Properties

Label 221.2.a.b.1.1
Level $221$
Weight $2$
Character 221.1
Self dual yes
Analytic conductor $1.765$
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] = [221,2,Mod(1,221)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(221, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("221.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 221 = 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 221.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: \(1.76469388467\)
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 \(\chi\) \(=\) 221.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -6.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +4.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} +4.00000 q^{21} -6.00000 q^{22} +6.00000 q^{23} -6.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} +4.00000 q^{30} -2.00000 q^{31} +5.00000 q^{32} -12.0000 q^{33} +1.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -6.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} +6.00000 q^{44} +2.00000 q^{45} +6.00000 q^{46} -4.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +1.00000 q^{52} +14.0000 q^{53} -4.00000 q^{54} -12.0000 q^{55} -6.00000 q^{56} +8.00000 q^{57} -6.00000 q^{58} +4.00000 q^{59} -4.00000 q^{60} +2.00000 q^{61} -2.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -12.0000 q^{66} -1.00000 q^{68} +12.0000 q^{69} +4.00000 q^{70} -10.0000 q^{71} -3.00000 q^{72} +10.0000 q^{73} +2.00000 q^{74} -2.00000 q^{75} -4.00000 q^{76} -12.0000 q^{77} -2.00000 q^{78} +14.0000 q^{79} -2.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} +2.00000 q^{85} -12.0000 q^{87} +18.0000 q^{88} -18.0000 q^{89} +2.00000 q^{90} -2.00000 q^{91} -6.00000 q^{92} -4.00000 q^{93} -4.00000 q^{94} +8.00000 q^{95} +10.0000 q^{96} +2.00000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 2.00000 0.816497
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −2.00000 −0.577350
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 4.00000 1.03280
\(16\) −1.00000 −0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) 4.00000 0.872872
\(22\) −6.00000 −1.27920
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −6.00000 −1.22474
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) −4.00000 −0.769800
\(28\) −2.00000 −0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 4.00000 0.730297
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 5.00000 0.883883
\(33\) −12.0000 −2.08893
\(34\) 1.00000 0.171499
\(35\) 4.00000 0.676123
\(36\) −1.00000 −0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) −6.00000 −0.948683
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 6.00000 0.904534
\(45\) 2.00000 0.298142
\(46\) 6.00000 0.884652
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −2.00000 −0.288675
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) 1.00000 0.138675
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −4.00000 −0.544331
\(55\) −12.0000 −1.61808
\(56\) −6.00000 −0.801784
\(57\) 8.00000 1.05963
\(58\) −6.00000 −0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −4.00000 −0.516398
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) −12.0000 −1.47710
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −1.00000 −0.121268
\(69\) 12.0000 1.44463
\(70\) 4.00000 0.478091
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −3.00000 −0.353553
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 2.00000 0.232495
\(75\) −2.00000 −0.230940
\(76\) −4.00000 −0.458831
\(77\) −12.0000 −1.36753
\(78\) −2.00000 −0.226455
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) −2.00000 −0.223607
\(81\) −11.0000 −1.22222
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −4.00000 −0.436436
