Properties

Label 91.10.a.c.1.1
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
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] = [91,10,Mod(1,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("91.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.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: \(46.8682610909\)
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: \( x^{14} - x^{13} - 4752 x^{12} + 9346 x^{11} + 8576824 x^{10} - 26923636 x^{9} - 7450416552 x^{8} + 31594524240 x^{7} + 3232668379296 x^{6} + \cdots - 24\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(41.8127\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-39.8127 q^{2} +138.193 q^{3} +1073.05 q^{4} +553.584 q^{5} -5501.81 q^{6} +2401.00 q^{7} -22336.8 q^{8} -585.818 q^{9} +O(q^{10})\) \(q-39.8127 q^{2} +138.193 q^{3} +1073.05 q^{4} +553.584 q^{5} -5501.81 q^{6} +2401.00 q^{7} -22336.8 q^{8} -585.818 q^{9} -22039.7 q^{10} -12620.6 q^{11} +148287. q^{12} -28561.0 q^{13} -95590.2 q^{14} +76501.2 q^{15} +339888. q^{16} +411254. q^{17} +23323.0 q^{18} +660516. q^{19} +594023. q^{20} +331800. q^{21} +502461. q^{22} +2.32926e6 q^{23} -3.08678e6 q^{24} -1.64667e6 q^{25} +1.13709e6 q^{26} -2.80100e6 q^{27} +2.57639e6 q^{28} -5.99627e6 q^{29} -3.04572e6 q^{30} +1.54887e6 q^{31} -2.09539e6 q^{32} -1.74408e6 q^{33} -1.63731e7 q^{34} +1.32916e6 q^{35} -628611. q^{36} +4.46126e6 q^{37} -2.62969e7 q^{38} -3.94692e6 q^{39} -1.23653e7 q^{40} -3.21595e6 q^{41} -1.32099e7 q^{42} +2.23767e7 q^{43} -1.35426e7 q^{44} -324299. q^{45} -9.27339e7 q^{46} +1.94726e7 q^{47} +4.69700e7 q^{48} +5.76480e6 q^{49} +6.55583e7 q^{50} +5.68323e7 q^{51} -3.06473e7 q^{52} +2.46033e7 q^{53} +1.11515e8 q^{54} -6.98659e6 q^{55} -5.36307e7 q^{56} +9.12784e7 q^{57} +2.38727e8 q^{58} +1.19261e8 q^{59} +8.20895e7 q^{60} -1.74738e8 q^{61} -6.16647e7 q^{62} -1.40655e6 q^{63} -9.05995e7 q^{64} -1.58109e7 q^{65} +6.94364e7 q^{66} -1.94831e8 q^{67} +4.41296e8 q^{68} +3.21886e8 q^{69} -5.29172e7 q^{70} +2.88148e8 q^{71} +1.30853e7 q^{72} +7.11716e7 q^{73} -1.77615e8 q^{74} -2.27557e8 q^{75} +7.08766e8 q^{76} -3.03021e7 q^{77} +1.57137e8 q^{78} +2.60602e8 q^{79} +1.88157e8 q^{80} -3.75547e8 q^{81} +1.28036e8 q^{82} +4.08082e8 q^{83} +3.56038e8 q^{84} +2.27664e8 q^{85} -8.90878e8 q^{86} -8.28639e8 q^{87} +2.81905e8 q^{88} +7.99234e7 q^{89} +1.29112e7 q^{90} -6.85750e7 q^{91} +2.49941e9 q^{92} +2.14043e8 q^{93} -7.75258e8 q^{94} +3.65651e8 q^{95} -2.89568e8 q^{96} +7.73718e8 q^{97} -2.29512e8 q^{98} +7.39339e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 27 q^{2} + 163 q^{3} + 2389 q^{4} + 2964 q^{5} + 471 q^{6} + 33614 q^{7} + 43263 q^{8} + 144129 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 27 q^{2} + 163 q^{3} + 2389 q^{4} + 2964 q^{5} + 471 q^{6} + 33614 q^{7} + 43263 q^{8} + 144129 q^{9} + 126524 q^{10} + 81825 q^{11} + 157399 q^{12} - 399854 q^{13} + 64827 q^{14} + 163856 q^{15} + 166361 q^{16} - 44922 q^{17} - 826396 q^{18} + 171756 q^{19} + 3899724 q^{20} + 391363 q^{21} + 917579 q^{22} + 1930479 q^{23} + 2992373 q^{24} + 8222344 q^{25} - 771147 q^{26} + 4139125 q^{27} + 5735989 q^{28} - 3799608 q^{29} - 5918004 q^{30} - 4392203 q^{31} + 3135663 q^{32} + 17499977 q^{33} - 20071132 q^{34} + 7116564 q^{35} + 2121398 q^{36} + 29198909 q^{37} - 44208366 q^{38} - 4655443 q^{39} + 134932928 q^{40} + 48410973 q^{41} + 1130871 q^{42} + 52650242 q^{43} - 14827353 q^{44} + 99215088 q^{45} - 34410455 q^{46} + 160580841 q^{47} + 227620515 q^{48} + 80707214 q^{49} + 149462949 q^{50} + 57114360 q^{51} - 68232229 q^{52} + 80753796 q^{53} + 301368833 q^{54} + 328919412 q^{55} + 103874463 q^{56} + 151101102 q^{57} + 335044204 q^{58} + 442445502 q^{59} + 561078360 q^{60} + 270199089 q^{61} + 543824517 q^{62} + 346053729 q^{63} + 223643137 q^{64} - 84654804 q^{65} + 317483345 q^{66} + 92500909 q^{67} + 255771204 q^{68} + 292017029 q^{69} + 303784124 q^{70} + 84383796 q^{71} + 1456696818 q^{72} + 367274315 q^{73} + 1091659407 q^{74} + 1154152501 q^{75} + 674789222 q^{76} + 196461825 q^{77} - 13452231 q^{78} + 434861545 q^{79} + 2644363752 q^{80} + 644207518 q^{81} + 634104331 q^{82} + 1013603934 q^{83} + 377914999 q^{84} + 1103701048 q^{85} + 2514069096 q^{86} + 1039292304 q^{87} + 1071310221 q^{88} + 1069739706 q^{89} - 1271572324 q^{90} - 960049454 q^{91} + 2301673917 q^{92} - 933838861 q^{93} + 2025486277 q^{94} + 2504029998 q^{95} - 116199027 q^{96} + 2839636281 q^{97} + 155649627 q^{98} + 5063037274 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −39.8127 −1.75949 −0.879744 0.475448i \(-0.842286\pi\)
−0.879744 + 0.475448i \(0.842286\pi\)
\(3\) 138.193 0.985006 0.492503 0.870311i \(-0.336082\pi\)
0.492503 + 0.870311i \(0.336082\pi\)
\(4\) 1073.05 2.09580
\(5\) 553.584 0.396113 0.198056 0.980191i \(-0.436537\pi\)
0.198056 + 0.980191i \(0.436537\pi\)
\(6\) −5501.81 −1.73311
\(7\) 2401.00 0.377964
\(8\) −22336.8 −1.92804
\(9\) −585.818 −0.0297626
\(10\) −22039.7 −0.696956
\(11\) −12620.6 −0.259905 −0.129952 0.991520i \(-0.541482\pi\)
−0.129952 + 0.991520i \(0.541482\pi\)
\(12\) 148287. 2.06437
\(13\) −28561.0 −0.277350
\(14\) −95590.2 −0.665024
\(15\) 76501.2 0.390174
\(16\) 339888. 1.29657
\(17\) 411254. 1.19424 0.597118 0.802153i \(-0.296312\pi\)
0.597118 + 0.802153i \(0.296312\pi\)
\(18\) 23323.0 0.0523670
\(19\) 660516. 1.16277 0.581383 0.813630i \(-0.302512\pi\)
0.581383 + 0.813630i \(0.302512\pi\)
\(20\) 594023. 0.830172
\(21\) 331800. 0.372297
\(22\) 502461. 0.457299
\(23\) 2.32926e6 1.73557 0.867785 0.496940i \(-0.165543\pi\)
0.867785 + 0.496940i \(0.165543\pi\)
\(24\) −3.08678e6 −1.89913
\(25\) −1.64667e6 −0.843095
\(26\) 1.13709e6 0.487994
\(27\) −2.80100e6 −1.01432
\(28\) 2.57639e6 0.792137
