Properties

Label 5520.2.a.bs.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
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] = [5520,2,Mod(1,5520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("5520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.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: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -3.12311 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -3.12311 q^{7} +1.00000 q^{9} +3.12311 q^{11} +2.00000 q^{13} +1.00000 q^{15} -1.12311 q^{17} -4.00000 q^{19} -3.12311 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} +3.12311 q^{33} -3.12311 q^{35} +1.12311 q^{37} +2.00000 q^{39} +2.00000 q^{41} +1.00000 q^{45} +8.00000 q^{47} +2.75379 q^{49} -1.12311 q^{51} +12.2462 q^{53} +3.12311 q^{55} -4.00000 q^{57} -2.24621 q^{59} +9.12311 q^{61} -3.12311 q^{63} +2.00000 q^{65} +8.00000 q^{67} -1.00000 q^{69} +10.2462 q^{71} -4.24621 q^{73} +1.00000 q^{75} -9.75379 q^{77} -3.12311 q^{79} +1.00000 q^{81} +13.3693 q^{83} -1.12311 q^{85} +2.00000 q^{87} -5.12311 q^{89} -6.24621 q^{91} -4.00000 q^{95} -16.2462 q^{97} +3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} - 8 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{45} + 16 q^{47} + 22 q^{49} + 6 q^{51} + 8 q^{53} - 2 q^{55} - 8 q^{57} + 12 q^{59} + 10 q^{61} + 2 q^{63} + 4 q^{65} + 16 q^{67} - 2 q^{69} + 4 q^{71} + 8 q^{73} + 2 q^{75} - 36 q^{77} + 2 q^{79} + 2 q^{81} + 2 q^{83} + 6 q^{85} + 4 q^{87} - 2 q^{89} + 4 q^{91} - 8 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.12311 −1.18042 −0.590211 0.807249i \(-0.700956\pi\)
−0.590211 + 0.807249i \(0.700956\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.12311 0.941652 0.470826 0.882226i \(-0.343956\pi\)
0.470826 + 0.882226i \(0.343956\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −1.12311 −0.272393 −0.136197 0.990682i \(-0.543488\pi\)
−0.136197 + 0.990682i \(0.543488\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −3.12311 −0.681518
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 3.12311 0.543663
\(34\) 0 0
\(35\) −3.12311 −0.527901
\(36\) 0 0
\(37\) 1.12311 0.184637 0.0923187 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 2.75379 0.393398
\(50\) 0 0
\(51\) −1.12311 −0.157266
\(52\) 0 0
\(53\) 12.2462 1.68215 0.841073 0.540921i \(-0.181924\pi\)
0.841073 + 0.540921i \(0.181924\pi\)
\(54\) 0 0
\(55\) 3.12311 0.421119
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −2.24621 −0.292432 −0.146216 0.989253i \(-0.546709\pi\)
−0.146216 + 0.989253i \(0.546709\pi\)
\(60\) 0 0
\(61\) 9.12311 1.16809 0.584047 0.811720i \(-0.301468\pi\)
0.584047 + 0.811720i \(0.301468\pi\)
\(62\) 0 0
\(63\) −3.12311 −0.393474
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.2462 1.21600 0.608001 0.793936i \(-0.291972\pi\)
0.608001 + 0.793936i \(0.291972\pi\)
\(72\) 0 0
\(73\) −4.24621 −0.496981 −0.248491 0.968634i \(-0.579935\pi\)
−0.248491 + 0.968634i \(0.579935\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −9.75379 −1.11155
\(78\) 0 0
\(79\) −3.12311 −0.351377 −0.175688 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.3693 1.46747 0.733737 0.679434i \(-0.237775\pi\)
0.733737 + 0.679434i \(0.237775\pi\)
\(84\) 0 0
\(85\) −1.12311 −0.121818
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −5.12311 −0.543048 −0.271524 0.962432i \(-0.587528\pi\)
−0.271524 + 0.962432i \(0.587528\pi\)
\(90\) 0 0
\(91\) −6.24621 −0.654781
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −16.2462 −1.64955 −0.824776 0.565459i \(-0.808699\pi\)
−0.824776 + 0.565459i \(0.808699\pi\)
