Properties

Label 2738.2.a.c.1.1
Level $2738$
Weight $2$
Character 2738.1
Self dual yes
Analytic conductor $21.863$
Analytic rank $2$
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] = [2738,2,Mod(1,2738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, 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("2738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.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: \(21.8630400734\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2738.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -6.00000 q^{11} -2.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +6.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -3.00000 q^{20} +8.00000 q^{21} -6.00000 q^{22} -6.00000 q^{23} -2.00000 q^{24} +4.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} -4.00000 q^{28} -3.00000 q^{29} +6.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +12.0000 q^{33} -3.00000 q^{34} +12.0000 q^{35} +1.00000 q^{36} -2.00000 q^{38} +4.00000 q^{39} -3.00000 q^{40} +3.00000 q^{41} +8.00000 q^{42} +4.00000 q^{43} -6.00000 q^{44} -3.00000 q^{45} -6.00000 q^{46} -6.00000 q^{47} -2.00000 q^{48} +9.00000 q^{49} +4.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +4.00000 q^{54} +18.0000 q^{55} -4.00000 q^{56} +4.00000 q^{57} -3.00000 q^{58} +6.00000 q^{60} +1.00000 q^{61} -2.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} +12.0000 q^{66} +2.00000 q^{67} -3.00000 q^{68} +12.0000 q^{69} +12.0000 q^{70} -12.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -8.00000 q^{75} -2.00000 q^{76} +24.0000 q^{77} +4.00000 q^{78} -14.0000 q^{79} -3.00000 q^{80} -11.0000 q^{81} +3.00000 q^{82} +6.00000 q^{83} +8.00000 q^{84} +9.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} -3.00000 q^{89} -3.00000 q^{90} +8.00000 q^{91} -6.00000 q^{92} +4.00000 q^{93} -6.00000 q^{94} +6.00000 q^{95} -2.00000 q^{96} +13.0000 q^{97} +9.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) −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 \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −2.00000 −0.816497
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −2.00000 −0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −4.00000 −1.06904
\(15\) 6.00000 1.54919
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −3.00000 −0.670820
\(21\) 8.00000 1.74574
\(22\) −6.00000 −1.27920
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −2.00000 −0.408248
\(25\) 4.00000 0.800000
\(26\) −2.00000 −0.392232
\(27\) 4.00000 0.769800
\(28\) −4.00000 −0.755929
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 6.00000 1.09545
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 12.0000 2.08893
\(34\) −3.00000 −0.514496
\(35\) 12.0000 2.02837
\(36\) 1.00000 0.166667
\(37\) 0 0
\(38\) −2.00000 −0.324443
\(39\) 4.00000 0.640513
\(40\) −3.00000 −0.474342
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 8.00000 1.23443
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −6.00000 −0.904534
\(45\) −3.00000 −0.447214
\(46\) −6.00000 −0.884652
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −2.00000 −0.288675
\(49\) 9.00000 1.28571
\(50\) 4.00000 0.565685
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 4.00000 0.544331
\(55\) 18.0000 2.42712
\(56\) −4.00000 −0.534522
\(57\) 4.00000 0.529813
\(58\) −3.00000 −0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 6.00000 0.774597
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −2.00000 −0.254000
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 12.0000 1.47710
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −3.00000 −0.363803
\(69\) 12.0000 1.44463
\(70\) 12.0000 1.43427
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) −2.00000 −0.229416
\(77\) 24.0000 2.73505
\(78\) 4.00000 0.452911
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) −3.00000 −0.335410
\(81\) −11.0000 −1.22222
\(82\) 3.00000 0.331295
