Properties

Label 25.10.a.c.1.1
Level $25$
Weight $10$
Character 25.1
Self dual yes
Analytic conductor $12.876$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.8758959041\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 652 x + 4000\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-27.7229\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-expansion

\(f(q)\) \(=\) \(q-31.7828 q^{2} -268.664 q^{3} +498.143 q^{4} +8538.88 q^{6} +637.237 q^{7} +440.406 q^{8} +52497.3 q^{9} +O(q^{10})\) \(q-31.7828 q^{2} -268.664 q^{3} +498.143 q^{4} +8538.88 q^{6} +637.237 q^{7} +440.406 q^{8} +52497.3 q^{9} -49042.6 q^{11} -133833. q^{12} +72726.2 q^{13} -20253.2 q^{14} -269047. q^{16} +67319.3 q^{17} -1.66851e6 q^{18} +341136. q^{19} -171203. q^{21} +1.55871e6 q^{22} -134355. q^{23} -118321. q^{24} -2.31144e6 q^{26} -8.81603e6 q^{27} +317435. q^{28} +4.45784e6 q^{29} +456520. q^{31} +8.32555e6 q^{32} +1.31760e7 q^{33} -2.13959e6 q^{34} +2.61512e7 q^{36} +1.30516e7 q^{37} -1.08422e7 q^{38} -1.95389e7 q^{39} -2.56667e7 q^{41} +5.44129e6 q^{42} +3.42710e6 q^{43} -2.44302e7 q^{44} +4.27016e6 q^{46} -3.39814e7 q^{47} +7.22831e7 q^{48} -3.99475e7 q^{49} -1.80863e7 q^{51} +3.62281e7 q^{52} -8.42456e7 q^{53} +2.80198e8 q^{54} +280643. q^{56} -9.16509e7 q^{57} -1.41682e8 q^{58} -7.46358e7 q^{59} +1.78017e8 q^{61} -1.45094e7 q^{62} +3.34533e7 q^{63} -1.26857e8 q^{64} -4.18769e8 q^{66} +6.94299e7 q^{67} +3.35347e7 q^{68} +3.60963e7 q^{69} -2.07860e8 q^{71} +2.31201e7 q^{72} -3.02516e8 q^{73} -4.14815e8 q^{74} +1.69935e8 q^{76} -3.12518e7 q^{77} +6.21000e8 q^{78} +3.72244e8 q^{79} +1.33524e9 q^{81} +8.15758e8 q^{82} +4.50079e8 q^{83} -8.52835e7 q^{84} -1.08923e8 q^{86} -1.19766e9 q^{87} -2.15987e7 q^{88} +5.82741e7 q^{89} +4.63438e7 q^{91} -6.69279e7 q^{92} -1.22650e8 q^{93} +1.08002e9 q^{94} -2.23678e9 q^{96} +7.85850e8 q^{97} +1.26964e9 q^{98} -2.57461e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 33q^{2} - 89q^{3} + 341q^{4} + 3421q^{6} - 5258q^{7} + 105q^{8} + 58234q^{9} + O(q^{10}) \) \( 3q - 33q^{2} - 89q^{3} + 341q^{4} + 3421q^{6} - 5258q^{7} + 105q^{8} + 58234q^{9} - 54699q^{11} - 141853q^{12} - 215884q^{13} + 272922q^{14} - 699247q^{16} - 334983q^{17} - 2571854q^{18} + 818845q^{19} - 2375394q^{21} - 72761q^{22} - 3526854q^{23} + 2805735q^{24} - 1280004q^{26} - 6633395q^{27} + 428134q^{28} + 2175480q^{29} + 4274066q^{31} + 9464577q^{32} + 22122137q^{33} - 6838963q^{34} + 26795798q^{36} + 10305042q^{37} + 24180495q^{38} - 50414092q^{39} + 5926311q^{41} + 47560254q^{42} + 24429956q^{43} - 21995703q^{44} + 14223246q^{46} - 66858708q^{47} + 29557871q^{48} - 6453929q^{49} - 25634699q^{51} + 57862932q^{52} - 132620514q^{53} + 282250495q^{54} - 169538130q^{56} - 252946415q^{57} - 201908320q^{58} + 5670960q^{59} + 125306926q^{61} + 39831174q^{62} - 284323404q^{63} + 167542401q^{64} - 556915843q^{66} - 88829483q^{67} + 71162559q^{68} - 314274942q^{69} + 297550596q^{71} + 552918990q^{72} - 181321729q^{73} - 251507358q^{74} + 89414865q^{76} - 561214086q^{77} + 1439424692q^{78} - 310025170q^{79} + 1398847363q^{81} + 1368322979q^{82} + 731088801q^{83} + 38267082q^{84} + 33947196q^{86} - 1046385560q^{87} + 943671285q^{88} - 1103860035q^{89} + 1183187656q^{91} + 190024242q^{92} - 107386758q^{93} + 1727891132q^{94} - 2567180449q^{96} + 332236842q^{97} - 457927581q^{98} - 892234522q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.7828 −1.40461 −0.702306 0.711875i \(-0.747846\pi\)
−0.702306 + 0.711875i \(0.747846\pi\)
\(3\) −268.664 −1.91498 −0.957489 0.288470i \(-0.906854\pi\)
−0.957489 + 0.288470i \(0.906854\pi\)
\(4\) 498.143 0.972936
\(5\) 0 0
\(6\) 8538.88 2.68980
\(7\) 637.237 0.100314 0.0501568 0.998741i \(-0.484028\pi\)
0.0501568 + 0.998741i \(0.484028\pi\)
\(8\) 440.406 0.0380144
\(9\) 52497.3 2.66714
\(10\) 0 0
\(11\) −49042.6 −1.00996 −0.504982 0.863130i \(-0.668501\pi\)
−0.504982 + 0.863130i \(0.668501\pi\)
\(12\) −133833. −1.86315
\(13\) 72726.2 0.706229 0.353115 0.935580i \(-0.385123\pi\)
0.353115 + 0.935580i \(0.385123\pi\)
\(14\) −20253.2 −0.140902
\(15\) 0 0
\(16\) −269047. −1.02633
\(17\) 67319.3 0.195488 0.0977439 0.995212i \(-0.468837\pi\)
0.0977439 + 0.995212i \(0.468837\pi\)
\(18\) −1.66851e6 −3.74630
\(19\) 341136. 0.600532 0.300266 0.953855i \(-0.402925\pi\)
0.300266 + 0.953855i \(0.402925\pi\)
\(20\) 0 0
\(21\) −171203. −0.192098
\(22\) 1.55871e6 1.41861
\(23\) −134355. −0.100110 −0.0500550 0.998746i \(-0.515940\pi\)
−0.0500550 + 0.998746i \(0.515940\pi\)
\(24\) −118321. −0.0727968
\(25\) 0 0
\(26\) −2.31144e6 −0.991978
\(27\) −8.81603e6 −3.19254
\(28\) 317435. 0.0975988
\(29\) 4.45784e6 1.17040 0.585199 0.810890i \(-0.301016\pi\)
0.585199 + 0.810890i \(0.301016\pi\)
\(30\) 0 0
\(31\) 456520. 0.0887834 0.0443917 0.999014i \(-0.485865\pi\)
0.0443917 + 0.999014i \(0.485865\pi\)
\(32\) 8.32555e6 1.40358
\(33\) 1.31760e7 1.93406
\(34\) −2.13959e6 −0.274585
\(35\) 0 0
\(36\) 2.61512e7 2.59496
\(37\) 1.30516e7 1.14487 0.572433 0.819951i \(-0.305999\pi\)
0.572433 + 0.819951i \(0.305999\pi\)
\(38\) −1.08422e7 −0.843515
\(39\) −1.95389e7 −1.35241
\(40\) 0 0
\(41\) −2.56667e7 −1.41854 −0.709272 0.704935i \(-0.750976\pi\)
−0.709272 + 0.704935i \(0.750976\pi\)
\(42\) 5.44129e6 0.269824
