Properties

Label 16.48.a.d.1.1
Level $16$
Weight $48$
Character 16.1
Self dual yes
Analytic conductor $223.852$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(223.852260248\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-124721.\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.52034e11 q^{3} +4.23962e16 q^{5} +3.90714e19 q^{7} -3.47448e21 q^{9} +O(q^{10})\) \(q-1.52034e11 q^{3} +4.23962e16 q^{5} +3.90714e19 q^{7} -3.47448e21 q^{9} -2.66558e24 q^{11} +2.26259e26 q^{13} -6.44566e27 q^{15} +8.41583e28 q^{17} -3.16735e29 q^{19} -5.94018e30 q^{21} -2.26985e31 q^{23} +1.08689e33 q^{25} +4.57064e33 q^{27} -3.39685e34 q^{29} -1.81646e35 q^{31} +4.05259e35 q^{33} +1.65648e36 q^{35} -7.13789e36 q^{37} -3.43991e37 q^{39} -2.78383e37 q^{41} +4.92777e37 q^{43} -1.47305e38 q^{45} -1.86825e39 q^{47} -3.71676e39 q^{49} -1.27949e40 q^{51} -2.48904e40 q^{53} -1.13011e41 q^{55} +4.81545e40 q^{57} -3.69239e40 q^{59} +3.52589e41 q^{61} -1.35753e41 q^{63} +9.59253e42 q^{65} +1.26183e42 q^{67} +3.45094e42 q^{69} +3.26023e43 q^{71} +3.59179e43 q^{73} -1.65245e44 q^{75} -1.04148e44 q^{77} +5.82596e44 q^{79} -6.02511e44 q^{81} +3.51619e44 q^{83} +3.56799e45 q^{85} +5.16437e45 q^{87} -2.92401e45 q^{89} +8.84027e45 q^{91} +2.76164e46 q^{93} -1.34284e46 q^{95} +2.17182e46 q^{97} +9.26152e45 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 38461494960q^{3} - 31114680242272200q^{5} + 39169218725888423200q^{7} - 17071972417358142200172q^{9} + O(q^{10}) \) \( 4q - 38461494960q^{3} - 31114680242272200q^{5} + 39169218725888423200q^{7} - \)\(17\!\cdots\!72\)\(q^{9} + \)\(19\!\cdots\!12\)\(q^{11} + \)\(12\!\cdots\!20\)\(q^{13} - \)\(12\!\cdots\!00\)\(q^{15} + \)\(21\!\cdots\!80\)\(q^{17} + \)\(10\!\cdots\!40\)\(q^{19} - \)\(26\!\cdots\!12\)\(q^{21} - \)\(13\!\cdots\!80\)\(q^{23} + \)\(10\!\cdots\!00\)\(q^{25} + \)\(73\!\cdots\!20\)\(q^{27} - \)\(22\!\cdots\!60\)\(q^{29} - \)\(75\!\cdots\!48\)\(q^{31} + \)\(26\!\cdots\!20\)\(q^{33} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(11\!\cdots\!60\)\(q^{37} - \)\(39\!\cdots\!36\)\(q^{39} + \)\(13\!\cdots\!28\)\(q^{41} + \)\(44\!\cdots\!00\)\(q^{43} + \)\(86\!\cdots\!00\)\(q^{45} - \)\(20\!\cdots\!20\)\(q^{47} + \)\(91\!\cdots\!72\)\(q^{49} + \)\(18\!\cdots\!52\)\(q^{51} + \)\(29\!\cdots\!60\)\(q^{53} - \)\(19\!\cdots\!00\)\(q^{55} + \)\(55\!\cdots\!60\)\(q^{57} - \)\(47\!\cdots\!80\)\(q^{59} + \)\(62\!\cdots\!88\)\(q^{61} - \)\(58\!\cdots\!40\)\(q^{63} + \)\(12\!\cdots\!00\)\(q^{65} - \)\(18\!\cdots\!80\)\(q^{67} - \)\(20\!\cdots\!04\)\(q^{69} - \)\(22\!\cdots\!68\)\(q^{71} + \)\(10\!\cdots\!80\)\(q^{73} - \)\(32\!\cdots\!00\)\(q^{75} - \)\(26\!\cdots\!00\)\(q^{77} + \)\(13\!\cdots\!60\)\(q^{79} - \)\(10\!\cdots\!16\)\(q^{81} + \)\(14\!\cdots\!60\)\(q^{83} - \)\(10\!\cdots\!00\)\(q^{85} + \)\(43\!\cdots\!60\)\(q^{87} - \)\(79\!\cdots\!80\)\(q^{89} - \)\(53\!\cdots\!68\)\(q^{91} + \)\(10\!\cdots\!20\)\(q^{93} - \)\(75\!\cdots\!00\)\(q^{95} + \)\(95\!\cdots\!20\)\(q^{97} - \)\(53\!\cdots\!16\)\(q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.52034e11 −0.932376 −0.466188 0.884686i \(-0.654373\pi\)
−0.466188 + 0.884686i \(0.654373\pi\)
\(4\) 0 0
\(5\) 4.23962e16 1.59049 0.795246 0.606286i \(-0.207341\pi\)
0.795246 + 0.606286i \(0.207341\pi\)
\(6\) 0 0
\(7\) 3.90714e19 0.539579 0.269790 0.962919i \(-0.413046\pi\)
0.269790 + 0.962919i \(0.413046\pi\)
\(8\) 0 0
\(9\) −3.47448e21 −0.130675
\(10\) 0 0
\(11\) −2.66558e24 −0.897561 −0.448781 0.893642i \(-0.648142\pi\)
−0.448781 + 0.893642i \(0.648142\pi\)
\(12\) 0 0
\(13\) 2.26259e26 1.50293 0.751463 0.659776i \(-0.229349\pi\)
0.751463 + 0.659776i \(0.229349\pi\)
\(14\) 0 0
\(15\) −6.44566e27 −1.48294
\(16\) 0 0
\(17\) 8.41583e28 1.02223 0.511114 0.859513i \(-0.329233\pi\)
0.511114 + 0.859513i \(0.329233\pi\)
\(18\) 0 0
\(19\) −3.16735e29 −0.281830 −0.140915 0.990022i \(-0.545004\pi\)
−0.140915 + 0.990022i \(0.545004\pi\)
\(20\) 0 0
\(21\) −5.94018e30 −0.503091
\(22\) 0 0
\(23\) −2.26985e31 −0.226669 −0.113335 0.993557i \(-0.536153\pi\)
−0.113335 + 0.993557i \(0.536153\pi\)
\(24\) 0 0
\(25\) 1.08689e33 1.52967
\(26\) 0 0
\(27\) 4.57064e33 1.05421
\(28\) 0 0
\(29\) −3.39685e34 −1.46125 −0.730624 0.682780i \(-0.760771\pi\)
−0.730624 + 0.682780i \(0.760771\pi\)
\(30\) 0 0
\(31\) −1.81646e35 −1.63015 −0.815074 0.579357i \(-0.803304\pi\)
−0.815074 + 0.579357i \(0.803304\pi\)
\(32\) 0 0
\(33\) 4.05259e35 0.836865
\(34\) 0 0
\(35\) 1.65648e36 0.858197
\(36\) 0 0
\(37\) −7.13789e36 −1.00191 −0.500956 0.865473i \(-0.667018\pi\)
