Properties

Label 9.74.a.a.1.4
Level 9
Weight 74
Character 9.1
Self dual yes
Analytic conductor 303.736
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 74 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(303.735576363\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 10073499617947743056 x^{3} + 1429272143092482488433869600 x^{2} + 7661214288514935343595600445215756800 x + 1722510836040319301450745177697157900206688000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{22}\cdot 5^{6}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.10442e9\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.14302e10 q^{2} -4.34246e21 q^{4} +5.37334e25 q^{5} +1.02465e31 q^{7} -9.84822e32 q^{8} +O(q^{10})\) \(q+7.14302e10 q^{2} -4.34246e21 q^{4} +5.37334e25 q^{5} +1.02465e31 q^{7} -9.84822e32 q^{8} +3.83819e36 q^{10} +4.39076e37 q^{11} +5.53509e40 q^{13} +7.31912e41 q^{14} -2.93327e43 q^{16} -9.40723e44 q^{17} +1.72590e46 q^{19} -2.33335e47 q^{20} +3.13633e48 q^{22} +5.77480e49 q^{23} +1.82849e51 q^{25} +3.95373e51 q^{26} -4.44951e52 q^{28} +8.69385e52 q^{29} +1.48414e54 q^{31} +7.20614e54 q^{32} -6.71960e55 q^{34} +5.50582e56 q^{35} -2.64558e57 q^{37} +1.23282e57 q^{38} -5.29179e58 q^{40} -3.07305e57 q^{41} +3.24223e59 q^{43} -1.90667e59 q^{44} +4.12496e60 q^{46} +8.51145e60 q^{47} +5.57697e61 q^{49} +1.30610e62 q^{50} -2.40359e62 q^{52} +8.36780e62 q^{53} +2.35931e63 q^{55} -1.00910e64 q^{56} +6.21004e63 q^{58} -2.35468e64 q^{59} +5.89372e64 q^{61} +1.06012e65 q^{62} +7.91776e65 q^{64} +2.97420e66 q^{65} +6.87121e66 q^{67} +4.08505e66 q^{68} +3.93282e67 q^{70} -8.24197e66 q^{71} -1.53931e68 q^{73} -1.88974e68 q^{74} -7.49465e67 q^{76} +4.49901e68 q^{77} -3.64169e68 q^{79} -1.57615e69 q^{80} -2.19509e68 q^{82} -1.69878e70 q^{83} -5.05483e70 q^{85} +2.31594e70 q^{86} -4.32412e70 q^{88} -7.76576e69 q^{89} +5.67155e71 q^{91} -2.50768e71 q^{92} +6.07975e71 q^{94} +9.27387e71 q^{95} +4.51236e71 q^{97} +3.98365e72 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 92089333488q^{2} + \)\(89\!\cdots\!60\)\(q^{4} - \)\(23\!\cdots\!50\)\(q^{5} - \)\(43\!\cdots\!08\)\(q^{7} + \)\(38\!\cdots\!80\)\(q^{8} + O(q^{10}) \) \( 5q + 92089333488q^{2} + \)\(89\!\cdots\!60\)\(q^{4} - \)\(23\!\cdots\!50\)\(q^{5} - \)\(43\!\cdots\!08\)\(q^{7} + \)\(38\!\cdots\!80\)\(q^{8} - \)\(10\!\cdots\!00\)\(q^{10} - \)\(50\!\cdots\!60\)\(q^{11} + \)\(47\!\cdots\!86\)\(q^{13} - \)\(26\!\cdots\!20\)\(q^{14} + \)\(57\!\cdots\!80\)\(q^{16} - \)\(66\!\cdots\!02\)\(q^{17} + \)\(31\!\cdots\!00\)\(q^{19} - \)\(68\!\cdots\!00\)\(q^{20} - \)\(94\!\cdots\!16\)\(q^{22} + \)\(41\!\cdots\!24\)\(q^{23} + \)\(32\!\cdots\!75\)\(q^{25} - \)\(44\!\cdots\!60\)\(q^{26} - \)\(37\!\cdots\!16\)\(q^{28} + \)\(21\!\cdots\!50\)\(q^{29} - \)\(39\!\cdots\!40\)\(q^{31} + \)\(94\!\cdots\!68\)\(q^{32} - \)\(32\!\cdots\!80\)\(q^{34} + \)\(10\!\cdots\!00\)\(q^{35} - \)\(67\!\cdots\!78\)\(q^{37} + \)\(20\!\cdots\!20\)\(q^{38} - \)\(87\!\cdots\!00\)\(q^{40} + \)\(89\!\cdots\!90\)\(q^{41} + \)\(11\!\cdots\!56\)\(q^{43} - \)\(36\!\cdots\!20\)\(q^{44} + \)\(13\!\cdots\!60\)\(q^{46} - \)\(26\!\cdots\!32\)\(q^{47} - \)\(47\!\cdots\!15\)\(q^{49} + \)\(19\!\cdots\!00\)\(q^{50} - \)\(18\!\cdots\!28\)\(q^{52} + \)\(22\!\cdots\!54\)\(q^{53} + \)\(52\!\cdots\!00\)\(q^{55} - \)\(17\!\cdots\!00\)\(q^{56} + \)\(63\!\cdots\!20\)\(q^{58} - \)\(49\!\cdots\!00\)\(q^{59} - \)\(20\!\cdots\!90\)\(q^{61} + \)\(45\!\cdots\!96\)\(q^{62} - \)\(26\!\cdots\!40\)\(q^{64} + \)\(22\!\cdots\!00\)\(q^{65} + \)\(17\!\cdots\!52\)\(q^{67} - \)\(25\!\cdots\!04\)\(q^{68} + \)\(60\!\cdots\!00\)\(q^{70} - \)\(29\!\cdots\!60\)\(q^{71} - \)\(23\!\cdots\!74\)\(q^{73} + \)\(38\!\cdots\!80\)\(q^{74} - \)\(40\!\cdots\!00\)\(q^{76} + \)\(11\!\cdots\!56\)\(q^{77} + \)\(12\!\cdots\!00\)\(q^{79} - \)\(76\!\cdots\!00\)\(q^{80} + \)\(19\!\cdots\!64\)\(q^{82} - \)\(10\!\cdots\!16\)\(q^{83} - \)\(28\!\cdots\!00\)\(q^{85} + \)\(13\!\cdots\!40\)\(q^{86} - \)\(22\!\cdots\!60\)\(q^{88} + \)\(44\!\cdots\!50\)\(q^{89} + \)\(50\!\cdots\!60\)\(q^{91} - \)\(10\!\cdots\!52\)\(q^{92} + \)\(12\!\cdots\!20\)\(q^{94} - \)\(11\!\cdots\!00\)\(q^{95} - \)\(47\!\cdots\!18\)\(q^{97} + \)\(76\!\cdots\!16\)\(q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.14302e10 0.735000 0.367500 0.930024i \(-0.380214\pi\)
0.367500 + 0.930024i \(0.380214\pi\)
\(3\) 0 0
\(4\) −4.34246e21 −0.459775
\(5\) 5.37334e25 1.65135 0.825676 0.564145i \(-0.190794\pi\)
0.825676 + 0.564145i \(0.190794\pi\)
\(6\) 0 0
\(7\) 1.02465e31 1.46049 0.730245 0.683185i \(-0.239406\pi\)
0.730245 + 0.683185i \(0.239406\pi\)
\(8\) −9.84822e32 −1.07293
\(9\) 0 0
\(10\) 3.83819e36 1.21374
\(11\) 4.39076e37 0.428260 0.214130 0.976805i \(-0.431308\pi\)
0.214130 + 0.976805i \(0.431308\pi\)
\(12\) 0 0
\(13\) 5.53509e40 1.21393 0.606963 0.794730i \(-0.292388\pi\)
0.606963 + 0.794730i \(0.292388\pi\)
\(14\) 7.31912e41 1.07346
\(15\) 0 0
\(16\) −2.93327e43 −0.328831
\(17\) −9.40723e44 −1.15365 −0.576827 0.816866i \(-0.695710\pi\)
−0.576827 + 0.816866i \(0.695710\pi\)
\(18\) 0 0
