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Properties

Label 1110.2.a.o.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} -2.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} -10.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +4.00000 q^{56} +2.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +4.00000 q^{61} +4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -2.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} -4.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} +1.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -8.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +4.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} +8.00000 q^{91} +4.00000 q^{93} -10.0000 q^{94} -2.00000 q^{95} +1.00000 q^{96} +12.0000 q^{97} +9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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 \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 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
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 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
\(22\) −2.00000 −0.426401
\(23\) 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
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) −2.00000 −0.342997
\(35\) −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
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −10.0000 −1.45865 −0.729325 − 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 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
\(55\) 2.00000 0.269680
\(56\) 4.00000 0.534522
\(57\) 2.00000 0.264906
\(58\) 2.00000 0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 4.00000 0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −2.00000 −0.246183
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 1.00000 0.116248
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) −8.00000 −0.911685
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 4.00000 0.436436
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 2.00000 0.214423
\(88\) −2.00000 −0.213201
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) −10.0000 −1.03142
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 9.00000 0.909137
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −2.00000 −0.198030
\(103\) 6.00000 0.591198 0.295599 − 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 2.00000 0.196116
\(105\) −4.00000 −0.390360
\(106\) 6.00000 0.582772
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 2.00000 0.190693
\(111\) 1.00000 0.0949158
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) −8.00000 −0.733359
\(120\) −1.00000 −0.0912871
\(121\) −7.00000 −0.636364
\(122\) 4.00000 0.362143
\(123\) −6.00000 −0.541002
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 4.00000 0.356348
\(127\) −12.0000 −1.06483 −0.532414 − 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −2.00000 −0.174078
\(133\) 8.00000 0.693688
\(134\) −12.0000 −1.03664
\(135\) −1.00000 −0.0860663
\(136\) −2.00000 −0.171499
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −4.00000 −0.338062
\(141\) −10.0000 −0.842152
\(142\) 8.00000 0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) −2.00000 −0.165521
\(147\) 9.00000 0.742307
\(148\) 1.00000 0.0821995
\(149\) 4.00000 0.327693 0.163846 − 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 2.00000 0.162221
\(153\) −2.00000 −0.161690
\(154\) −8.00000 −0.644658
\(155\) −4.00000 −0.321288
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −4.00000 −0.318223
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) 2.00000 0.155700
\(166\) 4.00000 0.310460
\(167\) −20.0000 −1.54765 −0.773823 − 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 4.00000 0.308607
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) 2.00000 0.152944
\(172\) −4.00000 −0.304997
\(173\) −10.0000 −0.760286 −0.380143 − 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 2.00000 0.151620
\(175\) 4.00000 0.302372
\(176\) −2.00000 −0.150756
