Properties

Label 197.14.a.a.1.5
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 1.5
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)
 
\(f(q)\) \(=\) \(q-167.056 q^{2} +2054.37 q^{3} +19715.7 q^{4} +12824.5 q^{5} -343195. q^{6} +34898.6 q^{7} -1.92511e6 q^{8} +2.62611e6 q^{9} +O(q^{10})\) \(q-167.056 q^{2} +2054.37 q^{3} +19715.7 q^{4} +12824.5 q^{5} -343195. q^{6} +34898.6 q^{7} -1.92511e6 q^{8} +2.62611e6 q^{9} -2.14242e6 q^{10} +5.93755e6 q^{11} +4.05034e7 q^{12} -1.86151e7 q^{13} -5.83001e6 q^{14} +2.63464e7 q^{15} +1.60089e8 q^{16} -1.38966e8 q^{17} -4.38708e8 q^{18} +7.18692e7 q^{19} +2.52845e8 q^{20} +7.16945e7 q^{21} -9.91903e8 q^{22} -3.24279e8 q^{23} -3.95488e9 q^{24} -1.05623e9 q^{25} +3.10977e9 q^{26} +2.11968e9 q^{27} +6.88050e8 q^{28} -4.79502e8 q^{29} -4.40132e9 q^{30} +2.49742e9 q^{31} -1.09734e10 q^{32} +1.21979e10 q^{33} +2.32150e10 q^{34} +4.47558e8 q^{35} +5.17757e10 q^{36} +1.94028e9 q^{37} -1.20062e10 q^{38} -3.82424e10 q^{39} -2.46886e10 q^{40} +2.36739e10 q^{41} -1.19770e10 q^{42} +3.90451e10 q^{43} +1.17063e11 q^{44} +3.36787e10 q^{45} +5.41727e10 q^{46} +7.26691e10 q^{47} +3.28883e11 q^{48} -9.56711e10 q^{49} +1.76450e11 q^{50} -2.85487e11 q^{51} -3.67011e11 q^{52} -7.11782e10 q^{53} -3.54105e11 q^{54} +7.61463e10 q^{55} -6.71834e10 q^{56} +1.47646e11 q^{57} +8.01036e10 q^{58} -5.26765e11 q^{59} +5.19437e11 q^{60} +1.73926e11 q^{61} -4.17210e11 q^{62} +9.16475e10 q^{63} +5.21724e11 q^{64} -2.38731e11 q^{65} -2.03774e12 q^{66} -6.11862e11 q^{67} -2.73980e12 q^{68} -6.66189e11 q^{69} -7.47673e10 q^{70} -1.18757e12 q^{71} -5.05554e12 q^{72} +1.79915e12 q^{73} -3.24135e11 q^{74} -2.16990e12 q^{75} +1.41695e12 q^{76} +2.07212e11 q^{77} +6.38862e12 q^{78} -4.18373e12 q^{79} +2.05307e12 q^{80} +1.67728e11 q^{81} -3.95486e12 q^{82} +2.99732e12 q^{83} +1.41351e12 q^{84} -1.78217e12 q^{85} -6.52272e12 q^{86} -9.85073e11 q^{87} -1.14304e13 q^{88} -9.94142e11 q^{89} -5.62623e12 q^{90} -6.49642e11 q^{91} -6.39339e12 q^{92} +5.13063e12 q^{93} -1.21398e13 q^{94} +9.21689e11 q^{95} -2.25435e13 q^{96} -3.98983e12 q^{97} +1.59824e13 q^{98} +1.55927e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q - 128 q^{2} - 8020 q^{3} + 409600 q^{4} - 99004 q^{5} - 83328 q^{6} - 2084037 q^{7} - 2111301 q^{8} + 51549776 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −167.056 −1.84573 −0.922863 0.385129i \(-0.874157\pi\)
−0.922863 + 0.385129i \(0.874157\pi\)
\(3\) 2054.37 1.62701 0.813505 0.581557i \(-0.197556\pi\)
0.813505 + 0.581557i \(0.197556\pi\)
\(4\) 19715.7 2.40670
\(5\) 12824.5 0.367060 0.183530 0.983014i \(-0.441248\pi\)
0.183530 + 0.983014i \(0.441248\pi\)
\(6\) −343195. −3.00302
\(7\) 34898.6 0.112117 0.0560583 0.998427i \(-0.482147\pi\)
0.0560583 + 0.998427i \(0.482147\pi\)
\(8\) −1.92511e6 −2.59639
\(9\) 2.62611e6 1.64716
\(10\) −2.14242e6 −0.677492
\(11\) 5.93755e6 1.01054 0.505271 0.862960i \(-0.331392\pi\)
0.505271 + 0.862960i \(0.331392\pi\)
\(12\) 4.05034e7 3.91573
\(13\) −1.86151e7 −1.06963 −0.534816 0.844968i \(-0.679619\pi\)
−0.534816 + 0.844968i \(0.679619\pi\)
\(14\) −5.83001e6 −0.206937
\(15\) 2.63464e7 0.597210
\(16\) 1.60089e8 2.38552
\(17\) −1.38966e8 −1.39633 −0.698167 0.715935i \(-0.746001\pi\)
−0.698167 + 0.715935i \(0.746001\pi\)
\(18\) −4.38708e8 −3.04021
\(19\) 7.18692e7 0.350465 0.175232 0.984527i \(-0.443932\pi\)
0.175232 + 0.984527i \(0.443932\pi\)
\(20\) 2.52845e8 0.883404
\(21\) 7.16945e7 0.182415
\(22\) −9.91903e8 −1.86518
\(23\) −3.24279e8 −0.456760 −0.228380 0.973572i \(-0.573343\pi\)
−0.228380 + 0.973572i \(0.573343\pi\)
\(24\) −3.95488e9 −4.22435
\(25\) −1.05623e9 −0.865267
\(26\) 3.10977e9 1.97425
\(27\) 2.11968e9 1.05294
\(28\) 6.88050e8 0.269832
\(29\) −4.79502e8 −0.149694 −0.0748468 0.997195i \(-0.523847\pi\)
−0.0748468 + 0.997195i \(0.523847\pi\)
\(30\) −4.40132e9 −1.10229
\(31\) 2.49742e9 0.505407 0.252703 0.967544i \(-0.418680\pi\)
0.252703 + 0.967544i \(0.418680\pi\)
\(32\) −1.09734e10 −1.80662
\(33\) 1.21979e10 1.64416
\(34\) 2.32150e10 2.57725
\(35\) 4.47558e8 0.0411535
\(36\) 5.17757e10 3.96424
\(37\) 1.94028e9 0.124323 0.0621617 0.998066i \(-0.480201\pi\)
0.0621617 + 0.998066i \(0.480201\pi\)
\(38\) −1.20062e10 −0.646862
\(39\) −3.82424e10 −1.74030
\(40\) −2.46886e10 −0.953030
\(41\) 2.36739e10 0.778348 0.389174 0.921164i \(-0.372760\pi\)
0.389174 + 0.921164i \(0.372760\pi\)
\(42\) −1.19770e10 −0.336688
\(43\) 3.90451e10 0.941936 0.470968 0.882150i \(-0.343905\pi\)
0.470968 + 0.882150i \(0.343905\pi\)
\(44\) 1.17063e11 2.43208
\(45\) 3.36787e10 0.604608
\(46\) 5.41727e10 0.843053
\(47\) 7.26691e10 0.983363 0.491682 0.870775i \(-0.336382\pi\)
0.491682 + 0.870775i \(0.336382\pi\)
\(48\) 3.28883e11 3.88126
\(49\) −9.56711e10 −0.987430
\(50\) 1.76450e11 1.59705
\(51\) −2.85487e11 −2.27185
\(52\) −3.67011e11 −2.57429
\(53\) −7.11782e10 −0.441117 −0.220559 0.975374i \(-0.570788\pi\)
−0.220559 + 0.975374i \(0.570788\pi\)
\(54\) −3.54105e11 −1.94344
\(55\) 7.61463e10 0.370930
\(56\) −6.71834e10 −0.291098
\(57\) 1.47646e11 0.570210
\(58\) 8.01036e10 0.276293
\(59\) −5.26765e11 −1.62584 −0.812922 0.582372i \(-0.802125\pi\)
−0.812922 + 0.582372i \(0.802125\pi\)
\(60\) 5.19437e11 1.43731
\(61\) 1.73926e11 0.432235 0.216117 0.976367i \(-0.430661\pi\)
0.216117 + 0.976367i \(0.430661\pi\)
\(62\) −4.17210e11 −0.932842
\(63\) 9.16475e10 0.184675
\(64\) 5.21724e11 0.949010
\(65\) −2.38731e11 −0.392619
\(66\) −2.03774e12 −3.03468