\(98\) 0 0
\(99\) 3.12311 0.313884
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) 9.36932 0.923186 0.461593 0.887092i \(-0.347278\pi\)
0.461593 + 0.887092i \(0.347278\pi\)
\(104\) 0 0
\(105\) −3.12311 −0.304784
\(106\) 0 0
\(107\) 13.3693 1.29246 0.646230 0.763142i \(-0.276345\pi\)
0.646230 + 0.763142i \(0.276345\pi\)
\(108\) 0 0
\(109\) 2.87689 0.275557 0.137778 0.990463i \(-0.456004\pi\)
0.137778 + 0.990463i \(0.456004\pi\)
\(110\) 0 0
\(111\) 1.12311 0.106600
\(112\) 0 0
\(113\) −9.12311 −0.858230 −0.429115 0.903250i \(-0.641174\pi\)
−0.429115 + 0.903250i \(0.641174\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 3.50758 0.321539
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 5.75379 0.510566 0.255283 0.966866i \(-0.417831\pi\)
0.255283 + 0.966866i \(0.417831\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 12.4924 1.08323
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −10.8769 −0.929276 −0.464638 0.885501i \(-0.653816\pi\)
−0.464638 + 0.885501i \(0.653816\pi\)
\(138\) 0 0
\(139\) −16.4924 −1.39887 −0.699435 0.714697i \(-0.746565\pi\)
−0.699435 + 0.714697i \(0.746565\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 6.24621 0.522334
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 2.75379 0.227129
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −6.24621 −0.508309 −0.254155 0.967164i \(-0.581797\pi\)
−0.254155 + 0.967164i \(0.581797\pi\)
\(152\) 0 0
\(153\) −1.12311 −0.0907977
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.12311 0.728103 0.364052 0.931379i \(-0.381393\pi\)
0.364052 + 0.931379i \(0.381393\pi\)
\(158\) 0 0
\(159\) 12.2462 0.971188
\(160\) 0 0
\(161\) 3.12311 0.246135
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 3.12311 0.243133
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 20.2462 1.53929 0.769645 0.638471i \(-0.220433\pi\)
0.769645 + 0.638471i \(0.220433\pi\)
\(174\) 0 0
\(175\) −3.12311 −0.236085
\(176\) 0 0
\(177\) −2.24621 −0.168836
\(178\) 0 0
\(179\) 8.49242 0.634753 0.317377 0.948300i \(-0.397198\pi\)
0.317377 + 0.948300i \(0.397198\pi\)
\(180\) 0 0
\(181\) 10.8769 0.808473 0.404237 0.914654i \(-0.367537\pi\)
0.404237 + 0.914654i \(0.367537\pi\)
\(182\) 0 0
\(183\) 9.12311 0.674399
\(184\) 0 0
\(185\) 1.12311 0.0825724
\(186\) 0 0
\(187\) −3.50758 −0.256499
\(188\) 0 0
\(189\) −3.12311 −0.227173
\(190\) 0 0
\(191\) −12.4924 −0.903920 −0.451960 0.892038i \(-0.649275\pi\)
−0.451960 + 0.892038i \(0.649275\pi\)
\(192\) 0 0
\(193\) 8.24621 0.593575 0.296788 0.954944i \(-0.404085\pi\)
0.296788 + 0.954944i \(0.404085\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −19.1231 −1.35560 −0.677801 0.735246i \(-0.737067\pi\)
−0.677801 + 0.735246i \(0.737067\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) −6.24621 −0.438398
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −12.4924 −0.864119
\(210\) 0 0
\(211\) 16.4924 1.13539 0.567693 0.823241i \(-0.307836\pi\)
0.567693 + 0.823241i \(0.307836\pi\)
\(212\) 0 0
\(213\) 10.2462 0.702059
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.24621 −0.286932
\(220\) 0 0
\(221\) −2.24621 −0.151097
\(222\) 0 0
\(223\) 10.2462 0.686137 0.343069 0.939310i \(-0.388534\pi\)
0.343069 + 0.939310i \(0.388534\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 5.36932 0.356374 0.178187 0.983997i \(-0.442977\pi\)
0.178187 + 0.983997i \(0.442977\pi\)
\(228\) 0 0
\(229\) 23.3693 1.54429 0.772144 0.635448i \(-0.219184\pi\)
0.772144 + 0.635448i \(0.219184\pi\)
\(230\) 0 0
\(231\) −9.75379 −0.641752
\(232\) 0 0
\(233\) 28.7386 1.88273 0.941365 0.337389i \(-0.109544\pi\)
