Properties

Label 81.2.a.a.1.1
Level $81$
Weight $2$
Character 81.1
Self dual yes
Analytic conductor $0.647$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 81.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 81.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} +2.00000 q^{7} +1.73205 q^{8} -3.00000 q^{10} +3.46410 q^{11} -1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} -5.19615 q^{17} +2.00000 q^{19} +1.73205 q^{20} -6.00000 q^{22} -3.46410 q^{23} -2.00000 q^{25} +1.73205 q^{26} +2.00000 q^{28} -1.73205 q^{29} +8.00000 q^{31} +5.19615 q^{32} +9.00000 q^{34} +3.46410 q^{35} -7.00000 q^{37} -3.46410 q^{38} +3.00000 q^{40} +6.92820 q^{41} +2.00000 q^{43} +3.46410 q^{44} +6.00000 q^{46} -6.92820 q^{47} -3.00000 q^{49} +3.46410 q^{50} -1.00000 q^{52} +6.00000 q^{55} +3.46410 q^{56} +3.00000 q^{58} -13.8564 q^{59} -7.00000 q^{61} -13.8564 q^{62} +1.00000 q^{64} -1.73205 q^{65} -10.0000 q^{67} -5.19615 q^{68} -6.00000 q^{70} +10.3923 q^{71} -7.00000 q^{73} +12.1244 q^{74} +2.00000 q^{76} +6.92820 q^{77} +2.00000 q^{79} -8.66025 q^{80} -12.0000 q^{82} +13.8564 q^{83} -9.00000 q^{85} -3.46410 q^{86} +6.00000 q^{88} +5.19615 q^{89} -2.00000 q^{91} -3.46410 q^{92} +12.0000 q^{94} +3.46410 q^{95} +2.00000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 4 q^{7} - 6 q^{10} - 2 q^{13} - 10 q^{16} + 4 q^{19} - 12 q^{22} - 4 q^{25} + 4 q^{28} + 16 q^{31} + 18 q^{34} - 14 q^{37} + 6 q^{40} + 4 q^{43} + 12 q^{46} - 6 q^{49} - 2 q^{52} + 12 q^{55} + 6 q^{58} - 14 q^{61} + 2 q^{64} - 20 q^{67} - 12 q^{70} - 14 q^{73} + 4 q^{76} + 4 q^{79} - 24 q^{82} - 18 q^{85} + 12 q^{88} - 4 q^{91} + 24 q^{94} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.73205 0.774597 0.387298 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) −3.00000 −0.948683
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −5.19615 −1.26025 −0.630126 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.73205 0.387298
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 1.73205 0.339683
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 9.00000 1.54349
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −3.46410 −0.561951
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 3.46410 0.522233
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 3.46410 0.489898
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) −13.8564 −1.80395 −0.901975 0.431788i \(-0.857883\pi\)
−0.901975 + 0.431788i \(0.857883\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −13.8564 −1.75977
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.73205 −0.214834
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −5.19615 −0.630126
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) 10.3923 1.23334 0.616670 0.787222i \(-0.288481\pi\)
0.616670 + 0.787222i \(0.288481\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 12.1244 1.40943
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) −8.66025 −0.968246
\(81\) 0 0
\(82\) −12.0000 −1.32518
\(83\) 13.8564 1.52094 0.760469 0.649374i \(-0.224969\pi\)
0.760469 + 0.649374i \(0.224969\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) −3.46410 −0.373544
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −3.46410 −0.361158
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 5.19615 0.524891
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) −6.92820 −0.689382 −0.344691 0.938716i \(-0.612016\pi\)
−0.344691 + 0.938716i \(0.612016\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −1.73205 −0.169842
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) −10.3923 −0.990867
\(111\) 0 0
\(112\) −10.0000 −0.944911
\(113\) 1.73205 0.162938 0.0814688 0.996676i \(-0.474039\pi\)
0.0814688 + 0.996676i \(0.474039\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) −1.73205 −0.160817
\(117\) 0 0
\(118\) 24.0000 2.20938
\(119\) −10.3923 −0.952661
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 12.1244 1.09769
