Properties

Label 91.10.a.a.1.12
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 1.12
Root \(43.4855\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+41.4855 q^{2} -265.461 q^{3} +1209.05 q^{4} -1043.84 q^{5} -11012.8 q^{6} +2401.00 q^{7} +28917.3 q^{8} +50786.7 q^{9} +O(q^{10})\) \(q+41.4855 q^{2} -265.461 q^{3} +1209.05 q^{4} -1043.84 q^{5} -11012.8 q^{6} +2401.00 q^{7} +28917.3 q^{8} +50786.7 q^{9} -43304.2 q^{10} +12300.7 q^{11} -320955. q^{12} +28561.0 q^{13} +99606.7 q^{14} +277099. q^{15} +580618. q^{16} -232027. q^{17} +2.10691e6 q^{18} -496842. q^{19} -1.26205e6 q^{20} -637373. q^{21} +510302. q^{22} -2.39174e6 q^{23} -7.67643e6 q^{24} -863522. q^{25} +1.18487e6 q^{26} -8.25683e6 q^{27} +2.90292e6 q^{28} -5.44610e6 q^{29} +1.14956e7 q^{30} +4.32926e6 q^{31} +9.28156e6 q^{32} -3.26536e6 q^{33} -9.62576e6 q^{34} -2.50626e6 q^{35} +6.14035e7 q^{36} +2.00548e7 q^{37} -2.06117e7 q^{38} -7.58184e6 q^{39} -3.01851e7 q^{40} -3.28800e7 q^{41} -2.64417e7 q^{42} +5.76698e6 q^{43} +1.48721e7 q^{44} -5.30132e7 q^{45} -9.92224e7 q^{46} +1.82697e7 q^{47} -1.54132e8 q^{48} +5.76480e6 q^{49} -3.58236e7 q^{50} +6.15942e7 q^{51} +3.45316e7 q^{52} -6.01983e7 q^{53} -3.42539e8 q^{54} -1.28400e7 q^{55} +6.94305e7 q^{56} +1.31892e8 q^{57} -2.25934e8 q^{58} +4.29026e7 q^{59} +3.35026e8 q^{60} -4.10519e7 q^{61} +1.79601e8 q^{62} +1.21939e8 q^{63} +8.77735e7 q^{64} -2.98131e7 q^{65} -1.35465e8 q^{66} -1.32771e8 q^{67} -2.80532e8 q^{68} +6.34913e8 q^{69} -1.03974e8 q^{70} -1.97237e8 q^{71} +1.46862e9 q^{72} -2.81790e8 q^{73} +8.31984e8 q^{74} +2.29232e8 q^{75} -6.00705e8 q^{76} +2.95340e7 q^{77} -3.14536e8 q^{78} +1.41312e8 q^{79} -6.06073e8 q^{80} +1.19223e9 q^{81} -1.36404e9 q^{82} -3.38531e8 q^{83} -7.70613e8 q^{84} +2.42199e8 q^{85} +2.39246e8 q^{86} +1.44573e9 q^{87} +3.55704e8 q^{88} -7.96320e7 q^{89} -2.19928e9 q^{90} +6.85750e7 q^{91} -2.89172e9 q^{92} -1.14925e9 q^{93} +7.57927e8 q^{94} +5.18623e8 q^{95} -2.46389e9 q^{96} +1.98650e8 q^{97} +2.39156e8 q^{98} +6.24713e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 21q^{2} - 323q^{3} + 2945q^{4} - 5202q^{5} - 9897q^{6} + 28812q^{7} - 25431q^{8} + 78519q^{9} + O(q^{10}) \) \( 12q - 21q^{2} - 323q^{3} + 2945q^{4} - 5202q^{5} - 9897q^{6} + 28812q^{7} - 25431q^{8} + 78519q^{9} - 65812q^{10} - 80061q^{11} - 184395q^{12} + 342732q^{13} - 50421q^{14} + 160096q^{15} + 385497q^{16} - 1493598q^{17} + 1520858q^{18} - 109038q^{19} - 622260q^{20} - 775523q^{21} + 4636975q^{22} - 3367443q^{23} - 5963895q^{24} - 51480q^{25} - 599781q^{26} - 8158937q^{27} + 7070945q^{28} - 13333098q^{29} + 2915424q^{30} - 3954765q^{31} + 4389297q^{32} - 5790219q^{33} + 14879968q^{34} - 12490002q^{35} + 80697058q^{36} + 580535q^{37} - 19134246q^{38} - 9225203q^{39} + 12365024q^{40} - 27018171q^{41} - 23762697q^{42} + 31237588q^{43} - 125053839q^{44} - 62765470q^{45} - 114008121q^{46} - 21983709q^{47} - 309724207q^{48} + 69177612q^{49} - 131331747q^{50} - 176522692q^{51} + 84112145q^{52} - 196548234q^{53} - 456152547q^{54} - 309055872q^{55} - 61059831q^{56} - 274411494q^{57} - 521980612q^{58} - 215907906q^{59} - 177006648q^{60} - 218340705q^{61} - 673289997q^{62} + 188524119q^{63} - 386667247q^{64} - 148574322q^{65} - 777397365q^{66} + 14544775q^{67} - 1246637448q^{68} - 65252625q^{69} - 158014612q^{70} - 552451776q^{71} + 369379470q^{72} - 349395159q^{73} + 73591023q^{74} + 329300747q^{75} - 1036299002q^{76} - 192226461q^{77} - 282668217q^{78} + 962249727q^{79} - 1494536184q^{80} + 874458108q^{81} - 1417698067q^{82} - 2032575912q^{83} - 442732395q^{84} - 411671064q^{85} - 2139249420q^{86} - 759642172q^{87} + 558651957q^{88} - 280821684q^{89} - 5764700804q^{90} + 822899532q^{91} - 4491569571q^{92} - 1729557923q^{93} - 1591372165q^{94} - 1282463328q^{95} - 2148993055q^{96} - 2115165937q^{97} - 121060821q^{98} - 3595669198q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 41.4855 1.83342 0.916709 0.399556i \(-0.130836\pi\)
0.916709 + 0.399556i \(0.130836\pi\)
\(3\) −265.461 −1.89215 −0.946075 0.323948i \(-0.894990\pi\)
−0.946075 + 0.323948i \(0.894990\pi\)
\(4\) 1209.05 2.36142
\(5\) −1043.84 −0.746912 −0.373456 0.927648i \(-0.621827\pi\)
−0.373456 + 0.927648i \(0.621827\pi\)
\(6\) −11012.8 −3.46910
\(7\) 2401.00 0.377964
\(8\) 28917.3 2.49605
\(9\) 50786.7 2.58023
\(10\) −43304.2 −1.36940
\(11\) 12300.7 0.253316 0.126658 0.991946i \(-0.459575\pi\)
0.126658 + 0.991946i \(0.459575\pi\)
\(12\) −320955. −4.46816
\(13\) 28561.0 0.277350
\(14\) 99606.7 0.692967
\(15\) 277099. 1.41327
\(16\) 580618. 2.21488
\(17\) −232027. −0.673781 −0.336890 0.941544i \(-0.609375\pi\)
−0.336890 + 0.941544i \(0.609375\pi\)
\(18\) 2.10691e6 4.73064
\(19\) −496842. −0.874635 −0.437317 0.899307i \(-0.644071\pi\)
−0.437317 + 0.899307i \(0.644071\pi\)
\(20\) −1.26205e6 −1.76377
\(21\) −637373. −0.715165
\(22\) 510302. 0.464435
\(23\) −2.39174e6 −1.78212 −0.891062 0.453881i \(-0.850039\pi\)
−0.891062 + 0.453881i \(0.850039\pi\)
\(24\) −7.67643e6 −4.72290
\(25\) −863522. −0.442123
\(26\) 1.18487e6 0.508498
\(27\) −8.25683e6 −2.99003
\(28\) 2.90292e6 0.892533
\(29\) −5.44610e6 −1.42986 −0.714932 0.699194i \(-0.753543\pi\)
−0.714932 + 0.699194i \(0.753543\pi\)
\(30\) 1.14956e7 2.59111
\(31\) 4.32926e6 0.841949 0.420974 0.907073i \(-0.361688\pi\)
0.420974 + 0.907073i \(0.361688\pi\)
\(32\) 9.28156e6 1.56475
\(33\) −3.26536e6 −0.479313
\(34\) −9.62576e6 −1.23532
\(35\) −2.50626e6 −0.282306
\(36\) 6.14035e7 6.09301
\(37\) 2.00548e7 1.75918 0.879591 0.475731i \(-0.157816\pi\)
0.879591 + 0.475731i \(0.157816\pi\)
\(38\) −2.06117e7 −1.60357
\(39\) −7.58184e6 −0.524788