−0.500956 + 0.865473i \(0.667018\pi\)
\(38\) 0 0
\(39\) −3.43991e37 −1.40129
\(40\) 0 0
\(41\) −2.78383e37 −0.350124 −0.175062 0.984557i \(-0.556013\pi\)
−0.175062 + 0.984557i \(0.556013\pi\)
\(42\) 0 0
\(43\) 4.92777e37 0.202368 0.101184 0.994868i \(-0.467737\pi\)
0.101184 + 0.994868i \(0.467737\pi\)
\(44\) 0 0
\(45\) −1.47305e38 −0.207837
\(46\) 0 0
\(47\) −1.86825e39 −0.948716 −0.474358 0.880332i \(-0.657320\pi\)
−0.474358 + 0.880332i \(0.657320\pi\)
\(48\) 0 0
\(49\) −3.71676e39 −0.708854
\(50\) 0 0
\(51\) −1.27949e40 −0.953100
\(52\) 0 0
\(53\) −2.48904e40 −0.750843 −0.375421 0.926854i \(-0.622502\pi\)
−0.375421 + 0.926854i \(0.622502\pi\)
\(54\) 0 0
\(55\) −1.13011e41 −1.42756
\(56\) 0 0
\(57\) 4.81545e40 0.262771
\(58\) 0 0
\(59\) −3.69239e40 −0.0895953 −0.0447976 0.998996i \(-0.514264\pi\)
−0.0447976 + 0.998996i \(0.514264\pi\)
\(60\) 0 0
\(61\) 3.52589e41 0.390857 0.195429 0.980718i \(-0.437390\pi\)
0.195429 + 0.980718i \(0.437390\pi\)
\(62\) 0 0
\(63\) −1.35753e41 −0.0705093
\(64\) 0 0
\(65\) 9.59253e42 2.39039
\(66\) 0 0
\(67\) 1.26183e42 0.154257 0.0771283 0.997021i \(-0.475425\pi\)
0.0771283 + 0.997021i \(0.475425\pi\)
\(68\) 0 0
\(69\) 3.45094e42 0.211341
\(70\) 0 0
\(71\) 3.26023e43 1.02018 0.510090 0.860121i \(-0.329612\pi\)
0.510090 + 0.860121i \(0.329612\pi\)
\(72\) 0 0
\(73\) 3.59179e43 0.585092 0.292546 0.956251i \(-0.405498\pi\)
0.292546 + 0.956251i \(0.405498\pi\)
\(74\) 0 0
\(75\) −1.65245e44 −1.42623
\(76\) 0 0
\(77\) −1.04148e44 −0.484305
\(78\) 0 0
\(79\) 5.82596e44 1.48297 0.741483 0.670972i \(-0.234123\pi\)
0.741483 + 0.670972i \(0.234123\pi\)
\(80\) 0 0
\(81\) −6.02511e44 −0.852250
\(82\) 0 0
\(83\) 3.51619e44 0.280373 0.140187 0.990125i \(-0.455230\pi\)
0.140187 + 0.990125i \(0.455230\pi\)
\(84\) 0 0
\(85\) 3.56799e45 1.62584
\(86\) 0 0
\(87\) 5.16437e45 1.36243
\(88\) 0 0
\(89\) −2.92401e45 −0.452182 −0.226091 0.974106i \(-0.572595\pi\)
−0.226091 + 0.974106i \(0.572595\pi\)
\(90\) 0 0
\(91\) 8.84027e45 0.810947
\(92\) 0 0
\(93\) 2.76164e46 1.51991
\(94\) 0 0
\(95\) −1.34284e46 −0.448248
\(96\) 0 0
\(97\) 2.17182e46 0.444311 0.222156 0.975011i \(-0.428691\pi\)
0.222156 + 0.975011i \(0.428691\pi\)
\(98\) 0 0
\(99\) 9.26152e45 0.117288
\(100\) 0 0
\(101\) −5.11557e46 −0.404894 −0.202447 0.979293i \(-0.564889\pi\)
−0.202447 + 0.979293i \(0.564889\pi\)
\(102\) 0 0
\(103\) −3.19569e47 −1.59547 −0.797736 0.603006i \(-0.793969\pi\)
−0.797736 + 0.603006i \(0.793969\pi\)
\(104\) 0 0
\(105\) −2.51841e47 −0.800162
\(106\) 0 0
\(107\) −2.00584e47 −0.409051 −0.204526 0.978861i \(-0.565565\pi\)
−0.204526 + 0.978861i \(0.565565\pi\)
\(108\) 0 0
\(109\) −5.86410e47 −0.773888 −0.386944 0.922103i \(-0.626469\pi\)
−0.386944 + 0.922103i \(0.626469\pi\)
\(110\) 0 0
\(111\) 1.08520e48 0.934158
\(112\) 0 0
\(113\) 3.11593e47 0.176297 0.0881484 0.996107i \(-0.471905\pi\)
0.0881484 + 0.996107i \(0.471905\pi\)
\(114\) 0 0
\(115\) −9.62329e47 −0.360516
\(116\) 0 0
\(117\) −7.86134e47 −0.196394
\(118\) 0 0
\(119\) 3.28818e48 0.551572
\(120\) 0 0
\(121\) −1.71442e48 −0.194384
\(122\) 0 0
\(123\) 4.23236e48 0.326447
\(124\) 0 0
\(125\) 1.59559e49 0.842432
\(126\) 0 0
\(127\) −2.66165e49 −0.967749 −0.483875 0.875137i \(-0.660771\pi\)
−0.483875 + 0.875137i \(0.660771\pi\)
\(128\) 0 0
\(129\) −7.49188e48 −0.188683
\(130\) 0 0
\(131\) −6.34924e49 −1.11390 −0.556950 0.830546i \(-0.688028\pi\)
−0.556950 + 0.830546i \(0.688028\pi\)
\(132\) 0 0
\(133\) −1.23753e49 −0.152069
\(134\) 0 0
\(135\) 1.93778e50 1.67672
\(136\) 0 0
\(137\) 1.99734e50 1.22326 0.611631 0.791143i \(-0.290514\pi\)
0.611631 + 0.791143i \(0.290514\pi\)
\(138\) 0 0
\(139\) 3.16608e50 1.37935 0.689674 0.724120i \(-0.257754\pi\)
0.689674 + 0.724120i \(0.257754\pi\)
\(140\) 0 0
\(141\) 2.84038e50 0.884560
\(142\) 0 0
\(143\) −6.03113e50 −1.34897
\(144\) 0 0
\(145\) −1.44014e51 −2.32410
\(146\) 0 0
\(147\) 5.65074e50 0.660919
\(148\) 0 0
\(149\) 1.00267e50 0.0853651 0.0426826 0.999089i \(-0.486410\pi\)
0.0426826 + 0.999089i \(0.486410\pi\)
\(150\) 0 0
\(151\) 2.40479e49 0.0149664 0.00748321 0.999972i \(-0.497618\pi\)
0.00748321 + 0.999972i \(0.497618\pi\)
\(152\) 0 0
\(153\) −2.92407e50 −0.133579
\(154\) 0 0
\(155\) −7.70111e51 −2.59274
\(156\) 0 0
\(157\) −1.42274e51 −0.354391 −0.177196 0.984176i \(-0.556702\pi\)
−0.177196 + 0.984176i \(0.556702\pi\)
\(158\) 0 0
\(159\) 3.78419e51 0.700068
\(160\) 0 0
\(161\) −8.86862e50 −0.122306
\(162\) 0 0
\(163\) 4.79256e51 0.494492 0.247246 0.968953i \(-0.420475\pi\)
0.247246 + 0.968953i \(0.420475\pi\)
\(164\) 0 0
