Properties

Label 197.14.a.a.1.12
Level $197$
Weight $14$
Character 197.1
Self dual yes
Analytic conductor $211.245$
Analytic rank $1$
Dimension $104$
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] = [197,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(211.244930035\)
Analytic rank: \(1\)
Dimension: \(104\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
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)
 
\(f(q)\) \(=\) \(q-147.901 q^{2} +534.557 q^{3} +13682.8 q^{4} -7800.19 q^{5} -79061.7 q^{6} +515509. q^{7} -812098. q^{8} -1.30857e6 q^{9} +O(q^{10})\) \(q-147.901 q^{2} +534.557 q^{3} +13682.8 q^{4} -7800.19 q^{5} -79061.7 q^{6} +515509. q^{7} -812098. q^{8} -1.30857e6 q^{9} +1.15366e6 q^{10} +2.66243e6 q^{11} +7.31424e6 q^{12} +1.65269e7 q^{13} -7.62445e7 q^{14} -4.16965e6 q^{15} +8.02079e6 q^{16} -8.87509e7 q^{17} +1.93540e8 q^{18} +3.31694e8 q^{19} -1.06729e8 q^{20} +2.75569e8 q^{21} -3.93778e8 q^{22} +1.26386e8 q^{23} -4.34112e8 q^{24} -1.15986e9 q^{25} -2.44434e9 q^{26} -1.55176e9 q^{27} +7.05361e9 q^{28} +5.69701e9 q^{29} +6.16696e8 q^{30} -7.98222e9 q^{31} +5.46642e9 q^{32} +1.42322e9 q^{33} +1.31264e10 q^{34} -4.02107e9 q^{35} -1.79049e10 q^{36} -1.46204e10 q^{37} -4.90580e10 q^{38} +8.83455e9 q^{39} +6.33452e9 q^{40} +1.90194e10 q^{41} -4.07570e10 q^{42} +1.26317e10 q^{43} +3.64296e10 q^{44} +1.02071e10 q^{45} -1.86927e10 q^{46} -1.26740e10 q^{47} +4.28757e9 q^{48} +1.68861e11 q^{49} +1.71545e11 q^{50} -4.74424e10 q^{51} +2.26134e11 q^{52} -2.90212e11 q^{53} +2.29508e11 q^{54} -2.07675e10 q^{55} -4.18644e11 q^{56} +1.77309e11 q^{57} -8.42595e11 q^{58} -1.35888e11 q^{59} -5.70525e10 q^{60} -7.22733e11 q^{61} +1.18058e12 q^{62} -6.74581e11 q^{63} -8.74197e11 q^{64} -1.28913e11 q^{65} -2.10496e11 q^{66} -1.06472e12 q^{67} -1.21436e12 q^{68} +6.75607e10 q^{69} +5.94722e11 q^{70} +5.87176e11 q^{71} +1.06269e12 q^{72} -2.24613e12 q^{73} +2.16238e12 q^{74} -6.20011e11 q^{75} +4.53851e12 q^{76} +1.37251e12 q^{77} -1.30664e12 q^{78} -2.25245e12 q^{79} -6.25637e10 q^{80} +1.25678e12 q^{81} -2.81300e12 q^{82} -2.98199e12 q^{83} +3.77056e12 q^{84} +6.92274e11 q^{85} -1.86825e12 q^{86} +3.04537e12 q^{87} -2.16216e12 q^{88} +1.98811e12 q^{89} -1.50965e12 q^{90} +8.51975e12 q^{91} +1.72932e12 q^{92} -4.26695e12 q^{93} +1.87451e12 q^{94} -2.58728e12 q^{95} +2.92211e12 q^{96} -5.54163e12 q^{97} -2.49747e13 q^{98} -3.48399e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q - 128 q^{2} - 8020 q^{3} + 409600 q^{4} - 99004 q^{5} - 83328 q^{6} - 2084037 q^{7} - 2111301 q^{8} + 51549776 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 104 q - 128 q^{2} - 8020 q^{3} + 409600 q^{4} - 99004 q^{5} - 83328 q^{6} - 2084037 q^{7} - 2111301 q^{8} + 51549776 q^{9} - 9626347 q^{10} - 10688800 q^{11} - 68157440 q^{12} - 94762650 q^{13} - 52465903 q^{14} - 186128761 q^{15} + 1543503872 q^{16} - 109572251 q^{17} - 449658263 q^{18} - 1495268031 q^{19} - 1194030741 q^{20} - 832275538 q^{21} - 2695120703 q^{22} - 2632554532 q^{23} - 190649667 q^{24} + 22696779124 q^{25} - 2460454754 q^{26} - 16267542277 q^{27} - 18137835250 q^{28} + 47556769 q^{29} - 9613748117 q^{30} - 24774628288 q^{31} - 29246249180 q^{32} - 42367036666 q^{33} - 39395206606 q^{34} - 26216433864 q^{35} + 190013501626 q^{36} - 82419664050 q^{37} - 34593622142 q^{38} - 39498281801 q^{39} - 150303128514 q^{40} - 25862853768 q^{41} - 138889965149 q^{42} - 258164897781 q^{43} - 197365738094 q^{44} - 240782892021 q^{45} - 218441397971 q^{46} - 208905762731 q^{47} - 664063830814 q^{48} + 1113758315863 q^{49} - 1516068087607 q^{50} - 1019253294393 q^{51} - 1162941441840 q^{52} + 161684855900 q^{53} + 1679137315823 q^{54} - 557952233701 q^{55} + 2842561845328 q^{56} + 801593765429 q^{57} - 7249760433 q^{58} - 775487110641 q^{59} - 57598844627 q^{60} - 36786715662 q^{61} + 681529132643 q^{62} - 4349115033663 q^{63} + 4600420238797 q^{64} - 2531819073161 q^{65} - 3958922748734 q^{66} - 7314072405766 q^{67} - 10297353950393 q^{68} - 2089200206275 q^{69} - 14268943913713 q^{70} - 6055724651085 q^{71} - 26860198563805 q^{72} - 7572811533391 q^{73} - 11175265675817 q^{74} - 24431657592434 q^{75} - 26425925198106 q^{76} - 9735327037686 q^{77} - 35310017230907 q^{78} - 8981210762721 q^{79} - 25913753436330 q^{80} + 15437375349812 q^{81} - 19721330628227 q^{82} - 22209909714532 q^{83} - 18837805278768 q^{84} - 5905005171430 q^{85} - 8913876797772 q^{86} - 7341562344401 q^{87} - 35154329886441 q^{88} - 4646484034146 q^{89} + 9564902968095 q^{90} - 22752431457047 q^{91} - 13601121953458 q^{92} - 9615114240293 q^{93} + 15352521272967 q^{94} + 5258551767043 q^{95} + 87277848810881 q^{96} - 42298840804040 q^{97} + 27000102375354 q^{98} - 8666459567773 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −147.901 −1.63409 −0.817047 0.576571i \(-0.804391\pi\)
−0.817047 + 0.576571i \(0.804391\pi\)
\(3\) 534.557 0.423356 0.211678 0.977339i \(-0.432107\pi\)
0.211678 + 0.977339i \(0.432107\pi\)
\(4\) 13682.8 1.67026
\(5\) −7800.19 −0.223255 −0.111627 0.993750i \(-0.535606\pi\)
−0.111627 + 0.993750i \(0.535606\pi\)
\(6\) −79061.7 −0.691804
\(7\) 515509. 1.65615 0.828074 0.560619i \(-0.189437\pi\)
0.828074 + 0.560619i \(0.189437\pi\)
\(8\) −812098. −1.09528
\(9\) −1.30857e6 −0.820770
\(10\) 1.15366e6 0.364819
\(11\) 2.66243e6 0.453134 0.226567 0.973996i \(-0.427250\pi\)
0.226567 + 0.973996i \(0.427250\pi\)
\(12\) 7.31424e6 0.707117
\(13\) 1.65269e7 0.949639 0.474820 0.880083i \(-0.342513\pi\)
0.474820 + 0.880083i \(0.342513\pi\)
\(14\) −7.62445e7 −2.70630
\(15\) −4.16965e6 −0.0945162
\(16\) 8.02079e6 0.119519
\(17\) −8.87509e7 −0.891774 −0.445887 0.895089i \(-0.647112\pi\)
−0.445887 + 0.895089i \(0.647112\pi\)
\(18\) 1.93540e8 1.34122
\(19\) 3.31694e8 1.61748 0.808741 0.588165i \(-0.200150\pi\)
