Properties

Label 285.10.a.f.1.3
Level $285$
Weight $10$
Character 285.1
Self dual yes
Analytic conductor $146.785$
Analytic rank $0$
Dimension $14$
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] = [285,10,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(146.785213307\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 5365 x^{12} + 7107 x^{11} + 10970098 x^{10} - 19024208 x^{9} - 10608934432 x^{8} + \cdots - 480881506516992 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(33.9313\) of defining polynomial
Character \(\chi\) \(=\) 285.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-33.9313 q^{2} -81.0000 q^{3} +639.331 q^{4} -625.000 q^{5} +2748.43 q^{6} +9880.04 q^{7} -4320.49 q^{8} +6561.00 q^{9} +21207.0 q^{10} -69660.6 q^{11} -51785.8 q^{12} -176048. q^{13} -335242. q^{14} +50625.0 q^{15} -180738. q^{16} -491581. q^{17} -222623. q^{18} +130321. q^{19} -399582. q^{20} -800283. q^{21} +2.36367e6 q^{22} -712914. q^{23} +349960. q^{24} +390625. q^{25} +5.97353e6 q^{26} -531441. q^{27} +6.31661e6 q^{28} -134373. q^{29} -1.71777e6 q^{30} -2.59229e6 q^{31} +8.34475e6 q^{32} +5.64251e6 q^{33} +1.66800e7 q^{34} -6.17502e6 q^{35} +4.19465e6 q^{36} -3.66220e6 q^{37} -4.42196e6 q^{38} +1.42599e7 q^{39} +2.70031e6 q^{40} -9.74654e6 q^{41} +2.71546e7 q^{42} +4.21021e6 q^{43} -4.45362e7 q^{44} -4.10062e6 q^{45} +2.41901e7 q^{46} -6.41652e7 q^{47} +1.46397e7 q^{48} +5.72616e7 q^{49} -1.32544e7 q^{50} +3.98181e7 q^{51} -1.12553e8 q^{52} -2.38522e7 q^{53} +1.80325e7 q^{54} +4.35379e7 q^{55} -4.26866e7 q^{56} -1.05560e7 q^{57} +4.55945e6 q^{58} -1.02156e8 q^{59} +3.23661e7 q^{60} -1.25148e8 q^{61} +8.79596e7 q^{62} +6.48229e7 q^{63} -1.90610e8 q^{64} +1.10030e8 q^{65} -1.91457e8 q^{66} -9.38601e6 q^{67} -3.14283e8 q^{68} +5.77460e7 q^{69} +2.09526e8 q^{70} +1.74680e8 q^{71} -2.83468e7 q^{72} +3.17964e8 q^{73} +1.24263e8 q^{74} -3.16406e7 q^{75} +8.33182e7 q^{76} -6.88250e8 q^{77} -4.83856e8 q^{78} +4.82600e7 q^{79} +1.12961e8 q^{80} +4.30467e7 q^{81} +3.30712e8 q^{82} +3.83115e8 q^{83} -5.11646e8 q^{84} +3.07238e8 q^{85} -1.42858e8 q^{86} +1.08842e7 q^{87} +3.00968e8 q^{88} -1.01770e9 q^{89} +1.39139e8 q^{90} -1.73936e9 q^{91} -4.55788e8 q^{92} +2.09975e8 q^{93} +2.17721e9 q^{94} -8.14506e7 q^{95} -6.75924e8 q^{96} -1.08495e9 q^{97} -1.94296e9 q^{98} -4.57043e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} - 1134 q^{3} + 3563 q^{4} - 8750 q^{5} + 81 q^{6} + 13054 q^{7} + 6249 q^{8} + 91854 q^{9} + 625 q^{10} + 43520 q^{11} - 288603 q^{12} + 256834 q^{13} + 250610 q^{14} + 708750 q^{15} + 866291 q^{16}+ \cdots + 285534720 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −33.9313 −1.49956 −0.749782 0.661685i \(-0.769842\pi\)
−0.749782 + 0.661685i \(0.769842\pi\)
\(3\) −81.0000 −0.577350
\(4\) 639.331 1.24869
\(5\) −625.000 −0.447214
\(6\) 2748.43 0.865774
\(7\) 9880.04 1.55531 0.777656 0.628690i \(-0.216409\pi\)
0.777656 + 0.628690i \(0.216409\pi\)
\(8\) −4320.49 −0.372931
\(9\) 6561.00 0.333333
\(10\) 21207.0 0.670626
\(11\) −69660.6 −1.43456 −0.717282 0.696782i \(-0.754614\pi\)
−0.717282 + 0.696782i \(0.754614\pi\)
\(12\) −51785.8 −0.720933
\(13\) −176048. −1.70956 −0.854782 0.518987i \(-0.826309\pi\)
−0.854782 + 0.518987i \(0.826309\pi\)
\(14\) −335242. −2.33229
\(15\) 50625.0 0.258199
\(16\) −180738. −0.689459
\(17\) −491581. −1.42750 −0.713748 0.700402i \(-0.753004\pi\)
−0.713748 + 0.700402i \(0.753004\pi\)
\(18\) −222623. −0.499855
\(19\) 130321. 0.229416
\(20\) −399582. −0.558432
\(21\) −800283. −0.897960
\(22\) 2.36367e6 2.15122
\(23\) −712914. −0.531205 −0.265602 0.964083i \(-0.585571\pi\)
−0.265602 + 0.964083i \(0.585571\pi\)
\(24\) 349960. 0.215312
\(25\) 390625. 0.200000
\(26\) 5.97353e6 2.56360
\(27\) −531441. −0.192450
\(28\) 6.31661e6 1.94211
\(29\) −134373. −0.0352794 −0.0176397 0.999844i \(-0.505615\pi\)
−0.0176397 + 0.999844i \(0.505615\pi\)
\(30\) −1.71777e6 −0.387186
\(31\) −2.59229e6 −0.504145 −0.252073 0.967708i \(-0.581112\pi\)
−0.252073 + 0.967708i \(0.581112\pi\)
\(32\) 8.34475e6 1.40682
\(33\) 5.64251e6 0.828246
\(34\) 1.66800e7 2.14062
\(35\) −6.17502e6 −0.695557
\(36\) 4.19465e6 0.416231
\(37\) −3.66220e6 −0.321243 −0.160622 0.987016i \(-0.551350\pi\)
−0.160622 + 0.987016i \(0.551350\pi\)
\(38\) −4.42196e6 −0.344024
\(39\) 1.42599e7 0.987018
\(40\) 2.70031e6 0.166780
\(41\) −9.74654e6 −0.538670 −0.269335 0.963047i \(-0.586804\pi\)
−0.269335 + 0.963047i \(0.586804\pi\)
\(42\) 2.71546e7 1.34655
\(43\) 4.21021e6 0.187800 0.0938999 0.995582i \(-0.470067\pi\)
0.0938999 + 0.995582i \(0.470067\pi\)
\(44\) −4.45362e7 −1.79133
\(45\) −4.10062e6 −0.149071
\(46\) 2.41901e7 0.796575
\(47\) −6.41652e7 −1.91805 −0.959024 0.283325i \(-0.908562\pi\)
−0.959024 + 0.283325i \(0.908562\pi\)
\(48\) 1.46397e7 0.398059
\(49\) 5.72616e7 1.41900
\(50\) −1.32544e7 −0.299913
\(51\) 3.98181e7 0.824166
\(52\) −1.12553e8 −2.13472
\(53\) −2.38522e7 −0.415228 −0.207614 0.978211i \(-0.566570\pi\)
−0.207614 + 0.978211i \(0.566570\pi\)
\(54\) 1.80325e7 0.288591
\(55\) 4.35379e7 0.641557
\(56\) −4.26866e7 −0.580024
\(57\) −1.05560e7 −0.132453
\(58\) 4.55945e6 0.0529037
\(59\) −1.02156e8 −1.09757 −0.548783 0.835965i \(-0.684909\pi\)
−0.548783 + 0.835965i \(0.684909\pi\)
\(60\) 3.23661e7 0.322411
\(61\) −1.25148e8 −1.15728 −0.578642 0.815582i \(-0.696417\pi\)
−0.578642 + 0.815582i \(0.696417\pi\)
\(62\) 8.79596e7 0.755998
\(63\) 6.48229e7 0.518437
\(64\) −1.90610e8 −1.42016
\(65\) 1.10030e8 0.764541
\(66\) −1.91457e8 −1.24201
