.183945
\(141\) 4.81778e28 0.466003
\(142\) 1.38464e28 0.121743
\(143\) −3.34387e29 −2.67432
\(144\) 2.16954e28 0.157934
\(145\) 6.57239e28 0.435768
\(146\) 4.53742e28 0.274186
\(147\) 7.49371e28 0.412965
\(148\) 1.44612e29 0.727231
\(149\) −1.05900e29 −0.486276 −0.243138 0.969992i \(-0.578177\pi\)
−0.243138 + 0.969992i \(0.578177\pi\)
\(150\) 4.30598e28 0.180651
\(151\) 3.29076e28 0.126214 0.0631070 0.998007i \(-0.479899\pi\)
0.0631070 + 0.998007i \(0.479899\pi\)
\(152\) −1.54308e29 −0.541376
\(153\) 2.27403e29 0.730233
\(154\) −3.46590e29 −1.01926
\(155\) −7.34887e28 −0.198035
\(156\) −3.94540e29 −0.974785
\(157\) −1.11931e29 −0.253692 −0.126846 0.991922i \(-0.540485\pi\)
−0.126846 + 0.991922i \(0.540485\pi\)
\(158\) 4.11298e28 0.0855629
\(159\) −4.13820e29 −0.790588
\(160\) −4.50360e28 −0.0790569
\(161\) 7.26487e29 1.17241
\(162\) 5.87186e29 0.871612
\(163\) 1.01040e30 1.38026 0.690128 0.723687i \(-0.257554\pi\)
0.690128 + 0.723687i \(0.257554\pi\)
\(164\) −3.85114e29 −0.484392
\(165\) 8.63917e29 1.00101
\(166\) −1.28118e30 −1.36821
\(167\) −8.58149e29 −0.845065 −0.422533 0.906348i \(-0.638859\pi\)
−0.422533 + 0.906348i \(0.638859\pi\)
\(168\) −4.08938e29 −0.371520
\(169\) 1.58525e30 1.32931
\(170\) −4.72049e29 −0.365533
\(171\) 1.35216e30 0.967340
\(172\) 3.17554e28 0.0209983
\(173\) 1.28043e30 0.782949 0.391474 0.920189i \(-0.371965\pi\)
0.391474 + 0.920189i \(0.371965\pi\)
\(174\) 1.55584e30 0.880137
\(175\) −3.14229e29 −0.164525
\(176\) −9.03566e29 −0.438065
\(177\) 2.45561e30 1.10286
\(178\) 5.18550e29 0.215834
\(179\) −4.74098e30 −1.82958 −0.914791 0.403928i \(-0.867645\pi\)
−0.914791 + 0.403928i \(0.867645\pi\)
\(180\) 3.94638e29 0.141260
\(181\) −3.96469e30 −1.31689 −0.658443 0.752631i \(-0.728785\pi\)
−0.658443 + 0.752631i \(0.728785\pi\)
\(182\) 2.87916e30 0.887772
\(183\) 5.86046e30 1.67819
\(184\) 1.89396e30 0.503885
\(185\) 2.63048e30 0.650455
\(186\) −1.73965e30 −0.399979
\(187\) −9.47082e30 −2.02547
\(188\) −9.16568e29 −0.182404
\(189\) −2.08892e30 −0.386979
\(190\) −2.80684e30 −0.484222
\(191\) 8.53219e30 1.37123 0.685615 0.727964i \(-0.259533\pi\)
0.685615 + 0.727964i \(0.259533\pi\)
\(192\) −1.06611e30 −0.159674
\(193\) −1.32910e31 −1.85582 −0.927908 0.372808i \(-0.878395\pi\)
−0.927908 + 0.372808i \(0.878395\pi\)
\(194\) 4.88201e30 0.635734
\(195\) −7.17664e30 −0.871874
\(196\) −1.42566e30 −0.161643
\(197\) −8.40332e30 −0.889522 −0.444761 0.895649i \(-0.646711\pi\)
−0.444761 + 0.895649i \(0.646711\pi\)
\(198\) 7.91769e30 0.782742
\(199\) 1.87522e31 1.73195 0.865975 0.500088i \(-0.166699\pi\)
0.865975 + 0.500088i \(0.166699\pi\)
\(200\) −8.19200e29 −0.0707107
\(201\) 2.42567e29 0.0195743
\(202\) 4.92959e30 0.372021
\(203\) −1.13538e31 −0.801573
\(204\) −1.11745e31 −0.738281
\(205\) −7.00518e30 −0.433253
\(206\) −4.38118e30 −0.253737
\(207\) −1.65963e31 −0.900349
\(208\) 7.50601e30 0.381552
\(209\) −5.63142e31 −2.68314
\(210\) −7.43854e30 −0.332297
\(211\) 4.59140e31 1.92367 0.961837 0.273623i \(-0.0882219\pi\)
0.961837 + 0.273623i \(0.0882219\pi\)
\(212\) 7.87281e30 0.309454
\(213\) 5.96225e30 0.219931
\(214\) 1.86324e31 0.645185
\(215\) 5.77627e29 0.0187814
\(216\) −5.44586e30 −0.166319
\(217\) 1.26951e31 0.364275
\(218\) 2.14348e31 0.578040
\(219\) 1.95380e31 0.495319
\(220\) −1.64358e31 −0.391817
\(221\) 7.86750e31 1.76417
\(222\) 6.22698e31 1.31375
\(223\) 5.48895e31 1.08987 0.544933 0.838479i \(-0.316555\pi\)
0.544933 + 0.838479i \(0.316555\pi\)
\(224\) 7.77993e30 0.145421
\(225\) 7.17842e30 0.126347
\(226\) 7.60639e31 1.26100
\(227\) −4.53524e31 −0.708355 −0.354178 0.935178i \(-0.615239\pi\)
−0.354178 + 0.935178i \(0.615239\pi\)
\(228\) −6.64446e31 −0.978001
\(229\) 3.94248e30 0.0547004 0.0273502 0.999626i \(-0.491293\pi\)
0.0273502 + 0.999626i \(0.491293\pi\)
\(230\) 3.44510e31 0.450688
\(231\) −1.49241e32 −1.84131
\(232\) −2.95994e31 −0.344505
\(233\) 4.71165e31 0.517449 0.258725 0.965951i \(-0.416698\pi\)
0.258725 + 0.965951i \(0.416698\pi\)
\(234\) −6.57730e31 −0.681764
\(235\) −1.66723e31 −0.163147
\(236\) −4.67174e31 −0.431684
\(237\) 1.77104e31 0.154570
\(238\) 8.15461e31 0.672379
\(239\) −5.90002e31 −0.459708 −0.229854 0.973225i \(-0.573825\pi\)
−0.229854 + 0.973225i \(0.573825\pi\)
\(240\) −1.93924e31 −0.142817
\(241\) −1.62357e32 −1.13042 −0.565212 0.824946i \(-0.691205\pi\)
−0.565212 + 0.824946i \(0.691205\pi\)
\(242\) −2.22357e32 −1.46401
\(243\) 1.77302e32 1.10415
