Properties

Label 9.48.a.c.1.3
Level $9$
Weight $48$
Character 9.1
Self dual yes
Analytic conductor $125.917$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(125.916896390\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{11}\cdot 5^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-124721.\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.54692e6 q^{2} -1.38345e14 q^{4} -4.23962e16 q^{5} -3.90714e19 q^{7} -4.31719e20 q^{8} +O(q^{10})\) \(q+1.54692e6 q^{2} -1.38345e14 q^{4} -4.23962e16 q^{5} -3.90714e19 q^{7} -4.31719e20 q^{8} -6.55837e22 q^{10} -2.66558e24 q^{11} +2.26259e26 q^{13} -6.04405e25 q^{14} +1.88024e28 q^{16} -8.41583e28 q^{17} +3.16735e29 q^{19} +5.86528e30 q^{20} -4.12345e30 q^{22} -2.26985e31 q^{23} +1.08689e33 q^{25} +3.50006e32 q^{26} +5.40532e33 q^{28} +3.39685e34 q^{29} +1.81646e35 q^{31} +8.98449e34 q^{32} -1.30186e35 q^{34} +1.65648e36 q^{35} -7.13789e36 q^{37} +4.89965e35 q^{38} +1.83032e37 q^{40} +2.78383e37 q^{41} -4.92777e37 q^{43} +3.68769e38 q^{44} -3.51128e37 q^{46} -1.86825e39 q^{47} -3.71676e39 q^{49} +1.68134e39 q^{50} -3.13017e40 q^{52} +2.48904e40 q^{53} +1.13011e41 q^{55} +1.68679e40 q^{56} +5.25467e40 q^{58} -3.69239e40 q^{59} +3.52589e41 q^{61} +2.80993e41 q^{62} -2.50722e42 q^{64} -9.59253e42 q^{65} -1.26183e42 q^{67} +1.16428e43 q^{68} +2.56245e42 q^{70} +3.26023e43 q^{71} +3.59179e43 q^{73} -1.10418e43 q^{74} -4.38186e43 q^{76} +1.04148e44 q^{77} -5.82596e44 q^{79} -7.97151e44 q^{80} +4.30637e43 q^{82} +3.51619e44 q^{83} +3.56799e45 q^{85} -7.62288e43 q^{86} +1.15078e45 q^{88} +2.92401e45 q^{89} -8.84027e45 q^{91} +3.14021e45 q^{92} -2.89005e45 q^{94} -1.34284e46 q^{95} +2.17182e46 q^{97} -5.74955e45 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 5785560q^{2} + 404807499161152q^{4} + 31114680242272200q^{5} - 39169218725888423200q^{7} - 2392716988073784337920q^{8} + O(q^{10}) \) \( 4q - 5785560q^{2} + 404807499161152q^{4} + 31114680242272200q^{5} - 39169218725888423200q^{7} - \)\(23\!\cdots\!20\)\(q^{8} + \)\(16\!\cdots\!00\)\(q^{10} + \)\(19\!\cdots\!12\)\(q^{11} + \)\(12\!\cdots\!20\)\(q^{13} - \)\(34\!\cdots\!04\)\(q^{14} + \)\(12\!\cdots\!84\)\(q^{16} - \)\(21\!\cdots\!80\)\(q^{17} - \)\(10\!\cdots\!40\)\(q^{19} + \)\(15\!\cdots\!00\)\(q^{20} - \)\(79\!\cdots\!80\)\(q^{22} - \)\(13\!\cdots\!80\)\(q^{23} + \)\(10\!\cdots\!00\)\(q^{25} - \)\(22\!\cdots\!88\)\(q^{26} - \)\(36\!\cdots\!80\)\(q^{28} + \)\(22\!\cdots\!60\)\(q^{29} + \)\(75\!\cdots\!48\)\(q^{31} - \)\(11\!\cdots\!60\)\(q^{32} - \)\(12\!\cdots\!16\)\(q^{34} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(11\!\cdots\!60\)\(q^{37} - \)\(29\!\cdots\!80\)\(q^{38} + \)\(70\!\cdots\!00\)\(q^{40} - \)\(13\!\cdots\!28\)\(q^{41} - \)\(44\!\cdots\!00\)\(q^{43} + \)\(15\!\cdots\!56\)\(q^{44} + \)\(12\!\cdots\!68\)\(q^{46} - \)\(20\!\cdots\!20\)\(q^{47} + \)\(91\!\cdots\!72\)\(q^{49} + \)\(19\!\cdots\!00\)\(q^{50} - \)\(55\!\cdots\!00\)\(q^{52} - \)\(29\!\cdots\!60\)\(q^{53} + \)\(19\!\cdots\!00\)\(q^{55} - \)\(27\!\cdots\!40\)\(q^{56} - \)\(73\!\cdots\!80\)\(q^{58} - \)\(47\!\cdots\!80\)\(q^{59} + \)\(62\!\cdots\!88\)\(q^{61} + \)\(68\!\cdots\!80\)\(q^{62} + \)\(55\!\cdots\!72\)\(q^{64} - \)\(12\!\cdots\!00\)\(q^{65} + \)\(18\!\cdots\!80\)\(q^{67} + \)\(27\!\cdots\!40\)\(q^{68} - \)\(97\!\cdots\!00\)\(q^{70} - \)\(22\!\cdots\!68\)\(q^{71} + \)\(10\!\cdots\!80\)\(q^{73} - \)\(18\!\cdots\!44\)\(q^{74} + \)\(35\!\cdots\!80\)\(q^{76} + \)\(26\!\cdots\!00\)\(q^{77} - \)\(13\!\cdots\!60\)\(q^{79} + \)\(94\!\cdots\!00\)\(q^{80} + \)\(38\!\cdots\!20\)\(q^{82} + \)\(14\!\cdots\!60\)\(q^{83} - \)\(10\!\cdots\!00\)\(q^{85} - \)\(95\!\cdots\!28\)\(q^{86} - \)\(15\!\cdots\!60\)\(q^{88} + \)\(79\!\cdots\!80\)\(q^{89} + \)\(53\!\cdots\!68\)\(q^{91} - \)\(13\!\cdots\!80\)\(q^{92} + \)\(34\!\cdots\!24\)\(q^{94} - \)\(75\!\cdots\!00\)\(q^{95} + \)\(95\!\cdots\!20\)\(q^{97} + \)\(29\!\cdots\!20\)\(q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.54692e6 0.130396 0.0651980 0.997872i \(-0.479232\pi\)
0.0651980 + 0.997872i \(0.479232\pi\)
\(3\) 0 0
\(4\) −1.38345e14 −0.982997
\(5\) −4.23962e16 −1.59049 −0.795246 0.606286i \(-0.792659\pi\)
−0.795246 + 0.606286i \(0.792659\pi\)
\(6\) 0 0
\(7\) −3.90714e19 −0.539579 −0.269790 0.962919i \(-0.586954\pi\)
−0.269790 + 0.962919i \(0.586954\pi\)
\(8\) −4.31719e20 −0.258575
\(9\) 0 0
\(10\) −6.55837e22 −0.207394
\(11\) −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 \(0.229349\pi\)
\(14\) −6.04405e25 −0.0703589
