Properties

Label 222.2.a.d.1.1
Level $222$
Weight $2$
Character 222.1
Self dual yes
Analytic conductor $1.773$
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] = [222,2,Mod(1,222)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(222, 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("222.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 222.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: \(1.77267892487\)
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 \(\chi\) \(=\) 222.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +3.00000 q^{19} -3.00000 q^{21} +1.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -4.00000 q^{29} -6.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +3.00000 q^{38} -1.00000 q^{39} -10.0000 q^{41} -3.00000 q^{42} +12.0000 q^{43} +1.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -5.00000 q^{50} +3.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -3.00000 q^{57} -4.00000 q^{58} +2.00000 q^{61} -6.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{67} -3.00000 q^{68} +1.00000 q^{69} +1.00000 q^{72} -3.00000 q^{73} -1.00000 q^{74} +5.00000 q^{75} +3.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} -10.0000 q^{82} +9.00000 q^{83} -3.00000 q^{84} +12.0000 q^{86} +4.00000 q^{87} +1.00000 q^{88} -3.00000 q^{89} +3.00000 q^{91} -1.00000 q^{92} +6.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} +2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 1.00000 0.213201
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 3.00000 0.486664
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −3.00000 −0.462910
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) −5.00000 −0.707107
\(51\) 3.00000 0.420084
\(52\) 1.00000 0.138675
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −3.00000 −0.397360
\(58\) −4.00000 −0.525226
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −6.00000 −0.762001
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −3.00000 −0.363803
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) −1.00000 −0.116248
\(75\) 5.00000 0.577350
\(76\) 3.00000 0.344124
\(77\) 3.00000 0.341882
\(78\) −1.00000 −0.113228
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 4.00000 0.428845
\(88\) 1.00000 0.106600
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) −1.00000 −0.104257
\(93\) 6.00000 0.622171
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 2.00000 0.202031
\(99\) 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 \(0.596490\pi\)
\(102\) 3.00000 0.297044
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) −13.0000 −1.25676 −0.628379 0.777908i \(-0.716281\pi\)
−0.628379 + 0.777908i \(0.716281\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 3.00000 0.283473
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 2.00000 0.181071
\(123\) 10.0000 0.901670
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 9.00000 0.780399
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 20.0000 1.70872 0.854358 0.519685i \(-0.173951\pi\)
0.854358 + 0.519685i \(0.173951\pi\)
\(138\) 1.00000 0.0851257
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −3.00000 −0.248282
\(147\) −2.00000 −0.164957
\(148\) −1.00000 −0.0821995
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 5.00000 0.408248
\(151\) 1.00000 0.0813788 0.0406894 0.999172i \(-0.487045\pi\)
0.0406894 + 0.999172i \(0.487045\pi\)
\(152\) 3.00000 0.243332
\(153\) −3.00000 −0.242536
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 14.0000 1.11378
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 1.00000 0.0785674
\(163\) 17.0000 1.33154 0.665771 0.746156i \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) −3.00000 −0.231455
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 3.00000 0.229416
\(172\) 12.0000 0.914991
\(173\) 1.00000 0.0760286 0.0380143 0.999277i \(-0.487897\pi\)
0.0380143 + 0.999277i \(0.487897\pi\)
\(174\) 4.00000 0.303239
\(175\) −15.0000 −1.13389
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −3.00000 −0.224860
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 3.00000 0.222375
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) −3.00000 −0.219382
\(188\) −6.00000 −0.437595
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 13.0000 0.926212 0.463106 0.886303i \(-0.346735\pi\)
0.463106 + 0.886303i \(0.346735\pi\)
\(198\) 1.00000 0.0710669
