Properties

Label 25.10.a.b.1.2
Level $25$
Weight $10$
Character 25.1
Self dual yes
Analytic conductor $12.876$
Analytic rank $1$
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] = [25,10,Mod(1,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, 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("25.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 25.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: \(12.8758959041\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1009}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 252 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-15.3824\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+36.7648 q^{2} -193.530 q^{3} +839.648 q^{4} -7115.07 q^{6} -7647.66 q^{7} +12045.9 q^{8} +17770.7 q^{9} +O(q^{10})\) \(q+36.7648 q^{2} -193.530 q^{3} +839.648 q^{4} -7115.07 q^{6} -7647.66 q^{7} +12045.9 q^{8} +17770.7 q^{9} -48361.0 q^{11} -162497. q^{12} -100456. q^{13} -281164. q^{14} +12964.5 q^{16} +201958. q^{17} +653335. q^{18} -58048.1 q^{19} +1.48005e6 q^{21} -1.77798e6 q^{22} +1.14078e6 q^{23} -2.33123e6 q^{24} -3.69324e6 q^{26} +370091. q^{27} -6.42134e6 q^{28} +1.56407e6 q^{29} -4.10744e6 q^{31} -5.69086e6 q^{32} +9.35929e6 q^{33} +7.42493e6 q^{34} +1.49211e7 q^{36} +1.70821e7 q^{37} -2.13412e6 q^{38} +1.94412e7 q^{39} -8.52969e6 q^{41} +5.44136e7 q^{42} -2.56254e7 q^{43} -4.06062e7 q^{44} +4.19406e7 q^{46} -4.58297e7 q^{47} -2.50902e6 q^{48} +1.81331e7 q^{49} -3.90848e7 q^{51} -8.43476e7 q^{52} +5.56483e7 q^{53} +1.36063e7 q^{54} -9.21228e7 q^{56} +1.12340e7 q^{57} +5.75026e7 q^{58} +2.15699e7 q^{59} -1.15309e8 q^{61} -1.51009e8 q^{62} -1.35904e8 q^{63} -2.15861e8 q^{64} +3.44092e8 q^{66} +7.69254e7 q^{67} +1.69573e8 q^{68} -2.20775e8 q^{69} +1.95870e8 q^{71} +2.14064e8 q^{72} -3.40942e8 q^{73} +6.28021e8 q^{74} -4.87399e7 q^{76} +3.69849e8 q^{77} +7.14751e8 q^{78} -5.92965e8 q^{79} -4.21404e8 q^{81} -3.13592e8 q^{82} -7.88477e8 q^{83} +1.24272e9 q^{84} -9.42111e8 q^{86} -3.02693e8 q^{87} -5.82552e8 q^{88} +8.40110e8 q^{89} +7.68253e8 q^{91} +9.57856e8 q^{92} +7.94910e8 q^{93} -1.68492e9 q^{94} +1.10135e9 q^{96} -2.35903e8 q^{97} +6.66658e8 q^{98} -8.59408e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{2} - 260 q^{3} + 1044 q^{4} - 5336 q^{6} - 1700 q^{7} + 20280 q^{8} + 2506 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{2} - 260 q^{3} + 1044 q^{4} - 5336 q^{6} - 1700 q^{7} + 20280 q^{8} + 2506 q^{9} + 23984 q^{11} - 176080 q^{12} - 115020 q^{13} - 440352 q^{14} - 312048 q^{16} - 412820 q^{17} + 1061890 q^{18} - 296520 q^{19} + 1084704 q^{21} - 3714280 q^{22} + 1049220 q^{23} - 2878560 q^{24} - 3303436 q^{26} + 2693080 q^{27} - 5205920 q^{28} - 3666980 q^{29} + 1613144 q^{31} - 1207840 q^{32} + 4550480 q^{33} + 23879308 q^{34} + 11801732 q^{36} + 21121940 q^{37} + 4248520 q^{38} + 20409272 q^{39} - 26957276 q^{41} + 64994880 q^{42} - 52889700 q^{43} - 25822352 q^{44} + 44391264 q^{46} - 58412180 q^{47} + 19094720 q^{48} + 13154114 q^{49} + 1779784 q^{51} - 87323800 q^{52} + 39035140 q^{53} - 48567920 q^{54} - 43149120 q^{56} + 27085360 q^{57} + 197510380 q^{58} - 54995560 q^{59} - 274579716 q^{61} - 304118880 q^{62} - 226693140 q^{63} - 169441216 q^{64} + 472798688 q^{66} + 318580 q^{67} + 43942040 q^{68} - 214688928 q^{69} - 7130936 q^{71} + 88372440 q^{72} - 120858180 q^{73} + 519897028 q^{74} - 97472240 q^{76} + 800132400 q^{77} + 688840400 q^{78} + 6877520 q^{79} - 275359358 q^{81} + 179618020 q^{82} - 1402348740 q^{83} + 1161929088 q^{84} - 212388136 q^{86} + 45016840 q^{87} + 13145760 q^{88} + 830088660 q^{89} + 681630904 q^{91} + 939144480 q^{92} + 414660480 q^{93} - 1348148912 q^{94} + 803360384 q^{96} - 638394580 q^{97} + 799918970 q^{98} - 1963732048 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 36.7648 1.62479 0.812394 0.583109i \(-0.198164\pi\)
0.812394 + 0.583109i \(0.198164\pi\)
\(3\) −193.530 −1.37944 −0.689718 0.724078i \(-0.742266\pi\)
−0.689718 + 0.724078i \(0.742266\pi\)
\(4\) 839.648 1.63994
\(5\) 0 0
\(6\) −7115.07 −2.24129
\(7\) −7647.66 −1.20389 −0.601946 0.798537i \(-0.705608\pi\)
−0.601946 + 0.798537i \(0.705608\pi\)
\(8\) 12045.9 1.03976
\(9\) 17770.7 0.902844
\(10\) 0 0
\(11\) −48361.0 −0.995929 −0.497965 0.867197i \(-0.665919\pi\)
−0.497965 + 0.867197i \(0.665919\pi\)
\(12\) −162497. −2.26219
\(13\) −100456. −0.975507 −0.487754 0.872981i \(-0.662184\pi\)
−0.487754 + 0.872981i \(0.662184\pi\)
\(14\) −281164. −1.95607
\(15\) 0 0
\(16\) 12964.5 0.0494557
\(17\) 201958. 0.586463 0.293231 0.956042i \(-0.405269\pi\)
0.293231 + 0.956042i \(0.405269\pi\)
\(18\) 653335. 1.46693
\(19\) −58048.1 −0.102187 −0.0510936 0.998694i \(-0.516271\pi\)
−0.0510936 + 0.998694i \(0.516271\pi\)
\(20\) 0 0
\(21\) 1.48005e6 1.66069
\(22\) −1.77798e6 −1.61817
\(23\) 1.14078e6 0.850017 0.425009 0.905189i \(-0.360271\pi\)
0.425009 + 0.905189i \(0.360271\pi\)
\(24\) −2.33123e6 −1.43428
\(25\) 0 0
\(26\) −3.69324e6 −1.58499
\(27\) 370091. 0.134021
\(28\) −6.42134e6 −1.97431
\(29\) 1.56407e6 0.410643 0.205322 0.978695i \(-0.434176\pi\)
0.205322 + 0.978695i \(0.434176\pi\)
\(30\) 0 0
\(31\) −4.10744e6 −0.798810 −0.399405 0.916775i \(-0.630783\pi\)
−0.399405 + 0.916775i \(0.630783\pi\)
\(32\) −5.69086e6 −0.959407
\(33\) 9.35929e6 1.37382
\(34\) 7.42493e6 0.952878
\(35\) 0 0
