Properties

Label 1350.2.a.x.1.1
Level $1350$
Weight $2$
Character 1350.1
Self dual yes
Analytic conductor $10.780$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1350,2,Mod(1,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.35890\) of defining polynomial
Character \(\chi\) \(=\) 1350.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.35890 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.35890 q^{7} +1.00000 q^{8} +4.35890 q^{11} -4.35890 q^{14} +1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{19} +4.35890 q^{22} -2.00000 q^{23} -4.35890 q^{28} +7.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +8.71780 q^{37} +6.00000 q^{38} -8.71780 q^{41} +8.71780 q^{43} +4.35890 q^{44} -2.00000 q^{46} +2.00000 q^{47} +12.0000 q^{49} -3.00000 q^{53} -4.35890 q^{56} +8.71780 q^{59} -4.00000 q^{61} +7.00000 q^{62} +1.00000 q^{64} -8.71780 q^{67} +4.00000 q^{68} -4.35890 q^{73} +8.71780 q^{74} +6.00000 q^{76} -19.0000 q^{77} -8.71780 q^{82} -5.00000 q^{83} +8.71780 q^{86} +4.35890 q^{88} -8.71780 q^{89} -2.00000 q^{92} +2.00000 q^{94} +4.35890 q^{97} +12.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} + 8 q^{17} + 12 q^{19} - 4 q^{23} + 14 q^{31} + 2 q^{32} + 8 q^{34} + 12 q^{38} - 4 q^{46} + 4 q^{47} + 24 q^{49} - 6 q^{53} - 8 q^{61} + 14 q^{62} + 2 q^{64} + 8 q^{68} + 12 q^{76} - 38 q^{77} - 10 q^{83} - 4 q^{92} + 4 q^{94} + 24 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −4.35890 −1.64751 −0.823754 0.566947i \(-0.808125\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 4.35890 1.31426 0.657129 0.753778i \(-0.271771\pi\)
0.657129 + 0.753778i \(0.271771\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −4.35890 −1.16496
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.35890 0.929320
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −4.35890 −0.823754
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) 8.71780 1.43320 0.716599 0.697486i \(-0.245698\pi\)
0.716599 + 0.697486i \(0.245698\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) −8.71780 −1.36149 −0.680746 0.732520i \(-0.738344\pi\)
−0.680746 + 0.732520i \(0.738344\pi\)
\(42\) 0 0
\(43\) 8.71780 1.32945 0.664726 0.747087i \(-0.268548\pi\)
0.664726 + 0.747087i \(0.268548\pi\)
\(44\) 4.35890 0.657129
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 12.0000 1.71429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.35890 −0.582482
\(57\) 0 0
\(58\) 0 0
\(59\) 8.71780 1.13496 0.567480 0.823387i \(-0.307918\pi\)
0.567480 + 0.823387i \(0.307918\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 7.00000 0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.71780 −1.06505 −0.532524 0.846415i \(-0.678756\pi\)
−0.532524 + 0.846415i \(0.678756\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −4.35890 −0.510171 −0.255085 0.966919i \(-0.582104\pi\)
−0.255085 + 0.966919i \(0.582104\pi\)
\(74\) 8.71780 1.01342
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) −19.0000 −2.16525
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −8.71780 −0.962720
\(83\) −5.00000 −0.548821 −0.274411 0.961613i \(-0.588483\pi\)
−0.274411 + 0.961613i \(0.588483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.71780 0.940064
\(87\) 0 0
\(88\) 4.35890 0.464660
\(89\) −8.71780 −0.924085 −0.462042 0.886858i \(-0.652883\pi\)
