Properties

Label 5610.2.a.o.1.1
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
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] = [5610,2,Mod(1,5610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, 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("5610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.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: \(44.7960755339\)
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 \(\chi\) \(=\) 5610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -2.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} -10.0000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -4.00000 q^{52} +12.0000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -2.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -1.00000 q^{66} +8.00000 q^{67} -1.00000 q^{68} -6.00000 q^{69} +2.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -16.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +2.00000 q^{76} +2.00000 q^{77} +4.00000 q^{78} -10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +1.00000 q^{85} +10.0000 q^{86} +6.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} -8.00000 q^{91} -6.00000 q^{92} -4.00000 q^{93} -2.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} +3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) −1.00000 −0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 1.00000 0.171499
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.00000 −0.324443
\(39\) −4.00000 −0.640513
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) −4.00000 −0.554700
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) −2.00000 −0.267261
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −1.00000 −0.123091
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −1.00000 −0.121268
\(69\) −6.00000 −0.722315
\(70\) 2.00000 0.239046
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 10.0000 1.16248
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) 2.00000 0.227921
\(78\) 4.00000 0.452911
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) 1.00000 0.108465
\(86\) 10.0000 1.07833
\(87\) 6.00000 0.643268
\(88\) −1.00000 −0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.00000 0.105409
\(91\) −8.00000 −0.838628
\(92\) −6.00000 −0.625543
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 3.00000 0.303046
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 1.00000 0.0990148
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 4.00000 0.392232
\(105\) −2.00000 −0.195180
\(106\) −12.0000 −1.16554
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 1.00000 0.0953463
\(111\) −10.0000 −0.949158
\(112\) 2.00000 0.188982
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −2.00000 −0.187317
\(115\) 6.00000 0.559503
\(116\) 6.00000 0.557086
\(117\) −4.00000 −0.369800
\(118\) −6.00000 −0.552345
\(119\) −2.00000 −0.183340
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) 6.00000 0.541002
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.0000 −0.880451
\(130\) −4.00000 −0.350823
\(131\) −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
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) 1.00000 0.0857493
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 6.00000 0.510754
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) 16.0000 1.32417
\(147\) −3.00000 −0.247436
\(148\) −10.0000 −0.821995
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −2.00000 −0.162221
\(153\) −1.00000 −0.0808452
\(154\) −2.00000 −0.161165
\(155\) 4.00000 0.321288
\(156\) −4.00000 −0.320256
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 10.0000 0.795557
\(159\) 12.0000 0.951662
\(160\) 1.00000 0.0790569
\(161\) −12.0000 −0.945732
\(162\) −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
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) −1.00000 −0.0766965
\(171\) 2.00000 0.152944
\(172\) −10.0000 −0.762493
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −6.00000 −0.454859
\(175\) 2.00000 0.151186
\(176\) 1.00000 0.0753778
\(177\) 6.00000 0.450988
\(178\) −6.00000 −0.449719
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 8.00000 0.592999
\(183\) −10.0000 −0.739221