\(15\) 0 0
\(16\) 1.88024e28 0.949280
\(17\) −8.41583e28 −1.02223 −0.511114 0.859513i \(-0.670767\pi\)
−0.511114 + 0.859513i \(0.670767\pi\)
\(18\) 0 0
\(19\) 3.16735e29 0.281830 0.140915 0.990022i \(-0.454996\pi\)
0.140915 + 0.990022i \(0.454996\pi\)
\(20\) 5.86528e30 1.56345
\(21\) 0 0
\(22\) −4.12345e30 −0.117038
\(23\) −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
\(26\) 3.50006e32 0.195975
\(27\) 0 0
\(28\) 5.40532e33 0.530405
\(29\) 3.39685e34 1.46125 0.730624 0.682780i \(-0.239229\pi\)
0.730624 + 0.682780i \(0.239229\pi\)
\(30\) 0 0
\(31\) 1.81646e35 1.63015 0.815074 0.579357i \(-0.196696\pi\)
0.815074 + 0.579357i \(0.196696\pi\)
\(32\) 8.98449e34 0.382357
\(33\) 0 0
\(34\) −1.30186e35 −0.133294
\(35\) 1.65648e36 0.858197
\(36\) 0 0
\(37\) −7.13789e36 −1.00191 −0.500956 0.865473i \(-0.667018\pi\)
−0.500956 + 0.865473i \(0.667018\pi\)
\(38\) 4.89965e35 0.0367494
\(39\) 0 0
\(40\) 1.83032e37 0.411261
\(41\) 2.78383e37 0.350124 0.175062 0.984557i \(-0.443987\pi\)
0.175062 + 0.984557i \(0.443987\pi\)
\(42\) 0 0
\(43\) −4.92777e37 −0.202368 −0.101184 0.994868i \(-0.532263\pi\)
−0.101184 + 0.994868i \(0.532263\pi\)
\(44\) 3.68769e38 0.882300
\(45\) 0 0
\(46\) −3.51128e37 −0.0295567
\(47\) −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
\(50\) 1.68134e39 0.199462
\(51\) 0 0
\(52\) −3.13017e40 −1.47737
\(53\) 2.48904e40 0.750843 0.375421 0.926854i \(-0.377498\pi\)
0.375421 + 0.926854i \(0.377498\pi\)
\(54\) 0 0
\(55\) 1.13011e41 1.42756
\(56\) 1.68679e40 0.139522
\(57\) 0 0
\(58\) 5.25467e40 0.190541
\(59\) −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 \(0.437390\pi\)
\(62\) 2.80993e41 0.212565
\(63\) 0 0
\(64\) −2.50722e42 −0.899422
\(65\) −9.59253e42 −2.39039
\(66\) 0 0
\(67\) −1.26183e42 −0.154257 −0.0771283 0.997021i \(-0.524575\pi\)
−0.0771283 + 0.997021i \(0.524575\pi\)
\(68\) 1.16428e43 1.00485
\(69\) 0 0
\(70\) 2.56245e42 0.111905
\(71\) 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 \(0.405498\pi\)
\(74\) −1.10418e43 −0.130645
\(75\) 0 0
\(76\) −4.38186e43 −0.277038
\(77\) 1.04148e44 0.484305
\(78\) 0 0
\(79\) −5.82596e44 −1.48297 −0.741483 0.670972i \(-0.765877\pi\)
−0.741483 + 0.670972i \(0.765877\pi\)
\(80\) −7.97151e44 −1.50982
\(81\) 0 0
\(82\) 4.30637e43 0.0456547
\(83\) 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
\(86\) −7.62288e43 −0.0263879
\(87\) 0 0
\(88\) 1.15078e45 0.232087
\(89\) 2.92401e45 0.452182 0.226091 0.974106i \(-0.427405\pi\)
0.226091 + 0.974106i \(0.427405\pi\)
\(90\) 0 0
\(91\) −8.84027e45 −0.810947
\(92\) 3.14021e45 0.222815
\(93\) 0 0
\(94\) −2.89005e45 −0.123709
\(95\) −1.34284e46 −0.448248
\(96\) 0 0
\(97\) 2.17182e46 0.444311 0.222156 0.975011i \(-0.428691\pi\)
0.222156 + 0.975011i \(0.428691\pi\)
\(98\) −5.74955e45 −0.0924317
\(99\) 0 0
\(100\) −1.50366e47 −1.50366
\(101\) 5.11557e46 0.404894 0.202447 0.979293i \(-0.435111\pi\)
0.202447 + 0.979293i \(0.435111\pi\)
\(102\) 0 0
\(103\) 3.19569e47 1.59547 0.797736 0.603006i \(-0.206031\pi\)
0.797736 + 0.603006i \(0.206031\pi\)
\(104\) −9.76803e46 −0.388618
\(105\) 0 0
\(106\) 3.85036e46 0.0979068
\(107\) −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 \(0.626469\pi\)
\(110\) 1.74819e47 0.186149
\(111\) 0 0
\(112\) −7.34637e47 −0.512212
\(113\) −3.11593e47 −0.176297 −0.0881484 0.996107i \(-0.528095\pi\)
−0.0881484 + 0.996107i \(0.528095\pi\)
\(114\) 0 0
\(115\) 9.62329e47 0.360516
\(116\) −4.69936e48 −1.43640
\(117\) 0 0
\(118\) −5.71184e46 −0.0116829
\(119\) 3.28818e48 0.551572
\(120\) 0 0
\(121\) −1.71442e48 −0.194384
\(122\) 5.45429e47 0.0509662
\(123\) 0 0
\(124\) −2.51298e49 −1.60243
\(125\) −1.59559e49 −0.842432
\(126\) 0 0
\(127\) 2.66165e49 0.967749 0.483875 0.875137i \(-0.339229\pi\)
0.483875 + 0.875137i \(0.339229\pi\)
\(128\) −1.65230e49 −0.499638
\(129\) 0 0
\(130\) −1.48389e49 −0.311697
\(131\) −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
\(134\) −1.95196e48 −0.0201144
\(135\) 0 0
\(136\) 3.63327e49 0.264322
\(137\) −1.99734e50 −1.22326 −0.611631 0.791143i \(-0.709486\pi\)
−0.611631 + 0.791143i \(0.709486\pi\)
\(138\) 0 0
\(139\) −3.16608e50 −1.37935 −0.689674 0.724120i \(-0.742246\pi\)
−0.689674 + 0.724120i \(0.742246\pi\)
\(140\) −2.29165e50 −0.843605
\(141\) 0 0
\(142\) 5.04332e49 0.133027
\(143\) −6.03113e50 −1.34897
\(144\) 0 0
\(145\) −1.44014e51 −2.32410
\(146\) 5.55623e49 0.0762936
\(147\) 0 0
\(148\) 9.87488e50 0.984876
\(149\) −1.00267e50 −0.0853651 −0.0426826 0.999089i \(-0.513590\pi\)
