Properties

Label 13.10.a.a.1.2
Level $13$
Weight $10$
Character 13.1
Self dual yes
Analytic conductor $6.695$
Analytic rank $1$
Dimension $4$
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] = [13,10,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 13.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: \(6.69546587013\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(16.5360\) of defining polynomial
Character \(\chi\) \(=\) 13.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-24.5360 q^{2} +49.9972 q^{3} +90.0171 q^{4} +1814.98 q^{5} -1226.73 q^{6} -8707.31 q^{7} +10353.8 q^{8} -17183.3 q^{9} -44532.3 q^{10} -82919.7 q^{11} +4500.61 q^{12} -28561.0 q^{13} +213643. q^{14} +90743.8 q^{15} -300130. q^{16} +374907. q^{17} +421610. q^{18} -361501. q^{19} +163379. q^{20} -435341. q^{21} +2.03452e6 q^{22} -2.31340e6 q^{23} +517661. q^{24} +1.34101e6 q^{25} +700774. q^{26} -1.84321e6 q^{27} -783807. q^{28} -649654. q^{29} -2.22649e6 q^{30} +4.32521e6 q^{31} +2.06285e6 q^{32} -4.14575e6 q^{33} -9.19873e6 q^{34} -1.58036e7 q^{35} -1.54679e6 q^{36} +1.21021e7 q^{37} +8.86981e6 q^{38} -1.42797e6 q^{39} +1.87919e7 q^{40} +2.59960e6 q^{41} +1.06816e7 q^{42} +3.08318e6 q^{43} -7.46419e6 q^{44} -3.11872e7 q^{45} +5.67617e7 q^{46} +2.60298e7 q^{47} -1.50057e7 q^{48} +3.54636e7 q^{49} -3.29032e7 q^{50} +1.87443e7 q^{51} -2.57098e6 q^{52} -1.01982e8 q^{53} +4.52251e7 q^{54} -1.50497e8 q^{55} -9.01536e7 q^{56} -1.80741e7 q^{57} +1.59399e7 q^{58} +1.37562e8 q^{59} +8.16850e6 q^{60} -4.29579e7 q^{61} -1.06124e8 q^{62} +1.49620e8 q^{63} +1.03052e8 q^{64} -5.18375e7 q^{65} +1.01720e8 q^{66} +5.54479e7 q^{67} +3.37480e7 q^{68} -1.15664e8 q^{69} +3.87757e8 q^{70} -1.43613e8 q^{71} -1.77912e8 q^{72} +3.37147e7 q^{73} -2.96937e8 q^{74} +6.70469e7 q^{75} -3.25413e7 q^{76} +7.22007e8 q^{77} +3.50367e7 q^{78} -2.66652e8 q^{79} -5.44728e8 q^{80} +2.46063e8 q^{81} -6.37839e7 q^{82} -9.89542e7 q^{83} -3.91882e7 q^{84} +6.80447e8 q^{85} -7.56490e7 q^{86} -3.24809e7 q^{87} -8.58533e8 q^{88} -5.08803e8 q^{89} +7.65211e8 q^{90} +2.48689e8 q^{91} -2.08246e8 q^{92} +2.16249e8 q^{93} -6.38667e8 q^{94} -6.56116e8 q^{95} +1.03137e8 q^{96} -4.53920e8 q^{97} -8.70137e8 q^{98} +1.42483e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 33 q^{2} - 163 q^{3} + 1429 q^{4} + 471 q^{5} - 4529 q^{6} - 11241 q^{7} - 45543 q^{8} - 29953 q^{9} - 67831 q^{10} - 40140 q^{11} - 155479 q^{12} - 114244 q^{13} - 277653 q^{14} + 83307 q^{15} + 726609 q^{16}+ \cdots + 2132181050 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.5360 −1.08435 −0.542175 0.840266i \(-0.682399\pi\)
−0.542175 + 0.840266i \(0.682399\pi\)
\(3\) 49.9972 0.356369 0.178185 0.983997i \(-0.442978\pi\)
0.178185 + 0.983997i \(0.442978\pi\)
\(4\) 90.0171 0.175815
\(5\) 1814.98 1.29869 0.649346 0.760493i \(-0.275043\pi\)
0.649346 + 0.760493i \(0.275043\pi\)
\(6\) −1226.73 −0.386429
\(7\) −8707.31 −1.37070 −0.685351 0.728213i \(-0.740351\pi\)
−0.685351 + 0.728213i \(0.740351\pi\)
\(8\) 10353.8 0.893705
\(9\) −17183.3 −0.873001
\(10\) −44532.3 −1.40824
\(11\) −82919.7 −1.70762 −0.853809 0.520587i \(-0.825713\pi\)
−0.853809 + 0.520587i \(0.825713\pi\)
\(12\) 4500.61 0.0626550
\(13\) −28561.0 −0.277350
\(14\) 213643. 1.48632
\(15\) 90743.8 0.462814
\(16\) −300130. −1.14490
\(17\) 374907. 1.08869 0.544344 0.838862i \(-0.316779\pi\)
0.544344 + 0.838862i \(0.316779\pi\)
\(18\) 421610. 0.946638
\(19\) −361501. −0.636383 −0.318191 0.948026i \(-0.603075\pi\)
−0.318191 + 0.948026i \(0.603075\pi\)
\(20\) 163379. 0.228329
\(21\) −435341. −0.488476
\(22\) 2.03452e6 1.85165
\(23\) −2.31340e6 −1.72376 −0.861878 0.507116i \(-0.830712\pi\)
−0.861878 + 0.507116i \(0.830712\pi\)
\(24\) 517661. 0.318489
\(25\) 1.34101e6 0.686599
\(26\) 700774. 0.300745
\(27\) −1.84321e6 −0.667480
\(28\) −783807. −0.240989
\(29\) −649654. −0.170565 −0.0852827 0.996357i \(-0.527179\pi\)
−0.0852827 + 0.996357i \(0.527179\pi\)
\(30\) −2.22649e6 −0.501852
\(31\) 4.32521e6 0.841162 0.420581 0.907255i \(-0.361826\pi\)
0.420581 + 0.907255i \(0.361826\pi\)
\(32\) 2.06285e6 0.347771
\(33\) −4.14575e6 −0.608542
\(34\) −9.19873e6 −1.18052
\(35\) −1.58036e7 −1.78012
\(36\) −1.54679e6 −0.153486
\(37\) 1.21021e7 1.06158 0.530790 0.847503i \(-0.321895\pi\)
0.530790 + 0.847503i \(0.321895\pi\)
\(38\) 8.86981e6 0.690062
\(39\) −1.42797e6 −0.0988391
\(40\) 1.87919e7 1.16065
\(41\) 2.59960e6 0.143674 0.0718372 0.997416i \(-0.477114\pi\)
0.0718372 + 0.997416i \(0.477114\pi\)
\(42\) 1.06816e7 0.529679
\(43\) 3.08318e6 0.137528 0.0687640 0.997633i \(-0.478094\pi\)
0.0687640 + 0.997633i \(0.478094\pi\)
\(44\) −7.46419e6 −0.300224
\(45\) −3.11872e7 −1.13376
\(46\) 5.67617e7 1.86915
\(47\) 2.60298e7 0.778090 0.389045 0.921219i \(-0.372805\pi\)
0.389045 + 0.921219i \(0.372805\pi\)
\(48\) −1.50057e7 −0.408009
\(49\) 3.54636e7 0.878822
\(50\) −3.29032e7 −0.744513
\(51\) 1.87443e7 0.387975
\(52\) −2.57098e6 −0.0487622
\(53\) −1.01982e8 −1.77535 −0.887673 0.460475i \(-0.847679\pi\)
−0.887673 + 0.460475i \(0.847679\pi\)
\(54\) 4.52251e7 0.723782
\(55\) −1.50497e8 −2.21767
\(56\) −9.01536e7 −1.22500
\(57\) −1.80741e7 −0.226787
\(58\) 1.59399e7 0.184953
\(59\) 1.37562e8 1.47797 0.738985 0.673722i \(-0.235305\pi\)
0.738985 + 0.673722i \(0.235305\pi\)
\(60\) 8.16850e6 0.0813695
\(61\) −4.29579e7 −0.397245 −0.198623 0.980076i \(-0.563647\pi\)
−0.198623 + 0.980076i \(0.563647\pi\)
\(62\) −1.06124e8 −0.912114
\(63\) 1.49620e8 1.19662
\(64\) 1.03052e8 0.767798
\(65\) −5.18375e7 −0.360192
\(66\) 1.01720e8 0.659873