\(67\) −6.11862e11 −0.826356 −0.413178 0.910650i \(-0.635581\pi\)
−0.413178 + 0.910650i \(0.635581\pi\)
\(68\) −2.73980e12 −3.36056
\(69\) −6.66189e11 −0.743153
\(70\) −7.47673e10 −0.0759581
\(71\) −1.18757e12 −1.10022 −0.550111 0.835092i \(-0.685415\pi\)
−0.550111 + 0.835092i \(0.685415\pi\)
\(72\) −5.05554e12 −4.27668
\(73\) 1.79915e12 1.39145 0.695726 0.718307i \(-0.255083\pi\)
0.695726 + 0.718307i \(0.255083\pi\)
\(74\) −3.24135e11 −0.229467
\(75\) −2.16990e12 −1.40780
\(76\) 1.41695e12 0.843464
\(77\) 2.07212e11 0.113299
\(78\) 6.38862e12 3.21212
\(79\) −4.18373e12 −1.93637 −0.968183 0.250243i \(-0.919489\pi\)
−0.968183 + 0.250243i \(0.919489\pi\)
\(80\) 2.05307e12 0.875627
\(81\) 1.67728e11 0.0659861
\(82\) −3.95486e12 −1.43662
\(83\) 2.99732e12 1.00630 0.503148 0.864200i \(-0.332175\pi\)
0.503148 + 0.864200i \(0.332175\pi\)
\(84\) 1.41351e12 0.439019
\(85\) −1.78217e12 −0.512538
\(86\) −6.52272e12 −1.73856
\(87\) −9.85073e11 −0.243553
\(88\) −1.14304e13 −2.62376
\(89\) −9.94142e11 −0.212038 −0.106019 0.994364i \(-0.533810\pi\)
−0.106019 + 0.994364i \(0.533810\pi\)
\(90\) −5.62623e12 −1.11594
\(91\) −6.49642e11 −0.119924
\(92\) −6.39339e12 −1.09929
\(93\) 5.13063e12 0.822302
\(94\) −1.21398e13 −1.81502
\(95\) 9.21689e11 0.128641
\(96\) −2.25435e13 −2.93939
\(97\) −3.98983e12 −0.486338 −0.243169 0.969984i \(-0.578187\pi\)
−0.243169 + 0.969984i \(0.578187\pi\)
\(98\) 1.59824e13 1.82252
\(99\) 1.55927e13 1.66453
\(100\) −2.08244e13 −2.08244
\(101\) 4.40567e12 0.412974 0.206487 0.978449i \(-0.433797\pi\)
0.206487 + 0.978449i \(0.433797\pi\)
\(102\) 4.76922e13 4.19321
\(103\) −4.67850e12 −0.386069 −0.193035 0.981192i \(-0.561833\pi\)
−0.193035 + 0.981192i \(0.561833\pi\)
\(104\) 3.58361e13 2.77718
\(105\) 9.19450e11 0.0669572
\(106\) 1.18908e13 0.814182
\(107\) −2.54486e13 −1.63934 −0.819669 0.572837i \(-0.805843\pi\)
−0.819669 + 0.572837i \(0.805843\pi\)
\(108\) 4.17909e13 2.53412
\(109\) 1.07009e13 0.611149 0.305575 0.952168i \(-0.401151\pi\)
0.305575 + 0.952168i \(0.401151\pi\)
\(110\) −1.27207e13 −0.684634
\(111\) 3.98605e12 0.202276
\(112\) 5.58689e12 0.267456
\(113\) −1.48463e13 −0.670825 −0.335412 0.942071i \(-0.608876\pi\)
−0.335412 + 0.942071i \(0.608876\pi\)
\(114\) −2.46651e13 −1.05245
\(115\) −4.15873e12 −0.167658
\(116\) −9.45371e12 −0.360268
\(117\) −4.88855e13 −1.76186
\(118\) 8.79993e13 3.00086
\(119\) −4.84970e12 −0.156552
\(120\) −5.07195e13 −1.55059
\(121\) 7.31770e11 0.0211968
\(122\) −2.90553e13 −0.797787
\(123\) 4.86349e13 1.26638
\(124\) 4.92385e13 1.21636
\(125\) −2.92007e13 −0.684665
\(126\) −1.53103e13 −0.340859
\(127\) −2.02281e13 −0.427791 −0.213896 0.976857i \(-0.568615\pi\)
−0.213896 + 0.976857i \(0.568615\pi\)
\(128\) 2.73713e12 0.0550085
\(129\) 8.02131e13 1.53254
\(130\) 3.98814e13 0.724667
\(131\) 7.46222e13 1.29004 0.645022 0.764164i \(-0.276848\pi\)
0.645022 + 0.764164i \(0.276848\pi\)
\(132\) 2.40491e14 3.95701
\(133\) 2.50813e12 0.0392929
\(134\) 1.02215e14 1.52523
\(135\) 2.71839e13 0.386493
\(136\) 2.67523e14 3.62542
\(137\) −3.22732e13 −0.417021 −0.208511 0.978020i \(-0.566862\pi\)
−0.208511 + 0.978020i \(0.566862\pi\)
\(138\) 1.11291e14 1.37166
\(139\) −4.96884e13 −0.584331 −0.292166 0.956368i \(-0.594376\pi\)
−0.292166 + 0.956368i \(0.594376\pi\)
\(140\) 8.82392e12 0.0990443
\(141\) 1.49289e14 1.59994
\(142\) 1.98391e14 2.03071
\(143\) −1.10528e14 −1.08091
\(144\) 4.20412e14 3.92934
\(145\) −6.14939e12 −0.0549465
\(146\) −3.00559e14 −2.56824
\(147\) −1.96544e14 −1.60656
\(148\) 3.82540e13 0.299210
\(149\) −6.11378e13 −0.457719 −0.228860 0.973459i \(-0.573500\pi\)
−0.228860 + 0.973459i \(0.573500\pi\)
\(150\) 3.62494e14 2.59841
\(151\) −1.40643e14 −0.965533 −0.482767 0.875749i \(-0.660368\pi\)
−0.482767 + 0.875749i \(0.660368\pi\)
\(152\) −1.38356e14 −0.909942
\(153\) −3.64939e14 −2.29999
\(154\) −3.46160e13 −0.209118
\(155\) 3.20283e13 0.185515
\(156\) −7.53976e14 −4.18839
\(157\) −2.45848e14 −1.31015 −0.655073 0.755566i \(-0.727362\pi\)
−0.655073 + 0.755566i \(0.727362\pi\)
\(158\) 6.98917e14 3.57400
\(159\) −1.46226e14 −0.717703
\(160\) −1.40729e14 −0.663138
\(161\) −1.13169e13 −0.0512104
\(162\) −2.80199e13 −0.121792
\(163\) −2.61548e14 −1.09227 −0.546137 0.837696i \(-0.683902\pi\)
−0.546137 + 0.837696i \(0.683902\pi\)
\(164\) 4.66747e14 1.87325
\(165\) 1.56433e14 0.603507
\(166\) −5.00720e14 −1.85735
\(167\) 2.72969e14 0.973771 0.486885 0.873466i \(-0.338133\pi\)
0.486885 + 0.873466i \(0.338133\pi\)
\(168\) −1.38020e14 −0.473620
\(169\) 4.36485e13 0.144114
\(170\) 2.97722e14 0.946004
\(171\) 1.88736e14 0.577273
\(172\) 7.69802e14 2.26696
\(173\) −3.14109e14 −0.890800 −0.445400 0.895332i \(-0.646938\pi\)
−0.445400 + 0.895332i \(0.646938\pi\)
\(174\) 1.64562e14 0.449532
\(175\) −3.68611e13 −0.0970109
\(176\) 9.50538e14 2.41067
\(177\) −1.08217e15 −2.64527
\(178\) 1.66077e14 0.391364
\(179\) −1.35863e14 −0.308713 −0.154357 0.988015i \(-0.549331\pi\)
−0.154357 + 0.988015i \(0.549331\pi\)
\(180\) 6.63999e14 1.45511
\(181\) 2.81928e14 0.595974 0.297987 0.954570i \(-0.403685\pi\)
0.297987 + 0.954570i \(0.403685\pi\)
\(182\) 1.08527e14 0.221346
\(183\) 3.57307e14 0.703250
\(184\) 6.24271e14 1.18593
\(185\) 2.48832e13 0.0456341
\(186\) −8.57103e14 −1.51774
\(187\) −8.25115e14 −1.41105
\(188\) 1.43272e15 2.36666
\(189\) 7.39736e13 0.118053
\(190\) −1.53974e14 −0.237437
\(191\) 1.04777e15 1.56153 0.780764 0.624825i \(-0.214830\pi\)
0.780764 + 0.624825i \(0.214830\pi\)
\(192\) 1.07181e15 1.54405
