Properties

Label 6042.2.a.n.1.1
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
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 \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +1.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +3.00000 q^{20} -1.00000 q^{21} +3.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -3.00000 q^{29} +3.00000 q^{30} +5.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +1.00000 q^{38} -4.00000 q^{39} +3.00000 q^{40} +12.0000 q^{41} -1.00000 q^{42} -7.00000 q^{43} +3.00000 q^{45} +3.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +4.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} +1.00000 q^{57} -3.00000 q^{58} +3.00000 q^{59} +3.00000 q^{60} -4.00000 q^{61} +5.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} -1.00000 q^{67} +6.00000 q^{68} +3.00000 q^{69} -3.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +2.00000 q^{74} +4.00000 q^{75} +1.00000 q^{76} -4.00000 q^{78} +8.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} +12.0000 q^{83} -1.00000 q^{84} +18.0000 q^{85} -7.00000 q^{86} -3.00000 q^{87} +15.0000 q^{89} +3.00000 q^{90} +4.00000 q^{91} +3.00000 q^{92} +5.00000 q^{93} +6.00000 q^{94} +3.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 3.00000 0.670820
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 3.00000 0.547723
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 1.00000 0.162221
\(39\) −4.00000 −0.640513
\(40\) 3.00000 0.474342
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) −1.00000 −0.154303
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 3.00000 0.442326
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 4.00000 0.565685
\(51\) 6.00000 0.840168
\(52\) −4.00000 −0.554700
\(53\) −1.00000 −0.137361
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 1.00000 0.132453
\(58\) −3.00000 −0.393919
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 3.00000 0.387298
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 5.00000 0.635001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −1.00000 −0.122169 −0.0610847 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) 6.00000 0.727607
\(69\) 3.00000 0.361158
\(70\) −3.00000 −0.358569
\(71\) −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 \(0.698986\pi\)
\(74\) 2.00000 0.232495
\(75\) 4.00000 0.461880
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −1.00000 −0.109109
\(85\) 18.0000 1.95237
\(86\) −7.00000 −0.754829
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 3.00000 0.316228
\(91\) 4.00000 0.419314
\(92\) 3.00000 0.312772
\(93\) 5.00000 0.518476
\(94\) 6.00000 0.618853
\(95\) 3.00000 0.307794
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 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
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) −4.00000 −0.392232
\(105\) −3.00000 −0.292770
\(106\) −1.00000 −0.0971286
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 1.00000 0.0962250
\(109\) 17.0000 1.62830 0.814152 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 1.00000 0.0936586
\(115\) 9.00000 0.839254
\(116\) −3.00000 −0.278543
\(117\) −4.00000 −0.369800
\(118\) 3.00000 0.276172
\(119\) −6.00000 −0.550019
\(120\) 3.00000 0.273861
\(121\) −11.0000 −1.00000
\(122\) −4.00000 −0.362143
\(123\) 12.0000 1.08200
\(124\) 5.00000 0.449013
\(125\) −3.00000 −0.268328
\(126\) −1.00000 −0.0890871
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.00000 −0.616316
\(130\) −12.0000 −1.05247
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) −1.00000 −0.0863868
\(135\) 3.00000 0.258199
\(136\) 6.00000 0.514496
\(137\) 15.0000 1.28154 0.640768 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 3.00000 0.255377
\(139\) −22.0000 −1.86602 −0.933008 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) −3.00000 −0.253546
\(141\) 6.00000 0.505291
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −9.00000 −0.747409
\(146\) −10.0000 −0.827606
\(147\) −6.00000 −0.494872
\(148\) 2.00000 0.164399