\(67\) −9.38601e6 −0.0569042 −0.0284521 0.999595i \(-0.509058\pi\)
−0.0284521 + 0.999595i \(0.509058\pi\)
\(68\) −3.14283e8 −1.78250
\(69\) 5.77460e7 0.306691
\(70\) 2.09526e8 1.04303
\(71\) 1.74680e8 0.815792 0.407896 0.913028i \(-0.366263\pi\)
0.407896 + 0.913028i \(0.366263\pi\)
\(72\) −2.83468e7 −0.124310
\(73\) 3.17964e8 1.31046 0.655232 0.755428i \(-0.272571\pi\)
0.655232 + 0.755428i \(0.272571\pi\)
\(74\) 1.24263e8 0.481725
\(75\) −3.16406e7 −0.115470
\(76\) 8.33182e7 0.286470
\(77\) −6.88250e8 −2.23120
\(78\) −4.83856e8 −1.48010
\(79\) 4.82600e7 0.139401 0.0697004 0.997568i \(-0.477796\pi\)
0.0697004 + 0.997568i \(0.477796\pi\)
\(80\) 1.12961e8 0.308335
\(81\) 4.30467e7 0.111111
\(82\) 3.30712e8 0.807771
\(83\) 3.83115e8 0.886090 0.443045 0.896499i \(-0.353898\pi\)
0.443045 + 0.896499i \(0.353898\pi\)
\(84\) −5.11646e8 −1.12128
\(85\) 3.07238e8 0.638396
\(86\) −1.42858e8 −0.281618
\(87\) 1.08842e7 0.0203686
\(88\) 3.00968e8 0.534994
\(89\) −1.01770e9 −1.71936 −0.859678 0.510836i \(-0.829336\pi\)
−0.859678 + 0.510836i \(0.829336\pi\)
\(90\) 1.39139e8 0.223542
\(91\) −1.73936e9 −2.65891
\(92\) −4.55788e8 −0.663311
\(93\) 2.09975e8 0.291068
\(94\) 2.17721e9 2.87624
\(95\) −8.14506e7 −0.102598
\(96\) −6.75924e8 −0.812227
\(97\) −1.08495e9 −1.24434 −0.622168 0.782883i \(-0.713748\pi\)
−0.622168 + 0.782883i \(0.713748\pi\)
\(98\) −1.94296e9 −2.12787
\(99\) −4.57043e8 −0.478188
\(100\) 2.49739e8 0.249739
\(101\) −2.54570e8 −0.243423 −0.121711 0.992566i \(-0.538838\pi\)
−0.121711 + 0.992566i \(0.538838\pi\)
\(102\) −1.35108e9 −1.23589
\(103\) 1.61531e6 0.00141413 0.000707063 1.00000i \(-0.499775\pi\)
0.000707063 1.00000i \(0.499775\pi\)
\(104\) 7.60613e8 0.637549
\(105\) 5.00177e8 0.401580
\(106\) 8.09335e8 0.622661
\(107\) 1.89093e9 1.39460 0.697299 0.716780i \(-0.254385\pi\)
0.697299 + 0.716780i \(0.254385\pi\)
\(108\) −3.39767e8 −0.240311
\(109\) 6.16231e8 0.418142 0.209071 0.977900i \(-0.432956\pi\)
0.209071 + 0.977900i \(0.432956\pi\)
\(110\) −1.47730e9 −0.962056
\(111\) 2.96638e8 0.185470
\(112\) −1.78569e9 −1.07232
\(113\) −3.36282e9 −1.94022 −0.970110 0.242666i \(-0.921978\pi\)
−0.970110 + 0.242666i \(0.921978\pi\)
\(114\) 3.58178e8 0.198622
\(115\) 4.45571e8 0.237562
\(116\) −8.59088e7 −0.0440531
\(117\) −1.15505e9 −0.569855
\(118\) 3.46629e9 1.64587
\(119\) −4.85684e9 −2.22020
\(120\) −2.18725e8 −0.0962903
\(121\) 2.49465e9 1.05798
\(122\) 4.24643e9 1.73542
\(123\) 7.89470e8 0.311001
\(124\) −1.65733e9 −0.629522
\(125\) −2.44141e8 −0.0894427
\(126\) −2.19952e9 −0.777430
\(127\) −2.51900e9 −0.859232 −0.429616 0.903012i \(-0.641351\pi\)
−0.429616 + 0.903012i \(0.641351\pi\)
\(128\) 2.19513e9 0.722797
\(129\) −3.41027e8 −0.108426
\(130\) −3.73345e9 −1.14648
\(131\) −5.00056e9 −1.48353 −0.741767 0.670657i \(-0.766012\pi\)
−0.741767 + 0.670657i \(0.766012\pi\)
\(132\) 3.60743e9 1.03423
\(133\) 1.28758e9 0.356813
\(134\) 3.18479e8 0.0853315
\(135\) 3.32151e8 0.0860663
\(136\) 2.12387e9 0.532358
\(137\) −2.10286e9 −0.509998 −0.254999 0.966941i \(-0.582075\pi\)
−0.254999 + 0.966941i \(0.582075\pi\)
\(138\) −1.95940e9 −0.459903
\(139\) −1.92992e9 −0.438503 −0.219251 0.975668i \(-0.570362\pi\)
−0.219251 + 0.975668i \(0.570362\pi\)
\(140\) −3.94788e9 −0.868537
\(141\) 5.19738e9 1.10739
\(142\) −5.92710e9 −1.22333
\(143\) 1.22636e10 2.45248
\(144\) −1.18582e9 −0.229820
\(145\) 8.39831e7 0.0157774
\(146\) −1.07889e10 −1.96512
\(147\) −4.63819e9 −0.819257
\(148\) −2.34136e9 −0.401134
\(149\) 7.40405e9 1.23064 0.615320 0.788277i \(-0.289027\pi\)
0.615320 + 0.788277i \(0.289027\pi\)
\(150\) 1.07361e9 0.173155
\(151\) 1.15986e9 0.181555 0.0907776 0.995871i \(-0.471065\pi\)
0.0907776 + 0.995871i \(0.471065\pi\)
\(152\) −5.63051e8 −0.0855562
\(153\) −3.22526e9 −0.475832
\(154\) 2.33532e10 3.34582
\(155\) 1.62018e9 0.225461
\(156\) 9.11678e9 1.23248
\(157\) −7.39247e9 −0.971049 −0.485525 0.874223i \(-0.661371\pi\)
−0.485525 + 0.874223i \(0.661371\pi\)
\(158\) −1.63752e9 −0.209041
\(159\) 1.93203e9 0.239732
\(160\) −5.21547e9 −0.629149
\(161\) −7.04362e9 −0.826189
\(162\) −1.46063e9 −0.166618
\(163\) 2.42581e9 0.269162 0.134581 0.990903i \(-0.457031\pi\)
0.134581 + 0.990903i \(0.457031\pi\)
\(164\) −6.23126e9 −0.672634
\(165\) −3.52657e9 −0.370403
\(166\) −1.29996e10 −1.32875
\(167\) −1.27031e10 −1.26382 −0.631911 0.775041i \(-0.717729\pi\)
−0.631911 + 0.775041i \(0.717729\pi\)
\(168\) 3.45762e9 0.334877
\(169\) 2.03883e10 1.92261
\(170\) −1.04250e10 −0.957316
\(171\) 8.55036e8 0.0764719
\(172\) 2.69171e9 0.234504
\(173\) 1.98216e10 1.68240 0.841202 0.540721i \(-0.181849\pi\)
0.841202 + 0.540721i \(0.181849\pi\)
\(174\) −3.69315e8 −0.0305440
\(175\) 3.85939e9 0.311062
\(176\) 1.25903e10 0.989074
\(177\) 8.27466e9 0.633680
\(178\) 3.45319e10 2.57828
\(179\) 1.27221e10 0.926230 0.463115 0.886298i \(-0.346732\pi\)
0.463115 + 0.886298i \(0.346732\pi\)
\(180\) −2.62166e9 −0.186144
\(181\) 6.13497e9 0.424873 0.212436 0.977175i \(-0.431860\pi\)
0.212436 + 0.977175i \(0.431860\pi\)
\(182\) 5.90187e10 3.98720
\(183\) 1.01370e10 0.668158
\(184\) 3.08014e9 0.198103
\(185\) 2.28888e9 0.143664
\(186\) −7.12473e9 −0.436476
\(187\) 3.42438e10 2.04784
\(188\) −4.10228e10 −2.39505
\(189\) −5.25066e9 −0.299320
\(190\) 2.76372e9 0.153852
\(191\) −3.51221e10 −1.90955 −0.954774 0.297331i \(-0.903903\pi\)
−0.954774 + 0.297331i \(0.903903\pi\)
\(192\) 1.54394e10 0.819928
\(193\) 3.31686e10 1.72075 0.860377 0.509657i \(-0.170228\pi\)
0.860377 + 0.509657i \(0.170228\pi\)