\(67\) 5.54479e7 0.336162 0.168081 0.985773i \(-0.446243\pi\)
0.168081 + 0.985773i \(0.446243\pi\)
\(68\) 3.37480e7 0.191407
\(69\) −1.15664e8 −0.614294
\(70\) 3.87757e8 1.93027
\(71\) −1.43613e8 −0.670706 −0.335353 0.942093i \(-0.608856\pi\)
−0.335353 + 0.942093i \(0.608856\pi\)
\(72\) −1.77912e8 −0.780205
\(73\) 3.37147e7 0.138952 0.0694762 0.997584i \(-0.477867\pi\)
0.0694762 + 0.997584i \(0.477867\pi\)
\(74\) −2.96937e8 −1.15112
\(75\) 6.70469e7 0.244683
\(76\) −3.25413e7 −0.111886
\(77\) 7.22007e8 2.34063
\(78\) 3.50367e7 0.107176
\(79\) −2.66652e8 −0.770236 −0.385118 0.922867i \(-0.625839\pi\)
−0.385118 + 0.922867i \(0.625839\pi\)
\(80\) −5.44728e8 −1.48688
\(81\) 2.46063e8 0.635132
\(82\) −6.37839e7 −0.155793
\(83\) −9.89542e7 −0.228867 −0.114433 0.993431i \(-0.536505\pi\)
−0.114433 + 0.993431i \(0.536505\pi\)
\(84\) −3.91882e7 −0.0858812
\(85\) 6.80447e8 1.41387
\(86\) −7.56490e7 −0.149128
\(87\) −3.24809e7 −0.0607843
\(88\) −8.58533e8 −1.52611
\(89\) −5.08803e8 −0.859597 −0.429798 0.902925i \(-0.641415\pi\)
−0.429798 + 0.902925i \(0.641415\pi\)
\(90\) 7.65211e8 1.22939
\(91\) 2.48689e8 0.380164
\(92\) −2.08246e8 −0.303062
\(93\) 2.16249e8 0.299764
\(94\) −6.38667e8 −0.843722
\(95\) −6.56116e8 −0.826465
\(96\) 1.03137e8 0.123935
\(97\) −4.53920e8 −0.520603 −0.260301 0.965527i \(-0.583822\pi\)
−0.260301 + 0.965527i \(0.583822\pi\)
\(98\) −8.70137e8 −0.952950
\(99\) 1.42483e9 1.49075
\(100\) 1.20714e8 0.120714
\(101\) 6.44663e8 0.616434 0.308217 0.951316i \(-0.400268\pi\)
0.308217 + 0.951316i \(0.400268\pi\)
\(102\) −4.59911e8 −0.420700
\(103\) −1.14825e9 −1.00524 −0.502620 0.864507i \(-0.667631\pi\)
−0.502620 + 0.864507i \(0.667631\pi\)
\(104\) −2.95715e8 −0.247869
\(105\) −7.90134e8 −0.634379
\(106\) 2.50224e9 1.92510
\(107\) 5.95299e8 0.439044 0.219522 0.975607i \(-0.429550\pi\)
0.219522 + 0.975607i \(0.429550\pi\)
\(108\) −1.65921e8 −0.117353
\(109\) 1.00898e9 0.684645 0.342322 0.939583i \(-0.388787\pi\)
0.342322 + 0.939583i \(0.388787\pi\)
\(110\) 3.69261e9 2.40473
\(111\) 6.05071e8 0.378315
\(112\) 2.61332e9 1.56932
\(113\) −1.59931e9 −0.922739 −0.461370 0.887208i \(-0.652642\pi\)
−0.461370 + 0.887208i \(0.652642\pi\)
\(114\) 4.43466e8 0.245917
\(115\) −4.19877e9 −2.23863
\(116\) −5.84800e7 −0.0299879
\(117\) 4.90772e8 0.242127
\(118\) −3.37524e9 −1.60264
\(119\) −3.26443e9 −1.49226
\(120\) 9.39542e8 0.413619
\(121\) 4.51772e9 1.91596
\(122\) 1.05402e9 0.430753
\(123\) 1.29973e8 0.0512011
\(124\) 3.89343e8 0.147889
\(125\) −1.11097e9 −0.407011
\(126\) −3.67108e9 −1.29756
\(127\) −1.58816e9 −0.541723 −0.270861 0.962618i \(-0.587308\pi\)
−0.270861 + 0.962618i \(0.587308\pi\)
\(128\) −3.58467e9 −1.18033
\(129\) 1.54151e8 0.0490107
\(130\) 1.27189e9 0.390574
\(131\) −3.54703e9 −1.05231 −0.526156 0.850388i \(-0.676367\pi\)
−0.526156 + 0.850388i \(0.676367\pi\)
\(132\) −3.73189e8 −0.106991
\(133\) 3.14770e9 0.872291
\(134\) −1.36047e9 −0.364517
\(135\) −3.34539e9 −0.866850
\(136\) 3.88170e9 0.972965
\(137\) −3.79917e9 −0.921395 −0.460697 0.887557i \(-0.652401\pi\)
−0.460697 + 0.887557i \(0.652401\pi\)
\(138\) 2.83793e9 0.666109
\(139\) 3.81526e9 0.866878 0.433439 0.901183i \(-0.357300\pi\)
0.433439 + 0.901183i \(0.357300\pi\)
\(140\) −1.42259e9 −0.312971
\(141\) 1.30142e9 0.277287
\(142\) 3.52370e9 0.727280
\(143\) 2.36827e9 0.473608
\(144\) 5.15721e9 0.999502
\(145\) −1.17911e9 −0.221512
\(146\) −8.27225e8 −0.150673
\(147\) 1.77308e9 0.313185
\(148\) 1.08940e9 0.186641
\(149\) 1.16149e10 1.93054 0.965269 0.261258i \(-0.0841374\pi\)
0.965269 + 0.261258i \(0.0841374\pi\)
\(150\) −1.64507e9 −0.265322
\(151\) 3.19196e9 0.499645 0.249822 0.968292i \(-0.419628\pi\)
0.249822 + 0.968292i \(0.419628\pi\)
\(152\) −3.74291e9 −0.568739
\(153\) −6.44213e9 −0.950425
\(154\) −1.77152e10 −2.53806
\(155\) 7.85016e9 1.09241
\(156\) −1.28542e8 −0.0173774
\(157\) −5.05861e9 −0.664481 −0.332240 0.943195i \(-0.607805\pi\)
−0.332240 + 0.943195i \(0.607805\pi\)
\(158\) 6.54260e9 0.835205
\(159\) −5.09883e9 −0.632679
\(160\) 3.74403e9 0.451647
\(161\) 2.01435e10 2.36275
\(162\) −6.03741e9 −0.688705
\(163\) −7.83438e9 −0.869282 −0.434641 0.900604i \(-0.643125\pi\)
−0.434641 + 0.900604i \(0.643125\pi\)
\(164\) 2.34009e8 0.0252601
\(165\) −7.52444e9 −0.790308
\(166\) 2.42794e9 0.248172
\(167\) −1.44048e8 −0.0143312 −0.00716559 0.999974i \(-0.502281\pi\)
−0.00716559 + 0.999974i \(0.502281\pi\)
\(168\) −4.50743e9 −0.436553
\(169\) 8.15731e8 0.0769231
\(170\) −1.66955e10 −1.53313
\(171\) 6.21178e9 0.555563
\(172\) 2.77539e8 0.0241794
\(173\) −1.36123e10 −1.15538 −0.577689 0.816257i \(-0.696045\pi\)
−0.577689 + 0.816257i \(0.696045\pi\)
\(174\) 7.96952e8 0.0659114
\(175\) −1.16766e10 −0.941122
\(176\) 2.48867e10 1.95506
\(177\) 6.87774e9 0.526703
\(178\) 1.24840e10 0.932104
\(179\) 4.98195e8 0.0362711 0.0181355 0.999836i \(-0.494227\pi\)
0.0181355 + 0.999836i \(0.494227\pi\)
\(180\) −2.80739e9 −0.199331
\(181\) 2.86758e9 0.198592 0.0992961 0.995058i \(-0.468341\pi\)
0.0992961 + 0.995058i \(0.468341\pi\)
\(182\) −6.10185e9 −0.412231
\(183\) −2.14777e9 −0.141566
\(184\) −2.39525e10 −1.54053
\(185\) 2.19650e10 1.37866
\(186\) −5.30588e9 −0.325049
\(187\) −3.10871e10 −1.85906
\(188\) 2.34312e9 0.136800
\(189\) 1.60494e10 0.914916
\(190\) 1.60985e10 0.896177
\(191\) −2.83331e10 −1.54044 −0.770218 0.637781i \(-0.779852\pi\)
−0.770218 + 0.637781i \(0.779852\pi\)
\(192\) 5.15232e9 0.273620
\(193\) 2.44273e10 1.26726 0.633632 0.773634i \(-0.281563\pi\)
0.633632 + 0.773634i \(0.281563\pi\)