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −5.00000 −0.353553
\(201\) −2.00000 −0.141069
\(202\) −6.00000 −0.422159
\(203\) −12.0000 −0.842235
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) −1.00000 −0.0695048
\(208\) 1.00000 0.0693375
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 0 0
\(214\) −13.0000 −0.888662
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −18.0000 −1.22192
\(218\) 11.0000 0.745014
\(219\) 3.00000 0.202721
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 1.00000 0.0671156
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 3.00000 0.200446
\(225\) −5.00000 −0.333333
\(226\) 18.0000 1.19734
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −3.00000 −0.198680
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) −4.00000 −0.262613
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) −9.00000 −0.583383
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 3.00000 0.190885
\(248\) −6.00000 −0.381000
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 3.00000 0.188982
\(253\) −1.00000 −0.0628695
\(254\) −7.00000 −0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 29.0000 1.80897 0.904485 0.426505i \(-0.140255\pi\)
0.904485 + 0.426505i \(0.140255\pi\)
\(258\) −12.0000 −0.747087
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 10.0000 0.617802
\(263\) 26.0000 1.60323 0.801614 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 9.00000 0.551825
\(267\) 3.00000 0.183597
\(268\) 2.00000 0.122169
\(269\) −7.00000 −0.426798 −0.213399 0.976965i \(-0.568453\pi\)
−0.213399 + 0.976965i \(0.568453\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) −3.00000 −0.181902
\(273\) −3.00000 −0.181568
\(274\) 20.0000 1.20824
\(275\) −5.00000 −0.301511
\(276\) 1.00000 0.0601929
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) −10.0000 −0.599760
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −31.0000 −1.84930 −0.924652 0.380812i \(-0.875644\pi\)
−0.924652 + 0.380812i \(0.875644\pi\)
\(282\) 6.00000 0.357295
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 1.00000 0.0591312
\(287\) −30.0000 −1.77084
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) −3.00000 −0.175562
\(293\) 31.0000 1.81104 0.905520 0.424304i \(-0.139481\pi\)
0.905520 + 0.424304i \(0.139481\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −1.00000 −0.0580259
\(298\) 18.0000 1.04271
\(299\) −1.00000 −0.0578315
\(300\) 5.00000 0.288675
\(301\) 36.0000 2.07501
\(302\) 1.00000 0.0575435
\(303\) 6.00000 0.344691
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 3.00000 0.170941
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 1.00000 0.0560772
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 13.0000 0.725589
\(322\) −3.00000 −0.167183
\(323\) −9.00000 −0.500773
\(324\) 1.00000 0.0555556
\(325\) −5.00000 −0.277350
\(326\) 17.0000 0.941543
\(327\) −11.0000 −0.608301
\(328\) −10.0000 −0.552158
\(329\) −18.0000 −0.992372
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 9.00000 0.493939
\(333\) −1.00000 −0.0547997
\(334\) 21.0000 1.14907
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) 21.0000 1.14394 0.571971 0.820274i \(-0.306179\pi\)
0.571971 + 0.820274i \(0.306179\pi\)
\(338\) −12.0000 −0.652714
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 3.00000 0.162221
\(343\) −15.0000 −0.809924
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 1.00000 0.0537603
\(347\) 22.0000 1.18102 0.590511 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) 4.00000 0.214423
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −15.0000 −0.801784
\(351\) −1.00000 −0.0533761
\(352\) 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
\(356\) −3.00000 −0.159000
\(357\) 9.00000 0.476331
\(358\) −24.0000 −1.26844
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −20.0000 −1.05118
\(363\) 10.0000 0.524864
\(364\) 3.00000 0.157243
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −21.0000 −1.09619 −0.548096 0.836416i \(-0.684647\pi\)
−0.548096 + 0.836416i \(0.684647\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) 6.00000 0.311086
\(373\) 20.0000 1.03556 0.517780 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) −3.00000 −0.155126
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −4.00000 −0.206010
\(378\) −3.00000 −0.154303
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 7.00000 0.358621
\(382\) 3.00000 0.153493
