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Properties

Label 1122.2.a.k.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +2.00000 q^{21} +1.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -8.00000 q^{38} +4.00000 q^{39} +10.0000 q^{41} +2.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} +6.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -5.00000 q^{50} -1.00000 q^{51} +4.00000 q^{52} +12.0000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -8.00000 q^{57} -2.00000 q^{58} +12.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +12.0000 q^{67} -1.00000 q^{68} +6.00000 q^{69} -10.0000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} -5.00000 q^{75} -8.00000 q^{76} +2.00000 q^{77} +4.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} +2.00000 q^{84} -4.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} +2.00000 q^{89} +8.00000 q^{91} +6.00000 q^{92} +4.00000 q^{93} -2.00000 q^{94} +1.00000 q^{96} -18.0000 q^{97} -3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 1.00000 0.213201
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −8.00000 −1.29777
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 2.00000 0.308607
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −2.00000 −0.291730 −0.145865 − 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −5.00000 −0.707107
\(51\) −1.00000 −0.140028
\(52\) 4.00000 0.554700
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) −8.00000 −1.05963
\(58\) −2.00000 −0.262613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 4.00000 0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 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
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −6.00000 −0.697486
\(75\) −5.00000 −0.577350
\(76\) −8.00000 −0.917663
\(77\) 2.00000 0.227921
\(78\) 4.00000 0.452911
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 1.00000 0.106600
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 6.00000 0.625543
\(93\) 4.00000 0.414781
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) −3.00000 −0.303046
\(99\) 1.00000 0.100504
\(100\) −5.00000 −0.500000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 12.0000 1.08643
\(123\) 10.0000 0.901670
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 10.0000 0.887357 0.443678 − 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 − 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 1.00000 0.0870388
\(133\) −16.0000 −1.38738
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 6.00000 0.510754
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) −10.0000 −0.839181
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) −3.00000 −0.247436
\(148\) −6.00000 −0.493197
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −5.00000 −0.408248
\(151\) −14.0000 −1.13930 −0.569652 − 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) −8.00000 −0.648886
\(153\) −1.00000 −0.0808452
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −10.0000 −0.795557
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) −4.00000 −0.304997
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −2.00000 −0.151620
\(175\) −10.0000 −0.755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 8.00000 0.592999
\(183\) 12.0000 0.887066
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −1.00000 −0.0731272
\(188\) −2.00000 −0.145865
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 − 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 1.00000 0.0710669
\(199\) 24.0000 1.70131 0.850657 − 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −5.00000 −0.353553
\(201\) 12.0000 0.846415
\(202\) −6.00000 −0.422159
\(203\) −4.00000 −0.280745
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 6.00000 0.417029
\(208\) 4.00000 0.277350
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 − 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 12.0000 0.824163
\(213\) −10.0000 −0.685189
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) −6.00000 −0.402694
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 2.00000 0.133631
\(225\) −5.00000 −0.333333
\(226\) −6.00000 −0.399114
\(227\) 28.0000 1.85843 0.929213 − 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) −8.00000 −0.529813
\(229\) 26.0000 1.71813 0.859064 − 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) −2.00000 −0.131306
\(233\) 26.0000 1.70332 0.851658 − 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) −2.00000 −0.129641
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) −32.0000 −2.03611
\(248\) 4.00000 0.254000
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 − 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 2.00000 0.125988
\(253\) 6.00000 0.377217
\(254\) 10.0000 0.627456
\(255\) 0 0
\(256\) 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
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 20.0000 1.23560
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 2.00000 0.122398
\(268\) 12.0000 0.733017
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −26.0000 −1.57939 −0.789694 − 0.613501i \(-0.789761\pi\)
−0.789694 + 0.613501i \(0.789761\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 8.00000 0.484182
\(274\) −2.00000 −0.120824
\(275\) −5.00000 −0.301511
\(276\) 6.00000 0.361158
\(277\) 4.00000 0.240337 0.120168 − 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −2.00000 −0.119098
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 20.0000 1.18056
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) −14.0000 −0.819288
\(293\) 30.0000 1.75262 0.876309 − 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 1.00000 0.0580259
\(298\) 6.00000 0.347571
\(299\) 24.0000 1.38796
\(300\) −5.00000 −0.288675
\(301\) −8.00000 −0.461112