\(43\) 3.42710e6 0.152869 0.0764344 0.997075i \(-0.475646\pi\)
0.0764344 + 0.997075i \(0.475646\pi\)
\(44\) −2.44302e7 −0.982631
\(45\) 0 0
\(46\) 4.27016e6 0.140616
\(47\) −3.39814e7 −1.01578 −0.507891 0.861421i \(-0.669575\pi\)
−0.507891 + 0.861421i \(0.669575\pi\)
\(48\) 7.22831e7 1.96540
\(49\) −3.99475e7 −0.989937
\(50\) 0 0
\(51\) −1.80863e7 −0.374355
\(52\) 3.62281e7 0.687116
\(53\) −8.42456e7 −1.46658 −0.733290 0.679916i \(-0.762016\pi\)
−0.733290 + 0.679916i \(0.762016\pi\)
\(54\) 2.80198e8 4.48428
\(55\) 0 0
\(56\) 280643. 0.00381337
\(57\) −9.16509e7 −1.15001
\(58\) −1.41682e8 −1.64396
\(59\) −7.46358e7 −0.801887 −0.400944 0.916103i \(-0.631318\pi\)
−0.400944 + 0.916103i \(0.631318\pi\)
\(60\) 0 0
\(61\) 1.78017e8 1.64618 0.823088 0.567913i \(-0.192249\pi\)
0.823088 + 0.567913i \(0.192249\pi\)
\(62\) −1.45094e7 −0.124706
\(63\) 3.34533e7 0.267551
\(64\) −1.26857e8 −0.945159
\(65\) 0 0
\(66\) −4.18769e8 −2.71661
\(67\) 6.94299e7 0.420930 0.210465 0.977601i \(-0.432502\pi\)
0.210465 + 0.977601i \(0.432502\pi\)
\(68\) 3.35347e7 0.190197
\(69\) 3.60963e7 0.191708
\(70\) 0 0
\(71\) −2.07860e8 −0.970753 −0.485376 0.874305i \(-0.661317\pi\)
−0.485376 + 0.874305i \(0.661317\pi\)
\(72\) 2.31201e7 0.101390
\(73\) −3.02516e8 −1.24680 −0.623398 0.781905i \(-0.714248\pi\)
−0.623398 + 0.781905i \(0.714248\pi\)
\(74\) −4.14815e8 −1.60809
\(75\) 0 0
\(76\) 1.69935e8 0.584279
\(77\) −3.12518e7 −0.101313
\(78\) 6.21000e8 1.89962
\(79\) 3.72244e8 1.07524 0.537621 0.843187i \(-0.319323\pi\)
0.537621 + 0.843187i \(0.319323\pi\)
\(80\) 0 0
\(81\) 1.33524e9 3.44650
\(82\) 8.15758e8 1.99250
\(83\) 4.50079e8 1.04097 0.520484 0.853872i \(-0.325752\pi\)
0.520484 + 0.853872i \(0.325752\pi\)
\(84\) −8.52835e7 −0.186899
\(85\) 0 0
\(86\) −1.08923e8 −0.214721
\(87\) −1.19766e9 −2.24129
\(88\) −2.15987e7 −0.0383932
\(89\) 5.82741e7 0.0984511 0.0492255 0.998788i \(-0.484325\pi\)
0.0492255 + 0.998788i \(0.484325\pi\)
\(90\) 0 0
\(91\) 4.63438e7 0.0708444
\(92\) −6.69279e7 −0.0974006
\(93\) −1.22650e8 −0.170018
\(94\) 1.08002e9 1.42678
\(95\) 0 0
\(96\) −2.23678e9 −2.68783
\(97\) 7.85850e8 0.901295 0.450647 0.892702i \(-0.351193\pi\)
0.450647 + 0.892702i \(0.351193\pi\)
\(98\) 1.26964e9 1.39048
\(99\) −2.57461e9 −2.69372
\(100\) 0 0
\(101\) −1.60460e8 −0.153433 −0.0767167 0.997053i \(-0.524444\pi\)
−0.0767167 + 0.997053i \(0.524444\pi\)
\(102\) 5.74832e8 0.525823
\(103\) −1.39454e9 −1.22085 −0.610425 0.792074i \(-0.709001\pi\)
−0.610425 + 0.792074i \(0.709001\pi\)
\(104\) 3.20291e7 0.0268469
\(105\) 0 0
\(106\) 2.67756e9 2.05998
\(107\) −1.56463e9 −1.15394 −0.576972 0.816764i \(-0.695766\pi\)
−0.576972 + 0.816764i \(0.695766\pi\)
\(108\) −4.39165e9 −3.10614
\(109\) 1.24703e9 0.846172 0.423086 0.906089i \(-0.360947\pi\)
0.423086 + 0.906089i \(0.360947\pi\)
\(110\) 0 0
\(111\) −3.50649e9 −2.19240
\(112\) −1.71447e8 −0.102955
\(113\) −1.81056e9 −1.04463 −0.522313 0.852754i \(-0.674931\pi\)
−0.522313 + 0.852754i \(0.674931\pi\)
\(114\) 2.91292e9 1.61531
\(115\) 0 0
\(116\) 2.22064e9 1.13872
\(117\) 3.81793e9 1.88361
\(118\) 2.37213e9 1.12634
\(119\) 4.28984e7 0.0196101
\(120\) 0 0
\(121\) 4.72275e7 0.0200291
\(122\) −5.65786e9 −2.31224
\(123\) 6.89572e9 2.71648
\(124\) 2.27412e8 0.0863806
\(125\) 0 0
\(126\) −1.06324e9 −0.375805
\(127\) −3.06491e9 −1.04545 −0.522723 0.852503i \(-0.675084\pi\)
−0.522723 + 0.852503i \(0.675084\pi\)
\(128\) −2.30815e8 −0.0760010
\(129\) −9.20739e8 −0.292740
\(130\) 0 0
\(131\) −1.83508e9 −0.544421 −0.272211 0.962238i \(-0.587755\pi\)
−0.272211 + 0.962238i \(0.587755\pi\)
\(132\) 6.56352e9 1.88172
\(133\) 2.17385e8 0.0602416
\(134\) −2.20667e9 −0.591243
\(135\) 0 0
\(136\) 2.96478e7 0.00743136
\(137\) −4.62426e9 −1.12150 −0.560751 0.827984i \(-0.689488\pi\)
−0.560751 + 0.827984i \(0.689488\pi\)
\(138\) −1.14724e9 −0.269276
\(139\) 2.57140e8 0.0584255 0.0292128 0.999573i \(-0.490700\pi\)
0.0292128 + 0.999573i \(0.490700\pi\)
\(140\) 0 0
\(141\) 9.12958e9 1.94520
\(142\) 6.60637e9 1.36353
\(143\) −3.56668e9 −0.713267
\(144\) −1.41242e10 −2.73737
\(145\) 0 0
\(146\) 9.61479e9 1.75127
\(147\) 1.07325e10 1.89571
\(148\) 6.50155e9 1.11388
\(149\) 3.14809e9 0.523250 0.261625 0.965170i \(-0.415742\pi\)
0.261625 + 0.965170i \(0.415742\pi\)
\(150\) 0 0
\(151\) −1.02772e10 −1.60871 −0.804356 0.594147i \(-0.797490\pi\)
−0.804356 + 0.594147i \(0.797490\pi\)
\(152\) 1.50238e8 0.0228289
\(153\) 3.53408e9 0.521393
\(154\) 9.93267e8 0.142306
\(155\) 0 0
\(156\) −9.73317e9 −1.31581
\(157\) 3.25077e9 0.427009 0.213505 0.976942i \(-0.431512\pi\)
0.213505 + 0.976942i \(0.431512\pi\)
\(158\) −1.18309e10 −1.51030
\(159\) 2.26338e10 2.80847
\(160\) 0 0
\(161\) −8.56158e7 −0.0100424
\(162\) −4.24377e10 −4.84100
\(163\) 3.40049e9 0.377309 0.188655 0.982043i \(-0.439587\pi\)
0.188655 + 0.982043i \(0.439587\pi\)
\(164\) −1.27857e10 −1.38015
\(165\) 0 0
\(166\) −1.43047e10 −1.46216
\(167\) 5.27785e9 0.525089 0.262544 0.964920i \(-0.415438\pi\)
0.262544 + 0.964920i \(0.415438\pi\)
\(168\) −7.53987e7 −0.00730251
\(169\) −5.31540e9 −0.501240
\(170\) 0 0
\(171\) 1.79087e10 1.60170
\(172\) 1.70719e9 0.148732
\(173\) 6.58861e9 0.559224 0.279612 0.960113i \(-0.409794\pi\)
0.279612 + 0.960113i \(0.409794\pi\)
\(174\) 3.80650e10 3.14814
\(175\) 0 0