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 18.0000 1.91881
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 2.00000 0.210819
\(91\) −2.00000 −0.209657
\(92\) −6.00000 −0.625543
\(93\) −4.00000 −0.414781
\(94\) −4.00000 −0.412568
\(95\) 8.00000 0.820783
\(96\) 10.0000 1.02062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −3.00000 −0.303046
\(99\) −6.00000 −0.603023
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 3.00000 0.294174
\(105\) 8.00000 0.780720
\(106\) 14.0000 1.35980
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 4.00000 0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −12.0000 −1.14416
\(111\) 4.00000 0.379663
\(112\) −2.00000 −0.188982
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 8.00000 0.749269
\(115\) 12.0000 1.11901
\(116\) 6.00000 0.557086
\(117\) −1.00000 −0.0924500
\(118\) 4.00000 0.368230
\(119\) 2.00000 0.183340
\(120\) −12.0000 −1.09545
\(121\) 25.0000 2.27273
\(122\) 2.00000 0.181071
\(123\) −12.0000 −1.08200
\(124\) 2.00000 0.179605
\(125\) −12.0000 −1.07331
\(126\) 2.00000 0.178174
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 12.0000 1.04447
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) −8.00000 −0.688530
\(136\) −3.00000 −0.257248
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 12.0000 1.02151
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) −4.00000 −0.338062
\(141\) −8.00000 −0.673722
\(142\) −10.0000 −0.839181
\(143\) 6.00000 0.501745
\(144\) −1.00000 −0.0833333
\(145\) −12.0000 −0.996546
\(146\) 10.0000 0.827606
\(147\) −6.00000 −0.494872
\(148\) −2.00000 −0.164399
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −2.00000 −0.163299
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −12.0000 −0.973329
\(153\) 1.00000 0.0808452
\(154\) −12.0000 −0.966988
\(155\) −4.00000 −0.321288
\(156\) 2.00000 0.160128
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 14.0000 1.11378
\(159\) 28.0000 2.22054
\(160\) 10.0000 0.790569
\(161\) 12.0000 0.945732
\(162\) −11.0000 −0.864242
\(163\) 18.0000 1.40987 0.704934 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 6.00000 0.468521
\(165\) −24.0000 −1.86840
\(166\) 12.0000 0.931381
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) −12.0000 −0.925820
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −12.0000 −0.909718
\(175\) −2.00000 −0.151186
\(176\) 6.00000 0.452267
\(177\) 8.00000 0.601317
\(178\) −18.0000 −1.34916
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −2.00000 −0.149071
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −2.00000 −0.148250
\(183\) 4.00000 0.295689
\(184\) −18.0000 −1.32698
\(185\) 4.00000 0.294086
\(186\) −4.00000 −0.293294
\(187\) −6.00000 −0.438763
\(188\) 4.00000 0.291730
\(189\) −8.00000 −0.581914
\(190\) 8.00000 0.580381
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 14.0000 1.01036
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) 3.00000 0.214286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −6.00000 −0.426401
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −12.0000 −0.842235
\(204\) −2.00000 −0.140028
\(205\) −12.0000 −0.838116
\(206\) 8.00000 0.557386
\(207\) 6.00000 0.417029
\(208\) 1.00000 0.0693375
\(209\) −24.0000 −1.66011
\(210\) 8.00000 0.552052
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) −14.0000 −0.961524
\(213\) −20.0000 −1.37038
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 12.0000 0.816497
\(217\) −4.00000 −0.271538
\(218\) 2.00000 0.135457
\(219\) 20.0000 1.35147
\(220\) 12.0000 0.809040
\(221\) −1.00000 −0.0672673
\(222\) 4.00000 0.268462
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 10.0000 0.668153
\(225\) −1.00000 −0.0666667
\(226\) 18.0000 1.19734
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −8.00000 −0.529813
