Properties

Label 50.12.a.f.1.2
Level $50$
Weight $12$
Character 50.1
Self dual yes
Analytic conductor $38.417$
Analytic rank $0$
Dimension $2$
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] = [50,12,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1969}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 492 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-21.6867\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32.0000 q^{2} +141.734 q^{3} +1024.00 q^{4} -4535.49 q^{6} -85586.9 q^{7} -32768.0 q^{8} -157058. q^{9} +767235. q^{11} +145136. q^{12} -220960. q^{13} +2.73878e6 q^{14} +1.04858e6 q^{16} +930719. q^{17} +5.02587e6 q^{18} -1.77341e7 q^{19} -1.21306e7 q^{21} -2.45515e7 q^{22} +3.99596e7 q^{23} -4.64434e6 q^{24} +7.07072e6 q^{26} -4.73683e7 q^{27} -8.76410e7 q^{28} +7.68554e7 q^{29} -2.96314e7 q^{31} -3.35544e7 q^{32} +1.08743e8 q^{33} -2.97830e7 q^{34} -1.60828e8 q^{36} -5.40911e7 q^{37} +5.67491e8 q^{38} -3.13176e7 q^{39} +1.26006e8 q^{41} +3.88179e8 q^{42} +2.88676e8 q^{43} +7.85648e8 q^{44} -1.27871e9 q^{46} +1.57008e9 q^{47} +1.48619e8 q^{48} +5.34780e9 q^{49} +1.31915e8 q^{51} -2.26263e8 q^{52} +4.09006e9 q^{53} +1.51579e9 q^{54} +2.80451e9 q^{56} -2.51353e9 q^{57} -2.45937e9 q^{58} +3.77882e9 q^{59} -9.64103e9 q^{61} +9.48205e8 q^{62} +1.34422e10 q^{63} +1.07374e9 q^{64} -3.47979e9 q^{66} +1.63819e10 q^{67} +9.53056e8 q^{68} +5.66364e9 q^{69} +1.03471e10 q^{71} +5.14649e9 q^{72} -4.27149e9 q^{73} +1.73091e9 q^{74} -1.81597e10 q^{76} -6.56653e10 q^{77} +1.00216e9 q^{78} -1.96636e10 q^{79} +2.11087e10 q^{81} -4.03220e9 q^{82} +1.35791e10 q^{83} -1.24217e10 q^{84} -9.23762e9 q^{86} +1.08930e10 q^{87} -2.51407e10 q^{88} +2.25058e10 q^{89} +1.89113e10 q^{91} +4.09186e10 q^{92} -4.19978e9 q^{93} -5.02424e10 q^{94} -4.75581e9 q^{96} +1.08976e11 q^{97} -1.71130e11 q^{98} -1.20501e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{2} - 604 q^{3} + 2048 q^{4} + 19328 q^{6} - 14092 q^{7} - 65536 q^{8} + 221914 q^{9} + 421584 q^{11} - 618496 q^{12} - 1730524 q^{13} + 450944 q^{14} + 2097152 q^{16} + 6323628 q^{17} - 7101248 q^{18}+ \cdots - 251492724912 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −32.0000 −0.707107
\(3\) 141.734 0.336750 0.168375 0.985723i \(-0.446148\pi\)
0.168375 + 0.985723i \(0.446148\pi\)
\(4\) 1024.00 0.500000
\(5\) 0 0
\(6\) −4535.49 −0.238118
\(7\) −85586.9 −1.92472 −0.962362 0.271772i \(-0.912390\pi\)
−0.962362 + 0.271772i \(0.912390\pi\)
\(8\) −32768.0 −0.353553
\(9\) −157058. −0.886599
\(10\) 0 0
\(11\) 767235. 1.43638 0.718188 0.695849i \(-0.244972\pi\)
0.718188 + 0.695849i \(0.244972\pi\)
\(12\) 145136. 0.168375
\(13\) −220960. −0.165054 −0.0825268 0.996589i \(-0.526299\pi\)
−0.0825268 + 0.996589i \(0.526299\pi\)
\(14\) 2.73878e6 1.36098
\(15\) 0 0
\(16\) 1.04858e6 0.250000
\(17\) 930719. 0.158983 0.0794913 0.996836i \(-0.474670\pi\)
0.0794913 + 0.996836i \(0.474670\pi\)
\(18\) 5.02587e6 0.626920
\(19\) −1.77341e7 −1.64310 −0.821551 0.570135i \(-0.806891\pi\)
−0.821551 + 0.570135i \(0.806891\pi\)
\(20\) 0 0
\(21\) −1.21306e7 −0.648151
\(22\) −2.45515e7 −1.01567
\(23\) 3.99596e7 1.29455 0.647274 0.762257i \(-0.275909\pi\)
0.647274 + 0.762257i \(0.275909\pi\)
\(24\) −4.64434e6 −0.119059
\(25\) 0 0
\(26\) 7.07072e6 0.116711
\(27\) −4.73683e7 −0.635312
\(28\) −8.76410e7 −0.962362
\(29\) 7.68554e7 0.695801 0.347901 0.937531i \(-0.386895\pi\)
0.347901 + 0.937531i \(0.386895\pi\)
\(30\) 0 0
\(31\) −2.96314e7 −0.185893 −0.0929465 0.995671i \(-0.529629\pi\)
−0.0929465 + 0.995671i \(0.529629\pi\)
\(32\) −3.35544e7 −0.176777
\(33\) 1.08743e8 0.483700
\(34\) −2.97830e7 −0.112418
\(35\) 0 0
\(36\) −1.60828e8 −0.443300
\(37\) −5.40911e7 −0.128238 −0.0641189 0.997942i \(-0.520424\pi\)
−0.0641189 + 0.997942i \(0.520424\pi\)
\(38\) 5.67491e8 1.16185
\(39\) −3.13176e7 −0.0555818
\(40\) 0 0
\(41\) 1.26006e8 0.169856 0.0849280 0.996387i \(-0.472934\pi\)
0.0849280 + 0.996387i \(0.472934\pi\)
\(42\) 3.88179e8 0.458312
\(43\) 2.88676e8 0.299457 0.149728 0.988727i \(-0.452160\pi\)
0.149728 + 0.988727i \(0.452160\pi\)
\(44\) 7.85648e8 0.718188
\(45\) 0 0
\(46\) −1.27871e9 −0.915384
\(47\) 1.57008e9 0.998579 0.499290 0.866435i \(-0.333594\pi\)
0.499290 + 0.866435i \(0.333594\pi\)
\(48\) 1.48619e8 0.0841875
\(49\) 5.34780e9 2.70456
\(50\) 0 0
\(51\) 1.31915e8 0.0535374
\(52\) −2.26263e8 −0.0825268
\(53\) 4.09006e9 1.34342 0.671711 0.740813i \(-0.265560\pi\)
0.671711 + 0.740813i \(0.265560\pi\)
\(54\) 1.51579e9 0.449234
\(55\) 0 0
\(56\) 2.80451e9 0.680492
\(57\) −2.51353e9 −0.553315
\(58\) −2.45937e9 −0.492006
\(59\) 3.77882e9 0.688129 0.344065 0.938946i \(-0.388196\pi\)
0.344065 + 0.938946i \(0.388196\pi\)
\(60\) 0 0
\(61\) −9.64103e9 −1.46154 −0.730768 0.682626i \(-0.760838\pi\)
−0.730768 + 0.682626i \(0.760838\pi\)
\(62\) 9.48205e8 0.131446
\(63\) 1.34422e10 1.70646
\(64\) 1.07374e9 0.125000
\(65\) 0 0
\(66\) −3.47979e9 −0.342028
\(67\) 1.63819e10 1.48236 0.741178 0.671308i \(-0.234267\pi\)
0.741178 + 0.671308i \(0.234267\pi\)
\(68\) 9.53056e8 0.0794913
\(69\) 5.66364e9 0.435939
\(70\) 0 0