\(36\) 1.49211e7 1.48061
\(37\) 1.70821e7 1.49842 0.749212 0.662330i \(-0.230432\pi\)
0.749212 + 0.662330i \(0.230432\pi\)
\(38\) −2.13412e6 −0.166033
\(39\) 1.94412e7 1.34565
\(40\) 0 0
\(41\) −8.52969e6 −0.471418 −0.235709 0.971824i \(-0.575741\pi\)
−0.235709 + 0.971824i \(0.575741\pi\)
\(42\) 5.44136e7 2.69827
\(43\) −2.56254e7 −1.14304 −0.571521 0.820587i \(-0.693646\pi\)
−0.571521 + 0.820587i \(0.693646\pi\)
\(44\) −4.06062e7 −1.63326
\(45\) 0 0
\(46\) 4.19406e7 1.38110
\(47\) −4.58297e7 −1.36996 −0.684978 0.728564i \(-0.740188\pi\)
−0.684978 + 0.728564i \(0.740188\pi\)
\(48\) −2.50902e6 −0.0682210
\(49\) 1.81331e7 0.449355
\(50\) 0 0
\(51\) −3.90848e7 −0.808988
\(52\) −8.43476e7 −1.59977
\(53\) 5.56483e7 0.968748 0.484374 0.874861i \(-0.339047\pi\)
0.484374 + 0.874861i \(0.339047\pi\)
\(54\) 1.36063e7 0.217755
\(55\) 0 0
\(56\) −9.21228e7 −1.25176
\(57\) 1.12340e7 0.140961
\(58\) 5.75026e7 0.667209
\(59\) 2.15699e7 0.231747 0.115873 0.993264i \(-0.463033\pi\)
0.115873 + 0.993264i \(0.463033\pi\)
\(60\) 0 0
\(61\) −1.15309e8 −1.06630 −0.533148 0.846022i \(-0.678991\pi\)
−0.533148 + 0.846022i \(0.678991\pi\)
\(62\) −1.51009e8 −1.29790
\(63\) −1.35904e8 −1.08693
\(64\) −2.15861e8 −1.60829
\(65\) 0 0
\(66\) 3.44092e8 2.23217
\(67\) 7.69254e7 0.466373 0.233186 0.972432i \(-0.425085\pi\)
0.233186 + 0.972432i \(0.425085\pi\)
\(68\) 1.69573e8 0.961762
\(69\) −2.20775e8 −1.17254
\(70\) 0 0
\(71\) 1.95870e8 0.914754 0.457377 0.889273i \(-0.348789\pi\)
0.457377 + 0.889273i \(0.348789\pi\)
\(72\) 2.14064e8 0.938742
\(73\) −3.40942e8 −1.40517 −0.702583 0.711601i \(-0.747970\pi\)
−0.702583 + 0.711601i \(0.747970\pi\)
\(74\) 6.28021e8 2.43462
\(75\) 0 0
\(76\) −4.87399e7 −0.167581
\(77\) 3.69849e8 1.19899
\(78\) 7.14751e8 2.18640
\(79\) −5.92965e8 −1.71280 −0.856401 0.516311i \(-0.827305\pi\)
−0.856401 + 0.516311i \(0.827305\pi\)
\(80\) 0 0
\(81\) −4.21404e8 −1.08772
\(82\) −3.13592e8 −0.765954
\(83\) −7.88477e8 −1.82363 −0.911817 0.410598i \(-0.865320\pi\)
−0.911817 + 0.410598i \(0.865320\pi\)
\(84\) 1.24272e9 2.72343
\(85\) 0 0
\(86\) −9.42111e8 −1.85720
\(87\) −3.02693e8 −0.566456
\(88\) −5.82552e8 −1.03553
\(89\) 8.40110e8 1.41932 0.709661 0.704543i \(-0.248848\pi\)
0.709661 + 0.704543i \(0.248848\pi\)
\(90\) 0 0
\(91\) 7.68253e8 1.17441
\(92\) 9.57856e8 1.39397
\(93\) 7.94910e8 1.10191
\(94\) −1.68492e9 −2.22589
\(95\) 0 0
\(96\) 1.10135e9 1.32344
\(97\) −2.35903e8 −0.270558 −0.135279 0.990808i \(-0.543193\pi\)
−0.135279 + 0.990808i \(0.543193\pi\)
\(98\) 6.66658e8 0.730106
\(99\) −8.59408e8 −0.899169
\(100\) 0 0
\(101\) −6.52957e8 −0.624365 −0.312182 0.950022i \(-0.601060\pi\)
−0.312182 + 0.950022i \(0.601060\pi\)
\(102\) −1.43694e9 −1.31443
\(103\) −9.53214e8 −0.834493 −0.417247 0.908793i \(-0.637005\pi\)
−0.417247 + 0.908793i \(0.637005\pi\)
\(104\) −1.21008e9 −1.01430
\(105\) 0 0
\(106\) 2.04590e9 1.57401
\(107\) 1.33363e9 0.983573 0.491787 0.870716i \(-0.336344\pi\)
0.491787 + 0.870716i \(0.336344\pi\)
\(108\) 3.10746e8 0.219785
\(109\) 1.93368e9 1.31210 0.656049 0.754718i \(-0.272226\pi\)
0.656049 + 0.754718i \(0.272226\pi\)
\(110\) 0 0
\(111\) −3.30590e9 −2.06698
\(112\) −9.91483e7 −0.0595393
\(113\) 9.74289e8 0.562128 0.281064 0.959689i \(-0.409313\pi\)
0.281064 + 0.959689i \(0.409313\pi\)
\(114\) 4.13016e8 0.229031
\(115\) 0 0
\(116\) 1.31327e9 0.673429
\(117\) −1.78517e9 −0.880731
\(118\) 7.93011e8 0.376539
\(119\) −1.54450e9 −0.706037
\(120\) 0 0
\(121\) −1.91571e7 −0.00812446
\(122\) −4.23929e9 −1.73250
\(123\) 1.65075e9 0.650290
\(124\) −3.44880e9 −1.31000
\(125\) 0 0
\(126\) −4.99648e9 −1.76602
\(127\) −1.25137e9 −0.426845 −0.213422 0.976960i \(-0.568461\pi\)
−0.213422 + 0.976960i \(0.568461\pi\)
\(128\) −5.02235e9 −1.65372
\(129\) 4.95927e9 1.57675
\(130\) 0 0
\(131\) 2.23060e9 0.661760 0.330880 0.943673i \(-0.392655\pi\)
0.330880 + 0.943673i \(0.392655\pi\)
\(132\) 7.85851e9 2.25298
\(133\) 4.43932e8 0.123022
\(134\) 2.82814e9 0.757757
\(135\) 0 0
\(136\) 2.43276e9 0.609781
\(137\) 3.91202e9 0.948764 0.474382 0.880319i \(-0.342672\pi\)
0.474382 + 0.880319i \(0.342672\pi\)
\(138\) −8.11675e9 −1.90514
\(139\) 1.64295e9 0.373299 0.186649 0.982427i \(-0.440237\pi\)
0.186649 + 0.982427i \(0.440237\pi\)
\(140\) 0 0
\(141\) 8.86939e9 1.88977
\(142\) 7.20110e9 1.48628
\(143\) 4.85815e9 0.971537
\(144\) 2.30388e8 0.0446508
\(145\) 0 0
\(146\) −1.25347e10 −2.28310
\(147\) −3.50929e9 −0.619856
\(148\) 1.43430e10 2.45732
\(149\) −6.33624e9 −1.05316 −0.526579 0.850126i \(-0.676525\pi\)
−0.526579 + 0.850126i \(0.676525\pi\)
\(150\) 0 0
\(151\) 5.67989e9 0.889086 0.444543 0.895757i \(-0.353366\pi\)
0.444543 + 0.895757i \(0.353366\pi\)
\(152\) −6.99241e8 −0.106250
\(153\) 3.58893e9 0.529484
\(154\) 1.35974e10 1.94811
\(155\) 0 0
\(156\) 1.63238e10 2.20678
\(157\) 4.23621e9 0.556454 0.278227 0.960515i \(-0.410253\pi\)
0.278227 + 0.960515i \(0.410253\pi\)
\(158\) −2.18002e10 −2.78294
\(159\) −1.07696e10 −1.33633
\(160\) 0 0
\(161\) −8.72432e9 −1.02333
\(162\) −1.54928e10 −1.76731
\(163\) −8.56450e9 −0.950293 −0.475147 0.879907i \(-0.657605\pi\)
−0.475147 + 0.879907i \(0.657605\pi\)
\(164\) −7.16193e9 −0.773095
\(165\) 0 0
\(166\) −2.89882e10 −2.96302
\(167\) −5.87750e9 −0.584748 −0.292374 0.956304i \(-0.594445\pi\)
−0.292374 + 0.956304i \(0.594445\pi\)