\(244\) −1.11494e32 −0.656882
\(245\) −2.59326e31 −0.144578
\(246\) −1.65829e32 −0.875058
\(247\) 4.67808e32 2.33700
\(248\) 3.30964e31 0.156561
\(249\) −5.51675e32 −2.47168
\(250\) −1.49012e31 −0.0632456
\(251\) 2.64625e32 1.06423 0.532115 0.846672i \(-0.321397\pi\)
0.532115 + 0.846672i \(0.321397\pi\)
\(252\) −6.81733e31 −0.259841
\(253\) 6.91197e32 2.49732
\(254\) −1.10444e32 −0.378343
\(255\) −2.03263e32 −0.660339
\(256\) 2.02824e31 0.0625000
\(257\) 2.74808e31 0.0803402 0.0401701 0.999193i \(-0.487210\pi\)
0.0401701 + 0.999193i \(0.487210\pi\)
\(258\) 1.36738e31 0.0379335
\(259\) −4.54414e32 −1.19648
\(260\) 1.36534e32 0.341271
\(261\) 2.59371e32 0.615567
\(262\) 1.00613e32 0.226771
\(263\) −6.36864e32 −1.36346 −0.681732 0.731602i \(-0.738773\pi\)
−0.681732 + 0.731602i \(0.738773\pi\)
\(264\) −3.89073e32 −0.791368
\(265\) 1.43206e32 0.276784
\(266\) 4.84879e32 0.890700
\(267\) 2.23286e32 0.389906
\(268\) −4.61478e30 −0.00766180
\(269\) −2.64799e32 −0.418080 −0.209040 0.977907i \(-0.567034\pi\)
−0.209040 + 0.977907i \(0.567034\pi\)
\(270\) −9.90596e31 −0.148760
\(271\) −9.06365e32 −1.29485 −0.647423 0.762131i \(-0.724153\pi\)
−0.647423 + 0.762131i \(0.724153\pi\)
\(272\) 2.12592e32 0.288979
\(273\) 1.23976e33 1.60377
\(274\) −6.36947e32 −0.784279
\(275\) −2.98965e32 −0.350452
\(276\) 8.15537e32 0.910271
\(277\) 6.76213e32 0.718798 0.359399 0.933184i \(-0.382982\pi\)
0.359399 + 0.933184i \(0.382982\pi\)
\(278\) 4.99117e32 0.505356
\(279\) −2.90014e32 −0.279745
\(280\) 1.41516e32 0.130069
\(281\) 9.77731e32 0.856415 0.428207 0.903680i \(-0.359145\pi\)
0.428207 + 0.903680i \(0.359145\pi\)
\(282\) −3.94672e32 −0.329514
\(283\) 1.47138e32 0.117113 0.0585567 0.998284i \(-0.481350\pi\)
0.0585567 + 0.998284i \(0.481350\pi\)
\(284\) −1.13430e32 −0.0860856
\(285\) −1.20862e33 −0.874750
\(286\) 2.73930e33 1.89103
\(287\) 1.21014e33 0.796947
\(288\) −1.77729e32 −0.111676
\(289\) 5.60594e32 0.336146
\(290\) −5.38410e32 −0.308135
\(291\) 2.10218e33 1.14846
\(292\) −3.71706e32 −0.193879
\(293\) −4.76752e32 −0.237454 −0.118727 0.992927i \(-0.537881\pi\)
−0.118727 + 0.992927i \(0.537881\pi\)
\(294\) −6.13885e32 −0.292010
\(295\) −8.49784e32 −0.386110
\(296\) −1.18466e33 −0.514230
\(297\) −1.98745e33 −0.824298
\(298\) 8.67536e32 0.343849
\(299\) −5.74184e33 −2.17515
\(300\) −3.52746e32 −0.127739
\(301\) −9.97845e31 −0.0345474
\(302\) −2.69579e32 −0.0892468
\(303\) 2.12267e33 0.672058
\(304\) 1.26409e33 0.382811
\(305\) −2.02806e33 −0.587533
\(306\) −1.86288e33 −0.516353
\(307\) −1.97700e33 −0.524370 −0.262185 0.965018i \(-0.584443\pi\)
−0.262185 + 0.965018i \(0.584443\pi\)
\(308\) 2.83926e33 0.720728
\(309\) −1.88653e33 −0.458377
\(310\) 6.02019e32 0.140032
\(311\) −4.03023e33 −0.897562 −0.448781 0.893642i \(-0.648142\pi\)
−0.448781 + 0.893642i \(0.648142\pi\)
\(312\) 3.23207e33 0.689277
\(313\) 1.11971e33 0.228696 0.114348 0.993441i \(-0.463522\pi\)
0.114348 + 0.993441i \(0.463522\pi\)
\(314\) 9.16941e32 0.179387
\(315\) −1.24007e33 −0.232409
\(316\) −3.36935e32 −0.0605021
\(317\) −6.02275e33 −1.03632 −0.518161 0.855283i \(-0.673383\pi\)
−0.518161 + 0.855283i \(0.673383\pi\)
\(318\) 3.39001e33 0.559030
\(319\) −1.08022e34 −1.70742
\(320\) 3.68935e32 0.0559017
\(321\) 8.02309e33 1.16553
\(322\) −5.95138e33 −0.829017
\(323\) 1.32497e34 1.76999
\(324\) −4.81023e33 −0.616323
\(325\) 2.48353e33 0.305242
\(326\) −8.27718e33 −0.975989
\(327\) 9.22980e33 1.04423
\(328\) 3.15485e33 0.342517
\(329\) 2.88012e33 0.300100
\(330\) −7.07721e33 −0.707821
\(331\) −3.93423e33 −0.377730 −0.188865 0.982003i \(-0.560481\pi\)
−0.188865 + 0.982003i \(0.560481\pi\)
\(332\) 1.04955e34 0.967470
\(333\) 1.03809e34 0.918834
\(334\) 7.02996e33 0.597551
\(335\) −8.39423e31 −0.00685292
\(336\) 3.35002e33 0.262704
\(337\) −1.77892e34 −1.34015 −0.670074 0.742294i \(-0.733738\pi\)
−0.670074 + 0.742294i \(0.733738\pi\)
\(338\) −1.29864e34 −0.939967
\(339\) 3.27529e34 2.27800
\(340\) 3.86703e33 0.258471
\(341\) 1.20784e34 0.775937
\(342\) −1.10769e34 −0.684013
\(343\) 1.83369e34 1.08857
\(344\) −2.60140e32 −0.0148480
\(345\) 1.48345e34 0.814172
\(346\) −1.04893e34 −0.553628
\(347\) −6.67131e32 −0.0338660 −0.0169330 0.999857i \(-0.505390\pi\)
−0.0169330 + 0.999857i \(0.505390\pi\)
\(348\) −1.27454e34 −0.622351
\(349\) −9.29683e33 −0.436708 −0.218354 0.975870i \(-0.570069\pi\)
−0.218354 + 0.975870i \(0.570069\pi\)
\(350\) 2.57416e33 0.116337
\(351\) 1.65099e34 0.717959
