Properties

Label 285.10.a.c.1.5
Level $285$
Weight $10$
Character 285.1
Self dual yes
Analytic conductor $146.785$
Analytic rank $1$
Dimension $12$
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] = [285,10,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(146.785213307\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 4398 x^{10} + 4080 x^{9} + 7026370 x^{8} + 7294322 x^{7} - 5023445596 x^{6} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{6}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-14.0719\) of defining polynomial
Character \(\chi\) \(=\) 285.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-17.0719 q^{2} -81.0000 q^{3} -220.549 q^{4} +625.000 q^{5} +1382.83 q^{6} +11804.8 q^{7} +12506.0 q^{8} +6561.00 q^{9} -10670.0 q^{10} +10723.5 q^{11} +17864.5 q^{12} +55804.2 q^{13} -201531. q^{14} -50625.0 q^{15} -100581. q^{16} +39049.4 q^{17} -112009. q^{18} +130321. q^{19} -137843. q^{20} -956188. q^{21} -183072. q^{22} +85590.2 q^{23} -1.01299e6 q^{24} +390625. q^{25} -952685. q^{26} -531441. q^{27} -2.60354e6 q^{28} -4.91333e6 q^{29} +864266. q^{30} -7.66372e6 q^{31} -4.68598e6 q^{32} -868607. q^{33} -666649. q^{34} +7.37800e6 q^{35} -1.44702e6 q^{36} -1.74637e7 q^{37} -2.22483e6 q^{38} -4.52014e6 q^{39} +7.81627e6 q^{40} +2.56765e7 q^{41} +1.63240e7 q^{42} -3.52977e7 q^{43} -2.36507e6 q^{44} +4.10062e6 q^{45} -1.46119e6 q^{46} -9.69853e6 q^{47} +8.14703e6 q^{48} +9.89996e7 q^{49} -6.66872e6 q^{50} -3.16300e6 q^{51} -1.23076e7 q^{52} -6.79387e7 q^{53} +9.07272e6 q^{54} +6.70222e6 q^{55} +1.47631e8 q^{56} -1.05560e7 q^{57} +8.38799e7 q^{58} -2.56493e7 q^{59} +1.11653e7 q^{60} -7.01660e6 q^{61} +1.30835e8 q^{62} +7.74513e7 q^{63} +1.31496e8 q^{64} +3.48776e7 q^{65} +1.48288e7 q^{66} -1.24277e8 q^{67} -8.61233e6 q^{68} -6.93281e6 q^{69} -1.25957e8 q^{70} +6.68509e7 q^{71} +8.20520e7 q^{72} +4.31812e8 q^{73} +2.98138e8 q^{74} -3.16406e7 q^{75} -2.87422e7 q^{76} +1.26589e8 q^{77} +7.71675e7 q^{78} -1.81702e8 q^{79} -6.28629e7 q^{80} +4.30467e7 q^{81} -4.38346e8 q^{82} -6.01036e8 q^{83} +2.10887e8 q^{84} +2.44059e7 q^{85} +6.02600e8 q^{86} +3.97979e8 q^{87} +1.34109e8 q^{88} -4.69609e8 q^{89} -7.00056e7 q^{90} +6.58757e8 q^{91} -1.88769e7 q^{92} +6.20762e8 q^{93} +1.65573e8 q^{94} +8.14506e7 q^{95} +3.79565e8 q^{96} -3.63557e8 q^{97} -1.69011e9 q^{98} +7.03572e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 33 q^{2} - 972 q^{3} + 2751 q^{4} + 7500 q^{5} + 2673 q^{6} - 3450 q^{7} - 18327 q^{8} + 78732 q^{9} - 20625 q^{10} - 8180 q^{11} - 222831 q^{12} - 54754 q^{13} - 198168 q^{14} - 607500 q^{15} + 319475 q^{16}+ \cdots - 53668980 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −17.0719 −0.754480 −0.377240 0.926116i \(-0.623127\pi\)
−0.377240 + 0.926116i \(0.623127\pi\)
\(3\) −81.0000 −0.577350
\(4\) −220.549 −0.430761
\(5\) 625.000 0.447214
\(6\) 1382.83 0.435599
\(7\) 11804.8 1.85831 0.929153 0.369695i \(-0.120538\pi\)
0.929153 + 0.369695i \(0.120538\pi\)
\(8\) 12506.0 1.07948
\(9\) 6561.00 0.333333
\(10\) −10670.0 −0.337413
\(11\) 10723.5 0.220837 0.110418 0.993885i \(-0.464781\pi\)
0.110418 + 0.993885i \(0.464781\pi\)
\(12\) 17864.5 0.248700
\(13\) 55804.2 0.541903 0.270952 0.962593i \(-0.412662\pi\)
0.270952 + 0.962593i \(0.412662\pi\)
\(14\) −201531. −1.40205
\(15\) −50625.0 −0.258199
\(16\) −100581. −0.383685
\(17\) 39049.4 0.113395 0.0566976 0.998391i \(-0.481943\pi\)
0.0566976 + 0.998391i \(0.481943\pi\)
\(18\) −112009. −0.251493
\(19\) 130321. 0.229416
\(20\) −137843. −0.192642
\(21\) −956188. −1.07289
\(22\) −183072. −0.166617
\(23\) 85590.2 0.0637748 0.0318874 0.999491i \(-0.489848\pi\)
0.0318874 + 0.999491i \(0.489848\pi\)
\(24\) −1.01299e6 −0.623238
\(25\) 390625. 0.200000
\(26\) −952685. −0.408855
\(27\) −531441. −0.192450
\(28\) −2.60354e6 −0.800485
\(29\) −4.91333e6 −1.28998 −0.644992 0.764189i \(-0.723139\pi\)
−0.644992 + 0.764189i \(0.723139\pi\)
\(30\) 864266. 0.194806
\(31\) −7.66372e6 −1.49043 −0.745216 0.666823i \(-0.767654\pi\)
−0.745216 + 0.666823i \(0.767654\pi\)
\(32\) −4.68598e6 −0.789997
\(33\) −868607. −0.127500
\(34\) −666649. −0.0855543
\(35\) 7.37800e6 0.831060
\(36\) −1.44702e6 −0.143587
\(37\) −1.74637e7 −1.53189 −0.765944 0.642907i \(-0.777728\pi\)
−0.765944 + 0.642907i \(0.777728\pi\)
\(38\) −2.22483e6 −0.173089
\(39\) −4.52014e6 −0.312868
\(40\) 7.81627e6 0.482758
\(41\) 2.56765e7 1.41908 0.709541 0.704664i \(-0.248902\pi\)
0.709541 + 0.704664i \(0.248902\pi\)
\(42\) 1.63240e7 0.809476
\(43\) −3.52977e7 −1.57449 −0.787243 0.616643i \(-0.788492\pi\)
−0.787243 + 0.616643i \(0.788492\pi\)
\(44\) −2.36507e6 −0.0951278
\(45\) 4.10062e6 0.149071
\(46\) −1.46119e6 −0.0481168
\(47\) −9.69853e6 −0.289912 −0.144956 0.989438i \(-0.546304\pi\)
−0.144956 + 0.989438i \(0.546304\pi\)
\(48\) 8.14703e6 0.221520
\(49\) 9.89996e7 2.45330
\(50\) −6.66872e6 −0.150896
\(51\) −3.16300e6 −0.0654687
\(52\) −1.23076e7 −0.233431
\(53\) −6.79387e7 −1.18270 −0.591352 0.806414i \(-0.701406\pi\)
−0.591352 + 0.806414i \(0.701406\pi\)
\(54\) 9.07272e6 0.145200
\(55\) 6.70222e6 0.0987612
\(56\) 1.47631e8 2.00600
\(57\) −1.05560e7 −0.132453
\(58\) 8.38799e7 0.973267
\(59\) −2.56493e7 −0.275576 −0.137788 0.990462i \(-0.543999\pi\)
−0.137788 + 0.990462i \(0.543999\pi\)
\(60\) 1.11653e7 0.111222
\(61\) −7.01660e6 −0.0648847 −0.0324424 0.999474i \(-0.510329\pi\)
−0.0324424 + 0.999474i \(0.510329\pi\)
\(62\) 1.30835e8 1.12450
\(63\) 7.74513e7 0.619435
\(64\) 1.31496e8 0.979721
\(65\) 3.48776e7 0.242347
\(66\) 1.48288e7 0.0961963