−0.462042 + 0.886858i \(0.652883\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) 0 0
\(97\) 4.35890 0.442579 0.221290 0.975208i \(-0.428973\pi\)
0.221290 + 0.975208i \(0.428973\pi\)
\(98\) 12.0000 1.21218
\(99\) 0 0
\(100\) 0 0
\(101\) −4.35890 −0.433727 −0.216863 0.976202i \(-0.569583\pi\)
−0.216863 + 0.976202i \(0.569583\pi\)
\(102\) 0 0
\(103\) −8.71780 −0.858990 −0.429495 0.903069i \(-0.641308\pi\)
−0.429495 + 0.903069i \(0.641308\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) 15.0000 1.45010 0.725052 0.688694i \(-0.241816\pi\)
0.725052 + 0.688694i \(0.241816\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.35890 −0.411877
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 8.71780 0.802538
\(119\) −17.4356 −1.59832
\(120\) 0 0
\(121\) 8.00000 0.727273
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) 7.00000 0.628619
\(125\) 0 0
\(126\) 0 0
\(127\) −4.35890 −0.386790 −0.193395 0.981121i \(-0.561950\pi\)
−0.193395 + 0.981121i \(0.561950\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 13.0767 1.14252 0.571258 0.820770i \(-0.306456\pi\)
0.571258 + 0.820770i \(0.306456\pi\)
\(132\) 0 0
\(133\) −26.1534 −2.26779
\(134\) −8.71780 −0.753103
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −4.35890 −0.360745
\(147\) 0 0
\(148\) 8.71780 0.716599
\(149\) −4.35890 −0.357095 −0.178547 0.983931i \(-0.557140\pi\)
−0.178547 + 0.983931i \(0.557140\pi\)
\(150\) 0 0
\(151\) −11.0000 −0.895167 −0.447584 0.894242i \(-0.647715\pi\)
−0.447584 + 0.894242i \(0.647715\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) −19.0000 −1.53106
\(155\) 0 0
\(156\) 0 0
\(157\) 17.4356 1.39151 0.695756 0.718278i \(-0.255069\pi\)
0.695756 + 0.718278i \(0.255069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.71780 0.687059
\(162\) 0 0
\(163\) 17.4356 1.36566 0.682831 0.730577i \(-0.260749\pi\)
0.682831 + 0.730577i \(0.260749\pi\)
\(164\) −8.71780 −0.680746
\(165\) 0 0
\(166\) −5.00000 −0.388075
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 8.71780 0.664726
\(173\) −19.0000 −1.44454 −0.722272 0.691609i \(-0.756902\pi\)
−0.722272 + 0.691609i \(0.756902\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.35890 0.328564
\(177\) 0 0
\(178\) −8.71780 −0.653427
\(179\) 21.7945 1.62900 0.814499 0.580166i \(-0.197012\pi\)
0.814499 + 0.580166i \(0.197012\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.00000 −0.147442
\(185\) 0 0
\(186\) 0 0
\(187\) 17.4356 1.27502
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 21.7945 1.56880 0.784401 0.620254i \(-0.212970\pi\)
0.784401 + 0.620254i \(0.212970\pi\)
\(194\) 4.35890 0.312951
\(195\) 0 0
\(196\) 12.0000 0.857143
\(197\) 5.00000 0.356235 0.178118 0.984009i \(-0.442999\pi\)
0.178118 + 0.984009i \(0.442999\pi\)
\(198\) 0 0
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −4.35890 −0.306691
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −8.71780 −0.607398
\(207\) 0 0
\(208\) 0 0
\(209\) 26.1534 1.80907
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) −3.00000 −0.206041
\(213\) 0 0
\(214\) 15.0000 1.02538
\(215\) 0 0
\(216\) 0 0
\(217\) −30.5123 −2.07131
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.71780 0.583787 0.291893 0.956451i \(-0.405715\pi\)
0.291893 + 0.956451i \(0.405715\pi\)
\(224\) −4.35890 −0.291241