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 12.0000 0.791257
\(231\) −24.0000 −1.57908
\(232\) 18.0000 1.18176
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −8.00000 −0.521862
\(236\) −4.00000 −0.260378
\(237\) 28.0000 1.81880
\(238\) 2.00000 0.129641
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −4.00000 −0.258199
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 25.0000 1.60706
\(243\) −10.0000 −0.641500
\(244\) −2.00000 −0.128037
\(245\) −6.00000 −0.383326
\(246\) −12.0000 −0.765092
\(247\) −4.00000 −0.254514
\(248\) 6.00000 0.381000
\(249\) 24.0000 1.52094
\(250\) −12.0000 −0.758947
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) −2.00000 −0.125988
\(253\) −36.0000 −2.26330
\(254\) 4.00000 0.250982
\(255\) 4.00000 0.250490
\(256\) −17.0000 −1.06250
\(257\) −10.0000 −0.623783 −0.311891 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 2.00000 0.124035
\(261\) −6.00000 −0.371391
\(262\) −6.00000 −0.370681
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 36.0000 2.21565
\(265\) 28.0000 1.72003
\(266\) 8.00000 0.490511
\(267\) −36.0000 −2.20316
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) −8.00000 −0.486864
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −4.00000 −0.242091
\(274\) −18.0000 −1.08742
\(275\) 6.00000 0.361814
\(276\) −12.0000 −0.722315
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 10.0000 0.599760
\(279\) −2.00000 −0.119737
\(280\) −12.0000 −0.717137
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −8.00000 −0.476393
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 10.0000 0.593391
\(285\) 16.0000 0.947758
\(286\) 6.00000 0.354787
\(287\) −12.0000 −0.708338
\(288\) 5.00000 0.294628
\(289\) 1.00000 0.0588235
\(290\) −12.0000 −0.704664
\(291\) 4.00000 0.234484
\(292\) −10.0000 −0.585206
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −6.00000 −0.349927
\(295\) 8.00000 0.465778
\(296\) −6.00000 −0.348743
\(297\) 24.0000 1.39262
\(298\) −18.0000 −1.04271
\(299\) −6.00000 −0.346989
\(300\) 2.00000 0.115470
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) −12.0000 −0.689382
\(304\) −4.00000 −0.229416
\(305\) 4.00000 0.229039
\(306\) 1.00000 0.0571662
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 12.0000 0.683763
\(309\) 16.0000 0.910208
\(310\) −4.00000 −0.227185
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 6.00000 0.339683
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 14.0000 0.790066
\(315\) 4.00000 0.225374
\(316\) −14.0000 −0.787562
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 28.0000 1.57016
\(319\) 36.0000 2.01561
\(320\) 14.0000 0.782624
\(321\) 20.0000 1.11629
\(322\) 12.0000 0.668734
\(323\) 4.00000 0.222566
\(324\) 11.0000 0.611111
\(325\) 1.00000 0.0554700
\(326\) 18.0000 0.996928
\(327\) 4.00000 0.221201
\(328\) 18.0000 0.993884
\(329\) −8.00000 −0.441054
\(330\) −24.0000 −1.32116
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 1.00000 0.0543928
\(339\) 36.0000 1.95525
\(340\) −2.00000 −0.108465
\(341\) 12.0000 0.649836
\(342\) 4.00000 0.216295
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 24.0000 1.29212
\(346\) 2.00000 0.107521
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 12.0000 0.643268
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −2.00000 −0.106904
\(351\) 4.00000 0.213504
\(352\) −30.0000 −1.59901
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 8.00000 0.425195
\(355\) −20.0000 −1.06149
\(356\) 18.0000 0.953998
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) 50.0000 2.62432
\(364\) 2.00000 0.104828
\(365\) 20.0000 1.04685
\(366\) 4.00000 0.209083
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −6.00000 −0.312772
\(369\) −6.00000 −0.312348
\(370\) 4.00000 0.207950
