Properties

Label 197.12.a.b.1.2
Level $197$
Weight $12$
Character 197.1
Self dual yes
Analytic conductor $151.364$
Analytic rank $0$
Dimension $92$
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] = [197,12,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 197.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: \(151.363606570\)
Analytic rank: \(0\)
Dimension: \(92\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 197.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-87.1440 q^{2} +786.167 q^{3} +5546.07 q^{4} +12634.2 q^{5} -68509.7 q^{6} -52867.9 q^{7} -304836. q^{8} +440912. q^{9} +O(q^{10})\) \(q-87.1440 q^{2} +786.167 q^{3} +5546.07 q^{4} +12634.2 q^{5} -68509.7 q^{6} -52867.9 q^{7} -304836. q^{8} +440912. q^{9} -1.10099e6 q^{10} -243497. q^{11} +4.36014e6 q^{12} +1.64401e6 q^{13} +4.60712e6 q^{14} +9.93256e6 q^{15} +1.52063e7 q^{16} -3.18554e6 q^{17} -3.84228e7 q^{18} -1.13223e6 q^{19} +7.00699e7 q^{20} -4.15630e7 q^{21} +2.12193e7 q^{22} -5.58534e7 q^{23} -2.39652e8 q^{24} +1.10794e8 q^{25} -1.43265e8 q^{26} +2.07363e8 q^{27} -2.93209e8 q^{28} +7.30013e7 q^{29} -8.65562e8 q^{30} +1.66906e8 q^{31} -7.00830e8 q^{32} -1.91429e8 q^{33} +2.77600e8 q^{34} -6.67941e8 q^{35} +2.44533e9 q^{36} -2.24431e8 q^{37} +9.86673e7 q^{38} +1.29246e9 q^{39} -3.85134e9 q^{40} -3.79319e8 q^{41} +3.62197e9 q^{42} +2.22500e8 q^{43} -1.35045e9 q^{44} +5.57055e9 q^{45} +4.86728e9 q^{46} +2.25481e9 q^{47} +1.19547e10 q^{48} +8.17688e8 q^{49} -9.65500e9 q^{50} -2.50437e9 q^{51} +9.11778e9 q^{52} -5.59241e7 q^{53} -1.80705e10 q^{54} -3.07638e9 q^{55} +1.61160e10 q^{56} -8.90124e8 q^{57} -6.36162e9 q^{58} +2.40144e9 q^{59} +5.50867e10 q^{60} -5.86229e9 q^{61} -1.45449e10 q^{62} -2.33101e10 q^{63} +2.99307e10 q^{64} +2.07706e10 q^{65} +1.66819e10 q^{66} +7.91950e9 q^{67} -1.76672e10 q^{68} -4.39101e10 q^{69} +5.82070e10 q^{70} +2.05210e10 q^{71} -1.34406e11 q^{72} +1.74831e10 q^{73} +1.95578e10 q^{74} +8.71024e10 q^{75} -6.27944e9 q^{76} +1.28732e10 q^{77} -1.12630e11 q^{78} +2.82858e10 q^{79} +1.92118e11 q^{80} +8.49161e10 q^{81} +3.30554e10 q^{82} +6.22953e10 q^{83} -2.30511e11 q^{84} -4.02466e10 q^{85} -1.93896e10 q^{86} +5.73912e10 q^{87} +7.42266e10 q^{88} -8.70641e10 q^{89} -4.85440e11 q^{90} -8.69152e10 q^{91} -3.09767e11 q^{92} +1.31216e11 q^{93} -1.96493e11 q^{94} -1.43048e10 q^{95} -5.50969e11 q^{96} +1.00198e11 q^{97} -7.12566e10 q^{98} -1.07361e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 92 q + 64 q^{2} + 2420 q^{3} + 98304 q^{4} + 16604 q^{5} + 46656 q^{6} + 305891 q^{7} + 234027 q^{8} + 5900444 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 92 q + 64 q^{2} + 2420 q^{3} + 98304 q^{4} + 16604 q^{5} + 46656 q^{6} + 305891 q^{7} + 234027 q^{8} + 5900444 q^{9} + 1074277 q^{10} + 595928 q^{11} + 4956160 q^{12} + 7463810 q^{13} + 4915769 q^{14} + 6749159 q^{15} + 109051904 q^{16} + 9869683 q^{17} + 15324721 q^{18} + 71500013 q^{19} + 79804779 q^{20} + 55741034 q^{21} + 120367289 q^{22} + 38597564 q^{23} + 104015637 q^{24} + 1039976880 q^{25} + 51802726 q^{26} + 665312351 q^{27} + 713160630 q^{28} - 2827541 q^{29} - 43013765 q^{30} + 939775728 q^{31} + 723479980 q^{32} + 978717002 q^{33} + 1063528738 q^{34} + 843197112 q^{35} + 6613742458 q^{36} + 2009031770 q^{37} + 918086794 q^{38} + 2279970607 q^{39} + 3093842646 q^{40} - 30014736 q^{41} + 3159549307 q^{42} + 6320365127 q^{43} + 3019176802 q^{44} + 5538230697 q^{45} + 4344764621 q^{46} + 2853606373 q^{47} + 15616060178 q^{48} + 31617715853 q^{49} - 28784136763 q^{50} - 7216660253 q^{51} + 19141188860 q^{52} + 3389732468 q^{53} + 14059623295 q^{54} + 23123173075 q^{55} + 75592019872 q^{56} + 27508365203 q^{57} + 42079603023 q^{58} + 40032802875 q^{59} + 99482549469 q^{60} + 31816702886 q^{61} + 35401555571 q^{62} + 79170604993 q^{63} + 168463042533 q^{64} + 50313809737 q^{65} + 93218912814 q^{66} + 129966589578 q^{67} + 73640879491 q^{68} - 9850635033 q^{69} + 62387700391 q^{70} + 32189826123 q^{71} + 47439469795 q^{72} + 60612500511 q^{73} - 110473527245 q^{74} + 48823210110 q^{75} + 48170982110 q^{76} - 9381499362 q^{77} - 251751253299 q^{78} + 40812909551 q^{79} - 36932560002 q^{80} + 294126355776 q^{81} + 29914326881 q^{82} + 112077199076 q^{83} - 242466547988 q^{84} - 65194167314 q^{85} - 384596985360 q^{86} + 30701235703 q^{87} + 102229734783 q^{88} - 47595308310 q^{89} - 957657692329 q^{90} + 217575098757 q^{91} - 762245602306 q^{92} - 47762776839 q^{93} - 251435906373 q^{94} - 84935429021 q^{95} - 1089429573223 q^{96} + 432665748880 q^{97} - 404245603446 q^{98} + 9542377031 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −87.1440 −1.92563 −0.962814 0.270165i \(-0.912922\pi\)
−0.962814 + 0.270165i \(0.912922\pi\)
\(3\) 786.167 1.86788 0.933938 0.357435i \(-0.116349\pi\)
0.933938 + 0.357435i \(0.116349\pi\)
\(4\) 5546.07 2.70804
\(5\) 12634.2 1.80805 0.904026 0.427477i \(-0.140597\pi\)
0.904026 + 0.427477i \(0.140597\pi\)
\(6\) −68509.7 −3.59683
\(7\) −52867.9 −1.18892 −0.594460 0.804125i \(-0.702634\pi\)
−0.594460 + 0.804125i \(0.702634\pi\)
\(8\) −304836. −3.28906
\(9\) 440912. 2.48896
\(10\) −1.10099e6 −3.48164
\(11\) −243497. −0.455863 −0.227931 0.973677i \(-0.573196\pi\)
−0.227931 + 0.973677i \(0.573196\pi\)
\(12\) 4.36014e6 5.05829
\(13\) 1.64401e6 1.22805 0.614024 0.789288i \(-0.289550\pi\)
0.614024 + 0.789288i \(0.289550\pi\)
\(14\) 4.60712e6 2.28942
\(15\) 9.93256e6 3.37722
\(16\) 1.52063e7 3.62545
\(17\) −3.18554e6 −0.544144 −0.272072 0.962277i \(-0.587709\pi\)
−0.272072 + 0.962277i \(0.587709\pi\)
\(18\) −3.84228e7 −4.79281
\(19\) −1.13223e6 −0.104904 −0.0524519 0.998623i \(-0.516704\pi\)
−0.0524519 + 0.998623i \(0.516704\pi\)
\(20\) 7.00699e7 4.89628
