Properties

Label 102.2.a.a.1.1
Level $102$
Weight $2$
Character 102.1
Self dual yes
Analytic conductor $0.814$
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] = [102,2,Mod(1,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, 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("102.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 102.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.814474100617\)
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 \(\chi\) \(=\) 102.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -1.00000 q^{12} -6.00000 q^{13} +2.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -4.00000 q^{20} +2.00000 q^{21} +6.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -4.00000 q^{29} -4.00000 q^{30} -6.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} +4.00000 q^{40} -10.0000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -4.00000 q^{45} -6.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -11.0000 q^{50} +1.00000 q^{51} -6.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -4.00000 q^{57} +4.00000 q^{58} +12.0000 q^{59} +4.00000 q^{60} -4.00000 q^{61} +6.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +24.0000 q^{65} -12.0000 q^{67} -1.00000 q^{68} -6.00000 q^{69} -8.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +4.00000 q^{74} -11.0000 q^{75} +4.00000 q^{76} -6.00000 q^{78} +10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{85} +4.00000 q^{86} +4.00000 q^{87} -2.00000 q^{89} +4.00000 q^{90} +12.0000 q^{91} +6.00000 q^{92} +6.00000 q^{93} -4.00000 q^{94} -16.0000 q^{95} +1.00000 q^{96} +6.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 0.534522
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −4.00000 −0.894427
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) 11.0000 2.20000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −4.00000 −0.730297
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 8.00000 1.35225
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −4.00000 −0.648886
\(39\) 6.00000 0.960769
\(40\) 4.00000 0.632456
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −4.00000 −0.596285
\(46\) −6.00000 −0.884652
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −11.0000 −1.55563
\(51\) 1.00000 0.140028
\(52\) −6.00000 −0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) −4.00000 −0.529813
\(58\) 4.00000 0.525226
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 4.00000 0.516398
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 6.00000 0.762001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 24.0000 2.97683
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −1.00000 −0.121268
\(69\) −6.00000 −0.722315
\(70\) −8.00000 −0.956183
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 4.00000 0.464991
\(75\) −11.0000 −1.27017
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) 4.00000 0.433861
\(86\) 4.00000 0.431331
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 4.00000 0.421637
\(91\) 12.0000 1.25794
\(92\) 6.00000 0.625543
\(93\) 6.00000 0.622171
\(94\) −4.00000 −0.412568
\(95\) −16.0000 −1.64157
\(96\) 1.00000 0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 11.0000 1.10000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 6.00000 0.588348
\(105\) −8.00000 −0.780720
\(106\) 2.00000 0.194257
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) −2.00000 −0.188982
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 4.00000 0.374634
\(115\) −24.0000 −2.23801
\(116\) −4.00000 −0.371391
\(117\) −6.00000 −0.554700
\(118\) −12.0000 −1.10469
\(119\) 2.00000 0.183340
\(120\) −4.00000 −0.365148
\(121\) −11.0000 −1.00000
\(122\) 4.00000 0.362143
\(123\) 10.0000 0.901670
\(124\) −6.00000 −0.538816
\(125\) −24.0000 −2.14663
\(126\) 2.00000 0.178174
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) −24.0000 −2.10494
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 12.0000 1.03664
\(135\) 4.00000 0.344265
\(136\) 1.00000 0.0857493
