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Properties

Label 1110.2.a.h.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
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] = [1110,2,Mod(1,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1110.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.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: \(8.86339462436\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} -6.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} +8.00000 q^{43} +6.00000 q^{44} +1.00000 q^{45} +6.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} +4.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} +8.00000 q^{61} -8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -6.00000 q^{66} -4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{70} -1.00000 q^{72} +14.0000 q^{73} -1.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -24.0000 q^{77} -2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -6.00000 q^{85} -8.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} +8.00000 q^{93} -6.00000 q^{94} +2.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -9.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −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
\(21\) −4.00000 −0.872872
\(22\) −6.00000 −1.27920
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) 6.00000 1.02899
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 6.00000 0.904534
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.00000 0.809040
\(56\) 4.00000 0.534522
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 1.00000 0.129099
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −8.00000 −1.01600
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −6.00000 −0.738549
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) −24.0000 −2.73505
\(78\) −2.00000 −0.226455
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 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 \(0.728845\pi\)
\(84\) −4.00000 −0.436436
\(85\) −6.00000 −0.650791
\(86\) −8.00000 −0.862662
\(87\) 6.00000 0.643268
\(88\) −6.00000 −0.639602
\(89\) 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
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) −6.00000 −0.618853
\(95\) 2.00000 0.205196
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −9.00000 −0.909137
\(99\) 6.00000 0.603023
\(100\) 1.00000 0.100000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 6.00000 0.594089
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −2.00000 −0.196116
\(105\) −4.00000 −0.390360
\(106\) −6.00000 −0.582772
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 20.0000 1.91565 0.957826 − 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) −6.00000 −0.572078
\(111\) 1.00000 0.0949158
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) 24.0000 2.20008
\(120\) −1.00000 −0.0912871
\(121\) 25.0000 2.27273
\(122\) −8.00000 −0.724286
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 4.00000 0.356348
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 6.00000 0.522233
\(133\) −8.00000 −0.693688
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −4.00000 −0.338062
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) −14.0000 −1.15865
\(147\) 9.00000 0.742307
\(148\) 1.00000 0.0821995
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −2.00000 −0.162221
\(153\) −6.00000 −0.485071
\(154\) 24.0000 1.93398
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 16.0000 1.27289
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −6.00000 −0.468521
\(165\) 6.00000 0.467099
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 4.00000 0.308607
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 2.00000 0.152944
\(172\) 8.00000 0.609994
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −6.00000 −0.454859
\(175\) −4.00000 −0.302372
\(176\) 6.00000 0.452267
\(177\) −12.0000 −0.901975
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 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
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) −8.00000 −0.586588
\(187\) −36.0000 −2.63258
\(188\) 6.00000 0.437595
\(189\) −4.00000 −0.290957
\(190\) −2.00000 −0.145095
