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Properties

Label 1155.2.a.f.1.1
Level $1155$
Weight $2$
Character 1155.1
Self dual yes
Analytic conductor $9.223$
Analytic rank $0$
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] = [1155,2,Mod(1,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, 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("1155.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.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: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1155.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} +3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} -5.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +3.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} +3.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +6.00000 q^{61} +1.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -1.00000 q^{70} +3.00000 q^{72} +10.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -1.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -3.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} -2.00000 q^{91} +4.00000 q^{95} -5.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 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
\(25\) 1.00000 0.200000
\(26\) 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 \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) 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
\(35\) 1.00000 0.169031
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 3.00000 0.474342
\(41\) −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 \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) 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
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 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 \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) 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
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 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 \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) −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 \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −1.00000 −0.109109
\(85\) 2.00000 0.216930
\(86\) 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 \(0.322192\pi\)
\(90\) −1.00000 −0.105409
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −5.00000 −0.510310
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) −1.00000 −0.100000
\(101\) −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 \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 1.00000 0.0975900
\(106\) 2.00000 0.194257
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 1.00000 0.0953463
\(111\) 6.00000 0.569495
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −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
\(120\) 3.00000 0.273861
\(121\) 1.00000 0.0909091
\(122\) −6.00000 −0.543214
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 3.00000 0.265165
\(129\) −4.00000 −0.352180
\(130\) 2.00000 0.175412
\(131\) −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
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) −1.00000 −0.0833333
\(145\) 6.00000 0.498273
\(146\) −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 \(0.763904\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −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 \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −8.00000 −0.636446
\(159\) −2.00000 −0.158610
\(160\) −5.00000 −0.395285
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) −1.00000 −0.0778499
\(166\) 4.00000 0.310460
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −6.00000 −0.454859
\(175\) 1.00000 0.0755929
\(176\) 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 \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −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
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) −4.00000 −0.290191
\(191\) 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 \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −10.0000 −0.717958
\(195\) −2.00000 −0.143223
\(196\) −1.00000 −0.0714286
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 1.00000 0.0710669
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 3.00000 0.212132
\(201\) 12.0000 0.846415
\(202\) 10.0000 0.703598
\(203\) 6.00000 0.421117
\(204\) −2.00000 −0.140028
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) −1.00000 −0.0690066
\(211\) −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
\(215\) −4.00000 −0.272798
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 10.0000 0.675737
\(220\) 1.00000 0.0674200
\(221\) −4.00000 −0.269069
\(222\) −6.00000 −0.402694
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −5.00000 −0.334077
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −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 \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 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 \(0.782863\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 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
\(245\) 1.00000 0.0638877
\(246\) 6.00000 0.382546
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) −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
\(255\) 2.00000 0.125245
\(256\) −17.0000 −1.06250
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 4.00000 0.249029
