Properties

Label 387.8.a.d.1.12
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 1.12
Root \(-20.1662\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q+19.1662 q^{2} +239.342 q^{4} +174.722 q^{5} -1126.43 q^{7} +2133.99 q^{8} +O(q^{10})\) \(q+19.1662 q^{2} +239.342 q^{4} +174.722 q^{5} -1126.43 q^{7} +2133.99 q^{8} +3348.75 q^{10} +8178.46 q^{11} -13500.8 q^{13} -21589.4 q^{14} +10264.6 q^{16} -37508.8 q^{17} -26617.7 q^{19} +41818.2 q^{20} +156750. q^{22} +20061.6 q^{23} -47597.3 q^{25} -258757. q^{26} -269602. q^{28} +27521.8 q^{29} -184114. q^{31} -76416.8 q^{32} -718900. q^{34} -196812. q^{35} +301128. q^{37} -510158. q^{38} +372855. q^{40} +507344. q^{41} -79507.0 q^{43} +1.95745e6 q^{44} +384503. q^{46} +137672. q^{47} +445307. q^{49} -912256. q^{50} -3.23129e6 q^{52} -577240. q^{53} +1.42896e6 q^{55} -2.40379e6 q^{56} +527487. q^{58} -1.70368e6 q^{59} +1.24127e6 q^{61} -3.52877e6 q^{62} -2.77849e6 q^{64} -2.35888e6 q^{65} -1.24429e6 q^{67} -8.97742e6 q^{68} -3.77214e6 q^{70} +3.10722e6 q^{71} -3.60450e6 q^{73} +5.77146e6 q^{74} -6.37071e6 q^{76} -9.21248e6 q^{77} -3.14376e6 q^{79} +1.79346e6 q^{80} +9.72384e6 q^{82} -347048. q^{83} -6.55361e6 q^{85} -1.52384e6 q^{86} +1.74527e7 q^{88} +4.60056e6 q^{89} +1.52077e7 q^{91} +4.80156e6 q^{92} +2.63864e6 q^{94} -4.65069e6 q^{95} -1.55843e7 q^{97} +8.53482e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 19.1662 1.69406 0.847032 0.531541i \(-0.178387\pi\)
0.847032 + 0.531541i \(0.178387\pi\)
\(3\) 0 0
\(4\) 239.342 1.86986
\(5\) 174.722 0.625104 0.312552 0.949901i \(-0.398816\pi\)
0.312552 + 0.949901i \(0.398816\pi\)
\(6\) 0 0
\(7\) −1126.43 −1.24126 −0.620629 0.784104i \(-0.713123\pi\)
−0.620629 + 0.784104i \(0.713123\pi\)
\(8\) 2133.99 1.47359
\(9\) 0 0
\(10\) 3348.75 1.05897
\(11\) 8178.46 1.85267 0.926333 0.376705i \(-0.122943\pi\)
0.926333 + 0.376705i \(0.122943\pi\)
\(12\) 0 0
\(13\) −13500.8 −1.70434 −0.852170 0.523265i \(-0.824714\pi\)
−0.852170 + 0.523265i \(0.824714\pi\)
\(14\) −21589.4 −2.10277
\(15\) 0 0
\(16\) 10264.6 0.626504
\(17\) −37508.8 −1.85166 −0.925832 0.377935i \(-0.876634\pi\)
−0.925832 + 0.377935i \(0.876634\pi\)
\(18\) 0 0
\(19\) −26617.7 −0.890292 −0.445146 0.895458i \(-0.646848\pi\)
−0.445146 + 0.895458i \(0.646848\pi\)
\(20\) 41818.2 1.16885
\(21\) 0 0
\(22\) 156750. 3.13854
\(23\) 20061.6 0.343809 0.171905 0.985114i \(-0.445008\pi\)
0.171905 + 0.985114i \(0.445008\pi\)
\(24\) 0 0
\(25\) −47597.3 −0.609245
\(26\) −258757. −2.88726
\(27\) 0 0
\(28\) −269602. −2.32097
\(29\) 27521.8 0.209548 0.104774 0.994496i \(-0.466588\pi\)
0.104774 + 0.994496i \(0.466588\pi\)
\(30\) 0 0
\(31\) −184114. −1.11000 −0.554999 0.831851i \(-0.687281\pi\)
−0.554999 + 0.831851i \(0.687281\pi\)
\(32\) −76416.8 −0.412253
\(33\) 0 0
\(34\) −718900. −3.13684
\(35\) −196812. −0.775915
\(36\) 0 0
\(37\) 301128. 0.977338 0.488669 0.872469i \(-0.337483\pi\)
0.488669 + 0.872469i \(0.337483\pi\)
\(38\) −510158. −1.50821
\(39\) 0 0
\(40\) 372855. 0.921148
\(41\) 507344. 1.14963 0.574817 0.818282i \(-0.305073\pi\)
0.574817 + 0.818282i \(0.305073\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) 1.95745e6 3.46422
\(45\) 0 0
\(46\) 384503. 0.582435
\(47\) 137672. 0.193420 0.0967102 0.995313i \(-0.469168\pi\)
0.0967102 + 0.995313i \(0.469168\pi\)
\(48\) 0 0
\(49\) 445307. 0.540721
\(50\) −912256. −1.03210
\(51\) 0 0
\(52\) −3.23129e6 −3.18687
\(53\) −577240. −0.532588 −0.266294 0.963892i \(-0.585799\pi\)
−0.266294 + 0.963892i \(0.585799\pi\)
\(54\) 0 0
\(55\) 1.42896e6 1.15811
\(56\) −2.40379e6 −1.82911
\(57\) 0 0
\(58\) 527487. 0.354988
\(59\) −1.70368e6 −1.07995 −0.539977 0.841680i \(-0.681567\pi\)
−0.539977 + 0.841680i \(0.681567\pi\)
\(60\) 0 0
\(61\) 1.24127e6 0.700182 0.350091 0.936716i \(-0.386151\pi\)
0.350091 + 0.936716i \(0.386151\pi\)
\(62\) −3.52877e6 −1.88041
\(63\) 0 0
\(64\) −2.77849e6 −1.32489
\(65\) −2.35888e6 −1.06539
\(66\) 0 0
\(67\) −1.24429e6 −0.505430 −0.252715 0.967541i \(-0.581323\pi\)
−0.252715 + 0.967541i \(0.581323\pi\)
\(68\) −8.97742e6 −3.46234
\(69\) 0 0
\(70\) −3.77214e6 −1.31445
\(71\) 3.10722e6 1.03031 0.515154 0.857098i \(-0.327735\pi\)
0.515154 + 0.857098i \(0.327735\pi\)
\(72\) 0 0
\(73\) −3.60450e6 −1.08446 −0.542232 0.840229i \(-0.682421\pi\)
−0.542232 + 0.840229i \(0.682421\pi\)
\(74\) 5.77146e6 1.65567
\(75\) 0 0
\(76\) −6.37071e6 −1.66472
\(77\) −9.21248e6 −2.29964
\(78\) 0 0
\(79\) −3.14376e6 −0.717390 −0.358695 0.933455i \(-0.616778\pi\)
−0.358695 + 0.933455i \(0.616778\pi\)
\(80\) 1.79346e6 0.391630
\(81\) 0 0
\(82\) 9.72384e6 1.94755