\(71\) 1.03471e10 0.680609 0.340304 0.940315i \(-0.389470\pi\)
0.340304 + 0.940315i \(0.389470\pi\)
\(72\) 5.14649e9 0.313460
\(73\) −4.27149e9 −0.241159 −0.120580 0.992704i \(-0.538475\pi\)
−0.120580 + 0.992704i \(0.538475\pi\)
\(74\) 1.73091e9 0.0906778
\(75\) 0 0
\(76\) −1.81597e10 −0.821551
\(77\) −6.56653e10 −2.76463
\(78\) 1.00216e9 0.0393023
\(79\) −1.96636e10 −0.718977 −0.359489 0.933149i \(-0.617049\pi\)
−0.359489 + 0.933149i \(0.617049\pi\)
\(80\) 0 0
\(81\) 2.11087e10 0.672658
\(82\) −4.03220e9 −0.120106
\(83\) 1.35791e10 0.378391 0.189196 0.981939i \(-0.439412\pi\)
0.189196 + 0.981939i \(0.439412\pi\)
\(84\) −1.24217e10 −0.324075
\(85\) 0 0
\(86\) −9.23762e9 −0.211748
\(87\) 1.08930e10 0.234311
\(88\) −2.51407e10 −0.507836
\(89\) 2.25058e10 0.427218 0.213609 0.976919i \(-0.431478\pi\)
0.213609 + 0.976919i \(0.431478\pi\)
\(90\) 0 0
\(91\) 1.89113e10 0.317683
\(92\) 4.09186e10 0.647274
\(93\) −4.19978e9 −0.0625994
\(94\) −5.02424e10 −0.706102
\(95\) 0 0
\(96\) −4.75581e9 −0.0595296
\(97\) 1.08976e11 1.28851 0.644255 0.764811i \(-0.277168\pi\)
0.644255 + 0.764811i \(0.277168\pi\)
\(98\) −1.71130e11 −1.91241
\(99\) −1.20501e11 −1.27349
\(100\) 0 0
\(101\) 1.63516e11 1.54807 0.774037 0.633140i \(-0.218234\pi\)
0.774037 + 0.633140i \(0.218234\pi\)
\(102\) −4.22127e9 −0.0378566
\(103\) 7.69876e10 0.654359 0.327180 0.944962i \(-0.393902\pi\)
0.327180 + 0.944962i \(0.393902\pi\)
\(104\) 7.24042e9 0.0583553
\(105\) 0 0
\(106\) −1.30882e11 −0.949943
\(107\) −1.19891e11 −0.826376 −0.413188 0.910646i \(-0.635585\pi\)
−0.413188 + 0.910646i \(0.635585\pi\)
\(108\) −4.85052e10 −0.317656
\(109\) −7.33426e10 −0.456573 −0.228287 0.973594i \(-0.573312\pi\)
−0.228287 + 0.973594i \(0.573312\pi\)
\(110\) 0 0
\(111\) −7.66655e9 −0.0431841
\(112\) −8.97444e10 −0.481181
\(113\) −2.18835e11 −1.11734 −0.558669 0.829391i \(-0.688688\pi\)
−0.558669 + 0.829391i \(0.688688\pi\)
\(114\) 8.04329e10 0.391253
\(115\) 0 0
\(116\) 7.86999e10 0.347901
\(117\) 3.47036e10 0.146337
\(118\) −1.20922e11 −0.486581
\(119\) −7.96574e10 −0.305997
\(120\) 0 0
\(121\) 3.03337e11 1.06318
\(122\) 3.08513e11 1.03346
\(123\) 1.78594e10 0.0571990
\(124\) −3.03426e10 −0.0929465
\(125\) 0 0
\(126\) −4.30149e11 −1.20665
\(127\) −3.93053e11 −1.05568 −0.527838 0.849345i \(-0.676997\pi\)
−0.527838 + 0.849345i \(0.676997\pi\)
\(128\) −3.43597e10 −0.0883883
\(129\) 4.09152e10 0.100842
\(130\) 0 0
\(131\) −2.63175e11 −0.596008 −0.298004 0.954565i \(-0.596321\pi\)
−0.298004 + 0.954565i \(0.596321\pi\)
\(132\) 1.11353e11 0.241850
\(133\) 1.51781e12 3.16252
\(134\) −5.24220e11 −1.04818
\(135\) 0 0
\(136\) −3.04978e10 −0.0562088
\(137\) −5.20521e11 −0.921457 −0.460728 0.887541i \(-0.652412\pi\)
−0.460728 + 0.887541i \(0.652412\pi\)
\(138\) −1.81237e11 −0.308255
\(139\) 6.82675e11 1.11592 0.557960 0.829868i \(-0.311584\pi\)
0.557960 + 0.829868i \(0.311584\pi\)
\(140\) 0 0
\(141\) 2.22533e11 0.336272
\(142\) −3.31107e11 −0.481263
\(143\) −1.69528e11 −0.237079
\(144\) −1.64688e11 −0.221650
\(145\) 0 0
\(146\) 1.36688e11 0.170525
\(147\) 7.57966e11 0.910761
\(148\) −5.53893e10 −0.0641189
\(149\) 1.01566e12 1.13299 0.566493 0.824066i \(-0.308300\pi\)
0.566493 + 0.824066i \(0.308300\pi\)
\(150\) 0 0
\(151\) 5.34040e11 0.553605 0.276803 0.960927i \(-0.410725\pi\)
0.276803 + 0.960927i \(0.410725\pi\)
\(152\) 5.81111e11 0.580924
\(153\) −1.46177e11 −0.140954
\(154\) 2.10129e12 1.95489
\(155\) 0 0
\(156\) −3.20692e10 −0.0277909
\(157\) 7.29675e11 0.610494 0.305247 0.952273i \(-0.401261\pi\)
0.305247 + 0.952273i \(0.401261\pi\)
\(158\) 6.29237e11 0.508394
\(159\) 5.79701e11 0.452397
\(160\) 0 0
\(161\) −3.42002e12 −2.49165
\(162\) −6.75479e11 −0.475641
\(163\) 2.12171e11 0.144429 0.0722143 0.997389i \(-0.476993\pi\)
0.0722143 + 0.997389i \(0.476993\pi\)
\(164\) 1.29030e11 0.0849280
\(165\) 0 0
\(166\) −4.34531e11 −0.267563
\(167\) 2.36613e11 0.140961 0.0704804 0.997513i \(-0.477547\pi\)
0.0704804 + 0.997513i \(0.477547\pi\)
\(168\) 3.97495e11 0.229156
\(169\) −1.74334e12 −0.972757
\(170\) 0 0
\(171\) 2.78529e12 1.45677
\(172\) 2.95604e11 0.149728
\(173\) 7.36983e11 0.361579 0.180790 0.983522i \(-0.442135\pi\)
0.180790 + 0.983522i \(0.442135\pi\)
\(174\) −3.48577e11 −0.165683
\(175\) 0 0
\(176\) 8.04504e11 0.359094
\(177\) 5.35588e11 0.231727
\(178\) −7.20185e11 −0.302089
\(179\) 1.35344e11 0.0550487 0.0275244 0.999621i \(-0.491238\pi\)
0.0275244 + 0.999621i \(0.491238\pi\)
\(180\) 0 0
\(181\) −1.35881e12 −0.519907 −0.259953 0.965621i \(-0.583707\pi\)
−0.259953 + 0.965621i \(0.583707\pi\)
\(182\) −6.05161e11 −0.224636
\(183\) −1.36646e12 −0.492172
\(184\) −1.30940e12 −0.457692
\(185\) 0 0
\(186\) 1.34393e11 0.0442645
\(187\) 7.14080e11 0.228359
\(188\) 1.60776e12 0.499290
\(189\) 4.05411e12 1.22280
\(190\) 0 0
\(191\) 1.10648e12 0.314964 0.157482 0.987522i \(-0.449662\pi\)
0.157482 + 0.987522i \(0.449662\pi\)
\(192\) 1.52186e11 0.0420938
\(193\) −6.17025e12 −1.65858 −0.829292 0.558816i \(-0.811256\pi\)
−0.829292 + 0.558816i \(0.811256\pi\)
\(194\) −3.48724e12 −0.911114
\(195\) 0 0