\(193\) 6.68990e14 0.931745 0.465873 0.884852i \(-0.345740\pi\)
0.465873 + 0.884852i \(0.345740\pi\)
\(194\) 6.66526e14 0.897647
\(195\) −4.90441e14 −0.638796
\(196\) −1.88622e15 −2.37645
\(197\) −5.84517e13 −0.0712470
\(198\) −2.60485e15 −3.07227
\(199\) 4.00401e14 0.457036 0.228518 0.973540i \(-0.426612\pi\)
0.228518 + 0.973540i \(0.426612\pi\)
\(200\) 2.03336e15 2.24657
\(201\) −1.25699e15 −1.34449
\(202\) −7.35994e14 −0.762238
\(203\) −1.67339e13 −0.0167831
\(204\) −5.62857e15 −5.46767
\(205\) 3.03606e14 0.285700
\(206\) 7.81572e14 0.712578
\(207\) −8.51593e14 −0.752358
\(208\) −2.98009e15 −2.55163
\(209\) 4.26727e14 0.354160
\(210\) −1.53600e14 −0.123585
\(211\) 8.38249e14 0.653939 0.326969 0.945035i \(-0.393973\pi\)
0.326969 + 0.945035i \(0.393973\pi\)
\(212\) −1.40333e15 −1.06164
\(213\) −2.43971e15 −1.79007
\(214\) 4.25133e15 3.02577
\(215\) 5.00736e14 0.345747
\(216\) −4.08060e15 −2.73385
\(217\) 8.71565e13 0.0566645
\(218\) −1.78765e15 −1.12801
\(219\) 3.69612e15 2.26391
\(220\) 1.50128e15 0.892718
\(221\) 2.58686e15 1.49356
\(222\) −6.65894e14 −0.373345
\(223\) −5.40493e14 −0.294312 −0.147156 0.989113i \(-0.547012\pi\)
−0.147156 + 0.989113i \(0.547012\pi\)
\(224\) −3.82956e14 −0.202552
\(225\) −2.77379e15 −1.42524
\(226\) 2.48017e15 1.23816
\(227\) 1.45466e15 0.705655 0.352828 0.935688i \(-0.385220\pi\)
0.352828 + 0.935688i \(0.385220\pi\)
\(228\) 2.91094e15 1.37233
\(229\) 9.69711e14 0.444336 0.222168 0.975008i \(-0.428687\pi\)
0.222168 + 0.975008i \(0.428687\pi\)
\(230\) 6.94741e14 0.309451
\(231\) 4.25690e14 0.184338
\(232\) 9.23091e14 0.388662
\(233\) −3.68613e15 −1.50923 −0.754617 0.656165i \(-0.772177\pi\)
−0.754617 + 0.656165i \(0.772177\pi\)
\(234\) 8.16661e15 3.25191
\(235\) 9.31949e14 0.360953
\(236\) −1.03856e16 −3.91293
\(237\) −8.59493e15 −3.15049
\(238\) 8.10171e14 0.288952
\(239\) 3.14117e15 1.09020 0.545098 0.838372i \(-0.316492\pi\)
0.545098 + 0.838372i \(0.316492\pi\)
\(240\) 4.21777e15 1.42465
\(241\) −2.69521e15 −0.886099 −0.443050 0.896497i \(-0.646103\pi\)
−0.443050 + 0.896497i \(0.646103\pi\)
\(242\) −1.22247e14 −0.0391234
\(243\) −3.03487e15 −0.945583
\(244\) 3.42907e15 1.04026
\(245\) −1.22694e15 −0.362446
\(246\) −8.12475e15 −2.33739
\(247\) −1.33785e15 −0.374868
\(248\) −4.80780e15 −1.31223
\(249\) 6.15761e15 1.63725
\(250\) 4.87815e15 1.26370
\(251\) 1.68949e15 0.426457 0.213229 0.977002i \(-0.431602\pi\)
0.213229 + 0.977002i \(0.431602\pi\)
\(252\) 1.80690e15 0.444457
\(253\) −1.92542e15 −0.461575
\(254\) 3.37923e15 0.789585
\(255\) −3.66123e15 −0.833905
\(256\) −4.73122e15 −1.05054
\(257\) 3.59727e15 0.778767 0.389383 0.921076i \(-0.372688\pi\)
0.389383 + 0.921076i \(0.372688\pi\)
\(258\) −1.34001e16 −2.82865
\(259\) 6.77130e13 0.0139387
\(260\) −4.70675e15 −0.944918
\(261\) −1.25922e15 −0.246570
\(262\) −1.24661e16 −2.38107
\(263\) −2.07601e15 −0.386828 −0.193414 0.981117i \(-0.561956\pi\)
−0.193414 + 0.981117i \(0.561956\pi\)
\(264\) −2.34823e16 −4.26889
\(265\) −9.12828e14 −0.161916
\(266\) −4.18998e14 −0.0725240
\(267\) −2.04234e15 −0.344988
\(268\) −1.20633e16 −1.98879
\(269\) −8.87878e15 −1.42877 −0.714387 0.699751i \(-0.753294\pi\)
−0.714387 + 0.699751i \(0.753294\pi\)
\(270\) −4.54123e15 −0.713360
\(271\) −1.70109e15 −0.260872 −0.130436 0.991457i \(-0.541638\pi\)
−0.130436 + 0.991457i \(0.541638\pi\)
\(272\) −2.22469e16 −3.33098
\(273\) −1.33460e15 −0.195117
\(274\) 5.39143e15 0.769707
\(275\) −6.27144e15 −0.874389
\(276\) −1.31344e16 −1.78855
\(277\) 1.01913e15 0.135554 0.0677771 0.997700i \(-0.478409\pi\)
0.0677771 + 0.997700i \(0.478409\pi\)
\(278\) 8.30075e15 1.07852
\(279\) 6.55851e15 0.832488
\(280\) −8.61596e14 −0.106851
\(281\) 1.98891e15 0.241004 0.120502 0.992713i \(-0.461550\pi\)
0.120502 + 0.992713i \(0.461550\pi\)
\(282\) −2.49397e16 −2.95306
\(283\) 4.15226e15 0.480477 0.240238 0.970714i \(-0.422774\pi\)
0.240238 + 0.970714i \(0.422774\pi\)
\(284\) −2.34138e16 −2.64791
\(285\) 1.89349e15 0.209301
\(286\) 1.84644e16 1.99506
\(287\) 8.26184e14 0.0872658
\(288\) −2.88174e16 −2.97580
\(289\) 9.40684e15 0.949746
\(290\) 1.02729e15 0.101416
\(291\) −8.19659e15 −0.791278
\(292\) 3.54715e16 3.34881
\(293\) −1.84819e16 −1.70651 −0.853254 0.521496i \(-0.825374\pi\)
−0.853254 + 0.521496i \(0.825374\pi\)
\(294\) 3.28338e16 2.96527
\(295\) −6.75552e15 −0.596782
\(296\) −3.73524e15 −0.322792
\(297\) 1.25857e16 1.06404
\(298\) 1.02134e16 0.844824
\(299\) 6.03650e15 0.488565
\(300\) −4.27810e16 −3.38815
\(301\) 1.36262e15 0.105607
\(302\) 2.34952e16 1.78211
\(303\) 9.05088e15 0.671914
\(304\) 1.15055e16 0.836039
\(305\) 2.23052e15 0.158656
\(306\) 6.09653e16 4.24515
\(307\) −1.04629e16 −0.713270 −0.356635 0.934244i \(-0.616076\pi\)
−0.356635 + 0.934244i \(0.616076\pi\)
\(308\) 4.08533e15 0.272676
\(309\) −9.61138e15 −0.628139
\(310\) −5.35052e15 −0.342409
\(311\) 1.46522e16 0.918248 0.459124 0.888372i \(-0.348163\pi\)
0.459124 + 0.888372i \(0.348163\pi\)
\(312\) 7.36206e16 4.51850
\(313\) −1.21812e16 −0.732235 −0.366118 0.930569i \(-0.619313\pi\)
−0.366118 + 0.930569i \(0.619313\pi\)
\(314\) 4.10704e16 2.41817
\(315\) 1.17534e15 0.0677866
\(316\) −8.24852e16 −4.66026
\(317\) −1.44293e16 −0.798655 −0.399328 0.916808i \(-0.630756\pi\)
−0.399328 + 0.916808i \(0.630756\pi\)
\(318\) 2.44280e16 1.32468
\(319\) −2.84706e15 −0.151272
\(320\) 6.69087e15 0.348344
\(321\) −5.22807e16 −2.66722
\(322\) 1.89055e15 0.0945203
\(323\) −9.98734e15 −0.489365
\(324\) 3.30687e15 0.158809