0.941365 + 0.337389i \(0.109544\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −3.12311 −0.202868
\(238\) 0 0
\(239\) 18.2462 1.18025 0.590125 0.807312i \(-0.299079\pi\)
0.590125 + 0.807312i \(0.299079\pi\)
\(240\) 0 0
\(241\) 30.4924 1.96419 0.982095 0.188387i \(-0.0603260\pi\)
0.982095 + 0.188387i \(0.0603260\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.75379 0.175933
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 13.3693 0.847246
\(250\) 0 0
\(251\) −17.3693 −1.09634 −0.548171 0.836366i \(-0.684676\pi\)
−0.548171 + 0.836366i \(0.684676\pi\)
\(252\) 0 0
\(253\) −3.12311 −0.196348
\(254\) 0 0
\(255\) −1.12311 −0.0703316
\(256\) 0 0
\(257\) 8.24621 0.514385 0.257192 0.966360i \(-0.417203\pi\)
0.257192 + 0.966360i \(0.417203\pi\)
\(258\) 0 0
\(259\) −3.50758 −0.217950
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 6.24621 0.385158 0.192579 0.981281i \(-0.438315\pi\)
0.192579 + 0.981281i \(0.438315\pi\)
\(264\) 0 0
\(265\) 12.2462 0.752279
\(266\) 0 0
\(267\) −5.12311 −0.313529
\(268\) 0 0
\(269\) 0.246211 0.0150118 0.00750588 0.999972i \(-0.497611\pi\)
0.00750588 + 0.999972i \(0.497611\pi\)
\(270\) 0 0
\(271\) 6.24621 0.379430 0.189715 0.981839i \(-0.439244\pi\)
0.189715 + 0.981839i \(0.439244\pi\)
\(272\) 0 0
\(273\) −6.24621 −0.378038
\(274\) 0 0
\(275\) 3.12311 0.188330
\(276\) 0 0
\(277\) 3.75379 0.225543 0.112772 0.993621i \(-0.464027\pi\)
0.112772 + 0.993621i \(0.464027\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −21.1231 −1.26010 −0.630049 0.776555i \(-0.716965\pi\)
−0.630049 + 0.776555i \(0.716965\pi\)
\(282\) 0 0
\(283\) −14.2462 −0.846849 −0.423425 0.905931i \(-0.639172\pi\)
−0.423425 + 0.905931i \(0.639172\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −6.24621 −0.368702
\(288\) 0 0
\(289\) −15.7386 −0.925802
\(290\) 0 0
\(291\) −16.2462 −0.952370
\(292\) 0 0
\(293\) 28.2462 1.65016 0.825081 0.565015i \(-0.191130\pi\)
0.825081 + 0.565015i \(0.191130\pi\)
\(294\) 0 0
\(295\) −2.24621 −0.130779
\(296\) 0 0
\(297\) 3.12311 0.181221
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0.246211 0.0141445
\(304\) 0 0
\(305\) 9.12311 0.522388
\(306\) 0 0
\(307\) 32.4924 1.85444 0.927220 0.374516i \(-0.122191\pi\)
0.927220 + 0.374516i \(0.122191\pi\)
\(308\) 0 0
\(309\) 9.36932 0.533002
\(310\) 0 0
\(311\) 22.7386 1.28939 0.644695 0.764440i \(-0.276984\pi\)
0.644695 + 0.764440i \(0.276984\pi\)
\(312\) 0 0
\(313\) −22.4924 −1.27135 −0.635673 0.771958i \(-0.719278\pi\)
−0.635673 + 0.771958i \(0.719278\pi\)
\(314\) 0 0
\(315\) −3.12311 −0.175967
\(316\) 0 0
\(317\) −28.7386 −1.61412 −0.807061 0.590468i \(-0.798943\pi\)
−0.807061 + 0.590468i \(0.798943\pi\)
\(318\) 0 0
\(319\) 6.24621 0.349721
\(320\) 0 0
\(321\) 13.3693 0.746203
\(322\) 0 0
\(323\) 4.49242 0.249965
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 2.87689 0.159093
\(328\) 0 0
\(329\) −24.9848 −1.37746
\(330\) 0 0
\(331\) −0.492423 −0.0270660 −0.0135330 0.999908i \(-0.504308\pi\)
−0.0135330 + 0.999908i \(0.504308\pi\)
\(332\) 0 0
\(333\) 1.12311 0.0615458
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 26.4924 1.44313 0.721567 0.692345i \(-0.243422\pi\)
0.721567 + 0.692345i \(0.243422\pi\)
\(338\) 0 0
\(339\) −9.12311 −0.495499
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.2614 0.716046
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −14.7386 −0.791211 −0.395606 0.918420i \(-0.629465\pi\)
−0.395606 + 0.918420i \(0.629465\pi\)
\(348\) 0 0
\(349\) 24.7386 1.32423 0.662114 0.749403i \(-0.269659\pi\)