\(184\) 6.00000 0.442326
\(185\) 10.0000 0.735215
\(186\) 4.00000 0.293294
\(187\) −1.00000 −0.0731272
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 2.00000 0.145095
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 10.0000 0.717958
\(195\) 4.00000 0.286446
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 12.0000 0.842235
\(204\) −1.00000 −0.0700140
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) −6.00000 −0.417029
\(208\) −4.00000 −0.277350
\(209\) 2.00000 0.138343
\(210\) 2.00000 0.138013
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 12.0000 0.824163
\(213\) −6.00000 −0.411113
\(214\) 12.0000 0.820303
\(215\) 10.0000 0.681994
\(216\) −1.00000 −0.0680414
\(217\) −8.00000 −0.543075
\(218\) −2.00000 −0.135457
\(219\) −16.0000 −1.08118
\(220\) −1.00000 −0.0674200
\(221\) 4.00000 0.269069
\(222\) 10.0000 0.671156
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −2.00000 −0.133631
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 2.00000 0.132453
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −6.00000 −0.395628
\(231\) 2.00000 0.131590
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −10.0000 −0.649570
\(238\) 2.00000 0.129641
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 3.00000 0.191663
\(246\) −6.00000 −0.382546
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 2.00000 0.125988
\(253\) −6.00000 −0.377217
\(254\) −20.0000 −1.25491
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 10.0000 0.622573
\(259\) −20.0000 −1.24274
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −12.0000 −0.737154
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) 8.00000 0.488678
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 1.00000 0.0608581
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −8.00000 −0.484182
\(274\) 6.00000 0.362473
\(275\) 1.00000 0.0603023
\(276\) −6.00000 −0.361158
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 4.00000 0.239904
\(279\) −4.00000 −0.239474
\(280\) 2.00000 0.119523
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −6.00000 −0.356034
\(285\) −2.00000 −0.118470
\(286\) 4.00000 0.236525
\(287\) 12.0000 0.708338
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 6.00000 0.352332
\(291\) −10.0000 −0.586210
\(292\) −16.0000 −0.936329
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 3.00000 0.174964
\(295\) −6.00000 −0.349334
\(296\) 10.0000 0.581238
\(297\) 1.00000 0.0580259
\(298\) 18.0000 1.04271
\(299\) 24.0000 1.38796
\(300\) 1.00000 0.0577350
\(301\) −20.0000 −1.15278
\(302\) −8.00000 −0.460348
\(303\) 18.0000 1.03407
\(304\) 2.00000 0.114708
\(305\) 10.0000 0.572598
\(306\) 1.00000 0.0571662
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) −4.00000 −0.227185
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 4.00000 0.226455
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 10.0000 0.564333
\(315\) −2.00000 −0.112687
\(316\) −10.0000 −0.562544
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −12.0000 −0.672927
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) 12.0000 0.668734
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 4.00000 0.221540
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −12.0000 −0.658586
\(333\) −10.0000 −0.547997
\(334\) −12.0000 −0.656611
\(335\) −8.00000 −0.437087
\(336\) 2.00000 0.109109
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) −3.00000 −0.163178
\(339\) −12.0000 −0.651751
\(340\) 1.00000 0.0542326
\(341\) −4.00000 −0.216612
\(342\) −2.00000 −0.108148
\(343\) −20.0000 −1.07990
\(344\) 10.0000 0.539164
\(345\) 6.00000 0.323029
\(346\) −18.0000 −0.967686
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 6.00000 0.321634
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) −2.00000 −0.106904
\(351\) −4.00000 −0.213504
\(352\) −1.00000 −0.0533002
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −6.00000 −0.318896
\(355\) 6.00000 0.318447
\(356\) 6.00000 0.317999
\(357\) −2.00000 −0.105851
\(358\) 6.00000 0.317110
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) 10.0000 0.525588
\(363\) 1.00000 0.0524864
\(364\) −8.00000 −0.419314
\(365\) 16.0000 0.837478