\(165\) 1.71814e52 1.33103
\(166\) 0 0
\(167\) 9.89736e51 0.577673 0.288837 0.957378i \(-0.406732\pi\)
0.288837 + 0.957378i \(0.406732\pi\)
\(168\) 0 0
\(169\) 2.85292e52 1.25878
\(170\) 0 0
\(171\) 1.10049e51 0.0368280
\(172\) 0 0
\(173\) −7.43991e52 −1.89446 −0.947228 0.320562i \(-0.896128\pi\)
−0.947228 + 0.320562i \(0.896128\pi\)
\(174\) 0 0
\(175\) 4.24665e52 0.825377
\(176\) 0 0
\(177\) 5.61369e51 0.0835365
\(178\) 0 0
\(179\) −3.60632e52 −0.412115 −0.206057 0.978540i \(-0.566063\pi\)
−0.206057 + 0.978540i \(0.566063\pi\)
\(180\) 0 0
\(181\) 6.19557e52 0.545299 0.272650 0.962113i \(-0.412100\pi\)
0.272650 + 0.962113i \(0.412100\pi\)
\(182\) 0 0
\(183\) −5.36056e52 −0.364426
\(184\) 0 0
\(185\) −3.02619e53 −1.59353
\(186\) 0 0
\(187\) −2.24331e53 −0.917511
\(188\) 0 0
\(189\) 1.78581e53 0.568832
\(190\) 0 0
\(191\) 3.09862e53 0.770699 0.385349 0.922771i \(-0.374081\pi\)
0.385349 + 0.922771i \(0.374081\pi\)
\(192\) 0 0
\(193\) 4.75475e53 0.925829 0.462914 0.886403i \(-0.346804\pi\)
0.462914 + 0.886403i \(0.346804\pi\)
\(194\) 0 0
\(195\) −1.45839e54 −2.22874
\(196\) 0 0
\(197\) −7.61644e53 −0.915790 −0.457895 0.889006i \(-0.651396\pi\)
−0.457895 + 0.889006i \(0.651396\pi\)
\(198\) 0 0
\(199\) 1.90631e54 1.80778 0.903890 0.427765i \(-0.140699\pi\)
0.903890 + 0.427765i \(0.140699\pi\)
\(200\) 0 0
\(201\) −1.91841e53 −0.143825
\(202\) 0 0
\(203\) −1.32720e54 −0.788459
\(204\) 0 0
\(205\) −1.18024e54 −0.556869
\(206\) 0 0
\(207\) 7.88655e52 0.0296199
\(208\) 0 0
\(209\) 8.44284e53 0.252959
\(210\) 0 0
\(211\) −1.25520e54 −0.300660 −0.150330 0.988636i \(-0.548034\pi\)
−0.150330 + 0.988636i \(0.548034\pi\)
\(212\) 0 0
\(213\) −4.95665e54 −0.951191
\(214\) 0 0
\(215\) 2.08918e54 0.321864
\(216\) 0 0
\(217\) −7.09718e54 −0.879594
\(218\) 0 0
\(219\) −5.46075e54 −0.545526
\(220\) 0 0
\(221\) 1.90416e55 1.53633
\(222\) 0 0
\(223\) 1.43141e54 0.0934545 0.0467272 0.998908i \(-0.485121\pi\)
0.0467272 + 0.998908i \(0.485121\pi\)
\(224\) 0 0
\(225\) −3.77639e54 −0.199889
\(226\) 0 0
\(227\) −3.38487e55 −1.45524 −0.727619 0.685982i \(-0.759373\pi\)
−0.727619 + 0.685982i \(0.759373\pi\)
\(228\) 0 0
\(229\) −4.65218e54 −0.162750 −0.0813752 0.996684i \(-0.525931\pi\)
−0.0813752 + 0.996684i \(0.525931\pi\)
\(230\) 0 0
\(231\) 1.58341e55 0.451555
\(232\) 0 0
\(233\) −6.87236e55 −1.60045 −0.800225 0.599700i \(-0.795287\pi\)
−0.800225 + 0.599700i \(0.795287\pi\)
\(234\) 0 0
\(235\) −7.92068e55 −1.50893
\(236\) 0 0
\(237\) −8.85744e55 −1.38268
\(238\) 0 0
\(239\) −1.09219e56 −1.39942 −0.699710 0.714427i \(-0.746687\pi\)
−0.699710 + 0.714427i \(0.746687\pi\)
\(240\) 0 0
\(241\) −4.82622e55 −0.508401 −0.254201 0.967152i \(-0.581812\pi\)
−0.254201 + 0.967152i \(0.581812\pi\)
\(242\) 0 0
\(243\) −2.99259e55 −0.259597
\(244\) 0 0
\(245\) −1.57577e56 −1.12743
\(246\) 0 0
\(247\) −7.16642e55 −0.423569
\(248\) 0 0
\(249\) −5.34580e55 −0.261414
\(250\) 0 0
\(251\) 4.06217e56 1.64599 0.822993 0.568052i \(-0.192303\pi\)
0.822993 + 0.568052i \(0.192303\pi\)
\(252\) 0 0
\(253\) 6.05047e55 0.203450
\(254\) 0 0
\(255\) −5.42456e56 −1.51590
\(256\) 0 0
\(257\) 1.93698e56 0.450501 0.225251 0.974301i \(-0.427680\pi\)
0.225251 + 0.974301i \(0.427680\pi\)
\(258\) 0 0
\(259\) −2.78887e56 −0.540611
\(260\) 0 0
\(261\) 1.18023e56 0.190948
\(262\) 0 0
\(263\) −8.33659e56 −1.12727 −0.563636 0.826023i \(-0.690598\pi\)
−0.563636 + 0.826023i \(0.690598\pi\)
\(264\) 0 0
\(265\) −1.05526e57 −1.19421
\(266\) 0 0
\(267\) 4.44548e56 0.421604
\(268\) 0 0
\(269\) −1.88490e57 −1.50006 −0.750032 0.661402i \(-0.769962\pi\)
−0.750032 + 0.661402i \(0.769962\pi\)
\(270\) 0 0
\(271\) −2.56612e57 −1.71593 −0.857963 0.513712i \(-0.828270\pi\)
−0.857963 + 0.513712i \(0.828270\pi\)
\(272\) 0 0
\(273\) −1.34402e57 −0.756108
\(274\) 0 0
\(275\) −2.89721e57 −1.37297
\(276\) 0 0
\(277\) −8.36679e56 −0.334414 −0.167207 0.985922i \(-0.553475\pi\)
−0.167207 + 0.985922i \(0.553475\pi\)
\(278\) 0 0
\(279\) 6.31127e56 0.213019
\(280\) 0 0
\(281\) −3.96974e57 −1.13283 −0.566414 0.824121i \(-0.691670\pi\)
−0.566414 + 0.824121i \(0.691670\pi\)
\(282\) 0 0
\(283\) 4.45361e57 1.07580 0.537900 0.843009i \(-0.319218\pi\)
0.537900 + 0.843009i \(0.319218\pi\)
\(284\) 0 0
\(285\) 2.04157e57 0.417936
\(286\) 0 0
\(287\) −1.08768e57 −0.188919
\(288\) 0 0
\(289\) 3.04657e56 0.0449481
\(290\) 0 0
\(291\) −3.30190e57 −0.414265
\(292\) 0 0
\(293\) 1.05167e58 1.12328 0.561641 0.827381i \(-0.310170\pi\)
0.561641 + 0.827381i \(0.310170\pi\)
\(294\) 0 0
\(295\) −1.56543e57 −0.142501
\(296\) 0 0
