Properties

Label 1155.2.a.b.1.1
Level $1155$
Weight $2$
Character 1155.1
Self dual yes
Analytic conductor $9.223$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +1.00000 q^{17} -2.00000 q^{18} -7.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} -2.00000 q^{22} +5.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +3.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} +8.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{35} +2.00000 q^{36} -2.00000 q^{37} +14.0000 q^{38} -2.00000 q^{39} -12.0000 q^{41} +2.00000 q^{42} -1.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -10.0000 q^{46} -4.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -2.00000 q^{50} +1.00000 q^{51} -4.00000 q^{52} -1.00000 q^{53} -2.00000 q^{54} -1.00000 q^{55} -7.00000 q^{57} -6.00000 q^{58} +9.00000 q^{59} -2.00000 q^{60} -11.0000 q^{61} +8.00000 q^{62} -1.00000 q^{63} -8.00000 q^{64} +2.00000 q^{65} -2.00000 q^{66} +2.00000 q^{67} +2.00000 q^{68} +5.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} +8.00000 q^{73} +4.00000 q^{74} +1.00000 q^{75} -14.0000 q^{76} -1.00000 q^{77} +4.00000 q^{78} +2.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +24.0000 q^{82} -9.00000 q^{83} -2.00000 q^{84} -1.00000 q^{85} +2.00000 q^{86} +3.00000 q^{87} -3.00000 q^{89} +2.00000 q^{90} +2.00000 q^{91} +10.0000 q^{92} -4.00000 q^{93} +8.00000 q^{94} +7.00000 q^{95} +8.00000 q^{96} -13.0000 q^{97} -2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214
\(6\) −2.00000 −0.816497
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) −4.00000 −1.00000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −2.00000 −0.471405
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −2.00000 −0.447214
\(21\) −1.00000 −0.218218
\(22\) −2.00000 −0.426401
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 2.00000 0.365148
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 8.00000 1.41421
\(33\) 1.00000 0.174078
\(34\) −2.00000 −0.342997
\(35\) 1.00000 0.169031
\(36\) 2.00000 0.333333
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 14.0000 2.27110
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 2.00000 0.308607
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 2.00000 0.301511
\(45\) −1.00000 −0.149071
\(46\) −10.0000 −1.47442
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) −2.00000 −0.282843
\(51\) 1.00000 0.140028
\(52\) −4.00000 −0.554700
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) −2.00000 −0.272166
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) −6.00000 −0.787839
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) −2.00000 −0.258199
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) 2.00000 0.248069
\(66\) −2.00000 −0.246183
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 2.00000 0.242536
\(69\) 5.00000 0.601929
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 4.00000 0.464991
\(75\) 1.00000 0.115470
\(76\) −14.0000 −1.60591
\(77\) −1.00000 −0.113961
\(78\) 4.00000 0.452911
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 24.0000 2.65036
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) −2.00000 −0.218218
\(85\) −1.00000 −0.108465
\(86\) 2.00000 0.215666
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 2.00000 0.210819
\(91\) 2.00000 0.209657
\(92\) 10.0000 1.04257
\(93\) −4.00000 −0.414781
\(94\) 8.00000 0.825137
\(95\) 7.00000 0.718185
\(96\) 8.00000 0.816497
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) −2.00000 −0.202031
\(99\) 1.00000 0.100504
\(100\) 2.00000 0.200000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −2.00000 −0.198030
\(103\) −9.00000 −0.886796 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 0 0
\(105\) 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 \(0.438066\pi\)
\(108\) 2.00000 0.192450
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 2.00000 0.190693
