Properties

Label 471.2.a.a.1.1
Level $471$
Weight $2$
Character 471.1
Self dual yes
Analytic conductor $3.761$
Analytic rank $1$
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] = [471,2,Mod(1,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, 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("471.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.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: \(3.76095393520\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 471.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +1.00000 q^{13} -3.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} -9.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} -2.00000 q^{30} -2.00000 q^{31} -5.00000 q^{32} +3.00000 q^{34} -6.00000 q^{35} -1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} -1.00000 q^{39} -6.00000 q^{40} -2.00000 q^{41} +3.00000 q^{42} +1.00000 q^{43} -2.00000 q^{45} +9.00000 q^{46} +1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +3.00000 q^{51} -1.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +9.00000 q^{56} +2.00000 q^{57} -1.00000 q^{59} -2.00000 q^{60} +8.00000 q^{61} +2.00000 q^{62} +3.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} +3.00000 q^{68} +9.00000 q^{69} +6.00000 q^{70} -12.0000 q^{71} +3.00000 q^{72} -14.0000 q^{73} -1.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} +1.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +3.00000 q^{84} +6.00000 q^{85} -1.00000 q^{86} -13.0000 q^{89} +2.00000 q^{90} +3.00000 q^{91} +9.00000 q^{92} +2.00000 q^{93} +4.00000 q^{95} +5.00000 q^{96} -2.00000 q^{98} +O(q^{100})\)

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 \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −3.00000 −0.801784
\(15\) 2.00000 0.516398
\(16\) −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 \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −3.00000 −0.566947
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −2.00000 −0.365148
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −6.00000 −1.01419
\(36\) −1.00000 −0.166667
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 2.00000 0.324443
\(39\) −1.00000 −0.160128
\(40\) −6.00000 −0.948683
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 3.00000 0.462910
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 9.00000 1.32698
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 3.00000 0.420084
\(52\) −1.00000 −0.138675
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 9.00000 1.20268
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) −2.00000 −0.258199
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 2.00000 0.254000
\(63\) 3.00000 0.377964
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 3.00000 0.363803
\(69\) 9.00000 1.08347
\(70\) 6.00000 0.717137
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 3.00000 0.353553
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 3.00000 0.327327
\(85\) 6.00000 0.650791
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) 0 0
\(89\) −13.0000 −1.37800 −0.688999 0.724763i \(-0.741949\pi\)
−0.688999 + 0.724763i \(0.741949\pi\)
\(90\) 2.00000 0.210819
\(91\) 3.00000 0.314485
\(92\) 9.00000 0.938315
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 5.00000 0.510310
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −13.0000 −1.29355 −0.646774 0.762682i \(-0.723882\pi\)
−0.646774 + 0.762682i \(0.723882\pi\)
\(102\) −3.00000 −0.297044
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 3.00000 0.294174
\(105\) 6.00000 0.585540
\(106\) 6.00000 0.582772
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.00000 −0.287348 −0.143674 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) −3.00000 −0.283473
\(113\) 7.00000 0.658505 0.329252 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) −2.00000 −0.187317
\(115\) 18.0000 1.67851
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 1.00000 0.0920575
\(119\) −9.00000 −0.825029
\(120\) 6.00000 0.547723
\(121\) −11.0000 −1.00000
\(122\) −8.00000 −0.724286