\(67\) −1.24277e8 −0.753451 −0.376725 0.926325i \(-0.622950\pi\)
−0.376725 + 0.926325i \(0.622950\pi\)
\(68\) −8.61233e6 −0.0488462
\(69\) −6.93281e6 −0.0368204
\(70\) −1.25957e8 −0.627018
\(71\) 6.68509e7 0.312208 0.156104 0.987741i \(-0.450106\pi\)
0.156104 + 0.987741i \(0.450106\pi\)
\(72\) 8.20520e7 0.359827
\(73\) 4.31812e8 1.77968 0.889839 0.456275i \(-0.150817\pi\)
0.889839 + 0.456275i \(0.150817\pi\)
\(74\) 2.98138e8 1.15578
\(75\) −3.16406e7 −0.115470
\(76\) −2.87422e7 −0.0988233
\(77\) 1.26589e8 0.410382
\(78\) 7.71675e7 0.236053
\(79\) −1.81702e8 −0.524853 −0.262427 0.964952i \(-0.584523\pi\)
−0.262427 + 0.964952i \(0.584523\pi\)
\(80\) −6.28629e7 −0.171589
\(81\) 4.30467e7 0.111111
\(82\) −4.38346e8 −1.07067
\(83\) −6.01036e8 −1.39011 −0.695055 0.718956i \(-0.744620\pi\)
−0.695055 + 0.718956i \(0.744620\pi\)
\(84\) 2.10887e8 0.462160
\(85\) 2.44059e7 0.0507118
\(86\) 6.02600e8 1.18792
\(87\) 3.97979e8 0.744773
\(88\) 1.34109e8 0.238389
\(89\) −4.69609e8 −0.793380 −0.396690 0.917953i \(-0.629841\pi\)
−0.396690 + 0.917953i \(0.629841\pi\)
\(90\) −7.00056e7 −0.112471
\(91\) 6.58757e8 1.00702
\(92\) −1.88769e7 −0.0274717
\(93\) 6.20762e8 0.860501
\(94\) 1.65573e8 0.218732
\(95\) 8.14506e7 0.102598
\(96\) 3.79565e8 0.456105
\(97\) −3.63557e8 −0.416966 −0.208483 0.978026i \(-0.566853\pi\)
−0.208483 + 0.978026i \(0.566853\pi\)
\(98\) −1.69011e9 −1.85097
\(99\) 7.03572e7 0.0736123
\(100\) −8.61521e7 −0.0861521
\(101\) −1.10033e9 −1.05215 −0.526074 0.850439i \(-0.676336\pi\)
−0.526074 + 0.850439i \(0.676336\pi\)
\(102\) 5.39985e7 0.0493948
\(103\) −4.43205e8 −0.388005 −0.194002 0.981001i \(-0.562147\pi\)
−0.194002 + 0.981001i \(0.562147\pi\)
\(104\) 6.97889e8 0.584974
\(105\) −5.97618e8 −0.479813
\(106\) 1.15984e9 0.892326
\(107\) −1.53314e9 −1.13072 −0.565360 0.824844i \(-0.691263\pi\)
−0.565360 + 0.824844i \(0.691263\pi\)
\(108\) 1.17209e8 0.0828999
\(109\) −1.91067e9 −1.29648 −0.648240 0.761436i \(-0.724495\pi\)
−0.648240 + 0.761436i \(0.724495\pi\)
\(110\) −1.14420e8 −0.0745133
\(111\) 1.41456e9 0.884436
\(112\) −1.18733e9 −0.713004
\(113\) 1.84040e9 1.06184 0.530919 0.847423i \(-0.321847\pi\)
0.530919 + 0.847423i \(0.321847\pi\)
\(114\) 1.80211e8 0.0999333
\(115\) 5.34939e7 0.0285209
\(116\) 1.08363e9 0.555675
\(117\) 3.66131e8 0.180634
\(118\) 4.37883e8 0.207916
\(119\) 4.60970e8 0.210723
\(120\) −6.33118e8 −0.278720
\(121\) −2.24295e9 −0.951231
\(122\) 1.19787e8 0.0489542
\(123\) −2.07979e9 −0.819308
\(124\) 1.69023e9 0.642020
\(125\) 2.44141e8 0.0894427
\(126\) −1.32224e9 −0.467351
\(127\) 1.51876e9 0.518049 0.259025 0.965871i \(-0.416599\pi\)
0.259025 + 0.965871i \(0.416599\pi\)
\(128\) 1.54333e8 0.0508177
\(129\) 2.85912e9 0.909030
\(130\) −5.95428e8 −0.182845
\(131\) 5.54699e9 1.64565 0.822823 0.568298i \(-0.192398\pi\)
0.822823 + 0.568298i \(0.192398\pi\)
\(132\) 1.91571e8 0.0549221
\(133\) 1.53841e9 0.426325
\(134\) 2.12165e9 0.568463
\(135\) −3.32151e8 −0.0860663
\(136\) 4.88353e8 0.122408
\(137\) 6.74188e8 0.163508 0.0817539 0.996653i \(-0.473948\pi\)
0.0817539 + 0.996653i \(0.473948\pi\)
\(138\) 1.18356e8 0.0277802
\(139\) 3.90541e9 0.887360 0.443680 0.896185i \(-0.353673\pi\)
0.443680 + 0.896185i \(0.353673\pi\)
\(140\) −1.62721e9 −0.357988
\(141\) 7.85581e8 0.167381
\(142\) −1.14127e9 −0.235555
\(143\) 5.98419e8 0.119672
\(144\) −6.59909e8 −0.127895
\(145\) −3.07083e9 −0.576899
\(146\) −7.37185e9 −1.34273
\(147\) −8.01897e9 −1.41641
\(148\) 3.85160e9 0.659877
\(149\) −3.80714e9 −0.632791 −0.316396 0.948627i \(-0.602473\pi\)
−0.316396 + 0.948627i \(0.602473\pi\)
\(150\) 5.40166e8 0.0871198
\(151\) −5.03611e9 −0.788314 −0.394157 0.919043i \(-0.628963\pi\)
−0.394157 + 0.919043i \(0.628963\pi\)
\(152\) 1.62980e9 0.247650
\(153\) 2.56203e8 0.0377984
\(154\) −2.16112e9 −0.309625
\(155\) −4.78983e9 −0.666542
\(156\) 9.96914e8 0.134771
\(157\) −1.27343e10 −1.67273 −0.836363 0.548175i \(-0.815323\pi\)
−0.836363 + 0.548175i \(0.815323\pi\)
\(158\) 3.10200e9 0.395991
\(159\) 5.50304e9 0.682834
\(160\) −2.92874e9 −0.353298
\(161\) 1.01038e9 0.118513
\(162\) −7.34890e8 −0.0838311
\(163\) −7.30036e9 −0.810028 −0.405014 0.914310i \(-0.632733\pi\)
−0.405014 + 0.914310i \(0.632733\pi\)
\(164\) −5.66293e9 −0.611285
\(165\) −5.42880e8 −0.0570198
\(166\) 1.02608e10 1.04881
\(167\) −1.73438e9 −0.172552 −0.0862758 0.996271i \(-0.527497\pi\)
−0.0862758 + 0.996271i \(0.527497\pi\)
\(168\) −1.19581e10 −1.15817
\(169\) −7.49039e9 −0.706341
\(170\) −4.16655e8 −0.0382611
\(171\) 8.55036e8 0.0764719
\(172\) 7.78489e9 0.678226
\(173\) −5.91992e9 −0.502468 −0.251234 0.967926i \(-0.580836\pi\)
−0.251234 + 0.967926i \(0.580836\pi\)
\(174\) −6.79427e9 −0.561916
\(175\) 4.61125e9 0.371661
\(176\) −1.07858e9 −0.0847317
\(177\) 2.07759e9 0.159104
\(178\) 8.01713e9 0.598589
\(179\) 1.13075e10 0.823241 0.411620 0.911355i \(-0.364963\pi\)
0.411620 + 0.911355i \(0.364963\pi\)
\(180\) −9.04391e8 −0.0642140
\(181\) −1.96730e10 −1.36244 −0.681219 0.732080i \(-0.738550\pi\)
−0.681219 + 0.732080i \(0.738550\pi\)
\(182\) −1.12463e10 −0.759778
\(183\) 5.68344e8 0.0374612
\(184\) 1.07039e9 0.0688436
\(185\) −1.09148e10 −0.685082
\(186\) −1.05976e10 −0.649231
\(187\) 4.18748e8 0.0250418
\(188\) 2.13901e9 0.124883
\(189\) −6.27355e9 −0.357631
\(190\) −1.39052e9 −0.0774080
\(191\) −8.57929e9 −0.466446 −0.233223 0.972423i \(-0.574927\pi\)
−0.233223 + 0.972423i \(0.574927\pi\)
\(192\) −1.06512e10 −0.565642
\(193\) 2.00369e10 1.03949 0.519747 0.854320i \(-0.326026\pi\)