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 4.00000 0.326599
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 1.00000 0.0811107
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 15.0000 1.20483
\(156\) −4.00000 −0.320256
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) −1.00000 −0.0793052
\(160\) 3.00000 0.237171
\(161\) −3.00000 −0.236433
\(162\) 1.00000 0.0785674
\(163\) 23.0000 1.80150 0.900750 0.434339i \(-0.143018\pi\)
0.900750 + 0.434339i \(0.143018\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 3.00000 0.230769
\(170\) 18.0000 1.38054
\(171\) 1.00000 0.0764719
\(172\) −7.00000 −0.533745
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) −3.00000 −0.227429
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 15.0000 1.12430
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 3.00000 0.223607
\(181\) −19.0000 −1.41226 −0.706129 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 4.00000 0.296500
\(183\) −4.00000 −0.295689
\(184\) 3.00000 0.221163
\(185\) 6.00000 0.441129
\(186\) 5.00000 0.366618
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) −1.00000 −0.0727393
\(190\) 3.00000 0.217643
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −10.0000 −0.717958
\(195\) −12.0000 −0.859338
\(196\) −6.00000 −0.428571
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 4.00000 0.282843
\(201\) −1.00000 −0.0705346
\(202\) 15.0000 1.05540
\(203\) 3.00000 0.210559
\(204\) 6.00000 0.420084
\(205\) 36.0000 2.51435
\(206\) 5.00000 0.348367
\(207\) 3.00000 0.208514
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) −3.00000 −0.207020
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) −1.00000 −0.0686803
\(213\) −12.0000 −0.822226
\(214\) −3.00000 −0.205076
\(215\) −21.0000 −1.43219
\(216\) 1.00000 0.0680414
\(217\) −5.00000 −0.339422
\(218\) 17.0000 1.15139
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 2.00000 0.134231
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.00000 0.266667
\(226\) 9.00000 0.598671
\(227\) −27.0000 −1.79205 −0.896026 0.444001i \(-0.853559\pi\)
−0.896026 + 0.444001i \(0.853559\pi\)
\(228\) 1.00000 0.0662266
\(229\) −19.0000 −1.25556 −0.627778 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 9.00000 0.593442
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −4.00000 −0.261488
\(235\) 18.0000 1.17419
\(236\) 3.00000 0.195283
\(237\) 8.00000 0.519656
\(238\) −6.00000 −0.388922
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 3.00000 0.193649
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −4.00000 −0.256074
\(245\) −18.0000 −1.14998
\(246\) 12.0000 0.765092
\(247\) −4.00000 −0.254514
\(248\) 5.00000 0.317500
\(249\) 12.0000 0.760469
\(250\) −3.00000 −0.189737
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) −13.0000 −0.815693
\(255\) 18.0000 1.12720
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) −7.00000 −0.435801
\(259\) −2.00000 −0.124274
\(260\) −12.0000 −0.744208
\(261\) −3.00000 −0.185695
\(262\) −12.0000 −0.741362
\(263\) 15.0000 0.924940 0.462470 0.886635i \(-0.346963\pi\)
0.462470 + 0.886635i \(0.346963\pi\)
\(264\) 0 0
\(265\) −3.00000 −0.184289
\(266\) −1.00000 −0.0613139
\(267\) 15.0000 0.917985
\(268\) −1.00000 −0.0610847
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 3.00000 0.182574
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 6.00000 0.363803
\(273\) 4.00000 0.242091
\(274\) 15.0000 0.906183
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −22.0000 −1.31947
\(279\) 5.00000 0.299342
\(280\) −3.00000 −0.179284
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 6.00000 0.357295
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −9.00000 −0.528498
\(291\) −10.0000 −0.586210
\(292\) −10.0000 −0.585206
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −6.00000 −0.349927
\(295\) 9.00000 0.524000
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 4.00000 0.230940
\(301\) 7.00000 0.403473
\(302\) −1.00000 −0.0575435
\(303\) 15.0000 0.861727
\(304\) 1.00000 0.0573539
\(305\) −12.0000 −0.687118
\(306\) 6.00000 0.342997