\(83\) −347048. −0.0666218 −0.0333109 0.999445i \(-0.510605\pi\)
−0.0333109 + 0.999445i \(0.510605\pi\)
\(84\) 0 0
\(85\) −6.55361e6 −1.15748
\(86\) −1.52384e6 −0.258342
\(87\) 0 0
\(88\) 1.74527e7 2.73007
\(89\) 4.60056e6 0.691745 0.345872 0.938282i \(-0.387583\pi\)
0.345872 + 0.938282i \(0.387583\pi\)
\(90\) 0 0
\(91\) 1.52077e7 2.11553
\(92\) 4.80156e6 0.642873
\(93\) 0 0
\(94\) 2.63864e6 0.327667
\(95\) −4.65069e6 −0.556525
\(96\) 0 0
\(97\) −1.55843e7 −1.73375 −0.866875 0.498526i \(-0.833875\pi\)
−0.866875 + 0.498526i \(0.833875\pi\)
\(98\) 8.53482e6 0.916016
\(99\) 0 0
\(100\) −1.13920e7 −1.13920
\(101\) 9.18925e6 0.887474 0.443737 0.896157i \(-0.353652\pi\)
0.443737 + 0.896157i \(0.353652\pi\)
\(102\) 0 0
\(103\) −1.58567e7 −1.42982 −0.714910 0.699216i \(-0.753533\pi\)
−0.714910 + 0.699216i \(0.753533\pi\)
\(104\) −2.88104e7 −2.51150
\(105\) 0 0
\(106\) −1.10635e7 −0.902238
\(107\) 833526. 0.0657773 0.0328886 0.999459i \(-0.489529\pi\)
0.0328886 + 0.999459i \(0.489529\pi\)
\(108\) 0 0
\(109\) 7.34548e6 0.543285 0.271642 0.962398i \(-0.412433\pi\)
0.271642 + 0.962398i \(0.412433\pi\)
\(110\) 2.73876e7 1.96191
\(111\) 0 0
\(112\) −1.15624e7 −0.777653
\(113\) −3.08279e6 −0.200988 −0.100494 0.994938i \(-0.532042\pi\)
−0.100494 + 0.994938i \(0.532042\pi\)
\(114\) 0 0
\(115\) 3.50519e6 0.214916
\(116\) 6.58710e6 0.391824
\(117\) 0 0
\(118\) −3.26530e7 −1.82951
\(119\) 4.22512e7 2.29839
\(120\) 0 0
\(121\) 4.74000e7 2.43237
\(122\) 2.37903e7 1.18615
\(123\) 0 0
\(124\) −4.40662e7 −2.07553
\(125\) −2.19664e7 −1.00595
\(126\) 0 0
\(127\) −1.91611e6 −0.0830057 −0.0415028 0.999138i \(-0.513215\pi\)
−0.0415028 + 0.999138i \(0.513215\pi\)
\(128\) −4.34716e7 −1.83219
\(129\) 0 0
\(130\) −4.52106e7 −1.80484
\(131\) 2.07138e7 0.805026 0.402513 0.915414i \(-0.368137\pi\)
0.402513 + 0.915414i \(0.368137\pi\)
\(132\) 0 0
\(133\) 2.99830e7 1.10508
\(134\) −2.38483e7 −0.856231
\(135\) 0 0
\(136\) −8.00434e7 −2.72860
\(137\) 3.16796e7 1.05259 0.526294 0.850303i \(-0.323581\pi\)
0.526294 + 0.850303i \(0.323581\pi\)
\(138\) 0 0
\(139\) −4.90696e7 −1.54975 −0.774873 0.632117i \(-0.782186\pi\)
−0.774873 + 0.632117i \(0.782186\pi\)
\(140\) −4.71054e7 −1.45085
\(141\) 0 0
\(142\) 5.95534e7 1.74541
\(143\) −1.10415e8 −3.15757
\(144\) 0 0
\(145\) 4.80866e6 0.130989
\(146\) −6.90844e7 −1.83715
\(147\) 0 0
\(148\) 7.20724e7 1.82748
\(149\) −8.04857e6 −0.199327 −0.0996637 0.995021i \(-0.531777\pi\)
−0.0996637 + 0.995021i \(0.531777\pi\)
\(150\) 0 0
\(151\) −9.45579e6 −0.223500 −0.111750 0.993736i \(-0.535646\pi\)
−0.111750 + 0.993736i \(0.535646\pi\)
\(152\) −5.68018e7 −1.31193
\(153\) 0 0
\(154\) −1.76568e8 −3.89573
\(155\) −3.21688e7 −0.693864
\(156\) 0 0
\(157\) 5.50699e7 1.13571 0.567853 0.823130i \(-0.307774\pi\)
0.567853 + 0.823130i \(0.307774\pi\)
\(158\) −6.02539e7 −1.21530
\(159\) 0 0
\(160\) −1.33517e7 −0.257701
\(161\) −2.25980e7 −0.426756
\(162\) 0 0
\(163\) 3.56380e7 0.644551 0.322275 0.946646i \(-0.395552\pi\)
0.322275 + 0.946646i \(0.395552\pi\)
\(164\) 1.21429e8 2.14965
\(165\) 0 0
\(166\) −6.65158e6 −0.112862
\(167\) 4.96680e7 0.825218 0.412609 0.910908i \(-0.364618\pi\)
0.412609 + 0.910908i \(0.364618\pi\)
\(168\) 0 0
\(169\) 1.19522e8 1.90477
\(170\) −1.25608e8 −1.96085
\(171\) 0 0
\(172\) −1.90293e7 −0.285150
\(173\) −5.97290e7 −0.877049 −0.438524 0.898719i \(-0.644499\pi\)
−0.438524 + 0.898719i \(0.644499\pi\)
\(174\) 0 0
\(175\) 5.36151e7 0.756230
\(176\) 8.39490e7 1.16070
\(177\) 0 0
\(178\) 8.81751e7 1.17186
\(179\) 5.19556e6 0.0677090 0.0338545 0.999427i \(-0.489222\pi\)
0.0338545 + 0.999427i \(0.489222\pi\)
\(180\) 0 0
\(181\) 9.47873e6 0.118816 0.0594080 0.998234i \(-0.481079\pi\)
0.0594080 + 0.998234i \(0.481079\pi\)
\(182\) 2.91473e8 3.58384
\(183\) 0 0
\(184\) 4.28111e7 0.506634
\(185\) 5.26136e7 0.610938
\(186\) 0 0
\(187\) −3.06764e8 −3.43052
\(188\) 3.29506e7 0.361668
\(189\) 0 0
\(190\) −8.91359e7 −0.942790
\(191\) −1.10607e8 −1.14859 −0.574295 0.818648i \(-0.694724\pi\)
−0.574295 + 0.818648i \(0.694724\pi\)
\(192\) 0 0
\(193\) −8.62108e7 −0.863199 −0.431599 0.902065i \(-0.642051\pi\)
−0.431599 + 0.902065i \(0.642051\pi\)
\(194\) −2.98691e8 −2.93708
\(195\) 0 0
\(196\) 1.06580e8 1.01107
\(197\) 1.14482e8 1.06686 0.533429 0.845845i \(-0.320903\pi\)
0.533429 + 0.845845i \(0.320903\pi\)
\(198\) 0 0
\(199\) −3.94085e7 −0.354490 −0.177245 0.984167i \(-0.556719\pi\)
−0.177245 + 0.984167i \(0.556719\pi\)
\(200\) −1.01572e8 −0.897778
\(201\) 0 0
\(202\) 1.76123e8 1.50344
\(203\) −3.10014e7 −0.260103
\(204\) 0 0
\(205\) 8.86442e7 0.718640
