Properties

Label 3.20.a.b.1.2
Level $3$
Weight $20$
Character 3.1
Self dual yes
Analytic conductor $6.865$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.86450089669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{87481}) \)
Defining polynomial: \( x^{2} - x - 21870 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-147.386\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q+1238.32 q^{2} -19683.0 q^{3} +1.00914e6 q^{4} +6.69930e6 q^{5} -2.43738e7 q^{6} +214621. q^{7} +6.00397e8 q^{8} +3.87420e8 q^{9} +O(q^{10})\) \(q+1238.32 q^{2} -19683.0 q^{3} +1.00914e6 q^{4} +6.69930e6 q^{5} -2.43738e7 q^{6} +214621. q^{7} +6.00397e8 q^{8} +3.87420e8 q^{9} +8.29585e9 q^{10} -1.30816e10 q^{11} -1.98629e10 q^{12} +1.34409e10 q^{13} +2.65768e8 q^{14} -1.31862e11 q^{15} +2.14402e11 q^{16} -3.48802e11 q^{17} +4.79749e11 q^{18} -6.20630e11 q^{19} +6.76052e12 q^{20} -4.22438e9 q^{21} -1.61991e13 q^{22} -2.65405e12 q^{23} -1.18176e13 q^{24} +2.58072e13 q^{25} +1.66441e13 q^{26} -7.62560e12 q^{27} +2.16582e11 q^{28} +1.27229e14 q^{29} -1.63287e14 q^{30} +2.32290e12 q^{31} -4.92835e13 q^{32} +2.57485e14 q^{33} -4.31927e14 q^{34} +1.43781e12 q^{35} +3.90961e14 q^{36} -7.19096e14 q^{37} -7.68535e14 q^{38} -2.64557e14 q^{39} +4.02224e15 q^{40} +1.72545e15 q^{41} -5.23112e12 q^{42} +2.48385e15 q^{43} -1.32011e16 q^{44} +2.59545e15 q^{45} -3.28655e15 q^{46} +1.39689e15 q^{47} -4.22007e15 q^{48} -1.13988e16 q^{49} +3.19574e16 q^{50} +6.86546e15 q^{51} +1.35637e16 q^{52} +1.05483e16 q^{53} -9.44290e15 q^{54} -8.76375e16 q^{55} +1.28858e14 q^{56} +1.22159e16 q^{57} +1.57550e17 q^{58} -2.80846e16 q^{59} -1.33067e17 q^{60} -3.60392e16 q^{61} +2.87648e15 q^{62} +8.31485e13 q^{63} -1.73437e17 q^{64} +9.00446e16 q^{65} +3.18848e17 q^{66} +2.91862e17 q^{67} -3.51989e17 q^{68} +5.22397e16 q^{69} +1.78046e15 q^{70} +3.20038e17 q^{71} +2.32606e17 q^{72} -3.15819e17 q^{73} -8.90467e17 q^{74} -5.07963e17 q^{75} -6.26301e17 q^{76} -2.80758e15 q^{77} -3.27605e17 q^{78} +1.83029e18 q^{79} +1.43634e18 q^{80} +1.50095e17 q^{81} +2.13665e18 q^{82} -1.64758e18 q^{83} -4.26298e15 q^{84} -2.33673e18 q^{85} +3.07579e18 q^{86} -2.50426e18 q^{87} -7.85414e18 q^{88} +5.75855e18 q^{89} +3.21398e18 q^{90} +2.88469e15 q^{91} -2.67830e18 q^{92} -4.57216e16 q^{93} +1.72979e18 q^{94} -4.15779e18 q^{95} +9.70047e17 q^{96} +6.23309e18 q^{97} -1.41154e19 q^{98} -5.06808e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 702 q^{2} - 39366 q^{3} + 772484 q^{4} + 6016140 q^{5} - 13817466 q^{6} + 113892064 q^{7} + 1008501624 q^{8} + 774840978 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 702 q^{2} - 39366 q^{3} + 772484 q^{4} + 6016140 q^{5} - 13817466 q^{6} + 113892064 q^{7} + 1008501624 q^{8} + 774840978 q^{9} + 8662242420 q^{10} - 6650071272 q^{11} - 15204802572 q^{12} - 44072356148 q^{13} - 60701219424 q^{14} - 118415683620 q^{15} + 119603620880 q^{16} + 336281471748 q^{17} + 271969183278 q^{18} + 602118925096 q^{19} + 6922191196440 q^{20} - 2241737495712 q^{21} - 19648459326744 q^{22} + 2368252165968 q^{23} - 19850337465192 q^{24} + 7200399078350 q^{25} + 47489309789364 q^{26} - 15251194969974 q^{27} - 26685589805888 q^{28} + 280977251970492 q^{29} - 170498917552860 q^{30} + 41610149253712 q^{31} - 212406109003296 q^{32} + 130893352846776 q^{33} - 799347349171332 q^{34} - 76222408017600 q^{35} + 299276129024676 q^{36} + 637994163989884 q^{37} - 14\!\cdots\!32 q^{38}+ \cdots - 25\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1238.32 1.71020 0.855099 0.518465i \(-0.173496\pi\)
0.855099 + 0.518465i \(0.173496\pi\)
\(3\) −19683.0 −0.577350
\(4\) 1.00914e6 1.92478
\(5\) 6.69930e6 1.53396 0.766981 0.641670i \(-0.221758\pi\)
0.766981 + 0.641670i \(0.221758\pi\)
\(6\) −2.43738e7 −0.987383
\(7\) 214621. 0.00201021 0.00100510 0.999999i \(-0.499680\pi\)
0.00100510 + 0.999999i \(0.499680\pi\)
\(8\) 6.00397e8 1.58155
\(9\) 3.87420e8 0.333333
\(10\) 8.29585e9 2.62338
\(11\) −1.30816e10 −1.67275 −0.836373 0.548161i \(-0.815328\pi\)
−0.836373 + 0.548161i \(0.815328\pi\)
\(12\) −1.98629e10 −1.11127
\(13\) 1.34409e10 0.351533 0.175766 0.984432i \(-0.443760\pi\)
0.175766 + 0.984432i \(0.443760\pi\)
\(14\) 2.65768e8 0.00343785
\(15\) −1.31862e11 −0.885633
\(16\) 2.14402e11 0.779990
\(17\) −3.48802e11 −0.713368 −0.356684 0.934225i \(-0.616093\pi\)
−0.356684 + 0.934225i \(0.616093\pi\)
\(18\) 4.79749e11 0.570066
\(19\) −6.20630e11 −0.441238 −0.220619 0.975360i \(-0.570808\pi\)
−0.220619 + 0.975360i \(0.570808\pi\)
\(20\) 6.76052e12 2.95253
\(21\) −4.22438e9 −0.00116059
\(22\) −1.61991e13 −2.86073
\(23\) −2.65405e12 −0.307252 −0.153626 0.988129i \(-0.549095\pi\)
−0.153626 + 0.988129i \(0.549095\pi\)
\(24\) −1.18176e13 −0.913109
\(25\) 2.58072e13 1.35304
\(26\) 1.66441e13 0.601191
\(27\) −7.62560e12 −0.192450
\(28\) 2.16582e11 0.00386920
\(29\) 1.27229e14 1.62857 0.814286 0.580464i \(-0.197129\pi\)
0.814286 + 0.580464i \(0.197129\pi\)
\(30\) −1.63287e14 −1.51461
\(31\) 2.32290e12 0.0157795 0.00788977 0.999969i \(-0.497489\pi\)
0.00788977 + 0.999969i \(0.497489\pi\)
\(32\) −4.92835e13 −0.247615
\(33\) 2.57485e14 0.965760
\(34\) −4.31927e14 −1.22000
\(35\) 1.43781e12 0.00308358
\(36\) 3.90961e14 0.641592
\(37\) −7.19096e14 −0.909642 −0.454821 0.890583i \(-0.650297\pi\)
−0.454821 + 0.890583i \(0.650297\pi\)
\(38\) −7.68535e14 −0.754605
\(39\) −2.64557e14 −0.202958
\(40\) 4.02224e15 2.42604
\(41\) 1.72545e15 0.823105 0.411553 0.911386i \(-0.364987\pi\)
0.411553 + 0.911386i \(0.364987\pi\)
\(42\) −5.23112e12 −0.00198484
\(43\) 2.48385e15 0.753660 0.376830 0.926282i \(-0.377014\pi\)
0.376830 + 0.926282i \(0.377014\pi\)
\(44\) −1.32011e16 −3.21966
\(45\) 2.59545e15 0.511321
\(46\) −3.28655e15 −0.525462