\(177\) 4.00000 0.300658
\(178\) −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
\(181\) −18.0000 −1.33793 −0.668965 − 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 8.00000 0.592999
\(183\) 4.00000 0.295689
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 4.00000 0.293294
\(187\) 4.00000 0.292509
\(188\) −10.0000 −0.729325
\(189\) 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
\(193\) 12.0000 0.863779 0.431889 − 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 12.0000 0.861550
\(195\) −2.00000 −0.143223
\(196\) 9.00000 0.642857
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −2.00000 −0.142134
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 8.00000 0.561490
\(204\) −2.00000 −0.140028
\(205\) 6.00000 0.419058
\(206\) 6.00000 0.418040
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) −4.00000 −0.276026
\(211\) 16.0000 1.10149 0.550743 − 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 6.00000 0.412082
\(213\) 8.00000 0.548151
\(214\) −8.00000 −0.546869
\(215\) 4.00000 0.272798
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −16.0000 −1.08366
\(219\) −2.00000 −0.135147
\(220\) 2.00000 0.134840
\(221\) −4.00000 −0.269069
\(222\) 1.00000 0.0671156
\(223\) 20.0000 1.33930 0.669650 − 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) 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
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 2.00000 0.131306
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 2.00000 0.130744
\(235\) 10.0000 0.652328
\(236\) 4.00000 0.260378
\(237\) −4.00000 −0.259828
\(238\) −8.00000 −0.518563
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 4.00000 0.256074
\(245\) −9.00000 −0.574989
\(246\) −6.00000 −0.382546
\(247\) 4.00000 0.254514
\(248\) 4.00000 0.254000
\(249\) 4.00000 0.253490
\(250\) −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
\(254\) −12.0000 −0.752947
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −4.00000 −0.249029
\(259\) 4.00000 0.248548
\(260\) −2.00000 −0.124035
\(261\) 2.00000 0.123797
\(262\) 4.00000 0.247121
\(263\) 26.0000 1.60323 0.801614 − 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) −2.00000 −0.123091
\(265\) −6.00000 −0.368577
\(266\) 8.00000 0.490511
\(267\) −6.00000 −0.367194
\(268\) −12.0000 −0.733017
\(269\) −24.0000 −1.46331 −0.731653 − 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −2.00000 −0.121268
\(273\) 8.00000 0.484182
\(274\) 12.0000 0.724947
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −20.0000 −1.19952
\(279\) 4.00000 0.239474
\(280\) −4.00000 −0.239046
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −10.0000 −0.595491
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 8.00000 0.474713
\(285\) −2.00000 −0.118470
\(286\) −4.00000 −0.236525
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 12.0000 0.703452
\(292\) −2.00000 −0.117041
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 9.00000 0.524891
\(295\) −4.00000 −0.232889
\(296\) 1.00000 0.0581238
\(297\) −2.00000 −0.116052
\(298\) 4.00000 0.231714
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −16.0000 −0.922225
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) −4.00000 −0.229039
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −8.00000 −0.455842
\(309\) 6.00000 0.341328
\(310\) −4.00000 −0.227185
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 2.00000 0.113228
\(313\) −4.00000 −0.226093 −0.113047 − 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) −2.00000 −0.112867
\(315\) −4.00000 −0.225374
\(316\) −4.00000 −0.225018
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) −4.00000 −0.223957
\(320\) −1.00000 −0.0559017
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) −16.0000 −0.884802
\(328\) −6.00000 −0.331295
\(329\) −40.0000 −2.20527
\(330\) 2.00000 0.110096
\(331\) −10.0000 −0.549650 −0.274825 − 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 4.00000 0.219529
\(333\) 1.00000 0.0547997
\(334\) −20.0000 −1.09435