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 8.00000 0.872872
\(85\) 9.00000 0.976187
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) −6.00000 −0.639602
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) −3.00000 −0.316228
\(91\) 8.00000 0.838628
\(92\) −6.00000 −0.625543
\(93\) 4.00000 0.414781
\(94\) −6.00000 −0.618853
\(95\) 6.00000 0.615587
\(96\) −2.00000 −0.204124
\(97\) 13.0000 1.31995 0.659975 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 9.00000 0.909137
\(99\) −6.00000 −0.603023
\(100\) 4.00000 0.400000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 6.00000 0.594089
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −2.00000 −0.196116
\(105\) −24.0000 −2.34216
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 4.00000 0.384900
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 18.0000 1.71623
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 4.00000 0.374634
\(115\) 18.0000 1.67851
\(116\) −3.00000 −0.278543
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 6.00000 0.547723
\(121\) 25.0000 2.27273
\(122\) 1.00000 0.0905357
\(123\) −6.00000 −0.541002
\(124\) −2.00000 −0.179605
\(125\) 3.00000 0.268328
\(126\) −4.00000 −0.356348
\(127\) −10.0000 −0.887357 −0.443678 0.896186i \(-0.646327\pi\)
−0.443678 + 0.896186i \(0.646327\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 6.00000 0.526235
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 12.0000 1.04447
\(133\) 8.00000 0.693688
\(134\) 2.00000 0.172774
\(135\) −12.0000 −1.03280
\(136\) −3.00000 −0.257248
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 12.0000 1.02151
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 12.0000 1.01419
\(141\) 12.0000 1.01058
\(142\) −12.0000 −1.00702
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 9.00000 0.747409
\(146\) −10.0000 −0.827606
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) −21.0000 −1.72039 −0.860194 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) −8.00000 −0.653197
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) −2.00000 −0.162221
\(153\) −3.00000 −0.242536
\(154\) 24.0000 1.93398
\(155\) 6.00000 0.481932
\(156\) 4.00000 0.320256
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) −14.0000 −1.11378
\(159\) 12.0000 0.951662
\(160\) −3.00000 −0.237171
\(161\) 24.0000 1.89146
\(162\) −11.0000 −0.864242
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 3.00000 0.234261
\(165\) −36.0000 −2.80260
\(166\) 6.00000 0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 8.00000 0.617213
\(169\) −9.00000 −0.692308
\(170\) 9.00000 0.690268
\(171\) −2.00000 −0.152944
\(172\) 4.00000 0.304997
\(173\) −21.0000 −1.59660 −0.798300 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) 6.00000 0.454859
\(175\) −16.0000 −1.20949
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) −3.00000 −0.224860
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −3.00000 −0.223607
\(181\) −1.00000 −0.0743294 −0.0371647 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) 8.00000 0.592999
\(183\) −2.00000 −0.147844
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 18.0000 1.31629
\(188\) −6.00000 −0.437595
\(189\) −16.0000 −1.16383
\(190\) 6.00000 0.435286
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) −2.00000 −0.144338
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) 13.0000 0.933346
\(195\) −12.0000 −0.859338
\(196\) 9.00000 0.642857
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) −6.00000 −0.426401
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 4.00000 0.282843
\(201\) −4.00000 −0.282138
\(202\) 3.00000 0.211079
\(203\) 12.0000 0.842235
\(204\) 6.00000 0.420084
\(205\) −9.00000 −0.628587
\(206\) 4.00000 0.278693
\(207\) −6.00000 −0.417029
\(208\) −2.00000 −0.138675
\(209\) 12.0000 0.830057
\(210\) −24.0000 −1.65616
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −6.00000 −0.412082