\(194\) 1.11374e10 0.564516
\(195\) −2.59173e9 −0.128361
\(196\) 3.19233e9 0.154510
\(197\) 1.37323e10 0.649601 0.324800 0.945783i \(-0.394703\pi\)
0.324800 + 0.945783i \(0.394703\pi\)
\(198\) −3.49597e10 −1.61650
\(199\) −8.56614e9 −0.387210 −0.193605 0.981080i \(-0.562018\pi\)
−0.193605 + 0.981080i \(0.562018\pi\)
\(200\) 1.38846e10 0.613617
\(201\) 2.77224e9 0.119798
\(202\) −1.58175e10 −0.668430
\(203\) 5.65673e9 0.233794
\(204\) 1.68731e9 0.0682117
\(205\) 4.71821e9 0.186589
\(206\) 2.81736e10 1.09003
\(207\) 3.97518e10 1.50484
\(208\) 8.57200e9 0.317539
\(209\) 2.99756e10 1.08670
\(210\) 1.93868e10 0.687889
\(211\) 2.70272e9 0.0938706 0.0469353 0.998898i \(-0.485055\pi\)
0.0469353 + 0.998898i \(0.485055\pi\)
\(212\) −9.18015e9 −0.312132
\(213\) −7.18027e9 −0.239019
\(214\) −1.46063e10 −0.476078
\(215\) 5.59590e9 0.178606
\(216\) −1.90842e10 −0.596530
\(217\) −3.76610e10 −1.15298
\(218\) −2.47565e10 −0.742394
\(219\) 1.68564e9 0.0495184
\(220\) −1.35473e10 −0.389899
\(221\) −1.07077e10 −0.301947
\(222\) −1.48460e10 −0.410225
\(223\) −3.57969e10 −0.969333 −0.484667 0.874699i \(-0.661059\pi\)
−0.484667 + 0.874699i \(0.661059\pi\)
\(224\) −1.79619e10 −0.476690
\(225\) −2.30430e10 −0.599401
\(226\) 3.92407e10 1.00057
\(227\) −7.55373e10 −1.88819 −0.944094 0.329675i \(-0.893061\pi\)
−0.944094 + 0.329675i \(0.893061\pi\)
\(228\) −1.62698e9 −0.0398726
\(229\) −5.69066e10 −1.36742 −0.683711 0.729753i \(-0.739635\pi\)
−0.683711 + 0.729753i \(0.739635\pi\)
\(230\) 1.03021e11 2.42745
\(231\) 3.60984e10 0.834130
\(232\) −6.72637e9 −0.152435
\(233\) −4.68000e10 −1.04026 −0.520132 0.854086i \(-0.674117\pi\)
−0.520132 + 0.854086i \(0.674117\pi\)
\(234\) −1.20416e10 −0.262550
\(235\) 4.72434e10 1.01050
\(236\) 1.23830e10 0.259849
\(237\) −1.33319e10 −0.274488
\(238\) 8.00961e10 1.61814
\(239\) −8.79993e9 −0.174457 −0.0872285 0.996188i \(-0.527801\pi\)
−0.0872285 + 0.996188i \(0.527801\pi\)
\(240\) −2.72349e10 −0.529877
\(241\) −1.76057e10 −0.336183 −0.168092 0.985771i \(-0.553760\pi\)
−0.168092 + 0.985771i \(0.553760\pi\)
\(242\) −1.10847e11 −2.07757
\(243\) 4.85824e10 0.893821
\(244\) −3.86695e9 −0.0698415
\(245\) 6.43656e10 1.14132
\(246\) −3.18902e9 −0.0555200
\(247\) 1.03248e10 0.176501
\(248\) 4.47823e10 0.751751
\(249\) −4.94743e9 −0.0815611
\(250\) 2.72588e10 0.441343
\(251\) −1.77113e10 −0.281655 −0.140828 0.990034i \(-0.544976\pi\)
−0.140828 + 0.990034i \(0.544976\pi\)
\(252\) 1.34684e10 0.210384
\(253\) 1.91826e11 2.94351
\(254\) 3.89671e10 0.587417
\(255\) 3.40204e10 0.503859
\(256\) 3.51910e10 0.512096
\(257\) −6.26733e10 −0.896156 −0.448078 0.893994i \(-0.647891\pi\)
−0.448078 + 0.893994i \(0.647891\pi\)
\(258\) −3.78224e9 −0.0531448
\(259\) −1.05377e11 −1.45511
\(260\) −4.66627e9 −0.0633271
\(261\) 1.11632e10 0.148904
\(262\) 8.70301e10 1.14107
\(263\) 4.49501e10 0.579335 0.289667 0.957127i \(-0.406455\pi\)
0.289667 + 0.957127i \(0.406455\pi\)
\(264\) −4.29243e10 −0.543857
\(265\) −1.85095e11 −2.30563
\(266\) −7.72322e10 −0.945869
\(267\) −2.54388e10 −0.306334
\(268\) 4.99126e9 0.0591023
\(269\) −8.65536e10 −1.00786 −0.503930 0.863745i \(-0.668113\pi\)
−0.503930 + 0.863745i \(0.668113\pi\)
\(270\) 8.20825e10 0.939969
\(271\) 8.36056e10 0.941615 0.470808 0.882236i \(-0.343963\pi\)
0.470808 + 0.882236i \(0.343963\pi\)
\(272\) −1.12521e11 −1.24644
\(273\) 1.24338e10 0.135479
\(274\) 9.32165e10 0.999114
\(275\) −1.11196e11 −1.17245
\(276\) −1.04117e10 −0.108002
\(277\) 1.73137e11 1.76698 0.883489 0.468452i \(-0.155188\pi\)
0.883489 + 0.468452i \(0.155188\pi\)
\(278\) −9.36114e10 −0.939999
\(279\) −7.43213e10 −0.734335
\(280\) −1.63627e11 −1.59090
\(281\) 7.57598e10 0.724871 0.362435 0.932009i \(-0.381945\pi\)
0.362435 + 0.932009i \(0.381945\pi\)
\(282\) −3.19316e10 −0.300676
\(283\) 1.68901e11 1.56528 0.782642 0.622472i \(-0.213872\pi\)
0.782642 + 0.622472i \(0.213872\pi\)
\(284\) −1.29277e10 −0.117920
\(285\) −3.28040e10 −0.294527
\(286\) −5.81079e10 −0.513557
\(287\) −2.26355e10 −0.196935
\(288\) −3.54466e10 −0.303605
\(289\) 2.19672e10 0.185240
\(290\) 2.89306e10 0.240196
\(291\) −2.26947e10 −0.185527
\(292\) 3.03490e9 0.0244299
\(293\) −1.18842e11 −0.942030 −0.471015 0.882125i \(-0.656112\pi\)
−0.471015 + 0.882125i \(0.656112\pi\)
\(294\) −4.35044e10 −0.339602
\(295\) 2.49672e11 1.91943
\(296\) 1.25302e11 0.948740
\(297\) 1.52839e11 1.13980
\(298\) −2.84984e11 −2.09338
\(299\) 6.60730e10 0.478084
\(300\) 6.03537e9 0.0430188
\(301\) −2.68462e10 −0.188510
\(302\) −7.83181e10 −0.541790
\(303\) 3.22313e10 0.219678
\(304\) 1.08497e11 0.728597
\(305\) −7.79675e10 −0.515899
\(306\) 1.58064e11 1.03059
\(307\) −1.99530e11 −1.28199 −0.640997 0.767543i \(-0.721479\pi\)
−0.640997 + 0.767543i \(0.721479\pi\)
\(308\) 6.49930e10 0.411518
\(309\) −5.74095e10 −0.358237
\(310\) −1.92612e11 −1.18455
\(311\) 1.90145e11 1.15256 0.576281 0.817252i \(-0.304503\pi\)
0.576281 + 0.817252i \(0.304503\pi\)
\(312\) −1.47849e10 −0.0883330
\(313\) −3.39674e10 −0.200038 −0.100019 0.994986i \(-0.531890\pi\)
−0.100019 + 0.994986i \(0.531890\pi\)
\(314\) 1.24118e11 0.720530
\(315\) 2.71557e11 1.55404
\(316\) −2.40033e10 −0.135419
\(317\) −8.53565e9 −0.0474756 −0.0237378 0.999718i \(-0.507557\pi\)
−0.0237378 + 0.999718i \(0.507557\pi\)
\(318\) 1.25105e11 0.686045
\(319\) 5.38691e10 0.291260
\(320\) 1.87037e11 0.997133
\(321\) 2.97633e10 0.156462
\(322\) −4.94242e11 −2.56205
\(323\) −1.35529e11 −0.692822
\(324\) 2.21499e10 0.111665
\(325\) −3.83007e10 −0.190428
\(326\) 1.92225e11 0.942606
\(327\) 5.04464e10 0.243986