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −16.0000 −1.15171 −0.575853 − 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −8.00000 −0.574367
\(195\) 2.00000 0.143223
\(196\) 9.00000 0.642857
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −6.00000 −0.426401
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −24.0000 −1.68447
\(204\) −6.00000 −0.420084
\(205\) −6.00000 −0.419058
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 12.0000 0.830057
\(210\) 4.00000 0.276026
\(211\) −16.0000 −1.10149 −0.550743 − 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) −1.00000 −0.0680414
\(217\) −32.0000 −2.17230
\(218\) −20.0000 −1.35457
\(219\) 14.0000 0.946032
\(220\) 6.00000 0.404520
\(221\) −12.0000 −0.807207
\(222\) −1.00000 −0.0671156
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 4.00000 0.267261
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 2.00000 0.132453
\(229\) 26.0000 1.71813 0.859064 − 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) −24.0000 −1.57908
\(232\) −6.00000 −0.393919
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) −2.00000 −0.130744
\(235\) 6.00000 0.391397
\(236\) −12.0000 −0.781133
\(237\) −16.0000 −1.03931
\(238\) −24.0000 −1.55569
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 1.00000 0.0645497
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −25.0000 −1.60706
\(243\) 1.00000 0.0641500
\(244\) 8.00000 0.512148
\(245\) 9.00000 0.574989
\(246\) 6.00000 0.382546
\(247\) 4.00000 0.254514
\(248\) −8.00000 −0.508001
\(249\) −12.0000 −0.760469
\(250\) −1.00000 −0.0632456
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) −8.00000 −0.498058
\(259\) −4.00000 −0.248548
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −6.00000 −0.369976 −0.184988 − 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) −6.00000 −0.369274
\(265\) 6.00000 0.368577
\(266\) 8.00000 0.490511
\(267\) 6.00000 0.367194
\(268\) −4.00000 −0.244339
\(269\) −24.0000 −1.46331 −0.731653 − 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −6.00000 −0.363803
\(273\) −8.00000 −0.484182
\(274\) 12.0000 0.724947
\(275\) 6.00000 0.361814
\(276\) 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
\(279\) 8.00000 0.478947
\(280\) 4.00000 0.239046
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) −6.00000 −0.357295
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) −12.0000 −0.709575
\(287\) 24.0000 1.41668
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) 8.00000 0.468968
\(292\) 14.0000 0.819288
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) −9.00000 −0.524891
\(295\) −12.0000 −0.698667
\(296\) −1.00000 −0.0581238
\(297\) 6.00000 0.348155
\(298\) 12.0000 0.695141
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −32.0000 −1.84445
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 8.00000 0.458079
\(306\) 6.00000 0.342997
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −24.0000 −1.36753
\(309\) 14.0000 0.796432
\(310\) −8.00000 −0.454369
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −2.00000 −0.113228
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) −14.0000 −0.790066
\(315\) −4.00000 −0.225374
\(316\) −16.0000 −0.900070
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −6.00000 −0.336463
\(319\) 36.0000 2.01561
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 16.0000 0.886158
\(327\) 20.0000 1.10600
\(328\) 6.00000 0.331295
\(329\) −24.0000 −1.32316
\(330\) −6.00000 −0.330289
\(331\) 14.0000 0.769510 0.384755 − 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) −12.0000 −0.658586
\(333\) 1.00000 0.0547997
\(334\) −12.0000 −0.656611
\(335\) −4.00000 −0.218543
\(336\) −4.00000 −0.218218
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) −6.00000 −0.325396
\(341\) 48.0000 2.59935
\(342\) −2.00000 −0.108148
\(343\) −8.00000 −0.431959
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −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 \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 4.00000 0.213809
\(351\) 2.00000 0.106752
\(352\) −6.00000 −0.319801
\(353\) 30.0000 1.59674 0.798369 − 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 24.0000 1.27021
\(358\) 12.0000 0.634220
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) 10.0000 0.525588
\(363\) 25.0000 1.31216
\(364\) −8.00000 −0.419314