\(176\) 1.31947e10 1.03656
\(177\) 2.00520e10 1.53560
\(178\) −1.85211e9 −0.138286
\(179\) −1.23663e10 −0.900326 −0.450163 0.892946i \(-0.648634\pi\)
−0.450163 + 0.892946i \(0.648634\pi\)
\(180\) 0 0
\(181\) −2.59914e10 −1.80002 −0.900009 0.435872i \(-0.856440\pi\)
−0.900009 + 0.435872i \(0.856440\pi\)
\(182\) −1.47293e9 −0.0995090
\(183\) −4.78267e10 −3.15239
\(184\) −5.91706e7 −0.00380562
\(185\) 0 0
\(186\) 3.89817e9 0.238810
\(187\) −3.30151e9 −0.197436
\(188\) −1.69276e10 −0.988292
\(189\) −5.61790e9 −0.320255
\(190\) 0 0
\(191\) 1.50506e10 0.818284 0.409142 0.912471i \(-0.365828\pi\)
0.409142 + 0.912471i \(0.365828\pi\)
\(192\) 3.40819e10 1.80996
\(193\) 1.40329e10 0.728014 0.364007 0.931396i \(-0.381408\pi\)
0.364007 + 0.931396i \(0.381408\pi\)
\(194\) −2.49765e10 −1.26597
\(195\) 0 0
\(196\) −1.98996e10 −0.963146
\(197\) −2.57285e10 −1.21707 −0.608536 0.793526i \(-0.708243\pi\)
−0.608536 + 0.793526i \(0.708243\pi\)
\(198\) 8.18280e10 3.78363
\(199\) −2.94367e10 −1.33061 −0.665303 0.746573i \(-0.731698\pi\)
−0.665303 + 0.746573i \(0.731698\pi\)
\(200\) 0 0
\(201\) −1.86533e10 −0.806072
\(202\) 5.09985e9 0.215515
\(203\) 2.84070e9 0.117407
\(204\) −9.00956e9 −0.364223
\(205\) 0 0
\(206\) 4.43222e10 1.71482
\(207\) −7.05326e9 −0.267007
\(208\) −1.95667e10 −0.724825
\(209\) −1.67302e10 −0.606516
\(210\) 0 0
\(211\) 1.17275e10 0.407319 0.203659 0.979042i \(-0.434716\pi\)
0.203659 + 0.979042i \(0.434716\pi\)
\(212\) −4.19664e10 −1.42689
\(213\) 5.58445e10 1.85897
\(214\) 4.97283e10 1.62084
\(215\) 0 0
\(216\) −3.88263e9 −0.121363
\(217\) 2.90911e8 0.00890619
\(218\) −3.96341e10 −1.18854
\(219\) 8.12751e10 2.38759
\(220\) 0 0
\(221\) 4.89588e9 0.138059
\(222\) 1.11446e11 3.07947
\(223\) 2.58340e10 0.699550 0.349775 0.936834i \(-0.386258\pi\)
0.349775 + 0.936834i \(0.386258\pi\)
\(224\) 5.30535e9 0.140799
\(225\) 0 0
\(226\) 5.75447e10 1.46729
\(227\) −2.50896e10 −0.627159 −0.313580 0.949562i \(-0.601528\pi\)
−0.313580 + 0.949562i \(0.601528\pi\)
\(228\) −4.56553e10 −1.11888
\(229\) −2.30463e10 −0.553785 −0.276892 0.960901i \(-0.589305\pi\)
−0.276892 + 0.960901i \(0.589305\pi\)
\(230\) 0 0
\(231\) 8.39622e9 0.194013
\(232\) 1.96326e9 0.0444920
\(233\) 3.17197e10 0.705062 0.352531 0.935800i \(-0.385321\pi\)
0.352531 + 0.935800i \(0.385321\pi\)
\(234\) −1.21344e11 −2.64575
\(235\) 0 0
\(236\) −3.71793e10 −0.780185
\(237\) −1.00009e11 −2.05906
\(238\) −1.36343e9 −0.0275446
\(239\) 2.23794e10 0.443668 0.221834 0.975084i \(-0.428796\pi\)
0.221834 + 0.975084i \(0.428796\pi\)
\(240\) 0 0
\(241\) −2.10823e10 −0.402569 −0.201285 0.979533i \(-0.564512\pi\)
−0.201285 + 0.979533i \(0.564512\pi\)
\(242\) −1.50102e9 −0.0281331
\(243\) −1.85206e11 −3.40743
\(244\) 8.86778e10 1.60162
\(245\) 0 0
\(246\) −2.19165e11 −3.81560
\(247\) 2.48095e10 0.424113
\(248\) 2.01054e8 0.00337505
\(249\) −1.20920e11 −1.99343
\(250\) 0 0
\(251\) −7.96509e10 −1.26666 −0.633329 0.773883i \(-0.718312\pi\)
−0.633329 + 0.773883i \(0.718312\pi\)
\(252\) 1.66645e10 0.260310
\(253\) 6.58910e9 0.101108
\(254\) 9.74114e10 1.46845
\(255\) 0 0
\(256\) 7.22868e10 1.05191
\(257\) −3.30815e10 −0.473027 −0.236514 0.971628i \(-0.576005\pi\)
−0.236514 + 0.971628i \(0.576005\pi\)
\(258\) 2.92636e10 0.411187
\(259\) 8.31695e9 0.114846
\(260\) 0 0
\(261\) 2.34025e11 3.12162
\(262\) 5.83240e10 0.764701
\(263\) 6.05028e10 0.779784 0.389892 0.920861i \(-0.372512\pi\)
0.389892 + 0.920861i \(0.372512\pi\)
\(264\) 5.80278e9 0.0735222
\(265\) 0 0
\(266\) −6.90908e9 −0.0846161
\(267\) −1.56562e10 −0.188532
\(268\) 3.45860e10 0.409538
\(269\) 2.61933e10 0.305004 0.152502 0.988303i \(-0.451267\pi\)
0.152502 + 0.988303i \(0.451267\pi\)
\(270\) 0 0
\(271\) 5.42857e10 0.611398 0.305699 0.952128i \(-0.401110\pi\)
0.305699 + 0.952128i \(0.401110\pi\)
\(272\) −1.81120e10 −0.200635
\(273\) −1.24509e10 −0.135666
\(274\) 1.46972e11 1.57528
\(275\) 0 0
\(276\) 1.79811e10 0.186520
\(277\) 1.76561e11 1.80192 0.900961 0.433900i \(-0.142863\pi\)
0.900961 + 0.433900i \(0.142863\pi\)
\(278\) −8.17261e9 −0.0820652
\(279\) 2.39661e10 0.236798
\(280\) 0 0
\(281\) −7.91126e9 −0.0756950 −0.0378475 0.999284i \(-0.512050\pi\)
−0.0378475 + 0.999284i \(0.512050\pi\)
\(282\) −2.90163e11 −2.73225
\(283\) −1.34806e11 −1.24931 −0.624657 0.780899i \(-0.714761\pi\)
−0.624657 + 0.780899i \(0.714761\pi\)
\(284\) −1.03544e11 −0.944480
\(285\) 0 0
\(286\) 1.13359e11 1.00186
\(287\) −1.63558e10 −0.142299
\(288\) 4.37069e11 3.74356
\(289\) −1.14056e11 −0.961785
\(290\) 0 0
\(291\) −2.11130e11 −1.72596
\(292\) −1.50696e11 −1.21305
\(293\) 1.02784e11 0.814741 0.407370 0.913263i \(-0.366446\pi\)
0.407370 + 0.913263i \(0.366446\pi\)
\(294\) −3.41107e11 −2.66273
\(295\) 0 0
\(296\) 5.74799e9 0.0435215
\(297\) 4.32361e11 3.22435
\(298\) −1.00055e11 −0.734964
\(299\) −9.77110e9 −0.0707006
\(300\) 0 0
\(301\) 2.18388e9 0.0153348
\(302\) 3.26638e11 2.25962
\(303\) 4.31098e10 0.293822
\(304\) −9.17815e10 −0.616345
\(305\) 0 0
\(306\) −1.12323e11 −0.732356
\(307\) −2.61472e11 −1.67998 −0.839988 0.542606i \(-0.817438\pi\)
−0.839988 + 0.542606i \(0.817438\pi\)
\(308\) −1.55679e10 −0.0985713
\(309\) 3.74662e11 2.33790
\(310\) 0 0
\(311\) −8.24828e10 −0.499967 −0.249984 0.968250i \(-0.580425\pi\)
−0.249984 + 0.968250i \(0.580425\pi\)