\(168\) 1.78285e10 1.72672
\(169\) −5.13100e8 −0.0483851
\(170\) 0 0
\(171\) −1.03155e9 −0.0922591
\(172\) −2.15163e10 −1.87452
\(173\) −1.48147e10 −1.25743 −0.628715 0.777635i \(-0.716419\pi\)
−0.628715 + 0.777635i \(0.716419\pi\)
\(174\) −1.11285e10 −0.920372
\(175\) 0 0
\(176\) −6.26978e8 −0.0492544
\(177\) −4.17441e9 −0.319680
\(178\) 3.08864e10 2.30610
\(179\) 5.95512e9 0.433563 0.216781 0.976220i \(-0.430444\pi\)
0.216781 + 0.976220i \(0.430444\pi\)
\(180\) 0 0
\(181\) 1.02707e10 0.711290 0.355645 0.934621i \(-0.384261\pi\)
0.355645 + 0.934621i \(0.384261\pi\)
\(182\) 2.82446e10 1.90816
\(183\) 2.23156e10 1.47089
\(184\) 1.37417e10 0.883815
\(185\) 0 0
\(186\) 2.92247e10 1.79036
\(187\) −9.76689e9 −0.584075
\(188\) −3.84808e10 −2.24664
\(189\) −2.83033e9 −0.161346
\(190\) 0 0
\(191\) 2.09899e9 0.114120 0.0570599 0.998371i \(-0.481827\pi\)
0.0570599 + 0.998371i \(0.481827\pi\)
\(192\) 4.17754e10 2.21853
\(193\) 1.95026e10 1.01178 0.505888 0.862599i \(-0.331165\pi\)
0.505888 + 0.862599i \(0.331165\pi\)
\(194\) −8.67293e9 −0.439600
\(195\) 0 0
\(196\) 1.52254e10 0.736913
\(197\) −6.38850e9 −0.302204 −0.151102 0.988518i \(-0.548282\pi\)
−0.151102 + 0.988518i \(0.548282\pi\)
\(198\) −3.15959e10 −1.46096
\(199\) 1.32140e10 0.597303 0.298652 0.954362i \(-0.403463\pi\)
0.298652 + 0.954362i \(0.403463\pi\)
\(200\) 0 0
\(201\) −1.48873e10 −0.643331
\(202\) −2.40058e10 −1.01446
\(203\) −1.19615e10 −0.494370
\(204\) −3.28174e10 −1.32669
\(205\) 0 0
\(206\) −3.50447e10 −1.35588
\(207\) 2.02725e10 0.767433
\(208\) −1.30236e9 −0.0482444
\(209\) 2.80727e9 0.101771
\(210\) 0 0
\(211\) 2.06410e9 0.0716901 0.0358450 0.999357i \(-0.488588\pi\)
0.0358450 + 0.999357i \(0.488588\pi\)
\(212\) 4.67250e10 1.58869
\(213\) −3.79065e10 −1.26184
\(214\) 4.90304e10 1.59810
\(215\) 0 0
\(216\) 4.45808e9 0.139350
\(217\) 3.14123e10 0.961680
\(218\) 7.10915e10 2.13188
\(219\) 6.59824e10 1.93834
\(220\) 0 0
\(221\) −2.02879e10 −0.572099
\(222\) −1.21541e11 −3.35840
\(223\) 1.59589e10 0.432145 0.216073 0.976377i \(-0.430675\pi\)
0.216073 + 0.976377i \(0.430675\pi\)
\(224\) 4.35217e10 1.15502
\(225\) 0 0
\(226\) 3.58195e10 0.913338
\(227\) −4.86308e10 −1.21561 −0.607807 0.794085i \(-0.707950\pi\)
−0.607807 + 0.794085i \(0.707950\pi\)
\(228\) 9.43262e9 0.231167
\(229\) −6.34109e10 −1.52372 −0.761858 0.647744i \(-0.775713\pi\)
−0.761858 + 0.647744i \(0.775713\pi\)
\(230\) 0 0
\(231\) −7.15767e10 −1.65393
\(232\) 1.88406e10 0.426971
\(233\) −2.45281e10 −0.545209 −0.272605 0.962126i \(-0.587885\pi\)
−0.272605 + 0.962126i \(0.587885\pi\)
\(234\) −6.56314e10 −1.43100
\(235\) 0 0
\(236\) 1.81111e10 0.380050
\(237\) 1.14756e11 2.36270
\(238\) −5.67833e10 −1.14716
\(239\) −7.82086e10 −1.55047 −0.775236 0.631671i \(-0.782369\pi\)
−0.775236 + 0.631671i \(0.782369\pi\)
\(240\) 0 0
\(241\) 6.59331e10 1.25900 0.629502 0.776999i \(-0.283259\pi\)
0.629502 + 0.776999i \(0.283259\pi\)
\(242\) −7.04304e8 −0.0132005
\(243\) 7.42696e10 1.36642
\(244\) −9.68186e10 −1.74866
\(245\) 0 0
\(246\) 6.06893e10 1.05658
\(247\) 5.83128e9 0.0996844
\(248\) −4.94777e10 −0.830571
\(249\) 1.52594e11 2.51559
\(250\) 0 0
\(251\) 1.83321e10 0.291527 0.145764 0.989319i \(-0.453436\pi\)
0.145764 + 0.989319i \(0.453436\pi\)
\(252\) −1.14112e11 −1.78249
\(253\) −5.51694e10 −0.846557
\(254\) −4.60064e10 −0.693533
\(255\) 0 0
\(256\) −7.41248e10 −1.07866
\(257\) −6.41469e10 −0.917227 −0.458613 0.888636i \(-0.651654\pi\)
−0.458613 + 0.888636i \(0.651654\pi\)
\(258\) 1.82326e11 2.56189
\(259\) −1.30638e11 −1.80394
\(260\) 0 0
\(261\) 2.77946e10 0.370747
\(262\) 8.20074e10 1.07522
\(263\) −2.20901e10 −0.284706 −0.142353 0.989816i \(-0.545467\pi\)
−0.142353 + 0.989816i \(0.545467\pi\)
\(264\) 1.12741e11 1.42845
\(265\) 0 0
\(266\) 1.63211e10 0.199885
\(267\) −1.62586e11 −1.95787
\(268\) 6.45902e10 0.764822
\(269\) 7.78901e10 0.906978 0.453489 0.891262i \(-0.350179\pi\)
0.453489 + 0.891262i \(0.350179\pi\)
\(270\) 0 0
\(271\) −9.36671e10 −1.05493 −0.527467 0.849576i \(-0.676858\pi\)
−0.527467 + 0.849576i \(0.676858\pi\)
\(272\) 2.61829e9 0.0290039
\(273\) −1.48680e11 −1.62002
\(274\) 1.43824e11 1.54154
\(275\) 0 0
\(276\) −1.85373e11 −1.92290
\(277\) 7.45925e10 0.761266 0.380633 0.924726i \(-0.375706\pi\)
0.380633 + 0.924726i \(0.375706\pi\)
\(278\) 6.04025e10 0.606532
\(279\) −7.29919e10 −0.721200
\(280\) 0 0
\(281\) −1.10771e11 −1.05986 −0.529930 0.848041i \(-0.677782\pi\)
−0.529930 + 0.848041i \(0.677782\pi\)
\(282\) 3.26081e11 3.07047
\(283\) −1.44123e11 −1.33566 −0.667830 0.744314i \(-0.732777\pi\)
−0.667830 + 0.744314i \(0.732777\pi\)
\(284\) 1.64461e11 1.50014
\(285\) 0 0
\(286\) 1.78609e11 1.57854
\(287\) 6.52321e10 0.567536
\(288\) −1.01130e11 −0.866194
\(289\) −7.78009e10 −0.656061
\(290\) 0 0
\(291\) 4.56542e10 0.373218
\(292\) −2.86271e11 −2.30438
\(293\) 1.39037e11 1.10211 0.551056 0.834469i \(-0.314225\pi\)
0.551056 + 0.834469i \(0.314225\pi\)
\(294\) −1.29018e11 −1.00713
\(295\) 0 0
\(296\) 2.05770e11 1.55800
\(297\) −1.78980e10 −0.133475
\(298\) −2.32950e11 −1.71116
\(299\) −1.14598e11 −0.829198
\(300\) 0 0
\(301\) 1.95974e11 1.37610
\(302\) 2.08820e11 1.44458
\(303\) 1.26366e11 0.861271
\(304\) −7.52566e8 −0.00505375
\(305\) 0 0
\(306\) 1.31946e11 0.860300
\(307\) 2.71765e10 0.174611 0.0873053 0.996182i \(-0.472174\pi\)