\(123\) 2.00000 0.180334
\(124\) 2.00000 0.179605
\(125\) 12.0000 1.07331
\(126\) −3.00000 −0.267261
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 3.00000 0.265165
\(129\) −1.00000 −0.0880451
\(130\) 2.00000 0.175412
\(131\) 5.00000 0.436852 0.218426 0.975854i \(-0.429908\pi\)
0.218426 + 0.975854i \(0.429908\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) −2.00000 −0.172774
\(135\) 2.00000 0.172133
\(136\) −9.00000 −0.771744
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −9.00000 −0.766131
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 6.00000 0.507093
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) −2.00000 −0.164957
\(148\) −1.00000 −0.0821995
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) −6.00000 −0.486664
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 1.00000 0.0800641
\(157\) −1.00000 −0.0798087
\(158\) 8.00000 0.636446
\(159\) 6.00000 0.475831
\(160\) 10.0000 0.790569
\(161\) −27.0000 −2.12790
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) −9.00000 −0.694365
\(169\) −12.0000 −0.923077
\(170\) −6.00000 −0.460179
\(171\) −2.00000 −0.152944
\(172\) −1.00000 −0.0762493
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) 1.00000 0.0751646
\(178\) 13.0000 0.974391
\(179\) 5.00000 0.373718 0.186859 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(180\) 2.00000 0.149071
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) −3.00000 −0.222375
\(183\) −8.00000 −0.591377
\(184\) −27.0000 −1.99047
\(185\) −2.00000 −0.147043
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) 0 0
\(189\) −3.00000 −0.218218
\(190\) −4.00000 −0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −7.00000 −0.505181
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) −2.00000 −0.142857
\(197\) 1.00000 0.0712470 0.0356235 0.999365i \(-0.488658\pi\)
0.0356235 + 0.999365i \(0.488658\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) −3.00000 −0.212132
\(201\) −2.00000 −0.141069
\(202\) 13.0000 0.914677
\(203\) 0 0
\(204\) −3.00000 −0.210042
\(205\) 4.00000 0.279372
\(206\) 8.00000 0.557386
\(207\) −9.00000 −0.625543
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) −6.00000 −0.414039
\(211\) 3.00000 0.206529 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 6.00000 0.412082
\(213\) 12.0000 0.822226
\(214\) −9.00000 −0.615227
\(215\) −2.00000 −0.136399
\(216\) −3.00000 −0.204124
\(217\) −6.00000 −0.407307
\(218\) 3.00000 0.203186
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 1.00000 0.0671156
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) −15.0000 −1.00223
\(225\) −1.00000 −0.0666667
\(226\) −7.00000 −0.465633
\(227\) −15.0000 −0.995585 −0.497792 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) −2.00000 −0.132453
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) −18.0000 −1.18688
\(231\) 0 0
\(232\) 0 0
\(233\) 25.0000 1.63780 0.818902 0.573933i \(-0.194583\pi\)
0.818902 + 0.573933i \(0.194583\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 1.00000 0.0650945
\(237\) 8.00000 0.519656
\(238\) 9.00000 0.583383
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) −2.00000 −0.129099
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 11.0000 0.707107
\(243\) −1.00000 −0.0641500
\(244\) −8.00000 −0.512148
\(245\) −4.00000 −0.255551
\(246\) −2.00000 −0.127515
\(247\) −2.00000 −0.127257
\(248\) −6.00000 −0.381000
\(249\) −4.00000 −0.253490
\(250\) −12.0000 −0.758947
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) −18.0000 −1.12942
\(255\) −6.00000 −0.375735
\(256\) −17.0000 −1.06250
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 1.00000 0.0622573
\(259\) 3.00000 0.186411
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −5.00000 −0.308901
\(263\) 26.0000 1.60323 0.801614 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 6.00000 0.367884
\(267\) 13.0000 0.795587
\(268\) −2.00000 −0.122169
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) −2.00000 −0.121716
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 3.00000 0.181902
