Properties

Label 91.10.a.a.1.10
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(29.7766\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+27.7766 q^{2} +192.937 q^{3} +259.538 q^{4} -1979.40 q^{5} +5359.14 q^{6} +2401.00 q^{7} -7012.53 q^{8} +17541.9 q^{9} +O(q^{10})\) \(q+27.7766 q^{2} +192.937 q^{3} +259.538 q^{4} -1979.40 q^{5} +5359.14 q^{6} +2401.00 q^{7} -7012.53 q^{8} +17541.9 q^{9} -54980.9 q^{10} -64493.1 q^{11} +50074.6 q^{12} +28561.0 q^{13} +66691.5 q^{14} -381900. q^{15} -327667. q^{16} -73609.6 q^{17} +487253. q^{18} +478701. q^{19} -513729. q^{20} +463243. q^{21} -1.79140e6 q^{22} -1.93873e6 q^{23} -1.35298e6 q^{24} +1.96490e6 q^{25} +793327. q^{26} -413105. q^{27} +623151. q^{28} -4.36298e6 q^{29} -1.06079e7 q^{30} -3.84855e6 q^{31} -5.51106e6 q^{32} -1.24431e7 q^{33} -2.04462e6 q^{34} -4.75254e6 q^{35} +4.55278e6 q^{36} -9.08052e6 q^{37} +1.32967e7 q^{38} +5.51049e6 q^{39} +1.38806e7 q^{40} +2.19095e7 q^{41} +1.28673e7 q^{42} +2.54984e7 q^{43} -1.67384e7 q^{44} -3.47223e7 q^{45} -5.38514e7 q^{46} +6.57905e6 q^{47} -6.32193e7 q^{48} +5.76480e6 q^{49} +5.45781e7 q^{50} -1.42021e7 q^{51} +7.41266e6 q^{52} +5.07901e6 q^{53} -1.14747e7 q^{54} +1.27657e8 q^{55} -1.68371e7 q^{56} +9.23593e7 q^{57} -1.21189e8 q^{58} -4.39918e6 q^{59} -9.91176e7 q^{60} -8.31432e7 q^{61} -1.06900e8 q^{62} +4.21180e7 q^{63} +1.46873e7 q^{64} -5.65336e7 q^{65} -3.45627e8 q^{66} +1.50394e8 q^{67} -1.91045e7 q^{68} -3.74054e8 q^{69} -1.32009e8 q^{70} +4.39196e7 q^{71} -1.23013e8 q^{72} +2.79088e7 q^{73} -2.52226e8 q^{74} +3.79102e8 q^{75} +1.24241e8 q^{76} -1.54848e8 q^{77} +1.53062e8 q^{78} +3.99467e8 q^{79} +6.48585e8 q^{80} -4.24980e8 q^{81} +6.08570e8 q^{82} -6.07691e8 q^{83} +1.20229e8 q^{84} +1.45703e8 q^{85} +7.08259e8 q^{86} -8.41783e8 q^{87} +4.52260e8 q^{88} +1.00679e9 q^{89} -9.64468e8 q^{90} +6.85750e7 q^{91} -5.03175e8 q^{92} -7.42530e8 q^{93} +1.82743e8 q^{94} -9.47540e8 q^{95} -1.06329e9 q^{96} -6.96867e8 q^{97} +1.60126e8 q^{98} -1.13133e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 27.7766 1.22756 0.613781 0.789476i \(-0.289648\pi\)
0.613781 + 0.789476i \(0.289648\pi\)
\(3\) 192.937 1.37522 0.687608 0.726082i \(-0.258661\pi\)
0.687608 + 0.726082i \(0.258661\pi\)
\(4\) 259.538 0.506910
\(5\) −1979.40 −1.41634 −0.708171 0.706041i \(-0.750480\pi\)
−0.708171 + 0.706041i \(0.750480\pi\)
\(6\) 5359.14 1.68816
\(7\) 2401.00 0.377964
\(8\) −7012.53 −0.605299
\(9\) 17541.9 0.891219
\(10\) −54980.9 −1.73865
\(11\) −64493.1 −1.32815 −0.664073 0.747668i \(-0.731174\pi\)
−0.664073 + 0.747668i \(0.731174\pi\)
\(12\) 50074.6 0.697111
\(13\) 28561.0 0.277350
\(14\) 66691.5 0.463975
\(15\) −381900. −1.94778
\(16\) −327667. −1.24995
\(17\) −73609.6 −0.213754 −0.106877 0.994272i \(-0.534085\pi\)
−0.106877 + 0.994272i \(0.534085\pi\)
\(18\) 487253. 1.09403
\(19\) 478701. 0.842700 0.421350 0.906898i \(-0.361556\pi\)
0.421350 + 0.906898i \(0.361556\pi\)
\(20\) −513729. −0.717958
\(21\) 463243. 0.519783
\(22\) −1.79140e6 −1.63038
\(23\) −1.93873e6 −1.44458 −0.722292 0.691588i \(-0.756911\pi\)
−0.722292 + 0.691588i \(0.756911\pi\)
\(24\) −1.35298e6 −0.832417
\(25\) 1.96490e6 1.00603
\(26\) 793327. 0.340465
\(27\) −413105. −0.149597
\(28\) 623151. 0.191594
\(29\) −4.36298e6 −1.14549 −0.572747 0.819732i \(-0.694122\pi\)
−0.572747 + 0.819732i \(0.694122\pi\)
\(30\) −1.06079e7 −2.39102
\(31\) −3.84855e6 −0.748463 −0.374231 0.927335i \(-0.622093\pi\)
−0.374231 + 0.927335i \(0.622093\pi\)
\(32\) −5.51106e6 −0.929096
\(33\) −1.24431e7 −1.82649
\(34\) −2.04462e6 −0.262397
\(35\) −4.75254e6 −0.535327
\(36\) 4.55278e6 0.451768
\(37\) −9.08052e6 −0.796531 −0.398266 0.917270i \(-0.630388\pi\)
−0.398266 + 0.917270i \(0.630388\pi\)
\(38\) 1.32967e7 1.03447
\(39\) 5.51049e6 0.381416
\(40\) 1.38806e7 0.857311
\(41\) 2.19095e7 1.21089 0.605444 0.795888i \(-0.292995\pi\)
0.605444 + 0.795888i \(0.292995\pi\)
\(42\) 1.28673e7 0.638066
\(43\) 2.54984e7 1.13738 0.568690 0.822552i \(-0.307450\pi\)
0.568690 + 0.822552i \(0.307450\pi\)
\(44\) −1.67384e7 −0.673251
\(45\) −3.47223e7 −1.26227
\(46\) −5.38514e7 −1.77332
\(47\) 6.57905e6 0.196663 0.0983315 0.995154i \(-0.468649\pi\)
0.0983315 + 0.995154i \(0.468649\pi\)
\(48\) −6.32193e7 −1.71895
\(49\) 5.76480e6 0.142857
\(50\) 5.45781e7 1.23496
\(51\) −1.42021e7 −0.293958
\(52\) 7.41266e6 0.140592
\(53\) 5.07901e6 0.0884174 0.0442087 0.999022i \(-0.485923\pi\)
0.0442087 + 0.999022i \(0.485923\pi\)
\(54\) −1.14747e7 −0.183640
\(55\) 1.27657e8 1.88111
\(56\) −1.68371e7 −0.228781
\(57\) 9.23593e7 1.15889
\(58\) −1.21189e8 −1.40616
\(59\) −4.39918e6 −0.0472648 −0.0236324 0.999721i \(-0.507523\pi\)
−0.0236324 + 0.999721i \(0.507523\pi\)
\(60\) −9.91176e7 −0.987348
\(61\) −8.31432e7 −0.768851 −0.384426 0.923156i \(-0.625600\pi\)
−0.384426 + 0.923156i \(0.625600\pi\)
\(62\) −1.06900e8 −0.918785
\(63\) 4.21180e7 0.336849
\(64\) 1.46873e7 0.109429
\(65\) −5.65336e7 −0.392823
\(66\) −3.45627e8 −2.24213
\(67\) 1.50394e8 0.911785 0.455893 0.890035i \(-0.349320\pi\)
0.455893 + 0.890035i \(0.349320\pi\)
\(68\) −1.91045e7 −0.108354
\(69\) −3.74054e8 −1.98661
\(70\) −1.32009e8 −0.657148
\(71\) 4.39196e7 0.205114 0.102557 0.994727i \(-0.467298\pi\)
0.102557 + 0.994727i \(0.467298\pi\)
\(72\) −1.23013e8 −0.539454
\(73\) 2.79088e7 0.115024 0.0575119 0.998345i \(-0.481683\pi\)
0.0575119 + 0.998345i \(0.481683\pi\)
\(74\) −2.52226e8 −0.977792