0.662114 + 0.749403i \(0.269659\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 10.2462 0.543812
\(356\) 0 0
\(357\) 3.50758 0.185641
\(358\) 0 0
\(359\) −20.4924 −1.08155 −0.540774 0.841168i \(-0.681869\pi\)
−0.540774 + 0.841168i \(0.681869\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −1.24621 −0.0654091
\(364\) 0 0
\(365\) −4.24621 −0.222257
\(366\) 0 0
\(367\) −4.87689 −0.254572 −0.127286 0.991866i \(-0.540627\pi\)
−0.127286 + 0.991866i \(0.540627\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) −38.2462 −1.98564
\(372\) 0 0
\(373\) −11.3693 −0.588681 −0.294340 0.955701i \(-0.595100\pi\)
−0.294340 + 0.955701i \(0.595100\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 5.75379 0.294776
\(382\) 0 0
\(383\) −30.2462 −1.54551 −0.772755 0.634705i \(-0.781122\pi\)
−0.772755 + 0.634705i \(0.781122\pi\)
\(384\) 0 0
\(385\) −9.75379 −0.497099
\(386\) 0 0
\(387\) 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
\(391\) 1.12311 0.0567979
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −3.12311 −0.157140
\(396\) 0 0
\(397\) −1.50758 −0.0756631 −0.0378316 0.999284i \(-0.512045\pi\)
−0.0378316 + 0.999284i \(0.512045\pi\)
\(398\) 0 0
\(399\) 12.4924 0.625403
\(400\) 0 0
\(401\) 7.36932 0.368006 0.184003 0.982926i \(-0.441094\pi\)
0.184003 + 0.982926i \(0.441094\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 3.50758 0.173864
\(408\) 0 0
\(409\) 0.246211 0.0121744 0.00608718 0.999981i \(-0.498062\pi\)
0.00608718 + 0.999981i \(0.498062\pi\)
\(410\) 0 0
\(411\) −10.8769 −0.536518
\(412\) 0 0
\(413\) 7.01515 0.345193
\(414\) 0 0
\(415\) 13.3693 0.656274
\(416\) 0 0
\(417\) −16.4924 −0.807637
\(418\) 0 0
\(419\) −39.6155 −1.93535 −0.967673 0.252210i \(-0.918843\pi\)
−0.967673 + 0.252210i \(0.918843\pi\)
\(420\) 0 0
\(421\) −11.3693 −0.554107 −0.277053 0.960855i \(-0.589358\pi\)
−0.277053 + 0.960855i \(0.589358\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −1.12311 −0.0544786
\(426\) 0 0
\(427\) −28.4924 −1.37884
\(428\) 0 0
\(429\) 6.24621 0.301570
\(430\) 0 0
\(431\) −9.75379 −0.469823 −0.234912 0.972017i \(-0.575480\pi\)
−0.234912 + 0.972017i \(0.575480\pi\)
\(432\) 0 0
\(433\) −22.4924 −1.08092 −0.540458 0.841371i \(-0.681749\pi\)
−0.540458 + 0.841371i \(0.681749\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 2.75379 0.131133
\(442\) 0 0
\(443\) −8.49242 −0.403487 −0.201744 0.979438i \(-0.564661\pi\)
−0.201744 + 0.979438i \(0.564661\pi\)
\(444\) 0 0
\(445\) −5.12311 −0.242858
\(446\) 0 0
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) 22.4924 1.06148 0.530742 0.847534i \(-0.321913\pi\)
0.530742 + 0.847534i \(0.321913\pi\)
\(450\) 0 0
\(451\) 6.24621 0.294123
\(452\) 0 0
\(453\) −6.24621 −0.293473
\(454\) 0 0
\(455\) −6.24621 −0.292827
\(456\) 0 0
\(457\) 16.7386 0.783000 0.391500 0.920178i \(-0.371956\pi\)
0.391500 + 0.920178i \(0.371956\pi\)
\(458\) 0 0
\(459\) −1.12311 −0.0524221
\(460\) 0 0
\(461\) −16.7386 −0.779596 −0.389798 0.920900i \(-0.627455\pi\)
−0.389798 + 0.920900i \(0.627455\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.3693 −0.988854 −0.494427 0.869219i \(-0.664622\pi\)
−0.494427 + 0.869219i \(0.664622\pi\)
\(468\) 0 0
\(469\) −24.9848 −1.15369
\(470\) 0 0
\(471\) 9.12311 0.420371
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 12.2462 0.560715
\(478\) 0 0
\(479\) −12.4924 −0.570793 −0.285397 0.958409i \(-0.592125\pi\)
−0.285397 + 0.958409i \(0.592125\pi\)
\(480\) 0 0
\(481\) 2.24621 0.102418
\(482\) 0 0
\(483\) 3.12311 0.142106
\(484\) 0 0
\(485\) −16.2462 −0.737702
\(486\) 0 0
\(487\) −7.50758 −0.340201 −0.170100 0.985427i \(-0.554409\pi\)