\(40\) −3.01851e7 −1.86433
\(41\) −3.28800e7 −1.81720 −0.908602 0.417662i \(-0.862849\pi\)
−0.908602 + 0.417662i \(0.862849\pi\)
\(42\) −2.64417e7 −1.31120
\(43\) 5.76698e6 0.257241 0.128621 0.991694i \(-0.458945\pi\)
0.128621 + 0.991694i \(0.458945\pi\)
\(44\) 1.48721e7 0.598186
\(45\) −5.30132e7 −1.92720
\(46\) −9.92224e7 −3.26738
\(47\) 1.82697e7 0.546123 0.273062 0.961997i \(-0.411964\pi\)
0.273062 + 0.961997i \(0.411964\pi\)
\(48\) −1.54132e8 −4.19089
\(49\) 5.76480e6 0.142857
\(50\) −3.58236e7 −0.810596
\(51\) 6.15942e7 1.27489
\(52\) 3.45316e7 0.654940
\(53\) −6.01983e7 −1.04796 −0.523978 0.851732i \(-0.675553\pi\)
−0.523978 + 0.851732i \(0.675553\pi\)
\(54\) −3.42539e8 −5.48198
\(55\) −1.28400e7 −0.189205
\(56\) 6.94305e7 0.943418
\(57\) 1.31892e8 1.65494
\(58\) −2.25934e8 −2.62154
\(59\) 4.29026e7 0.460946 0.230473 0.973079i \(-0.425973\pi\)
0.230473 + 0.973079i \(0.425973\pi\)
\(60\) 3.35026e8 3.33732
\(61\) −4.10519e7 −0.379620 −0.189810 0.981821i \(-0.560787\pi\)
−0.189810 + 0.981821i \(0.560787\pi\)
\(62\) 1.79601e8 1.54364
\(63\) 1.21939e8 0.975236
\(64\) 8.77735e7 0.653964
\(65\) −2.98131e7 −0.207156
\(66\) −1.35465e8 −0.878780
\(67\) −1.32771e8 −0.804945 −0.402473 0.915432i \(-0.631849\pi\)
−0.402473 + 0.915432i \(0.631849\pi\)
\(68\) −2.80532e8 −1.59108
\(69\) 6.34913e8 3.37205
\(70\) −1.03974e8 −0.517585
\(71\) −1.97237e8 −0.921140 −0.460570 0.887623i \(-0.652355\pi\)
−0.460570 + 0.887623i \(0.652355\pi\)
\(72\) 1.46862e9 6.44039
\(73\) −2.81790e8 −1.16137 −0.580687 0.814127i \(-0.697216\pi\)
−0.580687 + 0.814127i \(0.697216\pi\)
\(74\) 8.31984e8 3.22531
\(75\) 2.29232e8 0.836563
\(76\) −6.00705e8 −2.06538
\(77\) 2.95340e7 0.0957446
\(78\) −3.14536e8 −0.962155
\(79\) 1.41312e8 0.408185 0.204093 0.978952i \(-0.434576\pi\)
0.204093 + 0.978952i \(0.434576\pi\)
\(80\) −6.06073e8 −1.65432
\(81\) 1.19223e9 3.07736
\(82\) −1.36404e9 −3.33169
\(83\) −3.38531e8 −0.782972 −0.391486 0.920184i \(-0.628039\pi\)
−0.391486 + 0.920184i \(0.628039\pi\)
\(84\) −7.70613e8 −1.68881
\(85\) 2.42199e8 0.503254
\(86\) 2.39246e8 0.471631
\(87\) 1.44573e9 2.70552
\(88\) 3.55704e8 0.632291
\(89\) −7.96320e7 −0.134534 −0.0672671 0.997735i \(-0.521428\pi\)
−0.0672671 + 0.997735i \(0.521428\pi\)
\(90\) −2.19928e9 −3.53337
\(91\) 6.85750e7 0.104828
\(92\) −2.89172e9 −4.20834
\(93\) −1.14925e9 −1.59309
\(94\) 7.57927e8 1.00127
\(95\) 5.18623e8 0.653275
\(96\) −2.46389e9 −2.96075
\(97\) 1.98650e8 0.227833 0.113917 0.993490i \(-0.463660\pi\)
0.113917 + 0.993490i \(0.463660\pi\)
\(98\) 2.39156e8 0.261917
\(99\) 6.24713e8 0.653615
\(100\) −1.04404e9 −1.04404
\(101\) 1.06017e9 1.01375 0.506874 0.862020i \(-0.330801\pi\)
0.506874 + 0.862020i \(0.330801\pi\)
\(102\) 2.55527e9 2.33741
\(103\) 1.55417e9 1.36060 0.680301 0.732933i \(-0.261849\pi\)
0.680301 + 0.732933i \(0.261849\pi\)
\(104\) 8.25908e8 0.692280
\(105\) 6.65315e8 0.534165
\(106\) −2.49736e9 −1.92134
\(107\) −5.79937e8 −0.427715 −0.213857 0.976865i \(-0.568603\pi\)
−0.213857 + 0.976865i \(0.568603\pi\)
\(108\) −9.98289e9 −7.06073
\(109\) −4.44336e8 −0.301504 −0.150752 0.988572i \(-0.548169\pi\)
−0.150752 + 0.988572i \(0.548169\pi\)
\(110\) −5.32673e8 −0.346892
\(111\) −5.32377e9 −3.32864
\(112\) 1.39406e9 0.837147
\(113\) −1.29825e9 −0.749040 −0.374520 0.927219i \(-0.622192\pi\)
−0.374520 + 0.927219i \(0.622192\pi\)
\(114\) 5.47161e9 3.03420
\(115\) 2.49659e9 1.33109
\(116\) −6.58459e9 −3.37651
\(117\) 1.45052e9 0.715627
\(118\) 1.77984e9 0.845106
\(119\) −5.57097e8 −0.254665
\(120\) 8.01297e9 3.52759
\(121\) −2.20664e9 −0.935831
\(122\) −1.70306e9 −0.696001
\(123\) 8.72835e9 3.43842
\(124\) 5.23427e9 1.98819
\(125\) 2.94013e9 1.07714
\(126\) 5.05869e9 1.78801
\(127\) 3.78447e9 1.29089 0.645443 0.763808i \(-0.276673\pi\)
0.645443 + 0.763808i \(0.276673\pi\)
\(128\) −1.11083e9 −0.365765
\(129\) −1.53091e9 −0.486739
\(130\) −1.23681e9 −0.379803
\(131\) −2.77646e9 −0.823704 −0.411852 0.911251i \(-0.635118\pi\)
−0.411852 + 0.911251i \(0.635118\pi\)
\(132\) −3.94798e9 −1.13186
\(133\) −1.19292e9 −0.330581
\(134\) −5.50807e9 −1.47580
\(135\) 8.61881e9 2.23329
\(136\) −6.70960e9 −1.68179
\(137\) 3.18994e9 0.773642 0.386821 0.922155i \(-0.373573\pi\)
0.386821 + 0.922155i \(0.373573\pi\)
\(138\) 2.63397e10 6.18237
\(139\) −3.35059e9 −0.761298 −0.380649 0.924720i \(-0.624299\pi\)
−0.380649 + 0.924720i \(0.624299\pi\)
\(140\) −3.03019e9 −0.666643
\(141\) −4.84989e9 −1.03335
\(142\) −8.18247e9 −1.68883
\(143\) 3.51321e8 0.0702574
\(144\) 2.94877e10 5.71491
\(145\) 5.68486e9 1.06798
\(146\) −1.16902e10 −2.12928
\(147\) −1.53033e9 −0.270307
\(148\) 2.42472e10 4.15417
\(149\) −1.03113e10 −1.71386 −0.856932 0.515430i \(-0.827632\pi\)
−0.856932 + 0.515430i \(0.827632\pi\)
\(150\) 9.50979e9 1.53377
\(151\) 2.94643e9 0.461211 0.230606 0.973047i \(-0.425929\pi\)
0.230606 + 0.973047i \(0.425929\pi\)
\(152\) −1.43673e10 −2.18313
\(153\) −1.17839e10 −1.73851
\(154\) 1.22523e9 0.175540
\(155\) −4.51905e9 −0.628861
\(156\) −9.16680e9 −1.23924
\(157\) 4.71485e9 0.619326 0.309663 0.950846i \(-0.399784\pi\)
0.309663 + 0.950846i \(0.399784\pi\)
\(158\) 5.86240e9 0.748374
\(159\) 1.59803e10 1.98289
\(160\) −9.68846e9 −1.16873
\(161\) −5.74256e9 −0.673580
\(162\) 4.94604e10 5.64209
\(163\) −1.61057e10 −1.78704 −0.893521 0.449021i \(-0.851773\pi\)
−0.893521 + 0.449021i \(0.851773\pi\)
\(164\) −3.97534e10 −4.29118
\(165\) 3.40852e9 0.358004
\(166\) −1.40441e10 −1.43552
\(167\) 1.53870e10 1.53084 0.765420 0.643531i \(-0.222531\pi\)
0.765420 + 0.643531i \(0.222531\pi\)
\(168\) −1.84311e10 −1.78509
\(169\) 8.15731e8 0.0769231
\(170\) 1.00478e10 0.922675