0.808741 + 0.588165i \(0.200150\pi\)
\(20\) −1.06729e8 −0.372894
\(21\) 2.75569e8 0.701140
\(22\) −3.93778e8 −0.740463
\(23\) 1.26386e8 0.178020 0.0890101 0.996031i \(-0.471630\pi\)
0.0890101 + 0.996031i \(0.471630\pi\)
\(24\) −4.34112e8 −0.463691
\(25\) −1.15986e9 −0.950157
\(26\) −2.44434e9 −1.55180
\(27\) −1.55176e9 −0.770834
\(28\) 7.05361e9 2.76621
\(29\) 5.69701e9 1.77852 0.889262 0.457398i \(-0.151218\pi\)
0.889262 + 0.457398i \(0.151218\pi\)
\(30\) 6.16696e8 0.154448
\(31\) −7.98222e9 −1.61537 −0.807686 0.589613i \(-0.799280\pi\)
−0.807686 + 0.589613i \(0.799280\pi\)
\(32\) 5.46642e9 0.899970
\(33\) 1.42322e9 0.191837
\(34\) 1.31264e10 1.45724
\(35\) −4.02107e9 −0.369743
\(36\) −1.79049e10 −1.37090
\(37\) −1.46204e10 −0.936804 −0.468402 0.883516i \(-0.655170\pi\)
−0.468402 + 0.883516i \(0.655170\pi\)
\(38\) −4.90580e10 −2.64312
\(39\) 8.83455e9 0.402035
\(40\) 6.33452e9 0.244525
\(41\) 1.90194e10 0.625321 0.312660 0.949865i \(-0.398780\pi\)
0.312660 + 0.949865i \(0.398780\pi\)
\(42\) −4.07570e10 −1.14573
\(43\) 1.26317e10 0.304732 0.152366 0.988324i \(-0.451311\pi\)
0.152366 + 0.988324i \(0.451311\pi\)
\(44\) 3.64296e10 0.756853
\(45\) 1.02071e10 0.183241
\(46\) −1.86927e10 −0.290902
\(47\) −1.26740e10 −0.171506 −0.0857529 0.996316i \(-0.527330\pi\)
−0.0857529 + 0.996316i \(0.527330\pi\)
\(48\) 4.28757e9 0.0505991
\(49\) 1.68861e11 1.74283
\(50\) 1.71545e11 1.55265
\(51\) −4.74424e10 −0.377538
\(52\) 2.26134e11 1.58615
\(53\) −2.90212e11 −1.79855 −0.899273 0.437387i \(-0.855904\pi\)
−0.899273 + 0.437387i \(0.855904\pi\)
\(54\) 2.29508e11 1.25962
\(55\) −2.07675e10 −0.101164
\(56\) −4.18644e11 −1.81394
\(57\) 1.77309e11 0.684770
\(58\) −8.42595e11 −2.90628
\(59\) −1.35888e11 −0.419414 −0.209707 0.977764i \(-0.567251\pi\)
−0.209707 + 0.977764i \(0.567251\pi\)
\(60\) −5.70525e10 −0.157867
\(61\) −7.22733e11 −1.79612 −0.898058 0.439878i \(-0.855022\pi\)
−0.898058 + 0.439878i \(0.855022\pi\)
\(62\) 1.18058e12 2.63967
\(63\) −6.74581e11 −1.35932
\(64\) −8.74197e11 −1.59016
\(65\) −1.28913e11 −0.212011
\(66\) −2.10496e11 −0.313480
\(67\) −1.06472e12 −1.43797 −0.718987 0.695024i \(-0.755394\pi\)
−0.718987 + 0.695024i \(0.755394\pi\)
\(68\) −1.21436e12 −1.48950
\(69\) 6.75607e10 0.0753659
\(70\) 5.94722e11 0.604194
\(71\) 5.87176e11 0.543987 0.271994 0.962299i \(-0.412317\pi\)
0.271994 + 0.962299i \(0.412317\pi\)
\(72\) 1.06269e12 0.898969
\(73\) −2.24613e12 −1.73715 −0.868574 0.495560i \(-0.834963\pi\)
−0.868574 + 0.495560i \(0.834963\pi\)
\(74\) 2.16238e12 1.53083
\(75\) −6.20011e11 −0.402255
\(76\) 4.53851e12 2.70162
\(77\) 1.37251e12 0.750457
\(78\) −1.30664e12 −0.656964
\(79\) −2.25245e12 −1.04251 −0.521253 0.853402i \(-0.674535\pi\)
−0.521253 + 0.853402i \(0.674535\pi\)
\(80\) −6.25637e10 −0.0266832
\(81\) 1.25678e12 0.494433
\(82\) −2.81300e12 −1.02183
\(83\) −2.98199e12 −1.00115 −0.500575 0.865693i \(-0.666878\pi\)
−0.500575 + 0.865693i \(0.666878\pi\)
\(84\) 3.77056e12 1.17109
\(85\) 6.92274e11 0.199093
\(86\) −1.86825e12 −0.497960
\(87\) 3.04537e12 0.752949
\(88\) −2.16216e12 −0.496306
\(89\) 1.98811e12 0.424038 0.212019 0.977266i \(-0.431996\pi\)
0.212019 + 0.977266i \(0.431996\pi\)
\(90\) −1.50965e12 −0.299432
\(91\) 8.51975e12 1.57274
\(92\) 1.72932e12 0.297341
\(93\) −4.26695e12 −0.683878
\(94\) 1.87451e12 0.280257
\(95\) −2.58728e12 −0.361110
\(96\) 2.92211e12 0.381008
\(97\) −5.54163e12 −0.675494 −0.337747 0.941237i \(-0.609665\pi\)
−0.337747 + 0.941237i \(0.609665\pi\)
\(98\) −2.49747e13 −2.84794
\(99\) −3.48399e12 −0.371918
\(100\) −1.58701e13 −1.58701
\(101\) 1.55572e13 1.45828 0.729141 0.684363i \(-0.239920\pi\)
0.729141 + 0.684363i \(0.239920\pi\)
\(102\) 7.01679e12 0.616932
\(103\) −1.56888e13 −1.29464 −0.647318 0.762220i \(-0.724109\pi\)
−0.647318 + 0.762220i \(0.724109\pi\)
\(104\) −1.34214e13 −1.04012
\(105\) −2.14949e12 −0.156533
\(106\) 4.29227e13 2.93900
\(107\) 5.76474e12 0.371352 0.185676 0.982611i \(-0.440553\pi\)
0.185676 + 0.982611i \(0.440553\pi\)
\(108\) −2.12325e13 −1.28750
\(109\) 5.04693e12 0.288241 0.144120 0.989560i \(-0.453965\pi\)
0.144120 + 0.989560i \(0.453965\pi\)
\(110\) 3.07154e12 0.165312
\(111\) −7.81544e12 −0.396601
\(112\) 4.13479e12 0.197941
\(113\) 1.11960e13 0.505884 0.252942 0.967481i \(-0.418602\pi\)
0.252942 + 0.967481i \(0.418602\pi\)
\(114\) −2.62243e13 −1.11898
\(115\) −9.85838e11 −0.0397438
\(116\) 7.79511e13 2.97061
\(117\) −2.16266e13 −0.779435
\(118\) 2.00980e13 0.685362
\(119\) −4.57519e13 −1.47691
\(120\) 3.38616e12 0.103521
\(121\) −2.74342e13 −0.794670
\(122\) 1.06893e14 2.93502
\(123\) 1.01670e13 0.264733
\(124\) −1.09219e14 −2.69810
\(125\) 1.85689e13 0.435382
\(126\) 9.97714e13 2.22125
\(127\) 4.98283e13 1.05378 0.526892 0.849932i \(-0.323357\pi\)
0.526892 + 0.849932i \(0.323357\pi\)
\(128\) 8.45140e13 1.69849
\(129\) 6.75237e12 0.129010
\(130\) 1.90664e13 0.346446
\(131\) 7.78851e12 0.134645 0.0673226 0.997731i \(-0.478554\pi\)
0.0673226 + 0.997731i \(0.478554\pi\)
\(132\) 1.94737e13 0.320418
\(133\) 1.70991e14 2.67879
\(134\) 1.57474e14 2.34979
\(135\) 1.21040e13 0.172092
\(136\) 7.20744e13 0.976738
\(137\) 6.65711e13 0.860206 0.430103 0.902780i \(-0.358477\pi\)
0.430103 + 0.902780i \(0.358477\pi\)
\(138\) −9.99232e12 −0.123155
\(139\) 1.07241e13 0.126115 0.0630574 0.998010i \(-0.479915\pi\)
0.0630574 + 0.998010i \(0.479915\pi\)
\(140\) −5.50195e13 −0.617568
\(141\) −6.77499e12 −0.0726080
\(142\) −8.68441e13 −0.888927
\(143\) 4.40017e13 0.430314
\(144\) −1.04958e13 −0.0980976
\(145\) −4.44378e13 −0.397064
\(146\) 3.32206e14 2.83866
\(147\) 9.02657e13 0.737836
\(148\) −2.00048e14 −1.56471
\(149\) −3.82949e13 −0.286701 −0.143351 0.989672i \(-0.545788\pi\)
−0.143351 + 0.989672i \(0.545788\pi\)
\(150\) 9.17005e13 0.657322