−0.170100 + 0.985427i \(0.554409\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −2.24621 −0.101164
\(494\) 0 0
\(495\) 3.12311 0.140373
\(496\) 0 0
\(497\) −32.0000 −1.43540
\(498\) 0 0
\(499\) −36.9848 −1.65567 −0.827835 0.560972i \(-0.810427\pi\)
−0.827835 + 0.560972i \(0.810427\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) −42.7386 −1.90562 −0.952811 0.303565i \(-0.901823\pi\)
−0.952811 + 0.303565i \(0.901823\pi\)
\(504\) 0 0
\(505\) 0.246211 0.0109563
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 13.2614 0.586648
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 9.36932 0.412861
\(516\) 0 0
\(517\) 24.9848 1.09883
\(518\) 0 0
\(519\) 20.2462 0.888710
\(520\) 0 0
\(521\) 17.1231 0.750177 0.375088 0.926989i \(-0.377612\pi\)
0.375088 + 0.926989i \(0.377612\pi\)
\(522\) 0 0
\(523\) −4.49242 −0.196440 −0.0982200 0.995165i \(-0.531315\pi\)
−0.0982200 + 0.995165i \(0.531315\pi\)
\(524\) 0 0
\(525\) −3.12311 −0.136304
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.24621 −0.0974773
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) 13.3693 0.578006
\(536\) 0 0
\(537\) 8.49242 0.366475
\(538\) 0 0
\(539\) 8.60037 0.370444
\(540\) 0 0
\(541\) −32.2462 −1.38637 −0.693186 0.720758i \(-0.743794\pi\)
−0.693186 + 0.720758i \(0.743794\pi\)
\(542\) 0 0
\(543\) 10.8769 0.466772
\(544\) 0 0
\(545\) 2.87689 0.123233
\(546\) 0 0
\(547\) 32.4924 1.38928 0.694638 0.719360i \(-0.255565\pi\)
0.694638 + 0.719360i \(0.255565\pi\)
\(548\) 0 0
\(549\) 9.12311 0.389365
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 9.75379 0.414773
\(554\) 0 0
\(555\) 1.12311 0.0476732
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −3.50758 −0.148090
\(562\) 0 0
\(563\) 13.3693 0.563450 0.281725 0.959495i \(-0.409093\pi\)
0.281725 + 0.959495i \(0.409093\pi\)
\(564\) 0 0
\(565\) −9.12311 −0.383812
\(566\) 0 0
\(567\) −3.12311 −0.131158
\(568\) 0 0
\(569\) 18.8769 0.791361 0.395680 0.918388i \(-0.370509\pi\)
0.395680 + 0.918388i \(0.370509\pi\)
\(570\) 0 0
\(571\) 14.7386 0.616793 0.308396 0.951258i \(-0.400208\pi\)
0.308396 + 0.951258i \(0.400208\pi\)
\(572\) 0 0
\(573\) −12.4924 −0.521878
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −18.4924 −0.769850 −0.384925 0.922948i \(-0.625773\pi\)
−0.384925 + 0.922948i \(0.625773\pi\)
\(578\) 0 0
\(579\) 8.24621 0.342701
\(580\) 0 0
\(581\) −41.7538 −1.73224
\(582\) 0 0
\(583\) 38.2462 1.58400
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 26.2462 1.08330 0.541649 0.840605i \(-0.317800\pi\)
0.541649 + 0.840605i \(0.317800\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) 24.2462 0.995673 0.497836 0.867271i \(-0.334128\pi\)
0.497836 + 0.867271i \(0.334128\pi\)
\(594\) 0 0
\(595\) 3.50758 0.143797
\(596\) 0 0
\(597\) −19.1231 −0.782657
\(598\) 0 0
\(599\) −10.2462 −0.418649 −0.209324 0.977846i \(-0.567126\pi\)
−0.209324 + 0.977846i \(0.567126\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) −1.24621 −0.0506657
\(606\) 0 0
\(607\) 22.7386 0.922933 0.461466 0.887158i \(-0.347324\pi\)
0.461466 + 0.887158i \(0.347324\pi\)
\(608\) 0 0
\(609\) −6.24621 −0.253109
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) 34.8769 1.40866 0.704332 0.709870i \(-0.251247\pi\)
0.704332 + 0.709870i \(0.251247\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) −34.1080 −1.37313 −0.686567 0.727066i \(-0.740883\pi\)
−0.686567 + 0.727066i \(0.740883\pi\)
\(618\) 0 0
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −12.4924 −0.498899
\(628\) 0 0
\(629\) −1.26137 −0.0502940
\(630\) 0 0
\(631\) 14.6307 0.582438 0.291219 0.956656i \(-0.405939\pi\)
