Properties

Label 1122.2.a.c.1.1
Level $1122$
Weight $2$
Character 1122.1
Self dual yes
Analytic conductor $8.959$
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] = [1122,2,Mod(1,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, 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("1122.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.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: \(8.95921510679\)
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 \(\chi\) \(=\) 1122.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -4.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{21} -1.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} -1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -6.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} +4.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} -2.00000 q^{60} +2.00000 q^{61} +4.00000 q^{63} +1.00000 q^{64} +12.0000 q^{65} +1.00000 q^{66} +12.0000 q^{67} -1.00000 q^{68} -8.00000 q^{70} -1.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} +6.00000 q^{78} +4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -2.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} -14.0000 q^{89} -2.00000 q^{90} +24.0000 q^{91} -4.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} +18.0000 q^{97} -9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) −2.00000 −0.516398
\(16\) 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 \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) −4.00000 −0.872872
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 1.00000 0.171499
\(35\) 8.00000 1.35225
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) −6.00000 −0.960769
\(40\) −2.00000 −0.316228
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 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
\(55\) 2.00000 0.269680
\(56\) −4.00000 −0.534522
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −2.00000 −0.258199
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 12.0000 1.48842
\(66\) 1.00000 0.123091
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 10.0000 1.16248
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 4.00000 0.455842
\(78\) 6.00000 0.679366
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 2.00000 0.223607
\(81\) 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 \(0.728845\pi\)
\(84\) −4.00000 −0.436436
\(85\) −2.00000 −0.216930
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) −1.00000 −0.106600
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −2.00000 −0.210819
\(91\) 24.0000 2.51588
\(92\) 0 0
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 8.00000 0.820783
\(96\) 1.00000 0.102062
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) −9.00000 −0.909137
\(99\) 1.00000 0.100504
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −6.00000 −0.588348
\(105\) −8.00000 −0.780720
\(106\) 2.00000 0.194257
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −2.00000 −0.190693
\(111\) 10.0000 0.949158
\(112\) 4.00000 0.377964
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) −4.00000 −0.356348
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) −12.0000 −1.05247
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 16.0000 1.38738
\(134\) −12.0000 −1.03664
\(135\) −2.00000 −0.172133
\(136\) 1.00000 0.0857493
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 8.00000 0.676123
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 10.0000 0.827606
\(147\) −9.00000 −0.742307
\(148\) −10.0000 −0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −4.00000 −0.324443
\(153\) −1.00000 −0.0808452
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −4.00000 −0.318223
\(159\) 2.00000 0.158610
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −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
\(165\) −2.00000 −0.155700
\(166\) 12.0000 0.931381
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 4.00000 0.308607
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 2.00000 0.151620
\(175\) −4.00000 −0.302372
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 2.00000 0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −24.0000 −1.77900
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) −1.00000 −0.0731272
\(188\) 4.00000 0.291730
\(189\) −4.00000 −0.290957
\(190\) −8.00000 −0.580381
\(191\) −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 \(0.430717\pi\)
\(194\) −18.0000 −1.29232
\(195\) −12.0000 −0.859338
\(196\) 9.00000 0.642857
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) −2.00000 −0.140720
\(203\) 8.00000 0.561490
\(204\) 1.00000 0.0700140