\(196\) 5.47615e12 1.35228
\(197\) −2.00333e12 −0.481047 −0.240523 0.970643i \(-0.577319\pi\)
−0.240523 + 0.970643i \(0.577319\pi\)
\(198\) 3.85602e12 0.900494
\(199\) −7.73897e12 −1.75789 −0.878944 0.476925i \(-0.841751\pi\)
−0.878944 + 0.476925i \(0.841751\pi\)
\(200\) 0 0
\(201\) 2.32187e12 0.499183
\(202\) −5.23250e12 −1.09465
\(203\) −6.57782e12 −1.33922
\(204\) 1.35081e11 0.0267687
\(205\) 0 0
\(206\) −2.46360e12 −0.462702
\(207\) −6.27599e12 −1.14775
\(208\) −2.31693e11 −0.0412634
\(209\) −1.36062e13 −2.36011
\(210\) 0 0
\(211\) −9.01879e12 −1.48455 −0.742275 0.670096i \(-0.766253\pi\)
−0.742275 + 0.670096i \(0.766253\pi\)
\(212\) 4.18822e12 0.671711
\(213\) 1.46654e12 0.229195
\(214\) 3.83653e12 0.584336
\(215\) 0 0
\(216\) 1.55217e12 0.224617
\(217\) 2.53606e12 0.357792
\(218\) 2.34696e12 0.322846
\(219\) −6.05416e11 −0.0812103
\(220\) 0 0
\(221\) −2.05652e11 −0.0262407
\(222\) 2.45330e11 0.0305358
\(223\) −2.06620e12 −0.250897 −0.125448 0.992100i \(-0.540037\pi\)
−0.125448 + 0.992100i \(0.540037\pi\)
\(224\) 2.87182e12 0.340246
\(225\) 0 0
\(226\) 7.00271e12 0.790077
\(227\) 1.56806e13 1.72672 0.863359 0.504591i \(-0.168357\pi\)
0.863359 + 0.504591i \(0.168357\pi\)
\(228\) −2.57385e12 −0.276657
\(229\) 6.02309e12 0.632010 0.316005 0.948758i \(-0.397658\pi\)
0.316005 + 0.948758i \(0.397658\pi\)
\(230\) 0 0
\(231\) −9.30701e12 −0.930989
\(232\) −2.51840e12 −0.246003
\(233\) −1.43129e13 −1.36544 −0.682718 0.730682i \(-0.739202\pi\)
−0.682718 + 0.730682i \(0.739202\pi\)
\(234\) −1.11052e12 −0.103476
\(235\) 0 0
\(236\) 3.86951e12 0.344065
\(237\) −2.78701e12 −0.242116
\(238\) 2.54904e12 0.216373
\(239\) 2.22886e13 1.84881 0.924407 0.381407i \(-0.124560\pi\)
0.924407 + 0.381407i \(0.124560\pi\)
\(240\) 0 0
\(241\) 2.06638e13 1.63726 0.818629 0.574323i \(-0.194735\pi\)
0.818629 + 0.574323i \(0.194735\pi\)
\(242\) −9.70679e12 −0.751781
\(243\) 1.13830e13 0.861830
\(244\) −9.87241e12 −0.730768
\(245\) 0 0
\(246\) −5.71501e11 −0.0404458
\(247\) 3.91853e12 0.271200
\(248\) 9.70962e11 0.0657231
\(249\) 1.92462e12 0.127423
\(250\) 0 0
\(251\) −1.23384e13 −0.781721 −0.390861 0.920450i \(-0.627823\pi\)
−0.390861 + 0.920450i \(0.627823\pi\)
\(252\) 1.37648e13 0.853229
\(253\) 3.06584e13 1.85946
\(254\) 1.25777e13 0.746475
\(255\) 0 0
\(256\) 1.09951e12 0.0625000
\(257\) −1.19160e13 −0.662975 −0.331487 0.943460i \(-0.607550\pi\)
−0.331487 + 0.943460i \(0.607550\pi\)
\(258\) −1.30929e12 −0.0713061
\(259\) 4.62949e12 0.246822
\(260\) 0 0
\(261\) −1.20708e13 −0.616897
\(262\) 8.42159e12 0.421441
\(263\) −1.54645e12 −0.0757845 −0.0378923 0.999282i \(-0.512064\pi\)
−0.0378923 + 0.999282i \(0.512064\pi\)
\(264\) −3.56330e12 −0.171014
\(265\) 0 0
\(266\) −4.85699e13 −2.23624
\(267\) 3.18984e12 0.143866
\(268\) 1.67751e13 0.741178
\(269\) 1.43195e13 0.619855 0.309927 0.950760i \(-0.399695\pi\)
0.309927 + 0.950760i \(0.399695\pi\)
\(270\) 0 0
\(271\) 2.26776e13 0.942468 0.471234 0.882008i \(-0.343809\pi\)
0.471234 + 0.882008i \(0.343809\pi\)
\(272\) 9.75929e11 0.0397456
\(273\) 2.68038e12 0.106980
\(274\) 1.66567e13 0.651568
\(275\) 0 0
\(276\) 5.79957e12 0.217970
\(277\) −9.25283e12 −0.340907 −0.170454 0.985366i \(-0.554523\pi\)
−0.170454 + 0.985366i \(0.554523\pi\)
\(278\) −2.18456e13 −0.789074
\(279\) 4.65386e12 0.164813
\(280\) 0 0
\(281\) −2.26511e12 −0.0771265 −0.0385633 0.999256i \(-0.512278\pi\)
−0.0385633 + 0.999256i \(0.512278\pi\)
\(282\) −7.12107e12 −0.237780
\(283\) 3.60409e13 1.18024 0.590121 0.807315i \(-0.299080\pi\)
0.590121 + 0.807315i \(0.299080\pi\)
\(284\) 1.05954e13 0.340304
\(285\) 0 0
\(286\) 5.42490e12 0.167640
\(287\) −1.07845e13 −0.326926
\(288\) 5.27001e12 0.156730
\(289\) −3.34057e13 −0.974725
\(290\) 0 0
\(291\) 1.54457e13 0.433905
\(292\) −4.37401e12 −0.120580
\(293\) −1.63348e13 −0.441918 −0.220959 0.975283i \(-0.570919\pi\)
−0.220959 + 0.975283i \(0.570919\pi\)
\(294\) −2.42549e13 −0.644005
\(295\) 0 0
\(296\) 1.77246e12 0.0453389
\(297\) −3.63426e13 −0.912548
\(298\) −3.25012e13 −0.801142
\(299\) −8.82948e12 −0.213670
\(300\) 0 0
\(301\) −2.47069e13 −0.576371
\(302\) −1.70893e13 −0.391458
\(303\) 2.31758e13 0.521314
\(304\) −1.85956e13 −0.410776
\(305\) 0 0
\(306\) 4.67767e12 0.0996694
\(307\) 5.61322e13 1.17476 0.587382 0.809310i \(-0.300158\pi\)
0.587382 + 0.809310i \(0.300158\pi\)
\(308\) −6.72412e13 −1.38231
\(309\) 1.09118e13 0.220355
\(310\) 0 0
\(311\) 6.92340e13 1.34939 0.674694 0.738097i \(-0.264275\pi\)
0.674694 + 0.738097i \(0.264275\pi\)
\(312\) 1.02621e12 0.0196511
\(313\) 3.23822e13 0.609273 0.304637 0.952469i \(-0.401465\pi\)
0.304637 + 0.952469i \(0.401465\pi\)
\(314\) −2.33496e13 −0.431684
\(315\) 0 0
\(316\) −2.01356e13 −0.359489
\(317\) 1.84023e13 0.322885 0.161442 0.986882i \(-0.448385\pi\)
0.161442 + 0.986882i \(0.448385\pi\)
\(318\) −1.85504e13 −0.319893
\(319\) 5.89661e13 0.999433
\(320\) 0 0
\(321\) −1.69927e13 −0.278282
\(322\) 1.09441e14 1.76186
\(323\) −1.65055e13 −0.261224
\(324\) 2.16153e13 0.336329
\(325\) 0 0
\(326\) −6.78946e12 −0.102126
\(327\) −1.03951e13 −0.153751
\(328\) −4.12897e12 −0.0600532