\(206\) −3.03911e8 −2.42221
\(207\) 0 0
\(208\) −1.38580e8 −1.06778
\(209\) −2.17692e8 −1.64941
\(210\) 0 0
\(211\) −1.39696e8 −1.02375 −0.511876 0.859059i \(-0.671049\pi\)
−0.511876 + 0.859059i \(0.671049\pi\)
\(212\) −1.38158e8 −0.995862
\(213\) 0 0
\(214\) 1.59755e7 0.111431
\(215\) −1.38916e7 −0.0953275
\(216\) 0 0
\(217\) 2.07393e8 1.37779
\(218\) 1.40785e8 0.920360
\(219\) 0 0
\(220\) 3.42009e8 2.16550
\(221\) 5.06397e8 3.15586
\(222\) 0 0
\(223\) −1.20926e8 −0.730218 −0.365109 0.930965i \(-0.618968\pi\)
−0.365109 + 0.930965i \(0.618968\pi\)
\(224\) 8.60784e7 0.511712
\(225\) 0 0
\(226\) −5.90852e7 −0.340486
\(227\) 1.15625e8 0.656084 0.328042 0.944663i \(-0.393611\pi\)
0.328042 + 0.944663i \(0.393611\pi\)
\(228\) 0 0
\(229\) −1.39236e8 −0.766175 −0.383087 0.923712i \(-0.625139\pi\)
−0.383087 + 0.923712i \(0.625139\pi\)
\(230\) 6.71811e7 0.364082
\(231\) 0 0
\(232\) 5.87311e7 0.308788
\(233\) 3.66295e7 0.189708 0.0948539 0.995491i \(-0.469762\pi\)
0.0948539 + 0.995491i \(0.469762\pi\)
\(234\) 0 0
\(235\) 2.40543e7 0.120908
\(236\) −4.07761e8 −2.01936
\(237\) 0 0
\(238\) 8.09792e8 3.89363
\(239\) 3.71440e7 0.175993 0.0879967 0.996121i \(-0.471954\pi\)
0.0879967 + 0.996121i \(0.471954\pi\)
\(240\) 0 0
\(241\) 3.96773e7 0.182592 0.0912962 0.995824i \(-0.470899\pi\)
0.0912962 + 0.995824i \(0.470899\pi\)
\(242\) 9.08477e8 4.12060
\(243\) 0 0
\(244\) 2.97087e8 1.30924
\(245\) 7.78049e7 0.338007
\(246\) 0 0
\(247\) 3.59359e8 1.51736
\(248\) −3.92898e8 −1.63568
\(249\) 0 0
\(250\) −4.21012e8 −1.70414
\(251\) −6.08382e7 −0.242839 −0.121419 0.992601i \(-0.538745\pi\)
−0.121419 + 0.992601i \(0.538745\pi\)
\(252\) 0 0
\(253\) 1.64073e8 0.636963
\(254\) −3.67245e7 −0.140617
\(255\) 0 0
\(256\) −4.77537e8 −1.77896
\(257\) −9.91857e7 −0.364488 −0.182244 0.983253i \(-0.558336\pi\)
−0.182244 + 0.983253i \(0.558336\pi\)
\(258\) 0 0
\(259\) −3.39200e8 −1.21313
\(260\) −5.64577e8 −1.99212
\(261\) 0 0
\(262\) 3.97004e8 1.36377
\(263\) 2.73541e8 0.927209 0.463605 0.886042i \(-0.346556\pi\)
0.463605 + 0.886042i \(0.346556\pi\)
\(264\) 0 0
\(265\) −1.00857e8 −0.332923
\(266\) 5.74659e8 1.87208
\(267\) 0 0
\(268\) −2.97811e8 −0.945081
\(269\) −2.75672e8 −0.863494 −0.431747 0.901995i \(-0.642103\pi\)
−0.431747 + 0.901995i \(0.642103\pi\)
\(270\) 0 0
\(271\) 4.75651e8 1.45176 0.725882 0.687819i \(-0.241432\pi\)
0.725882 + 0.687819i \(0.241432\pi\)
\(272\) −3.85015e8 −1.16008
\(273\) 0 0
\(274\) 6.07177e8 1.78315
\(275\) −3.89272e8 −1.12873
\(276\) 0 0
\(277\) 2.38870e8 0.675279 0.337639 0.941276i \(-0.390372\pi\)
0.337639 + 0.941276i \(0.390372\pi\)
\(278\) −9.40475e8 −2.62537
\(279\) 0 0
\(280\) −4.19995e8 −1.14338
\(281\) 5.54874e8 1.49184 0.745920 0.666036i \(-0.232010\pi\)
0.745920 + 0.666036i \(0.232010\pi\)
\(282\) 0 0
\(283\) −2.03982e7 −0.0534981 −0.0267491 0.999642i \(-0.508516\pi\)
−0.0267491 + 0.999642i \(0.508516\pi\)
\(284\) 7.43686e8 1.92653
\(285\) 0 0
\(286\) −2.11624e9 −5.34913
\(287\) −5.71489e8 −1.42699
\(288\) 0 0
\(289\) 9.96573e8 2.42866
\(290\) 9.21635e7 0.221904
\(291\) 0 0
\(292\) −8.62707e8 −2.02779
\(293\) −2.34109e8 −0.543728 −0.271864 0.962336i \(-0.587640\pi\)
−0.271864 + 0.962336i \(0.587640\pi\)
\(294\) 0 0
\(295\) −2.97670e8 −0.675084
\(296\) 6.42603e8 1.44020
\(297\) 0 0
\(298\) −1.54260e8 −0.337673
\(299\) −2.70846e8 −0.585967
\(300\) 0 0
\(301\) 8.95593e7 0.189290
\(302\) −1.81231e8 −0.378624
\(303\) 0 0
\(304\) −2.73221e8 −0.557772
\(305\) 2.16877e8 0.437687
\(306\) 0 0
\(307\) 3.42975e8 0.676515 0.338258 0.941054i \(-0.390162\pi\)
0.338258 + 0.941054i \(0.390162\pi\)
\(308\) −2.20493e9 −4.29999
\(309\) 0 0
\(310\) −6.16553e8 −1.17545
\(311\) 7.24592e8 1.36594 0.682971 0.730446i \(-0.260688\pi\)
0.682971 + 0.730446i \(0.260688\pi\)
\(312\) 0 0
\(313\) 2.90144e8 0.534820 0.267410 0.963583i \(-0.413832\pi\)
0.267410 + 0.963583i \(0.413832\pi\)
\(314\) 1.05548e9 1.92396
\(315\) 0 0
\(316\) −7.52433e8 −1.34141
\(317\) −6.90711e8 −1.21784 −0.608918 0.793233i \(-0.708396\pi\)
−0.608918 + 0.793233i \(0.708396\pi\)
\(318\) 0 0
\(319\) 2.25086e8 0.388222
\(320\) −4.85463e8 −0.828193
\(321\) 0 0
\(322\) −4.33117e8 −0.722952
\(323\) 9.98398e8 1.64852
\(324\) 0 0
\(325\) 6.42599e8 1.03836
\(326\) 6.83044e8 1.09191
\(327\) 0 0
\(328\) 1.08267e9 1.69409
\(329\) −1.55078e8 −0.240085
\(330\) 0 0
\(331\) −1.29514e9 −1.96300 −0.981498 0.191470i \(-0.938675\pi\)
−0.981498 + 0.191470i \(0.938675\pi\)
\(332\) −8.30630e7 −0.124573
\(333\) 0 0
\(334\) 9.51944e8 1.39797
\(335\) −2.17405e8 −0.315946
\(336\) 0 0
\(337\) −4.49309e8 −0.639500 −0.319750 0.947502i \(-0.603599\pi\)