−0.0426826 + 0.999089i \(0.513590\pi\)
\(150\) 0 0
\(151\) −2.40479e49 −0.0149664 −0.00748321 0.999972i \(-0.502382\pi\)
−0.00748321 + 0.999972i \(0.502382\pi\)
\(152\) −1.36740e50 −0.0728740
\(153\) 0 0
\(154\) 1.61109e50 0.0631514
\(155\) −7.70111e51 −2.59274
\(156\) 0 0
\(157\) −1.42274e51 −0.354391 −0.177196 0.984176i \(-0.556702\pi\)
−0.177196 + 0.984176i \(0.556702\pi\)
\(158\) −9.01232e50 −0.193373
\(159\) 0 0
\(160\) −3.80908e51 −0.608136
\(161\) 8.86862e50 0.122306
\(162\) 0 0
\(163\) −4.79256e51 −0.494492 −0.247246 0.968953i \(-0.579525\pi\)
−0.247246 + 0.968953i \(0.579525\pi\)
\(164\) −3.85127e51 −0.344170
\(165\) 0 0
\(166\) 5.43927e50 0.0365596
\(167\) 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
\(170\) 5.51941e51 0.212004
\(171\) 0 0
\(172\) 6.81729e51 0.198927
\(173\) 7.43991e52 1.89446 0.947228 0.320562i \(-0.103872\pi\)
0.947228 + 0.320562i \(0.103872\pi\)
\(174\) 0 0
\(175\) −4.24665e52 −0.825377
\(176\) −5.01194e52 −0.852037
\(177\) 0 0
\(178\) 4.52321e51 0.0589627
\(179\) −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 \(0.412100\pi\)
\(182\) −1.36752e52 −0.105744
\(183\) 0 0
\(184\) 9.79935e51 0.0586109
\(185\) 3.02619e53 1.59353
\(186\) 0 0
\(187\) 2.24331e53 0.917511
\(188\) 2.58463e53 0.932585
\(189\) 0 0
\(190\) −2.07727e52 −0.0584497
\(191\) 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 \(0.346804\pi\)
\(194\) 3.35963e52 0.0579364
\(195\) 0 0
\(196\) 5.14194e53 0.696802
\(197\) 7.61644e53 0.915790 0.457895 0.889006i \(-0.348604\pi\)
0.457895 + 0.889006i \(0.348604\pi\)
\(198\) 0 0
\(199\) −1.90631e54 −1.80778 −0.903890 0.427765i \(-0.859301\pi\)
−0.903890 + 0.427765i \(0.859301\pi\)
\(200\) −4.69232e53 −0.395533
\(201\) 0 0
\(202\) 7.91339e52 0.0527965
\(203\) −1.32720e54 −0.788459
\(204\) 0 0
\(205\) −1.18024e54 −0.556869
\(206\) 4.94348e53 0.208043
\(207\) 0 0
\(208\) 4.25422e54 1.42670
\(209\) −8.44284e53 −0.252959
\(210\) 0 0
\(211\) 1.25520e54 0.300660 0.150330 0.988636i \(-0.451966\pi\)
0.150330 + 0.988636i \(0.451966\pi\)
\(212\) −3.44345e54 −0.738076
\(213\) 0 0
\(214\) −3.10288e53 −0.0533386
\(215\) 2.08918e54 0.321864
\(216\) 0 0
\(217\) −7.09718e54 −0.879594
\(218\) −9.07131e53 −0.100912
\(219\) 0 0
\(220\) −1.56344e55 −1.40329
\(221\) −1.90416e55 −1.53633
\(222\) 0 0
\(223\) −1.43141e54 −0.0934545 −0.0467272 0.998908i \(-0.514879\pi\)
−0.0467272 + 0.998908i \(0.514879\pi\)
\(224\) −3.51037e54 −0.206312
\(225\) 0 0
\(226\) −4.82011e53 −0.0229884
\(227\) −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 \(0.525931\pi\)
\(230\) 1.48865e54 0.0470098
\(231\) 0 0
\(232\) −1.46649e55 −0.377842
\(233\) 6.87236e55 1.60045 0.800225 0.599700i \(-0.204713\pi\)
0.800225 + 0.599700i \(0.204713\pi\)
\(234\) 0 0
\(235\) 7.92068e55 1.50893
\(236\) 5.10822e54 0.0880719
\(237\) 0 0
\(238\) 5.08657e54 0.0719228
\(239\) −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 \(0.581812\pi\)
\(242\) −2.65207e54 −0.0253468
\(243\) 0 0
\(244\) −4.87788e55 −0.384211
\(245\) 1.57577e56 1.12743
\(246\) 0 0
\(247\) 7.16642e55 0.423569
\(248\) −7.84201e55 −0.421515
\(249\) 0 0
\(250\) −2.46825e55 −0.109850
\(251\) 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
\(254\) 4.11736e55 0.126191
\(255\) 0 0
\(256\) 3.27300e56 0.834271
\(257\) −1.93698e56 −0.450501 −0.225251 0.974301i \(-0.572320\pi\)
−0.225251 + 0.974301i \(0.572320\pi\)
\(258\) 0 0
\(259\) 2.78887e56 0.540611
\(260\) 1.32707e57 2.34975
\(261\) 0 0
\(262\) −9.82178e55 −0.145248
\(263\) −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
\(266\) −1.91436e55 −0.0198292
\(267\) 0 0
\(268\) 1.74568e56 0.151634
\(269\) 1.88490e57 1.50006 0.750032 0.661402i \(-0.230038\pi\)
0.750032 + 0.661402i \(0.230038\pi\)
\(270\) 0 0
\(271\) 2.56612e57 1.71593 0.857963 0.513712i \(-0.171730\pi\)
0.857963 + 0.513712i \(0.171730\pi\)
\(272\) −1.58238e57 −0.970380
\(273\) 0 0
\(274\) −3.08974e56 −0.159508
\(275\) −2.89721e57 −1.37297
\(276\) 0 0
\(277\) −8.36679e56 −0.334414 −0.167207 0.985922i \(-0.553475\pi\)
−0.167207 + 0.985922i \(0.553475\pi\)
\(278\) −4.89768e56 −0.179861
\(279\) 0 0
\(280\) −7.15133e56 −0.221908
\(281\) 3.96974e57 1.13283 0.566414 0.824121i \(-0.308330\pi\)
0.566414 + 0.824121i \(0.308330\pi\)
\(282\) 0 0
\(283\) −4.45361e57 −1.07580 −0.537900 0.843009i \(-0.680782\pi\)
−0.537900 + 0.843009i \(0.680782\pi\)
\(284\) −4.51034e57 −1.00283
\(285\) 0 0
\(286\) −9.32969e56 −0.175900
\(287\) −1.08768e57 −0.188919
\(288\) 0 0
\(289\) 3.04657e56 0.0449481