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 6.00000 0.510754
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 8.00000 0.676123
\(141\) −4.00000 −0.336861
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 16.0000 1.32873
\(146\) −2.00000 −0.165521
\(147\) 3.00000 0.247436
\(148\) −4.00000 −0.328798
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 11.0000 0.898146
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) −4.00000 −0.324443
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 6.00000 0.480384
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −10.0000 −0.795557
\(159\) 2.00000 0.158610
\(160\) 4.00000 0.316228
\(161\) −12.0000 −0.945732
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) −2.00000 −0.154303
\(169\) 23.0000 1.76923
\(170\) −4.00000 −0.306786
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) −4.00000 −0.303239
\(175\) −22.0000 −1.66304
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 2.00000 0.149906
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −4.00000 −0.298142
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) −12.0000 −0.889499
\(183\) 4.00000 0.295689
\(184\) −6.00000 −0.442326
\(185\) 16.0000 1.17634
\(186\) −6.00000 −0.439941
\(187\) 0 0
\(188\) 4.00000 0.291730
\(189\) 2.00000 0.145479
\(190\) 16.0000 1.16076
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −6.00000 −0.430775
\(195\) −24.0000 −1.71868
\(196\) −3.00000 −0.214286
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) −11.0000 −0.777817
\(201\) 12.0000 0.846415
\(202\) −14.0000 −0.985037
\(203\) 8.00000 0.561490
\(204\) 1.00000 0.0700140
\(205\) 40.0000 2.79372
\(206\) −4.00000 −0.278693
\(207\) 6.00000 0.417029
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 8.00000 0.552052
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −2.00000 −0.137361
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 1.00000 0.0680414
\(217\) 12.0000 0.814613
\(218\) −16.0000 −1.08366
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) −4.00000 −0.268462
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 2.00000 0.133631
\(225\) 11.0000 0.733333
\(226\) −2.00000 −0.133038
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −4.00000 −0.264906
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 24.0000 1.58251
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 6.00000 0.392232
\(235\) −16.0000 −1.04372
\(236\) 12.0000 0.781133
\(237\) −10.0000 −0.649570
\(238\) −2.00000 −0.129641
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 4.00000 0.258199
\(241\) −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
\(244\) −4.00000 −0.256074
\(245\) 12.0000 0.766652
\(246\) −10.0000 −0.637577
\(247\) −24.0000 −1.52708
\(248\) 6.00000 0.381000
\(249\) 12.0000 0.760469
\(250\) 24.0000 1.51789
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −4.00000 −0.249029
\(259\) 8.00000 0.497096
\(260\) 24.0000 1.48842
\(261\) −4.00000 −0.247594
\(262\) 16.0000 0.988483
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 8.00000 0.490511
\(267\) 2.00000 0.122398
\(268\) −12.0000 −0.733017
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) −4.00000 −0.243432
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −12.0000 −0.726273
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −8.00000 −0.479808
\(279\) −6.00000 −0.359211
\(280\) −8.00000 −0.478091
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 4.00000 0.238197
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) −6.00000 −0.356034
\(285\) 16.0000 0.947758
\(286\) 0 0
\(287\) 20.0000 1.18056
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −16.0000 −0.939552
\(291\) −6.00000 −0.351726
\(292\) 2.00000 0.117041
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) −3.00000 −0.174964
\(295\) −48.0000 −2.79467
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −36.0000 −2.08193
\(300\) −11.0000 −0.635085
\(301\) 8.00000 0.461112
\(302\) 24.0000 1.38104
\(303\) −14.0000 −0.804279
\(304\) 4.00000 0.229416