\(371\) 28.0000 1.45369
\(372\) 4.00000 0.207390
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −6.00000 −0.310253
\(375\) −24.0000 −1.23935
\(376\) 12.0000 0.618853
\(377\) 6.00000 0.309016
\(378\) −8.00000 −0.411476
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) −8.00000 −0.410391
\(381\) 8.00000 0.409852
\(382\) 8.00000 0.409316
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −6.00000 −0.306186
\(385\) −24.0000 −1.22315
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) −4.00000 −0.202548
\(391\) 6.00000 0.303433
\(392\) 9.00000 0.454569
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) 28.0000 1.40883
\(396\) 6.00000 0.301511
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 2.00000 0.100251
\(399\) 16.0000 0.801002
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 6.00000 0.298511
\(405\) −22.0000 −1.09319
\(406\) −12.0000 −0.595550
\(407\) −12.0000 −0.594818
\(408\) −6.00000 −0.297044
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −12.0000 −0.592638
\(411\) −36.0000 −1.77575
\(412\) −8.00000 −0.394132
\(413\) 8.00000 0.393654
\(414\) 6.00000 0.294884
\(415\) 24.0000 1.17811
\(416\) −5.00000 −0.245145
\(417\) 20.0000 0.979404
\(418\) −24.0000 −1.17388
\(419\) 38.0000 1.85642 0.928211 0.372055i \(-0.121347\pi\)
0.928211 + 0.372055i \(0.121347\pi\)
\(420\) −8.00000 −0.390360
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −2.00000 −0.0973585
\(423\) −4.00000 −0.194487
\(424\) −42.0000 −2.03970
\(425\) −1.00000 −0.0485071
\(426\) −20.0000 −0.969003
\(427\) 4.00000 0.193574
\(428\) −10.0000 −0.483368
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 4.00000 0.192450
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) −4.00000 −0.192006
\(435\) −24.0000 −1.15071
\(436\) −2.00000 −0.0957826
\(437\) 24.0000 1.14808
\(438\) 20.0000 0.955637
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 36.0000 1.71623
\(441\) −3.00000 −0.142857
\(442\) −1.00000 −0.0475651
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) −4.00000 −0.189832
\(445\) −36.0000 −1.70656
\(446\) −24.0000 −1.13643
\(447\) −36.0000 −1.70274
\(448\) 14.0000 0.661438
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 36.0000 1.69517
\(452\) −18.0000 −0.846649
\(453\) −32.0000 −1.50349
\(454\) 18.0000 0.844782
\(455\) −4.00000 −0.187523
\(456\) −24.0000 −1.12390
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −6.00000 −0.280362
\(459\) −4.00000 −0.186704
\(460\) −12.0000 −0.559503
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) −24.0000 −1.11658
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 6.00000 0.278543
\(465\) −8.00000 −0.370991
\(466\) −22.0000 −1.01913
\(467\) −32.0000 −1.48078 −0.740392 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 28.0000 1.29017
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 28.0000 1.28608
\(475\) −4.00000 −0.183533
\(476\) −2.00000 −0.0916698
\(477\) 14.0000 0.641016
\(478\) 12.0000 0.548867
\(479\) 22.0000 1.00521 0.502603 0.864517i \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) 20.0000 0.912871
\(481\) −2.00000 −0.0911922
\(482\) −14.0000 −0.637683
\(483\) 24.0000 1.09204
\(484\) −25.0000 −1.13636
\(485\) 4.00000 0.181631
\(486\) −10.0000 −0.453609
\(487\) 34.0000 1.54069 0.770344 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) −6.00000 −0.271607
\(489\) 36.0000 1.62798
\(490\) −6.00000 −0.271052
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 12.0000 0.541002
\(493\) −6.00000 −0.270226
\(494\) −4.00000 −0.179969
\(495\) −12.0000 −0.539360
\(496\) 2.00000 0.0898027
\(497\) −20.0000 −0.897123
\(498\) 24.0000 1.07547
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) 12.0000 0.536656
\(501\) −4.00000 −0.178707
\(502\) 4.00000 0.178529