\(297\) −1.21834e58 −0.946222
\(298\) 0 0
\(299\) −5.13574e57 −0.340667
\(300\) 0 0
\(301\) 1.92535e57 0.109193
\(302\) 0 0
\(303\) 7.77740e57 0.377514
\(304\) 0 0
\(305\) 1.49484e58 0.621656
\(306\) 0 0
\(307\) −5.43218e57 −0.193741 −0.0968707 0.995297i \(-0.530883\pi\)
−0.0968707 + 0.995297i \(0.530883\pi\)
\(308\) 0 0
\(309\) 4.85853e58 1.48758
\(310\) 0 0
\(311\) 4.80030e58 1.26299 0.631493 0.775381i \(-0.282442\pi\)
0.631493 + 0.775381i \(0.282442\pi\)
\(312\) 0 0
\(313\) −2.50321e58 −0.566507 −0.283253 0.959045i \(-0.591414\pi\)
−0.283253 + 0.959045i \(0.591414\pi\)
\(314\) 0 0
\(315\) −5.75541e57 −0.112145
\(316\) 0 0
\(317\) −6.65301e58 −1.11719 −0.558593 0.829442i \(-0.688659\pi\)
−0.558593 + 0.829442i \(0.688659\pi\)
\(318\) 0 0
\(319\) 9.05460e58 1.31156
\(320\) 0 0
\(321\) 3.04956e58 0.381390
\(322\) 0 0
\(323\) −2.66559e58 −0.288094
\(324\) 0 0
\(325\) 2.45920e59 2.29898
\(326\) 0 0
\(327\) 8.91542e58 0.721555
\(328\) 0 0
\(329\) −7.29953e58 −0.511907
\(330\) 0 0
\(331\) 1.52984e58 0.0930438 0.0465219 0.998917i \(-0.485186\pi\)
0.0465219 + 0.998917i \(0.485186\pi\)
\(332\) 0 0
\(333\) 2.48005e58 0.130924
\(334\) 0 0
\(335\) 5.34969e58 0.245344
\(336\) 0 0
\(337\) −1.19696e59 −0.477284 −0.238642 0.971108i \(-0.576702\pi\)
−0.238642 + 0.971108i \(0.576702\pi\)
\(338\) 0 0
\(339\) −4.73727e58 −0.164375
\(340\) 0 0
\(341\) 4.84194e59 1.46316
\(342\) 0 0
\(343\) −3.50084e59 −0.922062
\(344\) 0 0
\(345\) 1.46307e59 0.336136
\(346\) 0 0
\(347\) 6.60768e59 1.32528 0.662638 0.748940i \(-0.269437\pi\)
0.662638 + 0.748940i \(0.269437\pi\)
\(348\) 0 0
\(349\) 7.13810e59 1.25079 0.625396 0.780308i \(-0.284937\pi\)
0.625396 + 0.780308i \(0.284937\pi\)
\(350\) 0 0
\(351\) 1.03415e60 1.58441
\(352\) 0 0
\(353\) 4.98386e58 0.0668129 0.0334064 0.999442i \(-0.489364\pi\)
0.0334064 + 0.999442i \(0.489364\pi\)
\(354\) 0 0
\(355\) 1.38221e60 1.62259
\(356\) 0 0
\(357\) −4.99916e59 −0.514273
\(358\) 0 0
\(359\) −2.08989e60 −1.88540 −0.942700 0.333641i \(-0.891723\pi\)
−0.942700 + 0.333641i \(0.891723\pi\)
\(360\) 0 0
\(361\) −1.16273e60 −0.920572
\(362\) 0 0
\(363\) 2.60649e59 0.181239
\(364\) 0 0
\(365\) 1.52278e60 0.930585
\(366\) 0 0
\(367\) 1.22059e59 0.0656021 0.0328011 0.999462i \(-0.489557\pi\)
0.0328011 + 0.999462i \(0.489557\pi\)
\(368\) 0 0
\(369\) 9.67236e58 0.0457523
\(370\) 0 0
\(371\) −9.72503e59 −0.405139
\(372\) 0 0
\(373\) −2.41910e60 −0.888172 −0.444086 0.895984i \(-0.646472\pi\)
−0.444086 + 0.895984i \(0.646472\pi\)
\(374\) 0 0
\(375\) −2.42583e60 −0.785463
\(376\) 0 0
\(377\) −7.68569e60 −2.19615
\(378\) 0 0
\(379\) −6.29795e60 −1.58920 −0.794598 0.607135i \(-0.792319\pi\)
−0.794598 + 0.607135i \(0.792319\pi\)
\(380\) 0 0
\(381\) 4.04661e60 0.902306
\(382\) 0 0
\(383\) −3.88964e60 −0.766898 −0.383449 0.923562i \(-0.625264\pi\)
−0.383449 + 0.923562i \(0.625264\pi\)
\(384\) 0 0
\(385\) −4.41548e60 −0.770284
\(386\) 0 0
\(387\) −1.71214e59 −0.0264443
\(388\) 0 0
\(389\) 5.15774e60 0.705739 0.352869 0.935673i \(-0.385206\pi\)
0.352869 + 0.935673i \(0.385206\pi\)
\(390\) 0 0
\(391\) −1.91027e60 −0.231707
\(392\) 0 0
\(393\) 9.65300e60 1.03857
\(394\) 0 0
\(395\) 2.46999e61 2.35865
\(396\) 0 0
\(397\) −1.92619e61 −1.63351 −0.816754 0.576986i \(-0.804229\pi\)
−0.816754 + 0.576986i \(0.804229\pi\)
\(398\) 0 0
\(399\) 1.88146e60 0.141786
\(400\) 0 0
\(401\) 5.06490e58 0.00339374 0.00169687 0.999999i \(-0.499460\pi\)
0.00169687 + 0.999999i \(0.499460\pi\)
\(402\) 0 0
\(403\) −4.10992e61 −2.44999
\(404\) 0 0
\(405\) −2.55442e61 −1.35550
\(406\) 0 0
\(407\) 1.90266e61 0.899277
\(408\) 0 0
\(409\) 1.50217e61 0.632733 0.316367 0.948637i \(-0.397537\pi\)
0.316367 + 0.948637i \(0.397537\pi\)
\(410\) 0 0
\(411\) −3.03664e61 −1.14054
\(412\) 0 0
\(413\) −1.44267e60 −0.0483437
\(414\) 0 0
\(415\) 1.49073e61 0.445932
\(416\) 0 0
\(417\) −4.81352e61 −1.28607
\(418\) 0 0
\(419\) −6.95944e61 −1.66167 −0.830835 0.556518i \(-0.812137\pi\)
−0.830835 + 0.556518i \(0.812137\pi\)
\(420\) 0 0
\(421\) −3.78966e61 −0.809042 −0.404521 0.914529i \(-0.632562\pi\)
−0.404521 + 0.914529i \(0.632562\pi\)
\(422\) 0 0
\(423\) 6.49121e60 0.123973
\(424\) 0 0
\(425\) 9.14712e61 1.56367
\(426\) 0 0
\(427\) 1.37762e61 0.210898
\(428\) 0 0
\(429\) 9.16936e61 1.25775
\(430\) 0 0
\(431\) −4.06747e60 −0.0500160 −0.0250080 0.999687i \(-0.507961\pi\)
−0.0250080 + 0.999687i \(0.507961\pi\)
\(432\) 0 0
\(433\) −2.59125e61 −0.285788 −0.142894 0.989738i \(-0.545641\pi\)
−0.142894 + 0.989738i \(0.545641\pi\)
\(434\) 0 0
\(435\) 2.18950e62 2.16694
\(436\) 0 0
