Properties

Label 91.10.a.a.1.9
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 1.9
Root \(24.5414\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+22.5414 q^{2} +66.2971 q^{3} -3.88662 q^{4} -391.554 q^{5} +1494.43 q^{6} +2401.00 q^{7} -11628.8 q^{8} -15287.7 q^{9} +O(q^{10})\) \(q+22.5414 q^{2} +66.2971 q^{3} -3.88662 q^{4} -391.554 q^{5} +1494.43 q^{6} +2401.00 q^{7} -11628.8 q^{8} -15287.7 q^{9} -8826.17 q^{10} +93595.7 q^{11} -257.672 q^{12} +28561.0 q^{13} +54121.8 q^{14} -25958.9 q^{15} -260139. q^{16} -385933. q^{17} -344606. q^{18} -1.01897e6 q^{19} +1521.82 q^{20} +159179. q^{21} +2.10978e6 q^{22} +1.82565e6 q^{23} -770955. q^{24} -1.79981e6 q^{25} +643804. q^{26} -2.31846e6 q^{27} -9331.78 q^{28} -5.33916e6 q^{29} -585149. q^{30} -6.13823e6 q^{31} +90052.9 q^{32} +6.20512e6 q^{33} -8.69946e6 q^{34} -940121. q^{35} +59417.5 q^{36} -1.31232e7 q^{37} -2.29690e7 q^{38} +1.89351e6 q^{39} +4.55330e6 q^{40} +7.87529e6 q^{41} +3.58812e6 q^{42} -4.05687e7 q^{43} -363771. q^{44} +5.98596e6 q^{45} +4.11527e7 q^{46} +5.94709e7 q^{47} -1.72465e7 q^{48} +5.76480e6 q^{49} -4.05702e7 q^{50} -2.55862e7 q^{51} -111006. q^{52} +5.09403e7 q^{53} -5.22612e7 q^{54} -3.66478e7 q^{55} -2.79207e7 q^{56} -6.75549e7 q^{57} -1.20352e8 q^{58} -616701. q^{59} +100892. q^{60} +5.11911e7 q^{61} -1.38364e8 q^{62} -3.67058e7 q^{63} +1.35221e8 q^{64} -1.11832e7 q^{65} +1.39872e8 q^{66} -1.58337e8 q^{67} +1.49998e6 q^{68} +1.21036e8 q^{69} -2.11916e7 q^{70} +9.17625e6 q^{71} +1.77777e8 q^{72} -3.70094e8 q^{73} -2.95815e8 q^{74} -1.19322e8 q^{75} +3.96036e6 q^{76} +2.24723e8 q^{77} +4.26823e7 q^{78} +2.66334e8 q^{79} +1.01858e8 q^{80} +1.47201e8 q^{81} +1.77520e8 q^{82} -4.26490e8 q^{83} -618670. q^{84} +1.51114e8 q^{85} -9.14474e8 q^{86} -3.53971e8 q^{87} -1.08840e9 q^{88} -1.11538e7 q^{89} +1.34932e8 q^{90} +6.85750e7 q^{91} -7.09563e6 q^{92} -4.06947e8 q^{93} +1.34056e9 q^{94} +3.98983e8 q^{95} +5.97025e6 q^{96} +4.95030e8 q^{97} +1.29947e8 q^{98} -1.43086e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 22.5414 0.996197 0.498099 0.867120i \(-0.334032\pi\)
0.498099 + 0.867120i \(0.334032\pi\)
\(3\) 66.2971 0.472551 0.236276 0.971686i \(-0.424073\pi\)
0.236276 + 0.971686i \(0.424073\pi\)
\(4\) −3.88662 −0.00759106
\(5\) −391.554 −0.280173 −0.140087 0.990139i \(-0.544738\pi\)
−0.140087 + 0.990139i \(0.544738\pi\)
\(6\) 1494.43 0.470754
\(7\) 2401.00 0.377964
\(8\) −11628.8 −1.00376
\(9\) −15287.7 −0.776695
\(10\) −8826.17 −0.279108
\(11\) 93595.7 1.92748 0.963738 0.266852i \(-0.0859835\pi\)
0.963738 + 0.266852i \(0.0859835\pi\)
\(12\) −257.672 −0.00358716
\(13\) 28561.0 0.277350
\(14\) 54121.8 0.376527
\(15\) −25958.9 −0.132396
\(16\) −260139. −0.992351
\(17\) −385933. −1.12071 −0.560353 0.828254i \(-0.689335\pi\)
−0.560353 + 0.828254i \(0.689335\pi\)
\(18\) −344606. −0.773742
\(19\) −1.01897e6 −1.79379 −0.896895 0.442244i \(-0.854182\pi\)
−0.896895 + 0.442244i \(0.854182\pi\)
\(20\) 1521.82 0.00212681
\(21\) 159179. 0.178608
\(22\) 2.10978e6 1.92015
\(23\) 1.82565e6 1.36033 0.680163 0.733061i \(-0.261909\pi\)
0.680163 + 0.733061i \(0.261909\pi\)
\(24\) −770955. −0.474328
\(25\) −1.79981e6 −0.921503
\(26\) 643804. 0.276295
\(27\) −2.31846e6 −0.839579
\(28\) −9331.78 −0.00286915
\(29\) −5.33916e6 −1.40179 −0.700894 0.713266i \(-0.747215\pi\)
−0.700894 + 0.713266i \(0.747215\pi\)
\(30\) −585149. −0.131893
\(31\) −6.13823e6 −1.19376 −0.596878 0.802332i \(-0.703592\pi\)
−0.596878 + 0.802332i \(0.703592\pi\)
\(32\) 90052.9 0.0151818
\(33\) 6.20512e6 0.910830
\(34\) −8.69946e6 −1.11645
\(35\) −940121. −0.105896
\(36\) 59417.5 0.00589594
\(37\) −1.31232e7 −1.15115 −0.575576 0.817748i \(-0.695222\pi\)
−0.575576 + 0.817748i \(0.695222\pi\)
\(38\) −2.29690e7 −1.78697
\(39\) 1.89351e6 0.131062
\(40\) 4.55330e6 0.281227
\(41\) 7.87529e6 0.435251 0.217625 0.976032i \(-0.430169\pi\)
0.217625 + 0.976032i \(0.430169\pi\)
\(42\) 3.58812e6 0.177928
\(43\) −4.05687e7 −1.80960 −0.904801 0.425835i \(-0.859980\pi\)
−0.904801 + 0.425835i \(0.859980\pi\)
\(44\) −363771. −0.0146316
\(45\) 5.98596e6 0.217609
\(46\) 4.11527e7 1.35515
\(47\) 5.94709e7 1.77772 0.888862 0.458176i \(-0.151497\pi\)
0.888862 + 0.458176i \(0.151497\pi\)
\(48\) −1.72465e7 −0.468937
\(49\) 5.76480e6 0.142857
\(50\) −4.05702e7 −0.917999
\(51\) −2.55862e7 −0.529591
\(52\) −111006. −0.00210538
\(53\) 5.09403e7 0.886789 0.443394 0.896327i \(-0.353774\pi\)
0.443394 + 0.896327i \(0.353774\pi\)
\(54\) −5.22612e7 −0.836387
\(55\) −3.66478e7 −0.540027
\(56\) −2.79207e7 −0.379385
\(57\) −6.75549e7 −0.847657
\(58\) −1.20352e8 −1.39646
\(59\) −616701. −0.00662583 −0.00331292 0.999995i \(-0.501055\pi\)
−0.00331292 + 0.999995i \(0.501055\pi\)
\(60\) 100892. 0.00100503
\(61\) 5.11911e7 0.473380 0.236690 0.971585i \(-0.423937\pi\)
0.236690 + 0.971585i \(0.423937\pi\)
\(62\) −1.38364e8 −1.18922
\(63\) −3.67058e7 −0.293563
\(64\) 1.35221e8 1.00748
\(65\) −1.11832e7 −0.0777061
\(66\) 1.39872e8 0.907367
\(67\) −1.58337e8 −0.959947 −0.479973 0.877283i \(-0.659354\pi\)
−0.479973 + 0.877283i \(0.659354\pi\)
\(68\) 1.49998e6 0.00850735
\(69\) 1.21036e8 0.642824
\(70\) −2.11916e7 −0.105493
\(71\) 9.17625e6 0.0428551 0.0214276 0.999770i \(-0.493179\pi\)
0.0214276 + 0.999770i \(0.493179\pi\)
\(72\) 1.77777e8 0.779615
\(73\) −3.70094e8 −1.52531 −0.762657 0.646803i \(-0.776106\pi\)
−0.762657 + 0.646803i \(0.776106\pi\)
\(74\) −2.95815e8 −1.14677
\(75\) −1.19322e8 −0.435457
\(76\) 3.96036e6 0.0136168