\(273\) −3.00000 −0.181568
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −9.00000 −0.541736
\(277\) 29.0000 1.74244 0.871221 0.490892i \(-0.163329\pi\)
0.871221 + 0.490892i \(0.163329\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) −18.0000 −1.07571
\(281\) 7.00000 0.417585 0.208792 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) 0 0
\(283\) 30.0000 1.78331 0.891657 0.452711i \(-0.149543\pi\)
0.891657 + 0.452711i \(0.149543\pi\)
\(284\) 12.0000 0.712069
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) −5.00000 −0.294628
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 2.00000 0.116642
\(295\) 2.00000 0.116445
\(296\) 3.00000 0.174371
\(297\) 0 0
\(298\) 0 0
\(299\) −9.00000 −0.520483
\(300\) −1.00000 −0.0577350
\(301\) 3.00000 0.172917
\(302\) −19.0000 −1.09333
\(303\) 13.0000 0.746830
\(304\) 2.00000 0.114708
\(305\) −16.0000 −0.916157
\(306\) 3.00000 0.171499
\(307\) 23.0000 1.31268 0.656340 0.754466i \(-0.272104\pi\)
0.656340 + 0.754466i \(0.272104\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) −4.00000 −0.227185
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −3.00000 −0.169842
\(313\) −27.0000 −1.52613 −0.763065 0.646322i \(-0.776306\pi\)
−0.763065 + 0.646322i \(0.776306\pi\)
\(314\) 1.00000 0.0564333
\(315\) −6.00000 −0.338062
\(316\) 8.00000 0.450035
\(317\) 5.00000 0.280828 0.140414 0.990093i \(-0.455157\pi\)
0.140414 + 0.990093i \(0.455157\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) −14.0000 −0.782624
\(321\) −9.00000 −0.502331
\(322\) 27.0000 1.50465
\(323\) 6.00000 0.333849
\(324\) −1.00000 −0.0555556
\(325\) −1.00000 −0.0554700
\(326\) 4.00000 0.221540
\(327\) 3.00000 0.165900
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −4.00000 −0.219529
\(333\) 1.00000 0.0547997
\(334\) −6.00000 −0.328305
\(335\) −4.00000 −0.218543
\(336\) 3.00000 0.163663
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 12.0000 0.652714
\(339\) −7.00000 −0.380188
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −15.0000 −0.809924
\(344\) 3.00000 0.161749
\(345\) −18.0000 −0.969087
\(346\) −6.00000 −0.322562
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) 27.0000 1.44528 0.722638 0.691226i \(-0.242929\pi\)
0.722638 + 0.691226i \(0.242929\pi\)
\(350\) 3.00000 0.160357
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −1.00000 −0.0531494
\(355\) 24.0000 1.27379
\(356\) 13.0000 0.688999
\(357\) 9.00000 0.476331
\(358\) −5.00000 −0.264258
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) −6.00000 −0.316228
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) 11.0000 0.577350
\(364\) −3.00000 −0.157243
\(365\) 28.0000 1.46559
\(366\) 8.00000 0.418167
\(367\) 25.0000 1.30499 0.652495 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) 9.00000 0.469157
\(369\) −2.00000 −0.104116
\(370\) 2.00000 0.103975
\(371\) −18.0000 −0.934513
\(372\) −2.00000 −0.103695
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 0 0
\(378\) 3.00000 0.154303
\(379\) −11.0000 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(380\) −4.00000 −0.205196
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 11.0000 0.562074 0.281037 0.959697i \(-0.409322\pi\)
0.281037 + 0.959697i \(0.409322\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) −2.00000 −0.101274
\(391\) 27.0000 1.36545
\(392\) 6.00000 0.303046
\(393\) −5.00000 −0.252217
\(394\) −1.00000 −0.0503793
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −2.00000 −0.100251
\(399\) 6.00000 0.300376
\(400\) 1.00000 0.0500000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 2.00000 0.0997509
\(403\) −2.00000 −0.0996271
\(404\) 13.0000 0.646774
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 9.00000 0.445566
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) −4.00000 −0.197546
\(411\) −10.0000 −0.493264
\(412\) 8.00000 0.394132
\(413\) −3.00000 −0.147620
\(414\) 9.00000 0.442326
\(415\) −8.00000 −0.392705
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) −6.00000 −0.292770