0.519747 + 0.854320i \(0.326026\pi\)
\(194\) 6.20663e9 0.314592
\(195\) −2.82509e9 −0.139919
\(196\) −2.18343e10 −1.05679
\(197\) −9.28019e9 −0.438994 −0.219497 0.975613i \(-0.570442\pi\)
−0.219497 + 0.975613i \(0.570442\pi\)
\(198\) −1.20113e9 −0.0555389
\(199\) 3.12211e10 1.41127 0.705635 0.708576i \(-0.250662\pi\)
0.705635 + 0.708576i \(0.250662\pi\)
\(200\) 4.88517e9 0.215896
\(201\) 1.00665e10 0.435005
\(202\) 1.87847e10 0.793824
\(203\) −5.80008e10 −2.39719
\(204\) 6.97599e8 0.0282013
\(205\) 1.60478e10 0.634633
\(206\) 7.56636e9 0.292742
\(207\) 5.61557e8 0.0212583
\(208\) −5.61282e9 −0.207920
\(209\) 1.39750e9 0.0506634
\(210\) 1.02025e10 0.362009
\(211\) 2.95630e10 1.02678 0.513391 0.858155i \(-0.328389\pi\)
0.513391 + 0.858155i \(0.328389\pi\)
\(212\) 1.49838e10 0.509462
\(213\) −5.41492e9 −0.180254
\(214\) 2.61737e10 0.853105
\(215\) −2.20611e10 −0.704131
\(216\) −6.64622e9 −0.207746
\(217\) −9.04687e10 −2.76968
\(218\) 3.26188e10 0.978167
\(219\) −3.49767e10 −1.02750
\(220\) −1.47817e9 −0.0425424
\(221\) 2.17912e9 0.0614492
\(222\) −2.41492e10 −0.667289
\(223\) 3.58409e10 0.970527 0.485263 0.874368i \(-0.338724\pi\)
0.485263 + 0.874368i \(0.338724\pi\)
\(224\) −5.53171e10 −1.46806
\(225\) 2.56289e9 0.0666667
\(226\) −3.14191e10 −0.801135
\(227\) −7.63678e10 −1.90895 −0.954474 0.298296i \(-0.903582\pi\)
−0.954474 + 0.298296i \(0.903582\pi\)
\(228\) 2.32812e9 0.0570556
\(229\) −7.53272e10 −1.81006 −0.905028 0.425351i \(-0.860151\pi\)
−0.905028 + 0.425351i \(0.860151\pi\)
\(230\) −9.13244e8 −0.0215185
\(231\) −1.02537e10 −0.236934
\(232\) −6.14462e10 −1.39251
\(233\) 5.23818e10 1.16434 0.582168 0.813068i \(-0.302205\pi\)
0.582168 + 0.813068i \(0.302205\pi\)
\(234\) −6.25057e9 −0.136285
\(235\) −6.06158e9 −0.129652
\(236\) 5.65694e9 0.118707
\(237\) 1.47179e10 0.303024
\(238\) −7.86965e9 −0.158986
\(239\) −2.49810e10 −0.495243 −0.247622 0.968857i \(-0.579649\pi\)
−0.247622 + 0.968857i \(0.579649\pi\)
\(240\) 5.09189e9 0.0990669
\(241\) 8.94691e10 1.70843 0.854213 0.519923i \(-0.174039\pi\)
0.854213 + 0.519923i \(0.174039\pi\)
\(242\) 3.82915e10 0.717684
\(243\) −3.48678e9 −0.0641500
\(244\) 1.54751e9 0.0279498
\(245\) 6.18748e10 1.09715
\(246\) 3.55061e10 0.618151
\(247\) 7.27246e9 0.124321
\(248\) −9.58427e10 −1.60889
\(249\) 4.86840e10 0.802581
\(250\) −4.16795e9 −0.0674827
\(251\) −3.78202e8 −0.00601440 −0.00300720 0.999995i \(-0.500957\pi\)
−0.00300720 + 0.999995i \(0.500957\pi\)
\(252\) −1.70818e10 −0.266828
\(253\) 9.17831e8 0.0140838
\(254\) −2.59281e10 −0.390858
\(255\) −1.97688e9 −0.0292785
\(256\) −6.99607e10 −1.01806
\(257\) −6.92550e10 −0.990266 −0.495133 0.868817i \(-0.664881\pi\)
−0.495133 + 0.868817i \(0.664881\pi\)
\(258\) −4.88106e10 −0.685844
\(259\) −2.06155e11 −2.84672
\(260\) −7.69224e9 −0.104393
\(261\) −3.22363e10 −0.429995
\(262\) −9.46977e10 −1.24161
\(263\) 1.22467e11 1.57840 0.789200 0.614136i \(-0.210495\pi\)
0.789200 + 0.614136i \(0.210495\pi\)
\(264\) −1.08628e10 −0.137634
\(265\) −4.24617e10 −0.528921
\(266\) −2.62637e10 −0.321653
\(267\) 3.80383e10 0.458058
\(268\) 2.74093e10 0.324557
\(269\) −2.06902e10 −0.240924 −0.120462 0.992718i \(-0.538438\pi\)
−0.120462 + 0.992718i \(0.538438\pi\)
\(270\) 5.67045e9 0.0649353
\(271\) −3.14415e10 −0.354113 −0.177057 0.984201i \(-0.556658\pi\)
−0.177057 + 0.984201i \(0.556658\pi\)
\(272\) −3.92761e9 −0.0435080
\(273\) −5.33593e10 −0.581405
\(274\) −1.15097e10 −0.123363
\(275\) 4.18889e9 0.0441674
\(276\) 1.52903e9 0.0158608
\(277\) 1.81200e11 1.84926 0.924631 0.380865i \(-0.124374\pi\)
0.924631 + 0.380865i \(0.124374\pi\)
\(278\) −6.66729e10 −0.669495
\(279\) −5.02817e10 −0.496811
\(280\) 9.22694e10 0.897112
\(281\) 1.51008e11 1.44485 0.722423 0.691451i \(-0.243028\pi\)
0.722423 + 0.691451i \(0.243028\pi\)
\(282\) −1.34114e10 −0.126285
\(283\) −2.02089e11 −1.87285 −0.936425 0.350867i \(-0.885887\pi\)
−0.936425 + 0.350867i \(0.885887\pi\)
\(284\) −1.47439e10 −0.134487
\(285\) −6.59750e9 −0.0592349
\(286\) −1.02162e10 −0.0902902
\(287\) 3.03105e11 2.63709
\(288\) −3.07447e10 −0.263332
\(289\) −1.17063e11 −0.987142
\(290\) 5.24249e10 0.435258
\(291\) 2.94482e10 0.240735
\(292\) −9.52358e10 −0.766615
\(293\) 1.59940e11 1.26781 0.633905 0.773411i \(-0.281451\pi\)
0.633905 + 0.773411i \(0.281451\pi\)
\(294\) 1.36899e11 1.06866
\(295\) −1.60308e10 −0.123241
\(296\) −2.18401e11 −1.65364
\(297\) −5.69893e9 −0.0425001
\(298\) 6.49952e10 0.477428
\(299\) 4.77629e9 0.0345598
\(300\) 6.97832e9 0.0497400
\(301\) −4.16682e11 −2.92588
\(302\) 8.59761e10 0.594766
\(303\) 8.91267e10 0.607458
\(304\) −1.31078e10 −0.0880233
\(305\) −4.38537e9 −0.0290173
\(306\) −4.37388e9 −0.0285181
\(307\) −2.20955e11 −1.41965 −0.709826 0.704378i \(-0.751226\pi\)
−0.709826 + 0.704378i \(0.751226\pi\)
\(308\) −2.79192e10 −0.176777
\(309\) 3.58996e10 0.224015
\(310\) 8.17716e10 0.502892
\(311\) −8.57467e10 −0.519752 −0.259876 0.965642i \(-0.583682\pi\)
−0.259876 + 0.965642i \(0.583682\pi\)
\(312\) −5.65290e10 −0.337735
\(313\) −1.99701e11 −1.17606 −0.588031 0.808838i \(-0.700097\pi\)
−0.588031 + 0.808838i \(0.700097\pi\)
\(314\) 2.17398e11 1.26204
\(315\) 4.84070e10 0.277020
\(316\) 4.00743e10 0.226086
\(317\) 1.01674e11 0.565513 0.282757 0.959192i \(-0.408751\pi\)
0.282757 + 0.959192i \(0.408751\pi\)
\(318\) −9.39474e10 −0.515184
\(319\) −5.26883e10 −0.284876
\(320\) 8.21850e10 0.438145
\(321\) 1.24184e11 0.652821
\(322\) −1.72490e10 −0.0894157
\(323\) 5.08896e9 0.0260146
\(324\) −9.49393e9 −0.0478623
\(325\) 2.17985e10 0.108381