\(47\) 1.39689e15 0.182067 0.0910336 0.995848i \(-0.470983\pi\)
0.0910336 + 0.995848i \(0.470983\pi\)
\(48\) −4.22007e15 −0.450327
\(49\) −1.13988e16 −0.999996
\(50\) 3.19574e16 2.31396
\(51\) 6.86546e15 0.411863
\(52\) 1.35637e16 0.676623
\(53\) 1.05483e16 0.439099 0.219550 0.975601i \(-0.429541\pi\)
0.219550 + 0.975601i \(0.429541\pi\)
\(54\) −9.44290e15 −0.329128
\(55\) −8.76375e16 −2.56593
\(56\) 1.28858e14 0.00317925
\(57\) 1.22159e16 0.254749
\(58\) 1.57550e17 2.78518
\(59\) −2.80846e16 −0.422060 −0.211030 0.977480i \(-0.567682\pi\)
−0.211030 + 0.977480i \(0.567682\pi\)
\(60\) −1.33067e17 −1.70465
\(61\) −3.60392e16 −0.394586 −0.197293 0.980345i \(-0.563215\pi\)
−0.197293 + 0.980345i \(0.563215\pi\)
\(62\) 2.87648e15 0.0269861
\(63\) 8.31485e13 0.000670069 0
\(64\) −1.73437e17 −1.20346
\(65\) 9.00446e16 0.539238
\(66\) 3.18848e17 1.65164
\(67\) 2.91862e17 1.31059 0.655296 0.755373i \(-0.272544\pi\)
0.655296 + 0.755373i \(0.272544\pi\)
\(68\) −3.51989e17 −1.37308
\(69\) 5.22397e16 0.177392
\(70\) 1.78046e15 0.00527353
\(71\) 3.20038e17 0.828414 0.414207 0.910183i \(-0.364059\pi\)
0.414207 + 0.910183i \(0.364059\pi\)
\(72\) 2.32606e17 0.527184
\(73\) −3.15819e17 −0.627873 −0.313936 0.949444i \(-0.601648\pi\)
−0.313936 + 0.949444i \(0.601648\pi\)
\(74\) −8.90467e17 −1.55567
\(75\) −5.07963e17 −0.781178
\(76\) −6.26301e17 −0.849286
\(77\) −2.80758e15 −0.00336256
\(78\) −3.27605e17 −0.347098
\(79\) 1.83029e18 1.71815 0.859077 0.511846i \(-0.171038\pi\)
0.859077 + 0.511846i \(0.171038\pi\)
\(80\) 1.43634e18 1.19647
\(81\) 1.50095e17 0.111111
\(82\) 2.13665e18 1.40767
\(83\) −1.64758e18 −0.967397 −0.483698 0.875235i \(-0.660707\pi\)
−0.483698 + 0.875235i \(0.660707\pi\)
\(84\) −4.26298e15 −0.00223388
\(85\) −2.33673e18 −1.09428
\(86\) 3.07579e18 1.28891
\(87\) −2.50426e18 −0.940256
\(88\) −7.85414e18 −2.64553
\(89\) 5.75855e18 1.74224 0.871120 0.491069i \(-0.163394\pi\)
0.871120 + 0.491069i \(0.163394\pi\)
\(90\) 3.21398e18 0.874460
\(91\) 2.88469e15 0.000706654 0
\(92\) −2.67830e18 −0.591392
\(93\) −4.57216e16 −0.00911032
\(94\) 1.72979e18 0.311371
\(95\) −4.15779e18 −0.676843
\(96\) 9.70047e17 0.142960
\(97\) 6.23309e18 0.832477 0.416238 0.909256i \(-0.363348\pi\)
0.416238 + 0.909256i \(0.363348\pi\)
\(98\) −1.41154e19 −1.71019
\(99\) −5.06808e18 −0.557582
\(100\) 2.60430e19 2.60430
\(101\) 8.80411e17 0.0801000 0.0400500 0.999198i \(-0.487248\pi\)
0.0400500 + 0.999198i \(0.487248\pi\)
\(102\) 8.50161e18 0.704368
\(103\) −1.96998e19 −1.48767 −0.743837 0.668361i \(-0.766996\pi\)
−0.743837 + 0.668361i \(0.766996\pi\)
\(104\) 8.06987e18 0.555968
\(105\) −2.83004e16 −0.00178031
\(106\) 1.30622e19 0.750946
\(107\) 2.58893e19 1.36136 0.680682 0.732579i \(-0.261684\pi\)
0.680682 + 0.732579i \(0.261684\pi\)
\(108\) −7.69528e18 −0.370424
\(109\) −3.42533e19 −1.51060 −0.755302 0.655377i \(-0.772510\pi\)
−0.755302 + 0.655377i \(0.772510\pi\)
\(110\) −1.08523e20 −4.38825
\(111\) 1.41540e19 0.525182
\(112\) 4.60151e16 0.00156794
\(113\) 5.49844e19 1.72185 0.860923 0.508735i \(-0.169887\pi\)
0.860923 + 0.508735i \(0.169887\pi\)
\(114\) 1.51271e19 0.435671
\(115\) −1.77803e19 −0.471313
\(116\) 1.28392e20 3.13464
\(117\) 5.20727e18 0.117178
\(118\) −3.47776e19 −0.721806
\(119\) −7.48601e16 −0.00143402
\(120\) −7.91697e19 −1.40068
\(121\) 1.09969e20 1.79808
\(122\) −4.46279e19 −0.674821
\(123\) −3.39620e19 −0.475220
\(124\) 2.34412e18 0.0303721
\(125\) 4.51110e19 0.541549
\(126\) 1.02964e17 0.00114595
\(127\) −8.76640e19 −0.905078 −0.452539 0.891745i \(-0.649482\pi\)
−0.452539 + 0.891745i \(0.649482\pi\)
\(128\) −1.88931e20 −1.81054
\(129\) −4.88897e19 −0.435126
\(130\) 1.11504e20 0.922204
\(131\) −1.23083e20 −0.946497 −0.473249 0.880929i \(-0.656919\pi\)
−0.473249 + 0.880929i \(0.656919\pi\)
\(132\) 2.59838e20 1.85887
\(133\) −1.33200e17 −0.000886980 0
\(134\) 3.61418e20 2.24137
\(135\) −5.10862e19 −0.295211
\(136\) −2.09419e20 −1.12823
\(137\) −1.10729e20 −0.556435 −0.278218 0.960518i \(-0.589744\pi\)
−0.278218 + 0.960518i \(0.589744\pi\)
\(138\) 6.46892e19 0.303376
\(139\) −2.36141e20 −1.03402 −0.517012 0.855978i \(-0.672956\pi\)
−0.517012 + 0.855978i \(0.672956\pi\)
\(140\) 1.45095e18 0.00593520
\(141\) −2.74949e19 −0.105117
\(142\) 3.96308e20 1.41675
\(143\) −1.75828e20 −0.588025
\(144\) 8.30637e19 0.259997
\(145\) 8.52349e20 2.49817
\(146\) −3.91084e20 −1.07379
\(147\) 2.24364e20 0.577348
\(148\) −7.25666e20 −1.75086
\(149\) 2.05810e20 0.465796 0.232898 0.972501i \(-0.425179\pi\)
0.232898 + 0.972501i \(0.425179\pi\)
\(150\) −6.29018e20 −1.33597
\(151\) −2.24449e20 −0.447546 −0.223773 0.974641i \(-0.571837\pi\)
−0.223773 + 0.974641i \(0.571837\pi\)
\(152\) −3.72624e20 −0.697842
\(153\) −1.35133e20 −0.237789
\(154\) −3.47667e18 −0.00575065
\(155\) 1.55618e19 0.0242052
\(156\) −2.66974e20 −0.390648
\(157\) 3.13212e20 0.431312 0.215656 0.976469i \(-0.430811\pi\)
0.215656 + 0.976469i \(0.430811\pi\)
\(158\) 2.26647e21 2.93838
\(159\) −2.07623e20 −0.253514
\(160\) −3.30165e20 −0.379832
\(161\) −5.69615e17 −0.000617640 0
\(162\) 1.85865e20 0.190022
\(163\) 2.00591e20 0.193432 0.0967160 0.995312i \(-0.469166\pi\)
0.0967160 + 0.995312i \(0.469166\pi\)
\(164\) 1.74122e21 1.58429
\(165\) 1.72497e21 1.48144
\(166\) −2.04022e21 −1.65444
\(167\) −2.56060e21 −1.96126 −0.980632 0.195860i \(-0.937250\pi\)
−0.980632 + 0.195860i \(0.937250\pi\)
\(168\) −2.53631e18 −0.00183554
\(169\) −1.28126e21 −0.876425
\(170\) −2.89361e21 −1.87144
\(171\) −2.40445e20 −0.147079
\(172\) 2.50655e21 1.45063
\(173\) 2.86308e21 1.56818 0.784089 0.620649i \(-0.213131\pi\)
0.784089 + 0.620649i \(0.213131\pi\)
\(174\) −3.10106e21 −1.60802
\(175\) 5.53876e18 0.00271989
\(176\) −2.80472e21 −1.30472