\(111\) −2.00000 −0.189832
\(112\) 4.00000 0.377964
\(113\) 11.0000 1.03479 0.517396 0.855746i \(-0.326901\pi\)
0.517396 + 0.855746i \(0.326901\pi\)
\(114\) 14.0000 1.31122
\(115\) −5.00000 −0.466252
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) −18.0000 −1.65703
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 22.0000 1.99179
\(123\) −12.0000 −1.08200
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) −4.00000 −0.350823
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.00000 0.174078
\(133\) 7.00000 0.606977
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) −10.0000 −0.851257
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 2.00000 0.169031
\(141\) −4.00000 −0.336861
\(142\) 16.0000 1.34269
\(143\) −2.00000 −0.167248
\(144\) −4.00000 −0.333333
\(145\) −3.00000 −0.249136
\(146\) −16.0000 −1.32417
\(147\) 1.00000 0.0824786
\(148\) −4.00000 −0.328798
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −2.00000 −0.163299
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 2.00000 0.161165
\(155\) 4.00000 0.321288
\(156\) −4.00000 −0.320256
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) −4.00000 −0.318223
\(159\) −1.00000 −0.0793052
\(160\) −8.00000 −0.632456
\(161\) −5.00000 −0.394055
\(162\) −2.00000 −0.157135
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −24.0000 −1.87409
\(165\) −1.00000 −0.0778499
\(166\) 18.0000 1.39707
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −7.00000 −0.535303
\(172\) −2.00000 −0.152499
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −6.00000 −0.454859
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) 9.00000 0.676481
\(178\) 6.00000 0.449719
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) −2.00000 −0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −4.00000 −0.296500
\(183\) −11.0000 −0.813143
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 8.00000 0.586588
\(187\) 1.00000 0.0731272
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) −14.0000 −1.01567
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −8.00000 −0.577350
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 26.0000 1.86669
\(195\) 2.00000 0.143223
\(196\) 2.00000 0.142857
\(197\) −16.0000 −1.13995 −0.569976 0.821661i \(-0.693048\pi\)
−0.569976 + 0.821661i \(0.693048\pi\)
\(198\) −2.00000 −0.142134
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 28.0000 1.97007
\(203\) −3.00000 −0.210559
\(204\) 2.00000 0.140028
\(205\) 12.0000 0.838116
\(206\) 18.0000 1.25412
\(207\) 5.00000 0.347524
\(208\) 8.00000 0.554700
\(209\) −7.00000 −0.484200
\(210\) −2.00000 −0.138013
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −2.00000 −0.137361
\(213\) −8.00000 −0.548151
\(214\) −8.00000 −0.546869
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 16.0000 1.08366
\(219\) 8.00000 0.540590
\(220\) −2.00000 −0.134840
\(221\) −2.00000 −0.134535
\(222\) 4.00000 0.268462
\(223\) −7.00000 −0.468755 −0.234377 0.972146i \(-0.575305\pi\)
−0.234377 + 0.972146i \(0.575305\pi\)
\(224\) −8.00000 −0.534522
\(225\) 1.00000 0.0666667
\(226\) −22.0000 −1.46342
\(227\) −9.00000 −0.597351 −0.298675 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) −14.0000 −0.927173
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 10.0000 0.659380
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 4.00000 0.261488
\(235\) 4.00000 0.260931
\(236\) 18.0000 1.17170
\(237\) 2.00000 0.129914
\(238\) 2.00000 0.129641
\(239\) 5.00000 0.323423 0.161712 0.986838i \(-0.448299\pi\)
0.161712 + 0.986838i \(0.448299\pi\)
\(240\) 4.00000 0.258199
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) −22.0000 −1.40841
\(245\) −1.00000 −0.0638877
\(246\) 24.0000 1.53018
\(247\) 14.0000 0.890799
\(248\) 0 0
\(249\) −9.00000 −0.570352
\(250\) 2.00000 0.126491
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −2.00000 −0.125988
