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Properties

Label 3822.2.a.a.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} +7.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} +9.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -9.00000 q^{29} -3.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -7.00000 q^{37} -7.00000 q^{38} +1.00000 q^{39} +3.00000 q^{40} -12.0000 q^{41} -1.00000 q^{43} +3.00000 q^{44} -3.00000 q^{45} -9.00000 q^{46} -1.00000 q^{48} -4.00000 q^{50} -3.00000 q^{51} -1.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -9.00000 q^{55} -7.00000 q^{57} +9.00000 q^{58} -12.0000 q^{59} +3.00000 q^{60} +1.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +3.00000 q^{66} +14.0000 q^{67} +3.00000 q^{68} -9.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} +7.00000 q^{73} +7.00000 q^{74} -4.00000 q^{75} +7.00000 q^{76} -1.00000 q^{78} -10.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} +6.00000 q^{83} -9.00000 q^{85} +1.00000 q^{86} +9.00000 q^{87} -3.00000 q^{88} +6.00000 q^{89} +3.00000 q^{90} +9.00000 q^{92} -4.00000 q^{93} -21.0000 q^{95} +1.00000 q^{96} +10.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 9.00000 1.87663 0.938315 − 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) 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 \(0.814897\pi\)
\(30\) −3.00000 −0.547723
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −7.00000 −1.13555
\(39\) 1.00000 0.160128
\(40\) 3.00000 0.474342
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 3.00000 0.452267
\(45\) −3.00000 −0.447214
\(46\) −9.00000 −1.32698
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) −3.00000 −0.420084
\(52\) −1.00000 −0.138675
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 9.00000 1.18176
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 3.00000 0.387298
\(61\) 1.00000 0.128037 0.0640184 − 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 3.00000 0.369274
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 3.00000 0.363803
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 7.00000 0.813733
\(75\) −4.00000 −0.461880
\(76\) 7.00000 0.802955
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 1.00000 0.107833
\(87\) 9.00000 0.964901
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) 9.00000 0.938315
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −21.0000 −2.15455
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 4.00000 0.400000
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 3.00000 0.297044
\(103\) 13.0000 1.28093 0.640464 − 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.0000 1.05361 0.526804 − 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 9.00000 0.858116
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 7.00000 0.655610
\(115\) −27.0000 −2.51776
\(116\) −9.00000 −0.835629
\(117\) −1.00000 −0.0924500
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −3.00000 −0.273861
\(121\) −2.00000 −0.181818
\(122\) −1.00000 −0.0905357
\(123\) 12.0000 1.08200
\(124\) 4.00000 0.359211
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.00000 0.0880451
\(130\) −3.00000 −0.263117
\(131\) 15.0000 1.31056 0.655278 − 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) −14.0000 −1.20942
\(135\) 3.00000 0.258199
\(136\) −3.00000 −0.257248
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 9.00000 0.766131
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −3.00000 −0.250873
\(144\) 1.00000 0.0833333
\(145\) 27.0000 2.24223
\(146\) −7.00000 −0.579324
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 4.00000 0.326599
\(151\) 17.0000 1.38344 0.691720 − 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) −7.00000 −0.567775
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) −12.0000 −0.963863
\(156\) 1.00000 0.0800641
\(157\) 13.0000 1.03751 0.518756 − 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 10.0000 0.795557
\(159\) 6.00000 0.475831
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −12.0000 −0.937043
\(165\) 9.00000 0.700649
\(166\) −6.00000 −0.465690
\(167\) −3.00000 −0.232147 −0.116073 − 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 9.00000 0.690268
\(171\) 7.00000 0.535303
\(172\) −1.00000 −0.0762493
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 12.0000 0.901975
\(178\) −6.00000 −0.449719
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −3.00000 −0.223607