0.291219 + 0.956656i \(0.405939\pi\)
\(632\) 0 0
\(633\) 16.4924 0.655515
\(634\) 0 0
\(635\) 5.75379 0.228332
\(636\) 0 0
\(637\) 5.50758 0.218218
\(638\) 0 0
\(639\) 10.2462 0.405334
\(640\) 0 0
\(641\) 15.3693 0.607052 0.303526 0.952823i \(-0.401836\pi\)
0.303526 + 0.952823i \(0.401836\pi\)
\(642\) 0 0
\(643\) −3.50758 −0.138325 −0.0691627 0.997605i \(-0.522033\pi\)
−0.0691627 + 0.997605i \(0.522033\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −44.4924 −1.74918 −0.874589 0.484865i \(-0.838869\pi\)
−0.874589 + 0.484865i \(0.838869\pi\)
\(648\) 0 0
\(649\) −7.01515 −0.275369
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22.4924 −0.880197 −0.440098 0.897950i \(-0.645056\pi\)
−0.440098 + 0.897950i \(0.645056\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) −4.24621 −0.165660
\(658\) 0 0
\(659\) −17.3693 −0.676613 −0.338306 0.941036i \(-0.609854\pi\)
−0.338306 + 0.941036i \(0.609854\pi\)
\(660\) 0 0
\(661\) −23.8617 −0.928114 −0.464057 0.885805i \(-0.653607\pi\)
−0.464057 + 0.885805i \(0.653607\pi\)
\(662\) 0 0
\(663\) −2.24621 −0.0872356
\(664\) 0 0
\(665\) 12.4924 0.484435
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) 0 0
\(669\) 10.2462 0.396141
\(670\) 0 0
\(671\) 28.4924 1.09994
\(672\) 0 0
\(673\) 18.9848 0.731812 0.365906 0.930652i \(-0.380759\pi\)
0.365906 + 0.930652i \(0.380759\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −13.5076 −0.519138 −0.259569 0.965725i \(-0.583581\pi\)
−0.259569 + 0.965725i \(0.583581\pi\)
\(678\) 0 0
\(679\) 50.7386 1.94717
\(680\) 0 0
\(681\) 5.36932 0.205753
\(682\) 0 0
\(683\) −34.2462 −1.31039 −0.655197 0.755458i \(-0.727415\pi\)
−0.655197 + 0.755458i \(0.727415\pi\)
\(684\) 0 0
\(685\) −10.8769 −0.415585
\(686\) 0 0
\(687\) 23.3693 0.891595
\(688\) 0 0
\(689\) 24.4924 0.933087
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) −9.75379 −0.370516
\(694\) 0 0
\(695\) −16.4924 −0.625593
\(696\) 0 0
\(697\) −2.24621 −0.0850813
\(698\) 0 0
\(699\) 28.7386 1.08699
\(700\) 0 0
\(701\) −19.7538 −0.746090 −0.373045 0.927813i \(-0.621686\pi\)
−0.373045 + 0.927813i \(0.621686\pi\)
\(702\) 0 0
\(703\) −4.49242 −0.169435
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) −0.768944 −0.0289191
\(708\) 0 0
\(709\) −11.3693 −0.426984 −0.213492 0.976945i \(-0.568484\pi\)
−0.213492 + 0.976945i \(0.568484\pi\)
\(710\) 0 0
\(711\) −3.12311 −0.117126
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 6.24621 0.233595
\(716\) 0 0
\(717\) 18.2462 0.681417
\(718\) 0 0
\(719\) 26.2462 0.978819 0.489409 0.872054i \(-0.337212\pi\)
0.489409 + 0.872054i \(0.337212\pi\)
\(720\) 0 0
\(721\) −29.2614 −1.08975
\(722\) 0 0
\(723\) 30.4924 1.13403
\(724\) 0 0
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 29.8617 1.10751 0.553755 0.832679i \(-0.313194\pi\)
0.553755 + 0.832679i \(0.313194\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 43.8617 1.62007 0.810035 0.586381i \(-0.199448\pi\)
0.810035 + 0.586381i \(0.199448\pi\)
\(734\) 0 0
\(735\) 2.75379 0.101575
\(736\) 0 0
\(737\) 24.9848 0.920329
\(738\) 0 0
\(739\) −7.50758 −0.276171 −0.138085 0.990420i \(-0.544095\pi\)
−0.138085 + 0.990420i \(0.544095\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) −34.7386 −1.27444 −0.637218 0.770683i \(-0.719915\pi\)
−0.637218 + 0.770683i \(0.719915\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) 13.3693 0.489158
\(748\) 0 0
\(749\) −41.7538 −1.52565
\(750\) 0 0
\(751\) 13.8617 0.505822 0.252911 0.967490i \(-0.418612\pi\)
0.252911 + 0.967490i \(0.418612\pi\)
\(752\) 0 0
\(753\) −17.3693 −0.632973
\(754\) 0 0