\(325\) 1.96620e16 0.925518
\(326\) 4.36931e16 2.01604
\(327\) 2.19836e16 0.994346
\(328\) −4.55747e16 −2.02089
\(329\) 2.53605e15 0.110251
\(330\) −2.61330e16 −1.11391
\(331\) 1.86828e16 0.780835 0.390418 0.920638i \(-0.372331\pi\)
0.390418 + 0.920638i \(0.372331\pi\)
\(332\) 5.90943e16 2.42186
\(333\) 5.09539e15 0.204781
\(334\) −4.56012e16 −1.79731
\(335\) −7.84685e15 −0.303322
\(336\) 1.14775e16 0.435154
\(337\) 2.95110e16 1.09746 0.548731 0.835999i \(-0.315111\pi\)
0.548731 + 0.835999i \(0.315111\pi\)
\(338\) −7.29174e15 −0.265994
\(339\) −3.04998e16 −1.09144
\(340\) −3.51367e16 −1.23353
\(341\) 1.48286e16 0.510735
\(342\) −3.15296e16 −1.06549
\(343\) −6.72007e15 −0.222824
\(344\) −7.51659e16 −2.44563
\(345\) −8.54357e15 −0.272782
\(346\) 5.24737e16 1.64417
\(347\) 4.68700e16 1.44130 0.720648 0.693301i \(-0.243845\pi\)
0.720648 + 0.693301i \(0.243845\pi\)
\(348\) −1.94214e16 −0.586160
\(349\) −1.98624e16 −0.588392 −0.294196 0.955745i \(-0.595052\pi\)
−0.294196 + 0.955745i \(0.595052\pi\)
\(350\) 6.15786e15 0.179055
\(351\) −3.94581e16 −1.12626
\(352\) −6.51552e16 −1.82567
\(353\) 1.12512e16 0.309502 0.154751 0.987953i \(-0.450542\pi\)
0.154751 + 0.987953i \(0.450542\pi\)
\(354\) 1.80783e17 4.88244
\(355\) −1.52301e16 −0.403847
\(356\) −1.96002e16 −0.510312
\(357\) −9.96307e15 −0.254712
\(358\) 2.26967e16 0.569800
\(359\) 3.80413e16 0.937867 0.468933 0.883234i \(-0.344638\pi\)
0.468933 + 0.883234i \(0.344638\pi\)
\(360\) −6.48350e16 −1.56980
\(361\) −3.68878e16 −0.877175
\(362\) −4.70978e16 −1.10001
\(363\) 1.50333e15 0.0344874
\(364\) −1.28081e16 −0.288621
\(365\) 2.30732e16 0.510746
\(366\) −5.96904e16 −1.29801
\(367\) 5.13474e14 0.0109696 0.00548478 0.999985i \(-0.498254\pi\)
0.00548478 + 0.999985i \(0.498254\pi\)
\(368\) −5.19136e16 −1.08961
\(369\) 6.21702e16 1.28207
\(370\) −4.15689e15 −0.0842281
\(371\) −2.48402e15 −0.0494566
\(372\) 1.01154e17 1.97904
\(373\) −4.59511e16 −0.883463 −0.441731 0.897147i \(-0.645636\pi\)
−0.441731 + 0.897147i \(0.645636\pi\)
\(374\) 1.37840e17 2.60442
\(375\) −5.99890e16 −1.11396
\(376\) −1.39896e17 −2.55319
\(377\) 8.92599e15 0.160117
\(378\) −1.23577e16 −0.217893
\(379\) −2.47235e16 −0.428504 −0.214252 0.976778i \(-0.568731\pi\)
−0.214252 + 0.976778i \(0.568731\pi\)
\(380\) 1.81718e16 0.309602
\(381\) −4.15561e16 −0.696021
\(382\) −1.75037e17 −2.88215
\(383\) 6.98369e16 1.13056 0.565279 0.824900i \(-0.308768\pi\)
0.565279 + 0.824900i \(0.308768\pi\)
\(384\) 5.62307e15 0.0894995
\(385\) 2.65740e15 0.0415874
\(386\) −1.11759e17 −1.71975
\(387\) 1.02537e17 1.55152
\(388\) −7.86624e16 −1.17047
\(389\) −3.13101e16 −0.458154 −0.229077 0.973408i \(-0.573571\pi\)
−0.229077 + 0.973408i \(0.573571\pi\)
\(390\) 8.19311e16 1.17904
\(391\) 4.50636e16 0.637789
\(392\) 1.84177e17 2.56375
\(393\) 1.53302e17 2.09892
\(394\) 9.76471e15 0.131503
\(395\) −5.36544e16 −0.710762
\(396\) 3.07421e17 4.00603
\(397\) 4.10001e16 0.525589 0.262794 0.964852i \(-0.415356\pi\)
0.262794 + 0.964852i \(0.415356\pi\)
\(398\) −6.68895e16 −0.843563
\(399\) 5.15263e15 0.0639300
\(400\) −1.69092e17 −2.06411
\(401\) 2.99900e16 0.360196 0.180098 0.983649i \(-0.442359\pi\)
0.180098 + 0.983649i \(0.442359\pi\)
\(402\) 2.09988e17 2.48156
\(403\) −4.64899e16 −0.540600
\(404\) 8.68610e16 0.993907
\(405\) 2.15103e15 0.0242208
\(406\) 2.79550e15 0.0309771
\(407\) 1.15205e16 0.125634
\(408\) 5.49592e17 5.89860
\(409\) −3.01326e16 −0.318299 −0.159149 0.987255i \(-0.550875\pi\)
−0.159149 + 0.987255i \(0.550875\pi\)
\(410\) −5.07193e16 −0.527325
\(411\) −6.63011e16 −0.678498
\(412\) −9.22400e16 −0.929154
\(413\) −1.83834e16 −0.182284
\(414\) 1.42264e17 1.38865
\(415\) 3.84393e16 0.369371
\(416\) 2.04272e17 1.93242
\(417\) −1.02078e17 −0.950713
\(418\) −7.12873e16 −0.653681
\(419\) 1.39179e17 1.25656 0.628278 0.777989i \(-0.283760\pi\)
0.628278 + 0.777989i \(0.283760\pi\)
\(420\) 1.81276e16 0.161146
\(421\) −1.85777e17 −1.62614 −0.813072 0.582163i \(-0.802207\pi\)
−0.813072 + 0.582163i \(0.802207\pi\)
\(422\) −1.40035e17 −1.20699
\(423\) 1.90837e17 1.61976
\(424\) 1.37026e17 1.14531
\(425\) 1.46780e17 1.20820
\(426\) 4.07568e17 3.30398
\(427\) 6.06975e15 0.0484607
\(428\) −5.01736e17 −3.94540
\(429\) −2.27066e17 −1.75865
\(430\) −8.36509e16 −0.638154
\(431\) 1.48837e17 1.11843 0.559217 0.829022i \(-0.311102\pi\)
0.559217 + 0.829022i \(0.311102\pi\)
\(432\) 3.39337e17 2.51181
\(433\) −1.54076e17 −1.12348 −0.561739 0.827314i \(-0.689868\pi\)
−0.561739 + 0.827314i \(0.689868\pi\)
\(434\) −1.45600e16 −0.104587
\(435\) −1.26331e16 −0.0893985
\(436\) 2.10975e17 1.47085
\(437\) −2.33057e16 −0.160078
\(438\) −6.17458e17 −4.17855
\(439\) −2.35755e17 −1.57196 −0.785981 0.618251i \(-0.787842\pi\)
−0.785981 + 0.618251i \(0.787842\pi\)
\(440\) −1.46590e17 −0.963077
\(441\) −2.51243e17 −1.62646
\(442\) −4.32151e17 −2.75671
\(443\) −1.61027e17 −1.01222 −0.506108 0.862470i \(-0.668916\pi\)
−0.506108 + 0.862470i \(0.668916\pi\)
\(444\) 7.85878e16 0.486817
\(445\) −1.27494e16 −0.0778306
\(446\) 9.02926e16 0.543220
\(447\) −1.25600e17 −0.744714
\(448\) 1.82074e16 0.106400
\(449\) −1.74061e17 −1.00254 −0.501268 0.865292i \(-0.667133\pi\)
−0.501268 + 0.865292i \(0.667133\pi\)
\(450\) 4.63378e17 2.63060
\(451\) 1.40565e17 0.786554
\(452\) −2.92706e17 −1.61448
\(453\) −2.88932e17 −1.57093
\(454\) −2.43009e17 −1.30245
\(455\) −8.33136e15 −0.0440192
\(456\) −2.84234e17 −1.48049
\(457\) −3.08125e17 −1.58224 −0.791119 0.611662i \(-0.790501\pi\)