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) −9.00000 −0.663489
\(185\) 21.0000 1.54395
\(186\) 4.00000 0.293294
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) 0 0
\(190\) 21.0000 1.52350
\(191\) −15.0000 −1.08536 −0.542681 − 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.00000 −0.287926 −0.143963 − 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −10.0000 −0.717958
\(195\) −3.00000 −0.214834
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 − 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) −3.00000 −0.213201
\(199\) 7.00000 0.496217 0.248108 − 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) −4.00000 −0.282843
\(201\) −14.0000 −0.987484
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) −3.00000 −0.210042
\(205\) 36.0000 2.51435
\(206\) −13.0000 −0.905753
\(207\) 9.00000 0.625543
\(208\) −1.00000 −0.0693375
\(209\) 21.0000 1.45260
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 − 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −6.00000 −0.412082
\(213\) −12.0000 −0.822226
\(214\) 6.00000 0.410152
\(215\) 3.00000 0.204598
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −11.0000 −0.745014
\(219\) −7.00000 −0.473016
\(220\) −9.00000 −0.606780
\(221\) −3.00000 −0.201802
\(222\) −7.00000 −0.469809
\(223\) −26.0000 −1.74109 −0.870544 − 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 18.0000 1.19734
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −7.00000 −0.463586
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 27.0000 1.78033
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 3.00000 0.193649
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) −7.00000 −0.445399
\(248\) −4.00000 −0.254000
\(249\) −6.00000 −0.380235
\(250\) −3.00000 −0.189737
\(251\) −21.0000 −1.32551 −0.662754 − 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 27.0000 1.69748
\(254\) −2.00000 −0.125491
\(255\) 9.00000 0.563602
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −1.00000 −0.0622573
\(259\) 0 0
\(260\) 3.00000 0.186052
\(261\) −9.00000 −0.557086
\(262\) −15.0000 −0.926703
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 3.00000 0.184637
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 14.0000 0.855186
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −3.00000 −0.182574
\(271\) −2.00000 −0.121491 −0.0607457 − 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) 12.0000 0.723627
\(276\) −9.00000 −0.541736
\(277\) 26.0000 1.56219 0.781094 − 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −4.00000 −0.239904
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000 0.712069
\(285\) 21.0000 1.24393
\(286\) 3.00000 0.177394
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) −27.0000 −1.58549
\(291\) −10.0000 −0.586210
\(292\) 7.00000 0.409644
\(293\) 18.0000 1.05157 0.525786 − 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) 7.00000 0.406867
\(297\) −3.00000 −0.174078
\(298\) 12.0000 0.695141
\(299\) −9.00000 −0.520483
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −17.0000 −0.978240
\(303\) 12.0000 0.689382
\(304\) 7.00000 0.401478
\(305\) −3.00000 −0.171780
\(306\) −3.00000 −0.171499
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 12.0000 0.681554
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) −6.00000 −0.336463
\(319\) −27.0000 −1.51171
\(320\) −3.00000 −0.167705
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 21.0000 1.16847
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −2.00000 −0.110770
\(327\) −11.0000 −0.608301
\(328\) 12.0000 0.662589
\(329\) 0 0
\(330\) −9.00000 −0.495434
\(331\) −10.0000 −0.549650 −0.274825 − 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 6.00000 0.329293
\(333\) −7.00000 −0.383598
\(334\) 3.00000 0.164153
\(335\) −42.0000 −2.29471
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 18.0000 0.977626
\(340\) −9.00000 −0.488094
\(341\) 12.0000 0.649836
\(342\) −7.00000 −0.378517
\(343\) 0 0
\(344\) 1.00000 0.0539164
\(345\) 27.0000 1.45363
\(346\) −6.00000 −0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 9.00000 0.482451
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −3.00000 −0.159901
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −12.0000 −0.637793
\(355\) −36.0000 −1.91068
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 3.00000 0.158114
\(361\) 30.0000 1.57895
\(362\) 2.00000 0.105118
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −21.0000 −1.09919