\(302\) −14.0000 −0.805609
\(303\) −6.00000 −0.344691
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) −32.0000 −1.82634 −0.913168 − 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 2.00000 0.113961
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 4.00000 0.226455
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 12.0000 0.673987 0.336994 − 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 12.0000 0.672927
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 12.0000 0.668734
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) −20.0000 −1.10940
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −4.00000 −0.219529
\(333\) −6.00000 −0.328798
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 10.0000 0.544735 0.272367 − 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 3.00000 0.163178
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) −8.00000 −0.432590
\(343\) −20.0000 −1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −2.00000 −0.107211
\(349\) −20.0000 −1.07058 −0.535288 − 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) −10.0000 −0.534522
\(351\) 4.00000 0.213504
\(352\) 1.00000 0.0533002
\(353\) 34.0000 1.80964 0.904819 − 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) −2.00000 −0.105851
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 − 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) 1.00000 0.0524864
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 12.0000 0.627250
\(367\) 4.00000 0.208798 0.104399 − 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 6.00000 0.312772
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 4.00000 0.207390
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) −8.00000 −0.412021
\(378\) 2.00000 0.102869
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 10.0000 0.512316
\(382\) 6.00000 0.306987
\(383\) 10.0000 0.510976 0.255488 − 0.966812i \(-0.417764\pi\)
0.255488 + 0.966812i \(0.417764\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −4.00000 −0.203331
\(388\) −18.0000 −0.913812
\(389\) 12.0000 0.608424 0.304212 − 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) −3.00000 −0.151523
\(393\) 20.0000 1.00887
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 24.0000 1.20301
\(399\) −16.0000 −0.801002
\(400\) −5.00000 −0.250000
\(401\) −2.00000 −0.0998752 −0.0499376 − 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 12.0000 0.598506
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) −6.00000 −0.297409
\(408\) −1.00000 −0.0495074
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) −8.00000 −0.391293
\(419\) 24.0000 1.17248 0.586238 − 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −28.0000 −1.36302
\(423\) −2.00000 −0.0972433
\(424\) 12.0000 0.582772
\(425\) 5.00000 0.242536
\(426\) −10.0000 −0.484502
\(427\) 24.0000 1.16144
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 − 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 − 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 0 0
\(437\) −48.0000 −2.29615
\(438\) −14.0000 −0.668946
\(439\) −34.0000 −1.62273 −0.811366 − 0.584539i \(-0.801275\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −4.00000 −0.190261
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 6.00000 0.283790
\(448\) 2.00000 0.0944911
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −5.00000 −0.235702
\(451\) 10.0000 0.470882
\(452\) −6.00000 −0.282216
\(453\) −14.0000 −0.657777
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 26.0000 1.21490
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 22.0000 1.02464 0.512321 − 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 2.00000 0.0930484
\(463\) 28.0000 1.30127 0.650635 − 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 28.0000 1.29569 0.647843 − 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 4.00000 0.184900
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) −10.0000 −0.459315
\(475\) 40.0000 1.83533
\(476\) −2.00000 −0.0916698
\(477\) 12.0000 0.549442
\(478\) −12.0000 −0.548867
\(479\) 4.00000 0.182765 0.0913823 − 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 22.0000 1.00207
\(483\) 12.0000 0.546019
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 12.0000 0.543214
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 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
\(494\) −32.0000 −1.43975
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −20.0000 −0.897123
\(498\) −4.00000 −0.179244
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) −8.00000 −0.357057
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 3.00000 0.133235
\(508\) 10.0000 0.443678
\(509\) 12.0000 0.531891 0.265945 − 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) −2.00000 −0.0879599
\(518\) −12.0000 −0.527250
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 − 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 20.0000 0.873704
\(525\) −10.0000 −0.436436
\(526\) −24.0000 −1.04645
\(527\) −4.00000 −0.174243
\(528\) 1.00000 0.0435194
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) −16.0000 −0.693688
\(533\) 40.0000 1.73259
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 − 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) −26.0000 −1.11680
\(543\) −2.00000 −0.0858282
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) −16.0000 −0.684111 −0.342055 − 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 12.0000 0.512148
\(550\) −5.00000 −0.213201
\(551\) 16.0000 0.681623
\(552\) 6.00000 0.255377
\(553\) −20.0000 −0.850487
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 4.00000 0.169334
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 6.00000 0.253095