\(305\) 16.0000 0.916157
\(306\) 1.00000 0.0571662
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) −24.0000 −1.36311
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) −6.00000 −0.339683
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −6.00000 −0.338600
\(315\) 8.00000 0.450749
\(316\) 10.0000 0.562544
\(317\) −16.0000 −0.898650 −0.449325 0.893368i \(-0.648335\pi\)
−0.449325 + 0.893368i \(0.648335\pi\)
\(318\) −2.00000 −0.112154
\(319\) 0 0
\(320\) −4.00000 −0.223607
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) −66.0000 −3.66102
\(326\) −12.0000 −0.664619
\(327\) −16.0000 −0.884802
\(328\) 10.0000 0.552158
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) −4.00000 −0.219199
\(334\) 2.00000 0.109435
\(335\) 48.0000 2.62252
\(336\) 2.00000 0.109109
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −23.0000 −1.25104
\(339\) −2.00000 −0.108625
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 20.0000 1.07990
\(344\) 4.00000 0.215666
\(345\) 24.0000 1.29212
\(346\) 4.00000 0.215041
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 4.00000 0.214423
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 22.0000 1.17595
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 12.0000 0.637793
\(355\) 24.0000 1.27379
\(356\) −2.00000 −0.106000
\(357\) −2.00000 −0.105851
\(358\) 12.0000 0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 4.00000 0.210819
\(361\) −3.00000 −0.157895
\(362\) 20.0000 1.05118
\(363\) 11.0000 0.577350
\(364\) 12.0000 0.628971
\(365\) −8.00000 −0.418739
\(366\) −4.00000 −0.209083
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) 6.00000 0.312772
\(369\) −10.0000 −0.520579
\(370\) −16.0000 −0.831800
\(371\) 4.00000 0.207670
\(372\) 6.00000 0.311086
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) −4.00000 −0.206284
\(377\) 24.0000 1.23606
\(378\) −2.00000 −0.102869
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −16.0000 −0.820783
\(381\) −8.00000 −0.409852
\(382\) 4.00000 0.204658
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) −4.00000 −0.203331
\(388\) 6.00000 0.304604
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 24.0000 1.21529
\(391\) −6.00000 −0.303433
\(392\) 3.00000 0.151523
\(393\) 16.0000 0.807093
\(394\) −8.00000 −0.403034
\(395\) −40.0000 −2.01262
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) −14.0000 −0.701757
\(399\) 8.00000 0.400501
\(400\) 11.0000 0.550000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −12.0000 −0.598506
\(403\) 36.0000 1.79329
\(404\) 14.0000 0.696526
\(405\) −4.00000 −0.198762
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) −1.00000 −0.0495074
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) −40.0000 −1.97546
\(411\) 6.00000 0.295958
\(412\) 4.00000 0.197066
\(413\) −24.0000 −1.18096
\(414\) −6.00000 −0.294884
\(415\) 48.0000 2.35623
\(416\) 6.00000 0.294174
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −8.00000 −0.390360
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −8.00000 −0.389434
\(423\) 4.00000 0.194487
\(424\) 2.00000 0.0971286
\(425\) −11.0000 −0.533578
\(426\) −6.00000 −0.290701
\(427\) 8.00000 0.387147
\(428\) 0 0
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) −12.0000 −0.576018
\(435\) −16.0000 −0.767141
\(436\) 16.0000 0.766261
\(437\) 24.0000 1.14808
\(438\) 2.00000 0.0955637
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −6.00000 −0.285391
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 4.00000 0.189832
\(445\) 8.00000 0.379236
\(446\) 4.00000 0.189405
\(447\) 6.00000 0.283790
\(448\) −2.00000 −0.0944911
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) −11.0000 −0.518545
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) 24.0000 1.12762
\(454\) −4.00000 −0.187729
\(455\) −48.0000 −2.25027
\(456\) 4.00000 0.187317
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 2.00000 0.0934539
\(459\) 1.00000 0.0466760
\(460\) −24.0000 −1.11901
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −4.00000 −0.185695