\(437\) 7.18941e60 0.0638821
\(438\) 0 0
\(439\) −3.51058e61 −0.280196 −0.140098 0.990138i \(-0.544742\pi\)
−0.140098 + 0.990138i \(0.544742\pi\)
\(440\) 0 0
\(441\) 1.29138e61 0.0926293
\(442\) 0 0
\(443\) 4.43894e61 0.286281 0.143140 0.989702i \(-0.454280\pi\)
0.143140 + 0.989702i \(0.454280\pi\)
\(444\) 0 0
\(445\) −1.23967e62 −0.719192
\(446\) 0 0
\(447\) −1.52440e61 −0.0795924
\(448\) 0 0
\(449\) 3.43198e62 1.61345 0.806725 0.590927i \(-0.201238\pi\)
0.806725 + 0.590927i \(0.201238\pi\)
\(450\) 0 0
\(451\) 7.42052e61 0.314257
\(452\) 0 0
\(453\) −3.65609e60 −0.0139543
\(454\) 0 0
\(455\) 3.74794e62 1.28981
\(456\) 0 0
\(457\) −9.64466e60 −0.0299403 −0.0149701 0.999888i \(-0.504765\pi\)
−0.0149701 + 0.999888i \(0.504765\pi\)
\(458\) 0 0
\(459\) 3.84658e62 1.07765
\(460\) 0 0
\(461\) −6.32143e61 −0.159898 −0.0799491 0.996799i \(-0.525476\pi\)
−0.0799491 + 0.996799i \(0.525476\pi\)
\(462\) 0 0
\(463\) −6.86213e62 −1.56785 −0.783927 0.620853i \(-0.786786\pi\)
−0.783927 + 0.620853i \(0.786786\pi\)
\(464\) 0 0
\(465\) 1.17083e63 2.41741
\(466\) 0 0
\(467\) 2.91371e62 0.543876 0.271938 0.962315i \(-0.412335\pi\)
0.271938 + 0.962315i \(0.412335\pi\)
\(468\) 0 0
\(469\) 4.93016e61 0.0832337
\(470\) 0 0
\(471\) 2.16304e62 0.330426
\(472\) 0 0
\(473\) −1.31354e62 −0.181637
\(474\) 0 0
\(475\) −3.44258e62 −0.431106
\(476\) 0 0
\(477\) 8.64813e61 0.0981161
\(478\) 0 0
\(479\) 9.39830e62 0.966417 0.483209 0.875505i \(-0.339471\pi\)
0.483209 + 0.875505i \(0.339471\pi\)
\(480\) 0 0
\(481\) −1.61501e63 −1.50580
\(482\) 0 0
\(483\) 1.34833e62 0.114035
\(484\) 0 0
\(485\) 9.20767e62 0.706674
\(486\) 0 0
\(487\) −2.39350e63 −1.66764 −0.833820 0.552037i \(-0.813851\pi\)
−0.833820 + 0.552037i \(0.813851\pi\)
\(488\) 0 0
\(489\) −7.28632e62 −0.461052
\(490\) 0 0
\(491\) −1.23586e63 −0.710481 −0.355241 0.934775i \(-0.615601\pi\)
−0.355241 + 0.934775i \(0.615601\pi\)
\(492\) 0 0
\(493\) −2.85874e63 −1.49373
\(494\) 0 0
\(495\) 3.92653e62 0.186546
\(496\) 0 0
\(497\) 1.27382e63 0.550468
\(498\) 0 0
\(499\) 2.94657e63 1.15865 0.579327 0.815095i \(-0.303316\pi\)
0.579327 + 0.815095i \(0.303316\pi\)
\(500\) 0 0
\(501\) −1.50473e63 −0.538609
\(502\) 0 0
\(503\) 1.27808e63 0.416591 0.208296 0.978066i \(-0.433208\pi\)
0.208296 + 0.978066i \(0.433208\pi\)
\(504\) 0 0
\(505\) −2.16880e63 −0.643981
\(506\) 0 0
\(507\) −4.33740e63 −1.17366
\(508\) 0 0
\(509\) 7.25177e63 1.78886 0.894429 0.447209i \(-0.147582\pi\)
0.894429 + 0.447209i \(0.147582\pi\)
\(510\) 0 0
\(511\) 1.40336e63 0.315704
\(512\) 0 0
\(513\) −1.44768e63 −0.297109
\(514\) 0 0
\(515\) −1.35485e64 −2.53759
\(516\) 0 0
\(517\) 4.97998e63 0.851531
\(518\) 0 0
\(519\) 1.13112e64 1.76634
\(520\) 0 0
\(521\) −9.84138e63 −1.40401 −0.702004 0.712173i \(-0.747711\pi\)
−0.702004 + 0.712173i \(0.747711\pi\)
\(522\) 0 0
\(523\) −4.91409e62 −0.0640698 −0.0320349 0.999487i \(-0.510199\pi\)
−0.0320349 + 0.999487i \(0.510199\pi\)
\(524\) 0 0
\(525\) −6.45635e63 −0.769561
\(526\) 0 0
\(527\) −1.52871e64 −1.66638
\(528\) 0 0
\(529\) −9.51264e63 −0.948621
\(530\) 0 0
\(531\) 1.28291e62 0.0117078
\(532\) 0 0
\(533\) −6.29866e63 −0.526210
\(534\) 0 0
\(535\) −8.50400e63 −0.650593
\(536\) 0 0
\(537\) 5.48283e63 0.384246
\(538\) 0 0
\(539\) 9.90734e63 0.636240
\(540\) 0 0
\(541\) 6.71742e61 0.00395428 0.00197714 0.999998i \(-0.499371\pi\)
0.00197714 + 0.999998i \(0.499371\pi\)
\(542\) 0 0
\(543\) −9.41938e63 −0.508424
\(544\) 0 0
\(545\) −2.48615e64 −1.23086
\(546\) 0 0
\(547\) −4.01645e62 −0.0182449 −0.00912243 0.999958i \(-0.502904\pi\)
−0.00912243 + 0.999958i \(0.502904\pi\)
\(548\) 0 0
\(549\) −1.22507e63 −0.0510751
\(550\) 0 0
\(551\) 1.07590e64 0.411823
\(552\) 0 0
\(553\) 2.27629e64 0.800177
\(554\) 0 0
\(555\) 4.60084e64 1.48577
\(556\) 0 0
\(557\) 6.03614e64 1.79127 0.895637 0.444785i \(-0.146720\pi\)
0.895637 + 0.444785i \(0.146720\pi\)
\(558\) 0 0
\(559\) 1.11495e64 0.304144
\(560\) 0 0
\(561\) 3.41059e64 0.855466
\(562\) 0 0
\(563\) 3.47404e63 0.0801470 0.0400735 0.999197i \(-0.487241\pi\)
0.0400735 + 0.999197i \(0.487241\pi\)
\(564\) 0 0
\(565\) 1.32104e64 0.280399
\(566\) 0 0
\(567\) −2.35409e64 −0.459856
\(568\) 0 0
\(569\) 9.25178e64 1.66375 0.831873 0.554966i \(-0.187269\pi\)
0.831873 + 0.554966i \(0.187269\pi\)
\(570\) 0 0
\(571\) 1.04188e65 1.72532 0.862658 0.505787i \(-0.168798\pi\)
0.862658 + 0.505787i \(0.168798\pi\)
\(572\) 0 0
\(573\) −4.71096e64 −0.718581
\(574\) 0 0
\(575\) −2.46708e64 −0.346728
\(576\) 0 0
\(577\) −8.68105e64 −1.12445 −0.562225 0.826984i \(-0.690055\pi\)