\(335\) 12.0000 0.655630
\(336\) 4.00000 0.218218
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) 2.00000 0.108465
\(341\) −8.00000 −0.433224
\(342\) 2.00000 0.108148
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 2.00000 0.107211
\(349\) 2.00000 0.107058 0.0535288 − 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 4.00000 0.213809
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 4.00000 0.212598
\(355\) −8.00000 −0.424596
\(356\) −6.00000 −0.317999
\(357\) −8.00000 −0.423405
\(358\) −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
\(362\) −18.0000 −0.946059
\(363\) −7.00000 −0.367405
\(364\) 8.00000 0.419314
\(365\) 2.00000 0.104685
\(366\) 4.00000 0.209083
\(367\) 28.0000 1.46159 0.730794 − 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) −1.00000 −0.0519875
\(371\) 24.0000 1.24602
\(372\) 4.00000 0.207390
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 4.00000 0.206835
\(375\) −1.00000 −0.0516398
\(376\) −10.0000 −0.515711
\(377\) 4.00000 0.206010
\(378\) 4.00000 0.205738
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −2.00000 −0.102598
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 20.0000 1.02195 0.510976 − 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 1.00000 0.0510310
\(385\) 8.00000 0.407718
\(386\) 12.0000 0.610784
\(387\) −4.00000 −0.203331
\(388\) 12.0000 0.609208
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) 4.00000 0.201773
\(394\) −6.00000 −0.302276
\(395\) 4.00000 0.201262
\(396\) −2.00000 −0.100504
\(397\) −6.00000 −0.301131 −0.150566 − 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 16.0000 0.802008
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −12.0000 −0.598506
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 8.00000 0.397033
\(407\) −2.00000 −0.0991363
\(408\) −2.00000 −0.0990148
\(409\) 38.0000 1.87898 0.939490 − 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 6.00000 0.296319
\(411\) 12.0000 0.591916
\(412\) 6.00000 0.295599
\(413\) 16.0000 0.787309
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 2.00000 0.0980581
\(417\) −20.0000 −0.979404
\(418\) −4.00000 −0.195646
\(419\) 6.00000 0.293119 0.146560 − 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) −4.00000 −0.195180
\(421\) −20.0000 −0.974740 −0.487370 − 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 16.0000 0.778868
\(423\) −10.0000 −0.486217
\(424\) 6.00000 0.291386
\(425\) −2.00000 −0.0970143
\(426\) 8.00000 0.387601
\(427\) 16.0000 0.774294
\(428\) −8.00000 −0.386695
\(429\) −4.00000 −0.193122
\(430\) 4.00000 0.192897
\(431\) 8.00000 0.385346 0.192673 − 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 16.0000 0.768025
\(435\) −2.00000 −0.0958927
\(436\) −16.0000 −0.766261
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 2.00000 0.0953463
\(441\) 9.00000 0.428571
\(442\) −4.00000 −0.190261
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 1.00000 0.0474579
\(445\) 6.00000 0.284427
\(446\) 20.0000 0.947027
\(447\) 4.00000 0.189194
\(448\) 4.00000 0.188982
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 12.0000 0.565058
\(452\) −6.00000 −0.282216
\(453\) −8.00000 −0.375873
\(454\) −12.0000 −0.563188
\(455\) −8.00000 −0.375046
\(456\) 2.00000 0.0936586
\(457\) −20.0000 −0.935561 −0.467780 − 0.883845i \(-0.654946\pi\)
−0.467780 + 0.883845i \(0.654946\pi\)
\(458\) −22.0000 −1.02799
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 − 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) −8.00000 −0.372194
\(463\) 6.00000 0.278844 0.139422 − 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 2.00000 0.0928477
\(465\) −4.00000 −0.185496
\(466\) 8.00000 0.370593
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 2.00000 0.0924500
\(469\) −48.0000 −2.21643
\(470\) 10.0000 0.461266
\(471\) −2.00000 −0.0921551
\(472\) 4.00000 0.184115
\(473\) 8.00000 0.367840
\(474\) −4.00000 −0.183726
\(475\) 2.00000 0.0917663
\(476\) −8.00000 −0.366679
\(477\) 6.00000 0.274721