\(326\) 1.24631e11 0.611150
\(327\) 1.54764e11 0.748523
\(328\) 3.21110e11 1.53187
\(329\) −1.14489e11 −0.538745
\(330\) 9.26800e9 0.0430203
\(331\) 6.55209e10 0.300022 0.150011 0.988684i \(-0.452069\pi\)
0.150011 + 0.988684i \(0.452069\pi\)
\(332\) 1.32558e11 0.598805
\(333\) −1.14579e11 −0.510630
\(334\) 2.96091e10 0.130187
\(335\) −7.76732e10 −0.336953
\(336\) 9.61740e10 0.411653
\(337\) −1.63149e11 −0.689050 −0.344525 0.938777i \(-0.611960\pi\)
−0.344525 + 0.938777i \(0.611960\pi\)
\(338\) 1.27875e11 0.532920
\(339\) −1.49072e11 −0.613052
\(340\) −5.38271e9 −0.0218447
\(341\) −8.21823e10 −0.329142
\(342\) −1.45971e10 −0.0576965
\(343\) 6.92304e11 2.70068
\(344\) −4.41434e11 −1.69962
\(345\) −4.33301e9 −0.0164666
\(346\) 1.01064e11 0.379102
\(347\) 4.60263e11 1.70421 0.852105 0.523370i \(-0.175325\pi\)
0.852105 + 0.523370i \(0.175325\pi\)
\(348\) −8.77741e10 −0.320819
\(349\) −1.56693e11 −0.565372 −0.282686 0.959212i \(-0.591225\pi\)
−0.282686 + 0.959212i \(0.591225\pi\)
\(350\) −7.87229e10 −0.280411
\(351\) −2.96566e10 −0.104289
\(352\) −5.02504e10 −0.174461
\(353\) 1.39647e11 0.478680 0.239340 0.970936i \(-0.423069\pi\)
0.239340 + 0.970936i \(0.423069\pi\)
\(354\) −3.54685e10 −0.120041
\(355\) 4.17818e10 0.139624
\(356\) 1.03572e11 0.341757
\(357\) −3.73386e10 −0.121661
\(358\) −1.93040e11 −0.621118
\(359\) 3.87102e11 1.22999 0.614993 0.788533i \(-0.289159\pi\)
0.614993 + 0.788533i \(0.289159\pi\)
\(360\) 5.12825e10 0.160919
\(361\) 1.69836e10 0.0526316
\(362\) 3.35856e11 1.02793
\(363\) 1.81679e11 0.549194
\(364\) −1.45289e11 −0.433786
\(365\) 2.69882e11 0.795896
\(366\) −9.70273e9 −0.0282637
\(367\) 3.70839e11 1.06706 0.533529 0.845782i \(-0.320866\pi\)
0.533529 + 0.845782i \(0.320866\pi\)
\(368\) −8.60872e9 −0.0244694
\(369\) 1.68463e11 0.473028
\(370\) 1.86336e11 0.516880
\(371\) −8.02003e11 −2.19783
\(372\) −1.36909e11 −0.370670
\(373\) −1.58356e11 −0.423590 −0.211795 0.977314i \(-0.567931\pi\)
−0.211795 + 0.977314i \(0.567931\pi\)
\(374\) −7.14884e9 −0.0188935
\(375\) −1.97754e10 −0.0516398
\(376\) −1.21290e11 −0.312954
\(377\) −2.74184e11 −0.699047
\(378\) 1.07102e11 0.269825
\(379\) −5.77139e11 −1.43682 −0.718412 0.695617i \(-0.755131\pi\)
−0.718412 + 0.695617i \(0.755131\pi\)
\(380\) −1.79639e10 −0.0441951
\(381\) −1.23019e11 −0.299096
\(382\) 1.46465e11 0.351924
\(383\) 3.76018e11 0.892923 0.446461 0.894803i \(-0.352684\pi\)
0.446461 + 0.894803i \(0.352684\pi\)
\(384\) −1.25010e10 −0.0293396
\(385\) 7.91183e10 0.183529
\(386\) −3.42068e11 −0.784277
\(387\) −2.31588e11 −0.524828
\(388\) 8.01824e10 0.179612
\(389\) 7.59614e11 1.68198 0.840988 0.541053i \(-0.181974\pi\)
0.840988 + 0.541053i \(0.181974\pi\)
\(390\) 4.82297e10 0.105566
\(391\) 3.34225e9 0.00723175
\(392\) 1.23809e12 2.64829
\(393\) −4.49306e11 −0.950114
\(394\) 1.58431e11 0.331212
\(395\) −1.13564e11 −0.234721
\(396\) −1.55172e10 −0.0317093
\(397\) 4.42109e11 0.893248 0.446624 0.894722i \(-0.352626\pi\)
0.446624 + 0.894722i \(0.352626\pi\)
\(398\) −5.33005e11 −1.06477
\(399\) −1.24611e11 −0.246139
\(400\) −3.92893e10 −0.0767369
\(401\) 2.21483e11 0.427752 0.213876 0.976861i \(-0.431391\pi\)
0.213876 + 0.976861i \(0.431391\pi\)
\(402\) −1.71854e11 −0.328202
\(403\) −4.27668e11 −0.807670
\(404\) 2.42677e11 0.453224
\(405\) 2.69042e10 0.0496904
\(406\) 9.90185e11 1.80863
\(407\) −1.87272e11 −0.338297
\(408\) −3.95566e10 −0.0706721
\(409\) −1.65746e11 −0.292879 −0.146439 0.989220i \(-0.546781\pi\)
−0.146439 + 0.989220i \(0.546781\pi\)
\(410\) −2.73967e11 −0.478818
\(411\) −5.46092e10 −0.0944012
\(412\) 9.77486e10 0.167137
\(413\) −3.02785e11 −0.512105
\(414\) −9.58687e9 −0.0160389
\(415\) −3.75648e11 −0.621676
\(416\) −2.61497e11 −0.428102
\(417\) −3.16338e11 −0.512318
\(418\) −2.38581e10 −0.0382245
\(419\) 1.50545e11 0.238619 0.119309 0.992857i \(-0.461932\pi\)
0.119309 + 0.992857i \(0.461932\pi\)
\(420\) 1.31804e11 0.206684
\(421\) 1.11582e12 1.73111 0.865553 0.500817i \(-0.166967\pi\)
0.865553 + 0.500817i \(0.166967\pi\)
\(422\) −5.04698e11 −0.774686
\(423\) −6.36321e10 −0.0966372
\(424\) −8.49643e11 −1.27670
\(425\) 1.52537e10 0.0226790
\(426\) 9.24431e10 0.135998
\(427\) −8.28295e10 −0.120576
\(428\) 3.38133e11 0.487070
\(429\) −4.84719e10 −0.0690928
\(430\) 3.76625e11 0.531253
\(431\) 5.20796e11 0.726976 0.363488 0.931599i \(-0.381586\pi\)
0.363488 + 0.931599i \(0.381586\pi\)
\(432\) 5.34527e10 0.0738401
\(433\) −7.81289e11 −1.06811 −0.534055 0.845450i \(-0.679333\pi\)
−0.534055 + 0.845450i \(0.679333\pi\)
\(434\) 1.54447e12 2.08967
\(435\) 2.48737e11 0.333073
\(436\) 4.21397e11 0.558472
\(437\) 1.11542e10 0.0146309
\(438\) 5.97120e11 0.775226
\(439\) −5.76901e11 −0.741328 −0.370664 0.928767i \(-0.620870\pi\)
−0.370664 + 0.928767i \(0.620870\pi\)
\(440\) 8.38181e10 0.106611
\(441\) 6.49536e11 0.817767
\(442\) −3.72018e10 −0.0463622
\(443\) −1.06242e11 −0.131063 −0.0655314 0.997851i \(-0.520874\pi\)
−0.0655314 + 0.997851i \(0.520874\pi\)
\(444\) −3.11979e11 −0.380980
\(445\) −2.93506e11 −0.354810
\(446\) −6.11874e11 −0.732243
\(447\) 3.08378e11 0.365342
\(448\) 1.55228e12 1.82062
\(449\) 3.68568e11 0.427966 0.213983 0.976837i \(-0.431356\pi\)
0.213983 + 0.976837i \(0.431356\pi\)
\(450\) −4.37535e10 −0.0502986
\(451\) 2.75343e11 0.313386
\(452\) −4.05898e11 −0.457398
\(453\) 4.07925e11 0.455133
\(454\) 1.30375e12 1.44026
\(455\) 4.11723e11 0.450354
\(456\) −1.32014e11 −0.142981
\(457\) −6.92422e11 −0.742588 −0.371294 0.928515i \(-0.621086\pi\)
−0.371294 + 0.928515i \(0.621086\pi\)
