Properties

Label 197.12.a.b.1.3
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 1.3
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)
 
\(f(q)\) \(=\) \(q-86.5721 q^{2} -287.043 q^{3} +5446.73 q^{4} -10768.4 q^{5} +24849.9 q^{6} -50234.8 q^{7} -294236. q^{8} -94753.4 q^{9} +O(q^{10})\) \(q-86.5721 q^{2} -287.043 q^{3} +5446.73 q^{4} -10768.4 q^{5} +24849.9 q^{6} -50234.8 q^{7} -294236. q^{8} -94753.4 q^{9} +932242. q^{10} +137694. q^{11} -1.56345e6 q^{12} +1.89680e6 q^{13} +4.34893e6 q^{14} +3.09099e6 q^{15} +1.43177e7 q^{16} +4.71231e6 q^{17} +8.20300e6 q^{18} -1.54798e7 q^{19} -5.86525e7 q^{20} +1.44195e7 q^{21} -1.19205e7 q^{22} -4.55820e7 q^{23} +8.44582e7 q^{24} +6.71300e7 q^{25} -1.64210e8 q^{26} +7.80471e7 q^{27} -2.73615e8 q^{28} +3.92624e7 q^{29} -2.67593e8 q^{30} +5.48803e7 q^{31} -6.36919e8 q^{32} -3.95241e7 q^{33} -4.07954e8 q^{34} +5.40948e8 q^{35} -5.16096e8 q^{36} +6.58234e8 q^{37} +1.34012e9 q^{38} -5.44464e8 q^{39} +3.16844e9 q^{40} +2.72100e8 q^{41} -1.24833e9 q^{42} +2.65418e8 q^{43} +7.49983e8 q^{44} +1.02034e9 q^{45} +3.94613e9 q^{46} -2.92021e9 q^{47} -4.10979e9 q^{48} +5.46206e8 q^{49} -5.81159e9 q^{50} -1.35263e9 q^{51} +1.03314e10 q^{52} -4.21817e9 q^{53} -6.75670e9 q^{54} -1.48274e9 q^{55} +1.47809e10 q^{56} +4.44336e9 q^{57} -3.39903e9 q^{58} +1.55089e9 q^{59} +1.68358e10 q^{60} -7.70506e9 q^{61} -4.75111e9 q^{62} +4.75992e9 q^{63} +2.58168e10 q^{64} -2.04255e10 q^{65} +3.42168e9 q^{66} +1.50199e10 q^{67} +2.56667e10 q^{68} +1.30840e10 q^{69} -4.68310e10 q^{70} +8.58511e9 q^{71} +2.78798e10 q^{72} -1.81116e10 q^{73} -5.69847e10 q^{74} -1.92692e10 q^{75} -8.43142e10 q^{76} -6.91703e9 q^{77} +4.71354e10 q^{78} -2.21343e10 q^{79} -1.54178e11 q^{80} -5.61758e9 q^{81} -2.35563e10 q^{82} +7.17506e9 q^{83} +7.85394e10 q^{84} -5.07439e10 q^{85} -2.29778e10 q^{86} -1.12700e10 q^{87} -4.05145e10 q^{88} +3.68534e10 q^{89} -8.83331e10 q^{90} -9.52855e10 q^{91} -2.48273e11 q^{92} -1.57530e10 q^{93} +2.52809e11 q^{94} +1.66692e11 q^{95} +1.82823e11 q^{96} +4.00765e9 q^{97} -4.72862e10 q^{98} -1.30470e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 92 q + 64 q^{2} + 2420 q^{3} + 98304 q^{4} + 16604 q^{5} + 46656 q^{6} + 305891 q^{7} + 234027 q^{8} + 5900444 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −86.5721 −1.91299 −0.956496 0.291746i \(-0.905764\pi\)
−0.956496 + 0.291746i \(0.905764\pi\)
\(3\) −287.043 −0.681993 −0.340996 0.940065i \(-0.610764\pi\)
−0.340996 + 0.940065i \(0.610764\pi\)
\(4\) 5446.73 2.65954
\(5\) −10768.4 −1.54105 −0.770523 0.637412i \(-0.780005\pi\)
−0.770523 + 0.637412i \(0.780005\pi\)
\(6\) 24849.9 1.30465
\(7\) −50234.8 −1.12971 −0.564853 0.825192i \(-0.691067\pi\)
−0.564853 + 0.825192i \(0.691067\pi\)
\(8\) −294236. −3.17468
\(9\) −94753.4 −0.534886
\(10\) 932242. 2.94801
\(11\) 137694. 0.257784 0.128892 0.991659i \(-0.458858\pi\)
0.128892 + 0.991659i \(0.458858\pi\)
\(12\) −1.56345e6 −1.81379
\(13\) 1.89680e6 1.41688 0.708441 0.705770i \(-0.249399\pi\)
0.708441 + 0.705770i \(0.249399\pi\)
\(14\) 4.34893e6 2.16112
\(15\) 3.09099e6 1.05098
\(16\) 1.43177e7 3.41360
\(17\) 4.71231e6 0.804942 0.402471 0.915433i \(-0.368151\pi\)
0.402471 + 0.915433i \(0.368151\pi\)
\(18\) 8.20300e6 1.02323
\(19\) −1.54798e7 −1.43423 −0.717116 0.696954i \(-0.754538\pi\)
−0.717116 + 0.696954i \(0.754538\pi\)
\(20\) −5.86525e7 −4.09847
\(21\) 1.44195e7 0.770451
\(22\) −1.19205e7 −0.493138
\(23\) −4.55820e7 −1.47669 −0.738347 0.674421i \(-0.764393\pi\)
−0.738347 + 0.674421i \(0.764393\pi\)
\(24\) 8.44582e7 2.16511
\(25\) 6.71300e7 1.37482
\(26\) −1.64210e8 −2.71048
\(27\) 7.80471e7 1.04678
\(28\) −2.73615e8 −3.00449
\(29\) 3.92624e7 0.355458 0.177729 0.984079i \(-0.443125\pi\)
0.177729 + 0.984079i \(0.443125\pi\)
\(30\) −2.67593e8 −2.01052
\(31\) 5.48803e7 0.344292 0.172146 0.985071i \(-0.444930\pi\)
0.172146 + 0.985071i \(0.444930\pi\)
\(32\) −6.36919e8 −3.35551
\(33\) −3.95241e7 −0.175807
\(34\) −4.07954e8 −1.53985
\(35\) 5.40948e8 1.74093
\(36\) −5.16096e8 −1.42255
\(37\) 6.58234e8 1.56052 0.780262 0.625452i \(-0.215086\pi\)
0.780262 + 0.625452i \(0.215086\pi\)
\(38\) 1.34012e9 2.74367
\(39\) −5.44464e8 −0.966304
\(40\) 3.16844e9 4.89233
\(41\) 2.72100e8 0.366790 0.183395 0.983039i \(-0.441291\pi\)
0.183395 + 0.983039i \(0.441291\pi\)
\(42\) −1.24833e9 −1.47387
\(43\) 2.65418e8 0.275330 0.137665 0.990479i \(-0.456040\pi\)
0.137665 + 0.990479i \(0.456040\pi\)
\(44\) 7.49983e8 0.685585
\(45\) 1.02034e9 0.824283
\(46\) 3.94613e9 2.82490
\(47\) −2.92021e9 −1.85727 −0.928637 0.370989i \(-0.879019\pi\)
−0.928637 + 0.370989i \(0.879019\pi\)
\(48\) −4.10979e9 −2.32805
\(49\) 5.46206e8 0.276235
\(50\) −5.81159e9 −2.63003
\(51\) −1.35263e9 −0.548965
\(52\) 1.03314e10 3.76825
\(53\) −4.21817e9 −1.38550 −0.692750 0.721177i \(-0.743601\pi\)
−0.692750 + 0.721177i \(0.743601\pi\)
\(54\) −6.75670e9 −2.00248
\(55\) −1.48274e9 −0.397256
\(56\) 1.47809e10 3.58646
\(57\) 4.44336e9 0.978136
\(58\) −3.39903e9 −0.679988
\(59\) 1.55089e9 0.282420 0.141210 0.989980i \(-0.454901\pi\)
0.141210 + 0.989980i \(0.454901\pi\)
\(60\) 1.68358e10 2.79513
\(61\) −7.70506e9 −1.16805 −0.584026 0.811735i \(-0.698523\pi\)
−0.584026 + 0.811735i \(0.698523\pi\)
\(62\) −4.75111e9 −0.658629
\(63\) 4.75992e9 0.604263
\(64\) 2.58168e10 3.00547
\(65\) −2.04255e10 −2.18348
\(66\) 3.42168e9 0.336317
\(67\) 1.50199e10 1.35912 0.679558 0.733622i \(-0.262171\pi\)
0.679558 + 0.733622i \(0.262171\pi\)
