Properties

Label 4400.2.a.bs.1.1
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
Analytic rank $1$
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] = [4400,2,Mod(1,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.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: \(35.1341768894\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.30278 q^{3} +4.30278 q^{7} -1.30278 q^{9} +O(q^{10})\) \(q-1.30278 q^{3} +4.30278 q^{7} -1.30278 q^{9} +1.00000 q^{11} -5.00000 q^{13} +3.90833 q^{17} +1.00000 q^{19} -5.60555 q^{21} -3.69722 q^{23} +5.60555 q^{27} -9.90833 q^{29} +4.21110 q^{31} -1.30278 q^{33} -9.60555 q^{37} +6.51388 q^{39} +1.60555 q^{41} -7.21110 q^{43} -3.00000 q^{47} +11.5139 q^{49} -5.09167 q^{51} -2.30278 q^{53} -1.30278 q^{57} -0.211103 q^{59} +2.90833 q^{61} -5.60555 q^{63} -4.00000 q^{67} +4.81665 q^{69} -4.60555 q^{71} -2.90833 q^{73} +4.30278 q^{77} +0.0916731 q^{79} -3.39445 q^{81} +14.5139 q^{83} +12.9083 q^{87} +5.30278 q^{89} -21.5139 q^{91} -5.48612 q^{93} -11.6972 q^{97} -1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51} - q^{53} + q^{57} + 14 q^{59} - 5 q^{61} - 4 q^{63} - 8 q^{67} - 12 q^{69} - 2 q^{71} + 5 q^{73} + 5 q^{77} + 11 q^{79} - 14 q^{81} + 11 q^{83} + 15 q^{87} + 7 q^{89} - 25 q^{91} - 29 q^{93} - 27 q^{97} + q^{99}+O(q^{100}) \) 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
\(3\) −1.30278 −0.752158 −0.376079 0.926588i \(-0.622728\pi\)
−0.376079 + 0.926588i \(0.622728\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.30278 1.62630 0.813148 0.582057i \(-0.197752\pi\)
0.813148 + 0.582057i \(0.197752\pi\)
\(8\) 0 0
\(9\) −1.30278 −0.434259
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.90833 0.947909 0.473954 0.880549i \(-0.342826\pi\)
0.473954 + 0.880549i \(0.342826\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −5.60555 −1.22323
\(22\) 0 0
\(23\) −3.69722 −0.770925 −0.385462 0.922724i \(-0.625958\pi\)
−0.385462 + 0.922724i \(0.625958\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.60555 1.07879
\(28\) 0 0
\(29\) −9.90833 −1.83993 −0.919965 0.392000i \(-0.871783\pi\)
−0.919965 + 0.392000i \(0.871783\pi\)
\(30\) 0 0
\(31\) 4.21110 0.756336 0.378168 0.925737i \(-0.376554\pi\)
0.378168 + 0.925737i \(0.376554\pi\)
\(32\) 0 0
\(33\) −1.30278 −0.226784
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.60555 −1.57914 −0.789571 0.613659i \(-0.789697\pi\)
−0.789571 + 0.613659i \(0.789697\pi\)
\(38\) 0 0
\(39\) 6.51388 1.04306
\(40\) 0 0
\(41\) 1.60555 0.250745 0.125372 0.992110i \(-0.459987\pi\)
0.125372 + 0.992110i \(0.459987\pi\)
\(42\) 0 0
\(43\) −7.21110 −1.09968 −0.549841 0.835269i \(-0.685312\pi\)
−0.549841 + 0.835269i \(0.685312\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 11.5139 1.64484
\(50\) 0 0
\(51\) −5.09167 −0.712977
\(52\) 0 0
\(53\) −2.30278 −0.316311 −0.158155 0.987414i \(-0.550555\pi\)
−0.158155 + 0.987414i \(0.550555\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.30278 −0.172557
\(58\) 0 0
\(59\) −0.211103 −0.0274832 −0.0137416 0.999906i \(-0.504374\pi\)
−0.0137416 + 0.999906i \(0.504374\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) 0 0
\(63\) −5.60555 −0.706233
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 4.81665 0.579857
\(70\) 0 0
\(71\) −4.60555 −0.546578 −0.273289 0.961932i \(-0.588112\pi\)
−0.273289 + 0.961932i \(0.588112\pi\)
\(72\) 0 0
\(73\) −2.90833 −0.340394 −0.170197 0.985410i \(-0.554440\pi\)
−0.170197 + 0.985410i \(0.554440\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.30278 0.490347
\(78\) 0 0
\(79\) 0.0916731 0.0103140 0.00515701 0.999987i \(-0.498358\pi\)
0.00515701 + 0.999987i \(0.498358\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) 0 0
\(83\) 14.5139 1.59311 0.796553 0.604569i \(-0.206655\pi\)
0.796553 + 0.604569i \(0.206655\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.9083 1.38392
\(88\) 0 0
\(89\) 5.30278 0.562093 0.281047 0.959694i \(-0.409318\pi\)