\(253\) 5.00000 0.314347
\(254\) 26.0000 1.63139
\(255\) −1.00000 −0.0626224
\(256\) 16.0000 1.00000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 2.00000 0.124515
\(259\) 2.00000 0.124274
\(260\) 4.00000 0.248069
\(261\) 3.00000 0.185695
\(262\) 36.0000 2.22409
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 1.00000 0.0614295
\(266\) −14.0000 −0.858395
\(267\) −3.00000 −0.183597
\(268\) 4.00000 0.244339
\(269\) 5.00000 0.304855 0.152428 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 2.00000 0.121716
\(271\) 31.0000 1.88312 0.941558 0.336851i \(-0.109362\pi\)
0.941558 + 0.336851i \(0.109362\pi\)
\(272\) −4.00000 −0.242536
\(273\) 2.00000 0.121046
\(274\) 20.0000 1.20824
\(275\) 1.00000 0.0603023
\(276\) 10.0000 0.601929
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 16.0000 0.959616
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 8.00000 0.476393
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) −16.0000 −0.949425
\(285\) 7.00000 0.414644
\(286\) 4.00000 0.236525
\(287\) 12.0000 0.708338
\(288\) 8.00000 0.471405
\(289\) −16.0000 −0.941176
\(290\) 6.00000 0.352332
\(291\) −13.0000 −0.762073
\(292\) 16.0000 0.936329
\(293\) 5.00000 0.292103 0.146052 0.989277i \(-0.453343\pi\)
0.146052 + 0.989277i \(0.453343\pi\)
\(294\) −2.00000 −0.116642
\(295\) −9.00000 −0.524000
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 20.0000 1.15857
\(299\) −10.0000 −0.578315
\(300\) 2.00000 0.115470
\(301\) 1.00000 0.0576390
\(302\) 36.0000 2.07157
\(303\) −14.0000 −0.804279
\(304\) 28.0000 1.60591
\(305\) 11.0000 0.629858
\(306\) −2.00000 −0.114332
\(307\) 34.0000 1.94048 0.970241 0.242140i \(-0.0778494\pi\)
0.970241 + 0.242140i \(0.0778494\pi\)
\(308\) −2.00000 −0.113961
\(309\) −9.00000 −0.511992
\(310\) −8.00000 −0.454369
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 9.00000 0.508710 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) −46.0000 −2.59593
\(315\) 1.00000 0.0563436
\(316\) 4.00000 0.225018
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 2.00000 0.112154
\(319\) 3.00000 0.167968
\(320\) 8.00000 0.447214
\(321\) 4.00000 0.223258
\(322\) 10.0000 0.557278
\(323\) −7.00000 −0.389490
\(324\) 2.00000 0.111111
\(325\) −2.00000 −0.110940
\(326\) 4.00000 0.221540
\(327\) −8.00000 −0.442401
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 2.00000 0.110096
\(331\) −11.0000 −0.604615 −0.302307 0.953211i \(-0.597757\pi\)
−0.302307 + 0.953211i \(0.597757\pi\)
\(332\) −18.0000 −0.987878
\(333\) −2.00000 −0.109599
\(334\) −32.0000 −1.75096
\(335\) −2.00000 −0.109272
\(336\) 4.00000 0.218218
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 18.0000 0.979071
\(339\) 11.0000 0.597438
\(340\) −2.00000 −0.108465
\(341\) −4.00000 −0.216612
\(342\) 14.0000 0.757033
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −5.00000 −0.269191
\(346\) −4.00000 −0.215041
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 6.00000 0.321634
\(349\) −11.0000 −0.588817 −0.294408 0.955680i \(-0.595123\pi\)
−0.294408 + 0.955680i \(0.595123\pi\)
\(350\) 2.00000 0.106904
\(351\) −2.00000 −0.106752
\(352\) 8.00000 0.426401
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −18.0000 −0.956689
\(355\) 8.00000 0.424596
\(356\) −6.00000 −0.317999
\(357\) −1.00000 −0.0529256
\(358\) −32.0000 −1.69125
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 20.0000 1.05118
\(363\) 1.00000 0.0524864
\(364\) 4.00000 0.209657
\(365\) −8.00000 −0.418739
\(366\) 22.0000 1.14996
\(367\) 7.00000 0.365397 0.182699 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) −20.0000 −1.04257
\(369\) −12.0000 −0.624695
\(370\) −4.00000 −0.207950
\(371\) 1.00000 0.0519174
\(372\) −8.00000 −0.414781
\(373\) −25.0000 −1.29445 −0.647225 0.762299i \(-0.724071\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) −2.00000 −0.103418