\(205\) −20.0000 −1.39686
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 4.00000 0.276686
\(210\) 8.00000 0.552052
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 10.0000 0.675737
\(220\) 2.00000 0.134840
\(221\) −6.00000 −0.403604
\(222\) −10.0000 −0.671156
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −4.00000 −0.267261
\(225\) −1.00000 −0.0666667
\(226\) −14.0000 −0.931266
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) −4.00000 −0.264906
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) −2.00000 −0.131306
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −6.00000 −0.392232
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 4.00000 0.259281
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −2.00000 −0.129099
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 18.0000 1.14998
\(246\) −10.0000 −0.637577
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 12.0000 0.758947
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 2.00000 0.125245
\(256\) 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
\(259\) −40.0000 −2.48548
\(260\) 12.0000 0.744208
\(261\) 2.00000 0.123797
\(262\) 4.00000 0.247121
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 1.00000 0.0615457
\(265\) −4.00000 −0.245718
\(266\) −16.0000 −0.981023
\(267\) 14.0000 0.856786
\(268\) 12.0000 0.733017
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 2.00000 0.121716
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −24.0000 −1.45255
\(274\) 6.00000 0.362473
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) −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
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) −6.00000 −0.354787
\(287\) −40.0000 −2.36113
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −4.00000 −0.234888
\(291\) −18.0000 −1.05518
\(292\) −10.0000 −0.585206
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 4.00000 0.229416
\(305\) 4.00000 0.229039
\(306\) 1.00000 0.0571662
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 4.00000 0.227921
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 6.00000 0.339683
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 18.0000 1.01580
\(315\) 8.00000 0.450749
\(316\) 4.00000 0.225018
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −2.00000 −0.112154
\(319\) 2.00000 0.111979
\(320\) 2.00000 0.111803
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) −12.0000 −0.664619
\(327\) −10.0000 −0.553001
\(328\) 10.0000 0.552158
\(329\) 16.0000 0.882109
\(330\) 2.00000 0.110096
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) −10.0000 −0.547997
\(334\) 8.00000 0.437741
\(335\) 24.0000 1.31126
\(336\) −4.00000 −0.218218
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −23.0000 −1.25104
\(339\) −14.0000 −0.760376
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 8.00000 0.431959
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −2.00000 −0.107211
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 4.00000 0.213809
\(351\) −6.00000 −0.320256
\(352\) −1.00000 −0.0533002
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 4.00000 0.211702
\(358\) −24.0000 −1.26844
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) −1.00000 −0.0524864
\(364\) 24.0000 1.25794
\(365\) −20.0000 −1.04685
\(366\) 2.00000 0.104542
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 20.0000 1.03975
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 1.00000 0.0517088
\(375\) 12.0000 0.619677
\(376\) −4.00000 −0.206284
\(377\) 12.0000 0.618031
\(378\) 4.00000 0.205738
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 8.00000 0.410391
\(381\) 16.0000 0.819705
\(382\) 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
\(385\) 8.00000 0.407718
\(386\) −6.00000 −0.305392
\(387\) −4.00000 −0.203331
\(388\) 18.0000 0.913812
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 12.0000 0.607644
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) 4.00000 0.201773
\(394\) 22.0000 1.10834
\(395\) 8.00000 0.402524
\(396\) 1.00000 0.0502519
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 16.0000 0.802008
\(399\) −16.0000 −0.801002
\(400\) −1.00000 −0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 2.00000 0.0993808
\(406\) −8.00000 −0.397033
\(407\) −10.0000 −0.495682
\(408\) −1.00000 −0.0495074
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 20.0000 0.987730
\(411\) 6.00000 0.295958
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) −6.00000 −0.294174
\(417\) 16.0000 0.783523
\(418\) −4.00000 −0.195646
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −8.00000 −0.390360