\(151\) −3.10744e13 −0.213330 −0.106665 0.994295i \(-0.534017\pi\)
−0.106665 + 0.994295i \(0.534017\pi\)
\(152\) −2.69368e14 −1.77159
\(153\) 1.16137e14 0.731941
\(154\) −2.02996e14 −1.22632
\(155\) 6.22628e13 0.360639
\(156\) 1.20881e14 0.671505
\(157\) 1.62776e14 0.867446 0.433723 0.901046i \(-0.357200\pi\)
0.433723 + 0.901046i \(0.357200\pi\)
\(158\) 3.33140e14 1.70355
\(159\) −1.55135e14 −0.761426
\(160\) −4.26391e13 −0.200922
\(161\) 6.51533e13 0.294828
\(162\) −1.85880e14 −0.807949
\(163\) −4.26116e13 −0.177954 −0.0889772 0.996034i \(-0.528360\pi\)
−0.0889772 + 0.996034i \(0.528360\pi\)
\(164\) 2.60239e14 1.04445
\(165\) −1.11014e13 −0.0428285
\(166\) 4.41041e14 1.63597
\(167\) 5.08677e14 1.81462 0.907309 0.420464i \(-0.138133\pi\)
0.907309 + 0.420464i \(0.138133\pi\)
\(168\) −2.23789e14 −0.767942
\(169\) −2.97380e13 −0.0981856
\(170\) −1.02388e14 −0.325336
\(171\) −4.34046e14 −1.32758
\(172\) 1.72837e14 0.508983
\(173\) 3.60888e14 1.02347 0.511733 0.859145i \(-0.329004\pi\)
0.511733 + 0.859145i \(0.329004\pi\)
\(174\) −4.50415e14 −1.23039
\(175\) −5.97919e14 −1.57360
\(176\) 2.13548e13 0.0541581
\(177\) −7.26399e13 −0.177561
\(178\) −2.94044e14 −0.692918
\(179\) −1.38990e14 −0.315818 −0.157909 0.987454i \(-0.550475\pi\)
−0.157909 + 0.987454i \(0.550475\pi\)
\(180\) 1.39662e14 0.306060
\(181\) −4.80398e14 −1.01553 −0.507763 0.861497i \(-0.669527\pi\)
−0.507763 + 0.861497i \(0.669527\pi\)
\(182\) −1.26008e15 −2.57001
\(183\) −3.86342e14 −0.760396
\(184\) −1.02638e14 −0.194981
\(185\) 1.14042e14 0.209146
\(186\) 6.31088e14 1.11752
\(187\) −2.36293e14 −0.404093
\(188\) −1.73416e14 −0.286460
\(189\) −7.99948e14 −1.27661
\(190\) 3.82662e14 0.590088
\(191\) 1.71838e13 0.0256095 0.0128048 0.999918i \(-0.495924\pi\)
0.0128048 + 0.999918i \(0.495924\pi\)
\(192\) −4.67308e14 −0.673202
\(193\) −4.23057e14 −0.589218 −0.294609 0.955618i \(-0.595189\pi\)
−0.294609 + 0.955618i \(0.595189\pi\)
\(194\) 8.19615e14 1.10382
\(195\) −6.89112e13 −0.0897562
\(196\) 2.31049e15 2.91098
\(197\) −5.84517e13 −0.0712470
\(198\) 5.15286e14 0.607750
\(199\) −5.89406e14 −0.672775 −0.336387 0.941724i \(-0.609205\pi\)
−0.336387 + 0.941724i \(0.609205\pi\)
\(200\) 9.41920e14 1.04068
\(201\) −5.69156e14 −0.608775
\(202\) −2.30093e15 −2.38297
\(203\) 2.93686e15 2.94550
\(204\) −6.49145e14 −0.630588
\(205\) −1.48355e14 −0.139606
\(206\) 2.32039e15 2.11556
\(207\) −1.65386e14 −0.146114
\(208\) 1.32558e14 0.113500
\(209\) 8.83114e14 0.732935
\(210\) 3.17913e14 0.255789
\(211\) −1.69305e15 −1.32079 −0.660396 0.750918i \(-0.729611\pi\)
−0.660396 + 0.750918i \(0.729611\pi\)
\(212\) −3.97091e15 −3.00405
\(213\) 3.13879e14 0.230300
\(214\) −8.52613e14 −0.606823
\(215\) −9.85299e13 −0.0680327
\(216\) 1.26018e15 0.844275
\(217\) −4.11491e15 −2.67530
\(218\) −7.46448e14 −0.471012
\(219\) −1.20069e15 −0.735432
\(220\) −2.84158e14 −0.168971
\(221\) −1.46677e15 −0.846863
\(222\) 1.15591e15 0.648084
\(223\) −1.03170e15 −0.561788 −0.280894 0.959739i \(-0.590631\pi\)
−0.280894 + 0.959739i \(0.590631\pi\)
\(224\) 2.81799e15 1.49048
\(225\) 1.51776e15 0.779860
\(226\) −1.65590e15 −0.826662
\(227\) 1.23090e14 0.0597109 0.0298554 0.999554i \(-0.490495\pi\)
0.0298554 + 0.999554i \(0.490495\pi\)
\(228\) 2.42609e15 1.14375
\(229\) 3.56608e15 1.63403 0.817015 0.576616i \(-0.195627\pi\)
0.817015 + 0.576616i \(0.195627\pi\)
\(230\) 1.45807e14 0.0649451
\(231\) 7.33684e14 0.317710
\(232\) −4.62653e15 −1.94797
\(233\) −3.74981e15 −1.53531 −0.767655 0.640864i \(-0.778576\pi\)
−0.767655 + 0.640864i \(0.778576\pi\)
\(234\) 3.19860e15 1.27367
\(235\) 9.88600e13 0.0382895
\(236\) −1.85933e15 −0.700533
\(237\) −1.20406e15 −0.441352
\(238\) 6.76677e15 2.41341
\(239\) −1.14364e15 −0.396920 −0.198460 0.980109i \(-0.563594\pi\)
−0.198460 + 0.980109i \(0.563594\pi\)
\(240\) −3.34438e13 −0.0112965
\(241\) 8.81238e14 0.289723 0.144861 0.989452i \(-0.453726\pi\)
0.144861 + 0.989452i \(0.453726\pi\)
\(242\) 4.05755e15 1.29857
\(243\) 3.14583e15 0.980155
\(244\) −9.88902e15 −2.99999
\(245\) −1.31715e15 −0.389094
\(246\) −1.50371e15 −0.432599
\(247\) 5.48186e15 1.53602
\(248\) 6.48234e15 1.76928
\(249\) −1.59404e15 −0.423843
\(250\) −2.74636e15 −0.711455
\(251\) −4.55421e15 −1.14957 −0.574783 0.818306i \(-0.694914\pi\)
−0.574783 + 0.818306i \(0.694914\pi\)
\(252\) −9.23016e15 −2.27042
\(253\) 3.36495e14 0.0806670
\(254\) −7.36967e15 −1.72198
\(255\) 3.70060e14 0.0842870
\(256\) −5.33831e15 −1.18534
\(257\) −5.44780e15 −1.17939 −0.589693 0.807627i \(-0.700751\pi\)
−0.589693 + 0.807627i \(0.700751\pi\)
\(258\) −9.98685e14 −0.210815
\(259\) −7.53696e15 −1.55149
\(260\) −1.76389e15 −0.354115
\(261\) −7.45495e15 −1.45976
\(262\) −1.15193e15 −0.220023
\(263\) 2.92179e14 0.0544423 0.0272211 0.999629i \(-0.491334\pi\)
0.0272211 + 0.999629i \(0.491334\pi\)
\(264\) −1.15580e15 −0.210114
\(265\) 2.26371e15 0.401534
\(266\) −2.52898e16 −4.37739
\(267\) 1.06276e15 0.179519
\(268\) −1.45684e16 −2.40180
\(269\) −6.76225e15 −1.08818 −0.544091 0.839026i \(-0.683125\pi\)
−0.544091 + 0.839026i \(0.683125\pi\)
\(270\) −1.79020e15 −0.281215
\(271\) 8.99311e15 1.37914 0.689572 0.724217i \(-0.257799\pi\)
0.689572 + 0.724217i \(0.257799\pi\)
\(272\) −7.11852e14 −0.106584
\(273\) 4.55429e15 0.665830
\(274\) −9.84596e15 −1.40566
\(275\) −3.08805e15 −0.430548
\(276\) 9.24420e14 0.125881
\(277\) −8.32255e14 −0.110697 −0.0553487 0.998467i \(-0.517627\pi\)
−0.0553487 + 0.998467i \(0.517627\pi\)
\(278\) −1.58611e15 −0.206083
\(279\) 1.04453e16 1.32585
\(280\) 3.26550e15 0.404970
\(281\) 1.32987e16 1.61146 0.805729 0.592284i \(-0.201774\pi\)
0.805729 + 0.592284i \(0.201774\pi\)
\(282\) 1.00203e15 0.118648
\(283\) 4.75010e15 0.549656 0.274828 0.961493i \(-0.411379\pi\)
0.274828 + 0.961493i \(0.411379\pi\)
\(284\) 8.03421e15 0.908603