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −12.1244 −1.08444
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0 0
\(130\) 3.00000 0.263117
\(131\) −3.46410 −0.302660 −0.151330 0.988483i \(-0.548356\pi\)
−0.151330 + 0.988483i \(0.548356\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 17.3205 1.49626
\(135\) 0 0
\(136\) −9.00000 −0.771744
\(137\) −1.73205 −0.147979 −0.0739895 0.997259i \(-0.523573\pi\)
−0.0739895 + 0.997259i \(0.523573\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 3.46410 0.292770
\(141\) 0 0
\(142\) −18.0000 −1.51053
\(143\) −3.46410 −0.289683
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 12.1244 1.00342
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) −8.66025 −0.709476 −0.354738 0.934966i \(-0.615430\pi\)
−0.354738 + 0.934966i \(0.615430\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 3.46410 0.280976
\(153\) 0 0
\(154\) −12.0000 −0.966988
\(155\) 13.8564 1.11297
\(156\) 0 0
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) −3.46410 −0.275589
\(159\) 0 0
\(160\) 9.00000 0.711512
\(161\) −6.92820 −0.546019
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 6.92820 0.541002
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 15.5885 1.19558
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 19.0526 1.44854 0.724270 0.689517i \(-0.242177\pi\)
0.724270 + 0.689517i \(0.242177\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −17.3205 −1.30558
\(177\) 0 0
\(178\) −9.00000 −0.674579
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 3.46410 0.256776
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) −12.1244 −0.891400
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) −6.92820 −0.505291
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) −17.3205 −1.25327 −0.626634 0.779314i \(-0.715568\pi\)
−0.626634 + 0.779314i \(0.715568\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) −3.46410 −0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 5.19615 0.370211 0.185105 0.982719i \(-0.440737\pi\)
0.185105 + 0.982719i \(0.440737\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −3.46410 −0.244949
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) −3.46410 −0.243132
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) −13.8564 −0.965422
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) 6.92820 0.479234
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.46410 0.236250
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) −19.0526 −1.29040
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) 5.19615 0.349531
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) −3.00000 −0.199557
\(227\) 3.46410 0.229920 0.114960 0.993370i \(-0.463326\pi\)
0.114960 + 0.993370i \(0.463326\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 10.3923 0.685248
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 25.9808 1.70206 0.851028 0.525120i \(-0.175980\pi\)
0.851028 + 0.525120i \(0.175980\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) −13.8564 −0.901975
\(237\) 0 0
\(238\) 18.0000 1.16677
\(239\) 27.7128 1.79259 0.896296 0.443455i \(-0.146248\pi\)
0.896296 + 0.443455i \(0.146248\pi\)
\(240\) 0 0
\(241\) 29.0000 1.86805 0.934027 0.357202i \(-0.116269\pi\)
0.934027 + 0.357202i \(0.116269\pi\)
\(242\) −1.73205 −0.111340
\(243\) 0 0
\(244\) −7.00000 −0.448129
\(245\) −5.19615 −0.331970
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 13.8564 0.879883
\(249\) 0 0
\(250\) 21.0000 1.32816
\(251\) −10.3923 −0.655956 −0.327978 0.944685i \(-0.606367\pi\)
−0.327978 + 0.944685i \(0.606367\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) −3.46410 −0.217357
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −8.66025 −0.540212 −0.270106 0.962831i \(-0.587059\pi\)
−0.270106 + 0.962831i \(0.587059\pi\)
\(258\) 0 0
\(259\) −14.0000 −0.869918
\(260\) −1.73205 −0.107417
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) −6.92820 −0.427211 −0.213606 0.976920i \(-0.568521\pi\)