0.281047 + 0.959694i \(0.409318\pi\)
\(90\) 0 0
\(91\) −21.5139 −2.25527
\(92\) 0 0
\(93\) −5.48612 −0.568884
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.6972 −1.18767 −0.593837 0.804586i \(-0.702387\pi\)
−0.593837 + 0.804586i \(0.702387\pi\)
\(98\) 0 0
\(99\) −1.30278 −0.130934
\(100\) 0 0
\(101\) 17.5139 1.74270 0.871348 0.490666i \(-0.163246\pi\)
0.871348 + 0.490666i \(0.163246\pi\)
\(102\) 0 0
\(103\) −7.90833 −0.779231 −0.389615 0.920978i \(-0.627392\pi\)
−0.389615 + 0.920978i \(0.627392\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −6.51388 −0.623916 −0.311958 0.950096i \(-0.600985\pi\)
−0.311958 + 0.950096i \(0.600985\pi\)
\(110\) 0 0
\(111\) 12.5139 1.18776
\(112\) 0 0
\(113\) −10.8167 −1.01755 −0.508773 0.860901i \(-0.669901\pi\)
−0.508773 + 0.860901i \(0.669901\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.51388 0.602208
\(118\) 0 0
\(119\) 16.8167 1.54158
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −2.09167 −0.188600
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −17.1194 −1.51910 −0.759552 0.650447i \(-0.774582\pi\)
−0.759552 + 0.650447i \(0.774582\pi\)
\(128\) 0 0
\(129\) 9.39445 0.827135
\(130\) 0 0
\(131\) −0.908327 −0.0793609 −0.0396804 0.999212i \(-0.512634\pi\)
−0.0396804 + 0.999212i \(0.512634\pi\)
\(132\) 0 0
\(133\) 4.30278 0.373098
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.09167 0.178704 0.0893518 0.996000i \(-0.471520\pi\)
0.0893518 + 0.996000i \(0.471520\pi\)
\(138\) 0 0
\(139\) −8.21110 −0.696457 −0.348228 0.937410i \(-0.613217\pi\)
−0.348228 + 0.937410i \(0.613217\pi\)
\(140\) 0 0
\(141\) 3.90833 0.329141
\(142\) 0 0
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −15.0000 −1.23718
\(148\) 0 0
\(149\) 2.78890 0.228475 0.114238 0.993453i \(-0.463557\pi\)
0.114238 + 0.993453i \(0.463557\pi\)
\(150\) 0 0
\(151\) 20.8167 1.69404 0.847018 0.531565i \(-0.178396\pi\)
0.847018 + 0.531565i \(0.178396\pi\)
\(152\) 0 0
\(153\) −5.09167 −0.411637
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −4.78890 −0.382196 −0.191098 0.981571i \(-0.561205\pi\)
−0.191098 + 0.981571i \(0.561205\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −15.9083 −1.25375
\(162\) 0 0
\(163\) 5.69722 0.446241 0.223121 0.974791i \(-0.428376\pi\)
0.223121 + 0.974791i \(0.428376\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.4222 −1.19341 −0.596703 0.802462i \(-0.703523\pi\)
−0.596703 + 0.802462i \(0.703523\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −1.30278 −0.0996257
\(172\) 0 0
\(173\) −16.8167 −1.27855 −0.639273 0.768980i \(-0.720765\pi\)
−0.639273 + 0.768980i \(0.720765\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.275019 0.0206717
\(178\) 0 0
\(179\) 5.51388 0.412127 0.206063 0.978539i \(-0.433935\pi\)
0.206063 + 0.978539i \(0.433935\pi\)
\(180\) 0 0
\(181\) −9.09167 −0.675779 −0.337889 0.941186i \(-0.609713\pi\)
−0.337889 + 0.941186i \(0.609713\pi\)
\(182\) 0 0
\(183\) −3.78890 −0.280083
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.90833 0.285805
\(188\) 0 0
\(189\) 24.1194 1.75443
\(190\) 0 0
\(191\) −6.69722 −0.484594 −0.242297 0.970202i \(-0.577901\pi\)
−0.242297 + 0.970202i \(0.577901\pi\)
\(192\) 0 0
\(193\) 1.21110 0.0871771 0.0435885 0.999050i \(-0.486121\pi\)
0.0435885 + 0.999050i \(0.486121\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.69722 −0.690899 −0.345449 0.938437i \(-0.612273\pi\)
−0.345449 + 0.938437i \(0.612273\pi\)
\(198\) 0 0
\(199\) 24.5139 1.73774 0.868871 0.495038i \(-0.164846\pi\)
0.868871 + 0.495038i \(0.164846\pi\)
\(200\) 0 0
\(201\) 5.21110 0.367563
\(202\) 0 0
\(203\) −42.6333 −2.99227
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.81665 0.334781
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −25.2389 −1.73751 −0.868757 0.495238i \(-0.835081\pi\)
−0.868757 + 0.495238i \(0.835081\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 18.1194 1.23003