\(383\) 1.00000 0.0510976 0.0255488 0.999674i \(-0.491867\pi\)
0.0255488 + 0.999674i \(0.491867\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) 12.0000 0.609994
\(388\) −10.0000 −0.507673
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 2.00000 0.101015
\(393\) −10.0000 −0.504433
\(394\) 13.0000 0.654931
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) −12.0000 −0.602263 −0.301131 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) −16.0000 −0.802008
\(399\) −9.00000 −0.450564
\(400\) −5.00000 −0.250000
\(401\) −11.0000 −0.549314 −0.274657 0.961542i \(-0.588564\pi\)
−0.274657 + 0.961542i \(0.588564\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −6.00000 −0.298881
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) −1.00000 −0.0495682
\(408\) 3.00000 0.148522
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) −20.0000 −0.986527
\(412\) −2.00000 −0.0985329
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 10.0000 0.489702
\(418\) 3.00000 0.146735
\(419\) −21.0000 −1.02592 −0.512959 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −8.00000 −0.389434
\(423\) −6.00000 −0.291730
\(424\) −1.00000 −0.0485643
\(425\) 15.0000 0.727607
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) −13.0000 −0.628379
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) 5.00000 0.240842 0.120421 0.992723i \(-0.461576\pi\)
0.120421 + 0.992723i \(0.461576\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 7.00000 0.336399 0.168199 0.985753i \(-0.446205\pi\)
0.168199 + 0.985753i \(0.446205\pi\)
\(434\) −18.0000 −0.864028
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) −3.00000 −0.143509
\(438\) 3.00000 0.143346
\(439\) 34.0000 1.62273 0.811366 0.584539i \(-0.198725\pi\)
0.811366 + 0.584539i \(0.198725\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) −3.00000 −0.142695
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) −18.0000 −0.851371
\(448\) 3.00000 0.141737
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) −5.00000 −0.235702
\(451\) −10.0000 −0.470882
\(452\) 18.0000 0.846649
\(453\) −1.00000 −0.0469841
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) −20.0000 −0.935561 −0.467780 0.883845i \(-0.654946\pi\)
−0.467780 + 0.883845i \(0.654946\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) −3.00000 −0.139573
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 1.00000 0.0462250
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) −14.0000 −0.643041
\(475\) −15.0000 −0.688247
\(476\) −9.00000 −0.412514
\(477\) −1.00000 −0.0457869
\(478\) −16.0000 −0.731823
\(479\) −19.0000 −0.868132 −0.434066 0.900881i \(-0.642922\pi\)
−0.434066 + 0.900881i \(0.642922\pi\)
\(480\) 0 0
\(481\) −1.00000 −0.0455961
\(482\) 8.00000 0.364390
\(483\) 3.00000 0.136505
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −6.00000 −0.271886 −0.135943 0.990717i \(-0.543406\pi\)
−0.135943 + 0.990717i \(0.543406\pi\)
\(488\) 2.00000 0.0905357
\(489\) −17.0000 −0.768767
\(490\) 0 0
\(491\) −11.0000 −0.496423 −0.248212 0.968706i \(-0.579843\pi\)
−0.248212 + 0.968706i \(0.579843\pi\)
\(492\) 10.0000 0.450835
\(493\) 12.0000 0.540453
\(494\) 3.00000 0.134976
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) −9.00000 −0.403300
\(499\) −15.0000 −0.671492 −0.335746 0.941953i \(-0.608988\pi\)
−0.335746 + 0.941953i \(0.608988\pi\)
\(500\) 0 0
\(501\) −21.0000 −0.938211
\(502\) −18.0000 −0.803379
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) −1.00000 −0.0444554
\(507\) 12.0000 0.532939
\(508\) −7.00000 −0.310575
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) −9.00000 −0.398137
\(512\) 1.00000 0.0441942
\(513\) −3.00000 −0.132453
\(514\) 29.0000 1.27914
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) −6.00000 −0.263880
\(518\) −3.00000 −0.131812
\(519\) −1.00000 −0.0438951
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −4.00000 −0.175075
\(523\) 32.0000 1.39926 0.699631 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) 10.0000 0.436852
\(525\) 15.0000 0.654654
\(526\) 26.0000 1.13365
\(527\) 18.0000 0.784092
\(528\) −1.00000 −0.0435194
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 9.00000 0.390199
\(533\) −10.0000 −0.433148
\(534\) 3.00000 0.129823
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 24.0000 1.03568
\(538\) −7.00000 −0.301791
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −27.0000 −1.16082 −0.580410 0.814324i \(-0.697108\pi\)
