Properties

Label 9.34.a.b.1.2
Level $9$
Weight $34$
Character 9.1
Self dual yes
Analytic conductor $62.085$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.0845459929\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 589050\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-766.996\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+171359. q^{2} +2.07741e10 q^{4} +2.01492e11 q^{5} +5.50153e13 q^{7} +2.08788e15 q^{8} +O(q^{10})\) \(q+171359. q^{2} +2.07741e10 q^{4} +2.01492e11 q^{5} +5.50153e13 q^{7} +2.08788e15 q^{8} +3.45276e16 q^{10} +8.18909e16 q^{11} -1.90399e18 q^{13} +9.42739e18 q^{14} +1.79329e20 q^{16} +3.33893e20 q^{17} -1.40494e20 q^{19} +4.18583e21 q^{20} +1.40328e22 q^{22} -3.12767e22 q^{23} -7.58162e22 q^{25} -3.26267e23 q^{26} +1.14289e24 q^{28} +1.50979e24 q^{29} +5.18762e23 q^{31} +1.27950e25 q^{32} +5.72158e25 q^{34} +1.10852e25 q^{35} -3.01507e25 q^{37} -2.40749e25 q^{38} +4.20691e26 q^{40} +2.18887e26 q^{41} +1.76701e27 q^{43} +1.70121e27 q^{44} -5.35955e27 q^{46} -3.25654e27 q^{47} -4.70431e27 q^{49} -1.29918e28 q^{50} -3.95538e28 q^{52} -9.17652e27 q^{53} +1.65004e28 q^{55} +1.14865e29 q^{56} +2.58716e29 q^{58} +1.18267e29 q^{59} -9.92930e27 q^{61} +8.88948e28 q^{62} +6.52116e29 q^{64} -3.83640e29 q^{65} +1.11293e30 q^{67} +6.93634e30 q^{68} +1.89955e30 q^{70} -7.58425e29 q^{71} -6.06835e30 q^{73} -5.16661e30 q^{74} -2.91863e30 q^{76} +4.50525e30 q^{77} -5.57890e30 q^{79} +3.61334e31 q^{80} +3.75083e31 q^{82} -4.13746e31 q^{83} +6.72769e31 q^{85} +3.02793e32 q^{86} +1.70978e32 q^{88} -6.21572e31 q^{89} -1.04749e32 q^{91} -6.49745e32 q^{92} -5.58039e32 q^{94} -2.83084e31 q^{95} +4.04003e32 q^{97} -8.06129e32 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 121680q^{2} + 14652233984q^{4} + 181061536500q^{5} - 67153080066800q^{7} + 2818750585098240q^{8} + O(q^{10}) \) \( 2q + 121680q^{2} + 14652233984q^{4} + 181061536500q^{5} - 67153080066800q^{7} + 2818750585098240q^{8} + 35542592216532000q^{10} - 133871815441914264q^{11} - 2981610478259443940q^{13} + 15496641262468340352q^{14} + \)\(19\!\cdots\!72\)\(q^{16} + 79361149261175525340q^{17} - \)\(13\!\cdots\!00\)\(q^{19} + \)\(43\!\cdots\!00\)\(q^{20} + \)\(24\!\cdots\!40\)\(q^{22} - \)\(26\!\cdots\!40\)\(q^{23} - \)\(19\!\cdots\!50\)\(q^{25} - \)\(27\!\cdots\!04\)\(q^{26} + \)\(18\!\cdots\!60\)\(q^{28} + \)\(16\!\cdots\!00\)\(q^{29} - \)\(62\!\cdots\!16\)\(q^{31} + \)\(57\!\cdots\!80\)\(q^{32} + \)\(69\!\cdots\!08\)\(q^{34} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(10\!\cdots\!20\)\(q^{37} + \)\(36\!\cdots\!60\)\(q^{38} + \)\(40\!\cdots\!00\)\(q^{40} - \)\(27\!\cdots\!44\)\(q^{41} + \)\(15\!\cdots\!00\)\(q^{43} + \)\(30\!\cdots\!12\)\(q^{44} - \)\(56\!\cdots\!76\)\(q^{46} - \)\(54\!\cdots\!40\)\(q^{47} + \)\(24\!\cdots\!14\)\(q^{49} - \)\(72\!\cdots\!00\)\(q^{50} - \)\(32\!\cdots\!00\)\(q^{52} + \)\(26\!\cdots\!20\)\(q^{53} + \)\(20\!\cdots\!00\)\(q^{55} + \)\(25\!\cdots\!00\)\(q^{56} + \)\(25\!\cdots\!60\)\(q^{58} + \)\(30\!\cdots\!00\)\(q^{59} - \)\(57\!\cdots\!36\)\(q^{61} + \)\(42\!\cdots\!60\)\(q^{62} + \)\(86\!\cdots\!24\)\(q^{64} - \)\(36\!\cdots\!00\)\(q^{65} + \)\(15\!\cdots\!60\)\(q^{67} + \)\(84\!\cdots\!20\)\(q^{68} + \)\(17\!\cdots\!00\)\(q^{70} + \)\(26\!\cdots\!76\)\(q^{71} + \)\(94\!\cdots\!40\)\(q^{73} - \)\(14\!\cdots\!28\)\(q^{74} + \)\(45\!\cdots\!00\)\(q^{76} + \)\(30\!\cdots\!00\)\(q^{77} - \)\(85\!\cdots\!00\)\(q^{79} + \)\(35\!\cdots\!00\)\(q^{80} + \)\(62\!\cdots\!40\)\(q^{82} - \)\(29\!\cdots\!20\)\(q^{83} + \)\(72\!\cdots\!00\)\(q^{85} + \)\(31\!\cdots\!36\)\(q^{86} + \)\(13\!\cdots\!20\)\(q^{88} - \)\(13\!\cdots\!00\)\(q^{89} + \)\(26\!\cdots\!44\)\(q^{91} - \)\(68\!\cdots\!60\)\(q^{92} - \)\(45\!\cdots\!12\)\(q^{94} - \)\(33\!\cdots\!00\)\(q^{95} - \)\(36\!\cdots\!60\)\(q^{97} - \)\(11\!\cdots\!40\)\(q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 171359. 1.84890 0.924449 0.381305i \(-0.124525\pi\)
0.924449 + 0.381305i \(0.124525\pi\)
\(3\) 0 0
\(4\) 2.07741e10 2.41843
\(5\) 2.01492e11 0.590545 0.295273 0.955413i \(-0.404589\pi\)
0.295273 + 0.955413i \(0.404589\pi\)
\(6\) 0 0
\(7\) 5.50153e13 0.625699 0.312850 0.949803i \(-0.398716\pi\)
0.312850 + 0.949803i \(0.398716\pi\)
\(8\) 2.08788e15 2.62253
\(9\) 0 0
\(10\) 3.45276e16 1.09186
\(11\) 8.18909e16 0.537349 0.268674 0.963231i \(-0.413414\pi\)
0.268674 + 0.963231i \(0.413414\pi\)
\(12\) 0 0
\(13\) −1.90399e18 −0.793597 −0.396798 0.917906i \(-0.629879\pi\)
−0.396798 + 0.917906i \(0.629879\pi\)
\(14\) 9.42739e18 1.15685
\(15\) 0 0
\(16\) 1.79329e20 2.43036
\(17\) 3.33893e20 1.66418 0.832090 0.554640i \(-0.187144\pi\)
0.832090 + 0.554640i \(0.187144\pi\)
\(18\) 0 0
\(19\) −1.40494e20 −0.111743 −0.0558717 0.998438i \(-0.517794\pi\)
−0.0558717 + 0.998438i \(0.517794\pi\)
\(20\) 4.18583e21 1.42819
\(21\) 0 0
\(22\) 1.40328e22 0.993503
\(23\) −3.12767e22 −1.06344 −0.531718 0.846922i \(-0.678453\pi\)
−0.531718 + 0.846922i \(0.678453\pi\)
\(24\) 0 0
\(25\) −7.58162e22 −0.651256
