Properties

Label 91.10.a.a.1.7
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 1.7
Root \(1.29652\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.703483 q^{2} -141.700 q^{3} -511.505 q^{4} -2244.71 q^{5} +99.6835 q^{6} +2401.00 q^{7} +720.018 q^{8} +395.868 q^{9} +O(q^{10})\) \(q-0.703483 q^{2} -141.700 q^{3} -511.505 q^{4} -2244.71 q^{5} +99.6835 q^{6} +2401.00 q^{7} +720.018 q^{8} +395.868 q^{9} +1579.12 q^{10} +73311.4 q^{11} +72480.2 q^{12} +28561.0 q^{13} -1689.06 q^{14} +318076. q^{15} +261384. q^{16} -455759. q^{17} -278.487 q^{18} +1.01843e6 q^{19} +1.14818e6 q^{20} -340222. q^{21} -51573.3 q^{22} -1.84063e6 q^{23} -102027. q^{24} +3.08562e6 q^{25} -20092.2 q^{26} +2.73299e6 q^{27} -1.22812e6 q^{28} -1.40083e6 q^{29} -223761. q^{30} -7.24656e6 q^{31} -552529. q^{32} -1.03882e7 q^{33} +320618. q^{34} -5.38956e6 q^{35} -202489. q^{36} +1.23016e7 q^{37} -716450. q^{38} -4.04709e6 q^{39} -1.61624e6 q^{40} +4.18002e6 q^{41} +239340. q^{42} +2.21662e7 q^{43} -3.74991e7 q^{44} -888612. q^{45} +1.29485e6 q^{46} -1.54254e7 q^{47} -3.70381e7 q^{48} +5.76480e6 q^{49} -2.17068e6 q^{50} +6.45810e7 q^{51} -1.46091e7 q^{52} +9.65923e7 q^{53} -1.92261e6 q^{54} -1.64563e8 q^{55} +1.72876e6 q^{56} -1.44312e8 q^{57} +985458. q^{58} -9.86408e7 q^{59} -1.62697e8 q^{60} +1.20146e8 q^{61} +5.09783e6 q^{62} +950480. q^{63} -1.33440e8 q^{64} -6.41113e7 q^{65} +7.30793e6 q^{66} -1.79789e8 q^{67} +2.33123e8 q^{68} +2.60817e8 q^{69} +3.79146e6 q^{70} +1.78657e8 q^{71} +285033. q^{72} +3.01174e8 q^{73} -8.65400e6 q^{74} -4.37232e8 q^{75} -5.20934e8 q^{76} +1.76021e8 q^{77} +2.84706e6 q^{78} -3.14221e8 q^{79} -5.86733e8 q^{80} -3.95056e8 q^{81} -2.94057e6 q^{82} -2.27502e8 q^{83} +1.74025e8 q^{84} +1.02305e9 q^{85} -1.55935e7 q^{86} +1.98497e8 q^{87} +5.27855e7 q^{88} -7.30202e8 q^{89} +625123. q^{90} +6.85750e7 q^{91} +9.41492e8 q^{92} +1.02684e9 q^{93} +1.08515e7 q^{94} -2.28609e9 q^{95} +7.82933e7 q^{96} +8.70061e7 q^{97} -4.05544e6 q^{98} +2.90217e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 21q^{2} - 323q^{3} + 2945q^{4} - 5202q^{5} - 9897q^{6} + 28812q^{7} - 25431q^{8} + 78519q^{9} + O(q^{10}) \) \( 12q - 21q^{2} - 323q^{3} + 2945q^{4} - 5202q^{5} - 9897q^{6} + 28812q^{7} - 25431q^{8} + 78519q^{9} - 65812q^{10} - 80061q^{11} - 184395q^{12} + 342732q^{13} - 50421q^{14} + 160096q^{15} + 385497q^{16} - 1493598q^{17} + 1520858q^{18} - 109038q^{19} - 622260q^{20} - 775523q^{21} + 4636975q^{22} - 3367443q^{23} - 5963895q^{24} - 51480q^{25} - 599781q^{26} - 8158937q^{27} + 7070945q^{28} - 13333098q^{29} + 2915424q^{30} - 3954765q^{31} + 4389297q^{32} - 5790219q^{33} + 14879968q^{34} - 12490002q^{35} + 80697058q^{36} + 580535q^{37} - 19134246q^{38} - 9225203q^{39} + 12365024q^{40} - 27018171q^{41} - 23762697q^{42} + 31237588q^{43} - 125053839q^{44} - 62765470q^{45} - 114008121q^{46} - 21983709q^{47} - 309724207q^{48} + 69177612q^{49} - 131331747q^{50} - 176522692q^{51} + 84112145q^{52} - 196548234q^{53} - 456152547q^{54} - 309055872q^{55} - 61059831q^{56} - 274411494q^{57} - 521980612q^{58} - 215907906q^{59} - 177006648q^{60} - 218340705q^{61} - 673289997q^{62} + 188524119q^{63} - 386667247q^{64} - 148574322q^{65} - 777397365q^{66} + 14544775q^{67} - 1246637448q^{68} - 65252625q^{69} - 158014612q^{70} - 552451776q^{71} + 369379470q^{72} - 349395159q^{73} + 73591023q^{74} + 329300747q^{75} - 1036299002q^{76} - 192226461q^{77} - 282668217q^{78} + 962249727q^{79} - 1494536184q^{80} + 874458108q^{81} - 1417698067q^{82} - 2032575912q^{83} - 442732395q^{84} - 411671064q^{85} - 2139249420q^{86} - 759642172q^{87} + 558651957q^{88} - 280821684q^{89} - 5764700804q^{90} + 822899532q^{91} - 4491569571q^{92} - 1729557923q^{93} - 1591372165q^{94} - 1282463328q^{95} - 2148993055q^{96} - 2115165937q^{97} - 121060821q^{98} - 3595669198q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.703483 −0.0310898 −0.0155449 0.999879i \(-0.504948\pi\)
−0.0155449 + 0.999879i \(0.504948\pi\)
\(3\) −141.700 −1.01001 −0.505003 0.863118i \(-0.668509\pi\)
−0.505003 + 0.863118i \(0.668509\pi\)
\(4\) −511.505 −0.999033
\(5\) −2244.71 −1.60619 −0.803094 0.595853i \(-0.796814\pi\)
−0.803094 + 0.595853i \(0.796814\pi\)
\(6\) 99.6835 0.0314009
\(7\) 2401.00 0.377964
\(8\) 720.018 0.0621496
\(9\) 395.868 0.0201122
\(10\) 1579.12 0.0499361
\(11\) 73311.4 1.50975 0.754874 0.655870i \(-0.227698\pi\)
0.754874 + 0.655870i \(0.227698\pi\)
\(12\) 72480.2 1.00903
\(13\) 28561.0 0.277350
\(14\) −1689.06 −0.0117509
\(15\) 318076. 1.62226
\(16\) 261384. 0.997101
\(17\) −455759. −1.32347 −0.661736 0.749737i \(-0.730180\pi\)
−0.661736 + 0.749737i \(0.730180\pi\)
\(18\) −278.487 −0.000625285 0
\(19\) 1.01843e6 1.79284 0.896420 0.443206i \(-0.146159\pi\)
0.896420 + 0.443206i \(0.146159\pi\)
\(20\) 1.14818e6 1.60463
\(21\) −340222. −0.381746
\(22\) −51573.3 −0.0469378
\(23\) −1.84063e6 −1.37149 −0.685743 0.727844i \(-0.740523\pi\)
−0.685743 + 0.727844i \(0.740523\pi\)
\(24\) −102027. −0.0627715
\(25\) 3.08562e6 1.57984
\(26\) −20092.2 −0.00862277
\(27\) 2.73299e6 0.989693
\(28\) −1.22812e6 −0.377599
\(29\) −1.40083e6 −0.367785 −0.183892 0.982946i \(-0.558870\pi\)
−0.183892 + 0.982946i \(0.558870\pi\)
\(30\) −223761. −0.0504358
\(31\) −7.24656e6 −1.40930 −0.704652 0.709553i \(-0.748897\pi\)
−0.704652 + 0.709553i \(0.748897\pi\)
\(32\) −552529. −0.0931494
\(33\) −1.03882e7 −1.52485
\(34\) 320618. 0.0411465
\(35\) −5.38956e6 −0.607082
\(36\) −202489. −0.0200928
\(37\) 1.23016e7 1.07908 0.539542 0.841958i \(-0.318597\pi\)
0.539542 + 0.841958i \(0.318597\pi\)
\(38\) −716450. −0.0557391
\(39\) −4.04709e6 −0.280125
\(40\) −1.61624e6 −0.0998240
\(41\) 4.18002e6 0.231021 0.115510 0.993306i \(-0.463150\pi\)