\(285\) −1.38305e15 −0.152878
\(286\) −6.50791e15 −0.703173
\(287\) 9.80470e15 1.03562
\(288\) −7.15320e15 −0.738668
\(289\) −2.02786e15 −0.204740
\(290\) 6.57241e15 0.648840
\(291\) −2.96232e15 −0.285974
\(292\) −3.07334e16 −2.90150
\(293\) −1.49169e16 −1.37733 −0.688665 0.725080i \(-0.741803\pi\)
−0.688665 + 0.725080i \(0.741803\pi\)
\(294\) −1.33504e16 −1.20569
\(295\) 1.05995e15 0.0936361
\(296\) 1.18732e16 1.02606
\(297\) −4.13147e15 −0.349291
\(298\) 5.66386e15 0.468497
\(299\) 2.08877e15 0.169055
\(300\) −8.48349e15 −0.671872
\(301\) 6.51177e15 0.504681
\(302\) 4.59594e15 0.348601
\(303\) 8.31619e15 0.617373
\(304\) 2.66045e15 0.193320
\(305\) 5.63746e15 0.400991
\(306\) −1.71768e16 −1.19606
\(307\) 1.89012e16 1.28852 0.644258 0.764808i \(-0.277166\pi\)
0.644258 + 0.764808i \(0.277166\pi\)
\(308\) 1.87798e16 1.25346
\(309\) −8.38656e15 −0.548092
\(310\) −9.20876e15 −0.589318
\(311\) 4.91406e13 0.00307962 0.00153981 0.999999i \(-0.499510\pi\)
0.00153981 + 0.999999i \(0.499510\pi\)
\(312\) −7.17452e15 −0.440340
\(313\) 1.31483e16 0.790374 0.395187 0.918601i \(-0.370680\pi\)
0.395187 + 0.918601i \(0.370680\pi\)
\(314\) −2.40748e16 −1.41749
\(315\) 5.26186e15 0.303474
\(316\) −3.08198e16 −1.74126
\(317\) 1.54783e16 0.856719 0.428360 0.903608i \(-0.359092\pi\)
0.428360 + 0.903608i \(0.359092\pi\)
\(318\) 2.29446e16 1.24424
\(319\) 1.51679e16 0.805910
\(320\) 6.81891e15 0.355009
\(321\) 3.08158e15 0.157214
\(322\) −9.63626e15 −0.481776
\(323\) −2.94381e16 −1.44243
\(324\) 1.71963e16 0.825833
\(325\) −1.91688e16 −0.902307
\(326\) 6.30232e15 0.290794
\(327\) 2.69787e15 0.122028
\(328\) −1.54456e16 −0.684898
\(329\) −6.53358e15 −0.284039
\(330\) 1.64191e15 0.0699858
\(331\) 1.39764e16 0.584134 0.292067 0.956398i \(-0.405657\pi\)
0.292067 + 0.956398i \(0.405657\pi\)
\(332\) −4.08020e16 −1.67218
\(333\) 1.91319e16 0.768900
\(334\) −7.52341e16 −2.96526
\(335\) 8.30506e15 0.321034
\(336\) 2.21028e15 0.0837996
\(337\) −4.61405e16 −1.71588 −0.857942 0.513747i \(-0.828257\pi\)
−0.857942 + 0.513747i \(0.828257\pi\)
\(338\) 4.39829e15 0.160445
\(339\) 5.98487e15 0.214169
\(340\) 9.47225e15 0.332537
\(341\) −2.12521e16 −0.731980
\(342\) 6.41959e16 2.16939
\(343\) 3.71021e16 1.23023
\(344\) −1.02582e16 −0.333765
\(345\) −5.26986e14 −0.0168258
\(346\) −5.33759e16 −1.67244
\(347\) −4.69919e16 −1.44505 −0.722523 0.691347i \(-0.757018\pi\)
−0.722523 + 0.691347i \(0.757018\pi\)
\(348\) 4.16693e16 1.25762
\(349\) −5.80573e16 −1.71985 −0.859927 0.510418i \(-0.829491\pi\)
−0.859927 + 0.510418i \(0.829491\pi\)
\(350\) 8.84330e16 2.57141
\(351\) −2.56458e16 −0.732014
\(352\) 1.45540e16 0.407807
\(353\) −3.84932e16 −1.05888 −0.529442 0.848346i \(-0.677599\pi\)
−0.529442 + 0.848346i \(0.677599\pi\)
\(354\) 1.07435e16 0.290152
\(355\) −4.58008e15 −0.121448
\(356\) 2.72029e16 0.708256
\(357\) −2.44570e16 −0.625258
\(358\) 2.05567e16 0.516077
\(359\) −3.36564e16 −0.829762 −0.414881 0.909876i \(-0.636177\pi\)
−0.414881 + 0.909876i \(0.636177\pi\)
\(360\) −8.28917e15 −0.200699
\(361\) 6.79680e16 1.61625
\(362\) 7.10515e16 1.65946
\(363\) −1.46651e16 −0.336428
\(364\) 1.16574e17 2.62690
\(365\) 1.75203e16 0.387826
\(366\) 5.71405e16 1.24256
\(367\) −3.11244e16 −0.664924 −0.332462 0.943117i \(-0.607879\pi\)
−0.332462 + 0.943117i \(0.607879\pi\)
\(368\) 1.01372e15 0.0212768
\(369\) −2.48883e16 −0.513244
\(370\) −1.68670e16 −0.341764
\(371\) −1.49607e17 −2.97866
\(372\) −5.83838e16 −1.14226
\(373\) 9.20915e16 1.77057 0.885283 0.465052i \(-0.153964\pi\)
0.885283 + 0.465052i \(0.153964\pi\)
\(374\) 3.49481e16 0.660326
\(375\) 9.92611e15 0.184321
\(376\) 1.02926e16 0.187846
\(377\) 9.41537e16 1.68896
\(378\) 1.18313e17 2.08611
\(379\) 7.13898e16 1.23732 0.618659 0.785660i \(-0.287676\pi\)
0.618659 + 0.785660i \(0.287676\pi\)
\(380\) −3.54012e16 −0.603149
\(381\) 2.66361e16 0.446126
\(382\) −2.54150e15 −0.0418484
\(383\) 1.61608e16 0.261620 0.130810 0.991407i \(-0.458242\pi\)
0.130810 + 0.991407i \(0.458242\pi\)
\(384\) 4.51775e16 0.719067
\(385\) −1.07058e16 −0.167543
\(386\) 6.25707e16 0.962838
\(387\) −1.65295e16 −0.250115
\(388\) −7.58251e16 −1.12825
\(389\) 8.20627e16 1.20081 0.600404 0.799697i \(-0.295007\pi\)
0.600404 + 0.799697i \(0.295007\pi\)
\(390\) 1.01921e16 0.146670
\(391\) −1.12169e16 −0.158754
\(392\) −1.37131e17 −1.90887
\(393\) 4.16340e15 0.0570029
\(394\) 8.64509e15 0.116424
\(395\) 1.75695e16 0.232744
\(396\) −4.76707e16 −0.621202
\(397\) 1.04725e17 1.34249 0.671247 0.741234i \(-0.265759\pi\)
0.671247 + 0.741234i \(0.265759\pi\)
\(398\) 8.71740e16 1.09938
\(399\) 9.14046e16 1.13408
\(400\) −9.30299e15 −0.113562
\(401\) −9.89387e16 −1.18830 −0.594152 0.804353i \(-0.702512\pi\)
−0.594152 + 0.804353i \(0.702512\pi\)
\(402\) 8.41789e16 0.994796
\(403\) −1.31921e17 −1.53402
\(404\) 2.12866e17 2.43572
\(405\) −9.80314e15 −0.110384
\(406\) −4.34366e17 −4.81323
\(407\) −3.89259e16 −0.424497
\(408\) 3.85279e16 0.413508
\(409\) −5.43168e16 −0.573764 −0.286882 0.957966i \(-0.592619\pi\)
−0.286882 + 0.957966i \(0.592619\pi\)
\(410\) 2.19420e16 0.228129
\(411\) 3.55861e16 0.364173
\(412\) −2.14667e17 −2.16239
\(413\) −7.00515e16 −0.694612
\(414\) 2.44608e16 0.238763
\(415\) 2.32601e16 0.223511
\(416\) 9.03428e16 0.854647
\(417\) 5.73266e15 0.0533914
\(418\) −1.30614e17 −1.19769
\(419\) 6.51208e16 0.587934 0.293967 0.955816i \(-0.405024\pi\)
0.293967 + 0.955816i \(0.405024\pi\)
\(420\) −2.94111e16 −0.261451
\(421\) −6.75894e16 −0.591623 −0.295811 0.955246i \(-0.595590\pi\)
−0.295811 + 0.955246i \(0.595590\pi\)
\(422\) 2.50405e17 2.15830
\(423\) 1.65849e16 0.140767
\(424\) 2.35680e17 1.96990
\(425\) 1.02939e17 0.847325
\(426\) −4.64231e16 −0.376332
\(427\) −3.72576e17 −2.97463
\(428\) 7.88778e16 0.620255
\(429\) 2.35214e16 0.182176
\(430\) 1.45727e16 0.111172