\(329\) −1.34378e14 −1.92199
\(330\) 0 0
\(331\) 6.74629e13 0.933277 0.466639 0.884448i \(-0.345465\pi\)
0.466639 + 0.884448i \(0.345465\pi\)
\(332\) 1.39050e13 0.189196
\(333\) 8.49546e12 0.113696
\(334\) −7.57163e12 −0.0996744
\(335\) 0 0
\(336\) −1.27198e13 −0.162038
\(337\) −1.07897e14 −1.35222 −0.676109 0.736802i \(-0.736335\pi\)
−0.676109 + 0.736802i \(0.736335\pi\)
\(338\) 5.57868e13 0.687843
\(339\) −3.10163e13 −0.376264
\(340\) 0 0
\(341\) −2.27342e13 −0.267012
\(342\) −8.91293e13 −1.03009
\(343\) −2.88468e14 −3.28081
\(344\) −9.45933e12 −0.105874
\(345\) 0 0
\(346\) −2.35834e13 −0.255675
\(347\) 1.29756e13 0.138458 0.0692288 0.997601i \(-0.477946\pi\)
0.0692288 + 0.997601i \(0.477946\pi\)
\(348\) 1.11545e13 0.117156
\(349\) −7.76358e13 −0.802643 −0.401321 0.915937i \(-0.631449\pi\)
−0.401321 + 0.915937i \(0.631449\pi\)
\(350\) 0 0
\(351\) 1.04665e13 0.104861
\(352\) −2.57441e13 −0.253918
\(353\) 4.21031e13 0.408839 0.204420 0.978883i \(-0.434469\pi\)
0.204420 + 0.978883i \(0.434469\pi\)
\(354\) −1.71388e13 −0.163856
\(355\) 0 0
\(356\) 2.30459e13 0.213609
\(357\) −1.12902e13 −0.103045
\(358\) −4.33101e12 −0.0389253
\(359\) 1.84963e14 1.63707 0.818533 0.574460i \(-0.194788\pi\)
0.818533 + 0.574460i \(0.194788\pi\)
\(360\) 0 0
\(361\) 1.98008e14 1.69978
\(362\) 4.34818e13 0.367630
\(363\) 4.29933e13 0.358025
\(364\) 1.93652e13 0.158841
\(365\) 0 0
\(366\) 4.37268e13 0.348018
\(367\) 1.68959e14 1.32470 0.662351 0.749194i \(-0.269559\pi\)
0.662351 + 0.749194i \(0.269559\pi\)
\(368\) 4.19007e13 0.323637
\(369\) −1.97903e13 −0.150594
\(370\) 0 0
\(371\) −3.50056e14 −2.58572
\(372\) −4.30058e12 −0.0312997
\(373\) 1.87187e14 1.34238 0.671191 0.741284i \(-0.265783\pi\)
0.671191 + 0.741284i \(0.265783\pi\)
\(374\) −2.28505e13 −0.161474
\(375\) 0 0
\(376\) −5.14483e13 −0.353051
\(377\) −1.69820e13 −0.114845
\(378\) −1.29732e14 −0.864651
\(379\) −7.47613e13 −0.491090 −0.245545 0.969385i \(-0.578967\pi\)
−0.245545 + 0.969385i \(0.578967\pi\)
\(380\) 0 0
\(381\) −5.57090e13 −0.355499
\(382\) −3.54074e13 −0.222713
\(383\) 1.55521e14 0.964264 0.482132 0.876099i \(-0.339863\pi\)
0.482132 + 0.876099i \(0.339863\pi\)
\(384\) −4.86995e12 −0.0297648
\(385\) 0 0
\(386\) 1.97448e14 1.17280
\(387\) −4.53390e13 −0.265498
\(388\) 1.11592e14 0.644255
\(389\) −8.76448e13 −0.498889 −0.249444 0.968389i \(-0.580248\pi\)
−0.249444 + 0.968389i \(0.580248\pi\)
\(390\) 0 0
\(391\) 3.71912e13 0.205811
\(392\) −1.75237e14 −0.956206
\(393\) −3.73009e13 −0.200706
\(394\) 6.41064e13 0.340151
\(395\) 0 0
\(396\) −1.23393e14 −0.636745
\(397\) −5.19767e13 −0.264521 −0.132261 0.991215i \(-0.542224\pi\)
−0.132261 + 0.991215i \(0.542224\pi\)
\(398\) 2.47647e14 1.24301
\(399\) 2.15125e14 1.06498
\(400\) 0 0
\(401\) −3.40753e14 −1.64114 −0.820570 0.571546i \(-0.806344\pi\)
−0.820570 + 0.571546i \(0.806344\pi\)
\(402\) −7.42999e13 −0.352976
\(403\) 6.54735e12 0.0306823
\(404\) 1.67440e14 0.774037
\(405\) 0 0
\(406\) 2.10490e14 0.946975
\(407\) −4.15005e13 −0.184198
\(408\) −4.32258e12 −0.0189283
\(409\) 3.34635e14 1.44575 0.722875 0.690979i \(-0.242820\pi\)
0.722875 + 0.690979i \(0.242820\pi\)
\(410\) 0 0
\(411\) −7.37756e13 −0.310301
\(412\) 7.88353e13 0.327180
\(413\) −3.23417e14 −1.32446
\(414\) 2.00832e14 0.811579
\(415\) 0 0
\(416\) 7.41419e12 0.0291776
\(417\) 9.67584e13 0.375786
\(418\) 4.35399e14 1.66885
\(419\) 1.37830e14 0.521396 0.260698 0.965420i \(-0.416047\pi\)
0.260698 + 0.965420i \(0.416047\pi\)
\(420\) 0 0
\(421\) 4.35453e14 1.60468 0.802342 0.596864i \(-0.203587\pi\)
0.802342 + 0.596864i \(0.203587\pi\)
\(422\) 2.88601e14 1.04974
\(423\) −2.46594e14 −0.885340
\(424\) −1.34023e14 −0.474972
\(425\) 0 0
\(426\) −4.69292e13 −0.162065
\(427\) 8.25146e14 2.81305
\(428\) −1.22769e14 −0.413188
\(429\) −2.40279e13 −0.0798365
\(430\) 0 0
\(431\) 1.51185e14 0.489647 0.244824 0.969568i \(-0.421270\pi\)
0.244824 + 0.969568i \(0.421270\pi\)
\(432\) −4.96693e13 −0.158828
\(433\) 3.75044e13 0.118413 0.0592065 0.998246i \(-0.481143\pi\)
0.0592065 + 0.998246i \(0.481143\pi\)
\(434\) −8.11539e13 −0.252997
\(435\) 0 0
\(436\) −7.51028e13 −0.228287
\(437\) −7.08648e14 −2.12707
\(438\) 1.93733e13 0.0574244
\(439\) −1.61934e14 −0.474004 −0.237002 0.971509i \(-0.576165\pi\)
−0.237002 + 0.971509i \(0.576165\pi\)
\(440\) 0 0
\(441\) −8.39917e14 −2.39786
\(442\) 6.58085e12 0.0185549
\(443\) 1.29342e14 0.360179 0.180090 0.983650i \(-0.442361\pi\)
0.180090 + 0.983650i \(0.442361\pi\)
\(444\) −7.85055e12 −0.0215920
\(445\) 0 0
\(446\) 6.61183e13 0.177411
\(447\) 1.43954e14 0.381533
\(448\) −9.18983e13 −0.240590
\(449\) 1.30255e13 0.0336852 0.0168426 0.999858i \(-0.494639\pi\)
0.0168426 + 0.999858i \(0.494639\pi\)
\(450\) 0 0
\(451\) 9.66764e13 0.243977
\(452\) −2.24087e14 −0.558669
\(453\) 7.56916e13 0.186427
\(454\) −5.01780e14 −1.22097
\(455\) 0 0
\(456\) 8.23633e13 0.195626
\(457\) −3.45724e14 −0.811318 −0.405659 0.914025i \(-0.632958\pi\)
−0.405659 + 0.914025i \(0.632958\pi\)
\(458\) −1.92739e14 −0.446899
\(459\) −4.40866e13 −0.101004
\(460\) 0 0