\(312\) −8.60505e9 −0.0514112
\(313\) −1.35766e11 −0.799540 −0.399770 0.916615i \(-0.630910\pi\)
−0.399770 + 0.916615i \(0.630910\pi\)
\(314\) −1.03318e11 −0.599783
\(315\) 0 0
\(316\) 1.85431e11 1.04614
\(317\) −9.06853e10 −0.504394 −0.252197 0.967676i \(-0.581153\pi\)
−0.252197 + 0.967676i \(0.581153\pi\)
\(318\) −7.19363e11 −3.94481
\(319\) −2.18624e11 −1.18206
\(320\) 0 0
\(321\) 4.20360e11 2.20978
\(322\) 2.72111e9 0.0141057
\(323\) 2.29650e10 0.117397
\(324\) 6.65143e11 3.35322
\(325\) 0 0
\(326\) −1.08077e11 −0.529973
\(327\) −3.35033e11 −1.62040
\(328\) −1.13038e10 −0.0539251
\(329\) −2.16542e10 −0.101897
\(330\) 0 0
\(331\) −3.45169e11 −1.58054 −0.790271 0.612758i \(-0.790060\pi\)
−0.790271 + 0.612758i \(0.790060\pi\)
\(332\) 2.24204e11 1.01279
\(333\) 6.85173e11 3.05352
\(334\) −1.67745e11 −0.737546
\(335\) 0 0
\(336\) 4.60615e10 0.197157
\(337\) 8.39418e10 0.354523 0.177261 0.984164i \(-0.443276\pi\)
0.177261 + 0.984164i \(0.443276\pi\)
\(338\) 1.68938e11 0.704048
\(339\) 4.86433e11 2.00043
\(340\) 0 0
\(341\) −2.23889e10 −0.0896681
\(342\) −5.69189e11 −2.24977
\(343\) −5.11709e10 −0.199618
\(344\) 1.50932e9 0.00581122
\(345\) 0 0
\(346\) −2.09404e11 −0.785493
\(347\) 1.56153e11 0.578188 0.289094 0.957301i \(-0.406646\pi\)
0.289094 + 0.957301i \(0.406646\pi\)
\(348\) −5.96607e11 −2.18063
\(349\) 6.31862e9 0.0227986 0.0113993 0.999935i \(-0.496371\pi\)
0.0113993 + 0.999935i \(0.496371\pi\)
\(350\) 0 0
\(351\) −6.41156e11 −2.25466
\(352\) −4.08307e11 −1.41757
\(353\) −2.83875e11 −0.973064 −0.486532 0.873663i \(-0.661738\pi\)
−0.486532 + 0.873663i \(0.661738\pi\)
\(354\) −6.37307e11 −2.15692
\(355\) 0 0
\(356\) 2.90288e10 0.0957866
\(357\) −1.15253e10 −0.0375529
\(358\) 3.93034e11 1.26461
\(359\) 6.00733e11 1.90878 0.954391 0.298558i \(-0.0965058\pi\)
0.954391 + 0.298558i \(0.0965058\pi\)
\(360\) 0 0
\(361\) −2.06314e11 −0.639361
\(362\) 8.26079e11 2.52833
\(363\) −1.26883e10 −0.0383552
\(364\) 2.30859e10 0.0689271
\(365\) 0 0
\(366\) 1.52006e12 4.42789
\(367\) 4.97260e11 1.43082 0.715412 0.698702i \(-0.246239\pi\)
0.715412 + 0.698702i \(0.246239\pi\)
\(368\) 3.61477e10 0.102746
\(369\) −1.34743e12 −3.78346
\(370\) 0 0
\(371\) −5.36845e10 −0.147118
\(372\) −6.10974e10 −0.165417
\(373\) 1.39183e11 0.372303 0.186151 0.982521i \(-0.440398\pi\)
0.186151 + 0.982521i \(0.440398\pi\)
\(374\) 1.04931e11 0.277321
\(375\) 0 0
\(376\) −1.49656e10 −0.0386144
\(377\) 3.24202e11 0.826569
\(378\) 1.78552e11 0.449834
\(379\) −5.65247e11 −1.40722 −0.703610 0.710587i \(-0.748430\pi\)
−0.703610 + 0.710587i \(0.748430\pi\)
\(380\) 0 0
\(381\) 8.23432e11 2.00201
\(382\) −4.78350e11 −1.14937
\(383\) −5.38864e11 −1.27963 −0.639816 0.768528i \(-0.720989\pi\)
−0.639816 + 0.768528i \(0.720989\pi\)
\(384\) 6.20117e10 0.145540
\(385\) 0 0
\(386\) −4.46004e11 −1.02258
\(387\) 1.79914e11 0.407723
\(388\) 3.91466e11 0.876902
\(389\) 7.67552e11 1.69955 0.849776 0.527144i \(-0.176737\pi\)
0.849776 + 0.527144i \(0.176737\pi\)
\(390\) 0 0
\(391\) −9.04466e9 −0.0195703
\(392\) −1.75931e10 −0.0376319
\(393\) 4.93021e11 1.04255
\(394\) 8.17722e11 1.70951
\(395\) 0 0
\(396\) −1.28252e12 −2.62082
\(397\) −6.25258e11 −1.26329 −0.631643 0.775259i \(-0.717619\pi\)
−0.631643 + 0.775259i \(0.717619\pi\)
\(398\) 9.35578e11 1.86899
\(399\) −5.84034e10 −0.115361
\(400\) 0 0
\(401\) −1.61329e11 −0.311574 −0.155787 0.987791i \(-0.549791\pi\)
−0.155787 + 0.987791i \(0.549791\pi\)
\(402\) 5.92853e11 1.13222
\(403\) 3.32009e10 0.0627014
\(404\) −7.99320e10 −0.149281
\(405\) 0 0
\(406\) −9.02853e10 −0.164911
\(407\) −6.40083e11 −1.15628
\(408\) −7.96531e9 −0.0142309
\(409\) −7.24004e11 −1.27934 −0.639670 0.768649i \(-0.720929\pi\)
−0.639670 + 0.768649i \(0.720929\pi\)
\(410\) 0 0
\(411\) 1.24237e12 2.14765
\(412\) −6.94679e11 −1.18781
\(413\) −4.75607e10 −0.0804403
\(414\) 2.24172e11 0.375042
\(415\) 0 0
\(416\) 6.05486e11 0.991252
\(417\) −6.90842e10 −0.111884
\(418\) 5.31731e11 0.851920
\(419\) 2.80040e11 0.443871 0.221936 0.975061i \(-0.428763\pi\)
0.221936 + 0.975061i \(0.428763\pi\)
\(420\) 0 0
\(421\) −6.55915e10 −0.101760 −0.0508801 0.998705i \(-0.516203\pi\)
−0.0508801 + 0.998705i \(0.516203\pi\)
\(422\) −3.72732e11 −0.572125
\(423\) −1.78393e12 −2.70924
\(424\) −3.71023e10 −0.0557512
\(425\) 0 0
\(426\) −1.77489e12 −2.61113
\(427\) 1.13439e11 0.165134
\(428\) −7.79410e11 −1.12271
\(429\) 9.58239e11 1.36589
\(430\) 0 0
\(431\) 1.00292e12 1.39997 0.699987 0.714156i \(-0.253189\pi\)
0.699987 + 0.714156i \(0.253189\pi\)
\(432\) 2.37192e12 3.27660
\(433\) −7.98482e11 −1.09162 −0.545808 0.837910i \(-0.683777\pi\)
−0.545808 + 0.837910i \(0.683777\pi\)
\(434\) −9.24596e9 −0.0125097
\(435\) 0 0
\(436\) 6.21201e11 0.823271
\(437\) −4.58332e10 −0.0601193
\(438\) −2.58315e12 −3.35363
\(439\) 7.98379e11 1.02593 0.512966 0.858409i \(-0.328547\pi\)
0.512966 + 0.858409i \(0.328547\pi\)
\(440\) 0 0
\(441\) −2.09714e12 −2.64030
\(442\) −1.55604e11 −0.193920
\(443\) 7.56642e11 0.933413 0.466707 0.884412i \(-0.345440\pi\)
0.466707 + 0.884412i \(0.345440\pi\)
\(444\) −1.74673e12 −2.13306
\(445\) 0 0
\(446\) −8.21074e11 −0.982597
\(447\) −8.45779e11 −1.00201
\(448\) −8.08381e10 −0.0948124
\(449\) −6.39419e11 −0.742467 −0.371234 0.928540i \(-0.621065\pi\)