\(19\) 1.72590e46 0.365182 0.182591 0.983189i \(-0.441552\pi\)
0.182591 + 0.983189i \(0.441552\pi\)
\(20\) −2.33335e47 −0.759251
\(21\) 0 0
\(22\) 3.13633e48 0.314771
\(23\) 5.77480e49 1.14412 0.572059 0.820212i \(-0.306145\pi\)
0.572059 + 0.820212i \(0.306145\pi\)
\(24\) 0 0
\(25\) 1.82849e51 1.72696
\(26\) 3.95373e51 0.892236
\(27\) 0 0
\(28\) −4.44951e52 −0.671497
\(29\) 8.69385e52 0.364490 0.182245 0.983253i \(-0.441664\pi\)
0.182245 + 0.983253i \(0.441664\pi\)
\(30\) 0 0
\(31\) 1.48414e54 0.545472 0.272736 0.962089i \(-0.412071\pi\)
0.272736 + 0.962089i \(0.412071\pi\)
\(32\) 7.20614e54 0.831244
\(33\) 0 0
\(34\) −6.71960e55 −0.847936
\(35\) 5.50582e56 2.41178
\(36\) 0 0
\(37\) −2.64558e57 −1.52461 −0.762304 0.647219i \(-0.775932\pi\)
−0.762304 + 0.647219i \(0.775932\pi\)
\(38\) 1.23282e57 0.268409
\(39\) 0 0
\(40\) −5.29179e58 −1.77179
\(41\) −3.07305e57 −0.0417791 −0.0208896 0.999782i \(-0.506650\pi\)
−0.0208896 + 0.999782i \(0.506650\pi\)
\(42\) 0 0
\(43\) 3.24223e59 0.774900 0.387450 0.921891i \(-0.373356\pi\)
0.387450 + 0.921891i \(0.373356\pi\)
\(44\) −1.90667e59 −0.196903
\(45\) 0 0
\(46\) 4.12496e60 0.840927
\(47\) 8.51145e60 0.791465 0.395732 0.918366i \(-0.370491\pi\)
0.395732 + 0.918366i \(0.370491\pi\)
\(48\) 0 0
\(49\) 5.57697e61 1.13303
\(50\) 1.30610e62 1.26932
\(51\) 0 0
\(52\) −2.40359e62 −0.558133
\(53\) 8.36780e62 0.969488 0.484744 0.874656i \(-0.338913\pi\)
0.484744 + 0.874656i \(0.338913\pi\)
\(54\) 0 0
\(55\) 2.35931e63 0.707207
\(56\) −1.00910e64 −1.56701
\(57\) 0 0
\(58\) 6.21004e63 0.267900
\(59\) −2.35468e64 −0.544293 −0.272147 0.962256i \(-0.587734\pi\)
−0.272147 + 0.962256i \(0.587734\pi\)
\(60\) 0 0
\(61\) 5.89372e64 0.403507 0.201753 0.979436i \(-0.435336\pi\)
0.201753 + 0.979436i \(0.435336\pi\)
\(62\) 1.06012e65 0.400922
\(63\) 0 0
\(64\) 7.91776e65 0.939795
\(65\) 2.97420e66 2.00462
\(66\) 0 0
\(67\) 6.87121e66 1.53216 0.766078 0.642747i \(-0.222206\pi\)
0.766078 + 0.642747i \(0.222206\pi\)
\(68\) 4.08505e66 0.530422
\(69\) 0 0
\(70\) 3.93282e67 1.77266
\(71\) −8.24197e66 −0.221361 −0.110681 0.993856i \(-0.535303\pi\)
−0.110681 + 0.993856i \(0.535303\pi\)
\(72\) 0 0
\(73\) −1.53931e68 −1.49983 −0.749913 0.661537i \(-0.769905\pi\)
−0.749913 + 0.661537i \(0.769905\pi\)
\(74\) −1.88974e68 −1.12059
\(75\) 0 0
\(76\) −7.49465e67 −0.167902
\(77\) 4.49901e68 0.625469
\(78\) 0 0
\(79\) −3.64169e68 −0.198570 −0.0992850 0.995059i \(-0.531656\pi\)
−0.0992850 + 0.995059i \(0.531656\pi\)
\(80\) −1.57615e69 −0.543016
\(81\) 0 0
\(82\) −2.19509e68 −0.0307076
\(83\) −1.69878e70 −1.52682 −0.763409 0.645915i \(-0.776476\pi\)
−0.763409 + 0.645915i \(0.776476\pi\)
\(84\) 0 0
\(85\) −5.05483e70 −1.90509
\(86\) 2.31594e70 0.569552
\(87\) 0 0
\(88\) −4.32412e70 −0.459495
\(89\) −7.76576e69 −0.0546322 −0.0273161 0.999627i \(-0.508696\pi\)
−0.0273161 + 0.999627i \(0.508696\pi\)
\(90\) 0 0
\(91\) 5.67155e71 1.77293
\(92\) −2.50768e71 −0.526038
\(93\) 0 0
\(94\) 6.07975e71 0.581726
\(95\) 9.27387e71 0.603044
\(96\) 0 0
\(97\) 4.51236e71 0.137162 0.0685812 0.997646i \(-0.478153\pi\)
0.0685812 + 0.997646i \(0.478153\pi\)
\(98\) 3.98365e72 0.832777
\(99\) 0 0
\(100\) −7.94014e72 −0.794014
\(101\) 9.01684e72 0.627082 0.313541 0.949575i \(-0.398485\pi\)
0.313541 + 0.949575i \(0.398485\pi\)
\(102\) 0 0
\(103\) −2.78483e73 −0.946762 −0.473381 0.880858i \(-0.656967\pi\)
−0.473381 + 0.880858i \(0.656967\pi\)
\(104\) −5.45108e73 −1.30246
\(105\) 0 0
\(106\) 5.97714e73 0.712573
\(107\) −7.06643e72 −0.0597988 −0.0298994 0.999553i \(-0.509519\pi\)
−0.0298994 + 0.999553i \(0.509519\pi\)
\(108\) 0 0
\(109\) −1.98041e74 −0.852486 −0.426243 0.904609i \(-0.640163\pi\)
−0.426243 + 0.904609i \(0.640163\pi\)
\(110\) 1.68526e74 0.519797
\(111\) 0 0
\(112\) −3.00559e74 −0.480255
\(113\) 1.25005e75 1.44400 0.721998 0.691896i \(-0.243224\pi\)
0.721998 + 0.691896i \(0.243224\pi\)
\(114\) 0 0
\(115\) 3.10300e75 1.88934
\(116\) −3.77527e74 −0.167583
\(117\) 0 0
\(118\) −1.68195e75 −0.400055
\(119\) −9.63915e75 −1.68490
\(120\) 0 0
\(121\) −8.58365e75 −0.816594
\(122\) 4.20990e75 0.296577
\(123\) 0 0
\(124\) −6.44481e75 −0.250795
\(125\) 4.13587e76 1.20047
\(126\) 0 0
\(127\) −7.12034e76 −1.15788 −0.578942 0.815369i \(-0.696534\pi\)
−0.578942 + 0.815369i \(0.696534\pi\)
\(128\) −1.15033e76 −0.140494
\(129\) 0 0
\(130\) 2.12447e77 1.47339
\(131\) −2.45112e77 −1.28517 −0.642587 0.766213i \(-0.722139\pi\)
−0.642587 + 0.766213i \(0.722139\pi\)
\(132\) 0 0
\(133\) 1.76845e77 0.533345
\(134\) 4.90812e77 1.12613
\(135\) 0 0
\(136\) 9.26444e77 1.23780
\(137\) 6.33470e77 0.647776 0.323888 0.946095i \(-0.395010\pi\)
0.323888 + 0.946095i \(0.395010\pi\)
\(138\) 0 0
\(139\) −1.79078e78 −1.07895 −0.539475 0.842002i \(-0.681377\pi\)
−0.539475 + 0.842002i \(0.681377\pi\)
\(140\) −2.39088e78 −1.10888
\(141\) 0 0
\(142\) −5.88726e77 −0.162700
\(143\) 2.43033e78 0.519876
\(144\) 0 0
\(145\) 4.67151e78 0.601901
\(146\) −1.09953e79 −1.10237
\(147\) 0 0
\(148\) 1.14883e79 0.700977
\(149\) 2.55369e79 1.21862 0.609310 0.792932i \(-0.291446\pi\)
