Properties

Label 43.8.a.b.1.11
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(16.3440\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+17.3440 q^{2} +73.8355 q^{3} +172.814 q^{4} -122.945 q^{5} +1280.60 q^{6} +247.011 q^{7} +777.246 q^{8} +3264.69 q^{9} +O(q^{10})\) \(q+17.3440 q^{2} +73.8355 q^{3} +172.814 q^{4} -122.945 q^{5} +1280.60 q^{6} +247.011 q^{7} +777.246 q^{8} +3264.69 q^{9} -2132.35 q^{10} +2503.52 q^{11} +12759.8 q^{12} +1376.69 q^{13} +4284.15 q^{14} -9077.68 q^{15} -8639.60 q^{16} -32082.0 q^{17} +56622.6 q^{18} +10245.1 q^{19} -21246.5 q^{20} +18238.2 q^{21} +43421.0 q^{22} -21850.6 q^{23} +57388.4 q^{24} -63009.6 q^{25} +23877.3 q^{26} +79571.5 q^{27} +42686.8 q^{28} +173813. q^{29} -157443. q^{30} +16044.0 q^{31} -249333. q^{32} +184849. q^{33} -556429. q^{34} -30368.6 q^{35} +564182. q^{36} -75677.9 q^{37} +177691. q^{38} +101649. q^{39} -95558.2 q^{40} +452656. q^{41} +316323. q^{42} -79507.0 q^{43} +432643. q^{44} -401375. q^{45} -378977. q^{46} +113517. q^{47} -637910. q^{48} -762529. q^{49} -1.09284e6 q^{50} -2.36879e6 q^{51} +237911. q^{52} -1.42412e6 q^{53} +1.38009e6 q^{54} -307794. q^{55} +191988. q^{56} +756453. q^{57} +3.01460e6 q^{58} +1.48283e6 q^{59} -1.56875e6 q^{60} -2.69078e6 q^{61} +278267. q^{62} +806413. q^{63} -3.21855e6 q^{64} -169257. q^{65} +3.20601e6 q^{66} +2.19036e6 q^{67} -5.54420e6 q^{68} -1.61335e6 q^{69} -526713. q^{70} +4.77162e6 q^{71} +2.53746e6 q^{72} +4.98637e6 q^{73} -1.31256e6 q^{74} -4.65235e6 q^{75} +1.77049e6 q^{76} +618397. q^{77} +1.76299e6 q^{78} +3.39256e6 q^{79} +1.06219e6 q^{80} -1.26466e6 q^{81} +7.85086e6 q^{82} +8.35875e6 q^{83} +3.15180e6 q^{84} +3.94431e6 q^{85} -1.37897e6 q^{86} +1.28336e7 q^{87} +1.94585e6 q^{88} -7.83169e6 q^{89} -6.96145e6 q^{90} +340058. q^{91} -3.77608e6 q^{92} +1.18462e6 q^{93} +1.96883e6 q^{94} -1.25958e6 q^{95} -1.84096e7 q^{96} +6.12777e6 q^{97} -1.32253e7 q^{98} +8.17321e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17.3440 1.53301 0.766503 0.642241i \(-0.221995\pi\)
0.766503 + 0.642241i \(0.221995\pi\)
\(3\) 73.8355 1.57885 0.789425 0.613847i \(-0.210379\pi\)
0.789425 + 0.613847i \(0.210379\pi\)
\(4\) 172.814 1.35011
\(5\) −122.945 −0.439860 −0.219930 0.975516i \(-0.570583\pi\)
−0.219930 + 0.975516i \(0.570583\pi\)
\(6\) 1280.60 2.42039
\(7\) 247.011 0.272190 0.136095 0.990696i \(-0.456545\pi\)
0.136095 + 0.990696i \(0.456545\pi\)
\(8\) 777.246 0.536715
\(9\) 3264.69 1.49277
\(10\) −2132.35 −0.674308
\(11\) 2503.52 0.567123 0.283561 0.958954i \(-0.408484\pi\)
0.283561 + 0.958954i \(0.408484\pi\)
\(12\) 12759.8 2.13162
\(13\) 1376.69 0.173794 0.0868970 0.996217i \(-0.472305\pi\)
0.0868970 + 0.996217i \(0.472305\pi\)
\(14\) 4284.15 0.417269
\(15\) −9077.68 −0.694473
\(16\) −8639.60 −0.527320
\(17\) −32082.0 −1.58376 −0.791881 0.610675i \(-0.790898\pi\)
−0.791881 + 0.610675i \(0.790898\pi\)
\(18\) 56622.6 2.28842
\(19\) 10245.1 0.342672 0.171336 0.985213i \(-0.445192\pi\)
0.171336 + 0.985213i \(0.445192\pi\)
\(20\) −21246.5 −0.593858
\(21\) 18238.2 0.429748
\(22\) 43421.0 0.869402
\(23\) −21850.6 −0.374469 −0.187235 0.982315i \(-0.559952\pi\)
−0.187235 + 0.982315i \(0.559952\pi\)
\(24\) 57388.4 0.847392
\(25\) −63009.6 −0.806523
\(26\) 23877.3 0.266427
\(27\) 79571.5 0.778009
\(28\) 42686.8 0.367486
\(29\) 173813. 1.32339 0.661696 0.749772i \(-0.269837\pi\)
0.661696 + 0.749772i \(0.269837\pi\)
\(30\) −157443. −1.06463
\(31\) 16044.0 0.0967270 0.0483635 0.998830i \(-0.484599\pi\)
0.0483635 + 0.998830i \(0.484599\pi\)
\(32\) −249333. −1.34510
\(33\) 184849. 0.895402
\(34\) −556429. −2.42792
\(35\) −30368.6 −0.119726
\(36\) 564182. 2.01540
\(37\) −75677.9 −0.245620 −0.122810 0.992430i \(-0.539191\pi\)
−0.122810 + 0.992430i \(0.539191\pi\)
\(38\) 177691. 0.525318
\(39\) 101649. 0.274395
\(40\) −95558.2 −0.236079
\(41\) 452656. 1.02571 0.512855 0.858475i \(-0.328588\pi\)
0.512855 + 0.858475i \(0.328588\pi\)
\(42\) 316323. 0.658806
\(43\) −79507.0 −0.152499
\(44\) 432643. 0.765676
\(45\) −401375. −0.656609
\(46\) −378977. −0.574063
\(47\) 113517. 0.159484 0.0797420 0.996816i \(-0.474590\pi\)
0.0797420 + 0.996816i \(0.474590\pi\)
\(48\) −637910. −0.832559
\(49\) −762529. −0.925912
\(50\) −1.09284e6 −1.23640
\(51\) −2.36879e6 −2.50052
\(52\) 237911. 0.234640
\(53\) −1.42412e6 −1.31395 −0.656976 0.753911i \(-0.728165\pi\)
−0.656976 + 0.753911i \(0.728165\pi\)
\(54\) 1.38009e6 1.19269
\(55\) −307794. −0.249455
\(56\) 191988. 0.146089
\(57\) 756453. 0.541028
\(58\) 3.01460e6 2.02877
\(59\) 1.48283e6 0.939960 0.469980 0.882677i \(-0.344261\pi\)
0.469980 + 0.882677i \(0.344261\pi\)
\(60\) −1.56875e6 −0.937612
\(61\) −2.69078e6 −1.51784 −0.758918 0.651187i \(-0.774271\pi\)
−0.758918 + 0.651187i \(0.774271\pi\)
\(62\) 278267. 0.148283
\(63\) 806413. 0.406318
\(64\) −3.21855e6 −1.53472
\(65\) −169257. −0.0764450
\(66\) 3.20601e6 1.37266
\(67\) 2.19036e6 0.889722 0.444861 0.895600i \(-0.353253\pi\)
0.444861 + 0.895600i \(0.353253\pi\)
\(68\) −5.54420e6 −2.13825
\(69\) −1.61335e6 −0.591231
\(70\) −526713. −0.183540
\(71\) 4.77162e6 1.58220 0.791101 0.611685i \(-0.209508\pi\)
0.791101 + 0.611685i \(0.209508\pi\)
\(72\) 2.53746e6 0.801191
\(73\) 4.98637e6 1.50022 0.750109 0.661314i \(-0.230001\pi\)
0.750109 + 0.661314i \(0.230001\pi\)
\(74\) −1.31256e6 −0.376536
\(75\) −4.65235e6 −1.27338