\(365\) 14.0000 0.732793
\(366\) −8.00000 −0.418167
\(367\) −28.0000 −1.46159 −0.730794 − 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) −1.00000 −0.0519875
\(371\) −24.0000 −1.24602
\(372\) 8.00000 0.414781
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 36.0000 1.86152
\(375\) 1.00000 0.0516398
\(376\) −6.00000 −0.309426
\(377\) 12.0000 0.618031
\(378\) 4.00000 0.205738
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 2.00000 0.102598
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −24.0000 −1.22315
\(386\) 16.0000 0.814379
\(387\) 8.00000 0.406663
\(388\) 8.00000 0.406138
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) −12.0000 −0.605320
\(394\) −18.0000 −0.906827
\(395\) −16.0000 −0.805047
\(396\) 6.00000 0.301511
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 4.00000 0.200502
\(399\) −8.00000 −0.400501
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 24.0000 1.19110
\(407\) 6.00000 0.297409
\(408\) 6.00000 0.297044
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 6.00000 0.296319
\(411\) −12.0000 −0.591916
\(412\) 14.0000 0.689730
\(413\) 48.0000 2.36193
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −2.00000 −0.0980581
\(417\) −4.00000 −0.195881
\(418\) −12.0000 −0.586939
\(419\) 6.00000 0.293119 0.146560 − 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) −4.00000 −0.195180
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 16.0000 0.778868
\(423\) 6.00000 0.291730
\(424\) −6.00000 −0.291386
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −32.0000 −1.54859
\(428\) 0 0
\(429\) 12.0000 0.579365
\(430\) −8.00000 −0.385794
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 38.0000 1.82616 0.913082 − 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) 32.0000 1.53605
\(435\) 6.00000 0.287678
\(436\) 20.0000 0.957826
\(437\) 0 0
\(438\) −14.0000 −0.668946
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −6.00000 −0.286039
\(441\) 9.00000 0.428571
\(442\) 12.0000 0.570782
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 1.00000 0.0474579
\(445\) 6.00000 0.284427
\(446\) 4.00000 0.189405
\(447\) −12.0000 −0.567581
\(448\) −4.00000 −0.188982
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −36.0000 −1.69517
\(452\) 6.00000 0.282216
\(453\) −16.0000 −0.751746
\(454\) 12.0000 0.563188
\(455\) −8.00000 −0.375046
\(456\) −2.00000 −0.0936586
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −26.0000 −1.21490
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 24.0000 1.11658
\(463\) −34.0000 −1.58011 −0.790057 − 0.613033i \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) 6.00000 0.278543
\(465\) 8.00000 0.370991
\(466\) 24.0000 1.11178
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 2.00000 0.0924500
\(469\) 16.0000 0.738811
\(470\) −6.00000 −0.276759
\(471\) 14.0000 0.645086
\(472\) 12.0000 0.552345
\(473\) 48.0000 2.20704
\(474\) 16.0000 0.734904
\(475\) 2.00000 0.0917663
\(476\) 24.0000 1.10004
\(477\) 6.00000 0.274721
\(478\) −24.0000 −1.09773
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 2.00000 0.0911922
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 8.00000 0.363261
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −8.00000 −0.362143
\(489\) −16.0000 −0.723545
\(490\) −9.00000 −0.406579
\(491\) −6.00000 −0.270776 −0.135388 − 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) −6.00000 −0.270501
\(493\) −36.0000 −1.62136
\(494\) −4.00000 −0.179969
\(495\) 6.00000 0.269680
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 14.0000 0.626726 0.313363 − 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 1.00000 0.0447214
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −4.00000 −0.177471
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 6.00000 0.265684
\(511\) −56.0000 −2.47729
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −30.0000 −1.32324
\(515\) 14.0000 0.616914
\(516\) 8.00000 0.352180
\(517\) 36.0000 1.58328
\(518\) 4.00000 0.175750
\(519\) 6.00000 0.263371
\(520\) −2.00000 −0.0877058
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −6.00000 −0.262613
\(523\) 32.0000 1.39926 0.699631 − 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) −12.0000 −0.524222
\(525\) −4.00000 −0.174574
\(526\) 6.00000 0.261612