\(328\) 2.69157e10 0.128403
\(329\) −2.26649e11 −1.06653
\(330\) 1.84620e11 0.856971
\(331\) 5.09528e10 0.233315 0.116657 0.993172i \(-0.462782\pi\)
0.116657 + 0.993172i \(0.462782\pi\)
\(332\) −8.90757e9 −0.0402381
\(333\) −2.07954e11 −0.926760
\(334\) 3.53436e9 0.0155400
\(335\) 1.00637e11 0.436571
\(336\) 1.30659e11 0.559258
\(337\) 2.31891e11 0.979375 0.489688 0.871898i \(-0.337111\pi\)
0.489688 + 0.871898i \(0.337111\pi\)
\(338\) −2.00148e10 −0.0834115
\(339\) −7.99609e10 −0.328836
\(340\) 6.12519e10 0.248579
\(341\) −3.58645e11 −1.43638
\(342\) −1.52412e11 −0.602425
\(343\) 4.25786e10 0.166099
\(344\) 3.19226e10 0.122909
\(345\) −2.09927e11 −0.797778
\(346\) 3.33992e11 1.25283
\(347\) −3.64138e10 −0.134829 −0.0674146 0.997725i \(-0.521475\pi\)
−0.0674146 + 0.997725i \(0.521475\pi\)
\(348\) −2.92384e9 −0.0106868
\(349\) 9.86839e10 0.356067 0.178034 0.984024i \(-0.443026\pi\)
0.178034 + 0.984024i \(0.443026\pi\)
\(350\) 2.86498e11 1.02051
\(351\) 5.26440e10 0.185126
\(352\) −1.71051e11 −0.593860
\(353\) −3.90458e11 −1.33841 −0.669204 0.743078i \(-0.733365\pi\)
−0.669204 + 0.743078i \(0.733365\pi\)
\(354\) −1.68752e11 −0.571131
\(355\) −2.60655e11 −0.871040
\(356\) −4.58010e10 −0.151130
\(357\) −1.63212e11 −0.531797
\(358\) −1.22237e10 −0.0393305
\(359\) −3.86316e11 −1.22749 −0.613744 0.789505i \(-0.710337\pi\)
−0.613744 + 0.789505i \(0.710337\pi\)
\(360\) −3.22906e11 −1.01325
\(361\) −1.92005e11 −0.595017
\(362\) −7.03591e10 −0.215343
\(363\) 2.25874e11 0.682788
\(364\) 2.23863e10 0.0668385
\(365\) 6.11914e10 0.180456
\(366\) 5.26979e10 0.153507
\(367\) 1.48155e11 0.426305 0.213152 0.977019i \(-0.431627\pi\)
0.213152 + 0.977019i \(0.431627\pi\)
\(368\) 6.94320e11 1.97353
\(369\) −4.46697e10 −0.125428
\(370\) −5.38934e11 −1.49495
\(371\) 8.87991e11 2.43347
\(372\) 1.94661e10 0.0527030
\(373\) −6.15338e11 −1.64598 −0.822989 0.568057i \(-0.807695\pi\)
−0.822989 + 0.568057i \(0.807695\pi\)
\(374\) 7.62755e11 2.01587
\(375\) −5.55453e10 −0.145046
\(376\) 2.69507e11 0.695383
\(377\) 1.85548e10 0.0473063
\(378\) −3.93789e11 −0.992089
\(379\) 2.48822e10 0.0619459 0.0309729 0.999520i \(-0.490139\pi\)
0.0309729 + 0.999520i \(0.490139\pi\)
\(380\) −5.90617e10 −0.145305
\(381\) −7.94035e10 −0.193053
\(382\) 6.95181e11 1.67037
\(383\) 4.82478e11 1.14573 0.572866 0.819649i \(-0.305832\pi\)
0.572866 + 0.819649i \(0.305832\pi\)
\(384\) −1.79224e11 −0.420634
\(385\) 1.31043e12 3.03976
\(386\) −5.99349e11 −1.37416
\(387\) −5.29792e10 −0.120062
\(388\) −4.08606e10 −0.0915297
\(389\) 3.80215e11 0.841890 0.420945 0.907086i \(-0.361698\pi\)
0.420945 + 0.907086i \(0.361698\pi\)
\(390\) 6.35909e10 0.139189
\(391\) −8.67310e11 −1.87663
\(392\) 3.67183e11 0.785408
\(393\) −1.77342e11 −0.375012
\(394\) −3.36937e11 −0.704394
\(395\) −4.83968e11 −1.00030
\(396\) 1.28259e11 0.262096
\(397\) 7.91759e11 1.59969 0.799845 0.600207i \(-0.204915\pi\)
0.799845 + 0.600207i \(0.204915\pi\)
\(398\) 2.10179e11 0.419871
\(399\) 1.57376e11 0.310858
\(400\) −4.02478e11 −0.786090
\(401\) −3.06132e11 −0.591234 −0.295617 0.955307i \(-0.595525\pi\)
−0.295617 + 0.955307i \(0.595525\pi\)
\(402\) −6.80199e10 −0.129903
\(403\) −1.23532e11 −0.233296
\(404\) 5.80307e10 0.108378
\(405\) 4.46598e11 0.824840
\(406\) −1.38794e11 −0.253515
\(407\) −1.00350e12 −1.81277
\(408\) 1.94074e11 0.346735
\(409\) 9.04916e11 1.59902 0.799509 0.600654i \(-0.205093\pi\)
0.799509 + 0.600654i \(0.205093\pi\)
\(410\) −1.15766e11 −0.202327
\(411\) −1.89948e11 −0.328357
\(412\) −1.03362e11 −0.176736
\(413\) −1.19780e12 −2.02586
\(414\) −9.75352e11 −1.63177
\(415\) −1.79599e11 −0.297227
\(416\) −5.89172e10 −0.0964544
\(417\) 1.90753e11 0.308929
\(418\) −7.35482e11 −1.17836
\(419\) 3.36210e11 0.532903 0.266451 0.963848i \(-0.414149\pi\)
0.266451 + 0.963848i \(0.414149\pi\)
\(420\) −7.11256e10 −0.111533
\(421\) −1.87100e11 −0.290271 −0.145135 0.989412i \(-0.546362\pi\)
−0.145135 + 0.989412i \(0.546362\pi\)
\(422\) −6.63140e10 −0.101789
\(423\) −4.47276e11 −0.679273
\(424\) −1.05590e12 −1.58664
\(425\) 5.02755e11 0.747491
\(426\) 1.76175e11 0.259180
\(427\) 3.74048e11 0.544504
\(428\) 5.35872e10 0.0771905
\(429\) 1.18407e11 0.168779
\(430\) −1.37301e11 −0.193672
\(431\) 9.60500e11 1.34076 0.670378 0.742020i \(-0.266132\pi\)
0.670378 + 0.742020i \(0.266132\pi\)
\(432\) 5.53203e11 0.764200
\(433\) 1.14812e12 1.56961 0.784806 0.619741i \(-0.212762\pi\)
0.784806 + 0.619741i \(0.212762\pi\)
\(434\) 9.24051e11 1.25024
\(435\) −5.89520e10 −0.0789400
\(436\) 9.08259e10 0.120371
\(437\) 8.36297e11 1.09697
\(438\) −4.13590e10 −0.0536953
\(439\) 5.44277e11 0.699407 0.349703 0.936860i \(-0.386282\pi\)
0.349703 + 0.936860i \(0.386282\pi\)
\(440\) −1.55822e12 −1.98194
\(441\) −6.09381e11 −0.767212
\(442\) 2.62725e11 0.327417
\(443\) −1.12065e11 −0.138247 −0.0691233 0.997608i \(-0.522020\pi\)
−0.0691233 + 0.997608i \(0.522020\pi\)
\(444\) 5.44668e10 0.0665133
\(445\) −9.23466e11 −1.11635
\(446\) 8.78313e11 1.05110
\(447\) 5.80714e11 0.687984
\(448\) −8.97307e11 −1.05242
\(449\) −1.11883e12 −1.29913 −0.649567 0.760304i \(-0.725050\pi\)
−0.649567 + 0.760304i \(0.725050\pi\)
\(450\) 5.65384e11 0.649961
\(451\) −2.15558e11 −0.245341
\(452\) −1.43965e11 −0.162231
\(453\) 1.59589e11 0.178058
\(454\) 1.85339e12 2.04746
\(455\) 4.51365e11 0.493716
\(456\) −1.87135e11 −0.202681
\(457\) −1.38914e12 −1.48978 −0.744890 0.667188i \(-0.767498\pi\)
−0.744890 + 0.667188i \(0.767498\pi\)
\(458\) 1.39626e12 1.48276
\(459\) −6.91032e11 −0.726677
\(460\) −3.77961e11 −0.393583