\(213\) 24.0000 1.64445
\(214\) −12.0000 −0.820303
\(215\) −12.0000 −0.818393
\(216\) 4.00000 0.272166
\(217\) 8.00000 0.543075
\(218\) 1.00000 0.0677285
\(219\) 20.0000 1.35147
\(220\) 18.0000 1.21356
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) −4.00000 −0.267261
\(225\) 4.00000 0.266667
\(226\) 18.0000 1.19734
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 4.00000 0.264906
\(229\) 11.0000 0.726900 0.363450 0.931614i \(-0.381599\pi\)
0.363450 + 0.931614i \(0.381599\pi\)
\(230\) 18.0000 1.18688
\(231\) −48.0000 −3.15817
\(232\) −3.00000 −0.196960
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) −2.00000 −0.130744
\(235\) 18.0000 1.17419
\(236\) 0 0
\(237\) 28.0000 1.81880
\(238\) 12.0000 0.777844
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 6.00000 0.387298
\(241\) −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
\(244\) 1.00000 0.0640184
\(245\) −27.0000 −1.72497
\(246\) −6.00000 −0.382546
\(247\) 4.00000 0.254514
\(248\) −2.00000 −0.127000
\(249\) −12.0000 −0.760469
\(250\) 3.00000 0.189737
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −4.00000 −0.251976
\(253\) 36.0000 2.26330
\(254\) −10.0000 −0.627456
\(255\) −18.0000 −1.12720
\(256\) 1.00000 0.0625000
\(257\) −27.0000 −1.68421 −0.842107 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) −8.00000 −0.498058
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) −3.00000 −0.185695
\(262\) −18.0000 −1.11204
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 12.0000 0.738549
\(265\) 18.0000 1.10573
\(266\) 8.00000 0.490511
\(267\) 6.00000 0.367194
\(268\) 2.00000 0.122169
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −12.0000 −0.730297
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) −3.00000 −0.181902
\(273\) −16.0000 −0.968364
\(274\) 3.00000 0.181237
\(275\) −24.0000 −1.44725
\(276\) 12.0000 0.722315
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) 2.00000 0.119952
\(279\) −2.00000 −0.119737
\(280\) 12.0000 0.717137
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 12.0000 0.714590
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −12.0000 −0.712069
\(285\) −12.0000 −0.710819
\(286\) 12.0000 0.709575
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 9.00000 0.528498
\(291\) −26.0000 −1.52415
\(292\) −10.0000 −0.585206
\(293\) 3.00000 0.175262 0.0876309 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) −18.0000 −1.04978
\(295\) 0 0
\(296\) 0 0
\(297\) −24.0000 −1.39262
\(298\) −21.0000 −1.21650
\(299\) 12.0000 0.693978
\(300\) −8.00000 −0.461880
\(301\) −16.0000 −0.922225
\(302\) −22.0000 −1.26596
\(303\) −6.00000 −0.344691
\(304\) −2.00000 −0.114708
\(305\) −3.00000 −0.171780
\(306\) −3.00000 −0.171499
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 24.0000 1.36753
\(309\) −8.00000 −0.455104
\(310\) 6.00000 0.340777
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 4.00000 0.226455
\(313\) 13.0000 0.734803 0.367402 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) 11.0000 0.620766
\(315\) 12.0000 0.676123
\(316\) −14.0000 −0.787562
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 12.0000 0.672927
\(319\) 18.0000 1.00781
\(320\) −3.00000 −0.167705
\(321\) 24.0000 1.33955
\(322\) 24.0000 1.33747
\(323\) 6.00000 0.333849
\(324\) −11.0000 −0.611111
\(325\) −8.00000 −0.443760
\(326\) 16.0000 0.886158
\(327\) −2.00000 −0.110600
\(328\) 3.00000 0.165647
\(329\) 24.0000 1.32316
\(330\) −36.0000 −1.98173
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) −6.00000 −0.327815
\(336\) 8.00000 0.436436
\(337\) −25.0000 −1.36184 −0.680918 0.732359i \(-0.738419\pi\)
−0.680918 + 0.732359i \(0.738419\pi\)
\(338\) −9.00000 −0.489535
\(339\) −36.0000 −1.95525
\(340\) 9.00000 0.488094
\(341\) 12.0000 0.649836
\(342\) −2.00000 −0.108148
\(343\) −8.00000 −0.431959
\(344\) 4.00000 0.215666
\(345\) −36.0000 −1.93817
\(346\) −21.0000 −1.12897