\(527\) −48.0000 −2.09091
\(528\) 6.00000 0.261116
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) −8.00000 −0.346844
\(533\) −12.0000 −0.519778
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −12.0000 −0.517838
\(538\) 24.0000 1.03471
\(539\) 54.0000 2.32594
\(540\) 1.00000 0.0430331
\(541\) 8.00000 0.343947 0.171973 − 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −8.00000 −0.343629
\(543\) −10.0000 −0.429141
\(544\) 6.00000 0.257248
\(545\) 20.0000 0.856706
\(546\) 8.00000 0.342368
\(547\) −4.00000 −0.171028 −0.0855138 − 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −12.0000 −0.512615
\(549\) 8.00000 0.341432
\(550\) −6.00000 −0.255841
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 64.0000 2.72156
\(554\) 10.0000 0.424859
\(555\) 1.00000 0.0424476
\(556\) −4.00000 −0.169638
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −8.00000 −0.338667
\(559\) 16.0000 0.676728
\(560\) −4.00000 −0.169031
\(561\) −36.0000 −1.51992
\(562\) 30.0000 1.26547
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 6.00000 0.252646
\(565\) 6.00000 0.252422
\(566\) 16.0000 0.672530
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −16.0000 −0.669579 −0.334790 − 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 20.0000 0.832611 0.416305 − 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) −19.0000 −0.790296
\(579\) −16.0000 −0.664937
\(580\) 6.00000 0.249136
\(581\) 48.0000 1.99138
\(582\) −8.00000 −0.331611
\(583\) 36.0000 1.49097
\(584\) −14.0000 −0.579324
\(585\) 2.00000 0.0826898
\(586\) 30.0000 1.23929
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 9.00000 0.371154
\(589\) 16.0000 0.659269
\(590\) 12.0000 0.494032
\(591\) 18.0000 0.740421
\(592\) 1.00000 0.0410997
\(593\) 12.0000 0.492781 0.246390 − 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) −6.00000 −0.246183
\(595\) 24.0000 0.983904
\(596\) −12.0000 −0.491539
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 32.0000 1.30422
\(603\) −4.00000 −0.162893
\(604\) −16.0000 −0.651031
\(605\) 25.0000 1.01639
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 − 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −2.00000 −0.0811107
\(609\) −24.0000 −0.972529
\(610\) −8.00000 −0.323911
\(611\) 12.0000 0.485468
\(612\) −6.00000 −0.242536
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 4.00000 0.161427
\(615\) −6.00000 −0.241943
\(616\) 24.0000 0.966988
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) −14.0000 −0.563163
\(619\) 32.0000 1.28619 0.643094 − 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 16.0000 0.639489
\(627\) 12.0000 0.479234
\(628\) 14.0000 0.558661
\(629\) −6.00000 −0.239236
\(630\) 4.00000 0.159364
\(631\) −28.0000 −1.11466 −0.557331 − 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 16.0000 0.636446
\(633\) −16.0000 −0.635943
\(634\) 18.0000 0.714871
\(635\) −4.00000 −0.158735
\(636\) 6.00000 0.237915
\(637\) 18.0000 0.713186
\(638\) −36.0000 −1.42525
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 12.0000 0.472134
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −72.0000 −2.82625
\(650\) −2.00000 −0.0784465
\(651\) −32.0000 −1.25418
\(652\) −16.0000 −0.626608
\(653\) 18.0000 0.704394 0.352197 − 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −20.0000 −0.782062
\(655\) −12.0000 −0.468879
\(656\) −6.00000 −0.234261
\(657\) 14.0000 0.546192
\(658\) 24.0000 0.935617
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 6.00000 0.233550
\(661\) 20.0000 0.777910 0.388955 − 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) −14.0000 −0.544125
\(663\) −12.0000 −0.466041
\(664\) 12.0000 0.465690
\(665\) −8.00000 −0.310227
\(666\) −1.00000 −0.0387492
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) −4.00000 −0.154649
\(670\) 4.00000 0.154533
\(671\) 48.0000 1.85302
\(672\) 4.00000 0.154303
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 10.0000 0.385186
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −6.00000 −0.230429
\(679\) −32.0000 −1.22805
\(680\) 6.00000 0.230089
\(681\) −12.0000 −0.459841
\(682\) −48.0000 −1.83801
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 2.00000 0.0764719
\(685\) −12.0000 −0.458496
\(686\) 8.00000 0.305441
\(687\) 26.0000 0.991962