\(177\) 5.52789e20 0.243676
\(178\) 7.13090e21 2.97958
\(179\) 3.79065e20 0.150179 0.0750896 0.997177i \(-0.476076\pi\)
0.0750896 + 0.997177i \(0.476076\pi\)
\(180\) 2.61916e21 0.984178
\(181\) −3.15578e21 −1.12502 −0.562510 0.826790i \(-0.690164\pi\)
−0.562510 + 0.826790i \(0.690164\pi\)
\(182\) 3.57216e18 0.00120852
\(183\) 7.09360e20 0.227814
\(184\) −1.59348e21 −0.485935
\(185\) −4.81744e21 −1.39536
\(186\) −5.66178e19 −0.0155805
\(187\) 4.56288e21 1.19328
\(188\) 1.40965e21 0.350439
\(189\) −1.63661e18 −0.000386864 0
\(190\) −5.14865e21 −1.15754
\(191\) 2.79667e21 0.598170 0.299085 0.954226i \(-0.403319\pi\)
0.299085 + 0.954226i \(0.403319\pi\)
\(192\) 3.41376e21 0.694818
\(193\) −7.34293e20 −0.142257 −0.0711287 0.997467i \(-0.522660\pi\)
−0.0711287 + 0.997467i \(0.522660\pi\)
\(194\) 7.71854e21 1.42370
\(195\) −1.77235e21 −0.311329
\(196\) −1.15030e22 −1.92477
\(197\) −5.32518e21 −0.848995 −0.424498 0.905429i \(-0.639549\pi\)
−0.424498 + 0.905429i \(0.639549\pi\)
\(198\) −6.27588e21 −0.953576
\(199\) −2.54553e21 −0.368701 −0.184350 0.982861i \(-0.559018\pi\)
−0.184350 + 0.982861i \(0.559018\pi\)
\(200\) 1.54945e22 2.13990
\(201\) −5.74472e21 −0.756670
\(202\) 1.09023e21 0.136987
\(203\) 2.73061e19 0.00327376
\(204\) 6.92820e21 0.792745
\(205\) 1.15593e22 1.26261
\(206\) −2.43945e22 −2.54422
\(207\) −1.02823e21 −0.102417
\(208\) 2.88175e21 0.274192
\(209\) 8.11882e21 0.738080
\(210\) −3.50448e19 −0.00304467
\(211\) 7.77610e21 0.645771 0.322886 0.946438i \(-0.395347\pi\)
0.322886 + 0.946438i \(0.395347\pi\)
\(212\) 1.06447e22 0.845168
\(213\) −6.29931e21 −0.478285
\(214\) 3.20591e22 2.32820
\(215\) 1.66401e22 1.15609
\(216\) −4.57838e21 −0.304370
\(217\) 4.98543e17 3.17201e−5 0
\(218\) −4.24164e22 −2.58343
\(219\) 6.21627e21 0.362503
\(220\) −8.84383e22 −4.93884
\(221\) −4.68820e21 −0.250772
\(222\) 1.75271e22 0.898165
\(223\) −1.89249e22 −0.929259 −0.464629 0.885505i \(-0.653812\pi\)
−0.464629 + 0.885505i \(0.653812\pi\)
\(224\) −1.05773e19 −0.000497757 0
\(225\) 9.99823e21 0.451013
\(226\) 6.80881e22 2.94470
\(227\) 2.42145e22 1.00422 0.502111 0.864803i \(-0.332557\pi\)
0.502111 + 0.864803i \(0.332557\pi\)
\(228\) 1.23275e22 0.490335
\(229\) −3.29966e22 −1.25902 −0.629510 0.776992i \(-0.716744\pi\)
−0.629510 + 0.776992i \(0.716744\pi\)
\(230\) −2.20176e22 −0.806039
\(231\) 5.52616e19 0.00194138
\(232\) 7.63882e22 2.57567
\(233\) −8.99479e21 −0.291145 −0.145573 0.989348i \(-0.546502\pi\)
−0.145573 + 0.989348i \(0.546502\pi\)
\(234\) 6.44825e21 0.200397
\(235\) 9.35817e21 0.279284
\(236\) −2.83412e22 −0.812371
\(237\) −3.60256e22 −0.991977
\(238\) −9.27005e19 −0.00245245
\(239\) 1.22851e22 0.312319 0.156159 0.987732i \(-0.450089\pi\)
0.156159 + 0.987732i \(0.450089\pi\)
\(240\) −2.82716e22 −0.690785
\(241\) −4.48313e22 −1.05298 −0.526489 0.850182i \(-0.676492\pi\)
−0.526489 + 0.850182i \(0.676492\pi\)
\(242\) 1.36176e23 3.07507
\(243\) −2.95431e21 −0.0641500
\(244\) −3.63685e22 −0.759491
\(245\) −7.63643e22 −1.53396
\(246\) −4.20557e22 −0.812720
\(247\) −8.34181e21 −0.155110
\(248\) 1.39466e21 0.0249562
\(249\) 3.24293e22 0.558527
\(250\) 5.58617e22 0.926155
\(251\) 3.85123e22 0.614750 0.307375 0.951588i \(-0.400549\pi\)
0.307375 + 0.951588i \(0.400549\pi\)
\(252\) 8.39083e19 0.00128973
\(253\) 3.47192e22 0.513955
\(254\) −1.08556e23 −1.54786
\(255\) 4.59938e22 0.631783
\(256\) −1.43025e23 −1.89292
\(257\) 9.07321e22 1.15717 0.578584 0.815623i \(-0.303606\pi\)
0.578584 + 0.815623i \(0.303606\pi\)
\(258\) −6.05408e22 −0.744151
\(259\) −1.54333e20 −0.00182857
\(260\) 9.08674e22 1.03791
\(261\) 4.92913e22 0.542857
\(262\) −1.52415e23 −1.61870
\(263\) −2.56086e21 −0.0262305 −0.0131152 0.999914i \(-0.504175\pi\)
−0.0131152 + 0.999914i \(0.504175\pi\)
\(264\) 1.54593e23 1.52740
\(265\) 7.06664e22 0.673561
\(266\) −1.64944e20 −0.00151691
\(267\) −1.13346e23 −1.00588
\(268\) 2.94529e23 2.52260
\(269\) 3.82577e22 0.316281 0.158140 0.987417i \(-0.449450\pi\)
0.158140 + 0.987417i \(0.449450\pi\)
\(270\) −6.32608e22 −0.504869
\(271\) 3.64857e22 0.281135 0.140567 0.990071i \(-0.455107\pi\)
0.140567 + 0.990071i \(0.455107\pi\)
\(272\) −7.47838e22 −0.556420
\(273\) −5.67794e19 −0.000407987 0
\(274\) −1.37117e23 −0.951615
\(275\) −3.37599e23 −2.26329
\(276\) 5.27170e22 0.341440
\(277\) 1.78571e23 1.11752 0.558758 0.829331i \(-0.311278\pi\)
0.558758 + 0.829331i \(0.311278\pi\)
\(278\) −2.92417e23 −1.76839
\(279\) 8.99939e20 0.00525985
\(280\) 8.63257e20 0.00487684
\(281\) −2.28261e23 −1.24659 −0.623293 0.781989i \(-0.714205\pi\)
−0.623293 + 0.781989i \(0.714205\pi\)
\(282\) −3.40474e22 −0.179770
\(283\) 1.87538e23 0.957453 0.478727 0.877964i \(-0.341098\pi\)
0.478727 + 0.877964i \(0.341098\pi\)
\(284\) 3.22962e23 1.59451
\(285\) 8.18377e22 0.390775
\(286\) −2.17731e23 −1.00564
\(287\) 3.70317e20 0.00165461
\(288\) −1.90934e22 −0.0825383
\(289\) −1.17410e23 −0.491106
\(290\) 1.05548e24 4.27236
\(291\) −1.22686e23 −0.480631
\(292\) −3.18705e23 −1.20852
\(293\) 2.82617e23 1.03742 0.518711 0.854950i \(-0.326412\pi\)
0.518711 + 0.854950i \(0.326412\pi\)
\(294\) 2.77833e23 0.987379
\(295\) −1.88147e23 −0.647424
\(296\) −4.31743e23 −1.43865
\(297\) 9.97549e22 0.321920
\(298\) 2.54857e23 0.796604
\(299\) −3.56728e22 −0.108009
\(300\) −5.12604e23 −1.50359
\(301\) 5.33087e20 0.00151501
\(302\) −2.77939e23 −0.765392
\(303\) −1.73291e22 −0.0462457
\(304\) −1.33064e23 −0.344162
\(305\) −2.41438e23 −0.605280
\(306\) −1.67337e23 −0.406667
\(307\) −2.47019e23 −0.581991 −0.290996 0.956724i \(-0.593986\pi\)
−0.290996 + 0.956724i \(0.593986\pi\)
\(308\) −2.83324e21 −0.00647219
\(309\) 3.87751e23 0.858909
\(310\) 1.92704e22 0.0413957