0.115510 + 0.993306i \(0.463150\pi\)
\(42\) 239340. 0.0118684
\(43\) 2.21662e7 0.988742 0.494371 0.869251i \(-0.335399\pi\)
0.494371 + 0.869251i \(0.335399\pi\)
\(44\) −3.74991e7 −1.50829
\(45\) −888612. −0.0323040
\(46\) 1.29485e6 0.0426393
\(47\) −1.54254e7 −0.461101 −0.230551 0.973060i \(-0.574053\pi\)
−0.230551 + 0.973060i \(0.574053\pi\)
\(48\) −3.70381e7 −1.00708
\(49\) 5.76480e6 0.142857
\(50\) −2.17068e6 −0.0491169
\(51\) 6.45810e7 1.33671
\(52\) −1.46091e7 −0.277082
\(53\) 9.65923e7 1.68152 0.840758 0.541411i \(-0.182110\pi\)
0.840758 + 0.541411i \(0.182110\pi\)
\(54\) −1.92261e6 −0.0307694
\(55\) −1.64563e8 −2.42494
\(56\) 1.72876e6 0.0234904
\(57\) −1.44312e8 −1.81078
\(58\) 985458. 0.0114344
\(59\) −9.86408e7 −1.05980 −0.529898 0.848061i \(-0.677770\pi\)
−0.529898 + 0.848061i \(0.677770\pi\)
\(60\) −1.62697e8 −1.62069
\(61\) 1.20146e8 1.11103 0.555515 0.831507i \(-0.312521\pi\)
0.555515 + 0.831507i \(0.312521\pi\)
\(62\) 5.09783e6 0.0438150
\(63\) 950480. 0.00760170
\(64\) −1.33440e8 −0.994205
\(65\) −6.41113e7 −0.445476
\(66\) 7.30793e6 0.0474075
\(67\) −1.79789e8 −1.09000 −0.544999 0.838437i \(-0.683470\pi\)
−0.544999 + 0.838437i \(0.683470\pi\)
\(68\) 2.33123e8 1.32219
\(69\) 2.60817e8 1.38521
\(70\) 3.79146e6 0.0188741
\(71\) 1.78657e8 0.834367 0.417183 0.908822i \(-0.363017\pi\)
0.417183 + 0.908822i \(0.363017\pi\)
\(72\) 285033. 0.00124997
\(73\) 3.01174e8 1.24127 0.620633 0.784101i \(-0.286876\pi\)
0.620633 + 0.784101i \(0.286876\pi\)
\(74\) −8.65400e6 −0.0335486
\(75\) −4.37232e8 −1.59564
\(76\) −5.20934e8 −1.79111
\(77\) 1.76021e8 0.570631
\(78\) 2.84706e6 0.00870905
\(79\) −3.14221e8 −0.907638 −0.453819 0.891094i \(-0.649939\pi\)
−0.453819 + 0.891094i \(0.649939\pi\)
\(80\) −5.86733e8 −1.60153
\(81\) −3.95056e8 −1.01971
\(82\) −2.94057e6 −0.00718241
\(83\) −2.27502e8 −0.526180 −0.263090 0.964771i \(-0.584742\pi\)
−0.263090 + 0.964771i \(0.584742\pi\)
\(84\) 1.74025e8 0.381377
\(85\) 1.02305e9 2.12574
\(86\) −1.55935e7 −0.0307398
\(87\) 1.98497e8 0.371465
\(88\) 5.27855e7 0.0938303
\(89\) −7.30202e8 −1.23364 −0.616819 0.787105i \(-0.711579\pi\)
−0.616819 + 0.787105i \(0.711579\pi\)
\(90\) 625123. 0.00100433
\(91\) 6.85750e7 0.104828
\(92\) 9.41492e8 1.37016
\(93\) 1.02684e9 1.42340
\(94\) 1.08515e7 0.0143356
\(95\) −2.28609e9 −2.87964
\(96\) 7.82933e7 0.0940814
\(97\) 8.70061e7 0.0997877 0.0498939 0.998755i \(-0.484112\pi\)
0.0498939 + 0.998755i \(0.484112\pi\)
\(98\) −4.05544e6 −0.00444141
\(99\) 2.90217e7 0.0303643
\(100\) −1.57831e9 −1.57831
\(101\) −1.71620e9 −1.64105 −0.820527 0.571608i \(-0.806320\pi\)
−0.820527 + 0.571608i \(0.806320\pi\)
\(102\) −4.54316e7 −0.0415583
\(103\) −9.00371e8 −0.788232 −0.394116 0.919061i \(-0.628949\pi\)
−0.394116 + 0.919061i \(0.628949\pi\)
\(104\) 2.05644e7 0.0172372
\(105\) 7.63700e8 0.613156
\(106\) −6.79510e7 −0.0522781
\(107\) 2.09084e9 1.54203 0.771015 0.636817i \(-0.219749\pi\)
0.771015 + 0.636817i \(0.219749\pi\)
\(108\) −1.39794e9 −0.988736
\(109\) −3.10773e8 −0.210875 −0.105437 0.994426i \(-0.533624\pi\)
−0.105437 + 0.994426i \(0.533624\pi\)
\(110\) 1.15767e8 0.0753909
\(111\) −1.74314e9 −1.08988
\(112\) 6.27583e8 0.376869
\(113\) 1.24372e9 0.717580 0.358790 0.933418i \(-0.383189\pi\)
0.358790 + 0.933418i \(0.383189\pi\)
\(114\) 1.01521e8 0.0562968
\(115\) 4.13169e9 2.20286
\(116\) 7.16530e8 0.367429
\(117\) 1.13064e7 0.00557812
\(118\) 6.93921e7 0.0329489
\(119\) −1.09428e9 −0.500225
\(120\) 2.29020e8 0.100823
\(121\) 3.01661e9 1.27934
\(122\) −8.45208e7 −0.0345417
\(123\) −5.92309e8 −0.233333
\(124\) 3.70665e9 1.40794
\(125\) −2.54213e9 −0.931327
\(126\) −668647. −0.000236336 0
\(127\) 1.40884e9 0.480557 0.240279 0.970704i \(-0.422761\pi\)
0.240279 + 0.970704i \(0.422761\pi\)
\(128\) 3.76767e8 0.124059
\(129\) −3.14095e9 −0.998635
\(130\) 4.51012e7 0.0138498
\(131\) 8.94955e8 0.265510 0.132755 0.991149i \(-0.457618\pi\)
0.132755 + 0.991149i \(0.457618\pi\)
\(132\) 5.31363e9 1.52338
\(133\) 2.44526e9 0.677629
\(134\) 1.26478e8 0.0338879
\(135\) −6.13477e9 −1.58963
\(136\) −3.28155e8 −0.0822533
\(137\) 1.21092e9 0.293680 0.146840 0.989160i \(-0.453090\pi\)
0.146840 + 0.989160i \(0.453090\pi\)
\(138\) −1.83480e8 −0.0430659
\(139\) 8.08378e9 1.83674 0.918370 0.395723i \(-0.129506\pi\)
0.918370 + 0.395723i \(0.129506\pi\)
\(140\) 2.75679e9 0.606495
\(141\) 2.18578e9 0.465715
\(142\) −1.25682e8 −0.0259403
\(143\) 2.09385e9 0.418729
\(144\) 1.03474e8 0.0200539
\(145\) 3.14446e9 0.590731
\(146\) −2.11871e8 −0.0385908
\(147\) −8.16872e8 −0.144287
\(148\) −6.29236e9 −1.07804
\(149\) −4.34320e9 −0.721891 −0.360946 0.932587i \(-0.617546\pi\)
−0.360946 + 0.932587i \(0.617546\pi\)
\(150\) 3.07585e8 0.0496084
\(151\) 2.65272e9 0.415235 0.207618 0.978210i \(-0.433429\pi\)
0.207618 + 0.978210i \(0.433429\pi\)
\(152\) 7.33291e8 0.111424
\(153\) −1.80420e8 −0.0266179
\(154\) −1.23828e8 −0.0177408
\(155\) 1.62665e10 2.26360
\(156\) 2.07011e9 0.279855
\(157\) −5.58264e9 −0.733316 −0.366658 0.930356i \(-0.619498\pi\)
−0.366658 + 0.930356i \(0.619498\pi\)
\(158\) 2.21049e8 0.0282183
\(159\) −1.36871e10 −1.69834
\(160\) 1.24027e9 0.149615
\(161\) −4.41935e9 −0.518373
\(162\) 2.77915e8 0.0317026
\(163\) −1.30367e10 −1.44652 −0.723261 0.690575i \(-0.757358\pi\)
−0.723261 + 0.690575i \(0.757358\pi\)
\(164\) −2.13810e9 −0.230798
\(165\) 2.33186e10 2.44920
\(166\) 1.60044e8 0.0163588
\(167\) 8.51514e9 0.847164 0.423582 0.905858i \(-0.360773\pi\)
0.423582 + 0.905858i \(0.360773\pi\)
\(168\) −2.44966e8 −0.0237254
\(169\) 8.15731e8 0.0769231
\(170\) −7.19697e8 −0.0660890
\(171\) 4.03166e8 0.0360579
\(172\) −1.13381e10 −0.987786
\(173\) −7.74574e9 −0.657439 −0.328720 0.944428i \(-0.606617\pi\)