\(29\) −5.99627e6 −1.57431 −0.787154 0.616756i \(-0.788447\pi\)
−0.787154 + 0.616756i \(0.788447\pi\)
\(30\) −3.04572e6 −0.686506
\(31\) 1.54887e6 0.301223 0.150611 0.988593i \(-0.451876\pi\)
0.150611 + 0.988593i \(0.451876\pi\)
\(32\) −2.09539e6 −0.353257
\(33\) −1.74408e6 −0.256008
\(34\) −1.63731e7 −2.10125
\(35\) 1.32916e6 0.149717
\(36\) −628611. −0.0623764
\(37\) 4.46126e6 0.391336 0.195668 0.980670i \(-0.437313\pi\)
0.195668 + 0.980670i \(0.437313\pi\)
\(38\) −2.62969e7 −2.04587
\(39\) −3.94692e6 −0.273192
\(40\) −1.23653e7 −0.763722
\(41\) −3.21595e6 −0.177739 −0.0888693 0.996043i \(-0.528325\pi\)
−0.0888693 + 0.996043i \(0.528325\pi\)
\(42\) −1.32099e7 −0.655053
\(43\) 2.23767e7 0.998134 0.499067 0.866563i \(-0.333676\pi\)
0.499067 + 0.866563i \(0.333676\pi\)
\(44\) −1.35426e7 −0.544708
\(45\) −324299. −0.0117894
\(46\) −9.27339e7 −3.05371
\(47\) 1.94726e7 0.582083 0.291041 0.956710i \(-0.405998\pi\)
0.291041 + 0.956710i \(0.405998\pi\)
\(48\) 4.69700e7 1.27713
\(49\) 5.76480e6 0.142857
\(50\) 6.55583e7 1.48341
\(51\) 5.68323e7 1.17633
\(52\) −3.06473e7 −0.581270
\(53\) 2.46033e7 0.428304 0.214152 0.976800i \(-0.431301\pi\)
0.214152 + 0.976800i \(0.431301\pi\)
\(54\) 1.11515e8 1.78469
\(55\) −6.98659e6 −0.102952
\(56\) −5.36307e7 −0.728732
\(57\) 9.12784e7 1.14533
\(58\) 2.38727e8 2.76998
\(59\) 1.19261e8 1.28135 0.640673 0.767814i \(-0.278656\pi\)
0.640673 + 0.767814i \(0.278656\pi\)
\(60\) 8.20895e7 0.817725
\(61\) −1.74738e8 −1.61586 −0.807930 0.589278i \(-0.799412\pi\)
−0.807930 + 0.589278i \(0.799412\pi\)
\(62\) −6.16647e7 −0.529998
\(63\) −1.40655e6 −0.0112492
\(64\) −9.05995e7 −0.675019
\(65\) −1.58109e7 −0.109862
\(66\) 6.94364e7 0.450442
\(67\) −1.94831e8 −1.18120 −0.590598 0.806966i \(-0.701108\pi\)
−0.590598 + 0.806966i \(0.701108\pi\)
\(68\) 4.41296e8 2.50288
\(69\) 3.21886e8 1.70955
\(70\) −5.29172e7 −0.263424
\(71\) 2.88148e8 1.34571 0.672857 0.739773i \(-0.265067\pi\)
0.672857 + 0.739773i \(0.265067\pi\)
\(72\) 1.30853e7 0.0573836
\(73\) 7.11716e7 0.293328 0.146664 0.989186i \(-0.453146\pi\)
0.146664 + 0.989186i \(0.453146\pi\)
\(74\) −1.77615e8 −0.688551
\(75\) −2.27557e8 −0.830454
\(76\) 7.08766e8 2.43692
\(77\) −3.03021e7 −0.0982347
\(78\) 1.57137e8 0.480677
\(79\) 2.60602e8 0.752759 0.376380 0.926466i \(-0.377169\pi\)
0.376380 + 0.926466i \(0.377169\pi\)
\(80\) 1.88157e8 0.513588
\(81\) −3.75547e8 −0.969352
\(82\) 1.28036e8 0.312729
\(83\) 4.08082e8 0.943834 0.471917 0.881643i \(-0.343562\pi\)
0.471917 + 0.881643i \(0.343562\pi\)
\(84\) 3.56038e8 0.780260
\(85\) 2.27664e8 0.473052
\(86\) −8.90878e8 −1.75621
\(87\) −8.28639e8 −1.55070
\(88\) 2.81905e8 0.501107
\(89\) 7.99234e7 0.135027 0.0675133 0.997718i \(-0.478494\pi\)
0.0675133 + 0.997718i \(0.478494\pi\)
\(90\) 1.29112e7 0.0207432
\(91\) −6.85750e7 −0.104828
\(92\) 2.49941e9 3.63740
\(93\) 2.14043e8 0.296707
\(94\) −7.75258e8 −1.02417
\(95\) 3.65651e8 0.460586
\(96\) −2.89568e8 −0.347960
\(97\) 7.73718e8 0.887380 0.443690 0.896180i \(-0.353669\pi\)
0.443690 + 0.896180i \(0.353669\pi\)
\(98\) −2.29512e8 −0.251355
\(99\) 7.39339e6 0.00773544
\(100\) −1.76696e9 −1.76696
\(101\) 1.62529e9 1.55412 0.777062 0.629425i \(-0.216709\pi\)
0.777062 + 0.629425i \(0.216709\pi\)
\(102\) −2.26265e9 −2.06974
\(103\) −1.32659e8 −0.116137 −0.0580685 0.998313i \(-0.518494\pi\)
−0.0580685 + 0.998313i \(0.518494\pi\)
\(104\) 6.37962e8 0.534743
\(105\) 1.83679e8 0.147472
\(106\) −9.79523e8 −0.753595
\(107\) −7.59386e8 −0.560062 −0.280031 0.959991i \(-0.590345\pi\)
−0.280031 + 0.959991i \(0.590345\pi\)
\(108\) −3.00561e9 −2.12582
\(109\) −6.18958e8 −0.419993 −0.209996 0.977702i \(-0.567345\pi\)
−0.209996 + 0.977702i \(0.567345\pi\)
\(110\) 2.78155e8 0.181142
\(111\) 6.16513e8 0.385468
\(112\) 8.16071e8 0.490057
\(113\) −4.93344e8 −0.284641 −0.142320 0.989821i \(-0.545456\pi\)
−0.142320 + 0.989821i \(0.545456\pi\)
\(114\) −3.63404e9 −2.01520
\(115\) 1.28944e9 0.687481
\(116\) −6.43428e9 −3.29943
\(117\) 1.67315e7 0.00825466
\(118\) −4.74812e9 −2.25451
\(119\) 9.87422e8 0.451379
\(120\) −1.70880e9 −0.752271
\(121\) −2.19867e9 −0.932450
\(122\) 6.95680e9 2.84309
\(123\) −4.44420e8 −0.175074
\(124\) 1.66202e9 0.631302
\(125\) −1.99279e9 −0.730073
\(126\) 5.59984e7 0.0197929
\(127\) 3.76640e9 1.28473 0.642363 0.766401i \(-0.277954\pi\)
0.642363 + 0.766401i \(0.277954\pi\)
\(128\) 4.67985e9 1.54094
\(129\) 3.09230e9 0.983168
\(130\) 6.29475e8 0.193301
\(131\) 1.18325e9 0.351039 0.175519 0.984476i \(-0.443840\pi\)
0.175519 + 0.984476i \(0.443840\pi\)
\(132\) −1.87148e9 −0.536540
\(133\) 1.58590e9 0.439484
\(134\) 7.75675e9 2.07830
\(135\) −1.55059e9 −0.401786
\(136\) −9.18612e9 −2.30254
\(137\) 4.33363e9 1.05102 0.525508 0.850788i \(-0.323875\pi\)
0.525508 + 0.850788i \(0.323875\pi\)
\(138\) −1.28151e10 −3.00793
\(139\) 6.24361e9 1.41863 0.709315 0.704892i \(-0.249005\pi\)
0.709315 + 0.704892i \(0.249005\pi\)
\(140\) 1.42625e9 0.313776
\(141\) 2.69097e9 0.573355
\(142\) −1.14719e10 −2.36777
\(143\) 3.60458e8 0.0720846
\(144\) −1.99112e8 −0.0385893
\(145\) −3.31944e9 −0.623604
\(146\) −2.83353e9 −0.516107
\(147\) 7.96653e8 0.140715
\(148\) 4.78715e9 0.820161
\(149\) 4.71550e9 0.783771 0.391885 0.920014i \(-0.371823\pi\)
0.391885 + 0.920014i \(0.371823\pi\)
\(150\) 9.05967e9 1.46117
\(151\) 1.78637e9 0.279625 0.139813 0.990178i \(-0.455350\pi\)
0.139813 + 0.990178i \(0.455350\pi\)
\(152\) −1.47538e10 −2.24186
\(153\) −2.40920e8 −0.0355436
\(154\) 1.20641e9 0.172843
\(155\) 8.57432e8 0.119318
\(156\) −4.23523e9 −0.572554
\(157\) 5.27823e9 0.693330 0.346665 0.937989i \(-0.387314\pi\)
0.346665 + 0.937989i \(0.387314\pi\)
\(158\) −1.03753e10 −1.32447
\(159\) 3.39999e9 0.421882
\(160\) −1.15998e9 −0.139930