−0.213606 + 0.976920i \(0.568521\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.92820 −0.424795
\(267\) 0 0
\(268\) −10.0000 −0.610847
\(269\) 15.5885 0.950445 0.475223 0.879866i \(-0.342368\pi\)
0.475223 + 0.879866i \(0.342368\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 25.9808 1.57532
\(273\) 0 0
\(274\) 3.00000 0.181237
\(275\) −6.92820 −0.417786
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −13.8564 −0.831052
\(279\) 0 0
\(280\) 6.00000 0.358569
\(281\) −12.1244 −0.723278 −0.361639 0.932318i \(-0.617783\pi\)
−0.361639 + 0.932318i \(0.617783\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 10.3923 0.616670
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 13.8564 0.817918
\(288\) 0 0
\(289\) 10.0000 0.588235
\(290\) 5.19615 0.305129
\(291\) 0 0
\(292\) −7.00000 −0.409644
\(293\) −19.0526 −1.11306 −0.556531 0.830827i \(-0.687868\pi\)
−0.556531 + 0.830827i \(0.687868\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) −12.1244 −0.704714
\(297\) 0 0
\(298\) 15.0000 0.868927
\(299\) 3.46410 0.200334
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −34.6410 −1.99337
\(303\) 0 0
\(304\) −10.0000 −0.573539
\(305\) −12.1244 −0.694239
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 6.92820 0.394771
\(309\) 0 0
\(310\) −24.0000 −1.36311
\(311\) 6.92820 0.392862 0.196431 0.980518i \(-0.437065\pi\)
0.196431 + 0.980518i \(0.437065\pi\)
\(312\) 0 0
\(313\) −25.0000 −1.41308 −0.706542 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(314\) −29.4449 −1.66167
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 8.66025 0.486408 0.243204 0.969975i \(-0.421801\pi\)
0.243204 + 0.969975i \(0.421801\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 1.73205 0.0968246
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) −10.3923 −0.578243
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 27.7128 1.53487
\(327\) 0 0
\(328\) 12.0000 0.662589
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 13.8564 0.760469
\(333\) 0 0
\(334\) −30.0000 −1.64153
\(335\) −17.3205 −0.946320
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 20.7846 1.13053
\(339\) 0 0
\(340\) −9.00000 −0.488094
\(341\) 27.7128 1.50073
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 3.46410 0.186772
\(345\) 0 0
\(346\) −33.0000 −1.77409
\(347\) −3.46410 −0.185963 −0.0929814 0.995668i \(-0.529640\pi\)
−0.0929814 + 0.995668i \(0.529640\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 6.92820 0.370328
\(351\) 0 0
\(352\) 18.0000 0.959403
\(353\) 13.8564 0.737502 0.368751 0.929528i \(-0.379785\pi\)
0.368751 + 0.929528i \(0.379785\pi\)
\(354\) 0 0
\(355\) 18.0000 0.955341
\(356\) 5.19615 0.275396
\(357\) 0 0
\(358\) 36.0000 1.90266
\(359\) −10.3923 −0.548485 −0.274242 0.961661i \(-0.588427\pi\)
−0.274242 + 0.961661i \(0.588427\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −3.46410 −0.182069
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −12.1244 −0.634618
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 17.3205 0.902894
\(369\) 0 0
\(370\) 21.0000 1.09174
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 31.1769 1.61212
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 1.73205 0.0892052
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 3.46410 0.177705
\(381\) 0 0
\(382\) 30.0000 1.53493
\(383\) 17.3205 0.885037 0.442518 0.896759i \(-0.354085\pi\)
0.442518 + 0.896759i \(0.354085\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 1.73205 0.0881591
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −27.7128 −1.40510 −0.702548 0.711637i \(-0.747954\pi\)
−0.702548 + 0.711637i \(0.747954\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) −5.19615 −0.262445
\(393\) 0 0
\(394\) −9.00000 −0.453413
\(395\) 3.46410 0.174298
\(396\) 0 0
\(397\) 29.0000 1.45547 0.727734 0.685859i \(-0.240573\pi\)
0.727734 + 0.685859i \(0.240573\pi\)
\(398\) −34.6410 −1.73640
\(399\) 0 0
\(400\) 10.0000 0.500000