−0.562225 + 0.826984i \(0.690055\pi\)
\(578\) 0 0
\(579\) −7.22883e64 −0.863221
\(580\) 0 0
\(581\) 1.37382e64 0.151284
\(582\) 0 0
\(583\) 6.63474e64 0.673928
\(584\) 0 0
\(585\) −3.33291e64 −0.312364
\(586\) 0 0
\(587\) −5.04453e64 −0.436340 −0.218170 0.975911i \(-0.570009\pi\)
−0.218170 + 0.975911i \(0.570009\pi\)
\(588\) 0 0
\(589\) 5.75338e64 0.459424
\(590\) 0 0
\(591\) 1.15796e65 0.853861
\(592\) 0 0
\(593\) 1.82757e65 1.24477 0.622385 0.782712i \(-0.286164\pi\)
0.622385 + 0.782712i \(0.286164\pi\)
\(594\) 0 0
\(595\) 1.39406e65 0.877272
\(596\) 0 0
\(597\) −2.89823e65 −1.68553
\(598\) 0 0
\(599\) −6.30818e64 −0.339136 −0.169568 0.985519i \(-0.554237\pi\)
−0.169568 + 0.985519i \(0.554237\pi\)
\(600\) 0 0
\(601\) 1.16613e65 0.579693 0.289847 0.957073i \(-0.406396\pi\)
0.289847 + 0.957073i \(0.406396\pi\)
\(602\) 0 0
\(603\) −4.38422e63 −0.0201574
\(604\) 0 0
\(605\) −7.26847e64 −0.309166
\(606\) 0 0
\(607\) 8.69736e64 0.342337 0.171168 0.985242i \(-0.445246\pi\)
0.171168 + 0.985242i \(0.445246\pi\)
\(608\) 0 0
\(609\) 2.01779e65 0.735141
\(610\) 0 0
\(611\) −4.22709e65 −1.42585
\(612\) 0 0
\(613\) 3.95467e65 1.23535 0.617674 0.786435i \(-0.288075\pi\)
0.617674 + 0.786435i \(0.288075\pi\)
\(614\) 0 0
\(615\) 1.79436e65 0.519212
\(616\) 0 0
\(617\) −4.10300e65 −1.10002 −0.550009 0.835159i \(-0.685376\pi\)
−0.550009 + 0.835159i \(0.685376\pi\)
\(618\) 0 0
\(619\) −1.39273e65 −0.346047 −0.173024 0.984918i \(-0.555354\pi\)
−0.173024 + 0.984918i \(0.555354\pi\)
\(620\) 0 0
\(621\) −1.03747e65 −0.238958
\(622\) 0 0
\(623\) −1.14245e65 −0.243988
\(624\) 0 0
\(625\) −9.58171e64 −0.189785
\(626\) 0 0
\(627\) −1.28360e65 −0.235853
\(628\) 0 0
\(629\) −6.00713e65 −1.02418
\(630\) 0 0
\(631\) −1.54026e65 −0.243728 −0.121864 0.992547i \(-0.538887\pi\)
−0.121864 + 0.992547i \(0.538887\pi\)
\(632\) 0 0
\(633\) 1.90833e65 0.280328
\(634\) 0 0
\(635\) −1.12844e66 −1.53920
\(636\) 0 0
\(637\) −8.40952e65 −1.06536
\(638\) 0 0
\(639\) −1.13276e65 −0.133312
\(640\) 0 0
\(641\) −1.65837e65 −0.181350 −0.0906752 0.995881i \(-0.528903\pi\)
−0.0906752 + 0.995881i \(0.528903\pi\)
\(642\) 0 0
\(643\) 1.22213e66 1.24211 0.621053 0.783768i \(-0.286705\pi\)
0.621053 + 0.783768i \(0.286705\pi\)
\(644\) 0 0
\(645\) −3.17627e65 −0.300099
\(646\) 0 0
\(647\) 9.62351e65 0.845440 0.422720 0.906260i \(-0.361075\pi\)
0.422720 + 0.906260i \(0.361075\pi\)
\(648\) 0 0
\(649\) 9.84237e64 0.0804173
\(650\) 0 0
\(651\) 1.07901e66 0.820112
\(652\) 0 0
\(653\) 3.91841e65 0.277109 0.138554 0.990355i \(-0.455754\pi\)
0.138554 + 0.990355i \(0.455754\pi\)
\(654\) 0 0
\(655\) −2.69183e66 −1.77165
\(656\) 0 0
\(657\) −1.24796e65 −0.0764567
\(658\) 0 0
\(659\) −2.16961e66 −1.23758 −0.618792 0.785555i \(-0.712378\pi\)
−0.618792 + 0.785555i \(0.712378\pi\)
\(660\) 0 0
\(661\) −9.87453e65 −0.524544 −0.262272 0.964994i \(-0.584472\pi\)
−0.262272 + 0.964994i \(0.584472\pi\)
\(662\) 0 0
\(663\) −2.89497e66 −1.43244
\(664\) 0 0
\(665\) −5.24665e65 −0.241865
\(666\) 0 0
\(667\) 7.71034e65 0.331220
\(668\) 0 0
\(669\) −2.17623e65 −0.0871347
\(670\) 0 0
\(671\) −9.39857e65 −0.350818
\(672\) 0 0
\(673\) −5.18176e66 −1.80353 −0.901765 0.432227i \(-0.857728\pi\)
−0.901765 + 0.432227i \(0.857728\pi\)
\(674\) 0 0
\(675\) 4.96780e66 1.61260
\(676\) 0 0
\(677\) 5.62905e66 1.70452 0.852259 0.523120i \(-0.175232\pi\)
0.852259 + 0.523120i \(0.175232\pi\)
\(678\) 0 0
\(679\) 8.48559e65 0.239741
\(680\) 0 0
\(681\) 5.14615e66 1.35683
\(682\) 0 0
\(683\) 2.17309e66 0.534800 0.267400 0.963586i \(-0.413835\pi\)
0.267400 + 0.963586i \(0.413835\pi\)
\(684\) 0 0
\(685\) 8.46798e66 1.94559
\(686\) 0 0
\(687\) 7.07289e65 0.151745
\(688\) 0 0
\(689\) −5.63168e66 −1.12846
\(690\) 0 0
\(691\) −3.80507e66 −0.712244 −0.356122 0.934439i \(-0.615901\pi\)
−0.356122 + 0.934439i \(0.615901\pi\)
\(692\) 0 0
\(693\) 3.61861e65 0.0632864
\(694\) 0 0
\(695\) 1.34230e67 2.19384
\(696\) 0 0
\(697\) −2.34282e66 −0.357906
\(698\) 0 0
\(699\) 1.04483e67 1.49222
\(700\) 0 0
\(701\) 4.48575e66 0.599048 0.299524 0.954089i \(-0.403172\pi\)
0.299524 + 0.954089i \(0.403172\pi\)
\(702\) 0 0
\(703\) 2.26082e66 0.282368
\(704\) 0 0
\(705\) 1.20421e67 1.40689
\(706\) 0 0
\(707\) −1.99872e66 −0.218472
\(708\) 0 0
\(709\) −1.23589e67 −1.26414 −0.632068 0.774913i \(-0.717794\pi\)
−0.632068 + 0.774913i \(0.717794\pi\)
\(710\) 0 0
\(711\) −2.02422e66 −0.193786
\(712\) 0 0
\(713\) 4.12310e66 0.369504
\(714\) 0 0
\(715\) −2.55697e67 −2.14552
\(716\) 0 0
\(717\) 1.66050e67 1.30479
\(718\) 0 0
\(719\) 5.05205e66 0.371825 0.185913 0.982566i \(-0.440476\pi\)