\(171\) −2.52329e10 −2.25676
\(172\) 6.97255e9 0.607455
\(173\) −1.79199e10 −1.52100 −0.760498 0.649340i \(-0.775045\pi\)
−0.760498 + 0.649340i \(0.775045\pi\)
\(174\) 5.99768e10 4.96034
\(175\) −2.07332e9 −0.167107
\(176\) 7.14202e9 0.561066
\(177\) −1.13890e10 −0.872179
\(178\) −3.30357e9 −0.246657
\(179\) 1.74217e10 1.26839 0.634194 0.773174i \(-0.281332\pi\)
0.634194 + 0.773174i \(0.281332\pi\)
\(180\) −6.40954e10 −4.55094
\(181\) 1.86466e10 1.29135 0.645677 0.763611i \(-0.276575\pi\)
0.645677 + 0.763611i \(0.276575\pi\)
\(182\) 2.84487e9 0.192194
\(183\) 1.08977e10 0.718297
\(184\) −6.91626e10 −4.44827
\(185\) −2.09340e10 −1.31395
\(186\) −4.76772e10 −2.92080
\(187\) −2.85410e9 −0.170680
\(188\) 2.20889e10 1.28963
\(189\) −1.98246e10 −1.13013
\(190\) 2.15154e10 1.19773
\(191\) 1.25238e10 0.680905 0.340452 0.940262i \(-0.389420\pi\)
0.340452 + 0.940262i \(0.389420\pi\)
\(192\) −2.33005e10 −1.23740
\(193\) 3.72135e10 1.93060 0.965300 0.261142i \(-0.0840991\pi\)
0.965300 + 0.261142i \(0.0840991\pi\)
\(194\) 8.24111e9 0.417713
\(195\) 7.91423e9 0.391970
\(196\) 6.96991e9 0.337346
\(197\) 2.82674e10 1.33717 0.668587 0.743634i \(-0.266899\pi\)
0.668587 + 0.743634i \(0.266899\pi\)
\(198\) 2.59165e10 1.19835
\(199\) 2.00504e10 0.906325 0.453162 0.891428i \(-0.350296\pi\)
0.453162 + 0.891428i \(0.350296\pi\)
\(200\) −2.49707e10 −1.10356
\(201\) 3.52455e10 1.52308
\(202\) 4.39817e10 1.85862
\(203\) −1.30761e10 −0.540438
\(204\) 7.44703e10 3.01056
\(205\) 3.43214e10 1.35729
\(206\) 6.44756e10 2.49455
\(207\) −1.21468e11 −4.59829
\(208\) 1.65830e10 0.614298
\(209\) −6.11151e9 −0.221559
\(210\) 2.76009e10 0.979348
\(211\) −3.63449e10 −1.26233 −0.631164 0.775649i \(-0.717423\pi\)
−0.631164 + 0.775649i \(0.717423\pi\)
\(212\) −7.27826e10 −2.47466
\(213\) 5.23588e10 1.74293
\(214\) −2.40590e10 −0.784180
\(215\) −6.01981e9 −0.192136
\(216\) −2.38765e11 −7.46328
\(217\) 1.03945e10 0.318227
\(218\) −1.84335e10 −0.552782
\(219\) 7.48043e10 2.19749
\(220\) −1.55241e10 −0.446792
\(221\) −6.62692e9 −0.186873
\(222\) −2.20859e11 −6.10278
\(223\) −2.36536e9 −0.0640510 −0.0320255 0.999487i \(-0.510196\pi\)
−0.0320255 + 0.999487i \(0.510196\pi\)
\(224\) 2.22850e10 0.591421
\(225\) −4.38554e10 −1.14078
\(226\) −5.38585e10 −1.37330
\(227\) −5.01901e10 −1.25459 −0.627295 0.778782i \(-0.715838\pi\)
−0.627295 + 0.778782i \(0.715838\pi\)
\(228\) 1.59464e11 3.90801
\(229\) −4.64619e10 −1.11645 −0.558223 0.829691i \(-0.688517\pi\)
−0.558223 + 0.829691i \(0.688517\pi\)
\(230\) 1.03572e11 2.44044
\(231\) −7.84014e9 −0.181163
\(232\) −1.57487e11 −3.56901
\(233\) −2.04647e9 −0.0454887 −0.0227444 0.999741i \(-0.507240\pi\)
−0.0227444 + 0.999741i \(0.507240\pi\)
\(234\) 6.01755e10 1.31204
\(235\) −1.90706e10 −0.407906
\(236\) 5.18713e10 1.08849
\(237\) −3.75129e10 −0.772348
\(238\) −2.31114e10 −0.466907
\(239\) 2.86697e10 0.568372 0.284186 0.958769i \(-0.408277\pi\)
0.284186 + 0.958769i \(0.408277\pi\)
\(240\) 1.60889e11 3.13022
\(241\) −4.99778e10 −0.954334 −0.477167 0.878813i \(-0.658336\pi\)
−0.477167 + 0.878813i \(0.658336\pi\)
\(242\) −9.15436e10 −1.71577
\(243\) −1.53973e11 −2.83280
\(244\) −4.96336e10 −0.896441
\(245\) −6.01753e9 −0.106702
\(246\) 3.62100e11 6.30407
\(247\) −1.41903e10 −0.242580
\(248\) 1.25191e11 2.10155
\(249\) 8.98668e10 1.48150
\(250\) 1.21973e11 1.97484
\(251\) −1.40276e10 −0.223075 −0.111537 0.993760i \(-0.535577\pi\)
−0.111537 + 0.993760i \(0.535577\pi\)
\(252\) 1.47430e11 2.30294
\(253\) −2.94201e10 −0.451441
\(254\) 1.57001e11 2.36673
\(255\) −6.42945e10 −0.952233
\(256\) −9.10233e10 −1.32456
\(257\) 8.33249e10 1.19145 0.595725 0.803189i \(-0.296865\pi\)
0.595725 + 0.803189i \(0.296865\pi\)
\(258\) −6.35106e10 −0.892396
\(259\) 4.81516e10 0.664908
\(260\) −3.60455e10 −0.489182
\(261\) −2.76590e11 −3.68938
\(262\) −1.15183e11 −1.51019
\(263\) 6.27252e10 0.808427 0.404214 0.914665i \(-0.367545\pi\)
0.404214 + 0.914665i \(0.367545\pi\)
\(264\) −9.44256e10 −1.19639
\(265\) 6.28375e10 0.782731
\(266\) −4.94887e10 −0.606093
\(267\) 2.11392e10 0.254559
\(268\) −1.60526e11 −1.90081
\(269\) −5.23885e10 −0.610029 −0.305015 0.952348i \(-0.598661\pi\)
−0.305015 + 0.952348i \(0.598661\pi\)
\(270\) 3.57556e11 4.09456
\(271\) −3.71531e10 −0.418440 −0.209220 0.977869i \(-0.567093\pi\)
−0.209220 + 0.977869i \(0.567093\pi\)
\(272\) −1.34719e11 −1.49234
\(273\) −1.82040e10 −0.198351
\(274\) 1.32336e11 1.41841
\(275\) −1.06219e10 −0.111997
\(276\) 7.67640e11 7.96282
\(277\) 7.16214e10 0.730944 0.365472 0.930822i \(-0.380908\pi\)
0.365472 + 0.930822i \(0.380908\pi\)
\(278\) −1.39001e11 −1.39578
\(279\) 2.19869e11 2.17242
\(280\) −7.24744e10 −0.704650
\(281\) −2.73356e10 −0.261548 −0.130774 0.991412i \(-0.541746\pi\)
−0.130774 + 0.991412i \(0.541746\pi\)
\(282\) −2.01200e11 −1.89456
\(283\) 8.49351e10 0.787133 0.393567 0.919296i \(-0.371241\pi\)
0.393567 + 0.919296i \(0.371241\pi\)
\(284\) −2.38469e11 −2.17520
\(285\) −1.37674e11 −1.23609
\(286\) 1.45747e10 0.128811
\(287\) −7.89448e10 −0.686839
\(288\) 4.71380e11 4.03743
\(289\) −6.47513e10 −0.546020
\(290\) 2.35839e11 1.95806
\(291\) −5.27340e10 −0.431094
\(292\) −3.40697e11 −2.74249
\(293\) −7.72191e10 −0.612097 −0.306048 0.952016i \(-0.599007\pi\)
−0.306048 + 0.952016i \(0.599007\pi\)
\(294\) −6.34866e10 −0.495586
\(295\) −4.47835e10 −0.344286
\(296\) 5.79932e11 4.39101
\(297\) −1.01565e11 −0.757425
\(298\) −4.27771e11 −3.14223
\(299\) −6.83104e10 −0.494272
\(300\) 2.77152e11 1.97548
\(301\) 1.38465e10 0.0972281
\(302\) 1.22234e11 0.845593
\(303\) −2.81435e11 −1.91816
\(304\) −2.88475e11 −1.93721
\(305\) 4.28516e10 0.283542
