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Properties

Label 4410.2.a.s.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4410,2,Mod(1,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, 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("4410.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.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: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4410.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +8.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -8.00000 q^{34} -10.0000 q^{37} -6.00000 q^{38} -1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{50} -2.00000 q^{52} -10.0000 q^{53} +4.00000 q^{55} -6.00000 q^{58} -14.0000 q^{59} +10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +8.00000 q^{68} -12.0000 q^{71} -4.00000 q^{73} +10.0000 q^{74} +6.00000 q^{76} +4.00000 q^{79} +1.00000 q^{80} -4.00000 q^{82} -2.00000 q^{83} +8.00000 q^{85} -4.00000 q^{86} -4.00000 q^{88} +8.00000 q^{89} +4.00000 q^{92} -4.00000 q^{94} +6.00000 q^{95} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 − 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −4.00000 −0.441726
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 14.0000 1.28880
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 − 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 18.0000 1.57267 0.786334 − 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −8.00000 −0.685994
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 − 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −8.00000 −0.599625
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 − 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) 32.0000 2.34007
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −22.0000 −1.44127 −0.720634 − 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) −14.0000 −0.911322
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 22.0000 1.38863 0.694314 − 0.719672i \(-0.255708\pi\)
0.694314 + 0.719672i \(0.255708\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 26.0000 1.58525 0.792624 − 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −10.0000 −0.599760
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 26.0000 1.54554 0.772770 − 0.634686i \(-0.218871\pi\)
0.772770 + 0.634686i \(0.218871\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) 16.0000 0.907277 0.453638 − 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 − 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −4.00000 −0.220863
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −2.00000 −0.109764
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −12.0000 −0.638696 −0.319348 − 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 8.00000 0.422224 0.211112 − 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 26.0000 1.36653
\(363\) 0 0
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) 0 0
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −32.0000 −1.65468
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 20.0000 1.02195 0.510976 − 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) 0 0
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) −26.0000 −1.30490 −0.652451 − 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 14.0000 0.699127 0.349563 − 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 − 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) −2.00000 −0.0981761
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −24.0000 −1.17388
\(419\) 6.00000 0.293119 0.146560 − 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 − 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 40.0000 1.92228 0.961139 − 0.276066i \(-0.0890309\pi\)
0.961139 + 0.276066i \(0.0890309\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) 16.0000 0.761042
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 − 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) −10.0000 −0.462745 −0.231372 − 0.972865i \(-0.574322\pi\)
−0.231372 + 0.972865i \(0.574322\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) 14.0000 0.644402
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 4.00000 0.182765 0.0913823 − 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) −44.0000 −1.99383 −0.996915 − 0.0784867i \(-0.974991\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 48.0000 2.16181
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 − 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −22.0000 −0.981908
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) −12.0000 −0.525730 −0.262865 − 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 − 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 0 0
\(527\) −32.0000 −1.39394
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −26.0000 −1.12094
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) −8.00000 −0.342997
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) 18.0000 0.758610 0.379305 − 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) −26.0000 −1.09286
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 38.0000 1.59304 0.796521 − 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 − 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 20.0000 0.832611 0.416305 − 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) −47.0000 −1.95494
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 2.00000 0.0825488 0.0412744 − 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 14.0000 0.576371
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −20.0000 −0.821302 −0.410651 − 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 28.0000 1.14405 0.572024 − 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) −40.0000 −1.63163 −0.815817 − 0.578310i \(-0.803712\pi\)
−0.815817 + 0.578310i \(0.803712\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 − 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 6.00000 0.241160 0.120580 − 0.992704i \(-0.461525\pi\)
0.120580 + 0.992704i \(0.461525\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −16.0000 −0.641542
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −80.0000 −3.18981
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 − 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) 0 0
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −18.0000 −0.710957 −0.355479 − 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −18.0000 −0.709851 −0.354925 − 0.934895i \(-0.615494\pi\)
−0.354925 + 0.934895i \(0.615494\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 28.0000 1.10079 0.550397 − 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) −56.0000 −2.19819