\(259\) 6.00000 0.372822
\(260\) 2.00000 0.124035
\(261\) 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
\(265\) −2.00000 −0.122859
\(266\) −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 \(0.684892\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −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
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) −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 \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) −2.00000 −0.118262
\(287\) −6.00000 −0.354169
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 10.0000 0.586210
\(292\) −10.0000 −0.585206
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 4.00000 0.232889
\(296\) 18.0000 1.04623
\(297\) −1.00000 −0.0580259
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −4.00000 −0.230556
\(302\) 16.0000 0.920697
\(303\) −10.0000 −0.574485
\(304\) −4.00000 −0.229416
\(305\) 6.00000 0.343559
\(306\) −2.00000 −0.114332
\(307\) −12.0000 −0.684876 −0.342438 − 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 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 \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −22.0000 −1.24153
\(315\) 1.00000 0.0563436
\(316\) −8.00000 −0.450035
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 2.00000 0.112154
\(319\) −6.00000 −0.335936
\(320\) 7.00000 0.391312
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) −1.00000 −0.0555556
\(325\) −2.00000 −0.110940
\(326\) 4.00000 0.221540
\(327\) −2.00000 −0.110600
\(328\) −18.0000 −0.993884
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) −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
\(335\) 12.0000 0.655630
\(336\) −1.00000 −0.0545545
\(337\) 18.0000 0.980522 0.490261 − 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) −2.00000 −0.108465
\(341\) 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 \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −6.00000 −0.321634
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) 5.00000 0.266501
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −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 \(0.361250\pi\)
\(360\) 3.00000 0.158114
\(361\) −3.00000 −0.157895
\(362\) 10.0000 0.525588
\(363\) 1.00000 0.0524864
\(364\) 2.00000 0.104828
\(365\) 10.0000 0.523424
\(366\) −6.00000 −0.313625
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) −6.00000 −0.311925
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 − 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 2.00000 0.103418
\(375\) 1.00000 0.0516398
\(376\) 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 \(0.532759\pi\)
\(380\) −4.00000 −0.205196
\(381\) 16.0000 0.819705
\(382\) −24.0000 −1.22795
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 3.00000 0.153093
\(385\) −1.00000 −0.0509647
\(386\) −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 \(0.451394\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) −12.0000 −0.605320
\(394\) 10.0000 0.503793
\(395\) 8.00000 0.402524
\(396\) 1.00000 0.0502519
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −8.00000 −0.401004
\(399\) 4.00000 0.200250
\(400\) −1.00000 −0.0500000
\(401\) 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
\(405\) 1.00000 0.0496904
\(406\) −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 \(0.612504\pi\)
\(410\) 6.00000 0.296319
\(411\) 2.00000 0.0986527
\(412\) −8.00000 −0.394132
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 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 \(0.405307\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 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
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) −4.00000 −0.193347
\(429\) 2.00000 0.0965609
\(430\) 4.00000 0.192897
\(431\) 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 \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 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 \(0.305885\pi\)
\(440\) −3.00000 −0.143019
\(441\) 1.00000 0.0476190
\(442\) 4.00000 0.190261
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −6.00000 −0.284747
\(445\) 10.0000 0.474045
\(446\) 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 \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 6.00000 0.282529
\(452\) 6.00000 0.282216
\(453\) −16.0000 −0.751746
\(454\) −12.0000 −0.563188
\(455\) −2.00000 −0.0937614
\(456\) 12.0000 0.561951
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −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 \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 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
\(475\) 4.00000 0.183533
\(476\) −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 \(0.760973\pi\)
\(480\) −5.00000 −0.228218
\(481\) −12.0000 −0.547153
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 18.0000 0.814822
\(489\) −4.00000 −0.180886
\(490\) −1.00000 −0.0451754
\(491\) 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
\(495\) −1.00000 −0.0449467
\(496\) 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 \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −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
\(505\) −10.0000 −0.444994
\(506\) 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 \(0.514113\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 10.0000 0.442374
\(512\) 11.0000 0.486136
\(513\) 4.00000 0.176604