\(290\) −2.22778e57 −0.303054
\(291\) 0 0
\(292\) −4.96905e57 −0.575144
\(293\) −1.05167e58 −1.12328 −0.561641 0.827381i \(-0.689830\pi\)
−0.561641 + 0.827381i \(0.689830\pi\)
\(294\) 0 0
\(295\) 1.56543e57 0.142501
\(296\) 3.08156e57 0.259069
\(297\) 0 0
\(298\) −1.55105e56 −0.0111313
\(299\) −5.13574e57 −0.340667
\(300\) 0 0
\(301\) 1.92535e57 0.109193
\(302\) −3.72002e55 −0.00195156
\(303\) 0 0
\(304\) 5.95539e57 0.267535
\(305\) −1.49484e58 −0.621656
\(306\) 0 0
\(307\) 5.43218e57 0.193741 0.0968707 0.995297i \(-0.469117\pi\)
0.0968707 + 0.995297i \(0.469117\pi\)
\(308\) −1.44083e58 −0.476071
\(309\) 0 0
\(310\) −1.19130e58 −0.338082
\(311\) 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 \(0.591414\pi\)
\(314\) −2.20087e57 −0.0462111
\(315\) 0 0
\(316\) 8.05990e58 1.45775
\(317\) 6.65301e58 1.11719 0.558593 0.829442i \(-0.311341\pi\)
0.558593 + 0.829442i \(0.311341\pi\)
\(318\) 0 0
\(319\) −9.05460e58 −1.31156
\(320\) 1.06297e59 1.43052
\(321\) 0 0
\(322\) 1.37191e57 0.0159482
\(323\) −2.66559e58 −0.288094
\(324\) 0 0
\(325\) 2.45920e59 2.29898
\(326\) −7.41372e57 −0.0644797
\(327\) 0 0
\(328\) −1.20183e58 −0.0905331
\(329\) 7.29953e58 0.511907
\(330\) 0 0
\(331\) −1.52984e58 −0.0930438 −0.0465219 0.998917i \(-0.514814\pi\)
−0.0465219 + 0.998917i \(0.514814\pi\)
\(332\) −4.86445e58 −0.275606
\(333\) 0 0
\(334\) 1.53105e58 0.0753262
\(335\) 5.34969e58 0.245344
\(336\) 0 0
\(337\) −1.19696e59 −0.477284 −0.238642 0.971108i \(-0.576702\pi\)
−0.238642 + 0.971108i \(0.576702\pi\)
\(338\) 4.41324e58 0.164140
\(339\) 0 0
\(340\) −4.93612e59 −1.59820
\(341\) −4.84194e59 −1.46316
\(342\) 0 0
\(343\) 3.50084e59 0.922062
\(344\) 2.12741e58 0.0523272
\(345\) 0 0
\(346\) 1.15090e59 0.247029
\(347\) 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 \(0.284937\pi\)
\(350\) −6.56924e58 −0.107626
\(351\) 0 0
\(352\) −2.39489e59 −0.343189
\(353\) −4.98386e58 −0.0668129 −0.0334064 0.999442i \(-0.510636\pi\)
−0.0334064 + 0.999442i \(0.510636\pi\)
\(354\) 0 0
\(355\) −1.38221e60 −1.62259
\(356\) −4.04520e59 −0.444493
\(357\) 0 0
\(358\) −5.57870e58 −0.0537381
\(359\) −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
\(362\) 9.58408e58 0.0711048
\(363\) 0 0
\(364\) 1.22300e60 0.797159
\(365\) −1.52278e60 −0.930585
\(366\) 0 0
\(367\) −1.22059e59 −0.0656021 −0.0328011 0.999462i \(-0.510443\pi\)
−0.0328011 + 0.999462i \(0.510443\pi\)
\(368\) −4.26786e59 −0.215173
\(369\) 0 0
\(370\) 4.68129e59 0.207790
\(371\) −9.72503e59 −0.405139
\(372\) 0 0
\(373\) −2.41910e60 −0.888172 −0.444086 0.895984i \(-0.646472\pi\)
−0.444086 + 0.895984i \(0.646472\pi\)
\(374\) 3.47023e59 0.119640
\(375\) 0 0
\(376\) 8.06560e59 0.245314
\(377\) 7.68569e60 2.19615
\(378\) 0 0
\(379\) 6.29795e60 1.58920 0.794598 0.607135i \(-0.207681\pi\)
0.794598 + 0.607135i \(0.207681\pi\)
\(380\) 1.85774e60 0.440626
\(381\) 0 0
\(382\) 4.79333e59 0.100496
\(383\) −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
\(386\) 7.35523e59 0.120724
\(387\) 0 0
\(388\) −3.00459e60 −0.436757
\(389\) −5.15774e60 −0.705739 −0.352869 0.935673i \(-0.614794\pi\)
−0.352869 + 0.935673i \(0.614794\pi\)
\(390\) 0 0
\(391\) 1.91027e60 0.231707
\(392\) 1.60460e60 0.183292
\(393\) 0 0
\(394\) 1.17821e60 0.119415
\(395\) 2.46999e61 2.35865
\(396\) 0 0
\(397\) −1.92619e61 −1.63351 −0.816754 0.576986i \(-0.804229\pi\)
−0.816754 + 0.576986i \(0.804229\pi\)
\(398\) −2.94891e60 −0.235727
\(399\) 0 0
\(400\) 2.04362e61 1.45208
\(401\) −5.06490e58 −0.00339374 −0.00169687 0.999999i \(-0.500540\pi\)
−0.00169687 + 0.999999i \(0.500540\pi\)
\(402\) 0 0
\(403\) 4.10992e61 2.44999
\(404\) −7.07710e60 −0.398010
\(405\) 0 0
\(406\) −2.05308e60 −0.102812
\(407\) 1.90266e61 0.899277
\(408\) 0 0
\(409\) 1.50217e61 0.632733 0.316367 0.948637i \(-0.397537\pi\)
0.316367 + 0.948637i \(0.397537\pi\)
\(410\) −1.82574e60 −0.0726135
\(411\) 0 0
\(412\) −4.42106e61 −1.56834
\(413\) 1.44267e60 0.0483437
\(414\) 0 0
\(415\) −1.49073e61 −0.445932
\(416\) 2.03282e61 0.574654
\(417\) 0 0
\(418\) −1.30604e60 −0.0329849
\(419\) −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 \(0.632562\pi\)
\(422\) 1.94170e60 0.0392049
\(423\) 0 0
\(424\) −1.07457e61 −0.194149
\(425\) −9.14712e61 −1.56367
\(426\) 0 0
\(427\) −1.37762e61 −0.210898
\(428\) 2.77497e61 0.402096
\(429\) 0 0
\(430\) 3.23181e60 0.0419698
\(431\) −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 \(0.545641\pi\)
\(434\) −1.09788e61 −0.114695
\(435\) 0 0
\(436\) 8.11265e61 0.760730