\(461\) 1.06041e12 1.09350 0.546752 0.837294i \(-0.315864\pi\)
0.546752 + 0.837294i \(0.315864\pi\)
\(462\) −8.85711e11 −0.904488
\(463\) 2.81576e11 0.284762 0.142381 0.989812i \(-0.454524\pi\)
0.142381 + 0.989812i \(0.454524\pi\)
\(464\) 1.94980e11 0.195281
\(465\) 3.92486e11 0.389301
\(466\) 1.14829e12 1.12801
\(467\) −9.91125e11 −0.964278 −0.482139 0.876095i \(-0.660140\pi\)
−0.482139 + 0.876095i \(0.660140\pi\)
\(468\) 4.41779e10 0.0425695
\(469\) −4.82802e11 −0.460778
\(470\) −1.15917e12 −1.09573
\(471\) −2.52916e11 −0.236801
\(472\) 1.42429e12 1.32087
\(473\) −2.55656e11 −0.234845
\(474\) 3.27112e11 0.297641
\(475\) −4.84778e11 −0.436940
\(476\) −2.93855e11 −0.262362
\(477\) 1.75239e12 1.54988
\(478\) 2.15915e11 0.189173
\(479\) −1.38074e12 −1.19841 −0.599203 0.800597i \(-0.704516\pi\)
−0.599203 + 0.800597i \(0.704516\pi\)
\(480\) 1.87191e11 0.160953
\(481\) −3.45648e11 −0.294429
\(482\) 4.31974e11 0.364541
\(483\) 1.00712e12 0.842013
\(484\) 4.06673e11 0.336853
\(485\) −8.23854e11 −0.676102
\(486\) −1.19202e12 −0.969215
\(487\) 9.65212e11 0.777575 0.388788 0.921327i \(-0.372894\pi\)
0.388788 + 0.921327i \(0.372894\pi\)
\(488\) −4.44777e11 −0.355020
\(489\) −3.91697e11 −0.309785
\(490\) −1.57928e12 −1.23759
\(491\) −1.10735e12 −0.859841 −0.429920 0.902867i \(-0.641458\pi\)
−0.429920 + 0.902867i \(0.641458\pi\)
\(492\) 1.16998e10 0.00900191
\(493\) −2.43559e11 −0.185692
\(494\) −2.53331e11 −0.191389
\(495\) 2.58604e12 1.93603
\(496\) −1.29812e12 −0.963050
\(497\) 1.25049e12 0.919338
\(498\) 1.21390e11 0.0884407
\(499\) −2.24583e12 −1.62153 −0.810763 0.585374i \(-0.800948\pi\)
−0.810763 + 0.585374i \(0.800948\pi\)
\(500\) −1.00006e11 −0.0715586
\(501\) −7.20198e9 −0.00510719
\(502\) 4.34564e11 0.305413
\(503\) 3.98259e11 0.277402 0.138701 0.990334i \(-0.455707\pi\)
0.138701 + 0.990334i \(0.455707\pi\)
\(504\) 1.54913e12 1.06943
\(505\) 1.17005e12 0.800557
\(506\) −4.70666e12 −3.19180
\(507\) 4.07843e10 0.0274130
\(508\) −1.42961e11 −0.0952428
\(509\) 2.66411e12 1.75923 0.879615 0.475687i \(-0.157800\pi\)
0.879615 + 0.475687i \(0.157800\pi\)
\(510\) −8.34727e11 −0.546360
\(511\) −2.93564e11 −0.190462
\(512\) 9.71905e11 0.625042
\(513\) 6.66323e11 0.424773
\(514\) 1.53775e12 0.971747
\(515\) −2.08405e12 −1.30550
\(516\) 1.38762e10 0.00861681
\(517\) −2.15838e12 −1.32868
\(518\) 2.58553e12 1.57785
\(519\) −6.80577e11 −0.411741
\(520\) −5.36715e11 −0.321906
\(521\) −3.11961e12 −1.85494 −0.927472 0.373894i \(-0.878022\pi\)
−0.927472 + 0.373894i \(0.878022\pi\)
\(522\) −2.73900e11 −0.161464
\(523\) −2.05566e12 −1.20142 −0.600709 0.799468i \(-0.705115\pi\)
−0.600709 + 0.799468i \(0.705115\pi\)
\(524\) −3.19294e11 −0.185012
\(525\) −5.83798e11 −0.335387
\(526\) −1.10290e12 −0.628202
\(527\) 1.62155e12 0.915762
\(528\) 1.24426e12 0.696722
\(529\) 3.55067e12 1.97133
\(530\) 4.54150e12 2.50011
\(531\) −2.36377e12 −1.29027
\(532\) 2.83347e11 0.153362
\(533\) −7.42472e10 −0.0398481
\(534\) 6.24166e11 0.332173
\(535\) 1.08045e12 0.570183
\(536\) 5.74096e11 0.300430
\(537\) 2.49083e10 0.0129259
\(538\) 2.12368e12 1.09287
\(539\) −2.94063e12 −1.50069
\(540\) −3.01142e11 −0.152405
\(541\) 8.06876e11 0.404966 0.202483 0.979286i \(-0.435099\pi\)
0.202483 + 0.979286i \(0.435099\pi\)
\(542\) −2.05135e12 −1.02104
\(543\) 1.43371e11 0.0707721
\(544\) 7.73378e11 0.378614
\(545\) 1.83128e12 0.889142
\(546\) −3.05076e11 −0.146906
\(547\) 1.93370e12 0.923521 0.461760 0.887005i \(-0.347218\pi\)
0.461760 + 0.887005i \(0.347218\pi\)
\(548\) −3.41990e11 −0.161995
\(549\) 7.38157e11 0.346795
\(550\) 2.72832e12 1.27134
\(551\) 2.34851e11 0.108545
\(552\) −1.19756e12 −0.548997
\(553\) 2.32183e12 1.05576
\(554\) −4.24810e12 −1.91602
\(555\) 1.09819e12 0.491314
\(556\) 3.43439e11 0.152410
\(557\) 3.06623e12 1.34976 0.674879 0.737928i \(-0.264196\pi\)
0.674879 + 0.737928i \(0.264196\pi\)
\(558\) 1.82355e12 0.796276
\(559\) −8.80587e10 −0.0381434
\(560\) 4.74312e12 2.03806
\(561\) −1.55427e12 −0.662512
\(562\) −1.85885e12 −0.786013
\(563\) −2.59944e12 −1.09042 −0.545208 0.838301i \(-0.683549\pi\)
−0.545208 + 0.838301i \(0.683549\pi\)
\(564\) 1.17150e11 0.0487512
\(565\) −2.90270e12 −1.19835
\(566\) −4.14416e12 −1.69732
\(567\) −2.14255e12 −0.870576
\(568\) −1.48694e12 −0.599413
\(569\) −1.63182e12 −0.652632 −0.326316 0.945261i \(-0.605807\pi\)
−0.326316 + 0.945261i \(0.605807\pi\)
\(570\) 8.04880e11 0.319370
\(571\) −1.05673e12 −0.416010 −0.208005 0.978128i \(-0.566697\pi\)
−0.208005 + 0.978128i \(0.566697\pi\)
\(572\) 2.13185e11 0.0832672
\(573\) −1.41658e12 −0.548964
\(574\) 5.55386e11 0.213546
\(575\) −3.10230e12 −1.18353
\(576\) −1.77077e12 −0.670289
\(577\) −8.51110e11 −0.319665 −0.159832 0.987144i \(-0.551095\pi\)
−0.159832 + 0.987144i \(0.551095\pi\)
\(578\) −5.38987e11 −0.200865
\(579\) 1.22130e12 0.451614
\(580\) −1.06140e11 −0.0389450
\(581\) 8.61624e11 0.313708
\(582\) 5.56839e11 0.201176
\(583\) 8.45633e12 3.03161
\(584\) 3.49075e11 0.124183
\(585\) 8.90739e11 0.314448
\(586\) 2.91591e12 1.02149
\(587\) 6.19927e11 0.215511 0.107755 0.994177i \(-0.465634\pi\)
0.107755 + 0.994177i \(0.465634\pi\)
\(588\) 1.59608e11 0.0550626
\(589\) −1.56357e12 −0.535301
\(590\) −6.12597e12 −2.08133
\(591\) 6.86579e11 0.231498
\(592\) −3.63220e12 −1.21541
\(593\) −3.79034e12 −1.25873 −0.629364 0.777110i \(-0.716685\pi\)
−0.629364 + 0.777110i \(0.716685\pi\)
\(594\) −3.75005e12 −1.23594
\(595\) −5.92486e12 −1.93799
\(596\) 1.04554e12 0.339417
\(597\) −4.28283e11 −0.137990
\(598\) −1.62117e12 −0.518410