\(306\) −4.88860e11 −3.18741
\(307\) −1.47700e11 −0.948981 −0.474491 0.880260i \(-0.657368\pi\)
−0.474491 + 0.880260i \(0.657368\pi\)
\(308\) 3.57080e10 0.226093
\(309\) −4.12572e11 −2.57446
\(310\) −1.87475e11 −1.15297
\(311\) 2.33473e11 1.41519 0.707596 0.706617i \(-0.249780\pi\)
0.707596 + 0.706617i \(0.249780\pi\)
\(312\) −2.19247e11 −1.30990
\(313\) 1.33403e11 0.785629 0.392815 0.919618i \(-0.371501\pi\)
0.392815 + 0.919618i \(0.371501\pi\)
\(314\) 1.95598e11 1.13548
\(315\) −1.27285e11 −0.728415
\(316\) 1.70853e11 0.963897
\(317\) −2.00761e11 −1.11664 −0.558320 0.829625i \(-0.688554\pi\)
−0.558320 + 0.829625i \(0.688554\pi\)
\(318\) 6.62952e11 3.63547
\(319\) −6.69910e10 −0.362208
\(320\) −9.16216e10 −0.488453
\(321\) 1.53951e11 0.809300
\(322\) −2.38233e11 −1.23495
\(323\) 1.15281e11 0.589312
\(324\) 1.44147e12 7.26694
\(325\) −2.46630e10 −0.122623
\(326\) −6.68152e11 −3.27640
\(327\) 1.17954e11 0.570490
\(328\) −9.50800e11 −4.53583
\(329\) 4.38655e10 0.206415
\(330\) 1.41404e11 0.656371
\(331\) 1.28413e11 0.588006 0.294003 0.955805i \(-0.405012\pi\)
0.294003 + 0.955805i \(0.405012\pi\)
\(332\) −4.09299e11 −1.84893
\(333\) 1.01852e12 4.53910
\(334\) 6.38337e11 2.80667
\(335\) 1.38592e11 0.601223
\(336\) −3.70070e11 −1.58401
\(337\) −2.39200e11 −1.01024 −0.505121 0.863048i \(-0.668552\pi\)
−0.505121 + 0.863048i \(0.668552\pi\)
\(338\) 3.38410e10 0.141032
\(339\) 3.44635e11 1.41730
\(340\) 2.92830e11 1.18839
\(341\) 5.32530e10 0.213279
\(342\) −1.04680e12 −4.13758
\(343\) 1.38413e10 0.0539949
\(344\) 1.66766e11 0.642087
\(345\) −6.62748e11 −2.51862
\(346\) −7.43416e11 −2.78862
\(347\) 4.47382e11 1.65652 0.828258 0.560347i \(-0.189332\pi\)
0.828258 + 0.560347i \(0.189332\pi\)
\(348\) 1.74795e12 6.38886
\(349\) 2.96596e11 1.07017 0.535083 0.844800i \(-0.320280\pi\)
0.535083 + 0.844800i \(0.320280\pi\)
\(350\) −8.60125e10 −0.306377
\(351\) −2.35823e11 −0.829286
\(352\) 1.14170e11 0.396378
\(353\) −2.62738e11 −0.900609 −0.450305 0.892875i \(-0.648685\pi\)
−0.450305 + 0.892875i \(0.648685\pi\)
\(354\) −4.72478e11 −1.59907
\(355\) 2.05884e11 0.688010
\(356\) −9.62788e10 −0.317692
\(357\) 1.47888e11 0.481865
\(358\) 7.22748e11 2.32548
\(359\) 4.54895e11 1.44539 0.722697 0.691165i \(-0.242902\pi\)
0.722697 + 0.691165i \(0.242902\pi\)
\(360\) −1.53300e12 −4.81040
\(361\) −7.58361e10 −0.235014
\(362\) 7.73562e11 2.36759
\(363\) 5.85777e11 1.77073
\(364\) 8.29103e10 0.247544
\(365\) 2.94144e11 0.867444
\(366\) 4.52096e11 1.31694
\(367\) 5.66047e11 1.62875 0.814376 0.580337i \(-0.197079\pi\)
0.814376 + 0.580337i \(0.197079\pi\)
\(368\) −1.38869e12 −3.94719
\(369\) −1.66986e12 −4.68881
\(370\) −8.68458e11 −2.40902
\(371\) −1.44536e11 −0.396090
\(372\) −1.38950e12 −3.76196
\(373\) −5.25567e11 −1.40585 −0.702923 0.711266i \(-0.748122\pi\)
−0.702923 + 0.711266i \(0.748122\pi\)
\(374\) −1.18404e11 −0.312927
\(375\) −7.80491e11 −2.03811
\(376\) 5.28310e11 1.36315
\(377\) −1.55546e11 −0.396573
\(378\) −8.22435e11 −2.07199
\(379\) −1.69380e11 −0.421683 −0.210842 0.977520i \(-0.567620\pi\)
−0.210842 + 0.977520i \(0.567620\pi\)
\(380\) 6.27040e11 1.54266
\(381\) −1.00463e12 −2.44255
\(382\) 5.19557e11 1.24838
\(383\) 2.00883e11 0.477032 0.238516 0.971139i \(-0.423339\pi\)
0.238516 + 0.971139i \(0.423339\pi\)
\(384\) 2.94882e11 0.692081
\(385\) −3.08288e10 −0.0715128
\(386\) 1.54382e12 3.53960
\(387\) 2.92886e11 0.663742
\(388\) 2.40178e11 0.538009
\(389\) −6.08173e11 −1.34665 −0.673323 0.739348i \(-0.735134\pi\)
−0.673323 + 0.739348i \(0.735134\pi\)
\(390\) 3.28326e11 0.718645
\(391\) 5.54947e11 1.20076
\(392\) 1.66703e11 0.356579
\(393\) 7.37044e11 1.55857
\(394\) 1.17269e12 2.45160
\(395\) −1.47507e11 −0.304878
\(396\) 7.55307e11 1.54346
\(397\) −4.18426e11 −0.845399 −0.422700 0.906270i \(-0.638917\pi\)
−0.422700 + 0.906270i \(0.638917\pi\)
\(398\) 8.31800e11 1.66167
\(399\) 3.16673e11 0.625509
\(400\) −5.01376e11 −0.979251
\(401\) −5.49160e11 −1.06059 −0.530297 0.847812i \(-0.677920\pi\)
−0.530297 + 0.847812i \(0.677920\pi\)
\(402\) 1.46218e12 2.79244
\(403\) 1.23648e11 0.233515
\(404\) 1.28180e12 2.39388
\(405\) −1.24450e12 −2.29852
\(406\) −5.42468e11 −0.990849
\(407\) 2.46689e11 0.445630
\(408\) 1.78114e12 3.18220
\(409\) −6.72213e10 −0.118782 −0.0593912 0.998235i \(-0.518916\pi\)
−0.0593912 + 0.998235i \(0.518916\pi\)
\(410\) 1.42384e12 2.48848
\(411\) −8.46806e11 −1.46385
\(412\) 1.87907e12 3.21295
\(413\) 1.03009e11 0.174221
\(414\) −5.03918e12 −8.43059
\(415\) 3.53372e11 0.584811
\(416\) 2.65091e11 0.433984
\(417\) 8.89452e11 1.44049
\(418\) −2.53539e11 −0.406211
\(419\) −5.58621e11 −0.885430 −0.442715 0.896662i \(-0.645985\pi\)
−0.442715 + 0.896662i \(0.645985\pi\)
\(420\) 8.04397e11 1.26139
\(421\) −9.31762e11 −1.44556 −0.722779 0.691079i \(-0.757136\pi\)
−0.722779 + 0.691079i \(0.757136\pi\)
\(422\) −1.50779e12 −2.31437
\(423\) 9.27857e11 1.40912
\(424\) −1.74078e12 −2.61575
\(425\) 2.00360e11 0.297894
\(426\) 2.17213e12 3.19553
\(427\) −9.85655e10 −0.143483
\(428\) −7.01172e11 −1.01001
\(429\) −9.32621e10 −0.132937
\(430\) −2.49735e11 −0.352266
\(431\) −3.44523e11 −0.480917 −0.240459 0.970659i \(-0.577298\pi\)
−0.240459 + 0.970659i \(0.577298\pi\)
\(432\) −4.79406e12 −6.62257
\(433\) −7.38412e11 −1.00949 −0.504747 0.863268i \(-0.668414\pi\)
−0.504747 + 0.863268i \(0.668414\pi\)
\(434\) 4.31223e11 0.583442
\(435\) −1.50911e12 −2.02078
\(436\) −5.37223e11 −0.711977
\(437\) 1.18831e12 1.55871
\(438\) 3.10329e12 4.02892
\(439\) 2.17589e10 0.0279606 0.0139803 0.999902i \(-0.495550\pi\)
0.0139803 + 0.999902i \(0.495550\pi\)
\(440\) −3.71298e11 −0.472265