\(311\) 3.27126e23 0.681539 0.340770 0.940147i \(-0.389312\pi\)
0.340770 + 0.940147i \(0.389312\pi\)
\(312\) −1.58839e23 −0.320988
\(313\) 3.73412e23 0.732011 0.366005 0.930613i \(-0.380725\pi\)
0.366005 + 0.930613i \(0.380725\pi\)
\(314\) 3.87855e23 0.737628
\(315\) 5.57037e20 0.00102786
\(316\) 1.84701e24 3.30706
\(317\) −5.83518e22 −0.101389 −0.0506945 0.998714i \(-0.516143\pi\)
−0.0506945 + 0.998714i \(0.516143\pi\)
\(318\) −2.57102e23 −0.433559
\(319\) −1.66436e24 −2.72419
\(320\) −1.16191e24 −1.84606
\(321\) −5.09579e23 −0.785984
\(322\) −7.05363e20 −0.00105629
\(323\) 2.16477e23 0.314766
\(324\) 1.51466e23 0.213864
\(325\) 3.46871e23 0.475638
\(326\) 2.48394e23 0.330807
\(327\) 6.74208e23 0.872148
\(328\) 1.03595e24 1.30178
\(329\) 2.99801e20 0.000365993 0
\(330\) 2.13606e24 2.53355
\(331\) 3.84253e23 0.442845 0.221422 0.975178i \(-0.428930\pi\)
0.221422 + 0.975178i \(0.428930\pi\)
\(332\) −1.66263e24 −1.86202
\(333\) −2.78592e23 −0.303214
\(334\) −3.17084e24 −3.35415
\(335\) 1.95527e24 2.01040
\(336\) −9.05716e20 −0.000905251 0
\(337\) 1.30128e24 1.26440 0.632202 0.774803i \(-0.282151\pi\)
0.632202 + 0.774803i \(0.282151\pi\)
\(338\) −1.58661e24 −1.49886
\(339\) −1.08226e24 −0.994109
\(340\) −2.35808e24 −2.10624
\(341\) −3.03872e22 −0.0263952
\(342\) −2.97746e23 −0.251535
\(343\) −4.89287e21 −0.00402040
\(344\) 1.49130e24 1.19195
\(345\) 3.49970e23 0.272113
\(346\) 3.54539e24 2.68189
\(347\) −8.56826e22 −0.0630613 −0.0315306 0.999503i \(-0.510038\pi\)
−0.0315306 + 0.999503i \(0.510038\pi\)
\(348\) −2.52714e24 −1.80978
\(349\) −1.11294e24 −0.775586 −0.387793 0.921746i \(-0.626763\pi\)
−0.387793 + 0.921746i \(0.626763\pi\)
\(350\) 6.85873e21 0.00465155
\(351\) −1.02495e23 −0.0676526
\(352\) 6.44706e23 0.414197
\(353\) −2.23748e24 −1.39926 −0.699631 0.714505i \(-0.746652\pi\)
−0.699631 + 0.714505i \(0.746652\pi\)
\(354\) 6.84527e23 0.416735
\(355\) 2.14403e24 1.27076
\(356\) 5.81117e24 3.35343
\(357\) 1.47347e21 0.000827930 0
\(358\) 4.69402e23 0.256836
\(359\) 2.02884e24 1.08106 0.540531 0.841324i \(-0.318223\pi\)
0.540531 + 0.841324i \(0.318223\pi\)
\(360\) 1.55830e24 0.808680
\(361\) −1.59324e24 −0.805309
\(362\) −3.90785e24 −1.92401
\(363\) −2.16452e24 −1.03812
\(364\) 2.91105e21 0.00136015
\(365\) −2.11577e24 −0.963133
\(366\) 8.78412e23 0.389608
\(367\) −2.03395e24 −0.879049 −0.439525 0.898230i \(-0.644853\pi\)
−0.439525 + 0.898230i \(0.644853\pi\)
\(368\) −5.69034e23 −0.239654
\(369\) 6.68474e23 0.274368
\(370\) −5.96551e24 −2.38633
\(371\) 2.26389e21 0.000882680 0
\(372\) −4.61394e22 −0.0175353
\(373\) −7.78333e23 −0.288358 −0.144179 0.989552i \(-0.546054\pi\)
−0.144179 + 0.989552i \(0.546054\pi\)
\(374\) 5.65029e24 2.04075
\(375\) −8.87920e23 −0.312663
\(376\) 8.38687e23 0.287949
\(377\) 1.71008e24 0.572497
\(378\) −2.02664e21 −0.000661615 0
\(379\) 3.15677e24 1.00501 0.502505 0.864575i \(-0.332412\pi\)
0.502505 + 0.864575i \(0.332412\pi\)
\(380\) −4.19578e24 −1.30277
\(381\) 1.72549e24 0.522547
\(382\) 3.46316e24 1.02299
\(383\) −3.09305e24 −0.891249 −0.445625 0.895220i \(-0.647018\pi\)
−0.445625 + 0.895220i \(0.647018\pi\)
\(384\) 3.71873e24 1.04532
\(385\) −1.88088e22 −0.00515804
\(386\) −9.09286e23 −0.243288
\(387\) 9.62295e23 0.251220
\(388\) 6.29005e24 1.60233
\(389\) −4.58505e24 −1.13978 −0.569892 0.821719i \(-0.693015\pi\)
−0.569892 + 0.821719i \(0.693015\pi\)
\(390\) −2.19473e24 −0.532435
\(391\) 9.25738e23 0.219184
\(392\) −6.84383e24 −1.58155
\(393\) 2.42263e24 0.546460
\(394\) −6.59426e24 −1.45195
\(395\) 1.22617e25 2.63558
\(396\) −5.11439e24 −1.07322
\(397\) 5.62169e24 1.15175 0.575873 0.817539i \(-0.304662\pi\)
0.575873 + 0.817539i \(0.304662\pi\)
\(398\) −3.15217e24 −0.630551
\(399\) 2.62178e21 0.000512098 0
\(400\) 5.53311e24 1.05536
\(401\) −8.34521e24 −1.55441 −0.777206 0.629247i \(-0.783364\pi\)
−0.777206 + 0.629247i \(0.783364\pi\)
\(402\) −7.11378e24 −1.29406
\(403\) 3.12218e22 0.00554703
\(404\) 8.88456e23 0.154175
\(405\) 1.00553e24 0.170440
\(406\) 3.38136e22 0.00559878
\(407\) 9.40691e24 1.52160
\(408\) 4.12200e24 0.651383
\(409\) −1.09120e25 −1.68474 −0.842370 0.538900i \(-0.818840\pi\)
−0.842370 + 0.538900i \(0.818840\pi\)
\(410\) 1.43141e25 2.15932
\(411\) 2.17947e24 0.321258
\(412\) −1.98798e25 −2.86344
\(413\) −6.02754e21 −0.000848427 0
\(414\) −1.27328e24 −0.175154
\(415\) −1.10376e25 −1.48395
\(416\) −6.62414e23 −0.0870448
\(417\) 4.64796e24 0.596994
\(418\) 1.00537e25 1.26226
\(419\) −1.35175e25 −1.65906 −0.829529 0.558463i \(-0.811391\pi\)
−0.829529 + 0.558463i \(0.811391\pi\)
\(420\) −2.85590e22 −0.00342669
\(421\) 9.91185e24 1.16272 0.581360 0.813647i \(-0.302521\pi\)
0.581360 + 0.813647i \(0.302521\pi\)
\(422\) 9.62927e24 1.10440
\(423\) 5.41183e23 0.0606891
\(424\) 6.33318e24 0.694458
\(425\) −9.00159e24 −0.965215
\(426\) −7.80053e24 −0.817962
\(427\) −7.73477e21 −0.000793200 0
\(428\) 2.61259e25 2.62032
\(429\) 3.46083e24 0.339497
\(430\) 2.06057e25 1.97714
\(431\) 1.91272e24 0.179522 0.0897610 0.995963i \(-0.471390\pi\)
0.0897610 + 0.995963i \(0.471390\pi\)
\(432\) −1.63494e24 −0.150109
\(433\) 5.77174e24 0.518408 0.259204 0.965823i \(-0.416540\pi\)
0.259204 + 0.965823i \(0.416540\pi\)
\(434\) 6.17353e20 5.42477e−5 0
\(435\) −1.67768e25 −1.44232
\(436\) −3.45663e25 −2.90758
\(437\) 1.64718e24 0.135571
\(438\) 7.69771e24 0.619951
\(439\) 8.11473e21 0.000639530 0 0.000319765 1.00000i \(-0.499898\pi\)
0.000319765 1.00000i \(0.499898\pi\)
\(440\) −5.26173e25 −4.05815
\(441\) −4.41615e24 −0.333332
\(442\) −5.80548e24 −0.428871
\(443\) 2.51728e24 0.182010 0.0910051 0.995850i \(-0.470992\pi\)
0.0910051 + 0.995850i \(0.470992\pi\)
\(444\) 1.42833e25 1.01086
\(445\) 3.85783e25 2.67253
\(446\) −2.34350e25 −1.58922