\(218\) 0 0
\(219\) 3.78890 0.256030
\(220\) 0 0
\(221\) −19.5416 −1.31451
\(222\) 0 0
\(223\) 20.6333 1.38171 0.690854 0.722994i \(-0.257235\pi\)
0.690854 + 0.722994i \(0.257235\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.30278 0.351958 0.175979 0.984394i \(-0.443691\pi\)
0.175979 + 0.984394i \(0.443691\pi\)
\(228\) 0 0
\(229\) 13.7250 0.906972 0.453486 0.891263i \(-0.350180\pi\)
0.453486 + 0.891263i \(0.350180\pi\)
\(230\) 0 0
\(231\) −5.60555 −0.368818
\(232\) 0 0
\(233\) −5.09167 −0.333567 −0.166783 0.985994i \(-0.553338\pi\)
−0.166783 + 0.985994i \(0.553338\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.119429 −0.00775778
\(238\) 0 0
\(239\) 4.11943 0.266464 0.133232 0.991085i \(-0.457465\pi\)
0.133232 + 0.991085i \(0.457465\pi\)
\(240\) 0 0
\(241\) −24.9361 −1.60627 −0.803137 0.595794i \(-0.796837\pi\)
−0.803137 + 0.595794i \(0.796837\pi\)
\(242\) 0 0
\(243\) −12.3944 −0.795104
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) 0 0
\(249\) −18.9083 −1.19827
\(250\) 0 0
\(251\) 3.90833 0.246691 0.123346 0.992364i \(-0.460638\pi\)
0.123346 + 0.992364i \(0.460638\pi\)
\(252\) 0 0
\(253\) −3.69722 −0.232443
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −41.3305 −2.56815
\(260\) 0 0
\(261\) 12.9083 0.799005
\(262\) 0 0
\(263\) −1.18335 −0.0729683 −0.0364841 0.999334i \(-0.511616\pi\)
−0.0364841 + 0.999334i \(0.511616\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.90833 −0.422783
\(268\) 0 0
\(269\) −23.7250 −1.44654 −0.723269 0.690567i \(-0.757361\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(270\) 0 0
\(271\) −14.2111 −0.863263 −0.431632 0.902050i \(-0.642062\pi\)
−0.431632 + 0.902050i \(0.642062\pi\)
\(272\) 0 0
\(273\) 28.0278 1.69632
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −21.6056 −1.29815 −0.649076 0.760724i \(-0.724844\pi\)
−0.649076 + 0.760724i \(0.724844\pi\)
\(278\) 0 0
\(279\) −5.48612 −0.328446
\(280\) 0 0
\(281\) −22.8167 −1.36113 −0.680564 0.732689i \(-0.738265\pi\)
−0.680564 + 0.732689i \(0.738265\pi\)
\(282\) 0 0
\(283\) 2.69722 0.160333 0.0801667 0.996781i \(-0.474455\pi\)
0.0801667 + 0.996781i \(0.474455\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.90833 0.407786
\(288\) 0 0
\(289\) −1.72498 −0.101469
\(290\) 0 0
\(291\) 15.2389 0.893318
\(292\) 0 0
\(293\) −15.2111 −0.888642 −0.444321 0.895868i \(-0.646555\pi\)
−0.444321 + 0.895868i \(0.646555\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.60555 0.325267
\(298\) 0 0
\(299\) 18.4861 1.06908
\(300\) 0 0
\(301\) −31.0278 −1.78841
\(302\) 0 0
\(303\) −22.8167 −1.31078
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.09167 −0.347670 −0.173835 0.984775i \(-0.555616\pi\)
−0.173835 + 0.984775i \(0.555616\pi\)
\(308\) 0 0
\(309\) 10.3028 0.586104
\(310\) 0 0
\(311\) 16.8167 0.953585 0.476792 0.879016i \(-0.341799\pi\)
0.476792 + 0.879016i \(0.341799\pi\)
\(312\) 0 0
\(313\) −21.8167 −1.23315 −0.616575 0.787296i \(-0.711480\pi\)
−0.616575 + 0.787296i \(0.711480\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.90833 −0.556507 −0.278254 0.960508i \(-0.589756\pi\)
−0.278254 + 0.960508i \(0.589756\pi\)
\(318\) 0 0
\(319\) −9.90833 −0.554760
\(320\) 0 0
\(321\) 3.90833 0.218142
\(322\) 0 0
\(323\) 3.90833 0.217465
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 8.48612 0.469284
\(328\) 0 0
\(329\) −12.9083 −0.711659
\(330\) 0 0
\(331\) 14.3944 0.791190 0.395595 0.918425i \(-0.370538\pi\)
0.395595 + 0.918425i \(0.370538\pi\)
\(332\) 0 0
\(333\) 12.5139 0.685756
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −26.8444 −1.46231 −0.731154 0.682212i \(-0.761018\pi\)
−0.731154 + 0.682212i \(0.761018\pi\)
\(338\) 0 0
\(339\) 14.0917 0.765355
\(340\) 0 0
\(341\) 4.21110 0.228044
\(342\) 0 0
\(343\) 19.4222 1.04870
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.51388 0.296000 0.148000 0.988987i \(-0.452716\pi\)
0.148000 + 0.988987i \(0.452716\pi\)