\(441\) 2.92775e11 0.368604
\(442\) −2.74921e11 −0.342616
\(443\) 6.68727e11 0.824959 0.412479 0.910967i \(-0.364663\pi\)
0.412479 + 0.910967i \(0.364663\pi\)
\(444\) −6.43669e12 −7.86031
\(445\) 8.31231e10 0.100485
\(446\) −9.81283e10 −0.117432
\(447\) 2.73726e12 3.24289
\(448\) 2.10744e11 0.247175
\(449\) 1.45910e12 1.69425 0.847123 0.531397i \(-0.178333\pi\)
0.847123 + 0.531397i \(0.178333\pi\)
\(450\) −1.81936e12 −2.09153
\(451\) −4.04447e11 −0.460328
\(452\) −1.56964e12 −1.76880
\(453\) −7.82163e11 −0.872681
\(454\) −2.08216e12 −2.30019
\(455\) −7.15813e10 −0.0782976
\(456\) 3.81397e12 4.13081
\(457\) 6.33648e11 0.679556 0.339778 0.940506i \(-0.389648\pi\)
0.339778 + 0.940506i \(0.389648\pi\)
\(458\) −1.92750e12 −2.04691
\(459\) 1.91581e12 2.01463
\(460\) 3.01850e12 3.14326
\(461\) 7.71863e10 0.0795950 0.0397975 0.999208i \(-0.487329\pi\)
0.0397975 + 0.999208i \(0.487329\pi\)
\(462\) −3.25252e11 −0.332148
\(463\) 1.14142e12 1.15433 0.577167 0.816626i \(-0.304158\pi\)
0.577167 + 0.816626i \(0.304158\pi\)
\(464\) −3.16211e12 −3.16698
\(465\) 1.19963e12 1.18990
\(466\) −8.48988e10 −0.0833998
\(467\) 1.78726e10 0.0173885 0.00869424 0.999962i \(-0.497233\pi\)
0.00869424 + 0.999962i \(0.497233\pi\)
\(468\) 1.75374e12 1.68990
\(469\) −3.18783e11 −0.304241
\(470\) −7.91155e11 −0.747861
\(471\) −1.25161e12 −1.17186
\(472\) 1.24063e12 1.15054
\(473\) 7.09380e10 0.0651635
\(474\) −1.55624e12 −1.41604
\(475\) 4.29034e11 0.386696
\(476\) −6.73556e11 −0.601371
\(477\) −3.05728e12 −2.70397
\(478\) 1.18938e12 1.04206
\(479\) −3.81823e11 −0.331400 −0.165700 0.986176i \(-0.552988\pi\)
−0.165700 + 0.986176i \(0.552988\pi\)
\(480\) 2.57191e12 2.21142
\(481\) 5.72785e11 0.487909
\(482\) −2.07335e12 −1.74969
\(483\) 1.52443e12 1.27451
\(484\) −2.66793e12 −2.20989
\(485\) −2.07359e11 −0.170171
\(486\) −6.38763e12 −5.19370
\(487\) 5.48407e11 0.441797 0.220898 0.975297i \(-0.429101\pi\)
0.220898 + 0.975297i \(0.429101\pi\)
\(488\) −1.18711e12 −0.947550
\(489\) 4.27544e12 3.38135
\(490\) −2.49640e11 −0.195629
\(491\) 6.35766e10 0.0493663 0.0246831 0.999695i \(-0.492142\pi\)
0.0246831 + 0.999695i \(0.492142\pi\)
\(492\) 1.05530e13 8.11956
\(493\) 1.26364e12 0.963415
\(494\) −5.88691e11 −0.444750
\(495\) −6.52101e11 −0.488193
\(496\) 2.51364e12 1.86482
\(497\) −4.73566e11 −0.348158
\(498\) 3.72817e12 2.71621
\(499\) 7.27190e11 0.525044 0.262522 0.964926i \(-0.415446\pi\)
0.262522 + 0.964926i \(0.415446\pi\)
\(500\) 3.55475e12 2.54358
\(501\) −4.08465e12 −2.89658
\(502\) −5.81941e11 −0.408989
\(503\) −6.31583e11 −0.439921 −0.219960 0.975509i \(-0.570593\pi\)
−0.219960 + 0.975509i \(0.570593\pi\)
\(504\) 3.52615e12 2.43424
\(505\) −1.10665e12 −0.757180
\(506\) −1.22051e12 −0.827681
\(507\) −2.16545e11 −0.145550
\(508\) 4.57560e12 3.04832
\(509\) −1.09628e12 −0.723921 −0.361961 0.932193i \(-0.617893\pi\)
−0.361961 + 0.932193i \(0.617893\pi\)
\(510\) −2.66729e12 −1.74584
\(511\) −6.76577e11 −0.438958
\(512\) −3.20740e12 −2.06271
\(513\) 4.10233e12 2.61519
\(514\) 3.45677e12 2.18442
\(515\) −1.62231e12 −1.01625
\(516\) −1.85094e12 −1.14940
\(517\) 2.24730e11 0.138342
\(518\) 1.99759e12 1.21905
\(519\) 4.75704e12 2.87795
\(520\) −8.62116e11 −0.517072
\(521\) 1.62813e12 0.968097 0.484049 0.875041i \(-0.339166\pi\)
0.484049 + 0.875041i \(0.339166\pi\)
\(522\) −1.14745e13 −6.76418
\(523\) 1.16573e12 0.681301 0.340650 0.940190i \(-0.389353\pi\)
0.340650 + 0.940190i \(0.389353\pi\)
\(524\) −3.35687e12 −1.94511
\(525\) 5.50385e11 0.316191
\(526\) 2.60218e12 1.48218
\(527\) −1.00450e12 −0.567289
\(528\) −1.89593e12 −1.06162
\(529\) 3.91925e12 2.17597
\(530\) 2.60684e12 1.43507
\(531\) 2.17888e12 1.18935
\(532\) −1.44229e12 −0.780640
\(533\) −9.39084e11 −0.504002
\(534\) 8.76971e11 0.466713
\(535\) 6.05362e11 0.319465
\(536\) −3.83938e12 −2.00918
\(537\) −4.62479e12 −2.39998
\(538\) −2.17336e12 −1.11844
\(539\) 7.09112e10 0.0361881
\(540\) 1.04205e13 5.27374
\(541\) 1.39519e12 0.700238 0.350119 0.936705i \(-0.386141\pi\)
0.350119 + 0.936705i \(0.386141\pi\)
\(542\) −1.54132e12 −0.767176
\(543\) −4.94994e12 −2.44344
\(544\) −2.15357e12 −1.05430
\(545\) 4.63816e11 0.225197
\(546\) −7.55202e11 −0.363661
\(547\) 2.04329e12 0.975858 0.487929 0.872883i \(-0.337752\pi\)
0.487929 + 0.872883i \(0.337752\pi\)
\(548\) 3.85679e12 1.82689
\(549\) −2.08489e12 −0.979507
\(550\) −4.40656e11 −0.205337
\(551\) 2.70585e12 1.25061
\(552\) 1.83600e13 8.41680
\(553\) 3.39290e11 0.154280
\(554\) 2.97125e12 1.34013
\(555\) 5.55717e12 2.48620
\(556\) −4.05102e12 −1.79774
\(557\) 1.80016e11 0.0792435 0.0396218 0.999215i \(-0.487385\pi\)
0.0396218 + 0.999215i \(0.487385\pi\)
\(558\) 9.12136e12 3.98296
\(559\) 1.64711e11 0.0713459
\(560\) −1.45518e12 −0.625275
\(561\) 7.57653e11 0.322952
\(562\) −1.13403e12 −0.479526
\(563\) 9.37714e9 0.00393353 0.00196677 0.999998i \(-0.499374\pi\)
0.00196677 + 0.999998i \(0.499374\pi\)
\(564\) −5.86375e12 −2.44017
\(565\) 1.35516e12 0.559466
\(566\) 3.52358e12 1.44314
\(567\) 2.86255e12 1.16313
\(568\) −5.70356e12 −2.29921
\(569\) −2.47739e12 −0.990806 −0.495403 0.868663i \(-0.664980\pi\)
−0.495403 + 0.868663i \(0.664980\pi\)
\(570\) −5.71149e12 −2.26628
\(571\) −6.10336e11 −0.240274 −0.120137 0.992757i \(-0.538333\pi\)
−0.120137 + 0.992757i \(0.538333\pi\)
\(572\) 4.24763e11 0.165907
\(573\) −3.32459e12 −1.28837
\(574\) −3.27506e12 −1.25926
\(575\) 2.06532e12 0.787918
\(576\) 4.45773e12 1.68738
\(577\) 1.80171e12 0.676698 0.338349 0.941021i \(-0.390132\pi\)
0.338349 + 0.941021i \(0.390132\pi\)
\(578\) −2.68624e12 −1.00108
\(579\) −9.87874e12 −3.65299
\(580\) 6.87327e12 2.52195
\(581\) −8.12812e11 −0.295936