\(68\) 2.56667e10 2.14077
\(69\) 1.30840e10 1.00709
\(70\) −4.68310e10 −3.33038
\(71\) 8.58511e9 0.564709 0.282354 0.959310i \(-0.408885\pi\)
0.282354 + 0.959310i \(0.408885\pi\)
\(72\) 2.78798e10 1.69809
\(73\) −1.81116e10 −1.02254 −0.511271 0.859419i \(-0.670825\pi\)
−0.511271 + 0.859419i \(0.670825\pi\)
\(74\) −5.69847e10 −2.98527
\(75\) −1.92692e10 −0.937620
\(76\) −8.43142e10 −3.81440
\(77\) −6.91703e9 −0.291220
\(78\) 4.71354e10 1.84853
\(79\) −2.21343e10 −0.809314 −0.404657 0.914469i \(-0.632609\pi\)
−0.404657 + 0.914469i \(0.632609\pi\)
\(80\) −1.54178e11 −5.26052
\(81\) −5.61758e9 −0.179012
\(82\) −2.35563e10 −0.701666
\(83\) 7.17506e9 0.199938 0.0999692 0.994991i \(-0.468126\pi\)
0.0999692 + 0.994991i \(0.468126\pi\)
\(84\) 7.85394e10 2.04904
\(85\) −5.07439e10 −1.24045
\(86\) −2.29778e10 −0.526705
\(87\) −1.12700e10 −0.242420
\(88\) −4.05145e10 −0.818381
\(89\) 3.68534e10 0.699571 0.349786 0.936830i \(-0.386254\pi\)
0.349786 + 0.936830i \(0.386254\pi\)
\(90\) −8.83331e10 −1.57685
\(91\) −9.52855e10 −1.60066
\(92\) −2.48273e11 −3.92732
\(93\) −1.57530e10 −0.234805
\(94\) 2.52809e11 3.55295
\(95\) 1.66692e11 2.21022
\(96\) 1.82823e11 2.28844
\(97\) 4.00765e9 0.0473855 0.0236927 0.999719i \(-0.492458\pi\)
0.0236927 + 0.999719i \(0.492458\pi\)
\(98\) −4.72862e10 −0.528435
\(99\) −1.30470e10 −0.137885
\(100\) 3.65639e11 3.65639
\(101\) −1.84185e11 −1.74376 −0.871878 0.489722i \(-0.837098\pi\)
−0.871878 + 0.489722i \(0.837098\pi\)
\(102\) 1.17100e11 1.05016
\(103\) 4.37962e10 0.372248 0.186124 0.982526i \(-0.440407\pi\)
0.186124 + 0.982526i \(0.440407\pi\)
\(104\) −5.58107e11 −4.49815
\(105\) −1.55275e11 −1.18730
\(106\) 3.65176e11 2.65045
\(107\) −1.62314e11 −1.11878 −0.559392 0.828903i \(-0.688965\pi\)
−0.559392 + 0.828903i \(0.688965\pi\)
\(108\) 4.25102e11 2.78395
\(109\) −2.64387e11 −1.64587 −0.822933 0.568139i \(-0.807664\pi\)
−0.822933 + 0.568139i \(0.807664\pi\)
\(110\) 1.28364e11 0.759948
\(111\) −1.88941e11 −1.06427
\(112\) −7.19246e11 −3.85637
\(113\) −1.69277e11 −0.864302 −0.432151 0.901801i \(-0.642245\pi\)
−0.432151 + 0.901801i \(0.642245\pi\)
\(114\) −3.84671e11 −1.87117
\(115\) 4.90845e11 2.27565
\(116\) 2.13852e11 0.945354
\(117\) −1.79728e11 −0.757870
\(118\) −1.34264e11 −0.540267
\(119\) −2.36722e11 −0.909347
\(120\) −9.09479e11 −3.33654
\(121\) −2.66352e11 −0.933548
\(122\) 6.67044e11 2.23447
\(123\) −7.81044e10 −0.250148
\(124\) 2.98919e11 0.915659
\(125\) −1.97082e11 −0.577619
\(126\) −4.12076e11 −1.15595
\(127\) −1.22516e11 −0.329057 −0.164528 0.986372i \(-0.552610\pi\)
−0.164528 + 0.986372i \(0.552610\pi\)
\(128\) −9.30604e11 −2.39392
\(129\) −7.61864e10 −0.187773
\(130\) 1.76828e12 4.17698
\(131\) −2.36127e11 −0.534754 −0.267377 0.963592i \(-0.586157\pi\)
−0.267377 + 0.963592i \(0.586157\pi\)
\(132\) −2.15277e11 −0.467564
\(133\) 7.77623e11 1.62026
\(134\) −1.30031e12 −2.59998
\(135\) −8.40441e11 −1.61314
\(136\) −1.38653e12 −2.55543
\(137\) −7.36038e11 −1.30298 −0.651489 0.758658i \(-0.725855\pi\)
−0.651489 + 0.758658i \(0.725855\pi\)
\(138\) −1.13271e12 −1.92656
\(139\) −1.63487e11 −0.267239 −0.133620 0.991033i \(-0.542660\pi\)
−0.133620 + 0.991033i \(0.542660\pi\)
\(140\) 2.94640e12 4.63006
\(141\) 8.38226e11 1.26665
\(142\) −7.43231e11 −1.08028
\(143\) 2.61178e11 0.365249
\(144\) −1.35665e12 −1.82589
\(145\) −4.22793e11 −0.547777
\(146\) 1.56796e12 1.95612
\(147\) −1.56785e11 −0.188390
\(148\) 3.58522e12 4.15027
\(149\) −1.43510e12 −1.60088 −0.800439 0.599414i \(-0.795400\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(150\) 1.66818e12 1.79366
\(151\) −8.46040e10 −0.0877036 −0.0438518 0.999038i \(-0.513963\pi\)
−0.0438518 + 0.999038i \(0.513963\pi\)
\(152\) 4.55470e12 4.55323
\(153\) −4.46507e11 −0.430552
\(154\) 5.98822e11 0.557101
\(155\) −5.90973e11 −0.530570
\(156\) −2.96555e12 −2.56992
\(157\) 7.09827e11 0.593888 0.296944 0.954895i \(-0.404033\pi\)
0.296944 + 0.954895i \(0.404033\pi\)
\(158\) 1.91621e12 1.54821
\(159\) 1.21080e12 0.944902
\(160\) 6.85859e12 5.17100
\(161\) 2.28980e12 1.66823
\(162\) 4.86326e11 0.342448
\(163\) 2.02859e12 1.38090 0.690449 0.723381i \(-0.257413\pi\)
0.690449 + 0.723381i \(0.257413\pi\)
\(164\) 1.48206e12 0.975492
\(165\) 4.25611e11 0.270926
\(166\) −6.21160e11 −0.382480
\(167\) 1.03298e12 0.615391 0.307695 0.951485i \(-0.400442\pi\)
0.307695 + 0.951485i \(0.400442\pi\)
\(168\) −4.24274e12 −2.44594
\(169\) 1.80570e12 1.00755
\(170\) 4.39301e12 2.37298
\(171\) 1.46676e12 0.767150
\(172\) 1.44566e12 0.732251
\(173\) −1.77073e12 −0.868756 −0.434378 0.900731i \(-0.643032\pi\)
−0.434378 + 0.900731i \(0.643032\pi\)
\(174\) 9.75668e11 0.463747
\(175\) −3.37226e12 −1.55315
\(176\) 1.97146e12 0.879971
\(177\) −4.45172e11 −0.192608
\(178\) −3.19047e12 −1.33827
\(179\) −1.80189e12 −0.732888 −0.366444 0.930440i \(-0.619425\pi\)
−0.366444 + 0.930440i \(0.619425\pi\)
\(180\) 5.55753e12 2.19221
\(181\) −4.45615e12 −1.70501 −0.852506 0.522718i \(-0.824918\pi\)
−0.852506 + 0.522718i \(0.824918\pi\)
\(182\) 8.24906e12 3.06205
\(183\) 2.21168e12 0.796603
\(184\) 1.34118e13 4.68803
\(185\) −7.08811e12 −2.40484
\(186\) 1.36377e12 0.449180
\(187\) 6.48856e11 0.207501
\(188\) −1.59056e13 −4.93949
\(189\) −3.92068e12 −1.18255
\(190\) −1.44309e13 −4.22813
\(191\) 1.41157e12 0.401809 0.200905 0.979611i \(-0.435612\pi\)
0.200905 + 0.979611i \(0.435612\pi\)
\(192\) −7.41052e12 −2.04971
\(193\) −3.62567e12 −0.974593 −0.487297 0.873236i \(-0.662017\pi\)