\(478\) 16.0000 0.731823
\(479\) 40.0000 1.82765 0.913823 − 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 2.00000 0.0911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −12.0000 −0.544892
\(486\) 1.00000 0.0453609
\(487\) 42.0000 1.90320 0.951601 − 0.307337i \(-0.0994378\pi\)
0.951601 + 0.307337i \(0.0994378\pi\)
\(488\) 4.00000 0.181071
\(489\) −4.00000 −0.180886
\(490\) −9.00000 −0.406579
\(491\) −22.0000 −0.992846 −0.496423 − 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) −6.00000 −0.270501
\(493\) −4.00000 −0.180151
\(494\) 4.00000 0.179969
\(495\) 2.00000 0.0898933
\(496\) 4.00000 0.179605
\(497\) 32.0000 1.43540
\(498\) 4.00000 0.179244
\(499\) 22.0000 0.984855 0.492428 − 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −20.0000 −0.893534
\(502\) 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
\(508\) −12.0000 −0.532414
\(509\) 24.0000 1.06378 0.531891 − 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 2.00000 0.0885615
\(511\) −8.00000 −0.353899
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −6.00000 −0.264649
\(515\) −6.00000 −0.264392
\(516\) −4.00000 −0.176090
\(517\) 20.0000 0.879599
\(518\) 4.00000 0.175750
\(519\) −10.0000 −0.438951
\(520\) −2.00000 −0.0877058
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 2.00000 0.0875376
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 4.00000 0.174741
\(525\) 4.00000 0.174574
\(526\) 26.0000 1.13365
\(527\) −8.00000 −0.348485
\(528\) −2.00000 −0.0870388
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) 4.00000 0.173585
\(532\) 8.00000 0.346844
\(533\) −12.0000 −0.519778
\(534\) −6.00000 −0.259645
\(535\) 8.00000 0.345870
\(536\) −12.0000 −0.518321
\(537\) −12.0000 −0.517838
\(538\) −24.0000 −1.03471
\(539\) −18.0000 −0.775315
\(540\) −1.00000 −0.0430331
\(541\) −4.00000 −0.171973 −0.0859867 − 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 16.0000 0.687259
\(543\) −18.0000 −0.772454
\(544\) −2.00000 −0.0857493
\(545\) 16.0000 0.685365
\(546\) 8.00000 0.342368
\(547\) 16.0000 0.684111 0.342055 − 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 12.0000 0.512615
\(549\) 4.00000 0.170716
\(550\) −2.00000 −0.0852803
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) −2.00000 −0.0849719
\(555\) −1.00000 −0.0424476
\(556\) −20.0000 −0.848189
\(557\) −34.0000 −1.44063 −0.720313 − 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) 4.00000 0.169334
\(559\) −8.00000 −0.338364
\(560\) −4.00000 −0.169031
\(561\) 4.00000 0.168880
\(562\) 6.00000 0.253095
\(563\) 4.00000 0.168580 0.0842900 − 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −10.0000 −0.421076
\(565\) 6.00000 0.252422
\(566\) 20.0000 0.840663
\(567\) 4.00000 0.167984
\(568\) 8.00000 0.335673
\(569\) 22.0000 0.922288 0.461144 − 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 8.00000 0.333044 0.166522 − 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) −13.0000 −0.540729
\(579\) 12.0000 0.498703
\(580\) −2.00000 −0.0830455
\(581\) 16.0000 0.663792
\(582\) 12.0000 0.497416
\(583\) −12.0000 −0.496989
\(584\) −2.00000 −0.0827606
\(585\) −2.00000 −0.0826898
\(586\) −14.0000 −0.578335
\(587\) 28.0000 1.15568 0.577842 − 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 9.00000 0.371154
\(589\) 8.00000 0.329634
\(590\) −4.00000 −0.164677
\(591\) −6.00000 −0.246807
\(592\) 1.00000 0.0410997
\(593\) 44.0000 1.80686 0.903432 − 0.428732i \(-0.141040\pi\)
0.903432 + 0.428732i \(0.141040\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 8.00000 0.327968
\(596\) 4.00000 0.163846
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 − 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 1.00000 0.0408248
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −16.0000 −0.652111
\(603\) −12.0000 −0.488678
\(604\) −8.00000 −0.325515
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) 38.0000 1.54237 0.771186 − 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) 2.00000 0.0811107
\(609\) 8.00000 0.324176