\(161\) 5.59255e9 0.655984
\(162\) 1.49515e10 1.70556
\(163\) 1.65396e10 1.83518 0.917592 0.397523i \(-0.130130\pi\)
0.917592 + 0.397523i \(0.130130\pi\)
\(164\) −3.45087e9 −0.372504
\(165\) −9.65494e8 −0.101408
\(166\) −1.62468e10 −1.66066
\(167\) 4.17251e9 0.415119 0.207560 0.978222i \(-0.433448\pi\)
0.207560 + 0.978222i \(0.433448\pi\)
\(168\) −7.41137e9 −0.717805
\(169\) 8.15731e8 0.0769231
\(170\) −9.06391e9 −0.832330
\(171\) −3.86942e8 −0.0346069
\(172\) 2.40113e10 2.09189
\(173\) −5.90018e9 −0.500792 −0.250396 0.968143i \(-0.580561\pi\)
−0.250396 + 0.968143i \(0.580561\pi\)
\(174\) 3.29903e10 2.72844
\(175\) −3.95365e9 −0.318660
\(176\) −4.28960e9 −0.336985
\(177\) 1.64810e10 1.26213
\(178\) −3.18197e9 −0.237578
\(179\) 1.17087e10 0.852453 0.426226 0.904617i \(-0.359843\pi\)
0.426226 + 0.904617i \(0.359843\pi\)
\(180\) −3.47989e8 −0.0247081
\(181\) −1.70008e10 −1.17738 −0.588689 0.808359i \(-0.700356\pi\)
−0.588689 + 0.808359i \(0.700356\pi\)
\(182\) 2.73015e9 0.184444
\(183\) −2.41475e10 −1.59163
\(184\) −5.20282e10 −3.34625
\(185\) 2.46968e9 0.155013
\(186\) −8.52161e9 −0.522052
\(187\) −5.19029e9 −0.310388
\(188\) 2.08951e10 1.21993
\(189\) −6.72520e9 −0.383378
\(190\) −1.45576e10 −0.810396
\(191\) −3.30137e9 −0.179492 −0.0897459 0.995965i \(-0.528605\pi\)
−0.0897459 + 0.995965i \(0.528605\pi\)
\(192\) −1.25202e10 −0.664898
\(193\) −9.15988e9 −0.475206 −0.237603 0.971362i \(-0.576362\pi\)
−0.237603 + 0.971362i \(0.576362\pi\)
\(194\) −3.08038e10 −1.56133
\(195\) −2.18495e9 −0.108215
\(196\) 6.18591e9 0.299400
\(197\) 2.22367e10 1.05190 0.525949 0.850516i \(-0.323710\pi\)
0.525949 + 0.850516i \(0.323710\pi\)
\(198\) −2.94350e8 −0.0136104
\(199\) −3.44398e10 −1.55676 −0.778380 0.627793i \(-0.783958\pi\)
−0.778380 + 0.627793i \(0.783958\pi\)
\(200\) 3.67814e10 1.62552
\(201\) −2.69242e10 −1.16349
\(202\) −6.47072e10 −2.73446
\(203\) −1.43970e10 −0.595033
\(204\) 6.09838e10 2.46535
\(205\) −1.78030e9 −0.0704045
\(206\) 5.28153e9 0.204342
\(207\) −1.36452e9 −0.0516551
\(208\) −9.70754e9 −0.359604
\(209\) −8.33613e9 −0.302208
\(210\) −7.31277e9 −0.259475
\(211\) 4.62942e10 1.60789 0.803944 0.594705i \(-0.202731\pi\)
0.803944 + 0.594705i \(0.202731\pi\)
\(212\) 2.64005e10 0.897638
\(213\) 3.98199e10 1.32554
\(214\) 3.02332e10 0.985422
\(215\) 1.23874e10 0.395374
\(216\) 6.25655e10 1.95566
\(217\) 3.71884e9 0.113852
\(218\) 2.46424e10 0.738972
\(219\) 9.83538e9 0.288930
\(220\) −7.49694e9 −0.215766
\(221\) −1.17458e10 −0.331222
\(222\) −2.45450e10 −0.678227
\(223\) −4.99563e10 −1.35275 −0.676376 0.736556i \(-0.736451\pi\)
−0.676376 + 0.736556i \(0.736451\pi\)
\(224\) −5.03104e9 −0.133519
\(225\) 9.64648e8 0.0250927
\(226\) 1.96413e10 0.500822
\(227\) 4.33225e10 1.08292 0.541462 0.840726i \(-0.317871\pi\)
0.541462 + 0.840726i \(0.317871\pi\)
\(228\) 9.79462e10 2.40038
\(229\) 2.15084e10 0.516832 0.258416 0.966034i \(-0.416800\pi\)
0.258416 + 0.966034i \(0.416800\pi\)
\(230\) −5.13361e10 −1.20962
\(231\) −4.18753e9 −0.0967618
\(232\) 1.33938e11 3.03533
\(233\) −5.05216e10 −1.12299 −0.561494 0.827481i \(-0.689773\pi\)
−0.561494 + 0.827481i \(0.689773\pi\)
\(234\) −6.66127e8 −0.0145240
\(235\) 1.07798e10 0.230570
\(236\) 1.27973e11 2.68544
\(237\) 3.60133e10 0.741473
\(238\) −3.93119e10 −0.794196
\(239\) 8.44479e10 1.67417 0.837083 0.547076i \(-0.184259\pi\)
0.837083 + 0.547076i \(0.184259\pi\)
\(240\) 2.60019e10 0.505887
\(241\) −6.36142e10 −1.21472 −0.607362 0.794426i \(-0.707772\pi\)
−0.607362 + 0.794426i \(0.707772\pi\)
\(242\) 8.75348e10 1.64063
\(243\) 3.23432e9 0.0595053
\(244\) −1.87503e11 −3.38652
\(245\) 3.19130e9 0.0565875
\(246\) 1.76936e10 0.308040
\(247\) −1.88650e10 −0.322493
\(248\) −3.45969e10 −0.580771
\(249\) 5.63938e10 0.929682
\(250\) 7.93383e10 1.28456
\(251\) −8.46103e10 −1.34552 −0.672762 0.739859i \(-0.734892\pi\)
−0.672762 + 0.739859i \(0.734892\pi\)
\(252\) −1.50929e9 −0.0235761
\(253\) −2.93967e10 −0.451083
\(254\) −1.49951e11 −2.26046
\(255\) 3.14615e10 0.465960
\(256\) −1.39930e11 −2.03625
\(257\) 5.29639e10 0.757322 0.378661 0.925535i \(-0.376385\pi\)
0.378661 + 0.925535i \(0.376385\pi\)
\(258\) −1.23113e11 −1.72987
\(259\) 1.07115e10 0.147911
\(260\) −1.69659e10 −0.230248
\(261\) 3.51272e9 0.0468555
\(262\) −4.71083e10 −0.617649
\(263\) 5.51620e10 0.710950 0.355475 0.934686i \(-0.384319\pi\)
0.355475 + 0.934686i \(0.384319\pi\)
\(264\) 3.89572e10 0.493594
\(265\) 1.36200e10 0.169657
\(266\) −6.31389e10 −0.773267
\(267\) 1.10448e10 0.133002
\(268\) −2.09063e11 −2.47555
\(269\) 7.68173e10 0.894486 0.447243 0.894412i \(-0.352406\pi\)
0.447243 + 0.894412i \(0.352406\pi\)
\(270\) 6.17331e10 0.706938
\(271\) 6.23826e10 0.702590 0.351295 0.936265i \(-0.385741\pi\)
0.351295 + 0.936265i \(0.385741\pi\)
\(272\) 1.39780e11 1.54841
\(273\) −9.47655e9 −0.103257
\(274\) −1.72533e11 −1.84925
\(275\) 2.07820e10 0.219124
\(276\) 3.45399e11 3.58287
\(277\) −8.55324e10 −0.872915 −0.436457 0.899725i \(-0.643767\pi\)
−0.436457 + 0.899725i \(0.643767\pi\)
\(278\) −2.48575e11 −2.49606
\(279\) −9.07357e8 −0.00896518
\(280\) −2.96891e10 −0.288660
\(281\) −1.43962e10 −0.137743 −0.0688713 0.997626i \(-0.521940\pi\)
−0.0688713 + 0.997626i \(0.521940\pi\)
\(282\) −1.07135e11 −1.00881
\(283\) −1.67936e11 −1.55635 −0.778173 0.628050i \(-0.783853\pi\)
−0.778173 + 0.628050i \(0.783853\pi\)
\(284\) 3.09196e11 2.82034
\(285\) 5.05303e10 0.453680
\(286\) −1.43508e10 −0.126832
\(287\) −7.72149e9 −0.0671789
\(288\) 1.22752e9 0.0105138
\(289\) 5.05424e10 0.426202
\(290\) 1.32156e11 1.09722
\(291\) 1.06922e11 0.874075
\(292\) 7.63705e10 0.614756
\(293\) −2.04851e10 −0.162381 −0.0811904 0.996699i \(-0.525872\pi\)
−0.0811904 + 0.996699i \(0.525872\pi\)
\(294\) −3.17169e10 −0.247587