\(599\) 2.62045e12 0.831679 0.415839 0.909438i \(-0.363488\pi\)
0.415839 + 0.909438i \(0.363488\pi\)
\(600\) 6.94190e11 0.218674
\(601\) 1.11042e12 0.347178 0.173589 0.984818i \(-0.444464\pi\)
0.173589 + 0.984818i \(0.444464\pi\)
\(602\) 6.58700e11 0.204411
\(603\) −9.52777e11 −0.293470
\(604\) 2.87331e11 0.0878449
\(605\) 8.19956e12 2.48824
\(606\) −7.90829e11 −0.238208
\(607\) 4.57077e12 1.36660 0.683298 0.730140i \(-0.260545\pi\)
0.683298 + 0.730140i \(0.260545\pi\)
\(608\) −7.45724e11 −0.221316
\(609\) 2.82821e11 0.0833171
\(610\) 1.91301e12 0.559415
\(611\) −7.43436e11 −0.215803
\(612\) −5.79902e11 −0.167099
\(613\) −5.61998e12 −1.60754 −0.803772 0.594938i \(-0.797177\pi\)
−0.803772 + 0.594938i \(0.797177\pi\)
\(614\) 4.89568e12 1.39013
\(615\) 2.35898e11 0.0664945
\(616\) 7.47551e12 2.09184
\(617\) 2.17420e12 0.603970 0.301985 0.953313i \(-0.402351\pi\)
0.301985 + 0.953313i \(0.402351\pi\)
\(618\) 1.40860e12 0.388454
\(619\) 5.91455e12 1.61925 0.809625 0.586947i \(-0.199670\pi\)
0.809625 + 0.586947i \(0.199670\pi\)
\(620\) 7.06649e11 0.192062
\(621\) 4.26409e12 1.15057
\(622\) −4.66542e12 −1.24978
\(623\) 4.43031e12 1.17825
\(624\) 4.28576e11 0.113161
\(625\) −4.63555e12 −1.21518
\(626\) 8.33426e11 0.216912
\(627\) 1.49869e12 0.387266
\(628\) −4.55361e11 −0.116826
\(629\) 4.53716e12 1.15573
\(630\) −6.66293e12 −1.68513
\(631\) −5.05658e12 −1.26977 −0.634884 0.772607i \(-0.718952\pi\)
−0.634884 + 0.772607i \(0.718952\pi\)
\(632\) −2.76086e12 −0.688364
\(633\) 1.35128e11 0.0334526
\(634\) 2.09431e11 0.0514801
\(635\) −2.88247e12 −0.703530
\(636\) −4.58982e11 −0.111234
\(637\) −1.01288e12 −0.243741
\(638\) −1.32173e12 −0.315828
\(639\) 2.46775e12 0.585527
\(640\) −6.50609e12 −1.53289
\(641\) 1.51370e12 0.354142 0.177071 0.984198i \(-0.443338\pi\)
0.177071 + 0.984198i \(0.443338\pi\)
\(642\) −7.30274e11 −0.169660
\(643\) 3.90178e12 0.900146 0.450073 0.892992i \(-0.351398\pi\)
0.450073 + 0.892992i \(0.351398\pi\)
\(644\) 1.81326e12 0.415407
\(645\) 2.79779e11 0.0636498
\(646\) 3.32535e12 0.751261
\(647\) 7.88115e12 1.76815 0.884077 0.467341i \(-0.154788\pi\)
0.884077 + 0.467341i \(0.154788\pi\)
\(648\) 2.54768e12 0.567620
\(649\) −1.14066e13 −2.52381
\(650\) 9.39747e11 0.206491
\(651\) −1.88294e12 −0.410887
\(652\) −7.05229e11 −0.152833
\(653\) −5.10697e12 −1.09914 −0.549571 0.835447i \(-0.685209\pi\)
−0.549571 + 0.835447i \(0.685209\pi\)
\(654\) −1.23776e12 −0.264567
\(655\) −6.43778e12 −1.36663
\(656\) −7.80217e11 −0.164493
\(657\) −5.79329e11 −0.121306
\(658\) 5.56107e12 1.15649
\(659\) −5.45308e11 −0.112631 −0.0563154 0.998413i \(-0.517935\pi\)
−0.0563154 + 0.998413i \(0.517935\pi\)
\(660\) −6.77329e11 −0.138948
\(661\) −3.05503e12 −0.622456 −0.311228 0.950335i \(-0.600740\pi\)
−0.311228 + 0.950335i \(0.600740\pi\)
\(662\) −1.25018e12 −0.252995
\(663\) −5.35356e11 −0.107605
\(664\) −1.02455e12 −0.204539
\(665\) 5.71301e12 1.13284
\(666\) 5.10236e12 1.00493
\(667\) 1.50291e12 0.294013
\(668\) −1.29668e10 −0.00251963
\(669\) −1.78974e12 −0.345441
\(670\) −2.46923e12 −0.473395
\(671\) 3.56205e12 0.678343
\(672\) −8.98046e11 −0.169878
\(673\) −1.57419e12 −0.295795 −0.147897 0.989003i \(-0.547251\pi\)
−0.147897 + 0.989003i \(0.547251\pi\)
\(674\) −5.68968e12 −1.06199
\(675\) −2.47177e12 −0.458291
\(676\) 7.34298e10 0.0135242
\(677\) −2.62249e11 −0.0479805 −0.0239903 0.999712i \(-0.507637\pi\)
−0.0239903 + 0.999712i \(0.507637\pi\)
\(678\) 1.96192e12 0.356573
\(679\) 3.95242e12 0.713591
\(680\) 7.04520e12 1.26358
\(681\) −3.77666e12 −0.672892
\(682\) 8.79973e12 1.55754
\(683\) −3.64995e12 −0.641791 −0.320896 0.947115i \(-0.603984\pi\)
−0.320896 + 0.947115i \(0.603984\pi\)
\(684\) 5.59166e11 0.0976762
\(685\) −6.89539e12 −1.19661
\(686\) −1.04471e12 −0.180110
\(687\) −2.84517e12 −0.487307
\(688\) −9.25354e11 −0.157456
\(689\) 2.91271e12 0.492392
\(690\) 5.15077e12 0.865070
\(691\) −1.69603e12 −0.282998 −0.141499 0.989938i \(-0.545192\pi\)
−0.141499 + 0.989938i \(0.545192\pi\)
\(692\) −1.22534e12 −0.203132
\(693\) −1.24065e13 −2.04337
\(694\) 8.93451e11 0.146202
\(695\) 6.92461e12 1.12581
\(696\) −3.36300e11 −0.0543232
\(697\) 9.74608e11 0.156416
\(698\) −2.42131e12 −0.386101
\(699\) −2.33987e12 −0.370718
\(700\) −1.05110e12 −0.165463
\(701\) 1.15906e13 1.81291 0.906454 0.422304i \(-0.138778\pi\)
0.906454 + 0.422304i \(0.138778\pi\)
\(702\) −1.29167e12 −0.200741
\(703\) −4.37492e12 −0.675571
\(704\) −8.54505e12 −1.31111
\(705\) 2.36204e12 0.360111
\(706\) 9.58030e12 1.45130
\(707\) −5.61328e12 −0.844946
\(708\) 6.19114e11 0.0926022
\(709\) −6.98541e12 −1.03821 −0.519104 0.854711i \(-0.673734\pi\)
−0.519104 + 0.854711i \(0.673734\pi\)
\(710\) 6.39544e12 0.944512
\(711\) 4.58196e12 0.672417
\(712\) −5.26804e12 −0.768226
\(713\) −1.00059e13 −1.44996
\(714\) 4.00459e12 0.576654
\(715\) 4.29835e12 0.615070
\(716\) 4.48461e10 0.00637699
\(717\) −4.39972e11 −0.0621711
\(718\) 9.47866e12 1.33103
\(719\) −3.43730e12 −0.479665 −0.239832 0.970814i \(-0.577092\pi\)
−0.239832 + 0.970814i \(0.577092\pi\)
\(720\) 9.36022e12 1.29804
\(721\) 9.99820e12 1.37788
\(722\) 4.71103e12 0.645206
\(723\) −8.80236e11 −0.119805
\(724\) 2.58131e11 0.0349154
\(725\) −8.71194e11 −0.117110
\(726\) −5.54205e12 −0.740381
\(727\) 1.25723e12 0.166920 0.0834601 0.996511i \(-0.473403\pi\)
0.0834601 + 0.996511i \(0.473403\pi\)
\(728\) 2.57488e12 0.339755
\(729\) −2.41427e12 −0.316601
\(730\) −1.50139e12 −0.195678
\(731\) 1.15591e12 0.149725
\(732\) −1.93337e11 −0.0248894
\(733\) −1.51950e12 −0.194416 −0.0972079 0.995264i \(-0.530991\pi\)