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) −3.00000 −0.146038
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) 3.00000 0.145521
\(426\) −12.0000 −0.581402
\(427\) 24.0000 1.16144
\(428\) −9.00000 −0.435031
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) 3.00000 0.143674
\(437\) 18.0000 0.861057
\(438\) −14.0000 −0.668946
\(439\) 37.0000 1.76591 0.882957 0.469454i \(-0.155549\pi\)
0.882957 + 0.469454i \(0.155549\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 3.00000 0.142695
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 1.00000 0.0474579
\(445\) 26.0000 1.23252
\(446\) 28.0000 1.32584
\(447\) 0 0
\(448\) 21.0000 0.992157
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −7.00000 −0.329252
\(453\) −19.0000 −0.892698
\(454\) 15.0000 0.703985
\(455\) −6.00000 −0.281284
\(456\) 6.00000 0.280976
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) 20.0000 0.934539
\(459\) 3.00000 0.140028
\(460\) −18.0000 −0.839254
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) 29.0000 1.34774 0.673872 0.738848i \(-0.264630\pi\)
0.673872 + 0.738848i \(0.264630\pi\)
\(464\) 0 0
\(465\) −4.00000 −0.185496
\(466\) −25.0000 −1.15810
\(467\) −26.0000 −1.20314 −0.601568 0.798821i \(-0.705457\pi\)
−0.601568 + 0.798821i \(0.705457\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) 1.00000 0.0460776
\(472\) −3.00000 −0.138086
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 2.00000 0.0917663
\(476\) 9.00000 0.412514
\(477\) −6.00000 −0.274721
\(478\) −16.0000 −0.731823
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) −10.0000 −0.456435
\(481\) 1.00000 0.0455961
\(482\) 18.0000 0.819878
\(483\) 27.0000 1.22854
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 24.0000 1.08643
\(489\) 4.00000 0.180886
\(490\) 4.00000 0.180702
\(491\) 27.0000 1.21849 0.609246 0.792981i \(-0.291472\pi\)
0.609246 + 0.792981i \(0.291472\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 0 0
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −36.0000 −1.61482
\(498\) 4.00000 0.179244
\(499\) −44.0000 −1.96971 −0.984855 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) −12.0000 −0.536656
\(501\) −6.00000 −0.268060
\(502\) −23.0000 −1.02654
\(503\) −15.0000 −0.668817 −0.334408 0.942428i \(-0.608537\pi\)
−0.334408 + 0.942428i \(0.608537\pi\)
\(504\) 9.00000 0.400892
\(505\) 26.0000 1.15698
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) −18.0000 −0.798621
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 6.00000 0.265684
\(511\) −42.0000 −1.85797
\(512\) 11.0000 0.486136
\(513\) 2.00000 0.0883022
\(514\) −14.0000 −0.617514
\(515\) 16.0000 0.705044
\(516\) 1.00000 0.0440225
\(517\) 0 0
\(518\) −3.00000 −0.131812
\(519\) −6.00000 −0.263371
\(520\) −6.00000 −0.263117
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −5.00000 −0.218426
\(525\) 3.00000 0.130931
\(526\) −26.0000 −1.13365
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) −12.0000 −0.521247
\(531\) −1.00000 −0.0433963
\(532\) 6.00000 0.260133
\(533\) −2.00000 −0.0866296
\(534\) −13.0000 −0.562565
\(535\) −18.0000 −0.778208
\(536\) 6.00000 0.259161
\(537\) −5.00000 −0.215766
\(538\) 24.0000 1.03471
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 7.00000 0.300676
\(543\) 20.0000 0.858282
\(544\) 15.0000 0.643120
\(545\) 6.00000 0.257012
\(546\) 3.00000 0.128388
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −10.0000 −0.427179
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 27.0000 1.14920
\(553\) −24.0000 −1.02058
\(554\) −29.0000 −1.23209
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 2.00000 0.0846668
\(559\) 1.00000 0.0422955
\(560\) 6.00000 0.253546
\(561\) 0 0
\(562\) −7.00000 −0.295277
\(563\) 15.0000 0.632175 0.316087 0.948730i \(-0.397631\pi\)
0.316087 + 0.948730i \(0.397631\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −30.0000 −1.26099
\(567\) 3.00000 0.125988
\(568\) −36.0000 −1.51053
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 4.00000 0.167542
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 9.00000 0.375326