\(76\) 1.77049e6 0.462644
\(77\) 618397. 0.154365
\(78\) 1.76299e6 0.420649
\(79\) 3.39256e6 0.774163 0.387081 0.922046i \(-0.373483\pi\)
0.387081 + 0.922046i \(0.373483\pi\)
\(80\) 1.06219e6 0.231947
\(81\) −1.26466e6 −0.264409
\(82\) 7.85086e6 1.57242
\(83\) 8.35875e6 1.60460 0.802302 0.596918i \(-0.203608\pi\)
0.802302 + 0.596918i \(0.203608\pi\)
\(84\) 3.15180e6 0.580205
\(85\) 3.94431e6 0.696634
\(86\) −1.37897e6 −0.233781
\(87\) 1.28336e7 2.08944
\(88\) 1.94585e6 0.304383
\(89\) −7.83169e6 −1.17758 −0.588790 0.808286i \(-0.700396\pi\)
−0.588790 + 0.808286i \(0.700396\pi\)
\(90\) −6.96145e6 −1.00659
\(91\) 340058. 0.0473051
\(92\) −3.77608e6 −0.505573
\(93\) 1.18462e6 0.152717
\(94\) 1.96883e6 0.244490
\(95\) −1.25958e6 −0.150728
\(96\) −1.84096e7 −2.12371
\(97\) 6.12777e6 0.681712 0.340856 0.940115i \(-0.389283\pi\)
0.340856 + 0.940115i \(0.389283\pi\)
\(98\) −1.32253e7 −1.41943
\(99\) 8.17321e6 0.846583
\(100\) −1.08889e7 −1.08889
\(101\) −1.48954e7 −1.43856 −0.719279 0.694721i \(-0.755528\pi\)
−0.719279 + 0.694721i \(0.755528\pi\)
\(102\) −4.10843e7 −3.83332
\(103\) 8.96048e6 0.807981 0.403990 0.914763i \(-0.367623\pi\)
0.403990 + 0.914763i \(0.367623\pi\)
\(104\) 1.07003e6 0.0932778
\(105\) −2.24229e6 −0.189029
\(106\) −2.46998e7 −2.01430
\(107\) −2.20290e6 −0.173841 −0.0869204 0.996215i \(-0.527703\pi\)
−0.0869204 + 0.996215i \(0.527703\pi\)
\(108\) 1.37510e7 1.05039
\(109\) 1.43255e7 1.05954 0.529770 0.848141i \(-0.322278\pi\)
0.529770 + 0.848141i \(0.322278\pi\)
\(110\) −5.33838e6 −0.382415
\(111\) −5.58772e6 −0.387797
\(112\) −2.13408e6 −0.143531
\(113\) 5.69345e6 0.371194 0.185597 0.982626i \(-0.440578\pi\)
0.185597 + 0.982626i \(0.440578\pi\)
\(114\) 1.31199e7 0.829399
\(115\) 2.68641e6 0.164714
\(116\) 3.00372e7 1.78672
\(117\) 4.49447e6 0.259434
\(118\) 2.57182e7 1.44096
\(119\) −7.92460e6 −0.431085
\(120\) −7.05559e6 −0.372734
\(121\) −1.32196e7 −0.678372
\(122\) −4.66689e7 −2.32685
\(123\) 3.34221e7 1.61944
\(124\) 2.77263e6 0.130592
\(125\) 1.73517e7 0.794617
\(126\) 1.39864e7 0.622887
\(127\) 2.89632e7 1.25468 0.627342 0.778744i \(-0.284143\pi\)
0.627342 + 0.778744i \(0.284143\pi\)
\(128\) −2.39079e7 −1.00764
\(129\) −5.87044e6 −0.240772
\(130\) −2.93559e6 −0.117191
\(131\) 1.30359e7 0.506632 0.253316 0.967384i \(-0.418479\pi\)
0.253316 + 0.967384i \(0.418479\pi\)
\(132\) 3.19444e7 1.20889
\(133\) 2.53065e6 0.0932721
\(134\) 3.79896e7 1.36395
\(135\) −9.78289e6 −0.342215
\(136\) −2.49356e7 −0.850029
\(137\) −3.16388e7 −1.05123 −0.525615 0.850723i \(-0.676165\pi\)
−0.525615 + 0.850723i \(0.676165\pi\)
\(138\) −2.79819e7 −0.906360
\(139\) −4.62889e7 −1.46193 −0.730963 0.682418i \(-0.760929\pi\)
−0.730963 + 0.682418i \(0.760929\pi\)
\(140\) −5.24811e6 −0.161642
\(141\) 8.38156e6 0.251801
\(142\) 8.27589e7 2.42552
\(143\) 3.44658e6 0.0985625
\(144\) −2.82056e7 −0.787166
\(145\) −2.13693e7 −0.582107
\(146\) 8.64835e7 2.29984
\(147\) −5.63017e7 −1.46188
\(148\) −1.30782e7 −0.331613
\(149\) 3.21433e7 0.796048 0.398024 0.917375i \(-0.369696\pi\)
0.398024 + 0.917375i \(0.369696\pi\)
\(150\) −8.06903e7 −1.95210
\(151\) 2.05927e7 0.486737 0.243369 0.969934i \(-0.421748\pi\)
0.243369 + 0.969934i \(0.421748\pi\)
\(152\) 7.96296e6 0.183917
\(153\) −1.04738e8 −2.36419
\(154\) 1.07255e7 0.236643
\(155\) −1.97253e6 −0.0425463
\(156\) 1.75663e7 0.370462
\(157\) −2.11067e7 −0.435283 −0.217641 0.976029i \(-0.569836\pi\)
−0.217641 + 0.976029i \(0.569836\pi\)
\(158\) 5.88404e7 1.18680
\(159\) −1.05150e8 −2.07453
\(160\) 3.06541e7 0.591655
\(161\) −5.39734e6 −0.101927
\(162\) −2.19343e7 −0.405341
\(163\) 983833. 0.0177936 0.00889682 0.999960i \(-0.497168\pi\)
0.00889682 + 0.999960i \(0.497168\pi\)
\(164\) 7.82251e7 1.38482
\(165\) −2.27262e7 −0.393851
\(166\) 1.44974e8 2.45987
\(167\) 9.39137e7 1.56035 0.780174 0.625563i \(-0.215131\pi\)
0.780174 + 0.625563i \(0.215131\pi\)
\(168\) 1.41755e7 0.230652
\(169\) −6.08532e7 −0.969796
\(170\) 6.84100e7 1.06794
\(171\) 3.34470e7 0.511530
\(172\) −1.37399e7 −0.205889
\(173\) 1.05361e8 1.54710 0.773551 0.633734i \(-0.218478\pi\)
0.773551 + 0.633734i \(0.218478\pi\)
\(174\) 2.22585e8 3.20312
\(175\) −1.55641e7 −0.219528
\(176\) −2.16294e7 −0.299055
\(177\) 1.09486e8 1.48406
\(178\) −1.35833e8 −1.80524
\(179\) −7.52808e7 −0.981067 −0.490533 0.871422i \(-0.663198\pi\)
−0.490533 + 0.871422i \(0.663198\pi\)
\(180\) −6.93631e7 −0.886492
\(181\) −2.35542e7 −0.295253 −0.147626 0.989043i \(-0.547163\pi\)
−0.147626 + 0.989043i \(0.547163\pi\)
\(182\) 5.89795e6 0.0725189
\(183\) −1.98676e8 −2.39643
\(184\) −1.69833e7 −0.200983
\(185\) 9.30419e6 0.108038
\(186\) 2.05460e7 0.234117
\(187\) −8.03180e7 −0.898188
\(188\) 1.96172e7 0.215320
\(189\) 1.96550e7 0.211767
\(190\) −2.18461e7 −0.231066
\(191\) 1.11671e8 1.15964 0.579820 0.814745i \(-0.303123\pi\)
0.579820 + 0.814745i \(0.303123\pi\)
\(192\) −2.37643e8 −2.42310
\(193\) −1.17636e8 −1.17785 −0.588926 0.808187i \(-0.700449\pi\)
−0.588926 + 0.808187i \(0.700449\pi\)
\(194\) 1.06280e8 1.04507
\(195\) −1.24972e7 −0.120695
\(196\) −1.31775e8 −1.25008
\(197\) −5.42883e7 −0.505912 −0.252956 0.967478i \(-0.581403\pi\)
−0.252956 + 0.967478i \(0.581403\pi\)
\(198\) 1.41756e8 1.29782
\(199\) 8.13364e7 0.731643 0.365822 0.930685i \(-0.380788\pi\)
0.365822 + 0.930685i \(0.380788\pi\)