0.0873053 + 0.996182i \(0.472174\pi\)
\(308\) 3.10543e11 1.96627
\(309\) 1.84475e11 1.15113
\(310\) 0 0
\(311\) 9.82604e10 0.595603 0.297802 0.954628i \(-0.403747\pi\)
0.297802 + 0.954628i \(0.403747\pi\)
\(312\) 2.34186e11 1.39916
\(313\) 1.82397e11 1.07416 0.537078 0.843533i \(-0.319528\pi\)
0.537078 + 0.843533i \(0.319528\pi\)
\(314\) 1.55743e11 0.904120
\(315\) 0 0
\(316\) −4.97882e11 −2.80889
\(317\) 1.52524e11 0.848341 0.424170 0.905582i \(-0.360566\pi\)
0.424170 + 0.905582i \(0.360566\pi\)
\(318\) −3.95942e11 −2.17125
\(319\) −7.56400e10 −0.408972
\(320\) 0 0
\(321\) −2.58096e11 −1.35678
\(322\) −3.20747e11 −1.66269
\(323\) −1.17233e10 −0.0599290
\(324\) −3.53831e11 −1.78379
\(325\) 0 0
\(326\) −3.14872e11 −1.54403
\(327\) −3.74225e11 −1.80996
\(328\) −1.02748e11 −0.490162
\(329\) 3.50490e11 1.64928
\(330\) 0 0
\(331\) 4.08043e11 1.86845 0.934223 0.356690i \(-0.116095\pi\)
0.934223 + 0.356690i \(0.116095\pi\)
\(332\) −6.62043e11 −2.99064
\(333\) 3.03561e11 1.35284
\(334\) −2.16085e11 −0.950092
\(335\) 0 0
\(336\) 1.91881e10 0.0821307
\(337\) 2.48513e11 1.04958 0.524789 0.851233i \(-0.324144\pi\)
0.524789 + 0.851233i \(0.324144\pi\)
\(338\) −1.88640e10 −0.0786156
\(339\) −1.88554e11 −0.775419
\(340\) 0 0
\(341\) 1.98640e11 0.795558
\(342\) −3.79248e10 −0.149902
\(343\) 1.69935e11 0.662917
\(344\) −3.08680e11 −1.18849
\(345\) 0 0
\(346\) −5.44657e11 −2.04306
\(347\) −3.44769e11 −1.27657 −0.638287 0.769799i \(-0.720356\pi\)
−0.638287 + 0.769799i \(0.720356\pi\)
\(348\) −2.54156e11 −0.928953
\(349\) 4.24804e11 1.53276 0.766380 0.642388i \(-0.222056\pi\)
0.766380 + 0.642388i \(0.222056\pi\)
\(350\) 0 0
\(351\) −3.71779e10 −0.130738
\(352\) 2.75216e11 0.955501
\(353\) −1.53549e11 −0.526334 −0.263167 0.964750i \(-0.584767\pi\)
−0.263167 + 0.964750i \(0.584767\pi\)
\(354\) −1.53471e11 −0.519412
\(355\) 0 0
\(356\) 7.05397e11 2.32760
\(357\) 2.98907e11 0.973933
\(358\) 2.18938e11 0.704447
\(359\) −3.08800e11 −0.981189 −0.490594 0.871388i \(-0.663220\pi\)
−0.490594 + 0.871388i \(0.663220\pi\)
\(360\) 0 0
\(361\) −3.19318e11 −0.989558
\(362\) 3.77600e11 1.15570
\(363\) 3.70745e9 0.0112072
\(364\) 6.45062e11 1.92595
\(365\) 0 0
\(366\) 8.20429e11 2.38988
\(367\) 1.26543e11 0.364116 0.182058 0.983288i \(-0.441724\pi\)
0.182058 + 0.983288i \(0.441724\pi\)
\(368\) 1.47897e10 0.0420382
\(369\) −1.51578e11 −0.425616
\(370\) 0 0
\(371\) −4.25579e11 −1.16627
\(372\) 6.67444e11 1.80706
\(373\) −2.88008e10 −0.0770397 −0.0385199 0.999258i \(-0.512264\pi\)
−0.0385199 + 0.999258i \(0.512264\pi\)
\(374\) −3.59077e11 −0.948999
\(375\) 0 0
\(376\) −5.52059e11 −1.42443
\(377\) −1.57120e11 −0.400586
\(378\) −1.04056e11 −0.262154
\(379\) 5.21974e11 1.29949 0.649744 0.760153i \(-0.274876\pi\)
0.649744 + 0.760153i \(0.274876\pi\)
\(380\) 0 0
\(381\) 2.42178e11 0.588805
\(382\) 7.71690e10 0.185421
\(383\) 1.77381e10 0.0421223 0.0210611 0.999778i \(-0.493296\pi\)
0.0210611 + 0.999778i \(0.493296\pi\)
\(384\) 9.71973e11 2.28120
\(385\) 0 0
\(386\) 7.17007e11 1.64392
\(387\) −4.55380e11 −1.03199
\(388\) −1.98076e11 −0.443699
\(389\) −2.05874e10 −0.0455856 −0.0227928 0.999740i \(-0.507256\pi\)
−0.0227928 + 0.999740i \(0.507256\pi\)
\(390\) 0 0
\(391\) 2.30390e11 0.498503
\(392\) 2.18429e11 0.467222
\(393\) −4.31686e11 −0.912855
\(394\) −2.34872e11 −0.491018
\(395\) 0 0
\(396\) −7.21600e11 −1.47458
\(397\) 8.47450e11 1.71221 0.856105 0.516803i \(-0.172878\pi\)
0.856105 + 0.516803i \(0.172878\pi\)
\(398\) 4.85809e11 0.970491
\(399\) −8.59139e10 −0.169701
\(400\) 0 0
\(401\) −2.81253e11 −0.543185 −0.271592 0.962412i \(-0.587550\pi\)
−0.271592 + 0.962412i \(0.587550\pi\)
\(402\) −5.47329e11 −1.04528
\(403\) 4.12616e11 0.779245
\(404\) −5.48254e11 −1.02392
\(405\) 0 0
\(406\) −4.39760e11 −0.803247
\(407\) −8.26111e11 −1.49232
\(408\) −4.70811e11 −0.841154
\(409\) −6.89579e11 −1.21851 −0.609255 0.792975i \(-0.708531\pi\)
−0.609255 + 0.792975i \(0.708531\pi\)
\(410\) 0 0
\(411\) −7.57091e11 −1.30876
\(412\) −8.00364e11 −1.36852
\(413\) −1.64959e11 −0.278998
\(414\) 7.45313e11 1.24692
\(415\) 0 0
\(416\) 5.71680e11 0.935908
\(417\) −3.17958e11 −0.514942
\(418\) 1.03208e11 0.165357
\(419\) −4.76873e11 −0.755857 −0.377928 0.925835i \(-0.623363\pi\)
−0.377928 + 0.925835i \(0.623363\pi\)
\(420\) 0 0
\(421\) 1.08180e12 1.67833 0.839165 0.543877i \(-0.183044\pi\)
0.839165 + 0.543877i \(0.183044\pi\)
\(422\) 7.58860e10 0.116481
\(423\) −8.14424e11 −1.23686
\(424\) 6.70333e11 1.00727
\(425\) 0 0
\(426\) −1.39363e12 −2.05023
\(427\) 8.81841e11 1.28370
\(428\) 1.11978e12 1.61300
\(429\) −9.40196e11 −1.34017
\(430\) 0 0
\(431\) −3.36212e11 −0.469316 −0.234658 0.972078i \(-0.575397\pi\)
−0.234658 + 0.972078i \(0.575397\pi\)
\(432\) 4.79806e9 0.00662809
\(433\) −1.26577e12 −1.73045 −0.865225 0.501383i \(-0.832825\pi\)
−0.865225 + 0.501383i \(0.832825\pi\)
\(434\) 1.15486e12 1.56253
\(435\) 0 0
\(436\) 1.62361e12 2.15176
\(437\) −6.62203e10 −0.0868609
\(438\) 2.42583e12 3.14939
\(439\) 2.16378e11 0.278050 0.139025 0.990289i \(-0.455603\pi\)
0.139025 + 0.990289i \(0.455603\pi\)
\(440\) 0 0
\(441\) 3.22237e11 0.405697
\(442\) −7.45878e11 −0.929539
\(443\) 3.36288e11 0.414853 0.207426 0.978251i \(-0.433491\pi\)
0.207426 + 0.978251i \(0.433491\pi\)
\(444\) −2.77579e12 −3.38972
\(445\) 0 0
\(446\) 5.86724e11 0.702145