\(366\) 10.0000 0.522708
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −6.00000 −0.312772
\(369\) 6.00000 0.312348
\(370\) −10.0000 −0.519875
\(371\) 24.0000 1.24602
\(372\) −4.00000 −0.207390
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 1.00000 0.0517088
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) −2.00000 −0.102869
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −2.00000 −0.102598
\(381\) 20.0000 1.02463
\(382\) 12.0000 0.613973
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −2.00000 −0.101929
\(386\) 4.00000 0.203595
\(387\) −10.0000 −0.508329
\(388\) −10.0000 −0.507673
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) −4.00000 −0.202548
\(391\) 6.00000 0.303433
\(392\) 3.00000 0.151523
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) 10.0000 0.503155
\(396\) 1.00000 0.0502519
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 4.00000 0.200502
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) −8.00000 −0.399004
\(403\) 16.0000 0.797017
\(404\) 18.0000 0.895533
\(405\) −1.00000 −0.0496904
\(406\) −12.0000 −0.595550
\(407\) −10.0000 −0.495682
\(408\) 1.00000 0.0495074
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 6.00000 0.296319
\(411\) −6.00000 −0.295958
\(412\) 8.00000 0.394132
\(413\) 12.0000 0.590481
\(414\) 6.00000 0.294884
\(415\) 12.0000 0.589057
\(416\) 4.00000 0.196116
\(417\) −4.00000 −0.195881
\(418\) −2.00000 −0.0978232
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) −1.00000 −0.0485071
\(426\) 6.00000 0.290701
\(427\) −20.0000 −0.967868
\(428\) −12.0000 −0.580042
\(429\) −4.00000 −0.193122
\(430\) −10.0000 −0.482243
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 8.00000 0.384012
\(435\) −6.00000 −0.287678
\(436\) 2.00000 0.0957826
\(437\) −12.0000 −0.574038
\(438\) 16.0000 0.764510
\(439\) 38.0000 1.81364 0.906821 0.421517i \(-0.138502\pi\)
0.906821 + 0.421517i \(0.138502\pi\)
\(440\) 1.00000 0.0476731
\(441\) −3.00000 −0.142857
\(442\) −4.00000 −0.190261
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) −8.00000 −0.378811
\(447\) −18.0000 −0.851371
\(448\) 2.00000 0.0944911
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 6.00000 0.282529
\(452\) −12.0000 −0.564433
\(453\) 8.00000 0.375873
\(454\) −12.0000 −0.563188
\(455\) 8.00000 0.375046
\(456\) −2.00000 −0.0936586
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −2.00000 −0.0934539
\(459\) −1.00000 −0.0466760
\(460\) 6.00000 0.279751
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 4.00000 0.185496
\(466\) 6.00000 0.277945
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −4.00000 −0.184900
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) −6.00000 −0.276172
\(473\) −10.0000 −0.459800
\(474\) 10.0000 0.459315
\(475\) 2.00000 0.0917663
\(476\) −2.00000 −0.0916698
\(477\) 12.0000 0.549442
\(478\) −24.0000 −1.09773
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 1.00000 0.0456435
\(481\) 40.0000 1.82384
\(482\) −8.00000 −0.364390
\(483\) −12.0000 −0.546019
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 10.0000 0.452679
\(489\) −4.00000 −0.180886
\(490\) −3.00000 −0.135526
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 6.00000 0.270501
\(493\) −6.00000 −0.270226
\(494\) 8.00000 0.359937
\(495\) −1.00000 −0.0449467
\(496\) −4.00000 −0.179605
\(497\) −12.0000 −0.538274
\(498\) 12.0000 0.537733
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 6.00000 0.267793
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −18.0000 −0.800989
\(506\) 6.00000 0.266733
\(507\) 3.00000 0.133235
\(508\) 20.0000 0.887357
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) −1.00000 −0.0442807
\(511\) −32.0000 −1.41560
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 30.0000 1.32324
\(515\) −8.00000 −0.352522
\(516\) −10.0000 −0.440225
\(517\) 0 0
\(518\) 20.0000 0.878750
\(519\) 18.0000 0.790112
\(520\) −4.00000 −0.175412
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) −6.00000 −0.262613
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) −12.0000 −0.524222
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 1.00000 0.0435194
\(529\) 13.0000 0.565217