−0.328720 + 0.944428i \(0.606617\pi\)
\(174\) −1.39639e8 −0.0115488
\(175\) 7.40857e9 0.597122
\(176\) 1.91624e10 1.50537
\(177\) 1.39774e10 1.07040
\(178\) 5.13685e8 0.0383536
\(179\) −1.12276e10 −0.817424 −0.408712 0.912663i \(-0.634022\pi\)
−0.408712 + 0.912663i \(0.634022\pi\)
\(180\) 4.54529e8 0.0322727
\(181\) −6.47798e9 −0.448627 −0.224314 0.974517i \(-0.572014\pi\)
−0.224314 + 0.974517i \(0.572014\pi\)
\(182\) −4.82413e7 −0.00325910
\(183\) −1.70247e10 −1.12215
\(184\) −1.32529e9 −0.0852374
\(185\) −2.76137e10 −1.73321
\(186\) −7.22363e8 −0.0442534
\(187\) −3.34123e10 −1.99811
\(188\) 7.89018e9 0.460656
\(189\) 6.56190e9 0.374069
\(190\) 1.60823e9 0.0895274
\(191\) 1.84477e10 1.00298 0.501490 0.865164i \(-0.332786\pi\)
0.501490 + 0.865164i \(0.332786\pi\)
\(192\) 1.89084e10 1.00415
\(193\) −6.56172e9 −0.340416 −0.170208 0.985408i \(-0.554444\pi\)
−0.170208 + 0.985408i \(0.554444\pi\)
\(194\) −6.12073e7 −0.00310239
\(195\) 9.08457e9 0.449934
\(196\) −2.94873e9 −0.142719
\(197\) 1.91473e9 0.0905752 0.0452876 0.998974i \(-0.485580\pi\)
0.0452876 + 0.998974i \(0.485580\pi\)
\(198\) −2.04162e7 −0.000944023 0
\(199\) −4.08821e10 −1.84797 −0.923983 0.382434i \(-0.875086\pi\)
−0.923983 + 0.382434i \(0.875086\pi\)
\(200\) 2.22170e9 0.0981863
\(201\) 2.54760e10 1.10090
\(202\) 1.20732e9 0.0510201
\(203\) −3.36338e9 −0.139009
\(204\) −3.30335e10 −1.33542
\(205\) −9.38296e9 −0.371063
\(206\) 6.33396e8 0.0245060
\(207\) −7.28648e8 −0.0275836
\(208\) 7.46539e9 0.276546
\(209\) 7.46627e10 2.70673
\(210\) −5.37250e8 −0.0190629
\(211\) 2.00944e9 0.0697917 0.0348959 0.999391i \(-0.488890\pi\)
0.0348959 + 0.999391i \(0.488890\pi\)
\(212\) −4.94075e10 −1.67989
\(213\) −2.53156e10 −0.842715
\(214\) −1.47087e9 −0.0479415
\(215\) −4.97568e10 −1.58810
\(216\) 1.96780e9 0.0615090
\(217\) −1.73990e10 −0.532667
\(218\) 2.18624e8 0.00655607
\(219\) −4.26764e10 −1.25369
\(220\) 8.41749e10 2.42259
\(221\) −1.30169e10 −0.367065
\(222\) 1.22627e9 0.0338843
\(223\) −2.67298e10 −0.723808 −0.361904 0.932215i \(-0.617873\pi\)
−0.361904 + 0.932215i \(0.617873\pi\)
\(224\) −1.32662e9 −0.0352072
\(225\) 1.22150e9 0.0317740
\(226\) −8.74937e8 −0.0223095
\(227\) −3.71464e10 −0.928540 −0.464270 0.885694i \(-0.653683\pi\)
−0.464270 + 0.885694i \(0.653683\pi\)
\(228\) 7.38163e10 1.80903
\(229\) 1.59863e10 0.384139 0.192069 0.981381i \(-0.438480\pi\)
0.192069 + 0.981381i \(0.438480\pi\)
\(230\) −2.90657e9 −0.0684867
\(231\) −2.49421e10 −0.576341
\(232\) −1.00862e9 −0.0228577
\(233\) −8.21114e10 −1.82516 −0.912582 0.408895i \(-0.865914\pi\)
−0.912582 + 0.408895i \(0.865914\pi\)
\(234\) −7.95386e6 −0.000173423 0
\(235\) 3.46257e10 0.740615
\(236\) 5.04553e10 1.05877
\(237\) 4.45250e10 0.916720
\(238\) 7.69805e8 0.0155519
\(239\) −4.54409e10 −0.900857 −0.450429 0.892812i \(-0.648729\pi\)
−0.450429 + 0.892812i \(0.648729\pi\)
\(240\) 8.31400e10 1.61756
\(241\) −2.45430e10 −0.468653 −0.234327 0.972158i \(-0.575289\pi\)
−0.234327 + 0.972158i \(0.575289\pi\)
\(242\) −2.12213e9 −0.0397744
\(243\) 2.18601e9 0.0402183
\(244\) −6.14554e10 −1.10996
\(245\) −1.29403e10 −0.229455
\(246\) 4.16679e8 0.00725427
\(247\) 2.90875e10 0.497244
\(248\) −5.21766e9 −0.0875877
\(249\) 3.22370e10 0.531445
\(250\) 1.78834e9 0.0289548
\(251\) 1.64140e10 0.261026 0.130513 0.991447i \(-0.458338\pi\)
0.130513 + 0.991447i \(0.458338\pi\)
\(252\) −4.86175e8 −0.00759435
\(253\) −1.34939e11 −2.07060
\(254\) −9.91096e8 −0.0149405
\(255\) −1.44966e11 −2.14701
\(256\) 6.80562e10 0.990348
\(257\) 5.71194e9 0.0816742 0.0408371 0.999166i \(-0.486998\pi\)
0.0408371 + 0.999166i \(0.486998\pi\)
\(258\) 2.20960e9 0.0310474
\(259\) 2.95363e10 0.407856
\(260\) 3.27933e10 0.445046
\(261\) −5.54543e8 −0.00739696
\(262\) −6.29586e8 −0.00825466
\(263\) 8.81123e10 1.13563 0.567813 0.823157i \(-0.307790\pi\)
0.567813 + 0.823157i \(0.307790\pi\)
\(264\) −7.47971e9 −0.0947691
\(265\) −2.16822e11 −2.70083
\(266\) −1.72020e9 −0.0210674
\(267\) 1.03470e11 1.24598
\(268\) 9.19628e10 1.08894
\(269\) −2.03394e10 −0.236839 −0.118420 0.992964i \(-0.537783\pi\)
−0.118420 + 0.992964i \(0.537783\pi\)
\(270\) 4.31571e9 0.0494214
\(271\) −1.48875e11 −1.67672 −0.838362 0.545114i \(-0.816486\pi\)
−0.838362 + 0.545114i \(0.816486\pi\)
\(272\) −1.19128e11 −1.31964
\(273\) −9.71707e9 −0.105877
\(274\) −8.51865e8 −0.00913047
\(275\) 2.26211e11 2.38515
\(276\) −1.33409e11 −1.38387
\(277\) −1.54759e11 −1.57942 −0.789709 0.613482i \(-0.789768\pi\)
−0.789709 + 0.613482i \(0.789768\pi\)
\(278\) −5.68680e9 −0.0571040
\(279\) −2.86869e9 −0.0283442
\(280\) −3.88058e9 −0.0377299
\(281\) 5.15488e10 0.493220 0.246610 0.969115i \(-0.420683\pi\)
0.246610 + 0.969115i \(0.420683\pi\)
\(282\) −1.53766e9 −0.0144790
\(283\) 1.24651e11 1.15520 0.577600 0.816320i \(-0.303989\pi\)
0.577600 + 0.816320i \(0.303989\pi\)
\(284\) −9.13838e10 −0.833560
\(285\) 3.23939e11 2.90845
\(286\) −1.47299e9 −0.0130182
\(287\) 1.00362e10 0.0873177
\(288\) −2.18729e8 −0.00187344
\(289\) 8.91280e10 0.751578
\(290\) −2.21207e9 −0.0183657
\(291\) −1.23288e10 −0.100786
\(292\) −1.54052e11 −1.24007
\(293\) 6.06702e10 0.480918 0.240459 0.970659i \(-0.422702\pi\)
0.240459 + 0.970659i \(0.422702\pi\)
\(294\) 5.74655e8 0.00448585
\(295\) 2.21420e11 1.70223
\(296\) 8.85741e9 0.0670647
\(297\) 2.00359e11 1.49419
\(298\) 3.05537e9 0.0224435
\(299\) −5.25703e10 −0.380382
\(300\) 2.23646e11 1.59410
\(301\) 5.32210e10 0.373709
\(302\) −1.86614e9 −0.0129096
\(303\) 2.43186e11 1.65747
\(304\) 2.66202e11 1.78764
\(305\) −2.69694e11 −1.78452
\(306\) 1.26923e8 0.000827547 0
\(307\) −7.23606e10 −0.464921 −0.232461 0.972606i \(-0.574678\pi\)
−0.232461 + 0.972606i \(0.574678\pi\)