\(514\) 14.0000 0.617514
\(515\) 8.00000 0.352522
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) 14.0000 0.614532
\(520\) −6.00000 −0.263117
\(521\) −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 \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) −23.0000 −1.00000
\(530\) 2.00000 0.0868744
\(531\) 4.00000 0.173585
\(532\) −4.00000 −0.173422
\(533\) 12.0000 0.519778
\(534\) −10.0000 −0.432742
\(535\) 4.00000 0.172935
\(536\) 36.0000 1.55496
\(537\) −4.00000 −0.172613
\(538\) 18.0000 0.776035
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) −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
\(545\) −2.00000 −0.0856706
\(546\) 2.00000 0.0855921
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 6.00000 0.256074
\(550\) 1.00000 0.0426401
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 10.0000 0.424859
\(555\) 6.00000 0.254686
\(556\) 4.00000 0.169638
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −1.00000 −0.0422577
\(561\) −2.00000 −0.0844401
\(562\) 30.0000 1.26547
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 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 \(0.486652\pi\)
\(570\) −4.00000 −0.167542
\(571\) −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 \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 13.0000 0.540729
\(579\) 2.00000 0.0831172
\(580\) −6.00000 −0.249136
\(581\) −4.00000 −0.165948
\(582\) −10.0000 −0.414513
\(583\) 2.00000 0.0828315
\(584\) 30.0000 1.24141
\(585\) −2.00000 −0.0826898
\(586\) −6.00000 −0.247858
\(587\) −36.0000 −1.48588 −0.742940 − 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) −10.0000 −0.411345
\(592\) −6.00000 −0.246598
\(593\) 2.00000 0.0821302 0.0410651 − 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 1.00000 0.0410305
\(595\) 2.00000 0.0819920
\(596\) 18.0000 0.737309
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 3.00000 0.122474
\(601\) 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
\(605\) 1.00000 0.0406558
\(606\) 10.0000 0.406222
\(607\) −48.0000 −1.94826 −0.974130 − 0.225989i \(-0.927439\pi\)
−0.974130 + 0.225989i \(0.927439\pi\)
\(608\) −20.0000 −0.811107
\(609\) 6.00000 0.243132
\(610\) −6.00000 −0.242933
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 6.00000 0.242338 0.121169 − 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 12.0000 0.484281
\(615\) −6.00000 −0.241943
\(616\) −3.00000 −0.120873
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −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
\(625\) 1.00000 0.0400000
\(626\) 14.0000 0.559553
\(627\) −4.00000 −0.159745
\(628\) −22.0000 −0.877896
\(629\) 12.0000 0.478471
\(630\) −1.00000 −0.0398410
\(631\) 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
\(635\) 16.0000 0.634941
\(636\) 2.00000 0.0793052
\(637\) −2.00000 −0.0792429
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) 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 \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −8.00000 −0.314756
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 3.00000 0.117851
\(649\) −4.00000 −0.157014
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 38.0000 1.48705 0.743527 − 0.668705i \(-0.233151\pi\)
0.743527 + 0.668705i \(0.233151\pi\)
\(654\) 2.00000 0.0782062
\(655\) −12.0000 −0.468879
\(656\) 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 \(0.627368\pi\)
\(660\) 1.00000 0.0389249
\(661\) −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
\(665\) 4.00000 0.155113
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) 16.0000 0.619059
\(669\) −16.0000 −0.618596
\(670\) −12.0000 −0.463600
\(671\) −6.00000 −0.231627
\(672\) −5.00000 −0.192879
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −18.0000 −0.693334
\(675\) 1.00000 0.0384900
\(676\) 9.00000 0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 6.00000 0.230429
\(679\) 10.0000 0.383765
\(680\) 6.00000 0.230089
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) −4.00000 −0.152944
\(685\) 2.00000 0.0764161
\(686\) −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
\(695\) −4.00000 −0.151729
\(696\) 18.0000 0.682288
\(697\) −12.0000 −0.454532
\(698\) 26.0000 0.984115
\(699\) −6.00000 −0.226941
\(700\) −1.00000 −0.0377964
\(701\) 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
\(715\) 2.00000 0.0747958
\(716\) 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 \(0.352306\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) 10.0000 0.371904
\(724\) 10.0000 0.371647
\(725\) 6.00000 0.222834
\(726\) −1.00000 −0.0371135
\(727\) 24.0000 0.890111 0.445055 − 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) −8.00000 −0.295891
\(732\) −6.00000 −0.221766
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 16.0000 0.590571
\(735\) 1.00000 0.0368856
\(736\) 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 \(0.429161\pi\)
\(740\) −6.00000 −0.220564
\(741\) −8.00000 −0.293887
\(742\) 2.00000 0.0734223
\(743\) −16.0000 −0.586983 −0.293492 − 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) −22.0000 −0.805477
\(747\) −4.00000 −0.146352
\(748\) 2.00000 0.0731272
\(749\) 4.00000 0.146157
\(750\) −1.00000 −0.0365148
\(751\) −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
\(755\) −16.0000 −0.582300
\(756\) −1.00000 −0.0363696
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) 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
\(765\) 2.00000 0.0723102