\(26\) −3.26267e23 −1.46728
\(27\) 0 0
\(28\) 1.14289e24 1.51321
\(29\) 1.50979e24 1.12034 0.560168 0.828379i \(-0.310736\pi\)
0.560168 + 0.828379i \(0.310736\pi\)
\(30\) 0 0
\(31\) 5.18762e23 0.128086 0.0640428 0.997947i \(-0.479601\pi\)
0.0640428 + 0.997947i \(0.479601\pi\)
\(32\) 1.27950e25 1.87096
\(33\) 0 0
\(34\) 5.72158e25 3.07690
\(35\) 1.10852e25 0.369504
\(36\) 0 0
\(37\) −3.01507e25 −0.401762 −0.200881 0.979616i \(-0.564380\pi\)
−0.200881 + 0.979616i \(0.564380\pi\)
\(38\) −2.40749e25 −0.206602
\(39\) 0 0
\(40\) 4.20691e26 1.54872
\(41\) 2.18887e26 0.536149 0.268075 0.963398i \(-0.413613\pi\)
0.268075 + 0.963398i \(0.413613\pi\)
\(42\) 0 0
\(43\) 1.76701e27 1.97246 0.986232 0.165369i \(-0.0528817\pi\)
0.986232 + 0.165369i \(0.0528817\pi\)
\(44\) 1.70121e27 1.29954
\(45\) 0 0
\(46\) −5.35955e27 −1.96618
\(47\) −3.25654e27 −0.837802 −0.418901 0.908032i \(-0.637585\pi\)
−0.418901 + 0.908032i \(0.637585\pi\)
\(48\) 0 0
\(49\) −4.70431e27 −0.608500
\(50\) −1.29918e28 −1.20411
\(51\) 0 0
\(52\) −3.95538e28 −1.91926
\(53\) −9.17652e27 −0.325182 −0.162591 0.986694i \(-0.551985\pi\)
−0.162591 + 0.986694i \(0.551985\pi\)
\(54\) 0 0
\(55\) 1.65004e28 0.317329
\(56\) 1.14865e29 1.64091
\(57\) 0 0
\(58\) 2.58716e29 2.07139
\(59\) 1.18267e29 0.714174 0.357087 0.934071i \(-0.383770\pi\)
0.357087 + 0.934071i \(0.383770\pi\)
\(60\) 0 0
\(61\) −9.92930e27 −0.0345920 −0.0172960 0.999850i \(-0.505506\pi\)
−0.0172960 + 0.999850i \(0.505506\pi\)
\(62\) 8.88948e28 0.236817
\(63\) 0 0
\(64\) 6.52116e29 1.02886
\(65\) −3.83640e29 −0.468655
\(66\) 0 0
\(67\) 1.11293e30 0.824583 0.412291 0.911052i \(-0.364729\pi\)
0.412291 + 0.911052i \(0.364729\pi\)
\(68\) 6.93634e30 4.02470
\(69\) 0 0
\(70\) 1.89955e30 0.683175
\(71\) −7.58425e29 −0.215849 −0.107924 0.994159i \(-0.534420\pi\)
−0.107924 + 0.994159i \(0.534420\pi\)
\(72\) 0 0
\(73\) −6.06835e30 −1.09205 −0.546026 0.837768i \(-0.683860\pi\)
−0.546026 + 0.837768i \(0.683860\pi\)
\(74\) −5.16661e30 −0.742818
\(75\) 0 0
\(76\) −2.91863e30 −0.270243
\(77\) 4.50525e30 0.336219
\(78\) 0 0
\(79\) −5.57890e30 −0.272711 −0.136355 0.990660i \(-0.543539\pi\)
−0.136355 + 0.990660i \(0.543539\pi\)
\(80\) 3.61334e31 1.43524
\(81\) 0 0
\(82\) 3.75083e31 0.991286
\(83\) −4.13746e31 −0.895252 −0.447626 0.894221i \(-0.647730\pi\)
−0.447626 + 0.894221i \(0.647730\pi\)
\(84\) 0 0
\(85\) 6.72769e31 0.982775
\(86\) 3.02793e32 3.64688
\(87\) 0 0
\(88\) 1.70978e32 1.40921
\(89\) −6.21572e31 −0.425164 −0.212582 0.977143i \(-0.568187\pi\)
−0.212582 + 0.977143i \(0.568187\pi\)
\(90\) 0 0
\(91\) −1.04749e32 −0.496553
\(92\) −6.49745e32 −2.57184
\(93\) 0 0
\(94\) −5.58039e32 −1.54901
\(95\) −2.83084e31 −0.0659896
\(96\) 0 0
\(97\) 4.04003e32 0.667808 0.333904 0.942607i \(-0.391634\pi\)
0.333904 + 0.942607i \(0.391634\pi\)
\(98\) −8.06129e32 −1.12506
\(99\) 0 0
\(100\) −1.57501e33 −1.57501
\(101\) 1.20576e33 1.02319 0.511596 0.859226i \(-0.329054\pi\)
0.511596 + 0.859226i \(0.329054\pi\)
\(102\) 0 0
\(103\) −1.69378e33 −1.04002 −0.520011 0.854159i \(-0.674072\pi\)
−0.520011 + 0.854159i \(0.674072\pi\)
\(104\) −3.97530e33 −2.08123
\(105\) 0 0
\(106\) −1.57248e33 −0.601228
\(107\) −7.07121e32 −0.231559 −0.115780 0.993275i \(-0.536937\pi\)
−0.115780 + 0.993275i \(0.536937\pi\)
\(108\) 0 0
\(109\) 4.42814e33 1.06828 0.534139 0.845397i \(-0.320636\pi\)
0.534139 + 0.845397i \(0.320636\pi\)
\(110\) 2.82750e33 0.586709
\(111\) 0 0
\(112\) 9.86582e33 1.52067
\(113\) −1.36496e34 −1.81687 −0.908437 0.418023i \(-0.862723\pi\)
−0.908437 + 0.418023i \(0.862723\pi\)
\(114\) 0 0
\(115\) −6.30200e33 −0.628007
\(116\) 3.13645e34 2.70945
\(117\) 0 0
\(118\) 2.02661e34 1.32043
\(119\) 1.83692e34 1.04128
\(120\) 0 0
\(121\) −1.65190e34 −0.711256
\(122\) −1.70148e33 −0.0639572
\(123\) 0 0
\(124\) 1.07768e34 0.309766
\(125\) −3.87332e34 −0.975142
\(126\) 0 0
\(127\) −7.12185e34 −1.37985 −0.689925 0.723881i \(-0.742356\pi\)
−0.689925 + 0.723881i \(0.742356\pi\)
\(128\) 1.83832e33 0.0312936
\(129\) 0 0
\(130\) −6.57403e34 −0.866496
\(131\) 2.95217e34 0.342898 0.171449 0.985193i \(-0.445155\pi\)
0.171449 + 0.985193i \(0.445155\pi\)
\(132\) 0 0
\(133\) −7.72929e33 −0.0699178
\(134\) 1.90711e35 1.52457
\(135\) 0 0
\(136\) 6.97128e35 4.36436
\(137\) −8.90834e34 −0.494205 −0.247103 0.968989i \(-0.579479\pi\)
−0.247103 + 0.968989i \(0.579479\pi\)
\(138\) 0 0
\(139\) −3.61707e35 −1.57984 −0.789920 0.613210i \(-0.789878\pi\)
−0.789920 + 0.613210i \(0.789878\pi\)
\(140\) 2.30284e35 0.893618
\(141\) 0 0
\(142\) −1.29963e35 −0.399083
\(143\) −1.55920e35 −0.426438
\(144\) 0 0
\(145\) 3.04210e35 0.661610
\(146\) −1.03987e36 −2.01909
\(147\) 0 0
\(148\) −6.26354e35 −0.971632
\(149\) −1.73500e35 −0.240839 −0.120419 0.992723i \(-0.538424\pi\)
−0.120419 + 0.992723i \(0.538424\pi\)
\(150\) 0 0
\(151\) −1.35744e36 −1.51217 −0.756085 0.654473i \(-0.772891\pi\)
−0.756085 + 0.654473i \(0.772891\pi\)
\(152\) −2.93333e35 −0.293050
\(153\) 0 0
\(154\) 7.72017e35 0.621634
\(155\) 1.04527e35 0.0756404
\(156\) 0 0