−0.487297 + 0.873236i \(0.662017\pi\)
\(194\) −3.46951e11 −0.0906481
\(195\) 5.86300e12 1.48912
\(196\) 2.97504e12 0.734657
\(197\) −2.96709e11 −0.0712470
\(198\) 1.12950e12 0.263772
\(199\) 7.07088e12 1.60613 0.803066 0.595890i \(-0.203200\pi\)
0.803066 + 0.595890i \(0.203200\pi\)
\(200\) −1.97520e13 −4.36463
\(201\) −4.31137e12 −0.926908
\(202\) 1.59453e13 3.33579
\(203\) −1.97234e12 −0.401563
\(204\) −7.36744e12 −1.45999
\(205\) −2.93008e12 −0.565240
\(206\) −3.79153e12 −0.712107
\(207\) 4.31905e12 0.789862
\(208\) 2.71578e13 4.83667
\(209\) −2.13147e12 −0.369722
\(210\) 1.34425e13 2.27130
\(211\) −1.19056e12 −0.195974 −0.0979870 0.995188i \(-0.531240\pi\)
−0.0979870 + 0.995188i \(0.531240\pi\)
\(212\) −2.29752e13 −3.68479
\(213\) −2.46429e12 −0.385128
\(214\) 1.40519e13 2.14023
\(215\) −2.85812e12 −0.424297
\(216\) −2.29642e13 −3.32320
\(217\) −2.75690e12 −0.388949
\(218\) 2.28885e13 3.14853
\(219\) 5.19881e12 0.697367
\(220\) −8.07610e12 −1.05652
\(221\) 8.93831e12 1.14051
\(222\) 1.63570e13 2.03593
\(223\) −9.73013e12 −1.18152 −0.590761 0.806846i \(-0.701173\pi\)
−0.590761 + 0.806846i \(0.701173\pi\)
\(224\) 3.19955e13 3.79074
\(225\) −6.36080e12 −0.735373
\(226\) 1.46546e13 1.65340
\(227\) −2.93570e12 −0.323273 −0.161636 0.986850i \(-0.551677\pi\)
−0.161636 + 0.986850i \(0.551677\pi\)
\(228\) 2.42018e13 2.60139
\(229\) −9.50474e12 −0.997344 −0.498672 0.866791i \(-0.666179\pi\)
−0.498672 + 0.866791i \(0.666179\pi\)
\(230\) −4.24935e13 −4.35330
\(231\) 1.98548e12 0.198610
\(232\) −1.15524e13 −1.12847
\(233\) 6.96508e12 0.664459 0.332230 0.943199i \(-0.392199\pi\)
0.332230 + 0.943199i \(0.392199\pi\)
\(234\) 1.55595e13 1.44980
\(235\) 3.14460e13 2.86215
\(236\) 8.44729e12 0.751106
\(237\) 6.35350e12 0.551946
\(238\) 2.04935e13 1.73957
\(239\) −9.66194e11 −0.0801449 −0.0400725 0.999197i \(-0.512759\pi\)
−0.0400725 + 0.999197i \(0.512759\pi\)
\(240\) 4.42558e13 3.58764
\(241\) 1.46440e13 1.16029 0.580143 0.814514i \(-0.302997\pi\)
0.580143 + 0.814514i \(0.302997\pi\)
\(242\) 2.30587e13 1.78587
\(243\) −1.22133e13 −0.924696
\(244\) −4.19674e13 −3.10648
\(245\) −5.88176e12 −0.425690
\(246\) 6.76166e12 0.478532
\(247\) −2.93621e13 −2.03214
\(248\) −1.61478e13 −1.09302
\(249\) −2.05955e12 −0.136357
\(250\) 1.70618e13 1.10498
\(251\) −2.23763e13 −1.41769 −0.708847 0.705362i \(-0.750784\pi\)
−0.708847 + 0.705362i \(0.750784\pi\)
\(252\) 2.59260e13 1.60706
\(253\) −6.27637e12 −0.380667
\(254\) 1.06064e13 0.629483
\(255\) 1.45657e13 0.845980
\(256\) 2.76916e13 1.57408
\(257\) 2.09807e13 1.16731 0.583656 0.812001i \(-0.301622\pi\)
0.583656 + 0.812001i \(0.301622\pi\)
\(258\) 6.59562e12 0.359209
\(259\) −3.30662e13 −1.76293
\(260\) −1.11252e14 −5.80705
\(261\) −3.72025e12 −0.190129
\(262\) 2.04421e13 1.02298
\(263\) 2.05205e13 1.00561 0.502807 0.864399i \(-0.332301\pi\)
0.502807 + 0.864399i \(0.332301\pi\)
\(264\) 1.16294e13 0.558130
\(265\) 4.54229e13 2.13512
\(266\) −6.73204e13 −3.09954
\(267\) −1.05785e13 −0.477103
\(268\) 8.18096e13 3.61462
\(269\) −3.41149e13 −1.47675 −0.738374 0.674391i \(-0.764406\pi\)
−0.738374 + 0.674391i \(0.764406\pi\)
\(270\) 7.27588e13 3.08592
\(271\) 3.17092e13 1.31781 0.658907 0.752224i \(-0.271019\pi\)
0.658907 + 0.752224i \(0.271019\pi\)
\(272\) 6.74694e13 2.74775
\(273\) 2.73510e13 1.09164
\(274\) 6.37204e13 2.49259
\(275\) 9.24340e12 0.354407
\(276\) 7.12650e13 2.67841
\(277\) 3.73768e13 1.37709 0.688546 0.725192i \(-0.258249\pi\)
0.688546 + 0.725192i \(0.258249\pi\)
\(278\) 1.41534e13 0.511227
\(279\) −5.20010e12 −0.184157
\(280\) −1.59166e14 −5.52690
\(281\) −5.20161e12 −0.177114 −0.0885571 0.996071i \(-0.528226\pi\)
−0.0885571 + 0.996071i \(0.528226\pi\)
\(282\) −7.25670e13 −2.42309
\(283\) 6.29189e11 0.0206042 0.0103021 0.999947i \(-0.496721\pi\)
0.0103021 + 0.999947i \(0.496721\pi\)
\(284\) 4.67608e13 1.50187
\(285\) −4.78478e13 −1.50735
\(286\) −2.26108e13 −0.698718
\(287\) −1.36689e13 −0.414365
\(288\) 6.03502e13 1.79482
\(289\) −1.20661e13 −0.352069
\(290\) 3.66021e13 1.04789
\(291\) −1.15037e12 −0.0323166
\(292\) −9.86491e13 −2.71949
\(293\) 1.47114e13 0.397998 0.198999 0.980000i \(-0.436231\pi\)
0.198999 + 0.980000i \(0.436231\pi\)
\(294\) 1.35732e13 0.360389
\(295\) −1.67006e13 −0.435222
\(296\) −1.93676e14 −4.95417
\(297\) 1.07466e13 0.269843
\(298\) 1.24240e14 3.06247
\(299\) −8.64601e13 −2.09230
\(300\) −1.04954e14 −2.49363
\(301\) −1.33332e13 −0.311042
\(302\) 7.32434e12 0.167776
\(303\) 5.28689e13 1.18923
\(304\) −2.21635e14 −4.89590
\(305\) 8.29711e13 1.80002
\(306\) 3.86551e13 0.823642
\(307\) 6.24695e13 1.30739 0.653697 0.756756i \(-0.273217\pi\)
0.653697 + 0.756756i \(0.273217\pi\)
\(308\) −3.76752e13 −0.774510
\(309\) −1.25714e13 −0.253870
\(310\) 5.11618e13 1.01498
\(311\) −3.35568e13 −0.654032 −0.327016 0.945019i \(-0.606043\pi\)
−0.327016 + 0.945019i \(0.606043\pi\)
\(312\) 1.60201e14 3.06771
\(313\) 8.44273e13 1.58851 0.794253 0.607587i \(-0.207862\pi\)
0.794253 + 0.607587i \(0.207862\pi\)
\(314\) −6.14513e13 −1.13610
\(315\) −5.12566e13 −0.931198
\(316\) −1.20560e14 −2.15240
\(317\) −7.78116e13 −1.36527 −0.682635 0.730759i \(-0.739166\pi\)
−0.682635 + 0.730759i \(0.739166\pi\)
\(318\) −1.04821e14 −1.80759
\(319\) 5.40620e12 0.0916312
\(320\) −2.78005e14 −4.63157
\(321\) 4.65912e13 0.763003
\(322\) −1.98233e14 −3.19131
\(323\) −7.29454e13 −1.15447
\(324\) −3.05975e13 −0.476089
\(325\) 1.27332e14 1.94796