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 6.00000 0.321634
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) −16.0000 −0.855236
\(351\) −8.00000 −0.427008
\(352\) −6.00000 −0.319801
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) 36.0000 1.91068
\(356\) −3.00000 −0.159000
\(357\) −24.0000 −1.27021
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −3.00000 −0.158114
\(361\) −15.0000 −0.789474
\(362\) −1.00000 −0.0525588
\(363\) −50.0000 −2.62432
\(364\) 8.00000 0.419314
\(365\) 30.0000 1.57027
\(366\) −2.00000 −0.104542
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −6.00000 −0.312772
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 4.00000 0.207390
\(373\) −25.0000 −1.29445 −0.647225 0.762299i \(-0.724071\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 18.0000 0.930758
\(375\) −6.00000 −0.309839
\(376\) −6.00000 −0.309426
\(377\) 6.00000 0.309016
\(378\) −16.0000 −0.822951
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 6.00000 0.307794
\(381\) 20.0000 1.02463
\(382\) 6.00000 0.306987
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −2.00000 −0.102062
\(385\) −72.0000 −3.66946
\(386\) −11.0000 −0.559885
\(387\) 4.00000 0.203331
\(388\) 13.0000 0.659975
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) −12.0000 −0.607644
\(391\) 18.0000 0.910299
\(392\) 9.00000 0.454569
\(393\) 36.0000 1.81596
\(394\) 3.00000 0.151138
\(395\) 42.0000 2.11325
\(396\) −6.00000 −0.301511
\(397\) 35.0000 1.75660 0.878300 0.478110i \(-0.158678\pi\)
0.878300 + 0.478110i \(0.158678\pi\)
\(398\) −8.00000 −0.401004
\(399\) −16.0000 −0.801002
\(400\) 4.00000 0.200000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −4.00000 −0.199502
\(403\) 4.00000 0.199254
\(404\) 3.00000 0.149256
\(405\) 33.0000 1.63978
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) −9.00000 −0.444478
\(411\) −6.00000 −0.295958
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −18.0000 −0.883585
\(416\) −2.00000 −0.0980581
\(417\) −4.00000 −0.195881
\(418\) 12.0000 0.586939
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −24.0000 −1.17108
\(421\) 37.0000 1.80327 0.901635 0.432498i \(-0.142368\pi\)
0.901635 + 0.432498i \(0.142368\pi\)
\(422\) −16.0000 −0.778868
\(423\) −6.00000 −0.291730
\(424\) −6.00000 −0.291386
\(425\) −12.0000 −0.582086
\(426\) 24.0000 1.16280
\(427\) −4.00000 −0.193574
\(428\) −12.0000 −0.580042
\(429\) −24.0000 −1.15873
\(430\) −12.0000 −0.578691
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 4.00000 0.192450
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 8.00000 0.384012
\(435\) −18.0000 −0.863034
\(436\) 1.00000 0.0478913
\(437\) 12.0000 0.574038
\(438\) 20.0000 0.955637
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 18.0000 0.858116
\(441\) 9.00000 0.428571
\(442\) 6.00000 0.285391
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 9.00000 0.426641
\(446\) 14.0000 0.662919
\(447\) 42.0000 1.98653
\(448\) −4.00000 −0.188982
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 4.00000 0.188562
\(451\) −18.0000 −0.847587
\(452\) 18.0000 0.846649
\(453\) 44.0000 2.06730
\(454\) 18.0000 0.844782
\(455\) −24.0000 −1.12514
\(456\) 4.00000 0.187317
\(457\) 37.0000 1.73079 0.865393 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) 11.0000 0.513996
\(459\) −12.0000 −0.560112
\(460\) 18.0000 0.839254
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) −48.0000 −2.23316
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −3.00000 −0.139272
\(465\) −12.0000 −0.556487
\(466\) −9.00000 −0.416917
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) 18.0000 0.830278
\(471\) −22.0000 −1.01371
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 28.0000 1.28608
\(475\) −8.00000 −0.367065
\(476\) 12.0000 0.550019
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 6.00000 0.273861
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) −48.0000 −2.18408
\(484\) 25.0000 1.13636