\(348\) 0 0
\(349\) −26.8167 −1.43546 −0.717731 0.696320i \(-0.754819\pi\)
−0.717731 + 0.696320i \(0.754819\pi\)
\(350\) 0 0
\(351\) −28.0278 −1.49601
\(352\) 0 0
\(353\) 24.6333 1.31110 0.655549 0.755152i \(-0.272437\pi\)
0.655549 + 0.755152i \(0.272437\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −21.9083 −1.15951
\(358\) 0 0
\(359\) −15.2111 −0.802811 −0.401406 0.915900i \(-0.631478\pi\)
−0.401406 + 0.915900i \(0.631478\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) −1.30278 −0.0683780
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −24.3028 −1.26859 −0.634297 0.773089i \(-0.718710\pi\)
−0.634297 + 0.773089i \(0.718710\pi\)
\(368\) 0 0
\(369\) −2.09167 −0.108888
\(370\) 0 0
\(371\) −9.90833 −0.514415
\(372\) 0 0
\(373\) 1.42221 0.0736390 0.0368195 0.999322i \(-0.488277\pi\)
0.0368195 + 0.999322i \(0.488277\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 49.5416 2.55152
\(378\) 0 0
\(379\) −24.8167 −1.27475 −0.637373 0.770555i \(-0.719979\pi\)
−0.637373 + 0.770555i \(0.719979\pi\)
\(380\) 0 0
\(381\) 22.3028 1.14261
\(382\) 0 0
\(383\) −21.6333 −1.10541 −0.552705 0.833377i \(-0.686404\pi\)
−0.552705 + 0.833377i \(0.686404\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.39445 0.477547
\(388\) 0 0
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) −14.4500 −0.730766
\(392\) 0 0
\(393\) 1.18335 0.0596919
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −25.3028 −1.26991 −0.634955 0.772549i \(-0.718981\pi\)
−0.634955 + 0.772549i \(0.718981\pi\)
\(398\) 0 0
\(399\) −5.60555 −0.280629
\(400\) 0 0
\(401\) −27.2111 −1.35886 −0.679429 0.733741i \(-0.737772\pi\)
−0.679429 + 0.733741i \(0.737772\pi\)
\(402\) 0 0
\(403\) −21.0555 −1.04885
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.60555 −0.476129
\(408\) 0 0
\(409\) 8.21110 0.406013 0.203006 0.979177i \(-0.434929\pi\)
0.203006 + 0.979177i \(0.434929\pi\)
\(410\) 0 0
\(411\) −2.72498 −0.134413
\(412\) 0 0
\(413\) −0.908327 −0.0446958
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 10.6972 0.523845
\(418\) 0 0
\(419\) −13.6056 −0.664675 −0.332337 0.943161i \(-0.607837\pi\)
−0.332337 + 0.943161i \(0.607837\pi\)
\(420\) 0 0
\(421\) 4.30278 0.209704 0.104852 0.994488i \(-0.466563\pi\)
0.104852 + 0.994488i \(0.466563\pi\)
\(422\) 0 0
\(423\) 3.90833 0.190029
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 12.5139 0.605589
\(428\) 0 0
\(429\) 6.51388 0.314493
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 0 0
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.69722 −0.176862
\(438\) 0 0
\(439\) −20.6972 −0.987825 −0.493912 0.869512i \(-0.664434\pi\)
−0.493912 + 0.869512i \(0.664434\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) −1.39445 −0.0662523 −0.0331261 0.999451i \(-0.510546\pi\)
−0.0331261 + 0.999451i \(0.510546\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3.63331 −0.171850
\(448\) 0 0
\(449\) 41.5139 1.95916 0.979581 0.201052i \(-0.0644361\pi\)
0.979581 + 0.201052i \(0.0644361\pi\)
\(450\) 0 0
\(451\) 1.60555 0.0756025
\(452\) 0 0
\(453\) −27.1194 −1.27418
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.3028 1.13684 0.568418 0.822740i \(-0.307556\pi\)
0.568418 + 0.822740i \(0.307556\pi\)
\(458\) 0 0
\(459\) 21.9083 1.02259
\(460\) 0 0
\(461\) 17.7889 0.828512 0.414256 0.910161i \(-0.364042\pi\)
0.414256 + 0.910161i \(0.364042\pi\)
\(462\) 0 0
\(463\) 26.2111 1.21813 0.609067 0.793119i \(-0.291544\pi\)
0.609067 + 0.793119i \(0.291544\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.6333 1.13989 0.569947 0.821682i \(-0.306964\pi\)
0.569947 + 0.821682i \(0.306964\pi\)
\(468\) 0 0
\(469\) −17.2111 −0.794735
\(470\) 0 0
\(471\) 6.23886 0.287471
\(472\) 0 0
\(473\) −7.21110 −0.331567
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) 13.1833 0.602362 0.301181 0.953567i \(-0.402619\pi\)
0.301181 + 0.953567i \(0.402619\pi\)
\(480\) 0 0
\(481\) 48.0278 2.18988
\(482\) 0 0