\(200\) −4.89740e7 −0.432873
\(201\) 1.61727e8 1.40474
\(202\) −2.58346e8 −2.20532
\(203\) 4.29336e7 0.360215
\(204\) −4.09359e8 −3.37597
\(205\) −5.56516e7 −0.451169
\(206\) 1.55410e8 1.23864
\(207\) −7.13354e7 −0.558996
\(208\) −1.18941e7 −0.0916450
\(209\) 2.56488e7 0.194337
\(210\) −3.88901e7 −0.289782
\(211\) 7.38401e7 0.541132 0.270566 0.962701i \(-0.412789\pi\)
0.270566 + 0.962701i \(0.412789\pi\)
\(212\) −2.46106e8 −1.77397
\(213\) 3.52315e8 2.49806
\(214\) −3.82071e7 −0.266499
\(215\) 9.77496e6 0.0670780
\(216\) 6.18466e7 0.417569
\(217\) 3.96305e6 0.0263282
\(218\) 2.48461e8 1.62428
\(219\) 3.68171e8 2.36862
\(220\) −5.31911e7 −0.336790
\(221\) −4.41670e7 −0.275248
\(222\) −9.69133e7 −0.594495
\(223\) −1.01645e8 −0.613790 −0.306895 0.951743i \(-0.599290\pi\)
−0.306895 + 0.951743i \(0.599290\pi\)
\(224\) −6.15878e7 −0.366123
\(225\) −2.05707e8 −1.20395
\(226\) 9.87471e7 0.569043
\(227\) −1.95225e8 −1.10776 −0.553880 0.832596i \(-0.686854\pi\)
−0.553880 + 0.832596i \(0.686854\pi\)
\(228\) 1.30725e8 0.730445
\(229\) −1.38246e8 −0.760727 −0.380363 0.924837i \(-0.624201\pi\)
−0.380363 + 0.924837i \(0.624201\pi\)
\(230\) 4.65931e7 0.252507
\(231\) 4.56597e7 0.243720
\(232\) 1.35095e8 0.710284
\(233\) −7.52341e7 −0.389645 −0.194822 0.980839i \(-0.562413\pi\)
−0.194822 + 0.980839i \(0.562413\pi\)
\(234\) 7.79519e7 0.397714
\(235\) −1.39563e7 −0.0701506
\(236\) 2.56253e8 1.26905
\(237\) 2.50491e8 1.22229
\(238\) −1.37444e8 −0.660856
\(239\) −5.37584e7 −0.254715 −0.127357 0.991857i \(-0.540649\pi\)
−0.127357 + 0.991857i \(0.540649\pi\)
\(240\) 7.84276e7 0.366209
\(241\) −4.33242e8 −1.99375 −0.996876 0.0789843i \(-0.974832\pi\)
−0.996876 + 0.0789843i \(0.974832\pi\)
\(242\) −2.29280e8 −1.03995
\(243\) −2.67400e8 −1.19547
\(244\) −4.65004e8 −2.04924
\(245\) 9.37488e7 0.407272
\(246\) 5.79672e8 2.48262
\(247\) 1.41043e7 0.0595544
\(248\) 1.24702e7 0.0519148
\(249\) 6.17173e8 2.53343
\(250\) 3.00948e8 1.21815
\(251\) −3.50742e8 −1.40001 −0.700003 0.714140i \(-0.746818\pi\)
−0.700003 + 0.714140i \(0.746818\pi\)
\(252\) 1.39359e8 0.548572
\(253\) −5.47035e7 −0.212370
\(254\) 5.02338e8 1.92344
\(255\) 2.91230e8 1.09988
\(256\) −2.68346e6 −0.00999666
\(257\) 3.82436e8 1.40538 0.702689 0.711497i \(-0.251983\pi\)
0.702689 + 0.711497i \(0.251983\pi\)
\(258\) −1.01817e8 −0.369106
\(259\) −1.86933e7 −0.0668554
\(260\) −2.92499e7 −0.103209
\(261\) 5.67444e8 1.97552
\(262\) 2.26095e8 0.776670
\(263\) −4.76819e8 −1.61625 −0.808125 0.589011i \(-0.799517\pi\)
−0.808125 + 0.589011i \(0.799517\pi\)
\(264\) 1.43673e8 0.480575
\(265\) 1.75087e8 0.577955
\(266\) 4.38916e7 0.142987
\(267\) −5.78257e8 −1.85922
\(268\) 3.78524e8 1.20122
\(269\) 4.56754e8 1.43070 0.715352 0.698765i \(-0.246267\pi\)
0.715352 + 0.698765i \(0.246267\pi\)
\(270\) −1.69674e8 −0.524617
\(271\) −3.80308e8 −1.16076 −0.580381 0.814345i \(-0.697096\pi\)
−0.580381 + 0.814345i \(0.697096\pi\)
\(272\) 2.77176e8 0.835149
\(273\) 2.51083e7 0.0746876
\(274\) −5.48742e8 −1.61154
\(275\) −1.57746e8 −0.457398
\(276\) −2.78809e8 −0.798225
\(277\) −1.74969e8 −0.494632 −0.247316 0.968935i \(-0.579549\pi\)
−0.247316 + 0.968935i \(0.579549\pi\)
\(278\) −8.02834e8 −2.24114
\(279\) 5.23787e7 0.144391
\(280\) −2.36039e7 −0.0642585
\(281\) −9.23019e7 −0.248164 −0.124082 0.992272i \(-0.539599\pi\)
−0.124082 + 0.992272i \(0.539599\pi\)
\(282\) 1.45370e8 0.386013
\(283\) −6.07622e8 −1.59361 −0.796803 0.604239i \(-0.793477\pi\)
−0.796803 + 0.604239i \(0.793477\pi\)
\(284\) 8.24601e8 2.13614
\(285\) −9.30018e7 −0.237977
\(286\) 5.97773e7 0.151097
\(287\) 1.11811e8 0.279189
\(288\) −8.13993e8 −2.00792
\(289\) 6.18915e8 1.50830
\(290\) −3.70629e8 −0.892373
\(291\) 4.52447e8 1.07632
\(292\) 8.61712e8 2.02545
\(293\) −4.12823e8 −0.958798 −0.479399 0.877597i \(-0.659145\pi\)
−0.479399 + 0.877597i \(0.659145\pi\)
\(294\) −9.76496e8 −2.24107
\(295\) −1.82306e8 −0.413451
\(296\) −5.88204e7 −0.131828
\(297\) 1.99209e8 0.441226
\(298\) 5.57493e8 1.22035
\(299\) −3.00815e7 −0.0650805
\(300\) −8.03989e8 −1.71920
\(301\) −1.96391e7 −0.0415087
\(302\) 3.57160e8 0.746171
\(303\) −1.09981e9 −2.27127
\(304\) −8.85136e7 −0.180698
\(305\) 3.30817e8 0.667635
\(306\) −1.81657e9 −3.62432
\(307\) 3.39020e8 0.668715 0.334358 0.942446i \(-0.391481\pi\)
0.334358 + 0.942446i \(0.391481\pi\)
\(308\) 1.06867e8 0.208410
\(309\) 6.61602e8 1.27568
\(310\) −3.42115e7 −0.0652237
\(311\) 3.14861e8 0.593550 0.296775 0.954947i \(-0.404089\pi\)
0.296775 + 0.954947i \(0.404089\pi\)
\(312\) 7.90061e7 0.147272
\(313\) −7.61170e7 −0.140306 −0.0701531 0.997536i \(-0.522349\pi\)
−0.0701531 + 0.997536i \(0.522349\pi\)
\(314\) −3.66074e8 −0.667291
\(315\) −9.91441e7 −0.178723
\(316\) 5.86280e8 1.04520
\(317\) 3.04947e6 0.00537671 0.00268835 0.999996i \(-0.499144\pi\)
0.00268835 + 0.999996i \(0.499144\pi\)
\(318\) −1.82372e9 −3.18027
\(319\) 4.35144e8 0.750525
\(320\) 3.95703e8 0.675064
\(321\) −1.62652e8 −0.274469
\(322\) −9.36113e7 −0.156255
\(323\) −3.28683e8 −0.542711
\(324\) −2.18551e8 −0.356981
\(325\) −8.67448e7 −0.140169
\(326\) 1.70636e7 0.0272777
\(327\) 1.05773e9 1.67286
\(328\) 3.51825e8 0.550514
\(329\) 2.80398e7 0.0434100
\(330\) −3.94162e8 −0.603776
\(331\) −6.34097e8 −0.961076 −0.480538 0.876974i \(-0.659559\pi\)
−0.480538 + 0.876974i \(0.659559\pi\)
\(332\) 1.44450e9 2.16639
\(333\) −2.47065e8 −0.366654