\(610\) −4.00000 −0.161955
\(611\) −20.0000 −0.809113
\(612\) −2.00000 −0.0808452
\(613\) −30.0000 −1.21169 −0.605844 − 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 12.0000 0.484281
\(615\) 6.00000 0.241943
\(616\) −8.00000 −0.322329
\(617\) 28.0000 1.12724 0.563619 − 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 6.00000 0.241355
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) −24.0000 −0.961540
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −4.00000 −0.159872
\(627\) −4.00000 −0.159745
\(628\) −2.00000 −0.0798087
\(629\) −2.00000 −0.0797452
\(630\) −4.00000 −0.159364
\(631\) −48.0000 −1.91085 −0.955425 − 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) −4.00000 −0.159111
\(633\) 16.0000 0.635943
\(634\) −18.0000 −0.714871
\(635\) 12.0000 0.476205
\(636\) 6.00000 0.237915
\(637\) 18.0000 0.713186
\(638\) −4.00000 −0.158362
\(639\) 8.00000 0.316475
\(640\) −1.00000 −0.0395285
\(641\) 34.0000 1.34292 0.671460 − 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) −8.00000 −0.315735
\(643\) 8.00000 0.315489 0.157745 − 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) −4.00000 −0.157378
\(647\) 8.00000 0.314512 0.157256 − 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 1.00000 0.0392837
\(649\) −8.00000 −0.314027
\(650\) 2.00000 0.0784465
\(651\) 16.0000 0.627089
\(652\) −4.00000 −0.156652
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −16.0000 −0.625650
\(655\) −4.00000 −0.156293
\(656\) −6.00000 −0.234261
\(657\) −2.00000 −0.0780274
\(658\) −40.0000 −1.55936
\(659\) 30.0000 1.16863 0.584317 − 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 2.00000 0.0778499
\(661\) −8.00000 −0.311164 −0.155582 − 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −10.0000 −0.388661
\(663\) −4.00000 −0.155347
\(664\) 4.00000 0.155230
\(665\) −8.00000 −0.310227
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) −20.0000 −0.773823
\(669\) 20.0000 0.773245
\(670\) 12.0000 0.463600
\(671\) −8.00000 −0.308837
\(672\) 4.00000 0.154303
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −14.0000 −0.538064 −0.269032 − 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) −6.00000 −0.230429
\(679\) 48.0000 1.84207
\(680\) 2.00000 0.0766965
\(681\) −12.0000 −0.459841
\(682\) −8.00000 −0.306336
\(683\) 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
\(687\) −22.0000 −0.839352
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 − 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) −10.0000 −0.380143
\(693\) −8.00000 −0.303895
\(694\) −12.0000 −0.455514
\(695\) 20.0000 0.758643
\(696\) 2.00000 0.0758098
\(697\) 12.0000 0.454532
\(698\) 2.00000 0.0757011
\(699\) 8.00000 0.302588
\(700\) 4.00000 0.151186
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 2.00000 0.0754851
\(703\) 2.00000 0.0754314
\(704\) −2.00000 −0.0753778
\(705\) 10.0000 0.376622
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −8.00000 −0.300446 −0.150223 − 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) −8.00000 −0.300235
\(711\) −4.00000 −0.150012
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −8.00000 −0.299392
\(715\) 4.00000 0.149592
\(716\) −12.0000 −0.448461
\(717\) 16.0000 0.597531
\(718\) −24.0000 −0.895672
\(719\) −20.0000 −0.745874 −0.372937 − 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 24.0000 0.893807
\(722\) −15.0000 −0.558242
\(723\) 10.0000 0.371904
\(724\) −18.0000 −0.668965
\(725\) 2.00000 0.0742781
\(726\) −7.00000 −0.259794
\(727\) 6.00000 0.222528 0.111264 − 0.993791i \(-0.464510\pi\)
0.111264 + 0.993791i \(0.464510\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 8.00000 0.295891
\(732\) 4.00000 0.147844
\(733\) −46.0000 −1.69905 −0.849524 − 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) 28.0000 1.03350
\(735\) −9.00000 −0.331970
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) −6.00000 −0.220863
\(739\) −32.0000 −1.17714 −0.588570 − 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) −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 \(0.535104\pi\)