\(352\) 7.40201e33 0.309759
\(353\) −2.79504e34 −1.12572 −0.562859 0.826553i \(-0.690299\pi\)
−0.562859 + 0.826553i \(0.690299\pi\)
\(354\) −2.01164e34 −0.779841
\(355\) −2.06328e33 −0.0769973
\(356\) −4.24796e33 −0.152618
\(357\) 3.51136e34 1.21466
\(358\) 3.88381e34 1.29371
\(359\) −4.42700e34 −1.42015 −0.710075 0.704126i \(-0.751339\pi\)
−0.710075 + 0.704126i \(0.751339\pi\)
\(360\) −3.23287e33 −0.0998860
\(361\) 4.51830e34 1.34471
\(362\) 3.24787e34 0.931179
\(363\) −9.57464e34 −2.64474
\(364\) −2.35860e34 −0.627749
\(365\) −6.76129e33 −0.173411
\(366\) −4.80089e34 −1.18666
\(367\) −4.21004e34 −1.00298 −0.501492 0.865162i \(-0.667215\pi\)
−0.501492 + 0.865162i \(0.667215\pi\)
\(368\) −1.55154e34 −0.356300
\(369\) −2.76451e34 −0.612015
\(370\) −2.15489e34 −0.459941
\(371\) −2.47386e34 −0.509129
\(372\) 1.42512e34 0.282828
\(373\) −5.44723e34 −1.04257 −0.521287 0.853382i \(-0.674548\pi\)
−0.521287 + 0.853382i \(0.674548\pi\)
\(374\) 7.75849e34 1.43222
\(375\) −6.41641e33 −0.114254
\(376\) 7.50853e33 0.128979
\(377\) 8.97352e34 1.48715
\(378\) 1.71124e34 0.273636
\(379\) 8.74120e34 1.34878 0.674391 0.738374i \(-0.264406\pi\)
0.674391 + 0.738374i \(0.264406\pi\)
\(380\) 2.29937e34 0.342397
\(381\) −4.75568e34 −0.683480
\(382\) −6.98957e34 −0.969606
\(383\) −7.39769e34 −0.990632 −0.495316 0.868713i \(-0.664948\pi\)
−0.495316 + 0.868713i \(0.664948\pi\)
\(384\) 8.73356e33 0.112907
\(385\) 5.16459e34 0.644638
\(386\) 1.08880e35 1.31226
\(387\) 2.27953e33 0.0265307
\(388\) −3.99934e34 −0.449532
\(389\) 6.86864e34 0.745677 0.372839 0.927896i \(-0.378385\pi\)
0.372839 + 0.927896i \(0.378385\pi\)
\(390\) 5.87910e34 0.616508
\(391\) −1.62626e35 −1.64741
\(392\) 1.16790e34 0.114299
\(393\) 4.33238e34 0.409664
\(394\) 6.88400e34 0.628987
\(395\) −6.12881e33 −0.0541147
\(396\) −6.48617e34 −0.553482
\(397\) 1.09834e35 0.905868 0.452934 0.891544i \(-0.350377\pi\)
0.452934 + 0.891544i \(0.350377\pi\)
\(398\) −1.53618e35 −1.22467
\(399\) 2.08788e35 1.60906
\(400\) 6.71089e33 0.0500000
\(401\) 1.31925e35 0.950336 0.475168 0.879895i \(-0.342387\pi\)
0.475168 + 0.879895i \(0.342387\pi\)
\(402\) −1.98711e33 −0.0138411
\(403\) −1.00337e35 −0.675837
\(404\) −4.03832e34 −0.263058
\(405\) −8.74976e34 −0.551256
\(406\) 9.30099e34 0.566797
\(407\) −4.32340e35 −2.54860
\(408\) 9.15417e34 0.522043
\(409\) 1.07033e35 0.590544 0.295272 0.955413i \(-0.404590\pi\)
0.295272 + 0.955413i \(0.404590\pi\)
\(410\) 5.73864e34 0.306356
\(411\) −2.74268e35 −1.41681
\(412\) 3.58906e34 0.179419
\(413\) 1.46799e35 0.710229
\(414\) 1.35957e35 0.636643
\(415\) 1.90911e35 0.865331
\(416\) −6.14892e34 −0.269798
\(417\) 2.14919e35 0.912930
\(418\) 4.61326e35 1.89727
\(419\) −2.62715e35 −1.04615 −0.523076 0.852286i \(-0.675216\pi\)
−0.523076 + 0.852286i \(0.675216\pi\)
\(420\) 6.09365e34 0.234970
\(421\) −1.82406e35 −0.681131 −0.340566 0.940221i \(-0.610619\pi\)
−0.340566 + 0.940221i \(0.610619\pi\)
\(422\) −3.76127e35 −1.36024
\(423\) −6.57951e34 −0.230462
\(424\) −6.44940e34 −0.218817
\(425\) 7.03408e34 0.231184
\(426\) −4.88427e34 −0.155514
\(427\) 3.50345e35 1.08074
\(428\) −1.52637e35 −0.456214
\(429\) 1.17954e36 3.41616
\(430\) −4.73192e33 −0.0132805
\(431\) 1.35228e35 0.367810 0.183905 0.982944i \(-0.441126\pi\)
0.183905 + 0.982944i \(0.441126\pi\)
\(432\) 4.46125e34 0.117605
\(433\) −3.26767e35 −0.834934 −0.417467 0.908692i \(-0.637082\pi\)
−0.417467 + 0.908692i \(0.637082\pi\)
\(434\) −1.03998e35 −0.257582
\(435\) −2.31838e35 −0.556648
\(436\) −1.75594e35 −0.408736
\(437\) −9.66985e35 −2.18233
\(438\) −1.60056e35 −0.350244
\(439\) −5.20602e35 −1.10468 −0.552338 0.833620i \(-0.686264\pi\)
−0.552338 + 0.833620i \(0.686264\pi\)
\(440\) 1.34642e35 0.277057
\(441\) −1.02340e35 −0.204232
\(442\) −6.44505e35 −1.24746
\(443\) −7.68224e35 −1.44224 −0.721120 0.692810i \(-0.756372\pi\)
−0.721120 + 0.692810i \(0.756372\pi\)
\(444\) −5.10114e35 −0.928960
\(445\) −7.72700e34 −0.136506
\(446\) −4.49655e35 −0.770652
\(447\) 3.73559e35 0.621166
\(448\) −6.37332e34 −0.102828
\(449\) 5.97023e35 0.934686 0.467343 0.884076i \(-0.345211\pi\)
0.467343 + 0.884076i \(0.345211\pi\)
\(450\) −5.88056e34 −0.0893408
\(451\) 1.15135e36 1.69756
\(452\) −6.23115e35 −0.891660
\(453\) −1.16080e35 −0.161225
\(454\) 3.71527e35 0.500883
\(455\) −4.29028e35 −0.561476
\(456\) 5.44314e35 0.691551
\(457\) 1.24389e36 1.53431 0.767154 0.641463i \(-0.221672\pi\)
0.767154 + 0.641463i \(0.221672\pi\)