0.185913 + 0.982566i \(0.440476\pi\)
\(720\) 0 0
\(721\) −1.24860e67 −0.860884
\(722\) 0 0
\(723\) 7.33749e66 0.474021
\(724\) 0 0
\(725\) −3.69202e67 −2.23522
\(726\) 0 0
\(727\) −2.10795e67 −1.19619 −0.598097 0.801424i \(-0.704076\pi\)
−0.598097 + 0.801424i \(0.704076\pi\)
\(728\) 0 0
\(729\) 2.05698e67 1.09429
\(730\) 0 0
\(731\) 4.14712e66 0.206866
\(732\) 0 0
\(733\) 2.34941e67 1.09905 0.549523 0.835478i \(-0.314809\pi\)
0.549523 + 0.835478i \(0.314809\pi\)
\(734\) 0 0
\(735\) 2.39570e67 1.05119
\(736\) 0 0
\(737\) −3.36352e66 −0.138455
\(738\) 0 0
\(739\) 8.69448e66 0.335814 0.167907 0.985803i \(-0.446299\pi\)
0.167907 + 0.985803i \(0.446299\pi\)
\(740\) 0 0
\(741\) 1.08954e67 0.394926
\(742\) 0 0
\(743\) −1.83903e67 −0.625679 −0.312840 0.949806i \(-0.601280\pi\)
−0.312840 + 0.949806i \(0.601280\pi\)
\(744\) 0 0
\(745\) 4.25094e66 0.135773
\(746\) 0 0
\(747\) −1.22169e66 −0.0366377
\(748\) 0 0
\(749\) −7.83710e66 −0.220715
\(750\) 0 0
\(751\) 2.92872e67 0.774709 0.387355 0.921931i \(-0.373389\pi\)
0.387355 + 0.921931i \(0.373389\pi\)
\(752\) 0 0
\(753\) −6.17588e67 −1.53468
\(754\) 0 0
\(755\) 1.01954e66 0.0238040
\(756\) 0 0
\(757\) −3.60671e67 −0.791330 −0.395665 0.918395i \(-0.629486\pi\)
−0.395665 + 0.918395i \(0.629486\pi\)
\(758\) 0 0
\(759\) −9.19877e66 −0.189691
\(760\) 0 0
\(761\) 6.36047e67 1.23296 0.616481 0.787370i \(-0.288558\pi\)
0.616481 + 0.787370i \(0.288558\pi\)
\(762\) 0 0
\(763\) −2.29118e67 −0.417574
\(764\) 0 0
\(765\) −1.23969e67 −0.212457
\(766\) 0 0
\(767\) −8.35437e66 −0.134655
\(768\) 0 0
\(769\) −1.58030e67 −0.239591 −0.119795 0.992799i \(-0.538224\pi\)
−0.119795 + 0.992799i \(0.538224\pi\)
\(770\) 0 0
\(771\) −2.94486e67 −0.420037
\(772\) 0 0
\(773\) −1.20430e68 −1.61628 −0.808142 0.588988i \(-0.799526\pi\)
−0.808142 + 0.588988i \(0.799526\pi\)
\(774\) 0 0
\(775\) −1.97430e68 −2.49358
\(776\) 0 0
\(777\) 4.24004e67 0.504052
\(778\) 0 0
\(779\) 8.81736e66 0.0986752
\(780\) 0 0
\(781\) −8.69040e67 −0.915674
\(782\) 0 0
\(783\) −1.55258e68 −1.54047
\(784\) 0 0
\(785\) −6.03186e67 −0.563656
\(786\) 0 0
\(787\) 1.72836e68 1.52134 0.760671 0.649137i \(-0.224870\pi\)
0.760671 + 0.649137i \(0.224870\pi\)
\(788\) 0 0
\(789\) 1.26745e68 1.05104
\(790\) 0 0
\(791\) 1.21744e67 0.0951261
\(792\) 0 0
\(793\) 7.97766e67 0.587429
\(794\) 0 0
\(795\) 1.60435e68 1.11345
\(796\) 0 0
\(797\) −5.17743e67 −0.338722 −0.169361 0.985554i \(-0.554170\pi\)
−0.169361 + 0.985554i \(0.554170\pi\)
\(798\) 0 0
\(799\) −1.57229e68 −0.969803
\(800\) 0 0
\(801\) 1.01594e67 0.0590887
\(802\) 0 0
\(803\) −9.57423e67 −0.525156
\(804\) 0 0
\(805\) −3.75995e67 −0.194527
\(806\) 0 0
\(807\) 2.86569e68 1.39862
\(808\) 0 0
\(809\) −1.34819e68 −0.620812 −0.310406 0.950604i \(-0.600465\pi\)
−0.310406 + 0.950604i \(0.600465\pi\)
\(810\) 0 0
\(811\) 1.29768e68 0.563869 0.281934 0.959434i \(-0.409024\pi\)
0.281934 + 0.959434i \(0.409024\pi\)
\(812\) 0 0
\(813\) 3.90137e68 1.59989
\(814\) 0 0
\(815\) 2.03186e68 0.786485
\(816\) 0 0
\(817\) −1.56080e67 −0.0570332
\(818\) 0 0
\(819\) −3.07154e67 −0.105970
\(820\) 0 0
\(821\) −3.05518e68 −0.995342 −0.497671 0.867366i \(-0.665811\pi\)
−0.497671 + 0.867366i \(0.665811\pi\)
\(822\) 0 0
\(823\) 3.54128e68 1.08959 0.544796 0.838568i \(-0.316607\pi\)
0.544796 + 0.838568i \(0.316607\pi\)
\(824\) 0 0
\(825\) 4.40474e68 1.28012
\(826\) 0 0
\(827\) −5.88899e68 −1.61682 −0.808410 0.588620i \(-0.799672\pi\)
−0.808410 + 0.588620i \(0.799672\pi\)
\(828\) 0 0
\(829\) 7.14204e68 1.85264 0.926319 0.376740i \(-0.122955\pi\)
0.926319 + 0.376740i \(0.122955\pi\)
\(830\) 0 0
\(831\) 1.27204e68 0.311799
\(832\) 0 0
\(833\) −3.12797e68 −0.724610
\(834\) 0 0
\(835\) 4.19610e68 0.918785
\(836\) 0 0
\(837\) −8.30241e68 −1.71852
\(838\) 0 0
\(839\) −4.08251e68 −0.798953 −0.399477 0.916743i \(-0.630808\pi\)
−0.399477 + 0.916743i \(0.630808\pi\)
\(840\) 0 0
\(841\) 6.13474e68 1.13525
\(842\) 0 0
\(843\) 6.03536e68 1.05622
\(844\) 0 0
\(845\) 1.20953e69 2.00209
\(846\) 0 0
\(847\) −6.69846e67 −0.104885
\(848\) 0 0
\(849\) −6.77100e68 −1.00305
\(850\) 0 0
\(851\) 1.62019e68 0.227102
\(852\) 0 0
\(853\) 6.69695e68 0.888331 0.444166 0.895945i \(-0.353500\pi\)
0.444166 + 0.895945i \(0.353500\pi\)
\(854\) 0 0
\(855\) 4.66566e67 0.0585746
\(856\) 0 0
\(857\) 1.01115e69 1.20162 0.600809 0.799393i \(-0.294845\pi\)
0.600809 + 0.799393i \(0.294845\pi\)
\(858\) 0 0
\(859\) 4.13647e68 0.465361 0.232681 0.972553i \(-0.425250\pi\)
0.232681 + 0.972553i \(0.425250\pi\)
\(860\) 0 0