−0.319750 + 0.947502i \(0.603599\pi\)
\(338\) 2.29077e9 3.22681
\(339\) 0 0
\(340\) −1.56855e9 −2.16433
\(341\) −1.50577e9 −2.05645
\(342\) 0 0
\(343\) 4.26057e8 0.570084
\(344\) −1.69667e8 −0.224721
\(345\) 0 0
\(346\) −1.14477e9 −1.48578
\(347\) −6.18982e7 −0.0795288 −0.0397644 0.999209i \(-0.512661\pi\)
−0.0397644 + 0.999209i \(0.512661\pi\)
\(348\) 0 0
\(349\) 7.40278e8 0.932193 0.466096 0.884734i \(-0.345660\pi\)
0.466096 + 0.884734i \(0.345660\pi\)
\(350\) 1.02760e9 1.28110
\(351\) 0 0
\(352\) −6.24972e8 −0.763767
\(353\) −1.53714e9 −1.85995 −0.929976 0.367620i \(-0.880173\pi\)
−0.929976 + 0.367620i \(0.880173\pi\)
\(354\) 0 0
\(355\) 5.42899e8 0.644050
\(356\) 1.10111e9 1.29346
\(357\) 0 0
\(358\) 9.95789e7 0.114704
\(359\) 1.14737e9 1.30880 0.654400 0.756149i \(-0.272921\pi\)
0.654400 + 0.756149i \(0.272921\pi\)
\(360\) 0 0
\(361\) −1.85371e8 −0.207380
\(362\) 1.81671e8 0.201282
\(363\) 0 0
\(364\) 3.63983e9 3.95573
\(365\) −6.29785e8 −0.677903
\(366\) 0 0
\(367\) −1.07850e9 −1.13890 −0.569452 0.822024i \(-0.692845\pi\)
−0.569452 + 0.822024i \(0.692845\pi\)
\(368\) 2.05925e8 0.215398
\(369\) 0 0
\(370\) 1.00840e9 1.03497
\(371\) 6.50222e8 0.661079
\(372\) 0 0
\(373\) 4.29210e8 0.428242 0.214121 0.976807i \(-0.431311\pi\)
0.214121 + 0.976807i \(0.431311\pi\)
\(374\) −5.87949e9 −5.81152
\(375\) 0 0
\(376\) 2.93790e8 0.285023
\(377\) −3.71565e8 −0.357141
\(378\) 0 0
\(379\) 4.96839e8 0.468790 0.234395 0.972141i \(-0.424689\pi\)
0.234395 + 0.972141i \(0.424689\pi\)
\(380\) −1.11310e9 −1.04062
\(381\) 0 0
\(382\) −2.11991e9 −1.94579
\(383\) 8.74354e7 0.0795228 0.0397614 0.999209i \(-0.487340\pi\)
0.0397614 + 0.999209i \(0.487340\pi\)
\(384\) 0 0
\(385\) −1.60962e9 −1.43751
\(386\) −1.65233e9 −1.46232
\(387\) 0 0
\(388\) −3.72997e9 −3.24186
\(389\) 1.92667e9 1.65953 0.829763 0.558116i \(-0.188476\pi\)
0.829763 + 0.558116i \(0.188476\pi\)
\(390\) 0 0
\(391\) −7.52485e8 −0.636619
\(392\) 9.50280e8 0.796802
\(393\) 0 0
\(394\) 2.19419e9 1.80733
\(395\) −5.49284e8 −0.448443
\(396\) 0 0
\(397\) 1.16563e9 0.934961 0.467481 0.884003i \(-0.345162\pi\)
0.467481 + 0.884003i \(0.345162\pi\)
\(398\) −7.55310e8 −0.600530
\(399\) 0 0
\(400\) −4.88569e8 −0.381694
\(401\) −7.50534e8 −0.581253 −0.290626 0.956837i \(-0.593864\pi\)
−0.290626 + 0.956837i \(0.593864\pi\)
\(402\) 0 0
\(403\) 2.48568e9 1.89181
\(404\) 2.19937e9 1.65945
\(405\) 0 0
\(406\) −5.94178e8 −0.440631
\(407\) 2.46276e9 1.81068
\(408\) 0 0
\(409\) 8.51195e8 0.615174 0.307587 0.951520i \(-0.400479\pi\)
0.307587 + 0.951520i \(0.400479\pi\)
\(410\) 1.69897e9 1.21742
\(411\) 0 0
\(412\) −3.79516e9 −2.67356
\(413\) 1.91908e9 1.34050
\(414\) 0 0
\(415\) −6.06369e7 −0.0416456
\(416\) 1.03168e9 0.702619
\(417\) 0 0
\(418\) −4.17231e9 −2.79421
\(419\) 1.60505e9 1.06596 0.532978 0.846129i \(-0.321073\pi\)
0.532978 + 0.846129i \(0.321073\pi\)
\(420\) 0 0
\(421\) −8.58777e8 −0.560910 −0.280455 0.959867i \(-0.590485\pi\)
−0.280455 + 0.959867i \(0.590485\pi\)
\(422\) −2.67743e9 −1.73430
\(423\) 0 0
\(424\) −1.23182e9 −0.784817
\(425\) 1.78532e9 1.12812
\(426\) 0 0
\(427\) −1.39820e9 −0.869107
\(428\) 1.99497e8 0.122994
\(429\) 0 0
\(430\) −2.66249e8 −0.161491
\(431\) −9.60074e8 −0.577610 −0.288805 0.957388i \(-0.593258\pi\)
−0.288805 + 0.957388i \(0.593258\pi\)
\(432\) 0 0
\(433\) −4.31741e8 −0.255574 −0.127787 0.991802i \(-0.540787\pi\)
−0.127787 + 0.991802i \(0.540787\pi\)
\(434\) 3.97492e9 2.33407
\(435\) 0 0
\(436\) 1.75808e9 1.01586
\(437\) −5.33992e8 −0.306090
\(438\) 0 0
\(439\) −1.43295e9 −0.808363 −0.404182 0.914679i \(-0.632444\pi\)
−0.404182 + 0.914679i \(0.632444\pi\)
\(440\) 3.04938e9 1.70658
\(441\) 0 0
\(442\) 9.70569e9 5.34624
\(443\) −1.62330e9 −0.887127 −0.443564 0.896243i \(-0.646286\pi\)
−0.443564 + 0.896243i \(0.646286\pi\)
\(444\) 0 0
\(445\) 8.03819e8 0.432413
\(446\) −2.31768e9 −1.23704
\(447\) 0 0
\(448\) 3.12978e9 1.64453
\(449\) 7.89867e8 0.411805 0.205903 0.978573i \(-0.433987\pi\)
0.205903 + 0.978573i \(0.433987\pi\)
\(450\) 0 0
\(451\) 4.14930e9 2.12989
\(452\) −7.37839e8 −0.375818
\(453\) 0 0
\(454\) 2.21608e9 1.11145
\(455\) 2.65712e9 1.32242
\(456\) 0 0
\(457\) 3.17411e9 1.55566 0.777831 0.628474i \(-0.216320\pi\)
0.777831 + 0.628474i \(0.216320\pi\)
\(458\) −2.66862e9 −1.29795
\(459\) 0 0
\(460\) 8.38938e8 0.401863
\(461\) 1.10733e9 0.526408 0.263204 0.964740i \(-0.415221\pi\)
0.263204 + 0.964740i \(0.415221\pi\)
\(462\) 0 0
\(463\) −1.45587e9 −0.681693 −0.340847 0.940119i \(-0.610714\pi\)
−0.340847 + 0.940119i \(0.610714\pi\)
\(464\) 2.82501e8 0.131283