\(461\) 2.96945e14 0.664234 0.332117 0.943238i \(-0.392237\pi\)
0.332117 + 0.943238i \(0.392237\pi\)
\(462\) 2.97824e14 0.658308
\(463\) −4.55258e14 −0.994402 −0.497201 0.867635i \(-0.665639\pi\)
−0.497201 + 0.867635i \(0.665639\pi\)
\(464\) 8.05887e13 0.173950
\(465\) 0 0
\(466\) 4.58014e14 0.965509
\(467\) 3.66499e14 0.763537 0.381768 0.924258i \(-0.375315\pi\)
0.381768 + 0.924258i \(0.375315\pi\)
\(468\) 3.55365e13 0.0731683
\(469\) −1.40208e15 −2.85313
\(470\) 0 0
\(471\) 1.03420e14 0.205584
\(472\) −1.23824e14 −0.243290
\(473\) 2.21482e14 0.430132
\(474\) 8.91843e13 0.171202
\(475\) 0 0
\(476\) −8.15691e13 −0.152999
\(477\) −6.42379e14 −1.19108
\(478\) −7.13234e14 −1.30731
\(479\) 5.78377e14 1.04801 0.524005 0.851715i \(-0.324437\pi\)
0.524005 + 0.851715i \(0.324437\pi\)
\(480\) 0 0
\(481\) 1.19520e13 0.0211661
\(482\) −6.61242e14 −1.15772
\(483\) −4.84734e14 −0.839062
\(484\) 3.10617e14 0.531589
\(485\) 0 0
\(486\) −3.64255e14 −0.609406
\(487\) 4.75203e14 0.786086 0.393043 0.919520i \(-0.371422\pi\)
0.393043 + 0.919520i \(0.371422\pi\)
\(488\) 3.15917e14 0.516731
\(489\) 3.00718e13 0.0486363
\(490\) 0 0
\(491\) 7.59641e14 1.20132 0.600662 0.799503i \(-0.294904\pi\)
0.600662 + 0.799503i \(0.294904\pi\)
\(492\) 1.82880e13 0.0285995
\(493\) 7.15307e13 0.110620
\(494\) −1.25393e14 −0.191767
\(495\) 0 0
\(496\) −3.10708e13 −0.0464732
\(497\) −8.85576e14 −1.30998
\(498\) −6.15879e13 −0.0901019
\(499\) −4.91039e14 −0.710499 −0.355249 0.934772i \(-0.615604\pi\)
−0.355249 + 0.934772i \(0.615604\pi\)
\(500\) 0 0
\(501\) 3.35362e13 0.0474686
\(502\) 3.94828e14 0.552760
\(503\) 4.25544e14 0.589278 0.294639 0.955609i \(-0.404801\pi\)
0.294639 + 0.955609i \(0.404801\pi\)
\(504\) −4.40472e14 −0.603324
\(505\) 0 0
\(506\) −9.81069e14 −1.31484
\(507\) −2.47090e14 −0.327576
\(508\) −4.02486e14 −0.527838
\(509\) −1.41466e15 −1.83529 −0.917644 0.397404i \(-0.869911\pi\)
−0.917644 + 0.397404i \(0.869911\pi\)
\(510\) 0 0
\(511\) 3.65584e14 0.464165
\(512\) −3.51844e13 −0.0441942
\(513\) 8.40035e14 1.04388
\(514\) 3.81311e14 0.468794
\(515\) 0 0
\(516\) 4.18972e13 0.0504210
\(517\) 1.20462e15 1.43434
\(518\) −1.48144e14 −0.174530
\(519\) 1.04456e14 0.121762
\(520\) 0 0
\(521\) 6.27267e14 0.715888 0.357944 0.933743i \(-0.383478\pi\)
0.357944 + 0.933743i \(0.383478\pi\)
\(522\) 3.86265e14 0.436212
\(523\) −2.23774e14 −0.250063 −0.125032 0.992153i \(-0.539903\pi\)
−0.125032 + 0.992153i \(0.539903\pi\)
\(524\) −2.69491e14 −0.298004
\(525\) 0 0
\(526\) 4.94865e13 0.0535877
\(527\) −2.75785e13 −0.0295537
\(528\) 1.14026e14 0.120925
\(529\) 6.43961e14 0.675855
\(530\) 0 0
\(531\) −5.93495e14 −0.610095
\(532\) 1.55424e15 1.58126
\(533\) −2.78424e13 −0.0280354
\(534\) −1.02075e14 −0.101728
\(535\) 0 0
\(536\) −5.36802e14 −0.524092
\(537\) 1.91829e13 0.0185377
\(538\) −4.58224e14 −0.438303
\(539\) 4.10302e15 3.88477
\(540\) 0 0
\(541\) 5.47756e14 0.508162 0.254081 0.967183i \(-0.418227\pi\)
0.254081 + 0.967183i \(0.418227\pi\)
\(542\) −7.25684e14 −0.666425
\(543\) −1.92589e14 −0.175079
\(544\) −3.12297e13 −0.0281044
\(545\) 0 0
\(546\) −8.57720e13 −0.0756460
\(547\) −4.33518e14 −0.378509 −0.189255 0.981928i \(-0.560607\pi\)
−0.189255 + 0.981928i \(0.560607\pi\)
\(548\) −5.33013e14 −0.460728
\(549\) 1.51420e15 1.29580
\(550\) 0 0
\(551\) −1.36296e15 −1.14327
\(552\) −1.85586e14 −0.154128
\(553\) 1.68295e15 1.38383
\(554\) 2.96091e14 0.241058
\(555\) 0 0
\(556\) 6.99060e14 0.557960
\(557\) 7.36061e14 0.581715 0.290857 0.956766i \(-0.406059\pi\)
0.290857 + 0.956766i \(0.406059\pi\)
\(558\) −1.48924e14 −0.116540
\(559\) −6.37858e13 −0.0494264
\(560\) 0 0
\(561\) 1.01209e14 0.0768998
\(562\) 7.24834e13 0.0545367
\(563\) 9.03199e14 0.672957 0.336478 0.941691i \(-0.390764\pi\)
0.336478 + 0.941691i \(0.390764\pi\)
\(564\) 2.27874e14 0.168136
\(565\) 0 0
\(566\) −1.15331e15 −0.834557
\(567\) −1.80663e15 −1.29468
\(568\) −3.39054e14 −0.240632
\(569\) −2.25786e15 −1.58701 −0.793503 0.608566i \(-0.791745\pi\)
−0.793503 + 0.608566i \(0.791745\pi\)
\(570\) 0 0
\(571\) 9.15100e14 0.630913 0.315457 0.948940i \(-0.397842\pi\)
0.315457 + 0.948940i \(0.397842\pi\)
\(572\) −1.73597e14 −0.118540
\(573\) 1.56826e14 0.106064
\(574\) 3.45104e14 0.231172
\(575\) 0 0
\(576\) −1.68640e14 −0.110825
\(577\) −2.84484e14 −0.185179 −0.0925893 0.995704i \(-0.529514\pi\)
−0.0925893 + 0.995704i \(0.529514\pi\)
\(578\) 1.06898e15 0.689234
\(579\) −8.74535e14 −0.558528
\(580\) 0 0
\(581\) −1.16219e15 −0.728299
\(582\) −4.94261e14 −0.306817
\(583\) 3.13804e15 1.92966
\(584\) 1.39968e14 0.0852626
\(585\) 0 0
\(586\) 5.22714e14 0.312484
\(587\) 1.99879e15 1.18374 0.591871 0.806032i \(-0.298389\pi\)
0.591871 + 0.806032i \(0.298389\pi\)
\(588\) 7.76157e14 0.455380
\(589\) 5.25486e14 0.305441
\(590\) 0 0
\(591\) −2.83940e14 −0.161992
\(592\) −5.67186e13 −0.0320595
\(593\) −8.52303e14 −0.477302 −0.238651 0.971105i \(-0.576705\pi\)
−0.238651 + 0.971105i \(0.576705\pi\)
\(594\) 1.16296e15 0.645269
\(595\) 0 0
\(596\) 1.04004e15 0.566493
\(597\) −1.09688e15 −0.591969
\(598\) 2.82543e14 0.151087