\(295\) 6.60213e10 0.507557
\(296\) −9.96504e10 −0.754512
\(297\) 3.53504e10 0.263627
\(298\) −1.87736e11 −1.37904
\(299\) −6.65259e10 −0.481361
\(300\) −2.44180e11 −1.74046
\(301\) 5.37266e10 0.377259
\(302\) −7.11203e10 −0.491997
\(303\) 2.24603e11 1.53082
\(304\) 2.24502e11 1.50761
\(305\) −9.67324e10 −0.640063
\(306\) 9.59167e9 0.0625386
\(307\) −1.23744e11 −0.795064 −0.397532 0.917588i \(-0.630133\pi\)
−0.397532 + 0.917588i \(0.630133\pi\)
\(308\) −3.25157e10 −0.205880
\(309\) −1.83325e10 −0.114396
\(310\) −3.41366e10 −0.209939
\(311\) −3.78281e10 −0.229294 −0.114647 0.993406i \(-0.536574\pi\)
−0.114647 + 0.993406i \(0.536574\pi\)
\(312\) 8.81616e10 0.526725
\(313\) −3.07459e11 −1.81066 −0.905332 0.424704i \(-0.860378\pi\)
−0.905332 + 0.424704i \(0.860378\pi\)
\(314\) −2.10140e11 −1.21990
\(315\) −7.78643e8 −0.00445596
\(316\) 2.79639e11 1.57763
\(317\) −2.01235e11 −1.11927 −0.559637 0.828738i \(-0.689059\pi\)
−0.559637 + 0.828738i \(0.689059\pi\)
\(318\) −1.35363e11 −0.742296
\(319\) 7.56767e10 0.409170
\(320\) −5.01545e10 −0.267384
\(321\) −1.04942e11 −0.551664
\(322\) −2.22654e11 −1.15420
\(323\) 2.71640e11 1.38862
\(324\) −4.02980e11 −2.03156
\(325\) 4.70305e10 0.233832
\(326\) −6.58484e11 −3.22899
\(327\) −8.55354e10 −0.413696
\(328\) 7.18341e10 0.342688
\(329\) 4.67538e10 0.220007
\(330\) 3.84389e10 0.178426
\(331\) 1.53397e11 0.702408 0.351204 0.936299i \(-0.385772\pi\)
0.351204 + 0.936299i \(0.385772\pi\)
\(332\) 4.37891e11 1.97809
\(333\) −2.61348e9 −0.0116472
\(334\) −1.66119e11 −0.730397
\(335\) −1.07856e11 −0.467887
\(336\) 1.12775e11 0.482710
\(337\) 1.76243e11 0.744350 0.372175 0.928163i \(-0.378612\pi\)
0.372175 + 0.928163i \(0.378612\pi\)
\(338\) −3.24764e10 −0.135345
\(339\) −6.81765e10 −0.280373
\(340\) 2.44295e11 0.991422
\(341\) −1.95478e10 −0.0782893
\(342\) 1.54052e10 0.0608905
\(343\) 1.38413e10 0.0539949
\(344\) −4.99826e11 −1.92445
\(345\) 1.78191e11 0.677173
\(346\) 2.34902e11 0.881138
\(347\) −1.75261e11 −0.648936 −0.324468 0.945897i \(-0.605185\pi\)
−0.324468 + 0.945897i \(0.605185\pi\)
\(348\) −8.89170e11 −3.24996
\(349\) −1.50786e11 −0.544059 −0.272030 0.962289i \(-0.587695\pi\)
−0.272030 + 0.962289i \(0.587695\pi\)
\(350\) 1.57405e11 0.560678
\(351\) 7.99994e10 0.281322
\(352\) 2.64452e10 0.0918131
\(353\) 1.26764e11 0.434519 0.217259 0.976114i \(-0.430288\pi\)
0.217259 + 0.976114i \(0.430288\pi\)
\(354\) −6.56154e11 −2.22071
\(355\) 1.59514e11 0.533054
\(356\) 8.57617e10 0.282988
\(357\) 1.36454e11 0.444611
\(358\) −4.66155e11 −1.49988
\(359\) −2.28942e10 −0.0727446 −0.0363723 0.999338i \(-0.511580\pi\)
−0.0363723 + 0.999338i \(0.511580\pi\)
\(360\) 7.24382e9 0.0227304
\(361\) 1.13594e11 0.352024
\(362\) 6.76848e11 2.07158
\(363\) −3.03839e11 −0.918469
\(364\) −7.35843e10 −0.219699
\(365\) 3.93995e10 0.116191
\(366\) 9.61378e11 2.80046
\(367\) −3.45619e11 −0.994489 −0.497244 0.867611i \(-0.665655\pi\)
−0.497244 + 0.867611i \(0.665655\pi\)
\(368\) 7.91687e11 2.25029
\(369\) 1.88396e9 0.00528997
\(370\) −9.83247e10 −0.272744
\(371\) 5.90725e10 0.161884
\(372\) 2.29678e11 0.621837
\(373\) 1.47684e11 0.395043 0.197521 0.980299i \(-0.436711\pi\)
0.197521 + 0.980299i \(0.436711\pi\)
\(374\) 2.06639e11 0.546123
\(375\) −2.75389e11 −0.719127
\(376\) −4.34957e11 −1.12228
\(377\) 1.71259e11 0.436635
\(378\) 2.67748e11 0.674549
\(379\) 5.23972e11 1.30446 0.652232 0.758020i \(-0.273833\pi\)
0.652232 + 0.758020i \(0.273833\pi\)
\(380\) 3.92362e11 0.965296
\(381\) 5.20489e11 1.26546
\(382\) 1.31436e11 0.315813
\(383\) −1.24106e11 −0.294712 −0.147356 0.989084i \(-0.547076\pi\)
−0.147356 + 0.989084i \(0.547076\pi\)
\(384\) 6.46720e11 1.51784
\(385\) −1.67748e10 −0.0389120
\(386\) 3.64679e11 0.836120
\(387\) −1.31087e10 −0.0297071
\(388\) 8.30237e11 1.85977
\(389\) −5.73191e11 −1.26919 −0.634594 0.772845i \(-0.718833\pi\)
−0.634594 + 0.772845i \(0.718833\pi\)
\(390\) 8.69888e10 0.190402
\(391\) 9.57917e11 2.07268
\(392\) −1.28767e11 −0.275435
\(393\) 1.63516e11 0.345776
\(394\) −8.85304e11 −1.85080
\(395\) 1.44265e11 0.298178
\(396\) 7.93346e9 0.0162119
\(397\) 3.61128e11 0.729632 0.364816 0.931080i \(-0.381132\pi\)
0.364816 + 0.931080i \(0.381132\pi\)
\(398\) 1.37114e12 2.73910
\(399\) 2.19159e11 0.432895
\(400\) −5.59683e11 −1.09313
\(401\) −2.67566e11 −0.516751 −0.258376 0.966045i \(-0.583187\pi\)
−0.258376 + 0.966045i \(0.583187\pi\)
\(402\) 1.07193e12 2.04714
\(403\) −4.42374e10 −0.0835442
\(404\) 1.74402e12 3.25713
\(405\) −2.07897e11 −0.383972
\(406\) 5.73184e11 1.04695
\(407\) −5.63039e10 −0.101710
\(408\) −1.26945e12 −2.26802
\(409\) −8.10480e11 −1.43215 −0.716073 0.698025i \(-0.754062\pi\)
−0.716073 + 0.698025i \(0.754062\pi\)
\(410\) 7.08785e10 0.123876
\(411\) 5.98876e11 1.03526
\(412\) −1.42350e11 −0.243400
\(413\) 2.86347e11 0.484303
\(414\) 5.43252e10 0.0908865
\(415\) 2.25908e11 0.373865
\(416\) 5.98465e10 0.0979758
\(417\) 8.62820e11 1.39736
\(418\) 3.31884e11 0.531732
\(419\) 5.82263e11 0.922904 0.461452 0.887165i \(-0.347329\pi\)
0.461452 + 0.887165i \(0.347329\pi\)
\(420\) 1.97097e11 0.309071
\(421\) −1.06151e12 −1.64685 −0.823427 0.567422i \(-0.807941\pi\)
−0.823427 + 0.567422i \(0.807941\pi\)
\(422\) −1.84310e12 −2.82906
\(423\) −1.14074e10 −0.0173243
\(424\) −5.49560e11 −0.825788
\(425\) −6.77200e11 −1.00685
\(426\) −1.58534e12 −2.33227
\(427\) −4.19547e11 −0.610738
\(428\) −8.14858e11 −1.17378
\(429\) 4.98126e10 0.0710038
\(430\) −4.93176e11 −0.695655
\(431\) 5.70015e11 0.795681 0.397840 0.917455i \(-0.369760\pi\)
0.397840 + 0.917455i \(0.369760\pi\)
\(432\) −9.52026e11 −1.31514
\(433\) −8.14009e11 −1.11284 −0.556421 0.830900i \(-0.687826\pi\)
−0.556421 + 0.830900i \(0.687826\pi\)
\(434\) −1.48057e11 −0.200320