\(157\) −2.78049e36 −1.62846 −0.814229 0.580543i \(-0.802840\pi\)
−0.814229 + 0.580543i \(0.802840\pi\)
\(158\) −9.55998e35 −0.504214
\(159\) 0 0
\(160\) 2.57809e36 1.10489
\(161\) −1.72069e36 −0.665390
\(162\) 0 0
\(163\) 2.81381e36 0.887565 0.443782 0.896135i \(-0.353636\pi\)
0.443782 + 0.896135i \(0.353636\pi\)
\(164\) 4.54718e36 1.29664
\(165\) 0 0
\(166\) −7.08993e36 −1.65523
\(167\) −6.77893e36 −1.43331 −0.716653 0.697430i \(-0.754327\pi\)
−0.716653 + 0.697430i \(0.754327\pi\)
\(168\) 0 0
\(169\) −2.13094e36 −0.370204
\(170\) 1.15285e37 1.81705
\(171\) 0 0
\(172\) 3.67080e37 4.77026
\(173\) 1.42570e37 1.68370 0.841852 0.539708i \(-0.181465\pi\)
0.841852 + 0.539708i \(0.181465\pi\)
\(174\) 0 0
\(175\) −4.17105e36 −0.407490
\(176\) 1.46854e37 1.30595
\(177\) 0 0
\(178\) −1.06512e37 −0.786085
\(179\) −7.37934e36 −0.496527 −0.248263 0.968693i \(-0.579860\pi\)
−0.248263 + 0.968693i \(0.579860\pi\)
\(180\) 0 0
\(181\) 2.39313e36 0.134051 0.0670254 0.997751i \(-0.478649\pi\)
0.0670254 + 0.997751i \(0.478649\pi\)
\(182\) −1.79497e37 −0.918076
\(183\) 0 0
\(184\) −6.53018e37 −2.78889
\(185\) −6.07513e36 −0.237259
\(186\) 0 0
\(187\) 2.73428e37 0.894245
\(188\) −6.76517e37 −2.02616
\(189\) 0 0
\(190\) −4.85091e36 −0.122008
\(191\) 6.98026e35 0.0160998 0.00804991 0.999968i \(-0.497438\pi\)
0.00804991 + 0.999968i \(0.497438\pi\)
\(192\) 0 0
\(193\) 6.67098e36 0.129567 0.0647835 0.997899i \(-0.479364\pi\)
0.0647835 + 0.997899i \(0.479364\pi\)
\(194\) 6.92298e37 1.23471
\(195\) 0 0
\(196\) −9.77280e37 −1.47161
\(197\) 8.89563e37 1.23164 0.615821 0.787886i \(-0.288825\pi\)
0.615821 + 0.787886i \(0.288825\pi\)
\(198\) 0 0
\(199\) 1.05106e38 1.23183 0.615916 0.787812i \(-0.288786\pi\)
0.615916 + 0.787812i \(0.288786\pi\)
\(200\) −1.58295e38 −1.70794
\(201\) 0 0
\(202\) 2.06618e38 1.89178
\(203\) 8.30613e37 0.700994
\(204\) 0 0
\(205\) 4.41040e37 0.316621
\(206\) −2.90245e38 −1.92290
\(207\) 0 0
\(208\) −3.41441e38 −1.92873
\(209\) −1.15051e37 −0.0600452
\(210\) 0 0
\(211\) −5.59870e37 −0.249705 −0.124852 0.992175i \(-0.539846\pi\)
−0.124852 + 0.992175i \(0.539846\pi\)
\(212\) −1.90634e38 −0.786428
\(213\) 0 0
\(214\) −1.21172e38 −0.428129
\(215\) 3.56038e38 1.16483
\(216\) 0 0
\(217\) 2.85398e37 0.0801431
\(218\) 7.58804e38 1.97514
\(219\) 0 0
\(220\) 3.42781e38 0.767436
\(221\) −6.35730e38 −1.32069
\(222\) 0 0
\(223\) 4.65676e38 0.833784 0.416892 0.908956i \(-0.363119\pi\)
0.416892 + 0.908956i \(0.363119\pi\)
\(224\) 7.03919e38 1.17066
\(225\) 0 0
\(226\) −2.33898e39 −3.35921
\(227\) 2.61802e38 0.349580 0.174790 0.984606i \(-0.444075\pi\)
0.174790 + 0.984606i \(0.444075\pi\)
\(228\) 0 0
\(229\) 9.20963e38 1.06404 0.532018 0.846733i \(-0.321434\pi\)
0.532018 + 0.846733i \(0.321434\pi\)
\(230\) −1.07991e39 −1.16112
\(231\) 0 0
\(232\) 3.15225e39 2.93811
\(233\) −7.13755e37 −0.0619693 −0.0309847 0.999520i \(-0.509864\pi\)
−0.0309847 + 0.999520i \(0.509864\pi\)
\(234\) 0 0
\(235\) −6.56167e38 −0.494760
\(236\) 2.45688e39 1.72718
\(237\) 0 0
\(238\) 3.14774e39 1.92522
\(239\) 1.83178e39 1.04546 0.522731 0.852498i \(-0.324913\pi\)
0.522731 + 0.852498i \(0.324913\pi\)
\(240\) 0 0
\(241\) −3.05313e39 −1.51867 −0.759336 0.650698i \(-0.774476\pi\)
−0.759336 + 0.650698i \(0.774476\pi\)
\(242\) −2.83069e39 −1.31504
\(243\) 0 0
\(244\) −2.06272e38 −0.0836583
\(245\) −9.47883e38 −0.359347
\(246\) 0 0
\(247\) 2.67499e38 0.0886793
\(248\) 1.08311e39 0.335908
\(249\) 0 0
\(250\) −6.63729e39 −1.80294
\(251\) −3.03407e39 −0.771631 −0.385815 0.922576i \(-0.626080\pi\)
−0.385815 + 0.922576i \(0.626080\pi\)
\(252\) 0 0
\(253\) −2.56127e39 −0.571435
\(254\) −1.22040e40 −2.55120
\(255\) 0 0
\(256\) −5.28662e39 −0.970999
\(257\) 2.91186e39 0.501503 0.250752 0.968051i \(-0.419322\pi\)
0.250752 + 0.968051i \(0.419322\pi\)
\(258\) 0 0
\(259\) −1.65875e39 −0.251382
\(260\) −7.96978e39 −1.13341
\(261\) 0 0
\(262\) 5.05883e39 0.633984
\(263\) 5.62907e39 0.662471 0.331236 0.943548i \(-0.392534\pi\)
0.331236 + 0.943548i \(0.392534\pi\)
\(264\) 0 0
\(265\) −1.84900e39 −0.192035
\(266\) −1.32449e39 −0.129271
\(267\) 0 0
\(268\) 2.31202e40 1.99419
\(269\) 9.77080e39 0.792533 0.396266 0.918136i \(-0.370306\pi\)
0.396266 + 0.918136i \(0.370306\pi\)
\(270\) 0 0
\(271\) 2.38779e40 1.71397 0.856985 0.515342i \(-0.172335\pi\)
0.856985 + 0.515342i \(0.172335\pi\)
\(272\) 5.98767e40 4.04456
\(273\) 0 0
\(274\) −1.52653e40 −0.913736
\(275\) −6.20865e39 −0.349952
\(276\) 0 0
\(277\) −1.39835e40 −0.699358 −0.349679 0.936870i \(-0.613709\pi\)
−0.349679 + 0.936870i \(0.613709\pi\)
\(278\) −6.19820e40 −2.92096
\(279\) 0 0
\(280\) 2.31444e40 0.969033
\(281\) 2.89927e40 1.14455 0.572274 0.820062i \(-0.306061\pi\)
0.572274 + 0.820062i \(0.306061\pi\)
\(282\) 0 0
\(283\) −1.50945e40 −0.530081 −0.265040 0.964237i \(-0.585385\pi\)
−0.265040 + 0.964237i \(0.585385\pi\)
\(284\) −1.57556e40 −0.522015
\(285\) 0 0
\(286\) −2.67183e40 −0.788441
\(287\) 1.20421e40 0.335468
\(288\) 0 0
\(289\) 7.12303e40 1.76950
\(290\) 5.21293e40 1.22325