\(582\) −2.18770e12 −0.790376
\(583\) −7.40483e11 −0.265465
\(584\) −8.14861e12 −2.89885
\(585\) −1.51411e12 −0.534510
\(586\) −3.20347e12 −1.12223
\(587\) 1.90732e12 0.663060 0.331530 0.943445i \(-0.392435\pi\)
0.331530 + 0.943445i \(0.392435\pi\)
\(588\) −1.85024e12 −0.638308
\(589\) −2.15095e12 −0.736398
\(590\) −1.85787e12 −0.631220
\(591\) −7.50391e12 −2.53013
\(592\) 1.16442e13 3.89638
\(593\) 2.10581e12 0.699315 0.349658 0.936878i \(-0.386298\pi\)
0.349658 + 0.936878i \(0.386298\pi\)
\(594\) −4.21347e12 −1.38868
\(595\) 5.81520e11 0.190212
\(596\) −1.24669e13 −4.04715
\(597\) −5.32260e12 −1.71490
\(598\) −2.83389e12 −0.906208
\(599\) 1.19625e11 0.0379664 0.0189832 0.999820i \(-0.493957\pi\)
0.0189832 + 0.999820i \(0.493957\pi\)
\(600\) 6.62877e12 2.08810
\(601\) −1.44004e12 −0.450236 −0.225118 0.974332i \(-0.572277\pi\)
−0.225118 + 0.974332i \(0.572277\pi\)
\(602\) 5.74430e11 0.178260
\(603\) −6.74299e12 −2.07694
\(604\) 3.56237e12 1.08911
\(605\) 2.30338e12 0.698983
\(606\) −1.16755e13 −3.51679
\(607\) 1.83769e12 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(608\) −4.61146e12 −1.36859
\(609\) 3.47120e12 1.02259
\(610\) 1.77772e12 0.519851
\(611\) 5.21800e11 0.151467
\(612\) −1.42473e13 −4.10535
\(613\) 1.57159e12 0.449538 0.224769 0.974412i \(-0.427837\pi\)
0.224769 + 0.974412i \(0.427837\pi\)
\(614\) −6.12741e12 −1.73988
\(615\) −9.11101e12 −2.56820
\(616\) 8.54045e11 0.238983
\(617\) −1.95775e12 −0.543844 −0.271922 0.962319i \(-0.587659\pi\)
−0.271922 + 0.962319i \(0.587659\pi\)
\(618\) −1.71158e13 −4.72007
\(619\) −2.72038e12 −0.744770 −0.372385 0.928078i \(-0.621460\pi\)
−0.372385 + 0.928078i \(0.621460\pi\)
\(620\) −5.46375e12 −1.48501
\(621\) 1.97481e13 5.32861
\(622\) 9.68575e12 2.59464
\(623\) −1.91197e11 −0.0508492
\(624\) −4.40215e12 −1.16234
\(625\) −1.38246e12 −0.362404
\(626\) 5.53431e12 1.44039
\(627\) 1.62237e12 0.419224
\(628\) 5.70048e12 1.46249
\(629\) −4.65326e12 −1.18530
\(630\) −5.28047e12 −1.33549
\(631\) −3.60508e12 −0.905279 −0.452640 0.891694i \(-0.649518\pi\)
−0.452640 + 0.891694i \(0.649518\pi\)
\(632\) 4.08637e12 1.01885
\(633\) 9.64816e12 2.38851
\(634\) −8.32869e12 −2.04727
\(635\) −3.95038e12 −0.964178
\(636\) 1.93210e13 4.68244
\(637\) 1.64648e11 0.0396214
\(638\) −2.77916e12 −0.664079
\(639\) −1.00170e13 −2.37675
\(640\) 1.15953e12 0.273194
\(641\) 3.15040e11 0.0737063 0.0368532 0.999321i \(-0.488267\pi\)
0.0368532 + 0.999321i \(0.488267\pi\)
\(642\) 6.38673e12 1.48379
\(643\) −6.05099e12 −1.39597 −0.697987 0.716111i \(-0.745921\pi\)
−0.697987 + 0.716111i \(0.745921\pi\)
\(644\) −6.94302e12 −1.59060
\(645\) 1.59803e12 0.363551
\(646\) 4.78248e12 1.08045
\(647\) 5.21258e12 1.16946 0.584728 0.811229i \(-0.301201\pi\)
0.584728 + 0.811229i \(0.301201\pi\)
\(648\) 3.44762e13 7.68125
\(649\) 5.27733e11 0.116765
\(650\) −1.02316e12 −0.224819
\(651\) −2.75935e12 −0.602133
\(652\) −1.94725e13 −4.21996
\(653\) −4.21135e12 −0.906383 −0.453192 0.891413i \(-0.649715\pi\)
−0.453192 + 0.891413i \(0.649715\pi\)
\(654\) 4.89339e12 1.04595
\(655\) 2.89819e12 0.615234
\(656\) −1.90907e13 −4.02489
\(657\) −1.43112e13 −2.99661
\(658\) 1.81978e12 0.378445
\(659\) 6.86390e12 1.41771 0.708853 0.705356i \(-0.249213\pi\)
0.708853 + 0.705356i \(0.249213\pi\)
\(660\) 4.12106e12 0.845398
\(661\) −6.38350e12 −1.30063 −0.650313 0.759666i \(-0.725362\pi\)
−0.650313 + 0.759666i \(0.725362\pi\)
\(662\) 5.32726e12 1.07806
\(663\) 1.75919e12 0.353592
\(664\) −9.78940e12 −1.95434
\(665\) 1.24521e12 0.246915
\(666\) 4.22537e13 8.32206
\(667\) 1.30256e13 2.54820
\(668\) 1.86036e13 3.61495
\(669\) 6.27912e11 0.121194
\(670\) 5.74954e12 1.10229
\(671\) −5.04968e11 −0.0961639
\(672\) −5.91581e12 −1.11906
\(673\) −5.23533e12 −0.983731 −0.491866 0.870671i \(-0.663685\pi\)
−0.491866 + 0.870671i \(0.663685\pi\)
\(674\) −9.92331e12 −1.85220
\(675\) 7.12995e12 1.32196
\(676\) 9.86257e11 0.181648
\(677\) −1.05540e13 −1.93093 −0.965466 0.260528i \(-0.916103\pi\)
−0.965466 + 0.260528i \(0.916103\pi\)
\(678\) 1.42973e13 2.59849
\(679\) 4.76960e11 0.0861128
\(680\) 7.00376e12 1.25615
\(681\) 1.33235e13 2.37387
\(682\) 2.20923e12 0.391030
\(683\) −2.39665e12 −0.421416 −0.210708 0.977549i \(-0.567577\pi\)
−0.210708 + 0.977549i \(0.567577\pi\)
\(684\) −3.05078e13 −5.32916
\(685\) −3.32979e12 −0.577842
\(686\) 5.74213e11 0.0989952
\(687\) 1.23338e13 2.11248
\(688\) 3.34841e12 0.569759
\(689\) −1.71933e12 −0.290651
\(690\) −2.74944e13 −4.61768
\(691\) 3.21963e12 0.537223 0.268611 0.963249i \(-0.413435\pi\)
0.268611 + 0.963249i \(0.413435\pi\)
\(692\) −2.16660e13 −3.59171
\(693\) 1.49994e12 0.247043
\(694\) 1.85599e13 3.03709
\(695\) 3.49748e12 0.568622
\(696\) 4.18066e13 6.75311
\(697\) 7.62904e12 1.22440
\(698\) 1.23044e13 1.96206
\(699\) 5.43259e11 0.0860715
\(700\) −2.50674e12 −0.394609
\(701\) −1.09193e12 −0.170790 −0.0853952 0.996347i \(-0.527215\pi\)
−0.0853952 + 0.996347i \(0.527215\pi\)
\(702\) −9.78324e12 −1.52043
\(703\) −9.96406e12 −1.53864
\(704\) 1.07968e12 0.165660
\(705\) 5.06252e12 0.771819
\(706\) −1.08998e13 −1.65119
\(707\) 2.54547e12 0.383161
\(708\) −1.37698e13 −2.05958
\(709\) −3.74025e12 −0.555895 −0.277948 0.960596i \(-0.589654\pi\)
−0.277948 + 0.960596i \(0.589654\pi\)
\(710\) 8.54120e12 1.26141
\(711\) 7.17677e12 1.05321
\(712\) −2.30275e12 −0.335804
\(713\) −1.03544e13 −1.50046
\(714\) 6.13519e12 0.883459
\(715\) −3.66723e11 −0.0524760
\(716\) 2.10637e13 2.99520
\(717\) −7.61069e12 −1.07544
\(718\) 1.88716e13 2.65001
\(719\) −1.33722e13 −1.86605 −0.933026 0.359810i \(-0.882842\pi\)
−0.933026 + 0.359810i \(0.882842\pi\)