−0.0972079 + 0.995264i \(0.530991\pi\)
\(734\) −3.63515e12 −0.462264
\(735\) 3.21810e12 0.406731
\(736\) −4.77221e12 −0.599473
\(737\) −4.59772e12 −0.574036
\(738\) 1.09602e12 0.136008
\(739\) 1.19661e13 1.47589 0.737944 0.674862i \(-0.235797\pi\)
0.737944 + 0.674862i \(0.235797\pi\)
\(740\) 1.97723e12 0.242390
\(741\) 5.16213e11 0.0628995
\(742\) −2.17878e13 −2.63873
\(743\) 1.52097e12 0.183093 0.0915465 0.995801i \(-0.470819\pi\)
0.0915465 + 0.995801i \(0.470819\pi\)
\(744\) 2.23899e12 0.267901
\(745\) 2.10808e13 2.50717
\(746\) 1.50980e13 1.78482
\(747\) 1.70036e12 0.199801
\(748\) −2.79838e12 −0.326850
\(749\) −5.18346e12 −0.601799
\(750\) 1.36286e12 0.157281
\(751\) −8.06595e12 −0.925286 −0.462643 0.886545i \(-0.653099\pi\)
−0.462643 + 0.886545i \(0.653099\pi\)
\(752\) −7.81230e12 −0.890838
\(753\) −8.85513e11 −0.100373
\(754\) −4.55260e11 −0.0512966
\(755\) 5.79333e12 0.648884
\(756\) 1.44472e12 0.160856
\(757\) 8.72882e12 0.966104 0.483052 0.875592i \(-0.339528\pi\)
0.483052 + 0.875592i \(0.339528\pi\)
\(758\) −6.10511e11 −0.0671710
\(759\) 9.59079e12 1.04898
\(760\) −6.79329e12 −0.738616
\(761\) 7.63407e11 0.0825135 0.0412568 0.999149i \(-0.486864\pi\)
0.0412568 + 0.999149i \(0.486864\pi\)
\(762\) 1.94825e12 0.209337
\(763\) −8.78554e12 −0.938443
\(764\) −2.55046e12 −0.270831
\(765\) −1.16923e13 −1.23431
\(766\) −1.18381e13 −1.24237
\(767\) −3.92892e12 −0.409915
\(768\) 1.75945e12 0.182495
\(769\) −5.30538e12 −0.547076 −0.273538 0.961861i \(-0.588194\pi\)
−0.273538 + 0.961861i \(0.588194\pi\)
\(770\) −3.21527e13 −3.29616
\(771\) −3.13349e12 −0.319363
\(772\) 2.19887e12 0.222804
\(773\) −1.53974e12 −0.155110 −0.0775550 0.996988i \(-0.524711\pi\)
−0.0775550 + 0.996988i \(0.524711\pi\)
\(774\) 1.29990e12 0.130189
\(775\) 5.80017e12 0.577541
\(776\) −4.69979e12 −0.465266
\(777\) −5.26854e12 −0.518556
\(778\) −9.32896e12 −0.912904
\(779\) −9.39759e11 −0.0914319
\(780\) −2.33300e11 −0.0225678
\(781\) 1.19084e13 1.14531
\(782\) 2.12803e13 2.03492
\(783\) 1.19745e12 0.113849
\(784\) −1.06437e13 −1.00617
\(785\) −9.18125e12 −0.862955
\(786\) 4.35127e12 0.406644
\(787\) −2.94759e12 −0.273893 −0.136946 0.990578i \(-0.543729\pi\)
−0.136946 + 0.990578i \(0.543729\pi\)
\(788\) 1.23615e12 0.114209
\(789\) 2.24738e12 0.206457
\(790\) 1.18747e13 1.08467
\(791\) 1.39257e13 1.26480
\(792\) 1.47524e13 1.33229
\(793\) 1.22692e12 0.110176
\(794\) −1.94266e13 −1.73462
\(795\) −9.25425e12 −0.821654
\(796\) −7.71099e11 −0.0680772
\(797\) −1.83481e13 −1.61075 −0.805376 0.592764i \(-0.798037\pi\)
−0.805376 + 0.592764i \(0.798037\pi\)
\(798\) −3.86139e12 −0.337079
\(799\) 9.75873e12 0.847096
\(800\) 2.76632e12 0.238779
\(801\) 8.74291e12 0.750429
\(802\) 7.51127e12 0.641104
\(803\) −2.79561e12 −0.237278
\(804\) 2.49549e11 0.0210622
\(805\) 3.65600e13 3.06849
\(806\) 3.03099e12 0.252975
\(807\) −4.32744e12 −0.359170
\(808\) 6.67470e12 0.550910
\(809\) −9.70701e12 −0.796741 −0.398371 0.917225i \(-0.630424\pi\)
−0.398371 + 0.917225i \(0.630424\pi\)
\(810\) −1.09578e13 −0.894415
\(811\) 6.14421e12 0.498738 0.249369 0.968409i \(-0.419777\pi\)
0.249369 + 0.968409i \(0.419777\pi\)
\(812\) 5.09203e11 0.0411045
\(813\) 4.18005e12 0.335563
\(814\) 2.46220e13 1.96568
\(815\) −1.42192e13 −1.12893
\(816\) −5.62572e12 −0.444194
\(817\) −1.11457e12 −0.0875205
\(818\) −2.22030e13 −1.73389
\(819\) −4.27330e12 −0.331884
\(820\) 4.24720e11 0.0328050
\(821\) 1.41092e13 1.08383 0.541913 0.840434i \(-0.317700\pi\)
0.541913 + 0.840434i \(0.317700\pi\)
\(822\) 4.66057e12 0.356054
\(823\) 7.99374e12 0.607366 0.303683 0.952773i \(-0.401784\pi\)
0.303683 + 0.952773i \(0.401784\pi\)
\(824\) −1.18888e13 −0.898389
\(825\) −5.55951e12 −0.417824
\(826\) 2.93892e13 2.19674
\(827\) −4.82994e12 −0.359060 −0.179530 0.983752i \(-0.557458\pi\)
−0.179530 + 0.983752i \(0.557458\pi\)
\(828\) 3.57834e12 0.264573
\(829\) 1.41648e13 1.04163 0.520816 0.853669i \(-0.325628\pi\)
0.520816 + 0.853669i \(0.325628\pi\)
\(830\) 4.40666e12 0.322298
\(831\) 8.65637e12 0.629697
\(832\) −2.94327e12 −0.212949
\(833\) 1.32956e13 0.956762
\(834\) −4.68031e12 −0.334987
\(835\) −2.61443e11 −0.0186118
\(836\) 2.69831e12 0.191058
\(837\) −7.97228e12 −0.561459
\(838\) −8.24927e12 −0.577853
\(839\) 1.79477e13 1.25049 0.625246 0.780428i \(-0.284999\pi\)
0.625246 + 0.780428i \(0.284999\pi\)
\(840\) −8.18088e12 −0.566948
\(841\) −1.40851e13 −0.970907
\(842\) 4.59068e12 0.314755
\(843\) 3.78778e12 0.258322
\(844\) 2.43291e11 0.0165038
\(845\) 1.48053e12 0.0998993
\(846\) 1.09744e13 0.736570
\(847\) −3.93372e13 −2.62620
\(848\) 3.06079e13 2.03260
\(849\) 8.44458e12 0.557819
\(850\) −1.23356e13 −0.810542
\(851\) −2.79970e13 −1.82990
\(852\) −6.46347e11 −0.0420231
\(853\) −8.65463e12 −0.559729 −0.279865 0.960039i \(-0.590290\pi\)
−0.279865 + 0.960039i \(0.590290\pi\)
\(854\) −9.17764e12 −0.590433
\(855\) 1.12742e13 0.721505
\(856\) 6.16360e12 0.392376
\(857\) −1.17199e13 −0.742185 −0.371092 0.928596i \(-0.621017\pi\)
−0.371092 + 0.928596i \(0.621017\pi\)
\(858\) −2.90524e12 −0.183016
\(859\) −2.70022e13 −1.69211 −0.846057 0.533092i \(-0.821030\pi\)
−0.846057 + 0.533092i \(0.821030\pi\)
\(860\) 5.03727e11 0.0314016
\(861\) −1.13171e12 −0.0701815
\(862\) −2.35669e13 −1.45385
\(863\) 2.11111e13 1.29557 0.647786 0.761823i \(-0.275695\pi\)
0.647786 + 0.761823i \(0.275695\pi\)
\(864\) −3.80228e12 −0.232130
\(865\) −2.47060e13 −1.50048
\(866\) −2.81704e13 −1.70201
\(867\) 1.09830e12 0.0660137
\(868\) −3.39013e12 −0.202711
\(869\) 2.21107e13 1.31527