\(366\) 1.00000 0.0522708
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 9.00000 0.469157
\(369\) −12.0000 −0.624695
\(370\) −21.0000 −1.09174
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −9.00000 −0.465379
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −21.0000 −1.07728
\(381\) −2.00000 −0.102463
\(382\) 15.0000 0.767467
\(383\) 21.0000 1.07305 0.536525 − 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) −1.00000 −0.0508329
\(388\) 10.0000 0.507673
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 3.00000 0.151911
\(391\) 27.0000 1.36545
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 12.0000 0.604551
\(395\) 30.0000 1.50946
\(396\) 3.00000 0.150756
\(397\) 34.0000 1.70641 0.853206 − 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) −7.00000 −0.350878
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 14.0000 0.698257
\(403\) −4.00000 −0.199254
\(404\) −12.0000 −0.597022
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) −21.0000 −1.04093
\(408\) 3.00000 0.148522
\(409\) 13.0000 0.642809 0.321404 − 0.946942i \(-0.395845\pi\)
0.321404 + 0.946942i \(0.395845\pi\)
\(410\) −36.0000 −1.77791
\(411\) −9.00000 −0.443937
\(412\) 13.0000 0.640464
\(413\) 0 0
\(414\) −9.00000 −0.442326
\(415\) −18.0000 −0.883585
\(416\) 1.00000 0.0490290
\(417\) −4.00000 −0.195881
\(418\) −21.0000 −1.02714
\(419\) −3.00000 −0.146560 −0.0732798 − 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −5.00000 −0.243396
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 12.0000 0.582086
\(426\) 12.0000 0.581402
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) 3.00000 0.144841
\(430\) −3.00000 −0.144673
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) −27.0000 −1.29455
\(436\) 11.0000 0.526804
\(437\) 63.0000 3.01370
\(438\) 7.00000 0.334473
\(439\) 1.00000 0.0477274 0.0238637 − 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 9.00000 0.429058
\(441\) 0 0
\(442\) 3.00000 0.142695
\(443\) 18.0000 0.855206 0.427603 − 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 7.00000 0.332205
\(445\) −18.0000 −0.853282
\(446\) 26.0000 1.23114
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 − 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) −4.00000 −0.188562
\(451\) −36.0000 −1.69517
\(452\) −18.0000 −0.846649
\(453\) −17.0000 −0.798730
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 7.00000 0.327805
\(457\) −28.0000 −1.30978 −0.654892 − 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 14.0000 0.654177
\(459\) −3.00000 −0.140028
\(460\) −27.0000 −1.25888
\(461\) −33.0000 −1.53696 −0.768482 − 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) 41.0000 1.90543 0.952716 − 0.303863i \(-0.0982765\pi\)
0.952716 + 0.303863i \(0.0982765\pi\)
\(464\) −9.00000 −0.417815
\(465\) 12.0000 0.556487
\(466\) −24.0000 −1.11178
\(467\) 3.00000 0.138823 0.0694117 − 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 12.0000 0.552345
\(473\) −3.00000 −0.137940
\(474\) −10.0000 −0.459315
\(475\) 28.0000 1.28473
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −24.0000 −1.09773
\(479\) 15.0000 0.685367 0.342684 − 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) −3.00000 −0.136931
\(481\) 7.00000 0.319173
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −30.0000 −1.36223
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 12.0000 0.541002
\(493\) −27.0000 −1.21602
\(494\) 7.00000 0.314945
\(495\) −9.00000 −0.404520
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 3.00000 0.134164
\(501\) 3.00000 0.134030
\(502\) 21.0000 0.937276
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) −27.0000 −1.20030
\(507\) −1.00000 −0.0444116
\(508\) 2.00000 0.0887357
\(509\) −9.00000 −0.398918 −0.199459 − 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) −9.00000 −0.398527
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −7.00000 −0.309058
\(514\) −18.0000 −0.793946
\(515\) −39.0000 −1.71855
\(516\) 1.00000 0.0440225
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) −3.00000 −0.131559
\(521\) 33.0000 1.44576 0.722878 − 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) 9.00000 0.393919
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 12.0000 0.522728
\(528\) −3.00000 −0.130558
\(529\) 58.0000 2.52174
\(530\) −18.0000 −0.781870
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 6.00000 0.259645
\(535\) 18.0000 0.778208