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 8.71780 0.567480
\(237\) 0 0
\(238\) −17.4356 −1.13018
\(239\) −8.71780 −0.563907 −0.281954 0.959428i \(-0.590982\pi\)
−0.281954 + 0.959428i \(0.590982\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 8.00000 0.514259
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 7.00000 0.444500
\(249\) 0 0
\(250\) 0 0
\(251\) −26.1534 −1.65079 −0.825394 0.564557i \(-0.809047\pi\)
−0.825394 + 0.564557i \(0.809047\pi\)
\(252\) 0 0
\(253\) −8.71780 −0.548083
\(254\) −4.35890 −0.273502
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) −38.0000 −2.36121
\(260\) 0 0
\(261\) 0 0
\(262\) 13.0767 0.807881
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −26.1534 −1.60357
\(267\) 0 0
\(268\) −8.71780 −0.532524
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −5.00000 −0.303728 −0.151864 0.988401i \(-0.548528\pi\)
−0.151864 + 0.988401i \(0.548528\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) −17.4356 −1.04760 −0.523802 0.851840i \(-0.675487\pi\)
−0.523802 + 0.851840i \(0.675487\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −8.71780 −0.518219 −0.259110 0.965848i \(-0.583429\pi\)
−0.259110 + 0.965848i \(0.583429\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 38.0000 2.24307
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) −4.35890 −0.255085
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.71780 0.506712
\(297\) 0 0
\(298\) −4.35890 −0.252504
\(299\) 0 0
\(300\) 0 0
\(301\) −38.0000 −2.19028
\(302\) −11.0000 −0.632979
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −19.0000 −1.08263
\(309\) 0 0
\(310\) 0 0
\(311\) −26.1534 −1.48302 −0.741511 0.670940i \(-0.765891\pi\)
−0.741511 + 0.670940i \(0.765891\pi\)
\(312\) 0 0
\(313\) 21.7945 1.23190 0.615949 0.787786i \(-0.288773\pi\)
0.615949 + 0.787786i \(0.288773\pi\)
\(314\) 17.4356 0.983948
\(315\) 0 0
\(316\) 0 0
\(317\) −17.0000 −0.954815 −0.477408 0.878682i \(-0.658423\pi\)
−0.477408 + 0.878682i \(0.658423\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 8.71780 0.485824
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 0 0
\(326\) 17.4356 0.965668
\(327\) 0 0
\(328\) −8.71780 −0.481360
\(329\) −8.71780 −0.480628
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −5.00000 −0.274411
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) −17.4356 −0.949777 −0.474889 0.880046i \(-0.657512\pi\)
−0.474889 + 0.880046i \(0.657512\pi\)
\(338\) −13.0000 −0.707107
\(339\) 0 0
\(340\) 0 0
\(341\) 30.5123 1.65233
\(342\) 0 0
\(343\) −21.7945 −1.17679
\(344\) 8.71780 0.470032
\(345\) 0 0
\(346\) −19.0000 −1.02145
\(347\) 27.0000 1.44944 0.724718 0.689046i \(-0.241970\pi\)
0.724718 + 0.689046i \(0.241970\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.35890 0.232330
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8.71780 −0.462042
\(357\) 0 0
\(358\) 21.7945 1.15187
\(359\) −26.1534 −1.38032 −0.690162 0.723655i \(-0.742461\pi\)
−0.690162 + 0.723655i \(0.742461\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 8.00000 0.420471
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.35890 −0.227533 −0.113766 0.993508i \(-0.536292\pi\)
−0.113766 + 0.993508i \(0.536292\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) 0 0
\(371\) 13.0767 0.678908