\(437\) −7.18941e60 −0.0638821
\(438\) 0 0
\(439\) 3.51058e61 0.280196 0.140098 0.990138i \(-0.455258\pi\)
0.140098 + 0.990138i \(0.455258\pi\)
\(440\) −4.87888e61 −0.369132
\(441\) 0 0
\(442\) −2.94559e61 −0.200331
\(443\) 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
\(446\) −2.21428e60 −0.0121861
\(447\) 0 0
\(448\) 9.79607e61 0.485309
\(449\) −3.43198e62 −1.61345 −0.806725 0.590927i \(-0.798762\pi\)
−0.806725 + 0.590927i \(0.798762\pi\)
\(450\) 0 0
\(451\) −7.42052e61 −0.314257
\(452\) 4.31072e61 0.173299
\(453\) 0 0
\(454\) −5.23613e61 −0.189757
\(455\) 3.74794e62 1.28981
\(456\) 0 0
\(457\) −9.64466e60 −0.0299403 −0.0149701 0.999888i \(-0.504765\pi\)
−0.0149701 + 0.999888i \(0.504765\pi\)
\(458\) −7.19656e60 −0.0212220
\(459\) 0 0
\(460\) −1.33133e62 −0.354386
\(461\) 6.32143e61 0.159898 0.0799491 0.996799i \(-0.474524\pi\)
0.0799491 + 0.996799i \(0.474524\pi\)
\(462\) 0 0
\(463\) 6.86213e62 1.56785 0.783927 0.620853i \(-0.213214\pi\)
0.783927 + 0.620853i \(0.213214\pi\)
\(464\) 6.38691e62 1.38713
\(465\) 0 0
\(466\) 1.06310e62 0.208692
\(467\) 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
\(470\) 1.22527e62 0.196758
\(471\) 0 0
\(472\) 1.59407e61 0.0231671
\(473\) 1.31354e62 0.181637
\(474\) 0 0
\(475\) 3.44258e62 0.431106
\(476\) −4.54902e62 −0.542194
\(477\) 0 0
\(478\) −1.68954e62 −0.182479
\(479\) 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
\(482\) −7.46579e61 −0.0662934
\(483\) 0 0
\(484\) 2.37180e62 0.191079
\(485\) −9.20767e62 −0.706674
\(486\) 0 0
\(487\) 2.39350e63 1.66764 0.833820 0.552037i \(-0.186149\pi\)
0.833820 + 0.552037i \(0.186149\pi\)
\(488\) −1.52219e62 −0.101066
\(489\) 0 0
\(490\) 2.43759e62 0.147012
\(491\) −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
\(494\) 1.10859e62 0.0552317
\(495\) 0 0
\(496\) 3.41539e63 1.54747
\(497\) −1.27382e63 −0.550468
\(498\) 0 0
\(499\) −2.94657e63 −1.15865 −0.579327 0.815095i \(-0.696684\pi\)
−0.579327 + 0.815095i \(0.696684\pi\)
\(500\) 2.20741e63 0.828108
\(501\) 0 0
\(502\) 6.28387e62 0.214630
\(503\) 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
\(506\) 9.35961e61 0.0265290
\(507\) 0 0
\(508\) −3.68224e63 −0.951294
\(509\) −7.25177e63 −1.78886 −0.894429 0.447209i \(-0.852418\pi\)
−0.894429 + 0.447209i \(0.852418\pi\)
\(510\) 0 0
\(511\) −1.40336e63 −0.315704
\(512\) 2.83172e63 0.608423
\(513\) 0 0
\(514\) −2.99635e62 −0.0587435
\(515\) −1.35485e64 −2.53759
\(516\) 0 0
\(517\) 4.97998e63 0.851531
\(518\) 4.31418e62 0.0704934
\(519\) 0 0
\(520\) 4.14127e63 0.618095
\(521\) 9.84138e63 1.40401 0.702004 0.712173i \(-0.252289\pi\)
0.702004 + 0.712173i \(0.252289\pi\)
\(522\) 0 0
\(523\) 4.91409e62 0.0640698 0.0320349 0.999487i \(-0.489801\pi\)
0.0320349 + 0.999487i \(0.489801\pi\)
\(524\) 8.78382e63 1.09496
\(525\) 0 0
\(526\) −1.28961e63 −0.146992
\(527\) −1.52871e64 −1.66638
\(528\) 0 0
\(529\) −9.51264e63 −0.948621
\(530\) −1.63240e63 −0.155720
\(531\) 0 0
\(532\) 1.71205e63 0.149484
\(533\) 6.29866e63 0.526210
\(534\) 0 0
\(535\) 8.50400e63 0.650593
\(536\) 5.44756e62 0.0398869
\(537\) 0 0
\(538\) 2.91580e63 0.195602
\(539\) 9.90734e63 0.636240
\(540\) 0 0
\(541\) 6.71742e61 0.00395428 0.00197714 0.999998i \(-0.499371\pi\)
0.00197714 + 0.999998i \(0.499371\pi\)
\(542\) 3.96959e63 0.223750
\(543\) 0 0
\(544\) −7.56119e63 −0.390856
\(545\) 2.48615e64 1.23086
\(546\) 0 0
\(547\) 4.01645e62 0.0182449 0.00912243 0.999958i \(-0.497096\pi\)
0.00912243 + 0.999958i \(0.497096\pi\)
\(548\) 2.76322e64 1.20246
\(549\) 0 0
\(550\) −4.48176e63 −0.179030
\(551\) 1.07590e64 0.411823
\(552\) 0 0
\(553\) 2.27629e64 0.800177
\(554\) −1.29428e63 −0.0436062
\(555\) 0 0
\(556\) 4.38010e64 1.35589
\(557\) −6.03614e64 −1.79127 −0.895637 0.444785i \(-0.853280\pi\)
−0.895637 + 0.444785i \(0.853280\pi\)
\(558\) 0 0
\(559\) −1.11495e64 −0.304144
\(560\) 3.11458e64 0.814669
\(561\) 0 0
\(562\) 6.14089e63 0.147716
\(563\) 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
\(566\) −6.88940e63 −0.140280
\(567\) 0 0
\(568\) −1.40750e64 −0.263793
\(569\) −9.25178e64 −1.66375 −0.831873 0.554966i \(-0.812731\pi\)
−0.831873 + 0.554966i \(0.812731\pi\)
\(570\) 0 0
\(571\) −1.04188e65 −1.72532 −0.862658 0.505787i \(-0.831202\pi\)
−0.862658 + 0.505787i \(0.831202\pi\)
\(572\) 8.34373e64 1.32603
\(573\) 0 0
\(574\) −1.68256e63 −0.0246343
\(575\) −2.46708e64 −0.346728
\(576\) 0 0
\(577\) −8.68105e64 −1.12445 −0.562225 0.826984i \(-0.690055\pi\)
−0.562225 + 0.826984i \(0.690055\pi\)