0.609310 + 0.792932i \(0.291446\pi\)
\(150\) 0 0
\(151\) −4.58400e79 −1.34458 −0.672289 0.740289i \(-0.734689\pi\)
−0.672289 + 0.740289i \(0.734689\pi\)
\(152\) −1.69971e79 −0.391817
\(153\) 0 0
\(154\) 3.21365e79 0.459719
\(155\) 7.97479e79 0.900767
\(156\) 0 0
\(157\) 2.23717e80 1.58257 0.791283 0.611450i \(-0.209414\pi\)
0.791283 + 0.611450i \(0.209414\pi\)
\(158\) −2.60126e79 −0.145949
\(159\) 0 0
\(160\) 3.87211e80 1.37268
\(161\) 5.91717e80 1.67097
\(162\) 0 0
\(163\) 1.41890e80 0.255331 0.127665 0.991817i \(-0.459252\pi\)
0.127665 + 0.991817i \(0.459252\pi\)
\(164\) 1.33446e79 0.0192090
\(165\) 0 0
\(166\) −1.21344e81 −1.12221
\(167\) −5.60731e80 −0.416489 −0.208244 0.978077i \(-0.566775\pi\)
−0.208244 + 0.978077i \(0.566775\pi\)
\(168\) 0 0
\(169\) 9.84674e80 0.473618
\(170\) −3.61067e81 −1.40024
\(171\) 0 0
\(172\) −1.40793e81 −0.356280
\(173\) 3.93500e81 0.805864 0.402932 0.915230i \(-0.367991\pi\)
0.402932 + 0.915230i \(0.367991\pi\)
\(174\) 0 0
\(175\) 1.87357e82 2.52221
\(176\) −1.28793e81 −0.140825
\(177\) 0 0
\(178\) −5.54710e80 −0.0401547
\(179\) 3.22411e82 1.90228 0.951140 0.308759i \(-0.0999138\pi\)
0.951140 + 0.308759i \(0.0999138\pi\)
\(180\) 0 0
\(181\) 3.64648e82 1.43419 0.717095 0.696976i \(-0.245472\pi\)
0.717095 + 0.696976i \(0.245472\pi\)
\(182\) 4.05120e82 1.30310
\(183\) 0 0
\(184\) −5.68715e82 −1.22756
\(185\) −1.42156e83 −2.51766
\(186\) 0 0
\(187\) −4.13049e82 −0.494064
\(188\) −3.69606e82 −0.363896
\(189\) 0 0
\(190\) 6.62435e82 0.443238
\(191\) 1.38052e83 0.762649 0.381325 0.924441i \(-0.375468\pi\)
0.381325 + 0.924441i \(0.375468\pi\)
\(192\) 0 0
\(193\) −2.46248e83 −0.930101 −0.465051 0.885284i \(-0.653964\pi\)
−0.465051 + 0.885284i \(0.653964\pi\)
\(194\) 3.22319e82 0.100814
\(195\) 0 0
\(196\) −2.42178e83 −0.520940
\(197\) 4.95791e83 0.885689 0.442844 0.896598i \(-0.353969\pi\)
0.442844 + 0.896598i \(0.353969\pi\)
\(198\) 0 0
\(199\) −4.06154e82 −0.0501826 −0.0250913 0.999685i \(-0.507988\pi\)
−0.0250913 + 0.999685i \(0.507988\pi\)
\(200\) −1.80074e84 −1.85292
\(201\) 0 0
\(202\) 6.44075e83 0.460905
\(203\) 8.90819e83 0.532334
\(204\) 0 0
\(205\) −1.65126e83 −0.0689920
\(206\) −1.98921e84 −0.695870
\(207\) 0 0
\(208\) −1.62359e84 −0.399177
\(209\) 7.57803e83 0.156393
\(210\) 0 0
\(211\) −9.12088e84 −1.32962 −0.664810 0.747013i \(-0.731487\pi\)
−0.664810 + 0.747013i \(0.731487\pi\)
\(212\) −3.63368e84 −0.445747
\(213\) 0 0
\(214\) −5.04757e83 −0.0439521
\(215\) 1.74216e85 1.27963
\(216\) 0 0
\(217\) 1.52073e85 0.796657
\(218\) −1.41461e85 −0.626577
\(219\) 0 0
\(220\) −1.02452e85 −0.325156
\(221\) −5.20699e85 −1.40045
\(222\) 0 0
\(223\) −1.10503e85 −0.213918 −0.106959 0.994263i \(-0.534111\pi\)
−0.106959 + 0.994263i \(0.534111\pi\)
\(224\) 7.38379e85 1.21402
\(225\) 0 0
\(226\) 8.92917e85 1.06134
\(227\) −9.88450e84 −0.100003 −0.0500013 0.998749i \(-0.515923\pi\)
−0.0500013 + 0.998749i \(0.515923\pi\)
\(228\) 0 0
\(229\) 6.28394e85 0.461569 0.230785 0.973005i \(-0.425871\pi\)
0.230785 + 0.973005i \(0.425871\pi\)
\(230\) 2.21648e86 1.38867
\(231\) 0 0
\(232\) −8.56190e85 −0.391074
\(233\) −7.37366e85 −0.287867 −0.143934 0.989587i \(-0.545975\pi\)
−0.143934 + 0.989587i \(0.545975\pi\)
\(234\) 0 0
\(235\) 4.57349e86 1.30699
\(236\) 1.02251e86 0.250253
\(237\) 0 0
\(238\) −6.88526e86 −1.23840
\(239\) −5.40038e86 −0.833491 −0.416746 0.909023i \(-0.636829\pi\)
−0.416746 + 0.909023i \(0.636829\pi\)
\(240\) 0 0
\(241\) 4.43408e86 0.504873 0.252436 0.967614i \(-0.418768\pi\)
0.252436 + 0.967614i \(0.418768\pi\)
\(242\) −6.13132e86 −0.600196
\(243\) 0 0
\(244\) −2.55932e86 −0.185522
\(245\) 2.99670e87 1.87103
\(246\) 0 0
\(247\) 9.55303e86 0.443305
\(248\) −1.46161e87 −0.585256
\(249\) 0 0
\(250\) 2.95426e87 0.882345
\(251\) 4.58295e86 0.118319 0.0591595 0.998249i \(-0.481158\pi\)
0.0591595 + 0.998249i \(0.481158\pi\)
\(252\) 0 0
\(253\) 2.53558e87 0.489980
\(254\) −5.08608e87 −0.851044
\(255\) 0 0
\(256\) −8.29980e87 −1.04306
\(257\) 5.46778e87 0.596008 0.298004 0.954565i \(-0.403679\pi\)
0.298004 + 0.954565i \(0.403679\pi\)
\(258\) 0 0
\(259\) −2.71080e88 −2.22667
\(260\) −1.29153e88 −0.921675
\(261\) 0 0
\(262\) −1.75084e88 −0.944603
\(263\) 5.24740e87 0.246354 0.123177 0.992385i \(-0.460692\pi\)
0.123177 + 0.992385i \(0.460692\pi\)
\(264\) 0 0
\(265\) 4.49631e88 1.60096
\(266\) 1.26321e88 0.392009
\(267\) 0 0
\(268\) −2.98379e88 −0.704448
\(269\) 6.27826e87 0.129384 0.0646920 0.997905i \(-0.479393\pi\)
0.0646920 + 0.997905i \(0.479393\pi\)
\(270\) 0 0
\(271\) 8.67497e88 1.36423 0.682117 0.731243i \(-0.261059\pi\)
0.682117 + 0.731243i \(0.261059\pi\)
\(272\) 2.75940e88 0.379358
\(273\) 0 0
\(274\) 4.52489e88 0.476116
\(275\) 8.02848e88 0.739588
\(276\) 0 0
\(277\) 1.05020e89 0.742610 0.371305 0.928511i \(-0.378910\pi\)
0.371305 + 0.928511i \(0.378910\pi\)
\(278\) −1.27916e89 −0.793027
\(279\) 0 0
\(280\) −5.42225e89 −2.58768
\(281\) 1.29679e89 0.543362 0.271681 0.962387i \(-0.412420\pi\)
0.271681 + 0.962387i \(0.412420\pi\)
\(282\) 0 0
\(283\) 5.99633e89 1.93946 0.969729 0.244182i \(-0.0785194\pi\)
