Properties

Label 5775.2.a.q.1.1
Level $5775$
Weight $2$
Character 5775.1
Self dual yes
Analytic conductor $46.114$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5775,2,Mod(1,5775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("5775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.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: \(46.1136071673\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5775.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{21} -1.00000 q^{22} +3.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +5.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} -1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{48} +1.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} +6.00000 q^{61} -1.00000 q^{63} +7.00000 q^{64} +1.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} -3.00000 q^{72} -10.0000 q^{73} -6.00000 q^{74} -4.00000 q^{76} +1.00000 q^{77} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -1.00000 q^{84} +4.00000 q^{86} -6.00000 q^{87} +3.00000 q^{88} +10.0000 q^{89} -2.00000 q^{91} -5.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 1.00000 0.174078
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) −2.00000 −0.226455
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) 3.00000 0.319801
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000 0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.00000 0.543214
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −3.00000 −0.265165
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −4.00000 −0.346844
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) −1.00000 −0.0824786
\(148\) 6.00000 0.493197
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −12.0000 −0.973329
\(153\) −2.00000 −0.161690
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −3.00000 −0.231455
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −4.00000 −0.300658
\(178\) 10.0000 0.749532
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −2.00000 −0.148250
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −7.00000 −0.505181
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) −10.0000 −0.703598
\(203\) −6.00000 −0.421117
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 6.00000 0.402694
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) −18.0000 −1.18176
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −8.00000 −0.519656
\(238\) 2.00000 0.129641
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −4.00000 −0.249029
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 2.00000 0.121268
\(273\) 2.00000 0.121046
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 6.00000 0.354169
\(288\) 5.00000 0.294628
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 10.0000 0.585206
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) 1.00000 0.0580259
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) −16.0000 −0.920697
\(303\) 10.0000 0.574485
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 6.00000 0.339683
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −2.00000 −0.112154
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 2.00000 0.110600
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −4.00000 −0.219529
\(333\) −6.00000 −0.328798
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −5.00000 −0.266501
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) −2.00000 −0.105851
\(358\) −4.00000 −0.211407
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 1.00000 0.0514344
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 24.0000 1.22795
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 4.00000 0.203331
\(388\) 10.0000 0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 12.0000 0.605320
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 8.00000 0.401004
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 6.00000 0.297409
\(408\) −6.00000 −0.297044
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 8.00000 0.394132
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) 4.00000 0.195881
\(418\) −4.00000 −0.195646
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 4.00000 0.193347
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000 0.0481125
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −4.00000 −0.190261
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 18.0000 0.851371
\(448\) −7.00000 −0.330719
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −6.00000 −0.282216
\(453\) 16.0000 0.751746
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 22.0000 1.02799
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) −12.0000 −0.552345
\(473\) −4.00000 −0.183920
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 2.00000 0.0915737
\(478\) −24.0000 −1.09773
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −18.0000 −0.814822
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −6.00000 −0.270501
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) −12.0000 −0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) −11.0000 −0.486136
\(513\) −4.00000 −0.176604
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 6.00000 0.263625
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 6.00000 0.262613
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 4.00000 0.173422
\(533\) −12.0000 −0.519778
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 36.0000 1.55496
\(537\) 4.00000 0.172613
\(538\) −18.0000 −0.776035
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −16.0000 −0.687259
\(543\) 10.0000 0.429141
\(544\) −10.0000 −0.428746
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 2.00000 0.0854358
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) −30.0000 −1.26547
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 2.00000 0.0836242
\(573\) −24.0000 −1.00261
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −13.0000 −0.540729
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 10.0000 0.414513
\(583\) −2.00000 −0.0828315
\(584\) 30.0000 1.24141
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 6.00000 0.246598
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −4.00000 −0.163028
\(603\) −12.0000 −0.488678