\(465\) 0 0
\(466\) 7.02046e8 0.321377
\(467\) 2.23002e9 1.01321 0.506606 0.862178i \(-0.330900\pi\)
0.506606 + 0.862178i \(0.330900\pi\)
\(468\) 0 0
\(469\) 1.40161e9 0.627369
\(470\) 4.61028e8 0.204826
\(471\) 0 0
\(472\) −3.63563e9 −1.59141
\(473\) −6.50245e8 −0.282529
\(474\) 0 0
\(475\) 1.26693e9 0.542406
\(476\) 1.01125e10 4.29766
\(477\) 0 0
\(478\) 7.11908e8 0.298144
\(479\) −9.33167e8 −0.387958 −0.193979 0.981006i \(-0.562139\pi\)
−0.193979 + 0.981006i \(0.562139\pi\)
\(480\) 0 0
\(481\) −4.06545e9 −1.66572
\(482\) 7.60461e8 0.309323
\(483\) 0 0
\(484\) 1.13448e10 4.54818
\(485\) −2.72292e9 −1.08377
\(486\) 0 0
\(487\) −1.72559e9 −0.676994 −0.338497 0.940967i \(-0.609919\pi\)
−0.338497 + 0.940967i \(0.609919\pi\)
\(488\) 2.64885e9 1.03178
\(489\) 0 0
\(490\) 1.49122e9 0.572606
\(491\) 6.20324e8 0.236501 0.118251 0.992984i \(-0.462271\pi\)
0.118251 + 0.992984i \(0.462271\pi\)
\(492\) 0 0
\(493\) −1.03231e9 −0.388012
\(494\) 6.88752e9 2.57051
\(495\) 0 0
\(496\) −1.88987e9 −0.695418
\(497\) −3.50007e9 −1.27888
\(498\) 0 0
\(499\) 3.98852e9 1.43701 0.718506 0.695521i \(-0.244826\pi\)
0.718506 + 0.695521i \(0.244826\pi\)
\(500\) −5.25748e9 −1.88097
\(501\) 0 0
\(502\) −1.16603e9 −0.411385
\(503\) −1.29341e9 −0.453157 −0.226579 0.973993i \(-0.572754\pi\)
−0.226579 + 0.973993i \(0.572754\pi\)
\(504\) 0 0
\(505\) 1.60556e9 0.554763
\(506\) 3.14464e9 1.07906
\(507\) 0 0
\(508\) −4.58605e8 −0.155209
\(509\) −2.09129e9 −0.702913 −0.351456 0.936204i \(-0.614313\pi\)
−0.351456 + 0.936204i \(0.614313\pi\)
\(510\) 0 0
\(511\) 4.06023e9 1.34610
\(512\) −3.58818e9 −1.18149
\(513\) 0 0
\(514\) −1.90101e9 −0.617466
\(515\) −2.77051e9 −0.893787
\(516\) 0 0
\(517\) 1.12594e9 0.358343
\(518\) −6.50116e9 −2.05512
\(519\) 0 0
\(520\) −5.03382e9 −1.56995
\(521\) −4.31681e9 −1.33731 −0.668653 0.743575i \(-0.733129\pi\)
−0.668653 + 0.743575i \(0.733129\pi\)
\(522\) 0 0
\(523\) 5.49978e9 1.68108 0.840542 0.541746i \(-0.182237\pi\)
0.840542 + 0.541746i \(0.182237\pi\)
\(524\) 4.95767e9 1.50528
\(525\) 0 0
\(526\) 5.24273e9 1.57075
\(527\) 6.90592e9 2.05534
\(528\) 0 0
\(529\) −3.00236e9 −0.881795
\(530\) −1.93303e9 −0.563993
\(531\) 0 0
\(532\) 7.17618e9 2.06634
\(533\) −6.84953e9 −1.95937
\(534\) 0 0
\(535\) 1.45635e8 0.0411176
\(536\) −2.65531e9 −0.744797
\(537\) 0 0
\(538\) −5.28357e9 −1.46281
\(539\) 3.64193e9 1.00178
\(540\) 0 0
\(541\) −5.48712e9 −1.48989 −0.744945 0.667126i \(-0.767524\pi\)
−0.744945 + 0.667126i \(0.767524\pi\)
\(542\) 9.11641e9 2.45938
\(543\) 0 0
\(544\) 2.86630e9 0.763354
\(545\) 1.28342e9 0.339610
\(546\) 0 0
\(547\) −5.58182e9 −1.45821 −0.729105 0.684402i \(-0.760063\pi\)
−0.729105 + 0.684402i \(0.760063\pi\)
\(548\) 7.58225e9 1.96819
\(549\) 0 0
\(550\) −7.46085e9 −1.91214
\(551\) −7.32566e8 −0.186559
\(552\) 0 0
\(553\) 3.54124e9 0.890466
\(554\) 4.57823e9 1.14397
\(555\) 0 0
\(556\) −1.17444e10 −2.89780
\(557\) 1.55205e9 0.380551 0.190276 0.981731i \(-0.439062\pi\)
0.190276 + 0.981731i \(0.439062\pi\)
\(558\) 0 0
\(559\) 1.07340e9 0.259909
\(560\) −2.02021e9 −0.486114
\(561\) 0 0
\(562\) 1.06348e10 2.52727
\(563\) 2.49961e9 0.590328 0.295164 0.955447i \(-0.404626\pi\)
0.295164 + 0.955447i \(0.404626\pi\)
\(564\) 0 0
\(565\) −5.38631e8 −0.125638
\(566\) −3.90954e8 −0.0906293
\(567\) 0 0
\(568\) 6.63076e9 1.51825
\(569\) −6.84845e9 −1.55847 −0.779236 0.626730i \(-0.784393\pi\)
−0.779236 + 0.626730i \(0.784393\pi\)
\(570\) 0 0
\(571\) −4.81071e7 −0.0108139 −0.00540696 0.999985i \(-0.501721\pi\)
−0.00540696 + 0.999985i \(0.501721\pi\)
\(572\) −2.64270e10 −5.90420
\(573\) 0 0
\(574\) −1.09532e10 −2.41742
\(575\) −9.54875e8 −0.209464
\(576\) 0 0
\(577\) −6.83896e9 −1.48209 −0.741045 0.671455i \(-0.765669\pi\)
−0.741045 + 0.671455i \(0.765669\pi\)
\(578\) 1.91005e10 4.11431
\(579\) 0 0
\(580\) 1.15091e9 0.244931
\(581\) 3.90926e8 0.0826948
\(582\) 0 0
\(583\) −4.72094e9 −0.986707
\(584\) −7.69196e9 −1.59806
\(585\) 0 0
\(586\) −4.48697e9 −0.921111
\(587\) −1.56312e9 −0.318976 −0.159488 0.987200i \(-0.550984\pi\)
−0.159488 + 0.987200i \(0.550984\pi\)
\(588\) 0 0
\(589\) 4.90070e9 0.988222
\(590\) −5.70519e9 −1.14364
\(591\) 0 0
\(592\) 3.09097e9 0.612306
\(593\) 3.33176e9 0.656118 0.328059 0.944657i \(-0.393605\pi\)
0.328059 + 0.944657i \(0.393605\pi\)
\(594\) 0 0
\(595\) 7.38220e9 1.43673
\(596\) −1.92636e9 −0.372713
\(597\) 0 0
\(598\) −5.19108e9 −0.992667
\(599\) −2.00026e9 −0.380271 −0.190136 0.981758i \(-0.560893\pi\)
−0.190136 + 0.981758i \(0.560893\pi\)
\(600\) 0 0
\(601\) −8.58111e9 −1.61244 −0.806218 0.591618i \(-0.798489\pi\)