\(483\) 20.7250 0.943019
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −10.2111 −0.462709 −0.231355 0.972869i \(-0.574316\pi\)
−0.231355 + 0.972869i \(0.574316\pi\)
\(488\) 0 0
\(489\) −7.42221 −0.335644
\(490\) 0 0
\(491\) 24.2111 1.09263 0.546316 0.837579i \(-0.316030\pi\)
0.546316 + 0.837579i \(0.316030\pi\)
\(492\) 0 0
\(493\) −38.7250 −1.74409
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −19.8167 −0.888898
\(498\) 0 0
\(499\) 21.5139 0.963093 0.481547 0.876420i \(-0.340075\pi\)
0.481547 + 0.876420i \(0.340075\pi\)
\(500\) 0 0
\(501\) 20.0917 0.897630
\(502\) 0 0
\(503\) 16.6056 0.740405 0.370202 0.928951i \(-0.379288\pi\)
0.370202 + 0.928951i \(0.379288\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −15.6333 −0.694300
\(508\) 0 0
\(509\) −26.3028 −1.16585 −0.582925 0.812526i \(-0.698092\pi\)
−0.582925 + 0.812526i \(0.698092\pi\)
\(510\) 0 0
\(511\) −12.5139 −0.553581
\(512\) 0 0
\(513\) 5.60555 0.247491
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.00000 −0.131940
\(518\) 0 0
\(519\) 21.9083 0.961669
\(520\) 0 0
\(521\) 23.4500 1.02736 0.513681 0.857981i \(-0.328282\pi\)
0.513681 + 0.857981i \(0.328282\pi\)
\(522\) 0 0
\(523\) −3.57779 −0.156446 −0.0782230 0.996936i \(-0.524925\pi\)
−0.0782230 + 0.996936i \(0.524925\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.4584 0.716938
\(528\) 0 0
\(529\) −9.33053 −0.405675
\(530\) 0 0
\(531\) 0.275019 0.0119348
\(532\) 0 0
\(533\) −8.02776 −0.347721
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −7.18335 −0.309984
\(538\) 0 0
\(539\) 11.5139 0.495938
\(540\) 0 0
\(541\) −6.72498 −0.289130 −0.144565 0.989495i \(-0.546178\pi\)
−0.144565 + 0.989495i \(0.546178\pi\)
\(542\) 0 0
\(543\) 11.8444 0.508292
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18.1194 0.774731 0.387365 0.921926i \(-0.373385\pi\)
0.387365 + 0.921926i \(0.373385\pi\)
\(548\) 0 0
\(549\) −3.78890 −0.161706
\(550\) 0 0
\(551\) −9.90833 −0.422109
\(552\) 0 0
\(553\) 0.394449 0.0167737
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.42221 0.399232 0.199616 0.979874i \(-0.436031\pi\)
0.199616 + 0.979874i \(0.436031\pi\)
\(558\) 0 0
\(559\) 36.0555 1.52499
\(560\) 0 0
\(561\) −5.09167 −0.214971
\(562\) 0 0
\(563\) 18.9083 0.796891 0.398445 0.917192i \(-0.369550\pi\)
0.398445 + 0.917192i \(0.369550\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −14.6056 −0.613375
\(568\) 0 0
\(569\) 15.1472 0.635003 0.317502 0.948258i \(-0.397156\pi\)
0.317502 + 0.948258i \(0.397156\pi\)
\(570\) 0 0
\(571\) 17.3305 0.725260 0.362630 0.931933i \(-0.381879\pi\)
0.362630 + 0.931933i \(0.381879\pi\)
\(572\) 0 0
\(573\) 8.72498 0.364491
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 31.3583 1.30546 0.652731 0.757589i \(-0.273623\pi\)
0.652731 + 0.757589i \(0.273623\pi\)
\(578\) 0 0
\(579\) −1.57779 −0.0655709
\(580\) 0 0
\(581\) 62.4500 2.59086
\(582\) 0 0
\(583\) −2.30278 −0.0953712
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −37.5416 −1.54951 −0.774755 0.632262i \(-0.782127\pi\)
−0.774755 + 0.632262i \(0.782127\pi\)
\(588\) 0 0
\(589\) 4.21110 0.173515
\(590\) 0 0
\(591\) 12.6333 0.519665
\(592\) 0 0
\(593\) 13.6056 0.558713 0.279357 0.960187i \(-0.409879\pi\)
0.279357 + 0.960187i \(0.409879\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −31.9361 −1.30706
\(598\) 0 0
\(599\) 14.0917 0.575770 0.287885 0.957665i \(-0.407048\pi\)
0.287885 + 0.957665i \(0.407048\pi\)
\(600\) 0 0
\(601\) 8.90833 0.363378 0.181689 0.983356i \(-0.441844\pi\)
0.181689 + 0.983356i \(0.441844\pi\)
\(602\) 0 0
\(603\) 5.21110 0.212213
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −7.21110 −0.292690 −0.146345 0.989234i \(-0.546751\pi\)
−0.146345 + 0.989234i \(0.546751\pi\)
\(608\) 0 0
\(609\) 55.5416 2.25066
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) 41.1194 1.66080 0.830399 0.557169i \(-0.188112\pi\)
