Properties

Label 3822.2.a.bf.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, 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("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} -7.00000 q^{17} +1.00000 q^{18} -7.00000 q^{19} -5.00000 q^{22} +2.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} -9.00000 q^{29} +1.00000 q^{32} -5.00000 q^{33} -7.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -7.00000 q^{38} +1.00000 q^{39} -4.00000 q^{41} +2.00000 q^{43} -5.00000 q^{44} +2.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} -5.00000 q^{50} -7.00000 q^{51} +1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -7.00000 q^{57} -9.00000 q^{58} -7.00000 q^{59} -13.0000 q^{61} +1.00000 q^{64} -5.00000 q^{66} +3.00000 q^{67} -7.00000 q^{68} +2.00000 q^{69} +9.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +4.00000 q^{74} -5.00000 q^{75} -7.00000 q^{76} +1.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} -4.00000 q^{82} -16.0000 q^{83} +2.00000 q^{86} -9.00000 q^{87} -5.00000 q^{88} +12.0000 q^{89} +2.00000 q^{92} +3.00000 q^{94} +1.00000 q^{96} -6.00000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.00000 −0.870388
\(34\) −7.00000 −1.20049
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −7.00000 −1.13555
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −5.00000 −0.707107
\(51\) −7.00000 −0.980196
\(52\) 1.00000 0.138675
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) −9.00000 −1.18176
\(59\) −7.00000 −0.911322 −0.455661 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) −7.00000 −0.848875
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 4.00000 0.464991
\(75\) −5.00000 −0.577350
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) 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
\(82\) −4.00000 −0.441726
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) −9.00000 −0.964901
\(88\) −5.00000 −0.533002
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) −5.00000 −0.500000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −7.00000 −0.693103
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 1.00000 0.0971286
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) −7.00000 −0.655610
\(115\) 0 0
\(116\) −9.00000 −0.835629
\(117\) 1.00000 0.0924500
\(118\) −7.00000 −0.644402
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −13.0000 −1.17696
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 22.0000 1.95218 0.976092 0.217357i \(-0.0697436\pi\)
0.976092 + 0.217357i \(0.0697436\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) −5.00000 −0.435194
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 2.00000 0.170251
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 9.00000 0.755263
\(143\) −5.00000 −0.418121
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −5.00000 −0.408248
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) −7.00000 −0.567775
\(153\) −7.00000 −0.565916
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 14.0000 1.11378
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −16.0000 −1.24184
\(167\) 5.00000 0.386912 0.193456 0.981109i \(-0.438030\pi\)
0.193456 + 0.981109i \(0.438030\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 2.00000 0.152499
\(173\) 13.0000 0.988372 0.494186 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) −7.00000 −0.526152
\(178\) 12.0000 0.899438
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) −13.0000 −0.960988
\(184\) 2.00000 0.147442
\(185\) 0 0
\(186\) 0 0
\(187\) 35.0000 2.55945
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −8.00000 −0.575853 −0.287926 0.957653i \(-0.592966\pi\)
−0.287926 + 0.957653i \(0.592966\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) −5.00000 −0.355335
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) −5.00000 −0.353553
\(201\) 3.00000 0.211604
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −7.00000 −0.490098
\(205\) 0 0
\(206\) 10.0000 0.696733
\(207\) 2.00000 0.139010
\(208\) 1.00000 0.0693375
\(209\) 35.0000 2.42100
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 1.00000 0.0686803
\(213\) 9.00000 0.616670
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 8.00000 0.541828
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −7.00000 −0.470871
\(222\) 4.00000 0.268462
\(223\) 1.00000 0.0669650 0.0334825 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 1.00000 0.0665190