\(465\) −24.0000 −1.11297
\(466\) 6.00000 0.277945
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −6.00000 −0.277350
\(469\) 24.0000 1.10822
\(470\) 16.0000 0.738025
\(471\) −6.00000 −0.276465
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 10.0000 0.459315
\(475\) 44.0000 2.01886
\(476\) 2.00000 0.0916698
\(477\) −2.00000 −0.0915737
\(478\) 20.0000 0.914779
\(479\) 10.0000 0.456912 0.228456 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) −4.00000 −0.182574
\(481\) 24.0000 1.09431
\(482\) 18.0000 0.819878
\(483\) 12.0000 0.546019
\(484\) −11.0000 −0.500000
\(485\) −24.0000 −1.08978
\(486\) 1.00000 0.0453609
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 4.00000 0.181071
\(489\) −12.0000 −0.542659
\(490\) −12.0000 −0.542105
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 10.0000 0.450835
\(493\) 4.00000 0.180151
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 12.0000 0.538274
\(498\) −12.0000 −0.537733
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −24.0000 −1.07331
\(501\) 2.00000 0.0893534
\(502\) −12.0000 −0.535586
\(503\) −26.0000 −1.15928 −0.579641 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(504\) 2.00000 0.0890871
\(505\) −56.0000 −2.49197
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 8.00000 0.354943
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 4.00000 0.177123
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) −16.0000 −0.705044
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 4.00000 0.175581
\(520\) −24.0000 −1.05247
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 4.00000 0.175075
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −16.0000 −0.698963
\(525\) 22.0000 0.960159
\(526\) 12.0000 0.523225
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −8.00000 −0.347498
\(531\) 12.0000 0.520756
\(532\) −8.00000 −0.346844
\(533\) 60.0000 2.59889
\(534\) −2.00000 −0.0865485
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 12.0000 0.517838
\(538\) 12.0000 0.517357
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) −12.0000 −0.515920 −0.257960 0.966156i \(-0.583050\pi\)
−0.257960 + 0.966156i \(0.583050\pi\)
\(542\) 16.0000 0.687259
\(543\) 20.0000 0.858282
\(544\) 1.00000 0.0428746
\(545\) −64.0000 −2.74146
\(546\) 12.0000 0.513553
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −6.00000 −0.256307
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) 6.00000 0.255377
\(553\) −20.0000 −0.850487
\(554\) 8.00000 0.339887
\(555\) −16.0000 −0.679162
\(556\) 8.00000 0.339276
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 6.00000 0.254000
\(559\) 24.0000 1.01509
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −4.00000 −0.168430
\(565\) −8.00000 −0.336563
\(566\) −32.0000 −1.34506
\(567\) −2.00000 −0.0839921
\(568\) 6.00000 0.251754
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −16.0000 −0.670166
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 4.00000 0.167102
\(574\) −20.0000 −0.834784
\(575\) 66.0000 2.75239
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −6.00000 −0.249351
\(580\) 16.0000 0.664364
\(581\) 24.0000 0.995688
\(582\) 6.00000 0.248708
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 24.0000 0.992278
\(586\) −2.00000 −0.0826192
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 3.00000 0.123718
\(589\) −24.0000 −0.988903
\(590\) 48.0000 1.97613
\(591\) −8.00000 −0.329076
\(592\) −4.00000 −0.164399
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) −6.00000 −0.245770
\(597\) −14.0000 −0.572982
\(598\) 36.0000 1.47215
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 11.0000 0.449073
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) −8.00000 −0.326056
\(603\) −12.0000 −0.488678
\(604\) −24.0000 −0.976546
\(605\) 44.0000 1.78885
\(606\) 14.0000 0.568711
\(607\) −38.0000 −1.54237 −0.771186 0.636610i \(-0.780336\pi\)
−0.771186 + 0.636610i \(0.780336\pi\)
\(608\) −4.00000 −0.162221
\(609\) −8.00000 −0.324176