\(21\) −4.15630e7 −2.22076
\(22\) 2.12193e7 0.877822
\(23\) −5.58534e7 −1.80945 −0.904724 0.425998i \(-0.859923\pi\)
−0.904724 + 0.425998i \(0.859923\pi\)
\(24\) −2.39652e8 −6.14355
\(25\) 1.10794e8 2.26905
\(26\) −1.43265e8 −2.36476
\(27\) 2.07363e8 2.78119
\(28\) −2.93209e8 −3.21965
\(29\) 7.30013e7 0.660909 0.330454 0.943822i \(-0.392798\pi\)
0.330454 + 0.943822i \(0.392798\pi\)
\(30\) −8.65562e8 −6.50327
\(31\) 1.66906e8 1.04709 0.523544 0.851999i \(-0.324610\pi\)
0.523544 + 0.851999i \(0.324610\pi\)
\(32\) −7.00830e8 −3.69222
\(33\) −1.91429e8 −0.851495
\(34\) 2.77600e8 1.04782
\(35\) −6.67941e8 −2.14963
\(36\) 2.44533e9 6.74021
\(37\) −2.24431e8 −0.532075 −0.266037 0.963963i \(-0.585714\pi\)
−0.266037 + 0.963963i \(0.585714\pi\)
\(38\) 9.86673e7 0.202006
\(39\) 1.29246e9 2.29384
\(40\) −3.85134e9 −5.94679
\(41\) −3.79319e8 −0.511321 −0.255660 0.966767i \(-0.582293\pi\)
−0.255660 + 0.966767i \(0.582293\pi\)
\(42\) 3.62197e9 4.27635
\(43\) 2.22500e8 0.230810 0.115405 0.993319i \(-0.463183\pi\)
0.115405 + 0.993319i \(0.463183\pi\)
\(44\) −1.35045e9 −1.23450
\(45\) 5.57055e9 4.50017
\(46\) 4.86728e9 3.48432
\(47\) 2.25481e9 1.43408 0.717038 0.697034i \(-0.245498\pi\)
0.717038 + 0.697034i \(0.245498\pi\)
\(48\) 1.19547e10 6.77190
\(49\) 8.17688e8 0.413532
\(50\) −9.65500e9 −4.36935
\(51\) −2.50437e9 −1.01639
\(52\) 9.11778e9 3.32561
\(53\) −5.59241e7 −0.0183689 −0.00918443 0.999958i \(-0.502924\pi\)
−0.00918443 + 0.999958i \(0.502924\pi\)
\(54\) −1.80705e10 −5.35555
\(55\) −3.07638e9 −0.824223
\(56\) 1.61160e10 3.91043
\(57\) −8.90124e8 −0.195947
\(58\) −6.36162e9 −1.27266
\(59\) 2.40144e9 0.437307 0.218653 0.975803i \(-0.429834\pi\)
0.218653 + 0.975803i \(0.429834\pi\)
\(60\) 5.50867e10 9.14565
\(61\) −5.86229e9 −0.888696 −0.444348 0.895854i \(-0.646565\pi\)
−0.444348 + 0.895854i \(0.646565\pi\)
\(62\) −1.45449e10 −2.01630
\(63\) −2.33101e10 −2.95918
\(64\) 2.99307e10 3.48439
\(65\) 2.07706e10 2.22037
\(66\) 1.66819e10 1.63966
\(67\) 7.91950e9 0.716616 0.358308 0.933603i \(-0.383354\pi\)
0.358308 + 0.933603i \(0.383354\pi\)
\(68\) −1.76672e10 −1.47357
\(69\) −4.39101e10 −3.37983
\(70\) 5.82070e10 4.13939
\(71\) 2.05210e10 1.34982 0.674911 0.737899i \(-0.264182\pi\)
0.674911 + 0.737899i \(0.264182\pi\)
\(72\) −1.34406e11 −8.18633
\(73\) 1.74831e10 0.987060 0.493530 0.869729i \(-0.335706\pi\)
0.493530 + 0.869729i \(0.335706\pi\)
\(74\) 1.95578e10 1.02458
\(75\) 8.71024e10 4.23831
\(76\) −6.27944e9 −0.284084
\(77\) 1.28732e10 0.541984
\(78\) −1.12630e11 −4.41708
\(79\) 2.82858e10 1.03423 0.517117 0.855915i \(-0.327005\pi\)
0.517117 + 0.855915i \(0.327005\pi\)
\(80\) 1.92118e11 6.55501
\(81\) 8.49161e10 2.70597
\(82\) 3.30554e10 0.984614
\(83\) 6.22953e10 1.73590 0.867952 0.496649i \(-0.165436\pi\)
0.867952 + 0.496649i \(0.165436\pi\)
\(84\) −2.30511e11 −6.01390
\(85\) −4.02466e10 −0.983841
\(86\) −1.93896e10 −0.444453
\(87\) 5.73912e10 1.23450
\(88\) 7.42266e10 1.49936
\(89\) −8.70641e10 −1.65270 −0.826350 0.563156i \(-0.809587\pi\)
−0.826350 + 0.563156i \(0.809587\pi\)
\(90\) −4.85440e11 −8.66566
\(91\) −8.69152e10 −1.46005
\(92\) −3.09767e11 −4.90006
\(93\) 1.31216e11 1.95583
\(94\) −1.96493e11 −2.76150
\(95\) −1.43048e10 −0.189671
\(96\) −5.50969e11 −6.89661
\(97\) 1.00198e11 1.18471 0.592356 0.805676i \(-0.298198\pi\)
0.592356 + 0.805676i \(0.298198\pi\)
\(98\) −7.12566e10 −0.796309
\(99\) −1.07361e11 −1.13462
\(100\) 6.14470e11 6.14470
\(101\) 1.32458e11 1.25404 0.627021 0.779002i \(-0.284274\pi\)
0.627021 + 0.779002i \(0.284274\pi\)
\(102\) 2.18240e11 1.95720
\(103\) 1.10119e11 0.935959 0.467979 0.883739i \(-0.344982\pi\)
0.467979 + 0.883739i \(0.344982\pi\)
\(104\) −5.01152e11 −4.03912
\(105\) −5.25113e11 −4.01524
\(106\) 4.87345e9 0.0353716
\(107\) 7.80172e10 0.537749 0.268875 0.963175i \(-0.413348\pi\)
0.268875 + 0.963175i \(0.413348\pi\)
\(108\) 1.15005e12 7.53160
\(109\) −3.81290e10 −0.237361 −0.118681 0.992932i \(-0.537866\pi\)
−0.118681 + 0.992932i \(0.537866\pi\)
\(110\) 2.68088e11 1.58715
\(111\) −1.76440e11 −0.993850
\(112\) −8.03923e11 −4.31038
\(113\) 2.06621e11 1.05498 0.527490 0.849561i \(-0.323133\pi\)
0.527490 + 0.849561i \(0.323133\pi\)
\(114\) 7.75690e10 0.377321
\(115\) −7.05660e11 −3.27158
\(116\) 4.04871e11 1.78977
\(117\) 7.24862e11 3.05656
\(118\) −2.09271e11 −0.842090
\(119\) 1.68413e11 0.646944
\(120\) −3.02780e12 −11.1079
\(121\) −2.26021e11 −0.792189
\(122\) 5.10863e11 1.71130
\(123\) −2.98208e11 −0.955084
\(124\) 9.25674e11 2.83556
\(125\) 7.82882e11 2.29452
\(126\) 2.03133e12 5.69827
\(127\) −5.46793e11 −1.46860 −0.734299 0.678826i \(-0.762489\pi\)
−0.734299 + 0.678826i \(0.762489\pi\)
\(128\) −1.17298e12 −3.01742
\(129\) 1.74922e11 0.431124
\(130\) −1.81004e12 −4.27561
\(131\) −3.13074e11 −0.709014 −0.354507 0.935053i \(-0.615351\pi\)
−0.354507 + 0.935053i \(0.615351\pi\)
\(132\) −1.06168e12 −2.30588
\(133\) 5.98588e10 0.124722
\(134\) −6.90137e11 −1.37994
\(135\) 2.61986e12 5.02855
\(136\) 9.71066e11 1.78972
\(137\) 5.60447e11 0.992136 0.496068 0.868284i \(-0.334777\pi\)
0.496068 + 0.868284i \(0.334777\pi\)
\(138\) 3.82650e12 6.50829
\(139\) 3.59980e11 0.588432 0.294216 0.955739i \(-0.404941\pi\)
0.294216 + 0.955739i \(0.404941\pi\)
\(140\) −3.70445e12 −5.82129
\(141\) 1.77266e12 2.67868
\(142\) −1.78828e12 −2.59926
\(143\) −4.00311e11 −0.559821
\(144\) 6.70462e12 9.02361
\(145\) 9.22310e11 1.19496
\(146\) −1.52355e12 −1.90071
\(147\) 6.42840e11 0.772427
\(148\) −1.24471e12 −1.44088
\(149\) −1.32672e12 −1.47998 −0.739990 0.672618i \(-0.765170\pi\)
−0.739990 + 0.672618i \(0.765170\pi\)
\(150\) −7.59045e12 −8.16141
\(151\) 1.54430e12 1.60088 0.800438 0.599416i \(-0.204600\pi\)
0.800438 + 0.599416i \(0.204600\pi\)
\(152\) 3.45145e11 0.345034
