Properties

Label 106.2.a.c.1.1
Level $106$
Weight $2$
Character 106.1
Self dual yes
Analytic conductor $0.846$
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] = [106,2,Mod(1,106)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(106, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("106.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 106 = 2 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 106.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: \(0.846414261426\)
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 \(\chi\) \(=\) 106.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -3.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +2.00000 q^{14} -6.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +3.00000 q^{20} -4.00000 q^{21} -3.00000 q^{22} -9.00000 q^{23} -2.00000 q^{24} +4.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} -6.00000 q^{30} +5.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +3.00000 q^{34} +6.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} +8.00000 q^{39} +3.00000 q^{40} +6.00000 q^{41} -4.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} +3.00000 q^{45} -9.00000 q^{46} -2.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} -6.00000 q^{51} -4.00000 q^{52} -1.00000 q^{53} +4.00000 q^{54} -9.00000 q^{55} +2.00000 q^{56} +8.00000 q^{57} +6.00000 q^{58} +15.0000 q^{59} -6.00000 q^{60} -10.0000 q^{61} +5.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} +6.00000 q^{66} -4.00000 q^{67} +3.00000 q^{68} +18.0000 q^{69} +6.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} +8.00000 q^{73} -10.0000 q^{74} -8.00000 q^{75} -4.00000 q^{76} -6.00000 q^{77} +8.00000 q^{78} +11.0000 q^{79} +3.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +9.00000 q^{85} -1.00000 q^{86} -12.0000 q^{87} -3.00000 q^{88} +9.00000 q^{89} +3.00000 q^{90} -8.00000 q^{91} -9.00000 q^{92} -10.0000 q^{93} -12.0000 q^{95} -2.00000 q^{96} -13.0000 q^{97} -3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −2.00000 −0.816497
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −2.00000 −0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 2.00000 0.534522
\(15\) −6.00000 −1.54919
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 3.00000 0.670820
\(21\) −4.00000 −0.872872
\(22\) −3.00000 −0.639602
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) −2.00000 −0.408248
\(25\) 4.00000 0.800000
\(26\) −4.00000 −0.784465
\(27\) 4.00000 0.769800
\(28\) 2.00000 0.377964
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −6.00000 −1.09545
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) 3.00000 0.514496
\(35\) 6.00000 1.01419
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 8.00000 1.28103
\(40\) 3.00000 0.474342
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −4.00000 −0.617213
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −3.00000 −0.452267
\(45\) 3.00000 0.447214
\(46\) −9.00000 −1.32698
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) −3.00000 −0.428571
\(50\) 4.00000 0.565685
\(51\) −6.00000 −0.840168
\(52\) −4.00000 −0.554700
\(53\) −1.00000 −0.137361
\(54\) 4.00000 0.544331
\(55\) −9.00000 −1.21356
\(56\) 2.00000 0.267261
\(57\) 8.00000 1.05963
\(58\) 6.00000 0.787839
\(59\) 15.0000 1.95283 0.976417 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) −6.00000 −0.774597
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 5.00000 0.635001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 6.00000 0.738549
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 3.00000 0.363803
\(69\) 18.0000 2.16695
\(70\) 6.00000 0.717137
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −10.0000 −1.16248
\(75\) −8.00000 −0.923760
\(76\) −4.00000 −0.458831
\(77\) −6.00000 −0.683763
\(78\) 8.00000 0.905822
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 3.00000 0.335410
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −4.00000 −0.436436
\(85\) 9.00000 0.976187
\(86\) −1.00000 −0.107833