\(458\) −3.22968e34 −0.0386790
\(459\) 4.67610e35 0.543767
\(460\) −2.82223e35 −0.318685
\(461\) −4.50718e35 −0.494245 −0.247122 0.968984i \(-0.579485\pi\)
−0.247122 + 0.968984i \(0.579485\pi\)
\(462\) 1.22258e36 1.30200
\(463\) 3.06062e35 0.316568 0.158284 0.987394i \(-0.449404\pi\)
0.158284 + 0.987394i \(0.449404\pi\)
\(464\) 2.42479e35 0.243602
\(465\) 2.59228e35 0.252969
\(466\) −3.85978e35 −0.365892
\(467\) 2.48619e35 0.228958 0.114479 0.993426i \(-0.463480\pi\)
0.114479 + 0.993426i \(0.463480\pi\)
\(468\) 5.38813e35 0.482080
\(469\) 1.45010e34 0.0126056
\(470\) 1.36579e35 0.115362
\(471\) 3.94833e35 0.324064
\(472\) 3.82709e35 0.305247
\(473\) −9.49374e34 −0.0735888
\(474\) −1.45083e35 −0.109298
\(475\) 4.18252e35 0.306249
\(476\) −6.68025e35 −0.475444
\(477\) 5.65143e35 0.390985
\(478\) 4.83330e35 0.325063
\(479\) −2.43474e36 −1.59193 −0.795964 0.605344i \(-0.793036\pi\)
−0.795964 + 0.605344i \(0.793036\pi\)
\(480\) 1.58863e35 0.100987
\(481\) 3.59149e36 2.21981
\(482\) 1.33003e36 0.799330
\(483\) −2.56265e36 −1.49763
\(484\) 1.82155e36 1.03521
\(485\) −7.27476e35 −0.402073
\(486\) −1.45246e36 −0.780755
\(487\) −1.21293e35 −0.0634154 −0.0317077 0.999497i \(-0.510095\pi\)
−0.0317077 + 0.999497i \(0.510095\pi\)
\(488\) 9.13356e35 0.464486
\(489\) −3.56414e36 −1.76313
\(490\) 2.12439e35 0.102232
\(491\) 6.41302e35 0.300236 0.150118 0.988668i \(-0.452035\pi\)
0.150118 + 0.988668i \(0.452035\pi\)
\(492\) 1.35847e36 0.618759
\(493\) 2.54156e36 1.12633
\(494\) −3.83228e36 −1.65251
\(495\) −1.17983e36 −0.495049
\(496\) −2.71125e35 −0.110705
\(497\) 3.56430e35 0.141633
\(498\) 4.51932e36 1.74774
\(499\) −4.09067e36 −1.53970 −0.769851 0.638223i \(-0.779670\pi\)
−0.769851 + 0.638223i \(0.779670\pi\)
\(500\) 1.22070e35 0.0447214
\(501\) 3.02708e36 1.07948
\(502\) −2.16781e36 −0.752524
\(503\) −5.43130e35 −0.183542 −0.0917710 0.995780i \(-0.529253\pi\)
−0.0917710 + 0.995780i \(0.529253\pi\)
\(504\) 5.58476e35 0.183735
\(505\) −7.34566e35 −0.235287
\(506\) −5.66229e36 −1.76587
\(507\) −5.59191e36 −1.69806
\(508\) 9.04756e35 0.267529
\(509\) −1.01633e36 −0.292646 −0.146323 0.989237i \(-0.546744\pi\)
−0.146323 + 0.989237i \(0.546744\pi\)
\(510\) 1.66513e36 0.466930
\(511\) 1.16801e36 0.318980
\(512\) −1.66153e35 −0.0441942
\(513\) 2.78045e36 0.720327
\(514\) −2.25123e35 −0.0568091
\(515\) 6.52846e35 0.160477
\(516\) −1.12016e35 −0.0268230
\(517\) 2.74022e36 0.639238
\(518\) 3.72256e36 0.846038
\(519\) −4.51666e36 −1.00013
\(520\) −1.11848e36 −0.241315
\(521\) −5.75177e36 −1.20918 −0.604592 0.796536i \(-0.706664\pi\)
−0.604592 + 0.796536i \(0.706664\pi\)
\(522\) −2.12477e36 −0.435272
\(523\) −7.30952e35 −0.145920 −0.0729602 0.997335i \(-0.523245\pi\)
−0.0729602 + 0.997335i \(0.523245\pi\)
\(524\) −8.24224e35 −0.160351
\(525\) 1.10843e36 0.210163
\(526\) 5.21719e36 0.964115
\(527\) −2.84183e36 −0.511864
\(528\) 3.18729e36 0.559582
\(529\) 6.02551e36 1.03120
\(530\) −1.17314e36 −0.195716
\(531\) −3.35357e36 −0.545420
\(532\) −3.97213e36 −0.629820
\(533\) −9.56441e36 −1.47857
\(534\) −1.82916e36 −0.275705
\(535\) −2.77645e36 −0.408051
\(536\) 3.78042e34 0.00541771
\(537\) 1.67236e37 2.33710
\(538\) 2.16923e36 0.295628
\(539\) 4.26221e36 0.566483
\(540\) 8.11496e35 0.105189
\(541\) 1.06277e37 1.34362 0.671812 0.740721i \(-0.265516\pi\)
0.671812 + 0.740721i \(0.265516\pi\)
\(542\) 7.42494e36 0.915594
\(543\) 1.39853e37 1.68218
\(544\) −1.74155e36 −0.204339
\(545\) −3.19404e36 −0.365584
\(546\) −1.01561e37 −1.13403
\(547\) 3.13391e36 0.341396 0.170698 0.985323i \(-0.445398\pi\)
0.170698 + 0.985323i \(0.445398\pi\)
\(548\) 5.21787e36 0.554569
\(549\) −8.00348e36 −0.829950
\(550\) 2.44912e36 0.247807
\(551\) 1.51123e37 1.49206
\(552\) −6.68088e36 −0.643659
\(553\) 1.05875e36 0.0995412
\(554\) −5.53954e36 −0.508267
\(555\) −9.27892e36 −0.830887
\(556\) −4.08877e36 −0.357341
\(557\) −1.82733e37 −1.55874 −0.779368 0.626566i \(-0.784460\pi\)
−0.779368 + 0.626566i \(0.784460\pi\)
\(558\) 2.37580e36 0.197810
\(559\) 7.88654e35 0.0640954
\(560\) −1.15930e36 −0.0919724
\(561\) 3.34079e37 2.58732
\(562\) −8.00958e36 −0.605577
\(563\) −6.65901e36 −0.491526 −0.245763 0.969330i \(-0.579038\pi\)
−0.245763 + 0.969330i \(0.579038\pi\)
\(564\) 3.23315e36 0.233001
\(565\) −1.13344e37 −0.797525
\(566\) −1.20535e36 −0.0828117
\(567\) 1.51151e37 1.01401
\(568\) 9.29219e35 0.0608717
\(569\) 1.71644e37 1.09802 0.549012 0.835814i \(-0.315004\pi\)