−0.580410 + 0.814324i \(0.697108\pi\)
\(542\) 32.0000 1.37452
\(543\) 20.0000 0.858282
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) −3.00000 −0.128388
\(547\) −23.0000 −0.983409 −0.491704 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) 20.0000 0.854358
\(549\) 2.00000 0.0853579
\(550\) −5.00000 −0.213201
\(551\) −12.0000 −0.511217
\(552\) 1.00000 0.0425628
\(553\) 42.0000 1.78602
\(554\) −1.00000 −0.0424859
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −6.00000 −0.254000
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 3.00000 0.126660
\(562\) −31.0000 −1.30766
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 13.0000 0.546431
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) 19.0000 0.796521 0.398261 0.917272i \(-0.369614\pi\)
0.398261 + 0.917272i \(0.369614\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 1.00000 0.0418121
\(573\) −3.00000 −0.125327
\(574\) −30.0000 −1.25218
\(575\) 5.00000 0.208514
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −8.00000 −0.332756
\(579\) 18.0000 0.748054
\(580\) 0 0
\(581\) 27.0000 1.12015
\(582\) 10.0000 0.414513
\(583\) −1.00000 −0.0414158
\(584\) −3.00000 −0.124141
\(585\) 0 0
\(586\) 31.0000 1.28060
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) −13.0000 −0.534749
\(592\) −1.00000 −0.0410997
\(593\) −16.0000 −0.657041 −0.328521 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 16.0000 0.654836
\(598\) −1.00000 −0.0408930
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 5.00000 0.204124
\(601\) −1.00000 −0.0407909 −0.0203954 0.999792i \(-0.506493\pi\)
−0.0203954 + 0.999792i \(0.506493\pi\)
\(602\) 36.0000 1.46725
\(603\) 2.00000 0.0814463
\(604\) 1.00000 0.0406894
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 3.00000 0.121666
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) −3.00000 −0.121268
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 2.00000 0.0804518
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 8.00000 0.320771
\(623\) −9.00000 −0.360577
\(624\) −1.00000 −0.0400320
\(625\) 25.0000 1.00000
\(626\) 30.0000 1.19904
\(627\) −3.00000 −0.119808
\(628\) −4.00000 −0.159617
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 14.0000 0.556890
\(633\) 8.00000 0.317971
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 1.00000 0.0396526
\(637\) 2.00000 0.0792429
\(638\) −4.00000 −0.158362
\(639\) 0 0
\(640\) 0 0
\(641\) −32.0000 −1.26392 −0.631962 0.774999i \(-0.717750\pi\)
−0.631962 + 0.774999i \(0.717750\pi\)
\(642\) 13.0000 0.513069
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) −9.00000 −0.354100
\(647\) 31.0000 1.21874 0.609368 0.792888i \(-0.291423\pi\)
0.609368 + 0.792888i \(0.291423\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −5.00000 −0.196116
\(651\) 18.0000 0.705476
\(652\) 17.0000 0.665771
\(653\) −8.00000 −0.313064 −0.156532 0.987673i \(-0.550031\pi\)
−0.156532 + 0.987673i \(0.550031\pi\)
\(654\) −11.0000 −0.430134
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) −3.00000 −0.117041
\(658\) −18.0000 −0.701713
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 0 0
\(661\) 1.00000 0.0388955 0.0194477 0.999811i \(-0.493809\pi\)
0.0194477 + 0.999811i \(0.493809\pi\)
\(662\) 28.0000 1.08825
\(663\) 3.00000 0.116510
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) 4.00000 0.154881
\(668\) 21.0000 0.812514
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) −3.00000 −0.115728
\(673\) −51.0000 −1.96591 −0.982953 0.183858i \(-0.941141\pi\)
−0.982953 + 0.183858i \(0.941141\pi\)
\(674\) 21.0000 0.808890
\(675\) 5.00000 0.192450
\(676\) −12.0000 −0.461538
\(677\) 13.0000 0.499631 0.249815 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) −18.0000 −0.691286
\(679\) −30.0000 −1.15129
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) −6.00000 −0.229752
\(683\) −2.00000 −0.0765279 −0.0382639 0.999268i \(-0.512183\pi\)
−0.0382639 + 0.999268i \(0.512183\pi\)
\(684\) 3.00000 0.114708
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) 12.0000 0.457496
\(689\) −1.00000 −0.0380970
\(690\) 0 0
\(691\) 24.0000 0.913003 0.456502 0.889723i \(-0.349102\pi\)
0.456502 + 0.889723i \(0.349102\pi\)
\(692\) 1.00000 0.0380143
\(693\) 3.00000 0.113961
\(694\) 22.0000 0.835109
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 30.0000 1.13633
\(698\) 10.0000 0.378506