\(861\) 1.65364e68 0.176144
\(862\) 0 0
\(863\) −6.85332e68 −0.691270 −0.345635 0.938369i \(-0.612336\pi\)
−0.345635 + 0.938369i \(0.612336\pi\)
\(864\) 0 0
\(865\) −3.15424e69 −3.01312
\(866\) 0 0
\(867\) −4.63182e67 −0.0419086
\(868\) 0 0
\(869\) −1.55296e69 −1.33105
\(870\) 0 0
\(871\) 2.85501e68 0.231836
\(872\) 0 0
\(873\) −7.54594e67 −0.0580602
\(874\) 0 0
\(875\) 6.23418e68 0.454559
\(876\) 0 0
\(877\) 6.43840e68 0.444925 0.222463 0.974941i \(-0.428590\pi\)
0.222463 + 0.974941i \(0.428590\pi\)
\(878\) 0 0
\(879\) −1.59889e69 −1.04732
\(880\) 0 0
\(881\) 8.86127e68 0.550253 0.275126 0.961408i \(-0.411280\pi\)
0.275126 + 0.961408i \(0.411280\pi\)
\(882\) 0 0
\(883\) −4.16643e68 −0.245294 −0.122647 0.992450i \(-0.539138\pi\)
−0.122647 + 0.992450i \(0.539138\pi\)
\(884\) 0 0
\(885\) 2.37999e68 0.132864
\(886\) 0 0
\(887\) −2.94393e69 −1.55856 −0.779278 0.626679i \(-0.784414\pi\)
−0.779278 + 0.626679i \(0.784414\pi\)
\(888\) 0 0
\(889\) −1.03994e69 −0.522177
\(890\) 0 0
\(891\) 1.60604e69 0.764946
\(892\) 0 0
\(893\) 5.91742e68 0.267376
\(894\) 0 0
\(895\) −1.52894e69 −0.655466
\(896\) 0 0
\(897\) 7.80807e68 0.317630
\(898\) 0 0
\(899\) 6.17026e69 2.38205
\(900\) 0 0
\(901\) −2.09473e69 −0.767532
\(902\) 0 0
\(903\) −2.92718e68 −0.101809
\(904\) 0 0
\(905\) 2.62669e69 0.867295
\(906\) 0 0
\(907\) −3.36233e69 −1.05407 −0.527034 0.849844i \(-0.676696\pi\)
−0.527034 + 0.849844i \(0.676696\pi\)
\(908\) 0 0
\(909\) 1.77739e68 0.0529094
\(910\) 0 0
\(911\) −4.29823e69 −1.21509 −0.607544 0.794286i \(-0.707845\pi\)
−0.607544 + 0.794286i \(0.707845\pi\)
\(912\) 0 0
\(913\) −9.37269e68 −0.251652
\(914\) 0 0
\(915\) −2.27267e69 −0.579617
\(916\) 0 0
\(917\) −2.48074e69 −0.601037
\(918\) 0 0
\(919\) −5.89124e69 −1.35610 −0.678050 0.735015i \(-0.737175\pi\)
−0.678050 + 0.735015i \(0.737175\pi\)
\(920\) 0 0
\(921\) 8.25876e68 0.180640
\(922\) 0 0
\(923\) 7.37656e69 1.53325
\(924\) 0 0
\(925\) −7.75813e69 −1.53259
\(926\) 0 0
\(927\) 1.11034e69 0.208488
\(928\) 0 0
\(929\) −3.81127e69 −0.680301 −0.340150 0.940371i \(-0.610478\pi\)
−0.340150 + 0.940371i \(0.610478\pi\)
\(930\) 0 0
\(931\) 1.17723e69 0.199776
\(932\) 0 0
\(933\) −7.29808e69 −1.17758
\(934\) 0 0
\(935\) −9.51078e69 −1.45930
\(936\) 0 0
\(937\) −3.18868e69 −0.465298 −0.232649 0.972561i \(-0.574739\pi\)
−0.232649 + 0.972561i \(0.574739\pi\)
\(938\) 0 0
\(939\) 3.80573e69 0.528197
\(940\) 0 0
\(941\) 7.20176e69 0.950785 0.475393 0.879774i \(-0.342306\pi\)
0.475393 + 0.879774i \(0.342306\pi\)
\(942\) 0 0
\(943\) 6.31886e68 0.0793623
\(944\) 0 0
\(945\) 7.57117e69 0.904723
\(946\) 0 0
\(947\) −7.19607e69 −0.818222 −0.409111 0.912485i \(-0.634161\pi\)
−0.409111 + 0.912485i \(0.634161\pi\)
\(948\) 0 0
\(949\) 8.12676e69 0.879350
\(950\) 0 0
\(951\) 1.01148e70 1.04164
\(952\) 0 0
\(953\) 1.03533e70 1.01483 0.507417 0.861700i \(-0.330600\pi\)
0.507417 + 0.861700i \(0.330600\pi\)
\(954\) 0 0
\(955\) 1.31370e70 1.22579
\(956\) 0 0
\(957\) −1.37661e70 −1.22287
\(958\) 0 0
\(959\) 7.80391e69 0.660047
\(960\) 0 0
\(961\) 2.05789e70 1.65738
\(962\) 0 0
\(963\) 6.96926e68 0.0534526
\(964\) 0 0
\(965\) 2.01583e70 1.47252
\(966\) 0 0
\(967\) 1.88655e70 1.31264 0.656321 0.754482i \(-0.272112\pi\)
0.656321 + 0.754482i \(0.272112\pi\)
\(968\) 0 0
\(969\) 4.05260e69 0.268612
\(970\) 0 0
\(971\) 1.54065e70 0.972862 0.486431 0.873719i \(-0.338299\pi\)
0.486431 + 0.873719i \(0.338299\pi\)
\(972\) 0 0
\(973\) 1.23703e70 0.744268
\(974\) 0 0
\(975\) −3.73882e70 −2.14351
\(976\) 0 0
\(977\) −1.28077e70 −0.699760 −0.349880 0.936795i \(-0.613778\pi\)
−0.349880 + 0.936795i \(0.613778\pi\)
\(978\) 0 0
\(979\) 7.79418e69 0.405861
\(980\) 0 0
\(981\) 2.03747e69 0.101128
\(982\) 0 0
\(983\) −1.22658e70 −0.580349 −0.290174 0.956974i \(-0.593713\pi\)
−0.290174 + 0.956974i \(0.593713\pi\)
\(984\) 0 0
\(985\) −3.22908e70 −1.45656
\(986\) 0 0
\(987\) 1.10978e70 0.477290
\(988\) 0 0
\(989\) −1.11853e69 −0.0458705
\(990\) 0 0
\(991\) −2.78865e70 −1.09059 −0.545296 0.838244i \(-0.683583\pi\)
−0.545296 + 0.838244i \(0.683583\pi\)
\(992\) 0 0
\(993\) −2.32587e69 −0.0867519
\(994\) 0 0
\(995\) 8.08202e70 2.87526
\(996\) 0 0
\(997\) 2.05189e70 0.696336 0.348168 0.937432i \(-0.386804\pi\)
0.348168 + 0.937432i \(0.386804\pi\)
\(998\) 0 0
\(999\) −3.26247e70 −1.05623
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 16.48.a.d.1.1 4
4.3 odd 2 1.48.a.a.1.2 4
12.11 even 2 9.48.a.c.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.48.a.a.1.2 4 4.3 odd 2
9.48.a.c.1.3 4 12.11 even 2
16.48.a.d.1.1 4 1.1 even 1 trivial