\(153\) −1.40454e12 −1.35435
\(154\) −1.12182e12 −1.04366
\(155\) 2.10872e12 1.89319
\(156\) 7.16810e12 6.21182
\(157\) 1.54023e12 1.28865 0.644327 0.764750i \(-0.277138\pi\)
0.644327 + 0.764750i \(0.277138\pi\)
\(158\) −2.46493e12 −1.99155
\(159\) −4.39657e10 −0.0343107
\(160\) −8.85439e12 −6.67573
\(161\) 2.95285e12 2.15129
\(162\) −7.39993e12 −5.21068
\(163\) −2.89330e11 −0.196952 −0.0984762 0.995139i \(-0.531397\pi\)
−0.0984762 + 0.995139i \(0.531397\pi\)
\(164\) −2.10373e12 −1.38468
\(165\) −2.41855e12 −1.53955
\(166\) −5.42866e12 −3.34270
\(167\) −2.72326e11 −0.162237 −0.0811183 0.996704i \(-0.525849\pi\)
−0.0811183 + 0.996704i \(0.525849\pi\)
\(168\) 1.26699e13 7.30419
\(169\) 9.10598e11 0.508100
\(170\) 3.50725e12 1.89451
\(171\) −4.99215e11 −0.261101
\(172\) 1.23400e12 0.625042
\(173\) −8.95594e11 −0.439397 −0.219699 0.975568i \(-0.570507\pi\)
−0.219699 + 0.975568i \(0.570507\pi\)
\(174\) −5.00130e12 −2.37718
\(175\) −5.85743e12 −2.69773
\(176\) −3.70268e12 −1.65271
\(177\) 1.88794e12 0.816835
\(178\) 7.58711e12 3.18249
\(179\) −1.68681e12 −0.686081 −0.343041 0.939321i \(-0.611457\pi\)
−0.343041 + 0.939321i \(0.611457\pi\)
\(180\) 3.08947e13 12.1867
\(181\) −1.70026e12 −0.650555 −0.325278 0.945619i \(-0.605458\pi\)
−0.325278 + 0.945619i \(0.605458\pi\)
\(182\) 7.57413e12 2.81151
\(183\) −4.60874e12 −1.65997
\(184\) 1.70261e13 5.95138
\(185\) −2.83549e12 −0.962019
\(186\) −1.14347e13 −3.76620
\(187\) 7.75669e11 0.248055
\(188\) 1.25053e13 3.88354
\(189\) −1.09629e13 −3.30662
\(190\) 1.24658e12 0.365237
\(191\) 2.77001e12 0.788493 0.394246 0.919005i \(-0.371006\pi\)
0.394246 + 0.919005i \(0.371006\pi\)
\(192\) 2.35305e13 6.50841
\(193\) 2.80210e11 0.0753214 0.0376607 0.999291i \(-0.488009\pi\)
0.0376607 + 0.999291i \(0.488009\pi\)
\(194\) −8.73162e12 −2.28132
\(195\) 1.63292e13 4.14738
\(196\) 4.53496e12 1.11986
\(197\) −2.96709e11 −0.0712470
\(198\) 9.35584e12 2.18486
\(199\) 2.56480e12 0.582588 0.291294 0.956634i \(-0.405914\pi\)
0.291294 + 0.956634i \(0.405914\pi\)
\(200\) −3.37739e13 −7.46304
\(201\) 6.22606e12 1.33855
\(202\) −1.15430e13 −2.41482
\(203\) −3.85943e12 −0.785768
\(204\) −1.38894e13 −2.75244
\(205\) −4.79237e12 −0.924495
\(206\) −9.59619e12 −1.80231
\(207\) −2.46264e13 −4.50365
\(208\) 2.49992e13 4.45223
\(209\) 2.75695e11 0.0478217
\(210\) 4.57605e13 7.73187
\(211\) −6.35815e11 −0.104659 −0.0523295 0.998630i \(-0.516665\pi\)
−0.0523295 + 0.998630i \(0.516665\pi\)
\(212\) −3.10159e11 −0.0497436
\(213\) 1.61329e13 2.52130
\(214\) −6.79873e12 −1.03550
\(215\) 2.81110e12 0.417316
\(216\) −6.32118e13 −9.14750
\(217\) −8.82398e12 −1.24490
\(218\) 3.32272e12 0.457070
\(219\) 1.37447e13 1.84371
\(220\) −1.70618e13 −2.23203
\(221\) −5.23705e12 −0.668235
\(222\) 1.53757e13 1.91378
\(223\) 8.89651e12 1.08030 0.540148 0.841570i \(-0.318368\pi\)
0.540148 + 0.841570i \(0.318368\pi\)
\(224\) 3.70514e13 4.38976
\(225\) 4.88503e13 5.64759
\(226\) −1.80058e13 −2.03150
\(227\) −7.22778e12 −0.795908 −0.397954 0.917405i \(-0.630280\pi\)
−0.397954 + 0.917405i \(0.630280\pi\)
\(228\) −4.93669e12 −0.530633
\(229\) −5.30547e12 −0.556709 −0.278355 0.960478i \(-0.589789\pi\)
−0.278355 + 0.960478i \(0.589789\pi\)
\(230\) 6.14940e13 6.29984
\(231\) 1.01205e13 1.01236
\(232\) −2.22534e13 −2.17377
\(233\) −9.75341e11 −0.0930462 −0.0465231 0.998917i \(-0.514814\pi\)
−0.0465231 + 0.998917i \(0.514814\pi\)
\(234\) −6.31674e13 −5.88580
\(235\) 2.84876e13 2.59288
\(236\) 1.33186e13 1.18425
\(237\) 2.22373e13 1.93182
\(238\) −1.46762e13 −1.24577
\(239\) −1.61817e13 −1.34226 −0.671129 0.741341i \(-0.734190\pi\)
−0.671129 + 0.741341i \(0.734190\pi\)
\(240\) 1.51037e14 12.2439
\(241\) 3.10447e10 0.00245976 0.00122988 0.999999i \(-0.499609\pi\)
0.00122988 + 0.999999i \(0.499609\pi\)
\(242\) 1.96964e13 1.52546
\(243\) 3.00244e13 2.27321
\(244\) −3.25127e13 −2.40663
\(245\) 1.03308e13 0.747688
\(246\) 2.59871e13 1.83914
\(247\) −1.86140e12 −0.128827
\(248\) −5.08790e13 −3.44393
\(249\) 4.89745e13 3.24245
\(250\) −6.82234e13 −4.41839
\(251\) −9.77956e12 −0.619604 −0.309802 0.950801i \(-0.600263\pi\)
−0.309802 + 0.950801i \(0.600263\pi\)
\(252\) −1.29279e14 −8.01358
\(253\) 1.36001e13 0.824860
\(254\) 4.76498e13 2.82797
\(255\) −3.16405e13 −1.83769
\(256\) 4.09200e13 2.32603
\(257\) −3.13864e13 −1.74626 −0.873132 0.487484i \(-0.837915\pi\)
−0.873132 + 0.487484i \(0.837915\pi\)
\(258\) −1.52434e13 −0.830184
\(259\) 1.18652e13 0.632595
\(260\) 1.15195e14 6.01287
\(261\) 3.21872e13 1.64498
\(262\) 2.72825e13 1.36530
\(263\) 1.19615e13 0.586177 0.293089 0.956085i \(-0.405317\pi\)
0.293089 + 0.956085i \(0.405317\pi\)
\(264\) 5.83546e13 2.80061
\(265\) −7.06554e11 −0.0332119
\(266\) −5.21633e12 −0.240169
\(267\) −6.84470e13 −3.08704
\(268\) 4.39221e13 1.94063
\(269\) −2.88848e13 −1.25035 −0.625175 0.780485i \(-0.714972\pi\)
−0.625175 + 0.780485i \(0.714972\pi\)
\(270\) −2.28305e14 −9.68311
\(271\) −1.75361e13 −0.728790 −0.364395 0.931245i \(-0.618724\pi\)
−0.364395 + 0.931245i \(0.618724\pi\)
\(272\) −4.84401e13 −1.97277
\(273\) −6.83299e13 −2.72719
\(274\) −4.88396e13 −1.91049
\(275\) −2.69779e13 −1.03438
\(276\) −2.43528e14 −9.15271
\(277\) −4.89346e13 −1.80292 −0.901462 0.432859i \(-0.857505\pi\)
−0.901462 + 0.432859i \(0.857505\pi\)
\(278\) −3.13701e13 −1.13310
\(279\) 7.35909e13 2.60616
\(280\) 2.03612e14 7.07026
\(281\) −6.93651e12 −0.236187 −0.118094 0.993002i \(-0.537678\pi\)
−0.118094 + 0.993002i \(0.537678\pi\)
\(282\) −1.54477e14 −5.15813
\(283\) 5.95228e13 1.94921 0.974604 0.223935i \(-0.0718905\pi\)
0.974604 + 0.223935i \(0.0718905\pi\)
\(284\) 1.13811e14 3.65538
\(285\) −1.12460e13 −0.354283
\(286\) 3.48847e13 1.07801
\(287\) 2.00538e13 0.607920