\(688\) 8.00000 0.304997
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 − 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 6.00000 0.228086
\(693\) −24.0000 −0.911685
\(694\) 12.0000 0.455514
\(695\) −4.00000 −0.151729
\(696\) −6.00000 −0.227429
\(697\) 36.0000 1.36360
\(698\) −26.0000 −0.984115
\(699\) −24.0000 −0.907763
\(700\) −4.00000 −0.151186
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 2.00000 0.0754314
\(704\) 6.00000 0.226134
\(705\) 6.00000 0.225973
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) 44.0000 1.65245 0.826227 − 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −24.0000 −0.898177
\(715\) 12.0000 0.448775
\(716\) −12.0000 −0.448461
\(717\) 24.0000 0.896296
\(718\) 24.0000 0.895672
\(719\) −36.0000 −1.34257 −0.671287 − 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 1.00000 0.0372678
\(721\) −56.0000 −2.08555
\(722\) 15.0000 0.558242
\(723\) 2.00000 0.0743808
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) −25.0000 −0.927837
\(727\) 38.0000 1.40934 0.704671 − 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) −48.0000 −1.77534
\(732\) 8.00000 0.295689
\(733\) 2.00000 0.0738717 0.0369358 − 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 28.0000 1.03350
\(735\) 9.00000 0.331970
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 6.00000 0.220863
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 1.00000 0.0367607
\(741\) 4.00000 0.146944
\(742\) 24.0000 0.881068
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) −8.00000 −0.293294
\(745\) −12.0000 −0.439646
\(746\) 22.0000 0.805477
\(747\) −12.0000 −0.439057
\(748\) −36.0000 −1.31629
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 32.0000 1.16770 0.583848 − 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) −16.0000 −0.582300
\(756\) −4.00000 −0.145479
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 4.00000 0.144905
\(763\) −80.0000 −2.89619
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 12.0000 0.433578
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 24.0000 0.864900
\(771\) 30.0000 1.08042
\(772\) −16.0000 −0.575853
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) −8.00000 −0.287554
\(775\) 8.00000 0.287368
\(776\) −8.00000 −0.287183
\(777\) −4.00000 −0.143499
\(778\) 30.0000 1.07555
\(779\) −12.0000 −0.429945
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 9.00000 0.321429
\(785\) 14.0000 0.499681
\(786\) 12.0000 0.428026
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 18.0000 0.641223
\(789\) −6.00000 −0.213606
\(790\) 16.0000 0.569254
\(791\) −24.0000 −0.853342
\(792\) −6.00000 −0.213201
\(793\) 16.0000 0.568177
\(794\) 22.0000 0.780751
\(795\) 6.00000 0.212798
\(796\) −4.00000 −0.141776
\(797\) 42.0000 1.48772 0.743858 − 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 8.00000 0.283197
\(799\) −36.0000 −1.27359
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) −18.0000 −0.635602
\(803\) 84.0000 2.96430
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −24.0000 −0.842235
\(813\) 8.00000 0.280572
\(814\) −6.00000 −0.210300
\(815\) −16.0000 −0.560456
\(816\) −6.00000 −0.210042
\(817\) 16.0000 0.559769
\(818\) 10.0000 0.349642
\(819\) −8.00000 −0.279543
\(820\) −6.00000 −0.209529
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 12.0000 0.418548
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −14.0000 −0.487713
\(825\) 6.00000 0.208893
\(826\) −48.0000 −1.67013
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) −28.0000 −0.972480 −0.486240 − 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 12.0000 0.416526
\(831\) −10.0000 −0.346896
\(832\) 2.00000 0.0693375
\(833\) −54.0000 −1.87099
\(834\) 4.00000 0.138509
\(835\) 12.0000 0.415277
\(836\) 12.0000 0.415029
\(837\) 8.00000 0.276520
\(838\) −6.00000 −0.207267
\(839\) −24.0000 −0.828572 −0.414286 − 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 4.00000 0.138013
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) −30.0000 −1.03325
\(844\) −16.0000 −0.550743
\(845\) −9.00000 −0.309609
\(846\) −6.00000 −0.206284
\(847\) −100.000 −3.43604
\(848\) 6.00000 0.206041
\(849\) −16.0000 −0.549119
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 32.0000 1.09502
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −12.0000 −0.409673