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) 11.0000 0.472927 0.236463 − 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 2.00000 0.0859074
\(543\) 2.00000 0.0858282
\(544\) −3.00000 −0.128624
\(545\) −33.0000 −1.41356
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 9.00000 0.384461
\(549\) 1.00000 0.0426790
\(550\) −12.0000 −0.511682
\(551\) −63.0000 −2.68389
\(552\) 9.00000 0.383065
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) −21.0000 −0.891400
\(556\) 4.00000 0.169638
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −4.00000 −0.169334
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 6.00000 0.253095
\(563\) 15.0000 0.632175 0.316087 − 0.948730i \(-0.397631\pi\)
0.316087 + 0.948730i \(0.397631\pi\)
\(564\) 0 0
\(565\) 54.0000 2.27180
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −21.0000 −0.879593
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −3.00000 −0.125436
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) 36.0000 1.50130
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 8.00000 0.332756
\(579\) 4.00000 0.166234
\(580\) 27.0000 1.12111
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) −18.0000 −0.745484
\(584\) −7.00000 −0.289662
\(585\) 3.00000 0.124035
\(586\) −18.0000 −0.743573
\(587\) 18.0000 0.742940 0.371470 − 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) 28.0000 1.15372
\(590\) −36.0000 −1.48210
\(591\) 12.0000 0.493614
\(592\) −7.00000 −0.287698
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) −7.00000 −0.286491
\(598\) 9.00000 0.368037
\(599\) −9.00000 −0.367730 −0.183865 − 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) 4.00000 0.163299
\(601\) −8.00000 −0.326327 −0.163163 − 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) 17.0000 0.691720
\(605\) 6.00000 0.243935
\(606\) −12.0000 −0.487467
\(607\) −23.0000 −0.933541 −0.466771 − 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(608\) −7.00000 −0.283887
\(609\) 0 0
\(610\) 3.00000 0.121466
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) 11.0000 0.444286 0.222143 − 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) −16.0000 −0.645707
\(615\) −36.0000 −1.45166
\(616\) 0 0
\(617\) 27.0000 1.08698 0.543490 − 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 13.0000 0.522937
\(619\) −17.0000 −0.683288 −0.341644 − 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) −12.0000 −0.481932
\(621\) −9.00000 −0.361158
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −29.0000 −1.16000
\(626\) 8.00000 0.319744
\(627\) −21.0000 −0.838659
\(628\) 13.0000 0.518756
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 − 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 10.0000 0.397779
\(633\) −5.00000 −0.198732
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 27.0000 1.06894
\(639\) 12.0000 0.474713
\(640\) 3.00000 0.118585
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −6.00000 −0.236801
\(643\) 49.0000 1.93237 0.966186 − 0.257847i \(-0.0830131\pi\)
0.966186 + 0.257847i \(0.0830131\pi\)
\(644\) 0 0
\(645\) −3.00000 −0.118125
\(646\) −21.0000 −0.826234
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −36.0000 −1.41312
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) 9.00000 0.352197 0.176099 − 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 11.0000 0.430134
\(655\) −45.0000 −1.75830
\(656\) −12.0000 −0.468521
\(657\) 7.00000 0.273096
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 9.00000 0.350325
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 10.0000 0.388661
\(663\) 3.00000 0.116510
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) −81.0000 −3.13633
\(668\) −3.00000 −0.116073
\(669\) 26.0000 1.00522
\(670\) 42.0000 1.62260
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 − 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 13.0000 0.500741
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −18.0000 −0.691286
\(679\) 0 0
\(680\) 9.00000 0.345134
\(681\) −18.0000 −0.689761
\(682\) −12.0000 −0.459504
\(683\) 33.0000 1.26271 0.631355 − 0.775494i \(-0.282499\pi\)
0.631355 + 0.775494i \(0.282499\pi\)
\(684\) 7.00000 0.267652
\(685\) −27.0000 −1.03162
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −1.00000 −0.0381246
\(689\) 6.00000 0.228582
\(690\) −27.0000 −1.02787
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −12.0000 −0.455186
\(696\) −9.00000 −0.341144
\(697\) −36.0000 −1.36360
\(698\) 26.0000 0.984115