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) −6.00000 −0.267261
\(505\) −12.0000 −0.533993
\(506\) −36.0000 −1.60040
\(507\) 2.00000 0.0888231
\(508\) −4.00000 −0.177471
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 4.00000 0.177123
\(511\) 20.0000 0.884748
\(512\) −11.0000 −0.486136
\(513\) −16.0000 −0.706417
\(514\) −10.0000 −0.441081
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 4.00000 0.175750
\(519\) 4.00000 0.175581
\(520\) 6.00000 0.263117
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −6.00000 −0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 6.00000 0.262111
\(525\) −4.00000 −0.174574
\(526\) 4.00000 0.174408
\(527\) −2.00000 −0.0871214
\(528\) 12.0000 0.522233
\(529\) 13.0000 0.565217
\(530\) 28.0000 1.21624
\(531\) 4.00000 0.173585
\(532\) −8.00000 −0.346844
\(533\) 6.00000 0.259889
\(534\) −36.0000 −1.55787
\(535\) 20.0000 0.864675
\(536\) 0 0
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 18.0000 0.775315
\(540\) 8.00000 0.344265
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 4.00000 0.171815
\(543\) −44.0000 −1.88822
\(544\) 5.00000 0.214373
\(545\) 4.00000 0.171341
\(546\) −4.00000 −0.171184
\(547\) 18.0000 0.769624 0.384812 0.922995i \(-0.374266\pi\)
0.384812 + 0.922995i \(0.374266\pi\)
\(548\) 18.0000 0.768922
\(549\) 2.00000 0.0853579
\(550\) 6.00000 0.255841
\(551\) −24.0000 −1.02243
\(552\) −36.0000 −1.53226
\(553\) 28.0000 1.19068
\(554\) 2.00000 0.0849719
\(555\) 8.00000 0.339581
\(556\) −10.0000 −0.424094
\(557\) 34.0000 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) −12.0000 −0.506640
\(562\) −22.0000 −0.928014
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 8.00000 0.336861
\(565\) 36.0000 1.51453
\(566\) 6.00000 0.252199
\(567\) −22.0000 −0.923913
\(568\) 30.0000 1.25877
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 16.0000 0.670166
\(571\) 38.0000 1.59025 0.795125 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) −6.00000 −0.250873
\(573\) 16.0000 0.668410
\(574\) −12.0000 −0.500870
\(575\) −6.00000 −0.250217
\(576\) 7.00000 0.291667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 1.00000 0.0415945
\(579\) 4.00000 0.166234
\(580\) 12.0000 0.498273
\(581\) 24.0000 0.995688
\(582\) 4.00000 0.165805
\(583\) −84.0000 −3.47892
\(584\) −30.0000 −1.24141
\(585\) −2.00000 −0.0826898
\(586\) 14.0000 0.578335
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 6.00000 0.247436
\(589\) −8.00000 −0.329634
\(590\) 8.00000 0.329355
\(591\) −12.0000 −0.493614
\(592\) −2.00000 −0.0821995
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 24.0000 0.984732
\(595\) 4.00000 0.163984
\(596\) 18.0000 0.737309
\(597\) 4.00000 0.163709
\(598\) −6.00000 −0.245358
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 6.00000 0.244949
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 50.0000 2.03279
\(606\) −12.0000 −0.487467
\(607\) 42.0000 1.70473 0.852364 0.522949i \(-0.175168\pi\)
0.852364 + 0.522949i \(0.175168\pi\)
\(608\) 20.0000 0.811107
\(609\) −24.0000 −0.972529
\(610\) 4.00000 0.161955
\(611\) 4.00000 0.161823
\(612\) −1.00000 −0.0404226
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 16.0000 0.645707
\(615\) −24.0000 −0.967773
\(616\) 36.0000 1.45048
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 16.0000 0.643614
\(619\) 18.0000 0.723481 0.361741 0.932279i \(-0.382183\pi\)
0.361741 + 0.932279i \(0.382183\pi\)
\(620\) 4.00000 0.160644
\(621\) −24.0000 −0.963087
\(622\) −6.00000 −0.240578
\(623\) −36.0000 −1.44231
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) −48.0000 −1.91694
\(628\) −14.0000 −0.558661
\(629\) 2.00000 0.0797452
\(630\) 4.00000 0.159364
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −42.0000 −1.67067
\(633\) −4.00000 −0.158986
