Properties

Label 387.8.a.d.1.3
Level 387
Weight 8
Character 387.1
Self dual yes
Analytic conductor 120.893
Analytic rank 1
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(120.893004862\)
Analytic rank: \(1\)
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: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(16.3440\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q-17.3440 q^{2} +172.814 q^{4} +122.945 q^{5} +247.011 q^{7} -777.246 q^{8} +O(q^{10})\) \(q-17.3440 q^{2} +172.814 q^{4} +122.945 q^{5} +247.011 q^{7} -777.246 q^{8} -2132.35 q^{10} -2503.52 q^{11} +1376.69 q^{13} -4284.15 q^{14} -8639.60 q^{16} +32082.0 q^{17} +10245.1 q^{19} +21246.5 q^{20} +43421.0 q^{22} +21850.6 q^{23} -63009.6 q^{25} -23877.3 q^{26} +42686.8 q^{28} -173813. q^{29} +16044.0 q^{31} +249333. q^{32} -556429. q^{34} +30368.6 q^{35} -75677.9 q^{37} -177691. q^{38} -95558.2 q^{40} -452656. q^{41} -79507.0 q^{43} -432643. q^{44} -378977. q^{46} -113517. q^{47} -762529. q^{49} +1.09284e6 q^{50} +237911. q^{52} +1.42412e6 q^{53} -307794. q^{55} -191988. q^{56} +3.01460e6 q^{58} -1.48283e6 q^{59} -2.69078e6 q^{61} -278267. q^{62} -3.21855e6 q^{64} +169257. q^{65} +2.19036e6 q^{67} +5.54420e6 q^{68} -526713. q^{70} -4.77162e6 q^{71} +4.98637e6 q^{73} +1.31256e6 q^{74} +1.77049e6 q^{76} -618397. q^{77} +3.39256e6 q^{79} -1.06219e6 q^{80} +7.85086e6 q^{82} -8.35875e6 q^{83} +3.94431e6 q^{85} +1.37897e6 q^{86} +1.94585e6 q^{88} +7.83169e6 q^{89} +340058. q^{91} +3.77608e6 q^{92} +1.96883e6 q^{94} +1.25958e6 q^{95} +6.12777e6 q^{97} +1.32253e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

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 \(-0.778005\pi\)
−0.766503 + 0.642241i \(0.778005\pi\)
\(3\) 0 0
\(4\) 172.814 1.35011
\(5\) 122.945 0.439860 0.219930 0.975516i \(-0.429417\pi\)
0.219930 + 0.975516i \(0.429417\pi\)
\(6\) 0 0
\(7\) 247.011 0.272190 0.136095 0.990696i \(-0.456545\pi\)
0.136095 + 0.990696i \(0.456545\pi\)
\(8\) −777.246 −0.536715
\(9\) 0 0
\(10\) −2132.35 −0.674308
\(11\) −2503.52 −0.567123 −0.283561 0.958954i \(-0.591516\pi\)
−0.283561 + 0.958954i \(0.591516\pi\)
\(12\) 0 0
\(13\) 1376.69 0.173794 0.0868970 0.996217i \(-0.472305\pi\)
0.0868970 + 0.996217i \(0.472305\pi\)
\(14\) −4284.15 −0.417269
\(15\) 0 0
\(16\) −8639.60 −0.527320
\(17\) 32082.0 1.58376 0.791881 0.610675i \(-0.209102\pi\)
0.791881 + 0.610675i \(0.209102\pi\)
\(18\) 0 0
\(19\) 10245.1 0.342672 0.171336 0.985213i \(-0.445192\pi\)
0.171336 + 0.985213i \(0.445192\pi\)
\(20\) 21246.5 0.593858
\(21\) 0 0
\(22\) 43421.0 0.869402
\(23\) 21850.6 0.374469 0.187235 0.982315i \(-0.440048\pi\)
0.187235 + 0.982315i \(0.440048\pi\)
\(24\) 0 0
\(25\) −63009.6 −0.806523
\(26\) −23877.3 −0.266427
\(27\) 0 0
\(28\) 42686.8 0.367486
\(29\) −173813. −1.32339 −0.661696 0.749772i \(-0.730163\pi\)
−0.661696 + 0.749772i \(0.730163\pi\)
\(30\) 0 0
\(31\) 16044.0 0.0967270 0.0483635 0.998830i \(-0.484599\pi\)
0.0483635 + 0.998830i \(0.484599\pi\)
\(32\) 249333. 1.34510
\(33\) 0 0
\(34\) −556429. −2.42792
\(35\) 30368.6 0.119726
\(36\) 0 0
\(37\) −75677.9 −0.245620 −0.122810 0.992430i \(-0.539191\pi\)
−0.122810 + 0.992430i \(0.539191\pi\)
\(38\) −177691. −0.525318
\(39\) 0 0
\(40\) −95558.2 −0.236079
\(41\) −452656. −1.02571 −0.512855 0.858475i \(-0.671412\pi\)
−0.512855 + 0.858475i \(0.671412\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) −432643. −0.765676
\(45\) 0 0
\(46\) −378977. −0.574063
\(47\) −113517. −0.159484 −0.0797420 0.996816i \(-0.525410\pi\)
−0.0797420 + 0.996816i \(0.525410\pi\)
\(48\) 0 0
\(49\) −762529. −0.925912
\(50\) 1.09284e6 1.23640
\(51\) 0 0
\(52\) 237911. 0.234640
\(53\) 1.42412e6 1.31395 0.656976 0.753911i \(-0.271835\pi\)
0.656976 + 0.753911i \(0.271835\pi\)
\(54\) 0 0
\(55\) −307794. −0.249455
\(56\) −191988. −0.146089
\(57\) 0 0
\(58\) 3.01460e6 2.02877
\(59\) −1.48283e6 −0.939960 −0.469980 0.882677i \(-0.655739\pi\)
−0.469980 + 0.882677i \(0.655739\pi\)
\(60\) 0 0
\(61\) −2.69078e6 −1.51784 −0.758918 0.651187i \(-0.774271\pi\)
−0.758918 + 0.651187i \(0.774271\pi\)
\(62\) −278267. −0.148283
\(63\) 0 0
\(64\) −3.21855e6 −1.53472
\(65\) 169257. 0.0764450
\(66\) 0 0
\(67\) 2.19036e6 0.889722 0.444861 0.895600i \(-0.353253\pi\)
0.444861 + 0.895600i \(0.353253\pi\)
\(68\) 5.54420e6 2.13825
\(69\) 0 0
\(70\) −526713. −0.183540
\(71\) −4.77162e6 −1.58220 −0.791101 0.611685i \(-0.790492\pi\)
−0.791101 + 0.611685i \(0.790492\pi\)
\(72\) 0 0
\(73\) 4.98637e6 1.50022 0.750109 0.661314i \(-0.230001\pi\)
0.750109 + 0.661314i \(0.230001\pi\)
\(74\) 1.31256e6 0.376536
\(75\) 0 0
\(76\) 1.77049e6 0.462644