\(288\) −3.09004e14 −9.18979
\(289\) −2.41242e13 −0.703907
\(290\) −8.03737e13 −2.30104
\(291\) 7.87721e13 2.21290
\(292\) 9.69627e13 2.67300
\(293\) 4.44871e12 0.120355 0.0601773 0.998188i \(-0.480833\pi\)
0.0601773 + 0.998188i \(0.480833\pi\)
\(294\) −5.60196e13 −1.48741
\(295\) 3.03402e13 0.790673
\(296\) 6.84145e13 1.75002
\(297\) −5.04924e13 −1.26784
\(298\) 1.15616e14 2.84989
\(299\) −9.18233e13 −2.22209
\(300\) 4.83076e14 11.4775
\(301\) −1.17631e13 −0.274414
\(302\) −1.34576e14 −3.08269
\(303\) 1.04134e14 2.34239
\(304\) −1.72170e13 −0.380324
\(305\) −7.40651e13 −1.60681
\(306\) 1.22397e14 2.60798
\(307\) 4.35234e13 0.910881 0.455441 0.890266i \(-0.349482\pi\)
0.455441 + 0.890266i \(0.349482\pi\)
\(308\) 7.13956e13 1.46772
\(309\) 8.65718e13 1.74825
\(310\) −1.83762e14 −3.64558
\(311\) 5.89792e12 0.114952 0.0574761 0.998347i \(-0.481695\pi\)
0.0574761 + 0.998347i \(0.481695\pi\)
\(312\) −3.93990e14 −7.54457
\(313\) 3.03098e13 0.570281 0.285141 0.958486i \(-0.407960\pi\)
0.285141 + 0.958486i \(0.407960\pi\)
\(314\) −1.34221e14 −2.48147
\(315\) −2.94503e14 −5.35035
\(316\) 1.56875e14 2.80075
\(317\) −8.45658e12 −0.148378 −0.0741889 0.997244i \(-0.523637\pi\)
−0.0741889 + 0.997244i \(0.523637\pi\)
\(318\) 3.83135e12 0.0660697
\(319\) −1.77756e13 −0.301284
\(320\) 3.78149e14 6.29996
\(321\) 6.13346e13 1.00445
\(322\) −2.57323e14 −4.14259
\(323\) 3.60677e12 0.0570827
\(324\) 4.70951e14 7.32787
\(325\) 1.82146e14 2.78651
\(326\) 2.52133e13 0.379257
\(327\) −2.99758e13 −0.443362
\(328\) 1.15630e14 1.68176
\(329\) −1.19207e14 −1.70500
\(330\) 2.10762e14 2.96460
\(331\) −1.08434e14 −1.50007 −0.750037 0.661396i \(-0.769964\pi\)
−0.750037 + 0.661396i \(0.769964\pi\)
\(332\) 3.45494e14 4.70090
\(333\) −9.89541e13 −1.32431
\(334\) 2.37316e13 0.312408
\(335\) 1.00056e14 1.29568
\(336\) −6.32018e14 −8.05125
\(337\) 1.08104e14 1.35480 0.677401 0.735613i \(-0.263106\pi\)
0.677401 + 0.735613i \(0.263106\pi\)
\(338\) −7.93531e13 −0.978412
\(339\) 1.62439e14 1.97057
\(340\) −2.23210e14 −2.66428
\(341\) −4.06412e13 −0.477328
\(342\) 4.35036e13 0.502784
\(343\) 6.13077e13 0.697264
\(344\) −6.78261e13 −0.759146
\(345\) −5.54767e14 −6.11090
\(346\) 7.80456e13 0.846116
\(347\) −6.13520e13 −0.654661 −0.327331 0.944910i \(-0.606149\pi\)
−0.327331 + 0.944910i \(0.606149\pi\)
\(348\) 3.18296e14 3.34307
\(349\) −6.29685e13 −0.651004 −0.325502 0.945541i \(-0.605533\pi\)
−0.325502 + 0.945541i \(0.605533\pi\)
\(350\) 5.10440e14 5.19482
\(351\) 3.40907e14 3.41544
\(352\) 1.70650e14 1.68314
\(353\) 3.83303e13 0.372204 0.186102 0.982530i \(-0.440415\pi\)
0.186102 + 0.982530i \(0.440415\pi\)
\(354\) −1.64522e14 −1.57292
\(355\) 2.59265e14 2.44055
\(356\) −4.82864e14 −4.47558
\(357\) 1.32401e14 1.20841
\(358\) 1.46996e14 1.32114
\(359\) 1.77156e14 1.56796 0.783982 0.620784i \(-0.213186\pi\)
0.783982 + 0.620784i \(0.213186\pi\)
\(360\) −1.69810e15 −14.8013
\(361\) −1.15208e14 −0.988995
\(362\) 1.48168e14 1.25273
\(363\) −1.77690e14 −1.47971
\(364\) −4.82038e14 −3.95388
\(365\) 2.20884e14 1.78466
\(366\) 4.01624e14 3.19649
\(367\) −5.67164e13 −0.444677 −0.222339 0.974969i \(-0.571369\pi\)
−0.222339 + 0.974969i \(0.571369\pi\)
\(368\) −8.49321e14 −6.56007
\(369\) −1.67246e14 −1.27266
\(370\) 2.47096e14 1.85249
\(371\) 2.95659e12 0.0218391
\(372\) 7.27734e14 5.29647
\(373\) 1.04076e14 0.746363 0.373182 0.927758i \(-0.378267\pi\)
0.373182 + 0.927758i \(0.378267\pi\)
\(374\) −6.75949e13 −0.477661
\(375\) 6.15476e14 4.28587
\(376\) −6.87347e14 −4.71675
\(377\) 1.20015e14 0.811628
\(378\) 9.55348e14 6.36732
\(379\) −9.10659e12 −0.0598191 −0.0299096 0.999553i \(-0.509522\pi\)
−0.0299096 + 0.999553i \(0.509522\pi\)
\(380\) −7.93354e13 −0.513638
\(381\) −4.29871e14 −2.74316
\(382\) −2.41390e14 −1.51834
\(383\) 4.60884e13 0.285758 0.142879 0.989740i \(-0.454364\pi\)
0.142879 + 0.989740i \(0.454364\pi\)
\(384\) −9.22157e14 −5.63616
\(385\) 1.62642e14 0.979936
\(386\) −2.44186e13 −0.145041
\(387\) 9.81030e13 0.574476
\(388\) 5.55703e14 3.20825
\(389\) −2.14867e14 −1.22306 −0.611530 0.791221i \(-0.709446\pi\)
−0.611530 + 0.791221i \(0.709446\pi\)
\(390\) −1.42299e15 −7.98632
\(391\) 1.77923e14 0.984600
\(392\) −2.49261e14 −1.36013
\(393\) −2.46129e14 −1.32435
\(394\) 2.58564e13 0.137195
\(395\) 3.57366e14 1.86995
\(396\) −5.95431e14 −3.07261
\(397\) −1.39664e14 −0.710783 −0.355392 0.934717i \(-0.615653\pi\)
−0.355392 + 0.934717i \(0.615653\pi\)
\(398\) −2.23507e14 −1.12185
\(399\) 4.70590e13 0.232966
\(400\) 1.68476e15 8.22635
\(401\) −3.58063e14 −1.72451 −0.862255 0.506474i \(-0.830949\pi\)
−0.862255 + 0.506474i \(0.830949\pi\)
\(402\) −5.42563e14 −2.57755
\(403\) 2.74395e14 1.28587
\(404\) 7.34624e14 3.39600
\(405\) 1.07284e15 4.89253
\(406\) 3.36326e14 1.51310
\(407\) 5.46482e13 0.242553
\(408\) 7.63421e14 3.34297
\(409\) 1.52082e14 0.657050 0.328525 0.944495i \(-0.393449\pi\)
0.328525 + 0.944495i \(0.393449\pi\)
\(410\) 4.17627e14 1.78023
\(411\) 4.40605e14 1.85319
\(412\) 6.10727e14 2.53462
\(413\) −1.26959e14 −0.519923
\(414\) 2.14604e15 8.67235
\(415\) 7.87048e14 3.13860
\(416\) −1.15217e15 −4.53422
\(417\) 2.83004e14 1.09912
\(418\) −2.40252e13 −0.0920868
\(419\) −2.33547e14 −0.883480 −0.441740 0.897143i \(-0.645639\pi\)
−0.441740 + 0.897143i \(0.645639\pi\)
\(420\) −2.91232e15 −10.8735
\(421\) 8.00503e13 0.294993 0.147496 0.989063i \(-0.452879\pi\)
0.147496 + 0.989063i \(0.452879\pi\)
\(422\) 5.54074e13 0.201534
\(423\) 9.94173e14 3.56936
\(424\) 1.70477e13 0.0604162
\(425\) −3.52937e14 −1.23469
\(426\) −1.40589e15 −4.85509
\(427\) 3.09927e14 1.05659
\(428\) 4.32689e14 1.45625
\(429\) −3.14711e14 −1.04568
\(430\) −2.44971e14 −0.803595
\(431\) 3.50083e14 1.13383 0.566913 0.823777i \(-0.308137\pi\)