\(291\) 0 0
\(292\) −1.26065e41 −2.64105
\(293\) 1.40305e40 0.277816 0.138908 0.990305i \(-0.455641\pi\)
0.138908 + 0.990305i \(0.455641\pi\)
\(294\) 0 0
\(295\) 2.38298e40 0.421752
\(296\) −6.29509e40 −1.05363
\(297\) 0 0
\(298\) −2.97309e40 −0.445286
\(299\) 5.95505e40 0.843939
\(300\) 0 0
\(301\) 9.72124e40 1.23417
\(302\) −2.32610e41 −2.79585
\(303\) 0 0
\(304\) −2.51945e40 −0.271577
\(305\) −2.00068e39 −0.0204282
\(306\) 0 0
\(307\) −2.92194e40 −0.267848 −0.133924 0.990992i \(-0.542758\pi\)
−0.133924 + 0.990992i \(0.542758\pi\)
\(308\) 9.35926e40 0.813120
\(309\) 0 0
\(310\) 1.79116e40 0.139851
\(311\) 2.24927e41 1.66531 0.832653 0.553794i \(-0.186821\pi\)
0.832653 + 0.553794i \(0.186821\pi\)
\(312\) 0 0
\(313\) 1.17836e41 0.784868 0.392434 0.919780i \(-0.371633\pi\)
0.392434 + 0.919780i \(0.371633\pi\)
\(314\) −4.76463e41 −3.01086
\(315\) 0 0
\(316\) −1.15897e41 −0.659530
\(317\) 9.02704e40 0.487604 0.243802 0.969825i \(-0.421605\pi\)
0.243802 + 0.969825i \(0.421605\pi\)
\(318\) 0 0
\(319\) 1.23638e41 0.602012
\(320\) 1.31396e41 0.607588
\(321\) 0 0
\(322\) −2.94857e41 −1.23024
\(323\) −4.69099e40 −0.185961
\(324\) 0 0
\(325\) 1.44353e41 0.516835
\(326\) 4.82174e41 1.64102
\(327\) 0 0
\(328\) 4.57008e41 1.40607
\(329\) −1.79159e41 −0.524212
\(330\) 0 0
\(331\) −2.91394e41 −0.771469 −0.385734 0.922610i \(-0.626052\pi\)
−0.385734 + 0.922610i \(0.626052\pi\)
\(332\) −8.59521e41 −2.16510
\(333\) 0 0
\(334\) −1.16163e42 −2.65004
\(335\) 2.24247e41 0.486954
\(336\) 0 0
\(337\) −5.69474e41 −1.12094 −0.560468 0.828176i \(-0.689379\pi\)
−0.560468 + 0.828176i \(0.689379\pi\)
\(338\) −3.65157e41 −0.684470
\(339\) 0 0
\(340\) 1.39762e42 2.37677
\(341\) 4.24819e40 0.0688266
\(342\) 0 0
\(343\) −6.84132e41 −1.00644
\(344\) 3.68929e42 5.17284
\(345\) 0 0
\(346\) 2.44306e42 3.11300
\(347\) −4.85020e41 −0.589283 −0.294641 0.955608i \(-0.595200\pi\)
−0.294641 + 0.955608i \(0.595200\pi\)
\(348\) 0 0
\(349\) 5.38533e41 0.595104 0.297552 0.954706i \(-0.403830\pi\)
0.297552 + 0.954706i \(0.403830\pi\)
\(350\) −7.14748e41 −0.753408
\(351\) 0 0
\(352\) 1.04779e42 1.00536
\(353\) 9.77578e41 0.895094 0.447547 0.894260i \(-0.352298\pi\)
0.447547 + 0.894260i \(0.352298\pi\)
\(354\) 0 0
\(355\) −1.52817e41 −0.127469
\(356\) −1.29126e42 −1.02823
\(357\) 0 0
\(358\) −1.26452e42 −0.918027
\(359\) −1.18143e42 −0.819124 −0.409562 0.912282i \(-0.634318\pi\)
−0.409562 + 0.912282i \(0.634318\pi\)
\(360\) 0 0
\(361\) −1.56103e42 −0.987513
\(362\) 4.10085e41 0.247846
\(363\) 0 0
\(364\) −2.17606e42 −1.20088
\(365\) −1.22273e42 −0.644907
\(366\) 0 0
\(367\) 7.16875e41 0.345504 0.172752 0.984965i \(-0.444734\pi\)
0.172752 + 0.984965i \(0.444734\pi\)
\(368\) −5.60881e42 −2.58453
\(369\) 0 0
\(370\) −1.04103e42 −0.438668
\(371\) −5.04849e41 −0.203466
\(372\) 0 0
\(373\) −6.80691e41 −0.251047 −0.125523 0.992091i \(-0.540061\pi\)
−0.125523 + 0.992091i \(0.540061\pi\)
\(374\) 4.68545e42 1.65337
\(375\) 0 0
\(376\) −6.79925e42 −2.19716
\(377\) −2.87462e42 −0.889096
\(378\) 0 0
\(379\) 5.14200e42 1.45742 0.728711 0.684821i \(-0.240120\pi\)
0.728711 + 0.684821i \(0.240120\pi\)
\(380\) −5.88082e41 −0.159591
\(381\) 0 0
\(382\) 1.19613e41 0.0297669
\(383\) 6.82496e42 1.62675 0.813375 0.581740i \(-0.197628\pi\)
0.813375 + 0.581740i \(0.197628\pi\)
\(384\) 0 0
\(385\) 9.07773e41 0.198552
\(386\) 1.14314e42 0.239556
\(387\) 0 0
\(388\) 8.39281e42 1.61504
\(389\) −3.03602e42 −0.559933 −0.279966 0.960010i \(-0.590323\pi\)
−0.279966 + 0.960010i \(0.590323\pi\)
\(390\) 0 0
\(391\) −1.04431e43 −1.76975
\(392\) −9.82202e42 −1.59581
\(393\) 0 0
\(394\) 1.52435e43 2.27718
\(395\) −1.12411e42 −0.161048
\(396\) 0 0
\(397\) 8.54761e42 1.12668 0.563341 0.826225i \(-0.309516\pi\)
0.563341 + 0.826225i \(0.309516\pi\)
\(398\) 1.80109e43 2.27753
\(399\) 0 0
\(400\) −1.35960e43 −1.58279
\(401\) −4.64638e42 −0.519078 −0.259539 0.965733i \(-0.583571\pi\)
−0.259539 + 0.965733i \(0.583571\pi\)
\(402\) 0 0
\(403\) −9.87719e41 −0.101648
\(404\) 2.50486e43 2.47452
\(405\) 0 0
\(406\) 1.42333e43 1.29607
\(407\) −2.46907e42 −0.215886
\(408\) 0 0
\(409\) 1.73477e43 1.39897 0.699483 0.714649i \(-0.253414\pi\)
0.699483 + 0.714649i \(0.253414\pi\)
\(410\) 7.55763e42 0.585399
\(411\) 0 0
\(412\) −3.51868e43 −2.51522
\(413\) 6.50647e42 0.446858
\(414\) 0 0
\(415\) −8.33667e42 −0.528687
\(416\) −2.43615e43 −1.48479
\(417\) 0 0
\(418\) −1.97151e42 −0.111017
\(419\) −1.59552e43 −0.863717 −0.431858 0.901941i \(-0.642142\pi\)
−0.431858 + 0.901941i \(0.642142\pi\)
\(420\) 0 0
\(421\) −2.07154e43 −1.03667 −0.518333 0.855179i \(-0.673447\pi\)
−0.518333 + 0.855179i \(0.673447\pi\)
\(422\) −9.59390e42 −0.461679
\(423\) 0 0
\(424\) −1.91594e43 −0.852797
\(425\) −2.53145e43 −1.08381
\(426\) 0 0
\(427\) −5.46263e41 −0.0216442
\(428\) −1.46898e43 −0.560008
\(429\) 0 0
\(430\) 6.10105e43 2.15365
\(431\) 2.15255e43 0.731272 0.365636 0.930758i \(-0.380851\pi\)
0.365636 + 0.930758i \(0.380851\pi\)
\(432\) 0 0
\(433\) −2.62411e43 −0.825908 −0.412954 0.910752i \(-0.635503\pi\)