\(431\) 1.42668e17 1.07207 0.536035 0.844196i \(-0.319922\pi\)
0.536035 + 0.844196i \(0.319922\pi\)
\(432\) −1.24464e16 −0.0921293
\(433\) −4.77089e16 −0.347879 −0.173939 0.984756i \(-0.555650\pi\)
−0.173939 + 0.984756i \(0.555650\pi\)
\(434\) 6.08600e17 4.37169
\(435\) −2.37545e16 −0.168099
\(436\) 6.90562e16 0.481438
\(437\) 4.19216e16 0.287944
\(438\) 1.77583e17 1.20176
\(439\) 1.78581e16 0.119074 0.0595369 0.998226i \(-0.481038\pi\)
0.0595369 + 0.998226i \(0.481038\pi\)
\(440\) 1.68652e16 0.110803
\(441\) −2.20966e17 −1.43046
\(442\) 2.16938e17 1.38385
\(443\) −2.16454e17 −1.36063 −0.680317 0.732918i \(-0.738158\pi\)
−0.680317 + 0.732918i \(0.738158\pi\)
\(444\) −1.06937e17 −0.662429
\(445\) −1.55076e16 −0.0946684
\(446\) 1.52590e17 0.918014
\(447\) −2.04708e16 −0.121377
\(448\) −4.50657e17 −2.63353
\(449\) 3.77058e15 0.0217173 0.0108587 0.999941i \(-0.496544\pi\)
0.0108587 + 0.999941i \(0.496544\pi\)
\(450\) −2.24479e17 −1.27437
\(451\) 5.06380e16 0.283354
\(452\) 1.53192e17 0.844960
\(453\) −1.66110e16 −0.0903145
\(454\) −1.82051e16 −0.0975732
\(455\) −6.64557e16 −0.351122
\(456\) −1.43992e17 −0.750012
\(457\) −3.18715e17 −1.63662 −0.818308 0.574780i \(-0.805087\pi\)
−0.818308 + 0.574780i \(0.805087\pi\)
\(458\) −5.27428e17 −2.67016
\(459\) 1.37720e17 0.687409
\(460\) −1.34890e16 −0.0663827
\(461\) −2.23376e17 −1.08388 −0.541938 0.840418i \(-0.682309\pi\)
−0.541938 + 0.840418i \(0.682309\pi\)
\(462\) −1.08513e17 −0.519169
\(463\) 1.33005e17 0.627468 0.313734 0.949511i \(-0.398420\pi\)
0.313734 + 0.949511i \(0.398420\pi\)
\(464\) 4.56945e16 0.212568
\(465\) 3.32830e16 0.152679
\(466\) 5.54602e17 2.50884
\(467\) −2.46819e17 −1.10108 −0.550540 0.834809i \(-0.685578\pi\)
−0.550540 + 0.834809i \(0.685578\pi\)
\(468\) −2.95912e17 −1.30186
\(469\) −5.48875e17 −2.38150
\(470\) −1.46215e16 −0.0625686
\(471\) 8.70129e16 0.367238
\(472\) 1.10354e17 0.459374
\(473\) 3.36311e16 0.138084
\(474\) 1.78082e17 0.721210
\(475\) −3.84719e17 −1.53686
\(476\) −6.26014e17 −2.46683
\(477\) 3.79763e17 1.47619
\(478\) 1.69146e17 0.648605
\(479\) −3.33942e17 −1.26325 −0.631627 0.775273i \(-0.717612\pi\)
−0.631627 + 0.775273i \(0.717612\pi\)
\(480\) −2.27930e16 −0.0850617
\(481\) −2.41630e17 −0.889625
\(482\) −1.30336e17 −0.473434
\(483\) 3.48282e16 0.124817
\(484\) −3.75376e17 −1.32731
\(485\) 4.32258e16 0.150807
\(486\) −4.65273e17 −1.60167
\(487\) 1.04619e17 0.355363 0.177682 0.984088i \(-0.443140\pi\)
0.177682 + 0.984088i \(0.443140\pi\)
\(488\) 5.86930e17 1.96724
\(489\) −2.27783e16 −0.0753381
\(490\) 1.94808e17 0.635816
\(491\) 5.50979e17 1.77462 0.887310 0.461174i \(-0.152572\pi\)
0.887310 + 0.461174i \(0.152572\pi\)
\(492\) 1.39113e17 0.442175
\(493\) −5.05614e17 −1.58604
\(494\) −8.10775e17 −2.51001
\(495\) 2.71758e16 0.0830325
\(496\) −6.40237e16 −0.193068
\(497\) 3.02694e17 0.900923
\(498\) 2.35761e17 0.692599
\(499\) 3.56426e17 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(500\) 2.54074e17 0.727202
\(501\) 2.71917e17 0.768230
\(502\) 6.73574e17 1.87850
\(503\) −3.83898e17 −1.05688 −0.528438 0.848972i \(-0.677222\pi\)
−0.528438 + 0.848972i \(0.677222\pi\)
\(504\) 5.47826e17 1.48883
\(505\) −1.21349e17 −0.325568
\(506\) −4.97681e16 −0.131817
\(507\) −1.58966e16 −0.0415675
\(508\) 6.81791e17 1.76010
\(509\) −4.18856e17 −1.06758 −0.533789 0.845618i \(-0.679232\pi\)
−0.533789 + 0.845618i \(0.679232\pi\)
\(510\) −5.47323e16 −0.137733
\(511\) −1.15790e18 −2.87697
\(512\) 9.72049e16 0.238470
\(513\) −5.14710e17 −1.24681
\(514\) 8.05737e17 1.92723
\(515\) 1.22376e17 0.289033
\(516\) 9.23914e16 0.215481
\(517\) −3.37438e16 −0.0777151
\(518\) 1.11473e18 2.53527
\(519\) 1.92915e17 0.433290
\(520\) 1.04690e17 0.232211
\(521\) −5.11992e17 −1.12155 −0.560774 0.827969i \(-0.689496\pi\)
−0.560774 + 0.827969i \(0.689496\pi\)
\(522\) 1.10260e18 2.38538
\(523\) 5.42076e17 1.15824 0.579121 0.815242i \(-0.303396\pi\)
0.579121 + 0.815242i \(0.303396\pi\)
\(524\) 1.06569e17 0.224893
\(525\) −3.19621e17 −0.666194
\(526\) −4.32136e16 −0.0889639
\(527\) 7.08429e17 1.44055
\(528\) 1.14154e16 0.0229282
\(529\) −4.88063e17 −0.968309
\(530\) −3.34805e17 −0.656144
\(531\) 1.77819e17 0.344242
\(532\) 2.33964e18 4.47429
\(533\) 3.14332e17 0.593829
\(534\) −1.57183e17 −0.293351
\(535\) −4.49661e16 −0.0829059
\(536\) 8.64660e17 1.57498
\(537\) −7.42978e16 −0.133704
\(538\) 1.00015e18 1.77819
\(539\) 4.49581e17 0.789733
\(540\) 1.65617e17 0.287439
\(541\) −6.73638e17 −1.15517 −0.577583 0.816332i \(-0.696004\pi\)
−0.577583 + 0.816332i \(0.696004\pi\)
\(542\) −1.33009e18 −2.25365
\(543\) −2.56800e17 −0.429929
\(544\) −4.85149e17 −0.802570
\(545\) −3.93670e16 −0.0643510
\(546\) −6.73586e17 −1.08803
\(547\) 8.73991e17 1.39505 0.697524 0.716561i \(-0.254285\pi\)
0.697524 + 0.716561i \(0.254285\pi\)
\(548\) 9.10880e17 1.43677
\(549\) 9.45749e17 1.47420
\(550\) 4.56727e17 0.703557
\(551\) 1.88966e18 2.87673
\(552\) −5.48659e16 −0.0825464
\(553\) −1.16116e18 −1.72655
\(554\) 1.23092e17 0.180890
\(555\) 6.09620e16 0.0885431
\(556\) 1.46736e17 0.210645
\(557\) 7.55036e17 1.07129 0.535647 0.844442i \(-0.320068\pi\)
0.535647 + 0.844442i \(0.320068\pi\)
\(558\) −1.54487e18 −2.16656
\(559\) 2.08763e17 0.289385
\(560\) −3.22522e16 −0.0441913
\(561\) −1.26312e17 −0.171075
\(562\) −1.96690e18 −2.63328
\(563\) −1.09722e17 −0.145208 −0.0726038 0.997361i \(-0.523131\pi\)
−0.0726038 + 0.997361i \(0.523131\pi\)
\(564\) −9.27009e16 −0.121275
\(565\) −8.73306e16 −0.112941
\(566\) −7.02547e17 −0.898190
\(567\) 6.47882e17 0.818854
\(568\) −4.76844e17 −0.595816
\(569\) −4.45545e17 −0.550378 −0.275189 0.961390i \(-0.588741\pi\)
−0.275189 + 0.961390i \(0.588741\pi\)
\(570\) 2.04555e17 0.249817
\(571\) 1.49536e18 1.80556 0.902781 0.430100i \(-0.141522\pi\)
0.902781 + 0.430100i \(0.141522\pi\)