\(334\) 1.62884e9 2.39202
\(335\) −2.69293e8 −0.391353
\(336\) −1.57571e8 −0.226615
\(337\) 2.48720e8 0.354003 0.177001 0.984211i \(-0.443360\pi\)
0.177001 + 0.984211i \(0.443360\pi\)
\(338\) −1.05544e9 −1.48670
\(339\) 4.20379e8 0.586060
\(340\) 6.81630e8 0.940529
\(341\) 4.01666e7 0.0548560
\(342\) 5.80105e8 0.784179
\(343\) −3.91777e8 −0.524215
\(344\) −6.17965e7 −0.0818482
\(345\) 1.98353e8 0.260059
\(346\) 1.82738e9 2.37172
\(347\) 7.91060e8 1.01638 0.508190 0.861245i \(-0.330315\pi\)
0.508190 + 0.861245i \(0.330315\pi\)
\(348\) 2.21781e9 2.82096
\(349\) −5.53018e8 −0.696386 −0.348193 0.937423i \(-0.613205\pi\)
−0.348193 + 0.937423i \(0.613205\pi\)
\(350\) −2.69943e8 −0.336538
\(351\) 1.09545e8 0.135213
\(352\) −6.24209e8 −0.762836
\(353\) 1.06776e9 1.29200 0.645999 0.763338i \(-0.276441\pi\)
0.645999 + 0.763338i \(0.276441\pi\)
\(354\) 1.89891e9 2.27507
\(355\) −5.86645e8 −0.695947
\(356\) −1.35342e9 −1.58986
\(357\) −5.85117e8 −0.680619
\(358\) −1.30567e9 −1.50398
\(359\) 1.46087e9 1.66641 0.833203 0.552967i \(-0.186504\pi\)
0.833203 + 0.552967i \(0.186504\pi\)
\(360\) −3.11967e8 −0.352412
\(361\) −7.88910e8 −0.882576
\(362\) −4.08524e8 −0.452624
\(363\) −9.76073e8 −1.07105
\(364\) 5.87666e7 0.0638669
\(365\) −6.13047e8 −0.659886
\(366\) −3.44582e9 −3.67375
\(367\) 1.28374e9 1.35564 0.677822 0.735226i \(-0.262924\pi\)
0.677822 + 0.735226i \(0.262924\pi\)
\(368\) 1.88781e8 0.197465
\(369\) 1.47778e9 1.53115
\(370\) 1.61372e8 0.165623
\(371\) −3.51772e8 −0.357645
\(372\) 2.04718e8 0.206185
\(373\) −4.86215e8 −0.485118 −0.242559 0.970137i \(-0.577987\pi\)
−0.242559 + 0.970137i \(0.577987\pi\)
\(374\) −1.39303e9 −1.37693
\(375\) 1.28117e9 1.25458
\(376\) 8.82303e7 0.0855974
\(377\) 2.39286e8 0.229998
\(378\) 3.40896e8 0.324639
\(379\) −1.35199e9 −1.27567 −0.637834 0.770174i \(-0.720169\pi\)
−0.637834 + 0.770174i \(0.720169\pi\)
\(380\) −2.17673e8 −0.203498
\(381\) 2.13852e9 1.98096
\(382\) 1.93682e9 1.77773
\(383\) −6.26582e8 −0.569878 −0.284939 0.958546i \(-0.591973\pi\)
−0.284939 + 0.958546i \(0.591973\pi\)
\(384\) −1.76525e9 −1.59092
\(385\) −7.60286e7 −0.0678991
\(386\) −2.04028e9 −1.80565
\(387\) −2.59565e8 −0.227645
\(388\) 1.05896e9 0.920384
\(389\) −1.29391e9 −1.11450 −0.557250 0.830345i \(-0.688144\pi\)
−0.557250 + 0.830345i \(0.688144\pi\)
\(390\) −2.16751e8 −0.185027
\(391\) 7.01011e8 0.593070
\(392\) −5.92672e8 −0.496951
\(393\) 9.62515e8 0.799896
\(394\) −9.41575e8 −0.775565
\(395\) −4.17096e8 −0.340523
\(396\) 1.41244e9 1.14298
\(397\) 2.44753e9 1.96319 0.981594 0.190981i \(-0.0611669\pi\)
0.981594 + 0.190981i \(0.0611669\pi\)
\(398\) 1.41070e9 1.12161
\(399\) 1.86852e8 0.147263
\(400\) 5.44378e8 0.425296
\(401\) −8.46869e8 −0.655860 −0.327930 0.944702i \(-0.606351\pi\)
−0.327930 + 0.944702i \(0.606351\pi\)
\(402\) 2.80498e9 2.15347
\(403\) 2.20877e7 0.0168106
\(404\) −2.57413e9 −1.94221
\(405\) 1.55483e8 0.116303
\(406\) 7.44640e8 0.552211
\(407\) −1.89461e8 −0.139297
\(408\) −1.84113e9 −1.34207
\(409\) −1.06925e9 −0.772763 −0.386381 0.922339i \(-0.626275\pi\)
−0.386381 + 0.922339i \(0.626275\pi\)
\(410\) −9.65221e8 −0.691645
\(411\) −2.33606e9 −1.65973
\(412\) 1.54849e9 1.09086
\(413\) 3.66275e8 0.255848
\(414\) −1.23724e9 −0.856944
\(415\) −1.02766e9 −0.705801
\(416\) −3.43254e8 −0.233770
\(417\) −3.41777e9 −2.30816
\(418\) 4.44853e8 0.297920
\(419\) 8.44198e8 0.560654 0.280327 0.959905i \(-0.409557\pi\)
0.280327 + 0.959905i \(0.409557\pi\)
\(420\) −3.87497e8 −0.255209
\(421\) −8.77632e8 −0.573225 −0.286612 0.958047i \(-0.592529\pi\)
−0.286612 + 0.958047i \(0.592529\pi\)
\(422\) 1.28068e9 0.829559
\(423\) 3.70596e8 0.238073
\(424\) −1.10689e9 −0.705217
\(425\) 2.02147e9 1.27734
\(426\) 6.11055e9 3.82954
\(427\) −6.64653e8 −0.413140
\(428\) −3.80691e8 −0.234704
\(429\) 2.54480e8 0.155615
\(430\) 1.69537e8 0.102831
\(431\) −4.03075e7 −0.0242502 −0.0121251 0.999926i \(-0.503860\pi\)
−0.0121251 + 0.999926i \(0.503860\pi\)
\(432\) −6.87467e8 −0.410259
\(433\) 1.00411e9 0.594391 0.297196 0.954817i \(-0.403949\pi\)
0.297196 + 0.954817i \(0.403949\pi\)
\(434\) 6.87351e7 0.0403612
\(435\) −1.57782e9 −0.919060
\(436\) 2.47564e9 1.43049
\(437\) −2.23862e8 −0.128320
\(438\) 6.38555e9 3.63111
\(439\) 6.13553e8 0.346120 0.173060 0.984911i \(-0.444635\pi\)
0.173060 + 0.984911i \(0.444635\pi\)
\(440\) −2.39232e8 −0.133886
\(441\) −2.48942e9 −1.38217
\(442\) −7.66031e8 −0.421957
\(443\) −1.76951e9 −0.967028 −0.483514 0.875337i \(-0.660640\pi\)
−0.483514 + 0.875337i \(0.660640\pi\)
\(444\) −9.65634e8 −0.523567
\(445\) 9.62864e8 0.517970
\(446\) −1.76293e9 −0.940944
\(447\) 2.37332e9 1.25684
\(448\) −7.95017e8 −0.417737
\(449\) −2.28688e9 −1.19229 −0.596145 0.802877i \(-0.703302\pi\)
−0.596145 + 0.802877i \(0.703302\pi\)
\(450\) −3.56777e9 −1.84567
\(451\) 1.13323e9 0.581704
\(452\) 9.83906e8 0.501152
\(453\) 1.52047e9 0.768485
\(454\) −3.38599e9 −1.69820
\(455\) −4.18083e7 −0.0208076
\(456\) 5.87950e8 0.290378
\(457\) 8.79369e8 0.430987 0.215494 0.976505i \(-0.430864\pi\)
0.215494 + 0.976505i \(0.430864\pi\)
\(458\) −2.39774e9 −1.16620
\(459\) −2.55281e9 −1.23218
\(460\) 4.64249e8 0.222381
\(461\) 1.43825e9 0.683724 0.341862 0.939750i \(-0.388942\pi\)
0.341862 + 0.939750i \(0.388942\pi\)
\(462\) 7.91920e8 0.373624
\(463\) −3.40869e9 −1.59608 −0.798040 0.602605i \(-0.794130\pi\)