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −14.0000 −0.547862 −0.273931 − 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 4.00000 0.156174
\(657\) 0 0
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 − 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 − 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −46.0000 −1.76792 −0.883962 − 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) 16.0000 0.612672
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) 46.0000 1.74992 0.874961 − 0.484193i \(-0.160887\pi\)
0.874961 + 0.484193i \(0.160887\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) 32.0000 1.21209
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 − 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) −60.0000 −2.26294
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) 42.0000 1.57734 0.788672 − 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) −8.00000 −0.299813
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 36.0000 1.34257 0.671287 − 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) −26.0000 −0.966282
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −20.0000 −0.741759 −0.370879 − 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 − 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) −10.0000 −0.367607
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 6.00000 0.219676
\(747\) 0 0
\(748\) 32.0000 1.17004
\(749\) 0 0
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 − 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) −12.0000 −0.435000 −0.217500 − 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) 28.0000 1.01102
\(768\) 0 0
\(769\) −28.0000 −1.00971 −0.504853 − 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) −32.0000 −1.14432
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) −6.00000 −0.213877 −0.106938 − 0.994266i \(-0.534105\pi\)
−0.106938 + 0.994266i \(0.534105\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) 0 0
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −14.0000 −0.494357
\(803\) −16.0000 −0.564628
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 2.00000 0.0703598
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 − 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) −20.0000 −0.699284
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 − 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 2.00000 0.0694210
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) −6.00000 −0.207267
\(839\) −36.0000 −1.24286 −0.621429 − 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) −8.00000 −0.274398
\(851\) −40.0000 −1.37118
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 − 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −24.0000 −0.819824 −0.409912 − 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) −38.0000 −1.29654 −0.648272 − 0.761409i \(-0.724508\pi\)
−0.648272 + 0.761409i \(0.724508\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) −40.0000 −1.35926
\(867\) 0 0
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −10.0000 −0.338643
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) −42.0000 −1.41824 −0.709120 − 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −32.0000 −1.07995
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −40.0000 −1.34763 −0.673817 − 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 4.00000 0.134307 0.0671534 − 0.997743i \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −8.00000 −0.268161
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −80.0000 −2.66519
\(902\) −16.0000 −0.532742
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) −26.0000 −0.864269
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 − 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −18.0000 −0.597351
\(909\) 0 0
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 − 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 − 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) 6.00000 0.197599
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −12.0000 −0.393707 −0.196854 − 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) 0 0
\(934\) 10.0000 0.327210
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) −12.0000 −0.392023 −0.196011 − 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 4.00000 0.130466
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) −14.0000 −0.455661
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 − 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −4.00000 −0.129234
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −20.0000 −0.644826
\(963\) 0 0
\(964\) −4.00000 −0.128831
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) −52.0000 −1.67221 −0.836104 − 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) −62.0000 −1.98967 −0.994837 − 0.101482i \(-0.967641\pi\)
−0.994837 + 0.101482i \(0.967641\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 44.0000 1.40985
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −54.0000 −1.72761 −0.863807 − 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) 32.0000 1.02272
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 20.0000 0.637901 0.318950 − 0.947771i \(-0.396670\pi\)
0.318950 + 0.947771i \(0.396670\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) −48.0000 −1.52863
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 − 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −28.0000 −0.886325
\(999\) 0 0
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 4410.2.a.s.1.1 1
3.2 odd 2 490.2.a.f.1.1 ✓ 1
7.6 odd 2 4410.2.a.i.1.1 1
12.11 even 2 3920.2.a.bg.1.1 1
15.2 even 4 2450.2.c.b.99.2 2
15.8 even 4 2450.2.c.b.99.1 2
15.14 odd 2 2450.2.a.n.1.1 1
21.2 odd 6 490.2.e.e.361.1 2
21.5 even 6 490.2.e.b.361.1 2
21.11 odd 6 490.2.e.e.471.1 2
21.17 even 6 490.2.e.b.471.1 2
21.20 even 2 490.2.a.i.1.1 yes 1
84.83 odd 2 3920.2.a.j.1.1 1
105.62 odd 4 2450.2.c.n.99.2 2
105.83 odd 4 2450.2.c.n.99.1 2
105.104 even 2 2450.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.f.1.1 ✓ 1 3.2 odd 2
490.2.a.i.1.1 yes 1 21.20 even 2
490.2.e.b.361.1 2 21.5 even 6
490.2.e.b.471.1 2 21.17 even 6
490.2.e.e.361.1 2 21.2 odd 6
490.2.e.e.471.1 2 21.11 odd 6
2450.2.a.d.1.1 1 105.104 even 2
2450.2.a.n.1.1 1 15.14 odd 2
2450.2.c.b.99.1 2 15.8 even 4
2450.2.c.b.99.2 2 15.2 even 4
2450.2.c.n.99.1 2 105.83 odd 4
2450.2.c.n.99.2 2 105.62 odd 4
3920.2.a.j.1.1 1 84.83 odd 2
3920.2.a.bg.1.1 1 12.11 even 2
4410.2.a.i.1.1 1 7.6 odd 2
4410.2.a.s.1.1 1 1.1 even 1 trivial