−0.371234 + 0.928540i \(0.621065\pi\)
\(450\) 0 0
\(451\) 1.25876e12 1.43268
\(452\) −9.01919e11 −1.01635
\(453\) 2.76111e12 3.08065
\(454\) 7.97418e11 0.880916
\(455\) 0 0
\(456\) −4.03636e10 −0.0437168
\(457\) 1.34276e12 1.44005 0.720024 0.693949i \(-0.244131\pi\)
0.720024 + 0.693949i \(0.244131\pi\)
\(458\) 7.32474e11 0.777853
\(459\) −5.93489e11 −0.624102
\(460\) 0 0
\(461\) 3.47112e11 0.357945 0.178972 0.983854i \(-0.442723\pi\)
0.178972 + 0.983854i \(0.442723\pi\)
\(462\) −2.66855e11 −0.272513
\(463\) −7.20214e11 −0.728362 −0.364181 0.931328i \(-0.618651\pi\)
−0.364181 + 0.931328i \(0.618651\pi\)
\(464\) −1.19937e12 −1.20122
\(465\) 0 0
\(466\) −1.00814e12 −0.990339
\(467\) 7.31553e10 0.0711737 0.0355869 0.999367i \(-0.488670\pi\)
0.0355869 + 0.999367i \(0.488670\pi\)
\(468\) 1.90188e12 1.83264
\(469\) 4.42433e10 0.0422250
\(470\) 0 0
\(471\) −8.73364e11 −0.817713
\(472\) −3.28701e10 −0.0304833
\(473\) −1.68074e11 −0.154392
\(474\) 3.17855e12 2.89219
\(475\) 0 0
\(476\) 2.13695e10 0.0190794
\(477\) −4.42267e12 −3.91158
\(478\) −7.11279e11 −0.623181
\(479\) −4.80307e10 −0.0416878 −0.0208439 0.999783i \(-0.506635\pi\)
−0.0208439 + 0.999783i \(0.506635\pi\)
\(480\) 0 0
\(481\) 9.49191e11 0.808539
\(482\) 6.70052e11 0.565454
\(483\) 2.30019e10 0.0192310
\(484\) 2.35261e10 0.0194870
\(485\) 0 0
\(486\) 5.88636e12 4.78612
\(487\) 1.24384e12 1.00204 0.501021 0.865435i \(-0.332958\pi\)
0.501021 + 0.865435i \(0.332958\pi\)
\(488\) 7.83997e10 0.0625785
\(489\) −9.13590e11 −0.722539
\(490\) 0 0
\(491\) −2.63145e11 −0.204328 −0.102164 0.994768i \(-0.532577\pi\)
−0.102164 + 0.994768i \(0.532577\pi\)
\(492\) 3.43506e12 2.64296
\(493\) 3.00099e11 0.228798
\(494\) −7.88515e11 −0.595715
\(495\) 0 0
\(496\) −1.22825e11 −0.0911212
\(497\) −1.32456e11 −0.0973798
\(498\) 3.84317e12 2.80000
\(499\) 3.75452e10 0.0271083 0.0135542 0.999908i \(-0.495685\pi\)
0.0135542 + 0.999908i \(0.495685\pi\)
\(500\) 0 0
\(501\) −1.41797e12 −1.00553
\(502\) 2.53153e12 1.77916
\(503\) 2.86189e11 0.199341 0.0996707 0.995020i \(-0.468221\pi\)
0.0996707 + 0.995020i \(0.468221\pi\)
\(504\) 1.47330e10 0.0101708
\(505\) 0 0
\(506\) −2.09420e11 −0.142017
\(507\) 1.42806e12 0.959864
\(508\) −1.52677e12 −1.01715
\(509\) −1.48505e12 −0.980646 −0.490323 0.871541i \(-0.663121\pi\)
−0.490323 + 0.871541i \(0.663121\pi\)
\(510\) 0 0
\(511\) −1.92774e11 −0.125071
\(512\) −2.17930e12 −1.40153
\(513\) −3.00746e12 −1.91722
\(514\) 1.05142e12 0.664420
\(515\) 0 0
\(516\) −4.58660e11 −0.284818
\(517\) 1.66654e12 1.02590
\(518\) −2.64336e11 −0.161314
\(519\) −1.77012e12 −1.07090
\(520\) 0 0
\(521\) −1.77521e12 −1.05556 −0.527778 0.849383i \(-0.676975\pi\)
−0.527778 + 0.849383i \(0.676975\pi\)
\(522\) −7.43795e12 −4.38466
\(523\) 1.20614e12 0.704922 0.352461 0.935826i \(-0.385345\pi\)
0.352461 + 0.935826i \(0.385345\pi\)
\(524\) −9.14135e11 −0.529687
\(525\) 0 0
\(526\) −1.92294e12 −1.09529
\(527\) 3.07326e10 0.0173561
\(528\) −3.54495e12 −1.98499
\(529\) −1.78310e12 −0.989978
\(530\) 0 0
\(531\) −3.91818e12 −2.13875
\(532\) 1.08289e11 0.0586112
\(533\) −1.86664e12 −1.00182
\(534\) 4.97596e11 0.264814
\(535\) 0 0
\(536\) 3.05773e10 0.0160014
\(537\) 3.32237e12 1.72410
\(538\) −8.32495e11 −0.428412
\(539\) 1.95913e12 0.999802
\(540\) 0 0
\(541\) 3.24195e12 1.62711 0.813557 0.581485i \(-0.197528\pi\)
0.813557 + 0.581485i \(0.197528\pi\)
\(542\) −1.72535e12 −0.858777
\(543\) 6.98296e12 3.44699
\(544\) 5.60471e11 0.274383
\(545\) 0 0
\(546\) 3.95725e11 0.190558
\(547\) 9.33463e11 0.445814 0.222907 0.974840i \(-0.428445\pi\)
0.222907 + 0.974840i \(0.428445\pi\)
\(548\) −2.30355e12 −1.09115
\(549\) 9.34540e12 4.39059
\(550\) 0 0
\(551\) 1.52073e12 0.702861
\(552\) 1.58970e10 0.00728769
\(553\) 2.37208e11 0.107861
\(554\) −5.61160e12 −2.53100
\(555\) 0 0
\(556\) 1.28092e11 0.0568443
\(557\) −1.31760e12 −0.580012 −0.290006 0.957025i \(-0.593657\pi\)
−0.290006 + 0.957025i \(0.593657\pi\)
\(558\) −7.61707e11 −0.332609
\(559\) 2.49240e11 0.107960
\(560\) 0 0
\(561\) 8.86998e11 0.378085
\(562\) 2.51442e11 0.106322
\(563\) −4.33363e12 −1.81788 −0.908938 0.416931i \(-0.863106\pi\)
−0.908938 + 0.416931i \(0.863106\pi\)
\(564\) 4.54784e12 1.89256
\(565\) 0 0
\(566\) 4.28452e12 1.75480
\(567\) 8.50868e11 0.345731
\(568\) −9.15429e10 −0.0369026
\(569\) 3.08940e12 1.23558 0.617788 0.786345i \(-0.288029\pi\)
0.617788 + 0.786345i \(0.288029\pi\)
\(570\) 0 0
\(571\) −7.43095e11 −0.292538 −0.146269 0.989245i \(-0.546726\pi\)
−0.146269 + 0.989245i \(0.546726\pi\)
\(572\) −1.77672e12 −0.693963
\(573\) −4.04356e12 −1.56700
\(574\) 5.19832e11 0.199875
\(575\) 0 0
\(576\) −6.65966e12 −2.52087
\(577\) −2.25852e12 −0.848269 −0.424134 0.905599i \(-0.639422\pi\)
−0.424134 + 0.905599i \(0.639422\pi\)
\(578\) 3.62501e12 1.35093
\(579\) −3.77013e12 −1.39413
\(580\) 0 0
\(581\) 2.86807e11 0.104423
\(582\) 6.71028e12 2.42430
\(583\) 4.13162e12 1.48119
\(584\) −1.33230e11 −0.0473962
\(585\) 0 0
\(586\) −3.26674e12 −1.14440
\(587\) −4.75794e12 −1.65405 −0.827023 0.562169i \(-0.809967\pi\)
−0.827023 + 0.562169i \(0.809967\pi\)
\(588\) 5.34630e12 1.84440
\(589\) 1.55735e11 0.0533173
\(590\) 0 0
\(591\) 6.91232e12 2.33067
\(592\) −3.51148e12 −1.17501
\(593\) −2.01077e12 −0.667755 −0.333878 0.942616i \(-0.608357\pi\)