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −7.00000 −0.463586
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) −3.00000 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −7.00000 −0.455661
\(237\) 14.0000 0.909398
\(238\) 0 0
\(239\) −1.00000 −0.0646846 −0.0323423 0.999477i \(-0.510297\pi\)
−0.0323423 + 0.999477i \(0.510297\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 14.0000 0.899954
\(243\) 1.00000 0.0641500
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) −4.00000 −0.255031
\(247\) −7.00000 −0.445399
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) −10.0000 −0.628695
\(254\) 22.0000 1.38040
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) −18.0000 −1.11204
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 3.00000 0.183254
\(269\) −23.0000 −1.40233 −0.701167 0.712997i \(-0.747337\pi\)
−0.701167 + 0.712997i \(0.747337\pi\)
\(270\) 0 0
\(271\) −3.00000 −0.182237 −0.0911185 0.995840i \(-0.529044\pi\)
−0.0911185 + 0.995840i \(0.529044\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 25.0000 1.50756
\(276\) 2.00000 0.120386
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 3.00000 0.178647
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 9.00000 0.534052
\(285\) 0 0
\(286\) −5.00000 −0.295656
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 10.0000 0.585206
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −5.00000 −0.290129
\(298\) 10.0000 0.579284
\(299\) 2.00000 0.115663
\(300\) −5.00000 −0.288675
\(301\) 0 0
\(302\) −17.0000 −0.978240
\(303\) 6.00000 0.344691
\(304\) −7.00000 −0.401478
\(305\) 0 0
\(306\) −7.00000 −0.400163
\(307\) −19.0000 −1.08439 −0.542194 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 0 0
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 1.00000 0.0566139
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −11.0000 −0.620766
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) 1.00000 0.0560772
\(319\) 45.0000 2.51952
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 49.0000 2.72643
\(324\) 1.00000 0.0555556
\(325\) −5.00000 −0.277350
\(326\) 13.0000 0.720003
\(327\) 8.00000 0.442401
\(328\) −4.00000 −0.220863
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −16.0000 −0.878114
\(333\) 4.00000 0.219199
\(334\) 5.00000 0.273588
\(335\) 0 0
\(336\) 0 0
\(337\) −33.0000 −1.79762 −0.898812 0.438334i \(-0.855569\pi\)
−0.898812 + 0.438334i \(0.855569\pi\)
\(338\) 1.00000 0.0543928
\(339\) 1.00000 0.0543125
\(340\) 0 0
\(341\) 0 0
\(342\) −7.00000 −0.378517
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −9.00000 −0.482451
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −5.00000 −0.266501
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) −7.00000 −0.372046
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) −10.0000 −0.528516
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 11.0000 0.578147
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) −13.0000 −0.679521
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 2.00000 0.104257
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −21.0000 −1.08734 −0.543669 0.839299i \(-0.682965\pi\)
−0.543669 + 0.839299i \(0.682965\pi\)
\(374\) 35.0000 1.80981
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 22.0000 1.12709
\(382\) −12.0000 −0.613973
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) 2.00000 0.101666
\(388\) −6.00000 −0.304604
\(389\) −23.0000 −1.16615 −0.583073 0.812420i \(-0.698150\pi\)
−0.583073 + 0.812420i \(0.698150\pi\)
\(390\) 0 0
\(391\) −14.0000 −0.708010
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −18.0000 −0.902258
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 3.00000 0.149626
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −20.0000 −0.991363
\(408\) −7.00000 −0.346552
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 10.0000 0.492665
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) −16.0000 −0.783523
\(418\) 35.0000 1.71191
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) −26.0000 −1.26566
\(423\) 3.00000 0.145865
\(424\) 1.00000 0.0485643
\(425\) 35.0000 1.69775
\(426\) 9.00000 0.436051
\(427\) 0 0
\(428\) 0 0
\(429\) −5.00000 −0.241402
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 1.00000 0.0481125
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) −14.0000 −0.669711
\(438\) 10.0000 0.477818
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7.00000 −0.332956