\(572\) 6.02066e17 0.718737
\(573\) 9.18569e15 0.0108419
\(574\) −1.45013e18 −1.69231
\(575\) −1.46590e17 −0.169147
\(576\) 1.14395e18 1.30515
\(577\) 9.27665e17 1.04652 0.523261 0.852172i \(-0.324715\pi\)
0.523261 + 0.852172i \(0.324715\pi\)
\(578\) 2.99923e17 0.334564
\(579\) −2.26148e17 −0.249449
\(580\) −6.08033e17 −0.663201
\(581\) −1.53724e18 −1.65805
\(582\) 4.38131e17 0.467309
\(583\) −7.72670e17 −0.814982
\(584\) 1.82408e18 1.90265
\(585\) 1.68692e17 0.174012
\(586\) 2.20622e18 2.25069
\(587\) 4.46667e17 0.450647 0.225323 0.974284i \(-0.427656\pi\)
0.225323 + 0.974284i \(0.427656\pi\)
\(588\) 1.23509e18 1.23238
\(589\) −2.64765e18 −2.61283
\(590\) −1.56768e17 −0.153010
\(591\) −3.12458e16 −0.0301629
\(592\) −1.17267e17 −0.111966
\(593\) −7.48723e17 −0.707075 −0.353537 0.935420i \(-0.615021\pi\)
−0.353537 + 0.935420i \(0.615021\pi\)
\(594\) 6.11049e17 0.570774
\(595\) 3.56874e17 0.329727
\(596\) −5.23981e17 −0.478867
\(597\) −3.15071e17 −0.284823
\(598\) −3.08932e17 −0.276252
\(599\) −1.54595e18 −1.36748 −0.683740 0.729726i \(-0.739648\pi\)
−0.683740 + 0.729726i \(0.739648\pi\)
\(600\) 5.03510e17 0.440580
\(601\) −6.29555e17 −0.544941 −0.272470 0.962164i \(-0.587841\pi\)
−0.272470 + 0.962164i \(0.587841\pi\)
\(602\) −9.63099e17 −0.824696
\(603\) 1.39327e18 1.18025
\(604\) −4.25184e17 −0.356318
\(605\) 2.13992e17 0.177414
\(606\) −1.22998e18 −1.00885
\(607\) −2.08261e18 −1.68998 −0.844991 0.534780i \(-0.820394\pi\)
−0.844991 + 0.534780i \(0.820394\pi\)
\(608\) 1.81318e18 1.45568
\(609\) 1.56992e18 1.24700
\(610\) −8.33788e17 −0.655257
\(611\) −2.09462e17 −0.162869
\(612\) 1.58908e18 1.22253
\(613\) −9.58740e17 −0.729806 −0.364903 0.931045i \(-0.618898\pi\)
−0.364903 + 0.931045i \(0.618898\pi\)
\(614\) −2.79551e18 −2.10556
\(615\) −7.93044e16 −0.0591029
\(616\) −1.11461e18 −0.821957
\(617\) 1.22241e18 0.891994 0.445997 0.895034i \(-0.352849\pi\)
0.445997 + 0.895034i \(0.352849\pi\)
\(618\) 1.24038e18 0.895634
\(619\) 1.55068e18 1.10798 0.553992 0.832522i \(-0.313104\pi\)
0.553992 + 0.832522i \(0.313104\pi\)
\(620\) 8.51930e17 0.602363
\(621\) −1.96122e17 −0.137224
\(622\) −7.26796e15 −0.00503240
\(623\) 1.02489e18 0.702270
\(624\) 7.08600e16 0.0480509
\(625\) 1.27100e18 0.852956
\(626\) −1.94466e18 −1.29155
\(627\) 4.72074e17 0.310293
\(628\) 2.22723e18 1.44886
\(629\) 1.29757e18 0.835417
\(630\) −7.78236e17 −0.495904
\(631\) −1.46698e18 −0.925194 −0.462597 0.886569i \(-0.653082\pi\)
−0.462597 + 0.886569i \(0.653082\pi\)
\(632\) 1.82921e18 1.14183
\(633\) −9.05032e17 −0.559165
\(634\) −2.28926e18 −1.39996
\(635\) −3.88670e17 −0.235262
\(636\) −2.12268e18 −1.27178
\(637\) 2.79074e18 1.65506
\(638\) −2.24335e18 −1.31693
\(639\) −7.68362e17 −0.446488
\(640\) −6.59226e17 −0.379196
\(641\) −3.15148e18 −1.79448 −0.897238 0.441546i \(-0.854430\pi\)
−0.897238 + 0.441546i \(0.854430\pi\)
\(642\) −4.55770e17 −0.256902
\(643\) 4.59086e16 0.0256167 0.0128083 0.999918i \(-0.495923\pi\)
0.0128083 + 0.999918i \(0.495923\pi\)
\(644\) 8.91480e17 0.492440
\(645\) −5.26698e16 −0.0288021
\(646\) 4.35394e18 2.35706
\(647\) −2.24460e18 −1.20299 −0.601493 0.798878i \(-0.705427\pi\)
−0.601493 + 0.798878i \(0.705427\pi\)
\(648\) −1.02063e18 −0.541540
\(649\) −3.61793e17 −0.190051
\(650\) 2.83510e18 1.47445
\(651\) −2.19965e18 −1.13260
\(652\) −5.83047e17 −0.297231
\(653\) −7.53849e17 −0.380495 −0.190247 0.981736i \(-0.560929\pi\)
−0.190247 + 0.981736i \(0.560929\pi\)
\(654\) −3.99019e17 −0.199406
\(655\) −6.07519e16 −0.0300602
\(656\) 1.52551e17 0.0747377
\(657\) 2.93923e18 1.42580
\(658\) 9.66326e17 0.464147
\(659\) 3.06841e18 1.45935 0.729673 0.683796i \(-0.239672\pi\)
0.729673 + 0.683796i \(0.239672\pi\)
\(660\) −1.51898e17 −0.0715349
\(661\) 1.33227e18 0.621273 0.310636 0.950529i \(-0.399458\pi\)
0.310636 + 0.950529i \(0.399458\pi\)
\(662\) −2.06713e18 −0.954531
\(663\) −7.84074e17 −0.358525
\(664\) 2.42167e18 1.09653
\(665\) −1.33377e18 −0.598052
\(666\) −2.82963e18 −1.25646
\(667\) 7.20024e17 0.316613
\(668\) 6.96013e18 3.03089
\(669\) −5.51503e17 −0.237836
\(670\) −1.22833e18 −0.524600
\(671\) −1.92423e18 −0.813880
\(672\) 1.50638e18 0.631005
\(673\) −1.45836e18 −0.605016 −0.302508 0.953147i \(-0.597824\pi\)
−0.302508 + 0.953147i \(0.597824\pi\)
\(674\) 6.82424e18 2.80392
\(675\) 1.79983e18 0.732413
\(676\) −4.06899e17 −0.163996
\(677\) −3.65197e18 −1.45781 −0.728905 0.684615i \(-0.759970\pi\)
−0.728905 + 0.684615i \(0.759970\pi\)
\(678\) −8.85171e17 −0.349973
\(679\) −2.85676e18 −1.11872
\(680\) −5.62194e17 −0.218061
\(681\) 6.57984e16 0.0252790
\(682\) 3.14322e18 1.19612
\(683\) 1.94656e18 0.733724 0.366862 0.930275i \(-0.380432\pi\)
0.366862 + 0.930275i \(0.380432\pi\)
\(684\) −5.93896e18 −2.21741
\(685\) −5.19268e17 −0.192045
\(686\) −5.48745e18 −2.01031
\(687\) 1.90627e18 0.691777
\(688\) 1.01316e17 0.0364212
\(689\) −4.79629e18 −1.70797
\(690\) 7.79420e16 0.0274949
\(691\) 1.92534e18 0.672822 0.336411 0.941715i \(-0.390787\pi\)
0.336411 + 0.941715i \(0.390787\pi\)
\(692\) 4.93797e18 1.70946
\(693\) −1.79603e18 −0.615952
\(694\) 6.95017e18 2.36134
\(695\) −8.36503e16 −0.0281557
\(696\) −2.47314e18 −0.824687
\(697\) −1.68799e18 −0.557644
\(698\) 8.58675e18 2.81040
\(699\) −2.00449e18 −0.649982
\(700\) −8.18120e18 −2.62833
\(701\) −2.24425e18 −0.714339 −0.357169 0.934040i \(-0.616258\pi\)
−0.357169 + 0.934040i \(0.616258\pi\)
\(702\) 3.79304e18 1.19618
\(703\) −4.84951e18 −1.51526
\(704\) −2.32749e18 −0.720553
\(705\) 5.28463e16 0.0162101
\(706\) 5.69320e18 1.73032
\(707\) 8.01986e18 2.41513
\(708\) −9.93917e17 −0.296575
\(709\) 1.91148e18 0.565156 0.282578 0.959244i \(-0.408810\pi\)
0.282578 + 0.959244i \(0.408810\pi\)
\(710\) 6.77400e17 0.198457
\(711\) 2.94749e18 0.855658
\(712\) −1.61454e18 −0.464438
\(713\) −1.00884e18 −0.287569