\(77\) −618397. −0.154365
\(78\) 0 0
\(79\) 3.39256e6 0.774163 0.387081 0.922046i \(-0.373483\pi\)
0.387081 + 0.922046i \(0.373483\pi\)
\(80\) −1.06219e6 −0.231947
\(81\) 0 0
\(82\) 7.85086e6 1.57242
\(83\) −8.35875e6 −1.60460 −0.802302 0.596918i \(-0.796392\pi\)
−0.802302 + 0.596918i \(0.796392\pi\)
\(84\) 0 0
\(85\) 3.94431e6 0.696634
\(86\) 1.37897e6 0.233781
\(87\) 0 0
\(88\) 1.94585e6 0.304383
\(89\) 7.83169e6 1.17758 0.588790 0.808286i \(-0.299604\pi\)
0.588790 + 0.808286i \(0.299604\pi\)
\(90\) 0 0
\(91\) 340058. 0.0473051
\(92\) 3.77608e6 0.505573
\(93\) 0 0
\(94\) 1.96883e6 0.244490
\(95\) 1.25958e6 0.150728
\(96\) 0 0
\(97\) 6.12777e6 0.681712 0.340856 0.940115i \(-0.389283\pi\)
0.340856 + 0.940115i \(0.389283\pi\)
\(98\) 1.32253e7 1.41943
\(99\) 0 0
\(100\) −1.08889e7 −1.08889
\(101\) 1.48954e7 1.43856 0.719279 0.694721i \(-0.244472\pi\)
0.719279 + 0.694721i \(0.244472\pi\)
\(102\) 0 0
\(103\) 8.96048e6 0.807981 0.403990 0.914763i \(-0.367623\pi\)
0.403990 + 0.914763i \(0.367623\pi\)
\(104\) −1.07003e6 −0.0932778
\(105\) 0 0
\(106\) −2.46998e7 −2.01430
\(107\) 2.20290e6 0.173841 0.0869204 0.996215i \(-0.472297\pi\)
0.0869204 + 0.996215i \(0.472297\pi\)
\(108\) 0 0
\(109\) 1.43255e7 1.05954 0.529770 0.848141i \(-0.322278\pi\)
0.529770 + 0.848141i \(0.322278\pi\)
\(110\) 5.33838e6 0.382415
\(111\) 0 0
\(112\) −2.13408e6 −0.143531
\(113\) −5.69345e6 −0.371194 −0.185597 0.982626i \(-0.559422\pi\)
−0.185597 + 0.982626i \(0.559422\pi\)
\(114\) 0 0
\(115\) 2.68641e6 0.164714
\(116\) −3.00372e7 −1.78672
\(117\) 0 0
\(118\) 2.57182e7 1.44096
\(119\) 7.92460e6 0.431085
\(120\) 0 0
\(121\) −1.32196e7 −0.678372
\(122\) 4.66689e7 2.32685
\(123\) 0 0
\(124\) 2.77263e6 0.130592
\(125\) −1.73517e7 −0.794617
\(126\) 0 0
\(127\) 2.89632e7 1.25468 0.627342 0.778744i \(-0.284143\pi\)
0.627342 + 0.778744i \(0.284143\pi\)
\(128\) 2.39079e7 1.00764
\(129\) 0 0
\(130\) −2.93559e6 −0.117191
\(131\) −1.30359e7 −0.506632 −0.253316 0.967384i \(-0.581521\pi\)
−0.253316 + 0.967384i \(0.581521\pi\)
\(132\) 0 0
\(133\) 2.53065e6 0.0932721
\(134\) −3.79896e7 −1.36395
\(135\) 0 0
\(136\) −2.49356e7 −0.850029
\(137\) 3.16388e7 1.05123 0.525615 0.850723i \(-0.323835\pi\)
0.525615 + 0.850723i \(0.323835\pi\)
\(138\) 0 0
\(139\) −4.62889e7 −1.46193 −0.730963 0.682418i \(-0.760929\pi\)
−0.730963 + 0.682418i \(0.760929\pi\)
\(140\) 5.24811e6 0.161642
\(141\) 0 0
\(142\) 8.27589e7 2.42552
\(143\) −3.44658e6 −0.0985625
\(144\) 0 0
\(145\) −2.13693e7 −0.582107
\(146\) −8.64835e7 −2.29984
\(147\) 0 0
\(148\) −1.30782e7 −0.331613
\(149\) −3.21433e7 −0.796048 −0.398024 0.917375i \(-0.630304\pi\)
−0.398024 + 0.917375i \(0.630304\pi\)
\(150\) 0 0
\(151\) 2.05927e7 0.486737 0.243369 0.969934i \(-0.421748\pi\)
0.243369 + 0.969934i \(0.421748\pi\)
\(152\) −7.96296e6 −0.183917
\(153\) 0 0
\(154\) 1.07255e7 0.236643
\(155\) 1.97253e6 0.0425463
\(156\) 0 0
\(157\) −2.11067e7 −0.435283 −0.217641 0.976029i \(-0.569836\pi\)
−0.217641 + 0.976029i \(0.569836\pi\)
\(158\) −5.88404e7 −1.18680
\(159\) 0 0
\(160\) 3.06541e7 0.591655
\(161\) 5.39734e6 0.101927
\(162\) 0 0
\(163\) 983833. 0.0177936 0.00889682 0.999960i \(-0.497168\pi\)
0.00889682 + 0.999960i \(0.497168\pi\)
\(164\) −7.82251e7 −1.38482
\(165\) 0 0
\(166\) 1.44974e8 2.45987
\(167\) −9.39137e7 −1.56035 −0.780174 0.625563i \(-0.784869\pi\)
−0.780174 + 0.625563i \(0.784869\pi\)
\(168\) 0 0
\(169\) −6.08532e7 −0.969796
\(170\) −6.84100e7 −1.06794
\(171\) 0 0
\(172\) −1.37399e7 −0.205889
\(173\) −1.05361e8 −1.54710 −0.773551 0.633734i \(-0.781522\pi\)
−0.773551 + 0.633734i \(0.781522\pi\)
\(174\) 0 0
\(175\) −1.55641e7 −0.219528
\(176\) 2.16294e7 0.299055
\(177\) 0 0
\(178\) −1.35833e8 −1.80524
\(179\) 7.52808e7 0.981067 0.490533 0.871422i \(-0.336802\pi\)
0.490533 + 0.871422i \(0.336802\pi\)
\(180\) 0 0
\(181\) −2.35542e7 −0.295253 −0.147626 0.989043i \(-0.547163\pi\)
−0.147626 + 0.989043i \(0.547163\pi\)
\(182\) −5.89795e6 −0.0725189
\(183\) 0 0
\(184\) −1.69833e7 −0.200983
\(185\) −9.30419e6 −0.108038
\(186\) 0 0
\(187\) −8.03180e7 −0.898188
\(188\) −1.96172e7 −0.215320
\(189\) 0 0
\(190\) −2.18461e7 −0.231066
\(191\) −1.11671e8 −1.15964 −0.579820 0.814745i \(-0.696877\pi\)
−0.579820 + 0.814745i \(0.696877\pi\)
\(192\) 0 0
\(193\) −1.17636e8 −1.17785 −0.588926 0.808187i \(-0.700449\pi\)
−0.588926 + 0.808187i \(0.700449\pi\)
\(194\) −1.06280e8 −1.04507
\(195\) 0 0
\(196\) −1.31775e8 −1.25008
\(197\) 5.42883e7 0.505912 0.252956 0.967478i \(-0.418597\pi\)
0.252956 + 0.967478i \(0.418597\pi\)
\(198\) 0 0
\(199\) 8.13364e7 0.731643 0.365822 0.930685i \(-0.380788\pi\)