−0.791119 + 0.611662i \(0.790501\pi\)
\(458\) −1.61996e17 −0.820123
\(459\) −2.94562e17 −1.47026
\(460\) −8.19923e16 −0.403503
\(461\) 1.77805e17 0.862755 0.431378 0.902171i \(-0.358028\pi\)
0.431378 + 0.902171i \(0.358028\pi\)
\(462\) −7.11140e16 −0.340238
\(463\) −4.07934e17 −1.92448 −0.962239 0.272204i \(-0.912247\pi\)
−0.962239 + 0.272204i \(0.912247\pi\)
\(464\) −7.67631e16 −0.357096
\(465\) 6.57980e16 0.301834
\(466\) 6.15789e17 2.78563
\(467\) −1.30093e17 −0.580356 −0.290178 0.956973i \(-0.593715\pi\)
−0.290178 + 0.956973i \(0.593715\pi\)
\(468\) −9.63811e17 −4.24028
\(469\) −2.13531e16 −0.0926483
\(470\) −1.55688e17 −0.666220
\(471\) −5.05063e17 −2.13162
\(472\) 1.01408e18 4.22132
\(473\) 2.31832e17 0.951867
\(474\) 1.43583e18 5.81494
\(475\) −7.59107e16 −0.303246
\(476\) −9.56152e16 −0.376775
\(477\) −1.86922e17 −0.726593
\(478\) −5.24751e17 −2.01220
\(479\) −4.32104e16 −0.163459 −0.0817293 0.996655i \(-0.526044\pi\)
−0.0817293 + 0.996655i \(0.526044\pi\)
\(480\) −2.89110e17 −1.07893
\(481\) −3.61186e16 −0.132980
\(482\) 4.50252e17 1.63550
\(483\) −2.32490e16 −0.0833199
\(484\) 1.44274e16 0.0510143
\(485\) −5.11678e16 −0.178515
\(486\) 5.06994e17 1.74529
\(487\) −4.32949e17 −1.47061 −0.735307 0.677734i \(-0.762962\pi\)
−0.735307 + 0.677734i \(0.762962\pi\)
\(488\) −3.34825e17 −1.12225
\(489\) −5.37316e17 −1.77714
\(490\) 2.04967e17 0.668976
\(491\) 3.56642e17 1.14869 0.574344 0.818614i \(-0.305257\pi\)
0.574344 + 0.818614i \(0.305257\pi\)
\(492\) 9.58871e17 3.04780
\(493\) 6.66342e16 0.209022
\(494\) 2.23497e17 0.691904
\(495\) 1.99969e17 0.610982
\(496\) 3.99811e17 1.20566
\(497\) −4.14445e16 −0.123353
\(498\) −1.02866e18 −3.02192
\(499\) −4.88532e17 −1.41657 −0.708287 0.705925i \(-0.750532\pi\)
−0.708287 + 0.705925i \(0.750532\pi\)
\(500\) −5.75712e17 −1.64778
\(501\) 5.60780e17 1.58434
\(502\) −2.82239e17 −0.787123
\(503\) −5.63176e17 −1.55043 −0.775216 0.631697i \(-0.782359\pi\)
−0.775216 + 0.631697i \(0.782359\pi\)
\(504\) −1.76431e17 −0.479487
\(505\) 5.65007e16 0.151586
\(506\) 3.21653e17 0.851941
\(507\) 8.96701e16 0.234475
\(508\) −3.98812e17 −1.02957
\(509\) 4.10438e17 1.04612 0.523060 0.852296i \(-0.324790\pi\)
0.523060 + 0.852296i \(0.324790\pi\)
\(510\) 6.11631e17 1.53916
\(511\) 6.27877e16 0.156005
\(512\) 7.67955e17 1.88400
\(513\) 1.52339e17 0.369019
\(514\) −6.00945e17 −1.43739
\(515\) −5.99997e16 −0.141710
\(516\) 1.58146e18 3.68837
\(517\) 4.31477e17 0.993731
\(518\) −1.13119e16 −0.0257271
\(519\) −6.45295e17 −1.44934
\(520\) 4.59582e17 1.01939
\(521\) −6.11645e16 −0.133984 −0.0669922 0.997753i \(-0.521340\pi\)
−0.0669922 + 0.997753i \(0.521340\pi\)
\(522\) 2.10361e17 0.455100
\(523\) 1.69083e17 0.361275 0.180638 0.983550i \(-0.442184\pi\)
0.180638 + 0.983550i \(0.442184\pi\)
\(524\) 1.47123e18 3.10475
\(525\) −7.57262e16 −0.157838
\(526\) 3.46810e17 0.713977
\(527\) −3.47056e17 −0.705716
\(528\) 1.95276e18 3.92218
\(529\) −3.98880e17 −0.791371
\(530\) 1.52493e17 0.298853
\(531\) −1.38334e18 −2.67803
\(532\) 4.94496e16 0.0945664
\(533\) −4.40692e17 −0.832547
\(534\) 3.41185e17 0.636753
\(535\) −3.26366e17 −0.601735
\(536\) 1.17790e18 2.14554
\(537\) −2.79112e17 −0.502280
\(538\) 1.48325e18 2.63713
\(539\) −5.68052e17 −0.997840
\(540\) 5.35949e17 0.930174
\(541\) 1.02602e18 1.75943 0.879715 0.475501i \(-0.157733\pi\)
0.879715 + 0.475501i \(0.157733\pi\)
\(542\) 2.84177e17 0.481497
\(543\) 5.79184e17 0.969657
\(544\) 1.52493e18 2.52264
\(545\) 1.37234e17 0.224328
\(546\) 2.22954e17 0.360133
\(547\) −4.33635e17 −0.692160 −0.346080 0.938205i \(-0.612487\pi\)
−0.346080 + 0.938205i \(0.612487\pi\)
\(548\) −6.36289e17 −1.00365
\(549\) 4.56748e17 0.711962
\(550\) 1.04768e18 1.61388
\(551\) −3.44614e16 −0.0524623
\(552\) 1.28248e18 1.92951
\(553\) −1.46006e17 −0.217099
\(554\) −1.70252e17 −0.250196
\(555\) 5.11193e16 0.0742472
\(556\) −9.79643e17 −1.40631
\(557\) 7.45475e17 1.05773 0.528864 0.848706i \(-0.322618\pi\)
0.528864 + 0.848706i \(0.322618\pi\)
\(558\) −1.09564e18 −1.53654
\(559\) −7.26830e17 −1.00753
\(560\) 7.16492e16 0.0981724
\(561\) −1.69509e18 −2.29580
\(562\) −3.32259e17 −0.444827
\(563\) −1.01184e18 −1.33908 −0.669541 0.742775i \(-0.733509\pi\)
−0.669541 + 0.742775i \(0.733509\pi\)
\(564\) 2.94334e18 3.85059
\(565\) −1.90397e17 −0.246233
\(566\) −6.93660e17 −0.886828
\(567\) 5.85346e15 0.00739814
\(568\) 2.28620e18 2.85660
\(569\) −3.05533e17 −0.377422 −0.188711 0.982033i \(-0.560431\pi\)
−0.188711 + 0.982033i \(0.560431\pi\)
\(570\) −3.16319e17 −0.386312
\(571\) −2.76855e17 −0.334285 −0.167143 0.985933i \(-0.553454\pi\)
−0.167143 + 0.985933i \(0.553454\pi\)
\(572\) −2.17914e18 −2.60143
\(573\) 2.15251e18 2.54062
\(574\) −1.38019e17 −0.161069
\(575\) 3.42515e17 0.395219
\(576\) 1.37011e18 1.56318
\(577\) 6.16642e17 0.695650 0.347825 0.937560i \(-0.386920\pi\)
0.347825 + 0.937560i \(0.386920\pi\)
\(578\) −1.57147e18 −1.75297
\(579\) 1.37435e18 1.51596
\(580\) −1.21240e17 −0.132240
\(581\) 1.04602e17 0.112823
\(582\) 1.36929e18 1.46048
\(583\) −4.22624e17 −0.445768
\(584\) −3.46355e18 −3.61275
\(585\) −6.26934e17 −0.646708
\(586\) 3.08752e18 3.14974
\(587\) 6.29563e17 0.635172 0.317586 0.948229i \(-0.397128\pi\)
0.317586 + 0.948229i \(0.397128\pi\)
\(588\) −3.87500e18 −3.86651
\(589\) 1.79488e17 0.177127
\(590\) 1.12855e18 1.10150
\(591\) −1.20081e17 −0.115920
\(592\) 3.10618e17 0.296576
\(593\) −2.72242e17 −0.257099 −0.128549 0.991703i \(-0.541032\pi\)
−0.128549 + 0.991703i \(0.541032\pi\)