\(563\) 4.00000 0.168580 0.0842900 − 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −2.00000 −0.0842152
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 2.00000 0.0839921
\(568\) −10.0000 −0.419591
\(569\) 38.0000 1.59304 0.796521 − 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 4.00000 0.167248
\(573\) 6.00000 0.250654
\(574\) 20.0000 0.834784
\(575\) −30.0000 −1.25109
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 − 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 1.00000 0.0415945
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) −18.0000 −0.746124
\(583\) 12.0000 0.496989
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) −40.0000 −1.65098 −0.825488 − 0.564419i \(-0.809100\pi\)
−0.825488 + 0.564419i \(0.809100\pi\)
\(588\) −3.00000 −0.123718
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) −6.00000 −0.246598
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 24.0000 0.982255
\(598\) 24.0000 0.981433
\(599\) −6.00000 −0.245153 −0.122577 − 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) −5.00000 −0.204124
\(601\) 18.0000 0.734235 0.367118 − 0.930175i \(-0.380345\pi\)
0.367118 + 0.930175i \(0.380345\pi\)
\(602\) −8.00000 −0.326056
\(603\) 12.0000 0.488678
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 6.00000 0.243532 0.121766 − 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) −8.00000 −0.324443
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) −1.00000 −0.0404226
\(613\) 24.0000 0.969351 0.484675 − 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −4.00000 −0.160904
\(619\) 36.0000 1.44696 0.723481 − 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) −18.0000 −0.721734
\(623\) 4.00000 0.160257
\(624\) 4.00000 0.160128
\(625\) 25.0000 1.00000
\(626\) −14.0000 −0.559553
\(627\) −8.00000 −0.319489
\(628\) −14.0000 −0.558661
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 − 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) −10.0000 −0.397779
\(633\) −28.0000 −1.11290
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) −12.0000 −0.475457
\(638\) −2.00000 −0.0791808
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) 38.0000 1.50091 0.750455 − 0.660922i \(-0.229834\pi\)
0.750455 + 0.660922i \(0.229834\pi\)
\(642\) −12.0000 −0.473602
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −22.0000 −0.864909 −0.432455 − 0.901656i \(-0.642352\pi\)
−0.432455 + 0.901656i \(0.642352\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −20.0000 −0.784465
\(651\) 8.00000 0.313545
\(652\) −12.0000 −0.469956
\(653\) −4.00000 −0.156532 −0.0782660 − 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) −14.0000 −0.546192
\(658\) −4.00000 −0.155936
\(659\) 20.0000 0.779089 0.389545 − 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 − 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 4.00000 0.155464
\(663\) −4.00000 −0.155347
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −12.0000 −0.464642
\(668\) −24.0000 −0.928588
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 2.00000 0.0771517
\(673\) −26.0000 −1.00223 −0.501113 − 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 10.0000 0.385186
\(675\) −5.00000 −0.192450
\(676\) 3.00000 0.115385
\(677\) −42.0000 −1.61419 −0.807096 − 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) −6.00000 −0.230429
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 4.00000 0.153168
\(683\) −8.00000 −0.306111 −0.153056 − 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 26.0000 0.991962
\(688\) −4.00000 −0.152499
\(689\) 48.0000 1.82865
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 2.00000 0.0759737
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) −10.0000 −0.378777
\(698\) −20.0000 −0.757011
\(699\) 26.0000 0.983410
\(700\) −10.0000 −0.377964
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 4.00000 0.150970
\(703\) 48.0000 1.81035
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 − 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 2.00000 0.0749532
\(713\) 24.0000 0.898807
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) −4.00000 −0.149279
\(719\) −2.00000 −0.0745874 −0.0372937 − 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 45.0000 1.67473
\(723\) 22.0000 0.818189
\(724\) −2.00000 −0.0743294
\(725\) 10.0000 0.371391
\(726\) 1.00000 0.0371135
\(727\) 28.0000 1.03846 0.519231 − 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 12.0000 0.443533
\(733\) −12.0000 −0.443230 −0.221615 − 0.975134i \(-0.571133\pi\)
−0.221615 + 0.975134i \(0.571133\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 12.0000 0.442026
\(738\) 10.0000 0.368105
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) 24.0000 0.881068
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) −4.00000 −0.146352
\(748\) −1.00000 −0.0365636
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 − 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) −2.00000 −0.0729325
\(753\) −8.00000 −0.291536
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −4.00000 −0.145287
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 − 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 10.0000 0.362262
\(763\) 0 0
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) 10.0000 0.361315
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −10.0000 −0.359908
\(773\) 20.0000 0.719350 0.359675 − 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) −4.00000 −0.143777
\(775\) −20.0000 −0.718421
\(776\) −18.0000 −0.646162
\(777\) −12.0000 −0.430498
\(778\) 12.0000 0.430221
\(779\) −80.0000 −2.86630
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) −6.00000 −0.214560
\(783\) −2.00000 −0.0714742
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −18.0000 −0.641223
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 1.00000 0.0355335