\(610\) −16.0000 −0.647821
\(611\) −24.0000 −0.970936
\(612\) −1.00000 −0.0404226
\(613\) 30.0000 1.21169 0.605844 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 12.0000 0.484281
\(615\) −40.0000 −1.61296
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 4.00000 0.160904
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 24.0000 0.963863
\(621\) −6.00000 −0.240772
\(622\) −30.0000 −1.20289
\(623\) 4.00000 0.160257
\(624\) 6.00000 0.240192
\(625\) 41.0000 1.64000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) 4.00000 0.159490
\(630\) −8.00000 −0.318728
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −10.0000 −0.397779
\(633\) −8.00000 −0.317971
\(634\) 16.0000 0.635441
\(635\) −32.0000 −1.26988
\(636\) 2.00000 0.0793052
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 4.00000 0.158114
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) −12.0000 −0.472866
\(645\) −16.0000 −0.629999
\(646\) 4.00000 0.157378
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 66.0000 2.58873
\(651\) −12.0000 −0.470317
\(652\) 12.0000 0.469956
\(653\) 20.0000 0.782660 0.391330 0.920250i \(-0.372015\pi\)
0.391330 + 0.920250i \(0.372015\pi\)
\(654\) 16.0000 0.625650
\(655\) 64.0000 2.50069
\(656\) −10.0000 −0.390434
\(657\) 2.00000 0.0780274
\(658\) 8.00000 0.311872
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −20.0000 −0.777322
\(663\) −6.00000 −0.233021
\(664\) 12.0000 0.465690
\(665\) 32.0000 1.24091
\(666\) 4.00000 0.154997
\(667\) −24.0000 −0.929284
\(668\) −2.00000 −0.0773823
\(669\) 4.00000 0.154649
\(670\) −48.0000 −1.85440
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 6.00000 0.231111
\(675\) −11.0000 −0.423390
\(676\) 23.0000 0.884615
\(677\) 8.00000 0.307465 0.153732 0.988113i \(-0.450871\pi\)
0.153732 + 0.988113i \(0.450871\pi\)
\(678\) 2.00000 0.0768095
\(679\) −12.0000 −0.460518
\(680\) −4.00000 −0.153393
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 4.00000 0.152944
\(685\) 24.0000 0.916993
\(686\) −20.0000 −0.763604
\(687\) 2.00000 0.0763048
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) −24.0000 −0.913664
\(691\) 16.0000 0.608669 0.304334 0.952565i \(-0.401566\pi\)
0.304334 + 0.952565i \(0.401566\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) −32.0000 −1.21383
\(696\) −4.00000 −0.151620
\(697\) 10.0000 0.378777
\(698\) 30.0000 1.13552
\(699\) 6.00000 0.226941
\(700\) −22.0000 −0.831522
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −6.00000 −0.226455
\(703\) −16.0000 −0.603451
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) −14.0000 −0.526897
\(707\) −28.0000 −1.05305
\(708\) −12.0000 −0.450988
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) −24.0000 −0.900704
\(711\) 10.0000 0.375029
\(712\) 2.00000 0.0749532
\(713\) −36.0000 −1.34821
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 20.0000 0.746914
\(718\) 0 0
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) −4.00000 −0.149071
\(721\) −8.00000 −0.297936
\(722\) 3.00000 0.111648
\(723\) 18.0000 0.669427
\(724\) −20.0000 −0.743294
\(725\) −44.0000 −1.63412
\(726\) −11.0000 −0.408248
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −12.0000 −0.444750
\(729\) 1.00000 0.0370370
\(730\) 8.00000 0.296093
\(731\) 4.00000 0.147945
\(732\) 4.00000 0.147844
\(733\) 18.0000 0.664845 0.332423 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) −10.0000 −0.369107
\(735\) −12.0000 −0.442627
\(736\) −6.00000 −0.221163
\(737\) 0 0
\(738\) 10.0000 0.368105
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 16.0000 0.588172
\(741\) 24.0000 0.881662
\(742\) −4.00000 −0.146845
\(743\) 38.0000 1.39408 0.697042 0.717030i \(-0.254499\pi\)
0.697042 + 0.717030i \(0.254499\pi\)
\(744\) −6.00000 −0.219971
\(745\) 24.0000 0.879292
\(746\) 14.0000 0.512576
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) −24.0000 −0.876356
\(751\) −34.0000 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(752\) 4.00000 0.145865
\(753\) −12.0000 −0.437304
\(754\) −24.0000 −0.874028