0.830399 + 0.557169i \(0.188112\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.6056 −0.426963 −0.213482 0.976947i \(-0.568480\pi\)
−0.213482 + 0.976947i \(0.568480\pi\)
\(618\) 0 0
\(619\) −17.4222 −0.700258 −0.350129 0.936702i \(-0.613862\pi\)
−0.350129 + 0.936702i \(0.613862\pi\)
\(620\) 0 0
\(621\) −20.7250 −0.831665
\(622\) 0 0
\(623\) 22.8167 0.914130
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.30278 −0.0520278
\(628\) 0 0
\(629\) −37.5416 −1.49688
\(630\) 0 0
\(631\) 39.9361 1.58983 0.794915 0.606721i \(-0.207515\pi\)
0.794915 + 0.606721i \(0.207515\pi\)
\(632\) 0 0
\(633\) 32.8806 1.30689
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −57.5694 −2.28098
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 42.2111 1.66724 0.833619 0.552340i \(-0.186265\pi\)
0.833619 + 0.552340i \(0.186265\pi\)
\(642\) 0 0
\(643\) −22.0000 −0.867595 −0.433798 0.901010i \(-0.642827\pi\)
−0.433798 + 0.901010i \(0.642827\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.2389 0.677729 0.338865 0.940835i \(-0.389957\pi\)
0.338865 + 0.940835i \(0.389957\pi\)
\(648\) 0 0
\(649\) −0.211103 −0.00828650
\(650\) 0 0
\(651\) −23.6056 −0.925174
\(652\) 0 0
\(653\) 19.1194 0.748201 0.374101 0.927388i \(-0.377951\pi\)
0.374101 + 0.927388i \(0.377951\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.78890 0.147819
\(658\) 0 0
\(659\) 20.0917 0.782660 0.391330 0.920250i \(-0.372015\pi\)
0.391330 + 0.920250i \(0.372015\pi\)
\(660\) 0 0
\(661\) 12.8167 0.498510 0.249255 0.968438i \(-0.419814\pi\)
0.249255 + 0.968438i \(0.419814\pi\)
\(662\) 0 0
\(663\) 25.4584 0.988721
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 36.6333 1.41845
\(668\) 0 0
\(669\) −26.8806 −1.03926
\(670\) 0 0
\(671\) 2.90833 0.112275
\(672\) 0 0
\(673\) 6.02776 0.232353 0.116176 0.993229i \(-0.462936\pi\)
0.116176 + 0.993229i \(0.462936\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.2389 1.00844 0.504221 0.863575i \(-0.331780\pi\)
0.504221 + 0.863575i \(0.331780\pi\)
\(678\) 0 0
\(679\) −50.3305 −1.93151
\(680\) 0 0
\(681\) −6.90833 −0.264728
\(682\) 0 0
\(683\) −9.84441 −0.376686 −0.188343 0.982103i \(-0.560312\pi\)
−0.188343 + 0.982103i \(0.560312\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −17.8806 −0.682186
\(688\) 0 0
\(689\) 11.5139 0.438644
\(690\) 0 0
\(691\) 26.5416 1.00969 0.504846 0.863210i \(-0.331549\pi\)
0.504846 + 0.863210i \(0.331549\pi\)
\(692\) 0 0
\(693\) −5.60555 −0.212937
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.27502 0.237683
\(698\) 0 0
\(699\) 6.63331 0.250895
\(700\) 0 0
\(701\) −26.7889 −1.01180 −0.505901 0.862591i \(-0.668840\pi\)
−0.505901 + 0.862591i \(0.668840\pi\)
\(702\) 0 0
\(703\) −9.60555 −0.362280
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 75.3583 2.83414
\(708\) 0 0
\(709\) 11.6333 0.436898 0.218449 0.975848i \(-0.429900\pi\)
0.218449 + 0.975848i \(0.429900\pi\)
\(710\) 0 0
\(711\) −0.119429 −0.00447895
\(712\) 0 0
\(713\) −15.5694 −0.583078
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −5.36669 −0.200423
\(718\) 0 0
\(719\) −28.8167 −1.07468 −0.537340 0.843366i \(-0.680571\pi\)
−0.537340 + 0.843366i \(0.680571\pi\)
\(720\) 0 0
\(721\) −34.0278 −1.26726
\(722\) 0 0
\(723\) 32.4861 1.20817
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.330532 0.0122588 0.00612938 0.999981i \(-0.498049\pi\)
0.00612938 + 0.999981i \(0.498049\pi\)
\(728\) 0 0
\(729\) 26.3305 0.975205
\(730\) 0 0
\(731\) −28.1833 −1.04240
\(732\) 0 0
\(733\) −12.3944 −0.457799 −0.228900 0.973450i \(-0.573513\pi\)
−0.228900 + 0.973450i \(0.573513\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −9.88057 −0.363463 −0.181731 0.983348i \(-0.558170\pi\)
−0.181731 + 0.983348i \(0.558170\pi\)
\(740\) 0 0
\(741\) 6.51388 0.239293
\(742\) 0 0
\(743\) 44.3028 1.62531 0.812656 0.582744i \(-0.198021\pi\)
0.812656 + 0.582744i \(0.198021\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −18.9083 −0.691820
\(748\) 0 0