\(77\) 2.24723e8 0.728517
\(78\) 4.26823e7 0.130564
\(79\) 2.66334e8 0.769317 0.384658 0.923059i \(-0.374319\pi\)
0.384658 + 0.923059i \(0.374319\pi\)
\(80\) 1.01858e8 0.278030
\(81\) 1.47201e8 0.379951
\(82\) 1.77520e8 0.433595
\(83\) −4.26490e8 −0.986411 −0.493205 0.869913i \(-0.664175\pi\)
−0.493205 + 0.869913i \(0.664175\pi\)
\(84\) −618670. −0.00135582
\(85\) 1.51114e8 0.313992
\(86\) −9.14474e8 −1.80272
\(87\) −3.53971e8 −0.662416
\(88\) −1.08840e9 −1.93472
\(89\) −1.11538e7 −0.0188438 −0.00942192 0.999956i \(-0.502999\pi\)
−0.00942192 + 0.999956i \(0.502999\pi\)
\(90\) 1.34932e8 0.216782
\(91\) 6.85750e7 0.104828
\(92\) −7.09563e6 −0.0103263
\(93\) −4.06947e8 −0.564111
\(94\) 1.34056e9 1.77096
\(95\) 3.98983e8 0.502572
\(96\) 5.97025e6 0.00717417
\(97\) 4.95030e8 0.567752 0.283876 0.958861i \(-0.408380\pi\)
0.283876 + 0.958861i \(0.408380\pi\)
\(98\) 1.29947e8 0.142314
\(99\) −1.43086e9 −1.49706
\(100\) 6.99518e6 0.00699518
\(101\) −9.32790e7 −0.0891944 −0.0445972 0.999005i \(-0.514200\pi\)
−0.0445972 + 0.999005i \(0.514200\pi\)
\(102\) −5.76749e8 −0.527577
\(103\) 1.32347e8 0.115863 0.0579317 0.998321i \(-0.481549\pi\)
0.0579317 + 0.998321i \(0.481549\pi\)
\(104\) −3.32130e8 −0.278393
\(105\) −6.23273e7 −0.0500411
\(106\) 1.14826e9 0.883416
\(107\) −5.92037e8 −0.436638 −0.218319 0.975877i \(-0.570057\pi\)
−0.218319 + 0.975877i \(0.570057\pi\)
\(108\) 9.01096e6 0.00637330
\(109\) −7.00685e8 −0.475449 −0.237724 0.971333i \(-0.576402\pi\)
−0.237724 + 0.971333i \(0.576402\pi\)
\(110\) −8.26091e8 −0.537973
\(111\) −8.70031e8 −0.543978
\(112\) −6.24594e8 −0.375074
\(113\) 1.52540e9 0.880100 0.440050 0.897973i \(-0.354961\pi\)
0.440050 + 0.897973i \(0.354961\pi\)
\(114\) −1.52278e9 −0.844434
\(115\) −7.14842e8 −0.381127
\(116\) 2.07513e7 0.0106411
\(117\) −4.36632e8 −0.215417
\(118\) −1.39013e7 −0.00660064
\(119\) −9.26626e8 −0.423587
\(120\) 3.01871e8 0.132894
\(121\) 6.40221e9 2.71516
\(122\) 1.15392e9 0.471580
\(123\) 5.22109e8 0.205678
\(124\) 2.38570e7 0.00906187
\(125\) 1.46948e9 0.538354
\(126\) −8.27398e8 −0.292447
\(127\) 3.21186e9 1.09557 0.547785 0.836619i \(-0.315471\pi\)
0.547785 + 0.836619i \(0.315471\pi\)
\(128\) 3.00196e9 0.988462
\(129\) −2.68959e9 −0.855129
\(130\) −2.52084e8 −0.0774106
\(131\) −4.15853e9 −1.23373 −0.616863 0.787070i \(-0.711597\pi\)
−0.616863 + 0.787070i \(0.711597\pi\)
\(132\) −2.41170e7 −0.00691417
\(133\) −2.44655e9 −0.677989
\(134\) −3.56914e9 −0.956296
\(135\) 9.07801e8 0.235228
\(136\) 4.48794e9 1.12492
\(137\) 7.05191e9 1.71027 0.855134 0.518407i \(-0.173475\pi\)
0.855134 + 0.518407i \(0.173475\pi\)
\(138\) 2.72831e9 0.640379
\(139\) −4.37330e9 −0.993670 −0.496835 0.867845i \(-0.665505\pi\)
−0.496835 + 0.867845i \(0.665505\pi\)
\(140\) 3.65390e6 0.000803859 0
\(141\) 3.94275e9 0.840065
\(142\) 2.06845e8 0.0426921
\(143\) 2.67319e9 0.534585
\(144\) 3.97693e9 0.770755
\(145\) 2.09057e9 0.392743
\(146\) −8.34243e9 −1.51951
\(147\) 3.82189e8 0.0675073
\(148\) 5.10050e7 0.00873846
\(149\) 1.47718e9 0.245525 0.122762 0.992436i \(-0.460825\pi\)
0.122762 + 0.992436i \(0.460825\pi\)
\(150\) −2.68969e9 −0.433801
\(151\) −6.20770e9 −0.971705 −0.485852 0.874041i \(-0.661491\pi\)
−0.485852 + 0.874041i \(0.661491\pi\)
\(152\) 1.18494e10 1.80053
\(153\) 5.90003e9 0.870448
\(154\) 5.06557e9 0.725747
\(155\) 2.40345e9 0.334458
\(156\) −7.35936e6 −0.000994900 0
\(157\) 2.93653e9 0.385732 0.192866 0.981225i \(-0.438222\pi\)
0.192866 + 0.981225i \(0.438222\pi\)
\(158\) 6.00354e9 0.766391
\(159\) 3.37719e9 0.419053
\(160\) −3.52606e7 −0.00425353
\(161\) 4.38339e9 0.514155
\(162\) 3.31811e9 0.378507
\(163\) 2.14103e7 0.00237563 0.00118782 0.999999i \(-0.499622\pi\)
0.00118782 + 0.999999i \(0.499622\pi\)
\(164\) −3.06083e7 −0.00330401
\(165\) −2.42964e9 −0.255190
\(166\) −9.61367e9 −0.982659
\(167\) 1.16485e10 1.15890 0.579448 0.815009i \(-0.303268\pi\)
0.579448 + 0.815009i \(0.303268\pi\)
\(168\) −1.85106e9 −0.179279
\(169\) 8.15731e8 0.0769231
\(170\) 3.40631e9 0.312798
\(171\) 1.55777e10 1.39323
\(172\) 1.57675e8 0.0137368
\(173\) −9.63553e9 −0.817839 −0.408920 0.912570i \(-0.634094\pi\)
−0.408920 + 0.912570i \(0.634094\pi\)
\(174\) −7.97899e9 −0.659897
\(175\) −4.32134e9 −0.348295
\(176\) −2.43479e10 −1.91273
\(177\) −4.08855e7 −0.00313105
\(178\) −2.51423e8 −0.0187722
\(179\) 1.02545e10 0.746578 0.373289 0.927715i \(-0.378230\pi\)
0.373289 + 0.927715i \(0.378230\pi\)
\(180\) −2.32652e7 −0.00165189
\(181\) 1.64816e10 1.14142 0.570711 0.821151i \(-0.306668\pi\)
0.570711 + 0.821151i \(0.306668\pi\)
\(182\) 1.54577e9 0.104430
\(183\) 3.39382e9 0.223696
\(184\) −2.12301e10 −1.36544
\(185\) 5.13845e9 0.322522
\(186\) −9.17313e9 −0.561965
\(187\) −3.61217e10 −2.16013
\(188\) −2.31141e8 −0.0134948
\(189\) −5.56661e9 −0.317331
\(190\) 8.99362e9 0.500661
\(191\) 9.70454e9 0.527624 0.263812 0.964574i \(-0.415020\pi\)
0.263812 + 0.964574i \(0.415020\pi\)
\(192\) 8.96476e9 0.476084
\(193\) −3.14916e10 −1.63375 −0.816877 0.576812i \(-0.804297\pi\)
−0.816877 + 0.576812i \(0.804297\pi\)
\(194\) 1.11587e10 0.565593
\(195\) −7.41412e8 −0.0367201
\(196\) −2.24056e7 −0.00108444
\(197\) 1.44276e10 0.682492 0.341246 0.939974i \(-0.389151\pi\)
0.341246 + 0.939974i \(0.389151\pi\)
\(198\) −3.22536e10 −1.49137
\(199\) 1.13380e10 0.512505 0.256252 0.966610i \(-0.417512\pi\)
0.256252 + 0.966610i \(0.417512\pi\)
\(200\) 2.09296e10 0.924967
\(201\) −1.04973e10 −0.453624
\(202\) −2.10264e9 −0.0888553