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 5.00000 0.284440
\(310\) 15.0000 0.851943
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −4.00000 −0.226455
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) −22.0000 −1.24153
\(315\) −3.00000 −0.169031
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 0 0
\(320\) 3.00000 0.167705
\(321\) −3.00000 −0.167444
\(322\) −3.00000 −0.167183
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) −16.0000 −0.887520
\(326\) 23.0000 1.27385
\(327\) 17.0000 0.940102
\(328\) 12.0000 0.662589
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 12.0000 0.658586
\(333\) 2.00000 0.109599
\(334\) 6.00000 0.328305
\(335\) −3.00000 −0.163908
\(336\) −1.00000 −0.0545545
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) 3.00000 0.163178
\(339\) 9.00000 0.488813
\(340\) 18.0000 0.976187
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) 13.0000 0.701934
\(344\) −7.00000 −0.377415
\(345\) 9.00000 0.484544
\(346\) 12.0000 0.645124
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −3.00000 −0.160817
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −4.00000 −0.213809
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 3.00000 0.159448
\(355\) −36.0000 −1.91068
\(356\) 15.0000 0.794998
\(357\) −6.00000 −0.317554
\(358\) −6.00000 −0.317110
\(359\) −27.0000 −1.42501 −0.712503 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 3.00000 0.158114
\(361\) 1.00000 0.0526316
\(362\) −19.0000 −0.998618
\(363\) −11.0000 −0.577350
\(364\) 4.00000 0.209657
\(365\) −30.0000 −1.57027
\(366\) −4.00000 −0.209083
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 3.00000 0.156386
\(369\) 12.0000 0.624695
\(370\) 6.00000 0.311925
\(371\) 1.00000 0.0519174
\(372\) 5.00000 0.259238
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 6.00000 0.309426
\(377\) 12.0000 0.618031
\(378\) −1.00000 −0.0514344
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 3.00000 0.153897
\(381\) −13.0000 −0.666010
\(382\) 12.0000 0.613973
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −7.00000 −0.355830
\(388\) −10.0000 −0.507673
\(389\) −3.00000 −0.152106 −0.0760530 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) −12.0000 −0.607644
\(391\) 18.0000 0.910299
\(392\) −6.00000 −0.303046
\(393\) −12.0000 −0.605320
\(394\) 12.0000 0.604551
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −7.00000 −0.350878
\(399\) −1.00000 −0.0500626
\(400\) 4.00000 0.200000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −1.00000 −0.0498755
\(403\) −20.0000 −0.996271
\(404\) 15.0000 0.746278
\(405\) 3.00000 0.149071
\(406\) 3.00000 0.148888
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 8.00000 0.395575 0.197787 0.980245i \(-0.436624\pi\)
0.197787 + 0.980245i \(0.436624\pi\)
\(410\) 36.0000 1.77791
\(411\) 15.0000 0.739895
\(412\) 5.00000 0.246332
\(413\) −3.00000 −0.147620
\(414\) 3.00000 0.147442
\(415\) 36.0000 1.76717
\(416\) −4.00000 −0.196116
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) −3.00000 −0.146385
\(421\) −25.0000 −1.21843 −0.609213 0.793007i \(-0.708514\pi\)
−0.609213 + 0.793007i \(0.708514\pi\)
\(422\) −22.0000 −1.07094
\(423\) 6.00000 0.291730
\(424\) −1.00000 −0.0485643
\(425\) 24.0000 1.16417
\(426\) −12.0000 −0.581402
\(427\) 4.00000 0.193574
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) −21.0000 −1.01271
\(431\) 27.0000 1.30054 0.650272 0.759701i \(-0.274655\pi\)
0.650272 + 0.759701i \(0.274655\pi\)
\(432\) 1.00000 0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −5.00000 −0.240008
\(435\) −9.00000 −0.431517
\(436\) 17.0000 0.814152
\(437\) 3.00000 0.143509
\(438\) −10.0000 −0.477818
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −24.0000 −1.14156
\(443\) −27.0000 −1.28281 −0.641404 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 2.00000 0.0949158
\(445\) 45.0000 2.13320
\(446\) −10.0000 −0.473514
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 4.00000 0.188562
\(451\) 0 0
\(452\) 9.00000 0.423324
\(453\) −1.00000 −0.0469841