\(576\) 7.00000 0.291667
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 8.00000 0.332756
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) −42.0000 −1.73797
\(585\) −2.00000 −0.0826898
\(586\) 4.00000 0.165238
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) 2.00000 0.0824786
\(589\) 4.00000 0.164817
\(590\) −2.00000 −0.0823387
\(591\) −1.00000 −0.0411345
\(592\) −1.00000 −0.0410997
\(593\) 35.0000 1.43728 0.718639 0.695383i \(-0.244765\pi\)
0.718639 + 0.695383i \(0.244765\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 0 0
\(597\) −2.00000 −0.0818546
\(598\) 9.00000 0.368037
\(599\) −39.0000 −1.59350 −0.796748 0.604311i \(-0.793448\pi\)
−0.796748 + 0.604311i \(0.793448\pi\)
\(600\) 3.00000 0.122474
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) −3.00000 −0.122271
\(603\) 2.00000 0.0814463
\(604\) −19.0000 −0.773099
\(605\) 22.0000 0.894427
\(606\) −13.0000 −0.528089
\(607\) 41.0000 1.66414 0.832069 0.554672i \(-0.187156\pi\)
0.832069 + 0.554672i \(0.187156\pi\)
\(608\) 10.0000 0.405554
\(609\) 0 0
\(610\) 16.0000 0.647821
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) −23.0000 −0.928204
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) 27.0000 1.08698 0.543490 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) −8.00000 −0.321807
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) −4.00000 −0.160644
\(621\) 9.00000 0.361158
\(622\) 18.0000 0.721734
\(623\) −39.0000 −1.56250
\(624\) 1.00000 0.0400320
\(625\) −19.0000 −0.760000
\(626\) 27.0000 1.07914
\(627\) 0 0
\(628\) 1.00000 0.0399043
\(629\) −3.00000 −0.119618
\(630\) 6.00000 0.239046
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) −24.0000 −0.954669
\(633\) −3.00000 −0.119239
\(634\) −5.00000 −0.198575
\(635\) −36.0000 −1.42862
\(636\) −6.00000 −0.237915
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) −6.00000 −0.237171
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 9.00000 0.355202
\(643\) −9.00000 −0.354925 −0.177463 0.984128i \(-0.556789\pi\)
−0.177463 + 0.984128i \(0.556789\pi\)
\(644\) 27.0000 1.06395
\(645\) 2.00000 0.0787499
\(646\) −6.00000 −0.236067
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) 6.00000 0.235159
\(652\) 4.00000 0.156652
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) −3.00000 −0.117309
\(655\) −10.0000 −0.390732
\(656\) 2.00000 0.0780869
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −33.0000 −1.28355 −0.641776 0.766892i \(-0.721802\pi\)
−0.641776 + 0.766892i \(0.721802\pi\)
\(662\) −10.0000 −0.388661
\(663\) 3.00000 0.116510
\(664\) 12.0000 0.465690
\(665\) 12.0000 0.465340
\(666\) −1.00000 −0.0387492
\(667\) 0 0
\(668\) −6.00000 −0.232147
\(669\) 28.0000 1.08254
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) 15.0000 0.578638
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) 12.0000 0.461538
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 7.00000 0.268833
\(679\) 0 0
\(680\) 18.0000 0.690268
\(681\) 15.0000 0.574801
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 2.00000 0.0764719
\(685\) −20.0000 −0.764161
\(686\) 15.0000 0.572703
\(687\) 20.0000 0.763048
\(688\) −1.00000 −0.0381246
\(689\) −6.00000 −0.228582
\(690\) 18.0000 0.685248
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) 0 0
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) −27.0000 −1.02197
\(699\) −25.0000 −0.945587
\(700\) 3.00000 0.113389
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 1.00000 0.0377426
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) −39.0000 −1.46675
\(708\) −1.00000 −0.0375823
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) −24.0000 −0.900704
\(711\) −8.00000 −0.300023
\(712\) −39.0000 −1.46159
\(713\) 18.0000 0.674105
\(714\) −9.00000 −0.336817
\(715\) 0 0
\(716\) −5.00000 −0.186859
\(717\) −16.0000 −0.597531
\(718\) 9.00000 0.335877
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 2.00000 0.0745356
\(721\) −24.0000 −0.893807
\(722\) 15.0000 0.558242
\(723\) 18.0000 0.669427