\(435\) −4.58722e11 −0.614254
\(436\) −6.64172e11 −0.880220
\(437\) 1.53851e12 2.01806
\(438\) −3.91573e11 −0.508369
\(439\) −4.20664e11 −0.540562 −0.270281 0.962781i \(-0.587117\pi\)
−0.270281 + 0.962781i \(0.587117\pi\)
\(440\) 1.56058e11 0.198495
\(441\) −3.37712e9 −0.00425180
\(442\) 4.67633e11 0.582781
\(443\) 3.07063e11 0.378800 0.189400 0.981900i \(-0.439346\pi\)
0.189400 + 0.981900i \(0.439346\pi\)
\(444\) 6.61548e11 0.807864
\(445\) 4.42444e10 0.0534857
\(446\) 1.98889e12 2.38015
\(447\) 6.51646e11 0.772019
\(448\) −2.17529e11 −0.255133
\(449\) −1.57334e12 −1.82690 −0.913448 0.406956i \(-0.866590\pi\)
−0.913448 + 0.406956i \(0.866590\pi\)
\(450\) −3.84052e10 −0.0441503
\(451\) 4.05873e10 0.0461951
\(452\) −5.29382e11 −0.596549
\(453\) 2.46864e11 0.275432
\(454\) −1.72479e12 −1.90539
\(455\) −3.79620e10 −0.0415239
\(456\) −2.03887e12 −2.20825
\(457\) −7.24916e11 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(458\) −8.56308e11 −0.909359
\(459\) −1.15192e12 −1.21134
\(460\) 1.38363e12 1.44082
\(461\) 1.66071e11 0.171254 0.0856268 0.996327i \(-0.472711\pi\)
0.0856268 + 0.996327i \(0.472711\pi\)
\(462\) 1.66717e11 0.170251
\(463\) 9.76413e11 0.987459 0.493729 0.869616i \(-0.335633\pi\)
0.493729 + 0.869616i \(0.335633\pi\)
\(464\) −2.03806e12 −2.04120
\(465\) 1.18491e11 0.117529
\(466\) 2.01140e12 1.97588
\(467\) 1.33384e12 1.29771 0.648854 0.760913i \(-0.275248\pi\)
0.648854 + 0.760913i \(0.275248\pi\)
\(468\) 1.79537e10 0.0173001
\(469\) −4.67790e11 −0.446450
\(470\) −4.29171e11 −0.405686
\(471\) 7.29412e11 0.682934
\(472\) −2.66392e12 −2.47049
\(473\) −2.82409e11 −0.259420
\(474\) −1.43378e12 −1.30461
\(475\) −1.08765e12 −0.980322
\(476\) 1.05955e12 0.945999
\(477\) −1.44130e10 −0.0127474
\(478\) −3.36210e12 −2.94567
\(479\) 6.18218e11 0.536576 0.268288 0.963339i \(-0.413542\pi\)
0.268288 + 0.963339i \(0.413542\pi\)
\(480\) −1.60300e11 −0.137831
\(481\) −1.27418e11 −0.108537
\(482\) 2.53265e12 2.13729
\(483\) 7.72848e11 0.646148
\(484\) −2.35928e12 −1.95423
\(485\) 4.28318e11 0.351503
\(486\) −1.28767e11 −0.104699
\(487\) −2.24518e12 −1.80872 −0.904359 0.426773i \(-0.859650\pi\)
−0.904359 + 0.426773i \(0.859650\pi\)
\(488\) 3.90310e12 3.11545
\(489\) 2.28564e12 1.80767
\(490\) −1.27054e11 −0.0995651
\(491\) 1.89681e12 1.47285 0.736423 0.676521i \(-0.236513\pi\)
0.736423 + 0.676521i \(0.236513\pi\)
\(492\) −4.76884e11 −0.366919
\(493\) −2.46599e12 −1.88010
\(494\) 7.51066e11 0.567423
\(495\) 4.09286e9 0.00306411
\(496\) 5.26443e11 0.390557
\(497\) 6.91843e11 0.508632
\(498\) −2.24519e12 −1.63577
\(499\) 1.20702e12 0.871491 0.435746 0.900070i \(-0.356485\pi\)
0.435746 + 0.900070i \(0.356485\pi\)
\(500\) −2.13836e12 −1.53009
\(501\) 5.76609e11 0.408895
\(502\) 3.36856e12 2.36743
\(503\) −1.00787e12 −0.702021 −0.351010 0.936372i \(-0.614162\pi\)
−0.351010 + 0.936372i \(0.614162\pi\)
\(504\) 3.14178e10 0.0216890
\(505\) 8.99737e11 0.615608
\(506\) 1.17036e12 0.793675
\(507\) 1.12728e11 0.0757697
\(508\) 4.04153e12 2.69252
\(509\) −2.32967e12 −1.53838 −0.769191 0.639019i \(-0.779340\pi\)
−0.769191 + 0.639019i \(0.779340\pi\)
\(510\) −1.25257e12 −0.819850
\(511\) 1.70883e11 0.110868
\(512\) 3.17492e12 2.04182
\(513\) −1.85011e12 −1.17942
\(514\) −2.10863e12 −1.33250
\(515\) −7.34382e10 −0.0460034
\(516\) 3.31819e12 2.06052
\(517\) −2.45757e11 −0.151286
\(518\) −4.26453e11 −0.260248
\(519\) −8.15361e11 −0.493284
\(520\) 3.53166e11 0.211818
\(521\) −1.71354e12 −1.01888 −0.509440 0.860506i \(-0.670148\pi\)
−0.509440 + 0.860506i \(0.670148\pi\)
\(522\) −1.39851e11 −0.0824418
\(523\) 2.58042e12 1.50811 0.754055 0.656812i \(-0.228095\pi\)
0.754055 + 0.656812i \(0.228095\pi\)
\(524\) 1.26968e12 0.735707
\(525\) −5.46365e11 −0.313882
\(526\) −2.19615e12 −1.25091
\(527\) 6.36981e11 0.359732
\(528\) −5.92791e11 −0.331932
\(529\) 3.62429e12 2.01220
\(530\) −5.42249e11 −0.298509
\(531\) −6.98655e10 −0.0381362
\(532\) 1.70175e12 0.921070
\(533\) 9.18507e10 0.0492958
\(534\) −4.39724e11 −0.234015
\(535\) −4.20384e11 −0.221848
\(536\) 4.35191e12 2.27740
\(537\) 1.61806e12 0.839671
\(538\) −3.05830e12 −1.57384
\(539\) −7.27554e10 −0.0371292
\(540\) −1.66386e12 −0.842062
\(541\) 1.61260e12 0.809356 0.404678 0.914459i \(-0.367384\pi\)
0.404678 + 0.914459i \(0.367384\pi\)
\(542\) −2.48362e12 −1.23620
\(543\) −2.34939e12 −1.15973
\(544\) −8.61740e11 −0.421872
\(545\) −3.42645e11 −0.166365
\(546\) 3.77287e11 0.181679
\(547\) −2.96173e12 −1.41450 −0.707249 0.706965i \(-0.750064\pi\)
−0.707249 + 0.706965i \(0.750064\pi\)
\(548\) 4.65020e12 2.20272
\(549\) 1.02365e11 0.0480922
\(550\) −8.27387e11 −0.385546
\(551\) −3.96063e12 −1.83055
\(552\) −7.18991e12 −3.29608
\(553\) 6.25706e11 0.284516
\(554\) 3.40527e12 1.53588
\(555\) 3.41292e11 0.152689
\(556\) 6.69969e12 2.97316
\(557\) −2.46069e12 −1.08320 −0.541600 0.840636i \(-0.682181\pi\)
−0.541600 + 0.840636i \(0.682181\pi\)
\(558\) 3.61243e10 0.0157741
\(559\) −6.39102e11 −0.276833
\(560\) 4.51764e11 0.194118
\(561\) −7.17260e11 −0.305734
\(562\) 5.73150e11 0.242356
\(563\) −2.01247e12 −0.844194 −0.422097 0.906551i \(-0.638706\pi\)
−0.422097 + 0.906551i \(0.638706\pi\)
\(564\) 2.88755e12 1.20164
\(565\) −2.73108e11 −0.112750
\(566\) 6.68600e12 2.73837
\(567\) −9.01688e11 −0.366380
\(568\) −6.43631e12 −2.59459
\(569\) −1.33203e11 −0.0532730 −0.0266365 0.999645i \(-0.508480\pi\)
−0.0266365 + 0.999645i \(0.508480\pi\)
\(570\) −2.01175e12 −0.798245
\(571\) 4.81051e12 1.89378 0.946888 0.321563i \(-0.104208\pi\)
0.946888 + 0.321563i \(0.104208\pi\)
\(572\) 3.86789e11 0.151075
\(573\) −4.56225e11 −0.176800
\(574\) 3.07413e11 0.118200
\(575\) −3.83552e12 −1.46325
\(576\) 5.30748e10 0.0200903
\(577\) 8.13533e11 0.305551 0.152776 0.988261i \(-0.451179\pi\)