0.566913 + 0.823777i \(0.308137\pi\)
\(432\) 3.15322e15 10.0831
\(433\) −1.57619e14 −0.497651 −0.248826 0.968548i \(-0.580045\pi\)
−0.248826 + 0.968548i \(0.580045\pi\)
\(434\) 7.68957e14 2.39722
\(435\) 7.25090e14 2.23203
\(436\) −2.11466e14 −0.642785
\(437\) 6.32390e13 0.189818
\(438\) −1.19776e15 −3.55029
\(439\) −5.00298e14 −1.46445 −0.732225 0.681063i \(-0.761518\pi\)
−0.732225 + 0.681063i \(0.761518\pi\)
\(440\) 9.37791e14 2.71092
\(441\) 3.60529e14 1.02927
\(442\) 4.56377e14 1.28677
\(443\) 2.78354e14 0.775133 0.387567 0.921842i \(-0.373316\pi\)
0.387567 + 0.921842i \(0.373316\pi\)
\(444\) −9.78549e14 −2.69139
\(445\) −1.09998e15 −2.98817
\(446\) −7.75277e14 −2.08025
\(447\) −1.04303e15 −2.76442
\(448\) −1.58237e15 −4.14266
\(449\) −4.73886e14 −1.22552 −0.612758 0.790270i \(-0.709940\pi\)
−0.612758 + 0.790270i \(0.709940\pi\)
\(450\) −4.25701e15 −10.8752
\(451\) 9.23631e13 0.233092
\(452\) 1.14594e15 2.85693
\(453\) 1.21408e15 2.99024
\(454\) 6.29857e14 1.53262
\(455\) −1.09810e15 −2.63985
\(456\) 2.71342e14 0.644481
\(457\) −4.20915e14 −0.987769 −0.493884 0.869528i \(-0.664423\pi\)
−0.493884 + 0.869528i \(0.664423\pi\)
\(458\) 4.62339e14 1.07201
\(459\) −6.60564e14 −1.51337
\(460\) −3.91364e15 −8.85957
\(461\) 5.26897e14 1.17861 0.589306 0.807910i \(-0.299401\pi\)
0.589306 + 0.807910i \(0.299401\pi\)
\(462\) −8.81938e14 −1.94943
\(463\) 4.35851e14 0.952012 0.476006 0.879442i \(-0.342084\pi\)
0.476006 + 0.879442i \(0.342084\pi\)
\(464\) 1.11008e15 2.39610
\(465\) 1.65781e15 3.53624
\(466\) 8.49951e13 0.179172
\(467\) 3.98633e14 0.830483 0.415241 0.909711i \(-0.363697\pi\)
0.415241 + 0.909711i \(0.363697\pi\)
\(468\) 4.02014e15 8.27730
\(469\) −4.18688e14 −0.852000
\(470\) −2.48253e15 −4.99293
\(471\) 1.21088e15 2.40705
\(472\) −7.32046e14 −1.43833
\(473\) −5.41781e13 −0.105217
\(474\) −1.93785e15 −3.71997
\(475\) −1.25444e14 −0.238032
\(476\) 9.34029e14 1.75195
\(477\) −2.46576e13 −0.0457194
\(478\) 1.41014e15 2.58469
\(479\) 1.15602e14 0.209468 0.104734 0.994500i \(-0.466601\pi\)
0.104734 + 0.994500i \(0.466601\pi\)
\(480\) −6.96103e15 −12.4694
\(481\) −3.68965e14 −0.653413
\(482\) −2.70536e12 −0.00473659
\(483\) 2.32143e15 4.01834
\(484\) −1.25353e15 −2.14528
\(485\) 1.26591e15 2.14202
\(486\) −2.61645e15 −4.37737
\(487\) 4.03307e14 0.667155 0.333577 0.942723i \(-0.391744\pi\)
0.333577 + 0.942723i \(0.391744\pi\)
\(488\) 1.78704e15 2.92297
\(489\) −2.27461e14 −0.367883
\(490\) −9.00267e14 −1.43977
\(491\) 4.64426e14 0.734461 0.367230 0.930130i \(-0.380306\pi\)
0.367230 + 0.930130i \(0.380306\pi\)
\(492\) −1.65388e15 −2.58641
\(493\) −2.32548e14 −0.359630
\(494\) 1.62210e14 0.248072
\(495\) −1.35641e15 −2.05146
\(496\) 2.53802e15 3.79617
\(497\) −1.08490e15 −1.60483
\(498\) −4.26783e15 −6.24376
\(499\) 3.73382e14 0.540257 0.270128 0.962824i \(-0.412934\pi\)
0.270128 + 0.962824i \(0.412934\pi\)
\(500\) 4.34192e15 6.21365
\(501\) −2.14094e14 −0.303038
\(502\) 8.52230e14 1.19313
\(503\) −9.19697e14 −1.27356 −0.636782 0.771044i \(-0.719735\pi\)
−0.636782 + 0.771044i \(0.719735\pi\)
\(504\) 7.10575e15 9.73290
\(505\) 1.67350e15 2.26737
\(506\) −1.18517e15 −1.58837
\(507\) 7.15882e14 0.949069
\(508\) −3.03256e15 −3.97703
\(509\) −5.01368e14 −0.650442 −0.325221 0.945638i \(-0.605439\pi\)
−0.325221 + 0.945638i \(0.605439\pi\)
\(510\) 2.75728e15 3.53871
\(511\) −9.24296e14 −1.17354
\(512\) −1.16367e15 −1.46166
\(513\) −2.34784e14 −0.291758
\(514\) 2.73514e15 3.36266
\(515\) 1.39126e15 1.69226
\(516\) 9.70132e14 1.16750
\(517\) −5.49040e14 −0.653741
\(518\) −1.03398e15 −1.21814
\(519\) −7.04086e14 −0.820740
\(520\) −6.33163e15 −7.30293
\(521\) −8.34015e14 −0.951845 −0.475922 0.879487i \(-0.657886\pi\)
−0.475922 + 0.879487i \(0.657886\pi\)
\(522\) −2.80492e15 −3.16761
\(523\) −7.13094e14 −0.796870 −0.398435 0.917197i \(-0.630447\pi\)
−0.398435 + 0.917197i \(0.630447\pi\)
\(524\) −1.73633e15 −1.92004
\(525\) −4.60492e15 −5.03902
\(526\) −1.04237e15 −1.12876
\(527\) −5.31686e14 −0.569766
\(528\) −2.91092e15 −3.08706
\(529\) 2.16679e15 2.27410
\(530\) 6.15719e13 0.0639537
\(531\) 1.05882e15 1.08844
\(532\) 3.31981e14 0.337753
\(533\) −6.23603e14 −0.627926
\(534\) 5.96474e15 5.94449
\(535\) 9.85681e14 0.972279
\(536\) −2.41415e15 −2.35699
\(537\) −1.32612e15 −1.28152
\(538\) 2.51713e15 2.40771
\(539\) −1.99105e14 −0.188514
\(540\) 1.45299e16 13.6175
\(541\) −1.58429e15 −1.46977 −0.734885 0.678192i \(-0.762764\pi\)
−0.734885 + 0.678192i \(0.762764\pi\)
\(542\) 1.52817e15 1.40338
\(543\) −1.33669e15 −1.21516
\(544\) 2.23252e15 2.00910
\(545\) −4.81728e14 −0.429162
\(546\) 5.95454e15 5.25156
\(547\) −2.40211e14 −0.209731 −0.104866 0.994486i \(-0.533441\pi\)
−0.104866 + 0.994486i \(0.533441\pi\)
\(548\) 3.10828e15 2.68675
\(549\) −2.58475e15 −2.21193
\(550\) 2.35096e15 1.99182
\(551\) −8.26545e13 −0.0693318
\(552\) 1.33854e16 11.1164
\(553\) −1.49541e15 −1.22962
\(554\) 4.26435e15 3.47176
\(555\) −2.22917e15 −1.79693
\(556\) 1.99647e15 1.59350
\(557\) 4.00785e14 0.316744 0.158372 0.987380i \(-0.449375\pi\)
0.158372 + 0.987380i \(0.449375\pi\)
\(558\) −6.41301e15 −5.01850
\(559\) 3.65792e14 0.283445
\(560\) −1.01569e16 −7.79339
\(561\) 6.09806e14 0.463336
\(562\) 6.04475e14 0.454809
\(563\) −1.30980e15 −0.975906 −0.487953 0.872870i \(-0.662256\pi\)
−0.487953 + 0.872870i \(0.662256\pi\)
\(564\) 9.83129e15 7.25397
\(565\) 2.61049e15 1.90746
\(566\) −5.18706e15 −3.75345
\(567\) −4.48934e15 −3.21718
\(568\) −6.25552e15 −4.43964
\(569\) −9.61821e14 −0.676047 −0.338024 0.941138i \(-0.609758\pi\)
−0.338024 + 0.941138i \(0.609758\pi\)
\(570\) 9.80018e14 0.682217
\(571\) 2.05240e15 1.41503 0.707513 0.706700i \(-0.249817\pi\)
0.707513 + 0.706700i \(0.249817\pi\)
\(572\) −2.22015e15 −1.51602