\(766\) 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 \(0.344689\pi\)
\(770\) 1.00000 0.0360375
\(771\) −14.0000 −0.504198
\(772\) −2.00000 −0.0719816
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 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
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) −1.00000 −0.0357143
\(785\) 22.0000 0.785214
\(786\) 12.0000 0.428026
\(787\) 20.0000 0.712923 0.356462 − 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −6.00000 −0.213335
\(792\) −3.00000 −0.106600
\(793\) −12.0000 −0.426132
\(794\) −22.0000 −0.780751
\(795\) −2.00000 −0.0709327
\(796\) −8.00000 −0.283552
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) −5.00000 −0.176777
\(801\) 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 \(0.397518\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −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
\(815\) −4.00000 −0.140114
\(816\) −2.00000 −0.0700140
\(817\) −16.0000 −0.559769
\(818\) 14.0000 0.489499
\(819\) −2.00000 −0.0698857
\(820\) 6.00000 0.209529
\(821\) 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 \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 24.0000 0.836080
\(825\) −1.00000 −0.0348155
\(826\) −4.00000 −0.139178
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 4.00000 0.138842
\(831\) −10.0000 −0.346896
\(832\) −14.0000 −0.485363
\(833\) 2.00000 0.0692959
\(834\) 4.00000 0.138509
\(835\) −16.0000 −0.553703
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 3.00000 0.103510
\(841\) 7.00000 0.241379
\(842\) −6.00000 −0.206774
\(843\) −30.0000 −1.03325
\(844\) 20.0000 0.688428
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 2.00000 0.0686803
\(849\) −20.0000 −0.686398
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) −6.00000 −0.205316
\(855\) 4.00000 0.136797
\(856\) 12.0000 0.410152
\(857\) 10.0000 0.341593 0.170797 − 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 4.00000 0.136399
\(861\) −6.00000 −0.204479
\(862\) −8.00000 −0.272481
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −5.00000 −0.170103
\(865\) 14.0000 0.476014
\(866\) 6.00000 0.203888
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) −6.00000 −0.203419
\(871\) −24.0000 −0.813209
\(872\) −6.00000 −0.203186
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −10.0000 −0.337869
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −24.0000 −0.809961
\(879\) 6.00000 0.202375
\(880\) 1.00000 0.0337100
\(881\) −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 \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 4.00000 0.134535
\(885\) 4.00000 0.134459
\(886\) −36.0000 −1.20944
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 18.0000 0.604040
\(889\) 16.0000 0.536623
\(890\) −10.0000 −0.335201
\(891\) −1.00000 −0.0335013
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) −4.00000 −0.133705
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −4.00000 −0.133259
\(902\) −6.00000 −0.199778
\(903\) −4.00000 −0.133112
\(904\) −18.0000 −0.598671
\(905\) −10.0000 −0.332411
\(906\) 16.0000 0.531564
\(907\) 36.0000 1.19536 0.597680 − 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) −12.0000 −0.398234
\(909\) −10.0000 −0.331679
\(910\) 2.00000 0.0662994
\(911\) −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
\(915\) 6.00000 0.198354
\(916\) −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
\(925\) 6.00000 0.197279
\(926\) −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
\(935\) −2.00000 −0.0654070
\(936\) −6.00000 −0.196116
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −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
\(945\) 1.00000 0.0325300
\(946\) −4.00000 −0.130051
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −8.00000 −0.259828
\(949\) −20.0000 −0.649227
\(950\) −4.00000 −0.129777
\(951\) 22.0000 0.713399
\(952\) 6.00000 0.194461
\(953\) 42.0000 1.36051 0.680257 − 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 2.00000 0.0647524
\(955\) 24.0000 0.776622
\(956\) 24.0000 0.776215
\(957\) −6.00000 −0.193952
\(958\) 32.0000 1.03387
\(959\) 2.00000 0.0645834
\(960\) 7.00000 0.225924
\(961\) −31.0000 −1.00000
\(962\) 12.0000 0.386896
\(963\) 4.00000 0.128898
\(964\) −10.0000 −0.322078
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 3.00000 0.0964237
\(969\) 8.00000 0.256997
\(970\) −10.0000 −0.321081
\(971\) −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
\(975\) −2.00000 −0.0640513
\(976\) −6.00000 −0.192055
\(977\) −38.0000 −1.21573 −0.607864 − 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 4.00000 0.127906
\(979\) −10.0000 −0.319601
\(980\) −1.00000 −0.0319438
\(981\) −2.00000 −0.0638551
\(982\) −20.0000 −0.638226
\(983\) −56.0000 −1.78612 −0.893061 − 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) −18.0000 −0.573819
\(985\) −10.0000 −0.318626
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 1.00000 0.0317821
\(991\) −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
\(995\) 8.00000 0.253617
\(996\) 4.00000 0.126745
\(997\) 6.00000 0.190022 0.0950110 − 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 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 1155.2.a.f.1.1 ✓ 1
3.2 odd 2 3465.2.a.n.1.1 1
5.4 even 2 5775.2.a.q.1.1 1
7.6 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 1.1 even 1 trivial
3465.2.a.n.1.1 1 3.2 odd 2
5775.2.a.q.1.1 1 5.4 even 2
8085.2.a.f.1.1 1 7.6 odd 2