\(720\) −3.07804e13 −4.26853
\(721\) 3.73156e12 0.514259
\(722\) −3.14610e12 −0.430879
\(723\) 1.32672e13 1.80574
\(724\) 2.25446e13 3.04943
\(725\) 4.70283e12 0.632176
\(726\) 2.43013e13 3.24649
\(727\) −8.09505e12 −1.07477 −0.537384 0.843338i \(-0.680587\pi\)
−0.537384 + 0.843338i \(0.680587\pi\)
\(728\) 1.98300e12 0.261657
\(729\) 1.74070e13 2.28271
\(730\) 1.22027e13 1.59039
\(731\) −1.33810e12 −0.173324
\(732\) 1.31758e13 1.69620
\(733\) −1.19185e13 −1.52494 −0.762469 0.647025i \(-0.776013\pi\)
−0.762469 + 0.647025i \(0.776013\pi\)
\(734\) 2.34827e13 2.98618
\(735\) 1.59742e12 0.201896
\(736\) −2.21990e13 −2.78858
\(737\) −1.63318e12 −0.203906
\(738\) −6.92751e13 −8.59654
\(739\) −3.85928e12 −0.475999 −0.238000 0.971265i \(-0.576492\pi\)
−0.238000 + 0.971265i \(0.576492\pi\)
\(740\) −2.53102e13 −3.10279
\(741\) 3.76697e12 0.458998
\(742\) −5.99616e12 −0.726199
\(743\) 1.06386e11 0.0128066 0.00640331 0.999979i \(-0.497962\pi\)
0.00640331 + 0.999979i \(0.497962\pi\)
\(744\) −3.32332e13 −3.97644
\(745\) 1.07634e13 1.28010
\(746\) −2.18034e13 −2.57750
\(747\) −1.71928e13 −2.02025
\(748\) −3.45074e12 −0.403046
\(749\) −1.39243e12 −0.161661
\(750\) −3.23790e13 −3.73670
\(751\) −9.28197e11 −0.106478 −0.0532390 0.998582i \(-0.516955\pi\)
−0.0532390 + 0.998582i \(0.516955\pi\)
\(752\) 1.06077e13 1.20960
\(753\) 3.72378e12 0.422091
\(754\) −6.45291e12 −0.727084
\(755\) −3.07560e12 −0.344484
\(756\) −2.39689e13 −2.66870
\(757\) 2.88351e12 0.319146 0.159573 0.987186i \(-0.448988\pi\)
0.159573 + 0.987186i \(0.448988\pi\)
\(758\) −7.02682e12 −0.773121
\(759\) 7.80989e12 0.854195
\(760\) 1.49972e13 1.63061
\(761\) −3.97237e12 −0.429357 −0.214679 0.976685i \(-0.568870\pi\)
−0.214679 + 0.976685i \(0.568870\pi\)
\(762\) −4.16776e13 −4.47822
\(763\) −1.06685e12 −0.113958
\(764\) 1.51419e13 1.60790
\(765\) 1.23005e13 1.29851
\(766\) 8.33371e12 0.874599
\(767\) 1.22534e12 0.127843
\(768\) 2.41632e13 2.50627
\(769\) −5.16621e12 −0.532725 −0.266363 0.963873i \(-0.585822\pi\)
−0.266363 + 0.963873i \(0.585822\pi\)
\(770\) −1.27895e12 −0.131113
\(771\) −2.21195e13 −2.25440
\(772\) 4.49928e13 4.55896
\(773\) 1.16416e13 1.17275 0.586376 0.810039i \(-0.300554\pi\)
0.586376 + 0.810039i \(0.300554\pi\)
\(774\) 1.21505e13 1.21692
\(775\) −3.73841e12 −0.372245
\(776\) 5.74444e12 0.568683
\(777\) −1.27824e13 −1.25811
\(778\) −2.52303e13 −2.46896
\(779\) 1.63361e13 1.58939
\(780\) 9.56868e12 0.925606
\(781\) −2.42616e12 −0.233340
\(782\) 2.30223e13 2.20150
\(783\) 4.49675e13 4.27535
\(784\) 3.34715e12 0.316412
\(785\) −4.92155e12 −0.462582
\(786\) 3.05766e13 2.85751
\(787\) 7.19806e12 0.668851 0.334425 0.942422i \(-0.391458\pi\)
0.334425 + 0.942422i \(0.391458\pi\)
\(788\) 3.41766e13 3.15763
\(789\) −1.66511e13 −1.52967
\(790\) −6.11941e12 −0.558969
\(791\) −3.11710e12 −0.283110
\(792\) 1.80650e13 1.63146
\(793\) −1.17248e12 −0.105288
\(794\) −1.73586e13 −1.54997
\(795\) −1.66809e13 −1.48104
\(796\) 2.42418e13 2.14021
\(797\) 1.62406e13 1.42574 0.712871 0.701296i \(-0.247395\pi\)
0.712871 + 0.701296i \(0.247395\pi\)
\(798\) 1.31373e13 1.14682
\(799\) −4.23906e12 −0.367967
\(800\) −8.01483e12 −0.691814
\(801\) −4.04425e12 −0.347129
\(802\) −2.27822e13 −1.94451
\(803\) −3.46622e12 −0.294195
\(804\) 4.26135e13 3.59662
\(805\) 5.99432e12 0.503104
\(806\) 5.12959e12 0.428130
\(807\) 1.39071e13 1.15427
\(808\) 3.06573e13 2.53037
\(809\) −1.25707e13 −1.03179 −0.515894 0.856653i \(-0.672540\pi\)
−0.515894 + 0.856653i \(0.672540\pi\)
\(810\) −5.16288e13 −4.21414
\(811\) 5.95715e12 0.483553 0.241777 0.970332i \(-0.422270\pi\)
0.241777 + 0.970332i \(0.422270\pi\)
\(812\) −1.58096e13 −1.27620
\(813\) 9.86272e12 0.791752
\(814\) 1.02340e13 0.817025
\(815\) 1.68118e13 1.33476
\(816\) 3.57627e13 2.82374
\(817\) −2.86528e12 −0.224992
\(818\) −2.78871e12 −0.217778
\(819\) 3.48270e12 0.270482
\(820\) 4.14962e13 3.20513
\(821\) −6.91775e12 −0.531399 −0.265699 0.964056i \(-0.585603\pi\)
−0.265699 + 0.964056i \(0.585603\pi\)
\(822\) −3.51302e13 −2.68384
\(823\) 7.32525e11 0.0556574 0.0278287 0.999613i \(-0.491141\pi\)
0.0278287 + 0.999613i \(0.491141\pi\)
\(824\) 4.49425e13 3.39613
\(825\) 2.81971e12 0.211915
\(826\) 4.27339e12 0.319420
\(827\) −1.70629e12 −0.126846 −0.0634232 0.997987i \(-0.520202\pi\)
−0.0634232 + 0.997987i \(0.520202\pi\)
\(828\) −1.46861e14 −10.8585
\(829\) 7.31092e12 0.537622 0.268811 0.963193i \(-0.413369\pi\)
0.268811 + 0.963193i \(0.413369\pi\)
\(830\) 1.46598e13 1.07220
\(831\) −1.90127e13 −1.38306
\(832\) 2.50690e12 0.181377
\(833\) −1.33759e12 −0.0962544
\(834\) 3.68993e13 2.64102
\(835\) −1.60616e13 −1.14340
\(836\) −7.38910e12 −0.523195
\(837\) −3.57459e13 −2.51746
\(838\) −2.31747e13 −1.62336
\(839\) −2.41292e13 −1.68118 −0.840588 0.541675i \(-0.817791\pi\)
−0.840588 + 0.541675i \(0.817791\pi\)
\(840\) 1.92391e13 1.33330
\(841\) 1.51529e13 1.04451
\(842\) −3.86546e13 −2.65031
\(843\) 7.25655e12 0.494887
\(844\) −4.39427e13 −2.98089
\(845\) −8.51493e11 −0.0574547
\(846\) 3.84926e13 2.58351
\(847\) −5.29814e12 −0.353711
\(848\) −3.49522e13 −2.32110
\(849\) −2.25470e13 −1.48937
\(850\) 8.31205e12 0.546164
\(851\) −4.79658e13 −3.13508
\(852\) 6.33042e13 4.11580
\(853\) 1.77234e13 1.14624 0.573121 0.819470i \(-0.305732\pi\)
0.573121 + 0.819470i \(0.305732\pi\)
\(854\) −4.08904e12 −0.263064
\(855\) 2.63392e13 1.68560
\(856\) −1.67702e13 −1.06760
\(857\) 1.70084e13 1.07708 0.538542 0.842599i \(-0.318975\pi\)
0.538542 + 0.842599i \(0.318975\pi\)
\(858\) −3.86902e12 −0.243730
\(859\) −9.01632e12 −0.565015 −0.282508 0.959265i \(-0.591166\pi\)
−0.282508 + 0.959265i \(0.591166\pi\)