\(75\) 3.79102e8 1.38350
\(76\) 1.24241e8 0.427173
\(77\) −1.54848e8 −0.501992
\(78\) 1.53062e8 0.468212
\(79\) 3.99467e8 1.15387 0.576937 0.816788i \(-0.304248\pi\)
0.576937 + 0.816788i \(0.304248\pi\)
\(80\) 6.48585e8 1.77036
\(81\) −4.24980e8 −1.09695
\(82\) 6.08570e8 1.48644
\(83\) −6.07691e8 −1.40550 −0.702751 0.711436i \(-0.748045\pi\)
−0.702751 + 0.711436i \(0.748045\pi\)
\(84\) 1.20229e8 0.263483
\(85\) 1.45703e8 0.302749
\(86\) 7.08259e8 1.39621
\(87\) −8.41783e8 −1.57530
\(88\) 4.52260e8 0.803925
\(89\) 1.00679e9 1.70092 0.850458 0.526043i \(-0.176325\pi\)
0.850458 + 0.526043i \(0.176325\pi\)
\(90\) −9.64468e8 −1.54952
\(91\) 6.85750e7 0.104828
\(92\) −5.03175e8 −0.732274
\(93\) −7.42530e8 −1.02930
\(94\) 1.82743e8 0.241416
\(95\) −9.47540e8 −1.19355
\(96\) −1.06329e9 −1.27771
\(97\) −6.96867e8 −0.799240 −0.399620 0.916681i \(-0.630858\pi\)
−0.399620 + 0.916681i \(0.630858\pi\)
\(98\) 1.60126e8 0.175366
\(99\) −1.13133e9 −1.18367
\(100\) 5.09965e8 0.509965
\(101\) −7.56951e8 −0.723805 −0.361902 0.932216i \(-0.617873\pi\)
−0.361902 + 0.932216i \(0.617873\pi\)
\(102\) −3.94484e8 −0.360852
\(103\) 2.12480e9 1.86016 0.930081 0.367354i \(-0.119736\pi\)
0.930081 + 0.367354i \(0.119736\pi\)
\(104\) −2.00285e8 −0.167880
\(105\) −9.16942e8 −0.736191
\(106\) 1.41078e8 0.108538
\(107\) −1.55627e9 −1.14778 −0.573890 0.818933i \(-0.694566\pi\)
−0.573890 + 0.818933i \(0.694566\pi\)
\(108\) −1.07217e8 −0.0758324
\(109\) −2.47100e9 −1.67669 −0.838345 0.545139i \(-0.816477\pi\)
−0.838345 + 0.545139i \(0.816477\pi\)
\(110\) 3.54589e9 2.30918
\(111\) −1.75197e9 −1.09540
\(112\) −7.86730e8 −0.472438
\(113\) −1.42851e9 −0.824197 −0.412099 0.911139i \(-0.635204\pi\)
−0.412099 + 0.911139i \(0.635204\pi\)
\(114\) 2.56543e9 1.42262
\(115\) 3.83753e9 2.04603
\(116\) −1.13236e9 −0.580662
\(117\) 5.01013e8 0.247180
\(118\) −1.22194e8 −0.0580205
\(119\) −1.76737e8 −0.0807914
\(120\) 2.67809e9 1.17899
\(121\) 1.80141e9 0.763972
\(122\) −2.30943e9 −0.943813
\(123\) 4.22716e9 1.66523
\(124\) −9.98846e8 −0.379403
\(125\) −2.32986e7 −0.00853561
\(126\) 1.16989e9 0.413503
\(127\) −3.20141e8 −0.109201 −0.0546003 0.998508i \(-0.517388\pi\)
−0.0546003 + 0.998508i \(0.517388\pi\)
\(128\) 3.22963e9 1.06343
\(129\) 4.91960e9 1.56414
\(130\) −1.57031e9 −0.482215
\(131\) −4.57241e9 −1.35651 −0.678257 0.734824i \(-0.737265\pi\)
−0.678257 + 0.734824i \(0.737265\pi\)
\(132\) −3.22946e9 −0.925865
\(133\) 1.14936e9 0.318511
\(134\) 4.17742e9 1.11927
\(135\) 8.17700e8 0.211881
\(136\) 5.16190e8 0.129385
\(137\) −2.27252e9 −0.551144 −0.275572 0.961280i \(-0.588867\pi\)
−0.275572 + 0.961280i \(0.588867\pi\)
\(138\) −1.03899e10 −2.43869
\(139\) 4.37921e9 0.995015 0.497507 0.867460i \(-0.334249\pi\)
0.497507 + 0.867460i \(0.334249\pi\)
\(140\) −1.23346e9 −0.271363
\(141\) 1.26934e9 0.270454
\(142\) 1.21994e9 0.251791
\(143\) −1.84199e9 −0.368361
\(144\) −5.74790e9 −1.11398
\(145\) 8.63609e9 1.62241
\(146\) 7.75210e8 0.141199
\(147\) 1.11225e9 0.196459
\(148\) −2.35674e9 −0.403770
\(149\) 3.70334e9 0.615539 0.307769 0.951461i \(-0.400418\pi\)
0.307769 + 0.951461i \(0.400418\pi\)
\(150\) 1.05302e10 1.69834
\(151\) 6.41403e9 1.00400 0.502001 0.864867i \(-0.332597\pi\)
0.502001 + 0.864867i \(0.332597\pi\)
\(152\) −3.35690e9 −0.510085
\(153\) −1.29125e9 −0.190502
\(154\) −4.30114e9 −0.616227
\(155\) 7.61783e9 1.06008
\(156\) 1.43018e9 0.193344
\(157\) −1.42115e9 −0.186677 −0.0933387 0.995634i \(-0.529754\pi\)
−0.0933387 + 0.995634i \(0.529754\pi\)
\(158\) 1.10958e10 1.41645
\(159\) 9.79932e8 0.121593
\(160\) 1.09086e10 1.31592
\(161\) −4.65490e9 −0.546001
\(162\) −1.18045e10 −1.34657
\(163\) 5.65290e9 0.627231 0.313615 0.949550i \(-0.398460\pi\)
0.313615 + 0.949550i \(0.398460\pi\)
\(164\) 5.68634e9 0.613812
\(165\) 2.46299e10 2.58693
\(166\) −1.68796e10 −1.72534
\(167\) −1.49764e10 −1.48999 −0.744995 0.667070i \(-0.767548\pi\)
−0.744995 + 0.667070i \(0.767548\pi\)
\(168\) −3.24850e9 −0.314624
\(169\) 8.15731e8 0.0769231
\(170\) 4.04712e9 0.371643
\(171\) 8.39730e9 0.751030
\(172\) 6.61781e9 0.576549
\(173\) 7.19515e9 0.610707 0.305353 0.952239i \(-0.401225\pi\)
0.305353 + 0.952239i \(0.401225\pi\)
\(174\) −2.33818e10 −1.93378
\(175\) 4.71771e9 0.380242
\(176\) 2.11323e10 1.66012
\(177\) −8.48767e8 −0.0649993
\(178\) 2.79651e10 2.08798
\(179\) −1.09510e10 −0.797290 −0.398645 0.917105i \(-0.630520\pi\)
−0.398645 + 0.917105i \(0.630520\pi\)
\(180\) −9.01177e9 −0.639858
\(181\) 6.47650e9 0.448525 0.224263 0.974529i \(-0.428003\pi\)
0.224263 + 0.974529i \(0.428003\pi\)
\(182\) 1.90478e9 0.128684
\(183\) −1.60414e10 −1.05734
\(184\) 1.35954e10 0.874405
\(185\) 1.79740e10 1.12816
\(186\) −2.06249e10 −1.26353
\(187\) 4.74731e9 0.283897
\(188\) 1.70751e9 0.0996904
\(189\) −9.91866e8 −0.0565425
\(190\) −2.63194e10 −1.46516
\(191\) −2.06585e10 −1.12318 −0.561590 0.827416i \(-0.689810\pi\)
−0.561590 + 0.827416i \(0.689810\pi\)
\(192\) 2.83373e9 0.150488
\(193\) 2.40074e10 1.24548 0.622741 0.782428i \(-0.286019\pi\)
0.622741 + 0.782428i \(0.286019\pi\)
\(194\) −1.93566e10 −0.981117
\(195\) −1.09075e10 −0.540216
\(196\) 1.49618e9 0.0724157
\(197\) 9.57908e9 0.453133 0.226567 0.973996i \(-0.427250\pi\)
0.226567 + 0.973996i \(0.427250\pi\)
\(198\) −3.14244e10 −1.45303
\(199\) −4.30848e10 −1.94753 −0.973767 0.227546i \(-0.926930\pi\)
−0.973767 + 0.227546i \(0.926930\pi\)
\(200\) −1.37789e10 −0.608947
\(201\) 2.90166e10 1.25390