0.152776 + 0.988261i \(0.451179\pi\)
\(578\) −2.01223e12 −0.749897
\(579\) −1.26583e12 −0.468081
\(580\) −3.56192e12 −1.30695
\(581\) 9.79804e11 0.356736
\(582\) −4.25685e12 −1.53792
\(583\) −3.10509e11 −0.111318
\(584\) −1.58975e12 −0.565549
\(585\) 9.26232e9 0.00326978
\(586\) 8.15568e11 0.285707
\(587\) −2.69805e12 −0.937947 −0.468973 0.883212i \(-0.655376\pi\)
−0.468973 + 0.883212i \(0.655376\pi\)
\(588\) 8.54847e11 0.294911
\(589\) 1.02306e12 0.350252
\(590\) −2.62848e12 −0.893041
\(591\) 3.07295e12 1.03613
\(592\) 1.51633e12 0.507395
\(593\) −3.52113e12 −1.16933 −0.584664 0.811276i \(-0.698774\pi\)
−0.584664 + 0.811276i \(0.698774\pi\)
\(594\) −1.40739e12 −0.463849
\(595\) 5.46621e11 0.178797
\(596\) 5.05996e12 1.64263
\(597\) −4.75932e12 −1.53342
\(598\) 2.64857e12 0.846948
\(599\) 1.64754e12 0.522895 0.261448 0.965218i \(-0.415800\pi\)
0.261448 + 0.965218i \(0.415800\pi\)
\(600\) 5.08291e12 1.60115
\(601\) 6.24386e12 1.95217 0.976085 0.217387i \(-0.0697535\pi\)
0.976085 + 0.217387i \(0.0697535\pi\)
\(602\) −2.13900e12 −0.663783
\(603\) 1.14136e11 0.0351555
\(604\) 1.91687e12 0.586037
\(605\) −1.21715e12 −0.369355
\(606\) −8.94206e12 −2.69346
\(607\) −4.61694e12 −1.38040 −0.690200 0.723618i \(-0.742478\pi\)
−0.690200 + 0.723618i \(0.742478\pi\)
\(608\) −1.38404e12 −0.410755
\(609\) −1.98956e12 −0.586111
\(610\) 3.85117e12 1.12618
\(611\) −5.56158e11 −0.161441
\(612\) −2.58519e11 −0.0744922
\(613\) 3.90806e12 1.11786 0.558932 0.829214i \(-0.311211\pi\)
0.558932 + 0.829214i \(0.311211\pi\)
\(614\) 4.92659e12 1.39891
\(615\) −2.46024e11 −0.0693489
\(616\) 6.76854e11 0.189401
\(617\) −2.71463e12 −0.754098 −0.377049 0.926193i \(-0.623061\pi\)
−0.377049 + 0.926193i \(0.623061\pi\)
\(618\) 7.29868e11 0.201278
\(619\) −1.75668e12 −0.480932 −0.240466 0.970658i \(-0.577300\pi\)
−0.240466 + 0.970658i \(0.577300\pi\)
\(620\) 9.20066e11 0.250067
\(621\) −6.52425e12 −1.76043
\(622\) 1.50604e12 0.403440
\(623\) 1.91896e11 0.0510352
\(624\) −1.34151e12 −0.354212
\(625\) 2.11297e12 0.553903
\(626\) 1.22408e13 3.18584
\(627\) −1.15199e12 −0.297677
\(628\) 5.66379e12 1.45308
\(629\) 1.83471e12 0.467348
\(630\) 3.09999e10 0.00784020
\(631\) −4.89370e12 −1.22887 −0.614434 0.788968i \(-0.710616\pi\)
−0.614434 + 0.788968i \(0.710616\pi\)
\(632\) −5.82103e12 −1.45135
\(633\) 6.39752e12 1.58378
\(634\) 8.01170e12 1.96935
\(635\) 2.08502e12 0.508896
\(636\) 3.64836e12 0.884179
\(637\) −1.64648e11 −0.0396214
\(638\) −3.01289e12 −0.719930
\(639\) −1.68802e11 −0.0400520
\(640\) 2.59069e12 0.610388
\(641\) −3.16251e12 −0.739896 −0.369948 0.929052i \(-0.620625\pi\)
−0.369948 + 0.929052i \(0.620625\pi\)
\(642\) 4.17800e12 0.970646
\(643\) 1.55515e12 0.358776 0.179388 0.983778i \(-0.442588\pi\)
0.179388 + 0.983778i \(0.442588\pi\)
\(644\) 6.00107e12 1.37481
\(645\) 1.71185e12 0.389446
\(646\) −1.08147e13 −2.44326
\(647\) −1.55484e12 −0.348832 −0.174416 0.984672i \(-0.555804\pi\)
−0.174416 + 0.984672i \(0.555804\pi\)
\(648\) 8.38852e12 1.86895
\(649\) −1.50516e12 −0.333028
\(650\) −1.87241e12 −0.411425
\(651\) 5.13916e11 0.112145
\(652\) 1.77478e13 3.84618
\(653\) −6.64297e11 −0.142973 −0.0714863 0.997442i \(-0.522774\pi\)
−0.0714863 + 0.997442i \(0.522774\pi\)
\(654\) 3.40539e12 0.727893
\(655\) 6.55028e11 0.139051
\(656\) −1.09306e12 −0.230451
\(657\) −4.16936e10 −0.00873021
\(658\) −1.86139e12 −0.387099
\(659\) −5.68774e12 −1.17478 −0.587388 0.809305i \(-0.699844\pi\)
−0.587388 + 0.809305i \(0.699844\pi\)
\(660\) −1.03602e12 −0.212530
\(661\) 6.80122e11 0.138574 0.0692868 0.997597i \(-0.477928\pi\)
0.0692868 + 0.997597i \(0.477928\pi\)
\(662\) −6.10713e12 −1.23588
\(663\) −1.62319e12 −0.326255
\(664\) −9.11525e12 −1.81975
\(665\) 8.77929e11 0.174085
\(666\) 1.04050e11 0.0204931
\(667\) −1.39668e13 −2.73232
\(668\) 4.47730e12 0.870006
\(669\) −6.90359e12 −1.33247
\(670\) 4.29402e12 0.823241
\(671\) 2.20531e12 0.419970
\(672\) −6.95252e11 −0.131517
\(673\) 1.80171e12 0.338545 0.169272 0.985569i \(-0.445858\pi\)
0.169272 + 0.985569i \(0.445858\pi\)
\(674\) −7.01670e12 −1.30967
\(675\) 4.61232e12 0.855170
\(676\) 8.75319e11 0.161215
\(677\) −9.29465e12 −1.70053 −0.850264 0.526356i \(-0.823558\pi\)
−0.850264 + 0.526356i \(0.823558\pi\)
\(678\) 2.71429e12 0.493313
\(679\) 1.85770e12 0.335398
\(680\) −5.08529e12 −0.912065
\(681\) 5.98685e12 1.06669
\(682\) 7.78248e11 0.137749
\(683\) −9.01059e12 −1.58438 −0.792191 0.610273i \(-0.791060\pi\)
−0.792191 + 0.610273i \(0.791060\pi\)
\(684\) −4.15207e11 −0.0725292
\(685\) 2.39903e12 0.416321
\(686\) −5.51059e11 −0.0950034
\(687\) 2.97231e12 0.509082
\(688\) 7.60559e12 1.29415
\(689\) −7.02695e11 −0.118790
\(690\) −7.09426e12 −1.19148
\(691\) −9.77754e12 −1.63147 −0.815734 0.578428i \(-0.803666\pi\)
−0.815734 + 0.578428i \(0.803666\pi\)
\(692\) −6.33118e12 −1.04956
\(693\) 1.77515e10 0.00292372
\(694\) 6.97759e12 1.14179
\(695\) 3.45636e12 0.561937
\(696\) 1.85092e13 2.98982
\(697\) −1.32257e12 −0.212262
\(698\) 6.00319e12 0.957266
\(699\) −6.98170e12 −1.10615
\(700\) −4.24246e12 −0.667847
\(701\) 8.29383e12 1.29725 0.648626 0.761108i \(-0.275344\pi\)
0.648626 + 0.761108i \(0.275344\pi\)
\(702\) −3.18499e12 −0.494984
\(703\) 2.94673e12 0.455032
\(704\) 1.14342e12 0.175441
\(705\) 1.48968e12 0.227113
\(706\) −5.04680e12 −0.764530
\(707\) 3.90233e12 0.587403
\(708\) 1.76850e13 2.64518
\(709\) 7.06044e12 1.04936 0.524679 0.851300i \(-0.324185\pi\)
0.524679 + 0.851300i \(0.324185\pi\)
\(710\) −6.35068e12 −0.937903
\(711\) −1.52665e11 −0.0224041
\(712\) −1.78524e12 −0.260337
\(713\) 3.60772e12 0.522794
\(714\) −5.43261e12 −0.782288
\(715\) 1.99544e11 0.0285536
\(716\) 1.25640e13 1.78657
\(717\) 1.16701e13 1.64906