\(326\) −1.75619e14 −2.64165
\(327\) 7.58904e13 1.12247
\(328\) −8.00616e13 −1.16444
\(329\) 1.46696e14 2.09817
\(330\) −3.68460e13 −0.518279
\(331\) 1.18497e14 1.63929 0.819643 0.572874i \(-0.194172\pi\)
0.819643 + 0.572874i \(0.194172\pi\)
\(332\) 3.90806e13 0.531744
\(333\) −6.23699e13 −0.834702
\(334\) −8.94272e13 −1.17724
\(335\) −1.61740e14 −2.09446
\(336\) 2.06455e14 2.63002
\(337\) 7.08465e13 0.887880 0.443940 0.896057i \(-0.353580\pi\)
0.443940 + 0.896057i \(0.353580\pi\)
\(338\) −1.56323e14 −1.92744
\(339\) 4.85897e13 0.589448
\(340\) −2.76389e14 −3.29903
\(341\) 7.55669e12 0.0887529
\(342\) −1.26981e14 −1.46755
\(343\) 7.18920e13 0.817642
\(344\) −7.80955e13 −0.874086
\(345\) −1.40893e14 −1.55198
\(346\) 1.53296e14 1.66192
\(347\) 1.33342e14 1.42284 0.711418 0.702769i \(-0.248053\pi\)
0.711418 + 0.702769i \(0.248053\pi\)
\(348\) −6.13847e13 −0.644724
\(349\) −1.62451e14 −1.67951 −0.839756 0.542964i \(-0.817302\pi\)
−0.839756 + 0.542964i \(0.817302\pi\)
\(350\) 2.91944e14 2.97115
\(351\) 1.48040e14 1.48317
\(352\) −8.76999e13 −0.864997
\(353\) −1.03309e14 −1.00318 −0.501588 0.865107i \(-0.667251\pi\)
−0.501588 + 0.865107i \(0.667251\pi\)
\(354\) 3.85395e13 0.368458
\(355\) −9.24477e13 −0.870243
\(356\) 2.00730e14 1.86054
\(357\) 6.79493e13 0.620168
\(358\) 1.55994e14 1.40201
\(359\) 1.35380e14 1.19821 0.599107 0.800669i \(-0.295523\pi\)
0.599107 + 0.800669i \(0.295523\pi\)
\(360\) −3.00221e14 −2.61684
\(361\) 1.23133e14 1.05702
\(362\) 3.85778e14 3.26167
\(363\) 7.64545e13 0.636673
\(364\) −5.18995e14 −4.25701
\(365\) 1.95033e14 1.57578
\(366\) −1.91470e14 −1.52390
\(367\) −2.12613e14 −1.66696 −0.833481 0.552549i \(-0.813655\pi\)
−0.833481 + 0.552549i \(0.813655\pi\)
\(368\) −6.52629e14 −5.04085
\(369\) −2.57824e13 −0.196191
\(370\) 6.13633e14 4.60044
\(371\) 2.11899e14 1.56521
\(372\) −8.58025e13 −0.624473
\(373\) −1.85839e14 −1.33272 −0.666360 0.745630i \(-0.732149\pi\)
−0.666360 + 0.745630i \(0.732149\pi\)
\(374\) −5.61729e13 −0.396947
\(375\) 5.65710e13 0.393932
\(376\) 8.59230e14 5.89626
\(377\) 7.44731e13 0.503642
\(378\) 3.39421e14 2.26222
\(379\) −1.50749e14 −0.990235 −0.495117 0.868826i \(-0.664875\pi\)
−0.495117 + 0.868826i \(0.664875\pi\)
\(380\) 9.07927e14 5.87816
\(381\) 3.51672e13 0.224414
\(382\) −1.22203e14 −0.768658
\(383\) −8.05851e13 −0.499645 −0.249823 0.968292i \(-0.580372\pi\)
−0.249823 + 0.968292i \(0.580372\pi\)
\(384\) 2.67123e14 1.63264
\(385\) 7.44852e13 0.448783
\(386\) 3.13882e14 1.86439
\(387\) −2.51493e13 −0.147270
\(388\) 2.18286e13 0.126024
\(389\) 2.01229e13 0.114543 0.0572714 0.998359i \(-0.481760\pi\)
0.0572714 + 0.998359i \(0.481760\pi\)
\(390\) −5.07572e14 −2.84867
\(391\) −2.14796e14 −1.18865
\(392\) −1.60713e14 −0.876957
\(393\) 6.77787e13 0.364699
\(394\) 2.56868e13 0.136295
\(395\) 2.38351e14 1.24719
\(396\) −7.10634e13 −0.366710
\(397\) 1.21538e14 0.618536 0.309268 0.950975i \(-0.399916\pi\)
0.309268 + 0.950975i \(0.399916\pi\)
\(398\) −6.12141e14 −3.07252
\(399\) −2.23211e14 −1.10501
\(400\) 9.61147e14 4.69310
\(401\) −1.70320e14 −0.820298 −0.410149 0.912019i \(-0.634523\pi\)
−0.410149 + 0.912019i \(0.634523\pi\)
\(402\) 3.73244e14 1.77317
\(403\) 1.04097e14 0.487822
\(404\) −1.00320e15 −4.63759
\(405\) 6.04922e13 0.275865
\(406\) 1.70750e14 0.768186
\(407\) 9.06348e13 0.402278
\(408\) 3.97993e14 1.74279
\(409\) −2.53767e14 −1.09637 −0.548186 0.836357i \(-0.684681\pi\)
−0.548186 + 0.836357i \(0.684681\pi\)
\(410\) 2.53663e14 1.08130
\(411\) 2.11274e14 0.888622
\(412\) 2.38546e14 0.990007
\(413\) −7.79087e13 −0.319051
\(414\) −3.73909e14 −1.51100
\(415\) −7.72638e13 −0.308114
\(416\) −1.20811e15 −4.75437
\(417\) 4.69276e13 0.182255
\(418\) 1.84526e14 0.707274
\(419\) −2.74281e14 −1.03757 −0.518786 0.854904i \(-0.673616\pi\)
−0.518786 + 0.854904i \(0.673616\pi\)
\(420\) −8.45742e14 −3.15767
\(421\) −3.68411e14 −1.35763 −0.678815 0.734310i \(-0.737506\pi\)
−0.678815 + 0.734310i \(0.737506\pi\)
\(422\) 1.03069e14 0.374897
\(423\) 2.76700e14 0.993429
\(424\) 1.24114e15 4.39853
\(425\) 3.16337e14 1.10665
\(426\) 2.13339e14 0.736746
\(427\) 3.87062e14 1.31955
\(428\) −8.84083e14 −2.97545
\(429\) −7.49694e13 −0.249097
\(430\) 2.47434e14 0.811676
\(431\) 2.94489e13 0.0953770 0.0476885 0.998862i \(-0.484815\pi\)
0.0476885 + 0.998862i \(0.484815\pi\)
\(432\) 1.11745e15 3.57330
\(433\) 4.56308e14 1.44070 0.720351 0.693610i \(-0.243981\pi\)
0.720351 + 0.693610i \(0.243981\pi\)
\(434\) 2.38671e14 0.744056
\(435\) 1.21360e14 0.373580
\(436\) −1.44005e15 −4.37724
\(437\) 7.05599e14 2.11792
\(438\) −4.50072e14 −1.33406
\(439\) −8.72119e13 −0.255282 −0.127641 0.991820i \(-0.540741\pi\)
−0.127641 + 0.991820i \(0.540741\pi\)
\(440\) 4.36276e14 1.26116
\(441\) −5.17549e13 −0.147754
\(442\) −7.73809e14 −2.18178
\(443\) 1.35402e14 0.377054 0.188527 0.982068i \(-0.439629\pi\)
0.188527 + 0.982068i \(0.439629\pi\)
\(444\) −1.02911e15 −2.83046
\(445\) −3.96851e14 −1.07807
\(446\) 8.42358e14 2.26024
\(447\) 4.11936e14 1.09179
\(448\) −1.29690e15 −3.39529
\(449\) −2.35281e14 −0.608460 −0.304230 0.952599i \(-0.598399\pi\)
−0.304230 + 0.952599i \(0.598399\pi\)
\(450\) 5.50668e14 1.40676
\(451\) 3.74666e13 0.0945525
\(452\) −9.22005e14 −2.29864
\(453\) 2.42850e13 0.0598132
\(454\) 2.54149e14 0.618418
\(455\) 1.02607e15 2.46669
\(456\) −1.30739e15 −3.10527
\(457\) −5.78531e14 −1.35765 −0.678825 0.734300i \(-0.737510\pi\)
−0.678825 + 0.734300i \(0.737510\pi\)