\(594\) −2.10251e18 −1.96393
\(595\) −6.21951e16 −0.0574640
\(596\) −1.20538e18 −1.10159
\(597\) 8.22572e17 0.743603
\(598\) −1.00843e18 −0.901757
\(599\) 1.71240e18 1.51472 0.757360 0.652998i \(-0.226489\pi\)
0.757360 + 0.652998i \(0.226489\pi\)
\(600\) 4.17728e18 3.65519
\(601\) −4.70950e17 −0.407653 −0.203826 0.979007i \(-0.565338\pi\)
−0.203826 + 0.979007i \(0.565338\pi\)
\(602\) −2.27633e17 −0.194921
\(603\) −1.60682e18 −1.36114
\(604\) −2.77287e18 −2.32375
\(605\) 9.38461e15 0.00778048
\(606\) −1.51200e18 −1.24017
\(607\) 3.83443e17 0.311153 0.155576 0.987824i \(-0.450276\pi\)
0.155576 + 0.987824i \(0.450276\pi\)
\(608\) −7.88650e17 −0.633157
\(609\) −3.43776e16 −0.0273064
\(610\) −3.72621e17 −0.292835
\(611\) −1.35275e18 −1.05184
\(612\) −7.19503e18 −5.53539
\(613\) 4.81387e16 0.0366439 0.0183219 0.999832i \(-0.494168\pi\)
0.0183219 + 0.999832i \(0.494168\pi\)
\(614\) 1.74790e18 1.31650
\(615\) 6.23720e17 0.464838
\(616\) −3.98905e17 −0.294167
\(617\) 1.06313e17 0.0775772 0.0387886 0.999247i \(-0.487650\pi\)
0.0387886 + 0.999247i \(0.487650\pi\)
\(618\) 1.60564e18 1.15937
\(619\) −2.28357e18 −1.63164 −0.815820 0.578305i \(-0.803714\pi\)
−0.815820 + 0.578305i \(0.803714\pi\)
\(620\) 6.31461e17 0.446478
\(621\) −6.87366e17 −0.480942
\(622\) −2.44774e18 −1.69483
\(623\) −3.46941e16 −0.0237730
\(624\) −6.12220e18 −4.15152
\(625\) 9.14863e17 0.613954
\(626\) 2.03494e18 1.35151
\(627\) 8.76654e17 0.576221
\(628\) −4.84707e18 −3.15313
\(629\) −2.69632e17 −0.173597
\(630\) −1.96347e17 −0.125116
\(631\) 2.54479e18 1.60495 0.802474 0.596688i \(-0.203517\pi\)
0.802474 + 0.596688i \(0.203517\pi\)
\(632\) 8.05412e18 5.02756
\(633\) 1.72207e18 1.06397
\(634\) 2.41050e18 1.47410
\(635\) −2.59417e17 −0.157025
\(636\) −2.88296e18 −1.72730
\(637\) 1.78093e18 1.05619
\(638\) 4.75619e17 0.279206
\(639\) −3.11869e18 −1.81225
\(640\) 3.51024e16 0.0201914
\(641\) −2.72353e18 −1.55080 −0.775399 0.631471i \(-0.782451\pi\)
−0.775399 + 0.631471i \(0.782451\pi\)
\(642\) 8.73381e18 4.92296
\(643\) −7.92017e17 −0.441940 −0.220970 0.975281i \(-0.570922\pi\)
−0.220970 + 0.975281i \(0.570922\pi\)
\(644\) −2.23120e17 −0.123248
\(645\) 1.02870e18 0.562534
\(646\) 1.66844e18 0.903234
\(647\) −6.42858e17 −0.344538 −0.172269 0.985050i \(-0.555110\pi\)
−0.172269 + 0.985050i \(0.555110\pi\)
\(648\) −3.22894e17 −0.171325
\(649\) −3.12769e18 −1.64299
\(650\) −3.28465e18 −1.70825
\(651\) 1.79052e17 0.0921938
\(652\) −5.15660e18 −2.62878
\(653\) −6.71995e17 −0.339180 −0.169590 0.985515i \(-0.554244\pi\)
−0.169590 + 0.985515i \(0.554244\pi\)
\(654\) −3.67249e18 −1.83529
\(655\) 9.56996e17 0.473524
\(656\) 3.78993e18 1.85676
\(657\) 4.72476e18 2.29195
\(658\) −4.23662e17 −0.203494
\(659\) 3.36471e18 1.60027 0.800133 0.599822i \(-0.204762\pi\)
0.800133 + 0.599822i \(0.204762\pi\)
\(660\) 3.08418e18 1.45246
\(661\) −5.83150e17 −0.271939 −0.135969 0.990713i \(-0.543415\pi\)
−0.135969 + 0.990713i \(0.543415\pi\)
\(662\) −3.12107e18 −1.44121
\(663\) 5.31437e18 2.43004
\(664\) −5.77016e18 −2.61273
\(665\) 3.21656e16 0.0144229
\(666\) −8.51216e17 −0.377970
\(667\) 1.55492e17 0.0683740
\(668\) 5.38178e18 2.34358
\(669\) −1.11037e18 −0.478850
\(670\) 1.31086e18 0.559849
\(671\) 1.03269e18 0.436792
\(672\) −7.86734e17 −0.329555
\(673\) −1.96018e18 −0.813202 −0.406601 0.913606i \(-0.633286\pi\)
−0.406601 + 0.913606i \(0.633286\pi\)
\(674\) −4.92999e18 −2.02561
\(675\) −2.23887e18 −0.911077
\(676\) 8.60561e17 0.346839
\(677\) 2.44960e18 0.977842 0.488921 0.872328i \(-0.337391\pi\)
0.488921 + 0.872328i \(0.337391\pi\)
\(678\) 5.09518e18 2.01450
\(679\) −1.39239e17 −0.0545266
\(680\) 3.43086e18 1.33075
\(681\) 2.98840e18 1.14811
\(682\) −2.47720e18 −0.942677
\(683\) 3.28820e18 1.23944 0.619718 0.784824i \(-0.287247\pi\)
0.619718 + 0.784824i \(0.287247\pi\)
\(684\) 3.72107e18 1.38932
\(685\) −4.13889e17 −0.153072
\(686\) 1.12263e18 0.411272
\(687\) 1.99215e18 0.722940
\(688\) 6.25070e18 2.24700
\(689\) 1.32499e18 0.471833
\(690\) 1.42725e18 0.503480
\(691\) 4.97808e18 1.73962 0.869810 0.493387i \(-0.164242\pi\)
0.869810 + 0.493387i \(0.164242\pi\)
\(692\) −6.19287e18 −2.14389
\(693\) 5.44162e17 0.186622
\(694\) −7.82991e18 −2.66024
\(695\) −6.37231e17 −0.214485
\(696\) 1.89637e18 0.632358
\(697\) −3.28985e18 −1.08683
\(698\) 3.31814e18 1.08601
\(699\) −7.57266e18 −2.45554
\(700\) −7.26742e17 −0.233476
\(701\) 2.36642e17 0.0753225 0.0376613 0.999291i \(-0.488009\pi\)
0.0376613 + 0.999291i \(0.488009\pi\)
\(702\) 6.59171e18 2.07877
\(703\) 1.39446e17 0.0435710
\(704\) 3.09776e18 0.959015
\(705\) 1.91457e18 0.587275
\(706\) −1.87958e18 −0.571257
\(707\) 1.53752e17 0.0463013
\(708\) −2.13358e19 −6.36637
\(709\) −3.99621e18 −1.18154 −0.590769 0.806840i \(-0.701176\pi\)
−0.590769 + 0.806840i \(0.701176\pi\)
\(710\) 2.54427e18 0.745391
\(711\) −1.09869e19 −3.18951
\(712\) 1.91383e18 0.550533
\(713\) −8.09862e17 −0.230850
\(714\) 1.66439e18 0.470129
\(715\) −1.41748e18 −0.396758
\(716\) −2.67863e18 −0.742982
\(717\) 6.45312e18 1.77376
\(718\) −6.35502e18 −1.73104
\(719\) 5.57196e18 1.50408 0.752038 0.659120i \(-0.229071\pi\)
0.752038 + 0.659120i \(0.229071\pi\)
\(720\) 5.39160e18 1.44230
\(721\) −1.63273e17 −0.0432848
\(722\) 6.16233e18 1.61902
\(723\) −5.53696e18 −1.44169
\(724\) 5.55841e18 1.43433
\(725\) 5.06466e17 0.129525
\(726\) −2.51140e17 −0.0636542
\(727\) −3.91166e18 −0.982625 −0.491313 0.870983i \(-0.663483\pi\)
−0.491313 + 0.870983i \(0.663483\pi\)