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 9.00000 0.333562
\(729\) 1.00000 0.0370370
\(730\) −28.0000 −1.03633
\(731\) −3.00000 −0.110959
\(732\) 8.00000 0.295689
\(733\) −10.0000 −0.369358 −0.184679 0.982799i \(-0.559125\pi\)
−0.184679 + 0.982799i \(0.559125\pi\)
\(734\) −25.0000 −0.922767
\(735\) 4.00000 0.147542
\(736\) 45.0000 1.65872
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 2.00000 0.0735215
\(741\) 2.00000 0.0734718
\(742\) 18.0000 0.660801
\(743\) −10.0000 −0.366864 −0.183432 0.983032i \(-0.558721\pi\)
−0.183432 + 0.983032i \(0.558721\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 27.0000 0.986559
\(750\) 12.0000 0.438178
\(751\) 41.0000 1.49611 0.748056 0.663636i \(-0.230988\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 0 0
\(753\) −23.0000 −0.838167
\(754\) 0 0
\(755\) −38.0000 −1.38296
\(756\) 3.00000 0.109109
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 11.0000 0.399538
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 18.0000 0.652071
\(763\) −9.00000 −0.325822
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) −11.0000 −0.397446
\(767\) −1.00000 −0.0361079
\(768\) 17.0000 0.613435
\(769\) −39.0000 −1.40638 −0.703188 0.711004i \(-0.748241\pi\)
−0.703188 + 0.711004i \(0.748241\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 14.0000 0.503871
\(773\) 19.0000 0.683383 0.341691 0.939812i \(-0.389000\pi\)
0.341691 + 0.939812i \(0.389000\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) −3.00000 −0.107624
\(778\) 9.00000 0.322666
\(779\) 4.00000 0.143315
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) −27.0000 −0.965518
\(783\) 0 0
\(784\) −2.00000 −0.0714286
\(785\) 2.00000 0.0713831
\(786\) 5.00000 0.178344
\(787\) −27.0000 −0.962446 −0.481223 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(788\) −1.00000 −0.0356235
\(789\) −26.0000 −0.925625
\(790\) −16.0000 −0.569254
\(791\) 21.0000 0.746674
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) −8.00000 −0.283909
\(795\) −12.0000 −0.425596
\(796\) −2.00000 −0.0708881
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −6.00000 −0.212398
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) −13.0000 −0.459332
\(802\) 12.0000 0.423735
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) 54.0000 1.90325
\(806\) 2.00000 0.0704470
\(807\) 24.0000 0.844840
\(808\) −39.0000 −1.37202
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 2.00000 0.0702728
\(811\) 13.0000 0.456492 0.228246 0.973604i \(-0.426701\pi\)
0.228246 + 0.973604i \(0.426701\pi\)
\(812\) 0 0
\(813\) 7.00000 0.245501
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) −3.00000 −0.105021
\(817\) −2.00000 −0.0699711
\(818\) −4.00000 −0.139857
\(819\) 3.00000 0.104828
\(820\) −4.00000 −0.139686
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 10.0000 0.348790
\(823\) −27.0000 −0.941161 −0.470580 0.882357i \(-0.655955\pi\)
−0.470580 + 0.882357i \(0.655955\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) −42.0000 −1.46048 −0.730242 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) 9.00000 0.312772
\(829\) 13.0000 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) 8.00000 0.277684
\(831\) −29.0000 −1.00600
\(832\) 7.00000 0.242681
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 20.0000 0.690889
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 18.0000 0.621059
\(841\) −29.0000 −1.00000
\(842\) −32.0000 −1.10279
\(843\) −7.00000 −0.241093
\(844\) −3.00000 −0.103264
\(845\) 24.0000 0.825625
\(846\) 0 0
\(847\) −33.0000 −1.13389
\(848\) 6.00000 0.206041
\(849\) −30.0000 −1.02960
\(850\) −3.00000 −0.102899
\(851\) −9.00000 −0.308516
\(852\) −12.0000 −0.411113
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) −24.0000 −0.821263
\(855\) 4.00000 0.136797
\(856\) 27.0000 0.922841
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −13.0000 −0.443554 −0.221777 0.975097i \(-0.571186\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) 2.00000 0.0681994
\(861\) 6.00000 0.204479
\(862\) 10.0000 0.340601