0.969729 + 0.244182i \(0.0785194\pi\)
\(284\) 3.57904e88 0.101776
\(285\) 0 0
\(286\) 1.73599e89 0.382109
\(287\) −3.14881e88 −0.0610180
\(288\) 0 0
\(289\) 2.20036e89 0.330920
\(290\) 3.33687e89 0.442397
\(291\) 0 0
\(292\) 6.68437e89 0.689583
\(293\) 3.90288e89 0.355400 0.177700 0.984085i \(-0.443134\pi\)
0.177700 + 0.984085i \(0.443134\pi\)
\(294\) 0 0
\(295\) −1.26525e90 −0.898819
\(296\) 2.60543e90 1.63580
\(297\) 0 0
\(298\) 1.82410e90 0.895686
\(299\) 3.19641e90 1.38888
\(300\) 0 0
\(301\) 3.32217e90 1.13173
\(302\) −3.27436e90 −0.988265
\(303\) 0 0
\(304\) −5.06254e89 −0.120083
\(305\) 3.16690e90 0.666331
\(306\) 0 0
\(307\) −5.03208e90 −0.834059 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(308\) −1.95368e90 −0.287575
\(309\) 0 0
\(310\) 5.69641e90 0.662063
\(311\) −5.56953e90 −0.575525 −0.287763 0.957702i \(-0.592911\pi\)
−0.287763 + 0.957702i \(0.592911\pi\)
\(312\) 0 0
\(313\) −1.93772e91 −1.58461 −0.792305 0.610125i \(-0.791119\pi\)
−0.792305 + 0.610125i \(0.791119\pi\)
\(314\) 1.59802e91 1.16319
\(315\) 0 0
\(316\) 1.58139e90 0.0912976
\(317\) 1.45832e91 0.750221 0.375110 0.926980i \(-0.377605\pi\)
0.375110 + 0.926980i \(0.377605\pi\)
\(318\) 0 0
\(319\) 3.81727e90 0.156096
\(320\) 4.25448e91 1.55193
\(321\) 0 0
\(322\) 4.22665e91 1.22817
\(323\) −1.62360e91 −0.421294
\(324\) 0 0
\(325\) 1.01209e92 2.09640
\(326\) 1.01352e91 0.187668
\(327\) 0 0
\(328\) 3.02641e90 0.0448263
\(329\) 8.72128e91 1.15593
\(330\) 0 0
\(331\) −5.05919e91 −0.537476 −0.268738 0.963213i \(-0.586607\pi\)
−0.268738 + 0.963213i \(0.586607\pi\)
\(332\) 7.37688e91 0.701993
\(333\) 0 0
\(334\) −4.00531e91 −0.306119
\(335\) 3.69214e92 2.53013
\(336\) 0 0
\(337\) 1.39888e92 0.771419 0.385710 0.922620i \(-0.373957\pi\)
0.385710 + 0.922620i \(0.373957\pi\)
\(338\) 7.03355e91 0.348109
\(339\) 0 0
\(340\) 2.19504e92 0.875913
\(341\) 6.51650e91 0.233604
\(342\) 0 0
\(343\) 6.70944e91 0.194290
\(344\) −3.19302e92 −0.831417
\(345\) 0 0
\(346\) 2.81078e92 0.592310
\(347\) 9.06972e91 0.172016 0.0860078 0.996294i \(-0.472589\pi\)
0.0860078 + 0.996294i \(0.472589\pi\)
\(348\) 0 0
\(349\) 4.50813e92 0.693216 0.346608 0.938010i \(-0.387333\pi\)
0.346608 + 0.938010i \(0.387333\pi\)
\(350\) 1.33830e93 1.85382
\(351\) 0 0
\(352\) 3.16405e92 0.355988
\(353\) 1.31440e93 1.33337 0.666685 0.745339i \(-0.267712\pi\)
0.666685 + 0.745339i \(0.267712\pi\)
\(354\) 0 0
\(355\) −4.42869e92 −0.365545
\(356\) 3.37225e91 0.0251185
\(357\) 0 0
\(358\) 2.30299e93 1.39818
\(359\) 1.25513e93 0.688245 0.344122 0.938925i \(-0.388176\pi\)
0.344122 + 0.938925i \(0.388176\pi\)
\(360\) 0 0
\(361\) −1.93576e93 −0.866642
\(362\) 2.60469e93 1.05413
\(363\) 0 0
\(364\) −2.46285e93 −0.815148
\(365\) −8.27122e93 −2.47674
\(366\) 0 0
\(367\) 5.71510e93 1.40189 0.700943 0.713217i \(-0.252763\pi\)
0.700943 + 0.713217i \(0.252763\pi\)
\(368\) −1.69391e93 −0.376222
\(369\) 0 0
\(370\) −1.01542e94 −1.85048
\(371\) 8.57410e93 1.41593
\(372\) 0 0
\(373\) −3.74173e93 −0.507809 −0.253905 0.967229i \(-0.581715\pi\)
−0.253905 + 0.967229i \(0.581715\pi\)
\(374\) −2.95042e93 −0.363137
\(375\) 0 0
\(376\) −8.38226e93 −0.849190
\(377\) 4.81213e93 0.442464
\(378\) 0 0
\(379\) 2.02793e94 1.53717 0.768586 0.639747i \(-0.220961\pi\)
0.768586 + 0.639747i \(0.220961\pi\)
\(380\) −4.02714e93 −0.277265
\(381\) 0 0
\(382\) 9.86109e93 0.560547
\(383\) 8.75423e93 0.452337 0.226169 0.974088i \(-0.427380\pi\)
0.226169 + 0.974088i \(0.427380\pi\)
\(384\) 0 0
\(385\) 2.41747e94 1.03287
\(386\) −1.75895e94 −0.683624
\(387\) 0 0
\(388\) −1.95947e93 −0.0630639
\(389\) 1.08943e94 0.319182 0.159591 0.987183i \(-0.448982\pi\)
0.159591 + 0.987183i \(0.448982\pi\)
\(390\) 0 0
\(391\) −5.43249e94 −1.31992
\(392\) −5.49233e94 −1.21567
\(393\) 0 0
\(394\) 3.54144e94 0.650981
\(395\) −1.95680e94 −0.327909
\(396\) 0 0
\(397\) −1.01644e95 −1.41654 −0.708268 0.705944i \(-0.750523\pi\)
−0.708268 + 0.705944i \(0.750523\pi\)
\(398\) −2.90116e93 −0.0368842
\(399\) 0 0
\(400\) −5.36347e94 −0.567879
\(401\) −1.22596e95 −1.18497 −0.592483 0.805583i \(-0.701852\pi\)
−0.592483 + 0.805583i \(0.701852\pi\)
\(402\) 0 0
\(403\) 8.21484e94 0.662163
\(404\) −3.91552e94 −0.288317
\(405\) 0 0
\(406\) 6.36314e94 0.391265
\(407\) −1.16161e95 −0.652928
\(408\) 0 0
\(409\) 2.38071e94 0.111894 0.0559469 0.998434i \(-0.482182\pi\)
0.0559469 + 0.998434i \(0.482182\pi\)
\(410\) −1.17950e94 −0.0507091
\(411\) 0 0
\(412\) 1.20930e95 0.435298
\(413\) −2.41273e95 −0.794935
\(414\) 0 0
\(415\) −9.12814e95 −2.52131
\(416\) 3.98866e95 1.00907
\(417\) 0 0
\(418\) 5.41300e94 0.114949
\(419\) −6.44324e95 −1.25399 −0.626993 0.779025i \(-0.715714\pi\)
−0.626993 + 0.779025i \(0.715714\pi\)
\(420\) 0 0
\(421\) −6.80537e95 −1.11315 −0.556577 0.830796i \(-0.687886\pi\)
−0.556577 + 0.830796i \(0.687886\pi\)
\(422\) −6.51507e95 −0.977270
\(423\) 0 0
\(424\) −8.24080e95 −1.04020
\(425\) −1.72010e96 −1.99232
\(426\) 0 0
\(427\) 6.03902e95 0.589318
\(428\) 3.06857e94 0.0274940
\(429\) 0 0
\(430\) 1.24443e96 0.940530
\(431\) 2.50780e96 1.74130 0.870649 0.491904i \(-0.163699\pi\)