\(203\) −1.28193e10 −0.529826
\(204\) 9.94441e7 0.00402016
\(205\) −3.08360e9 −0.121946
\(206\) 2.98328e9 0.115423
\(207\) −2.79100e10 −1.05656
\(208\) −7.42983e9 −0.275229
\(209\) −9.53715e10 −3.45748
\(210\) −1.40494e9 −0.0498508
\(211\) 1.04589e10 0.363257 0.181628 0.983367i \(-0.441863\pi\)
0.181628 + 0.983367i \(0.441863\pi\)
\(212\) −1.97986e8 −0.00673167
\(213\) 6.08359e8 0.0202512
\(214\) −1.33453e10 −0.434978
\(215\) 1.58848e10 0.507002
\(216\) 2.69608e10 0.842736
\(217\) −1.47379e10 −0.451197
\(218\) −1.57944e10 −0.473641
\(219\) −2.45362e10 −0.720789
\(220\) 1.42436e8 0.00409938
\(221\) −1.10226e10 −0.310828
\(222\) −1.96117e10 −0.541909
\(223\) −2.88573e10 −0.781418 −0.390709 0.920514i \(-0.627770\pi\)
−0.390709 + 0.920514i \(0.627770\pi\)
\(224\) 2.16217e8 0.00573818
\(225\) 2.75150e10 0.715727
\(226\) 3.43847e10 0.876753
\(227\) −1.01009e10 −0.252490 −0.126245 0.991999i \(-0.540293\pi\)
−0.126245 + 0.991999i \(0.540293\pi\)
\(228\) 2.62561e8 0.00643462
\(229\) −6.00090e10 −1.44197 −0.720986 0.692949i \(-0.756311\pi\)
−0.720986 + 0.692949i \(0.756311\pi\)
\(230\) −1.61135e10 −0.379678
\(231\) 1.48985e10 0.344262
\(232\) 6.20880e10 1.40706
\(233\) −1.49600e10 −0.332530 −0.166265 0.986081i \(-0.553171\pi\)
−0.166265 + 0.986081i \(0.553171\pi\)
\(234\) −9.84228e9 −0.214597
\(235\) −2.32861e10 −0.498071
\(236\) 2.39688e6 5.02971e−5 0
\(237\) 1.76572e10 0.363541
\(238\) −2.08874e10 −0.421977
\(239\) −5.33741e10 −1.05813 −0.529066 0.848581i \(-0.677458\pi\)
−0.529066 + 0.848581i \(0.677458\pi\)
\(240\) 6.75292e9 0.131384
\(241\) 1.32320e10 0.252667 0.126334 0.991988i \(-0.459679\pi\)
0.126334 + 0.991988i \(0.459679\pi\)
\(242\) 1.44314e11 2.70484
\(243\) 5.53931e10 1.01913
\(244\) −1.98961e8 −0.00359346
\(245\) −2.25723e9 −0.0400248
\(246\) 1.17690e10 0.204896
\(247\) −2.91029e10 −0.497508
\(248\) 7.13802e10 1.19824
\(249\) −2.82751e10 −0.466129
\(250\) 3.31240e10 0.536307
\(251\) 5.18625e9 0.0824748 0.0412374 0.999149i \(-0.486870\pi\)
0.0412374 + 0.999149i \(0.486870\pi\)
\(252\) 1.42661e8 0.00222846
\(253\) 1.70873e11 2.62200
\(254\) 7.23997e10 1.09140
\(255\) 1.00184e10 0.148377
\(256\) −1.56487e9 −0.0227719
\(257\) −8.01772e10 −1.14644 −0.573221 0.819401i \(-0.694306\pi\)
−0.573221 + 0.819401i \(0.694306\pi\)
\(258\) −6.06270e10 −0.851877
\(259\) −3.15088e10 −0.435094
\(260\) 4.34648e7 0.000589872 0
\(261\) 8.16235e10 1.08876
\(262\) −9.37389e10 −1.22903
\(263\) 6.08567e8 0.00784346 0.00392173 0.999992i \(-0.498752\pi\)
0.00392173 + 0.999992i \(0.498752\pi\)
\(264\) −7.21581e10 −0.914255
\(265\) −1.99459e10 −0.248455
\(266\) −5.51487e10 −0.675410
\(267\) −7.39467e8 −0.00890467
\(268\) 6.15398e8 0.00728701
\(269\) −6.55756e10 −0.763584 −0.381792 0.924248i \(-0.624693\pi\)
−0.381792 + 0.924248i \(0.624693\pi\)
\(270\) 2.04631e10 0.234333
\(271\) 2.61259e10 0.294245 0.147123 0.989118i \(-0.452999\pi\)
0.147123 + 0.989118i \(0.452999\pi\)
\(272\) 1.00396e11 1.11214
\(273\) 4.54632e9 0.0495368
\(274\) 1.58960e11 1.70376
\(275\) −1.68454e11 −1.77617
\(276\) −4.70419e8 −0.00487971
\(277\) 6.74230e10 0.688096 0.344048 0.938952i \(-0.388202\pi\)
0.344048 + 0.938952i \(0.388202\pi\)
\(278\) −9.85801e10 −0.989892
\(279\) 9.38394e10 0.927185
\(280\) 1.09325e10 0.106294
\(281\) 4.36337e10 0.417487 0.208744 0.977970i \(-0.433063\pi\)
0.208744 + 0.977970i \(0.433063\pi\)
\(282\) 8.88749e10 0.836871
\(283\) −1.65395e11 −1.53279 −0.766395 0.642370i \(-0.777951\pi\)
−0.766395 + 0.642370i \(0.777951\pi\)
\(284\) −3.56646e7 −0.000325316 0
\(285\) 2.64514e10 0.237491
\(286\) 6.02573e10 0.532553
\(287\) 1.89086e10 0.164509
\(288\) −1.37670e9 −0.0117916
\(289\) 3.03566e10 0.255984
\(290\) 4.71243e10 0.391250
\(291\) 3.28190e10 0.268292
\(292\) 1.43842e9 0.0115788
\(293\) 5.54864e10 0.439827 0.219914 0.975519i \(-0.429422\pi\)
0.219914 + 0.975519i \(0.429422\pi\)
\(294\) 8.61507e9 0.0672506
\(295\) 2.41472e8 0.00185638
\(296\) 1.52607e11 1.15548
\(297\) −2.16997e11 −1.61827
\(298\) 3.32977e10 0.244591
\(299\) 5.21425e10 0.377287
\(300\) 4.63760e8 0.00330558
\(301\) −9.74054e10 −0.683965
\(302\) −1.39930e11 −0.968009
\(303\) −6.18413e9 −0.0421489
\(304\) 2.65075e11 1.78007
\(305\) −2.00441e10 −0.132629
\(306\) 1.32995e11 0.867138
\(307\) −2.95750e11 −1.90021 −0.950105 0.311932i \(-0.899024\pi\)
−0.950105 + 0.311932i \(0.899024\pi\)
\(308\) −8.73415e8 −0.00553022
\(309\) 8.77421e9 0.0547514
\(310\) 5.41770e10 0.333187
\(311\) 2.21826e11 1.34460 0.672298 0.740281i \(-0.265308\pi\)
0.672298 + 0.740281i \(0.265308\pi\)
\(312\) −2.20192e10 −0.131555
\(313\) −3.12860e11 −1.84247 −0.921236 0.389004i \(-0.872819\pi\)
−0.921236 + 0.389004i \(0.872819\pi\)
\(314\) 6.61933e10 0.384265
\(315\) 1.43723e10 0.0822486
\(316\) −1.03514e9 −0.00583993
\(317\) 1.52625e11 0.848903 0.424451 0.905451i \(-0.360467\pi\)
0.424451 + 0.905451i \(0.360467\pi\)
\(318\) 7.61266e10 0.417459
\(319\) −4.99723e11 −2.70191
\(320\) −5.29464e10 −0.282268
\(321\) −3.92503e10 −0.206334
\(322\) 9.88077e10 0.512200
\(323\) 3.93256e11 2.01031
\(324\) −5.72115e8 −0.00288423
\(325\) −5.14044e10 −0.255579
\(326\) 4.82618e8 0.00236660
\(327\) −4.64534e10 −0.224674
\(328\) −9.15801e10 −0.436887
\(329\) 1.42790e11 0.671916
\(330\) −5.47674e10 −0.254220
\(331\) −2.19954e11 −1.00718 −0.503589 0.863943i \(-0.667987\pi\)
−0.503589 + 0.863943i \(0.667987\pi\)
\(332\) 1.65761e9 0.00748790
\(333\) 2.00624e11 0.894094
\(334\) 2.62572e11 1.15449
\(335\) 6.19977e10 0.268951