0.549012 + 0.835814i \(0.315004\pi\)
\(570\) 9.90101e36 0.618542
\(571\) −1.61696e37 −0.986535 −0.493267 0.869878i \(-0.664198\pi\)
−0.493267 + 0.869878i \(0.664198\pi\)
\(572\) −2.24403e37 −1.33716
\(573\) −3.00969e37 −1.75160
\(574\) −9.91345e36 −0.563526
\(575\) −5.13360e36 −0.285040
\(576\) 1.45596e36 0.0789668
\(577\) 2.68745e37 1.42386 0.711929 0.702251i \(-0.247822\pi\)
0.711929 + 0.702251i \(0.247822\pi\)
\(578\) −4.59238e36 −0.237691
\(579\) 4.68835e37 2.37061
\(580\) 4.41066e36 0.217884
\(581\) −3.29797e37 −1.59173
\(582\) −1.72211e37 −0.812082
\(583\) −2.35369e37 −1.08449
\(584\) 3.04501e36 0.137093
\(585\) 9.80094e36 0.431185
\(586\) 3.90555e36 0.167905
\(587\) 2.21802e37 0.931860 0.465930 0.884822i \(-0.345720\pi\)
0.465930 + 0.884822i \(0.345720\pi\)
\(588\) 5.02894e36 0.206482
\(589\) −1.68977e37 −0.678066
\(590\) 6.96143e36 0.273021
\(591\) 2.96424e37 1.13627
\(592\) 9.70477e36 0.363615
\(593\) −3.27250e37 −1.19851 −0.599255 0.800558i \(-0.704536\pi\)
−0.599255 + 0.800558i \(0.704536\pi\)
\(594\) 1.62812e37 0.582867
\(595\) −1.21513e37 −0.425250
\(596\) −7.10686e36 −0.243138
\(597\) −6.61475e37 −2.21238
\(598\) 4.70372e37 1.53807
\(599\) −5.26063e37 −1.68180 −0.840902 0.541187i \(-0.817975\pi\)
−0.840902 + 0.541187i \(0.817975\pi\)
\(600\) 2.88969e36 0.0903254
\(601\) −3.25919e37 −0.996102 −0.498051 0.867148i \(-0.665951\pi\)
−0.498051 + 0.867148i \(0.665951\pi\)
\(602\) 8.17435e35 0.0244287
\(603\) −3.31268e35 −0.00968045
\(604\) 2.20839e36 0.0631070
\(605\) 3.31338e37 0.925919
\(606\) −1.73889e37 −0.475217
\(607\) −1.02382e37 −0.273637 −0.136818 0.990596i \(-0.543688\pi\)
−0.136818 + 0.990596i \(0.543688\pi\)
\(608\) −1.03554e37 −0.270688
\(609\) 4.00499e37 1.02392
\(610\) 1.66138e37 0.415449
\(611\) −2.27632e37 −0.556773
\(612\) 1.52607e37 0.365117
\(613\) −2.22273e37 −0.520201 −0.260100 0.965582i \(-0.583756\pi\)
−0.260100 + 0.965582i \(0.583756\pi\)
\(614\) 1.61956e37 0.370786
\(615\) 2.47105e37 0.553435
\(616\) −2.32593e37 −0.509631
\(617\) −3.05546e37 −0.654978 −0.327489 0.944855i \(-0.606202\pi\)
−0.327489 + 0.944855i \(0.606202\pi\)
\(618\) 1.54544e37 0.324122
\(619\) −1.32801e37 −0.272507 −0.136254 0.990674i \(-0.543506\pi\)
−0.136254 + 0.990674i \(0.543506\pi\)
\(620\) −4.93174e36 −0.0990177
\(621\) −3.41270e37 −0.670443
\(622\) 3.30156e37 0.634672
\(623\) 1.33483e37 0.251095
\(624\) −2.64771e37 −0.487392
\(625\) 2.22045e36 0.0400000
\(626\) −9.17269e36 −0.161712
\(627\) 1.98646e38 3.42742
\(628\) −7.51158e36 −0.126846
\(629\) 1.01722e38 1.68124
\(630\) 1.01586e37 0.164338
\(631\) 8.40893e37 1.33151 0.665755 0.746171i \(-0.268110\pi\)
0.665755 + 0.746171i \(0.268110\pi\)
\(632\) 2.76017e36 0.0427815
\(633\) −1.61959e38 −2.45729
\(634\) 4.93384e37 0.732790
\(635\) 1.64574e37 0.239285
\(636\) −2.77710e37 −0.395294
\(637\) −3.54066e37 −0.493403
\(638\) 8.84919e37 1.20733
\(639\) −8.14249e36 −0.108767
\(640\) −3.02231e36 −0.0395285
\(641\) 7.86398e37 1.00707 0.503534 0.863975i \(-0.332033\pi\)
0.503534 + 0.863975i \(0.332033\pi\)
\(642\) −6.57252e37 −0.824155
\(643\) 1.16284e38 1.42781 0.713904 0.700243i \(-0.246925\pi\)
0.713904 + 0.700243i \(0.246925\pi\)
\(644\) 4.87537e37 0.586204
\(645\) −2.03755e36 −0.0239913
\(646\) −1.08541e38 −1.25157
\(647\) −7.65469e37 −0.864410 −0.432205 0.901775i \(-0.642264\pi\)
−0.432205 + 0.901775i \(0.642264\pi\)
\(648\) 3.94054e37 0.435806
\(649\) 1.39669e38 1.51285
\(650\) −2.03451e37 −0.215839
\(651\) −4.47814e37 −0.465323
\(652\) 6.78067e37 0.690128
\(653\) −1.03403e38 −1.03087 −0.515436 0.856928i \(-0.672370\pi\)
−0.515436 + 0.856928i \(0.672370\pi\)
\(654\) −7.56105e37 −0.738384
\(655\) −1.49925e37 −0.143423
\(656\) −2.58445e37 −0.242196
\(657\) −2.66826e37 −0.244960
\(658\) −2.35939e37 −0.212203
\(659\) −5.24833e37 −0.462453 −0.231227 0.972900i \(-0.574274\pi\)
−0.231227 + 0.972900i \(0.574274\pi\)
\(660\) 5.79765e37 0.500505
\(661\) −1.14190e38 −0.965846 −0.482923 0.875663i \(-0.660425\pi\)
−0.482923 + 0.875663i \(0.660425\pi\)
\(662\) 3.22292e37 0.267096
\(663\) −2.77523e38 −2.25354
\(664\) −8.59788e37 −0.684104
\(665\) −7.22527e37 −0.563328
\(666\) −8.50402e37 −0.649714
\(667\) −1.85488e38 −1.38873
\(668\) −5.75894e37 −0.422533
\(669\) −1.93620e38 −1.39219
\(670\) 6.87655e35 0.00484575
\(671\) 3.33327e38 2.30206
\(672\) −2.74434e37 −0.185760
\(673\) 3.00048e36 0.0199061 0.00995307 0.999950i \(-0.496832\pi\)
0.00995307 + 0.999950i \(0.496832\pi\)