\(194\) 3.68138e10 1.86596
\(195\) −8.91242e9 −0.441408
\(196\) 3.66091e10 1.77189
\(197\) 1.17394e10 0.555324 0.277662 0.960679i \(-0.410441\pi\)
0.277662 + 0.960679i \(0.410441\pi\)
\(198\) 1.55081e10 0.717074
\(199\) −3.84124e10 −1.73633 −0.868166 0.496273i \(-0.834701\pi\)
−0.868166 + 0.496273i \(0.834701\pi\)
\(200\) −1.68769e9 −0.0745862
\(201\) 7.60267e8 0.0328536
\(202\) 8.63789e9 0.365028
\(203\) −1.32761e9 −0.0548705
\(204\) 2.54569e10 1.02913
\(205\) 6.09159e9 0.240901
\(206\) −5.48095e7 −0.00212057
\(207\) −4.67743e9 −0.177068
\(208\) 3.18185e10 1.17867
\(209\) −9.07824e9 −0.329112
\(210\) −1.69716e10 −0.602195
\(211\) 9.53224e9 0.331073 0.165537 0.986204i \(-0.447064\pi\)
0.165537 + 0.986204i \(0.447064\pi\)
\(212\) −1.52494e10 −0.518492
\(213\) −1.41490e10 −0.470998
\(214\) −6.41618e10 −2.09129
\(215\) −2.63138e9 −0.0839867
\(216\) 2.29609e9 0.0717706
\(217\) −2.56119e10 −0.784103
\(218\) −2.09095e10 −0.627031
\(219\) −2.57551e10 −0.756596
\(220\) 2.78351e10 0.801108
\(221\) 8.65418e10 2.44040
\(222\) −1.00653e10 −0.278124
\(223\) −3.45762e10 −0.936280 −0.468140 0.883654i \(-0.655076\pi\)
−0.468140 + 0.883654i \(0.655076\pi\)
\(224\) 8.24464e10 2.18804
\(225\) 2.56289e9 0.0666667
\(226\) 1.14105e11 2.90948
\(227\) 2.09866e10 0.524597 0.262299 0.964987i \(-0.415519\pi\)
0.262299 + 0.964987i \(0.415519\pi\)
\(228\) −6.74878e9 −0.165393
\(229\) −5.45785e10 −1.31148 −0.655741 0.754986i \(-0.727643\pi\)
−0.655741 + 0.754986i \(0.727643\pi\)
\(230\) −1.51188e10 −0.356239
\(231\) 5.57482e10 1.28818
\(232\) 5.80558e8 0.0131568
\(233\) 2.43534e10 0.541324 0.270662 0.962674i \(-0.412757\pi\)
0.270662 + 0.962674i \(0.412757\pi\)
\(234\) 3.91923e10 0.854534
\(235\) 4.01033e10 0.857777
\(236\) −6.53116e10 −1.37052
\(237\) −3.90906e9 −0.0804831
\(238\) 1.64799e11 3.32934
\(239\) −9.34512e10 −1.85265 −0.926327 0.376720i \(-0.877052\pi\)
−0.926327 + 0.376720i \(0.877052\pi\)
\(240\) −9.14984e9 −0.178018
\(241\) −7.79516e10 −1.48850 −0.744250 0.667902i \(-0.767193\pi\)
−0.744250 + 0.667902i \(0.767193\pi\)
\(242\) −8.46468e10 −1.58650
\(243\) −3.48678e9 −0.0641500
\(244\) −8.00109e10 −1.44509
\(245\) −3.57885e10 −0.634594
\(246\) −2.67877e10 −0.466367
\(247\) −2.29427e10 −0.392201
\(248\) 1.12000e10 0.188011
\(249\) −3.10323e10 −0.511584
\(250\) 8.28400e9 0.134125
\(251\) −8.60114e10 −1.36781 −0.683903 0.729573i \(-0.739719\pi\)
−0.683903 + 0.729573i \(0.739719\pi\)
\(252\) 4.14433e10 0.647369
\(253\) 4.96620e10 0.762048
\(254\) 8.54727e10 1.28847
\(255\) −2.48863e10 −0.368578
\(256\) 2.31087e10 0.336276
\(257\) 3.35045e10 0.479076 0.239538 0.970887i \(-0.423004\pi\)
0.239538 + 0.970887i \(0.423004\pi\)
\(258\) 1.15715e10 0.162592
\(259\) −3.61827e10 −0.499634
\(260\) 7.03455e10 0.954676
\(261\) −8.81621e8 −0.0117598
\(262\) 1.69675e11 2.22466
\(263\) −1.25298e11 −1.61490 −0.807449 0.589938i \(-0.799152\pi\)
−0.807449 + 0.589938i \(0.799152\pi\)
\(264\) −2.43784e10 −0.308879
\(265\) 1.49076e10 0.185696
\(266\) −4.36891e10 −0.535064
\(267\) 8.24339e10 0.992671
\(268\) −6.00076e9 −0.0710559
\(269\) −3.18538e10 −0.370917 −0.185458 0.982652i \(-0.559377\pi\)
−0.185458 + 0.982652i \(0.559377\pi\)
\(270\) −1.12703e10 −0.129062
\(271\) −3.80191e10 −0.428193 −0.214097 0.976812i \(-0.568681\pi\)
−0.214097 + 0.976812i \(0.568681\pi\)
\(272\) 8.88472e10 0.984201
\(273\) 1.40888e11 1.53512
\(274\) 7.13529e10 0.764775
\(275\) −2.72112e10 −0.286913
\(276\) 3.69188e10 0.382963
\(277\) 1.05646e11 1.07819 0.539095 0.842245i \(-0.318766\pi\)
0.539095 + 0.842245i \(0.318766\pi\)
\(278\) 6.54846e10 0.657563
\(279\) −1.70080e10 −0.168048
\(280\) 2.66792e10 0.259395
\(281\) −4.31314e10 −0.412682 −0.206341 0.978480i \(-0.566156\pi\)
−0.206341 + 0.978480i \(0.566156\pi\)
\(282\) −1.76354e11 −1.66060
\(283\) 1.39197e11 1.29000 0.645002 0.764181i \(-0.276857\pi\)
0.645002 + 0.764181i \(0.276857\pi\)
\(284\) 1.11678e11 1.01867
\(285\) 6.59750e9 0.0592349
\(286\) −4.16119e11 −3.67765
\(287\) −9.62962e10 −0.837800
\(288\) 5.47499e10 0.468940
\(289\) 1.23064e11 1.03775
\(290\) −2.84965e9 −0.0236593
\(291\) 8.78812e10 0.718418
\(292\) 2.03284e11 1.63637
\(293\) 1.18645e11 0.940473 0.470236 0.882541i \(-0.344169\pi\)
0.470236 + 0.882541i \(0.344169\pi\)
\(294\) 1.57380e11 1.22853
\(295\) 6.38477e10 0.490847
\(296\) 1.58225e10 0.119802
\(297\) 3.70205e10 0.276082
\(298\) −2.51229e11 −1.84542
\(299\) 1.25507e11 0.908129
\(300\) −2.02288e10 −0.144187
\(301\) 4.15970e10 0.292087
\(302\) −3.93555e10 −0.272254
\(303\) 2.06202e10 0.140540
\(304\) −2.35539e10 −0.158173
\(305\) 7.82175e10 0.517553
\(306\) 1.09437e11 0.713541
\(307\) 1.54568e11 0.993109 0.496554 0.868006i \(-0.334598\pi\)
0.496554 + 0.868006i \(0.334598\pi\)
\(308\) −4.40019e11 −2.78608
\(309\) −1.30840e8 −0.000816446 0
\(310\) −5.49747e10 −0.338093
\(311\) −5.00657e10 −0.303472 −0.151736 0.988421i \(-0.548486\pi\)
−0.151736 + 0.988421i \(0.548486\pi\)
\(312\) −6.16097e10 −0.368089
\(313\) 1.63498e11 0.962858 0.481429 0.876485i \(-0.340118\pi\)
0.481429 + 0.876485i \(0.340118\pi\)
\(314\) 2.50836e11 1.45615
\(315\) −4.05143e10 −0.231852
\(316\) 3.08541e10 0.174069
\(317\) 1.23719e11 0.688129 0.344064 0.938946i \(-0.388196\pi\)
0.344064 + 0.938946i \(0.388196\pi\)
\(318\) −6.55561e10 −0.359494
\(319\) 9.36051e9 0.0506106
\(320\) 1.19131e11 0.635113
\(321\) −1.53166e11 −0.805172
\(322\) 2.38999e11 1.23892
\(323\) −6.40634e10 −0.327490
\(324\) 2.75211e10 0.138744
\(325\) −6.87687e10 −0.341913
\(326\) −8.23109e10 −0.403625