\(87\) −12.0000 −1.28654
\(88\) −3.00000 −0.319801
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 3.00000 0.316228
\(91\) −8.00000 −0.838628
\(92\) −9.00000 −0.938315
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) −12.0000 −1.23117
\(96\) −2.00000 −0.204124
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) −3.00000 −0.303046
\(99\) −3.00000 −0.301511
\(100\) 4.00000 0.400000
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) −6.00000 −0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −4.00000 −0.392232
\(105\) −12.0000 −1.17108
\(106\) −1.00000 −0.0971286
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 4.00000 0.384900
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) −9.00000 −0.858116
\(111\) 20.0000 1.89832
\(112\) 2.00000 0.188982
\(113\) 3.00000 0.282216 0.141108 0.989994i \(-0.454933\pi\)
0.141108 + 0.989994i \(0.454933\pi\)
\(114\) 8.00000 0.749269
\(115\) −27.0000 −2.51776
\(116\) 6.00000 0.557086
\(117\) −4.00000 −0.369800
\(118\) 15.0000 1.38086
\(119\) 6.00000 0.550019
\(120\) −6.00000 −0.547723
\(121\) −2.00000 −0.181818
\(122\) −10.0000 −0.905357
\(123\) −12.0000 −1.08200
\(124\) 5.00000 0.449013
\(125\) −3.00000 −0.268328
\(126\) 2.00000 0.178174
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) −12.0000 −1.05247
\(131\) −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
\(135\) 12.0000 1.03280
\(136\) 3.00000 0.257248
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 18.0000 1.53226
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 6.00000 0.507093
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 18.0000 1.49482
\(146\) 8.00000 0.662085
\(147\) 6.00000 0.494872
\(148\) −10.0000 −0.821995
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −8.00000 −0.653197
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) 3.00000 0.242536
\(154\) −6.00000 −0.483494
\(155\) 15.0000 1.20483
\(156\) 8.00000 0.640513
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 11.0000 0.875113
\(159\) 2.00000 0.158610
\(160\) 3.00000 0.237171
\(161\) −18.0000 −1.41860
\(162\) −11.0000 −0.864242
\(163\) −7.00000 −0.548282 −0.274141 0.961689i \(-0.588394\pi\)
−0.274141 + 0.961689i \(0.588394\pi\)
\(164\) 6.00000 0.468521
\(165\) 18.0000 1.40130
\(166\) −6.00000 −0.465690
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) −4.00000 −0.308607
\(169\) 3.00000 0.230769
\(170\) 9.00000 0.690268
\(171\) −4.00000 −0.305888
\(172\) −1.00000 −0.0762493
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) −12.0000 −0.909718
\(175\) 8.00000 0.604743
\(176\) −3.00000 −0.226134
\(177\) −30.0000 −2.25494
\(178\) 9.00000 0.674579
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 3.00000 0.223607
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −8.00000 −0.592999
\(183\) 20.0000 1.47844
\(184\) −9.00000 −0.663489
\(185\) −30.0000 −2.20564
\(186\) −10.0000 −0.733236
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) 8.00000 0.581914
\(190\) −12.0000 −0.870572
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) −2.00000 −0.144338
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) −13.0000 −0.933346
\(195\) 24.0000 1.71868
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −3.00000 −0.213201
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 4.00000 0.282843
\(201\) 8.00000 0.564276
\(202\) 15.0000 1.05540
\(203\) 12.0000 0.842235
\(204\) −6.00000 −0.420084
\(205\) 18.0000 1.25717
\(206\) 8.00000 0.557386
\(207\) −9.00000 −0.625543
\(208\) −4.00000 −0.277350
\(209\) 12.0000 0.830057
\(210\) −12.0000 −0.828079
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) −1.00000 −0.0686803
\(213\) −24.0000 −1.64445
\(214\) 0 0
\(215\) −3.00000 −0.204598
\(216\) 4.00000 0.272166
\(217\) 10.0000 0.678844
\(218\) −7.00000 −0.474100
\(219\) −16.0000 −1.08118
\(220\) −9.00000 −0.606780
\(221\) −12.0000 −0.807207
\(222\) 20.0000 1.34231
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 2.00000 0.133631