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 2.00000 0.102869
\(379\) 37.0000 1.90056 0.950281 0.311393i \(-0.100796\pi\)
0.950281 + 0.311393i \(0.100796\pi\)
\(380\) 14.0000 0.718185
\(381\) −13.0000 −0.666010
\(382\) −48.0000 −2.45589
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) −12.0000 −0.610784
\(387\) −1.00000 −0.0508329
\(388\) −26.0000 −1.31995
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) −4.00000 −0.202548
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 32.0000 1.61214
\(395\) −2.00000 −0.100631
\(396\) 2.00000 0.100504
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 8.00000 0.401004
\(399\) 7.00000 0.350438
\(400\) −4.00000 −0.200000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) −4.00000 −0.199502
\(403\) 8.00000 0.398508
\(404\) −28.0000 −1.39305
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −24.0000 −1.18528
\(411\) −10.0000 −0.493264
\(412\) −18.0000 −0.886796
\(413\) −9.00000 −0.442861
\(414\) −10.0000 −0.491473
\(415\) 9.00000 0.441793
\(416\) −16.0000 −0.784465
\(417\) −8.00000 −0.391762
\(418\) 14.0000 0.684762
\(419\) 35.0000 1.70986 0.854931 0.518742i \(-0.173599\pi\)
0.854931 + 0.518742i \(0.173599\pi\)
\(420\) 2.00000 0.0975900
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 20.0000 0.973585
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 16.0000 0.775203
\(427\) 11.0000 0.532327
\(428\) 8.00000 0.386695
\(429\) −2.00000 −0.0965609
\(430\) −2.00000 −0.0964486
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −4.00000 −0.192450
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −8.00000 −0.384012
\(435\) −3.00000 −0.143839
\(436\) −16.0000 −0.766261
\(437\) −35.0000 −1.67428
\(438\) −16.0000 −0.764510
\(439\) −15.0000 −0.715911 −0.357955 0.933739i \(-0.616526\pi\)
−0.357955 + 0.933739i \(0.616526\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 4.00000 0.190261
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −4.00000 −0.189832
\(445\) 3.00000 0.142214
\(446\) 14.0000 0.662919
\(447\) −10.0000 −0.472984
\(448\) 8.00000 0.377964
\(449\) 40.0000 1.88772 0.943858 0.330350i \(-0.107167\pi\)
0.943858 + 0.330350i \(0.107167\pi\)
\(450\) −2.00000 −0.0942809
\(451\) −12.0000 −0.565058
\(452\) 22.0000 1.03479
\(453\) −18.0000 −0.845714
\(454\) 18.0000 0.844782
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) −8.00000 −0.373815
\(459\) 1.00000 0.0466760
\(460\) −10.0000 −0.466252
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 2.00000 0.0930484
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) −12.0000 −0.557086
\(465\) 4.00000 0.185496
\(466\) −28.0000 −1.29707
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) −4.00000 −0.184900
\(469\) −2.00000 −0.0923514
\(470\) −8.00000 −0.369012
\(471\) 23.0000 1.05978
\(472\) 0 0
\(473\) −1.00000 −0.0459800
\(474\) −4.00000 −0.183726
\(475\) −7.00000 −0.321182
\(476\) −2.00000 −0.0916698
\(477\) −1.00000 −0.0457869
\(478\) −10.0000 −0.457389
\(479\) −38.0000 −1.73626 −0.868132 0.496333i \(-0.834679\pi\)
−0.868132 + 0.496333i \(0.834679\pi\)
\(480\) −8.00000 −0.365148
\(481\) 4.00000 0.182384
\(482\) −20.0000 −0.910975
\(483\) −5.00000 −0.227508
\(484\) 2.00000 0.0909091
\(485\) 13.0000 0.590300
\(486\) −2.00000 −0.0907218
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 2.00000 0.0903508
\(491\) 43.0000 1.94056 0.970281 0.241979i \(-0.0777966\pi\)
0.970281 + 0.241979i \(0.0777966\pi\)
\(492\) −24.0000 −1.08200
\(493\) 3.00000 0.135113
\(494\) −28.0000 −1.25978
\(495\) −1.00000 −0.0449467
\(496\) 16.0000 0.718421
\(497\) 8.00000 0.358849
\(498\) 18.0000 0.806599
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) −2.00000 −0.0894427
\(501\) 16.0000 0.714827
\(502\) 24.0000 1.07117
\(503\) 5.00000 0.222939 0.111469 0.993768i \(-0.464444\pi\)