\(863\) −3.00000 −0.102121 −0.0510606 0.998696i \(-0.516260\pi\)
−0.0510606 + 0.998696i \(0.516260\pi\)
\(864\) 5.00000 0.170103
\(865\) −12.0000 −0.408012
\(866\) 26.0000 0.883516
\(867\) 8.00000 0.271694
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) −9.00000 −0.304778
\(873\) 0 0
\(874\) −18.0000 −0.608859
\(875\) 36.0000 1.21702
\(876\) −14.0000 −0.473016
\(877\) −56.0000 −1.89099 −0.945493 0.325643i \(-0.894419\pi\)
−0.945493 + 0.325643i \(0.894419\pi\)
\(878\) −37.0000 −1.24869
\(879\) 4.00000 0.134917
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 25.0000 0.841317 0.420658 0.907219i \(-0.361799\pi\)
0.420658 + 0.907219i \(0.361799\pi\)
\(884\) 3.00000 0.100901
\(885\) −2.00000 −0.0672293
\(886\) 16.0000 0.537531
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −3.00000 −0.100673
\(889\) 54.0000 1.81110
\(890\) −26.0000 −0.871522
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) 0 0
\(894\) 0 0
\(895\) −10.0000 −0.334263
\(896\) 9.00000 0.300669
\(897\) 9.00000 0.300501
\(898\) −34.0000 −1.13459
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) −3.00000 −0.0998337
\(904\) 21.0000 0.698450
\(905\) 40.0000 1.32964
\(906\) 19.0000 0.631233
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 15.0000 0.497792
\(909\) −13.0000 −0.431183
\(910\) 6.00000 0.198898
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) −3.00000 −0.0992312
\(915\) 16.0000 0.528944
\(916\) 20.0000 0.660819
\(917\) 15.0000 0.495344
\(918\) −3.00000 −0.0990148
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 54.0000 1.78033
\(921\) −23.0000 −0.757876
\(922\) 15.0000 0.493999
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) −29.0000 −0.952999
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −49.0000 −1.60764 −0.803819 0.594874i \(-0.797202\pi\)
−0.803819 + 0.594874i \(0.797202\pi\)
\(930\) 4.00000 0.131165
\(931\) −4.00000 −0.131095
\(932\) −25.0000 −0.818902
\(933\) 18.0000 0.589294
\(934\) 26.0000 0.850746
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) −6.00000 −0.195907
\(939\) 27.0000 0.881112
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) −1.00000 −0.0325818
\(943\) 18.0000 0.586161
\(944\) 1.00000 0.0325472
\(945\) 6.00000 0.195180
\(946\) 0 0
\(947\) 27.0000 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(948\) −8.00000 −0.259828
\(949\) −14.0000 −0.454459
\(950\) −2.00000 −0.0648886
\(951\) −5.00000 −0.162136
\(952\) −27.0000 −0.875075
\(953\) −27.0000 −0.874616 −0.437308 0.899312i \(-0.644068\pi\)
−0.437308 + 0.899312i \(0.644068\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −21.0000 −0.678479
\(959\) 30.0000 0.968751
\(960\) 14.0000 0.451848
\(961\) −27.0000 −0.870968
\(962\) −1.00000 −0.0322413
\(963\) 9.00000 0.290021
\(964\) 18.0000 0.579741
\(965\) 28.0000 0.901352
\(966\) −27.0000 −0.868711
\(967\) −30.0000 −0.964735 −0.482367 0.875969i \(-0.660223\pi\)
−0.482367 + 0.875969i \(0.660223\pi\)
\(968\) −33.0000 −1.06066
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 1.00000 0.0320256
\(976\) −8.00000 −0.256074
\(977\) 45.0000 1.43968 0.719839 0.694141i \(-0.244216\pi\)
0.719839 + 0.694141i \(0.244216\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 4.00000 0.127775
\(981\) −3.00000 −0.0957826
\(982\) −27.0000 −0.861605
\(983\) −27.0000 −0.861166 −0.430583 0.902551i \(-0.641692\pi\)
−0.430583 + 0.902551i \(0.641692\pi\)
\(984\) 6.00000 0.191273
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) −9.00000 −0.286183
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 10.0000 0.317500
\(993\) −10.0000 −0.317340
\(994\) 36.0000 1.14185
\(995\) −4.00000 −0.126809
\(996\) 4.00000 0.126745
\(997\) 4.00000 0.126681 0.0633406 0.997992i \(-0.479825\pi\)
0.0633406 + 0.997992i \(0.479825\pi\)
\(998\) 44.0000 1.39280
\(999\) −1.00000 −0.0316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 471.2.a.a.1.1 1
3.2 odd 2 1413.2.a.a.1.1 1
4.3 odd 2 7536.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
471.2.a.a.1.1 1 1.1 even 1 trivial
1413.2.a.a.1.1 1 3.2 odd 2
7536.2.a.h.1.1 1 4.3 odd 2