\(225\) 4.00000 0.266667
\(226\) 3.00000 0.199557
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 8.00000 0.529813
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −27.0000 −1.78033
\(231\) 12.0000 0.789542
\(232\) 6.00000 0.393919
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 15.0000 0.976417
\(237\) −22.0000 −1.42905
\(238\) 6.00000 0.388922
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) −6.00000 −0.387298
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) −2.00000 −0.128565
\(243\) 10.0000 0.641500
\(244\) −10.0000 −0.640184
\(245\) −9.00000 −0.574989
\(246\) −12.0000 −0.765092
\(247\) 16.0000 1.01806
\(248\) 5.00000 0.317500
\(249\) 12.0000 0.760469
\(250\) −3.00000 −0.189737
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 2.00000 0.125988
\(253\) 27.0000 1.69748
\(254\) −7.00000 −0.439219
\(255\) −18.0000 −1.12720
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 2.00000 0.124515
\(259\) −20.0000 −1.24274
\(260\) −12.0000 −0.744208
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 6.00000 0.369274
\(265\) −3.00000 −0.184289
\(266\) −8.00000 −0.490511
\(267\) −18.0000 −1.10158
\(268\) −4.00000 −0.244339
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 12.0000 0.730297
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 3.00000 0.181902
\(273\) 16.0000 0.968364
\(274\) 12.0000 0.724947
\(275\) −12.0000 −0.723627
\(276\) 18.0000 1.08347
\(277\) 17.0000 1.02143 0.510716 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) 8.00000 0.479808
\(279\) 5.00000 0.299342
\(280\) 6.00000 0.358569
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 12.0000 0.712069
\(285\) 24.0000 1.42164
\(286\) 12.0000 0.709575
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 18.0000 1.05700
\(291\) 26.0000 1.52415
\(292\) 8.00000 0.468165
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 6.00000 0.349927
\(295\) 45.0000 2.62000
\(296\) −10.0000 −0.581238
\(297\) −12.0000 −0.696311
\(298\) −18.0000 −1.04271
\(299\) 36.0000 2.08193
\(300\) −8.00000 −0.461880
\(301\) −2.00000 −0.115278
\(302\) 8.00000 0.460348
\(303\) −30.0000 −1.72345
\(304\) −4.00000 −0.229416
\(305\) −30.0000 −1.71780
\(306\) 3.00000 0.171499
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) −6.00000 −0.341882
\(309\) −16.0000 −0.910208
\(310\) 15.0000 0.851943
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 8.00000 0.452911
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 23.0000 1.29797
\(315\) 6.00000 0.338062
\(316\) 11.0000 0.618798
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 2.00000 0.112154
\(319\) −18.0000 −1.00781
\(320\) 3.00000 0.167705
\(321\) 0 0
\(322\) −18.0000 −1.00310
\(323\) −12.0000 −0.667698
\(324\) −11.0000 −0.611111
\(325\) −16.0000 −0.887520
\(326\) −7.00000 −0.387694
\(327\) 14.0000 0.774202
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 18.0000 0.990867
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −6.00000 −0.329293
\(333\) −10.0000 −0.547997
\(334\) 9.00000 0.492458
\(335\) −12.0000 −0.655630
\(336\) −4.00000 −0.218218
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 3.00000 0.163178
\(339\) −6.00000 −0.325875
\(340\) 9.00000 0.488094
\(341\) −15.0000 −0.812296
\(342\) −4.00000 −0.216295
\(343\) −20.0000 −1.07990
\(344\) −1.00000 −0.0539164
\(345\) 54.0000 2.90726
\(346\) 3.00000 0.161281
\(347\) 27.0000 1.44944 0.724718 0.689046i \(-0.241970\pi\)
0.724718 + 0.689046i \(0.241970\pi\)
\(348\) −12.0000 −0.643268
\(349\) 35.0000 1.87351 0.936754 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 8.00000 0.427618
\(351\) −16.0000 −0.854017
\(352\) −3.00000 −0.159901
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) −30.0000 −1.59448
\(355\) 36.0000 1.91068
\(356\) 9.00000 0.476999
\(357\) −12.0000 −0.635107
\(358\) −6.00000 −0.317110
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 3.00000 0.158114
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) 4.00000 0.209946
\(364\) −8.00000 −0.419314