\(401\) 12.1244 0.605461 0.302731 0.953076i \(-0.402102\pi\)
0.302731 + 0.953076i \(0.402102\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −6.92820 −0.344691
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −24.2487 −1.20196
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) −20.7846 −1.02648
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −27.7128 −1.36366
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) −5.19615 −0.254762
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) 6.92820 0.338465 0.169232 0.985576i \(-0.445871\pi\)
0.169232 + 0.985576i \(0.445871\pi\)
\(420\) 0 0
\(421\) −25.0000 −1.21843 −0.609213 0.793007i \(-0.708514\pi\)
−0.609213 + 0.793007i \(0.708514\pi\)
\(422\) 17.3205 0.843149
\(423\) 0 0
\(424\) 0 0
\(425\) 10.3923 0.504101
\(426\) 0 0
\(427\) −14.0000 −0.677507
\(428\) 0 0
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) −27.7128 −1.33026
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) −6.92820 −0.331421
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 10.3923 0.495434
\(441\) 0 0
\(442\) −9.00000 −0.428086
\(443\) 34.6410 1.64584 0.822922 0.568154i \(-0.192342\pi\)
0.822922 + 0.568154i \(0.192342\pi\)
\(444\) 0 0
\(445\) 9.00000 0.426641
\(446\) −3.46410 −0.164030
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −20.7846 −0.980886 −0.490443 0.871473i \(-0.663165\pi\)
−0.490443 + 0.871473i \(0.663165\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 1.73205 0.0814688
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) −3.46410 −0.162400
\(456\) 0 0
\(457\) 29.0000 1.35656 0.678281 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 1.73205 0.0809334
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 13.8564 0.645357 0.322679 0.946509i \(-0.395417\pi\)
0.322679 + 0.946509i \(0.395417\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 8.66025 0.402042
\(465\) 0 0
\(466\) −45.0000 −2.08458
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 20.7846 0.958723
\(471\) 0 0
\(472\) −24.0000 −1.10469
\(473\) 6.92820 0.318559
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −10.3923 −0.476331
\(477\) 0 0
\(478\) −48.0000 −2.19547
\(479\) 24.2487 1.10795 0.553976 0.832533i \(-0.313110\pi\)
0.553976 + 0.832533i \(0.313110\pi\)
\(480\) 0 0
\(481\) 7.00000 0.319173
\(482\) −50.2295 −2.28789
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 3.46410 0.157297
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −12.1244 −0.548844
\(489\) 0 0
\(490\) 9.00000 0.406579
\(491\) 17.3205 0.781664 0.390832 0.920462i \(-0.372187\pi\)
0.390832 + 0.920462i \(0.372187\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 3.46410 0.155857
\(495\) 0 0
\(496\) −40.0000 −1.79605
\(497\) 20.7846 0.932317
\(498\) 0 0
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) −12.1244 −0.542218
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) −20.7846 −0.926740 −0.463370 0.886165i \(-0.653360\pi\)
−0.463370 + 0.886165i \(0.653360\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 20.7846 0.923989
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 27.7128 1.22835 0.614174 0.789170i \(-0.289489\pi\)
0.614174 + 0.789170i \(0.289489\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) −8.66025 −0.382733
\(513\) 0 0
\(514\) 15.0000 0.661622
\(515\) 13.8564 0.610586
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 24.2487 1.06543
\(519\) 0 0
\(520\) −3.00000 −0.131559
\(521\) 20.7846 0.910590 0.455295 0.890341i \(-0.349534\pi\)
0.455295 + 0.890341i \(0.349534\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) −3.46410 −0.151330
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −41.5692 −1.81078
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) −6.92820 −0.300094
\(534\) 0 0
\(535\) 0 0
\(536\) −17.3205 −0.748132
\(537\) 0 0
\(538\) −27.0000 −1.16405
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) −3.46410 −0.148796
\(543\) 0 0
\(544\) −27.0000 −1.15762