\(530\) 12.0000 0.521247
\(531\) 6.00000 0.260378
\(532\) 4.00000 0.173422
\(533\) −24.0000 −1.03956
\(534\) −6.00000 −0.259645
\(535\) 12.0000 0.518805
\(536\) −8.00000 −0.345547
\(537\) −6.00000 −0.258919
\(538\) 30.0000 1.29339
\(539\) −3.00000 −0.129219
\(540\) −1.00000 −0.0430331
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 28.0000 1.20270
\(543\) −10.0000 −0.429141
\(544\) 1.00000 0.0428746
\(545\) −2.00000 −0.0856706
\(546\) 8.00000 0.342368
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) −1.00000 −0.0426401
\(551\) 12.0000 0.511217
\(552\) 6.00000 0.255377
\(553\) −20.0000 −0.850487
\(554\) −26.0000 −1.10463
\(555\) 10.0000 0.424476
\(556\) −4.00000 −0.169638
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 4.00000 0.169334
\(559\) 40.0000 1.69182
\(560\) −2.00000 −0.0845154
\(561\) −1.00000 −0.0422200
\(562\) 30.0000 1.26547
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 4.00000 0.168133
\(567\) 2.00000 0.0839921
\(568\) 6.00000 0.251754
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 2.00000 0.0837708
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −4.00000 −0.167248
\(573\) −12.0000 −0.501307
\(574\) −12.0000 −0.500870
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −4.00000 −0.166234
\(580\) −6.00000 −0.249136
\(581\) −24.0000 −0.995688
\(582\) 10.0000 0.414513
\(583\) 12.0000 0.496989
\(584\) 16.0000 0.662085
\(585\) 4.00000 0.165380
\(586\) −18.0000 −0.743573
\(587\) −6.00000 −0.247647 −0.123823 0.992304i \(-0.539516\pi\)
−0.123823 + 0.992304i \(0.539516\pi\)
\(588\) −3.00000 −0.123718
\(589\) −8.00000 −0.329634
\(590\) 6.00000 0.247016
\(591\) 6.00000 0.246807
\(592\) −10.0000 −0.410997
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 2.00000 0.0819920
\(596\) −18.0000 −0.737309
\(597\) −4.00000 −0.163709
\(598\) −24.0000 −0.981433
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 20.0000 0.815139
\(603\) 8.00000 0.325785
\(604\) 8.00000 0.325515
\(605\) −1.00000 −0.0406558
\(606\) −18.0000 −0.731200
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 12.0000 0.486265
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) 22.0000 0.887848
\(615\) −6.00000 −0.241943
\(616\) −2.00000 −0.0805823
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) −8.00000 −0.321807
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 4.00000 0.160644
\(621\) −6.00000 −0.240772
\(622\) 6.00000 0.240578
\(623\) 12.0000 0.480770
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 2.00000 0.0798723
\(628\) −10.0000 −0.399043
\(629\) 10.0000 0.398726
\(630\) 2.00000 0.0796819
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 10.0000 0.397779
\(633\) 20.0000 0.794929
\(634\) 18.0000 0.714871
\(635\) −20.0000 −0.793676
\(636\) 12.0000 0.475831
\(637\) 12.0000 0.475457
\(638\) −6.00000 −0.237542
\(639\) −6.00000 −0.237356
\(640\) 1.00000 0.0395285
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 12.0000 0.473602
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −12.0000 −0.472866
\(645\) 10.0000 0.393750
\(646\) 2.00000 0.0786889
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 6.00000 0.235521
\(650\) 4.00000 0.156893
\(651\) −8.00000 −0.313545
\(652\) −4.00000 −0.156652
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 12.0000 0.468879
\(656\) 6.00000 0.234261
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 28.0000 1.08825
\(663\) 4.00000 0.155347
\(664\) 12.0000 0.465690
\(665\) −4.00000 −0.155113
\(666\) 10.0000 0.387492
\(667\) −36.0000 −1.39393
\(668\) 12.0000 0.464294
\(669\) 8.00000 0.309298
\(670\) 8.00000 0.309067
\(671\) −10.0000 −0.386046
\(672\) −2.00000 −0.0771517
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 16.0000 0.616297
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 12.0000 0.460857
\(679\) −20.0000 −0.767530
\(680\) −1.00000 −0.0383482
\(681\) 12.0000 0.459841
\(682\) 4.00000 0.153168
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 2.00000 0.0764719
\(685\) 6.00000 0.229248
\(686\) 20.0000 0.763604
\(687\) 2.00000 0.0763048
\(688\) −10.0000 −0.381246
\(689\) −48.0000 −1.82865
\(690\) −6.00000 −0.228416
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 18.0000 0.684257