\(634\) −22.0000 −0.873732
\(635\) 8.00000 0.317470
\(636\) −28.0000 −1.11027
\(637\) 3.00000 0.118864
\(638\) 36.0000 1.42525
\(639\) −10.0000 −0.395594
\(640\) −6.00000 −0.237171
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 20.0000 0.789337
\(643\) 38.0000 1.49857 0.749287 0.662246i \(-0.230396\pi\)
0.749287 + 0.662246i \(0.230396\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 33.0000 1.29636
\(649\) −24.0000 −0.942082
\(650\) 1.00000 0.0392232
\(651\) −8.00000 −0.313545
\(652\) −18.0000 −0.704934
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 4.00000 0.156412
\(655\) −12.0000 −0.468879
\(656\) 6.00000 0.234261
\(657\) 10.0000 0.390137
\(658\) −8.00000 −0.311872
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 24.0000 0.934199
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 28.0000 1.08825
\(663\) −2.00000 −0.0776736
\(664\) −36.0000 −1.39707
\(665\) 16.0000 0.620453
\(666\) 2.00000 0.0774984
\(667\) −36.0000 −1.39393
\(668\) 2.00000 0.0773823
\(669\) −48.0000 −1.85579
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 20.0000 0.771517
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 26.0000 1.00148
\(675\) 4.00000 0.153960
\(676\) −1.00000 −0.0384615
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 36.0000 1.38257
\(679\) 4.00000 0.153506
\(680\) −6.00000 −0.230089
\(681\) 36.0000 1.37952
\(682\) 12.0000 0.459504
\(683\) 30.0000 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) −4.00000 −0.152944
\(685\) −36.0000 −1.37549
\(686\) −20.0000 −0.763604
\(687\) −12.0000 −0.457829
\(688\) 0 0
\(689\) −14.0000 −0.533358
\(690\) 24.0000 0.913664
\(691\) 18.0000 0.684752 0.342376 0.939563i \(-0.388768\pi\)
0.342376 + 0.939563i \(0.388768\pi\)
\(692\) −2.00000 −0.0760286
\(693\) −12.0000 −0.455842
\(694\) −26.0000 −0.986947
\(695\) 20.0000 0.758643
\(696\) 36.0000 1.36458
\(697\) −6.00000 −0.227266
\(698\) −2.00000 −0.0757011
\(699\) −44.0000 −1.66423
\(700\) 2.00000 0.0755929
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 4.00000 0.150970
\(703\) 8.00000 0.301726
\(704\) −42.0000 −1.58293
\(705\) −16.0000 −0.602595
\(706\) −14.0000 −0.526897
\(707\) −12.0000 −0.451306
\(708\) −8.00000 −0.300658
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) −20.0000 −0.750587
\(711\) 14.0000 0.525041
\(712\) 54.0000 2.02374
\(713\) −12.0000 −0.449404
\(714\) 4.00000 0.149696
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 24.0000 0.896296
\(718\) 8.00000 0.298557
\(719\) −38.0000 −1.41716 −0.708580 0.705630i \(-0.750664\pi\)
−0.708580 + 0.705630i \(0.750664\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) −28.0000 −1.04133
\(724\) 22.0000 0.817624
\(725\) 6.00000 0.222834
\(726\) 50.0000 1.85567
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 6.00000 0.222375
\(729\) 13.0000 0.481481
\(730\) 20.0000 0.740233
\(731\) 0 0
\(732\) −4.00000 −0.147844
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) −18.0000 −0.664392
\(735\) −12.0000 −0.442627
\(736\) 30.0000 1.10581
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −4.00000 −0.147043
\(741\) −8.00000 −0.293887
\(742\) 28.0000 1.02791
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 12.0000 0.439941
\(745\) −36.0000 −1.31894
\(746\) −22.0000 −0.805477
\(747\) 12.0000 0.439057
\(748\) 6.00000 0.219382
\(749\) 20.0000 0.730784
\(750\) −24.0000 −0.876356
\(751\) −46.0000 −1.67856 −0.839282 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) 4.00000 0.145865
\(753\) 8.00000 0.291536
\(754\) 6.00000 0.218507
\(755\) −32.0000 −1.16460
\(756\) 8.00000 0.290957
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) −2.00000 −0.0726433
\(759\) −72.0000 −2.61343
\(760\) −24.0000 −0.870572
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 8.00000 0.289809