\(365\) 24.0000 1.25622
\(366\) 20.0000 1.04542
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −9.00000 −0.469157
\(369\) 6.00000 0.312348
\(370\) −30.0000 −1.55963
\(371\) −2.00000 −0.103835
\(372\) −10.0000 −0.518476
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) −9.00000 −0.465379
\(375\) 6.00000 0.309839
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 8.00000 0.411476
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −12.0000 −0.615587
\(381\) 14.0000 0.717242
\(382\) 9.00000 0.460480
\(383\) −27.0000 −1.37964 −0.689818 0.723983i \(-0.742309\pi\)
−0.689818 + 0.723983i \(0.742309\pi\)
\(384\) −2.00000 −0.102062
\(385\) −18.0000 −0.917365
\(386\) 8.00000 0.407189
\(387\) −1.00000 −0.0508329
\(388\) −13.0000 −0.659975
\(389\) 33.0000 1.67317 0.836583 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 24.0000 1.21529
\(391\) −27.0000 −1.36545
\(392\) −3.00000 −0.151523
\(393\) 24.0000 1.21064
\(394\) −18.0000 −0.906827
\(395\) 33.0000 1.66041
\(396\) −3.00000 −0.150756
\(397\) −25.0000 −1.25471 −0.627357 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) −4.00000 −0.200502
\(399\) 16.0000 0.801002
\(400\) 4.00000 0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 8.00000 0.399004
\(403\) −20.0000 −0.996271
\(404\) 15.0000 0.746278
\(405\) −33.0000 −1.63978
\(406\) 12.0000 0.595550
\(407\) 30.0000 1.48704
\(408\) −6.00000 −0.297044
\(409\) −19.0000 −0.939490 −0.469745 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 18.0000 0.888957
\(411\) −24.0000 −1.18383
\(412\) 8.00000 0.394132
\(413\) 30.0000 1.47620
\(414\) −9.00000 −0.442326
\(415\) −18.0000 −0.883585
\(416\) −4.00000 −0.196116
\(417\) −16.0000 −0.783523
\(418\) 12.0000 0.586939
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) −12.0000 −0.585540
\(421\) −7.00000 −0.341159 −0.170580 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) 11.0000 0.535472
\(423\) 0 0
\(424\) −1.00000 −0.0485643
\(425\) 12.0000 0.582086
\(426\) −24.0000 −1.16280
\(427\) −20.0000 −0.967868
\(428\) 0 0
\(429\) −24.0000 −1.15873
\(430\) −3.00000 −0.144673
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 4.00000 0.192450
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 10.0000 0.480015
\(435\) −36.0000 −1.72607
\(436\) −7.00000 −0.335239
\(437\) 36.0000 1.72211
\(438\) −16.0000 −0.764510
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) −9.00000 −0.429058
\(441\) −3.00000 −0.142857
\(442\) −12.0000 −0.570782
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 20.0000 0.949158
\(445\) 27.0000 1.27992
\(446\) 8.00000 0.378811
\(447\) 36.0000 1.70274
\(448\) 2.00000 0.0944911
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 4.00000 0.188562
\(451\) −18.0000 −0.847587
\(452\) 3.00000 0.141108
\(453\) −16.0000 −0.751746
\(454\) −12.0000 −0.563188
\(455\) −24.0000 −1.12514
\(456\) 8.00000 0.374634
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 14.0000 0.654177
\(459\) 12.0000 0.560112
\(460\) −27.0000 −1.25888
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 12.0000 0.558291
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 6.00000 0.278543
\(465\) −30.0000 −1.39122
\(466\) −12.0000 −0.555889
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −4.00000 −0.184900
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −46.0000 −2.11957
\(472\) 15.0000 0.690431
\(473\) 3.00000 0.137940
\(474\) −22.0000 −1.01049
\(475\) −16.0000 −0.734130
\(476\) 6.00000 0.275010
\(477\) −1.00000 −0.0457869
\(478\) −21.0000 −0.960518
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −6.00000 −0.273861
\(481\) 40.0000 1.82384
\(482\) −25.0000 −1.13872
\(483\) 36.0000 1.63806
\(484\) −2.00000 −0.0909091
\(485\) −39.0000 −1.77090
\(486\) 10.0000 0.453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −10.0000 −0.452679
\(489\) 14.0000 0.633102
\(490\) −9.00000 −0.406579
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) −12.0000 −0.541002
\(493\) 18.0000 0.810679
\(494\) 16.0000 0.719874