\(202\) −2.10255e10 −0.888516
\(203\) −1.04755e10 −0.432956
\(204\) −3.68597e9 −0.149010
\(205\) −4.33676e10 −1.71503
\(206\) 5.90197e10 2.28347
\(207\) −3.40090e10 −1.28744
\(208\) −9.35851e9 −0.346674
\(209\) −3.08729e10 −1.11923
\(210\) −2.54695e10 −0.903720
\(211\) −3.49320e10 −1.21325 −0.606627 0.794986i \(-0.707478\pi\)
−0.606627 + 0.794986i \(0.707478\pi\)
\(212\) 1.31820e9 0.0448197
\(213\) 8.47374e9 0.282076
\(214\) −4.32279e10 −1.40897
\(215\) −5.04716e10 −1.61092
\(216\) 2.89691e9 0.0905511
\(217\) −9.24038e9 −0.282892
\(218\) −6.86358e10 −2.05824
\(219\) 5.38465e9 0.158183
\(220\) 3.31320e10 0.953554
\(221\) −2.10236e9 −0.0592847
\(222\) −4.86638e10 −1.34468
\(223\) −3.68024e10 −0.996561 −0.498280 0.867016i \(-0.666035\pi\)
−0.498280 + 0.867016i \(0.666035\pi\)
\(224\) −1.32321e10 −0.351165
\(225\) 3.44679e10 0.896590
\(226\) −3.96792e10 −1.01175
\(227\) 5.61234e10 1.40290 0.701451 0.712718i \(-0.252536\pi\)
0.701451 + 0.712718i \(0.252536\pi\)
\(228\) 2.39707e10 0.587455
\(229\) 8.03052e9 0.192967 0.0964837 0.995335i \(-0.469240\pi\)
0.0964837 + 0.995335i \(0.469240\pi\)
\(230\) 1.06593e11 2.51162
\(231\) −2.98759e10 −0.690348
\(232\) 3.05956e10 0.693366
\(233\) 2.42669e10 0.539403 0.269701 0.962944i \(-0.413075\pi\)
0.269701 + 0.962944i \(0.413075\pi\)
\(234\) 1.39164e10 0.303429
\(235\) −1.30226e10 −0.278542
\(236\) −1.14175e9 −0.0239590
\(237\) 7.70721e10 1.58683
\(238\) −4.90914e9 −0.0991766
\(239\) 9.28157e10 1.84006 0.920028 0.391853i \(-0.128166\pi\)
0.920028 + 0.391853i \(0.128166\pi\)
\(240\) 1.25136e11 2.43463
\(241\) −9.33767e10 −1.78304 −0.891521 0.452979i \(-0.850361\pi\)
−0.891521 + 0.452979i \(0.850361\pi\)
\(242\) 5.00369e10 0.937824
\(243\) −7.38634e10 −1.35894
\(244\) −2.15788e10 −0.389738
\(245\) −1.14108e10 −0.202335
\(246\) 1.17416e11 2.04418
\(247\) 1.36722e10 0.233723
\(248\) 2.69881e10 0.453043
\(249\) −1.17246e11 −1.93287
\(250\) −6.47155e8 −0.0104780
\(251\) −6.85408e10 −1.08998 −0.544989 0.838443i \(-0.683466\pi\)
−0.544989 + 0.838443i \(0.683466\pi\)
\(252\) 1.09312e10 0.170752
\(253\) 1.25035e11 1.91862
\(254\) −8.89243e9 −0.134051
\(255\) 2.81115e10 0.416345
\(256\) 8.21881e10 1.19599
\(257\) 1.14839e11 1.64206 0.821029 0.570886i \(-0.193400\pi\)
0.821029 + 0.570886i \(0.193400\pi\)
\(258\) 1.36650e11 1.92008
\(259\) −2.18023e10 −0.301061
\(260\) −1.46726e10 −0.199126
\(261\) −7.65349e10 −1.02089
\(262\) −1.27006e11 −1.66521
\(263\) −1.31479e11 −1.69455 −0.847274 0.531155i \(-0.821758\pi\)
−0.847274 + 0.531155i \(0.821758\pi\)
\(264\) 8.72578e10 1.10557
\(265\) −1.00534e10 −0.125229
\(266\) 3.19253e10 0.390992
\(267\) 1.94247e11 2.33913
\(268\) 3.90328e10 0.462193
\(269\) −1.56870e11 −1.82664 −0.913321 0.407240i \(-0.866491\pi\)
−0.913321 + 0.407240i \(0.866491\pi\)
\(270\) 2.27129e10 0.260097
\(271\) −8.62663e10 −0.971582 −0.485791 0.874075i \(-0.661468\pi\)
−0.485791 + 0.874075i \(0.661468\pi\)
\(272\) 2.41195e10 0.267182
\(273\) 1.32307e10 0.144162
\(274\) −6.31228e10 −0.676564
\(275\) −1.26722e11 −1.33615
\(276\) −9.70813e10 −1.00704
\(277\) 1.73311e10 0.176875 0.0884376 0.996082i \(-0.471813\pi\)
0.0884376 + 0.996082i \(0.471813\pi\)
\(278\) 1.21640e11 1.22144
\(279\) −6.75108e10 −0.667044
\(280\) 3.33273e10 0.324033
\(281\) 1.30591e11 1.24949 0.624747 0.780827i \(-0.285202\pi\)
0.624747 + 0.780827i \(0.285202\pi\)
\(282\) 3.52580e10 0.331999
\(283\) 7.69394e10 0.713033 0.356516 0.934289i \(-0.383964\pi\)
0.356516 + 0.934289i \(0.383964\pi\)
\(284\) 1.13988e10 0.103974
\(285\) −1.82816e11 −1.64139
\(286\) −5.11641e10 −0.452187
\(287\) 5.26046e10 0.457673
\(288\) −9.66743e10 −0.828028
\(289\) −1.13170e11 −0.954309
\(290\) 2.39881e11 1.99161
\(291\) −1.34452e11 −1.09913
\(292\) 7.24339e9 0.0583067
\(293\) −1.95672e11 −1.55105 −0.775523 0.631319i \(-0.782514\pi\)
−0.775523 + 0.631319i \(0.782514\pi\)
\(294\) 3.08944e10 0.241166
\(295\) 8.70773e9 0.0669431
\(296\) 6.36774e10 0.482139
\(297\) 2.66424e10 0.198687
\(298\) 1.02866e11 0.755612
\(299\) −5.53722e10 −0.400655
\(300\) 9.83913e10 0.701312
\(301\) 6.12218e10 0.429889
\(302\) 1.78160e11 1.23248
\(303\) −1.46044e11 −0.995388
\(304\) −1.56855e11 −1.05333
\(305\) 1.64574e11 1.08896
\(306\) −3.58665e10 −0.233853
\(307\) −6.88012e10 −0.442052 −0.221026 0.975268i \(-0.570941\pi\)
−0.221026 + 0.975268i \(0.570941\pi\)
\(308\) −4.01889e10 −0.254465
\(309\) 4.09954e11 2.55813
\(310\) 2.11597e11 1.30131
\(311\) −2.21637e11 −1.34345 −0.671723 0.740803i \(-0.734445\pi\)
−0.671723 + 0.740803i \(0.734445\pi\)
\(312\) −3.86425e10 −0.230871
\(313\) −3.29620e10 −0.194117 −0.0970585 0.995279i \(-0.530943\pi\)
−0.0970585 + 0.995279i \(0.530943\pi\)
\(314\) −3.94747e10 −0.229158
\(315\) −8.33684e10 −0.477094
\(316\) 1.03677e11 0.584911
\(317\) 1.04622e11 0.581912 0.290956 0.956736i \(-0.406027\pi\)
0.290956 + 0.956736i \(0.406027\pi\)
\(318\) 2.72191e10 0.149263
\(319\) 2.81382e11 1.52138
\(320\) −2.90720e10 −0.154989
\(321\) −3.00263e11 −1.57844
\(322\) −1.29297e11 −0.670251
\(323\) −3.52370e10 −0.180131
\(324\) −1.10298e11 −0.556054
\(325\) 5.61194e10 0.279022
\(326\) 1.57018e11 0.769965
\(327\) −4.76748e11 −2.30581
\(328\) −1.53641e11 −0.732950
\(329\) 1.57963e10 0.0743316
\(330\) 6.84135e11 3.17562
\(331\) 2.72908e10 0.124966 0.0624828 0.998046i \(-0.480098\pi\)
0.0624828 + 0.998046i \(0.480098\pi\)
\(332\) −1.57719e11 −0.712463
\(333\) −1.59289e11 −0.709884
\(334\) −4.15993e11 −1.82906