−0.412954 + 0.910752i \(0.635503\pi\)
\(434\) 4.89057e42 0.148176
\(435\) 0 0
\(436\) 9.19908e43 2.58355
\(437\) 4.39417e42 0.118832
\(438\) 0 0
\(439\) 1.50091e43 0.376434 0.188217 0.982127i \(-0.439729\pi\)
0.188217 + 0.982127i \(0.439729\pi\)
\(440\) 3.44507e43 0.832203
\(441\) 0 0
\(442\) −1.08938e44 −2.44182
\(443\) −1.03686e43 −0.223902 −0.111951 0.993714i \(-0.535710\pi\)
−0.111951 + 0.993714i \(0.535710\pi\)
\(444\) 0 0
\(445\) −1.25242e43 −0.251079
\(446\) 7.97979e43 1.54158
\(447\) 0 0
\(448\) 3.58764e43 0.643756
\(449\) 3.05007e42 0.0527528 0.0263764 0.999652i \(-0.491603\pi\)
0.0263764 + 0.999652i \(0.491603\pi\)
\(450\) 0 0
\(451\) 1.79248e43 0.288099
\(452\) −2.83558e44 −4.39397
\(453\) 0 0
\(454\) 4.48623e43 0.646338
\(455\) −2.11061e43 −0.293237
\(456\) 0 0
\(457\) −8.15124e43 −1.05343 −0.526716 0.850042i \(-0.676577\pi\)
−0.526716 + 0.850042i \(0.676577\pi\)
\(458\) 1.57816e44 1.96730
\(459\) 0 0
\(460\) −1.30919e44 −1.51879
\(461\) −9.49311e43 −1.06253 −0.531267 0.847204i \(-0.678284\pi\)
−0.531267 + 0.847204i \(0.678284\pi\)
\(462\) 0 0
\(463\) −6.84730e43 −0.713564 −0.356782 0.934188i \(-0.616126\pi\)
−0.356782 + 0.934188i \(0.616126\pi\)
\(464\) 2.70748e44 2.72282
\(465\) 0 0
\(466\) −1.22309e43 −0.114575
\(467\) −1.13867e44 −1.02961 −0.514803 0.857308i \(-0.672135\pi\)
−0.514803 + 0.857308i \(0.672135\pi\)
\(468\) 0 0
\(469\) 6.12282e43 0.515941
\(470\) −1.12440e44 −0.914762
\(471\) 0 0
\(472\) 2.46926e44 1.87294
\(473\) 1.44702e44 1.05990
\(474\) 0 0
\(475\) 1.06517e43 0.0727736
\(476\) 3.81605e44 2.51825
\(477\) 0 0
\(478\) 3.13893e44 1.93295
\(479\) 2.43710e44 1.44990 0.724948 0.688804i \(-0.241864\pi\)
0.724948 + 0.688804i \(0.241864\pi\)
\(480\) 0 0
\(481\) 5.74067e43 0.318837
\(482\) −5.23183e44 −2.80787
\(483\) 0 0
\(484\) −3.43169e44 −1.72012
\(485\) 8.14036e43 0.394371
\(486\) 0 0
\(487\) −1.55812e44 −0.705300 −0.352650 0.935755i \(-0.614719\pi\)
−0.352650 + 0.935755i \(0.614719\pi\)
\(488\) −2.07311e43 −0.0907185
\(489\) 0 0
\(490\) −1.62429e44 −0.664397
\(491\) 2.64492e43 0.104609 0.0523044 0.998631i \(-0.483343\pi\)
0.0523044 + 0.998631i \(0.483343\pi\)
\(492\) 0 0
\(493\) 5.04108e44 1.86444
\(494\) 4.58384e43 0.163959
\(495\) 0 0
\(496\) 9.30290e43 0.311294
\(497\) −4.17250e43 −0.135057
\(498\) 0 0
\(499\) 3.31565e44 1.00441 0.502203 0.864750i \(-0.332523\pi\)
0.502203 + 0.864750i \(0.332523\pi\)
\(500\) −8.04647e44 −2.35831
\(501\) 0 0
\(502\) −5.19917e44 −1.42667
\(503\) 4.39485e44 1.16700 0.583502 0.812112i \(-0.301682\pi\)
0.583502 + 0.812112i \(0.301682\pi\)
\(504\) 0 0
\(505\) 2.42951e44 0.604242
\(506\) −4.38898e44 −1.05653
\(507\) 0 0
\(508\) −1.47950e45 −3.33706
\(509\) 7.37330e44 1.60997 0.804986 0.593294i \(-0.202173\pi\)
0.804986 + 0.593294i \(0.202173\pi\)
\(510\) 0 0
\(511\) −3.33852e44 −0.683296
\(512\) −9.21704e44 −1.82657
\(513\) 0 0
\(514\) 4.98975e44 0.927229
\(515\) −3.41284e44 −0.614181
\(516\) 0 0
\(517\) −2.66681e44 −0.450192
\(518\) −2.84242e44 −0.464780
\(519\) 0 0
\(520\) −8.00992e44 −1.22906
\(521\) −1.67201e43 −0.0248552 −0.0124276 0.999923i \(-0.503956\pi\)
−0.0124276 + 0.999923i \(0.503956\pi\)
\(522\) 0 0
\(523\) 1.06261e45 1.48284 0.741420 0.671041i \(-0.234153\pi\)
0.741420 + 0.671041i \(0.234153\pi\)
\(524\) 6.13288e44 0.829275
\(525\) 0 0
\(526\) 9.64595e44 1.22484
\(527\) 1.73211e44 0.213158
\(528\) 0 0
\(529\) 1.13224e44 0.130894
\(530\) −3.16843e44 −0.355052
\(531\) 0 0
\(532\) −1.60569e44 −0.169091
\(533\) −4.16759e44 −0.425486
\(534\) 0 0
\(535\) −1.42480e44 −0.136746
\(536\) 2.32366e45 2.16249
\(537\) 0 0
\(538\) 1.67432e45 1.46531
\(539\) −3.85240e44 −0.326977
\(540\) 0 0
\(541\) −1.73107e45 −1.38217 −0.691084 0.722774i \(-0.742867\pi\)
−0.691084 + 0.722774i \(0.742867\pi\)
\(542\) 4.09171e45 3.16896
\(543\) 0 0
\(544\) 4.27216e45 3.11362
\(545\) 8.92237e44 0.630867
\(546\) 0 0
\(547\) 1.88809e45 1.25670 0.628350 0.777931i \(-0.283731\pi\)
0.628350 + 0.777931i \(0.283731\pi\)
\(548\) −1.85063e45 −1.19520
\(549\) 0 0
\(550\) −1.06391e45 −0.647025
\(551\) −2.12115e44 −0.125190
\(552\) 0 0
\(553\) −3.06925e44 −0.170635
\(554\) −2.39620e45 −1.29304
\(555\) 0 0
\(556\) −7.51415e45 −3.82073
\(557\) −1.82254e45 −0.899636 −0.449818 0.893120i \(-0.648511\pi\)
−0.449818 + 0.893120i \(0.648511\pi\)
\(558\) 0 0
\(559\) −3.36437e45 −1.56534
\(560\) 1.98789e45 0.898027
\(561\) 0 0
\(562\) 4.96817e45 2.11615
\(563\) −3.54966e45 −1.46824 −0.734122 0.679018i \(-0.762406\pi\)
−0.734122 + 0.679018i \(0.762406\pi\)
\(564\) 0 0
\(565\) −2.75028e45 −1.07295
\(566\) −2.58658e45 −0.980066
\(567\) 0 0
\(568\) −1.58350e45 −0.566070
\(569\) 3.23817e45 1.12447 0.562234 0.826978i \(-0.309942\pi\)
0.562234 + 0.826978i \(0.309942\pi\)
\(570\) 0 0
\(571\) 1.75221e45 0.574237 0.287118 0.957895i \(-0.407303\pi\)
0.287118 + 0.957895i \(0.407303\pi\)
\(572\) −3.23909e45 −1.03131
\(573\) 0 0
\(574\) 2.06353e45 0.620247
\(575\) 2.37128e45 0.692569
\(576\) 0 0
\(577\) −2.22828e45 −0.614566 −0.307283 0.951618i \(-0.599420\pi\)