\(308\) −9.00355e10 −0.570079
\(309\) 1.27583e11 0.796119
\(310\) −1.14432e10 −0.0703751
\(311\) −1.18116e11 −0.715957 −0.357978 0.933730i \(-0.616534\pi\)
−0.357978 + 0.933730i \(0.616534\pi\)
\(312\) −2.91398e9 −0.0174097
\(313\) 2.46753e11 1.45316 0.726579 0.687083i \(-0.241109\pi\)
0.726579 + 0.687083i \(0.241109\pi\)
\(314\) 3.92729e9 0.0227987
\(315\) −2.13356e9 −0.0122097
\(316\) 1.60725e11 0.906761
\(317\) 2.14744e11 1.19441 0.597207 0.802087i \(-0.296277\pi\)
0.597207 + 0.802087i \(0.296277\pi\)
\(318\) 9.62866e9 0.0528012
\(319\) −1.02697e11 −0.555262
\(320\) 2.99535e11 1.59688
\(321\) −2.96271e11 −1.55746
\(322\) 3.10894e9 0.0161161
\(323\) −4.64160e11 −2.37277
\(324\) 2.02073e11 1.01872
\(325\) 8.81284e10 0.438168
\(326\) 9.17113e9 0.0449721
\(327\) 4.40366e10 0.212985
\(328\) 3.00969e9 0.0143579
\(329\) −3.70364e10 −0.174280
\(330\) −1.64042e10 −0.0761453
\(331\) 2.05061e11 0.938983 0.469492 0.882937i \(-0.344437\pi\)
0.469492 + 0.882937i \(0.344437\pi\)
\(332\) 1.16368e11 0.525671
\(333\) 4.86983e9 0.0217028
\(334\) −5.99025e9 −0.0263382
\(335\) 4.03574e11 1.75074
\(336\) −8.89285e10 −0.380640
\(337\) 1.80378e11 0.761813 0.380907 0.924613i \(-0.375612\pi\)
0.380907 + 0.924613i \(0.375612\pi\)
\(338\) −5.73853e8 −0.00239153
\(339\) −1.76235e11 −0.724760
\(340\) −5.23294e11 −2.12369
\(341\) −5.31256e11 −2.12769
\(342\) −2.83620e8 −0.00112104
\(343\) 1.38413e10 0.0539949
\(344\) 1.59601e10 0.0614499
\(345\) −5.85460e11 −2.22491
\(346\) 5.44900e9 0.0204397
\(347\) −2.67709e11 −0.991245 −0.495622 0.868538i \(-0.665060\pi\)
−0.495622 + 0.868538i \(0.665060\pi\)
\(348\) −1.01532e11 −0.371106
\(349\) −5.69032e10 −0.205316 −0.102658 0.994717i \(-0.532735\pi\)
−0.102658 + 0.994717i \(0.532735\pi\)
\(350\) −5.21180e9 −0.0185644
\(351\) 7.80568e10 0.274491
\(352\) −4.05066e10 −0.140632
\(353\) −1.19566e11 −0.409847 −0.204923 0.978778i \(-0.565695\pi\)
−0.204923 + 0.978778i \(0.565695\pi\)
\(354\) −9.83286e9 −0.0332786
\(355\) −4.01033e11 −1.34015
\(356\) 3.73502e11 1.23245
\(357\) 1.55059e11 0.505231
\(358\) 7.89841e9 0.0254136
\(359\) −3.86131e11 −1.22690 −0.613451 0.789733i \(-0.710219\pi\)
−0.613451 + 0.789733i \(0.710219\pi\)
\(360\) −6.39817e8 −0.00200768
\(361\) 7.14518e11 2.21427
\(362\) 4.55715e9 0.0139478
\(363\) −4.27454e11 −1.29214
\(364\) −3.50764e10 −0.104727
\(365\) −6.76050e11 −1.99371
\(366\) 1.19766e10 0.0348874
\(367\) 6.76922e10 0.194779 0.0973893 0.995246i \(-0.468951\pi\)
0.0973893 + 0.995246i \(0.468951\pi\)
\(368\) −4.81112e11 −1.36751
\(369\) 1.65474e9 0.00464634
\(370\) 1.94258e10 0.0538853
\(371\) 2.31918e11 0.635554
\(372\) −5.25233e11 −1.42203
\(373\) 2.15189e11 0.575612 0.287806 0.957689i \(-0.407074\pi\)
0.287806 + 0.957689i \(0.407074\pi\)
\(374\) 2.35050e10 0.0621209
\(375\) 3.60219e11 0.940646
\(376\) −1.11066e10 −0.0286573
\(377\) −4.00090e10 −0.102005
\(378\) −4.61618e9 −0.0116297
\(379\) 3.89471e11 0.969613 0.484806 0.874622i \(-0.338890\pi\)
0.484806 + 0.874622i \(0.338890\pi\)
\(380\) 1.16935e12 2.87685
\(381\) −1.99633e11 −0.485366
\(382\) −1.29776e10 −0.0311825
\(383\) −6.99200e11 −1.66038 −0.830188 0.557483i \(-0.811767\pi\)
−0.830188 + 0.557483i \(0.811767\pi\)
\(384\) −5.33879e10 −0.125300
\(385\) −3.95116e11 −0.916540
\(386\) 4.61606e9 0.0105835
\(387\) 8.77489e9 0.0198858
\(388\) −4.45041e10 −0.0996913
\(389\) −3.57806e11 −0.792272 −0.396136 0.918192i \(-0.629649\pi\)
−0.396136 + 0.918192i \(0.629649\pi\)
\(390\) −6.39084e9 −0.0139884
\(391\) 8.38883e11 1.81512
\(392\) 4.15076e9 0.00887852
\(393\) −1.26815e11 −0.268166
\(394\) −1.34698e9 −0.00281597
\(395\) 7.05335e11 1.45784
\(396\) −1.48447e10 −0.0303350
\(397\) −2.85387e11 −0.576603 −0.288301 0.957540i \(-0.593090\pi\)
−0.288301 + 0.957540i \(0.593090\pi\)
\(398\) 2.87598e10 0.0574530
\(399\) −3.46493e11 −0.684410
\(400\) 8.06532e11 1.57526
\(401\) −2.28809e11 −0.441900 −0.220950 0.975285i \(-0.570916\pi\)
−0.220950 + 0.975285i \(0.570916\pi\)
\(402\) −1.79220e10 −0.0342270
\(403\) −2.06969e11 −0.390870
\(404\) 8.77847e11 1.63947
\(405\) 8.86787e11 1.63784
\(406\) 2.36608e9 0.00432178
\(407\) 9.01851e11 1.62915
\(408\) 4.64995e10 0.0830763
\(409\) −9.95423e11 −1.75895 −0.879473 0.475948i \(-0.842105\pi\)
−0.879473 + 0.475948i \(0.842105\pi\)
\(410\) 6.60075e9 0.0115363
\(411\) −1.71588e11 −0.296619
\(412\) 4.60545e11 0.787470
\(413\) −2.36837e11 −0.400565
\(414\) 5.12591e8 0.000857570 0
\(415\) 5.10677e11 0.845143
\(416\) −1.57808e10 −0.0258350
\(417\) −1.14547e12 −1.85512
\(418\) −5.25240e10 −0.0841520
\(419\) −8.11072e11 −1.28557 −0.642786 0.766046i \(-0.722221\pi\)
−0.642786 + 0.766046i \(0.722221\pi\)
\(420\) −3.90637e11 −0.612563
\(421\) −6.61188e9 −0.0102578 −0.00512892 0.999987i \(-0.501633\pi\)
−0.00512892 + 0.999987i \(0.501633\pi\)
\(422\) −1.41361e9 −0.00216981
\(423\) −6.10643e9 −0.00927376
\(424\) 6.95482e10 0.104506
\(425\) −1.40630e12 −2.09087
\(426\) 1.78091e10 0.0261999
\(427\) 2.88471e11 0.419930
\(428\) −1.06947e12 −1.54054
\(429\) −2.96698e11 −0.422918
\(430\) 3.50030e10 0.0493739
\(431\) 8.38756e10 0.117081 0.0585407 0.998285i \(-0.481355\pi\)
0.0585407 + 0.998285i \(0.481355\pi\)
\(432\) 7.14359e11 0.986824
\(433\) −4.93551e10 −0.0674740 −0.0337370 0.999431i \(-0.510741\pi\)
−0.0337370 + 0.999431i \(0.510741\pi\)
\(434\) 1.22399e10 0.0165605
\(435\) −4.45569e11 −0.596642
\(436\) 1.58962e11 0.210671
\(437\) −1.87456e12 −2.45885
\(438\) 3.00221e10 0.0389769
\(439\) −1.02700e12 −1.31971 −0.659856 0.751392i \(-0.729383\pi\)
−0.659856 + 0.751392i \(0.729383\pi\)
\(440\) −1.18488e11 −0.150709
\(441\) 2.28210e9 0.00287317
\(442\) 9.15718e9 0.0114120
\(443\) 7.86570e11 0.970332 0.485166 0.874422i \(-0.338759\pi\)