\(599\) 1.08984e14 0.0577449 0.0288725 0.999583i \(-0.490808\pi\)
0.0288725 + 0.999583i \(0.490808\pi\)
\(600\) 0 0
\(601\) 3.21315e14 0.167156 0.0835778 0.996501i \(-0.473365\pi\)
0.0835778 + 0.996501i \(0.473365\pi\)
\(602\) 7.90620e14 0.407556
\(603\) −2.57291e15 −1.31426
\(604\) 5.46856e14 0.276803
\(605\) 0 0
\(606\) −7.41624e14 −0.368625
\(607\) 3.08680e15 1.52044 0.760222 0.649664i \(-0.225090\pi\)
0.760222 + 0.649664i \(0.225090\pi\)
\(608\) 5.95058e14 0.290462
\(609\) −9.32301e14 −0.450984
\(610\) 0 0
\(611\) −3.46924e14 −0.164819
\(612\) −1.49685e14 −0.0704769
\(613\) −8.70867e14 −0.406368 −0.203184 0.979141i \(-0.565129\pi\)
−0.203184 + 0.979141i \(0.565129\pi\)
\(614\) −1.79623e15 −0.830684
\(615\) 0 0
\(616\) 2.15172e15 0.977444
\(617\) −3.06108e15 −1.37818 −0.689091 0.724675i \(-0.741990\pi\)
−0.689091 + 0.724675i \(0.741990\pi\)
\(618\) −3.49177e14 −0.155815
\(619\) −1.21317e15 −0.536566 −0.268283 0.963340i \(-0.586456\pi\)
−0.268283 + 0.963340i \(0.586456\pi\)
\(620\) 0 0
\(621\) −1.89282e15 −0.822442
\(622\) −2.21549e15 −0.954162
\(623\) −1.92620e15 −0.822276
\(624\) −3.28389e13 −0.0138955
\(625\) 0 0
\(626\) −1.03623e15 −0.430821
\(627\) −1.92847e15 −0.794768
\(628\) 7.47187e14 0.305247
\(629\) −5.03436e13 −0.0203876
\(630\) 0 0
\(631\) 2.95354e15 1.17539 0.587693 0.809084i \(-0.300036\pi\)
0.587693 + 0.809084i \(0.300036\pi\)
\(632\) 6.44338e14 0.254197
\(633\) −1.27827e15 −0.499922
\(634\) −5.88875e14 −0.228314
\(635\) 0 0
\(636\) 5.93614e14 0.226199
\(637\) −1.18165e15 −0.446398
\(638\) −1.88691e15 −0.706706
\(639\) −1.62510e15 −0.603427
\(640\) 0 0
\(641\) −1.24519e15 −0.454483 −0.227241 0.973838i \(-0.572971\pi\)
−0.227241 + 0.973838i \(0.572971\pi\)
\(642\) 5.43767e14 0.196775
\(643\) −2.82493e14 −0.101356 −0.0506778 0.998715i \(-0.516138\pi\)
−0.0506778 + 0.998715i \(0.516138\pi\)
\(644\) −3.50210e15 −1.24582
\(645\) 0 0
\(646\) 5.28175e14 0.184714
\(647\) 3.45042e15 1.19646 0.598230 0.801324i \(-0.295871\pi\)
0.598230 + 0.801324i \(0.295871\pi\)
\(648\) −6.91691e14 −0.237821
\(649\) 2.89924e15 0.988413
\(650\) 0 0
\(651\) 3.59446e14 0.120487
\(652\) 2.17263e14 0.0722143
\(653\) −3.55303e15 −1.17105 −0.585527 0.810653i \(-0.699112\pi\)
−0.585527 + 0.810653i \(0.699112\pi\)
\(654\) 3.32645e14 0.108718
\(655\) 0 0
\(656\) 1.32127e14 0.0424640
\(657\) 6.70874e14 0.213812
\(658\) 4.30010e15 1.35905
\(659\) 1.76602e14 0.0553511 0.0276756 0.999617i \(-0.491189\pi\)
0.0276756 + 0.999617i \(0.491189\pi\)
\(660\) 0 0
\(661\) 3.46137e15 1.06694 0.533470 0.845819i \(-0.320888\pi\)
0.533470 + 0.845819i \(0.320888\pi\)
\(662\) −2.15881e15 −0.659927
\(663\) −2.91479e13 −0.00883654
\(664\) −4.44960e14 −0.133782
\(665\) 0 0
\(666\) −2.71855e14 −0.0803949
\(667\) 3.07111e15 0.900748
\(668\) 2.42292e14 0.0704804
\(669\) −2.92851e14 −0.0844895
\(670\) 0 0
\(671\) −7.39693e15 −2.09932
\(672\) 4.07035e14 0.114578
\(673\) −6.92295e15 −1.93290 −0.966448 0.256864i \(-0.917311\pi\)
−0.966448 + 0.256864i \(0.917311\pi\)
\(674\) 3.45272e15 0.956162
\(675\) 0 0
\(676\) −1.78518e15 −0.486379
\(677\) 3.89173e15 1.05173 0.525865 0.850568i \(-0.323742\pi\)
0.525865 + 0.850568i \(0.323742\pi\)
\(678\) 9.92523e14 0.266058
\(679\) −9.32695e15 −2.48002
\(680\) 0 0
\(681\) 2.22248e15 0.581472
\(682\) 7.27496e14 0.188806
\(683\) −4.79007e15 −1.23318 −0.616592 0.787283i \(-0.711487\pi\)
−0.616592 + 0.787283i \(0.711487\pi\)
\(684\) 2.85214e15 0.728387
\(685\) 0 0
\(686\) 9.23099e15 2.31988
\(687\) 8.53677e14 0.212829
\(688\) 3.02698e14 0.0748641
\(689\) −9.03740e14 −0.221737
\(690\) 0 0
\(691\) 1.60990e15 0.388749 0.194374 0.980927i \(-0.437732\pi\)
0.194374 + 0.980927i \(0.437732\pi\)
\(692\) 7.54670e14 0.180790
\(693\) 1.03133e16 2.45112
\(694\) −4.15220e14 −0.0979043
\(695\) 0 0
\(696\) −3.56943e14 −0.0828415
\(697\) 1.17276e14 0.0270041
\(698\) 2.48435e15 0.567554
\(699\) −2.02863e15 −0.459810
\(700\) 0 0
\(701\) −2.46356e15 −0.549686 −0.274843 0.961489i \(-0.588626\pi\)
−0.274843 + 0.961489i \(0.588626\pi\)
\(702\) −3.34928e14 −0.0741477
\(703\) 9.59257e14 0.210708
\(704\) 8.23812e14 0.179547
\(705\) 0 0
\(706\) −1.34730e15 −0.289093
\(707\) −1.39948e16 −2.97962
\(708\) 5.48442e14 0.115864
\(709\) 6.21323e15 1.30246 0.651228 0.758882i \(-0.274254\pi\)
0.651228 + 0.758882i \(0.274254\pi\)
\(710\) 0 0
\(711\) 3.08834e15 0.637445
\(712\) −7.37470e14 −0.151044
\(713\) −1.18406e15 −0.240647
\(714\) 3.61285e14 0.0728635
\(715\) 0 0
\(716\) 1.38592e14 0.0275244
\(717\) 3.15905e15 0.622588
\(718\) −5.91883e15 −1.15758
\(719\) 7.12825e15 1.38348 0.691742 0.722145i \(-0.256844\pi\)
0.691742 + 0.722145i \(0.256844\pi\)
\(720\) 0 0
\(721\) −6.58914e15 −1.25946
\(722\) −6.33627e15 −1.20193
\(723\) 2.92877e15 0.551346
\(724\) −1.39142e15 −0.259953
\(725\) 0 0
\(726\) −1.37578e15 −0.253162
\(727\) −5.59273e15 −1.02137 −0.510686 0.859767i \(-0.670609\pi\)
−0.510686 + 0.859767i \(0.670609\pi\)
\(728\) −6.19685e14 −0.112318
\(729\) −2.12599e15 −0.382437
\(730\) 0 0
\(731\) 2.68676e14 0.0476084
\(732\) −1.39926e15 −0.246086
\(733\) 2.75558e15 0.480995 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(734\) −5.40669e15 −0.936705