0.111469 + 0.993768i \(0.464444\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) −10.0000 −0.444554
\(507\) −9.00000 −0.399704
\(508\) −26.0000 −1.15356
\(509\) −33.0000 −1.46270 −0.731350 0.682003i \(-0.761109\pi\)
−0.731350 + 0.682003i \(0.761109\pi\)
\(510\) 2.00000 0.0885615
\(511\) −8.00000 −0.353899
\(512\) −32.0000 −1.41421
\(513\) −7.00000 −0.309058
\(514\) −28.0000 −1.23503
\(515\) 9.00000 0.396587
\(516\) −2.00000 −0.0880451
\(517\) −4.00000 −0.175920
\(518\) −4.00000 −0.175750
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 11.0000 0.481919 0.240959 0.970535i \(-0.422538\pi\)
0.240959 + 0.970535i \(0.422538\pi\)
\(522\) −6.00000 −0.262613
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) −36.0000 −1.57267
\(525\) −1.00000 −0.0436436
\(526\) 32.0000 1.39527
\(527\) −4.00000 −0.174243
\(528\) −4.00000 −0.174078
\(529\) 2.00000 0.0869565
\(530\) −2.00000 −0.0868744
\(531\) 9.00000 0.390567
\(532\) 14.0000 0.606977
\(533\) 24.0000 1.03956
\(534\) 6.00000 0.259645
\(535\) −4.00000 −0.172935
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) −10.0000 −0.431131
\(539\) 1.00000 0.0430730
\(540\) −2.00000 −0.0860663
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) −62.0000 −2.66313
\(543\) −10.0000 −0.429141
\(544\) 8.00000 0.342997
\(545\) 8.00000 0.342682
\(546\) −4.00000 −0.171184
\(547\) −23.0000 −0.983409 −0.491704 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) −20.0000 −0.854358
\(549\) −11.0000 −0.469469
\(550\) −2.00000 −0.0852803
\(551\) −21.0000 −0.894630
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 20.0000 0.849719
\(555\) 2.00000 0.0848953
\(556\) −16.0000 −0.678551
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 8.00000 0.338667
\(559\) 2.00000 0.0845910
\(560\) −4.00000 −0.169031
\(561\) 1.00000 0.0422200
\(562\) −12.0000 −0.506189
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) −8.00000 −0.336861
\(565\) −11.0000 −0.462773
\(566\) −44.0000 −1.84946
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 7.00000 0.293455 0.146728 0.989177i \(-0.453126\pi\)
0.146728 + 0.989177i \(0.453126\pi\)
\(570\) −14.0000 −0.586395
\(571\) 34.0000 1.42286 0.711428 0.702759i \(-0.248049\pi\)
0.711428 + 0.702759i \(0.248049\pi\)
\(572\) −4.00000 −0.167248
\(573\) 24.0000 1.00261
\(574\) −24.0000 −1.00174
\(575\) 5.00000 0.208514
\(576\) −8.00000 −0.333333
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 32.0000 1.33102
\(579\) 6.00000 0.249351
\(580\) −6.00000 −0.249136
\(581\) 9.00000 0.373383
\(582\) 26.0000 1.07773
\(583\) −1.00000 −0.0414158
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) −10.0000 −0.413096
\(587\) −10.0000 −0.412744 −0.206372 0.978474i \(-0.566166\pi\)
−0.206372 + 0.978474i \(0.566166\pi\)
\(588\) 2.00000 0.0824786
\(589\) 28.0000 1.15372
\(590\) 18.0000 0.741048
\(591\) −16.0000 −0.658152
\(592\) 8.00000 0.328798
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 1.00000 0.0409960
\(596\) −20.0000 −0.819232
\(597\) −4.00000 −0.163709
\(598\) 20.0000 0.817861
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 0 0
\(601\) 47.0000 1.91717 0.958585 0.284807i \(-0.0919294\pi\)
0.958585 + 0.284807i \(0.0919294\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 2.00000 0.0814463
\(604\) −36.0000 −1.46482
\(605\) −1.00000 −0.0406558
\(606\) 28.0000 1.13742
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) −56.0000 −2.27110
\(609\) −3.00000 −0.121566
\(610\) −22.0000 −0.890754
\(611\) 8.00000 0.323645
\(612\) 2.00000 0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −68.0000 −2.74426
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 18.0000 0.724066
\(619\) 22.0000 0.884255 0.442127 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 8.00000 0.321288
\(621\) 5.00000 0.200643
\(622\) −24.0000 −0.962312
\(623\) 3.00000 0.120192
\(624\) 8.00000 0.320256
\(625\) 1.00000 0.0400000