\(674\) 1.45729e38 0.947628
\(675\) 1.47610e37 0.0940840
\(676\) 1.06384e38 0.664657
\(677\) 2.27686e38 1.39441 0.697204 0.716873i \(-0.254427\pi\)
0.697204 + 0.716873i \(0.254427\pi\)
\(678\) −2.68312e38 −1.61079
\(679\) 1.25671e38 0.739593
\(680\) −3.16787e37 −0.182767
\(681\) 1.59979e38 0.904848
\(682\) −9.89465e37 −0.548670
\(683\) −1.59944e37 −0.0869538 −0.0434769 0.999054i \(-0.513843\pi\)
−0.0434769 + 0.999054i \(0.513843\pi\)
\(684\) 9.07416e37 0.483670
\(685\) 9.49125e37 0.496022
\(686\) −1.50216e38 −0.769735
\(687\) −1.39069e37 −0.0698739
\(688\) 2.13107e36 0.0104991
\(689\) 1.95524e38 0.944582
\(690\) −1.21524e38 −0.575706
\(691\) 1.56832e38 0.728586 0.364293 0.931284i \(-0.381311\pi\)
0.364293 + 0.931284i \(0.381311\pi\)
\(692\) 8.59282e37 0.391474
\(693\) 2.03814e38 0.910617
\(694\) 5.46514e36 0.0239469
\(695\) −7.43743e37 −0.319616
\(696\) 1.04411e38 0.440069
\(697\) −2.70892e38 −1.11983
\(698\) 7.61596e37 0.308799
\(699\) −1.66201e38 −0.660987
\(700\) −2.10875e37 −0.0822626
\(701\) −2.65673e38 −1.01661 −0.508304 0.861178i \(-0.669727\pi\)
−0.508304 + 0.861178i \(0.669727\pi\)
\(702\) −1.35249e38 −0.507674
\(703\) 6.04844e38 2.22714
\(704\) −6.06373e37 −0.219033
\(705\) 5.88107e37 0.208403
\(706\) 2.28970e38 0.796004
\(707\) 1.26896e38 0.432797
\(708\) 1.64793e38 0.551431
\(709\) −3.14013e38 −1.03092 −0.515458 0.856915i \(-0.672378\pi\)
−0.515458 + 0.856915i \(0.672378\pi\)
\(710\) 1.69024e37 0.0544453
\(711\) −2.41866e37 −0.0764426
\(712\) 3.47993e37 0.107917
\(713\) 2.07402e38 0.631108
\(714\) −2.87650e38 −0.858893
\(715\) −4.08187e38 −1.19599
\(716\) −3.18162e38 −0.914791
\(717\) 2.08121e38 0.587228
\(718\) 3.62660e38 1.00420
\(719\) 1.32586e38 0.360295 0.180147 0.983640i \(-0.442343\pi\)
0.180147 + 0.983640i \(0.442343\pi\)
\(720\) 2.64837e37 0.0706301
\(721\) −1.12779e38 −0.295189
\(722\) −3.70139e38 −0.950852
\(723\) 5.72708e38 1.44400
\(724\) −2.66066e38 −0.658443
\(725\) 8.02294e37 0.194881
\(726\) 7.84354e38 1.87011
\(727\) −3.02965e38 −0.709051 −0.354525 0.935046i \(-0.615358\pi\)
−0.354525 + 0.935046i \(0.615358\pi\)
\(728\) 1.93217e38 0.443886
\(729\) −7.88385e37 −0.177794
\(730\) 5.53884e37 0.122620
\(731\) 2.23370e37 0.0485445
\(732\) 3.93289e38 0.839097
\(733\) −6.47887e38 −1.35705 −0.678525 0.734578i \(-0.737380\pi\)
−0.678525 + 0.734578i \(0.737380\pi\)
\(734\) 3.44886e38 0.709217
\(735\) 9.14760e37 0.184683
\(736\) 1.27102e38 0.251942
\(737\) 1.37966e37 0.0268510
\(738\) 2.26468e38 0.432760
\(739\) −5.25007e38 −0.985065 −0.492533 0.870294i \(-0.663929\pi\)
−0.492533 + 0.870294i \(0.663929\pi\)
\(740\) 1.76529e38 0.325228
\(741\) −1.65017e39 −2.98527
\(742\) 2.02659e38 0.360009
\(743\) 1.07441e39 1.87423 0.937113 0.349027i \(-0.113488\pi\)
0.937113 + 0.349027i \(0.113488\pi\)
\(744\) −1.16746e38 −0.199990
\(745\) −1.29273e38 −0.217469
\(746\) 4.46237e38 0.737211
\(747\) 7.53408e38 1.22237
\(748\) −6.35576e38 −1.01273
\(749\) 4.79629e38 0.750587
\(750\) 5.25632e37 0.0807895
\(751\) −6.49718e38 −0.980812 −0.490406 0.871494i \(-0.663152\pi\)
−0.490406 + 0.871494i \(0.663152\pi\)
\(752\) −6.15098e37 −0.0912019
\(753\) −9.33453e38 −1.35944
\(754\) −7.35111e38 −1.05157
\(755\) 4.01704e37 0.0564446
\(756\) −1.40185e38 −0.193490
\(757\) −1.41452e38 −0.191785 −0.0958926 0.995392i \(-0.530571\pi\)
−0.0958926 + 0.995392i \(0.530571\pi\)
\(758\) −7.16079e38 −0.953733
\(759\) −2.43817e39 −3.19007
\(760\) −1.88364e38 −0.242111
\(761\) 1.02263e39 1.29130 0.645650 0.763634i \(-0.276587\pi\)
0.645650 + 0.763634i \(0.276587\pi\)
\(762\) 3.89586e38 0.483293
\(763\) 5.51768e38 0.672473
\(764\) 5.72586e38 0.685615
\(765\) 2.77591e38 0.326570
\(766\) 6.06019e38 0.700483
\(767\) −1.16024e39 −1.31768
\(768\) −7.15453e37 −0.0798371
\(769\) 2.43418e38 0.266899 0.133449 0.991056i \(-0.457395\pi\)
0.133449 + 0.991056i \(0.457395\pi\)
\(770\) −4.23083e38 −0.455828
\(771\) −9.69375e37 −0.102626
\(772\) −8.91946e38 −0.927908
\(773\) 1.87371e39 1.91548 0.957741 0.287632i \(-0.0928679\pi\)
0.957741 + 0.287632i \(0.0928679\pi\)
\(774\) −1.86739e37 −0.0187600
\(775\) −8.97079e37 −0.0885641
\(776\) 3.27626e38 0.317867
\(777\) 1.60292e39 1.52837
\(778\) −5.62679e38 −0.527273
\(779\) −1.61075e39 −1.48344
\(780\) −4.81616e38 −0.435937
\(781\) 3.39116e38 0.301689
\(782\) 1.33223e39 1.16490
\(783\) 5.33347e38 0.458380
\(784\) −9.56742e37 −0.0808217
\(785\) −1.36635e38 −0.113454
\(786\) −3.54909e38 −0.289676