\(699\) 6.00000 0.226941
\(700\) −15.0000 −0.566947
\(701\) −16.0000 −0.604312 −0.302156 0.953259i \(-0.597706\pi\)
−0.302156 + 0.953259i \(0.597706\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −3.00000 −0.113147
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) −18.0000 −0.676960
\(708\) 0 0
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) −3.00000 −0.112430
\(713\) 6.00000 0.224702
\(714\) 9.00000 0.336817
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 16.0000 0.597531
\(718\) 30.0000 1.11959
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) −10.0000 −0.372161
\(723\) −8.00000 −0.297523
\(724\) −20.0000 −0.743294
\(725\) 20.0000 0.742781
\(726\) 10.0000 0.371135
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 3.00000 0.111187
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) −2.00000 −0.0739221
\(733\) 18.0000 0.664845 0.332423 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) −21.0000 −0.775124
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 2.00000 0.0736709
\(738\) −10.0000 −0.368105
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) −3.00000 −0.110208
\(742\) −3.00000 −0.110133
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) 20.0000 0.732252
\(747\) 9.00000 0.329293
\(748\) −3.00000 −0.109691
\(749\) −39.0000 −1.42503
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) −6.00000 −0.218797
\(753\) 18.0000 0.655956
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) 3.00000 0.109037 0.0545184 0.998513i \(-0.482638\pi\)
0.0545184 + 0.998513i \(0.482638\pi\)
\(758\) −4.00000 −0.145287
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 7.00000 0.253583
\(763\) 33.0000 1.19468
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 1.00000 0.0361315
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −4.00000 −0.144244 −0.0721218 0.997396i \(-0.522977\pi\)
−0.0721218 + 0.997396i \(0.522977\pi\)
\(770\) 0 0
\(771\) −29.0000 −1.04441
\(772\) −18.0000 −0.647834
\(773\) −35.0000 −1.25886 −0.629431 0.777056i \(-0.716712\pi\)
−0.629431 + 0.777056i \(0.716712\pi\)
\(774\) 12.0000 0.431331
\(775\) 30.0000 1.07763
\(776\) −10.0000 −0.358979
\(777\) 3.00000 0.107624
\(778\) 30.0000 1.07555
\(779\) −30.0000 −1.07486
\(780\) 0 0
\(781\) 0 0
\(782\) 3.00000 0.107280
\(783\) 4.00000 0.142948
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) −38.0000 −1.35455 −0.677277 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) 13.0000 0.463106
\(789\) −26.0000 −0.925625
\(790\) 0 0
\(791\) 54.0000 1.92002
\(792\) 1.00000 0.0355335
\(793\) 2.00000 0.0710221
\(794\) −12.0000 −0.425864
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 16.0000 0.566749 0.283375 0.959009i \(-0.408546\pi\)
0.283375 + 0.959009i \(0.408546\pi\)
\(798\) −9.00000 −0.318597
\(799\) 18.0000 0.636794
\(800\) −5.00000 −0.176777
\(801\) −3.00000 −0.106000
\(802\) −11.0000 −0.388424
\(803\) −3.00000 −0.105868
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) 7.00000 0.246412
\(808\) −6.00000 −0.211079
\(809\) 21.0000 0.738321 0.369160 0.929366i \(-0.379645\pi\)
0.369160 + 0.929366i \(0.379645\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −12.0000 −0.421117
\(813\) −32.0000 −1.12229
\(814\) −1.00000 −0.0350500
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 36.0000 1.25948
\(818\) −6.00000 −0.209785
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) −20.0000 −0.697580
\(823\) 29.0000 1.01088 0.505438 0.862863i \(-0.331331\pi\)
0.505438 + 0.862863i \(0.331331\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) 2.00000 0.0695468 0.0347734 0.999395i \(-0.488929\pi\)
0.0347734 + 0.999395i \(0.488929\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −17.0000 −0.590434 −0.295217 0.955430i \(-0.595392\pi\)
−0.295217 + 0.955430i \(0.595392\pi\)
\(830\) 0 0
\(831\) 1.00000 0.0346896
\(832\) 1.00000 0.0346688
\(833\) −6.00000 −0.207888
\(834\) 10.0000 0.346272
\(835\) 0 0
\(836\) 3.00000 0.103757
\(837\) 6.00000 0.207390
\(838\) −21.0000 −0.725433
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 26.0000 0.896019
\(843\) 31.0000 1.06770
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) −30.0000 −1.03081
\(848\) −1.00000 −0.0343401
\(849\) −13.0000 −0.446159
\(850\) 15.0000 0.514496
\(851\) 1.00000 0.0342796
\(852\) 0 0
\(853\) −35.0000 −1.19838 −0.599189 0.800608i \(-0.704510\pi\)