\(335\) −2.97689e11 −1.29140
\(336\) −1.51790e11 −0.649704
\(337\) −2.15225e11 −0.908988 −0.454494 0.890750i \(-0.650180\pi\)
−0.454494 + 0.890750i \(0.650180\pi\)
\(338\) 2.26582e10 0.0944279
\(339\) −2.75614e11 −1.13345
\(340\) 3.78154e10 0.153467
\(341\) 2.48205e11 0.994068
\(342\) 2.33248e11 0.921937
\(343\) 1.38413e10 0.0539949
\(344\) −1.78809e11 −0.688455
\(345\) 7.40403e11 2.81373
\(346\) 1.99857e11 0.749681
\(347\) 4.07226e11 1.50783 0.753915 0.656972i \(-0.228163\pi\)
0.753915 + 0.656972i \(0.228163\pi\)
\(348\) −2.18475e11 −0.798536
\(349\) 1.30805e11 0.471965 0.235982 0.971757i \(-0.424169\pi\)
0.235982 + 0.971757i \(0.424169\pi\)
\(350\) 1.31042e11 0.466771
\(351\) −1.17987e10 −0.0414908
\(352\) 3.55425e11 1.23398
\(353\) −1.80701e11 −0.619405 −0.309703 0.950833i \(-0.600229\pi\)
−0.309703 + 0.950833i \(0.600229\pi\)
\(354\) −2.35758e10 −0.0797907
\(355\) −8.69344e10 −0.290512
\(356\) 2.61300e11 0.862211
\(357\) −3.40991e10 −0.111106
\(358\) −3.04182e11 −0.978724
\(359\) 5.64868e11 1.79482 0.897412 0.441193i \(-0.145445\pi\)
0.897412 + 0.441193i \(0.145445\pi\)
\(360\) 2.43492e11 0.764051
\(361\) −9.35332e10 −0.289857
\(362\) 1.79895e11 0.550593
\(363\) 3.47559e11 1.05063
\(364\) 1.77978e10 0.0531386
\(365\) −5.52426e10 −0.162913
\(366\) −4.45576e11 −1.29795
\(367\) −5.88038e11 −1.69203 −0.846015 0.533159i \(-0.821005\pi\)
−0.846015 + 0.533159i \(0.821005\pi\)
\(368\) 6.35260e11 1.80566
\(369\) 3.84333e11 1.07917
\(370\) 4.99255e11 1.38489
\(371\) 1.21947e10 0.0334186
\(372\) −1.92715e11 −0.521761
\(373\) −5.55684e11 −1.48641 −0.743204 0.669065i \(-0.766695\pi\)
−0.743204 + 0.669065i \(0.766695\pi\)
\(374\) 1.31864e11 0.348501
\(375\) −4.49517e9 −0.0117383
\(376\) −4.61358e10 −0.119040
\(377\) −1.24611e11 −0.317703
\(378\) −2.75506e10 −0.0694094
\(379\) −6.02427e11 −1.49978 −0.749890 0.661562i \(-0.769894\pi\)
−0.749890 + 0.661562i \(0.769894\pi\)
\(380\) −2.45923e11 −0.605023
\(381\) −6.17673e10 −0.150174
\(382\) −5.73823e11 −1.37877
\(383\) −1.10425e10 −0.0262224 −0.0131112 0.999914i \(-0.504174\pi\)
−0.0131112 + 0.999914i \(0.504174\pi\)
\(384\) 6.23116e11 1.46244
\(385\) 3.06506e11 0.710993
\(386\) 6.66844e11 1.52891
\(387\) 4.47290e11 1.01365
\(388\) −1.80863e11 −0.405143
\(389\) −3.07348e11 −0.680544 −0.340272 0.940327i \(-0.610519\pi\)
−0.340272 + 0.940327i \(0.610519\pi\)
\(390\) −3.02972e11 −0.663149
\(391\) 1.42709e11 0.308786
\(392\) −4.04258e10 −0.0864713
\(393\) −8.82190e11 −1.86550
\(394\) 2.66074e11 0.556250
\(395\) −7.90704e11 −1.63428
\(396\) −2.93623e11 −0.600014
\(397\) −1.77069e11 −0.357754 −0.178877 0.983871i \(-0.557246\pi\)
−0.178877 + 0.983871i \(0.557246\pi\)
\(398\) −1.19675e12 −2.39072
\(399\) 2.21755e11 0.438021
\(400\) −6.43832e11 −1.25749
\(401\) −3.76120e11 −0.726402 −0.363201 0.931711i \(-0.618316\pi\)
−0.363201 + 0.931711i \(0.618316\pi\)
\(402\) 8.05980e11 1.53924
\(403\) −1.09919e11 −0.207586
\(404\) −1.96457e11 −0.366904
\(405\) 8.41205e11 1.55365
\(406\) −2.90974e11 −0.531480
\(407\) 5.85630e11 1.05791
\(408\) 9.95923e10 0.177932
\(409\) 9.86468e10 0.174312 0.0871561 0.996195i \(-0.472222\pi\)
0.0871561 + 0.996195i \(0.472222\pi\)
\(410\) −1.20460e12 −2.10531
\(411\) −4.38454e11 −0.757942
\(412\) 5.51467e11 0.942935
\(413\) −1.05624e10 −0.0178644
\(414\) −9.44653e11 −1.58041
\(415\) 1.20286e12 1.99067
\(416\) −1.57401e11 −0.257685
\(417\) 8.44914e11 1.36836
\(418\) −8.57543e11 −1.37392
\(419\) −6.51066e11 −1.03196 −0.515979 0.856601i \(-0.672572\pi\)
−0.515979 + 0.856601i \(0.672572\pi\)
\(420\) −2.37981e11 −0.373182
\(421\) 7.49947e11 1.16349 0.581743 0.813373i \(-0.302371\pi\)
0.581743 + 0.813373i \(0.302371\pi\)
\(422\) −9.70290e11 −1.48935
\(423\) 1.15409e11 0.175270
\(424\) −3.56167e10 −0.0535190
\(425\) −1.44635e11 −0.215042
\(426\) 2.35371e11 0.346267
\(427\) −1.99627e11 −0.290598
\(428\) −4.03912e11 −0.581821
\(429\) −3.55388e11 −0.506577
\(430\) −1.40193e12 −1.97750
\(431\) 8.99608e11 1.25576 0.627879 0.778311i \(-0.283923\pi\)
0.627879 + 0.778311i \(0.283923\pi\)
\(432\) 1.35361e11 0.186990
\(433\) 1.39165e12 1.90254 0.951271 0.308356i \(-0.0997788\pi\)
0.951271 + 0.308356i \(0.0997788\pi\)
\(434\) −2.56666e11 −0.347268
\(435\) 1.66622e12 2.23117
\(436\) −6.41317e11 −0.849931
\(437\) −9.28073e11 −1.21735
\(438\) 1.49567e11 0.194179
\(439\) 6.42407e11 0.825506 0.412753 0.910843i \(-0.364567\pi\)
0.412753 + 0.910843i \(0.364567\pi\)
\(440\) −8.95202e11 −1.13863
\(441\) 1.01125e11 0.127317
\(442\) −5.83965e10 −0.0727757
\(443\) 5.17969e11 0.638979 0.319490 0.947590i \(-0.396489\pi\)
0.319490 + 0.947590i \(0.396489\pi\)
\(444\) −4.54703e11 −0.555271
\(445\) −1.99283e12 −2.40908
\(446\) −1.02224e12 −1.22334
\(447\) 7.14513e11 0.846498
\(448\) 3.52642e10 0.0413602
\(449\) −1.18674e12 −1.37800 −0.688999 0.724762i \(-0.741950\pi\)
−0.688999 + 0.724762i \(0.741950\pi\)
\(450\) 9.57401e11 1.10062
\(451\) −1.41301e12 −1.60824
\(452\) −3.70753e11 −0.417794
\(453\) 1.23751e12 1.38072
\(454\) 1.55891e12 1.72215
\(455\) −1.35737e11 −0.148473
\(456\) −6.47673e11 −0.701477
\(457\) 2.63210e11 0.282279 0.141140 0.989990i \(-0.454923\pi\)
0.141140 + 0.989990i \(0.454923\pi\)
\(458\) 2.23060e11 0.236880
\(459\) 3.04085e10 0.0319770
\(460\) 9.95984e11 1.03715
\(461\) 1.22231e12 1.26046 0.630229 0.776409i \(-0.282961\pi\)
0.630229 + 0.776409i \(0.282961\pi\)
\(462\) −8.29851e11 −0.847445