\(728\) 1.25063e18 0.311368
\(729\) −6.50216e18 −1.60446
\(730\) −3.85453e18 −0.942697
\(731\) −5.42592e18 −1.31526
\(732\) 7.04457e18 1.69252
\(733\) −1.03950e18 −0.247542 −0.123771 0.992311i \(-0.539499\pi\)
−0.123771 + 0.992311i \(0.539499\pi\)
\(734\) −8.57789e16 −0.0202468
\(735\) −2.52058e18 −0.589703
\(736\) 3.55845e18 0.825192
\(737\) −3.63296e18 −0.835068
\(738\) −1.03859e19 −2.36634
\(739\) 3.88010e18 0.876303 0.438152 0.898901i \(-0.355633\pi\)
0.438152 + 0.898901i \(0.355633\pi\)
\(740\) 4.90590e17 0.109828
\(741\) −2.74845e18 −0.609915
\(742\) 4.14970e17 0.0912833
\(743\) −6.89829e18 −1.50423 −0.752115 0.659032i \(-0.770966\pi\)
−0.752115 + 0.659032i \(0.770966\pi\)
\(744\) −9.87700e18 −2.13502
\(745\) −7.84064e17 −0.168010
\(746\) 7.67640e18 1.63063
\(747\) 7.87130e18 1.65753
\(748\) −1.62677e19 −3.39599
\(749\) −8.88118e17 −0.183797
\(750\) 1.00215e19 2.05606
\(751\) 8.43457e18 1.71555 0.857775 0.514025i \(-0.171846\pi\)
0.857775 + 0.514025i \(0.171846\pi\)
\(752\) 1.16336e19 2.34583
\(753\) 3.47083e18 0.693850
\(754\) −1.49114e18 −0.295532
\(755\) −1.80368e18 −0.354408
\(756\) 1.45844e18 0.284117
\(757\) 2.82123e18 0.544898 0.272449 0.962170i \(-0.412166\pi\)
0.272449 + 0.962170i \(0.412166\pi\)
\(758\) 4.13021e18 0.790902
\(759\) −3.95553e18 −0.750988
\(760\) −1.77435e18 −0.334003
\(761\) 3.49050e18 0.651459 0.325730 0.945463i \(-0.394390\pi\)
0.325730 + 0.945463i \(0.394390\pi\)
\(762\) 6.94219e18 1.28466
\(763\) 3.73445e17 0.0685200
\(764\) 2.06576e19 3.75814
\(765\) −4.68018e18 −0.844234
\(766\) −1.16667e19 −2.08670
\(767\) 9.80581e18 1.73906
\(768\) −9.71967e18 −1.70924
\(769\) 1.08383e19 1.88991 0.944956 0.327197i \(-0.106104\pi\)
0.944956 + 0.327197i \(0.106104\pi\)
\(770\) −4.43934e17 −0.0767589
\(771\) 7.39012e18 1.26706
\(772\) 1.31896e19 2.24243
\(773\) −5.87557e18 −0.990566 −0.495283 0.868732i \(-0.664936\pi\)
−0.495283 + 0.868732i \(0.664936\pi\)
\(774\) −1.71294e19 −2.86369
\(775\) −2.63786e18 −0.437312
\(776\) 7.68085e18 1.26272
\(777\) 1.39107e17 0.0226785
\(778\) 5.23053e18 0.845627
\(779\) 1.70142e18 0.272784
\(780\) −9.66940e18 −1.53739
\(781\) −7.05126e18 −1.11182
\(782\) −7.52814e18 −1.17718
\(783\) −1.01639e18 −0.157619
\(784\) −1.53159e19 −2.35553
\(785\) −3.15289e18 −0.480902
\(786\) −2.56100e19 −3.87402
\(787\) 1.16087e19 1.74160 0.870798 0.491642i \(-0.163603\pi\)
0.870798 + 0.491642i \(0.163603\pi\)
\(788\) −1.15242e18 −0.171470
\(789\) −4.26490e18 −0.629373
\(790\) 8.96329e18 1.31187
\(791\) −5.18115e17 −0.0752107
\(792\) −3.00175e19 −4.32177
\(793\) −3.23765e18 −0.462332
\(794\) −6.84931e18 −0.970093
\(795\) −1.87529e18 −0.263440
\(796\) 7.89420e18 1.09995
\(797\) −6.97871e17 −0.0964486 −0.0482243 0.998837i \(-0.515356\pi\)
−0.0482243 + 0.998837i \(0.515356\pi\)
\(798\) −8.60777e17 −0.117997
\(799\) −1.00985e19 −1.37310
\(800\) 1.15905e19 1.56321
\(801\) −2.61073e18 −0.349261
\(802\) −5.01002e18 −0.664823
\(803\) 1.06825e19 1.40612
\(804\) −2.47825e19 −3.23579
\(805\) −1.45134e17 −0.0187973
\(806\) 7.76642e18 0.997799
\(807\) −1.82403e19 −2.32463
\(808\) −8.48138e18 −1.07224
\(809\) −4.03649e18 −0.506219 −0.253109 0.967438i \(-0.581453\pi\)
−0.253109 + 0.967438i \(0.581453\pi\)
\(810\) −3.59343e17 −0.0447050
\(811\) −1.94069e18 −0.239508 −0.119754 0.992804i \(-0.538211\pi\)
−0.119754 + 0.992804i \(0.538211\pi\)
\(812\) −3.29921e17 −0.0403920
\(813\) −3.49467e18 −0.424441
\(814\) −1.92457e18 −0.231886
\(815\) −3.35423e18 −0.400930
\(816\) −4.57033e19 −5.41953
\(817\) 2.80614e18 0.330115
\(818\) 5.03383e18 0.587492
\(819\) −1.70603e18 −0.197534
\(820\) 5.98582e18 0.687596
\(821\) −5.70883e18 −0.650604 −0.325302 0.945610i \(-0.605466\pi\)
−0.325302 + 0.945610i \(0.605466\pi\)
\(822\) 1.10760e19 1.25232
\(823\) 1.09771e19 1.23137 0.615683 0.787994i \(-0.288880\pi\)
0.615683 + 0.787994i \(0.288880\pi\)
\(824\) 9.00661e18 1.00239
\(825\) −1.28839e19 −1.42264
\(826\) 3.07105e18 0.336447
\(827\) −1.38129e19 −1.50141 −0.750706 0.660637i \(-0.770286\pi\)
−0.750706 + 0.660637i \(0.770286\pi\)
\(828\) −1.67898e19 −1.81070
\(829\) −4.07843e18 −0.436404 −0.218202 0.975904i \(-0.570019\pi\)
−0.218202 + 0.975904i \(0.570019\pi\)
\(830\) −6.42151e18 −0.681757
\(831\) 2.09368e18 0.220548
\(832\) −9.71197e18 −1.01509
\(833\) 1.32950e19 1.37878
\(834\) 1.70528e19 1.75476
\(835\) 3.50071e18 0.357432
\(836\) 8.41322e18 0.852357
\(837\) 5.29373e18 0.532165
\(838\) −2.32506e19 −2.31926
\(839\) 1.78304e18 0.176486 0.0882428 0.996099i \(-0.471875\pi\)
0.0882428 + 0.996099i \(0.471875\pi\)
\(840\) −1.77004e18 −0.173847
\(841\) −1.00307e19 −0.977592
\(842\) 3.10352e19 3.00142
\(843\) 4.08595e18 0.392116
\(844\) 1.65267e19 1.57384
\(845\) 5.59772e17 0.0528984
\(846\) −3.18805e19 −2.98963
\(847\) 2.55377e16 0.00237651
\(848\) −1.13949e19 −1.05229
\(849\) 8.53027e18 0.781741
\(850\) −2.45205e19 −2.23001
\(851\) −6.29192e17 −0.0567859
\(852\) −4.81006e19 −4.30817
\(853\) 3.94325e17 0.0350498 0.0175249 0.999846i \(-0.494421\pi\)
0.0175249 + 0.999846i \(0.494421\pi\)
\(854\) −1.01399e18 −0.0894452
\(855\) 2.42046e18 0.211894
\(856\) 4.89912e19 4.25636
\(857\) 1.99892e19 1.72353 0.861767 0.507305i \(-0.169358\pi\)
0.861767 + 0.507305i \(0.169358\pi\)
\(858\) 3.79327e19 3.24599
\(859\) −1.78398e18 −0.151508 −0.0757539 0.997127i \(-0.524136\pi\)
−0.0757539 + 0.997127i \(0.524136\pi\)
\(860\) 9.87236e18 0.832110
\(861\) 1.69729e18 0.141982
\(862\) −2.48642e19 −2.06432
\(863\) −1.46602e19 −1.20801 −0.604004 0.796982i \(-0.706429\pi\)