\(626\) −18.0000 −0.719425
\(627\) −7.00000 −0.279553
\(628\) 46.0000 1.83560
\(629\) −2.00000 −0.0797452
\(630\) −2.00000 −0.0796819
\(631\) −19.0000 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) 0 0
\(633\) −10.0000 −0.397464
\(634\) −12.0000 −0.476581
\(635\) 13.0000 0.515889
\(636\) −2.00000 −0.0793052
\(637\) −2.00000 −0.0792429
\(638\) −6.00000 −0.237542
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −8.00000 −0.315735
\(643\) −21.0000 −0.828159 −0.414080 0.910241i \(-0.635896\pi\)
−0.414080 + 0.910241i \(0.635896\pi\)
\(644\) −10.0000 −0.394055
\(645\) 1.00000 0.0393750
\(646\) 14.0000 0.550823
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 4.00000 0.156893
\(651\) 4.00000 0.156772
\(652\) −4.00000 −0.156652
\(653\) 13.0000 0.508729 0.254365 0.967108i \(-0.418134\pi\)
0.254365 + 0.967108i \(0.418134\pi\)
\(654\) 16.0000 0.625650
\(655\) 18.0000 0.703318
\(656\) 48.0000 1.87409
\(657\) 8.00000 0.312110
\(658\) −8.00000 −0.311872
\(659\) −9.00000 −0.350590 −0.175295 0.984516i \(-0.556088\pi\)
−0.175295 + 0.984516i \(0.556088\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 22.0000 0.855054
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) −7.00000 −0.271448
\(666\) 4.00000 0.154997
\(667\) 15.0000 0.580802
\(668\) 32.0000 1.23812
\(669\) −7.00000 −0.270636
\(670\) 4.00000 0.154533
\(671\) −11.0000 −0.424650
\(672\) −8.00000 −0.308607
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) −46.0000 −1.77185
\(675\) 1.00000 0.0384900
\(676\) −18.0000 −0.692308
\(677\) −35.0000 −1.34516 −0.672580 0.740025i \(-0.734814\pi\)
−0.672580 + 0.740025i \(0.734814\pi\)
\(678\) −22.0000 −0.844905
\(679\) 13.0000 0.498894
\(680\) 0 0
\(681\) −9.00000 −0.344881
\(682\) 8.00000 0.306336
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) −14.0000 −0.535303
\(685\) 10.0000 0.382080
\(686\) 2.00000 0.0763604
\(687\) 4.00000 0.152610
\(688\) 4.00000 0.152499
\(689\) 2.00000 0.0761939
\(690\) 10.0000 0.380693
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 4.00000 0.152057
\(693\) −1.00000 −0.0379869
\(694\) 32.0000 1.21470
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 22.0000 0.832712
\(699\) 14.0000 0.529529
\(700\) −2.00000 −0.0755929
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 4.00000 0.150970
\(703\) 14.0000 0.528020
\(704\) −8.00000 −0.301511
\(705\) 4.00000 0.150649
\(706\) −28.0000 −1.05379
\(707\) 14.0000 0.526524
\(708\) 18.0000 0.676481
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) −16.0000 −0.600469
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) −20.0000 −0.749006
\(714\) 2.00000 0.0748481
\(715\) 2.00000 0.0747958
\(716\) 32.0000 1.19590
\(717\) 5.00000 0.186728
\(718\) −50.0000 −1.86598
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 4.00000 0.149071
\(721\) 9.00000 0.335178
\(722\) −60.0000 −2.23297
\(723\) 10.0000 0.371904
\(724\) −20.0000 −0.743294
\(725\) 3.00000 0.111417
\(726\) −2.00000 −0.0742270
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 16.0000 0.592187
\(731\) −1.00000 −0.0369863
\(732\) −22.0000 −0.813143
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −14.0000 −0.516749
\(735\) −1.00000 −0.0368856
\(736\) 40.0000 1.47442
\(737\) 2.00000 0.0736709
\(738\) 24.0000 0.883452
\(739\) 42.0000 1.54499 0.772497 0.635018i \(-0.219007\pi\)
0.772497 + 0.635018i \(0.219007\pi\)
\(740\) 4.00000 0.147043
\(741\) 14.0000 0.514303
\(742\) −2.00000 −0.0734223
\(743\) 14.0000 0.513610 0.256805 0.966463i \(-0.417330\pi\)
0.256805 + 0.966463i \(0.417330\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 50.0000 1.83063
\(747\) −9.00000 −0.329293
\(748\) 2.00000 0.0731272
\(749\) −4.00000 −0.146157
\(750\) 2.00000 0.0730297
\(751\) −15.0000 −0.547358 −0.273679 0.961821i \(-0.588241\pi\)
−0.273679 + 0.961821i \(0.588241\pi\)
\(752\) 16.0000 0.583460