\(454\) −27.0000 −1.26717
\(455\) 12.0000 0.562569
\(456\) 1.00000 0.0468293
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −19.0000 −0.887812
\(459\) 6.00000 0.280056
\(460\) 9.00000 0.419627
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −3.00000 −0.139272
\(465\) 15.0000 0.695608
\(466\) 6.00000 0.277945
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −4.00000 −0.184900
\(469\) 1.00000 0.0461757
\(470\) 18.0000 0.830278
\(471\) −22.0000 −1.01371
\(472\) 3.00000 0.138086
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) 4.00000 0.183533
\(476\) −6.00000 −0.275010
\(477\) −1.00000 −0.0457869
\(478\) 24.0000 1.09773
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 3.00000 0.136931
\(481\) −8.00000 −0.364769
\(482\) 26.0000 1.18427
\(483\) −3.00000 −0.136505
\(484\) −11.0000 −0.500000
\(485\) −30.0000 −1.36223
\(486\) 1.00000 0.0453609
\(487\) 38.0000 1.72194 0.860972 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) −4.00000 −0.181071
\(489\) 23.0000 1.04010
\(490\) −18.0000 −0.813157
\(491\) −21.0000 −0.947717 −0.473858 0.880601i \(-0.657139\pi\)
−0.473858 + 0.880601i \(0.657139\pi\)
\(492\) 12.0000 0.541002
\(493\) −18.0000 −0.810679
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 12.0000 0.538274
\(498\) 12.0000 0.537733
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −3.00000 −0.134164
\(501\) 6.00000 0.268060
\(502\) −12.0000 −0.535586
\(503\) 39.0000 1.73892 0.869462 0.494000i \(-0.164466\pi\)
0.869462 + 0.494000i \(0.164466\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 45.0000 2.00247
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) −13.0000 −0.576782
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 18.0000 0.797053
\(511\) 10.0000 0.442374
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −24.0000 −1.05859
\(515\) 15.0000 0.660979
\(516\) −7.00000 −0.308158
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 12.0000 0.526742
\(520\) −12.0000 −0.526235
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −3.00000 −0.131306
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) −12.0000 −0.524222
\(525\) −4.00000 −0.174574
\(526\) 15.0000 0.654031
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −3.00000 −0.130312
\(531\) 3.00000 0.130189
\(532\) −1.00000 −0.0433555
\(533\) −48.0000 −2.07911
\(534\) 15.0000 0.649113
\(535\) −9.00000 −0.389104
\(536\) −1.00000 −0.0431934
\(537\) −6.00000 −0.258919
\(538\) 3.00000 0.129339
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) 35.0000 1.50477 0.752384 0.658725i \(-0.228904\pi\)
0.752384 + 0.658725i \(0.228904\pi\)
\(542\) −1.00000 −0.0429537
\(543\) −19.0000 −0.815368
\(544\) 6.00000 0.257248
\(545\) 51.0000 2.18460
\(546\) 4.00000 0.171184
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 15.0000 0.640768
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) 3.00000 0.127688
\(553\) −8.00000 −0.340195
\(554\) −16.0000 −0.679775
\(555\) 6.00000 0.254686
\(556\) −22.0000 −0.933008
\(557\) 15.0000 0.635570 0.317785 0.948163i \(-0.397061\pi\)
0.317785 + 0.948163i \(0.397061\pi\)
\(558\) 5.00000 0.211667
\(559\) 28.0000 1.18427
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −15.0000 −0.632737
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 6.00000 0.252646
\(565\) 27.0000 1.13590
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) −12.0000 −0.503509
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 3.00000 0.125656
\(571\) 2.00000 0.0836974 0.0418487 0.999124i \(-0.486675\pi\)
0.0418487 + 0.999124i \(0.486675\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) −12.0000 −0.500870
\(575\) 12.0000 0.500435
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 19.0000 0.790296
\(579\) 14.0000 0.581820
\(580\) −9.00000 −0.373705
\(581\) −12.0000 −0.497844
\(582\) −10.0000 −0.414513
\(583\) 0 0
\(584\) −10.0000 −0.413803
\(585\) −12.0000 −0.496139
\(586\) −6.00000 −0.247858
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) −6.00000 −0.247436
\(589\) 5.00000 0.206021