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 17.4356 0.901573
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) 0 0
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 21.7945 1.10931
\(387\) 0 0
\(388\) 4.35890 0.221290
\(389\) 4.35890 0.221005 0.110502 0.993876i \(-0.464754\pi\)
0.110502 + 0.993876i \(0.464754\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 12.0000 0.606092
\(393\) 0 0
\(394\) 5.00000 0.251896
\(395\) 0 0
\(396\) 0 0
\(397\) 34.8712 1.75013 0.875067 0.484001i \(-0.160817\pi\)
0.875067 + 0.484001i \(0.160817\pi\)
\(398\) −3.00000 −0.150376
\(399\) 0 0
\(400\) 0 0
\(401\) 34.8712 1.74138 0.870692 0.491828i \(-0.163671\pi\)
0.870692 + 0.491828i \(0.163671\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −4.35890 −0.216863
\(405\) 0 0
\(406\) 0 0
\(407\) 38.0000 1.88359
\(408\) 0 0
\(409\) −31.0000 −1.53285 −0.766426 0.642333i \(-0.777967\pi\)
−0.766426 + 0.642333i \(0.777967\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.71780 −0.429495
\(413\) −38.0000 −1.86986
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 26.1534 1.27920
\(419\) −8.71780 −0.425892 −0.212946 0.977064i \(-0.568306\pi\)
−0.212946 + 0.977064i \(0.568306\pi\)
\(420\) 0 0
\(421\) 16.0000 0.779792 0.389896 0.920859i \(-0.372511\pi\)
0.389896 + 0.920859i \(0.372511\pi\)
\(422\) 6.00000 0.292075
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 0 0
\(427\) 17.4356 0.843768
\(428\) 15.0000 0.725052
\(429\) 0 0
\(430\) 0 0
\(431\) −8.71780 −0.419922 −0.209961 0.977710i \(-0.567334\pi\)
−0.209961 + 0.977710i \(0.567334\pi\)
\(432\) 0 0
\(433\) 21.7945 1.04738 0.523688 0.851910i \(-0.324556\pi\)
0.523688 + 0.851910i \(0.324556\pi\)
\(434\) −30.5123 −1.46464
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) 9.00000 0.429547 0.214773 0.976664i \(-0.431099\pi\)
0.214773 + 0.976664i \(0.431099\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 8.71780 0.412800
\(447\) 0 0
\(448\) −4.35890 −0.205939
\(449\) −26.1534 −1.23425 −0.617127 0.786863i \(-0.711704\pi\)
−0.617127 + 0.786863i \(0.711704\pi\)
\(450\) 0 0
\(451\) −38.0000 −1.78935
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) −4.35890 −0.203901 −0.101950 0.994789i \(-0.532508\pi\)
−0.101950 + 0.994789i \(0.532508\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) 0 0
\(461\) −39.2301 −1.82713 −0.913564 0.406696i \(-0.866681\pi\)
−0.913564 + 0.406696i \(0.866681\pi\)
\(462\) 0 0
\(463\) −13.0767 −0.607726 −0.303863 0.952716i \(-0.598276\pi\)
−0.303863 + 0.952716i \(0.598276\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 0 0
\(469\) 38.0000 1.75468
\(470\) 0 0
\(471\) 0 0
\(472\) 8.71780 0.401269
\(473\) 38.0000 1.74724
\(474\) 0 0
\(475\) 0 0
\(476\) −17.4356 −0.799159
\(477\) 0 0
\(478\) −8.71780 −0.398743
\(479\) −8.71780 −0.398326 −0.199163 0.979966i \(-0.563822\pi\)
−0.199163 + 0.979966i \(0.563822\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 8.00000 0.363636
\(485\) 0 0
\(486\) 0 0
\(487\) 26.1534 1.18512 0.592562 0.805525i \(-0.298117\pi\)
0.592562 + 0.805525i \(0.298117\pi\)
\(488\) −4.00000 −0.181071
\(489\) 0 0
\(490\) 0 0
\(491\) 13.0767 0.590143 0.295072 0.955475i \(-0.404657\pi\)
0.295072 + 0.955475i \(0.404657\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) 0 0
\(498\) 0 0
\(499\) 30.0000 1.34298 0.671492 0.741012i \(-0.265654\pi\)