−0.599189 + 0.800608i \(0.704510\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −13.0000 −0.444331
\(857\) −45.0000 −1.53717 −0.768585 0.639747i \(-0.779039\pi\)
−0.768585 + 0.639747i \(0.779039\pi\)
\(858\) −1.00000 −0.0341394
\(859\) 49.0000 1.67186 0.835929 0.548837i \(-0.184929\pi\)
0.835929 + 0.548837i \(0.184929\pi\)
\(860\) 0 0
\(861\) 30.0000 1.02240
\(862\) 5.00000 0.170301
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 7.00000 0.237870
\(867\) 8.00000 0.271694
\(868\) −18.0000 −0.610960
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) 11.0000 0.372507
\(873\) −10.0000 −0.338449
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) 3.00000 0.101361
\(877\) −30.0000 −1.01303 −0.506514 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(878\) 34.0000 1.14744
\(879\) −31.0000 −1.04560
\(880\) 0 0
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 2.00000 0.0673435
\(883\) 21.0000 0.706706 0.353353 0.935490i \(-0.385041\pi\)
0.353353 + 0.935490i \(0.385041\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −22.0000 −0.738688 −0.369344 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 1.00000 0.0335578
\(889\) −21.0000 −0.704317
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 16.0000 0.535720
\(893\) −18.0000 −0.602347
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 1.00000 0.0333890
\(898\) 6.00000 0.200223
\(899\) 24.0000 0.800445
\(900\) −5.00000 −0.166667
\(901\) 3.00000 0.0999445
\(902\) −10.0000 −0.332964
\(903\) −36.0000 −1.19800
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −1.00000 −0.0332228
\(907\) 5.00000 0.166022 0.0830111 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) −12.0000 −0.398234
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) −3.00000 −0.0993399
\(913\) 9.00000 0.297857
\(914\) −20.0000 −0.661541
\(915\) 0 0
\(916\) 0 0
\(917\) 30.0000 0.990687
\(918\) 3.00000 0.0990148
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) −3.00000 −0.0986928
\(925\) 5.00000 0.164399
\(926\) −26.0000 −0.854413
\(927\) −2.00000 −0.0656886
\(928\) −4.00000 −0.131306
\(929\) −10.0000 −0.328089 −0.164045 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −6.00000 −0.196537
\(933\) −8.00000 −0.261908
\(934\) −6.00000 −0.196326
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 6.00000 0.195907
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 4.00000 0.130327
\(943\) 10.0000 0.325645
\(944\) 0 0
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 10.0000 0.324956 0.162478 0.986712i \(-0.448051\pi\)
0.162478 + 0.986712i \(0.448051\pi\)
\(948\) −14.0000 −0.454699
\(949\) −3.00000 −0.0973841
\(950\) −15.0000 −0.486664
\(951\) −18.0000 −0.583690
\(952\) −9.00000 −0.291692
\(953\) −28.0000 −0.907009 −0.453504 0.891254i \(-0.649826\pi\)
−0.453504 + 0.891254i \(0.649826\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 4.00000 0.129302
\(958\) −19.0000 −0.613862
\(959\) 60.0000 1.93750
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −1.00000 −0.0322413
\(963\) −13.0000 −0.418919
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −10.0000 −0.321412
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −30.0000 −0.961756
\(974\) −6.00000 −0.192252
\(975\) 5.00000 0.160128
\(976\) 2.00000 0.0640184
\(977\) −43.0000 −1.37569 −0.687846 0.725857i \(-0.741444\pi\)
−0.687846 + 0.725857i \(0.741444\pi\)
\(978\) −17.0000 −0.543600
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) −11.0000 −0.351024
\(983\) 46.0000 1.46717 0.733586 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 18.0000 0.572946
\(988\) 3.00000 0.0954427
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −6.00000 −0.190500
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) 27.0000 0.855099 0.427549 0.903992i \(-0.359377\pi\)
0.427549 + 0.903992i \(0.359377\pi\)
\(998\) −15.0000 −0.474817
\(999\) 1.00000 0.0316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 222.2.a.d.1.1 1
3.2 odd 2 666.2.a.b.1.1 1
4.3 odd 2 1776.2.a.h.1.1 1
5.4 even 2 5550.2.a.n.1.1 1
8.3 odd 2 7104.2.a.f.1.1 1
8.5 even 2 7104.2.a.v.1.1 1
12.11 even 2 5328.2.a.h.1.1 1
37.36 even 2 8214.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
222.2.a.d.1.1 1 1.1 even 1 trivial
666.2.a.b.1.1 1 3.2 odd 2
1776.2.a.h.1.1 1 4.3 odd 2
5328.2.a.h.1.1 1 12.11 even 2
5550.2.a.n.1.1 1 5.4 even 2
7104.2.a.f.1.1 1 8.3 odd 2
7104.2.a.v.1.1 1 8.5 even 2
8214.2.a.b.1.1 1 37.36 even 2