\(327\) −4.99147e10 −0.241415
\(328\) 4.21098e10 0.200887
\(329\) −6.33955e11 −2.98316
\(330\) 1.19661e11 0.555443
\(331\) −3.51714e11 −1.61051 −0.805255 0.592929i \(-0.797972\pi\)
−0.805255 + 0.592929i \(0.797972\pi\)
\(332\) 2.44937e11 1.10645
\(333\) −2.40277e10 −0.107081
\(334\) 4.31033e11 1.89518
\(335\) 5.86625e9 0.0254483
\(336\) 1.44641e11 0.619106
\(337\) 4.10219e11 1.73253 0.866267 0.499582i \(-0.166513\pi\)
0.866267 + 0.499582i \(0.166513\pi\)
\(338\) −6.91802e11 −2.88308
\(339\) 2.72389e11 1.12019
\(340\) 1.96427e11 0.797160
\(341\) 1.80580e11 0.723229
\(342\) −2.90125e10 −0.114675
\(343\) 1.67051e11 0.651668
\(344\) −1.81902e10 −0.0700364
\(345\) −3.60913e10 −0.137156
\(346\) −6.72570e11 −2.52287
\(347\) 4.08881e11 1.51396 0.756979 0.653439i \(-0.226674\pi\)
0.756979 + 0.653439i \(0.226674\pi\)
\(348\) 6.95861e9 0.0254341
\(349\) −4.04337e11 −1.45891 −0.729457 0.684027i \(-0.760227\pi\)
−0.729457 + 0.684027i \(0.760227\pi\)
\(350\) −1.30954e11 −0.466458
\(351\) 9.35590e10 0.329006
\(352\) −5.81300e11 −2.01817
\(353\) −1.14276e11 −0.391714 −0.195857 0.980633i \(-0.562749\pi\)
−0.195857 + 0.980633i \(0.562749\pi\)
\(354\) −2.80770e11 −0.950245
\(355\) −1.09175e11 −0.364833
\(356\) −6.50649e11 −2.14695
\(357\) 3.93404e11 1.28183
\(358\) −4.31676e11 −1.38894
\(359\) 2.73301e11 0.868393 0.434197 0.900818i \(-0.357032\pi\)
0.434197 + 0.900818i \(0.357032\pi\)
\(360\) 1.77167e10 0.0555933
\(361\) 1.69836e10 0.0526316
\(362\) −2.08167e11 −0.637124
\(363\) −2.02067e11 −0.610823
\(364\) −1.11203e12 −3.32016
\(365\) −1.98727e11 −0.586057
\(366\) −3.43961e11 −1.00195
\(367\) 3.97112e11 1.14266 0.571328 0.820722i \(-0.306428\pi\)
0.571328 + 0.820722i \(0.306428\pi\)
\(368\) 1.28850e11 0.366244
\(369\) −6.39470e10 −0.179557
\(370\) −7.76644e10 −0.215434
\(371\) −2.35661e11 −0.645809
\(372\) 1.34244e11 0.363455
\(373\) −2.90603e11 −0.777338 −0.388669 0.921378i \(-0.627065\pi\)
−0.388669 + 0.921378i \(0.627065\pi\)
\(374\) −1.16194e12 −3.07086
\(375\) 1.97754e10 0.0516398
\(376\) 2.77225e11 0.715299
\(377\) 2.36561e10 0.0603124
\(378\) 1.78161e11 0.448849
\(379\) 3.42610e11 0.852949 0.426474 0.904500i \(-0.359755\pi\)
0.426474 + 0.904500i \(0.359755\pi\)
\(380\) −5.20739e10 −0.128113
\(381\) 2.04039e11 0.496078
\(382\) 1.19174e12 2.86349
\(383\) −1.05333e11 −0.250133 −0.125066 0.992148i \(-0.539914\pi\)
−0.125066 + 0.992148i \(0.539914\pi\)
\(384\) −1.77806e11 −0.417307
\(385\) 4.30156e11 0.997821
\(386\) −1.12545e12 −2.58038
\(387\) 2.76232e10 0.0626000
\(388\) −6.93643e11 −1.55379
\(389\) −7.35118e11 −1.62774 −0.813868 0.581050i \(-0.802642\pi\)
−0.813868 + 0.581050i \(0.802642\pi\)
\(390\) 3.02410e11 0.661919
\(391\) 3.50455e11 0.758293
\(392\) −2.47398e11 −0.529187
\(393\) 4.05045e11 0.856519
\(394\) −3.98331e11 −0.832743
\(395\) −3.01625e10 −0.0623420
\(396\) −2.92202e11 −0.597110
\(397\) −1.30600e11 −0.263868 −0.131934 0.991258i \(-0.542119\pi\)
−0.131934 + 0.991258i \(0.542119\pi\)
\(398\) 1.30338e12 2.60374
\(399\) −1.04294e11 −0.206006
\(400\) −7.06006e10 −0.137892
\(401\) 3.26468e11 0.630509 0.315254 0.949007i \(-0.397910\pi\)
0.315254 + 0.949007i \(0.397910\pi\)
\(402\) −2.57968e10 −0.0492662
\(403\) 4.56366e11 0.861869
\(404\) −1.62755e11 −0.303960
\(405\) −2.69042e10 −0.0496904
\(406\) 4.50475e10 0.0822818
\(407\) 2.55111e11 0.460845
\(408\) −1.72034e11 −0.307357
\(409\) 9.96939e11 1.76163 0.880813 0.473464i \(-0.156997\pi\)
0.880813 + 0.473464i \(0.156997\pi\)
\(410\) −2.06695e11 −0.361246
\(411\) 1.70332e11 0.294448
\(412\) 1.03272e9 0.00176581
\(413\) −1.00931e12 −1.70706
\(414\) 1.58711e11 0.265525
\(415\) −2.39447e11 −0.396272
\(416\) −1.46907e12 −2.40505
\(417\) 1.56323e11 0.253170
\(418\) 3.08036e11 0.493524
\(419\) 1.41597e11 0.224435 0.112217 0.993684i \(-0.464205\pi\)
0.112217 + 0.993684i \(0.464205\pi\)
\(420\) 3.19779e11 0.501450
\(421\) 7.96738e11 1.23608 0.618039 0.786147i \(-0.287927\pi\)
0.618039 + 0.786147i \(0.287927\pi\)
\(422\) −3.23441e11 −0.496465
\(423\) −4.20988e11 −0.639349
\(424\) 1.03053e11 0.154851
\(425\) −1.92024e11 −0.285499
\(426\) 4.80095e11 0.706291
\(427\) −1.23647e12 −1.79994
\(428\) 1.20893e12 1.74143
\(429\) −9.93352e11 −1.41594
\(430\) 8.92860e10 0.125943
\(431\) −4.11727e11 −0.574727 −0.287364 0.957822i \(-0.592779\pi\)
−0.287364 + 0.957822i \(0.592779\pi\)
\(432\) 9.60513e10 0.132686
\(433\) −5.25038e11 −0.717787 −0.358893 0.933379i \(-0.616846\pi\)
−0.358893 + 0.933379i \(0.616846\pi\)
\(434\) 8.69044e11 1.17581
\(435\) −6.80263e9 −0.00910910
\(436\) 3.93975e11 0.522131
\(437\) −9.29077e10 −0.121867
\(438\) 8.73902e11 1.13456
\(439\) 5.92349e11 0.761180 0.380590 0.924744i \(-0.375721\pi\)
0.380590 + 0.924744i \(0.375721\pi\)
\(440\) −1.88105e11 −0.239256
\(441\) 3.75693e11 0.472998
\(442\) −2.93647e12 −3.65953
\(443\) −3.46221e11 −0.427107 −0.213553 0.976931i \(-0.568504\pi\)
−0.213553 + 0.976931i \(0.568504\pi\)
\(444\) 1.89650e11 0.231595
\(445\) 6.36064e11 0.768919
\(446\) 1.17322e12 1.40401
\(447\) −5.99728e11 −0.710511
\(448\) −1.88324e12 −2.20879
\(449\) −1.90995e11 −0.221775 −0.110888 0.993833i \(-0.535369\pi\)
−0.110888 + 0.993833i \(0.535369\pi\)
\(450\) −8.69621e10 −0.0999709
\(451\) 6.78950e11 0.772758
\(452\) −2.14996e12 −2.42274
\(453\) −9.39485e10 −0.104821
\(454\) −7.12102e11 −0.786667
\(455\) 1.08710e12 1.18910
\(456\) 4.56071e10 0.0493959
\(457\) −8.99317e11 −0.964472 −0.482236 0.876041i \(-0.660175\pi\)
−0.482236 + 0.876041i \(0.660175\pi\)