−0.307283 + 0.951618i \(0.599420\pi\)
\(578\) 1.22060e46 3.27162
\(579\) 0 0
\(580\) 6.31970e45 1.60005
\(581\) −2.27624e45 −0.560159
\(582\) 0 0
\(583\) −7.51473e44 −0.174736
\(584\) −1.26700e46 −2.86394
\(585\) 0 0
\(586\) 2.40426e45 0.513654
\(587\) −1.91072e45 −0.396889 −0.198444 0.980112i \(-0.563589\pi\)
−0.198444 + 0.980112i \(0.563589\pi\)
\(588\) 0 0
\(589\) −7.28827e43 −0.0143127
\(590\) 4.08346e45 0.779777
\(591\) 0 0
\(592\) −5.40689e45 −0.976426
\(593\) −8.87275e45 −1.55832 −0.779158 0.626827i \(-0.784353\pi\)
−0.779158 + 0.626827i \(0.784353\pi\)
\(594\) 0 0
\(595\) 3.70126e45 0.614921
\(596\) −3.60431e45 −0.582451
\(597\) 0 0
\(598\) 1.02045e46 1.56036
\(599\) 1.96939e45 0.292947 0.146474 0.989215i \(-0.453208\pi\)
0.146474 + 0.989215i \(0.453208\pi\)
\(600\) 0 0
\(601\) 6.49113e45 0.913885 0.456942 0.889496i \(-0.348945\pi\)
0.456942 + 0.889496i \(0.348945\pi\)
\(602\) 1.66583e46 2.28185
\(603\) 0 0
\(604\) −2.81996e46 −3.65707
\(605\) −3.32846e45 −0.420029
\(606\) 0 0
\(607\) −1.08656e46 −1.29849 −0.649247 0.760577i \(-0.724916\pi\)
−0.649247 + 0.760577i \(0.724916\pi\)
\(608\) −1.79761e45 −0.209068
\(609\) 0 0
\(610\) −3.42835e44 −0.0377696
\(611\) 6.20043e45 0.664877
\(612\) 0 0
\(613\) 8.70542e45 0.884488 0.442244 0.896895i \(-0.354182\pi\)
0.442244 + 0.896895i \(0.354182\pi\)
\(614\) −5.00703e45 −0.495224
\(615\) 0 0
\(616\) 9.40640e45 0.881742
\(617\) 1.12685e46 1.02840 0.514199 0.857671i \(-0.328089\pi\)
0.514199 + 0.857671i \(0.328089\pi\)
\(618\) 0 0
\(619\) −1.07685e46 −0.931663 −0.465831 0.884874i \(-0.654245\pi\)
−0.465831 + 0.884874i \(0.654245\pi\)
\(620\) 2.17145e45 0.182931
\(621\) 0 0
\(622\) 3.85433e46 3.07898
\(623\) −3.41959e45 −0.266025
\(624\) 0 0
\(625\) 1.02173e45 0.0753905
\(626\) 2.01923e46 1.45114
\(627\) 0 0
\(628\) −5.77622e46 −3.93831
\(629\) −1.00671e46 −0.668605
\(630\) 0 0
\(631\) 2.47581e46 1.56039 0.780196 0.625535i \(-0.215119\pi\)
0.780196 + 0.625535i \(0.215119\pi\)
\(632\) −1.16481e46 −0.715190
\(633\) 0 0
\(634\) 1.54687e46 0.901531
\(635\) −1.43500e46 −0.814864
\(636\) 0 0
\(637\) 8.95698e45 0.482904
\(638\) 2.11865e46 1.11306
\(639\) 0 0
\(640\) 3.70407e44 0.0184803
\(641\) 6.66844e45 0.324240 0.162120 0.986771i \(-0.448167\pi\)
0.162120 + 0.986771i \(0.448167\pi\)
\(642\) 0 0
\(643\) −8.88748e42 −0.000410485 0 −0.000205243 1.00000i \(-0.500065\pi\)
−0.000205243 1.00000i \(0.500065\pi\)
\(644\) −3.57459e46 −1.60920
\(645\) 0 0
\(646\) −8.03845e45 −0.343824
\(647\) 1.36920e46 0.570884 0.285442 0.958396i \(-0.407860\pi\)
0.285442 + 0.958396i \(0.407860\pi\)
\(648\) 0 0
\(649\) 9.68495e45 0.383760
\(650\) 2.47363e46 0.955575
\(651\) 0 0
\(652\) 5.84545e46 2.14651
\(653\) 1.32140e46 0.473115 0.236557 0.971618i \(-0.423981\pi\)
0.236557 + 0.971618i \(0.423981\pi\)
\(654\) 0 0
\(655\) 5.94840e45 0.202497
\(656\) 3.92527e46 1.30304
\(657\) 0 0
\(658\) −3.07007e46 −0.969215
\(659\) −3.14029e46 −0.966852 −0.483426 0.875385i \(-0.660608\pi\)
−0.483426 + 0.875385i \(0.660608\pi\)
\(660\) 0 0
\(661\) −5.62366e46 −1.64701 −0.823503 0.567313i \(-0.807983\pi\)
−0.823503 + 0.567313i \(0.807983\pi\)
\(662\) −4.99331e46 −1.42637
\(663\) 0 0
\(664\) −8.63851e46 −2.34782
\(665\) −1.55739e45 −0.0412896
\(666\) 0 0
\(667\) −4.72211e46 −1.19141
\(668\) −1.40826e47 −3.46635
\(669\) 0 0
\(670\) 3.84269e46 0.900328
\(671\) −8.13119e44 −0.0185880
\(672\) 0 0
\(673\) 1.86687e45 0.0406317 0.0203159 0.999794i \(-0.493533\pi\)
0.0203159 + 0.999794i \(0.493533\pi\)
\(674\) −9.75848e46 −2.07250
\(675\) 0 0
\(676\) −4.42685e46 −0.895311
\(677\) 3.19535e46 0.630674 0.315337 0.948980i \(-0.397882\pi\)
0.315337 + 0.948980i \(0.397882\pi\)
\(678\) 0 0
\(679\) 2.22264e46 0.417847
\(680\) 1.40466e47 2.57735
\(681\) 0 0
\(682\) 7.27967e45 0.127253
\(683\) 9.67204e46 1.65035 0.825176 0.564876i \(-0.191076\pi\)
0.825176 + 0.564876i \(0.191076\pi\)
\(684\) 0 0
\(685\) −1.79496e46 −0.291851
\(686\) −1.17232e47 −1.86080
\(687\) 0 0
\(688\) 3.16875e47 4.79379
\(689\) 1.74720e46 0.258063
\(690\) 0 0
\(691\) −4.21417e46 −0.593368 −0.296684 0.954976i \(-0.595881\pi\)
−0.296684 + 0.954976i \(0.595881\pi\)
\(692\) 2.96176e47 4.07192
\(693\) 0 0
\(694\) −8.31128e46 −1.08952
\(695\) −7.28813e46 −0.932968
\(696\) 0 0
\(697\) 7.30848e46 0.892249
\(698\) 9.22827e46 1.10029
\(699\) 0 0
\(700\) −8.66499e46 −0.985485
\(701\) −1.07382e47 −1.19284 −0.596422 0.802671i \(-0.703411\pi\)
−0.596422 + 0.802671i \(0.703411\pi\)
\(702\) 0 0
\(703\) 4.23598e45 0.0448943
\(704\) 5.34024e46 0.552856
\(705\) 0 0
\(706\) 1.67517e47 1.65494
\(707\) 6.63351e46 0.640211
\(708\) 0 0
\(709\) 9.99645e46 0.920837 0.460419 0.887702i \(-0.347699\pi\)
0.460419 + 0.887702i \(0.347699\pi\)
\(710\) −2.61866e46 −0.235677
\(711\) 0 0
\(712\) −1.29776e47 −1.11500
\(713\) −1.62251e46 −0.136211
\(714\) 0 0
\(715\) −3.14166e46 −0.251831
\(716\) −1.53299e47 −1.20081
\(717\) 0 0
\(718\) −2.02449e47 −1.51448
\(719\) 7.93598e46 0.580195 0.290097 0.956997i \(-0.406312\pi\)