0.365822 + 0.930685i \(0.380788\pi\)
\(200\) 4.89740e7 0.432873
\(201\) 0 0
\(202\) −2.58346e8 −2.20532
\(203\) −4.29336e7 −0.360215
\(204\) 0 0
\(205\) −5.56516e7 −0.451169
\(206\) −1.55410e8 −1.23864
\(207\) 0 0
\(208\) −1.18941e7 −0.0916450
\(209\) −2.56488e7 −0.194337
\(210\) 0 0
\(211\) 7.38401e7 0.541132 0.270566 0.962701i \(-0.412789\pi\)
0.270566 + 0.962701i \(0.412789\pi\)
\(212\) 2.46106e8 1.77397
\(213\) 0 0
\(214\) −3.82071e7 −0.266499
\(215\) −9.77496e6 −0.0670780
\(216\) 0 0
\(217\) 3.96305e6 0.0263282
\(218\) −2.48461e8 −1.62428
\(219\) 0 0
\(220\) −5.31911e7 −0.336790
\(221\) 4.41670e7 0.275248
\(222\) 0 0
\(223\) −1.01645e8 −0.613790 −0.306895 0.951743i \(-0.599290\pi\)
−0.306895 + 0.951743i \(0.599290\pi\)
\(224\) 6.15878e7 0.366123
\(225\) 0 0
\(226\) 9.87471e7 0.569043
\(227\) 1.95225e8 1.10776 0.553880 0.832596i \(-0.313146\pi\)
0.553880 + 0.832596i \(0.313146\pi\)
\(228\) 0 0
\(229\) −1.38246e8 −0.760727 −0.380363 0.924837i \(-0.624201\pi\)
−0.380363 + 0.924837i \(0.624201\pi\)
\(230\) −4.65931e7 −0.252507
\(231\) 0 0
\(232\) 1.35095e8 0.710284
\(233\) 7.52341e7 0.389645 0.194822 0.980839i \(-0.437587\pi\)
0.194822 + 0.980839i \(0.437587\pi\)
\(234\) 0 0
\(235\) −1.39563e7 −0.0701506
\(236\) −2.56253e8 −1.26905
\(237\) 0 0
\(238\) −1.37444e8 −0.660856
\(239\) 5.37584e7 0.254715 0.127357 0.991857i \(-0.459351\pi\)
0.127357 + 0.991857i \(0.459351\pi\)
\(240\) 0 0
\(241\) −4.33242e8 −1.99375 −0.996876 0.0789843i \(-0.974832\pi\)
−0.996876 + 0.0789843i \(0.974832\pi\)
\(242\) 2.29280e8 1.03995
\(243\) 0 0
\(244\) −4.65004e8 −2.04924
\(245\) −9.37488e7 −0.407272
\(246\) 0 0
\(247\) 1.41043e7 0.0595544
\(248\) −1.24702e7 −0.0519148
\(249\) 0 0
\(250\) 3.00948e8 1.21815
\(251\) 3.50742e8 1.40001 0.700003 0.714140i \(-0.253182\pi\)
0.700003 + 0.714140i \(0.253182\pi\)
\(252\) 0 0
\(253\) −5.47035e7 −0.212370
\(254\) −5.02338e8 −1.92344
\(255\) 0 0
\(256\) −2.68346e6 −0.00999666
\(257\) −3.82436e8 −1.40538 −0.702689 0.711497i \(-0.748017\pi\)
−0.702689 + 0.711497i \(0.748017\pi\)
\(258\) 0 0
\(259\) −1.86933e7 −0.0668554
\(260\) 2.92499e7 0.103209
\(261\) 0 0
\(262\) 2.26095e8 0.776670
\(263\) 4.76819e8 1.61625 0.808125 0.589011i \(-0.200483\pi\)
0.808125 + 0.589011i \(0.200483\pi\)
\(264\) 0 0
\(265\) 1.75087e8 0.577955
\(266\) −4.38916e7 −0.142987
\(267\) 0 0
\(268\) 3.78524e8 1.20122
\(269\) −4.56754e8 −1.43070 −0.715352 0.698765i \(-0.753733\pi\)
−0.715352 + 0.698765i \(0.753733\pi\)
\(270\) 0 0
\(271\) −3.80308e8 −1.16076 −0.580381 0.814345i \(-0.697096\pi\)
−0.580381 + 0.814345i \(0.697096\pi\)
\(272\) −2.77176e8 −0.835149
\(273\) 0 0
\(274\) −5.48742e8 −1.61154
\(275\) 1.57746e8 0.457398
\(276\) 0 0
\(277\) −1.74969e8 −0.494632 −0.247316 0.968935i \(-0.579549\pi\)
−0.247316 + 0.968935i \(0.579549\pi\)
\(278\) 8.02834e8 2.24114
\(279\) 0 0
\(280\) −2.36039e7 −0.0642585
\(281\) 9.23019e7 0.248164 0.124082 0.992272i \(-0.460401\pi\)
0.124082 + 0.992272i \(0.460401\pi\)
\(282\) 0 0
\(283\) −6.07622e8 −1.59361 −0.796803 0.604239i \(-0.793477\pi\)
−0.796803 + 0.604239i \(0.793477\pi\)
\(284\) −8.24601e8 −2.13614
\(285\) 0 0
\(286\) 5.97773e7 0.151097
\(287\) −1.11811e8 −0.279189
\(288\) 0 0
\(289\) 6.18915e8 1.50830
\(290\) 3.70629e8 0.892373
\(291\) 0 0
\(292\) 8.61712e8 2.02545
\(293\) 4.12823e8 0.958798 0.479399 0.877597i \(-0.340855\pi\)
0.479399 + 0.877597i \(0.340855\pi\)
\(294\) 0 0
\(295\) −1.82306e8 −0.413451
\(296\) 5.88204e7 0.131828
\(297\) 0 0
\(298\) 5.57493e8 1.22035
\(299\) 3.00815e7 0.0650805
\(300\) 0 0
\(301\) −1.96391e7 −0.0415087
\(302\) −3.57160e8 −0.746171
\(303\) 0 0
\(304\) −8.85136e7 −0.180698
\(305\) −3.30817e8 −0.667635
\(306\) 0 0
\(307\) 3.39020e8 0.668715 0.334358 0.942446i \(-0.391481\pi\)
0.334358 + 0.942446i \(0.391481\pi\)
\(308\) −1.06867e8 −0.208410
\(309\) 0 0
\(310\) −3.42115e7 −0.0652237
\(311\) −3.14861e8 −0.593550 −0.296775 0.954947i \(-0.595911\pi\)
−0.296775 + 0.954947i \(0.595911\pi\)
\(312\) 0 0
\(313\) −7.61170e7 −0.140306 −0.0701531 0.997536i \(-0.522349\pi\)
−0.0701531 + 0.997536i \(0.522349\pi\)
\(314\) 3.66074e8 0.667291
\(315\) 0 0
\(316\) 5.86280e8 1.04520
\(317\) −3.04947e6 −0.00537671 −0.00268835 0.999996i \(-0.500856\pi\)
−0.00268835 + 0.999996i \(0.500856\pi\)
\(318\) 0 0
\(319\) 4.35144e8 0.750525
\(320\) −3.95703e8 −0.675064
\(321\) 0 0
\(322\) −9.36113e7 −0.156255
\(323\) 3.28683e8 0.542711
\(324\) 0 0
\(325\) −8.67448e7 −0.140169
\(326\) −1.70636e7 −0.0272777
\(327\) 0 0
\(328\) 3.51825e8 0.550514
\(329\) −2.80398e7 −0.0434100
\(330\) 0 0
\(331\) −6.34097e8 −0.961076 −0.480538 0.876974i \(-0.659559\pi\)