\(458\) 1.85192e12 1.96665
\(459\) 2.61246e11 0.274722
\(460\) 2.84867e11 0.296642
\(461\) −2.78718e11 −0.287416 −0.143708 0.989620i \(-0.545903\pi\)
−0.143708 + 0.989620i \(0.545903\pi\)
\(462\) −1.89161e12 −1.93171
\(463\) 7.59495e11 0.768087 0.384044 0.923315i \(-0.374531\pi\)
0.384044 + 0.923315i \(0.374531\pi\)
\(464\) 2.42863e10 0.0243237
\(465\) −1.31235e11 −0.130170
\(466\) −8.26341e11 −0.811751
\(467\) 9.77081e11 0.950615 0.475307 0.879820i \(-0.342337\pi\)
0.475307 + 0.879820i \(0.342337\pi\)
\(468\) −7.38459e11 −0.711574
\(469\) −9.27341e10 −0.0885038
\(470\) −1.36075e12 −1.28629
\(471\) 5.98790e11 0.560636
\(472\) 4.41365e11 0.409317
\(473\) −2.93286e11 −0.269411
\(474\) 1.32639e11 0.120690
\(475\) 5.09066e10 0.0458831
\(476\) −3.10513e12 −2.77235
\(477\) −1.56494e11 −0.138409
\(478\) 3.17092e12 2.77817
\(479\) −1.93121e12 −1.67617 −0.838086 0.545538i \(-0.816325\pi\)
−0.838086 + 0.545538i \(0.816325\pi\)
\(480\) 4.22453e11 0.363239
\(481\) 6.44722e11 0.549187
\(482\) 2.64500e12 2.23210
\(483\) 5.70533e11 0.477000
\(484\) 1.59491e12 1.32109
\(485\) 6.78095e11 0.556484
\(486\) 1.18311e11 0.0961971
\(487\) 1.99512e12 1.60727 0.803634 0.595124i \(-0.202897\pi\)
0.803634 + 0.595124i \(0.202897\pi\)
\(488\) 5.40701e11 0.431587
\(489\) −1.96491e11 −0.155401
\(490\) 1.21435e12 0.951614
\(491\) 1.79495e12 1.39375 0.696876 0.717192i \(-0.254573\pi\)
0.696876 + 0.717192i \(0.254573\pi\)
\(492\) 5.04732e11 0.388345
\(493\) 6.60553e10 0.0503612
\(494\) 7.78476e11 0.588131
\(495\) 2.85652e11 0.213852
\(496\) 4.68524e11 0.347587
\(497\) 1.72584e12 1.26881
\(498\) 1.05297e12 0.767154
\(499\) −5.00717e11 −0.361526 −0.180763 0.983527i \(-0.557857\pi\)
−0.180763 + 0.983527i \(0.557857\pi\)
\(500\) −1.56087e11 −0.111686
\(501\) 1.02895e12 0.729669
\(502\) 2.91848e12 2.05111
\(503\) −2.44897e12 −1.70580 −0.852900 0.522074i \(-0.825158\pi\)
−0.852900 + 0.522074i \(0.825158\pi\)
\(504\) −2.80067e11 −0.193341
\(505\) 1.59106e11 0.108862
\(506\) −1.68510e12 −1.14274
\(507\) −1.65146e12 −1.11002
\(508\) −1.61047e12 −1.07292
\(509\) 1.39357e12 0.920235 0.460117 0.887858i \(-0.347807\pi\)
0.460117 + 0.887858i \(0.347807\pi\)
\(510\) 8.44424e11 0.552706
\(511\) 3.14150e12 2.03818
\(512\) −1.90802e12 −1.22706
\(513\) −6.92579e10 −0.0441511
\(514\) −1.13685e12 −0.718405
\(515\) −1.00957e9 −0.000632417 0
\(516\) −2.18029e11 −0.135391
\(517\) 4.46979e12 2.75156
\(518\) 1.22772e12 0.749233
\(519\) −1.60555e12 −0.971336
\(520\) −4.75383e11 −0.285121
\(521\) 1.18713e12 0.705875 0.352937 0.935647i \(-0.385183\pi\)
0.352937 + 0.935647i \(0.385183\pi\)
\(522\) 2.99145e10 0.0176346
\(523\) 7.35893e11 0.430088 0.215044 0.976604i \(-0.431011\pi\)
0.215044 + 0.976604i \(0.431011\pi\)
\(524\) −3.19701e12 −1.85248
\(525\) −3.12611e11 −0.179592
\(526\) 4.25154e12 2.42164
\(527\) 1.27432e12 0.719665
\(528\) −1.01981e12 −0.571042
\(529\) −1.29291e12 −0.717822
\(530\) −5.05834e11 −0.278463
\(531\) −6.70247e11 −0.365856
\(532\) 8.23187e11 0.445550
\(533\) 1.71586e12 0.920892
\(534\) −2.79709e12 −1.48857
\(535\) −1.18183e12 −0.623683
\(536\) 4.05522e10 0.0212213
\(537\) −1.03049e12 −0.534759
\(538\) 1.08084e12 0.556213
\(539\) −3.98888e12 −2.03564
\(540\) 2.12354e11 0.107470
\(541\) 3.42601e12 1.71950 0.859748 0.510719i \(-0.170621\pi\)
0.859748 + 0.510719i \(0.170621\pi\)
\(542\) 1.29004e12 0.642103
\(543\) −4.96932e11 −0.245300
\(544\) −4.10212e12 −2.00823
\(545\) −3.85144e11 −0.186999
\(546\) −4.78051e12 −2.30201
\(547\) −2.19990e11 −0.105065 −0.0525327 0.998619i \(-0.516729\pi\)
−0.0525327 + 0.998619i \(0.516729\pi\)
\(548\) −1.34443e12 −0.636831
\(549\) −8.21096e11 −0.385761
\(550\) 9.23310e11 0.430244
\(551\) −1.75116e10 −0.00809365
\(552\) −2.49491e11 −0.114375
\(553\) 4.76811e11 0.216812
\(554\) −3.58471e12 −1.61681
\(555\) −1.85399e11 −0.0829447
\(556\) −1.23386e12 −0.547556
\(557\) −4.44924e11 −0.195856 −0.0979281 0.995193i \(-0.531222\pi\)
−0.0979281 + 0.995193i \(0.531222\pi\)
\(558\) 5.77103e11 0.251999
\(559\) −7.41198e11 −0.321056
\(560\) 1.11606e12 0.479558
\(561\) −2.77375e12 −1.18232
\(562\) 1.46350e12 0.618843
\(563\) −2.85252e12 −1.19658 −0.598289 0.801281i \(-0.704152\pi\)
−0.598289 + 0.801281i \(0.704152\pi\)
\(564\) 3.32285e12 1.38278
\(565\) 2.10176e12 0.867693
\(566\) −4.72313e12 −1.93444
\(567\) 4.25303e11 0.172812
\(568\) −7.54702e11 −0.304234
\(569\) 3.03910e12 1.21546 0.607729 0.794144i \(-0.292081\pi\)
0.607729 + 0.794144i \(0.292081\pi\)
\(570\) −2.23862e11 −0.0888265
\(571\) 8.39653e11 0.330550 0.165275 0.986248i \(-0.447149\pi\)
0.165275 + 0.986248i \(0.447149\pi\)
\(572\) 7.84050e12 3.06240
\(573\) 2.84489e12 1.10248
\(574\) 3.26745e12 1.25634
\(575\) −2.78482e11 −0.106241
\(576\) −1.25059e12 −0.473385
\(577\) −2.51851e12 −0.945914 −0.472957 0.881085i \(-0.656813\pi\)
−0.472957 + 0.881085i \(0.656813\pi\)
\(578\) −4.17572e12 −1.55617
\(579\) −2.68666e12 −0.993478
\(580\) 5.36930e10 0.0197012
\(581\) 3.78519e12 1.37815
\(582\) −2.98192e12 −1.07731
\(583\) 1.66156e12 0.595672
\(584\) −1.37376e12 −0.488712
\(585\) 7.21906e11 0.254847
\(586\) −4.02578e12 −1.41030
\(587\) −7.40507e11 −0.257429 −0.128715 0.991682i \(-0.541085\pi\)
−0.128715 + 0.991682i \(0.541085\pi\)
\(588\) −2.96534e12 −1.02300
\(589\) −3.37829e11 −0.115659
\(590\) −2.16643e12 −0.736056
\(591\) −9.50887e11 −0.320616
\(592\) 6.61897e11 0.221484
\(593\) 1.87921e12 0.624066 0.312033 0.950071i \(-0.398990\pi\)
0.312033 + 0.950071i \(0.398990\pi\)
\(594\) −1.25615e12 −0.414003
\(595\) 3.03553e12 0.992905