\(755\) 96.0000 3.49380
\(756\) 2.00000 0.0727393
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 16.0000 0.580381
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 8.00000 0.289809
\(763\) −32.0000 −1.15848
\(764\) −4.00000 −0.144715
\(765\) 4.00000 0.144620
\(766\) −28.0000 −1.01168
\(767\) −72.0000 −2.59977
\(768\) −1.00000 −0.0360844
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 6.00000 0.215945
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 4.00000 0.143777
\(775\) −66.0000 −2.37079
\(776\) −6.00000 −0.215387
\(777\) −8.00000 −0.286998
\(778\) 14.0000 0.501924
\(779\) −40.0000 −1.43315
\(780\) −24.0000 −0.859338
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) 4.00000 0.142948
\(784\) −3.00000 −0.107143
\(785\) −24.0000 −0.856597
\(786\) −16.0000 −0.570701
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 8.00000 0.284988
\(789\) 12.0000 0.427211
\(790\) 40.0000 1.42314
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) −20.0000 −0.709773
\(795\) −8.00000 −0.283731
\(796\) 14.0000 0.496217
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) −8.00000 −0.283197
\(799\) −4.00000 −0.141510
\(800\) −11.0000 −0.388909
\(801\) −2.00000 −0.0706665
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) 48.0000 1.69178
\(806\) −36.0000 −1.26805
\(807\) 12.0000 0.422420
\(808\) −14.0000 −0.492518
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 4.00000 0.140546
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 8.00000 0.280745
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) −48.0000 −1.68137
\(816\) 1.00000 0.0350070
\(817\) −16.0000 −0.559769
\(818\) 26.0000 0.909069
\(819\) 12.0000 0.419314
\(820\) 40.0000 1.39686
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −6.00000 −0.209274
\(823\) 18.0000 0.627441 0.313720 0.949515i \(-0.398425\pi\)
0.313720 + 0.949515i \(0.398425\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 6.00000 0.208514
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) −48.0000 −1.66610
\(831\) 8.00000 0.277517
\(832\) −6.00000 −0.208013
\(833\) 3.00000 0.103944
\(834\) 8.00000 0.277017
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 12.0000 0.414533
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 8.00000 0.276026
\(841\) −13.0000 −0.448276
\(842\) −34.0000 −1.17172
\(843\) 18.0000 0.619953
\(844\) 8.00000 0.275371
\(845\) −92.0000 −3.16490
\(846\) −4.00000 −0.137523
\(847\) 22.0000 0.755929
\(848\) −2.00000 −0.0686803
\(849\) −32.0000 −1.09824
\(850\) 11.0000 0.377297
\(851\) −24.0000 −0.822709
\(852\) 6.00000 0.205557
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) −8.00000 −0.273754
\(855\) −16.0000 −0.547188
\(856\) 0 0
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 16.0000 0.545595
\(861\) −20.0000 −0.681598
\(862\) −14.0000 −0.476842
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 1.00000 0.0340207
\(865\) 16.0000 0.544016
\(866\) 18.0000 0.611665
\(867\) −1.00000 −0.0339618
\(868\) 12.0000 0.407307
\(869\) 0 0
\(870\) 16.0000 0.542451
\(871\) 72.0000 2.43963
\(872\) −16.0000 −0.541828
\(873\) 6.00000 0.203069
\(874\) −24.0000 −0.811812
\(875\) 48.0000 1.62270
\(876\) −2.00000 −0.0675737
\(877\) −24.0000 −0.810422 −0.405211 0.914223i \(-0.632802\pi\)
−0.405211 + 0.914223i \(0.632802\pi\)
\(878\) 10.0000 0.337484
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 3.00000 0.101015
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 6.00000 0.201802
\(885\) 48.0000 1.61350
\(886\) −12.0000 −0.403148
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) −4.00000 −0.134231
\(889\) −16.0000 −0.536623
\(890\) −8.00000 −0.268161
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 16.0000 0.535420
\(894\) −6.00000 −0.200670
\(895\) 48.0000 1.60446
\(896\) 2.00000 0.0668153
\(897\) 36.0000 1.20201
\(898\) 26.0000 0.867631
\(899\) 24.0000 0.800445
\(900\) 11.0000 0.366667
\(901\) 2.00000 0.0666297
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) −2.00000 −0.0665190