−0.806218 + 0.591618i \(0.798489\pi\)
\(602\) 1.71651e9 0.320670
\(603\) 0 0
\(604\) −2.26316e9 −0.417914
\(605\) 8.28183e9 1.52049
\(606\) 0 0
\(607\) 2.24609e9 0.407631 0.203815 0.979009i \(-0.434666\pi\)
0.203815 + 0.979009i \(0.434666\pi\)
\(608\) 2.03404e9 0.367026
\(609\) 0 0
\(610\) 4.15669e9 0.741470
\(611\) −1.85867e9 −0.329654
\(612\) 0 0
\(613\) −7.36697e9 −1.29175 −0.645873 0.763445i \(-0.723507\pi\)
−0.645873 + 0.763445i \(0.723507\pi\)
\(614\) 6.57351e9 1.14606
\(615\) 0 0
\(616\) −1.96593e10 −3.38872
\(617\) −8.07677e9 −1.38433 −0.692165 0.721739i \(-0.743343\pi\)
−0.692165 + 0.721739i \(0.743343\pi\)
\(618\) 0 0
\(619\) 4.11289e9 0.696995 0.348498 0.937310i \(-0.386692\pi\)
0.348498 + 0.937310i \(0.386692\pi\)
\(620\) −7.69934e9 −1.29743
\(621\) 0 0
\(622\) 1.38876e10 2.31399
\(623\) −5.18222e9 −0.858634
\(624\) 0 0
\(625\) −1.19481e8 −0.0195757
\(626\) 5.56094e9 0.906020
\(627\) 0 0
\(628\) 1.31805e10 2.12360
\(629\) −1.12949e10 −1.80970
\(630\) 0 0
\(631\) −1.93829e9 −0.307126 −0.153563 0.988139i \(-0.549075\pi\)
−0.153563 + 0.988139i \(0.549075\pi\)
\(632\) −6.70876e9 −1.05714
\(633\) 0 0
\(634\) −1.32383e10 −2.06309
\(635\) −3.34787e8 −0.0518872
\(636\) 0 0
\(637\) −6.01198e9 −0.921572
\(638\) 4.31403e9 0.657674
\(639\) 0 0
\(640\) −7.59545e9 −1.14531
\(641\) 4.63296e9 0.694793 0.347397 0.937718i \(-0.387066\pi\)
0.347397 + 0.937718i \(0.387066\pi\)
\(642\) 0 0
\(643\) 4.81785e9 0.714685 0.357342 0.933973i \(-0.383683\pi\)
0.357342 + 0.933973i \(0.383683\pi\)
\(644\) −5.40864e9 −0.797971
\(645\) 0 0
\(646\) 1.91354e10 2.79270
\(647\) 4.86975e9 0.706873 0.353437 0.935458i \(-0.385013\pi\)
0.353437 + 0.935458i \(0.385013\pi\)
\(648\) 0 0
\(649\) −1.39335e10 −2.00080
\(650\) 1.23161e10 1.75905
\(651\) 0 0
\(652\) 8.52966e9 1.20522
\(653\) 4.32144e9 0.607340 0.303670 0.952777i \(-0.401788\pi\)
0.303670 + 0.952777i \(0.401788\pi\)
\(654\) 0 0
\(655\) 3.61915e9 0.503225
\(656\) 5.20771e9 0.720250
\(657\) 0 0
\(658\) −2.97225e9 −0.406719
\(659\) 1.84953e9 0.251746 0.125873 0.992046i \(-0.459827\pi\)
0.125873 + 0.992046i \(0.459827\pi\)
\(660\) 0 0
\(661\) 1.67946e9 0.226186 0.113093 0.993584i \(-0.463924\pi\)
0.113093 + 0.993584i \(0.463924\pi\)
\(662\) −2.48229e10 −3.32544
\(663\) 0 0
\(664\) −7.40597e8 −0.0981733
\(665\) 5.23869e9 0.690791
\(666\) 0 0
\(667\) 5.52130e8 0.0720445
\(668\) 1.18876e10 1.54304
\(669\) 0 0
\(670\) −4.16682e9 −0.535234
\(671\) 1.01517e10 1.29720
\(672\) 0 0
\(673\) −3.83952e9 −0.485540 −0.242770 0.970084i \(-0.578056\pi\)
−0.242770 + 0.970084i \(0.578056\pi\)
\(674\) −8.61153e9 −1.08335
\(675\) 0 0
\(676\) 2.86065e10 3.56165
\(677\) −1.14031e10 −1.41241 −0.706206 0.708006i \(-0.749595\pi\)
−0.706206 + 0.708006i \(0.749595\pi\)
\(678\) 0 0
\(679\) 1.75547e10 2.15203
\(680\) −1.39853e10 −1.70566
\(681\) 0 0
\(682\) −2.88599e10 −3.48377
\(683\) −2.62449e9 −0.315190 −0.157595 0.987504i \(-0.550374\pi\)
−0.157595 + 0.987504i \(0.550374\pi\)
\(684\) 0 0
\(685\) 5.53512e9 0.657977
\(686\) 8.16588e9 0.965759
\(687\) 0 0
\(688\) −8.16111e8 −0.0955410
\(689\) 7.79318e9 0.907711
\(690\) 0 0
\(691\) 1.64230e9 0.189356 0.0946778 0.995508i \(-0.469818\pi\)
0.0946778 + 0.995508i \(0.469818\pi\)
\(692\) −1.42956e10 −1.63995
\(693\) 0 0
\(694\) −1.18635e9 −0.134727
\(695\) −8.57353e9 −0.968752
\(696\) 0 0
\(697\) −1.90299e10 −2.12873
\(698\) 1.41883e10 1.57919
\(699\) 0 0
\(700\) 1.28323e10 1.41404
\(701\) 5.25977e9 0.576705 0.288352 0.957524i \(-0.406893\pi\)
0.288352 + 0.957524i \(0.406893\pi\)
\(702\) 0 0
\(703\) −8.01532e9 −0.870116
\(704\) −2.27238e10 −2.45457
\(705\) 0 0
\(706\) −2.94610e10 −3.15088
\(707\) −1.03511e10 −1.10158
\(708\) 0 0
\(709\) −9.05667e9 −0.954348 −0.477174 0.878809i \(-0.658339\pi\)
−0.477174 + 0.878809i \(0.658339\pi\)
\(710\) 1.04053e10 1.09106
\(711\) 0 0
\(712\) 9.81755e9 1.01935
\(713\) −3.69362e9 −0.381627
\(714\) 0 0
\(715\) −1.92920e10 −1.97381
\(716\) 1.24351e9 0.126606
\(717\) 0 0
\(718\) 2.19907e10 2.21719
\(719\) −1.23375e10 −1.23787 −0.618935 0.785443i \(-0.712435\pi\)
−0.618935 + 0.785443i \(0.712435\pi\)
\(720\) 0 0
\(721\) 1.78615e10 1.77478
\(722\) −3.55285e9 −0.351315
\(723\) 0 0
\(724\) 2.26865e9 0.222169
\(725\) −1.30996e9 −0.127666
\(726\) 0 0
\(727\) 8.00292e8 0.0772464 0.0386232 0.999254i \(-0.487703\pi\)
0.0386232 + 0.999254i \(0.487703\pi\)
\(728\) 3.24530e10 3.11742
\(729\) 0 0
\(730\) −1.20706e10 −1.14841
\(731\) 2.98221e9 0.282376
\(732\) 0 0
\(733\) −8.12305e9 −0.761825 −0.380912 0.924611i \(-0.624390\pi\)
−0.380912 + 0.924611i \(0.624390\pi\)
\(734\) −2.06706e10 −1.92938