\(749\) −12.9083 −0.471660
\(750\) 0 0
\(751\) 5.66947 0.206882 0.103441 0.994636i \(-0.467015\pi\)
0.103441 + 0.994636i \(0.467015\pi\)
\(752\) 0 0
\(753\) −5.09167 −0.185551
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0555 0.837967 0.418983 0.907994i \(-0.362387\pi\)
0.418983 + 0.907994i \(0.362387\pi\)
\(758\) 0 0
\(759\) 4.81665 0.174833
\(760\) 0 0
\(761\) −42.4222 −1.53780 −0.768902 0.639367i \(-0.779197\pi\)
−0.768902 + 0.639367i \(0.779197\pi\)
\(762\) 0 0
\(763\) −28.0278 −1.01467
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.05551 0.0381124
\(768\) 0 0
\(769\) −26.8167 −0.967033 −0.483517 0.875335i \(-0.660641\pi\)
−0.483517 + 0.875335i \(0.660641\pi\)
\(770\) 0 0
\(771\) −23.4500 −0.844530
\(772\) 0 0
\(773\) 22.1194 0.795581 0.397790 0.917476i \(-0.369777\pi\)
0.397790 + 0.917476i \(0.369777\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 53.8444 1.93166
\(778\) 0 0
\(779\) 1.60555 0.0575248
\(780\) 0 0
\(781\) −4.60555 −0.164800
\(782\) 0 0
\(783\) −55.5416 −1.98490
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −4.21110 −0.150110 −0.0750548 0.997179i \(-0.523913\pi\)
−0.0750548 + 0.997179i \(0.523913\pi\)
\(788\) 0 0
\(789\) 1.54163 0.0548836
\(790\) 0 0
\(791\) −46.5416 −1.65483
\(792\) 0 0
\(793\) −14.5416 −0.516389
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.5139 0.514108 0.257054 0.966397i \(-0.417248\pi\)
0.257054 + 0.966397i \(0.417248\pi\)
\(798\) 0 0
\(799\) −11.7250 −0.414800
\(800\) 0 0
\(801\) −6.90833 −0.244094
\(802\) 0 0
\(803\) −2.90833 −0.102633
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 30.9083 1.08802
\(808\) 0 0
\(809\) −3.63331 −0.127740 −0.0638701 0.997958i \(-0.520344\pi\)
−0.0638701 + 0.997958i \(0.520344\pi\)
\(810\) 0 0
\(811\) 54.8722 1.92682 0.963411 0.268028i \(-0.0863719\pi\)
0.963411 + 0.268028i \(0.0863719\pi\)
\(812\) 0 0
\(813\) 18.5139 0.649310
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −7.21110 −0.252285
\(818\) 0 0
\(819\) 28.0278 0.979369
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) −10.4222 −0.363295 −0.181648 0.983364i \(-0.558143\pi\)
−0.181648 + 0.983364i \(0.558143\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −7.81665 −0.271812 −0.135906 0.990722i \(-0.543394\pi\)
−0.135906 + 0.990722i \(0.543394\pi\)
\(828\) 0 0
\(829\) −38.7527 −1.34594 −0.672969 0.739671i \(-0.734981\pi\)
−0.672969 + 0.739671i \(0.734981\pi\)
\(830\) 0 0
\(831\) 28.1472 0.976415
\(832\) 0 0
\(833\) 45.0000 1.55916
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 23.6056 0.815927
\(838\) 0 0
\(839\) −16.1194 −0.556505 −0.278252 0.960508i \(-0.589755\pi\)
−0.278252 + 0.960508i \(0.589755\pi\)
\(840\) 0 0
\(841\) 69.1749 2.38534
\(842\) 0 0
\(843\) 29.7250 1.02378
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.30278 0.147845
\(848\) 0 0
\(849\) −3.51388 −0.120596
\(850\) 0 0
\(851\) 35.5139 1.21740
\(852\) 0 0
\(853\) −19.7250 −0.675370 −0.337685 0.941259i \(-0.609644\pi\)
−0.337685 + 0.941259i \(0.609644\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −48.6056 −1.65840 −0.829200 0.558952i \(-0.811204\pi\)
−0.829200 + 0.558952i \(0.811204\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) −19.6056 −0.667381 −0.333690 0.942683i \(-0.608294\pi\)
−0.333690 + 0.942683i \(0.608294\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.24726 0.0763210
\(868\) 0 0
\(869\) 0.0916731 0.00310980
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) 15.2389 0.515757
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 0 0
\(879\) 19.8167 0.668399
\(880\) 0 0
\(881\) −34.5416 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(882\) 0 0
\(883\) −12.4500 −0.418975 −0.209487 0.977811i \(-0.567179\pi\)
−0.209487 + 0.977811i \(0.567179\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.2389 1.58613 0.793063 0.609140i \(-0.208485\pi\)
0.793063 + 0.609140i \(0.208485\pi\)
\(888\) 0 0
\(889\) −73.6611 −2.47051