\(578\) 4.71281e62 0.00586105
\(579\) 0 0
\(580\) 1.99235e65 2.28459
\(581\) −1.37382e64 −0.151284
\(582\) 0 0
\(583\) −6.63474e64 −0.673928
\(584\) −1.55064e64 −0.151290
\(585\) 0 0
\(586\) −1.62685e64 −0.146471
\(587\) −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
\(590\) 2.42160e63 0.0185815
\(591\) 0 0
\(592\) −1.34210e65 −0.951094
\(593\) −1.82757e65 −1.24477 −0.622385 0.782712i \(-0.713836\pi\)
−0.622385 + 0.782712i \(0.713836\pi\)
\(594\) 0 0
\(595\) −1.39406e65 −0.877272
\(596\) 1.38714e64 0.0839137
\(597\) 0 0
\(598\) −7.94459e63 −0.0444216
\(599\) −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 \(0.406396\pi\)
\(602\) 2.97837e63 0.0142384
\(603\) 0 0
\(604\) 3.32689e63 0.0147119
\(605\) 7.26847e64 0.309166
\(606\) 0 0
\(607\) −8.69736e64 −0.342337 −0.171168 0.985242i \(-0.554754\pi\)
−0.171168 + 0.985242i \(0.554754\pi\)
\(608\) 2.84570e64 0.107760
\(609\) 0 0
\(610\) −2.31241e64 −0.0810614
\(611\) −4.22709e65 −1.42585
\(612\) 0 0
\(613\) 3.95467e65 1.23535 0.617674 0.786435i \(-0.288075\pi\)
0.617674 + 0.786435i \(0.288075\pi\)
\(614\) 8.40317e63 0.0252631
\(615\) 0 0
\(616\) −4.49627e64 −0.125229
\(617\) 4.10300e65 1.10002 0.550009 0.835159i \(-0.314624\pi\)
0.550009 + 0.835159i \(0.314624\pi\)
\(618\) 0 0
\(619\) 1.39273e65 0.346047 0.173024 0.984918i \(-0.444646\pi\)
0.173024 + 0.984918i \(0.444646\pi\)
\(620\) 1.06541e66 2.54865
\(621\) 0 0
\(622\) 7.42569e64 0.164688
\(623\) −1.14245e65 −0.243988
\(624\) 0 0
\(625\) −9.58171e64 −0.189785
\(626\) −3.87227e64 −0.0738702
\(627\) 0 0
\(628\) 1.96828e65 0.348365
\(629\) 6.00713e65 1.02418
\(630\) 0 0
\(631\) 1.54026e65 0.243728 0.121864 0.992547i \(-0.461113\pi\)
0.121864 + 0.992547i \(0.461113\pi\)
\(632\) 2.51518e65 0.383457
\(633\) 0 0
\(634\) 1.02917e65 0.145677
\(635\) −1.12844e66 −1.53920
\(636\) 0 0
\(637\) −8.40952e65 −1.06536
\(638\) −1.40068e65 −0.171022
\(639\) 0 0
\(640\) 7.00513e65 0.794670
\(641\) 1.65837e65 0.181350 0.0906752 0.995881i \(-0.471097\pi\)
0.0906752 + 0.995881i \(0.471097\pi\)
\(642\) 0 0
\(643\) −1.22213e66 −1.24211 −0.621053 0.783768i \(-0.713295\pi\)
−0.621053 + 0.783768i \(0.713295\pi\)
\(644\) −1.22692e65 −0.120226
\(645\) 0 0
\(646\) −4.12346e64 −0.0375663
\(647\) 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
\(650\) 3.80419e65 0.299777
\(651\) 0 0
\(652\) 6.63024e65 0.486084
\(653\) −3.91841e65 −0.277109 −0.138554 0.990355i \(-0.544246\pi\)
−0.138554 + 0.990355i \(0.544246\pi\)
\(654\) 0 0
\(655\) 2.69183e66 1.77165
\(656\) 5.23427e65 0.332365
\(657\) 0 0
\(658\) 1.12918e65 0.0667506
\(659\) −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 \(0.584472\pi\)
\(662\) −2.36654e64 −0.0121325
\(663\) 0 0
\(664\) −1.51800e65 −0.0724975
\(665\) 5.24665e65 0.241865
\(666\) 0 0
\(667\) −7.71034e65 −0.331220
\(668\) −1.36925e66 −0.567851
\(669\) 0 0
\(670\) 8.27556e64 0.0319919
\(671\) −9.39857e65 −0.350818
\(672\) 0 0
\(673\) −5.18176e66 −1.80353 −0.901765 0.432227i \(-0.857728\pi\)
−0.901765 + 0.432227i \(0.857728\pi\)
\(674\) −1.85161e65 −0.0622359
\(675\) 0 0
\(676\) −3.94685e66 −1.23738
\(677\) −5.62905e66 −1.70452 −0.852259 0.523120i \(-0.824768\pi\)
−0.852259 + 0.523120i \(0.824768\pi\)
\(678\) 0 0
\(679\) −8.48559e65 −0.239741
\(680\) −1.54037e66 −0.420402
\(681\) 0 0
\(682\) −7.49010e65 −0.190790
\(683\) 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
\(686\) 5.41553e65 0.120233
\(687\) 0 0
\(688\) −9.26539e65 −0.192104
\(689\) 5.63168e66 1.12846
\(690\) 0 0
\(691\) 3.80507e66 0.712244 0.356122 0.934439i \(-0.384099\pi\)
0.356122 + 0.934439i \(0.384099\pi\)
\(692\) −1.02927e67 −1.86224
\(693\) 0 0
\(694\) 1.02216e66 0.172811
\(695\) 1.34230e67 2.19384
\(696\) 0 0
\(697\) −2.34282e66 −0.357906
\(698\) 1.10421e66 0.163098
\(699\) 0 0
\(700\) 5.87500e66 0.811343
\(701\) −4.48575e66 −0.599048 −0.299524 0.954089i \(-0.596828\pi\)
−0.299524 + 0.954089i \(0.596828\pi\)
\(702\) 0 0
\(703\) −2.26082e66 −0.282368
\(704\) 6.68321e66 0.807286
\(705\) 0 0
\(706\) −7.70965e64 −0.00871212
\(707\) −1.99872e66 −0.218472
\(708\) 0 0
\(709\) −1.23589e67 −1.26414 −0.632068 0.774913i \(-0.717794\pi\)
−0.632068 + 0.774913i \(0.717794\pi\)
\(710\) −2.13818e66 −0.211579
\(711\) 0 0
\(712\) −1.26235e66 −0.116923
\(713\) −4.12310e66 −0.369504
\(714\) 0 0
\(715\) 2.55697e67 2.14552
\(716\) 4.98915e66 0.405108
\(717\) 0 0
\(718\) −3.23290e66 −0.245848
\(719\) 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
\(722\) −1.79865e66 −0.120039
\(723\) 0 0