\(336\) −4.14087e10 −0.177241
\(337\) −1.30279e11 −0.550224 −0.275112 0.961412i \(-0.588715\pi\)
−0.275112 + 0.961412i \(0.588715\pi\)
\(338\) 1.83877e10 0.0766306
\(339\) 1.01130e11 0.415892
\(340\) −5.87322e8 −0.00238353
\(341\) −5.74512e11 −2.30093
\(342\) 3.51144e11 1.38793
\(343\) 1.38413e10 0.0539949
\(344\) 4.71765e11 1.81640
\(345\) −4.73919e10 −0.180102
\(346\) −2.17198e11 −0.814729
\(347\) −2.65960e11 −0.984768 −0.492384 0.870378i \(-0.663874\pi\)
−0.492384 + 0.870378i \(0.663874\pi\)
\(348\) 1.37575e9 0.00502844
\(349\) −2.10746e11 −0.760405 −0.380203 0.924903i \(-0.624146\pi\)
−0.380203 + 0.924903i \(0.624146\pi\)
\(350\) −9.74090e10 −0.346971
\(351\) −6.62174e10 −0.232857
\(352\) 8.42857e9 0.0292625
\(353\) 2.15031e11 0.737081 0.368540 0.929612i \(-0.379858\pi\)
0.368540 + 0.929612i \(0.379858\pi\)
\(354\) −9.21615e8 −0.00311914
\(355\) −3.59300e9 −0.0120069
\(356\) 4.33508e7 0.000143045 0
\(357\) −6.14326e10 −0.200167
\(358\) 2.31150e11 0.743739
\(359\) −2.50831e11 −0.796995 −0.398497 0.917169i \(-0.630468\pi\)
−0.398497 + 0.917169i \(0.630468\pi\)
\(360\) −6.96095e10 −0.218427
\(361\) 7.15618e11 2.21768
\(362\) 3.71518e11 1.13708
\(363\) 4.24448e11 1.28305
\(364\) −2.66525e8 −0.000795759 0
\(365\) 1.44912e11 0.427352
\(366\) 7.65014e10 0.222846
\(367\) −4.69761e11 −1.35170 −0.675849 0.737040i \(-0.736223\pi\)
−0.675849 + 0.737040i \(0.736223\pi\)
\(368\) −4.74924e11 −1.34992
\(369\) −1.20395e11 −0.338057
\(370\) 1.15828e11 0.321295
\(371\) 1.22308e11 0.335175
\(372\) 1.58165e9 0.00428220
\(373\) 3.42061e11 0.914984 0.457492 0.889214i \(-0.348748\pi\)
0.457492 + 0.889214i \(0.348748\pi\)
\(374\) −8.14232e11 −2.15192
\(375\) 9.74220e10 0.254400
\(376\) −6.91575e11 −1.78441
\(377\) −1.52492e11 −0.388786
\(378\) −1.25479e11 −0.316124
\(379\) 2.27292e11 0.565859 0.282929 0.959141i \(-0.408694\pi\)
0.282929 + 0.959141i \(0.408694\pi\)
\(380\) −1.55070e9 −0.00381505
\(381\) 2.12937e11 0.517712
\(382\) 2.18754e11 0.525618
\(383\) 1.97764e11 0.469627 0.234814 0.972040i \(-0.424552\pi\)
0.234814 + 0.972040i \(0.424552\pi\)
\(384\) 1.99021e11 0.467099
\(385\) −8.79913e10 −0.204111
\(386\) −7.09864e11 −1.62754
\(387\) 6.20202e11 1.40551
\(388\) −1.92399e9 −0.00430984
\(389\) 7.21867e11 1.59839 0.799197 0.601069i \(-0.205258\pi\)
0.799197 + 0.601069i \(0.205258\pi\)
\(390\) −1.67124e10 −0.0365805
\(391\) −7.04581e11 −1.52453
\(392\) −6.70377e10 −0.143394
\(393\) −2.75698e11 −0.582999
\(394\) 3.25219e11 0.679897
\(395\) −1.04284e11 −0.215542
\(396\) 5.56122e9 0.0113643
\(397\) 8.41609e9 0.0170041 0.00850204 0.999964i \(-0.497294\pi\)
0.00850204 + 0.999964i \(0.497294\pi\)
\(398\) 2.55574e11 0.510556
\(399\) −1.62199e11 −0.320384
\(400\) 4.68201e11 0.914455
\(401\) −6.75074e10 −0.130377 −0.0651886 0.997873i \(-0.520765\pi\)
−0.0651886 + 0.997873i \(0.520765\pi\)
\(402\) −2.36624e11 −0.451899
\(403\) −1.75314e11 −0.331088
\(404\) 3.62540e8 0.000677080 0
\(405\) −5.76371e10 −0.106452
\(406\) −2.88965e11 −0.527811
\(407\) −1.22828e12 −2.21882
\(408\) 2.97537e11 0.531582
\(409\) 2.10564e11 0.372074 0.186037 0.982543i \(-0.440436\pi\)
0.186037 + 0.982543i \(0.440436\pi\)
\(410\) −6.95086e10 −0.121482
\(411\) 4.67521e11 0.808189
\(412\) −5.14382e8 −0.000879526 0
\(413\) −1.48070e9 −0.00250433
\(414\) −6.29131e11 −1.05254
\(415\) 1.66994e11 0.276366
\(416\) 2.57200e9 0.00421067
\(417\) −2.89937e11 −0.469560
\(418\) −2.14980e12 −3.44434
\(419\) 9.95463e10 0.157784 0.0788918 0.996883i \(-0.474862\pi\)
0.0788918 + 0.996883i \(0.474862\pi\)
\(420\) 2.42243e8 0.000379865 0
\(421\) −4.42260e10 −0.0686133 −0.0343067 0.999411i \(-0.510922\pi\)
−0.0343067 + 0.999411i \(0.510922\pi\)
\(422\) 2.35757e11 0.361875
\(423\) −9.09173e11 −1.38075
\(424\) −5.92374e11 −0.890123
\(425\) 6.94607e11 1.03273
\(426\) 1.37132e10 0.0201742
\(427\) 1.22910e11 0.178921
\(428\) 2.30102e9 0.00331455
\(429\) 1.77224e11 0.252619
\(430\) 3.58066e11 0.505074
\(431\) −8.65731e11 −1.20847 −0.604234 0.796807i \(-0.706521\pi\)
−0.604234 + 0.796807i \(0.706521\pi\)
\(432\) 6.03120e11 0.833158
\(433\) 1.08066e12 1.47738 0.738689 0.674046i \(-0.235445\pi\)
0.738689 + 0.674046i \(0.235445\pi\)
\(434\) −3.32212e11 −0.449481
\(435\) 1.38599e11 0.185591
\(436\) 2.72330e9 0.00360916
\(437\) −1.86029e12 −2.44014
\(438\) −5.53079e11 −0.718048
\(439\) 7.82505e10 0.100553 0.0502767 0.998735i \(-0.483990\pi\)
0.0502767 + 0.998735i \(0.483990\pi\)
\(440\) 4.26169e11 0.542057
\(441\) −8.81305e10 −0.110956
\(442\) −2.48465e11 −0.309646
\(443\) 8.02517e11 0.990005 0.495002 0.868892i \(-0.335167\pi\)
0.495002 + 0.868892i \(0.335167\pi\)
\(444\) 3.38148e9 0.00412937
\(445\) 4.36733e9 0.00527954
\(446\) −6.50483e11 −0.778447
\(447\) 9.79327e10 0.116023
\(448\) 3.24666e11 0.380790
\(449\) 7.23243e11 0.839800 0.419900 0.907570i \(-0.362065\pi\)
0.419900 + 0.907570i \(0.362065\pi\)
\(450\) 6.20225e11 0.713005
\(451\) 7.37093e11 0.838935
\(452\) −5.92867e9 −0.00668089
\(453\) −4.11552e11 −0.459180
\(454\) −2.27688e11 −0.251530
\(455\) −2.68508e10 −0.0293701
\(456\) 7.85582e11 0.850844
\(457\) 2.75313e11 0.295260 0.147630 0.989043i \(-0.452836\pi\)
0.147630 + 0.989043i \(0.452836\pi\)
\(458\) −1.35269e12 −1.43649
\(459\) 8.94769e11 0.940922
\(460\) 2.77832e9 0.00289316
\(461\) −6.14394e10 −0.0633567 −0.0316783 0.999498i \(-0.510085\pi\)
−0.0316783 + 0.999498i \(0.510085\pi\)
\(462\) 3.35833e11 0.342952
\(463\) 1.05539e12 1.06733 0.533666 0.845695i \(-0.320814\pi\)