\(485\) −39.0000 −1.77090
\(486\) 10.0000 0.453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 1.00000 0.0452679
\(489\) −32.0000 −1.44709
\(490\) −27.0000 −1.21974
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) −6.00000 −0.270501
\(493\) 9.00000 0.405340
\(494\) 4.00000 0.179969
\(495\) 18.0000 0.809040
\(496\) −2.00000 −0.0898027
\(497\) 48.0000 2.15309
\(498\) −12.0000 −0.537733
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 3.00000 0.134164
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) −4.00000 −0.178174
\(505\) −9.00000 −0.400495
\(506\) 36.0000 1.60040
\(507\) 18.0000 0.799408
\(508\) −10.0000 −0.443678
\(509\) 27.0000 1.19675 0.598377 0.801215i \(-0.295813\pi\)
0.598377 + 0.801215i \(0.295813\pi\)
\(510\) −18.0000 −0.797053
\(511\) 40.0000 1.76950
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) −27.0000 −1.19092
\(515\) −12.0000 −0.528783
\(516\) −8.00000 −0.352180
\(517\) 36.0000 1.58328
\(518\) 0 0
\(519\) 42.0000 1.84360
\(520\) 6.00000 0.263117
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −3.00000 −0.131306
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −18.0000 −0.786334
\(525\) 32.0000 1.39659
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 12.0000 0.522233
\(529\) 13.0000 0.565217
\(530\) 18.0000 0.781870
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −6.00000 −0.259889
\(534\) 6.00000 0.259645
\(535\) 36.0000 1.55642
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) −54.0000 −2.32594
\(540\) −12.0000 −0.516398
\(541\) −35.0000 −1.50477 −0.752384 0.658725i \(-0.771096\pi\)
−0.752384 + 0.658725i \(0.771096\pi\)
\(542\) −10.0000 −0.429537
\(543\) 2.00000 0.0858282
\(544\) −3.00000 −0.128624
\(545\) −3.00000 −0.128506
\(546\) −16.0000 −0.684737
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 3.00000 0.128154
\(549\) 1.00000 0.0426790
\(550\) −24.0000 −1.02336
\(551\) 6.00000 0.255609
\(552\) 12.0000 0.510754
\(553\) 56.0000 2.38136
\(554\) 1.00000 0.0424859
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 33.0000 1.39825 0.699127 0.714997i \(-0.253572\pi\)
0.699127 + 0.714997i \(0.253572\pi\)
\(558\) −2.00000 −0.0846668
\(559\) −8.00000 −0.338364
\(560\) 12.0000 0.507093
\(561\) −36.0000 −1.51992
\(562\) −15.0000 −0.632737
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 12.0000 0.505291
\(565\) −54.0000 −2.27180
\(566\) 16.0000 0.672530
\(567\) 44.0000 1.84783
\(568\) −12.0000 −0.503509
\(569\) −3.00000 −0.125767 −0.0628833 0.998021i \(-0.520030\pi\)
−0.0628833 + 0.998021i \(0.520030\pi\)
\(570\) −12.0000 −0.502625
\(571\) 38.0000 1.59025 0.795125 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) 12.0000 0.501745
\(573\) −12.0000 −0.501307
\(574\) −12.0000 −0.500870
\(575\) −24.0000 −1.00087
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −8.00000 −0.332756
\(579\) 22.0000 0.914289
\(580\) 9.00000 0.373705
\(581\) −24.0000 −0.995688
\(582\) −26.0000 −1.07773
\(583\) 36.0000 1.49097
\(584\) −10.0000 −0.413803
\(585\) 6.00000 0.248069
\(586\) 3.00000 0.123929
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) −18.0000 −0.742307
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 27.0000 1.10876 0.554379 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) −24.0000 −0.984732
\(595\) −36.0000 −1.47586
\(596\) −21.0000 −0.860194
\(597\) 16.0000 0.654836
\(598\) 12.0000 0.490716
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) −8.00000 −0.326599
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) −16.0000 −0.652111
\(603\) 2.00000 0.0814463
\(604\) −22.0000 −0.895167
\(605\) −75.0000 −3.04918
\(606\) −6.00000 −0.243733
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) −2.00000 −0.0811107
\(609\) −24.0000 −0.972529
\(610\) −3.00000 −0.121466
\(611\) 12.0000 0.485468
\(612\) −3.00000 −0.121268
\(613\) −25.0000 −1.00974 −0.504870 0.863195i \(-0.668460\pi\)
−0.504870 + 0.863195i \(0.668460\pi\)