\(458\) 1.28598e12 1.36565
\(459\) −2.07525e10 −0.0218229
\(460\) −1.17980e10 −0.0122857
\(461\) 1.23633e11 0.127491 0.0637455 0.997966i \(-0.479695\pi\)
0.0637455 + 0.997966i \(0.479695\pi\)
\(462\) 1.75051e11 0.178762
\(463\) −5.65668e11 −0.572067 −0.286033 0.958220i \(-0.592337\pi\)
−0.286033 + 0.958220i \(0.592337\pi\)
\(464\) 4.94185e11 0.494947
\(465\) 3.87976e11 0.384828
\(466\) −8.94257e11 −0.878468
\(467\) 1.11034e12 1.08026 0.540130 0.841582i \(-0.318375\pi\)
0.540130 + 0.841582i \(0.318375\pi\)
\(468\) −8.07501e10 −0.0778102
\(469\) −1.46707e12 −1.40014
\(470\) 1.03483e11 0.0978201
\(471\) 1.03147e12 0.965749
\(472\) −3.20771e11 −0.297479
\(473\) −3.78517e11 −0.347704
\(474\) −2.51262e11 −0.228625
\(475\) 5.09066e10 0.0458831
\(476\) −1.01667e11 −0.0907711
\(477\) −4.45746e11 −0.394235
\(478\) 4.26473e11 0.373651
\(479\) −4.59356e11 −0.398694 −0.199347 0.979929i \(-0.563882\pi\)
−0.199347 + 0.979929i \(0.563882\pi\)
\(480\) 2.37228e11 0.203976
\(481\) −9.74545e11 −0.830136
\(482\) −1.52741e12 −1.28897
\(483\) −8.18404e10 −0.0684236
\(484\) 4.94682e11 0.409753
\(485\) −2.27223e11 −0.186473
\(486\) 5.95261e10 0.0483999
\(487\) 2.77447e11 0.223511 0.111756 0.993736i \(-0.464353\pi\)
0.111756 + 0.993736i \(0.464353\pi\)
\(488\) −8.77498e10 −0.0700417
\(489\) 5.91329e11 0.467670
\(490\) −1.05632e12 −0.827777
\(491\) −8.86757e11 −0.688554 −0.344277 0.938868i \(-0.611876\pi\)
−0.344277 + 0.938868i \(0.611876\pi\)
\(492\) 4.58697e11 0.352926
\(493\) −1.91863e11 −0.146278
\(494\) −1.24155e11 −0.0937978
\(495\) 4.39733e10 0.0329204
\(496\) 7.70822e11 0.571856
\(497\) 7.89161e11 0.580179
\(498\) −8.31129e11 −0.605531
\(499\) −6.77785e11 −0.489372 −0.244686 0.969602i \(-0.578685\pi\)
−0.244686 + 0.969602i \(0.578685\pi\)
\(500\) −5.38451e10 −0.0385284
\(501\) 1.40484e11 0.0996228
\(502\) 6.45664e9 0.00453774
\(503\) −8.11800e11 −0.565449 −0.282724 0.959201i \(-0.591238\pi\)
−0.282724 + 0.959201i \(0.591238\pi\)
\(504\) 9.68608e11 0.668668
\(505\) −6.87706e11 −0.470535
\(506\) −1.56691e10 −0.0106260
\(507\) 6.06722e11 0.407806
\(508\) −3.34961e11 −0.223155
\(509\) 1.61730e11 0.106797 0.0533986 0.998573i \(-0.482995\pi\)
0.0533986 + 0.998573i \(0.482995\pi\)
\(510\) 3.37491e10 0.0220900
\(511\) 5.09745e12 3.30719
\(512\) 1.11535e12 0.717290
\(513\) −6.92579e10 −0.0441511
\(514\) 1.18232e12 0.747136
\(515\) −2.77003e11 −0.173521
\(516\) −6.30576e11 −0.391574
\(517\) −1.04003e11 −0.0640232
\(518\) 3.51946e12 2.14779
\(519\) 4.79513e11 0.290100
\(520\) 4.36181e11 0.261608
\(521\) −2.87943e12 −1.71213 −0.856065 0.516868i \(-0.827098\pi\)
−0.856065 + 0.516868i \(0.827098\pi\)
\(522\) 5.50336e11 0.324422
\(523\) −1.88151e12 −1.09963 −0.549817 0.835285i \(-0.685303\pi\)
−0.549817 + 0.835285i \(0.685303\pi\)
\(524\) −1.22338e12 −0.708879
\(525\) −3.73511e11 −0.214579
\(526\) −2.09074e12 −1.19087
\(527\) −2.99264e11 −0.169008
\(528\) 8.73651e10 0.0489199
\(529\) −1.79383e12 −0.995933
\(530\) 7.24903e11 0.399060
\(531\) −1.68285e11 −0.0918587
\(532\) −3.39296e11 −0.183644
\(533\) 1.43285e12 0.769006
\(534\) −6.49388e11 −0.345596
\(535\) −9.58213e11 −0.505673
\(536\) −1.55421e12 −0.813335
\(537\) −9.15905e11 −0.475298
\(538\) 3.53222e11 0.181772
\(539\) 1.06163e12 0.541779
\(540\) 7.32556e10 0.0370740
\(541\) −1.62100e12 −0.813568 −0.406784 0.913524i \(-0.633350\pi\)
−0.406784 + 0.913524i \(0.633350\pi\)
\(542\) 5.36767e11 0.267171
\(543\) 1.59351e12 0.786604
\(544\) −1.82985e11 −0.0895819
\(545\) −1.19417e12 −0.579803
\(546\) 9.10946e11 0.438658
\(547\) 1.76631e12 0.843575 0.421787 0.906695i \(-0.361403\pi\)
0.421787 + 0.906695i \(0.361403\pi\)
\(548\) −1.48692e11 −0.0704327
\(549\) −4.60359e10 −0.0216282
\(550\) −7.15123e10 −0.0333234
\(551\) −6.40309e11 −0.295943
\(552\) −8.67019e10 −0.0397469
\(553\) −2.14496e12 −0.975338
\(554\) −3.09343e12 −1.39523
\(555\) 8.84097e11 0.395532
\(556\) −8.61336e11 −0.382240
\(557\) 2.60533e12 1.14687 0.573435 0.819251i \(-0.305610\pi\)
0.573435 + 0.819251i \(0.305610\pi\)
\(558\) 8.58405e11 0.374834
\(559\) −1.96976e12 −0.853219
\(560\) −7.42084e11 −0.318865
\(561\) −3.39186e10 −0.0144579
\(562\) −2.57800e12 −1.09011
\(563\) 4.01223e12 1.68306 0.841528 0.540214i \(-0.181657\pi\)
0.841528 + 0.540214i \(0.181657\pi\)
\(564\) −1.73260e11 −0.0721010
\(565\) 1.15025e12 0.474868
\(566\) 3.45004e12 1.41303
\(567\) 5.08158e11 0.206478
\(568\) 8.36039e11 0.337023
\(569\) −3.21451e12 −1.28561 −0.642806 0.766029i \(-0.722230\pi\)
−0.642806 + 0.766029i \(0.722230\pi\)
\(570\) 1.12632e11 0.0446915
\(571\) −6.41308e11 −0.252467 −0.126233 0.992001i \(-0.540289\pi\)
−0.126233 + 0.992001i \(0.540289\pi\)
\(572\) −1.31981e11 −0.0515501
\(573\) 6.94922e11 0.269303
\(574\) −5.17459e12 −1.98963
\(575\) 3.34337e10 0.0127550
\(576\) 8.62745e11 0.326574
\(577\) −3.87706e12 −1.45617 −0.728085 0.685487i \(-0.759589\pi\)
−0.728085 + 0.685487i \(0.759589\pi\)
\(578\) 1.99849e12 0.744778
\(579\) −1.62299e12 −0.600152
\(580\) 6.77270e11 0.248505
\(581\) −7.09511e12 −2.58325
\(582\) −5.02737e11 −0.181630
\(583\) −7.28544e11 −0.261184
\(584\) 5.40025e12 1.92113
\(585\) 2.28832e11 0.0807822
\(586\) −2.73049e12 −0.956536
\(587\) 3.11635e12 1.08336 0.541682 0.840583i \(-0.317788\pi\)
0.541682 + 0.840583i \(0.317788\pi\)
\(588\) 1.76858e12 0.610136
\(589\) −9.98744e11 −0.341929
\(590\) 2.73677e11 0.0929831
\(591\) 7.51696e11 0.253453
\(592\) 1.75650e12 0.587762
\(593\) −1.96777e12 −0.653475 −0.326738 0.945115i \(-0.605949\pi\)
−0.326738 + 0.945115i \(0.605949\pi\)