\(714\) 3.61722e18 1.02173
\(715\) −3.43222e17 −0.0960695
\(716\) −1.90177e18 −0.527500
\(717\) −6.11340e17 −0.168038
\(718\) 4.97783e18 1.35591
\(719\) −3.58784e18 −0.968491 −0.484246 0.874932i \(-0.660906\pi\)
−0.484246 + 0.874932i \(0.660906\pi\)
\(720\) 8.18691e16 0.0219007
\(721\) −8.08772e18 −2.14411
\(722\) −1.00526e19 −2.64110
\(723\) 4.71072e17 0.122656
\(724\) −6.57320e18 −1.69620
\(725\) −6.60773e18 −1.68988
\(726\) 2.16899e18 0.549756
\(727\) −1.76161e18 −0.442524 −0.221262 0.975214i \(-0.571018\pi\)
−0.221262 + 0.975214i \(0.571018\pi\)
\(728\) −6.91887e18 −1.72259
\(729\) −3.22089e17 −0.0794781
\(730\) −2.59127e18 −0.633744
\(731\) −1.12108e18 −0.271752
\(732\) −5.28624e18 −1.27006
\(733\) −4.58156e18 −1.09103 −0.545516 0.838101i \(-0.683666\pi\)
−0.545516 + 0.838101i \(0.683666\pi\)
\(734\) 4.60334e18 1.08655
\(735\) −7.04090e17 −0.164725
\(736\) 6.90881e17 0.160213
\(737\) −2.83476e18 −0.651595
\(738\) 3.68101e18 0.838689
\(739\) 2.02965e18 0.458387 0.229193 0.973381i \(-0.426391\pi\)
0.229193 + 0.973381i \(0.426391\pi\)
\(740\) 1.56042e18 0.349329
\(741\) 2.93037e18 0.650285
\(742\) 2.21270e19 4.86741
\(743\) 4.10138e18 0.894340 0.447170 0.894449i \(-0.352432\pi\)
0.447170 + 0.894449i \(0.352432\pi\)
\(744\) 3.46518e18 0.749034
\(745\) 2.98707e17 0.0640074
\(746\) −1.36205e19 −2.89327
\(747\) 3.90215e18 0.821713
\(748\) −3.23316e18 −0.674942
\(749\) 2.97178e18 0.615013
\(750\) −1.46808e18 −0.301199
\(751\) −2.47765e18 −0.503943 −0.251971 0.967735i \(-0.581079\pi\)
−0.251971 + 0.967735i \(0.581079\pi\)
\(752\) −1.01656e17 −0.0204982
\(753\) −2.43449e18 −0.486676
\(754\) −1.39255e19 −2.75991
\(755\) 2.42386e17 0.0476269
\(756\) −1.09455e19 −2.13228
\(757\) −4.82152e18 −0.931239 −0.465619 0.884985i \(-0.654168\pi\)
−0.465619 + 0.884985i \(0.654168\pi\)
\(758\) −1.05586e19 −2.02189
\(759\) 1.79876e17 0.0341508
\(760\) 2.10112e18 0.395515
\(761\) 1.03824e19 1.93775 0.968873 0.247559i \(-0.0796283\pi\)
0.968873 + 0.247559i \(0.0796283\pi\)
\(762\) −3.93951e18 −0.729012
\(763\) 2.60174e18 0.477369
\(764\) 2.35122e17 0.0427746
\(765\) −9.05890e17 −0.163409
\(766\) −2.39020e18 −0.427512
\(767\) −2.24580e18 −0.398292
\(768\) −2.85363e18 −0.501822
\(769\) −8.18910e18 −1.42796 −0.713978 0.700168i \(-0.753109\pi\)
−0.713978 + 0.700168i \(0.753109\pi\)
\(770\) 1.58341e18 0.273781
\(771\) −2.91216e18 −0.499300
\(772\) −5.78860e18 −0.984150
\(773\) 4.20651e18 0.709178 0.354589 0.935022i \(-0.384621\pi\)
0.354589 + 0.935022i \(0.384621\pi\)
\(774\) 2.44474e18 0.408711
\(775\) 9.25826e18 1.53486
\(776\) 4.50035e18 0.739852
\(777\) −4.02893e18 −0.656831
\(778\) −1.21372e19 −1.96223
\(779\) 6.30864e18 1.01144
\(780\) −9.42898e17 −0.149917
\(781\) 1.56332e18 0.246499
\(782\) 1.65899e18 0.259419
\(783\) −8.84040e18 −1.37095
\(784\) 1.35440e18 0.208301
\(785\) −1.26968e18 −0.193661
\(786\) −6.15773e17 −0.0931480
\(787\) −1.12250e19 −1.68404 −0.842020 0.539447i \(-0.818633\pi\)
−0.842020 + 0.539447i \(0.818633\pi\)
\(788\) −7.99784e17 −0.119001
\(789\) 1.56186e17 0.0230485
\(790\) −2.59856e18 −0.380326
\(791\) 5.77162e18 0.837819
\(792\) 2.82934e18 0.407353
\(793\) −1.19445e19 −1.70566
\(794\) −1.54890e19 −2.19376
\(795\) 1.21008e18 0.169992
\(796\) −8.06473e18 −1.12371
\(797\) 4.39876e18 0.607927 0.303964 0.952684i \(-0.401690\pi\)
0.303964 + 0.952684i \(0.401690\pi\)
\(798\) −1.35189e19 −1.85320
\(799\) 1.12483e18 0.152944
\(800\) −6.34028e18 −0.855113
\(801\) −2.60158e18 −0.348038
\(802\) 1.46332e19 1.94180
\(803\) −5.98018e18 −0.787160
\(804\) −7.78765e18 −1.01682
\(805\) −5.08209e17 −0.0658216
\(806\) 1.95113e19 2.50673
\(807\) −3.61481e18 −0.460688
\(808\) −1.26339e19 −1.59722
\(809\) 2.90564e18 0.364399 0.182199 0.983262i \(-0.441678\pi\)
0.182199 + 0.983262i \(0.441678\pi\)
\(810\) 1.44990e18 0.180378
\(811\) 5.29580e18 0.653575 0.326788 0.945098i \(-0.394034\pi\)
0.326788 + 0.945098i \(0.394034\pi\)
\(812\) 4.01845e19 4.91976
\(813\) 4.80733e18 0.583869
\(814\) 5.75719e18 0.693669
\(815\) 3.32379e17 0.0397291
\(816\) −3.80525e17 −0.0451229
\(817\) 4.18987e18 0.492898
\(818\) 8.03353e18 0.937584
\(819\) −1.11487e19 −1.29086
\(820\) −2.02992e18 −0.233178
\(821\) 6.82729e18 0.778068 0.389034 0.921223i \(-0.372809\pi\)
0.389034 + 0.921223i \(0.372809\pi\)
\(822\) −5.26323e18 −0.595093
\(823\) −6.91413e18 −0.775602 −0.387801 0.921743i \(-0.626765\pi\)
−0.387801 + 0.921743i \(0.626765\pi\)
\(824\) 1.27408e19 1.41798
\(825\) −1.65074e18 −0.182275
\(826\) 1.03607e19 1.13506
\(827\) 5.09534e18 0.553843 0.276922 0.960892i \(-0.410686\pi\)
0.276922 + 0.960892i \(0.410686\pi\)
\(828\) −2.26294e18 −0.244048
\(829\) −8.76965e18 −0.938378 −0.469189 0.883098i \(-0.655454\pi\)
−0.469189 + 0.883098i \(0.655454\pi\)
\(830\) −3.44020e18 −0.365238
\(831\) −4.44887e17 −0.0468644
\(832\) −1.44477e19 −1.51007
\(833\) −1.49865e19 −1.55421
\(834\) −8.47867e17 −0.0872466
\(835\) −3.96778e18 −0.405122
\(836\) 1.20835e19 1.22420
\(837\) 1.23865e19 1.24518
\(838\) −9.63146e18 −0.960740
\(839\) 1.11763e18 0.110623 0.0553113 0.998469i \(-0.482385\pi\)
0.0553113 + 0.998469i \(0.482385\pi\)
\(840\) 1.74560e18 0.171446
\(841\) 2.21953e19 2.16315
\(842\) 9.99656e18 0.966767
\(843\) 7.10893e18 0.682221
\(844\) −2.31657e19 −2.20607
\(845\) 2.31962e17 0.0219204
\(846\) −2.45293e18 −0.230026
\(847\) −1.41426e19 −1.31609
\(848\) −2.32773e18 −0.214961
\(849\) 2.53920e18 0.232700
\(850\) −1.52248e19 −1.38461
\(851\) −1.84782e18 −0.166770
\(852\) 4.29474e18 0.384662
\(853\) −4.03242e17 −0.0358424 −0.0179212 0.999839i \(-0.505705\pi\)
−0.0179212 + 0.999839i \(0.505705\pi\)
\(854\) 5.51044e19 4.86083
\(855\) 3.38564e18 0.296388
\(856\) −4.68153e18 −0.406732
\(857\) 1.64069e18 0.141466 0.0707330 0.997495i \(-0.477466\pi\)
0.0707330 + 0.997495i \(0.477466\pi\)