\(463\) −5.00249e11 −0.505908 −0.252954 0.967478i \(-0.581402\pi\)
−0.252954 + 0.967478i \(0.581402\pi\)
\(464\) 1.42961e12 1.43181
\(465\) 1.46976e12 1.45784
\(466\) 6.74052e11 0.662150
\(467\) −5.37954e11 −0.523382 −0.261691 0.965152i \(-0.584280\pi\)
−0.261691 + 0.965152i \(0.584280\pi\)
\(468\) 1.30032e11 0.125298
\(469\) 3.61095e11 0.344622
\(470\) −3.61722e11 −0.341928
\(471\) −2.74193e11 −0.256722
\(472\) 3.08494e10 0.0286093
\(473\) −1.64447e12 −1.51061
\(474\) 2.14080e12 1.94793
\(475\) 9.40597e11 0.847778
\(476\) −4.58699e10 −0.0409540
\(477\) 8.90953e10 0.0787993
\(478\) 2.57810e12 2.25878
\(479\) −1.04910e12 −0.910561 −0.455280 0.890348i \(-0.650461\pi\)
−0.455280 + 0.890348i \(0.650461\pi\)
\(480\) 2.10468e12 1.80967
\(481\) −2.59349e11 −0.220918
\(482\) −2.59368e12 −2.18880
\(483\) −8.98104e11 −0.750870
\(484\) 4.67533e11 0.387265
\(485\) 1.37938e12 1.13200
\(486\) −2.05167e12 −1.66819
\(487\) −5.76259e11 −0.464235 −0.232117 0.972688i \(-0.574565\pi\)
−0.232117 + 0.972688i \(0.574565\pi\)
\(488\) 5.83044e11 0.465385
\(489\) 1.09066e12 0.862578
\(490\) −3.16954e11 −0.248378
\(491\) 1.27331e12 0.988708 0.494354 0.869261i \(-0.335405\pi\)
0.494354 + 0.869261i \(0.335405\pi\)
\(492\) 1.09711e12 0.844124
\(493\) 3.21158e11 0.244854
\(494\) 3.79766e11 0.286910
\(495\) 2.23935e12 1.67648
\(496\) 1.26105e12 0.935542
\(497\) 1.05451e11 0.0775259
\(498\) −3.25670e12 −2.37272
\(499\) −2.04231e12 −1.47459 −0.737293 0.675574i \(-0.763896\pi\)
−0.737293 + 0.675574i \(0.763896\pi\)
\(500\) −6.04687e9 −0.00432679
\(501\) −2.88951e12 −2.04906
\(502\) −1.90383e12 −1.33802
\(503\) −3.91060e11 −0.272387 −0.136194 0.990682i \(-0.543487\pi\)
−0.136194 + 0.990682i \(0.543487\pi\)
\(504\) −2.95354e11 −0.203894
\(505\) 1.49831e12 1.02516
\(506\) 3.47304e12 2.35522
\(507\) 1.57385e11 0.105786
\(508\) −8.30888e10 −0.0553549
\(509\) −3.45834e11 −0.228370 −0.114185 0.993460i \(-0.536426\pi\)
−0.114185 + 0.993460i \(0.536426\pi\)
\(510\) 7.80842e11 0.511090
\(511\) 6.70090e10 0.0434749
\(512\) 6.29334e11 0.404731
\(513\) −1.97754e11 −0.126066
\(514\) 3.18982e12 2.01573
\(515\) −4.20583e12 −2.63463
\(516\) 1.27682e12 0.792880
\(517\) −4.24303e11 −0.261197
\(518\) −6.05594e11 −0.369571
\(519\) 1.38821e12 0.839853
\(520\) 3.96444e11 0.237775
\(521\) 1.07313e12 0.638093 0.319047 0.947739i \(-0.396637\pi\)
0.319047 + 0.947739i \(0.396637\pi\)
\(522\) −2.12588e12 −1.25320
\(523\) 1.56232e12 0.913087 0.456544 0.889701i \(-0.349087\pi\)
0.456544 + 0.889701i \(0.349087\pi\)
\(524\) −1.18671e12 −0.687631
\(525\) 9.10224e11 0.522915
\(526\) −3.65202e12 −2.08016
\(527\) 2.83291e11 0.159987
\(528\) 4.07721e12 2.28302
\(529\) 1.95753e12 1.08682
\(530\) −2.79249e11 −0.153727
\(531\) −7.71698e10 −0.0421233
\(532\) 2.98303e11 0.161456
\(533\) 6.25756e11 0.335840
\(534\) 5.39552e12 2.87142
\(535\) 3.08048e12 1.62565
\(536\) −1.05464e12 −0.551903
\(537\) −2.11287e12 −1.09645
\(538\) −4.35730e12 −2.24232
\(539\) −3.71790e11 −0.189735
\(540\) 2.12224e11 0.107405
\(541\) −1.00334e12 −0.503571 −0.251786 0.967783i \(-0.581018\pi\)
−0.251786 + 0.967783i \(0.581018\pi\)
\(542\) −2.39618e12 −1.19268
\(543\) 1.24956e12 0.616819
\(544\) 4.05667e11 0.198598
\(545\) 4.89109e12 2.37477
\(546\) 3.67503e11 0.176968
\(547\) −3.62889e12 −1.73313 −0.866565 0.499064i \(-0.833677\pi\)
−0.866565 + 0.499064i \(0.833677\pi\)
\(548\) −5.89805e11 −0.279380
\(549\) −1.45849e12 −0.685215
\(550\) −3.51991e12 −1.64021
\(551\) −2.08856e12 −0.965307
\(552\) 2.62307e12 1.20250
\(553\) 9.59119e11 0.436124
\(554\) 4.81398e11 0.217125
\(555\) 3.46785e12 1.55147
\(556\) 1.13657e12 0.504383
\(557\) 5.36804e11 0.236302 0.118151 0.992996i \(-0.462303\pi\)
0.118151 + 0.992996i \(0.462303\pi\)
\(558\) −1.87522e12 −0.818838
\(559\) 7.28261e11 0.315452
\(560\) 1.55725e12 0.669133
\(561\) 9.15934e11 0.390419
\(562\) 3.62736e12 1.53383
\(563\) −2.01120e12 −0.843660 −0.421830 0.906675i \(-0.638612\pi\)
−0.421830 + 0.906675i \(0.638612\pi\)
\(564\) 3.29443e11 0.137096
\(565\) 2.82760e12 1.16735
\(566\) 2.13711e12 0.875293
\(567\) −1.02038e12 −0.414607
\(568\) −3.07988e11 −0.124155
\(569\) 2.86573e12 1.14612 0.573060 0.819513i \(-0.305756\pi\)
0.573060 + 0.819513i \(0.305756\pi\)
\(570\) −5.07800e12 −2.01491
\(571\) −5.55697e10 −0.0218764 −0.0109382 0.999940i \(-0.503482\pi\)
−0.0109382 + 0.999940i \(0.503482\pi\)
\(572\) −4.78065e11 −0.186726
\(573\) −3.98581e12 −1.54462
\(574\) 1.46118e12 0.561822
\(575\) −3.80941e12 −1.45329
\(576\) 2.57643e11 0.0975251
\(577\) 9.50499e11 0.356994 0.178497 0.983940i \(-0.442877\pi\)
0.178497 + 0.983940i \(0.442877\pi\)
\(578\) −3.14346e12 −1.17147
\(579\) 4.63193e12 1.71281
\(580\) 2.24139e12 0.822417
\(581\) −1.45907e12 −0.531230
\(582\) −3.73461e12 −1.34925
\(583\) −3.27561e11 −0.117431
\(584\) −1.95711e11 −0.0696238
\(585\) −9.91705e11 −0.350091
\(586\) −5.43510e12 −1.90401
\(587\) −3.29219e12 −1.14449 −0.572246 0.820082i \(-0.693928\pi\)
−0.572246 + 0.820082i \(0.693928\pi\)
\(588\) 2.88670e11 0.0995873
\(589\) −1.84231e12 −0.630729
\(590\) 2.41871e11 0.0821769
\(591\) 1.84816e12 0.623156
\(592\) 2.97539e12 0.995626
\(593\) −2.67476e12 −0.888256 −0.444128 0.895963i \(-0.646487\pi\)
−0.444128 + 0.895963i \(0.646487\pi\)
\(594\) 7.40035e11 0.243901
\(595\) 3.49832e11 0.114428
\(596\) 9.61157e11 0.312023
\(597\) −8.31267e12 −2.67828
\(598\) −1.53805e12 −0.491830
\(599\) 3.68582e12 1.16980 0.584902 0.811104i \(-0.301133\pi\)