\(793\) 48.0000 1.70453
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) 4.00000 0.141687 0.0708436 − 0.997487i \(-0.477431\pi\)
0.0708436 + 0.997487i \(0.477431\pi\)
\(798\) −16.0000 −0.566394
\(799\) 2.00000 0.0707549
\(800\) −5.00000 −0.176777
\(801\) 2.00000 0.0706665
\(802\) −2.00000 −0.0706225
\(803\) −14.0000 −0.494049
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) −46.0000 −1.61727 −0.808637 − 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −4.00000 −0.140372
\(813\) −26.0000 −0.911860
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 32.0000 1.11954
\(818\) −18.0000 −0.629355
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 − 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 20.0000 0.697156 0.348578 − 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −4.00000 −0.139347
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 − 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 6.00000 0.208514
\(829\) 46.0000 1.59765 0.798823 − 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) 4.00000 0.138675
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 4.00000 0.138260
\(838\) 24.0000 0.829066
\(839\) −30.0000 −1.03572 −0.517858 − 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −10.0000 −0.344623
\(843\) 6.00000 0.206651
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) −2.00000 −0.0687614
\(847\) 2.00000 0.0687208
\(848\) 12.0000 0.412082
\(849\) −20.0000 −0.686398
\(850\) 5.00000 0.171499
\(851\) −36.0000 −1.23406
\(852\) −10.0000 −0.342594
\(853\) 4.00000 0.136957 0.0684787 − 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 4.00000 0.136558
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) −28.0000 −0.953684
\(863\) 14.0000 0.476566 0.238283 − 0.971196i \(-0.423415\pi\)
0.238283 + 0.971196i \(0.423415\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 1.00000 0.0339618
\(868\) 8.00000 0.271538
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) −18.0000 −0.609208
\(874\) −48.0000 −1.62362
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 8.00000 0.270141 0.135070 − 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) −34.0000 −1.14744
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) −3.00000 −0.101015
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 32.0000 1.07445 0.537227 − 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) −6.00000 −0.201347
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 24.0000 0.803579
\(893\) 16.0000 0.535420
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 24.0000 0.801337
\(898\) −18.0000 −0.600668
\(899\) −8.00000 −0.266815
\(900\) −5.00000 −0.166667
\(901\) −12.0000 −0.399778
\(902\) 10.0000 0.332964
\(903\) −8.00000 −0.266223
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −14.0000 −0.465119
\(907\) 4.00000 0.132818 0.0664089 − 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 28.0000 0.929213
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −42.0000 −1.39152 −0.695761 − 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) −8.00000 −0.264906
\(913\) −4.00000 −0.132381
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) 40.0000 1.32092
\(918\) −1.00000 −0.0330049
\(919\) 22.0000 0.725713 0.362857 − 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) 0 0
\(921\) −32.0000 −1.05444
\(922\) 22.0000 0.724531
\(923\) −40.0000 −1.31662
\(924\) 2.00000 0.0657952
\(925\) 30.0000 0.986394
\(926\) 28.0000 0.920137
\(927\) −4.00000 −0.131377
\(928\) −2.00000 −0.0656532
\(929\) −42.0000 −1.37798 −0.688988 − 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 26.0000 0.851658
\(933\) −18.0000 −0.589294
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 24.0000 0.783628
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 − 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −14.0000 −0.456145
\(943\) 60.0000 1.95387
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −10.0000 −0.324785
\(949\) −56.0000 −1.81784
\(950\) 40.0000 1.29777
\(951\) 12.0000 0.389127
\(952\) −2.00000 −0.0648204
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 12.0000 0.388514
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) −2.00000 −0.0646508
\(958\) 4.00000 0.129234
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −24.0000 −0.773791
\(963\) −12.0000 −0.386695
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) 34.0000 1.09337 0.546683 − 0.837340i \(-0.315890\pi\)
0.546683 + 0.837340i \(0.315890\pi\)
\(968\) 1.00000 0.0321412
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −32.0000 −1.02693 −0.513464 − 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) −20.0000 −0.640513
\(976\) 12.0000 0.384111
\(977\) −58.0000 −1.85558 −0.927792 − 0.373097i \(-0.878296\pi\)
−0.927792 + 0.373097i \(0.878296\pi\)
\(978\) −12.0000 −0.383718
\(979\) 2.00000 0.0639203
\(980\) 0 0
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) 38.0000 1.21201 0.606006 − 0.795460i \(-0.292771\pi\)
0.606006 + 0.795460i \(0.292771\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) −4.00000 −0.127321
\(988\) −32.0000 −1.01806
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 − 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 4.00000 0.127000
\(993\) 4.00000 0.126936
\(994\) −20.0000 −0.634361
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −32.0000 −1.01345 −0.506725 − 0.862108i \(-0.669144\pi\)
−0.506725 + 0.862108i \(0.669144\pi\)
\(998\) 4.00000 0.126618
\(999\) −6.00000 −0.189832
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.k.1.1 ✓ 1
3.2 odd 2 3366.2.a.g.1.1 1
4.3 odd 2 8976.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1122.2.a.k.1.1 ✓ 1 1.1 even 1 trivial
3366.2.a.g.1.1 1 3.2 odd 2
8976.2.a.i.1.1 1 4.3 odd 2