\(495\) −9.00000 −0.404520
\(496\) 5.00000 0.224507
\(497\) 24.0000 1.07655
\(498\) 12.0000 0.537733
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −3.00000 −0.134164
\(501\) −18.0000 −0.804181
\(502\) 6.00000 0.267793
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 2.00000 0.0890871
\(505\) 45.0000 2.00247
\(506\) 27.0000 1.20030
\(507\) −6.00000 −0.266469
\(508\) −7.00000 −0.310575
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −18.0000 −0.797053
\(511\) 16.0000 0.707798
\(512\) 1.00000 0.0441942
\(513\) −16.0000 −0.706417
\(514\) 6.00000 0.264649
\(515\) 24.0000 1.05757
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) −20.0000 −0.878750
\(519\) −6.00000 −0.263371
\(520\) −12.0000 −0.526235
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 6.00000 0.262613
\(523\) −7.00000 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) −12.0000 −0.524222
\(525\) −16.0000 −0.698297
\(526\) −15.0000 −0.654031
\(527\) 15.0000 0.653410
\(528\) 6.00000 0.261116
\(529\) 58.0000 2.52174
\(530\) −3.00000 −0.130312
\(531\) 15.0000 0.650945
\(532\) −8.00000 −0.346844
\(533\) −24.0000 −1.03956
\(534\) −18.0000 −0.778936
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 12.0000 0.516398
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −4.00000 −0.171815
\(543\) 20.0000 0.858282
\(544\) 3.00000 0.128624
\(545\) −21.0000 −0.899541
\(546\) 16.0000 0.684737
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 12.0000 0.512615
\(549\) −10.0000 −0.426790
\(550\) −12.0000 −0.511682
\(551\) −24.0000 −1.02243
\(552\) 18.0000 0.766131
\(553\) 22.0000 0.935535
\(554\) 17.0000 0.722261
\(555\) 60.0000 2.54686
\(556\) 8.00000 0.339276
\(557\) 33.0000 1.39825 0.699127 0.714997i \(-0.253572\pi\)
0.699127 + 0.714997i \(0.253572\pi\)
\(558\) 5.00000 0.211667
\(559\) 4.00000 0.169182
\(560\) 6.00000 0.253546
\(561\) 18.0000 0.759961
\(562\) −3.00000 −0.126547
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 9.00000 0.378633
\(566\) −22.0000 −0.924729
\(567\) −22.0000 −0.923913
\(568\) 12.0000 0.503509
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 24.0000 1.00525
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 12.0000 0.501745
\(573\) −18.0000 −0.751961
\(574\) 12.0000 0.500870
\(575\) −36.0000 −1.50130
\(576\) 1.00000 0.0416667
\(577\) −19.0000 −0.790980 −0.395490 0.918470i \(-0.629425\pi\)
−0.395490 + 0.918470i \(0.629425\pi\)
\(578\) −8.00000 −0.332756
\(579\) −16.0000 −0.664937
\(580\) 18.0000 0.747409
\(581\) −12.0000 −0.497844
\(582\) 26.0000 1.07773
\(583\) 3.00000 0.124247
\(584\) 8.00000 0.331042
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 6.00000 0.247436
\(589\) −20.0000 −0.824086
\(590\) 45.0000 1.85262
\(591\) 36.0000 1.48084
\(592\) −10.0000 −0.410997
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −12.0000 −0.492366
\(595\) 18.0000 0.737928
\(596\) −18.0000 −0.737309
\(597\) 8.00000 0.327418
\(598\) 36.0000 1.47215
\(599\) −18.0000 −0.735460 −0.367730 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) −8.00000 −0.326599
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −2.00000 −0.0815139
\(603\) −4.00000 −0.162893
\(604\) 8.00000 0.325515
\(605\) −6.00000 −0.243935
\(606\) −30.0000 −1.21867
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) −4.00000 −0.162221
\(609\) −24.0000 −0.972529
\(610\) −30.0000 −1.21466
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) −19.0000 −0.767403 −0.383701 0.923457i \(-0.625351\pi\)
−0.383701 + 0.923457i \(0.625351\pi\)
\(614\) −7.00000 −0.282497
\(615\) −36.0000 −1.45166
\(616\) −6.00000 −0.241747
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) −16.0000 −0.643614
\(619\) −13.0000 −0.522514 −0.261257 0.965269i \(-0.584137\pi\)
−0.261257 + 0.965269i \(0.584137\pi\)
\(620\) 15.0000 0.602414
\(621\) −36.0000 −1.44463
\(622\) 12.0000 0.481156
\(623\) 18.0000 0.721155
\(624\) 8.00000 0.320256
\(625\) −29.0000 −1.16000
\(626\) −10.0000 −0.399680
\(627\) −24.0000 −0.958468