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 16.0000 0.778868
\(423\) 4.00000 0.194487
\(424\) 2.00000 0.0971286
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 12.0000 0.580042
\(429\) −6.00000 −0.289683
\(430\) 8.00000 0.385794
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 9.00000 0.428571
\(442\) 6.00000 0.285391
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 10.0000 0.474579
\(445\) −28.0000 −1.32733
\(446\) −8.00000 −0.378811
\(447\) 6.00000 0.283790
\(448\) 4.00000 0.188982
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 1.00000 0.0471405
\(451\) −10.0000 −0.470882
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −28.0000 −1.31411
\(455\) 48.0000 2.25027
\(456\) 4.00000 0.187317
\(457\) 34.0000 1.59045 0.795226 0.606313i \(-0.207352\pi\)
0.795226 + 0.606313i \(0.207352\pi\)
\(458\) 26.0000 1.21490
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 4.00000 0.186097
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 6.00000 0.277350
\(469\) 48.0000 2.21643
\(470\) −8.00000 −0.369012
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) −4.00000 −0.183340
\(477\) −2.00000 −0.0915737
\(478\) 8.00000 0.365911
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 2.00000 0.0912871
\(481\) −60.0000 −2.73576
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 36.0000 1.63468
\(486\) 1.00000 0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −12.0000 −0.542659
\(490\) −18.0000 −0.813157
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 10.0000 0.450835
\(493\) −2.00000 −0.0900755
\(494\) −24.0000 −1.07981
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −12.0000 −0.536656
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −4.00000 −0.178174
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) −16.0000 −0.709885
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −40.0000 −1.76950
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) 32.0000 1.41009
\(516\) 4.00000 0.176090
\(517\) 4.00000 0.175920
\(518\) 40.0000 1.75750
\(519\) −2.00000 −0.0877903
\(520\) −12.0000 −0.526235
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −4.00000 −0.174741
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) 4.00000 0.173749
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) −60.0000 −2.59889
\(534\) −14.0000 −0.605839
\(535\) 24.0000 1.03761
\(536\) −12.0000 −0.518321
\(537\) −24.0000 −1.03568
\(538\) 30.0000 1.29339
\(539\) 9.00000 0.387657
\(540\) −2.00000 −0.0860663
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) −32.0000 −1.37452
\(543\) 2.00000 0.0858282
\(544\) 1.00000 0.0428746
\(545\) 20.0000 0.856706
\(546\) 24.0000 1.02711
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −6.00000 −0.256307
\(549\) 2.00000 0.0853579
\(550\) 1.00000 0.0426401
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) −10.0000 −0.424859
\(555\) 20.0000 0.848953
\(556\) −16.0000 −0.678551
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 8.00000 0.338062
\(561\) 1.00000 0.0422200
\(562\) 18.0000 0.759284
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) −4.00000 −0.168430
\(565\) 28.0000 1.17797
\(566\) 16.0000 0.672530
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 8.00000 0.335083
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 6.00000 0.250873
\(573\) 4.00000 0.167102
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −6.00000 −0.249351
\(580\) 4.00000 0.166091
\(581\) −48.0000 −1.99138
\(582\) 18.0000 0.746124
\(583\) −2.00000 −0.0828315
\(584\) 10.0000 0.413803
\(585\) 12.0000 0.496139
\(586\) −26.0000 −1.07405
\(587\) −48.0000 −1.98117 −0.990586 0.136892i \(-0.956289\pi\)
−0.990586 + 0.136892i \(0.956289\pi\)
\(588\) −9.00000 −0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) −10.0000 −0.410997
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 1.00000 0.0410305
\(595\) −8.00000 −0.327968
\(596\) −6.00000 −0.245770
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −28.0000 −1.14405 −0.572024 0.820237i \(-0.693842\pi\)
−0.572024 + 0.820237i \(0.693842\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 16.0000 0.652111
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 2.00000 0.0812444
\(607\) 4.00000 0.162355 0.0811775 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) −4.00000 −0.162221
\(609\) −8.00000 −0.324176
\(610\) −4.00000 −0.161955
\(611\) 24.0000 0.970936
\(612\) −1.00000 −0.0404226
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) −12.0000 −0.484281
\(615\) 20.0000 0.806478
\(616\) −4.00000 −0.161165
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 16.0000 0.643614
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −56.0000 −2.24359