\(755\) −6.24621 −0.227323
\(756\) 0 0
\(757\) −47.8617 −1.73956 −0.869782 0.493436i \(-0.835741\pi\)
−0.869782 + 0.493436i \(0.835741\pi\)
\(758\) 0 0
\(759\) −3.12311 −0.113362
\(760\) 0 0
\(761\) −31.7538 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(762\) 0 0
\(763\) −8.98485 −0.325273
\(764\) 0 0
\(765\) −1.12311 −0.0406060
\(766\) 0 0
\(767\) −4.49242 −0.162212
\(768\) 0 0
\(769\) 28.7386 1.03634 0.518171 0.855277i \(-0.326613\pi\)
0.518171 + 0.855277i \(0.326613\pi\)
\(770\) 0 0
\(771\) 8.24621 0.296980
\(772\) 0 0
\(773\) −19.7538 −0.710494 −0.355247 0.934772i \(-0.615603\pi\)
−0.355247 + 0.934772i \(0.615603\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −3.50758 −0.125834
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 9.12311 0.325618
\(786\) 0 0
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 0 0
\(789\) 6.24621 0.222371
\(790\) 0 0
\(791\) 28.4924 1.01307
\(792\) 0 0
\(793\) 18.2462 0.647942
\(794\) 0 0
\(795\) 12.2462 0.434328
\(796\) 0 0
\(797\) 10.4924 0.371661 0.185830 0.982582i \(-0.440503\pi\)
0.185830 + 0.982582i \(0.440503\pi\)
\(798\) 0 0
\(799\) −8.98485 −0.317861
\(800\) 0 0
\(801\) −5.12311 −0.181016
\(802\) 0 0
\(803\) −13.2614 −0.467983
\(804\) 0 0
\(805\) 3.12311 0.110075
\(806\) 0 0
\(807\) 0.246211 0.00866705
\(808\) 0 0
\(809\) −12.2462 −0.430554 −0.215277 0.976553i \(-0.569065\pi\)
−0.215277 + 0.976553i \(0.569065\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) 6.24621 0.219064
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −6.24621 −0.218260
\(820\) 0 0
\(821\) −36.2462 −1.26500 −0.632501 0.774560i \(-0.717971\pi\)
−0.632501 + 0.774560i \(0.717971\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 3.12311 0.108733
\(826\) 0 0
\(827\) −8.87689 −0.308680 −0.154340 0.988018i \(-0.549325\pi\)
−0.154340 + 0.988018i \(0.549325\pi\)
\(828\) 0 0
\(829\) 15.7538 0.547152 0.273576 0.961850i \(-0.411794\pi\)
0.273576 + 0.961850i \(0.411794\pi\)
\(830\) 0 0
\(831\) 3.75379 0.130217
\(832\) 0 0
\(833\) −3.09280 −0.107159
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.75379 0.336738 0.168369 0.985724i \(-0.446150\pi\)
0.168369 + 0.985724i \(0.446150\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −21.1231 −0.727518
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 3.89205 0.133732
\(848\) 0 0
\(849\) −14.2462 −0.488929
\(850\) 0 0
\(851\) −1.12311 −0.0384996
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −24.7386 −0.845056 −0.422528 0.906350i \(-0.638857\pi\)
−0.422528 + 0.906350i \(0.638857\pi\)
\(858\) 0 0
\(859\) −15.5076 −0.529112 −0.264556 0.964370i \(-0.585225\pi\)
−0.264556 + 0.964370i \(0.585225\pi\)
\(860\) 0 0
\(861\) −6.24621 −0.212870
\(862\) 0 0
\(863\) 16.9848 0.578171 0.289085 0.957303i \(-0.406649\pi\)
0.289085 + 0.957303i \(0.406649\pi\)
\(864\) 0 0
\(865\) 20.2462 0.688392
\(866\) 0 0
\(867\) −15.7386 −0.534512
\(868\) 0 0
\(869\) −9.75379 −0.330875
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) −16.2462 −0.549851
\(874\) 0 0
\(875\) −3.12311 −0.105580
\(876\) 0 0
\(877\) 52.7386 1.78086 0.890429 0.455123i \(-0.150405\pi\)
0.890429 + 0.455123i \(0.150405\pi\)
\(878\) 0 0
\(879\) 28.2462 0.952721
\(880\) 0 0
\(881\) −9.61553 −0.323955 −0.161978 0.986794i \(-0.551787\pi\)
−0.161978 + 0.986794i \(0.551787\pi\)
\(882\) 0 0
\(883\) −44.9848 −1.51386 −0.756930 0.653496i \(-0.773302\pi\)
−0.756930 + 0.653496i \(0.773302\pi\)
\(884\) 0 0
\(885\) −2.24621 −0.0755056
\(886\) 0 0
\(887\) −19.5076 −0.655000 −0.327500 0.944851i \(-0.606206\pi\)
−0.327500 + 0.944851i \(0.606206\pi\)