\(735\) 0 0
\(736\) −1.53304e9 −0.141736
\(737\) −1.01764e10 −0.936393
\(738\) 0 0
\(739\) −1.58853e10 −1.44791 −0.723954 0.689848i \(-0.757677\pi\)
−0.723954 + 0.689848i \(0.757677\pi\)
\(740\) 1.25926e10 1.14237
\(741\) 0 0
\(742\) 1.24623e10 1.11991
\(743\) −1.84265e10 −1.64810 −0.824048 0.566521i \(-0.808289\pi\)
−0.824048 + 0.566521i \(0.808289\pi\)
\(744\) 0 0
\(745\) −1.40626e9 −0.124600
\(746\) 8.22631e9 0.725469
\(747\) 0 0
\(748\) −7.34215e10 −6.41457
\(749\) −9.38910e8 −0.0816465
\(750\) 0 0
\(751\) 4.18559e9 0.360592 0.180296 0.983612i \(-0.442294\pi\)
0.180296 + 0.983612i \(0.442294\pi\)
\(752\) 1.41315e9 0.121179
\(753\) 0 0
\(754\) −7.12147e9 −0.605020
\(755\) −1.65213e9 −0.139711
\(756\) 0 0
\(757\) 1.67312e10 1.40182 0.700908 0.713252i \(-0.252778\pi\)
0.700908 + 0.713252i \(0.252778\pi\)
\(758\) 9.52250e9 0.794161
\(759\) 0 0
\(760\) −9.92452e9 −0.820091
\(761\) 1.16945e10 0.961911 0.480955 0.876745i \(-0.340290\pi\)
0.480955 + 0.876745i \(0.340290\pi\)
\(762\) 0 0
\(763\) −8.27419e9 −0.674356
\(764\) −2.64728e10 −2.14770
\(765\) 0 0
\(766\) 1.67580e9 0.134717
\(767\) 2.30009e10 1.84061
\(768\) 0 0
\(769\) −1.70901e10 −1.35519 −0.677597 0.735433i \(-0.736979\pi\)
−0.677597 + 0.735433i \(0.736979\pi\)
\(770\) −3.08503e10 −2.43524
\(771\) 0 0
\(772\) −2.06338e10 −1.61406
\(773\) 1.12070e9 0.0872690 0.0436345 0.999048i \(-0.486106\pi\)
0.0436345 + 0.999048i \(0.486106\pi\)
\(774\) 0 0
\(775\) 8.76334e9 0.676260
\(776\) −3.32567e10 −2.55484
\(777\) 0 0
\(778\) 3.69269e10 2.81134
\(779\) −1.35043e10 −1.02351
\(780\) 0 0
\(781\) 2.54122e10 1.90882
\(782\) −1.44223e10 −1.07847
\(783\) 0 0
\(784\) 4.57092e9 0.338764
\(785\) 9.62192e9 0.709934
\(786\) 0 0
\(787\) 3.19989e9 0.234004 0.117002 0.993132i \(-0.462672\pi\)
0.117002 + 0.993132i \(0.462672\pi\)
\(788\) 2.74004e10 1.99487
\(789\) 0 0
\(790\) −1.05277e10 −0.759692
\(791\) 3.47255e9 0.249477
\(792\) 0 0
\(793\) −1.67580e10 −1.19335
\(794\) 2.23406e10 1.58388
\(795\) 0 0
\(796\) −9.43209e9 −0.662846
\(797\) −1.93896e10 −1.35664 −0.678322 0.734765i \(-0.737292\pi\)
−0.678322 + 0.734765i \(0.737292\pi\)
\(798\) 0 0
\(799\) −5.16390e9 −0.358150
\(800\) 3.63723e9 0.251163
\(801\) 0 0
\(802\) −1.43849e10 −0.984680
\(803\) −2.94793e10 −2.00915
\(804\) 0 0
\(805\) −3.94836e9 −0.266767
\(806\) 4.76410e10 3.20485
\(807\) 0 0
\(808\) 1.96098e10 1.30777
\(809\) 2.20959e10 1.46721 0.733604 0.679577i \(-0.237837\pi\)
0.733604 + 0.679577i \(0.237837\pi\)
\(810\) 0 0
\(811\) 1.03080e10 0.678583 0.339292 0.940681i \(-0.389813\pi\)
0.339292 + 0.940681i \(0.389813\pi\)
\(812\) −7.41993e9 −0.486355
\(813\) 0 0
\(814\) 4.72017e10 3.06741
\(815\) 6.22675e9 0.402911
\(816\) 0 0
\(817\) 2.11629e9 0.135768
\(818\) 1.63141e10 1.04214
\(819\) 0 0
\(820\) 2.12162e10 1.34375
\(821\) 1.08963e10 0.687192 0.343596 0.939117i \(-0.388355\pi\)
0.343596 + 0.939117i \(0.388355\pi\)
\(822\) 0 0
\(823\) 1.58099e10 0.988622 0.494311 0.869285i \(-0.335420\pi\)
0.494311 + 0.869285i \(0.335420\pi\)
\(824\) −3.38379e10 −2.10697
\(825\) 0 0
\(826\) 3.67813e10 2.27090
\(827\) −6.58783e9 −0.405017 −0.202508 0.979281i \(-0.564909\pi\)
−0.202508 + 0.979281i \(0.564909\pi\)
\(828\) 0 0
\(829\) 1.31751e10 0.803180 0.401590 0.915819i \(-0.368458\pi\)
0.401590 + 0.915819i \(0.368458\pi\)
\(830\) −1.16218e9 −0.0705503
\(831\) 0 0
\(832\) 3.75117e10 2.25806
\(833\) −1.67029e10 −1.00123
\(834\) 0 0
\(835\) 8.67808e9 0.515847
\(836\) −5.21026e10 −3.08417
\(837\) 0 0
\(838\) 3.07626e10 1.80580
\(839\) −1.44377e10 −0.843981 −0.421990 0.906600i \(-0.638668\pi\)
−0.421990 + 0.906600i \(0.638668\pi\)
\(840\) 0 0
\(841\) −1.64924e10 −0.956090
\(842\) −1.64595e10 −0.950218
\(843\) 0 0
\(844\) −3.34350e10 −1.91427
\(845\) 2.08831e10 1.19068
\(846\) 0 0
\(847\) −5.33930e10 −3.01920
\(848\) −5.92517e9 −0.333668
\(849\) 0 0
\(850\) 3.42177e10 1.91110
\(851\) 6.04109e9 0.336018
\(852\) 0 0
\(853\) −1.27917e10 −0.705679 −0.352840 0.935684i \(-0.614784\pi\)
−0.352840 + 0.935684i \(0.614784\pi\)
\(854\) −2.67982e10 −1.47232
\(855\) 0 0
\(856\) 1.77873e9 0.0969288
\(857\) 2.68071e10 1.45485 0.727423 0.686189i \(-0.240718\pi\)
0.727423 + 0.686189i \(0.240718\pi\)
\(858\) 0 0
\(859\) 7.43216e9 0.400073 0.200036 0.979788i \(-0.435894\pi\)
0.200036 + 0.979788i \(0.435894\pi\)
\(860\) −3.32484e9 −0.178249
\(861\) 0 0
\(862\) −1.84009e10 −0.978508
\(863\) −2.49200e10 −1.31980 −0.659902 0.751352i \(-0.729402\pi\)
−0.659902 + 0.751352i \(0.729402\pi\)
\(864\) 0 0
\(865\) −1.04360e10 −0.548247
\(866\) −8.27482e9 −0.432958
\(867\) 0 0
\(868\) 4.96376e10 2.57627