\(718\) 9.11480e11 0.127993
\(719\) 6.47046e12 0.902932 0.451466 0.892288i \(-0.350901\pi\)
0.451466 + 0.892288i \(0.350901\pi\)
\(720\) −1.10226e11 −0.0152857
\(721\) −3.18515e11 −0.0438957
\(722\) −4.52247e12 −0.619382
\(723\) −8.79101e12 −1.19651
\(724\) −1.82427e13 −2.46755
\(725\) 9.87387e12 1.32729
\(726\) 1.20967e13 1.61603
\(727\) 4.63198e12 0.614981 0.307490 0.951551i \(-0.400511\pi\)
0.307490 + 0.951551i \(0.400511\pi\)
\(728\) 1.53175e12 0.202114
\(729\) 7.83884e12 1.02796
\(730\) −1.56860e12 −0.204437
\(731\) 9.20254e12 1.19201
\(732\) −2.59115e13 −3.33574
\(733\) 2.64815e12 0.338824 0.169412 0.985545i \(-0.445813\pi\)
0.169412 + 0.985545i \(0.445813\pi\)
\(734\) 1.37600e13 1.74979
\(735\) 4.41014e11 0.0557391
\(736\) −4.88071e12 −0.613102
\(737\) 2.45889e12 0.306998
\(738\) −7.50055e10 −0.00930763
\(739\) −3.40925e12 −0.420493 −0.210247 0.977648i \(-0.567427\pi\)
−0.210247 + 0.977648i \(0.567427\pi\)
\(740\) 2.65009e12 0.324876
\(741\) −2.60700e12 −0.317658
\(742\) −2.35183e12 −0.284832
\(743\) −1.42937e13 −1.72066 −0.860328 0.509741i \(-0.829741\pi\)
−0.860328 + 0.509741i \(0.829741\pi\)
\(744\) −4.78104e12 −0.572063
\(745\) 2.61042e12 0.310462
\(746\) −5.87970e12 −0.695073
\(747\) −2.39061e11 −0.0280910
\(748\) −5.56943e12 −0.650510
\(749\) −1.82329e12 −0.211683
\(750\) 1.09640e13 1.26529
\(751\) −2.17272e12 −0.249243 −0.124622 0.992204i \(-0.539772\pi\)
−0.124622 + 0.992204i \(0.539772\pi\)
\(752\) 6.61852e12 0.754711
\(753\) −1.16925e13 −1.32535
\(754\) −6.81829e12 −0.768253
\(755\) 9.88908e11 0.110763
\(756\) −7.21647e12 −0.803483
\(757\) −1.09645e12 −0.121355 −0.0606776 0.998157i \(-0.519326\pi\)
−0.0606776 + 0.998157i \(0.519326\pi\)
\(758\) −2.08607e13 −2.29519
\(759\) −4.06240e12 −0.444319
\(760\) −8.16749e12 −0.888030
\(761\) −1.37438e12 −0.148551 −0.0742754 0.997238i \(-0.523664\pi\)
−0.0742754 + 0.997238i \(0.523664\pi\)
\(762\) −2.07221e13 −2.22657
\(763\) −1.48612e12 −0.158742
\(764\) −3.54253e12 −0.376178
\(765\) −1.33370e11 −0.0140793
\(766\) 4.94099e12 0.518542
\(767\) −3.40623e12 −0.355381
\(768\) −1.93373e13 −2.00572
\(769\) 3.02105e12 0.311523 0.155761 0.987795i \(-0.450217\pi\)
0.155761 + 0.987795i \(0.450217\pi\)
\(770\) 6.67849e11 0.0684652
\(771\) 7.31922e12 0.745967
\(772\) −9.82900e12 −0.995936
\(773\) −1.04927e13 −1.05701 −0.528504 0.848931i \(-0.677247\pi\)
−0.528504 + 0.848931i \(0.677247\pi\)
\(774\) 5.21892e11 0.0522693
\(775\) −2.55048e12 −0.253959
\(776\) −1.72824e13 −1.71091
\(777\) 1.48025e12 0.145693
\(778\) 2.28203e13 2.23312
\(779\) −2.12419e12 −0.206668
\(780\) −2.34456e12 −0.226796
\(781\) −3.63661e12 −0.349757
\(782\) −3.81372e13 −3.64686
\(783\) 1.67955e13 1.59686
\(784\) 1.95939e12 0.185224
\(785\) 2.92194e12 0.274637
\(786\) −6.51002e12 −0.608388
\(787\) 7.81781e12 0.726439 0.363219 0.931704i \(-0.381678\pi\)
0.363219 + 0.931704i \(0.381678\pi\)
\(788\) 2.38611e13 2.20456
\(789\) 7.62298e12 0.700291
\(790\) −5.74359e12 −0.524640
\(791\) −1.18452e12 −0.107584
\(792\) −1.65145e11 −0.0149143
\(793\) 4.99070e12 0.448159
\(794\) −1.43775e13 −1.28378
\(795\) 1.88218e12 0.167113
\(796\) −3.69556e13 −3.26265
\(797\) 8.53286e12 0.749087 0.374543 0.927209i \(-0.377799\pi\)
0.374543 + 0.927209i \(0.377799\pi\)
\(798\) −8.72532e12 −0.761673
\(799\) 8.00821e12 0.695145
\(800\) 3.45042e12 0.297829
\(801\) −4.68205e10 −0.00401874
\(802\) 1.06525e13 0.909217
\(803\) −8.98230e11 −0.0762373
\(804\) −2.88910e13 −2.43843
\(805\) 3.09595e12 0.259844
\(806\) 1.76121e12 0.146995
\(807\) 1.06156e13 0.881075
\(808\) −3.63039e13 −2.99642
\(809\) 1.10666e13 0.908338 0.454169 0.890916i \(-0.349936\pi\)
0.454169 + 0.890916i \(0.349936\pi\)
\(810\) 8.27692e12 0.675595
\(811\) 5.03979e12 0.409090 0.204545 0.978857i \(-0.434429\pi\)
0.204545 + 0.978857i \(0.434429\pi\)
\(812\) −1.54487e13 −1.24707
\(813\) 8.62082e12 0.692056
\(814\) 2.24161e12 0.178958
\(815\) 9.15604e12 0.726940
\(816\) 1.93166e13 1.52520
\(817\) 1.47802e13 1.16060
\(818\) 3.22674e13 2.51984
\(819\) 4.01724e10 0.00311997
\(820\) −1.91035e12 −0.147554
\(821\) 8.10308e12 0.622452 0.311226 0.950336i \(-0.399260\pi\)
0.311226 + 0.950336i \(0.399260\pi\)
\(822\) −2.38428e13 −1.82152
\(823\) −1.29467e13 −0.983692 −0.491846 0.870682i \(-0.663678\pi\)
−0.491846 + 0.870682i \(0.663678\pi\)
\(824\) 2.96319e12 0.223917
\(825\) 2.87192e12 0.215839
\(826\) −1.14002e13 −0.852125
\(827\) 3.68205e12 0.273725 0.136862 0.990590i \(-0.456298\pi\)
0.136862 + 0.990590i \(0.456298\pi\)
\(828\) −1.46420e12 −0.108259
\(829\) 1.62662e13 1.19617 0.598084 0.801434i \(-0.295929\pi\)
0.598084 + 0.801434i \(0.295929\pi\)
\(830\) −8.99398e12 −0.657810
\(831\) −1.18199e13 −0.859826
\(832\) 2.58761e12 0.187217
\(833\) 2.37080e12 0.170605
\(834\) −3.43512e13 −2.45864
\(835\) 2.30983e12 0.164434
\(836\) −8.94507e12 −0.633367
\(837\) −4.33839e12 −0.305537
\(838\) −2.31815e13 −1.62384
\(839\) −1.27324e13 −0.887115 −0.443558 0.896246i \(-0.646284\pi\)
−0.443558 + 0.896246i \(0.646284\pi\)
\(840\) −4.10282e12 −0.284332
\(841\) 2.14481e13 1.47845
\(842\) 4.22616e13 2.89762
\(843\) −1.98944e12 −0.135677
\(844\) 4.96760e13 3.36981
\(845\) 4.51576e11 0.0304702
\(846\) 4.54160e11 0.0304819
\(847\) −5.27900e12 −0.352433
\(848\) 8.36237e12 0.555326
\(849\) −2.32076e13 −1.53301
\(850\) 2.69611e13 1.77155
\(851\) 1.03914e13 0.679191
\(852\) 4.27287e13 2.77806
\(853\) 2.12089e13 1.37166 0.685831 0.727761i \(-0.259439\pi\)
0.685831 + 0.727761i \(0.259439\pi\)
\(854\) 1.67033e13 1.07459
\(855\) −2.14205e11 −0.0137083
\(856\) 1.69623e13 1.07982
\(857\) 2.44933e13 1.55108 0.775540 0.631299i \(-0.217478\pi\)
0.775540 + 0.631299i \(0.217478\pi\)
\(858\) −1.98317e12 −0.124930
\(859\) −2.91849e13 −1.82889 −0.914447 0.404705i \(-0.867374\pi\)