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) 48.0000 1.94826 0.974130 0.225989i \(-0.0725612\pi\)
0.974130 + 0.225989i \(0.0725612\pi\)
\(608\) 20.0000 0.811107
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 8.00000 0.321807
\(619\) 12.0000 0.482321 0.241160 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −32.0000 −1.28308
\(623\) −10.0000 −0.400642
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 14.0000 0.559553
\(627\) 4.00000 0.159745
\(628\) 22.0000 0.877896
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) −24.0000 −0.954669
\(633\) 20.0000 0.794929
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 2.00000 0.0792429
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −3.00000 −0.117851
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −38.0000 −1.48705 −0.743527 0.668705i \(-0.766849\pi\)
−0.743527 + 0.668705i \(0.766849\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −4.00000 −0.155464
\(663\) 4.00000 0.155347
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) −16.0000 −0.619059
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 5.00000 0.192879
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −6.00000 −0.230429
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −22.0000 −0.839352
\(688\) −4.00000 −0.152499
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) 14.0000 0.532200
\(693\) 1.00000 0.0379869
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 18.0000 0.682288
\(697\) 12.0000 0.454532
\(698\) −26.0000 −0.984115
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −24.0000 −0.905177
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 10.0000 0.376089
\(708\) 4.00000 0.150329
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −30.0000 −1.12430
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 24.0000 0.896296
\(718\) 16.0000 0.597115
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) −10.0000 −0.371904
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −1.00000 −0.0371135
\(727\) −24.0000 −0.890111 −0.445055 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 6.00000 0.221766
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) −6.00000 −0.220863
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) −2.00000 −0.0734223
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 4.00000 0.146352
\(748\) −2.00000 −0.0731272
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 16.0000 0.579619
\(763\) 2.00000 0.0724049
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 8.00000 0.288863
\(768\) 17.0000 0.613435
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 2.00000 0.0719816
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 30.0000 1.07694
\(777\) −6.00000 −0.215249
\(778\) 6.00000 0.215110
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 3.00000 0.106600
\(793\) 12.0000 0.426132
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 34.0000 1.20058
\(803\) 10.0000 0.352892
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 30.0000 1.05540
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −36.0000 −1.26413 −0.632065 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 6.00000 0.210559
\(813\) 16.0000 0.561144
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 16.0000 0.559769
\(818\) −14.0000 −0.489499
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 14.0000 0.488603 0.244302 0.969699i \(-0.421441\pi\)
0.244302 + 0.969699i \(0.421441\pi\)
\(822\) 2.00000 0.0697580
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 14.0000 0.485363
\(833\) −2.00000 −0.0692959
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) 30.0000 1.03325
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −2.00000 −0.0686803
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 2.00000 0.0682789
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 8.00000 0.272481
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 6.00000 0.203888
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 6.00000 0.203186
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 24.0000 0.809961
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 1.00000 0.0336718
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −18.0000 −0.604040
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) 0 0
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 6.00000 0.199778
\(903\) 4.00000 0.133112
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 12.0000 0.398234
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 4.00000 0.132453
\(913\) −4.00000 −0.132381
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 12.0000 0.396275
\(918\) 2.00000 0.0660098
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) −2.00000 −0.0658665
\(923\) 0 0
\(924\) 1.00000 0.0328976
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) −8.00000 −0.262754
\(928\) 30.0000 0.984798
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) −6.00000 −0.196537
\(933\) 32.0000 1.04763
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 12.0000 0.391814
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 22.0000 0.716799
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 8.00000 0.259828
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) −6.00000 −0.194461
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 6.00000 0.193952
\(958\) −32.0000 −1.03387
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −12.0000 −0.386896
\(963\) −4.00000 −0.128898
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) 4.00000 0.128234
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) −4.00000 −0.127906
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 20.0000 0.638226
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) −18.0000 −0.573819
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −28.0000 −0.886325
\(999\) 6.00000 0.189832
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 5775.2.a.q.1.1 1
5.4 even 2 1155.2.a.f.1.1 1
15.14 odd 2 3465.2.a.n.1.1 1
35.34 odd 2 8085.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.f.1.1 1 5.4 even 2
3465.2.a.n.1.1 1 15.14 odd 2
5775.2.a.q.1.1 1 1.1 even 1 trivial
8085.2.a.f.1.1 1 35.34 odd 2