0.485166 + 0.874422i \(0.338759\pi\)
\(444\) 8.91626e11 1.08883
\(445\) 1.63909e12 1.98145
\(446\) 1.88039e10 0.0225031
\(447\) 6.15431e11 0.729115
\(448\) −3.20389e11 −0.375774
\(449\) −7.97033e11 −0.925482 −0.462741 0.886494i \(-0.653134\pi\)
−0.462741 + 0.886494i \(0.653134\pi\)
\(450\) −8.59304e8 −0.000987849 0
\(451\) 3.06443e11 0.348783
\(452\) −6.36170e11 −0.716886
\(453\) −3.75890e11 −0.419390
\(454\) 2.61319e10 0.0288682
\(455\) −1.53931e11 −0.168374
\(456\) −1.03907e11 −0.112539
\(457\) −2.73199e11 −0.292993 −0.146496 0.989211i \(-0.546800\pi\)
−0.146496 + 0.989211i \(0.546800\pi\)
\(458\) −1.12461e10 −0.0119428
\(459\) −1.24558e12 −1.30983
\(460\) −2.11338e12 −2.20073
\(461\) 1.35636e12 1.39868 0.699342 0.714787i \(-0.253476\pi\)
0.699342 + 0.714787i \(0.253476\pi\)
\(462\) 1.75463e10 0.0179183
\(463\) 1.53935e12 1.55677 0.778383 0.627790i \(-0.216040\pi\)
0.778383 + 0.627790i \(0.216040\pi\)
\(464\) −3.66154e11 −0.366718
\(465\) −2.30496e12 −2.28625
\(466\) 5.77640e10 0.0567440
\(467\) 1.62368e12 1.57969 0.789847 0.613304i \(-0.210160\pi\)
0.789847 + 0.613304i \(0.210160\pi\)
\(468\) −5.78328e9 −0.00557273
\(469\) −4.31673e11 −0.411981
\(470\) −2.43586e10 −0.0230256
\(471\) 7.91060e11 0.740653
\(472\) −7.10232e10 −0.0658660
\(473\) 1.62503e12 1.49275
\(474\) −3.13226e10 −0.0285007
\(475\) 3.14250e12 2.83239
\(476\) 5.59728e11 0.499742
\(477\) 3.82378e10 0.0338190
\(478\) 3.19669e10 0.0280075
\(479\) −7.39673e11 −0.641993 −0.320996 0.947080i \(-0.604018\pi\)
−0.320996 + 0.947080i \(0.604018\pi\)
\(480\) −1.75746e11 −0.151112
\(481\) 3.51347e11 0.299284
\(482\) 1.72656e10 0.0145704
\(483\) 6.26222e11 0.523560
\(484\) −1.54301e12 −1.27810
\(485\) −1.95304e11 −0.160278
\(486\) −1.53782e9 −0.00125038
\(487\) 5.14104e11 0.414163 0.207081 0.978324i \(-0.433604\pi\)
0.207081 + 0.978324i \(0.433604\pi\)
\(488\) 8.65074e10 0.0690501
\(489\) 1.84731e12 1.46100
\(490\) 9.10330e9 0.00713373
\(491\) 2.14326e12 1.66421 0.832104 0.554619i \(-0.187136\pi\)
0.832104 + 0.554619i \(0.187136\pi\)
\(492\) 3.02969e11 0.233107
\(493\) 6.38439e11 0.486752
\(494\) −2.04625e10 −0.0154592
\(495\) −6.51453e10 −0.0487708
\(496\) −1.89414e12 −1.40522
\(497\) 4.28955e11 0.315361
\(498\) −2.26782e10 −0.0165225
\(499\) 1.08262e12 0.781672 0.390836 0.920460i \(-0.372186\pi\)
0.390836 + 0.920460i \(0.372186\pi\)
\(500\) 1.30031e12 0.930426
\(501\) −1.20659e12 −0.855641
\(502\) −1.15470e10 −0.00811525
\(503\) −1.84969e12 −1.28838 −0.644190 0.764866i \(-0.722805\pi\)
−0.644190 + 0.764866i \(0.722805\pi\)
\(504\) 6.84363e8 0.000472443 0
\(505\) 3.85239e12 2.63584
\(506\) 9.49274e10 0.0643746
\(507\) −1.15589e11 −0.0776928
\(508\) −7.20630e11 −0.480093
\(509\) −1.30612e12 −0.862487 −0.431244 0.902236i \(-0.641925\pi\)
−0.431244 + 0.902236i \(0.641925\pi\)
\(510\) 1.01981e11 0.0667503
\(511\) 7.23120e11 0.469155
\(512\) −2.40781e11 −0.154849
\(513\) 2.78336e12 1.77436
\(514\) −4.01826e9 −0.00253924
\(515\) 2.02108e12 1.26605
\(516\) 1.60661e12 0.997670
\(517\) −1.13086e12 −0.696147
\(518\) −2.07783e10 −0.0126802
\(519\) 1.09757e12 0.664017
\(520\) −4.61613e10 −0.0276862
\(521\) −9.94836e11 −0.591537 −0.295768 0.955260i \(-0.595576\pi\)
−0.295768 + 0.955260i \(0.595576\pi\)
\(522\) 3.90112e8 0.000229970 0
\(523\) −8.42112e11 −0.492167 −0.246083 0.969249i \(-0.579144\pi\)
−0.246083 + 0.969249i \(0.579144\pi\)
\(524\) −4.57774e11 −0.265253
\(525\) −1.04979e12 −0.603097
\(526\) −6.19855e10 −0.0353065
\(527\) 3.30268e12 1.86517
\(528\) −2.71531e12 −1.52043
\(529\) 1.58677e12 0.880975
\(530\) 1.52531e11 0.0839684
\(531\) −3.90488e10 −0.0213148
\(532\) −1.25076e12 −0.676975
\(533\) 1.19386e11 0.0640737
\(534\) −7.27891e10 −0.0387374
\(535\) −4.69333e12 −2.47679
\(536\) −1.29451e11 −0.0677430
\(537\) 1.59095e12 0.825603
\(538\) 1.43084e10 0.00736329
\(539\) 4.22626e11 0.215678
\(540\) 3.13797e12 1.58809
\(541\) −1.66119e12 −0.833741 −0.416870 0.908966i \(-0.636873\pi\)
−0.416870 + 0.908966i \(0.636873\pi\)
\(542\) 1.04731e11 0.0521291
\(543\) 9.17929e11 0.453116
\(544\) 2.51820e11 0.123281
\(545\) 6.97598e11 0.338704
\(546\) 6.83579e9 0.00329171
\(547\) −1.43942e12 −0.687455 −0.343727 0.939069i \(-0.611690\pi\)
−0.343727 + 0.939069i \(0.611690\pi\)
\(548\) −6.19394e11 −0.293396
\(549\) 4.75621e10 0.0223452
\(550\) −1.59136e11 −0.0741541
\(551\) −1.42665e12 −0.659378
\(552\) 1.87793e11 0.0860903
\(553\) −7.54443e11 −0.343055
\(554\) 1.08870e11 0.0491039
\(555\) 3.91286e12 1.75055
\(556\) −4.13489e12 −1.83496
\(557\) 1.25789e12 0.553724 0.276862 0.960910i \(-0.410706\pi\)
0.276862 + 0.960910i \(0.410706\pi\)
\(558\) 2.01807e9 0.000881217 0
\(559\) 6.33088e11 0.274228
\(560\) −1.40875e12 −0.605322
\(561\) 4.73452e12 2.01810
\(562\) −3.62637e10 −0.0153341
\(563\) −2.10121e12 −0.881418 −0.440709 0.897650i \(-0.645273\pi\)
−0.440709 + 0.897650i \(0.645273\pi\)
\(564\) −1.11804e12 −0.465265
\(565\) −2.79180e12 −1.15257
\(566\) −8.76899e10 −0.0359150
\(567\) −9.48529e11 −0.385413
\(568\) 1.28636e11 0.0518556
\(569\) 3.41528e11 0.136591 0.0682953 0.997665i \(-0.478244\pi\)
0.0682953 + 0.997665i \(0.478244\pi\)
\(570\) −2.27886e11 −0.0904232
\(571\) −1.41778e12 −0.558145 −0.279073 0.960270i \(-0.590027\pi\)
−0.279073 + 0.960270i \(0.590027\pi\)
\(572\) −1.07101e12 −0.418324
\(573\) −2.61404e12 −1.01302
\(574\) −7.06032e9 −0.00271469
\(575\) −5.67949e12 −2.16672
\(576\) −5.28247e10 −0.0199957
\(577\) 3.62201e12 1.36037 0.680186 0.733039i \(-0.261899\pi\)
0.680186 + 0.733039i \(0.261899\pi\)
\(578\) −6.27001e10 −0.0233664
\(579\) 9.29795e11 0.343822
\(580\) −1.60841e12 −0.590160
\(581\) −5.46233e11 −0.198877
\(582\) 8.67307e9 0.00313343
\(583\) 7.08132e12 2.53867
\(584\) 2.16851e11 0.0771443