\(744\) 4.00000 0.146647
\(745\) −4.00000 −0.146549
\(746\) 10.0000 0.366126
\(747\) 4.00000 0.146352
\(748\) 4.00000 0.146254
\(749\) −32.0000 −1.16925
\(750\) −1.00000 −0.0365148
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −10.0000 −0.364662
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 8.00000 0.291150
\(756\) 4.00000 0.145479
\(757\) 30.0000 1.09037 0.545184 − 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) −50.0000 −1.81250 −0.906249 − 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) −12.0000 −0.434714
\(763\) −64.0000 −2.31696
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 20.0000 0.722629
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 8.00000 0.288300
\(771\) −6.00000 −0.216085
\(772\) 12.0000 0.431889
\(773\) 54.0000 1.94225 0.971123 − 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) −4.00000 −0.143777
\(775\) 4.00000 0.143684
\(776\) 12.0000 0.430775
\(777\) 4.00000 0.143499
\(778\) −18.0000 −0.645331
\(779\) −12.0000 −0.429945
\(780\) −2.00000 −0.0716115
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 9.00000 0.321429
\(785\) 2.00000 0.0713831
\(786\) 4.00000 0.142675
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −6.00000 −0.213741
\(789\) 26.0000 0.925625
\(790\) 4.00000 0.142314
\(791\) −24.0000 −0.853342
\(792\) −2.00000 −0.0710669
\(793\) 8.00000 0.284088
\(794\) −6.00000 −0.212932
\(795\) −6.00000 −0.212798
\(796\) 16.0000 0.567105
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 8.00000 0.283197
\(799\) 20.0000 0.707549
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) 30.0000 1.05934
\(803\) 4.00000 0.141157
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 8.00000 0.280745
\(813\) 16.0000 0.561144
\(814\) −2.00000 −0.0701000
\(815\) 4.00000 0.140114
\(816\) −2.00000 −0.0700140
\(817\) −8.00000 −0.279885
\(818\) 38.0000 1.32864
\(819\) 8.00000 0.279543
\(820\) 6.00000 0.209529
\(821\) 16.0000 0.558404 0.279202 − 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(822\) 12.0000 0.418548
\(823\) −56.0000 −1.95204 −0.976019 − 0.217687i \(-0.930149\pi\)
−0.976019 + 0.217687i \(0.930149\pi\)
\(824\) 6.00000 0.209020
\(825\) −2.00000 −0.0696311
\(826\) 16.0000 0.556711
\(827\) −4.00000 −0.139094 −0.0695468 − 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) −48.0000 −1.66711 −0.833554 − 0.552437i \(-0.813698\pi\)
−0.833554 + 0.552437i \(0.813698\pi\)
\(830\) −4.00000 −0.138842
\(831\) −2.00000 −0.0693792
\(832\) 2.00000 0.0693375
\(833\) −18.0000 −0.623663
\(834\) −20.0000 −0.692543
\(835\) 20.0000 0.692129
\(836\) −4.00000 −0.138343
\(837\) 4.00000 0.138260
\(838\) 6.00000 0.207267
\(839\) 48.0000 1.65714 0.828572 − 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) −4.00000 −0.138013
\(841\) −25.0000 −0.862069
\(842\) −20.0000 −0.689246
\(843\) 6.00000 0.206651
\(844\) 16.0000 0.550743
\(845\) 9.00000 0.309609
\(846\) −10.0000 −0.343807
\(847\) −28.0000 −0.962091
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 16.0000 0.547509
\(855\) −2.00000 −0.0683986
\(856\) −8.00000 −0.273434
\(857\) 10.0000 0.341593 0.170797 − 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) −4.00000 −0.136558
\(859\) 14.0000 0.477674 0.238837 − 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 4.00000 0.136399
\(861\) −24.0000 −0.817918
\(862\) 8.00000 0.272481
\(863\) −26.0000 −0.885050 −0.442525 − 0.896756i \(-0.645917\pi\)
−0.442525 + 0.896756i \(0.645917\pi\)
\(864\) 1.00000 0.0340207
\(865\) 10.0000 0.340010
\(866\) 30.0000 1.01944
\(867\) −13.0000 −0.441503
\(868\) 16.0000 0.543075
\(869\) 8.00000 0.271381
\(870\) −2.00000 −0.0678064
\(871\) −24.0000 −0.813209
\(872\) −16.0000 −0.541828
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) −4.00000 −0.135225
\(876\) −2.00000 −0.0675737
\(877\) −14.0000 −0.472746 −0.236373 − 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) −8.00000 −0.269987
\(879\) −14.0000 −0.472208