0.533666 + 0.845695i \(0.320814\pi\)
\(464\) 1.38892e12 1.39107
\(465\) 1.59342e11 0.158049
\(466\) −3.37220e11 −0.331266
\(467\) 1.58916e12 1.54612 0.773059 0.634334i \(-0.218725\pi\)
0.773059 + 0.634334i \(0.218725\pi\)
\(468\) 1.69702e9 0.00163524
\(469\) −3.80168e11 −0.362826
\(470\) −5.24900e11 −0.496177
\(471\) 1.94683e11 0.182278
\(472\) 7.17149e9 0.00665074
\(473\) −3.79705e12 −3.48796
\(474\) 3.98017e11 0.362159
\(475\) 1.83396e12 1.65298
\(476\) 3.60145e9 0.00321548
\(477\) −7.78760e11 −0.688765
\(478\) −1.20312e12 −1.05411
\(479\) −1.20954e12 −1.04981 −0.524906 0.851160i \(-0.675900\pi\)
−0.524906 + 0.851160i \(0.675900\pi\)
\(480\) −2.33767e9 −0.00201001
\(481\) −3.74812e11 −0.319272
\(482\) 2.98267e11 0.251706
\(483\) 2.90606e11 0.242965
\(484\) −2.48830e10 −0.0206109
\(485\) −1.93831e11 −0.159069
\(486\) 1.24864e12 1.01525
\(487\) −8.91824e11 −0.718454 −0.359227 0.933250i \(-0.616960\pi\)
−0.359227 + 0.933250i \(0.616960\pi\)
\(488\) −5.95291e11 −0.475160
\(489\) 1.41944e9 0.00112261
\(490\) −5.08811e10 −0.0398725
\(491\) −7.46715e10 −0.0579813 −0.0289906 0.999580i \(-0.509229\pi\)
−0.0289906 + 0.999580i \(0.509229\pi\)
\(492\) −2.02924e9 −0.00156131
\(493\) 2.06056e12 1.57099
\(494\) −6.56019e11 −0.495616
\(495\) 5.60260e11 0.419437
\(496\) 1.59679e12 1.18462
\(497\) 2.20322e10 0.0161977
\(498\) −6.37359e11 −0.464357
\(499\) 1.33765e12 0.965808 0.482904 0.875673i \(-0.339582\pi\)
0.482904 + 0.875673i \(0.339582\pi\)
\(500\) −5.71130e9 −0.00408668
\(501\) 7.72259e11 0.547638
\(502\) 1.16905e11 0.0821612
\(503\) 4.15314e11 0.289281 0.144641 0.989484i \(-0.453797\pi\)
0.144641 + 0.989484i \(0.453797\pi\)
\(504\) 4.26844e11 0.294667
\(505\) 3.65238e10 0.0249899
\(506\) 3.85172e12 2.61202
\(507\) 5.40806e10 0.0363501
\(508\) −1.24833e10 −0.00831653
\(509\) −1.12069e12 −0.740039 −0.370020 0.929024i \(-0.620649\pi\)
−0.370020 + 0.929024i \(0.620649\pi\)
\(510\) 2.25828e11 0.147813
\(511\) −8.88596e11 −0.576515
\(512\) −1.57228e12 −1.01115
\(513\) 2.36244e12 1.50603
\(514\) −1.80730e12 −1.14208
\(515\) −5.18210e10 −0.0324618
\(516\) 1.04534e10 0.00649134
\(517\) 5.56622e12 3.42652
\(518\) −7.10252e11 −0.433440
\(519\) −6.38807e11 −0.386471
\(520\) 1.30047e11 0.0779982
\(521\) −8.83586e11 −0.525387 −0.262693 0.964879i \(-0.584611\pi\)
−0.262693 + 0.964879i \(0.584611\pi\)
\(522\) 1.83991e12 1.08462
\(523\) −2.69371e11 −0.157432 −0.0787161 0.996897i \(-0.525082\pi\)
−0.0787161 + 0.996897i \(0.525082\pi\)
\(524\) 1.61626e10 0.00936529
\(525\) −2.86493e11 −0.164587
\(526\) 1.37179e10 0.00781364
\(527\) 2.36895e12 1.33785
\(528\) −1.61419e12 −0.903864
\(529\) 1.53186e12 0.850488
\(530\) −4.49608e11 −0.247510
\(531\) 9.42794e9 0.00514626
\(532\) 9.50883e9 0.00514665
\(533\) 2.24926e11 0.120717
\(534\) −1.66686e10 −0.00887081
\(535\) 2.31814e11 0.122334
\(536\) 1.84127e12 0.963556
\(537\) 6.79842e11 0.352796
\(538\) −1.47816e12 −0.760680
\(539\) 5.39561e11 0.275354
\(540\) −3.52828e9 −0.00178563
\(541\) −7.05511e11 −0.354092 −0.177046 0.984203i \(-0.556654\pi\)
−0.177046 + 0.984203i \(0.556654\pi\)
\(542\) 5.88914e11 0.293127
\(543\) 1.09268e12 0.539380
\(544\) −3.47544e10 −0.0170143
\(545\) 2.74356e11 0.133208
\(546\) 1.02480e11 0.0493484
\(547\) −2.49219e12 −1.19025 −0.595126 0.803633i \(-0.702898\pi\)
−0.595126 + 0.803633i \(0.702898\pi\)
\(548\) −2.74081e10 −0.0129828
\(549\) −7.82594e11 −0.367672
\(550\) −3.79720e12 −1.76942
\(551\) 5.44046e12 2.51451
\(552\) −1.40750e12 −0.645240
\(553\) 6.39468e11 0.290774
\(554\) 1.51981e12 0.685479
\(555\) 3.40664e11 0.152408
\(556\) 1.69974e10 0.00754301
\(557\) −2.05323e12 −0.903836 −0.451918 0.892060i \(-0.649260\pi\)
−0.451918 + 0.892060i \(0.649260\pi\)
\(558\) 2.11527e12 0.923659
\(559\) −1.15868e12 −0.501893
\(560\) 2.44562e11 0.105086
\(561\) −2.39476e12 −1.02077
\(562\) 9.83562e11 0.415900
\(563\) −4.20623e12 −1.76443 −0.882216 0.470845i \(-0.843949\pi\)
−0.882216 + 0.470845i \(0.843949\pi\)
\(564\) −1.53240e10 −0.00637698
\(565\) −5.97278e11 −0.246580
\(566\) −3.72822e12 −1.52696
\(567\) 3.53429e11 0.143608
\(568\) −1.06709e11 −0.0430162
\(569\) −1.08108e12 −0.432367 −0.216184 0.976353i \(-0.569361\pi\)
−0.216184 + 0.976353i \(0.569361\pi\)
\(570\) 5.96251e11 0.236588
\(571\) −2.15704e12 −0.849173 −0.424586 0.905387i \(-0.639581\pi\)
−0.424586 + 0.905387i \(0.639581\pi\)
\(572\) −1.03897e10 −0.00405807
\(573\) 6.43383e11 0.249329
\(574\) 4.26225e11 0.163884
\(575\) −3.28583e12 −1.25354
\(576\) −2.06722e12 −0.782502
\(577\) 1.24361e12 0.467083 0.233541 0.972347i \(-0.424969\pi\)
0.233541 + 0.972347i \(0.424969\pi\)
\(578\) 6.84280e11 0.255011
\(579\) −2.08780e12 −0.772032
\(580\) −8.12526e9 −0.00298134
\(581\) −1.02400e12 −0.372828
\(582\) 7.39786e11 0.267271
\(583\) 4.76779e12 1.70926
\(584\) 4.30375e12 1.53105
\(585\) 1.70965e11 0.0603540
\(586\) 1.25074e12 0.438154
\(587\) −9.64442e11 −0.335278 −0.167639 0.985848i \(-0.553614\pi\)
−0.167639 + 0.985848i \(0.553614\pi\)
\(588\) −1.48543e9 −0.000512452 0
\(589\) 6.25469e12 2.14135
\(590\) 5.44311e9 0.00184932
\(591\) 9.56511e11 0.322512
\(592\) 3.41386e12 1.14235
\(593\) −4.96432e12 −1.64859 −0.824297 0.566158i \(-0.808429\pi\)
−0.824297 + 0.566158i \(0.808429\pi\)
\(594\) −4.89142e12 −1.61211
\(595\) 3.62824e11 0.118678
\(596\) −5.74124e9 −0.00186379
\(597\) 7.51677e11 0.242185
\(598\) 1.17536e12 0.375852
\(599\) 2.35087e12 0.746120 0.373060 0.927807i \(-0.378309\pi\)
0.373060 + 0.927807i \(0.378309\pi\)
\(600\) 1.38757e12 0.437094