\(614\) −28.0000 −1.12999
\(615\) 18.0000 0.725830
\(616\) 24.0000 0.966988
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −8.00000 −0.321807
\(619\) −40.0000 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 6.00000 0.240966
\(621\) −24.0000 −0.963087
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 4.00000 0.160128
\(625\) −29.0000 −1.16000
\(626\) 13.0000 0.519584
\(627\) −24.0000 −0.958468
\(628\) 11.0000 0.438948
\(629\) 0 0
\(630\) 12.0000 0.478091
\(631\) 10.0000 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) −14.0000 −0.556890
\(633\) 32.0000 1.27189
\(634\) −21.0000 −0.834017
\(635\) 30.0000 1.19051
\(636\) 12.0000 0.475831
\(637\) −18.0000 −0.713186
\(638\) 18.0000 0.712627
\(639\) −12.0000 −0.474713
\(640\) −3.00000 −0.118585
\(641\) −21.0000 −0.829450 −0.414725 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) 24.0000 0.947204
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 24.0000 0.945732
\(645\) 24.0000 0.944999
\(646\) 6.00000 0.236067
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) −8.00000 −0.313786
\(651\) −16.0000 −0.627089
\(652\) 16.0000 0.626608
\(653\) −3.00000 −0.117399 −0.0586995 0.998276i \(-0.518695\pi\)
−0.0586995 + 0.998276i \(0.518695\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 54.0000 2.10995
\(656\) 3.00000 0.117130
\(657\) −10.0000 −0.390137
\(658\) 24.0000 0.935617
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −36.0000 −1.40130
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 4.00000 0.155464
\(663\) −12.0000 −0.466041
\(664\) 6.00000 0.232845
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) −12.0000 −0.464294
\(669\) −28.0000 −1.08254
\(670\) −6.00000 −0.231800
\(671\) −6.00000 −0.231627
\(672\) 8.00000 0.308607
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −25.0000 −0.962964
\(675\) 16.0000 0.615840
\(676\) −9.00000 −0.346154
\(677\) 3.00000 0.115299 0.0576497 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) −36.0000 −1.38257
\(679\) −52.0000 −1.99558
\(680\) 9.00000 0.345134
\(681\) −36.0000 −1.37952
\(682\) 12.0000 0.459504
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −9.00000 −0.343872
\(686\) −8.00000 −0.305441
\(687\) −22.0000 −0.839352
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) −36.0000 −1.37050
\(691\) 50.0000 1.90209 0.951045 0.309053i \(-0.100012\pi\)
0.951045 + 0.309053i \(0.100012\pi\)
\(692\) −21.0000 −0.798300
\(693\) 24.0000 0.911685
\(694\) −6.00000 −0.227757
\(695\) −6.00000 −0.227593
\(696\) 6.00000 0.227429
\(697\) −9.00000 −0.340899
\(698\) −1.00000 −0.0378506
\(699\) 18.0000 0.680823
\(700\) −16.0000 −0.604743
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −8.00000 −0.301941
\(703\) 0 0
\(704\) −6.00000 −0.226134
\(705\) −36.0000 −1.35584
\(706\) 21.0000 0.790345
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 36.0000 1.35106
\(711\) −14.0000 −0.525041
\(712\) −3.00000 −0.112430
\(713\) 12.0000 0.449404
\(714\) −24.0000 −0.898177
\(715\) −36.0000 −1.34632
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) −3.00000 −0.111803
\(721\) −16.0000 −0.595871
\(722\) −15.0000 −0.558242
\(723\) 28.0000 1.04133
\(724\) −1.00000 −0.0371647
\(725\) −12.0000 −0.445669
\(726\) −50.0000 −1.85567
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 8.00000 0.296500
\(729\) 13.0000 0.481481
\(730\) 30.0000 1.11035
\(731\) −12.0000 −0.443836
\(732\) −2.00000 −0.0739221
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −4.00000 −0.147643
\(735\) 54.0000 1.99182
\(736\) −6.00000 −0.221163
\(737\) −12.0000 −0.442026
\(738\) 3.00000 0.110432
\(739\) 38.0000 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 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
\(745\) 63.0000 2.30814
\(746\) −25.0000 −0.915315
\(747\) 6.00000 0.219529
\(748\) 18.0000 0.658145
\(749\) 48.0000 1.75388
\(750\) −6.00000 −0.219089