\(447\) 1.22625e12 1.45276
\(448\) 1.65083e12 1.93620
\(449\) 1.40309e12 1.62922 0.814608 0.580012i \(-0.196952\pi\)
0.814608 + 0.580012i \(0.196952\pi\)
\(450\) 0 0
\(451\) 4.12505e11 0.469499
\(452\) 8.18060e11 0.921854
\(453\) −1.09923e12 −1.22644
\(454\) −1.78790e12 −1.97511
\(455\) 0 0
\(456\) 1.35324e11 0.146566
\(457\) −1.26840e12 −1.36030 −0.680148 0.733075i \(-0.738084\pi\)
−0.680148 + 0.733075i \(0.738084\pi\)
\(458\) −2.33129e12 −2.47572
\(459\) 7.47428e10 0.0785981
\(460\) 0 0
\(461\) −3.57771e11 −0.368936 −0.184468 0.982839i \(-0.559056\pi\)
−0.184468 + 0.982839i \(0.559056\pi\)
\(462\) −2.63150e12 −2.68729
\(463\) 3.72773e11 0.376990 0.188495 0.982074i \(-0.439639\pi\)
0.188495 + 0.982074i \(0.439639\pi\)
\(464\) 2.02774e10 0.0203087
\(465\) 0 0
\(466\) −9.01771e11 −0.885849
\(467\) 1.30696e12 1.27155 0.635777 0.771873i \(-0.280680\pi\)
0.635777 + 0.771873i \(0.280680\pi\)
\(468\) −1.49891e12 −1.44434
\(469\) −5.88299e11 −0.561462
\(470\) 0 0
\(471\) −8.19832e11 −0.767593
\(472\) 2.59828e11 0.240961
\(473\) 1.23927e12 1.13839
\(474\) 4.21898e12 3.83889
\(475\) 0 0
\(476\) −1.29684e12 −1.15786
\(477\) 9.88908e11 0.874628
\(478\) −2.87532e12 −2.51919
\(479\) 3.77216e10 0.0327402 0.0163701 0.999866i \(-0.494789\pi\)
0.0163701 + 0.999866i \(0.494789\pi\)
\(480\) 0 0
\(481\) −1.71600e12 −1.46172
\(482\) 2.42401e12 2.04561
\(483\) 1.68841e12 1.41162
\(484\) −1.60852e10 −0.0133236
\(485\) 0 0
\(486\) 2.73050e12 2.22014
\(487\) −1.89303e12 −1.52503 −0.762513 0.646973i \(-0.776035\pi\)
−0.762513 + 0.646973i \(0.776035\pi\)
\(488\) −1.38899e12 −1.10869
\(489\) 1.65748e12 1.31087
\(490\) 0 0
\(491\) −7.57823e11 −0.588438 −0.294219 0.955738i \(-0.595060\pi\)
−0.294219 + 0.955738i \(0.595060\pi\)
\(492\) 1.38605e12 1.06644
\(493\) 3.15876e11 0.240827
\(494\) 2.14385e11 0.161966
\(495\) 0 0
\(496\) −5.32510e10 −0.0395057
\(497\) −1.49794e12 −1.10126
\(498\) 5.61006e12 4.08729
\(499\) 3.90008e11 0.281593 0.140796 0.990039i \(-0.455034\pi\)
0.140796 + 0.990039i \(0.455034\pi\)
\(500\) 0 0
\(501\) 1.13747e12 0.806623
\(502\) 6.73973e11 0.473670
\(503\) 3.44868e11 0.240213 0.120107 0.992761i \(-0.461676\pi\)
0.120107 + 0.992761i \(0.461676\pi\)
\(504\) −1.63708e12 −1.13014
\(505\) 0 0
\(506\) −2.02829e12 −1.37548
\(507\) 9.93001e10 0.0667442
\(508\) −1.05071e12 −0.699999
\(509\) −1.33209e12 −0.879636 −0.439818 0.898087i \(-0.644957\pi\)
−0.439818 + 0.898087i \(0.644957\pi\)
\(510\) 0 0
\(511\) 2.60741e12 1.69167
\(512\) −1.53738e11 −0.0988703
\(513\) −2.14831e10 −0.0136952
\(514\) −2.35835e12 −1.49030
\(515\) 0 0
\(516\) 4.16404e12 2.58578
\(517\) 2.21637e12 1.36438
\(518\) −4.80289e12 −2.93102
\(519\) 2.86707e12 1.73455
\(520\) 0 0
\(521\) −1.31345e12 −0.780985 −0.390493 0.920606i \(-0.627695\pi\)
−0.390493 + 0.920606i \(0.627695\pi\)
\(522\) 1.02186e12 0.602385
\(523\) −1.60667e12 −0.939006 −0.469503 0.882931i \(-0.655567\pi\)
−0.469503 + 0.882931i \(0.655567\pi\)
\(524\) 1.87292e12 1.08524
\(525\) 0 0
\(526\) −8.12136e11 −0.462587
\(527\) −8.29529e11 −0.468472
\(528\) 1.21339e11 0.0679433
\(529\) −4.99767e11 −0.277471
\(530\) 0 0
\(531\) 3.83311e11 0.209231
\(532\) 3.72746e11 0.201749
\(533\) 8.56858e11 0.459871
\(534\) −5.97744e12 −3.18112
\(535\) 0 0
\(536\) 9.26634e11 0.484916
\(537\) −1.15249e12 −0.598072
\(538\) 2.86361e12 1.47365
\(539\) −8.76935e11 −0.447525
\(540\) 0 0
\(541\) −1.22163e12 −0.613132 −0.306566 0.951849i \(-0.599180\pi\)
−0.306566 + 0.951849i \(0.599180\pi\)
\(542\) −3.44365e12 −1.71404
\(543\) −1.98769e12 −0.981179
\(544\) −1.14931e12 −0.562656
\(545\) 0 0
\(546\) −5.46617e12 −2.63218
\(547\) −1.65203e11 −0.0788996 −0.0394498 0.999222i \(-0.512561\pi\)
−0.0394498 + 0.999222i \(0.512561\pi\)
\(548\) 3.28472e12 1.55591
\(549\) −2.04911e12 −0.962698
\(550\) 0 0
\(551\) −9.07912e10 −0.0419625
\(552\) −2.65943e12 −1.21917
\(553\) 4.53479e12 2.06203
\(554\) 2.74238e12 1.23690
\(555\) 0 0
\(556\) 1.37950e12 0.612186
\(557\) −3.45024e12 −1.51880 −0.759401 0.650623i \(-0.774508\pi\)
−0.759401 + 0.650623i \(0.774508\pi\)
\(558\) −2.68353e12 −1.17180
\(559\) 2.57422e12 1.11505
\(560\) 0 0
\(561\) 1.89018e12 0.805695
\(562\) −4.07248e12 −1.72205
\(563\) −4.38944e11 −0.184129 −0.0920643 0.995753i \(-0.529347\pi\)
−0.0920643 + 0.995753i \(0.529347\pi\)
\(564\) 7.44717e12 3.09910
\(565\) 0 0
\(566\) −5.29867e12 −2.17016
\(567\) 3.22275e12 1.30949
\(568\) 2.35942e12 0.951126
\(569\) −2.03068e12 −0.812150 −0.406075 0.913840i \(-0.633103\pi\)
−0.406075 + 0.913840i \(0.633103\pi\)
\(570\) 0 0
\(571\) 1.25554e12 0.494276 0.247138 0.968980i \(-0.420510\pi\)
0.247138 + 0.968980i \(0.420510\pi\)
\(572\) 4.07914e12 1.59326
\(573\) −4.06217e11 −0.157421
\(574\) 2.39824e12 0.922125
\(575\) 0 0
\(576\) −3.83599e12 −1.45203
\(577\) 1.45596e12 0.546838 0.273419 0.961895i \(-0.411845\pi\)
0.273419 + 0.961895i \(0.411845\pi\)
\(578\) −2.86033e12 −1.06596
\(579\) −3.77432e12 −1.39568
\(580\) 0 0
\(581\) 6.03000e12 2.19546
\(582\) 1.67847e12 0.606400
\(583\) −2.69121e12 −0.964805
\(584\) −4.10695e12 −1.46104
\(585\) 0 0
\(586\) 5.11165e12 1.79070
\(587\) 7.51503e11 0.261252 0.130626 0.991432i \(-0.458301\pi\)
0.130626 + 0.991432i \(0.458301\pi\)
\(588\) −2.94656e12 −1.01652
\(589\) 2.38429e11 0.0816281
\(590\) 0 0
\(591\) 1.23636e12 0.416872
\(592\) 2.21462e11 0.0741057