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −49.0000 −1.84807
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 36.0000 1.35106
\(711\) −10.0000 −0.375029
\(712\) −6.00000 −0.224860
\(713\) 36.0000 1.34821
\(714\) 0 0
\(715\) 9.00000 0.336581
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) −24.0000 −0.895672
\(719\) −18.0000 −0.671287 −0.335643 − 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) −30.0000 −1.11648
\(723\) −10.0000 −0.371904
\(724\) −2.00000 −0.0743294
\(725\) −36.0000 −1.33701
\(726\) −2.00000 −0.0742270
\(727\) 1.00000 0.0370879 0.0185440 − 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 21.0000 0.777245
\(731\) −3.00000 −0.110959
\(732\) −1.00000 −0.0369611
\(733\) −32.0000 −1.18195 −0.590973 − 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) 42.0000 1.54709
\(738\) 12.0000 0.441726
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 21.0000 0.771975
\(741\) 7.00000 0.257151
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 − 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 4.00000 0.146647
\(745\) 36.0000 1.31894
\(746\) 4.00000 0.146450
\(747\) 6.00000 0.219529
\(748\) 9.00000 0.329073
\(749\) 0 0
\(750\) 3.00000 0.109545
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 21.0000 0.765283
\(754\) −9.00000 −0.327761
\(755\) −51.0000 −1.85608
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 16.0000 0.581146
\(759\) −27.0000 −0.980038
\(760\) 21.0000 0.761750
\(761\) 12.0000 0.435000 0.217500 − 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 2.00000 0.0724524
\(763\) 0 0
\(764\) −15.0000 −0.542681
\(765\) −9.00000 −0.325396
\(766\) −21.0000 −0.758761
\(767\) 12.0000 0.433295
\(768\) −1.00000 −0.0360844
\(769\) 13.0000 0.468792 0.234396 − 0.972141i \(-0.424689\pi\)
0.234396 + 0.972141i \(0.424689\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −4.00000 −0.143963
\(773\) −33.0000 −1.18693 −0.593464 − 0.804861i \(-0.702240\pi\)
−0.593464 + 0.804861i \(0.702240\pi\)
\(774\) 1.00000 0.0359443
\(775\) 16.0000 0.574737
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −84.0000 −3.00961
\(780\) −3.00000 −0.107417
\(781\) 36.0000 1.28818
\(782\) −27.0000 −0.965518
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −39.0000 −1.39197
\(786\) 15.0000 0.535032
\(787\) 31.0000 1.10503 0.552515 − 0.833503i \(-0.313668\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) −12.0000 −0.427482
\(789\) −12.0000 −0.427211
\(790\) −30.0000 −1.06735
\(791\) 0 0
\(792\) −3.00000 −0.106600
\(793\) −1.00000 −0.0355110
\(794\) −34.0000 −1.20661
\(795\) −18.0000 −0.638394
\(796\) 7.00000 0.248108
\(797\) −36.0000 −1.27519 −0.637593 − 0.770374i \(-0.720070\pi\)
−0.637593 + 0.770374i \(0.720070\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) 6.00000 0.212000
\(802\) −18.0000 −0.635602
\(803\) 21.0000 0.741074
\(804\) −14.0000 −0.493742
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 6.00000 0.211210
\(808\) 12.0000 0.422159
\(809\) 54.0000 1.89854 0.949269 − 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 3.00000 0.105409
\(811\) 43.0000 1.50993 0.754967 − 0.655763i \(-0.227653\pi\)
0.754967 + 0.655763i \(0.227653\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 21.0000 0.736050
\(815\) −6.00000 −0.210171
\(816\) −3.00000 −0.105021
\(817\) −7.00000 −0.244899
\(818\) −13.0000 −0.454534
\(819\) 0 0
\(820\) 36.0000 1.25717
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 9.00000 0.313911
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −13.0000 −0.452876
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 27.0000 0.938882 0.469441 − 0.882964i \(-0.344455\pi\)
0.469441 + 0.882964i \(0.344455\pi\)
\(828\) 9.00000 0.312772
\(829\) 25.0000 0.868286 0.434143 − 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 18.0000 0.624789
\(831\) −26.0000 −0.901930
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) 9.00000 0.311458
\(836\) 21.0000 0.726300
\(837\) −4.00000 −0.138260
\(838\) 3.00000 0.103633
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 10.0000 0.344623
\(843\) 6.00000 0.206651
\(844\) 5.00000 0.172107
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −4.00000 −0.137280
\(850\) −12.0000 −0.411597
\(851\) −63.0000 −2.15961
\(852\) −12.0000 −0.411113
\(853\) −44.0000 −1.50653 −0.753266 − 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 0 0
\(855\) −21.0000 −0.718185
\(856\) 6.00000 0.205076