0.584902 + 0.811104i \(0.301133\pi\)
\(600\) −2.65846e12 −0.837433
\(601\) 5.13356e12 1.60503 0.802516 0.596631i \(-0.203494\pi\)
0.802516 + 0.596631i \(0.203494\pi\)
\(602\) 1.70053e12 0.527716
\(603\) 2.63818e12 0.812600
\(604\) 1.66468e12 0.508939
\(605\) −3.56570e12 −1.08205
\(606\) −4.05661e12 −1.22190
\(607\) −1.31730e12 −0.393855 −0.196928 0.980418i \(-0.563096\pi\)
−0.196928 + 0.980418i \(0.563096\pi\)
\(608\) −2.63815e12 −0.782949
\(609\) −2.02112e12 −0.595408
\(610\) 4.57129e12 1.33676
\(611\) 1.87904e11 0.0545445
\(612\) −3.35128e11 −0.0965672
\(613\) 1.79649e12 0.513870 0.256935 0.966429i \(-0.417287\pi\)
0.256935 + 0.966429i \(0.417287\pi\)
\(614\) −1.91106e12 −0.542646
\(615\) −8.36723e12 −2.35854
\(616\) 1.08588e12 0.303855
\(617\) 4.76490e12 1.32364 0.661822 0.749661i \(-0.269784\pi\)
0.661822 + 0.749661i \(0.269784\pi\)
\(618\) 1.13871e13 3.14026
\(619\) 3.72505e12 1.01982 0.509911 0.860227i \(-0.329678\pi\)
0.509911 + 0.860227i \(0.329678\pi\)
\(620\) 1.97711e12 0.537365
\(621\) 8.00901e11 0.216106
\(622\) −6.15631e12 −1.64916
\(623\) 2.41730e12 0.642886
\(624\) −1.80561e12 −0.476752
\(625\) −3.79157e12 −0.993937
\(626\) −9.15570e11 −0.238291
\(627\) −5.95654e12 −1.53918
\(628\) −3.68843e11 −0.0946286
\(629\) 6.68413e11 0.170262
\(630\) −2.31569e12 −0.585662
\(631\) 3.82136e12 0.959591 0.479796 0.877380i \(-0.340711\pi\)
0.479796 + 0.877380i \(0.340711\pi\)
\(632\) −2.80127e12 −0.698439
\(633\) −6.73968e12 −1.66849
\(634\) 2.90605e12 0.714334
\(635\) 6.33688e11 0.154665
\(636\) 2.54329e11 0.0616367
\(637\) 1.64648e11 0.0396214
\(638\) 7.81583e12 1.86759
\(639\) 7.70432e11 0.182802
\(640\) −6.39272e12 −1.50618
\(641\) −4.37610e12 −1.02383 −0.511913 0.859037i \(-0.671063\pi\)
−0.511913 + 0.859037i \(0.671063\pi\)
\(642\) −8.34028e12 −1.93764
\(643\) 7.58510e12 1.74989 0.874947 0.484218i \(-0.160896\pi\)
0.874947 + 0.484218i \(0.160896\pi\)
\(644\) −1.20812e12 −0.276774
\(645\) −9.73786e12 −2.21536
\(646\) −9.78763e11 −0.221122
\(647\) −3.06082e12 −0.686703 −0.343351 0.939207i \(-0.611562\pi\)
−0.343351 + 0.939207i \(0.611562\pi\)
\(648\) 2.98019e12 0.663981
\(649\) 2.83717e11 0.0627745
\(650\) 1.55880e12 0.342516
\(651\) −1.78282e12 −0.389038
\(652\) 1.46714e12 0.317950
\(653\) −6.66671e12 −1.43484 −0.717418 0.696643i \(-0.754676\pi\)
−0.717418 + 0.696643i \(0.754676\pi\)
\(654\) −1.32424e13 −2.83053
\(655\) 9.05063e12 1.92129
\(656\) −7.17902e12 −1.51355
\(657\) 4.89572e11 0.102511
\(658\) 4.38767e11 0.0912467
\(659\) −9.07263e11 −0.187391 −0.0936955 0.995601i \(-0.529868\pi\)
−0.0936955 + 0.995601i \(0.529868\pi\)
\(660\) 6.39240e12 1.31134
\(661\) −2.34030e12 −0.476832 −0.238416 0.971163i \(-0.576628\pi\)
−0.238416 + 0.971163i \(0.576628\pi\)
\(662\) 7.58046e11 0.153403
\(663\) −4.05625e11 −0.0815293
\(664\) 4.26145e12 0.850748
\(665\) −2.27504e12 −0.451120
\(666\) −4.42451e12 −0.871427
\(667\) 8.45866e12 1.65476
\(668\) −3.88694e12 −0.755291
\(669\) −7.10055e12 −1.37049
\(670\) −8.26878e12 −1.58528
\(671\) 5.36216e12 1.02115
\(672\) −2.55296e12 −0.482928
\(673\) 6.84623e12 1.28642 0.643211 0.765689i \(-0.277602\pi\)
0.643211 + 0.765689i \(0.277602\pi\)
\(674\) −5.97822e12 −1.11584
\(675\) −8.11709e11 −0.150499
\(676\) 2.11713e11 0.0389931
\(677\) 7.20374e12 1.31798 0.658991 0.752151i \(-0.270984\pi\)
0.658991 + 0.752151i \(0.270984\pi\)
\(678\) −7.65560e12 −1.39138
\(679\) −1.67318e12 −0.302084
\(680\) −1.02175e12 −0.183254
\(681\) 1.08283e13 1.92929
\(682\) 6.89429e12 1.22028
\(683\) −9.42073e11 −0.165650 −0.0828250 0.996564i \(-0.526394\pi\)
−0.0828250 + 0.996564i \(0.526394\pi\)
\(684\) 2.17942e12 0.380705
\(685\) 4.49822e12 0.780609
\(686\) 3.84464e11 0.0662822
\(687\) 1.54939e12 0.265372
\(688\) −8.35501e12 −1.42167
\(689\) 1.45062e11 0.0245226
\(690\) 2.05658e13 3.45403
\(691\) 6.19915e12 1.03438 0.517191 0.855870i \(-0.326978\pi\)
0.517191 + 0.855870i \(0.326978\pi\)
\(692\) 1.86742e12 0.309573
\(693\) −2.71632e12 −0.447385
\(694\) 1.13113e13 1.85096
\(695\) −8.66821e12 −1.40928
\(696\) 5.90303e12 0.953528
\(697\) −1.61275e12 −0.258832
\(698\) 3.63331e12 0.579366
\(699\) 4.68200e12 0.741795
\(700\) 1.22443e12 0.192749
\(701\) 7.77996e12 1.21688 0.608438 0.793602i \(-0.291796\pi\)
0.608438 + 0.793602i \(0.291796\pi\)
\(702\) −3.27728e11 −0.0509326
\(703\) −4.34685e12 −0.671237
\(704\) −9.47229e11 −0.145338
\(705\) −2.51254e12 −0.383056
\(706\) −5.01926e12 −0.760359
\(707\) −1.81744e12 −0.273572
\(708\) −2.20287e11 −0.0329488
\(709\) 2.22942e12 0.331348 0.165674 0.986181i \(-0.447020\pi\)
0.165674 + 0.986181i \(0.447020\pi\)
\(710\) −2.41474e12 −0.356622
\(711\) 7.00739e12 1.02835
\(712\) −7.06013e12 −1.02956
\(713\) 7.46132e12 1.08122
\(714\) −9.47157e11 −0.136389
\(715\) 3.64603e12 0.521726
\(716\) −2.84221e12 −0.404155
\(717\) 1.79076e13 2.53047
\(718\) 1.56901e13 2.20326
\(719\) 6.79515e11 0.0948242 0.0474121 0.998875i \(-0.484903\pi\)
0.0474121 + 0.998875i \(0.484903\pi\)
\(720\) 1.13774e13 1.57778
\(721\) 5.10165e12 0.703075
\(722\) −2.59803e12 −0.355817
\(723\) −1.80159e13 −2.45207
\(724\) 1.68090e12 0.227362
\(725\) −8.57281e12 −1.15240
\(726\) 9.65399e12 1.28971
\(727\) 1.31466e13 1.74546 0.872729 0.488205i \(-0.162348\pi\)
0.872729 + 0.488205i \(0.162348\pi\)
\(728\) −4.80884e11 −0.0634526
\(729\) −5.88614e12 −0.771892
\(730\) −1.53445e12 −0.199986
\(731\) −1.87693e12 −0.243120
\(732\) −4.16336e12 −0.535975