0.671492 + 0.741012i \(0.265654\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −26.1534 −1.16728
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.71780 −0.387553
\(507\) 0 0
\(508\) −4.35890 −0.193395
\(509\) 13.0767 0.579614 0.289807 0.957085i \(-0.406409\pi\)
0.289807 + 0.957085i \(0.406409\pi\)
\(510\) 0 0
\(511\) 19.0000 0.840511
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) 8.71780 0.383408
\(518\) −38.0000 −1.66962
\(519\) 0 0
\(520\) 0 0
\(521\) −17.4356 −0.763867 −0.381934 0.924190i \(-0.624742\pi\)
−0.381934 + 0.924190i \(0.624742\pi\)
\(522\) 0 0
\(523\) −34.8712 −1.52481 −0.762405 0.647100i \(-0.775982\pi\)
−0.762405 + 0.647100i \(0.775982\pi\)
\(524\) 13.0767 0.571258
\(525\) 0 0
\(526\) −2.00000 −0.0872041
\(527\) 28.0000 1.21970
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) −26.1534 −1.13389
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −8.71780 −0.376552
\(537\) 0 0
\(538\) 0 0
\(539\) 52.3068 2.25301
\(540\) 0 0
\(541\) 40.0000 1.71973 0.859867 0.510518i \(-0.170546\pi\)
0.859867 + 0.510518i \(0.170546\pi\)
\(542\) −5.00000 −0.214768
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) −17.4356 −0.745492 −0.372746 0.927933i \(-0.621584\pi\)
−0.372746 + 0.927933i \(0.621584\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −17.4356 −0.740767
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −33.0000 −1.39825 −0.699127 0.714997i \(-0.746428\pi\)
−0.699127 + 0.714997i \(0.746428\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.71780 −0.366436
\(567\) 0 0
\(568\) 0 0
\(569\) 17.4356 0.730938 0.365469 0.930823i \(-0.380909\pi\)
0.365469 + 0.930823i \(0.380909\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 38.0000 1.58609
\(575\) 0 0
\(576\) 0 0
\(577\) −17.4356 −0.725853 −0.362927 0.931818i \(-0.618222\pi\)
−0.362927 + 0.931818i \(0.618222\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) 21.7945 0.904188
\(582\) 0 0
\(583\) −13.0767 −0.541581
\(584\) −4.35890 −0.180373
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) 42.0000 1.73058
\(590\) 0 0
\(591\) 0 0
\(592\) 8.71780 0.358299
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.35890 −0.178547
\(597\) 0 0
\(598\) 0 0
\(599\) 26.1534 1.06860 0.534299 0.845295i \(-0.320576\pi\)
0.534299 + 0.845295i \(0.320576\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) −38.0000 −1.54876
\(603\) 0 0
\(604\) −11.0000 −0.447584
\(605\) 0 0
\(606\) 0 0
\(607\) −8.71780 −0.353845 −0.176922 0.984225i \(-0.556614\pi\)
−0.176922 + 0.984225i \(0.556614\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.71780 0.352109 0.176054 0.984380i \(-0.443667\pi\)
0.176054 + 0.984380i \(0.443667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −19.0000 −0.765532
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −26.1534 −1.04866
\(623\) 38.0000 1.52244
\(624\) 0 0
\(625\) 0 0
\(626\) 21.7945 0.871083
\(627\) 0 0
\(628\) 17.4356 0.695756
\(629\) 34.8712 1.39041
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −17.0000 −0.675156
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.71780 −0.344332 −0.172166 0.985068i \(-0.555077\pi\)
−0.172166 + 0.985068i \(0.555077\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 8.71780 0.343529
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 38.0000 1.49163