\(858\) −3.47885e18 −0.297692
\(859\) 1.86876e19 1.58708 0.793538 0.608521i \(-0.208237\pi\)
0.793538 + 0.608521i \(0.208237\pi\)
\(860\) −1.34817e18 −0.113633
\(861\) 5.24117e18 0.438437
\(862\) −2.11007e19 −1.75186
\(863\) 1.18272e19 0.974563 0.487282 0.873245i \(-0.337989\pi\)
0.487282 + 0.873245i \(0.337989\pi\)
\(864\) −8.48258e18 −0.693727
\(865\) −2.81500e18 −0.228493
\(866\) 7.05621e18 0.568467
\(867\) −1.08401e18 −0.0866778
\(868\) −5.63035e19 −4.46845
\(869\) −5.99700e18 −0.472395
\(870\) 3.51332e18 0.274690
\(871\) −1.75966e19 −1.36556
\(872\) −4.09860e18 −0.315703
\(873\) 7.25162e18 0.554425
\(874\) −6.20026e18 −0.470528
\(875\) 9.57241e18 0.721056
\(876\) −1.64287e19 −1.22837
\(877\) 5.36616e18 0.398260 0.199130 0.979973i \(-0.436188\pi\)
0.199130 + 0.979973i \(0.436188\pi\)
\(878\) −2.64124e18 −0.194578
\(879\) −7.97391e18 −0.583101
\(880\) −1.66572e17 −0.0120910
\(881\) −2.33175e19 −1.68011 −0.840057 0.542498i \(-0.817479\pi\)
−0.840057 + 0.542498i \(0.817479\pi\)
\(882\) 3.26812e19 2.33751
\(883\) −8.04431e17 −0.0571142 −0.0285571 0.999592i \(-0.509091\pi\)
−0.0285571 + 0.999592i \(0.509091\pi\)
\(884\) −2.00696e19 −1.41449
\(885\) 5.66605e17 0.0396414
\(886\) 3.20138e19 2.22340
\(887\) 6.74205e18 0.464824 0.232412 0.972617i \(-0.425338\pi\)
0.232412 + 0.972617i \(0.425338\pi\)
\(888\) 6.34690e18 0.434388
\(889\) 2.56869e19 1.74522
\(890\) 2.29360e18 0.154697
\(891\) 3.34610e18 0.224044
\(892\) −1.41166e19 −0.938334
\(893\) −4.20390e18 −0.277408
\(894\) 3.02766e18 0.198341
\(895\) 1.08415e18 0.0705079
\(896\) 4.35677e19 2.81296
\(897\) 1.11657e18 0.0715704
\(898\) −5.57674e17 −0.0354882
\(899\) −4.54748e19 −2.87298
\(900\) 2.07672e19 1.30257
\(901\) 2.57565e19 1.60390
\(902\) −7.48943e18 −0.463027
\(903\) 3.48091e18 0.213660
\(904\) −9.09221e18 −0.554083
\(905\) 3.74720e18 0.226721
\(906\) 2.45679e18 0.147582
\(907\) −5.92732e17 −0.0353517 −0.0176759 0.999844i \(-0.505627\pi\)
−0.0176759 + 0.999844i \(0.505627\pi\)
\(908\) 1.68421e18 0.0997330
\(909\) −2.03577e19 −1.19691
\(910\) 9.82889e18 0.573767
\(911\) −3.49053e18 −0.202312 −0.101156 0.994871i \(-0.532254\pi\)
−0.101156 + 0.994871i \(0.532254\pi\)
\(912\) 1.42216e18 0.0818431
\(913\) −7.93936e18 −0.453655
\(914\) 4.71383e19 2.67438
\(915\) 3.01354e18 0.169762
\(916\) 4.87939e19 2.72926
\(917\) 4.01505e18 0.222992
\(918\) −2.03690e19 −1.12329
\(919\) −2.99518e19 −1.64011 −0.820054 0.572286i \(-0.806057\pi\)
−0.820054 + 0.572286i \(0.806057\pi\)
\(920\) 8.00597e17 0.0435304
\(921\) 1.01038e19 0.545501
\(922\) 3.30376e19 1.77116
\(923\) 9.70417e18 0.516592
\(924\) 1.00389e19 0.530660
\(925\) 1.69576e19 0.890111
\(926\) −1.96716e19 −1.02534
\(927\) 2.05299e19 1.06260
\(928\) 3.11422e19 1.60062
\(929\) −4.24088e17 −0.0216448 −0.0108224 0.999941i \(-0.503445\pi\)
−0.0108224 + 0.999941i \(0.503445\pi\)
\(930\) −4.92260e18 −0.249492
\(931\) 5.60101e19 2.81899
\(932\) −5.13079e19 −2.56437
\(933\) 2.62684e16 0.00130378
\(934\) 3.65049e19 1.79927
\(935\) 1.84313e18 0.0902155
\(936\) 1.75629e19 0.853696
\(937\) −1.16943e19 −0.564502 −0.282251 0.959341i \(-0.591081\pi\)
−0.282251 + 0.959341i \(0.591081\pi\)
\(938\) 8.11794e19 3.89159
\(939\) 7.02854e18 0.334610
\(940\) 1.35268e18 0.0639535
\(941\) 3.34650e19 1.57130 0.785648 0.618674i \(-0.212330\pi\)
0.785648 + 0.618674i \(0.212330\pi\)
\(942\) −1.28693e19 −0.600102
\(943\) 2.40380e18 0.111320
\(944\) −1.08993e18 −0.0501280
\(945\) 6.23975e18 0.285010
\(946\) −4.97409e18 −0.225643
\(947\) −2.95774e19 −1.33255 −0.666276 0.745705i \(-0.732113\pi\)
−0.666276 + 0.745705i \(0.732113\pi\)
\(948\) −1.64750e19 −0.737174
\(949\) −3.71215e19 −1.64966
\(950\) 5.69004e19 2.51138
\(951\) 8.27404e18 0.362697
\(952\) 3.71550e19 1.61762
\(953\) −1.82956e19 −0.791120 −0.395560 0.918440i \(-0.629450\pi\)
−0.395560 + 0.918440i \(0.629450\pi\)
\(954\) −5.61674e19 −2.41224
\(955\) −1.34037e17 −0.00571744
\(956\) −1.56482e19 −0.662961
\(957\) 8.10811e18 0.341187
\(958\) 4.93905e19 2.06428
\(959\) 3.43180e19 1.42463
\(960\) 3.64509e18 0.150295
\(961\) 3.92983e19 1.60943
\(962\) 3.57373e19 1.45373
\(963\) −7.54358e18 −0.304794
\(964\) 1.20578e19 0.483913
\(965\) 3.29992e18 0.131546
\(966\) −5.15113e18 −0.203963
\(967\) 1.97967e19 0.778610 0.389305 0.921109i \(-0.372715\pi\)
0.389305 + 0.921109i \(0.372715\pi\)
\(968\) 2.22792e19 0.870382
\(969\) −1.57364e19 −0.610660
\(970\) −6.39315e18 −0.246433
\(971\) −2.43674e19 −0.933005 −0.466503 0.884520i \(-0.654486\pi\)
−0.466503 + 0.884520i \(0.654486\pi\)
\(972\) 4.30438e19 1.63712
\(973\) 5.52839e18 0.208865
\(974\) −1.54733e19 −0.580698
\(975\) −1.02468e19 −0.381997
\(976\) −5.79689e18 −0.214670
\(977\) 3.07695e19 1.13189 0.565947 0.824442i \(-0.308511\pi\)
0.565947 + 0.824442i \(0.308511\pi\)
\(978\) 3.36895e18 0.123110
\(979\) 5.29321e18 0.192146
\(980\) −1.80223e19 −0.649890
\(981\) −6.60427e18 −0.236579
\(982\) −8.14905e19 −2.89990
\(983\) −3.07549e19 −1.08722 −0.543609 0.839339i \(-0.682943\pi\)
−0.543609 + 0.839339i \(0.682943\pi\)
\(984\) −8.25658e18 −0.289956
\(985\) 4.55935e17 0.0159062
\(986\) 7.47811e19 2.59174
\(987\) −3.49257e18 −0.120250
\(988\) 7.50073e19 2.56557
\(989\) 1.59648e18 0.0542484
\(990\) −4.01933e18 −0.135683
\(991\) −2.75978e19 −0.925541 −0.462770 0.886478i \(-0.653145\pi\)
−0.462770 + 0.886478i \(0.653145\pi\)
\(992\) −4.36342e19 −1.45379
\(993\) 7.47117e18 0.247297
\(994\) −4.47689e19 −1.47219
\(995\) 4.59748e18 0.150200
\(996\) −2.18110e19 −0.707930
\(997\) −1.21160e19 −0.390697 −0.195348 0.980734i \(-0.562584\pi\)
−0.195348 + 0.980734i \(0.562584\pi\)
\(998\) −5.27159e19 −1.68886
\(999\) 2.26874e19 0.722120
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 197.14.a.a.1.12 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.14.a.a.1.12 104 1.1 even 1 trivial