\(594\) 9.72918e10 0.0320654
\(595\) 2.88107e11 0.0942382
\(596\) 8.39663e11 0.272582
\(597\) −2.52891e12 −0.814797
\(598\) −8.15405e10 −0.0260746
\(599\) −2.44774e12 −0.776864 −0.388432 0.921477i \(-0.626983\pi\)
−0.388432 + 0.921477i \(0.626983\pi\)
\(600\) −3.95699e11 −0.124648
\(601\) 8.75163e11 0.273624 0.136812 0.990597i \(-0.456314\pi\)
0.136812 + 0.990597i \(0.456314\pi\)
\(602\) 7.11357e12 2.20751
\(603\) −8.15383e11 −0.251150
\(604\) 1.11071e12 0.339574
\(605\) −1.40185e12 −0.425403
\(606\) −1.52156e12 −0.458314
\(607\) −1.87732e12 −0.561293 −0.280647 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(608\) −6.10682e11 −0.181238
\(609\) 4.69807e12 1.38402
\(610\) 7.48668e10 0.0218930
\(611\) −5.41219e11 −0.157104
\(612\) −5.65055e10 −0.0162821
\(613\) 2.33539e12 0.668017 0.334009 0.942570i \(-0.391599\pi\)
0.334009 + 0.942570i \(0.391599\pi\)
\(614\) 3.77213e12 1.07110
\(615\) −1.29987e12 −0.366406
\(616\) 1.58313e12 0.442999
\(617\) 4.28203e12 1.18951 0.594753 0.803909i \(-0.297250\pi\)
0.594753 + 0.803909i \(0.297250\pi\)
\(618\) −6.12875e11 −0.169014
\(619\) −5.25474e12 −1.43861 −0.719305 0.694694i \(-0.755540\pi\)
−0.719305 + 0.694694i \(0.755540\pi\)
\(620\) 1.05639e12 0.287120
\(621\) −4.54862e10 −0.0122735
\(622\) 1.46386e12 0.392142
\(623\) −5.54364e12 −1.47434
\(624\) 4.54638e11 0.120043
\(625\) 1.52588e11 0.0400000
\(626\) 3.40928e12 0.887315
\(627\) −1.13198e11 −0.0292505
\(628\) 2.80853e12 0.720545
\(629\) −6.81945e11 −0.173709
\(630\) −8.26401e11 −0.209006
\(631\) −1.73310e12 −0.435202 −0.217601 0.976038i \(-0.569823\pi\)
−0.217601 + 0.976038i \(0.569823\pi\)
\(632\) −2.27237e12 −0.566568
\(633\) −2.39461e12 −0.592813
\(634\) −1.73577e12 −0.426668
\(635\) 9.49222e11 0.231679
\(636\) −1.21369e12 −0.294138
\(637\) 5.52459e12 1.32945
\(638\) 8.99490e11 0.214933
\(639\) 4.38609e11 0.104069
\(640\) 9.64583e10 0.0227263
\(641\) 9.42113e11 0.220415 0.110208 0.993909i \(-0.464848\pi\)
0.110208 + 0.993909i \(0.464848\pi\)
\(642\) −2.12007e12 −0.492540
\(643\) 1.07698e11 0.0248462 0.0124231 0.999923i \(-0.496046\pi\)
0.0124231 + 0.999923i \(0.496046\pi\)
\(644\) −2.22838e11 −0.0510508
\(645\) 1.78695e12 0.406530
\(646\) −8.68783e10 −0.0196275
\(647\) 6.89733e12 1.54743 0.773717 0.633532i \(-0.218395\pi\)
0.773717 + 0.633532i \(0.218395\pi\)
\(648\) 5.38343e11 0.119942
\(649\) −2.75051e11 −0.0608573
\(650\) −3.72143e11 −0.0817710
\(651\) 7.32796e12 1.59908
\(652\) 1.61009e12 0.348928
\(653\) −6.08371e12 −1.30936 −0.654681 0.755906i \(-0.727197\pi\)
−0.654681 + 0.755906i \(0.727197\pi\)
\(654\) −2.64212e12 −0.564745
\(655\) 3.46687e12 0.735955
\(656\) −2.58255e12 −0.544480
\(657\) 2.83312e12 0.593226
\(658\) 1.95455e12 0.406472
\(659\) −7.58358e12 −1.56635 −0.783177 0.621799i \(-0.786402\pi\)
−0.783177 + 0.621799i \(0.786402\pi\)
\(660\) 1.19732e11 0.0245619
\(661\) −4.44191e12 −0.905030 −0.452515 0.891757i \(-0.649473\pi\)
−0.452515 + 0.891757i \(0.649473\pi\)
\(662\) −1.11857e12 −0.226361
\(663\) −1.76509e11 −0.0354777
\(664\) −7.51658e12 −1.50060
\(665\) 9.61508e11 0.190658
\(666\) 1.95608e12 0.385260
\(667\) −4.20533e11 −0.0822685
\(668\) 3.82516e11 0.0743285
\(669\) −2.90312e12 −0.560334
\(670\) 1.32603e12 0.254224
\(671\) −7.52428e10 −0.0143289
\(672\) 4.48068e12 0.847583
\(673\) −5.95654e12 −1.11925 −0.559624 0.828747i \(-0.689054\pi\)
−0.559624 + 0.828747i \(0.689054\pi\)
\(674\) 2.78527e12 0.519874
\(675\) −2.07594e11 −0.0384900
\(676\) 1.65200e12 0.304264
\(677\) 3.79342e12 0.694036 0.347018 0.937858i \(-0.387194\pi\)
0.347018 + 0.937858i \(0.387194\pi\)
\(678\) 2.54495e12 0.462535
\(679\) −4.29172e12 −0.774850
\(680\) 3.05221e11 0.0547424
\(681\) 6.18579e12 1.10213
\(682\) 1.40301e12 0.248331
\(683\) −8.63237e12 −1.51788 −0.758939 0.651162i \(-0.774282\pi\)
−0.758939 + 0.651162i \(0.774282\pi\)
\(684\) −1.88578e11 −0.0329411
\(685\) 4.21367e11 0.0731229
\(686\) −1.18190e13 −2.03761
\(687\) 6.10150e12 1.04504
\(688\) 3.55027e12 0.604106
\(689\) −3.79127e12 −0.640911
\(690\) 7.39727e10 0.0124237
\(691\) 8.23079e12 1.37338 0.686689 0.726951i \(-0.259063\pi\)
0.686689 + 0.726951i \(0.259063\pi\)
\(692\) 1.30563e12 0.216443
\(693\) 8.30552e11 0.136794
\(694\) −7.85757e12 −1.28579
\(695\) 2.44088e12 0.396840
\(696\) 4.97714e12 0.803967
\(697\) 1.00265e12 0.160917
\(698\) 2.67505e12 0.426562
\(699\) −4.24292e12 −0.672230
\(700\) −1.01701e12 −0.160097
\(701\) 5.13274e12 0.802819 0.401410 0.915899i \(-0.368520\pi\)
0.401410 + 0.915899i \(0.368520\pi\)
\(702\) 5.06296e11 0.0786842
\(703\) −2.27588e12 −0.351439
\(704\) 1.41010e12 0.216359
\(705\) 4.90988e11 0.0748549
\(706\) −2.38404e12 −0.361154
\(707\) −1.29892e13 −1.95521
\(708\) −4.58212e11 −0.0685357
\(709\) 3.81960e12 0.567688 0.283844 0.958870i \(-0.408390\pi\)
0.283844 + 0.958870i \(0.408390\pi\)
\(710\) −7.13296e11 −0.105343
\(711\) −1.19215e12 −0.174951
\(712\) −5.87294e12 −0.856438
\(713\) −6.55940e11 −0.0950520
\(714\) 6.37442e11 0.0917907
\(715\) 3.74012e11 0.0535190
\(716\) −2.49386e12 −0.354620
\(717\) 2.02346e12 0.285929
\(718\) −6.60857e12 −0.927999
\(719\) −1.11913e13 −1.56172 −0.780859 0.624708i \(-0.785218\pi\)
−0.780859 + 0.624708i \(0.785218\pi\)
\(720\) −4.12443e11 −0.0571963
\(721\) −5.23194e12 −0.721032
\(722\) −2.89942e11 −0.0397094
\(723\) −7.24700e12 −0.986360
\(724\) 4.33887e12 0.586885
\(725\) −1.91927e12 −0.257997
\(726\) −3.10161e12 −0.414355
\(727\) −8.48783e12 −1.12692 −0.563458 0.826145i \(-0.690529\pi\)
−0.563458 + 0.826145i \(0.690529\pi\)
\(728\) 8.23844e12 1.08706