\(650\) 0 0
\(651\) 0 0
\(652\) 17.4356 0.682831
\(653\) −35.0000 −1.36966 −0.684828 0.728705i \(-0.740123\pi\)
−0.684828 + 0.728705i \(0.740123\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −8.71780 −0.340373
\(657\) 0 0
\(658\) −8.71780 −0.339855
\(659\) −4.35890 −0.169799 −0.0848993 0.996390i \(-0.527057\pi\)
−0.0848993 + 0.996390i \(0.527057\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) −5.00000 −0.194038
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) 0 0
\(671\) −17.4356 −0.673094
\(672\) 0 0
\(673\) 13.0767 0.504070 0.252035 0.967718i \(-0.418900\pi\)
0.252035 + 0.967718i \(0.418900\pi\)
\(674\) −17.4356 −0.671594
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 0 0
\(679\) −19.0000 −0.729153
\(680\) 0 0
\(681\) 0 0
\(682\) 30.5123 1.16838
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −21.7945 −0.832118
\(687\) 0 0
\(688\) 8.71780 0.332363
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −19.0000 −0.722272
\(693\) 0 0
\(694\) 27.0000 1.02491
\(695\) 0 0
\(696\) 0 0
\(697\) −34.8712 −1.32084
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) 0 0
\(701\) −21.7945 −0.823167 −0.411583 0.911372i \(-0.635024\pi\)
−0.411583 + 0.911372i \(0.635024\pi\)
\(702\) 0 0
\(703\) 52.3068 1.97279
\(704\) 4.35890 0.164282
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) 19.0000 0.714569
\(708\) 0 0
\(709\) 12.0000 0.450669 0.225335 0.974281i \(-0.427652\pi\)
0.225335 + 0.974281i \(0.427652\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −8.71780 −0.326713
\(713\) −14.0000 −0.524304
\(714\) 0 0
\(715\) 0 0
\(716\) 21.7945 0.814499
\(717\) 0 0
\(718\) −26.1534 −0.976036
\(719\) −34.8712 −1.30048 −0.650238 0.759731i \(-0.725331\pi\)
−0.650238 + 0.759731i \(0.725331\pi\)
\(720\) 0 0
\(721\) 38.0000 1.41519
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) 0 0
\(727\) 39.2301 1.45496 0.727482 0.686127i \(-0.240691\pi\)
0.727482 + 0.686127i \(0.240691\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 34.8712 1.28976
\(732\) 0 0
\(733\) −8.71780 −0.321999 −0.161000 0.986954i \(-0.551472\pi\)
−0.161000 + 0.986954i \(0.551472\pi\)
\(734\) −4.35890 −0.160890
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) −38.0000 −1.39975
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 13.0767 0.480061
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 17.4356 0.637509
\(749\) −65.3835 −2.38906
\(750\) 0 0
\(751\) 35.0000 1.27717 0.638584 0.769552i \(-0.279520\pi\)
0.638584 + 0.769552i \(0.279520\pi\)
\(752\) 2.00000 0.0729325
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.71780 0.316854 0.158427 0.987371i \(-0.449358\pi\)
0.158427 + 0.987371i \(0.449358\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) −52.3068 −1.89612 −0.948060 0.318092i \(-0.896958\pi\)
−0.948060 + 0.318092i \(0.896958\pi\)
\(762\) 0 0
\(763\) −43.5890 −1.57803
\(764\) 0 0
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 0 0
\(768\) 0 0
\(769\) 9.00000 0.324548 0.162274 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 21.7945 0.784401
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 4.35890 0.156475
\(777\) 0 0
\(778\) 4.35890 0.156274
\(779\) −52.3068 −1.87409
\(780\) 0 0
\(781\) 0 0
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) 12.0000 0.428571