\(545\) 19.0526 0.816122
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −1.73205 −0.0739895
\(549\) 0 0
\(550\) 12.0000 0.511682
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −3.46410 −0.147176
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) −36.3731 −1.54118 −0.770588 0.637333i \(-0.780037\pi\)
−0.770588 + 0.637333i \(0.780037\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) −17.3205 −0.731925
\(561\) 0 0
\(562\) 21.0000 0.885832
\(563\) −34.6410 −1.45994 −0.729972 0.683477i \(-0.760467\pi\)
−0.729972 + 0.683477i \(0.760467\pi\)
\(564\) 0 0
\(565\) 3.00000 0.126211
\(566\) 48.4974 2.03850
\(567\) 0 0
\(568\) 18.0000 0.755263
\(569\) −32.9090 −1.37962 −0.689808 0.723993i \(-0.742305\pi\)
−0.689808 + 0.723993i \(0.742305\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) −3.46410 −0.144841
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 6.92820 0.288926
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) −17.3205 −0.720438
\(579\) 0 0
\(580\) −3.00000 −0.124568
\(581\) 27.7128 1.14972
\(582\) 0 0
\(583\) 0 0
\(584\) −12.1244 −0.501709
\(585\) 0 0
\(586\) 33.0000 1.36322
\(587\) −38.1051 −1.57277 −0.786383 0.617739i \(-0.788049\pi\)
−0.786383 + 0.617739i \(0.788049\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 41.5692 1.71138
\(591\) 0 0
\(592\) 35.0000 1.43849
\(593\) −15.5885 −0.640141 −0.320071 0.947394i \(-0.603707\pi\)
−0.320071 + 0.947394i \(0.603707\pi\)
\(594\) 0 0
\(595\) −18.0000 −0.737928
\(596\) −8.66025 −0.354738
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) −13.8564 −0.566157 −0.283079 0.959097i \(-0.591356\pi\)
−0.283079 + 0.959097i \(0.591356\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) −6.92820 −0.282372
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 1.73205 0.0704179
\(606\) 0 0
\(607\) 26.0000 1.05531 0.527654 0.849460i \(-0.323072\pi\)
0.527654 + 0.849460i \(0.323072\pi\)
\(608\) 10.3923 0.421464
\(609\) 0 0
\(610\) 21.0000 0.850265
\(611\) 6.92820 0.280285
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 27.7128 1.11840
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) 12.1244 0.488108 0.244054 0.969762i \(-0.421523\pi\)
0.244054 + 0.969762i \(0.421523\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 13.8564 0.556487
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 10.3923 0.416359
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 43.3013 1.73067
\(627\) 0 0
\(628\) 17.0000 0.678374
\(629\) 36.3731 1.45029
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 3.46410 0.137795
\(633\) 0 0
\(634\) −15.0000 −0.595726
\(635\) 3.46410 0.137469
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 10.3923 0.411435
\(639\) 0 0
\(640\) −21.0000 −0.830098
\(641\) −22.5167 −0.889355 −0.444677 0.895691i \(-0.646682\pi\)
−0.444677 + 0.895691i \(0.646682\pi\)
\(642\) 0 0
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) −6.92820 −0.273009
\(645\) 0 0
\(646\) 18.0000 0.708201
\(647\) 31.1769 1.22569 0.612845 0.790203i \(-0.290025\pi\)
0.612845 + 0.790203i \(0.290025\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) −3.46410 −0.135873
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −13.8564 −0.542243 −0.271122 0.962545i \(-0.587395\pi\)
−0.271122 + 0.962545i \(0.587395\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) −34.6410 −1.35250
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) 3.46410 0.134942 0.0674711 0.997721i \(-0.478507\pi\)
0.0674711 + 0.997721i \(0.478507\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) −3.46410 −0.134636
\(663\) 0 0
\(664\) 24.0000 0.931381
\(665\) 6.92820 0.268664
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) 17.3205 0.670151
\(669\) 0 0
\(670\) 30.0000 1.15900
\(671\) −24.2487 −0.936111
\(672\) 0 0
\(673\) −25.0000 −0.963679 −0.481840 0.876259i \(-0.660031\pi\)
−0.481840 + 0.876259i \(0.660031\pi\)
\(674\) −45.0333 −1.73462