\(729\) 2.82430e11 0.0370370
\(730\) −4.60741e12 −0.600487
\(731\) −1.37836e12 −0.178539
\(732\) −1.25348e11 −0.0161368
\(733\) 4.12823e12 0.528197 0.264098 0.964496i \(-0.414926\pi\)
0.264098 + 0.964496i \(0.414926\pi\)
\(734\) −6.33093e12 −0.805073
\(735\) −5.01185e12 −0.633440
\(736\) −4.01074e11 −0.0503819
\(737\) −1.33269e12 −0.166390
\(738\) −2.87599e12 −0.356890
\(739\) 3.52537e12 0.434815 0.217408 0.976081i \(-0.430240\pi\)
0.217408 + 0.976081i \(0.430240\pi\)
\(740\) 2.40725e12 0.295106
\(741\) −5.89069e11 −0.0717768
\(742\) 1.36917e13 1.65821
\(743\) −6.43206e12 −0.774284 −0.387142 0.922020i \(-0.626538\pi\)
−0.387142 + 0.922020i \(0.626538\pi\)
\(744\) 7.76326e12 0.928894
\(745\) −2.37946e12 −0.282993
\(746\) 2.70345e12 0.319590
\(747\) −3.94340e12 −0.463370
\(748\) −9.23547e10 −0.0107870
\(749\) −1.80984e13 −2.10122
\(750\) 3.37604e11 0.0389612
\(751\) 1.02373e12 0.117437 0.0587184 0.998275i \(-0.481299\pi\)
0.0587184 + 0.998275i \(0.481299\pi\)
\(752\) 9.75485e11 0.111235
\(753\) 3.06344e10 0.00347242
\(754\) 4.68085e12 0.527417
\(755\) −3.14757e12 −0.352545
\(756\) 1.38363e12 0.154053
\(757\) 5.86577e12 0.649222 0.324611 0.945848i \(-0.394767\pi\)
0.324611 + 0.945848i \(0.394767\pi\)
\(758\) 9.85287e12 1.08405
\(759\) −7.43443e10 −0.00813130
\(760\) 1.01862e12 0.110752
\(761\) 6.18495e12 0.668506 0.334253 0.942483i \(-0.391516\pi\)
0.334253 + 0.942483i \(0.391516\pi\)
\(762\) 2.10017e12 0.225662
\(763\) −2.25550e13 −2.40926
\(764\) 1.89216e12 0.200926
\(765\) 1.60127e11 0.0169039
\(766\) −6.41934e12 −0.673692
\(767\) −1.43134e12 −0.149336
\(768\) 5.66682e12 0.587779
\(769\) −1.87533e11 −0.0193379 −0.00966895 0.999953i \(-0.503078\pi\)
−0.00966895 + 0.999953i \(0.503078\pi\)
\(770\) −1.35070e12 −0.138469
\(771\) 5.60965e12 0.571731
\(772\) −4.41912e12 −0.447773
\(773\) −3.37329e12 −0.339818 −0.169909 0.985460i \(-0.554347\pi\)
−0.169909 + 0.985460i \(0.554347\pi\)
\(774\) 3.95366e12 0.395972
\(775\) −2.99364e12 −0.298086
\(776\) −4.54666e12 −0.450106
\(777\) 1.66985e13 1.64355
\(778\) −1.29681e13 −1.26902
\(779\) 3.34618e12 0.325560
\(780\) 6.23072e11 0.0602715
\(781\) 7.16879e11 0.0689471
\(782\) −5.70586e10 −0.00545621
\(783\) 2.61114e12 0.248258
\(784\) −9.95744e12 −0.941294
\(785\) −7.95891e12 −0.748066
\(786\) 7.67052e12 0.716841
\(787\) −3.45820e12 −0.321340 −0.160670 0.987008i \(-0.551365\pi\)
−0.160670 + 0.987008i \(0.551365\pi\)
\(788\) 2.04674e12 0.189101
\(789\) −9.91980e12 −0.911290
\(790\) 1.93875e12 0.177093
\(791\) 2.17255e13 1.97322
\(792\) 8.79889e11 0.0794629
\(793\) −3.91556e11 −0.0351612
\(794\) −7.54765e12 −0.673937
\(795\) 3.43940e12 0.305373
\(796\) −6.88580e12 −0.607919
\(797\) −1.50198e13 −1.31857 −0.659284 0.751894i \(-0.729141\pi\)
−0.659284 + 0.751894i \(0.729141\pi\)
\(798\) 2.12736e12 0.185707
\(799\) −3.78722e11 −0.0328746
\(800\) −1.83046e12 −0.157999
\(801\) −3.08111e12 −0.264460
\(802\) −3.78115e12 −0.322730
\(803\) 4.63055e12 0.393018
\(804\) −2.22015e12 −0.187383
\(805\) 6.31485e11 0.0530007
\(806\) 7.30111e12 0.609371
\(807\) 1.67591e12 0.139097
\(808\) −1.37608e13 −1.13577
\(809\) −5.19090e12 −0.426063 −0.213032 0.977045i \(-0.568334\pi\)
−0.213032 + 0.977045i \(0.568334\pi\)
\(810\) −4.59306e11 −0.0374904
\(811\) 5.64885e11 0.0458528 0.0229264 0.999737i \(-0.492702\pi\)
0.0229264 + 0.999737i \(0.492702\pi\)
\(812\) 1.27920e13 1.03261
\(813\) 2.54676e12 0.204447
\(814\) 3.19710e12 0.255238
\(815\) −4.56273e12 −0.362256
\(816\) 3.18137e11 0.0251193
\(817\) −4.60003e12 −0.361212
\(818\) 2.82960e12 0.220971
\(819\) 4.32211e12 0.335674
\(820\) −3.53933e12 −0.273375
\(821\) 2.42921e13 1.86604 0.933020 0.359826i \(-0.117164\pi\)
0.933020 + 0.359826i \(0.117164\pi\)
\(822\) 9.32284e11 0.0712238
\(823\) −1.33695e13 −1.01582 −0.507910 0.861410i \(-0.669582\pi\)
−0.507910 + 0.861410i \(0.669582\pi\)
\(824\) −5.54273e12 −0.418843
\(825\) −3.39300e11 −0.0255000
\(826\) 5.16912e12 0.386372
\(827\) 4.50524e12 0.334922 0.167461 0.985879i \(-0.446443\pi\)
0.167461 + 0.985879i \(0.446443\pi\)
\(828\) −1.23851e11 −0.00915722
\(829\) −8.89486e12 −0.654099 −0.327050 0.945007i \(-0.606054\pi\)
−0.327050 + 0.945007i \(0.606054\pi\)
\(830\) 6.41303e12 0.469042
\(831\) −1.46772e13 −1.06767
\(832\) 7.33803e12 0.530914
\(833\) 3.86588e12 0.278193
\(834\) 5.40050e12 0.386533
\(835\) −1.08399e12 −0.0771675
\(836\) −3.08219e11 −0.0218238
\(837\) 4.07282e12 0.286834
\(838\) −2.57010e12 −0.180033
\(839\) 1.18724e13 0.827199 0.413599 0.910459i \(-0.364271\pi\)
0.413599 + 0.910459i \(0.364271\pi\)
\(840\) −7.47382e12 −0.517948
\(841\) 9.63362e12 0.664060
\(842\) −1.90492e13 −1.30608
\(843\) −1.22317e13 −0.834183
\(844\) −6.52011e12 −0.442297
\(845\) −4.68149e12 −0.315885
\(846\) 1.08632e12 0.0729108
\(847\) −2.64776e13 −1.76768
\(848\) 6.83332e12 0.453785
\(849\) 1.63692e13 1.08129
\(850\) −2.60410e11 −0.0171109
\(851\) −1.49472e12 −0.0976959
\(852\) 1.19426e12 0.0776462
\(853\) 8.65613e12 0.559826 0.279913 0.960025i \(-0.409694\pi\)
0.279913 + 0.960025i \(0.409694\pi\)
\(854\) 1.41406e12 0.0909719
\(855\) 5.34398e11 0.0341993
\(856\) −1.91735e13 −1.22059
\(857\) −2.58026e12 −0.163399 −0.0816997 0.996657i \(-0.526035\pi\)
−0.0816997 + 0.996657i \(0.526035\pi\)
\(858\) 8.27509e11 0.0521291
\(859\) −1.20094e13 −0.752581 −0.376291 0.926502i \(-0.622801\pi\)
−0.376291 + 0.926502i \(0.622801\pi\)
\(860\) 4.86556e12 0.303312
\(861\) −2.45515e13 −1.52252
\(862\) −8.89099e12 −0.548489
\(863\) 2.57711e13 1.58156 0.790779 0.612102i \(-0.209676\pi\)