\(628\) 23.0000 0.917800
\(629\) −30.0000 −1.19618
\(630\) 6.00000 0.239046
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 11.0000 0.437557
\(633\) −22.0000 −0.874421
\(634\) −12.0000 −0.476581
\(635\) −21.0000 −0.833360
\(636\) 2.00000 0.0793052
\(637\) 12.0000 0.475457
\(638\) −18.0000 −0.712627
\(639\) 12.0000 0.474713
\(640\) 3.00000 0.118585
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −13.0000 −0.512670 −0.256335 0.966588i \(-0.582515\pi\)
−0.256335 + 0.966588i \(0.582515\pi\)
\(644\) −18.0000 −0.709299
\(645\) 6.00000 0.236250
\(646\) −12.0000 −0.472134
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) −11.0000 −0.432121
\(649\) −45.0000 −1.76640
\(650\) −16.0000 −0.627572
\(651\) −20.0000 −0.783862
\(652\) −7.00000 −0.274141
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 14.0000 0.547443
\(655\) −36.0000 −1.40664
\(656\) 6.00000 0.234261
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) 18.0000 0.701180 0.350590 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 18.0000 0.700649
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 20.0000 0.777322
\(663\) 24.0000 0.932083
\(664\) −6.00000 −0.232845
\(665\) −24.0000 −0.930680
\(666\) −10.0000 −0.387492
\(667\) −54.0000 −2.09089
\(668\) 9.00000 0.348220
\(669\) −16.0000 −0.618596
\(670\) −12.0000 −0.463600
\(671\) 30.0000 1.15814
\(672\) −4.00000 −0.154303
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 2.00000 0.0770371
\(675\) 16.0000 0.615840
\(676\) 3.00000 0.115385
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −6.00000 −0.230429
\(679\) −26.0000 −0.997788
\(680\) 9.00000 0.345134
\(681\) 24.0000 0.919682
\(682\) −15.0000 −0.574380
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) −4.00000 −0.152944
\(685\) 36.0000 1.37549
\(686\) −20.0000 −0.763604
\(687\) −28.0000 −1.06827
\(688\) −1.00000 −0.0381246
\(689\) 4.00000 0.152388
\(690\) 54.0000 2.05574
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 3.00000 0.114043
\(693\) −6.00000 −0.227921
\(694\) 27.0000 1.02491
\(695\) 24.0000 0.910372
\(696\) −12.0000 −0.454859
\(697\) 18.0000 0.681799
\(698\) 35.0000 1.32477
\(699\) 24.0000 0.907763
\(700\) 8.00000 0.302372
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −16.0000 −0.603881
\(703\) 40.0000 1.50863
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 30.0000 1.12827
\(708\) −30.0000 −1.12747
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 36.0000 1.35106
\(711\) 11.0000 0.412532
\(712\) 9.00000 0.337289
\(713\) −45.0000 −1.68526
\(714\) −12.0000 −0.449089
\(715\) 36.0000 1.34632
\(716\) −6.00000 −0.224231
\(717\) 42.0000 1.56852
\(718\) −15.0000 −0.559795
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 3.00000 0.111803
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) 50.0000 1.85952
\(724\) −10.0000 −0.371647
\(725\) 24.0000 0.891338
\(726\) 4.00000 0.148454
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) −8.00000 −0.296500
\(729\) 13.0000 0.481481
\(730\) 24.0000 0.888280
\(731\) −3.00000 −0.110959
\(732\) 20.0000 0.739221
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) −16.0000 −0.590571
\(735\) 18.0000 0.663940
\(736\) −9.00000 −0.331744
\(737\) 12.0000 0.442026
\(738\) 6.00000 0.220863
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −30.0000 −1.10282
\(741\) −32.0000 −1.17555
\(742\) −2.00000 −0.0734223
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) −10.0000 −0.366618
\(745\) −54.0000 −1.97841
\(746\) 29.0000 1.06177
\(747\) −6.00000 −0.219529
\(748\) −9.00000 −0.329073
\(749\) 0 0
\(750\) 6.00000 0.219089
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) −24.0000 −0.874028
\(755\) 24.0000 0.873449
\(756\) 8.00000 0.290957
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) −28.0000 −1.01701
\(759\) −54.0000 −1.96008
\(760\) −12.0000 −0.435286
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 14.0000 0.507166
\(763\) −14.0000 −0.506834
\(764\) 9.00000 0.325609
\(765\) 9.00000 0.325396