\(763\) 4.00000 0.144810
\(764\) −8.00000 −0.289430
\(765\) 2.00000 0.0723102
\(766\) −24.0000 −0.867155
\(767\) −4.00000 −0.144432
\(768\) −34.0000 −1.22687
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) −24.0000 −0.864900
\(771\) −20.0000 −0.720282
\(772\) −2.00000 −0.0719816
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) −6.00000 −0.215387
\(777\) 8.00000 0.286998
\(778\) 10.0000 0.358517
\(779\) −24.0000 −0.859889
\(780\) 4.00000 0.143223
\(781\) 60.0000 2.14697
\(782\) 6.00000 0.214560
\(783\) 24.0000 0.857690
\(784\) 3.00000 0.107143
\(785\) 28.0000 0.999363
\(786\) −12.0000 −0.428026
\(787\) 14.0000 0.499046 0.249523 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) 6.00000 0.213741
\(789\) 8.00000 0.284808
\(790\) 28.0000 0.996195
\(791\) 36.0000 1.28001
\(792\) 18.0000 0.639602
\(793\) −2.00000 −0.0710221
\(794\) −14.0000 −0.496841
\(795\) 56.0000 1.98612
\(796\) −2.00000 −0.0708881
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 16.0000 0.566394
\(799\) −4.00000 −0.141510
\(800\) −5.00000 −0.176777
\(801\) −18.0000 −0.635999
\(802\) −6.00000 −0.211867
\(803\) −60.0000 −2.11735
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 2.00000 0.0704470
\(807\) −60.0000 −2.11210
\(808\) 18.0000 0.633238
\(809\) 50.0000 1.75791 0.878953 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) −22.0000 −0.773001
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 12.0000 0.421117
\(813\) 8.00000 0.280572
\(814\) −12.0000 −0.420600
\(815\) 36.0000 1.26102
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 10.0000 0.349642
\(819\) −2.00000 −0.0698857
\(820\) 12.0000 0.419058
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) −36.0000 −1.25564
\(823\) −18.0000 −0.627441 −0.313720 0.949515i \(-0.601575\pi\)
−0.313720 + 0.949515i \(0.601575\pi\)
\(824\) −24.0000 −0.836080
\(825\) 12.0000 0.417786
\(826\) 8.00000 0.278356
\(827\) 18.0000 0.625921 0.312961 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) −6.00000 −0.208514
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 24.0000 0.833052
\(831\) 4.00000 0.138758
\(832\) −7.00000 −0.242681
\(833\) −3.00000 −0.103944
\(834\) 20.0000 0.692543
\(835\) −4.00000 −0.138426
\(836\) 24.0000 0.830057
\(837\) 8.00000 0.276520
\(838\) 38.0000 1.31269
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) −24.0000 −0.828079
\(841\) 7.00000 0.241379
\(842\) −22.0000 −0.758170
\(843\) −44.0000 −1.51544
\(844\) 2.00000 0.0688428
\(845\) 2.00000 0.0688021
\(846\) −4.00000 −0.137523
\(847\) 50.0000 1.71802
\(848\) −14.0000 −0.480762
\(849\) 12.0000 0.411839
\(850\) −1.00000 −0.0342997
\(851\) 12.0000 0.411355
\(852\) 20.0000 0.685189
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 4.00000 0.136877
\(855\) 8.00000 0.273594
\(856\) −30.0000 −1.02538
\(857\) −14.0000 −0.478231 −0.239115 0.970991i \(-0.576857\pi\)
−0.239115 + 0.970991i \(0.576857\pi\)
\(858\) 12.0000 0.409673
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) −6.00000 −0.204361
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) −20.0000 −0.680414
\(865\) 4.00000 0.136004
\(866\) 6.00000 0.203888
\(867\) 2.00000 0.0679236
\(868\) 4.00000 0.135769
\(869\) −84.0000 −2.84950
\(870\) −24.0000 −0.813676
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) 2.00000 0.0676897
\(874\) 24.0000 0.811812
\(875\) −24.0000 −0.811348
\(876\) −20.0000 −0.675737
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 2.00000 0.0674967
\(879\) 28.0000 0.944417
\(880\) 12.0000 0.404520
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) −3.00000 −0.101015
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 1.00000 0.0336336
\(885\) 16.0000 0.537834
\(886\) −28.0000 −0.940678
\(887\) 34.0000 1.14161 0.570804 0.821086i \(-0.306632\pi\)