\(787\) 1.05821e39 0.849007 0.424503 0.905426i \(-0.360449\pi\)
0.424503 + 0.905426i \(0.360449\pi\)
\(788\) −5.63937e38 −0.444761
\(789\) 2.24651e39 1.74168
\(790\) 5.02072e37 0.0382649
\(791\) 1.95801e39 1.46701
\(792\) 5.31347e38 0.391371
\(793\) −2.76898e39 −2.00508
\(794\) −8.99760e38 −0.640546
\(795\) −5.05151e38 −0.353562
\(796\) 1.25844e39 0.865975
\(797\) −1.98365e39 −1.34208 −0.671041 0.741421i \(-0.734152\pi\)
−0.671041 + 0.741421i \(0.734152\pi\)
\(798\) −1.71039e39 −1.13778
\(799\) −6.44722e38 −0.421687
\(800\) −5.49756e37 −0.0353553
\(801\) −3.04936e38 −0.192828
\(802\) −1.08073e39 −0.671989
\(803\) 1.11127e39 0.679453
\(804\) 1.62784e37 0.00978714
\(805\) 8.86825e38 0.524316
\(806\) 8.21958e38 0.477889
\(807\) 9.34066e38 0.534053
\(808\) 3.30819e38 0.186010
\(809\) 2.34082e39 1.29438 0.647192 0.762327i \(-0.275943\pi\)
0.647192 + 0.762327i \(0.275943\pi\)
\(810\) 7.16780e38 0.389797
\(811\) −1.86049e39 −0.995051 −0.497525 0.867449i \(-0.665758\pi\)
−0.497525 + 0.867449i \(0.665758\pi\)
\(812\) −7.61937e38 −0.400786
\(813\) 3.19716e39 1.65403
\(814\) 3.54173e39 1.80213
\(815\) 1.23340e39 0.617270
\(816\) −7.49909e38 −0.369140
\(817\) 1.32818e38 0.0643069
\(818\) −8.76812e38 −0.417577
\(819\) −1.69310e39 −0.793142
\(820\) −4.70109e38 −0.216627
\(821\) 2.17461e39 0.985710 0.492855 0.870112i \(-0.335953\pi\)
0.492855 + 0.870112i \(0.335953\pi\)
\(822\) 2.24680e39 1.00183
\(823\) −1.47840e39 −0.648477 −0.324239 0.945975i \(-0.605108\pi\)
−0.324239 + 0.945975i \(0.605108\pi\)
\(824\) −2.94016e38 −0.126868
\(825\) 1.05459e39 0.447665
\(826\) −1.20258e39 −0.502208
\(827\) −1.81409e39 −0.745307 −0.372654 0.927971i \(-0.621552\pi\)
−0.372654 + 0.927971i \(0.621552\pi\)
\(828\) −1.11376e39 −0.450174
\(829\) 3.68398e39 1.46498 0.732489 0.680779i \(-0.238359\pi\)
0.732489 + 0.680779i \(0.238359\pi\)
\(830\) −1.56395e39 −0.611881
\(831\) −2.38531e39 −0.918188
\(832\) 5.03720e38 0.190776
\(833\) −1.00282e39 −0.373693
\(834\) −1.76061e39 −0.645539
\(835\) −1.04754e39 −0.377925
\(836\) −3.77918e39 −1.34157
\(837\) −5.96358e38 −0.208311
\(838\) 2.15216e39 0.739742
\(839\) 3.62449e39 1.22591 0.612957 0.790116i \(-0.289980\pi\)
0.612957 + 0.790116i \(0.289980\pi\)
\(840\) −4.99192e38 −0.166149
\(841\) −1.54276e38 −0.0505304
\(842\) 1.49427e39 0.481632
\(843\) −3.44891e39 −1.09398
\(844\) 3.08123e39 0.961837
\(845\) 1.93512e39 0.594487
\(846\) 5.38993e38 0.162961
\(847\) −5.72383e39 −1.70318
\(848\) 5.28335e38 0.154727
\(849\) −5.19021e38 −0.149600
\(850\) −5.76232e38 −0.163471
\(851\) −7.42382e39 −2.07290
\(852\) 4.00120e38 0.109965
\(853\) 7.16984e39 1.93953 0.969767 0.244031i \(-0.0784699\pi\)
0.969767 + 0.244031i \(0.0784699\pi\)
\(854\) −2.87003e39 −0.764196
\(855\) 1.65058e39 0.432608
\(856\) 1.25040e39 0.322592
\(857\) −5.79545e39 −1.47179 −0.735896 0.677095i \(-0.763239\pi\)
−0.735896 + 0.677095i \(0.763239\pi\)
\(858\) −9.66276e39 −2.41559
\(859\) 1.97147e39 0.485156 0.242578 0.970132i \(-0.422007\pi\)
0.242578 + 0.970132i \(0.422007\pi\)
\(860\) 3.87639e37 0.00939070
\(861\) −4.26871e39 −1.01801
\(862\) −1.10779e39 −0.260081
\(863\) −1.32958e39 −0.307305 −0.153652 0.988125i \(-0.549104\pi\)
−0.153652 + 0.988125i \(0.549104\pi\)
\(864\) −3.65465e38 −0.0831593
\(865\) 1.56302e39 0.350145
\(866\) 2.67688e39 0.590387
\(867\) −1.97747e39 −0.429390
\(868\) 8.51954e38 0.182138
\(869\) 1.00732e39 0.212031
\(870\) 1.89922e39 0.393609
\(871\) −1.14609e38 −0.0233870
\(872\) 1.43847e39 0.289020
\(873\) −2.87090e39 −0.567970
\(874\) 7.92154e39 1.54314
\(875\) −3.83580e38 −0.0735779
\(876\) 1.31118e39 0.247660
\(877\) −7.27571e39 −1.35326 −0.676629 0.736324i \(-0.736560\pi\)
−0.676629 + 0.736324i \(0.736560\pi\)
\(878\) 4.26477e39 0.781124
\(879\) 1.68172e39 0.303322
\(880\) −1.10299e39 −0.195909
\(881\) −6.56252e39 −1.14788 −0.573939 0.818898i \(-0.694585\pi\)
−0.573939 + 0.818898i \(0.694585\pi\)
\(882\) 8.38366e38 0.144414
\(883\) 6.35223e39 1.07760 0.538799 0.842434i \(-0.318878\pi\)
0.538799 + 0.842434i \(0.318878\pi\)
\(884\) 5.27979e39 0.882086
\(885\) 2.99758e39 0.493215
\(886\) 6.29329e39 1.01982
\(887\) 4.10862e39 0.655732 0.327866 0.944724i \(-0.393670\pi\)
0.327866 + 0.944724i \(0.393670\pi\)
\(888\) 4.17885e39 0.656874
\(889\) −2.84300e39 −0.440153
\(890\) 6.32996e38 0.0965241
\(891\) 1.43809e40 2.15992
\(892\) 3.68357e39 0.544933
\(893\) −3.83357e39 −0.558609
\(894\) −3.06020e39 −0.439231