\(573\) 2.17769e15 1.47281
\(574\) −1.74757e15 −1.17063
\(575\) −6.18820e15 −4.10574
\(576\) 1.31968e16 8.67251
\(577\) 5.64715e14 0.367589 0.183794 0.982965i \(-0.441162\pi\)
0.183794 + 0.982965i \(0.441162\pi\)
\(578\) 2.10228e15 1.35546
\(579\) 2.20292e14 0.140691
\(580\) 5.11520e15 3.23600
\(581\) −3.29342e15 −2.06385
\(582\) −6.86451e15 −4.26121
\(583\) 1.36174e13 0.00837367
\(584\) −5.32949e15 −3.24649
\(585\) 9.15802e15 5.52643
\(586\) −3.87679e14 −0.231758
\(587\) 4.21382e14 0.249555 0.124778 0.992185i \(-0.460178\pi\)
0.124778 + 0.992185i \(0.460178\pi\)
\(588\) 3.56524e15 2.09177
\(589\) −1.88977e14 −0.109843
\(590\) −2.64396e15 −1.52254
\(591\) −2.33263e14 −0.133081
\(592\) −3.41275e15 −1.92901
\(593\) −4.78325e14 −0.267869 −0.133934 0.990990i \(-0.542761\pi\)
−0.133934 + 0.990990i \(0.542761\pi\)
\(594\) 4.40011e15 2.44139
\(595\) 2.12775e15 1.16971
\(596\) −7.35810e15 −4.00785
\(597\) 2.01636e15 1.08820
\(598\) 8.00185e15 4.27892
\(599\) −5.93499e14 −0.314465 −0.157232 0.987562i \(-0.550257\pi\)
−0.157232 + 0.987562i \(0.550257\pi\)
\(600\) −2.65519e16 −13.9400
\(601\) 3.22160e15 1.67596 0.837978 0.545704i \(-0.183738\pi\)
0.837978 + 0.545704i \(0.183738\pi\)
\(602\) 1.02509e15 0.528420
\(603\) 3.49180e15 1.78363
\(604\) 8.56478e15 4.33524
\(605\) −2.85558e15 −1.43232
\(606\) −9.07469e15 −4.51058
\(607\) −3.13500e15 −1.54419 −0.772093 0.635509i \(-0.780790\pi\)
−0.772093 + 0.635509i \(0.780790\pi\)
\(608\) 7.93502e14 0.387328
\(609\) −3.03415e15 −1.46772
\(610\) 6.45432e15 3.09412
\(611\) 3.70692e15 1.76111
\(612\) −7.78969e15 −3.66765
\(613\) 1.54585e15 0.721332 0.360666 0.932695i \(-0.382549\pi\)
0.360666 + 0.932695i \(0.382549\pi\)
\(614\) −3.79280e15 −1.75402
\(615\) −3.76761e15 −1.72684
\(616\) −3.92421e15 −1.78262
\(617\) −1.75213e15 −0.788855 −0.394428 0.918927i \(-0.629057\pi\)
−0.394428 + 0.918927i \(0.629057\pi\)
\(618\) −7.54421e15 −3.36649
\(619\) 4.13897e15 1.83060 0.915300 0.402773i \(-0.131954\pi\)
0.915300 + 0.402773i \(0.131954\pi\)
\(620\) 1.16951e16 5.12684
\(621\) −1.15819e16 −5.03243
\(622\) −5.13969e14 −0.221355
\(623\) 4.60290e15 1.96493
\(624\) 1.96535e16 8.31621
\(625\) 4.48120e15 1.87955
\(626\) −2.64131e15 −1.09815
\(627\) 2.16743e14 0.0893250
\(628\) 8.54220e15 3.48973
\(629\) 7.14932e14 0.289525
\(630\) 2.56642e16 10.3028
\(631\) 1.34196e15 0.534046 0.267023 0.963690i \(-0.413960\pi\)
0.267023 + 0.963690i \(0.413960\pi\)
\(632\) −8.62251e15 −3.40165
\(633\) −4.99857e14 −0.195490
\(634\) 7.36940e14 0.285721
\(635\) −6.90827e15 −2.65530
\(636\) −2.43837e14 −0.0929150
\(637\) 1.34428e15 0.507837
\(638\) 1.54904e15 0.580160
\(639\) 9.04794e15 3.35966
\(640\) −1.48196e16 −5.45565
\(641\) 2.56166e15 0.934978 0.467489 0.883999i \(-0.345159\pi\)
0.467489 + 0.883999i \(0.345159\pi\)
\(642\) −5.34494e15 −1.93419
\(643\) 1.86299e15 0.668420 0.334210 0.942499i \(-0.391531\pi\)
0.334210 + 0.942499i \(0.391531\pi\)
\(644\) 1.63767e16 5.82579
\(645\) 2.21000e15 0.779495
\(646\) −3.14308e14 −0.109920
\(647\) −2.31615e15 −0.803144 −0.401572 0.915827i \(-0.631536\pi\)
−0.401572 + 0.915827i \(0.631536\pi\)
\(648\) −2.58855e16 −8.90007
\(649\) −5.84744e14 −0.199352
\(650\) −1.58729e16 −5.36577
\(651\) −6.93712e15 −2.32533
\(652\) −1.60464e15 −0.533355
\(653\) 3.45678e15 1.13933 0.569665 0.821877i \(-0.307073\pi\)
0.569665 + 0.821877i \(0.307073\pi\)
\(654\) 2.61221e15 0.853749
\(655\) −3.95542e15 −1.28193
\(656\) −5.76802e15 −1.85377
\(657\) 7.70852e15 2.45675
\(658\) 1.03882e16 3.28320
\(659\) −1.88198e15 −0.589856 −0.294928 0.955519i \(-0.595296\pi\)
−0.294928 + 0.955519i \(0.595296\pi\)
\(660\) −1.34134e16 −4.16916
\(661\) 1.30744e15 0.403007 0.201504 0.979488i \(-0.435417\pi\)
0.201504 + 0.979488i \(0.435417\pi\)
\(662\) 9.44940e15 2.88858
\(663\) −4.11719e15 −1.24818
\(664\) −1.89898e16 −5.70948
\(665\) 7.56265e14 0.225504
\(666\) 8.62326e15 2.55013
\(667\) −4.07737e15 −1.19588
\(668\) −1.51034e15 −0.439344
\(669\) 6.99415e15 2.01786
\(670\) −8.71930e15 −2.49500
\(671\) 1.42745e15 0.405123
\(672\) 2.91286e16 8.19952
\(673\) −4.80307e15 −1.34102 −0.670512 0.741899i \(-0.733925\pi\)
−0.670512 + 0.741899i \(0.733925\pi\)
\(674\) −9.42059e15 −2.60885
\(675\) 2.29746e16 6.31068
\(676\) 5.05024e15 1.37596
\(677\) −6.54600e15 −1.76904 −0.884521 0.466500i \(-0.845515\pi\)
−0.884521 + 0.466500i \(0.845515\pi\)
\(678\) −1.41556e16 −3.79459
\(679\) −5.29724e15 −1.40853
\(680\) 1.22686e16 3.23591
\(681\) −5.68224e15 −1.48666
\(682\) 3.54163e15 0.919156
\(683\) −3.91112e15 −1.00690 −0.503451 0.864024i \(-0.667937\pi\)
−0.503451 + 0.864024i \(0.667937\pi\)
\(684\) −2.76868e15 −0.707073
\(685\) 7.08077e15 1.79383
\(686\) −5.34259e15 −1.34267
\(687\) −4.17098e15 −1.03986
\(688\) 3.38340e15 0.836790
\(689\) −9.19397e13 −0.0225578
\(690\) 4.83446e16 11.7673
\(691\) 6.39806e14 0.154497 0.0772483 0.997012i \(-0.475387\pi\)
0.0772483 + 0.997012i \(0.475387\pi\)
\(692\) −4.96703e15 −1.18991
\(693\) 5.67594e15 1.34898
\(694\) 5.34646e15 1.26063
\(695\) 4.54804e15 1.06392
\(696\) −1.74949e16 −4.06033
\(697\) 1.20834e15 0.278232
\(698\) 5.48733e15 1.25359
\(699\) −7.66781e14 −0.173799
\(700\) −3.24857e16 −7.30556
\(701\) −6.15819e15 −1.37405 −0.687027 0.726631i \(-0.741085\pi\)
−0.687027 + 0.726631i \(0.741085\pi\)
\(702\) −2.97080e16 −6.57687
\(703\) 2.54108e14 0.0558166
\(704\) −7.28803e15 −1.58840
\(705\) 2.23960e16 4.84319
\(706\) −3.34025e15 −0.716726
\(707\) −7.00280e15 −1.49096
\(708\) 1.04706e16 2.21202
\(709\) 4.62591e15 0.969713 0.484857 0.874594i \(-0.338872\pi\)
0.484857 + 0.874594i \(0.338872\pi\)
\(710\) −2.25934e16 −4.69959
\(711\) 1.24715e16 2.57417
\(712\) 2.65403e16 5.43582
\(713\) −9.32227e15 −1.89465
\(714\) −1.15379e16 −2.32695