\(590\) 9.00000 0.370524
\(591\) 12.0000 0.493614
\(592\) 2.00000 0.0821995
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) −18.0000 −0.737928
\(596\) 0 0
\(597\) −7.00000 −0.286491
\(598\) −12.0000 −0.490716
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 4.00000 0.163299
\(601\) 11.0000 0.448699 0.224350 0.974509i \(-0.427974\pi\)
0.224350 + 0.974509i \(0.427974\pi\)
\(602\) 7.00000 0.285299
\(603\) −1.00000 −0.0407231
\(604\) −1.00000 −0.0406894
\(605\) −33.0000 −1.34164
\(606\) 15.0000 0.609333
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 1.00000 0.0405554
\(609\) 3.00000 0.121566
\(610\) −12.0000 −0.485866
\(611\) −24.0000 −0.970936
\(612\) 6.00000 0.242536
\(613\) −40.0000 −1.61558 −0.807792 0.589467i \(-0.799338\pi\)
−0.807792 + 0.589467i \(0.799338\pi\)
\(614\) −16.0000 −0.645707
\(615\) 36.0000 1.45166
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 5.00000 0.201129
\(619\) 35.0000 1.40677 0.703384 0.710810i \(-0.251671\pi\)
0.703384 + 0.710810i \(0.251671\pi\)
\(620\) 15.0000 0.602414
\(621\) 3.00000 0.120386
\(622\) −24.0000 −0.962312
\(623\) −15.0000 −0.600962
\(624\) −4.00000 −0.160128
\(625\) −29.0000 −1.16000
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 12.0000 0.478471
\(630\) −3.00000 −0.119523
\(631\) −10.0000 −0.398094 −0.199047 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(632\) 8.00000 0.318223
\(633\) −22.0000 −0.874421
\(634\) 6.00000 0.238290
\(635\) −39.0000 −1.54767
\(636\) −1.00000 −0.0396526
\(637\) 24.0000 0.950915
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 3.00000 0.118585
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) −3.00000 −0.118401
\(643\) −49.0000 −1.93237 −0.966186 0.257847i \(-0.916987\pi\)
−0.966186 + 0.257847i \(0.916987\pi\)
\(644\) −3.00000 −0.118217
\(645\) −21.0000 −0.826874
\(646\) 6.00000 0.236067
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −16.0000 −0.627572
\(651\) −5.00000 −0.195965
\(652\) 23.0000 0.900750
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 17.0000 0.664753
\(655\) −36.0000 −1.40664
\(656\) 12.0000 0.468521
\(657\) −10.0000 −0.390137
\(658\) −6.00000 −0.233904
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) −10.0000 −0.388661
\(663\) −24.0000 −0.932083
\(664\) 12.0000 0.465690
\(665\) −3.00000 −0.116335
\(666\) 2.00000 0.0774984
\(667\) −9.00000 −0.348481
\(668\) 6.00000 0.232147
\(669\) −10.0000 −0.386622
\(670\) −3.00000 −0.115900
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 17.0000 0.654816
\(675\) 4.00000 0.153960
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 9.00000 0.345643
\(679\) 10.0000 0.383765
\(680\) 18.0000 0.690268
\(681\) −27.0000 −1.03464
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 1.00000 0.0382360
\(685\) 45.0000 1.71936
\(686\) 13.0000 0.496342
\(687\) −19.0000 −0.724895
\(688\) −7.00000 −0.266872
\(689\) 4.00000 0.152388
\(690\) 9.00000 0.342624
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −66.0000 −2.50352
\(696\) −3.00000 −0.113715
\(697\) 72.0000 2.72719
\(698\) 2.00000 0.0757011
\(699\) 6.00000 0.226941
\(700\) −4.00000 −0.151186
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) −4.00000 −0.150970
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 30.0000 1.12906
\(707\) −15.0000 −0.564133
\(708\) 3.00000 0.112747
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) −36.0000 −1.35106
\(711\) 8.00000 0.300023
\(712\) 15.0000 0.562149
\(713\) 15.0000 0.561754
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 24.0000 0.896296
\(718\) −27.0000 −1.00763
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 3.00000 0.111803
\(721\) −5.00000 −0.186210
\(722\) 1.00000 0.0372161
\(723\) 26.0000 0.966950
\(724\) −19.0000 −0.706129
\(725\) −12.0000 −0.445669
\(726\) −11.0000 −0.408248
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) −30.0000 −1.11035
\(731\) −42.0000 −1.55343
\(732\) −4.00000 −0.147844
\(733\) −7.00000 −0.258551 −0.129275 0.991609i \(-0.541265\pi\)
−0.129275 + 0.991609i \(0.541265\pi\)
\(734\) −16.0000 −0.590571