\(601\) −2.37861e12 −0.743682 −0.371841 0.928296i \(-0.621273\pi\)
−0.371841 + 0.928296i \(0.621273\pi\)
\(602\) −2.19565e12 −0.681364
\(603\) 2.42062e12 0.745586
\(604\) 2.41270e10 0.00737627
\(605\) −2.50681e12 −0.760715
\(606\) −1.39399e11 −0.0419886
\(607\) −5.88382e12 −1.75918 −0.879590 0.475733i \(-0.842183\pi\)
−0.879590 + 0.475733i \(0.842183\pi\)
\(608\) −9.17615e10 −0.0272329
\(609\) −8.49884e11 −0.250370
\(610\) −4.51821e11 −0.132124
\(611\) 1.69855e12 0.493052
\(612\) −2.29312e10 −0.00660762
\(613\) −2.45729e12 −0.702884 −0.351442 0.936210i \(-0.614309\pi\)
−0.351442 + 0.936210i \(0.614309\pi\)
\(614\) −6.66660e12 −1.89298
\(615\) −2.04434e11 −0.0576255
\(616\) −2.61326e12 −0.731256
\(617\) 1.18050e12 0.327931 0.163966 0.986466i \(-0.447571\pi\)
0.163966 + 0.986466i \(0.447571\pi\)
\(618\) 1.97783e11 0.0545432
\(619\) −5.82546e12 −1.59486 −0.797430 0.603412i \(-0.793807\pi\)
−0.797430 + 0.603412i \(0.793807\pi\)
\(620\) −9.34130e9 −0.00253889
\(621\) −4.23270e12 −1.14210
\(622\) 5.00027e12 1.33948
\(623\) −2.67804e10 −0.00712230
\(624\) −4.92576e11 −0.130060
\(625\) 2.93987e12 0.770671
\(626\) −7.05230e12 −1.83547
\(627\) −6.32285e12 −1.63384
\(628\) −1.14132e10 −0.00292811
\(629\) 5.06469e12 1.29010
\(630\) 3.23971e11 0.0819358
\(631\) 5.20483e12 1.30700 0.653498 0.756928i \(-0.273301\pi\)
0.653498 + 0.756928i \(0.273301\pi\)
\(632\) −3.09715e12 −0.772209
\(633\) 6.93392e11 0.171657
\(634\) 3.44037e12 0.845674
\(635\) −1.25762e12 −0.306949
\(636\) −1.31259e10 −0.00318106
\(637\) 1.64648e11 0.0396214
\(638\) −1.12644e13 −2.69164
\(639\) −1.40284e11 −0.0332854
\(640\) −1.17543e12 −0.276941
\(641\) −4.11424e12 −0.962561 −0.481281 0.876567i \(-0.659828\pi\)
−0.481281 + 0.876567i \(0.659828\pi\)
\(642\) −8.84756e11 −0.205549
\(643\) 2.85804e12 0.659354 0.329677 0.944094i \(-0.393060\pi\)
0.329677 + 0.944094i \(0.393060\pi\)
\(644\) −1.70366e10 −0.00390298
\(645\) 1.05312e12 0.239584
\(646\) 8.86452e12 2.00267
\(647\) −8.06581e12 −1.80958 −0.904792 0.425855i \(-0.859973\pi\)
−0.904792 + 0.425855i \(0.859973\pi\)
\(648\) −1.71177e12 −0.381380
\(649\) −5.77206e10 −0.0127711
\(650\) −1.15873e12 −0.254607
\(651\) −9.77079e11 −0.213214
\(652\) −8.32139e7 −1.80336e−5 0
\(653\) −3.40144e12 −0.732072 −0.366036 0.930601i \(-0.619285\pi\)
−0.366036 + 0.930601i \(0.619285\pi\)
\(654\) −1.04712e12 −0.223819
\(655\) 1.62829e12 0.345657
\(656\) −2.04867e12 −0.431921
\(657\) 5.65789e12 1.18470
\(658\) 3.21867e12 0.669361
\(659\) 3.72618e12 0.769625 0.384812 0.922995i \(-0.374266\pi\)
0.384812 + 0.922995i \(0.374266\pi\)
\(660\) 9.44310e9 0.00193717
\(661\) 6.25787e12 1.27503 0.637515 0.770438i \(-0.279963\pi\)
0.637515 + 0.770438i \(0.279963\pi\)
\(662\) −4.95807e12 −1.00335
\(663\) −7.30769e11 −0.146882
\(664\) 4.95957e12 0.990119
\(665\) 9.57958e11 0.189954
\(666\) 4.52233e12 0.890694
\(667\) −9.74746e12 −1.90689
\(668\) −4.52732e10 −0.00879725
\(669\) −1.91315e12 −0.369260
\(670\) 1.39751e12 0.267929
\(671\) 4.79127e12 0.912429
\(672\) 1.43346e10 0.00271158
\(673\) 1.01140e13 1.90044 0.950220 0.311579i \(-0.100858\pi\)
0.950220 + 0.311579i \(0.100858\pi\)
\(674\) −2.93667e12 −0.548132
\(675\) 4.17278e12 0.773675
\(676\) −3.17044e9 −0.000583928 0
\(677\) −8.92555e11 −0.163300 −0.0816500 0.996661i \(-0.526019\pi\)
−0.0816500 + 0.996661i \(0.526019\pi\)
\(678\) 2.27960e12 0.414310
\(679\) 1.18857e12 0.214590
\(680\) −1.75727e12 −0.315173
\(681\) −6.69660e11 −0.119314
\(682\) −1.29503e13 −2.29218
\(683\) 1.05244e12 0.185056 0.0925282 0.995710i \(-0.470505\pi\)
0.0925282 + 0.995710i \(0.470505\pi\)
\(684\) −6.05448e10 −0.0105761
\(685\) −2.76120e12 −0.479172
\(686\) 3.12002e11 0.0537896
\(687\) −3.97842e12 −0.681406
\(688\) 1.05535e13 1.79576
\(689\) 1.45491e12 0.245951
\(690\) −1.06828e12 −0.179417
\(691\) 9.39665e12 1.56791 0.783956 0.620816i \(-0.213199\pi\)
0.783956 + 0.620816i \(0.213199\pi\)
\(692\) 3.74497e10 0.00620827
\(693\) −3.43550e12 −0.565836
\(694\) −5.99511e12 −0.981023
\(695\) 1.71238e12 0.278400
\(696\) 4.11625e12 0.664906
\(697\) −3.03934e12 −0.487788
\(698\) −4.75051e12 −0.757514
\(699\) −9.91807e11 −0.157138
\(700\) 1.67954e10 0.00264393
\(701\) −6.60089e11 −0.103246 −0.0516228 0.998667i \(-0.516439\pi\)
−0.0516228 + 0.998667i \(0.516439\pi\)
\(702\) −1.49263e12 −0.231972
\(703\) 1.33722e13 2.06492
\(704\) 1.26561e13 1.94188
\(705\) −1.54380e12 −0.235364
\(706\) 4.84709e12 0.734278
\(707\) −2.23963e11 −0.0337123
\(708\) 1.58906e8 2.37680e−5 0
\(709\) 1.24774e12 0.185446 0.0927228 0.995692i \(-0.470443\pi\)
0.0927228 + 0.995692i \(0.470443\pi\)
\(710\) −8.09911e10 −0.0119612
\(711\) −4.07164e12 −0.597525
\(712\) 1.29706e11 0.0189147
\(713\) −1.12063e13 −1.62390
\(714\) −1.38477e12 −0.199406
\(715\) −1.04670e12 −0.149777
\(716\) −3.98553e10 −0.00566732
\(717\) −3.53855e12 −0.500021
\(718\) −5.65406e12 −0.793964
\(719\) 9.24139e12 1.28961 0.644803 0.764349i \(-0.276939\pi\)
0.644803 + 0.764349i \(0.276939\pi\)
\(720\) −1.55718e12 −0.215945
\(721\) 3.17765e11 0.0437922
\(722\) 1.61310e13 2.20925
\(723\) 8.77243e11 0.119398
\(724\) −6.40578e10 −0.00866460
\(725\) 9.60948e12 1.29175
\(726\) 9.56763e12 1.27817
\(727\) 2.56696e12 0.340811 0.170406 0.985374i \(-0.445492\pi\)
0.170406 + 0.985374i \(0.445492\pi\)
\(728\) −7.97444e11 −0.105223
\(729\) 7.75047e11 0.101638
\(730\) 3.26651e12 0.425727
\(731\) 1.56568e13 2.02803
\(732\) −1.31905e10 −0.00169809
\(733\) −1.35156e13 −1.72929 −0.864647 0.502380i \(-0.832458\pi\)