−0.798040 + 0.602605i \(0.794130\pi\)
\(464\) −1.50167e9 −0.697850
\(465\) −1.45643e8 −0.0671743
\(466\) −1.30486e9 −0.597327
\(467\) 3.01086e9 1.36798 0.683992 0.729490i \(-0.260242\pi\)
0.683992 + 0.729490i \(0.260242\pi\)
\(468\) 7.76705e8 0.350264
\(469\) 5.41043e8 0.242174
\(470\) −2.42057e8 −0.107541
\(471\) −1.55842e9 −0.687247
\(472\) 1.15252e9 0.504490
\(473\) −1.99047e8 −0.0864854
\(474\) 4.34451e9 1.87377
\(475\) −6.45540e8 −0.276373
\(476\) −1.36948e9 −0.582011
\(477\) −4.64929e9 −1.96143
\(478\) −9.32385e8 −0.390479
\(479\) 3.75213e7 0.0155992 0.00779961 0.999970i \(-0.497517\pi\)
0.00779961 + 0.999970i \(0.497517\pi\)
\(480\) 2.26336e9 0.934135
\(481\) −1.04185e8 −0.0426872
\(482\) −7.51414e9 −3.05643
\(483\) −3.98515e8 −0.160927
\(484\) −2.28452e9 −0.915874
\(485\) −7.53376e8 −0.299858
\(486\) −4.63778e9 −1.83266
\(487\) 1.34816e8 0.0528921 0.0264460 0.999650i \(-0.491581\pi\)
0.0264460 + 0.999650i \(0.491581\pi\)
\(488\) −2.09140e9 −0.814644
\(489\) 7.26418e7 0.0280935
\(490\) 1.62598e9 0.624350
\(491\) 1.26229e9 0.481253 0.240627 0.970618i \(-0.422647\pi\)
0.240627 + 0.970618i \(0.422647\pi\)
\(492\) 5.77579e9 2.18642
\(493\) −5.57626e9 −2.09594
\(494\) 2.44625e8 0.0912972
\(495\) −1.00485e9 −0.372378
\(496\) −1.38614e8 −0.0510060
\(497\) 1.17864e9 0.430660
\(498\) 1.07042e10 3.88376
\(499\) −1.34095e9 −0.483128 −0.241564 0.970385i \(-0.577660\pi\)
−0.241564 + 0.970385i \(0.577660\pi\)
\(500\) 2.99862e9 1.07282
\(501\) 6.93417e9 2.46356
\(502\) −6.08326e9 −2.14622
\(503\) 5.65419e9 1.98099 0.990495 0.137546i \(-0.0439215\pi\)
0.990495 + 0.137546i \(0.0439215\pi\)
\(504\) 6.26781e8 0.218077
\(505\) 1.83131e9 0.632764
\(506\) −9.48776e8 −0.325564
\(507\) −4.49313e9 −1.53116
\(508\) 5.00524e9 1.69396
\(509\) 4.83455e9 1.62497 0.812483 0.582985i \(-0.198115\pi\)
0.812483 + 0.582985i \(0.198115\pi\)
\(510\) 5.05109e9 1.68612
\(511\) 1.23169e9 0.408345
\(512\) 3.01367e9 0.992317
\(513\) 8.15218e8 0.266602
\(514\) 6.63297e9 2.15445
\(515\) −1.10164e9 −0.355398
\(516\) −1.01449e9 −0.325068
\(517\) 2.84191e8 0.0904470
\(518\) −3.24216e8 −0.102490
\(519\) 7.77940e9 2.44264
\(520\) −1.31554e8 −0.0410292
\(521\) 1.47590e9 0.457218 0.228609 0.973518i \(-0.426582\pi\)
0.228609 + 0.973518i \(0.426582\pi\)
\(522\) 9.84173e9 3.02848
\(523\) −3.13778e9 −0.959106 −0.479553 0.877513i \(-0.659201\pi\)
−0.479553 + 0.877513i \(0.659201\pi\)
\(524\) 2.25279e9 0.684007
\(525\) −1.14918e9 −0.346602
\(526\) −8.26994e9 −2.47772
\(527\) −5.14724e8 −0.153193
\(528\) −1.59702e9 −0.472163
\(529\) −2.92738e9 −0.859773
\(530\) 3.03671e9 0.886008
\(531\) 4.84097e9 1.40314
\(532\) 4.37331e8 0.125927
\(533\) 6.23168e8 0.178262
\(534\) −1.00293e10 −2.85020
\(535\) 2.70835e8 0.0764656
\(536\) 1.70245e9 0.477527
\(537\) −5.55840e9 −1.54896
\(538\) 7.92193e9 2.19328
\(539\) −1.90901e9 −0.525106
\(540\) −1.69062e9 −0.462026
\(541\) −2.43519e9 −0.661214 −0.330607 0.943768i \(-0.607254\pi\)
−0.330607 + 0.943768i \(0.607254\pi\)
\(542\) −6.59606e9 −1.77945
\(543\) −1.73914e9 −0.466160
\(544\) 7.99909e9 2.13032
\(545\) −1.76124e9 −0.466049
\(546\) 4.35479e8 0.114497
\(547\) −4.11336e9 −1.07458 −0.537292 0.843396i \(-0.680553\pi\)
−0.537292 + 0.843396i \(0.680553\pi\)
\(548\) −5.46761e9 −1.41927
\(549\) −8.78457e9 −2.26578
\(550\) −2.73594e9 −0.701193
\(551\) 1.78073e9 0.453489
\(552\) −1.25397e9 −0.317322
\(553\) 8.37998e8 0.210720
\(554\) −3.03466e9 −0.758274
\(555\) 6.86980e8 0.170576
\(556\) −7.99935e9 −1.97375
\(557\) 5.89179e9 1.44462 0.722311 0.691568i \(-0.243080\pi\)
0.722311 + 0.691568i \(0.243080\pi\)
\(558\) 9.08456e8 0.221352
\(559\) −1.09457e8 −0.0265033
\(560\) 2.62373e8 0.0631337
\(561\) −5.93032e9 −1.41810
\(562\) −1.60088e9 −0.380437
\(563\) −3.72337e9 −0.879341 −0.439671 0.898159i \(-0.644905\pi\)
−0.439671 + 0.898159i \(0.644905\pi\)
\(564\) 1.44845e9 0.339959
\(565\) −6.99979e8 −0.163273
\(566\) −1.05386e10 −2.44301
\(567\) −3.12385e8 −0.0719697
\(568\) 3.70872e9 0.849191
\(569\) 5.89440e9 1.34136 0.670682 0.741745i \(-0.266001\pi\)
0.670682 + 0.741745i \(0.266001\pi\)
\(570\) −1.61302e9 −0.364819
\(571\) −4.09269e9 −0.919988 −0.459994 0.887922i \(-0.652148\pi\)
−0.459994 + 0.887922i \(0.652148\pi\)
\(572\) 5.95615e8 0.133070
\(573\) 8.24528e9 1.83090
\(574\) 1.93925e9 0.427998
\(575\) 1.37680e9 0.302018
\(576\) −1.05076e10 −2.29099
\(577\) 6.73874e9 1.46037 0.730185 0.683249i \(-0.239434\pi\)
0.730185 + 0.683249i \(0.239434\pi\)
\(578\) 1.07345e10 2.31224
\(579\) −8.68574e9 −1.85965
\(580\) −3.69291e9 −0.785906
\(581\) 2.06470e9 0.436758
\(582\) 7.84723e9 1.65001
\(583\) −3.56530e9 −0.745172
\(584\) 3.87563e9 0.805189
\(585\) −5.52570e8 −0.114115
\(586\) −7.15999e9 −1.46984
\(587\) 6.12315e9 1.24951 0.624757 0.780819i \(-0.285198\pi\)
0.624757 + 0.780819i \(0.285198\pi\)
\(588\) −9.72970e9 −1.97369
\(589\) 1.64373e8 0.0331456
\(590\) −3.16191e9 −0.633822
\(591\) −4.00841e9 −0.798759
\(592\) 6.53827e8 0.129520
\(593\) −5.62188e9 −1.10711 −0.553554 0.832813i \(-0.686729\pi\)
−0.553554 + 0.832813i \(0.686729\pi\)
\(594\) 3.45508e9 0.676403
\(595\) 9.74287e8 0.189617
\(596\) 5.55481e9 1.07475
\(597\) 6.00552e9 1.15516
\(598\) −5.21734e8 −0.0997688
\(599\) 2.62393e9 0.498836 0.249418 0.968396i \(-0.419761\pi\)
0.249418 + 0.968396i \(0.419761\pi\)
\(600\) −3.61602e9 −0.683442