\(693\) 2.00000 0.0759737
\(694\) 36.0000 1.36654
\(695\) 4.00000 0.151729
\(696\) −6.00000 −0.227429
\(697\) −6.00000 −0.227266
\(698\) −8.00000 −0.302804
\(699\) −6.00000 −0.226941
\(700\) 2.00000 0.0755929
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 4.00000 0.150970
\(703\) −20.0000 −0.754314
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 36.0000 1.35392
\(708\) 6.00000 0.225494
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) −6.00000 −0.225176
\(711\) −10.0000 −0.375029
\(712\) −6.00000 −0.224860
\(713\) 24.0000 0.898807
\(714\) 2.00000 0.0748481
\(715\) 4.00000 0.149592
\(716\) −6.00000 −0.224231
\(717\) 24.0000 0.896296
\(718\) −24.0000 −0.895672
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.0000 0.595871
\(722\) 15.0000 0.558242
\(723\) 8.00000 0.297523
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) −1.00000 −0.0371135
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) −16.0000 −0.592187
\(731\) 10.0000 0.369863
\(732\) −10.0000 −0.369611
\(733\) 44.0000 1.62518 0.812589 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) −32.0000 −1.18114
\(735\) 3.00000 0.110657
\(736\) 6.00000 0.221163
\(737\) 8.00000 0.294684
\(738\) −6.00000 −0.220863
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 10.0000 0.367607
\(741\) −8.00000 −0.293887
\(742\) −24.0000 −0.881068
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 4.00000 0.146647
\(745\) 18.0000 0.659469
\(746\) 4.00000 0.146450
\(747\) −12.0000 −0.439057
\(748\) −1.00000 −0.0365636
\(749\) −24.0000 −0.876941
\(750\) 1.00000 0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) 24.0000 0.874028
\(755\) −8.00000 −0.291150
\(756\) 2.00000 0.0727393
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −20.0000 −0.726433
\(759\) −6.00000 −0.217786
\(760\) 2.00000 0.0725476
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −20.0000 −0.724524
\(763\) 4.00000 0.144810
\(764\) −12.0000 −0.434145
\(765\) 1.00000 0.0361551
\(766\) −12.0000 −0.433578
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 2.00000 0.0720750
\(771\) −30.0000 −1.08042
\(772\) −4.00000 −0.143963
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 10.0000 0.359443
\(775\) −4.00000 −0.143684
\(776\) 10.0000 0.358979
\(777\) −20.0000 −0.717496
\(778\) −12.0000 −0.430221
\(779\) 12.0000 0.429945
\(780\) 4.00000 0.143223
\(781\) −6.00000 −0.214697
\(782\) −6.00000 −0.214560
\(783\) 6.00000 0.214423
\(784\) −3.00000 −0.107143
\(785\) 10.0000 0.356915
\(786\) 12.0000 0.428026
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −10.0000 −0.355784
\(791\) −24.0000 −0.853342
\(792\) −1.00000 −0.0355335
\(793\) 40.0000 1.42044
\(794\) 22.0000 0.780751
\(795\) −12.0000 −0.425596
\(796\) −4.00000 −0.141776
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −16.0000 −0.564628
\(804\) 8.00000 0.282138
\(805\) 12.0000 0.422944
\(806\) −16.0000 −0.563576
\(807\) −30.0000 −1.05605
\(808\) −18.0000 −0.633238
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 1.00000 0.0351364
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 12.0000 0.421117
\(813\) −28.0000 −0.982003
\(814\) 10.0000 0.350500
\(815\) 4.00000 0.140114
\(816\) −1.00000 −0.0350070
\(817\) −20.0000 −0.699711
\(818\) 10.0000 0.349642
\(819\) −8.00000 −0.279543
\(820\) −6.00000 −0.209529
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 6.00000 0.209274
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −8.00000 −0.278693
\(825\) 1.00000 0.0348155
\(826\) −12.0000 −0.417533
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) −12.0000 −0.416526
\(831\) 26.0000 0.901930
\(832\) −4.00000 −0.138675
\(833\) 3.00000 0.103944
\(834\) 4.00000 0.138509
\(835\) −12.0000 −0.415277
\(836\) 2.00000 0.0691714
\(837\) −4.00000 −0.138260
\(838\) −12.0000 −0.414533
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −30.0000 −1.03325
\(844\) 20.0000 0.688428
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 12.0000 0.412082
\(849\) −4.00000 −0.137280
\(850\) 1.00000 0.0342997
\(851\) 60.0000 2.05677
\(852\) −6.00000 −0.205557
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 20.0000 0.684386