\(458\) 8.22846e14 1.90791
\(459\) 3.67782e14 0.842598
\(460\) 2.67350e15 6.05218
\(461\) −2.60112e14 −0.581842 −0.290921 0.956747i \(-0.593962\pi\)
−0.290921 + 0.956747i \(0.593962\pi\)
\(462\) −1.71888e14 −0.379939
\(463\) 5.07415e14 1.10833 0.554164 0.832408i \(-0.313038\pi\)
0.554164 + 0.832408i \(0.313038\pi\)
\(464\) 5.62147e14 1.21339
\(465\) 1.69635e14 0.361845
\(466\) −6.02981e14 −1.27110
\(467\) −8.80591e14 −1.83456 −0.917279 0.398245i \(-0.869619\pi\)
−0.917279 + 0.398245i \(0.869619\pi\)
\(468\) −9.78933e14 −2.01558
\(469\) −7.54523e14 −1.53540
\(470\) −2.72234e15 −5.47526
\(471\) −2.03751e14 −0.405027
\(472\) −4.56327e14 −0.896593
\(473\) 3.65465e13 0.0709756
\(474\) −5.50036e14 −1.05587
\(475\) −1.03916e15 −1.97182
\(476\) −1.28936e15 −2.41844
\(477\) 3.99686e14 0.741085
\(478\) 8.36455e13 0.153317
\(479\) −6.61023e14 −1.19776 −0.598882 0.800837i \(-0.704388\pi\)
−0.598882 + 0.800837i \(0.704388\pi\)
\(480\) −1.96871e15 −3.52659
\(481\) 1.24854e15 2.21108
\(482\) −1.26776e15 −2.21962
\(483\) −6.57271e14 −1.13772
\(484\) −1.45075e15 −2.48281
\(485\) −4.31559e13 −0.0730232
\(486\) 1.05733e15 1.76894
\(487\) 6.47398e14 1.07093 0.535466 0.844557i \(-0.320136\pi\)
0.535466 + 0.844557i \(0.320136\pi\)
\(488\) 2.26710e15 3.70819
\(489\) −5.82291e14 −0.941763
\(490\) 5.09196e14 0.814342
\(491\) 6.23604e14 0.986190 0.493095 0.869975i \(-0.335865\pi\)
0.493095 + 0.869975i \(0.335865\pi\)
\(492\) −4.25414e14 −0.665279
\(493\) 1.85017e14 0.286123
\(494\) 2.54194e15 3.88746
\(495\) 1.40495e14 0.212487
\(496\) 7.85760e14 1.17528
\(497\) −4.31271e14 −0.637955
\(498\) 1.78300e14 0.260849
\(499\) −5.85885e14 −0.847734 −0.423867 0.905724i \(-0.639328\pi\)
−0.423867 + 0.905724i \(0.639328\pi\)
\(500\) −1.07345e15 −1.53620
\(501\) −2.96509e14 −0.419692
\(502\) 1.93716e15 2.71204
\(503\) −7.59409e13 −0.105160 −0.0525801 0.998617i \(-0.516744\pi\)
−0.0525801 + 0.998617i \(0.516744\pi\)
\(504\) −1.40054e15 −1.91834
\(505\) 1.98337e15 2.68721
\(506\) 5.43359e14 0.728214
\(507\) −5.18313e14 −0.687145
\(508\) −6.67310e14 −0.875139
\(509\) 1.24817e15 1.61929 0.809645 0.586920i \(-0.199660\pi\)
0.809645 + 0.586920i \(0.199660\pi\)
\(510\) −1.26098e15 −1.61835
\(511\) 9.09833e14 1.15517
\(512\) −4.91443e14 −0.617289
\(513\) −1.20815e15 −1.50133
\(514\) −1.81634e15 −2.23306
\(515\) −4.71615e14 −0.573651
\(516\) −4.14967e14 −0.499390
\(517\) −4.02096e14 −0.478775
\(518\) 2.86261e15 3.37248
\(519\) 5.08275e14 0.592486
\(520\) 6.00991e15 6.93186
\(521\) −6.33330e14 −0.722808 −0.361404 0.932409i \(-0.617702\pi\)
−0.361404 + 0.932409i \(0.617702\pi\)
\(522\) 3.22070e14 0.363716
\(523\) 1.66720e15 1.86306 0.931531 0.363663i \(-0.118474\pi\)
0.931531 + 0.363663i \(0.118474\pi\)
\(524\) −1.28612e15 −1.42220
\(525\) 9.67984e14 1.05923
\(526\) −1.77650e15 −1.92373
\(527\) 2.58613e14 0.277135
\(528\) −5.65894e14 −0.600134
\(529\) 1.12491e15 1.18062
\(530\) −3.93236e15 −4.08447
\(531\) −1.46952e14 −0.151062
\(532\) 4.23550e15 4.30914
\(533\) 5.16120e14 0.519698
\(534\) 9.15803e14 0.912694
\(535\) 1.74786e15 1.72410
\(536\) −4.41940e15 −4.31476
\(537\) 5.17221e14 0.499824
\(538\) 2.95340e15 2.82501
\(539\) 7.52093e13 0.0712088
\(540\) −4.57766e15 −4.29020
\(541\) 6.70038e14 0.621605 0.310802 0.950475i \(-0.399402\pi\)
0.310802 + 0.950475i \(0.399402\pi\)
\(542\) −2.74513e15 −2.52097
\(543\) 1.27911e15 1.16281
\(544\) −3.00136e15 −2.70099
\(545\) 2.84702e15 2.53636
\(546\) −2.36784e15 −2.08830
\(547\) −2.04406e15 −1.78470 −0.892348 0.451348i \(-0.850943\pi\)
−0.892348 + 0.451348i \(0.850943\pi\)
\(548\) −4.00900e15 −3.46532
\(549\) 7.30081e14 0.624774
\(550\) −8.00221e14 −0.677977
\(551\) −6.07773e14 −0.509809
\(552\) −3.84978e15 −3.19721
\(553\) 1.11191e15 0.914286
\(554\) −3.23579e15 −2.63437
\(555\) 2.03459e15 1.64008
\(556\) −8.90468e14 −0.710733
\(557\) −7.15674e14 −0.565603 −0.282801 0.959179i \(-0.591264\pi\)
−0.282801 + 0.959179i \(0.591264\pi\)
\(558\) 4.50184e14 0.352291
\(559\) 5.03446e14 0.390111
\(560\) 7.74512e15 5.94284
\(561\) −1.86250e14 −0.141514
\(562\) 4.50315e14 0.338818
\(563\) −3.38439e14 −0.252164 −0.126082 0.992020i \(-0.540240\pi\)
−0.126082 + 0.992020i \(0.540240\pi\)
\(564\) 4.56559e15 3.36870
\(565\) 1.82284e15 1.33193
\(566\) −5.44702e13 −0.0394156
\(567\) 2.82198e14 0.202231
\(568\) −2.52604e15 −1.79277
\(569\) 1.78757e14 0.125645 0.0628225 0.998025i \(-0.479990\pi\)
0.0628225 + 0.998025i \(0.479990\pi\)
\(570\) 4.14228e15 2.88355
\(571\) 3.91659e14 0.270028 0.135014 0.990844i \(-0.456892\pi\)
0.135014 + 0.990844i \(0.456892\pi\)
\(572\) 1.42257e15 0.971394
\(573\) −4.05182e14 −0.274031
\(574\) 1.18334e15 0.792676
\(575\) −3.05992e15 −2.03019
\(576\) −2.44623e15 −1.60758
\(577\) 5.05916e14 0.329315 0.164658 0.986351i \(-0.447348\pi\)
0.164658 + 0.986351i \(0.447348\pi\)
\(578\) 1.04458e15 0.673505
\(579\) 1.04072e15 0.664666
\(580\) −2.30284e15 −1.45683
\(581\) −3.60438e14 −0.225871
\(582\) 9.95898e13 0.0618213
\(583\) −5.80817e14 −0.357159
\(584\) 5.32908e15 3.24625
\(585\) 1.93539e15 1.16791
\(586\) −1.27359e15 −0.761367
\(587\) 1.73810e15 1.02936 0.514678 0.857384i \(-0.327912\pi\)
0.514678 + 0.857384i \(0.327912\pi\)
\(588\) −8.53964e14 −0.501031
\(589\) −8.49535e14 −0.493795
\(590\) 1.44581e15 0.832576
\(591\) 8.51683e13 0.0485900
\(592\) 9.42439e15 5.32701
\(593\) −1.47703e15 −0.827156 −0.413578 0.910469i \(-0.635721\pi\)
−0.413578 + 0.910469i \(0.635721\pi\)
\(594\) −9.30357e14 −0.516208