−0.480538 + 0.876974i \(0.659559\pi\)
\(332\) −1.44450e9 −2.16639
\(333\) 0 0
\(334\) 1.62884e9 2.39202
\(335\) 2.69293e8 0.391353
\(336\) 0 0
\(337\) 2.48720e8 0.354003 0.177001 0.984211i \(-0.443360\pi\)
0.177001 + 0.984211i \(0.443360\pi\)
\(338\) 1.05544e9 1.48670
\(339\) 0 0
\(340\) 6.81630e8 0.940529
\(341\) −4.01666e7 −0.0548560
\(342\) 0 0
\(343\) −3.91777e8 −0.524215
\(344\) 6.17965e7 0.0818482
\(345\) 0 0
\(346\) 1.82738e9 2.37172
\(347\) −7.91060e8 −1.01638 −0.508190 0.861245i \(-0.669685\pi\)
−0.508190 + 0.861245i \(0.669685\pi\)
\(348\) 0 0
\(349\) −5.53018e8 −0.696386 −0.348193 0.937423i \(-0.613205\pi\)
−0.348193 + 0.937423i \(0.613205\pi\)
\(350\) 2.69943e8 0.336538
\(351\) 0 0
\(352\) −6.24209e8 −0.762836
\(353\) −1.06776e9 −1.29200 −0.645999 0.763338i \(-0.723559\pi\)
−0.645999 + 0.763338i \(0.723559\pi\)
\(354\) 0 0
\(355\) −5.86645e8 −0.695947
\(356\) 1.35342e9 1.58986
\(357\) 0 0
\(358\) −1.30567e9 −1.50398
\(359\) −1.46087e9 −1.66641 −0.833203 0.552967i \(-0.813496\pi\)
−0.833203 + 0.552967i \(0.813496\pi\)
\(360\) 0 0
\(361\) −7.88910e8 −0.882576
\(362\) 4.08524e8 0.452624
\(363\) 0 0
\(364\) 5.87666e7 0.0638669
\(365\) 6.13047e8 0.659886
\(366\) 0 0
\(367\) 1.28374e9 1.35564 0.677822 0.735226i \(-0.262924\pi\)
0.677822 + 0.735226i \(0.262924\pi\)
\(368\) −1.88781e8 −0.197465
\(369\) 0 0
\(370\) 1.61372e8 0.165623
\(371\) 3.51772e8 0.357645
\(372\) 0 0
\(373\) −4.86215e8 −0.485118 −0.242559 0.970137i \(-0.577987\pi\)
−0.242559 + 0.970137i \(0.577987\pi\)
\(374\) 1.39303e9 1.37693
\(375\) 0 0
\(376\) 8.82303e7 0.0855974
\(377\) −2.39286e8 −0.229998
\(378\) 0 0
\(379\) −1.35199e9 −1.27567 −0.637834 0.770174i \(-0.720169\pi\)
−0.637834 + 0.770174i \(0.720169\pi\)
\(380\) 2.17673e8 0.203498
\(381\) 0 0
\(382\) 1.93682e9 1.77773
\(383\) 6.26582e8 0.569878 0.284939 0.958546i \(-0.408027\pi\)
0.284939 + 0.958546i \(0.408027\pi\)
\(384\) 0 0
\(385\) −7.60286e7 −0.0678991
\(386\) 2.04028e9 1.80565
\(387\) 0 0
\(388\) 1.05896e9 0.920384
\(389\) 1.29391e9 1.11450 0.557250 0.830345i \(-0.311856\pi\)
0.557250 + 0.830345i \(0.311856\pi\)
\(390\) 0 0
\(391\) 7.01011e8 0.593070
\(392\) 5.92672e8 0.496951
\(393\) 0 0
\(394\) −9.41575e8 −0.775565
\(395\) 4.17096e8 0.340523
\(396\) 0 0
\(397\) 2.44753e9 1.96319 0.981594 0.190981i \(-0.0611669\pi\)
0.981594 + 0.190981i \(0.0611669\pi\)
\(398\) −1.41070e9 −1.12161
\(399\) 0 0
\(400\) 5.44378e8 0.425296
\(401\) 8.46869e8 0.655860 0.327930 0.944702i \(-0.393649\pi\)
0.327930 + 0.944702i \(0.393649\pi\)
\(402\) 0 0
\(403\) 2.20877e7 0.0168106
\(404\) 2.57413e9 1.94221
\(405\) 0 0
\(406\) 7.44640e8 0.552211
\(407\) 1.89461e8 0.139297
\(408\) 0 0
\(409\) −1.06925e9 −0.772763 −0.386381 0.922339i \(-0.626275\pi\)
−0.386381 + 0.922339i \(0.626275\pi\)
\(410\) 9.65221e8 0.691645
\(411\) 0 0
\(412\) 1.54849e9 1.09086
\(413\) −3.66275e8 −0.255848
\(414\) 0 0
\(415\) −1.02766e9 −0.705801
\(416\) 3.43254e8 0.233770
\(417\) 0 0
\(418\) 4.44853e8 0.297920
\(419\) −8.44198e8 −0.560654 −0.280327 0.959905i \(-0.590443\pi\)
−0.280327 + 0.959905i \(0.590443\pi\)
\(420\) 0 0
\(421\) −8.77632e8 −0.573225 −0.286612 0.958047i \(-0.592529\pi\)
−0.286612 + 0.958047i \(0.592529\pi\)
\(422\) −1.28068e9 −0.829559
\(423\) 0 0
\(424\) −1.10689e9 −0.705217
\(425\) −2.02147e9 −1.27734
\(426\) 0 0
\(427\) −6.64653e8 −0.413140
\(428\) 3.80691e8 0.234704
\(429\) 0 0
\(430\) 1.69537e8 0.102831
\(431\) 4.03075e7 0.0242502 0.0121251 0.999926i \(-0.496140\pi\)
0.0121251 + 0.999926i \(0.496140\pi\)
\(432\) 0 0
\(433\) 1.00411e9 0.594391 0.297196 0.954817i \(-0.403949\pi\)
0.297196 + 0.954817i \(0.403949\pi\)
\(434\) −6.87351e7 −0.0403612
\(435\) 0 0
\(436\) 2.47564e9 1.43049
\(437\) 2.23862e8 0.128320
\(438\) 0 0
\(439\) 6.13553e8 0.346120 0.173060 0.984911i \(-0.444635\pi\)
0.173060 + 0.984911i \(0.444635\pi\)
\(440\) 2.39232e8 0.133886
\(441\) 0 0
\(442\) −7.66031e8 −0.421957
\(443\) 1.76951e9 0.967028 0.483514 0.875337i \(-0.339360\pi\)
0.483514 + 0.875337i \(0.339360\pi\)
\(444\) 0 0
\(445\) 9.62864e8 0.517970
\(446\) 1.76293e9 0.940944
\(447\) 0 0
\(448\) −7.95017e8 −0.417737
\(449\) 2.28688e9 1.19229 0.596145 0.802877i \(-0.296698\pi\)
0.596145 + 0.802877i \(0.296698\pi\)
\(450\) 0 0
\(451\) 1.13323e9 0.581704
\(452\) −9.83906e8 −0.501152
\(453\) 0 0
\(454\) −3.38599e9 −1.69820
\(455\) 4.18083e7 0.0208076
\(456\) 0 0
\(457\) 8.79369e8 0.430987 0.215494 0.976505i \(-0.430864\pi\)
0.215494 + 0.976505i \(0.430864\pi\)
\(458\) 2.39774e9 1.16620
\(459\) 0 0
\(460\) 4.64249e8 0.222381
\(461\) −1.43825e9 −0.683724 −0.341862 0.939750i \(-0.611058\pi\)
−0.341862 + 0.939750i \(0.611058\pi\)