\(724\) −8.57124e66 −0.536028
\(725\) 3.69202e67 2.23522
\(726\) 0 0
\(727\) 2.10795e67 1.19619 0.598097 0.801424i \(-0.295924\pi\)
0.598097 + 0.801424i \(0.295924\pi\)
\(728\) 3.81651e66 0.209690
\(729\) 0 0
\(730\) −2.35563e66 −0.121344
\(731\) 4.14712e66 0.206866
\(732\) 0 0
\(733\) 2.34941e67 1.09905 0.549523 0.835478i \(-0.314809\pi\)
0.549523 + 0.835478i \(0.314809\pi\)
\(734\) −1.88816e65 −0.00855425
\(735\) 0 0
\(736\) −2.03934e66 −0.0866685
\(737\) 3.36352e66 0.138455
\(738\) 0 0
\(739\) −8.69448e66 −0.335814 −0.167907 0.985803i \(-0.553701\pi\)
−0.167907 + 0.985803i \(0.553701\pi\)
\(740\) −4.18657e67 −1.56644
\(741\) 0 0
\(742\) −1.50439e66 −0.0528285
\(743\) −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
\(746\) −3.74217e66 −0.115814
\(747\) 0 0
\(748\) −3.10350e67 −0.901911
\(749\) 7.83710e66 0.220715
\(750\) 0 0
\(751\) −2.92872e67 −0.774709 −0.387355 0.921931i \(-0.626611\pi\)
−0.387355 + 0.921931i \(0.626611\pi\)
\(752\) −3.51277e67 −0.900597
\(753\) 0 0
\(754\) 1.18892e67 0.286369
\(755\) 1.01954e66 0.0238040
\(756\) 0 0
\(757\) −3.60671e67 −0.791330 −0.395665 0.918395i \(-0.629486\pi\)
−0.395665 + 0.918395i \(0.629486\pi\)
\(758\) 9.74245e66 0.207225
\(759\) 0 0
\(760\) 5.79727e66 0.115906
\(761\) −6.36047e67 −1.23296 −0.616481 0.787370i \(-0.711442\pi\)
−0.616481 + 0.787370i \(0.711442\pi\)
\(762\) 0 0
\(763\) 2.29118e67 0.417574
\(764\) −4.28678e67 −0.757595
\(765\) 0 0
\(766\) −6.01697e66 −0.100000
\(767\) −8.35437e66 −0.134655
\(768\) 0 0
\(769\) −1.58030e67 −0.239591 −0.119795 0.992799i \(-0.538224\pi\)
−0.119795 + 0.992799i \(0.538224\pi\)
\(770\) −6.83041e66 −0.100442
\(771\) 0 0
\(772\) −6.57793e67 −0.910087
\(773\) 1.20430e68 1.61628 0.808142 0.588988i \(-0.200474\pi\)
0.808142 + 0.588988i \(0.200474\pi\)
\(774\) 0 0
\(775\) 1.97430e68 2.49358
\(776\) −9.37613e66 −0.114888
\(777\) 0 0
\(778\) −7.97864e66 −0.0920255
\(779\) 8.81736e66 0.0986752
\(780\) 0 0
\(781\) −8.69040e67 −0.915674
\(782\) 2.95503e66 0.0302137
\(783\) 0 0
\(784\) −6.98842e67 −0.672901
\(785\) 6.03186e67 0.563656
\(786\) 0 0
\(787\) −1.72836e68 −1.52134 −0.760671 0.649137i \(-0.775130\pi\)
−0.760671 + 0.649137i \(0.775130\pi\)
\(788\) −1.05369e68 −0.900219
\(789\) 0 0
\(790\) 3.82088e67 0.307558
\(791\) 1.21744e67 0.0951261
\(792\) 0 0
\(793\) 7.97766e67 0.587429
\(794\) −2.97966e67 −0.213003
\(795\) 0 0
\(796\) 2.63727e68 1.77704
\(797\) 5.17743e67 0.338722 0.169361 0.985554i \(-0.445830\pi\)
0.169361 + 0.985554i \(0.445830\pi\)
\(798\) 0 0
\(799\) 1.57229e68 0.969803
\(800\) 9.76519e67 0.584879
\(801\) 0 0
\(802\) −7.83502e64 −0.000442530 0
\(803\) −9.57423e67 −0.525156
\(804\) 0 0
\(805\) −3.75995e67 −0.194527
\(806\) 6.35773e67 0.319469
\(807\) 0 0
\(808\) −2.20848e67 −0.104695
\(809\) 1.34819e68 0.620812 0.310406 0.950604i \(-0.399535\pi\)
0.310406 + 0.950604i \(0.399535\pi\)
\(810\) 0 0
\(811\) −1.29768e68 −0.563869 −0.281934 0.959434i \(-0.590976\pi\)
−0.281934 + 0.959434i \(0.590976\pi\)
\(812\) 1.83611e68 0.775053
\(813\) 0 0
\(814\) 2.94328e67 0.117262
\(815\) 2.03186e68 0.786485
\(816\) 0 0
\(817\) −1.56080e67 −0.0570332
\(818\) 2.32374e67 0.0825059
\(819\) 0 0
\(820\) 1.63279e68 0.547401
\(821\) 3.05518e68 0.995342 0.497671 0.867366i \(-0.334189\pi\)
0.497671 + 0.867366i \(0.334189\pi\)
\(822\) 0 0
\(823\) −3.54128e68 −1.08959 −0.544796 0.838568i \(-0.683393\pi\)
−0.544796 + 0.838568i \(0.683393\pi\)
\(824\) −1.37964e68 −0.412549
\(825\) 0 0
\(826\) 2.23170e66 0.00630383
\(827\) −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 \(0.122955\pi\)
\(830\) −2.30604e67 −0.0581477
\(831\) 0 0
\(832\) −5.67282e68 −1.35176
\(833\) 3.12797e68 0.724610
\(834\) 0 0
\(835\) −4.19610e68 −0.918785
\(836\) 1.16802e68 0.248658
\(837\) 0 0
\(838\) −1.07657e68 −0.216675
\(839\) −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
\(842\) −5.86231e67 −0.105496
\(843\) 0 0
\(844\) −1.73650e68 −0.295548
\(845\) −1.20953e69 −2.00209
\(846\) 0 0
\(847\) 6.69846e67 0.104885
\(848\) 4.68000e68 0.712760
\(849\) 0 0
\(850\) −1.41499e68 −0.203896
\(851\) 1.62019e68 0.227102
\(852\) 0 0
\(853\) 6.69695e68 0.888331 0.444166 0.895945i \(-0.353500\pi\)
0.444166 + 0.895945i \(0.353500\pi\)
\(854\) −2.13107e67 −0.0275003
\(855\) 0 0
\(856\) 8.65958e67 0.105770
\(857\) −1.01115e69 −1.20162 −0.600809 0.799393i \(-0.705155\pi\)
−0.600809 + 0.799393i \(0.705155\pi\)
\(858\) 0 0
\(859\) −4.13647e68 −0.465361 −0.232681 0.972553i \(-0.574750\pi\)
−0.232681 + 0.972553i \(0.574750\pi\)