\(766\) −27.0000 −0.975550
\(767\) −60.0000 −2.16647
\(768\) −2.00000 −0.0721688
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) −18.0000 −0.648675
\(771\) −12.0000 −0.432169
\(772\) 8.00000 0.287926
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 20.0000 0.718421
\(776\) −13.0000 −0.466673
\(777\) 40.0000 1.43499
\(778\) 33.0000 1.18311
\(779\) −24.0000 −0.859889
\(780\) 24.0000 0.859338
\(781\) −36.0000 −1.28818
\(782\) −27.0000 −0.965518
\(783\) 24.0000 0.857690
\(784\) −3.00000 −0.107143
\(785\) 69.0000 2.46272
\(786\) 24.0000 0.856052
\(787\) −10.0000 −0.356462 −0.178231 0.983989i \(-0.557037\pi\)
−0.178231 + 0.983989i \(0.557037\pi\)
\(788\) −18.0000 −0.641223
\(789\) 30.0000 1.06803
\(790\) 33.0000 1.17409
\(791\) 6.00000 0.213335
\(792\) −3.00000 −0.106600
\(793\) 40.0000 1.42044
\(794\) −25.0000 −0.887217
\(795\) 6.00000 0.212798
\(796\) −4.00000 −0.141776
\(797\) 21.0000 0.743858 0.371929 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) 16.0000 0.566394
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) 9.00000 0.317999
\(802\) 30.0000 1.05934
\(803\) −24.0000 −0.846942
\(804\) 8.00000 0.282138
\(805\) −54.0000 −1.90325
\(806\) −20.0000 −0.704470
\(807\) 0 0
\(808\) 15.0000 0.527698
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −33.0000 −1.15950
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) 12.0000 0.421117
\(813\) 8.00000 0.280572
\(814\) 30.0000 1.05150
\(815\) −21.0000 −0.735598
\(816\) −6.00000 −0.210042
\(817\) 4.00000 0.139942
\(818\) −19.0000 −0.664319
\(819\) −8.00000 −0.279543
\(820\) 18.0000 0.628587
\(821\) 3.00000 0.104701 0.0523504 0.998629i \(-0.483329\pi\)
0.0523504 + 0.998629i \(0.483329\pi\)
\(822\) −24.0000 −0.837096
\(823\) −10.0000 −0.348578 −0.174289 0.984695i \(-0.555763\pi\)
−0.174289 + 0.984695i \(0.555763\pi\)
\(824\) 8.00000 0.278693
\(825\) 24.0000 0.835573
\(826\) 30.0000 1.04383
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) −9.00000 −0.312772
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) −18.0000 −0.624789
\(831\) −34.0000 −1.17945
\(832\) −4.00000 −0.138675
\(833\) −9.00000 −0.311832
\(834\) −16.0000 −0.554035
\(835\) 27.0000 0.934374
\(836\) 12.0000 0.415029
\(837\) 20.0000 0.691301
\(838\) −18.0000 −0.621800
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) −12.0000 −0.414039
\(841\) 7.00000 0.241379
\(842\) −7.00000 −0.241236
\(843\) 6.00000 0.206651
\(844\) 11.0000 0.378636
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) −1.00000 −0.0343401
\(849\) 44.0000 1.51008
\(850\) 12.0000 0.411597
\(851\) 90.0000 3.08516
\(852\) −24.0000 −0.822226
\(853\) 38.0000 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) −20.0000 −0.684386
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) −9.00000 −0.307434 −0.153717 0.988115i \(-0.549124\pi\)
−0.153717 + 0.988115i \(0.549124\pi\)
\(858\) −24.0000 −0.819346
\(859\) 35.0000 1.19418 0.597092 0.802173i \(-0.296323\pi\)
0.597092 + 0.802173i \(0.296323\pi\)
\(860\) −3.00000 −0.102299
\(861\) −24.0000 −0.817918
\(862\) −12.0000 −0.408722
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 4.00000 0.136083
\(865\) 9.00000 0.306009
\(866\) 2.00000 0.0679628
\(867\) 16.0000 0.543388
\(868\) 10.0000 0.339422
\(869\) −33.0000 −1.11945
\(870\) −36.0000 −1.22051
\(871\) 16.0000 0.542139
\(872\) −7.00000 −0.237050
\(873\) −13.0000 −0.439983
\(874\) 36.0000 1.21772
\(875\) −6.00000 −0.202837
\(876\) −16.0000 −0.540590
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −10.0000 −0.337484
\(879\) 0 0
\(880\) −9.00000 −0.303390
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) −3.00000 −0.101015
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −12.0000 −0.403604
\(885\) −90.0000 −3.02532
\(886\) 12.0000 0.403148
\(887\) −9.00000 −0.302190 −0.151095 0.988519i \(-0.548280\pi\)
−0.151095 + 0.988519i \(0.548280\pi\)
\(888\) 20.0000 0.671156
\(889\) −14.0000 −0.469545
\(890\) 27.0000 0.905042
\(891\) 33.0000 1.10554