\(601\) 5.85029e9 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(602\) −3.40620e8 −0.0636330
\(603\) 7.15085e9 1.32815
\(604\) 3.55870e9 0.657147
\(605\) 1.62527e9 0.298389
\(606\) −1.90751e10 −3.48187
\(607\) 3.51051e9 0.637103 0.318552 0.947905i \(-0.396804\pi\)
0.318552 + 0.947905i \(0.396804\pi\)
\(608\) −2.55444e9 −0.460928
\(609\) 3.17003e9 0.568725
\(610\) 5.73769e9 1.02349
\(611\) 1.56277e8 0.0277174
\(612\) −1.81001e10 −3.19191
\(613\) −6.95815e9 −1.22006 −0.610032 0.792377i \(-0.708843\pi\)
−0.610032 + 0.792377i \(0.708843\pi\)
\(614\) 5.87996e9 1.02514
\(615\) −4.10907e9 −0.712329
\(616\) 4.80646e8 0.0828501
\(617\) −7.18978e8 −0.123230 −0.0616151 0.998100i \(-0.519625\pi\)
−0.0616151 + 0.998100i \(0.519625\pi\)
\(618\) 1.14748e10 1.95563
\(619\) −1.57508e9 −0.266922 −0.133461 0.991054i \(-0.542609\pi\)
−0.133461 + 0.991054i \(0.542609\pi\)
\(620\) −3.40879e8 −0.0574420
\(621\) −1.73869e9 −0.291340
\(622\) 5.46094e9 0.909916
\(623\) −1.93451e9 −0.320526
\(624\) −8.78205e8 −0.144694
\(625\) 2.78932e9 0.457003
\(626\) −1.32017e9 −0.215090
\(627\) 1.89380e9 0.306829
\(628\) −3.64753e9 −0.587678
\(629\) 2.42790e9 0.389003
\(630\) −1.71955e9 −0.273983
\(631\) 8.53967e9 1.35313 0.676564 0.736384i \(-0.263468\pi\)
0.676564 + 0.736384i \(0.263468\pi\)
\(632\) 2.63685e9 0.415504
\(633\) 5.45202e9 0.854367
\(634\) 5.28899e7 0.00824252
\(635\) −3.56087e9 −0.551885
\(636\) −1.81714e10 −2.80084
\(637\) −1.04977e9 −0.160918
\(638\) 7.54712e9 1.15056
\(639\) 1.55779e10 2.36186
\(640\) 2.93934e9 0.443221
\(641\) −1.19699e9 −0.179509 −0.0897547 0.995964i \(-0.528608\pi\)
−0.0897547 + 0.995964i \(0.528608\pi\)
\(642\) −2.82104e9 −0.420762
\(643\) 1.19758e10 1.77651 0.888254 0.459353i \(-0.151919\pi\)
0.888254 + 0.459353i \(0.151919\pi\)
\(644\) −9.32733e8 −0.137612
\(645\) 7.21739e8 0.105906
\(646\) −5.70068e9 −0.831979
\(647\) −2.44410e9 −0.354776 −0.177388 0.984141i \(-0.556765\pi\)
−0.177388 + 0.984141i \(0.556765\pi\)
\(648\) −9.82953e8 −0.141912
\(649\) 3.71230e9 0.533072
\(650\) −1.50450e9 −0.214880
\(651\) 2.92614e8 0.0415682
\(652\) 1.70020e8 0.0240233
\(653\) 1.08778e10 1.52878 0.764389 0.644755i \(-0.223041\pi\)
0.764389 + 0.644755i \(0.223041\pi\)
\(654\) 1.83453e10 2.56450
\(655\) −1.60270e9 −0.222847
\(656\) −3.91077e9 −0.540877
\(657\) 1.62789e10 2.23948
\(658\) 4.86322e8 0.0665478
\(659\) −9.90053e9 −1.34760 −0.673798 0.738916i \(-0.735338\pi\)
−0.673798 + 0.738916i \(0.735338\pi\)
\(660\) −3.92739e9 −0.531741
\(661\) −1.16773e9 −0.157267 −0.0786337 0.996904i \(-0.525056\pi\)
−0.0786337 + 0.996904i \(0.525056\pi\)
\(662\) −1.09978e10 −1.47334
\(663\) −3.26109e9 −0.434576
\(664\) 6.49680e9 0.861214
\(665\) −3.11130e8 −0.0410266
\(666\) −4.28509e9 −0.562082
\(667\) −3.79791e9 −0.495570
\(668\) 1.62296e10 2.10664
\(669\) −7.50503e9 −0.969083
\(670\) −4.67062e9 −0.599946
\(671\) −6.73644e9 −0.860799
\(672\) −4.54737e9 −0.578053
\(673\) −3.21526e9 −0.406596 −0.203298 0.979117i \(-0.565166\pi\)
−0.203298 + 0.979117i \(0.565166\pi\)
\(674\) 4.31380e9 0.542688
\(675\) −5.01377e9 −0.627482
\(676\) −1.05163e10 −1.30933
\(677\) 1.11460e10 1.38057 0.690287 0.723536i \(-0.257484\pi\)
0.690287 + 0.723536i \(0.257484\pi\)
\(678\) 7.29105e9 0.898434
\(679\) 1.51362e9 0.185556
\(680\) 3.06570e9 0.373894
\(681\) −1.44146e10 −1.74899
\(682\) 6.96648e8 0.0840946
\(683\) 1.36119e10 1.63473 0.817364 0.576121i \(-0.195434\pi\)
0.817364 + 0.576121i \(0.195434\pi\)
\(684\) 5.78010e9 0.690620
\(685\) 3.88981e9 0.462394
\(686\) −6.79497e9 −0.803624
\(687\) −1.02075e10 −1.20107
\(688\) 6.86909e8 0.0804155
\(689\) −1.96057e9 −0.228357
\(690\) 3.44023e9 0.398672
\(691\) 5.11725e9 0.590016 0.295008 0.955495i \(-0.404678\pi\)
0.295008 + 0.955495i \(0.404678\pi\)
\(692\) 1.82078e10 2.08875
\(693\) 2.01887e9 0.230432
\(694\) 1.37201e10 1.55812
\(695\) 5.69097e9 0.643042
\(696\) 9.97482e9 1.12143
\(697\) −1.45221e10 −1.62448
\(698\) −9.59153e9 −1.06756
\(699\) −5.55495e9 −0.615191
\(700\) −2.68968e9 −0.296386
\(701\) −5.39916e9 −0.591988 −0.295994 0.955190i \(-0.595651\pi\)
−0.295994 + 0.955190i \(0.595651\pi\)
\(702\) 1.89995e9 0.207283
\(703\) −7.75328e8 −0.0841670
\(704\) −8.05771e9 −0.870377
\(705\) −1.03047e9 −0.110757
\(706\) 1.85192e10 1.98064
\(707\) −3.67933e9 −0.391562
\(708\) 1.89206e10 2.00363
\(709\) −1.04570e9 −0.110191 −0.0550954 0.998481i \(-0.517546\pi\)
−0.0550954 + 0.998481i \(0.517546\pi\)
\(710\) −1.01748e10 −1.06689
\(711\) 1.10756e10 1.15565
\(712\) −6.08715e9 −0.632025
\(713\) −3.50572e8 −0.0362213
\(714\) −1.01483e10 −1.04339
\(715\) −4.23738e8 −0.0433537
\(716\) −1.30095e10 −1.32454
\(717\) −3.96928e9 −0.402156
\(718\) 2.53373e10 2.55461
\(719\) −6.61572e8 −0.0663783 −0.0331892 0.999449i \(-0.510566\pi\)
−0.0331892 + 0.999449i \(0.510566\pi\)
\(720\) 3.46773e9 0.346243
\(721\) 2.21334e9 0.219925
\(722\) −1.36828e10 −1.35299
\(723\) −3.19887e10 −3.14784
\(724\) −4.07049e9 −0.398623
\(725\) −1.09519e10 −1.06735
\(726\) −1.69290e10 −1.64192
\(727\) 1.73518e9 0.167484 0.0837421 0.996487i \(-0.473313\pi\)
0.0837421 + 0.996487i \(0.473313\pi\)
\(728\) 2.64308e8 0.0253893
\(729\) −1.69778e10 −1.62306
\(730\) −1.06327e10 −1.01161
\(731\) 2.55074e9 0.241522
\(732\) −3.43338e10 −3.23544
\(733\) −6.25015e9 −0.586174 −0.293087 0.956086i \(-0.594683\pi\)