\(624\) −6.00000 −0.240192
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) −4.00000 −0.159745
\(628\) −18.0000 −0.718278
\(629\) 10.0000 0.398726
\(630\) −8.00000 −0.318728
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −4.00000 −0.159111
\(633\) 16.0000 0.635943
\(634\) 22.0000 0.873732
\(635\) −32.0000 −1.26988
\(636\) 2.00000 0.0793052
\(637\) 54.0000 2.13956
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 12.0000 0.473602
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 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
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 10.0000 0.391031
\(655\) −8.00000 −0.312586
\(656\) −10.0000 −0.390434
\(657\) −10.0000 −0.390137
\(658\) −16.0000 −0.623745
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −20.0000 −0.777322
\(663\) 6.00000 0.233021
\(664\) 12.0000 0.465690
\(665\) 32.0000 1.24091
\(666\) 10.0000 0.387492
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) −8.00000 −0.309298
\(670\) −24.0000 −0.927201
\(671\) 2.00000 0.0772091
\(672\) 4.00000 0.154303
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) 26.0000 0.999261 0.499631 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 14.0000 0.537667
\(679\) 72.0000 2.76311
\(680\) 2.00000 0.0766965
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 4.00000 0.152944
\(685\) −12.0000 −0.458496
\(686\) −8.00000 −0.305441
\(687\) 26.0000 0.991962
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 2.00000 0.0760286
\(693\) 4.00000 0.151947
\(694\) 4.00000 0.151838
\(695\) −32.0000 −1.21383
\(696\) 2.00000 0.0758098
\(697\) 10.0000 0.378777
\(698\) 18.0000 0.681310
\(699\) −6.00000 −0.226941
\(700\) −4.00000 −0.151186
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 6.00000 0.226455
\(703\) −40.0000 −1.50863
\(704\) 1.00000 0.0376889
\(705\) −8.00000 −0.301297
\(706\) −2.00000 −0.0752710
\(707\) 8.00000 0.300871
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) −4.00000 −0.149696
\(715\) 12.0000 0.448775
\(716\) 24.0000 0.896922
\(717\) 8.00000 0.298765
\(718\) 24.0000 0.895672
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 2.00000 0.0745356
\(721\) 64.0000 2.38348
\(722\) 3.00000 0.111648
\(723\) 18.0000 0.669427
\(724\) −2.00000 −0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 1.00000 0.0371135
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) −24.0000 −0.889499
\(729\) 1.00000 0.0370370
\(730\) 20.0000 0.740233
\(731\) 4.00000 0.147945
\(732\) −2.00000 −0.0739221
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 8.00000 0.295285
\(735\) −18.0000 −0.663940
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 10.0000 0.368105
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) −20.0000 −0.735215
\(741\) −24.0000 −0.881662
\(742\) 8.00000 0.293689
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 26.0000 0.951928
\(747\) −12.0000 −0.439057
\(748\) −1.00000 −0.0365636
\(749\) 48.0000 1.75388
\(750\) −12.0000 −0.438178
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −16.0000 −0.579619
\(763\) 40.0000 1.44810
\(764\) −4.00000 −0.144715
\(765\) −2.00000 −0.0723102
\(766\) −28.0000 −1.01168
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −8.00000 −0.288300
\(771\) −18.0000 −0.648254
\(772\) 6.00000 0.215945
\(773\) 54.0000 1.94225 0.971123 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −18.0000 −0.646162
\(777\) 40.0000 1.43499
\(778\) 2.00000 0.0717035
\(779\) −40.0000 −1.43315
\(780\) −12.0000 −0.429669
\(781\) 0 0
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 9.00000 0.321429
\(785\) −36.0000 −1.28490
\(786\) −4.00000 −0.142675
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −22.0000 −0.783718
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 56.0000 1.99113
\(792\) −1.00000 −0.0355335
\(793\) 12.0000 0.426132
\(794\) 10.0000 0.354887
\(795\) 4.00000 0.141865
\(796\) −16.0000 −0.567105
\(797\) −50.0000 −1.77109 −0.885545 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 16.0000 0.566394
\(799\) −4.00000 −0.141510
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) −6.00000 −0.211867
\(803\) −10.0000 −0.352892
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) −2.00000 −0.0703598
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 8.00000 0.280745
\(813\) −32.0000 −1.12229
\(814\) 10.0000 0.350500
\(815\) 24.0000 0.840683
\(816\) 1.00000 0.0350070
\(817\) −16.0000 −0.559769
\(818\) 14.0000 0.489499
\(819\) 24.0000 0.838628
\(820\) −20.0000 −0.698430
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −6.00000 −0.209274
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −16.0000 −0.557386