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 13.8564 0.532545 0.266272 0.963898i \(-0.414208\pi\)
0.266272 + 0.963898i \(0.414208\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) −15.5885 −0.597790
\(681\) 0 0
\(682\) −48.0000 −1.83801
\(683\) 20.7846 0.795301 0.397650 0.917537i \(-0.369826\pi\)
0.397650 + 0.917537i \(0.369826\pi\)
\(684\) 0 0
\(685\) −3.00000 −0.114624
\(686\) 34.6410 1.32260
\(687\) 0 0
\(688\) −10.0000 −0.381246
\(689\) 0 0
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 19.0526 0.724270
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) 13.8564 0.525603
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −3.46410 −0.131118
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −46.7654 −1.76630 −0.883152 0.469087i \(-0.844583\pi\)
−0.883152 + 0.469087i \(0.844583\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 3.46410 0.130558
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) −13.8564 −0.521124
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) −31.1769 −1.17005
\(711\) 0 0
\(712\) 9.00000 0.337289
\(713\) −27.7128 −1.03785
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) −20.7846 −0.776757
\(717\) 0 0
\(718\) 18.0000 0.671754
\(719\) 10.3923 0.387568 0.193784 0.981044i \(-0.437924\pi\)
0.193784 + 0.981044i \(0.437924\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 25.9808 0.966904
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 3.46410 0.128654
\(726\) 0 0
\(727\) −34.0000 −1.26099 −0.630495 0.776193i \(-0.717148\pi\)
−0.630495 + 0.776193i \(0.717148\pi\)
\(728\) −3.46410 −0.128388
\(729\) 0 0
\(730\) 21.0000 0.777245
\(731\) −10.3923 −0.384373
\(732\) 0 0
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) −34.6410 −1.27862
\(735\) 0 0
\(736\) −18.0000 −0.663489
\(737\) −34.6410 −1.27602
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −12.1244 −0.445700
\(741\) 0 0
\(742\) 0 0
\(743\) 6.92820 0.254171 0.127086 0.991892i \(-0.459438\pi\)
0.127086 + 0.991892i \(0.459438\pi\)
\(744\) 0 0
\(745\) −15.0000 −0.549557
\(746\) 17.3205 0.634149
\(747\) 0 0
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 34.6410 1.26323
\(753\) 0 0
\(754\) −3.00000 −0.109254
\(755\) 34.6410 1.26072
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 27.7128 1.00657
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) −29.4449 −1.06738 −0.533688 0.845682i \(-0.679194\pi\)
−0.533688 + 0.845682i \(0.679194\pi\)
\(762\) 0 0
\(763\) 22.0000 0.796453
\(764\) −17.3205 −0.626634
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) 13.8564 0.500326
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) −20.7846 −0.749025
\(771\) 0 0
\(772\) −1.00000 −0.0359908
\(773\) 25.9808 0.934463 0.467232 0.884135i \(-0.345251\pi\)
0.467232 + 0.884135i \(0.345251\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 3.46410 0.124354
\(777\) 0 0
\(778\) 48.0000 1.72088
\(779\) 13.8564 0.496457
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) −31.1769 −1.11488
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) 29.4449 1.05093
\(786\) 0 0
\(787\) 26.0000 0.926800 0.463400 0.886149i \(-0.346629\pi\)
0.463400 + 0.886149i \(0.346629\pi\)
\(788\) 5.19615 0.185105
\(789\) 0 0
\(790\) −6.00000 −0.213470
\(791\) 3.46410 0.123169
\(792\) 0 0
\(793\) 7.00000 0.248577
\(794\) −50.2295 −1.78258
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 53.6936 1.90192 0.950962 0.309308i \(-0.100097\pi\)
0.950962 + 0.309308i \(0.100097\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) −10.3923 −0.367423
\(801\) 0 0
\(802\) −21.0000 −0.741536
\(803\) −24.2487 −0.855718
\(804\) 0 0
\(805\) −12.0000 −0.422944
\(806\) 13.8564 0.488071
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) 46.7654 1.64418 0.822091 0.569355i \(-0.192807\pi\)
0.822091 + 0.569355i \(0.192807\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −3.46410 −0.121566
\(813\) 0 0
\(814\) 42.0000 1.47210
\(815\) −27.7128 −0.970737
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 32.9090 1.15063