0.290097 + 0.956997i \(0.406312\pi\)
\(720\) 0 0
\(721\) −9.31838e46 −0.650741
\(722\) −2.67498e47 −1.82581
\(723\) 0 0
\(724\) 4.97151e46 0.324192
\(725\) −1.14466e47 −0.729626
\(726\) 0 0
\(727\) 1.33485e47 0.813046 0.406523 0.913641i \(-0.366741\pi\)
0.406523 + 0.913641i \(0.366741\pi\)
\(728\) −2.18702e47 −1.30222
\(729\) 0 0
\(730\) −2.09526e47 −1.19237
\(731\) 5.89992e47 3.28254
\(732\) 0 0
\(733\) 8.41792e46 0.447702 0.223851 0.974623i \(-0.428137\pi\)
0.223851 + 0.974623i \(0.428137\pi\)
\(734\) 1.22843e47 0.638802
\(735\) 0 0
\(736\) −4.00184e47 −1.98965
\(737\) 9.11389e46 0.443088
\(738\) 0 0
\(739\) 2.39720e47 1.11448 0.557239 0.830352i \(-0.311861\pi\)
0.557239 + 0.830352i \(0.311861\pi\)
\(740\) −1.26206e47 −0.573793
\(741\) 0 0
\(742\) −8.65106e46 −0.376188
\(743\) −4.51636e47 −1.92076 −0.960380 0.278695i \(-0.910098\pi\)
−0.960380 + 0.278695i \(0.910098\pi\)
\(744\) 0 0
\(745\) −3.49589e46 −0.142226
\(746\) −1.16643e47 −0.464160
\(747\) 0 0
\(748\) 5.68023e47 2.16267
\(749\) −3.89025e46 −0.144886
\(750\) 0 0
\(751\) −2.31769e46 −0.0826030 −0.0413015 0.999147i \(-0.513150\pi\)
−0.0413015 + 0.999147i \(0.513150\pi\)
\(752\) −5.83991e47 −2.03616
\(753\) 0 0
\(754\) −4.92594e47 −1.64385
\(755\) −2.73514e47 −0.893006
\(756\) 0 0
\(757\) −6.14297e46 −0.191998 −0.0959989 0.995381i \(-0.530605\pi\)
−0.0959989 + 0.995381i \(0.530605\pi\)
\(758\) 8.81131e47 2.69463
\(759\) 0 0
\(760\) −5.91044e46 −0.173059
\(761\) 4.18126e47 1.19801 0.599004 0.800746i \(-0.295563\pi\)
0.599004 + 0.800746i \(0.295563\pi\)
\(762\) 0 0
\(763\) 2.43616e47 0.668421
\(764\) 1.45009e46 0.0389362
\(765\) 0 0
\(766\) 1.16952e48 3.00769
\(767\) −2.25179e47 −0.566766
\(768\) 0 0
\(769\) −2.07493e47 −0.500286 −0.250143 0.968209i \(-0.580478\pi\)
−0.250143 + 0.968209i \(0.580478\pi\)
\(770\) 1.55555e47 0.367103
\(771\) 0 0
\(772\) 1.38584e47 0.313348
\(773\) 3.82656e47 0.846930 0.423465 0.905912i \(-0.360814\pi\)
0.423465 + 0.905912i \(0.360814\pi\)
\(774\) 0 0
\(775\) −3.93306e46 −0.0834165
\(776\) 8.43509e47 1.75134
\(777\) 0 0
\(778\) −5.20250e47 −1.03526
\(779\) −3.07522e46 −0.0599112
\(780\) 0 0
\(781\) −6.21081e46 −0.115986
\(782\) −1.78952e48 −3.27209
\(783\) 0 0
\(784\) −8.43619e47 −1.47887
\(785\) −5.60247e47 −0.961679
\(786\) 0 0
\(787\) −6.26036e47 −1.03043 −0.515213 0.857062i \(-0.672287\pi\)
−0.515213 + 0.857062i \(0.672287\pi\)
\(788\) 1.84799e48 2.97863
\(789\) 0 0
\(790\) −1.92626e47 −0.297761
\(791\) −7.50935e47 −1.13682
\(792\) 0 0
\(793\) 1.89053e46 0.0274521
\(794\) 1.46471e48 2.08312
\(795\) 0 0
\(796\) 2.18348e48 2.97910
\(797\) −1.56391e47 −0.209002 −0.104501 0.994525i \(-0.533325\pi\)
−0.104501 + 0.994525i \(0.533325\pi\)
\(798\) 0 0
\(799\) −1.08734e48 −1.39425
\(800\) −9.70066e47 −1.21847
\(801\) 0 0
\(802\) −7.96201e47 −0.959722
\(803\) −4.96943e47 −0.586813
\(804\) 0 0
\(805\) −3.46706e47 −0.392943
\(806\) −1.69255e47 −0.187937
\(807\) 0 0
\(808\) 2.51747e48 2.68335
\(809\) −4.01144e47 −0.418938 −0.209469 0.977815i \(-0.567174\pi\)
−0.209469 + 0.977815i \(0.567174\pi\)
\(810\) 0 0
\(811\) 2.34209e47 0.234833 0.117417 0.993083i \(-0.462539\pi\)
0.117417 + 0.993083i \(0.462539\pi\)
\(812\) 1.72553e48 1.69530
\(813\) 0 0
\(814\) −4.23098e47 −0.399152
\(815\) 5.66962e47 0.524147
\(816\) 0 0
\(817\) −2.48253e47 −0.220410
\(818\) 2.97269e48 2.58655
\(819\) 0 0
\(820\) 9.16221e47 0.765723
\(821\) −4.03504e47 −0.330510 −0.165255 0.986251i \(-0.552845\pi\)
−0.165255 + 0.986251i \(0.552845\pi\)
\(822\) 0 0
\(823\) 1.82505e48 1.43608 0.718039 0.696002i \(-0.245040\pi\)
0.718039 + 0.696002i \(0.245040\pi\)
\(824\) −3.53640e48 −2.72749
\(825\) 0 0
\(826\) 1.11494e48 0.826195
\(827\) −7.52706e47 −0.546744 −0.273372 0.961908i \(-0.588139\pi\)
−0.273372 + 0.961908i \(0.588139\pi\)
\(828\) 0 0
\(829\) −2.19354e48 −1.53107 −0.765537 0.643392i \(-0.777526\pi\)
−0.765537 + 0.643392i \(0.777526\pi\)
\(830\) −1.42857e48 −0.977489
\(831\) 0 0
\(832\) −1.24162e48 −0.816499
\(833\) −1.57074e48 −1.01265
\(834\) 0 0
\(835\) −1.36590e48 −0.846433
\(836\) −2.39009e47 −0.145215
\(837\) 0 0
\(838\) −2.73408e48 −1.59693
\(839\) 2.86394e48 1.64018 0.820091 0.572233i \(-0.193923\pi\)
0.820091 + 0.572233i \(0.193923\pi\)
\(840\) 0 0
\(841\) 4.63381e47 0.255155
\(842\) −3.54978e48 −1.91669
\(843\) 0 0
\(844\) −1.16308e48 −0.603892
\(845\) −4.29368e47 −0.218622
\(846\) 0 0
\(847\) −9.08800e47 −0.445033
\(848\) −1.64561e48 −0.790308
\(849\) 0 0
\(850\) −4.33788e48 −2.00385
\(851\) 9.43013e47 0.427248
\(852\) 0 0
\(853\) 4.28657e48 1.86832 0.934158 0.356859i \(-0.116152\pi\)
0.934158 + 0.356859i \(0.116152\pi\)
\(854\) −9.36073e46 −0.0400180
\(855\) 0 0
\(856\) −1.47638e48 −0.607270
\(857\) 8.64522e47 0.348813 0.174406 0.984674i \(-0.444199\pi\)
0.174406 + 0.984674i \(0.444199\pi\)
\(858\) 0 0
\(859\) 2.35883e48 0.915817 0.457909 0.888999i \(-0.348599\pi\)
0.457909 + 0.888999i \(0.348599\pi\)
\(860\) 7.39639e48 2.81705
\(861\) 0 0
\(862\) 3.68860e48 1.35205
\(863\) 6.27316e46 0.0225584 0.0112792 0.999936i \(-0.496410\pi\)