\(595\) 2.54911e15 1.40135
\(596\) −7.81662e15 −4.25760
\(597\) −2.02965e15 −1.09537
\(598\) 7.48503e15 4.00255
\(599\) −2.58595e15 −1.37016 −0.685081 0.728467i \(-0.740233\pi\)
−0.685081 + 0.728467i \(0.740233\pi\)
\(600\) 5.66968e15 2.97664
\(601\) −2.84322e15 −1.47911 −0.739555 0.673096i \(-0.764964\pi\)
−0.739555 + 0.673096i \(0.764964\pi\)
\(602\) 1.15429e15 0.595021
\(603\) −1.42319e15 −0.726972
\(604\) −4.60815e14 −0.233251
\(605\) 2.86818e15 1.43864
\(606\) −4.57697e15 −2.27499
\(607\) −5.45615e14 −0.268750 −0.134375 0.990931i \(-0.542903\pi\)
−0.134375 + 0.990931i \(0.542903\pi\)
\(608\) 9.85935e15 4.81259
\(609\) 5.66146e14 0.273863
\(610\) −7.18298e15 −3.44343
\(611\) −5.53907e15 −2.63154
\(612\) −2.43200e15 −1.14507
\(613\) −3.67185e15 −1.71338 −0.856688 0.515834i \(-0.827482\pi\)
−0.856688 + 0.515834i \(0.827482\pi\)
\(614\) −5.40811e15 −2.50104
\(615\) 8.41058e14 0.385490
\(616\) 2.03524e15 0.924530
\(617\) 3.26263e15 1.46893 0.734463 0.678648i \(-0.237434\pi\)
0.734463 + 0.678648i \(0.237434\pi\)
\(618\) 1.08833e15 0.485652
\(619\) 1.64075e15 0.725676 0.362838 0.931852i \(-0.381808\pi\)
0.362838 + 0.931852i \(0.381808\pi\)
\(620\) −3.21887e15 −1.41107
\(621\) −3.55754e15 −1.54577
\(622\) 2.90509e15 1.25116
\(623\) −1.85132e15 −0.790310
\(624\) −7.79546e15 −3.29858
\(625\) −1.15558e15 −0.484685
\(626\) −7.30905e15 −3.03880
\(627\) 6.11824e14 0.252148
\(628\) 3.86624e15 1.57947
\(629\) 3.10180e15 1.25613
\(630\) 4.43739e15 1.78137
\(631\) −3.46678e15 −1.37964 −0.689818 0.723983i \(-0.742310\pi\)
−0.689818 + 0.723983i \(0.742310\pi\)
\(632\) 6.51270e15 2.56931
\(633\) 3.41742e14 0.133653
\(634\) 6.73632e15 2.61175
\(635\) 1.31929e15 0.507092
\(636\) 6.59488e15 2.51300
\(637\) 1.03605e15 0.391392
\(638\) −4.68026e14 −0.175290
\(639\) −8.13468e14 −0.302055
\(640\) 1.00211e16 3.68914
\(641\) −1.65515e15 −0.604111 −0.302056 0.953290i \(-0.597673\pi\)
−0.302056 + 0.953290i \(0.597673\pi\)
\(642\) −4.03350e15 −1.45962
\(643\) 2.68040e14 0.0961701 0.0480850 0.998843i \(-0.484688\pi\)
0.0480850 + 0.998843i \(0.484688\pi\)
\(644\) 1.24719e16 4.43672
\(645\) 8.20404e14 0.289367
\(646\) 6.31504e15 2.20850
\(647\) 1.15109e15 0.399151 0.199575 0.979882i \(-0.436044\pi\)
0.199575 + 0.979882i \(0.436044\pi\)
\(648\) 1.65289e15 0.568305
\(649\) 2.13548e14 0.0728032
\(650\) −1.10234e16 −3.72644
\(651\) 7.91349e14 0.265261
\(652\) 1.10492e16 3.67255
\(653\) −2.70126e15 −0.890316 −0.445158 0.895452i \(-0.646852\pi\)
−0.445158 + 0.895452i \(0.646852\pi\)
\(654\) −6.57000e15 −2.14727
\(655\) 2.54271e15 0.824081
\(656\) 3.89585e15 1.25208
\(657\) 1.71614e15 0.546943
\(658\) −1.26998e16 −4.01379
\(659\) −2.06804e14 −0.0648170 −0.0324085 0.999475i \(-0.510318\pi\)
−0.0324085 + 0.999475i \(0.510318\pi\)
\(660\) 2.31819e15 0.720538
\(661\) 4.36563e15 1.34567 0.672836 0.739792i \(-0.265076\pi\)
0.672836 + 0.739792i \(0.265076\pi\)
\(662\) −1.02586e16 −3.13594
\(663\) −2.56568e15 −0.777818
\(664\) −2.11116e15 −0.634741
\(665\) −8.37374e15 −2.49690
\(666\) 5.39949e15 1.59678
\(667\) −1.78966e15 −0.524902
\(668\) 5.62636e15 1.63666
\(669\) 2.79297e15 0.805790
\(670\) 1.40022e16 4.00669
\(671\) −1.06094e15 −0.301105
\(672\) −9.18407e15 −2.58526
\(673\) −2.70502e15 −0.755244 −0.377622 0.925960i \(-0.623258\pi\)
−0.377622 + 0.925960i \(0.623258\pi\)
\(674\) −6.13334e15 −1.69851
\(675\) 5.23930e15 1.43914
\(676\) 9.83517e15 2.67963
\(677\) −3.32057e14 −0.0897376 −0.0448688 0.998993i \(-0.514287\pi\)
−0.0448688 + 0.998993i \(0.514287\pi\)
\(678\) −4.20651e15 −1.12761
\(679\) −2.01323e14 −0.0535317
\(680\) 1.49307e16 3.93804
\(681\) 8.42671e14 0.220470
\(682\) −6.54199e14 −0.169784
\(683\) 1.95787e15 0.504047 0.252023 0.967721i \(-0.418904\pi\)
0.252023 + 0.967721i \(0.418904\pi\)
\(684\) 7.98905e15 2.04027
\(685\) 7.92594e15 2.00795
\(686\) −6.22385e15 −1.56414
\(687\) 2.72827e15 0.680182
\(688\) 3.80018e15 0.939869
\(689\) −8.00104e15 −1.96309
\(690\) 1.21974e16 2.96892
\(691\) −2.61062e15 −0.630397 −0.315199 0.949026i \(-0.602071\pi\)
−0.315199 + 0.949026i \(0.602071\pi\)
\(692\) −9.64468e15 −2.31049
\(693\) 6.55412e14 0.155769
\(694\) −1.15437e16 −2.72187
\(695\) 1.76049e15 0.411828
\(696\) 3.31603e15 0.769606
\(697\) 1.28222e15 0.295245
\(698\) 1.40637e16 3.21289
\(699\) −1.99928e15 −0.453156
\(700\) −1.83678e16 −4.13065
\(701\) 4.45543e15 0.994125 0.497062 0.867715i \(-0.334412\pi\)
0.497062 + 0.867715i \(0.334412\pi\)
\(702\) −1.28161e16 −2.83728
\(703\) −1.01893e16 −2.23815
\(704\) 3.55482e15 0.774761
\(705\) −9.02634e15 −1.95196
\(706\) 8.94368e15 1.91907
\(707\) 9.25248e15 1.96993
\(708\) −2.42473e15 −0.512249
\(709\) 3.60048e14 0.0754756 0.0377378 0.999288i \(-0.487985\pi\)
0.0377378 + 0.999288i \(0.487985\pi\)
\(710\) 8.00340e15 1.66477
\(711\) 2.09730e15 0.432890
\(712\) −1.08436e16 −2.22092
\(713\) −2.50156e15 −0.508414
\(714\) −5.88251e15 −1.18638
\(715\) −2.81247e15 −0.562866
\(716\) −9.81444e15 −1.94914
\(717\) 2.77339e14 0.0546583
\(718\) −1.17201e16 −2.29217
\(719\) 7.57262e14 0.146973 0.0734864 0.997296i \(-0.476587\pi\)
0.0734864 + 0.997296i \(0.476587\pi\)
\(720\) 1.46089e16 2.81378
\(721\) −2.20009e15 −0.420530
\(722\) −1.06599e16 −2.02208
\(723\) −4.20345e15 −0.791307
\(724\) −2.42714e16 −4.53454
\(725\) 2.63569e15 0.488692
\(726\) −6.61882e15 −1.21795
\(727\) −4.77912e15 −0.872789 −0.436394 0.899756i \(-0.643745\pi\)
−0.436394 + 0.899756i \(0.643745\pi\)
\(728\) 2.80364e16 5.08159