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) −12.1244 −0.423143 −0.211571 0.977363i \(-0.567858\pi\)
−0.211571 + 0.977363i \(0.567858\pi\)
\(822\) 0 0
\(823\) −28.0000 −0.976019 −0.488009 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) 13.8564 0.482711
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) 10.3923 0.361376 0.180688 0.983540i \(-0.442168\pi\)
0.180688 + 0.983540i \(0.442168\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) −41.5692 −1.44289
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 15.5885 0.540108
\(834\) 0 0
\(835\) 30.0000 1.03819
\(836\) 6.92820 0.239617
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 45.0333 1.55472 0.777361 0.629054i \(-0.216558\pi\)
0.777361 + 0.629054i \(0.216558\pi\)
\(840\) 0 0
\(841\) −26.0000 −0.896552
\(842\) 43.3013 1.49226
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) −20.7846 −0.715012
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) −18.0000 −0.617395
\(851\) 24.2487 0.831235
\(852\) 0 0
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 24.2487 0.829774
\(855\) 0 0
\(856\) 0 0
\(857\) −22.5167 −0.769154 −0.384577 0.923093i \(-0.625653\pi\)
−0.384577 + 0.923093i \(0.625653\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 3.46410 0.118125
\(861\) 0 0
\(862\) 0 0
\(863\) −31.1769 −1.06127 −0.530637 0.847599i \(-0.678047\pi\)
−0.530637 + 0.847599i \(0.678047\pi\)
\(864\) 0 0
\(865\) 33.0000 1.12203
\(866\) −19.0526 −0.647432
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 6.92820 0.235023
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) 19.0526 0.645201
\(873\) 0 0
\(874\) 12.0000 0.405906
\(875\) −24.2487 −0.819756
\(876\) 0 0
\(877\) 53.0000 1.78968 0.894841 0.446384i \(-0.147289\pi\)
0.894841 + 0.446384i \(0.147289\pi\)
\(878\) −34.6410 −1.16908
\(879\) 0 0
\(880\) −30.0000 −1.01130
\(881\) −20.7846 −0.700251 −0.350126 0.936703i \(-0.613861\pi\)
−0.350126 + 0.936703i \(0.613861\pi\)
\(882\) 0 0
\(883\) 56.0000 1.88455 0.942275 0.334840i \(-0.108682\pi\)
0.942275 + 0.334840i \(0.108682\pi\)
\(884\) 5.19615 0.174766
\(885\) 0 0
\(886\) −60.0000 −2.01574
\(887\) −3.46410 −0.116313 −0.0581566 0.998307i \(-0.518522\pi\)
−0.0581566 + 0.998307i \(0.518522\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) −15.5885 −0.522526
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) −13.8564 −0.463687
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) 36.0000 1.20134
\(899\) −13.8564 −0.462137
\(900\) 0 0
\(901\) 0 0
\(902\) −41.5692 −1.38410
\(903\) 0 0
\(904\) 3.00000 0.0997785
\(905\) 3.46410 0.115151
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 3.46410 0.114960
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) 24.2487 0.803396 0.401698 0.915772i \(-0.368420\pi\)
0.401698 + 0.915772i \(0.368420\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) −50.2295 −1.66144
\(915\) 0 0
\(916\) −1.00000 −0.0330409
\(917\) −6.92820 −0.228789
\(918\) 0 0
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) −10.3923 −0.342624
\(921\) 0 0
\(922\) −24.0000 −0.790398
\(923\) −10.3923 −0.342067
\(924\) 0 0
\(925\) 14.0000 0.460317
\(926\) −13.8564 −0.455350
\(927\) 0 0
\(928\) −9.00000 −0.295439
\(929\) 50.2295 1.64798 0.823988 0.566608i \(-0.191744\pi\)
0.823988 + 0.566608i \(0.191744\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 25.9808 0.851028
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) −31.1769 −1.01959
\(936\) 0 0
\(937\) −25.0000 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(938\) 34.6410 1.13107
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) −50.2295 −1.63743 −0.818717 0.574197i \(-0.805314\pi\)
−0.818717 + 0.574197i \(0.805314\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 69.2820 2.25494
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −17.3205 −0.562841 −0.281420 0.959585i \(-0.590806\pi\)
−0.281420 + 0.959585i \(0.590806\pi\)
\(948\) 0 0