\(855\) −2.00000 −0.0683986
\(856\) 12.0000 0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 4.00000 0.136558
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 10.0000 0.340997
\(861\) 12.0000 0.408959
\(862\) −12.0000 −0.408722
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) −14.0000 −0.475739
\(867\) 1.00000 0.0339618
\(868\) −8.00000 −0.271538
\(869\) −10.0000 −0.339227
\(870\) 6.00000 0.203419
\(871\) −32.0000 −1.08428
\(872\) −2.00000 −0.0677285
\(873\) −10.0000 −0.338449
\(874\) 12.0000 0.405906
\(875\) −2.00000 −0.0676123
\(876\) −16.0000 −0.540590
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) −38.0000 −1.28244
\(879\) 18.0000 0.607125
\(880\) −1.00000 −0.0337100
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 3.00000 0.101015
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 4.00000 0.134535
\(885\) −6.00000 −0.201688
\(886\) 6.00000 0.201574
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 10.0000 0.335578
\(889\) 40.0000 1.34156
\(890\) 6.00000 0.201120
\(891\) 1.00000 0.0335013
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 6.00000 0.200558
\(896\) −2.00000 −0.0668153
\(897\) 24.0000 0.801337
\(898\) −12.0000 −0.400445
\(899\) −24.0000 −0.800445
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) −6.00000 −0.199778
\(903\) −20.0000 −0.665558
\(904\) 12.0000 0.399114
\(905\) 10.0000 0.332411
\(906\) −8.00000 −0.265782
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 12.0000 0.398234
\(909\) 18.0000 0.597022
\(910\) −8.00000 −0.265197
\(911\) −54.0000 −1.78910 −0.894550 0.446968i \(-0.852504\pi\)
−0.894550 + 0.446968i \(0.852504\pi\)
\(912\) 2.00000 0.0662266
\(913\) −12.0000 −0.397142
\(914\) 10.0000 0.330771
\(915\) 10.0000 0.330590
\(916\) 2.00000 0.0660819
\(917\) −24.0000 −0.792550
\(918\) 1.00000 0.0330049
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −6.00000 −0.197814
\(921\) −22.0000 −0.724925
\(922\) −30.0000 −0.987997
\(923\) 24.0000 0.789970
\(924\) 2.00000 0.0657952
\(925\) −10.0000 −0.328798
\(926\) −8.00000 −0.262896
\(927\) 8.00000 0.262754
\(928\) −6.00000 −0.196960
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) −4.00000 −0.131165
\(931\) −6.00000 −0.196642
\(932\) −6.00000 −0.196537
\(933\) −6.00000 −0.196431
\(934\) 6.00000 0.196326
\(935\) 1.00000 0.0327035
\(936\) 4.00000 0.130744
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) −16.0000 −0.522419
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 10.0000 0.325818
\(943\) −36.0000 −1.17232
\(944\) 6.00000 0.195283
\(945\) −2.00000 −0.0650600
\(946\) 10.0000 0.325128
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) −10.0000 −0.324785
\(949\) 64.0000 2.07753
\(950\) −2.00000 −0.0648886
\(951\) −18.0000 −0.583690
\(952\) 2.00000 0.0648204
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −12.0000 −0.388514
\(955\) 12.0000 0.388311
\(956\) 24.0000 0.776215
\(957\) 6.00000 0.193952
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) −40.0000 −1.28965
\(963\) −12.0000 −0.386695
\(964\) 8.00000 0.257663
\(965\) 4.00000 0.128765
\(966\) 12.0000 0.386094
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −2.00000 −0.0642493
\(970\) −10.0000 −0.321081
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 1.00000 0.0320750
\(973\) −8.00000 −0.256468
\(974\) 16.0000 0.512673
\(975\) −4.00000 −0.128103
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 4.00000 0.127906
\(979\) 6.00000 0.191761
\(980\) 3.00000 0.0958315
\(981\) 2.00000 0.0638551
\(982\) −12.0000 −0.382935
\(983\) −30.0000 −0.956851 −0.478426 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) −6.00000 −0.191273
\(985\) −6.00000 −0.191176
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 60.0000 1.90789
\(990\) 1.00000 0.0317821
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 4.00000 0.127000
\(993\) −28.0000 −0.888553
\(994\) 12.0000 0.380617
\(995\) 4.00000 0.126809
\(996\) −12.0000 −0.380235
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 28.0000 0.886325
\(999\) −10.0000 −0.316386
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 5610.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.o.1.1 1 1.1 even 1 trivial