−0.604004 + 0.796982i \(0.706429\pi\)
\(864\) −2.32601e19 −1.90227
\(865\) −4.02830e18 −0.326977
\(866\) 2.57394e19 2.07363
\(867\) 1.93251e19 1.54525
\(868\) 1.71835e18 0.136375
\(869\) −2.48411e19 −1.95678
\(870\) 2.11044e18 0.165005
\(871\) 1.13899e19 0.883897
\(872\) −2.06003e19 −1.58678
\(873\) −1.04777e19 −0.801079
\(874\) 3.89335e18 0.295460
\(875\) −1.01906e18 −0.0767623
\(876\) 7.28715e19 5.44855
\(877\) −1.83740e19 −1.36366 −0.681830 0.731511i \(-0.738816\pi\)
−0.681830 + 0.731511i \(0.738816\pi\)
\(878\) 3.93843e19 2.90141
\(879\) −3.79688e19 −2.77651
\(880\) 1.21902e19 0.884859
\(881\) 9.20911e17 0.0663551 0.0331776 0.999449i \(-0.489437\pi\)
0.0331776 + 0.999449i \(0.489437\pi\)
\(882\) 4.19717e19 3.00200
\(883\) 3.66478e18 0.260198 0.130099 0.991501i \(-0.458471\pi\)
0.130099 + 0.991501i \(0.458471\pi\)
\(884\) 5.10018e19 3.59456
\(885\) −1.38783e19 −0.970971
\(886\) 2.69005e19 1.86827
\(887\) −1.17105e19 −0.807371 −0.403685 0.914898i \(-0.632271\pi\)
−0.403685 + 0.914898i \(0.632271\pi\)
\(888\) −7.67357e18 −0.525186
\(889\) −7.05933e17 −0.0479625
\(890\) 2.12987e18 0.143654
\(891\) 9.95892e17 0.0666818
\(892\) −1.06562e19 −0.708323
\(893\) 5.22267e18 0.344634
\(894\) 2.09822e19 1.37454
\(895\) −1.74238e18 −0.113316
\(896\) 9.55218e16 0.00616737
\(897\) 1.24012e19 0.794901
\(898\) 2.90779e19 1.85041
\(899\) −1.19752e18 −0.0756561
\(900\) −5.46872e19 −3.43012
\(901\) 9.89132e18 0.615947
\(902\) −2.34822e19 −1.45176
\(903\) 2.79932e18 0.171823
\(904\) 2.85807e19 1.74172
\(905\) 3.61560e18 0.218758
\(906\) 4.82679e19 2.89951
\(907\) −1.17036e19 −0.698028 −0.349014 0.937118i \(-0.613483\pi\)
−0.349014 + 0.937118i \(0.613483\pi\)
\(908\) 2.86796e19 1.69830
\(909\) 1.15698e19 0.680237
\(910\) 1.39180e18 0.0812473
\(911\) 2.36394e19 1.37014 0.685072 0.728475i \(-0.259771\pi\)
0.685072 + 0.728475i \(0.259771\pi\)
\(912\) 2.36365e19 1.36024
\(913\) 1.77967e19 1.01690
\(914\) 5.14742e19 2.92038
\(915\) 4.58230e18 0.258135
\(916\) 1.91185e19 1.06939
\(917\) 2.60421e18 0.144636
\(918\) 4.92083e19 2.71370
\(919\) 2.18682e19 1.19746 0.598731 0.800950i \(-0.295672\pi\)
0.598731 + 0.800950i \(0.295672\pi\)
\(920\) 8.00599e18 0.435306
\(921\) −2.14947e19 −1.16050
\(922\) −2.97034e19 −1.59241
\(923\) 2.21068e19 1.17683
\(924\) 8.39278e18 0.443647
\(925\) −2.04939e18 −0.107573
\(926\) 6.81478e19 3.55206
\(927\) −1.22863e19 −0.635919
\(928\) 5.26177e18 0.270439
\(929\) −2.63865e19 −1.34673 −0.673364 0.739311i \(-0.735151\pi\)
−0.673364 + 0.739311i \(0.735151\pi\)
\(930\) −1.09919e19 −0.557103
\(931\) −6.87580e18 −0.346059
\(932\) −7.26746e19 −3.63228
\(933\) 3.01010e19 1.49400
\(934\) 2.17328e19 1.07118
\(935\) −1.05817e19 −0.517941
\(936\) 9.41096e19 4.57447
\(937\) −1.73731e19 −0.838627 −0.419313 0.907842i \(-0.637729\pi\)
−0.419313 + 0.907842i \(0.637729\pi\)
\(938\) 3.56716e18 0.171003
\(939\) −2.50246e19 −1.19135
\(940\) 1.83740e19 0.868707
\(941\) 2.45945e19 1.15480 0.577398 0.816463i \(-0.304068\pi\)
0.577398 + 0.816463i \(0.304068\pi\)
\(942\) 8.43739e19 3.93439
\(943\) −7.67693e18 −0.355518
\(944\) −8.43295e19 −3.87848
\(945\) 9.48678e17 0.0433323
\(946\) −3.87290e19 −1.75689
\(947\) 2.67363e19 1.20456 0.602278 0.798287i \(-0.294260\pi\)
0.602278 + 0.798287i \(0.294260\pi\)
\(948\) −1.69455e20 −7.58229
\(949\) −3.34914e19 −1.48834
\(950\) 1.26813e19 0.559708
\(951\) −2.96431e19 −1.29942
\(952\) 9.33618e18 0.406470
\(953\) 1.30292e19 0.563394 0.281697 0.959503i \(-0.409103\pi\)
0.281697 + 0.959503i \(0.409103\pi\)
\(954\) 3.12265e19 1.34109
\(955\) 1.34372e19 0.573175
\(956\) 6.19304e19 2.62378
\(957\) −5.84892e18 −0.246121
\(958\) 7.21856e18 0.301700
\(959\) −1.12629e18 −0.0467551
\(960\) 1.37455e19 0.566759
\(961\) −1.81804e19 −0.744564
\(962\) 6.03383e18 0.245445
\(963\) −6.68308e19 −2.70026
\(964\) −5.31380e19 −2.13258
\(965\) 8.57949e18 0.342006
\(966\) 3.88389e18 0.153786
\(967\) −1.57944e18 −0.0621198 −0.0310599 0.999518i \(-0.509888\pi\)
−0.0310599 + 0.999518i \(0.509888\pi\)
\(968\) −1.40873e18 −0.0550350
\(969\) −2.05177e19 −0.796203
\(970\) 8.54789e18 0.329490
\(971\) 3.02093e19 1.15668 0.578342 0.815794i \(-0.303700\pi\)
0.578342 + 0.815794i \(0.303700\pi\)
\(972\) −5.98347e19 −2.27574
\(973\) −1.73405e18 −0.0655133
\(974\) 7.23268e19 2.71435
\(975\) 4.03929e19 1.50583
\(976\) 2.78436e19 1.03110
\(977\) −1.37769e19 −0.506802 −0.253401 0.967361i \(-0.581549\pi\)
−0.253401 + 0.967361i \(0.581549\pi\)
\(978\) 8.97619e19 3.28012
\(979\) −5.90277e18 −0.214273
\(980\) −2.41900e19 −0.872299
\(981\) 2.81017e19 1.00666
\(982\) −5.95791e19 −2.12016
\(983\) −2.41783e19 −0.854727 −0.427363 0.904080i \(-0.640558\pi\)
−0.427363 + 0.904080i \(0.640558\pi\)
\(984\) −9.36272e19 −3.28802
\(985\) −7.49617e17 −0.0261519
\(986\) −1.11316e19 −0.385797
\(987\) 5.20998e18 0.179380
\(988\) −2.63768e19 −0.902197
\(989\) −1.26615e19 −0.430239
\(990\) −3.34060e19 −1.12771
\(991\) 3.20449e19 1.07468 0.537341 0.843365i \(-0.319429\pi\)
0.537341 + 0.843365i \(0.319429\pi\)
\(992\) −2.74053e19 −0.913079
\(993\) 3.83813e19 1.27043
\(994\) 6.92355e18 0.227676
\(995\) 5.13496e18 0.167760
\(996\) 1.21402e20 3.94038
\(997\) 4.68142e19 1.50959 0.754795 0.655961i \(-0.227737\pi\)
0.754795 + 0.655961i \(0.227737\pi\)
\(998\) 8.16122e19 2.61461
\(999\) 4.11276e18 0.130906
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.1.5 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.14.a.a.1.5 104 1.1 even 1 trivial