−0.293087 + 0.956086i \(0.594683\pi\)
\(734\) 2.22651e10 2.07821
\(735\) 6.92199e9 0.643021
\(736\) 5.44807e9 0.503698
\(737\) 5.48362e9 0.504581
\(738\) 2.56306e10 2.34726
\(739\) 9.00137e9 0.820452 0.410226 0.911984i \(-0.365450\pi\)
0.410226 + 0.911984i \(0.365450\pi\)
\(740\) 1.60789e9 0.145863
\(741\) 1.04140e9 0.0940274
\(742\) −6.10112e9 −0.548272
\(743\) −1.37996e10 −1.23426 −0.617129 0.786862i \(-0.711704\pi\)
−0.617129 + 0.786862i \(0.711704\pi\)
\(744\) 9.20741e8 0.0819657
\(745\) −3.95185e9 −0.350149
\(746\) −8.43290e9 −0.743689
\(747\) 2.72887e10 2.39530
\(748\) −1.38800e10 −1.21265
\(749\) −5.44141e8 −0.0473178
\(750\) 2.22207e10 1.92328
\(751\) 2.50869e9 0.216126 0.108063 0.994144i \(-0.465535\pi\)
0.108063 + 0.994144i \(0.465535\pi\)
\(752\) −9.80739e8 −0.0840990
\(753\) −2.58972e10 −2.21040
\(754\) 4.15018e9 0.352588
\(755\) −2.53176e9 −0.214096
\(756\) 3.39666e9 0.285907
\(757\) −1.59957e10 −1.34020 −0.670099 0.742272i \(-0.733748\pi\)
−0.670099 + 0.742272i \(0.733748\pi\)
\(758\) −2.34490e10 −1.95561
\(759\) −4.03906e9 −0.335300
\(760\) −9.79003e8 −0.0808978
\(761\) −1.66651e10 −1.37076 −0.685378 0.728187i \(-0.740363\pi\)
−0.685378 + 0.728187i \(0.740363\pi\)
\(762\) 3.70904e10 3.03682
\(763\) 3.53856e9 0.288397
\(764\) 1.92982e10 1.56564
\(765\) 1.28769e10 1.03991
\(766\) −1.08674e10 −0.873627
\(767\) 2.04140e9 0.163359
\(768\) −1.98134e8 −0.0157832
\(769\) 9.23584e9 0.732376 0.366188 0.930541i \(-0.380663\pi\)
0.366188 + 0.930541i \(0.380663\pi\)
\(770\) −1.31864e9 −0.104090
\(771\) 2.82374e10 2.21888
\(772\) −2.03291e10 −1.59022
\(773\) −2.96075e9 −0.230554 −0.115277 0.993333i \(-0.536776\pi\)
−0.115277 + 0.993333i \(0.536776\pi\)
\(774\) −4.50190e9 −0.348981
\(775\) −1.01093e9 −0.0780126
\(776\) 4.76278e9 0.365885
\(777\) −1.38023e9 −0.105555
\(778\) −2.24415e10 −1.70854
\(779\) 4.63751e9 0.351482
\(780\) −2.15968e9 −0.162951
\(781\) 1.19459e10 0.897303
\(782\) 1.21583e10 0.909180
\(783\) 1.38305e10 1.02961
\(784\) 6.58795e9 0.488252
\(785\) 2.59496e9 0.191464
\(786\) 1.66938e10 1.22625
\(787\) 9.32429e9 0.681874 0.340937 0.940086i \(-0.389256\pi\)
0.340937 + 0.940086i \(0.389256\pi\)
\(788\) −9.38176e9 −0.683034
\(789\) −3.52062e10 −2.55182
\(790\) −7.23411e9 −0.522024
\(791\) 1.40634e9 0.101036
\(792\) 6.35260e9 0.454374
\(793\) −3.70438e9 −0.263791
\(794\) 4.24500e10 3.00958
\(795\) 1.29277e10 0.912504
\(796\) 1.40560e10 0.987796
\(797\) −2.22955e10 −1.55996 −0.779978 0.625806i \(-0.784770\pi\)
−0.779978 + 0.625806i \(0.784770\pi\)
\(798\) 3.24076e9 0.225754
\(799\) −3.64184e9 −0.252585
\(800\) 1.57104e10 1.08485
\(801\) −2.55680e10 −1.75786
\(802\) −1.46881e10 −1.00544
\(803\) 1.24835e10 0.850808
\(804\) 2.79486e10 1.89654
\(805\) 6.63573e8 0.0448336
\(806\) 3.83088e8 0.0257707
\(807\) 3.37247e10 2.25887
\(808\) −1.15774e10 −0.772096
\(809\) −5.80413e9 −0.385405 −0.192703 0.981257i \(-0.561725\pi\)
−0.192703 + 0.981257i \(0.561725\pi\)
\(810\) 2.69670e9 0.178293
\(811\) −1.62210e10 −1.06783 −0.533917 0.845537i \(-0.679280\pi\)
−0.533917 + 0.845537i \(0.679280\pi\)
\(812\) 7.41951e9 0.486328
\(813\) −2.80803e10 −1.83267
\(814\) −3.28601e9 −0.213542
\(815\) −1.20957e8 −0.00782671
\(816\) 2.04654e10 1.31858
\(817\) −8.14557e8 −0.0522570
\(818\) −1.85450e10 −1.18465
\(819\) 1.11018e9 0.0706156
\(820\) −9.61736e9 −0.609126
\(821\) 6.33227e9 0.399354 0.199677 0.979862i \(-0.436011\pi\)
0.199677 + 0.979862i \(0.436011\pi\)
\(822\) −4.05166e10 −2.54438
\(823\) −2.22297e10 −1.39006 −0.695031 0.718980i \(-0.744609\pi\)
−0.695031 + 0.718980i \(0.744609\pi\)
\(824\) 6.96450e9 0.433655
\(825\) −1.16473e10 −0.722162
\(826\) 6.35267e9 0.392216
\(827\) −4.64860e9 −0.285794 −0.142897 0.989738i \(-0.545642\pi\)
−0.142897 + 0.989738i \(0.545642\pi\)
\(828\) −1.23277e10 −0.754704
\(829\) −8.14688e9 −0.496650 −0.248325 0.968677i \(-0.579880\pi\)
−0.248325 + 0.968677i \(0.579880\pi\)
\(830\) −1.78238e10 −1.08200
\(831\) −1.29189e10 −0.780950
\(832\) −4.43095e9 −0.266726
\(833\) 2.44634e10 1.46643
\(834\) −5.92777e10 −3.53842
\(835\) −1.15462e10 −0.686335
\(836\) 4.43247e9 0.262376
\(837\) 1.27665e9 0.0752544
\(838\) 1.46417e10 0.859486
\(839\) 1.95604e10 1.14343 0.571715 0.820452i \(-0.306278\pi\)
0.571715 + 0.820452i \(0.306278\pi\)
\(840\) −1.74281e9 −0.101455
\(841\) 1.29610e10 0.751366
\(842\) −1.52216e10 −0.878757
\(843\) −6.81516e9 −0.391814
\(844\) 1.27606e10 0.730586
\(845\) 7.48158e9 0.426574
\(846\) 6.42761e9 0.364967
\(847\) −3.26537e9 −0.184646
\(848\) 1.23038e10 0.692873
\(849\) −4.48641e10 −2.51607
\(850\) 3.50604e10 1.95817
\(851\) 1.65361e9 0.0919770
\(852\) 6.08849e10 3.37265
\(853\) 1.58440e10 0.874061 0.437031 0.899447i \(-0.356030\pi\)
0.437031 + 0.899447i \(0.356030\pi\)
\(854\) −1.15277e10 −0.633346
\(855\) −4.11213e9 −0.225002
\(856\) −1.71220e9 −0.0933029
\(857\) 2.56284e10 1.39088 0.695438 0.718586i \(-0.255210\pi\)
0.695438 + 0.718586i \(0.255210\pi\)
\(858\) 4.41369e9 0.238559
\(859\) −3.28663e9 −0.176919 −0.0884596 0.996080i \(-0.528194\pi\)
−0.0884596 + 0.996080i \(0.528194\pi\)
\(860\) 1.68925e9 0.0905624
\(861\) 8.25562e9 0.440797
\(862\) −6.99092e8 −0.0371757
\(863\) 1.66288e10 0.880688 0.440344 0.897829i \(-0.354857\pi\)
0.440344 + 0.897829i \(0.354857\pi\)
\(864\) −1.98398e10 −1.04650
\(865\) −1.29536e10 −0.680508
\(866\) 1.74152e10 0.911205