\(859\) 14.0000 0.477674 0.238837 − 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 8.00000 0.272798
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) 30.0000 1.02121 0.510606 − 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −38.0000 −1.29129
\(867\) 19.0000 0.645274
\(868\) −32.0000 −1.08615
\(869\) −96.0000 −3.25658
\(870\) −6.00000 −0.203419
\(871\) −8.00000 −0.271070
\(872\) −20.0000 −0.677285
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) −4.00000 −0.135225
\(876\) 14.0000 0.473016
\(877\) 2.00000 0.0675352 0.0337676 − 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −20.0000 −0.674967
\(879\) −30.0000 −1.01187
\(880\) 6.00000 0.202260
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) −9.00000 −0.303046
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −12.0000 −0.403604
\(885\) −12.0000 −0.403376
\(886\) 12.0000 0.403148
\(887\) −6.00000 −0.201460 −0.100730 − 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 16.0000 0.536623
\(890\) −6.00000 −0.201120
\(891\) 6.00000 0.201008
\(892\) −4.00000 −0.133930
\(893\) 12.0000 0.401565
\(894\) 12.0000 0.401340
\(895\) −12.0000 −0.401116
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 36.0000 1.19867
\(903\) −32.0000 −1.06489
\(904\) −6.00000 −0.199557
\(905\) −10.0000 −0.332411
\(906\) 16.0000 0.531564
\(907\) −40.0000 −1.32818 −0.664089 − 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 2.00000 0.0662266
\(913\) −72.0000 −2.38285
\(914\) −8.00000 −0.264616
\(915\) 8.00000 0.264472
\(916\) 26.0000 0.859064
\(917\) 48.0000 1.58510
\(918\) 6.00000 0.198030
\(919\) −4.00000 −0.131948 −0.0659739 − 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 6.00000 0.197599
\(923\) 0 0
\(924\) −24.0000 −0.789542
\(925\) 1.00000 0.0328798
\(926\) 34.0000 1.11731
\(927\) 14.0000 0.459820
\(928\) −6.00000 −0.196960
\(929\) −42.0000 −1.37798 −0.688988 − 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) −8.00000 −0.262330
\(931\) 18.0000 0.589926
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) −36.0000 −1.17733
\(936\) −2.00000 −0.0653720
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) −16.0000 −0.522419
\(939\) −16.0000 −0.522140
\(940\) 6.00000 0.195698
\(941\) −48.0000 −1.56476 −0.782378 − 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) −14.0000 −0.456145
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) −4.00000 −0.130120
\(946\) −48.0000 −1.56061
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −16.0000 −0.519656
\(949\) 28.0000 0.908918
\(950\) −2.00000 −0.0648886
\(951\) −18.0000 −0.583690
\(952\) −24.0000 −0.777844
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 36.0000 1.16371
\(958\) 24.0000 0.775405
\(959\) 48.0000 1.55000
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) −46.0000 −1.47926 −0.739630 − 0.673014i \(-0.765000\pi\)
−0.739630 + 0.673014i \(0.765000\pi\)
\(968\) −25.0000 −0.803530
\(969\) −12.0000 −0.385496
\(970\) −8.00000 −0.256865
\(971\) −54.0000 −1.73294 −0.866471 − 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000 0.512936
\(974\) −2.00000 −0.0640841
\(975\) 2.00000 0.0640513
\(976\) 8.00000 0.256074
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 16.0000 0.511624
\(979\) 36.0000 1.15056
\(980\) 9.00000 0.287494
\(981\) 20.0000 0.638551
\(982\) 6.00000 0.191468
\(983\) −30.0000 −0.956851 −0.478426 − 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) 6.00000 0.191273
\(985\) 18.0000 0.573528
\(986\) 36.0000 1.14647
\(987\) −24.0000 −0.763928
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) −6.00000 −0.190693
\(991\) 56.0000 1.77890 0.889449 − 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) −8.00000 −0.254000
\(993\) 14.0000 0.444277
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) −12.0000 −0.380235
\(997\) 26.0000 0.823428 0.411714 − 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −14.0000 −0.443162
\(999\) 1.00000 0.0316386
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 1110.2.a.h.1.1 ✓ 1
3.2 odd 2 3330.2.a.m.1.1 1
4.3 odd 2 8880.2.a.n.1.1 1
5.4 even 2 5550.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.h.1.1 ✓ 1 1.1 even 1 trivial
3330.2.a.m.1.1 1 3.2 odd 2
5550.2.a.bd.1.1 1 5.4 even 2
8880.2.a.n.1.1 1 4.3 odd 2