\(735\) 0 0
\(736\) −1.34082e15 −0.228846
\(737\) 1.25688e16 2.12922
\(738\) 6.33291e14 0.106486
\(739\) 1.99004e15 0.332138 0.166069 0.986114i \(-0.446893\pi\)
0.166069 + 0.986114i \(0.446893\pi\)
\(740\) 0 0
\(741\) 5.55389e14 0.0913266
\(742\) 1.12018e16 1.82838
\(743\) 7.01045e15 1.13581 0.567907 0.823093i \(-0.307753\pi\)
0.567907 + 0.823093i \(0.307753\pi\)
\(744\) 1.37618e14 0.0221322
\(745\) 0 0
\(746\) −5.98997e15 −0.949208
\(747\) −2.13271e15 −0.335481
\(748\) 7.31218e14 0.114179
\(749\) 1.02611e16 1.59054
\(750\) 0 0
\(751\) −9.37589e15 −1.43216 −0.716082 0.698016i \(-0.754066\pi\)
−0.716082 + 0.698016i \(0.754066\pi\)
\(752\) 1.64634e15 0.249645
\(753\) −1.74877e15 −0.263245
\(754\) 5.43423e14 0.0812074
\(755\) 0 0
\(756\) 4.15141e15 0.611400
\(757\) −1.88289e15 −0.275294 −0.137647 0.990481i \(-0.543954\pi\)
−0.137647 + 0.990481i \(0.543954\pi\)
\(758\) 2.39236e15 0.347253
\(759\) 4.34534e15 0.626173
\(760\) 0 0
\(761\) 6.71695e15 0.954018 0.477009 0.878898i \(-0.341721\pi\)
0.477009 + 0.878898i \(0.341721\pi\)
\(762\) 1.78269e15 0.251375
\(763\) 6.27717e15 0.878777
\(764\) 1.13304e15 0.157482
\(765\) 0 0
\(766\) −4.97667e15 −0.681837
\(767\) −8.34968e14 −0.113578
\(768\) 1.55838e14 0.0210469
\(769\) −5.72271e15 −0.767374 −0.383687 0.923463i \(-0.625346\pi\)
−0.383687 + 0.923463i \(0.625346\pi\)
\(770\) 0 0
\(771\) −1.68890e15 −0.223257
\(772\) −6.31833e15 −0.829292
\(773\) 3.24936e15 0.423458 0.211729 0.977328i \(-0.432091\pi\)
0.211729 + 0.977328i \(0.432091\pi\)
\(774\) 1.45085e15 0.187735
\(775\) 0 0
\(776\) −3.57094e15 −0.455557
\(777\) 6.56157e14 0.0831174
\(778\) 2.80463e15 0.352767
\(779\) −2.23461e15 −0.279091
\(780\) 0 0
\(781\) 7.93865e15 0.977611
\(782\) −1.19012e15 −0.145530
\(783\) −3.64051e15 −0.442051
\(784\) 5.60757e15 0.676140
\(785\) 0 0
\(786\) 1.19363e15 0.141920
\(787\) −1.29754e16 −1.53201 −0.766003 0.642838i \(-0.777757\pi\)
−0.766003 + 0.642838i \(0.777757\pi\)
\(788\) −2.05140e15 −0.240523
\(789\) −2.19185e14 −0.0255204
\(790\) 0 0
\(791\) 1.87294e16 2.15057
\(792\) 3.94857e15 0.450247
\(793\) 2.13028e15 0.241232
\(794\) 1.66325e15 0.187045
\(795\) 0 0
\(796\) −7.92470e15 −0.878944
\(797\) 1.21467e16 1.33795 0.668974 0.743286i \(-0.266734\pi\)
0.668974 + 0.743286i \(0.266734\pi\)
\(798\) −6.88401e15 −0.753053
\(799\) 1.46130e15 0.158757
\(800\) 0 0
\(801\) −3.53473e15 −0.378771
\(802\) 1.09041e16 1.16046
\(803\) −3.27724e15 −0.346395
\(804\) 2.37760e15 0.249592
\(805\) 0 0
\(806\) −2.09515e14 −0.0216957
\(807\) 2.02956e15 0.208736
\(808\) −5.35808e15 −0.547327
\(809\) 3.06008e15 0.310468 0.155234 0.987878i \(-0.450387\pi\)
0.155234 + 0.987878i \(0.450387\pi\)
\(810\) 0 0
\(811\) 1.63765e15 0.163910 0.0819552 0.996636i \(-0.473884\pi\)
0.0819552 + 0.996636i \(0.473884\pi\)
\(812\) −6.73568e15 −0.669612
\(813\) 3.21419e15 0.317376
\(814\) 1.32802e15 0.130248
\(815\) 0 0
\(816\) 1.38323e14 0.0133843
\(817\) −5.11941e15 −0.492038
\(818\) −1.07083e16 −1.02230
\(819\) −2.97018e15 −0.281657
\(820\) 0 0
\(821\) 1.07044e15 0.100156 0.0500778 0.998745i \(-0.484053\pi\)
0.0500778 + 0.998745i \(0.484053\pi\)
\(822\) 2.36082e15 0.219416
\(823\) 3.24575e15 0.299651 0.149825 0.988712i \(-0.452129\pi\)
0.149825 + 0.988712i \(0.452129\pi\)
\(824\) −2.52273e15 −0.231351
\(825\) 0 0
\(826\) 1.03494e16 0.936533
\(827\) 2.05440e16 1.84674 0.923369 0.383913i \(-0.125424\pi\)
0.923369 + 0.383913i \(0.125424\pi\)
\(828\) −6.42662e15 −0.573873
\(829\) 5.85872e15 0.519700 0.259850 0.965649i \(-0.416327\pi\)
0.259850 + 0.965649i \(0.416327\pi\)
\(830\) 0 0
\(831\) −1.31144e15 −0.114800
\(832\) −2.37254e14 −0.0206317
\(833\) 4.97730e15 0.429978
\(834\) −3.09627e15 −0.265721
\(835\) 0 0
\(836\) −1.39328e16 −1.18006
\(837\) 1.40359e15 0.118100
\(838\) −4.41057e15 −0.368683
\(839\) −2.25383e16 −1.87168 −0.935838 0.352430i \(-0.885355\pi\)
−0.935838 + 0.352430i \(0.885355\pi\)
\(840\) 0 0
\(841\) −6.29376e15 −0.515861
\(842\) −1.39345e16 −1.13468
\(843\) −3.21043e14 −0.0259724
\(844\) −9.23524e15 −0.742275
\(845\) 0 0
\(846\) 7.89100e15 0.626030
\(847\) −2.59617e16 −2.04632
\(848\) 4.28874e15 0.335856
\(849\) 5.10823e15 0.397446
\(850\) 0 0
\(851\) −2.16146e15 −0.166010
\(852\) 1.50173e15 0.114597
\(853\) −1.84471e16 −1.39865 −0.699326 0.714803i \(-0.746516\pi\)
−0.699326 + 0.714803i \(0.746516\pi\)
\(854\) −2.64047e16 −1.98913
\(855\) 0 0
\(856\) 3.92860e15 0.292168
\(857\) 1.53957e15 0.113764 0.0568820 0.998381i \(-0.481884\pi\)
0.0568820 + 0.998381i \(0.481884\pi\)
\(858\) 7.68894e14 0.0564529
\(859\) −2.14973e16 −1.56827 −0.784136 0.620589i \(-0.786893\pi\)
−0.784136 + 0.620589i \(0.786893\pi\)
\(860\) 0 0
\(861\) −1.52853e15 −0.110092
\(862\) −4.83792e15 −0.346233
\(863\) 1.16923e16 0.831462 0.415731 0.909488i \(-0.363526\pi\)
0.415731 + 0.909488i \(0.363526\pi\)
\(864\) 1.58942e15 0.112308
\(865\) 0 0
\(866\) −1.20014e15 −0.0837306
\(867\) −4.73472e15 −0.328239
\(868\) 2.59693e15 0.178896
\(869\) −1.50866e16 −1.03272
\(870\) 0 0
\(871\) −3.61974e15 −0.244668