\(888\) 0 0
\(889\) −17.9697 −0.602684
\(890\) 0 0
\(891\) 3.12311 0.104628
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) 8.49242 0.283870
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −13.7538 −0.458205
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.8769 0.361560
\(906\) 0 0
\(907\) −54.2462 −1.80122 −0.900608 0.434632i \(-0.856878\pi\)
−0.900608 + 0.434632i \(0.856878\pi\)
\(908\) 0 0
\(909\) 0.246211 0.00816631
\(910\) 0 0
\(911\) 30.2462 1.00210 0.501051 0.865418i \(-0.332947\pi\)
0.501051 + 0.865418i \(0.332947\pi\)
\(912\) 0 0
\(913\) 41.7538 1.38185
\(914\) 0 0
\(915\) 9.12311 0.301601
\(916\) 0 0
\(917\) −12.4924 −0.412536
\(918\) 0 0
\(919\) −19.1231 −0.630813 −0.315407 0.948957i \(-0.602141\pi\)
−0.315407 + 0.948957i \(0.602141\pi\)
\(920\) 0 0
\(921\) 32.4924 1.07066
\(922\) 0 0
\(923\) 20.4924 0.674516
\(924\) 0 0
\(925\) 1.12311 0.0369275
\(926\) 0 0
\(927\) 9.36932 0.307729
\(928\) 0 0
\(929\) 34.9848 1.14782 0.573908 0.818920i \(-0.305427\pi\)
0.573908 + 0.818920i \(0.305427\pi\)
\(930\) 0 0
\(931\) −11.0152 −0.361007
\(932\) 0 0
\(933\) 22.7386 0.744429
\(934\) 0 0
\(935\) −3.50758 −0.114710
\(936\) 0 0
\(937\) 32.7386 1.06952 0.534762 0.845003i \(-0.320401\pi\)
0.534762 + 0.845003i \(0.320401\pi\)
\(938\) 0 0
\(939\) −22.4924 −0.734012
\(940\) 0 0
\(941\) 26.4924 0.863628 0.431814 0.901963i \(-0.357874\pi\)
0.431814 + 0.901963i \(0.357874\pi\)
\(942\) 0 0
\(943\) −2.00000 −0.0651290
\(944\) 0 0
\(945\) −3.12311 −0.101595
\(946\) 0 0
\(947\) −5.75379 −0.186973 −0.0934865 0.995621i \(-0.529801\pi\)
−0.0934865 + 0.995621i \(0.529801\pi\)
\(948\) 0 0
\(949\) −8.49242 −0.275676
\(950\) 0 0
\(951\) −28.7386 −0.931914
\(952\) 0 0
\(953\) 23.8617 0.772958 0.386479 0.922298i \(-0.373691\pi\)
0.386479 + 0.922298i \(0.373691\pi\)
\(954\) 0 0
\(955\) −12.4924 −0.404245
\(956\) 0 0
\(957\) 6.24621 0.201911
\(958\) 0 0
\(959\) 33.9697 1.09694
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 13.3693 0.430820
\(964\) 0 0
\(965\) 8.24621 0.265455
\(966\) 0 0
\(967\) −60.9848 −1.96114 −0.980570 0.196168i \(-0.937150\pi\)
−0.980570 + 0.196168i \(0.937150\pi\)
\(968\) 0 0
\(969\) 4.49242 0.144317
\(970\) 0 0
\(971\) −43.1231 −1.38389 −0.691943 0.721952i \(-0.743245\pi\)
−0.691943 + 0.721952i \(0.743245\pi\)
\(972\) 0 0
\(973\) 51.5076 1.65126
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) −35.8617 −1.14732 −0.573659 0.819094i \(-0.694477\pi\)
−0.573659 + 0.819094i \(0.694477\pi\)
\(978\) 0 0
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) 2.87689 0.0918522
\(982\) 0 0
\(983\) 9.75379 0.311098 0.155549 0.987828i \(-0.450285\pi\)
0.155549 + 0.987828i \(0.450285\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) 0 0
\(987\) −24.9848 −0.795276
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −61.4773 −1.95289 −0.976445 0.215767i \(-0.930775\pi\)
−0.976445 + 0.215767i \(0.930775\pi\)
\(992\) 0 0
\(993\) −0.492423 −0.0156266
\(994\) 0 0
\(995\) −19.1231 −0.606243
\(996\) 0 0
\(997\) 3.75379 0.118884 0.0594418 0.998232i \(-0.481068\pi\)
0.0594418 + 0.998232i \(0.481068\pi\)
\(998\) 0 0
\(999\) 1.12311 0.0355335
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 5520.2.a.bs.1.1 2
4.3 odd 2 690.2.a.l.1.2 2
12.11 even 2 2070.2.a.t.1.2 2
20.3 even 4 3450.2.d.v.2899.1 4
20.7 even 4 3450.2.d.v.2899.4 4
20.19 odd 2 3450.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.l.1.2 2 4.3 odd 2
2070.2.a.t.1.2 2 12.11 even 2
3450.2.a.bi.1.1 2 20.19 odd 2
3450.2.d.v.2899.1 4 20.3 even 4
3450.2.d.v.2899.4 4 20.7 even 4
5520.2.a.bs.1.1 2 1.1 even 1 trivial