0.870649 + 0.491904i \(0.163699\pi\)
\(432\) 0 0
\(433\) −1.03865e96 −0.609063 −0.304531 0.952502i \(-0.598500\pi\)
−0.304531 + 0.952502i \(0.598500\pi\)
\(434\) 1.08626e96 0.585543
\(435\) 0 0
\(436\) 8.59983e95 0.391952
\(437\) 9.96675e95 0.417812
\(438\) 0 0
\(439\) 4.08395e96 1.44919 0.724596 0.689174i \(-0.242027\pi\)
0.724596 + 0.689174i \(0.242027\pi\)
\(440\) −2.32350e96 −0.758787
\(441\) 0 0
\(442\) −3.71936e96 −1.02933
\(443\) −2.68993e96 −0.685496 −0.342748 0.939427i \(-0.611358\pi\)
−0.342748 + 0.939427i \(0.611358\pi\)
\(444\) 0 0
\(445\) −4.17281e95 −0.0902170
\(446\) −7.89325e95 −0.157229
\(447\) 0 0
\(448\) 8.11296e96 1.37256
\(449\) 8.90575e96 1.38893 0.694463 0.719528i \(-0.255642\pi\)
0.694463 + 0.719528i \(0.255642\pi\)
\(450\) 0 0
\(451\) −1.34931e95 −0.0178923
\(452\) −5.42831e96 −0.663913
\(453\) 0 0
\(454\) −7.06052e95 −0.0735019
\(455\) 3.04752e97 2.92773
\(456\) 0 0
\(457\) 1.95894e97 1.60354 0.801769 0.597634i \(-0.203892\pi\)
0.801769 + 0.597634i \(0.203892\pi\)
\(458\) 4.48863e96 0.339253
\(459\) 0 0
\(460\) −1.34746e97 −0.868673
\(461\) 1.16420e97 0.693335 0.346667 0.937988i \(-0.387313\pi\)
0.346667 + 0.937988i \(0.387313\pi\)
\(462\) 0 0
\(463\) −2.97083e97 −1.51068 −0.755338 0.655336i \(-0.772527\pi\)
−0.755338 + 0.655336i \(0.772527\pi\)
\(464\) −2.55014e96 −0.119856
\(465\) 0 0
\(466\) −5.26702e96 −0.211582
\(467\) −3.83937e97 −1.42624 −0.713122 0.701040i \(-0.752719\pi\)
−0.713122 + 0.701040i \(0.752719\pi\)
\(468\) 0 0
\(469\) 7.04061e97 2.23770
\(470\) 3.26686e97 0.960635
\(471\) 0 0
\(472\) 2.31894e97 0.583991
\(473\) 1.42359e97 0.331859
\(474\) 0 0
\(475\) 3.15580e97 0.630656
\(476\) 4.18576e97 0.774676
\(477\) 0 0
\(478\) −3.85751e97 −0.612616
\(479\) −7.62727e93 −0.000112234 0 −5.61168e−5 1.00000i \(-0.500018\pi\)
−5.61168e−5 1.00000i \(0.500018\pi\)
\(480\) 0 0
\(481\) −1.46435e98 −1.85076
\(482\) 3.16728e97 0.371081
\(483\) 0 0
\(484\) 3.72741e97 0.375450
\(485\) 2.42464e97 0.226503
\(486\) 0 0
\(487\) −7.46183e97 −0.599844 −0.299922 0.953964i \(-0.596961\pi\)
−0.299922 + 0.953964i \(0.596961\pi\)
\(488\) −5.80427e97 −0.432936
\(489\) 0 0
\(490\) 2.14055e98 1.37521
\(491\) −8.28295e97 −0.493982 −0.246991 0.969018i \(-0.579442\pi\)
−0.246991 + 0.969018i \(0.579442\pi\)
\(492\) 0 0
\(493\) −8.17850e97 −0.420495
\(494\) 6.82375e97 0.325829
\(495\) 0 0
\(496\) −4.35338e97 −0.179368
\(497\) −8.44516e97 −0.323296
\(498\) 0 0
\(499\) 1.76754e98 0.584387 0.292193 0.956359i \(-0.405615\pi\)
0.292193 + 0.956359i \(0.405615\pi\)
\(500\) −1.79598e98 −0.551946
\(501\) 0 0
\(502\) 3.27361e97 0.0869644
\(503\) 6.60146e98 1.63083 0.815414 0.578878i \(-0.196509\pi\)
0.815414 + 0.578878i \(0.196509\pi\)
\(504\) 0 0
\(505\) 4.84506e98 1.03553
\(506\) 1.81117e98 0.360135
\(507\) 0 0
\(508\) 3.09198e98 0.532366
\(509\) −4.28277e98 −0.686319 −0.343159 0.939277i \(-0.611497\pi\)
−0.343159 + 0.939277i \(0.611497\pi\)
\(510\) 0 0
\(511\) −1.57726e99 −2.19048
\(512\) −4.84211e98 −0.626153
\(513\) 0 0
\(514\) 3.90565e98 0.438066
\(515\) −1.49639e99 −1.56344
\(516\) 0 0
\(517\) 3.73718e98 0.338952
\(518\) −1.93633e99 −1.63661
\(519\) 0 0
\(520\) −2.92905e99 −2.15083
\(521\) −9.97828e98 −0.683091 −0.341546 0.939865i \(-0.610950\pi\)
−0.341546 + 0.939865i \(0.610950\pi\)
\(522\) 0 0
\(523\) −5.97094e98 −0.355411 −0.177706 0.984084i \(-0.556867\pi\)
−0.177706 + 0.984084i \(0.556867\pi\)
\(524\) 1.06439e99 0.590891
\(525\) 0 0
\(526\) 3.74823e98 0.181070
\(527\) −1.39616e99 −0.629287
\(528\) 0 0
\(529\) 7.87231e98 0.309008
\(530\) 3.21172e99 1.17671
\(531\) 0 0
\(532\) −7.67942e98 −0.245219
\(533\) −1.70096e98 −0.0507168
\(534\) 0 0
\(535\) −3.79704e98 −0.0987488
\(536\) −6.76692e99 −1.64390
\(537\) 0 0
\(538\) 4.48457e98 0.0950973
\(539\) 2.44872e99 0.485231
\(540\) 0 0
\(541\) 6.65195e99 1.15146 0.575730 0.817640i \(-0.304718\pi\)
0.575730 + 0.817640i \(0.304718\pi\)
\(542\) 6.19655e99 1.00271
\(543\) 0 0
\(544\) −6.77898e99 −0.958968
\(545\) −1.06414e100 −1.40775
\(546\) 0 0
\(547\) −1.55974e100 −1.80515 −0.902576 0.430531i \(-0.858326\pi\)
−0.902576 + 0.430531i \(0.858326\pi\)
\(548\) −2.75081e99 −0.297832
\(549\) 0 0
\(550\) 5.73476e99 0.543597
\(551\) 1.50047e99 0.133105
\(552\) 0 0
\(553\) −3.73147e99 −0.290009
\(554\) 7.50161e99 0.545818
\(555\) 0 0
\(556\) 7.77636e99 0.496074
\(557\) 2.03966e100 1.21855 0.609275 0.792959i \(-0.291460\pi\)
0.609275 + 0.792959i \(0.291460\pi\)
\(558\) 0 0
\(559\) 1.79461e100 0.940672
\(560\) −1.61501e100 −0.793070
\(561\) 0 0
\(562\) 9.26301e99 0.399371
\(563\) 3.81014e100 1.53952 0.769758 0.638336i \(-0.220377\pi\)
0.769758 + 0.638336i \(0.220377\pi\)
\(564\) 0 0
\(565\) 6.71697e100 2.38454
\(566\) 4.28319e100 1.42550
\(567\) 0 0
\(568\) 8.11687e99 0.237506
\(569\) −2.20282e100 −0.604480 −0.302240 0.953232i \(-0.597734\pi\)
−0.302240 + 0.953232i \(0.597734\pi\)
\(570\) 0 0
\(571\) 7.36336e99 0.177770 0.0888850 0.996042i \(-0.471670\pi\)
0.0888850 + 0.996042i \(0.471670\pi\)
\(572\) −1.05536e100 −0.239026
\(573\) 0 0