\(585\) −2.53796e10 −0.00895950
\(586\) −4.26805e10 −0.0149517
\(587\) −1.31047e12 −0.455570 −0.227785 0.973711i \(-0.573148\pi\)
−0.227785 + 0.973711i \(0.573148\pi\)
\(588\) 4.17834e11 0.144147
\(589\) −7.38014e12 −2.52665
\(590\) −1.55765e11 −0.0529221
\(591\) −2.71317e11 −0.0914815
\(592\) 3.21546e12 1.07596
\(593\) 5.56196e11 0.184706 0.0923531 0.995726i \(-0.470561\pi\)
0.0923531 + 0.995726i \(0.470561\pi\)
\(594\) −1.40949e11 −0.0464540
\(595\) 2.45634e12 0.803456
\(596\) 2.22157e12 0.721193
\(597\) 5.79298e12 1.86646
\(598\) 3.69823e10 0.0118260
\(599\) −4.06127e12 −1.28897 −0.644483 0.764619i \(-0.722927\pi\)
−0.644483 + 0.764619i \(0.722927\pi\)
\(600\) −3.14815e11 −0.0991688
\(601\) 2.27689e12 0.711880 0.355940 0.934509i \(-0.384161\pi\)
0.355940 + 0.934509i \(0.384161\pi\)
\(602\) −3.74401e10 −0.0116186
\(603\) −7.11727e10 −0.0219223
\(604\) −1.35688e12 −0.414834
\(605\) −6.77143e12 −2.05486
\(606\) −1.71077e11 −0.0515306
\(607\) 2.87860e12 0.860661 0.430331 0.902671i \(-0.358397\pi\)
0.430331 + 0.902671i \(0.358397\pi\)
\(608\) −5.62713e11 −0.167002
\(609\) 4.76591e11 0.140400
\(610\) 1.89725e11 0.0554805
\(611\) −4.40565e11 −0.127887
\(612\) 9.22860e10 0.0265922
\(613\) 5.36912e12 1.53579 0.767893 0.640578i \(-0.221305\pi\)
0.767893 + 0.640578i \(0.221305\pi\)
\(614\) 5.09044e10 0.0144543
\(615\) 1.32956e12 0.374776
\(616\) 1.26738e11 0.0354645
\(617\) 2.82930e12 0.785951 0.392976 0.919549i \(-0.371446\pi\)
0.392976 + 0.919549i \(0.371446\pi\)
\(618\) −8.97521e10 −0.0247512
\(619\) 4.26897e12 1.16873 0.584366 0.811490i \(-0.301343\pi\)
0.584366 + 0.811490i \(0.301343\pi\)
\(620\) −8.32038e12 −2.26142
\(621\) −5.03042e12 −1.35735
\(622\) 8.30926e10 0.0222590
\(623\) −1.75321e12 −0.466271
\(624\) −1.05785e12 −0.279313
\(625\) −3.20252e11 −0.0839521
\(626\) −1.73587e11 −0.0451785
\(627\) −1.05797e13 −2.73382
\(628\) 2.85555e12 0.732607
\(629\) −5.60658e12 −1.42814
\(630\) 1.50092e9 0.000379599 0
\(631\) −4.12711e12 −1.03637 −0.518183 0.855269i \(-0.673392\pi\)
−0.518183 + 0.855269i \(0.673392\pi\)
\(632\) −2.26245e11 −0.0564094
\(633\) −2.84737e11 −0.0704901
\(634\) −1.51069e11 −0.0371342
\(635\) −3.16245e12 −0.771865
\(636\) 7.00103e12 1.69670
\(637\) 1.64648e11 0.0396214
\(638\) 7.22453e10 0.0172630
\(639\) 7.07246e10 0.0167809
\(640\) −8.45735e11 −0.199262
\(641\) −1.60585e12 −0.375702 −0.187851 0.982197i \(-0.560152\pi\)
−0.187851 + 0.982197i \(0.560152\pi\)
\(642\) 2.08422e11 0.0484212
\(643\) −4.64927e12 −1.07259 −0.536297 0.844030i \(-0.680177\pi\)
−0.536297 + 0.844030i \(0.680177\pi\)
\(644\) 2.26052e12 0.517872
\(645\) 7.05053e12 1.60399
\(646\) 3.26528e11 0.0737691
\(647\) −2.76021e12 −0.619260 −0.309630 0.950857i \(-0.600205\pi\)
−0.309630 + 0.950857i \(0.600205\pi\)
\(648\) −2.84447e11 −0.0633745
\(649\) −7.23149e12 −1.60003
\(650\) −6.19968e10 −0.0136226
\(651\) 2.46544e12 0.537996
\(652\) 6.66836e12 1.44512
\(653\) −1.95541e12 −0.420852 −0.210426 0.977610i \(-0.567485\pi\)
−0.210426 + 0.977610i \(0.567485\pi\)
\(654\) −3.09790e10 −0.00662167
\(655\) −2.00892e12 −0.426458
\(656\) 1.09259e12 0.230351
\(657\) 1.19225e11 0.0249646
\(658\) 2.60545e10 0.00541834
\(659\) −4.06055e12 −0.838688 −0.419344 0.907827i \(-0.637740\pi\)
−0.419344 + 0.907827i \(0.637740\pi\)
\(660\) −1.19276e13 −2.44683
\(661\) 7.81284e12 1.59185 0.795926 0.605394i \(-0.206985\pi\)
0.795926 + 0.605394i \(0.206985\pi\)
\(662\) −1.44257e11 −0.0291929
\(663\) 1.84450e12 0.370738
\(664\) −1.63806e11 −0.0327019
\(665\) −5.48891e12 −1.08840
\(666\) −3.42585e9 −0.000674736 0
\(667\) 2.57841e12 0.504411
\(668\) −4.35554e12 −0.846345
\(669\) 3.78761e12 0.731051
\(670\) −2.83908e11 −0.0544303
\(671\) 8.80808e12 1.67737
\(672\) 1.87982e11 0.0355594
\(673\) 2.99372e11 0.0562526 0.0281263 0.999604i \(-0.491046\pi\)
0.0281263 + 0.999604i \(0.491046\pi\)
\(674\) −1.26893e11 −0.0236847
\(675\) 8.43295e12 1.56355
\(676\) −4.17250e11 −0.0768487
\(677\) 1.54230e12 0.282175 0.141088 0.989997i \(-0.454940\pi\)
0.141088 + 0.989997i \(0.454940\pi\)
\(678\) 1.23979e11 0.0225327
\(679\) 2.08902e11 0.0377162
\(680\) 7.36613e11 0.132114
\(681\) 5.26364e12 0.937831
\(682\) 3.73729e11 0.0661496
\(683\) 4.69286e12 0.825171 0.412586 0.910919i \(-0.364626\pi\)
0.412586 + 0.910919i \(0.364626\pi\)
\(684\) −2.06221e11 −0.0360231
\(685\) −2.71818e12 −0.471705
\(686\) −9.73711e9 −0.00167869
\(687\) −2.26526e12 −0.387983
\(688\) 5.79389e12 0.985875
\(689\) 2.75877e12 0.466369
\(690\) 4.11861e11 0.0691720
\(691\) −7.96390e12 −1.32885 −0.664423 0.747357i \(-0.731322\pi\)
−0.664423 + 0.747357i \(0.731322\pi\)
\(692\) 3.96199e12 0.656804
\(693\) 6.96810e10 0.0114766
\(694\) 1.88329e11 0.0308177
\(695\) −1.81458e13 −2.95015
\(696\) 1.42922e11 0.0230864
\(697\) −1.90508e12 −0.305750
\(698\) 4.00305e10 0.00638324
\(699\) 1.16352e13 1.84343
\(700\) −3.78952e12 −0.596545
\(701\) −2.66464e12 −0.416780 −0.208390 0.978046i \(-0.566822\pi\)
−0.208390 + 0.978046i \(0.566822\pi\)
\(702\) −5.49116e10 −0.00853389
\(703\) 1.25284e13 1.93463
\(704\) −9.78267e12 −1.50100
\(705\) −4.90645e12 −0.748026
\(706\) 8.41127e10 0.0127421
\(707\) −4.12061e12 −0.620260
\(708\) −7.14951e12 −1.06937
\(709\) 1.01201e13 1.50410 0.752052 0.659103i \(-0.229064\pi\)
0.752052 + 0.659103i \(0.229064\pi\)
\(710\) 2.82120e11 0.0416650
\(711\) −1.24390e11 −0.0182546
\(712\) −5.25759e11 −0.0766702
\(713\) 1.33382e13 1.93284
\(714\) −1.09081e11 −0.0157075
\(715\) −4.70009e12 −0.672556
\(716\) 5.74296e12 0.816634
\(717\) 6.43897e12 0.909871
\(718\) 2.71637e11 0.0381442
\(719\) −2.63749e12 −0.368054 −0.184027 0.982921i \(-0.558913\pi\)
−0.184027 + 0.982921i \(0.558913\pi\)
\(720\) −2.32269e11 −0.0322103