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) 1.00000 0.0473514
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) −5.00000 −0.235702
\(451\) 20.0000 0.941763
\(452\) 1.00000 0.0470360
\(453\) −17.0000 −0.798730
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −7.00000 −0.327805
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −18.0000 −0.841085
\(459\) −7.00000 −0.326732
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) −3.00000 −0.138972
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −11.0000 −0.506853
\(472\) −7.00000 −0.322201
\(473\) −10.0000 −0.459800
\(474\) 14.0000 0.643041
\(475\) 35.0000 1.60591
\(476\) 0 0
\(477\) 1.00000 0.0457869
\(478\) −1.00000 −0.0457389
\(479\) 11.0000 0.502603 0.251301 0.967909i \(-0.419141\pi\)
0.251301 + 0.967909i \(0.419141\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −5.00000 −0.226572 −0.113286 0.993562i \(-0.536138\pi\)
−0.113286 + 0.993562i \(0.536138\pi\)
\(488\) −13.0000 −0.588482
\(489\) 13.0000 0.587880
\(490\) 0 0
\(491\) −22.0000 −0.992846 −0.496423 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) −4.00000 −0.180334
\(493\) 63.0000 2.83738
\(494\) −7.00000 −0.314945
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 0 0
\(501\) 5.00000 0.223384
\(502\) −18.0000 −0.803379
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −10.0000 −0.444554
\(507\) 1.00000 0.0444116
\(508\) 22.0000 0.976092
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −7.00000 −0.309058
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) −15.0000 −0.659699
\(518\) 0 0
\(519\) 13.0000 0.570637
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) −9.00000 −0.393919
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) −5.00000 −0.217597
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −7.00000 −0.303774
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 3.00000 0.129580
\(537\) −10.0000 −0.431532
\(538\) −23.0000 −0.991600
\(539\) 0 0
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) −3.00000 −0.128861
\(543\) 11.0000 0.472055
\(544\) −7.00000 −0.300123
\(545\) 0 0
\(546\) 0 0
\(547\) −38.0000 −1.62476 −0.812381 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) −12.0000 −0.512615
\(549\) −13.0000 −0.554826
\(550\) 25.0000 1.06600
\(551\) 63.0000 2.68389
\(552\) 2.00000 0.0851257
\(553\) 0 0
\(554\) 19.0000 0.807233
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 35.0000 1.47770
\(562\) −24.0000 −1.01238
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 3.00000 0.126323
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) 0 0
\(568\) 9.00000 0.377632
\(569\) −31.0000 −1.29959 −0.649794 0.760111i \(-0.725145\pi\)
−0.649794 + 0.760111i \(0.725145\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −5.00000 −0.209061
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) −10.0000 −0.417029
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 32.0000 1.33102
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) −5.00000 −0.207079
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) 45.0000 1.85735 0.928674 0.370896i \(-0.120949\pi\)
0.928674 + 0.370896i \(0.120949\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 4.00000 0.164399
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −18.0000 −0.736691
\(598\) 2.00000 0.0817861
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) −5.00000 −0.204124
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 0 0
\(603\) 3.00000 0.122169
\(604\) −17.0000 −0.691720
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −7.00000 −0.283887
\(609\) 0 0
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) −7.00000 −0.282958
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −19.0000 −0.766778
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 10.0000 0.402259
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) −20.0000 −0.801927
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 25.0000 1.00000
\(626\) 6.00000 0.239808
\(627\) 35.0000 1.39777
\(628\) −11.0000 −0.438948
\(629\) −28.0000 −1.11643
\(630\) 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
\(633\) −26.0000 −1.03341
\(634\) 26.0000 1.03259
\(635\) 0 0
\(636\) 1.00000 0.0396526
\(637\) 0 0
\(638\) 45.0000 1.78157
\(639\) 9.00000 0.356034
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 9.00000 0.354925 0.177463 0.984128i \(-0.443211\pi\)
0.177463 + 0.984128i \(0.443211\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 49.0000 1.92788