−0.333878 + 0.942616i \(0.608357\pi\)
\(594\) −1.37416e13 −4.52896
\(595\) 0 0
\(596\) 1.56820e12 0.509089
\(597\) 7.90857e12 2.54808
\(598\) 3.10552e11 0.0993070
\(599\) −3.29427e12 −1.04554 −0.522768 0.852475i \(-0.675101\pi\)
−0.522768 + 0.852475i \(0.675101\pi\)
\(600\) 0 0
\(601\) −1.98887e12 −0.621830 −0.310915 0.950438i \(-0.600635\pi\)
−0.310915 + 0.950438i \(0.600635\pi\)
\(602\) −6.94096e10 −0.0215395
\(603\) 3.64488e12 1.12268
\(604\) −5.11952e12 −1.56517
\(605\) 0 0
\(606\) −1.37015e12 −0.412706
\(607\) −4.11631e12 −1.23072 −0.615360 0.788246i \(-0.710989\pi\)
−0.615360 + 0.788246i \(0.710989\pi\)
\(608\) 2.84015e12 0.842897
\(609\) −7.63194e11 −0.224832
\(610\) 0 0
\(611\) −2.47134e12 −0.717376
\(612\) 1.76048e12 0.507282
\(613\) 2.06864e12 0.591715 0.295858 0.955232i \(-0.404395\pi\)
0.295858 + 0.955232i \(0.404395\pi\)
\(614\) 8.31031e12 2.35971
\(615\) 0 0
\(616\) −1.37635e10 −0.00385137
\(617\) 5.42058e12 1.50578 0.752891 0.658145i \(-0.228659\pi\)
0.752891 + 0.658145i \(0.228659\pi\)
\(618\) −1.19078e13 −3.28385
\(619\) 3.87240e12 1.06016 0.530081 0.847947i \(-0.322162\pi\)
0.530081 + 0.847947i \(0.322162\pi\)
\(620\) 0 0
\(621\) 1.18447e12 0.319605
\(622\) 2.62153e12 0.702260
\(623\) 3.71344e10 0.00987599
\(624\) 5.25688e12 1.38802
\(625\) 0 0
\(626\) 4.31500e12 1.12304
\(627\) 4.49480e12 1.16147
\(628\) 1.61935e12 0.415453
\(629\) 8.78623e11 0.223807
\(630\) 0 0
\(631\) −3.22937e12 −0.810933 −0.405467 0.914110i \(-0.632891\pi\)
−0.405467 + 0.914110i \(0.632891\pi\)
\(632\) 1.63939e11 0.0408747
\(633\) −3.15076e12 −0.780006
\(634\) 2.88223e12 0.708478
\(635\) 0 0
\(636\) 1.12749e13 2.73246
\(637\) −2.90523e12 −0.699123
\(638\) 6.94847e12 1.66034
\(639\) −1.09121e13 −2.58913
\(640\) 0 0
\(641\) 1.10935e12 0.259543 0.129771 0.991544i \(-0.458576\pi\)
0.129771 + 0.991544i \(0.458576\pi\)
\(642\) −1.33602e13 −3.10388
\(643\) 2.83993e12 0.655177 0.327588 0.944821i \(-0.393764\pi\)
0.327588 + 0.944821i \(0.393764\pi\)
\(644\) −4.26489e10 −0.00977061
\(645\) 0 0
\(646\) −7.29892e11 −0.164897
\(647\) 2.66962e12 0.598935 0.299467 0.954107i \(-0.403191\pi\)
0.299467 + 0.954107i \(0.403191\pi\)
\(648\) 5.88050e11 0.131017
\(649\) 3.66033e12 0.809878
\(650\) 0 0
\(651\) −7.81574e10 −0.0170552
\(652\) 1.69393e12 0.367098
\(653\) −4.72156e12 −1.01619 −0.508097 0.861300i \(-0.669651\pi\)
−0.508097 + 0.861300i \(0.669651\pi\)
\(654\) 1.06483e13 2.27604
\(655\) 0 0
\(656\) 6.90554e12 1.45590
\(657\) −1.58813e13 −3.32538
\(658\) 6.88231e11 0.143126
\(659\) 4.01448e12 0.829173 0.414586 0.910010i \(-0.363926\pi\)
0.414586 + 0.910010i \(0.363926\pi\)
\(660\) 0 0
\(661\) −8.01591e12 −1.63323 −0.816613 0.577186i \(-0.804151\pi\)
−0.816613 + 0.577186i \(0.804151\pi\)
\(662\) 1.09704e13 2.22005
\(663\) −1.31535e12 −0.264380
\(664\) 1.98218e11 0.0395718
\(665\) 0 0
\(666\) −2.17767e13 −4.28901
\(667\) −5.98931e11 −0.117168
\(668\) 2.62912e12 0.510878
\(669\) −6.94065e12 −1.33962
\(670\) 0 0
\(671\) −8.73040e12 −1.66258
\(672\) −1.42536e12 −0.269626
\(673\) −3.55147e12 −0.667330 −0.333665 0.942692i \(-0.608286\pi\)
−0.333665 + 0.942692i \(0.608286\pi\)
\(674\) −2.66790e12 −0.497967
\(675\) 0 0
\(676\) −2.64783e12 −0.487675
\(677\) −6.06872e12 −1.11032 −0.555160 0.831744i \(-0.687343\pi\)
−0.555160 + 0.831744i \(0.687343\pi\)
\(678\) −1.54602e13 −2.80983
\(679\) 5.00773e11 0.0904122
\(680\) 0 0
\(681\) 6.74068e12 1.20100
\(682\) 7.11581e11 0.125949
\(683\) 9.96514e11 0.175223 0.0876113 0.996155i \(-0.472077\pi\)
0.0876113 + 0.996155i \(0.472077\pi\)
\(684\) 8.92111e12 1.55836
\(685\) 0 0
\(686\) 1.62635e12 0.280386
\(687\) 6.19170e12 1.06049
\(688\) −9.22050e11 −0.156894
\(689\) −6.12686e12 −1.03574
\(690\) 0 0
\(691\) 1.02702e13 1.71367 0.856835 0.515591i \(-0.172428\pi\)
0.856835 + 0.515591i \(0.172428\pi\)
\(692\) 3.28207e12 0.544090
\(693\) −1.64063e12 −0.270217
\(694\) −4.96299e12 −0.812130
\(695\) 0 0
\(696\) −5.27457e11 −0.0852012
\(697\) −1.72786e12 −0.277308
\(698\) −2.00823e11 −0.0320232
\(699\) −8.52194e12 −1.35018
\(700\) 0 0
\(701\) −8.26127e12 −1.29216 −0.646079 0.763270i \(-0.723592\pi\)
−0.646079 + 0.763270i \(0.723592\pi\)
\(702\) 2.03777e13 3.16693
\(703\) 4.45236e12 0.687529
\(704\) 6.22140e12 0.954578
\(705\) 0 0
\(706\) 9.02234e12 1.36678
\(707\) −1.02251e11 −0.0153915
\(708\) 9.98875e12 1.49404
\(709\) −8.11428e12 −1.20598 −0.602992 0.797747i \(-0.706025\pi\)
−0.602992 + 0.797747i \(0.706025\pi\)
\(710\) 0 0
\(711\) 1.95418e13 2.86782
\(712\) 2.56643e10 0.00374256
\(713\) −6.13355e10 −0.00888810
\(714\) 3.66304e11 0.0527473
\(715\) 0 0
\(716\) −6.16017e12 −0.875960
\(717\) −6.01254e12 −0.849614
\(718\) −1.90929e13 −2.68110
\(719\) 1.06399e13 1.48476 0.742380 0.669979i \(-0.233697\pi\)
0.742380 + 0.669979i \(0.233697\pi\)
\(720\) 0 0
\(721\) −8.88651e11 −0.122468
\(722\) 6.55723e12 0.898055
\(723\) 5.66405e12 0.770911
\(724\) −1.29475e13 −1.75130
\(725\) 0 0
\(726\) 4.03270e11 0.0538742
\(727\) −1.35040e13 −1.79290 −0.896450 0.443145i \(-0.853863\pi\)
−0.896450 + 0.443145i \(0.853863\pi\)
\(728\) 2.04101e10 0.00269311
\(729\) 2.34766e13 3.07866
\(730\) 0 0
\(731\) 2.30710e11 0.0298840
\(732\) −2.38245e13 −3.06708
\(733\) 3.03764e12 0.388659 0.194329 0.980936i \(-0.437747\pi\)