\(574\) −2.24921e99 −0.0448482
\(575\) 1.05592e101 1.97585
\(576\) 0 0
\(577\) −1.01125e101 −1.66702 −0.833512 0.552501i \(-0.813674\pi\)
−0.833512 + 0.552501i \(0.813674\pi\)
\(578\) 1.57172e100 0.243226
\(579\) 0 0
\(580\) −2.02858e100 −0.276739
\(581\) −1.74066e101 −2.22990
\(582\) 0 0
\(583\) 3.67411e100 0.415192
\(584\) 1.51594e101 1.60921
\(585\) 0 0
\(586\) 2.78784e100 0.261219
\(587\) 5.72321e100 0.503906 0.251953 0.967740i \(-0.418927\pi\)
0.251953 + 0.967740i \(0.418927\pi\)
\(588\) 0 0
\(589\) 2.56148e100 0.199197
\(590\) −9.03770e100 −0.660632
\(591\) 0 0
\(592\) 7.76021e100 0.501339
\(593\) 3.32917e100 0.202227 0.101114 0.994875i \(-0.467759\pi\)
0.101114 + 0.994875i \(0.467759\pi\)
\(594\) 0 0
\(595\) −5.17945e101 −2.78236
\(596\) −1.10893e101 −0.560292
\(597\) 0 0
\(598\) 2.28320e101 1.02082
\(599\) 2.64261e100 0.111162 0.0555808 0.998454i \(-0.482299\pi\)
0.0555808 + 0.998454i \(0.482299\pi\)
\(600\) 0 0
\(601\) 8.61412e100 0.320843 0.160422 0.987049i \(-0.448715\pi\)
0.160422 + 0.987049i \(0.448715\pi\)
\(602\) 2.37303e101 0.831824
\(603\) 0 0
\(604\) 1.99058e101 0.618204
\(605\) −4.61229e101 −1.34848
\(606\) 0 0
\(607\) −1.93297e101 −0.501001 −0.250501 0.968116i \(-0.580595\pi\)
−0.250501 + 0.968116i \(0.580595\pi\)
\(608\) 1.24371e101 0.303556
\(609\) 0 0
\(610\) 2.26212e101 0.489753
\(611\) 4.71116e101 0.960780
\(612\) 0 0
\(613\) 4.68691e101 0.848358 0.424179 0.905578i \(-0.360563\pi\)
0.424179 + 0.905578i \(0.360563\pi\)
\(614\) −3.59443e101 −0.613033
\(615\) 0 0
\(616\) −4.43073e101 −0.671087
\(617\) −1.24724e102 −1.78049 −0.890246 0.455479i \(-0.849468\pi\)
−0.890246 + 0.455479i \(0.849468\pi\)
\(618\) 0 0
\(619\) −5.63173e101 −0.714386 −0.357193 0.934031i \(-0.616266\pi\)
−0.357193 + 0.934031i \(0.616266\pi\)
\(620\) −3.46302e101 −0.414150
\(621\) 0 0
\(622\) −3.97833e101 −0.423011
\(623\) −7.95721e100 −0.0797898
\(624\) 0 0
\(625\) 2.86353e101 0.255436
\(626\) −1.38412e102 −1.16469
\(627\) 0 0
\(628\) −9.71481e101 −0.727625
\(629\) 2.48876e102 1.75887
\(630\) 0 0
\(631\) −1.04720e102 −0.659111 −0.329555 0.944136i \(-0.606899\pi\)
−0.329555 + 0.944136i \(0.606899\pi\)
\(632\) 3.58641e101 0.213053
\(633\) 0 0
\(634\) 1.04168e102 0.551412
\(635\) −3.82600e102 −1.91207
\(636\) 0 0
\(637\) 3.08691e102 1.37542
\(638\) 2.72668e101 0.114731
\(639\) 0 0
\(640\) −6.18112e101 −0.232006
\(641\) −1.40637e102 −0.498635 −0.249317 0.968422i \(-0.580206\pi\)
−0.249317 + 0.968422i \(0.580206\pi\)
\(642\) 0 0
\(643\) −1.43530e102 −0.454195 −0.227098 0.973872i \(-0.572924\pi\)
−0.227098 + 0.973872i \(0.572924\pi\)
\(644\) −2.56951e102 −0.768273
\(645\) 0 0
\(646\) −1.15974e102 −0.309651
\(647\) 4.09180e102 1.03254 0.516271 0.856425i \(-0.327320\pi\)
0.516271 + 0.856425i \(0.327320\pi\)
\(648\) 0 0
\(649\) −1.03388e102 −0.233099
\(650\) 7.22936e102 1.54086
\(651\) 0 0
\(652\) −6.16150e101 −0.117395
\(653\) −6.20506e102 −1.11793 −0.558965 0.829191i \(-0.688802\pi\)
−0.558965 + 0.829191i \(0.688802\pi\)
\(654\) 0 0
\(655\) −1.31707e103 −2.12227
\(656\) 9.01411e100 0.0137383
\(657\) 0 0
\(658\) 6.22963e102 0.849605
\(659\) −4.77744e102 −0.616422 −0.308211 0.951318i \(-0.599730\pi\)
−0.308211 + 0.951318i \(0.599730\pi\)
\(660\) 0 0
\(661\) −6.60511e102 −0.763004 −0.381502 0.924368i \(-0.624593\pi\)
−0.381502 + 0.924368i \(0.624593\pi\)
\(662\) −3.61379e102 −0.395045
\(663\) 0 0
\(664\) 1.67300e103 1.63818
\(665\) 9.50250e102 0.880740
\(666\) 0 0
\(667\) 5.02053e102 0.417020
\(668\) 2.43495e102 0.191491
\(669\) 0 0
\(670\) 2.63730e103 1.85964
\(671\) 2.58780e102 0.172806
\(672\) 0 0
\(673\) −1.40758e103 −0.843186 −0.421593 0.906785i \(-0.638529\pi\)
−0.421593 + 0.906785i \(0.638529\pi\)
\(674\) 9.99226e102 0.566993
\(675\) 0 0
\(676\) −4.27590e102 −0.217758
\(677\) 1.99664e103 0.963417 0.481708 0.876332i \(-0.340016\pi\)
0.481708 + 0.876332i \(0.340016\pi\)
\(678\) 0 0
\(679\) 4.62360e102 0.200324
\(680\) 4.97810e103 2.04404
\(681\) 0 0
\(682\) 4.65475e102 0.171699
\(683\) −9.40230e102 −0.328760 −0.164380 0.986397i \(-0.552562\pi\)
−0.164380 + 0.986397i \(0.552562\pi\)
\(684\) 0 0
\(685\) 3.40385e103 1.06971
\(686\) 4.79257e102 0.142803
\(687\) 0 0
\(688\) −9.51036e102 −0.254811
\(689\) 4.63166e103 1.17689
\(690\) 0 0
\(691\) 2.43467e103 0.556534 0.278267 0.960504i \(-0.410240\pi\)
0.278267 + 0.960504i \(0.410240\pi\)
\(692\) −1.70875e103 −0.370516
\(693\) 0 0
\(694\) 6.47852e102 0.126431
\(695\) −9.62245e103 −1.78172
\(696\) 0 0
\(697\) 2.89089e102 0.0481987
\(698\) 3.22017e103 0.509514
\(699\) 0 0
\(700\) −8.13590e103 −1.15965
\(701\) −5.37926e103 −0.727805 −0.363902 0.931437i \(-0.618556\pi\)
−0.363902 + 0.931437i \(0.618556\pi\)
\(702\) 0 0
\(703\) −4.56602e103 −0.556760
\(704\) 3.47650e103 0.402476
\(705\) 0 0
\(706\) 9.38879e103 0.980027
\(707\) 9.23913e103 0.915846
\(708\) 0 0
\(709\) −1.07535e104 −0.961530 −0.480765 0.876849i \(-0.659641\pi\)
−0.480765 + 0.876849i \(0.659641\pi\)
\(710\) −3.16343e103 −0.268675
\(711\) 0 0
\(712\) 7.64789e102 0.0586168
\(713\) 8.57061e103 0.624085
\(714\) 0 0
\(715\) 1.30590e104 0.858498