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −3.00000 −0.102418
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 3.00000 0.102299
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.0000 −0.612018
\(866\) −16.0000 −0.543702
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −30.0000 −1.01768
\(870\) 27.0000 0.915386
\(871\) −14.0000 −0.474372
\(872\) −11.0000 −0.372507
\(873\) 10.0000 0.338449
\(874\) −63.0000 −2.13101
\(875\) 0 0
\(876\) −7.00000 −0.236508
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −1.00000 −0.0337484
\(879\) −18.0000 −0.607125
\(880\) −9.00000 −0.303390
\(881\) 3.00000 0.101073 0.0505363 − 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) −25.0000 −0.841317 −0.420658 − 0.907219i \(-0.638201\pi\)
−0.420658 + 0.907219i \(0.638201\pi\)
\(884\) −3.00000 −0.100901
\(885\) −36.0000 −1.21013
\(886\) −18.0000 −0.604722
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) −7.00000 −0.234905
\(889\) 0 0
\(890\) 18.0000 0.603361
\(891\) 3.00000 0.100504
\(892\) −26.0000 −0.870544
\(893\) 0 0
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) 9.00000 0.300501
\(898\) −21.0000 −0.700779
\(899\) −36.0000 −1.20067
\(900\) 4.00000 0.133333
\(901\) −18.0000 −0.599667
\(902\) 36.0000 1.19867
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 6.00000 0.199447
\(906\) 17.0000 0.564787
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 18.0000 0.597351
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −33.0000 −1.09334 −0.546669 − 0.837349i \(-0.684105\pi\)
−0.546669 + 0.837349i \(0.684105\pi\)
\(912\) −7.00000 −0.231793
\(913\) 18.0000 0.595713
\(914\) 28.0000 0.926158
\(915\) 3.00000 0.0991769
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 3.00000 0.0990148
\(919\) −34.0000 −1.12156 −0.560778 − 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 27.0000 0.890164
\(921\) −16.0000 −0.527218
\(922\) 33.0000 1.08680
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) −41.0000 −1.34734
\(927\) 13.0000 0.426976
\(928\) 9.00000 0.295439
\(929\) −12.0000 −0.393707 −0.196854 − 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) −12.0000 −0.393496
\(931\) 0 0
\(932\) 24.0000 0.786146
\(933\) 18.0000 0.589294
\(934\) −3.00000 −0.0981630
\(935\) −27.0000 −0.882994
\(936\) 1.00000 0.0326860
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 13.0000 0.423563
\(943\) −108.000 −3.51696
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) −57.0000 −1.85225 −0.926126 − 0.377215i \(-0.876882\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(948\) 10.0000 0.324785
\(949\) −7.00000 −0.227230
\(950\) −28.0000 −0.908440
\(951\) 0 0
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 − 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 6.00000 0.194257
\(955\) 45.0000 1.45617
\(956\) 24.0000 0.776215
\(957\) 27.0000 0.872786
\(958\) −15.0000 −0.484628
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) −15.0000 −0.483871
\(962\) −7.00000 −0.225689
\(963\) −6.00000 −0.193347
\(964\) 10.0000 0.322078
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −13.0000 −0.418052 −0.209026 − 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) 2.00000 0.0642824
\(969\) −21.0000 −0.674617
\(970\) 30.0000 0.963242
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 4.00000 0.128103
\(976\) 1.00000 0.0320092
\(977\) 15.0000 0.479893 0.239946 − 0.970786i \(-0.422870\pi\)
0.239946 + 0.970786i \(0.422870\pi\)
\(978\) 2.00000 0.0639529
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) −30.0000 −0.957338
\(983\) 39.0000 1.24391 0.621953 − 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) −12.0000 −0.382546
\(985\) 36.0000 1.14706
\(986\) 27.0000 0.859855
\(987\) 0 0
\(988\) −7.00000 −0.222700
\(989\) −9.00000 −0.286183
\(990\) 9.00000 0.286039
\(991\) 38.0000 1.20711 0.603555 − 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) −4.00000 −0.127000
\(993\) 10.0000 0.317340
\(994\) 0 0
\(995\) −21.0000 −0.665745
\(996\) −6.00000 −0.190117
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −32.0000 −1.01294
\(999\) 7.00000 0.221470
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.a.1.1 1
7.6 odd 2 546.2.a.d.1.1 ✓ 1
21.20 even 2 1638.2.a.l.1.1 1
28.27 even 2 4368.2.a.l.1.1 1
91.90 odd 2 7098.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.d.1.1 ✓ 1 7.6 odd 2
1638.2.a.l.1.1 1 21.20 even 2
3822.2.a.a.1.1 1 1.1 even 1 trivial
4368.2.a.l.1.1 1 28.27 even 2
7098.2.a.w.1.1 1 91.90 odd 2