0.194329 + 0.980936i \(0.437747\pi\)
\(734\) −1.58043e13 −2.00975
\(735\) 0 0
\(736\) −1.11858e12 −0.140513
\(737\) −3.40502e12 −0.425124
\(738\) 4.28251e13 5.31429
\(739\) 7.29483e12 0.899736 0.449868 0.893095i \(-0.351471\pi\)
0.449868 + 0.893095i \(0.351471\pi\)
\(740\) 0 0
\(741\) −6.66542e12 −0.812168
\(742\) 1.70624e12 0.206644
\(743\) −1.18034e13 −1.42088 −0.710442 0.703756i \(-0.751505\pi\)
−0.710442 + 0.703756i \(0.751505\pi\)
\(744\) −5.40160e10 −0.00646315
\(745\) 0 0
\(746\) −4.42362e12 −0.522941
\(747\) 2.36279e13 2.77641
\(748\) −1.64463e12 −0.192092
\(749\) −9.97041e11 −0.115756
\(750\) 0 0
\(751\) 1.20855e13 1.38639 0.693197 0.720748i \(-0.256202\pi\)
0.693197 + 0.720748i \(0.256202\pi\)
\(752\) 9.14258e12 1.04253
\(753\) 2.13993e13 2.42562
\(754\) −1.03040e13 −1.16101
\(755\) 0 0
\(756\) −2.79852e12 −0.311588
\(757\) 1.48077e13 1.63891 0.819455 0.573144i \(-0.194276\pi\)
0.819455 + 0.573144i \(0.194276\pi\)
\(758\) 1.79651e13 1.97660
\(759\) −1.77025e12 −0.193619
\(760\) 0 0
\(761\) −1.75185e13 −1.89350 −0.946749 0.321972i \(-0.895654\pi\)
−0.946749 + 0.321972i \(0.895654\pi\)
\(762\) −2.61709e13 −2.81204
\(763\) 7.94656e11 0.0848826
\(764\) 7.49736e12 0.796138
\(765\) 0 0
\(766\) 1.71266e13 1.79739
\(767\) −5.42798e12 −0.566316
\(768\) −1.94209e13 −2.01439
\(769\) 4.16729e12 0.429720 0.214860 0.976645i \(-0.431070\pi\)
0.214860 + 0.976645i \(0.431070\pi\)
\(770\) 0 0
\(771\) 8.88781e12 0.905837
\(772\) 6.99039e12 0.708311
\(773\) −1.05896e13 −1.06677 −0.533386 0.845872i \(-0.679081\pi\)
−0.533386 + 0.845872i \(0.679081\pi\)
\(774\) −5.71815e12 −0.572692
\(775\) 0 0
\(776\) 3.46093e11 0.0342622
\(777\) −2.23446e12 −0.219927
\(778\) −2.43949e13 −2.38721
\(779\) −8.75583e12 −0.851881
\(780\) 0 0
\(781\) 1.01940e13 0.980426
\(782\) 2.87464e11 0.0274887
\(783\) −3.93004e13 −3.73654
\(784\) 1.07478e13 1.01600
\(785\) 0 0
\(786\) −1.56696e13 −1.46439
\(787\) 2.50861e12 0.233102 0.116551 0.993185i \(-0.462816\pi\)
0.116551 + 0.993185i \(0.462816\pi\)
\(788\) −1.28165e13 −1.18413
\(789\) −1.62549e13 −1.49327
\(790\) 0 0
\(791\) −1.15376e12 −0.104790
\(792\) −1.13387e12 −0.102400
\(793\) 1.29465e13 1.16258
\(794\) 1.98724e13 1.77443
\(795\) 0 0
\(796\) −1.46637e13 −1.29460
\(797\) 1.85822e13 1.63131 0.815653 0.578541i \(-0.196378\pi\)
0.815653 + 0.578541i \(0.196378\pi\)
\(798\) 1.85622e12 0.162038
\(799\) −2.28760e12 −0.198573
\(800\) 0 0
\(801\) 3.05923e12 0.262583
\(802\) 5.12746e12 0.437641
\(803\) 1.48362e13 1.25922
\(804\) −9.29202e12 −0.784256
\(805\) 0 0
\(806\) −1.05522e12 −0.0880712
\(807\) −7.03720e12 −0.584075
\(808\) −7.06675e10 −0.00583268
\(809\) −4.47333e12 −0.367166 −0.183583 0.983004i \(-0.558770\pi\)
−0.183583 + 0.983004i \(0.558770\pi\)
\(810\) 0 0
\(811\) −3.49582e12 −0.283763 −0.141881 0.989884i \(-0.545315\pi\)
−0.141881 + 0.989884i \(0.545315\pi\)
\(812\) 1.41508e12 0.114229
\(813\) −1.45846e13 −1.17081
\(814\) 2.03436e13 1.62412
\(815\) 0 0
\(816\) 4.86605e12 0.384212
\(817\) 1.16911e12 0.0918026
\(818\) 2.30108e13 1.79698
\(819\) 2.43293e12 0.188952
\(820\) 0 0
\(821\) −6.84197e12 −0.525578 −0.262789 0.964853i \(-0.584642\pi\)
−0.262789 + 0.964853i \(0.584642\pi\)
\(822\) −3.94860e13 −3.01662
\(823\) 8.11058e12 0.616244 0.308122 0.951347i \(-0.400300\pi\)
0.308122 + 0.951347i \(0.400300\pi\)
\(824\) −6.14163e11 −0.0464099
\(825\) 0 0
\(826\) 1.51161e12 0.112987
\(827\) −5.06349e10 −0.00376422 −0.00188211 0.999998i \(-0.500599\pi\)
−0.00188211 + 0.999998i \(0.500599\pi\)
\(828\) −3.51353e12 −0.259781
\(829\) −1.50474e13 −1.10654 −0.553269 0.833002i \(-0.686620\pi\)
−0.553269 + 0.833002i \(0.686620\pi\)
\(830\) 0 0
\(831\) −4.74356e13 −3.45064
\(832\) −9.22584e12 −0.667499
\(833\) −2.68924e12 −0.193521
\(834\) 2.19569e12 0.157153
\(835\) 0 0
\(836\) −8.33403e12 −0.590102
\(837\) −4.02469e12 −0.283444
\(838\) −8.90044e12 −0.623467
\(839\) 2.68352e13 1.86972 0.934859 0.355020i \(-0.115526\pi\)
0.934859 + 0.355020i \(0.115526\pi\)
\(840\) 0 0
\(841\) 5.36519e12 0.369831
\(842\) 2.08468e12 0.142934
\(843\) 2.12547e12 0.144954
\(844\) 5.84197e12 0.396295
\(845\) 0 0
\(846\) 5.66983e13 3.80543
\(847\) 3.00951e10 0.00200919
\(848\) 2.26660e13 1.50520
\(849\) 3.62176e13 2.39241
\(850\) 0 0
\(851\) −1.75354e12 −0.114613
\(852\) 2.78186e13 1.80866
\(853\) 4.22107e12 0.272993 0.136497 0.990641i \(-0.456416\pi\)
0.136497 + 0.990641i \(0.456416\pi\)
\(854\) −3.60540e12 −0.231949
\(855\) 0 0
\(856\) −6.89073e11 −0.0438665
\(857\) −7.57740e12 −0.479851 −0.239926 0.970791i \(-0.577123\pi\)
−0.239926 + 0.970791i \(0.577123\pi\)
\(858\) −3.04555e13 −1.91855
\(859\) 1.41463e13 0.886486 0.443243 0.896401i \(-0.353828\pi\)
0.443243 + 0.896401i \(0.353828\pi\)
\(860\) 0 0
\(861\) 4.39421e12 0.272500
\(862\) −3.18757e13 −1.96642
\(863\) 1.52498e13 0.935868 0.467934 0.883763i \(-0.344998\pi\)
0.467934 + 0.883763i \(0.344998\pi\)
\(864\) −7.33983e13 −4.48099
\(865\) 0 0
\(866\) 2.53780e13 1.53330
\(867\) 3.06427e13 1.84180
\(868\) 1.44915e11 0.00866515
\(869\) −1.82558e13 −1.08596
\(870\) 0 0
\(871\) 5.04937e12 0.297273
\(872\) 5.49201e11 0.0321668
\(873\) 4.12550e13 2.40388
\(874\) 1.45671e12 0.0844443
\(875\) 0 0
\(876\) 4.04867e13 2.32297
\(877\) 3.30350e12 0.188572 0.0942858 0.995545i \(-0.469943\pi\)