\(785\) 0 0
\(786\) 0 0
\(787\) −17.4356 −0.621512 −0.310756 0.950490i \(-0.600582\pi\)
−0.310756 + 0.950490i \(0.600582\pi\)
\(788\) 5.00000 0.178118
\(789\) 0 0
\(790\) 0 0
\(791\) 78.4602 2.78972
\(792\) 0 0
\(793\) 0 0
\(794\) 34.8712 1.23753
\(795\) 0 0
\(796\) −3.00000 −0.106332
\(797\) 37.0000 1.31061 0.655304 0.755366i \(-0.272541\pi\)
0.655304 + 0.755366i \(0.272541\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 34.8712 1.23134
\(803\) −19.0000 −0.670495
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −4.35890 −0.153346
\(809\) −26.1534 −0.919504 −0.459752 0.888047i \(-0.652062\pi\)
−0.459752 + 0.888047i \(0.652062\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 38.0000 1.33190
\(815\) 0 0
\(816\) 0 0
\(817\) 52.3068 1.82998
\(818\) −31.0000 −1.08389
\(819\) 0 0
\(820\) 0 0
\(821\) 34.8712 1.21701 0.608506 0.793549i \(-0.291769\pi\)
0.608506 + 0.793549i \(0.291769\pi\)
\(822\) 0 0
\(823\) 39.2301 1.36747 0.683737 0.729728i \(-0.260353\pi\)
0.683737 + 0.729728i \(0.260353\pi\)
\(824\) −8.71780 −0.303699
\(825\) 0 0
\(826\) −38.0000 −1.32219
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 0 0
\(829\) −42.0000 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 48.0000 1.66310
\(834\) 0 0
\(835\) 0 0
\(836\) 26.1534 0.904534
\(837\) 0 0
\(838\) −8.71780 −0.301151
\(839\) 34.8712 1.20389 0.601944 0.798539i \(-0.294393\pi\)
0.601944 + 0.798539i \(0.294393\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 16.0000 0.551396
\(843\) 0 0
\(844\) 6.00000 0.206529
\(845\) 0 0
\(846\) 0 0
\(847\) −34.8712 −1.19819
\(848\) −3.00000 −0.103020
\(849\) 0 0
\(850\) 0 0
\(851\) −17.4356 −0.597685
\(852\) 0 0
\(853\) −8.71780 −0.298492 −0.149246 0.988800i \(-0.547685\pi\)
−0.149246 + 0.988800i \(0.547685\pi\)
\(854\) 17.4356 0.596634
\(855\) 0 0
\(856\) 15.0000 0.512689
\(857\) 48.0000 1.63965 0.819824 0.572615i \(-0.194071\pi\)
0.819824 + 0.572615i \(0.194071\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −8.71780 −0.296929
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 21.7945 0.740607
\(867\) 0 0
\(868\) −30.5123 −1.03565
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) −8.71780 −0.294379 −0.147190 0.989108i \(-0.547023\pi\)
−0.147190 + 0.989108i \(0.547023\pi\)
\(878\) 9.00000 0.303735
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 26.1534 0.880132 0.440066 0.897965i \(-0.354955\pi\)
0.440066 + 0.897965i \(0.354955\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 19.0000 0.637240
\(890\) 0 0
\(891\) 0 0
\(892\) 8.71780 0.291893
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) −4.35890 −0.145621
\(897\) 0 0
\(898\) −26.1534 −0.872750
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) −38.0000 −1.26526
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) −26.1534 −0.868409 −0.434205 0.900814i \(-0.642971\pi\)
−0.434205 + 0.900814i \(0.642971\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) 0 0
\(911\) 26.1534 0.866501 0.433250 0.901274i \(-0.357367\pi\)
0.433250 + 0.901274i \(0.357367\pi\)
\(912\) 0 0
\(913\) −21.7945 −0.721292
\(914\) −4.35890 −0.144180
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −57.0000 −1.88231
\(918\) 0 0
\(919\) 39.0000 1.28649 0.643246 0.765660i \(-0.277587\pi\)