\(462\) 0 0
\(463\) −3.40869e9 −1.59608 −0.798040 0.602605i \(-0.794130\pi\)
−0.798040 + 0.602605i \(0.794130\pi\)
\(464\) 1.50167e9 0.697850
\(465\) 0 0
\(466\) −1.30486e9 −0.597327
\(467\) −3.01086e9 −1.36798 −0.683992 0.729490i \(-0.739758\pi\)
−0.683992 + 0.729490i \(0.739758\pi\)
\(468\) 0 0
\(469\) 5.41043e8 0.242174
\(470\) 2.42057e8 0.107541
\(471\) 0 0
\(472\) 1.15252e9 0.504490
\(473\) 1.99047e8 0.0864854
\(474\) 0 0
\(475\) −6.45540e8 −0.276373
\(476\) 1.36948e9 0.582011
\(477\) 0 0
\(478\) −9.32385e8 −0.390479
\(479\) −3.75213e7 −0.0155992 −0.00779961 0.999970i \(-0.502483\pi\)
−0.00779961 + 0.999970i \(0.502483\pi\)
\(480\) 0 0
\(481\) −1.04185e8 −0.0426872
\(482\) 7.51414e9 3.05643
\(483\) 0 0
\(484\) −2.28452e9 −0.915874
\(485\) 7.53376e8 0.299858
\(486\) 0 0
\(487\) 1.34816e8 0.0528921 0.0264460 0.999650i \(-0.491581\pi\)
0.0264460 + 0.999650i \(0.491581\pi\)
\(488\) 2.09140e9 0.814644
\(489\) 0 0
\(490\) 1.62598e9 0.624350
\(491\) −1.26229e9 −0.481253 −0.240627 0.970618i \(-0.577353\pi\)
−0.240627 + 0.970618i \(0.577353\pi\)
\(492\) 0 0
\(493\) −5.57626e9 −2.09594
\(494\) −2.44625e8 −0.0912972
\(495\) 0 0
\(496\) −1.38614e8 −0.0510060
\(497\) −1.17864e9 −0.430660
\(498\) 0 0
\(499\) −1.34095e9 −0.483128 −0.241564 0.970385i \(-0.577660\pi\)
−0.241564 + 0.970385i \(0.577660\pi\)
\(500\) −2.99862e9 −1.07282
\(501\) 0 0
\(502\) −6.08326e9 −2.14622
\(503\) −5.65419e9 −1.98099 −0.990495 0.137546i \(-0.956078\pi\)
−0.990495 + 0.137546i \(0.956078\pi\)
\(504\) 0 0
\(505\) 1.83131e9 0.632764
\(506\) 9.48776e8 0.325564
\(507\) 0 0
\(508\) 5.00524e9 1.69396
\(509\) −4.83455e9 −1.62497 −0.812483 0.582985i \(-0.801885\pi\)
−0.812483 + 0.582985i \(0.801885\pi\)
\(510\) 0 0
\(511\) 1.23169e9 0.408345
\(512\) −3.01367e9 −0.992317
\(513\) 0 0
\(514\) 6.63297e9 2.15445
\(515\) 1.10164e9 0.355398
\(516\) 0 0
\(517\) 2.84191e8 0.0904470
\(518\) 3.24216e8 0.102490
\(519\) 0 0
\(520\) −1.31554e8 −0.0410292
\(521\) −1.47590e9 −0.457218 −0.228609 0.973518i \(-0.573418\pi\)
−0.228609 + 0.973518i \(0.573418\pi\)
\(522\) 0 0
\(523\) −3.13778e9 −0.959106 −0.479553 0.877513i \(-0.659201\pi\)
−0.479553 + 0.877513i \(0.659201\pi\)
\(524\) −2.25279e9 −0.684007
\(525\) 0 0
\(526\) −8.26994e9 −2.47772
\(527\) 5.14724e8 0.153193
\(528\) 0 0
\(529\) −2.92738e9 −0.859773
\(530\) −3.03671e9 −0.886008
\(531\) 0 0
\(532\) 4.37331e8 0.125927
\(533\) −6.23168e8 −0.178262
\(534\) 0 0
\(535\) 2.70835e8 0.0764656
\(536\) −1.70245e9 −0.477527
\(537\) 0 0
\(538\) 7.92193e9 2.19328
\(539\) 1.90901e9 0.525106
\(540\) 0 0
\(541\) −2.43519e9 −0.661214 −0.330607 0.943768i \(-0.607254\pi\)
−0.330607 + 0.943768i \(0.607254\pi\)
\(542\) 6.59606e9 1.77945
\(543\) 0 0
\(544\) 7.99909e9 2.13032
\(545\) 1.76124e9 0.466049
\(546\) 0 0
\(547\) −4.11336e9 −1.07458 −0.537292 0.843396i \(-0.680553\pi\)
−0.537292 + 0.843396i \(0.680553\pi\)
\(548\) 5.46761e9 1.41927
\(549\) 0 0
\(550\) −2.73594e9 −0.701193
\(551\) −1.78073e9 −0.453489
\(552\) 0 0
\(553\) 8.37998e8 0.210720
\(554\) 3.03466e9 0.758274
\(555\) 0 0
\(556\) −7.99935e9 −1.97375
\(557\) −5.89179e9 −1.44462 −0.722311 0.691568i \(-0.756920\pi\)
−0.722311 + 0.691568i \(0.756920\pi\)
\(558\) 0 0
\(559\) −1.09457e8 −0.0265033
\(560\) −2.62373e8 −0.0631337
\(561\) 0 0
\(562\) −1.60088e9 −0.380437
\(563\) 3.72337e9 0.879341 0.439671 0.898159i \(-0.355095\pi\)
0.439671 + 0.898159i \(0.355095\pi\)
\(564\) 0 0
\(565\) −6.99979e8 −0.163273
\(566\) 1.05386e10 2.44301
\(567\) 0 0
\(568\) 3.70872e9 0.849191
\(569\) −5.89440e9 −1.34136 −0.670682 0.741745i \(-0.733999\pi\)
−0.670682 + 0.741745i \(0.733999\pi\)
\(570\) 0 0
\(571\) −4.09269e9 −0.919988 −0.459994 0.887922i \(-0.652148\pi\)
−0.459994 + 0.887922i \(0.652148\pi\)
\(572\) −5.95615e8 −0.133070
\(573\) 0 0
\(574\) 1.93925e9 0.427998
\(575\) −1.37680e9 −0.302018
\(576\) 0 0
\(577\) 6.73874e9 1.46037 0.730185 0.683249i \(-0.239434\pi\)
0.730185 + 0.683249i \(0.239434\pi\)
\(578\) −1.07345e10 −2.31224
\(579\) 0 0
\(580\) −3.69291e9 −0.785906
\(581\) −2.06470e9 −0.436758
\(582\) 0 0
\(583\) −3.56530e9 −0.745172
\(584\) −3.87563e9 −0.805189
\(585\) 0 0
\(586\) −7.15999e9 −1.46984
\(587\) −6.12315e9 −1.24951 −0.624757 0.780819i \(-0.714802\pi\)
−0.624757 + 0.780819i \(0.714802\pi\)
\(588\) 0 0
\(589\) 1.64373e8 0.0331456
\(590\) 3.16191e9 0.633822
\(591\) 0 0
\(592\) 6.53827e8 0.129520
\(593\) 5.62188e9 1.10711 0.553554 0.832813i \(-0.313271\pi\)
0.553554 + 0.832813i \(0.313271\pi\)
\(594\) 0 0
\(595\) 9.74287e8 0.189617
\(596\) −5.55481e9 −1.07475
\(597\) 0 0
\(598\) −5.21734e8 −0.0997688
\(599\) −2.62393e9 −0.498836 −0.249418 0.968396i \(-0.580239\pi\)