\(729\) 4.50088e15 0.809648
\(730\) −1.68844e16 −3.01446
\(731\) 1.25073e15 0.221625
\(732\) 1.20465e16 2.11860
\(733\) 1.92867e14 0.0336656 0.0168328 0.999858i \(-0.494642\pi\)
0.0168328 + 0.999858i \(0.494642\pi\)
\(734\) 1.84063e16 3.18888
\(735\) 1.68832e15 0.290318
\(736\) 2.90320e16 4.95507
\(737\) 2.06816e15 0.350358
\(738\) 2.23204e15 0.375311
\(739\) 8.60815e15 1.43670 0.718349 0.695683i \(-0.244898\pi\)
0.718349 + 0.695683i \(0.244898\pi\)
\(740\) −3.86071e16 −6.39576
\(741\) 8.42817e15 1.38590
\(742\) −1.83445e16 −2.99423
\(743\) 5.66593e15 0.917979 0.458990 0.888442i \(-0.348212\pi\)
0.458990 + 0.888442i \(0.348212\pi\)
\(744\) 4.63510e15 0.745431
\(745\) 1.54537e16 2.46703
\(746\) 1.60885e16 2.54948
\(747\) −6.79861e14 −0.106944
\(748\) 3.53415e15 0.551856
\(749\) 8.15383e15 1.26390
\(750\) −4.89747e15 −0.753589
\(751\) −1.28689e16 −1.96572 −0.982862 0.184341i \(-0.940985\pi\)
−0.982862 + 0.184341i \(0.940985\pi\)
\(752\) −4.18107e16 −6.34000
\(753\) 6.42296e15 0.966858
\(754\) −6.44729e15 −0.963463
\(755\) 9.11048e14 0.135155
\(756\) −2.13549e16 −3.14505
\(757\) −4.18693e15 −0.612164 −0.306082 0.952005i \(-0.599018\pi\)
−0.306082 + 0.952005i \(0.599018\pi\)
\(758\) 1.30506e16 1.89431
\(759\) 1.80159e15 0.259612
\(760\) −4.90468e16 −7.01674
\(761\) −6.59676e15 −0.936948 −0.468474 0.883477i \(-0.655196\pi\)
−0.468474 + 0.883477i \(0.655196\pi\)
\(762\) −3.04450e15 −0.429303
\(763\) 1.32814e16 1.85934
\(764\) 7.68846e15 1.06863
\(765\) 4.80816e15 0.663500
\(766\) 6.97642e15 0.955817
\(767\) 2.94173e15 0.400156
\(768\) −7.94867e15 −1.07351
\(769\) −3.82006e15 −0.512242 −0.256121 0.966645i \(-0.582445\pi\)
−0.256121 + 0.966645i \(0.582445\pi\)
\(770\) −6.44835e15 −0.858518
\(771\) −6.02235e15 −0.796099
\(772\) −1.97481e16 −2.59197
\(773\) 6.98362e15 0.910108 0.455054 0.890464i \(-0.349620\pi\)
0.455054 + 0.890464i \(0.349620\pi\)
\(774\) 2.17723e15 0.281727
\(775\) 3.68412e15 0.473341
\(776\) −1.17919e15 −0.150434
\(777\) 9.49142e15 1.20231
\(778\) −1.74208e15 −0.219120
\(779\) −4.21205e15 −0.526062
\(780\) 3.19342e16 3.96037
\(781\) 1.18212e15 0.145573
\(782\) 1.85954e16 2.27388
\(783\) 3.06432e15 0.372087
\(784\) 7.82041e15 0.942956
\(785\) −7.64369e15 −0.915209
\(786\) −5.86775e15 −0.697666
\(787\) 1.25624e16 1.48324 0.741618 0.670823i \(-0.234059\pi\)
0.741618 + 0.670823i \(0.234059\pi\)
\(788\) −1.61610e15 −0.189484
\(789\) −5.89026e15 −0.685822
\(790\) −2.06345e16 −2.38586
\(791\) 8.50358e15 0.976407
\(792\) 3.83888e15 0.437740
\(793\) −1.46150e16 −1.65499
\(794\) −1.05218e16 −1.18325
\(795\) −1.30383e16 −1.45614
\(796\) 3.85132e16 4.27157
\(797\) 9.12350e15 1.00494 0.502471 0.864594i \(-0.332424\pi\)
0.502471 + 0.864594i \(0.332424\pi\)
\(798\) 1.93239e16 2.11387
\(799\) −1.37609e16 −1.49500
\(800\) −4.27564e16 −4.61324
\(801\) −3.49198e15 −0.374191
\(802\) 1.47450e16 1.56922
\(803\) −2.49386e15 −0.263595
\(804\) −2.34829e16 −2.46515
\(805\) −2.46575e16 −2.57082
\(806\) −9.01191e15 −0.933199
\(807\) 9.79244e15 1.00713
\(808\) 5.41937e16 5.53587
\(809\) 1.40131e16 1.42174 0.710868 0.703326i \(-0.248303\pi\)
0.710868 + 0.703326i \(0.248303\pi\)
\(810\) −5.23694e15 −0.527728
\(811\) −9.89235e15 −0.990114 −0.495057 0.868861i \(-0.664853\pi\)
−0.495057 + 0.868861i \(0.664853\pi\)
\(812\) −1.07428e16 −1.06797
\(813\) −9.10191e15 −0.898741
\(814\) −7.84645e15 −0.769554
\(815\) −2.18446e16 −2.12803
\(816\) −1.93666e16 −1.87395
\(817\) −4.10861e15 −0.394888
\(818\) 2.19692e16 2.09735
\(819\) 9.02862e15 0.856170
\(820\) −1.59594e16 −1.50328
\(821\) −4.49746e15 −0.420804 −0.210402 0.977615i \(-0.567477\pi\)
−0.210402 + 0.977615i \(0.567477\pi\)
\(822\) −1.82905e16 −1.69993
\(823\) −6.75910e15 −0.624007 −0.312004 0.950081i \(-0.601000\pi\)
−0.312004 + 0.950081i \(0.601000\pi\)
\(824\) −1.28864e16 −1.18177
\(825\) −2.65325e15 −0.241703
\(826\) 6.74472e15 0.610342
\(827\) 9.35357e14 0.0840808 0.0420404 0.999116i \(-0.486614\pi\)
0.0420404 + 0.999116i \(0.486614\pi\)
\(828\) 2.35247e16 2.10067
\(829\) −2.13470e16 −1.89359 −0.946797 0.321830i \(-0.895702\pi\)
−0.946797 + 0.321830i \(0.895702\pi\)
\(830\) 6.68889e15 0.589420
\(831\) −1.07287e16 −0.939168
\(832\) 4.89693e16 4.25839
\(833\) 2.57389e15 0.222353
\(834\) −4.06263e15 −0.348653
\(835\) −1.11235e16 −0.948346
\(836\) −1.16096e16 −0.983289
\(837\) 4.28325e15 0.360399
\(838\) 2.37451e16 1.98487
\(839\) −5.95088e15 −0.494186 −0.247093 0.968992i \(-0.579475\pi\)
−0.247093 + 0.968992i \(0.579475\pi\)
\(840\) 4.56875e16 3.76930
\(841\) −1.06590e16 −0.873650
\(842\) 3.18941e16 2.59713
\(843\) 1.49309e15 0.120791
\(844\) −6.48467e15 −0.521200
\(845\) −1.94445e16 −1.55269
\(846\) −2.39545e16 −1.90042
\(847\) 1.33801e16 1.05463
\(848\) −6.03945e16 −4.72955
\(849\) −1.80604e14 −0.0140519
\(850\) −2.73860e16 −2.11702
\(851\) −3.00036e16 −2.30442
\(852\) −1.34223e16 −1.02426
\(853\) −3.56735e15 −0.270474 −0.135237 0.990813i \(-0.543180\pi\)
−0.135237 + 0.990813i \(0.543180\pi\)
\(854\) −3.35088e16 −2.52430
\(855\) −1.57946e16 −1.18221
\(856\) 4.77587e16 3.55178
\(857\) −2.04044e16 −1.50775 −0.753873 0.657020i \(-0.771817\pi\)
−0.753873 + 0.657020i \(0.771817\pi\)
\(858\) 6.49026e15 0.476521
\(859\) −8.33941e15 −0.608377 −0.304189 0.952612i \(-0.598385\pi\)
−0.304189 + 0.952612i \(0.598385\pi\)
\(860\) −1.55674e16 −1.12843
\(861\) 3.92356e15 0.282594
\(862\) −2.54945e15 −0.182455
\(863\) 2.61014e16 1.85611 0.928057 0.372438i \(-0.121478\pi\)