−0.864647 + 0.502380i \(0.832458\pi\)
\(734\) −1.05891e13 −1.34656
\(735\) −1.49648e11 −0.0189137
\(736\) 1.64406e11 0.0206522
\(737\) −1.48197e13 −1.85027
\(738\) −2.71387e12 −0.336772
\(739\) 8.83237e12 1.08937 0.544687 0.838639i \(-0.316648\pi\)
0.544687 + 0.838639i \(0.316648\pi\)
\(740\) −1.99712e10 −0.00244828
\(741\) −1.92944e12 −0.235098
\(742\) 2.75698e12 0.333900
\(743\) 1.24941e13 1.50402 0.752012 0.659149i \(-0.229083\pi\)
0.752012 + 0.659149i \(0.229083\pi\)
\(744\) 4.73230e12 0.566231
\(745\) −5.78396e11 −0.0687894
\(746\) 7.71052e12 0.911505
\(747\) 6.52005e12 0.766141
\(748\) 1.40391e11 0.0163977
\(749\) −1.42148e12 −0.165034
\(750\) 2.19603e12 0.253432
\(751\) 9.68723e11 0.111127 0.0555636 0.998455i \(-0.482304\pi\)
0.0555636 + 0.998455i \(0.482304\pi\)
\(752\) −1.54707e13 −1.76413
\(753\) 3.43833e11 0.0389736
\(754\) −3.43737e12 −0.387307
\(755\) 2.43065e12 0.272246
\(756\) 2.16353e10 0.00240888
\(757\) −4.86384e11 −0.0538330 −0.0269165 0.999638i \(-0.508569\pi\)
−0.0269165 + 0.999638i \(0.508569\pi\)
\(758\) 5.12348e12 0.563707
\(759\) 1.13284e13 1.23903
\(760\) −4.63969e12 −0.504461
\(761\) −2.85734e12 −0.308838 −0.154419 0.988005i \(-0.549351\pi\)
−0.154419 + 0.988005i \(0.549351\pi\)
\(762\) 4.79989e12 0.515744
\(763\) −1.68235e12 −0.179703
\(764\) −3.77179e10 −0.00400523
\(765\) −2.31018e12 −0.243876
\(766\) 4.45788e12 0.467841
\(767\) −1.76136e10 −0.00183768
\(768\) −1.03746e11 −0.0107609
\(769\) −1.72096e13 −1.77461 −0.887303 0.461187i \(-0.847424\pi\)
−0.887303 + 0.461187i \(0.847424\pi\)
\(770\) −1.98344e12 −0.203335
\(771\) −5.31551e12 −0.541752
\(772\) 1.22396e11 0.0124019
\(773\) −1.26196e12 −0.127127 −0.0635636 0.997978i \(-0.520247\pi\)
−0.0635636 + 0.997978i \(0.520247\pi\)
\(774\) 1.39802e13 1.40016
\(775\) 1.10476e13 1.10005
\(776\) −5.75660e12 −0.569886
\(777\) −2.08894e12 −0.205604
\(778\) 1.62719e13 1.59232
\(779\) −8.02471e12 −0.780748
\(780\) 2.88159e9 0.000278744 0
\(781\) 8.58857e11 0.0826022
\(782\) −1.58822e13 −1.51873
\(783\) 1.23786e13 1.17691
\(784\) −1.49965e12 −0.141764
\(785\) −1.14981e12 −0.108072
\(786\) −6.21462e12 −0.580782
\(787\) −3.34979e12 −0.311265 −0.155633 0.987815i \(-0.549742\pi\)
−0.155633 + 0.987815i \(0.549742\pi\)
\(788\) −5.60748e10 −0.00518084
\(789\) 4.03462e10 0.00370644
\(790\) −2.35071e12 −0.214722
\(791\) 3.66249e12 0.332646
\(792\) 1.66392e13 1.50269
\(793\) 1.46207e12 0.131292
\(794\) 1.89710e11 0.0169394
\(795\) −1.32235e12 −0.117407
\(796\) −4.40666e10 −0.00389046
\(797\) −1.97846e13 −1.73686 −0.868429 0.495813i \(-0.834870\pi\)
−0.868429 + 0.495813i \(0.834870\pi\)
\(798\) −3.65620e12 −0.319166
\(799\) −2.29518e13 −1.99231
\(800\) −1.62078e11 −0.0139901
\(801\) 1.70516e11 0.0146359
\(802\) −1.52171e12 −0.129881
\(803\) −3.46392e13 −2.94001
\(804\) 4.07991e10 0.00344349
\(805\) −1.71634e12 −0.144053
\(806\) −3.95182e12 −0.329829
\(807\) −4.34747e12 −0.360833
\(808\) 1.08472e12 0.0895298
\(809\) 1.17947e12 0.0968093 0.0484046 0.998828i \(-0.484586\pi\)
0.0484046 + 0.998828i \(0.484586\pi\)
\(810\) −1.29922e12 −0.106047
\(811\) 2.03470e13 1.65161 0.825803 0.563959i \(-0.190723\pi\)
0.825803 + 0.563959i \(0.190723\pi\)
\(812\) 4.98239e10 0.00402194
\(813\) 1.73207e12 0.139046
\(814\) −2.76870e13 −2.21038
\(815\) −8.38330e9 −0.000665589 0
\(816\) 6.65598e12 0.525541
\(817\) 4.13384e13 3.24604
\(818\) 4.74640e12 0.370659
\(819\) −1.04835e12 −0.0814198
\(820\) 1.19848e10 0.000925696 0
\(821\) −3.20868e12 −0.246480 −0.123240 0.992377i \(-0.539329\pi\)
−0.123240 + 0.992377i \(0.539329\pi\)
\(822\) 1.05386e13 0.805116
\(823\) −2.87132e12 −0.218163 −0.109082 0.994033i \(-0.534791\pi\)
−0.109082 + 0.994033i \(0.534791\pi\)
\(824\) −1.53903e12 −0.116299
\(825\) −1.11680e13 −0.839333
\(826\) −3.33770e10 −0.00249481
\(827\) 1.28110e13 0.952377 0.476188 0.879343i \(-0.342018\pi\)
0.476188 + 0.879343i \(0.342018\pi\)
\(828\) 1.08476e11 0.00802041
\(829\) −5.07698e12 −0.373345 −0.186672 0.982422i \(-0.559770\pi\)
−0.186672 + 0.982422i \(0.559770\pi\)
\(830\) 3.76427e12 0.275315
\(831\) 4.46995e12 0.325160
\(832\) 3.86205e12 0.279423
\(833\) −2.22483e12 −0.160101
\(834\) −6.53557e12 −0.467774
\(835\) −4.56100e12 −0.324692
\(836\) 3.70673e11 0.0262460
\(837\) 1.42312e13 1.00225
\(838\) 2.24391e12 0.157184
\(839\) −7.11401e12 −0.495662 −0.247831 0.968803i \(-0.579718\pi\)
−0.247831 + 0.968803i \(0.579718\pi\)
\(840\) 7.24791e11 0.0502292
\(841\) 1.39995e13 0.965007
\(842\) −9.96916e11 −0.0683524
\(843\) 2.89278e12 0.197284
\(844\) −4.06497e10 −0.00275750
\(845\) −3.19403e11 −0.0215518
\(846\) −2.04940e13 −1.37550
\(847\) 1.53717e13 1.02623
\(848\) −1.32516e13 −0.880006
\(849\) −1.09652e13 −0.724321
\(850\) 1.56574e13 1.02881
\(851\) −2.39584e13 −1.56594
\(852\) −2.36446e9 −0.000153728 0
\(853\) 1.39575e13 0.902690 0.451345 0.892350i \(-0.350944\pi\)
0.451345 + 0.892350i \(0.350944\pi\)
\(854\) 2.77056e12 0.178241
\(855\) −6.09953e12 −0.390345
\(856\) 6.88467e12 0.438280
\(857\) 2.19454e13 1.38973 0.694865 0.719141i \(-0.255464\pi\)
0.694865 + 0.719141i \(0.255464\pi\)
\(858\) 3.99488e12 0.251658
\(859\) −2.71253e13 −1.69983 −0.849916 0.526918i \(-0.823347\pi\)
−0.849916 + 0.526918i \(0.823347\pi\)
\(860\) −6.17384e10 −0.00384868
\(861\) 1.25358e12 0.0777390
\(862\) −1.95148e13 −1.20387
\(863\) −1.34740e13 −0.826893 −0.413447 0.910528i \(-0.635675\pi\)
−0.413447 + 0.910528i \(0.635675\pi\)
\(864\) −2.08784e11 −0.0127463