\(870\) 1.44645e12 0.0855986
\(871\) −1.58365e12 −0.0932346
\(872\) 1.04468e13 0.611870
\(873\) 7.79984e12 0.454487
\(874\) −2.05194e13 −1.18950
\(875\) 9.67354e12 0.557891
\(876\) 1.51737e11 0.00870606
\(877\) −1.48840e13 −0.849614 −0.424807 0.905284i \(-0.639658\pi\)
−0.424807 + 0.905284i \(0.639658\pi\)
\(878\) −1.33544e13 −0.758402
\(879\) −5.94176e12 −0.335711
\(880\) 4.51687e13 2.53902
\(881\) 8.21496e11 0.0459424 0.0229712 0.999736i \(-0.492687\pi\)
0.0229712 + 0.999736i \(0.492687\pi\)
\(882\) 1.49518e13 0.831926
\(883\) −5.06069e12 −0.280147 −0.140074 0.990141i \(-0.544734\pi\)
−0.140074 + 0.990141i \(0.544734\pi\)
\(884\) −9.63878e11 −0.0530868
\(885\) 1.24829e13 0.684025
\(886\) 2.74964e12 0.149908
\(887\) −4.00796e12 −0.217404 −0.108702 0.994074i \(-0.534669\pi\)
−0.108702 + 0.994074i \(0.534669\pi\)
\(888\) 6.26478e12 0.338102
\(889\) 1.38286e13 0.742540
\(890\) 2.26582e13 1.21051
\(891\) −2.04035e13 −1.08456
\(892\) −3.22233e12 −0.170423
\(893\) −9.40979e12 −0.495163
\(894\) −1.42484e13 −0.746016
\(895\) 9.04211e11 0.0471049
\(896\) 3.12129e13 1.61788
\(897\) 3.30347e12 0.170374
\(898\) 2.74516e13 1.40872
\(899\) −2.80989e12 −0.143473
\(900\) −2.07427e12 −0.105384
\(901\) −3.82338e13 −1.93280
\(902\) 5.28894e12 0.266035
\(903\) −1.34224e12 −0.0671791
\(904\) −1.65589e13 −0.824657
\(905\) 5.20459e12 0.257910
\(906\) −3.91569e12 −0.193077
\(907\) 4.52189e12 0.221864 0.110932 0.993828i \(-0.464616\pi\)
0.110932 + 0.993828i \(0.464616\pi\)
\(908\) −6.79966e12 −0.331971
\(909\) −1.10774e13 −0.538147
\(910\) −1.10747e13 −0.535361
\(911\) −5.68437e12 −0.273432 −0.136716 0.990610i \(-0.543655\pi\)
−0.136716 + 0.990610i \(0.543655\pi\)
\(912\) 5.42456e12 0.259650
\(913\) 8.20525e12 0.390817
\(914\) 3.40839e13 1.61544
\(915\) −3.89816e12 −0.183850
\(916\) −5.12257e12 −0.240413
\(917\) 3.08851e13 1.44241
\(918\) 1.69552e13 0.787972
\(919\) 3.28047e13 1.51711 0.758555 0.651609i \(-0.225906\pi\)
0.758555 + 0.651609i \(0.225906\pi\)
\(920\) −4.34731e13 −2.00067
\(921\) −9.97597e12 −0.456864
\(922\) −2.60183e13 −1.18574
\(923\) 4.10174e12 0.186020
\(924\) 3.24947e12 0.146652
\(925\) 1.62291e13 0.728880
\(926\) −6.90877e12 −0.308782
\(927\) 1.97308e13 0.877576
\(928\) −1.34014e12 −0.0593177
\(929\) 1.82868e13 0.805504 0.402752 0.915309i \(-0.368054\pi\)
0.402752 + 0.915309i \(0.368054\pi\)
\(930\) −9.63005e12 −0.422139
\(931\) −1.28201e13 −0.559267
\(932\) −4.21280e12 −0.182894
\(933\) 9.50675e12 0.410738
\(934\) 2.43183e13 1.04562
\(935\) −5.64224e13 −2.41435
\(936\) 5.08134e12 0.216390
\(937\) 2.71335e13 1.14995 0.574974 0.818171i \(-0.305012\pi\)
0.574974 + 0.818171i \(0.305012\pi\)
\(938\) 1.18461e13 0.499644
\(939\) −1.69828e12 −0.0712875
\(940\) 4.25271e12 0.177660
\(941\) −9.48833e12 −0.394490 −0.197245 0.980354i \(-0.563200\pi\)
−0.197245 + 0.980354i \(0.563200\pi\)
\(942\) 6.20556e12 0.256775
\(943\) −6.01392e12 −0.247660
\(944\) −4.12866e13 −1.69213
\(945\) 2.91293e13 1.18819
\(946\) 6.27279e12 0.254654
\(947\) 1.99262e13 0.805100 0.402550 0.915398i \(-0.368124\pi\)
0.402550 + 0.915398i \(0.368124\pi\)
\(948\) −1.20010e12 −0.0482591
\(949\) −9.62925e11 −0.0385385
\(950\) 1.18945e13 0.473796
\(951\) −4.26759e11 −0.0169188
\(952\) −3.37992e13 −1.33364
\(953\) 2.48922e13 0.977565 0.488782 0.872406i \(-0.337441\pi\)
0.488782 + 0.872406i \(0.337441\pi\)
\(954\) −4.29967e13 −1.68061
\(955\) −5.14238e13 −2.00055
\(956\) −7.92144e11 −0.0306721
\(957\) 2.69330e12 0.103796
\(958\) 3.38780e13 1.29949
\(959\) 3.30805e13 1.26296
\(960\) 9.35134e12 0.355347
\(961\) −7.73217e12 −0.292446
\(962\) 8.48083e12 0.319264
\(963\) −1.02292e13 −0.383286
\(964\) −1.58481e12 −0.0591060
\(965\) 4.43349e13 1.64579
\(966\) −2.47107e13 −0.913037
\(967\) −4.20642e13 −1.54701 −0.773505 0.633790i \(-0.781499\pi\)
−0.773505 + 0.633790i \(0.781499\pi\)
\(968\) 4.67756e13 1.71230
\(969\) −6.77609e12 −0.246900
\(970\) 2.02141e13 0.733132
\(971\) −1.34740e12 −0.0486418 −0.0243209 0.999704i \(-0.507742\pi\)
−0.0243209 + 0.999704i \(0.507742\pi\)
\(972\) 4.37325e12 0.157147
\(973\) −3.32207e13 −1.18823
\(974\) −2.36825e13 −0.843164
\(975\) −1.91493e12 −0.0678628
\(976\) 1.28929e13 0.454807
\(977\) −2.05822e13 −0.722714 −0.361357 0.932428i \(-0.617686\pi\)
−0.361357 + 0.932428i \(0.617686\pi\)
\(978\) 9.61070e12 0.335916
\(979\) 4.21898e13 1.46786
\(980\) 5.79401e12 0.200661
\(981\) −1.73377e13 −0.597695
\(982\) 2.71700e13 0.932368
\(983\) −5.72322e13 −1.95501 −0.977506 0.210906i \(-0.932359\pi\)
−0.977506 + 0.210906i \(0.932359\pi\)
\(984\) 1.34571e12 0.0457587
\(985\) 2.49239e13 0.843631
\(986\) 5.97598e12 0.201355
\(987\) −1.13318e13 −0.380078
\(988\) 9.29412e11 0.0310315
\(989\) −7.13263e12 −0.237065
\(990\) −6.34511e13 −2.09933
\(991\) 1.46766e13 0.483386 0.241693 0.970353i \(-0.422297\pi\)
0.241693 + 0.970353i \(0.422297\pi\)
\(992\) 8.92228e12 0.292532
\(993\) 2.54750e12 0.0831462
\(994\) −3.06820e13 −0.996884
\(995\) −1.55473e13 −0.502866
\(996\) −4.45354e11 −0.0143396
\(997\) −5.31286e13 −1.70294 −0.851471 0.524402i \(-0.824289\pi\)
−0.851471 + 0.524402i \(0.824289\pi\)
\(998\) 5.51037e13 1.75830
\(999\) −2.23067e13 −0.708583
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 13.10.a.a.1.2 4
3.2 odd 2 117.10.a.c.1.3 4
4.3 odd 2 208.10.a.g.1.1 4
5.4 even 2 325.10.a.a.1.3 4
13.12 even 2 169.10.a.a.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.a.1.2 4 1.1 even 1 trivial
117.10.a.c.1.3 4 3.2 odd 2
169.10.a.a.1.3 4 13.12 even 2
208.10.a.g.1.1 4 4.3 odd 2
325.10.a.a.1.3 4 5.4 even 2