\(890\) 0 0
\(891\) −3.39445 −0.113718
\(892\) 0 0
\(893\) −3.00000 −0.100391
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −24.0833 −0.804117
\(898\) 0 0
\(899\) −41.7250 −1.39161
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 40.4222 1.34517
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 0 0
\(909\) −22.8167 −0.756781
\(910\) 0 0
\(911\) 39.2111 1.29912 0.649561 0.760310i \(-0.274953\pi\)
0.649561 + 0.760310i \(0.274953\pi\)
\(912\) 0 0
\(913\) 14.5139 0.480339
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.90833 −0.129064
\(918\) 0 0
\(919\) −41.2111 −1.35943 −0.679714 0.733477i \(-0.737896\pi\)
−0.679714 + 0.733477i \(0.737896\pi\)
\(920\) 0 0
\(921\) 7.93608 0.261503
\(922\) 0 0
\(923\) 23.0278 0.757968
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 10.3028 0.338388
\(928\) 0 0
\(929\) 46.3944 1.52215 0.761076 0.648662i \(-0.224671\pi\)
0.761076 + 0.648662i \(0.224671\pi\)
\(930\) 0 0
\(931\) 11.5139 0.377352
\(932\) 0 0
\(933\) −21.9083 −0.717246
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.21110 −0.170239 −0.0851196 0.996371i \(-0.527127\pi\)
−0.0851196 + 0.996371i \(0.527127\pi\)
\(938\) 0 0
\(939\) 28.4222 0.927524
\(940\) 0 0
\(941\) 52.3944 1.70801 0.854005 0.520265i \(-0.174167\pi\)
0.854005 + 0.520265i \(0.174167\pi\)
\(942\) 0 0
\(943\) −5.93608 −0.193305
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.6333 −1.19042 −0.595211 0.803569i \(-0.702932\pi\)
−0.595211 + 0.803569i \(0.702932\pi\)
\(948\) 0 0
\(949\) 14.5416 0.472041
\(950\) 0 0
\(951\) 12.9083 0.418581
\(952\) 0 0
\(953\) 49.2666 1.59590 0.797951 0.602722i \(-0.205917\pi\)
0.797951 + 0.602722i \(0.205917\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 12.9083 0.417267
\(958\) 0 0
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) −13.2666 −0.427955
\(962\) 0 0
\(963\) 3.90833 0.125944
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.09167 0.131579 0.0657897 0.997834i \(-0.479043\pi\)
0.0657897 + 0.997834i \(0.479043\pi\)
\(968\) 0 0
\(969\) −5.09167 −0.163568
\(970\) 0 0
\(971\) 30.3583 0.974244 0.487122 0.873334i \(-0.338047\pi\)
0.487122 + 0.873334i \(0.338047\pi\)
\(972\) 0 0
\(973\) −35.3305 −1.13264
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.9722 0.510997 0.255499 0.966809i \(-0.417760\pi\)
0.255499 + 0.966809i \(0.417760\pi\)
\(978\) 0 0
\(979\) 5.30278 0.169477
\(980\) 0 0
\(981\) 8.48612 0.270941
\(982\) 0 0
\(983\) −48.8444 −1.55789 −0.778947 0.627089i \(-0.784246\pi\)
−0.778947 + 0.627089i \(0.784246\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 16.8167 0.535280
\(988\) 0 0
\(989\) 26.6611 0.847773
\(990\) 0 0
\(991\) 6.09167 0.193508 0.0967542 0.995308i \(-0.469154\pi\)
0.0967542 + 0.995308i \(0.469154\pi\)
\(992\) 0 0
\(993\) −18.7527 −0.595100
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −14.2750 −0.452094 −0.226047 0.974116i \(-0.572580\pi\)
−0.226047 + 0.974116i \(0.572580\pi\)
\(998\) 0 0
\(999\) −53.8444 −1.70356
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 4400.2.a.bs.1.1 2
4.3 odd 2 275.2.a.e.1.1 2
5.2 odd 4 4400.2.b.y.4049.3 4
5.3 odd 4 4400.2.b.y.4049.2 4
5.4 even 2 4400.2.a.bh.1.2 2
12.11 even 2 2475.2.a.t.1.2 2
20.3 even 4 275.2.b.c.199.4 4
20.7 even 4 275.2.b.c.199.1 4
20.19 odd 2 275.2.a.f.1.2 yes 2
44.43 even 2 3025.2.a.n.1.2 2
60.23 odd 4 2475.2.c.k.199.1 4
60.47 odd 4 2475.2.c.k.199.4 4
60.59 even 2 2475.2.a.o.1.1 2
220.219 even 2 3025.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 2 4.3 odd 2
275.2.a.f.1.2 yes 2 20.19 odd 2
275.2.b.c.199.1 4 20.7 even 4
275.2.b.c.199.4 4 20.3 even 4
2475.2.a.o.1.1 2 60.59 even 2
2475.2.a.t.1.2 2 12.11 even 2
2475.2.c.k.199.1 4 60.23 odd 4
2475.2.c.k.199.4 4 60.47 odd 4
3025.2.a.h.1.1 2 220.219 even 2
3025.2.a.n.1.2 2 44.43 even 2
4400.2.a.bh.1.2 2 5.4 even 2
4400.2.a.bs.1.1 2 1.1 even 1 trivial
4400.2.b.y.4049.2 4 5.3 odd 4
4400.2.b.y.4049.3 4 5.2 odd 4