\(716\) −1.40006e104 −0.874622
\(717\) 0 0
\(718\) 8.96545e103 0.505860
\(719\) −3.21953e104 −1.72659 −0.863293 0.504703i \(-0.831602\pi\)
−0.863293 + 0.504703i \(0.831602\pi\)
\(720\) 0 0
\(721\) −2.85349e104 −1.38274
\(722\) −1.38272e104 −0.636982
\(723\) 0 0
\(724\) −1.58347e104 −0.659405
\(725\) 1.58966e104 0.629460
\(726\) 0 0
\(727\) 1.43449e104 0.513678 0.256839 0.966454i \(-0.417319\pi\)
0.256839 + 0.966454i \(0.417319\pi\)
\(728\) −5.58547e104 −1.90224
\(729\) 0 0
\(730\) −5.90815e104 −1.82040
\(731\) −3.05004e104 −0.893968
\(732\) 0 0
\(733\) −3.44849e104 −0.914817 −0.457408 0.889257i \(-0.651222\pi\)
−0.457408 + 0.889257i \(0.651222\pi\)
\(734\) 4.08231e104 1.03039
\(735\) 0 0
\(736\) 4.16140e104 0.951042
\(737\) 3.01699e104 0.656161
\(738\) 0 0
\(739\) −6.34155e104 −1.24932 −0.624660 0.780897i \(-0.714762\pi\)
−0.624660 + 0.780897i \(0.714762\pi\)
\(740\) 6.17307e104 1.15756
\(741\) 0 0
\(742\) 6.12450e104 1.04071
\(743\) −4.62751e104 −0.748611 −0.374305 0.927305i \(-0.622119\pi\)
−0.374305 + 0.927305i \(0.622119\pi\)
\(744\) 0 0
\(745\) 1.37218e105 2.01237
\(746\) −2.67273e104 −0.373240
\(747\) 0 0
\(748\) 1.79365e104 0.227158
\(749\) −7.24065e103 −0.0873355
\(750\) 0 0
\(751\) 4.35926e104 0.477040 0.238520 0.971138i \(-0.423338\pi\)
0.238520 + 0.971138i \(0.423338\pi\)
\(752\) −2.49664e104 −0.260258
\(753\) 0 0
\(754\) 3.43731e104 0.325211
\(755\) −2.46314e105 −2.22037
\(756\) 0 0
\(757\) −7.98775e104 −0.653770 −0.326885 0.945064i \(-0.605999\pi\)
−0.326885 + 0.945064i \(0.605999\pi\)
\(758\) 1.44856e105 1.12982
\(759\) 0 0
\(760\) −9.13311e104 −0.647027
\(761\) 2.56065e105 1.72906 0.864529 0.502583i \(-0.167617\pi\)
0.864529 + 0.502583i \(0.167617\pi\)
\(762\) 0 0
\(763\) −2.02923e105 −1.24505
\(764\) −5.99485e104 −0.350647
\(765\) 0 0
\(766\) 6.25316e104 0.332468
\(767\) −1.30333e105 −0.660732
\(768\) 0 0
\(769\) −6.72521e104 −0.310023 −0.155012 0.987913i \(-0.549542\pi\)
−0.155012 + 0.987913i \(0.549542\pi\)
\(770\) 1.72681e105 0.759158
\(771\) 0 0
\(772\) 1.06932e105 0.427638
\(773\) 2.46049e105 0.938578 0.469289 0.883045i \(-0.344510\pi\)
0.469289 + 0.883045i \(0.344510\pi\)
\(774\) 0 0
\(775\) 2.71374e105 0.942010
\(776\) −4.44387e104 −0.147166
\(777\) 0 0
\(778\) 7.78182e104 0.234599
\(779\) −5.30379e103 −0.0152570
\(780\) 0 0
\(781\) −3.61885e104 −0.0948000
\(782\) −3.88044e105 −0.970140
\(783\) 0 0
\(784\) −1.63588e105 −0.372576
\(785\) 1.20211e106 2.61337
\(786\) 0 0
\(787\) 2.71217e105 0.537328 0.268664 0.963234i \(-0.413418\pi\)
0.268664 + 0.963234i \(0.413418\pi\)
\(788\) −2.15295e105 −0.407218
\(789\) 0 0
\(790\) −1.39775e105 −0.241013
\(791\) 1.28087e106 2.10894
\(792\) 0 0
\(793\) 3.26223e105 0.489828
\(794\) −7.26044e105 −1.04115
\(795\) 0 0
\(796\) 1.76370e104 0.0230727
\(797\) −1.45163e106 −1.81395 −0.906977 0.421179i \(-0.861616\pi\)
−0.906977 + 0.421179i \(0.861616\pi\)
\(798\) 0 0
\(799\) −8.00691e105 −0.913077
\(800\) 1.31764e106 1.43553
\(801\) 0 0
\(802\) −8.75705e105 −0.870950
\(803\) −6.75873e105 −0.642315
\(804\) 0 0
\(805\) 3.17950e106 2.75937
\(806\) 5.86788e105 0.486690
\(807\) 0 0
\(808\) −8.87998e105 −0.672818
\(809\) −3.21507e104 −0.0232847 −0.0116423 0.999932i \(-0.503706\pi\)
−0.0116423 + 0.999932i \(0.503706\pi\)
\(810\) 0 0
\(811\) 8.87148e105 0.587131 0.293566 0.955939i \(-0.405158\pi\)
0.293566 + 0.955939i \(0.405158\pi\)
\(812\) −3.86834e105 −0.244754
\(813\) 0 0
\(814\) −8.29742e105 −0.479902
\(815\) 7.62423e105 0.421641
\(816\) 0 0
\(817\) 5.59578e105 0.282980
\(818\) 1.70055e105 0.0822419
\(819\) 0 0
\(820\) 7.17051e104 0.0317208
\(821\) −1.04180e105 −0.0440817 −0.0220409 0.999757i \(-0.507016\pi\)
−0.0220409 + 0.999757i \(0.507016\pi\)
\(822\) 0 0
\(823\) −1.21303e106 −0.469653 −0.234827 0.972037i \(-0.575452\pi\)
−0.234827 + 0.972037i \(0.575452\pi\)
\(824\) 2.74257e106 1.01581
\(825\) 0 0
\(826\) −1.72342e106 −0.584277
\(827\) −2.35778e106 −0.764808 −0.382404 0.923995i \(-0.624904\pi\)
−0.382404 + 0.923995i \(0.624904\pi\)
\(828\) 0 0
\(829\) 1.61461e106 0.479543 0.239771 0.970829i \(-0.422928\pi\)
0.239771 + 0.970829i \(0.422928\pi\)
\(830\) −6.52025e106 −1.85316
\(831\) 0 0
\(832\) 4.38255e106 1.14084
\(833\) −5.24639e106 −1.30713
\(834\) 0 0
\(835\) −3.01300e106 −0.687769
\(836\) −3.29073e105 −0.0719056
\(837\) 0 0
\(838\) −4.60242e106 −0.921679
\(839\) 5.26994e106 1.01040 0.505201 0.863002i \(-0.331418\pi\)
0.505201 + 0.863002i \(0.331418\pi\)
\(840\) 0 0
\(841\) −4.93340e106 −0.867147
\(842\) −4.86109e106 −0.818168
\(843\) 0 0
\(844\) 3.96070e106 0.611326
\(845\) 5.29099e106 0.782109
\(846\) 0 0
\(847\) −8.79527e106 −1.19263
\(848\) −2.45451e106 −0.318798
\(849\) 0 0
\(850\) −1.22867e107 −1.46435
\(851\) −1.52777e107 −1.74433
\(852\) 0 0
\(853\) 3.38806e105 0.0355068 0.0177534 0.999842i \(-0.494349\pi\)
0.0177534 + 0.999842i \(0.494349\pi\)
\(854\) 4.31369e106 0.433148
\(855\) 0 0
\(856\) 6.95918e105 0.0641602
\(857\) −1.33280e107 −1.17751 −0.588754 0.808313i \(-0.700381\pi\)
−0.588754 + 0.808313i \(0.700381\pi\)
\(858\) 0 0