\(721\) −2.16179e12 −0.297924
\(722\) −5.02651e11 −0.0688414
\(723\) 3.47775e12 0.473343
\(724\) 3.31352e12 0.448194
\(725\) −4.32242e12 −0.581040
\(726\) 3.00706e11 0.0401724
\(727\) 2.50725e12 0.332884 0.166442 0.986051i \(-0.446772\pi\)
0.166442 + 0.986051i \(0.446772\pi\)
\(728\) 4.93752e10 0.00651505
\(729\) 7.46612e12 0.979087
\(730\) 4.75590e11 0.0619840
\(731\) −1.01024e13 −1.30857
\(732\) 8.70822e12 1.12106
\(733\) 6.24013e12 0.798410 0.399205 0.916862i \(-0.369286\pi\)
0.399205 + 0.916862i \(0.369286\pi\)
\(734\) −4.76203e10 −0.00605564
\(735\) 1.83364e12 0.231751
\(736\) 1.01700e12 0.127753
\(737\) −1.31806e13 −1.64562
\(738\) −1.16408e9 −0.000144454 0
\(739\) −5.21085e12 −0.642700 −0.321350 0.946961i \(-0.604137\pi\)
−0.321350 + 0.946961i \(0.604137\pi\)
\(740\) 1.41245e13 1.73154
\(741\) −4.12169e12 −0.502220
\(742\) −1.63150e11 −0.0197593
\(743\) −8.14479e12 −0.980461 −0.490231 0.871593i \(-0.663087\pi\)
−0.490231 + 0.871593i \(0.663087\pi\)
\(744\) 7.39342e11 0.0884641
\(745\) 9.74925e12 1.15949
\(746\) −1.51382e11 −0.0178957
\(747\) −9.00609e10 −0.0105826
\(748\) 1.70906e13 1.99618
\(749\) 5.02010e12 0.582833
\(750\) −2.53408e11 −0.0292445
\(751\) 1.14917e13 1.31827 0.659133 0.752026i \(-0.270923\pi\)
0.659133 + 0.752026i \(0.270923\pi\)
\(752\) −4.03196e12 −0.459765
\(753\) −2.32587e12 −0.263637
\(754\) 2.81457e10 0.00317132
\(755\) −5.95459e12 −0.666946
\(756\) −3.35644e12 −0.373707
\(757\) −1.29516e13 −1.43349 −0.716743 0.697338i \(-0.754368\pi\)
−0.716743 + 0.697338i \(0.754368\pi\)
\(758\) −2.73986e11 −0.0301451
\(759\) 1.91209e13 2.09132
\(760\) −1.64603e12 −0.178968
\(761\) 5.53759e12 0.598536 0.299268 0.954169i \(-0.403258\pi\)
0.299268 + 0.954169i \(0.403258\pi\)
\(762\) 1.40438e11 0.0150900
\(763\) −7.46167e11 −0.0797032
\(764\) −9.43609e12 −1.00201
\(765\) 4.04992e11 0.0427534
\(766\) 4.91875e11 0.0516209
\(767\) −2.81728e12 −0.293935
\(768\) −9.64356e12 −1.00026
\(769\) −6.46078e12 −0.666218 −0.333109 0.942888i \(-0.608098\pi\)
−0.333109 + 0.942888i \(0.608098\pi\)
\(770\) 2.77957e11 0.0284951
\(771\) −8.09382e11 −0.0824914
\(772\) 3.35635e12 0.340087
\(773\) −1.01691e13 −1.02441 −0.512207 0.858862i \(-0.671172\pi\)
−0.512207 + 0.858862i \(0.671172\pi\)
\(774\) −6.17299e9 −0.000618245 0
\(775\) −2.23601e13 −2.22647
\(776\) 6.26460e10 0.00620177
\(777\) −4.18529e12 −0.411937
\(778\) 2.51710e11 0.0246316
\(779\) 4.25707e12 0.414183
\(780\) −4.64680e12 −0.449499
\(781\) 1.30976e13 1.25968
\(782\) −5.90140e11 −0.0564319
\(783\) −3.82844e12 −0.363994
\(784\) 1.50683e12 0.142443
\(785\) 1.25314e13 1.17784
\(786\) 8.92123e10 0.00833725
\(787\) −1.96391e13 −1.82488 −0.912441 0.409208i \(-0.865805\pi\)
−0.912441 + 0.409208i \(0.865805\pi\)
\(788\) −9.79394e11 −0.0904877
\(789\) −1.24855e13 −1.14699
\(790\) −4.96191e11 −0.0453239
\(791\) 2.98618e12 0.271220
\(792\) 2.08961e10 0.00188713
\(793\) 3.43149e12 0.308144
\(794\) 2.00765e11 0.0179265
\(795\) 3.07237e13 2.72785
\(796\) 2.09114e13 1.84618
\(797\) −4.90622e12 −0.430709 −0.215355 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(798\) 2.43752e11 0.0212782
\(799\) 7.03026e12 0.610255
\(800\) −1.70489e12 −0.147161
\(801\) −2.89064e11 −0.0248112
\(802\) 1.60963e11 0.0137386
\(803\) 2.20795e13 1.87400
\(804\) −1.30311e13 −1.09984
\(805\) 9.92019e12 0.832604
\(806\) 1.45599e11 0.0121521
\(807\) 2.88210e12 0.239209
\(808\) −1.23570e12 −0.101991
\(809\) −1.27268e13 −1.04460 −0.522301 0.852761i \(-0.674926\pi\)
−0.522301 + 0.852761i \(0.674926\pi\)
\(810\) −6.23840e11 −0.0509202
\(811\) −5.10271e11 −0.0414197 −0.0207098 0.999786i \(-0.506593\pi\)
−0.0207098 + 0.999786i \(0.506593\pi\)
\(812\) 1.72039e12 0.138875
\(813\) 2.10956e13 1.69350
\(814\) −6.34437e11 −0.0506499
\(815\) 2.92638e13 2.32338
\(816\) 1.68804e13 1.33284
\(817\) 2.25748e13 1.77265
\(818\) 7.00263e11 0.0546854
\(819\) 2.71467e10 0.00210833
\(820\) 4.79943e12 0.370704
\(821\) −8.85712e12 −0.680375 −0.340187 0.940358i \(-0.610491\pi\)
−0.340187 + 0.940358i \(0.610491\pi\)
\(822\) 1.20709e11 0.00922183
\(823\) 1.45720e13 1.10718 0.553592 0.832788i \(-0.313257\pi\)
0.553592 + 0.832788i \(0.313257\pi\)
\(824\) −6.48284e11 −0.0489883
\(825\) −3.20541e13 −2.40902
\(826\) 1.66610e11 0.0124535
\(827\) −3.86291e12 −0.287170 −0.143585 0.989638i \(-0.545863\pi\)
−0.143585 + 0.989638i \(0.545863\pi\)
\(828\) 3.72707e11 0.0275569
\(829\) 8.59011e12 0.631689 0.315845 0.948811i \(-0.397712\pi\)
0.315845 + 0.948811i \(0.397712\pi\)
\(830\) −3.59253e11 −0.0262754
\(831\) 2.19293e13 1.59522
\(832\) −3.81118e12 −0.275743
\(833\) −2.62736e12 −0.189067
\(834\) 8.05819e11 0.0576754
\(835\) −1.91141e13 −1.36070
\(836\) −3.81904e13 −2.70412
\(837\) −1.98047e13 −1.39478
\(838\) 5.70575e11 0.0399682
\(839\) −1.34013e13 −0.933720 −0.466860 0.884331i \(-0.654615\pi\)
−0.466860 + 0.884331i \(0.654615\pi\)
\(840\) 5.49878e11 0.0381074
\(841\) −1.25448e13 −0.864735
\(842\) 4.65135e9 0.000318914 0
\(843\) −7.30447e12 −0.498155
\(844\) −1.02784e12 −0.0697243
\(845\) −1.83108e12 −0.123553
\(846\) 4.29577e9 0.000288320 0
\(847\) 7.24288e12 0.483544
\(848\) 2.52477e13 1.67664
\(849\) −1.76630e13 −1.16676
\(850\) 9.89306e11 0.0650048
\(851\) −2.26428e13 −1.47995
\(852\) 1.29491e13 0.841901
\(853\) −2.74955e13 −1.77824 −0.889120 0.457675i \(-0.848682\pi\)
−0.889120 + 0.457675i \(0.848682\pi\)
\(854\) −2.02934e11 −0.0130555
\(855\) −9.04992e11 −0.0579158
\(856\) 1.50544e12 0.0958367
\(857\) 2.16233e13 1.36933 0.684664 0.728859i \(-0.259949\pi\)
0.684664 + 0.728859i \(0.259949\pi\)
\(858\) 2.08722e11 0.0131485
\(859\) −4.48311e12 −0.280938 −0.140469 0.990085i \(-0.544861\pi\)
−0.140469 + 0.990085i \(0.544861\pi\)
\(860\) 2.54508e13 1.58657