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) 66.0000 2.40199
\(756\) −16.0000 −0.581914
\(757\) −35.0000 −1.27210 −0.636048 0.771649i \(-0.719432\pi\)
−0.636048 + 0.771649i \(0.719432\pi\)
\(758\) −16.0000 −0.581146
\(759\) −72.0000 −2.61343
\(760\) 6.00000 0.217643
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 20.0000 0.724524
\(763\) −4.00000 −0.144810
\(764\) 6.00000 0.217072
\(765\) 9.00000 0.325396
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) −72.0000 −2.59470
\(771\) 54.0000 1.94476
\(772\) −11.0000 −0.395899
\(773\) −21.0000 −0.755318 −0.377659 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) 4.00000 0.143777
\(775\) −8.00000 −0.287368
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) 9.00000 0.322666
\(779\) −6.00000 −0.214972
\(780\) −12.0000 −0.429669
\(781\) 72.0000 2.57636
\(782\) 18.0000 0.643679
\(783\) −12.0000 −0.428845
\(784\) 9.00000 0.321429
\(785\) −33.0000 −1.17782
\(786\) 36.0000 1.28408
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 3.00000 0.106871
\(789\) 0 0
\(790\) 42.0000 1.49429
\(791\) −72.0000 −2.56003
\(792\) −6.00000 −0.213201
\(793\) −2.00000 −0.0710221
\(794\) 35.0000 1.24210
\(795\) −36.0000 −1.27679
\(796\) −8.00000 −0.283552
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) −16.0000 −0.566394
\(799\) 18.0000 0.636794
\(800\) 4.00000 0.141421
\(801\) −3.00000 −0.106000
\(802\) −6.00000 −0.211867
\(803\) 60.0000 2.11735
\(804\) −4.00000 −0.141069
\(805\) −72.0000 −2.53767
\(806\) 4.00000 0.140894
\(807\) −36.0000 −1.26726
\(808\) 3.00000 0.105540
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 33.0000 1.15950
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 12.0000 0.421117
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) −48.0000 −1.68137
\(816\) 6.00000 0.210042
\(817\) −8.00000 −0.279885
\(818\) −11.0000 −0.384606
\(819\) 8.00000 0.279543
\(820\) −9.00000 −0.314294
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) −6.00000 −0.209274
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) 4.00000 0.139347
\(825\) 48.0000 1.67115
\(826\) 0 0
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) −6.00000 −0.208514
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) −18.0000 −0.624789
\(831\) −2.00000 −0.0693792
\(832\) −2.00000 −0.0693375
\(833\) −27.0000 −0.935495
\(834\) −4.00000 −0.138509
\(835\) 36.0000 1.24583
\(836\) 12.0000 0.415029
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) −24.0000 −0.828079
\(841\) −20.0000 −0.689655
\(842\) 37.0000 1.27510
\(843\) 30.0000 1.03325
\(844\) −16.0000 −0.550743
\(845\) 27.0000 0.928828
\(846\) −6.00000 −0.206284
\(847\) −100.000 −3.43604
\(848\) −6.00000 −0.206041
\(849\) −32.0000 −1.09824
\(850\) −12.0000 −0.411597
\(851\) 0 0
\(852\) 24.0000 0.822226
\(853\) 37.0000 1.26686 0.633428 0.773802i \(-0.281647\pi\)
0.633428 + 0.773802i \(0.281647\pi\)
\(854\) −4.00000 −0.136877
\(855\) 6.00000 0.205196
\(856\) −12.0000 −0.410152
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) −24.0000 −0.819346
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) −12.0000 −0.409197
\(861\) 24.0000 0.817918
\(862\) −6.00000 −0.204361
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 4.00000 0.136083
\(865\) 63.0000 2.14206
\(866\) −25.0000 −0.849535
\(867\) 16.0000 0.543388
\(868\) 8.00000 0.271538
\(869\) 84.0000 2.84950
\(870\) −18.0000 −0.610257
\(871\) −4.00000 −0.135535
\(872\) 1.00000 0.0338643
\(873\) 13.0000 0.439983
\(874\) 12.0000 0.405906
\(875\) −12.0000 −0.405674
\(876\) 20.0000 0.675737
\(877\) −25.0000 −0.844190 −0.422095 0.906552i \(-0.638705\pi\)
−0.422095 + 0.906552i \(0.638705\pi\)
\(878\) −32.0000 −1.07995
\(879\) −6.00000 −0.202375
\(880\) 18.0000 0.606780
\(881\) 51.0000 1.71823 0.859117 0.511780i \(-0.171014\pi\)
0.859117 + 0.511780i \(0.171014\pi\)
\(882\) 9.00000 0.303046
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 0 0
\(889\) 40.0000 1.34156