\(860\) −2.89027e68 −0.316392
\(861\) 0 0
\(862\) −6.29207e66 −0.00652188
\(863\) −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
\(866\) −4.00847e67 −0.0372656
\(867\) 0 0
\(868\) 9.81856e68 0.864638
\(869\) 1.55296e69 1.33105
\(870\) 0 0
\(871\) −2.85501e68 −0.231836
\(872\) 2.53164e68 0.200108
\(873\) 0 0
\(874\) −1.11215e67 −0.00832997
\(875\) 6.23418e68 0.454559
\(876\) 0 0
\(877\) 6.43840e68 0.444925 0.222463 0.974941i \(-0.428590\pi\)
0.222463 + 0.974941i \(0.428590\pi\)
\(878\) 5.43059e67 0.0365364
\(879\) 0 0
\(880\) 2.12487e69 1.35516
\(881\) −8.86127e68 −0.550253 −0.275126 0.961408i \(-0.588720\pi\)
−0.275126 + 0.961408i \(0.588720\pi\)
\(882\) 0 0
\(883\) 4.16643e68 0.245294 0.122647 0.992450i \(-0.460862\pi\)
0.122647 + 0.992450i \(0.460862\pi\)
\(884\) 2.63430e69 1.51021
\(885\) 0 0
\(886\) 6.86670e67 0.0373298
\(887\) −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
\(890\) −1.91767e68 −0.0937797
\(891\) 0 0
\(892\) 1.98028e68 0.0918654
\(893\) −5.91742e68 −0.267376
\(894\) 0 0
\(895\) 1.52894e69 0.655466
\(896\) 6.45578e68 0.269594
\(897\) 0 0
\(898\) −5.30901e68 −0.210387
\(899\) 6.17026e69 2.38205
\(900\) 0 0
\(901\) −2.09473e69 −0.767532
\(902\) −1.14790e68 −0.0409779
\(903\) 0 0
\(904\) 1.34521e68 0.0455859
\(905\) −2.62669e69 −0.867295
\(906\) 0 0
\(907\) 3.36233e69 1.05407 0.527034 0.849844i \(-0.323304\pi\)
0.527034 + 0.849844i \(0.323304\pi\)
\(908\) 4.68278e69 1.43049
\(909\) 0 0
\(910\) 5.79777e68 0.168185
\(911\) −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
\(914\) −1.49195e67 −0.00390409
\(915\) 0 0
\(916\) 6.43603e68 0.159983
\(917\) 2.48074e69 0.601037
\(918\) 0 0
\(919\) 5.89124e69 1.35610 0.678050 0.735015i \(-0.262825\pi\)
0.678050 + 0.735015i \(0.262825\pi\)
\(920\) −4.15455e68 −0.0932202
\(921\) 0 0
\(922\) 9.77877e67 0.0208501
\(923\) 7.37656e69 1.53325
\(924\) 0 0
\(925\) −7.75813e69 −1.53259
\(926\) 1.06152e69 0.204442
\(927\) 0 0
\(928\) 3.05190e69 0.558718
\(929\) 3.81127e69 0.680301 0.340150 0.940371i \(-0.389522\pi\)
0.340150 + 0.940371i \(0.389522\pi\)
\(930\) 0 0
\(931\) −1.17723e69 −0.199776
\(932\) −9.50754e69 −1.57324
\(933\) 0 0
\(934\) 4.50729e68 0.0709192
\(935\) −9.51078e69 −1.45930
\(936\) 0 0
\(937\) −3.18868e69 −0.465298 −0.232649 0.972561i \(-0.574739\pi\)
−0.232649 + 0.972561i \(0.574739\pi\)
\(938\) 7.62658e67 0.0108533
\(939\) 0 0
\(940\) −1.09578e70 −1.48327
\(941\) −7.20176e69 −0.950785 −0.475393 0.879774i \(-0.657694\pi\)
−0.475393 + 0.879774i \(0.657694\pi\)
\(942\) 0 0
\(943\) −6.31886e68 −0.0793623
\(944\) −6.94259e68 −0.0850510
\(945\) 0 0
\(946\) 2.03194e68 0.0236848
\(947\) −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
\(950\) 5.32540e68 0.0562144
\(951\) 0 0
\(952\) −1.41957e69 −0.142623
\(953\) −1.03533e70 −1.01483 −0.507417 0.861700i \(-0.669400\pi\)
−0.507417 + 0.861700i \(0.669400\pi\)
\(954\) 0 0
\(955\) −1.31370e70 −1.22579
\(956\) 1.51099e70 1.37563
\(957\) 0 0
\(958\) 1.45384e69 0.126017
\(959\) 7.80391e69 0.660047
\(960\) 0 0
\(961\) 2.05789e70 1.65738
\(962\) −2.49830e69 −0.196350
\(963\) 0 0
\(964\) 6.67681e69 0.499757
\(965\) −2.01583e70 −1.47252
\(966\) 0 0
\(967\) −1.88655e70 −1.31264 −0.656321 0.754482i \(-0.727888\pi\)
−0.656321 + 0.754482i \(0.727888\pi\)
\(968\) 7.40145e68 0.0502627
\(969\) 0 0
\(970\) −1.42436e69 −0.0921474
\(971\) 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
\(974\) 3.70256e69 0.217453
\(975\) 0 0
\(976\) 6.62954e69 0.371033
\(977\) 1.28077e70 0.699760 0.349880 0.936795i \(-0.386222\pi\)
0.349880 + 0.936795i \(0.386222\pi\)
\(978\) 0 0
\(979\) −7.79418e69 −0.405861
\(980\) −2.17999e70 −1.10826
\(981\) 0 0
\(982\) −1.91177e69 −0.0926438
\(983\) −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
\(986\) −4.42225e69 −0.194776
\(987\) 0 0
\(988\) −9.91435e69 −0.416367
\(989\) 1.11853e69 0.0458705
\(990\) 0 0
\(991\) 2.78865e70 1.09059 0.545296 0.838244i \(-0.316417\pi\)
0.545296 + 0.838244i \(0.316417\pi\)
\(992\) 1.63200e70 0.623298
\(993\) 0 0
\(994\) −1.97050e69 −0.0717787
\(995\) 8.08202e70 2.87526
\(996\) 0 0
\(997\) 2.05189e70 0.696336 0.348168 0.937432i \(-0.386804\pi\)
0.348168 + 0.937432i \(0.386804\pi\)
\(998\) −4.55812e69 −0.151084
\(999\) 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 9.48.a.c.1.3 4
3.2 odd 2 1.48.a.a.1.2 4
12.11 even 2 16.48.a.d.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.48.a.a.1.2 4 3.2 odd 2
9.48.a.c.1.3 4 1.1 even 1 trivial
16.48.a.d.1.1 4 12.11 even 2