\(869\) −2.57111e10 −1.32908
\(870\) 0 0
\(871\) 1.67989e10 0.861424
\(872\) 1.56752e10 0.800580
\(873\) 0 0
\(874\) −1.02346e10 −0.518537
\(875\) 2.47437e10 1.24864
\(876\) 0 0
\(877\) −3.16610e10 −1.58499 −0.792493 0.609881i \(-0.791217\pi\)
−0.792493 + 0.609881i \(0.791217\pi\)
\(878\) −2.74642e10 −1.36942
\(879\) 0 0
\(880\) 1.46677e10 0.725560
\(881\) −9.18139e9 −0.452369 −0.226185 0.974084i \(-0.572625\pi\)
−0.226185 + 0.974084i \(0.572625\pi\)
\(882\) 0 0
\(883\) −2.18382e10 −1.06746 −0.533732 0.845653i \(-0.679211\pi\)
−0.533732 + 0.845653i \(0.679211\pi\)
\(884\) 1.21202e11 5.90101
\(885\) 0 0
\(886\) −3.11124e10 −1.50285
\(887\) 3.51106e10 1.68929 0.844647 0.535324i \(-0.179811\pi\)
0.844647 + 0.535324i \(0.179811\pi\)
\(888\) 0 0
\(889\) 2.15837e9 0.103031
\(890\) 1.54061e10 0.732535
\(891\) 0 0
\(892\) −2.89426e10 −1.36540
\(893\) −3.66450e9 −0.172201
\(894\) 0 0
\(895\) 9.07778e8 0.0423252
\(896\) 4.89679e10 2.27422
\(897\) 0 0
\(898\) 1.51387e10 0.697624
\(899\) −5.06716e9 −0.232598
\(900\) 0 0
\(901\) 2.16516e10 0.986174
\(902\) 7.95261e10 3.60817
\(903\) 0 0
\(904\) −6.57863e9 −0.296174
\(905\) 1.65614e9 0.0742724
\(906\) 0 0
\(907\) 1.81270e10 0.806676 0.403338 0.915051i \(-0.367850\pi\)
0.403338 + 0.915051i \(0.367850\pi\)
\(908\) 2.76738e10 1.22678
\(909\) 0 0
\(910\) 5.09267e10 2.24027
\(911\) −4.43842e10 −1.94498 −0.972488 0.232952i \(-0.925161\pi\)
−0.972488 + 0.232952i \(0.925161\pi\)
\(912\) 0 0
\(913\) −2.83832e9 −0.123428
\(914\) 6.08354e10 2.63539
\(915\) 0 0
\(916\) −3.33250e10 −1.43264
\(917\) −2.33327e10 −0.999245
\(918\) 0 0
\(919\) −3.80235e10 −1.61602 −0.808012 0.589166i \(-0.799456\pi\)
−0.808012 + 0.589166i \(0.799456\pi\)
\(920\) 7.48004e9 0.316699
\(921\) 0 0
\(922\) 2.12232e10 0.891769
\(923\) −4.19497e10 −1.75600
\(924\) 0 0
\(925\) −1.43329e10 −0.595438
\(926\) −2.79034e10 −1.15483
\(927\) 0 0
\(928\) −2.10313e9 −0.0863868
\(929\) 3.00377e10 1.22917 0.614585 0.788851i \(-0.289323\pi\)
0.614585 + 0.788851i \(0.289323\pi\)
\(930\) 0 0
\(931\) −1.18530e10 −0.481400
\(932\) 8.76696e9 0.354726
\(933\) 0 0
\(934\) 4.27409e10 1.71645
\(935\) −5.35985e10 −2.14443
\(936\) 0 0
\(937\) −4.22202e10 −1.67661 −0.838303 0.545204i \(-0.816452\pi\)
−0.838303 + 0.545204i \(0.816452\pi\)
\(938\) 2.68635e10 1.06280
\(939\) 0 0
\(940\) 5.75718e9 0.226080
\(941\) −3.37621e10 −1.32089 −0.660444 0.750875i \(-0.729632\pi\)
−0.660444 + 0.750875i \(0.729632\pi\)
\(942\) 0 0
\(943\) 1.01781e10 0.395254
\(944\) −1.74876e10 −0.676596
\(945\) 0 0
\(946\) −1.24627e10 −0.478622
\(947\) 2.01458e10 0.770832 0.385416 0.922743i \(-0.374058\pi\)
0.385416 + 0.922743i \(0.374058\pi\)
\(948\) 0 0
\(949\) 4.86635e10 1.84830
\(950\) 2.42821e10 0.918871
\(951\) 0 0
\(952\) 9.01635e10 3.38689
\(953\) 2.24395e10 0.839825 0.419913 0.907565i \(-0.362061\pi\)
0.419913 + 0.907565i \(0.362061\pi\)
\(954\) 0 0
\(955\) −1.93254e10 −0.717989
\(956\) 8.89011e9 0.329082
\(957\) 0 0
\(958\) −1.78852e10 −0.657226
\(959\) −3.56850e10 −1.30653
\(960\) 0 0
\(961\) 6.38552e9 0.232094
\(962\) −7.79191e10 −2.82183
\(963\) 0 0
\(964\) 9.49642e9 0.341421
\(965\) −1.50629e10 −0.539589
\(966\) 0 0
\(967\) −3.16478e10 −1.12551 −0.562757 0.826622i \(-0.690259\pi\)
−0.562757 + 0.826622i \(0.690259\pi\)
\(968\) 1.01151e11 3.58432
\(969\) 0 0
\(970\) −5.21879e10 −1.83598
\(971\) 2.82843e10 0.991468 0.495734 0.868474i \(-0.334899\pi\)
0.495734 + 0.868474i \(0.334899\pi\)
\(972\) 0 0
\(973\) 5.52736e10 1.92363
\(974\) −3.30728e10 −1.14687
\(975\) 0 0
\(976\) 1.27412e10 0.438667
\(977\) −2.65176e10 −0.909709 −0.454855 0.890566i \(-0.650309\pi\)
−0.454855 + 0.890566i \(0.650309\pi\)
\(978\) 0 0
\(979\) 3.76255e10 1.28157
\(980\) 1.86219e10 0.632024
\(981\) 0 0
\(982\) 1.18892e10 0.400648
\(983\) −4.89517e10 −1.64373 −0.821866 0.569681i \(-0.807067\pi\)
−0.821866 + 0.569681i \(0.807067\pi\)
\(984\) 0 0
\(985\) 2.00026e10 0.666898
\(986\) −1.97854e10 −0.657318
\(987\) 0 0
\(988\) 8.60094e10 2.83724
\(989\) −1.59503e9 −0.0524304
\(990\) 0 0
\(991\) −1.75057e10 −0.571374 −0.285687 0.958323i \(-0.592222\pi\)
−0.285687 + 0.958323i \(0.592222\pi\)
\(992\) 1.40694e10 0.457600
\(993\) 0 0
\(994\) −6.70829e10 −2.16650
\(995\) −6.88553e9 −0.221593
\(996\) 0 0
\(997\) −1.49917e10 −0.479090 −0.239545 0.970885i \(-0.576998\pi\)
−0.239545 + 0.970885i \(0.576998\pi\)
\(998\) 7.64447e10 2.43439
\(999\) 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.1.12 13
3.2 odd 2 43.8.a.b.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.2 13 3.2 odd 2
387.8.a.d.1.12 13 1.1 even 1 trivial