\(596\) 4.73364e12 1.53669
\(597\) 3.11141e12 1.00247
\(598\) −4.25861e12 −1.36180
\(599\) 5.78020e11 0.183452 0.0917259 0.995784i \(-0.470762\pi\)
0.0917259 + 0.995784i \(0.470762\pi\)
\(600\) 1.36703e11 0.0430623
\(601\) −3.96250e12 −1.23889 −0.619447 0.785038i \(-0.712643\pi\)
−0.619447 + 0.785038i \(0.712643\pi\)
\(602\) −1.41144e12 −0.438004
\(603\) −6.15816e10 −0.0189681
\(604\) 7.41533e11 0.226707
\(605\) −1.55916e12 −0.473142
\(606\) −6.99669e11 −0.210749
\(607\) −3.06642e12 −0.916817 −0.458408 0.888742i \(-0.651580\pi\)
−0.458408 + 0.888742i \(0.651580\pi\)
\(608\) 1.08750e12 0.322746
\(609\) 1.07536e11 0.0316795
\(610\) −2.65402e12 −0.776104
\(611\) 1.12961e13 3.27903
\(612\) −2.06201e12 −0.594168
\(613\) −4.26173e12 −1.21903 −0.609514 0.792775i \(-0.708636\pi\)
−0.609514 + 0.792775i \(0.708636\pi\)
\(614\) −5.24469e12 −1.48923
\(615\) −4.93419e11 −0.139084
\(616\) 2.97358e12 0.832082
\(617\) 6.21898e12 1.72757 0.863785 0.503860i \(-0.168087\pi\)
0.863785 + 0.503860i \(0.168087\pi\)
\(618\) 4.43957e9 0.00122431
\(619\) 1.11768e12 0.305992 0.152996 0.988227i \(-0.451108\pi\)
0.152996 + 0.988227i \(0.451108\pi\)
\(620\) 1.03583e12 0.281531
\(621\) 3.78872e11 0.102230
\(622\) 1.69879e12 0.455076
\(623\) −1.00549e13 −2.67414
\(624\) −2.57729e12 −0.680508
\(625\) 1.52588e11 0.0400000
\(626\) −5.54768e12 −1.44387
\(627\) 7.35338e11 0.190013
\(628\) −4.72624e12 −1.21254
\(629\) 1.80027e12 0.458574
\(630\) 1.37470e12 0.347677
\(631\) −2.69935e12 −0.677839 −0.338920 0.940815i \(-0.610061\pi\)
−0.338920 + 0.940815i \(0.610061\pi\)
\(632\) −2.08507e11 −0.0519869
\(633\) −7.72111e11 −0.191145
\(634\) −4.19794e12 −1.03189
\(635\) 1.57437e12 0.384260
\(636\) 1.23520e12 0.299352
\(637\) −1.00808e13 −2.42586
\(638\) −3.17614e11 −0.0758938
\(639\) 1.14607e12 0.271931
\(640\) −1.37196e12 −0.323244
\(641\) −1.19746e12 −0.280157 −0.140078 0.990140i \(-0.544735\pi\)
−0.140078 + 0.990140i \(0.544735\pi\)
\(642\) 5.19710e12 1.20741
\(643\) −7.98005e11 −0.184101 −0.0920505 0.995754i \(-0.529342\pi\)
−0.0920505 + 0.995754i \(0.529342\pi\)
\(644\) −4.50320e12 −1.03166
\(645\) 2.13142e11 0.0484897
\(646\) 2.17375e12 0.491093
\(647\) 2.32481e12 0.521577 0.260788 0.965396i \(-0.416017\pi\)
0.260788 + 0.965396i \(0.416017\pi\)
\(648\) −1.85983e11 −0.0414368
\(649\) 7.11627e12 1.57453
\(650\) 2.33341e12 0.512720
\(651\) 2.07456e12 0.452702
\(652\) 1.55090e12 0.336100
\(653\) −2.42366e12 −0.521630 −0.260815 0.965389i \(-0.583991\pi\)
−0.260815 + 0.965389i \(0.583991\pi\)
\(654\) 1.69367e12 0.362017
\(655\) 3.12535e12 0.663457
\(656\) 1.76157e12 0.371391
\(657\) 2.08616e12 0.436821
\(658\) 2.15109e13 4.47344
\(659\) 2.04543e12 0.422474 0.211237 0.977435i \(-0.432251\pi\)
0.211237 + 0.977435i \(0.432251\pi\)
\(660\) −2.25464e12 −0.462520
\(661\) −1.14535e12 −0.233363 −0.116682 0.993169i \(-0.537226\pi\)
−0.116682 + 0.993169i \(0.537226\pi\)
\(662\) 1.19341e13 2.41506
\(663\) −7.00989e12 −1.40896
\(664\) −1.65525e12 −0.330450
\(665\) −8.04735e11 −0.159572
\(666\) 8.15290e11 0.160575
\(667\) 9.57964e10 0.0187406
\(668\) −8.12149e12 −1.57813
\(669\) 2.80067e12 0.540561
\(670\) −1.99049e11 −0.0381614
\(671\) 8.71789e12 1.66020
\(672\) −6.67816e12 −1.26327
\(673\) 4.53266e12 0.851697 0.425848 0.904795i \(-0.359976\pi\)
0.425848 + 0.904795i \(0.359976\pi\)
\(674\) −1.39193e13 −2.59805
\(675\) −2.07594e11 −0.0384900
\(676\) 1.30349e13 2.40075
\(677\) 3.07988e12 0.563489 0.281744 0.959489i \(-0.409087\pi\)
0.281744 + 0.959489i \(0.409087\pi\)
\(678\) −9.24249e12 −1.67979
\(679\) −1.07194e13 −1.93533
\(680\) −1.32742e12 −0.238078
\(681\) −1.69992e12 −0.302876
\(682\) −6.12732e12 −1.08453
\(683\) 8.81574e12 1.55012 0.775060 0.631887i \(-0.217719\pi\)
0.775060 + 0.631887i \(0.217719\pi\)
\(684\) 5.46651e11 0.0954899
\(685\) 1.31429e12 0.228078
\(686\) −5.66827e12 −0.977219
\(687\) 4.42086e12 0.757184
\(688\) −7.60942e11 −0.129480
\(689\) 4.19912e12 0.709859
\(690\) 1.22462e12 0.205675
\(691\) −1.13198e12 −0.188880 −0.0944401 0.995531i \(-0.530106\pi\)
−0.0944401 + 0.995531i \(0.530106\pi\)
\(692\) 1.26725e13 2.10081
\(693\) −4.51561e12 −0.743732
\(694\) −1.38738e13 −2.27028
\(695\) 1.20620e12 0.196104
\(696\) −4.70252e10 −0.00759607
\(697\) 4.79121e12 0.768950
\(698\) 1.37197e13 2.18773
\(699\) −1.97262e12 −0.312534
\(700\) 2.46743e12 0.388421
\(701\) 9.40714e12 1.47138 0.735692 0.677316i \(-0.236857\pi\)
0.735692 + 0.677316i \(0.236857\pi\)
\(702\) −3.17458e12 −0.493365
\(703\) −4.77262e11 −0.0736983
\(704\) 1.32780e13 2.03731
\(705\) −3.24836e12 −0.495238
\(706\) 3.87753e12 0.587400
\(707\) −2.51516e12 −0.378598
\(708\) 5.29024e12 0.791272
\(709\) −6.66779e12 −0.991001 −0.495500 0.868608i \(-0.665015\pi\)
−0.495500 + 0.868608i \(0.665015\pi\)
\(710\) 3.70444e12 0.547091
\(711\) 3.16634e11 0.0464670
\(712\) 4.39698e12 0.641201
\(713\) 1.84808e12 0.267804
\(714\) −1.33487e13 −1.92219
\(715\) −7.66475e12 −1.09678
\(716\) 8.13361e12 1.15658
\(717\) 7.56955e12 1.06963
\(718\) −9.27346e12 −1.30221
\(719\) 2.41689e12 0.337269 0.168635 0.985679i \(-0.446064\pi\)
0.168635 + 0.985679i \(0.446064\pi\)
\(720\) 7.41137e11 0.102778
\(721\) 1.59593e10 0.00219941
\(722\) −5.76274e11 −0.0789244
\(723\) 6.31408e12 0.859385
\(724\) 3.92227e12 0.530535
\(725\) −5.24895e10 −0.00705588
\(726\) 6.85639e12 0.915968
\(727\) −7.53146e12 −0.999941 −0.499970 0.866043i \(-0.666656\pi\)
−0.499970 + 0.866043i \(0.666656\pi\)
\(728\) 7.51489e12 0.991588
\(729\) 2.82430e11 0.0370370