\(447\) −4.05095e24 −0.268928
\(448\) −3.72232e22 −0.00241920
\(449\) −1.07451e25 −0.683706 −0.341853 0.939753i \(-0.611055\pi\)
−0.341853 + 0.939753i \(0.611055\pi\)
\(450\) 1.23810e25 0.771322
\(451\) −2.25716e25 −1.37685
\(452\) 5.54869e25 3.31417
\(453\) 4.41784e24 0.258391
\(454\) 2.99852e25 1.71742
\(455\) 1.93254e22 0.00108398
\(456\) 7.33436e24 0.402899
\(457\) 8.09140e24 0.435331 0.217665 0.976023i \(-0.430156\pi\)
0.217665 + 0.976023i \(0.430156\pi\)
\(458\) −4.08603e25 −2.15317
\(459\) 2.65982e24 0.137288
\(460\) −1.79428e25 −0.907173
\(461\) 1.48075e25 0.733368 0.366684 0.930346i \(-0.380493\pi\)
0.366684 + 0.930346i \(0.380493\pi\)
\(462\) 6.84313e22 0.00332014
\(463\) −2.65292e25 −1.26097 −0.630485 0.776202i \(-0.717144\pi\)
−0.630485 + 0.776202i \(0.717144\pi\)
\(464\) 2.72783e25 1.27027
\(465\) −3.06303e23 −0.0139749
\(466\) −1.11384e25 −0.497916
\(467\) 3.57468e25 1.56576 0.782882 0.622170i \(-0.213749\pi\)
0.782882 + 0.622170i \(0.213749\pi\)
\(468\) 5.25486e24 0.225541
\(469\) 6.26397e22 0.00263456
\(470\) 1.15884e25 0.477631
\(471\) −6.16494e24 −0.249018
\(472\) −1.68619e25 −0.667510
\(473\) −3.24927e25 −1.26068
\(474\) −4.46110e25 −1.69648
\(475\) −1.60167e25 −0.597013
\(476\) −7.55442e22 −0.00276016
\(477\) 4.08664e24 0.146366
\(478\) 1.52128e25 0.534127
\(479\) −2.88220e24 −0.0992056 −0.0496028 0.998769i \(-0.515796\pi\)
−0.0496028 + 0.998769i \(0.515796\pi\)
\(480\) 6.49864e24 0.219296
\(481\) −9.66528e24 −0.319769
\(482\) −5.55152e25 −1.80080
\(483\) 1.12117e22 0.000356595 0
\(484\) 1.10974e26 3.46090
\(485\) 4.17574e25 1.27699
\(486\) −3.65837e24 −0.109709
\(487\) 6.03233e25 1.77403 0.887014 0.461743i \(-0.152776\pi\)
0.887014 + 0.461743i \(0.152776\pi\)
\(488\) −2.16378e25 −0.624059
\(489\) −3.94822e24 −0.111678
\(490\) −9.45632e25 −2.62337
\(491\) 4.20599e25 1.14444 0.572222 0.820099i \(-0.306082\pi\)
0.572222 + 0.820099i \(0.306082\pi\)
\(492\) −3.42723e25 −0.914693
\(493\) −4.43779e25 −1.16177
\(494\) −1.03298e25 −0.265269
\(495\) −3.39526e25 −0.855310
\(496\) 4.98034e23 0.0123079
\(497\) 6.86869e22 0.00166528
\(498\) 4.01577e25 0.955191
\(499\) −3.96358e25 −0.924979 −0.462490 0.886625i \(-0.653044\pi\)
−0.462490 + 0.886625i \(0.653044\pi\)
\(500\) 4.55232e25 1.04236
\(501\) 5.04004e25 1.13234
\(502\) 4.76904e25 1.05134
\(503\) 5.09940e25 1.10312 0.551561 0.834135i \(-0.314032\pi\)
0.551561 + 0.834135i \(0.314032\pi\)
\(504\) 4.99221e22 0.00105975
\(505\) 5.89814e24 0.122870
\(506\) 4.29933e25 0.878965
\(507\) 2.52191e25 0.506004
\(508\) −8.84651e25 −1.74207
\(509\) −4.45056e25 −0.860193 −0.430096 0.902783i \(-0.641520\pi\)
−0.430096 + 0.902783i \(0.641520\pi\)
\(510\) 5.69549e25 1.08047
\(511\) −6.77814e22 −0.00126215
\(512\) −7.80561e25 −1.42673
\(513\) 4.73267e24 0.0849164
\(514\) 1.12355e26 1.97899
\(515\) −1.31975e26 −2.28204
\(516\) −4.93364e25 −0.837520
\(517\) −1.82735e25 −0.304552
\(518\) −1.91113e23 −0.00312721
\(519\) −5.63540e25 −0.905388
\(520\) 5.40625e25 0.852833
\(521\) −1.65819e24 −0.0256847 −0.0128424 0.999918i \(-0.504088\pi\)
−0.0128424 + 0.999918i \(0.504088\pi\)
\(522\) 6.10382e25 0.928393
\(523\) −4.69898e25 −0.701839 −0.350920 0.936406i \(-0.614131\pi\)
−0.350920 + 0.936406i \(0.614131\pi\)
\(524\) −1.24207e26 −1.82180
\(525\) −1.09019e23 −0.00157033
\(526\) −3.17115e24 −0.0448593
\(527\) −8.10231e23 −0.0112566
\(528\) 5.52053e25 0.753283
\(529\) −6.75715e25 −0.905596
\(530\) 8.75073e25 1.15192
\(531\) −1.08805e25 −0.140687
\(532\) −1.34417e23 −0.00170724
\(533\) 2.31916e25 0.289349
\(534\) −1.40358e26 −1.72026
\(535\) 1.73440e26 2.08828
\(536\) 1.75233e26 2.07277
\(537\) −7.46114e24 −0.0867060
\(538\) 4.73752e25 0.540903
\(539\) 1.49115e26 1.67274
\(540\) −5.15530e25 −0.568216
\(541\) 6.20242e25 0.671718 0.335859 0.941912i \(-0.390973\pi\)
0.335859 + 0.941912i \(0.390973\pi\)
\(542\) 4.51808e25 0.480796
\(543\) 6.21152e25 0.649531
\(544\) 1.71902e25 0.176641
\(545\) −2.29473e26 −2.31721
\(546\) −7.03109e22 −0.000697738 0
\(547\) 7.76624e25 0.757410 0.378705 0.925517i \(-0.376370\pi\)
0.378705 + 0.925517i \(0.376370\pi\)
\(548\) −1.11740e26 −1.07101
\(549\) −1.39623e25 −0.131529
\(550\) −4.18054e26 −3.87068
\(551\) −7.89624e25 −0.718588
\(552\) 3.13646e25 0.280555
\(553\) 3.92818e23 0.00345384
\(554\) 2.21128e26 1.91117
\(555\) 9.48217e25 0.805609
\(556\) −2.38299e26 −1.99027
\(557\) 3.67420e25 0.301675 0.150837 0.988559i \(-0.451803\pi\)
0.150837 + 0.988559i \(0.451803\pi\)
\(558\) 1.11441e24 0.00899538
\(559\) 3.33852e25 0.264936
\(560\) 3.08269e23 0.00240516
\(561\) −8.98112e25 −0.688943
\(562\) −2.82659e26 −2.13191
\(563\) 1.24531e26 0.923526 0.461763 0.887003i \(-0.347217\pi\)
0.461763 + 0.887003i \(0.347217\pi\)
\(564\) −2.77462e25 −0.202326
\(565\) 3.68357e26 2.64125
\(566\) 2.32231e26 1.63743
\(567\) 3.22134e22 0.000223356 0
\(568\) 1.92150e26 1.31018
\(569\) −5.63513e25 −0.377866 −0.188933 0.981990i \(-0.560503\pi\)
−0.188933 + 0.981990i \(0.560503\pi\)
\(570\) 1.01341e26 0.668303
\(571\) −1.23813e26 −0.803012 −0.401506 0.915856i \(-0.631513\pi\)
−0.401506 + 0.915856i \(0.631513\pi\)
\(572\) −1.77435e26 −1.13182
\(573\) −5.50469e25 −0.345354
\(574\) 4.58570e23 0.00282971
\(575\) −6.84936e25 −0.415724
\(576\) −6.71930e25 −0.401153
\(577\) −3.01722e26 −1.77189 −0.885945 0.463790i \(-0.846489\pi\)
−0.885945 + 0.463790i \(0.846489\pi\)
\(578\) −1.45390e26 −0.839888
\(579\) 1.44531e25 0.0821323
\(580\) 8.60137e26 4.80841
\(581\) −3.53605e23 −0.00194467
\(582\) −1.51924e26 −0.821974
\(583\) −1.37989e26 −0.734501
\(584\) −1.89617e26 −0.993014
\(585\) 3.48851e25 0.179746
\(586\) 3.49968e26 1.77420
\(587\) −6.00743e25 −0.299659 −0.149829 0.988712i \(-0.547872\pi\)
−0.149829 + 0.988712i \(0.547872\pi\)