\(593\) 1.95284e12 0.648514 0.324257 0.945969i \(-0.394886\pi\)
0.324257 + 0.945969i \(0.394886\pi\)
\(594\) −6.58016e11 −0.216869
\(595\) 0 0
\(596\) −5.32021e12 −1.72711
\(597\) −2.55730e12 −0.823942
\(598\) −4.21318e12 −1.34727
\(599\) −4.92766e12 −1.56394 −0.781969 0.623317i \(-0.785785\pi\)
−0.781969 + 0.623317i \(0.785785\pi\)
\(600\) 0 0
\(601\) −4.36498e12 −1.36473 −0.682366 0.731011i \(-0.739049\pi\)
−0.682366 + 0.731011i \(0.739049\pi\)
\(602\) 7.20494e12 2.23587
\(603\) 1.36702e12 0.421062
\(604\) 4.76911e12 1.45805
\(605\) 0 0
\(606\) 4.64583e12 1.39938
\(607\) 1.49883e12 0.448128 0.224064 0.974574i \(-0.428068\pi\)
0.224064 + 0.974574i \(0.428068\pi\)
\(608\) 3.30343e11 0.0980391
\(609\) 2.31490e12 0.681952
\(610\) 0 0
\(611\) 4.60386e12 1.33640
\(612\) 3.01343e12 0.868321
\(613\) 3.19261e12 0.913216 0.456608 0.889668i \(-0.349064\pi\)
0.456608 + 0.889668i \(0.349064\pi\)
\(614\) 9.99137e11 0.283705
\(615\) 0 0
\(616\) 4.45516e12 1.24666
\(617\) 1.58167e12 0.439371 0.219685 0.975571i \(-0.429497\pi\)
0.219685 + 0.975571i \(0.429497\pi\)
\(618\) 6.78218e12 1.87034
\(619\) 2.53778e12 0.694777 0.347389 0.937721i \(-0.387068\pi\)
0.347389 + 0.937721i \(0.387068\pi\)
\(620\) 0 0
\(621\) 4.22194e11 0.113920
\(622\) 3.61252e12 0.967729
\(623\) −6.42488e12 −1.70871
\(624\) 2.52046e11 0.0665501
\(625\) 0 0
\(626\) 6.70576e12 1.74527
\(627\) −5.43289e11 −0.140387
\(628\) 3.55693e12 0.912550
\(629\) 3.44987e12 0.878770
\(630\) 0 0
\(631\) −6.66150e12 −1.67278 −0.836392 0.548131i \(-0.815339\pi\)
−0.836392 + 0.548131i \(0.815339\pi\)
\(632\) −7.14279e12 −1.78091
\(633\) −3.99464e11 −0.0988918
\(634\) 5.60749e12 1.37837
\(635\) 0 0
\(636\) −9.04266e12 −2.19149
\(637\) −1.82158e12 −0.438349
\(638\) −2.78089e12 −0.664493
\(639\) 3.48073e12 0.825880
\(640\) 0 0
\(641\) −6.22257e12 −1.45582 −0.727911 0.685671i \(-0.759509\pi\)
−0.727911 + 0.685671i \(0.759509\pi\)
\(642\) −9.48883e12 −2.20447
\(643\) 4.55187e11 0.105012 0.0525061 0.998621i \(-0.483279\pi\)
0.0525061 + 0.998621i \(0.483279\pi\)
\(644\) −7.32535e12 −1.67819
\(645\) 0 0
\(646\) −4.31003e11 −0.0973719
\(647\) 2.50192e12 0.561312 0.280656 0.959808i \(-0.409448\pi\)
0.280656 + 0.959808i \(0.409448\pi\)
\(648\) −5.07618e12 −1.13097
\(649\) −1.04314e12 −0.230803
\(650\) 0 0
\(651\) −6.07920e12 −1.32658
\(652\) −7.19116e12 −1.55842
\(653\) −3.54499e12 −0.762966 −0.381483 0.924376i \(-0.624587\pi\)
−0.381483 + 0.924376i \(0.624587\pi\)
\(654\) −1.37583e13 −2.94080
\(655\) 0 0
\(656\) −1.10583e11 −0.0233143
\(657\) −6.05877e12 −1.26865
\(658\) 1.28857e13 2.67973
\(659\) 5.93850e12 1.22657 0.613285 0.789862i \(-0.289848\pi\)
0.613285 + 0.789862i \(0.289848\pi\)
\(660\) 0 0
\(661\) −6.68673e12 −1.36241 −0.681204 0.732093i \(-0.738543\pi\)
−0.681204 + 0.732093i \(0.738543\pi\)
\(662\) 1.50016e13 3.03583
\(663\) 3.92630e12 0.789174
\(664\) −9.49790e12 −1.89614
\(665\) 0 0
\(666\) 1.11604e13 2.19808
\(667\) 1.78426e12 0.349054
\(668\) −4.93503e12 −0.958950
\(669\) −3.08851e12 −0.596117
\(670\) 0 0
\(671\) 5.57645e12 1.06196
\(672\) −8.42274e12 −1.59328
\(673\) −8.43444e12 −1.58485 −0.792425 0.609969i \(-0.791182\pi\)
−0.792425 + 0.609969i \(0.791182\pi\)
\(674\) 9.13652e12 1.70534
\(675\) 0 0
\(676\) −4.30823e11 −0.0793486
\(677\) −9.48091e12 −1.73461 −0.867303 0.497781i \(-0.834148\pi\)
−0.867303 + 0.497781i \(0.834148\pi\)
\(678\) −6.93213e12 −1.25989
\(679\) 1.80411e12 0.325723
\(680\) 0 0
\(681\) 9.41150e12 1.67686
\(682\) 7.30295e12 1.29261
\(683\) 1.00045e13 1.75914 0.879571 0.475768i \(-0.157830\pi\)
0.879571 + 0.475768i \(0.157830\pi\)
\(684\) −8.66142e11 −0.151299
\(685\) 0 0
\(686\) 6.24762e12 1.07710
\(687\) 1.22719e13 2.10187
\(688\) −3.32221e11 −0.0565300
\(689\) −5.59021e12 −0.945021
\(690\) 0 0
\(691\) 5.59146e12 0.932983 0.466491 0.884526i \(-0.345518\pi\)
0.466491 + 0.884526i \(0.345518\pi\)
\(692\) −1.24391e13 −2.06211
\(693\) 6.57246e12 1.08250
\(694\) −1.26754e13 −2.07416
\(695\) 0 0
\(696\) −3.64621e12 −0.588980
\(697\) −1.72264e12 −0.276469
\(698\) 1.56178e13 2.49041
\(699\) 4.74692e12 0.752081
\(700\) 0 0
\(701\) −6.41878e12 −1.00397 −0.501986 0.864876i \(-0.667397\pi\)
−0.501986 + 0.864876i \(0.667397\pi\)
\(702\) −1.36684e12 −0.212422
\(703\) −9.91586e11 −0.153120
\(704\) 1.04393e13 1.60174
\(705\) 0 0
\(706\) −5.64520e12 −0.855181
\(707\) 4.99359e12 0.751667
\(708\) −3.50503e12 −0.524255
\(709\) 1.98300e12 0.294724 0.147362 0.989083i \(-0.452922\pi\)
0.147362 + 0.989083i \(0.452922\pi\)
\(710\) 0 0
\(711\) −1.05374e13 −1.54639
\(712\) 1.01199e13 1.47576
\(713\) −4.68569e12 −0.679002
\(714\) 1.09892e13 1.58244
\(715\) 0 0
\(716\) 5.00020e12 0.711015
\(717\) 1.51357e13 2.13878
\(718\) −1.13530e13 −1.59422
\(719\) −9.53823e12 −1.33103 −0.665515 0.746385i \(-0.731788\pi\)
−0.665515 + 0.746385i \(0.731788\pi\)
\(720\) 0 0
\(721\) 7.28986e12 1.00464
\(722\) −1.17397e13 −1.60782
\(723\) −1.27600e13 −1.73671
\(724\) 8.62377e12 1.16647
\(725\) 0 0
\(726\) 1.36304e11 0.0182093
\(727\) 6.30135e11 0.0836621 0.0418310 0.999125i \(-0.486681\pi\)
0.0418310 + 0.999125i \(0.486681\pi\)
\(728\) 9.25429e12 1.22110
\(729\) −6.07886e12 −0.797165
\(730\) 0 0
\(731\) −5.17524e12 −0.670352
\(732\) 1.87373e13 2.41216
\(733\) −2.79069e12 −0.357061 −0.178531 0.983934i \(-0.557134\pi\)
−0.178531 + 0.983934i \(0.557134\pi\)