−0.914447 + 0.404705i \(0.867374\pi\)
\(860\) 1.32923e13 0.828623
\(861\) −1.06705e12 −0.0661716
\(862\) −2.26938e13 −1.39999
\(863\) 9.87328e12 0.605917 0.302958 0.953004i \(-0.402026\pi\)
0.302958 + 0.953004i \(0.402026\pi\)
\(864\) 5.86919e12 0.358316
\(865\) −3.26625e12 −0.198370
\(866\) 3.24079e13 1.95803
\(867\) 6.98458e12 0.419811
\(868\) 3.99050e12 0.238610
\(869\) −3.28896e12 −0.195646
\(870\) 1.82629e13 1.08077
\(871\) 5.56458e12 0.327605
\(872\) 1.38256e13 0.809764
\(873\) −4.53257e11 −0.0264108
\(874\) −6.12523e13 −3.55075
\(875\) −4.78469e12 −0.275942
\(876\) 1.05538e13 0.605539
\(877\) 2.00460e13 1.14427 0.572137 0.820158i \(-0.306115\pi\)
0.572137 + 0.820158i \(0.306115\pi\)
\(878\) 1.67478e13 0.951112
\(879\) −2.83089e12 −0.159946
\(880\) −2.37466e12 −0.133484
\(881\) −1.74200e13 −0.974218 −0.487109 0.873341i \(-0.661949\pi\)
−0.487109 + 0.873341i \(0.661949\pi\)
\(882\) 1.34452e11 0.00748099
\(883\) −1.98206e13 −1.09722 −0.548609 0.836079i \(-0.684842\pi\)
−0.548609 + 0.836079i \(0.684842\pi\)
\(884\) −1.26039e13 −0.694174
\(885\) 9.12365e12 0.499947
\(886\) −1.22250e13 −0.666495
\(887\) −2.43482e11 −0.0132072 −0.00660361 0.999978i \(-0.502102\pi\)
−0.00660361 + 0.999978i \(0.502102\pi\)
\(888\) −1.37709e13 −0.743199
\(889\) 9.04314e12 0.485580
\(890\) −1.76149e12 −0.0941075
\(891\) 4.73964e12 0.251939
\(892\) −5.36055e13 −2.83510
\(893\) 1.28620e13 0.676826
\(894\) −2.59438e13 −1.35836
\(895\) 6.48176e12 0.337667
\(896\) 1.12363e13 0.582422
\(897\) −9.19339e12 −0.474143
\(898\) 6.26388e13 3.21440
\(899\) −9.28745e12 −0.474218
\(900\) 1.03511e12 0.0525892
\(901\) 1.01182e13 0.511496
\(902\) −1.61589e12 −0.0812797
\(903\) 7.42461e12 0.371603
\(904\) 1.10197e13 0.548799
\(905\) −9.41138e12 −0.466375
\(906\) −9.82830e12 −0.484620
\(907\) 3.25572e13 1.59740 0.798702 0.601727i \(-0.205520\pi\)
0.798702 + 0.601727i \(0.205520\pi\)
\(908\) 4.64872e13 2.26959
\(909\) −9.52125e11 −0.0462548
\(910\) 1.51137e12 0.0730608
\(911\) 3.73752e13 1.79784 0.898921 0.438112i \(-0.144353\pi\)
0.898921 + 0.438112i \(0.144353\pi\)
\(912\) 3.10244e13 1.48500
\(913\) −5.15025e12 −0.245307
\(914\) 2.88608e13 1.36789
\(915\) −1.33677e13 −0.630466
\(916\) 2.30796e13 1.08317
\(917\) 2.84098e12 0.132680
\(918\) 4.58612e13 2.13134
\(919\) −1.71224e13 −0.791854 −0.395927 0.918282i \(-0.629577\pi\)
−0.395927 + 0.918282i \(0.629577\pi\)
\(920\) −2.88020e13 −1.32549
\(921\) −1.71005e13 −0.783143
\(922\) −6.61173e12 −0.301319
\(923\) −8.22979e12 −0.373234
\(924\) −4.49342e12 −0.202793
\(925\) −7.34622e12 −0.329933
\(926\) −3.88736e13 −1.73742
\(927\) 7.77142e10 0.00345654
\(928\) 1.25645e13 0.556135
\(929\) 3.61729e13 1.59336 0.796678 0.604404i \(-0.206589\pi\)
0.796678 + 0.604404i \(0.206589\pi\)
\(930\) −4.71743e12 −0.206791
\(931\) 3.80774e12 0.166109
\(932\) −5.42121e13 −2.35356
\(933\) −5.22756e12 −0.225856
\(934\) −5.31037e13 −2.28330
\(935\) −2.87326e12 −0.122949
\(936\) −3.73729e11 −0.0159153
\(937\) −2.74411e13 −1.16298 −0.581492 0.813552i \(-0.697531\pi\)
−0.581492 + 0.813552i \(0.697531\pi\)
\(938\) 1.86240e13 0.785524
\(939\) −4.24886e13 −1.78352
\(940\) 1.15672e13 0.483229
\(941\) −3.63435e13 −1.51103 −0.755515 0.655131i \(-0.772613\pi\)
−0.755515 + 0.655131i \(0.772613\pi\)
\(942\) −2.90398e13 −1.20161
\(943\) −7.49077e12 −0.308478
\(944\) 4.05356e13 1.66135
\(945\) −3.72297e12 −0.151861
\(946\) 1.12434e13 0.456446
\(947\) −3.42860e13 −1.38530 −0.692648 0.721276i \(-0.743556\pi\)
−0.692648 + 0.721276i \(0.743556\pi\)
\(948\) 3.86440e13 1.55398
\(949\) −2.03273e12 −0.0813546
\(950\) 4.33023e13 1.72486
\(951\) −2.78092e13 −1.10249
\(952\) −2.20559e13 −0.870278
\(953\) 3.00422e13 1.17981 0.589907 0.807471i \(-0.299165\pi\)
0.589907 + 0.807471i \(0.299165\pi\)
\(954\) 5.73822e11 0.0224290
\(955\) −1.82759e12 −0.0710989
\(956\) 9.06167e13 3.50871
\(957\) 1.04580e13 0.403035
\(958\) −2.46129e13 −0.944100
\(959\) 1.04051e13 0.397247
\(960\) −6.93098e12 −0.263375
\(961\) −2.40406e13 −0.909265
\(962\) 5.07285e12 0.190970
\(963\) 4.44862e11 0.0166689
\(964\) −6.82611e13 −2.54581
\(965\) −5.07077e12 −0.188235
\(966\) −3.07691e13 −1.13689
\(967\) −2.34806e12 −0.0863555 −0.0431778 0.999067i \(-0.513748\pi\)
−0.0431778 + 0.999067i \(0.513748\pi\)
\(968\) 4.91113e13 1.79780
\(969\) 3.75387e13 1.36780
\(970\) −1.70525e13 −0.618465
\(971\) 1.90529e11 0.00687818 0.00343909 0.999994i \(-0.498905\pi\)
0.00343909 + 0.999994i \(0.498905\pi\)
\(972\) 3.47059e12 0.124711
\(973\) 1.49909e13 0.536192
\(974\) 8.93866e13 3.18242
\(975\) 6.49927e12 0.230326
\(976\) −5.93915e13 −2.09508
\(977\) −4.08114e13 −1.43303 −0.716516 0.697571i \(-0.754264\pi\)
−0.716516 + 0.697571i \(0.754264\pi\)
\(978\) −9.09976e13 −3.18057
\(979\) −1.00868e12 −0.0350940
\(980\) 3.42442e12 0.118596
\(981\) 3.62596e11 0.0125001
\(982\) −7.55171e13 −2.59146
\(983\) −4.98504e13 −1.70285 −0.851427 0.524473i \(-0.824262\pi\)
−0.851427 + 0.524473i \(0.824262\pi\)
\(984\) 9.92694e12 0.337550
\(985\) 1.23099e13 0.416670
\(986\) 9.81777e13 3.30801
\(987\) 6.46103e12 0.216708
\(988\) −2.02431e13 −0.675880
\(989\) 5.21212e13 1.73233
\(990\) −1.62948e11 −0.00539126
\(991\) −2.22205e13 −0.731852 −0.365926 0.930644i \(-0.619248\pi\)
−0.365926 + 0.930644i \(0.619248\pi\)
\(992\) −3.24550e12 −0.106409
\(993\) 2.11983e13 0.691877
\(994\) −2.75441e13 −0.894932
\(995\) −1.90653e13 −0.616653
\(996\) 6.05133e13 1.94843
\(997\) −6.99200e12 −0.224116 −0.112058 0.993702i \(-0.535744\pi\)
−0.112058 + 0.993702i \(0.535744\pi\)
\(998\) −4.80548e13 −1.53338
\(999\) −1.24960e13 −0.396941
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 91.10.a.c.1.1 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.c.1.1 14 1.1 even 1 trivial