\(715\) −5.05759e15 −1.01219
\(716\) −9.35520e15 −1.85794
\(717\) −1.27215e16 −2.50717
\(718\) −1.54381e16 −3.01931
\(719\) −2.95761e14 −0.0574027 −0.0287013 0.999588i \(-0.509137\pi\)
−0.0287013 + 0.999588i \(0.509137\pi\)
\(720\) 8.47072e16 16.3152
\(721\) −5.82175e15 −1.11278
\(722\) 1.00397e16 1.90444
\(723\) 2.44063e13 0.00459453
\(724\) −9.42978e15 −1.76173
\(725\) 8.08808e15 1.49964
\(726\) 1.54846e16 2.84937
\(727\) −6.30502e15 −1.15146 −0.575728 0.817642i \(-0.695281\pi\)
−0.575728 + 0.817642i \(0.695281\pi\)
\(728\) 2.64949e16 4.80219
\(729\) 8.56161e15 1.54012
\(730\) −1.92488e16 −3.43658
\(731\) −7.08783e14 −0.125594
\(732\) −2.55604e16 −4.49528
\(733\) 5.25600e15 0.917452 0.458726 0.888578i \(-0.348306\pi\)
0.458726 + 0.888578i \(0.348306\pi\)
\(734\) 4.94249e15 0.856283
\(735\) 8.12174e15 1.39659
\(736\) 3.91437e16 6.68088
\(737\) −1.92838e15 −0.326678
\(738\) 1.45745e16 2.45067
\(739\) −9.12100e15 −1.52229 −0.761146 0.648580i \(-0.775363\pi\)
−0.761146 + 0.648580i \(0.775363\pi\)
\(740\) −1.57258e16 −2.60519
\(741\) −1.46337e15 −0.240632
\(742\) −2.57649e14 −0.0420540
\(743\) −2.18359e15 −0.353780 −0.176890 0.984231i \(-0.556604\pi\)
−0.176890 + 0.984231i \(0.556604\pi\)
\(744\) −3.99994e16 −6.43283
\(745\) −1.67620e16 −2.67588
\(746\) −9.06956e15 −1.43722
\(747\) 2.74667e16 4.32060
\(748\) 4.30192e15 0.671743
\(749\) −4.12461e15 −0.639341
\(750\) −5.36350e16 −8.25300
\(751\) 1.00530e16 1.53560 0.767798 0.640692i \(-0.221353\pi\)
0.767798 + 0.640692i \(0.221353\pi\)
\(752\) 3.42872e16 5.19917
\(753\) −7.68837e15 −1.15734
\(754\) −1.04586e16 −1.56289
\(755\) 1.95109e16 2.89447
\(756\) −6.08009e16 −8.95447
\(757\) 4.05739e15 0.593225 0.296613 0.954998i \(-0.404143\pi\)
0.296613 + 0.954998i \(0.404143\pi\)
\(758\) 7.93584e14 0.115189
\(759\) 1.06920e16 1.54074
\(760\) 4.36062e15 0.623840
\(761\) −2.25831e15 −0.320751 −0.160375 0.987056i \(-0.551270\pi\)
−0.160375 + 0.987056i \(0.551270\pi\)
\(762\) 3.74607e16 5.28230
\(763\) 2.01580e15 0.282204
\(764\) 1.53627e16 2.13527
\(765\) −1.77452e16 −2.44874
\(766\) −4.01633e15 −0.550264
\(767\) 3.94799e15 0.537033
\(768\) 3.21699e16 4.34474
\(769\) −3.92462e15 −0.526263 −0.263131 0.964760i \(-0.584755\pi\)
−0.263131 + 0.964760i \(0.584755\pi\)
\(770\) −1.41732e16 −1.88699
\(771\) −2.46750e16 −3.26181
\(772\) 1.55406e15 0.203974
\(773\) −5.34422e15 −0.696462 −0.348231 0.937409i \(-0.613218\pi\)
−0.348231 + 0.937409i \(0.613218\pi\)
\(774\) −8.54909e15 −1.10623
\(775\) 1.84921e16 2.37590
\(776\) −3.05438e16 −3.89658
\(777\) 9.32801e15 1.18161
\(778\) 1.87244e16 2.35516
\(779\) 4.29477e14 0.0536395
\(780\) 9.05629e16 11.2313
\(781\) −4.99679e15 −0.615334
\(782\) −1.55049e16 −1.89597
\(783\) 1.51378e16 1.83812
\(784\) 1.24340e16 1.49924
\(785\) 1.94594e16 2.32995
\(786\) 2.14486e16 2.55021
\(787\) 9.57063e15 1.13000 0.565002 0.825090i \(-0.308876\pi\)
0.565002 + 0.825090i \(0.308876\pi\)
\(788\) −1.64557e15 −0.192940
\(789\) 9.40374e15 1.09491
\(790\) −3.11423e16 −3.60083
\(791\) −1.09236e16 −1.25429
\(792\) 3.27274e16 3.73184
\(793\) −9.63764e15 −1.09136
\(794\) 1.21709e16 1.36870
\(795\) −5.55470e14 −0.0620356
\(796\) 1.42246e16 1.57767
\(797\) 2.69107e15 0.296418 0.148209 0.988956i \(-0.452649\pi\)
0.148209 + 0.988956i \(0.452649\pi\)
\(798\) −4.10091e15 −0.448605
\(799\) −7.18279e15 −0.780343
\(800\) −7.76475e16 −8.37785
\(801\) −3.83876e16 −4.11351
\(802\) 3.12031e16 3.32077
\(803\) −4.25709e15 −0.449964
\(804\) 3.45302e16 3.62485
\(805\) 3.73068e16 3.88965
\(806\) −2.39119e16 −2.47611
\(807\) −2.27083e16 −2.33550
\(808\) −4.03781e16 −4.12461
\(809\) −1.22996e16 −1.24788 −0.623942 0.781471i \(-0.714470\pi\)
−0.623942 + 0.781471i \(0.714470\pi\)
\(810\) −9.34918e16 −9.42119
\(811\) 1.42153e16 1.42279 0.711395 0.702792i \(-0.248064\pi\)
0.711395 + 0.702792i \(0.248064\pi\)
\(812\) −2.14047e16 −2.12789
\(813\) −1.37863e16 −1.36129
\(814\) −4.76226e15 −0.467067
\(815\) −3.65543e15 −0.356100
\(816\) −3.80820e16 −3.68489
\(817\) −2.51922e14 −0.0242128
\(818\) −1.32530e16 −1.26523
\(819\) −3.83219e16 −3.63401
\(820\) −2.65789e16 −2.50357
\(821\) 1.42313e16 1.33155 0.665773 0.746154i \(-0.268102\pi\)
0.665773 + 0.746154i \(0.268102\pi\)
\(822\) −3.83961e16 −3.56855
\(823\) −3.08929e15 −0.285206 −0.142603 0.989780i \(-0.545547\pi\)
−0.142603 + 0.989780i \(0.545547\pi\)
\(824\) −3.35682e16 −3.07842
\(825\) −2.12092e16 −1.93209
\(826\) 1.10637e16 1.00118
\(827\) 1.91274e16 1.71940 0.859698 0.510802i \(-0.170652\pi\)
0.859698 + 0.510802i \(0.170652\pi\)
\(828\) −1.36580e17 −12.1961
\(829\) 2.34185e15 0.207735 0.103867 0.994591i \(-0.466878\pi\)
0.103867 + 0.994591i \(0.466878\pi\)
\(830\) −6.85865e16 −6.04378
\(831\) −3.84708e16 −3.36764
\(832\) 4.92062e16 4.27899
\(833\) −2.60478e15 −0.225021
\(834\) −2.46621e16 −2.11649
\(835\) −3.44061e15 −0.293332
\(836\) 1.52903e15 0.129503
\(837\) 3.46102e16 2.91216
\(838\) 2.03522e16 1.70125
\(839\) 2.07683e16 1.72468 0.862342 0.506326i \(-0.168997\pi\)
0.862342 + 0.506326i \(0.168997\pi\)
\(840\) 1.60073e17 13.2064
\(841\) −6.87132e15 −0.563199
\(842\) −6.97590e15 −0.568047
\(843\) −5.45326e15 −0.441168
\(844\) −3.52627e15 −0.283421
\(845\) 1.15046e16 0.918672
\(846\) −8.66362e16 −6.87326
\(847\) 1.19492e16 0.941850
\(848\) −8.50397e14 −0.0665954
\(849\) 4.67949e16 3.64088
\(850\) 3.07564e16 2.37756
\(851\) 1.25352e16 0.962762
\(852\) 8.94742e16 6.82779
\(853\) −1.94025e16 −1.47108 −0.735541 0.677480i \(-0.763072\pi\)
−0.735541 + 0.677480i \(0.763072\pi\)
\(854\) −2.70083e16 −2.03460
\(855\) −6.30716e15 −0.472085
\(856\) −2.37825e16 −1.76869
\(857\) 3.13237e15 0.231462 0.115731 0.993281i \(-0.463079\pi\)
0.115731 + 0.993281i \(0.463079\pi\)
\(858\) 2.74252e16 2.01358