\(734\) 4.65232e12 0.591612
\(735\) 0 0
\(736\) −6.49203e12 −0.815512
\(737\) −3.72019e12 −0.464474
\(738\) −5.57274e12 −0.691537
\(739\) 7.26759e12 0.896376 0.448188 0.893939i \(-0.352069\pi\)
0.448188 + 0.893939i \(0.352069\pi\)
\(740\) 0 0
\(741\) −1.12852e12 −0.137508
\(742\) −1.56463e13 −1.89494
\(743\) 9.19047e12 1.10634 0.553169 0.833069i \(-0.313418\pi\)
0.553169 + 0.833069i \(0.313418\pi\)
\(744\) 9.57540e12 1.14572
\(745\) 0 0
\(746\) −1.05885e12 −0.125173
\(747\) −1.40118e13 −1.64646
\(748\) −8.20074e12 −0.957847
\(749\) −1.01991e13 −1.18412
\(750\) 0 0
\(751\) −5.21242e11 −0.0597943 −0.0298972 0.999553i \(-0.509518\pi\)
−0.0298972 + 0.999553i \(0.509518\pi\)
\(752\) −5.94160e11 −0.0677521
\(753\) −3.54779e12 −0.402143
\(754\) −5.77648e12 −0.650867
\(755\) 0 0
\(756\) −2.37648e12 −0.264598
\(757\) −1.15123e13 −1.27417 −0.637087 0.770792i \(-0.719861\pi\)
−0.637087 + 0.770792i \(0.719861\pi\)
\(758\) 1.91903e13 2.11139
\(759\) 1.06769e13 1.16777
\(760\) 0 0
\(761\) 1.43517e13 1.55121 0.775607 0.631217i \(-0.217444\pi\)
0.775607 + 0.631217i \(0.217444\pi\)
\(762\) 8.90360e12 0.956684
\(763\) −1.47882e13 −1.57962
\(764\) 1.76242e12 0.187149
\(765\) 0 0
\(766\) 6.52136e11 0.0684398
\(767\) −2.16682e12 −0.226071
\(768\) 1.43453e13 1.48794
\(769\) −7.42318e12 −0.765458 −0.382729 0.923861i \(-0.625016\pi\)
−0.382729 + 0.923861i \(0.625016\pi\)
\(770\) 0 0
\(771\) 1.24143e13 1.26526
\(772\) 1.63753e13 1.65925
\(773\) −5.82774e12 −0.587074 −0.293537 0.955948i \(-0.594832\pi\)
−0.293537 + 0.955948i \(0.594832\pi\)
\(774\) −1.67419e13 −1.67676
\(775\) 0 0
\(776\) −2.84166e12 −0.281316
\(777\) 2.52824e13 2.48842
\(778\) −7.56890e11 −0.0740670
\(779\) 4.95132e11 0.0481729
\(780\) 0 0
\(781\) −9.47246e12 −0.911031
\(782\) 8.47023e12 0.809962
\(783\) 5.78848e11 0.0550347
\(784\) 2.35087e11 0.0222232
\(785\) 0 0
\(786\) −1.58708e13 −1.48320
\(787\) −7.20033e11 −0.0669061 −0.0334531 0.999440i \(-0.510650\pi\)
−0.0334531 + 0.999440i \(0.510650\pi\)
\(788\) −5.36409e12 −0.495596
\(789\) 4.27508e12 0.392733
\(790\) 0 0
\(791\) −7.45103e12 −0.676741
\(792\) −1.03523e13 −0.934921
\(793\) 1.15834e13 1.04018
\(794\) 3.11563e13 2.78198
\(795\) 0 0
\(796\) 1.10951e13 0.979539
\(797\) −9.03491e12 −0.793161 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(798\) −3.15861e12 −0.275729
\(799\) −9.25566e12 −0.803428
\(800\) 0 0
\(801\) 1.49293e13 1.28143
\(802\) −1.03402e13 −0.882560
\(803\) 1.64883e13 1.39945
\(804\) −1.25001e13 −1.05502
\(805\) 0 0
\(806\) 1.51697e13 1.26611
\(807\) −1.50740e13 −1.25112
\(808\) −7.86544e12 −0.649190
\(809\) −2.33217e13 −1.91422 −0.957111 0.289721i \(-0.906438\pi\)
−0.957111 + 0.289721i \(0.906438\pi\)
\(810\) 0 0
\(811\) 1.16365e13 0.944555 0.472277 0.881450i \(-0.343432\pi\)
0.472277 + 0.881450i \(0.343432\pi\)
\(812\) −1.00434e13 −0.810736
\(813\) 1.81273e13 1.45521
\(814\) −3.03718e13 −2.42471
\(815\) 0 0
\(816\) −5.06716e11 −0.0400091
\(817\) 1.48750e12 0.116804
\(818\) −2.53522e13 −1.97982
\(819\) 1.36524e13 1.06030
\(820\) 0 0
\(821\) −1.36931e13 −1.05186 −0.525931 0.850527i \(-0.676283\pi\)
−0.525931 + 0.850527i \(0.676283\pi\)
\(822\) −2.78343e13 −2.12646
\(823\) −8.13478e12 −0.618083 −0.309041 0.951049i \(-0.600008\pi\)
−0.309041 + 0.951049i \(0.600008\pi\)
\(824\) −1.14823e13 −0.867674
\(825\) 0 0
\(826\) −6.06468e12 −0.453313
\(827\) −9.52838e12 −0.708344 −0.354172 0.935180i \(-0.615237\pi\)
−0.354172 + 0.935180i \(0.615237\pi\)
\(828\) 1.70217e13 1.25854
\(829\) 2.36561e13 1.73960 0.869798 0.493407i \(-0.164249\pi\)
0.869798 + 0.493407i \(0.164249\pi\)
\(830\) 0 0
\(831\) −1.44359e13 −1.05012
\(832\) 2.16845e13 1.56890
\(833\) 3.66212e12 0.263530
\(834\) −1.16897e13 −0.836671
\(835\) 0 0
\(836\) 2.35711e12 0.166898
\(837\) −1.52013e12 −0.107057
\(838\) −1.75321e13 −1.22811
\(839\) −1.43311e12 −0.0998509 −0.0499255 0.998753i \(-0.515898\pi\)
−0.0499255 + 0.998753i \(0.515898\pi\)
\(840\) 0 0
\(841\) −1.20608e13 −0.831372
\(842\) 3.97721e13 2.72693
\(843\) 2.14375e13 1.46201
\(844\) 1.73311e12 0.117567
\(845\) 0 0
\(846\) −2.99421e13 −2.00963
\(847\) 1.46507e11 0.00978097
\(848\) 7.21454e11 0.0479101
\(849\) 2.78921e13 1.84246
\(850\) 0 0
\(851\) 1.94870e13 1.27369
\(852\) −3.18281e13 −2.06935
\(853\) −2.03077e13 −1.31338 −0.656690 0.754161i \(-0.728044\pi\)
−0.656690 + 0.754161i \(0.728044\pi\)
\(854\) 3.24207e13 2.08575
\(855\) 0 0
\(856\) 1.60647e13 1.02268
\(857\) 1.23092e12 0.0779498 0.0389749 0.999240i \(-0.487591\pi\)
0.0389749 + 0.999240i \(0.487591\pi\)
\(858\) −3.45661e13 −2.17750
\(859\) −1.69941e13 −1.06495 −0.532476 0.846445i \(-0.678738\pi\)
−0.532476 + 0.846445i \(0.678738\pi\)
\(860\) 0 0
\(861\) −1.26243e13 −0.782879
\(862\) −1.23607e13 −0.762538
\(863\) 3.86270e12 0.237051 0.118526 0.992951i \(-0.462183\pi\)
0.118526 + 0.992951i \(0.462183\pi\)
\(864\) −2.10614e12 −0.128580
\(865\) 0 0
\(866\) −4.65357e13 −2.81162
\(867\) 1.50568e13 0.904995
\(868\) 2.63752e13 1.57709
\(869\) 2.86764e13 1.70583
\(870\) 0 0
\(871\) −7.72761e12 −0.454950
\(872\) 2.32929e13 1.36427
\(873\) −4.19216e12 −0.244272
\(874\) −2.43457e12 −0.141131
\(875\) 0 0
\(876\) 5.54019e13 3.17875
\(877\) 3.44305e12 0.196538 0.0982688 0.995160i \(-0.468670\pi\)
0.0982688 + 0.995160i \(0.468670\pi\)
\(878\) 7.95508e12 0.451772