\(872\) 2.40329e15 0.161423
\(873\) −1.71156e16 −1.14239
\(874\) 2.26767e16 1.50407
\(875\) 0 0
\(876\) −6.19946e14 −0.0406052
\(877\) −1.13573e16 −0.739225 −0.369613 0.929186i \(-0.620510\pi\)
−0.369613 + 0.929186i \(0.620510\pi\)
\(878\) 5.18188e15 0.335172
\(879\) −2.31520e15 −0.148816
\(880\) 0 0
\(881\) 1.12847e16 0.716346 0.358173 0.933655i \(-0.383400\pi\)
0.358173 + 0.933655i \(0.383400\pi\)
\(882\) 2.68773e16 1.69554
\(883\) 5.36286e15 0.336211 0.168106 0.985769i \(-0.446235\pi\)
0.168106 + 0.985769i \(0.446235\pi\)
\(884\) −2.10587e14 −0.0131203
\(885\) 0 0
\(886\) −4.13894e15 −0.254685
\(887\) −8.12685e15 −0.496984 −0.248492 0.968634i \(-0.579935\pi\)
−0.248492 + 0.968634i \(0.579935\pi\)
\(888\) 2.51218e14 0.0152679
\(889\) 3.36402e16 2.03188
\(890\) 0 0
\(891\) 1.61953e16 0.966190
\(892\) −2.11579e15 −0.125448
\(893\) −2.78439e16 −1.64077
\(894\) −4.60653e15 −0.269785
\(895\) 0 0
\(896\) 2.94075e15 0.170123
\(897\) −1.25144e15 −0.0719534
\(898\) −4.16815e14 −0.0238190
\(899\) −2.27733e15 −0.129345
\(900\) 0 0
\(901\) 3.80670e15 0.213581
\(902\) −3.09364e15 −0.172518
\(903\) −3.50181e15 −0.194093
\(904\) 7.17077e15 0.395039
\(905\) 0 0
\(906\) −2.42213e15 −0.131823
\(907\) 1.18191e16 0.639361 0.319680 0.947525i \(-0.396424\pi\)
0.319680 + 0.947525i \(0.396424\pi\)
\(908\) 1.60570e16 0.863359
\(909\) −2.56815e16 −1.37252
\(910\) 0 0
\(911\) 1.50443e16 0.794366 0.397183 0.917739i \(-0.369988\pi\)
0.397183 + 0.917739i \(0.369988\pi\)
\(912\) −2.63563e15 −0.138329
\(913\) 1.04183e16 0.543513
\(914\) 1.10632e16 0.573688
\(915\) 0 0
\(916\) 6.16764e15 0.316005
\(917\) 2.25243e16 1.14715
\(918\) 1.41077e15 0.0714203
\(919\) −3.98208e15 −0.200389 −0.100195 0.994968i \(-0.531947\pi\)
−0.100195 + 0.994968i \(0.531947\pi\)
\(920\) 0 0
\(921\) 7.95585e15 0.395602
\(922\) −9.50225e15 −0.469685
\(923\) −2.28629e15 −0.112337
\(924\) −9.53038e15 −0.465494
\(925\) 0 0
\(926\) 1.45682e16 0.703148
\(927\) −1.20916e16 −0.580154
\(928\) −2.57884e15 −0.123001
\(929\) 4.06546e16 1.92763 0.963814 0.266575i \(-0.0858919\pi\)
0.963814 + 0.266575i \(0.0858919\pi\)
\(930\) 0 0
\(931\) −9.48384e16 −4.44387
\(932\) −1.46564e16 −0.682718
\(933\) 9.81282e15 0.454407
\(934\) −1.17280e16 −0.539902
\(935\) 0 0
\(936\) −1.13717e15 −0.0517378
\(937\) 1.18458e15 0.0535792 0.0267896 0.999641i \(-0.491472\pi\)
0.0267896 + 0.999641i \(0.491472\pi\)
\(938\) 4.48664e16 2.01746
\(939\) 4.58966e15 0.205173
\(940\) 0 0
\(941\) −1.02223e16 −0.451652 −0.225826 0.974168i \(-0.572508\pi\)
−0.225826 + 0.974168i \(0.572508\pi\)
\(942\) −3.30944e15 −0.145370
\(943\) 5.03516e15 0.219887
\(944\) 3.96238e15 0.172032
\(945\) 0 0
\(946\) −7.08743e15 −0.304150
\(947\) 1.68932e16 0.720755 0.360378 0.932807i \(-0.382648\pi\)
0.360378 + 0.932807i \(0.382648\pi\)
\(948\) −2.85390e15 −0.121058
\(949\) 9.43829e14 0.0398042
\(950\) 0 0
\(951\) 2.60824e15 0.108731
\(952\) 2.61021e15 0.108186
\(953\) 3.76078e16 1.54977 0.774884 0.632103i \(-0.217808\pi\)
0.774884 + 0.632103i \(0.217808\pi\)
\(954\) 2.05561e16 0.842219
\(955\) 0 0
\(956\) 2.28235e16 0.924407
\(957\) 8.35751e15 0.336559
\(958\) −1.85081e16 −0.741055
\(959\) 4.45498e16 1.77355
\(960\) 0 0
\(961\) −2.45305e16 −0.965444
\(962\) −3.82463e14 −0.0149667
\(963\) 1.88300e16 0.732664
\(964\) 2.11598e16 0.818629
\(965\) 0 0
\(966\) 1.55115e16 0.593306
\(967\) 3.52262e16 1.33974 0.669869 0.742479i \(-0.266350\pi\)
0.669869 + 0.742479i \(0.266350\pi\)
\(968\) −9.93976e15 −0.375890
\(969\) −2.33939e15 −0.0879674
\(970\) 0 0
\(971\) 3.56726e16 1.32626 0.663131 0.748504i \(-0.269227\pi\)
0.663131 + 0.748504i \(0.269227\pi\)
\(972\) 1.16562e16 0.430915
\(973\) −5.84281e16 −2.14784
\(974\) −1.52065e16 −0.555847
\(975\) 0 0
\(976\) −1.01094e16 −0.365384
\(977\) 4.79343e16 1.72277 0.861383 0.507957i \(-0.169599\pi\)
0.861383 + 0.507957i \(0.169599\pi\)
\(978\) −9.62298e14 −0.0343911
\(979\) 1.72672e16 0.613646
\(980\) 0 0
\(981\) 1.15191e16 0.404798
\(982\) −2.43085e16 −0.849464
\(983\) −2.05131e16 −0.712833 −0.356416 0.934327i \(-0.616002\pi\)
−0.356416 + 0.934327i \(0.616002\pi\)
\(984\) −5.85217e14 −0.0202229
\(985\) 0 0
\(986\) −2.28898e15 −0.0782203
\(987\) −1.90460e16 −0.647230
\(988\) 4.01257e15 0.135600
\(989\) 1.15354e16 0.387661
\(990\) 0 0
\(991\) −2.11314e16 −0.702301 −0.351150 0.936319i \(-0.614209\pi\)
−0.351150 + 0.936319i \(0.614209\pi\)
\(992\) 9.94265e14 0.0328615
\(993\) 9.56179e15 0.314281
\(994\) 2.83384e16 0.926298
\(995\) 0 0
\(996\) 1.97081e15 0.0637116
\(997\) −4.57566e16 −1.47106 −0.735531 0.677491i \(-0.763067\pi\)
−0.735531 + 0.677491i \(0.763067\pi\)
\(998\) 1.57133e16 0.502398
\(999\) 2.56220e15 0.0814711
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 50.12.a.f.1.2 2
5.2 odd 4 50.12.b.f.49.1 4
5.3 odd 4 50.12.b.f.49.4 4
5.4 even 2 10.12.a.d.1.1 2
15.14 odd 2 90.12.a.l.1.2 2
20.19 odd 2 80.12.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.12.a.d.1.1 2 5.4 even 2
50.12.a.f.1.2 2 1.1 even 1 trivial
50.12.b.f.49.1 4 5.2 odd 4
50.12.b.f.49.4 4 5.3 odd 4
80.12.a.g.1.2 2 20.19 odd 2
90.12.a.l.1.2 2 15.14 odd 2