\(733\) 4.85417e12 0.621080 0.310540 0.950560i \(-0.399490\pi\)
0.310540 + 0.950560i \(0.399490\pi\)
\(734\) −1.63337e13 −2.07707
\(735\) −2.20158e12 −0.278254
\(736\) 1.06845e13 1.34216
\(737\) −9.69934e12 −1.21098
\(738\) 1.06754e13 1.32475
\(739\) 1.10976e13 1.36876 0.684380 0.729126i \(-0.260073\pi\)
0.684380 + 0.729126i \(0.260073\pi\)
\(740\) 4.66493e12 0.571876
\(741\) 2.63787e12 0.321419
\(742\) 3.38727e11 0.0410235
\(743\) 2.14410e12 0.258104 0.129052 0.991638i \(-0.458807\pi\)
0.129052 + 0.991638i \(0.458807\pi\)
\(744\) 5.20702e12 0.623033
\(745\) −7.33039e12 −0.871813
\(746\) −1.54350e13 −1.82466
\(747\) −1.06600e13 −1.25261
\(748\) 1.23211e12 0.143910
\(749\) −3.73661e12 −0.433820
\(750\) −1.24860e11 −0.0144095
\(751\) 7.97081e12 0.914372 0.457186 0.889371i \(-0.348857\pi\)
0.457186 + 0.889371i \(0.348857\pi\)
\(752\) −2.15574e12 −0.245819
\(753\) −1.32241e13 −1.49895
\(754\) −3.46127e12 −0.390000
\(755\) −1.26959e13 −1.42201
\(756\) −2.57427e11 −0.0286620
\(757\) 9.54234e12 1.05615 0.528073 0.849199i \(-0.322915\pi\)
0.528073 + 0.849199i \(0.322915\pi\)
\(758\) −1.67333e13 −1.84107
\(759\) 2.41239e13 2.63852
\(760\) 6.64465e12 0.722456
\(761\) 5.85311e12 0.632639 0.316319 0.948653i \(-0.397553\pi\)
0.316319 + 0.948653i \(0.397553\pi\)
\(762\) −1.71568e12 −0.184349
\(763\) −5.93286e12 −0.633730
\(764\) −5.36167e12 −0.569351
\(765\) 2.55590e12 0.269816
\(766\) −3.06723e11 −0.0321897
\(767\) −1.25645e11 −0.0131089
\(768\) 1.58572e13 1.64475
\(769\) −1.04494e13 −1.07751 −0.538755 0.842463i \(-0.681105\pi\)
−0.538755 + 0.842463i \(0.681105\pi\)
\(770\) 8.51368e12 0.872788
\(771\) 2.21567e13 2.25818
\(772\) 6.23084e12 0.631347
\(773\) −8.71660e12 −0.878091 −0.439045 0.898465i \(-0.644683\pi\)
−0.439045 + 0.898465i \(0.644683\pi\)
\(774\) 1.24242e13 1.24432
\(775\) −7.56201e12 −0.752973
\(776\) 4.88680e12 0.483779
\(777\) −4.20649e12 −0.414023
\(778\) −8.53706e12 −0.835411
\(779\) 1.04881e13 1.02042
\(780\) −2.83090e12 −0.273841
\(781\) −2.83251e12 −0.272422
\(782\) 3.96398e12 0.379054
\(783\) 1.80237e12 0.171363
\(784\) −1.88894e12 −0.178565
\(785\) 2.81302e12 0.264399
\(786\) −2.45042e13 −2.29002
\(787\) 6.09638e11 0.0566481 0.0283241 0.999599i \(-0.490983\pi\)
0.0283241 + 0.999599i \(0.490983\pi\)
\(788\) 2.48614e12 0.229698
\(789\) −2.53671e13 −2.33037
\(790\) −2.19630e13 −2.00618
\(791\) −3.42986e12 −0.311517
\(792\) 7.93348e12 0.716474
\(793\) −2.37465e12 −0.213241
\(794\) −4.91836e12 −0.439165
\(795\) −1.93968e12 −0.172217
\(796\) −1.11821e13 −0.987225
\(797\) 9.57599e12 0.840662 0.420331 0.907371i \(-0.361914\pi\)
0.420331 + 0.907371i \(0.361914\pi\)
\(798\) 6.15959e12 0.537698
\(799\) −4.84281e11 −0.0420375
\(800\) −1.08287e13 −0.934695
\(801\) 1.76609e13 1.51589
\(802\) −1.04473e13 −0.891704
\(803\) −1.79992e12 −0.152768
\(804\) 7.53090e12 0.635615
\(805\) 9.21390e12 0.773325
\(806\) −3.05316e12 −0.254825
\(807\) −3.02660e13 −2.51203
\(808\) 5.30814e12 0.438118
\(809\) 6.93398e11 0.0569133 0.0284567 0.999595i \(-0.490941\pi\)
0.0284567 + 0.999595i \(0.490941\pi\)
\(810\) 2.33658e13 1.90721
\(811\) −4.60004e12 −0.373394 −0.186697 0.982417i \(-0.559778\pi\)
−0.186697 + 0.982417i \(0.559778\pi\)
\(812\) −2.71880e12 −0.219470
\(813\) −1.66440e13 −1.33613
\(814\) 1.62668e13 1.29865
\(815\) −1.11893e13 −0.888374
\(816\) 4.65355e12 0.367433
\(817\) 1.22061e13 0.958470
\(818\) 2.74007e12 0.213979
\(819\) 1.20293e12 0.0934251
\(820\) −1.12555e13 −0.869368
\(821\) 2.66819e12 0.204962 0.102481 0.994735i \(-0.467322\pi\)
0.102481 + 0.994735i \(0.467322\pi\)
\(822\) −1.21788e13 −0.930421
\(823\) −2.42814e13 −1.84491 −0.922454 0.386108i \(-0.873819\pi\)
−0.922454 + 0.386108i \(0.873819\pi\)
\(824\) −1.49002e13 −1.12595
\(825\) −2.44494e13 −1.83750
\(826\) −2.93388e11 −0.0219297
\(827\) 2.27850e13 1.69385 0.846923 0.531715i \(-0.178452\pi\)
0.846923 + 0.531715i \(0.178452\pi\)
\(828\) −8.82662e12 −0.652617
\(829\) −2.09926e13 −1.54373 −0.771865 0.635786i \(-0.780676\pi\)
−0.771865 + 0.635786i \(0.780676\pi\)
\(830\) 3.34114e13 2.44367
\(831\) 3.34382e12 0.243242
\(832\) 4.19484e11 0.0303501
\(833\) −4.24345e11 −0.0305363
\(834\) 2.34688e13 1.67975
\(835\) 2.96443e13 2.11034
\(836\) −8.01268e12 −0.567348
\(837\) 1.58986e12 0.111968
\(838\) −1.80844e13 −1.26679
\(839\) 1.45733e13 1.01538 0.507692 0.861539i \(-0.330499\pi\)
0.507692 + 0.861539i \(0.330499\pi\)
\(840\) 6.43009e12 0.445615
\(841\) 4.52848e12 0.312155
\(842\) 2.08310e13 1.42825
\(843\) 2.51958e13 1.71832
\(844\) −9.06617e12 −0.615011
\(845\) −1.61466e12 −0.108949
\(846\) 3.20566e12 0.215155
\(847\) 4.32518e12 0.288754
\(848\) −1.66423e12 −0.110518
\(849\) 1.48445e13 0.980574
\(850\) −4.01747e12 −0.263978
\(851\) 1.76047e13 1.15066
\(852\) 2.19926e12 0.142987
\(853\) −1.67650e13 −1.08426 −0.542130 0.840295i \(-0.682382\pi\)
−0.542130 + 0.840295i \(0.682382\pi\)
\(854\) −5.54495e12 −0.356728
\(855\) −1.66216e13 −1.06372
\(856\) 1.09134e13 0.694750
\(857\) −7.86341e11 −0.0497963 −0.0248981 0.999690i \(-0.507926\pi\)
−0.0248981 + 0.999690i \(0.507926\pi\)
\(858\) −9.87146e12 −0.621855
\(859\) −5.85873e12 −0.367142 −0.183571 0.983006i \(-0.558766\pi\)
−0.183571 + 0.983006i \(0.558766\pi\)
\(860\) −1.30993e13 −0.816591
\(861\) 1.01494e13 0.629399
\(862\) 2.49880e13 1.54152
\(863\) −2.88458e13 −1.77025 −0.885123 0.465357i \(-0.845926\pi\)
−0.885123 + 0.465357i \(0.845926\pi\)
\(864\) 2.27665e12 0.138990