\(588\) 2.26414e26 1.11127
\(589\) −1.44166e24 −0.00696254
\(590\) −2.32986e26 −1.10722
\(591\) 1.04816e26 0.490168
\(592\) −1.54176e26 −0.709511
\(593\) −9.94196e25 −0.450248 −0.225124 0.974330i \(-0.572279\pi\)
−0.225124 + 0.974330i \(0.572279\pi\)
\(594\) 1.23528e26 0.550547
\(595\) −5.01511e23 −0.00219973
\(596\) 2.07690e26 0.896554
\(597\) 5.01037e25 0.212869
\(598\) −4.41742e25 −0.184717
\(599\) −1.95677e26 −0.805350 −0.402675 0.915343i \(-0.631920\pi\)
−0.402675 + 0.915343i \(0.631920\pi\)
\(600\) −3.04979e26 −1.23547
\(601\) −4.73445e26 −1.88782 −0.943912 0.330197i \(-0.892885\pi\)
−0.943912 + 0.330197i \(0.892885\pi\)
\(602\) 6.60129e23 0.00259097
\(603\) 1.13073e26 0.436864
\(604\) −2.26500e26 −0.861426
\(605\) 7.36715e26 2.75818
\(606\) −2.14589e25 −0.0790894
\(607\) −1.22707e26 −0.445222 −0.222611 0.974907i \(-0.571458\pi\)
−0.222611 + 0.974907i \(0.571458\pi\)
\(608\) 3.05868e25 0.109257
\(609\) −5.37466e23 −0.00189011
\(610\) −2.98976e26 −1.03515
\(611\) 1.87754e25 0.0640026
\(612\) −1.36368e26 −0.457692
\(613\) 3.20354e26 1.05866 0.529329 0.848417i \(-0.322444\pi\)
0.529329 + 0.848417i \(0.322444\pi\)
\(614\) −3.05888e26 −0.995320
\(615\) −2.27522e26 −0.728969
\(616\) −1.68566e24 −0.00531807
\(617\) 9.79781e25 0.304383 0.152191 0.988351i \(-0.451367\pi\)
0.152191 + 0.988351i \(0.451367\pi\)
\(618\) 4.80158e26 1.46890
\(619\) −1.22713e25 −0.0369682 −0.0184841 0.999829i \(-0.505884\pi\)
−0.0184841 + 0.999829i \(0.505884\pi\)
\(620\) 1.57040e25 0.0465896
\(621\) 2.02387e25 0.0591307
\(622\) 4.05085e26 1.16557
\(623\) 1.23591e24 0.00350226
\(624\) −5.67215e25 −0.158305
\(625\) −1.90020e26 −0.522324
\(626\) 4.62402e26 1.25188
\(627\) −1.59803e26 −0.426131
\(628\) 3.16074e26 0.830179
\(629\) 2.50822e26 0.648909
\(630\) 6.89788e23 0.00175784
\(631\) 4.30803e26 1.08143 0.540716 0.841205i \(-0.318153\pi\)
0.540716 + 0.841205i \(0.318153\pi\)
\(632\) 1.09890e27 2.71735
\(633\) −1.53057e26 −0.372836
\(634\) −7.22579e25 −0.173395
\(635\) −5.87288e26 −1.38836
\(636\) −2.09520e26 −0.487958
\(637\) −1.53211e26 −0.351532
\(638\) −2.06101e27 −4.65890
\(639\) 1.23989e26 0.276138
\(640\) −1.26571e27 −2.77730
\(641\) 5.54923e26 1.19972 0.599862 0.800104i \(-0.295222\pi\)
0.599862 + 0.800104i \(0.295222\pi\)
\(642\) −6.31020e26 −1.34419
\(643\) 6.72911e26 1.41239 0.706193 0.708019i \(-0.250411\pi\)
0.706193 + 0.708019i \(0.250411\pi\)
\(644\) −5.74820e23 −0.00118882
\(645\) −3.27527e26 −0.667466
\(646\) 2.68066e26 0.538311
\(647\) −3.57860e26 −0.708146 −0.354073 0.935218i \(-0.615204\pi\)
−0.354073 + 0.935218i \(0.615204\pi\)
\(648\) 9.01163e25 0.175728
\(649\) 3.67391e26 0.705999
\(650\) 4.29536e26 0.813435
\(651\) −9.81282e21 −1.83136e−5 0
\(652\) 2.02423e26 0.372313
\(653\) 5.10990e26 0.926269 0.463135 0.886288i \(-0.346725\pi\)
0.463135 + 0.886288i \(0.346725\pi\)
\(654\) 8.34882e26 1.49155
\(655\) −8.24568e26 −1.45189
\(656\) 3.69940e26 0.642014
\(657\) −1.22355e26 −0.209291
\(658\) 3.71249e23 0.000625920 0
\(659\) −6.67000e26 −1.10844 −0.554222 0.832369i \(-0.686984\pi\)
−0.554222 + 0.832369i \(0.686984\pi\)
\(660\) 1.74073e27 2.85144
\(661\) 2.27178e26 0.366819 0.183410 0.983037i \(-0.441287\pi\)
0.183410 + 0.983037i \(0.441287\pi\)
\(662\) 4.75827e26 0.757352
\(663\) 9.22779e25 0.144784
\(664\) −9.89202e26 −1.52999
\(665\) −8.92348e23 −0.00136059
\(666\) −3.44985e26 −0.518556
\(667\) −3.37674e26 −0.500382
\(668\) −2.58400e27 −3.77500
\(669\) 3.72498e26 0.536508
\(670\) 2.42125e27 3.43818
\(671\) 4.71450e26 0.660043
\(672\) 2.08192e23 0.000287380 0
\(673\) 2.72017e26 0.370214 0.185107 0.982718i \(-0.440737\pi\)
0.185107 + 0.982718i \(0.440737\pi\)
\(674\) 1.61139e27 2.16238
\(675\) −1.96795e26 −0.260393
\(676\) −1.29297e27 −1.68692
\(677\) 4.27674e26 0.550200 0.275100 0.961416i \(-0.411289\pi\)
0.275100 + 0.961416i \(0.411289\pi\)
\(678\) −1.34018e27 −1.70012
\(679\) 1.33775e24 0.00167345
\(680\) −1.40296e27 −1.73066
\(681\) −4.76614e26 −0.579788
\(682\) −3.76290e25 −0.0451409
\(683\) −1.11866e26 −0.132344 −0.0661719 0.997808i \(-0.521079\pi\)
−0.0661719 + 0.997808i \(0.521079\pi\)
\(684\) −2.42642e26 −0.283095
\(685\) −7.41805e26 −0.853551
\(686\) −6.05892e24 −0.00687569
\(687\) 6.49473e26 0.726896
\(688\) 5.32543e26 0.587847
\(689\) 1.41779e26 0.154358
\(690\) 4.33373e26 0.465367
\(691\) 1.82536e27 1.93333 0.966667 0.256037i \(-0.0824171\pi\)
0.966667 + 0.256037i \(0.0824171\pi\)
\(692\) 2.88924e27 3.01839
\(693\) −1.08771e24 −0.00112085
\(694\) −1.06102e26 −0.107847
\(695\) −1.58198e27 −1.58615
\(696\) −1.50355e27 −1.48706
\(697\) −6.01839e26 −0.587177
\(698\) −1.37817e27 −1.32641
\(699\) 1.77045e26 0.168093
\(700\) 5.58937e24 0.00523518
\(701\) −2.03227e27 −1.87785 −0.938924 0.344125i \(-0.888176\pi\)
−0.938924 + 0.344125i \(0.888176\pi\)
\(702\) −1.26921e26 −0.115699
\(703\) 4.46292e26 0.401369
\(704\) 2.26883e27 2.01308
\(705\) −1.84197e26 −0.161245
\(706\) −2.77070e27 −2.39301
\(707\) 1.88955e23 0.000161017 0
\(708\) 5.57840e26 0.469023
\(709\) −1.86039e27 −1.54335 −0.771675 0.636017i \(-0.780581\pi\)
−0.771675 + 0.636017i \(0.780581\pi\)
\(710\) 2.65499e27 2.17324
\(711\) 7.09091e26 0.572718
\(712\) 3.45742e27 2.75544
\(713\) −6.16510e24 −0.00484830
\(714\) 1.82462e24 0.00141592
\(715\) −1.17793e27 −0.902009
\(716\) 3.82529e26 0.289062
\(717\) −2.41807e26 −0.180317
\(718\) 2.51234e27 1.84883
\(719\) 1.92780e27 1.40003 0.700014 0.714129i \(-0.253177\pi\)
0.700014 + 0.714129i \(0.253177\pi\)
\(720\) 5.56469e26 0.398825
\(721\) −4.22798e24 −0.00299053
\(722\) −1.97293e27 −1.37724
\(723\) 8.82414e26 0.607937
\(724\) −3.18462e27 −2.16541
\(725\) 3.28343e27 2.20352
\(726\) −2.68036e27 −1.77539