0.0942858 + 0.995545i \(0.469943\pi\)
\(878\) −2.53747e13 −1.44104
\(879\) −2.76142e13 −1.56021
\(880\) 0 0
\(881\) 3.14728e13 1.76013 0.880063 0.474856i \(-0.157500\pi\)
0.880063 + 0.474856i \(0.157500\pi\)
\(882\) 6.66528e13 3.70860
\(883\) 1.10669e13 0.612637 0.306318 0.951929i \(-0.400903\pi\)
0.306318 + 0.951929i \(0.400903\pi\)
\(884\) 2.43885e12 0.134323
\(885\) 0 0
\(886\) −2.40482e13 −1.31108
\(887\) −6.92035e12 −0.375381 −0.187690 0.982228i \(-0.560100\pi\)
−0.187690 + 0.982228i \(0.560100\pi\)
\(888\) −1.54428e12 −0.0833426
\(889\) −1.95308e12 −0.104872
\(890\) 0 0
\(891\) −6.54838e13 −3.48084
\(892\) 1.28690e13 0.680618
\(893\) −1.15923e13 −0.610010
\(894\) 2.68812e13 1.40744
\(895\) 0 0
\(896\) −1.47084e11 −0.00762394
\(897\) 2.62514e12 0.135390
\(898\) 2.03225e13 1.04288
\(899\) 2.03509e12 0.103912
\(900\) 0 0
\(901\) −5.67136e12 −0.286699
\(902\) −4.00069e13 −2.01236
\(903\) −5.86729e11 −0.0293659
\(904\) −7.97383e11 −0.0397108
\(905\) 0 0
\(906\) −8.77557e13 −4.32712
\(907\) −2.00486e13 −0.983676 −0.491838 0.870687i \(-0.663675\pi\)
−0.491838 + 0.870687i \(0.663675\pi\)
\(908\) −1.24982e13 −0.610186
\(909\) −8.42371e12 −0.409229
\(910\) 0 0
\(911\) 2.53798e13 1.22083 0.610416 0.792081i \(-0.291002\pi\)
0.610416 + 0.792081i \(0.291002\pi\)
\(912\) 2.46584e13 1.18029
\(913\) −2.20730e13 −1.05134
\(914\) −4.26768e13 −2.02271
\(915\) 0 0
\(916\) −1.14803e13 −0.538797
\(917\) −1.16938e12 −0.0546129
\(918\) 1.88627e13 0.876622
\(919\) −6.15926e12 −0.284845 −0.142423 0.989806i \(-0.545489\pi\)
−0.142423 + 0.989806i \(0.545489\pi\)
\(920\) 0 0
\(921\) 7.02482e13 3.21712
\(922\) −1.10322e13 −0.502774
\(923\) −1.51169e13 −0.685574
\(924\) 4.18252e12 0.188762
\(925\) 0 0
\(926\) 2.28904e13 1.02307
\(927\) −7.32095e13 −3.25618
\(928\) 3.71140e13 1.64275
\(929\) 2.20000e13 0.969061 0.484531 0.874774i \(-0.338990\pi\)
0.484531 + 0.874774i \(0.338990\pi\)
\(930\) 0 0
\(931\) −1.36275e13 −0.594489
\(932\) 1.58010e13 0.685980
\(933\) 2.21601e13 0.957426
\(934\) −2.32508e12 −0.0999715
\(935\) 0 0
\(936\) 1.68144e12 0.0716045
\(937\) 2.49274e12 0.105645 0.0528225 0.998604i \(-0.483178\pi\)
0.0528225 + 0.998604i \(0.483178\pi\)
\(938\) −1.40617e12 −0.0593098
\(939\) 3.64753e13 1.53110
\(940\) 0 0
\(941\) −1.26366e13 −0.525382 −0.262691 0.964880i \(-0.584610\pi\)
−0.262691 + 0.964880i \(0.584610\pi\)
\(942\) 2.77579e13 1.14857
\(943\) 3.44844e12 0.142010
\(944\) 2.00805e13 0.823002
\(945\) 0 0
\(946\) 5.34185e12 0.216861
\(947\) −2.05809e13 −0.831551 −0.415775 0.909467i \(-0.636490\pi\)
−0.415775 + 0.909467i \(0.636490\pi\)
\(948\) −4.98186e13 −2.00334
\(949\) −2.20008e13 −0.880524
\(950\) 0 0
\(951\) 2.43639e13 0.965904
\(952\) 1.88927e10 0.000745466 0
\(953\) −2.97887e13 −1.16986 −0.584929 0.811084i \(-0.698878\pi\)
−0.584929 + 0.811084i \(0.698878\pi\)
\(954\) 1.40565e14 5.49425
\(955\) 0 0
\(956\) 1.11481e13 0.431660
\(957\) 5.87364e13 2.26362
\(958\) 1.52655e12 0.0585553
\(959\) −2.94675e12 −0.112502
\(960\) 0 0
\(961\) −2.62312e13 −0.992118
\(962\) −3.01679e13 −1.13568
\(963\) −8.21389e13 −3.07773
\(964\) −1.05020e13 −0.391674
\(965\) 0 0
\(966\) −7.31063e11 −0.0270121
\(967\) 1.74473e13 0.641664 0.320832 0.947136i \(-0.396037\pi\)
0.320832 + 0.947136i \(0.396037\pi\)
\(968\) 2.07993e10 0.000761394 0
\(969\) −6.16988e12 −0.224812
\(970\) 0 0
\(971\) −2.53844e13 −0.916390 −0.458195 0.888852i \(-0.651504\pi\)
−0.458195 + 0.888852i \(0.651504\pi\)
\(972\) −9.22592e13 −3.31521
\(973\) 1.63859e11 0.00586088
\(974\) −3.95328e13 −1.40748
\(975\) 0 0
\(976\) −4.78948e13 −1.68952
\(977\) 3.23029e13 1.13427 0.567135 0.823625i \(-0.308052\pi\)
0.567135 + 0.823625i \(0.308052\pi\)
\(978\) 2.90364e13 1.01489
\(979\) −2.85791e12 −0.0994321
\(980\) 0 0
\(981\) 6.54659e13 2.25686
\(982\) 8.36348e12 0.287002
\(983\) −1.78014e13 −0.608083 −0.304041 0.952659i \(-0.598336\pi\)
−0.304041 + 0.952659i \(0.598336\pi\)
\(984\) 3.03692e12 0.103265
\(985\) 0 0
\(986\) −9.53796e12 −0.321373
\(987\) 5.81771e12 0.195130
\(988\) 1.23587e13 0.412635
\(989\) −4.60447e11 −0.0153037
\(990\) 0 0
\(991\) −3.38638e12 −0.111533 −0.0557666 0.998444i \(-0.517760\pi\)
−0.0557666 + 0.998444i \(0.517760\pi\)
\(992\) 3.80078e12 0.124615
\(993\) 9.27345e13 3.02670
\(994\) 4.20982e12 0.136781
\(995\) 0 0
\(996\) −6.02355e13 −1.93948
\(997\) 4.17766e13 1.33908 0.669538 0.742778i \(-0.266492\pi\)
0.669538 + 0.742778i \(0.266492\pi\)
\(998\) −1.19329e12 −0.0380767
\(999\) −1.15063e14 −3.65503
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 25.10.a.c.1.1 3
3.2 odd 2 225.10.a.p.1.3 3
4.3 odd 2 400.10.a.y.1.3 3
5.2 odd 4 25.10.b.c.24.1 6
5.3 odd 4 25.10.b.c.24.6 6
5.4 even 2 25.10.a.d.1.3 yes 3
15.2 even 4 225.10.b.m.199.6 6
15.8 even 4 225.10.b.m.199.1 6
15.14 odd 2 225.10.a.m.1.1 3
20.3 even 4 400.10.c.q.49.6 6
20.7 even 4 400.10.c.q.49.1 6
20.19 odd 2 400.10.a.u.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.10.a.c.1.1 3 1.1 even 1 trivial
25.10.a.d.1.3 yes 3 5.4 even 2
25.10.b.c.24.1 6 5.2 odd 4
25.10.b.c.24.6 6 5.3 odd 4
225.10.a.m.1.1 3 15.14 odd 2
225.10.a.p.1.3 3 3.2 odd 2
225.10.b.m.199.1 6 15.8 even 4
225.10.b.m.199.6 6 15.2 even 4
400.10.a.u.1.1 3 20.19 odd 2
400.10.a.y.1.3 3 4.3 odd 2
400.10.c.q.49.1 6 20.7 even 4
400.10.c.q.49.6 6 20.3 even 4