0.570804 + 0.821086i \(0.306632\pi\)
\(888\) −12.0000 −0.402694
\(889\) 8.00000 0.268311
\(890\) −36.0000 −1.20672
\(891\) 66.0000 2.21108
\(892\) 24.0000 0.803579
\(893\) −16.0000 −0.535420
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) −12.0000 −0.400668
\(898\) 34.0000 1.13459
\(899\) 12.0000 0.400222
\(900\) 1.00000 0.0333333
\(901\) 14.0000 0.466408
\(902\) 36.0000 1.19867
\(903\) 0 0
\(904\) −54.0000 −1.79601
\(905\) −44.0000 −1.46261
\(906\) −32.0000 −1.06313
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) −18.0000 −0.597351
\(909\) −6.00000 −0.199007
\(910\) −4.00000 −0.132599
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) −8.00000 −0.264906
\(913\) −72.0000 −2.38285
\(914\) −26.0000 −0.860004
\(915\) 8.00000 0.264472
\(916\) 6.00000 0.198246
\(917\) −12.0000 −0.396275
\(918\) −4.00000 −0.132020
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −36.0000 −1.18688
\(921\) 32.0000 1.05444
\(922\) 14.0000 0.461065
\(923\) 10.0000 0.329154
\(924\) 24.0000 0.789542
\(925\) −2.00000 −0.0657596
\(926\) −28.0000 −0.920137
\(927\) 8.00000 0.262754
\(928\) −30.0000 −0.984798
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) −8.00000 −0.262330
\(931\) −12.0000 −0.393284
\(932\) 22.0000 0.720634
\(933\) −12.0000 −0.392862
\(934\) −32.0000 −1.04707
\(935\) −12.0000 −0.392442
\(936\) 3.00000 0.0980581
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 52.0000 1.69696
\(940\) 8.00000 0.260931
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 28.0000 0.912289
\(943\) −36.0000 −1.17232
\(944\) −4.00000 −0.130189
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) 14.0000 0.454939 0.227469 0.973785i \(-0.426955\pi\)
0.227469 + 0.973785i \(0.426955\pi\)
\(948\) −28.0000 −0.909398
\(949\) −10.0000 −0.324614
\(950\) −4.00000 −0.129777
\(951\) −44.0000 −1.42680
\(952\) −6.00000 −0.194461
\(953\) −10.0000 −0.323932 −0.161966 0.986796i \(-0.551783\pi\)
−0.161966 + 0.986796i \(0.551783\pi\)
\(954\) 14.0000 0.453267
\(955\) 16.0000 0.517748
\(956\) −12.0000 −0.388108
\(957\) 72.0000 2.32743
\(958\) 22.0000 0.710788
\(959\) −36.0000 −1.16250
\(960\) 28.0000 0.903696
\(961\) −27.0000 −0.870968
\(962\) −2.00000 −0.0644826
\(963\) 10.0000 0.322245
\(964\) 14.0000 0.450910
\(965\) 4.00000 0.128765
\(966\) 24.0000 0.772187
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −75.0000 −2.41059
\(969\) 8.00000 0.256997
\(970\) 4.00000 0.128432
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 10.0000 0.320750
\(973\) 20.0000 0.641171
\(974\) 34.0000 1.08943
\(975\) 2.00000 0.0640513
\(976\) −2.00000 −0.0640184
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 36.0000 1.15115
\(979\) 108.000 3.45169
\(980\) 6.00000 0.191663
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 46.0000 1.46717 0.733586 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 36.0000 1.14764
\(985\) −12.0000 −0.382352
\(986\) −6.00000 −0.191079
\(987\) −16.0000 −0.509286
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) −12.0000 −0.381385
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −10.0000 −0.317500
\(993\) 56.0000 1.77711
\(994\) −20.0000 −0.634361
\(995\) 4.00000 0.126809
\(996\) −24.0000 −0.760469
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −22.0000 −0.696398
\(999\) −8.00000 −0.253109
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 221.2.a.b.1.1 1
3.2 odd 2 1989.2.a.a.1.1 1
4.3 odd 2 3536.2.a.c.1.1 1
5.4 even 2 5525.2.a.c.1.1 1
13.12 even 2 2873.2.a.a.1.1 1
17.16 even 2 3757.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
221.2.a.b.1.1 1 1.1 even 1 trivial
1989.2.a.a.1.1 1 3.2 odd 2
2873.2.a.a.1.1 1 13.12 even 2
3536.2.a.c.1.1 1 4.3 odd 2
3757.2.a.d.1.1 1 17.16 even 2
5525.2.a.c.1.1 1 5.4 even 2