0.0112792 + 0.999936i \(0.496410\pi\)
\(864\) 0 0
\(865\) 2.87267e48 0.994304
\(866\) −4.49666e48 −1.52702
\(867\) 0 0
\(868\) 5.92890e47 0.193820
\(869\) −4.56861e47 −0.146541
\(870\) 0 0
\(871\) −2.11901e48 −0.654386
\(872\) 9.24541e48 2.80159
\(873\) 0 0
\(874\) 7.52982e47 0.219708
\(875\) −2.13092e48 −0.610145
\(876\) 0 0
\(877\) −2.00783e48 −0.553647 −0.276823 0.960921i \(-0.589282\pi\)
−0.276823 + 0.960921i \(0.589282\pi\)
\(878\) 2.57194e48 0.695988
\(879\) 0 0
\(880\) 2.95899e48 0.771223
\(881\) 4.20296e48 1.07511 0.537555 0.843229i \(-0.319348\pi\)
0.537555 + 0.843229i \(0.319348\pi\)
\(882\) 0 0
\(883\) 7.94997e48 1.95891 0.979453 0.201672i \(-0.0646376\pi\)
0.979453 + 0.201672i \(0.0646376\pi\)
\(884\) −1.32067e49 −3.19399
\(885\) 0 0
\(886\) −1.77675e48 −0.413971
\(887\) 7.43922e48 1.70133 0.850664 0.525710i \(-0.176200\pi\)
0.850664 + 0.525710i \(0.176200\pi\)
\(888\) 0 0
\(889\) −3.91811e48 −0.863371
\(890\) −2.14614e48 −0.464219
\(891\) 0 0
\(892\) 9.67400e48 2.01644
\(893\) 4.57523e47 0.0936189
\(894\) 0 0
\(895\) −1.48688e48 −0.293222
\(896\) 1.01136e47 0.0195804
\(897\) 0 0
\(898\) 5.22659e47 0.0975345
\(899\) 7.83220e47 0.143499
\(900\) 0 0
\(901\) −3.06398e48 −0.541161
\(902\) 3.07159e48 0.532666
\(903\) 0 0
\(904\) −2.84986e49 −4.76480
\(905\) 4.82197e47 0.0791631
\(906\) 0 0
\(907\) 8.96944e48 1.41986 0.709929 0.704273i \(-0.248727\pi\)
0.709929 + 0.704273i \(0.248727\pi\)
\(908\) 5.43871e48 0.845433
\(909\) 0 0
\(910\) −3.61672e48 −0.542166
\(911\) −2.25421e48 −0.331850 −0.165925 0.986138i \(-0.553061\pi\)
−0.165925 + 0.986138i \(0.553061\pi\)
\(912\) 0 0
\(913\) −3.38820e48 −0.481063
\(914\) −1.39679e49 −1.94769
\(915\) 0 0
\(916\) 1.91322e49 2.57329
\(917\) 1.62415e48 0.214551
\(918\) 0 0
\(919\) −7.04277e48 −0.897507 −0.448754 0.893656i \(-0.648132\pi\)
−0.448754 + 0.893656i \(0.648132\pi\)
\(920\) −1.31578e49 −1.64696
\(921\) 0 0
\(922\) −1.62673e49 −1.96452
\(923\) 1.44404e48 0.171297
\(924\) 0 0
\(925\) 2.28591e48 0.261650
\(926\) −1.17335e49 −1.31931
\(927\) 0 0
\(928\) 1.93177e49 2.09611
\(929\) −1.56732e49 −1.67070 −0.835348 0.549721i \(-0.814734\pi\)
−0.835348 + 0.549721i \(0.814734\pi\)
\(930\) 0 0
\(931\) 6.60926e47 0.0679959
\(932\) −1.48276e48 −0.149868
\(933\) 0 0
\(934\) −1.95123e49 −1.90364
\(935\) 5.50937e48 0.528093
\(936\) 0 0
\(937\) 4.17295e48 0.386136 0.193068 0.981185i \(-0.438156\pi\)
0.193068 + 0.981185i \(0.438156\pi\)
\(938\) 1.04920e49 0.953922
\(939\) 0 0
\(940\) −1.36313e49 −1.19654
\(941\) 5.21719e48 0.449995 0.224998 0.974359i \(-0.427763\pi\)
0.224998 + 0.974359i \(0.427763\pi\)
\(942\) 0 0
\(943\) −6.84604e48 −0.570160
\(944\) 2.12086e49 1.73570
\(945\) 0 0
\(946\) 2.47960e49 1.95965
\(947\) 1.05918e49 0.822616 0.411308 0.911496i \(-0.365072\pi\)
0.411308 + 0.911496i \(0.365072\pi\)
\(948\) 0 0
\(949\) 1.15541e49 0.866650
\(950\) 1.82527e48 0.134551
\(951\) 0 0
\(952\) 3.83527e49 2.73078
\(953\) −1.49599e49 −1.04688 −0.523438 0.852064i \(-0.675351\pi\)
−0.523438 + 0.852064i \(0.675351\pi\)
\(954\) 0 0
\(955\) 1.40647e47 0.00950767
\(956\) 3.80536e49 2.52837
\(957\) 0 0
\(958\) 4.17619e49 2.68071
\(959\) −4.90095e48 −0.309224
\(960\) 0 0
\(961\) −1.61344e49 −0.983594
\(962\) 9.83718e48 0.589498
\(963\) 0 0
\(964\) −6.34261e49 −3.67280
\(965\) 1.34415e48 0.0765152
\(966\) 0 0
\(967\) −2.39619e49 −1.31821 −0.659106 0.752050i \(-0.729065\pi\)
−0.659106 + 0.752050i \(0.729065\pi\)
\(968\) −3.44897e49 −1.86529
\(969\) 0 0
\(970\) 1.39493e49 0.729151
\(971\) 1.35152e47 0.00694552 0.00347276 0.999994i \(-0.498895\pi\)
0.00347276 + 0.999994i \(0.498895\pi\)
\(972\) 0 0
\(973\) −1.98994e49 −0.988505
\(974\) −2.66999e49 −1.30403
\(975\) 0 0
\(976\) −1.78061e48 −0.0840711
\(977\) −1.84460e48 −0.0856333 −0.0428166 0.999083i \(-0.513633\pi\)
−0.0428166 + 0.999083i \(0.513633\pi\)
\(978\) 0 0
\(979\) −5.09011e48 −0.228461
\(980\) −1.96914e49 −0.869055
\(981\) 0 0
\(982\) 4.53233e48 0.193411
\(983\) 6.69809e48 0.281072 0.140536 0.990076i \(-0.455117\pi\)
0.140536 + 0.990076i \(0.455117\pi\)
\(984\) 0 0
\(985\) 1.79240e49 0.727340
\(986\) 8.63836e49 3.44717
\(987\) 0 0
\(988\) 5.55705e48 0.214464
\(989\) −5.52661e49 −2.09759
\(990\) 0 0
\(991\) 1.17168e49 0.430123 0.215062 0.976600i \(-0.431005\pi\)
0.215062 + 0.976600i \(0.431005\pi\)
\(992\) 6.63755e48 0.239643
\(993\) 0 0
\(994\) −7.14997e48 −0.249706
\(995\) 2.11780e49 0.727453
\(996\) 0 0
\(997\) 2.44007e49 0.810834 0.405417 0.914132i \(-0.367126\pi\)
0.405417 + 0.914132i \(0.367126\pi\)
\(998\) 5.68168e49 1.85705
\(999\) 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 9.34.a.b.1.2 2
3.2 odd 2 1.34.a.a.1.1 2
12.11 even 2 16.34.a.b.1.1 2
15.2 even 4 25.34.b.a.24.1 4
15.8 even 4 25.34.b.a.24.4 4
15.14 odd 2 25.34.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.34.a.a.1.1 2 3.2 odd 2
9.34.a.b.1.2 2 1.1 even 1 trivial
16.34.a.b.1.1 2 12.11 even 2
25.34.a.a.1.2 2 15.14 odd 2
25.34.b.a.24.1 4 15.2 even 4
25.34.b.a.24.4 4 15.8 even 4