\(859\) −1.17251e107 −0.951405 −0.475702 0.879606i \(-0.657806\pi\)
−0.475702 + 0.879606i \(0.657806\pi\)
\(860\) −7.56527e106 −0.588344
\(861\) 0 0
\(862\) 1.79133e107 1.27985
\(863\) 2.09858e107 1.43725 0.718623 0.695400i \(-0.244773\pi\)
0.718623 + 0.695400i \(0.244773\pi\)
\(864\) 0 0
\(865\) 2.11441e107 1.33076
\(866\) −7.41912e106 −0.447661
\(867\) 0 0
\(868\) −6.60369e106 −0.366283
\(869\) −1.59898e106 −0.0850395
\(870\) 0 0
\(871\) 3.80328e107 1.85993
\(872\) 1.95035e107 0.914661
\(873\) 0 0
\(874\) 7.11927e106 0.307092
\(875\) 4.23783e107 1.75327
\(876\) 0 0
\(877\) 1.41996e107 0.540493 0.270246 0.962791i \(-0.412895\pi\)
0.270246 + 0.962791i \(0.412895\pi\)
\(878\) 2.91718e107 1.06516
\(879\) 0 0
\(880\) −6.92050e106 −0.232552
\(881\) −7.66883e106 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(882\) 0 0
\(883\) −3.35873e107 −0.996802 −0.498401 0.866947i \(-0.666079\pi\)
−0.498401 + 0.866947i \(0.666079\pi\)
\(884\) 2.26111e107 0.643893
\(885\) 0 0
\(886\) −1.92142e107 −0.503839
\(887\) −6.18714e107 −1.55696 −0.778481 0.627669i \(-0.784009\pi\)
−0.778481 + 0.627669i \(0.784009\pi\)
\(888\) 0 0
\(889\) −7.29588e107 −1.69108
\(890\) −2.98065e106 −0.0663094
\(891\) 0 0
\(892\) 4.79854e106 0.0983540
\(893\) 1.46899e107 0.289029
\(894\) 0 0
\(895\) 1.73243e108 3.14133
\(896\) −1.17869e107 −0.205191
\(897\) 0 0
\(898\) 6.36140e107 1.02086
\(899\) 1.29029e107 0.198819
\(900\) 0 0
\(901\) −7.87178e107 −1.11845
\(902\) −9.63812e105 −0.0131508
\(903\) 0 0
\(904\) −1.23108e108 −1.54931
\(905\) 1.95938e108 2.36835
\(906\) 0 0
\(907\) 7.84800e107 0.875171 0.437586 0.899177i \(-0.355834\pi\)
0.437586 + 0.899177i \(0.355834\pi\)
\(908\) 4.29230e106 0.0459787
\(909\) 0 0
\(910\) 2.17685e108 2.15188
\(911\) 1.50467e108 1.42896 0.714482 0.699653i \(-0.246662\pi\)
0.714482 + 0.699653i \(0.246662\pi\)
\(912\) 0 0
\(913\) −7.45895e107 −0.653874
\(914\) 1.39927e108 1.17860
\(915\) 0 0
\(916\) −2.72877e107 −0.212218
\(917\) −2.51155e108 −1.87698
\(918\) 0 0
\(919\) −1.19946e108 −0.827887 −0.413944 0.910303i \(-0.635849\pi\)
−0.413944 + 0.910303i \(0.635849\pi\)
\(920\) −3.05590e108 −2.02714
\(921\) 0 0
\(922\) 8.31589e107 0.509601
\(923\) −4.56200e107 −0.268716
\(924\) 0 0
\(925\) −4.83742e108 −2.63294
\(926\) −2.12207e108 −1.11035
\(927\) 0 0
\(928\) 6.26491e107 0.302980
\(929\) −3.42076e107 −0.159056 −0.0795278 0.996833i \(-0.525341\pi\)
−0.0795278 + 0.996833i \(0.525341\pi\)
\(930\) 0 0
\(931\) 9.62531e107 0.413763
\(932\) 3.20198e107 0.132354
\(933\) 0 0
\(934\) −2.74247e108 −1.04829
\(935\) −2.21946e108 −0.815873
\(936\) 0 0
\(937\) 5.24351e108 1.78290 0.891450 0.453119i \(-0.149689\pi\)
0.891450 + 0.453119i \(0.149689\pi\)
\(938\) 5.02913e108 1.64471
\(939\) 0 0
\(940\) −1.98602e108 −0.600920
\(941\) 5.99656e107 0.174534 0.0872672 0.996185i \(-0.472187\pi\)
0.0872672 + 0.996185i \(0.472187\pi\)
\(942\) 0 0
\(943\) −1.77463e107 −0.0478003
\(944\) 6.90691e107 0.178981
\(945\) 0 0
\(946\) 1.01687e108 0.243916
\(947\) −6.02710e108 −1.39102 −0.695511 0.718515i \(-0.744822\pi\)
−0.695511 + 0.718515i \(0.744822\pi\)
\(948\) 0 0
\(949\) −8.52020e108 −1.82068
\(950\) 2.25419e108 0.463532
\(951\) 0 0
\(952\) 9.49284e108 1.80779
\(953\) −7.10042e108 −1.30135 −0.650674 0.759357i \(-0.725513\pi\)
−0.650674 + 0.759357i \(0.725513\pi\)
\(954\) 0 0
\(955\) 7.41801e108 1.25940
\(956\) 2.34509e108 0.383219
\(957\) 0 0
\(958\) −5.44818e104 −8.24916e−5 0
\(959\) 6.49087e108 0.946071
\(960\) 0 0
\(961\) −5.20026e108 −0.702460
\(962\) −1.04599e109 −1.36031
\(963\) 0 0
\(964\) −1.92548e108 −0.232128
\(965\) −1.32317e109 −1.53592
\(966\) 0 0
\(967\) −3.03350e108 −0.326495 −0.163248 0.986585i \(-0.552197\pi\)
−0.163248 + 0.986585i \(0.552197\pi\)
\(968\) 8.45337e108 0.876152
\(969\) 0 0
\(970\) 1.73193e108 0.166480
\(971\) 2.07397e109 1.91999 0.959997 0.280012i \(-0.0903384\pi\)
0.959997 + 0.280012i \(0.0903384\pi\)
\(972\) 0 0
\(973\) −1.83492e109 −1.57579
\(974\) −5.33000e108 −0.440885
\(975\) 0 0
\(976\) −1.72879e108 −0.132686
\(977\) −1.80962e109 −1.33794 −0.668969 0.743290i \(-0.733264\pi\)
−0.668969 + 0.743290i \(0.733264\pi\)
\(978\) 0 0
\(979\) −3.40976e107 −0.0233968
\(980\) −1.30130e109 −0.860254
\(981\) 0 0
\(982\) −5.91653e108 −0.363076
\(983\) 1.37476e109 0.812875 0.406438 0.913679i \(-0.366771\pi\)
0.406438 + 0.913679i \(0.366771\pi\)
\(984\) 0 0
\(985\) 2.66405e109 1.46258
\(986\) −5.84192e108 −0.309064
\(987\) 0 0
\(988\) −4.14836e108 −0.203820
\(989\) 1.87233e109 0.886578
\(990\) 0 0
\(991\) 2.85911e109 1.25760 0.628800 0.777567i \(-0.283546\pi\)
0.628800 + 0.777567i \(0.283546\pi\)
\(992\) 1.06949e109 0.453420
\(993\) 0 0
\(994\) −6.03240e108 −0.237622
\(995\) −2.18240e108 −0.0828691
\(996\) 0 0
\(997\) −8.40964e108 −0.296759 −0.148380 0.988930i \(-0.547406\pi\)
−0.148380 + 0.988930i \(0.547406\pi\)
\(998\) 1.26256e109 0.429524
\(999\) 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 9.74.a.a.1.4 5
3.2 odd 2 1.74.a.a.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.74.a.a.1.2 5 3.2 odd 2
9.74.a.a.1.4 5 1.1 even 1 trivial