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 1.00000 0.0392837
\(649\) 35.0000 1.37387
\(650\) −5.00000 −0.196116
\(651\) 0 0
\(652\) 13.0000 0.509119
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 4.00000 0.155464
\(663\) −7.00000 −0.271857
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −18.0000 −0.696963
\(668\) 5.00000 0.193456
\(669\) 1.00000 0.0386622
\(670\) 0 0
\(671\) 65.0000 2.50930
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) −33.0000 −1.27111
\(675\) −5.00000 −0.192450
\(676\) 1.00000 0.0384615
\(677\) −25.0000 −0.960828 −0.480414 0.877042i \(-0.659514\pi\)
−0.480414 + 0.877042i \(0.659514\pi\)
\(678\) 1.00000 0.0384048
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −40.0000 −1.53056 −0.765279 0.643699i \(-0.777399\pi\)
−0.765279 + 0.643699i \(0.777399\pi\)
\(684\) −7.00000 −0.267652
\(685\) 0 0
\(686\) 0 0
\(687\) −18.0000 −0.686743
\(688\) 2.00000 0.0762493
\(689\) 1.00000 0.0380970
\(690\) 0 0
\(691\) −5.00000 −0.190209 −0.0951045 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) 13.0000 0.494186
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 28.0000 1.06058
\(698\) −2.00000 −0.0757011
\(699\) −3.00000 −0.113470
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 1.00000 0.0377426
\(703\) −28.0000 −1.05604
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 0 0
\(708\) −7.00000 −0.263076
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 12.0000 0.449719
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) −1.00000 −0.0373457
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) 11.0000 0.408812
\(725\) 45.0000 1.67126
\(726\) 14.0000 0.519589
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −14.0000 −0.517809
\(732\) −13.0000 −0.480494
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) −15.0000 −0.552532
\(738\) −4.00000 −0.147242
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) −7.00000 −0.257151
\(742\) 0 0
\(743\) −5.00000 −0.183432 −0.0917161 0.995785i \(-0.529235\pi\)
−0.0917161 + 0.995785i \(0.529235\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −21.0000 −0.768865
\(747\) −16.0000 −0.585409
\(748\) 35.0000 1.27973
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 3.00000 0.109399
\(753\) −18.0000 −0.655956
\(754\) −9.00000 −0.327761
\(755\) 0 0
\(756\) 0 0
\(757\) 29.0000 1.05402 0.527011 0.849858i \(-0.323312\pi\)
0.527011 + 0.849858i \(0.323312\pi\)
\(758\) −4.00000 −0.145287
\(759\) −10.0000 −0.362977
\(760\) 0 0
\(761\) 8.00000 0.290000 0.145000 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) 22.0000 0.796976
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) −7.00000 −0.252755
\(768\) 1.00000 0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −8.00000 −0.287926
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) −23.0000 −0.824590
\(779\) 28.0000 1.00320
\(780\) 0 0
\(781\) −45.0000 −1.61023
\(782\) −14.0000 −0.500639
\(783\) −9.00000 −0.321634
\(784\) 0 0
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) −17.0000 −0.605985 −0.302992 0.952993i \(-0.597986\pi\)
−0.302992 + 0.952993i \(0.597986\pi\)
\(788\) 22.0000 0.783718
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) 0 0
\(792\) −5.00000 −0.177667
\(793\) −13.0000 −0.461644
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −18.0000 −0.637993
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) −21.0000 −0.742927
\(800\) −5.00000 −0.176777
\(801\) 12.0000 0.423999
\(802\) −38.0000 −1.34183
\(803\) −50.0000 −1.76446
\(804\) 3.00000 0.105802
\(805\) 0 0
\(806\) 0 0
\(807\) −23.0000 −0.809638
\(808\) 6.00000 0.211079
\(809\) 25.0000 0.878953 0.439477 0.898254i \(-0.355164\pi\)
0.439477 + 0.898254i \(0.355164\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) −3.00000 −0.105215
\(814\) −20.0000 −0.701000
\(815\) 0 0
\(816\) −7.00000 −0.245049
\(817\) −14.0000 −0.489798
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) 0 0
\(821\) −20.0000 −0.698005 −0.349002 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) −12.0000 −0.418548
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 10.0000 0.348367
\(825\) 25.0000 0.870388
\(826\) 0 0
\(827\) 21.0000 0.730242 0.365121 0.930960i \(-0.381028\pi\)
0.365121 + 0.930960i \(0.381028\pi\)
\(828\) 2.00000 0.0695048
\(829\) 41.0000 1.42399 0.711994 0.702185i \(-0.247792\pi\)
0.711994 + 0.702185i \(0.247792\pi\)
\(830\) 0 0
\(831\) 19.0000 0.659103