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) 36.0000 1.20402
\(895\) −18.0000 −0.601674
\(896\) 2.00000 0.0668153
\(897\) −72.0000 −2.40401
\(898\) 18.0000 0.600668
\(899\) 30.0000 1.00056
\(900\) 4.00000 0.133333
\(901\) −3.00000 −0.0999445
\(902\) −18.0000 −0.599334
\(903\) 4.00000 0.133112
\(904\) 3.00000 0.0997785
\(905\) −30.0000 −0.997234
\(906\) −16.0000 −0.531564
\(907\) 53.0000 1.75984 0.879918 0.475125i \(-0.157597\pi\)
0.879918 + 0.475125i \(0.157597\pi\)
\(908\) −12.0000 −0.398234
\(909\) 15.0000 0.497519
\(910\) −24.0000 −0.795592
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 8.00000 0.264906
\(913\) 18.0000 0.595713
\(914\) 26.0000 0.860004
\(915\) 60.0000 1.98354
\(916\) 14.0000 0.462573
\(917\) −24.0000 −0.792550
\(918\) 12.0000 0.396059
\(919\) −43.0000 −1.41844 −0.709220 0.704988i \(-0.750953\pi\)
−0.709220 + 0.704988i \(0.750953\pi\)
\(920\) −27.0000 −0.890164
\(921\) 14.0000 0.461316
\(922\) 18.0000 0.592798
\(923\) −48.0000 −1.57994
\(924\) 12.0000 0.394771
\(925\) −40.0000 −1.31519
\(926\) 17.0000 0.558655
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) −27.0000 −0.885841 −0.442921 0.896561i \(-0.646058\pi\)
−0.442921 + 0.896561i \(0.646058\pi\)
\(930\) −30.0000 −0.983739
\(931\) 12.0000 0.393284
\(932\) −12.0000 −0.393073
\(933\) −24.0000 −0.785725
\(934\) 12.0000 0.392652
\(935\) −27.0000 −0.882994
\(936\) −4.00000 −0.130744
\(937\) 5.00000 0.163343 0.0816714 0.996659i \(-0.473974\pi\)
0.0816714 + 0.996659i \(0.473974\pi\)
\(938\) −8.00000 −0.261209
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −46.0000 −1.49876
\(943\) −54.0000 −1.75848
\(944\) 15.0000 0.488208
\(945\) 24.0000 0.780720
\(946\) 3.00000 0.0975384
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −22.0000 −0.714527
\(949\) −32.0000 −1.03876
\(950\) −16.0000 −0.519109
\(951\) 24.0000 0.778253
\(952\) 6.00000 0.194461
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 27.0000 0.873699
\(956\) −21.0000 −0.679189
\(957\) 36.0000 1.16371
\(958\) 24.0000 0.775405
\(959\) 24.0000 0.775000
\(960\) −6.00000 −0.193649
\(961\) −6.00000 −0.193548
\(962\) 40.0000 1.28965
\(963\) 0 0
\(964\) −25.0000 −0.805196
\(965\) 24.0000 0.772587
\(966\) 36.0000 1.15828
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 24.0000 0.770991
\(970\) −39.0000 −1.25221
\(971\) −9.00000 −0.288824 −0.144412 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 10.0000 0.320750
\(973\) 16.0000 0.512936
\(974\) −4.00000 −0.128168
\(975\) 32.0000 1.02482
\(976\) −10.0000 −0.320092
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 14.0000 0.447671
\(979\) −27.0000 −0.862924
\(980\) −9.00000 −0.287494
\(981\) −7.00000 −0.223493
\(982\) −18.0000 −0.574403
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) −12.0000 −0.382546
\(985\) −54.0000 −1.72058
\(986\) 18.0000 0.573237
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) 9.00000 0.286183
\(990\) −9.00000 −0.286039
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 5.00000 0.158750
\(993\) −40.0000 −1.26936
\(994\) 24.0000 0.761234
\(995\) −12.0000 −0.380426
\(996\) 12.0000 0.380235
\(997\) 62.0000 1.96356 0.981780 0.190022i \(-0.0608559\pi\)
0.981780 + 0.190022i \(0.0608559\pi\)
\(998\) −4.00000 −0.126618
\(999\) −40.0000 −1.26554
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 106.2.a.c.1.1 1
3.2 odd 2 954.2.a.a.1.1 1
4.3 odd 2 848.2.a.f.1.1 1
5.2 odd 4 2650.2.b.a.849.2 2
5.3 odd 4 2650.2.b.a.849.1 2
5.4 even 2 2650.2.a.e.1.1 1
7.6 odd 2 5194.2.a.p.1.1 1
8.3 odd 2 3392.2.a.c.1.1 1
8.5 even 2 3392.2.a.o.1.1 1
12.11 even 2 7632.2.a.c.1.1 1
53.52 even 2 5618.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
106.2.a.c.1.1 1 1.1 even 1 trivial
848.2.a.f.1.1 1 4.3 odd 2
954.2.a.a.1.1 1 3.2 odd 2
2650.2.a.e.1.1 1 5.4 even 2
2650.2.b.a.849.1 2 5.3 odd 4
2650.2.b.a.849.2 2 5.2 odd 4
3392.2.a.c.1.1 1 8.3 odd 2
3392.2.a.o.1.1 1 8.5 even 2
5194.2.a.p.1.1 1 7.6 odd 2
5618.2.a.d.1.1 1 53.52 even 2
7632.2.a.c.1.1 1 12.11 even 2