\(735\) −18.0000 −0.663940
\(736\) 3.00000 0.110581
\(737\) 0 0
\(738\) 12.0000 0.441726
\(739\) 32.0000 1.17714 0.588570 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) 6.00000 0.220564
\(741\) −4.00000 −0.146944
\(742\) 1.00000 0.0367112
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 5.00000 0.183309
\(745\) 0 0
\(746\) 23.0000 0.842090
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 3.00000 0.109618
\(750\) −3.00000 −0.109545
\(751\) −46.0000 −1.67856 −0.839282 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) 6.00000 0.218797
\(753\) −12.0000 −0.437304
\(754\) 12.0000 0.437014
\(755\) −3.00000 −0.109181
\(756\) −1.00000 −0.0363696
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 3.00000 0.108821
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −13.0000 −0.470940
\(763\) −17.0000 −0.615441
\(764\) 12.0000 0.434145
\(765\) 18.0000 0.650791
\(766\) −6.00000 −0.216789
\(767\) −12.0000 −0.433295
\(768\) 1.00000 0.0360844
\(769\) −4.00000 −0.144244 −0.0721218 0.997396i \(-0.522977\pi\)
−0.0721218 + 0.997396i \(0.522977\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 14.0000 0.503871
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) −7.00000 −0.251610
\(775\) 20.0000 0.718421
\(776\) −10.0000 −0.358979
\(777\) −2.00000 −0.0717496
\(778\) −3.00000 −0.107555
\(779\) 12.0000 0.429945
\(780\) −12.0000 −0.429669
\(781\) 0 0
\(782\) 18.0000 0.643679
\(783\) −3.00000 −0.107211
\(784\) −6.00000 −0.214286
\(785\) −66.0000 −2.35564
\(786\) −12.0000 −0.428026
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) 12.0000 0.427482
\(789\) 15.0000 0.534014
\(790\) 24.0000 0.853882
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 8.00000 0.283909
\(795\) −3.00000 −0.106399
\(796\) −7.00000 −0.248108
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −1.00000 −0.0353996
\(799\) 36.0000 1.27359
\(800\) 4.00000 0.141421
\(801\) 15.0000 0.529999
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) −1.00000 −0.0352673
\(805\) −9.00000 −0.317208
\(806\) −20.0000 −0.704470
\(807\) 3.00000 0.105605
\(808\) 15.0000 0.527698
\(809\) −51.0000 −1.79306 −0.896532 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) 3.00000 0.105409
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 3.00000 0.105279
\(813\) −1.00000 −0.0350715
\(814\) 0 0
\(815\) 69.0000 2.41696
\(816\) 6.00000 0.210042
\(817\) −7.00000 −0.244899
\(818\) 8.00000 0.279713
\(819\) 4.00000 0.139771
\(820\) 36.0000 1.25717
\(821\) −21.0000 −0.732905 −0.366453 0.930437i \(-0.619428\pi\)
−0.366453 + 0.930437i \(0.619428\pi\)
\(822\) 15.0000 0.523185
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 5.00000 0.174183
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 3.00000 0.104257
\(829\) 47.0000 1.63238 0.816189 0.577785i \(-0.196083\pi\)
0.816189 + 0.577785i \(0.196083\pi\)
\(830\) 36.0000 1.24958
\(831\) −16.0000 −0.555034
\(832\) −4.00000 −0.138675
\(833\) −36.0000 −1.24733
\(834\) −22.0000 −0.761798
\(835\) 18.0000 0.622916
\(836\) 0 0
\(837\) 5.00000 0.172825
\(838\) −36.0000 −1.24360
\(839\) −9.00000 −0.310715 −0.155357 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(840\) −3.00000 −0.103510
\(841\) −20.0000 −0.689655
\(842\) −25.0000 −0.861557
\(843\) −15.0000 −0.516627
\(844\) −22.0000 −0.757271
\(845\) 9.00000 0.309609
\(846\) 6.00000 0.206284
\(847\) 11.0000 0.377964
\(848\) −1.00000 −0.0343401
\(849\) −4.00000 −0.137280
\(850\) 24.0000 0.823193
\(851\) 6.00000 0.205677
\(852\) −12.0000 −0.411113
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 4.00000 0.136877
\(855\) 3.00000 0.102598
\(856\) −3.00000 −0.102538
\(857\) −57.0000 −1.94708 −0.973541 0.228510i \(-0.926614\pi\)
−0.973541 + 0.228510i \(0.926614\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) −21.0000 −0.716094
\(861\) −12.0000 −0.408959
\(862\) 27.0000 0.919624
\(863\) −9.00000 −0.306364 −0.153182 0.988198i \(-0.548952\pi\)
−0.153182 + 0.988198i \(0.548952\pi\)
\(864\) 1.00000 0.0340207
\(865\) 36.0000 1.22404
\(866\) −34.0000 −1.15537