\(753\) −12.0000 −0.437304
\(754\) 12.0000 0.437014
\(755\) 18.0000 0.655087
\(756\) −2.00000 −0.0727393
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −74.0000 −2.68780
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 26.0000 0.941881
\(763\) 8.00000 0.289619
\(764\) 48.0000 1.73658
\(765\) −1.00000 −0.0361551
\(766\) −36.0000 −1.30073
\(767\) −18.0000 −0.649942
\(768\) 16.0000 0.577350
\(769\) 9.00000 0.324548 0.162274 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 14.0000 0.504198
\(772\) 12.0000 0.431889
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 2.00000 0.0718885
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 2.00000 0.0717496
\(778\) 28.0000 1.00385
\(779\) 84.0000 3.00961
\(780\) 4.00000 0.143223
\(781\) −8.00000 −0.286263
\(782\) −10.0000 −0.357599
\(783\) 3.00000 0.107211
\(784\) −4.00000 −0.142857
\(785\) −23.0000 −0.820905
\(786\) 36.0000 1.28408
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −32.0000 −1.13995
\(789\) −16.0000 −0.569615
\(790\) 4.00000 0.142314
\(791\) −11.0000 −0.391115
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) −28.0000 −0.993683
\(795\) 1.00000 0.0354663
\(796\) −8.00000 −0.283552
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −14.0000 −0.495595
\(799\) −4.00000 −0.141510
\(800\) 8.00000 0.282843
\(801\) −3.00000 −0.106000
\(802\) 48.0000 1.69494
\(803\) 8.00000 0.282314
\(804\) 4.00000 0.141069
\(805\) 5.00000 0.176227
\(806\) −16.0000 −0.563576
\(807\) 5.00000 0.176008
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 2.00000 0.0702728
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −6.00000 −0.210559
\(813\) 31.0000 1.08722
\(814\) 4.00000 0.140200
\(815\) 2.00000 0.0700569
\(816\) −4.00000 −0.140028
\(817\) 7.00000 0.244899
\(818\) −20.0000 −0.699284
\(819\) 2.00000 0.0698857
\(820\) 24.0000 0.838116
\(821\) 13.0000 0.453703 0.226852 0.973929i \(-0.427157\pi\)
0.226852 + 0.973929i \(0.427157\pi\)
\(822\) 20.0000 0.697580
\(823\) 38.0000 1.32460 0.662298 0.749240i \(-0.269581\pi\)
0.662298 + 0.749240i \(0.269581\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 18.0000 0.626300
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 10.0000 0.347524
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) −18.0000 −0.624789
\(831\) −10.0000 −0.346896
\(832\) 16.0000 0.554700
\(833\) 1.00000 0.0346479
\(834\) 16.0000 0.554035
\(835\) −16.0000 −0.553703
\(836\) −14.0000 −0.484200
\(837\) −4.00000 −0.138260
\(838\) −70.0000 −2.41811
\(839\) −3.00000 −0.103572 −0.0517858 0.998658i \(-0.516491\pi\)
−0.0517858 + 0.998658i \(0.516491\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −2.00000 −0.0689246
\(843\) 6.00000 0.206651
\(844\) −20.0000 −0.688428
\(845\) 9.00000 0.309609
\(846\) 8.00000 0.275046
\(847\) −1.00000 −0.0343604
\(848\) 4.00000 0.137361
\(849\) 22.0000 0.755038
\(850\) −2.00000 −0.0685994
\(851\) −10.0000 −0.342796
\(852\) −16.0000 −0.548151
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) −22.0000 −0.752825
\(855\) 7.00000 0.239395
\(856\) 0 0
\(857\) −50.0000 −1.70797 −0.853984 0.520300i \(-0.825820\pi\)
−0.853984 + 0.520300i \(0.825820\pi\)
\(858\) 4.00000 0.136558
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 2.00000 0.0681994
\(861\) 12.0000 0.408959
\(862\) −48.0000 −1.63489
\(863\) 21.0000 0.714848 0.357424 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 8.00000 0.272166
\(865\) −2.00000 −0.0680020
\(866\) 52.0000 1.76703
\(867\) −16.0000 −0.543388
\(868\) 8.00000 0.271538
\(869\) 2.00000 0.0678454
\(870\) 6.00000 0.203419
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) −13.0000 −0.439983
\(874\) 70.0000 2.36779
\(875\) 1.00000 0.0338062
\(876\) 16.0000 0.540590
\(877\) −19.0000 −0.641584 −0.320792 0.947150i \(-0.603949\pi\)
−0.320792 + 0.947150i \(0.603949\pi\)
\(878\) 30.0000 1.01245
\(879\) 5.00000 0.168646