\(727\) −1.11559e27 −0.729339 −0.364670 0.931137i \(-0.618818\pi\)
−0.364670 + 0.931137i \(0.618818\pi\)
\(728\) 1.73196e24 0.00111761
\(729\) 5.81497e25 0.0370370
\(730\) −2.61999e27 −1.64715
\(731\) −8.66372e26 −0.537637
\(732\) 7.15842e26 0.438492
\(733\) 2.04762e27 1.23812 0.619059 0.785344i \(-0.287514\pi\)
0.619059 + 0.785344i \(0.287514\pi\)
\(734\) −2.51868e27 −1.50335
\(735\) 1.50308e27 0.885630
\(736\) 1.30801e26 0.0760802
\(737\) −3.81802e27 −2.19229
\(738\) 8.27782e26 0.469224
\(739\) 1.50373e27 0.841486 0.420743 0.907180i \(-0.361769\pi\)
0.420743 + 0.907180i \(0.361769\pi\)
\(740\) −4.86146e27 −2.68575
\(741\) 1.64192e26 0.0895527
\(742\) 2.80341e24 0.00150956
\(743\) 1.88836e27 1.00390 0.501950 0.864897i \(-0.332616\pi\)
0.501950 + 0.864897i \(0.332616\pi\)
\(744\) −2.74511e25 −0.0144084
\(745\) 1.37878e27 0.714514
\(746\) −9.63821e26 −0.493149
\(747\) −6.38306e26 −0.322466
\(748\) 4.60457e27 2.29681
\(749\) 5.55638e24 0.00273662
\(750\) −1.09953e27 −0.534716
\(751\) 8.19244e26 0.393400 0.196700 0.980464i \(-0.436978\pi\)
0.196700 + 0.980464i \(0.436978\pi\)
\(752\) 2.99495e26 0.142011
\(753\) −7.58037e26 −0.354926
\(754\) 2.11762e27 0.979082
\(755\) −1.50365e27 −0.686518
\(756\) −1.65157e24 −0.000744628 0
\(757\) 1.94096e27 0.864184 0.432092 0.901829i \(-0.357776\pi\)
0.432092 + 0.901829i \(0.357776\pi\)
\(758\) 3.90907e27 1.71876
\(759\) −6.83378e26 −0.296732
\(760\) −2.49632e27 −1.07046
\(761\) 2.40135e27 1.01695 0.508477 0.861076i \(-0.330209\pi\)
0.508477 + 0.861076i \(0.330209\pi\)
\(762\) 2.13670e27 0.893659
\(763\) −7.35147e24 −0.00303663
\(764\) 2.82223e27 1.15134
\(765\) −9.05296e26 −0.364760
\(766\) −3.83018e27 −1.52421
\(767\) −3.77482e26 −0.148368
\(768\) 2.81516e27 1.09288
\(769\) −3.49979e27 −1.34197 −0.670984 0.741472i \(-0.734128\pi\)
−0.670984 + 0.741472i \(0.734128\pi\)
\(770\) −2.32913e25 −0.00882128
\(771\) −1.78588e27 −0.668091
\(772\) −7.41002e26 −0.273814
\(773\) −3.84483e27 −1.40337 −0.701685 0.712487i \(-0.747569\pi\)
−0.701685 + 0.712487i \(0.747569\pi\)
\(774\) 1.19163e27 0.429636
\(775\) 5.99475e25 0.0213503
\(776\) 3.74233e27 1.31661
\(777\) 3.03774e24 0.00105572
\(778\) −5.67774e27 −1.94926
\(779\) −1.07086e27 −0.363186
\(780\) −1.78854e27 −0.599240
\(781\) −4.18661e27 −1.38573
\(782\) 1.14636e27 0.374848
\(783\) −9.70201e26 −0.313419
\(784\) −2.44394e27 −0.779987
\(785\) 2.09830e27 0.661615
\(786\) 2.99999e27 0.934556
\(787\) 4.22212e27 1.29948 0.649741 0.760155i \(-0.274877\pi\)
0.649741 + 0.760155i \(0.274877\pi\)
\(788\) −5.37384e27 −1.63413
\(789\) 5.04054e25 0.0151442
\(790\) 1.51838e28 4.50737
\(791\) 1.18008e25 0.00346127
\(792\) −3.04286e27 −0.881845
\(793\) −4.84399e26 −0.138710
\(794\) 6.96142e27 1.96971
\(795\) −1.39093e27 −0.388881
\(796\) −2.56879e27 −0.709667
\(797\) −2.10706e27 −0.575205 −0.287602 0.957750i \(-0.592858\pi\)
−0.287602 + 0.957750i \(0.592858\pi\)
\(798\) 3.24659e24 0.000875789 0
\(799\) −4.87237e26 −0.129881
\(800\) −1.27187e27 −0.335033
\(801\) 2.23098e27 0.580747
\(802\) −1.03340e28 −2.65835
\(803\) 4.13142e27 1.05027
\(804\) −5.79722e27 −1.45642
\(805\) −3.81602e24 −0.000947436 0
\(806\) 3.86625e25 0.00948652
\(807\) −7.53027e26 −0.182605
\(808\) 5.28596e26 0.126682
\(809\) 2.31386e27 0.548056 0.274028 0.961722i \(-0.411644\pi\)
0.274028 + 0.961722i \(0.411644\pi\)
\(810\) 1.24516e27 0.291487
\(811\) 2.32427e27 0.537760 0.268880 0.963174i \(-0.413347\pi\)
0.268880 + 0.963174i \(0.413347\pi\)
\(812\) 2.75556e25 0.00630127
\(813\) −7.18148e26 −0.162313
\(814\) 1.16487e28 2.60224
\(815\) 1.34382e27 0.296717
\(816\) 1.47197e27 0.321249
\(817\) −1.54155e27 −0.332544
\(818\) −1.35125e28 −2.88124
\(819\) 1.11759e24 0.000235551 0
\(820\) 1.16649e28 2.43025
\(821\) 6.08231e27 1.25259 0.626295 0.779587i \(-0.284571\pi\)
0.626295 + 0.779587i \(0.284571\pi\)
\(822\) 2.69888e27 0.549415
\(823\) −3.84390e27 −0.773523 −0.386762 0.922180i \(-0.626406\pi\)
−0.386762 + 0.922180i \(0.626406\pi\)
\(824\) −1.18277e28 −2.35283
\(825\) 6.64496e27 1.30671
\(826\) −7.46400e24 −0.00145098
\(827\) 5.47093e27 1.05138 0.525689 0.850677i \(-0.323807\pi\)
0.525689 + 0.850677i \(0.323807\pi\)
\(828\) −1.03763e27 −0.197131
\(829\) 8.53655e26 0.160330 0.0801649 0.996782i \(-0.474455\pi\)
0.0801649 + 0.996782i \(0.474455\pi\)
\(830\) −1.36681e28 −2.53785
\(831\) −3.51482e27 −0.645198
\(832\) −2.33115e27 −0.423056
\(833\) 3.97594e27 0.713365
\(834\) 5.75564e27 1.02098
\(835\) −1.71543e28 −3.00850
\(836\) 8.19301e27 1.42064
\(837\) −1.77135e25 −0.00303677
\(838\) −1.67389e28 −2.83732
\(839\) −8.76504e27 −1.46898 −0.734489 0.678621i \(-0.762578\pi\)
−0.734489 + 0.678621i \(0.762578\pi\)
\(840\) −1.69915e25 −0.00281565
\(841\) 1.00841e28 1.65224
\(842\) 1.22740e28 1.98848
\(843\) 4.49287e27 0.719716
\(844\) 7.84716e27 1.24297
\(845\) −8.58357e27 −1.34440
\(846\) 6.70155e26 0.103790
\(847\) 2.36016e25 0.00361451
\(848\) 2.26158e27 0.342493
\(849\) −3.69131e27 −0.552786
\(850\) −1.11468e28 −1.65071
\(851\) 1.90852e27 0.279489
\(852\) −6.35687e27 −0.920592
\(853\) 4.55754e27 0.652702 0.326351 0.945249i \(-0.394181\pi\)
0.326351 + 0.945249i \(0.394181\pi\)
\(854\) −9.57809e24 −0.00135653
\(855\) −1.61081e27 −0.225614
\(856\) 1.55438e28 2.15307
\(857\) 8.72531e27 1.19526 0.597631 0.801771i \(-0.296109\pi\)
0.597631 + 0.801771i \(0.296109\pi\)
\(858\) 4.28559e27 0.580606
\(859\) −6.82608e27 −0.914610 −0.457305 0.889310i \(-0.651185\pi\)
−0.457305 + 0.889310i \(0.651185\pi\)
\(860\) 1.67921e28 2.22521
\(861\) −7.28895e24 −0.000955290 0
\(862\) 2.36855e27 0.307018
\(863\) 1.70088e27 0.218058 0.109029 0.994039i \(-0.465226\pi\)
0.109029 + 0.994039i \(0.465226\pi\)
\(864\) 3.75816e26 0.0476535
\(865\) 1.91806e28 2.40552
\(866\) 7.14723e27 0.886580