\(860\) −7.27823e12 −0.453715
\(861\) 2.09568e13 1.29960
\(862\) −1.42927e13 −0.881722
\(863\) 1.21027e13 0.742735 0.371368 0.928486i \(-0.378889\pi\)
0.371368 + 0.928486i \(0.378889\pi\)
\(864\) −7.66362e13 −4.67867
\(865\) 1.87055e13 1.13605
\(866\) −3.06334e13 −1.85082
\(867\) 1.71890e13 1.03315
\(868\) 1.25675e13 0.751467
\(869\) 1.73824e12 0.103400
\(870\) −6.26062e13 −3.70494
\(871\) −3.79207e12 −0.223252
\(872\) −1.28490e13 −0.752568
\(873\) 1.00888e13 0.587862
\(874\) 4.92978e13 2.85776
\(875\) 7.05925e12 0.407120
\(876\) 9.04418e13 5.18921
\(877\) −6.47514e12 −0.369616 −0.184808 0.982775i \(-0.559166\pi\)
−0.184808 + 0.982775i \(0.559166\pi\)
\(878\) 9.02680e11 0.0512635
\(879\) 2.04987e13 1.15818
\(880\) −7.45513e12 −0.419067
\(881\) −1.58670e13 −0.887368 −0.443684 0.896183i \(-0.646329\pi\)
−0.443684 + 0.896183i \(0.646329\pi\)
\(882\) 1.21459e13 0.675806
\(883\) −2.11575e13 −1.17123 −0.585614 0.810590i \(-0.699147\pi\)
−0.585614 + 0.810590i \(0.699147\pi\)
\(884\) −8.01226e12 −0.441286
\(885\) 1.18883e13 0.651440
\(886\) 2.77425e13 1.51249
\(887\) −1.93467e13 −1.04942 −0.524711 0.851281i \(-0.675827\pi\)
−0.524711 + 0.851281i \(0.675827\pi\)
\(888\) −1.53949e14 −8.30844
\(889\) 9.08651e12 0.487909
\(890\) 3.44841e12 0.184231
\(891\) 1.46653e13 0.779547
\(892\) −2.85983e12 −0.151251
\(893\) −9.07714e12 −0.477658
\(894\) 1.13557e14 5.94557
\(895\) −1.81855e13 −0.947374
\(896\) −2.66710e12 −0.138246
\(897\) 1.81338e13 0.935237
\(898\) 6.05315e13 3.10626
\(899\) −2.35776e13 −1.20387
\(900\) −5.30232e13 −2.69386
\(901\) 1.39676e13 0.706093
\(902\) −1.67787e13 −0.843973
\(903\) −3.67572e12 −0.183970
\(904\) −3.75419e13 −1.86964
\(905\) −1.94640e13 −0.964527
\(906\) −3.24484e13 −1.59999
\(907\) −3.61259e13 −1.77250 −0.886249 0.463210i \(-0.846698\pi\)
−0.886249 + 0.463210i \(0.846698\pi\)
\(908\) −6.06822e13 −2.96261
\(909\) 5.38426e13 2.61570
\(910\) −2.96959e12 −0.143552
\(911\) 2.05288e13 0.987484 0.493742 0.869608i \(-0.335629\pi\)
0.493742 + 0.869608i \(0.335629\pi\)
\(912\) 7.65790e13 3.66550
\(913\) −4.16417e12 −0.198340
\(914\) 2.62872e13 1.24591
\(915\) −1.13754e13 −0.536505
\(916\) −5.61747e13 −2.63640
\(917\) −6.66629e12 −0.311331
\(918\) 7.94782e13 3.69365
\(919\) 1.05080e13 0.485962 0.242981 0.970031i \(-0.421875\pi\)
0.242981 + 0.970031i \(0.421875\pi\)
\(920\) 7.21948e13 3.32246
\(921\) 3.92086e13 1.79562
\(922\) 3.20211e12 0.145931
\(923\) −5.63328e12 −0.255478
\(924\) −9.47910e12 −0.427802
\(925\) −1.73178e13 −0.777775
\(926\) 4.73524e13 2.11638
\(927\) 7.89312e13 3.51067
\(928\) −5.05483e13 −2.23739
\(929\) −1.62101e13 −0.714026 −0.357013 0.934099i \(-0.616205\pi\)
−0.357013 + 0.934099i \(0.616205\pi\)
\(930\) 4.97674e13 2.18158
\(931\) −2.86419e12 −0.124948
\(932\) −2.47428e12 −0.107418
\(933\) −6.19781e13 −2.67775
\(934\) 7.41454e11 0.0318803
\(935\) 2.97922e12 0.127483
\(936\) 4.19451e13 1.78624
\(937\) 2.51993e12 0.106797 0.0533986 0.998573i \(-0.482995\pi\)
0.0533986 + 0.998573i \(0.482995\pi\)
\(938\) −1.32249e13 −0.557800
\(939\) −3.54135e13 −1.48653
\(940\) −2.30573e13 −0.963236
\(941\) −5.25553e12 −0.218506 −0.109253 0.994014i \(-0.534846\pi\)
−0.109253 + 0.994014i \(0.534846\pi\)
\(942\) −5.19237e13 −2.14851
\(943\) 7.86402e13 3.23848
\(944\) 2.49100e13 1.02094
\(945\) 2.06938e13 0.844105
\(946\) 2.94290e12 0.119472
\(947\) −1.13066e13 −0.456833 −0.228417 0.973563i \(-0.573355\pi\)
−0.228417 + 0.973563i \(0.573355\pi\)
\(948\) −4.53548e13 −1.82384
\(949\) −8.04820e12 −0.322107
\(950\) 1.77987e13 0.708976
\(951\) 5.32944e13 2.11285
\(952\) −1.61098e13 −0.635657
\(953\) −7.11604e12 −0.279460 −0.139730 0.990190i \(-0.544624\pi\)
−0.139730 + 0.990190i \(0.544624\pi\)
\(954\) −1.26833e14 −4.95750
\(955\) −1.30729e13 −0.508576
\(956\) 3.46630e13 1.34216
\(957\) 1.77835e13 0.685352
\(958\) −1.58401e13 −0.607595
\(959\) 7.65905e12 0.292409
\(960\) 2.43220e13 0.924227
\(961\) −7.69716e12 −0.291122
\(962\) 2.37623e13 0.894541
\(963\) −2.94531e13 −1.10360
\(964\) −6.04255e13 −2.25358
\(965\) −3.88449e13 −1.44199
\(966\) 6.32416e13 2.33672
\(967\) −3.19433e13 −1.17479 −0.587395 0.809300i \(-0.699846\pi\)
−0.587395 + 0.809300i \(0.699846\pi\)
\(968\) −6.38101e13 −2.33588
\(969\) −3.06026e13 −1.11507
\(970\) −8.60241e12 −0.311995
\(971\) −3.41655e13 −1.23339 −0.616696 0.787201i \(-0.711529\pi\)
−0.616696 + 0.787201i \(0.711529\pi\)
\(972\) −1.86160e14 −6.68942
\(973\) −8.04476e12 −0.287743
\(974\) 2.27509e13 0.809998
\(975\) 6.54708e12 0.232021
\(976\) −2.38355e13 −0.840813
\(977\) −1.11381e13 −0.391099 −0.195549 0.980694i \(-0.562649\pi\)
−0.195549 + 0.980694i \(0.562649\pi\)
\(978\) 1.77369e14 6.19943
\(979\) −9.79531e11 −0.0340797
\(980\) −7.27548e12 −0.251967
\(981\) −2.25664e13 −0.777949
\(982\) 2.63751e12 0.0905090
\(983\) −3.28240e12 −0.112125 −0.0560623 0.998427i \(-0.517855\pi\)
−0.0560623 + 0.998427i \(0.517855\pi\)
\(984\) 2.52401e14 8.58248
\(985\) −2.95067e13 −0.998751
\(986\) 5.24229e13 1.76634
\(987\) −1.16446e13 −0.390568
\(988\) −1.71567e13 −0.572833
\(989\) −1.37931e13 −0.458436
\(990\) −2.70527e13 −0.895061
\(991\) 2.19446e13 0.722763 0.361382 0.932418i \(-0.382305\pi\)
0.361382 + 0.932418i \(0.382305\pi\)
\(992\) 4.01822e13 1.31744
\(993\) −3.40886e13 −1.11260
\(994\) −1.96461e13 −0.638319
\(995\) −2.09294e13 −0.676944
\(996\) 1.08653e14 3.49845
\(997\) 1.06807e13 0.342352 0.171176 0.985240i \(-0.445243\pi\)
0.171176 + 0.985240i \(0.445243\pi\)
\(998\) 3.01679e13 0.962625
\(999\) −1.65589e14 −5.26001
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.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.12 12 1.1 even 1 trivial