−0.249418 + 0.968396i \(0.580239\pi\)
\(600\) 0 0
\(601\) 5.85029e9 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(602\) 3.40620e8 0.0636330
\(603\) 0 0
\(604\) 3.55870e9 0.657147
\(605\) −1.62527e9 −0.298389
\(606\) 0 0
\(607\) 3.51051e9 0.637103 0.318552 0.947905i \(-0.396804\pi\)
0.318552 + 0.947905i \(0.396804\pi\)
\(608\) 2.55444e9 0.460928
\(609\) 0 0
\(610\) 5.73769e9 1.02349
\(611\) −1.56277e8 −0.0277174
\(612\) 0 0
\(613\) −6.95815e9 −1.22006 −0.610032 0.792377i \(-0.708843\pi\)
−0.610032 + 0.792377i \(0.708843\pi\)
\(614\) −5.87996e9 −1.02514
\(615\) 0 0
\(616\) 4.80646e8 0.0828501
\(617\) 7.18978e8 0.123230 0.0616151 0.998100i \(-0.480375\pi\)
0.0616151 + 0.998100i \(0.480375\pi\)
\(618\) 0 0
\(619\) −1.57508e9 −0.266922 −0.133461 0.991054i \(-0.542609\pi\)
−0.133461 + 0.991054i \(0.542609\pi\)
\(620\) 3.40879e8 0.0574420
\(621\) 0 0
\(622\) 5.46094e9 0.909916
\(623\) 1.93451e9 0.320526
\(624\) 0 0
\(625\) 2.78932e9 0.457003
\(626\) 1.32017e9 0.215090
\(627\) 0 0
\(628\) −3.64753e9 −0.587678
\(629\) −2.42790e9 −0.389003
\(630\) 0 0
\(631\) 8.53967e9 1.35313 0.676564 0.736384i \(-0.263468\pi\)
0.676564 + 0.736384i \(0.263468\pi\)
\(632\) −2.63685e9 −0.415504
\(633\) 0 0
\(634\) 5.28899e7 0.00824252
\(635\) 3.56087e9 0.551885
\(636\) 0 0
\(637\) −1.04977e9 −0.160918
\(638\) −7.54712e9 −1.15056
\(639\) 0 0
\(640\) 2.93934e9 0.443221
\(641\) 1.19699e9 0.179509 0.0897547 0.995964i \(-0.471392\pi\)
0.0897547 + 0.995964i \(0.471392\pi\)
\(642\) 0 0
\(643\) 1.19758e10 1.77651 0.888254 0.459353i \(-0.151919\pi\)
0.888254 + 0.459353i \(0.151919\pi\)
\(644\) 9.32733e8 0.137612
\(645\) 0 0
\(646\) −5.70068e9 −0.831979
\(647\) 2.44410e9 0.354776 0.177388 0.984141i \(-0.443235\pi\)
0.177388 + 0.984141i \(0.443235\pi\)
\(648\) 0 0
\(649\) 3.71230e9 0.533072
\(650\) 1.50450e9 0.214880
\(651\) 0 0
\(652\) 1.70020e8 0.0240233
\(653\) −1.08778e10 −1.52878 −0.764389 0.644755i \(-0.776959\pi\)
−0.764389 + 0.644755i \(0.776959\pi\)
\(654\) 0 0
\(655\) −1.60270e9 −0.222847
\(656\) 3.91077e9 0.540877
\(657\) 0 0
\(658\) 4.86322e8 0.0665478
\(659\) 9.90053e9 1.34760 0.673798 0.738916i \(-0.264662\pi\)
0.673798 + 0.738916i \(0.264662\pi\)
\(660\) 0 0
\(661\) −1.16773e9 −0.157267 −0.0786337 0.996904i \(-0.525056\pi\)
−0.0786337 + 0.996904i \(0.525056\pi\)
\(662\) 1.09978e10 1.47334
\(663\) 0 0
\(664\) 6.49680e9 0.861214
\(665\) 3.11130e8 0.0410266
\(666\) 0 0
\(667\) −3.79791e9 −0.495570
\(668\) −1.62296e10 −2.10664
\(669\) 0 0
\(670\) −4.67062e9 −0.599946
\(671\) 6.73644e9 0.860799
\(672\) 0 0
\(673\) −3.21526e9 −0.406596 −0.203298 0.979117i \(-0.565166\pi\)
−0.203298 + 0.979117i \(0.565166\pi\)
\(674\) −4.31380e9 −0.542688
\(675\) 0 0
\(676\) −1.05163e10 −1.30933
\(677\) −1.11460e10 −1.38057 −0.690287 0.723536i \(-0.742516\pi\)
−0.690287 + 0.723536i \(0.742516\pi\)
\(678\) 0 0
\(679\) 1.51362e9 0.185556
\(680\) −3.06570e9 −0.373894
\(681\) 0 0
\(682\) 6.96648e8 0.0840946
\(683\) −1.36119e10 −1.63473 −0.817364 0.576121i \(-0.804566\pi\)
−0.817364 + 0.576121i \(0.804566\pi\)
\(684\) 0 0
\(685\) 3.88981e9 0.462394
\(686\) 6.79497e9 0.803624
\(687\) 0 0
\(688\) 6.86909e8 0.0804155
\(689\) 1.96057e9 0.228357
\(690\) 0 0
\(691\) 5.11725e9 0.590016 0.295008 0.955495i \(-0.404678\pi\)
0.295008 + 0.955495i \(0.404678\pi\)
\(692\) −1.82078e10 −2.08875
\(693\) 0 0
\(694\) 1.37201e10 1.55812
\(695\) −5.69097e9 −0.643042
\(696\) 0 0
\(697\) −1.45221e10 −1.62448
\(698\) 9.59153e9 1.06756
\(699\) 0 0
\(700\) −2.68968e9 −0.296386
\(701\) 5.39916e9 0.591988 0.295994 0.955190i \(-0.404349\pi\)
0.295994 + 0.955190i \(0.404349\pi\)
\(702\) 0 0
\(703\) −7.75328e8 −0.0841670
\(704\) 8.05771e9 0.870377
\(705\) 0 0
\(706\) 1.85192e10 1.98064
\(707\) 3.67933e9 0.391562
\(708\) 0 0
\(709\) −1.04570e9 −0.110191 −0.0550954 0.998481i \(-0.517546\pi\)
−0.0550954 + 0.998481i \(0.517546\pi\)
\(710\) 1.01748e10 1.06689
\(711\) 0 0
\(712\) −6.08715e9 −0.632025
\(713\) 3.50572e8 0.0362213
\(714\) 0 0
\(715\) −4.23738e8 −0.0433537
\(716\) 1.30095e10 1.32454
\(717\) 0 0
\(718\) 2.53373e10 2.55461
\(719\) 6.61572e8 0.0663783 0.0331892 0.999449i \(-0.489434\pi\)
0.0331892 + 0.999449i \(0.489434\pi\)
\(720\) 0 0
\(721\) 2.21334e9 0.219925
\(722\) 1.36828e10 1.35299
\(723\) 0 0
\(724\) −4.07049e9 −0.398623
\(725\) 1.09519e10 1.06735
\(726\) 0 0
\(727\) 1.73518e9 0.167484 0.0837421 0.996487i \(-0.473313\pi\)
0.0837421 + 0.996487i \(0.473313\pi\)
\(728\) −2.64308e8 −0.0253893
\(729\) 0 0
\(730\) −1.06327e10 −1.01161
\(731\) −2.55074e9 −0.241522
\(732\) 0 0
\(733\) −6.25015e9 −0.586174 −0.293087 0.956086i \(-0.594683\pi\)