\(867\) 2.31098e27 0.283540
\(868\) 5.03098e23 6.10542e−5 0
\(869\) −2.39431e28 −2.87404
\(870\) −2.07750e28 −2.46665
\(871\) 3.92289e27 0.460716
\(872\) −2.05656e28 −2.38910
\(873\) 2.41483e27 0.277492
\(874\) 2.03973e27 0.231854
\(875\) 9.68176e24 0.00108862
\(876\) 6.27308e27 0.697737
\(877\) −9.69466e27 −1.06669 −0.533343 0.845899i \(-0.679064\pi\)
−0.533343 + 0.845899i \(0.679064\pi\)
\(878\) 1.00486e25 0.00109372
\(879\) −5.56274e27 −0.598955
\(880\) −1.87897e28 −2.00140
\(881\) −2.20901e27 −0.232769 −0.116385 0.993204i \(-0.537131\pi\)
−0.116385 + 0.993204i \(0.537131\pi\)
\(882\) −5.46858e27 −0.570064
\(883\) −5.45945e27 −0.563018 −0.281509 0.959559i \(-0.590835\pi\)
−0.281509 + 0.959559i \(0.590835\pi\)
\(884\) −4.73104e27 −0.482681
\(885\) 3.70330e27 0.373790
\(886\) 3.11718e27 0.311273
\(887\) 1.69664e28 1.67616 0.838081 0.545546i \(-0.183678\pi\)
0.838081 + 0.545546i \(0.183678\pi\)
\(888\) 8.49799e27 0.830602
\(889\) −1.88145e25 −0.00181939
\(890\) 4.77721e28 4.57056
\(891\) −1.96348e27 −0.185861
\(892\) −1.90978e28 −1.78862
\(893\) −8.66950e26 −0.0803351
\(894\) −5.01636e27 −0.459920
\(895\) 2.53947e27 0.230369
\(896\) −4.05485e25 −0.00363956
\(897\) 7.02148e26 0.0623592
\(898\) −1.33058e28 −1.16927
\(899\) 2.95541e26 0.0256981
\(900\) 1.00896e28 0.868100
\(901\) −3.67927e27 −0.313239
\(902\) −2.79508e28 −2.35468
\(903\) −1.04927e25 −0.000874692 0
\(904\) 3.30125e28 2.72319
\(905\) −2.11415e28 −1.72574
\(906\) 5.47068e27 0.441899
\(907\) 1.06963e28 0.854996 0.427498 0.904016i \(-0.359395\pi\)
0.427498 + 0.904016i \(0.359395\pi\)
\(908\) 2.44357e28 1.93290
\(909\) 3.41089e26 0.0267000
\(910\) 2.39310e25 0.00185382
\(911\) −2.36272e28 −1.81129 −0.905645 0.424036i \(-0.860613\pi\)
−0.905645 + 0.424036i \(0.860613\pi\)
\(912\) 2.61910e27 0.198702
\(913\) 2.15530e28 1.61821
\(914\) 1.00197e28 0.744502
\(915\) 4.75222e27 0.349459
\(916\) −3.32982e28 −2.42333
\(917\) −2.64161e25 −0.00190265
\(918\) 3.29370e27 0.234789
\(919\) 4.79736e27 0.338458 0.169229 0.985577i \(-0.445872\pi\)
0.169229 + 0.985577i \(0.445872\pi\)
\(920\) −1.06752e28 −0.745406
\(921\) 4.86208e27 0.336013
\(922\) 1.83363e28 1.25420
\(923\) 4.30160e27 0.291215
\(924\) 5.57666e25 0.00373672
\(925\) −1.85578e28 −1.23078
\(926\) −3.28515e28 −2.15651
\(927\) −7.63210e27 −0.495891
\(928\) −6.27031e27 −0.403258
\(929\) −4.77499e27 −0.303965 −0.151982 0.988383i \(-0.548566\pi\)
−0.151982 + 0.988383i \(0.548566\pi\)
\(930\) −3.79300e26 −0.0238998
\(931\) 7.07446e27 0.441237
\(932\) −9.07699e27 −0.560390
\(933\) −6.43881e27 −0.393487
\(934\) 4.42658e28 2.67777
\(935\) 3.05681e28 1.83045
\(936\) 3.12643e27 0.185323
\(937\) −3.08658e27 −0.181114 −0.0905568 0.995891i \(-0.528865\pi\)
−0.0905568 + 0.995891i \(0.528865\pi\)
\(938\) 7.75677e25 0.00450562
\(939\) −7.34987e27 −0.422627
\(940\) 9.44368e27 0.537560
\(941\) 2.51804e27 0.141893 0.0709466 0.997480i \(-0.477398\pi\)
0.0709466 + 0.997480i \(0.477398\pi\)
\(942\) −7.63414e27 −0.425870
\(943\) −4.57943e27 −0.252901
\(944\) −6.02139e27 −0.329202
\(945\) −1.09642e25 −0.000593435 0
\(946\) −4.02363e28 −2.15602
\(947\) 8.85552e27 0.469774 0.234887 0.972023i \(-0.424528\pi\)
0.234887 + 0.972023i \(0.424528\pi\)
\(948\) −3.63547e28 −1.90933
\(949\) −4.24489e27 −0.220718
\(950\) −1.98337e28 −1.02101
\(951\) 1.14854e27 0.0585370
\(952\) −4.49458e25 −0.00226797
\(953\) 1.35516e28 0.677030 0.338515 0.940961i \(-0.390075\pi\)
0.338515 + 0.940961i \(0.390075\pi\)
\(954\) 5.06055e27 0.250315
\(955\) 1.87358e28 0.917570
\(956\) 1.23973e28 0.601144
\(957\) 3.27597e28 1.57281
\(958\) −3.56907e27 −0.169661
\(959\) −2.37647e25 −0.00111855
\(960\) 2.28698e28 1.06582
\(961\) −2.16653e28 −0.999751
\(962\) −1.19687e28 −0.546868
\(963\) 1.00300e28 0.453788
\(964\) −4.52409e28 −2.02675
\(965\) −4.91925e27 −0.218217
\(966\) 1.38837e25 0.000609848 0
\(967\) 5.51108e27 0.239710 0.119855 0.992791i \(-0.461757\pi\)
0.119855 + 0.992791i \(0.461757\pi\)
\(968\) 6.60250e28 2.84376
\(969\) −4.26091e27 −0.181730
\(970\) 5.17088e28 2.18390
\(971\) 2.37023e28 0.991308 0.495654 0.868520i \(-0.334928\pi\)
0.495654 + 0.868520i \(0.334928\pi\)
\(972\) −2.98131e27 −0.123475
\(973\) −5.06808e25 −0.00207860
\(974\) 7.46993e28 3.03394
\(975\) −6.82747e27 −0.274610
\(976\) −7.72688e27 −0.307773
\(977\) −6.97540e27 −0.275151 −0.137575 0.990491i \(-0.543931\pi\)
−0.137575 + 0.990491i \(0.543931\pi\)
\(978\) −4.88915e27 −0.190992
\(979\) −7.53310e28 −2.91433
\(980\) −7.70621e28 −2.95252
\(981\) −1.32704e28 −0.503535
\(982\) 5.20835e28 1.95722
\(983\) 1.96330e28 0.730683 0.365341 0.930874i \(-0.380952\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(984\) −2.03907e28 −0.751585
\(985\) −3.56750e28 −1.30233
\(986\) −5.49538e28 −1.98686
\(987\) −5.90099e24 −0.000211306 0
\(988\) −8.41804e27 −0.298552
\(989\) −6.59227e27 −0.231564
\(990\) −4.20440e28 −1.46275
\(991\) 4.20556e28 1.44918 0.724592 0.689178i \(-0.242028\pi\)
0.724592 + 0.689178i \(0.242028\pi\)
\(992\) −1.14481e26 −0.00390725
\(993\) −7.56325e27 −0.255676
\(994\) 8.50560e25 0.00284796
\(995\) −1.70533e28 −0.565573
\(996\) 3.27256e28 1.07504
\(997\) −1.15418e28 −0.375551 −0.187776 0.982212i \(-0.560128\pi\)
−0.187776 + 0.982212i \(0.560128\pi\)
\(998\) −4.90816e28 −1.58190
\(999\) 5.48353e27 0.175061
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 3.20.a.b.1.2 2
3.2 odd 2 9.20.a.c.1.1 2
4.3 odd 2 48.20.a.j.1.2 2
5.2 odd 4 75.20.b.b.49.4 4
5.3 odd 4 75.20.b.b.49.1 4
5.4 even 2 75.20.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.20.a.b.1.2 2 1.1 even 1 trivial
9.20.a.c.1.1 2 3.2 odd 2
48.20.a.j.1.2 2 4.3 odd 2
75.20.a.b.1.1 2 5.4 even 2
75.20.b.b.49.1 4 5.3 odd 4
75.20.b.b.49.4 4 5.2 odd 4