\(890\) 9.00000 0.301681
\(891\) 66.0000 2.21108
\(892\) 14.0000 0.468755
\(893\) 12.0000 0.401565
\(894\) 42.0000 1.40469
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) −24.0000 −0.801337
\(898\) −6.00000 −0.200223
\(899\) 6.00000 0.200111
\(900\) 4.00000 0.133333
\(901\) 18.0000 0.599667
\(902\) −18.0000 −0.599334
\(903\) 32.0000 1.06489
\(904\) 18.0000 0.598671
\(905\) 3.00000 0.0997234
\(906\) 44.0000 1.46180
\(907\) 58.0000 1.92586 0.962929 0.269754i \(-0.0869425\pi\)
0.962929 + 0.269754i \(0.0869425\pi\)
\(908\) 18.0000 0.597351
\(909\) 3.00000 0.0995037
\(910\) −24.0000 −0.795592
\(911\) 42.0000 1.39152 0.695761 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(912\) 4.00000 0.132453
\(913\) −36.0000 −1.19143
\(914\) 37.0000 1.22385
\(915\) 6.00000 0.198354
\(916\) 11.0000 0.363450
\(917\) 72.0000 2.37765
\(918\) −12.0000 −0.396059
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 18.0000 0.593442
\(921\) 56.0000 1.84526
\(922\) 30.0000 0.987997
\(923\) 24.0000 0.789970
\(924\) −48.0000 −1.57908
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 4.00000 0.131377
\(928\) −3.00000 −0.0984798
\(929\) −9.00000 −0.295280 −0.147640 0.989041i \(-0.547168\pi\)
−0.147640 + 0.989041i \(0.547168\pi\)
\(930\) −12.0000 −0.393496
\(931\) −18.0000 −0.589926
\(932\) −9.00000 −0.294805
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) −54.0000 −1.76599
\(936\) −2.00000 −0.0653720
\(937\) −1.00000 −0.0326686 −0.0163343 0.999867i \(-0.505200\pi\)
−0.0163343 + 0.999867i \(0.505200\pi\)
\(938\) −8.00000 −0.261209
\(939\) −26.0000 −0.848478
\(940\) 18.0000 0.587095
\(941\) 27.0000 0.880175 0.440087 0.897955i \(-0.354947\pi\)
0.440087 + 0.897955i \(0.354947\pi\)
\(942\) −22.0000 −0.716799
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) 48.0000 1.56144
\(946\) −24.0000 −0.780307
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) 28.0000 0.909398
\(949\) 20.0000 0.649227
\(950\) −8.00000 −0.259554
\(951\) 42.0000 1.36194
\(952\) 12.0000 0.388922
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −6.00000 −0.194257
\(955\) −18.0000 −0.582466
\(956\) 0 0
\(957\) −36.0000 −1.16371
\(958\) −24.0000 −0.775405
\(959\) −12.0000 −0.387500
\(960\) 6.00000 0.193649
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −14.0000 −0.450910
\(965\) 33.0000 1.06231
\(966\) −48.0000 −1.54437
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 25.0000 0.803530
\(969\) −12.0000 −0.385496
\(970\) −39.0000 −1.25221
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) 10.0000 0.320750
\(973\) −8.00000 −0.256468
\(974\) −2.00000 −0.0640841
\(975\) 16.0000 0.512410
\(976\) 1.00000 0.0320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −32.0000 −1.02325
\(979\) 18.0000 0.575282
\(980\) −27.0000 −0.862483
\(981\) 1.00000 0.0319275
\(982\) 6.00000 0.191468
\(983\) −30.0000 −0.956851 −0.478426 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) −6.00000 −0.191273
\(985\) −9.00000 −0.286764
\(986\) 9.00000 0.286618
\(987\) −48.0000 −1.52786
\(988\) 4.00000 0.127257
\(989\) −24.0000 −0.763156
\(990\) 18.0000 0.572078
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −8.00000 −0.253872
\(994\) 48.0000 1.52247
\(995\) 24.0000 0.760851
\(996\) −12.0000 −0.380235
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −14.0000 −0.443162
\(999\) 0 0
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 2738.2.a.c.1.1 1
37.11 even 6 74.2.c.b.47.1 2
37.27 even 6 74.2.c.b.63.1 yes 2
37.36 even 2 2738.2.a.a.1.1 1
111.11 odd 6 666.2.f.d.343.1 2
111.101 odd 6 666.2.f.d.433.1 2
148.11 odd 6 592.2.i.a.417.1 2
148.27 odd 6 592.2.i.a.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.c.b.47.1 2 37.11 even 6
74.2.c.b.63.1 yes 2 37.27 even 6
592.2.i.a.417.1 2 148.11 odd 6
592.2.i.a.433.1 2 148.27 odd 6
666.2.f.d.343.1 2 111.11 odd 6
666.2.f.d.433.1 2 111.101 odd 6
2738.2.a.a.1.1 1 37.36 even 2
2738.2.a.c.1.1 1 1.1 even 1 trivial