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 35.0000 1.21050
\(837\) 0 0
\(838\) −26.0000 −0.898155
\(839\) −5.00000 −0.172619 −0.0863096 0.996268i \(-0.527507\pi\)
−0.0863096 + 0.996268i \(0.527507\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 38.0000 1.30957
\(843\) −24.0000 −0.826604
\(844\) −26.0000 −0.894957
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 0 0
\(848\) 1.00000 0.0343401
\(849\) 10.0000 0.343199
\(850\) 35.0000 1.20049
\(851\) 8.00000 0.274236
\(852\) 9.00000 0.308335
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −13.0000 −0.444072 −0.222036 0.975039i \(-0.571270\pi\)
−0.222036 + 0.975039i \(0.571270\pi\)
\(858\) −5.00000 −0.170697
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 11.0000 0.373795
\(867\) 32.0000 1.08678
\(868\) 0 0
\(869\) −70.0000 −2.37459
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 8.00000 0.270914
\(873\) −6.00000 −0.203069
\(874\) −14.0000 −0.473557
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) −12.0000 −0.405211 −0.202606 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 22.0000 0.742464
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −7.00000 −0.235435
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 4.00000 0.134231
\(889\) 0 0
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 1.00000 0.0334825
\(893\) −21.0000 −0.702738
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) 4.00000 0.133482
\(899\) 0 0
\(900\) −5.00000 −0.166667
\(901\) −7.00000 −0.233204
\(902\) 20.0000 0.665927
\(903\) 0 0
\(904\) 1.00000 0.0332595
\(905\) 0 0
\(906\) −17.0000 −0.564787
\(907\) 54.0000 1.79304 0.896520 0.443003i \(-0.146087\pi\)
0.896520 + 0.443003i \(0.146087\pi\)
\(908\) −12.0000 −0.398234
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) −7.00000 −0.231793
\(913\) 80.0000 2.64761
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −18.0000 −0.594737
\(917\) 0 0
\(918\) −7.00000 −0.231034
\(919\) −14.0000 −0.461817 −0.230909 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) −19.0000 −0.626071
\(922\) 8.00000 0.263466
\(923\) 9.00000 0.296239
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 32.0000 1.05159
\(927\) 10.0000 0.328443
\(928\) −9.00000 −0.295439
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −3.00000 −0.0982683
\(933\) −20.0000 −0.654771
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 1.00000 0.0326686 0.0163343 0.999867i \(-0.494800\pi\)
0.0163343 + 0.999867i \(0.494800\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 8.00000 0.260793 0.130396 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) −11.0000 −0.358399
\(943\) −8.00000 −0.260516
\(944\) −7.00000 −0.227831
\(945\) 0 0
\(946\) −10.0000 −0.325128
\(947\) −5.00000 −0.162478 −0.0812391 0.996695i \(-0.525888\pi\)
−0.0812391 + 0.996695i \(0.525888\pi\)
\(948\) 14.0000 0.454699
\(949\) 10.0000 0.324614
\(950\) 35.0000 1.13555
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 1.00000 0.0323762
\(955\) 0 0
\(956\) −1.00000 −0.0323423
\(957\) 45.0000 1.45464
\(958\) 11.0000 0.355394
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.00000 0.0321578 0.0160789 0.999871i \(-0.494882\pi\)
0.0160789 + 0.999871i \(0.494882\pi\)
\(968\) 14.0000 0.449977
\(969\) 49.0000 1.57411
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −5.00000 −0.160210
\(975\) −5.00000 −0.160128
\(976\) −13.0000 −0.416120
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 13.0000 0.415694
\(979\) −60.0000 −1.91761
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) −22.0000 −0.702048
\(983\) −59.0000 −1.88181 −0.940904 0.338674i \(-0.890022\pi\)
−0.940904 + 0.338674i \(0.890022\pi\)
\(984\) −4.00000 −0.127515
\(985\) 0 0
\(986\) 63.0000 2.00633
\(987\) 0 0
\(988\) −7.00000 −0.222700
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 0 0
\(996\) −16.0000 −0.506979
\(997\) −21.0000 −0.665077 −0.332538 0.943090i \(-0.607905\pi\)
−0.332538 + 0.943090i \(0.607905\pi\)
\(998\) 40.0000 1.26618
\(999\) 4.00000 0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3822.2.a.bf.1.1 1
7.3 odd 6 546.2.i.d.79.1 2
7.5 odd 6 546.2.i.d.235.1 yes 2
7.6 odd 2 3822.2.a.u.1.1 1
21.5 even 6 1638.2.j.i.235.1 2
21.17 even 6 1638.2.j.i.1171.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.d.79.1 2 7.3 odd 6
546.2.i.d.235.1 yes 2 7.5 odd 6
1638.2.j.i.235.1 2 21.5 even 6
1638.2.j.i.1171.1 2 21.17 even 6
3822.2.a.u.1.1 1 7.6 odd 2
3822.2.a.bf.1.1 1 1.1 even 1 trivial