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 24.0000 0.833052
\(831\) −10.0000 −0.346896
\(832\) 6.00000 0.208013
\(833\) −9.00000 −0.311832
\(834\) −16.0000 −0.554035
\(835\) −16.0000 −0.553703
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 8.00000 0.276026
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 18.0000 0.619953
\(844\) −16.0000 −0.550743
\(845\) 46.0000 1.58245
\(846\) −4.00000 −0.137523
\(847\) 4.00000 0.137442
\(848\) −2.00000 −0.0686803
\(849\) 16.0000 0.549119
\(850\) −1.00000 −0.0342997
\(851\) 0 0
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) −8.00000 −0.273754
\(855\) 8.00000 0.273594
\(856\) −12.0000 −0.410152
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 6.00000 0.204837
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −8.00000 −0.272798
\(861\) 40.0000 1.36320
\(862\) −32.0000 −1.08992
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 1.00000 0.0340207
\(865\) 4.00000 0.136004
\(866\) −18.0000 −0.611665
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 4.00000 0.135613
\(871\) 72.0000 2.43963
\(872\) −10.0000 −0.338643
\(873\) 18.0000 0.609208
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 10.0000 0.337869
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 28.0000 0.944954
\(879\) −26.0000 −0.876958
\(880\) 2.00000 0.0674200
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) −9.00000 −0.303046
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −10.0000 −0.335578
\(889\) −64.0000 −2.14649
\(890\) 28.0000 0.938562
\(891\) 1.00000 0.0335013
\(892\) 8.00000 0.267860
\(893\) 16.0000 0.535420
\(894\) −6.00000 −0.200670
\(895\) 48.0000 1.60446
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) −22.0000 −0.734150
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 2.00000 0.0666297
\(902\) 10.0000 0.332964
\(903\) 16.0000 0.532447
\(904\) −14.0000 −0.465633
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 28.0000 0.929213
\(909\) 2.00000 0.0663358
\(910\) −48.0000 −1.59118
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −4.00000 −0.132453
\(913\) −12.0000 −0.397142
\(914\) −34.0000 −1.12462
\(915\) −4.00000 −0.132236
\(916\) −26.0000 −0.859064
\(917\) −16.0000 −0.528367
\(918\) −1.00000 −0.0330049
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) −2.00000 −0.0658665
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) 10.0000 0.328798
\(926\) −32.0000 −1.05159
\(927\) 16.0000 0.525509
\(928\) −2.00000 −0.0656532
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 6.00000 0.196537
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) −2.00000 −0.0654070
\(936\) −6.00000 −0.196116
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) −48.0000 −1.56726
\(939\) −10.0000 −0.326338
\(940\) 8.00000 0.260931
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −18.0000 −0.586472
\(943\) 0 0
\(944\) 0 0
\(945\) −8.00000 −0.260240
\(946\) 4.00000 0.130051
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) −4.00000 −0.129914
\(949\) −60.0000 −1.94768
\(950\) 4.00000 0.129777
\(951\) 22.0000 0.713399
\(952\) 4.00000 0.129641
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 2.00000 0.0647524
\(955\) −8.00000 −0.258874
\(956\) −8.00000 −0.258738
\(957\) −2.00000 −0.0646508
\(958\) 8.00000 0.258468
\(959\) −24.0000 −0.775000
\(960\) −2.00000 −0.0645497
\(961\) −31.0000 −1.00000
\(962\) 60.0000 1.93448
\(963\) 12.0000 0.386695
\(964\) −18.0000 −0.579741
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 4.00000 0.128499
\(970\) −36.0000 −1.15589
\(971\) 32.0000 1.02693 0.513464 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −64.0000 −2.05175
\(974\) −16.0000 −0.512673
\(975\) 6.00000 0.192154
\(976\) 2.00000 0.0640184
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 12.0000 0.383718
\(979\) −14.0000 −0.447442
\(980\) 18.0000 0.574989
\(981\) 10.0000 0.319275
\(982\) −28.0000 −0.893516
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) −10.0000 −0.318788
\(985\) −44.0000 −1.40196
\(986\) 2.00000 0.0636930
\(987\) −16.0000 −0.509286
\(988\) 24.0000 0.763542
\(989\) 0 0
\(990\) −2.00000 −0.0635642
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) 12.0000 0.380235
\(997\) −62.0000 −1.96356 −0.981780 0.190022i \(-0.939144\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(998\) 28.0000 0.886325
\(999\) 10.0000 0.316386
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 1122.2.a.c.1.1 1
3.2 odd 2 3366.2.a.m.1.1 1
4.3 odd 2 8976.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1122.2.a.c.1.1 1 1.1 even 1 trivial
3366.2.a.m.1.1 1 3.2 odd 2
8976.2.a.ba.1.1 1 4.3 odd 2