\(949\) 7.00000 0.227230
\(950\) 6.92820 0.224781
\(951\) 0 0
\(952\) −18.0000 −0.583383
\(953\) 5.19615 0.168320 0.0841599 0.996452i \(-0.473179\pi\)
0.0841599 + 0.996452i \(0.473179\pi\)
\(954\) 0 0
\(955\) −30.0000 −0.970777
\(956\) 27.7128 0.896296
\(957\) 0 0
\(958\) −42.0000 −1.35696
\(959\) −3.46410 −0.111862
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −12.1244 −0.390905
\(963\) 0 0
\(964\) 29.0000 0.934027
\(965\) −1.73205 −0.0557567
\(966\) 0 0
\(967\) −46.0000 −1.47926 −0.739630 0.673014i \(-0.765000\pi\)
−0.739630 + 0.673014i \(0.765000\pi\)
\(968\) 1.73205 0.0556702
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) −31.1769 −1.00051 −0.500257 0.865877i \(-0.666761\pi\)
−0.500257 + 0.865877i \(0.666761\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 27.7128 0.887976
\(975\) 0 0
\(976\) 35.0000 1.12032
\(977\) 48.4974 1.55157 0.775785 0.630997i \(-0.217354\pi\)
0.775785 + 0.630997i \(0.217354\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) −5.19615 −0.165985
\(981\) 0 0
\(982\) −30.0000 −0.957338
\(983\) 34.6410 1.10488 0.552438 0.833554i \(-0.313697\pi\)
0.552438 + 0.833554i \(0.313697\pi\)
\(984\) 0 0
\(985\) 9.00000 0.286764
\(986\) −15.5885 −0.496438
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) −6.92820 −0.220304
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 41.5692 1.31982
\(993\) 0 0
\(994\) −36.0000 −1.14185
\(995\) 34.6410 1.09819
\(996\) 0 0
\(997\) −7.00000 −0.221692 −0.110846 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(998\) 17.3205 0.548271
\(999\) 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 81.2.a.a.1.1 2
3.2 odd 2 inner 81.2.a.a.1.2 yes 2
4.3 odd 2 1296.2.a.o.1.2 2
5.2 odd 4 2025.2.b.k.649.2 4
5.3 odd 4 2025.2.b.k.649.3 4
5.4 even 2 2025.2.a.j.1.2 2
7.6 odd 2 3969.2.a.i.1.1 2
8.3 odd 2 5184.2.a.bq.1.1 2
8.5 even 2 5184.2.a.br.1.1 2
9.2 odd 6 81.2.c.b.28.1 4
9.4 even 3 81.2.c.b.55.2 4
9.5 odd 6 81.2.c.b.55.1 4
9.7 even 3 81.2.c.b.28.2 4
11.10 odd 2 9801.2.a.v.1.2 2
12.11 even 2 1296.2.a.o.1.1 2
15.2 even 4 2025.2.b.k.649.4 4
15.8 even 4 2025.2.b.k.649.1 4
15.14 odd 2 2025.2.a.j.1.1 2
21.20 even 2 3969.2.a.i.1.2 2
24.5 odd 2 5184.2.a.br.1.2 2
24.11 even 2 5184.2.a.bq.1.2 2
27.2 odd 18 729.2.e.o.568.1 12
27.4 even 9 729.2.e.o.406.1 12
27.5 odd 18 729.2.e.o.649.2 12
27.7 even 9 729.2.e.o.325.1 12
27.11 odd 18 729.2.e.o.82.2 12
27.13 even 9 729.2.e.o.163.2 12
27.14 odd 18 729.2.e.o.163.1 12
27.16 even 9 729.2.e.o.82.1 12
27.20 odd 18 729.2.e.o.325.2 12
27.22 even 9 729.2.e.o.649.1 12
27.23 odd 18 729.2.e.o.406.2 12
27.25 even 9 729.2.e.o.568.2 12
33.32 even 2 9801.2.a.v.1.1 2
36.7 odd 6 1296.2.i.s.433.1 4
36.11 even 6 1296.2.i.s.433.2 4
36.23 even 6 1296.2.i.s.865.2 4
36.31 odd 6 1296.2.i.s.865.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.a.a.1.1 2 1.1 even 1 trivial
81.2.a.a.1.2 yes 2 3.2 odd 2 inner
81.2.c.b.28.1 4 9.2 odd 6
81.2.c.b.28.2 4 9.7 even 3
81.2.c.b.55.1 4 9.5 odd 6
81.2.c.b.55.2 4 9.4 even 3
729.2.e.o.82.1 12 27.16 even 9
729.2.e.o.82.2 12 27.11 odd 18
729.2.e.o.163.1 12 27.14 odd 18
729.2.e.o.163.2 12 27.13 even 9
729.2.e.o.325.1 12 27.7 even 9
729.2.e.o.325.2 12 27.20 odd 18
729.2.e.o.406.1 12 27.4 even 9
729.2.e.o.406.2 12 27.23 odd 18
729.2.e.o.568.1 12 27.2 odd 18
729.2.e.o.568.2 12 27.25 even 9
729.2.e.o.649.1 12 27.22 even 9
729.2.e.o.649.2 12 27.5 odd 18
1296.2.a.o.1.1 2 12.11 even 2
1296.2.a.o.1.2 2 4.3 odd 2
1296.2.i.s.433.1 4 36.7 odd 6
1296.2.i.s.433.2 4 36.11 even 6
1296.2.i.s.865.1 4 36.31 odd 6
1296.2.i.s.865.2 4 36.23 even 6
2025.2.a.j.1.1 2 15.14 odd 2
2025.2.a.j.1.2 2 5.4 even 2
2025.2.b.k.649.1 4 15.8 even 4
2025.2.b.k.649.2 4 5.2 odd 4
2025.2.b.k.649.3 4 5.3 odd 4
2025.2.b.k.649.4 4 15.2 even 4
3969.2.a.i.1.1 2 7.6 odd 2
3969.2.a.i.1.2 2 21.20 even 2
5184.2.a.bq.1.1 2 8.3 odd 2
5184.2.a.bq.1.2 2 24.11 even 2
5184.2.a.br.1.1 2 8.5 even 2
5184.2.a.br.1.2 2 24.5 odd 2
9801.2.a.v.1.1 2 33.32 even 2
9801.2.a.v.1.2 2 11.10 odd 2