−0.293087 + 0.956086i \(0.594683\pi\)
\(734\) −2.22651e10 −2.07821
\(735\) 0 0
\(736\) 5.44807e9 0.503698
\(737\) −5.48362e9 −0.504581
\(738\) 0 0
\(739\) 9.00137e9 0.820452 0.410226 0.911984i \(-0.365450\pi\)
0.410226 + 0.911984i \(0.365450\pi\)
\(740\) −1.60789e9 −0.145863
\(741\) 0 0
\(742\) −6.10112e9 −0.548272
\(743\) 1.37996e10 1.23426 0.617129 0.786862i \(-0.288296\pi\)
0.617129 + 0.786862i \(0.288296\pi\)
\(744\) 0 0
\(745\) −3.95185e9 −0.350149
\(746\) 8.43290e9 0.743689
\(747\) 0 0
\(748\) −1.38800e10 −1.21265
\(749\) 5.44141e8 0.0473178
\(750\) 0 0
\(751\) 2.50869e9 0.216126 0.108063 0.994144i \(-0.465535\pi\)
0.108063 + 0.994144i \(0.465535\pi\)
\(752\) 9.80739e8 0.0840990
\(753\) 0 0
\(754\) 4.15018e9 0.352588
\(755\) 2.53176e9 0.214096
\(756\) 0 0
\(757\) −1.59957e10 −1.34020 −0.670099 0.742272i \(-0.733748\pi\)
−0.670099 + 0.742272i \(0.733748\pi\)
\(758\) 2.34490e10 1.95561
\(759\) 0 0
\(760\) −9.79003e8 −0.0808978
\(761\) 1.66651e10 1.37076 0.685378 0.728187i \(-0.259637\pi\)
0.685378 + 0.728187i \(0.259637\pi\)
\(762\) 0 0
\(763\) 3.53856e9 0.288397
\(764\) −1.92982e10 −1.56564
\(765\) 0 0
\(766\) −1.08674e10 −0.873627
\(767\) −2.04140e9 −0.163359
\(768\) 0 0
\(769\) 9.23584e9 0.732376 0.366188 0.930541i \(-0.380663\pi\)
0.366188 + 0.930541i \(0.380663\pi\)
\(770\) 1.31864e9 0.104090
\(771\) 0 0
\(772\) −2.03291e10 −1.59022
\(773\) 2.96075e9 0.230554 0.115277 0.993333i \(-0.463224\pi\)
0.115277 + 0.993333i \(0.463224\pi\)
\(774\) 0 0
\(775\) −1.01093e9 −0.0780126
\(776\) −4.76278e9 −0.365885
\(777\) 0 0
\(778\) −2.24415e10 −1.70854
\(779\) −4.63751e9 −0.351482
\(780\) 0 0
\(781\) 1.19459e10 0.897303
\(782\) −1.21583e10 −0.909180
\(783\) 0 0
\(784\) 6.58795e9 0.488252
\(785\) −2.59496e9 −0.191464
\(786\) 0 0
\(787\) 9.32429e9 0.681874 0.340937 0.940086i \(-0.389256\pi\)
0.340937 + 0.940086i \(0.389256\pi\)
\(788\) 9.38176e9 0.683034
\(789\) 0 0
\(790\) −7.23411e9 −0.522024
\(791\) −1.40634e9 −0.101036
\(792\) 0 0
\(793\) −3.70438e9 −0.263791
\(794\) −4.24500e10 −3.00958
\(795\) 0 0
\(796\) 1.40560e10 0.987796
\(797\) 2.22955e10 1.55996 0.779978 0.625806i \(-0.215230\pi\)
0.779978 + 0.625806i \(0.215230\pi\)
\(798\) 0 0
\(799\) −3.64184e9 −0.252585
\(800\) −1.57104e10 −1.08485
\(801\) 0 0
\(802\) −1.46881e10 −1.00544
\(803\) −1.24835e10 −0.850808
\(804\) 0 0
\(805\) 6.63573e8 0.0448336
\(806\) −3.83088e8 −0.0257707
\(807\) 0 0
\(808\) −1.15774e10 −0.772096
\(809\) 5.80413e9 0.385405 0.192703 0.981257i \(-0.438275\pi\)
0.192703 + 0.981257i \(0.438275\pi\)
\(810\) 0 0
\(811\) −1.62210e10 −1.06783 −0.533917 0.845537i \(-0.679280\pi\)
−0.533917 + 0.845537i \(0.679280\pi\)
\(812\) −7.41951e9 −0.486328
\(813\) 0 0
\(814\) −3.28601e9 −0.213542
\(815\) 1.20957e8 0.00782671
\(816\) 0 0
\(817\) −8.14557e8 −0.0522570
\(818\) 1.85450e10 1.18465
\(819\) 0 0
\(820\) −9.61736e9 −0.609126
\(821\) −6.33227e9 −0.399354 −0.199677 0.979862i \(-0.563989\pi\)
−0.199677 + 0.979862i \(0.563989\pi\)
\(822\) 0 0
\(823\) −2.22297e10 −1.39006 −0.695031 0.718980i \(-0.744609\pi\)
−0.695031 + 0.718980i \(0.744609\pi\)
\(824\) −6.96450e9 −0.433655
\(825\) 0 0
\(826\) 6.35267e9 0.392216
\(827\) 4.64860e9 0.285794 0.142897 0.989738i \(-0.454358\pi\)
0.142897 + 0.989738i \(0.454358\pi\)
\(828\) 0 0
\(829\) −8.14688e9 −0.496650 −0.248325 0.968677i \(-0.579880\pi\)
−0.248325 + 0.968677i \(0.579880\pi\)
\(830\) 1.78238e10 1.08200
\(831\) 0 0
\(832\) −4.43095e9 −0.266726
\(833\) −2.44634e10 −1.46643
\(834\) 0 0
\(835\) −1.15462e10 −0.686335
\(836\) −4.43247e9 −0.262376
\(837\) 0 0
\(838\) 1.46417e10 0.859486
\(839\) −1.95604e10 −1.14343 −0.571715 0.820452i \(-0.693722\pi\)
−0.571715 + 0.820452i \(0.693722\pi\)
\(840\) 0 0
\(841\) 1.29610e10 0.751366
\(842\) 1.52216e10 0.878757
\(843\) 0 0
\(844\) 1.27606e10 0.730586
\(845\) −7.48158e9 −0.426574
\(846\) 0 0
\(847\) −3.26537e9 −0.184646
\(848\) −1.23038e10 −0.692873
\(849\) 0 0
\(850\) 3.50604e10 1.95817
\(851\) −1.65361e9 −0.0919770
\(852\) 0 0
\(853\) 1.58440e10 0.874061 0.437031 0.899447i \(-0.356030\pi\)
0.437031 + 0.899447i \(0.356030\pi\)
\(854\) 1.15277e10 0.633346
\(855\) 0 0
\(856\) −1.71220e9 −0.0933029
\(857\) −2.56284e10 −1.39088 −0.695438 0.718586i \(-0.744790\pi\)
−0.695438 + 0.718586i \(0.744790\pi\)
\(858\) 0 0
\(859\) −3.28663e9 −0.176919 −0.0884596 0.996080i \(-0.528194\pi\)
−0.0884596 + 0.996080i \(0.528194\pi\)
\(860\) −1.68925e9 −0.0905624
\(861\) 0 0
\(862\) −6.99092e8 −0.0371757
\(863\) −1.66288e10 −0.880688 −0.440344 0.897829i \(-0.645143\pi\)
−0.440344 + 0.897829i \(0.645143\pi\)
\(864\) 0 0
\(865\) −1.29536e10 −0.680508