\(865\) −1.42421e13 −0.864970
\(866\) 3.86552e13 2.33549
\(867\) −2.18346e13 −1.31238
\(868\) −2.39823e12 −0.143401
\(869\) −2.57628e13 −1.53251
\(870\) 4.62820e13 2.73890
\(871\) 4.29539e12 0.252884
\(872\) 1.73279e13 1.01490
\(873\) −1.22243e13 −0.712298
\(874\) −2.57787e13 −1.49437
\(875\) −5.59399e10 −0.00322616
\(876\) 1.39752e12 0.0801844
\(877\) −3.16984e13 −1.80942 −0.904709 0.426030i \(-0.859912\pi\)
−0.904709 + 0.426030i \(0.859912\pi\)
\(878\) 1.78439e13 1.01336
\(879\) −3.77525e13 −2.13302
\(880\) −4.18292e13 −2.35130
\(881\) 5.70895e12 0.319275 0.159637 0.987176i \(-0.448967\pi\)
0.159637 + 0.987176i \(0.448967\pi\)
\(882\) 2.80892e12 0.156290
\(883\) −9.90298e11 −0.0548205 −0.0274102 0.999624i \(-0.508726\pi\)
−0.0274102 + 0.999624i \(0.508726\pi\)
\(884\) −5.45643e11 −0.0300520
\(885\) 1.68005e12 0.0920613
\(886\) 1.43874e13 0.784387
\(887\) −3.47830e13 −1.88674 −0.943368 0.331748i \(-0.892362\pi\)
−0.943368 + 0.331748i \(0.892362\pi\)
\(888\) 1.22858e13 0.663046
\(889\) −7.68659e11 −0.0412740
\(890\) −5.53541e13 −2.95730
\(891\) 2.74083e13 1.45691
\(892\) −9.55161e12 −0.505167
\(893\) 3.14940e12 0.165728
\(894\) 1.98467e13 1.03913
\(895\) 2.16765e13 1.12924
\(896\) 7.75434e12 0.401938
\(897\) −1.06834e13 −0.550988
\(898\) −3.29637e13 −1.69158
\(899\) 1.67912e13 0.857359
\(900\) 8.94574e12 0.454490
\(901\) −3.73864e11 −0.0188996
\(902\) −3.92485e13 −1.97421
\(903\) 1.18120e13 0.591191
\(904\) 1.00175e13 0.498885
\(905\) −1.28196e13 −0.635265
\(906\) 3.43737e13 1.69492
\(907\) −5.69371e12 −0.279359 −0.139680 0.990197i \(-0.544607\pi\)
−0.139680 + 0.990197i \(0.544607\pi\)
\(908\) 1.45661e13 0.711145
\(909\) −1.32783e13 −0.645069
\(910\) −3.77031e12 −0.182260
\(911\) 2.05264e13 0.987369 0.493685 0.869641i \(-0.335650\pi\)
0.493685 + 0.869641i \(0.335650\pi\)
\(912\) −3.02631e13 −1.44856
\(913\) 3.91918e13 1.86671
\(914\) 7.31106e12 0.346515
\(915\) 3.17524e13 1.49755
\(916\) 2.08422e12 0.0978171
\(917\) −1.09784e13 −0.512714
\(918\) 8.44645e11 0.0392538
\(919\) 3.23659e13 1.49682 0.748408 0.663238i \(-0.230818\pi\)
0.748408 + 0.663238i \(0.230818\pi\)
\(920\) −2.69108e13 −1.23846
\(921\) −1.32743e13 −0.607917
\(922\) 3.39517e13 1.54729
\(923\) 1.25439e12 0.0568885
\(924\) −7.75394e12 −0.349944
\(925\) −1.78423e13 −0.801332
\(926\) −1.38952e13 −0.621034
\(927\) 3.72730e13 1.65781
\(928\) 2.40447e13 1.06427
\(929\) 1.27974e13 0.563705 0.281853 0.959458i \(-0.409051\pi\)
0.281853 + 0.959458i \(0.409051\pi\)
\(930\) 4.08250e13 1.78959
\(931\) 2.75961e12 0.120386
\(932\) 6.29819e12 0.273429
\(933\) −4.27620e13 −1.84753
\(934\) −1.49425e13 −0.642484
\(935\) −9.39682e12 −0.402095
\(936\) −3.51337e12 −0.149618
\(937\) −1.95690e13 −0.829353 −0.414677 0.909969i \(-0.636105\pi\)
−0.414677 + 0.909969i \(0.636105\pi\)
\(938\) 1.00300e13 0.423046
\(939\) −6.35960e12 −0.266953
\(940\) −3.37985e12 −0.141196
\(941\) 2.32300e13 0.965820 0.482910 0.875670i \(-0.339580\pi\)
0.482910 + 0.875670i \(0.339580\pi\)
\(942\) −7.61615e12 −0.315142
\(943\) −4.24766e13 −1.74923
\(944\) 1.44147e12 0.0590787
\(945\) 1.96330e12 0.0800835
\(946\) −4.56778e13 −1.85436
\(947\) 1.92596e13 0.778164 0.389082 0.921203i \(-0.372792\pi\)
0.389082 + 0.921203i \(0.372792\pi\)
\(948\) 2.00031e13 0.804378
\(949\) 7.97103e11 0.0319019
\(950\) 2.61266e13 1.04070
\(951\) 2.01856e13 0.800255
\(952\) 1.23937e12 0.0489030
\(953\) −2.29516e13 −0.901352 −0.450676 0.892688i \(-0.648817\pi\)
−0.450676 + 0.892688i \(0.648817\pi\)
\(954\) 2.47476e12 0.0967311
\(955\) 4.08915e13 1.59081
\(956\) 2.40892e13 0.932743
\(957\) 5.42892e13 2.09223
\(958\) −2.91405e13 −1.11777
\(959\) −5.45632e12 −0.208313
\(960\) −5.60908e12 −0.213143
\(961\) −1.16282e13 −0.439804
\(962\) −7.20382e12 −0.271191
\(963\) −2.72999e13 −1.02292
\(964\) −2.42348e13 −0.903842
\(965\) −4.75203e13 −1.76403
\(966\) −2.49463e13 −0.921740
\(967\) 1.18375e12 0.0435351 0.0217675 0.999763i \(-0.493071\pi\)
0.0217675 + 0.999763i \(0.493071\pi\)
\(968\) −1.26324e13 −0.462432
\(969\) −6.79853e12 −0.247718
\(970\) 3.83144e13 1.38960
\(971\) −8.71642e12 −0.314667 −0.157334 0.987545i \(-0.550290\pi\)
−0.157334 + 0.987545i \(0.550290\pi\)
\(972\) −1.91704e13 −0.688862
\(973\) 1.05145e13 0.376080
\(974\) −1.60065e13 −0.569877
\(975\) 1.08275e13 0.383715
\(976\) 2.72433e13 0.961027
\(977\) 2.35038e13 0.825302 0.412651 0.910889i \(-0.364603\pi\)
0.412651 + 0.910889i \(0.364603\pi\)
\(978\) 3.02947e13 1.05887
\(979\) −6.49308e13 −2.25906
\(980\) −2.96155e12 −0.102565
\(981\) −4.33459e13 −1.49430
\(982\) 3.53682e13 1.21370
\(983\) 6.71499e12 0.229380 0.114690 0.993401i \(-0.463413\pi\)
0.114690 + 0.993401i \(0.463413\pi\)
\(984\) −2.96431e13 −1.00796
\(985\) −1.89608e13 −0.641792
\(986\) 8.92066e12 0.300573
\(987\) 3.04770e12 0.102222
\(988\) 3.54845e12 0.118476
\(989\) −4.94347e13 −1.64304
\(990\) 6.22015e13 2.05799
\(991\) −4.15350e13 −1.36799 −0.683994 0.729487i \(-0.739759\pi\)
−0.683994 + 0.729487i \(0.739759\pi\)
\(992\) 2.12096e13 0.695393
\(993\) 5.26542e12 0.171855
\(994\) 2.92907e12 0.0951679
\(995\) 8.52820e13 2.75838
\(996\) −3.04299e13 −0.979790
\(997\) −5.00307e13 −1.60364 −0.801822 0.597563i \(-0.796136\pi\)
−0.801822 + 0.597563i \(0.796136\pi\)
\(998\) −5.67285e13 −1.81015
\(999\) 3.75121e12 0.119159
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.a.1.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.10 12 1.1 even 1 trivial