\(730\) 6.74307e12 0.878830
\(731\) −2.06966e12 −0.268084
\(732\) 6.48089e12 0.834324
\(733\) −2.65885e11 −0.0340194 −0.0170097 0.999855i \(-0.505415\pi\)
−0.0170097 + 0.999855i \(0.505415\pi\)
\(734\) −1.34745e13 −1.71349
\(735\) 2.89887e12 0.366383
\(736\) −5.94909e12 −0.747309
\(737\) 6.53835e11 0.0816328
\(738\) 2.16980e12 0.269257
\(739\) −7.54408e12 −0.930478 −0.465239 0.885185i \(-0.654032\pi\)
−0.465239 + 0.885185i \(0.654032\pi\)
\(740\) 1.46335e12 0.179393
\(741\) 1.85836e12 0.226437
\(742\) 7.99626e12 0.968432
\(743\) −9.27540e12 −1.11656 −0.558281 0.829652i \(-0.688539\pi\)
−0.558281 + 0.829652i \(0.688539\pi\)
\(744\) −9.07197e11 −0.108548
\(745\) −4.62753e12 −0.550359
\(746\) 9.86051e12 1.16567
\(747\) 2.51362e12 0.295363
\(748\) 2.18931e13 2.55712
\(749\) 1.86825e13 2.16904
\(750\) −6.71004e11 −0.0774372
\(751\) −3.97286e12 −0.455747 −0.227874 0.973691i \(-0.573177\pi\)
−0.227874 + 0.973691i \(0.573177\pi\)
\(752\) 1.15971e13 1.32242
\(753\) 6.96693e12 0.789703
\(754\) −8.02681e11 −0.0904423
\(755\) −7.24912e11 −0.0811940
\(756\) −3.35691e12 −0.373759
\(757\) −3.95644e12 −0.437899 −0.218949 0.975736i \(-0.570263\pi\)
−0.218949 + 0.975736i \(0.570263\pi\)
\(758\) −1.16252e13 −1.27905
\(759\) −4.02262e12 −0.439968
\(760\) 3.51907e11 0.0382619
\(761\) 4.02381e12 0.434917 0.217458 0.976070i \(-0.430223\pi\)
0.217458 + 0.976070i \(0.430223\pi\)
\(762\) −6.92329e12 −0.743901
\(763\) 6.08838e12 0.650342
\(764\) −2.24547e13 −2.38444
\(765\) 2.01579e12 0.212799
\(766\) 3.57409e12 0.375090
\(767\) 1.79844e13 1.87636
\(768\) −1.87181e12 −0.194149
\(769\) 1.47012e13 1.51595 0.757976 0.652283i \(-0.226188\pi\)
0.757976 + 0.652283i \(0.226188\pi\)
\(770\) −1.45957e13 −1.49630
\(771\) −2.71387e12 −0.276595
\(772\) 2.12057e13 2.14869
\(773\) 5.27223e12 0.531113 0.265556 0.964095i \(-0.414444\pi\)
0.265556 + 0.964095i \(0.414444\pi\)
\(774\) −9.37289e11 −0.0938727
\(775\) −1.01261e12 −0.100829
\(776\) 4.68753e12 0.464052
\(777\) 2.93080e12 0.288464
\(778\) 2.49435e13 2.44090
\(779\) −1.27018e12 −0.123579
\(780\) −5.69798e12 −0.551183
\(781\) −1.21683e13 −1.17031
\(782\) −1.18914e13 −1.13711
\(783\) 7.14113e10 0.00678952
\(784\) −1.03493e13 −0.978339
\(785\) 4.62030e12 0.434267
\(786\) −1.37437e13 −1.28441
\(787\) 1.14758e13 1.06634 0.533169 0.846009i \(-0.321001\pi\)
0.533169 + 0.846009i \(0.321001\pi\)
\(788\) 7.50533e12 0.693428
\(789\) 1.01492e13 0.932362
\(790\) 1.02345e12 0.0934858
\(791\) −3.32248e13 −3.01765
\(792\) 1.97465e12 0.178331
\(793\) 2.20320e13 1.97845
\(794\) 4.43144e12 0.395688
\(795\) −1.20752e12 −0.107211
\(796\) −2.45582e13 −2.16815
\(797\) −8.04453e12 −0.706217 −0.353108 0.935582i \(-0.614875\pi\)
−0.353108 + 0.935582i \(0.614875\pi\)
\(798\) 3.53882e12 0.308919
\(799\) 3.15424e13 2.73801
\(800\) 3.25967e12 0.281364
\(801\) −6.67715e12 −0.573119
\(802\) −1.10775e13 −0.945489
\(803\) −2.21496e13 −1.87995
\(804\) 4.86062e11 0.0410241
\(805\) 4.40226e12 0.369483
\(806\) −1.54851e13 −1.29243
\(807\) 2.58016e12 0.214149
\(808\) 1.09987e12 0.0907799
\(809\) −1.46887e13 −1.20563 −0.602814 0.797882i \(-0.705954\pi\)
−0.602814 + 0.797882i \(0.705954\pi\)
\(810\) 9.12894e11 0.0745139
\(811\) 5.18047e12 0.420509 0.210255 0.977647i \(-0.432571\pi\)
0.210255 + 0.977647i \(0.432571\pi\)
\(812\) −8.48782e11 −0.0685164
\(813\) 3.07955e12 0.247218
\(814\) −8.65624e12 −0.691066
\(815\) −1.51613e12 −0.120373
\(816\) −7.19662e12 −0.568228
\(817\) 5.48678e11 0.0430842
\(818\) −3.38274e13 −2.64167
\(819\) −1.14119e13 −0.886302
\(820\) 3.89454e12 0.300811
\(821\) 2.10676e13 1.61834 0.809172 0.587572i \(-0.199916\pi\)
0.809172 + 0.587572i \(0.199916\pi\)
\(822\) −5.77958e12 −0.441543
\(823\) −1.23011e12 −0.0934644 −0.0467322 0.998907i \(-0.514881\pi\)
−0.0467322 + 0.998907i \(0.514881\pi\)
\(824\) −6.97893e9 −0.000527371 0
\(825\) 2.20411e12 0.165649
\(826\) 3.42471e13 2.55984
\(827\) −1.85555e12 −0.137942 −0.0689712 0.997619i \(-0.521972\pi\)
−0.0689712 + 0.997619i \(0.521972\pi\)
\(828\) −2.99042e12 −0.221104
\(829\) 9.09884e12 0.669099 0.334550 0.942378i \(-0.391416\pi\)
0.334550 + 0.942378i \(0.391416\pi\)
\(830\) 8.12474e12 0.594235
\(831\) −8.55734e12 −0.622493
\(832\) 3.35565e13 2.42785
\(833\) −2.81487e13 −2.02561
\(834\) −5.30425e12 −0.379644
\(835\) 7.93945e12 0.565199
\(836\) −5.80400e12 −0.410960
\(837\) 1.37765e12 0.0970228
\(838\) −4.80455e12 −0.336554
\(839\) 1.12716e13 0.785341 0.392670 0.919679i \(-0.371551\pi\)
0.392670 + 0.919679i \(0.371551\pi\)
\(840\) −2.16101e12 −0.149762
\(841\) −1.44891e13 −0.998755
\(842\) −2.70343e13 −1.85358
\(843\) 3.49364e12 0.238262
\(844\) 6.09425e12 0.413409
\(845\) −1.27427e13 −0.859818
\(846\) 1.42847e13 0.958745
\(847\) 2.46473e13 1.64548
\(848\) 4.31098e12 0.286283
\(849\) −1.12750e13 −0.744784
\(850\) 6.51561e12 0.428125
\(851\) 2.61083e12 0.170646
\(852\) −9.04592e12 −0.588131
\(853\) 9.70339e12 0.627556 0.313778 0.949496i \(-0.398405\pi\)
0.313778 + 0.949496i \(0.398405\pi\)
\(854\) 4.19549e13 2.69912
\(855\) −5.34398e11 −0.0341993
\(856\) −8.16976e12 −0.520089
\(857\) 2.75330e13 1.74357 0.871787 0.489885i \(-0.162961\pi\)
0.871787 + 0.489885i \(0.162961\pi\)
\(858\) 3.37057e13 2.12329
\(859\) −1.21276e13 −0.759989 −0.379994 0.924989i \(-0.624074\pi\)
−0.379994 + 0.924989i \(0.624074\pi\)
\(860\) −1.68232e12 −0.104874
\(861\) 7.79999e12 0.483704
\(862\) 1.39704e13 0.861841
\(863\) 1.58553e13 0.973032 0.486516 0.873672i \(-0.338268\pi\)
0.486516 + 0.873672i \(0.338268\pi\)