\(895\) −5.78732e39 −0.818214
\(896\) 5.22102e38 0.0727105
\(897\) 2.02541e40 2.77853
\(898\) −4.89082e39 −0.660923
\(899\) −3.24134e39 −0.431487
\(900\) 4.81735e38 0.0631735
\(901\) 5.53780e39 0.715406
\(902\) −9.43190e39 −1.20036
\(903\) 3.51986e38 0.0441306
\(904\) 5.10456e39 0.630499
\(905\) −4.83971e39 −0.588929
\(906\) 9.50929e38 0.114003
\(907\) 1.26175e40 1.49030 0.745152 0.666895i \(-0.232377\pi\)
0.745152 + 0.666895i \(0.232377\pi\)
\(908\) −3.04355e39 −0.354178
\(909\) −2.89887e39 −0.332366
\(910\) 3.51459e39 0.397024
\(911\) −8.55940e39 −0.952676 −0.476338 0.879262i \(-0.658036\pi\)
−0.476338 + 0.879262i \(0.658036\pi\)
\(912\) −4.45902e39 −0.489000
\(913\) −3.13777e40 −3.39052
\(914\) −1.01899e40 −1.08492
\(915\) 7.15389e39 0.750511
\(916\) 2.64575e38 0.0273502
\(917\) 2.58995e39 0.263818
\(918\) −3.83066e39 −0.384501
\(919\) 4.48874e39 0.443982 0.221991 0.975049i \(-0.428744\pi\)
0.221991 + 0.975049i \(0.428744\pi\)
\(920\) 2.31197e39 0.225344
\(921\) 6.97378e39 0.669828
\(922\) 3.69228e39 0.349484
\(923\) −2.81707e39 −0.262769
\(924\) −1.00154e40 −0.920653
\(925\) 3.21104e39 0.290892
\(926\) −2.50726e39 −0.223847
\(927\) 2.57638e39 0.226690
\(928\) −1.98638e39 −0.172253
\(929\) 1.47342e40 1.25926 0.629631 0.776895i \(-0.283206\pi\)
0.629631 + 0.776895i \(0.283206\pi\)
\(930\) −2.12360e39 −0.178876
\(931\) −5.96284e39 −0.495031
\(932\) 3.16193e39 0.258725
\(933\) 1.42165e40 1.14654
\(934\) −2.03669e39 −0.161898
\(935\) −1.15611e40 −0.905817
\(936\) −4.41395e39 −0.340882
\(937\) −1.33104e40 −1.01323 −0.506614 0.862173i \(-0.669103\pi\)
−0.506614 + 0.862173i \(0.669103\pi\)
\(938\) −1.18792e38 −0.00891350
\(939\) −3.94974e39 −0.292135
\(940\) −1.11886e39 −0.0815734
\(941\) −1.28931e40 −0.926612 −0.463306 0.886198i \(-0.653337\pi\)
−0.463306 + 0.886198i \(0.653337\pi\)
\(942\) −3.23447e39 −0.229148
\(943\) 1.97702e40 1.38071
\(944\) −3.13515e39 −0.215842
\(945\) −2.54995e39 −0.173062
\(946\) 7.77727e38 0.0520352
\(947\) −1.41966e40 −0.936396 −0.468198 0.883624i \(-0.655097\pi\)
−0.468198 + 0.883624i \(0.655097\pi\)
\(948\) 1.18852e39 0.0772850
\(949\) −9.23142e39 −0.591800
\(950\) −3.42632e39 −0.216551
\(951\) 2.12450e40 1.32379
\(952\) 5.47246e39 0.336190
\(953\) 1.53298e40 0.928501 0.464251 0.885704i \(-0.346324\pi\)
0.464251 + 0.885704i \(0.346324\pi\)
\(954\) −4.62965e39 −0.276468
\(955\) 1.04153e40 0.613233
\(956\) −3.95944e39 −0.229854
\(957\) 3.81044e40 2.18104
\(958\) 1.99454e40 1.12566
\(959\) −1.63961e40 −0.912406
\(960\) −1.30140e39 −0.0714085
\(961\) −1.48584e40 −0.803910
\(962\) −2.94215e40 −1.56964
\(963\) −1.09569e40 −0.576413
\(964\) −1.08956e40 −0.565212
\(965\) −1.62244e40 −0.829946
\(966\) 2.09932e40 1.05898
\(967\) −6.60284e38 −0.0328453 −0.0164226 0.999865i \(-0.505228\pi\)
−0.0164226 + 0.999865i \(0.505228\pi\)
\(968\) −1.49221e40 −0.732004
\(969\) −4.67377e40 −2.26098
\(970\) 5.95949e39 0.284309
\(971\) 1.76228e40 0.829117 0.414558 0.910023i \(-0.363936\pi\)
0.414558 + 0.910023i \(0.363936\pi\)
\(972\) 1.18986e40 0.552077
\(973\) 1.28481e40 0.587916
\(974\) 9.93633e38 0.0448415
\(975\) −8.76054e39 −0.389914
\(976\) −7.48221e39 −0.328441
\(977\) 2.50038e39 0.108250 0.0541251 0.998534i \(-0.482763\pi\)
0.0541251 + 0.998534i \(0.482763\pi\)
\(978\) 2.91974e40 1.24672
\(979\) 1.26999e40 0.534853
\(980\) −1.74030e39 −0.0722892
\(981\) −1.26049e40 −0.516425
\(982\) −5.25355e39 −0.212299
\(983\) 2.38859e40 0.952069 0.476034 0.879427i \(-0.342074\pi\)
0.476034 + 0.879427i \(0.342074\pi\)
\(984\) −1.11286e40 −0.437529
\(985\) −1.02580e40 −0.397807
\(986\) −2.08205e40 −0.796439
\(987\) −1.01595e40 −0.383346
\(988\) 3.13940e40 1.16850
\(989\) −1.63019e39 −0.0598535
\(990\) 9.66515e39 0.350053
\(991\) −1.88335e40 −0.672879 −0.336440 0.941705i \(-0.609223\pi\)
−0.336440 + 0.941705i \(0.609223\pi\)
\(992\) 2.22106e39 0.0782803
\(993\) 1.38778e40 0.482510
\(994\) −2.91987e39 −0.100149
\(995\) 2.28908e40 0.774551
\(996\) −3.70223e40 −1.23584
\(997\) −3.07289e40 −1.01196 −0.505979 0.862546i \(-0.668869\pi\)
−0.505979 + 0.862546i \(0.668869\pi\)
\(998\) 3.35107e40 1.08873
\(999\) 2.13463e40 0.684208
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 10.28.a.a.1.1 2
5.2 odd 4 50.28.b.e.49.2 4
5.3 odd 4 50.28.b.e.49.3 4
5.4 even 2 50.28.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.28.a.a.1.1 2 1.1 even 1 trivial
50.28.a.e.1.2 2 5.4 even 2
50.28.b.e.49.2 4 5.2 odd 4
50.28.b.e.49.3 4 5.3 odd 4