\(867\) 19.0000 0.645274
\(868\) −5.00000 −0.169711
\(869\) 0 0
\(870\) −9.00000 −0.305129
\(871\) 4.00000 0.135535
\(872\) 17.0000 0.575693
\(873\) −10.0000 −0.338449
\(874\) 3.00000 0.101477
\(875\) 3.00000 0.101419
\(876\) −10.0000 −0.337869
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 26.0000 0.877457
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −45.0000 −1.51609 −0.758044 0.652203i \(-0.773845\pi\)
−0.758044 + 0.652203i \(0.773845\pi\)
\(882\) −6.00000 −0.202031
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −24.0000 −0.807207
\(885\) 9.00000 0.302532
\(886\) −27.0000 −0.907083
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 2.00000 0.0671156
\(889\) 13.0000 0.436006
\(890\) 45.0000 1.50840
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) −1.00000 −0.0334077
\(897\) −12.0000 −0.400668
\(898\) −6.00000 −0.200223
\(899\) −15.0000 −0.500278
\(900\) 4.00000 0.133333
\(901\) −6.00000 −0.199889
\(902\) 0 0
\(903\) 7.00000 0.232945
\(904\) 9.00000 0.299336
\(905\) −57.0000 −1.89474
\(906\) −1.00000 −0.0332228
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) −27.0000 −0.896026
\(909\) 15.0000 0.497519
\(910\) 12.0000 0.397796
\(911\) 3.00000 0.0993944 0.0496972 0.998764i \(-0.484174\pi\)
0.0496972 + 0.998764i \(0.484174\pi\)
\(912\) 1.00000 0.0331133
\(913\) 0 0
\(914\) −28.0000 −0.926158
\(915\) −12.0000 −0.396708
\(916\) −19.0000 −0.627778
\(917\) 12.0000 0.396275
\(918\) 6.00000 0.198030
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 9.00000 0.296721
\(921\) −16.0000 −0.527218
\(922\) 18.0000 0.592798
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −4.00000 −0.131448
\(927\) 5.00000 0.164222
\(928\) −3.00000 −0.0984798
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 15.0000 0.491869
\(931\) −6.00000 −0.196642
\(932\) 6.00000 0.196537
\(933\) −24.0000 −0.785725
\(934\) −6.00000 −0.196326
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 1.00000 0.0326512
\(939\) −28.0000 −0.913745
\(940\) 18.0000 0.587095
\(941\) −3.00000 −0.0977972 −0.0488986 0.998804i \(-0.515571\pi\)
−0.0488986 + 0.998804i \(0.515571\pi\)
\(942\) −22.0000 −0.716799
\(943\) 36.0000 1.17232
\(944\) 3.00000 0.0976417
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 8.00000 0.259828
\(949\) 40.0000 1.29845
\(950\) 4.00000 0.129777
\(951\) 6.00000 0.194563
\(952\) −6.00000 −0.194461
\(953\) −21.0000 −0.680257 −0.340128 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 36.0000 1.16493
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −15.0000 −0.484628
\(959\) −15.0000 −0.484375
\(960\) 3.00000 0.0968246
\(961\) −6.00000 −0.193548
\(962\) −8.00000 −0.257930
\(963\) −3.00000 −0.0966736
\(964\) 26.0000 0.837404
\(965\) 42.0000 1.35203
\(966\) −3.00000 −0.0965234
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −11.0000 −0.353553
\(969\) 6.00000 0.192748
\(970\) −30.0000 −0.963242
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 1.00000 0.0320750
\(973\) 22.0000 0.705288
\(974\) 38.0000 1.21760
\(975\) −16.0000 −0.512410
\(976\) −4.00000 −0.128037
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 23.0000 0.735459
\(979\) 0 0
\(980\) −18.0000 −0.574989
\(981\) 17.0000 0.542768
\(982\) −21.0000 −0.670137
\(983\) −15.0000 −0.478426 −0.239213 0.970967i \(-0.576889\pi\)
−0.239213 + 0.970967i \(0.576889\pi\)
\(984\) 12.0000 0.382546
\(985\) 36.0000 1.14706
\(986\) −18.0000 −0.573237
\(987\) −6.00000 −0.190982
\(988\) −4.00000 −0.127257
\(989\) −21.0000 −0.667761
\(990\) 0 0
\(991\) 50.0000 1.58830 0.794151 0.607720i \(-0.207916\pi\)
0.794151 + 0.607720i \(0.207916\pi\)
\(992\) 5.00000 0.158750
\(993\) −10.0000 −0.317340
\(994\) 12.0000 0.380617
\(995\) −21.0000 −0.665745
\(996\) 12.0000 0.380235
\(997\) −55.0000 −1.74187 −0.870934 0.491400i \(-0.836485\pi\)
−0.870934 + 0.491400i \(0.836485\pi\)
\(998\) −40.0000 −1.26618
\(999\) 2.00000 0.0632772
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 6042.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.n.1.1 1 1.1 even 1 trivial