0.928057 + 0.372438i \(0.121478\pi\)
\(864\) −4.97096e16 −3.51249
\(865\) 1.90679e16 1.33879
\(866\) −3.95035e16 −2.75605
\(867\) 3.46348e15 0.240108
\(868\) −1.50161e16 −1.03442
\(869\) −3.04776e15 −0.208628
\(870\) −1.05064e16 −0.714655
\(871\) 2.84899e16 1.92571
\(872\) 7.77921e16 5.22510
\(873\) −3.79739e14 −0.0253458
\(874\) −6.10852e16 −4.05157
\(875\) 9.90037e15 0.652540
\(876\) 2.83165e16 1.85467
\(877\) −5.51813e15 −0.359165 −0.179583 0.983743i \(-0.557475\pi\)
−0.179583 + 0.983743i \(0.557475\pi\)
\(878\) 7.55012e15 0.488353
\(879\) −4.22279e15 −0.271432
\(880\) −2.12295e16 −1.35608
\(881\) 9.48757e14 0.0602265 0.0301132 0.999546i \(-0.490413\pi\)
0.0301132 + 0.999546i \(0.490413\pi\)
\(882\) 4.48053e15 0.282652
\(883\) −1.39775e16 −0.876284 −0.438142 0.898906i \(-0.644363\pi\)
−0.438142 + 0.898906i \(0.644363\pi\)
\(884\) 4.86846e16 3.03322
\(885\) 4.79379e15 0.296818
\(886\) −1.17220e16 −0.721302
\(887\) −1.77585e16 −1.08599 −0.542996 0.839735i \(-0.682710\pi\)
−0.542996 + 0.839735i \(0.682710\pi\)
\(888\) 5.55933e16 3.37871
\(889\) 6.15454e15 0.371737
\(890\) 3.43563e16 2.06234
\(891\) −7.73507e14 −0.0461463
\(892\) −5.29974e16 −3.14230
\(893\) 4.52042e16 2.66376
\(894\) −3.56622e16 −2.08858
\(895\) 1.94035e16 1.12941
\(896\) 4.67487e16 2.70443
\(897\) 2.48177e16 1.42693
\(898\) 2.03688e16 1.16398
\(899\) 2.15473e15 0.122381
\(900\) −3.46456e16 −1.95575
\(901\) −1.98773e16 −1.11525
\(902\) −3.24356e15 −0.180878
\(903\) 3.82721e15 0.212129
\(904\) 4.98072e16 2.74389
\(905\) 4.79855e16 2.62750
\(906\) −2.10240e15 −0.114422
\(907\) −9.03137e15 −0.488555 −0.244278 0.969705i \(-0.578551\pi\)
−0.244278 + 0.969705i \(0.578551\pi\)
\(908\) −1.59900e16 −0.859756
\(909\) 1.74521e16 0.932710
\(910\) −8.88291e16 −4.71876
\(911\) −1.47302e16 −0.777784 −0.388892 0.921283i \(-0.627142\pi\)
−0.388892 + 0.921283i \(0.627142\pi\)
\(912\) 6.36186e16 3.33897
\(913\) 9.87963e14 0.0515408
\(914\) 5.00847e16 2.59717
\(915\) −2.38163e16 −1.22760
\(916\) −5.17698e16 −2.65247
\(917\) 1.18618e16 0.604115
\(918\) −3.18396e16 −1.61188
\(919\) 2.51043e16 1.26332 0.631659 0.775247i \(-0.282374\pi\)
0.631659 + 0.775247i \(0.282374\pi\)
\(920\) −1.44424e17 −7.22447
\(921\) −1.79314e16 −0.891634
\(922\) 2.25184e16 1.11306
\(923\) 1.62842e16 0.800126
\(924\) 1.08144e16 0.528210
\(925\) 4.41872e16 2.14544
\(926\) −4.39280e16 −2.12022
\(927\) −4.14984e15 −0.199110
\(928\) −2.50070e16 −1.19274
\(929\) −3.40423e16 −1.61411 −0.807054 0.590477i \(-0.798940\pi\)
−0.807054 + 0.590477i \(0.798940\pi\)
\(930\) −1.46856e16 −0.692207
\(931\) −8.45514e15 −0.396185
\(932\) 3.79369e16 1.76715
\(933\) 9.63225e15 0.446045
\(934\) 7.62346e16 3.50949
\(935\) −6.98714e15 −0.319768
\(936\) 5.28825e16 2.40600
\(937\) −1.60071e16 −0.724010 −0.362005 0.932176i \(-0.617908\pi\)
−0.362005 + 0.932176i \(0.617908\pi\)
\(938\) 6.53207e16 2.93721
\(939\) −2.42342e16 −1.08335
\(940\) 1.71278e17 7.61199
\(941\) 1.79408e16 0.792683 0.396341 0.918103i \(-0.370280\pi\)
0.396341 + 0.918103i \(0.370280\pi\)
\(942\) 1.76391e16 0.774814
\(943\) −1.24029e16 −0.541636
\(944\) 2.22052e16 0.964069
\(945\) 4.22194e16 1.82237
\(946\) −3.16391e15 −0.135776
\(947\) 1.31442e16 0.560800 0.280400 0.959883i \(-0.409533\pi\)
0.280400 + 0.959883i \(0.409533\pi\)
\(948\) 3.46058e16 1.46792
\(949\) −3.43541e16 −1.44882
\(950\) 8.99620e16 3.77207
\(951\) 2.23353e16 0.931105
\(952\) 6.96519e16 2.88689
\(953\) −4.39785e16 −1.81230 −0.906150 0.422958i \(-0.860992\pi\)
−0.906150 + 0.422958i \(0.860992\pi\)
\(954\) −3.46017e16 −1.41769
\(955\) −1.52004e16 −0.619207
\(956\) −5.26260e15 −0.213148
\(957\) −1.55181e15 −0.0624918
\(958\) 5.72262e16 2.29131
\(959\) 3.69747e16 1.47198
\(960\) 7.97994e16 3.15869
\(961\) −2.23966e16 −0.881463
\(962\) −1.08089e17 −4.22978
\(963\) 1.53798e16 0.598422
\(964\) 7.97618e16 3.08583
\(965\) 3.90427e16 1.50189
\(966\) 5.69014e16 2.17645
\(967\) 8.85236e15 0.336677 0.168339 0.985729i \(-0.446160\pi\)
0.168339 + 0.985729i \(0.446160\pi\)
\(968\) 7.83703e16 2.96372
\(969\) 2.09385e16 0.787343
\(970\) 3.73610e15 0.139693
\(971\) 2.16728e16 0.805766 0.402883 0.915251i \(-0.368008\pi\)
0.402883 + 0.915251i \(0.368008\pi\)
\(972\) −6.65227e16 −2.45927
\(973\) 8.21271e15 0.301902
\(974\) −5.60466e16 −2.04868
\(975\) −3.65499e16 −1.32850
\(976\) −1.10319e17 −3.98727
\(977\) 2.25522e16 0.810531 0.405265 0.914199i \(-0.367179\pi\)
0.405265 + 0.914199i \(0.367179\pi\)
\(978\) 5.04102e16 1.80158
\(979\) 5.07449e15 0.180338
\(980\) −3.20364e16 −1.13214
\(981\) 2.50516e16 0.880350
\(982\) −5.39867e16 −1.88657
\(983\) −1.07495e16 −0.373547 −0.186773 0.982403i \(-0.559803\pi\)
−0.186773 + 0.982403i \(0.559803\pi\)
\(984\) 2.29811e16 0.794141
\(985\) 3.19508e15 0.109795
\(986\) −1.60173e16 −0.547351
\(987\) −4.21081e16 −1.43094
\(988\) −1.59927e17 −5.40455
\(989\) −1.20983e16 −0.406578
\(990\) −1.21629e16 −0.406485
\(991\) 4.13282e16 1.37354 0.686770 0.726874i \(-0.259028\pi\)
0.686770 + 0.726874i \(0.259028\pi\)
\(992\) −3.49543e16 −1.15528
\(993\) −3.40138e16 −1.11798
\(994\) 3.73360e16 1.22040
\(995\) −7.61420e16 −2.47512
\(996\) −1.12178e16 −0.362645
\(997\) 4.17879e16 1.34347 0.671734 0.740793i \(-0.265550\pi\)
0.671734 + 0.740793i \(0.265550\pi\)
\(998\) 5.07214e16 1.62171
\(999\) 5.13732e16 1.63353
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.1.3 92
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.12.a.b.1.3 92 1.1 even 1 trivial