\(865\) 3.77283e12 0.229137
\(866\) 2.43595e13 1.47176
\(867\) 2.01255e12 0.120966
\(868\) 5.72806e10 0.00342507
\(869\) 2.49277e13 1.48284
\(870\) 3.12420e12 0.184886
\(871\) −4.52228e12 −0.266241
\(872\) 8.14812e12 0.477236
\(873\) −7.56787e12 −0.440970
\(874\) −4.19335e13 −2.43086
\(875\) 3.52821e12 0.203479
\(876\) 9.53628e10 0.00547155
\(877\) −2.38700e13 −1.36255 −0.681277 0.732026i \(-0.738575\pi\)
−0.681277 + 0.732026i \(0.738575\pi\)
\(878\) 1.76387e12 0.100171
\(879\) 3.67858e12 0.207841
\(880\) 9.53351e12 0.535897
\(881\) −6.58056e12 −0.368020 −0.184010 0.982924i \(-0.558908\pi\)
−0.184010 + 0.982924i \(0.558908\pi\)
\(882\) −1.98658e12 −0.110535
\(883\) 1.75753e12 0.0972925 0.0486462 0.998816i \(-0.484509\pi\)
0.0486462 + 0.998816i \(0.484509\pi\)
\(884\) 4.28408e10 0.00235952
\(885\) 1.60089e10 0.000877235 0
\(886\) 1.80898e13 0.986240
\(887\) 1.35188e12 0.0733299 0.0366649 0.999328i \(-0.488327\pi\)
0.0366649 + 0.999328i \(0.488327\pi\)
\(888\) 1.01174e13 0.546023
\(889\) 7.71167e12 0.414086
\(890\) 9.84456e10 0.00525946
\(891\) 1.37774e13 0.732347
\(892\) 1.12157e11 0.00593179
\(893\) −6.05992e13 −3.18886
\(894\) 2.20754e12 0.115582
\(895\) −4.01518e12 −0.209171
\(896\) 7.20771e12 0.373604
\(897\) 3.45690e12 0.178287
\(898\) 1.63029e13 0.836606
\(899\) 3.27730e13 1.67339
\(900\) −1.06940e11 −0.00543313
\(901\) −1.96596e13 −0.993830
\(902\) 1.66151e13 0.835744
\(903\) −6.45770e12 −0.323208
\(904\) −1.77386e13 −0.883408
\(905\) −6.45345e12 −0.319796
\(906\) −9.27695e12 −0.457434
\(907\) −1.78688e13 −0.876726 −0.438363 0.898798i \(-0.644442\pi\)
−0.438363 + 0.898798i \(0.644442\pi\)
\(908\) 3.92584e10 0.00191666
\(909\) 1.42602e12 0.0692769
\(910\) −6.05254e11 −0.0292585
\(911\) −2.20038e13 −1.05843 −0.529217 0.848486i \(-0.677514\pi\)
−0.529217 + 0.848486i \(0.677514\pi\)
\(912\) 1.75737e13 0.841174
\(913\) −3.99177e13 −1.90128
\(914\) 6.20593e12 0.294137
\(915\) −1.32886e12 −0.0626738
\(916\) 2.33233e11 0.0109461
\(917\) −9.98463e12 −0.466305
\(918\) 2.01693e13 0.937344
\(919\) 4.13313e12 0.191144 0.0955718 0.995423i \(-0.469532\pi\)
0.0955718 + 0.995423i \(0.469532\pi\)
\(920\) 8.31275e12 0.382560
\(921\) −1.96073e13 −0.897946
\(922\) −1.38493e12 −0.0631158
\(923\) 2.62083e11 0.0118859
\(924\) −5.79048e10 −0.00261331
\(925\) 2.36193e13 1.06079
\(926\) 2.37900e13 1.06327
\(927\) −2.02328e12 −0.0899906
\(928\) −4.80807e11 −0.0212816
\(929\) 3.08089e12 0.135708 0.0678541 0.997695i \(-0.478385\pi\)
0.0678541 + 0.997695i \(0.478385\pi\)
\(930\) 3.59178e12 0.157448
\(931\) −5.87418e12 −0.256256
\(932\) 5.81441e10 0.00252426
\(933\) 1.47064e13 0.635390
\(934\) 3.58219e13 1.54024
\(935\) 1.41436e13 0.605212
\(936\) 5.07750e12 0.216226
\(937\) −1.74551e13 −0.739766 −0.369883 0.929078i \(-0.620602\pi\)
−0.369883 + 0.929078i \(0.620602\pi\)
\(938\) −8.56951e12 −0.361446
\(939\) −2.07417e13 −0.870662
\(940\) 9.05042e10 0.00378088
\(941\) −8.49435e12 −0.353165 −0.176582 0.984286i \(-0.556504\pi\)
−0.176582 + 0.984286i \(0.556504\pi\)
\(942\) 4.38842e12 0.181585
\(943\) 1.43776e13 0.592083
\(944\) 1.60428e11 0.00657516
\(945\) 2.17963e12 0.0889077
\(946\) −8.55908e13 −3.47470
\(947\) 6.95169e12 0.280876 0.140438 0.990089i \(-0.455149\pi\)
0.140438 + 0.990089i \(0.455149\pi\)
\(948\) −6.86268e10 −0.00275966
\(949\) −1.05703e13 −0.423046
\(950\) 4.13399e13 1.64670
\(951\) 1.01186e13 0.401150
\(952\) 1.07755e13 0.425180
\(953\) 9.03759e12 0.354923 0.177462 0.984128i \(-0.443211\pi\)
0.177462 + 0.984128i \(0.443211\pi\)
\(954\) −1.75543e13 −0.686146
\(955\) −3.79985e12 −0.147826
\(956\) 2.07445e11 0.00803234
\(957\) −3.31301e13 −1.27679
\(958\) −2.72648e13 −1.04582
\(959\) 1.69316e13 0.646421
\(960\) −3.51019e12 −0.133386
\(961\) 1.12382e13 0.425053
\(962\) −8.44878e12 −0.318058
\(963\) 9.05088e12 0.339135
\(964\) −5.14278e10 −0.00191801
\(965\) 1.23307e13 0.457734
\(966\) 6.55066e12 0.242041
\(967\) −2.87825e12 −0.105854 −0.0529272 0.998598i \(-0.516855\pi\)
−0.0529272 + 0.998598i \(0.516855\pi\)
\(968\) −7.44499e13 −2.72537
\(969\) 2.60717e13 0.949975
\(970\) −4.36922e12 −0.158464
\(971\) 1.36281e13 0.491980 0.245990 0.969272i \(-0.420887\pi\)
0.245990 + 0.969272i \(0.420887\pi\)
\(972\) −2.15292e11 −0.00773625
\(973\) −1.05003e13 −0.375572
\(974\) −2.01029e13 −0.715722
\(975\) −3.40796e12 −0.120774
\(976\) −1.33168e13 −0.469760
\(977\) −7.83200e12 −0.275009 −0.137505 0.990501i \(-0.543908\pi\)
−0.137505 + 0.990501i \(0.543908\pi\)
\(978\) 3.19962e10 0.00111834
\(979\) −1.04395e12 −0.0363210
\(980\) 8.77301e9 0.000303830 0
\(981\) 1.07119e13 0.369279
\(982\) −1.68320e12 −0.0577608
\(983\) 5.00319e13 1.70906 0.854528 0.519405i \(-0.173846\pi\)
0.854528 + 0.519405i \(0.173846\pi\)
\(984\) −6.07149e12 −0.206451
\(985\) −5.64920e12 −0.191216
\(986\) 4.64478e13 1.56502
\(987\) 9.46654e12 0.317515
\(988\) 1.13112e11 0.00377661
\(989\) −7.40644e13 −2.46165
\(990\) 1.26290e13 0.417842
\(991\) 5.33015e13 1.75553 0.877765 0.479091i \(-0.159034\pi\)
0.877765 + 0.479091i \(0.159034\pi\)
\(992\) −5.52766e11 −0.0181233
\(993\) −1.45823e13 −0.475943
\(994\) 4.96635e11 0.0161361
\(995\) −4.43944e12 −0.143590
\(996\) 1.09894e11 0.00353842
\(997\) −3.21504e13 −1.03052 −0.515262 0.857033i \(-0.672305\pi\)
−0.515262 + 0.857033i \(0.672305\pi\)
\(998\) 3.01525e13 0.962136
\(999\) 3.04256e13 0.966483
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.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.9 12 1.1 even 1 trivial