\(880\) 2.00000 0.0674200
\(881\) 14.0000 0.471672 0.235836 − 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 9.00000 0.303046
\(883\) 32.0000 1.07689 0.538443 − 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) −4.00000 −0.134535
\(885\) −4.00000 −0.134459
\(886\) 36.0000 1.20944
\(887\) −22.0000 −0.738688 −0.369344 − 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 1.00000 0.0335578
\(889\) −48.0000 −1.60987
\(890\) 6.00000 0.201120
\(891\) −2.00000 −0.0670025
\(892\) 20.0000 0.669650
\(893\) −20.0000 −0.669274
\(894\) 4.00000 0.133780
\(895\) 12.0000 0.401116
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 8.00000 0.266815
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) 12.0000 0.399556
\(903\) −16.0000 −0.532447
\(904\) −6.00000 −0.199557
\(905\) 18.0000 0.598340
\(906\) −8.00000 −0.265782
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) −40.0000 −1.32526 −0.662630 − 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 2.00000 0.0662266
\(913\) −8.00000 −0.264761
\(914\) −20.0000 −0.661541
\(915\) −4.00000 −0.132236
\(916\) −22.0000 −0.726900
\(917\) 16.0000 0.528367
\(918\) −2.00000 −0.0660098
\(919\) 24.0000 0.791687 0.395843 − 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 14.0000 0.461065
\(923\) 16.0000 0.526646
\(924\) −8.00000 −0.263181
\(925\) 1.00000 0.0328798
\(926\) 6.00000 0.197172
\(927\) 6.00000 0.197066
\(928\) 2.00000 0.0656532
\(929\) −10.0000 −0.328089 −0.164045 − 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) −4.00000 −0.131165
\(931\) 18.0000 0.589926
\(932\) 8.00000 0.262049
\(933\) 8.00000 0.261908
\(934\) −20.0000 −0.654420
\(935\) −4.00000 −0.130814
\(936\) 2.00000 0.0653720
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) −48.0000 −1.56726
\(939\) −4.00000 −0.130535
\(940\) 10.0000 0.326164
\(941\) 8.00000 0.260793 0.130396 − 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) −4.00000 −0.130120
\(946\) 8.00000 0.260102
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −4.00000 −0.129914
\(949\) −4.00000 −0.129845
\(950\) 2.00000 0.0648886
\(951\) −18.0000 −0.583690
\(952\) −8.00000 −0.259281
\(953\) −36.0000 −1.16615 −0.583077 − 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) −4.00000 −0.129302
\(958\) 40.0000 1.29234
\(959\) 48.0000 1.55000
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) −8.00000 −0.257796
\(964\) 10.0000 0.322078
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) −46.0000 −1.47926 −0.739630 − 0.673014i \(-0.765000\pi\)
−0.739630 + 0.673014i \(0.765000\pi\)
\(968\) −7.00000 −0.224989
\(969\) −4.00000 −0.128499
\(970\) −12.0000 −0.385297
\(971\) −6.00000 −0.192549 −0.0962746 − 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 1.00000 0.0320750
\(973\) −80.0000 −2.56468
\(974\) 42.0000 1.34577
\(975\) 2.00000 0.0640513
\(976\) 4.00000 0.128037
\(977\) −10.0000 −0.319928 −0.159964 − 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) −4.00000 −0.127906
\(979\) 12.0000 0.383522
\(980\) −9.00000 −0.287494
\(981\) −16.0000 −0.510841
\(982\) −22.0000 −0.702048
\(983\) −14.0000 −0.446531 −0.223265 − 0.974758i \(-0.571672\pi\)
−0.223265 + 0.974758i \(0.571672\pi\)
\(984\) −6.00000 −0.191273
\(985\) 6.00000 0.191176
\(986\) −4.00000 −0.127386
\(987\) −40.0000 −1.27321
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 2.00000 0.0635642
\(991\) 4.00000 0.127064 0.0635321 − 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 4.00000 0.127000
\(993\) −10.0000 −0.317340
\(994\) 32.0000 1.01498
\(995\) −16.0000 −0.507234
\(996\) 4.00000 0.126745
\(997\) −30.0000 −0.950110 −0.475055 − 0.879956i \(-0.657572\pi\)
−0.475055 + 0.879956i \(0.657572\pi\)
\(998\) 22.0000 0.696398
\(999\) 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.o.1.1 ✓ 1
3.2 odd 2 3330.2.a.l.1.1 1
4.3 odd 2 8880.2.a.b.1.1 1
5.4 even 2 5550.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.o.1.1 ✓ 1 1.1 even 1 trivial
3330.2.a.l.1.1 1 3.2 odd 2
5550.2.a.b.1.1 1 5.4 even 2
8880.2.a.b.1.1 1 4.3 odd 2