\(864\) −4.43474e12 −0.270742
\(865\) −1.23885e13 −0.752394
\(866\) 1.78152e13 1.07637
\(867\) −9.96820e12 −0.599143
\(868\) −1.63745e13 −0.979104
\(869\) −3.36182e12 −0.199980
\(870\) 2.30822e11 0.0136597
\(871\) 1.65239e12 0.0972814
\(872\) −2.66242e12 −0.155938
\(873\) −7.11837e12 −0.414779
\(874\) 3.15247e12 0.182747
\(875\) −2.41212e12 −0.139111
\(876\) −1.64660e13 −0.944757
\(877\) −2.86146e13 −1.63339 −0.816695 0.577069i \(-0.804196\pi\)
−0.816695 + 0.577069i \(0.804196\pi\)
\(878\) −2.00992e13 −1.14144
\(879\) −9.61027e12 −0.542982
\(880\) −7.86893e12 −0.442327
\(881\) −1.23867e13 −0.692729 −0.346365 0.938100i \(-0.612584\pi\)
−0.346365 + 0.938100i \(0.612584\pi\)
\(882\) −1.27477e13 −0.709292
\(883\) −3.23596e12 −0.179135 −0.0895673 0.995981i \(-0.528548\pi\)
−0.0895673 + 0.995981i \(0.528548\pi\)
\(884\) 5.53288e13 3.04731
\(885\) −5.17166e12 −0.283391
\(886\) 1.17477e13 0.640474
\(887\) −2.35148e11 −0.0127551 −0.00637757 0.999980i \(-0.502030\pi\)
−0.00637757 + 0.999980i \(0.502030\pi\)
\(888\) −1.28162e12 −0.0691675
\(889\) −2.48878e13 −1.33637
\(890\) −2.15825e13 −1.15304
\(891\) −2.99866e12 −0.159396
\(892\) −2.21056e13 −1.16913
\(893\) −8.36208e12 −0.440030
\(894\) 2.03495e13 1.06546
\(895\) −7.95129e12 −0.414223
\(896\) 2.16880e13 1.12417
\(897\) −1.01661e13 −0.524308
\(898\) 6.48070e12 0.332566
\(899\) 3.48333e11 0.0177859
\(900\) 1.63853e12 0.0832462
\(901\) 1.17253e13 0.592737
\(902\) −2.30376e13 −1.15880
\(903\) −3.36936e12 −0.168637
\(904\) 1.45291e13 0.723568
\(905\) −3.83435e12 −0.190009
\(906\) 3.18779e12 0.157186
\(907\) 3.22682e13 1.58322 0.791611 0.611026i \(-0.209243\pi\)
0.791611 + 0.611026i \(0.209243\pi\)
\(908\) 1.34174e13 0.655061
\(909\) −1.67023e12 −0.0811409
\(910\) −3.68867e13 −1.78313
\(911\) 2.50945e13 1.20711 0.603553 0.797323i \(-0.293751\pi\)
0.603553 + 0.797323i \(0.293751\pi\)
\(912\) 1.90787e12 0.0913211
\(913\) −2.66880e13 −1.27115
\(914\) 3.05149e13 1.44629
\(915\) −6.33562e12 −0.298809
\(916\) −3.48937e13 −1.63764
\(917\) −4.94057e13 −2.30736
\(918\) −8.86442e12 −0.411963
\(919\) 1.22135e13 0.564835 0.282418 0.959292i \(-0.408864\pi\)
0.282418 + 0.959292i \(0.408864\pi\)
\(920\) −1.92509e12 −0.0885942
\(921\) −1.25200e13 −0.573372
\(922\) 9.45727e12 0.430999
\(923\) −3.07519e13 −1.39465
\(924\) 3.56416e13 1.60854
\(925\) −1.43055e12 −0.0642487
\(926\) −2.57706e13 −1.15180
\(927\) 1.05980e10 0.000471376 0
\(928\) −1.12131e12 −0.0496317
\(929\) −2.81613e13 −1.24046 −0.620228 0.784422i \(-0.712960\pi\)
−0.620228 + 0.784422i \(0.712960\pi\)
\(930\) 4.45295e12 0.195198
\(931\) 7.46239e12 0.325540
\(932\) 1.55699e13 0.675948
\(933\) 4.05532e12 0.175210
\(934\) −3.31536e13 −1.42551
\(935\) −2.14024e13 −0.915820
\(936\) 4.99038e12 0.212516
\(937\) 4.52888e12 0.191939 0.0959695 0.995384i \(-0.469405\pi\)
0.0959695 + 0.995384i \(0.469405\pi\)
\(938\) 3.14659e12 0.132717
\(939\) −1.32433e13 −0.555906
\(940\) 2.56393e13 1.07110
\(941\) 3.51517e13 1.46148 0.730741 0.682655i \(-0.239175\pi\)
0.730741 + 0.682655i \(0.239175\pi\)
\(942\) −2.03177e13 −0.840709
\(943\) 6.94844e12 0.286144
\(944\) 1.84635e13 0.756727
\(945\) 3.28166e12 0.133860
\(946\) 9.95155e12 0.403999
\(947\) −1.47283e13 −0.595083 −0.297541 0.954709i \(-0.596167\pi\)
−0.297541 + 0.954709i \(0.596167\pi\)
\(948\) −2.49918e12 −0.100499
\(949\) −5.59768e13 −2.24032
\(950\) −1.72733e12 −0.0688047
\(951\) −1.00212e13 −0.397291
\(952\) 2.09839e13 0.827982
\(953\) 7.97246e12 0.313094 0.156547 0.987671i \(-0.449964\pi\)
0.156547 + 0.987671i \(0.449964\pi\)
\(954\) 5.31004e12 0.207554
\(955\) 2.19513e13 0.853976
\(956\) −5.97462e13 −2.31340
\(957\) −7.58201e11 −0.0292200
\(958\) 6.55282e13 2.51353
\(959\) −2.07764e13 −0.793207
\(960\) −9.64964e12 −0.366683
\(961\) −1.97197e13 −0.745838
\(962\) −2.18762e13 −0.823541
\(963\) 1.24064e13 0.464866
\(964\) −4.98369e13 −1.85868
\(965\) −2.07304e13 −0.769545
\(966\) −1.93589e13 −0.715293
\(967\) −4.87770e13 −1.79389 −0.896945 0.442143i \(-0.854218\pi\)
−0.896945 + 0.442143i \(0.854218\pi\)
\(968\) −1.07781e13 −0.394552
\(969\) 5.18913e12 0.189077
\(970\) −2.30086e13 −0.834484
\(971\) −4.14096e13 −1.49491 −0.747454 0.664314i \(-0.768724\pi\)
−0.747454 + 0.664314i \(0.768724\pi\)
\(972\) −2.22921e12 −0.0801037
\(973\) −1.90677e13 −0.682009
\(974\) −6.76969e13 −2.41020
\(975\) 5.57026e12 0.197404
\(976\) 2.26189e13 0.797899
\(977\) −4.15877e13 −1.46029 −0.730145 0.683292i \(-0.760547\pi\)
−0.730145 + 0.683292i \(0.760547\pi\)
\(978\) 6.66718e12 0.233033
\(979\) 7.08938e13 2.46653
\(980\) −2.28807e13 −0.792413
\(981\) 4.04309e12 0.139381
\(982\) −6.09049e13 −2.09002
\(983\) 3.82096e13 1.30522 0.652608 0.757696i \(-0.273675\pi\)
0.652608 + 0.757696i \(0.273675\pi\)
\(984\) −3.41090e12 −0.115982
\(985\) −7.33709e12 −0.248348
\(986\) −2.24134e12 −0.0755199
\(987\) 5.13504e13 1.72233
\(988\) −1.46680e13 −0.489739
\(989\) −3.00151e12 −0.0997602
\(990\) −9.69254e12 −0.320685
\(991\) 3.49657e13 1.15163 0.575813 0.817582i \(-0.304686\pi\)
0.575813 + 0.817582i \(0.304686\pi\)
\(992\) −2.16320e13 −0.709241
\(993\) 2.84888e13 0.929828
\(994\) −5.85600e13 −1.90266
\(995\) 2.40078e13 0.776512
\(996\) −1.98399e13 −0.638812
\(997\) −2.24998e13 −0.721193 −0.360596 0.932722i \(-0.617427\pi\)
−0.360596 + 0.932722i \(0.617427\pi\)
\(998\) 1.69900e13 0.542132
\(999\) 1.94624e12 0.0618233
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 285.10.a.f.1.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.10.a.f.1.3 14 1.1 even 1 trivial