\(866\) −1.74152e10 −0.911205
\(867\) 0 0
\(868\) 6.84869e8 0.0355458
\(869\) −8.49334e9 −0.439045
\(870\) 0 0
\(871\) 3.01545e9 0.154628
\(872\) −1.11344e10 −0.568671
\(873\) 0 0
\(874\) −3.88265e9 −0.196715
\(875\) −4.28607e9 −0.216287
\(876\) 0 0
\(877\) −6.52780e9 −0.326790 −0.163395 0.986561i \(-0.552244\pi\)
−0.163395 + 0.986561i \(0.552244\pi\)
\(878\) −1.06415e10 −0.530603
\(879\) 0 0
\(880\) 2.65922e9 0.131542
\(881\) −2.61062e10 −1.28626 −0.643128 0.765759i \(-0.722364\pi\)
−0.643128 + 0.765759i \(0.722364\pi\)
\(882\) 0 0
\(883\) 9.02672e9 0.441232 0.220616 0.975361i \(-0.429193\pi\)
0.220616 + 0.975361i \(0.429193\pi\)
\(884\) 7.63266e9 0.371615
\(885\) 0 0
\(886\) −3.06903e10 −1.48246
\(887\) −2.03347e9 −0.0978372 −0.0489186 0.998803i \(-0.515577\pi\)
−0.0489186 + 0.998803i \(0.515577\pi\)
\(888\) 0 0
\(889\) 7.15423e9 0.341513
\(890\) −1.66999e10 −0.794051
\(891\) 0 0
\(892\) −1.75657e10 −0.828682
\(893\) −1.16299e9 −0.0546507
\(894\) 0 0
\(895\) 9.25537e9 0.431532
\(896\) 5.90551e9 0.274270
\(897\) 0 0
\(898\) −3.96637e10 −1.82779
\(899\) −2.78866e9 −0.128008
\(900\) 0 0
\(901\) 4.56884e10 2.08099
\(902\) −1.96548e10 −0.891755
\(903\) 0 0
\(904\) 4.42521e9 0.199225
\(905\) −2.89587e9 −0.129870
\(906\) 0 0
\(907\) 3.47024e10 1.54431 0.772155 0.635435i \(-0.219179\pi\)
0.772155 + 0.635435i \(0.219179\pi\)
\(908\) 3.37376e10 1.49559
\(909\) 0 0
\(910\) −7.25121e8 −0.0318982
\(911\) 3.83459e10 1.68037 0.840183 0.542302i \(-0.182447\pi\)
0.840183 + 0.542302i \(0.182447\pi\)
\(912\) 0 0
\(913\) 2.09263e10 0.910007
\(914\) −1.52518e10 −0.660706
\(915\) 0 0
\(916\) −2.38908e10 −1.02706
\(917\) −3.22002e9 −0.137900
\(918\) 0 0
\(919\) 7.63731e9 0.324591 0.162295 0.986742i \(-0.448110\pi\)
0.162295 + 0.986742i \(0.448110\pi\)
\(920\) −2.08800e9 −0.0884044
\(921\) 0 0
\(922\) 2.49450e10 1.04815
\(923\) −6.56905e9 −0.274977
\(924\) 0 0
\(925\) 4.76844e9 0.198098
\(926\) 5.91203e10 2.44680
\(927\) 0 0
\(928\) −4.33372e10 −1.78009
\(929\) 1.44540e10 0.591469 0.295734 0.955270i \(-0.404436\pi\)
0.295734 + 0.955270i \(0.404436\pi\)
\(930\) 0 0
\(931\) −7.81218e9 −0.317284
\(932\) 1.30015e10 0.526062
\(933\) 0 0
\(934\) 5.22202e10 2.09713
\(935\) −9.87466e9 −0.395077
\(936\) 0 0
\(937\) −2.76136e10 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(938\) −9.38384e9 −0.371254
\(939\) 0 0
\(940\) −2.41183e9 −0.0947108
\(941\) −1.88863e10 −0.738894 −0.369447 0.929252i \(-0.620453\pi\)
−0.369447 + 0.929252i \(0.620453\pi\)
\(942\) 0 0
\(943\) −9.89081e9 −0.384097
\(944\) 1.28111e10 0.495659
\(945\) 0 0
\(946\) −3.45228e9 −0.132583
\(947\) 1.55375e10 0.594506 0.297253 0.954799i \(-0.403930\pi\)
0.297253 + 0.954799i \(0.403930\pi\)
\(948\) 0 0
\(949\) 6.86469e9 0.260729
\(950\) 1.11962e10 0.423681
\(951\) 0 0
\(952\) −6.15936e9 −0.231370
\(953\) 1.05814e10 0.396021 0.198011 0.980200i \(-0.436552\pi\)
0.198011 + 0.980200i \(0.436552\pi\)
\(954\) 0 0
\(955\) −1.37293e10 −0.510079
\(956\) 9.29019e9 0.343892
\(957\) 0 0
\(958\) 6.50768e8 0.0239137
\(959\) 7.81511e9 0.286134
\(960\) 0 0
\(961\) −2.72552e10 −0.990644
\(962\) 1.80699e9 0.0654398
\(963\) 0 0
\(964\) −7.48701e10 −2.69178
\(965\) −1.44627e10 −0.518090
\(966\) 0 0
\(967\) −1.98827e9 −0.0707102 −0.0353551 0.999375i \(-0.511256\pi\)
−0.0353551 + 0.999375i \(0.511256\pi\)
\(968\) 1.02748e10 0.364092
\(969\) 0 0
\(970\) −1.30665e10 −0.459684
\(971\) 1.06982e10 0.375012 0.187506 0.982263i \(-0.439960\pi\)
0.187506 + 0.982263i \(0.439960\pi\)
\(972\) 0 0
\(973\) −1.14339e10 −0.397922
\(974\) −2.33825e9 −0.0810838
\(975\) 0 0
\(976\) 2.32473e10 0.800384
\(977\) 3.25937e10 1.11816 0.559078 0.829115i \(-0.311155\pi\)
0.559078 + 0.829115i \(0.311155\pi\)
\(978\) 0 0
\(979\) −1.96068e10 −0.667832
\(980\) −1.62011e10 −0.549860
\(981\) 0 0
\(982\) 2.18931e10 0.737764
\(983\) 3.26564e10 1.09656 0.548278 0.836296i \(-0.315284\pi\)
0.548278 + 0.836296i \(0.315284\pi\)
\(984\) 0 0
\(985\) 6.67446e9 0.222530
\(986\) 9.67145e10 3.21309
\(987\) 0 0
\(988\) 2.43742e9 0.0804047
\(989\) −1.73728e9 −0.0571060
\(990\) 0 0
\(991\) 1.86068e10 0.607316 0.303658 0.952781i \(-0.401792\pi\)
0.303658 + 0.952781i \(0.401792\pi\)
\(992\) 4.00030e9 0.130107
\(993\) 0 0
\(994\) 2.04424e10 0.660205
\(995\) 9.99987e9 0.321821
\(996\) 0 0
\(997\) 5.77600e10 1.84584 0.922920 0.384993i \(-0.125796\pi\)
0.922920 + 0.384993i \(0.125796\pi\)
\(998\) 2.32575e10 0.740638
\(999\) 0 0
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 387.8.a.d.1.3 13
3.2 odd 2 43.8.a.b.1.11 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.11 13 3.2 odd 2
387.8.a.d.1.3 13 1.1 even 1 trivial