\(879\) −2.69077e13 −1.52029
\(880\) 0 0
\(881\) 2.37092e13 1.32594 0.662971 0.748645i \(-0.269295\pi\)
0.662971 + 0.748645i \(0.269295\pi\)
\(882\) 1.18470e13 0.659172
\(883\) 3.07625e13 1.70294 0.851468 0.524407i \(-0.175713\pi\)
0.851468 + 0.524407i \(0.175713\pi\)
\(884\) −1.70347e13 −0.938206
\(885\) 0 0
\(886\) 1.23635e13 0.674048
\(887\) 1.39503e13 0.756706 0.378353 0.925661i \(-0.376491\pi\)
0.378353 + 0.925661i \(0.376491\pi\)
\(888\) −3.98225e13 −2.14917
\(889\) 9.57008e12 0.513875
\(890\) 0 0
\(891\) 2.03795e13 1.08329
\(892\) 1.33998e13 0.708691
\(893\) 2.66032e12 0.139992
\(894\) 4.50828e13 2.36043
\(895\) 0 0
\(896\) 3.84092e13 1.99090
\(897\) 2.21782e13 1.14383
\(898\) 5.15844e13 2.64713
\(899\) −6.42431e12 −0.328026
\(900\) 0 0
\(901\) 1.12386e13 0.568134
\(902\) 1.51656e13 0.762836
\(903\) −3.79268e13 −1.89824
\(904\) 1.17362e13 0.584479
\(905\) 0 0
\(906\) −4.04128e13 −1.99270
\(907\) −9.36080e12 −0.459283 −0.229641 0.973275i \(-0.573755\pi\)
−0.229641 + 0.973275i \(0.573755\pi\)
\(908\) −4.08328e13 −1.99353
\(909\) −1.16035e13 −0.563704
\(910\) 0 0
\(911\) 6.17441e12 0.297004 0.148502 0.988912i \(-0.452555\pi\)
0.148502 + 0.988912i \(0.452555\pi\)
\(912\) 1.45644e11 0.00697132
\(913\) 3.81316e13 1.81621
\(914\) −4.66324e13 −2.21019
\(915\) 0 0
\(916\) −5.32428e13 −2.49880
\(917\) −1.70588e13 −0.796687
\(918\) 2.74790e12 0.127705
\(919\) 3.99328e12 0.184676 0.0923380 0.995728i \(-0.470566\pi\)
0.0923380 + 0.995728i \(0.470566\pi\)
\(920\) 0 0
\(921\) −5.25945e12 −0.240864
\(922\) −1.31534e13 −0.599443
\(923\) −1.96763e13 −0.892350
\(924\) −6.00992e13 −2.71234
\(925\) 0 0
\(926\) 1.37049e13 0.612529
\(927\) −1.69393e13 −0.753417
\(928\) −8.90089e12 −0.393974
\(929\) 1.84456e13 0.812496 0.406248 0.913763i \(-0.366837\pi\)
0.406248 + 0.913763i \(0.366837\pi\)
\(930\) 0 0
\(931\) −1.05259e12 −0.0459183
\(932\) −2.05950e13 −0.894108
\(933\) −1.90163e13 −0.821596
\(934\) 4.80499e13 2.06601
\(935\) 0 0
\(936\) −2.15040e13 −0.915750
\(937\) 8.90163e12 0.377261 0.188630 0.982048i \(-0.439595\pi\)
0.188630 + 0.982048i \(0.439595\pi\)
\(938\) −2.16287e13 −0.912257
\(939\) −3.52991e13 −1.48173
\(940\) 0 0
\(941\) −1.64341e13 −0.683270 −0.341635 0.939833i \(-0.610981\pi\)
−0.341635 + 0.939833i \(0.610981\pi\)
\(942\) −3.01409e13 −1.24718
\(943\) −9.73052e12 −0.400713
\(944\) 2.79643e11 0.0114612
\(945\) 0 0
\(946\) 4.55615e13 1.84964
\(947\) −4.26042e13 −1.72138 −0.860691 0.509128i \(-0.829968\pi\)
−0.860691 + 0.509128i \(0.829968\pi\)
\(948\) 9.63548e13 3.87468
\(949\) 3.42497e13 1.37075
\(950\) 0 0
\(951\) −2.95178e13 −1.17023
\(952\) −1.86049e13 −0.734111
\(953\) −3.52200e12 −0.138316 −0.0691578 0.997606i \(-0.522031\pi\)
−0.0691578 + 0.997606i \(0.522031\pi\)
\(954\) 3.63570e13 1.42109
\(955\) 0 0
\(956\) −6.56677e13 −2.54268
\(957\) 1.46386e13 0.564151
\(958\) 1.38683e12 0.0531958
\(959\) −2.99178e13 −1.14221
\(960\) 0 0
\(961\) −9.56859e12 −0.361903
\(962\) −6.30885e13 −2.37499
\(963\) 2.36994e13 0.888013
\(964\) 5.53606e13 2.06469
\(965\) 0 0
\(966\) 6.20741e13 2.29358
\(967\) 3.34889e13 1.23163 0.615816 0.787890i \(-0.288826\pi\)
0.615816 + 0.787890i \(0.288826\pi\)
\(968\) −2.30764e11 −0.00844750
\(969\) 2.26880e12 0.0826682
\(970\) 0 0
\(971\) −3.77122e13 −1.36143 −0.680715 0.732548i \(-0.738331\pi\)
−0.680715 + 0.732548i \(0.738331\pi\)
\(972\) 6.23603e13 2.24083
\(973\) −1.25647e13 −0.449411
\(974\) −6.95968e13 −2.47784
\(975\) 0 0
\(976\) −1.49492e12 −0.0527344
\(977\) 4.00215e13 1.40530 0.702648 0.711537i \(-0.252001\pi\)
0.702648 + 0.711537i \(0.252001\pi\)
\(978\) 6.09370e13 2.12988
\(979\) −4.06286e13 −1.41355
\(980\) 0 0
\(981\) 3.43629e13 1.18462
\(982\) −2.78612e13 −0.956088
\(983\) −3.31096e13 −1.13100 −0.565500 0.824748i \(-0.691317\pi\)
−0.565500 + 0.824748i \(0.691317\pi\)
\(984\) 1.98847e13 0.676147
\(985\) 0 0
\(986\) 1.16131e13 0.391293
\(987\) −6.78301e13 −2.27507
\(988\) 4.89622e12 0.163476
\(989\) −2.92330e13 −0.971606
\(990\) 0 0
\(991\) 1.90619e12 0.0627820 0.0313910 0.999507i \(-0.490006\pi\)
0.0313910 + 0.999507i \(0.490006\pi\)
\(992\) 2.33748e13 0.766383
\(993\) −7.89684e13 −2.57740
\(994\) −5.50715e13 −1.78932
\(995\) 0 0
\(996\) 1.28125e14 4.12540
\(997\) 2.13028e13 0.682823 0.341411 0.939914i \(-0.389095\pi\)
0.341411 + 0.939914i \(0.389095\pi\)
\(998\) 1.43386e13 0.457529
\(999\) 6.32195e12 0.200820
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 25.10.a.b.1.2 2
3.2 odd 2 225.10.a.h.1.1 2
4.3 odd 2 400.10.a.t.1.2 2
5.2 odd 4 25.10.b.b.24.4 4
5.3 odd 4 25.10.b.b.24.1 4
5.4 even 2 5.10.a.b.1.1 2
15.2 even 4 225.10.b.h.199.1 4
15.8 even 4 225.10.b.h.199.4 4
15.14 odd 2 45.10.a.f.1.2 2
20.3 even 4 400.10.c.p.49.4 4
20.7 even 4 400.10.c.p.49.1 4
20.19 odd 2 80.10.a.f.1.1 2
35.34 odd 2 245.10.a.d.1.1 2
40.19 odd 2 320.10.a.s.1.2 2
40.29 even 2 320.10.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.10.a.b.1.1 2 5.4 even 2
25.10.a.b.1.2 2 1.1 even 1 trivial
25.10.b.b.24.1 4 5.3 odd 4
25.10.b.b.24.4 4 5.2 odd 4
45.10.a.f.1.2 2 15.14 odd 2
80.10.a.f.1.1 2 20.19 odd 2
225.10.a.h.1.1 2 3.2 odd 2
225.10.b.h.199.1 4 15.2 even 4
225.10.b.h.199.4 4 15.8 even 4
245.10.a.d.1.1 2 35.34 odd 2
320.10.a.k.1.1 2 40.29 even 2
320.10.a.s.1.2 2 40.19 odd 2
400.10.a.t.1.2 2 4.3 odd 2
400.10.c.p.49.1 4 20.7 even 4
400.10.c.p.49.4 4 20.3 even 4