0.643246 + 0.765660i \(0.277587\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −39.2301 −1.29197
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −13.0767 −0.429727
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 72.0000 2.35970
\(932\) 14.0000 0.458585
\(933\) 0 0
\(934\) −21.0000 −0.687141
\(935\) 0 0
\(936\) 0 0
\(937\) −47.9479 −1.56639 −0.783195 0.621777i \(-0.786411\pi\)
−0.783195 + 0.621777i \(0.786411\pi\)
\(938\) 38.0000 1.24074
\(939\) 0 0
\(940\) 0 0
\(941\) −47.9479 −1.56306 −0.781528 0.623870i \(-0.785560\pi\)
−0.781528 + 0.623870i \(0.785560\pi\)
\(942\) 0 0
\(943\) 17.4356 0.567781
\(944\) 8.71780 0.283740
\(945\) 0 0
\(946\) 38.0000 1.23549
\(947\) 27.0000 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) −17.4356 −0.565091
\(953\) 32.0000 1.03658 0.518291 0.855204i \(-0.326568\pi\)
0.518291 + 0.855204i \(0.326568\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.71780 −0.281954
\(957\) 0 0
\(958\) −8.71780 −0.281659
\(959\) 78.4602 2.53361
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 30.5123 0.981209 0.490605 0.871382i \(-0.336776\pi\)
0.490605 + 0.871382i \(0.336776\pi\)
\(968\) 8.00000 0.257130
\(969\) 0 0
\(970\) 0 0
\(971\) −39.2301 −1.25895 −0.629477 0.777019i \(-0.716731\pi\)
−0.629477 + 0.777019i \(0.716731\pi\)
\(972\) 0 0
\(973\) −17.4356 −0.558960
\(974\) 26.1534 0.838009
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) −38.0000 −1.21449
\(980\) 0 0
\(981\) 0 0
\(982\) 13.0767 0.417294
\(983\) −34.0000 −1.08443 −0.542216 0.840239i \(-0.682414\pi\)
−0.542216 + 0.840239i \(0.682414\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −17.4356 −0.554420
\(990\) 0 0
\(991\) −53.0000 −1.68360 −0.841800 0.539789i \(-0.818504\pi\)
−0.841800 + 0.539789i \(0.818504\pi\)
\(992\) 7.00000 0.222250
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −52.3068 −1.65657 −0.828286 0.560305i \(-0.810684\pi\)
−0.828286 + 0.560305i \(0.810684\pi\)
\(998\) 30.0000 0.949633
\(999\) 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 1350.2.a.x.1.1 2
3.2 odd 2 1350.2.a.w.1.1 2
5.2 odd 4 270.2.c.c.109.4 yes 4
5.3 odd 4 270.2.c.c.109.2 yes 4
5.4 even 2 1350.2.a.w.1.2 2
15.2 even 4 270.2.c.c.109.1 4
15.8 even 4 270.2.c.c.109.3 yes 4
15.14 odd 2 inner 1350.2.a.x.1.2 2
20.3 even 4 2160.2.f.m.1729.4 4
20.7 even 4 2160.2.f.m.1729.3 4
45.2 even 12 810.2.i.h.109.2 8
45.7 odd 12 810.2.i.h.109.3 8
45.13 odd 12 810.2.i.h.379.3 8
45.22 odd 12 810.2.i.h.379.1 8
45.23 even 12 810.2.i.h.379.2 8
45.32 even 12 810.2.i.h.379.4 8
45.38 even 12 810.2.i.h.109.4 8
45.43 odd 12 810.2.i.h.109.1 8
60.23 odd 4 2160.2.f.m.1729.1 4
60.47 odd 4 2160.2.f.m.1729.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.2.c.c.109.1 4 15.2 even 4
270.2.c.c.109.2 yes 4 5.3 odd 4
270.2.c.c.109.3 yes 4 15.8 even 4
270.2.c.c.109.4 yes 4 5.2 odd 4
810.2.i.h.109.1 8 45.43 odd 12
810.2.i.h.109.2 8 45.2 even 12
810.2.i.h.109.3 8 45.7 odd 12
810.2.i.h.109.4 8 45.38 even 12
810.2.i.h.379.1 8 45.22 odd 12
810.2.i.h.379.2 8 45.23 even 12
810.2.i.h.379.3 8 45.13 odd 12
810.2.i.h.379.4 8 45.32 even 12
1350.2.a.w.1.1 2 3.2 odd 2
1350.2.a.w.1.2 2 5.4 even 2
1350.2.a.x.1.1 2 1.1 even 1 trivial
1350.2.a.x.1.2 2 15.14 odd 2 inner
2160.2.f.m.1729.1 4 60.23 odd 4
2160.2.f.m.1729.2 4 60.47 odd 4
2160.2.f.m.1729.3 4 20.7 even 4
2160.2.f.m.1729.4 4 20.3 even 4