0.790779 + 0.612102i \(0.209676\pi\)
\(864\) 2.49032e12 0.152035
\(865\) −3.69995e12 −0.224710
\(866\) 1.33381e13 0.805867
\(867\) 9.48210e12 0.569926
\(868\) 1.99528e13 1.19307
\(869\) −1.94849e12 −0.115907
\(870\) −4.24642e12 −0.251296
\(871\) −6.93519e12 −0.408297
\(872\) −2.38949e13 −1.39952
\(873\) −2.38530e12 −0.138989
\(874\) −1.90424e11 −0.0110387
\(875\) 2.88203e12 0.166212
\(876\) 7.71410e12 0.442605
\(877\) −4.15440e12 −0.237143 −0.118571 0.992946i \(-0.537831\pi\)
−0.118571 + 0.992946i \(0.537831\pi\)
\(878\) 9.84880e12 0.559317
\(879\) −1.29552e13 −0.731970
\(880\) −6.74113e11 −0.0378932
\(881\) 2.65371e13 1.48409 0.742047 0.670348i \(-0.233855\pi\)
0.742047 + 0.670348i \(0.233855\pi\)
\(882\) −1.10888e13 −0.616989
\(883\) 1.41263e13 0.781998 0.390999 0.920391i \(-0.372130\pi\)
0.390999 + 0.920391i \(0.372130\pi\)
\(884\) −4.80604e11 −0.0264699
\(885\) 1.29850e12 0.0711534
\(886\) 1.81376e12 0.0988843
\(887\) −1.50302e13 −0.815283 −0.407642 0.913142i \(-0.633649\pi\)
−0.407642 + 0.913142i \(0.633649\pi\)
\(888\) 1.76905e13 0.954731
\(889\) 1.79286e13 0.962694
\(890\) 5.01071e12 0.267697
\(891\) 4.61614e11 0.0245374
\(892\) −7.90470e12 −0.418065
\(893\) −1.26392e12 −0.0665103
\(894\) −5.26461e12 −0.275643
\(895\) 7.06717e12 0.368164
\(896\) 1.82187e12 0.0944348
\(897\) −3.86880e11 −0.0199531
\(898\) −6.29217e12 −0.322892
\(899\) 3.76544e13 1.92263
\(900\) −5.65244e11 −0.0287174
\(901\) −2.65297e12 −0.134113
\(902\) −4.70063e12 −0.236443
\(903\) 3.37513e13 1.68926
\(904\) 2.30160e13 1.14623
\(905\) −1.22956e13 −0.609301
\(906\) −6.96406e12 −0.343389
\(907\) 2.29165e13 1.12439 0.562193 0.827006i \(-0.309958\pi\)
0.562193 + 0.827006i \(0.309958\pi\)
\(908\) 1.68429e13 0.822299
\(909\) −7.21926e12 −0.350716
\(910\) −7.02891e12 −0.339783
\(911\) −7.02130e12 −0.337742 −0.168871 0.985638i \(-0.554012\pi\)
−0.168871 + 0.985638i \(0.554012\pi\)
\(912\) 1.06173e12 0.0508203
\(913\) −6.44524e12 −0.306988
\(914\) 1.18210e13 0.560267
\(915\) 3.55215e11 0.0167532
\(916\) 1.66134e13 0.779701
\(917\) 6.54810e13 3.05811
\(918\) 3.54284e11 0.0164649
\(919\) 1.14513e13 0.529584 0.264792 0.964306i \(-0.414697\pi\)
0.264792 + 0.964306i \(0.414697\pi\)
\(920\) 6.68996e11 0.0307878
\(921\) 1.78974e13 0.819636
\(922\) −2.11065e12 −0.0961893
\(923\) 3.73056e12 0.169187
\(924\) 2.26146e12 0.102062
\(925\) −6.82174e12 −0.306378
\(926\) 9.65703e12 0.431613
\(927\) −2.90787e12 −0.129335
\(928\) 2.30238e13 1.01908
\(929\) −4.07798e13 −1.79628 −0.898141 0.439708i \(-0.855082\pi\)
−0.898141 + 0.439708i \(0.855082\pi\)
\(930\) −6.62350e12 −0.290345
\(931\) 1.29017e13 0.562826
\(932\) −1.15528e13 −0.501550
\(933\) 6.94548e12 0.300079
\(934\) −1.89556e13 −0.815034
\(935\) 2.61718e11 0.0111990
\(936\) 4.57885e12 0.194991
\(937\) 2.76611e12 0.117231 0.0586153 0.998281i \(-0.481331\pi\)
0.0586153 + 0.998281i \(0.481331\pi\)
\(938\) 2.50456e13 1.05638
\(939\) 1.61758e13 0.679000
\(940\) 1.33688e12 0.0558492
\(941\) 1.33465e13 0.554900 0.277450 0.960740i \(-0.410511\pi\)
0.277450 + 0.960740i \(0.410511\pi\)
\(942\) −1.76093e13 −0.728638
\(943\) 2.19765e12 0.0905017
\(944\) 2.57982e12 0.105734
\(945\) −3.92097e12 −0.159938
\(946\) 6.46201e12 0.262336
\(947\) −4.79853e13 −1.93880 −0.969402 0.245480i \(-0.921054\pi\)
−0.969402 + 0.245480i \(0.921054\pi\)
\(948\) −3.24602e12 −0.130531
\(949\) 2.40969e13 0.964413
\(950\) −8.69074e11 −0.0346179
\(951\) −8.23558e12 −0.326499
\(952\) 5.76491e12 0.227471
\(953\) 4.46482e13 1.75342 0.876710 0.481020i \(-0.159733\pi\)
0.876710 + 0.481020i \(0.159733\pi\)
\(954\) 7.60974e12 0.297442
\(955\) −5.36205e12 −0.208601
\(956\) 5.50954e12 0.213331
\(957\) 4.26775e12 0.164473
\(958\) 7.84209e12 0.300806
\(959\) 7.95865e12 0.303847
\(960\) −6.65698e12 −0.252963
\(961\) 3.22930e13 1.22139
\(962\) 1.66374e13 0.626320
\(963\) −1.00589e13 −0.376907
\(964\) −1.97324e13 −0.735923
\(965\) 1.25230e13 0.464876
\(966\) 1.39717e12 0.0516242
\(967\) −2.18351e13 −0.803039 −0.401519 0.915851i \(-0.631518\pi\)
−0.401519 + 0.915851i \(0.631518\pi\)
\(968\) −2.80504e13 −1.02683
\(969\) −4.12206e11 −0.0150196
\(970\) 3.87914e12 0.140690
\(971\) −1.25175e13 −0.451888 −0.225944 0.974140i \(-0.572547\pi\)
−0.225944 + 0.974140i \(0.572547\pi\)
\(972\) 7.69008e11 0.0276333
\(973\) 4.61026e13 1.64899
\(974\) −4.73655e12 −0.168635
\(975\) −1.76568e12 −0.0625736
\(976\) 7.05734e11 0.0248953
\(977\) 3.20249e13 1.12451 0.562253 0.826965i \(-0.309935\pi\)
0.562253 + 0.826965i \(0.309935\pi\)
\(978\) −1.00951e13 −0.352847
\(979\) −5.03588e12 −0.175208
\(980\) −1.36464e13 −0.472609
\(981\) −1.25359e13 −0.432160
\(982\) 1.51386e13 0.519500
\(983\) 5.40250e13 1.84546 0.922728 0.385452i \(-0.125954\pi\)
0.922728 + 0.385452i \(0.125954\pi\)
\(984\) −2.60099e13 −0.884426
\(985\) −5.80012e12 −0.196324
\(986\) 3.27546e12 0.110364
\(987\) 9.27363e12 0.311044
\(988\) −1.60394e12 −0.0535527
\(989\) −3.02114e12 −0.100412
\(990\) −7.50708e11 −0.0248378
\(991\) −8.74099e12 −0.287892 −0.143946 0.989586i \(-0.545979\pi\)
−0.143946 + 0.989586i \(0.545979\pi\)
\(992\) 3.59121e13 1.17744
\(993\) −5.30719e12 −0.173218
\(994\) −1.34725e13 −0.437733
\(995\) 1.95132e13 0.631139
\(996\) −1.07372e13 −0.345720
\(997\) −1.29294e13 −0.414429 −0.207214 0.978296i \(-0.566440\pi\)
−0.207214 + 0.978296i \(0.566440\pi\)
\(998\) 1.15711e13 0.369221
\(999\) 9.28090e12 0.294812
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 285.10.a.c.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.10.a.c.1.5 12 1.1 even 1 trivial