\(861\) −1.42213e12 −0.0881914
\(862\) −5.90050e10 −0.00364004
\(863\) 8.55165e11 0.0524809 0.0262405 0.999656i \(-0.491646\pi\)
0.0262405 + 0.999656i \(0.491646\pi\)
\(864\) −1.51005e12 −0.0921892
\(865\) 1.73870e13 1.05597
\(866\) 3.47205e10 0.00209776
\(867\) −1.26294e13 −0.759098
\(868\) 8.89968e12 0.532152
\(869\) −2.30359e13 −1.37030
\(870\) 3.13450e11 0.0185495
\(871\) −5.13494e12 −0.302311
\(872\) −2.23763e11 −0.0131058
\(873\) 3.44430e10 0.00200695
\(874\) 1.31872e12 0.0764454
\(875\) −6.10364e12 −0.352008
\(876\) 2.18292e13 1.25247
\(877\) −1.15519e12 −0.0659409 −0.0329705 0.999456i \(-0.510497\pi\)
−0.0329705 + 0.999456i \(0.510497\pi\)
\(878\) 7.22476e11 0.0410297
\(879\) −8.59696e12 −0.485730
\(880\) −4.30142e13 −2.41791
\(881\) −1.67352e13 −0.935922 −0.467961 0.883749i \(-0.655011\pi\)
−0.467961 + 0.883749i \(0.655011\pi\)
\(882\) −1.60542e9 −8.93265e−5 0
\(883\) −2.70703e13 −1.49855 −0.749273 0.662262i \(-0.769597\pi\)
−0.749273 + 0.662262i \(0.769597\pi\)
\(884\) 6.65822e12 0.366710
\(885\) −3.13753e13 −1.71926
\(886\) −5.53338e11 −0.0301675
\(887\) 1.08621e13 0.589192 0.294596 0.955622i \(-0.404815\pi\)
0.294596 + 0.955622i \(0.404815\pi\)
\(888\) −1.25509e12 −0.0677358
\(889\) 3.38263e12 0.181634
\(890\) −1.15308e12 −0.0616031
\(891\) −2.89621e13 −1.53950
\(892\) 1.36724e13 0.723109
\(893\) −1.57098e13 −0.826681
\(894\) −4.32945e11 −0.0226681
\(895\) 2.52027e13 1.31294
\(896\) 9.04619e11 0.0468899
\(897\) 7.44920e12 0.384188
\(898\) 5.60699e11 0.0287731
\(899\) 1.01512e13 0.518320
\(900\) −6.24803e11 −0.0317433
\(901\) −4.40228e13 −2.22544
\(902\) −2.15578e11 −0.0108436
\(903\) −7.54141e12 −0.377449
\(904\) 8.95503e11 0.0445973
\(905\) 1.45412e13 0.720580
\(906\) 2.64432e11 0.0130388
\(907\) −1.93922e13 −0.951468 −0.475734 0.879589i \(-0.657817\pi\)
−0.475734 + 0.879589i \(0.657817\pi\)
\(908\) 1.90006e13 0.927642
\(909\) −6.79391e11 −0.0330052
\(910\) 1.08288e11 0.00523473
\(911\) −3.28650e13 −1.58089 −0.790445 0.612533i \(-0.790151\pi\)
−0.790445 + 0.612533i \(0.790151\pi\)
\(912\) −3.77208e13 −1.80553
\(913\) −1.66785e13 −0.794398
\(914\) 1.92191e11 0.00910910
\(915\) 3.82156e13 1.80238
\(916\) −8.17708e12 −0.383768
\(917\) 2.14879e12 0.100353
\(918\) 8.76245e11 0.0407224
\(919\) −2.15486e13 −0.996552 −0.498276 0.867019i \(-0.666033\pi\)
−0.498276 + 0.867019i \(0.666033\pi\)
\(920\) 2.97489e12 0.136907
\(921\) 1.02535e13 0.469573
\(922\) −9.54174e11 −0.0434849
\(923\) 5.10262e12 0.231412
\(924\) 1.27580e13 0.575784
\(925\) 3.79582e13 1.70478
\(926\) −1.08291e12 −0.0483996
\(927\) −3.56429e11 −0.0158531
\(928\) 7.73997e11 0.0342589
\(929\) −1.01040e13 −0.445066 −0.222533 0.974925i \(-0.571433\pi\)
−0.222533 + 0.974925i \(0.571433\pi\)
\(930\) 1.62150e12 0.0710793
\(931\) 5.87106e12 0.256120
\(932\) 4.20004e13 1.82340
\(933\) 1.67370e13 0.723121
\(934\) −1.14223e12 −0.0491125
\(935\) 7.50011e13 3.20934
\(936\) 8.14081e9 0.000346678 0
\(937\) 2.70666e13 1.14711 0.573555 0.819167i \(-0.305564\pi\)
0.573555 + 0.819167i \(0.305564\pi\)
\(938\) 3.03674e11 0.0128084
\(939\) −3.49649e13 −1.46770
\(940\) −1.77112e13 −0.739899
\(941\) −5.95239e12 −0.247479 −0.123739 0.992315i \(-0.539489\pi\)
−0.123739 + 0.992315i \(0.539489\pi\)
\(942\) −5.56497e11 −0.0230268
\(943\) −7.69388e12 −0.316842
\(944\) −2.57831e13 −1.05672
\(945\) −1.47296e13 −0.600824
\(946\) −1.14318e12 −0.0464094
\(947\) −6.70567e12 −0.270936 −0.135468 0.990782i \(-0.543254\pi\)
−0.135468 + 0.990782i \(0.543254\pi\)
\(948\) −2.27748e13 −0.915834
\(949\) 8.60184e12 0.344265
\(950\) −2.21069e12 −0.0880587
\(951\) −3.04293e13 −1.20637
\(952\) −7.87899e11 −0.0310888
\(953\) −4.18180e13 −1.64227 −0.821136 0.570732i \(-0.806659\pi\)
−0.821136 + 0.570732i \(0.806659\pi\)
\(954\) −2.68997e10 −0.00105143
\(955\) −4.14098e13 −1.61097
\(956\) 2.32432e13 0.899987
\(957\) 1.45521e13 0.560818
\(958\) 5.20348e11 0.0199595
\(959\) 2.90743e12 0.111001
\(960\) −4.24440e13 −1.61286
\(961\) 2.60731e13 0.986136
\(962\) −2.47167e11 −0.00930470
\(963\) 8.27696e11 0.0310136
\(964\) 1.25539e13 0.468200
\(965\) 1.47292e13 0.546771
\(966\) −4.40537e11 −0.0162774
\(967\) 1.81685e13 0.668191 0.334095 0.942539i \(-0.391569\pi\)
0.334095 + 0.942539i \(0.391569\pi\)
\(968\) 2.17202e12 0.0795104
\(969\) 6.57714e13 2.39651
\(970\) 1.37393e11 0.00498301
\(971\) −8.57355e11 −0.0309510 −0.0154755 0.999880i \(-0.504926\pi\)
−0.0154755 + 0.999880i \(0.504926\pi\)
\(972\) −1.11816e12 −0.0401795
\(973\) 1.94091e13 0.694223
\(974\) −3.61664e11 −0.0128763
\(975\) −1.24878e13 −0.442552
\(976\) 3.14043e13 1.10781
\(977\) 2.01702e13 0.708247 0.354124 0.935199i \(-0.384779\pi\)
0.354124 + 0.935199i \(0.384779\pi\)
\(978\) −1.29955e12 −0.0454221
\(979\) −5.35321e13 −1.86248
\(980\) 6.61905e12 0.229234
\(981\) −1.23025e11 −0.00424116
\(982\) −1.50775e12 −0.0517400
\(983\) 1.17099e13 0.400001 0.200001 0.979796i \(-0.435906\pi\)
0.200001 + 0.979796i \(0.435906\pi\)
\(984\) −4.26473e11 −0.0145015
\(985\) −4.29802e12 −0.145481
\(986\) −4.49131e11 −0.0151331
\(987\) 5.24806e12 0.176024
\(988\) −1.48784e13 −0.496763
\(989\) −4.07998e13 −1.35605
\(990\) 4.58286e10 0.00151628
\(991\) 1.71996e13 0.566485 0.283242 0.959048i \(-0.408590\pi\)
0.283242 + 0.959048i \(0.408590\pi\)
\(992\) 4.00393e12 0.131276
\(993\) −2.90572e13 −0.948379
\(994\) −3.01762e11 −0.00980452
\(995\) 9.17686e13 2.96818
\(996\) −1.64894e13 −0.530931
\(997\) −2.94981e13 −0.945508 −0.472754 0.881194i \(-0.656740\pi\)
−0.472754 + 0.881194i \(0.656740\pi\)
\(998\) −7.61607e11 −0.0243021
\(999\) 3.36202e13 1.06796
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.1.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.7 12 1.1 even 1 trivial