\(880\) 4.00000 0.134840
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −4.00000 −0.134535
\(885\) −9.00000 −0.302532
\(886\) 24.0000 0.806296
\(887\) 51.0000 1.71241 0.856206 0.516634i \(-0.172815\pi\)
0.856206 + 0.516634i \(0.172815\pi\)
\(888\) 0 0
\(889\) 13.0000 0.436006
\(890\) −6.00000 −0.201120
\(891\) 1.00000 0.0335013
\(892\) −14.0000 −0.468755
\(893\) 28.0000 0.936984
\(894\) 20.0000 0.668900
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) −10.0000 −0.333890
\(898\) −80.0000 −2.66963
\(899\) −12.0000 −0.400222
\(900\) 2.00000 0.0666667
\(901\) −1.00000 −0.0333148
\(902\) 24.0000 0.799113
\(903\) 1.00000 0.0332779
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 36.0000 1.19602
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) −18.0000 −0.597351
\(909\) −14.0000 −0.464351
\(910\) 4.00000 0.132599
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 28.0000 0.927173
\(913\) −9.00000 −0.297857
\(914\) −6.00000 −0.198462
\(915\) 11.0000 0.363649
\(916\) 8.00000 0.264327
\(917\) 18.0000 0.594412
\(918\) −2.00000 −0.0660098
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 34.0000 1.12034
\(922\) 4.00000 0.131733
\(923\) 16.0000 0.526646
\(924\) −2.00000 −0.0657952
\(925\) −2.00000 −0.0657596
\(926\) 28.0000 0.920137
\(927\) −9.00000 −0.295599
\(928\) 24.0000 0.787839
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) −8.00000 −0.262330
\(931\) −7.00000 −0.229416
\(932\) 28.0000 0.917170
\(933\) 12.0000 0.392862
\(934\) 48.0000 1.57061
\(935\) −1.00000 −0.0327035
\(936\) 0 0
\(937\) 4.00000 0.130674 0.0653372 0.997863i \(-0.479188\pi\)
0.0653372 + 0.997863i \(0.479188\pi\)
\(938\) 4.00000 0.130605
\(939\) 9.00000 0.293704
\(940\) 8.00000 0.260931
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) −46.0000 −1.49876
\(943\) −60.0000 −1.95387
\(944\) −36.0000 −1.17170
\(945\) 1.00000 0.0325300
\(946\) 2.00000 0.0650256
\(947\) −35.0000 −1.13735 −0.568674 0.822563i \(-0.692543\pi\)
−0.568674 + 0.822563i \(0.692543\pi\)
\(948\) 4.00000 0.129914
\(949\) −16.0000 −0.519382
\(950\) 14.0000 0.454220
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 2.00000 0.0647524
\(955\) −24.0000 −0.776622
\(956\) 10.0000 0.323423
\(957\) 3.00000 0.0969762
\(958\) 76.0000 2.45545
\(959\) 10.0000 0.322917
\(960\) 8.00000 0.258199
\(961\) −15.0000 −0.483871
\(962\) −8.00000 −0.257930
\(963\) 4.00000 0.128898
\(964\) 20.0000 0.644157
\(965\) −6.00000 −0.193147
\(966\) 10.0000 0.321745
\(967\) 35.0000 1.12552 0.562762 0.826619i \(-0.309739\pi\)
0.562762 + 0.826619i \(0.309739\pi\)
\(968\) 0 0
\(969\) −7.00000 −0.224872
\(970\) −26.0000 −0.834810
\(971\) 17.0000 0.545556 0.272778 0.962077i \(-0.412058\pi\)
0.272778 + 0.962077i \(0.412058\pi\)
\(972\) 2.00000 0.0641500
\(973\) 8.00000 0.256468
\(974\) 52.0000 1.66619
\(975\) −2.00000 −0.0640513
\(976\) 44.0000 1.40841
\(977\) 27.0000 0.863807 0.431903 0.901920i \(-0.357842\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 4.00000 0.127906
\(979\) −3.00000 −0.0958804
\(980\) −2.00000 −0.0638877
\(981\) −8.00000 −0.255420
\(982\) −86.0000 −2.74437
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) −6.00000 −0.191079
\(987\) 4.00000 0.127321
\(988\) 28.0000 0.890799
\(989\) −5.00000 −0.158991
\(990\) 2.00000 0.0635642
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) −32.0000 −1.01600
\(993\) −11.0000 −0.349074
\(994\) −16.0000 −0.507489
\(995\) 4.00000 0.126809
\(996\) −18.0000 −0.570352
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 50.0000 1.58272
\(999\) −2.00000 −0.0632772
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.b.1.1 1
3.2 odd 2 3465.2.a.t.1.1 1
5.4 even 2 5775.2.a.z.1.1 1
7.6 odd 2 8085.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.b.1.1 1 1.1 even 1 trivial
3465.2.a.t.1.1 1 3.2 odd 2
5775.2.a.z.1.1 1 5.4 even 2
8085.2.a.c.1.1 1 7.6 odd 2