\(867\) 4.56980e10 2.38139
\(868\) 6.84869e8 0.0355458
\(869\) 8.49334e9 0.439045
\(870\) −2.73656e10 −1.40892
\(871\) 3.01545e9 0.154628
\(872\) 1.11344e10 0.568671
\(873\) 2.00052e10 1.01764
\(874\) −3.88265e9 −0.196715
\(875\) 4.28607e9 0.216287
\(876\) 6.36250e10 3.19789
\(877\) −6.52780e9 −0.326790 −0.163395 0.986561i \(-0.552244\pi\)
−0.163395 + 0.986561i \(0.552244\pi\)
\(878\) 1.06415e10 0.530603
\(879\) −3.04810e10 −1.51380
\(880\) 2.65922e9 0.131542
\(881\) 2.61062e10 1.28626 0.643128 0.765759i \(-0.277636\pi\)
0.643128 + 0.765759i \(0.277636\pi\)
\(882\) −4.31764e10 −2.11888
\(883\) 9.02672e9 0.441232 0.220616 0.975361i \(-0.429193\pi\)
0.220616 + 0.975361i \(0.429193\pi\)
\(884\) −7.63266e9 −0.371615
\(885\) −1.34607e10 −0.652777
\(886\) −3.06903e10 −1.48246
\(887\) 2.03347e9 0.0978372 0.0489186 0.998803i \(-0.484423\pi\)
0.0489186 + 0.998803i \(0.484423\pi\)
\(888\) −4.34303e9 −0.208136
\(889\) 7.15423e9 0.341513
\(890\) 1.66999e10 0.794051
\(891\) −3.16611e9 −0.149953
\(892\) −1.75657e10 −0.828682
\(893\) 1.16299e9 0.0546507
\(894\) 4.11628e10 1.92674
\(895\) 9.25537e9 0.431532
\(896\) −5.90551e9 −0.274270
\(897\) −2.22109e9 −0.102752
\(898\) −3.96637e10 −1.82779
\(899\) 2.78866e9 0.128008
\(900\) −3.55489e10 −1.62546
\(901\) 4.56884e10 2.08099
\(902\) 1.96548e10 0.891755
\(903\) −1.45006e9 −0.0655360
\(904\) 4.42521e9 0.199225
\(905\) 2.89587e9 0.129870
\(906\) 2.63711e10 1.17809
\(907\) 3.47024e10 1.54431 0.772155 0.635435i \(-0.219179\pi\)
0.772155 + 0.635435i \(0.219179\pi\)
\(908\) −3.37376e10 −1.49559
\(909\) −4.86288e10 −2.14744
\(910\) −7.25121e8 −0.0318982
\(911\) −3.83459e10 −1.68037 −0.840183 0.542302i \(-0.817553\pi\)
−0.840183 + 0.542302i \(0.817553\pi\)
\(912\) −6.53545e9 −0.285295
\(913\) 2.09263e10 0.910007
\(914\) 1.52518e10 0.660706
\(915\) 2.44261e10 1.05410
\(916\) −2.38908e10 −1.02706
\(917\) 3.22002e9 0.137900
\(918\) −4.42759e10 −1.88894
\(919\) 7.63731e9 0.324591 0.162295 0.986742i \(-0.448110\pi\)
0.162295 + 0.986742i \(0.448110\pi\)
\(920\) 2.08800e9 0.0884044
\(921\) 2.50317e10 1.05580
\(922\) 2.49450e10 1.04815
\(923\) 6.56905e9 0.274977
\(924\) 7.89061e9 0.329048
\(925\) 4.76844e9 0.198098
\(926\) −5.91203e10 −2.44680
\(927\) 2.92532e10 1.20613
\(928\) −4.33372e10 −1.78009
\(929\) −1.44540e10 −0.591469 −0.295734 0.955270i \(-0.595564\pi\)
−0.295734 + 0.955270i \(0.595564\pi\)
\(930\) −2.52602e9 −0.102979
\(931\) −7.81218e9 −0.317284
\(932\) −1.30015e10 −0.526062
\(933\) 2.32479e10 0.937127
\(934\) 5.22202e10 2.09713
\(935\) 9.87466e9 0.395077
\(936\) 3.49330e9 0.139242
\(937\) −2.76136e10 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(938\) 9.38384e9 0.371254
\(939\) −5.62014e9 −0.221522
\(940\) −2.41183e9 −0.0947108
\(941\) 1.88863e10 0.738894 0.369447 0.929252i \(-0.379547\pi\)
0.369447 + 0.929252i \(0.379547\pi\)
\(942\) −2.70293e10 −1.05355
\(943\) −9.89081e9 −0.384097
\(944\) −1.28111e10 −0.495659
\(945\) −2.41648e9 −0.0931476
\(946\) −3.45228e9 −0.132583
\(947\) −1.55375e10 −0.594506 −0.297253 0.954799i \(-0.596070\pi\)
−0.297253 + 0.954799i \(0.596070\pi\)
\(948\) 4.32883e10 1.65022
\(949\) 6.86469e9 0.260729
\(950\) −1.11962e10 −0.423681
\(951\) 2.25159e8 0.00848902
\(952\) −6.15936e9 −0.231370
\(953\) −1.05814e10 −0.396021 −0.198011 0.980200i \(-0.563448\pi\)
−0.198011 + 0.980200i \(0.563448\pi\)
\(954\) −8.06372e10 −3.00688
\(955\) −1.37293e10 −0.510079
\(956\) −9.29019e9 −0.343892
\(957\) 3.21291e10 1.18497
\(958\) 6.50768e8 0.0239137
\(959\) −7.81511e9 −0.286134
\(960\) 2.92170e10 1.06582
\(961\) −2.72552e10 −0.990644
\(962\) −1.80699e9 −0.0654398
\(963\) −7.19178e9 −0.259504
\(964\) −7.48701e10 −2.69178
\(965\) 1.44627e10 0.518090
\(966\) −6.91184e9 −0.246703
\(967\) −1.98827e9 −0.0707102 −0.0353551 0.999375i \(-0.511256\pi\)
−0.0353551 + 0.999375i \(0.511256\pi\)
\(968\) −1.02748e10 −0.364092
\(969\) −2.42685e10 −0.856860
\(970\) −1.30665e10 −0.459684
\(971\) −1.06982e10 −0.375012 −0.187506 0.982263i \(-0.560040\pi\)
−0.187506 + 0.982263i \(0.560040\pi\)
\(972\) −4.62103e10 −1.61401
\(973\) −1.14339e10 −0.397922
\(974\) 2.33825e9 0.0810838
\(975\) −6.40485e9 −0.221306
\(976\) 2.32473e10 0.800384
\(977\) −3.25937e10 −1.11816 −0.559078 0.829115i \(-0.688845\pi\)
−0.559078 + 0.829115i \(0.688845\pi\)
\(978\) 1.25990e9 0.0430675
\(979\) −1.96068e10 −0.667832
\(980\) 1.62011e10 0.549860
\(981\) 4.67683e10 1.58165
\(982\) 2.18931e10 0.737764
\(983\) −3.26564e10 −1.09656 −0.548278 0.836296i \(-0.684716\pi\)
−0.548278 + 0.836296i \(0.684716\pi\)
\(984\) 2.59772e10 0.869179
\(985\) 6.67446e9 0.222530
\(986\) −9.67145e10 −3.21309
\(987\) 2.07034e9 0.0685379
\(988\) 2.43742e9 0.0804047
\(989\) 1.73728e9 0.0571060
\(990\) −1.74281e10 −0.570858
\(991\) 1.86068e10 0.607316 0.303658 0.952781i \(-0.401792\pi\)
0.303658 + 0.952781i \(0.401792\pi\)
\(992\) −4.00030e9 −0.130107
\(993\) −4.68189e10 −1.51740
\(994\) 2.04424e10 0.660205
\(995\) −9.99987e9 −0.321821
\(996\) 1.06656e11 3.42040
\(997\) 5.77600e10 1.84584 0.922920 0.384993i \(-0.125796\pi\)
0.922920 + 0.384993i \(0.125796\pi\)
\(998\) −2.32575e10 −0.740638
\(999\) −6.02181e9 −0.191094
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 43.8.a.b.1.11 13
3.2 odd 2 387.8.a.d.1.3 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.11 13 1.1 even 1 trivial
387.8.a.d.1.3 13 3.2 odd 2