\(859\) −1.69679e15 −0.123785 −0.0618923 0.998083i \(-0.519714\pi\)
−0.0618923 + 0.998083i \(0.519714\pi\)
\(860\) 1.55906e16 1.13011
\(861\) 1.57656e16 1.13552
\(862\) −3.05077e16 −2.18333
\(863\) −1.62531e16 −1.15578 −0.577891 0.816114i \(-0.696124\pi\)
−0.577891 + 0.816114i \(0.696124\pi\)
\(864\) −1.45326e17 −10.2688
\(865\) −1.13151e16 −0.794454
\(866\) 1.37355e16 0.958291
\(867\) −1.89657e16 −1.31481
\(868\) −4.89384e16 −3.37125
\(869\) −6.88750e15 −0.471468
\(870\) −6.31872e16 −4.29807
\(871\) 1.30197e16 0.880039
\(872\) 1.16231e16 0.780695
\(873\) 4.41783e16 2.94870
\(874\) −5.51090e15 −0.365519
\(875\) −4.13893e16 −2.72800
\(876\) 7.62289e16 4.99283
\(877\) 8.57996e15 0.558454 0.279227 0.960225i \(-0.409922\pi\)
0.279227 + 0.960225i \(0.409922\pi\)
\(878\) 4.35980e16 2.81998
\(879\) 3.49743e15 0.224807
\(880\) −4.67802e16 −2.98818
\(881\) 1.89430e16 1.20249 0.601246 0.799064i \(-0.294671\pi\)
0.601246 + 0.799064i \(0.294671\pi\)
\(882\) −3.14179e16 −1.98198
\(883\) −1.01854e16 −0.638551 −0.319275 0.947662i \(-0.603440\pi\)
−0.319275 + 0.947662i \(0.603440\pi\)
\(884\) −2.90450e16 −1.80961
\(885\) 2.38525e16 1.47688
\(886\) −2.42568e16 −1.49262
\(887\) 3.01644e15 0.184465 0.0922326 0.995737i \(-0.470600\pi\)
0.0922326 + 0.995737i \(0.470600\pi\)
\(888\) 5.37852e16 3.26883
\(889\) 2.89078e16 1.74605
\(890\) 9.58567e16 5.75410
\(891\) −2.06768e16 −1.23355
\(892\) 4.93407e16 2.92549
\(893\) −2.55297e15 −0.150440
\(894\) 9.08934e16 5.32324
\(895\) −2.13115e16 −1.24047
\(896\) 6.20129e16 3.58747
\(897\) −7.21885e16 −4.15059
\(898\) 4.12963e16 2.35989
\(899\) 1.21844e16 0.692030
\(900\) 2.70927e17 15.2939
\(901\) 1.78148e14 0.00999530
\(902\) −8.04889e15 −0.448849
\(903\) −9.24778e15 −0.512572
\(904\) −6.29856e16 −3.46989
\(905\) −2.14814e16 −1.17624
\(906\) −1.05799e17 −5.75809
\(907\) −5.94995e15 −0.321865 −0.160932 0.986965i \(-0.551450\pi\)
−0.160932 + 0.986965i \(0.551450\pi\)
\(908\) −4.00858e16 −2.15535
\(909\) 5.84025e16 3.12126
\(910\) 9.56928e16 5.08337
\(911\) 3.48799e15 0.184172 0.0920861 0.995751i \(-0.470647\pi\)
0.0920861 + 0.995751i \(0.470647\pi\)
\(912\) −1.35355e16 −0.710397
\(913\) −1.51687e16 −0.791333
\(914\) 3.66802e16 1.90208
\(915\) −5.82275e16 −3.00132
\(916\) −2.94245e16 −1.50759
\(917\) 1.65516e16 0.842961
\(918\) 5.75642e16 2.91419
\(919\) −9.47651e15 −0.476885 −0.238442 0.971157i \(-0.576637\pi\)
−0.238442 + 0.971157i \(0.576637\pi\)
\(920\) 2.15110e17 10.7604
\(921\) 3.42167e16 1.70141
\(922\) −4.59159e16 −2.26957
\(923\) 3.37366e16 1.65765
\(924\) 5.61289e16 2.74151
\(925\) −2.48655e16 −1.20731
\(926\) −3.79818e16 −1.83322
\(927\) 4.85527e16 2.32956
\(928\) −5.11615e16 −2.44022
\(929\) −1.03025e16 −0.488492 −0.244246 0.969713i \(-0.578540\pi\)
−0.244246 + 0.969713i \(0.578540\pi\)
\(930\) −1.44468e17 −6.80949
\(931\) −9.25813e14 −0.0433811
\(932\) −5.40931e15 −0.251973
\(933\) 4.63675e15 0.214716
\(934\) −3.47385e16 −1.59920
\(935\) 9.79992e15 0.448496
\(936\) −2.20964e17 −10.0532
\(937\) 2.53627e15 0.114717 0.0573584 0.998354i \(-0.481732\pi\)
0.0573584 + 0.998354i \(0.481732\pi\)
\(938\) 3.64861e16 1.64063
\(939\) 2.38286e16 1.06521
\(940\) 1.57994e17 7.02164
\(941\) 3.43149e15 0.151614 0.0758072 0.997122i \(-0.475847\pi\)
0.0758072 + 0.997122i \(0.475847\pi\)
\(942\) −1.05520e17 −4.63508
\(943\) 2.11862e16 0.925209
\(944\) 3.65169e16 1.58544
\(945\) −1.38507e17 −5.97854
\(946\) 4.72130e15 0.202610
\(947\) 1.77905e16 0.759038 0.379519 0.925184i \(-0.376090\pi\)
0.379519 + 0.925184i \(0.376090\pi\)
\(948\) 1.23330e17 5.23145
\(949\) 2.87424e16 1.21216
\(950\) 1.09317e16 0.458361
\(951\) −6.64829e15 −0.277151
\(952\) −5.13382e16 −2.12783
\(953\) 1.39263e16 0.573886 0.286943 0.957948i \(-0.407361\pi\)
0.286943 + 0.957948i \(0.407361\pi\)
\(954\) 2.14876e15 0.0880385
\(955\) 3.49967e16 1.42564
\(956\) −8.97449e16 −3.63489
\(957\) −1.39746e16 −0.562761
\(958\) −1.00740e16 −0.403358
\(959\) −2.96297e16 −1.17957
\(960\) 2.97288e17 11.7675
\(961\) 2.44920e15 0.0963930
\(962\) 3.21531e16 1.25823
\(963\) 3.43987e16 1.33844
\(964\) 1.72176e14 0.00666115
\(965\) 3.54021e15 0.136185
\(966\) −2.02299e17 −7.73784
\(967\) −4.21314e15 −0.160236 −0.0801180 0.996785i \(-0.525530\pi\)
−0.0801180 + 0.996785i \(0.525530\pi\)
\(968\) 6.88993e16 2.60555
\(969\) 2.83552e15 0.106623
\(970\) −1.10317e17 −4.12474
\(971\) −1.89268e16 −0.703674 −0.351837 0.936061i \(-0.614443\pi\)
−0.351837 + 0.936061i \(0.614443\pi\)
\(972\) 1.66518e17 6.15596
\(973\) −1.90314e16 −0.699599
\(974\) −3.51458e16 −1.28469
\(975\) 1.43197e17 5.20485
\(976\) −8.91435e16 −3.22193
\(977\) 1.55524e16 0.558956 0.279478 0.960152i \(-0.409839\pi\)
0.279478 + 0.960152i \(0.409839\pi\)
\(978\) 1.98219e16 0.708405
\(979\) 2.11999e16 0.753404
\(980\) 5.72953e16 2.02477
\(981\) −1.68115e16 −0.590783
\(982\) −4.04719e16 −1.41430
\(983\) −4.23979e16 −1.47333 −0.736665 0.676258i \(-0.763600\pi\)
−0.736665 + 0.676258i \(0.763600\pi\)
\(984\) 9.09046e16 3.14132
\(985\) −3.74867e15 −0.128818
\(986\) 2.02652e16 0.692513
\(987\) −9.37168e16 −3.18473
\(988\) −1.03234e16 −0.348868
\(989\) −1.24274e16 −0.417638
\(990\) 1.18203e17 3.95035
\(991\) −1.03132e16 −0.342757 −0.171379 0.985205i \(-0.554822\pi\)
−0.171379 + 0.985205i \(0.554822\pi\)
\(992\) −1.16973e17 −3.86608
\(993\) −8.52475e16 −2.80195
\(994\) 9.45425e16 3.09031
\(995\) 3.24041e16 1.05335
\(996\) 2.71616e17 8.78070
\(997\) 2.96304e16 0.952609 0.476304 0.879280i \(-0.341976\pi\)
0.476304 + 0.879280i \(0.341976\pi\)
\(998\) −3.25380e16 −1.04033
\(999\) −4.65387e16 −1.47980
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 197.12.a.b.1.2 92
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.12.a.b.1.2 92 1.1 even 1 trivial