\(905\) 80.0000 2.65929
\(906\) −24.0000 −0.797347
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 4.00000 0.132745
\(909\) 14.0000 0.464351
\(910\) 48.0000 1.59118
\(911\) 26.0000 0.861418 0.430709 0.902491i \(-0.358263\pi\)
0.430709 + 0.902491i \(0.358263\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −16.0000 −0.528944
\(916\) −2.00000 −0.0660819
\(917\) 32.0000 1.05673
\(918\) −1.00000 −0.0330049
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 24.0000 0.791257
\(921\) 12.0000 0.395413
\(922\) 10.0000 0.329332
\(923\) 36.0000 1.18495
\(924\) 0 0
\(925\) −44.0000 −1.44671
\(926\) 4.00000 0.131448
\(927\) 4.00000 0.131377
\(928\) 4.00000 0.131306
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 24.0000 0.786991
\(931\) −12.0000 −0.393284
\(932\) −6.00000 −0.196537
\(933\) −30.0000 −0.982156
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) −24.0000 −0.783628
\(939\) 26.0000 0.848478
\(940\) −16.0000 −0.521862
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 6.00000 0.195491
\(943\) −60.0000 −1.95387
\(944\) 12.0000 0.390567
\(945\) −8.00000 −0.260240
\(946\) 0 0
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) −10.0000 −0.324785
\(949\) −12.0000 −0.389536
\(950\) −44.0000 −1.42755
\(951\) 16.0000 0.518836
\(952\) −2.00000 −0.0648204
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 2.00000 0.0647524
\(955\) 16.0000 0.517748
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) −10.0000 −0.323085
\(959\) 12.0000 0.387500
\(960\) 4.00000 0.129099
\(961\) 5.00000 0.161290
\(962\) −24.0000 −0.773791
\(963\) 0 0
\(964\) −18.0000 −0.579741
\(965\) −24.0000 −0.772587
\(966\) −12.0000 −0.386094
\(967\) −44.0000 −1.41494 −0.707472 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) 11.0000 0.353553
\(969\) 4.00000 0.128499
\(970\) 24.0000 0.770594
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.0000 −0.512936
\(974\) 38.0000 1.21760
\(975\) 66.0000 2.11369
\(976\) −4.00000 −0.128037
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) 12.0000 0.383326
\(981\) 16.0000 0.510841
\(982\) 20.0000 0.638226
\(983\) −38.0000 −1.21201 −0.606006 0.795460i \(-0.707229\pi\)
−0.606006 + 0.795460i \(0.707229\pi\)
\(984\) −10.0000 −0.318788
\(985\) −32.0000 −1.01960
\(986\) −4.00000 −0.127386
\(987\) 8.00000 0.254643
\(988\) −24.0000 −0.763542
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 6.00000 0.190500
\(993\) −20.0000 −0.634681
\(994\) −12.0000 −0.380617
\(995\) −56.0000 −1.77532
\(996\) 12.0000 0.380235
\(997\) −20.0000 −0.633406 −0.316703 0.948525i \(-0.602576\pi\)
−0.316703 + 0.948525i \(0.602576\pi\)
\(998\) −32.0000 −1.01294
\(999\) 4.00000 0.126554
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 102.2.a.a.1.1 1
3.2 odd 2 306.2.a.d.1.1 1
4.3 odd 2 816.2.a.h.1.1 1
5.2 odd 4 2550.2.d.q.2449.1 2
5.3 odd 4 2550.2.d.q.2449.2 2
5.4 even 2 2550.2.a.be.1.1 1
7.6 odd 2 4998.2.a.x.1.1 1
8.3 odd 2 3264.2.a.p.1.1 1
8.5 even 2 3264.2.a.bf.1.1 1
12.11 even 2 2448.2.a.t.1.1 1
15.14 odd 2 7650.2.a.z.1.1 1
17.2 even 8 1734.2.f.g.1483.1 4
17.4 even 4 1734.2.b.d.577.2 2
17.8 even 8 1734.2.f.g.829.2 4
17.9 even 8 1734.2.f.g.829.1 4
17.13 even 4 1734.2.b.d.577.1 2
17.15 even 8 1734.2.f.g.1483.2 4
17.16 even 2 1734.2.a.h.1.1 1
24.5 odd 2 9792.2.a.a.1.1 1
24.11 even 2 9792.2.a.b.1.1 1
51.50 odd 2 5202.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.a.1.1 1 1.1 even 1 trivial
306.2.a.d.1.1 1 3.2 odd 2
816.2.a.h.1.1 1 4.3 odd 2
1734.2.a.h.1.1 1 17.16 even 2
1734.2.b.d.577.1 2 17.13 even 4
1734.2.b.d.577.2 2 17.4 even 4
1734.2.f.g.829.1 4 17.9 even 8
1734.2.f.g.829.2 4 17.8 even 8
1734.2.f.g.1483.1 4 17.2 even 8
1734.2.f.g.1483.2 4 17.15 even 8
2448.2.a.t.1.1 1 12.11 even 2
2550.2.a.be.1.1 1 5.4 even 2
2550.2.d.q.2449.1 2 5.2 odd 4
2550.2.d.q.2449.2 2 5.3 odd 4
3264.2.a.p.1.1 1 8.3 odd 2
3264.2.a.bf.1.1 1 8.5 even 2
4998.2.a.x.1.1 1 7.6 odd 2
5202.2.a.g.1.1 1 51.50 odd 2
7650.2.a.z.1.1 1 15.14 odd 2
9792.2.a.a.1.1 1 24.5 odd 2
9792.2.a.b.1.1 1 24.11 even 2